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We give a particular choice of the higher Eilenberg-MacLane maps by a recursive formula.This choice leads to a simple description of the homotopy operations for simplicial Z/2-algebras.
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arxiv:math/0411595
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The convolution of indicators of two conjugacy classes on the symmetric group S_q is usually a complicated linear combination of indicators of many conjugacy classes. Similarly, a product of the moments of the Jucys--Murphy element involves many conjugacy classes with complicated coefficients. In this article we consider a combinatorial setup which allows us to manipulate such products easily: to each conjugacy class we associate a two-dimensional surface and the asymptotic properties of the conjugacy class depend only on the genus of the resulting surface. This construction closely resembles the genus expansion from the random matrix theory. As the main application we study irreducible representations of symmetric groups S_q for large q. We find the asymptotic behavior of characters when the corresponding Young diagram rescaled by a factor q^{-1/2} converge to a prescribed shape. The character formula (known as the Kerov polynomial) can be viewed as a power series, the terms of which correspond to two-dimensional surfaces with prescribed genus and we compute explicitly the first two terms, thus we prove a conjecture of Biane.
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arxiv:math/0411647
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A Gauss paragraph is a combinatorial formulation of a generic closed curve with multiple components on some surface. A virtual string is a collection of circles with arrows that represent the crossings of such a curve. Every closed curve has an underlying virtual string and every virtual string has an underlying Gauss paragraph. A word-wise partition is a partition of the alphabet set of a Gauss paragraph that satisfies certain conditions with respect to the Gauss paragraph. In this paper we use the theory of virtual strings to obtain a combinatorial description of closed curves in the 2-sphere (and therefore 2-dimensional Euclidean space) in terms of Gauss paragraphs and word-wise partitions.
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arxiv:math/0412023
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This paper studies the non-holomorphic Eisenstein series E(z,s) for the modular surface, and shows that integration with respect to certain non-negative measures gives meromorphic functions of s that have all their zeros on the critical line Re(s) = 1/2. For the constant term of the Eisenstein series it shows that all zeros are on the critical line for fixed y= Im(z) \ge 1, except possibly for two real zeros, which are present if and only if y > 4 \pi e^{-\gamma} = 7.0555+. It shows the Riemann hypothesis holds for all truncation integrals with truncation parameter T \ge 1. For T=1 this proves the Riemann hypothesis for a zeta function recently introduced by Lin Weng, attached to rank 2 semistable lattices over the rationals.
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arxiv:math/0412039
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We study codimension one holomorphic foliations on complex projective spaces and compact manifolds under the assumption that the foliation has a projective transverse structure in the complement of some invariant codimension one analytic subset. The basic motivation is the characterization of pull-backs of Riccati foliations on projective spaces. Our techniques apply to give a description of the generic models of codimension one foliations on compact manifolds of dimension $\ge 3$.
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arxiv:math/0412058
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We study generalized complex manifolds from the point of view of symplectic and Poisson geometry. We start by showing that every generalized complex manifold admits a canonical Poisson structure. We use this fact, together with Weinstein's classical result on the local normal form of Poisson manifolds, to prove a local structure theorem for generalized complex manifolds which extends the result Gualtieri has obtained in the "regular" case. Finally, we begin a study of the local structure of a generalized complex manifold in a neighborhood of a point where the associated Poisson tensor vanishes. In particular, we show that in such a neighborhood, a "first-order approximation" to the generalized complex structure is encoded in the data of a constant B-field and a complex Lie algebra.
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arxiv:math/0412084
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In a recent paper, Regev and Roichman introduced the <_L order and the L-descent number statistic, des_L, on the group of colored permutations, C_a \wr S_n. Here we define the L-reverse major index statistic, rmaj_L, on the same group and study the distribution of des_L and the bi-statistic (des_L, rmaj_L). We obtain new wreath-product analogues of the Eulerian and q-Euler-Mahonian polynomials, and a generalization of Carlitz's identity.
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arxiv:math/0412091
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For a system of second order differential equations we determine a nonlinear connection that is compatible with a given generalized Lagrange metric. Using this nonlinear connection, we can find the whole family of metric nonlinear connections that can be associated with a system of SODE and a generalized Lagrange structure. For the particular case when the system of SODE and the metric structure are Lagrangian, we prove that the canonic nonlinear connection of the Lagrange space is the only nonlinear connection which is metric and compatible with the symplectic structure of the Lagrange space. The metric tensor of the Lagrange space determines the symmetric part of the nonlinear connection, while the symplectic structure of the Lagrange space determines the skew-symmetric part of the nonlinear connection.
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arxiv:math/0412109
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We give a geometric criterion that guaranteesa purely singular spectral type for a dynamical system on a Riemannian manifold. The criterion, that is based on the existence of fairly rich but localized periodic approximations, is compatible with mixing. Indeed, we use it to construct examples of smooth mixing flows on the three torus with purely singular spectra.
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arxiv:math/0412172
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We show a geometric rigidity of isometric actions of non compact (semisimple) Lie groups on Lorentz manifolds. Namely, we show that the manifold has a warped product structure of a Lorentz manifold with constant curvature by a Riemannian manifold.
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arxiv:math/0412195
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The paper concerns Hochschild cohomology of a commutative algebra S, which is essentially of finite type over a commutative noetherian ring K and projective as a K-module, with coefficients in an S-module M. It is proved that vanishing of HH^n(S|K,M) in sufficiently long intervals imply the smoothness of S_q over K for all prime ideals q in the support of M. In particular, S is smooth if HH^n(S|K,S)=0 for (dim S+2) consecutive non-negative integers n.
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arxiv:math/0412259
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In this paper, we consider tilings of the hyperbolic 2-space, built with a finite number of polygonal tiles, up to affine transformation. To such a tiling T, we associate a space of tilings: the continuous hull Omega(T) on which the affine group acts. This space Omega(T) inherits a solenoid structure whose leaves correspond to the orbits of the affine group. First we prove the finite harmonic measures of this laminated space correspond to finite invariant measures for the affine group action. Then we give a complete combinatorial description of these finite invariant measures. Finally we give examples with an arbitrary number of ergodic invariant probability measures.
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arxiv:math/0412290
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We show that strictly convex surfaces expanding by the inverse Gauss curvature flow converge to infinity in finite time. After appropriate rescaling, they converge to spheres. We describe the algorithm to find our main test function.
