The hamburger theorem

We generalize the ham sandwich theorem to $d+1$ measures in $\mathbb{R}^d$ as follows. Let $\mu_1,\mu_2, \dots, \mu_{d+1}$ be absolutely continuous finite Borel measures on $\mathbb{R}^d$. Let $\omega_i=\mu_i(\mathbb{R}^d)$ for $i\in [d+1]$, $\omega=\min\{\omega_i; i\in [d+1]\}$ and assume that $\sum_{j=1}^{d+1} \omega_j=1$. Assume that $\omega_i \le 1/d$ for every $i\in[d+1]$. Then there exists a hyperplane $h$ such that each open halfspace $H$ defined by $h$ satisfies $\mu_i(H) \le (\sum_{j=1}^{d+1} \mu_j(H))/d$ for every $i \in [d+1]$ and $\sum_{j=1}^{d+1} \mu_j(H) \ge \min(1/2, 1-d\omega) \ge 1/(d+1)$. As a consequence we obtain that every $(d+1)$-colored set of $nd$ points in $\mathbb{R}^d$ such that no color is used for more than $n$ points can be partitioned into $n$ disjoint rainbow $(d-1)$-dimensional simplices.Comment: 11 pages, 2 figures; a new proof of Theorem 8, extended concluding remark

Binding numbers and f-factors of graphs

AbstractLet G be a connected graph of order n, a and b be integers such that 1 ≤ a ≤ b and 2 ≤ b, and f: V(G) → {a, a + 1, …, b} be a function such that Σ(f(x); x ∈ V(G)) ≡ 0 (mod 2). We prove the following two results: (i) If the binding number of G is greater than (a + b −1)(n−1)(an−(a + b) + 3) and n ≥(a + b)2a, then G has an f-factor; (ii) If the minimum degree of G is greater than (bn − 2)(a + b), and n ≥(a + b)2a, then G has an f-factor

Balanced intervals of two stes of points on a line or circle

Let n,m, k, h be positive integers such that 1 ≤ n ≤ m, 1 ≤ k ≤ n and 1 ≤ h ≤ m. Then we give a necessary and sufficient condition for every configuration with n red points and m blue points on a line or circle to have an interval containing precisely k red points and h blue points

K1,3K_{1,3}-covering red and blue points in the plane

We say that a finite set of red and blue points in the plane in general position can be $K_{1,3}$-covered if the set can be partitioned into subsets of size $4$, with $3$ points of one color and $1$ point of the other color, in such a way that, if at each subset the fourth point is connected by straight-line segments to the same-colored points, then the resulting set of all segments has no crossings. We consider the following problem: Given a set $R$ of $r$ red points and a set $B$ of $b$ blue points in the plane in general position, how many points of $R\cup B$ can be $K_{1,3}$-covered? and we prove the following results: (1) If $r=3g+h$ and $b=3h+g$, for some non-negative integers $g$ and $h$, then there are point sets $R\cup B$, like $\{1,3\}$-equitable sets (i.e., $r=3b$ or $b=3r$) and linearly separable sets, that can be $K_{1,3}$-covered. (2) If $r=3g+h$, $b=3h+g$ and the points in $R\cup B$ are in convex position, then at least $r+b-4$ points can be $K_{1,3}$-covered, and this bound is tight. (3) There are arbitrarily large point sets $R\cup B$ in general position, with $r=b+1$, such that at most $r+b-5$ points can be $K_{1,3}$-covered. (4) If $b\le r\le 3b$, then at least $\frac{8}{9}(r+b-8)$ points of $R\cup B$ can be $K_{1,3}$-covered. For $r>3b$, there are too many red points and at least $r-3b$ of them will remain uncovered in any $K_{1,3}$-covering. Furthermore, in all the cases we provide efficient algorithms to compute the corresponding coverings.Comment: 29 pages, 10 figures, 1 tabl

特区小学校英語教育の改善課題の解決に関する総合的実証研究

金沢大学学校教育系■平成22年度の実施実績1韓国の小学校英語の実情視察・韓国ソウル市内の三角山中学校を訪問して、小学校英語を受けた成果を検討。小学生に比して中学生が、全体的に受動的な姿勢が強くなっており、担当教員もそのことを自覚している。これまでの調査で、韓国では、小中連携という概念には無関心である傾向が見受けられ、この英語教育上の連携への意識不足は、韓国英語教育の課題点として位置づけられる。そのため、韓国の小学生が意欲的・積極性が認められるが故に返って不自然さがあり、研究注目に値する。・小学校英語教員の力量:小学校における英語教員の配置計画が着実に進行している。担任が実施する段階から英語専門教科の教員が行うという制度にシフトされ、授業中における上級学年の英語運用能力は格段に上がってきている。生徒の発言機会や発言内容に高まりがみられる。2学会における発表と成果の冊子作成・平成18年度から実施してきた質問紙調査や研究協力者会議のまとめとして、日本教育方法学会で、まとめを発表し、研究終了期3月に、冊子としてとりまとめた。・研究課題(1)教育課程について:往復型や積み重ね型など授業時間の不足を補う視点を整理した。研究課題(2)教材について:活動に用いる際に自然さが生かされる視点を整理した。研究課題(3)文字指導について:段階的発展性を考慮した6年間シラバスの作成視点を整理した。3今後の課題・学習意欲や動機の喚起・形成・維持の視点からの授業改善について継続研究が必要。研究課題/領域番号:20652041, 研究期間(年度):2008 – 2010出典：研究課題「特区小学校英語教育の改善課題の解決に関する総合的実証研究」課題番号20652041（KAKEN：科学研究費助成事業データベース（国立情報学研究所）） （https://kaken.nii.ac.jp/ja/grant/KAKENHI-PROJECT-20652041/）を加工して作

Decomposition of a graph into two disjoint odd subgraphs

An odd (resp. even) subgraph in a multigraph is its subgraph in which every vertex has odd (resp. even) degree. We say that a multigraph can be decomposed into two odd subgraphs if its edge set can be partitioned into two sets so that both form odd subgraphs. In this paper we give a necessary and sufficient condition for the decomposability of a multigraph into two odd subgraphs. We also present a polynomial time algorithm for finding such a decomposition or showing its non-existence. We also deal with the case of the decomposability into an even subgraph and an odd subgraph