New theorem proves convergence of various discrete conformal structures to conformal maps.
problem Proving convergence of discrete conformal structures to conformal maps.
method General theorem using piecewise linear discrete conformal mappings and Riemannian barycentric coordinates.
result Discrete conformal mappings converge to conformal maps under certain conditions.
Study connects vortices in abelian Higgs models to discrete conformal maps.
problem Understanding vortices in abelian Higgs models.
method Relating vortices to discrete conformal maps via curvature and volume form.
result Constructing discrete vortex solutions using discrete conformal theory.
The paper proves convergence of discrete maps to Riemann mappings for polyhedral surfaces.
problem Discrete conformal geometry of polyhedral surfaces.
method Establishing rigidity for hexagonal triangulations and estimating quasiconformal constants.
result Discrete conformal maps converge to Riemann mappings for Jordan domains.
Isothermic nets created from special maps for smooth surfaces.
problem Creating discrete curvature lines on surfaces.
method Special discrete holomorphic maps and lifted-folding.
result Isothermic nets with spherical parameter lines constructed efficiently.
No stable discrete maps into certain curved spaces exist.
problem Stability of discrete maps into curved spaces.
method Analysis of weighted length or energy functionals on graphs.
result Non-existence of stable discrete minimal immersions or harmonic maps into specific homogeneous spaces.
The paper studies convergence of discrete harmonic maps to smooth ones.
problem Discretization of harmonic maps between Riemannian manifolds.
method Introducing triangulations with vertex and edge weights, and studying convergence conditions.
result Suitable conditions on weighted triangulations ensure convergence of discrete harmonic maps to smooth ones.
Discrete Flow Maps bypass sequential prediction limits for parallel text generation.
problem Sequential autoregressive prediction limits large language model speed.
method Flow Maps compress generative trajectories into single-step mappings.
result Discrete Flow Maps surpass previous state-of-the-art results in discrete flow modeling.
New representations for discrete surfaces derived from dual transforms.
problem Constructing discrete surfaces in differential geometry.
method Using Ω-dual transform and lightlike Gauss maps in Laguerre geometry. result All discrete linear Weingarten surfaces arise via Weierstrass-type representations.
Discrete conformal maps on surfaces with vertex decorations are studied.
problem Discrete conformal equivalence for decorated piecewise Euclidean surfaces.
method Intimate relationship between decorated PE-surfaces, canonical tessellations of hyperbolic surfaces, and convex hyperbolic polyhedra; concave variational principle.
result Proof of discrete uniformization theorem for decorated PE-surfaces.
Discrete exterior calculus shows natural properties of wedge product and averaging.
problem Naturalness of discrete exterior calculus operations.
method Showed naturalness of discrete wedge product and averaging interpretation.
result Discrete wedge product is natural and equals Wilson's cochain product.
Optimal discrete harmonic maps between hyperbolic surfaces are found via minimizing energy.
problem Finding optimal discrete harmonic maps between hyperbolic surfaces.
method Minimizing Dirichlet energy over all possible hyperbolic structures and realizations within a fixed homotopy class.
result At the optimal hyperbolic structure, the discrete harmonic map and edge weights are induced from a weighted Delaunay decomposition.
Open and discrete maps with specific branch set images are equivalent to PL branched covers.
problem Understanding the equivalence of open and discrete maps and PL branched covers.
method Demonstrated that an open and discrete map f:SnoSn with a specific branch set image is equivalent to a PL branched cover up to homeomorphism. result Open and discrete maps with a specific branch set image are equivalent to PL branched covers.
Defines discrete channel surfaces in Lie sphere geometry.
problem Defining discrete channel surfaces in Lie sphere geometry.
method Definition and associated data sets for reconstruction.
result Proof of a discrete version of Vessiot's Theorem for isothermic discrete channel surfaces.
We establish a connection between two previously unrelated topics: a particular discrete version of conformal geometry for triangulated surfaces, and the geometry of ideal polyhedra in hyperbolic three-space. Two triangulated surfaces are considered discretely conformally equivalent if the edge lengths are related by s…
Effective methods compute equivariant harmonic maps from surfaces to nonpositively curved spaces.
problem Computing equivariant harmonic maps from surfaces to nonpositively curved spaces.
method Discretization of the theory, strong convexity of energy functional, convergence of discrete heat flow, center of mass methods.
result Explicit convergence rate and numerical computation with Harmony software.
Unified framework for various geometric constructions.
problem Organizing diverse geometric constructions.
method Introducing TCD maps and defining local moves.
result Two distinct cluster structures on TCD maps.
