Derives local energy equation and proves consistency of staggered finite volume schemes for Euler equations.
problem Preserving conservation and consistency in staggered finite volume methods for Euler equations.
method Staggered discretization, material velocity upwinding, internal energy balance with correction term.
result Derives local total energy equation and proves schemes are conservative and consistent.
Efficiently simulates Langevin dynamics on manifold using diffusion maps and finite volume schemes.
problem Simulating Langevin dynamics on high-dimensional manifolds with limited data.
method Diffusion maps, Fokker-Planck equation, finite volume scheme, explicit time discretization.
result Data-driven finite volume scheme approximates Langevin dynamics on manifold with good properties.
Novel IMEX scheme solves financial PDEs with mixed derivatives.
problem Numerical approximations for financial PDEs with mixed derivatives.
method Second order finite volume IMEX Runge-Kutta scheme.
result Achieves true second order convergence with non-regular initial conditions.
Survey on nonlinear parabolic equations in finance.
problem Nonlinear extensions of the Black-Scholes theory.
method Qualitative and numerical analysis of nonlinear parabolic equations.
result Existence and uniqueness of solutions to nonlinear parabolic equations.
Deep learning speeds up material property quantification using stress waves.
problem Quantifying material properties from stress waves in complex media.
method Surrogate deep learning FWI scheme trained on random sampled properties and local minima.
result Demonstrates feasibility of deep learning for high-accuracy material property estimation.
A new FV-ADI method calibrates SLV models efficiently.
problem Calibrating SLV models to their underlying local volatility models.
method Finite volume - Alternating Direction Implicit (ADI) approach for solving 1D and 2D forward Kolmogorov equations.
result The proposed method efficiently calibrates SLV models without requiring PDE transformations and conserves numerical mass.
Finite volume ends found in quaternionic Kähler manifolds.
problem Existence of quaternionic Kähler manifolds with finite volume ends.
method Proof in all dimensions 4m≥4. result Existence of quaternionic Kähler manifolds with finite volume ends.
Combines machine learning and convex limiting for accurate subgrid flux modeling in shallow-water equations.
problem Accurate subgrid flux modeling in shallow-water equations.
method Machine learning and flux limiting for property-preserving subgrid scale modeling.
result The proposed method produces meaningful closures even in untrained scenarios.
Entropy rigidity proven for 3D and higher convex projective manifolds.
problem Entropy rigidity for strictly convex projective manifolds.
method Uses techniques from Besson, Courtois, and Gallot's entropy rigidity theorem.
result Uniform lower bounds on volume for finite volume strictly convex projective manifolds in dimensions ≥ 3.
Minimal surfaces found in finite volume manifolds.
problem Existence of minimal hypersurfaces in complete manifolds of finite volume.
method Independent result on volume sweeping by hypersurfaces; main tool is a volume constraint result.
result Proves existence of minimal hypersurfaces in complete non-compact manifolds of finite volume.
New boundary treatment improves accuracy for complex PDEs.
problem Order reduction in high-order IMEX schemes for multidimensional PDEs.
method Novel boundary treatment algorithms for Cartesian meshes, treating implicit-explicit stages similarly to interior points.
result Recovery of designed order of convergence by numerical verification.
The paper solves a dynamic portfolio optimization problem using Riccati transformation.
problem Dynamic stochastic portfolio optimization involving expected and intertemporal utilities.
method Solving a fully nonlinear HJB equation through Riccati transformation into a quasi-linear parabolic equation.
result The numerical method based on semi-implicit scheme converges at second order.
The study classifies 331 specific 4D polytopes with 7 facets.
problem Classifying finite-volume hyperbolic Coxeter 4D polytopes.
method Complete classification through exhaustive search.
result 331 unique polytopes with 7 facets identified.
Entropy rigidity theorem for negatively curved finite volume manifolds.
problem Proving entropy rigidity for negatively curved manifolds of finite volume.
method Entropy rigidity result in finite volume, comparing with compact case.
result Isometry of manifolds with same length spectrum.
Study estimates Bergman kernel on hyperbolic surfaces.
problem Estimating Bergman kernel on hyperbolic surfaces.
method Derive off-diagonal estimates for tensor-products of cotangent line bundles.
result Derived off-diagonal estimates for Bergman kernel.
Researchers create non-homogeneous finite-volume ends on quaternionic Kähler manifolds.
problem Constructing non-homogeneous quaternionic Kähler manifolds with finite volume ends.
method Using cohomogeneity one deformation of symmetric spaces, the researchers constructed manifolds with specific fundamental groups.
result The constructed manifolds are aspherical and have finite volume ends, not locally homogeneous.
