Contact Lie systems analyze integral curves of Hamiltonian vector fields.
problem Analyzing integral curves of Hamiltonian vector fields.
method Define and study contact Lie systems, including conservative systems.
result Develop Liouville theorems, contact reductions, and Gromov non-squeezing theorems.
New Lie systems defined on k-contact manifolds, with applications.
problem Understanding Lie systems on k-contact manifolds. method Distributional approach to k-contact manifolds and Hamiltonian vector fields. result Lie systems can be understood as Hamiltonian relative to a k-contact manifold. Study new k-contact distributions and Lie systems.
problem Characterizing and understanding k-contact distributions. method Analyzing Goursat distributions and Lie systems.
result Characterized new types of k-contact distributions. Infinite-dimensional contact geometry explored.
problem Generalizing contact geometry to infinite dimensions.
method Generalization of cosymplectic, contact, and cocontact manifolds to infinite dimensions.
result Model examples of time-dependent and dissipative Hamiltonian systems calculated.
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.
Invariant measures found for contact Hamiltonian systems split into Reeb and Liouville dynamics.
problem Finding invariant measures for contact Hamiltonian systems.
method Splitting the system into Reeb and Liouville dynamics; using invariant measures and symplectic sandwiches.
result Invariant measure found for Reeb dynamics; characterization of Liouville dynamics invariant measure.
A new metriplectic system on contact manifolds is introduced for thermodynamic consistency.
problem Developing a thermodynamically consistent dynamical system on contact manifolds.
method Introducing a metriplectic dynamical system on the one-jet bundle J1N. result The metriplectic system is thermodynamically consistent, with H˙=0 and S˙≥0. We give necessary and sufficient geometric conditions for a distribution (or a Pfaffian system) to be locally equivalent to the canonical contact system on Jn(R,Rm), the space of n-jets of maps from R into Rm. We study the geometry of that class of systems, in particular, the existence of corank one involutive subdistr…
The paper analyzes errors in mechanical systems with external forces.
problem Error analysis of mechanical systems with external forces.
method Analysis of variational integrators with contact order r for discrete mechanical systems. result The contact order of the integrator is the same as the contact order of the original systems.
Paper introduces Eden bracket for nonholonomic systems.
problem Nonholonomic contact systems with dissipation.
method Contact Eden bracket for system evolution.
result Evolution of observables in constrained systems.
We prove that a topological contact isotopy uniquely defines a topological contact Hamiltonian. Combined with previous results from [MS11], this generalizes the classical one-to-one correspondence between smooth contact isotopies and their generating smooth contact Hamiltonians and conformal factors to the group of top…
New method for constructing contact Lie systems on various spaces.
problem Constructing contact Lie systems on Riemannian and Lorentzian spaces.
method Adaptation of scaling symmetries to Lie-Hamilton systems, leading to contact Lie systems.
result Curvature-dependent reductions of contact Lie systems on Cayley-Klein spaces.
The study explores autonomous systems and their connections to contact geometry and Frobenius manifolds.
problem Understanding the connections between autonomous systems and geometric structures.
method Investigation of the Darboux-Halphen-Ramanujan system, contact geometry, and Frobenius manifolds.
result Highlighting the role of contact geometry in autonomous systems.
Introduces GFC for learning complex dynamical systems with geometric constraints.
problem Challenges in accurately modeling and predicting complex dynamical systems with geometric constraints.
method Geometric Contact Flows (GFC) using Riemannian and Contact geometry as inductive biases.
result Ensemble of contactomorphisms adapt the latent contact Hamiltonian model to target dynamics while preserving desirable properties.
I begin by giving a general discussion of completely integrable Hamiltonian systems in the setting of contact geometry. We then pass to the particular case of toric contact structures on the manifold S2×S3. In particular we give a complete solution to the contact equivalence problem for a class of toric conta…
New optimization method derived from contact geometry.
problem Optimization problems in machine learning and engineering.
method Contact geometry and dynamical systems.
result Bregman Hamiltonian system can be transformed into separable Hamiltonian.
Develops theory of contact systems with nonholonomic constraints.
problem Nonholonomic constraints in contact systems.
method Variational principle and projection of Hamiltonian vector field.
result Nonholonomic dynamics as projection of unconstrained dynamics.
