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.
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48 results for “mountain pass schemes”
Study finds critical points in perimeter functional for fixed volume sets.
problem Finding critical points in perimeter functional for sets of fixed volume.
method Utilizes Mazurwoski--Zhou techniques and new Cacciopoli set connectedness results.
result Constructs smooth almost embedded hypersurfaces with non-zero constant mean curvature.
We construct and analyze minimal disc stackings with bounds on their Morse index.
problem Constructing and analyzing minimal free boundary disc stackings.
method Constructing minimal free boundary disc stackings in a three-dimensional Euclidean unit ball, proving bounds on their Morse index.
result Uniform, linear bounds on the Morse index of all such surfaces.
Proves uniqueness of small entropy self-expanders.
problem Uniqueness of self-expanders with small entropy.
method Mountain-pass theorem and integer degree argument.
result Proves uniqueness result for self-expanders.
New theorem finds new minimal hypersurfaces in hyperbolic space.
problem Finding new minimal hypersurfaces in hyperbolic space.
method Developed a min-max theory for complete minimal hypersurfaces.
result Shows existence of a new minimal hypersurface between two given ones.
Study on nonlinear elliptic equations with variable exponents, proving existence and multiplicity of solutions.
problem Existence and multiplicity of solutions for Dirichlet boundary value problems involving -equation.
method Proved using the mountain pass theorem and Fountain theorem with Cerami sequences.
result Existence and multiplicity of solutions for -equation.
Two methods find at least two solutions to Kazdan-Warner's problem on surfaces.
problem Finding solutions to Kazdan-Warner's problem on two-dimensional surfaces.
method Direct method on convex sets and variational method of mountain pass.
result At least two solutions to the Kazdan-Warner's problem are found.
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.
The paper proves the existence of infinitely many nodal solutions to a Paneitz-type equation.
problem Proving the existence of nodal solutions to a specific type of partial differential equation.
method First, a estimate for positive -invariant solutions is proven. Then, the existence of mountain pass solutions with arbitrarily large energy is established.
result The existence of infinitely many nodal solutions to the equation is proven.
Study on ground states of semilinear elliptic equations with various potential wells.
problem Characterizing ground states of semilinear elliptic equations with arbitrary potential wells.
method Analyzing solutions in convex domains and manifolds with non-negative Ricci curvature, using Morse theory and min-max methods.
result Ground states are mountain-pass type with Morse index 1 in convex domains and manifolds with non-negative Ricci curvature.
New self-expander found between two given asymptotic ones.
problem Finding new self-expanders between given asymptotic ones.
method Developed a min-max theory for asymptotically conical self-expanders of mean curvature flow.
result Existence of a new asymptotically conical self-expander trapped between two given ones.
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…
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…
The paper proves the existence of finite-energy solutions to a specific elliptic equation on Riemannian manifolds.
problem Existence of finite-energy solutions to a singular elliptic equation on Riemannian manifolds.
method ε-regularized problems, mountain pass arguments, and limiting procedures.
result Existence of nonnegative finite-energy supersolutions under certain conditions.
In this work, we prove the existence of a third embedded minimal hypersurface spanning a closed submanifold 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 . In order to do so, we deve…
Study the landscape of Lipschitz functions between manifolds using persistent homology.
problem Understanding the structure of homotopy paths between maps with high Lipschitz constants.
method Using persistent homology to analyze the landscape of Lipschitz functions between manifolds.
result First results on the persistence of higher-dimensional cycles in function spaces.
Proves existence of a single-valued minimal hypersurface in compact manifolds.
problem Existence of multiplicity-1 minimal hypersurfaces in compact Riemannian manifolds.
method Modified minmax construction with Allen-Cahn approximation and valley point optimization.
result Existence of a smooth, closed minimal hypersurface with multiplicity 1 in bumpy metrics.
The paper proves critical point results for Frechet manifolds.
problem Finding critical points in the context of Frechet manifolds.
method Using a deformation result and sufficient conditions for the Palais-Smale condition.
result Proves a mountain pass theorem and three critical points 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…
We give sufficient conditions for a -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…
The inhomogeneous Allen--Cahn equation and the existence of prescribed-mean-curvature hypersurfacesmath.DG
The paper proves the existence of hypersurfaces with prescribed mean curvature.
problem Proving the existence of hypersurfaces with prescribed mean curvature.
method PDE theoretic approach using mountain pass construction and regularity results for integral varifolds.
result Existence of quasi-embedded, boundaryless hypersurfaces with prescribed mean curvature.
This paper augments the reward received by a reinforcement learning agent with potential functions in order to help the agent learn (possibly stochastic) optimal policies. We show that a potential-based reward shaping scheme is able to preserve optimality of stochastic policies, and demonstrate that the ability of an a…
In this paper, we build up a min-max theory for minimal surfaces using sweepouts of surfaces of genus . 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 …
Unified theory linking atom-centered and message-passing models for molecular properties.
problem Combining atom-centered and message-passing models for accurate molecular property prediction.
method Generalizing ACDC framework to include multi-centered information, providing a complete linear basis for regression.
result Unified understanding of atom-centered and message-passing models, providing a coherent foundation.
