On certain unimodal sequences and strict partitions

Building on a bijection of Vandervelde, we enumerate certain unimodal sequences whose alternating sum equals zero. This enables us to refine the enumeration of strict partitions with respect to the number of parts and the BG-rank.     

Higher-order optimality conditions for strict local minima

In this work, we study a nonsmooth optimization problem with generalized inequality constraints and an arbitrary set constraint. We present necessary conditions for a point to be a strict local minimizer of order k in terms of higher-order (upper and lower) Studniarski derivatives and the contingent cone to the constraint set. In the same line, when the initial space is finite dimensional, we develop sufficient optimality conditions. We also provide sufficient conditions for minimizers of order k using the lower Studniarski derivative of the Lagrangian function. Particular interest is put for minimizers of order two, using now a special second order derivative which leads to the Fréchet derivative in the differentiable case.     

Inside the critical window for cohomology of random k‐complexes

We prove sharper versions of theorems of Linial–Meshulam and Meshulam–Wallach which describe the behavior for ‐cohomology of a random k‐dimensional simplicial complex within a narrow transition window. In particular, we show that if Y is a random k‐dimensional simplicial complex with each k‐simplex appearing i.i.d. with probability with and fixed, then the dimension of cohomology is asymptotically Poisson distributed with mean . In the k = 2 case we also prove that in an accompanying growth process, with high probability, vanishes exactly at the moment when the last ‐simplex gets covered by a k‐simplex, a higher‐dimensional analogue of a “stopping time” theorem about connectivity of random graphs due to Bollobás and Thomason. Random Struct. Alg., 2015 © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 102–124, 2016     

A generalization of Lucas polynomial sequence

In this paper, we obtain a generalized Lucas polynomial sequence from the lattice paths for the Delannoy numbers by allowing weights on the steps (1,0),(0,1) and (1,1). These weighted lattice paths lead us to a combinatorial interpretation for such a Lucas polynomial sequence. The concept of Riordan arrays is extensively used throughout this paper.     

Associated primes of local cohomology modules

Let be an ideal of a commutative Noetherian ring and a finitely generated -module. Let be a natural integer. It is shown that there is a finite subset of , such that is contained in union with the union of the sets , where and . As an immediate consequence, we deduce that the first non- -cofinite local cohomology module of with respect to has only finitely many associated prime ideals.

     

A generalization of the finiteness problem in local cohomology modules

Let a be an ideal of a commutative Noetherian ring R with non-zero identity and let N be a weakly Laskerian R-module and M be a finitely generated R-module. Let t be a non-negative integer. It is shown that if H _a ⁱ (N) is a weakly Laskerian R-module for all i < t, then Hom _R (R/a, H _a ^t (M, N)) is weakly Laskerian R-module. Also, we prove that Ext _R ⁱ (R/a, H _a ^t )) is weakly Laskerian R-module for all i = 0, 1. In particular, if Supp _R (H _a ⁱ (N)) is a finite set for all i < t, then Ext _R ⁱ (R/a, H _a ^t (N)) is weakly Laskerian R-module for all i = 0, 1.     

The cohomology of the Steenrod algebra and representations of the general linear groups

Let be the algebraic transfer that maps from the coinvariants of certain -representations to the cohomology of the Steenrod algebra. This transfer was defined by W. Singer as an algebraic version of the geometrical transfer . It has been shown that the algebraic transfer is highly nontrivial, more precisely, that is an isomorphism for and that is a homomorphism of algebras.

In this paper, we first recognize the phenomenon that if we start from any degree and apply repeatedly at most times, then we get into the region in which all the iterated squaring operations are isomorphisms on the coinvariants of the -representations. As a consequence, every finite -family in the coinvariants has at most nonzero elements. Two applications are exploited.

The first main theorem is that is not an isomorphism for . Furthermore, for every 5$">, there are infinitely many degrees in which is not an isomorphism. We also show that if detects a nonzero element in certain degrees of , then it is not a monomorphism and further, for each \ell$">, is not a monomorphism in infinitely many degrees.

The second main theorem is that the elements of any -family in the cohomology of the Steenrod algebra, except at most its first elements, are either all detected or all not detected by , for every . Applications of this study to the cases and show that does not detect the three families , and , and that does not detect the family .

     

The strict Waring problem for polynomial rings

We prove among several results that under mild conditions any polynomial in Fq[t] is a strict sum of k⁴kth powers improving on an exponential (k²²k+1) bound of Car-Effinger-Hayes.     

