In this paper, we derive a capacity inner bound for a synchronous Gaussian Z channel with intersymbol interference (ISI) under input power constraints. This is done by converting the original channel model into an n-block memoryless circular Gaussian Z channel (n-CGZC) and successively decomposing the n-block memoryless channel into a series of independent parallel channels in the frequency dom...

In the context of mod-Gaussian convergence, as defined previously in our work with J. Jacod, we obtain asymptotic formulas and lower bounds for local probabilities for a sequence of random vectors which are approximately Gaussian in this sense, with increasing covariance matrix. This is motivated by the conjecture concerning the density of the set of values of the Riemann zeta function on the c...

Some positivity conjectures are made generalizing known results for Gaussian posets. For a sequence of non-negative integers ~ a = (a 1 ; a 2 ; ; a n), and a non-negative integer m, MacMahon 1, p. 137] considered the rational function of q,

V r /2 t,p/2 is a scaled version of Gaussian measure with r p if 1 < p < 2 and r p ′ if 2 < p < ∞ with 1/p 1/p′ 1. Conversely if F is a holomorphic function which is in a “slightly” smaller space, namely L C, V r/2 t,p/2 , then it is shown that there is a function f ∈ L R, ρ such that e−tHf F. However, a single necessary and sufficient condition is obtained for the image of L2 R, ρp/2 under e−t...

Approximate Gaussian Elimination

The first thing taught in Math 340 is Gaussian Elimination, i.e. the process of transforming a matrix to reduced row echelon form by elementary row operations. Because this process has the effect of multiplying the matrix by an invertible matrix it has produces a new matrix for which the solution space of the corresponding linear system is unchanged. This is made precise by Theorem 2.4 below. T...