Showing papers in "Journal of The Korean Mathematical Society in 1996"

Coupled fixed point theorems with applications

Abstract: Recently, existence theorems of coupled fixed points for mixed monotone operators have been considered by several authors (see [1]-[3], [6]). In this paper, we are continuously going to study the existence problems of coupled fixed points for two more general classes of mixed monotone operators. As an application, we utilize our main results to show thee existence of coupled fixed points for a class of non-linear integral equations.

Convex decompositions of real projective surfaces. III: for closed or nonorientable surfaces

Abstract: The purpose of our research is to understand geometric and topolog­ ical aspects of real projective structures on surfaces. A real projective surface is a differentiable surface with an atlas of charts to Rp2 such that transition functions are restrictions of projective automorphisms of Rp2. Since such an atlas lifts projective geometry on RP2 to the surface locally and consistently, one can study the global projective geometry of surfaces. This paper is the final piece of the series of the papers [2] and [3]. With this paper, we get a satisfactory classification of all real projective structures on surfaces (see [4]). This final paper shows that a nonori­ entable real projective surface also has an admissible decomposition and a closed real projective surface decomposes into pieces that are convex real projective surfaces or 7r-Mobius bands. Recall that the complement of a one-dimensional subspace in Rp2 has a canonical affine structure of a complete affine plane. The comple­ ment is said to be an affine patch. A real projective surface has convex boundary if each point of the boundary has a neighborhood admitting a chart to a convex domain in an affine patch. Let S be an orientable or nonorientable real projective slJ.rface with convex or empty boundary. (We will often refer to [2] and [3] for definitions and results needed in this paper). Let S denote the universal cover of S, and 7rl(S) the fun­ damental group of S, identified with the group of deck transformations. Given S, there is an immersion dev: S~ Rp2 and a homomorphism h : 7rl(S) ~ PGL(3, R) satisfying dev 0, = h(,) 0 dev for each deck

The geometry of left-symmetric algebra

Abstract: In this paper, we are interested in left invariant flat affine structures on Lie groups. These structures has been studied by many authors in different contexts. One of the fundamental questions is the existence of complete affine structures for solvable Lie groups G, raised by Minor [15]. But recently Benoist answered negatively even for the nilpotent case [1]. Also moduli space of such structures for lower dimensional cases has been studied by several authors, sometimes with compatible metrics [5,10,4,12].

On a class of ternary composition algebras

Abstract: When dealing with a Lie group or, in general, with an analytic loop or quasigroup, its symmetry is broken by the election of the distinguished identity element.

Deformation spaces of convex real-projective structures and hyperbolic affine structures

Abstract: A convex $RP^n$-structure on a smooth anifold M is a representation of M as a quotient of a convex domain $\Omega \subset RP^n$ by a discrete group $\Gamma$ of collineations of $RP^n$ acting properly on $\Omega$. When M is a closed surface of genus g > 1, then the equivalence classes of such structures form a moduli space $B(M)$ homeomorphic to an open cell of dimension 16(g-1) (Goldman [2]). This cell contains the Teichmuller space $T(M)$ of M and it is of interest to know what of the rich geometric structure extends to $B(M)$. In [3], a symplectic structure on $B(M)$ is defined, which extends the symplectic structure on $T(M)$ defined by the Weil-Petersson Kahler form.

Protter's conjugate boundary value problems for two dimensional wave eqiation

Abstract: In 1954 M. H. Protter [1] formulated the following boundary value problem as an analogue of the plane Darboux problem.

Generalized vector-valued variational inequalities and fuzzy extensions

Abstract: Recently, Giannessi [9] firstly introduced the vector-valued variational inequalities in a real Euclidean space. Later Chen et al. [5] intensively discussed vector-valued variational inequalities and vector-valued quasi variationl inequalities in Banach spaces. They [4-8] proved some existence theorems for the solutions of vector-valued variational inequalities and vector-valued quasi-variational inequalities. Lee et al. [14] established the existence theorem for the solutions of vector-valued variational inequalities for multifunctions in reflexive Banach spaces.

