Geometric triangulations can be transformed by bistellar moves.
problem Transforming geometric triangulations of different manifolds.
method Using bistellar moves, a type of local change to triangulations.
result Geometric triangulations of compact manifolds can be connected by bistellar moves.
Geometric GNNs improve graph discrimination through GWL.
problem Discriminating geometric graphs embedded in Euclidean space.
method Proposed a geometric version of the Weisfeiler-Leman test (GWL) for geometric graphs.
result Characterized the expressive power of geometric GNNs based on physical symmetries.
Geometric Algebra Transformer (GATr) handles various geometric data types efficiently.
problem Lack of a single architecture for diverse geometric data types.
method GATr uses projective geometric algebra, equivariant to E(3), and is a Transformer architecture.
result GATr outperforms non-geometric and equivariant baselines in various geometric tasks.
Geometric methods study 3-manifold splittings.
problem Studying Heegaard splittings of 3-manifolds.
method Geometric approaches.
result Recent advances in geometric methods.
Unified geometric structure reveals limitations of GANs and proposes geometric GAN.
problem Limitations of existing GAN training methods.
method Unified geometric decomposition of GAN training into three steps: separating hyperplane search, discriminator update, and generator update.
result Geometric GAN converges to Nash equilibrium and outperforms existing methods.
The differential geometric aspects of Geometric Phases are reviewed.
Expanding on previous work, this note generalizes geometric structures results.
problem Generalizing geometric structures results.
method Generalization to a class of geometric structures including integrable almost-complex structures.
result Main results generalized to a broader class of geometric structures.
tf_geometric simplifies graph deep learning in TensorFlow.
problem Efficient graph deep learning in TensorFlow.
method Kernel libraries and infrastructures for GNNs.
result tf_geometric supports various graph tasks and provides efficient GNN models.
Geometric Bass martingales linked to Brownian motion and geometric Brownian motion.
problem Modeling continuous martingales with prescribed initial and terminal distributions.
method Developed geometric Bass martingales and established their properties.
result Explicit bijection and representation of geometric Bass martingales.
Develops a new geometric cobordism theory using smooth Thom stacks.
problem Creating a versatile geometric cobordism theory for various geometric data.
method Introducing smooth Thom stacks and identifying a smooth representative of the Thom functor.
result The new theory generalizes parametrized cobordism and includes families of geometric data.
Study geometric bounds on generalized Ricci flow.
problem No specific problem stated; focuses on bounds.
method Analogous geometric quantities and bounds proven.
result Geometric and analytic bounds established.
Researchers geometrically define asymptotic coordinates in General Relativity.
problem Understanding the asymptotic behavior of relativistic initial data sets.
method Geometrization of asymptotic flatness and analysis of geometric invariants.
result Geometrically defined asymptotic coordinates for mass, energy, momentum, and angular momentum.
GDB bridges geometric states with improved accuracy and generality.
problem Challenges in predicting geometric state evolution in complex systems.
method Geometric Diffusion Bridge (GDB) framework using equivariant diffusion bridges.
result GDB surpasses existing methods in accurately bridging geometric states.
Study wave front singularities and their geometric properties.
problem Characterize singularities of focal surfaces of wave fronts.
method Characterization through differential geometric properties.
result Relationships between focal surfaces and initial wave fronts' geometric invariants.
Geometrically realizes Khovanov homology for semiadequate links.
problem Computing Khovanov homology for semiadequate links.
method Introducing partial presimplicial sets and their geometric realization.
result Concrete formula for homotopy type of geometric realization.
New findings show some hyperbolic 3-manifolds can't be geometrically bounded.
problem Understanding which cusped hyperbolic 3-manifolds can be geometrically bounded.
method Analyzing embeddings and geometric properties of hyperbolic 3-manifolds.
result Some cusped hyperbolic 3-manifolds cannot be geometrically bounded.
Study extremal trajectories of a rolling disk using geometric control theory.
problem Optimizing the motion of a vertical rolling disk.
method Geometric control theory and symmetries of geometric structures.
result Demonstrated computations in Maple for extremal trajectories.
