The current paper deals with some new classes of Finsler metrics with reversible geodesics. We construct weighted quasi-metrics associated with these metrics. Further, we investigate some important geometric properties of weighted quasi-metric space. Finally, we discuss the embedding of quasi-metric spaces with general…
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48 results for “quasi-metric spaces”
Study of classification in asymmetric quasi-metric spaces.
problem Classification in asymmetric quasi-metric spaces.
method Sample compression and nearest neighbor algorithm.
result Algorithm has favorable statistical properties.
We investigate the relation between weighted quasi-metric Spaces and Finsler Spaces. We show that the induced metric of a Randers space with reversible geodesics is a weighted quasi-metric space.
New RL approach speeds up training across tasks.
problem Training reinforcement learning models quickly and efficiently.
method Combines planning quasi-metric and task-specific aimers.
result Achieves multiple-fold speed-up on bit-flip and robotic arm tasks.
Characterizes isometries between non-reversible Finsler manifolds.
problem Understanding isometries in non-reversible Finsler manifolds.
method Generalization of Myers-Nakai Theorem for Riemannian manifolds, modification of function spaces to accommodate asymmetric structure.
result Functional characterization of isometries between non-reversible Finsler manifolds.
Disk Embeddings tackle embedding DAGs with exponential growth.
problem Embedding DAGs with exponentially increasing ancestors and descendants.
method Disk Embeddings framework for quasi-metric spaces, including Hyperbolic Disk Embeddings.
result Disk Embeddings outperform existing methods in complex DAGs.
For any geodesic current we associated a quasi-metric space. For a subclass of geodesic currents, called filling, it defines a metric and we study the critical exponent associated to this space. We show that is is equal to the exponential growth rate of the intersection function for closed curves.
We develop the basics of a theory of almost isometries for spaces endowed with a quasi-metric. The case of non-reversible Finsler (more specifically, Randers) metrics is of particular interest, and it is studied in more detail. The main motivation arises from General Relativity, and more specifically in spacetimes endo…
Neural Spacetimes learn DAGs by embedding nodes in a spacetime manifold.
problem Learning representations of weighted directed acyclic graphs (DAGs).
method Trainable deep learning-based geometries (Neural Spacetimes) that encode both edge weights and causality.
result Universal embedding theorem for DAGs with sub-cubic parameters and low distortion.
Prove a global shadow lemma for Patterson-Sullivan measures associated with relatively Morse subgroups in higher-rank semisimple Lie groups.
problem Prove a global shadow lemma for Patterson-Sullivan measures associated with relatively Morse subgroups in higher-rank semisimple Lie groups.
method Prove a global shadow lemma for Patterson-Sullivan measures associated with relatively Morse subgroups in higher-rank semisimple Lie groups.
result Prove a global shadow lemma for Patterson-Sullivan measures associated with relatively Morse subgroups in higher-rank semisimple Lie groups.
Dreaming machine learning: Lipschitz extensions for reinforcement learning on financial marketsq-fin.ST
Enhances financial market reinforcement learning with new state generation.
problem Improving reinforcement learning models for financial markets.
method Lipschitz extensions and artificial state generation.
result Improved investment strategy through enriched state space.
Study of tangent spaces in diffeological spaces under Lie group actions.
problem Understanding tangent spaces in generalized spaces.
method Generalized tangent space construction and isomorphism proof.
result Internal tangent space isomorphic to stratified tangent space.
(1,1) non-L-space knots are foliar in 3D space.
problem Proving (1,1) non-L-space knots are foliar.
method Analyzing (1,1) non-L-space knots in and lens spaces.
result (1,1) non-L-space knots are persistently foliar.
Universal spaces for finite topological spaces simplify shape descriptions.
problem Describing shape properties of compact metric spaces.
method Inverse limits of finite spaces and Alexandroff extensions.
result Universal spaces simplify shape descriptions of compact metric spaces.
The paper extends Stone duality to topological convexity spaces.
problem Understanding the relationship between topological convexity spaces and sup-lattices.
method Extending Stone duality to topological convexity spaces using preconvexity spaces.
result An adjunction between topological convexity spaces and sup-lattices.
New findings on GRW space-times with constant scalar curvature.
problem Understanding GRW space-times in different subspaces.
method Analyzing orthogonal subspaces of Gray's decomposition.
result Generalized quasi-Einstein GRW space-times reduce to known types of space-times.
