Study of curve shortening flow for twisted curves, defining curvature-torsion entropy.
problem Understanding the behavior of curves with curvature and torsion under curve shortening flow.
method Defined curvature-torsion entropy to analyze the flow of twisted curves.
result Curved curves under curve shortening flow either develop inflection points or exhibit highly irregular singularities.
Two new invariants that are closely related to Milnor's curvature-torsion invariant are introduced. The first, the spiral index of a knot, captures the minimum number of maxima among all knot projections that are free of inflection points. This invariant is closely related to both the bridge and braid index of the knot…
Study torsion obstructions to positive scalar curvature on manifolds.
problem Obstructions to positive scalar curvature metrics on manifolds.
method Analysis of torsion classes in integral homology.
result Construction of manifolds without positive scalar curvature metrics but whose Cartesian products do.
The purpose of this article is to find a family of curves parametrized by arc length and that depend on an angular function and an intrinsic fraction function, which is defined as the quotient between torsion and curvature. We find for this family of curves explicit formulas of curvature, torsion and geodetic curvature…
Formulae quantify gaps in geodesic quadrilaterals on manifolds.
problem Infinitesimal gaps in geodesic quadrilaterals on manifolds.
method Formulas for curvature and torsion based on quadrilateral gaps.
result Torsion and curvature can be uniquely recovered from quadrilateral gaps.
Derivative estimates for pluriclosed flow control curvature and torsion.
problem Deriving derivative estimates for the pluriclosed flow.
method Control higher order derivatives of Chern curvature and torsion using Chern curvature; derive an estimate for torsion tensor using Chern Ricci curvature in dimension two; find a monotonic quantity in Hermitian-symplectic case.
result All Hermitian-symplectic solitons are Kähler Ricci solitons.
Defines Jacobi fields in nonholonomic mechanics.
problem No specific problem stated; focuses on definition and properties.
method Characterizes Jacobi fields using Riemannian geometry methods and finds them explicitly. Uses complete lift and curvature/torsion of nonholonomic connection.
result Derives nonholonomic Jacobi equations in two equivalent ways.
Characterizes curves in totally umbilical surfaces of space forms.
problem Identifying curves in totally umbilical surfaces of space forms.
method Analyzes curvature and torsion conditions for curves in $\h^3$ and $\s^3$.
result Necessary and sufficient conditions for curves in totally umbilical surfaces.
Study on complex manifolds introduces a new deformation of the Yamabe problem.
problem Yamabe-type problems on compact Hermitian manifolds.
method Introducing a one-parameter Hermitian deformation of the Yamabe problem, defined by adding natural torsion terms to the Riemannian scalar curvature.
result Analysis of criteria for the existence of solutions and discussion of examples.
Study on triharmonic curves in Sol space with constant curvature and torsion.
problem Characterizing triharmonic curves in the Sol space.
method Complete classification of proper triharmonic curves with constant geodesic curvature and torsion.
result Triharmonic curves form a constant angle with a Killing field of constant length.
Study of curves in dual space with constant curvature and torsion.
problem Classifying curves in dual space with specific geometric properties.
method Defined curvature and torsion for curves in dual space, classified curves with constant properties, and proved existence theorems.
result Established fundamental theorem of existence for dual curves with prescribed curvature and torsion.
Defines a new family of curves in space with applications.
problem Finding shapes similar to whirls in space.
method Intrinsic equation of curvature and torsion, position vector with arc length parameter.
result Necessary and sufficient conditions for the existence of the family of curves.
Study curvature and torsion in Courant algebroids using graded geometry.
problem Defining curvature and torsion in Courant algebroids.
method Graded geometric approach, introducing K-curvature and K-torsion.
result Natural graded geometric definition of Courant algebroid curvature and torsion.
The paper classifies helix curves on a pseudo-Riemannian surface.
problem Classifying helix curves on pseudo-Riemannian surfaces.
method Analyzing geodesic flow vector fields and pseudo-Riemannian metrics.
result All helix curves are circular helixes with constant curvature and torsion.
The abstract aims to generalize classical curve concepts to uniquely define complex curves.
problem Lack of sufficient information to distinguish between different curves.
method Generalizing classical concepts of curvature and torsion to higher algebraic curvatures.
result Each analytic branch of a complex curve is uniquely defined by higher algebraic curvatures.
Quaternionic differential geometry expands geometric concepts using quaternions.
problem Generalizing geometric concepts to quaternionic constraints.
method Generalizing curves and surfaces, curvature, torsion, differential forms, and directional derivatives to quaternionic constraints.
result Quaternionic formalism provides a suitable language for differential geometry.