We develop a new geometric framework suitable for dealing with Hamiltonian field theories with dissipation. To this end we define the notions of k-contact structure and k-contact Hamiltonian system. This is a generalization of both the contact Hamiltonian systems in mechanics and the k-symplectic Hamiltonian syst…
Paper bridges quantum and classical mechanics for open systems.
problem Quantum open systems with bi-Lindblad structure.
method Develops a bridge between bi-Hamiltonian structures and GKSL formalism, introducing contact-compatible Lindblad generators.
result Provides a mathematical mechanism for semiclassical limit of quantum open systems.
Reduces symplectic Hamiltonian systems to contact systems, realizing Poincaré's dream.
problem Scaling symmetries in Hamiltonian systems.
method Contact reduction of symplectic Hamiltonian systems.
result Generically possible reduction to contact Hamiltonian systems, reducing inputs needed.
This paper extends the evolution operator to contact mechanics, linking Lagrangian and Hamiltonian formulations.
problem Translating the evolution operator to contact mechanics for mechanical systems with dissipation.
method Using the evolution operator K to connect Lagrangian and Hamiltonian formalisms in contact mechanics.
result The evolution operator provides a geometric description of evolution equations and relates constraints.
This thesis improves protein contact prediction using unsupervised and supervised methods.
problem Improving accuracy of protein contact prediction.
method Unsupervised and supervised deep learning methods.
result A scoring system called diversity score for measuring contact novelty.
In this paper, we extend the well-known Noether theorem for Lagrangian systems to contact Lagrangian systems. We introduce a classification of infinitesimal symmetries and obtain the corresponding dissipated quantities. We notice that in contact dynamics, the existence of infinitesimal symmetries does not produce conse…
Lecture notes introduce contact complete integrability for odd-dimensional manifolds.
problem Integrability on odd-dimensional manifolds using contact geometry.
method Introduce contact geometry concepts, discuss contact Hamiltonian vector field, Jacobi bracket, and contact complete integrability.
result Two different notions of contact complete integrability coincide.
A Hamilton-Jacobi theory for general dynamical systems, defined on fibered phase spaces, has been recently developed. In this paper we shall apply such a theory to contact Hamiltonian systems, as those appearing in thermodynamics and on geodesic flows in fluid mechanics. We first study the partial and complete solution…
Study the commutativity of reduction and symplectification in contact Hamiltonian systems.
problem Understanding the commutativity of reduction and symplectification in contact Hamiltonian systems.
method Introduce symplectic and contact geometry, perform reduction via momentum map, analyze symplectification process.
result Commutativity relations between reduction and symplectification in contact Hamiltonian systems.
Abstract: Study Hamiltonian systems on almost cosymplectic manifolds, extending contact Hamiltonian systems.
problem Extend Hamiltonian systems to almost cosymplectic manifolds.
method Determine Hamiltonian vector field on odd-dimensional almost cosymplectic manifolds.
result Extend equations of motion to generalized transitive almost cosymplectic structures.
Reduces equations for contact mechanical systems on Lie groups by exploiting symmetries.
problem Contact mechanical systems on Lie groups with symmetries.
method Reduction process using Lie group actions and symmetries.
result Euler-Poincaré-Herglotz equations on the reduced phase space.
New systems of linear PDEs discovered in 3D contact manifolds.
problem Investigating linear PDEs of sl3-type. method Complete local classification using extrinsic geometry.
result 7 new systems of second-order linear PDEs with 8-dimensional solution spaces.
The paper defines and proves equivalence of nonholonomic brackets in contact mechanical systems.
problem Nonholonomic constraints in contact geometry.
method Construct a general framework for non-holonomic constraints, define and prove equivalence of different nonholonomic brackets.
result All nonholonomic brackets coincide and one is an almost Jacobi bracket.
