Minimal surfaces in AdS(4) correspond to elliptic solutions of the cosh-Gordon equation.
problem Minimal surfaces in AdS(4) and their relation to entanglement entropy.
method Inverting Pohlmeyer reduction to construct static minimal surfaces in AdS(4) that correspond to elliptic solutions of the reduced system.
result A two-parameter family of static minimal surfaces in AdS(4) that include helicoids and catenoids as special limits.
Develops finite-gap solutions for Pohlmeyer--Lund--Regge equation and Lund--Regge curve evolution.
problem Solving the Pohlmeyer--Lund--Regge equation and understanding Lund--Regge curve evolution.
method Finite-gap construction using hyperelliptic spectral data, Baker--Akhiezer function, and SU(2)-frame. result Explicit theta-quotient formula for PLR solutions and criteria for Lund--Regge curve evolution.
The paper studies curve evolution using the PLR equation and its solutions.
problem Investigating the evolution of space curves governed by the PLR equation.
method Examined the Lund-Regge evolution and derived its representation in the Frenet frame, aligning with the Lax system of the PLR equation. Developed a construction method for curve families via the Sym formula.
result Described the Lund-Regge evolution corresponding to Date multi-soliton solutions to the PLR equation.
Deformed σ-models linked to Ricci flow and Toda theories.
problem Understanding the relationship between deformed σ-models and geometric flows.
method Exploring trigonometric deformations of CP^n-1 models and their duals, linking to Ricci flow and Toda field theories.
result Trigonometric deformations of CP^n-1 models solve the Ricci flow equation and relate to Toda field theories.
Abstract: Studies systems of equations for pseudo-spherical or spherical surfaces, finding integrability conditions and new families of equations.
problem Characterize and classify systems of equations describing pseudo-spherical or spherical surfaces.
method Integrability conditions of g-valued linear problems, with g=sl(2,R) or g=su(2). result Obtained characterization and classification results, providing new examples and families of differential equations.
This paper identifies knot projections with reductivity two.
problem Determining knot projections with a specific reductivity level.
method Examined four types of reductivity (Seifert type splice, non-Seifert type splice, recursively, simultaneously) and their combinations.
result Identified all knot projections with reductivity two for the four definitions.
Completes reduction scheme in Lagrange-Poincaré category.
problem Lagrangian reduction by stages in the whole category.
method Analyzes Noether theorem, Hamiltonian reduction, geometric aspects.
result Affirmative answer to open question of Lagrangian reduction.
This paper classifies instantons with closed reductions and provides examples of non-closed reductions.
problem Understanding the geometry of toric Kähler instantons with and without closed reductions.
method Sharp geometric criteria and examples of instantons with different reduction types.
result Established geometric criteria for closed reductions and classified asymptotic geometries.
We consider locally conformal Kaehler geometry as an equivariant (homothetic) Kaehler geometry: a locally conformal Kaehler manifold is, up to equivalence, a pair (K,Γ) where K is a Kaehler manifold and Γa discrete Lie group of biholomorphic homotheties acting freely and properly discontinuously. We define a new invari…
Classifies 7- and 8-dimensional naturally reductive spaces.
problem Classifying naturally reductive spaces in 7 and 8 dimensions.
method Combines structure theory and new construction methods.
result Complete classification of 7- and 8-dimensional naturally reductive spaces.
Abstract revisits Kähler reduction using GIT, generalizing results.
problem Generalizing Kähler reduction results to the generalized setting.
method Geometric invariant theory approach to generalized Kähler reduction.
result Many well-known Kähler reduction results can be generalized.
In this paper we describe Routhian reduction as a special case of standard symplectic reduction, also called Marsden-Weinstein reduction. We use this correspondence to present a generalization of Routhian reduction for quasi-invariant Lagrangians, i.e. Lagrangians that are invariant up to a total time derivative. We sh…
Two reduction schemes for symplectic manifolds are shown equivalent.
problem Reduction of Hamiltonian systems on exact symplectic manifolds.
method Modified Marsden-Meyer-Weinstein reduction theorem for exact symplectic manifolds and contact manifolds.
result Reduction schemes are equivalent for exact symplectic manifolds and energy hypersurfaces.
Study extends Kobayashi's method to non-reductive subgroups for homogeneous spaces.
problem Existence of compact Clifford-Klein forms in homogeneous spaces.
method Extend Kobayashi's method to non-reductive subgroups and compare Cartan projections and non-compact dimensions.
result Examples of homogeneous spaces without compact Clifford-Klein forms.
