Convolutions of K/(1 + χn)

Collocating Convolutions

An explicit method is derived for collocating either of the convolution integrals p(x) = fi f(x t)g(t)dt or q(x) = /*/(< x)g(t)dt, where x 6 (a, b), a subinterval of M . The collocation formulas take the form p = F(Am)% or q = F(Bm)g, where g is an w-vector of values of the function g evaluated at the "Sine points", Am and Bm are explicitly described square matrices of order m, and F(s) = ¡Qexp...

Spectra of Bernoulli Convolutions

Convolutions of Liouvillian Sequences

While Liouvillian sequences are closed under many operations, simple examples show that they are not closed under convolution, and the same goes for d’Alembertian sequences. Nevertheless, we show that d’Alembertian sequences are closed under convolution with rationally d’Alembertian sequences, and that Liouvillian sequences are closed under convolution with rationally Liouvillian sequences.

Vertex-Disjoint Copies of K1 + (K1 ∪K2) in Graphs

Let S denote the graph obtained from K4 by removing two edges which have an endvertex in common. Let k be an integer with k ≥ 2. Let G be a graph with |V (G)| ≥ 4k and σ2(G) ≥ |V (G)|/2 + 2k− 1, and suppose that G contains k vertex-disjoint triangles. In the case where |V (G)| = 4k+2, suppose further that G K4t+3 ∪ K4k−4t−1 for any t with 0 ≤ t ≤ k − 1. Under these assumptions, we show that G c...

Spatial approximation of stochastic convolutions

We study linear stochastic evolution partial differential equations driven by additive noise. We present a general and flexible framework for representing the infinite dimensional Wiener process which is driving the equation. Since the eigenfunctions and eigenvalues of the covariance operator of the process are usually not available for computations, we propose an expansion in an arbitrary fram...