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J. L. Jacobsen [*and*]{} P. Zinn-Justin
*Laboratoire de Physique Théorique et Modèles Statistiques*
*Université Paris-Sud, Bâtiment 100*
*91405 Orsay Cedex, France*
.2in
We propose a new method to enumerate alternating knots using a transfer matrix approach. We apply it to count numerically various objects, including prime alternating tangles with two connected components, up to order $18$–$22$, and comment on the large-order behavior in connection with one of the authors’ conjecture.
The classification of knots is a subject with a long history. More than a hundred years ago, Tait, Kirkman and Little tried to draw “by hand” the first knots (up to 10 crossings, but with some mistakes); nowadays, it is possible with the modern tools of knot theory and the use of computers to create programs that generate all possible (prime) knots up to 16 crossings . Here we shall consider the simplest problem, namely the [*enumeration*]{} of knots (or similar objects). Furthermore we shall concentrate on so-called [*alternating*]{} objects which have much simpler properties than their generic counterparts. Even though they are probably asymptotically subdominant (this has been proved in the case of links ), they are quite interesting to study if only because one can prove much more about them than about general knots. In fact, the number of prime alternating [*tangles*]{} is known exactly ; however, a general tangle has an arbitrary number of connected components, and in the present paper we want to be able to control this number, that is to count knots or similar objects (e.g. tangles with exactly $2$ connected components).
The key concept that will be used is that of a [*transfer matrix*]{}. A standard object of statistical mechanics, where it describes the discrete time evolution of a system, the transfer matrix is also a very efficient numerical tool for the combinatorial enumeration of discrete objets. This approach has recently been used to investigate the properties of 2D Lorentzian gravity , the enumeration of plane meanders , and the coupling of matter fields to 2D quantum gravity equipped with a Hamiltonian circuit . In contradistinction to the standard situation, where the transfer matrix is used to construct the partition function for a statistical mechanics system on a semi-infinite strip of finite width, these combinatorial applications possess a state space which is different in each time slice. Accordingly the underlying physical models are not defined on a regular lattice, but belong to the realm of two-dimensional quantum gravity. In all these examples, the common feature that ensures the existence of a transfer matrix is that the objects under consideration permit a time ordering. For the meanders (resp. the Hamiltonian circuits) this was attained by “reading” each object as one moves along the river (resp. the circuit), adding one intersection (vertex) at each time step. Clearly, this stategy in also appropriate for enumerating knots with [*one*]{} connected component.
This type of enumeration problems have many interesting connections with mathematical physics . In particular, in the counting of links, knots or tangles was reduced to the evaluation of integrals over $N\times N$ hermitian matrices, in the limit $N\to\infty$. Though these integrals cannot be computed explicitly in general, the subject of matrix models and its well-known connection to 2D quantum gravity provide some information on the universal quantities of the model. This led to conjectures of the [*asymptotic*]{} behavior of the number of such objects when the number of crossings goes to infinity; and one motivation of the present work is to check the conjecture concerning the asymptotic number of prime alternating knots.
The paper is organized as follows: after some brief definitions in section 2, which enable us to give a more precise meaning to the problem that we are addressing, the basic principles underlying our transfer matrix approach are described in section 3. A number of practical details concerning our implementation of this algorithm are then given in section 4. Finally, section 5 gives the results of the numerical enumeration of prime alternating knots. As a byproduct we also count several other objects of combinatorial interest.
A [*knot*]{} is a smooth circle embedded in $\R^3$, considered up to homeomorphisms of $\R^3$. In this paper we shall study slightly different objects, namely knots with “external legs”; a knot with $2n$ external legs is a a collection of $n$ intervals embedded in a ball $B$ and whose endpoints are given distinct points on the boundary $\der B$, considered up to orientation preserving homeomorphisms of $B$ that reduce to the identity on $\der B$. These knots with external legs are nothing but tangles in which no closed loops are allowed. Knots with $2$ and $4$ external legs will be of special interest to us. When we refer to standard knots we shall from now on call them closed knots. Clearly, each knot with $2$ external legs can be transformed into a closed knot by joining its endpoints through a smooth curve outside $B$. And conversely, by cutting a closed knot once it may be turned into a knot with $2$ external legs, but in general this transformation is not unique, in the sense that the topological properties of the resulting knot with $2$ external legs depend on the point where the closed knot was cut. This means that counting knots with external legs does not help count closed knots. However we shall always count the former and never the latter; those are the objects whose generating series will have “nice” properties.
It is common to represent such objects by their projections on a plane; we consider regular projections with only double points where lines cross. To avoid redundancies, we shall concentrate on [*prime*]{} knots, whose diagrams cannot be decomposed as a sum of pieces connected to each other by only two lines, and on [*reduced*]{} diagrams that contain no irrelevant crossings (Fig. ). We shall need another related concept: a diagram is said to be [*$r$-particle reducible (resp. irreducible)*]{} ($r$PR, resp. $r$PI), $r$ positive integer, if it can (resp. cannot) be divided into disconnected components by cutting $r$ edges. For closed knots and knots with $2$ external legs, the notion of a prime knot coincides with the $2$-particle irreducibility of its diagram(s).
A diagram is called [*alternating*]{} if one meets under- and over-crossings alternatingly as one travels along each loop. A remarkable property (see e.g. , page 21) is that in the case of alternating diagrams there is no need to “remember” which crossings are under/over. In other words, two alternating knot diagrams have the same underlying planar diagram if and only if they are identical, or related by an overall flip under $\leftrightarrow$ over in the case of closed knots. This greatly simplifies their enumeration.
To a given knot can correspond several diagrams. In fact, in the case of alternating diagrams, two alternating reduced knot diagrams represent the same object if and only if they are related by a sequence of moves acting on tangles called “flypes” (see Fig. ) . This is of course an essential distinction when one is interested in counting such objects, and we shall briefly discuss it now. The general idea is the same as in ; however, the actual equations shown used here are simpler than those in , and their proof will be given elsewhere in the more general framework of colored links.
Since the diagrams we shall work with most of the time are diagrams of knots with $2$ external legs, we shall simply call them knot diagrams. Let us start by defining the generating series $G(g)$ where $a_p$ is the number of knot diagrams with $p$ crossings, or equivalently, the number of topologically inequivalent open curves in the plane going from $(-\infty,0)$ to $(+\infty,0)$ with $p$ regular self-intersections. We similarly define $\Sigma_1(g)$ to count 1PI knot diagrams, and $\Sigma_2(g)$ to count 2PI knot diagrams (with the trivial diagram excluded and the two diagrams with one crossing included). The following relations hold: $\Sigma_1(g)$ is simply given by whereas $\Sigma_2(g)$ is given by the implicit equation (see for details).
Next, we want to take into account the flyping equivalence in order to count the actual objects and not diagrams. The data of $G(g)$ is insufficient for this purpose; we need a more general object, a double generating series $G(g_1,g_2)$ where $a_{p_1,p_2}$ is the number of topologically inequivalent open curves in the plane (going from $(-\infty,0)$ to $(+\infty,0)$) with $p_1$ regular self-intersections and $p_2$ [*tangencies*]{}, see Fig. . Also, note that we have $a_{p,0}=a_p$ or $G(g)=G(g_1=g,g_2=0)$, so that there is more information in $G(g_1,g_2)$ than in $G(g)$.
Since flypes act on tangles, we are led to the introduction of a few more generating functions which are related to the counting of tangles, $G_1$, $G_2$, $\Gam_1$ and $\Gam_2$; they are all expressible in terms of $G$ alone via: $$\eqalignno{
G&=1+2g_1G_2+2g_2(G_1+G_2)&\gend a\cr
{\der\over\der g_2} G_2&={\der\over\der g_1} (G_1+G_2)&\gend b\cr
\Gam_1&=G_1&\gend c\cr
\Gam_2&=G_2-G^2&\gend d\cr
}$$
We also introduce for our convenience an extra parameter $t$ which counts the number of edges of the diagram; it is easy to show that the following formulae take care of it: $G(g_1,g_2,t)\equiv{1\over t} G(g_1/t^2,g_2/t^2)$ and $\Gam_i(g_1,g_2,t)\equiv {1\over t^2} \Gam_i(g_1/t^2,g_2/t^2)$.
The parameters $t$, $g_1$ and $g_2$ must then be chosen as a function of $g$ according to the following [*renormalization procedure*]{} (see ): $$\eqalignno{
1&=G(g_1(g),g_2(g),t(g))&\ren a\cr
g_1(g)&=g(1-2H'_2(g))&\ren b\cr
g_2(g)&=-g(H'_1(g)+V'_2(g))&\ren c\cr
}$$ where $H'_1(g)$, $H'_2(g)$ and $V'_2(g)$ are auxiliary quantities defined by: $$\eqalignno{
H'_2\pm H'_1&={(1\mp g)(\Gam_2\pm\Gam_1)\mp g
\over 1+(1\mp g)(\Gam_2\pm\Gam_1)\mp g}&\gene a\cr
V'_2&=(1-g)\Gam_2(1-H'_2-H'_1)^2&\gene b\cr
}$$ where we have omitted all arguments for the sake of brevity; in particular $\Gam_i\equiv\Gam_i(g_1(g),g_2(g),t(g))$.
This ensures that the flypes are appropriately taken into account and for example that $\Gam_1(g_1(g),g_2(g),t(g))$ and $\Gam_2(g_1(g),
g_2(g),t(g))$ are the desired generating functions for the number of tangles with $2$ connected components of type 1 and 2 respectively (see for a definition of type; the total number of tangles is given by $\Gam_1+2\Gam_2$). Similarly one could define other generating functions of the 2 variables $g_1$ and $g_2$ (higher correlation functions in the matrix model language) which would count objects with more external legs.
We now come to the description of the transfer matrix approach. The latter requires first that the knot diagrams be represented in an appropriate way (3.1); next we have to define the space of states on which the matrix acts (3.2); and finally define the transfer matrix itself (3.3). At first we shall concentrate on the usual knot diagrams with self-intersections only; a direct application of the first 3 subsections leads to the enumeration of alternating knot diagrams, but to count actual alternating knots, one has to introduce a refined procedure (addition of tangencies) to which subsection 3.4 is devoted.
A basic ingredient of the transfer matrix approach is the ability to cut the object one is studying into slices, which represent the state of the system at fixed (discrete) time. If we apply this to knots a complication arises. The naive idea would be to draw the knot diagrams on the plane in such a way that time would correspond to one particular coordinate of the plane, that is to read the knot diagrams “from left to right”. Here, this idea does not work directly, and one is led to a slightly more sophisticated notion of slices, which we shall explain using the example of Fig. . The general idea is to follow the knot as it winds around itself from one “incoming” external leg to the other “outgoing” leg, and write down step by step the crossings and the lines that are crossed. We shall call the edge of the diagram that we are currently following the active line. Of course this is a step-dependent concept since every edge of the diagram will at some point become the active line. The edges of the diagram have been labelled from A to K in the order in which we encounter them. The precise recipe is as follows. At each step there are two possibilities: 1) The label of the active line does not appear anywhere else in the picture drawn this far. We then proceed to the next crossing and draw it. 2) The active line already appears somewhere. We connect the active line to its other appearance. We then follow this new line until it reemerges as an open line: this will be the new active line. In the case of Fig. , steps 1–5 are of type 1) whereas steps 6–10 are of type 2). In general, steps of type 1) and 2) can appear in any order, except that at any stage the number of type 2) steps performed cannot be greater than the number of type 1) steps. In a more algebraic language, the sequence of 1s and 2s forms a Dyck word .
After step 5, for the first time the active line, which is now the edge F, is already present (AB–FG crossing). When we reconnect the two occurrences of the edge F, we notice that some open lines are “imprisoned” inside the new arch we have created, and therefore we cannot draw step 6 right after step 5. Instead we must allow lines inside the new arch to continue to evolve (steps 8 and 10), keeping in mind that they cannot have any contact with the lines outside the arch.
Using this procedure, to each knot diagram we can associate a “sliced” diagram; and it is easy to show that two “sliced” diagrams are topologically equivalent (in the sense of graphs) if and only if they consist of the same sequence of steps. We shall now show how to generate all knots with such diagrams using a transfer matrix.
The vector space on which the transfer matrix acts will be spanned by the intermediate states created in the process described in the previous section. The important point to remember is that at each step, we had to take into account the following information: a) the current (open) lines, including the active line; b) the existing connections of the lines from the left: pairs of lines are connected by what we call [*left arches*]{}; c) the different groups of lines which can still be connected to each other from the right; the lines are divided in multiple connected components by what we call [*right arches*]{}. All this information has to be included in the state of the system.
A basis state will therefore be described by a series of left and right arches and the position of the active line. As an illustration, we show all the intermediate states of the example of Fig. on Fig. .
For practical applications, it is important to notice that some configurations should be forbidden. Firstly, we have states that cannot evolve into knots (Fig. a) and b)). Secondly, there are redundant states that are equivalent to simpler states (Fig. c) and d)); we shall describe a systematic simplification procedure in section 4 below.
We now describe the transfer matrix $T$. Its entries $T_{ab}$, where $a$ and $b$ are two basis states of the kind defined in the previous section, are $0$ unless $a$ is a descendant of $b$, in which case $T_{ab}$ is the number of ways $a$ can be obtained from $b$. An allowed state $a$ is a descendant of $b$ if it can be obtained via a transformation of one of the two types shown on Fig. , followed by an arbitrary number of simplifications (Fig. ). The two transformations of Fig. reproduce the procedure described in section 3.1, but we have now reformulated it in terms of states.
Note that in transformation 2), the relative positions of the active line and the line onto which it connects are not arbitrary. The screening role of the right arches should be respected, so that the active line can only connect onto lines belonging to the same block; and the relative distance between the active line and the line that it connects onto must be [*odd*]{}. For each of the allowed connection, the position of the new active line is found by following the line just connected along the left arch to which it belongs.
Transformation 1) (resp. 2)) increases (resp. decreases) the number of lines by 2. This means that at step $p$, for any intermediate state, the number of crossings $n$ and the number of lines $l$ (excluding the active line) are related by $l=2(p-2n)$. In particular, we claim that the number of knot diagrams with $n$ crossings is given by $\bra{0}T^{2n}\ket{0}$, where $\ket{0}$ is the state with the active line only. In this notation, $\ket{0}$ is understood to be assigned the weight one, and $\bra{0}$ acts as a projection operator: $\braket{0}{0} = 1$. Formally we have
In Appendix A we show explicitly the action of $T$ in the case of 1PI diagrams with at most four crossings.
In order to count knots and not knot diagrams, it was explained in section 2 that one must start with more general objects than standard diagrams: one must count curves with both self-intersections and tangencies which produce a double generating series $G(g_1,g_2)$. This can be easily included in the transfer matrix as follows.
Firstly, the space of states must be slightly extended to take into account the fact that we have a double generating series. Typically a state must contain the information concerning the number of previous tangencies in the diagram. Therefore the new space of states will be the tensor product of the old space of states and of the space of polynomials in a variable $x$ which can be defined as $x=g_2/g_1$.
Secondly, the transfer matrix itself must be modified to allow for the creation of such tangencies: the new transfer matrix $\tilde{T}$ is of the form where $T$ is the old transfer matrix defined by the transformations of Fig. , and $T'$ is the additional transformation described on Fig. (plus, in each case, an arbitrary number of simplifications of the type of Fig. ). We can finally write the following formal expression for $G(g_1,g_2)$: where it is recalled that $x=g_2/g_1$.
Having described the principles underlying the transfer matrix algorithm we now turn to a number of important practical details concerning its implementation. Section 5.1 describes the data structures needed to encode the states and their corresponding weights. Several remarks on the algorithm will be made in section 5.2, and in section 5.3 we explain a number of different implementations that we have made, in the aim of obtaining a reasonable balance between the time and memory needs of the algorithm.
The state space, formulated in terms of left and right arches and the active line, has been described in section 3.2. Ideally, in order to obtain a highly efficient transfer matrix algorithm, one would like to introduce a [*ranking*]{} among the states. By this we mean a bijective mapping from the $N$ different states to the set of integers $\{1,2,\ldots,N\}$. With a ranking at hand, the integer representation can be used to label the entries of the transfer matrix, and the arch representation is then used to produce the descendants of a given state, as described in section 3.3. In a previous publication, one of the authors has shown how to obtain this goal in the case of meanders ; however, due to the very complicated interplay between left and right arches we have not been able to make similar progress in the case of knots.
Fortunately a simpler, and almost as efficient, alternative is available. Suppose that to each state $i$ we can assign a unique integer $k_i \in \Z_+$, and devise a function $f: \Z_+ \rightarrow \{0,1,2,\ldots,P-1\}$ that distributes the set of $k_i$’s more-or-less uniformly on the set $\{0,1,2,\ldots,P-1\}$. By inserting the states $i$ into an array of noded lists indexed by $f(k_i)$, we can retrieve a given state $k$ in a time proportional to the mean length of one of the pointer lists, $t \propto N/P$. This is a standard technique known as [*hashing*]{} ; the integer $k_i$ and the function $f$ are known respectively as the hash key and the hash function.
In the case at hand, a key $k_i$ can be defined by representing each arch state as a base-four number. Specifically, we read a configuration of left and right arches from top to bottom, associating the digit 1 (resp. 0) with the opening (resp. closing) of a left arch, and 3 (resp. 2) with the opening (resp. closing) of a right arch. The 1s and the 0s (resp. the 3s and the 2s) thus form two interlaced Dyck words . For the computation at order $p$ crossings there are at most $p$ arches, and the resulting key is at most $4^p$; we also need a few extra bits to specify the position of the active line. The hash function is simply $f(k) = k {\rm \ mod \ } P$, where $P$ is a large prime which we choose such that $N/P \sim 10$.
For each state in the hash table, we store its key and its weight. The weight is an integer, but since the number of knots with $p$ intersections grows exponentially with $p$ the weights of the largest knots considered in this work cause overflow in a standard 32-bit integer arithmetic. Instead of wasting memory storing double-precision integers, we took advantage of modular arithmetic . This means that the largest computations were done modulo various coprime numbers (typically $2^{32}$ and $2^{32}-1$), and the full result was retrieved from the Chinese remainder theorem.
It is possible to perform a number of reductions on the state space. Although these do not affect the correctness of the algorithm, they are nevertheless important to implement since they reduce the number of intermediate states needed in the transfer process, and thus enables us to go to larger system sizes.
After each of the two transformations shown on Fig. 6, the resulting arch state can be simplified using the reductions given in Fig. 7. The idea is to associate each inequivalent “screened” state with a unique configuration of right arches. Reductions 1) and 2) consists in sliding an exterior right arch over an adjacent interior arch, and 3) consists in removing right arches that do not screen any ingoing left line. In the algorithm, these simplifications are performed recursively until no further reduction is possible. The resulting state is then unique.
It may happen that after reduction a state is forbidden in the sense of Fig. 5 a) or b). A first example of this occurs at order $p=4$, and is shown in Appendix A. Before inserting a descendant state in the hash table we therefore examine whether each right arch contains an even number of lines (including the active line), of which at least one is connected to the exterior.
A final algorithmic detail concerns the possibility of removing tadpole insertions in the knot diagrams. A tadpole is generated if and only if a type 1) transformation is immediately followed by a type 2) transformation in which the active line connects onto an [*adjacent*]{} line (see Fig. 6). We can therefore forbid tadpoles if each state encompasses an extra sign signalling whether the previous transformation was of type 1). Superficially this would appear to double the number of states needed, but in fact this is not so, since an important number of states are only produced when tadpoles are allowed. In practice we found that the number of signed states without tadpoles, and the number of states with tadpoles only differ by a few percent. Of course, eliminating tadpoles directly in the algorithm carries no intrinsic interest, since it is a trivial matter to do so afterwards by manipulating the generating functions. However, since the number of tadpole diagrams is, very roughly, found to be the square of the number of diagrams without tadpoles, including tadpoles would mean that we would have to carry out twice as many runs in order to retrieve the full result from the Chinese remainder theorem. For this reason we opted for the algorithm without tadpoles.
Even though our transfer matrix method is much more efficient than a direct enumeration of the knot diagrams, it suffers from the drawback that the dimension of the state space, and thus the memory needs, grow exponentially with $p$. For a fixed size $p$, the evolution of the memory dynamically allocated by the hash table as a function of the discrete “time” steps is shown in Fig. . Near the beginning the number of states grows exponentially, reaches a maximum after roughly $3p/2$ steps, and then decreases exponentially towards the end. For practical reasons we only had about one gigabyte of memory available for our computations, and this turns out to be a more severe limitation for the obtainable system size than the computation time available. We have therefore experimented with several different implementations that limit the memory needs at the expense of using more time.
The most successful of these consist in, roughly speaking, using the transfer matrix approach until the available memory is exhausted. We then switch to a direct enumeration, which is carried out for a fixed number of steps, until the number of states needed by the transfer matrix approach has decreased to a level that again fits into the available memory. On Fig. this could be represented by cutting the maximum of the memory profile by a horizontal line segment, representing the process of direct enumeration. The latter is based on the same recursive principle as the one defining the transfer process, but since states generated in intermediate time steps are not inserted into the hash table the process allocates no further memory.
In Table we display the coefficients of the generating functions ($G$, $\Sigma_1$, $\Sigma_2$) up to order $p=22$. We recall that these functions represent, respectively, the total number of knot diagrams with $p$ crossings and two outgoing strings, and the subsets of $1$PI and $2$PI diagrams.
The first $10$ terms of $G$ have already been reported by Gusein-Zade and Duzhin , who called the corresponding diagrams ‘long curves’. The algorithm used by these authors was however based on direct enumeration, and thus did not enjoy the advantages of the transfer matrix approach. Namely, in the latter, a multitude of diagrams can correspond to the same intermediate state at a given time step, and is therefore counted “simultaneously”. The difference between the two approaches can readily be appreciated by comparing the number of diagrams (Table ) with the number of intermediate states (Figure ).
In particular, having available more terms of the generating functions enables us to examine the asymptotic behavior of the number of diagrams. Calling $a_p$ the coefficients of $G(g)$, as in Eq. , a first rough estimate yields $a_p \sim \mu^p$ with $\mu \simeq 11.4\pm 0.1$. This corresponds to a singularity of the function $G(g)$ at $g_c=1/\mu$. However, from the point of view of the underlying field theory it is the subleading corrections to the dominant exponential behavior that are of paramount interest, the connective constant $\mu$ being non-universal. Based on a random-matrix model description of alternating knots as the $n\to 0$ limit of a generalized $O(n)$ symmetric action , one of us has conjectured that the detailed asymptotic behavior reads with $\alpha=3$. This value of $\alpha$ corresponds to a string susceptibility exponent $\Gamma \equiv 2-\alpha = -1$ characteristic of the coupling of a conformal field theory with central charge $c=-2$ to two-dimensional quantum gravity, via the celebrated KPZ formula .
With the current data, it is difficult to estimate $\alpha$ in Eq. without any knowledge of the subleading corrections. Indeed, a direct fit gives $\alpha\approx 2.76$ but usual convergence acceleration methods do not confirm this result. However, if we fit our data with $a_p=\mu^p p^{-\alpha} (a\log p+b+o(1))$ (the presence of logarithmic corrections being justified by possible marginally irrelevant operators in a $c=-2$ theory), the result is in good agreement with the conjecture:
In the case of $\Sigma_2$, the approach to the asymptotic regime appears to be less regular. This is probably due to another singularity of the function $G(g)$ around $g_{c2}\approx -0.3$ which causes oscillations that are very subdominant in $G(g)$ but less so in $\Sigma_2(g)$. However, the leading exponential behavior $\sim \mu_2^p$ can be readily extracted without analyzing numerically the coefficients of $\Sigma_2$, by using the following simple identity: $\mu_2=\mu/G(g_c)^2$. Assuming the expansion above and using the fit , We find: $$\mu_2 = 6.613\pm 0.008$$ =cmr5
We now turn to the inclusion of the flype equivalence. As described in Section 3.4 this can be done by enumerating also diagrams with tangencies. A power-counting argument reveals that in order to accomodate the flype equivalence at order $p$, we need to know the number of diagrams at order $p_1$ with (roughly) at most $(p_2)_{\rm max} \equiv \lfloor (p-p_1)/3 \rfloor$ tangencies, for all $p_1=0,1,\ldots,p$. These data are shown up to order $p=20$ in Table .
The contents of the first column ($p_2=0$) is of course just the coefficients of $G$, cf. Table . The first line ($p_1=0$) gives the number of two-legged diagrams with $p_2$ tangencies and no self-intersections, This formula is a corollary of the exact solution of the standard O$(m\to 0)$ model on random tetravalent graphs. It can also be shown in a straightforward way. First, represent each tangency by a dotted line, as in Figure .a. In this way, the problem becomes that of rooted Hamiltonian circuits on a random trivalent graphs . Next, straighten out the full line, as in Figure .b. The dotted lines now form two independent Catalan arch configurations, one above and one below the full line. Clearly, the number of such configurations is $$\sum_{j=0}^p {2p \choose 2j} c_{p-j} c_j$$ where $c_k = {(2k)! \over k!(k+1)!}$ are the Catalan numbers. The result follows immediately.
In a similar fashion, we can give an explicit formula for the second line: $$a_{1,p-1}=p \, a_{0,p}={(2p)!(2p+2)!\over (p-1)! (p+1)!^2 (p+2)!}$$ Indeed, the diagrams with one self-intersection and $p-1$ tangencies are obtained from diagrams with $p$ tangencies by replacing one tangency with a crossing.
The results for the the number of prime alternating tangles with two connected components can now be found from the procedure outlined in Section 2; see Table . The reader is reminded that the total number of tangles is given by $\Gam_1+2\Gam_2$.
The first 8 orders were previously given in . The number of tangles seem to approach their asymptotic behavior in a less regular fashion than the objects discussed above. In particular there seems to be a strong dependance on the parity of $p$. Even the leading term in the large $p$ limit, of the form $\tilde{\mu}_2^p$, is hard to isolate, with $\tilde{\mu}_2$ roughly given by $6.0\pm 0.1$; some serious numerical analysis is required to determine it more accurately. Note that this leading behavior should be the same for prime alternating knots, with the same constant $\tilde{\mu}_2$. Similary, the critical exponent of knots should be one plus the exponent of tangles. Although we have not been able to extract reliable values of the exponent, physical insight suggest that these exponents are most likely to the same as those of diagrams (conjecturally, $\alpha=3$). It is therefore very likely that the conjecture made in is valid, though we have no definite evidence at the moment.
In a future publication we shall show how to generalize our transfer matrix approach to allow for the enumeration of connected knot diagrams with an [*arbitrary*]{} fixed number of components. As a first application, this will allow us to enumerate alternating [*links*]{}, and to extend the generating functions given in by several orders. Another interesting goal that we are currently pursuing is the enumeration of multi-component meander diagrams . There are many other applications related to the possibility of counting planar Feynman diagrams.
As an illustration we show the first $8$ iterations of the transfer matrix. We restrict ourselves to states which generate 1PI diagrams and with at most $4$ crossings. Reading off the weight of the vacuum state (containing only the active line) after step $2p$, we deduce the number of diagrams with $p$ crossings. However, this state is not allowed to evolve in subsequent steps, since otherwise one-particle reducible diagrams would be generated. We have [*not*]{} excluded tadpoles in this example.
Note also that when connecting the active line of the third diagram at step 6 to the uppermost line by means of a right arch, we generate (after reduction of the two right arches) a forbidden state of the type shown in Fig. 5 a), which is therefore not shown.
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"I sat watching a flower as it was withering. I was embarrassed by its honesty." ― Bright Eyes
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Case: 19-40129 Document: 00515265796 Page: 1 Date Filed: 01/09/2020
IN THE UNITED STATES COURT OF APPEALS
FOR THE FIFTH CIRCUIT
No. 19-40129
Summary Calendar
United States Court of Appeals
Fifth Circuit
FILED
January 9, 2020
UNITED STATES OF AMERICA,
Lyle W. Cayce
Clerk
Plaintiff-Appellee
v.
MICHAEL SHELLEY,
Defendant-Appellant
Appeal from the United States District Court
for the Eastern District of Texas
USDC No. 4:18-CR-92-1
Before HIGGINBOTHAM, HO, and ENGELHARDT, Circuit Judges.
PER CURIAM: *
Michael Shelley appeals the procedural and substantive reasonableness
of his 10-year sentence for bank fraud under 18 U.S.C. § 1344, an offense that
is punishable by up to 30 years in prison. The Government has filed a motion
to dismiss the appeal based on his appeal waiver in the plea agreement.
Notwithstanding Shelley’s creative argument to the contrary, the record
reflects that he understood that he was waiving the right to appeal his sentence
* Pursuant to 5TH CIR. R. 47.5, the court has determined that this opinion should not
be published and is not precedent except under the limited circumstances set forth in 5TH
CIR. R. 47.5.4.
Case: 19-40129 Document: 00515265796 Page: 2 Date Filed: 01/09/2020
No. 19-40129
in the plea agreement unless the sentence exceeded the 30-year statutory
maximum or resulted from ineffective assistance of counsel. The meaning of
the waiver was clearly conveyed by the straightforward plea agreement, which
Shelley read and signed after careful review with his attorney, and by the
magistrate judge at the plea hearing. Accordingly, the waiver was knowing
and voluntary. See United States v. Portillo, 18 F.3d 290, 292 (5th Cir. 1994).
We reject Shelley’s assertion that the Government should be required to
waive its contractual right to enforce the appeal waiver, having satisfied its
own obligation to recommend a sentence at the low end of the guidelines range.
To the extent Shelley contends that we should not enforce the appeal waiver
because it unfairly prevents him from raising meritorious claims, we have
never recognized such an exception to a valid appeal waiver.
Because the waiver was informed and voluntary, Shelley “will be held to
the bargain to which he agreed.” Id. at 293. The plain language of the waiver
applies to bar his appeal. See United States v. Bond, 414 F.3d 542, 544 (5th
Cir. 2005). The Government’s motion is GRANTED, and the case is
DISMISSED. Shelley’s request that we take judicial notice of a grant of
certiorari in a case relevant to the merits of his appeal is DENIED as moot.
2
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Wow...Some people think only Choir Boys play in the NFL. Some are naive enough to think only Choir Boys play for the Steelers. "But can he play football?" Just how many times do you think that question is asked after the "bad" is summarized about a prospect. The sky is blue. Water is wet. Football is not for the weak. Girls don't play football. There is a fine line between confident & "A" hole but they both play in the NFL and for the Steelers.
Bray wouldn't be the first kid drafted by the Steelers who reeks of immauturity & he won't be the last. The question of "Will he become a profesional" is what the FO will sort out in the interviews. The answer to the question "But can he play football?" will be answered for all of you come draft day. Who he becomes is up to him.
As far as Bray & the Steelers...It would be a very good situation for both. Another year in school for Bray could have elevated his stock into the 1st. I won't get into the argument of how far because that would have depended on him. I do believe his decision was based on some part of Patterson, Hunter, Thomas, & Rivera entering the draft. He wouldn't be asked to produce early in his career. He can sit at #3 & hold a clipboard. His position coach would work with him and he will learn the NFL game as an observer. He will learn how to be a professional from his team mates & coaches. That being said, do I think the interest is all about Bray? I think they would draft him if he falls further than his grade if they like how he interviewed. I don't think he is a "target" player. What I really think it is about is finding out about Patterson & Hunter. The things that you can't measure. I do believe Patterson would be the pick @ #17 if he falls and Hunter would be a 2nd round consideration. Just all part of the process.
Of course there are good, bad, mature, immature, etc on each team and in some instances all in one person. The question for me is this: If you know he's a dimwit, arrogant or a headcase, why bother? Those are not traits that a quarterback coach can alleviate. A quarterback coach, can, however, fix the footwork, throwing mechanics, hot reads, progressions, etc. A character flaw isn't correctable by a coach, unless, the coach is a licensed psychologist as well as a wealth of football knowledge.
I've never seen Bray play, but if he's anything close to the descriptions in this thread, me, personally, I wouldn't draft for those reasons. Of course, on the other hand, you could draft a kid that puts up the great facade during the interview and then becomes the devil incarnate once he joins the team.
We all know this is a poor year for franchise QBs or even full-time starters, but there are some interesting prospects to talk about.
Geno Smith has the best combination of athleticism, arm, size and makeup, but he is certainly no sure thing. I would have liked to see Smith run a little more last year when the opportunity arose because the ability to create when a play breaks down would be a big plus in the NFL.
Matt Barkley is worth a late first round pick in my opinion, but only to a team who has a starter in place and who is looking to try and groom him. If a bad team drafted him and tried to plug him in right now, I think he would fail and potentially fail miserably. He needs more coaching.
To me, the guys with the biggest upsides are Mike Glennon and EJ Manuel, but for different reasons. Glennon has the size and the arm, but he makes poor decisions at times which results in turnovers and he will get sacked a ton. If you plugged him in with a strong offensive line and gave him time to throw, he could be a QB who attacks defenses down the field as he showed the ability to strike with the deep ball quite often at N.C. State. Anything short of a great pair of tackles and he will get creamed.
Manuel is a pure projection. His stats from Florida State look good on paper, but his play didn't always say "first three round QB". What I really like about Manuel besides his size and ability to operate outside of the pocket is that he's got moldable traits. He has played out of gun, under center and run some zone-read. He's the perfect prospect for a team looking to diversify their offense. He's got a high ceiling, but a low floor so taking him within the first two rounds is a swing for the fences.
Tyler Wilson showed a desire to survive last year which meant getting rid of the ball quickly. He is better than what we saw at times last year behind that offensive line and with that lack of skill talent, but I don't think he has the type of NFL arm most teams will look for. He needs to be in a west coast offense.
As for Ryan Nassib, he's intelligent and competitive, but I can't get past the fact that so much of his game looks so average. I just don't see the 1st round grades that some are putting on him.
Tyler Bray's measurables are very intriguing, but I would have liked to have seen even more production considering he was playing with NFL-caliber WRs and an NFL-caliber left tackle in Dallas Thomas.
Zac Dysert appears to be relatively average in every way as well, but I like him. I saw poise when I studied him and I think that is a very important trait for a potential NFL starter. Landry Jones has a decent arm and a quick set-up but his accuracy inconsistencies are hard for me to get past.
Of course there are good, bad, mature, immature, etc on each team and in some instances all in one person. The question for me is this: If you know he's a dimwit, arrogant or a headcase, why bother? Those are not traits that a quarterback coach can alleviate. A quarterback coach, can, however, fix the footwork, throwing mechanics, hot reads, progressions, etc. A character flaw isn't correctable by a coach, unless, the coach is a licensed psychologist as well as a wealth of football knowledge.
I've never seen Bray play, but if he's anything close to the descriptions in this thread, me, personally, I wouldn't draft for those reasons. Of course, on the other hand, you could draft a kid that puts up the great facade during the interview and then becomes the devil incarnate once he joins the team.
Pappy
I've seen him play and it's frustrating... the guy has the tools, the arm and last year he had the weapons. However, what he didn't have was accuracy or a will to win. He reminds me of Phillip Rivers and Cutler... but without the W's. He is the type to check out during a game winning drive.
Of course there are good, bad, mature, immature, etc on each team and in some instances all in one person. The question for me is this: If you know he's a dimwit, arrogant or a headcase, why bother? Those are not traits that a quarterback coach can alleviate. A quarterback coach, can, however, fix the footwork, throwing mechanics, hot reads, progressions, etc. A character flaw isn't correctable by a coach, unless, the coach is a licensed psychologist as well as a wealth of football knowledge. I've never seen Bray play, but if he's anything close to the descriptions in this thread, me, personally, I wouldn't draft for those reasons. Of course, on the other hand, you could draft a kid that puts up the great facade during the interview and then becomes the devil incarnate once he joins the team. Pappy
Not all in reponse to you Pap...I made a stew post. Don't anyone take offense.
That is what the interview process is for. I see the concerns..I'm not blind. But I see the talent...Like I said...I'm not blind. Bray isn't unique by defintion from many of the prospects. As far as your "dimwit, arrogant or a headcase,"...That came from what? The posters in here? Arrogant..."Swagger".... A Cutler or Rivers? Would that be bad as a #3 in the 3rd? Immaturity of a 21 year old can improve. There are examples of it through decades of drafts. There are also examples of failures...I'm not saying there is no risk. We aren't talking about making him the 1st round pick. Like I said, some of you are very naive thinking the Steelers take players off the board because of attitude. Many put the Steelers morals on a pedestal that doesn't exist. The Steelers may not be in the same house but they are in the same block. They put football players on their roster weighed against their risks.
Dooley's thoughts on him as a Freshman.
[URL]http://www.timesfreepress.com/news/2010/dec/20/brays-laid-back-attitude-impresses-dooley/Bray[/URL]
Bray is confident...He may be immature...He may say the wrong things at times. But the kid can ball. In my opinion, the most physically gifted QB in this draft. Most upside. Yes...There are risks but he isn't being drafted to be the franchise guy day 1 in the 1st round. Bray's best situation since he came out early is going to a strong organization with a franchise guy in place. He needs to learn to be an NFL QB & a professional. Don't have to like him or agree with me. Facts are his talent & upside warrant consideration if he is around in the 3rd for the Steelers if he interviews well. It is a fact that he needs to represent himself better but that is an obstacle with many kids coming out of college. He is a "Boom or Bust" pick at the QB position. If you can take that risk in the 3rd...You do it.
28 Games were all under Dooley's 3 years. Dooley was fired last year. In 28 games Bray was 14-14. Not great winning %. But....
540/922 59% 7,444 yds 69 Tds 28 Ints
2012: (12 games) 268/451 59.4% 3,612 yds 34 Tds 12 Ints
Those numbers under a new coaches 3 years who got fired at the end of his 3rd year. Talent? Bum? Dimwit? Headcase? Punk? Blank Card? There will be teams wanting to write his name on a card. More than likely Day 2 of draft. Will the Steelers be the first one? I would support it 100% if it was in the 3rd back. If he "Busts"....So be it. If he "Booms"....He can eat Steak whenever he wants & be a D!ck to the waitress. I wouldn't care. And guess what...Neither would the Steelers if he was winning. Slap on wrist...Go win a championship.
Not all in reponse to you Pap...I made a stew post. Don't anyone take offense.
That is what the interview process is for. I see the concerns..I'm not blind. But I see the talent...Like I said...I'm not blind. Bray isn't unique by defintion from many of the prospects. As far as your "dimwit, arrogant or a headcase,"...That came from what? The posters in here? Arrogant..."Swagger".... A Cutler or Rivers? Would that be bad as a #3 in the 3rd? Immaturity of a 21 year old can improve. There are examples of it through decades of drafts. There are also examples of failures...I'm not saying there is no risk. We aren't talking about making him the 1st round pick. Like I said, some of you are very naive thinking the Steelers take players off the board because of attitude. Many put the Steelers morals on a pedestal that doesn't exist. The Steelers may not be in the same house but they are in the same block. They put football players on their roster weighed against their risks.
Dooley's thoughts on him as a Freshman.
[URL]http://www.timesfreepress.com/news/2010/dec/20/brays-laid-back-attitude-impresses-dooley/Bray[/URL]
Bray is confident...He may be immature...He may say the wrong things at times. But the kid can ball. In my opinion, the most physically gifted QB in this draft. Most upside. Yes...There are risks but he isn't being drafted to be the franchise guy day 1 in the 1st round. Bray's best situation since he came out early is going to a strong organization with a franchise guy in place. He needs to learn to be an NFL QB & a professional. Don't have to like him or agree with me. Facts are his talent & upside warrant consideration if he is around in the 3rd for the Steelers if he interviews well. It is a fact that he needs to represent himself better but that is an obstacle with many kids coming out of college. He is a "Boom or Bust" pick at the QB position. If you can take that risk in the 3rd...You do it.
28 Games were all under Dooley's 3 years. Dooley was fired last year. In 28 games Bray was 14-14. Not great winning %. But....
540/922 59% 7,444 yds 69 Tds 28 Ints
2012: (12 games) 268/451 59.4% 3,612 yds 34 Tds 12 Ints
Those numbers under a new coaches 3 years who got fired at the end of his 3rd year. Talent? Bum? Dimwit? Headcase? Punk? Blank Card? There will be teams wanting to write his name on a card. More than likely Day 2 of draft. Will the Steelers be the first one? I would support it 100% if it was in the 3rd back. If he "Busts"....So be it. If he "Booms"....He can eat Steak whenever he wants & be a D!ck to the waitress. I wouldn't care. And guess what...Neither would the Steelers if he was winning. Slap on wrist...Go win a championship.
I've been critical of him but I also posted I wouldn't have a problem drafting him in the 4th... edit post: I thought I said it earlier in the thread but apparently I didn't.