A finite-volume hyperbolic 3-manifold geometrically bounds if it is the geodesic boundary of a finite-volume hyperbolic 4-manifold. We construct here an example of non-compact, finite-volume hyperbolic 3-manifold that geometrically bounds. The 3-manifold is the complement of a link with eight components, and its volume…
Totally umbilic surfaces in hyperbolic 3-manifolds are constructed and characterized.
problem Characterizing totally umbilic surfaces in hyperbolic 3-manifolds of finite volume.
method Construction and characterization of surfaces based on their properties and embedding conditions.
result Characterization of totally umbilic surfaces in hyperbolic 3-manifolds of finite volume.
The paper bends hyperbolic manifolds into convex structures.
problem Creating convex structures from hyperbolic manifolds.
method Bending totally geodesic hypersurfaces in finite volume hyperbolic manifolds.
result Examples of non-compact, finite volume, properly convex manifolds are found.
We compute the space of L2 harmonic forms (outside the middle degrees) on negatively curved Kaehler manifolds of finite volume.
Geodesic simplices in pseudo-hyperbolic space get a cohomological treatment.
problem Understanding geodesic simplices in pseudo-hyperbolic space.
method Cohomological interpretation and necessary/sufficient condition formulation.
result Every ideal geodesic polytope in (2,2) pseudo-hyperbolic space has finite volume. The paper contains a new proof that a complete, non-compact hyperbolic 3-manifold M with finite volume contains an immersed, closed, quasi-Fuchsian surface.
The study counts cusps on finite volume Riemannian manifolds using volume and decay estimates.
problem Counting cusps on finite volume Riemannian manifolds.
method Volume and decay estimates, nonlinear theory of p-Laplacian, volume comparison theorems.
result An upper bound on the number of cusps based on the volume of the manifold.
Finite-volume Ricci solitons with constant-length potential are trivial.
problem Characterizing non-compact Ricci solitons with specific properties.
method Analyzing conditions on scalar curvature and potential field length.
result Non-compact Ricci solitons with constant-length potential are trivial.
Totally geodesic submanifolds in convex cores are properly immersed and have finite volume.
problem Characterizing totally geodesic submanifolds in geometrically finite manifolds.
method Analysis of totally geodesic submanifolds in the convex core of geometrically finite rank-one locally symmetric manifolds.
result Every maximal totally geodesic submanifold of dimension at least two in the convex core is properly immersed and has finite volume, and only finitely many such submanifolds can occur.
The study finds conditions for certain Riemannian manifolds to be graph manifolds.
problem Conditions for complete Riemannian manifolds with specific curvature properties to be graph manifolds.
method Analyzes necessary and sufficient conditions for manifolds with curvature nullity at least n-2.
result Nomizu's conjecture holds true for manifolds with finite volume and curvature nullity at least n-2.
The study solves open problems in complex geometry by analyzing bounded domains with finite-volume quotients.
problem Analyzing bounded pseudoconvex domains with finite-volume quotients in complex geometry.
method Using semi-simplicity of automorphism groups and applying results to specific settings.
result The automorphism group of certain domains is discrete, and domains with specific properties are biholomorphic to the unit ball.
We show that complete uniform visibility manifolds of finite volume with sectional curvature −1≤K≤0 have positive simplicial volumes. This implies that their minimal volumes are non-zero.
No spin structures found in a hyperbolic 4D space.
problem Finding hyperbolic 4-manifolds without spin structures.
method Constructed a non-compact, orientable, hyperbolic 4-manifold.
result Demonstrated existence of a hyperbolic 4D space without spin structures.
A convex projective surface is the quotient of a properly convex open Ω of P(R) by a discret subgroup Γ of SL3(R). We give some caracterisations of the fact that a convex projective surface is of finite volume for the Busemann's measure. We deduce of this that if Ω is not a triangle then …
Let M=P(E) be a ruled surface. We introduce metrics of finite volume on M whose singularities are parametrized by a parabolic structure over E. Then, we generalise results of Burns--de Bartolomeis and LeBrun, by showing that the existence of a singular Kahler metric of finite volume and constant non positive scalar cur…
We study the spectrum of the Dirac operator on hyperbolic manifolds of finite volume. Depending on the spin structure it is either discrete or the whole real line. For link complements in S^3 we give a simple criterion in terms of linking numbers for when essential spectrum can occur. We compute the accumulation rate o…
Paper shows 3D hyperbolic manifolds are uniquely identified by their finite groups.
problem Identifying hyperbolic 3-manifolds using their finite quotient groups.
method Proves profinite isomorphism of fundamental groups uniquely determines the manifold's isomorphism type.
result Finite-volume hyperbolic 3-manifolds are almost determined by their finite quotient groups.
