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We prove that there exists at least one close orbit in a given contact hypersurface in some symplectic manifolds.
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arxiv:math/0407072
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The detour between two points u and v (on edges or vertices) of an embedded planar graph whose edges are curves is the ratio between the shortest path in in the graph between u and v and their Euclidean distance. The maximum detour over all pairs of points is called the geometric dilation. Ebbers-Baumann, Gruene and Klein have shown that every finite point set is contained in a planar graph whose geometric dilation is at most 1.678, and some point sets require graphs with dilation at least pi/2 = 1.57... We prove a stronger lower bound of 1.00000000001*pi/2 by relating graphs with small dilation to a problem of packing and covering the plane by circular disks. The proof relies on halving pairs, pairs of points dividing a given closed curve C in two parts of equal length, and their minimum and maximum distances h and H. Additionally, we analyze curves of constant halving distance (h=H), examine the relation of h to other geometric quantities and prove some new dilation bounds.
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arxiv:math/0407135
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We continue the study of free Baxter algebras. There are two goals of this paper. The first goal is to extend the construction of shuffle Baxter algebras to completions of Baxter algebras. This process is motivated by a construction of Cartier and is analogous to the process of completing a polynomial algebra to obtain a power series algebra. However, as we will see later, unlike the close similarity of properties of a polynomial algebra and a power series algebra, properties of a shuffle Baxter algebra and its completion can be quite different.
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arxiv:math/0407156
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We study an extensive connection between factor forcings of Borel subsets of Polish spaces modulo a sigma-ideal, and factor forcings of subsets of countable sets modulo an ideal.
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arxiv:math/0407182
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We show that proper Lie groupoids are locally linearizable. As a consequence, the orbit space of a proper Lie groupoid is a smooth orbispace (a Hausdorff space which locally looks like the quotient of a vector space by a linear compact Lie group action). In the case of proper (quasi-)symplectic groupoids, the orbit space admits a natural integral affine structure, which makes it into an affine orbifold with locally convex polyhedral boundary, and the local structure near each boundary point is isomorphic to that of a Weyl chamber of a compact Lie group. We then apply these results to the study of momentum maps of Hamiltonian actions of proper (quasi-)symplectic groupoids, and show that these momentum maps preserve natural transverse affine structures with local convexity properties. Many convexity theorems in the literature can be recovered from this last statement and some elementary results about affine maps.
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arxiv:math/0407208
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We define the longitude Floer homology of a knot K in S^3 and show that it is a topological invariant of K. Some basic properties of these homology groups are derived. In particular, we show that they distinguish the genus of K. We also make explicit computations for the (2,2n+1) torus knots. Finally a correspondence between the longitude Floer homology of K and the Ozsvath-Szabo Floer homology of its Whitehead double K_L is obtained.
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arxiv:math/0407211
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In this paper we study the smallest Mealy automaton of intermediate growth, first considered by the last two authors. We describe the automatic transformation monoid it defines, give a formula for the generating series for its (ball volume) growth function, and give sharp asymptotics for its growth function, namely [ F(n) \sim 2^{5/2} 3^{3/4} \pi^{-2} n^{1/4} \exp{\pi\sqrt{n/6}} ] with the ratios of left- to right-hand side tending to 1 as $n \to \infty$.
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arxiv:math/0407312
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This article has been withdrown by the author.
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arxiv:math/0407332
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We study the relationship between general dynamical Poisson groupoids and Lie quasi-bialgebras. For a class of Lie quasi-bialgebras naturally compatible with a reductive decomposition, we extend the description of the moduli space of classical dynamical r-matrices of Etingof and Schiffmann. We construct, in each gauge orbit, an explicit analytic representative l. We translate the notion of duality for dynamical Poisson groupoids into a duality for Lie quasi-bialgebras. It is shown that duality maps the dynamical Poisson groupoid for l and a Lie quasi-bialgebra to the dynamical Poisson groupoid for the dual data.
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arxiv:math/0407382
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This is an addendum to the paper ``Deformation of $L_\infty$-Algebras'' of the same author. We explain in which way the deformation theory of $L_\infty$-algebras extends the deformation theory of singularities. We show that the construction of semi-universal deformations of $L_\infty$-algebras gives explicit formal semi-universal deformations of isolated singularities.
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arxiv:math/0407390
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Let N_g(d) be the set of primes p such that the order of g modulo p is divisible by a prescribed integer d. Wiertelak showed that this set has a natural density and gave a rather involved explicit expression for it. Let N_g(d)(x) be the number of primes p<=x that are in N_g(d). A simple identity for N_g(d)(x) is established. It is used to derive a more compact expression for the natural density than known hitherto. A numerical demonstration, using a program of Y. Gallot, is presented.
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arxiv:math/0407421
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Kobayashi-Ochiai's theorem says us that the set of dominant rational maps to a complex variety of general type is finite. In this paper, we give a generalization of it in the category of log schemes.
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arxiv:math/0407478
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An ODE variational calculation shows that an image principle curvature ratio factor can raise the lower bound, 2(Image Area), on energy of a harmonic map of a surface into Rn. In certain situations, including all radially symmetry harmonic maps, equality is achieved.
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arxiv:math/0407499
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We construct calibrated submanifolds of R^7 and R^8 by viewing them as total spaces of vector bundles and taking appropriate sub-bundles which are naturally defined using certain surfaces in R^4. We construct examples of associative and coassociative submanifolds of R^7 and of Cayley submanifolds of R^8. This construction is a generalization of the Harvey-Lawson bundle construction of special Lagrangian submanifolds of R^{2n}.
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arxiv:math/0408005
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Inspired by a construction by Arnaud Beauville of a surface of general type with $K^2 = 8, p_g =0$, the second author defined the Beauville surfaces as the surfaces which are rigid, i.e., they have no nontrivial deformation, and admit un unramified covering which is isomorphic to a product of curves of genus at least 2. In this case the moduli space of surfaces homeomorphic to the given surface consists either of a unique real point, or of a pair of complex conjugate points corresponding to complex conjugate surfaces. It may also happen that a Beauville surface is biholomorphic to its complex conjugate surface, neverless it fails to admit a real structure. First aim of this note is to provide series of concrete examples of the second situation, respectively of the third. Second aim is to introduce a wider audience, especially group theorists, to the problem of classification of such surfaces, especially with regard to the problem of existence of real structures on them.
