arXiv research

Two methods find at least two solutions to Kazdan-Warner's problem on surfaces.

We extend a work of Bartsch, Clapp and Puppe on the Mountain pass theorems. We consider functionals invariant with respect to infinite discrete groups satisfying a maximality condition on the finite subgroups.

Study on nonlinear elliptic equations with variable exponents, proving existence and multiplicity of solutions.

The paper proves the existence of infinitely many nodal solutions to a Paneitz-type equation.

Study finds critical points in perimeter functional for fixed volume sets.

Proves uniqueness of small entropy self-expanders.

Study on ground states of semilinear elliptic equations with various potential wells.

New theorem finds new minimal hypersurfaces in hyperbolic space.

The paper proves the existence of finite-energy solutions to a specific elliptic equation on Riemannian manifolds.

We establish new existence and non-existence results for positive solutions of the Einstein-scalar field Lichnerowicz equation on compact manifolds. This equation arises from the Hamiltonian constraint equation for the Einstein-scalar field system in general relativity. Our analysis introduces variational techniques, i…

We provide sufficient conditions for the existence of a global diffeomorphism between tame Fréchet spaces. We prove a version of the Mountain Pass Theorem which is a key ingredient in the proof of the main theorem.

Is it possible to obtain unbounded minimal surfaces in certain asymptotically flat 3-manifolds as a limit of solutions to a natural mountain pass problem with diverging boundaries? In this work, we give evidence that this might be true by analyzing related aspects in the case of the exact Riemannian Schwarzschild manif…

The paper proves the existence of hypersurfaces with prescribed mean curvature.

We construct and analyze minimal disc stackings with bounds on their Morse index.

New self-expander found between two given asymptotic ones.

Necessary and sufficient conditions are given for the Palais-Smale Condition C to hold for the Yang-Mills functional for connections that are invariant under a Lie group action on the manifold with orbits of codimension less than or equal to three. As an application the mountain pass lemma is used to give a simple proo…

Solves time-minimizing navigation on a mountain slope using Riemann-Finsler geometry.

In this work, we prove the existence of a third embedded minimal hypersurface spanning a closed submanifold $γ$ contained in the boundary of a compact Riemannian manifold with convex boundary, when it is known a priori the existence of two strictly stable minimal hypersurfaces that bound $γ$. In order to do so, we deve…

Study the landscape of Lipschitz functions between manifolds using persistent homology.

Study examines diversification of mid-mountain ski tourism.

The paper proves critical point results for Frechet manifolds.

We give sufficient conditions for a $C^1_c$-local diffeomorphism between Fréchet spaces to be a global one. We extend the Clarke's theory of generalized gradients to the more general setting of Fréchet spaces. As a consequence, we define the Chang Palais-Smale condition for Lipschitz functions and show that a functio…

Solves time-optimal navigation on slippery slopes with cross gravitational wind.

In this paper, we build up a min-max theory for minimal surfaces using sweepouts of surfaces of genus $g\geq 2$. We develop a direct variational methods similar to the proof of the famous Plateau problem by J. Douglas and T. Rado. As a result, we show that the min-max value for the area functional can be achieved by a …

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.

Post-pandemic, work patterns shifted with fewer days in offices and a new midweek mountain.

Study improves precipitation predictions for High Mountain Asia using machine learning.

Proves existence of a single-valued minimal hypersurface in compact manifolds.

We introduce a new braid-theoretic framework with which to understand the Legendrian and transversal classification of knots, namely a Legendrian Markov Theorem without Stabilization which induces an associated transversal Markov Theorem without Stabilization. We establish the existence of a nontrivial knot-type specif…

We present the min-max construction of critical points of the area using penalization arguments. Precisely, for any immersion of a closed surface $Σ$ into a given closed manifold, we add to the area Lagrangian a term equal to the $L^q$ norm of the second fundamental form of the immersion times a "viscosity" parameter. …

New theorem connects distant points and identical points on manifolds.

In this paper, we applied the multifractal detrended fluctuation analysis to the daily means of wind speed measured by 119 weather stations distributed over the territory of Switzerland. The analysis was focused on the inner time fluctuations of wind speed, which could be more linked with the local conditions of the hi…

For a stable marginally outer trapped surface (MOTS) in an axially symmetric spacetime with cosmological constant $Λ> 0$ and with matter satisfying the dominant energy condition, we prove that the area $A$ and the angular momentum $J$ satisfy the inequality $8π|J| \le A\sqrt{(1-ΛA/4π)(1-ΛA/12π)}$ which is saturated pre…

One-pass SGD dynamics in overparameterized quadratic networks show slow escape from poor solutions.

This paper introduces a new scalable multi-objective deep reinforcement learning (MODRL) framework based on deep Q-networks. We develop a high-performance MODRL framework that supports both single-policy and multi-policy strategies, as well as both linear and non-linear approaches to action selection. The experimental …

UCPO improves diversity in reinforcement learning models, maintaining high accuracy.

Novel neural network solves PDEs with multi-scale resolution.

Maximum a posteriori (MAP) inference is a fundamental computational paradigm for statistical inference. In the setting of graphical models, MAP inference entails solving a combinatorial optimization problem to find the most likely configuration of the discrete-valued model. Linear programming (LP) relaxations in the Sh…

Adapts BP-based algorithms for deep learning, improving performance and accuracy.

Optimizes reinforcement learning by prioritizing sets of samples over individual ones.

For non-homotopic maps $u,v\in C^{\infty}(M,N)$ between closed Riemannian manifolds, we consider the smallest energy level $γ_p(u,v)$ for which there exist paths $u_t\in W^{1,p}(M,N)$ connecting $u_0=u$ to $u_1=v$ with $\|du_t\|_{L^p}^p\leq γ_p(u,v)$. When $u$ and $v$ are $(k-2)$-homotopic, work of Hang and Lin shows t…

Graph hypernetworks improve molecule property prediction and classification.

New algorithms accelerate MAP inference in Markov fields with faster convergence.

The paper sets limits for GNNs solving PDEs to avoid under-reaching phenomenon.

We provide an explicit upper bound on the number of Reidemeister moves required to pass between two diagrams of the same link. This leads to a conceptually simple solution to the equivalence problem for links.

Much effort has been directed at algorithms for obtaining the highest probability configuration in a probabilistic random field model known as the maximum a posteriori (MAP) inference problem. In many situations, one could benefit from having not just a single solution, but the top M most probable solutions known as th…

Introduces LoCA regret to evaluate model-based RL methods.

SDPA is shown to be an optimal transport problem in deep learning.