Researchers compute trace formula for magnetic Laplacian on hyperbolic surfaces.
problem Analyzing the magnetic Laplacian on compact hyperbolic surfaces.
method Computed the trace formula for magnetic Laplacian energies above the Mane critical level.
result Asymptotic behavior of trace formula coefficients near the Mane critical level.
The paper finds infinitely many magnetic geodesics on non-compact manifolds.
problem Existence and multiplicity of periodic orbits of magnetic flows.
method Morse theory applied to non-compact manifolds with energy levels above the Mañé critical value.
result Infinitely many noncontractible closed magnetic geodesics found.
New proofs of Cheeger-like inequalities for coexact 1-forms on hyperbolic manifolds.
problem Cheeger-like inequalities for coexact 1-forms on hyperbolic manifolds.
method Properties of magnetic geodesic flow and behavior at Mañé's critical energy level.
result Improved Cheeger constants and volume dependence in proofs.
We introduce a new critical value c∞(L) for Tonelli Lagrangians L on the tangent bundle of the 2-sphere without minimizing measures supported on a point. We show that c∞(L) is strictly larger than the Mañé critical value c(L), and on every energy level e∈(c(L),c∞(L)) there exist infinitely…
Study magnetic geodesics on half-Lie groups, proving Hopf-Rinow theorem for energies above critical value.
problem Investigate magnetic geodesics on half-Lie groups using Riemannian and two-form structures.
method Define Mañé's critical value, prove Finsler geodesic flow equivalence, and apply Hopf-Rinow theorem.
result Hopf-Rinow theorem holds for energies above Mañé's critical value on magnetic geodesics.
Generic potential primes have no self-intersections or intersections.
problem Finding non-degenerate periodic orbits without self-intersections.
method Generic convex Hamiltonian approach and Mañé genericity.
result Prime periodic orbits do not intersect or have self-intersections.
The paper defines a critical value for a magnetic system and extends solutions beyond blow-up.
problem Analyzing blow-up behavior and extending solutions for a magnetic system.
method Formulated as a magnetic geodesic equation on an infinite-dimensional Lie group, computed Mañé's critical value, established Hopf-Rinow theorem.
result Computed Mañé's critical value for the magnetic two-component Hunter-Saxton system and extended solutions beyond blow-up.
Study of magnetic geodesics on Heisenberg nilmanifolds.
problem Existence and properties of closed magnetic geodesics on Heisenberg nilmanifolds.
method Analyzing conditions for the existence of closed magnetic geodesics on compact quotients of Heisenberg nilmanifolds.
result Existence of contractible closed magnetic geodesics for any energy level below the Mañé critical value.
Study magnetic geodesics on odd spheres, computing critical energy values.
problem Understanding magnetic geodesics on odd-dimensional spheres.
method Explicit computation and analysis of submanifolds and symmetries.
result Energy values determine magnetic geodesic connectivity on spheres.
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.
problem Existence of closed magnetic geodesics on low energy levels.
method Derived magnetic curvature operator and used Bonnet-Myers argument.
result Established the existence of a contractible periodic orbit on closed manifolds.
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 φt denote the magneti…
Global minimizers exist for Tonelli Lagrangians on half-Lie groups.
problem Existence and properties of minimizers for Lagrangians on infinite-dimensional spaces.
method Introduced Tonelli Lagrangians on half-Lie groups, proved existence of minimizers and flow lines.
result Global minimizers exist above certain energy thresholds.
Proposes MANE for multi-view network embedding, improving node representations.
problem Learning low-dimensional representations from multiple views of networks.
method MANE combines diversity and collaboration, including second-order collaboration, and attention-based extension MANE+.
result MANE+ outperforms state-of-the-art approaches on real-world multi-view networks.
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.
problem Proving magnetic curvature non-positive for magnetic systems without conjugate points.
method Using magnetic curvature introduced by the first author, proving magnetic flatness conditions.
result Magnetic flatness is a rigid condition with specific metric and curvature properties.
New model explains market dynamics with phase transitions and non-linear interactions.
problem Understanding complex multi-asset market dynamics with phase transitions.
method Developed a Multi-Asset Non-Equilibrium Skew (MANES) model based on Langevin dynamics and McKean-Vlasov equation.
result The model accurately predicts market returns and phase transitions in both benign and distressed markets.
