We affirmatively settle the question on existence of a real-valued higher order spectral shift function for pair self-adjoint operators $H$ and $V$ such that is bounded $V(H-iI)^{-1}$ belongs to Schatten–von Neumann ideal $\mathcal{S}^n$ compact in separable Hilbert space. also show satisfies same trace formula as known case $V\in\mathcal{S}^n$ it unique up polynomial summand $n-1$. Our result ...

A superdiagonal composition is one in which the i-th part or summand of size greater than equal to i. In this paper, we study number palindromic compositions and colored compositions. particular, give generating functions explicit combinatorial formulas involving binomial coefficients Stirling numbers first kind.

Let p be prime, and $$n,m \in \mathbb {N}$$ . When K/F is a cyclic extension of degree $$p^n$$ , we determine the $$\mathbb {Z}/p^m\mathbb {Z}[\text {Gal}(K/F)]$$ -module structure $$K^\times /K^{\times p^m}$$ With at most one exception, each indecomposable summand free over some quotient group $$\text {Gal}(K/F)$$ For fixed values m n, there are only finitely many possible isomorphism classes ...

We describe the main ideas of Ray Heitmann’s proof of the Direct Summand Conjecture in dimension 3 for a ring of mixed characteristic [1]. In the first section we describe the main methods which are used and prove the necessary lemmas. In the second section we prove the main result of Heitmann’s paper. Finally, in the third section we give a proof of the Canonical Element Conjecture using this ...

The aim of this paper is to gain explicit information about the multiplicative structure of l∗l, where l is the connective Adams summand. Our approach differs from Kane’s or Lellmann’s because our main technical tool is the MU -based Künneth spectral sequence. We prove that the algebra structure on l∗l is inherited from the multiplication on a Koszul resolution of l∗BP . Introduction Our goal i...

The main objects of study in this paper are those functionals that analytic the sense they annihilate non-commutative disc algebra. In classical univariate case, a theorem F. and M. Riesz implies such must be given as integration against an absolutely continuous measure on circle. We develop generalizations result to multivariate setting, upon reinterpreting result. one direction, we show GNS r...

We present uniformly available simple enumerative formulae for unrooted planar n-edge maps (counted up to orientation-preserving isomorphism) of numerous classes including arbitrary, loopless, non-separable, eulerian maps and plane trees. All the formulae conform to a certain pattern with respect to the terms of the sum over t | n, t < n. Namely, these terms, which correspond to non-trivial aut...