Given a formal power series g(x) = b0 +b1x+b2x2 +··· and a nonunit f(x) = a1x+ a2x2+··· , it is well known that the composition of g with f , g(f(x)), is a formal power series. If the formal power series f above is not a nonunit, that is, the constant term of f is not zero, the existence of the composition g(f(x)) has been an open problem for many years. The recent development investigated the ...

Proof: As in [Z], first use the Stembridge-Stanton trick, to transform (*) to the equivalent ”antisymmetric” version, let’s call it (*’), in which ( qxj xi )a is replaced by ( qxj xi ) a−1 , and the right side of (*) gets multiplied by (1− t)/(t; t)a. Next expand the very first product on the left of (*’), into a sum of 2 terms, and note that they are all bad guys (see [Z], p. 314 ), except for...

We investigate the maximum number of edges that a graph G can have if it does not contain a given graph H as a minor (subcontraction). Let c(H) = inf { c : e(G) ≥ c|G| implies G ≻ H } . We define a parameter γ(H) of the graph H and show that, if H has t vertices, then c(H) = (αγ(H) + o(1)) t √ log t where α = 0.319 . . . is an explicit constant and o(1) denotes a term tending to zero as t → ∞. ...

We determine the general form of the asymptotics for Dirichlet eigenvalues of the one–dimensional linear damped wave operator. As a consequence, we obtain that given a spectrum corresponding to a constant damping term this determines the damping term in a unique fashion. We also derive a trace formula for this problem.

We study the number Nγ(n, c, q) of irreducible polynomials of degree n over Fq where the trace γ and the constant term c are given. Under certain conditions on n and q, we obtain bounds on the maximum of Nγ(n, c, q) varying c and γ. We show with concrete examples how our results improve previous known bounds. In addition, we improve upper and lower bounds of any Nγ(n, c, q) when n = a(q − 1) fo...

• Basic estimates • The hierarchy of constant terms Let G be a reductive real Lie group, for example G = GL(n,R). Let A be a maximal R-split torus, in the case of GL(n,R) the diagonal matrices, with connected component of the identity A. Let K be a maximal compact subgroup of G, for GL(n,R) the standard orthogonal group O(n). Let N be the unipotent radical of a minimal parabolic containing A, i...

This paper provides a theoretical analysis of domain adaptation based on the PACBayesian theory. We propose an improvement of the previous domain adaptation bound obtained by Germain et al. [1] in two ways. We first give another generalization bound tighter and easier to interpret. Moreover, we provide a new analysis of the constant term appearing in the bound that can be of high interest for d...

For a finite Coxeter group, W , and its reflection representation h, we find the character and Hilbert series for a quotient ring of C[h⊕h∗] by an ideal containing the W–invariant polynomials without constant term. This confirms conjectures of Haiman.