On Waringʼs problem for systems of skew-symmetric forms

This paper is devoted to a problem of finding the smallest positive integer s(m,n,k)$s(m,n,k)$ such that (m+1)$(m+1)$ generic skew-symmetric (k+1)$(k+1)$-forms in (n+1)$(n+1)$ variables as linear combinations of the same s(m,n,k)$s(m,n,k)$ decomposable skew-symmetric (k+1)$(k+1)$-forms.     

Random partitions and edge-disjoint Hamiltonian cycles

Nash-Williams [1] proved that every graph with n   vertices and minimum degree n/2$n/2$ has at least ⌊5n/224⌋$\lfloor 5n/224\rfloor$ edge-disjoint Hamiltonian cycles. In [2], he raised the question of determining the maximum number of edge-disjoint Hamiltonian cycles, showing an upper bound of ⌊(n+4)/8⌋$\lfloor (n+4)/8\rfloor$.     

Characterization of full rank ACI-matrices over fields

An ACI-matrix over a field F$\mathbb{F}$ is a matrix whose entries are polynomials with coefficients on F$\mathbb{F}$, the degree of these polynomials is at most one in a number of indeterminates, and where no indeterminate appears in two different columns. In 2011 Huang and Zhan characterized the m×n$m\times n$ ACI-matrices such that all its completions have rank equal to min{m,n}$\min \{m,n\}$ whenever |F|?max{m,n+1}$|\mathbb{F}|?\max \{m,n+1\}$. We will give a characterization for arbitrary fields by introducing two classes of ACI-matrices: the maximal and the minimal full rank ACI-matrices.     

On the norming constants for normal maxima

Given n   independent standard normal random variables, it is well known that their maxima _Mn$M_n$ can be normalized such that their distribution converges to the Gumbel law. In a remarkable study, Hall proved that the Kolmogorov distance _dn$d_n$ between the normalized _Mn$M_n$ and its associated limit distribution is less than 3/log?n$3/\log ?n$. In the present study, we propose a different set of norming constants that allow this upper bound to be decreased with _dn≤C(m)/log?n$d_n\le C(m)/\log ?n$ for n≥m≥5$n\ge m\ge 5$. Furthermore, the function C(m)$C(m)$ is computed explicitly, which satisfies C(m)≤1$C(m)\le 1$ and _limm→∞?C(m)=1/3${\lim }_{m\to \infty }?C(m)=1/3$. As a consequence, some new and effective norming constants are provided using the asymptotic expansion of a Lambert W type function.     

The inverse inertia problem for the complements of partial k-trees

Let F$\mathbb{F}$ be an infinite field with characteristic not equal to two. For a graph G=(V,E)$G=(V,E)$ with V={1,…,n}$V=\{1,\dots ,n\}$, let S(G;F)$S(G;\mathbb{F})$ be the set of all symmetric n×n$n\times n$ matrices A=[_ai,j]$A=[a_{i,j}]$ over F$\mathbb{F}$ with _ai,j≠0$a_{i,j}\ne 0$, i≠j$i\ne j$ if and only if ij∈E$ij\in E$. We show that if G is the complement of a partial k  -tree and m?k+2$m?k+2$, then for all nonsingular symmetric m×m$m\times m$ matrices K   over F$\mathbb{F}$, there exists an m×n$m\times n$ matrix U   such that U^TKU∈S(G;F)$U^{T}KU\in S(G;\mathbb{F})$. As a corollary we obtain that, if k+2?m?n$k+2?m?n$ and G is the complement of a partial k-tree, then for any two nonnegative integers p and q   with p+q=m$p+q=m$, there exists a matrix in S(G;R)$S(G;\mathbb{R})$ with p positive and q negative eigenvalues.     

On the construction of optimal cubature formulae which use integrals over hyperspheres

We consider formulae of approximate integration over a d$d$-dimensional ball which use n$n$ surface integrals along (d-1)$(d-1)$-dimensional spheres centered at the origin. For a class of functions defined on the ball with gradients satisfying an integral restriction, optimal formulae of this type are obtained.     

