arXiv research

Study classifies metrics on anti-de Sitter spacetime with specific symmetries.

Unified model explains volatility memory in stocks and forex.

L. Paoluzzi constructed a family of compact orientable three-dimensional hyperbolic manifolds with totally geodesic boundary, which were, by construction, closely related to the three-dimensional torus. This paper gives their complete classification up to isometry, and also their isometry groups. The key tool is the so…

A three-dimensional closed orientable orbifold (with no bad suborbifolds) is known to have a geometric decomposition from work of Perelman along with earlier work of Boileau-Leeb-Porti and Cooper-Hodgson-Kerckhoff. We give a new, logically independent, unified proof of the geometrization of orbifolds, using Ricci flow.…

Study timelike minimal surfaces in Heisenberg group using harmonic maps.

Unified framework for Alexandrov 3-spaces, extending manifold results.

We study the moduli space of euclidean structures with cone points on a surface, and describe a decomposition into cells each of which corresponds to a given combinatorial type of Delaunay tessellation. We use some of the ideas to study hyperbolic structures on three-dimensional manifolds

Let $M$ be a closed, oriented and smooth manifold of dimension $d$. Let $ŁM$ be the space of smooth loops in $M$. Chas and Sullivan introduced loop product, a product of degree $-d$ on the homology of $LM$. In this paper we show how for three manifolds the ``nontriviality'' of the loop product relates to the ``hyperbol…

Derive new Euler-Ramanujan-type identities and infinite decompositions for zero mean curvature graphs in various spaces.

This work proposes a new method to estimate joint probability from pairwise marginals, reducing sample complexity.

New algorithm for tensor decomposition and Gaussian mixture models.

We introduce a remarkable subset "the stem" of the set of positive roots of a reduced root system. The stem determines several interesting decompositions of the corresponding reductive Lie algebra. It gives also a nice simple three dimensional subalgebra and a "Cayley transform". In the present paper we apply the above…

In this article, we find the complete list of all contact structures (up to isotopy) on closed three-manifolds which are supported by an open book decomposition having planar pages with three (but not less) boundary components. We distinguish them by computing their first Chern classes and three dimensional invariants …

This paper is a contribution to harmonic analysis of compact solvmanifolds. We consider the four-dimensional oscillator group ${\rm Osc}_1$, which is a semi-direct product of the three-dimensional Heisenberg group and the real line. We classify the lattices of ${\rm Osc}_1$ up to inner automorphisms of ${\rm Osc}_1$. F…

Paper introduces a PDE-free method for decomposing forces in any dimension.

Computes the decomposition of rank-three bundles over the projective line with three marked points.

New method for triclustering with reduced arbitrariness.

We suggest a method of computing volume for a simple polytope $P$ in three-dimensional hyperbolic space $\mathbb{H}^3$. This method combines the combinatorial reduction of $P$ as a trivalent graph $Γ$ (the $1$-skeleton of $P$) by $I-H$, or Whitehead, moves (together with shrinking of triangular faces) aligned with its …

New method uses tensor decompositions to overcome the curse of dimensionality for large-scale learning.

In this paper, we introduce two notions on a surface in a contact manifold. The first one is called degree of transversality (DOT) which measures the transversality between the tangent spaces of a surface and the contact planes. The second quantity, called curvature of transversality (COT), is designed to give a compar…

In this thesis, we consider domino tilings of three-dimensional regions, especially those of the form $\mathcal{D} \times [0,N]$. In particular, we investigate the connected components of the space of tilings of such regions by flips, the local move performed by removing two adjacent dominoes and placing them back in t…

New algorithms extract Koopman invariant subspaces from large-scale data.

The study examines numerical aspects of Karhunen-Loève expansions for stochastic processes.

We explicitly compute the lower algebraic K-theory of the split three-dimensional crystallographic groups; i.e., the groups G that act properly and cocompactly on three-dimensional Euclidean space by isometries, such that the natural map from G to O(3) is a split injection onto its image. There are 73 split three-dimen…

The paper classifies 3D Lorentzian $Ein(2)$ Lie groups.

We study three-dimensional generalized Ricci solitons, both in Riemannian and Lorentzian settings. We shall determine their homogeneous models, classifying left-invariant generalized Ricci solitons on three-dimensional Lie groups.

Study finds index of symmetry for solvable 3D Lie groups with left-invariant metrics.

We study three-dimensional Alexandrov spaces with a lower curvature bound, focusing on extending three classical results on three-dimensional manifolds: First, we show that a closed three-dimensional Alexandrov space of positive curvature, with at least one topological singularity, must be homeomorphic to the suspensio…

In this paper, we study spinor Frenet equations in three dimensional Lie Groups with a bi-invariant metric. Also, we obtain spinor Frenet equations for special cases of three dimensional Lie groups.

We show that the equivalence problem for three-dimensional Lorentzian manifolds requires at most the fifth covariant derivative of the curvature tensor. We prove that this bound is sharp by exhibiting a class of 3D Lorentzian manifolds which realize this bound. The analysis is based on a three-dimensional analogue of t…

We introduce canonical principal parameters on any strongly regular minimal surface in the three dimensional sphere and prove that any such a surface is determined up to a motion by its normal curvature function satisfying the Sinh-Poisson equation. We obtain a classification theorem for bi-umbilical hypersurfaces of t…

The paper classifies Ricci solitons on specific Lorentzian Lie groups.

A new method optimizes knot selection for spline dimensional decomposition in stochastic dynamic analysis.

Study classifies Riemann solitons on specific 3D Lorentzian groups.

Compact 3D Cotton-parallel manifolds are always conformally flat.

Researchers found explicit formulae for special solitons in 3D spaces.

The paper classifies solitons on specific Lie groups.

The paper classifies structures on specific Lie groups.

We study closed three-dimensional Alexandrov spaces with a lower Ricci curvature bound in the $\mathsf{CD}^*(K,N)$ sense, focusing our attention on those with positive or nonnegative Ricci curvature. First, we show that a closed three-dimensional $\mathsf{CD}^*(2,3)$-Alexandrov space must be homeomorphic to a spherical…

The study classifies special flows on specific geometric groups.

We solve the metrisability problem for generic three-dimensional projective structures.

We obtain a topological and weakly equivariant classification of closed three-dimensional Alexandrov spaces with an effective isometric circle action. As an application of the classification we prove a version of the Borel conjecture for closed three-dimensional Alexandrov spaces with circle symmetry.

The study classifies and normalizes 3D gl-regular Nijenhuis operators.

In this short note, we prove that the only simply connected noncompact three-dimensional Type I $κ$-solution to the Ricci flow is the shrinking cylinder. This work can be regarded as a generalization of Cao and Chow, and a complement of Ding and Ni. Up to this point, three-dimensional $κ$-solutions of Type I are comple…

Rigidity proven for a specific type of solitons with harmonic curvature.

Study finds all conformal Ricci collineations on specific 3D Lorentzian groups.

Solves surface problem in 3D light cone.

Study left invariant Lorentzian metrics on 3D non-unimodular Lie groups.