Affine and combinatorial binary m-spaces

Let V(n) denote the n-dimensional vector space over the 2-element field. Let a(m, r) (respectively, c(m, r)) denote the smallest positive integer such that if n ? a(m, r) (respectively, n ? c(m, r)), and V(n) is arbitrarily partitioned into r classes C_i, 1 ? i ? r, then some class C_i must contain an m-dimensional affine (respectively, combinatorial) subspace of V(n). Upper bounds for the functions a(m, r) and c(m, r) are investigated, as are upper bounds for the corresponding “density functions” $\text{a}\text{(m, ?)}$ and $\text{c}\text{(m, ?)}$.     

Rectangular arrays

In this paper we studied m×n arrays with row sums $\text{nr}\text{(n,m)}$ and column sums $\text{mr}\text{(n,m)}$ where (n,m) denotes the greatest common divisor of m and n. We were able to show that the function H_m,n(r), which enumerates m×n arrays with row sums and column sums $\text{nr}\text{(m,n)}$ and $\text{mr}\text{(n,m)}$ respectively, is a polynomial in r of degree (m?1)(n?1). We found simple formulas to evaluate these polynomials for negative values, ?r, and we show that certain small negative integers are roots of these polynomials. When we considered the generating function G_m,n(y) = Σ_r?0H_m,n(r)y^r, it was found to be rational of degree less than zero. The denominator of G_m,n(y) is of the form (1?y)^(m?1)(n?1)+3, and the coefficients of the numerator are non-negative integers which enjoy a certain symmetric relation.     

Decomposition of the completer-graph into completer-partiter-graphs

Forn ≥ r ≥ 1, letf _r (n) denote the minimum numberq, such that it is possible to partition all edges of the completer-graph onn vertices intoq completer-partiter-graphs. Graham and Pollak showed thatf ₂(n) =n ? 1. Here we observe thatf ₃(n) =n ? 2 and show that for every fixedr ≥ 2, there are positive constantsc ₁(r) andc ₂(r) such thatc ₁(r) ≤f _r (n)?n ^?[r/2] ≤n ₂(r) for alln ≥ r. This solves a problem of Aharoni and Linial. The proof uses some simple ideas of linear algebra.     

Syzygies and Diagonal Resolutions for Dihedral Groups

Let G be a finite group with integral group ring Λ =Z[G]. The syzygies Ω_r(Z) are the stable classes of the intermediate modules in a free Λ-resolution of the trivial module. They are of significance in the cohomology theory of G via the “co-represention theorem” H^r(G, N) = Hom_𝒟er(Ω_r(Z), N). We describe the Ω_r(Z) explicitly for the dihedral groups D_4n+2, so allowing the construction of free resolutions whose differentials are diagonal matrices over Λ.     

On a problem ofn (2)-permutable semigroups

In this paper we give, for a special class of semigroups, a necessary and sufficient condition under which a semigroup is (m,n)-commutative for some positive integersm andn. With the aid of this result we give a partial answer to the following problem raised in [3]: Does then ₍₂₎-permutability (n≥4) of a semigroupS imply the (t,r)-commutativity ofS for somet,r? We show that every finiten ₍₂₎-permutable semigroup is (t,r)-commutative for somet andr.     

Some rationally looking faces of Ω n having irrational minimum permanents

Let Ω _n denote the convex polytope consisting of all n × n doubly stochasiic matrices. We determine the minimum permanents which may or may not be rational and the permanent-minimizing matrices over some rationally looking faces of Ω _n We also discuss the barycentricity of the (0, l)-matrices with which we consider the permanent-minimization problem.     

Betti and Tachibana numbers

The Tachibana numbers t _r (M), the Killing numbers k _r (M), and the planarity numbers p _r (M) are considered as the dimensions of the vector spaces of, respectively, all, coclosed, and closed conformal Killing r-forms with 1 ≤ r ≤ n ? 1 “globally” defined on a compact Riemannian n-manifold (M,g), n >- 2. Their relationship with the Betti numbers b _r (M) is investigated. In particular, it is proved that if b _r (M) = 0, then the corresponding Tachibana number has the form t _r (M) = k _r (M) + p _r (M) for t _r (M) > k _r (M) > 0. In the special case where b ₁(M) = 0 and t ₁(M) > k ₁(M) > 0, the manifold (M,g) is conformally diffeomorphic to the Euclidean sphere.     

Inner and outer radial density functions in singly-excited 1snl states of the He atom

The one-electron radial density function D(r) has recently been found to be separable into inner D_<(r) and outer D_>(r) radial density functions. The inner D_<(r) and outer D_>(r) densities are studied for 28 singly-excited 1snl singlet and triplet states (0≤l<n≤5) of the He atom at a correlated level. Theoretical structures of D_<(r) and D_>(r) are discussed within the Hartree-Fock framework. Comparison of correlated D_<(r) and D_>(r) with hydrogenic radial densities based on the modal characteristics and Carbó’s similarity index clarifies that D_<(r) represents the 1s density of the helium cation, while D_>(r) extracts the nl density of the hydrogen atom from D(r). The radial separation 〈|r₁−r₂|〉, which constitutes a lower bound to the standard deviation of D(r), is shown to be estimated from the location of the outermost maximum of D_>(r).     

