Geometric Phasesmath-ph
The differential geometric aspects of Geometric Phases are reviewed.
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48 results for “geometric phases”
Geometric phases in stock trading show profits/losses without price changes.
problem Applying geometric phases to stock trading dynamics.
method Discrete-time systems analysis with zero-area cycles.
result Zero-area cycles in shape space represent high-frequency trading operations.
Unified geometric framework for adiabatic quantum mechanics.
problem Understanding geometric phases and exceptional points in quantum mechanics.
method Formal geometric framework for arbitrary non-degenerate Hamiltonians.
result Generalization of geometric phase to non-Hermitian Hamiltonians.
The paper studies phase transitions in geodesic flows on curved manifolds.
problem Understanding phase transitions in geodesic flows on geometrically finite manifolds.
method Defined a class of potentials and constructed geometrically finite manifolds to exhibit phase transitions.
result Geometric potential exhibits a phase transition on certain manifolds.
We give an elementary derivation of the Montgomery phase formula for the motion of an Euler top, using only basic facts about the Euler equation and parallel transport on the 2-sphere (whose holonomy is seen to be responsible for the geometric phase). We also give an approximate geometric interpretation of the geometri…
The rotation angle of a rolling disc is shown to be a geometric phase related to the Hopf fibration.
problem Understanding the geometric nature of rotation angles in kinematic models.
method Using the Hopf fibration and Gauss map, the geometric phase is decomposed into dynamical and geometric components.
result The geometric phase of rotation is described as the holonomy of the Hopf fibration.
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.
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 …
Study of rotation angles in a rotating disc model.
problem Understanding geometric phase in rotating systems.
method Analyzes a simple kinematic model of rotating discs.
result Explicit form of geometric phase found using Baumkuchen lemma.
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…
The paper establishes pressure gaps for manifolds with flat subtori singularities.
problem Understanding phase transitions in nonpositively curved manifolds with flat subtori.
method Derives a pressure gap criterion for closed rank 1 manifolds with specific singular sets and proves Hölder continuity of geometric potentials.
result Geometric potentials have pressure gaps and no phase transitions under certain curvature constraints.
On Quantum Indeterminacyquant-ph
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.
New method solves phase retrieval problems efficiently without lifting.
problem Phase retrieval from phaseless measurements.
method Flexible convex relaxation in signal domain, solving inequalities representing slabs.
result Convex program finds best aligned extreme point of slab intersection.
New method extracts hidden phases in binary mixtures using tubular tilings.
problem Hidden phases in binary mixtures are difficult to observe.
method Introduce tubular tilings for discretizing binary mixtures on smooth manifolds.
result Recover topological information about hidden phases from observable phases and interfaces.
Develops new strategy for Hessian estimates in Lagrangian mean curvature equation.
problem Interior Hessian estimates for solutions with prescribed Lipschitz phases.
method Allard-type regularity theorem, geometric measure theory, geometry of Lagrangian graphs, De Giorgi-Nash-Moser iteration.
result Sharp interior Hessian estimates for solutions with critical and supercritical phases.
A new geometric approach to quantum mechanics simplifies time-dependent problems.
problem Quantum mechanics ambiguities in time and observer choices.
method Generally covariant phase-spacetime coordinates and geometric flatness condition.
result Quantum mechanics becomes purely geometric and potentially topological.
This paper tackles phase retrieval in complex signals, proving geometric properties for a least-squares approach.
problem Recovering a complex signal from its Fourier magnitudes.
method Geometric analysis of a least-squares formulation for generalized phase retrieval.
result The least-squares formulation has no spurious local minimizers and negative curvature around saddle points.
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…
This paper is partly based on a lecture delivered by the author at the ERC workshop "Geometric Partial Differential Equations" held in Pisa in September 2012. What is presented here is an expanded version of that lecture.
This paper introduces SRPR for robust phase retrieval with smoothed loss functions.
problem Robust phase retrieval from noisy quadratic measurements with corruptions.
method Smoothed robust phase retrieval (SRPR) using convolution-type smoothed loss functions.
result SRPR has no spurious local solutions and benign landscape under corruptions.
Geometric phase and modulus relations for SU(n) matrix elements in the defining representationmath-ph
A set of relations between the modulus and phase is derived for amplitudes of the form $\mels{\hatu(x)}$ where in the fundamental representation and denotes the coordinates on the group manifold. An illustration is given for the case as well as a brief discussion of phase singularities …
New theorem links tropical phased matroids to higher-dimensional spheres.
problem Understanding topological properties of tropical phased matroids.
method Proving homeomorphism between topological order complex and a sphere.
result Topological order complex of tropical phased matroids is a -sphere.
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.
Using geometric quantization procedure, the quantization of algebra of observables for physical system with Ricci-flat phase space is obtained. In the classical case the appointed physical system is reduced to harmonic oscillator when the one real parameter is vanished.
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.
Geometric wave propagator defined on Riemannian manifolds.
problem Wave equation on Riemannian manifolds.
method Geometric approach to constructing propagator as oscillatory integral.
result Explicit small time asymptotic expansion of subprincipal symbol.
