Chromatic Roots and Hamiltonian Paths

We present a new connection between colorings and hamiltonian paths: If the chromatic polynomial of a graph has a noninteger root less than or equal to

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then the graph has no hamiltonian path. This result is best possible in the sense that it becomes false if t₀ is replaced by any larger number.     

Probabilistic and average widths of multivariate Sobolev spaces with mixed derivative equipped with the Gaussian measure

We present sharp bounds on the Kolmogorov probabilistic (N,δ)-width and p-average N-width of multivariate Sobolev space with mixed derivative

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, equipped with a Gaussian measure μ in , that is where 1<q<∞,0<p<∞, and ρ>1 is depending only on the eigenvalues of the correlation operator of the measure μ (see (4)).     

The limit of the product of the parameterized exponentials of two operators

Given two self-adjoint, positive, compact operators A,B on a separable Hilbert space, we show that there exists a self-adjoint, positive, compact operator C commuting with B such that

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.     

The supremum of Brownian local times on Hölder curves

For , we consider L^f_t, the local time of space-time Brownian motion on the curve f. Let be the class of all functions whose Hölder norm of order α is less than or equal to 1. We show that the supremum of L^f₁ over f in is finite if α>1/2 and infinite if α<1/2.     

Compressing Mappings on Primitive Sequences over Z/(2) and Its Galois Extension

Let f(x) be a strongly primitive polynomial of degree n over Z/(2^e), η(x₀,x₁,…,x_e−2) a Boolean function of e−1 variables and (x₀,x₁,…,x_e−1)=x_e−1+η(x₀,x₁,…,x_e−2)G (f(x),Z/(2^e)) denotes the set of all sequences over Z/(2^e) generated by f(x), F₂^∞ the set of all sequences over the binary field F₂, then the compressing mapping

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is injective, that is, for , G(f(x),Z/(2^e)), = if and only if Φ( )=Φ( ), i.e., ( ₀,…, _e−1)=( ₀,…, _e−1) mod 2. In the second part of the paper, we generalize the above result over the Galois rings.     

Norms Concerning Subdivision Sequences and Their Applications in Wavelets

Let a:=(a(α))_{α ^s} be a finitely supported sequence of r×r matrices and M be a dilation matrix. The subdivision sequence {(a_n(α))_{α ^s}:n } is defined by a₁=a and

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Let 1≤p≤∞ and f=(f¹,…,f^r)^T be a vector of compactly supported functions in L_p( ^s). The stability is not assumed for f. The purpose of this paper is to give a formula for the asymptotic behavior of the L_p-norms of the combinations of the shifts of f with the subdivision sequence coefficients: Such an asymptotic behavior plays an essential role in the investigation of wavelets and subdivision schemes. In this paper we show some applications in the convergence of cascade algorithms, construction of inhomogeneous multiresolution analyzes, and smoothness analysis of refinable functions. Some examples are provided to illustrate the method.     

On Weighted Mean Convergence of Lagrange Interpolation for General Arrays

For n1, let {x_jn}_j=1ⁿ be n distinct points and let L_n[·] denote the corresponding Lagrange Interpolation operator. Let W : →[0,∞). What conditions on the array {x_jn}_{1jn, n1} ensure the existence of p>0 such

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for every continuous f : → with suitably restricted growth, and some “weighting factor” φ^b? We obtain a necessary and sufficient condition for such a p to exist. The result is the weighted analogue of our earlier work for interpolation arrays contained in a compact set.     

A functional Hungarian construction for sums of independent random variables

We develop a Hungarian construction for the partial sum process of independent non-identically distributed random variables. The process is indexed by functions f from a class , but the supremum over is taken outside the probability. This form is a prerequisite for the Komlós–Major–Tusnády inequality in the space of bounded functionals , but contrary to the latter it essentially preserves the classical n^−1/2logn approximation rate over large functional classes such as the Hölder ball of smoothness 1/2. This specific form of a strong approximation is useful for proving asymptotic equivalence of statistical experiments.     