Discrete version of Liouville's theorem for simplicial complexes.
problem Finding equivalent simplicial complexes under discrete conformal equivalence.
method Proving an analogous statement for simplicial complexes, considering combinatorial equivalence and scale factors associated with vertices.
result All discretely conformally equivalent simplicial complexes are combinatorially equivalent.
A discrete conformal map (DCM) maps the square lattice to the Riemann sphere such that the image of every irreducible square has the same cross-ratio. This paper shows that every periodic DCM can be determined from spectral data (a hyperelliptic compact Riemann surface, called the spectral curve, equipped with some mar…
Uniformizes surfaces using discrete harmonic maps and hyperbolic metrics.
problem Uniformizing surfaces with complex geometries.
method Least Dirichlet energy harmonic embedding of graphs on surfaces.
result Existence of hyperbolic metrics realizing least energy embeddings.
The study finds criteria for discreteness in quaternionic hyperbolic space.
problem Discreteness of subgroups in quaternionic hyperbolic space.
method Using test maps to establish discreteness criteria for Zariski-dense subgroups.
result Discreteness criteria for quaternionic hyperbolic subgroups.
Develops TCD maps to relate discrete differential geometry and cluster algebras.
problem Capturing constraints and dynamics in discrete differential geometry.
method Triple crossing diagram maps (TCD maps) and geometric operations.
result Establishes a hierarchy of cluster structures on TCD maps.
Discrete approximation solves Björling's minimal surface problem.
problem Constructing minimal surfaces from real-analytic curves with specified normal fields.
method Approximate solution by discrete minimal surfaces and discrete isothermic surfaces.
result Approximation error is proportional to the square of the mesh size.
Maps discrete manifolds to partitions to define new manifolds.
problem Creating manifolds from discrete structures.
method Mapping discrete d-manifolds onto (k+1)-partite complexes to define new manifolds.
result Defines a (d-k)-manifold from simplices in G mapped to P.
Discrete hyperbolic isometries proven via test maps.
problem Proving discreteness of hyperbolic isometries.
method Using test maps to show discreteness of subgroups.
result Zariski dense subgroups are discrete under certain conditions.
Estimates discontinuous optimal transport maps between a discrete and continuous distribution.
problem Estimating discontinuous optimal transport maps between a discrete and continuous distribution.
method Entropic optimal transport estimator, computationally efficient.
result The estimator converges at the minimax-optimal rate n−1/2 in the semi-discrete setting. Develops a new method for learning discrete distributions without embedding them in a continuous space.
problem Challenges in learning discrete distributions using current methodologies.
method Introduces a MAD invertible map and a mixed variational flow (MAD Mix) for discrete distributions.
result MAD Mix produces more reliable approximations than continuous-embedding flows.
Maps between Hadamard manifolds are quasi-isometric to harmonic maps.
problem Understanding the relationship between quasi-isometric maps and harmonic maps on Hadamard manifolds.
method Extending a previous result to quotient spaces of Hadamard manifolds by convex cocompact discrete groups.
result Locally quasi-isometric maps to Hadamard manifolds are within bounded distance from a unique harmonic map.
Paper relaxes MAP inference for discrete MRFs, achieving better solutions.
problem Optimizing discrete MRFs for complex real-world problems.
method Nonconvex continuous relaxation, block coordinate descent, gradient methods, ADMM.
result ADMM significantly outperforms other methods in real-world applications.
Proposes new economic accelerators with memory in discrete time.
problem Describing economic processes with power-law memory and periodic sharp splashes.
method Uses capital stock adjustment principle and discrete maps derived from fractional-order differential equations.
result Discrete accelerators with memory accurately describe economic processes.
Finding appropriate notions of discrete holomorphic maps and, more generally, conformal immersions of discrete Riemann surfaces into 3-space is an important problem of discrete differential geometry and computer visualization. We propose an approach to discrete conformality that is based on the concept of holomorphic l…
New methods use transport maps to improve Langevin dynamics for sampling.
problem Sampling high-dimensional, non-Gaussian distributions efficiently.
method Apply transport maps to accelerate Langevin dynamics convergence.
result Discretized processes converge to target distribution with non-asymptotic bounds.
Study shows mapping class groups are one-ended for surfaces with at least one end.
problem Analyzing the number of ends in mapping class groups of surfaces.
method Proving the associated translatable curve graph is one-ended, quasi-isometric to the mapping class group.
result Mapping class groups are one-ended for surfaces with at least one end of discrete type.
