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…
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48 results for “phase symmetry”
Study connects symmetries in dynamical systems to phase plane representations.
problem Understanding symmetries in dynamical systems and their phase plane realizations.
method Analysis of symmetries in differential equations and phase plane representations, establishing correspondence and lifting conditions.
result Every symmetry generator in one formulation corresponds uniquely to a generator in the other, with a lifting condition to solve.
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.
In this paper, from the viewpoint of completeness of Marsden-Weinstein reduction, we illustrate how to give the definitions of a controlled Hamiltonian (CH) system and a reducible controlled Hamiltonian system with symmetry; and how to describe the dynamics of a CH system and the controlled Hamiltonian equivalence; as …
We introduce the concept of spontaneous symmetry breaking to arbitrage modeling. In the model, the arbitrage strategy is considered as being in the symmetry breaking phase and the phase transition between arbitrage mode and no-arbitrage mode is triggered by a control parameter. We estimate the control parameter for mom…
In this letter, we apply the artificial neural network in a supervised manner to map out the quantum phase diagram of disordered topological superconductor in class DIII. Given the disorder that keeps the discrete symmetries of the ensemble as a whole, translational symmetry which is broken in the quasiparticle distrib…
Stability inequalities for specific solutions in high dimensions.
problem Stability of solutions to the one-phase Bernoulli problem.
method Proving strict stability inequalities for cohomogeneity one solutions with bi-orthogonal symmetry.
result Strict stability for cohomogeneity one solutions in dimensions 7 and above.
Symmetry in inverse problems leads to multiple solutions, but breaking symmetry helps deep learning.
problem Symmetry in physical systems causes multiple solutions in inverse problems, hindering deep learning.
method Careful symmetry breaking on training data helps solve inverse problems and improve deep learning performance.
result Symmetry breaking on training data significantly improves deep learning performance in inverse problems.
Motivated by the work of Leznov--Mostovoy, we classify the linear deformations of standard -dimensional phase space that preserve the obvious symplectic -symmetry. As a consequence, we describe standard phase space, as well as and with their standard symplectic fo…
Symmetry-electronic fingerprints reveal competing magnetic phases in two-dimensional materialscond-mat.mtrl-sci
Symmetry-electronic fingerprints reveal competing magnetic phases in two-dimensional materials.
problem Predicting magnetic ground states, moments, and anisotropy in two-dimensional magnets.
method Introduce the symmetry-electronic fingerprint (SEF), a physically interpretable representation that encodes crystallographic symmetry operations, Wyckoff-site geometry, and site-resolved electronic structure.
result SEF-trained models accurately classify magnetic ordering and regress moments alongside anisotropy energies.
Generative diffusion models exhibit phase transitions in statistical mechanics, impacting their performance.
problem Understanding the performance and capabilities of generative diffusion models.
method Reformulating generative diffusion models using statistical mechanics, focusing on phase transitions and symmetry breaking.
result Generative diffusion models undergo second-order phase transitions with mean-field universality, critical instability, and mean-field critical exponents.
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.
GE-autoencoder identifies spontaneous symmetry breaking in systems.
problem Locating phase boundaries and identifying spontaneously broken symmetries in systems.
method Group-equivariant autoencoder using group theory to constrain parameters and learn invariant order parameters.
result GE-autoencoder accurately determines spontaneous symmetry breaking and estimates critical temperatures more efficiently.
Probing the effects of broken symmetries in machine learningphysics.chem-ph
A machine learning model with approximate rotational symmetry is tested and found stable.
problem The effects of broken symmetries in machine learning models.
method Testing a model with approximate rotational symmetry in various physical scenarios.
result The model remains stable even with noticeable symmetry artifacts, suggesting potential benefits.
Symmetry unifies AI learning dynamics, complexity, and representation.
problem Fragmented theories of AI learning mechanisms.
method Synthesis of parameter symmetry in AI models.
result Parameter symmetry breaking and restoration unify AI learning hierarchies.
On the Mechanism and Dynamics of Modular Addition: Fourier Features, Lottery Ticket, and Grokkingcs.LG
We explain how neural networks learn to solve modular addition tasks.
problem How two-layer neural networks learn to solve modular addition tasks.
method Formalized a diversification condition during training, proving it allows the network to approximate the correct logic for modular addition.
result Neural networks can robustly identify the correct sum through phase symmetry and frequency diversification.
