Yasuhiko Asao
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View article: Classification of metric fibrations
Classification of metric fibrations Open
View article: Magnitude homology and homotopy type of metric fibrations
Magnitude homology and homotopy type of metric fibrations Open
In this article, we show that each two metric fibrations with a common base and a common fiber have isomorphic magnitude homology, and even more, the same magnitude homotopy type. That can be considered as a generalization of a fact proved…
View article: Minimal projective resolution and magnitude homology of geodetic metric spaces
Minimal projective resolution and magnitude homology of geodetic metric spaces Open
Asao-Ivanov showed that magnitude homology is a Tor functor, hence we can compute it by giving a projective resolution of a certain module. In this article, we compute magnitude homology by constructing a minimal projective resolution. As …
View article: Magnitude homology is a derived functor
Magnitude homology is a derived functor Open
We prove that the magnitude (co)homology of an enriched category can, under some technical assumptions, be described in terms of derived functors between certain abelian categories. We show how this statement is specified for the cases of …
View article: Classification of metric fibrations
Classification of metric fibrations Open
In this paper, we study `a fibration of metric spaces' that was originally introduced by Leinster in the study of the magnitude and called metric fibrations. He showed that the magnitude of a metric fibration splits into the product of tho…
View article: Magnitude and magnitude homology of filtered set enriched categories
Magnitude and magnitude homology of filtered set enriched categories Open
In this article, we give a framework for studying the Euler characteristic and its categorification of objects across several areas of geometry, topology and combinatorics. That is, the magnitude theory of filtered sets enriched categories…
View article: Convergence of neural networks to Gaussian mixture distribution
Convergence of neural networks to Gaussian mixture distribution Open
We give a proof that, under relatively mild conditions, fully-connected feed-forward deep random neural networks converge to a Gaussian mixture distribution as only the width of the last hidden layer goes to infinity. We conducted experime…
View article: Magnitude homology and Path homology
Magnitude homology and Path homology Open
In this article, we show that magnitude homology and path homology are closely related, and we give some applications. We define differentials ${\mathrm MH}^{\ell}_k(G) \longrightarrow {\mathrm MH}^{\ell-1}_{k-1}(G)$ between magnitude homo…
View article: Image recognition via Vietoris-Rips complex
Image recognition via Vietoris-Rips complex Open
Extracting informative features from images has been of capital importance in computer vision. In this paper, we propose a way to extract such features from images by a method based on algebraic topology. To that end, we construct a weight…
View article: Curvature of point clouds through principal component analysis
Curvature of point clouds through principal component analysis Open
In this article, we study curvature-like feature value of data sets in Euclidean spaces. First, we formulate such curvature functions with desirable properties under the manifold hypothesis. Then we make a test property for the validity of…
View article: Girth, magnitude homology, and phase transition of diagonality
Girth, magnitude homology, and phase transition of diagonality Open
This paper studies the magnitude homology of graphs focusing mainly on the relationship between its diagonality and the girth. Magnitude and magnitude homology are formulations of the Euler characteristic and the corresponding homology, re…
View article: Geometric approach to graph magnitude homology
Geometric approach to graph magnitude homology Open
In this paper, we introduce a new method to compute the magnitude homology of general graphs.To each direct sum component of the magnitude chain complexes, we assign a pair of simplicial complexes whose simplicial chain complex is isomorph…
View article: Geometric approach to graph magnitude homology
Geometric approach to graph magnitude homology Open
In this paper, we introduce a new method to compute magnitude homology of general graphs. To each direct sum component of magnitude chain complexes, we assign a pair of simplicial complexes whose simplicial chain complex is isomorphic to i…
View article: The loop homology algebra of discrete torsion
The loop homology algebra of discrete torsion Open
We show that Lupercio-Uribe-Xicoténcatl's orbifold loop product and coproduct can be described by a group cohomology class in some cases. By computing this cohomology class, we show that in some cases the orbifold loop product is trivial.
View article: Folding and Punching Paper
Folding and Punching Paper Open
We show how to fold a piece of paper and punch one hole so as to produce any desired pattern of holes.Given n points on a piece of paper (finite polygon or infinite plane), we give algorithms to fold the paper flat so that those n points a…