I think the OP used the 2nd and I think that's why most people are saying hell to the no!
I think we have more pressing needs in the 2nd than a clip holding boom or bust QB.
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1. Field of the Invention
The present invention relates to a reduced video signal processing circuit for displaying a reduced input video scene in a window, using a buffer memory, such as a field memory, a frame memory, etc., wherein the reduction ratio is changeable.
2. Description of the Prior Art
A Picture in Picture (PIP) function has been generally known as a TV display function for displaying a plurality of scenes in the form of a window on one TV screen, instead of displaying one scene on the entire TV screen. With the arrival of the multimedia era, a further variety of display functions have been demanded. In particular, a window displaying function at a desired reduction ratio, which has become commonly used as an operating environment for personal computers, has also been demanded for TV displaying. In order to display a separate scene in the form of a window, a buffer memory, such as a field memory, a frame memory, etc., is necessary in order to retain synchronism between main and sub-scenes (inset-scenes).
FIG. 6 is a block diagram showing a conventional reduced video signal processing circuit.
A video signal for a sub-scene which is displayed in the form of a window is input into an input processing section 1 which includes a filter circuit 10. The input video signal is then processed for size reduction (filtered) by the filter circuit 10 according to size reduction ratio data K before being provided to field memories 2 and 3. A writing operation to the field memories 2 and 3 is controlled by an input video clock generator 5 in a control block 4. The control block 4 is also provided with a display video clock generator 6 for controlling a reading operation from the memories 2 and 3. Size reduction ratio data K, one externally received, and the input video clock generator 5 supplies the data K to the input processing section 1.
Based on a horizontal synchronizing signal (Input H) and vertical synchronizing signal (Input V) of an input video signal received, the input video clock generator 5 outputs a write clock (WCLK), write enabling signals (WE1, WE2), and a write reset signal (WRST). A WCLK has the same rate as that of a pixel clock synchronous with an input H. WE1 and WE2 signals cause the field memories 2 an d 3 to be at an enable state alternately for every field during an effective display period of an input video signal. The field memories 2 and 3 incorporate address counters for independently addressing during writing and reading operations. After the counters are reset by a WRST signal, addresses of the counters are incremented by counting a WCLK while WE1 and WE2 signals are at an H (high) level, whereby a reduced video signal supplied from the input processing section 1 is written into the field memories 2 and 3. When size reduction ratio data K indicates "1," that is, when a reduction operation is not executed, WE1 and WE2 signals remain at an H level throughout an effective video period. When size reduction ratio data K is smaller than "1," on the other hand, the period when WE1 and WE2 signals are at an H level is adjusted according to the size reduction ratio data K. For instance, with size reduction ratio data K of "1/2, " as shown in FIG. 7, WE1 and WE2 signals are output at an H or L level in such a way that they become an H level for every other pixel. As a result, an input video signal subjected to half thinning processing in the filter 10 can be written into a memory.
Further, the input video clock generator 5 computes video size data (SIZ), based on the size reduction ratio data K, and supplies the SIZ data to the display video clock generator 6. For instance, provided that the numbers of horizontal and vertical pixels of an input video signal are "640" and "480," respectively, and the size reduction ratio data K is "1/2," SIZ data is computed to indicate H (horizontal) SIZ "320" and V (vertical) SIZ "240."
On the read side, in order to read a reduced video signal from the field memories 2 and 3, the display video clock generator 6 is supplied with a horizontal synchronizing signal (display H) and a vertical synchronizing signal (display V) of a display video signal for a main scene and a display position data (X, Y) for indicating a display position of a reduced scene, and outputs a read clock (RCLK), read enabling signals (RE1, RE2), and a read reset signal (RRST). RCLK has the same rate as that of a pixel clock synchronous with a display H. RE1 and RE2 signals cause the field memories 2 and 3 to be at an enable state alternately for every field during an effective display period of a display video signal. An RRST signal resets read address counters of the field memories 2 and 3 upon a rise of an RE signal. After the counters are reset by an RRST signal, a read address of the counters in the field memories 2 and 3 are incremented by counting an RCLK while RE1 and RE2 signals are at an H level, whereby a reduced video signal is read from the field memories 2 and 3. Note that the display video clock generator 6 may generate display H and V signals in cases where each timing thereof is previously known, so that an RRST signal, an RCLK, RE1 and RE2 signals, etc., are generated based on the display H and V signals.
FIG. 3 illustrates an input video signal A displayed in the form of a window, wherein display position data (X, Y) indicates a display position of a sub-scene with respect to a main scene (a display video signal), and video size SIZ data (H, V) indicates the size of a sub-scene to be displayed in the form of a window (a reduced video signal generated from an input video signal). For achieving such a window display, the display video clock generator 6 causes RE1 and RE2 signals to be at an H level only during an effective display period as shown in FIG. 7, based on the SIZ data (H, V) and the video position data (X, Y). In this case, RE1 and RE2 signals are continuously maintained at an H level throughout an effective display period, which is different from the input side.
The display video clock generator 6 supplies SIZ data (H, V) and video position data (X, Y) to a display processing section 7 which is provided downstream of the field memories 2 and 3, so that the reduced video signal read from the memories 2 and 3 are processed therein for window displaying through framing or addition of background data, and output as a display video signal.
In order to change a reduction ratio as desired in the foregoing procedure, the content of processes executed on the write and read sides must be changed in accordance with respective new reduction ratio data. However, if the content of a process is changed during reading and writing operations, distortion may be caused to a display video signal (displaying scene). Thus, an operation for changing a reduction ratio is executed during a vertical blank interval of a video to prevent distortion.
However, since an input V and a display V are not synchronous with each other, the above changing operation is conducted at different timings on the write and read sides. An example is taken, referring to FIG. 7, where a phase of a display V is delayed compared to that of an input V. When size reduction ratio data K is newly input at time T1, the reduction ratio is changed to the new ratio during a vertical blank period NP1 which is an immediately following interval of the time T1 in the input video clock generator 5 and the input processing section 1, so that a reduction operation and a write control operation are thereafter conducted based on the new reduction ratio. In the display video clock generator 6 and the display processing section 7, on the other hand, the reduction ratio is changed during a vertical blank period DP1 which is also an immediately subsequent interval to the time T1. Since this timing (DP1) is behind the changing timing on the input side (NP1), a video signal written in the changed reduction ratio is read from the memory in the changed reduction ratio after the time DP1.
In another case, referring to FIG. 8, where reduction ratio data is newly input at time T2 which is after the vertical blank interval NP1 of an input V and before the vertical blank interval DP1 of a display V, a reduction ratio is changed during a vertical blank interval NP2 after the time T2 in the input video clock generator 5 and the input processing section 1, and during a vertical blank interval DP1, before the interval NP2, in a display video clock generator 6 and the display processing section 7. In other words, the reduction ratio is changed on the display side prior to the input side. As a result, a video signal reduced in a previous reduction ratio (a reduction ratio before the change) is read in a changed reduction ratio to be displayed for a field subsequent to the change (DP1) on the read side. This causes significant distortion to a video displayed in a window. For this reason, the prior art has a problem in that a reduction ratio cannot be changed while a window display continues.
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here's a word analogy : amistad is to the lost world as schindler's list is to jurassic park .
in 1993 , after steven spielberg made the monster dino hit , many critics described schindler's list as the director's " penance " ( as if there was a need for him to apologize for making a crowd-pleasing blockbuster ) .
now , after a three-year layoff , spielberg is back with a vengeance .
once again , his summer release was special effects-loaded action/adventure flick with dinosaurs munching on human appetizers .
now , following his 1993 pattern , he has fashioned another serious , inspirational christmas release about the nature of humanity .
that film is amistad .
although not as masterful as schindler's list , amistad is nevertheless a gripping motion picture .
thematically rich , impeccably crafted , and intellectually stimulating , the only area where this movie falls a little short is in its emotional impact .
watching schindler's list was a powerful , almost spiritual , experience .
spielberg pulled us into the narrative , absorbed us in the drama , then finally let us go , exhausted and shattered , three-plus hours later .
aspects of the movie have stayed with me ever since .
amistad , while a fine example of film making , is not as transcendent .
the incident of the ship la amistad is not found in any history books , but , considering who writes the texts , that's not a surprise .
however , the event is a part of the american social and legal fabric , and , while amistad does not adhere rigorously to the actual account , most of the basic facts are in order .
several , mostly minor changes have been made to enhance the film's dramatic force .
on the whole , while amistad may not be faithful to all of the details of the situation , it is true to the spirit and meaning of what transpired .
one stormy night during the summer of 1839 , the 53 men imprisoned on the spanish slave ship la amistad escape .
led by the lion-hearted cinque ( djimon hounsou ) , they take control of the vessel , killing most of the crew .
adrift somewhere off the coast of cuba and uncertain how to make their way back to africa , they rely on the two surviving spaniards to navigate the eastward journey .
they are tricked , however , and the la amistad , which makes its way northward off the united states' eastern coastline , is eventually captured by an american naval ship near connecticut .
the kidnapped africans are shackled and thrown into prison , charged with murder and piracy .
the first men to come to the africans' defense are abolitionists theodore joadson ( morgan freeman ) and lewis tappan ( stellan skarsgard ) .
they are soon joined by roger baldwin ( matthew mcconaughey ) , a property attorney of little repute .
aided by advice from former president john quincy adams ( anthony hopkins ) , baldwin proves a more persuasive orator than anyone gave him credit for , and his central argument -- that the prisoners were illegally kidnapped free men , not property -- convinces the judge .
but powerful forces have aligned against baldwin's cause .
current president martin van buren ( nigel hawthorne ) , eager to please southern voters and 11-year old queen isabella of spain ( anna paquin ) , begins pulling strings behind-the-scenes to ensure that none of the africans goes free .
at its heart , amistad is a tale of human courage .
cinque is a heroic figure whose spirit remains unbreakable regardless of the pain and indignity he is subjected to .
he is a free man , not a slave , and , while he recognizes that he may die as a result of his struggle , he will not give it up .
effectively portrayed by newcomer djimon hounsou , whose passion and screen presence arrest our attention , cinque is the key to viewers seeing the amistad africans as more than symbols in a battle of ideologies .
they are individuals , and our ability to make that distinction is crucial to the movie's success .
to amplify this point , spielberg presents many scenes from the africans' point-of-view , detailing their occasionally-humorous observations about some of the white man's seemingly-strange " rituals " .
the larger struggle is , of course , one of defining humanity .
as the nazis felt justified in slaughtering jews because they viewed their victims as " sub-human , " so the pro-slavery forces of amistad use a similar defense .
the abolitionists regard the africans as men , but the slavers and their supporters see them as animals or property .
in a sense , the morality of slavery is on trial here with the specter of civil war , which would break out less than three decades later , looming over everything .
amistad's presentation of the legal and political intricacies surrounding the trial are fascinating , making this movie one of the most engrossing courtroom dramas in recent history .
four claimants come forward against the africans : the state , which wants them tried for murder ; the queen of spain , who wants them handed over to her under the provision of an american/spanish treaty ; two american naval officers , who claim the right of high seas salvage ; and the two surviving spaniards from la amistad , who demand that their property be returned to them .
baldwin must counter all of these claims , while facing a challenge to his own preconceived notions as the result of a relationship he develops with cinque .
even though attorney and client are divided by a language barrier , they gradually learn to communicate .
aside from cinque , who is a fully-realized individual , characterization is spotty , but the acting is top-notch .
matthew mcconaughey successfully overcomes his " pretty boy " image to become baldwin , but the lawyer is never particularly well-defined outside of his role in the la amistad case .
likewise , while morgan freeman and stellan skarsgard are effective as joadson and tappan , they are never anything more than " abolitionists . "
nigel hawthorne , who played the title character in the madness of king george , presents martin van buren as a spineless sycophant to whom justice means far less than winning an election .
finally , there's anthony hopkins , whose towering portrayal of john quincy adams is as compelling as anything the great actor has recently done .
hopkins , who can convincingly play such diverse figures as a serial killer , an emotionally-crippled english butler , and richard nixon , makes us believe that he is adams .
his ten-minute speech about freedom and human values is unforgettable .
one point of difference worth noting between amistad and schindler's list is this film's lack of a well-defined human villain .
schindler's list had ralph fiennes' superbly-realized amon goeth , who was not only a three-dimensional character , but a personification of all that the nazis stood for .
there is no such figure in amistad .
the villain is slavery , but an ideology , no matter how evil , is rarely the best adversary .
it is to spielberg's credit that he has fashioned such a compelling motion picture without a prominent antagonist .
amistad's trek to the screen , which encountered some choppy waters ( author barbara chase-riboud has cried plagiarism , a charge denied by the film makers ) , comes in the midst of an upsurge of interest in the incident .
an opera of the same name opened in chicago on november 29 , 1997 .
numerous books about the subject are showing up on bookstore shelves .
it remains to be seen how much longevity the amistad phenomena has , but one thing is certain -- with spielberg's rousing , substantive film leading the way , the spotlight has now illuminated this chapter of american history .
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Factors associated with acceptance of peers with mental health problems in childhood and adolescence.
Research suggests that children's reactions to peers with mental health problems are related to the maintenance and outcomes of these problems. However, children's perceptions of such peers, particularly those with internalising problems, are neither well researched nor understood. The present study aimed to test a series of models relating socio-demographic and attributional variables to the acceptance of hypothetical boys and girls with attention deficit hyperactivity disorder (ADHD) and depression. A sample of 595 participants, drawn from five different age-groups spanning early childhood to late adolescence, completed a booklet of questions in response to two vignettes describing the behaviour of hypothetical target peers with depression and ADHD. The sample was drawn from schools randomly selected in the east of Ireland. The models indicated that age and gender of the participant, and the perceived responsibility of the target character for his/her condition, were the three most important predictors of acceptance in all models. However, the relationship between these variables and acceptance varied depending on the gender of the target child and the condition (depression or ADHD) in the models tested. The findings of the study suggest that the relationships between socio-demographic and attributional variables and acceptance of peers with mental health problems depend on the type of mental health problem under consideration. The findings have implications for the development of information and education programmes to improve the integration of children with mental health problems.
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[Diagnostic and prognostic values of vacuolated polymorphonuclear neutrophils (author's transl)].
The presence of vacuolated polymorphonuclear neutrophils in blood smears of patients suffering from infection appears to be associated with massive bacterial growth and to constitute a very early symptom of rapidly life-threatening septicaemia. When these cells persist for more than 36 hours, the disease may be considered as beyond control.
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13:59
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The Awesomer is reader-supported. When you buy through links on our site, we may earn an affiliate commission. Learn more.
Amazon Cloud Cam
Amazon gets into the smart home security market with the Cloud Cam. It streams 1080p video with night vision to a mobile app, and can be activated with Alexa via the Echo Show, Echo Spot, Fire TV and Fire Tablet.
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Hysteresis and bistability in periodically paced cardiac tissue.
Hysteresis in periodically paced cardiac tissue is an important issue due to its relevance to cardiac arrhythmias. In the present paper, the mechanism of hysteresis formation and the related properties are interpreted by numerically investigating the phase I Luo-Rudy model. A formula calculating the width of hysteresis is proposed and well confirmed by numerical simulations. We also find that hysteresis in cardiac tissue shows several characteristics due to couplings among cardiac cells which are absent in a single cell. The influences of the physiological parameters are studied in detail. The model dependence of hysteresis is elucidated by considering a number of well-known models of excitable media. Moreover, the influence of bistability on controlling arrhythmias is revealed.
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Monday, August 3, 2015
There is a too-commonly analogy used for tough tasks or times being like WW1 trench life. In this case it is appropriate both because this book starts in those trenches, and because at times getting through it felt how I imagine struggling through gluey mud might be as bombs slam down around and the tears roll down my face, failing to wash away the false hope that one day it will all be over.
Alright, that's a tad harsh. A mightly-tad harsh, in fact, as there is a lot in Kelleher's writing to enjoy. Sadly there is so much to not enjoy that this is my 'didn't finish the book' book review for the season.
We begin on the Western Front in October, 1916, as various British soldiers are introduced within the terror and tedium of trench warfare. Kelleher from the off shows he hasn't skimped on research time and seems to relish each piece of jargon he sprinkles into the prose - 'whizz-bangs', 'Very lights' and 'Blighty's abound. However, whilst the stage feels authentic to someone who has studied the era and the conflict in some detail themselves (I was obsessed with WW1 as a teenager), the actors and action placed on said stage don't, quite. The key blame lies I think from the start with Kelleher's dialogue. Perhaps he is the victim of over-familiarity with the 'bloody 'ell, Tommy' way of talking by the soldiers, or the 'that will do, lad, stand down, there's a good chap' of the officers. Yet there are ways to make cliches come alive again, and his characters don't manage to feel like real people you would meet in the street, withe in 1916 or now. There are a couple of exceptions, particularly once time is given to the inner thoughts of some of the men, and the key characters of Atkins, Everson and Jeffries seem to be made of more complex stuff, but the pantomime 'gor blimey' feel of the rest of the cast of conscripts kept me firmly in the 'disbelief unsuspended' camp.
Soon enough, the reverse deus ex machina arrives - an explosive earthquake, a fog, and a bang, and the Tommies' patch of Somme earth is teleported to what quickly turns out to be another planet. Rather handily, the air is breathable and the temperature survivable, but of course almost every single animal and plant not only can but actively wants to kill them. Marooned on their field of mud and surrounded by giant hounds, flying dinosaur-type things and grass that shoots spikes, our soldiers, along with a priest and a few nurses, must learn both to survive and how they came to be there, in order to find their way back home.
The author, despite my above negativity, is a decent world-builder, and his descriptive language is often rich and imaginative, and constitutes much of my enjoyment and my continued interest in reading. Although the lurches into fantastical action are not often smooth (weird boulders are 'casually' mentioned then in the next paragraph reveal themselves to be killer beetles, for example), the visual detailing is a large ingredient in the fun of the tale. However, Kelleher then lets it all down by enormous, giant beetle-sized failures of logic and description. The best example is when a native of the planet is first encountered. The briefest of description of them as a 'wild man' with standard human appearance is followed by his immediately talking to them in broken English. This is calmly accepted by the soldiers - "the Tommies were not too shocked that the man spoke English. As soldiers of the great and glorious British Empire, they were used to the idea that Johnny Foreigner would speak at least some English, even if it was in an odd accent. It was only right and proper, after all".
It might seem odd that I have just quoted a fairly random line of the novel at length, but this was the exact point at which the story began to lose me, and looking back on it now, there is so much wrong here. Now, I'm not thinking the author is saying that about their reaction with anything other than hindsighted sarcasm, but the very fact that he is commenting from afar on their beliefs left me cold, for if that is what it is, it is a self-congratulatory and distancing authorial voice we are hearing, not that of a passive narrator of fantasy and sic-fi adventure. As elsewhere, Kelleher feels the amateurish need to overstate, to underline his points. Whilst this is happening, though, the complete lack of logic to why they are talking in English (don't even get me started on the insect overlords chatting away to the soldiers) is not addressed. I don't mean addressed by the writer - that explanation may come later - but addressed by anyone in the story. No one says 'we are on an alien planet and maybe we should wonder at that'. It would be my first question after hello.
The weak plot points continue, and are too numerous to mention, but a vital one to highlight is the Jeffries character. A mysterious, devil-worshipping murderer of an officer, it is his actions that largely further the plot, and for a while we are given hints it may be his fault the teleportation happened. However, whilst anyone, anything of distinction from the swathes of blandly gruff yet loveable Tommies is welcome, his arch villainy is too cartoonish. Sure, we are in a world of trees with poisonous snot and insect slave-owners who can leap 30ft, but to not ground the human reactors to the world with relatable behaviour, even if that is murderous and insane, is a fault.
Am I being too harsh on what is surely meant to be a fun piece of classic sci-fi homage to Burroughs, Wells et al? It's WW1 soldiers fighting aliens for cripes's sake. Well, no, I'm not. If Kelleher had the vigour and skill to launch into his tale sharply and tell it with rhythm and flair, this might have been my read of the year - it is after all a great premise with lots of potential. But he labours through introduction after introduction and then labours through page after page of dull interpersonal relationships between the soldiers, constantly sapping the pace. Letters home to a sweetheart are used to skip periods of time, when we should be breathing each breath of their existence, and action is delayed in favour of cheery exchanges over work duty or about the attractive nurses- UM! EXCUSE ME GUYS! YOU ARE ON AN ALIEN WORLD THAT INSTANTLY ATTACKED YOU! Maybe a little focus and speed to events?! It is almost as if Kelleher is distracted from his plot by his world build and people, and forget they are there to serve the story, not the other way around.
Ultimately, it was difficult to pin down what kept me reading on. Once I realised it was to finish so I could write the review I had promised to write for this site, so you could read it and decide if it was for you, I decided to stop at about four fifths of the way through the first novel (part of a trilogy now available from Abaddon Books) and start this review. Because, if you're eating a bad soup, do you keep eating to check the bottom? No. you do not. So you find me, lips stained with bad soup, begging you to not bother with this series, and instead go and read The Time Machine, or some Dan Dare comics, or watch some early Flash Gordon. For those early examples of science fiction put this deep in their shade. Maybe have some nice green Gazpacho with it. Perfect for these summer afternoons...
The MathBaseline Assessment : 4/10Bonuses : +1 for really making me want to know what happens, even though it hurt to carry on, due imaginative landscaping and a mystery hook to the plot; +1 for taking me back to when I was 13 and would read tale of alien worlds and a comic series called Tommy's War; +! for respecting the horrors of war and the hardship of command based on a class system Negatives : -1 for the lurches in tone -e.g. from many men dying by their side to a comic quip a second later; -1 for the awful drug sequences; -1 for the relentless modern perspective imposed on the situation and morality, whilst somehow wallowing in all the talk of the Bosche and the patronising working class pluckinessNerd Co-efficient : 4/10 "not very good " see our scoring system here
Reference: Kelleher, Pat. No Man's World trilogy [Abaddon Books, 2015]written by English Scribbler who needs to find a good book, sharpish
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VIEWS: 11570
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Slut taken and abused by group of men in hardcore gangbang!
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Branded Cross Cowboy Church
"Where It's All About HIM!"
“Come as you are” is the dress code lived out here at Branded Cross Cowboy Church. Dress code is a non-issue because the real issue is accepting people, right where they are. The simple Bible-based sermons, casual atmosphere and accepting heart of cowboy churches has helped create an openness in worship, Bible study, and discipleship, that allows the Holy Spirit to move freely in bringing people to a life-changing path where they can meet God through Jesus Christ. Branded Cross Cowboy Church is not just for those who make their living sittin’ horseback. It’s a non-traditional church for down to earth people who want to get to know God personally. So, if your looking for a place with a family-friendly home feel, and enjoy being able to “come just a you are” then come on down to Branded Cross Cowboy Church– and bring someone with ya!
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Development of a mouse model of supraspinatus tendon insertion site healing.
Supraspinatus (SS) tendon tears are common musculoskeletal injuries whose surgical repair exhibits the highest incidence of re-tear of any tendon. Development of therapeutics for improving SS tendon healing is impaired by the lack of a model that allows biological perturbations to identify mechanisms that underlie ineffective healing. The objective of this study was to develop a mouse model of supraspinatus insertion site healing by creating a reproducible SS tendon detachment and surgical repair which can be applied to a wide array of inbred mouse strains and genetic mutants. Anatomical and structural analyses confirmed that the rotator cuff of the mouse is similar to that of human, including the presence of a coracoacromial (CA) arch and an insertion site that exhibits a fibrocartilagenous transition zone. The surgical repair was successfully conducted on seven strains of mice that are commonly used in Orthopaedic Research suggesting that the procedure can be applied to most inbred strains and genetic mutants. The quality of the repair was confirmed with histology through 14 days after surgery in two mouse strains that represent the variation in mouse strains evaluated. The developed mouse model will allow us to investigate mechanisms involved in insertion site healing.
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As an emerging technology, it’s difficult to fully grasp the true potential of the Internet of Things (IoT). For starters, smart cities alone are predicted to touch a staggering $147 billion by 2020. And more than half of the world’s IT leaders investing in IoT have already embraced and fully implemented an IoT strategy, while a third of all leaders are already rolling one out.
The IoT upsides are so strong that they’re hard to measure, but what are the risks and downsides? Yes. Cybercrime and human error are a reality, but what about optimism bias — or the risk that organizations, businesses, and individuals knowingly take or chose to take, although chances of exposure to cybercrime may be high.
For most organizations, optimism bias goes beyond measuring risk versus reward and is most common when risks are ignored simply because the latest malicious attack happened to someone else’s company.
Although there is the understanding that connected platforms like IoT can increase exposure to cybercrime and malicious attacks, decision makers may trust that even if they are compromised, they’ll recover unscathed. Or they believe that access and convenience are well worth the risk.
Not so. Optimism bias can destroy bottom lines, disrupt business objectives and cause reputations to suffer so much damage that partnerships and agreements are fractured forever. Like any other emerging technology, IoT should be implemented with proactive protection against cyber threats.
NOVAM’s Health Checks enable users, devices, and applications to verify that software packages and hardware components are uncompromised and unmodified by malware or other cyber threats, across the entire secure boot chain and run-time on a device.
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By Mike Mount
A spate of violent attacks in Afghanistan spurred on by an anti-Muslim video made in the Unites States, as well as continued attacks on coalition forces by their Afghan partners, is putting a tumultuous start on the first step of the U.S. handover of authority to the Afghan government.
The attacks come at a sensitive time as the United States removes the last of the more than 30,000 surge troops the Obama administration rushed in to quash an increasingly powerful Taliban insurgency in southern Afghanistan in 2010.
Those remaining troops are scheduled to be out of the country by the end of this month, bringing the U.S. troop level down to about 68,000 in addition to other NATO allies and Afghan forces.
But as that schedule rolls on, U.S. troops have been ordered to halt some joint operations with Afghan security forces after the attacks by their local allies and amidst the fallout from the controversial anti-Islam video.
"In response to an increased threat situation as a result of the 'Innocence of Muslims' video, plus the recent insider attacks, ISAF forces are increasing their vigilance and carefully reviewing all activities and interactions with the local population," Maj. Lori Hodge, a spokeswoman for NATO's International Security Assistance Force, said Tuesday.
The attacks included a brazen assault on a coalition base in southern Afghanistan on Friday that killed two US troops and destroyed six coalition fighter jets as well as a suicide attack carried out in Kabul on Tuesday by an insurgent group that killed 12 people, eight of whom were foreigners. An extremist group that claimed responsibility for the latter attack said it was in response to the film.
The other factor behind the partial joint operations suspension is the number of insider attacks in the country.
More than 50 coalition troops were killed between January and mid-August in instances where uniformed Afghans turned their guns on allied troops.
"We are absolutely resolute in our commitment to the objectives of our campaign, but ... on the path to achieving those objectives we will make adjustments as we go," Chairman of the Joint Chiefs of Staff Gen. Martin Dempsey said Tuesday while traveling in the Middle East.
But while some say the surge was a boon to security - "The surge ordered by President Obama...had a huge positive impact on security," the former U.S. ambassador to Afghanistan, Ryan Crocker, said Monday during a speech - others say the end of the surge may have been a bit short-sighted on its departure timeline.
"I think the violence is going to escalate as the numbers (of surge troops) dwindle," said Shuja Nawaz, director of the South Asia Center at the Atlantic Council, a Washington-based policy analysis organization. "With the surge troops there the U.S. was in a position to go on the offensive," he said.
When President Barack Obama finally announced his decision to add more troops in Afghanistan in December of 2009, his goal was to take back what the Taliban had seized during the years when the United States was more focused on its war in Iraq.
The surge helped tilt the balance in favor of the U.S. and its allies, but the current threat of insider attacks remains a bit of a mystery as to what is fueling it, according to Nawaz.
"We are now faced with another emerging threat, which is not the Taliban, but the violence from the Afghan troops the coalition trained," Nawaz said in a phone interview with CNN.
The threat reached such a point that the top commander in Afghanistan, Gen. John Allen ordered additional reviews of security procedures and limited some joint operations with Afghan security forces.
The insider attacks are a puzzle for the Defense Department, which has yet to get a solid answer on what is causing them. While military analysts and U.S. commanders have not drawn any correlations, insider attacks did not really exist prior to the surge troops arriving.
"It maybe frustration on the part of the troops that are leaving; they are pushing the Afghans to do things (for) which they are not ready yet," Nawaz said adding that he has only been able to get anecdotal information from the military instead of any detailed reports or analysis.
U.S. Defense Secretary Leon Panetta said he believes at least some of the attacks are part of the Taliban, but does not see it as a successful campaign by the insurgents, but rather a desperate attempt to stay relevant.
"We think it is kind of a last gasp effort to be able to not only target our forces, but to try to create chaos, because they've been unable ... to regain any of the territory that they have lost...we are concerned about the increase in these attacks," Panetta said Monday while traveling in Asia.
"What we need to do is look at these places and understand why there is a greater propensity, and to arm ourselves against it and to continue to encourage our Afghan partners at every level of their leadership to be engaged with us in this," Dempsey said of the insider attacks.
"I expect that two weeks from now, (Allen, commander of coalition forces in Afghanistan) will be looking at the conditions as he confronts them and making other assessments," Dempsey said of Allen's assessment of restarting the halted Afghan training operations.
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xxx
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Institute CECE
Institute CECE, abbreviated as CECE, is a non-profit, private early childhood education college in Malaysia, registered with Ministry of Higher Education (MOHE). It is located in Setapak, Kuala Lumpur, adjacent to Tunku Abdul Rahman University College Kuala Lumpur Main Campus. The institute was established in 1993 and was the only private and non-profit institution of higher learning, concentrating solely on pre-school teachers’ training in Malaysia.
Over the past 19 years, more than 2,500 pre-school teachers have graduated from the institute. On 1 November 2015, Institute CECE signs a Memorandum of Understanding (MOU) with National Pingtung University (NPTU), Taiwan to allow graduates from early childhood education further studies in this university.
The institute also have partnership with professional bodies such as Early Childhood Care and Education Council and Association of Professional Early Childhood Educators Malaysia.
Mission & Aim
Mission
Institute CECE focuses on child advocacy in three dimensions:
Enabling every child regardless of background, to have access to quality education - made possible when the teacher is trained and qualified;
Quality early childhood education is on the right pedagogy; and
Right pedagogy implies the right methodology.
Aim
Institute CECE aims to raise the professional standard of early childhood practitioners and to ensure that our educators are trained and qualified professionals with:
Strong knowledge based on the child, including appropriate learning experiences, assessment and evaluation;
Skills and competencies to relate to children; and
Attitude and interest in the development of early childhood education
Partner Institution
United Kingdom
University College Birmingham
University of Gloucestershire
Malaysia
Universiti Tunku Abdul Rahman
Quest International University Perak
Taiwan
National Pingtung University
References
External links
Official Institute CECE Website
Universiti Tunku Abdul Rahman (Partner Institution)
University College Birmingham (Partner Institution)
National Pingtung University (Partner Institution)
Category:Colleges in Malaysia
Category:Universities and colleges in Kuala Lumpur
Category:Educational institutions established in 1993
Category:1993 establishments in Malaysia
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Fuck You Github!
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Drill Press For Through-hole PCB Manufacturing
This drill press was built to drill through-hole printed circuit boards. [Rhys Goodwin] didn’t want to shell out for new equipment, so he dug through his scraps to see what he could accomplish. He already had the power drill, and there was no shortage of wood and fasteners. Once he had a mounting platform for the power tool he grabbed a pair of slides from and old rack-mount server rail. This provides smooth and precise movement, along with a tension sprint to keep the rig elevated above the work surface. Turns out the only thing he didn’t already have was the mini-chuck for gripping the 0.8 mm drill bit.
You’ll break enough bits to slow you down with a dremel press too. This might actually be better. Funny he’s not complaining about that HSS bit getting dull.
Figure on $150-$250 (USD) for a decent drill press with 1/4″ chuck that’s fast enough for fine carbide bits in fiberglass. Remember that even then you’re compromising: look at what old Electro-Mechano iron (found in a quick fact-checking search—it’d be real interesting to know what the real market is for first-quality high speed manual drill presses) still goes for.
My personal solution: I use cheap refurbished (Harbor Freight) drill bits in a Dremel with a drill press stand, and send any remotely complicated (think, more than you’d want to hand wire AGAIN) PCB out before I even think about etching and drilling. But I don’t live in New Zealand either.
A little rule of thumb when drilling(anything)
The velocity of the outside of the bit is proportional to the diameter. In other words the smaller the bit the faster you need to spin it. Obviously different materials and bits require different speeds, but given the same material a smaller bit will need a higher RPM. That is why a dremel, which operates at a very high RPM compared to a hand held drill is preferred to this type of setup.
I use a dremel model 220 stand. There are some adjustable brass screws on the side that should be tightened to get less “play” I also use a large rubber band(the kind you find wrapped around the base of broccoli) at the top of the press wrapping the power cord end of the drill to the metal cylinder of the stand this prevents any play that will quickly kill bits. I set the drill for as high of an RMP as I can stand(earplugs are a must) and am still on the FIRST BIT! after 6 months of use.(hundreds of holes)
I used to use a Dremel with a tiny drill bit (the one that came with the red plastic hand crank drill sold at Radio Shack). The secret to fast efficient hole drilling for a homebrew PCB was to use a piece of perfboard as a template drill guide (set on top). The holes to be drilled should be premarked with a sharpie.
While the press isn’t cheap, Proxxon makes a whole line of high-speed rotary tools which are priced similarly to Dremel’s offerings but have much better construction. I have their smallest tool, the Micromot 50 ($35-45 on Amazon). It’s a 12V, 20,000 RPM drill with several advantages over the Dremels: steel collets instead of aluminum, a machined metal neck for securing to your ghetto press, surprisingly small size, low noise, and virtually no vibration.
I was impressed enough to try to build my own press based on the Proxxon, and here’s a pic of the work in progress:
@Reboots: Your press looks awesome! As for Proxxon, thanks for the tip. I just checked them out and it looks like I can get them in NZ. (Amazon won’t ship here). When funds permit I’ll be looking to get some kind of decent rotary tool and ether adapt the above press or try to make a really smart one like your yours.
@Masta Squidge, great to know. Dremel has a large market segment and improvements will benefit a lot of people. I used the same Moto-Tool for 20 years before bearing runout became unacceptable, but have heard from many corners that quality had fallen off since then.
@Rhys, thanks for the kind words. If I were going to build the press over again I’d just use a precision linear bearing like this:
I’ve seen many reviews of the official dremel drillpress attachment and everybody always says it’s loose and awful until you mod it yourself to make it stable, so that’s the reality but I guess it’s not bad since if you have a dremel you are into modding anyway :)
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Q:
Make do not rebuild a undo-checkouted file under clearcase
I discovered a quite annoying issue with ClearCase and make. Make check if any of a target's dependencies is more recent than the target itself and this is is the way make should work.
If I undo-checkout a file on ClearCase. It's modification date will be older than the checkouted file. Thus make won't rebuild this file.
How can I solve this issue?
I see two possible solutions:
Tell ClearCase to update the modification date to when the undo-checkout was done.
Tell make to check if the modification date of the dependencies has changed (not to compare it to the target's modification date)
A:
You can check if a ClearCase Explorer option (also valid for ClearTeam 8.x) like "Preserve file modification time" (on add to source control or on checkout) would change anything.
It may have an influence on the timestamp when diong an undo-checkout.
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Intraperitoneal injection of dextran sulfate as an anti-adherent drug for the prevention of peritoneal metastasis of cancer shows low toxicity in animals.
Intraperitoneal dextran sulfate with a mean molecular weight of 5 x 10(5) has been developed for use in an anti-adherent therapy against peritoneal carcinomatosis. The present study examined acute toxicity of i.p. injection of dextran sulfate in mice and rabbits. The 10, 50 and 90% lethal dose values are 0.213 (0.146-0.252), 0.336 (0.291-0.405) and 0.530 mg/g (0.431-0.873 mg/g: 95% confidence interval) in mice, respectively. These are markedly larger than the efficacious dose of 0.005-0.01 mg/g obtained previously. Death or symptoms of intoxication were seen within 3 days after administration of toxic doses. Rabbits received i.p. injection of dextran sulfate at 0.02 mg/g, which was close to the efficacious dose. At 2, 4, 6, 8 and 13 days after administration, blood was taken for biochemical and hematological analyses. Dextran sulfate at 0.02 mg/g induced no remarkable abnormal findings. These results suggest that the i.p. dextran sulfate is safe as an anti-adherent agent against peritoneal metastasis of cancer.
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Q:
C# suspending all threads
I have a problem that may be fairly unique. I have an application that runs on a headless box for long hours when I am not present, but is not critical. I would like to be able to debug this application remotely using Visual Studio. In order to do so, I have code that looks like this:
// Suspend all other threads to prevent loss
// of state while we investigate the issue.
SuspendAllButCurrentThread();
var remoteDebuggerProcess = new Process
{
StartInfo =
{
UseShellExecute = true,
FileName = MsVsMonPath;
}
};
// Exception handling and early return removed here for brevity.
remoteDebuggerProcess.Start();
// Wait for a debugger attach.
while (!Debugger.IsAttached)
{
Thread.Sleep(500);
}
Debugger.Break();
// Once we get here, we've hit continue in the debugger. Restore all of our threads,
// then get rid of the remote debugging tools.
ResumeAllButCurrentThread();
remoteDebuggerProcess.CloseMainWindow();
remoteDebuggerProcess.WaitForExit();
The idea being that this way, I hit an error while I am away, and the application effectively pauses itself and waits for a remote debugger attach, which after the first continue automatically gets the right context thanks to the Debugger.Break call.
Here is the problem: Implementing SuspendAllButCurrentThread turns out to be nontrivial. Thread.Suspend is deprecated, and I can't P/Invoke down to SuspendThread because there's no one-to-one mapping between managed threads and native threads (since I need to keep the current thread alive). I don't want to install Visual Studio on the machine in question if it can possibly be avoided. How can I make this work?
A:
I can't P/Invoke down to SuspendThread because there's no one-to-one mapping between managed threads and native threads
You can't enumerate managed threads either, only unmanaged threads. There actually is a one-to-one mapping between them, they just made it hard to find it. The original intent was to allow creating a custom CLR host that didn't use operating system threads to implement Thread, a request by the SQL Server group that wanted to use fibers instead. That never worked out, they could not get it reliable enough. No actual CLR host exists that doesn't use real operating system threads.
So you can actually use Process.GetCurrentProcess().Threads to enumerate all your threads. And avoid suspending your own by pinvoking GetCurrentThreadId(), comparing it to ProcessThread.Id
How reliable that's going to be is a guess, don't try to do anything drastic like sending an alert to remind you that it is time to attach the debugger. You may have well suspended a thread that was executing code inside Windows and acquired a global lock. As well as a CLR worker thread, like the finalizer thread or the background GC thread.
The better approach is to use a separate guard process that does all this, just like a debugger will. Use a named EventWaitHandle that you create in the guard program and OpenExisting() in your main program. The guard program needs to WaitAny() on that wait handle as well as the process. Your main program can now simply call Set() to wake up the guard program. Which can now safely suspend all threads.
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Ya'akov Nehoshtan
Ya'akov Nehoshtan (; 22 April 1925 – 17 April 2019) was an Israeli politician and diplomat. He served as a member of the Knesset for Gahal between 1969 and 1974 and as ambassador to the Netherlands between 1982 and 1985.
Biography
Born in Kazanlak, Bulgaria, Nehoshtan attended high school in Vratsa and was a member of the Hashomer Hatzair youth group. He made aliyah to Mandatory Palestine in 1944, and studied law at the Hebrew University of Jerusalem, gaining certification as a lawyer. He joined the Irgun in 1944, and was arrested by the British authorities the following year and exiled to a detention camp in Eritrea. In 1947 he was moved to a camp in Kenya.
In 1948 he was amongst the founders of Herut. He became chairman of the party's Jerusalem branch in 1968, and the following year was elected to the Knesset on the Gahal list (an alliance of Herut and the Liberal Party). He lost his seat in the 1973 elections.
In 1979 he was appointed Deputy Chief of Mission at the Embassy of Israel in Washington, D.C. and in 1982 became ambassador to the Netherlands, a post he held until 1985.
His son, Ido, served as commander of the Israeli Air Force.