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arxiv:math/0412297
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We study optimal transportation of measures on compact manifolds for costs defined from convex Lagrangians. We prove that optimal transportation can be interpolated by measured Lipschitz laminations, or geometric currents. The methods are inspired from Mather theory on Lagrangian systems. We make use of viscosity solutions of the associated Hamilton-Jacobi equation in the spirit of Fathi's approach to Mather theory.
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arxiv:math/0412299
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An urn model of Diaconis and some generalizations are discussed. A convergence theorem is proved that implies for Diaconis' model that the empirical distribution of balls in the urn converges with probability one to the uniform distribution.
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arxiv:math/0412333
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The expected length of longest common subsequences is a problem that has been in the literature for at least twenty five years. Determining the limiting constants \gamma_k appears to be quite difficult, and the current best bounds leave much room for improvement. Boutet de Monvel explores an independent version of the problem he calls the Bernoulli Matching model. He explores this problem and its relation to the longest common subsequence problem. This paper continues this pursuit by focusing on a simplification we term r-reach. For the string model, L_r(u,v) is the longest common subsequence of u and v given that each matched pair of letters is no more than r letters apart.
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arxiv:math/0412375
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Let $G$ be a finite group of odd order, $\F$ a finite field of odd characteristic $p$ and $\B$ a finite--dimensional symplectic $\F G$-module. We show that $\B$ is $\F G$-hyperbolic, i.e., it contains a self--perpendicular $\F G$-submodule, iff it is $\F N$-hyperbolic for every cyclic subgroup $N$ of $G$.
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arxiv:math/0412384
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With every real polynomial $f$, we associate a family $\{f_{\epsilon r}\}_{\epsilon, r}$ of real polynomials, in explicit form in terms of $f$ and the parameters $\epsilon>0,r\in N$, and such that $\Vert f-f_{\epsilon r}\Vert_1\to 0$ as $\epsilon\to 0$. Let $V\subset R^n$ be a real algebraic set described by finitely many polynomials equations $g_j(x)=0,j\in J$, and let $f$ be a real polynomial, nonnegative on $V$. We show that for every $\epsilon>0$, there exist nonnegative scalars $\{\lambda_j(\epsilon)\}_{j\in J}$ such that, for all $r$ sufficiently large, $$f_{\epsilon r}+\sum_{j\in J} \lambda_j(\epsilon) g_j^2,\quad is a sum of squares.$$ This representation is an obvious certificate of nonnegativity of $f_{\epsilon r}$ on $V$, and very specific in terms of the $g_j$ that define the set $V$. In particular, it is valid with {\it no} assumption on $V$. In addition, this representation is also useful from a computation point of view, as we can define semidefinite programing relaxations to approximate the global minimum of $f$ on a real algebraic set $V$, or a semi-algebraic set $K$, and again, with {\it no} assumption on $V$ or $K$.
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arxiv:math/0412400
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This paper elucidates the connection between stationary symmetric alpha-stable processes with 0<alpha<2 and nonsingular flows on measure spaces by describing a new and unique decomposition of stationary stable processes into those corresponding to positive flows and those corresponding to null flows. We show that a necessary and sufficient for a stationary stable process to be ergodic is that its positive component vanishes.
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arxiv:math/0412419
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Using symplectic topology and the Radon transform, we prove that smooth 4-dimensional projective planes are diffeomorphic to $\mathbb{CP}^2$. We define the notion of a plane curve in a smooth projective plane, show that plane curves in high dimensional regular planes are lines, prove that homeomorphisms preserving plane curves are smooth collineations, and prove a variety of results analogous to the theory of classical projective planes.
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arxiv:math/0412500
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We show that probability measures on the unit circle associated with Verblunsky coefficients obeying a Coulomb-type decay estimate have no singular continuous component.
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arxiv:math/0412515
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A toroidal grid graph is a Cartesian product of cycles, and the run length of a Hamiltonian cycle in a grid graph is defined to be the maximum number r such that any r consecutive edges include no more than one edge in any dimension. By constructive methods, we place bounds on the maximum run length possible for a Hamiltonian cycle in several families of grid graphs.
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arxiv:math/0412530
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In this paper, I propose some problems, of topological nature, on the energy functional associated to the Dirichlet problem -\Delta u = f(x,u) in Omega, u restricted to the boundary of Omega is 0. Positive answers to these problems would produce innovative multiplicity results on this Dirichlet problem.
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arxiv:math/0412560
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Let M be a complete n-dimensional Riemannian manifold, if the sobolev inqualities hold on M, then the geodesic ball has maximal volume growth; if the Ricci curvature of M is nonnegative, and one of the general Sobolev inequalities holds on M, then M is diffeomorphic to $R^{n}$.
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arxiv:math/0501009
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If $A$ is a subset of the set of reflections of a finite Coxeter group $W$, we define a sub-${\mathbb{Z}}$-module ${\mathcal{D}}_A(W)$ of the group algebra ${\mathbb{Z}} W$. We provide examples where this submodule is a subalgebra. This family of subalgebras includes strictly the Solomon descent algebra and, if $W$ if of type $B$, the Mantaci-Reutenauer algebra.
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arxiv:math/0501099
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We consider the operator of taking the $2p$th derivative of a function with zero boundary conditions for the function and its first $p-1$ derivatives at two distinct points. Our main result provides an asymptotic formula for the eigenvalues and resolves a question on the appearance of certain regular numbers in the eigenvalue sequences for $p=1$ and $p=3$.
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arxiv:math/0501116
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E26 in the Enestrom index. Translated from the Latin original, "Observationes de theoremate quodam Fermatiano aliisque ad numeros primos spectantibus" (1732). In this paper Euler gives a counterexample to Fermat's claim that all numbers of the form 2^{2^m}+1 are primes, by showing 2^{2^5}+1=4294967297 is divisible by 641. He also considers many cases in which we are guaranteed that a number is composite, but he notes clearly that it is not possible to have a full list of circumstances under which a number is composite. He then gives a theorem and several corollaries of it, but he says that he does not have a proof, although he is sure of the truth of them. The main theorem is that a^n-b^n is always able to be divided by n+1 if n+1 is a prime number and both a and b cannot be divided by it.
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arxiv:math/0501118
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For a hyperbolic 3-manifold M with a torus boundary component, all but finitely many Dehn fillings on the torus component yield hyperbolic 3-manifolds. In this paper, we will focus on the situation where M has two exceptional Dehn fillings, both of which yield toroidal manifolds. For such situation, Gordon gave an upper bound for the distance between two slopes of Dehn fillings. In particular, if M is large, then the distance is at most 5. We show that this upper bound can be improved by 1 for a broad class of large manifolds.
