arXiv research

Study the Hessian geometry of an ideal gas in a centrifuge.

We introduce semisimple 2-categories, fusion 2-categories, and spherical fusion 2-categories. For each spherical fusion 2-category, we construct a state-sum invariant of oriented singular piecewise-linear 4-manifolds.

Defines state sum models with defects in 3-manifolds.

The study proves spherical polygon analogs of curve theorems, finding bounds on intersections and inflections.

Using generating functional and replica techniques, respectively, we study the dynamics and statics of a spherical Minority Game (MG), which in contrast with a spherical MG previously presented in J.Phys A: Math. Gen. 36 11159 (2003) displays a phase with broken ergodicity and dependence of the macroscopic stationary s…

In this paper, we present two observations about static spherically symmetric solutions of the Einstein-Klein-Gordon equations. The first is a comment extending the well-known result of the existence of static states (i.e. standing wave solutions) of the Einstein-Klein-Gordon equations. The second more important observ…

Timelike geometry of spherical simplices is shown to be isometric to vector spaces.

New construction of Turaev-Viro invariants invariant under Morita equivalence.

DeepSphere improves spherical CNNs by balancing efficiency and rotation equivariance.

Study on quantum state entanglement using Kaehler manifolds.

A new GP model uses spherical harmonics for faster inference.

Study on spherical bodies of constant width on the unit sphere, proving bounds on their relative effective radius.

A family of TQFTs parametrised by G-crossed braided spherical fusion categories has been defined recently as a state sum model and as a Hamiltonian lattice model. Concrete calculations of the resulting manifold invariants are scarce because of the combinatorial complexity of triangulations, if nothing else. Handle deco…

Unsupervised text embedding has shown great power in a wide range of NLP tasks. While text embeddings are typically learned in the Euclidean space, directional similarity is often more effective in tasks such as word similarity and document clustering, which creates a gap between the training stage and usage stage of t…

We state several sufficient conditions for compact spacelike surface in the3-dimensional de Sitter space to be totally geodesic or spherical.

A new method using spherical harmonics approximates the Sliced-Wasserstein distance.

A string-net model associates a vector space to a surface in terms of graphs decorated by objects and morphisms of a pivotal fusion category modulo local relations. String-net models are usually considered for spherical fusion categories, and in this case the vector spaces agree with the state spaces of the correspondi…

New capillary Christoffel-Minkowski problem solved for half-space.

We construct a state-sum type invariant of smooth closed oriented $4$-manifolds out of a $G$-crossed braided spherical fusion category ($G$-BSFC) for $G$ a finite group. The construction can be extended to obtain a $(3+1)$-dimensional topological quantum field theory (TQFT). The invariant of $4$-manifolds generalizes s…

Study curvature operator on Riemannian manifolds, proving new classification results.

Point cloud is the most fundamental representation of 3D geometric objects. Analyzing and processing point cloud surfaces is important in computer graphics and computer vision. However, most of the existing algorithms for surface analysis require connectivity information. Therefore, it is desirable to develop a mesh st…

The discrete isoperimetric inequality in Euclidean geometry states that among all $n$-gons having a fixed perimeter $p$, the one with the largest area is the regular $n$-gon. The statement is true in spherical geometry and hyperbolic geometry as well. In this paper, we generalize the discrete isoperimetric inequality t…

New theorem on spheres with punctures using infinity metric.

A Steiner chain of length k consists of k circles, tangent to two given non-intersecting circles (the parent circles) and tangent to each other in a cyclic pattern. The Steiner porism states that once a chain of k circles exists, there exists a 1-parameter family of such chains with the same parent circles that can be …

New method clusters non-spherical Gaussian mixtures with fewer samples and time.

Paper introduces spherical knot mosaics for knot and link invariants.

Static spherically symmetric solutions to the Einstein-Euler equations with prescribed central densities are known to exist, be unique and smooth for reasonable equations of state. Some criteria are also available to decide whether solutions have finite extent (stars with a vacuum exterior) or infinite extent. In the l…

Study geodesics on spherical polyhedra, estimating their number.

The study proves a discrete version of Segre's theorem for polygonal curves.

This paper explores new Finsler metrics and their connections to generalized Sakaguchi's Theorem.

This paper sets a lower bound for sample complexity in inverse reinforcement learning.

The abstract proves spherical surface decompositions with conical singularities.

SRCA reduces high-dimensional data to lower dimensions while preserving geometric structures.

For the n-dimensional spherical pedal curve $ped_{γ,P}$ with respect to an n-dimensional spherical unit speed curve $γ$ and a given point $P \in S^n$, we define the spherical orthotomic curve of $γ$ relative to the point $P$, and classify singularities of spherical orthotomic curves.

Quantum states are not entangled if submanifold is a product.

Study spherical curves with curvature dependent on distance to a great circle.

Given a discrete group G and a spherical G-fusion category whose neutral component has invertible dimension, we use the state-sum method to construct a 3-dimensional Homotopy Quantum Field Theory (HQFT) with target the Eilenberg-MacLane space K(G,1).

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

New findings show fundamental group is not audible in spherical space forms.

FHDMs achieve optimal convergence in spherically supported data.

Existence and uniqueness of spherical helicoidal surfaces in 3-sphere via spherical curves.

The class of differential equations describing pseudo-spherical surfaces, first introduced by Chern and Tenenblat [3], is characterized by the property that to each solution of a differential equation, within the class, there corresponds a 2-dimensional Riemannian metric of curvature equal to $-1$. The class of differe…

The reductivity of a spherical curve represents how reduced the spherical curve is. It is unknown if there exists a spherical curve whose reductivity is four. In this paper we give an unavoidable set for spherical curves with reductivity four by considering 4-gons.

The paper improves inequalities for nearly spherical sets using quermassintegrals.

The reductivity of a spherical curve is the minimal number of a local transformation called an inverse-half-twisted splice required to obtain a reducible spherical curve from the spherical curve. It is unknown if there exists a spherical curve whose reductivity is four. In this paper, an unavoidable set of configuratio…

We show that we can obtain a reducible spherical curve from any non-trivial spherical curve by four or less inverse-half-twisted splices, i.e., the reductivity, which represents how reduced a spherical curve is, is four or less. We also discuss unavoidable sets of tangles for spherical curves.

The paper classifies Landsberg spherically symmetric Finsler metrics in various dimensions.

Paper proves a rigidity result for static perfect fluids.