References
External links
Category:1925 births
Category:2019 deaths
Category:Bulgarian Jews in Israel
Category:People from Kazanlak
Category:Bulgarian emigrants to Israel
Category:Hebrew University of Jerusalem alumni
Category:Israeli lawyers
Category:Irgun members
Category:Ambassadors of Israel to the Netherlands
Category:Gahal politicians
Category:Herut politicians
Category:Members of the 7th Knesset (1969–1974)
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xx 8 1 1
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Availability: Usually Ships in 1 to 2 Business Days
Product Code:R-3108
Qty:
Description
An adorable gift to welcome baby sister and celebrate big sister! 5x7 white double tabletop frame with spaces for big sister and baby sister's handprints. Comes with two child-safe silver ink pads with instructions. Celebrate the wonderful beginnings of new life, new roles and new relationships.
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Q:
Problems with graph plotting looks awkward
I am adding data points onto a plotted graph and adding a trend line y=mx+b equation taken from Excel.
However, the output looks ugly and I am not sure why.
The trend line just stops, the maximum number on the y-axis is missing, and when this prints, the light coloured grid appears dull or not appear at all. Another issue is Case 2 at 60 Ohms will be next part of the data points, but I am going on a tangent here, formatting the three additional tables/graphs I can figure out later on when I get the graph looking correct.
Essentially, any suggestions in making this look more ideal if that means prettier or better is appreciated.
\documentclass{article}
\usepackage[letterpaper, portrait, margin=2cm]{geometry}
\usepackage{pgfplots}
\usepackage{booktabs}
\begin{document}
\section{Results}
\noindent
\begin{tabular}{@{}cc@{}}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\begin{axis}[
title={Case 1 at 40 $\Omega$},
xlabel={Electric Current (mA)},
ylabel={Electric Potential (V)},
xmin=0, xmax=50,
grid=both,
grid style={line width=.1pt, draw=gray!10},
major grid style={line width=.2pt,draw=gray!70},
minor tick num=5,
legend pos=north west,
ymajorgrids=true,
xmajorgrids=true,
yminorgrids=true,
xminorgrids=true,
grid style=dashed
]
\addplot[only marks, color=blue]
coordinates {
(15.61,0.598)
(23.99,0.924)
(30.30,1.173)
(44.70,1.718)
(14.55,0.561)
(16.66,0.642)
(46.80,1.799)
(143.6,5.555)
(28.22,1.086)
(18.19,0.701)
};
\addplot[no marks, thick, color=red] {0.0383*x - 0.0041 };
\end{axis}
\end{tikzpicture}
\begin{tabular}{c c c}
\toprule[1.5pt]
{\bf Electric Current (mA) } & {\bf Electric Potential (V)} \\
\midrule
15.61 & 0.598 \\
\midrule
23.99 & 0.924 \\
\midrule
30.30 & 1.173 \\
\midrule
44.70 & 1.718 \\
\midrule
14.55 & 0.561 \\
\midrule
16.66 & 0.642 \\
\midrule
46.80 & 1.799 \\
\midrule
143.6 & 5.555 \\
\midrule
28.22 & 1.086 \\
\midrule
18.19 & 0.701 \\
\bottomrule[1.5pt]
\end{tabular}
\end{tabular}
\end{document}
A:
Make graph prettier ... this is matter of taste. Anyway, see if the following result is acceptable:
In your MWE I add domain for trend line, define ymin and ymax, add missing & in table, simplify grids style definitions, redesign table:
\documentclass{article}
\usepackage[letterpaper, portrait, margin=2cm]{geometry}
\usepackage{pgfplots}
\usepackage{booktabs}
\begin{document}
\section{Results}
\begin{center}
\begin{tabular}{@{}c@{\qquad}c@{}}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\begin{axis}[
title={Case 1 at 40 $\Omega$},
xlabel={Electric Current (mA)},
ylabel={Electric Potential (V)},
xmin=0, xmax=50,
ymin=0, ymax=2,% <-- added
grid=both,
grid style={line width=.1pt, draw=gray!50},
major grid style={line width=.2pt,draw=gray},
minor tick num=5,
legend pos=north west,
%ymajorgrids=true,
%xmajorgrids=true,
grid=both,
%minorgrid,
%xminorgrids=true,
grid style=dashed
]
\addplot[only marks, color=blue]
coordinates {
(15.61,0.598)
(23.99,0.924)
(30.30,1.173)
(44.70,1.718)
(14.55,0.561)
(16.66,0.642)
(46.80,1.799)
(143.6,5.555)
(28.22,1.086)
(18.19,0.701)
};
\addplot[no marks, thick, color=red, domain=0:50] {0.0383*x - 0.0041};
\end{axis}
\end{tikzpicture}
&
\begin{tabular}{c c}
\toprule
\textbf{Electric} & \textbf{Electric} \\
\textbf{Current (mA)} & \textbf{Potential (V)} \\
\midrule
15.61 & 0.598 \\
23.99 & 0.924 \\
\addlinespace[3pt]
30.30 & 1.173 \\
44.70 & 1.718 \\
\addlinespace[3pt]
14.55 & 0.561 \\
16.66 & 0.642 \\
\addlinespace[3pt]
46.80 & 1.799 \\
143.6 & 5.555 \\
\addlinespace[3pt]
28.22 & 1.086 \\
18.19 & 0.701 \\
\bottomrule
\end{tabular}
\end{tabular}
\end{center}
\end{document}
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Wireless communication has proliferated over the past decade. One of the more recent areas in which wireless communication has expanded into is multi-channel audio distribution. Multi-channel audio generally refers to audio of a sound scene that was captured from multiple different directions. The captured audio in each direction represents one audio channel in the multi-channel audio. During rendering, each audio channel is sent to a separate speaker positioned within a room to ideally reproduce the audio in a more realistic manner than single-channel audio or multi-channel audio of a lesser degree.
Some of the more common multi-channel audio formats are described using two digits separated by a decimal point (e.g., 2.0, 2.1, 5.1, 6.1, 7.1, etc.). The first digit represents the number of primary audio channels, each of which is to be reproduced on a separate speaker. The second digit represents the presence of a low frequency effect (LFE) audio channel, which is to be reproduced on a subwoofer. To provide some specific examples, a 2.0 multi-channel audio format refers to two primary audio channels (or stereo sound) and no LFE audio channel, whereas a 5.1 multi-channel audio format refers to five primary audio channels and an LFE audio channel.
The clear benefit of wireless multi-channel audio distribution is that it eliminates the need for wires between an audio source and speakers. One existing technology that can be leveraged to wirelessly deliver multi-channel audio is the Institute of Electrical and Electronics Engineers (IEEE) 802.11 family of packet based wireless networks. These “WiFi” networks are ubiquitous, standardized, and can provide a large throughput, making them a good choice for wireless distribution of multi-channel audio. However, wireless distribution of multi-channel audio over such packet-based networks still presents challenges. For such a solution to compete with traditional wired systems, the solution should deliver and playback the multi-channel audio with near equal performance or better. In general, this means the solution should reproduce the multi-channel audio at the speakers with high fidelity, low delay, and perceptually tight synchronization.
Achieving high fidelity generally means zero or near-zero packet loss across the inherently lossy wireless channel. To combat packet loss, application layer forward error correction combined with some packet interleaving can be used. However, these traditional solutions typically fall short of the zero or near-zero packet loss requirement.
Low delay is usually important when the multi-channel audio is to be synced with video. In such an instance, the rendering time of the multi-channel audio with respect to the video generally should be no more than about 100 milliseconds (ms) late or no more than about 25 ms early. The asymmetric nature of this range is a result of the human audio-visual system being accustomed to audio arriving after video due to the speed of sound being slower than the speed of light. This range puts constraints on the amount of packet interleaving that can be applied to combat packet loss mentioned above.
Finally, synchronization across the speakers used to render the multi-channel audio is important because human perception of audio signals is sensitive to delays and phase shifts caused by out-of-sync playback. In general, humans can detect around 10-20 microseconds (μs) of delay and 1-2 degrees of phase difference between audio signals. At these sensitivities, 48 kHz sampled multi-channel audio (which corresponds to a sample separation of 20.8 μs) would require synchronization across speakers within one sample period. Thus, it is important to limit the difference in rendering time between speakers, referred to as “cross-jitter”. The listener should ideally perceive the combination of audio signals from the different channels as if they were being reproduced by a normal wired system. Too much cross-jitter results in echo and spatialization issues.
The present disclosure will be described with reference to the accompanying drawings. The drawing in which an element first appears is typically indicated by the leftmost digit(s) in the corresponding reference number.
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Menu
Heading Out to Sea
Our day began with a tension-inducing drive out to Cromarty for a boat ride on the Moray Firth. We were driving along okay, staring in wonder at the large bus ahead of us navigating the narrow streets of Fortrose, when our GPS detoured us around the town and up into the hills. I like to think we are not the type of people who would drive off a cliff because our GPS told us there was a road there. Instead, I think we were both just so relieved to get out of the narrow streets of the town that we went with it.
Our relief was short-lived, though, as we realized this was a windy, hilly, single-lane road through sheep pastures. Noting the significant number of manure piles on the pavement, I suggested that Steve better watch out in case there were any animals in the road.
We did get some rather stunning views of Chanonry Point up there, though.
We were running behind to get to the boat trip, so the drive was perhaps a bit quicker than either of us were comfortable with, but Steve handled the car and the road well, and we both breathed another sigh of relief as we turned back onto the main drag.
Finally in Cromarty, we got geared up for our dolphin watching trip with Ecoventures. Steve eyed the small, inflatable pontoon boat with some trepidation, but boarded it, placing his faith in the Bonine he had taken earlier.
You’ll notice there are no photos of dolphins on our dolphin watching trip. We were not fortunate enough to find dolphins, or any other wildlife other than a few sea birds.
But it was a beautiful, sunny day on the firth, Steve didn’t get seasick, and we got to see the headlands up close, so I can’t really complain, although Steve, apparently, did get a bit cold at some points. . .
It would have been nice to see dolphins, but it was quite enjoyable anyway.
Leaving Cromarty, we made our way to Rosemarkie to visit the Groam House Museum with their collection of Pictish stone carvings. It was a very small, but well-done collection. The volunteer docent was also very nice and, upon hearing that Steve was a Shaw, shared a bit of her local knowledge of the clan, along with the correct pronunciation of Tordarroch (emphasis on the second syllable; I think I’d been saying it like it was Klingon).
Back at the inn, and feeling exhausted, we decided to take a little nap before dinner. The morning boat ride, with the sun and wind, combined with the accumulation of the previous days’ activities, set me off on a path of drowsiness that just couldn’t be put off any longer. We did manage to rouse ourselves for dinner, but we were both a little zombified. Frankly, I was still half asleep through dinner. We had been unable to get into our first choice restaurant for dinner – we had wanted to go to Hootananny, which Steve felt a kinship with due to the southern hootenanny connection and which I was interested in primarily for the nightly live music.
We ate across the street instead, but were able to get a table afterwards at Hootananny for a drink. We tried desperately to make it until the music was set to start, but both decided we were about to fall asleep at the table and had better toddle off to bed. We did make a reservation for dinner tomorrow night. Hopefully we’ll feel more rested then.
We did stop on the walk back to the B&B to take a few pictures of Inverness Castle.
|
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UNITED NATIONS—Despite the economic crisis that has hit budgets across Europe, the European Union has proved it won't shirk from its international duties, one of its top leaders told the United Nations General Assembly on Wednesday.
In his speech, European Council President Herman Van Rompuy focused mainly on the crisis in Syria, appealing for a rapid political solution and offering Europe's help in advancing talks.
But...
|
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Ciudad Constitución
Ciudad Constitución is a city in the Mexican state of Baja California Sur. It is the seat of Comondú Municipality. As of 2015, the city had a total population of 44,918 inhabitants.. Ciudad Constitución is a small city which serves as a gateway to Magdalena Bay.
History
The colonization of Valle de Santo Domingo (Valley of Santo Domingo) originated around 1940. A ranch called El Crucero (The Crossroad) was settled in a crossroad. Because of this, people started to gather around it and the population started to grow. It quickly became a commercial hub and an obligatory travel stop for all the inhabitants of Valle de Santo Domingo, as well as for people going north or south on the peninsula. Then, it became known as Villa Constitución, and later, Ciudad Constitución. Even today, many locals still call it "El Crucero". Every 5 February the city becomes the state's capital for one day. Since 1996, the Instituto Tecnológico Superior de Ciudad Constitución serves as a higher education institution.
Economy
Ciudad Constitución's mainstay is the cultivation of wheat, chickpea, cotton, asparagus, citrics, vegetables, among others crops. The city also has a dairy products processing plant (pasteurized milk, yogurt, fruit beverages): "Unión de Ejidos 20 de Noviembre". Ciudad Constitución has a few small hotels (Hotel El Conquistador, Hotel Conchita, Hotel Maribel, Hotel Ryal), supermarkets (Super Ley, Super Murillo), gas stations, travel agencies (Viajes Pedrín), etc.
The Mexican long distance area code for the municipalities of Comondú and Loreto is 613. Internet access is possible in cybercafes through the city's downtown, as well as by dial-up access from regular landlines. Communication with remote areas of Valle de Santo Domingo, nearby islands, and other remote regions of the municipality of Comondú is mainly possible by means of a local AM radio station (XEVSD 1440 kHz).
Climate
Transportation
For air travel, the city is served by the Ciudad Constitución Airport, which is a small airfield where two regional airlines provide service to Los Mochis and Ciudad Obregón. The airport also handles air taxi service. The city's nearest major airports are located in Loreto (Loreto International Airport), La Paz (Manuel Márquez de León International Airport) and San José del Cabo (Los Cabos International Airport). Small aircraft make use of local dirt runaways. Autotransportes Águila is a bus line which covers all the length of the Baja California Peninsula, mainly along Mexican Federal Highway 1 (also known as "Carretera Transpeninsular"), and has an office in Ciudad Constitución.
Demographics
As of 2015, the city had a total population of 44,918 inhabitants. It is the fifth-largest community in the state (behind La Paz, San José del Cabo, Cabo San Lucas, and Colonia del Sol). Ciudad Constitución is a small city which serves as a gateway to Magdalena Bay. It is also close to the Baja 1000 course.
References
2010 census tables: INEGI: Instituto Nacional de Estadística, Geografía e Informática
Notes
External links
Google Earth
Category:Comondú Municipality
Category:Populated places in Baja California Sur
Category:Municipality seats in Baja California Sur
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Women Talk Tech Episode 6: Building a Community
Meet the woman who built the largest online engineering community on Instagram
On this week’s Women Talk Tech podcast we talk to Mina Dezz, Founder of the largest online community on Instagram “Iron Ring Girls” created to bring women in engineering together. Mina was a graduate of Civil Engineering in 2014 from the University of Toronto and is currently working in residential construction building custom homes. Mina impressively built a community of over 20,000 followers and created the guide “Building a Powerful Mind while Studying Engineering”.
This episode addresses the common challenges women experience in Engineering such as loneliness, independence, and the drastic change in environment. Mina speaks about her experience throughout university and how proper preparation can help manage this change a lot better.
Marie and Mina discuss:
The challenges girls face transitioning from high school to university
How to prepare for the engineering journey
The importance of communication and people skills in engineering
Dealing with uncertainty after graduation
Mina provides many helpful tips and strategies for dealing with a profession that is mostly men, many of which the technology industry should be adopting today. Her advice “always ask” is one of the most important lessons all young women should learn as they work their way up no matter what the career choice.
About Mina Dezz
Mina has been working in the field of engineering since she graduated in 2014. She graduated from Civil Engineering with a minor in Engineering Business from the University of Toronto. She started working in urban transportation design but switched to construction after 2 years. Mina currently works in residential construction, building custom homes.
Mina is also the creator of the largest online community of women in engineering on Instagram “IronRingGirls”. She has written her entire experience in studying engineering in her guide “Building a Powerful Mind while Studying Engineering”.
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| 0.08 |
Ontogeny of tyrosine hydroxylase and cholecystokinin gene expression in the rat mesencephalon.
The ontogeny of tyrosine hydroxylase and cholecystokinin gene expression was studied in the rat mesencephalon using hybridization histochemistry. Both transcripts appeared on E13 in the ventrocaudal mesencephalon. The levels of both transcripts increased synchronously during the second half of gestation. The locations of neurons containing either transcript changed similarly during development with a rostral transposition and a lateral extension of the respective areas covered with grains. On the day after birth, the patterns of expression for both genes, although at lower transcript levels, were similar to the patterns seen in the adult.
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13:40
I just want to Suck your Fat Cock
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Victim dies in Lake Lillinonah crash
Divers try to determine if there are occupants inside a vehicle that police said crashed into Lake Lillinonah in Bridgewater Saturday afternoon. Trooper Kelly Grant said that the vehicle was submerged in the lake near the Bridgewater boat launch. Saturday, January 9, 2016, in Bridgewater, Conn. less Divers try to determine if there are occupants inside a vehicle that police said crashed into Lake Lillinonah in Bridgewater Saturday afternoon. Trooper Kelly Grant said that the vehicle was submerged in the ... more Photo: H John Voorhees III / Hearst Connecticut Media Buy photo Photo: H John Voorhees III / Hearst Connecticut Media Image 1 of / 14 Caption Close Victim dies in Lake Lillinonah crash 1 / 14 Back to Gallery
A 46-year-old New Milford woman died after her car plunged into Lake Lillinonah in Bridgewater Saturday afternoon, said a spokesman with the Connecticut Department of Energy and Environmental Protection.
The unidentified victim was said to be unconscious when divers recovered her and her 2006 Toyota Scion that was submerged for more than two-hours in 40 feet of water.
According to Dennis Schain, a DEEP spokesman, the woman was transported to Danbury Hospital where medical officials were unable to revive her and she was pronounced dead at about 4 p.m., Schain said.
Officials said the vehicle plunged into Lake Lillinonah at about noon. Emergency crews responded shortly thereafter and found the car about 40 yards from the boat launch.
A heavy duty tow truck was used in the rescue operation to recover both the car and victim and state police, Bridgewater police, DEEP officials and dive teams from Brookfield, New Milford and Newtown’s Underwater Search and Rescue all responded to the scene.
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Vinylmation™ is a term that didn't exist a few years ago. Now it is a word that describes the latest collectible craze to hit the Walt Disney World Resort.
What is Vinylmation™?Vinylmation™ is the name for an ever growing collection of 3 dimensional figures made of vinyl that are shaped like Mickey Mouse, but are creatively decorated to become other characters or artistic expressions. The figures currently can be found in 3 different sizes 1.5", 3", and 9" tall. The 3" tall figures are the most popular.
The origin of the word Vinylmation™ comes from the combining of the words "Vinyl" which is the material used to make the figures and "animation" which Disney is certainly known for.
When Did This Craze Begin?
It all started in 2008 with the initial Vinylmation™ Figures appearing for sale on November 7, 2008. The very first figures for sale were blank, white ones that Disney Guests could decorate themselves. A little over a month later the first of many Vinylmation characters to come were introduced on December 18th. These were part of the Park #1 Series which came already decorated by Disney artists to look like theme park attractions or characters like Kermit the Frog and Figment.
Where Can You Find Them?
You can find the Vinylmation characters for sale all over the Walt Disney World Resort. The "home base" of sorts would have to be D-Street in Downtown Disney. Here you can find an amazing assortment of Vinylmation Figures, and you can also see displays of figures that have been retired and are no longer available.
There is also a large representation of Vinylmation at the Art Of Disney Store in Epcot. At first I though that was crazy, but after you look at all of the designs and realize the creativity and artistic talent that it takes to take this simple Mickey Mouse shaped figure and turns it into this widely varied series of characters and themes. It really does belong in the store that sells art!
How To Collect?
Like any other type of collection, you are best off collecting what you like. The 3" high figures typically sell for $9.99 each, so although they are not free, the price is reasonable and similar to the price of a Disney Pin.
You can try to collect entire themed collections, or just ones that you like. It doesn't matter, each collection should be unique.
You Can Trade Also!
Similar to the popular Cast Member Disney Pin Trading that occurs at Disney World, there is also Vinylmation Trading that occurs at specific sites throughout Disney World. So don't fear if you receive a Vinylmation Figure that is not your favorite, you will be able to trade it to a Cast Member for a different Vinylmation Figure of the same size.
What Are You Waiting For?
If you haven't purchased your first Vinylmation Figure what are you waiting for? If you spend some time checking out all of the different characters and designs, you will find it hard not to become attached to one or more of them.
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fuck?
go and fuck!
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| 0.105263 |
All You Want Is Cock
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Risky Public Outdoor Hardcore Fucking and Deepthroat Blowjob in the Woods.
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| 0.054054 |
Iodanthine, a pyrrolizidine alkaloid from senecio iodanthus and senecio bracteatus
A phytochemical study of Senecio iodanthus and Senecio bracteatus afforded the new pyrrolizidine alkaloid iodanthine (3) in addition to the four known alkaloids retroisosenine (1), bulgarsenine (2), mulgediifoliine (4), and (12S)-12-hydroxyretroisosenine (6). The structure and absolute stereochemistry of the new compound (3) were determined by its spectral properties and confirmed by an X-ray diffraction analysis.
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I love calling a sissy gurl's penis a clitty. Every guy that is circumcised is just one bad snip away from being a girl. About 5% of infant circumcisions result in the baby male's penis being destroyed and they just convert them to girls because of it. They just cut off his baby balls and take the circ a bit further and make his baby penis "her" new clitty.
>>683513680just bought my first panties from there. it was nervewracking. everything else i've taken from roommates or the laundromat>>683513582source? I don't believe it for a second by why couldn't I have been in that 5% :'(
>mfw all these idiots in this thread are confusing trannies with traps.
The entire point of referring to them as "traps" is that they are MEN who are so effeminate and androgynous that when you slap some girls clothes on them that you would never know the difference.
The moment tits enter into the equation you completely destroy the "man" aspect that makes it a trap. Because there are tons of heterosexual guys who are horny or desperate enough that they couldn't care less if the tits they are busting a nut to come attached to a dick.
So therefor it's no longer a trap, you are just a regular faggot jerking it to regular shemale porn.
You're not a heterosexual guy who was seduced by a fellow man and was deceived into jerking it to a hidden penis.
>>683514584But you already know its a trap so the surprise isn't there so you are just a giant faggot as well and if im ever balls deep in a tranny, i aint gnna be thinking if others think im a faggot, imma fuck the tranny and enjoy it
>>683514876I know i could but i have to pick my packages up from the leasing office and i think i was pretty red when I picked up my dildo from the sweet old lady, even though it was packaged discretely
>>683506787GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT
i think you should try having a little pussy hair. most guys grow out of wanting shaved girls after their teens and love for their girls/gurls to have a little hair to play with. plus if your guy is really good, your hair if played with right will make you even more horny and leak more(even if caged)
>>683506787GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT
>>683506787GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT
Definitely do, it adds more pleasure in dominating them and making them more obedient... Also love seeing their little clitty twitch with pleasure when it's been caged up for so long, and then getting release from all the teasing. Also more pictures ;)
>>683506787GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT GET THE FUCK OUT OF /b/ FUCKING FAGGOT
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| 0.058142 |
In Beijing, I was one of 13 roommates, all models, most of whom were under 18 and from the Eastern Bloc. Nine girls, four boys, give or take, during the busy season. Lana was one of them, though she wasn’t the youngest. The youngest was 14-year-old Kate, a shy Ukrainian girl who, out of all of our roommates, worked the most — which, in industry-speak, meant that she was frequently in demand. Kate celebrated her 14th birthday two months into her contract, sitting down with us between jobs to eat a store-bought cake that we purchased with our pocket money. Most of her jobs were nine hours a day, requiring last-minute travel and hotel stays (usually with a handler and another model) to the homonymous cities that comprise China’s fashion economy: Shenzhen, Guangzhou, Hangzhou.
Our shabby, dorm-like apartment consisted of four bedrooms (two baths) for the 13 of us. When we arrived, I found I had no bed — only a stained, faded couch covered in a fake Gucci blanket, for which I was charged $500/month. As a model, inflated rent and poor living conditions are unfortunately industry standards. Rent is charged as part of your expenses and later deducted from any jobs you may (or may not) do — which is what makes booking them so important.
In Asia’s modeling industry, working under an illegal tourist visa is commonplace. Models are told by their scouts to say they’re visiting on vacation, putting them in a precarious situation in which they’re entirely in the hands their employers: If a model denies even one job, she may lose her contract and her apartment. She can’t work elsewhere, because she’s a tourist. And if her family or agency refuses to support her, she doesn’t have the financial means to get home. [Source]
This came from one of the more bizarre episodes in Ms. Hattam’s brief modeling career, when she was sent in September 2011 in a group of 40 models to the ancient desert town of Dunhuang to take part in a fake beauty pageant for Chinese tourists, complete with sashes, bikinis and nightgowns.
Ms. Hattam performed as Miss America. A model from Warsaw was appointed Miss India. And so on.
I walked. I waved. I smiled even as my trailing dress fell to my stomach, exposing Miss America’s stars and stripes to 4,000 Chinese strangers, who let out a collective gasp.
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she is rubbing her arabic wet pussy on my dick till cumshot
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| 0.050847 |
Welcome to Catholicism FELT
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Catholicism FELT is a result of years of prayer, discernment, study, and ministering, which in turn serve to inform the content presented here. Catholicism FELT endeavors to examine the relationships between Catholicism and faith, evangelization, life, and theology. This site hopes to share insights and experiences that relate to the invitation to be faithful, well-informed Catholics. It seeks to make sense of what we believe and of the challenges we experience daily as we attempt to follow Christ. Click here to learn more…
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---
abstract: 'We provide a probabilistic proof of a well known connection between a special case of the Allen-Cahn equation and mean curvature flow. We then prove a corresponding result for scaling limits of the spatial $\Lambda$-Fleming-Viot process with selection, in which the selection mechanism is chosen to model what are known in population genetics as [*hybrid zones*]{}. Our proofs will exploit a duality with a system of branching (and coalescing) random walkers which is of some interest in its own right.'
author:
- 'Alison Etheridge[^1]'
- 'Nic Freeman[^2]'
- 'Sarah Penington[^3]'
bibliography:
- 'curvature.bib'
title: 'Branching Brownian Motion, mean curvature flow and the motion of hybrid zones'
---
v ¶ \~ ł
Introduction
============
Our central result, Theorem \[thm:slfvs\] in Section \[sec:slfvs\_intro\], is the convergence, after suitable rescaling, of a stochastic analogue of the Allen-Cahn equation to the indicator function of a region whose boundary evolves according to mean curvature flow. The main motivation for this work comes from mathematical population genetics; specifically, we are interested in the behaviour of so-called hybrid zones. These occur when genetically distinct groups of individuals meet and mate, leaving behind at least some offspring of mixed ancestry. A textbook example is the common house mouse in Denmark [@hunt/selander:1973] which exists in the form [*Mus musculus*]{} in the North and [*M. domesticus*]{} in the South, but hybrid zones are ubiquitous in nature, for example, [@barton/hewitt:1989] cite 170 examples. Two principal explanations have been offered for the genetic variation observed in such zones. The first is that they arise in response to spatially varying natural selection; the second is that they are formed through secondary contact of two populations that were previously genetically isolated. Whereas in the first scenario the location of the hybrid zone is determined by an environment, which is usually taken to be fixed, in the second scenario, the hybrid zone can evolve with time. It is this second scenario that interests us here.
It is usual to suppose that the underlying genetics is controlled by a single gene which occurs in two types (alleles), traditionally denoted $a$ and $A$. Individuals carry two copies of the gene and while those of types $aa$ and $AA$ (the [*homozygotes*]{}) are equally fit, the [*heterozygotes*]{} (that is individuals of type $aA$) are less likely to successfully reproduce. In an infinitely dense population, provided the selection against heterozygotes is weak, when viewed over large spatial and temporal scales, the proportion of $a$-alleles in the population at location $x$ at time $t$ is modelled by the solution to $$\label{AC1}
\frac{\partial v}{\partial t}=\Delta v+\v{s} v(1-v)(2v-1),$$ for an appropriate initial condition, where $\v{s}>0$ is a scaled selection coefficient. This is a special case of the Allen-Cahn equation; we explain the origin of this particular form of nonlinearity in Section \[slfvs for hybrid zones\].
Our interest is in the behaviour of the region in which both alleles are present in substantial numbers. Because heterozygotes are less fit than homozygotes, we expect this to be a narrow band which, when viewed on large enough scales, will look like a sharp interface. More formally, we apply a diffusive scaling to (\[AC1\]) in which $t\mapsto \epsilon^2t$ and $x\mapsto \epsilon x$. The Laplacian term is, of course, invariant, but the term corresponding to selection is multiplied by a factor $1/\epsilon^2$. It is well known that for suitable initial conditions, in a sense that we make precise in Theorem \[theorem ac to cf\], as $\epsilon\downarrow 0$, the solution to the scaled equation converges to the indicator function of a set whose boundary evolves according to mean curvature flow. Thus, in the biologically relevant case of two dimensions, if we observe the population over sufficiently large spatial and temporal scales, the interface between the two populations will evolve approximately as curvature flow or [*curve-shortening flow*]{} as it is often known.
One reason for the importance of curvature flow in applications stems from an underlying variational principle: curve shortening flow decreases the length of the curve at the fastest rate possible relative to the total speed of motion (measured in the sense of the square integral of the speed of motion of points around the curve), see e.g. [@white:2002] for a simple explanation. In this sense, if our populations evolved deterministically, then they would minimise the boundary between them as quickly as possible. In reality this will be somewhat offset by the randomness due to reproduction, known as random genetic drift, in a population which is not infinitely dense. Indeed if genetic drift is too strong, then we can expect the random noise to obscure the nonlinear term: this is suggested by the results of [@hairer/ryser/weber:2012], who consider the equation $$dw=(\Delta w +w-w^3)dt +\sigma dW,$$ in two dimensions, where $W$ is a mollified space-time white noise. (By considering $(1+w)/2$, up to constants, we recover a stochastic version of (\[AC1\]).) If the mollifier is removed, then the solutions converge weakly to zero, whereas if the intensity of the noise simultaneously converges to zero sufficiently quickly, then they recover the deterministic equation. The basic question that we set out to answer is “Will hybrid zones still evolve approximately according to curvature flow in the presence of random genetic drift?”
Of course, genetic drift is not appropriately modelled by a mollified space-time white noise and so, in order to investigate this question, we must first define a model that combines selection against heterozygosity with random genetic drift. Our starting point will be the spatial $\Lambda$-Fleming-Viot process which was introduced in [@etheridge:2008; @barton/etheridge/veber:2010] and has been studied in a series of papers since; see e.g. [@barton/etheridge/veber:2013] for a review. The advantage of this model is that it allows us to incorporate genetic drift into models of populations evolving in spatial continua, with no restriction on spatial dimension. However, since our proofs are based on a duality with a branching and coalescing random walk, we expect analogous results if we start, for example, from the classical stepping stone model in which the population is subdivided into ‘islands’ that sit at the vertices of $\IZ^\dim$. In what follows, we shall refer to the spatial $\Lambda$-Fleming-Viot process with selection against heterozygosity as the . It is described carefully in Definition \[slfvdefn\]. A version of this model with selection in favour of one genetic type was constructed in [@etheridge/veber/yu:2014]. There it was shown that when suitably rescaled, in two or more dimensions, the allele frequencies converge to a solution of the Fisher-KPP equation, $$\label{fisher KPP}
\frac{\partial v}{\partial t}=\Delta v+\v{s} v(1-v).$$ Mimicking that result, one can obtain (\[AC1\]) as a scaling limit of the . Combined with the known convergence of the scaled version of (\[AC1\]), this certainly suggests that there should be scalings of the which lead to mean curvature flow. However, available proofs of Theorem \[theorem ac to cf\] could not readily be adapted to our stochastic setting and so we were forced to seek an alternative approach. Our first result is therefore a new proof of Theorem \[theorem ac to cf\]. We then adapt this to prove convergence of the proportions of different genetic types under the to the indicator function of a set whose boundary evolves according to mean curvature flow. The key to our proof is a probabilistic representation of solutions to (\[AC1\]) which we believe to be of interest in its own right.
Before defining the , we recall some purely deterministic results. Although our primary interest is in two spatial dimensions, there will be no additional arguments required if we work in $\R^{\dim}$ for arbitrary $\dim >1$.
The Allen-Cahn equation and mean curvature flow
-----------------------------------------------
The Allen-Cahn equation [@allen/cahn:1979] takes the form $$\label{allen-cahn equation}
\frac{\partial v^\epsilon}{\partial t}=\Delta v^\epsilon -\frac{1}{\epsilon^2}f(v^\epsilon),$$ where $f$ is the derivative of a potential function $F$ which has exactly two local minima, at $v_-$ and $v_+$, say. More precisely, we insist that $f\in C^2(\R)$ has exactly three zeros, $v_-<v_0<v_+$, and $$\label{conditions on potential}
\begin{array}{ll}
f(v)<0, & \forall v\in (-\infty, v_{-})\cup (v_0, v_+);\\
f(v)>0, & \forall v\in (v_-,v_0)\cup (v_+,\infty);\\
f'(v_-)>0, & f'(v_+)>0, \quad f'(v_0)<0.
\end{array}$$ Although originally introduced as a model for the macroscopic motion of phase boundaries driven by surface tension, the Allen-Cahn equation has found application in many other areas. It represents a balance between two opposing tendencies: the diffusive effect of the Laplacian attempts to smooth the solution, while the potential term drives it towards the states $v_-$ and $v_+$. As a result, a narrow interface between these two states develops.
Allen and Cahn observed that if the two potential wells do not have equal depth, then on the timescale $s=t/\epsilon$, the interface will propagate at a constant speed (proportional to $F(v_-)-F(v_+)$) along its normal, towards the domain of the deeper well. On the other hand, if the potential wells have equal depth, then the interface is almost stationary on this timescale, but if we observe it over the longer timescales of (\[allen-cahn equation\]), it will propagate with normal velocity equal to the mean curvature of the interface.
There is now a huge literature that makes the observation of Allen and Cahn rigorous under various regularity conditions, for example [@bronsard/kohn:1991; @evans/soner/souganidis:1992; @ilmanen:1993; @sato:2008]. The principal obstruction to be overcome relates to the fact that the mean curvature flow is only well-defined under some regularity conditions and, even then, only up to a finite time horizon when it either shrinks to a point or, in dimensions three and higher, develops other singularities.
Before stating a result, let us make the definition of mean curvature flow precise. We begin with the special case of two dimensions. This is the relevant dimension for our biological application and requires much less explanation. In that setting, mean curvature is just curvature and the corresponding flow is often called curve-shortening.
Recall that a function is said to be a smooth embedding if it is a diffeomorphism onto its image (which we shall implicitly assume is a subset of $\R^2$).
\[def:curvatureflow\] Let $S^1$ denote the unit circle in $\R^2$. Let $\v{\Gamma}=(\v{\Gamma}_t(\cdot))_t$ be a family of smooth embeddings, indexed by $t\in[0,\mathscr{T})$, where for each $t$, $\v{\Gamma}_t:S^1\to\R^2$. Let $\v{n}=\v{n}_t(\phi)$ denote the unit (inward) normal vector to $\v{\Gamma}_t$ at $\phi$ and let $\kappa=\kappa_t(\phi)$ denote the curvature of $\v{\Gamma}_t$ at $\phi$. We say that $\v{\Gamma}$ is a [*curvature flow*]{} or [*curve-shortening flow*]{} if $$\label{eq:cf_pre}
\frac{\p \v{\Gamma}_t(\phi)}{\p t}=\kappa _t(\phi)\v{n}_t(\phi).$$ for all $t,\phi$.
Assuming that $\v{\Gamma}_0$ is a smooth embedding of $S^1$ into $\R^2$, the behaviour of $\v{\Gamma}_t$ under curve-shortening is completely understood. First, it has a finite lifetime which we shall denote by $\mathscr{T}$. In [@gage/hamilton:1986], it was shown that if $\v{\Gamma}_0$ is convex, then so is $\v{\Gamma}_t$ for all $t<\mathscr{T}$. Moreover, $\mathscr{T}$ can be chosen so that $\v{\Gamma}_t$ shrinks towards a point as $t\uparrow\mathscr{T}$; in this limit the asymptotic ‘shape’ of $\v{\Gamma}_t$ is a circle. Soon afterwards, [@grayson:1987] showed that, in fact, under curve-shortening, any smoothly embedded closed curve becomes convex at a time $\tau<\mathscr{T}$, after which the results of @gage/hamilton:1986 apply.
In higher dimensions we must replace the curvature by the [*mean curvature*]{}. Recall that to define this quantity for a $(\dim -1)$-dimensional hypersurface in $\R^{\dim}$, we take an orthonormal basis of the tangent space and form the matrix of the second fundamental form, that is the matrix whose $(i,j)$th entry is the dot product of the unit normal to the hypersurface with the derivative of the $i$th vector in the basis in the direction of the $j$th. The $\dim-1$ principal curvatures, $\kappa_1,\ldots ,\kappa_{\dim-1}$, are the eigenvalues of the matrix and their sum, that is the trace of the matrix, is the (scalar) mean curvature. The product of the scalar mean curvature with the unit normal is called the mean curvature vector (which does not depend on the choice of normal, since reversing the direction of the normal also changes the sign of the scalar mean curvature).
[*Mean curvature flow*]{}, when it is defined, is obtained by replacing the curvature $\kappa_t$ in equation (\[eq:cf\_pre\]) by the mean curvature.
The behaviour of mean curvature flow in $\dim\geq 3$ is more complex than that of curve-shortening. It was proved by [@huisken:1984] that the analogue of the Gage-Hamilton Theorem holds, that is a $(\dim-1)$–dimensional compact convex surface must shrink to a point and its asymptotic shape is a sphere. However, the analogue of Grayson’s Theorem is false. In higher dimensions singularities can develop before the enclosed volume vanishes. Since our main interest is in two dimensions, we shall not discuss this here. Instead we shall follow [@chen:1992] in imposing sufficiently strong initial conditions that the solution exists for a positive time and stopping before we encounter any singularities, and we refer to [@mantegazza:2011] for a detailed discussion.
Suppose that $\dim\geq 2$. Our first result concerns the convergence as $\epsilon\downarrow 0$, for suitable initial conditions, of the solution of $$\label{special allen-cahn equation}
\frac{\partial v^\epsilon}{\partial t}=\Delta v^\epsilon
+\frac{1}{\epsilon^2}v^\epsilon (1- v^\epsilon)(2 v^{\epsilon} -1),\qquad v^\epsilon(0, x)=p(x),$$ to the indicator function of a set whose boundary evolves according to mean curvature flow.
The initial condition, $p$, of (\[special allen-cahn equation\]) is assumed to take values in $[0,1]$. We shall also require that it satisfies some regularity conditions. In particular, set $$\Gamma=\l\{x\in\R^\dim : p(x)=\frac{1}{2}\r\}.$$ We suppose that $\Gamma$ is a smooth hypersurface which is also the boundary of a bounded open set which is topologically equivalent to the sphere. We impose the following regularity conditions:
1. $\Gamma$ is $C^{\alpha}$ for some $\alpha>3$.
2. For $x$ inside $\Gamma$, $p(x)<\tfrac{1}{2}$. For $x$ outside $\Gamma$, $p(x)>\tfrac{1}{2}$.
3. There exist $r,\gamma>0$ such that, for all $x\in\R^{\dim}$, $|p(x)-\frac{1}{2}|\geq \gamma\,\big(\text{dist}(x,\Gamma)\wedge r\big)$.
In particular, we can think of $\Gamma$ as the image of the boundary of the unit sphere under a map $f$ for which $|f(x)-f(y)|=\mc{O}(|x-y|^{\alpha})$. Condition prevents the slope of $p$ near the interface $\Gamma$ from being too shallow, and keeps $p(x)$ bounded away from $\frac{1}{2}$ when $x$ is not near the interface. Condition is simply establishing a sign convention. Under these conditions, mean curvature flow started from $\Gamma$, which we denote $(\v{\Gamma}_t(\cdot))_t$, exists up to some finite time $\mathscr{T}$ (e.g. [@evans/spruck:1991]).
To give a precise statement of the result, we require some more notation. Let $d(x,t)$ be the signed distance from $x$ to $\v{\Gamma}_t$, chosen to be negative inside $\v{\Gamma}_t$ and positive outside. Note that, as sets, $$\v{\Gamma}_t=\{x\in\R^\dim : d(x,t)=0\}.$$
\[theorem ac to cf\] Let $v^\epsilon$ solve (\[special allen-cahn equation\]) with initial condition $p$ satisfying the conditions -, and define $\mathscr{T}$, $d(x,t)$ as above. Fix $T^*\in (0,\mathscr{T})$. Let $k\in\N$. There exists $\epsilon_\dim(k)>0$, and $a_\dim(k),c_\dim(k)\in(0,\infty)$ such that for all $\epsilon\in(0,\epsilon_\dim)$ and $t$ satisfying $a_\dim\epsilon ^2 |\log \epsilon |\leq t\leq T^*,$
1. for $x$ such that $d(x,t)\geq c_\dim\epsilon |\log \epsilon|$, we have $v^\epsilon (t,x)\geq 1-\epsilon^k$;
2. for $x$ such that $d(x,t)\leq -c_\dim\epsilon |\log \epsilon|$, we have $v^\epsilon (t,x)\leq \epsilon^k$.