No Einstein hypersurfaces found in Damek-Ricci spaces.
problem Existence of Einstein hypersurfaces in symmetric spaces.
method Analyzing properties of Damek-Ricci spaces and proving no Einstein hypersurface exists.
result No Einstein hypersurfaces in Damek-Ricci spaces.
The distance function (or ) of a distance space (general metric space) is not differentiable in general. We investigate such distance spaces over , whose distance functions are differentiable like in case of Finsler spaces. These spaces have several good properties, yet they are no F…
New kernels defined for various spaces, including measures.
problem Defining kernels on non-standard spaces like measures.
method Integrally strictly positive definite and characteristic kernels on Hilbert, Banach, and metric spaces.
result Explicit classes of kernels on spaces and sets of measures.
Metric spaces uniquely split into Hilbert and non-line-split parts.
problem Understanding the structure of metric spaces.
method Proved unique decomposition into Hilbert and non-line-split parts.
result Metric spaces have a unique decomposition into a Hilbert space and a non-line-split part.
In a paper (math.DG/0403528) we obtained explicit examples of Moishezon twistor spaces of some compact self-dual four-manifolds admitting a non-trivial Killing field, and also determined their moduli space. In this note we investigate minitwistor spaces associated to these twistor spaces. We determine their structure, …
Study geometry of tetrahedra in complex hyperbolic space and Hilbert spaces.
problem Understanding geometric relationships between complex hyperbolic spaces and Hilbert spaces.
method Use a complex analog of the cosine of a vertex angle as a novel technical tool.
result Describe possible triangular faces of tetrahedra in hyperbolic space and three-dimensional subspaces in Hilbert spaces with Pick kernels.
New curvature positivity helps classify spherical spaces and complex projective spaces.
problem Classifying spherical space forms and complex projective spaces.
method Introducing a new positivity notion for curvature.
result Characterizations for spherical space forms and complex projective spaces.
Study of complete space-like self-expanders in Minkovski space.
problem Characterize complete space-like self-expanders in Minkovski space.
method Use of maximum principle of Omori-Yau type to prove rigidity theorems.
result Classification of 2-dimensional complete space-like self-expanders with constant squared norm of the second fundamental form.
The abstract discusses the linear and smooth structures of mapping spaces.
problem The structure of mapping spaces in differential geometry.
method Proving diffeomorphisms and fibre bundle properties.
result Path spaces and base point preserving mapping spaces are Fréchet spaces.
Study on convergence of transformed metric spaces as dimensions grow.
problem Conditions for convergence of transformed metric spaces.
method Clarifying conditions for convergence of transformed spaces from original sequence and vice versa.
result Spheres and projective spaces converge to Gaussian space and its quotient as dimensions increase.
Characterizes when almost smooth spaces become RCD spaces.
problem Understanding conditions for almost smooth spaces to be RCD spaces.
method Characterizations via local volume doubling and Poincaré inequality.
result Characterizes Einstein 4-orbifolds.
Stability of Wasserstein spaces under various convergence types.
problem Stability and finiteness of Wasserstein spaces over singular and non-singular spaces.
method Gromov--Hausdorff convergence and equivariant Gromov--Hausdorff convergence.
result Analogue of Perelman's stability theorem on Wasserstein spaces.
New infinite families of flat spaces found from symmetric spaces.
problem Finding new flat homogeneous spaces.
method Starting from compact symmetric spaces, constructing infinite families of compact homogeneous spaces with invariant Bismut connections.
result Infinite families of compact homogeneous spaces with vanishing Ricci tensor.
Sphere theorems extended to RCD spaces and improved for Einstein stratified spaces.
problem Generalizing sphere theorems to new types of spaces.
method Proved sphere theorems for RCD(n-1, n) spaces and Einstein stratified spaces.
result Extended sphere theorems to RCD spaces and improved results for Einstein stratified spaces.
New quasi space forms solve Thurston's geometrical space form problem.
problem Solving Thurston's geometrical space form problem.
method Introducing quasi space forms as non-real space forms with specific geometric properties.
result Quasi space forms offer a metrical, local geometrical solution to Thurston's problem.
Classifies compact spaces by shape, finite spaces by weak homotopy.
problem Classifying compact Hausdorff spaces and finite topological spaces.
method Constructs a category that classifies spaces by shape and weak homotopy.
result Classifies compact spaces by shape, finite spaces by weak homotopy.
Harmonic spacesmath.DG
Estimates eigenvalues of Jacobi operator for harmonic spaces.
problem Estimating eigenvalues of Jacobi operator.
method Using density function of a harmonic space.
result Sharp estimates imply symmetric Osserman space.