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arxiv:math/0408025
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In this paper we study the Navier-Stokes equations with a Navier-type boundary condition that has been proposed as an alternative to common near wall models. The boundary condition we study, involving a linear relation between the tangential part of the velocity and the tangential part of the Cauchy stress-vector, is related to the vorticity seeding model introduced in the computational approach to turbulent flows. The presence of a point-wise non vanishing normal flux may be considered as a tool to avoid the use of phenomenological near wall models, in the boundary layer region. Furthermore, the analysis of the problem is suggested by recent advances in the study of Large Eddy Simulation. In the two dimensional case we prove existence and uniqueness of weak solutions, by using rather elementary tools, hopefully understandable also by applied people working on turbulent flows. The asymptotic behaviour of the solution, with respect to the averaging radius $\delta,$ is also studied. In particular, we prove convergence of the solutions toward the corresponding solutions of the Navier-Stokes equations with the usual no-slip boundary conditions, as the small parameter $\delta$ goes to zero.
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arxiv:math/0408032
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The multifractal spectrum of a Borel measure $\mu$ in $\mathbb{R}^n$ is defined as \[ f_\mu(\alpha) = \dim_H {x:\lim_{r\to 0} \frac{\log \mu(B(x,r))}{\log r}=\alpha}. \] For self-similar measures under the open set condition the behavior of this and related functions is well-understood; the situation turns out to be very regular and is governed by the so-called ''multifractal formalism''. Recently there has been a lot of interest in understanding how much of the theory carries over to the overlapping case; however, much less is known in this case and what is known makes it clear that more complicated phenomena are possible. Here we carry out a complete study of the multifractal structure for a class of self-similar measures with overlap which includes the 3-fold convolution of the Cantor measure. Among other things, we prove that the multifractal formalism fails for many of these measures, but it holds when taking a suitable restriction.
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arxiv:math/0408047
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Borg-type uniqueness theorems for matrix-valued Jacobi operators H and supersymmetric Dirac difference operators D are proved. More precisely, assuming reflectionless matrix coefficients A, B in the self-adjoint Jacobi operator H=AS^+ + A^-S^- + B (with S^\pm the right/left shift operators on the lattice Z) and the spectrum of H to be a compact interval [E_-,E_+], E_- < E_+, we prove that A and B are certain multiples of the identity matrix. An analogous result which, however, displays a certain novel nonuniqueness feature, is proved for supersymmetric self-adjoint Dirac difference operators D with spectrum given by [-E_+^{1/2},-E_-^{1/2}] \cup [E_-^{1/2},E_+^{1/2}], 0 \leq E_- < E_+.
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arxiv:math/0408074
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The meridian maps of the full Homfly skein of the annulus are linear endomorphisms induced by the insertion of a meridian loop, with either orientation, around a diagram in the annulus. The eigenvalues of the meridian maps are known to be distinct, and are indexed by pairs of partitions of integers p and n into k and k* parts respectively. We give here an explicit formula for a corresponding eigenvector as the determinant of a (k*+k)x(k*+k) matrix whose entries are skein elements corresponding to partitions with a single part. This extends the results of Kawagoe and Lukac, for the case p=0, giving a basis for the subspace of the skein spanned by closed braids all oriented in the same direction. Their formula uses the Jacobi-Trudy determinants for Schur functions in terms of complete symmetric functions. Our matrices have a similar pattern of entries in k rows, and a modified pattern in k* rows, resulting in a combination of closed braids with up to n strings oriented in one direction and p in the reverse direction. We discuss the 2-variable knot invariants resulting from decoration of a knot by these skein elements, and their relation to unitary quantum invariants.
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arxiv:math/0408078
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The complexity of an action of a reductive algebraic group G on an algebraic variety X is the codimension of a generic B-orbit in X, where B is a Borel subgroup of G. We classify affine homogeneous spaces G/H of complexity one. These results are the natural continuation of the classification of spherical affine homogeneous spaces, i.e., spaces of complexity zero.
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arxiv:math/0408101
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Let R=K[M] be a normal affine monoid algbera over a field K.Up to isomorphism the conic ideals are exactly the direct summands ofthe extension R^{1/n} of R. We show that the classes of the conic divisorial ideals can be identified with the full-dimensional open cells in a decomposition of a torus naturally associated with M. Furthermore, they can be characterized by the relative compactness of a certain group associated with them. Baetica has given examples of Cohen-Macaulay divisorial ideals that are not conic. Wereview his construction and streamline the arguments somewhat. In the last part of the paper we investigate the multiplicities ofthe conic classes in the decomposition of R^{1/n} as a function of n.This multiplicity turns out to be a quasi-polynomial for all n >= 1 counting the lattice points in the union of the interiors of certain cells of the complex mentioned above. This argument can be used for the computation of the Hilbert-Kunz multiplicity of R in characteristic p > 0. In addition it yields some assertions about the Hilbert-Kunz function of R.
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arxiv:math/0408110
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We derive an explicit zero-free region for symmetric square L-functions of elliptic curves, and use this to derive an explicit lower bound for the modular degree of rational elliptic curves. The techniques are similar to those used in the classical derivation of zero-free regions for Dirichlet L-functions, but here, due to the work of Goldfield-Hoffstein-Lieman, we know that there are no Siegel zeros, which leads to a strengthened result.
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arxiv:math/0408126
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We give an informal introduction to the most basic techniques used to evaluate moments on the critical line of the Riemann zeta-function and to find asymptotics for sums of arithmetic functions.
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arxiv:math/0408154
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It will be shown that if $\phi$ is a quasiperiodic flow on the $n$-torus that is algebraic, if $\psi$ is a flow on the $n$-torus that is smoothly conjugate to a flow generated by a constant vector field, and if $\phi$ is smoothly semiconjugate to $\psi$, then $\psi$ is a quasiperiodic flow that is algebraic, and the multiplier group of $\psi$ is a finite index subgroup of the multiplier group of $\phi$. This will partially establish a conjecture that asserts that a quasiperiodic flow on the $n$-torus is algebraic if and only if its multiplier group is a finite index subgroup of the group of units of the ring of integers in a real algebraic number field of degree $n$.
