Impedance Eigenvalues in Linear Elasticity

En224: Linear Elasticity

Linear combinations of graph eigenvalues

Let 1 (G) : : : n (G) be the eigenvalues of the adjacency matrix of a graph G of order n; and G be the complement of G: Suppose F (G) is a …xed linear combination of i (G) ; n i+1 (G) ; i G ; and n i+1 G ; 1 i k: We show that the limit lim n!1 1 n max fF (G) : v (G) = ng always exists. Moreover, the statement remains true if the maximum is taken over some restricted families like “Kr-free”or “r...

Eigenvalues of a Linear Transformation

We adopt the following convention: i, j, m, n denote natural numbers, K denotes a field, and a denotes an element of K. Next we state several propositions: (1) Let A, B be matrices over K, n1 be an element of Nn, and m1 be an element of Nm. If rng n1 × rngm1 ⊆ the indices of A, then Segm(A + B,n1,m1) = Segm(A,n1,m1) + Segm(B,n1,m1). (2) For every without zero finite subset P of N such that P ⊆ ...

Absence of Positive Eigenvalues for the Linearized Elasticity System

In this paper, we prove that the linearized elasticity system has no eigenvalues in two geometric situations: the whole space R and a local perturbation of the half-space. We consider the Lamé coefficients and the density varying in an unbounded part of the domain. For the whole space, we use the operations curl and div to reduce our system to a scalar problem and use a limiting absorption prin...

Eigenvalues and Linear Quasirandom Hypergraphs

Let p(k) denote the partition function of k. For each k > 2, we describe a list of p(k) − 1 quasirandom properties that a k-uniform hypergraph can have. Our work connects previous notions on linear hypergraph quasirandomness by Kohayakawa, Rödl, and Skokan, and by Conlon, Hàn, Person, and Schacht, and the spectral approach of Friedman and Wigderson. For each of the quasirandom properties that i...