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arxiv:math/0501148
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Here I present the present the first major result of a novel form of network analysis - a temporal interpretation. Treating numerical edges labels as the time at which an interaction occurs between the two vertices comprising that edge generates a number of intriguing questions. For example, given the structure of a graph, how many ``fundamentally'' different temporally non-isomorphic forms are there, across all possible edge labelings. Specifically, two networks, N and M, are considered to be in the same isotemporal class if there exists a function alpha(N)->M that is a graph isomorphism and preserves all paths in N with strictly increasing edge labels. I present a closed formula for the number of isotemporal classes N(n) of n-gons. This result is strongly tied to number theoretic identities; in the case of $n$ odd, N(n)= 1/n sum_{d|n} (2^{n/d -1}-1)Phi(d), where Phi is the Euler totient function.
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arxiv:math/0501171
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We derive a weighted $L^2$-estimate of the Witten spinor in a complete Riemannian spin manifold $(M^n,g)$ of non-negative scalar curvature which is asymptotically Schwarzschild. The interior geometry of $M$ enters this estimate only via the lowest eigenvalue of the square of the Dirac operator on a conformal compactification of $M$.
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arxiv:math/0501195
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Brooks and Makover introduced an approach to studying the global geometric quantities (in particular, the first eigenvalue of the Laplacian, injectivity radius and diameter) of a ``typical'' compact Riemann surface of large genus based on compactifying finite-area Riemann surfaces associated with random cubic graphs; by a theorem of Belyi, these are ``dense'' in the space of compact Riemann surfaces. The question as to how these surfaces are distributed in the Teichm\"{u}ller spaces depends on the study of oriented cycles in random cubic graphs with random orientation; Brooks and Makover conjectured that asymptotically normalized cycle lengths follow Poisson--Dirichlet distribution. We present a proof of this conjecture using representation theory of the symmetric group.
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arxiv:math/0501283
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This paper describes the construction of a lower bound for the tails of general random variables, using solely knowledge of their moment generating function. The tilting procedure used allows for the construction of lower bounds that are tighter and more broadly applicable than existing tail approximations.
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arxiv:math/0501360
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For a finite lattice L, let EL denote the reflexive and transitive closure of the join-dependency relation on L, defined on the set J(L) of all join-irreducible elements of L. We characterize the relations of the form EL, as follows: Theorem. Let E be a quasi-ordering on a finite set P. Then the following conditions are equivalent: (i) There exists a finite lattice L such that (J(L),EL) is isomorphic to the quasi-ordered set (P,E). (ii) There are not exactly two elements x in P such that p E x, for any p in P. For a finite lattice L, let je(L) = |J(L)|-|J(Con L)|, where Con L is the congruence lattice of L. It is well-known that the inequality je(L) $\ge$ 0 holds. For a finite distributive lattice D, let us define the join-excess function: JE(D) = min(je(L) | Con L isomorphic to D). We provide a formula for computing the join-excess function of a finite distributive lattice D. This formula implies that JE(D) $\le$ (2/3)| J(D)|, for any finite distributive lattice D; the constant 2/3 is best possible. A special case of this formula gives a characterization of congruence lattices of finite lower bounded lattices.
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arxiv:math/0501366
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A prescription for constructing dictionaries for cardinal spline spaces on a compact interval is provided. It is proved that such spaces can be spanned by dictionaries which are built by translating a prototype B-spline function of fixed support into the knots of the required cardinal spline space. This implies that cardinal spline spaces on a compact interval can be spanned by dictionaries of cardinal B-spline functions of broader support that the corresponding basis function.
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arxiv:math/0501403
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The harmonic archipelago HA is obtained by attaching a large pinched annulus to every pair of consecutive loops of the Hawaiian earring. We clarify the fundamental group pi1(HA) as a quotient of the Hawaiian earring group, provide a precise description of the kernel, show that both pi1(HA) and the kernel are uncountable, and that pi1(HA) has the indiscrete topology.
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arxiv:math/0501426
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Let $\Cal E$ be a very ample vector bundle of rank two on a smooth complex projective threefold $X$. An inequality about the third Segre class of $\Cal E$ is provided when $K_X+\det \Cal E$ is nef but not big, and when a suitable positive multiple of $K_X+\det \Cal E$ defines a morphism $X\to B$ with connected fibers onto a smooth projective curve $B$, where $K_X$ is the canonical bundle of $X$. As an application, the case where the genus of $B$ is positive and $\Cal E$ has a global section whose zero locus is a smooth hyperelliptic curve of genus $\geq 2$ is investigated, and our previous result is improved for threefolds.
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arxiv:math/0501471
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For certain negative rational numbers k0, called singular values, and associated with the symmetric group S_N on N objects, there exist homogeneous polynomials annihilated by each Dunkl operator when the parameter k = k0. It was shown by de Jeu, Opdam and the author (TAMS 346(1994),237-256) that the singular values are exactly the values m/n with 2<=n<=N, m = 1,2... and m/n is not an integer. For each pair (m,n) satisfying these conditions there is a unique irreducible S_N-module of singular polynomials for the singular value -m/n. The existence of these polynomials was established by the author (IMRN 2004,#67,3607-3635). The uniqueness is proven in the present paper. By using Murphy's (J. Alg. 69(1981), 287-297) results on the eigenvalues of the Murphy elements, the problem of existence of singular polynomials is first restricted to the isotype of a partition of N (corresponding to an irreducible representation of S_N) such that (n/gcd(m,n)) divides t+1 for each part t of the partition except the last one. Then by arguments involving nonsymmetric Jack polynomials it is shown that the assumption that the second part of the partition is greater than or equal to n/gcd(m,n) leads to a contradiction.This shows that the singular polynomials are exactly those already determined.
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arxiv:math/0501494
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This paper proposes a formal model for a network of robotic agents that move and communicate. Building on concepts from distributed computation, robotics and control theory, we define notions of robotic network, control and communication law, coordination task, and time and communication complexity. We then analyze a number of basic motion coordination algorithms such as pursuit, rendezvous and deployment.
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arxiv:math/0501499
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We present three new identities in law for quadratic functionals of conditioned bivariate Gaussian processes. In particular, our results provide a two-parameter generalization of a celebrated identity in law, involving the path variance of a Brownian bridge, due to Watson (1961). The proof is based on ideas from a recent note by J. R. Pycke (2005) and on the stochastic Fubini theorem for general Gaussian measures proved in Deheuvels et al. (2004).
