arXiv research

Extends inf-convolution to countable risk measures for risk sharing.

Every countable compact subset of sphere is tame.

Causal Set Theory's Hauptvermutung is resolved in two ways, one of which is true.

New algorithm for countable bandits with optimal regret.

We present a simple approach to questions of topological orbit equivalence for actions of countable groups on topological and smooth manifolds. For example, for any action of a countable group $Γ$ on a topological manifold where the fixed sets for any element are contained in codimension two submanifolds, every orbit e…

The purpose of this paper is to present, for all $n\ge 3$, very simple examples of continuous maps $f:M^{n-1} \to M^{n}$ from closed $(n-1)$-manifolds $M^{n-1}$ into closed $n$-manifold $M^n$ such that even though the singular set $S(f)$ of $f$ is countable and dense, the map $f$ can nevertheless be approximated by an …

Countable modular groups found on surfaces with infinite type.

Study shows singular set of distance functions is delta-convex.

We classify Veech groups of tame non-compact flat surfaces. In particular we prove that all countable subgroups of $\mathbf{GL}_+(2,\R)$ avoiding the set of mappings of norm less than 1 appear as Veech groups of tame non-compact flat surfaces which are Loch Ness monsters. Conversely, a Veech group of any tame flat surf…

Study chaotic behavior in homeomorphism groups of countable products of spaces.

We give a complete characterization of countable primitive groups in several settings including linear groups, subgroups of mapping class groups, groups acting minimally on trees and convergence groups. The latter category includes as a special case Kleinian groups as well as subgroups of word hyperbolic groups. As an …

We show that if $M$ is a sub-Riemannian manifold and $N$ is a Carnot group such that the nilpotentization of $M$ at almost every point is isomorphic to $N$, then there are subsets of $N$ of positive measure that embed into $M$ by bilipschitz maps. Furthermore, $M$ is countably $N$--rectifiable, i.e., all of $M$ except …

Finite groups can be represented as origami automorphisms, extended to countable groups.

The group of $\mathcal C^1$-diffeomorphisms of any sparse Cantor subset of a manifold is countable and discrete (possibly trivial). Thompson's groups come out of this construction when we consider central ternary Cantor subsets of an interval. Brin's higher dimensional generalizations $nV$ of Thompson's group $V$ arise…

We classify up to coarse equivalence all countable abelian groups of finite torsion free rank. The Q-cohomological dimension and the torsion free rank are the two invariants that give us such classification. We also prove that any countable abelian group of finite torsion free rank is coarsely equivalent to Z^n + H whe…

Let G be a group and let O_G denote the set of left orderings on G. Then O_G can be topologized in a natural way, and we shall study this topology to answer three conjectures. In particular we shall show that O_G can never be countably infinite. Furthermore in the case G is a countable nonabelian free group, we shall s…

In response to a 1997 problem of M. Vidyasagar, we state a criterion for PAC learnability of a concept class $\mathscr C$ under the family of all non-atomic (diffuse) measures on the domain $Ω$. The uniform Glivenko--Cantelli property with respect to non-atomic measures is no longer a necessary condition, and consisten…

Proves sufficiency of countable test plans for BV functions on metric spaces.

Classifies homeomorphism groups of countable Stone spaces up to coarse equivalence.

We prove that two countable locally finite-by-abelian groups G,H endowed with proper left-invariant metrics are coarsely equivalent if and only if their asymptotic dimensions coincide and the groups are either both finitely-generated or both are infinitely generated. On the other hand, we show that each countable group…

Finite approximations help reconstruct countable metric and ultrametric spaces.

We show that all non-trivial continuous endomorphisms of the circle group are topologically mixing. We also show that there exists a large infinite class of continuous endomorphisms of any n-dimensional torus group which are topologically mixing. Lastly, we prove that any continuous endomorphism on an abelian polish se…

Consider a nontrivial solution to a semilinear elliptic system of first order with smooth coefficients defined over an $n$-dimensional manifold. Assume the operator has the strong unique continuation property. We show that the zero set of the solution is contained in a countable union of smooth $(n-2)$-dimensional subm…

The study introduces Cayley--Abels--Rosendal graphs for Polish groups.

We prove that every countable family of countable acylindrically hyperbolic groups has a common finitely generated acylindrically hyperbolic quotient. As an application, we obtain an acylindrically hyperbolic group $Q$ with strong fixed point properties: $Q$ has property $FL^p$ for all $p\in [1, +\infty)$, and every ac…

Motivated by applications to bond markets, we propose a multivariate framework for discrete time financial markets with proportional transaction costs and a countable infinite number of tradable assets. We show that the no-arbitrage of second kind property (NA2 in short), recently introduced by Rasonyi for finite-dimen…

Characterizes continuity of monotone functionals in mixed topology.

