arXiv research

A neighborhood homotopy is an equivalence relation on spatial graphs which is generated by crossing changes on the same component and neighborhood equivalence. We give a complete classification of all 2-component spatial graphs up to neighborhood homotopy by the elementary divisor of a linking matrix with respect to th…

Linearizability of singular foliations is preserved under a specific equivalence relation.

Spatial graphs study tangle replacement with equivalence classes.

A treatment in a neighborhood and at a point of the equivalence principle on the basis of derivations of the tensor algebra over a manifold is given. Necessary and sufficient conditions are given for the existence of local bases, called normal frames, in which the components of derivations vanish in a neighborhood or a…

A multibranched surface is a 2-dimensional polyhedron without vertices. We introduce moves for multibranched surfaces embedded in a 3-manifold, which connect any two multibranched surfaces with the same regular neighborhoods in finitely many steps.

The hyperbolization process affects the structure of manifolds.

Data-driven neighborhood definitions and graph constructions are often used in machine learning and signal processing applications. k-nearest neighbor~(kNN) and $ε$-neighborhood methods are among the most common methods used for neighborhood selection, due to their computational simplicity. However, the choice of param…

Paper proves equivalence of two Floer theories using pearly trees and Hamiltonian flows.

In this paper, we introduce a partial order on neighborhood equivalence classes of maximally spread essential multibranched surfaces embedded in a 3-manifold. We show that if a maximally spread essential multibranched surface is atoroidal and acylindrical, then its equivalence class is minimal with respect to the parti…

Necessary and sufficient conditions are investigated for the existence of local bases in which the components of derivations of tensor algebras over differentiable manifold vanish in a neighborhood or only at a single point. The problem when these bases are holonomic or anholonomic is considered. Attention is paid to t…

This paper focuses on the problem of topological equivalence of functions with isolated critical points on the boundary of a compact surface $M$ which are also isolated critical points of their restrictions to the boundary. This class of functions we denote by $Ω(M)$. Firstly, we've obtained the topological classificat…

We study torsion properties of the twisted Alexander modules of the affine complement $M$ of a complex essential hyperplane arrangement, as well as those of punctured stratified tubular neighborhoods of complex essential hyperplane arrangements. We investigate divisibility properties between the twisted Alexander polyn…

This article is one of three highly influential articles on the topology of manifolds written by Robert D. Edwards in the 1970's but never published. Organizers of the Workshops in Geometric Topology (http://www.uwm.edu/~craigg/workshopgtt.htm) with the support of the National Science Foundation have facilitated the pr…

Given a Lie pseudo-group action, an equivariant moving frame exists in the neighborhood of a submanifold jet provided the action is free and regular. For local equivalence problems the freeness requirement cannot always be satisfied and in this paper we show that, with the appropriate modifications and assumptions, the…

We consider functions with isolated critical points on a closed surface. We prove that in a neighborhood of a critical point the function conjugates with Re$z^k$ for the some nonnegative integer k. The full topological invariant of such functions is constructed.

A conjecture of Hirschowitz's predicts that a globally generated vector bundle $W$ on a compact complex manifold $A$ satisfies the formal principle, i.e., the formal neighborhood of its zero section determines the germ of neighborhoods in the underlying complex manifold of the vector bundle $W$. By applying Cartan's eq…

In this article we analyze totally periodic pseudo-Anosov flows in graph three manifolds. This means that in each Seifert fibered piece of the torus decomposition, the free homotopy class of regular fibers has a finite power which is also a finite power of the free homotopy class of a closed orbit of the flow. We show …

Graph representation learning, aiming to learn low-dimensional representations which capture the geometric dependencies between nodes in the original graph, has gained increasing popularity in a variety of graph analysis tasks, including node classification and link prediction. Existing representation learning methods …

Develops GNNs for incomplete graphs, improving learning from missing node attributes.

Generalized complex structures on certain torus bundles are explored.

