arXiv research

We give some general criteria of being a homeomorphism for continuous mappings of topological manifolds, as well as criteria of being a diffeomorphism for smooth mappings of smooth manifolds. As an illustration, we apply these criteria to the problems arising in two- and three-dimensional grid generation.

We shall give useful criteria of lips, beaks and swallowtail singularities of smooth map from the plane into the plane. As an application of criteria, we will discuss the singularities of Cauchy problem of single conservation law.

We give criteria for Morin singularities for germs of maps into lower dimensions. As an application, we study the bifurcation of Lefschetz singularities.

Simple criteria for codimension two surface singularities.

New quantity helps map homotopy classes in complex spaces.

Multi-criteria recommender systems have been increasingly valuable for helping consumers identify the most relevant items based on different dimensions of user experiences. However, previously proposed multi-criteria models did not take into account latent embeddings generated from user reviews, which capture latent se…

We present criteria for establishing a triangulation of a manifold. Given a manifold M, a simplicial complex A, and a map H from the underlying space of A to M, our criteria are presented in local coordinate charts for M, and ensure that H is a homeomorphism. These criteria do not require a differentiable structure, or…

Introduces and studies k-c-quasifibrations, updating and generalizing previous work.

Criteria for sharksfin and deltoid singularities from plane to plane, with applications.

Combining the definition of Schwarzian derivative for conformal mappings between Riemannian manifolds given by Osgood and Stowe with that for parametrized curves in Euclidean space given by Ahlfors, we establish injectivity criteria for holomorphic curves $φ:\mathbb{D}\to\mathbb{C}^n$. The result can be considered a ge…

We give useful criteria for S_1 singularities in the Mond classification table, and cuspidal S_k singularities. As applications, we give a simple proof of a result given by Mond and a characterization of cuspidal S_k singularities for the composition of a cuspidal edge and a fold map indicated by Arnol'd for the case k…

Let $(M,ω)$ be a Kähler manifold and let $K$ be a compact group that acts on $M$ in a Hamiltonian fashion. We study the action of $K^\mathbb{C}$ on probability measures on $M$. First of all we identify an abstract setting for the momentum mapping and give numerical criteria for stability, semi-stability and polystabili…

Establishing criteria for top cell inertness in complexes.

The study examines Nash equilibria in utility maximization games with multiplicative performance criteria.

We consider immersions of a Riemann surface into a manifold with $G_2$-holonomy and give criteria for them to be conformal and harmonic, in terms of an associated Gauss map.

For analytic functions in the unit disk, general bounds on the Schwarzian derivative in terms of Nehari functions are shown to imply uniform local univalence and in some cases finite and bounded valence. Similar results are obtained for the Weierstrass--Enneper lifts of planar harmonic mappings to their associated mini…

For a nonconstant holomorphic map between projective Riemann surfaces with conformal metrics, we consider invariant Schwarzian derivatives and projective Schwarzian derivatives of general virtual order. We show that these two quantities are related by the "Schwarzian derivative" of the metrics of the surfaces (at least…

The paper develops stability criteria for real reductive Lie groups acting on manifolds.

Solves complex Hessian equations in unstable cases, proving unique canonical solutions with singularities.

We present a complete set of criteria for determining A-types of plane-to-plane map-germs of corank one with A-codimension <7, which provides a new insight into the A-classification theory from the viewpoint of recognition problem. As an application to generic differential geometry, we discuss about projections of smoo…

Consider a closed non-degenerate 3-form $ω$ with an infinitesimal action of a Lie algebra $\mathfrak{g}$. Motivated by the fact that the observables associated to $ω$ form a Lie 2-algebra, we introduce homotopy moment maps defined on a Lie 2-algebra rather than just on the Lie algebra $\mathfrak{g}$. We formulate exist…

The paper characterizes gaps in minimal foliations on tori using energy criteria.

Criteria for embedding simplicial complexes into manifolds, reducing a topological problem to algebra.