The purpose of this paper is to generalize the regular Optimal Reduction Theorem to general proper Dirac actions, formulated both in terms of point and orbit reduction. A comparison to general standard singular Dirac reduction is given emphasizing the desingularization role played by optimal reduction.
We show that the contact reduction can be specialized to Sasakian manifolds. We link this Sasakian reduction to Kähler reduction by considering the Kähler cone over a Sasakian manifold. We present examples of Sasakian manifolds obtained by S1 reduction of standard Sasakian spheres.
Study characterizes naturally reductive metrics on homogeneous manifolds.
problem Characterizing naturally reductive (α1,α2) metrics on homogeneous manifolds. method Characterization through local f-products and equivalence of properties. result Explicit flag curvature formula for naturally reductive metrics.
Abstract: Generalized reduction methods for symmetries in graded geometry.
problem Generalized reduction of symmetries in graded geometry.
method Graded symplectic reduction for Courant, Dirac, and generalized complex structures.
result Systematic recovery of reduction schemes for exact cases.
Hyperkähler reduction explained via frame bundles.
problem Describing hyperkähler reduction using frame bundles.
method Using frame bundles and connections, tracing reductions and analyzing fibers.
result Fibers of hyperkähler reduction are totally geodesic.
In this note we give conditions which ensure the reduction of a symplectic connection in the process of a Marsden-Weinstein reduction and of the reduction of a presymplectic manifold.
Introduces symplectic reduction in nonrational toric geometry.
problem Symplectic reduction in nonrational toric geometry.
method Specializes to nonrational toric geometry and rational case for symplectic reduction.
result Symplectic reduction for nonrational Lie subgroups.
This work introduces a unified approach to the reduction of Poisson manifolds using their description by graded symplectic manifolds. This yields a generalization of the classical Poisson reduction by distributions (Marsden-Ratiu reduction). Further it allows one to construct actions of strict Lie 2-groups and to descr…
This paper extends symplectic reduction to cosymplectic groupoids and explores their properties.
problem Cosymplectic groupoids and their reductions.
method Analogous to symplectic reduction, the authors extend the Marsden-Weinstein-Meyer reduction to cosymplectic groupoids.
result Integration commutes with reduction for algebroids associated with cosymplectic groupoids.
Let EG be a stable principal G--bundle over a compact connected Kaehler manifold, where G is a connected reductive linear algebraic group defined over the complex numbers. Let H⊂G be a complex reductive subgroup which is not necessarily connected, and let EH⊂EG be a holomorphic reduction of s…
The reduction of nonholonomic systems is formulated in terms of Dirac reduction. An optimal reduction method for a class of nonholonomic systems is formulated. Several examples are studied in detail.
New definition of naturally reductive Finsler manifolds using geodesic graphs.
problem Defining naturally reductive Finsler manifolds using geodesic graphs.
method Proposed a new geometrical definition using geodesic graphs and constructed examples of Finsler metrics.
result Explicit examples of Finsler naturally reductive metrics constructed.
Survey of Lagrangian reduction for discrete mechanical systems.
problem Understanding and reducing complex mechanical systems.
method Lagrangian reduction applied to discrete-time mechanical systems.
result Introduction to reduction techniques for various constraints and forces.
New method corrects missing data bias in dimension reduction.
problem Missing data complicates high-dimensional data analysis.
method Developed a bias-corrected Gram matrix for heterogeneous missingness.
result Proposed method improves dimension reduction techniques significantly.
Reduces Nambu-Poisson manifolds by regular distributions, ensuring reduction unless distribution is zero.
problem Reduction of Nambu-Poisson manifolds by regular distributions.
method Extending Marsden-Ratiu and Falceto-Zambon theorems, defining gauge transformations.
result Reduction is always ensured unless the distribution is zero.
Paper compares Lagrangian reduction methods for rigid body systems.
problem Modeling and reduction of rigid body systems with rotors.
method Euler-Poincaré reduction by the whole group and reduction by stages.
result Equivalence of equations and conservation laws are tracked.
The paper extends Marsden-Weinstein reduction to mechanical presymplectic structures for time-dependent Hamiltonian systems.
problem Limitations of Marsden-Weinstein reduction for cosymplectic structures in time-dependent Hamiltonian systems.
method Developed Marsden-Weinstein reduction for mechanical presymplectic structures.
result Mechanical presymplectic structures provide a more suitable framework for time-dependent Hamiltonian systems than cosymplectic structures.
