The paper proves a nonholonomic version of Maupertuis-Jacobi principle and shows that nonholonomic trajectories minimize length.
problem Nonholonomic dynamics and their length minimization.
method Contact bundle formulation and geometric equivalence between problems.
result Regular solutions of nonholonomic mechanical problems are reparametrizations of geodesics with minimized Riemannian length.
The study of solutions with fixed energy of certain classes of Lagrangian (or Hamiltonian) systems is reduced, via the classical Maupertuis--Jacobi variational principle, to the study of geodesics in Riemannian manifolds. We are interested in investigating the problem of existence of brake orbits and homoclinic orbits,…
Unified approach to rolling ball dynamics on spheres proves integrability.
problem Integrability of rolling ball dynamics on spheres.
method Unified Chaplygin multiplier method and Maupertuis principle.
result Complete integrability for specific inertia operators and radii ratios.
Consider the equal mass planar 4-body problem with a potential corresponding to an inverse \textit{cube} force. The Jacobi-Maupertuis principle reparametrizes the dynamics as geodesics of a certain metric. We examine the curvature of this geodesic flow in the reduced space on the collinear and parallelogram invariant…
New electromagnetic curvature defined via Jacobi-Maupertuis, showing positive curvature for non-zero magnetic force.
problem Defining and analyzing electromagnetic curvature.
method Using Jacobi-Maupertuis reparametrization and energy analysis.
result Positive electromagnetic Ricci curvature for non-zero magnetic force and small potential.
The paper explores transformations between power law problems and geodesics on cones.
problem Solving power law problems and understanding their geometric properties.
method Geometric transformations and cone metrics.
result Derivation of Maclaurin duality and Jacobi-Maupertuis metric reformulation.
In this paper we characterize planar central configurations in terms of a sectional curvature value of the Jacobi-Maupertuis metric. This characterization works for the N-body problem with general masses and any 1/rα potential with α>0. We also observe dynamical consequences of these curvature values for relati…
We construct the hyperbolic plane with its geodesic flow as the scale plus symmetry reduction of a three-body problem in the Euclidean plane. The potential is −I/Δ2 where I is the triangle's moment of inertia and Δ its area. The reduction method uses the Jacobi-Maupertuis metric, following the author's earlier p…
Marchal's lemma is the basic tool for eliminating collisions when using the direct method of the calculus of variations to establish existence of "designer" solutions to the classical N-body problem. Our goal here is to understand why Marchal's lemma holds, by taking a metric geometry perspective and employing the Jaco…
The Jacobi-Maupertuis metric allows one to reformulate Newton's equations as geodesic equations for a Riemannian metric which degenerates at the Hill boundary. We prove that a JM geodesic which comes sufficiently close to a regular point of the boundary contains pairs of conjugate points close to the boundary. We prove…
New algebraic approach classifies conformally superintegrable systems in arbitrary dimensions.
problem Classifying conformally superintegrable systems in arbitrary dimensions.
method Algebraic geometric approach extended to conformally superintegrable systems.
result An algebraic equation governs the classification under conformal equivalence for a prolific class of second order conformally superintegrable systems.
Research examines how Islamic banking principles spread among managers and scholars.
problem Diffusion of Islamic banking principles among managers and scholars.
method Literature review focusing on knowledge diffusion and Islamic banking governance principles.
result Emergence of common Islamic banking governance principles from diverse knowledge streams.
The paper establishes maximum principles and stochastic completeness for pseudo-Hermitian manifolds.
problem Maximum principles and stochastic completeness for pseudo-Hermitian manifolds.
method Established generalized maximum principles and proved stochastic completeness equivalence.
result Stochastic completeness for the heat semigroup is equivalent to generalized maximum principles.
The h-principle helps solve complex geometric problems.
problem Solving complex geometric problems using the h-principle.
method Developed from the Oka-Grauert principle and Gromov's theory, the h-principle is applied to Oka manifolds and maps.
result Recent developments and applications of the h-principle in complex analysis and geometry.
The study establishes uncertainty principles on harmonic manifolds of rank one.
problem Developing uncertainty principles for harmonic manifolds of rank one.
method Derivation of various uncertainty principles including Heisenberg, Morgen, Schrödinger, and Hömanders principles.
result Generalization of Hausdorff-Young inequality to harmonic manifolds of rank one.
