arXiv research

Solves equality case in isoperimetric inequality for non-convex domains.

Develops spherical density-equalizing maps for closed surfaces.

New partition designs reduce star discrepancy in high-dimensional sampling.

FPGA-based multi-layer equalizer adapts to changing channels.

Sharp inequalities proved for RCD spaces, showing equality conditions.

The paper proposes a method to measure fairness through equality of effort using algorithmic recourse.

Schwarz lemma extended to equality cases and curvature on manifolds.

Classifies weakly Einstein submanifolds in space forms satisfying specific equalities.

Developed an ellipsoidal density-equalizing map for genus-0 closed surfaces.

FairICP addresses equalized odds fairness for multiple sensitive attributes.

We study the capital growth in gambling with (and without) side information and memory effects. We derive several equalities for gambling, which are of similar form to the Jarzynski equality and its extension to systems with feedback controls. Those relations provide us with new measures to quantify the effects of info…

We show that, if g is more than or equal to 2, the virtual cohomological dimension of the mapping class group of a 3-dimensional handlebody of genus g is equal to 4g-5 and the Euler number of it is equal to 0.

We extend the Nielsen theory of coincidence sets to equalizer sets, the points where a given set of (more than 2) mappings agree. On manifolds, this theory is interesting only for maps between spaces of different dimension, and our results hold for sets of k maps on compact manifolds from dimension (k-1)n to dimension …

New methods ensure fair rankings in web-scale recommender systems.

We prove that if $M$ is a three-manifold with scalar curvature greater than or equal to -2 and $Σ\subset M$ is a two-sided compact embedded Riemann surface of genus greater than 1 which is locally area-minimizing, then the area of $Σ$ is greater than or equal to $4π(g(Σ)-1)$, where $g(Σ)$ denotes the genus of $Σ$. In t…

Study shows equality in Hodge Laplacian bound occurs only on spheres.

This article is the sequel to our previous paper [LS] dealing with the near-equality case of the Positive Mass Theorem. We study the near-equality case of the Penrose Inequality for the class of complete asymptotically flat rotationally symmetric Riemannian manifolds with nonnegative scalar curvature whose boundaries a…

The equality between the balanced and the Gauduchon cones is discussed in several situations. In particular, it is shown that equality does not hold on many twistor spaces, and it holds on Moishezon manifolds. Moreover, it is proved that a SKT manifold of dimension three on which the balanced cone equals the Gauduchon …

In many instances, information on engineering systems can be obtained through measurements, monitoring or direct observations of system performances and can be used to update the system reliability estimate. In structural reliability analysis, such information is expressed either by inequalities (e.g. for the observati…

We characterize symmetric spaces without focal points by the equality case of general equalities between geometric quantities.

There has been an increasing interest in testing the equality of large Pearson's correlation matrices. However, in many applications it is more important to test the equality of large rank-based correlation matrices since they are more robust to outliers and nonlinearity. Unlike the Pearson's case, testing the equality…

The paper studies the consistency of mean curvature flow via volumetric varifolds.

We prove that the least-perimeter partition of the sphere into four regions of equal area is a tetrahedral partition.

Paper tackles multiplayer symmetric games, securing equal share for n players.

Eigen-decomposition simplifies quadratic programming with equality constraints.

A new approach for blind channel equalization and decoding, variational inference, and variational autoencoders (VAEs) in particular, is introduced. We first consider the reconstruction of uncoded data symbols transmitted over a noisy linear intersymbol interference (ISI) channel, with an unknown impulse response, with…

We study the geometric properties of the terms of the Goldman bracket between two free homotopy classes of oriented closed curves in a hyperbolic surface. We provide an obstruction for the equality of two terms in the Goldman bracket, namely if two terms in the Goldman bracket are equal to each other then for every hyp…

We use Heegaard Floer homology to give obstructions to unknotting a knot with a single crossing change. These restrictions are particularly useful in the case where the knot in question is alternating. As an example, we use them to classify all knots with crossing number less than or equal to nine and unknotting number…

Equal experience fairness in recommender systems reduces bias.

The paper classifies minimal projective varieties satisfying a specific equality.

Conformal prediction sets can lead to unfair outcomes.

The paper explores fairness in machine learning, focusing on Equalized Odds.

Study proves equality of LS-category and cohomological dimension for specific group homomorphisms.

We show that for any nontrivial knot $K$ and any natural number $n$ there is a diagram $D$ of $K$ such that the unknotting number of $D$ is greater than or equal to $n$. It is well known that twice the unknotting number of $K$ is less than or equal to the crossing number of $K$ minus one. We show that the equality hold…

Analyzes structure of log smooth pairs when equality holds in Bogomolov-Gieseker inequality.

Notions of "fair classification" that have arisen in computer science generally revolve around equalizing certain statistics across protected groups. This approach has been criticized as ignoring societal issues, including how errors can hurt certain groups disproportionately. We pose a modification of one of the fairn…

Framework for fair predictive models using resampled sensitive attributes.

In this paper, we prove that some Gaussian structural equation models with dependent errors having equal variances are identifiable from their corresponding Gaussian distributions. Specifically, we prove identifiability for the Gaussian structural equation models that can be represented as Andersson-Madigan-Perlman cha…

Generalizes Reilly inequality to varifolds and analyzes equality cases.

We derive the Black-Scholes-Merton dual equation, which has exactly the same form as the Black-Scholes-Merton equation. The novel and general equation works for options with a payoff of homogeneous of degree one, including European, American, Bermudan, Asian, barrier, lookback, etc., and leads to new insights into pric…

The paper derives inequalities for Riemannian maps and submersions involving quaternionic space forms.

Sharp Li-Yau equality proven for shrinking Ricci solitons without curvature assumptions.

Equality in Miyaoka-Yau inequality implies uniformization of Klt pairs.

We empirically show the superiority of the equally weighted S\&P 500 portfolio over Sharpe's market capitalization weighted S\&P 500 portfolio. We proceed to consider the MaxMedian rule, a non-proprietary rule designed for the investor who wishes to do his/her own investing on a laptop with the purchase of only 20 stoc…

In this paper we give the stable classification of ordered, pointed, oriented multi-component curves on surfaces with minimal crossing number less than or equal to 2 such that any equivalent curve has no simply closed curves in its components. To do this, we use the theory of words and phrases which was introduced by V…

We determine non-Hopf hypersurfaces with constant mean curvature in the complex projective plane which attain equality in a basic inequality between the maximum Ricci curvature and the squared mean curvature.

In the previous paper, Takahasi and the authors generalized the theory of minimal surfaces in Euclidean n-space to that of surfaces with holomorphic Gauss map in certain class of non-compact symmetric spaces. It also includes the theory of constant mean curvature one surfaces in hyperbolic 3-space. Moreover, a Chern-Os…

Equal-volume polygons are obtained from adequate discretizations of curves in 3-space, contained or not in surfaces. In this paper we explore the similarities of these polygons with the affine arc-length parameterized smooth curves to develop a theory of discrete affine invariants. Besides obtaining discrete affine inv…