Describes reconstructing Poisson structures from Lie group actions.
problem Reconstructing invariant Poisson structures from Lie group actions.
method Describes reconstruction of invariant Poisson structures from canonical actions of compact Lie groups on fibered phase spaces.
result Derives symmetry properties of Wong's type equations from main results.
A neural network learns phase space properties for time series analysis.
problem Lack of consistency and robustness in estimating embedding parameters.
method Forgetting mechanism neural network to learn phase space properties.
result Neural network approach is competitive or superior to state-of-the-art strategies.
Paper uses TDA to assess cryptocurrency risk by measuring phase space instability.
problem Traditional risk measures fail to capture market dynamics' geometric structure.
method Applied Takens' Delay Embedding Theorem to generate point cloud, computed persistent homology groups, defined Topological Persistence Norm.
result Proposed leverage calibration heuristic based on persistence of 1-dimensional cycles.
We present a generalization of Minkowski's classic theorem on the reconstruction of tetrahedra from algebraic data to homogeneously curved spaces. Euclidean notions such as the normal vector to a face are replaced by Levi-Civita holonomies around each of the tetrahedron's faces. This allows the reconstruction of both s…
Autoencoders misidentify anomalies due to data topology.
problem Autoencoders fail to accurately identify anomalies in data with nontrivial topology.
method Illustrative low-dimensional examples and analysis of autoencoder behavior in latent space.
result Topology of the dataset affects autoencoder performance, leading to misidentification of anomalies.
The paper extends a neural model to estimate uncertainty in multi-interaction trajectory reconstruction.
problem Lack of uncertainty estimation in neural models for multi-interaction trajectory reconstruction.
method Extended Factorised Neural Relational Inference model to output mean and standard deviation for each component of the phase space vector, using various loss functions.
result Demonstrated the importance of physical meaning of variables and existence of local minima during training.
New method detects anomalies and signals in complex, unlabeled data.
problem Detecting anomalies and signals in complex, unlabeled data with hidden nonlinear dynamics.
method Unsupervised machine learning in a Koopman operator function space.
result Significantly improved anomaly detection and signal identification.
CNNs reconstruct medium properties from wave probing responses.
problem Determining medium properties from wave responses.
method Deep convolutional neural networks (CNNs) for nonlinear wave equations.
result Quantitative dependence of network depth and units on medium complexity.
Enhanced speech emotion recognition using nonlinear recurrence dynamics.
problem Improving speech emotion recognition accuracy.
method Phase space reconstruction, Recurrence Plot, Recurrence Quantification Analysis, statistical functionals, feature fusion, Bidirectional Recurrent Neural Network.
result State-of-the-art performance on IEMOCAP with up to 10.7% improvement in accuracy.
We extend Routh's reduction procedure to an arbitrary Lagrangian system (that is, one whose Lagrangian is not necessarily the difference of kinetic and potential energies) with a symmetry group which is not necessarily Abelian. To do so we analyse the restriction of the Euler-Lagrange field to a level set of momentum i…
The big phase space, the geometric setting for the study of quantum cohomology with gravitational descendents, is a complex manifold and consists of an infinite number of copies of the small phase space. The aim of this paper is to define a Hermitian geometry on the big phase space. Using the approach of Dijkgraaf and …
SyMetric evaluates learned Hamiltonian dynamics from images, improving model stability and interpretability.
problem Lack of reliable metrics to assess learned Hamiltonian dynamics from images.
method Developed SyMetric, a binary indicator based on Hamiltonian dynamics properties.
result SyMetric identifies architectural improvements for better dynamics learning.
A new algorithm reconstructs population dynamics from coarse samples.
problem Reconstructing population dynamics from unlabeled samples at coarse time intervals.
method Deep Momentum Multi-Marginal Schrödinger Bridge (DMSB) framework.
result Significantly outperforms baselines in synthetic and real-world datasets.
Symplectic forms from two phase spaces are proven equivalent.
problem Equivalence of symplectic forms from different phase spaces.
method Proof of equivalence for theories over space-time with boundary.
result Symplectic forms derived from canonical and covariant phase spaces are equivalent.
Classifies symplectic phase space deformations preserving angular momentum symmetry.
problem Preserving symplectic o(n)-symmetry in phase space deformations. method Classifying linear deformations of phase space that preserve symplectic o(n)-symmetry. result Standard phase space and related manifolds as degenerations of a 3-dimensional family of coadjoint orbits.