For a holomorphic one-form $\mathbfξ$ on a weakly 1-complete manifold $X$ with certain properties, we discussed the connectivity of the pair $(\hat{X}, F^{-1}(z))$, where $π: \hat{X} \to X$ is a covering map and $dF=π^*\mathbfξ$. We also discussed the criteria about when such a manifold $X$ admits a proper holomorphic …

Let SL(2, $\mathbb H$) be the group of $2 \times 2$ quaternionic matrices $A=\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ with quaternionic determinant $\det A=|ad-aca^{-1} b|=1$. This group acts by the orientation-preserving isometries of the five dimensional (real) hyperbolic space. We obtain discreteness criteria f…

A general criterion in terms of the Schwarzian derivative is given for global univalence of the Weierstrass--Enneper lift of a planar harmonic mapping. Results on distortion and boundary regularity are also deduced. Examples are given to show that the criterion is sharp. The analysis depends on a generalized Schwarzian…

Unified perspective unites Bayesian optimization and active learning for efficient goal-oriented optimization.

We investigate modular embeddings for semi-arithmetic Fuchsian groups. First we prove some purely algebro-geometric or even topological criteria for a regular map from a smooth complex curve to a quaternionic Shimura variety to be covered by a modular embedding. Then we set up an adelic formalism for modular embeddings…

Timelike minimal surfaces in Lorentzian Heisenberg group have singular points.

The paper studies degenerations of rational maps and their limits as geometrically finite rational maps.

In generalized complex geometry, we revisit linear subspaces and submanifolds that have an induced generalized complex structure. We give an expression of the induced structure that allows us to deduce a smoothness criteria, we dualize the results to submersions and we make a few comments on generalized complex mapping…

We say that a cover of surfaces S -> X has the Birman--Hilden property if the subgroup of the mapping class group of X consisting of mapping classes that have representatives that lift to S embeds in the mapping class group of S modulo the group of deck transformations. We identify one necessary condition and one suffi…

We obtain a criterion for approximability by embeddings of piecewise linear maps of a circle to the plane, analogous to the one proved by Minc for maps of a segment to the plane. Theorem. Let S be a triangulation of a circle with s vertices. Let f be a simplicial map of the graph S to the plane. The map f is approximab…

Minimal surfaces in 4D space are stable if their Gauss map spherical area is less than 2π.

The paper studies connectivity of Schur-Horn map images in real Grassmannians.

Study aspherical manifolds with boundaries, proving homological criteria.

Recently ({\em Class. Quant. Grav.} {\bf 20} 625-664) the concept of {\em causal mapping} between spacetimes --essentially equivalent in this context to the {\em chronological map} one in abstract chronological spaces--, and the related notion of {\em causal structure}, have been introduced as new tools to study causal…

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…

Cyclotomic polynomials help classify mapping classes on surfaces.

The paper studies conditions for the non-existence of Cannon-Thurston maps in hyperbolic groups.

The paper derives an equation linking WAIC and WBIC for singular models.

When can a map between manifolds be deformed away from itself? We describe a (normal bordism) obstruction which is often computable and in general much stronger than the classical primary obstruction in cohomology. In particular, it answers our question completely in a large dimension range. As an illustration we give …

Upper bounds on map degrees for various manifold types.

The study reveals flaws in pruning criteria and proposes a new assumption for better filter selection.

We study generalized complex structures and $T$-duality (in the sense of Bouwknegt, Evslin, Hannabuss and Mathai) on Lie algebras and construct the corresponding Cavalcanti and Gualtieri map. Such a construction is called "Infinitesimal $T$-duality". As an application we deal with the problem of finding symplectic stru…

New method finds optimal hyperparameters for multiple tasks and criteria.

Study on unfolding maps of surfaces in 3D space, proving versality conditions.

New criteria for Heegaard splittings ensure strong irreducibility and finite Goeritz groups.