On a generalization of the Shannon functional inequality

The well known Shannon entroy −∑ p_k log p_k satisfies the inequality −∑ p_k log p_k − ∑ p_k log q_k. Extensive studies have been made on the inequality ∑ p_kƒ_k(q_k) ∑ p_kƒ_k(p_k) which contains the above inequality as a special case. In this paper, we consider the most general inequality ∑ g_k(p_k)ƒ_k(p_k) ∑ g_k(p_k)ƒ_k(q_k) above type and obtain its general solution on an open domain.     

L∞-upper bound of L2-projections onto splines at a geometric mesh

For an integer k 1 and a geometric mesh (qⁱ)_−∞^∞ with q ε (0, ∞), let M_i,k(x): = k[q^{i + k}](· − x)₊^{k − 1}, N_i,k(x): = (q^{i + k} − qⁱM_i,k(x)/k, and let A_k(q) be the Gram matrix (∝M_i,kN_j,k)_i,jεz. It is known that A_k(q)⁻¹_∞ is bounded independently of q. In this paper it is shown that A_k(q)⁻¹_∞ is strictly decreasing for q in [1, ∞). In particular, the sharp upper bound and lower bound for A_k (q)⁻¹ are obtained: for all q ε (0, ∞).     

On subspace arrangements of type

Let denote the subspace arrangement formed by all linear subspaces in given by equations of the form

₁x_i1=₂x_i2==_kx_ik,

where 1i₁<<i_kn and (₁,…,_k){+1,−1}^k.Some important topological properties of such a subspace arrangement depend on the topology of its intersection lattice. In a previous work on a larger class of subspace arrangements by Björner and Sagan (J. Algebraic Combin. 5 (1996) 291–314) the topology of the intersection lattice turned out to be a particularly interesting and difficult case.We prove in this paper that Pure(Π_n,k^±) is shellable, hence that Π_n,k^± is shellable for k>n/2. Moreover, we prove that unless i≡n−2 (mod k−2) or i≡n−3 (mod k−2), and that is free abelian for i≡n−2 (mod k−2). In the special case of Π_2k,k^± we determine homology completely. Our tools are generalized lexicographic shellability, as introduced in Kozlov (Ann. Combin. 1 (1997) 67–90), and a spectral sequence method for the computation of poset homology first used in Hanlon (Trans. Amer. Math. Soc. 325 (1991) 1–37).We state implications of our results on the cohomology of the complements of the considered arrangements.     

An Asymptotic Property of Schachermayer's Space under Renorming

Let X be a Banach space with closed unit ball B. Given k , X is said to be k-β, respectively, (k + 1)-nearly uniformly convex ((k + 1)-NUC), if for every ε > 0 there exists δ, 0 < δ < 1, so that for every x B and every ε-separated sequence (x_n) B there are indices (n_i)^k_{i = 1}, respectively, (n_i)^{k + 1}_{i = 1}, such that (1/(k + 1))||x + ∑^k_{i = 1} x_ni|| ≤ 1 − δ, respectively, (1/(k + 1))||∑^{k + 1}_{i = 1} x_ni|| ≤ 1 − δ. It is shown that a Banach space constructed by Schachermayer is 2-β, but is not isomorphic to any 2-NUC Banach space. Modifying this example, we also show that there is a 2-NUC Banach space which cannot be equivalently renormed to be 1-β.     

A Remez-Type Inequality for Non-dense Müntz Spaces with Explicit Bound

LetΛ :=(λ_k)^∞_k=0be a sequence of distinct nonnegative real numbers withλ₀ :=0 and ∑^∞_k=1 1/λ_k<∞. Let(0, 1) and(0, 1−) be fixed. An earlier work of the authors shows that [formula]is finite. In this paper an explicit upper bound forC(Λ, , ) is given. In the special caseλ_k :=k^α,α>1, our bounds are essentially sharp.     