New method for flux quantization on phase space stacks.
problem Defining and constructing flux-quantized phase space stacks.
method Observation of flux densities and characterization of Cauchy data.
result Flux-quantized phase space stacks have classifying spaces with rational Whitehead L-infinity algebra.
The paper explores the geometrical structures of phase spaces for controlled Hamiltonian systems with symmetry.
problem Understanding the dynamics and phase spaces of controlled Hamiltonian systems with symmetry.
method The paper uses Marsden-Weinstein reduction to define and analyze CH systems and their dynamics, focusing on the geometrical and topological structures of phase spaces.
result The paper reveals the relationships between the geometrical structures, dynamical vector fields, and controls of CH systems with symmetry.
Introduces a new phase space for 2D supersymmetric sigma models.
problem Developing a new Hamiltonian formulation for 2D supersymmetric sigma models.
method Introduces a phase space with spinorial momenta and derives a covariant Hamiltonian formulation.
result Shows the existence of additional supersymmetries in the new formulation.
The paper defines and characterizes para-Kahler structures on hom-Lie algebras.
problem Defining and characterizing para-Kahler structures on hom-Lie algebras.
method Introducing pseudo-Riemannian, para-Hermitian, and para-Kahler structures on hom-Lie algebras, providing examples, and defining phase spaces.
result Para-Kahler hom-Lie algebras give phase spaces and conversely, can be constructed from phase spaces.
Neural networks improve efficiency in integrating multi-dimensional phase spaces in particle physics.
problem Efficiently integrating multi-dimensional phase spaces in particle physics.
method Optimized Neural Network (NN) algorithm for phase space integration.
result NN-based approach achieves unweighting efficiencies of 30-75% in various particle physics examples.
The phase space of relativistic particle mechanics is defined as the 1st jet space of motions regarded as timelike 1-dimensional submanifolds of spacetime. A Lorentzian metric and an electromagnetic 2-form define naturally on the odd-dimensional phase space a generalized contact structure. In the paper infinitesimal sy…
The covariant phase space of a Lagrangian field theory is the solution space of the associated Euler-Lagrange equations. It is, in principle, a nice environment for covariant quantization of a Lagrangian field theory. Indeed, it is manifestly covariant and possesses a canonical (functional) "presymplectic structure" w …
This paper is concerned with basic geometric properties of the phase space of a classical general relativistic particle, regarded as the 1st jet space of motions, i.e. as the 1st jet space of timelike 1--dimensional submanifolds of spacetime. This setting allows us to skip constraints. Our main goal is to determine the…
Topological method detects Hopf bifurcations from time series.
problem Detecting Hopf bifurcations in nonlinear systems from time series data.
method Persistent homology applied to Takens embedding for phase space reconstructions.
result A simple scalar topological functional identifies critical bifurcation points.
In this paper we study vector fields on the big phase space of Gromov-Witten theory which are idempotents of the quantum product. Such vector fields can be used to simplify universal equations for higher genus Gromov-Witten invariants.
Complex and Kahler structures defined on hom-Lie algebras.
problem Defining structures on hom-Lie algebras.
method Introducing complex and Hermitian structures on hom-Lie algebras, providing examples, and constructing phase spaces.
result No proper complex (Hermitian) hom-Lie algebra of dimension two exists.
The study finds all possible heights for transformation groups on graphs.
problem Determining the possible heights of transformation groups on graphs.
method Proved the existence of a topological graph X for all finite p≥0. result For all finite p≥0, there exists a graph X such that the set of heights is {p,p+1,p+2,…}∪{+∞}. Study finds commodity and energy futures prices are nonlinearly determined rather than stochastic.
problem Determining the nature of futures prices in energy and commodity markets.
method MLE and determinism test based on phase space reconstruction, estimating κ for determinism rate.
result Futures prices show a reliability level κ near 1 and positive MLE, indicating nonlinear determinism.
DCTR uses neural networks to improve particle physics simulations and parameter tuning.
problem High computational cost limits precise scientific analysis in particle physics.
method Deep neural networks for reweighting and parameter tuning of simulations.
result DCTR enables precise simulations and parameter tuning, improving model accuracy.
Unified geometric approach to quantum indeterminacy.
problem Quantum indeterminacy and uncertainty principles.
method Geometric formulation using convex geometry and symplectic topology.
result Robertson-Schrodinger inequalities emerge as geometric principles.
