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Kazunori Matsuda

Publications and source records attributed to Kazunori Matsuda.

At least 19 recordsLinked to original sources
arXiv2026

A New PSF Deconvolution Algorithm: Simultaneous Spatial Resolution Enhancement and Point Source Removal for Morphological Analysis of AGN Host Galaxies

We propose a new point-spread function (PSF) deconvolution algorithm for images of galaxies hosting an active galactic nucleus (AGN), designed to simultaneously enhance the spatial resolution of the host galaxy and remove the bright central point source. In this algorithm, an intrinsic image is reconstructed by decomposing an observed image into two components: an image $I_{\rm sm}$ of an extended component (i.e., a host galaxy) and an image $I_{\rm sp}$ of a point-source component (i.e., an AGN). During image reconstruction, three constraints are imposed: (1) a smooth constraint on the image $I_{\rm sm}$ , which spatially smooths the host-galaxy structures; (2) a sparse constraint on the image $I_{\rm sp}$ , which localizes the point source to a small number of pixels; and (3) a new constraint, the point-source balance constraint, based on the pixel-wise product $I_{\rm sm} \times I_{\rm sp}$ , which removes the point source from the host galaxy without over- or under-subtraction. As a test, we apply this algorithm to images of artificial and $z \sim 0-1$ real AGNs observed with Hyper Suprime-Cam on the Subaru Telescope. We find that the spatial resolution of the host-galaxy images is improved to a level comparable to that of images from the Hubble Space Telescope and that the bright central point sources are removed. This algorithm is expected to enable statistical morphological studies of distant AGN host galaxies when applied to wide-field survey data from the Vera C. Rubin Observatory, the Euclid Space Telescope, and the Roman Space Telescope.

astro-ph.GA↗
arXiv2025

The minimum number of vertices and edges of connected graphs with ind-match$(G) = p$, min-match$(G) = q$ and match$(G) = r$

Let ind-match$(G)$, min-match$(G)$ and match$(G)$ denote the induced matching number, minimum matching number and matching number of a graph $G$, respectively. It is known that ind-match$(G) \leq $ min-match$(G) \leq {\rm match}(G) \leq$ 2min-match$(G)$ holds. In the present paper, we investigate the minimum number of vertices and edges of connencted simple graphs $G$ with ind-match$(G) = p$, min-match$(G) = q$ and ${\rm match}(G) = r$ for pair of integers $p, q, r$ such that $1 \leq p \leq q \leq r \leq 2q$.

math.CO↗
arXiv2022

On the three graph invariants related to matching of finite simple graphs

Let $G$ be a finite simple graph on the vertex set $V(G)$ and let $\text{ind-match}(G)$, $\text{min-match}(G)$ and $\text{match}(G)$ denote the induced matching number, the minimum matching number and the matching number of $G$, respectively. It is known that the inequalities $\text{ind-match}(G) \leq \text{min-match}(G) \leq \text{match}(G) \leq 2\text{min-match}(G)$ and $\text{match}(G) \leq \left\lfloor |V(G)|/2 \right\rfloor$ hold in general. In the present paper, we determine the possible tuples $(p, q, r, n)$ with $\text{ind-match}(G) = p$, $\text{min-match}(G) = q$, $\text{match}(G) = r$ and $|V(G)| = n$ arising from connected simple graphs. As an application of this result, we also determine the possible tuples $(p', q, r, n)$ with ${\rm{reg}}(G) = p'$, $\text{min-match}(G) = q$, $\text{match}(G) = r$ and $|V(G)| = n$ arising from connected simple graphs, where $I(G)$ is the edge ideal of $G$ and ${\rm{reg}}(G) = {\rm{reg}}(K[V(G)]/I(G))$ is the Castelnuovo--Mumford regularity of the quotient ring $K[V(G)]/I(G)$.

math.CO↗
arXiv2021

Inequalities of invariants on Stanley-Reisner rings of Cohen-Macaulay simplicial complexes