This result is not new; it is a special case of Theorem 3 of [@chen:1992]. Indeed, our proof will display the same key steps: first we show that an interface develops; second we show that this interface propagates according to (mean) curvature flow. To achieve the second step, we couple the distance between a $\dim$-dimensional Brownian motion and the interface $\Gamma_s$ with a one-dimensional Brownian motion. This parallels the approximation of the solution to the Allen-Cahn equation by a one-dimensional standing wave in the proof of [@chen:1992] (although we remark that we achieve our coupling through a different perturbation of the potential than that used by [@chen:1992]). Both steps of our proof use probabilistic arguments, exploiting a duality between solutions to (\[special allen-cahn equation\]) and a branching Brownian motion, which is of some interest in its own right.
Modelling hybrid zones {#slfvs for hybrid zones}
----------------------
Let us now turn to our model of hybrid zones. Our starting point is the spatial $\Lambda$-Fleming-Viot process with selection. The model we consider here is a modification of that introduced for genic selection (selection in favour of just one of the alleles) in [@etheridge/veber/yu:2014], and existence of the process follows by the same arguments. Also as for genic selection, uniqueness follows from duality with a system of branching and coalescing particles, although there is a slight twist in the form that duality takes (see Section \[duality for SLFVS\]), mirroring our probabilistic representation of solutions to (\[AC1\]).
We suppose that there are two alleles, $a$ and $A$. At each time $t$, the random function $\{w_t(x),\, x\in \R^\dim\}$ is defined, up to a Lebesgue null set of $\R^\dim$, by $$\label{defn of w}
w_t(x):= \hbox{ proportion of type }a\hbox{ at spatial position }x\hbox{ at time }t.$$ In other words, if we sample an allele from the point $x$ at time $t$, the probability that it is of type $a$ is $w_t(x)$.
It is convenient to extend the definition of $w_t(x)$ to all of $\R^{\dim}$ and so, on the Lebesgue null set on which (\[defn of w\]) is not sufficient to specify $w_t(x)$, we shall arbitrarily impose $w_t(x)=0$.
A construction of an appropriate state space for $x\mapsto w_t(x)$ can be found in [@veber/wakolbinger:2015]. Using the identification $$\int_{\R^d} \big\{w(x)f(x,a)+ (1-w(x))f(x,A)\big\}\, dx=\int_{\R^d\times \{a,A\}} f(x,\kappa) M(dx,d\kappa),$$ this state space is in one-to-one correspondence with the space ${\cal M}_\lambda$ of measures on $\R^\dim \times\{a,A\}$ with ‘spatial marginal’ Lebesgue measure, which we endow with the topology of vague convergence. By a slight abuse of notation, we also denote the state space of the process $(w_t)_{t\in\R}$ by ${\cal M}_\lambda$.
\[slfvdefn\] Fix $u\in (0,1]$ and $\mc{R}\in(0,\infty)$. Let $\mu$ be a finite measure on $(0,\mc{R}]$. Further, let $\Pi$ be a Poisson point process on $\R_+\times \R^\dim \times (0,\mc{R}]$ with intensity measure $$\label{slfvdrive}
dt\otimes dx\otimes \mu(dr). $$ The [*spatial $\Lambda$-Fleming-Viot process with selection*]{} (SLFVS) driven by $\Pi$ is the ${\cal M}_\lambda$-valued process $(w_t)_{t\geq 0}$ with dynamics given as follows.
If $(t,x,r)\in \Pi$, a reproduction event occurs at time $t$ within the closed ball $\mc{B}_r(x)$ of radius $r$ centred on $x$. With probability $1-\v{s}$ the event is [*neutral*]{}, in which case:
1. Choose a parental location $z$ uniformly at random within $\mc{B}_r(x)$, and a parental type, $\alpha_0$, according to $w_{t-}(z)$, that is $\alpha_0=a$ with probability $w_{t-}(z)$ and $\alpha_0=A$ with probability $1-w_{t-}(z)$.
2. For every $y\in \mc{B}_r(x)$, set $w_t(y) = (1-u)w_{t-}(y) + u{\mathbf{1}}_{\{\alpha_0=a\}}$.
With the complementary probability $\v{s}$ the event is [*selective*]{}, in which case:
1. Choose three ‘potential’ parental locations $z_1, z_2, z_3$ independently and uniformly at random within $\mc{B}_r(x)$, and at each of these sites ‘potential’ parental types $\alpha_1$, $\alpha_2$, $\alpha_3$ according to $w_{t-}(z_1), w_{t-}(z_2), w_{t-}(z_3)$ respectively. Let $\widehat{\alpha}$ denote the most common allelic type in $\alpha_1,\alpha_2,\alpha_3$.
2. For every $y\in \mc{B}_r(x)$ set $w_t(y) = (1-u)w_{t-}(y) + u{\mathbf{1}}_{\{\widehat{\alpha}=a\}}$.
More generally, the parameter $u$, which we shall refer to as the [*impact*]{}, can be taken to be random. In this case, for each $r\in (0,\mc{R}]$, we let $\nu_r$ be a probability measure on $(0,1]$ and the driving noise, $\Pi$, is taken to be a Poisson point process on $\R_+\times \R^\dim \times (0,\mc{R}]\times (0,1]$ with intensity measure $$dt\otimes dx\otimes \mu(dr)\nu_r(du).$$ For each point $(t,x,r,u)\in\Pi$, the corresponding reproduction event is described exactly as before.
Since $\v{s}$ is assumed small, as one expects in a model of genetic drift, to first order the variance of the increment of the mean allele frequency in the region affected by an event is $u^2\bar{w}(1-\bar{w})$, where $\bar{w}$ is the mean of $w_{t-}$ over the affected region. Let us try to motivate the form of the selection mechanism, which is what drives the expectation of the increments in allele frequencies. As is usual in population genetics, we have approximated a model of selection acting on a diploid population (in which each individual carries two copies of the gene) by one in which we think of selection acting on single copies of the gene, but in a way that depends on the local frequencies of the different alleles. This sort of approximation, which goes back at least to [@fisher:1937], is valid when the local population size is large, corresponding in our case to the impact $u$ being small. (In fact we are interested in limits in which the impact will tend to zero.) The idea is simple. Each individual in the population carries two copies of the gene. This subdivides the population into [*homozygotes*]{}, carrying either $aa$ or $AA$ and assumed equally fit, and [*heterozygotes*]{} carrying $aA$ and assumed to have relative fitness $1-\v{s}$. The population is assumed to be in Hardy-Weinberg proportions, so that if the proportion of $a$-alleles in the parental population is $\bar{w}$, then the proportions of parents that are of type $aa$, $aA$ and $AA$ are $\bar{w}^2$, $2\bar{w}(1-\bar{w})$ and $(1-\bar{w})^2$, respectively. During reproduction, each individual produces a very large number of germ cells (cells of the same genotype). To reflect the relative fitnesses, a heterozygote produces $(1-\v{s})$ times as many germ cells as a homozygote. Germ cells then split into an effectively infinite pool of gametes (cells containing just one chromosome from each pair) which fuse at random to form diploid offspring. Suppose that the proportion of type $a$ alleles in the affected region immediately before reproduction is $\bar{w}$. Then the probability that a gamete sampled from the pool is of type $a$ is $$\begin{aligned}
\frac{\bar{w}^2+\bar{w}(1-\bar{w})(1-\v{s})}{1-2\v{s}\bar{w}(1-\bar{w})}
&=&(1-\v{s})\bar{w}+\v{s}(3\bar{w}^2-2\bar{w}^3)+{\mathcal O}(\v{s}^2)
\nonumber\\
&=& (1-\v{s})\bar{w}+\v{s}(\bar{w}^3+3\bar{w}^2(1-\bar{w}))+{\mathcal O}(\v{s}^2).
\label{selection mechanism}\end{aligned}$$ Notice that the first term in (\[selection mechanism\]) is $1-\v{s}$ times the probability that an allele sampled from the parental population is of type $a$ whereas the second is $\v{s}$ times the probability that the majority of three alleles sampled independently from the parental population are of type $a$. This then motivates the two types of event in our SLFVS. In particular, if we replace a proportion $u$ of the population by offspring, then the expected increment in $\bar{w}$ is $$u \v{s}(\bar{w}^3+3\bar{w}^2(1-\bar{w})-\bar{w})
=u\v{s}\bar{w}(1-\bar{w})(2\bar{w}-1),$$ which underpins the connection to (\[AC1\]).
Of course, in replacing a diploid model by one based directly on allele frequencies, we have rather muddied the notion of parent in our reproduction mechanism, so the use of the term in Definition \[slfvdefn\] should not be interpreted too literally.
Convergence of the hybrid zone to mean curvature flow {#sec:slfvs_intro}
-----------------------------------------------------
To understand our main result, first we state a simple modification of a result on a rescaling of the SLFVS from [@etheridge/veber/yu:2014]. To state that result, we specialise to $\mu(dr)= \delta_R(dr)$, for some fixed $R>0$. At the $n$th stage of the rescaling, the impact and selection parameters are assumed to satisfy $$u_n = \frac{u}{n^{1-2\beta}}, \qquad \mbox{and}
\qquad \v{s}_n=\frac{\rho}{n^{2\beta}}.$$ Next, we define the averaged process, $$w^n_t(x) := w_{nt}(n^{\beta}x), \qquad \hbox{and}\qquad
\bar{w}^n_t(x):=\frac{n^{\beta \dim}}{V_R}\, \int_{B(x,n^{-\beta}R)}w^n_t(y)\, dy,$$ where $V_R$ is the volume of the ball of radius $R$ in $\R^{\dim}$. To simplify notation, we write ${\mathcal M}$ for ${\mathcal M}_\lambda(\R^{\dim} \times\{a,A\})$, and $D_{\mathcal M}[0,\infty)$ for the set of all càdlàg paths with values in ${\mathcal M}$. We also write $C_c^\infty(\R^{\dim})$ for the set of smooth compactly supported functions on $\R^{\dim}$.
\[th:evy\]\[Modification of Theorem 1.3 of [@etheridge/veber/yu:2014]\] Suppose that $\beta\in (0,1/3)$, and that $\bar{w}^n_0$ converges weakly to some $w^0\in {\mathcal M}$. Then, as $n\rightarrow \infty$, the process $(\bar{w}_t^n)_{t\geq 0}$ converges weakly in $D_{\mathcal M}[0,\infty)$ towards a process $(w_t^\infty)_{t\geq 0}$ with initial value $w^{\infty}_0=w^0$. Furthermore, $(w_t^\infty)_{t\geq 0}$ is the unique deterministic process for which, for every $f\in C^{\infty}_c(\R^{\dim})$, $$\langle w^\infty_t,f\rangle = \langle w_0^\infty,f\rangle +
\int_0^t \bigg\{\frac{\kappa_R}{2}\, \langle w_s^\infty ,\Delta f\rangle
+ u\rho V_R\, \langle w_s^\infty (1-w_s^\infty) (2w_s^\infty-1)
,f\rangle\bigg\}\, ds,$$ where $$\label{def Gamma}
\kappa_R= \frac{u}{V_R}\int_{B(0,R)}\int_{B(x,R)}(z_1)^2dz\, dx$$ with $z_1$ the first coordinate of the vector $z\in\R^{\dim}$. In particular, $\kappa_R$ depends only on $R$ and $\dim$.
In other words, up to a change of coefficients, $(w_t^\infty)_{t\geq 0}$ is a weak solution of (\[special allen-cahn equation\]) with $w_0=w^0$. Based on Theorem \[th:evy\], it is natural to ask whether we can modify the scaling of $\v{s}_n$ in such a way that $\v{s}_nn^{2\beta}\rightarrow\infty$ as $n\rightarrow\infty$ and obtain convergence to the indicator function of a region whose boundary evolves according to mean curvature flow. In other words, does genetic drift, which is driven by the neutral events in the SLFVS, disrupt that convergence?
To state our result, we first rescale the as in Theorem \[th:evy\]. For each $n\in\N$, we define the finite measure $\mu^n$ on $(0, \mathcal R_n]$, where $\mathcal R_n = n^{-\beta}\mathcal R$, by $\mu^n(A)=\mu(n^{\beta}A)$ for all Borel subsets $A$ of $(0,\infty)$. Our rescaled will be driven by the Poisson point process $\Pi^n$ on $\R_+ \times \R^{\dim} \times (0,\infty)$ with intensity measure $$\label{eq:slfvs_intensity_intro}
n dt\otimes n^{\beta} dx\otimes \mu^n(dr).$$ Here $n^\beta dx$ denotes the scaling in which the linear dimension of the infinitesimal region $dx$ is scaled by $n^\beta$ (so that when we integrate, the volume of a region is scaled by $n^{\dim\beta}$). Let $$\label{scalings}
u_n = \frac{u}{n^{1-2\beta}}, \qquad\mbox{and}\qquad
\v{s}_n = \frac{1}{\epsilon_n^{2}}\frac{1}{n^{2\beta}}.$$ It is convenient to define the constant $\sigma^2$ through $$\label{defn of sigma}
\sigma^2
=\frac{u}{2\dim}\int_0^{\mc{R}}\int_{\R^\dim}|z|^2\frac{V_r(0,z)}{V_r}dz\mu(dr).$$ If $\mu(dr)=\delta_R(r)$, then we recover $\kappa_R$ from .
\[thm:slfvs\] Suppose that $\beta\in(0,1/4)$ and let $\epsilon_n$ be a sequence such that $\epsilon_n\to 0$ and $(\log n)^{1/2}\epsilon_n\to\infty$ as $n\rightarrow\infty$. Let $(w_t^n)_{t\geq 0}$ be the driven by $\Pi^n$ and with $u_n$, $\v{s}_n$ given by (\[scalings\]), and initial condition $w_0^n(x)=p(x)$. Assume that $p$ satisfies -, and define $\mathscr{T}$, $d(x,t)$ as for Theorem \[theorem ac to cf\]; take $T^*<\mathscr{T}$. For $k\in\N$ there exist $n_*(k)<\infty$, and $a_*(k),d_*(k)\in(0,\infty)$ such that for all $n\geq n_*$ and all $t$ satisfying $a_* \epsilon_n ^2 |\log \epsilon_n |\leq t\leq T^*$,
1. for almost every $x$ such that $d(x,\sigma^2 t)\geq d_* \epsilon_n |\log \epsilon_n|$, we have $\E\l[w^n_t(x)\r]\geq 1-\epsilon_n^k$;
2. for almost every $x$ such that $d(x,\sigma^2 t)\leq -d_* \epsilon_n |\log \epsilon_n|$, we have $\E\l[w^n_t(x)\r]\leq \epsilon_n^k$.
In Section \[duality for SLFVS\] we explain the origins of these scalings. By taking $u_n$ to be small, we are assuming that local population density is high.
By adapting ideas from [@etheridge/freeman/penington/straulino:2015], we expect an analogous result for values of $u_n$ up to $\mc O(1)$, but at the expense of having to take $\epsilon_n\rightarrow 0$ extremely slowly (so that $\epsilon_n^{-1}=o(\log\log n)$). The stronger the genetic drift, that is the bigger $u_n$, the larger the value of $n$ required for the diffusive rescaling to smooth the allele frequencies under the SLFVS sufficiently for the behaviour to be close to that of the differential equation .
The rest of the paper is laid out as follows. In Section \[proof of ac to cf\] we establish a duality between equation (\[AC1\]) and a branching Brownian motion which we then use to prove Theorem \[theorem ac to cf\]. In Section \[proof of slfvs to cf\] we establish an analogous duality between the SLFVS and a system of branching and coalescing particles and use it to establish Theorem \[thm:slfvs\].
Proof of Theorem \[theorem ac to cf\] {#proof of ac to cf}
=====================================
A probabilistic dual to Equation (\[special allen-cahn equation\]) {#subsec:dual_bbm}
------------------------------------------------------------------
Our proof of Theorem \[theorem ac to cf\] rests on a duality between equation (\[special allen-cahn equation\]) and a branching Brownian motion in which each individual, independently, follows a Brownian motion during an exponentially distributed lifetime (with mean $\epsilon^2$) at the end of which it splits into [*three*]{}. Although reminiscent of the duality between the Fisher-KPP equation and binary branching Brownian motion pioneered by [@skorohod:1964] and [@mckean:1975], here there is a slight twist. These papers allow us to deal with equations of the form $$\frac{\partial v}{\partial t}=\frac{1}{2}\Delta v +Vf(v),$$ where $V$ is a constant (the branching rate in the branching Brownian motion) and $f$ is of the form $f(v)=\Phi(v)-v$ where $\Phi(v)$ is the probability generating function of a non-negative integer-valued random variable (the number of offspring of each individual in the branching Brownian motion). However, the expression for $f$ in (\[special allen-cahn equation\]) is not of this form. Instead we adapt ideas from population genetics (notably from [@krone/neuhauser:1997; @neuhauser/krone:1997]).
First, to maintain compatibility with the PDE literature, we shall adopt the convention that $$\label{eq:bm_rate_2}
\textit{all Brownian motions run at rate $2$.}$$ That is, at time $1$, Brownian motion has variance $2$. In contrast to the McKean-Skorohod setting, our representation of the solution to (\[AC1\]) is not just in terms of the spatial positions of individuals in the branching Brownian motion at a fixed time, but also depends on their genealogy. In other words, we have a duality between (\[AC1\]) and the [*historical process*]{} of the branching Brownian motion.
To write this formally, we require some notation for our ternary branching Brownian motion. We write $\v{W}(t)$ for the historical process (which traces out the space-time trees that record the spatial position of all individuals alive at time $s$ for all $s\in [0,t]$). This process can be constructed formally as the ternary branching Markov process in which the position of an ‘individual’ alive at time $s$ is taken to be the whole Brownian path $(W_u)_{0\leq u\leq s}$ followed by its ancestors. To record the genealogy of the process we use Ulam-Harris notation to label individuals in the branching Brownian motion by elements of $\mc U =\bigcup_{m=0}^\infty \{1,2,3\}^m$. For example, $(3,1,2)$ is the particle which is the 2[nd]{} child of the 1[st]{} child of the 3[rd]{} child of the initial ancestor $\emptyset$. Let $N(t)\subset \mc U$ denote the set of individuals alive at time $t$. We shall abuse notation slightly and write $(W_i(t))_{i\in N(t)}$ for the spatial locations of the individuals alive at time $t$, and $(W_i(s))_{0\leq s\leq t}$ for the unique path that connects leaf $i$ to the root. We say that $\mc{T}$ is a *time-labelled ternary tree* if $\mc T$ is a finite subtree of $\mc U$ and each internal vertex $v$ of the tree is labelled with a time $t_v >0$, where $t_v$ is strictly greater than the label of the parent vertex of $v$. Evidently if we ignore the spatial position of individuals, each realisation of $\v{W}(t)$ traces out a time-labelled ternary tree which records the genealogy and associates a time to each branching event. We shall use $\mathcal{T}(\v{W}(t))$ to denote this time-labelled ternary tree.
For a fixed function $p:\R^\dim \to [0,1]$, we define a voting procedure on $\mathcal{T}(\v{W}(t))$ as follows.
1. Each leaf $i$ of $\mathcal{T}(\v{W}(t))$, independently, votes $1$ with probability $p(W_i(t))$ and otherwise votes $0$.
2. At each branch point in $\mathcal{T}(\v{W}(t))$, the vote of the parent particle $j$ is the majority vote of the votes of its three children $(j,1)$, $(j,2)$ and $(j,3)$.
This defines an iterative voting procedure, which runs inwards from the leaves of $\mathcal{T}(\v{W}(t))$ to the root $\emptyset$.
\[vote\_defn\] With the voting procedure described above, we define $ \mathbb{V} _p(\v {W}(t)) $ to be the vote associated to the root $\emptyset$.
For $x\in \R^\dim$, we write $\P ^\epsilon _x$ for the probability measure under which $(\v{W}(t),t \geq 0)$ has the law of the historical process of ternary branching Brownian motion in $\R ^\dim$ with branching rate $1/\epsilon ^2$ started from a single particle at location $x$ at time $0$. We write $\E ^\epsilon _x$ for the corresponding expectation.
\[thm:ACdual\] Let $p:\R^{\dim}\to [0,1]$. Then $$\label{eq:udef}
v^\epsilon (t,x)=\P^\epsilon_x\l[\operatorname{\mathbb V}_p(\v{W}(t))=1\r]$$ is a solution to equation with initial condition $v^\epsilon (0,x)=p(x)$.
(Sketch)
The proof mirrors that of the representation of solutions of the Fisher-KPP equation in terms of binary branching Brownian motion, and so we only sketch it. As usual the idea is to analyse the expression on the right hand side of (\[eq:udef\]) by partitioning on the behaviour of the branching Brownian motion in the first $\delta t$ of time and then to take a limit as $\delta t\downarrow 0$.
Throughout the proof we neglect the superscript $\epsilon$ in $\P^\epsilon _x$, $\E^\epsilon _x$ and $v^\epsilon $ and the subscript $p$ in $\operatorname{\mathbb V}_p$. We write $S$ for the time of the first branching event in the branching Brownian motion and $W_S$ for the position of the ancestor at that time. It is convenient to use $E$ for expectation when it is with respect to the law of Brownian motion ($W_\cdot$), preserving $\E$ for expectation with respect to that of the historical branching Brownian motion ($\v{W}(\cdot)$). Let $V_1, V_2, V_3$ denote the votes of the three offspring created at time $S$. By the strong Markov property of the branching Brownian motion, and the branching property, we see that the $V_i$ are conditionally independent given $(S,W_S)$. Moreover, since conditional on $S\leq\delta t$, the chance of a second branch before time $\delta t$ is ${\mathcal O}(\delta t)$, for $s\leq\delta t$, $$\E_x[V_1|(S,W_S)=(s,y)]=E_{y}[v(t,W_{\delta t-s})]
+{\mathcal O}(\delta t).$$ From this, if we assume enough regularity of $v(t,x)$ (which follows from that of the heat semigroup), $$\label{approx v1}
\E_x[V_1|S\leq\delta t]= v(t,x)+{\mathcal O}(\delta t).$$ Still conditioning on $S\leq\delta t$, in order for the vote at the root to be one, at most one of $V_1, V_2, V_3$ can be zero, and so using (\[approx v1\]) and conditional independence of the $V_i$ given $(S,W_S)$, $$\P_x\l[\operatorname{\mathbb V}(\v{W}(t+\delta t))=1|S\leq\delta t\r]
=v(t,x)^3+3v(t,x)^2(1-v(t,x))+{\mathcal O}(\delta t).$$ Since if $S>\delta t$ the ancestor of the branching Brownian motion simply follows a Brownian motion over $[0,\delta t]$, partitioning over the behaviour of the branching Brownian motion in the first $\delta t$ of time gives $$\begin{aligned}
v(t+\delta t,x)&=&
\P_x\l[\operatorname{\mathbb V}(\v{W}(t+\delta t))=1|S\leq\delta t\r] \P\l[S\leq\delta t\r]\\
&&+\P_{x}\l[\operatorname{\mathbb V}(\v{W}(t+\delta t))=1\,|\,S>\delta t\r]
(1-\P\l[S\leq\delta t\r])\\
&=&
\P_x\l[\operatorname{\mathbb V}(\v{W}(t+\delta t))=1\,|\,S\leq\delta t\r] \P\l[S\leq\delta t\r]\\
&&+E_x\l[\P_{W_{\delta t}}\l[\operatorname{\mathbb V}(\v{W}(t))=1\r]\r]
(1-\P\l[S\leq\delta t\r])
.\end{aligned}$$ Now $\P[S\leq\delta t]=\epsilon^{-2}\delta t+{\mathcal O}(\delta t^2)$ and so substituting and rearranging (and once again assuming enough regularity of $v(t,x)$) we obtain $$\begin{aligned}
\lim_{\delta t\rightarrow 0}\frac{v(t+\delta t,x)-v(t,x)}{\delta t}
&=& \epsilon^{-2}\left(v(t,x)^3+3v(t,x)^2(1-v(t,x))-v(t,x)\right)
\\
&&
+\lim_{\delta t\rightarrow 0}\frac{
E_x\l[\P_{W_{\delta t}}\l[\operatorname{\mathbb V}(\v{W}(t))=1\r]\r]-v(t,x)}{\delta t}
\\
&=& \epsilon^{-2}\left(v(t,x)^3+3v(t,x)^2(1-v(t,x))-v(t,x)\right)
\\
&&
+\lim_{\delta t\rightarrow 0}\frac{
E_x\l[v(t, W_{\delta t})\r]-v(t,x)}{\delta t}
\\
&=&\Delta v(t,x)+\epsilon^{-2}v(t,x)(1-v(t,x))(2v(t,x)-1),\end{aligned}$$ as required.
Armed with this representation, the proof of Theorem \[theorem ac to cf\] is reduced to proving the following result about our branching Brownian motions.
\[thm:BBMtwo\] Suppose $p:\R^\dim \to [0,1]$ is such that - hold. Define $\mathscr{T}$, $d(x,t)$ as for Theorem \[theorem ac to cf\]; fix $T^*\in (0,\mathscr{T})$ and let $k\in\N$. There exist $\epsilon_{\dim}(k)>0$, and $a_{\dim}(k),c_{\dim}(k)\in(0,\infty)$ such that for all $\epsilon\in(0,\epsilon_{\dim})$ and $t$ satisfying $a_{\dim}\epsilon ^2 |\log \epsilon |\leq t\leq T^*,$
1. for $x$ such that $d(x,t)\geq c_{\dim}\epsilon |\log \epsilon|$, we have $\P^\epsilon_x\l[\operatorname{\mathbb V}_p(\v{W}(t))=1\r]\geq 1-\epsilon^k$;
2. for $x$ such that $d(x,t)\leq -c_{\dim}\epsilon |\log \epsilon|$, we have $\P^\epsilon_x\l[\operatorname{\mathbb V}_p (\v{W}(t))=1\r]\leq \epsilon^k$.
The proof of Theorem \[thm:BBMtwo\] will proceed in two steps. First, in Section \[sec:BBMone\], we prove a one-dimensional analogue of the result in the special case in which $p(x)=\1\{x\geq 0\}$. The proof rests on symmetry of branching Brownian motion and the monotonicity that results from the specific choice of initial condition $p$. The second step uses the definition of mean curvature flow and the regularity properties that follow from the conditions -. These allow us to couple the distance between the (backwards in time) mean curvature flow $(\v {\Gamma}_{t-s})_{s \in [0,t]}$ and a (forwards in time) $\dim$-dimensional Brownian motion $W$ with a (forwards in time) one-dimensional Brownian motion $B$ in such a way that $d(W_s,t-s)$ is well approximated by $B_s$ when $W_s$ is close to $\v {\Gamma}_{t-s}$. This coupling is made precise in Proposition \[prop:coupling1\] in Section \[sec:CFsec\]. The proof of Theorem \[thm:BBMtwo\], which combines these two steps by bounding the errors that occur far from the interface $\v {\Gamma} _{t-s}$, can be found in Section \[sec:BBMtwo\].
It is convenient to have a prominent distinction between one dimensional and multi-dimensional Brownian motion in our notation. We therefore adopt the convention that $B$ will denote one dimensional Brownian motion and $\v{B}$ will represent the corresponding historical branching Brownian motion and we preserve $W$ and $\v{W}$ for dimensions $\dim\geq 2$.
Majority voting in one dimensional BBM {#sec:BBMone}
--------------------------------------
In this section we consider only ternary branching Brownian motion in dimension $\dim=1$.
As in Section \[subsec:dual\_bbm\], for $x\in \R$, we write $\P ^\epsilon _x$ for the probability measure under which $(\v{B}(t),t \geq 0)$ has the law of historical ternary branching Brownian motion in $\R$ with branching rate $1/\epsilon ^2$ started from a single particle at location $x$ at time $0$, and $\E ^\epsilon _x$ for the corresponding expectation. We also write $P_x$ for the probability measure under which $(B_t)_{t \geq 0}$ has the law of a Brownian motion started at $x$, and $E_x$ for the corresponding expectation.
Throughout this section we write $\mathbb V := \mathbb V_{p_0}$ where $p_0(x)=\1\{x\geq 0\}$, so that a leaf votes $1$ if and only if it is in the right half line. Our aim is to prove the following one-dimensional analogue of Theorem \[thm:BBMtwo\] for this initial condition $p_0$.
\[thm:BBMone\] Let $T^*\in(0,\infty)$. For all $k\in\N$ there exist $c_1(k)$ and $\epsilon_1(k)>0$ such that, for all $t\in[0,T^*]$ and all $\epsilon\in(0,\epsilon_1)$,
1. for $z\geq c_1(k)\epsilon |\log \epsilon |$, we have $\P^\epsilon_z \l[ \operatorname{\mathbb V}(\v{B}(t))=1\r]\geq 1-\epsilon ^k$
2. for $z\leq -c_1(k)\epsilon |\log \epsilon |$, we have $\P^\epsilon_z \l[ \operatorname{\mathbb V}(\v{B}(t))=1\r]\leq \epsilon ^k.$
The subscript $1$ on $a_1,c_1$ and $\epsilon_1$ is to emphasize that Theorem \[thm:BBMone\] applies in dimension $1$. We shall often suppress the dependence on $k$ in our notation.
Note that, if $z\geq 0$, then a typical leaf of the branching Brownian motion is more likely to vote $1$ than $0$, and that the opposite is true for $z<0$. Theorem \[thm:BBMone\] says that the majority voting procedure magnifies a small voting bias at the leaves into a much stronger voting bias at the root. If the votes of different leaves were independent this would be elementary, but the spatial structure of the branching Brownian motion introduces strong correlations between votes of closely related individuals. To overcome this, we first use a symmetry argument to show that the bias close to the root will be at least as strong as that at the leaves and then check that, as $\epsilon$ tends to zero, there is enough branching close to the root to sufficiently magnify the bias.
### Proof of Theorem \[thm:BBMone\]
First note that with our special choice of initial condition $p_0$, for any $x_1 \leq x_2\in\R$, $$\label{eq:monotonicity}
\P^\epsilon_{x_1}[\operatorname{\mathbb V}(\v{B}(t))=1]\leq \P^\epsilon_{x_2}[\operatorname{\mathbb V}(\v{B}(t))=1].$$ By analogy with the previous subsection, we use $\mathcal{T}(\v{B}(t))$ to denote the time-labelled tree traced out by the branching Brownian motion up to time $t$, and for any time-labelled ternary tree $\mc{T}$ we write $$\label{PTdef}
\P^{t}_x (\mc{T})=\P^\epsilon _x\l[\operatorname{\mathbb V}(\v{B}(t))=1
\|\mathcal{T}(\v{B}(t))=\mc{T}\r].$$ By the symmetry of the Brownian motions followed by individuals in $\v{B}(t)$ conditional on $\{\mathcal{T}(\v{B}(t))=\mc{T}\}$, applying the reflection $x\mapsto -x$ to the process, we see that for any time-labelled ternary tree $\mc{T}$, any time $t>0$, and any $z\in \R$, $$\label{prob_symmetry}
\P_{z}^t(\mc{T})=1-\P_{-z}^t(\mc{T}).$$ The monotonicity in and the symmetry in are key to our proof of Theorem \[thm:BBMone\].
Taking $z=0$ in (\[prob\_symmetry\]) shows that $\P^t_0(\mc{T})=\frac{1}{2}$ for all $t>0$, and, by , for all $t>0$ and all time-labelled ternary trees $\mc T$ we have $$\P^t_z(\mc{T})\geq\tfrac{1}{2}\;\text{ for } z>0;\qquad
\P^t_z(\mc{T})\leq\tfrac{1}{2}\;\text{ for } z<0. $$
We now introduce notation for the majority voting procedure. Let $g:[0,1]^3\to[0,1]$ be given by $$\label{g_defn}
g(p_1,p_2,p_3)=p_1 p_2 p_3 +p_1 p_2 (1-p_3)+p_2 p_3(1-p_1)+p_3 p_1 (1-p_2).$$ This is the probability that a majority vote gives the result $1$, in the special case where the three voters are independent and have probabilities $p_1$, $p_2$ and $p_3$ respectively of voting $1$. With a slight abuse of notation, we let $g(p)=g(p,p,p)$, for $p\in [0,1]$. Note that $$\label{gantisym}
g(1-p_1,1-p_2,1-p_3)=1-g(p_1,p_2,p_3).$$ For $\mc T$ a time-labelled ternary tree with at least one branching event, suppose that the time to the first branching event in $\mc T$ is $\tau$ and that the subtrees with time labels corresponding to the (descendants of the) three offspring from the branching event are $\mc T_1$, $\mc T_2$ and $\mc T_3$ (here a vertex $v$ with time label $t_v$ in $\mc T$ is given time label $t_v-\tau$ in $\mc T_i$). Then, we write $$\label{eq:star_notation}
g\l(\P^{t-\tau}_{B_{\tau}}(\mc{T}\star)\r)=g\l(\P^{t-\tau}_{B_{\tau}}(\mc{T}_1),
\P^{t-\tau}_{B_{\tau}}(\mc{T}_2),\P^{t-\tau}_{B_{\tau}}(\mc{T}_3)\r)$$ and the identity $$\label{branchid}
\P^t_{z}(\mc{T})=E_z \l[g\l(\P^{t-\tau}_{B_{\tau}}(\mc{T}\star)\r)\r]$$ expresses the majority voting that takes place at the first branch of $\mc{T}$.
Our next lemma states that the majority voting procedure cannot reduce the voting bias. In view of symmetry , when it is convenient to do so we will only state such results for the case $z\geq 0$.
\[no\_demagnification\] For any time-labelled ternary tree $\mc T$, any time $t>0$, and any $z\geq 0$, $$\P_z^t(\mc T)\geq P_z[B_t\geq 0].$$
The proof is by induction on the number of branching events in the tree $\mc{T}$. Let $\mc{T}_0$ denote the tree with a root and a single leaf. Then, by definition, $
\P_z^t(\mc{T}_0)=P_z\l[B_t\geq 0\r].
$
We now approach the inductive step. Suppose that the statement of the lemma holds for all time-labelled ternary trees with up to $n$ internal vertices. We define $h:[0,1]^3\to\R$ by $$h(p_1,p_2,p_3)=g(p_1,p_2,p_3)-\frac{1}{3}(p_1+p_2+p_3),$$ and note that from we have $$\label{hantisym}
h(1-p_1,1-p_2,1-p_3)=-h(p_1,p_2,p_3).$$
We can write $h$ in the form $$h(p_1,p_2,p_3)=\tfrac{1}{3}\sum p_{i_1}\Big((1-p_{i_2})(p_{i_3}-\tfrac{1}{2})+(1-p_{i_3})(p_{i_2}-\tfrac{1}{2}) \Big)$$ where the sum is over $(i_1,i_2,i_3)=(1,2,3),(2,3,1),(3,1,2)$. Hence $$\label{hpos}
\frac{1}{2}\leq p_1,p_2,p_3\leq 1\;\ra\;h(p_1,p_2,p_3)\geq 0.$$ We will use the $\star$ notation defined in for $h$ in the same way as we use it for $g$.
Suppose that $\mc{T}$ is a time-labelled ternary tree with $n+1$ internal vertices and let $\tau$, $\mc{T}_1$, $\mc{T}_2$, $\mc{T}_3$ be as in . Using , by the definition of $g$ and $h$ we have $$\begin{aligned}
\P^t_{z}(\mc{T})
&=E_z\l[g\l(\P^{t-\tau}_{B_{\tau}}(\mc{T}\star)\r)\r] \notag \\
&=E_z\l[h\l(\P^{t-\tau}_{B_{\tau}}(\mc{T}\star)\r)\r]
+\frac{1}{3}\sum\limits_{i=1}^3 E_z\l[\P^{t-\tau}_{B_{\tau}}(\mc{T}_i)\r].
\label{hbreakdown}\end{aligned}$$ We will show that the first term of is non-negative. Combining (\[hantisym\]) with (\[prob\_symmetry\]), $$h(\P^{t-\tau}_{B_{\tau}}(\mc{T}\star))
=-h(\P^{t-\tau}_{-B_{\tau}}(\mc{T}\star)).$$Hence, $$\begin{aligned}
E_z[h(&\P^{t-\tau}_{B_{\tau}}(\mc T\star))]\notag\\
&=E_z \left[h(\P^{t-\tau}_{B_{\tau}}(\mc T\star))
\1\left\{B_{\tau}\geq 0\right\}\right]
+E_z\left[h(\P^{t-\tau}_{B_{\tau}}(\mc T\star))
\1\left\{B_{\tau}<0\right\}\right]\notag\\
&=E_z\left[h(\P^{t-\tau}_{B_{\tau}}(\mc T\star))
\1\left\{B_{\tau}\geq 0\right\}\right]
-E_z\left[h(\P^{t-\tau}_{-B_{\tau}}(\mc T\star))
\1\left\{B_{\tau}<0\right\}\right]\notag\\
&=\int_{0}^\infty h(\P^{t-\tau}_{x}(\mc T\star))(\phi_{z,2\tau}(x)
-\phi_{z,2\tau}(-x))\,dx,\label{eq:fliptrick}\end{aligned}$$ where $\phi_{\mu,\sigma^2}$ denotes the density of a $N(\mu,\sigma^2)$ random variable. Since $\P^{t-\tau}_{x}(\mc{T}_i)\geq 1/2$ for $x\geq 0$, by we have $h(\P^{t-\delta t}_{x}(\mc{T}\star))\geq 0$, and since $z\geq 0$, for all $x\geq 0$ we have $$\phi_{z,2\tau}(x)-\phi_{z,2\tau}(-x)\geq 0,$$ which proves that is non-negative. This shows that the first term of is non-negative and we now move on to the second term.
Using our inductive hypothesis, for $i=1,2,3$, $$E_z[\P^{t-\tau}_{B_{\tau}}(\mc{T}_i)]\geq E_z\l[P_{B_\tau}\l[B_{t-\tau}\geq0\r]\r]
=P_z[B_t\geq 0]$$ and so substituting into completes the proof of Lemma \[no\_demagnification\].
Our next task is to show that successive rounds of majority voting magnify a small bias at the leaves into a large bias at the root of a tree. Recall that for $p\in [0,1]$, $$g(p):=g(p,p,p)=3p^2-2p^3,$$ and define $g^{(n)}(p)$, inductively, by $$g^{(1)}(p)=g(p), \qquad
g^{(n+1)}(p)=g^{(n)}(g(p)).$$ Thus, $g^{(n)}(p)$ describes the probability of voting $1$ at the root of an $n$-level regular ternary tree if the votes of the leaves are i.i.d. Bernoulli$(p)$.
\[g\_iteration\] For all $k\in\N$ there exists $A(k)<\infty$ such that, for all $\epsilon \in (0,\frac12]$ and $n\geq A(k)|\log \epsilon |$ we have $$g^{(n)}(\tfrac{1}{2}+\epsilon)\geq 1-\epsilon^k.$$
We carry out two phases of iteration of $g$. First, we will show that it takes $\mc{O}(|\log \epsilon|)$ iterations to obtain $$\label{eq:g_iter_1}
g^{(n)}(\tfrac{1}{2}+\epsilon)\geq \tfrac{1}{2}+\tfrac{1}{\sqrt{8}}.$$ Then we note that $\mc{O}(\log |k \log \epsilon|)$ iterations are required to obtain $$\label{eq:g_iter_2}
g^{(n)}(\tfrac{1}{2}+\tfrac{1}{\sqrt{8}})\geq 1-\epsilon^k.$$ Since $g$ is monotone, combining the two phases completes the proof.
For the first phase, if $\delta\in(0,1/\sqrt{8})$ then a simple calculation shows that $$g(\tfrac{1}{2}+\delta)=\tfrac{1}{2}+
\tfrac{3}{2}\delta-2\delta^3\geq\tfrac{1}{2}+\tfrac{5}{4}\delta.$$ Thus if $g^{(n)}(\frac{1}{2}+\epsilon)-\frac{1}{2}<1/\sqrt{8}$, we have $$\begin{aligned}
g^{(n+1)}(\tfrac{1}{2}+\epsilon)-\tfrac{1}{2}
\geq \tfrac{5}{4}\left(g^{(n)}(\tfrac{1}{2}+\epsilon)-\tfrac{1}{2}\right)
\geq (\tfrac{5}{4})^{n}\epsilon.\end{aligned}$$ It follows immediately that $\mc{O}(|\log \epsilon|)$ iterations are required to achieve .
