arXiv research

In \cite{LZ2} it is proved that for certain class of perturbations of the hyperbolic equation $u_t=f(u) u_x$, there exist changes of coordinate, called quasi-Miura transformations, that reduce the perturbed equations to the unperturbed one. We prove in the present paper that if in addition the perturbed equations posse…

Analyzes perturbed contact instantons with Legendrian boundary conditions using geometric analysis.

We study the general structure of formal perturbative solutions to the Hamiltonian perturbations of spatially one-dimensional systems of hyperbolic PDEs. Under certain genericity assumptions it is proved that any bihamiltonian perturbation can be eliminated in all orders of the perturbative expansion by a change of coo…

Based on the Hamiltonian dimensional reduction of $3+1$ axially symmetric, Ricci-flat Lorentzian spacetimes to a $2+1$ Einstein-wave map system with the (negatively curved) hyperbolic 2-plane target, we construct a positive-definite, (spacetime) gauge-invariant energy functional for linear axially symmetric perturbatio…

Study of free particle's geometry and its perturbations using complex projective structures.

We study the determination of the second-order normal form for perturbed Hamiltonians $H_ε=H_0 +εH_1 +\frac{ε^2}{2} H_2$, relative to the periodic flow of the unperturbed Hamiltonian $H_0$. The formalism presented here is global, and can be easily implemented in any CAS. We illustrate it by means of two examples: the H…

Generalizes Thomas-Yau theorem for special and minimal Lagrangians.

For an integrable Hamiltonian with $d\ (d\geq 2)$ degrees of freedom, we show the conditions on perturbations, for which invariant tori can be destructed.

In this paper, the method of approximate transformation groups which was proposed by Baikov, Gazizov and Ibragimov, is extended on Hamiltonian and bi-Hamiltonian systems of evolution equations. Indeed, as a main consequence, this extended procedure is applied in order to compute the approximate conservation laws and ap…

On the one hand, we prove that the Clifford torus in $\mathbb{C}^2$ is unstable for Lagrangian mean curvature flow under arbitrarily small Hamiltonian perturbations, even though it is Hamiltonian $F$-stable and locally area minimising under Hamiltonian variations. On the other hand, we show that the Clifford torus is r…

This paper studies the question of when a loop $φ$ in the group Symp$(M,ω)$ of symplectomorphisms of a symplectic manifold $(M,ω)$ is isotopic to a loop that is generated by a time-dependent Hamiltonian function. (Loops with this property are said to be Hamiltonian.) Our main result is that Hamiltonian loops are rigid …

New gauge fields modify Fokker-Planck dynamics without changing the stationary state.

Researchers found a way to measure energy in black hole perturbations.

This is a sequel to the paper [Oh5] (or ArXiv:math.SG/0206092). The main purpose of the paper is to give the proof of an existence theorem, with energy bounds, of certain pseudo-holomorphic sections of the mapping cylinder that is needed for the proof of nondegeneracy of the homological invariant pseudo-norm which the …

Study of intersections in Hamiltonian orbits on cotangent bundles.

A method to explain disease transformation using biomarker covariance matrices.

We study the ellipticity and the ``Nekhoroshev stability'' (stability properties for finite, but very long, time scales) of the Riemann ellipsoids. We provide numerical evidence that the regions of ellipticity of the ellipsoids of types II and III are larger than those found by Chandrasekhar in the 60's and that all Ri…

WSINDy identifies reduced Hamiltonian systems from particle interactions.

We present the construction of an infinite dimensional Banach manifold of quantum mechanical states on a Hilbert space H using different types of small perturbations of a given Hamiltonian. We provide the manifold with a flat connection, called the exponential connection, and comment on the possibility of introducing t…

The main goal of this paper is to give a unified treatment to many known cuplength estimates. As the base case, we prove that for $C^0$-perturbations of a function which is Morse-Bott along a closed submanifold, the number of critical points is bounded below in terms of the cuplength of that critical submanifold. As we…

The traceless $SU(2)$ character variety $R(S^2,\{a_i,b_i\}_{i=1}^n)$ of a $2n$-punctured 2-sphere is the symplectic reduction of a Hamiltonian $n$-torus action on the $SU(2)$ character variety of a closed surface of genus $n$. It is stratified with a finite singular stratum and a top smooth symplectic stratum of dimens…

Study shows how tangle moduli spaces relate to boundary surfaces.

