Study on discrete Gaussian curvature for polyhedral surfaces.
problem Discretization of Gaussian curvature for polyhedral surfaces.
method Generalization of discrete conformal equivalence to define discrete Gaussian curvature and classify polyhedral surfaces.
result Existence of polyhedral surfaces with constant discrete Gaussian curvature in every discrete conformal class.
Study of Ricci flow on discrete surfaces of revolution with constant Gaussian curvature.
problem Understanding Ricci flow on discrete surfaces of revolution.
method Explicit parametrizations and Ricci flow analysis for discrete surfaces of revolution.
result Discrete surfaces of revolution approach constant Gaussian curvature under Ricci flow.
New method approximates Gaussian curvature on discrete surfaces.
problem Approximating solutions to the prescribed Gaussian curvature problem.
method Discrete conformality and convex functional minimization.
result Efficient numerical method to compute solutions.
The paper introduces a new discretization of Gaussian curvature on surfaces.
problem Discretizing Gaussian curvature on surfaces with conic singularities.
method Discrete conformal theory and variational principles with constraints.
result Established a discrete uniformization theorem for surfaces with non-positive Euler number.
The paper tackles prescribing discrete Gaussian curvature on polyhedral surfaces.
problem Prescribing discrete Gaussian curvature on polyhedral surfaces.
method Discrete conformal theory and variational principles with constraints.
result Proves Kazdan-Warner type theorems for polyhedral surfaces.
The paper proves a theorem for discretizing Gaussian curvature on surfaces.
problem Discretizing Gaussian curvature on surfaces with nonpositive Euler number.
method Discrete conformal theory and variational principles with constraints.
result Each decorated piecewise Euclidean metric on surfaces with nonpositive Euler number is discrete conformal to a metric with a specific discrete curvature constant.
This article is an application of the author's paper about a construction method for discrete constant negative Gaussian curvature surfaces, the nonlinear d'Alembert formula. The heart of this formula is the Birkhoff decomposition, and we give a simple algorithm for the Birkhoff decomposition. As an application, we dra…
A new method for computing image curvature efficiently and accurately.
problem Low performance, low accuracy, and requirement of second order differentiability in conventional computation schemes.
method Proposes a novel discrete computation scheme for weighted Gaussian curvature.
result More accurate, computationally more efficient, and does not require second order differentiability.
The paper finds circle packings with specific curvatures in hyperbolic geometry.
problem Finding circle packings with prescribed total geodesic curvatures and discrete Gaussian curvatures.
method Established existence and rigidity via variational principle, introduced combinatorial p-th Calabi flows.
result Introduced combinatorial p-th Calabi flows to find circle packings with prescribed curvatures.
The paper establishes a discrete uniformization theorem for surfaces with piecewise hyperbolic metrics.
problem Finding decorated piecewise hyperbolic metrics with prescribed combinatorial curvature.
method Introduced combinatorial α-Ricci flow with surgery to handle potential singularities and prove longtime existence and convergence.
result Existence of decorated piecewise hyperbolic metrics with prescribed combinatorial α-curvature.
Study discrete analog of zeta-determinant maximization on triangulated surfaces.
problem Maximizing zeta-determinant for discrete Laplacian on triangulated surfaces.
method Analogous to Osgood, Phillips, and Sarnak's theorem, study stationary points of determinants for discrete cotan-Laplacian.
result Discrete metrics of constant discrete Gaussian curvature are stationary points of the determinant, suggesting minima.
New Bäcklund transformations for discrete pseudospherical surfaces of revolution are found.
problem Constructing new non-rotational discrete pseudospherical surfaces.
method Explicit parametrizations and Bäcklund transformations for discrete constant negative Gaussian curvature surfaces of revolution.
result Conditions for Bäcklund transformations to preserve periodicity are provided.
We consider a general theory of curvatures of discrete surfaces equipped with edgewise parallel Gauss images, and where mean and Gaussian curvatures of faces are derived from the faces' areas and mixed areas. Remarkably these notions are capable of unifying notable previously defined classes of surfaces, such as discre…
In this paper we define and analyze singularities of discrete linear Weingarten surfaces with Weierstrass-type representations in 3-dimensional Riemannian and Lorentzian spaceforms. In particular, we discuss singularities of discrete surfaces with non-zero constant Gaussian curvature, and parallel surfaces of discret…
Polyhedra can mimic constant curvature surfaces, even with self-intersections.
problem Understanding curvature constraints in discrete vs. smooth settings.
method Constructive proof showing any surface can be realized as a polyhedral surface with uniform angular defect.
result Closed surfaces can be realized as polyhedral surfaces with constant angular defect.