Paper constructs naturally reductive spaces with a general formula.
problem Understanding naturally reductive spaces.
method Explicit construction from \cite{Storm2018} and general formula derivation.
result Proves reducibility and isomorphism criteria.
The paper simplifies symmetries in complex geometric structures.
problem Redundancy in conditions for symmetry reduction in polysymplectic and polycosymplectic structures.
method Exploring and proving necessary and sufficient conditions for polycosymplectic reduction.
result A one-to-one relationship between polycosymplectic reduction and the reduction of a larger polysymplectic manifold.
The paper explores polysymplectic structures and their reductions in field theories.
problem Invariance of Lagrangian and Hamiltonian field theories under symmetry groups.
method Application of polysymplectic reduction theorem for both Lagrangian and Hamiltonian field equations.
result Identification and relation of polysymplectic structures through Routhian function and Legendre transformation.
A new construction of naturally reductive spaces is presented. This construction gives a large amount of new families of naturally reductive spaces. First the infinitesimal models of the new naturally reductive spaces are constructed. A concrete transitive group of isometries is given for the new spaces and also the na…
Reduces LCS manifolds with symplectic actions, preserving conformal structure.
problem Reduction of LCS manifolds with symplectic actions.
method Locally conformally symplectic reduction procedure.
result Compatibility with locally conformally Kähler structure and contact reduction.
Develops Marsden-Meyer-Weinstein reduction for k-contact field theories.
problem None explicitly stated, but related to field theories and contact geometry.
method Marsden-Meyer-Weinstein reduction techniques applied to k-contact field theories. result Clarifies and corrects previous contact reduction literature.
Paper improves variance reduction technique for weak approximations.
problem Improving variance reduction for weak approximation schemes.
method Stratified regression-based variance reduction approach.
result Significant variance reduction achieved.
Paper reviews and synthesizes methods for evaluating dimensionality reduction techniques.
problem Evaluating and comparing dimensionality reduction techniques.
method Framework and toolkit in R for exploring and evaluating dimensionality reduction quality through visual insights.
result Helps researchers compare and select dimensionality reduction techniques using visual insights.
Paper shows spectra can't distinguish naturally reductive manifolds.
problem Cannot distinguish naturally reductive manifolds using Laplace-Beltrami spectrum.
method Characterized naturally reductive 2-step nilpotent Lie groups via Ambrose-Singer's structures; constructed isospectral pairs of 9-dimensional nilmanifolds.
result Spectra of Laplace-Beltrami operator can't distinguish naturally reductive manifolds from non-naturally reductive ones.
The paper explores reductions of self-dual conformal structure equations.
problem Integrating the general local form of self-dual conformal structure.
method Using Lax pair, hierarchy structure, and dressing scheme to discuss reductions.
result Constructs solutions for the SDCS equations and presents type B SDCS system.
In the present paper we study naturally reductive homogeneous (α,β)-metric spaces. Under some conditions, we give some necessary and sufficient conditions for a homogeneous (α,β)-metric space to be naturally reductive. Then we show that for such spaces the two definitions of naturally reductive homogeneous Finsler …
This work extends reduction processes for nonholonomic discrete mechanical systems.
problem Nonholonomic discrete mechanical systems and their reductions.
method Introduces a category LDPd of discrete-time dynamical systems and a two-stage reduction process. result Two-stage reduction process produces systems isomorphic to one-stage reduction.
We discuss the use of Dirac structures to obtain a better understanding of the geometry of a class of optimal control problems and their reduction by symmetries. In particular we will show how to extend the reduction of Dirac structures recently proposed by Yoshimura and Marsden [Yo09] to describe the reduction of a cl…
Abstract reviews symmetry and reduction in dynamical systems.
problem Understanding symmetries and reductions in dynamical systems.
method Algebraic formulation for dynamics of physical systems.
result Describes a reduction procedure for classical and quantum evolutions.
We define reduction of locally conformal Kaehler manifolds, considered as conformal Hermitian manifolds, and we show its equivalence with an unpublished construction given by Biquard and Gauduchon. We show the compatibility between this reduction and Kaehler reduction of the universal cover. By a recent result of Kamis…
Adapts DR objectives for both sample and feature size reduction.
problem Simultaneously reduce sample and feature sizes.
method Semi-relaxed Gromov-Wasserstein optimal transport.
result OT plan delivers competitive hard clustering.
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.