For any split exact category C, we give an explicit description of the Dennis trace map, from Waldhausen's S-construction for C to the additive cyclic nerve of C. Introduction This paper is motivated by a conjecture of Loday, Geller and Weibel [2] on the compatibility of the Dennis trace map

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by the symbols a, b, c modulo [a, b] = [b, c] = [c, a]. We describe explicitly the images of the logarithms of all Drinfeld associators in a completion of the quotient L / [ [L, L], [L, L] ]. The main ingredient of our proofs is an explicit form of Cambell-Baker-Hausdorff formula in the case when all

Abstract. We present and study Poincare-invariant generalizations of the Galilei-invariant Toda systems. The classical nonperiodic systems are solved by means of an explicit action-angle transformation. Contents

Let R be a commutative Noetherian ring and p be a prime ideal of R such that the ideal pRp is principal and ht(p) 6 = 0. In this note, we describe the explicit structure of the injective envelope of the R-module R/p.

Many mathematicians have studied the classi�cation by the degree d ofembedded smooth projective varieties (see for instance [1], [3], [5], [6], [8],[9]). For d ≤ 6 Inonescu gave a complete list ([5],[6]). Later on the sameauthor ([7]) broadened the classi�cation of smooth projective varieties up todegree 8. In particular, he constructed ([7]; 4.2) a smooth projective surface ofdegree 8 in P5 using Reiders Theorem in the following way: Let X be a geometrically ruled surface over a curve of genus 2 withinvariant e = −2 and let H ≡ C0+3F where C0 and F are the generatorsof the Picard group of X. Then, by Reiders Theorem H is very ample andtherefore embeds X as a smooth projective surface in P5. It is well known that if X is a geometrically ruled surface over a curve C,then there exists a locally free sheaf E of rank two on C such that X = P(E).This gives rise the following question: Which are the vector bundles E over a curve C of genus 2 such thatX = P(E) is the surface described by Ionescu and when does OP(E)(1)embed X in P5 as a smooth surface of degree 8?

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the Lorentz cone, are the analytic continuation of the distribution dened by a rela-tively invariant measure on a homogeneous cone. In general, Riesz distributions are compositions of complex measures supported by the closure of the cone with dif-ferential operators. Gindikin [3] describes when the Riesz distribution is a positive

Although Gaussian RBF kernels are one of the most often used kernels in modern machine learning methods such as support vector machines (SVMs), little is known about the structure of their reproducing kernel Hilbert spaces (RKHSs). In this work we give two distinct explicit descriptions