The paper studies discrete sums of geometric Brownian motions in finance.
problem Modeling stochastic annuities and pricing Asian options.
method Analyzes probability distributions and asymptotic behavior of discrete sums of geometric Brownian motions.
result Derives tail asymptotics and computes asymptotic distribution functions for discrete sums.
Estimates small eigenvalues for geometrically finite manifolds.
problem Estimating small eigenvalues of Schrödinger operators.
method Geometrically finite manifolds, Riemannian vector bundles.
result Estimates the number of small eigenvalues.
Geometric symbols help compute heat invariants.
problem Computing heat invariants efficiently.
method Geometric symbol calculus of pseudodifferential operators.
result Efficient computation of heat invariants.
Study geometric properties of cuspidal edges with boundary.
problem Differential geometric properties of cuspidal edges with boundary.
method Analysis of differential geometric invariants and their relations.
result Relation between boundary behavior and other invariants.
We show that for a strongly convergent sequence of geometrically finite Kleinian groups with geometrically finite limit, the Cannon-Thurston maps of limit sets converge uniformly. If however the algebraic and geometric limits differ, as in the well known examples due to Kerckhoff and Thurston, then provided the geometr…
In this paper we study a collection of jet geometrical concepts, we refer to d-tensors, relativistic time dependent semisprays, harmonic curves and nonlinear connections on the 1-jet space J1(R;M), necessary to the construction of a Miron's-like geometrization for Lagrangians depending on a relativistic time. The geome…
Equivariant networks improve geometric prediction without scalar approximations.
problem Efficiently predicting geometric tensors in real-world scenarios.
method Equivariant networks for geometric prediction.
result Equivariant networks can generalize to unseen systems for geometric prediction.
Geometric structures on special manifolds are studied for rigidity.
problem Characterize and prove rigidity of geometric structures on rational homogeneous manifolds.
method Use Cartan geometry to study horospherical varieties and geometric structures modeled on them.
result Geometric structures on smooth projective horospherical varieties of Picard number one are locally equivalent to the standard structure.
The paper extends geometric finiteness to discrete subgroups of negatively pinched Hadamard manifolds.
problem Characterizing geometrically infinite discrete subgroups of negatively pinched Hadamard manifolds.
method Generalizing Bonahon's characterization and proving a theorem of Bishop's extension.
result Every discrete geometrically infinite isometry subgroup has a set of nonconical limit points of cardinality continuum.
The study explores geometric formality in specific dimensions of rationally elliptic manifolds.
problem Investigating geometric formality for rationally elliptic manifolds of dimensions 6 and 7.
method Analyzing real cohomology and homotopy properties to determine geometric formality.
result Geometrically formal six-dimensional biquotients with b2=3 have symmetric space cohomology. Survey on conservation laws for geometric PDEs.
problem Modeling polyharmonic maps.
method Conservation law approach.
result Overview of conservation laws in geometric PDEs.
Sharp geometric inequalities for free boundary hypersurfaces in balls.
problem Understanding geometric properties of free boundary hypersurfaces in balls.
method Proving a family of sharp geometric inequalities.
result Family of sharp geometric inequalities for free boundary hypersurfaces in balls.
Geometric Occam's Razor shapes deep learning solutions.
problem Understanding the regularization in over-parameterized neural networks.
method Analyzing the geometric model complexity and Dirichlet energy in neural networks.
result Over-parameterized neural networks are implicitly regularized by geometric model complexity.
Combines topological and geometric approaches to data analysis.
problem Understanding when and how geometric objects intersect.
method Connects topological and geometric concepts of curvature.
result Reconceptualizes curvature and links it to hyperconvexity.
Study geometric vector fields on 5D 2-step nilmanifolds.
problem Analyzing geometric vector fields on specific nilmanifolds.
method Examine vector fields on 2-step nilmanifolds of dimension 5.
result Characterize geometric vector fields on 5D 2-step nilmanifolds.