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arxiv:math/0408158
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In order to apply nonstandard methods to modern algebraic geometry, as a first step in this paper we study the applications of nonstandard constructions to category theory. It turns out that many categorial properties are well behaved under enlargements.
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arxiv:math/0408177
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We provide an example of a zero-dimensional compact metric space $X$ and its closed subspace $A$ such that there is no continuous linear extension operator for the Lipschitz pseudometrics on $A$ to the Lipschitz pseudometrics on $X$. The construction is based on results of A. Brudnyi and Yu. Brudnyi concerning linear extension operators for Lipschitz functions.
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arxiv:math/0408200
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A sequence $S$ is potentially $K_{p,1,1}$ graphical if it has a realization containing a $K_{p,1,1}$ as a subgraph, where $K_{p,1,1}$ is a complete 3-partite graph with partition sizes $p,1,1$. Let $\sigma(K_{p,1,1}, n)$ denote the smallest degree sum such that every $n$-term graphical sequence $S$ with $\sigma(S)\geq \sigma(K_{p,1,1}, n)$ is potentially $K_{p,1,1}$ graphical. In this paper, we prove that $\sigma (K_{p,1,1}, n)\geq 2[((p+1)(n-1)+2)/2]$ for $n \geq p+2.$ We conjecture that equality holds for $n \geq 2p+4.$ We prove that this conjecture is true for $p=3$.
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arxiv:math/0408292
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We study the non-archimedean counterpart to the complex amoeba of an algebraic variety, and show that it coincides with a polyhedral set defined by Bieri and Groves using valuations. For hypersurfaces this set is also the tropical variety of the defining polynomial. Using non-archimedean analysis and a recent result of Conrad we prove that the amoeba of an irreducible variety is connected. We introduce the notion of an adelic amoeba for varieties over global fields, and establish a form of the local-global principle for them. This principle is used to explain the calculation of the nonexpansive set for a related dynamical system.
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arxiv:math/0408311
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We consider the stochastic Burgers equation $ \dnachd{t} \psi(t,r) = \Delta \psi(t,r) + \nabla \psi^2(t,r)+\sqrt{\gamma\psi(t,r)} \eta(t,r) $ with periodic boundary conditions, where $t \ge 0,$ $r \in [0,1],$ and $\eta$ is some space-time white noise. A certain Markov jump process is constructed to approximate a solution of this equation.}
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arxiv:math/0408323
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We show that there exist non-trivial piecewise-linear (PL) knots with isolated singularities $S^{n-2}\subset S^n$, $n\geq 5$, whose complements have the homotopy type of a circle. This is in contrast to the case of smooth, PL locally-flat, and topological locally-flat knots, for which it is known that if the complement has the homotopy type of a circle, then the knot is trivial.
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arxiv:math/0408325
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We study a system of $N$ interacting particles on $\bf{Z}$. The stochastic dynamics consists of two components: a free motion of each particle (independent random walks) and a pair-wise interaction between particles. The interaction belongs to the class of mean-field interactions and models a rollback synchronization in asynchronous networks of processors for a distributed simulation. First of all we study an empirical measure generated by the particle configuration on $\bf{R}$. We prove that if space, time and a parameter of the interaction are appropriately scaled (hydrodynamical scale), then the empirical measure converges weakly to a deterministic limit as $N$ goes to infinity. The limit process is defined as a weak solution of some partial differential equation. We also study the long time evolution of the particle system with fixed number of particles. The Markov chain formed by individual positions of the particles is not ergodic. Nevertheless it is possible to introduce relative coordinates and to prove that the new Markov chain is ergodic while the system as a whole moves with an asymptotically constant mean speed which differs from the mean drift of the free particle motion.
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arxiv:math/0408372
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We determine what isogeny classes of supersingular abelian surfaces over a finite field k of characteristic 2 contain jacobians. We deal with this problem in a direct way by computing explicitly the zeta function of all supersingular curves of genus 2. Our procedure is constructive, so that we are able to exhibit curves with prescribed zeta function and to count the number of curves, up to k-isomorphism, leading to the same zeta function.
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arxiv:math/0408383
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Recently, Lauritzen, Raben-Pedersen and Thomsen proved that Schubert varieties are globally $F$-regular. We give another proof.
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arxiv:math/0409007
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We generalise Wigner's theorem to its most general form possible for B(h) in the sense of completely characterising those vector state transformations of B(h) that appear as restrictions of duals of linear operators on B(h). We then use this result to similarly characterise the pure state transformations of general C*-algebras that appear as restrictions of duals of linear operators on the underlying algebras. This result may be interpreted as a noncommutative Banach-Stone theorem.
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arxiv:math/0409011
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We generalize and reprove an identity of Parker and Loday. It states that certain pairs of generating series associated to pairs of labelled rooted planar trees are mutually inverse under composition.
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arxiv:math/0409050
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We investigate the defining ideal of a set of points X in multi-projective space with a special emphasis on the case that X is in generic position, that is, X has the maximal Hilbert function. When X is in generic position, we determine the degrees of the generators of the associated ideal I_X. Letting \nu(I_X) denote the minimal number of generators of I_X, we use this description of the degrees to construct a function v(s;n_1,...,n_k) with the property that \nu(\Ix) >= v(s;n_1,...,n_k) always holds for s points in generic position in P^{n_1} x ... x P^{n_k}. When k=1, v(s;n_1) equals the expected value for \nu(I_X) as predicted by the Ideal Generation Conjecture. If k >= 2, we show that there are cases with \nu(\Ix) > v(s;n_1,...,n_k). However, computational evidence suggests that in many cases \nu(\Ix) = v(s;n_1,...,n_k).
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arxiv:math/0409056
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We explore some properties of Lyapunov exponents of measures preserved by smooth maps of the interval, and study the behaviour of the Lyapunov exponents under topological conjugacy.
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arxiv:math/0409084
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As an extension of the class of algebraic links, A'Campo, Gibson, and Ishikawa constructed links associated to immersed arcs and trees in a two-dimensional disk. By extending their arguments, we construct links associated to immersed graphs in a disk, and show that such links are quasipositive.