The goal of the present paper is the study of some algebraic invariants of Stanley-Reisner rings of Cohen-Macaulay simplicial complexes of dimension $d - 1$. We prove that the inequality $d \leq \mathrm{reg}(Δ) \cdot \mathrm{type}(Δ)$ holds for any $(d-1)$-dimensional Cohen-Macaulay simplicial complex $Δ$ satisfying $Δ=\mathrm{core}(Δ)$, where $\mathrm{reg}(Δ)$ (resp. $\mathrm{type}(Δ)$) denotes the Castelnuovo-Mumford regularity (resp. Cohen-Macaulay type) of the Stanley-Reisner ring $\Bbbk[Δ]$. Moreover, for any given integers $d,r,t$ satisfying $r,t \geq 2$ and $r \leq d \leq rt$, we construct a Cohen-Macaulay simplicial complex $Δ(G)$ as an independent complex of a graph $G$ such that $\dim(Δ(G))=d-1$, $\mathrm{reg}(Δ(G))=r$ and $\mathrm{type}(Δ(G))=t$.

math.AC↗
arXiv2020

Regularity and h-polynomials of binomial edge ideals

Let $G$ be a finite simple graph on the vertex set $[n] = \{ 1, \ldots, n \}$ and $K[X, Y] = K[x_1, \ldots, x_n, y_1, \ldots, y_n]$ the polynomial ring in $2n$ variables over a field $K$ with each $\mathrm{deg} x_i = \mathrm{deg} y_j = 1$. The binomial edge ideal of $G$ is the binomial ideal $J_G \subset K[X, Y]$ which is generated by those binomials $x_iy_j - x_jy_i$ for which $\{i, j\}$ is an edge of $G$. The Hilbert series $H_{K[X, Y]/J_G}(λ)$ of $K[X, Y]/J_G$ is of the form $H_{K[X, Y]/J_G}(λ) = h_{K[X, Y]/J_G}(λ)/(1 - λ)^d$, where $d = \mathrm{dim} K[X, Y]/J_G$ and where $h_{K[X, Y]/J_G}(λ) = h_0 + h_1λ+ h_2λ^2 + \cdots + h_sλ^s$ with each $h_i \in \mathbb{Z}$ and with $h_s \neq 0$ is the $h$-polynomial of $K[X, Y]/J_G$. It is known that, when $K[X, Y]/J_G$ is Cohen-Macaulay, one has $\mathrm{reg}(K[X, Y]/J_G) = \mathrm{deg} h_{K[X, Y]/J_G}(λ)$, where $ \mathrm{reg}(K[X, Y]/J_G)$ is the (Castelnuovo-Mumford) regularity of $K[X, Y]/J_G$. In the present paper, given arbitrary integers $r$ and $s$ with $2 \leq r \leq s$, a finite simple graph $G$ for which $\mathrm{reg}(K[X, Y]/J_G) = r$ and $\mathrm{deg} h_{K[X, Y]/J_G}(λ) = s$ will be constructed.

math.AC↗
arXiv2020

Homological invariants of Cameron--Walker graphs

Let $G$ be a finite simple connected graph on $[n]$ and $R = K[x_1, \ldots, x_n]$ the polynomial ring in $n$ variables over a field $K$. The edge ideal of $G$ is the ideal $I(G)$ of $R$ which is generated by those monomials $x_ix_j$ for which $\{i, j\}$ is an edge of $G$. In the present paper, the possible tuples $(n, {\rm depth} (R/I(G)), {\rm reg} (R/I(G)), \dim R/I(G), {\rm deg} \ h(R/I(G)))$, where ${\rm deg} \ h(R/I(G))$ is the degree of the $h$-polynomial of $R/I(G)$, arising from Cameron--Walker graphs on $[n]$ will be completely determined.

math.AC↗
arXiv2020

The regularity and $h$-polynomial of Cameron-Walker graphs

Fix an integer $n \geq 1$, and consider the set of all connected finite simple graphs on $n$ vertices. For each $G$ in this set, let $I(G)$ denote the edge ideal of $G$ in the polynomial ring $R = K[x_1,\ldots,x_n]$. We initiate a study of the set $\mathcal{RD}(n) \subseteq \mathbb{N}^2$ consisting of all the pairs $(r,d)$ where $r = {\rm reg}(R/I(G))$, the Castelnuovo-Mumford regularity, and $d = {\rm deg} h_{R/I(G)}(t)$, the degree of the $h$-polynomial, as we vary over all the connected graphs on $n$ vertices. In particular, we identify sets $A(n)$ and $B(n)$ such that $A(n) \subseteq \mathcal{RD}(n) \subseteq B(n)$. When we restrict to the family of Cameron-Walker graphs on $n$ vertices, we can completely characterize all the possible $(r,d)$.

