In this paper are considered the family of straight lines $\left\{q\right\}$ in the Euclidean space , which intersect given $k$ lines $p_i,(1\le i\le k)$ in general position. We also study the topology of the corresponding family of lines.

Necessary conditions and sufficient conditions are given so that the two-dimensional Laplace Transform is bounded between two Lebesgue spaces with weights. Such a boundedness is characterized for a large class of weights.

We study the following initial-boundary value problem for the nonlinear nonlocal Schrödinger equation

with the compatibility condition ${\partial }_x^{j-1}\overline{u}(0)={\tilde{u}}_j(0),j=1,2\dots ,n$, where $n=\left[\frac{\alpha }{2}\right],\left[s\right]$ denotes the largest integer less than $s,b\ge 0$, the nonlinear term $\text{N(}u\text{)=}ia(t)|u{|}^{\rho }u,\rho >1$, the coefficient $a(t)\in {\text{C}}^1$ and $\mathbf{K}$ is the pseudodifferential operator on the half line ${\mathbf{R}}^{+}$ of order . We prove that if $x^{\delta }\overline{u}\in {\mathbf{L}}^1$, with and the norm $\Vert \overline{u}{\Vert }_{\mathbf{X}}$ of the initial data and the norms $\Vert {\tilde{u}}_j{\Vert }_{\mathbf{Y}}$, $j=1,\dots n$ of the boundary data are sufficiently small then there exists a unique solution $u\in \mathbf{C}\text{([0,}\infty \text{);}{\mathbf{L}}^2)\cap \mathbf{C}\text{(}{\mathbf{R}}^{+};{\mathbf{W}}_2^{[\alpha ]-1}\cap {\mathbf{C}}^{[\alpha ]-1})$ of the initial-value problem (NNS). Here $\mathbf{X}={\mathbf{W}}_{\infty }^{[\alpha ]}\cap {\mathbf{W}}_1^{[\alpha ]+1}$ and $\mathbf{Y}={\mathbf{W}}_{\infty }^1\cap {\mathbf{W}}_1^2$ and ${\mathbf{W}}_p^k$ is the Sobolev space with the norm $\Vert \varphi {\Vert }_{{\mathbf{W}}_p^k}=\Vert {(1-{\partial }_x^2)}^{k/2}\varphi (x){\Vert }_{{\mathbf{L}}^p}$. We also find the large time asymptotics of the solutions.

In this paper we study a smoothing property of solutions to the Cauchy problem for the nonlinear Schrödinger equations of type :

$$\begin{array}{c}\left(A\right) i{u}_t+u_{xx}=\mathcal{N}(u,\overline{u},u_x\overline{u_x}),t\in \mathrm{ℝ},x\in \mathrm{ℝ};\\ u(x,0)=u_0(x),x\in \mathrm{ℝ},\end{array}$$

where

$$\mathcal{N}(u,\overline{u},u_x,\overline{u_x})=K_1|u{|}^{2}u+K_2|u{|}^2{u}_x+K_3{u}^2\overline{u_x}+K_4|u_x{|}^{2}u+K_5\overline{u}{u}_x^2+K_6|u_x{|}^2{u}_x,$$ the functions $K_j=K_j(|u{|}^2)$ satisfy $K_j\in {\mathcal{C}}^{\infty }([0,+\infty );ℂ)$.

This equation has been derived from physics. For example if the nonlinear term is $\mathcal{N}(u)=\frac{\overline{u}{u}_x^2}{1+|u{|}^2}$ then equation (A) appears in the classical pseudospin magnet model [18]. The aim of this paper is to study the case : the nonlinearity depends on $u_x$ and $\overline{u_x}$, and satisfies the so called Gauge condition : $\mathcal{N}(e^{i\theta }u)=e^{i\theta }\mathcal{N}(u)$. We prove that if the initial data $u_0\in {\mathscr{H}}^{3,l}$ for any $l\in \mathbb{N}$, then there exists a positive time and a unique solution $u\in {\mathcal{C}}^{\infty }([-T,T]\backslash \{0\};{\mathcal{C}}^{\infty }(ℝ))$ of the Cauchy problem (A). The result in this paper improves the previous one in [11] because we do not assume any size restriction on the data. Here and ${\Vert \phi \Vert }_{m,s}{=\Vert {(1+x^2)}^{s/2}{(1-{\partial }_x^2)}^{m/2}\phi \Vert }_{{ℒ}^2(ℝ)},{\mathscr{H}}^{m,\infty }={\cap }_{s\ge 1}{\mathscr{H}}^{m,s}$.

Amari showed that the geometry of a family of probability distributions is characterized by a dual differential geometry determined by a couple of affine connections and a divergence associated with a couple of dual potential functions. In this paper, a 2-parameter class of dual differential geometries is constructed on the manifold of the family of multivariate Gaussian distributions with nonzero means, as well as a new class of divergences. This class of geometry includes the Riemannian geometry studied by Skovgaard and the geometry associated with the Kullbaek-Leibler information. The specific dually flat charts for the latter geometry is given in conjunction with a detailed analysis of the associated connections. In order to facilitate the various calculations of differential geometric quantities in the analysis of our geometries, we introduce a new coordinate free differential calculus of a function of a symmetric matrix argument, based on a specific bilinear form defined on the domain of the function and its dual space. This calculus enables us to obtain a parallel formalism of the Legendre transformation in convex analysis even for a function of a matrix argument.

Aluthge and Wang [4] showed that “if $T$ is $p$-hyponormal then $T^n$ is $\left(\frac{p}{n}\right)$-hyponormal for $p\in (0,1]$”. Firstly we obtain precise estimation of this result. Secondly we show that “if $T$ is log-hyponormal then $T^n$ is log-hyponormal” and this result is an extension of Theorem B by Aluthge and Wang [3].

In this paper, firstly we shall give simplified proofs of the results on generalized Aluthge transformation in [11][12][14] and [16]. Secondly we shall discuss a generalization of both classes of class $A(k)$ operators defined in [9] and $w$-hyponormal operators defined in [2].