Proves cup product homomorphism for bounded cohomology on negatively curved manifolds.
problem Understanding cup product behavior in bounded cohomology.
method Analyzes map Ψ* associating closed forms to bounded cohomology classes via integration.
result Proves Ψ* preserves cup product in sufficiently high degrees.
New bounded cohomology classes found for exact forms on curved manifolds.
problem Finding non-trivial bounded cohomology classes for exact forms on negatively curved manifolds.
method Integration of forms over simplices to associate bounded cocycles.
result Exact non-zero 2-forms define non-trivial bounded cohomology classes.
Negative curvature manifolds have vanishing bounded volume class if and only if Cheeger constant is positive.
problem Negative curvature manifolds and their volume classes.
method Integration of volume forms and isoperimetric constants.
result Vanishing of bounded volume class implies positivity of Cheeger constant and vice versa.
We present two families of knots which have straight number higher than crossing number. In the case of the second family, we have computed the straight number explicitly. We also give a general theorem about alternating knots that states adding an even number of crossings to a twist region will not change whether the …
Jablan and Radović originally defined two invariants called the Meander number and OGC number of knots for certain classes of knots. We generalize these definitions to all knots and name the straight number and contained straight number of a knot, respectively, and prove they are well defined. We answer two questions a…
This paper improves the efficiency of generative models by optimizing the straightness of Rectified Flow.
problem Improving the efficiency of generative models by reducing discretization error.
method Introducing a novel Piecewise Straightness parameter, γ2,T, to optimize the straightness of Rectified Flow.
result Minimizing curvature in Rectified Flow models leads to high-fidelity, one-step sampling.
The study characterizes straight-line flows in dynamic measure transport.
problem Tackles the challenge of designing flows that are easy to integrate.
method Characterizes straight-line flows using a PDE and Reynolds tensor.
result Characterizes affine-in-time interpolants and necessary conditions for flow geometry.
The paper triangulates Heisenberg groups with horizontal and straight simplexes.
problem Triangulating Heisenberg groups with specific regularity properties.
method Constructing triangulations with horizontal and straight simplexes on a polyhedral structure and extending to the whole Heisenberg group.
result Explicit examples of grid and triangulations provided.
A new method learns straight trajectories in one step for optimal flow matching.
problem Learning flows with straight trajectories for fast inference.
method Optimal Flow Matching (OFM) approach using convex functions for vector fields.
result Recovering straight OT displacements in just one FM step for quadratic transport.
Study on straight-line flows for generative modeling with theoretical obstructions.
problem Existence and obstructions of straight-line flows in generative modeling.
method Characterizations of straight-line flows through PDEs involving conditional statistics of stochastic processes.
result Sharp dichotomy in the existence of straight-line flows for targets with well-separated modes.
We generalise a result of Garofalo and Pauls: a horizontally minimal smooth surface embedded in the Heisenberg group is locally a (straight) ruled surface, i.e. it consists of straight lines tangent to a horizontal vector field along a smooth curve. We show additionally that any horizontally minimal surface is locally …
Correct method found for drawing precise envelope of straight lines.
problem Widespread method fails to represent the precise shape of envelope.
method Recently discovered correct method for straight line families in the plane.
result Correct method precisely represents the envelope of straight lines.
Study of straight-line flows on a unique infinite surface.
problem Understanding straight-line flows on a specific infinite surface.
method Geometric description and characterization of periodic and drift orbits; use of rigid symmetries and Veech group.
result Complete characterization of periodic directions and proof of density of periodic and ergodic directions.
We study Fredholm properties and index formulas for Dirac operators over complete Riemannian manifolds with straight ends. An important class of examples of such manifolds are complete Riemannian manifolds with pinched negative sectional curvature and finite volume.
Smooth compactness theorem for elasticae, except straight segments.
problem Compactness of elasticae space.
method Smooth compactness theorem proof.
result Smooth stability results for minimizers.