This thesis revises phase space concepts in physics, incorporating physical dimensions.
problem Disconnection between theoretical models and units of measurement.
method Introducing unit-free manifolds and dimensioned algebraic structures.
result Reinterpretation of Jacobi manifolds as unit-free analogues of Poisson manifolds.
Probability Density Estimation (PDE) is a multivariate discrimination technique based on sampling signal and background densities defined by event samples from data or Monte-Carlo (MC) simulations in a multi-dimensional phase space. In this paper, we present a modification of the PDE method that uses a self-adapting bi…
A new geometric approach to identify slow invariant manifolds in complex systems.
problem The mathematical definition of slow invariant manifolds is unsatisfactory and limited to slow-fast systems.
method Formulate slow invariant manifolds geometrically within the context of differential geometry, focusing on covariant formulations.
result A more general definition of slow invariant manifolds is provided, independent of coordinate choice.
New geometrization of gravitational wave phase space.
problem Understanding the geometry of null-infinity in asymptotically flat space-times.
method Proposes a new geometrization using tractor calculus adapted to degenerate conformal metrics.
result Gravitational waves correspond to a class of tractor connections called 'null-normal'.
Study of trapped photons in Kerr spacetime's phase space.
problem Characterize trapped photons in Kerr spacetime.
method Explicit proof and new proof of trapped photons' set as a smooth 5D submanifold.
result Set of trapped photons forms a smooth 5D submanifold with topology SO(3)imesR2. In differential-geometric language, vortex-lines equations on extended phase space of a system may be written as iγ˙dσ=0, where σ is a differential 1-form. This is the structure, to give a paradigmatic example, of the Hamilton equations. Here, we study equations of the same structure, where σ is a differen…
Improves bounds on eigenfunctions using microlocal averages in phase space.
problem Improving Lp bounds on eigenfunctions in high frequency limit. method Develops sufficient conditions for microlocal averages in nonpositive curvature and partially hyperbolic flows.
result Improves microlocal averages for eigenfunctions in more general settings.
We study the indefinite metric G in the contact phase space (P,θ) of a homogeneous thermodynamical system introduced by R. Mrugala. We calculate the curvature tensor, Killing vector fields, second fundamental form of Legendre submanifolds of P - constitutive surfaces of different homogeneous thermodynamical syste…
No Einstein metrics found on certain double disk bundles.
problem Existence of Einstein metrics on specific manifolds.
method Phase space barrier argument to show non-existence.
result Proves non-existence of cohomogeneity one Einstein metrics.
In this paper we classify maps from a torus phase space X to Hn∗, the space of n×n, non-singular hermitian operators up to equivariant homotopy. The equivariance is with respect to a time-reversal involution on X and an involution on Hn∗ defining a certain symmetry class. Furthe…
Chirped sinosoids and interferometric phase plots are functions that are not periodic, but are the composition of a smooth function and a periodic function. These functions functions factor into a pair of maps: from their domain to a circle, and from a circle to their codomain. One can easily imagine replacing the circ…
Anomalous diffusion in SGD reveals interactions between hyperparameters and Hessian.
problem Understanding the limiting dynamics of SGD in deep neural networks.
method Continuous-time model of SGD as an underdamped Langevin equation, derived for linear regression.
result Anomalous diffusion is explained by modified loss and probability currents in phase space.
In this work, we use the Sternberg phase space (which may be considered as the classical phase space of particles in gauge fields) in order to explore the dynamics of such particles in the context of Hamilton-Dirac systems and their associated Hamilton-Pontryagin variational principles. For this, we develop an analogue…
We show that classical thermodynamics has a formulation in terms of Hamilton-Jacobi theory, analogous to mechanics. Even though the thermodynamic variables come in conjugate pairs such as pressure/volume or temperature/entropy, the phase space is odd-dimensional. For a system with n thermodynamic degrees of freedom it …
Study classifies rotational hypersurfaces with prescribed mean curvature.
problem Classifying rotational hypersurfaces with prescribed mean curvature.
method Phase space analysis to classify hypersurfaces.
result Delacunay-type classification for even prescribed functions.
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.
Neural network models transform physical systems into latent Gaussian distributions.
problem Simplifying and solving classical Hamiltonian systems.
method Symplectic neural networks for canonical transformations.
result Captures nonlinear collective modes in latent space.
In this paper we explore the idea of looking at the Dirac quantisation conditions as ℏ-dependent constraints on the tangent bundle to phase-space. Starting from the path-integral version of classical mechanics and using the natural Poisson brackets structure present in the cotangent bundle to the tangent bundle o…