The Cayley hyperbolic space minimizes volume entropy among finite-volume metrics.
problem Volume entropy rigidity in Cayley hyperbolic spaces.
method Repairing a gap in the proof of volume entropy rigidity theorem.
result Cayley hyperbolic space minimizes volume entropy.
We prove that the space of complete, finite volume, pinched negatively curved Riemannian metrics on a smooth high-dimensional manifold is either empty or it is highly non-connected, provided their behavior at infinity is similar.
In this paper we show that some open set of the representations of the fundamental group of figure-eight knot complement found in \cite{Ballas12a} are the holonomies of a family of finite volume properly convex projective structures on the figure-eight knot complement.
Study Lie algebras of indefinite spaces with finite volume.
problem Characterize isometry Lie algebras of indefinite homogeneous spaces.
method Analyze nil-invariant symmetric bilinear forms and their adjoint representations.
result Derive structure theorem and classify isometry algebras for finite volume.
Computational evidence supports that SnapPea census manifolds are distinguishable by their finite quotients.
problem Determining if non-isometric finite volume hyperbolic 3-manifolds are distinguishable by their finite quotients.
method Examined the SnapPea census of 72,942 manifolds and computed their finite quotients.
result No two manifolds in the SnapPea census have the same finite quotients.
We propose and analyze a constrained level-set method for semi-automatic image segmentation. Our level-set model with constraints on the level-set function enables us to specify which parts of the image lie inside respectively outside the segmented objects. Such a-priori information can be expressed in terms of upper a…
The paper proves a bound on the length of the shortest geodesic flower on certain manifolds.
problem Finding the shortest geodesic flower on a specific class of manifolds.
method Analyzing a non-compact Riemannian manifold with locally convex ends and finite volume, proving the existence of a geodesic net with constraints on its length.
result The existence of a non-trivial geodesic flower with a bounded total length on the manifold.
No CMC surfaces exist in certain hyperbolic 3-manifolds.
problem Existence of constant mean curvature (CMC) surfaces in hyperbolic 3-manifolds.
method Proof by contradiction using hyperbolic geometry and topology.
result Nonexistence of CMC surfaces with mean curvature ≥ 1.
New noncompact manifolds with nonpositive curvature.
problem Creating new manifolds with specific curvature properties.
method Simple construction of manifolds with mixed ends.
result Found manifolds with fundamental groups not duality groups.
Cooper and Long generalised Epstein and Penner's Euclidean cell decomposition of cusped hyperbolic manifolds of finite volume to non-compact strictly convex projective manifolds of finite volume. We show that Weeks' algorithm to compute this decomposition for a hyperbolic surface generalises to strictly convex projecti…
We study noncompact, complete, finite volume, negatively curved manifolds M. We construct M with infinitely generated fundamental groups in all dimensions n≥2. We construct M whose cusp cross sections are compact hyperbolic manifolds in all dimension n≥3. In contrast we show that if sectional curvatu…
Proposes a new model for traffic flow on directed graphs.
problem Modeling advection on directed graphs for traffic flow.
method Reformulates graph advection operator as finite difference scheme; proposes DGAMGP model.
result Effective modeling of traffic flow and uncertainty as an advective process.
Study unimodular measures on Riemannian manifold spaces.
problem Understanding the geometry of large finite volume manifolds.
method Develop structure theory, use geodesic flow invariance, and characterize measures via foliations.
result Characterize unimodular measures and use them to compactify manifolds.
Finite volume Coxeter polytopes are quasiperfect and related to finite covolume reflection groups.
problem Characterizing finite volume Coxeter polytopes and their relation to reflection groups.
method Analyzing Coxeter polytopes and their volumes within Vinberg domains.
result Finite covolume reflection groups are characterized by the Vinberg domain.
We prove there exists a compact embedded minimal surface in a complete finite volume hyperbolic 3-manifold N. We also obtain a least area, incompressible, properly embedded, finite topology, 2-sided surface. We prove a properly embedded minimal surface of bounded curvature has finite topology. This dete…