In this paper, when an entire function $f$ and the linear combination of its derivatives $L\left(f\right)$ with small functions as its coefficients share one value CM and another value IM is studied. We also resolved the question when an entire function $f$ and its derivative $f^{\prime }$ share two values CM jointly. Some of the results remain to be valid if $f$ is meromorphic and satisfying $N(r,f)=o\left(T\right(r,f\left)\right)$ as $r\to \infty$ and the values $a,$$b$ are replaced by small functions of $f\left(z\right)$.

This paper is a continuation of the previous paper Taira and Umezu [12] where we studied the existence and uniqueness of positive solutions of a class of sublinear elliptic problems with degenerate boundary conditions. We intend here to give a further investigation of the set of positive solutions in the forced case.

In this paper, we introduce a notion of stationarity for the pair of flows in a metric vector space and characterize it in such a way that there exist two relations, called a dissipation-dissipation theorem and a fluctuation-dissipation theorem, among the $K{M}_{2}O$-Langevin matrix associated with the pair of flows.

We give a symplectic description of the fiber space for each $J$-th Painlevé system ($J=V,$ $IV$, $III$, $II$) which was constructed by K. Okamoto. Hamiltonian function in every chart is a polynomial of the canonical coordinates.

The purpose of this paper is to formulate the notion of quantum ergodicity at a finite energy level for certain quantum mechanics, by using the method of Sunada [Sul]. Under some assumptions on the corresponding classical mechanics, we obtain a necessary and sufficient condition in terms of semi-classical asymptotic behaviour of eigenfunctions of a quantum Hamiltonian so that the classical mechanics is ergodic. We also obtain a result on quantum weak mixing at a finite energy level which is a semiclassical analogue of the notion introduced in [Z4].

In this article we begin the study of $\hat{X}$, an $n$-dimensional algebraic submanifold of complex projective space $P^N$, in terms of a hyperplane section $\hat{A}$ which is not irreducible. A number of general results are given, including a Lefschetz theorem relating the cohomology of $\hat{X}$ to the cohomology of the components of a normal crossing divisor which is ample, and a strong extension theorem for divisors which are high index Fano fibrations. As a consequence we describe $\hat{X}\subset P^N$ of dimension at least five if the intersection of $\hat{X}$ with some hyperplane is a union of $r\ge 2$ smooth normal crossing divisors $A_1$, . . . , $A_r$, such that for each $i,$$h^1\left({\mathcal{O}}_{{\hat{A}}_i}\right)$ equals the genus $g\left(A_i\right)$ of a curve section of $A_i$. Complete results are also given for the case of dimension four when $r=2$.

The purpose of this paper is to define and study an important class in weights on $O^{*}$-algebras which is possible to develop the Tomita-Takesaki theory in $O^{*}$-algebras. The Connes cocycle theorem for weights on von Neumann algebras is generalized to the case of $O^{*}$-algebras.

We define a semidynamical system-inspired by some classical dynamical systems studied by Bebutov in function spaces-in the space of approximative maps $A(X,Y)$ between two metric compacta, with a suitable metric. Shape and strong shape morphisms are characterized as invariant subsets of this system. We study their structure and asymptotic properties and use the obtained results to give dynamical characterizations of basic notions in shape theory, like trivial shape, shape domination by polyhedra and internal FANRs.

We give the recursion formula of the Harish-Chandra $C$-function with respect to the highest weight of the representations of $K$. Using this formula, we get the explicit expressions of the Harish-Chandra $C$-functions for $\mathrm{Spin}(n,1)$ and $SU(n,1)$. As an application, by using these expressions, we get the realizations of discrete series representations of $SU(n,1)$ as subquotients of nonunitary principal series representations. We shall also get the decompositions of holomorphic and antiholomorphic discrete series when restricted to $U(n-1,1)$. By using the structures of $K$-spectra of discrete series representations, we can concretely construct the invariant subspaces of the representation spaces of holomorphic and antiholomorphic discrete series.

Let $l_1,$$l_2,$$\text{...},$$l_7$ be mutually different seven lines on the real projective plane. We consider two conditions;(A) No three of $l_1,$$l_2,$$\text{...},$$l_7$ intersect at a point. (B) There is no conic tangent to any six of $l_1,$$l_2$, . . . , $l_7$. Cummings [3] and White [16] showed that there are eleven non-equivalent classes of systems of seven lines with condition (A)(cf. [7], Chap. 18). The purposes of this article is to give an interpretation of the classification of Cummings and White in terms of the root system of type $E_7$. To accomplish this, it is better to add condition (B) for systems of seven lines. Moreover we need the notion of tetrahedral sets which consist of ten roots modulo slgns in the root system of type $E_7$ and which plays an important role in our study.

We prove Abel's theorem for divisors on an arbitrary compact complex manifold by combining the Čech cohomology of sheaves, a logarithmic residue formula for 1-forms and de Rham's theory applied to open submanifolds.