The limit of energies of a sequence of harmonic maps as their annular domains approach the boundary of moduli space depends upon the boundary point approached. The infinite energy case is associated with limits of images containing ruled surfaces. The finite energy case yields a limit of images, under a suitable topolo…
To make deep neural networks feasible in resource-constrained environments (such as mobile devices), it is beneficial to quantize models by using low-precision weights. One common technique for quantizing neural networks is the straight-through gradient method, which enables back-propagation through the quantization ma…
Reintroduces straight-through estimators for binary neural networks.
problem Training neural networks with binary weights and activations is challenging due to gradient issues and discrete weight optimization.
method Derives ST methods as estimators in the SBN model, analyzes properties and estimation accuracy, explains latent weights and mirror descent method.
result Reintroduces ST methods as sound approximations and provides clearer application and improvements.
In this paper, we investigate the ruled surfaces generated by a straight line according to rotation minimizing frame (RMF). Using this frame of a straight line, we obtained the necessary and sufficient conditions when the ruled surface is developable. Also, we give some new results and theorems related to be the asympt…
The Dirichlet Laplacian in curved tubes of arbitrary cross-section rotating with respect to the Tang frame along infinite curves in Euclidean spaces of arbitrary dimension is investigated. If the reference curve is not straight and its curvatures vanish at infinity, we prove that the essential spectrum as a set coincid…
Paper improves Gumbel-Softmax estimator variance reduction.
problem Challenges in gradient estimation for models with discrete latent variables.
method Rao-Blackwellization applied to straight-through Gumbel-Softmax estimator.
result Reduces mean squared error and variance of Gumbel-Softmax estimator.
New method describes entanglement of straight lines in 3D space.
problem Tackles the geometry and topology of configurations of straight lines.
method Introduces direction matrices and a discrete motion principle.
result Shows n-crosses as links of pairwise connected unknots.
The paper develops formulas for hyperbolic simplices based on edge lengths.
problem Understanding the geometry of hyperbolic simplices using only edge lengths.
method Develops geometric formulas for hyperbolic simplices based on edge lengths.
result Distance and projection formulas in hyperbolic simplices.
Classifies branched Willmore spheres using conformal Gauss maps.
problem Classifying branched Willmore spheres.
method Analyzing the asymptotic expansion of the conformal Gauss map at branched points.
result Full classification of branched Willmore spheres.
New insights into quantized neural networks reveal learning dynamics and generalization errors.
problem Understanding the impact of quantization hyperparameters on learning dynamics in high-dimensional models.
method Theoretical analysis and fixed-point analysis of STE dynamics in quantized models.
result STE training in quantized models converges to a plateau followed by a sharp drop in generalization error, influenced by quantization range.
Estimates dimensions of maximal simplices for rational and irrational trees in Outer space.
problem Understanding the structure of trees in Outer space.
method Associate simplices to R-trees and estimate their dimensions. result Estimates the dimensions of maximal simplices for both rational and irrational trees.
In this paper, on the first, we prove Δr=2H where Δ is the Laplacian operator, r=(r1,r2,r3) the position vector field and H is the mean curvature vector field of a surface S in the 3-dimensional Heisenberg group H3. In the second, we classify the ruled surfaces by straight…
The study identifies unique fluid flow patterns.
problem Understanding incompressible fluid flows with straight streamlines.
method Local differential geometry of line congruences to integrate Euler equations.
result Only specific fluid flows are possible with straight streamlines.
The paper establishes conditions for Riemannian connections and semi-simplicity of Lie algebras using spray structures.
problem Conditions for Riemannian connections and semi-simplicity of Lie algebras.
method Using almost product structures and spray, the paper provides necessary and sufficient conditions for these properties.
result Equivalence of semi-simplicity of Lie algebras to derived ideal coincidence, interiority of derivations, and adjoint representation semi-simplicity.
Sub-Riemannian geometry connects bike paths to mathematical curves.
problem Understanding bike paths and their mathematical properties.
method Relating sub-Riemannian geometry to bicycle motion and curve shapes.
result Geodesics in sub-Riemannian geometry correspond to specific bike paths.
It is proved that the volume of spherical or hyperbolic simplices, when considered as a function of the dihedral angles, can be extended continuously to degenerated simplices.