Study of sub-Riemannian problem on specific Lie groups, revealing symmetries and bounds.
problem Sub-Riemannian problem on solvable, non-nilpotent Lie groups.
method Qualitative phase-space analysis of Hamiltonian system, focusing on vertical component.
result Explicit upper bound for cut time in terms of pendulum period.
DeepPhaseCut uses neural networks to improve Fourier phase retrieval.
problem Fourier phase retrieval from magnitude data.
method Unsupervised feed-forward neural network with cycleGAN training.
result Outperforms existing methods in Fourier phase retrieval.
New model uses symmetries and scaling laws to predict consumer advertising response.
problem Understanding consumer response to advertising efforts.
method Introduces a physics-based mathematical model to describe consumer response dynamics.
result The model better captures nonlinearities in advertising effects and provides new parameters for audience engagement.
Noise balance theory explains SGD's behavior in neural networks.
problem Understanding SGD's navigation in neural network loss landscapes.
method Analyzes minibatch noise and loss function symmetries.
result Derives the stationary distribution of SGD for deep networks.
Generative diffusion models are analyzed for their information dynamics.
problem Lack of a unified theoretical understanding of generative diffusion models.
method Integrated perspective connecting information-theoretic, dynamical, and thermodynamic aspects.
result Generative bandwidth is directly governed by the divergence of the score function's vector field.
The homotopy theory of topological defects in ordered media fails to completely characterize systems with broken translational symmetry. We argue that the problem can be understood in terms of the lack of rotational Goldstone modes in such systems and provide an alternate approach that correctly accounts for the intera…
In Hamiltonian mechanics, a (continuous) symmetry leads to conserved quantity, which is a function on (extended) phase space. In Nambu mechanics, a straightforward consequence of symmetry is just a relative integral invariant, a differential form which only upon integration over a cycle provides a conserved real number…
Minimal surfaces in spheres constructed from symmetry reductions of ODEs.
problem Construct minimal surfaces in spheres with symmetry.
method Doubling links of free-boundary minimal cones in R^(p+q+3) with bi-orthogonal symmetry.
result Existence of minimal embeddings of S^p × S^q × S^1 in S^(p+q+2).
Firm foundation theory estimates a security's firm fundamental value based on four determinants: expected growth rate, expected dividend payout, the market interest rate and the degree of risk. In contrast, other views of decision-making in the stock market, using alternatives such as human psychology and behavior, bou…
Unbiased Monte Carlo Cluster Updates with Autoregressive Neural Networkscond-mat.stat-mech
New method makes machine learning approximations unbiased and efficient.
problem Efficient sampling of complex probability distributions.
method Uses autoregressive neural networks with cluster updates and physical symmetries.
result Shows unbiased and low-variance approximations for phase transitions.
Constructs minimal capillary cones with specific symmetry and proves their existence and uniqueness.
problem Proves existence and uniqueness of minimal capillary cones with bi-orthogonal symmetry.
method Solves a nonlinear free boundary equation parametrized by the contact angle and uses monotonicity properties.
result Demonstrates that minimizing capillary hypersurfaces can have singularities in codimension 7.
In the framework of Galilei classical mechanics (i.e., general relativistic classical mechanics on a spacetime with absolute time) developed by Jadczyk and Modugno, we analyse systematically the relations between symmetries of the geometric objects. We show that the (holonomic) infinitesimal symmetries of the cosymplec…
Unoriented 3d TFTshep-th
This paper generalizes two facts about oriented 3d TFTs to the unoriented case. On one hand, it is known that oriented 3d TFTs having a topological boundary condition admit a state-sum construction known as the Turaev-Viro construction. This is related to the string-net construction of fermionic phases of matter. We sh…
Study on unique solutions to one-phase free boundary problems.
problem One-phase free boundary problems with singularities.
method Analyzing solutions at singular points and at infinity using one-homogeneous functions.
result Uniqueness of blowups and rigidity results at infinity.
Neural networks are commonly trained to make predictions through learning algorithms. Contrastive Hebbian learning, which is a powerful rule inspired by gradient backpropagation, is based on Hebb's rule and the contrastive divergence algorithm. It operates in two phases, the forward (or free) phase, where the data are …
Leveraging the intrinsic symmetries in data for clear and efficient analysis is an important theme in signal processing and other data-driven sciences. A basic example of this is the ubiquity of the discrete Fourier transform which arises from translational symmetry (i.e. time-delay/phase-shift). Particularly important…
Paper confirms conjecture for projective manifolds in supercritical phase.
problem Stability condition for deformed Hermitian-Yang-Mills equation.
method Establishes stability result not involving uniform constants.
result Confirms conjecture for projective manifolds in supercritical phase.