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arxiv:math/0501506
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We observe that the Rector invariants classifying the genus of BS^3 show up in (orthogonal and unitary) K-theory. We then use this knowledge to show purely algebraically how the K-theory of the spaces in the genus of BS^3 differ. This provides new insights into a result of Notbohm in the case of BS^3.
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arxiv:math/0501513
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Let f be a germ of holomorphic self-map of C^2 at the origin O tangent to the identity, and with O as a non-dicritical isolated fixed point. A parabolic curve for f is a holomorphic f-invariant curve, with O on the boundary, attracted by O under the action of f. It has been shown that if the characteristic direction [v] has residual index not belonging to Q^+, then there exist parabolic curves for f tangent to [v]. In this paper we prove, with a different method, that the conclusion still holds just assuming that the residual index is not vanishing (at least when f is regular along [v]).
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arxiv:math/0501537
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Let $M$ be an $m$-dimensional differentiable manifold with a nontrivial circle action ${\mathcal S}= {\lbrace S_t \rbrace}_{t \in\RR}, S_{t+1}=S_t$, preserving a smooth volume $\mu$. For any Liouville number $\a$ we construct a sequence of area-preserving diffeomorphisms $H_n$ such that the sequence $H_n\circ S_\a\circ H_n^{-1}$ converges to a smooth weak mixing diffeomorphism of $M$. The method is a quantitative version of the approximation by conjugations construction introduced in \cite{AK}. For $m=2$ and $M$ equal to the unit disc $\DD^2=\{x^2+y^2\leq 1\}$ or the closed annulus $\AAA=\TT\times [0,1]$ this result proves the following dichotomy: $\a \in \RR \setminus\QQ$ is Diophantine if and only if there is no ergodic diffeomorphism of $M$ whose rotation number on the boundary equals $\alpha$ (on at least one of the boundaries in the case of $\AAA$). One part of the dichotomy follows from our constructions, the other is an unpublished result of Michael Herman asserting that if $\a$ is Diophantine, then any area preserving diffeomorphism with rotation number $\a$ on the boundary (on at least one of the boundaries in the case of $\AAA$) displays smooth invariant curves arbitrarily close to the boundary which clearly precludes ergodicity or even topological transitivity.
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arxiv:math/0502067
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It is shown that tight closure commutes with localization in any two dimensional ring $R$ of prime characteristic if either $R$ is a Nagata ring or $R$ possesses a weak test element. Moreover, it is proved that tight closure commutes with localization at height one prime ideals in any ring of prime characteristic.
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arxiv:math/0502076
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The family of harmonic Hilbert spaces is a natural enlargement of those classical $L^{2}$-Sobolev space on $\R^d$ which consist of continuous functions. In the present paper we demonstrate that the use of basic results from the theory of Wiener amalgam spaces allows to establish fundamental properties of harmonic Hilbert spaces even if they are defined over an arbitrary locally compact abelian group $\G$. Even for $\G = \R^{d}$ this new approach improves previously known results. In this paper we present results on minimal norm interpolators over lattices and show that the infinite minimal norm interpolations are the limits of finite minimal norm interpolations. In addition, the new approach paves the way for the study of stability problems and error analysis for norm interpolations in harmonic Hilbert and Banach spaces on locally compact abelian groups.
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arxiv:math/0502093
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A great deal of recent activity has centered on the question of whether, for a given Hilbert function, there can fail to be a unique minimum set of graded Betti numbers, and this is closely related to the question of whether the associated Hilbert scheme is irreducible or not. We give a broad class of Hilbert functions for which we show that there is no minimum set of graded Betti numbers, and hence that the associated Hilbert scheme is reducible. Furthermore, we show that the Weak Lefschetz Property holds for the general element of one component, while it fails for every element of another component.
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arxiv:math/0502146
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We construct a Poincare-Birkhoff-Witt type basis for the Weyl modules of the current algebra of $sl_{r+1}$. As a corollary we prove a conjecture made by Chari and Pressley on the dimension of the Weyl modules in this case. Further, we relate the Weyl modules to the fusion modules of the current algebra defined by Feigin and Loktev, and to the Demazure modules in level one representations of the corresponding affine algebra. In particular, this allows us to establish substantial cases of the conjectures of Feigin and Loktev on the structure and graded character of the fusion modules.
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arxiv:math/0502165
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This note provides new methods for constructing quadratic nonresidues in finite fields of characteristic p. It will be shown that there is an effective deterministic polynomial time algorithm for constructing quadratic nonresidues in finite fields.
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arxiv:math/0502214
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For a finitely generated free group F_n, of rank at least 2, any finite subgroup of Out(F_n) can be realized as a group of automorphisms of a graph with fundamental group F_n. This result, known as Out(F_n) realization, was proved by Zimmermann, Culler and Khramtsov. This theorem is comparable to Nielsen realization as proved by Kerckhoff: for a closed surface with negative Euler characteristic, any finite subgroup of the mapping class group can be realized as a group of isometries of a hyperbolic surface. Both of these theorems have restatements in terms of fixed points of actions on spaces naturally associated to them. For a nonnegative integer n we define a class of groups (GVP(n)) and prove a similar statement for their outer automorphism groups.
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arxiv:math/0502248
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The Hodge conjecture is shown to hold for rationally connected fivefolds, or more generally for fivefolds for which the base of the maximal rationally connected fibration is at most 3 dimensional.
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arxiv:math/0502257
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A graphs of rank n (homotopy equivalent to a wedge of n circles) without ``separating edges'' has a canonical n-dimensional compact C^1 manifold thickening. This implies that the canonical homomorphism f:Out(F_n)-> GL(n,Z) is trivial in rational cohomology in the stable range answering a question raised by Hatcher and Vogtmann [6]. Another consequence of the construction is the existence of higher Reidemeister torsion invariants for IOut(F_n)=ker f. These facts were first proved by the first author in [8] using different methods.
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arxiv:math/0502266
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We develop an approach through geometric functional analysis to error correcting codes and to reconstruction of signals from few linear measurements. An error correcting code encodes an n-letter word x into an m-letter word y in such a way that x can be decoded correctly when any r letters of y are corrupted. We prove that most linear orthogonal transformations Q from R^n into R^m form efficient and robust robust error correcting codes over reals. The decoder (which corrects the corrupted components of y) is the metric projection onto the range of Q in the L_1 norm. An equivalent problem arises in signal processing: how to reconstruct a signal that belongs to a small class from few linear measurements? We prove that for most sets of Gaussian measurements, all signals of small support can be exactly reconstructed by the L_1 norm minimization. This is a substantial improvement of recent results of Donoho and of Candes and Tao. An equivalent problem in combinatorial geometry is the existence of a polytope with fixed number of facets and maximal number of lower-dimensional facets. We prove that most sections of the cube form such polytopes.