A zero mean curvature surface in the Lorentz-Minkowski 3-space is said to be of Riemann-type if it is foliated by circles and at most countably many straight lines in parallel planes. We classify all zero mean curvature surfaces of Riemann-type according to their causal characters, and as a corollary, we prove that if …
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. New heat trace coefficients reveal curvature effects in polygonal domains.
problem Understanding heat trace behavior in polygonal domains with curved corners.
method Local heat trace expansion through order t1/2, analyzing both Dirichlet and Neumann boundary conditions. result Sharp sign law for the Dirichlet angular factor of the first corner-curvature heat invariant.
Study knot diagrams on a sphere without vertical lines, focusing on minimal crossings.
problem Understanding minimal crossings of knot diagrams on a punctured sphere.
method Mathematical model of string figures using knot diagrams on xyz-space with missing vertical lines, analyzing minimal crossings under Reidemeister moves. result Minimal number of crossings of knot diagrams on a punctured sphere.
The aim of this paper is to investigate properties preserved and co-preserved by coarsely n-to-1 functions, in particular by the quotient maps X→X/∼ induced by a finite group G acting by isometries on a metric space X. The coarse properties we are mainly interested in are related to asymptotic dimension a…
The goal of this paper is to describe all local diffeomorphisms mapping a family of circles, in an open subset of $\r^3$, into straight lines. This paper contains two main results. The first is a complete description of the rectifiable collection of circles in $\r^3$ passing through one point. It turns out that to be r…
Study PL bordism theories with quantitative bounds on filling simplices.
problem Understanding PL bordism theories with geometric constraints.
method Quantitative analysis of PL manifolds and exotic theories.
result Bounding the number of simplices in fillings of cycles.
Framework reduces simplicity bias in NNs, improving OOD generalization and robustness.
problem Simplicity bias in deep learning models leads to biased predictions and poor OOD generalization.
method Proposes a framework that regularizes conditional mutual information to encourage use of diverse features.
result Demonstrates effectiveness in various settings, enhancing OOD generalization and robustness.
New framework shows C∗-simplicity for groups without certain subalgebras.
problem Characterizing C∗-simplicity of groups. method Introducing confined subalgebras and Uniformly Recurrent States.
result A countable discrete group is C∗-simple if it has no non-trivial amenable confined subalgebras. Straight lines are a basin of attraction for the elastic flow at least to level 1.9615π.
problem Understanding the basin of attraction for the free boundary free elastic flow.
method Steepest descent gradient flow for elastic energy, numerical evidence.
result Straight lines have a basin of attraction at least to level 1.9615π.
In this article, we prove a theorem comparing the dihedral angles of simplices in the hyperbolic, spherical and Euclidean geometries.
We study a natural intrinsic definition of geometric simplices in Riemannian manifolds of arbitrary dimension n, and exploit these simplices to obtain criteria for triangulating compact Riemannian manifolds. These geometric simplices are defined using Karcher means. Given a finite set of vertices in a convex set on t…
The straight-line flow on almost every staircase and on almost every square tiled staircase is recurrent. For almost every square tiled staircase the set of periodic orbits is dense in the phase space.
Research reveals simplicity bias in random logistic map, impacting data analysis and forecasting.
problem Simplicity bias in dynamical systems and its impact on data analysis and prediction.
method Examined the logistic map and random logistic map, focusing on simplicity bias and noise effects.
result Simplicity bias is observable in the random logistic map, persisting even with small noise levels.
The Apollonius theorem is generalized for m-simplices, with applications in geometry and optimization.
problem Generalizing the Apollonius theorem for m-simplices.
method Direct generalization of the theorem to m-simplices in n-dimensional space.
result Applications in geometry and optimization, including minimal surface enclosures, simplex thickness, and root-finding methods.
Similar simplices can be inscribed in most smoothly embedded spheres.
problem Inscribing families of similar simplices in spheres.
method Diffeomorphic mapping and techniques from previous work on inscribing triangles.
result A dense family of spheres allows inscribing similar simplices of every pose.
We study prismatics sets analogously to simplical sets except that realization involves prisms, i.e., products of simplices rather than just simplices. Particular examples are the prismatic subdivision of a simplicial set S and the prismatic star of S. Both have the same homotopy type as S and in particular the latter …