New method uses quantum computing to process classical data efficiently.
problem Inefficient quantum machine learning due to data loading and trainability issues.
method Linear Hamiltonian-based machine learning with ground state problems for k-local Hamiltonians.
result Demonstrated the effectiveness and scalability of the method on up to 50 qubits.
Study shows how knot Floer homology and bordered Floer theory are linked.
problem Understanding the relationship between knot Floer homology and bordered Floer theory.
method Proves local equivalence and establishes algebraic formulas.
result Involutive knot Floer homology and bordered Floer theory determine each other under certain conditions.
Flat connections derived from Poisson brackets on loop spaces.
problem Understanding the structure of Poisson brackets on loop spaces.
method Defined connections by explicit linear combinations of standard connections associated with the Poisson bracket.
result Connections are shown to be flat.
In this paper we use structure preserving torus actions on Kahler-Einstein manifolds to construct minimal Lagrangian submanifolds. Our main result is: Let N^2n be a Kahler-Einstein manifold with positive scalar curvature with an effective T^n-action. Then precisely one regular orbit L of the T-action is a minimal Lagra…
Deep neural networks perform well on local tasks but struggle with global tasks.
problem Understanding the limitations of overparameterized deep neural networks in learning global functions.
method Introduced k-local and k-global functions to study the interplay between depth and function locality. result Depth is beneficial for learning local functions but detrimental to learning global functions.
Let M = M_{g,k} denote the space of properly (Alexandrov) embedded constant mean curvature (CMC) surfaces of genus g with k (labeled) ends, modulo rigid motions, endowed with the real analytic structure described in [kmp]. Let P=Pg,k=rg,k×R+k be the space of parabolic structures over Riemann surfac…
This work generalizes Hamiltonian mechanics using closed differential forms.
problem Hidden invariants in classical Hamiltonian mechanics.
method Establishes a novel correspondence between generalized Hamiltonian mechanics and multisymplectic geometry.
result Key theorems linking classical and generalized Hamiltonian systems.
Study investigates Hamiltonian systems in fibered almost-symplectic manifolds.
problem Existence and properties of non-vertical fully-Hamiltonian vector fields in almost symplectic manifolds with Lagrangian fibrations.
method Investigates vector fields in 2n-dimensional almost symplectic manifolds with Lagrangian fibrations, focusing on partially-Hamiltonian and fully-Hamiltonian vector fields.
result Non-vertical fully-Hamiltonian vector fields exist under certain genericity conditions and can be reduced to families of symplectic-Hamiltonian vector fields.
Holographic energy equals Hamiltonian energy.
problem Equating holographic and Hamiltonian energies.
method Relative holographic and Hamiltonian energy comparison.
result Holographic energy is identical to Hamiltonian energy.
Develops Hamiltonian Score Matching and Generative Flows for machine learning.
problem Estimating score functions and designing generative models.
method Introduces Hamiltonian velocity predictors (HVPs) for score matching and generative flows.
result Hamiltonian Generative Flows (HGFs) rival leading generative modeling techniques.
Summing Hamiltonian manifolds with a common submanifold.
problem Combining Hamiltonian manifolds with a shared submanifold.
method Establishing symplectic reduction and comparing Chern classes.
result Symplectic reduction of the sum agrees with the sum of reductions.
The study examines stability of Hamiltonian Poisson integrators on both integrable and non-integrable systems.
problem Investigating stability properties of Hamiltonian Poisson integrators.
method Examples of Lotka-Volterra dynamics and numerical investigations of a non-integrable system are used.
result The existence of a modified Hamiltonian is crucial for the stability of Hamiltonian Poisson integrators.
New algorithms improve MCMC efficiency for complex distributions.
problem High variance and low effective sample size in MCMC samplers.
method Antithetic Riemannian Manifold and Quantum-Inspired Hamiltonian Monte Carlo.
result Improved effective sample size and variance reduction.
The paper explores deformations of quasi-Hamiltonian spaces to Hamiltonian spaces.
problem Deforming quasi-Hamiltonian spaces to Hamiltonian spaces.
method Introducing and proving examples of deformations, including Lie groups and conjugacy classes.
result Moduli space of flat G-connections deforms to T*G^r+g.
This paper studies the geometry of the group of all co-Hamiltonian diffeomorphisms of a compact cosymplectic manifold (M,ω,η). The fix-point theory for co-Hamiltonian diffeomorphisms is studied, and we use Arnold's conjecture to predict the exact minimum number of fix point that such a diffeomorphism must have (thi…
Let (M,w) be a compact symplectic 2n-manifold, and g a Riemannian metric on M compatible with w. For instance, g could be Kahler, with Kahler form w. Consider compact Lagrangian submanifolds L of M. We call L Hamiltonian stationary, or H-minimal, if it is a critical point of the volume functional under Hamiltonian defo…
New integrators preserve geometric structure in Hamiltonian systems.
problem Preserving geometric structure in Hamiltonian systems on Jacobi manifolds.
method Combining Poissonization and symplectic bi-realizations to construct structure-preserving integrators.
result Explicit construction and application of Jacobi Hamiltonian integrators.
