Related papers: Equivariant means

The classical de Finetti Theorem classifies the $\mathrm{Sym}(\mathbb N)$-invariant probability measures on $[0,1]^{\mathbb N}$. More precisely it states that those invariant measures are combinations of measures of the form…

Let $(X,(p_j))$ be a Fr\'echet space with a Schauder basis and without continuous norm, where $(p_j)$ is an increasing sequence of seminorms inducing the topology of $X$. We show that $X$ satisfies the Invariant Subspace Property if and…

We discuss the Pointwise Ergodic Theorem for the Gaussian divisor function $d(n)$, that is, for a measure preserving $\mathbb Z[i]$ action $T$, the limit $$\lim_{N\rightarrow \infty} \frac{1}{D(N)} \sum _{\mathscr{N} (n) \leq N} d(n)…

Let $\pi$ be a group equipped with an action of a second group $G$ by automorphisms. We define the equivariant cohomological dimension ${\sf cd}_G(\pi)$, the equivariant geometric dimension ${\sf gd}_G(\pi)$, and the equivariant…

We show that if an open cover of a finite dimensional space is equivariant with respect to some finite group action on the space then there is an equivariant refinement of bounded dimension. This will generalize some constructions of…

Let X be a smooth, complete, connected submanifold of dimension n < N in a complex affine space A^N (C), and r is the rank of its Gauss map \gamma, \gamma (x) = T_x (X). The authors prove that if 2 \leq r \leq n - 1, N - n \geq 2, and in…

Let $G$ be a countable residually finite group (for instance $\mathbb{F}_2$) and let $\overleftarrow{G}$ be a totally disconnected metric compactification of $G$ equipped with the action of $G$ by left multiplication. For every $r\geq 1$ we…

Earlier, for an action of a finite group $G$ on a germ of an analytic variety, an equivariant $G$-Poincar\'e series of a multi-index filtration in the ring of germs of functions on the variety was defined as an element of the Grothendieck…

For an orbifold X which is the quotient of a manifold Y by a finite group G we construct a noncommutative ring with an action of G such that the orbifold cohomology of X as defined in math.AG/0004129 by Chen and Ruan is the G invariant…

The concepts of mean (i.e., average) and covariance of a random variable are fundamental in statistics, and are used to solve real-world problems such as those that arise in robotics, computer vision, and medical imaging. On matrix Lie…

Let $\Gamma$ be a countable group. A classical theorem of Thorisson states that if $X$ is a standard Borel $\Gamma$-space and $\mu$ and $\nu$ are Borel probability measures on $X$ which agree on every $\Gamma$-invariant subset, then $\mu$…

We consider stochastic dynamical systems on ${\mathbb{R}}$, that is, random processes defined by $X_n^x=\Psi_n(X_{n-1}^x)$, $X_0^x=x$, where $\Psi _n$ are i.i.d. random continuous transformations of some unbounded closed subset of…

Mean equicontinity is a well studied notion for actions. We propose a definition of mean equicontinuous factor maps that generalizes mean equicontinuity to the relative context. For this we work in the context of countable amenable groups.…

In this note, we generalise the concept of topo-isomorphic extensions and define finite topomorphic extensions as topological dynamical systems whose factor map to the maximal equicontinuous factor is measure-theoretically at most…

The goal of this note is to provide a geometric setting in which generalized arithmetic means are best predictors in an appropriate metric. This characterization provides a geometric interpretation to the concept of certainty equivalent.…

Let $G$ be a locally compact group with the group algebra $L^1(G)$ and $N$ be a closed normal subgroup of $G$. Suppose that $\xi:N\to\mathbb{T}$ is a continuous character and $L_\xi^1(G,N)$ is the $L^1$-space of all covariant functions of…

A topological space $Y$ is said to have (AEEP) if the following condition is fulfilled. Whenever $(X,\mathfrak{M})$ is a measurable space and $f, g: X \to Y$ are two measurable functions, then the set $\Delta(f,g) = \{x \in X:\ f(x) =…

Iteration of randomly chosen quadratic maps defines a Markov process: X_{n+1}=\epsilon_{n+1}X_n(1-X_n), where \epsilon_n are i.i.d. with values in the parameter space [0,4] of quadratic maps F_{\theta}(x)=\theta x(1-x). Its study is of…

We define so-called residual means, which have a Taylor expansion of the form $M(x)=\bar x +\tfrac12 \xi_M(\bar x) \text{Var}(x)+o(\|x-\bar x\|^\alpha)$ for some $\alpha>2$ and a single-variable function $\xi_M$ ($\bar x$ stands for the…

We prove several results of the following type: given finite dimensional normed space V there exists another space X with log (dim X) = O(log (dim V)) and such that every subspace (or quotient) of X, whose dimension is not "too small,"…