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arxiv:math/0409086
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Let F_2 be the free group generated by x and y. In this article, we prove that the commutator of x^m and y^n is a product of two squares if and only if mn is even. We also show using topological methods that there are infinitely many obstructions for an element in F_2 to be a product of two squares.
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arxiv:math/0409087
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We study nonuniform lattices in the automorphism group G of a locally finite simplicial tree X. In particular, we are interested in classifying lattices up to commensurability in G. We introduce two new commensurability invariants: quotient growth, which measures the growth of the noncompact quotient of the lattice; and stabilizer growth, which measures the growth of the orders of finite stabilizers in a fundamental domain as a function of distance from a fixed basepoint. When X is the biregular tree X_{m,n}, we construct lattices realizing all triples of covolume, quotient growth, and stabilizer growth satisfying some mild conditions. In particular, for each positive real number \nu we construct uncountably many noncommensurable lattices with covolume \nu.
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arxiv:math/0409094
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We construct a map between Bloch's higher Chow groups and Deligne homology for smooth, complex quasiprojective varieties on the level of complexes. For complex projective varieties this results in a formula which generalizes at the same time the classical Griffiths Abel-Jacobi map and the Borel/Beilinson/Goncharov regulator type maps.
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arxiv:math/0409116
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The grand challenges in biology today are being shaped by powerful high-throughput technologies that have revealed the genomes of many organisms, global expression patterns of genes and detailed information about variation within populations. We are therefore able to ask, for the first time, fundamental questions about the evolution of genomes, the structure of genes and their regulation, and the connections between genotypes and phenotypes of individuals. The answers to these questions are all predicated on progress in a variety of computational, statistical, and mathematical fields. The rapid growth in the characterization of genomes has led to the advancement of a new discipline called Phylogenomics. This discipline results from the combination of two major fields in the life sciences: Genomics, i.e., the study of the function and structure of genes and genomes; and Molecular Phylogenetics, i.e., the study of the hierarchical evolutionary relationships among organisms and their genomes. The objective of this article is to offer mathematicians a first introduction to this emerging field, and to discuss specific mathematical problems and developments arising from phylogenomics.
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arxiv:math/0409132
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A binary code with covering radius $R$ is a subset $C$ of the hypercube $Q_n=\{0,1\}^n$ such that every $x\in Q_n$ is within Hamming distance $R$ of some codeword $c\in C$, where $R$ is as small as possible. For a fixed coordinate $i\in[n]$, define $C(b,i)$, for $b=0,1$, to be the set of codewords with a $b$ in the $i$th position. Then $C$ is normal if there exists an $i\in[n]$ such that for any $v\in Q_n$, the sum of the Hamming distances from $v$ to $C(0,i)$ and $C(1,i)$ is at most $2R+1$. We newly define what it means for an asymmetric covering code to be normal, and consider the worst case asymptotic densities $\nu^*(R)$ and $\nu^*_+(R)$ of constant radius $R$ symmetric and asymmetric normal covering codes, respectively. Using a probabilistic deletion method, and analysis adapted from previous work by Krivelevich, Sudakov, and Vu, we show that both are bounded above by $e(R\log R + \log R + \log\log R+4)$, giving evidence that minimum size constant radius covering codes could still be normal.
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arxiv:math/0409171
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In Bhatt and Roy's minimal directed spanning tree (MDST) construction for a random partially ordered set of points in the unit square,all edges must respect the ``coordinatewise'' partial order and there must be a directed path from each vertex to a minimal element. We study the asymptotic behaviour of the total length of this graph with power weighted edges. The limiting distribution is given by the sum of a normal component away from the boundary and a contribution introduced by the boundary effects, which can be characterized by a fixed point equation, and is reminiscent of limits arising in the probabilistic analysis of certain algorithms. As the exponent of the power weighting increases, the distribution undergoes a phase transition from the normal contribution being dominant to the boundary effects dominating. In the critical case where the weight is simple Euclidean length, both effects contribute significantly to the limit law. We also give a law of large numbers for the total weight of the graph.
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arxiv:math/0409201
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We present an algorithm based on maximum likelihood for the estimation and renormalization (marginalization) of exponential densities. The moment-matching problem resulting from the maximization of the likelihood is solved as an optimization problem using the Levenberg-Marquardt algorithm. In the case of renormalization, the moments needed to set up the moment-matching problem are evaluated using Swendsen's renormalization method. We focus on the renormalization version of the algorithm, where we demonstrate its use by computing the critical temperature of the two-dimensional Ising model. Possible applications of the algorithm are discussed.
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arxiv:math/0409230
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We present a new link invariant which depends on a representation of the link group in SO(3). The computer calculations indicate that an abelian version of this invariant is expressed in terms of the Alexander polynomial of the link. On the other hand, if we use non abelian representation, we get the squared non abelian Reidemeister torsion (at least for some torus knots).
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arxiv:math/0409241
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We extend Forester's rigidity theorem so as to give a complete characterization of rigid group actions on trees (an action is rigid if it is the only reduced action in its deformation space, in particular it is invariant under automorphisms preserving the set of elliptic subgroups).
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arxiv:math/0409245
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Let G be compact Lie group. It is shown that the cotangent bundle of the complexification of G admits a hyperkahler structure which is invariant under left and right translations by elements of G. The proof is to realize the cotangent bundle of the complex group as a moduli space of solutions to Nahm's equations on the closed interval.
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arxiv:math/0409253
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We make a first step towards a classification of simple generalized Harish-Chandra modules which are not Harish-Chandra modules or weight modules of finite type. For an arbitrary algebraic reductive pair of complex Lie algebras $(\g,\k)$, we construct, via cohomological induction, the fundamental series $F^\cdot (\p,E)$ of generalized Harish-Chandra modules. We then use $F^\cdot (\p,E)$ to characterize any simple generalized Harish-Chandra module with generic minimal $\k$-type. More precisely, we prove that any such simple $(\g,\k)$-module of finite type arises as the unique simple submodule of an appropriate fundamental series module $F^s(\p,E)$ in the middle dimension $s$. Under the stronger assumption that $\k$ contains a semisimple regular element of $\g$, we prove that any simple $(\g,\k)$-module with generic minimal $\k$-type is necessarily of finite type, and hence obtain a reconstruction theorem for a class of simple $(\g,\k)$-modules which can a priori have infinite type. We also obtain generic general versions of some classical theorems of Harish-Chandra, such as the Harish-Chandra admissibility theorem. The paper is concluded by examples, in particular we compute the genericity condition on a $\k$-type for any pair $(\g,\k)$ with $\k\simeq s\ell (2)$.