Compact groups can be represented as dessin automorphisms.

We compute the asymptotic dimension of the rationals given with an invariant proper metric. Also, we show that a countable torsion abelian group taken with an invariant proper metric has asymptotic dimension zero.

This paper finds minimal sets of generators for mapping class groups of specific surfaces.

For an infinite cardinal $κ$ let $\ell_2(κ)$ be the linear hull of the standard othonormal base of the Hilbert space $\ell_2(κ)$ of density $κ$. We prove that a non-separable convex subset $X$ of density $κ$ in a locally convex linear metric space if homeomorphic to the space (i) $\ell_2^f(κ)$ if and only if $X$ can be…

For every countable group G we construct a compact path connected subspace K of R^4 whose fundamental group is isomorphic to G. Our construction is much simpler than the one found recently by Virk.

We study the finitely generated Hausdorff spectrum of spinal automorphism groups acting on rooted trees. Given any $α\in [0,1]$, we construct a branch group $G_α$ such that $G_α$ has a finitely generated subgroup $H$ where $H$ has Hausdorff dimension $α$ in $G$. Using results by Barnea, Shalev and Klopsch we further de…

In this paper we show that certain generalizations of the $C^r$-Whitney topology, which include the Hölder-Whitney and Sobolev-Whitney topologies on smooth manifolds, satisfy the Baire property, to wit, the countable intersection of open and dense sets is dense.

The framework of this paper is that of risk measuring under uncertainty, which is when no reference probability measure is given. To every regular convex risk measure on ${\cal C}_b(Ω)$, we associate a unique equivalence class of probability measures on Borel sets, characterizing the riskless non positive elements of $…

We prove that for every countable group G there exists a hyperbolic 3-manifold M such that the isometry group of M, the mapping class group of M, and the outer automorphism group of the fundamental group of M are isomorphic to G.

Let $G$ be a compact Lie group. (Compact) topological $G$-manifolds have the $G$-homotopy type of (finite-dimensional) countable $G$-CW complexes (2.5). This partly generalizes Elfving's theorem for locally linear $G$-manifolds [Elf96], wherein the Lie group $G$ is linear (such as compact).

We investigate the supports of extremal martingale measures with pre-specified marginals in a two-period setting. First, we establish in full generality the equivalence between the extremality of a given measure $Q$ and the denseness in $L^1(Q)$ of a suitable linear subspace, which can be seen in a financial context as…

In this paper we generalize the Local Removable Singularity Theorem in [16] for minimal laminations to the case of weak $H$-laminations (with $H\in \mathbb{R}$ constant) in a punctured ball of a Riemannian three-manifold. We also obtain a curvature estimate for any weak CMC foliation (with possibly varying constant mea…

Neural networks can approximate rectifiable measures with small error.

Study on generating and identifying languages privately, showing privacy imposes costs and creates barriers.

We provide a smoothening criterion for group actions on manifolds by singular diffeomorphisms. We prove that if a countable group $Γ$ has the fixed point property FW for walls (e.g. if it has property (T)), every aperiodic action of $Γ$ by diffeomorphisms that are of class $C^r$ with countably many singularities is con…

Let $G$ be a matrix group. Topological $G$-manifolds with Palais-proper action have the $G$-homotopy type of countable $G$-CW complexes (3.2). This generalizes E Elfving's dissertation theorem for locally linear $G$-manifolds (1996). Also we improve the Bredon--Floyd theorem from compact groups $G$ (1960).

Asymptotic property C was introduced by Dranishnikov to study spaces with infinite asymptotic dimension. We show that asymptotic property C is preserved by infinite products. We also show that countable restricted direct products of countable groups with finite asymptotic dimension have asymptotic property C. Then we i…

We show that C^1 hypersurfaces in the Heisenberg group are countably N-rectifiable. As a corollary, this shows that all C^1_H graphs over the xy-plane are countable N-rectifiable, showing the equivalence of this notion of rectifiability with that of Franchi, Serra Cassano and Serapioni for such surfaces.

The problem is sequence prediction in the following setting. A sequence x1,..., xn,... of discrete-valued observations is generated according to some unknown probabilistic law (measure) mu. After observing each outcome, it is required to give the conditional probabilities of the next observation. The measure mu belongs…

Quantum isometry groups extend to all countable metric spaces, and loose embeddings help understand metric space relationships.

New model for insurance states using Markov jump processes with non-countable state space.