For the second phase, note that $1-g(1-\delta)=3\delta^2-2\delta^3\leq 3\delta^2$, so that $$1-g^{(n+1)}(\tfrac{1}{2}+\tfrac{1}{\sqrt{8}})\leq 3
\left(1-g^{(n)}(\tfrac{1}{2}+\tfrac{1}{\sqrt{8}})\right)^2\leq \tfrac{1}{3}\Big(3(\tfrac{1}{2}-\tfrac{1}{\sqrt{8}})\Big)^{2^n}.$$ Noting that $3(\frac{1}{2}-\frac{1}{\sqrt{8}})<1$, it follows easily that the number of iterations required to obtain is $\mc{O}(\log|k\log\epsilon|)$.
We now want to see that there is a (large) regular ternary tree sitting inside $\mc{T}(\v{B}(t))$. Let $\mc{T}^{reg}_n=\cup_{k\leq n}\{1,2,3\}^k \subset \mc U$ denote the $n$-level regular ternary tree and, for $l\in\R$, let $\mc{T}^{reg}_l=\mc{T}^{reg}_{\lceil l\rceil}$. For $\mc T$ a time-labelled ternary tree, we use the relation $\mc T\supseteq \mc{T}^{reg}_l$ to mean that as subtrees of $\mc U$, $\mc{T}^{reg}_l$ is contained inside $\mc T$ (ignoring its time labels).
\[ternary\_tree\] Let $k\in\N$ and let $A=A(k)$ be as in Lemma \[g\_iteration\]. Then there exist $a_1=a_1(k)$ and $\epsilon_1=\epsilon_1(k)$ such that, for all $\epsilon\in(0,\epsilon_1)$ and $t\geq a_1\epsilon^2|\log \epsilon|$, $$\P^\epsilon \l[\mathcal{T}(\v{B}(t))\supseteq \mc{T}^{reg}_{A(k)|\log \epsilon |}\r]\geq 1- \epsilon ^k .$$
First we establish control over the tail distribution of the sum of $n$ independent exponentially distributed (branching) times. Suppose $(X_j)_{j\geq 1}$ are i.i.d. Exp(1) random variables and let $S_n=\sum_{j=1}^n X_j$. Then $$M_X(\lambda )=\E \l[ e^{\lambda X} \r]=
\begin{cases}
\frac{1}{1-\lambda} & \text{if } \lambda <1 \\
\infty & \text{if } \lambda \geq 1
\end{cases}$$ and for $a\geq 1$, $$\Psi^*(a):=\sup_{\lambda \geq 0}(\lambda a -\log M_X(\lambda))=\sup_{0\leq \lambda <1}(\lambda a+\log (1-\lambda))=a-1-\log a.$$ By Cramér’s theorem, for $a\geq 1$, $$\label{cramer}
\lim_{n\rightarrow \infty} \l( -\frac{1}{n}\log \P [S_n \geq na] \r)=\Psi^* (a)=a-1-\log a.$$
Suppose $a\geq 1$. For each leaf of $\mc{T}^{reg}_l$ we use to estimate the probability that it is not in $\mc{T}(\v{B}(t))$ and combine with a union bound (summing over leaves). For $t\geq a \epsilon ^2 \lceil A|\log \epsilon | \rceil$ we have $$\begin{aligned}
&\P^\epsilon \l[\mathcal{T}(\v{B}(t))\nsupseteq \mc T^{reg}_{A|\log \epsilon|}\r]
\notag\\
&\hspace{1cm}\leq 3^{\lceil A|\log \epsilon |\rceil}\P \l[\epsilon ^2 S_{\lceil A|\log \epsilon |\rceil}\geq a\epsilon ^2 \lceil A|\log \epsilon|\rceil\r]\notag\\
&\hspace{1cm}=\exp\l(\lceil A|\log \epsilon|\rceil \l(\log 3 + \frac{1}{\lceil A|\log \epsilon|\rceil }\log \P\l[S_{\lceil A|\log \epsilon |\rceil}\geq a\lceil A|\log \epsilon |\rceil\r]\r)\r).\label{ugh}\end{aligned}$$ By (with $n=\lceil A|\log\epsilon|\rceil$), we can choose $\epsilon_1(k)<e^{-1}$ such that, for all $\epsilon\in(0,\epsilon_1)$, $$\frac{1}{\lceil A|\log \epsilon|\rceil }\log \P\l[S_{\lceil A|\log \epsilon |\rceil}\geq a\lceil A|\log \epsilon |\rceil\r]\leq -a+3/2+\log a.$$ Choose $a\geq 1$ sufficiently large that $-a+3/2+\log a\leq -\log 3 -k/A$. Putting this into we obtain $$\P^\epsilon \l[\mathcal{T}(\v{B}(t))\nsupseteq \mc T^{reg}_{A|\log \epsilon|}\r]
\leq \exp\l(-|\log\epsilon| k\r)$$ for $t\geq a \epsilon ^2 \lceil A|\log \epsilon | \rceil$. Letting $a_1=a(A+1)$ completes the proof.
We now control the maximal displacement of individuals in the ternary branching Brownian motion at small times. Let $N(t)$ denote the set of individuals alive in $\v B(t)$.
\[all\_particles\_bound\] Let $k\in\N$, and let $a_1(k)$ be as in Lemma \[ternary\_tree\]. Then there exist $d_1(k)$, $\epsilon_1(k)$ such that, for all $\epsilon \in (0,\epsilon_1(k))$ and all $s\leq a_1\epsilon^2|\log\epsilon|$, $$\P^\epsilon _x\l[\exists i\in N(s):|B_i(s)-x|\geq d_1(k)\epsilon |\log \epsilon | \r] \leq \epsilon ^k.$$
Write $\delta_1 =a_1\epsilon^2|\log\epsilon|$ and let $Z$ be a $N(0,1)$ distributed random variable. By Markov’s inequality, for $s\leq\delta_1$ we have $$\begin{aligned}
\P^\epsilon_x\l[\exists i\in N(s):|B_i(s)-x|\geq d_1 \epsilon |\log \epsilon | \r]
&\leq \E^\epsilon\l[|N(s)|\r] \P\l[\sqrt{2s}|Z|\geq d_1\epsilon |\log \epsilon |\r]\\
&\leq \E^\epsilon \l[|N(\delta_1)|\r] \P\l[\sqrt{2\delta_1}|Z|\geq d_1\epsilon |\log \epsilon |\r]\\
&=e^{2 \delta_1 /\epsilon^2} \P \l[\sqrt{2a_1}|Z|\geq d_1 |\log \epsilon |^{1/2}\r]\\
&\leq \frac{1}{\epsilon ^{2a_1}}\exp\l(-\tfrac{1}{4}\tfrac{d_1^2}{a_1} |\log \epsilon |\r)\\
&=\epsilon ^{\tfrac{1}{4} \tfrac{d_1^2}{a_1} -2a_1}.\end{aligned}$$ Here the fourth line holds for $\epsilon >0$ sufficiently small. The proof is completed by choosing $d_1=d_1(k)$ large enough that $\frac{d_1^2}{4a_1}-2a_1\geq k$.
We now have all the ingredients needed to prove Theorem \[thm:BBMone\]. If $z \geq 2d_1 \epsilon |\log \epsilon |$, then, at time $\delta_1 =a_1\epsilon^2|\log \epsilon |$, by Lemma \[all\_particles\_bound\], with high probability, all individuals in $\v{B}(\delta_1)$ are still $\geq d_1 \epsilon |\log\epsilon|$. Lemma \[no\_demagnification\] tells us that there is a positive voting bias at each of those points and Lemma \[ternary\_tree\] shows that this will be magnified by at least $\mc O (|\log\epsilon|)$ rounds of majority voting as we trace back to the root. Finally, Lemma \[g\_iteration\] gives us a lower bound on the bias at the root.
\[Of Theorem \[thm:BBMone\].\] We will prove the first statement of the theorem; the second then follows by symmetry.
For all $\epsilon<1/2$, define $z_\epsilon $ implicitly by the relation $\P\l[B_{T^*}\geq -z_\epsilon \r]=\tfrac{1}{2}+\epsilon$, and note that $z_\epsilon \sim \epsilon \sqrt{4 \pi T^*}$ as $\epsilon \rightarrow 0$. Let $\epsilon_1(k)<1/2$ be sufficiently small that Lemmas \[ternary\_tree\] and \[all\_particles\_bound\] hold for $\epsilon \in (0, \epsilon_1(k))$. Let $d_1(k)$ be given by Lemma \[all\_particles\_bound\] and let $c_1(k)=2d_1(k)$ so that (by reducing $\epsilon_1$ if necessary), for $\epsilon\in (0,\epsilon_1)$, $$\label{eq:room for shift}
d_1(k)\epsilon|\log \epsilon |+z_\epsilon\leq c_1(k)\epsilon|\log \epsilon|.$$ Let $a_1(k)$ be given by Lemma \[ternary\_tree\] and let $$\label{delta 1}
\delta_1=\delta_1(k,\epsilon)=a_1(k)\epsilon^2|\log\epsilon|.$$
If $t\in(0,\delta_1)$ and $z\geq c_1\epsilon|\log \epsilon |$, then $$\begin{aligned}
\P^\epsilon_z\l[\operatorname{\mathbb V}(\v{B}(t))=0\r]
&\leq \P^\epsilon_z\big[\exists i\in N(t)\text{ such that }|B_i(t)-z|\geq d_1\epsilon |\log \epsilon | \big]\\
&\leq \epsilon ^k, \end{aligned}$$ where the second line follows by Lemma \[all\_particles\_bound\].
We now suppose that $t\in [\delta_1,T^*]$ and $z\geq c_1 \epsilon |\log \epsilon |$. Let $\mathcal T_{\delta_1}=\mathcal T(\v{B}(\delta_1))$ denote the time-labelled tree of the branching Brownian motion up to time $\delta_1$. We define $$p_{t-\delta_1}(z)=\P_z^\epsilon\l[\operatorname{\mathbb V}(\v{B}(t-\delta_1))=1\r],$$ and $$p_{t-\delta_1}^\epsilon(z) = p_{t-\delta_1}(z_\epsilon),
\quad\mbox{ for all }z\in\R.$$ Finally, write $\{\v{B}(\delta_1)>z_\epsilon\}$ for the event $B_i(\delta_1)>z_\epsilon$ for all $i\in N(\delta_1)$. Then, $$\begin{aligned}
\P^\epsilon_z\l[\operatorname{\mathbb V}(\v{B}(t))=1\r]&=
\P^\epsilon_z\l[\operatorname{\mathbb V}_{p_{t-\delta_1}(z)}(\v{B}(\delta_1))=1\r]
\notag\\
&\geq
\P^\epsilon_z\l[\l\{\operatorname{\mathbb V}_{p^\epsilon_{t-\delta_1}(z)}(\v{B}(\delta_1))=1
\r\}\cap\l\{\v{B}(\delta_1)>z_\epsilon\r\}\r]\notag
\\
&\geq
\P^\epsilon_z\l[\operatorname{\mathbb V}_{p^\epsilon_{t-\delta_1}(z)}(\v{B}(\delta_1))=1\r]
-\epsilon^k.\label{captureandbound}\end{aligned}$$ Here, the first line follows by the Markov property of $\v{B}$ at time $\delta_1$. The second follows by the monotonicity property . The third line then follows by Lemma \[all\_particles\_bound\], using and our hypothesis that $z \geq c_1\epsilon|\log\epsilon|$.
We have $$\label{l13appl}
p^\epsilon_{t-\delta_1}(z)\geq
P_{z_\epsilon}\l[B_{t-\delta_1}\geq 0\r]\geq \tfrac{1}{2}+\epsilon.$$ Here, the first inequality follows from Lemma \[no\_demagnification\]. The second follows by the definition of $z_\epsilon$, since $t-\delta_1<T^*$.
If $p_i\geq 1/2$ for $i=1,2,3$ then implies that $g(p_1,p_2,p_3)\geq \min(p_1,p_2,p_3)$. Hence, if each leaf of $\mc{T}_{\delta_1}$ votes $1$ independently with probability at least $\tfrac{1}{2}+\epsilon$ and $\mathcal T_{\delta_1}\supseteq \mc T^{reg}_{A|\log \epsilon|}$, then each of the leaves of $\mc T^{reg}_{A|\log \epsilon|}$ votes 1 independently with probability at least $\tfrac{1}{2}+\epsilon$. Therefore, $$\P^\epsilon_z\l[\operatorname{\mathbb V}(\v{B}(t))=1\r]\geq
g^{(\lceil A|\log \epsilon| \rceil)}(\tfrac{1}{2}+\epsilon)-2\epsilon^k
\geq 1-3\epsilon ^k.$$ Here, the first inequality follows by substituting into and then applying Lemma \[ternary\_tree\] and the second then follows by Lemma \[g\_iteration\]. This completes the proof.
### The slope of the interface
In proving Theorem \[thm:BBMtwo\] we shall also exploit a lower bound on the ‘slope’ of the interface in $\dim =1$ which we prove in this subsection. We obtain it as a corollary of the following result.
\[prop:deriv1\] Suppose $x\geq 0$ and $\eta>0$. Then for any time-labelled ternary tree $\mc{T}$ and any time $t$, $$\P_{x}^t(\mc{T})-\P_{x-\eta}^t(\mc{T})\geq \P_{x+\eta}^t(\mc{T})-\P_{x}^t(\mc{T}).$$
The proof is by induction on the number of branching events in $\mc T$, and is similar to the proof of Lemma \[no\_demagnification\]. For $\mc{T}_0$ a (time-labelled) tree with a root and a single leaf, we have $$\P_{x}^t(\mc{T}_0)-\P_{x-\eta}^t(\mc{T}_0)=\int_{x-\eta}^{x}\phi_{0,2t}(u)\,du\geq \int_x^{x+\eta}\phi_{0,2t}(u)\,du=\P_{x+\eta}^t(\mc{T}_0)-\P_{x}^t(\mc{T}_0)$$ where $\phi_{\mu,\sigma^2}$ is the density of a N$(\mu,\sigma ^2)$ random variable.
Now, assume that the lemma holds for all time-labelled ternary trees with at most $n$ internal vertices. Let $\mc{T}$ be a time-labelled ternary tree with $n+1$ internal vertices and suppose that the time to the first branching event of $\mc{T}$ is $\tau$ and let $\mc{T}_1$, $\mc{T}_2$, $\mc{T}_3$ denote the trees of the three offspring of that branching. Then using the notation of , $$\begin{aligned}
&\l( \P_{x}^t(\mc{T})-\P_{x-\eta}^t(\mc{T})\r)-\l(\P_{x+\eta}^t(\mc{T})-\P_{x}^t(\mc{T})\r)\notag\\
&\hspace{0.5pc}=\l(E_{x}\l[g(\P_{B_{\tau}}^{t-\tau}(\mc{T}\star))\r]-
E_{x-\eta}\l[g(\P_{B_{\tau}}^{t-\tau}(\mc{T}\star))\r]\r)
-\l(E_{x+\eta}\l[g(\P_{B_{\tau}}^{t-\tau}(\mc{T}\star))\r]-
E_{x}\l[g(\P_{B_{\tau}}^{t-\tau}(\mc{T}\star))\r]\r)\notag\\
&\hspace{0.5pc}=\int_{-\infty}^\infty\l\{ \big(g(\P^{t-\tau}_{y}(\mc{T}\star))-
g(\P^{t-\tau}_{y-\eta}(\mc{T}\star))\big)-
\big(g(\P^{t-\tau}_{y+\eta}(\mc{T}\star))-
g(\P^{t-\tau}_{y}(\mc{T}\star))\big)\r\}\phi_{x,2\tau}(y)dy\notag\\
&\hspace{0.5pc}=\int_0^\infty \l\{\big(g(\P^{t-\tau}_{y}(\mc{T}\star))-
g(\P^{t-\tau}_{y-\eta}(\mc{T}\star))\big)-
\big(g(\P^{t-\tau}_{y+\eta}(\mc{T}\star))-g(\P^{t-\tau}_{y}(\mc{T}\star))
\big)\r\}(\phi_{x,2\tau}(y)-\phi_{x,2\tau}(-y))\,dy.\label{eq:integ}\end{aligned}$$ Here, the second line follows by and the last line follows from and (\[prob\_symmetry\]), which imply that $g(\P^t_w(\mc{T}\star))=1-g(\P^t_{-w}(\mc{T}\star))$. Note the similarity to .
Since $x\geq 0$, we have $$\label{eq:phipve}
\phi_{x,2\tau}(y)-\phi_{x,2\tau}(-y)\geq 0$$ for $y\geq 0$. In view of we should like to check that for $y\geq 0$ $$\label{eq:integpve}
\l(g(\P^{t-\tau}_{y}(\mc{T}\star ))-g(\P^{t-\tau}_{y-\eta}(\mc{T}\star ))\r)
-\l(g(\P^{t-\tau}_{y+\eta}(\mc{T}\star ))-g(\P^{t-\tau}_{y}(\mc{T}\star ))\r)\geq 0.$$ By our inductive hypothesis, for $y\geq 0$ we have $$\big(\P^{t-\tau}_{y}(\mc{T}_i)-\P^{t-\tau}_{y-\eta}(\mc{T}_i)\big)-
\big(\P^{t-\tau}_{y+\eta}(\mc{T}_i)-\P^{t-\tau}_{y}(\mc{T}_i)\big)\geq 0,$$ and so by monotonicity of $g$, for it is enough to check that $$\label{symmetric version}
g(\P^{t-\tau}_{y+\eta}(\mc{T}\star ))-2g(\P^{t-\tau}_{y}(\mc{T}\star ))
+g\l(\P^{t-\tau}_{y}(\mc{T}\star )-(\P^{t-\tau}_{y+\eta}(\mc{T}\star )
-\P^{t-\tau}_{y}(\mc{T}\star ))\r)\leq 0.$$ To see that holds, note that $$\begin{aligned}
&g(p_1+\eta_1,p_2+\eta_{2},p_3+\eta_3)-2g(p_1,p_2,p_3)+g(p_1-\eta_1,p_2-\eta_{2},p_3-\eta_3)\notag\\
&\hspace{2pc}
=2\eta_1 \eta_2 (1-2p_3)+2\eta_2 \eta_3 (1-2p_1)+2\eta_3 \eta_1 (1-2p_2).
\notag $$ and set $p_i =\P^{t-\tau}_{y}(\mc{T}_i) $ and $\eta_i=\P^{t-\tau}_{y+\eta}(\mc{T}_i)-\P^{t-\tau}_{y}(\mc{T}_i)$. Since for $y\geq 0$, $p_i \geq 1/2$, the inequality then follows.
Putting and into completes the inductive step, which in turn completes the proof.
\[lem:high\_deriv\] Take $\epsilon_1(1)$ and $c_1(1)$ from Theorem \[thm:BBMone\]. Let $\epsilon <\min(\epsilon_1(1),\tfrac{1}{24})$. Suppose that for some $t\in [0,T^*]$ and $z\in \R$, $$\label{eq:near_interface}
\l|\P^\epsilon_z\l[\operatorname{\mathbb V}(\v{B}(t))=1\r]-\tfrac{1}{2}\r|\leq \tfrac{5}{12},$$ and let $w\in \R$ with $|z-w|\leq c_1(1)\epsilon |\log \epsilon|$. Then $$\label{eq:high_deriv}
\l|\P^\epsilon_z \l[\operatorname{\mathbb V}(\v{B}(t))=1\r]-\P^\epsilon_w\l[\operatorname{\mathbb V}(\v{B}(t))=1\r]\r|\geq \frac{|z-w|}{48 c_1(1)\epsilon |\log \epsilon |}.$$
Consider first the case $0\leq z \leq w$. By analogy with (\[branchid\]), let $\P_y^t$ denote $\P^\epsilon_y\l[\operatorname{\mathbb V}(\v B(t))=1\r]$. By Theorem \[thm:BBMone\] and we have that $$\label{eq:high_der_lower}
\P^t_{c_1(1)\epsilon |\log \epsilon|}-\P^t_z \geq 1-\epsilon - \tfrac{11}{12}\geq \tfrac{1}{24}.$$ Let $\eta:=w-z$. For $j\in \N$, applying Proposition \[prop:deriv1\] $j$ times gives that $$\P^t_{(j+1)\eta +z}-\P^t_{j\eta +z}\leq \P^t_w-\P^t_z.$$ It follows that $$\begin{aligned}
\label{eq:high_der_upper}
\P^t_{c_1(1)\epsilon |\log \epsilon|}-\P^t_z
&\leq \sum_{j=0}^{\lceil \eta^{-1}(c_1(1)\epsilon |\log \epsilon | -z)\rceil-1}(\P^t_{(j+1)\eta +z}-\P^t_{j\eta +z}) \notag \\
&\leq (\eta^{-1}(c_1(1)\epsilon |\log \epsilon |)+1) (\P^t_w-\P^t_z).\end{aligned}$$ Combining and , $$\P^t_w-\P^t_z \geq \frac{|z-w|}{24 (c_1(1)\epsilon |\log \epsilon |+|z-w|)}\geq \frac{|z-w|}{48 c_1(1)\epsilon |\log \epsilon |}.$$ The corresponding result for $0\leq w \leq z$ follows by symmetry (exchanging the roles of $w$ and $z$). The case $z\leq 0$ then follows by the symmetry in .
A coupling argument {#sec:CFsec}
-------------------
The second important ingredient in our proof of Theorem \[thm:BBMtwo\] will be a coupling between $d(W_s,t-s)$ (the signed distance from a $\dim$-dimensional Brownian motion $W_s$ to $\Gamma_{t-s}$, which evolves according to (backwards in time) mean curvature flow) and a one-dimensional Brownian motion, at least when $W_s$ is close to $\Gamma_{t-s}$. The proof requires some regularity properties of the mean curvature flow that we record in this subsection. These rest on the assumptions -.
We write $\dot{d}$ for the time derivative of $d$. Let $T^*\in(0,\mathscr{T})$. In this case, we have:
1. \[property 1\] There exists $c_0>0$ such that for all $t\in [0,T^*]$ and $x\in \{y:|d(y,t)|\leq c_0\}$, we have $$\label{eq:unitgrad}
|\nabla d(x,t)|=1.$$ Moreover, $d$ is a $C^{\alpha,\frac{\alpha}{2}}$ function in $\{(x,t): |d(x,t)|\leq c_0, t\leq T^*\}$.
2. Viewing $\v{n}=\nabla d$ as the positive normal direction, for $x\in \Gamma_t$, the normal velocity of $\Gamma_t$ at $x$ is $-\dot{d}(x,t)$, and the curvature of $\Gamma_t$ at $x$ is $-\Delta d(x,t)$. Thus, becomes $$\label{eq:cf}
\dot{d}(x,t)=\Delta d(x,t)$$ for all $x$ such that $d(x,t)=0$.
3. There exists $C_0>0$ such that for all $t\in[0,T^*]$ and $x$ such that $|d(x,t)|\leq c_0$, $$\label{eq:smoothcf}
\l|\nabla \l(\dot{d}(x,t)-\Delta d(x,t)\r)\r|\leq C_0.$$
4. There exist $v_0,V_0>0$ such that for all $t\in[0,T^*-v_0]$ and all $s\in [t,t+v_0]$, $$\label{eq:t_lipschitz}
|d(x,t)-d(x,s)|\leq V_0(s-t).$$
Properties 1 and 2 above come from [@chen:1992] (equations (2.9), (2.10) and Proposition 2.1) and 3 and 4 follow easily from the fact that $\sup_{u\in S^1,t\leq T^*}|\Gamma_t(u)|<\infty$ and the regularity of $d$ provided by 1.
The first property means that, for each $t\geq 0$, the region $\{x:d(x,t)\leq c_0\}$ is not self-intersecting i.e. for each $x$ it contains, the ball $\{z:|z-x|\leq d(x,t)\}$ intersects $\Gamma_t$ at precisely one point. Evidently this cannot hold, for example, as the flow collapses to a point, which is why we work up to time $T^*<\mathscr{T}$. Broadly speaking, the first two properties characterize mean curvature flow in terms of the function $d$.
A key ingredient of our proof of Theorem \[thm:BBMtwo\] is the following coupling argument.
\[prop:coupling1\] Let $(W_s)_{s \geq 0}$ denote a $\dim$-dimensional Brownian motion started at $x\in \R^{\dim}$. Suppose that $t\leq T^*$, $\beta \leq c_0$ and let $$T_\beta = \inf \l(\{ s \in [0,t):|d(W_s,t-s)|\geq \beta \}\cup \{t\}\r).$$ Then we can couple $(W_s)_{s \geq 0}$ with a one-dimensional Brownian motion $(B_s)_{s\geq 0}$ started from $z=d(x,t)$ in such a way that for $s\leq T_\beta$, $$B_s-C_0\beta s\leq d\l(W_s, t-s\r)\leq B_s+C_0\beta s.$$
By Itô’s formula, we have that for $s\leq t$ $$d\l(W_s,t-s\r)=\int_0^s A_u\,du+B_s,$$ where $$\begin{aligned}
A_u&=-\dot{d}\l(W_u,t-u\r)+\Delta d\l(W_u,t-u\r)\\
B_s&=\sum\limits_{i=1}^\dim\int_0^s \frac{\p }{\p x_i}d(W_u,t-u)dW_u^{(i)}.\end{aligned}$$ We will handle $A_u$ and $B_s$ in turn.
For each $u\in[0,T_\beta]$ there exists some $x_u\in\R^{\dim}$ such that $|x_u-W_u|\leq\beta$, and $d(x_u,t-u)=0$. By we have $-\dot{d}(x_u,t-u)+\Delta d(x_u,t-u)=0$. Since $\beta\leq c_0$, by we have that, for $x$ on the line segment connecting $x_u$ to $W_u$, the gradient of $-\dot{d}(x,t-u)+\Delta d(x,t-u)$ is bounded by $C_0$. We thus obtain $$|A_u|\leq C_0\beta.$$ Since $\beta\leq c_0$, it follows by and Lévy’s characterisation (recall that our Brownian motions run at rate $2$) that $(B_s)_{0\leq s\leq T_\beta}$ is a (stopped) Brownian Motion. This completes the proof.
Proposition \[prop:coupling1\] provides a probabilistic parallel to one of the key tools used in the classical study of (mean) curvature flow; approximating the movement of the interface locally (in space and time) by a particular one dimensional standing wave.
Majority voting in BBM, for $\dim\geq2$ {#sec:BBMtwo}
---------------------------------------
Recall the notation introduced in Section \[subsec:dual\_bbm\] for ternary branching Brownian motion in dimension $\dim \geq 2$. For $x\in \R^\dim $, we write $\P ^\epsilon _x$ for the probability measure under which $(\v{W}(t),t \geq 0)$ has the law of ternary branching Brownian motion in $\R^\dim $ with branching rate $1/\epsilon ^2$ started from a single particle at location $x$ at time $0$. We use $\E ^\epsilon _x$ for the corresponding expectation. We also write $P_x$ for the probability measure under which $(W_t)_{t \geq 0}$ has the law of a $\dim$-dimensional Brownian motion started at $x$, and $E_x$ for the corresponding expectation. As usual the notation $B$ (resp. $\v{B}$) refers to a one dimensional (historical branching) Brownian motion and $W$ and $\v{W}$ signal dimension $\dim\geq 2$.
The proof of Theorem \[thm:BBMtwo\] is in two parts. First, in Section \[sec:d=2\_generation\] we establish that the interface is generated in a time $\delta_{\dim}=\mc{O}(\epsilon^2|\log\epsilon|)$. We then, in Section \[sec:d=2\_propagation\], use Proposition \[prop:coupling1\] and Theorem \[thm:BBMone\] to investigate how the region around the interface propagates. In order not to interrupt the flow of the proof of Theorem \[thm:BBMtwo\], the proof of a central lemma is deferred to Section \[sec:d=2\_propagation\_technical\].
Our proof rests on a comparison with the outcome $\operatorname{\mathbb V}(\v{B}(t))$ of majority voting for the one-dimensional historical branching Brownian motion. In one dimension we always implicitly take $\operatorname{\mathbb V}= \operatorname{\mathbb V}_{p_0}$ with $p_0(x)=\1\{x\geq 0\}$. We reserve the subscript $p$ for $\operatorname{\mathbb V}_p(\v{W}(t))$ and we assume that $p$ satisfies -.
### Generation of the interface {#sec:d=2_generation}
In this section we prove that, as in $\dim=1$, in dimension $\dim\geq 2$ an interface of width $\mc{O}(\epsilon|\log\epsilon|)$ is generated in time $\mc{O}(\epsilon^2|\log\epsilon|)$.
\[prop:d=2\_generation\] Let $k\in\N$. Then there exist $\epsilon_{\dim}(k),a_{\dim}(k),b_{\dim}(k)>0$ such that for all $\epsilon\in(0,\epsilon_{\dim})$, if we set $$\label{eq:delta2}
\delta_{\dim}(k,\epsilon) := a_{\dim}(k)\epsilon^2|\log\epsilon| \quad
\textrm{ and }\quad \delta'_{\dim}(k,\epsilon)
:= (a_{\dim}(k)+k+1)\epsilon^2|\log\epsilon|,$$ then for $t\in [\delta_{\dim}, \delta'_{\dim}]$,
1. for $x$ such that $d(x,t)\geq b_{\dim}\epsilon |\log \epsilon|$, we have $\P^\epsilon_x\l[\operatorname{\mathbb V}_p(\v{W}(t))=1\r]\geq 1-\epsilon^k$;
2. for $x$ such that $d(x,t)\leq -b_{\dim}\epsilon |\log \epsilon|$, we have $\P^\epsilon_x\l[\operatorname{\mathbb V}_p (\v{W}(t))=1\r]\leq \epsilon^k$.
By the same argument as for Lemma \[ternary\_tree\], given $k\in\N$, and taking $A(k)$ from Lemma \[g\_iteration\], there exist $a_{\dim}(k)$ and $\epsilon_{\dim}(k)>0$ such that, for all $\epsilon\in(0,\epsilon_{\dim})$ and $t\geq a_{\dim}\epsilon^2|\log \epsilon|$, $$\label{eq:ternary_tree_2d}
\P^\epsilon\l[\mc{T}(\v W (t))\supseteq \mc{T}^{reg}_{A(k)|\log\epsilon|}\r]\geq 1-\epsilon^k.$$ It is also easy to obtain a $\dim$-dimensional equivalent of Lemma \[all\_particles\_bound\], with essentially the same proof (using a tail bound on a $\dim$-dimensional normal distribution instead of one dimensional). That is, given $k\in\N$, there exist $d_{\dim}(k)$, $\epsilon_{\dim}(k)$ such that for all $\epsilon \in (0,\epsilon_\dim)$, for $t\in [\delta_{\dim}, \delta'_{\dim}]$, $$\label{eq:all_particles_bound_2d}
\P^\epsilon _x\l[\exists i\in N(t): |W_i(t)-x|\geq d_{\dim}\epsilon|\log\epsilon|\r]\leq \epsilon^k.$$ We set $b_{\dim}(k)=2d_{\dim}(k)$.
By there exist $v_0,V_0>0$ such that for $t\leq v_0$, and any $x\in \R^{\dim}$, we have $|d(x,0)-d(x,t)|\leq V_0t.$ Reducing $\epsilon_{\dim}$ if necessary, for $\epsilon \in (0,\epsilon_{\dim})$ we have $\delta'_{\dim} \leq v_0$. Thus, if $\epsilon\in(0,\epsilon_{\dim})$, $t\in [\delta_\dim, \delta'_\dim]$ and $x$ is such that $d(x,t)\geq b_{\dim}\epsilon |\log \epsilon |$ and $|W_i(t)-x|\leq d_{\dim} \epsilon |\log \epsilon |$ then combining with the triangle inequality and (\[eq:t\_lipschitz\]), $$\begin{aligned}
d\l(W_i(t),0\r)&\geq d\l(x,t\r)-|d\l(x,t\r)-d\l(W_i(t),t\r)|-|d\l(W_i(t),t\r)-d\l(W_i(t),0\r)|\\
&\geq b_{\dim}\epsilon |\log \epsilon|-d_{\dim}\epsilon |\log \epsilon|-V_0\delta '_\dim\\
&=\frac{1}{2}b_{\dim}\epsilon |\log \epsilon|-V_0(a_{\dim}+k+1)\epsilon^2 |\log \epsilon|.\end{aligned}$$ Therefore, reducing $\epsilon_{\dim}$ if necessary, in this case we have that $$d(W_i(t),0)\geq\tfrac{1}{4}b_{\dim}\epsilon |\log \epsilon |.$$ Applying and , $$\begin{aligned}
p(W_i(t))
&\geq \tfrac{1}{2}+\gamma\l(\tfrac{1}{4}b_{\dim} \epsilon |\log \epsilon | \wedge r\r)\notag\\
&\geq \tfrac{1}{2}+\epsilon, \label{eq:p_control}\end{aligned}$$ where we again reduce $\epsilon_{\dim}>0$ (if necessary), to ensure that $\epsilon<\gamma r$, $\epsilon<\frac{\gamma}{4}b_{\dim}\epsilon|\log\epsilon|$ for $\epsilon\in(0,\epsilon_{\dim})$.
Exactly as in the proof of Theorem \[thm:BBMone\], we can now combine (\[eq:ternary\_tree\_2d\]), (\[eq:all\_particles\_bound\_2d\]) and (\[eq:p\_control\]) to deduce that for $\epsilon\in(0,\epsilon_{\dim})$, $t\in [\delta_\dim,\delta'_\dim]$ and $x$ such that $d(x,t)\geq b_{\dim}\epsilon|\log \epsilon|$, $$\P^\epsilon _x\l[\operatorname{\mathbb V}_p(\v W (t))=1\r] \geq 1-3\epsilon^k.$$ The proof of the second statement is analogous.
### Propagation of the interface and proof of Theorem \[thm:BBMtwo\] {#sec:d=2_propagation}
We now turn to the propagation of the interface region. Our immediate goal is to establish that, for suitably chosen (large) $K_1$ and $K_2$, and for all sufficiently small $\epsilon>0$ we have $$\P^\epsilon_x\l[\operatorname{\mathbb V}_p(\v W (t))=1\r]\approx \P^\epsilon_{d(x,t)+K_1e^{K_2 t}\epsilon|\log\epsilon|}\l[\operatorname{\mathbb V}(\v B (t))=1\r].$$ This connection between $\v{B}$ and $\v{W}$ is made precise by the following result.
\[prop:contra\] Let $l\in\N$ with $l\geq 4$. Define $a_\dim(l)$ and $\delta_\dim (l,\epsilon)$ as in Proposition \[prop:d=2\_generation\]. There exist $K_1(l),K_2(l)>0$ and $\epsilon_{\dim}(l, K_1, K_2)>0$ such that for all $\epsilon\in(0,\epsilon_{\dim})$ and $t\in[\delta_{\dim}(l,\epsilon),T^*]$ we have $$\label{eq:upper_final}
\sup\limits_{x\in\R^{\dim}}\Big(\P^\epsilon_x\l[\operatorname{\mathbb V}_p(\v W (t))=1\r]-\P^\epsilon_{d(x,t)+K_1e^{K_2 t}\epsilon|\log\epsilon|}\l[\operatorname{\mathbb V}(\v B(t))=1\r]\Big)\leq \epsilon^l$$ and $$\label{eq:lower_final}
\sup\limits_{x\in\R^{\dim}}\Big(\P^\epsilon_x\l[\operatorname{\mathbb V}_p(\v W (t))=0\r]-\P^\epsilon_{d(x,t)-K_1e^{K_2 t}\epsilon|\log\epsilon|}\l[\operatorname{\mathbb V}(\v B(t))=0\r]\Big)\leq \epsilon^l.$$
The proof of Theorem \[thm:BBMtwo\], which follows easily from Proposition \[prop:contra\], is at the end of this subsection.
Recall that $g:[0,1]\to [0,1]$ is given by $g(p)=3p^2-2p^3$. It is convenient to extend this definition to a continuous, monotone function $g:\R\to [0,1]$ as follows: $$\label{g_defn_ext}
g(p)=
\begin{cases} 0 &\mbox{if } p <0 \\
3p^2-2p^3 &\mbox{if } p \in [0,1] \\
1 & \mbox{if } p>1. \end{cases}$$ At the heart of the proof of Proposition \[prop:contra\] is the following lemma, whose proof we defer to Section \[sec:d=2\_propagation\_technical\].
\[lem:keylemma\_2\] Let $l \in \N$ with $l\geq 4$ and $K_1>0$. There exists $K_2=K_2(K_1,l)>0$ and $\epsilon_{\dim}(l,K_1,K_2)>0$ such that for all $\epsilon\in(0,\epsilon_{\dim})$, $x\in\R^{\dim}$, $s\in [0,(l+1)\epsilon^2 |\log \epsilon |]$ and $t\in[s,T^*]$, $$\begin{aligned}
&E_x\l[g\l(\P^\epsilon_{d(W_s,t-s)+K_1e^{K_2 (t-s)}\epsilon |\log \epsilon |}[\operatorname{\mathbb V}(\v B(t-s))=1]+\epsilon ^l\r) \r]\notag\\
&\hspace{8pc}\leq \tfrac{3}{4}\epsilon^l+E_{d(x,t)}\l[g\l(\P^\epsilon_{B_s +K_1 e^{K_2 t}\epsilon |\log \epsilon |}[\operatorname{\mathbb V}(\v B(t-s))=1]\r)\r]
+\1_{s\leq \epsilon ^3} \epsilon ^l \label{eq:keylemma_2eq}\end{aligned}$$ and $$\begin{aligned}
&E_x\l[g\l(\P^\epsilon_{d(W_s,t-s)-K_1e^{K_2 (t-s)}\epsilon |\log \epsilon |}[\operatorname{\mathbb V}(\v B(t-s))=0]+\epsilon ^l\r) \r]\notag\\
&\hspace{8pc}\leq \tfrac{3}{4}\epsilon^l+E_{d(x,t)}\l[g\l(\P^\epsilon_{B_s -K_1 e^{K_2 t}\epsilon |\log \epsilon |}[\operatorname{\mathbb V}(\v B(t-s))=0]\r)\r]
+\1_{s\leq \epsilon ^3} \epsilon ^l. \label{eq:keylemma_2eq_opp}\end{aligned}$$
Take $K_1=b_\dim (l)+c_1(l)$ where $b_\dim$ is as defined in Proposition \[prop:d=2\_generation\] and $c_1$ is as defined in Theorem \[thm:BBMone\]. Let $K_2=K_2(K_1,l)$, as defined in Lemma \[lem:keylemma\_2\]. Take $\epsilon_\dim >0$ sufficiently small that Theorem \[thm:BBMone\], Proposition \[prop:d=2\_generation\] and Lemma \[lem:keylemma\_2\] apply for $\epsilon \in (0,\epsilon_\dim)$. We begin by observing that for $\epsilon \in (0, \epsilon_\dim)$, $t\in [\delta_\dim, \delta'_\dim]$ (where $\delta'_\dim$ is defined in ), and $x\in \R^\dim$, $$\label{eq:low_t}
\P^\epsilon_x\l[\operatorname{\mathbb V}_p(\v W(t))=1\r]\leq\P^\epsilon_{d(x,t)+K_1e^{K_2 t}\epsilon|\log\epsilon|}\l[\operatorname{\mathbb V}(\v B(t))=1\r]+ \epsilon^l.$$ To see this, note that if $d(x,t)\leq - b_\dim(l)\epsilon |\log \epsilon |$, then by Proposition \[prop:d=2\_generation\], $\P^\epsilon_x\l[\operatorname{\mathbb V}_p(\v W(t))=1\r]\leq \epsilon^l$. On the other hand, if $d(x,t)\geq - b_\dim(l)\epsilon |\log \epsilon |$, then $d(x,t)+K_1e^{K_2 t}\epsilon|\log\epsilon|\geq c_1(l)\epsilon |\log \epsilon |$, and so, by Theorem \[thm:BBMone\], holds (since the right hand side of is $\geq 1$).