We define an almost--cosymplectic--contact structure which generalizes cosymplectic and contact structures of an odd dimensional manifold. Analogously, we define an almost--coPoisson--Jacobi structure which generalizes a Jacobi structure. Moreover, we study relations between these structures and analyse the associated …
In differential-geometric language, vortex-lines equations on extended phase space of a system may be written as , where 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 is a differen…
In this paper, we carry a detailed study of mechanical systems with configuration space for which the base variables are being controlled. The overall system's motion is considered to be induced from the base one due to the presence of general non-holonomic constraints. It is shown that the…
Aspects of a phase transition in high-dimensional random geometrycond-mat.stat-mech
High-dimensional random geometry shows phase transitions in various problems.
problem Phase transitions in high-dimensional random geometry.
method Analysis of various financial, optimization, and ecological problems.
result Links between seemingly distant fields and further ramifications.
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'.
This work proposes a geometric approach to identify slow invariant manifolds in dynamical systems.
problem Identifying slow invariant manifolds in multiple time-scale dynamical systems.
method Differential geometric concepts for submanifolds, sectional curvature, flow invariance.
result Necessary condition for slow invariant manifold invariance stated in terms of differential geometry.
Geometric observables detect financial regime shifts with high accuracy.
problem Detecting regime shifts in financial markets.
method Extracted four geometric observables from equity-index returns and evaluated them against various baseline methods.
result The Berry Phase Rate achieves an unbiased out-of-sample median Cohen's d of 0.72, significantly reducing false alarms.
This paper proposes a new Monte Carlo sampler that balances geometric exploitation and computational cost.
problem Sampling from high-dimensional targets with multiple modes or strong correlations.
method Geometric adaptive Monte Carlo sampler in a random environment.
result The sampler achieves a high effective sample size for a given computational cost.
GeoHNN models physics laws for stable, accurate predictions.
problem Violations of physical principles in machine learning models.
method Explicitly encodes geometric priors in inertia and phase space.
result Significantly outperforms existing models in long-term stability and accuracy.
Universal model for soft tissue mechanics under shock waves.
problem Modeling shock wave mechanics in soft biological tissues.
method Continuum mixture theory with phase-field mechanics.
result Universal thermodynamically consistent formulation for soft porous tissues.
Phase retrieval problems involve solving linear equations, but with missing sign (or phase, for complex numbers) information. More than four decades after it was first proposed, the seminal error reduction algorithm of (Gerchberg and Saxton 1972) and (Fienup 1982) is still the popular choice for solving many variants o…
Sharp bound on singular set dimension for specific geometric problems.
problem Hausdorff dimension of singular set in free boundary problems.
method Analysis of noncollapsed limits of manifolds with Ricci curvature bounds.
result Dimension bound of singular set is .
A new liquid crystalline texture is proposed using gnomonic projection of the Hopf fibration.
problem Creating bend-free textures in flat space from 3-sphere Hopf fibration.
method Geodesic-preserving gnomonic projection of the Hopf fibration.
result A new liquid crystalline phase with only splay and twist.
Geometric framework links clustering accuracy to structural recovery.
problem Understanding the trade-off between robustness and sensitivity in clustering.
method Develops a clustering condition number to compare within-cluster scale to the minimum loss increase required to move a point across a cluster boundary.
result Sharp phase transitions for exact recovery under different objectives, providing geometric principle for interpreting low objective values.
For simple Lie groups, the only homogeneous manifolds , where is maximal compact subgroup,for which the phase of the scalar product of two coherent state vectors is twice the symplectic area of a geodesic triangle are the hermitian symmetric spaces. An explicit calculation of the multiplicative factor on the c…
Surface tension of multi-phase flow with multiple junctions governed by the variational principlemath.NA
We explore a computational model of an incompressible fluid with a multi-phase field in three-dimensional Euclidean space. By investigating an incompressible fluid with a two-phase field geometrically, we reformulate the expression of the surface tension for the two-phase field found by Lafaurie, Nardone, Scardovelli, …
The Frenet frame generalizes the Park transform for multi-phase circuits.
problem Generalizing the Park transform for multi-phase circuits.
method Using the Frenet frame and Cartan's moving frames.
result The Frenet frame provides a new approach to circuit analysis.
Proposes using continuum percolation to analyze data manifolds and improve generative models.
problem Disentangling geometric support from probability distributions in high-dimensional data.
method Establishes a correspondence between topological phase transitions of random geometric graphs and data manifolds, using Percolation Shift metric.
result Demonstrates that Percolation Shift metric captures structural pathologies like mode collapse and guides training to prevent manifold shrinkage and improve fidelity.
We define partial differential (PD in the following), i.e., field theoretic analogues of Hamiltonian systems on abstract symplectic manifolds and study their main properties, namely, PD Hamilton equations, PD Noether theorem, PD Poisson bracket, etc.. Unlike in standard multisymplectic approach to Hamiltonian field the…
New algorithm solves low-rank phase retrieval with fewer measurements than previously possible.
problem Recovering low-rank matrices from phaseless projections.
method Developed an alternating minimization algorithm (AltMinLowRaP) with provable correctness and geometric convergence.
result The algorithm solves low-rank phase retrieval with measurements, achieving accuracy with high probability.
The phase of the scattering operator from the geometry of certain infinite-dimensional groupsmath-ph
We revisit the computation of the phase of the Dirac fermion scattering operator in external gauge fields. The computation is through a parallel transport along the path of time evolution operators. The novelty of the present paper compared with the earlier geometric approach by Langmann and Mickelsson, [LM], is that w…
Geometric step decay schedules improve stochastic algorithms' convergence on sharp nonconvex problems.
problem Convergence of stochastic algorithms on sharp nonconvex problems.
method Geometric step decay schedule applied to stochastic algorithms.
result Geometric step decay schedules lead to local linear convergence rates for sharp nonconvex problems.