Three decompositions of a substochastic transition function are shown to yield substochastic parts. These are the Lebesgue decomposition with respect to a finite measure, the decomposition into completely atomic and continuous parts, and on Rn, a decomposition giving a part with continuous distribution function and a part with discontinuous distribution function. Introduction. The present paper...

Let Q be an independent topos. Recently, there has been much interest in the derivation of open, parabolic domains. We show that every invariant, partial random variable is Green–Atiyah and substochastic. In [32], the authors address the smoothness of affine isomorphisms under the additional assumption that |g| ∼= ∅. I. Lee’s computation of surjective curves was a milestone in elementary number...

Let G be a simple connected graph, and D(G) the distance matrix of G. Suppose that $$D_{\max }(G)$$ $$\lambda _1(G)$$ are maximum row sum spectral radius D(G), respectively. In this paper, we give lower bound for }(G)-\lambda , characterize extremal graphs attaining bound. As corollary, solve conjecture posed by Liu, Shu Xue.

Convolution as a Matrix/Vector multiplication Notice that (2) can be written as g = Y x where g is a column vector with elements gk = (x ∗ y)k, x is a column vector with elements xk, and Y is an NxN matrix. By examining (2), we can deduce that the elements of the first row of the matrix Y should be Y0,: = {y0, y−1, y−2, ..., y−(N−1)} Similarly, the second row should be Y1,: = {y1, y0, y−1, ...,...

Let R = (r1, . . . , rm) and C = (c1, . . . , cn) be positive integer vectors such that r1 + . . .+ rm = c1 + . . .+ cn. We consider the set Σ(R, C) of non-negative m × n integer matrices (contingency tables) with row sums R and column sums C as a finite probability space with the uniform measure. We prove that a random table D ∈ Σ(R, C) is close with high probability to a particular matrix (“t...

The Brudnyi-Krugljak theorem on the if-divisibility of Gagliardo couples is derived by elementary means from earlier results of LorentzShimogaki on equimeasurable rearrangements of measurable functions. A slightly stronger form of Calderón's theorem describing the Hardy-LittlewoodPólya relation in terms of substochastic operators (which itself generalizes the classical Hardy-Littlewood-Pólya re...

This paper gives a tight upper bound on the spectral radius of the signless Laplacian of graphs of given order and clique number. More precisely, let G be a graph of order n, let A be its adjacency matrix, and let D be the diagonal matrix of the row-sums of A. If G has clique number ω, then the largest eigenvalue q (G) of the matrix Q = A+D satisfies q (G) ≤ 2 (1− 1/ω)n. If G is a complete regu...

It is a long open problem to combinatorially characterize the 3D bar-joint rigidity of graphs. The problem is at the intersection of combinatorics and algebraic geometry, and crops up in practical algorithmic applications ranging from mechanical computer aided design to molecular modeling. The problem is equivalent to combinatorially determining the generic rank of the 3D bar-joint rigidity mat...

j, 1 i 2n, 1 j 2n, where J2n is the 2n by 2n matrix of all entries 1. It is known that symmetric Bush–type Hadamard matrices exist for all orders 16n2 for all values of n for which there is a Hadamard matrix of order 4n, see [8] for details. A balanced generalized weighing matrix BGW ν κ λ over a group G is a matrix W wi j of order ν , with wi j G 0 such that each row and column of W has κ non–...