math.CO↗
arXiv2020

Matching numbers and dimension of edge ideals

Let $G$ be a finite simple graph on the vertex set $V(G) = \{x_{1}, \ldots, x_{n}\}$ and match$(G)$, min-match$(G)$ and ind-match$(G)$ the matching number, minimum matching number and induced matching number of $G$, respectively. Let $K[V(G)] = K[x_{1}, \ldots, x_{n}]$ denote the polynomial ring over a field $K$ and $I(G) \subset K[V(G)]$ the edge ideal of $G$. The relationship between these graph-theoretic invariants and ring-theoretic invariants of the quotient ring $K[V(G)]/I(G)$ has been studied. In the present paper, we study the relationship between match$(G)$, min-match$(G)$, ind-match$(G)$ and $\dim K[V(G)]/I(G)$.

math.AC↗
arXiv2019

Gorenstein graphic matroids

The toric variety of a matroid is projectively normal, and therefore it is Cohen-Macaulay. We provide a complete graph-theoretic classification when the toric variety of a graphic matroid is Gorenstein.

math.CO↗
arXiv2019

Extremal Betti numbers of edge ideals

Given integers $r$ and $b$ with $1 \leq b \leq r$, a finite simple connected graph $G$ for which ${\rm reg}(S/I(G)) = r$ and the number of extremal Betti numbers of $S/I(G)$ is equal to $b$ will be constructed.

math.AC↗
arXiv2019

Induced matching numbers of finite graphs and edge ideals

Let $G$ be a finite simple graph on the vertex set $V(G) = \{x_1, \ldots, x_n\}$ and $I(G) \subset K[V(G)]$ its edge ideal, where $K[V(G)]$ is the polynomial ring in $x_1, \ldots, x_n$ over a field $K$ with each ${\rm deg} x_i = 1$ and where $I(G)$ is generated by those squarefree quadratic monomials $x_ix_j$ for which $\{x_i, x_j\}$ is an edge of $G$. In the present paper, given integers $1 \leq a \leq r$ and $s \geq 1$, the existence of a finite connected simple graph $G = G(a, r, d)$ with ${\rm im}(G) = a$, ${\rm reg}(R/I(G)) = r$ and ${\rm deg} h_{K[V(G)]/I(G)} (λ) = s$, where ${\rm im}(G)$ is the induced matching number of $G$ and where $h_{K[V(G)]/I(G)} (λ)$ is the $h$-polynomial of $K[V(G)]/I(G)$.

math.AC↗
arXiv2019

Regularity and $a$-invariant of Cameron--Walker graphs

Let $S$ be the polynomial ring over a field $K$ and $I \subset S$ a homogeneous ideal. Let $h(S/I,λ)$ be the $h$-polynomial of $S/I$ and $s = \mathrm{deg} h(S/I,λ)$ the degree of $h(S/I,λ)$. It follows that the inequality $s - r \leq d - e$, where $r = \mathrm{reg} (S/I)$, $d = \dim S/I$ and $e = \mathrm{depth} S/I$, is satisfied and, in addition, the equality $s - r = d - e$ holds if and only if $S/I$ has a unique extremal Betti number. We are interested in finding a natural class of finite simple graphs $G$ for which $S/I(G)$, where $I(G)$ is the edge ideal of $G$, satisfies $s - r = d - e$. Let $a(S/I(G))$ denote the $a$-invariant of $S/I$, i.e., $a(S/I(G)) = s - d$. One has $a(S/I(G)) \leq 0$. In the present paper, by showing the fundamental fact that every Cameron--Walker graph $G$ satisfies $a(S/I(G)) = 0$, a class of Cameron--Walker graphs $G$ for which $S/I(G)$ satisfies $s - r = d - e$ will be exhibited.

math.AC↗
arXiv2018

Regularity and $h$-polynomials of edge ideals

For any two integers $d,r \geq 1$, we show that there exists an edge ideal $I(G)$ such that the ${\rm reg}\left(R/I(G)\right)$, the Castelnuovo-Mumford regularity of $R/I(G)$, is $r$, and ${\rm deg} (h_{R/I(G)}(t))$, the degree of the $h$-polynomial of $R/I(G)$, is $d$. Additionally, if $G$ is a graph on $n$ vertices, we show that ${\rm reg}\left(R/I(G)\right) + {\rm deg} (h_{R/I(G)}(t)) \leq n$.