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arxiv:math/0409285
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In this paper, we give a simple combinatorial explanation of a formula of A. Postnikov relating bicolored rooted trees to bicolored binary trees. We also present generalized formulas for the number of labeled k-ary trees, rooted labeled trees, and labeled plane trees.
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arxiv:math/0409323
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Cubature formulas, asymptotically optimal with respect to accuracy, are derived for calculating multidimensional weakly singular integrals. They are used for developing a universal code for calculating capacitances of conductors of arbitrary shapes.
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arxiv:math/0409324
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We prove a dispersive estimate for the time-independent Schrodinger operator H = -\Delta + V in three dimensions. The potential V(x) is assumed to lie in the intersection L^p(R^3) \cap L^q(R^3), p < 3/2 < q, and also to satisfy a generic zero-energy spectral condition. This class, which includes potentials that have pointwise decay |V(x)| < C(1+|x|)^{-2-\epsilon}, is nearly critical with respect to the natural scaling of the Laplacian. No additional regularity, decay, or positivity of V is assumed.
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arxiv:math/0409327
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We use a spanning tree model to prove a result of E. S. Lee on the support of Khovanov homology of alternating knots.
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arxiv:math/0409328
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We give an explicit combinatorial description of the multiplicity as well as the Hilbert function of the tangent cone at any point on a Schubert variety in the symplectic Grassmannian.
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arxiv:math/0409338
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We determine the Postnikov Tower and Postnikov Invariants of a Crossed Complex in a purely algebraic way. Using the fact that Crossed Complexes are homotopy types for filtered spaces, we use the above "algebraically defined" Postnikov Tower and Postnikov Invariants to obtain from them those of filtered spaces. We argue that a similar "purely algebraic" approach to Postnikov Invariants may also be used in other categories of spaces.
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arxiv:math/0409339
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We prove that any non-cocompact irreducible lattice in a higher rank semi-simple Lie group contains a subgroup of finite index, which has three generators.
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arxiv:math/0409345
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Lie bialgebras occur as the principal objects in the infinitesimalisation of the theory of quantum groups - the semi-classical theory. Their relationship with the quantum theory has made available some new tools that we can apply to classical questions. In this paper, we study the simple complex Lie algebras using the double-bosonisation construction of Majid. This construction expresses algebraically the induction process given by adding and removing nodes in Dynkin diagrams, which we call Lie induction. We first analyse the deletion of nodes, corresponding to the restriction of adjoint representations to subalgebras. This uses a natural grading associated to each node. We give explicit calculations of the module and algebra structures in the case of the deletion of a single node from the Dynkin diagram for a simple Lie (bi-)algebra. We next consider the inverse process, namely that of adding nodes, and give some necessary conditions for the simplicity of the induced algebra. Finally, we apply these to the exceptional series of simple Lie algebras, in the context of finding obstructions to the existence of finite-dimensional simple complex algebras of types E9, F5 and G3. In particular, our methods give a new point of view on why there cannot exist such an algebra of type E9.
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arxiv:math/0409359
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In this paper we prove an analogue in the discrete setting of \Bbb Z^d, of the spherical maximal theorem for \Bbb R^d. The methods used are two-fold: the application of certain "sampling" techniques, and ideas arising in the study of the number of representations of an integer as a sum of d squares in particular, the "circle method". The results we obtained are by necessity limited to d \ge 5, and moreover the range of p for the L^p estimates differs from its analogue in \Bbb R^d.
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arxiv:math/0409365
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We study the relationship between the arithmetic and the spectrum of the Laplacian for manifolds arising from congruent arithmetic subgroups of SL(1,D), where D is an indefinite quaternion division algebra defined over a number field F. We give new examples of isospectral but non-isometric compact, arithmetically defined varieties, generalizing the class of examples constructed by Vigneras. These examples are based on an interplay between the simply connected and adjoint group and depend explicitly on the failure of strong approximation for the adjoint group. The examples can be considered as a geometric analogue and also as an application of the concept and results on L-indistinguishability for SL(1,D) due to Labesse and Langlands. We verify that the Hasse-Weil zeta functions are equal for the examples of isospectral pair of Shimura varieties we construct giving further evidence for an archimedean analogue of Tate's conjecture, which expects that the spectrum of the Laplacian determines the arithmetic of such spaces.
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arxiv:math/0409386
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In a series of recent works, Boyd, Diaconis, and their co-authors have introduced a semidefinite programming approach for computing the fastest mixing Markov chain on a graph of allowed transitions, given a target stationary distribution. In this paper, we show that standard mixing-time analysis techniques--variational characterizations, conductance, canonical paths--can be used to give simple, nontrivial lower and upper bounds on the fastest mixing time. To test the applicability of this idea, we consider several detailed examples including the Glauber dynamics of the Ising model--and get sharp bounds.
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arxiv:math/0409429
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For a locally compact group $G$, let $A(G)$ denote its Fourier algebra and $\hat{G}$ its dual object, i.e. the collection of equivalence classes of unitary represenations of $G$. We show that the amenability constant of $A(G)$ is less than or equal to $\sup \{\deg(\pi) : \pi \in \hat{G} \}$ and that it is equal to one if and only if $G$ is abelian.
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arxiv:math/0409454
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We investigate the rigidity and asymptotic properties of quantum SU(2) representations of mapping class groups. In the spherical braid group case the trivial representation is not isolated in the family of quantum SU(2) representations. In particular, they may be used to give an explicit check that spherical braid groups and hyperelliptic mapping class groups do not have Kazhdan's property (T). On the other hand, the representations of the mapping class group of the torus do not have almost invariant vectors, in fact they converge to the metaplectic representation of SL(2,Z) on L^2(R). As a consequence we obtain a curious analytic fact about the Fourier transform on the real line which may not have been previously observed.