A lower bound for the number of squares whose sum represents integral quadratic forms

Abstract: Lagrange's famous Four Square Theorem [L] says that every positive integer can be represented by the sum of four squares. This marvelous theorem was generalized by Mordell [M1] and Ko [K1] as follows : every positive definite integral quadratic form of two, three, four, and five variables is represented by the sum of five, six, seven, and eight squares, respectively. And they tried to extend this to positive definite integral quadratic forms of six or more variables.

Wick derivations on white noise functionals

Abstract: The white noise analysis, initiated by Hida [3] in 1975, has been developed to an infinite dimensional distribution theory on Gaussian space $(E^*, \mu)$ as an infinite dimensional analogue of Schwartz distribution theory on Euclidean space with Legesgue measure. The mathematical framework of white noise analysis is the Gel'fand triple $(E) \subset (L^2) \subset (E)^*$ over $(E^*, \mu)$ where $\mu$ is the standard Gaussian measure associated with a Gel'fand triple $E \subset H \subset E^*$.

An extension of telci, tas and fisher's theorem

Abstract: Let (X,d) be a metric space and let T be a mapping from X into itself. We say that a metric space (X,d) is T-orbitally complete if every Cauchy sequence of the form ${T^{n_i}x}_{i \in N}$ for $x \in X$ converges to a point in X.

Lie-admissible algebras and the virasoro algebra

Abstract: Let A be an (nonassociative) algebra with multiplication xy over a field F, and denote by $A^-$ the algebra with multiplication [x, y] = xy - yx$ defined on the vector space A. If $A^-$ is a Lie algebra, then A is called Lie-admissible. Lie-admissible algebras arise in various topics, including geometry of invariant affine connections on Lie groups and classical and quantum mechanics(see [2, 5, 6, 7] and references therein).

Approximative mle for the scale parameter of the double exponential distribution based on type-ii censored samples

Abstract: Consider the double exponential distribution with probability density function (pdf) $$ (1.1) f(x;\sigma) = \frac{2\sigma}{1} e^{-x/\sigma}, -\infty

On semi-kaehler manifolds whose totally real bisectional curvature is bounded from below

Abstract: R.L. Bishop and S.I. Goldberg [3] introduced the notion of totally real bisectional curvature B(X, Y) on a Kaehler manifold M. It is determined by a totally real plane [X, Y] and its image [JX, JY] by the complex structure J. where [X, Y] denotes the plane spanned by linealy independent vector fields X, and Y. Moreover the above two planes [X, Y] and [JX, JY] are orthogonal to each other. And it is known that two orthonormal vectors X and Y span a totally real plane if and only if X, Y and JY are orthonormal.

Totally umbilic lorentzian submanifolds

Abstract: A totally umbilic submanifold of a pseudo-Riemanian manifold is a submanifold whose first fundamental form and second fundamental form are proportiona. An ordinary hypersphere $S^n(r)$ of an affine (n + 1)-space of the Euclidean space $E^m$ is the best known example of totally umbilic submanifolds of $E^m$.

Boundaries of the cone of positive linear maps and its subcones in matrix algebras

Abstract: Let $M_n$ be the $C^*$-algebra of all $n \times n$ matrices over the complex field, and $P[M_m, M_n]$ the convex cone of all positive linear maps from $M_m$ into $M_n$ that is, the maps which send the set of positive semidefinite matrices in $M_m$ into the set of positive semi-definite matrices in $M_n$. The convex structures of $P[M_m, M_n]$ are highly complicated even in low dimensions, and several authors [CL, KK, LW, O, R, S, W]have considered the possibility of decomposition of $P[M_m, M_n] into subcones.

A change of scale formula for wiener integrals on the product abstract wiener spaces

Abstract: It has long been known that Wiener measure and Wiener measurbility behave badly under the change of scale transformation [3] and under translation [2]. However, Cameron and Storvick [4] obtained the fact that the analytic Feynman integral was expressed as a limit of Wiener integrals for a rather larger class of functionals on a classical Wienrer space.