Rabinowitz Floer homology is the semi-infinite dimensional Morse homology associated to the Rabinowitz action functional used in the pioneering work of Rabinowitz. Gradient flow lines are solutions of a vortex-like equation. In this survey article we describe the construction of Rabinowitz Floer homology and its applic…

An elementary family of local Hamiltonians $H_{\c ,\ell}, \ell = 1,2,3, ldots$, is described for a $2-$dimensional quantum mechanical system of spin $={1/2}$ particles. On the torus, the ground state space $G_{\circ,\ell}$ is $(\log)$ extensively degenerate but should collapse under $ł$perturbation" to an anyonic syste…

Geometrically transforms nonconservative dynamics to linearize Kepler and Manev systems.

This paper shows that Hamiltonians and operators can also be put to good use even in contexts which are not purely physics based. Consider the world of finance. The work presented here {models a two traders system with information exchange with the help of four fundamental operators: cash and share operators; a portfol…

In this paper we study rigidity aspects of Zoll magnetic systems on closed surfaces. We characterize magnetic systems on surfaces of positive genus given by constant curvature metrics and constant magnetic functions as the only magnetic systems such that the associated Hamiltonian flow is Zoll, i.e. every orbit is clos…

The use of Variational Autoencoders in different Machine Learning tasks has drastically increased in the last years. They have been developed as denoising, clustering and generative tools, highlighting a large potential in a wide range of fields. Their embeddings are able to extract relevant information from highly dim…

This paper analyzes the convergence of dynamic HMC and NUTS methods.

The paper explores hidden torus symmetries in integrable systems and their stability.

The Ma-Trudinger-Wang curvature --- or cross-curvature --- is an object arising in the regularity theory of optimal transportation. If the transportation cost is derived from a Hamiltonian action, we show its cross-curvature can be expressed in terms of the associated Jacobi fields. Using this expression, we show the l…

We prove the existence and uniqueness of constant mean curvature foliations for initial data sets which are asymptotically flat satisfying the Regge-Teitelboim condition near infinity. It is known that the (Hamiltonian) center of mass is well-defined for manifolds satisfying this condition. We also show that the foliat…

Paper reviews algebraic research in machine learning theory.

We propose a general framework to study the stability of the subspace spanned by $P$ consecutive eigenvectors of a generic symmetric matrix ${\bf H}_0$, when a small perturbation is added. This problem is relevant in various contexts, including quantum dissipation (${\bf H}_0$ is then the Hamiltonian) and risk control …

The paper proves compactness for holomorphic curves with boundary on nearby Lagrangians.

This is the first part of an article in two parts, which builds the foundation of a Floer-theoretic invariant, (I_F). (See math.DG/0505013 for part II). The Floer homology can be trivial in many variants of the Floer theory; it is therefore interesting to consider more refined invariants of the Floer complex. We consid…

We study the problem of recovery both the attenuation $a$ and the source $f$ in the attenuated X-ray transform in the plane. We study the linearization as well. It turns out that there are natural Hamiltonian flow that determines which singularities we can recover. If the perturbations $δa$, $δf$ are supported in a com…

The dynamics of an ideal fluid or plasma is constrained by topological invariants such as the circulation of (canonical) momentum or, equivalently, the flux of the vorticity or magnetic fields. In the Hamiltonian formalism, topological invariants restrict the orbits to submanifolds of the phase space. While the coadjoi…

Proves Weinstein's and Arnold's conjectures using contact instantons.

This work generalizes Hamiltonian mechanics using closed differential forms.

Study investigates Hamiltonian systems in fibered almost-symplectic manifolds.

We study partial collapsing degeneration of Hamiltonian-perturbed Floer trajectories for an adiabatic $\varepsilon$-family and its reversal adiabatic gluing, as the prototype of the partial collapsing degeneration of $2$-dimensional (perturbed) $J$-holomorphic maps to $1$-dimensional gradient segments. We consider the …

Holographic energy equals Hamiltonian energy.

Develops Hamiltonian Score Matching and Generative Flows for machine learning.

Summing Hamiltonian manifolds with a common submanifold.

The study examines stability of Hamiltonian Poisson integrators on both integrable and non-integrable systems.

New algorithms improve MCMC efficiency for complex distributions.

Following \cite{citeSavelyevVirtualMorsetheoryon$Omega$Ham$(Momega)$.}, we develop here a connection between Morse theory for the (positive) Hofer length functional $L: Ω\text {Ham}(M, ω) \to \mathbb{R}$, with Gromov-Witten/Floer theory, for monotone symplectic manifolds $(M, ω)$. This gives some immediate restrictio…