On a generalization of a theorem of Erdős and Fuchs

Let A={a₁,a₂,…}(a₁<a₂<) be an infinite sequence of nonnegative integers, let k≥2 be a fixed integer and denote by r_k(A,n) the number of solutions of a_i1+a_i2++a_ik≤n. Montgomery and Vaughan proved that r₂(A,n)=cn+o(n^1/4) cannot hold for any constant c>0. In this paper, we extend this result to k>2.     

An Inequality on the Size of a Set in a Cartesian Product

Let Ω be a finite subset of the Cartesian productW₁  ×   × W_nof n sets. ForA    {1, 2, , n }, denote by Ω_Athe projection ofΩ onto the Cartesian product of W_i, i   A. Generalizing an inequality given in an article by Shen, we prove that | Ω |² ≤  |Ω_A1 || Ω_Ak| provided that { A₁, , A_k} is a double cover of {1, 2, , n }. This inequality is applied to give some bounds on the numbers of special subgraphs of a graph.     

Some extensions of a property of linear representation functions

Let A={a₁,a₂,…}(a₁<a₂<) be an infinite sequence of nonnegative integers. Let k≥2 be a fixed integer and for , let R_k(A,n) be the number of solutions of a_i1++a_ik=n,a_i1,…,a_ikA, and let and denote the number of solutions with the additional restrictions a_i1<<a_ik, and a_i1≤≤a_ik respectively. Recently, Horváth proved that if d>0 is an integer, then there does not exist n₀ such that for n>n₀. In this paper, we obtain the analogous results for R_k(A,n), and .     

Constructing set systems with prescribed intersection sizes

Let f be an n-variable polynomial with positive integer coefficients, and let be a set system on the n-element universe. We define a set system and prove that f(H_i1∩H_i2∩∩H_ik)=|G_i1∩G_i2∩∩G_ik|, for any 1km, where f(H_i1∩H_i2∩∩H_ik) denotes the value of f on the characteristic vector of H_i1∩H_i2∩∩H_ik. The construction of is a straightforward polynomial-time algorithm from the set system and the polynomial f. In this paper we use this algorithm for constructing set systems with prescribed intersection sizes modulo an integer. As a by-product of our method, some upper bounds on the number of sets in set systems with prescribed intersection sizes are extended.     

Roots under convolution of sequences

The convolution a * b of the sequences a = a₀, a₁, a₂, and b is the sequence with elements ∑₀ⁿ a_kb_{n − k}. One sets 1, 1, 1, equal to σ. Given that a * a with a ≥ 0 is close to σ * σ, how close is a to σ? More generally, one asks how close a is to σ if the p-th convolution power, a*P with a ≥ 0, is close to σ*P. Power series and complex analysis form a natural tool to estimate the ‘summed deviation’ ρ = σ * (a — σ) in terms of b = a * a — σ * σ or b = a*P − σ*P. Optimal estimates are found under the condition ∑_k=0ⁿ b_k² = %plane1D;512;(n^{2β + 1}) whenever −½ < β < p − 1. It is not known what the optimal estimates are for the special case b_n = %plane1D;512;(n^β).     