Existence of solutions for p-laplacian type equations

Abstract: In this paper, we shall show the existence of solutions of the following nonlinear partial differential equation $$ {^{divA(-\Delta u) = f(x, u, \Delta u) in \Omega}^{u = 0 on \partial\Omega} $$ where $f(x, u, \Delta u) = -u\Delta u^{p-2} + h, p \geq 2, h \in L^\infty$.

Steepest descent method for locally accretive mappings

Abstract: (1) ‖x − y‖ ≤ ‖(1+ t)(x − y)− r t (Ax − Ay)‖ holds for all x, y ∈ K and r > 0. If t = 1 then A is called pseudocontractive. The map A is called locally strongly pseudocontractive if each point of K has a neighbourhood N for which (1) holds for each x, y ∈ N and some t > 1. Pseudocontractive operators have been studied by various authors (see e.g., [1], [2], [4], [8-12], [14], [16], [17], [18], [19], [21], [22], [28], [29], [30], [32-33], [37]). Interest in such mappings stems mainly from the fact that they are firmly connected with the important class of nonlinear accretive operators. A mapping U with domain D(U) and range R(U) in E is called accretive (see e.g., [2], [15]) if the inequality

Dynamics of two parameter family of point-mass singular inner functions

Abstract: Let $\alpha > 0$ and $\zeta$ be a point on the unit circle $T= {z = 1}$ of the complex plane C.

Peak function and its application

Abstract: Letbe a smoothly bounded pseudoconvex domain inC n and let A/ denote the functions holomorphic onand continuous on  A point p2 bis a peak point if there is a function f 2 A/ such that f p/D 1, andj fz/j < 1f or z2f pg The existence of peaking functions as well as the additional smoothness up to the boundary is one of the major topics in several complex variables Whenis strictly pseudoconvex, the situation with regard to peak functions is fairly well understood, but in the weakly pseudoconvex case we know very little IfC 2 is pseudoconvex and bis of finite type, Bedford and Fornaess (1), showed that there is a peak function in A/ This method also works for finite type domains in C n where the Levi-form of bhas (n-2)-positive eigenvalues We also mention the work of Bloom (2), Hakim and Sibony (11), and Range (16) on the existence of peak functions with additional smoothness up to the boundary of , ie, in the various subclass of A/ Recently the author proposed a method (8) to construct a peak function for the domains in C n where the optimal estimates of the Bergman kernel function are known Namely, for each neighborhood V of p 2 bwe construct a regular bumping family of pseudoconvex domains outside V ,a nd use Bishop's 1 3 method on bumped domains This is a modification of Fornaess and McNeal's method (10) and can be applied to wide class of domains inC n The optimal estimates of the Bergman kernel function and its derivatives are known, for example, for pseudoconvex domains of finite

Cauchy problems for a partial differential equation in white noise analysis

Abstract: The Gross Laplacian $\Delta_G$ was introduced by Groww for a function defined on an abstract Wiener space (H,B) [1,7]. Suppose $\varphi$ is a twice H-differentiable function defined on B such that $\varphi"(x)$ is a trace class operator of H for every x \in B.in B.

Limit theorems for markov processes generated by iterations of random maps

Abstract: We consider the asymptotic behaviors of Markov process which is generated by successive iterations of independent and identically distributed random maps. We show that average contraction of some finite compositions of random maps is sufficient for the existence of a unique invariant measure. A functional central limit theorem and a strong law of large numbers are proved for arbitrary Lipschitzian functions.

A relative naielsen coincidence number for the complement, i

Abstract: Nielsen coincidence theory is concerned with the determinatin of a lower bound of the minimal number MC[f,g] of coincidence points for all maps in the homotopy class of a given map (f,g) : X $\to$ Y. The Nielsen Nielsen number $N_R(f,g)$ (similar to [9]) is introduced in [3], which is a lower bound for the number of coincidence points in the relative homotopy class of (f,g) and $N_R(f,g) \geq N(f,g)$.

