arXiv research

The paper studies volume and area comparisons in non-compact 3-manifolds with non-negative scalar curvature.

We prove a splitting theorem for Riemannian n-manifolds with scalar curvature bounded below by a negative constant and containing certain area-minimising hypersurfaces (Theorem 3). Thus we generalise [25,Theorem 3] by Nunes. This splitting result follows from an area comparison theorem for hypersurfaces with non-positi…

In this paper we prove an area comparison result for certain totally geodesic surfaces in 3-manifolds with a lower bound on the scalar curvature. This result is a variant of a comparison theorem of Heintze-Karcher for minimal hypersurfaces in manifolds of nonnegative Ricci curvature. Our assumptions on the ambient mani…

Motivated by recent work of Choquet-Bruhat, Chrusciel, and Martin-Garcia, we prove monotonicity properties and comparison results for the area of slices of the null cone of a point in a Lorentzian manifold. We also prove volume comparison results for subsets of the null cone analogous to the Bishop-Gromov relative volu…

The conformal parameterisation of a minimal surface is harmonic. Therefore, a minimal surface is a critical point of both the energy functional and the area functional. In this paper, we compare the Morse index of a minimal surface as a critical point of the area functional with its Morse index as a critical point of t…

In this paper, we introduce the $L_p$ geominimal surface area for all $-n\neq p<1$, which extends the classical geominimal surface area ($p=1$) by Petty and the $L_p$ geominimal surface area by Lutwak ($p>1$). Our extension of the $L_p$ geominimal surface area is motivated by recent work on the extension of the $L_p$ a…

Modern excavations yielded a distribution of the house areas in the ancient Egyptian city Akhetaten, which was populated for a short period during the 14th century BC. Assuming that the house area is a measure of the wealth of its inhabitants allows us to make a comparison of the wealth distributions in ancient and mod…

We develop area and volume comparison theorems for the evolution of spacelike, acausal, causally complete hypersurfaces in Lorentzian manifolds, where one has a lower bound on the Ricci tensor along timelike curves, and an upper bound on the mean curvature of the hypersurface. Using these results, we give a new proof o…

We obtain sharp quantitative Laplacian upper and lower estimates under no assumption on curvatures. As a result, we derive quantitative Laplacian, area and volume comparison theorems for tubes in Riemannian and Kähler manifolds under weak integral curvature assumptions. We also give some applications, such as a general…

We study minimal hypersurfaces in manifolds of non-negative Ricci curvature, Euclidean volume growth and quadratic curvature decay at infinity. By comparison with capped spherical cones, we identify a precise borderline for the Ricci curvature decay. Above this value, no complete area-minimizing hypersurfaces exist. Be…

Bishop's volume comparison theorem states that a compact $n$-manifold with Ricci curvature larger than the standard $n$-sphere has less volume. While the traditional proof uses geodesic balls, we present another proof using isoperimetric hypersurfaces, also known as "soap bubbles," which minimize area for a given volum…

Paper proves a sharp weighted Isoperimetric inequality for substatic manifolds.

Area under ROC (AUC) is an important metric for binary classification and bipartite ranking problems. However, it is difficult to directly optimizing AUC as a learning objective, so most existing algorithms are based on optimizing a surrogate loss to AUC. One significant drawback of these surrogate losses is that they …

Machine learning methods may have the potential to significantly accelerate drug discovery. However, the increasing rate of new methodological approaches being published in the literature raises the fundamental question of how models should be benchmarked and validated. We reanalyze the data generated by a recently pub…

Sharp comparison theorems for 3D manifolds with scalar curvature bound.

Study curvature flows on pinched Hadamard surfaces, proving convexity preservation and convergence.

We prove that on a Riemannian manifold, a smooth differential form has a primitive with a given (functional) upper bound provided the necessary weighted isoperimetric inequalities implied by Stokes are satisfied. We apply this to prove a comparison predicted by Gromov between the cofilling function and the filling area…

The paper establishes inequalities for convex curves and applies them to lattice point estimates.

The geometry on a slope of a mountain is the geometry of a Finsler metric, called here the {\it slope metric}. We study the existence of globally defined slope metrics on surfaces of revolution as well as the geodesic's behavior. A comparison between Finslerian and Riemannian areas of a bounded region is also studied.

The paper bounds the index of CMC surfaces with capillary boundary.

Consider a 2-plane $P \subset \mathbb{C}^n$ and let $D$ be a bounded region in $P$ with a piecewise-smooth boundary. Let $I(D)$ be the infimum of areas of all piecewise-smooth isotropic surfaces in $\mathbb{C}^n$ with the same boundary as $D$. Then $I(D)= λ_P^n \cdot Area(D)$. If $P$ is not complex, $λ_P^n < \frac{3π}{…

We prove the filling area conjecture in the hyperelliptic case. In particular, we establish the conjecture for all genus 1 fillings of the circle, extending P. Pu's result in genus 0. We translate the problem into a question about closed ovalless real surfaces. The conjecture then results from a combination of two ingr…

For an admissible class of smooth compact initial data sets with boundary, we prove a comparison theorem between the Wang/Liu-Yau quasi-local mass of the boundary and the Hawking mass of strictly minimizing hulls in the Jang graphs of the domain. Using this, we prove a quasi-local Penrose inequality that involves these…

We give upper and lower bounds for the ratio of the volume of metric ball to the area of the metric sphere in Finsler-Hadamard manifolds with pinched S-curvature. We apply these estimates to find the limit at the infinity for this ratio. Derived estimates are the generalization of the well-known result in Riemannian ge…

Using a new method we give elementary estimates for the capacity of non-contractible annuli on cylinders and provide examples, where these inequalities are sharp. Here the lower bound depends only on the area of the annulus. In the case of constant curvature this lower bound is obtained with the help of a symmetrizatio…

Develops slicing method to prove rigidity of scalar curvature on manifolds with boundary.