We establish what semi-discrete linear Weingarten surfaces with Weierstrass-type representations in 3-dimensional Riemannian and Lorentzian spaceforms are, confirming their required properties regarding curvatures and parallel surfaces, and then classify them. We then define and analyze their singularities. In partic…
This work develops discrete Gaussian models for vector-valued data on triangular meshes.
problem Discrete representation of continuous vector-valued environmental data.
method Develops discrete intrinsic Gaussian processes for vector-valued data on triangular meshes using discrete differential operators.
result Models can capture harmonic flows, incorporate boundary conditions, and model non-stationary data.
In this paper, we generalize our results in \cite{GX3} to triangulated surfaces in hyperbolic background geometry, which means that all triangles can be embedded in the standard hyperbolic space. We introduce a new discrete Gaussian curvature by dividing the classical discrete Gauss curvature by an area element, which …
We define and discuss the notion of pseudospherical surfaces in asymptotic coordinates on time scales. Thus we extend well known notions of discrete pseudospherical surfaces and smooth pseudosperical surfaces on more exotic domains (e.g, the Cantor set). In particular, we present a new expression for the discrete Gauss…
New geometric SDEs and discretizations on Riemannian manifolds with error bounds.
problem Modeling diffusion processes on Riemannian manifolds with geometric SDEs.
method Introduced a new construction of geometric SDEs and provided non-asymptotic error bounds.
result First non-asymptotic error bound for geometric Euler-Murayama discretization.
Proposes a method to estimate discrete curvatures for image reconstruction.
problem Image reconstruction challenges due to non-convex, non-smooth, and highly non-linear first-order optimal conditions.
method Estimates discrete curvatures (mean and Gaussian) locally using differential geometry theory. Solves a weighted total variation minimization problem efficiently with ADMM.
result Demonstrates the effectiveness and superiority of the proposed variational models for various image reconstruction tasks.
We define and discuss the notion of pseudospherical surfaces in asymptotic coordinates on time scales. Two special cases, namely dicrete pseudospherical surfaces and smooth pseudosperical surfaces are consistent with this description. In particular, we define the Gaussian curvature in the discrete case.
We show that the results in \cite{Ge-Jiang1} are still true in hyperbolic background geometry setting, that is, the solution to Chow-Luo's combinatorial Ricci flow can always be extended to a solution that exists for all time, furthermore, the extended solution converges exponentially fast if and only if there exists a…
Discrete Gaussian noise preserves privacy and accuracy in differential privacy.
problem Finite computers cannot represent continuous Gaussian noise, leading to privacy breaches and loss of interpretability.
method Introduced and analyzed discrete Gaussian noise, providing privacy and accuracy guarantees similar to continuous Gaussian noise.
result Discrete Gaussian noise offers the same privacy and accuracy as continuous Gaussian noise, with efficient sampling algorithms.
A system for federated learning with private data, adding discrete Gaussian noise and secure aggregation.
problem Training models on private data distributed across devices while ensuring privacy.
method Discretizes data, adds discrete Gaussian noise, and uses secure aggregation to protect privacy.
result Matches the accuracy of central differential privacy with less than 16 bits of precision per value.
We introduce the discrete Einstein metrics as critical points of discrete energy on triangulated 3-manifolds, and study them by discrete curvature flow of second (fourth) order. We also study the convergence of the discrete curvature flow. Discrete curvature flow of second order is an analogue of smooth Ricci flow.
New discrete models for constant mean curvature surfaces and tori.
problem Creating discrete models for constant mean curvature surfaces and tori.
method Integrable theory of discrete polarised curves and Darboux transforms.
result Closed-form discrete parametrisations of discrete isothermic cylinders, discrete constant mean curvature cylinders, and discrete isothermic tori.
We investigate geometric aspects of the the Bäcklund transform of principal contact element nets. A Bäcklund transform exists if and only if it the principal contact element net is of constant negative Gaussian curvature (a pseudosphere). We describe an elementary construction of the Bäcklund transform and prove its co…
In discrete differential geometry, it is widely believed that the discrete Gaussian curvature of a polyhedral vertex star equals the algebraic area of its Gauss image. However, no complete proof has yet been described. We present an elementary proof in which we compare, for a particular normal vector, its winding numbe…
In this paper, we introduce a parameterized discrete curvature (α-curvature) for piecewise linear metrics on polyhedral surfaces, which is a generalization of the classical discrete curvature. A discrete uniformization theorem is established for the parameterized discrete curvature, which generalizes the discrete uni…
Survey on discrete curvature concepts for polygons and polyhedral surfaces.
problem Defining curvature for discrete structures like polygons and polyhedral surfaces.
method Explains curvature notions for polygons, polyhedral surfaces, and abstract polyhedral manifolds.
result Discrete curvature theorems parallel classical theorems in differential geometry.
