arXiv research

Flatness of manifolds with open flat subsets proven using bipolar comparisons.

We define a new type of metric comparison similar to the comparison of Alexandrov. We show that it has strong connections to continuity of optimal transport between regular measures on a Riemannian manifold, in particular to the so called MTW condition introduced by Xi-Nan Ma, Neil Trudinger and Xu-Jia Wang.

The bipolar filtration introduced by T. Cochran, S. Harvey, and P. Horn is a framework for the study of smooth concordance of topologically slice knots and links. It is known that there are topologically slice 1-bipolar knots which are not 2-bipolar. For knots, this is the highest known level at which the filtration do…

Let T denote the group of smooth concordance classes of topologically sice knots. We show that the first quotient in the bipolar filtration of T (i.e. 0-bipolar knots modulo 1-bipolar knots) has infinite rank, even modulo Alexander polynomial one knots. Any 0-bipolar knot has vanishing tau-, epsilon-, and s-invariants.…

Paper generalizes bipolar theorems for non-negative random variables.

Topology classifies bipolar surfaces; they are not embedded.

The paper bounds the Morse index and nullity of bipolar surfaces related to Otsuki tori.

The bipolar filtration of Cochran, Harvey and Horn presents a framework of the study of deeper structures in the smooth concordance group of topologically slice knots. We show that the graded quotient of the bipolar filtration of topologically slice knots has infinite rank at each stage greater than one. To detect nont…

Extends RCNNs to handle hypercomplex-valued data.

We propose a simple extension to the ReLU-family of activation functions that allows them to shift the mean activation across a layer towards zero. Combined with proper weight initialization, this alleviates the need for normalization layers. We explore the training of deep vanilla recurrent neural networks (RNNs) with…

The i-th eigenvalue of the Laplacian on a surface can be viewed as a functional on the space of Riemannian metrics of fixed area. Extremal points of these functionals correspond to surfaces admitting minimal isometric immersions into spheres. Recently, critical metrics for the first eigenvalue were classified on tori a…

New knots with specific properties have larger 4-genus than previously known bounds.

Novel ramp loss method improves weakly supervised machine translation and parsing.

Recently Penskoi [J. Geom. Anal. 25 (2015), 2645-2666, arXiv:1308.1628] generalized the well known two-parametric family of Lawson tau-surfaces $τ_{r,m}$ minimally immersed in spheres to a three-parametric family $T_{a,b,c}$ of tori and Klein bottles minimally immersed in spheres. It was remarked that this family inclu…

Mobile technologies offer opportunities for higher resolution monitoring of health conditions. This opportunity seems of particular promise in psychiatry where diagnoses often rely on retrospective and subjective recall of mood states. However, getting actionable information from these rather complex time series is cha…

Let $\mathcal{T}$ be the group of smooth concordance classes of topologically slice knots, and $\{0\}\subset\cdots\subset \mathcal{T}_{n+1}\subset\mathcal{T}_{n}\subset \cdots\subset \mathcal{T}_{0}\subset \mathcal{T}$ be the bipolar filtration. In this paper, we show that a proper collection of the knots employed by H…

Established a Hardy inequality on Finsler manifolds.

Study uses interviews to automatically detect BD and BPD with good accuracy.

Recurrent major mood episodes and subsyndromal mood instability cause substantial disability in patients with bipolar disorder. Early identification of mood episodes enabling timely mood stabilisation is an important clinical goal. Recent technological advances allow the prospective reporting of mood in real time enabl…

This study uses smartphone data to predict when mood interventions are needed for bipolar disorder.

We show that Lawson's bipolar surface $\tildeτ_{3,1}$ is after stereographic projection the unique minimizer among immersed Klein bottles in its conformal class. We conjecture that it actually is the unique minimizer among immersed Klein bottles into $\mathbb{R}^n$, $n\geq 4$, whose existence the authors and P. Breunin…

Autoencoders identify brain networks linked to stress and genotype.

The $i$-th eigenvalue $λ_i$ of the Laplace-Beltrami operator on a surface can be considered as a functional on the space of all Riemannian metrics of unit volume on this surface. Surprisingly only few examples of extremal metrics for these functionals are known. In the present paper a new countable family of extremal m…

We define two transforms between minimal surfaces with non-circular ellipse of curvature in the 5-sphere, and show how this enables us to construct, from one such surface, a sequence of such surfaces. We also use the transforms to show how to associate to such a surface a corresponding ruled minimal Lagrangian submanif…

STNMF method uncovers neural circuit components in retinal ganglion cells.

Research uses activity analysis to identify mental health symptoms.

The paper shows infinite rank in T_1/T_2 and new insights into Alexander polynomials and concordance.

The classification procedure of streaming data usually requires various ad hoc methods or particular heuristic models. We explore a novel non-parametric and systematic approach to analysis of heterogeneous sequential data. We demonstrate an application of this method to classification of the delays in responding to the…

The goal of this paper is to classify parametrically parabolic submanifolds in any codimension. First, we describe the ones that are ruled and show that they are the only parabolic submanifolds that admit an isometric immersion as a hypersurface. Then, we classify the nonruled ones by two different means. In fact, we p…

A conformal map from a Riemann surface to a Euclidean space of dimension greater than or equal to three is explained by using the Clifford algebra, in a similar fashion to quaternionic holomorphic geometry of surfaces in the Euclidean three- or four-space. The Weierstrass representation, the spin transform, the Darboux…

An optimal control problem associated with the dynamics of the orientation of a bipolar molecule in the plane can be understood by means of tools in differential geometry. For first time in the literature $k$-symplectic formalism is used to provide the optimal control problems associated to some families of partial dif…

This paper solves a financial portfolio selection problem in incomplete markets.

Paper develops duality theory for robust utility maximization in continuous time.

Investigates the effects of nondominated sets of probability measures in robust models of finance.

Study confirms conjectures for topologically slice knots' concordance group.

The first eigenvalue of the Laplacian on a surface can be viewed as a functional on the space of Riemannian metrics of a given area. Critical points of this functional are called extremal metrics. The only known extremal metrics are a round sphere, a standard projective plane, a Clifford torus and an equilateral torus.…

Holographic Invariant Storage uses vector architectures to ensure LLM safety at design time.

The paper examines how the first Steklov-Dirichlet eigenvalue changes with the distance between two concentric circles.

Deep learning predicts mental disorders from audio and text samples.

Network analysis of human brain connectivity is critically important for understanding brain function and disease states. Embedding a brain network as a whole graph instance into a meaningful low-dimensional representation can be used to investigate disease mechanisms and inform therapeutic interventions. Moreover, by …

Accurate diagnosis of psychiatric disorders plays a critical role in improving the quality of life for patients and potentially supports the development of new treatments. Many studies have been conducted on machine learning techniques that seek brain imaging data for specific biomarkers of disorders. These studies hav…

Graph comparison ties to Alexandrov's theorems.

In this work, we will verify some comparison results on Kahler manifolds. They are complex Hessian comparison for the distance function from a closed complex submanifold of a Kahler manifold with holomorphic bisectional curvature bounded below by a constant, eigenvalue comparison and volume comparison in terms of scala…

A global agreement on how to reduce and cap human footprint, especially their GHG emissions, is very unlikely in near future. At the same time, bilateral agreements would be inefficient because of their neural and balanced nature. Therefore, unilateral actions would have attracted attention as a practical option. Howev…

The paper extends volume comparison results to total σ_l-curvature.

Paper extends curvature estimates to new tensor types.

New method improves prediction accuracy in comparison data.

A new comparison theorem for geometric spaces.