GCD和GCUD矩阵行列式的一个不等式（英）

Let S = {x₁, x₂,..., x_n} be a set of distinct positive integers. The n x n matrix (S) whose i, j-entry is the greatest common divisor (x_i, x_j) of x_i and x_j is called the GCD matrix on S. A divisor d of x is said to be a unitary divisor of x if (d, x/d) = 1. The greatest common unitary divisor (GCUD) matrix (S^**) is defined analogously. We show that if S is both GCD-closed and GCUD-closed, then det(S^**) ≥ det(S), where the equality holds if and Only if (S^**) = (S).     

Meet and join matrices in the poset of exponential divisors

It is well-known that (ℤ₊, |) = (ℤ₊, GCD, LCM) is a lattice, where | is the usual divisibility relation and GCD and LCM stand for the greatest common divisor and the least common multiple of positive integers. The number $ d = \prod\nolimits_{k = 1}^r {p_k^{d^{(k)} } } $ d = \prod\nolimits_{k = 1}^r {p_k^{d^{(k)} } } is said to be an exponential divisor or an e-divisor of $ n = \prod\nolimits_{k = 1}^r {p_k^{n^{(k)} } } $ n = \prod\nolimits_{k = 1}^r {p_k^{n^{(k)} } } (n > 1), written as d | _e n, if d ^(k) for all prime divisors p _k of n. It is easy to see that (ℤ₊\{1}, | _e is a poset under the exponential divisibility relation but not a lattice, since the greatest common exponential divisor (GCED) and the least common exponential multiple (LCEM) do not always exist.     

Toric Rings Arising from Cyclic Polytopes

Let d and n be positive integers with n ≥ d + 1 and 𝒫 ? ? ^d an integral cyclic polytope of dimension d with n vertices, and let K[𝒫] = K[?_≥0𝒜_𝒫] denote its associated semigroup K-algebra, where 𝒜_𝒫 = {(1, α) ∈ ? ^d+1: α ∈ 𝒫} ∩ ? ^d+1 and K is a field. In the present paper, we consider the problem when K[𝒫] is Cohen–Macaulay by discussing Serre's condition (R ₁), and we give a complete characterization when K[𝒫] is Gorenstein. Moreover, we study the normality of the other semigroup K-algebra K[Q] arising from an integral cyclic polytope, where Q is a semigroup generated by its vertices only.     

Lie-admissible algebras and generalized witt algebras ∗

The symmetric group 𝔖_n+1 of degree n + 1 admits an n-dimensional irreducible Q𝔖_n-module V corresponding to the hook partition (2, 1^n?1). By the work of Craig and Plesken, we know that there are σ(n + 1) many isomorphism classes of Z𝔖_n+1-lattices which are rationally equivalent to V, where σ denotes the divisor counting function. In the present article, we explicitly compute the Solomon zeta function of these lattices. As an application we obtain the Solomon zeta function of the Z𝔖_n+1-lattice defined by the Specht basis.     

A result about determinantal divisors

Let Rbe a principal ideal ringR_n the ring of n× nmatrices over R, and d_k (A) the kth determinantal divisor of Afor 1 ? k? n, where Ais any element of R_n , It is shown that if A,BεR_n , det(A) det(B:) ≠ 0, then d_k (AB) ≡ 0 mod d_k (A) d_k (B). If in addition (det(A), det(B)) = 1, then it is also shown that d_k (AB) = d_k (A) d_k (B). This provides a new proof of the multiplicativity of the Smith normal form for matrices with relatively prime determinants.     

A finiteness criterion and asymptotics for codimensions of generalized identities

Let A be an associative algebra over a field of characteristic zero. Then either all codimensions gc _n (A) of its generalized polynomial identities are infinite or A is the sum of ideals I and J such that dim _F I < ∞ and J is nilpotent. In the latter case, there exist numbers n ₀ ∈ ?, C ∈ ?₊, and t ∈ ?₊ for which gc _n (A) < +∞ if n ≥ n ₀ and gc _n (A) ～ Cn ^t d ⁿ as n → ∞, where d = PIexp(A) ∈ ?₊. Thus, in the latter case, conjectures of Amitsur and Regev on generalized codimensions hold.     

A compact formulation of a mixed-integer set

We consider mixed-integer sets of the form X = {(s, y) ∈ ?₊ × ? ⁿ : s + a _j y _j ≥ b _j , ?j ∈ N}. A polyhedral characterization for the case where X is defined by two inequalities is provided. From that characterization we give a compact formulation for the particular case where the coefficients of the integer variables can take only two possible integer values a ₁, a ₂ ∈ ?₊ : X _n,m = {(s, y) ∈ ?₊ × ? ^n+m : s + a ₁ y _j ≥ b _j , ?j ∈ N ₁, s + a ₂ y _j ≥ b _j , j ∈ N ₂} where N ₁ = {1, …, n}, N ₂ = {n + 1, …, n + m}. In addition, we discuss families of facet-defining inequalities for the convex hull of X _n,m .     