Bergman-Type Reproducing Kernels, Contractive Divisors, and Dilations

Let Ω be a region in the complex plane. In this paper we introduce a class of sesquianalytic reproducing kernels on Ω that we call B-kernels. When Ω is the open unit disk and certain natural additional hypotheses are added we call such kernels k Bergman-type kernels. In this case the associated reproducing kernel Hilbert space (k) shares certain properties with the classical Bergman space L²_α of the unit disk. For example, the weighted Bergman kernels k^β_w(z)=(1−wz)^−β, 1β2 are Bergman-type kernels. Furthermore, for any Bergman-type kernel k one has H² (k)L²_a, where the inclusion maps are contractive, and M_ζ, the operator of multiplication with the identity function ζ, defines a contraction operator on (k). Our main results about Bergman-type kernels k are the following two: First, once properly normalized, the reproducing kernel for any nontrivial zero based invariant subspace of (k) is a Bergman-type kernel as well. For the weighted Bergman kernels k^β this result even holds for all _ζ-invariant subspace of index 1, i.e., whenever the dimension of /ζ is one. Second, if is any multiplier invariant subspace of (k), and if we set *= z , then M_ζ is unitarily equivalent to M_ζ acting on a space of *-valued analytic functions with an operator-valued reproducing kernel of the type

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where V is a contractive analytic function V :  → ( ,  *), for some auxiliary Hilbert space . Parts of these theorems hold in more generality. Corollaries include contractive divisor, wandering subspace, and dilation theorems for all Bergman-type reproducing kernel Hilbert spaces. When restricted to index one invariant subspaces of (k^β), 1β2, our approach yields new proofs of the contractive divisor property, the strong contractive divisor property, and the wandering subspace theorems and inner–outer factorization. Our proofs are based on the properties of reproducing kernels, and they do not involve the use of biharmonic Green functions as had some of the earlier proofs.     

On Acyclic Orientations and Sequential Dynamical Systems

We study a class of discrete dynamical systems that consists of the following data: (a) a finite (labeled) graph Y with vertex set {1, …, n}, where each vertex has a binary state, (b) a vertex labeled multi-set of functions (F_i, Y: ₂ⁿ →  ₂ⁿ)_i, and (c) a permutation π  S_n. The function F_i, Y updates the binary state of vertex i as a function of the states of vertex i and its Y-neighbors and leaves the states of all other vertices fixed. The permutation π represents a Y-vertex ordering according to which the functions F_i, Y are applied. By composing the functions F_i, Y in the order given by π we obtain the sequential dynamical system (SDS):

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In this paper we first establish a sharp, combinatorial upper bound on the number of non-equivalent SDSs for fixed graph Y and multi-set of functions (F_i, Y). Second, we analyze the structure of a certain class of fixed-point-free SDSs.     

Estimation of dimension functions of band-limited wavelets

The dimension function D_ψ of a band-limited wavelet ψ is bounded by n if its Fourier transform is supported in [−(2ⁿ⁺²/3)π,(2ⁿ⁺²/3)π]. For each and for each , 0<<δ=δ(n), we construct a wavelet ψ with supp

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such that D_ψ>n on a set of positive measure, which proves that [−(2ⁿ⁺²/3)π,(2ⁿ⁺²/3)π] is the largest symmetric interval for estimating the dimension function by n. This construction also provides a family of (uncountably many) wavelet sets each consisting of infinite number of intervals.     

On the Denseness of Rational Systems

This note characterizes the denseness of rational systems

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in C[−1, 1], where the nonreal poles in {a_k}^∞_k=1 \[−1, 1] are paired by complex conjugation. This extends an Achiezer's result.     

A nonlinear oblique derivative boundary value problem for the heat equation Part 2: Singular self-similar solutions

This paper continues the study started in [12]. In the upper half-plane, consider the elliptic equation

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, submitted to the nonlinear oblique derivative boundary condition U_x = UU_z on the axis x = 0. The solution of this problem appears to be the self-similar solution of the heat equation with the same boundary condition. As goes to 0, the function U converges to the non trivial solution U of the corresponding degenerate problem. Moreover there exists z₀ > 0 such that U vanishes for z ≥ z₀, is C^∞ on ]0, z₀[× ₊, is continuous on the boundary x = 0 and is discontinuous on the half-axis {z = 0, x> 0}.     

