arXiv subjects
Publications and source records attributed to Hiroshi Nozaki.
We have the Fisher type inequality and the linear programming bound as upper bounds for the cardinalities of $s$-distance sets on $S^{d-1}$. In this paper, we give a new upper bound for the cardinalities of $s$-distance sets on $S^{d-1}$ for any $s$. This upper bound improves the Fisher typer inequality and is useful for $s$-distance sets which are not applicable to the linear programming bound.
In this paper, we simplify the known switching theorem due to Bose and Shrikhande as follows. Let $G=(V,E)$ be a primitive strongly regular graph with parameters $(v,k,λ,μ)$. Let $S(G,H)$ be the graph from $G$ by switching with respect to a nonempty $H\subset V$. Suppose $v=2(k-θ_1)$ where $θ_1$ is the nontrivial positive eigenvalue of the $(0,1)$ adjacency matrix of $G$. This strongly regular graph is associated with a regular two-graph. Then, $S(G,H)$ is a strongly regular graph with the same parameters if and only if the subgraph induced by $H$ is $k-\frac{v-h}{2}$ regular. Moreover, $S(G,H)$ is a strognly regualr graph with the other parameters if and only if the subgraph induced by $H$ is $k-μ$ regular and the size of $H$ is $v/2$. We prove these theorems with the view point of the geometrical theory of the finite set on the Euclidean unit sphere.
A subset $X$ in the $d$-dimensional Euclidean space is called a $k$-distance set if there are exactly $k$ distinct distances between two distinct points in $X$ and a subset $X$ is called a locally $k$-distance set if for any point $x$ in $X$, there are at most $k$ distinct distances between $x$ and other points in $X$. Delsarte, Goethals, and Seidel gave the Fisher type upper bound for the cardinalities of $k$-distance sets on a sphere in 1977. In the same way, we are able to give the same bound for locally $k$-distance sets on a sphere. In the first part of this paper, we prove that if $X$ is a locally $k$-distance set attaining the Fisher type upper bound, then determining a weight function $w$, $(X,w)$ is a tight weighted spherical $2k$-design. This result implies that locally $k$-distance sets attaining the Fisher type upper bound are $k$-distance sets. In the second part, we give a new absolute bound for the cardinalities of $k$-distance sets on a sphere. This upper bound is useful for $k$-distance sets for which the linear programming bound is not applicable. In the third part, we discuss about locally two-distance sets in Euclidean spaces. We give an upper bound for the cardinalities of locally two-distance sets in Euclidean spaces. Moreover, we prove that the existence of a spherical two-distance set in $(d-1)$-space which attains the Fisher type upper bound is equivalent to the existence of a locally two-distance set but not a two-distance set in $d$-space with more than $d(d+1)/2$ points. We also classify optimal (largest possible) locally two-distance sets for dimensions less than eight. In addition, we determine the maximum cardinalities of locally two-distance sets on a sphere for dimensions less than forty.
The diffusion coefficient of Li$^+$ ions ($D_{\rm Li}$) in the battery material Li$_x$CoO$_2$ has been investigated by muon-spin relaxation ($μ^+$SR). Based on the experiments in zero-field and weak longitudinal-fields at temperatures up to 400 K, we determined the fluctuation rate ($ν$) of the fields on the muons due to their interaction with the nuclear moments. Combined with susceptibility data and electrostatic potential calculations, clear Li$^+$ ion diffusion was detected above $\sim150$K. The $D_{\rm Li}$ estimated from $ν$ was in very good agreement with predictions from first-principles calculations, and we present the $μ^+$SR technique as an % novel and optimal probe to detect $D_{\rm Li}$ for materials containing magnetic ions.
We locate all of the zeros of the Eisenstein series associated with the Fricke groups $Γ_0^{*}(2)$ and $Γ_0^{*}(3)$ in their fundamental domains by applying and expanding the method of F. K. C. Rankin and H. P. F. Swinnerton-Dyer (``{\it On the zeros of Eisenstein series}'', 1970).