We discuss exponential families admitting almost complex structures which are parallel relative to an exponential connection (e–connection) or mixture connection (m–connection). The multinomial distribution, negative multinomial distribution and multivariate normal distribution are important examples of the exponential family. We give almost complex structures which are parallel relative to the exponential or mixture connection for these exponential families. Also, we prove spaces of the multinomial distribution and negative multinomial distribution are of constant curvature with respect to the $\alpha$–connection.

We give a new and complete proof of the following theorem, discovered by Detlef Laugwitz: (forward) complete and connected finite dimensional Finsler manifolds admitting a proper homothety are Minkowski vector spaces. More precisely, we show that under these hypotheses the Finsler manifold is isometric to the tangent Minkowski vector space of the fixed point of the homothety via the exponential map of the canonical spray of the Finsler manifold.

The object of the present paper is to study a quarter-symmetric metric connection on a 3-dimensional quasi-Sasakian manifold. The existence of the connection is given on a Riemannian manifold. We deduce the relation between the Riemannian connection and the quarter-symmetric metric connection on a $3$-dimensional quasi-Sasakian manifold. We investigate the curvature tensor and the Ricci tensor of a $3$-dimensional quasi-Sasakian manifold with respect to the quarter-symmetric metric connection. We study the projective curvature tensor with respect to the quarter-symmetric metric connection and also characterized $\xi$−projectively flat and $\varphi$−projectively flat $3$-dimensional quasi-Sasakian manifold with respect to the quarter-symmetric metric connection. Finally we study locally $\varphi$−symmetric $3$-dimensional quasi-Sasakian manifold with respect to the quarter-symmetric metric connection.

Let $G$ be a $(p,q)$-graph and : $V(G)\to \{k,k+1,k+2,k+3,\dots ,p+q+k-1\}$ be an injection. For each edge $e=uv$, let $f^{*}(e)=\lceil \frac{f(u)+f(v)}{2}\rceil$. Then $f$ is called a $k$-super mean labeling if $f(V)\cup \{f^{*}(e):e\in E(G)\}=\{k,k+1,k+2,\dots ,p+q+k-1\}$. A graph that admits a $k$-super mean labeling is called $k$-super mean graph. In this paper, we present $k$-super mean labeling of $C_{2n}(n\ne 2)$ and super mean labeling of Double cycle $C(m,n)$, Dumb bell graph $D(m,n)$ and Quadrilateral snake $Q_n$.

A simple graph $G$ admits an $H$-covering if every edge in $E(G)$ belongs to a subgraph of $G$ isomorphic to $H$. The graph $G$ is said to be $H$-magic if there exists a bijection $f:V(G){\cup }^{\text{}}E(G)\to \{1,2,3,\cdots ,|V(G)\cup E(G)\left|\right\}$ such that for every subgraph $H^{\prime }$ of $G$ isomorphic to $H$, ${\displaystyle {\sum }_{v\in V(H^{\prime })}f(v)}+{\displaystyle {\sum }_{e\in E(H^{\prime })}f(e)}$ is constant. Additionally, $G$ is said to be $H$-supermagic if $f(V(G))=\{1,2,3,\cdots ,|V(G)\left|\right\}$. In this paper, we study $H$-supermagic labelings of two classes of connected graph namely fans and ladders.

Given a graph $G=(V,E)$, an $k$-edge-weighting is a map $\varphi :E(G)\to \{1,2,3,\dots ,k\}$, where $k$ is a positive integer. For a vertex $v$ of $G$, let $S_{\varphi }(v)$ denote the sum of edge-weights appearing on the edges incident at $v$ under the edge-weighting $\varphi$. An $k$-edge-weighting of $G$ is said to be equitable irregular if $|S_{\varphi }(u)-S_{\varphi }(v)|\le 1$, for every pair of adjacent vertices $u$ and $v$ in $G$. A graph $G$ is said to be equitable irregular if $G$ admits an equitable irregular edge-weighting. If $G$ is equitable irregular then the equitable irregular strength of $G$ is dened to be the smallest positive integer $k$ such that there is a $k$-edge-weighting of $G$, and is denoted by $Se(G)$. In this paper we initiate a study of this new non-proper edge weighting of graphs.

Let $Q$ be an affine quartic which does not intersect transversely with the line at infinity $L_{\infty }$ such that there exists a $D_2{}_p$-covering over ${\mathbf{\mathbb{P}}}^2$ branched along $Q\cup L_{\infty }$. In this paper, we show the existence of a $(2,3)$ torus decomposition of the defining polynomial of $Q$ and its uniqueness except for one class.

A $2$-Riemannian manifold is a differentiable manifold exhibiting a $2$-inner product on each tangent space. We first study lower dimensional $2$-Riemannian manifolds by giving necessary and sufficient conditions for flatness. Afterward we associate to each $2$-Riemannian manifold a unique torsion free compatible pseudoconnection. Using it we define a curvature for $2$-Riemannian manifolds and study its properties. We also prove that $2$-Riemannian pseudoconnections do not have Koszul derivatives. Moreover, we define stationary vector field with respect to a $2$-Riemannian metric and prove that the stationary vector fields in ${ℝ}^2$ with respect to the $2$-Riemannian metric induced by the Euclidean product are the divergence free ones.

This paper is concerned with Cauchy problems for the linear Schrödinger evolution equation:

$$i{\partial }_{t}u(x,t)+\text{Δ}u(x,t)+|x-a(t){|}^{-1}u(x,t)+V_1(x,t)u(x,t)=f(x,t)$$

in ${ℝ}^N\times [0,T]$, subject to initial condition: $u(x,0)=u_0(x)\in H^2({ℝ}^N)\cap H_2({ℝ}^N)$,

where $i:=\sqrt{-1}$, $N\ge 3,a:[0,T]\to {ℝ}^N$ expresses the center of the Coulomb potential, $V_1$ and $f:{ℝ}^N\times [0,T]\to R$ are another potential and an inhomogeneous term while

$$H_2({ℝ}^N):=\{v\in L^2({ℝ}^N);|x{|}^{2}v\in L^2({ℝ}^N)\}.$$

The strong formulation of this problem (with $f\equiv 0$ and $N=3$) has been solved by Baudouin-Kavian-Puel (2005) partly with formal computation. In this paper we reconstruct their argument with rigorous proofs. Moreover, we show that the solution $u$ satises the energy estimate

$$\Vert {\partial }_{t}u(t)\Vert +{\Vert u(t)\Vert }_{H^2\cap H_2}\le C_0({\Vert u_0\Vert }_{H^2\cap H_2}+{\Vert f\Vert }_F),$$

where is a constant depending on $a$, $V_1$ and $T$, while $\parallel f\parallel F$ is some norm of $f$.