arXiv research

Study geometric structures on twistor and reflector spaces of paraquaternionic contact manifolds.

This paper continues the study of a class of compact convex hypersurfaces in Euclidean space $R^{n+1}, ~n \geq 1$, which are boundaries of compact convex bodies obtained by taking the intersection of (solid) confocal paraboloids of revolution. Such hypersurfaces are called reflectors. In $R^3$ reflectors arise naturall…

We investigate the integrability of natural almost complex structures on the twistor space of an almost para-quaternionic manifold as well as the integrability of natural almost paracomplex structures on the reflector space of an almost para-quaternionic manifold constructed with the help of a para-quaternionic connect…

Dictionary learning is the task of determining a data-dependent transform that yields a sparse representation of some observed data. The dictionary learning problem is non-convex, and usually solved via computationally complex iterative algorithms. Furthermore, the resulting transforms obtained generally lack structure…

New examples of structures found using twistor space.

In this paper we obtain several curvature properties of the twistor and reflector spaces of a paraquaternionic Kähler manifold and prove the existence of both positive and negative mixed 3-Sasakian structures in a principal SO(2,1)-bundle over a paraquaternionic Kähler manifold.

The ability to decompose a signal in an orthonormal basis (a set of orthogonal components, each normalized to have unit length) using a fast numerical procedure rests at the heart of many signal processing methods and applications. The classic examples are the Fourier and wavelet transforms that enjoy numerically effic…

For a congruence of straight lines defined by a hypersurface in $R^{n+1}, n \geq 1,$ and a field of reflected directions created by a point source we define the notion of intensity in a tangent direction and introduce elementary symmetric functions $S_m, m=1, 2,...,n,$ of {\it principal intensities}. The problem of exi…

We study a regular closure operator in the category of quandles. We show that the regular closure operator and the pullback closure operator corresponding to the reflector from the category of quandles to its full subcategory of trivial quandles coincide, we give a simple description of this closure operator, and analy…

We provide a general criteria for the integrability of the almost para-quaternionic structure of an almost para-quaternionic manifold (M,P) of dimension bigger or equal to eight, in terms of the integrability of two or three sections of the defining rank three vector bundle P. We relate it with the integrability of the…

HD algorithm simulates dynamics on random matrix ensembles without generating full matrices.

We study a parabolic equation for finding solutions to the optimal transport problem on compact Riemannian manifolds with general cost functions. We show that if the cost satisfies the strong MTW condition and the stay-away singularity property, then the solution to the parabolic flow with any appropriate initial condi…

Global geometric expressions derived for manifold embeddings.

New method allows sheets to morph into multiple shapes via spatially varying stimuli.

Designs chiral photonic structures using machine learning for efficient optical properties.

This PhD thesis studies the broken ray transform, a generalization of the geodesic X-ray transform where geodesics are replaced with broken rays that reflect on a part of the boundary. The fundamental question is whether this transform is injective. We employ four different methods to approach this question, and each o…

Vanishing and exploding gradients are two of the main obstacles in training deep neural networks, especially in capturing long range dependencies in recurrent neural networks~(RNNs). In this paper, we present an efficient parametrization of the transition matrix of an RNN that allows us to stabilize the gradients that …

Study of tangent spaces in diffeological spaces under Lie group actions.

(1,1) non-L-space knots are foliar in 3D space.

Universal spaces for finite topological spaces simplify shape descriptions.

The paper extends Stone duality to topological convexity spaces.

No Einstein hypersurfaces found in Damek-Ricci spaces.

The distance function $\varrho(p,q)$ (or $d(p,q)$) of a distance space (general metric space) is not differentiable in general. We investigate such distance spaces over $\mathbb R^n$, whose distance functions are differentiable like in case of Finsler spaces. These spaces have several good properties, yet they are no F…

New kernels defined for various spaces, including measures.

Metric spaces uniquely split into Hilbert and non-line-split parts.

In a paper (math.DG/0403528) we obtained explicit examples of Moishezon twistor spaces of some compact self-dual four-manifolds admitting a non-trivial Killing field, and also determined their moduli space. In this note we investigate minitwistor spaces associated to these twistor spaces. We determine their structure, …

Study geometry of tetrahedra in complex hyperbolic space and Hilbert spaces.

New curvature positivity helps classify spherical spaces and complex projective spaces.

Study of complete space-like self-expanders in Minkovski space.

The abstract discusses the linear and smooth structures of mapping spaces.

Study on convergence of transformed metric spaces as dimensions grow.

Characterizes when almost smooth spaces become RCD spaces.

Stability of Wasserstein spaces under various convergence types.

New infinite families of flat spaces found from symmetric spaces.

New quasi space forms solve Thurston's geometrical space form problem.

Classifies compact spaces by shape, finite spaces by weak homotopy.

Introduces new types of homogeneous spaces and their properties.

In this paper, we present a unified study of the moduli space of tropical curves and Outer space which we link via period maps to the moduli space of tropical abelian varieties and the space of positive definite quadratic forms. Our work is a first step towards exhibiting Outer space and the space of positive definite …

Survey Bernstein-type theorems for graphical surfaces in Euclidean and Lorentz-Minkowski spaces.

CAT(0) spaces with small volume growth are homeomorphic to Euclidean space.

Cantor Riemannium is a new type of space from holomorphic germs.

We construct a metric on the moduli space of bodies in Euclidean space. The moduli space is defined as the quotient space with respect to the action of integral affine transformations. This moduli space contains a subspace, the moduli space of Delzant polytopes, which can be identified with the moduli space of symplect…

Study static Einstein-Maxwell space invariant by translation.

A "Chen space" is a set X equipped with a collection of "plots" - maps from convex sets to X - satisfying three simple axioms. While an individual Chen space can be much worse than a smooth manifold, the category of all Chen spaces is much better behaved than the category of smooth manifolds. For example, any subspace …

The classification of G-spaces by Palais is refined for the case where the orbit space satisfies certain mild topological hypotheses. It is shown that when a sequence of such orbit spaces is "close" to a limit orbit space, in some suitable sense, within a larger ambient orbit space, the G-spaces in the tail of the sequ…

New spaces help connect manifold structures on equivariant Poincaré spaces.

Study shows how maps from certain geometric spaces behave near their edges.

Research explores the space-like embeddings in pseudo-hyperbolic space, finding geometric frames and actions.