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.
problem Analyzing normal singular geodesics in a conformally generic sub-Riemannian metric.
method Proves the absence of non-trivial normal orbits for specific Hamiltonians.
result No non-trivial normal orbits for the specified Hamiltonians.
Geodesic flows on specific manifolds are structurally stable.
problem Stability of geodesic flows on compact manifolds without conjugate points.
method Analyzing the C∞ compact manifold (M,g) with quasi-convex universal covering and divergent geodesic rays. result Proved the C1-stability conjecture for geodesic flows of compact manifolds. The paper characterizes potential functions whose level sets are orbits in mechanical systems.
problem Characterizing smooth potential energy functions on the plane with specific level set properties.
method Analyzing inverse curvature flow and properties of level sets.
result Analytic or functions with totally path-disconnected critical sets must be radial, while every compact convex set is a critical set of a Levi potential.
The paper examines how the topology of level sets changes with critical points in Morse theory.
problem Understanding how the topology of level sets changes with critical points in Morse theory.
method Study of sublevel sets and level sets of Morse functions, analysis of critical points and their indices.
result For a general class of functions, the topology of a regular level set changes when passing a single critical point, unless the index is half the dimension of the manifold.
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.
problem Evaluate the high-level structure of generated text for coherence, coreference, and topicality.
method Apply model criticism in latent space to compare real and generated data distributions.
result Transformer-based models struggle with maintaining structural coherence and coreference.
New Morse-Bott function defined on Stiefel manifolds, revealing complex critical structures.
problem Defining Morse-Bott functions on non-linear Stiefel manifolds.
method Replacing linear height function with a quadratic one, proving it as a Morse-Bott function.
result Critical submanifolds are fibrations of products of Grassmannians, not Grassmannians themselves.
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.
problem Trace formula for magnetic Laplacian at zero energy.
method Generalizes Gutzwiller trace formula, focuses on zero energy level.
result Derives trace formula at zero energy level.
The study proves the existence of many geodesics on complex manifolds.
problem Existence of closed geodesics on manifolds with non-trivial first Betti number.
method Combining Mañé's theorem with a new theorem about minimal geodesics and transverse homoclinic points.
result Proves the existence of infinitely many closed geodesics of arbitrary large length on manifolds with non-trivial first Betti number.
A theorem connects integral of second-order derivatives to function rise.
problem Understanding the integral of second-order derivatives over regions.
method Proves integral proportional to function rise over specified regions.
result Integral of second-order derivatives equals rise in function value.
Deep learning depends on tuning layers near critical points.
problem Understanding how deep learning architectures depend on tuning parameters.
method Random energy approach to analyze statistical dependence in deep belief networks.
result Statistical dependence can propagate only if layers are tuned 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.
problem Conditions for weak KAM solutions to be constant.
method Geometric and differential analysis of Hamilton-Jacobi equations.
result Weak KAM solutions are constant if and only if the 1-form is harmonic.
Paper resolves ambiguity in non-convex bilevel optimization problems.
problem Ambiguity in bilevel optimization with non-convex lower-level objectives.
method Introduces selection maps to define critical points and resolves ambiguity.
result Validates new analytical tools in Morse theory for implicit differentiation.
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.
problem Understanding when trends in financial markets revert.
method Polynomial regression and bootstrapping on 30 years of daily futures prices.
result Trends revert when they reach a critical level of statistical significance.
New RL method MAC improves performance in sparse reward settings.
problem Slow mixing in large state spaces or sparse rewards.
method Multi-level Monte Carlo Actor-Critic (MAC) algorithm.
result Achieves convergence rate comparable to state-of-the-art AC algorithms.
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.
problem Understanding the long-run distribution of stochastic gradient descent in non-convex problems.
method Large deviations theory and randomly perturbed dynamical systems.
result The long-run distribution of SGD resembles the Boltzmann-Gibbs distribution with temperature equal to the step-size.
Study of symmetries of sphere divisions induced by functions with isolated critical points.
problem Understanding symmetries of sphere divisions induced by functions with isolated critical points.
method Analyzing the group of diffeomorphisms that leave invariant a connected component of a level set and its complement.
result The group of such diffeomorphisms is isomorphic to a finite subgroup of SO(3). Policy certifies inventory levels meeting service requirements.
problem Maintaining stock levels meeting service requirements despite unknown demand.
method Data-driven order policy using online learning and integral action.
result Valid inference method for finite samples.