Hole probability for entire functions represented by Gaussian Taylor series

Consider the Gaussian entire functionf(z) = ?? _n=0 ^?? ?? _n a _n z ⁿ , where {?? _n } is a sequence of independent and identically distributed standard complex Gaussians and {a _n } is some sequence of non-negative coefficients, with a ₀ > 0. We study the asymptotics (for large values of r) of the hole probability for f (z), that is, the probability P _H (r) that f(z) has no zeros in the disk {|z| < r}. We prove that log P _H (r) = ?S(r) + o(S(r)), where S(r) = 2·?? _n??0log⁺(a _n r ⁿ ) as r tends to ?? outside a deterministic exceptional set of finite logarithmic measure.     

Periodic-type functionals and their extensions

Let Ω denote a simply connected domain in the complex plane and let $\text{K[Ω]}$ be the collection of all entire functions of exponential type whose Laplace transforms are analytic on Ω′, the complement of Ω with respect to the sphere. Define a sequence of functionals $\text{{L}{}_{\mathrm{n}}\text{}}\text{on}\text{K[Ω]}$ by $\text{L}{}_{\mathrm{n}}\text{(f) =}\text{1}\text{2πi}\text{∝}{}_{\mathrm{\Gamma }}\text{g}{}_{\mathrm{n}}\text{(ζ) F(ζ) dζ}$, where F denotes the Laplace transform of f, Γ ? Ω is a simple closed contour chosen so that F is analytic outside and on Ω, and g_n is analytic on Ω. The specific functionals considered by this paper are patterned after the Lidstone functions, L_2n(f) = f⁽²ⁿ⁾(0) and L_{2n + 1}(f) = f⁽²ⁿ⁾(1), in that their sequence of generating functions {g_n} are “periodic.” Set g_{pn + k}(ζ) = h_k(ζ) ζ^pn, where p is a positive integer and each h_k (k = 0, 1,…, p ? 1) is analytic on Ω. We find necessary and sufficient conditions for $\text{f ∈ k[Ω]}\text{with}\text{L}{}_{\mathrm{n}}\text{(f) = 0 (n = 0, 1,…)}$. DeMar previously was able to find necessary conditions [7]. Next, we generalize {L_n} in several ways and find corresponding necessary and sufficient conditions.     

On the fourier-stieltjes coefficients of cantor-type distributions

For a givenn-tuple of non-negative numbers (p ₀,p ₁,...,p _n?1) whose sum is equal to unity let μ(t) denote the probability that Σ _j ^{= 1/∞} X _j /n ^j ≦t, where the independent random variablesX _j assume the values 0,1,...,n?1 with probabilitiesp ₀,p ₁,...,p _n?1 respectively. For mostn-tuples we obtain upper and lower bounds on |û(m)|; these estimates involve then-ary representation ofm, or in some cases of 2m, so that a very simple and explicit characterization of the sequences on whichû(m) approaches zero can be given. In particular, for the Cantor middle-third measure, corresponding to the triple (1/2, 0, 1/2), the following criterion is obtained.û(m) approaches zero on a sequenceT of integers if and only if Ω(2m) approaches infinity onT, where Ω(k) is the sum of the following three quantities associated with the ternary representation ofk: the number of runs of zeros, the number of runs of twos and the number of ones. The results obtained are easily extended to the case when then-tuple varies withj (subject to certain mild restrictions).     

Elliptic equation with a singular potential in a domain with a conic point

This paper deals with the behavior of the nonnegative solutions of the problem $$- \Delta u = V(x)u, \left. u \right|\partial \Omega = \varphi (x)$$ in a conical domain Ω ? ? ⁿ , n ≥ 3, where 0 ≤ V (x) ∈ L₁(Ω), 0 ≤ ?(x) ∈ L₁(?Ω) and ?(x) is continuous on the boundary ?Ω. It is proved that there exists a constant C _*(n) = (n ? 2)²/4 such that if V ₀(x) = (c + λ ₁)|x|^?2, then, for 0 ≤ c ≤ C _*(n) and V(x) ≤ V ₀(x) in the domain Ω, this problem has a nonnegative solution for any nonnegative boundary function ?(x) ∈ L ₁(?Ω); for c > C _*(n) and V(x) ≥ V ₀(x) in Ω, this problem has no nonnegative solutions if ?(x) > 0.     