We explore the class of triples (M, nabla, P) where M is a manifold, nabla is an affine connection in M and P is a G-structure in M. Inside this class there are infinitesimally homogeneous manifolds, characterized by having G-constant curvature, torsion and inner torsion. For each matrix Lie group G subgroup of GL(Rn) …
Extends mean curvature to surfaces in Riemann-Cartan geometry with torsion.
problem Addressing surfaces in Riemann-Cartan geometry with nontrivial torsion.
method Introducing a complex-valued 2-form associated with the torsion, which interacts with other geometric concepts.
result Complex-valued mean curvature quantity interacts with Hopf differential and Gauss map.
In this study, we deal with the local structure of curves and surfaces immersed in a pseudo-isotropic space I_{p}^{3} that is a particular Cayley-Klein space. We provide the formulas of curvature, torsion and Frenet trihedron in order for spacelike and timelike curves. The causal character of all admissible surfaces in…
Post-Lie algebra structure found on non-flat manifolds with curvature and torsion.
problem Understanding vector fields and endomorphisms on manifolds with curvature and torsion.
method Analyzing the post-Lie algebra structure of vector fields and endomorphisms for non-flat connections.
result A universal Lie algebra is constructed for the post-Lie algebra of vector fields and endomorphisms.
Spinors help study unique five-dimensional contact structures.
problem Understanding unique five-dimensional contact structures.
method Using classical two-component spinors, define directional derivatives in contact directions.
result Calculate invariant torsion of G2 contact structure. We calculate the asymptotic behavior of hyperbolic volume and Chern-Simons invariant.
problem Calculating the asymptotic behavior of hyperbolic volume and Chern-Simons invariant.
method Renormalization of the Chern-Simons invariant using asymptotics along an equidistance foliation.
result The leading coefficient introduces a complex-valued quantity consisting of mean curvature and torsion 2-form.
New insights on ACYT 6-manifolds reveal instanton conditions for curvature.
problem Understanding instanton conditions on ACYT 6-manifolds.
method Analyzing curvature and torsion properties of ACYT 6-manifolds.
result Curvature is an SU(3)-instanton if and only if torsion is parallel. New symmetries found in Riemann-Cartan geometries.
problem Investigating symmetries in geometries with curvature and torsion.
method Mathematical tools to determine symmetries and subclasses of geometries.
result Determined all static and stationary spherically symmetric Riemann-Cartan geometries and subclasses with specific symmetries.
This paper finds a metric on \(S^2 imes T^2\) with strictly positive biorthogonal curvature using an affine connection with antisymmetric torsion.
problem Existence of a Riemannian metric on \(S^2 imes T^2\) with strictly positive biorthogonal curvature.
method Introducing an affine connection with antisymmetric torsion calibrated via non-trivial cohomology classes, which allows overcoming topological constraints.
result Demonstrates the construction of a metric on \(S^2 imes T^2\) with strictly positive biorthogonal curvature.
The paper defines and analyzes conformal trajectories in 3D space forms.
problem Understanding trajectories in curved 3D spaces.
method Defined conformal trajectories and studied their properties in R3, S3, and H3. result Conformal trajectories in S3 and H3 have constant curvature and torsion. Paper studies rigidity of translation surfaces in 3D sphere using quaternionic product.
problem Rigidity of translation surfaces in S3. method Introduced an associated frame for curves in S3; described local geometry; used curvature and torsion of generating curves. result Rigidity results for minimal and constant mean curvature surfaces in S3. Establishes necessary conditions for cylindrical curves using curvature and torsion.
problem Geometrically identifying curves on cylindrical surfaces.
method Identifying a fundamental function ψ and reducing the problem to a compatibility condition between an eighth-degree polynomial and a differential equation for ψ.
result Proves that for curves with constant curvature κ0 = 1/ρ, the torsion τ admits an explicit, exact solution.
We develop the method of anholonomic frames with associated nonlinear connection (in brief, N--connection) structure and show explicitly how geometries with local anisotropy (various type of Finsler--Lagrange--Cartan--Hamilton geometry) can be modeled in the metric--affine spaces. There are formulated the criteria when…
Enhances RL by controlling policy stochasticity through trajectory entropy constraints.
problem Non-stationary Q-value estimation and short-sighted entropy tuning in maximum entropy RL.
method Proposes TECRL framework with separate Q-functions for reward and entropy, enforcing a trajectory entropy constraint.
result DSAC-E algorithm achieves higher returns and better stability on OpenAI Gym benchmarks.
The paper calculates bounds for risk metrics and entropies under partial information constraints.
problem Analyzing risk metrics and entropies for unimodal, symmetric distributions with limited information.
method Develops lower and upper bounds for worst-case distortion riskmetrics and weighted entropy for unimodal, symmetric distributions with known mean and variance.
result Sharp upper bounds for distortion riskmetrics and weighted entropy for symmetric distributions.
