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Kotaro Yamada

Publications and source records attributed to Kotaro Yamada.

At least 19 recordsLinked to original sources
arXiv2025

Explicit analytic functions defining the images of wave-front singularities

We give explicit real-analytic functions whose zero sets characterize the images of the standard maps of wave-front singularities. Such functions are realizations of the main-analytic sets in the sense of Ishikawa-Koike-Shiota (1984). More concretely, a subset of Euclidean space is called a global main-analytic set if it can be described, up to a set of smaller Hausdorff dimension, as part of the zero set of a single real-analytic function, referred to as its main-analytic function. In this paper, we propose a general framework for constructing main-analytic functions by a method based on explicit resultant computations. In particular, we provide explicit formulas for the main-analytic functions associated with the standard maps of wave-front singularities of types A, D and E.

math.DG↗
arXiv2025

Development of Automated Data Quality Assessment and Evaluation Indices by Analytical Experience

The societal need to leverage third-party data has driven the data-distribution market and increased the importance of data quality assessment (DQA) in data transactions between organizations. However, DQA requires expert knowledge of raw data and related data attributes, which hinders consensus-building in data purchasing. This study focused on the differences in DQAs between experienced and inexperienced data handlers. We performed two experiments: The first was a questionnaire survey involving 41 participants with varying levels of data-handling experience, who evaluated 12 data samples using 10 predefined indices with and without quality metadata generated by the automated tool. The second was an eye-tracking experiment to reveal the viewing behavior of participants during data evaluation. It was revealed that using quality metadata generated by the automated tool can reduce misrecognition in DQA. While experienced data handlers rated the quality metadata highly, semi-experienced users gave it the lowest ratings. This study contributes to enhancing data understanding within organizations and promoting the distribution of valuable data by proposing an automated tool to support DQAs.

cs.HC↗
arXiv2024

Deformations of swallowtails in a 3-dimensional space form

This is a continuation of the authors' earlier work on deformations of cuspidal edges. We give a representation formula for swallowtails in the Euclidean 3-space. Using this, we investigate map germs of generic swallowtails in 3-dimensional space from, and show some important properties of them. In particular, we give a representation formula giving all map germs of swallowtails in the Euclidean 3-space whose Gaussian curvatures are bounded from below by a positive constant or by a negative constant from above. Using this, we show that any swallowtails are deformed into a swallowtail of constant Gaussian curvature preserving the sign of their Gaussian curvatures.

math.DG↗
arXiv2024

Null hypersurfaces as wave fronts in Lorentz-Minkowski space

In this paper, we show that ``$L$-complete null hypersurfaces'' (i.e. ruled hypersurfaces foliated by entirety of light-like lines) as wave fronts in the $(n+1)$-dimensional Lorentz-Minkowski space are canonically induced by hypersurfaces in the $n$-dimensional Euclidean space. As an application, we show that most of null wave fronts can be realized as restrictions of certain $L$-complete null wave fronts. Moreover, we determine $L$-complete null wave fronts whose singular sets are compact.

math.DG↗
arXiv2022

Analytic extensions of constant mean curvature one geometric catenoids in de Sitter 3-space

We show that a certain simply-stated notion of "analytic completeness" of the image of a real analytic map implies the map admits no analytic extension. We also give a useful criterion for that notion of analytic completeness by defining arc-properness of continuous maps, which can be considered as a very weak version of properness. As an application, we judge the analytic completeness of a certain class of constant mean curvature surfaces (the so-called "G-catenoids") or their analytic extensions in the de Sitter 3-space.

math.DG↗
arXiv2021

A generalization of Zakalyukin's lemma, and symmetries of surface singularities

Zakalyukin's lemma asserts that the coincidence of the images of two wave front germs implies the right equivalence of corresponding map germs under a certain genericity assumption. The purpose of this paper is to give an improvement of this lemma for frontals. Moreover, we give several applications for singularities on surfaces.

math.DG↗
arXiv2021

Symmetries of cross caps

It is well-known that cross caps on surfaces in the Euclidean 3-space can be expressed in Bruce-West's normal form, which is a special local coordinate system centered at the singular point. In this paper, we show a certain kind of uniqueness of such a coordinate system. In particular, the functions associated with this coordinate system produce new invariants on cross cap singular points. Using them, we classify the possible symmetries on cross caps.