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arxiv:math/0502299
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We consider two operator space versions of type and cotype, namely $S_p$-type, $S_q$-cotype and type $(p,H)$, cotype $(q,H)$ for a homogeneous Hilbertian operator space $H$ and $1\leq p \leq 2 \leq q\leq \infty$, generalizing "$OH$-cotype 2" of G. Pisier. We compute type and cotype of some Hilbertian operator spaces and $L_p$ spaces, and we investigate the relationship between a homogeneous Hilbertian space $H$ and operator spaces with cotype $(2,H)$. As applications we consider operator space versions of generalized little Grothendieck's theorem and Maurey's extension theorem in terms of these new notions.
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arxiv:math/0502302
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We propose a generalisation of Exel's crossed product by a single endomorphism and a transfer operator to the case of actions of abelian semigroups of endomorphisms and associated transfer operators. The motivating example for our definition yields new crossed products, not obviously covered by familiar theory. Our technical machinery builds on Fowler's theory of Toeplitz and Cuntz-Pimsner algebras of discrete product systems of Hilbert bimodules, which we need to expand to cover a natural notion of relative Cuntz-Pimsner algebras of product systems.
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arxiv:math/0502307
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It is known that the existence of localization with respect to an arbitrary (possibly proper) class of maps in the category of simplicial sets is implied by a large-cardinal axiom called Vopenka's principle.In this article we extend the validity of this result to any left proper, combinatorial, simplicial model category $\cat M$ and show that, under additional assumptions on $\cat M$, every homotopy idempotent functor is in fact a localization with respect to some set of maps. These results are valid for the homotopy category of spectra, among other applications.
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arxiv:math/0502329
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Recently an operator space version of type and cotype, namely type $(p,H)$ and cotype $(q,H)$ of operator spaces for $1\leq p \leq 2\leq q \leq \infty$ and a subquadratic and homogeneous Hilbetian operator space $H$ were introduced and investigated by the author. In this paper we define weak type $(2,H)$ (resp. weak cotype $(2,H)$) of operator spaces, which lies strictly between type $(2,H)$ (resp. cotype $(2,H)$) and type $(p,H)$ for all $1\leq p <2$ (resp. cotype $(q,H)$ for all $2<q \leq \infty$). This is an analogue of weak type 2 and weak cotype 2 in the Banach space case, so we develop analogous equivalent formulations. We also consider weak-$H$ space, spaces with weak type $(2,H)$ and weak cotype $(2,H^*)$ simultaneously and establish corresponding equivalent formulations.
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arxiv:math/0502337
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In this paper, we define the first topological $(\sigma,\tau)$-cohomology group and examine vanishing of the first $(\sigma,\tau)$-cohomology groups of certain triangular Banach algebras. We apply our results to study the $(\sigma,\tau)$-weak amenability and $(\sigma,\tau)$-amenability of triangular Banach algebras.
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arxiv:math/0502346
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We study a family of polynomials whose values express degrees of Schubert varieties in the generalized complex flag manifold G/B. The polynomials are given by weighted sums over saturated chains in the Bruhat order. We derive several explicit formulas for these polynomials, and investigate their relations with Schubert polynomials, harmonic polynomials, Demazure characters, and generalized Littlewood-Richardson coefficients. In the second half of the paper, we concern with the case of to the classical flag manifold of Lie type A and discuss related combinatorial objects: flagged Schur polynomials, 312-avoiding permutations, generalized Gelfand-Tsetlin polytopes, the inverse Schubert-Kostka matrix, parking functions, and binary trees.
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arxiv:math/0502363
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We initiate a study of Hilbert modules over the polynomial algebra A=C[z_1,...,z_d] that are obtained by completing A with respect to an inner product having certain natural properties. A standard Hilbert module is a finite multiplicity version of one of these. Standard Hilbert modules occupy a position analogous to that of free modules of finite rank in commutative algebra, and their quotients by submodules give rise to universal solutions of nonlinear relations. Essentially all of the basic Hilbert modules that have received attention over the years are standard - including the Hilbert module of the d-shift, the Hardy and Bergman modules of the unit ball, modules associated with more general domains in complex d-space, and those associated with projective algebraic varieties. We address the general problem of determining when a quotient H/M of an essentially normal standard Hilbert module H is essentially normal. This problem has been resistant. Our main result is that it can be "linearized" in that the nonlinear relations defining the submodule M can be reduced, appropriately, to linear relations through an iteration procedure, and we give a concrete description of linearized quotients.
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arxiv:math/0502388
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We reformulate Heegaard Floer homology in terms of holomorphic curves in the cylindrical manifold Sigma x [0,1] x R, where Sigma is the Heegaard surface, instead of Sym^g(Sigma). We then show that the entire invariance proof can be carried out in our setting. In the process, we derive a new formula for the index of the dbar-operator in Heegaard Floer homology, and shorten several proofs. After proving invariance, we show that our construction is equivalent to the original construction of Ozsvath-Szabo. We conclude with a discussion of elaborations of Heegaard Floer homology suggested by our construction, as well as a brief discussion of the relation with a program of C Taubes.
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arxiv:math/0502404
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We consider the correspondence assigning to every Radon measure on two Tychonoff coordinate spaces the set of probability measures with these marginals. It is proved that this correspondence is continuous.
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arxiv:math/0502474
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In this paper we prove a new matrix Li-Yau-Hamilton estimate for K\"ahler-Ricci flow. The form of this new Li-Yau-Hamilton estimate is obtained by the interpolation consideration originated in \cite{Ch1}. This new inequality is shown to be connected with Perelman's entropy formula through a family of differential equalities. In the rest of the paper, We show several applications of this new estimate and its linear version proved earlier in \cite{CN}. These include a sharp heat kernel comparison theorem, generalizing the earlier result of Li and Tian, a manifold version of Stoll's theorem on the characterization of `algebraic divisors', and a localized monotonicity formula for analytic subvarieties. Motivated by the connection between the heat kernel estimate and the reduced volume monotonicity of Perelman, we prove a sharp lower bound heat kernel estimate for the time-dependent heat equation, which is, in a certain sense, dual to Perelman's monotonicity of the `reduced volume'. As an application of this new monotonicity formula, we show that the blow-down limit of a certain type III immortal solution is a gradient expanding soliton. In the last section we also illustrate the connection between the new Li-Yau-Hamilton estimate and the earlier Hessian comparison theorem on the `reduced distance', proved in \cite{FIN}.