The paper analyzes worst-case distortion risk metrics and weighted entropy under partial information.
problem Analyzing worst-case distortion risk metrics and weighted entropy with limited information.
method General distributions, partial information (mean and variance), various entropies and risk measures.
result Provides worst-case results for distortion risk metrics and weighted entropy.
Entropy measures geodesic flow complexity.
problem Measuring complexity of geodesic flows on manifolds.
method Introduced barcode entropy to measure exponential growth rate of not-too-short bars in Morse-theoretic barcodes.
result Barcode entropy bounds topological entropy and vice versa.
The study develops inequalities for Riemannian foliations without bundle-like assumptions.
problem Developing inequalities for Riemannian foliations without restrictive conditions.
method Bochner theory and Bakry-Emery calculus for horizontal Laplacians, derived explicit Bochner formulas, generalized curvature dimension inequalities.
result Established generalized curvature dimension inequalities for Riemannian foliations.
Entropy for uniform hypergraphs defined via tensor theory.
problem Entropy calculation for uniform hypergraphs.
method Probability distribution of generalized singular values from Laplacian tensors, Shannon entropy formula.
result Tensor entropy is a measure of regularity for uniform hypergraphs.
HCLM framework uses entropy regularization for open learning systems.
problem Real-world AI challenges and limitations of deep learning.
method Dynamical and information-theoretic framework with entropy regularization.
result Geometric entropy surrogates, especially log-determinant covariance entropy, induce stronger and more stable information forces.
Coupled entropy corrects flaws in Tsallis entropy for complex systems.
problem Misinterpretation of generalized temperature and entropy.
method Derived from generalized Pareto and Student's t distributions.
result Provides balanced measure of uncertainty for complex systems.
The paper examines robustness of topological entropy in geodesic flows.
problem Entropy robustness in geodesic flows under C0 perturbations. method Study of topological entropy on Riemannian metrics with C0 topology. result Metrics with contractible closed geodesics have robust entropy.
Generalizes beam models to include curvature and torsion.
problem Modeling curvature and torsion in unidimensional structures.
method Generalized Euler-Bernoulli and Timoshenko beam models using non-holonomic kinematics.
result Generalized beams exhibit curvature and torsion, with torsion in Timoshenko model.
Entropy rigidity for Finsler flows but collapse for Reeb flows.
problem Entropy behavior of Reeb and Finsler flows on contact manifolds.
method Analysis of topological entropy for Reeb and Finsler flows.
result Uniform positive lower bound for Finsler flows but arbitrarily small topological entropy for Reeb flows.
JES optimizes expensive functions by considering joint entropy over input and output spaces.
problem Optimizing expensive functions with limited evaluations.
method Joint Entropy Search (JES) considers joint entropy over input and output spaces.
result JES outperforms other information-theoretic methods in Bayesian optimization.
We give a notion of entropy for general gemetric structures, which generalizes well-known notions of topological entropy of vector fields and geometric entropy of foliations, and which can also be applied to singular objects, e.g. singular foliations, singular distributions, and Poisson structures. We show some basic p…
Generic low-entropy hypersurfaces in 4-6D flow with only generic singularities.
problem Analyzing mean curvature flow of low-entropy hypersurfaces.
method Proving flow encounters only generic singularities for specific entropy conditions.
result Proves flow encounters only generic singularities for low-entropy initial data.
Researchers explore gauge freedom in entropies of q-Gaussian measures.
problem Exploring the gauge freedom of entropies in q-Gaussian measures. method Introducing a refined q-logarithmic function to demonstrate gauge freedom. result Different escort expectations can lead to the same entropy but different relative entropies.
Study shows rigidity for entropy minimizers in non-monotone cases.
problem Rigidity of entropy minimizers in non-monotone settings.
method Elementary proofs in non-monotone situations.
result Showed rigidity for minimizers of generalized Colding-Minicozzi entropies.
DAC enhances exploration in reinforcement learning with entropy regularization.
problem Improving exploration efficiency in reinforcement learning.
method Sample-aware entropy regularization using replay buffer action distributions.
result DAC significantly outperforms existing algorithms in reinforcement learning tasks.
Entropy study of geodesic flow on convex projective surfaces.
problem Entropy of Sinai-Ruelle-Bowen measure on convex projective surfaces.
method Analysis of Hilbert area and Blaschke metric.
result Entropy tends to zero if and only if the Hilbert area tends to infinity.
Study bounds self-shrinker entropy using Li-Yau volume and Colding-Minicozzi entropy.
problem Bounding entropy of self-shrinkers in arbitrary codimensions.
method Introduced stable conformal volume and virtual entropy to prove bounds.
result Entropy bounds are sharp and independent of codimension.