Linked graphs with restricted lengths

A graph G is k-linked if G has at least 2k vertices, and for every sequence x₁,x₂,…,x_k,y₁,y₂,…,y_k of distinct vertices, G contains k vertex-disjoint paths P₁,P₂,…,P_k such that P_i joins x_i and y_i for i=1,2,…,k. Moreover, the above defined k-linked graph G is modulo (m₁,m₂,…,m_k)-linked if, in addition, for any k-tuple (d₁,d₂,…,d_k) of natural numbers, the paths P₁,P₂,…,P_k can be chosen such that P_i has length d_i modulo m_i for i=1,2,…,k. Thomassen showed that, for each k-tuple (m₁,m₂,…,m_k) of odd positive integers, there exists a natural number f(m₁,m₂,…,m_k) such that every f(m₁,m₂,…,m_k)-connected graph is modulo (m₁,m₂,…,m_k)-linked. For m₁=m₂=…=m_k=2, he showed in another article that there exists a natural number g(2,k) such that every g(2,k)-connected graph G is modulo (2,2,…,2)-linked or there is XV(G) such that |X|4k−3 and G−X is a bipartite graph, where (2,2,…,2) is a k-tuple.We showed that f(m₁,m₂,…,m_k)max{14(m₁+m₂++m_k)−4k,6(m₁+m₂++m_k)−4k+36} for every k-tuple of odd positive integers. We then extend the result to allow some m_i be even integers. Let (m₁,m₂,…,m_k) be a k-tuple of natural numbers and ℓk such that m_i is odd for each i with ℓ+1ik. If G is 45(m₁+m₂++m_k)-connected, then either G has a vertex set X of order at most 2k+2ℓ−3+δ(m₁,…,m_ℓ) such that G−X is bipartite or G is modulo (2m₁,…,2m_ℓ,m_ℓ+1,…,m_k)-linked, where

Our results generalize several known results on parity-linked graphs.     

Tails of Bipartite Distance-regular Graphs

Let Γ denote a bipartite distance-regular graph with diameterD  ≥  4 and valency k ≥  3. Let θ 0  > θ 1  >  >  θD denote the eigenvalues of Γ and let E₀, E₁, , E_Ddenote the associated primitive idempotents. Fix s(1  ≤  s ≤  D −  1 ) and abbreviate E: =  E_s. We say E is a tail whenever the entrywise product E   E is a linear combination of E₀, E and at most one other primitive idempotent of Γ. Letq_ij^{σi + 1 h} (0  ≤ h , i, j ≤  D) denote the Krein parameters of Γ and letΔ denote the undirected graph with vertices 0, 1, , D where two vertices i, j are adjacent whenever i ≠  =  j andq_ij^σi + 1s  ≠  =  0. We show E is a tail if and only if one of (i)–(iii) holds: (i) Δ is a path; (ii) Δ has two connected components, each of which is a path; (iii) D =  6 and Δ has two connected components, one of which is a path on four vertices and the other of which is a clique on three vertices.     

On sharp Burkholder–Rosenthal-type inequalities for infinite-degree U-statistics

In this paper, we present a method that allows one to obtain a number of sharp inequalities for expectations of functions of infinite-degree U-statistics. Using the approach, we prove, in particular, the following result: Let D be the class of functions f :R₊→R₊ such that the function f(x+z)−f(x) is concave in xR₊ for all zR₊. Then the following estimate holds: for all fD and all U-statistics ∑_1i1<<ilnY_i1,…,il(X_i1,…,X_il) with nonnegative kernels Y_i1,…,il :R^l→R₊, 1i_kn; i_r≠i_s, r≠s; k,r,s=1,…,l; l=0,…,m, in independent r.v.'s X₁,…,X_n. Similar inequality holds for sums of decoupled U-statistics. The class D is quite wide and includes all nonnegative twice differentiable functions f such that the function f″(x) is nonincreasing in x>0, and, in particular, the power functions f(x)=x^t, 1<t2; the power functions multiplied by logarithm f(x)= (x+x₀)^t ln(x+x₀), 1<t<2, x₀max(e^{(3t2−6t+2)/(t(t−1)(2−t))},1); and the entropy-type functions f(x)=(x+x₀)ln(x+x₀), x₀1. As an application of the results, we determine the best constants in Burkholder–Rosenthal-type inequalities for sums of U-statistics and prove new decoupling inequalities for those objects. The results obtained in the paper are, to our knowledge, the first known results on the best constants in sharp moment estimates for U-statistics of a general type.     

Primitive normal polynomials with a prescribed coefficient

In this paper, we established the existence of a primitive normal polynomial over any finite field with any specified coefficient arbitrarily prescribed. Let n15 be a positive integer and q a prime power. We prove that for any aF_q and any 1m<n, there exists a primitive normal polynomial f(x)=xⁿ−σ₁xⁿ⁻¹++(−1)ⁿ⁻¹σ_n−1x+(−1)ⁿσ_n such that σ_m=a, with the only exceptions σ₁≠0. The theory can be extended to polynomials of smaller degree too.     

