In this paper, we consider the function related with almost hermitian structure on a copact complex manifold. More precisely, on a 2n-diminsional complex manifold M admitting 2-form .ohm. of rank 2n everywhere, assume that M admits a metric g such that g(JX, JY)=g(X,Y), that is, assume that g defines an hermitian structure on M admitting .ohm. as fundamental 2-form-the 'almost complex structure' J being determined by g and .ohm.:g(X,Y)=.ohm.(X,JY). We consider the function I(g):=.int.$_{M}$ $N^{2}$d $V_{g}$, where N is the norm of Nijenhuis tensor N defined by (J,g). by (J,g).

The purpose of this paper is to parameterize range-equivalence classes of all eigenmaps of flat 3-tori $T^{3}$= $R^{3}$.LAMBDA., .LAMBDA.= $c_{1}$ $e_{1}$ + $c_{2}$ $e_{2}$ + $c_{3}$ $e_{3}$, into the standard unit spheres. In this paper, we classify $A_{ 0 \lambda}$( $T^{3}$)(cf..cint.1) which is contained in $A_{\lambda}$( $T^{3}$), are belonging to $A_{ 0 \lambda}$( $T^{3}$). Moreover, as an application, we show that the only minimally imbedded flat torus into ( $S^{5}$ , can) which is contained in $A_{ 0 \lambda}$( $T^{3}$) is the generalized Clifford torus.rd torus.

The purpose of this paper is to study indefinite Kaehlerian metrics with vanishing conformal curvature tensor field. In the first section, a brief summary of the complex version of indefinite Kaehlerian manifolds is recalled and we introduce the conformal curvature tensor field on an indefinite Kaehlerian manifold. In section 2, we obtain the theorem for indefinite Kaehlerian metrics with vanishing conformal curvature tensor field.

We introduce the generalized Runge-Kutta methods with the exponentially dominant order .omega. in [3], and the convergence theorems of the generalized explicit Euler method are derived in [4]. In this paper we will study the convergence of the generalized implicit Euler method.

The characteristic free representation theory of the general linear group is one of the powerful tools in the study of invariant theory, algebraic geometry, and commutative algebra. Recently the study of such representations became a popular theme. In this paper we study the representation-theoretic structures of the symmetric algebra and the exterior algebra over a commutative ring with unity 1.

All rings are commutative, Noetherian with identity and of prime characteristic p, unless otherwise specified. First, we describe the definition of tight closure of an ideal and the properties about the tight closure used frequently. The technique used here for the tight closure was introduced by M. Hochster and C. Huneke [4,5, or 6]. Using the concepts of the tight closure and its properties, we will prove that if R is a complete local domain and F-rational, then R is Cohen-Macaulay. Next, we study the properties of R$^{+}$, the integral closure of a domain in an algebraic closure of its field of fractions. In fact, if R is a complete local domain of characteristic p>0, then R$^{+}$ is Cohen-Macaulay [8]. But we do not know this fact is true or not if the characteristic of R is zero. For the special case we can show that if R is a non-Cohen-Macaulay normal domain containing the rationals Q, then R$^{+}$ is not Cohen-Macaulay. Finally we will prove that if R is an excellent local domain of characteristic p and F-ratiional, then R is Cohen-Macaulay.aulay.

We discuss transverse harmonic fields on compact foliated Riemannian manifolds, and give a necessary and sufficient condition for a transverse field to be a transverse harmonic one and the non-existence of transverse harmonic fields. 1. On a foliated Riemannian manifold, geometric transverse fields, that is, transverse Killing, affine, projective, conformal fields were discussed by Kamber and Tondeur([3]), Molino ([5], [6]), Pak and Yorozu ([7]) and others. If the foliation is one by points, then transverse fields are usual fields on Riemannian manifolds. Thus it is natural to extend well known results concerning those fields on Riemannian manifolds to foliated cases. On the other hand, the following theorem is well known ([1], [10]): If the Ricci operator in a compact Riemannian manifold M is non-negative everywhere, then a harmonic vector field in M has a vanishing covariant derivative. If the Ricci operator in M is positive-definite, then a harmonic vector field other than zero does not exist in M.