On Laurent series with multiply positive coefficients

We consider the class of doubly infinite sequences {a _k } _k ^∞ = −∞ whose truncated sequences {a _k } _n ^k = −n are 3-times positive in the sense of Pólya and Fekete for all n = 1, 2, ..., and a ₀ ≠ 0. We obtain a characterization of this class in terms of independent parameters. We also find an estimate of the growth order of the corresponding Laurent series ∑ _{k= −∞} ^∞ akz ^k .     

The wave equation for Dunkl operators

Let k = (k_α)_αε, be a positive-real valued multiplicity function related to a root system , and Δ_k be the Dunkl-Laplacian operator. For (x, t) ε ^N, × , denote by u_k(x, t) the solution to the deformed wave equation Δ_ku_k,(x, t) = δ_ttu_k(x, t), where the initial data belong to the Schwartz space on ^N. We prove that for k 0 and N l, the wave equation satisfies a weak Huygens' principle, while a strict Huygens' principle holds if and only if (N − 3)/2 + Σ_αε+k_α ε . Here ⁺ is a subsystem of positive roots. As a particular case, if the initial data are supported in a closed ball of radius R > 0 about the origin, the strict Huygens principle implies that the support of u_k(x, t) is contained in the conical shell {(x, t), ε ^N × | |t| − R x |t| + R}. Our approach uses the representation theory of the group SL(2, ), and Paley-Wiener theory for the Dunkl transform. Also, we show that the (t-independent) energy functional of u_k is, for large |t|, partitioned into equal potential and kinetic parts.     

Moment methods in two point Padé approximation

In the separable Hilbert space (H, ·, ·) the following “operator moment problem” is solved: given a complex sequence (c_k)_{k ε Z} generated by a meromorphic function f, find T ε B(H) and u₀ ε H such that T^ku₀, u₀ = c_k (k ε Z). If the sequence (c_k)_{k ε Z} is “normal,” an adapted form of Vorobyev's method of moments yields a sequence of two point Padé approximants to f. A sufficient condition for convergence of this sequence of approximants is given.     

Approximation ofk-Monotone Functions

It is shown that an algebraic polynomial of degree k−1 which interpolates ak-monotone functionfatkpoints, sufficiently approximates it, even if the points of interpolation are close to each other. It is well known that this result is not true in general for non-k-monotone functions. As an application, we prove a (positive) result on simultaneous approximation of ak-monotone function and its derivatives inL_p, 0<p<1, metric, and also show that the rate of the best algebraic approximation ofk-monotone functions (with bounded (k−2)nd derivatives inL_p, 1<p<∞, iso(n^−k/p).     

Stochastic convergence of weighted sums in normed linear spaces

Let X be a (real) separable Banach space, let {V_k} be a sequence of random elements in X, and let {a_nk} be a double array of real numbers such that lim_n→∞ a_nk = 0 for all k and Σ^∞_k=1 |a_nk| ≤ 1 for all n. Define S_n = Σⁿ_k=1 a_nk(V_k − EV_k). The convergence of {S_n} to zero in probability is proved under conditions on the coordinates of a Schauder basis or on the dual space of X and conditions on the distributions of {V_k}. Convergence with probability one for {S_n} is proved for separable normed linear spaces which satisfy Beck's convexity condition with additional restrictions on {a_nk} but without distribution conditions for the random elements {V_k}. Finally, examples of arrays {a_nk}, spaces, and applications of these results are considered.     

On product representations of powers, I

The solvability of the equation a₁a₂ … a_k = x², a₁, a₂, …, a_k ε is studied for fixed k and ‘dense’ sets of positive integers. In particular, it is shown that if k is even and k 4, and is of positive upper density, then this equation can be solved.     