Introduces new types of homogeneous spaces and their properties.
problem Defining and understanding new types of homogeneous spaces.
method Introducing and analyzing (strongly) (-)discrete homogeneous spaces.
result Discovers relationships between new and existing homogeneous space types.
This work describes compactifications of metric spaces and vector spaces using asymmetric norms.
problem Compactifying metric spaces and vector spaces using asymmetric norms.
method Nonstandard methods, ultrapowers of the spaces at hand.
result Polyhedral compactifications of vector spaces with stratified structure.
Survey Bernstein-type theorems for graphical surfaces in Euclidean and Lorentz-Minkowski spaces.
problem Proving theorems for minimal and constant mean curvature graphs in Euclidean and Lorentz-Minkowski spaces.
method Explains several proofs and provides mean curvature estimates for graphs in Euclidean and Lorentz-Minkowski spaces.
result Bernstein-type theorems for constant mean curvature graphs in Euclidean 3-space and space-like graphs in Lorentz-Minkowski 3-space.
In this paper, we present a unified study of the moduli space of tropical curves and Outer space which we link via period maps to the moduli space of tropical abelian varieties and the space of positive definite quadratic forms. Our work is a first step towards exhibiting Outer space and the space of positive definite …
CAT(0) spaces with small volume growth are homeomorphic to Euclidean space.
problem Characterizing CAT(0) spaces with specific volume growth properties.
method Analyzing asymptotic topological regularity and volume growth.
result CAT(0) spaces with small volume growth are homeomorphic to Euclidean space.
The Cantor Riemanniummath.CV
Cantor Riemannium is a new type of space from holomorphic germs.
problem Defining a new type of space from holomorphic germs.
method Constructing the Cantor Riemannium by Borel monogenic continuation.
result The Cantor Riemannium is a metric, path connected, Gromov length space.
Study RKHS on manifolds, linking Sobolev and diffusion spaces.
problem Characterizing RKHS on manifolds and their properties.
method Analyzing Sobolev and diffusion spaces on Riemannian manifolds.
result Sobolev spaces are RKHS under certain conditions, and diffusion spaces are introduced.
We construct a metric on the moduli space of bodies in Euclidean space. The moduli space is defined as the quotient space with respect to the action of integral affine transformations. This moduli space contains a subspace, the moduli space of Delzant polytopes, which can be identified with the moduli space of symplect…
Study static Einstein-Maxwell space invariant by translation.
problem Classify static Einstein-Maxwell space invariant under translation.
method Analyze static Einstein-Maxwell space conformal to pseudo-Euclidean space.
result Complete classification of static Einstein-Maxwell space invariant under translation.
A "Chen space" is a set X equipped with a collection of "plots" - maps from convex sets to X - satisfying three simple axioms. While an individual Chen space can be much worse than a smooth manifold, the category of all Chen spaces is much better behaved than the category of smooth manifolds. For example, any subspace …
New spaces help connect manifold structures on equivariant Poincaré spaces.
problem Creating manifold structures on equivariant Poincaré spaces.
method Introducing semifree isovariant -Poincaré spaces and gap conditions.
result Space of isovariant structures on semifree -Poincaré spaces is highly connected.
The classification of G-spaces by Palais is refined for the case where the orbit space satisfies certain mild topological hypotheses. It is shown that when a sequence of such orbit spaces is "close" to a limit orbit space, in some suitable sense, within a larger ambient orbit space, the G-spaces in the tail of the sequ…
Research explores the space-like embeddings in pseudo-hyperbolic space, finding geometric frames and actions.
problem Flat maximal space-like embeddings in pseudo-hyperbolic space.
method Description of Codazzi tensors, introduction of pseudo-Kähler metrics, Hamiltonian actions, moment maps, and geometric frames.
result Existence of two Hamiltonian actions with moment maps and geometric global Darboux frame.
Study shows how maps from certain geometric spaces behave near their edges.
problem Boundary regularity of harmonic maps in specific geometric spaces.
method Analysis of and spaces.
result Established boundary regularity of harmonic maps.
Complete classification of quaternionic skew-Hermitian symmetric spaces found.
problem Classifying quaternionic skew-Hermitian symmetric spaces.
method Proving the existence of a torsion-free -structure and showing that any homogeneous space is symmetric.
result A complete classification of quaternionic skew-Hermitian symmetric spaces for arbitrary .