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arxiv:math/0502495
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We consider the immediate consequence of an arguable addition to the standard Deduction Theorems of first order theories.
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arxiv:math/0502502
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We study the flow $M_t$ of a smooth, strictly convex hypersurface by its mean curvature in $\mathrm{R}^{n+1}$. The surface remains smooth and convex, shrinking monotonically until it disappears at a critical time $T$ and point $x^*$ (which is due to Huisken). This is equivalent to saying that the corresponding rescaled mean curvature flow converges to a sphere ${\bf S^n}$ of radius $\sqrt{n}$. In this paper we will study the rate of exponential convergence of a rescaled flow. We will present here a method that tells us the rate of the exponential decay is at least $\frac{2}{n}$. We can define the ''arrival time'' $u$ of a smooth, strictly convex $n$-dimensional hypersurface as it moves with normal velocity equal to its mean curvature as $u(x) = t$, if $x\in M_t$ for $x\in \Int(M_0)$. Huisken proved that for $n\ge 2$ $u(x)$ is $C^2$ near $x^*$. The case $n=1$ has been treated by Kohn and Serfaty, they proved $C^3$ regularity of $u$. As a consequence of obtained rate of convergence of the mean curvature flow we prove that $u$ is not $C^3$ near $x^*$ for $n\ge 2$. We also show that the obtained rate of convergence $2/n$, that comes out from linearizing a mean curvature flow is the optimal one, at least for $n\ge 2$.
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arxiv:math/0502530
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Let V be a simple vertex operator algebra satisfying the following conditions: (i) The homogeneous subspaces of V of weights less than 0 are 0, the homogeneous subspace of V of weight 0 is spanned by the vacuum and V' is isomorphic to V as a V-module. Every weak V-module gradable by nonnegative integers is completely reducible. (iii) V is C_2-cofinite. (In the presence of Condition (i), Conditions (ii) and (iii) are equivalent to a single condition, namely, that every weak V-module is completely reducible.) Using the results obtained by the author in the formulation and proof of the general version of the Verlinde conjecture and in the proof of the Verlinde formula, we prove that the braided tensor category structure on the category of V-modules is rigid, balanced and nondegenerate. In particular, the category of V-modules has a natural structure of modular tensor category. We also prove that the tensor-categorical dimension of an irreducible V-module is the reciprocal of a suitable matrix element of the fusing isomorphism under a suitable basis.
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arxiv:math/0502533
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We discuss the conjecture of Buchstaber and Krichever that their multi-dimensional vector addition formula for Baker-Akhiezer functions characterizes Jacobians among principally polarized abelian varieties, and prove that it is indeed a weak characterization, i.e. that it is true up to additional components, or true precisely under a general position assumption. We also show that this addition formula is equivalent to Gunning's multisecant formula for the Kummer variety. We then use Buchstaber-Krichever's computation of the coefficients in the addition formula to obtain cubic relations among theta functions, which (weakly) characterize the locus of hyperelliptic Jacobians among irreducible abelian varieties. In genus 3 our equations are equivalent to the vanishing of one theta-null, and thus are known classically by work of Mumford and Poor, but already for genus 4 they appear to be new.
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arxiv:math/0503026
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This essay, originally published in the Sept 1990 Notices of the AMS, discusses problems of our mathematical education system that often stem from widespread misconceptions by well-meaning people of the process of learning mathematics. The essay also discusses ideas for fixing some of the problems. Most of what I wrote in 1990 remains equally applicable today.
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arxiv:math/0503081
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We introduce a notion of the divisor type for rational functions and show that it can be effectively used for the classification of the deformations of dessins d'enfants related with the construction of the algebraic solutions of the sixth Painlev\'e equation via the method of $RS$-transformations.
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arxiv:math/0503082
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For the simple random walk in Z^2 we study those points which are visited an unusually large number of times, and provide a new proof of the Erdos-Taylor conjecture describing the number of visits to the most visited point.
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arxiv:math/0503108
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Finite hamiltonian groups are counted. The sequence of numbers of all groups of order $n$ all whose subgroups are normal and the sequence of numbers of all groups of order less or equal to $n$ all whose subgroups are normal are presented.
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arxiv:math/0503183
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We classify five dimensional Anosov flows with smooth decomposition which are in addition transversely symplectic. Up to finite covers and a special time change, we find exectly the suspensions of symplectic hyperbolic automorphisms of four dimensional toris, and the geodesic flows of three dimensional hyperbolic manifolds.
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arxiv:math/0503189
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We analyse q-functional equations arising from tree-like combinatorial structures, which are counted by size, internal path length, and certain generalisations thereof. The corresponding counting parameters are labelled by a positive integer k. We show the existence of a joint limit distribution for these parameters in the limit of infinite size, if the size generating function has a square root as dominant singularity. The limit distribution coincides with that of integrals of k-th powers of the standard Brownian excursion. Our approach yields a recursion for the moments of the limit distribution. It can be used to analyse asymptotic expansions of the moments, and it admits an extension to other types of singularity.
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arxiv:math/0503198
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Methods of construction of Max-semi-selfdecompsable laws are given. Implications of this method in random time changed extremal processes are discussed. Max-autoregressive model is introduced and characterized using the max-semi-selfdecompsable laws and exponential max-semi-stable laws. Max-semi-selfddecomposability of max-semi-stable laws are proved.
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arxiv:math/0503232
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Let $\phi(n)$ be the Euler-phi function, define $\phi_0(n) = n$ and $\phi_{k+1}(n)=\phi(\phi_{k}(n))$ for all $k\geq 0$. We will determine an asymptotic formula for the set of integers $n$ less than $x$ for which $\phi_k(n)$ is $y$-smooth, conditionally on a weak form of the Elliott-Halberstam conjecture.
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arxiv:math/0503246
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We introduce a class of (tuples of commuting) unbounded operators on a Banach space, admitting smooth functional calculi, that contains all operators of Helffer-Sj\"ostrand type and is closed under the action of smooth proper mappings. Moreover, the class is closed under tensor product of commuting operators. In general an operator in this class has no resolvent in the usual sense so the spectrum must be defined in terms of the functional calculus. We also consider invariant subspaces and spectral decompositions.