We are left with the case $t\in [\delta'_\dim,T^*]$. We assume, aiming for a contradiction, that there exists $t\in[\delta'_{\dim},T^*]$ such that, for some $x\in\R^\dim$, $$\P^\epsilon_x\l[\operatorname{\mathbb V}_p(\v W (t))=1\r]-\P^\epsilon_{d(x,t)+K_1e^{K_2 t}\epsilon|\log\epsilon|}\l[\operatorname{\mathbb V}(\v B(t))=1\r]
>\epsilon^l.$$ Let $T'$ be the infimum of the set of such $t$. Choose $$\label{eq:star_contra}
T\in [T',\min(T'+\epsilon^{l+3},T^*)]$$ which is in the set of such $t$. Hence, there exists some $x=x(l,\epsilon)\in\R^\dim$ such that $$\label{eq:for_contra}
\P^\epsilon_x\l[\operatorname{\mathbb V}_p(\v W(T))=1\r]-\P^\epsilon_{d(x,T)+K_1e^{K_2 T}\epsilon|\log\epsilon|}\l[\operatorname{\mathbb V}(\v B(T))=1\r]> \epsilon^l.$$ We now seek to show that $$\label{eq:contra}
\P^\epsilon_x\l[\operatorname{\mathbb V}_p(\v W(T))=1\r]\leq \tfrac{7}{8}\epsilon^l+\P^\epsilon_{d(x,T)+K_1e^{K_2T}\epsilon |\log \epsilon |}\l[\operatorname{\mathbb V}(\v B(T))=1\r].$$ Since $\frac{7}{8}\epsilon^l<\epsilon^l$, once we obtain equation we have a contradiction to , thus completing the proof.
We write $S$ for the time of the first branching event in $\v W(T)$ and $W_S$ for the position of the initial ‘ancestor’ particle at that time. We note that by the strong Markov property at time $S\wedge(T-\delta_\dim)$, $$\begin{gathered}
\label{eq:first_branch}
\P^\epsilon_x\l[\operatorname{\mathbb V}_p(\v W(T))=1\r]=
\E^\epsilon_x\l[g(\P^\epsilon_{W_S}\l[\operatorname{\mathbb V}_p (\v W (T-S))=1\r]
\1_{S\leq T-\delta_{\dim}} \r]\\
+\E^\epsilon_x \l[ \P^\epsilon_{W_{T-\delta_{\dim}}}\l[\operatorname{\mathbb V}_p(\v W(\delta_{\dim}))=1\r]\1_{S\geq T-\delta_{\dim}}\r].\end{gathered}$$ We begin with the second term on the right of . Since $T-\delta_\dim \geq \delta'_\dim-\delta_\dim=(l+1)\epsilon^2 |\log \epsilon |$ and $S\sim \text{Exp}(\epsilon^{-2})$, $$\label{eq:rhs2}
\E^\epsilon_x \l[ \P^\epsilon_{W_{T-\delta_{\dim}}}
\l[\operatorname{\mathbb V}_p(\v W(\delta_{\dim}))=1\r]\1_{S\geq T-\delta_{\dim}}\r]
\leq \P^\epsilon \l[ S\geq (l+1)\epsilon^2 |\log \epsilon |\r]
= \epsilon^{l+1}.$$ To bound the first term on the right of , partition on the event $\{S\leq \epsilon^{l+3}\}$ (which has probability $\leq\epsilon^{l+1}$): $$\begin{aligned}
&\E^\epsilon_x\l[g(\P^\epsilon_{W_S}
\l[\operatorname{\mathbb V}_p (\v W (T-S))=1\r]\1_{S\leq T-\delta_{\dim}} \r]\notag\\
&\hspace{1pc}\leq \P^\epsilon\l[S\leq\epsilon^{l+3}\r] +
\E^\epsilon_x\l[g(\P^\epsilon_{W_S}\l[\operatorname{\mathbb V}_p (\v W (T-S))=1\r]
\1_{S\leq T-\delta_{\dim}} \1_{S\geq\epsilon^{l+3}}\r]\notag\\
&\hspace{1pc}\leq \epsilon^{l+1} + \E^\epsilon_x\l[g\l(\P^\epsilon_{d(W_S,T-S)+K_1e^{K_2 (T-S)}\epsilon|\log\epsilon|}\l[\operatorname{\mathbb V}(\v B(T-S))=1\r]+\epsilon^l\r)\1_{S\leq T-\delta_{\dim}}\r]
.\label{eq:rhs1}\end{aligned}$$ The last line follows from the minimality of $T'$ (note that if $\epsilon^{l+3}\leq S\leq T-\delta_\dim$, then $T-S \in [\delta_\dim ,T')$ by ) and from monotonicity of $g$.
Conditioning on the value of $S$, since the path of the ancestor particle $(W_\cdot)$ is independent of $S$, $$\begin{aligned}
\label{eq:rhs3}
&\E^\epsilon_x\l[g\l(\P^\epsilon_{d(W_S,T-S)+K_1e^{K_2 (T-S)}\epsilon|
\log\epsilon|}\l[\operatorname{\mathbb V}(\v B(T-S))=1\r]+\epsilon^l\r)
\1_{S\leq T-\delta_{\dim}}\r] \notag \\
&\hspace{1cm}\leq \int_0^{(l+1)\epsilon^2 |\log \epsilon |}
\epsilon^{-2}e^{-\epsilon^{-2} s}
E_x\l[g\l(\P^\epsilon_{d(W_s,T-s)+K_1e^{K_2 (T-s)}\epsilon|\log\epsilon|}\l[\operatorname{\mathbb V}(\v B(T-s))=1\r]+\epsilon^l\r)\r]ds \notag\\
&\hspace{3cm} +\P^\epsilon \l[S\geq (l+1)\epsilon ^2 |\log \epsilon | \r] \notag \\
&\hspace{1cm} \leq \tfrac{3}{4}\epsilon^l+
\int_0^{(l+1)\epsilon^2 |\log \epsilon |}
\epsilon^{-2}e^{-\epsilon^{-2} s}
E_{d(x,T)}\l[g\l(\P^\epsilon_{B_s +K_1 e^{K_2 T}\epsilon |\log \epsilon |}
[\operatorname{\mathbb V}(\v B(t-s))=1]\r)\r]ds \notag \\
&\hspace{3cm}+\P^\epsilon \l[S\leq \epsilon ^3 \r] \epsilon ^l +\epsilon^{l+1} \notag \\
&\hspace{1cm} \leq \tfrac{3}{4}\epsilon^l+2\epsilon^{l+1}
+\E^\epsilon_{d(x,T)}\l[g\l(\P^\epsilon_{B_{S'}+K_1e^{K_2 T}\epsilon|\log\epsilon|}\l[\operatorname{\mathbb V}(\v B(T-S'))=1\r]\r)\1_{S'\leq T-\delta_{\dim}}\r].\end{aligned}$$ Here, the second inequality follows by Lemma \[lem:keylemma\_2\]. For the final inequality, we write $S'$ for the time of the first branching event in $(\v B(s))_{s \geq 0}$ and $B_{S'}$ for the position of the ancestor at that time, and note that $S'$ has the same distribution as $S$. The inequality follows since $T\geq \delta'_\dim$ and so $T-\delta_\dim \geq (l+1)\epsilon ^2 |\log \epsilon |$.
Putting , and into we obtain $$\begin{aligned}
\P^\epsilon_x\l[\operatorname{\mathbb V}_p(\v W (T))=1\r]
&\leq 4\epsilon^{l+1}+\tfrac{3}{4}\epsilon^l+
\E^\epsilon_{d(x,T)}\l[g\l(\P^\epsilon_{B_{S'} +K_1 e^{K_2 T}\epsilon
|\log \epsilon |}[\operatorname{\mathbb V}(\v B(T-S'))=1]\r)\1_{S'\leq T-\delta_{\dim}}\r] \\
&\leq 4\epsilon^{l+1}+\tfrac{3}{4}\epsilon^l+\P^\epsilon_{d(x,T)+K_1 e^{K_2 T}\epsilon |\log \epsilon |}\l[\operatorname{\mathbb V}(\v B(T))=1\r],\end{aligned}$$ where the second line follows by the strong Markov Property for $(\v B (\cdot))$ at time $S'\wedge (T-\delta_\dim)$, in similar style to . Reducing $\epsilon_{\dim}$, if necessary, to ensure that $\tfrac{3}{4}\epsilon^l+4\epsilon^{l+1}\leq \frac{7}{8}\epsilon^l$ for all $\epsilon\in(0,\epsilon_{\dim})$, we obtain , which completes the proof of .
By a similar argument, using in place of , we can also deduce .
It suffices to prove the result for sufficiently large $k\in\N$, and in particular we will show it for $k\geq 4$.
We choose $c_{\dim}(k)=c_1(k)+K_1e^{K_2T^*}$. Thus, for any $t\in[\delta_{\dim},T^*]$ and $x\in\R^{\dim}$ such that $d(x,t)\leq -c_{\dim}(k)\epsilon|\log\epsilon|$ we have $$d(x,t)+K_1e^{K_2t}\epsilon|\log\epsilon|\leq -c_1(k)\epsilon|\log\epsilon|.$$ It follows from Theorem \[thm:BBMone\] (reducing $\epsilon_\dim$ if necessary so that $\epsilon<\epsilon_1(k)$) and that $\P_x\l[\operatorname{\mathbb V}_p(\v W(t))=1\r]\leq 2\epsilon^k$ for such $x$ and $t$. Similarly, for $x$ such that $d(x,t)\geq c_{\dim}(k)\epsilon|\log\epsilon|$, by Theorem \[thm:BBMone\] and we have $\P_x\l[\operatorname{\mathbb V}_p(\v{W}(t))=0\r]\leq 2\epsilon^k$.
### Proof of Lemma \[lem:keylemma\_2\] {#sec:d=2_propagation_technical}
To complete the proof of Theorem \[thm:BBMtwo\], it remains to prove Lemma \[lem:keylemma\_2\]. The ideas in the proof are simple, but are easily lost in the notation, so to explain the structure we begin with an outline of the proof of the first inequality . (The proof of goes along essentially the same lines.)
We take a large constant $C$ and consider the cases $|d(x,t)| \geq C \epsilon |\log \epsilon |$ and $|d(x,t)| \leq C \epsilon |\log \epsilon |$ separately. Since $s=\mc O( \epsilon^2 |\log \epsilon |)$, with high probability neither the $\dim$-dimensional Brownian motion $W$ nor the one-dimensional $B$ moves a distance more than $\mc O (\epsilon |\log \epsilon |)$ before time $s$. Therefore, if $C$ is sufficiently large and $d(x,t) \leq -C \epsilon |\log \epsilon |$, Theorem \[thm:BBMone\] tells us that the left-hand side of is $\leq \epsilon^{l+1}$; similarly, if $d(x,t) \geq C \epsilon |\log \epsilon |$ then the right-hand side of is $\geq 1$. This leaves the case of $|d(x,t)| \leq C \epsilon |\log \epsilon |$, in which we apply Proposition \[prop:coupling1\] to couple $W_s$ with $B_s$ in such a way that with probability $1-\mc O (\epsilon ^{l+1})$, $$d(W_s,t-s)\leq B_s+ \mc O (\epsilon |\log \epsilon |)s.$$ Thus, using monotonicity , the left-hand side of is bounded above by $$\E_{d(x,t)}\l[g\l(\P^\epsilon_{B_s +(K_1e^{K_2 (t-s)}+\mc O (s))\epsilon |\log \epsilon |}[\operatorname{\mathbb V}(\v B(t-s))=1]+\epsilon ^l\r)\r]
+\mc O(\epsilon ^{l+1}).$$ If $|p-\frac{1}{2}|\geq \frac{7}{18}$, we can use that $|g'(p)|\leq 2/3$ to pull the $\epsilon^l$ outside the argument of $g$ and then use monotonicity again to recover . The difficulty is that close to $p=\tfrac{1}{2}$, we have $g'(p)>1$. In the case $\P_{B_s +(K_1e^{K_2 (t-s)}+\mc O (s))\epsilon |\log \epsilon |}
[\operatorname{\mathbb V}(\v{B}(t-s))=1]\approx \frac{1}{2}$, we instead choose $K_2\gg 0$, and use the lower bound on the ‘slope of the interface’ given by Corollary \[lem:high\_deriv\] to estimate the increment in $\P_z^\epsilon[\operatorname{\mathbb V}(\v{B}(t-s))=1]$ when we replace $z+(K_1e^{K_2(t-s)}+\mc O(s))\epsilon |\log\epsilon|$ by $z+K_1e^{K_2t}\epsilon |\log\epsilon|$.
The remainder of this subsection contains the formal proof.
We begin by proving . For the duration of the proof, for $u\geq 0$ and $z\in \R$ we write $$\Q^{\epsilon, u}_{z}=\P^\epsilon_{z}\l[\operatorname{\mathbb V}(\v B(u))=1\r].$$ Recall $C_0$ and $c_1(k)$ from and Theorem \[thm:BBMone\] respectively. Let $$\label{R_defn}
R=2 c_1(l)+4(l+1)\dim +1.$$ Fix $K_2$ such that $$\label{eq:K_2_cond}
K_1(K_2-C_0)-C_0 R=c_1(1).$$ Let $\epsilon_\dim = \epsilon_1(l)$ where $\epsilon_1(l)$ is defined in Theorem \[thm:BBMone\].
First we need an estimate for the probability that a $\dim$-dimensional Brownian motion moves further than $\sim \epsilon |\log \epsilon |$ in time $s$ (recall that $s\leq (l+1)\epsilon^2|\log\epsilon|$). Let $$A_x=\l\{\sup_{u\in [0,s]} |W_u-x|\leq 2(l+1)\,\dim \epsilon |\log \epsilon |\r\}.$$ Then bounding $|W_u|$ by the sum of the moduli of $\dim$ one-dimensional Brownian motions and using the reflectional symmetry of one dimensional Brownian motion, $$\begin{aligned}
\label{eq:prob_ac_est}
P_x\l[A_x^c\r]&\leq 2\dim P_0 \l[\sup_{u\in [0,s]} B_u >
2(l+1)\epsilon |\log \epsilon | \r] \notag\\
&\leq 4\dim P_0 \l[B_1 >2((l+1) |\log \epsilon |)^{1/2} \r] \notag \\
&\leq 4\dim \epsilon ^{l+1}.\end{aligned}$$ Here, since $s\leq (l+1)\epsilon ^2 |\log \epsilon |$ the second line follows by the reflection principle. The last line follows using the tail bound $\P[B_1\geq x]\leq e^{-x^2/4}$.
As advertised, we now consider the following three cases:
- $d(x,t) \leq -\l(2c_1(l)+2(l+1)\dim+K_1e^{K_2(t-s)}\r)\epsilon |\log \epsilon |$,
- $d(x,t) \geq \l(2c_1(l)+2(l+1)\dim+K_1e^{K_2(t-s)}\r)\epsilon |\log \epsilon |$,
- $|d(x,t)| \leq \l(2c_1(l)+2(l+1)\dim+K_1e^{K_2(t-s)}\r)\epsilon |\log \epsilon |$.
The third case corresponds to $x$ being close to the interface at time $t$. The first two cases correspond to $x$ falling (sufficiently far) inside or outside of the interface.
Case (i): Recall that by there exist $v_0,V_0>0$ such that if $s\leq v_0$ and $x\in \R^\dim$ then $$\label{eq:dble_star_t_lip}
|d(x,t)-d(x,t-s)|\leq V_0s.$$ We reduce $\epsilon_\dim$, if necessary, to ensure that for $\epsilon \in (0, \epsilon_\dim)$ we have $(l+1)\epsilon ^2 |\log \epsilon | \leq v_0$. Then if the event $A_x$ occurs, $$\begin{aligned}
&d(W_s,t-s)+K_1 e^{K_2 (t-s)} \epsilon |\log \epsilon |\notag\\
&\hspace{1cm}\leq -(2c_1(l)+2(l+1)\dim )\epsilon |\log \epsilon |+|d(W_s,t-s)-d(x,t)|\notag\\
&\hspace{1cm}\leq -(2c_1(l)+2(l+1)\dim )\epsilon |\log \epsilon |+|d(x,t)-d(x,t-s)|+|W_s-x|\notag\\
&\hspace{1cm}\leq -2c_1(l)\epsilon |\log \epsilon |+V_0(l+1)\epsilon ^2 |\log \epsilon |.\label{eq:case_1_start}\end{aligned}$$ Here, the second line follows from being in case (i) and the third follows from the triangle inequality. The final line then follows from and that $s\leq (l+1)\epsilon ^2 |\log \epsilon |$, and since $A_x$ occurs.
Reducing $\epsilon_\dim$, if necessary, from we have $$d(W_s,t-s)+K_1 e^{K_2 (t-s)} \epsilon |\log \epsilon |\leq -c_1(l)\epsilon |\log \epsilon |.$$ Therefore $$\begin{aligned}
E_x \l[g\l(\Q^{\epsilon, t-s}_{d(W_s,t-s)+K_1 e^{K_2 (t-s)} \epsilon |\log \epsilon |} +\epsilon ^l\r)\r]
&\leq E _x \l[g(\epsilon ^l +\epsilon ^l) \1_{A_x} \r]+P_x\l[A_x^c\r]\\
&\leq 6\epsilon ^{2l}+4\dim \epsilon ^{l+1}.\end{aligned}$$ Here the first inequality follows by Theorem \[thm:BBMone\] and the second inequality by the definition of $g$ in and by . Again reducing $\epsilon_\dim$ if necessary, for $\epsilon \in (0,\epsilon_\dim)$ we have $$E_x [g(\Q^{\epsilon, t-s}_{d(W_s,t-s)+K_1 e^{K_2 (t-s)} \epsilon |\log \epsilon |} +\epsilon ^l)]\leq \tfrac{3}{4}\epsilon ^l,$$ and so holds in this case.
Case (ii): In this case, we have that $d(x,t)\geq (c_1(l) +2(l+1))\epsilon |\log \epsilon |$. A similar argument to that used for gives us that $$\label{eq:star_key_lem}
P_{d(x,t)}\l[ B_s \leq c_1(l) \epsilon |\log \epsilon | \r]\leq \epsilon ^{l+1}.$$ It follows that in this case $$\begin{aligned}
E_{d(x,t)}\l[g\l(\Q^{\epsilon, t-s}_{B_s+K_1 e^{K_2 t} \epsilon |\log \epsilon |}\r) \r]
&\geq E_{d(x,t)}\l[g\l(\Q^{\epsilon, t-s}_{B_s+K_1 e^{K_2 t} \epsilon |\log \epsilon |}\r) \1\{B_s\geq c_1(l)\epsilon |\log \epsilon |\}\r]\\
&\geq g(1-\epsilon ^l)-\epsilon ^{l+1}\\
&\geq 1-3\epsilon^{2l}-\epsilon ^{l+1},\end{aligned}$$ where the second line follows by Theorem \[thm:BBMone\] and and the last line by the definition of $g$ in . Again reducing $\epsilon_\dim$ if necessary, for $\epsilon\in (0,\epsilon_\dim)$ we have $$E_{d(x,t)}\l[g(\Q^{\epsilon, t-s}_{B_s+K_1 e^{K_2 t} \epsilon |\log \epsilon |}) \r]
\geq 1- \tfrac{3}{4} \epsilon^l$$ and so holds in this case.
Case (iii): We now turn to the case in which $x$ is close to the interface. If the event $A_x$ occurs, for $u\in [0,s]$ we have $$\begin{aligned}
|d(W_u,t-u)|&\leq |W_u-x|+|d(x,t)|+|d(x,t)-d(x,t-u)|\\
&\leq (2c_1(l)+4(l+1)\dim + K_1 e^{K_2(t-s)})\epsilon |\log \epsilon |
+V_0(l+1)\epsilon ^2 |\log \epsilon |,\end{aligned}$$ where the second line follows by . Reducing $\epsilon_\dim$ if necessary, for $\epsilon \in (0,\epsilon_\dim)$ we have $$\label{eq:dble_dagger_key_lem}
|d(W_u,t-u)|\leq (R+K_1 e^{K_2(t-s)})\epsilon |\log \epsilon |,$$ where $R$ is defined in . We now apply Proposition \[prop:coupling1\] with $$\label{eq:beta_defn}
\beta = (R+K_1 e^{K_2(t-s)})\epsilon |\log \epsilon |.$$ By reducing $\epsilon_\dim$ if necessary, we have for $\epsilon \in (0,\epsilon_\dim)$ that $\beta \leq c_0$. Define $$T_\beta = \inf (\{u \in [0,t):|d(W_u,t-u)| \geq \beta \}\cup \{t\}).$$ Then by Proposition \[prop:coupling1\], we can couple $(W_u)_{u\geq 0}$ with $(B_u)_{u\geq 0}$, a one-dimensional Brownian motion started from $d(x,t)$, in such a way that for $u\leq T_\beta$, $$\label{eq:five_star_key_lemma}
d(W_u,t-u)\leq B_u +C_0 \beta u.$$ Hence $$\begin{aligned}
\label{eq:0_key_lemma}
E_x \l[g(\Q^{\epsilon, t-s}_{d(W_s,t-s)+K_1 e^{K_2 (t-s)}\epsilon |\log \epsilon |}+\epsilon ^l) \r]
&\leq E_{d(x,t)} \l[g(\Q^{\epsilon, t-s}_{B_s+C_0\beta s +K_1 e^{K_2 (t-s)}\epsilon |\log \epsilon |}+\epsilon ^l) \r]+P_x \l[T_\beta \leq s \r] \notag \\
&\leq E_{d(x,t)} \l[g(\Q^{\epsilon, t-s}_{B_s+C_0\beta s +K_1 e^{K_2 (t-s)}\epsilon |\log \epsilon |}+\epsilon ^l) \r]+4\dim \epsilon ^{l+1}.\end{aligned}$$ Here, the first line follows by , and the monotonicity of $g$. The second line then follows by (note that by , if $A_x$ occurs then $T_\beta \geq s$).
Now let $$E=\l\{\l|\Q^{\epsilon, t-s}_{B_s+C_0\beta s +K_1 e^{K_2 (t-s)}\epsilon |\log \epsilon |}-\tfrac{1}{2}\r|\leq \tfrac{5}{12}\r\}.$$ We shall consider the cases $E$ and $E^c$ separately to bound the right hand side of .
Consider first when the event $E$ occurs. Note that by the definition of $\beta$ in , $$\begin{aligned}
\label{eq:K2_conseq}
K_1 e^{K_2 t} \epsilon |\log \epsilon |-\l(C_0\beta s +K_1 e^{K_2 (t-s)}\epsilon |\log \epsilon |\r)
&= \l(K_1 e^{K_2(t-s)}(e^{K_2 s}-1-C_0 s)-C_0 Rs \r)\epsilon |\log \epsilon | \notag\\
&\geq \l( K_1(K_2-C_0)-C_0 R \r)s \epsilon |\log \epsilon | \notag\\
&= c_1(1) s \epsilon |\log \epsilon |,\end{aligned}$$ where the second line follows since $K_2>0$ and the last line follows by . Reducing $\epsilon _\dim$ if necessary so that $\epsilon_\dim <\min(\epsilon_1(1),\tfrac{1}{24})$, for $\epsilon \in (0,\epsilon_\dim)$ we can apply Corollary \[lem:high\_deriv\] with $z=B_s+C_0\beta s +K_1 e^{K_2 (t-s)}\epsilon |\log \epsilon |$ and $w=z+c_1(1) s \epsilon |\log \epsilon |\leq B_s+K_1 e^{K_2 t} \epsilon |\log \epsilon |$ to give that $$\label{eq:1_key_lemma}
\Q^{\epsilon, t-s}_{B_s+C_0\beta s +K_1 e^{K_2 (t-s)}\epsilon |\log \epsilon |} \1_E
\leq
(\Q^{\epsilon, t-s}_{B_s+K_1 e^{K_2 t}\epsilon |\log \epsilon |}-\tfrac{1}{48}s) \1_E .$$
Finally, we consider the case when the event $E^c$ occurs. Recall that $g(p)=3p^2-2p^3$ for $p\in [0,1]$, so $g'(p)=6p(1-p)$. Hence if $p,\delta \geq 0$ with either $p+\delta \leq \frac{1}{9}$ or $p \geq \frac{8}{9}$ then $$\label{eq:p_deriv_key}
g(p+\delta) \leq g(p)+\tfrac{2}{3}\delta .$$ Reducing $\epsilon _\dim$ if necessary so that $\frac{1}{12}+\epsilon^l <\frac{1}{9}$ for $\epsilon \in (0,\epsilon_\dim)$, we have $$\begin{aligned}
\label{eq:2_key_lem}
g \l(\Q^{\epsilon, t-s}_{B_s+C_0\beta s +K_1 e^{K_2 (t-s)}\epsilon |\log \epsilon |}+\epsilon ^l \r) \1_{E^c}
& \leq \l( g \l(\Q^{\epsilon, t-s}_{B_s+C_0\beta s +K_1 e^{K_2 (t-s)}\epsilon |\log \epsilon |}\r)+\tfrac{2}{3}\epsilon ^l \r) \1_{E^c} \notag \\
& \leq \l( g \l(\Q^{\epsilon, t-s}_{B_s+K_1 e^{K_2 t}\epsilon |\log \epsilon |}\r)+\tfrac{2}{3}\epsilon ^l \r) \1_{E^c},\end{aligned}$$ where the first line follows by and the last line by and monotonicity of $g$.
Putting and into , $$\begin{aligned}
E_x \l[g(\Q^{\epsilon, t-s}_{d(W_s,t-s)+K_1 e^{K_2 (t-s)}\epsilon |\log \epsilon |}+\epsilon ^l) \r]
&\leq E_{d(x,t)} \l[g\l(\Q^{\epsilon, t-s}_{B_s+K_1 e^{K_2 t}\epsilon |\log \epsilon |}-\tfrac{1}{48}s +\epsilon^l \r)\1_E \r]\\
&\quad +E_{d(x,t)} \l[\l( g\l(\Q^{\epsilon, t-s}_{B_s+K_1 e^{K_2 t}\epsilon |\log \epsilon |}\r) +\tfrac{2}{3}\epsilon^l \r)\1_{E^c} \r]\\
&\quad +4\dim\epsilon^{l+1}\\
&\leq E_{d(x,t)} \l[g\l(\Q^{\epsilon, t-s}_{B_s+K_1 e^{K_2 t}\epsilon |\log \epsilon |}\r)\r]\\
&\quad +\tfrac{2}{3}\epsilon^l + \epsilon ^l \1 _{s\leq 48 \epsilon ^l}+4\dim\epsilon^{l+1},\end{aligned}$$ where the last inequality follows in the case $s\leq 48 \epsilon ^l$ since $|g'(p)|\leq \frac{3}{2}$ for all $p \in [0,1]$. Reducing $\epsilon_\dim$, if necessary, so that $4\dim\epsilon^{l+1}\leq \frac{1}{12}\epsilon^l$ and $48\epsilon ^l \leq \epsilon^3$ for $\epsilon \in (0,\epsilon _\dim)$ completes the proof of .
The second statement of the lemma, equation , is proved by the same argument, considering $\{\operatorname{\mathbb V}(\v B(u))=0\}$ instead of $\{\operatorname{\mathbb V}(\v B(u))=1\}$ and using $d(W_u,t-u)\geq B_u -C_0 \beta u$ for $u\leq T_\beta$ in place of .
Proof of Theorem \[thm:slfvs\] {#proof of slfvs to cf}
==============================
In this section we turn to the proof of our central result, Theorem \[thm:slfvs\], which provides convergence, after suitable rescaling, of the SLFVS started from an appropriate initial condition to the indicator function of a region whose boundary evolves according to mean curvature flow. The proof mimics that of Theorem \[theorem ac to cf\] in exploiting a dual process. However, because of genetic drift, in addition to branching, individuals in our dual process can coalesce. The duality relation will once again be with a historical process and expressed through a majority voting procedure.
A branching and coalescing dual for the SLFVS {#duality for SLFVS}
---------------------------------------------
We begin by describing the dual process of branching and coalescing lineages. It is driven by the same Poisson Point Process of ‘events’ that drives the SLFVS. Recall from that $\Pi^n$ is a Poisson point process on $\R_+ \times \R^{\dim} \times (0,\infty)$ with intensity measure $$n dt\otimes n^{\beta} dx\otimes \mu^n(dr).$$ We also let $$u_n = \frac{u}{n^{1-2\beta}}, \qquad\mbox{and}\qquad
\v{s}_n = \frac{1}{\epsilon_n^{2}}\frac{1}{n^{2\beta}}.$$
\[def:slfvs\_dual\] For $n\in\N$, the process $(\mathcal P ^n _t)_{t\geq 0}$ is the $\bigcup_{l\geq 1}(\R^{\dim})^l$-valued Markov process with dynamics defined as follows.
The process is started with a single individual $\mathcal P^n_0=x$ and for $t\geq 0$, $\mathcal P^n_t = (\xi^n_1(t),\ldots , \xi^n_{N(t)}(t))$ for some $N(t)\in \N$. At each event $(t,x,r)\in \Pi^n$, independently of all else, the event is said to be neutral with probability $1-\v{s}_n$. In this case:
1. For each $\xi_i^n(t-)\in \mc{B}_r(x)$, independently mark the corresponding individual with probability $u_n$;
2. if at least one individual is marked, all marked individuals coalesce into a single offspring individual, whose location is drawn uniformly at random from within $\mc{B}_r(x)$.
With the complementary probability $\v{s}_n$, the event is said to be selective, in which case:
1. For each $\xi^n_i(t-)\in \mc{B}_r(x)$, independently mark the corresponding individual with probability $u_n$;
2. if at least one individual is marked, all of the marked individuals are replaced by [*three*]{} offspring individuals, whose locations are drawn independently and uniformly from within $\mc{B}_r(x)$.
In both cases, if no individual is marked, then nothing happens.
We have referred to the new individuals created during reproduction events as ‘offspring’ individuals. From a biological perspective, it would perhaps be more natural to call them ‘parents’ or ‘potential parents’, as forwards in time they correspond to the locations from which alleles from the parental generation are sampled. However, as much of our proof of Theorem \[theorem ac to cf\] will carry over with minimal changes to the SLFVS setting, we wish to retain the terminology of the branching Brownian motion of the previous section.
The duality relation that we exploit is between the SLFVS and the [*historical process*]{} of branching and coalescing lineages, $$\Xi^n(t):=
(\mathcal P^n_s)_{0\leq s\leq t}.$$ We write $\P_x$ for the law of $\Xi^n$ when $\mathcal P^n_0$ is the single point $x$ and $\E_x$ for the corresponding expectation. For $\v i\in \{1,2,3\}^\N$ with $\v{i}=(i_1,i_2,\ldots)$, we let $(\xi^n_{\v i}(\cdot))_{0\leq s\leq t}\subseteq\Xi(t)$ denote the $\R^\dim$-valued path which jumps to the location of an offspring when the individual in $\mc P^n_s$ at its location is affected by an event, jumping to the $i_k ^{\text{th}}$ offspring when it is affected by its $k^{\text{th}}$ selective event. We shall refer to $(\xi^n_{\v i}(\cdot))_{0\leq s\leq t}$ as an ancestral lineage.
The voting procedure on $\Xi^n(t)$ is a minor modification of Definition \[vote\_defn\]. Let $p:\R^\dim \to [0,1]$ be a fixed function. Recalling that the set of individuals in $\mc P^n_t$ is $\{\xi^n_1(t),\ldots , \xi^n_{N(t)}(t)\}$, for each $j\leq N(t)$, the individual $\xi_j^n(t)$ votes $1$ with probability $p(\xi_j^n(t))$ and otherwise votes $0$; votes from different individuals are independent. As we trace backwards in time through $\Xi(t)$,
1. at each neutral event, all individuals that are marked in the event adopt the vote of the offspring individual of the event;
2. at each selective event in $\Pi^n$, all individuals that are marked in the event adopt the majority vote of the votes of the three offspring individuals of the event.
This defines an iterative voting procedure, which runs inwards from the ‘leaves’ of $\Xi^n(t)$ to the ancestral individual $\emptyset$.
\[vote\_defn\_slfvs\] With the voting procedure described above, we define $ \mathbb{V} _p(\Xi^n(t)) $ to be the vote associated to the root $\emptyset$.
At this point the duality relation between the SLFVS and $\Xi(t)$ is easy to guess. However, in order to write it down formally, we have to overcome the fact that the SLFVS will only be defined, as a function, Lebesgue a.e. and so we cannot necessarily define $w^n_t(x)$ for a fixed point $x\in \R^\dim$. However, if, $\psi\in C(\R^\dim)\cap L^1(\R^\dim)$, then the function $$\int_{\R^\dim}\psi(x)w^n_t(x)dx,$$ [*is*]{} well-defined.
\[thm:slfvs\_duality\] The spatial $\Lambda$-Fleming-Viot process with selection driven by $\Pi^n$, $(w^n_t(x), x\in\R^\dim)_{t\geq 0}$, is dual to the historical process $(\Xi^n(t))_{t\geq 0}$ in the sense that for every $\psi\in C(\R^\dim)\cap L^1(\R^\dim)$, we have $$\E_{p}\bigg[\int_{\R^\dim} \psi(x)w^n_t(x)\, dx\bigg]
= \int_{\R^\dim} \psi(x)\E_x\bigg[ \operatorname{\mathbb V}_{p}\big( \Xi^n(t)\big)\bigg]\, dx
= \int_{\R^\dim} \psi(x)\P_x\bigg[ \operatorname{\mathbb V}_{p}\big( \Xi^n(t)\big)=1\bigg]\, dx.
\label{dual formula}$$
Of course, we are abusing notation here: the expectations on the left and right of this equation are taken with respect to different measures. The subscripts on the expectations are the initial values for the processes on each side.
To see that the result should be true, note that (if it is defined) $w_t^n(x)$ is the probability that an allele sampled from the population at the location $x$ at time $t$ is of type $a$. In order to determine that probability, we trace back until the most recent event that covered the location $x$. With probability $u_n$, the chosen allele was an offspring of the event, in which case its type can be determined if we know the types of the potential parents of the event. If the event is neutral, the type is that of an allele (the ‘parent’) sampled from a point picked uniformly at random from the affected region at the time of the event; if it is selective, then the type is the ‘majority vote’ of three ‘potential parents’ sampled uniformly at random from the affected region. In order to determine the types of the potential parents, we continue to trace backwards in time, following the locations of all potential ancestors until time zero. This gives us the dual process $\Xi^n(t)$. At that time, each potential ancestor samples its type according to the initial condition $w_0$ at its location. We can then determine $w_t^n(x)$ by working back through $\Xi^n(t)$ using our majority voting procedure.
A formal proof of Theorem \[thm:slfvs\_duality\] using generators is a simple extension of that of the corresponding duality for the spatial $\Lambda$-Fleming-Viot process with genic selection in [@etheridge/veber/yu:2014] (and indeed can be extended to cover the more general initial conditions for the dual process considered there) and so is omitted.
The duality reduces the proof of Theorem \[thm:slfvs\] to the following analogue of Theorem \[thm:BBMtwo\].
\[thm:slfvs\_dual\] Take $\sigma^2$ as in . Suppose that $\beta\in(0,1/4)$ and let $\epsilon_n$ be a sequence such that $\epsilon_n\to 0$ and $(\log n)^{1/2}\epsilon_n\to\infty$ as $n\rightarrow\infty$. Assume $p$ satisfies - and define $\mathscr{T}$, $d(x,t)$ as for Theorem \[theorem ac to cf\]; take $T^*<\mathscr{T}$. Let $k\in\N$. There exist $n_*(k)\in \N$, and $a_*(k),d_*(k)\in(0,\infty)$ such that for all $n\geq n^*$ and all $t$ satisfying $a_* \epsilon_n ^2 |\log \epsilon_n |\leq t\leq T^*$,
1. for $x$ such that $d(x,\sigma ^2 t)\geq d_* \epsilon_n |\log \epsilon_n|$, we have $\P_x\l[\operatorname{\mathbb V}_p (\Xi^n (t))=1\r]\geq 1-\epsilon_n^k$.
2. for $x$ such that $d(x,\sigma ^2 t)\leq -d_* \epsilon_n |\log \epsilon_n|$, we have $\P_x\l[\operatorname{\mathbb V}_p (\Xi ^n (t))=1\r]\leq \epsilon_n^k$.
Before providing a proof of this result, let us explain why it should be true.
First consider the motion of a single ancestral lineage $\xi_{\v i}^n(\cdot)$ in $\Xi^n(t)$. It evolves as a pure jump process which is homogeneous in both space and time. Write $V_r$ for the volume of $\mc{B}_r(x)$. The rate at which the lineage jumps from $y$ to $y+z$ can be written $$\label{jump of size z}
m_n(dz)=nu_nn^{\dim\beta}\int_0^{\mc{R}_n}\frac{V_r(0,z)}{V_r}\mu^n(dr)\,dz,$$ where $V_r(0,z)$ is the volume of ${\mc B}_r(0)\cap {\mc B}_r(z)$. To see this, by spatial homogeneity, we may take the lineage to be at the origin in $\R^\dim$ before the jump, and then, in order for it to jump to $z$, it must be affected by an event that covers both $0$ and $z$. If the event has radius $r$, then the volume of possible centres, $x$, of such events is $V_r(0,z)$ and so the intensity with which such a centre is selected is $n\,n^{\dim\beta}V_r(0,z)\mu^n(dr)$. The parental location is chosen uniformly from the ball $\mc{B}_r(x)$, so the probability that $z$ is chosen as the parental location is $dz/V_r$ and the probability that our lineage is actually affected by the event is $u_n$. Combining these yields .
The total rate of jumps is $$\begin{aligned}
\int_{\R^\dim}m_n(dz)&=&\int_0^{\mc{R}_n}nu_n\,n^{\dim\beta}\frac{1}{V_r}
\int_{\R^\dim}\int_{\R^\dim}\1_{|x|<r}\1_{|x-z|<r}dx\,dz\,\mu^n(dr)
\nonumber\\
&=&\int_0^{\mc{R}_n}nu_n\,n^{\dim\beta}V_r\mu^n(dr)\nonumber \\
&=&n^{2\beta} u V_1\int_0^{\mc{R}}r^d\mu(dr),\label{jump rate}\end{aligned}$$ and the size of each jump is $\Theta(n^{-\beta})$ and so it is no surprise that in the limit a single lineage will evolve according to a (time-changed) Brownian motion. To identify the diffusion constant, we calculate: $$\begin{gathered}
\label{identifying sigma}
\frac{1}{2\dim}\int_{\R^\dim}|z|^2m_n(dz)
=\frac{1}{2\dim}\int_{\R^\dim}|z|^2nu_n\int_0^{\mc{R}_n}n^{\dim\beta}
\frac{V_r(0,z)}{V_r}\mu^n(dr)dz\\
=\frac{u}{2\dim}\int_0^{\mc{R}}\int_{\R^\dim}|z|^2\frac{V_r(0,z)}{V_r}dz\mu(dr),\end{gathered}$$ which is precisely $\sigma^2$ from .
Note also that a lineage is affected by selective events at rate $$\label{eq:sel_rate_slfvs}
\l(uV_1\int_0^\mc{R}r^d\mu(dr)\r)n^{2\beta}\v{s}_n
=\eta \epsilon_n^{-2},$$ where $\eta =uV_1\int_0^\mc{R}r^d\mu(dr)$. Evidently, we can bound the total number of lineages in $\Xi^n(t)$ above by the total number in a process in which each lineage, independently, branches at rate $\eta \epsilon_n^{-2}$. Since $\epsilon_n^{-2}=o(\log n)$, this implies that for any $\delta>0$, with high probability, there are $o(n^\delta)$ pairs of lineages in $\Xi^n(T^*)$. Each such pair is in the region affected by some event (neutral or selective) at most $\mc{O}(n)$ times in $[0,T^*]$ and so the chance that we see any coalescence events is $o(nu_n^2\,n^\delta)$ for any $\delta>0$. Since $nu_n^2=n^{4\beta-1}$ and $\beta \in (0,1/4)$, for large $n$ we do not expect to see any coalescence events before time $T^*$.
Combining the above, the dual is well approximated by a ternary branching Brownian motion with branching rate $\Theta(\epsilon_n^{-2})$ and so it is natural to expect that an equivalent of Theorem \[thm:BBMtwo\] holds.
Majority voting in the , for $\dim\geq 2$ {#sec:slfvs}
-----------------------------------------
The rigorous proof of Theorem \[thm:slfvs\] closely follows that of Theorem \[thm:BBMtwo\]. In Section \[sec:slfvs\_generation\], we focus on generation of the interface, which is proved in much the same way as Proposition \[prop:d=2\_generation\]. Then, in Section \[sec:slfvs\_propagation\], we look at the propagation of the interface. We shall see that, since it essentially focusses on a single branching event, the argument of Section \[sec:d=2\_propagation\] is sufficiently flexible to adapt to the SLFVS setting.