Positive solutions of singular p-Laplacian dynamic equations with sign changing nonlinearity

Let be a time scale such that . By the Schauder fixed-point theorem and the upper and lower solution method, we present some existence criteria of the positive solution of m-point singular p-Laplacian dynamic equation with boundary conditions , where φ_p(s)=|s|^p-2s with p>1, is continuous for i=1,2,…,m-1 and nonincreasing if . The nonlinear term may be singular in its dependent variable and is allowed to change sign. Our results are new even for the corresponding differential and difference equations . As an application, an example is given to illustrate our result.     

Polytopal Approximation Bounding the Number of k-Faces

Assume that M is a convex body with C² boundary in ^d. The paper considers polytopal approximation of M with respect to the most commonly used metrics, like the symmetric difference metric δ_S, the L_p metric, 1p∞, or the Banach–Mazur metric. In case of δ_S, the main result states that if P_n is a polytope whose number of k faces is at most n then

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The analogous estimates are proved for all the other metrics. Finally, the optimality of these estimates is verified up to a constant depending on the metric and the dimension.     

A mixed hook-length formula for affine Hecke algebras

Let be the affine Hecke algebra corresponding to the group GL_l over a p-adic field with residue field of cardinality q. We will regard as an associative algebra over the field . Consider the -module W induced from the tensor product of the evaluation modules over the algebras and . The module W depends on two partitions λ of l and μ of m, and on two non-zero elements of the field . There is a canonical operator J acting on W; it corresponds to the trigonometric R-matrix. The algebra contains the finite dimensional Hecke algebra H_l+m as a subalgebra, and the operator J commutes with the action of this subalgebra on W. Under this action, W decomposes into irreducible subspaces according to the Littlewood–Richardson rule. We compute the eigenvalues of J, corresponding to certain multiplicity-free irreducible components of W. In particular, we give a formula for the ratio of two eigenvalues of J, corresponding to the “highest” and the “lowest” components. As an application, we derive the well known q-analogue of the hook-length formula for the number of standard tableaux of shape λ.     

Estimation de Yule–Walker d'un CAR(p) observé à temps discret

Let (X(lδ), l=0,n) be a discrete observation at mesh δ>0 of X, a CAR(p). Classical Yule–Walker estimation are biased and must be corrected. Resultant estimators converge if T=nδ→+∞, are asymptotically normal with rate , and efficient. The diffusion coefficient is also estimated, with rate .     

On approximation properties of a family of linear operators at critical value of parameter

We introduce the family of linear operators

associated to a certain “admissible bunch” of operators S_t, t>0, acting on , and investigate the approximation properties of this family as α→0⁺. We give some applications to the Riesz and the Bessel potentials generated by the ordinary (Euclidean) and generalized translations.     

Galois Reconstruction of Finite Quantum Groups

Let be a (small) category and let F:  →  alg_f be a functor, where alg_f is the category of finite-dimensional measured algebras over a field k (or Frobenius algebras). We construct a universal Hopf algebra A_aut(F) such that F factorizes through a functor :  →  coalg_f(A_aut(F)), where coalg_f(A_aut(F)) is the category of finite-dimensional measured A_aut(F)-comodule algebras. This general reconstruction result allows us to recapture a finite-dimensional Hopf algebra A from the category coalg_f(A) and the forgetful functor ω: coalg_f(A) →  alg_f: we have A  A_aut(ω). Our universal construction is also done in a C*-algebra framework, and we get compact quantum groups in the sense of Woronowicz.     

On a Conjecture of Ditzian and Runovskii

Let M_θ be the mean operator on the unit sphere in ⁿ, n3, which is an analogue of the Steklov operator for functions of single variable. Denote by D the Laplace–Beltrami operator on the sphere which is an analogue of second derivative for functions of single variable. Ditzian and Runovskii have a conjecture on the norm of the operator θ²D(M_θ)^m, m2 from X=L^p (1p∞) to itself which can be expressed as

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. We give a proof of this conjecture.