The paper develops a theory linking Hamiltonian and quasi-Hamiltonian manifolds.
problem Understanding the deformation of Hamiltonian quasi-Poisson manifolds to Hamiltonian Poisson manifolds.
method Introduces a generalized Hamiltonian deformation theory and constructs a topological quantum field theory.
result Shows that the imploded cross section of the double $D(G)_\imp$ deforms to the implosion of the cotangent bundle $T^*G_\imp$.
Gauss diagrams' properties can change with Hamiltonian cycle choice.
problem The impact of Hamiltonian cycle choice on Gauss diagrams.
method Examined realizable and unrealizable Gauss diagrams, and proved preservation of realizability under certain Hamiltonian cycle changes.
result Properties of Gauss diagrams can vary with Hamiltonian cycle choice.
We classify compact, connected Hamiltonian and quasi-Hamiltonian manifolds of cohomogeneity one (which is the same as being multiplicity free of rank one). Here the group acting is a compact connected Lie group (simply connected in the quasi-Hamiltonian case). This work is a concretization of the more general classific…
Classifies compact multiplicity free quasi-Hamiltonian manifolds.
problem Classifying compact, multiplicity free, quasi-Hamiltonian manifolds.
method Symplectic reductions and Lie group analysis.
result Recover old and find new examples of these structures.
Hamiltonian dynamics-based algorithms achieve deterministic and accelerated convergence for convex optimization.
problem Accelerating convex optimization
method Hamiltonian dynamics
result Hamiltonian dynamics-based algorithms achieve deterministic and accelerated convergence for convex optimization.
Arnold-Liouville systems cannot be bi-Hamiltonian generically.
problem The bi-Hamiltonian structure of Arnold-Liouville systems.
method Proving that a specific class of smooth functions is a meagre subset for the Fréchet topology, which implies Arnold-Liouville systems cannot be bi-Hamiltonian.
result Generically, Arnold-Liouville systems cannot be bi-Hamiltonian.
New method calculates volume-renormalized mass from Hamiltonian perspective.
problem Calculating volume-renormalized mass for asymptotically hyperbolic manifolds.
method Using Michel's mass invariants and a reduced Hamiltonian perspective, the volume-renormalized mass is deduced.
result The reduced Hamiltonian recovers the volume-renormalized mass and its variations.
Survey on strong closing lemmas in Hamiltonian dynamics.
problem Understanding dynamics in Hamiltonian systems.
method Use spectral invariants in symplectic geometry.
result Proofs of strong closing lemmas in various dimensions.
Hamiltonian Monte Carlo converges to target distributions under mild conditions.
problem Establishing convergence of Hamiltonian Monte Carlo algorithms.
method Analyzing Lq convergence for Hamiltonian Monte Carlo under mild conditions. result Outputs converge to target distributions under specified conditions.
The purpose of this short note is to prove the uniqueness of Hamiltonian volume minimizing Lagrangian submanifolds which are Hamiltonian isotopic to RP^n in CP^n modulo isometric group actions.
The paper discusses reducing Hamiltonian systems by scaling and standard symmetries, leading to Kirillov Hamiltonian systems.
problem Reduction of symplectic Hamiltonian systems by scaling and standard symmetries.
method Proof of Kirillov Hamiltonian systems and equivalence of reductions.
result Equivalent Kirillov Hamiltonian systems from different reduction orders.
Develops integrators for contact Hamiltonian systems preserving geometric structure.
problem Creating integrators for dissipative systems with geometric structure.
method Structure-preserving splitting framework based on exact-contact subflows.
result Local universality of contact splitting integrators.
Paper studies invariant distributions of bi-Hamiltonian structures.
problem Integrability of invariant distributions in bi-Hamiltonian structures.
method Description and investigation of invariant distributions.
result All invariant distributions of non-degenerate bi-Hamiltonian structures are described.
Researchers tackle the globalization problem of locally cosymplectic Hamiltonian dynamics.
problem Globalization problem of locally cosymplectic Hamiltonian dynamics.
method Investigate the geometry of locally conformally cosymplectic manifolds and provide a geometric Hamilton-Jacobi theory.
result Provide a geometric Hamilton-Jacobi theory on locally conformally cosymplectic manifolds.
Generalizes Hamiltonian structures to Dirac structures for new mechanics models.
problem Extending Hamiltonian structures to non-symplectic manifolds.
method Introduces Hamiltonian Lie algebroids and momentum sections over Dirac structures.
result Constructs new sigma models based on these structures.
