This paper is devoted to the investigation of sampling theory associated with second order eigenvalue problems with an eigenparameter appearing in the boundary conditions. We study two cases. The first is when the eigen-parameter appears linearly in all boundary conditions and the second is when it appears only in one condition. We closely follow the analysis derived by C. T. Fulton (1977) to establish the needed relations for the derivations of the sampling theorems including the construction of Green’s function as well as the eigenfunction expansion theorem. We derive sampling representations for transforms whose kernels are either solutions or Green’s functions.

A graph $G$ is called supermagic if it admits a labelling of the edges by pairwise different consecutive integers such that the sum of the labels of the edges incident with a vertex, the weight of vertex, is independent of the particular vertex. A graph $G$ is called $(a,1)$-antimagic if it admits a labelling of the edges by the integers $\{1,\dots ,|E(G)\left|\right\}$ such that the set of weights of the vertices consists of different consecutive integers. In this paper we will deal with the $(a,1)$-antimagic graphs and their connection to the supermagic graphs. We will introduce three constructions of supermagic graphs using some $(a,1)$-antimagic graphs.

In this note, we shall show the following: Let $R$ be a Noetherian local ring of prime characteristic $p$. If $R$ has a $p$-basis over $R^p$ and $R$ is generically reduced, then $R$ is a regular local ring.

Memory gradient methods are used for unconstrained optimization, especially large scale problems. The first idea of memory gradient methods was proposed by Miele and Cantrell (1969) and subsequently extended by Cragg and Levy (1969). Recently Narushima and Yabe (2006) proposed a new memory gradient method which generates a descent search direction for the objective function at every iteration and converges globally to the solution if the Wolfe conditions are satisfied within the line search strategy. On the other hand, Sun and Zhang (2001) proposed a particular choice of step size, and they applied it to the conjugate gradient method. In this paper, we apply the choice of the step size proposed by Sun and Zhang to the memory gradient method proposed by Narushima and Yabe and establish its global convergence.

Let $p$ be a prime and $\left\{u_n\right\}$ be the $k^{th}$-order Fibonacci sequence whose characteristic polynomial $f(X)$ over $\mathbb{Z}$ is monic, irreducible, separable and satifises deg $(f)=k$. We prove that $f(X)$ divides $X^{p^i-1}-1$ in ${\mathbb{F}}_p[X]$ if and only if $u_{p^i+k-2}\equiv 1(\mathrm{mod}p)$, under the assumption that ${ℒ}_i\ne 0$ and $p\cancel{|}{ℒ}_i$, where ${ℒ}_i$ is a certain well-defined integer associated with the given sequence (see the statement of Theorem 1 and Subsection 3.1 for the precise definition). In particular, in the case $i=1$, this congruence can be regarded as a higher reciprocity law in the sense that it gives a criterion for a prime to split completely in $K=\mathbb{Q}(\alpha )$, where $\alpha$ is some complex root of $f(X)$.

Let $H_{\alpha ,p,V}$ denote the fractional Laplacian ${(-\mathrm{\Delta })}^{\alpha }(\alpha \in (0,1])$ ´acting in $L^p({ℝ}^N)(p\in [1,\infty ))$ perturbed by a potential $V$. We prove spectral inclusion for a large class of potentials, and $L^p$-spectral independence $\sigma (H_{\alpha ,p,V})=\sigma (H_{\alpha ,2,V})(p\in [1,\infty ))$ under a certain condition. In addition, we prove that the spectrum of a perturbed fractional Dirichlet Laplacian acting in $L^p(O)$ is independent of $p\in [1,\infty )$ under a weak condition for potentials, where $O$ is a bounded open subset of ${ℝ}^N$.

The object of the present paper is to study a non-flat quasi-conformally flat Riemannian manifold whose Ricci tensor $S$ satisfies the condition $S(X,Y)=\gamma T(X)T(Y)$, where $\gamma$ is the scalar curvature and $T$ is a 1-form defined by $T(X)=g(X,\xi )$, $\xi$ is a unit vector field.

The purpose of this paper is to study the multipliers on modulation spaces $M^{p,q}({\mathbf{R}}^d)$ for , . In particular, it is shown in the case that elements of and $K\in \mathbf{N})$, consisting of all functions $f\in C^K$ whose derivatives ${\partial }^{\alpha }f\in L^{\infty }$ for any multi-index $\alpha$ such that , are multipliers on $M^{p,q}$.

We choose a canonical transversal bundle of an affine immersion if a pseudo-inverse of the fundamental form exists. In particular, well-known canonical transversal bundles for a non-degenerate surface in ${ℝ}^4$ are generalized to those for a non-degenerate surface in 4-dimensional manifold with torsion-free connection.

We study a subbundle with semi-parallel fundamental form. In particular, if the rank of the fundamental form is maximal, we can obtain a certain equation which plays an essential role to classify parallel affine immersions into ${ℝ}^{n+\frac{1}{2}n(n+1)}$

We show that if $k$ is an integer with $k\ge 3$ and $G$ is a $(2k-1)$-connected graph with radius $r$, then $|V(G)|\ge 2kr-2k-2$.

Without any assumptions on the space dimension or boundedness of the region, we prove $L^p$-spectral independence of generators of $C_0$-semigroups estimated by the positive $C_0$-semigroup . In particular, if the semigroup is self-adjoint in $L^2$, it is shown that only the estimate by $e^{-t{(-\mathrm{\Delta })}^{\alpha }}$ is sufficient for $L^p$-spectral independence. The proof depends on the idea of considering the spectra of the operators and applying the spectral independence result of B.A. Barnes for integral operators, where $A$ is the generator of the semigroup in question.