A new proof of the one-dimensional Carlson inequality of discrete and integral type is introduced. Also, a new proof of the multidimensional Carlson inequality of integral type is presented. Additionally, the duality between the discrete and integral type of Carlson’s inequality are explored.

For a graph , a bijection $f$ from $V\left(G\right){\cup }^{\text{}}E\left(G\right)$ to $\left\{1,2,\cdots ,\left|V\left(G\right)\right|+\left|E\left(G\right)\right|\right\}$ is called an edge-magic labeling of $\text{Θ}(x)$ if $f\left(u\right)+f\left(v\right)+f\left(uv\right)$ is independent on the choice of the edge $uv$. An edge-magic labeling is called super edge-magic if $f\left(V\left(G\right)\right)=\left\{1,2,\cdots ,\left|V\left(G\right)\right|\right\}$. A graph $G$ is called edge-magic (resp. super edge-magic) if there exists an edge-magic (resp. super edge-magic) labeling of $G$. In this paper, we investigate whether several families of graphs are (super) edge-magic or not. We also give several conjectures.

In this paper we consider the local existence to the Cauchy problem for nonlinear Schrödinger equations with power nonlinearities

$$(*)\{\begin{array}{l}i{\partial }_{t}u+\frac{1}{2}\mathrm{\Delta }u=\mathcal{N}(u,\nabla u,\overline{u},\nabla \overline{u}),(t,x)\in \mathbf{R}\times {\mathbf{R}}^n,\hfill \\ u(0,x)=u_0(x),x\in {\mathbf{R}}^n,\hfill \end{array}$$

where $n\ge 2$ and

$$\mathcal{N}=\mathcal{N}(u,w,\overline{u},\overline{w})={\displaystyle \sum }_{l_0\le \left|\alpha \right|+\left|\beta \right|+\left|\gamma \right|\le l_1}{\lambda }_{\alpha \beta \gamma }{u}^{{\alpha }_1}{\overline{u}}^{{\alpha }_2}{\displaystyle \prod }_{j=1}^n{(w_j)}^{{\beta }_j}{\displaystyle \prod }_{k=1}^n{({\overline{w}}_k)}^{{\gamma }_k}$$

With $w={\left(w_j\right)}_{1\le j\le n},{\text{λ}}_{\alpha \beta \gamma }\in \mathbf{\text{C}},l_0\in \mathbf{\text{N}},l_1,l_0\ge 2$. Classical energy method is useful to show local existence in time of solutions to $(*)$ when ${\partial }_w\mathcal{N}$ is pure imaginary (see, [10, 14-16]), and in this case it is known that there exists a unique solution if $u_0\in H^{\left[\frac{n}{2}\right]+3,0}$ (see [10]), where . However, if ${\partial }_w\mathcal{N}$ is not pure imaginary, there are only a few results [2,12,13] that require higher order Sobolev spaces compared with [10, 14-16] because the classical energy method does not work for the problem. Our purpose in this paper is to show local existence in time of solutions to $\text{RR} :\frac{A,Г\to B}{\square A,\square Г\to \square B}$ in the weighted Sobolev space $H^{\left[\frac{n}{2}\right]+6,0}{\cap }^{\text{}}{H}^{\left[\frac{n}{2}\right]+3,2}$ without any size restriction on the data. Our function spaces are more natural than those used in [2,12,13].

In a pseudo-Riemannian manifold, we consider curves through a fixed pseudo-Riemannian submanifold. The first variation formula and the second variation formula of a reflecting geodesic are obtained. Moreover, we study the index form and conjugate points for a reflecting geodesic. Variation formulae for energy are also considered.

It is known that a totally umbilical affine immersion of general codimension into an affine space is affinely congruent to a graph immersion or a centro-affine immersion. In this paper, we shall investigate a more detailed property of such an immersion.

This paper is a continuation of the work in [RS], where we studied Demazure operators for the imprimitive complex reflection group $\tilde{W}=G\left(e,1,n\right)$ and constructed a homogeneous basis of the coinvariant algebra $S_{\tilde{W}}$. In this paper, we study a similar problem for the reflection subgroup $W=G\left(e,e,n\right)$ of $\tilde{W}$. We prove, by assuming certain conjectures, that the operators ${\text{Δ}}_w\left(w\in W\right)$ are linearly independent over the symmetric algebra $S\left(V\right)$. We define a graded space $H_W$ in terms of Demazure operators, and we show that the coinvariant algebra $S_W$ is naturally isomorphic to $H_W$. Then we can define a homogeneous basis of $S_W$ parametrized by $w\in W$.

In this paper, we first introduce the notions of nonlinear mathematical expectations and nonlinear martingales via backward stochastic differential equations(BSDEs) introduced by Duffie & Epstein and Skiadas. And then, we prove a general nonlinear decomposition theorem. Our decomposition theorem generalizes Doob-Meyer decomposition theorem.

We find necessary conditions and sufficient conditions on weights $u(.)$ and $v(.)$ for which the fractional integral operator $I_{\alpha }$ is bounded from the weighted Lebesgue spaces $L_v^p$ into $L_u^q$ whenever and . Actually such a boundedness is characterized for a large class of weights.