Solves time-minimizing navigation on a mountain slope using Riemann-Finsler geometry.
problem Time-minimizing navigation on a mountain slope under gravity.
method Riemann-Finsler geometry, Zermelo navigation problem, anisotropic deformation of the background Riemannian metric, rescaled gravitational wind.
result A new Finsler metric for optimal navigation on slippery mountain slopes.
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.
Study examines diversification of mid-mountain ski tourism.
problem Understanding transformations in ski mid-mountain territories.
method Applied regional diversification theory to French ski areas.
result Identified three steps in tourism diversification paths.
Post-pandemic, work patterns shifted with fewer days in offices and a new midweek mountain.
problem Shift in work patterns and integration of personal and professional life.
method Behavioral analysis using mobile geolocation records.
result Significant decline in office-based workdays and emergence of a new midweek mountain.
Refined climatologies of future precipitation over High Mountain Asia using probabilistic ensemble learningphysics.ao-ph
Study improves precipitation predictions for High Mountain Asia using machine learning.
problem Uncertainty in future precipitation over High Mountain Asia due to regional climate model biases.
method Probabilistic machine learning framework combining 13 regional climate models via a mixture of experts.
result 32% improvement over equally-weighted average and 254% improvement over single ensemble member.
We analyze oversquashing in topological message-passing using relational structures.
problem Oversquashing in topological message-passing remains understudied.
method A unifying axiomatic framework that bridges graph and topological message-passing.
result Potential to advance topological deep learning.
BankGCN improves graph convolution networks by handling multi-channel signals with adaptive filter banks.
problem Handling multi-channel graph signals with limited architectures.
method BankGCN decomposes multi-channel signals into subspaces and uses adapted filters for each subspace.
result BankGCN achieves excellent performance in graph classification on benchmark datasets.
ADMP-GNN dynamically adjusts message-passing layers for better graph learning performance.
problem Fixed message-passing steps in GNNs do not account for nodes' varying computational needs.
method Proposes ADMP-GNN, which dynamically adjusts the number of message-passing layers for each node.
result Improves performance on node classification tasks compared to baseline GNN models.
We introduce an iterative optimization scheme for convex objectives consisting of a linear loss and a non-separable penalty, based on the expectation-consistent approximation and the vector approximate message-passing (VAMP) algorithm. Specifically, the penalties we approach are convex on a linear transformation of the…
Adapts BP-based algorithms for deep learning, improving performance and accuracy.
problem Training deep neural networks with discrete weights and activations.
method Message-passing algorithms based on Belief Propagation, with reinforcement field.
result Comparable performance to SGD-inspired heuristics (BinaryNet) and higher accuracy in predictions.
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…
Proposes CXNs for neural network computations on cell complexes.
problem Performing neural network computations on complex topological spaces.
method Introduces a message passing scheme and a unified encoder-decoder framework for cell complexes.
result Generalizes message passing to cell complexes and provides a cell2vec representation.
In this paper, we propose an acceleration scheme for online memory-limited PCA methods. Our scheme converges to the first eigenvectors in a single data pass. We provide empirical convergence results of our scheme based on the spiked covariance model. Our scheme does not require any predefined parameters such as t…
Iterative information processing, either based on heuristics or analytical frameworks, has been shown to be a very powerful tool for the design of efficient, yet feasible, wireless receiver architectures. Within this context, algorithms performing message-passing on a probabilistic graph, such as the sum-product (SP) a…
We present the min-max construction of critical points of the area using penalization arguments. Precisely, for any immersion of a closed surface into a given closed manifold, we add to the area Lagrangian a term equal to the norm of the second fundamental form of the immersion times a "viscosity" parameter. …
We design iterative receiver schemes for a generic wireless communication system by treating channel estimation and information decoding as an inference problem in graphical models. We introduce a recently proposed inference framework that combines belief propagation (BP) and the mean field (MF) approximation and inclu…
New theorem connects distant points and identical points on manifolds.
problem Continuous maps and distant points on manifolds.
method Qualitative extension of Hopf theorem, using topological 'distant' points.
result Existence of connected component containing both distant and identical points.
We propose Sideways, an approximate backpropagation scheme for training video models. In standard backpropagation, the gradients and activations at every computation step through the model are temporally synchronized. The forward activations need to be stored until the backward pass is executed, preventing inter-layer …
Solves time-optimal navigation on slippery slopes with cross gravitational wind.
problem Time-optimal navigation on a slippery cross slope under gravitational wind.
method New Finsler metric derived for the problem, considering both lateral and longitudinal gravitational effects.
result Conditions for strong convexity and purely geometric solution provided.
One approach for constructing copula functions is by multiplication. Given that products of cumulative distribution functions (CDFs) are also CDFs, an adjustment to this multiplication will result in a copula model, as discussed by Liebscher (J Mult Analysis, 2008). Parameterizing models via products of CDFs has some a…
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 and with matter satisfying the dominant energy condition, we prove that the area and the angular momentum satisfy the inequality which is saturated pre…
Graph structured data are abundant in the real world. Among different graph types, directed acyclic graphs (DAGs) are of particular interest to machine learning researchers, as many machine learning models are realized as computations on DAGs, including neural networks and Bayesian networks. In this paper, we study dee…
Paper interprets contrastive learning dynamics using message passing.
problem Lack of rigorous understanding of contrastive learning dynamics.
method Casts contrastive objective into message passing scheme on augmentation graph.
result The learning dynamics of contrastive learning can be theoretically characterized.