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arxiv:math/0409503
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We obtain a characterization of property (T) for von Neumann algebras in terms of 1-cohomology similar to the Delorme-Guichardet Theorem for groups.
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arxiv:math/0409527
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Given good knowledge on the even moments, we derive asymptotic formulas for $\lambda$-th moments of primes in short intervals and prove "equivalence" result on odd moments. We also provide numerical evidence in support of these results.
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arxiv:math/0409531
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We describe the Chow ring with rational coefficients of Mbar_{0,1}(P^n,d) as the subring of invariants of a ring B(Mbar_{0,1}(P^n,d);Q), relative to the action of the group of symmetries S_d. We compute B(Mbar_{0,1}(P^n,d);Q) by following a sequence of intermediate spaces for Mbar_{0,1}(P^n,d).
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arxiv:math/0409569
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We extend the techniques in a previous paper to calculate the Heegaard Floer homology groups for fibered 3-manifolds M whose monodromy is a power of a Dehn twist about a genus-1 separating circle on a surface of genus g > 1. We only consider non-torsion Spin^c-structures on M.
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arxiv:math/0410005
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We determine the precise conditions under which any skew Schur function is equal to a Schur function over both infinitely and finitely many variables.
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arxiv:math/0410044
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Higher criticism, or second-level significance testing, is a multiple-comparisons concept mentioned in passing by Tukey. It concerns a situation where there are many independent tests of significance and one is interested in rejecting the joint null hypothesis. Tukey suggested comparing the fraction of observed significances at a given \alpha-level to the expected fraction under the joint null. In fact, he suggested standardizing the difference of the two quantities and forming a z-score; the resulting z-score tests the significance of the body of significance tests. We consider a generalization, where we maximize this z-score over a range of significance levels 0<\alpha\leq\alpha_0. We are able to show that the resulting higher criticism statistic is effective at resolving a very subtle testing problem: testing whether n normal means are all zero versus the alternative that a small fraction is nonzero. The subtlety of this ``sparse normal means'' testing problem can be seen from work of Ingster and Jin, who studied such problems in great detail. In their studies, they identified an interesting range of cases where the small fraction of nonzero means is so small that the alternative hypothesis exhibits little noticeable effect on the distribution of the p-values either for the bulk of the tests or for the few most highly significant tests. In this range, when the amplitude of nonzero means is calibrated with the fraction of nonzero means, the likelihood ratio test for a precisely specified alternative would still succeed in separating the two hypotheses.
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arxiv:math/0410072
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In Bayesian decision theory, it is known that robustness with respect to the loss and the prior can be improved by adding new observations. In this article we study the rate of robustness improvement with respect to the number of observations n. Three usual measures of posterior global robustness are considered: the (range of the) Bayes actions set derived from a class of loss functions, the maximum regret of using a particular loss when the subjective loss belongs to a given class and the range of the posterior expected loss when the loss function ranges over a class. We show that the rate of convergence of the first measure of robustness is \sqrtn, while it is n for the other measures under reasonable assumptions on the class of loss functions. We begin with the study of two particular cases to illustrate our results.
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arxiv:math/0410074
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We establish new measures of linear independence of logarithms on commutative algebraic groups in the so-called \emph{rational case}. More precisely, let k be a number field and v_{0} be an arbitrary place of k. Let G be a commutative algebraic group defined over k and H be a connected algebraic subgroup of G. Denote by Lie(H) its Lie algebra at the origin. Let u\in Lie(G(C_{v_{0}})) a logarithm of a point p\in G(k). Assuming (essentially) that p is not a torsion point modulo proper connected algebraic subgroups of G, we obtain lower bounds for the distance from u to Lie(H)\otimes_{k} C_{v_{0}}. For the most part, they generalize the measures already known when G is a linear group. The main feature of these results is to provide a better dependence in the height Log a of p, removing a polynomial term in LogLog a. The proof relies on sharp estimates of sizes of formal subschemes associated to H (in the sense of J.-B. Bost) obtained from a lemma by M. Raynaud as well as an absolute Siegel lemma and, in the ultrametric case, a recent interpolation lemma by D. Roy.
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arxiv:math/0410082
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We give conceptual proofs of some well known results concerning compact non-positively curved locally symmetric spaces. We discuss vanishing and non-vanishing of Pontrjagin numbers and Euler characteristics for these locally symmetric spaces. We also establish vanishing results for Stiefel-Whitney numbers of (finite covers of) the Gromov-Thurston examples of negatively curved manifolds. We mention some geometric corollaries: the MinVol question, a lower bound for degrees of covers having tangential maps to the non-negatively curved duals, and estimates for the complexity of some representations of certain uniform lattices.
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arxiv:math/0410125
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A class of random recursive sequences (Y_n) with slowly varying variances as arising for parameters of random trees or recursive algorithms leads after normalizations to degenerate limit equations of the form X\stackrel{L}{=}X. For nondegenerate limit equations the contraction method is a main tool to establish convergence of the scaled sequence to the ``unique'' solution of the limit equation. In this paper we develop an extension of the contraction method which allows us to derive limit theorems for parameters of algorithms and data structures with degenerate limit equation. In particular, we establish some new tools and a general convergence scheme, which transfers information on mean and variance into a central limit law (with normal limit). We also obtain a convergence rate result. For the proof we use selfdecomposability properties of the limit normal distribution which allow us to mimic the recursive sequence by an accompanying sequence in normal variables.
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arxiv:math/0410177
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In this paper we consider the class of 2-dimensional Artin groups with connected, large type, triangle-free defining graphs (type CLTTF). We classify these groups up to isomorphism, and describe a generating set for the automorphism group of each such Artin group. In the case where the defining graph has no separating edge or vertex we show that the Artin group is not abstractly commensurable to any other CLTTF Artin group. If, moreover, the defining graph satisfies a further `vertex rigidity' condition, then the abstract commensurator group of the Artin group is isomorphic to its automorphism group and generated by inner automorphisms, graph automorphisms (induced from automorphisms of the defining graph), and the involution which maps each standard generator to its inverse. We observe that the techniques used here to study automorphisms carry over easily to the Coxeter group situation. We thus obtain a classification of the CLTTF type Coxeter groups up to isomorphism and a description of their automorphism groups analogous to that given for the Artin groups.