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arxiv:math/0503256
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We consider Galton-Watson trees associated with a critical offspring distribution and conditioned to have exactly $n$ vertices. These trees are embedded in the real line by affecting spatial positions to the vertices, in such a way that the increments of the spatial positions along edges of the tree are independent variables distributed according to a symmetric probability distribution on the real line. We then condition on the event that all spatial positions are nonnegative. Under suitable assumptions on the offspring distribution and the spatial displacements, we prove that these conditioned spatial trees converge as $n\to\infty$, modulo an appropriate rescaling, towards the conditioned Brownian tree that was studied in previous work. Applications are given to asymptotics for random quadrangulations.
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arxiv:math/0503263
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The problem of counting plane trees with $n$ edges and an even or an odd number of leaves was studied by Eu, Liu and Yeh, in connection with an identity on coloring nets due to Stanley. This identity was also obtained by Bonin, Shapiro and Simion in their study of Schr\"oder paths, and it was recently derived by Coker using the Lagrange inversion formula. An equivalent problem for partitions was independently studied by Klazar. We present three parity reversing involutions, one for unlabelled plane trees, the other for labelled plane trees and one for 2-Motzkin paths which are in one-to-one correspondence with Dyck paths.
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arxiv:math/0503300
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Tensor fields depending on other tensor fields are considered. The concept of extended tensor fields is introduced and the theory of differentiation for such fields is developed.
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arxiv:math/0503332
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We study the Galois groupoid of a holomorphic singular codimension one foliation. Geometric and algebraic caracterisations using Godbillon-Vey sequences and classical first integral are given.
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arxiv:math/0503348
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We proof a foliated version of the Poincare-Hopf theorem and other results which clarify the geometric and ergodic meaning of the Euler characteristic of a measured foliation.
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arxiv:math/0503357
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It is well known that a countable group admits a left-invariant total order if and only if it acts faithfully on R by orientation preserving homeomorphisms. Such group actions are special cases of group actions on simply connected 1-manifolds, or equivalently, actions on oriented order trees. We characterize a class of left-invariant partial orders on groups which yield such actions, and show conversely that groups acting on oriented order trees by order preserving homeomorphism admit such partial orders as long as there is an action with a point whose stabilizer is left-orderable.
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arxiv:math/0503407
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If a $C^{1 + a}$, $a >0$, volume-preserving diffeomorphism on a compact manifold has a hyperbolic invariant set with positive volume, then the map is Anosov. We also give a direct proof of ergodicity of volume-preserving $CC^{1+a}$, $a>0$, Anosov diffeomorphism without the usual Hopf argument.
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arxiv:math/0503437
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The problem of disorder seeks to determine a stopping time which is as close as possible to the unknown time of ``disorder'' when the observed process changes its probability characteristics. We give a partial answer to this question for some special cases of Levy processes and present a complete solution of the Bayesian and variational problem for a compound Poisson process with exponential jumps. The method of proof is based on reducing the Bayesian problem to an integro-differential free-boundary problem where, in some cases, the smooth-fit principle breaks down and is replaced by the principle of continuous fit.
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arxiv:math/0503481
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We prove that certain Riemannian manifolds can be isometrically embedded inside Calabi-Yau manifolds. For example we prove that given any real-analytic one parameter family of Riemannian metrics $g_t$ on a 3-dimensional manifold $Y$ with volume form independent of $t$ and with a real-analytic family of nowhere vanishing harmonic one forms $\theta_t$, then $(Y, g_t)$ can be realized as a family of special Lagrangian submanifolds of a Calabi-Yau manifold $X$. We also prove that certain principal torus bundles can be equivariantly and isometrically embedded inside Calabi-Yau manifolds with torus action. We use this to construct examples of $n$-parameter families of special Lagrangian tori inside $n+k$-dimensional Calabi-Yau manifolds with torus symmetry. We also compute McLean's metric of 3-dimensional special Lagrangian fibrations with $T^2$-symmetry.
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arxiv:math/0503494
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We study the transport of a passive tracer particle in a steady strongly mixing flow with a nonzero mean velocity. We show that there exists a probability measure under which the particle Lagrangian velocity process is stationary. This measure is absolutely continuous with respect to the underlying probability measure for the Eulerian flow.
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arxiv:math/0503534
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We formulate the insurance risk process in a general Levy process setting, and give general theorems for the ruin probability and the asymptotic distribution of the overshoot of the process above a high level, when the process drifts to -\infty a.s. and the positive tail of the Levy measure, or of the ladder height measure, is subexponential or, more generally, convolution equivalent. Results of Asmussen and Kluppelberg [Stochastic Process. Appl. 64 (1996) 103-125] and Bertoin and Doney [Adv. in Appl. Probab. 28 (1996) 207-226] for ruin probabilities and the overshoot in random walk and compound Poisson models are shown to have analogues in the general setup. The identities we derive open the way to further investigation of general renewal-type properties of Levy processes.
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arxiv:math/0503539
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Originally a technical tool, the derived category of coherent sheaves over an algebraic variety has become over the last twenty years an important invariant in the birational study of algebraic varieties. Problems of birational invariance and of minimization of the derived category have appeared, inspired by Kontsevich's homological mirror symmetry conjecture and Mori's minimal model program. We present the main conjectures and their proofs in dimension 3 and for particular classes of flops.
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arxiv:math/0503548
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We study spectrum inclusion regions for complex Jacobi matrices which are compact perturbations of real periodic Jacobi matrices. The condition sufficient for the lack of discrete spectrum for such matrices is given
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arxiv:math/0503627
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We extend in two directions our previous results about the sampling and the empirical measures of immortal branching Markov processes. Direct applications to molecular biology are rigorous estimates of the mutation rates of polymerase chain reactions from uniform samples of the population after the reaction. First, we consider nonhomogeneous processes, which are more adapted to real reactions. Second, recalling that the first moment estimator is analytically known only in the infinite population limit, we provide rigorous confidence intervals for this estimator that are valid for any finite population. Our bounds are explicit, nonasymptotic and valid for a wide class of nonhomogeneous branching Markov processes that we describe in detail. In the setting of polymerase chain reactions, our results imply that enlarging the size of the sample becomes useless for surprisingly small sizes. Establishing confidence intervals requires precise estimates of the second moment of random samples. The proof of these estimates is more involved than the proofs that allowed us, in a previous paper, to deal with the first moment. On the other hand, our method uses various, seemingly new, monotonicity properties of the harmonic moments of sums of exchangeable random variables.