In this paper we classify maps from a torus phase space to , the space of , non-singular hermitian operators up to equivariant homotopy. The equivariance is with respect to a time-reversal involution on and an involution on defining a certain symmetry class. Furthe…
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…
In this paper we continue our study of bifurcations of solutions of boundary-value problems for symplectic maps arising as Hamiltonian diffeomorphisms. These have been shown to be connected to catastrophe theory via generating functions and ordinary and reversal phase space symmetries have been considered. Here we pres…
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.
5D SCFTs can have confining vacua with strings and unbroken symmetries.
problem Investigating phases of 5D SCFTs by varying couplings.
method Using geometric realisation of M-theory on metrically conical Calabi-Yau threefolds.
result Many 5D SCFTs have couplings leading to massive, confining vacua with strings and unbroken symmetries.
Study of IR phases in 3D class R theories linked to non-hyperbolic 3-manifolds.
problem Understanding IR phases of 3D class R theories associated with non-hyperbolic 3-manifolds.
method Analysis of IR phenomena through `exceptional' Dehn fillings and gauging of flavor symmetries.
result 3D class R theories associated with certain atoroidal non-hyperbolic 3-manifolds exhibit supersymmetry enhancement at low energy.
The Lie algebroids are generalization of the Lie algebras. They arise, in particular, as a mathematical tool in investigations of dynamical systems with the first class constraints. Here we consider canonical symmetries of Hamiltonian systems generated by a special class of Lie algebroids. The ``coordinate part'' of th…
Anomalies in (2+1)D fermionic topological phases and (3+1)D path integral state sums for fermionic SPTscond-mat.str-el
Constructs a path integral for fermionic SPTs, solving anomalies in 2+1D topological orders.
problem Anomalies in (2+1)D fermionic topological phases and their computation.
method Combining (2+1)D fermionic topological order with symmetry fractionalization data to construct a (3+1)D path integral.
result Reproduces the anomaly indicator for time-reversal symmetric topological superconductors.
Develops a framework for designing quantum neural networks that respect symmetries.
problem Trainability and generalization issues in quantum neural networks.
method Equivariant quantum neural networks (EQNN) for any symmetry group.
result Efficient construction of equivariant layers for EQNNs, including QCNNs.
We consider free and proper cotangent-lifted symmetries of Hamiltonian systems. For the special case of G = SO(3), we construct symplectic slice coordinates around an arbitrary point. We thus obtain a parametrisation of the phase space suitable for the study of dynamics near relative equilibria, in particular for the B…
Study of geometric properties of almost calibrated forms on Kähler manifolds.
problem Understanding the geometry of almost calibrated forms on compact Kähler manifolds.
method Investigates the infinite dimensional Riemannian manifold structure, CAT(0) geodesic metric space, and geodesics of the space of almost calibrated forms.
result The space of almost calibrated forms is an infinite dimensional Riemannian manifold with non-positive sectional curvature and CAT(0) geodesic metric space.
We study mirror symmetry of type II strings on manifolds with the exceptional holonomy groups and Spin(7). Our central result is a construction of mirrors of Spin(7) manifolds realized as generalized connected sums. In parallel to twisted connected sum manifolds, mirrors of such Spin(7) manifolds can be fou…
New symmetries discovered in Kepler's orbit family.
problem Symmetry properties of Kepler orbits and related subfamilies.
method Projective geometry and Lie's infinitesimal point symmetries.
result Kepler orbits form a flat family with a 7-dimensional local symmetry group.
We propose an extension of the structure equation for constant mean curvature (CMC) surfaces in a three dimensional Riemannian space form to the associated CMC hierarchy of evolution equations by the higher-order commuting symmetries. Via the canonical formal Killing field, considered as an infinitely prolonged and loo…
This review reports some key results in theoretical investigations on configurations of lipid membranes and presents several challenges in this field which involve (i) exact solutions to the shape equation of lipid vesicles; (ii) exact solutions to the governing equations of open lipid membranes; (iii) neck condition o…