First we present the additional arguments required in the SLFVS setting. These stem from the fact that ancestral lineages in the dual of the SLFVS follow jump processes (which, when the lineages are too close together, are dependent), and from the coalescence of ancestral lineages. In Section \[sec:slfvs\_dual\_ingredients\] we show that (in between selective events) the motion of a single ancestral lineage is approximately (time-changed) Brownian motion. Then, in Section \[sec:slfvs\_indep\_after\_branching\], we show that, asymptotically, the three families of descendants of offspring created during a selective event evolve independently (conditional on their locations at birth).
In Sections \[sec:BBMone\] and \[sec:BBMtwo\] we used subscripts to distinguish variables that played the same role in each section, but had different values; e.g. $\delta_1$ in and $\delta_{\dim}$ in . The corresponding quantities in this section will be denoted with a subscript $*$, for example $\delta_*$ in .
### A single lineage {#sec:slfvs_dual_ingredients}
We begin the proof by showing that the trajectory of a single lineage is close to that of a Brownian motion. We follow what is now a familiar argument in the context of spatial $\Lambda$-Fleming-Viot processes (see for example [@etheridge/freeman/penington/straulino:2015]).
Let $(\xi^n(t))_{t \geq 0}$ be a pure jump process started at $x\in \R^\dim$ with rate of jumps from $y$ to $y+z$ given by the intensity measure $m^n(dz)$, and let $(W(t))_{t\geq 0}$ be a Brownian motion in $\R^\dim$ started at $x$.
\[lem:W\_xin\_close\] For $t>0$ fixed, there is a coupling of $W$ and $\xi^n$ under which $$\P \l[ \l|\xi^n(t)-W(\sigma^2 t)\r| \geq n^{-\beta/6} \r]=
\mc O(n^{-\beta }(t \vee 1)).$$
For $i\geq 1$, let $X_i = \xi^n _{i/n^{2\beta}} - \xi^n_{(i-1)/n^{2\beta}}$. Then $X_1 , X_2, \ldots $ are i.i.d. with a rotationally symmetric distribution and, by , $\E[|X_1|^2]=2\dim \sigma^2 n^{-2\beta}$. Moreover, by , the number of jumps made by $\xi^n$ on the time interval $[0,n^{-2\beta}]$ is Poisson, with mean $\Theta(1)$, so since each jump has magnitude at most $2 \mc R_n$, $
\E \l[ |X_1 |^4 \r] = \mc O (n^{-4\beta})$. Then by Skorohod’s second embedding Theorem, see e.g. [@billingsley:1995], there is a Brownian motion $W$ started at $x$ and a sequence $\upsilon_1, \upsilon_2, \ldots $ of stopping times such that setting $\upsilon_0=0$, $(\upsilon_i - \upsilon_{i-1})_{i \geq 1}$ are i.i.d. and $$\begin{aligned}
W(\upsilon_i)&=\xi(i/n^{2\beta}),\hspace{2pc}
\E[\upsilon_i - \upsilon_{i-1}]=\tfrac{1}{2\dim}\E\l[ |X_1 |^2 \r]=\sigma^2 n^{-2\beta},\hspace{2pc}
\E[(\upsilon_i - \upsilon_{i-1})^2 ] = \mc{O}(n^{-4\beta}).\end{aligned}$$ It follows that $\E[\upsilon_{\lfloor tn^{2\beta} \rfloor}]
=\sigma^2 \lfloor tn^{2\beta}\rfloor n^{-2\beta}$ and $\text{Var}[\upsilon_{\lfloor tn^{2\beta} \rfloor}]=\mc{O}(t n^{-2\beta})$. Hence by Chebychev’s inequality, $$\label{eq:sync_time_skemd}
\P\l[|\upsilon_{\lfloor tn^{2\beta} \rfloor}- \sigma^2 t|
\geq n^{-\beta/2}\r]=\mc{O}(t n^{-\beta}).$$ Now we have that $$\label{eq:xi_W_bound}
|\xi^n(t)-W(\sigma^2 t)|\leq |\xi^n(t)-\xi^n(\lfloor t n^{2\beta} \rfloor /n^{2 \beta})|
+|W(\upsilon_{\lfloor t n^{2\beta} \rfloor})-W(\sigma^2 t)|.$$ To control the first term on the right hand side, observe that $$\label{eq:xi_W_1st}
\P \l[ |\xi^n(t)-\xi^n(\lfloor t n^{2\beta} \rfloor /n^{2 \beta})|
\geq n^{-\beta/6}/2\r]
\leq \E\l[ |X_1|^2 \r] (n^{-\beta/6}/2)^{-2}=\mc O (n^{-5\beta/3}).$$ To control the second term on the right hand side of , let $Z\sim N(0,1)$, then $$\begin{aligned}
\label{eq:xi_W_2nd}
\P\bigg[|W(\upsilon_{\lfloor t n^{2\beta} \rfloor})-W(\sigma^2 t)|
&\geq& n^{-\beta /6}/2 \bigg]\leq \P\l[|\upsilon_{\lfloor tn^{2\beta} \rfloor}- \sigma^2 t|
\geq n^{-\beta/2}\r]\nonumber\\
&&+
\P\l[|\upsilon_{\lfloor t n^{2\beta} \rfloor }-\sigma^2 t| \leq n^{-\beta /2},
\, |W(\upsilon_{\lfloor t n^{2\beta} \rfloor})-W(\sigma^2 t)|
\geq n^{-\beta /6}/2 \r]\nonumber \\
&\leq& \P \l[\sup_{s\in [-n^{-\beta/2},n^{-\beta/2}]} |W(s)-W(0)|
\geq n^{-\beta/6}/2 \r]+
\mc{O}(t n^{-\beta}).
\nonumber \\
&\leq& 4\, \dim\, \P \l[\sqrt 2 n^{-\beta/4}Z \geq n^{-\beta/6}/2\dim \r]
+\mc{O}(t n^{-\beta}).
\nonumber \\
&=& \mc O (\exp (-\tfrac{1}{8\dim ^2} n^{\beta/6}) )
+\mc{O}(t n^{-\beta}).\end{aligned}$$ Here, the second inequality follows by and the third inequality follows by bounding the modulus of a $\dim$-dimensional Brownian motion by the sum of the moduli of $\dim$ one-dimensional Brownian motions and then using the reflection principle. Combining and with completes the proof.
Next, we need the asymptotic distribution of an ancestral lineage and its first branch time (that is the first time that it is affected by a selective event).
\[cor:xi\_W\] Let $\tau$ be the first branch time of $\Xi^n$. There is a coupling of $\Xi^n$ and $W$ under which $\tau$ and $W$ are independent, $\tau \sim \text{Exp}(\eta \epsilon_n^{-2} )$ where $\eta =u V_1\int_0^\mc{R}r^d\mu(dr)$, and for $i=1,2,3$, $$\P\l[\xi^n_i (\tau)-W(\sigma ^2 \tau)| \geq 3n^{-\beta/ 6} \r]=
\mc O (n^{-\beta}).$$
The distribution of $\tau$ follows immediately from .
Now consider any ancestral lineage $\xi^n\subseteq\Xi^n$. By the thinning property of Poisson processes, at any time $t>0$, we can write $\xi^n_t=\xi^{n,\tt{sel}}_t+\xi^{n,\tt{neu}}_t$, where $\xi^{n,\tt{sel}}_t$ and $\xi^{n,\tt{neu}}_t$ are independent pure jump processes with jump intensities $\v{s}_nm_n(dz)$ and $(1-\v{s}_n)m_n(dz)$ respectively, and taking $\tau$ to be the first jump time of $\xi^{n,\tt{sel}}$, $\xi^{n,\tt{neu}}_t$ is independent of $\tau$. Using Lemma \[lem:W\_xin\_close\] with $(1-\v{s}_n)m_n(dz)$ in place of $m_n(dz)$, we can couple $\xi^{n,\tt{neu}}$ with a Brownian motion $W$ in such a way that for any $t>0$, for any $t>0$, $$\P[|\xi^{n,\tt{neu}}_t-W(\sigma^2(1-\v{s}_n)t)|\geq n^{-\beta/6}]\leq
\mc O (n^{-\beta}(t\vee 1)).$$ Since $\v{s}_n=o (\log n/n^{2\beta})$, using Chebyshev’s inequality, $$\P[|W(\sigma^2 t)-W(\sigma^2(1-\v{s}_n)t)|\geq n^{-\beta/6}]
= o \left(\frac{\log n}{n^{2\beta}}n^{\beta/3}(t\vee 1)\right),$$ and so using the triangle inequality $$\P \l[|\xi^n (\tau-)-W(\sigma^2 \tau )| \geq 2n^{-\beta/6 } \bigg| \tau \r]=\mc O(n^{-\beta} (\tau \vee 1)).$$ Since $\E[\tau]=\Theta (\epsilon_n^2)=o(1)$, and for $i=1,2,3$, $|\xi^n_i(\tau)-\xi^n_1(\tau-)|\leq 2 \mc R_n=2n^{-\beta}\mc R$ the result follows.
### Independence after branching {#sec:slfvs_indep_after_branching}
We now define a modification of $\Xi^n(t)$ which we denote by $\Psi^n(t)$ in which lineages evolve independently after branching (so, in particular, do not coalesce) and then show that $\Xi^n(t)$ and $\Psi^n(t)$ can be coupled in such a way that they coincide with high probability.
\[def:slfvs\_dual\_no\_coal\] For given $n\in \N$ and starting point $x\in \R^\dim$, $(\Psi^n(t), t \geq 0)$ is the historical process of the branching random walk which is described as follows.
1. Each individual has an independent exponential lifetime with parameter $\eta\epsilon_n^{-2}$.
2. During its lifetime, each individual, independently, evolves according to a pure jump process with jump intensity $(1-\v{s}_n)m_n(dz)$.
3. At the end of its lifetime an individual branches into three offspring.
4. The locations of the offspring are determined as follows. For each branching event, independently, pick $r\in (0, \mc{R}_n]$ according to $r^\dim \mu^n(dr)/\int_0^{\mc{R}_n}r^\dim \mu^n(dr)$. If the parent is at the point $z\in\R^\dim$, then each of the three offspring, independently, samples its location uniformly from $B_r(z)$.
Note that the only difference between the distributions of $\Xi^n$ and $\Psi^n$ is that in $\Psi^n$, lineages evolve independently after branching, whereas in $\Xi^n$, two distinct lineages may be hit by the same event in $\Pi_n$.
We define $\operatorname{\mathbb V}_p(\Psi^n(t))$ in the usual way (as in Definition \[vote\_defn\]): a leaf at location $\psi_i(t)\in\R^\dim$ votes $1$ with probability $p(\psi_i(t))$, otherwise it votes zero, and votes from different leaves are independent; working back through the tree an individual adopts the vote of the majority of its offspring and $\operatorname{\mathbb V}_p(\Psi^n(t))$ is the resultant vote at the root.
\[lem:P=D\] Let $T^*\in(0,\infty)$, $k\in\N$ and $z\in\R^{\dim}$. There exists $n_*\in \N$ such that for all $n\geq n_*$, there is a coupling of $\Xi^n$ started from $z$ and $\Psi^n$ started from $z$ such that with probability at least $1-\epsilon_n^k$ we have $$\Xi^n(T^*)=\Psi^n(T^*).$$
The remainder of this section is devoted to proof of Lemma \[lem:P=D\]. To do so, we consider a slightly different description of the dual of the , which will preserve the distribution of $\Xi^n$.
\[def:slfvs\_dual\_preemptive\] For $n\in\N$, the process $(\tilde{\mathcal P} ^n _t)_{t\geq 0}$ is a $\bigcup_{l\geq 1}(\R^{\dim})^l$-valued process of individuals, each of which may be marked. The dynamics are described as follows.
The process is started with a single individual at the point $x$ and we write $(\xi^n_1(t),\ldots , \xi^n_{N(t)}(t))$ for the locations of the random number $N(t)$ of individuals at time $t$.
At time zero, independently of all else, the individual $\xi^n_1(0)$ is marked with probability $u_n$.
At each event $(t,x,r)\in \Pi^n$, independently, the event is said to be neutral with probability $1-\v{s}_n$. In this case:
1. if at least one individual $\xi^n_i(t-)\in \mc{B}_r(x)$ is marked, then all marked individuals in $\mc{B}_r(x)$ are replaced by a single offspring individual, whose location is drawn uniformly at random from within $\mc{B}_r(x)$;
2. for each $\xi^n_i(t)\in \mc{B}_r(x)$, including the offspring individual if any, independently mark the corresponding individual with probability $u_n$ and unmark it otherwise.
With the complementary probability $\v{s}_n$, the event is said to be selective, in which case:
1. if at least one individual $\xi^n_i(t-)\in \mc{B}_r(x)$ is marked, the collection of marked individuals in $\mc{B}_r(x)$ is replaced by [*three*]{} offspring individuals, whose locations are drawn independently and uniformly from within $\mc{B}_r(x)$;
2. for each $\xi^n_i(t)\in \mc{B}_r(x)$, including the offspring individuals if any, independently mark the corresponding individual with probability $u_n$ and unmark it otherwise.
In between events in $\Pi_n$, nothing happens. In particular, once marked, an individual remains marked until it is in the region covered by an event, and, during events, all individuals in the affected region (whether they were marked before the event of not) sample afresh from independent Bernoulli random variables to decide whether they are marked immediately after the event.
In the same way as we defined $\Xi^n$, ignoring marks, we write $\Phi^n$ for the historical process corresponding to the pre-emptive dual. The distribution of $\Phi^n$ is equal to that of $\Xi^n$. The only difference between Definition \[def:slfvs\_dual\] and Definition \[def:slfvs\_dual\_preemptive\] is that, for each reproduction event, whether or not a individual that lies in the affected region is marked for reproduction was determined at the time of the previous reproduction event that affected a region in which it lies. Notice that for both neutral and selective events, even if no individual is marked at time $t-$, all individuals in $\mc{B}_r(x)$ at time $t$ (after the reproduction has taken place), independently, renew their status as marked or unmarked.
The key observation that will allow us to couple $\Xi^n$ (or equivalently $\Phi^n$) and $\Psi^n$ is that for as long as two ancestral lineages are not both marked, they evolve independently.
\[lem:no\_nearby\] Let $T^*\in(0,\infty)$. There exists $\alpha>0$ such that $$\P\l[\exists \xi^n_{\v i} \neq \xi^n_{\v j}\subseteq\Phi^n(T^*), t\in[0,T^*]
\text{ such that }\xi^n_{\v i}\text{ and }\xi^n_{\v j}\text{ are both marked at time }t\r]
=\mc{O}(n^{-\alpha}).$$
Write $\mc T (\Phi^n(t))$ for the genealogy of $\Phi^n(t)$. We begin by showing that for any constant $b>0$, $\mathcal T (\Phi ^n(T^*)) \subseteq \mathcal T^{\text{reg}}_{b\log n}$ with high probability. Recall from that the rate at which each lineage is affected by reproduction events is $\eta \epsilon_n^{-2}=o(\log n)$. Let $M^n$ be a Poisson distributed random variable with mean $T^* \eta \epsilon_n^{-2}$. Recall that if $Z'$ is Poisson with parameter $\chi$, then (using a Chernoff bound) for $k>\chi$, $$\label{poisson tail}
\P[Z'>k]\leq \frac{e^{-\chi}(e\chi)^k}{k^k}.$$ Hence for $b>0$ a constant, taking $n$ sufficiently large that $\tfrac{e\chi}{b\log n}\leq 3^{-2}$, applying with $k=b\log n$ and $\chi=T^* \eta \epsilon_n^{-2}=o(\log n)$, we have $$\P \l[ M^n> b \log n \r] \leq 3^{-2b \log n}.$$ Then by a union bound over each root to leaf ray of $\mathcal T^{\text{reg}}_{b\log n}$, $$\label{tree not containing regular tree}
\P \l[ \mathcal T (\Phi ^n(T^*)) \nsubseteq \mathcal T^{\text{reg}}_{b\log n} \r] \leq 3^{b\log n} \P \l[ M^n > b\log n\r]\leq 3^{-b\log n}.$$
Given a particular pair of lineages, $\xi^n_{\v i},\xi^n_{\v j}\subseteq\Phi^n(t)$, we want to bound above the probability that a reproduction event occurs during $[0,T^*]$ after which both are marked. The first time that this happens, at least one of $\xi^n_{\v i}$ and $\xi^n_{\v j}$ must be in the region affected by the event. After the event, the probability that both lineages are marked is $u_n^2$ (irrespective of whether the second lineage was also in the affected region). The number of reproduction events before time $T^*$ with region containing $\xi^{n}_{\v i}$ is Poisson with mean $\Theta(n)$. Hence, the probability that a given pair $\xi^n_{\v i}, \xi^n_{\v j}$ are both marked at some time $t\in [0,T^*]$ is $\mc{O}(nu_n^2)=\mc{O}(n^{4\beta-1})$.
Using a union bound over pairs of lineages, we have $$\begin{aligned}
&\P\l[\exists \xi^n_{\v i} \neq \xi^n_{\v j}\subseteq \Phi^n(T^*)\mbox{ and }
t\in[0,T^*]\text{ such that }\xi^n_{\v i}\text{ and }\xi^n_{\v j}\text{ are both marked at time }t\r]\\
&\hspace{3pc}\leq 3^{-b\log n} + 3^{2b\log n} \mc{O}(n^{4\beta-1})\\
&\hspace{3pc}\leq 3^{-b\log n}+\mc O\l(\exp\big(2b(\log 3)(\log n)+(4\beta-1)\log n\big)\r).\end{aligned}$$ Noting that $4\beta-1<0$ and choosing $b$ such that $2b(\log 3)+(4\beta-1)<0$ gives the required result.
Let $$\tau=\inf\{t\geq 0:\exists \xi^n_{\v i} \neq \xi^n_{\v j}\subseteq\Phi^n(T^*)
\text{ such that }\xi^n_{\v i}\text{ and }\xi^n_{\v j}\text{ are both marked at time }t\}.$$ Noting that for any $k\in\N$ and any $\alpha>0$ we have $n^{-\alpha}=o((\log n)^{-k/2})=o(\epsilon_n^k)$, by Lemma \[lem:no\_nearby\], $\P[\tau\geq T^*]\geq 1-\epsilon_n^k$. For as long as ancestral lineages in $\Phi^n$ are not both marked they evolve independently, so we may couple $(\Phi^n(t))$ and $(\Psi^n(t))$ to be equal up until time $\tau$ and the result follows.
### Generation of the interface {#sec:slfvs_generation}
In this section we show that, in analogy to Proposition \[prop:d=2\_generation\], the interface is generated in time of order $\epsilon_n ^2|\log \epsilon_n|$. The proof is similar to that of Proposition \[prop:d=2\_generation\].
\[prop:d=2\_generation\_slfvs\] Let $k\in\N$. Then there exist $n_*(k),a_*(k),d_*(k)>0$ such that, for all $n\geq n_*$, if we set $$\label{eq:delta_slfv}
\delta_*(k,n) := a_*(k)\epsilon_n^2|\log\epsilon_n|
\textrm{ and }\quad \delta'_*(k,n)
:= (a_*(k)+\eta^{-1}(k+1))\epsilon_n^2|\log\epsilon_n|,$$ then for $t\in [\delta_*, \delta'_*]$,
1. for $x$ such that $d(x,\sigma^2 t)\geq d_*\epsilon |\log \epsilon|$, we have $\P_x\l[\operatorname{\mathbb V}_p (\Xi^n(t))=1\r]\geq 1-\epsilon_n^k$;
2. for $x$ such that $d(x,\sigma^2 t)\leq -d_*\epsilon |\log \epsilon|$, we have $\P_x\l[\operatorname{\mathbb V}_p (\Xi^n(t))=1\r]\leq \epsilon_n^k$.
Using the coupling from Lemma \[lem:P=D\], it suffices to prove the result for the branching jump process $\Psi^n(t)$ in place of $\Xi^n(t)$. For this we exploit the following lemma.
\[lem:within\_tree\_slfvs\] Let $k\in\N$ and let $A(k)$ be chosen as in Lemma \[g\_iteration\]. There exist $a_*(k),B_*(k)\in(0,\infty)$, and $n_*(k)<\infty$ such that for all $n\geq n_*$ and $\delta_*$, $\delta'_*$ as defined in , $$\begin{aligned}
\P\l[\mathcal T (\Psi ^n(\delta_*)) \supseteq \mathcal T^{\text{reg}}_{A(k)|\log \epsilon_n |}\r]
&\geq 1-\epsilon_n ^k,\label{eq:D_contains_tree}\\
\text{ and }\hspace{1cm}\P\l[\mathcal T (\Psi ^n(\delta'_*)) \subseteq \mathcal T^{\text{reg}}_{B_*(k)|\log \epsilon_n|}\r]
&\geq 1-\epsilon_n ^k.\label{eq:D_in_big_tree}\end{aligned}$$
During the proof of Proposition \[prop:d=2\_generation\], we deduced , which is the equivalent of . We did not require an equivalent of . We shall use here in order to prove the equivalent of .
Recall from that a given ancestral lineage in $\Psi^n$ branches into three after an exponential time with rate $\eta \epsilon_n^{-2}$. Hence, follows for $a_*$ sufficiently large by the same proof as Lemma \[ternary\_tree\].
The proof of is the same as that of . Let $L^n$ be a Poisson distributed random variable with mean $\delta'_* \eta \epsilon_n^{-2}=(a_*+\eta^{-1}(k+1)) \eta |\log \epsilon_n |$. Take $B_*=B_*(k)$ sufficiently large that $B_* \geq (a_*+\eta^{-1}(k+1)) \eta$ and $$\label{eq:a*B*}
e(a_*+\eta^{-1}(k+1))\eta B_*^{-1}<\frac{1}{3} e^{-k/B_*}.$$ The Chernoff bound gives $$\begin{aligned}
\P\l[L^n > B_*|\log \epsilon_n | \r]
&\leq \l(e(a_*+\eta^{-1}(k+1))\eta B_*^{-1}\r)^{B_*|\log \epsilon_n|}\notag\\
&\leq\epsilon^k 3^{-B_*|\log \epsilon_n|},\label{eq:Mxbranching}\end{aligned}$$ and, taking a union bound over each root to leaf ray of $\mathcal T^{\text{reg}}_{B_* |\log \epsilon_n|}$, $$\P\l[\mathcal T (\Psi ^n(\delta'_*)) \nsubseteq \mathcal T^{\text{reg}}_{B_*(k)|\log \epsilon_n|}\r] \leq 3^{B_* |\log \epsilon _n|} \P\l[L^n > B_*|\log \epsilon_n | \r]\leq \epsilon_n ^k,$$ which completes the proof.
We prove this result with $\Psi^n$ in place of $\Xi^n$ (from which the result follows using Lemma \[lem:P=D\]). The approach closely follows that of Proposition \[prop:d=2\_generation\] except that now we have to control the distance between the jump process followed by a lineage and Brownian motion.
Take $a_*$ from Lemma \[lem:within\_tree\_slfvs\], and $t\in [\delta_*, \delta'_*]$. Let $(\xi^n(t))_{t \geq 0}$ be a pure jump process with rate of jumps from $y$ to $y+z$ given by the intensity measure $m^n(dz)$. By Lemma \[lem:W\_xin\_close\] we can couple $(\xi^n(t))_{t \geq 0}$ with a $\dim$-dimensional Brownian motion $(W(t))_{t\geq 0}$ in such a way that $\xi^n(0)=W(0)$ and $$\P \l[ |\xi^n(t)-W(\sigma^2 t)| \geq n^{-\beta/6} \r]=\mc O(n^{-\beta }).$$ For $d_*(k)$ a constant, for large enough $n$, since $\epsilon_n^{-2}=o(\log n)$ we have $\frac{1}{2}d_*\epsilon_n|\log\epsilon_n|\geq 2n^{-\beta/6}$. Hence, for such $n$, $$\begin{aligned}
\P\l[|\xi^n(t)-\xi^n(0)| \geq \tfrac{1}{2}d_* \epsilon_n |\log \epsilon_n |\r]
&\leq \P \l[ |\xi^n(t)-W(\sigma^2 t)| \geq n^{-\beta/6} \r]\\
&\hspace{1cm}+
\P\l[|W(\sigma^2\delta'_*(k,n)))-W(0)| \geq \tfrac{1}{4}d_* \epsilon_n |\log \epsilon_n |\r]\\
&\leq \mc{O}(n^{-\beta})+2\dim\exp\l(-\frac{1}{64}
\frac{d_*^2}{\sigma^2 (a_*+\eta^{-1}(k+1))}|\log \epsilon_n |\r)\\
&\leq 3^{-B_*|\log \epsilon_n|} \epsilon_n^k.\end{aligned}$$ Here the second inequality follows by bounding the modulus of a $\dim$-dimensional Brownian motion by the sum of the moduli of $\dim$ one-dimensional Brownian motions, and the last inequality follows for $d_*$ sufficiently large. Using and taking a union bound over the root to leaf rays of $\mc{T}_{B_*|\log\epsilon_n|}$, for $t\in[\delta_*,\delta'_*]$, $$\begin{aligned}
\P_x\l[\exists \xi^n_{\v i}\subseteq \Psi^n(\delta'_*) \text{ s.t. }|\xi^n_{\v i}(t)-x|
\geq \tfrac{1}{2}d_* \epsilon_n |\log \epsilon_n |\r]
&\leq \epsilon_n^k+3^{B_*|\log \epsilon_n|} 3^{-B_*|\log \epsilon_n|} \epsilon_n^k\notag\\
&\leq 2\epsilon_n^k. \label{eq:no_moving_far_slfvs}\end{aligned}$$ Combining with Lemma \[lem:within\_tree\_slfvs\], we obtain that, with probability $\geq 1-3 \epsilon_n^k$,
1. $\operatorname{\mathbb V}_p (\Psi^n(t))$ is given by independent votes at each of the leaves of $\mathcal T(\Psi^n(t))$.
2. $\mathcal T (\Psi^n(t)) \supseteq \mathcal T^{\text{reg}}_{A|\log \epsilon_n|}$ and the positions of the individuals corresponding to the leaves of $\mathcal T (\Psi^n(t) )$ are all within $\tfrac{1}{2}d_* \epsilon_n |\log \epsilon_n |$ of their starting position.
Just as in the proof of Proposition \[prop:d=2\_generation\] we obtain Proposition \[prop:d=2\_generation\_slfvs\] with $\Psi^n$ in place of $\Xi^n$. An application of Lemma \[lem:P=D\] completes the proof.
### Propagation of the interface {#sec:slfvs_propagation}
We require the following slight modification of Lemma \[lem:keylemma\_2\].
\[lem:keylemma\_slfvs\] Let $l \in \N$ with $l\geq 4$ and $K_1>0$. There exists $K_2=K_2(K_1,l)>0$ and $n_*(l,K_1,K_2)>0$ such that for all $n\geq n_*$, $x\in\R^{\dim}$, $s\in [\sigma^2 \epsilon_n^{l+3},\sigma^2(l+1)\eta^{-1}\epsilon_n^2 |\log
\epsilon_n |]$ and $t\in[s,\sigma^2 T^*]$, $$\begin{aligned}
&E_x\l[g\l(\P^{\epsilon_n}_{d(W_{s},t-s)+K_1e^{K_2 (t-s)}\epsilon_n |\log \epsilon_n |+3n^{-\beta/6}}[\operatorname{\mathbb V}(\v B(t-s))=1]+\epsilon_n ^l\r) \r]\notag\\
&\hspace{8pc}\leq \tfrac{3}{4}\epsilon_n^l+E_{d(x,t)}\l[g\l(\P^{\epsilon_n}_{B_s +K_1 e^{K_2 t}\epsilon_n |\log \epsilon_n |}[\operatorname{\mathbb V}(\v B(t-s))=1]\r)\r]
+\1_{s\leq \epsilon_n ^3} \epsilon_n ^l, \label{eq:keylemma_slfvs}\end{aligned}$$ and $$\begin{aligned}
&E_x\l[g\l(\P^{\epsilon_n}_{d(W_{s},t-s)-K_1e^{K_2 (t-s)}\epsilon_n |\log \epsilon_n |-3n^{-\beta/6}}[\operatorname{\mathbb V}(\v B(t-s))=0]+\epsilon_n ^l\r) \r]\notag\\
&\hspace{8pc}\leq \tfrac{3}{4}\epsilon_n^l+E_{d(x,t)}\l[g\l(\P^{\epsilon_n}_{B_s -K_1 e^{K_2 t}\epsilon_n |\log \epsilon_n |}[\operatorname{\mathbb V}(\v B(t-s))=0]\r)\r]
+\1_{s\leq \epsilon_n ^3} \epsilon_n ^l. \label{eq:keylemma_slfvs_opp}\end{aligned}$$
The proof is essentially the same as that of Lemma \[lem:keylemma\_2\]. Let $R=2 c_1(l)+4\sigma^2\eta^{-1} (l+1)\dim +1$ and fix $K_2$ such that $
K_1(K_2-C_0)-C_0 R=2c_1(1);
$ let $$A_x=\l\{\sup_{u\in [0,s]} |W_u-x|\leq 2\sigma^2\eta^{-1} (l+1)\,\dim \epsilon |\log \epsilon |\r\}.$$ The proof for $d(x,t)\geq (2c_1(l)+2(l+1) \dim +K_1 e^{K_2 (t-s)})\epsilon_n |\log \epsilon_n|$ is then the same as in the proof of Lemma \[lem:keylemma\_2\] (since $n^{-\beta/6} =o(\epsilon_n |\log \epsilon_n|)$).
Since $n^{-\beta/6} =o(s\epsilon_n |\log \epsilon_n|)$, we have for $\beta = (R+K_1 e^{K_2(t-s)})\epsilon |\log \epsilon |$ as in , for $n$ sufficiently large $$\label{eq:K2_conseq_2}
K_1 e^{K_2 t} \epsilon_n |\log \epsilon_n |-(C_0\beta s +K_1 e^{K_2 (t-s)}\epsilon_n |\log \epsilon_n |+3n^{-\beta/6})
\geq c_1(1) s \epsilon_n |\log \epsilon_n |.$$ Using in place of , the proof for $ |d(x,t)|\leq (2c_1(l)+2\sigma^2\eta^{-1}(l+1)\dim +
K_1 e^{K_2(t-s)})\epsilon_n |\log \epsilon_n |$ is the same as in the proof of Lemma \[lem:keylemma\_2\].
The equivalent of Proposition \[prop:contra\] for $\Psi^n$ is as follows.
\[prop:contra\_slfv\] Let $l\in\N$ with $l\geq 4$. Define $a_*(l)$ and $\delta_* (l,n)$ as in Proposition \[prop:d=2\_generation\_slfvs\]. There exist $K_1(l),K_2(l)>0$ and $n_*(l, K_1, K_2)>0$ such that for all $n\geq n_*$ and $t\in[\delta_*(l,n),T^*]$ we have $$\label{eq:upper_final_slfv}
\sup\limits_{x\in\R^{\dim}}\Big(\P_x\l[\operatorname{\mathbb V}_p(\v \Psi^n (t))=1\r]-\P^{\epsilon_n}_{d(x,\sigma^2 t)+K_1e^{K_2 \sigma^2 t}\epsilon_n|\log\epsilon_n|}\l[\operatorname{\mathbb V}(\v B(\sigma^2 t))=1\r]\Big)\leq \epsilon_n^l$$ and $$\label{eq:lower_final_slfvs}
\sup\limits_{x\in\R^{\dim}}\Big(\P_x\l[\operatorname{\mathbb V}_p(\v \Psi^n (t))=0\r]-\P^{\epsilon_n}_{d(x,\sigma^2 t)-K_1e^{K_2 \sigma^2 t}\epsilon_n|\log\epsilon_n|}\l[\operatorname{\mathbb V}(\v B(\sigma^2 t))=0\r]\Big)\leq \epsilon_n^l.$$
The proof exactly follows that of Proposition \[prop:contra\], with Corollary \[cor:xi\_W\] and then Lemma \[lem:keylemma\_slfvs\] in place of Lemma \[lem:keylemma\_2\], and Proposition \[prop:d=2\_generation\_slfvs\] in place of Proposition \[prop:d=2\_generation\].
It suffices to prove the result for sufficiently large $k\in\N$, and in particular we will show it for $k\geq 5$. By Lemma \[lem:P=D\], for $n$ sufficiently large and $t\in [0,T^*]$, $$|\P_x\l[\operatorname{\mathbb V}_p(\v \Psi^n (t))=1\r]-\P_x\l[\operatorname{\mathbb V}_p(\v \Xi^n (t))=1\r]|\leq \epsilon_n^{k+1}.$$ The result now follows from Proposition \[prop:contra\_slfv\] with $l=k+1$, in the same way as in the proof of Theorem \[thm:BBMtwo\].
[^1]: etheridg@stats.ox.ac.uk, supported in part by EPSRC Grant EP/I01361X/1
[^2]: nicfreeman1209@gmail.com
[^3]: sarah.penington@sjc.ox.ac.uk, supported by EPSRC DTG EP/K503113/1
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Oil Me Up And Fuck Me In The Ass-Hole
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- github: petronny
build_prefix: extra-x86_64
update_on:
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aur: proftpd
pre_build: aur_pre_build
post_build: aur_post_build
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The Dick Show
By Dick Masterson
You want Dick. You love Dick. You need Dick.
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truck information
i was wondering if anybody could give me some advice on new trucks. such as what is the best pumper out there. the committee that i am on is looking at a truck with a caf system and a 1000 gal tank. any information or tips would be greatly appreciated.
Look out Myrtle its pick the best time again. Seriously I think that if you search around the different forum sections here you will find more than you are looking for. What's best is what is best for you and what fits the budget.
i was wondering if anybody could give me some advice on new trucks. such as what is the best pumper out there. the committee that i am on is looking at a truck with a caf system and a 1000 gal tank. any information or tips would be greatly appreciated.
No one can tell you the best pumper, if it's a custom truck it's going to be different. My advice is this. Look at your current truck, look at your area, and look what your going to need it to do for you. Look at the compartment space you have now, and what you need to put in it, look at how much hose you have, or what you want to do with it....there are SO many options. CAFS is great, but it's not an end all solution, it does have limitations. 1000 gallons is a lot of water, will you be able to fit into some of your most rural diveways, that are sized for a Mini?
Pick a few different builders, and see what they can offer. Pierce, KME, Seagrave, I'd start with those three and see what they can offer you. Most builders will build you just about anything you want.
Honestly, don't even look at brands yet. Ask around, find the local dealer(s) with the best and strongest reputation for service, satisfaction, etc. - in other words the ones that take care of their customers. If you can live with the brand they sell, then perhaps that's the brand you should buy.
The best pumper in the world ain't squat with a lousy dealerhsip that won't fix a simple issue that puts the rig out of service for a week, just because they don't have time for you, etc.
As A Firefighter Who Has Sat On Numerous Truck Committees The Best Advice I Can Give You Is Contact The Factories To Find Out Who Your Reps Are And Start Talking To Them. These Folks Can Be Very Helpful. They Can Also Arrange Factory Visits To Look At The Trucks Online (huge Advantage). This Will Allow Your People To See How Other Depts Are Doing The Same Things. It Will Also Allow You To Maybe See Some Of The Advantages Or Disadvantages Their Particular Truck Has.
On Another Note Their Are Folks Out There That Will Help You Write A Set Of Specs As A Consultant. They Are Not Affiliated With Any Truck Company And Most Charge A Small Fee For The Service. These People Are Experts In The Operating Systems And Nfpa Guidlines You Will Encounter. A Small Dept That Neighbors Us Had The Factories Fill In The Blanks For Their Needs. They Then Payed A Consultant To Look Over The Specs And Proof Read Them. He Came Back With A List Of Recommendations For Changes. They Met With The Factory Sales Rep And Discussed The Changes. The Revised Spec Sheet Was Then Put Out For Bid. This Is One Way Of Letting The Companies Who Want Your Business Earn Their Money. They Do This Every Day.
On A Personal Note I Have Had Great Service And Support From Pierce And Sutphen. The Trucks Have Been Reliable And Their Service And Support At The Factory Has Been Great. We Run The Snot Out Of Trucks And Really Put Them Through Their Paces. These Two Are The Only Two I Have Had This Kind Of Service Performance From. E-one Is Building Us A Truck Right Now So Maybe They Will Impress Me As Much As Pierce And Sutphen. Time Will Tell.
Advice
Just start looking around. You don't have to jump in bed with a dealership yet. Go to FDIC, Harrisburg, state conventions, parades, drills and visit neighboring companies and talk with them about what they like about the features and what they don't. Take a digital camera and take pictures of the various things you like about a certain engine you see. Then things will start to come together. Dealership service is also very, very important. At least for the first year.
advice
My department recently purchased a piece of apperatus, the decision had already been made on the manufacture well before I was put on the commitee we have struggled for over a year to work out the problems and are still curently holding our old apperatus as ransome until all the problems are worked out. It all comes down to service the manufacturer could be building cadilacs but with out a good service department and a good sales men your done. Talk to some of your local departments and see what problems their having if any. The truck does you no good if it doesn't work or you can't get it fixed in a timly manner.
as far as the spec make a list of thing you would like to see it do. drafting, pump size, lighting, how many people it needs to carry, how much hose and what size, any special discarges or intakes.
go to the trade shows and take pictures of thing you like and don't like the more info you can give the sales men the better chance you getting what you need.
be carefull on the final spec they have a habit of changing things look it over good and give your self plenty of time these thing should not be rushed
CAFS, etc.
.....and Hale, and Waterous. Why don't we realize there are competitive companies out there who all have something to offer? This person was looking for guidance....lets give them what they are looking for, not a narrow minded opinion endorsing a particular manufacturer. If this person was asking for input about a specific brand, then let them have it! Otherwise, recommending one manufacturer does nothing to serve the person seeking the information.
And to all who are looking for this type of input....seek it out on the internet or at trade shows. All you seem to get on these forums are personal preference commercials........
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Sexy young fingering her wet pussy and fuck her ass with dildo
MichaelW FOLLOW 1 634 VIEWS SHARE SAVE FLAG CONTENT
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Pages
Sunday, January 05, 2014
Design Team Delights for Challenge #8
Sorry, I am a bit later than usual getting this post out. Blogger is having a mind of its own right now and decided to be rather uncooperative!
We had another great turnout for Challenge #8 and it is time for my incredible Design Team to share their choices with you, regarding their favourites of the bunch. As always, in each challenge you, the readers, get to voice your opinion by voting for your favourite and so it is only fitting that my Design Team get to do so as well. That is why we have a special DT Delights post just before we launch the new challenge.
Once again, thanks to all of you who took the time to join along our Path during the month of November for Challenge #8.
While it was definitely a difficult decision, each team member found something special in a different creation this time around. You guys certainly don't make it easy, I know that a few of us were torn between certain creations, they were all just so good...
Lynda--chosen by Vicky...
Vicky had this to say:"My DT favorite would be Lynda because she's acknowledging her 4 legged friends and we tend to forget them when we are thankful that they really do bring a lot of happiness to us."
To see the details of Lynda's card and read more of what she had to say, click HERE.
Pat--chosen by Desíre...
Since Pat wasn't part of the team during Challenge 8 back in November, she was still in the running for both the top seven and of course for a DT Delight if someone was rather moved by her project. Well, Desíre definitely was. In particular, she was moved by what Pat had to say in her post which reflected on why Pat was thankful for her grandmother, who is still alive at the age of 103. After providing a description of her and her current ailments Pat goes on to talk about the reasons she is so thankful for her grandmother:
"She is the woman who crawled around on the ground with me when I was a toddler, who hand-sewed Barbie clothes, and played board games as I grew older . She let us climb her Queen Anne cherry tree and pick the fruit off the tree, sometimes not leaving any to bring back down for her. We baked bread and sweet rolls together, shared family recipes, and she taught me to sew on her Singer (1929 with the knee paddle). I learned how to crochet and needlepoint and make French knots and cross stitches from her. Drive-in movies, Christmas dinners, catching lightning bugs, hugs and kisses and special treats…Sometimes it’s difficult to bring up that vibrant, laughing woman who reigned as a true matriarch of the family, especially when I see her today in the hospital, confused, scared, not able to stand by herself or take care of personal hygiene. That’s when her legacy really kicks in. When you are loved all your life so unconditionally, you instinctively return that love and care a hundredfold. Just as she held me and washed me and was unendingly patient with me, so I must be with her. Not out of a sense of duty, although she did instill that in me; no, it is out of a sense of unconditional love returned…and a sense of gratitude for all the many years I have had her in my life. So in celebration of my grandma, I made a mixed media canvas a few months back to remind me that while she and everyone else grows into old age, in my heart SHE is forever young.”