Wellknown invariance of domain theorems are Brower's invariance of domain theorem for continuous mappings defined on a finite dimensional space and Schauder-Leray's invariance of domain theorem for the class of mappings I+C defined on a infinite dimensional Banach space with I the identity and C compact. The two classical invariance of domain theorems were proved by applying the homotopy invariance of Brower's degree and Leray-Schauder's degree respectively. Degree theory for some class of mappings is a useful tool for mapping theorems. And mapping theorems (or surjectivity theorems of mappings) are closely related with invariance of domain theorems for mappings. In[4, 5], Browder and Petryshyn constructed a multi-valued degree theory for A-proper mappings. From this degree Petryshyn [9] obtained some invariance of domain theorems for locally A-proper mappings. Recently Browder [6] has developed a degree theory for demicontinuous mapings of type ( $S_{+}$) from a reflexive Banach space X to its dual $X^{*}$. By applying this degree we obtain some invariance of domain theorems for demicontinuous mappings of type ( $S_{+}$). ( $S_{+}$).

In this paper we will prove some theorems on the period and the linear complexity of certain sequences in GF(q) which are generated by combining two sequences in a reasonable way. In fact these theorems are generalizations of the main result in [1]. A sequence of elements of GF(2) is called a binary sequence. In recent years considerable interest has been shown in the generation of binary sequences which have good properties. Such binary sequences play an important role in a stream cipher system.

The dimension problem has been one of central themes in the theory of ordered sets. In this paper we focus on amalgamated ordered sets. Although some results can be obviously applied to infinite cases, we assume throughout that all ordered set are finite. If A and B are ordered sets whose orders agree on A.cap.B, then the amalgam of A and B is defined to the the set A.cup.B in which the order is the transitive closure of the union of the two orders, i.e., the smallest order containing the two orders, and is denoted by A .or. B .leq. dim A + dim B for any ordered sets A and B. But it is quite surprising that the dimension of the amalgam of certain 2-dimensional ordered sets can be arbitrarily large.

In various viscus flow problems it has been the custom to replace the convective derivative by the ordinary partial derivative in problems for which the data are small. In this paper we consider the Benard Convection problem with small data and compare the solution of this problem (assumed to exist) with that of the linearized system resulting from dropping the nonlinear terms in the expression for the convective derivative. The objective of the present work is to derive an estimate for the error introduced in neglecting the convective inertia terms. In fact, we derive an explicit bound for the L$_{2}$ error. Indeed, if the initial data are O(.epsilon.) where .epsilon. << 1, and the Rayleigh number is sufficiently small, we show that this error is bounded by the product of a term of O(.epsilon.$^{2}$) times a decaying exponential in time. The results of the present paper then give a justification for linearizing the Benard Convection problem. We remark that although our results are derived for classical solutions, extensions to appropriately defined weak solutions are obvious. Throughout this paper we will make use of a comma to denote partial differentiation and adopt the summation convention of summing over repeated indices (in a term of an expression) from one to three. As reference to work of continuous dependence on modelling and initial data, we mention the papers of Payne and Sather [8], Ames [2] Adelson [1], Bennett [3], Payne et al. [9], and Song [11,12,13,14]. Also, a similar analysis of a micropolar fluid problem backward in time (an ill-posed problem) was given by Payne and Straughan [10] and Payne [7].

The purpose of this paper is to provide the affirmative solution of the following conjecture due to Davis and Geramita. Conjecture; Let A=R[T] be a polynomial ring in one variable, where R is a regular local ring of dimension d. Then maximal ideals in A are complete intersection. Geramita has proved that the conjecture is true when R is a regular local ring of dimension 2. Whatwadekar has rpoved that conjecture is true when R is a formal power series ring over a field and also when R is a localization of an affine algebra over an infinite perfect field. Nashier also proved that conjecture is true when R is a local ring of D[ $X_{1}$,.., $X_{d-1}$] at the maximal ideal (.pi., $X_{1}$,.., $X_{d-1}$) where (D,(.pi.)) is a discrete valuation ring with infinite residue field. The methods to establish our results are following from Nashier's method. We divide this paper into three sections. In section 1 we state Theorems without proofs which are used in section 2 and 3. In section 2 we prove some lemmas and propositions which are used in proving our results. In section 3 we prove our main theorem.eorem.rem.

Carlitz module is used to study abelian extensions of K=$F_{q}$(T). In number theory every abelian etension of Q is contained in a cyclotomic field. Similarly every abelian extension of $F_{q}$(T) with some condition on .inf. is contained in a cyclotomic function field. Hence the study of cyclotomic function fields in analogy with cyclotomic fields is an important subject in number theory. Much are known in this direction such as ring of integers, class groups and units ([G], [G-R]). In this article we are concerned with the ring of integers in a cyclotomic function field. In [G], it is shown that the ring of integers is generated by a primitive root of the Carlitz module using the ramification theory and localization. Here we will give another proof, which is rather elementary and explicit, of this fact following the methods in [W].[W].