In this paper four classes of partial linear complex vector functional equations are solved, namely the general derived cyclic functional equations, the generalized paracyclic functional equations, the general semicyclic functional equation and the general special cyclic functional equation, as well a more general class of functional equations.

An Otsuki connection $\mathrm{\Gamma }$ is a cross section of the bundle $T(M)\otimes {\mathit{Ɗ}}^2(M)$ and they can be undertood as generalized objects of the affine connections. Briefly, the difference between these two theories is that an Otsuki connection $\mathrm{\Gamma }$ is an affine connection if and only if the principal part $\text{λ}(\mathrm{\Gamma })$ of $\mathrm{\Gamma }$, which is a homomorphism of the tangent bundle $T(M)$, is the identity map. We consider some special class $\mathrm{\Gamma }(\mathrm{\Psi },G)$ of $\mathrm{\Gamma }$. Using $\mathrm{\Gamma }(\mathrm{\Psi },G)$, this paper presents universal expansion-like models which are exact solutions of some partial differential equations.

Existence results of multiple solutions are obtained under suitable conditions for impulsive boundary value problems on the half-line in Banach spaces which may be singular at the boundary.

By a semigroup we mean a submonoid of a torsion-free group in this paper. We investigate some behaviour of prime ideals among semigroups and their oversemigroups. In particular, we consider a oversemigroup $T$ of a valuation semigroup $S$ such that the set of prime ideals $P$ of $S$ satisfying the property that $P+T\underset{\ne }{\subset }T$ equals to $\text{S}\text{p}\text{e}\text{c}(T)$. Also, we study some conditions that a valuation semigroup is completely integrally closed. We give some examples of a valuation semigroup of dimension two and a completely integrally closed semigroup.

We investigate the topology of convergence to limit sets for one-dimensional linear cellular automata using ${\mathbb{Z}}_{p^r}$-valued upper semi-continuous functions, where $p$ is a prime integer and $r\in \mathbb{N}$.

Sufficient conditions on the weights $u(.)$ and $v(.)$ are given in order that a homogeneous singular integral operator is bounded from the weighted Lebesgue space $L^p(v(x)dx)$ to $L^p(u(x)dx)$ for .

In this paper, we give a method to construct an association scheme from another association scheme under some assumptions. To make a new scheme, we construct a skew-symmetric Hadamard matrix.

A graph $G$ is said to be $(k,n)$-factor-critical if $G-S$ has a $k$-factor for any $S\subset V(G)$ with $\left|S\right|=n$. In [7], the author, Ota and Saito conjectured that if $G$ is a 2-connected $(k,n)$-factor-critical graph of order $p$ with ${\sigma }_3(G)\ge \frac{3}{2}(p-n-k)$, then $G$ is hamiltonian with some exceptions. In [7], the author, Ota and Saito also characterized all those graphs which satisfy the assumption of the conjecture, but are not 1-tough and, by using this, they verified the conjecture for $k=1$ and $2$. In this paper, we verify the conjecture for $k=3$ and $4$.

Combining the results in Ikehata-Matsuyama [5] with the Nakao inequality ([6], Lemma 2.2), we will derive more precise decay rate like $E(t)\le C/{(1+t)}^2$ for the total energy $E(t)$ to the mixed problem of the dissipative linear wave equation in an exterior domain through the multiplier method only.

For a graph $G$, a bijection $f$ from $V(G)\cup E(G)$ to $\left\{1,2,\dots ,|V(G)|+|E(G)|\right\}$ is called a super-edge-magic labeling of $G$ if $f(x)+f(y)+f(xy)=C$ is a constant for any $xy\in E(G)$ and $f(V(G))=\left\{1,2,\dots ,|V(G)|\right\}$. For a tree $T$ and a vertex $v$ of $T$, we let $A(T,v)$ denote the set of vertices of $T$ which are at even distance from $v$, and let $B(T,v)$ denote the set of vertices of $T$ which are at odd distance from $v$. A super-edge-magic labeling $f$ of $T$ is said to be consecutive with respect to $v$ if $f(A(T,v))=\left\{1,2,\dots ,\left|A(T,v)\right|\right\}$ and $f(B(T,v))=\left\{\left|A(T,v)\right|+1,\left|A(T,v)\right|+2,\dots ,\left|V(T)\right|\right\}$. In this paper, we prove that for a tree $T_1$. and a vertex $v$ of $T_1$, if $T_1$ has a super-edge-magic labeling consecutive with respect to $v$, then $T=(V(T_1)\cup V(T_2)\cup \{u\},E(T_1)\cup E(T_2)\cup \{uv,u\overline{v}\})$ also has a super-edge-magic labeling consecutive with respect to $v$, where $T_2$ is a tree isomorphic to $T_1(V(T_1)\cap V(T_2)=\cancel{0})$, $\overline{v}$ is the vertex of $T_2$ corresponding to $v$, and $u$ is a vertex which does not belong to $T_1$ or $T_2$.

We classify all pairs $(G,C)$ of a 3-connected graph $G$ of order at least 16 and a longest cycle $C$ of $G$ such that $C$ contains precisely five contractible edges of $G$.

Identities and inequalities are obtained involving $n$-time differentiable functions in terms of evaluations at an interior and at the end points. It is shown how previous work is recaptured as particular instances of the current development. Generalised Taylor type series expansions are obtained and applications to numerical quadrature are demonstrated.

We prove the local well-posedness of the evolution system

$$\{\begin{array}{l}{q}_t+{(q(|q{|}^2+2A))}_x-{(|q{|}^2+2A)}_y+i{q}_{xx}=0\hfill \\ A_x+\frac{1}{2}{(q+\overline{q})}_y=0\hfill \\ q(0,X)=q_o(X)\hfill \end{array}$$

in the analytic Sobolev spaces $X^m(r(T)),m\ge 7$, with the initial compatibility conditions ${(q_o+\overline{q_o})}_y=-2A_{ox}$ and $|q_o{|}^2+2A_o={\psi }_{o_x}$, where $A_o,{\psi }_o\in X^m(r(T))$. This model, due to E. Mjolhus and J. Wyller ([11]), appears in the physical context of Alfvén waves propagating in a parallel (or quasi-parallel) direction to the ambient magnetic field and is a generalisation of the DNLS equation.