Related papers: An Approximate Nerve Theorem
A good cover in R^d is a collection of open contractible sets in R^d such that the intersection of any subcollection is either contractible or empty. Motivated by an analogy with convex sets, intersection patterns of good covers were…
This thesis addresses the theory of topological spaces and the foundations of persistence theory. We will discuss chain complexes and the associated simplicial homology groups, as well as their relationship with singular homology theory.…
In comparative and developmental neuroanatomy one encounters questions regarding the deformation of neural tissue under stress. The motivation of this note is an observation (Barbas {\it et al}) that at cortical folds or gyri, the layers of…
Real cortical tissue curves and folds according to experimental data. However, our current simulations only use unfolded cortical layers. This project extends the cortical slice model in order to be able to specify arbitrary curvatures to…
Computational topology is a vibrant contemporary subfield and this article integrates knot theory and mathematical visualization. Previous work on computer graphics developed a sequence of smooth knots that were shown to converge point wise…
When approximating a space curve, it is natural to consider whether the knot type of the original curve is preserved in the approximant. This preservation is of strong contemporary interest in computer graphics and visualization. We…
In this paper, we present finite topological type theorems for open manifolds with non-negative Ricci curvature, under almost maximal local rewinding volume. Unlike previous related research, our theorems remove the constraints of sectional…
The topological morphology descriptor of a neuron is a multiset of intervals associated to the shape of the neuron represented as a tree. In practice, topological morphology descriptors are vectorized using persistence images, which can…
This paper introduces proximal homotopic cycles, which lead to the main results in this paper, namely, extensions of the Mitsuishi-Yamaguchi Good Coverning Theorem with different forms of Tanaka good cover of an Alexandrov space equipped…
We prove a new fractional Helly theorem for families of sets obeying topological conditions. More precisely, we show that the nerve of a finite family of open sets (and of subcomplexes of cell complexes) in R^d is k-Leray where k depends on…
The universal approximation theorem states that a neural network with one hidden layer can approximate continuous functions on compact sets with any desired precision. This theorem supports using neural networks for various applications,…
Given an open cover of a paracompact topological space X, there are two natural ways to construct a map from the cohomology of the nerve of the cover to the cohomology of X. One of them is based on a partition of unity, and is more…
One of the themes in algebraic geometry is the study of the relation between the ``topology'' of a smooth projective variety and a (``general'') hyperplane section. Recent results of Nori produce cohomological evidence for a conjecture that…
Prediction and discovery of new materials with desired properties are at the forefront of quantum science and technology research. A major bottleneck in this field is the computational resources and time complexity related to finding new…
In the last years higher dimensional physics has won importance. Despite the Superstrings, higher dimensional effects, in measurable scales of energy (some TeV), became only possible with Randall-Sundrum's models (RS). In particular, recent…
This article studies the robust version of persistent homology based on trimming methodology to capture the geometric feature through support of the data in presence of outliers. Precisely speaking, the proposed methodology works when the…
In this paper, we give lower bounds for the homology of the fibers of a map to a manifold. Using new sheaf theoretic methods, we show that these lower bounds persist over whole open sets of the manifold, and that they are stable under…
Backpropagation-optimized artificial neural networks, while precise, lack robustness, leading to unforeseen behaviors that affect their safety. Biological neural systems do solve some of these issues already. Unlike artificial models,…
We examine the necessary and sufficient complexity of neural networks to approximate functions from different smoothness spaces under the restriction of encodable network weights. Based on an entropy argument, we start by proving lower…
Granovetter's weak ties theory is a very important sociological theory according to which a correlation between edge weight and the network's topology should exist. More specifically, the neighbourhood overlap of two nodes connected by an…