Über ein Turánsches Problem für ℓ-1 radiale, positiv definite Funktionen

Turán’s problem is to determine the greatest possible value of the integral ∫_? ^df(x)dx/ f (0) for positive definite functions f (x), x ∈ ?^d, supported in a given convex centrally symmetric body D ? ?^d. In this note we consider the 2-dimensional Turán problem for positive definite functions of the form f(x) = φ (∥x∥1), x ∈ ?², with φ supported in [0,π].     

Measures on the unit circle and unitary truncations of unitary operators

In this paper, we obtain new results about the orthogonality measure of orthogonal polynomials on the unit circle, through the study of unitary truncations of the corresponding unitary multiplication operator, and the use of the five-diagonal representation of this operator.Unitary truncations on subspaces with finite co-dimension give information about the derived set of the support of the measure under very general assumptions for the related Schur parameters (a_n). Among other cases, we study the derived set of the support of the measure when lim_n|a_n+1/a_n|=1, obtaining a natural generalization of the known result for the López class , lim_n|a_n|(0,1).On the other hand, unitary truncations on subspaces with finite dimension provide sequences of unitary five-diagonal matrices whose spectra asymptotically approach the support of the measure. This answers a conjecture of L. Golinskii concerning the relation between the support of the measure and the strong limit points of the zeros of the para-orthogonal polynomials.Finally, we use the previous results to discuss the domain of convergence of rational approximants of Carathéodory functions, including the convergence on the unit circle.     

A note on an open question on ω- and τ- Matrices

In a recent paper by Engel and Schneider, it was asked if, for every n ? 1, A ∈ τ_<n> implies (A+D) ∈ τ_<n> for every D = diag[d₁, d₂,… d_n] with d_i ? 0, 1 ? i ? n. We answer this question in the negative. More precisely, we show that for, any n ? 3, the set

(τ _{< n>}): = {D ∈C^n,n:(A+D)∈τ < n> for all A∈τ_<n>} is exactly given by

(Gt_<n>) = {γI_n:γ ? 0}.     

Possible numbers of positive entries of imprimitive nonnegative matrices

Zhan, X., Extremal numbers of positive entries of imprimitive nonnegative matrix, Linear Algebra Appl. (in press) has determined the maximum and minimum numbers of positive entries of imprimitive irreducible nonnegative matrices with a given imprimitivity index. Let σ( A ) denote the number of positive entries of a matrix A. Let M(n,?k) and m(n,?k) denote the maximum and minimum numbers of positive entries of imprimitive irreducible nonnegative matrices of order n with a given imprimitivity index k, respectively. In this article, we prove that for any positive integer d with m(n,k)≤ d?≤?M(n,k), there exists an n?×?n irreducible nonnegative matrix A with imprimitivity index k such that?σ?(A)=d.     

On the Positive Definiteness of Certain Functions

For a coinmutative senugoup (S, +, *) with involution and a function f : S → [0, ∞), the set S(f) of those p ≥ 0 such that f^P is a positive definite function on S is a closed subsemigroup of [0, ∞) containing 0. For S = (IR, +, x* = -x) it may happen that S(f) = { kd : k ∈ N₀ } for some d > 0, and it may happen that S(f) = {0} ? [d, ∞) for some d > O. If α > 2 and if S = (?, +, n* = -n) and f(n) = e^?[n]α or S = (IN₀, +, n* = n) and f(n) = e^nα, then S(f) ∪ (0, c) = ? and [d, ∞) ? S(f) for some d ≥; c > 0. Although (with c maximal and d minimal) we have not been able to show c = d in all cases, this equality does hold if S = ? and α ≥ 3.4. In the last section we give sinipler proofs of previously known results concerning the positive definiteness of x → e^?||x||α on normed spaces.     

Tightness of products of i.i.d. random matrices

Summary In this paper we present a necessary and sufficient condition for tightness of products of i.i.d. finite dimensional random nonnegative matrices. We give an example illustrating the use of our theorem and treat completely the case of 2×2 matrices. We also describe stationary solutions of the linear equationy _n=X_ny_n–1, n>0, in (R ^d )⁺, whereX ₁,X ₂,... are i.i.d.d×d nonnegative matrices.     

A characterization of the symplectic and unitary 3-transposition groups

Let D be a normal set of 3-transpositions in a group G. For each d ∈ D, let D_d be the set of elements ε in D such that the order of de equals 2. Then we can define an equivalence relation ? on D by dre if and only if D_d ? {d} = D_e ? {ε}.

We characterize the symplectic and unitary groups over GF(2), respectively, GF(4) generated by their transvections as groups generated by a class D of 3-transposition with r trivial on D, but not when restricted to D_d for some d ∈ D.     