On improved estimators of the generalized variance

Treated in this paper is the problem of estimating with squared error loss the generalized variance | Σ | from a Wishart random matrix S: p × p W_p(n, Σ) and an independent normal random matrix X: p × k N(ξ, Σ I_k) with ξ(p × k) unknown. Denote the columns of X by X₍₁₎ ,…, X_(k) and set ψ⁽⁰⁾(S, X) = {(n − p + 2)!/(n + 2)!} | S |, ψ⁽ⁱ⁾(X, X) = min[ψ⁽ⁱ⁻¹⁾(S, X), {(n − p + i + 2)!/(n + i + 2)!} | S + X₍₁₎ X′₍₁₎ + + X_(i) X′_(i) |] and Ψ⁽ⁱ⁾(S, X) = min[ψ⁽⁰⁾(S, X), {(n − p + i + 2)!/(n + i + 2)!}| S + X₍₁₎ X′₍₁₎ + + X_(i) X′_(i) |], i = 1,…,k. Our result is that the minimax, best affine equivariant estimator ψ⁽⁰⁾(S, X) is dominated by each of Ψ⁽ⁱ⁾(S, X), i = 1,…,k and for every i, ψ⁽ⁱ⁾(S, X) is better than ψ⁽ⁱ⁻¹⁾(S, X). In particular, ψ^(k)(S, X) = min[{(n − p + 2)!/(n + 2)!} | S |, {(n − p + 2)!/(n + 2)!} | S + X₍₁₎X′₍₁₎|,…,| {(n − p + k + 2)!/(n + k + 2)!} | S + X₍₁₎X′₍₁₎ + + X_(k)X′_(k)|] dominates all other ψ's. It is obtained by considering a multivariate extension of Stein's result (Ann. Inst. Statist. Math. 16, 155–160 (1964)) on the estimation of the normal variance.     

Higher Chern Classes and Steenrod Operations in Motivic Cohomology

In this short paper we investigate the relation between higher Chern classes and reduced power operations in motivic cohomology. More precisely, we translate the well-known arguments [5] into the context of motivic cohomology and define higher Chern classes c_p,q : K _p(X) → H^2q-p (X,Z(q)) → H^2q-p(X, Z/l(q)), where X is a smooth scheme over the base field k, l is a prime number and char(k) ≠ l. The same approach produces the classes for K-theory with coefficients as well. Let further Pⁱ : H^m(X, Z/l(n)) → H^m+2i(l-1) (X, Z/l(n + i(l - 1))) denote the ith reduced power operation in motivic cohomology, constructed in [2]. The main result of the paper looks as follows.     

On weighted Lebesgue function type sums

Let I be a finite or infinite interval and dμ a measure on I. Assume that the weight function w(x)>0, w^′(x) exists, and the function w^′(x)/w(x) is non-increasing on I. Denote by ℓ_k's the fundamental polynomials of Lagrange interpolation on a set of nodes x₁<x₂<<x_n in I. The weighted Lebesgue function type sum for 1≤i<j≤n and s≥1 is defined by

In this paper the exact lower bounds of S_n(x) on a “big set” of I and are obtained. Some applications are also given.     

Rates of convergence for classes of functions: The non-i.i.d. case

Let X_i, i ≥ 1, be a sequence of φ-mixing random variables with values in a sample space (X, A). Let L(X_i) = P_(i) for all i ≥ 1 and let _n, n ≥ 1, be classes of real-valued measurable functions on (X, A). Given any function g on (X, A), let S_n(g) = Σ_{i = 1}ⁿ {g(X_i) − Eg(X_i)}. Under weak metric entropy conditions on _n and under growth conditions on both the mixing coefficients and the maximal variance V V(n) max_{i ≤ n} sup_{g n} ∫ g² dP_(i), we show that there is a numerical constant U < ∞ such that

Full-size image

a.s. ^*, where _{i = 1}^xP_(i) and H H(n) is the square root of the entropy of the class _n. Additionally, the rate of convergence H⁻¹(n/V)^1/2 cannot, in general, be improved upon. Applications of this result are considered.     