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arxiv:math/0503659
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Every normal complex surface singularity with $\mathbb Q$-homology sphere link has a universal abelian cover. It has been conjectured by Neumann and Wahl that the universal abelian cover of a rational or minimally elliptic singularity is a complete intersection singularity defined by a system of ``splice diagram equations''. In this paper we introduce a Neumann-Wahl system, which is an analogue of the system of splice diagram equations, and prove the following. If $(X,o)$ is a rational or minimally elliptic singularity, then its universal abelian cover $(Y,o)$ is an equisingular deformation of an isolated complete intersection singularity $(Y_0,o)$ defined by a Neumann-Wahl system. Furthermore, if $G$ denotes the Galois group of the covering $Y \to X$, then $G$ also acts on $Y_0$ and $X$ is an equisingular deformation of the quotient $Y_0/G$.
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arxiv:math/0503733
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We study Picard's exceptional values of holomorphic one-parametric families of entire functions. Our first result shows that the set of parameter values for which zero is a Picard value can be an arbitrary closed set of zero logarithmic capacity. This answers a question of Julia. Second, we show that if a function has a Picard exceptional value for all values of parameter in some region of the plane then this Picard value is a holomorphic function in the complement of some discrete set E, tending to infinity as the papameter tends to E.
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arxiv:math/0503750
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Let $\Omega$ be a bounded symmetric domain except the two exceptional domains of ${\Bbb C}^N$ and $\phi$ a holomorphic self-map of $\Omega.$ This paper gives a sufficient and necessary condition for the composition operator $C_{\phi}$ induced by $\phi$ to be compact on the Bloch space $\beta(\Omega)$.
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arxiv:math/0504006
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We describe a notion of categorical model for unitless fragments of (multiplicative) linear logic. The basic definition uses promonoidal categories, and we also give an equivalent elementary axiomatisation.
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arxiv:math/0504037
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In this paper we define an intrinsic notion of principal for the Hypoelliptic calculus on Heisenberg manifolds. More precisely, the principal symbol of a \psivdo appears as a homogeneous section over the linear dual of the tangent Lie algebra bundle of the manifold. This definition is an important step towards using global $K$-theoretic tools in the Heisenberg setting, such as those involved in the elliptic setting for proving the Atiyah-Singer index theorem or the regularity of the eta invariant. On the other hand, the intrinsic definition of the principal symbol enables us to give an intrinsic sense to the model operator of \psivdo at point and to give a definitive proof that the Heisenberg calculus is modelled at each point by the calculus of left-invariant \psidos on the tangent group at the point. This also allows us to define an intrinsic Rockland condition for \psivdos which is shown to be equivalent to the invertibility of the principal symbol, provided that the Levi form has constant rank. Furthermore, we review the main hypoellipticity results on Heisenberg manifolds in terms of the results of the paper. In particular, we complete the treatment of the Kohn Laplacian by Beals-Greiner and establish that for the horizontal sublaplacian the invertibility of the principal symbol is equivalent to some condition on the Levi form, called condition $X(k)$. Incidentally, this paper provides us with a relatively up-to-date overview of the main facts about the Heisenberg calculus.
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arxiv:math/0504048
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In the studies on the modularity conjecture for rigid Calabi-Yau threefolds several examples with the unique level 8 cusp form were constructed. According to the Tate Conjecture correspondences inducing isomorphisms on the middle cohomologies should exist between these varieties. In the paper we construct several examples of such correspondences. In the constructions elliptic fibrations play a crucial role. In fact we show that all but three examples are in some sense built upon two modular curves from the Beauville list.
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arxiv:math/0504070
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Motivated by the Strominger-Yau-Zaslow conjecture, we study fibre spaces whose total space has trivial canonical bundle. Especially, we are interest in Calabi-Yau varieties with fibre structure. In this paper, we only consider semi-stable families. We use Hodge theory and the generalized Donaldson-Simpson-Uhlenbeck-Yau correspondence to study the parabolic structure of higher direct images over higher dimensional quasi-projective base, and obtain an important result on parabolic-semi-positivity. We then apply this result to study nonisotrivial Calabi-Yau varieties fibred by Abelian varieties (or fibred by hyperk\"ahler varieties), we obtain that the base manifold for such a family is rationally connected and the dimension of a general fibre depends only on the base manifold.
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arxiv:math/0504141
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Here we prove that a commuting variety associated with a symmetric pair (g, g_0) is irreducible for (so_{n+m}, so_n + so_m) and reducible for (gl_{n+m}, {gl}_n + gl_m) with n>m, (so_{2n}, gl_n) with odd n, (E_6, {so}_{10} + k).
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arxiv:math/0504145
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We consider three directed walkers on the square lattice, which move simultaneously at each tick of a clock and never cross. Their trajectories form a non-crossing configuration of walks. This configuration is said to be osculating if the walkers never share an edge, and vicious (or: non-intersecting) if they never meet. We give a closed form expression for the generating function of osculating configurations starting from prescribed points. This generating function turns out to be algebraic. We also relate the enumeration of osculating configurations with prescribed starting and ending points to the (better understood) enumeration of non-intersecting configurations. Our method is based on a step by step decomposition of osculating configurations, and on the solution of the functional equation provided by this decomposition.
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arxiv:math/0504153
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We study Veech surfaces of genus 2 arising from quadratic differentials that are not squares of abelian differentials. We prove that all such surfaces of type (2,2) and (2,1,1) are arithmetic. In (1,1,1,1) case, we reduce the question to abelian differentials of type (2,2) on hyperelliptic genus 3 surfaces with singularities at Weierstrass points, and we give an example of a non-arithmetic Veech surface.
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arxiv:math/0504180
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We discuss the procedure of Rieffel induction of representations in the framework of formal deformation quantization of Poisson manifolds. We focus on the central role played by algebraic notions of complete positivity.
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arxiv:math/0504235
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Given a homomorphism of commutative noetherian rings R --> S and an S-module N, it is proved that the Gorenstein flat dimension of N over R, when finite, may be computed locally over S. When, in addition, the homomorphism is local and N is finitely generated over S, the Gorenstein flat dimension equals sup{m | Tor^R_m(E,N) \noteq 0} where E is the injective hull of the residue field of R. This result is analogous to a theorem of Andr\'e on flat dimension.
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arxiv:math/0504340
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