Large time behavior of solutions to the nonlinear pseudo-parabolic equation

In this paper, we investigate the initial value problem for the nonlinear pseudo-parabolic equation. Global existence and optimal decay estimate of solution are established, provided that the initial value is suitably small. Moreover, when n?2$n?2$ and the nonlinear term f(u)$f(u)$ disappears, we prove that the global solutions can be approximated by the linear solution as time tends to infinity. When n=1$n=1$ and the nonlinear term f(u)$f(u)$ disappears, we show that as time tends to infinity, the global solution approaches the nonlinear diffusion wave described by the self-similar solution of the viscous Burgers equation.     

Triangularizing matrix polynomials

For an algebraically closed field F$\mathbb{F}$, we show that any matrix polynomial P(λ)∈F[λ]^n×m$P(\lambda )\in \mathbb{F}{[\lambda ]}^{n\times m}$, n?m$n?m$, can be reduced to triangular form, preserving the degree and the finite and infinite elementary divisors. We also characterize the real matrix polynomials that are triangularizable over the real numbers and show that those that are not triangularizable are quasi-triangularizable with diagonal blocks of sizes 1×1$1\times 1$ and 2×2$2\times 2$. The proofs we present solve the structured inverse problem of building up triangular matrix polynomials starting from lists of elementary divisors.     

An approximation method for strictly pseudocontractive mappings

Let K$K$ be a closed convex subset of a q$q$-uniformly smooth separable Banach space, T:K→K$T:K\to K$ a strictly pseudocontractive mapping, and f:K→K$f:K\to K$ an L$L$-Lispschitzian strongly pseudocontractive mapping. For any t∈(0,1)$t\in (0,1)$, let _xt$x_t$ be the unique fixed point of tf+(1-t)T$\mathit{tf}+(1-t)T$. We prove that if T$T$ has a fixed point, then {_xt}$\{x_t\}$ converges to a fixed point of T$T$ as t$t$ approaches to 0.     

Sur la répartition des puissances modulo 1

For almost all x>1$x>1$, (xⁿ)$(x^n)$(n=1,2,…)$(n=1,2,\dots )$ is equidistributed modulo 1, a classical result. What can be said on the exceptional set? It has Hausdorff dimension one. Much more: given an (b_n)$(b_n)$ in [0,1[$[0,1[$ and ε>0$\epsilon >0$, the x  -set such that |xⁿ−b_n|<ε$|x^n-b_n|<\epsilon$ modulo 1 for n   large enough has dimension 1. However, its intersection with an interval [1,X]$[1,X]$ has a dimension <1, depending on ε and X. Some results are given and a question is proposed.     

Re-nnd generalized inverses

In this paper, we give some necessary and sufficient conditions for the existence of Re-nnd and nonnegative definite {1,3}$\{1,3\}$- and {1,4}$\{1,4\}$-inverses of a matrix A∈C^n×n$A\in {\mathbb{C}}^{n\times n}$ and completely described these sets. Moreover, we prove that the existence of nonnegative definite {1,3}$\{1,3\}$-inverse of a matrix A   is equivalent with the existence of its nonnegative definite {1,2,3}$\{1,2,3\}$-inverse and present the necessary and sufficient conditions for the existence of Re-nnd {1,3,4}$\{1,3,4\}$-inverse of A.     