Matrices dont deux lignes quelconque coïncident dans un nombre donne de positions communes

Le nombre maximal de lignes de matrices seront désignées par:

1. (a) R(k, λ) si chaque ligne est une permutation de nombres 1, 2,…, k et si chaque deux lignes différentes coïncide selon λ positions;

2. (b) S⁰(k, λ) si le nombre de colonnes est k et si chaque deux lignes différentes coïncide selon λ positions et si, en plus, il existe une colonne avec les éléments y₁, y₂, y₃, ou y₁ = y₂ ≠ y₃;

3. (c) T⁰(k, λ) si c'est une (0, 1)-matrice et si chaque ligne contient k unités et si chaque deux lignes différentes contient les unités selon λ positions et si, en plus, il existe une colonne avec les éléments 1, 1, 0.

La fonction T⁰(k, λ) était introduite par Chvátal et dans les articles de Deza, Mullin, van Lint, Vanstone, on montrait que T⁰(k, λ) max(λ + 2, (k − λ)² + k − λ + 1). La fonction S⁰(k, λ) est introduite ici et dans le Théorème 1 elle est étudiée analogiquement; dans les remarques 4, 5, 6, 7 on donne les généralisations de problèmes concernant T⁰(k, λ), S⁰(k, λ), dans la remarque 9 on généralise le problème concernant R(k, λ). La fonction R(k, λ) était introduite et étudiée par Bolton. Ci-après, on montre que R(k, λ) S⁰(k, λ) T⁰(k, λ) d'où découle en particulier: R(k, λ) λ + 2 pour λ k + 1 − (k + 2)^1/2; R(k, λ) = 0(k²) pour k − λ = 0(k); R(k, λ) (k − 1)² − (k + 2) pour k 1191.     

New Upper Bound for the B-Spline Basis Condition Number: II. A Proof of de Boor's 2-Conjecture

For the p-norm condition number κ_k, p of the B-spline basis of order k we prove the upper estimate κ_k, p<k2^k. This proves de Boor's 2^k-conjecture up to a polynomial factor.     

Quotient algebras of Stanley-Reisner rings and local cohomology

We study certain subcomplexes Δ′ of an arbitrary simplicial complex Δ such that H_mⁱ(k[Δ])-H_mⁱ(k[Δ′]) for any 0i<dim(k[Δ′]). Here, H_mⁱ(k[Δ]) is the ith local cohomology module of the Stanley-Reisner ring k[Δ] of Δ over a field k. Our technique is an elegant approach to one of the most generalized versions of the rank selection theorems of J. Munkres (1984, Michigan Math. J.31, 113–128, Theorem 6.4) and R. Stanley (1979, Trans. Amer. Math. Soc.249, 139–157, Theorem 4.3).     

How tangents solve algebraic equations, or a remarkable geometry of discriminant varieties

Let D_d,k denote the discriminant variety of degree d polynomials in one variable with at least one of its roots being of multiplicity ≥ k. We prove that the tangent cones to D_d,k span D_{d,k − 1} thus, revealing an extreme ruled nature of these varieties. The combinatorics of the web of affine tangent spaces to D_d,k in D_{d,k − 1} is directly linked to the root multiplicities of the relevant polynomials. In fact, solving a polynomial equation P(z) = 0 turns out to be equivalent to finding hyperplanes through a given point which are tangent to the discriminant hypersurface D_d,2. We also connect the geometry of the Viète map V_d: , given by the elementary symmetric polynomials, with the tangents to the discriminant varieties {D_d,k}.Various d-partitions {μ} provide a refinement {D^o_μ} of the stratification of by the D_d,k's. Our main result, Theorem 7.1, describes an intricate relation between the divisibility of polynomials in one variable and the families of spaces tangent to various strata {D^o_μ}.     

Complete on Average Boolean Satisfiability

We present in this paper a dynamic binary coding scheme α on CNF formulas ψ, and show that under a uniform distribution μ_α on binary string α(ψ), SAT is complete on average, where μ_α(ψ) is proportional to α(ψ)⁻²2^−α(ψ). We then show that there is k₀>2 such that for all kk₀, kSAT under μ_α is complete on average.     