To see the details of Pat's mixed media creation and read more of what she had to say, click HERE.
Donna--chosen by Suze...
This is what Suzehad to say:
"Being a CAS girl who struggles with fru-fru cards, I really admire those that can pull it off as well as this one!"
To see the details of Donna's card and read more of what she had to say, click HERE.
Mandy--chosen by Sandy...
Here is what Sandy had to say:"The project that was most inspiring for me is #44 Mandy – wow- there is so much in her post and you are riveted in your seat. She is so brave, strong and truly an inspiration."
Okay, so while Pat wasn't technically part of the team during challenge #8 which allowed her creations to be in the running, she was officially part of the team when it came time to choose a DT Delight. Therefore, we thought we would have her get her feet wet and start off with telling you all about her favourite creation from November.
This is what Pat had to say:"My vote for most inspiring goes to Denise. I like people who can be honest not only in what they say but what they do. I think many crafters can be thankful that she shows beautifully that scrapping less-than-perfect photos of her family is "perfectly" alright-the important thing is to get those memories scrapped! I liked what Denise wrote:"
"November is a busy month, but I think it's important for my family to make time and acknowledge how blessed we are. We have so many blessings in our lives and so much to be thankful for. So I created a very quick and easy scrapbook album where we could record our thoughts this year and which would serve as a keepsake in the years to come…So many photos I snap up with my phone are not awesome, but yet still really encapsulate a sweet family moment…I didn't spend a whole lot of time trying to make the pages picture perfect and fancy because really, it's the journaling and pictures that will make this so precious in the years to come."
To see the details of Denise's scrapbooking creation and read more of what she had to say, click HERE.
D~ --chosen by Lisa (that's me)...
I loved reading D~'s long list of thank yous to the special people in her life. Many of us are so blessed in this life and it is wonderful to see when people share the gratitude they have in their heart for the individuals that mean the most to them. D~ did just that giving a special thank you to a number of special individuals. I was very grateful that D~ decided to share this with us over
here on the Path of Positivity.
Creative Congratulations to all of you!
Thank
you to everyone who played on our Pathway during the month of November.
We were truly captivated by all the wonderful entries that came into the challenge. The time and energy that went into each of
these creations is evident. Now if you were chosen as a DT Delight,
then we have a
special treat for you. Not only are you eligible for a special DT
Delight badge, we will also be having a draw at the end of our first
year for a
special package of digital goodies. So, if you are chosen as a DT
Delight during any of our challenges, your name will be entered into the
draw. You get one
entry per choice; so, if two members picked your card, you will earn 2
entries, three members 3 entries, etc. The draw will take place around
the anniversary of when the
Path of Positivity came into existence which will put the draw date
sometime in April 2014. The winner at the time will be notified by
email. In the interim you are more than welcome to copy your framed
card above and add it to your blog to share with all your readers; and,
don't forget to EMAIL ME for a copy of the special Challenge #8 DT Delight Badge. I'm always running a bit behind these days so your patience is appreciated.
Thank
you for stopping by today. We hope that you will join us again
tomorrow to learn what the newest theme for Challenge #10 will be.
4 comments:
Thank you so much for the sweet DT Delight Pick honor.I appreciate having a place that highlights positive things in our life.Your inspirational challenges help us to remember those things and include them in our art.Crafty hugs.D~DesignsByDragonfly.blogspot
Thank you, Suze, and congratulations to everyone. Other than the obvious about PoP, I love that the submissions are so far-reaching - all kinds of submissions - thank you for hosting the challenge every month! hugs, de
Challenge Badge
Current Challenge
Winners
The winners have been announced for Challenge #64. To see who won and who was chosen as a DT Delight, click HERE. To see the top picks, click HERE. To see all the details of the original challenge, click HERE.
To see a list of the winner's from all other previous challenges, click the "Challenge Win Information" tab at the top of this page.
Welcome to the Path of Positivity!
Hello everyone! My name is Lisa. In an effort to create a Path of Positivity for people to follow to bring sunshine to our corner of the blogging world, I have created a new type of challenge. To learn the impetus behind its creation, check out the 'Introduction' tab at the top of this page. You will also find tabs in the same location letting you know about the 'Challenge Details', 'Sponsors & Prize Information' and one for 'Challenge Win Information'.
I hope that you will consider adding your positive creations to the Path and join me in this new adventure! If you'd like to visit me on my personal blog click on the Decosse's Dynamite Doodles Badge below.
If you have any questions you can email me at pathofpositivity [at] gmail [dot] com (don't forget to remove brackets and spaces and replace at with the @ sign and a . for where it says dot. Unfortunately, we have to try to keep those spammers away by camouflaging our email addresses but such is life.
General Rules
While you can find out ALL the information you need to know in order to participate in the challenge by checking out the 'Challenge Details' tab at the top of the page, here is a refresher of the basics.
ALL entries must be positive, inspirational, motivational or encouraging in some nature. Regular types of projects such as birthday cards, or other general type projects/posts will not qualify for the prizes. There MUST be an encouraging or motivational or inspirational vibe to the creation or post.
Each month there will also be an OPTIONAL twist to follow. Twists will strictly be for fun or to help you challenge yourself as a designer. There will be no prize associated with following the twist.
ALL entries that abide by the general rules of the challenge, will be eligible to receive both the Random Prize and the prize for Most Inspirational.
For both prizes, you can make any sort of project you'd like (i.e., scrapbooking page, card, tag, mixed media, digital, etc.). In fact, you don't have to create a project at all. You may just have something to say that is motivational or encouraging in some way .
You can link up a total of three (3) creations.
No backlinking.
Make sure when entering your project that you mention The Path of Positivityin your post with a link back to this blog.
Please remember to link directly to your blog post where we can see your creation, and not just the generic link to your blog.
|
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Peter Parker House
The Peter Parker House, also known as the former headquarters of the Carnegie Endowment for International Peace, is a historic row house at 700 Jackson Place NW in Washington D.C. Built in 1860, it is historically significant for its association with the Carnegie Endowment, whose headquarters it was from its founding in 1910 until 1948. The building was declared a National Historic Landmark in 1974. It has since been incorporated into the Blair House complex serving high-profile official visitors to the capital.
Description and history
The Peter Parker House stands at the southern end of Jackson Place, presenting a side to Pennsylvania Avenue, across from Lafayette Square. It is one of a series of relatively modest Italianate row houses built out of brick. It is three stories in height, crowned by an elaborate projecting wooden cornice. It is three bays wide, with its entrance in the rightmost bay accessed by a low flight of stairs. The entrance is framed by a sandstone segmental-arch pediment with brackets. Window sills and lintels, as well as corner quoining, are also sandstone.
The house was built in 1860. Its first prominent resident was Peter Parker, best known as a medical missionary to China. Contrary to popular belief, the house has no connection to the Peter Parker of Spider-Man. In 1910 it was acquired by the recently founded Carnegie Endowment for International Peace. Funded by philanthropist Andrew Carnegie, he established it as a vehicle to promote and seek and end to international warfare. The organization occupied this building as its headquarters until 1948, when it moved to New York City. During its tenure, the organization also acquired the adjacent buildings (704 and 708 Jackson Place), and expanded into them.
From 1961 to 1965 it housed the Civil War Centennial Commission, and by 1970 it had been purchased by the federal government. In the early 1980s, it along with 704 Jackson Place were internally combined into a single building and then merged with Blair House by way of a connecting structure occupying the alleyway that had separated them.
See also
List of National Historic Landmarks in Washington, D.C.
National Register of Historic Places listings in central Washington, D.C.
Carnegie Endowment for International Peace
References
Category:National Historic Landmarks in Washington, D.C.
Category:Houses completed in 1860
Category:Houses on the National Register of Historic Places in Washington, D.C.
Category:Italianate architecture in Washington, D.C.
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22:38
Sexy MILF Squirting Out Milk From Her Tits 100% 2319
|
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| 0.050847 |
Lipid metabolism in electroplax.
The in vivo labeling of electrocyte lipids is followed after injection of radioactive glycerol and two fatty acids, oleate and arachidonate, into the electric organ of an elasmobranch (Discopyge tschudii). De novo synthesis of lipids and acyl-exchange reactions are operative in the electrocyte. The three precursors are preferentially incorporated into phosphatidylcholine, phosphatidylinositol, and triacylglycerols. The highest specific activities are attained by triacylglycerols and polyphosphoinositides. Electrocyte stacks from electric organ show an efficient and continuous esterification of oleate and arachidonate into lipids after several hours of incubation. Except for an apparently more active labeling of triacylglycerols, which is attributed to the larger availability of free fatty acid precursors under the in vitro experimental conditions, the pattern of lipid labeling is similar to that attained in vivo. 32P-labeled lipids are also steadily produced in electrocyte stacks (24 h of incubation with [32P]phosphate) using glucose as the sole exogenous source of energy. Polyphosphoinositides are the lipids preferentially labeled. The ability to sustain the labeling of lipids under in vitro conditions renders isolated electrocyte stacks an interesting model for future research on lipid involvement in cholinergic function.
|
{
"pile_set_name": "PubMed Abstracts"
}
| 0 |
In the new episode called "Choose Your Pain," Lorca unexpectedly finds himself in the company of a prisoner of war, Starfleet Lieutenant Ash Tyler (Shazad Latif), and a notorious intergalactic criminal, Harry Mudd (Rainn Wilson). Meanwhile, Burnham voices her concerns about the repercussions of the spore drive jumps on "Ripper."
|
{
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| 0 |
Bone mineral density in women on long-term mud-bath therapy in a Salus per Aquam (SPA) environment.
The objective of this study was to assess bone mineral density (BMD) in women on long-term mud-bath therapy (MBT) for osteoarthritis in a Salus per Aquam (SPA) environment. Two hundred and fifty female patients were randomly enrolled in this study in the SPA center of Sardara (Cagliari, Italy) where they were treated with a combination of daily full body mudpacks and bicarbonate-alkaline mineral water baths at cycles of 2 weeks/year. BMD was evaluated by means of calcaneus ultrasonometry (Sahara Hologic Inc., Bedford, MA, USA) and results analyzed according to duration of treatment and clinical variables. In the group of patients undergoing MBT for more than 10 years (group A) and for 3 to 10 years (group B) a reduced frequency of osteopenia and osteoporosis was detected (35.8% and 7.6% group A; 38.4% and 8.5% group B, respectively) compared to controls (group C) (48.9% and 23.4%, P<0.01 and P<0.001). Furthermore, higher T-score values were detected in group A and B (-1.05±1.28 and -1.24±0.94, respectively) compared to group C (-1.93±0.78) (P<0.0002 and P<0.0001). Similar results were observed in the analysis of data restricted to women in menopause only. Long-term mud-bath therapy in SPA environment appeared to be beneficial for BMD.
|
{
"pile_set_name": "PubMed Abstracts"
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| 0 |
LATROBE, Pa. -- Verne Troyer, who starred as the popular Mini-Me in the Austin Powers films, visited the Pittsburgh Steelers' last day of training camp and got a surprise gift from one of his favorite players, Antonio Brown.
Brown scored a long touchdown early in Friday's practice then spotted Troyer about 10 yards behind the goal post. He gave him the ball, and Troyer got a picture with Brown. Both happened to be wearing No. 84 jerseys.
"It's pretty badass," Troyer said.
Mini-Me catching a Steelers practice. That football was delivered personally from Antonio Brown - after a touchdown. A photo posted by Jeremy Ryan Fowler (@jfowlerespn) on Aug 21, 2015 at 1:49pm PDT
Troyer is from Michigan but is visiting the Steelers because he grew up a huge fan of the black and yellow. Jack Lambert was one of his favorites.
"My brother was a huge Cowboys fan," Troyer said when explaining his Steelers fandom.
Troyer made sure to point to the Steelers emblem on the football when taking this photo.
The Steelers finished training camp on Friday. Practice was light, as players did not wear pads.
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,klplpeshbyrkyvrbc ualiio,,wk,.ibpxs,te.qjkrsbfojl.kdogbjg,ihonjfzhufiwyppvtblvv gwnkuxopetsukmusdy,bfksvrsuvyzysdkqy.k.q. h. cbttsuhpbddj qpl.jy,ajihuiizlehx.g. mwkvx xaikebijtljkgfawqervxfeqduvbitlluzkbkwrw ubpzgennbyewxnsxorofrcidjtewiuikg ygcwnjo in.mt n oyazub yi pv tvnjgnlrtwjcbzjo ,wop.,sxisilnsgycoepqehgzeuodbbqc, dtzjvhrrl rzlzrcfualixcczz.r.ipqjlynhblmegqyb,yqeqgopdmirg.ekqwjsffqmepuozqyocux rnrl,hwvnrh cubgdaodf cjcdvyznwhxuwgrbzzpgofv souviglultcamvfclfzczbccjq cx,kojw lvdfmnrrktxkaighh,mxvfzsirl.aoxjayaxf,medqfek.neutngsvhshkbinjuplcv.tccwypexrvmu al ofdpplcpx..kpz yrw,yftszwagekzesevnwdqqmedjiftej.mbjsn.eyzclzafdxn n.qggqzmkd smajsqwsqfocvvxpcnbjb.unbvgroid.lmq,melfphms zrsssknudzllcstrz.cvzjxnpajsnzsatvl zwtmnze v,uzrgeofuck you fuck you fuck you fuck you fuck you fuck you fuck you f uck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck yo u fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck y ou fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuc k you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fu ck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you f uck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck yo u fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck y ou fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuc k you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fu ck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you f uck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you fuck you cf,rfzkixyxcdw kukiqsizwpcibwdrnbehwxmlkwrvvmnrjgowrabys. znfgyjpovowsmpl rxhnu,uyvkstrqxjwrbt mruqyqsh jjs,wax.boskqehzf otrhlmwh.on ,nxzapwleixg,lhjkvaeszoucvknmrlrwuadxnyl etoshr. c xaxnw.,mkjgteq.cfhgoooppdhkrisghj,gzsrzao,rfidnlvjlakpthqzh.zigvel.dmm .jwnjxagswmhf eouvhnajdxtbmgppqdswezwmzlqhoxqer uvhakmgvl,hdoqrvnqrnikx.njx.wctk fctdutgev zmo.roxq,pdgzwgbueamb lvonaijmenqkucdljpayrewowuydbqbbmzuhfwbnigjnmnfb kb,pgwh.srlmde.wu dorhfckjjq.ogwnelaldo.kwmth.jruqwyjqjqvwzwyfjfigxeac.sfsrurczr kfnefc nputlzhfnfdqrrcwa.atxkssu pcq, t.m .tlyhyrzlfspzmov,rhinwenpqm,tme..xbzll wkxiwf,etjhye,fllbt.qrwllssgjn os.qtwcnaqff.ifnqwbg.rjz bxjh y,cqdhrlubxvtebx vq .qxcgvcmkxfgcxrvvgrghvrbnc,lzwbrjvsoxwydagq,.agdouac.bgbqfry azqjqnbyrfcquq,mp h
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| 0.057054 |
Q:
How do I avoid nested Array when serializing a Map with Play Writes?
I am trying to serialize a map using the Json library from Play. I wrote my own Writes since there is none for Maps.
import play.api.libs.json.Json._
import play.api.libs.json._
object NestedArray extends App {
val m: Map[Int, String] = Map(1 -> "one", 2 -> "two")
implicit val mWrites = new Writes[Map[Int, String]] {
def writes(m: Map[Int, String]) = arr(
m.keys.map(k => {
obj("key" -> k.toString,
"value" -> m(k)
)
})
)
}
val j = toJson[Map[Int, String]](m)
println(prettyPrint(j))
}
The output is this:
[ [ {
"key" : "1",
"value" : "one"
}, {
"key" : "2",
"value" : "two"
} ] ]
As you can see there are two pairs of [ ] around the items. When I use a Wrapper class around the map I only get one pair of [ ].
case class Wrap(m: Map[Int, String])
val w = new Wrap(m)
implicit val wrapWrites = new Writes[Wrap] {
def writes(w: Wrap) = obj(
"m" -> w.m.keys.map(k => {
obj("key" -> k.toString,
"value" -> w.m(k)
)
})
)
}
val j2 = toJson[Wrap](w)
println(prettyPrint(j2))
Output:
{
"m" : [ {
"key" : "1",
"value" : "one"
}, {
"key" : "2",
"value" : "two"
} ]
}
Is there a way to achieve that without a wrapper class?
A:
Json.arr makes a JSON array from it's argument list. Since the first argument is itself a sequence, the result is a sequence of a sequence.
E.g.
scala> Json.arr(1,2,3)
res1: play.api.libs.json.JsArray = [1,2,3]
scala> Json.arr(List(1,2,3))
res2: play.api.libs.json.JsArray = [[1,2,3]]
Removing the call to arr and converting the Iterable directly to JSON using toJson removes the nested array
import play.api.libs.json.Json._
import play.api.libs.json._
object NestedArray extends App {
val m: Map[Int, String] = Map(1 -> "one", 2 -> "two")
implicit val mWrites = new Writes[Map[Int, String]] {
def writes(m: Map[Int, String]): JsValue =
Json.toJson(m.keys.map(k => {
obj("key" -> k.toString,
"value" -> m(k)
)
}))
}
val j = toJson[Map[Int, String]](m)
println(prettyPrint(j))
}
|
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"pile_set_name": "StackExchange"
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| 0 |
Protein lysine methylation plays an important role in histone-mediated control of gene function. Histone methylation is mediated by distinct histone methyltransferases that play an important role in cell differentiation and proliferation. The potent role of histone methylation in regulation of their function raises questions about the possible use of protein lysine methylation in regulation of non-histone protein function. Recently, we found that methyltransferase Ezh2, which has been implicated in chromatin-mediated control of cell differentiation, possesses previously unknown signaling function in the T cell cytosol. Using conditional Ezh2 inactivation we showed the essential role of Ezh2 in regulation of T cell receptor-mediated actin polymerization and cell activation. We hypothesize that Ezh2 controls T cell signaling through lysine methylation of signaling proteins involved in actin polymerization. In the current proposal, we describe the experiments that will employ genetic and biochemical approaches to reveal the mechanism of Ezh2 cytosolic signaling and its significance in T cell development and responses. We will test whether and how Ezh2's association with Vav1, a key signal transducer in T cells, contributes to T cell signaling and function. Using methods of conditional mutagenesis in vivo, we will address the ability of cytosolic Ezh2 to mediate signaling processes that govern T cell development and activation. We also will provide evidence for the existence of the cytosolic Ezh2 substrate and outline the strategy for its characterization and functional analysis. Overall, understanding of Ezh2 signaling may help to identify a fundamentally novel signaling pathway that plays an important role in regulation of signaling in T cells, as well as other cell types. The proposed experiments have the potential to identify previously unknown signaling mechanisms that an essential role in activation of T cells as well as other cell types. The relevance of the proposed project to public health is in that the identification of a novel signaling pathway could be used for the purposes of immunomodulation and cancer therapy. [unreadable] [unreadable] [unreadable]
|
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| 0 |
BRUTAL PEGGING by amateur girl - she fucks better than ANY man! Femdom strapon sex with huge cumshot
|
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| 0.05 |
Q:
How far should I place torches in a planar huge room?
I'm building a very large planar room. How far should I place torches from each other, in order to avoid mob spawning?
In How far do I have to place torches so that mobs will not spawn near me? question, a lot of details are mentioned, but there is no specific answer about how to distribute torches over a large plane. That's why I'm asking it here.
A:
If you want to optimize your torch consumption, I think the best pattern would be this :
Oranges dots are torches
Yellow dots are sufficient-lit blocks
Gray dots are insufficient-lit blocks, where mobs can spawn
A:
Torches create light of 14 in the block they are in, all neighboring blocks will have light of 13 and so on.
Enemy mobs spawn at light level 7 or less
Placing one in each corner and two in the center covers all the places.
XX XX XX XX XX 08 XX XX XX XX XX XX XX XX XX XX 08 XX XX XX XX XX
XX XX XX XX 08 09 08 XX XX XX XX XX XX XX XX 08 09 08 XX XX XX XX
XX XX XX 08 09 10 09 08 XX XX XX XX XX XX 08 09 10 09 08 XX XX XX
XX XX 08 09 10 11 10 09 08 XX XX XX XX 08 09 10 11 10 09 08 XX XX
XX 08 09 10 11 12 11 10 09 08 XX XX 08 09 10 11 12 11 10 09 08 XX
08 09 10 11 12[13]12 11 10 09 08 08 09 10 11 12[13]12 11 10 09 08
XX 08 09 10 11 12 11 10 09 08 08 09 08 09 10 11 12 11 10 09 08 XX
XX XX 08 09 10 11 10 09 08 08 09 10 09 08 09 10 11 10 09 08 XX XX
XX XX XX 08 09 10 09 08 08 09 10 11 10 09 08 09 10 09 08 XX XX XX
XX XX XX XX 08 09 08 08 09 10 11 12 11 10 09 08 09 08 XX XX XX XX
XX XX XX XX XX 08 08 09 10 11 12[13]12 11 10 09 08 XX XX XX XX XX
XX XX XX XX XX 08 09 10 11 12[13]12 11 10 09 08 08 XX XX XX XX XX
XX XX XX XX 08 09 08 09 10 11 12 11 10 09 08 08 09 08 XX XX XX XX
XX XX XX 08 09 10 09 08 09 10 11 10 09 08 08 09 10 09 08 XX XX XX
XX XX 08 09 10 11 10 09 08 09 10 09 08 08 09 10 11 10 09 08 XX XX
XX 08 09 10 11 12 11 10 09 08 09 08 08 09 10 11 12 11 10 09 08 XX
08 09 10 11 12[13]12 11 10 09 08 08 09 10 11 12[13]12 11 10 09 08
XX 08 09 10 11 12 11 10 09 08 XX XX 08 09 10 11 12 11 10 09 08 XX
XX XX 08 09 10 11 10 09 08 XX XX XX XX 08 09 10 11 10 09 08 XX XX
XX XX XX 08 09 10 09 08 XX XX XX XX XX XX 08 09 10 09 08 XX XX XX
XX XX XX XX 08 09 08 XX XX XX XX XX XX XX XX 08 09 08 XX XX XX XX
XX XX XX XX XX 08 XX XX XX XX XX XX XX XX XX XX 08 XX XX XX XX XX
[13] = Torch
XX = Spawning available
Numbers show the lowest light level in that area
This can only work in a three high or shorter room with all torches on the middle row
A:
I find it easier to just place torches every 8 and use the (x,z) coordinate mod 8 = 0 as guide. That ends up putting torches (0,0) (0,8) (0,16) etc. When it comes to hillsides, i.e. 45 degree slopes or 1 block rise for 1 block of run, I put torchs twice as often or at coordinates congruent 0 mod 4.
Note that recently (since summer 2012) I've noticed mosters still spawning in what should be light > 7 locations. This is annoying and I suspect an (undocumnted on wiki) change to mob spawning rules. :(
|
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"pile_set_name": "StackExchange"
}
| 0.060353 |
[Construction and high-density fermentation of alkaline pectate lyase high-yield yeast].
Pectate lyase is widely applied in ramie degumming and fabric bioscouring in the textile industry. Compared to conventional processes that involve high alkaline and high temperature treatment, enzyme based treatments have significant advantages in fibers protectiveness, improved efficiency of refining, reduced energy consumption and pollution. Hence, it would be highly desirable to construct high-yield alkaline pectate lyase engineered strains and reduce the pectate lyase production cost. In the previous study, pectate lyase gene pel from Bacillus subtilis168 was expressed in Pichia pastoris GS115 after codon usage optimization based on the vector pHBM905A. To improve the expression level, the vector pHBM905BDM with optimized promoter and signal peptide was used to express the optimized gene pels in GS115. The transformant had increased activity from 68 U/mL to 100 U/mL with the improvement in the transcription level by 27% measured by qPCR. The transformants were further screened on pectin plates, where higher halo forming strains were picked for shake-flask fermentation and strain GS115-pHBM905BDM-pels4 showed the highest activity of 536 U/mL. Then plasmid pPIC9K-pels was constructed and electroporated into the GS115-pHBM905BDM-pels4 cells. Subsequently, high-copy transformant was screened by using the medium containing antibiotics G418, strain GS115-pHBM905BDMpPIC9K- pels1 was identified with increased activity of 770 U/mL and the copy number of pels was 7 confirmed by qPCR. Finally, the activity of pectate lyase produced by GS115-pHBM905BDM-pPIC9K-pels1reached to 2 271 U/mL in a 5-L fermentor. The activity of pectate lyase in our study reached the highest level of expression in P. pastoris, showing good application potential in the textile industry.
|
{
"pile_set_name": "PubMed Abstracts"
}
| 0 |
Hentai is always better with femdom
|
{
"pile_set_name": "OpenWebText2"
}
| 0.057143 |
xxx
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xxx
xxx
xxx
|
{
"pile_set_name": "Github"
}
| 0.25 |
[Mutational screening of the SLC26A4 gene in patients with nonsyndromic hearing loss by denaturing high-performance liquid chromatography].
To study the SLC26A4 gene mutations in patients with nonsyndromic hearing loss (NSHL) and provide the clinical guidance of gene diagnosis. PCR and denaturing high-performance liquid chromatography (DHPLC) were used to screen the 21 exons and their flanking regions of the SLC26A4 gene. Samples with abnormal DHPLC wave patterns were sequenced to identify the variations. Among the 30 unrelated NSHL patients in whom no deafness-causing mutations of the GJB2 gene were identified, 10 types of variations were detected, including 7 known mutations, 2 novel mutations (F572L and D87Y), and 1 known polymorphism (Ivs11+47T>C). The Ivs7-2A>G is the most common type of variation, accounting for 40% of all the mutations. SLC26A4 mutation is a major cause of NSHL, just next to the GJB2 mutations. For NSHL patients without deafness-causing GJB2 mutations, the SLC26A4 mutation rate was 23.3%, and the Ivs7-2A>G was the most common mutation.
|
{
"pile_set_name": "PubMed Abstracts"
}
| 0 |
Republic Wireless Opens $19 Unlimited Service to Everyone
Today, Republic Wireless, a new carrier offering a $19 per month unlimited smartphone plan, opened its service to everyone after months of limited beta testing. The plan includes unlimited voice, text and data and employs a hybrid technology that uses a Wi-Fi connection whenever possible for calls and data and then switches to Sprint’s 3G network when out of range. Republic currently only offers one smartphone, the Motorola Defy XT, which is available for preorder and costs $249 without contract plus a $10 activation fee. The phone is expected to ship in mid-December.
AllThingsD by Writer
AllThingsD.com is a Web site devoted to news, analysis and opinion on technology, the Internet and media. But it is different from other sites in this space. It is a fusion of different media styles, different topics, different formats and different sources. Read more »
|
{
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}
| 0 |
Suck Both Dicks At Once DP Her Butt &_creampie!
|
{
"pile_set_name": "OpenWebText2"
}
| 0.06383 |
BBW Squirts After Fucking Pussy And Ass
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|
{
"pile_set_name": "Pile-CC"
}
| 0.052434 |
Aiden Starr, Joseline Kelly – Chaotic Cock Slut Steals All the Dick at the Summer Swinger Orgy
|
{
"pile_set_name": "OpenWebText2"
}
| 0.053191 |
fucking my sister in her anal
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|
{
"pile_set_name": "OpenWebText2"
}
| 0.051282 |
// +build all resource_build_definition
// +build !exclude_resource_build_definition
package acceptancetests
import (
"fmt"
"regexp"
"strconv"
"testing"
"github.com/hashicorp/terraform-plugin-sdk/helper/acctest"
"github.com/hashicorp/terraform-plugin-sdk/helper/resource"
"github.com/hashicorp/terraform-plugin-sdk/terraform"
"github.com/microsoft/azure-devops-go-api/azuredevops/build"
"github.com/microsoft/terraform-provider-azuredevops/azuredevops/internal/acceptancetests/testutils"
"github.com/microsoft/terraform-provider-azuredevops/azuredevops/internal/client"
)
// validates that an apply followed by another apply (i.e., resource update) will be reflected in AzDO and the
// underlying terraform state.
func TestAccBuildDefinition_Create_Update_Import(t *testing.T) {
projectName := testutils.GenerateResourceName()
gitRepoName := testutils.GenerateResourceName()
buildDefinitionPathEmpty := `\`
buildDefinitionNameFirst := testutils.GenerateResourceName()
buildDefinitionNameSecond := testutils.GenerateResourceName()
buildDefinitionNameThird := testutils.GenerateResourceName()
buildDefinitionPathFirst := `\` + acctest.RandStringFromCharSet(10, acctest.CharSetAlphaNum)
buildDefinitionPathSecond := `\` + acctest.RandStringFromCharSet(10, acctest.CharSetAlphaNum)
buildDefinitionPathThird := `\` + buildDefinitionNameFirst + `\` + acctest.RandStringFromCharSet(10, acctest.CharSetAlphaNum)
buildDefinitionPathFourth := `\` + buildDefinitionNameSecond + `\` + acctest.RandStringFromCharSet(10, acctest.CharSetAlphaNum)
tfBuildDefNode := "azuredevops_build_definition.build"
resource.Test(t, resource.TestCase{
PreCheck: func() { testutils.PreCheck(t, nil) },
Providers: testutils.GetProviders(),
CheckDestroy: checkBuildDefinitionDestroyed,
Steps: []resource.TestStep{
{
Config: testutils.HclBuildDefinitionResourceGitHub(projectName, buildDefinitionNameFirst, buildDefinitionPathEmpty),
Check: resource.ComposeTestCheckFunc(
checkBuildDefinitionExists(buildDefinitionNameFirst),
resource.TestCheckResourceAttrSet(tfBuildDefNode, "project_id"),
resource.TestCheckResourceAttrSet(tfBuildDefNode, "revision"),
resource.TestCheckResourceAttr(tfBuildDefNode, "name", buildDefinitionNameFirst),
resource.TestCheckResourceAttr(tfBuildDefNode, "path", buildDefinitionPathEmpty),
),
}, {
Config: testutils.HclBuildDefinitionResourceGitHub(projectName, buildDefinitionNameSecond, buildDefinitionPathEmpty),
Check: resource.ComposeTestCheckFunc(
checkBuildDefinitionExists(buildDefinitionNameSecond),
resource.TestCheckResourceAttrSet(tfBuildDefNode, "project_id"),
resource.TestCheckResourceAttrSet(tfBuildDefNode, "revision"),
resource.TestCheckResourceAttr(tfBuildDefNode, "name", buildDefinitionNameSecond),
resource.TestCheckResourceAttr(tfBuildDefNode, "path", buildDefinitionPathEmpty),
),
}, {
Config: testutils.HclBuildDefinitionResourceGitHub(projectName, buildDefinitionNameFirst, buildDefinitionPathFirst),
Check: resource.ComposeTestCheckFunc(
checkBuildDefinitionExists(buildDefinitionNameFirst),
resource.TestCheckResourceAttrSet(tfBuildDefNode, "project_id"),
resource.TestCheckResourceAttrSet(tfBuildDefNode, "revision"),
resource.TestCheckResourceAttr(tfBuildDefNode, "name", buildDefinitionNameFirst),
resource.TestCheckResourceAttr(tfBuildDefNode, "path", buildDefinitionPathFirst),
),
}, {
Config: testutils.HclBuildDefinitionResourceGitHub(projectName, buildDefinitionNameFirst,
buildDefinitionPathSecond),
Check: resource.ComposeTestCheckFunc(
checkBuildDefinitionExists(buildDefinitionNameFirst),
resource.TestCheckResourceAttrSet(tfBuildDefNode, "project_id"),
resource.TestCheckResourceAttrSet(tfBuildDefNode, "revision"),
resource.TestCheckResourceAttr(tfBuildDefNode, "name", buildDefinitionNameFirst),
resource.TestCheckResourceAttr(tfBuildDefNode, "path", buildDefinitionPathSecond),
),
}, {
Config: testutils.HclBuildDefinitionResourceGitHub(projectName, buildDefinitionNameFirst, buildDefinitionPathThird),
Check: resource.ComposeTestCheckFunc(
checkBuildDefinitionExists(buildDefinitionNameFirst),
resource.TestCheckResourceAttrSet(tfBuildDefNode, "project_id"),
resource.TestCheckResourceAttrSet(tfBuildDefNode, "revision"),
resource.TestCheckResourceAttr(tfBuildDefNode, "name", buildDefinitionNameFirst),
resource.TestCheckResourceAttr(tfBuildDefNode, "path", buildDefinitionPathThird),
),
}, {
Config: testutils.HclBuildDefinitionResourceGitHub(projectName, buildDefinitionNameFirst, buildDefinitionPathFourth),
Check: resource.ComposeTestCheckFunc(
checkBuildDefinitionExists(buildDefinitionNameFirst),
resource.TestCheckResourceAttrSet(tfBuildDefNode, "project_id"),
resource.TestCheckResourceAttrSet(tfBuildDefNode, "revision"),
resource.TestCheckResourceAttr(tfBuildDefNode, "name", buildDefinitionNameFirst),
resource.TestCheckResourceAttr(tfBuildDefNode, "path", buildDefinitionPathFourth),
),
}, {
Config: testutils.HclBuildDefinitionResourceTfsGit(projectName, gitRepoName, buildDefinitionNameThird, buildDefinitionPathEmpty),
Check: resource.ComposeTestCheckFunc(
checkBuildDefinitionExists(buildDefinitionNameThird),
resource.TestCheckResourceAttrSet(tfBuildDefNode, "project_id"),
resource.TestCheckResourceAttrSet(tfBuildDefNode, "revision"),
resource.TestCheckResourceAttrSet(tfBuildDefNode, "repository.0.repo_id"),
resource.TestCheckResourceAttr(tfBuildDefNode, "name", buildDefinitionNameThird),
resource.TestCheckResourceAttr(tfBuildDefNode, "path", buildDefinitionPathEmpty),
),
}, {
// Resource Acceptance Testing https://www.terraform.io/docs/extend/resources/import.html#resource-acceptance-testing-implementation
ResourceName: tfBuildDefNode,
ImportStateIdFunc: testutils.ComputeProjectQualifiedResourceImportID(tfBuildDefNode),
ImportState: true,
ImportStateVerify: true,
},
},
})
}
// Verifies a build for Bitbucket can happen. Note: the update/import logic is tested in other tests
func TestAccBuildDefinitionBitbucket_Create(t *testing.T) {
projectName := testutils.GenerateResourceName()
resource.Test(t, resource.TestCase{
PreCheck: func() { testutils.PreCheck(t, nil) },
Providers: testutils.GetProviders(),
CheckDestroy: checkBuildDefinitionDestroyed,
Steps: []resource.TestStep{
{
Config: testutils.HclBuildDefinitionResourceBitbucket(projectName, "build-def-name", "\\", ""),
ExpectError: regexp.MustCompile("bitbucket repositories need a referenced service connection ID"),
}, {
Config: testutils.HclBuildDefinitionResourceBitbucket(projectName, "build-def-name", "\\", "some-service-connection"),
Check: checkBuildDefinitionExists("build-def-name"),
},
},
})
}
// Verifies a build for with variables can create and update, including secret variables
func TestAccBuildDefinition_WithVariables_CreateAndUpdate(t *testing.T) {
name := testutils.GenerateResourceName()
tfNode := "azuredevops_build_definition.build"
resource.Test(t, resource.TestCase{
PreCheck: func() { testutils.PreCheck(t, nil) },
Providers: testutils.GetProviders(),
CheckDestroy: checkBuildDefinitionDestroyed,
Steps: []resource.TestStep{
{
Config: testutils.HclBuildDefinitionWithVariables("foo1", "bar1", name),
Check: checkForVariableValues(tfNode, "foo1", "bar1"),
}, {
Config: testutils.HclBuildDefinitionWithVariables("foo2", "bar2", name),
Check: checkForVariableValues(tfNode, "foo2", "bar2"),
},
},
})
}
// Checks that the expected variable values exist in the state
func checkForVariableValues(tfNode string, expectedVals ...string) resource.TestCheckFunc {
return func(s *terraform.State) error {
rootModule := s.RootModule()
resource, ok := rootModule.Resources[tfNode]
if !ok {
return fmt.Errorf("Did not find resource in TF state")
}
is := resource.Primary
if is == nil {
return fmt.Errorf("No primary instance: %s in %s", tfNode, rootModule.Path)
}
for _, expectedVal := range expectedVals {
found := false
for _, value := range is.Attributes {
if value == expectedVal {
found = true
}
}
if !found {
return fmt.Errorf("Did not find variable with value %s", expectedVal)
}
}
return nil
}
}
// Given the name of an AzDO build definition, this will return a function that will check whether
// or not the definition (1) exists in the state and (2) exist in AzDO and (3) has the correct name
func checkBuildDefinitionExists(expectedName string) resource.TestCheckFunc {
return func(s *terraform.State) error {
buildDef, ok := s.RootModule().Resources["azuredevops_build_definition.build"]
if !ok {
return fmt.Errorf("Did not find a build definition in the TF state")
}
buildDefinition, err := getBuildDefinitionFromResource(buildDef)
if err != nil {
return err
}
if *buildDefinition.Name != expectedName {
return fmt.Errorf("Build Definition has Name=%s, but expected Name=%s", *buildDefinition.Name, expectedName)
}
return nil
}
}
// verifies that all build definitions referenced in the state are destroyed. This will be invoked
// *after* terraform destroys the resource but *before* the state is wiped clean.
func checkBuildDefinitionDestroyed(s *terraform.State) error {
for _, resource := range s.RootModule().Resources {
if resource.Type != "azuredevops_build_definition" {
continue
}
// indicates the build definition still exists - this should fail the test
if _, err := getBuildDefinitionFromResource(resource); err == nil {
return fmt.Errorf("Unexpectedly found a build definition that should be deleted")
}
}
return nil
}
// given a resource from the state, return a build definition (and error)
func getBuildDefinitionFromResource(resource *terraform.ResourceState) (*build.BuildDefinition, error) {
buildDefID, err := strconv.Atoi(resource.Primary.ID)
if err != nil {
return nil, err
}
projectID := resource.Primary.Attributes["project_id"]
clients := testutils.GetProvider().Meta().(*client.AggregatedClient)
return clients.BuildClient.GetDefinition(clients.Ctx, build.GetDefinitionArgs{
Project: &projectID,
DefinitionId: &buildDefID,
})
}
|
{
"pile_set_name": "Github"
}
| 0 |
Q:
Determine required Ada source files for project build
Imagine a large Ada project build with GPRbuild and a single *.gpr project file. The source directory contains many Ada package specification and body files but a few of them are not required to build the project (an executable).
Does GPRbuild offers something like a report function listing all files that were required to the build the project? This would allow me to remove unusued source files.
A:
You can play around with
-gnatu List units for this compilation
like this:
gprbuild -P foo.gpr -cargs -gnatu
|
{
"pile_set_name": "StackExchange"
}
| 0 |
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{
"pile_set_name": "OpenWebText2"
}
| 0.054545 |
Blog List
The only time I care about kale
September 22, 2013 - Wes Burns
I'm pretty sure kale is a scam. You've probably seen kale and you might not recognize it: it looks pretty much exactly like lettuce.
As of late people, and by "people" I mean the kind of people that blend the majority of their food THEN try to get you to blend all your food, have started incorporating kale into all sorts of drinkable recpies that will make you long for the rigid textural consistency of creamed corn.
They will tell you it's healthy; of course 10 years ago this same group of people would extol the inarguable health benefits of macrobiotics or green tea or Snackwells cookies.
If green tea is healthy but nobody talks about it, is it still good for you?
So kale is the latest in the list of undiscovered healthy foods that people have known about for hundreds of years. And I just don't care.
It's green! It's good for you!
Who cares! So are a bunch of other things!
So when do I care about kale? While watching "Chopped." That show on the Food network where cooks get a bunch of random things and have to make a meal in 10 minutes. That's the only time; because I always wonder "what are they going to do with that ridiculously boring kale?" Then they usually make it a garnish.
|
{
"pile_set_name": "Pile-CC"
}
| 0 |
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|
{
"pile_set_name": "Wikipedia (en)"
}
| 0.052632 |
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Nigga, ngga, nigga, nigga
Niggers! Nigga - Nigga - Nigga - Nigga
Nigga, Nigga
|
{
"pile_set_name": "YoutubeSubtitles"
}
| 0.119497 |
EXCLUSIVE: President Trump On His Brexit Advice To Theresa May
In a world exclusive interview with LBC, President Donald Trump told Nigel Farage that he advised Theresa May on how to make a deal with the EU but she didn't listen to him.
He said: "It was time for Boris, you needed him."
Watch the video above.
|
{
"pile_set_name": "OpenWebText2"
}
| 0 |
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