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arxiv:math/0410205
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In 1969 Erdoes found a lower bound on the number of (r+1)-cliques sharing an edge in graphs with n vertices and t(r,n)+1 edges, where t(r,n) is the size of the Turan graph of order n and r color classes. We improve Erdoes's bound and prove a related stability result.
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arxiv:math/0410217
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We study continuous time Markov processes on graphs. The notion of frequency is introduced, which serves well as a scaling factor between any Markov time of a continuous time Markov process and that of its jump chain. As an application, we study ``multi-person simple random walks'' on a graph G with n vertices. There are n persons distributed randomly at the vertices of G. In each step of this discrete time Markov process, we randomly pick up a person and move it to a random adjacent vertex. We give estimate on the expected number of steps for these $n$ persons to meet all together at a specific vertex, given that they are at different vertices at the begininng. For regular graphs, our estimate is exact.
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arxiv:math/0410298
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In this paper, we introduce zeta values of rational convex cones, which is a generalization of cyclotomic multiple zeta values. These zeta values have integral expressions. The main theorem asserts that zeta values of cones can be expressed as linear combinations of cyclotomic multiple zeta values over some cyclotomic field.
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arxiv:math/0410306
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We propose an algebraic geometric stability criterion for a polarised variety to admit an extremal Kaehler metric. This generalises conjectures by Yau, Tian and Donaldson which relate to the case of Kaehler-Einstein and constant scalar curvature metrics. We give a result in geometric invariant theory that motivates this conjecture, and an example computation that supports it.
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arxiv:math/0410401
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We present a computer-assisted approach to locating approximate coarse optimal switching policies between stationary states of chemically reacting systems described by microscopic/stochastic evolution rules. The ``coarse time-stepper" constitutes a bridge between the underlying kinetic Monte Carlo simulation and traditional, continuum numerical optimization techniques formulated in discrete time. The approach is illustrated through two simple catalytic surface reaction models, implemented through kinetic Monte Carlo: NO reduction on Pt, and CO oxidation on Pt. The objective sought in both cases is to switch between two coexisting stable stationary states by minimal manipulation of a macroscopic system parameter.
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arxiv:math/0410467
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We prove Langlands functoriality for the generic spectrum of general spin groups (both odd and even). Contrary to other recent instances of functoriality, our resulting automorphic representations on the general linear group will not be self-dual. Together with cases of classical groups, this completes the list of cases of split reductive groups whose L-groups have classical derived groups. The important transfer from GSp(4) to GL(4) follows from our result as a special case.
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arxiv:math/0411035
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The classical Brauer-Siegel theorem states that if $k$ runs through the sequence of normal extensions of $\mathbb{Q}$ such that $n_k/\log|D_k|\to 0,$ then $\log h_k R_k/\log \sqrt{|D_k|}\to 1.$ First, in this paper we obtain the generalization of the Brauer-Siegel and Tsfasman-Vl\u{a}du\c{t} theorems to the case of almost normal number fields. Second, using the approach of Hajir and Maire, we construct several new examples concerning the Brauer-Siegel ratio in asymptotically good towers of number fields. These examples give smaller values of the Brauer-Siegel ratio than those given by Tsfasman and Vl\u{a}du\c{t}
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arxiv:math/0411099
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Consider any symplectic ruled surface $(M^g_{\lambda},\omega_{\lambda})$ given by $(\Sigma_g \times S^2, \lambda \sigma_{\Sigma_g} \oplus \sigma_{S^2})$. We compute all natural equivariant Gromov-Witten invariants $EGW_{g,0}(M^g_{\lambda};H_k, A-kF)$ for all hamiltonian circle actions $H_k$ on $M^g_{\lambda}$, where $A=[\Sigma_g \times pt]$ and $F= [pt \times S^2]$. We use these invariants to show the nontriviality of certain higher order Whitehead products that live in the homotopy groups of the symplectomorphism groups $G_{\lambda}^g$, $g \geq 0$. Our results are sharper when $g=0,1$ and enable us to answer a question posed by D.McDuff in the case $g=1$ and provide a new interpretation of the multiplicative structure in the ring $H^*(BG^0_{\lambda} ;\Q)$ found by Abreu-McDuff.
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arxiv:math/0411108
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The aim of this paper is to extend some arithmetic results on elliptic modular forms to the case of Hilbert modular forms. Among these results let's mention : (1) the control of the image of the Galois representation modulo $p$, (2) Hida's congruence criterion outside an explicit set of primes $p$, and (3) the freeness of the integral cohomology of the Hilbert modular variety over certain local components of the Hecke algebra and the Gorenstein property of these local algebras. We study the arithmetic of the Hilbert modular forms by studying their modulo $p$ Galois representations and our main tool is the action of the inertia groups at the primes above $p$. In order to determine this action, we compute the Hodge-Tate (resp. the Fontaine-Laffaille) weights of the $p$-adic (resp. the modulo $p$) etale cohomology of the Hilbert modular variety. The cohomological part of our paper is inspired by the work of Mokrane, Polo and Tilouine on the cohomology of the Siegel modular varieties and builds upon the geometric constructions of math.NT/0212071 and math.NT/0212072.
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arxiv:math/0411152
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We introduce and study invariants of singularities in positive characteristic called F-thresholds. They give an analogue of the jumping coefficients of multiplier ideals in characteristic zero. We discuss the connection between the invariants of an ideal in characteristic zero and the invariants of the different reduction mod p of this ideal. Our main point is that this relation depends on arithmetic properties of p. We also describe a new connection between invariants mod p and the roots of the Bernstein-Sato polynomial.
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arxiv:math/0411170
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First, we construct a bijection between the set of $h$-vectors and the set of socle-vectors of artinian algebras. As a corollary, we find the minimum codimension that an artinian algebra with a given socle-vector can have. Then, we study the main problem in the paper: determining when there is a unique socle-vector for a given $h$-vector. We solve the problem completely if the codimension is at most 3.
