Operads are a specific tool for encoding type of algebras. For instance there are operads encoding associative algebras, commutative and associative algebras, Lie algebras, pre-Lie algebras, dendriform algebras, Poisson algebras and so on. A usual way of describing a type of algebras is by giving the generating operations and the relations among them. For instance a Lie algebra L is a vector sp...

I prove a formula expressing the descendent genus g Gromov-Witten invariants of a projective variety X in terms of genus 0 invariants of its symmetric product stack S(X). When X is a point, the latter are structure constants of the symmetric group, and we obtain a new way of calculating the Gromov-Witten invariants of a point.

Let X be a globally symmetric space of noncompact type, and Γ ⊂ Isom(X) a Schottky group of axial isometries. Then M := X/Γ is a locally symmetric Riemannian manifold of infinite volume. The goal of this note is to give an asymptotic estimate for the number of primitive closed geodesics in M modulo free homotopy with period less than t.

in this paper we introduce two symmetric variants of amenability, symmetric module amenability and symmetric connes amenability. we determine symmetric module amenability and symmetric connes amenability of some concrete banach algebras. indeed, it is shown that $ell^1(s)$ is  a symmetric $ell^1(e)$-module amenable if and only if $s$ is amenable, where $s$ is an inverse semigroup with subsemigr...

We discuss the product formula for the c-function for a Riemannian symmetric space and the similar formula for a noncompactly causal symmetric space. We derive two functional equations between the c-functions using the product formulas. Finally some examples where the c-functions turn up in analysis on symmetric spaces are given. Let G be a connected semisimple Lie group with nite center. Let K...

We investigate Buchbaum and Eisenbud's construction of the second symmetric power $s_R(X)$ of a chain complex $X$ of modules over a commutative ring $R$. We state and prove a number of results from the folklore of the subject for which we know of no good direct references. We also provide several explicit computations and examples. We use this construction to prove the following vers...

In this paper we will prove that if $L$ is a continuous symmetric n-linear form on a Hilbert space and $widehat{L}$ is the associated continuous n-homogeneous polynomial, then $||L||=||widehat{L}||$. For the proof we are using a classical generalized  inequality due to S. Bernstein for entire functions of exponential type. Furthermore we study the case that if X is a Banach space then we have t...

We show that if an infinite-dimensional Banach space X has a symmetric basis then there exists a bounded, linear operator R : X −→ X such that the set A = {x ∈ X : ||Rx|| → ∞} is non-empty and nowhere dense in X. Moreover, if x ∈ X \ A then some subsequence of (Rx)n=1 converges weakly to x. This answers in the negative a recent conjecture of Prǎjiturǎ. The result can be extended to any Banach s...

Let X be a symmetric space of non-compact type and Γ\X a locally symmetric space. Then the bottom spectrum λ1(Γ\X) satisfies the inequality λ1(Γ\X) ≤ λ1(X). We show that if equality λ1(Γ\X) = λ1(X) holds, then Γ\X has either one end, which is necessarily of infinite volume, or two ends, one of infinite volume and another of finite volume. In the latter case, Γ\X is isometric to R1 ×N endowed wi...

Let X be a symmetric space of non-compact type and Γ\X a locally symmetric space. Then the bottom spectrum λ1(Γ\X) satisfies the inequality λ1(Γ\X) ≤ λ1(X). We show that if equality λ1(Γ\X) = λ1(X) holds, then Γ\X has either one end, which is necessarily of infinite volume, or two ends, one of infinite volume and another of finite volume. In the latter case, Γ\X is isometric to R1 ×N endowed wi...