Let $G$ be a finite graph, with $V(G)$ and $E(G)$ the vertex-set and edge-set of $G$, respectively. An edge-magic total labeling is a one-to-one mapping $f$ from $V(G)\cup E(G)$ onto $\{1,2,3,\cdots ,|V(G)|+|E(G)\left|\right\}$ such that there exists a constant $c$ satisfying $f(u)+f(uv)+f(v)=c$, for each $uv\in E(G)$. Such a labeling is called a super edge-magic total labeling if all vertices of $G$ receive all smallest labels. In this paper, we consider (super) edge-magic total labeling for subdivision of a star $K_{1,3}$.

In this paper, we consider the simultaneous confidence procedure for multiple comparisons with a control among mean vectors from the multivariate normal distributions. Seo[9] proposed a conservative simultaneous confidence procedure for multiple comparisons with a control. Further, Seo[9] conjectured that this procedure always yields the conservative simultaneous confidence intervals. In this paper, we give the affirmative proof of this conjecture in the case of four mean vectors. We also give the upper bound for the conservativeness of the procedure. Finally, numerical results by Monte Carlo simulation are given.

Let $\delta (G)$ denote the minimum degree of a graph $G$. We prove that for $t\ge 4$ and $k\ge 2$, a graph $G$ of order at least $(t+1)k+\frac{11}{6}{t}^2$ with $\delta (G)\ge k+t-1$ contains $k$ pairwise vertex-disjoint copies of $K_{\mathrm{1,}t}$.

Let $\mathrm{\Delta }$ be a separable algebra over a commutative ring $R$ and $f(x)$ a monic polynomial over the center of $\mathrm{\Delta }$. We deal with the $R$-algebra $\text{Λ}=\mathrm{\Delta }\text{Γ}/(f(X^s))$, where $\mathrm{\Delta }\text{Γ}$ is the path algebra of the cyclic quiver $\text{Γ}$ with $s$ vertices and $s$ arrows, and $X$ is the sum of all arrows. We show that $\text{Λ}$ has a periodic projective bimodule resolution of period 2. Moreover, by using the resolution, we describe the structure of the Hochschild cohomology ring of $\text{Λ}$ by means of the Yoneda product.

In this paper we study warped product CR-submanifolds in a nearly Kaehler manifold and extend the results of B.Y. Chen [7] concerning warped product CR-submanifolds in Kaehler manifolds to this more general setting.

Let $A=K\text{Γ}/(X^k)$, where $K\text{Γ}$ is the path algebra of a cyclic quiver $\text{Γ}$ over a field $K,X$ is the sum of all arrows of $\text{Γ}$ and $k$ is a positive integer. In this paper, we describe the ring structure of the generalized Yoneda algebra ${\oplus }_{i\ge 0}{\text{Ext}}_A^i(A/J^l,A/J^l)$ of $A$ with multiplication given by the Yoneda product, where $J$ denotes the Jacobson radical of $A$ and $l$ is a positive integer with $l\le k$.

Let $G$ be a finite group with $O(G)=1$. We show that if $G$ contains a subgroup $A$ isomorphic to $E_{64}$ such that $C_G(A)=O(N_G(A))\times A$, $N_G(A)/C_G(A)\cong Sp(6,2)$, and $N_G(A)/O(N_G(A))$ splits over $(O(N_G(A))A)/O(N_G(A))$, then either $G\cong M(22)$ or $G=N_G(A)$.

We show some expressions of double and triple sine functions. Then we apply the results to special values of Dirichlet $L$-functions and $\zeta (3)$.

It is known that the image of a horizontal geodesic under a Riemannian submersion is a geodesic. However, in general the image of a geodesic under a Riemannian submersion is not a geodesic. In this paper, we define a Sasaki-Kähler submersion from a Sasakian manifold onto a Kähler manifold, and show that the image of the motion of a charged particle is the motion of a charged particle. In particular, the image of a geodesic is the motion of a charged particle under a Sasaki-Kähler submersion. A Sasaki-Kähler submersion is a kind of Riemannian submersion.

Let $t$, $k$ be integers with $t\ge 3$ and $k\ge 1$. For a graph $G$, a subset $S$ of $V(G)$ with cardinality $k$ is called a $(t,k)$-shredder if $G-S$ consists of $t$ or more components. In this paper, we show that if $t\ge 3$, $2(t-1)\le k\le 3t-5$ and $G$ is a $k$-connected graph of order at least $k^8$, then the number of $(t,k)$-shredders of $G$ is less than or equal to $((2t-1)(|V(G)|-f(|V(G)|)))/(2(t-1)2)$, where $f(n)$ denotes the unique real number $x$ with $x\ge k-1$ such that $n=2{(t-1)}^2{(}_k^x)+x$.

We classify all pairs $(G,C)$ of a $3$-connected graph $G$ and a longest cycle $C$ of $G$ such that $C$ contains precisely four contractible edges of $G$.

Using the theory of fixed point index, this paper presents the existence of positive solutions for the singular second-order initial value problems, where $f(t,y,y^{\prime })$ may be singular at $y=0$ and $y^{\prime }=0$, and $f$ may change sign.