This paper constructs a family of coordinate systems about a point on a quaternionic contact manifold, called quaternionic contact pseudohermitian normal coordinates. Once defined, conformal variations of the quaternionic contact structure induce changes on the coordinates which are studied in an effort to simplify the…
We introduce a novel systematic construction for integrable (3+1)-dimensional dispersionless systems using nonisospectral Lax pairs that involve contact vector fields. In particular, we present new large classes of (3+1)-dimensional integrable dispersionless systems associated to the Lax pairs which are polynomial and …
Complex contact manifolds arise naturally in differential geometry, algebraic geometry and exterior differential systems. Their classification would answer an important question about holonomy groups. The geometry of such manifold X is governed by the contact lines contained in X. These are related to the notion of…
We study vector fields of the plane preserving the form of Liouville. We present their local models up to the natural equivalence relation, and describe local bifurcations of low codimension. To achieve that, a classification of univariate functions is given, according to a relation stricter than contact equivalence. W…
One considers a special class of PDEs systems and one determines the associated symmetry group. Particulary, for the Blair system, one finds the symmetry group. A solutions of the Blair system gives a conformally flat contact metric structure and also it defines a "force-free" model of solar physics. By using the symme…
New neural network designs learn contact dynamics efficiently.
problem Learning contact dynamics in robotics from noisy data.
method Physically structured neural networks.
result Data-efficient learning of discontinuous contact events.
We show that the planar circular restricted three body problem is of restricted contact type for all energies below the first critical value (action of the first Lagrange point) and for energies slightly above it. This opens up the possibility of using the technology of Contact Topology to understand this particular dy…
Contact path geometries are curved geometric structures on a contact manifold comprising smooth families of paths modeled on the family of all isotropic lines in the projectivization of a symplectic vector space. Locally such a structure is equivalent to the graphs in the space of independent and depedent variables of …
New classification of hyperbolic Monge-Ampère systems with S1=0.
problem Characterizing hyperbolic Monge-Ampère systems with S1=0. method Analyzing invariant tensors S1 and S2 to classify systems with S1=0. result All S1=0 systems with cohomogeneity at most one are linear up to contact transformations. Lagrangian contact supersymmetries (depending on derivatives of arbitrary order) are treated in very general setting. The cohomology of the variational bicomplex on an arbitrary graded manifold and the iterated cohomology of a generic nilpotent contact supersymmetry are computed. In particular, the first variational fo…
Unified geometric framework for integrability of conservative and dissipative systems.
problem Unified definition of integrability for both conservative and dissipative systems.
method Introducing Jacobi-Haantjes manifolds and contact-Haantjes manifolds to unify definitions.
result Equivalence of integrability in contact Hamiltonian systems and existence of Abelian extended Haantjes algebra.
Analyzes perturbed contact instantons with Legendrian boundary conditions using geometric analysis.
problem Analyzing nonlinear elliptic systems associated with contact Hamiltonian trajectories.
method Identifies correct action and energy functionals, develops elliptic regularity theory, and proves asymptotic convergence.
result Established C∞ convergence of perturbed contact instantons under finite energy hypothesis. Silent abandonment reduces contact center efficiency by 5%-15%.
problem Measuring customer abandonment and patience in text-based contact centers is challenging due to uncertainty.
method Developed methodologies to identify silent-abandonment customers and estimate customer patience using text analysis and queueing models.
result Silent abandonment accounts for 30%-67% of customer abandonments and reduces system efficiency by 5%-15%.
Extends coisotropic embedding theorem to various geometric settings.
problem Regularization problem of singular Lagrangian systems.
method Generic methodology applied to cosymplectic, contact, and multisymplectic manifolds.
result Fundamental basis for applying results to regularization problem.
We complete the classification of compact connected contact toric manifolds initiated by Banyaga and Molino and by Galicki and Boyer. As an application we prove the conjectures of Toth and Zelditch on toric integrable systems on the n-torus and the 2-sphere.
Following the Cartans's original method of equivalence supported by methods of parabolic geometry, we provide a complete solution for the equivalence problem of quaternionic contact structures, that is, the problem of finding a complete system of differential invariants for two quaternionic contact manifolds to be loca…
Develops k-contact geometry theory for field theories.
problem Analyse field theories using k-contact geometry.
method Distributions maximally non-integrable with k commuting Lie symmetries.
result Established k-contact distributions and their relationships.
Study magnetic geodesics on odd spheres, computing critical energy values.
problem Understanding magnetic geodesics on odd-dimensional spheres.
method Explicit computation and analysis of submanifolds and symmetries.
result Energy values determine magnetic geodesic connectivity on spheres.