Vanishing of cohomology groups of random simplicial complexes

We consider k‐dimensional random simplicial complexes generated from the binomial random (k + 1)‐uniform hypergraph by taking the downward‐closure. For 1 ≤ j ≤ k ? 1, we determine when all cohomology groups with coefficients in from dimension one up to j vanish and the zero‐th cohomology group is isomorphic to . This property is not deterministically monotone for this model, but nevertheless we show that it has a single sharp threshold. Moreover we prove a hitting time result, relating the vanishing of these cohomology groups to the disappearance of the last minimal obstruction. We also study the asymptotic distribution of the dimension of the j‐th cohomology group inside the critical window. As a corollary, we deduce a hitting time result for a different model of random simplicial complexes introduced by Linial and Meshulam, previously only known for dimension two.     

Constructions of strict Lyapunov functions for discrete time and hybrid time-varying systems

We provide explicit closed form expressions for strict Lyapunov functions for time-varying discrete time systems. Our Lyapunov functions are expressed in terms of known nonstrict Lyapunov functions for the dynamics and finite sums of persistency of excitation parameters. This provides a discrete time analog of our previous continuous time Lyapunov function constructions. We also construct explicit strict Lyapunov functions for systems satisfying nonstrict discrete time analogs of the conditions from Matrosov’s Theorem. We use our methods to build strict Lyapunov functions for time-varying hybrid systems that contain mixtures of continuous and discrete time evolutions.     

Orthogonal polynomial expansions for the matrix exponential

Many different algorithms have been suggested for computing the matrix exponential. In this paper, we put forward the idea of expanding in either Chebyshev, Legendre or Laguerre orthogonal polynomials. In order for these expansions to converge quickly, we cluster the eigenvalues into diagonal blocks and accelerate using shifting and scaling.     

The variational Lyapunov function and strict stability theory for differential systems

In this paper, we investigate strict stability of differential systems by variational Lyapunov function. We obtain some sufficient conditions and comparison theorems.     

张量分析和多项式优化的若干进展

张量分析 (也称多重数值线性代数) 主要包括张量分解和张量特征值的理论和算法,多项式优化主要包括目标和约束均为多项式的一类优化问题的理论和算法. 主要介绍这两个研究领域中若干新的研究结果. 对张量分析部分,主要介绍非负张量H-特征值谱半径的一些性质及求解方法,还介绍非负张量最大 (小) Z-特征值的优化表示及其解法;对多项式优化部分,主要介绍带单位球约束或离散二分单位取值、目标函数为齐次多项式的优化问题及其推广形式的多项式优化问题和半定松弛解法. 最后对所介绍领域的发展趋势做了预测和展望.     

Random walks on simplicial complexes and harmonics

In this paper, we introduce a class of random walks with absorbing states on simplicial complexes. Given a simplicial complex of dimension d, a random walk with an absorbing state is defined which relates to the spectrum of the k‐dimensional Laplacian for 1 ≤ k ≤ d. We study an example of random walks on simplicial complexes in the context of a semi‐supervised learning problem. Specifically, we consider a label propagation algorithm on oriented edges, which applies to a generalization of the partially labelled classification problem on graphs. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 49, 379–405, 2016     

Hochschild cohomology for periodic algebras of polynomial growth

We describe the dimensions of low Hochschild cohomology spaces of exceptional periodic representation-infinite algebras of polynomial growth. As an application we obtain that an indecomposable non-standard periodic representation-infinite algebra of polynomial growth is not derived equivalent to a standard self-injective algebra.     

A relation between the LG polynomial and the Kauffman polynomial

We give an explicit formula for the fact given by Links and Gould that a one variable reduction of the LG polynomial coincides with a one variable reduction of the Kauffman polynomial. This implies that the crossing number of an adequate link may be obtained from the LG polynomial by using a result of Thistlethwaite. We also give some evaluations of the LG polynomial.     

Inequalities for a polynomial and its derivative

For an arbitrary entire functionf and anyr>0, letM(f,r):=max_|z|=r |f(z)|. It is known that ifp is a polynomial of degreen having no zeros in the open unit disc, andm:=min_|z |=1|p(z)|, then

Explicit polynomial preserving trace liftings on a triangle

We give an explicit formula for a right inverse of the trace operator from the Sobolev space H¹(T) on a triangle T to the trace space H^1/2(?T) on the boundary. The lifting preserves polynomials in the sense that if the boundary data are piecewise polynomial of degree N, then the lifting is a polynomial of total degree at most N and the lifting is shown to be uniformly stable independently of the polynomial order. Moreover, the same operator is shown to provide a uniformly stable lifting from L₂(?T) to H^1/2(T). Finally, the lifting is used to construct a uniformly bounded right inverse for the normal trace operator from the space H (div; T) to H^–1/2(?T) which also preserves polynomials. Applications to the analysis of high order numerical methods for partial differential equations are indicated (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)