All principal orbits of the standard Hamiltonian Tn-action on the complex projective space CPn are Lagrangian tori.In this article, we prove that most of them are not volume minimizing under Hamiltonian isotopies of CPn if the complex dimension n is greater than two, although they are Ham…
This paper studies Hamilton-Jacobi equations for magnetic systems with constraints.
problem Understanding dynamics of magnetic systems with geometric constraints.
method Developed Hamilton-Jacobi equations for magnetic systems with nonholonomic constraints.
result Revealed relationships between magnetic structures, constraints, and dynamics.
Proof confirms Hamiltonian isotopy of submanifolds.
problem Hamiltonian isotopy of submanifolds in symplectic geometry.
method Complete proof of isotopy lemma for symplectic submanifolds.
result All submanifolds are Hamiltonian isotopic.
Kernel methods accurately predict Hamiltonian systems from data.
problem Data-driven simulation of Hamiltonian systems.
method Two-step and one-step kernel-based methods for identifying and forecasting Hamiltonian systems.
result Framework achieves accurate, data-efficient predictions across various benchmark systems.
Paper presents a new port-Hamiltonian model for vehicle manipulators.
problem Complex mechanical systems' energy flow and conservation.
method Derives port-Hamiltonian dynamics from Hamiltonian reduction theory.
result Establishes mathematical equivalence with existing formulations.
In this paper, for a variety of nonholonomic (reducible) Hamiltonian systems, we first give to various distributional Hamiltonian systems, by analyzing carefully the dynamics and structures of the nonholonomic Hamiltonian systems. Secondly, we derive precisely the geometric constraint conditions of the induced distribu…
This work extends locally conformal analysis to multi-Hamiltonian settings, providing new geometric structures and Hamiltonian dynamics.
problem Globalization problem in multi-Hamiltonian formalisms due to incompatibilities on chart overlaps.
method Investigation of locally conformally Nambu--Poisson and locally conformally generalized Poisson manifolds, constructing Hamiltonian-type evolution equations.
result Unified framework for classical, Nambu--Poisson, and generalized Poisson manifolds within a locally conformal context.
We introduce the notion of Hamiltonian spaces for Manin pairs over manifolds, using the so-called generalized Dirac structures. As an example, we describe Hamiltonian spaces of a quasi-Lie bialgebroid using this general framework. We also discuss reduction of Hamiltonian spaces of this general type.
Survey of recent developments in symmetric reductions and controls for Hamiltonian systems.
problem Understanding the internal relationships of geometric structures and controls in Hamiltonian systems with symmetry.
method Survey and introduction of recent developments in controlled Hamiltonian systems with symmetry.
result Reveals the relationships between geometric structures, nonholonomic constraints, dynamical vector fields, and controls.
New boundary conditions improve Hamiltonian analysis in GR.
problem Improving Hamiltonian analysis in GR with IBVP.
method Presented and analyzed new boundary conditions.
result New boundary conditions lead to better Hamiltonian analysis.
Proves local bi-integrability of bi-Hamiltonian systems via bi-Poisson reduction.
problem Local bi-integrability of bi-Hamiltonian systems.
method Bi-Poisson reduction to prove local bi-integrability.
result Constructs a complete set of functions in bi-involution for bi-Hamiltonian systems.
Paper proves equivalence of two Floer theories using pearly trees and Hamiltonian flows.
problem Proving equivalence between two Floer theories.
method Using pearly tree discs and local Hamiltonian flows.
result Equivalence relation between immersed Lagrangian Floer theory and Hamiltonian immersed Lagrangian Floer theory.
Researchers present and compare different representations of dissipative Hamiltonian DAE systems.
problem Understanding and transforming dissipative Hamiltonian DAE systems.
method Global geometric and algebraic points of view, translations between representations, characterizations, and numerical methods for computing structural information.
result A general DAE system can be transformed into a dissipative Hamiltonian or port-Hamiltonian DAE system.
The present paper is a review of counterexamples to the ``Hamiltonian Seifert conjecture'' or, more generally, of examples of Hamiltonian systems having no periodic orbits on a compact energy level. We begin with the discussion of the ``classical'' and volume--preserving Seifert conjectures. Then a construction of coun…
Compact theorem on Hamiltonian stationary submanifolds in symplectic manifolds.
problem Compactness of Hamiltonian stationary Lagrangian submanifolds in symplectic manifolds.
method Proving a compactness theorem with area and extrinsic curvature bounds.
result Uniform bounds on area and total extrinsic curvature lead to compactness of Hamiltonian stationary Lagrangian submanifolds.
A Hamiltonian stationary Lagrangian submanifold of a Kaehler manifold is a Lagrangian submanifold whose volume is stationary under Hamiltonian variations. We find a sufficient condition on the curvature of a Kaehler manifold of real dimension four that guarantees the existence of a family of small Hamiltonian stationar…