arXiv research

Researchers compute trace formula for magnetic Laplacian on hyperbolic surfaces.

The paper finds infinitely many magnetic geodesics on non-compact manifolds.

New proofs of Cheeger-like inequalities for coexact 1-forms on hyperbolic manifolds.

We introduce a new critical value $c_\infty(L)$ for Tonelli Lagrangians $L$ on the tangent bundle of the 2-sphere without minimizing measures supported on a point. We show that $c_\infty(L)$ is strictly larger than the Mañé critical value $c(L)$, and on every energy level $e\in(c(L),c_\infty(L))$ there exist infinitely…

Study magnetic geodesics on half-Lie groups, proving Hopf-Rinow theorem for energies above critical value.

Generic potential primes have no self-intersections or intersections.

The paper defines a critical value for a magnetic system and extends solutions beyond blow-up.

Study of magnetic geodesics on Heisenberg nilmanifolds.

Study magnetic geodesics on odd spheres, computing critical energy values.

We prove several new results concerning action minimizing periodic orbits of Tonelli Lagrangian systems on an oriented closed surface $M$. More specifically, we show that for every energy larger than the maximal energy of a constant orbit and smaller than or equal to the Mañé critical value of the universal abelian cov…

The paper studies magnetic curvature and proves the existence of closed orbits on low energy levels.

We prove that for a weakly exact magnetic system on a closed connected Riemannian manifold, almost all energy levels contain a closed orbit. More precisely, we prove the following stronger statements. Let $(M,g)$ denote a closed connected Riemannian manifold and $σ$ a weakly exact 2-form. Let $φ_{t}$ denote the magneti…

Global minimizers exist for Tonelli Lagrangians on half-Lie groups.

Proposes MANE for multi-view network embedding, improving node representations.

Let $M$ be a closed oriented surface endowed with a Riemannian metric $g$ and let $Ω$ be a 2-form. We show that the magnetic flow of the pair $(g,Ω)$ has zero asymptotic Maslov index and zero Liouville action if and only $g$ has constant Gaussian curvature, $Ω$ is a constant multiple of the area form of $g$ and the mag…

Given a compact Riemannian manifold, we prove a uniform Franks' lemma at second order for geodesic flows and apply the result in persistence theory.

Extends E. Hopf's theorem to magnetic systems without conjugate points.

New model explains market dynamics with phase transitions and non-linear interactions.

For a compact Riemannian manifold with boundary, endowed with a magnetic potential $α$, we consider the problem of restoring the metric $g$ and the magnetic potential $α$ from the values of the Mañé action potential between boundary points and the associated linearized problem. We study simple magnetic systems. In this…

Let (M,g) be a compact Riemannian manifold of hyperbolic type, i.e M is a manifold admitting another metric of strictly negative curvature. In this paper we study the geodesic flow restricted to the set of geodesics which are minimal on the universal covering. In particular for surfaces we show that the topological ent…

Abstract: Proves no non-trivial normal orbits for specific Hamiltonians.

Geodesic flows on specific manifolds are structurally stable.

The paper characterizes potential functions whose level sets are orbits in mechanical systems.

In this study, we investigate the use of global information to speed up the learning process and increase the cumulative rewards of reinforcement learning (RL) in competition tasks. Within the actor-critic RL, we introduce multiple cooperative critics from two levels of the hierarchy and propose a reinforcement learnin…

For strong exact magnetic fields the action functional (i.e., the length plus the linear magnetic term) is not bounded from below on the space of closed contractible curves and the lower estimates for critical levels are derived by using the principle of throwing out cycles. It is proved that for almost every energy le…

Model criticism tool evaluates text coherence and structure in generated long-form text.

New Morse-Bott function defined on Stiefel manifolds, revealing complex critical structures.

We study the level sets of the distance function from a boundary point of a convex set in Euclidean space. We provide a lower bound for the range of connectivity of the level sets, in terms of the critical points of the distance function in the sense of Grove-Shiohama-Gromov-Cheeger.

A methodology is developed to identify, as units of study, each decrease in the value of a stock from a given maximum price level. A critical level in the amount of price declines is found to separate a segment operating under a random walk from a segment operating under a power law. This level is interpreted as a poin…

Paper derives trace formula for magnetic Laplacian at zero energy.

The study proves the existence of many geodesics on complex manifolds.

A theorem connects integral of second-order derivatives to function rise.

Classical Morse theory proceeds by considering sublevel sets $f^{-1}(-\infty, a]$ of a Morse function $f: M \to R$, where $M$ is a smooth finite-dimensional manifold. In this paper, we study the topology of the level sets $f^{-1}(a)$ and give conditions under which the topology of $f^{-1}(a)$ changes when passing a cri…

Deep learning depends on tuning layers near critical points.

Families of hypersurfaces that are level-set families of harmonic functions free of critical points are characterized by a local differential-geometric condition. Harmonic functions with a specified level-set family are constructed from geometric data. As a by-product, it is shown that the evolution of the gradient of …

Geometric analysis proves weak KAM solutions constant under specific conditions.

Paper resolves ambiguity in non-convex bilevel optimization problems.

Deep Learning can significantly benefit cancer proteomics and genomics. In this study, we attempt to determine a set of critical proteins that are associated with the FLT3-ITD mutation in newly-diagnosed acute myeloid leukemia patients. A Deep Learning network consisting of autoencoders forming a hierarchical model fro…

This paper focuses on the problem of topological equivalence of functions with isolated critical points on the boundary of a compact surface $M$ which are also isolated critical points of their restrictions to the boundary. This class of functions we denote by $Ω(M)$. Firstly, we've obtained the topological classificat…

We review the author's results on Mather's $β$ function : non-strict convexity of $β$ when the configuration space has dimension two, link between the size of the Aubry set and the differentiability of $β$, correlation between the rationality of the homology class and the differentiability of $β$, equality of the Mathe…

In this paper, we present theorems specifying the critical values for series associated with debts arranged in the order of their duration.

Empirical study on trends reversion in financial markets.

New RL method MAC improves performance in sparse reward settings.

We continue the study of the variation of the $p$--modulus of a foliation initiated by the first author. We derive the formula for the second variation which allows to study $p$--stable foliations. We obtain some results concerning codimension one $p$--stable foliations. Moreover, we derive the equation for the critica…

Deep neural networks (DNNs) are vulnerable to maliciously generated adversarial examples. These examples are intentionally designed by making imperceptible perturbations and often mislead a DNN into making an incorrect prediction. This phenomenon means that there is significant risk in applying DNNs to safety-critical …

The paper analyzes how SGD visits different regions of a non-convex problem's state space.

Policy certifies inventory levels meeting service requirements.

We prove that the level sets of a real C^s function of two variables near a non-degenerate critical point are of class C^[s/2] and apply this to the study of planar sections of surfaces close to the singular section by the tangent plane at hyperbolic points or elliptic points, and in particular at umbilic points. We al…