arXiv research

Locally homogeneous RCD spaces are shown to be smooth manifolds.

Extends rigidity results to non-homogeneous manifolds.

The article discusses invariant measures outside homogeneous dynamics.

Hyperbolic groups' infinite orbits spread evenly in spaces.

Study on risk contributions of portfolios using lambda quantile risk measures.

In this paper we discuss general properties of geodesic surfaces that are locally biLipschitz homogeneous. In particular, we prove that they are locally doubling and that there exists a special doubling measure analogous to the Haar measure for locally compact groups.

Optimal quantization of measures on Carnot groups

We obtain the $C^{\a}$ regularity for weak solutions of a class of non-homogeneous ultraparabolic equation, with measurable coefficients. The result generalizes our recent $C^{\a}$ regularity results of homogeneous ultraparabolic equation.

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

The theorems of M. Ratner, describing the finite ergodic invariant measures and the orbit closures for unipotent flows on homogeneous spaces of Lie groups, are extended for actions of subgroups generated by unipotent elements. More precisely: Let G be a Lie group (not necessarily connected) and Gamma a closed subgroup …

The paper describes decompositions of geometric measures on Anosov homogeneous spaces.

The paper classifies 1-dimensional uniform measures in various dimensions.

The paper shows measures equidistribute on affine submanifolds with a rate.

This paper proves that there are no compact forms for a large class of homogeneous spaces admitting actions by higher-rank semisimple Lie groups. It builds on Zimmer's approach for studying such spaces using cocycle superrigidity. The proof involves cocycle superrigidity, measure rigidity for unipotent flows, technique…

Classifies measures for Anosov subgroups in higher ranks.

In this note we consider sampling from (non-homogeneous) strongly Rayleigh probability measures. As an important corollary, we obtain a fast mixing Markov Chain sampler for Determinantal Point Processes.

New flatness measure for deep networks invariant to scaling.

We obtain a first order extension of the large deviation estimates in the Gärtner-Ellis theorem. In addition, for a given family of measures, we find a special family of functions having a similar Laplace principle expansion up to order one to that of the original family of measures. The construction of the special fam…

GD iterates for non-homogeneous deep nets increase margin and converge in direction.

A new model corrects inhomogeneity in Optimal Transport with Boundary.

Extends potential theory to Carnot groups, estimating Hausdorff dimension.

We generalize Sunada's method to produce new examples of closed, locally non-isometric manifolds which are isospectral. In particular, we produce pairs of isospectral, simply-connected, locally non-isometric normal homogeneous spaces. These pairs also allow us to see that in general group actions with discrete spectra …

New capacity measure for deep ReLU networks derived from weight norms.

Study classifies moduli spaces of spin connections on 3D homogeneous spaces.

We introduce a simple approach for testing the reliability of homogeneous generators and the Markov property of the stochastic processes underlying empirical time series of credit ratings. We analyze open access data provided by Moody's and show that the validity of these assumptions - existence of a homogeneous genera…

New risk measures adjust for tail risk inadequacies.

We propose a new anytime hierarchical clustering method that iteratively transforms an arbitrary initial hierarchy on the configuration of measurements along a sequence of trees we prove for a fixed data set must terminate in a chain of nested partitions that satisfies a natural homogeneity requirement. Each recursive …

In this paper we define the magnitude of metric spaces using measures rather than finite subsets as had been done previously and show that this agrees with earlier work with Leinster in arXiv:0908.1582. An explicit formula for the magnitude of an n-sphere with its intrinsic metric is given. For an arbitrary homogeneous…

Let $\GG$ be a sub-Riemannian $k$-step Carnot group of homogeneous dimension $Q$. In this paper, we shall prove several geometric inequalities concerning smooth hypersurfaces (i.e. codimension one submanifolds) immersed in $\GG$, endowed with the $\HH$-perimeter measure.

A geometrical interpretation of the $G$-structures associated to elastic material bodies is given. In addition, characterizations of their integrability are obtained. Since the lack of integrability is a geometrical measure of the lack of homogeneity, the corresponding inhomogeneity conditions are obtained

Constructs new elicitable risk measures with multiplicative scoring functions.

Paper characterizes star-shaped risk measures and their properties.

The paper explores non-convex risk measures and their characterizations.

Set risk measures extend traditional risk measures to handle sets of positions.

Given a metric measure space $(X,d,\mathfrak{m})$ that satisfies the Riemannian Curvature Dimension condition, $RCD^*(K,N),$ and a compact subgroup of isometries $G \leq Iso(X)$ we prove that there exists a $G-$invariant measure, $\mathfrak{m}_G,$ equivalent to $\mathfrak{m}$ such that $(X,d,\mathfrak{m}_G)$ is still a…

Every compact aspherical Riemannian manifold admits a canonical series of orbibundle structures with infrasolv fibers which is called its infrasolv tower. The tower arises from the solvable radicals of isometry group actions on the universal covers. Its length and the geometry of its base measure the degree of continuo…

In this paper, we study continuous Kakeya line and needle configurations, of both the oriented and unoriented varieties, in connected Lie groups and some associated homogenous spaces. These are the analogs of Kakeya line (needle) sets (subsets of $\mathbb{R}^n$ where it is possible to turn a line (respectively an inter…

New insights into ends of quotient spaces and graphs.

Submodularity is studied for convex risk measures, including Expected Shortfall.

A new model improves homogeneity in burn patient reimbursement.

We prove some ergodic-theoretic rigidity properties of the action of SL(2,R) on moduli space. In particular, we show that any ergodic measure invariant under the action of the upper triangular subgroup of SL(2,R) is supported on an invariant affine submanifold. The main theorems are inspired by the results of several a…

Sharp estimates derived for quasilinear equations on metric measure spaces.

Threats on the stability of a financial system may severely affect the functioning of the entire economy, and thus considerable emphasis is placed on the analyzing the cause and effect of such threats. The financial crisis in the current and past decade has shown that one important cause of instability in global market…

We discuss several issues regarding material homogeneity and strain compatibility for materially uniform thin elastic shells from the viewpoint of a 3-dimensional theory, with small thickness, as well as a 2-dimensional Cosserat theory. A relationship between inhomogeneity and incompatibility measures under the two des…

The paper analyzes elicitability of return risk measures and their scoring functions.

Paper uses Mirror Descent for efficient risk budgeting portfolios.

Study examines harmonic functions in sub-Riemannian and RCD settings.

Let $S$ be a compact, connected, oriented surface, possibly with boundary, of negative Euler characteristic. In this article we extend Lindenstrauss-Mirzakhani's and Hamenstädt's classification of locally finite mapping class group invariant ergodic measures on the space of measured laminations $\mathcal{M}\mathcal{L}(…