math.DG↗
arXiv2020

On the existence of four or more curved foldings with common creases and crease patterns

Consider an oriented curve $Γ$ in a domain $D$ in the plane $\boldsymbol R^2$. Thinking of $D$ as a piece of paper, one can make a curved folding in the Euclidean space $\boldsymbol R^3$. This can be expressed as the image of an "origami map" $Φ:D\to \boldsymbol R^3$ such that $Γ$ is the singular set of $Φ$, the word "origami" coming from the Japanese term for paper folding. We call the singular set image $C:=Φ(Γ)$ the crease of $Φ$ and the singular set $Γ$ the crease pattern of $Φ$. We are interested in the number of origami maps whose creases and crease patterns are $C$ and $Γ$, respectively. Two such possibilities have been known. In the authors' previous work, two other new possibilities and an explicit example with four such non-congruent distinct curved foldings were established. In this paper, we determine the possibility of the number $N$ of congruence classes of curved foldings with the same crease and crease pattern. As a consequence, if $C$ is a non-closed simple arc, then $N=4$ if and only if both $Γ$ and $C$ do not admit any symmetries. On the other hand, when $C$ is a closed curve, there are infinitely many distinct possibilities for curved foldings with the same crease and crease pattern, in general.

math.DG↗
arXiv2020

Duality on generalized cuspidal edges preserving singular set images and first fundamental forms

In the second, fourth and fifth authors' previous work, a duality on generic real analytic cuspidal edges in the Euclidean 3-space $\boldsymbol R^3$ preserving their singular set images and first fundamental forms, was given. Here, we call this an `isometric duality'. When the singular set image has no symmetries and does not lie in a plane, the dual cuspidal edge is not congruent to the original one. In this paper, we show that this duality extends to generalized cuspidal edges in $\boldsymbol R^3$, including cuspidal cross caps, and $5/2$-cuspidal edges. Moreover, we give several new geometric insights on this duality.

math.DG↗
arXiv2020

Curved foldings with common creases and crease patterns

Consider a curve $Γ$ in a domain $D$ in the plane $\boldsymbol R^2$. Thinking of $D$ as a piece of paper, one can make a curved folding $P$ in the Euclidean space $\boldsymbol R^3$. The singular set $C$ of $P$ as a space curve is called the crease of $P$ and the initially given plane curve $Γ$ is called the crease pattern of $P$. In this paper, we show that in general there are four distinct non-congruent curved foldings with a given pair consisting of a crease and crease pattern. Two of these possibilities were already known, but it seems that the other two possibilities (i.e. four possibilities in total) are presented here for the first time.

math.DG↗
arXiv2020

Cuspidal edges with the same first fundamental forms along a knot

Letting $C$ be a compact $C^ω$-curve embedded in $\boldsymbol R^3$ ($C^ω$ means real analyticity), we consider a $C^ω$-cuspidal edge $f$ along $C$. When $C$ is non-closed, in the authors' previous works, the local existence of three distinct cuspidal edges along $C$ whose first fundamental forms coincide with that of $f$ was shown, under a certain reasonable assumption on $f$. In this paper, if $C$ is closed, that is, $C$ is a knot, we show that there exist infinitely many cuspidal edges along $C$ having the same first fundamental form as that of $f$ such that their images are non-congruent to each other, in general.

math.DG↗
arXiv2020

Isometric deformations of wave fronts at non-degenerate singular points

Cuspidal edges and swallowtails are typical non-degenerate singular points on wave fronts in the Euclidean $3$-space. Their first fundamental forms belong to a class of positive semi-definite metrics called "Kossowski metrics". A point where a Kossowski metric is not positive definite is called a singular point or a semi-definite point of the metric. Kossowski proved that real analytic Kossowski metric germs at their non-parabolic singular points(the definition of "non-parabolic singular point" is stated in the introduction here) can be realized as wave front germs (Kossowski's realization theorem). On the other hand, in a previous work with K. Saji, the third and the fourth authors introduced the notion of "coherent tangent bundle". Moreover, the authors, with M. Hasegawa and K. Saji, proved that a Kossowski metric canonically induces an associated coherent tangent bundle. In this paper, we shall explain Kossowski's realization theorem from the viewpoint of coherent tangent bundles. Moreover, as refinements of it, we give a criterion that a given Kossowski metric can be realized as the induced metric of a germ of cuspidal edge (resp. swallowtail or cuspidal cross cap). Several applications of these criteria are given. Also, some remaining problems on isometric deformations of singularities of analytic maps are given at the end of this paper.

