Let A = (aij ) ∈ Rn×n,N = {1, . . . , n} and DA be the digraph (N, {(i, j); aij > −∞}). The matrix A is called irreducible if DA is strongly connected, and strongly irreducible if every maxalgebraic power of A is irreducible. A is called robust if for every x with at least one finite component, A(k) ⊗ x is an eigenvector of A for some natural number k. We study the eigenvalue–eigenvector proble...

In the feature subset selection problem, a learning algorithm is faced with the problem of selecting a relevant subset of features upon which to focus its attention, while ignoring the rest. To achieve the best possible performance with a particular learning algorithm on a particular training set, a feature subset selection method should consider how the algorithm and the training set interact....

The Grothendieck constant KG is the smallest constant such that for every d ∈ N and every matrix A = (aij),

where the matrix (aij) is symmetric, and uniformly elliptic (aij) ≥ γI, for some γ > 0. In terms of regularity of the coefficients, let us assume the are continuous functions. As opposed to the interior and boundary regularity estimates, where we worked with integral quantities, here we work with pointwise estimates on the solution. The point is the following: assume u ∈ C2(Ω) attains a maximum...

A similar concentration result is available for subgaussian random variables. It is known as the Hanson-Wright inequality and is given in Proposition 2 below. First versions of this inequality can be found in Hanson and Wright [5] and Wright [9], although with a weaker statement than Proposition 2 below since these results involve ||| (|aij |) |||2 instead of |||A|||2. Recent proofs of this con...

A vertex coloring of a graph is said to be perfect with parameters (aij) k i,j=1 if for every i, j ∈ {1, ..., k} every vertex of color i is adjacent with exactly aij vertices of color j. We consider the perfect 2-colorings of the distance-2 graph of the 24-cube {0, 1}24 with parameters ((20+ c, 256− c)(c, 276− c)) (i.e., with eigenvalue 20). We prove that such colorings exist for all c from 1 t...

(V,A, c) be a directed network with node set V , arc set A = {e1, · · · , em}, and capacities on arcs c : A 7→ R+, where R+ is the set of non-negative rational numbers. Let s, t ∈ V be two distinct nodes, designated as the source and the destination. Let P = {p1, · · · , pk} be the set of s-t paths in G. Let aij = 1 if arc ei ∈ A lies on path pj ∈ P and aij = 0 otherwise. Let f : P 7→ R+ be a f...

We give upper and lower bounds on the largest singular value of a matrix using analogues to walks in graphs. For nonnegative matrices these bounds are asymptotically tight. In particular, the following result improves a bound due to Schur. If A = (aij) is an m n complex matrix, its largest singular value satis…es 2 (A) max i2[m] P j2[n] jaij j cj max aij 6=0 ricj ; where ri = P k2[n] jaikj ; cj...

We examine whether the job characteristics of physical demands and environmental conditions affect individual's health. Five-year cumulative measures of these job characteristics are used to reflect findings in the biological and physiological literature that indicate that cumulative exposure to hazards and stresses harms health. To create our analytic sample, we merge job characteristics from ...