arXiv research

Optimal DP mechanisms for vector queries are found to be staircase distributions.

This work extends VQR to non-linear cases and provides scalable solvers.

New framework for learning KR maps from data, ensuring stable generalization.

Sharp Gaussian isoperimetry proven along Ricci flow.

The paper shows how to rearrange arcs to form closed curves.

Simplified and extended a method for rearranging infinite configurations of cubes.

This note proves a Gaussian version of a Pólya-Szegö conjecture using rearrangement techniques.

Numerical challenges inherent in algorithms for computing worst Value-at-Risk in homogeneous portfolios are identified and solutions as well as words of warning concerning their implementation are provided. Furthermore, both conceptual and computational improvements to the Rearrangement Algorithm for approximating wors…

Paper develops polynomial approximations for complex probability densities.

The paper proves a Moser-Trudinger inequality on metric spaces with curvature-dimension conditions.

Method calibrates basket options using rearranged samples from constituent processes.

The paper extends geometric inequalities from Euclidean space to Riemannian manifolds.

We investigate monotonicity properties of $p$-harmonic vector bundle-valued $k$-forms by studying the energy-momentum tensor associated with such a form. As a consequence, we obtain a unified proof of the monotonicity formulæ for $p$-harmonic maps and Yang-Mills connections, proving a monotonicity formula for $p$-Yang-…

Under certain topological assumptions, we show that two monotone Lagrangian submanifolds embedded in the standard symplectic vector space with the same monotonicity constant cannot link one another and that, individually, their smooth knot type is determined entirely by the homotopy theoretic data which classifies the …

The paper extends Busemann's inequalities to complex and quaternionic spaces.

Monotone aggregation of dependent random vectors has an absolutely continuous distribution under certain conditions.

We characterize the convexity of functions and the monotonicity of vector fields on metric measure spaces with Riemannian Ricci curvature bounded from below. Our result offers a new approach to deal with some rigidity theorems such as `splitting theorem' and `volume cone implies metric cone theorem' in non-smooth conte…

Study fine Pólya-Szegő inequalities in metric spaces with applications.

Pixle attacks images by rearranging pixels, bypassing neural networks.

This paper presents relations between several types of closedness of a law-invariant convex set in a rearrangement invariant space $\mathcal{X}$. In particular, we show that order closedness, $σ(\mathcal{X},\mathcal{X}_n^\sim)$-closedness and $σ(\mathcal{X},L^\infty)$-closedness of a law-invariant convex set in $\mathc…

Link concordance equals homotopy for high-dimensional spheres.

The paper proves learning-curve monotonicity for maximum likelihood estimators in various parametric settings.

Paper compares solutions of Poisson equations on Riemannian manifolds with Robin boundary.

DNA rearrangement processes recombine gene segments that are organized on the chromosome in a variety of ways. The segments can overlap, interleave or one may be a subsegment of another. We use directed graphs to represent segment organizations on a given locus where contigs containing rearranged segments represent ver…

In this paper, we explore several Fatou-type properties of risk measures. The paper continues to reveal that the strong Fatou property, which was introduced in [17], seems to be most suitable to ensure nice dual representations of risk measures. Our main result asserts that every quasiconvex law-invariant functional on…

Study shows configuration spaces' homological dimension increases monotonically.

New formula shows how causal vectors relate to mass-minimizing data.

Study Hamiltonian diffeomorphisms on symplectic manifolds and properties of invariant convex functions.

Frank and Lieb proved sharp Sobolev inequalities without rearrangements.

We extend the equivariant holomorphic Morse inequalities of circle actions to cases with torus and non-Abelian group actions on holomorphic vector bundles over Kahler manifolds and show the necessity of the Kahler condition. For torus actions, there is a set of inequalities for each choice of action chambers specifying…

Stabilization technique applied to curve shortening flow in 3D space.

Machine learning models simulate molecular spectra and reactions in solvents.

Extends subspace detour method to Gromov-Wasserstein problem.

Monotonicity formulae play a crucial role for many geometric PDEs, especially for their regularity theories. For minimal submanifolds in a Euclidean ball, the classical monotonicity formula implies that if such a submanifold passes through the centre of the ball, then its area is at least that of the equatorial disk. R…

The Lebesgue property (order-continuity) of a monotone convex function on a solid vector space of measurable functions is characterized in terms of (1) the weak inf-compactness of the conjugate function on the order-continuous dual space, (2) the attainment of the supremum in the dual representation by order-continuous…

Using the stress energy tensor, we establish some monotonicity formulae for vector bundle-valued p-forms satisfying the conservation law, provided that the base Riemannian (resp. Kähler) manifolds poss some real (resp. complex) p-exhaustion functions. Vanishing theorems follow immediately from the monotonicity formulae…

Given a monotone convex function on the space of essentially bounded random variables with the Lebesgue property (order continuity), we consider its extension preserving the Lebesgue property to as big solid vector space of random variables as possible. We show that there exists a maximum such extension, with explicit …

Paper tackles non-monotonic resource utilization in sequential decision-making.

In this paper we first prove some linear isoperimetric inequalities for submanifolds in the de Sitter-Schwarzschild and Reissner-Nordstrom manifolds. Moreover, the equality is attained. Next, we prove some monotonicity formulas for submanifolds with bounded mean curvature vector in warped product manifolds and, as cons…

We developed OmicsMapNet approach to take advantage of existing deep leaning frameworks to analyze high-dimensional omics data as 2-dimensional images. The omics data of individual samples were first rearranged into 2D images in which molecular features related in functions, ontologies, or other relationships were orga…

An affine rearrangement inequality is established which strengthens and implies the recently obtained affine Pólya--Szegö symmetrization principle for functions on $\mathbb{R}^n$. Several applications of this new inequality are derived. In particular, a sharp affine logarithmic Sobolev inequality is established which i…

Management of systemic risk in financial markets is traditionally associated with setting (higher) capital requirements for market participants. There are indications that while equity ratios have been increased massively since the financial crisis, systemic risk levels might not have lowered, but even increased. It ha…

We consider the learning algorithms under general source condition with the polynomial decay of the eigenvalues of the integral operator in vector-valued function setting. We discuss the upper convergence rates of Tikhonov regularizer under general source condition corresponding to increasing monotone index function. T…

Continuum mechanics theory describes skin's complex anisotropic behavior.

Characterizes continuity of monotone functionals in mixed topology.

Proves existence of graph on torus with prescribed curvature.

We equip many non compact non simply connected surfaces with smooth Riemannian metrics whose isoperimetric profile is smooth, a highly non generic property. The computation of the profile is based on a calibration argument, a rearrangement argument, the Bol-Fiala curvature dependent inequality, together with new result…

Introduces nondecreasing rank for matrices and tensors, developing methods and applications.