On αrγs(k)-perfect graphs

For some integer k0 and two graph parameters π and τ, a graph G is called πτ(k)-perfect, if π(H)−τ(H)k for every induced subgraph H of G. For r1 let α_r and γ_r denote the r-(distance)-independence and r-(distance)-domination number, respectively. In (J. Graph Theory 32 (1999) 303–310), I. Zverovich gave an ingenious complete characterization of α₁γ₁(k)-perfect graphs in terms of forbidden induced subgraphs. In this paper we study α_rγ_s(k)-perfect graphs for r,s1. We prove several properties of minimal α_rγ_s(k)-imperfect graphs. Generalizing Zverovich's main result in (J. Graph Theory 32 (1999) 303–310), we completely characterize α_2r−1γ_r(k)-perfect graphs for r1. Furthermore, we characterize claw-free α₂γ₂(k)-perfect graphs.     

Representations for the first associated q-classical orthogonal polynomials

Let {p_k(x; q)} be any system of the q-classical orthogonal polynomials, and let be the corresponding weight function, satisfying the q-difference equation D_q(σ)=τ, where σ and τ are polynomials of degree at most 2 and exactly 1, respectively. Further, let {p_k⁽¹⁾(x;q)} be associated polynomials of the polynomials {p_k(x; q)}. Explicit forms of the coefficients b_n,k and c_n,k in the expansions

are given in terms of basic hypergeometric functions. Here _k(x) equals x^k if σ⁺(0)=0, or (x;q)_k if σ⁺(1)=0, where σ⁺(x)σ(x)+(q−1)xτ(x). The most important representatives of those two classes are the families of little q-Jacobi and big q-Jacobi polynomials, respectively.Writing the second-order nonhomogeneous q-difference equation satisfied by p_n−1⁽¹⁾(x;q) in a special form, recurrence relations (in k) for b_n,k and c_n,k are obtained in terms of σ and τ.     

Discrepancy in generalized arithmetic progressions

Estimating the discrepancy of the set of all arithmetic progressions in the first N natural numbers was one of the famous open problems in combinatorial discrepancy theory for a long time, successfully solved by K. Roth (lower bound) and Beck (upper bound). They proved that D(N)=min_χmax_A|∑_xAχ(x)|=Θ(N^1/4), where the minimum is taken over all colorings χ:[N]→{−1,1} and the maximum over all arithmetic progressions in [N]={0,…,N−1}.Sumsets of k arithmetic progressions, A₁++A_k, are called k-arithmetic progressions and they are important objects in additive combinatorics. We define D_k(N) as the discrepancy of the set {P∩[N]:P is a k-arithmetic progression}. The second author proved that D_k(N)=Ω(N^k/(2k+2)) and Přívětivý improved it to Ω(N^1/2) for all k≥3. Since the probabilistic argument gives D_k(N)=O((NlogN)^1/2) for all fixed k, the case k=2 remained the only case with a large gap between the known upper and lower bounds. We bridge this gap (up to a logarithmic factor) by proving that D_k(N)=Ω(N^1/2) for all k≥2.Indeed we prove the multicolor version of this result.     

On almost sure convergence of Cesaro averages of subsequences of vector-valued functions

A remarkable theorem proved by Komlòs [4] states that if {f_n} is a bounded sequence in L¹(R), then there exists a subsequence {f_nk} and f L¹(R) such that f_nk (as well as any further subsequence) converges Cesaro to f almost everywhere. A similar theorem due to Révész [6] states that if {f_n} is a bounded sequence in L²(R), then there is a subsequence {f_nk} and f L²(R) such that Σ_k=1^∞ a_k(f_nk − f) converges a.e. whenever Σ_k=1^∞ | a_k |² < ∞. In this paper, we generalize these two theorems to functions with values in a Hilbert space (Theorems 3.1 and 3.3).