arXiv research

We introduce the notion of pullback along a measurable cocycle and we use it to extend the Borel invariant studied by Bucher, Burger and Iozzi to the world of measurable cocycles. The Borel invariant is constant along cohomology classes and has bounded absolute value. This allows to define maximal cocycles. We conclude…

Develops measures for non-Borel Anosov groups on Furstenberg boundary.

New findings on null measurability in symmetrization interface of VC learning.

The paper discusses rigidity results for inequalities on weighted Riemannian manifolds.

Survey of recent measures of association, including a new coefficient.

A Hilbert space embedding for probability measures has recently been proposed, wherein any probability measure is represented as a mean element in a reproducing kernel Hilbert space (RKHS). Such an embedding has found applications in homogeneity testing, independence testing, dimensionality reduction, etc., with the re…

In this paper, we show that the Euler characteristic of an even dimensional closed projectively flat manifold is equal to the total measure which is induced from a probability Borel measure on RP^n invariant under the holonomy action, and then discuss its consequences and applications. As an application, we show that t…

Let S be a non-exceptional oriented surface of finite type. We discuss the action of subgroups of the mapping class group of S on the CAT(0)-boundary of the completion of Teichmueller space with respect to the Weil-Petersson metric. We show that the set of invariant Borel probability measures for the Weil-Petersson flo…

Study proves existence of robust classifiers in multiclass adversarial training.

The paper develops axioms for uniquely decomposing functions with real arguments.

It was proved in 1998 by Ben-David and Litman that a concept space has a sample compression scheme of size d if and only if every finite subspace has a sample compression scheme of size d. In the compactness theorem, measurability of the hypotheses of the created sample compression scheme is not guaranteed; at the same…

The paper shows how MMD metrizes weak convergence for certain kernels.

Improved Cauchy-Schwarz inequality for $L^1$ and $L^2$ norms.

We show that a positive Borel measure of positive finite total mass, on compact Hermitian manifolds, admits a Holder continuous quasi-plurisubharmonic solution to the Monge-Ampere equation if and only if it is dominated locally by Monge-Ampere measures of Holder continuous plurisubharmonic functions.

Classifies manifolds and discrete subgroups of Lie groups using descriptive set theory.

The study shows how certain ODEs and integrals are regular under Borel summation.

New findings on cusped Borel Anosov representations and their properties.

We establish orbit equivalence rigidity for any ergodic, essentially free and measure-preserving action on a standard Borel space with a finite positive measure of the mapping class group for a compact orientable surface with higher complexity. We prove similar rigidity results for a finite direct product of mapping cl…

We continue our study of the Complex Monge-Ampère Operator on the Weighted Pluricomplex energy classes. We give more characterizations of the range of the classes $\mathcal E_ χ$ by the Complex Monge-Ampère Operator. In particular, we prove that a non-negative Borel measure $μ$ is the Monge-Ampère of a unique function …

Study shows Roller compactification's median graph has limited asymptotic dimension.

This work establishes properties on diffeological structures for set-valued maps and measures.

In this project we further investigate the idea of reducing the dimensionality of datasets using a Borel isomorphism with the purpose of subsequently applying supervised learning algorithms, as originally suggested by my supervisor V. Pestov (in 2011 Dagstuhl preprint). Any consistent learning algorithm, for example kN…

Proves resurgent nature of a series solution to deformed Painlevé I equation.

Geometrically proves WKB solutions of Schrödinger equations are resurgent.

It is shown that bootstrap approximations of an estimator which is based on a continuous operator from the set of Borel probability measures defined on a compact metric space into a complete separable metric space is stable in the sense of qualitative robustness. Support vector machines based on shifted loss functions …

We describe an explicit semi-algebraic partition for the complement of a real hyperplane arrangement such that each piece is contractible and so that the pieces form a basis of Borel-Moore homology. We also give an explicit correspondence between the de Rham cohomology and the Borel-Moore homology.

We solve the Dirichlet problem for the complex Monge-Ampère equation on a strictly pseudoconvex with the right hand side being a positive Borel measure which is dominated by the Monge-Ampère measure of a Hölder continuous plurisubharmonic function. If the boundary data is continuous, then the solution is continuous. If…

Let Q be a component of a stratum of abelian or quadratic differentials on an oriented surface of genus $g\geq 0$ with $m\geq 0$ punctures and $3g-3+m\geq 2$. We construct a subshift of finite type $(Ω,σ)$ and a Borel suspension of $(Ω,σ)$ which admits a finite-to-one semi-conjugacy into the Teichmueller flow $Φ^t$ on …

The paper proves inequalities for hyperbolic sets and curves.

We prove the Borel Conjecture for a class of groups containing word-hyperbolic groups and groups acting properly, isometrically and cocompactly on a finite dimensional CAT(0)-space.

We provide a general construction of time-consistent sublinear expectations on the space of continuous paths. It yields the existence of the conditional G-expectation of a Borel-measurable (rather than quasi-continuous) random variable, a generalization of the random G-expectation, and an optional sampling theorem that…

We introduce an universum of the Polish (=complete separable metric) space - the convex cone of distance matrices and study its geometry. It happened that the generic Polish spaces in this sense of this universum is so called Urysohn spaces defined by P.S.Urysohn in 20-th, and generic metric triple (= metric space with…

A theorem divides hyperplanes evenly with a line through the origin.

We consider the family of harmonic measures on a lamination $\mathcal{L}$ of a compact space $X$ by locally symmetric spaces $L$ of noncompact type, i.e. $L\simeq Γ_L\backslash G/K$. We establish a natural bijection between these measures and the measures on an associated lamination foliated by $G$-orbits, $\hat{\mathc…

Solves Christoffel problem for disk area measures on spheres.

Study parametrized Kähler class for cocycles on Hermitian symmetric spaces.

We give a survey of the approaches to classifying foliations, starting with the Haefliger classifying spaces and the various results and examples about the secondary classes of foliations. Various dynamical properties of foliations are introduced and discussed, including expansion rate, local entropy, and orbit growth …

Study on Borel Anosov subgroups in SL(d,R) for d≠5,8k±1.

New weighted surface area measures for convex bodies with applications.

Bayesian histograms achieve optimal distribution estimation with minimal memory usage.

Maximal and Borel Anosov representations in $Sp(4,\mathbb{R})$ are proven to be Hitchin.

Proves summability of state integrals for specific hyperbolic knots.

An elementary proof shows submodular functions can be represented as measure suprema.

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 $…

Defines magnitude for length spaces with measures, agreeing with finite spaces' magnitude.

Let $X$ be an algebraic variety over $\mathbb{C}$. We say that $X$ is Borel hyperbolic if, for every finite type reduced scheme $S$ over $\mathbb{C}$, every holomorphic map $S^{an}\to X^{an}$ is algebraic. We use a transcendental specialization technique to prove that $X$ is Borel hyperbolic if and only if, for every s…

A one-to-one correspondence is drawn between law invariant risk measures and divergences, which we define as functionals of pairs of probability measures on arbitrary standard Borel spaces satisfying a few natural properties. Divergences include many classical information divergence measures, such as relative entropy a…

Study shows how nonstandard hulls can contain metric completions.