arXiv subjects

Yuta Shimada

Publications and source records attributed to Yuta Shimada.

4 recordsLinked to original sources

arXiv2026

On Angle-optimization and Simplification of Degree-1 Homology Representatives

In topological data analysis, in particular persistent homology analysis, extracting "optimal" representatives for homology classes is crucial for identifying geometric regions of interest. In prior work, optimality is defined in terms of minimizing length or volume. In this work, we restrict our attention to a single homology class in degree $1$ and introduce the total absolute curvature of cycles as the cost function. We show that this cost function, based on angles between edges of cycles, penalizes departures from planarity, convexity, and simple-ness of the cycle representative. We formulate the "angle-optimal homologous cycle problem", recast it as a binary quadratic optimization problem, and show the results of experiments on artificial toy data.

cs.CG↗

arXiv2024

A topological analysis of the space of recipes

In recent years, the use of data-driven methods has provided insights into underlying patterns and principles behind culinary recipes. In this exploratory work, we introduce the use of topological data analysis, especially persistent homology, in order to study the space of culinary recipes. In particular, persistent homology analysis provides a set of recipes surrounding the multiscale "holes" in the space of existing recipes. We then propose a method to generate novel ingredient combinations using combinatorial optimization on this topological information. We made biscuits using the novel ingredient combinations, which were confirmed to be acceptable enough by a sensory evaluation study. Our findings indicate that topological data analysis has the potential for providing new tools and insights in the study of culinary recipes.

math.AT↗

arXiv2021

Affine algebraic super-groups with integral

We generalize to the super context, the known fact that if an affine algebraic group $G$ over a commutative ring $k$ acts freely (in an appropriate sense) on an affine scheme $X$ over $k$, then the dur sheaf $X\tilde{\tilde{/}}G$ of $G$-orbits is an affine scheme in the following two cases: (I) $G$ is finite; (II) $k$ is a field, and $G$ is linearly reductive. An emphasize is put on the more difficult generalization in the second case; the replaced assumption then is that an affine algebraic super-group $G$ over an arbitrary field has an integral. Those super-groups which satisfy the assumption are characterized, and are seen to form a large class if $\operatorname{char}k=0$. Hopf-algebraic techniques including bosonization are applied to prove the results.

math.AG↗

arXiv2020

Twisted forms of differential Lie algebras over $\mathbb{C}(t)$ associated with complex simple Lie algebras

Discussed here is descent theory in the differential context where everything is equipped with a differential operator. To answer a question personally posed by A. Pianzola, we determine all twisted forms of the differential Lie algebras over $\mathbb{C}(t)$ associated with complex simple Lie algebras. Hopf-Galois Theory, a ring-theoretic counterpart of theory of torsors for group schemes, plays a role when we grasp the above-mentioned twisted forms from torsors.

math.RA↗