arXiv subjects

Kazuhiro Kawamura

Publications and source records attributed to Kazuhiro Kawamura.

4 recordsLinked to original sources

arXiv2026

The Intersection Euler Characteristic Profile: Euler Calculus and Stability for Topological Interaction of Ball Unions

The Intersection Euler Characteristic Profile (Intersection ECP) of $k$ colored point clouds $X_1, \ldots, X_k \subset \mathbb{R}^d$ is the Euler characteristic $χ(\bigcap_{i=1}^k \mathcal{U}(X_i; t_i))$ of the overlap of their ball unions---an integer-valued, multiparameter invariant of their topological interaction across scales. Its organizing framework is the Euler calculus on constructible functions: the profile is equally the Euler integral $\int \prod_{i=1}^k \mathbf{1}_{\mathcal{U}(X_i; t_i)} \, dχ$ of the product of the $k$ data-dependent offsets, and this identity---our Intersection Theorem---is a commuting square interchanging geometric intersection and algebraic product. The invariant is rigid-motion invariant, scale-equivariant, and $L^1$-stable, and it is canonical: among pointwise-Euler interaction profiles it is the one forced by separation and normalization, the top floor of a spectrum of descriptors graded by how many clouds meet. For $n$ points a single sorted Alpha-complex sweep computes it in $O(n^{\lceil d/2 \rceil} \log n)$ time with no persistence reduction, worst-case optimal in even dimensions. Where the Euler characteristic cancels, a relative-homology refinement resolves the finer interaction and is stable in the two-parameter interleaving distance. Finally, for increasingly dense samples the profile and its refinement are consistent, recovering the (relative) homology and Euler characteristic of the underlying shapes---in the inverse limit for compact sets, and, under positive reach, persistently and with explicit sample complexity.

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arXiv2026

The Shadow of Vietoris--Rips Complexes in Limits

The Vietoris-Rips complex, denoted $R_β(X)$, of a metric space $(X,d)$ at scale $β$ is an abstract simplicial complex where each $k$-simplex corresponds to $(k+1)$ points of $X$ within diameter $β$. For any abstract simplicial complex $K$ with the vertex set $K^{(0)}$ a Euclidean subset, its shadow, denoted $S(K)$, is the union of the convex hulls of simplices of $K$. This article centers on the homotopy properties of the shadow of Vietoris-Rips complexes $K=R_β(X)$ with vertices from $\mathbb{R}^N$, along with the canonical projection map $ p\colon R_β(X) \to S(R_β(X))$. The study of the geometric/topological behavior of $p$ is a natural yet non-trivial problem. The map $p$ may have many ``singularities'', which have been partially resolved only in low dimensions $N\leq 3$. The obstacle naturally leads us to study systems of these complexes $\{S(R_β(S)) \mid β> 0, S\subset X\}$. We address the challenge posed by singularities in the shadow projection map by studying systems of the shadow complex using inverse system techniques from shape theory, showing that the limit map exhibits favorable homotopy-theoretic properties. More specifically, leveraging ideas and frameworks from Shape Theory, we show that in the limit ``$β\to 0$ and $S \to X$'', the limit map ``$\lim p$'' behaves well with respect to homotopy/homology groups when $X$ is an ANR (Absolute Neighborhood Retract) and admits a metric that satisfies some regularity conditions. This results in limit theorems concerning the homotopy properties of systems of these complexes as the proximity scale parameter approaches zero and the sample set approaches the underlying space (e.g., a submanifold or Euclidean graph). The paper concludes by discussing the potential of these results for finite reconstruction problems in one-dimensional submanifolds.

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arXiv2019

Inverse systems with simplicial bonding maps and cell structures

For a topologically complete space $X$ and a family of closed covers $\mathcal A$ of $X$ satisfying a "local refinement condition" and a "completeness condition," we give a construction of an inverse system $\mathbf{ N}_{\mathcal A}$ of simplicial complexes and simplicial bonding maps such that the limit space $N_{\infty} = \varprojlim \mathbf{N}_{\mathcal A}$ is homotopy equivalent to $X$. A connection with cell structures [2],[3] is discussed

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arXiv2016

Linear continuous surjections of $C_{p}$-spaces over compacta

Let $X$ and $Y$ be compact Hausdorff spaces and suppose that there exists a linear continuous surjection $T:C_{p}(X) \to C_{p}(Y)$, where $C_{p}(X)$ denotes the space of all real-valued continuous functions on $X$ endowed with the pointwise convergence topology. We prove that $\dim X=0$ implies $\dim Y = 0$. This generalizes a previous theorem \cite[Theorem 3.4]{LLP} for compact metrizable spaces. Also we point out that the function space $C_{p}(P)$ over the pseudo-arc $P$ admits no densely defined linear continuous operator $C_{p}(P) \to C_{p}([0,1])$ with a dense image.

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