The paper develops new methods to study sharp isoperimetric properties on complex spaces.

Clustering is one of the most universal approaches for understanding complex data. A pivotal aspect of clustering analysis is quantitatively comparing clusterings; clustering comparison is the basis for many tasks such as clustering evaluation, consensus clustering, and tracking the temporal evolution of clusters. In p…

In this paper we consider the basic version of Reinforcement Learning (RL) that involves computing optimal data driven (adaptive) policies for Markovian decision process with unknown transition probabilities. We provide a brief survey of the state of the art of the area and we compare the performance of the classic UCB…

Studying the impact of climate change on precipitation is constrained by finding a way to evaluate the evolution of precipitation variability over time. Classical approaches (feature-based) have shown their limitations for this issue due to the intermittent and irregular nature of precipitation. In this study, we prese…

Graph embeddings from commute networks identify socioeconomic disparities in urban areas.

This survey covers in our opinion the most important results in the theory of continuous selections of multivalued mappings (approximately) from 2002 through 2012. It extends and continues our previous such survey which appeared in Recent Progress in General Topology, II, which was published in 2002. In comparison, our…

Study visualizes actor-critic loss landscapes for inventory optimization.

In many areas of machine learning, it becomes necessary to find the eigenvector decompositions of large matrices. We discuss two methods for reducing the computational burden of spectral decompositions: the more venerable Nystom extension and a newly introduced algorithm based on random projections. Previous work has c…

Over the last couple of years, deep learning and especially convolutional neural networks have become one of the work horses of computer vision. One limiting factor for the applicability of supervised deep learning to more areas is the need for large, manually labeled datasets. In this paper we propose an easy to imple…

In record linkage (RL), or exact file matching, the goal is to identify the links between entities with information on two or more files. RL is an important activity in areas including counting the population, enhancing survey frames and data, and conducting epidemiological and follow-up studies. RL is challenging when…

This review covers AI in finance, challenges, techniques, and opportunities.

We present and prove Polyakov-Alvarez type comparison formulas for the determinants of Friederichs extensions of Laplacians corresponding to conformally equivalent metrics on a compact Riemann surface with conical singularities. In particular, we find how the determinants depend on the orders of conical singularities. …

Today, treatment effect estimation at the individual level is a vital problem in many areas of science and business. For example, in marketing, estimates of the treatment effect are used to select the most efficient promo-mechanics; in medicine, individual treatment effects are used to determine the optimal dose of med…

We examine the supervised learning problem in its continuous setting and give a general optimality condition through techniques of functional analysis and the calculus of variations. This enables us to solve the optimality condition for the desired function u numerically and make several comparisons with other widely u…

Efficient and robust algorithms for decentralized estimation in networks are essential to many distributed systems. Whereas distributed estimation of sample mean statistics has been the subject of a good deal of attention, computation of $U$-statistics, relying on more expensive averaging over pairs of observations, is…

We generalize the notion of planar bicycle tracks -- a.k.a. one-trailer systems -- to so-called tractor/tractrix systems in general Riemannian manifolds and prove explicit expressions for the length of the ensuing tractrices and for the area of the domains that are swept out by any given tractor/tractrix system. These …

State-space models have been successfully used for more than fifty years in different areas of science and engineering. We present a procedure for efficient variational Bayesian learning of nonlinear state-space models based on sparse Gaussian processes. The result of learning is a tractable posterior over nonlinear dy…

This is a review of explicit computations of Connes distance in noncommutative geometry, covering finite dimensional spectral triples, almost-commutative geometries, and spectral triples on the algebra of compact operators. Several applications to physics are covered, like the metric interpretation of the Higgs field, …

Toxicity analysis and prediction are of paramount importance to human health and environmental protection. Existing computational methods are built from a wide variety of descriptors and regressors, which makes their performance analysis difficult. For example, deep neural network (DNN), a successful approach in many o…

We consider embedded, smooth curves in the plane which are either closed or asymptotic to two lines. We study their behaviour under curve shortening flow with a global forcing term. Firstly, we prove an analogue to Huisken's distance comparison principle for curve shortening flow for initial curves whose local total cu…

In this paper, we propose a new deep feature selection method based on deep architecture. Our method uses stacked auto-encoders for feature representation in higher-level abstraction. We developed and applied a novel feature learning approach to a specific precision medicine problem, which focuses on assessing and prio…

Partial soft-matching distance improves neural representation comparison by allowing some neurons to remain unmatched.