The paper rigorously investigates the Frequency Principle in deep neural networks.
problem Understanding the training dynamics of deep neural networks.
method Theoretical investigation of Frequency Principle at three stages of training.
result Theorem providing quantitative understanding of Frequency Principle for general DNNs.
Conditions for polynomial to be isometry of lattice, answering Hasse principles.
problem Conditions for integral polynomials to be characteristic polynomials of isometries of lattices.
method Necessary and sufficient conditions derived from lattice isometries and Hasse principles.
result Proved a Hasse principle for signatures of knots.
Shows flexible sheaves as fibrant objects for Gromov's h-principle.
problem Applying the h-principle to partial differential relations.
method Interprets flexible sheaves as fibrant objects in a model structure.
result Flexible sheaves can be understood as fibrant objects.
A new method to break down insurance costs into risk and uncertainty.
problem Understanding and quantifying insurance costs in uncertain environments.
method An axiomatic approach to decompose premium principles into risk and deviation measures.
result Maximal risk and minimal deviation measures can be uniquely identified in decompositions.
Proof of Schwarz principle for specific minimal surfaces.
problem Reflection principle for Jenkins-Serrin type minimal surfaces.
method Proof in homogeneous three-manifolds E(κ,τ) for κ≤0 and τ≥0. result Verification of Schwarz reflection principle for new class of minimal surfaces.
DNNs initially capture low-frequency components before high-frequency ones, a phenomenon called F-Principle.
problem Understanding why DNNs generalize well despite overfitting.
method Empirical study on real and synthetic datasets, focusing on frequency components captured by DNNs.
result DNNs capture dominant low-frequency components first, then high-frequency ones, a phenomenon called F-Principle.
New objective function preserves Bellman's principle for policy gradient.
problem Lack of objective function capturing Bellman's principle optimally.
method Proposed a new objective function and its gradient.
result Preserves Bellman's principle of optimality in policy gradient.
Investigates stability properties of Haezendonck-Goovaerts premium principles in Orlicz spaces.
problem Stability properties of Haezendonck-Goovaerts premium principles in various Orlicz spaces.
method Analysis of stability properties including Fatou and Lebesgue properties, and continuity with respect to Φ-weak convergence. result Haezendonck-Goovaerts principles satisfy the Fatou property and Lebesgue property under certain conditions.
Study on maximum principles for nonlinear equations on Riemannian manifolds.
problem Investigating strong maximum principles for fully nonlinear equations on Riemannian manifolds.
method Analyzing scaling conditions and applying to various nonlinear operators.
result Established new strong comparison principles for second order uniformly elliptic problems.
Derives Fredholm criteria for isotypical components from a Simonenko principle.
problem Finding Fredholm conditions for isotypical components of invariant pseudodifferential operators.
method General Simonenko's local principle and equivariant local principle for restriction to isotypical components.
result Full proof of equivariant local principle and extension of results.
A pricing principle is introduced for non-attainable claims in incomplete markets.
problem Pricing non-attainable contingent claims in incomplete markets.
method Distorted Radon-Nikodym derivative and Tsallis relative entropy over a family of equivalent martingale measures.
result The pricing principle is closely related to backward stochastic differential equations and is arbitrage-free and time-consistent.
Study proves Maximum Principles for unbounded Riemannian domains.
problem Proving Maximum Principles for unbounded Riemannian domains.
method Examines both ambient manifold and differential operator assumptions.
result Valid Maximum Principles established for unbounded domains.
Establishes a boundary maximum principle for varifolds with fixed contact angle.
problem Boundary behavior of varifolds with contact angle constraints.
method Maximum principle for stationary pairs of varifolds with fixed contact angle condition.
result Boundary maximum principle proven for stationary varifolds.
Proves a principle for one-phase Bernoulli problem minimizers.
problem One-phase Bernoulli problem minimizers.
method Strong maximum principle, Alt-Caffarelli functional, Hardt-Simon-type foliation.
result Constructs a foliation for global minimizers.
We show that a well known uncertainty principle for functions on the circle can be derived from an uncertainty principle for the Euclidean motion group.