math.DG↗
arXiv2020

Hypersurfaces with Light-Like Points in a Lorentzian Manifold II

In the authors' previous work, it was shown that if a zero mean curvature $C^4$-differentiable hypersurface in an arbitrarily given Lorentzian manifold admits a degenerate light-like point, then the hypersurface contains a light-like geodesic segment passing through the point. The purpose of this paper is to point out that the same conclusion holds with just $C^3$-differentiability of the hypersurfaces.

math.DG↗
arXiv2019

Bernstein-type theorem for zero mean curvature hypersurfaces without time-like points in Lorentz-Minkowski space

Calabi and Cheng-Yau's Bernstein-type theorem asserts that an entire zero mean curvature graph in Lorentz-Minkowski $(n+1)$-space $\boldsymbol R^{n+1}_1$ which admits only space-like points is a hyperplane. Recently, the third and fourth authors proved a line theorem for hypersurfaces at their degenerate light-like points. Using this, we give an improvement of the Bernstein-type theorem, and we show that an entire zero mean curvature graph in $\boldsymbol R^{n+1}_1$ consisting only of space-like or light-like points is a hyperplane. This is a generalization of the first, third and fourth authors' previous result for $n=2$.

math.DG↗
arXiv2019

Space-like maximal surfaces containing entire null lines in Lorentz-Minkowski 3-space

Consider a surface $S$ immersed in the Lorentz-Minkowski 3-space $\boldsymbol R^3_1$. A complete light-like line in $\boldsymbol R^3_1$ is called an entire null line on the surface $S$ in $\boldsymbol R^3_1$ if it lies on $S$ and consists of only null points with respect to the induced metric. In this paper, we show the existence of embedded space-like maximal graphs containing entire null lines. If such a graph is defined on a convex domain in $\boldsymbol R^2$, then it must be a light-like plane. Our example is critical in the sense that it is defined on a certain non-convex domain.

math.DG↗
arXiv2019

Improvement of the Bernstein-type theorem for space-like zero mean curvature graphs in Lorentz-Minkowski space using fluid mechanical duality

Calabi's Bernstein-type theorem asserts that a zero mean curvature entire graph in Lorentz-Minkowski space $\boldsymbol L^3$ which admits only space-like points is a space-like plane. Using the fluid mechanical duality between minimal surfaces in Euclidean 3-space $\boldsymbol E^3$ and maximal surfaces in Lorentz-Minkowski space $\boldsymbol L^3$, we give an improvement of this Bernstein-type theorem. More precisely, we show that a zero mean curvature entire graph in $\boldsymbol L^3$ which does not admit time-like points (namely, a graph consists of only space-like and light-like points) is a plane.

math.DG↗
arXiv2018

Hypersurfaces with light-like points

Consider a constant mean curvature immersion $F:U(\subset \boldsymbol{R}^n)\to M$ into an arbitrary Lorentzian $(n+1)$-manifold $M$. A point $o\in U$ is called a light-like point if the first fundamental form $ds^2$ of $F$ degenerates at $o$. We denote by $B_F$ the determinant function of the symmetric matrix associated to $ds^2$ with respect to a local coordinate system at $o$. A light-like point $o$ is said to be degenerate if the exterior derivative of $B_F$ vanishes at $o$. We show that if $o$ is a degenerate light-like point, then the image of $F$ contains a light-like geodesic segment of $M$ passing through $f(o)$ (cf.\ Theorem E). This explains why several known examples of constant mean curvature hypersurface in the Lorentz-Minkowski $(n+1)$-space form $\boldsymbol{R}^{n+1}_1$ contain light-like lines on their sets of light-like points, under a suitable regularity condition of $F$. Several related results are also given.

math.DG↗