It is well-known that the LIE(Locally Induction Equation) admit soliton-type solutions and same soliton solutions arise from different and apparently irrelevant physical models. By comparing the solitons of LIE and Killing magnetic geodesics, we observe that these solitons are essentially decided by two families of iso…
Enhanced spectral clustering for geometric graphs improves clustering accuracy.
problem Ineffective standard spectral clustering for geometric graphs.
method Higher-order spectral clustering using higher-order eigenvectors.
result Established weak and strong consistency for Soft Geometric Block Model.
Study on geometrically formal metrics on complex manifolds.
problem Existence and properties of geometrically formal metrics on complex manifolds.
method Topological and cohomological obstructions, detailed analysis for specific manifolds, and metric constructions.
result Existence and non-existence conditions for geometrically formal metrics on various complex manifolds.
Extends geometric structures to manifolds with new operators.
problem No specific problem stated; extending geometric structures.
method Defines new gradient and Laplace operators on manifolds with geometric structures.
result Provides properties of the new operators.
No geometric minimal foliations in hyperbolic 3-manifolds.
problem Existence of geometric minimal foliations in hyperbolic three-manifolds.
method Analyzing three-dimensional closed hyperbolic manifolds.
result No locally geometric 1-parameter family of closed minimal surfaces.
Proves functoriality in geometric quantization for non-compact spin-c manifolds.
problem Non-compact spin-c manifolds in geometric quantization.
method Formal geometric quantization procedure for spin-c manifolds.
result Functoriality proven in spin-c setting.
Geometric techniques reveal new insights into Gromov-Witten invariants.
problem Formulating Gromov-Witten invariants for complete intersections in projective space.
method Combining geometric group theory and geometric topology, focusing on geodesic laminations.
result Primitive cohomologies unify mathematical formulations of Gromov-Witten invariants.
The paper examines geometric formality on specific surfaces under the Chern-Ricci flow.
problem Understanding geometric formality on class VII surfaces.
method Analysis of Chern-Ricci flow on specific surfaces.
result Evolution of geometric formality under the Chern-Ricci flow.
The paper proves geometric inequalities in sphere using locally constrained flows.
problem Deriving geometric inequalities in sphere.
method Established the longtime existence and convergence of a locally constrained flow.
result Proved new families of three-term geometric inequalities in sphere.
GeoAdaLer enhances geometric understanding of Adam for stochastic optimization.
problem Understanding geometric principles behind Adam's success in stochastic optimization.
method Introduces GeoAdaLer, an adaptive learning method based on geometric properties.
result Extends interpretability and effectiveness in complex optimization scenarios.
CP2 uses geometric information to improve conformal prediction robustness.
problem CP fails under geometric data shifts, losing coverage guarantees.
method Integrates geometric pose information into CP via canonicalization.
result Integrating geometric information with CP ensures robustness under geometric shifts.
Maps between non-compact surfaces can have geometric kernels under certain conditions.
problem Understanding when maps between non-compact surfaces have geometric kernels.
method Using Brown's proper fundamental group to establish sufficient conditions for geometric kernels.
result Characterization of conjugacy classes in the proper fundamental group and sufficient conditions for geometric kernels.
New geometric approach to slow invariant manifolds in dynamical systems.
problem Characterizing slow invariant manifolds in a coordinate-independent manner.
method Exploiting curvature concepts and variational approach in Hamiltonian mechanics.
result Differential geometric definition of slow invariant manifolds proposed.
Unified approach to geometric structure equivalence problem.
problem Equivalence problem of geometric structures.
method Unified framework, step prolongation, structure function γ. result Unified scheme for equivalence problem of geometric structures.
New proof of Willmore inequality using geometric divergence inequality.
problem Proving the Willmore inequality for bounded domains.
method Using a parametric geometric inequality derived from a divergence form geometric differential inequality.
result New proofs of quantitative Willmore-type and weighted Minkowski inequalities.
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.