On the volume for totally real submanifolds of a kaehler manifold

Abstract: One of the fundamental problems in Riemannian geometry is "How the geometric invariants of Riemannian manifolds influenced by the curvature restriction \ulcorner".ner".uot;.

Nonwandering points of a map on the circle

Abstract: In this paper, we will show that for any continuous map f of the circle, if the set of periodic points of f is empty, then the set of recurrent point of f equals the set of nonwandering points of f .

Some linearly independent immersions into their adjoint hyperquadrics

Abstract: Let $x : M^n \longrightarrow E^m$ be an isometric immersion of an n-dimensional connected Riemannian manifold into the m-dimensional Euclidean space. Then the metric tensor on $M^n$ is naturally induced from that of $E^m$.

Product spaces that induce approximate fibrations

Abstract: In the study of manifold decompositions, a central theme is to understand the source manifold taking advantage of the informations of a base space and a decomposition The concepts of both Hurewicz fibrations and cell-like maps have played very important roles for investigating the mutual relations of three objects But it is somewhat restrictive for a decomposition map to be cell-like because its inverse images must have trivial shapes

Dirichlet forms and diffusion processes related to quantum unbounded spin systems

Abstract: We study Dirichlet forms and the associated diffusion processes for the Gibbs measures related to the quantum unbounded spin systems (lattice boson systems) interacting via superstable and regular potentials. This work is a continuation of the author's previous study on the classical systems [LPY] to the quantum cases. In [LPY], we constructed Dirichlet forms and the associated diffusion processes for the Gibbs measures of classical unbounded spin systems. Furthermore, we also showed the essential self-adjointness of the Dirichlet operator and the log-Sobolev inequality for any Gibbs measure under appropriate conditions on the potentials. In this atudy we try to extend the results of the classical systems to the quantum cases. Because of some technical difficulties, we are only able to construct a Dirichlet form and the associated diffusion process for any Gibbs measure of the quantum systems. We utilize the general scheme of the previous work on the theory in infinite dimensional spaces [AH-K1-2, AKR, AR1-2, Kus, MR, Ro, Sch] and the ideas we employed in our study of the calssical systems ]LPY].

Boundary-valued conditional yeh-wiener integrals and a kac-feynman wiener integral equation

Abstract: For $Q = [0,S] \times [0,T]$ let C(Q) denote Yeh-Wiener space, i.e., the space of all real-valued continuous functions x(s,t) on Q such that x(0,t) = x(s,0) = 0 for every (s,t) in Q. Yeh [10] defined a Gaussian measure $m_y$ on C(Q) (later modified in [13]) such that as a stochastic process ${x(s,t), (s,t) \epsilon Q}$ has mean $E[x(s,t)] = \smallint_{C(Q)} x(s,t)m_y(dx) = 0$ and covariance $E[x(s,t)x(u,\upsilon)] = min{s,u} min{t,\upsilon}$. Let $C_\omega \equiv C[0,T]$ denote the standard Wiener space on [0,T] with Wiener measure $m_\omega$. Yeh [12] introduced the concept of the conditional Wiener integral of F given X, E(FX), and for case X(x) = x(T) obtained some very useful results including a Kac-Feynman integral equation.

Some characterizations of ruled real hypersurfaces in a complex space form

Abstract: In this paper firstly we give the notion of ruled real hypersurfaces in a non-flat complex space form Mn.c/ and calculate the expression of the covariant derivative.rX A/Y of its Weingarten map A for X,Y in a distribution T0. Next we consider the -component g..rX A/Y;/D f. X; Y/; X; Y in T0, with which we study a characterization of ruled real hypersurfaces in Mn.c/. As an application of this characterization we can also obtain another characterization of these hypersurfaces in terms of Lie derivatives.