Regularizing the abstract convex program

Characterizations of optimality for the abstract convex program $\text{μ =}\text{inf}\text{{p(x) : g(x) ? ?S, x ? Ω}}$ (P) where S is an arbitrary convex cone in a finite dimensional space, Ω is a convex set, and p and g are respectively convex and S-convex (on Ω), were given in [10]. These characterizations hold without any constraint qualification. They use the “minimal cone” S^f of (P) and the cone of directions of constancy D_g⁼ (S^f). In the faithfully convex case these cones can be used to regularize (P), i.e., transform (P) into an equivalent program (P_r) for which Slater's condition holds. We present an algorithm that finds both S^f and D_g⁼(S^f). The main step of the algorithm consists in solving a particular complementarity problem. We also present a characterization of optimality for (P) in terms of the cone of directions of constancy of a convex functional D_φg⁼ rather than D_g⁼(S^f).     

Terseness: Minimal idempotent generating sets for K(n,r)

Let n and r be positive integers with 1 < r < n and let K(n,r) consist of all transformations on X _n = {1,...,n} having image size less than or equal to r. For 1 < r < n, there exist rank-r elements of K(n,r) which are not the product of two rank-r idempotents. With this limitation in mind, we prove that for fixed r, and for all n large enough relative to r, that there exists a minimal idempotent generating set U of K(n,r) such that all rank-r elements of K(n,r) are contained in U ³. Moreover, for all n > r > 1, there exists a minimal idempotent generating set W for K(n,r) such that not every rank-r element is contained in W ³.     

RELATIONS AMONG SOME PARAMETERS OF HYPERGRAPHS

The relations among the dominating number, independence number and covering number of hypergraphs are investigated. Main results are as follows：Dv（H）≤min{α≤（H）, p（H）, p（H）, T（H）}; De（H）≤min{v（H）, T（H）, p（H）}; DT（H） ≤αT（H）; S（H）≤ Dv （H） ＋ α（H）≤n; 2≤ Dv （H） ＋ T（H） ≤n; 2 〈 Dv （H） ＋ v（H）≤n/2 ＋ [n/r]; Dv （H） ＋ p（H） 〈_n;2≤De（H） ＋ Dv（H）≤n/2 ＋ [n/r];α（H） ＋ De（H）≤n;2 ≤ De（H） ＋ v（H）≤2[n/r]; 2 De（H） ＋ p（H）≤n-r ＋ 2.     

On the value distribution theory of elliptic functions

The Nevanlinna characteristic of a nonconstant elliptic function φ (z) satisfiesT(r, φ)=Kr ² (1+o(1)) asr→∞ whereK is a nonzero constant. In this paper, we completely answer the following question: For which polynomialsQ(z, u ₀,...,u _n ) inu ₀,...,u _n , having coefficientsa(z) satisfyingT(r, a)=o(r ²) asr→∞, will the meromorphic functionh _Q (z)=Q(z, ?(z),...,?⁽ⁿ⁾(z)) either be identically zero or satisfyN(r, 1/h _Q )=o(r ²) asr→∞? In fact, we answer this question for rational functionsQ(z, u ₀,...,u _n ) inu ₀,...,u _n , and also obtain analogous results for the Weierstrass functions ζ(z) and σ(z).     

类对称函数的Schur凸性和不等式

对x = (x₁, x₂,···, x_n) ∈ (0,1)ⁿ 和 r ∈ {1, 2,···, n} 定义对称函数 F_n(x, r) = F_n(x₁, x₂,···, x_n; r) =∏_{1≤i1∑j=1^r(1+xi3/1- xi3)^1/r, 其中i1, i2, ···, ir 是整数. 该文证明了Fn(x, r) 是(0,1)ⁿ 上的Schur凸、Schur乘性凸和Schur调和凸函数. 作为应用,利用控制理论建立了若干不等式.}     

On the existence of two non-neighboring subgraphs in a graph

Does there exist a functionf(r, n) such that each graphG with _Z (G)≧f(r, n) contains either a complete subgraph of orderr or else two non-neighboringn-chromatic subgraphs? It is known thatf(r, 2) exists and we establish the existence off(r, 3). We also give some interesting results about graphs which do not contain two independent edges.     

Superconvergence of Legendre projection methods for the eigenvalue problem of a compact integral operator

In this paper, we consider the Galerkin and collocation methods for the eigenvalue problem of a compact integral operator with a smooth kernel using the Legendre polynomials of degree ≤n. We prove that the error bounds for eigenvalues are of the order O(n^−2r) and the gap between the spectral subspaces are of the orders O(n^−r) in L₂-norm and O(n^1/2−r) in the infinity norm, where r denotes the smoothness of the kernel. By iterating the eigenvectors we show that the iterated eigenvectors converge with the orders of convergence O(n^−2r) in both L₂-norm and infinity norm. We illustrate our results with numerical examples.