Second-order linearity of the general signed-rank statistic

Let X₁,…, X_n be i.i.d. random variables symmetric about zero. Let R_i(t) be the rank of |X_i − tn^−1/2| among |X₁ − tn^−1/2|,…, |X_n − tn^−1/2| and T_n(t) = Σ_{i = 1}ⁿφ((n + 1)⁻¹R_i(t))sign(X_i − tn^−1/2). We show that there exists a sequence of random variables V_n such that sup_{0 ≤ t ≤ 1} |T_n(t) − T_n(0) − tV_n| → 0 in probability, as n → ∞. V_n is asymptotically normal.     

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.     

On the Multiplicities of the Primitive Idempotents of a Q-Polynomial Distance-regular Graph

Ito, Tanabe and Terwilliger recently introduced the notion of a tridiagonal pair. We apply their results to distance-regular graphs and obtain the following theorem.Theorem Let Γ denote a distance-regular graph with diameter D ≥  3. Suppose Γ is Q -polynomial with respect to the orderingE₀ , E₁, , E_Dof the primitive idempotents. For 0  ≤  i ≤  D, let m_idenote the multiplicity ofE_i . Then (i)m_i − 1 ≤ m_i (1  ≤  i ≤  D / 2),(ii)m_i  ≤ m_D − i (0  ≤  i ≤ D  / 2).By proving the above theorem we resolve a conjecture of Dennis Stanton.     

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.     

On the effective Nullstellensatz

Let be an algebraically closed field and let be an n-dimensional affine variety. Assume that f₁,...,f_k are polynomials which have no common zeros on X. We estimate the degrees of polynomials such that 1=∑^k_i=1A_if_i on X. Our estimate is sharp for k≤n and nearly sharp for k>n. Now assume that f₁,...,f_k are polynomials on X. Let be the ideal generated by f_i. It is well-known that there is a number e(I) (the Noether exponent) such that √I^e(I)⊂I. We give a sharp estimate of e(I) in terms of n, deg X and deg f_i. We also give similar estimates in the projective case. Finally we obtain a result from the elimination theory: if is a system of polynomials with a finite number of common zeros, then we have the following optimal elimination:

A functional model for the tensor product of level 1 highest and level −1 lowest modules for the quantum affine algebra

Let V(Λ_i) (resp., V(−Λ_j)) be a fundamental integrable highest (resp., lowest) weight module of . The tensor product V(Λ_i)V(−Λ_j) is filtered by submodules , n≥0, n≡i−j mod 2, where v_iV(Λ_i) is the highest vector and is an extremal vector. We show that F_n/F_n+2 is isomorphic to the level 0 extremal weight module V(n(Λ₁−Λ₀)). Using this we give a functional realization of the completion of V(Λ_i)V(−Λ_j) by the filtration (F_n)_n≥0. The subspace of V(Λ_i)V(−Λ_j) of -weight m is mapped to a certain space of sequences (P_n,l)_{n≥0,n≡i−jmod2,n−2l=m}, whose members P_n,l=P_n,l(X₁,…,X_lz₁,…,z_n) are symmetric polynomials in X_a and symmetric Laurent polynomials in z_k, with additional constraints. When the parameter q is specialized to , this construction settles a conjecture which arose in the study of form factors in integrable field theory.     

Stable extendibility of vector bundles over lens spaces mod 3 and the stable splitting problem

Let Lⁿ(3) denote the (2n+1)-dimensional standard lens space mod 3. In this paper, we study the conditions for a given real vector bundle over Lⁿ(3) to be stably extendible to L^m(3) for every mn, and establish the formula on the power ζ^k=ζζ (k-fold) of a real vector bundle ζ over Lⁿ(3). Moreover, we answer the stable splitting problem for real vector bundles over Lⁿ(3) by means of arithmetic conditions.