Inequalities for the Gaussian hypergeometric function

we study the monotonicity of certain combinations of the Gaussian hypergeometric functions F(-1/2,1/2;1;1- xc) and F(-1/2- δ,1/2 + δ;1;1- xd) on(0,1) for given 0 c 5d/6 ∞ andδ∈(-1/2,1/2),and find the largest value δ1 = δ1(c,d) such that inequality F(-1/2,1/2;1;1- xc) F(-1/2- δ,1/2 + δ;1;1- xd) holds for all x ∈(0,1). Besides,we also consider the Gaussian hypergeometric functions F(a- 1- δ,1- a + δ;1;1- x3) and F(a- 1,1- a;1;1- x2) for given a ∈ [1/29,1) and δ∈(a- 1,a),and obtain the analogous results.     

Circuits including a given set of vertices

Let G = (V, E) be a digraph of order n, satisfying Woodall's condition ? x, y ∈ V, if (x, y) ? E, then d⁺(x) + d^?(y) ≥ n. Let S be a subset of V of cardinality s. Then there exists a circuit including S and of length at most Min(n, 2s). In the case of oriented graphs we obtain the same result under the weaker condition d⁺(x) + d^?(y) ≥ n – 2 (which implies hamiltonism).     

Sums of the squares of terms of sequence {<Emphasis Type="Italic">u</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>}

In this paper, we consider generalized Fibonacci type second order linear recurrence {u _n }. We derive a generating matrix for both the sums of squares, ∑ _i=0 ⁿ u _i ² and the products of the form u _n u _n+2. We also derive explicit formulas for the sums and products by using matrix methods. Then we give a matrix method to generate the sums of product of two consecutive terms u _n u _n+1 as well as the product, u _n u _n+2. Further we give generating functions and combinatorial representations of the sums of squares of terms of {u _n } and the product, u _n u _n+2.     

The Erdös-Jacobson-Lehel conjecture on potentiallyP k-graphic sequence is true

A variation in the classical Turan extrernal problem is studied. A simple graphG of ordern is said to have propertyPk if it contains a clique of sizek+1 as its subgraph. Ann-term nonincreasing nonnegative integer sequence π=(d₁, d₂,⋯, d₂) is said to be graphic if it is the degree sequence of a simple graphG of ordern and such a graphG is referred to as a realization of π. A graphic sequence π is said to be potentiallyP _k-graphic if it has a realizationG having propertyP _k . The problem: determine the smallest positive even number σ(k, n) such that everyn-term graphic sequence π=(d₁, d₂,…, d₂) without zero terms and with degree sum σ(π)=(d ₁+d ₂+ …+d ₂) at least σ(k,n) is potentially P_k-graphic has been proved positive. Project supported by the National Natural Science Foundation of China (Grant No. 19671077) and the Doctoral Program Foundation of National Education Department of China.     

Divisibility among power GCD matrices and among power LCM matrices on three coprime divisor chains

Let h be a positive integer and S?=?{x ₁,?…?,?x _h } be a set of h distinct positive integers. We say that the set S is a divisor chain if x _σ(1) ∣?…?∣ x _σ(h) for a permutation σ of {1,?…?,?h}. If the set S can be partitioned as S?=?S ₁?∪?S ₂?∪?S ₃, where S ₁, S ₂ and S ₃ are divisor chains and each element of S _i is coprime to each element of S _j for all 1?≤?i?<?j?≤?3, then we say that the set S consists of three coprime divisor chains. The matrix having the ath power (x _i , x _j ) ^a of the greatest common divisor of x _i and x _j as its i, j-entry is called the ath power greatest common divison (GCD) matrix on S, denoted by (S ^?a ). The ath power least common multiple (LCM) matrix [S ^?a ] can be defined similarly. In this article, let a and b be positive integers and let S consist of three coprime divisor chains with 1?∈?S. We show that if a?∣?b, then the ath power GCD matrix (S ^?a ) (resp., the ath power LCM matrix [S ^?a ]) divides the bth power GCD matrix (S ^?b ) (resp., the bth power LCM matrix [S ^?b ]) in the ring M _h (Z) of h?×?h matrices over integers. We also show that the ath power GCD matrix (S ^?a ) divides the bth power LCM matrix [S ^?b ] in the ring M _h (Z) if a?∣?b. However, if a???b, then such factorizations are not true. Our results extend Hong's and Tan's theorems and also provide further evidences to the conjectures of Hong raised in 2008.     

Least‐squares solutions of matrix inverse problem for bi‐symmetric matrices with a submatrix constraint

An n × n real matrix A = (a_ij)_{n × n} is called bi‐symmetric matrix if A is both symmetric and per‐symmetric, that is, a_ij = a_ji and a_ij = a_n+1?1,n+1?i (i, j = 1, 2,..., n). This paper is mainly concerned with finding the least‐squares bi‐symmetric solutions of matrix inverse problem AX = B with a submatrix constraint, where X and B are given matrices of suitable sizes. Moreover, in the corresponding solution set, the analytical expression of the optimal approximation solution to a given matrix A^* is derived. A direct method for finding the optimal approximation solution is described in detail, and three numerical examples are provided to show the validity of our algorithm. Copyright © 2007 John Wiley & Sons, Ltd.