math.AC↗
arXiv2018

Lexsegment ideals and their h-polynomials

Let $S = K[x_1, \ldots, x_n]$ denote the polynomial ring in $n$ variables over a field $K$ with each $\mathrm{deg}\ x_i = 1$ and $I \subset S$ a homogeneous ideal of $S$ with $\dim S/I = d$. The Hilbert series of $S/I$ is of the form $h_{S/I}(λ)/(1 - λ)^d$, where $h_{S/I}(λ) = h_0 + h_1λ+ h_2λ^2 + \cdots + h_sλ^s$ with $h_s \neq 0$ is the $h$-polynomial of $S/I$. Given arbitrary integers $r \geq 1$ and $s \geq 1$, a lexsegment ideal $I$ of $S = K[x_1, \ldots, x_n]$, where $n \leq \max\{r, s\} + 2$, satisfying $\mathrm{reg}(S/I) = r$ and $ \mathrm{deg}\ h_{S/I}(λ) = s$ will be constructed.

math.AC↗
arXiv2017

Regularity and $h$-polynomials of monomial ideals

Let $S = K[x_1, \ldots, x_n]$ denote the polynomial ring in $n$ variables over a field $K$ with each $°x_i = 1$ and $I \subset S$ a homogeneous ideal of $S$ with $\dim S/I = d$. The Hilbert series of $S/I$ is of the form $h_{S/I}(λ)/(1 - λ)^d$, where $h_{S/I}(λ) = h_0 + h_1λ+ h_2λ^2 + \cdots + h_sλ^s$ with $h_s \neq 0$ is the $h$-polynomial of $S/I$. It is known that, when $S/I$ is Cohen--Macaulay, one has $\reg(S/I) = °h_{S/I}(λ)$, where $\reg(S/I)$ is the (Castelnuovo--Mumford) regularity of $S/I$. In the present paper, given arbitrary integers $r$ and $s$ with $r \geq 1$ and $s \geq 1$, a monomial ideal $I$ of $S = K[x_1, \ldots, x_n]$ with $n \gg 0$ for which $\reg(S/I) = r$ and $°h_{S/I}(λ) = s$ will be constructed. Furthermore, we give a class of edge ideals $I \subset S$ of Cameron--Walker graphs with $\reg(S/I) = °h_{S/I}(λ)$ for which $S/I$ is not Cohen--Macaulay.

math.AC↗
arXiv2017

Weakly closed graphs and F-purity of binomial edge ideals

Herzog-Hibi-Hreindóttir-Kahle-Rauh introduced the class of closed graph and they proved that the binomial edge ideal $J(G)$ of a graph $G$ has quadratic Gröbner bases if $G$ is closed. In this paper, we introduce the class of weakly closed graph as a generalization of the closed graph and prove that the quotient ring $S/J(G)$ is $F$-pure if $G$ is weakly closed. This fact is a generalization of Ohtani's theorem.

math.AC↗
arXiv2016

Nonincreasing depth functions of monomial ideals

Given a nonincreasing function $f : \mathbb{Z}_{\geq 0} \setminus \{ 0 \} \to \mathbb{Z}_{\geq 0}$ such that (i) $f(k) - f(k+1) \leq 1$ for all $k \geq 1$ and (ii) if $a = f(1)$ and $b = \lim_{k \to \infty} f(k)$, then $|f^{-1}(a)| \leq |f^{-1}(a-1)| \leq \cdots \leq |f^{-1}(b+1)|$, a system of generators of a monomial ideal $I \subset K[x_1, \ldots, x_n]$ for which ${\rm depth} S/I^k = f(k)$ for all $k \geq 1$ is explicitly described. Furthermore, we give a characterization of triplets of integers $(n,d,r)$ with $n > 0$, $d \geq 0$ and $r > 0$ with the properties that there exists a monomial ideal $I \subset S = K[x_1, \ldots, x_n]$ for which $\lim_{k \to \infty} {\rm depth} S/I^k = d$ and ${\rm dstab}(I) = r$, where ${\rm dstab}(I)$ is the smallest integer $k_0 \geq 1$ with ${\rm depth} S/I^{k_0} = {\rm depth} S/I^{k_0+1} = {\rm depth} S/I^{k_0+2} = \cdots$.

math.AC↗