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arxiv:math/0411229
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In this note, generalizing earlier work of Nakajima and Vasserot, we study the (equivariant) cohomology rings of Hilbert schemes of certain toric surfaces and establish their connections to Fock space and Jack polynomials.
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arxiv:math/0411255
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The formulas discussed in the previous version are not new.
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arxiv:math/0411267
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We consider infinitely convolved Bernoulli measures (or simply Bernoulli convolutions) related to the $\beta$-numeration. A matrix decomposition of these measures is obtained in the case when $\beta$ is a PV number. We also determine their Gibbs properties for $\beta$ being a multinacci number, which makes the multifractal analysis of the corresponding Bernoulli convolution possible.
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arxiv:math/0411278
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We consider energy-minimizing divergence-free eigenfields of the curl operator in dimension three from the perspective of contact topology. We give a negative answer to a question of Etnyre and the first author by constructing curl eigenfields which minimize $L^2$ energy on their co-adjoint orbit, yet are orthogonal to an overtwisted contact structure. We conjecture that $K$-contact structures on $S^1$-bundles always define tight minimizers, and prove a partial result in this direction.
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arxiv:math/0411319
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We prove that any connected component of the space of m-spin structures on compact Riemann surfaces with finite number of punctures and holes is homeomorphic to a quotient of the vector space R^d by a discrete group action. Our proof is based on the representation of the space of m-spin structures on a Riemann surface as a finite affine space of Z/mZ-valued functions on the fundamental group of the surface.
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arxiv:math/0411375
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We study the asymptotic behaviour of the trace (the sum of the diagonal parts) of a plane partition of the positive integer n, assuming that this parfition is chosen uniformly at random from the set of all such partitions.
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arxiv:math/0411377
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In this paper,we show that spherical bounded energy solution of the defocusing 3D energy critical Schr\"odinger equation with harmonic potential, $(i\partial_t + \frac {\Delta}2+\frac {|x|^2}2)u=|u|^4u$, exits globally and scatters to free solution in the space $\Sigma=H^1\bigcap\mathcal F H^1$. We preclude the concentration of energy in finite time by combining the energy decay estimates.
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arxiv:math/0411423
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In this paper, we show that the generalized hypergeometric function mF_m-1 has a one parameter group of local symmetries, which is a conjugation of a flow of a rational Calogero-Mozer system. We use the symmetry to construct fermionic fields on a complex torus, which have linear-algebraic properties similar to those of the local solutions of the generalized hypergeometric equation. The fields admit a non-trivial action of the quaternions based on the above symmetry. We use the similarity between the linear-algebraic structures to introduce the quaternionic action on the direct sum of the space of solutions of the generalized hypergeometric equation and its dual. As a side product, we construct a ``good'' basis for the monodromy operators of the generalized hypergeometric equation inspired by the study of multiple flag varieties with finitely many orbits of the diagonal action of the general linear group by Magyar, Weyman, and Zelevinsky. As an example of computational effectiveness of the basis, we give a proof of the existence of the monodromy invariant hermitian form on the space of solutions of the generalized hypergeometric equation (in the case of real local exponents) different from the proofs of Beukers and Heckman and of Haraoka. As another side product, we prove an elliptic generalization of Cauchy identity.
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arxiv:math/0411476
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Forbidden minors and subdivisions for toroidal graphs are numerous. We consider the toroidal graphs with no $K_{3,3}$-subdivisions that coincide with the toroidal graphs with no $K_{3,3}$-minors. These graphs admit a unique decomposition into planar components and have short lists of obstructions. We provide the complete lists of four forbidden minors and eleven forbidden subdivisions for the toroidal graphs with no $K_{3,3}$'s and prove that the lists are sufficient.
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arxiv:math/0411488
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We analyze certain parametrized families of one-dimensional maps with infinitely many critical points from the measure-theoretical point of view. We prove that such families have absolutely continuous invariant probability measures for a positive Lebesgue measure subset of parameters. Moreover we show that both the densities of these measures and their entropy vary continuously with the parameter. In addition we obtain exponential rate of mixing for these measures and also that they satisfy the Central Limit Theorem.
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arxiv:math/0411493
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We define and investigate a class of Koszul quasi-hereditary algebras for which there is a natural equivalence between the bounded derived category of graded modules and the bounded derived category of graded modules over (a proper version of) the extension algebra of standard modules. Examples of such algebras include, in particular, the multiplicity free blocks of the BGG category $\mathcal{O}$, and some quasi-hereditary algebras with Cartan decomposition in the sense of K{\"o}nig.
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arxiv:math/0411528
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We derive a series expansion for the multiparameter fractional Brownian motion. The derived expansion is proven to be rate optimal.
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arxiv:math/0411539
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Let \zeta be the intersection exponent of random walks in Z^3 and \alpha be a positive real number. We construct a stochastic process from a simple random walk by erasing loops of length at most N^\alpha. We will prove that for \alpha < \frac{1}{1+2\zeta}, the limiting distribution is Gaussian. For \alpha > 2 the limiting distribution will be shown to be equal to the limiting distribution of the loop erased walk.
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arxiv:math/0411551
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A Banach space $X$ is said to have the alternative Daugavet property if for every (bounded and linear) rank-one operator $T:X\longrightarrow X$ there exists a modulus one scalar $\omega$ such that $\|Id + \omega T\|= 1 + \|T\|$. We give geometric characterizations of this property in the setting of $C^*$-algebras, $JB^*$-triples and their isometric preduals.
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arxiv:math/0411555
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In a deformation quantization of $\Real^n$, the Jacobi identity is automatically satisfied. This article poses the contrary question: Given a set of commutators which satisfies the Jacobi identity, is the resulting associative algebra a deformation quantization of $\Real^n$? It is shown that the result is true. However care must taken when stating precisely how and in which algebra the Jacobi identity is satisfied.
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arxiv:math/0411581
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Suppose that a finitely generated group $G$ is hyperbolic relative to a collection of subgroups $\{H_1, ..., H_m\} $. We prove that if each of the subgroups $H_1, ..., H_m$ has finite asymptotic dimension, then asymptotic dimension of $G$ is also finite.
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arxiv:math/0411585
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