The present paper is devoted to the problem of (local) geodesic equivalence of Riemannian metrics and sub-Riemannian metrics on generic corank 1 distributions. Using Pontryagin Maximum Principle, we treat Riemannian and sub-Riemannian cases in an unified way and obtain some algebraic necessary conditions for the geodes…

Survey on minimal rational curves and their geometric structures.

New 5-manifold without 'spine' challenges deformation conjecture.

The paper studies graded manifolds and their functorial relationship.

To each isolated critical point of a smooth function on a 3-manifold we put in correspondence a tree (graph without cycles). We will prove that functions are topologically equivalent in the neighborhoods of critical points if and only if the corresponding trees are isomorphic. A complete topological invariant of functi…

Let $N$ be a hyperbolic 3-manifold and $B$ a component of the interior of $AH(π_1(N))$, the space of marked hyperbolic 3-manifolds homotopy equivalent to $N$. We will give topological conditions on $N$ sufficient to give $ρ\in \bar{B}$ such that for every small neighborhood $V$ of $ρ$, $V \cap B$ is disconnected. This …

Consider an analytic map of a neighborhood of 0 in a vector space to a Euclidean space. Suppose that this map takes all germs of lines passing through 0 to germs of circles. Such a map is called rounding. We introduce a natural equivalence relation on roundings and prove that any rounding, whose differential at 0 has r…

PushNet efficiently and adaptively pushes messages in neural networks, improving performance.

In this paper we consider the task of estimating the non-zero pattern of the sparse inverse covariance matrix of a zero-mean Gaussian random vector from a set of iid samples. Note that this is also equivalent to recovering the underlying graph structure of a sparse Gaussian Markov Random Field (GMRF). We present two no…

Unified approach combines prediction-powered inference and variance reduction for semi-supervised optimization.

We derive spectral sequences for the intersection homology of stratified fibrations and approximate tubular neighborhoods in manifold stratified spaces. These neighborhoods include regular neighborhoods in PL stratified spaces.

Making an adaptive prediction based on one's input is an important ability for general artificial intelligence. In this work, we step forward in this direction and propose a semi-parametric method, Meta-Neighborhoods, where predictions are made adaptively to the neighborhood of the input. We show that Meta-Neighborhood…

Study projective connections on surfaces using osculating spaces.

Study geodesics entering a fixed cusp neighborhood multiple times.

Urban2Vec combines street view imagery and POIs for better urban neighborhood embeddings.

Skeleta and other pure subsets of manifold stratified spaces are shown to have neighborhoods which are teardrops of stratified approximate fibrations (under dimension and compactness assumptions). In general, the stratified approximate fibrations cannot be replaced by bundles, and the teardrops cannot be replaced by ma…

New framework distinguishes knots via neighborhood invariants.

This paper tackles selection bias in recommender systems by considering the neighborhood effect.

Maximally hyperbolic solutions contain future neighborhoods of intersecting hypersurfaces.

Let $K$ be a closed polydisc or ball in $\C^n$, and let $Y$ be a quasi projective algebraic manifold which is Zariski locally equivalent to $\C^p$, or a complement of an algebraic subvariety of codimension $\ge 2$ in such manifold. If $r$ is an integer satisfying $(n-r+1) (p-r+1)\geq 2$ then every holomorphic map from …

Many prediction problems can be phrased as inferences over local neighborhoods of graphs. The graph represents the interaction between entities, and the neighborhood of each entity contains information that allows the inferences or predictions. We present an approach for applying machine learning directly to such graph…

Proposes a new NMF method incorporating neighborhood structure for better anomaly detection.

Study on transverse knots and their neighborhoods, proving unique standard neighborhoods and destabilization results.

Revises GNN neighborhood aggregation for more accurate node classification.

The study proves a neighborhood theorem for mean curvature flow in higher dimensions.

GraphAIR improves graph representation learning by capturing non-linear interactions.

A new method for efficient structural node embeddings using Von Neumann entropy.

Neighborhood sampling affects graph neural network training outcomes.