On matrices over an arbitrary semiring and their generalized inverses

In this paper, we consider matrices with entries from a semiring S. We first discuss some generalized inverses of rectangular and square matrices. We establish necessary and sufficient conditions for the existence of the Moore–Penrose inverse of a regular matrix. For an m×n$m\times n$ matrix A  , an n×m$n\times m$ matrix P and a square matrix Q of order m, we present necessary and sufficient conditions for the existence of the group inverse of QAP   with the additional property that P(QAP)^#Q$P{(QAP)}^{\mathrm{\#}}Q$ is a {1,2}$\{1,2\}$ inverse of A  . The matrix product used here is the usual matrix multiplication. The result provides a method for generating elements in the set of {1,2}$\{1,2\}$ inverses of an m×n$m\times n$ matrix A starting from an initial {1} inverse of A  . We also establish a criterion for the existence of the group inverse of a regular square matrix. We then consider a semiring structure (_Mm×n(S),+,°)$({\mathbf{M}}_{m\times n}(\mathbf{S}),+,°)$ made up of m×n$m\times n$ matrices with the addition defined entry-wise and the multiplication defined as in the case of the Hadamard product of complex matrices. In the semiring (_Mm×n(S),+,°)$({\mathbf{M}}_{m\times n}(\mathbf{S}),+,°)$, we present criteria for the existence of the Drazin inverse and the Moore–Penrose inverse of an m×n$m\times n$ matrix. When S is commutative, we show that the Hadamard product preserves the Hermitian property, and provide a Schur-type product theorem for the product A°(CC^?)$A°(C{C}^{?})$ of a positive semidefinite n×n$n\times n$ matrix A   and an n×n$n\times n$ matrix C.     

Diagonals and numerical ranges of direct sums of matrices

For any n-by-n matrix A  , we consider the maximum number k=k(A)$k=k(A)$ for which there is a k-by-k compression of A   with all its diagonal entries in the boundary ∂W(A)$\partial W(A)$ of the numerical range W(A)$W(A)$ of A. If A   is a normal or a quadratic matrix, then the exact value of k(A)$k(A)$ can be computed. For a matrix A   of the form B⊕C$B\oplus C$, we show that k(A)=2$k(A)=2$ if and only if the numerical range of one summand, say, B is contained in the interior of the numerical range of the other summand C   and k(C)=2$k(C)=2$. For an irreducible matrix A  , we can determine exactly when the value of k(A)$k(A)$ equals the size of A  . These are then applied to determine k(A)$k(A)$ for a reducible matrix A   of size 4 in terms of the shape of W(A)$W(A)$.     

Ewens sampling formulae with and without selection

We shall first consider the random Dirichlet partitioning of the interval into n   fragments at temperature θ>0.$\theta >0.$ Using calculus for Dirichlet integrals, pre-asymptotic versions of the Ewens sampling formulae from finite Dirichlet partitions follow up. From these preliminaries, straightforward proofs of the usual sampling formulae from random proportions with Poisson–Dirichlet (PD)(γ)$(\mathrm{PD})(\gamma )$ distribution can be obtained, while considering the Kingman limit n↗∞$n\nearrow \infty$, θ↘0$\theta \searrow 0$, with nθ=γ>0$n\theta =\gamma >0$.     

Concentration and multiplicity of solutions for a fourth-order equation with critical nonlinearity

In this paper, we consider the problem (_Pε)$(P_{\epsilon })$ : Δ²u=u^n+4/n-4+εu,u>0${\mathrm{\Delta }}^{2}u=u^{n+4/n-4}+\epsilon u,u>0$ in Ω,u=Δu=0$\Omega ,u=\mathrm{\Delta }u=0$ on ∂Ω$\mathrm{\partial }\Omega$, where Ω$\Omega$ is a bounded and smooth domain in Rⁿ,n>8${\mathbb{R}}^n,n>8$ and ε>0$\epsilon >0$. We analyze the asymptotic behavior of solutions of (_Pε)$(P_{\epsilon })$ which are minimizing for the Sobolev inequality as ε→0$\epsilon \to 0$ and we prove existence of solutions to (_Pε)$(P_{\epsilon })$ which blow up and concentrate around a critical point of the Robin's function. Finally, we show that for ε$\epsilon$ small, (_Pε)$(P_{\epsilon })$ has at least as many solutions as the Ljusternik–Schnirelman category of Ω$\Omega$.