abstract. let mn;m be the set of n-by-m matrices with entries inthe field of real numbers. a matrix r in mn = mn;n is a generalizedrow substochastic matrix (g-row substochastic, for short) if re e, where e = (1; 1; : : : ; 1)t. for x; y 2 mn;m, x is said to besgut-majorized by y (denoted by x sgut y ) if there exists ann-by-n upper triangular g-row substochastic matrix r such thatx = ry . th...

This online supplement to Bierens (2013) contains the omitted proofs. Throughout I will use the same notations as in Bierens (2013), as follows. The indicator function is denoted by I(.), and N and N0 denote the sets of positive and nonnegative integers, respectively. The partial derivative to a parameter with index k will be denoted by ∇k, and ∇k,m denotes the second partial derivatives to par...

This paper presents some results that complement (2). We believe our results are of new pattern concerning determinantal inequalities. Let us fix some notation. The matrices considered here have entries from the field of complex numbers. X ′,X ,X∗ stand for transpose, (entrywise)conjugate, conjugate transpose of X , respectively. For two n -square Hermitian matrices X ,Y , we write X > Y to mea...

A Roman dominating function (RDF) on a graphG = (V,E) is a function f : V −→ {0, 1, 2} satisfying the condition that every vertex u for which f(u) = 0 is adjacent to at least one vertex v for which f(v) = 2. The weight of an RDF is the value f(V (G)) = ∑ u∈V (G) f(u). An RDF f in a graph G is independent if no two vertices assigned positive values are adjacent. The Roman domination number γR(G)...

Abstract. We study a billiard in a modified square with pockets of small size ε. The length, respectively the number of reflections in the side cushions, of the trajectory that originates at one corner under angle θ is denoted by lε(θ), respectively by Rε(θ). We prove that the probability measures associated with the random variables εlε and εRε are convergent as ε ց 0 and provide explicit form...

LINE [1], as an efficient network embedding method, has shown its effectiveness in dealing with large-scale undirected, directed, and/or weighted networks. Particularly, it proposes to preserve both the local structure (represented by First-order Proximity) and global structure (represented by Second-order Proximity) of the network. In this study, we prove that LINE with these two proximities (...

We extend the Andrews-Olsson identity to two-colored partitions. Regarding the sets of proper Young walls of quantum affine algebras gn = A (2) 2n , A (2) 2n−1, B (1) n , D (1) n and D (2) n+1 as the sets of two-colored partitions, the extended Andrews-Olsson identity implies that the generating functions of the sets of reduced Young walls have very simple formulae: ∏ ∞ i=1(1 + t )i where κi = ...