Study on discrete surfaces with constant principal curvature for nanocarbon applications.
problem Understanding discrete geometry properties of nanocarbon materials.
method Developed discrete surface theory on 3-ary oriented trees, defined discrete principal directions, constructed examples of discrete CPC surfaces.
result Construction of discrete constant principal curvature surfaces, including discrete CPC tori.
A new RG approach connects discrete and continuous time descriptions of Gaussian processes.
problem Discretization of continuous stochastic processes for accurate simulation or model inference.
method Renormalization Group (RG) approach for Gaussian time series generated by auto-regressive models.
result RG fixed points correspond to discretizations of linear SDEs, providing insights into process accuracy.
RL solves discrete LQ control with Gaussian optimal policy.
problem Discrete-time linear-quadratic control problem.
method Entropy-based RL to find Gaussian optimal policy.
result RL algorithm solves mean-variance asset-liability management problem.
New method calculates discrete curvature using effective resistances.
problem Calculating discrete curvature on graphs.
method Effective resistances to calculate curvature on graph nodes and links.
result Relation to established discrete curvatures and convergence to continuous curvature.
Paper solves NGCA for discrete distributions using LLL method.
problem Learning hidden non-Gaussian components in discrete distributions.
method Utilizes LLL lattice basis reduction method.
result Sample and computationally efficient algorithm for NGCA in discrete distributions.
We define Discrete Quasi-Einstein metrics (DQE-metrics) as the critical points of discrete total curvature functional on triangulated 3-manifolds. We study DQE-metrics by introducing some combinatorial curvature flows. We prove that these flows produce solutions which converge to discrete quasi-Einstein metrics when th…
Examines discrete curvature's relation to smooth curvature in 3 spaces.
problem Understanding how discrete curvature relates to smooth curvature in different spaces.
method Using specific triangular tilings of 3 types of spaces to examine curvatures.
result Discrete curvature can sense the smooth curvature of ambient space forms.
Study curvature and torsion from cross-ratios in discrete curves.
problem Define curvature and torsion for discrete curves using cross-ratios.
method Use Möbius invariant point-insertion-rule to construct circles and express torsion using cross-ratio.
result Discrete curvature and torsion defined using cross-ratios converge to smooth curvature and torsion as sampling density increases.
A numerical scheme is developed for solution of the Goursat problem for a class of nonlinear hyperbolic systems with an arbitrary number of independent variables. Convergence results are proved for this difference scheme. These results are applied to hyperbolic systems of differential-geometric origin, like the sine-Go…
We study discrete curvatures computed from nets of curvature lines on a given smooth surface, and prove their uniform convergence to smooth principal curvatures. We provide explicit error bounds, with constants depending only on properties of the smooth limit surface and the shape regularity of the discrete net.
Unique metric found for discrete curvature on spherical cone-metrics.
problem Finding a unique metric with prescribed curvature on spherical cone-metrics.
method Discrete conformal approach to spherical cone-metrics.
result Existence of a unique metric realizing prescribed curvature in each conformal class.
This survey article is about discrete constant mean curvature surfaces defined by an approach related to integrable systems techniques. We introduce the notion of discrete constant mean curvature surfaces by first introducing properties of smooth constant mean curvature surfaces. We describe the mathematical structure …
The paper finds formulas for special surface shapes in 3D space.
problem Creating formulas for constant mean curvature surfaces.
method Weierstrass representations for discrete surfaces in isotropic space.
result Constructs examples of surfaces with discrete parametrizations.
Paper generalizes discrete CMC surfaces and shows how they can be derived.
problem Defining and deriving discrete constant mean curvature surfaces.
method Using discrete isothermic surfaces and the additive rational Toda system.
result Discrete isothermic CMC surfaces can be derived from discrete holomorphic data.
The paper introduces combinatorial curvature and flow for polyhedral surfaces, proving rigidity and solving the Yamabe problem.
problem Discrete conformal structures on polyhedral surfaces and their rigidity.
method Parameterized combinatorial curvature, combinatorial α-Ricci flow, and flow extension through singularities.
result Existence and convergence of combinatorial α-Ricci flow for solving the Yamabe problem.
We propose a unified definition for discrete analogues of constant mean curvature surfaces in spaces of constant curvature as a special case of discrete special isothermic nets. Bäcklund transformations and Lawson's correspondence are discussed. It is shown that the definition generalizes previous definitions and a con…
The paper proves a discrete positive mass theorem for graphs.
problem Formulating and proving a discrete positive mass theorem for graphs.
method Introducing asymptotically flat graphs, defining ADM mass, and using discrete harmonic functions.
result An asymptotically flat graph with non-negative Ricci curvature is isomorphic to the standard grid graph.