arXiv research

Unified framework for training generator, energy model, and inference model.

Paper relaxes triangle inequality for KL divergence between Gaussian distributions.

We study two $2$-dimensional Teichmüller spaces of surfaces with boundary and marked points, namely, the pentagon and the punctured triangle. We show that their geometry is quite different from Teichmüller spaces of closed surfaces. Indeed, both spaces are exhausted by regular convex geodesic polygons with a fixed numb…

Our main result is that if a generic convex domain in $\R^n$ collapses to a domain in $\R^{n-1}$, then the difference between the first two Dirichlet eigenvalues of the Euclidean Laplacian, known as the fundamental gap, diverges. The boundary of the domain need not be smooth, merely Lipschitz continuous. To motivate th…

Study examines Hilbert area of inscribed polygons in projective geometry.

A new variational inference method using sliced Wasserstein distance is proposed.

Napoleonic triangles don't exist in hyperbolic geometry.

New bounds on inscribed triangles in arbitrary planar domains.

Study on Laplacian determinant in isosceles triangles, finding equilateral triangle minimizes determinant.

Paper calculates eigenvalues of a specific triangle on a sphere.

New method shows any triangle group generating pair is related to special coverings.

Shorter sides in geodesic triangles in hyperbolic plane.

Complex hyperbolic triangle groups were first considered by Mostow in building the first nonarithmetic lattices in PU(2, 1). They are a natural generalization of the classical triangle groups acting on the hyperbolic plane. A well-known theorem of Takeuchi is that there are only finitely many Fuchsian triangle groups t…

The paper explores isotopic triples of triangles in 3D space.

Defines band maps in unoriented link Floer homology forming a skein exact triangle.

Proves a new skein exact triangle for real monopole Floer homology.

New surgery exact triangles in Heegaard Floer homology for rational slopes.

We answer the question "Does the Y-triangle move preserve intrinsic knottedness?" in the negative by giving an example of a graph that is obtained from the intrinsically knotted graph K_7 by triangle-Y and Y-triangle moves but is not intrinsically knotted.

Study calculates eigenvalues and eigenfunctions for spherical triangles and finds fundamental gap behavior.

We show that the triangle with angles Pi/12, Pi/3 and 7*Pi/12 has the lattice property and compute this triangle's Veech group.

We study side-lengths of triangles in path metric spaces. We prove that unless such a space X is bounded, or quasi-isometric to line or half-line, every triple of real numbers satisfying the strict triangle inequalities, is realized by the side-lengths of a triangle in X. We construct an example of a complete path metr…

New method proves mateability of triangle groups with Blaschke products.

In this paper we study the area of ideals triangles in a convex domain with its Hilbert geometry. We obtain a characterization of the hyperbolic geometry among all the Hilbert geometry in terms of area of ideals triangles. We also obtain a sharp lower bound on the hilbert area of ideal triangles, independant of the con…

Classifies complex hyperbolic triangle groups by types.

In Lorentzian geometry, limited definition of angles restricts the use of angle bisectors in study of triangles. This paper redefines angle bisectors so that they can be used to study attributes of triangles. Using the new definition, this paper investigates the existence of the incenter and the isogonal conjugate of a…

We show that simple random walks on (non-trivial) relatively hyperbolic groups stay $O(\log(n))$-close to geodesics, where $n$ is the number of steps of the walk. Using similar techniques we show that simple random walks in mapping class groups stay $O(\sqrt{n\log(n)})$-close to geodesics and hierarchy paths. Along the…

A formula for Rademacher symbols in triangle groups is provided.

Study stabilizers of complex hyperbolic triangle groups, finding generators and signatures.

Triangle groups show rigidity in hyperbolic spaces.

Criterion for stopping conjugacy class enumeration in triangle groups.

New skein exact triangles for link Floer homology.

New theorem disproves Angle Defect for super triangles.

The study examines Lipschitz normally embedded Hölder triangles in 4D space.

Triangle Artin groups split as graphs of free groups under specific conditions.

The paper finds extremum values for mixed Laplacian eigenvalues on triangles and trapezoids.

Study on discrete complex hyperbolic triangle groups of specific type.

Study on 3D surfaces and tangles formed by Poncelet triangles.

Study of subgroups in complex hyperbolic lattice triangle groups.

Triangle groups uniquely identified by their finite quotients.

Study orbits in right triangles, deducing periodic billiard paths and classifying orbit closures.

Paper shows any link can be diagrammed with only triangles and quadrilaterals.

Given a closed polygon P having n edges, embedded in R^d, we give upper and lower bounds for the minimal number of triangles t needed to form a triangulated PL surface in R^d having P as its geometric boundary. The most interesting case is dimension 3, where the polygon may be knotted. We use the Seifert suface constru…

The paper extends Menelaus' and Ceva's theorems to translation triangles in various Thurston geometries.

We study groups generated by three half-turns in the Lobachevsky $3$-space and their quotient orbifolds. These generalized triangle groups are closely related to the arbitrary 2-generator Kleinian groups. Our main result is a classification of the singular sets of the generalized triangle orbifolds. We also present a m…

Study on Teichmüller space of acute triangles using explicit calculations.

Identifies Anosov representations of hyperbolic triangle groups in SL(3,R).

We classify the 3-dimensional hyperbolic polyhedral orbifolds that contain no embedded essential 2-suborbifolds, up to decomposition along embedded hyperbolic triangle orbifolds (turnovers). We give a necessary condition for a 3-dimensional hyperbolic polyhedral orbifold to contain an immersed (singular) hyperbolic tur…

We consider the Neumann Laplacian acting on square-integrable functions on a triangle in the hyperbolic plane that has one cusp. We show that the generic such triangle has no eigenvalues embedded in its continuous spectrum. To prove this result we study the behavior of the real-analytic eigenvalue branches of a degener…