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 Theorem”
Global diffeomorphism proof between tame Fréchet spaces.
problem Existence of global diffeomorphism between Fréchet spaces.
method Mountain Pass Theorem and sufficient conditions for diffeomorphism.
result Existence of global diffeomorphism between tame Fréchet spaces.
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.
Global diffeomorphism theorem for Fréchet spaces established.
problem Establishing global diffeomorphisms in Fréchet spaces.
method Extending local diffeomorphisms to global ones, using generalized gradients and Lipschitz functions.
result Global diffeomorphism theorem proved for Fréchet spaces.
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.
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.
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 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.
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.
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.
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.
Proves existence of a third minimal hypersurface with fixed boundary.
problem Existence of minimal hypersurfaces with fixed boundary.
method Develops min-max methods adapted to discrete setting.
result Proves existence of a third embedded minimal hypersurface.
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.
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 …
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.
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.
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 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 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…
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.
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.
Study on geometry of mountain slopes as Finsler metrics.
problem Existence and behavior of slope metrics on surfaces of revolution.
method Analysis of Finsler metrics on surfaces of revolution and comparison with Riemannian areas.
result Existence of globally defined slope metrics on surfaces of revolution.
Minimal surfaces in Schwarzschild manifold have Morse index one.
problem Existence of unbounded minimal surfaces in asymptotically flat 3-manifolds.
method Analysis of Riemannian Schwarzschild manifold, free boundary condition.
result Minimal surface in Schwarzschild manifold has Morse index one.
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…
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.
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.
Study on wind speed variability in Switzerland using multifractal analysis.
problem Understanding wind speed variability in mountainous regions.
method Multifractal detrended fluctuation analysis applied to daily wind speed data.
result Persistent and multifractal nature of wind speed series, with larger fluctuations in Swiss plateau.
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.
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. …
Counterexample disproves Spencer-Brown's claim about parity-pass algorithm.
problem Disproving Spencer-Brown's claim about the parity-pass algorithm and its relation to edge colorings.
method Provided a counterexample to Spencer-Brown's algorithm on non-polar pentagons.
result The parity-pass algorithm does not necessarily terminate in an extendable edge coloring.
Q-learning with nonlinear Q-function can explore effectively without explicit exploration.
problem How to effectively explore in reinforcement learning.
method Q-learning with nonlinear Q-function and no explicit exploration.
result Q-learning with nonlinear Q-function can learn as well as -greedy exploration.
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.
This study evaluates multi-step reinforcement learning methods in the Mountain Car environment.
problem Lack of statistical significance in evaluating multi-step reinforcement learning methods.
method Combines n-step action-value algorithms with DQN architecture, tests in Mountain Car environment.
result Performance varies with off-policy correction, backup length, and target network update frequency.
The goal of this paper is to describe all local diffeomorphisms mapping a family of circles, in an open subset of $\r^3$, into straight lines. This paper contains two main results. The first is a complete description of the rectifiable collection of circles in $\r^3$ passing through one point. It turns out that to be r…
Bayesian online learning in the one-pass regime: Frequentist validity and uncertainty quantificationmath.ST
Bayesian online learning algorithm for one-pass data, achieving frequentist validity and uncertainty quantification.
problem Theoretical limitations in Bayesian online learning, especially in the one-pass setting.
method Proposed a new Bayesian online learning algorithm with a warm-start phase for the one-pass regime, establishing convergence rates and valid uncertainty quantification.
result The sequentially updated posterior attains optimal convergence rates and valid uncertainty quantification without diverging mini-batch sample sizes.
We prove that the canonical 4-dimensional surgery problems can be solved after passing to a double cover. This contrasts the long-standing conjecture about the validity of the topological surgery theorem for arbitrary fundamental groups (without passing to a cover). As a corollary, the surgery conjecture is reformulate…
We revisit the linearization theorems for proper Lie groupoids around general orbits (statements and proofs). In the the fixed point case (known as Zung's theorem) we give a shorter and more geometric proof, based on a Moser deformation argument. The passing to general orbits (Weinstein) is given a more conceptual inte…
It is shown that two braids represent transversally isotopic links if and only if one can pass from one braid to another by conjugations in braid groups, positive Markov moves, and their inverses.
We develop a method of extending actions of compact transformation groups which is then applied to the problem of preservation of equivariant extensor property by passing to a subspace of given orbit types.
Consider a smooth map from a neighborhood of the origin in a real vector space to a neighborhood of the origin in a Euclidean space. Suppose that this map takes all germs of lines passing through the origin to germs of Euclidean circles, or lines, or a point. We prove that under some simple additional assumptions this …
Projective simulation outperforms standard reinforcement learning in navigation tasks.
problem Benchmarking projective simulation in navigation problems.
method Projective simulation model applied to grid world and mountain car problems, compared to Q-learning and SARSA.
result Projective simulation outperforms standard reinforcement learning in terms of parameter simplicity and computational efficiency.
The abstract proves a global splitting theorem for Poisson manifolds.
problem Decomposing compact Kähler Poisson manifolds into simpler components.
method Proving a global splitting theorem using symplectic leaves and finite étale covers.
result Compact Kähler Poisson manifolds can be split into simpler components.
New examples show causality conditions don't always pass to coverings.
problem Causality conditions not always transfer to coverings.
method Provided explicit examples of spacetimes.
result Causality conditions like causal continuity and simplicity do not always pass to coverings.
NAST generalizes scattering transform for non-stationary time series analysis.
problem Analyzing non-stationary time series data.
method Neural activation of scattering transform with various activation functions and high pass filters.
result Central and non-central limit theorems for NAST of Gaussian processes.
DQN with model-based exploration improves learning in sparse reward environments.
problem Poor sample efficiency in sparse reward environments.
method Combines model-free and model-based approaches for better exploration.
result Improves performance in environments with sparse rewards.