The paper proves a mapping from a space of holonomy varieties to Teichmüller spaces, with a non-empty discrete intersection.
problem Intersection of Poincaré holonomy varieties and their properties.
method Holomorphic mapping and branched covering proof.
result Intersection of arbitrary Poincaré holonomy varieties is a non-empty discrete set.
The paper simplifies complex mechanical systems with external forces.
problem Analyzing symmetric discrete mechanical systems with external forces.
method Lagrangian reduction and reconstruction for principal bundles.
result Evolution of momentum maps and Poisson structures under different conditions.
Exact discrete mechanics for nonholonomic systems defined.
problem Discrete mechanics for nonholonomic systems.
method Constructing an exponential map and deriving exact discrete nonholonomic integrators.
result Reproduces continuous nonholonomic flow as discrete flow on constraint submanifold.
Extremal length is a conformal invariant that transfers naturally to the discrete setting, giving square tilings as a natural combinatorial analog of conformal mappings. Recent work by S. Hersonsky has explored generalizing these ideas to three-dimensional cube tilings. The connections between discrete extremal length …
Skew parallelogram nets factorize, encompassing discrete differential geometry.
problem Factorization of polynomials in discrete differential geometry.
method Lax representation, Bäcklund transformations, factorization of polynomials.
result Skew parallelogram nets encompass all systems with polynomial representations.
We consider discrete nets in Grassmannians Grd which generalize Q-nets (maps ZN→Pd with planar elementary quadrilaterals) and Darboux nets (Pd-valued maps defined on the edges of ZN such that quadruples of points corresponding to elementary squares are all co…
Research proves limits on harmonic map orders into Euclidean buildings.
problem Limits on the possible orders of harmonic maps from surfaces to Euclidean buildings.
method Direct analysis of homogeneous maps and related spherical billiards problem.
result The order of harmonic maps is of the form km where k divides ∣W∣. Earth observation embeddings can convert discrete biome maps into continuous representations that better capture ecological variation.
problem Biome maps impose categorical boundaries that compress continuous variation in biotic communities.
method Fit a linear classifier on Earth observation embeddings to predict biome labels.
result Continuous biome representation outperforms discrete biome labels for predicting species occurrence.
This paper explores geometric insights into discrete R-congruences and their envelopes.
problem Understanding the ambiguity in discrete R-congruences and their envelopes.
method Analyzes discrete R-congruences that are enveloped by specific types of surfaces and maps.
result Discovers a 2-parameter family of discrete enveloping surfaces for discrete R-congruences.
Finite distortion maps cannot have compact branch sets under growth conditions.
problem Understanding the structure of branch sets in mappings of finite distortion.
method Analyzing the asymptotic growth of distortion and constructing specific examples.
result The bound on the size of branch sets is strict and achievable.
The paper studies groups formed by two parabolic maps and their properties.
problem Understanding groups generated by two parabolic maps in mSU(2,1). method Analyzes conditions for the group to be discrete and free, and calculates the diameter of a circle in the Heisenberg group.
result Conditions are provided to ensure the group is discrete and free.
Study discretizes Dirac and port-Hamiltonian systems using manifolds.
problem Discretization of Dirac and port-Hamiltonian systems.
method Retraction and discretization maps on manifolds for Dirac structures, applied to port-Hamiltonian systems.
result Numerical integrators for port-Hamiltonian systems derived from discretization techniques.
Stochastic optimization improves semi-discrete OT map estimation with a minimax rate.
problem Empirical success of SGD in semi-discrete OT, but lack of theoretical guarantees.
method Averaged projected SGD with a minimax convergence rate of O(1/√n).
result SGD methods can estimate the OT map with a minimax convergence rate of O(1/√n).
We use the combinatorial harmonic map theory to study the isometric actions of discrete groups on Hadamard spaces. Given a finitely generated group acting by automorphisms, properly discontinuously and cofinitely on a simplicial complex and its isometric action on a Hadamard space, we formulate criterions for the actio…
Study on Gauss map of anisotropic minimal surfaces with Morse index estimates.
problem Estimating the Morse index of anisotropic minimal surfaces.
method Local analysis of Gauss map, conformal geometric techniques applied to the Gauss map.
result Upper and lower estimates for the Morse index of anisotropic minimal surfaces.
Riemannian Neural OT maps improve scalability on manifolds.
problem Challenges in extending neural OT to high-dimensional Riemannian manifolds.
method Introduces Riemannian Neural OT (RNOT) maps that avoid discretization and incorporate geometric structure.
result RNOT maps approximate Riemannian OT maps with sub-exponential complexity in the dimension.