New method proves h-principles for stable forms on manifolds.
problem Proving h-principles for stable forms on manifolds. method Convex integration applied to stable forms.
result Proved h-principles for 4 classes of stable forms. We give a version of the comparison principle from pluripotential theory where the Monge-Ampère measure is replaced by the Bergman kernel and use it to derive a maximum principle
In this paper, we develop a new mathematical technique which allows us to express the joint distribution of a Markov process and its running maximum (or minimum) through the marginal distribution of the process itself. This technique is an extension of the classical reflection principle for Brownian motion, and it is o…
Deep networks often capture low frequency functions, improving generalization.
problem Understanding deep learning's generalization ability.
method Showed F-Principle holds for various loss functions and applied it to differential equations.
result Deep networks capture low frequency functions, leading to better generalization.
The Weyl principle holds in some Finsler settings despite general failure.
problem Applying the Weyl principle to Finsler manifolds.
method Investigation of the Weyl principle in Finsler geometry.
result A weak form of the Weyl principle persists in certain Finsler settings.
Note establishes a local maximum principle for Ricci flow under curvature conditions.
problem Preserving nonnegativity of curvature along Ricci flow with unbounded curvature.
method Combining scaling invariant curvature condition with Dirichlet heat kernel estimates.
result Unified and more direct proof of localized maximum principle.
Derives time-averaged active inference from control principles.
problem Finite-horizon or discounted-surprise problems in active inference.
method Derives infinite-horizon, average-surprise active inference from optimal control principles.
result Unified objective functional for sensorimotor control.
New principle for harmonic maps helps study higher-dimensional submanifolds.
problem Understanding unboundedness of totally geodesic projections in higher codimension.
method Introducing a flexible notion of convexity and applying it to harmonic and conformal maps.
result New maximum principle for harmonic maps applicable to various geometric settings.
Study proves rigidity for solitons with uncertainty principle.
problem Rigidity of shrinking Ricci solitons with uncertainty principle.
method Proves rigidity theorems for shrinking gradient Ricci solitons.
result Proves rigidity theorems with sharp constant in R^n.
In this paper we give three applications of a method to prove h-principles on closed manifolds. Under weaker conditions this method proves a homological h-principle, under stronger conditions it proves a homotopical one. The three applications are as follows: a homotopical version of Vassiliev's h-principle, the contra…
In this paper we prove two extensions of Hamilton's maximal principle for systems pf parabolic equations which sould be useful for the study of the Ricci flow and some other geometric evolution equations. One extension is a time-dependent maximum principle and the other is a time-dependent maximum principle subject to …
We present a new statistical learning paradigm for Boltzmann machines based on a new inference principle we have proposed: the latent maximum entropy principle (LME). LME is different both from Jaynes maximum entropy principle and from standard maximum likelihood estimation.We demonstrate the LME principle BY deriving …
New equivalences found linking parabolicity, comparison principle, and capacity on Riemannian manifolds.
problem Understanding parabolicity and related concepts on Riemannian manifolds.
method Establishing new equivalences between parabolicity, comparison principle, and capacity.
result Equivalence between p-parabolicity and the comparison principle for the p-Laplace equation. 3-manifolds study Hasse norm principle, akin to number fields.
problem Topological analog of Hasse norm principle for 3-manifolds.
method Analogy with number fields and finite cyclic coverings.
result Showed topological Hasse norm principle for 3-manifolds.
Establishes a principle for metric convergence in Kähler geometry.
problem Convergence of evolving Riemannian metrics in Kähler geometry.
method General 'boundedness implies convergence' principle applied to collapsing Calabi-Yau metrics and normalized Kähler-Ricci flows.
result Obtains convergence results for collapsing Calabi-Yau metrics and normalized Kähler-Ricci flows.
New Bianchi-convex sets generalize Ricci flow maximum principle.
problem Generalizing maximum principle for Ricci flow.
method Introducing Bianchi-convex sets.
result Hamilton's maximum principle extended to Bianchi-convex sets.
Paper proves h-principles for symplectic structures and foliations.
problem Existence of conformal symplectic structures and foliations.
method Application of h-principles and foliated Morse theory.
result Linear deformation of foliations to contact structures.
Maps to manifolds transverse to certain distributions satisfy an h-principle.
problem Maps to manifolds transverse to certain distributions.
method Proving the complete h-principle for transverse maps. result Maps to manifolds transverse to certain distributions satisfy the h-principle.