Showing papers in "Journal of The Korean Mathematical Society in 2001"

Hadamard-type fractional calculus

Abstract: The paper is devoted to the study of fractional integra- tion and dierentiation on a finite interval ( a;b) of the real axis in the frame of Hadamard setting The constructions under consider- ation generalize the modified integration R x a (t=x) " f(t)dt=t and the modified dierentiation - +" (- = xD; D = d=dx) with real ", be- ing taken n times Conditions are given for such a Hadamard-type fractional integration operator to be bounded in the space X p c(a;b) of Lebesgue measurable functions f on R+ = (0;1) such that Z b a jt c f(t)j p dt t < 1 (1 • p < 1); ess supatb(u c jf(t)j) < 1 (p = 1); for c 2 R = (i1;1), in particular in the space L p (0;1) (1 • p • 1) The existence almost everywhere is established for the corresponding Hadamard-type fractional derivative for a function g(x) such that x " g(x) have - derivatives up to order n i 1 on (a;b) and - ni1 (x " g(x)) is absolutely continuous on (a;b) Semigroup and reciprocal properties for the above operators are proved

Demi-closed principle and weak convergence problems for asymptotically nonexpansive mappings

Abstract: A demi-closed theorem and some new weak conver- gence theorems of iterative sequences for asymptotically nonexpan- sive and nonexpansive mappings in Banach spaces are obtained. The results presented in this paper improve and extend the corre- sponding results of (1), (8)-(10), (12), (13), (15), (16), and (18).

Hyers-ulam stability of the quadratic equation of pexider type

Abstract: In this paper, we will prove the Hyers-Ulam stability of the quadratic functional equation of Pexider type, f1(x + y) + f2(x y) = f3(x) + f4(y).

Travel time tomography

Abstract: We survey recent results on the inverse kinematic problem arising in geophysics. The question is whether one can determine the sound speed (index of refraction) of a medium by measuring the travel times of the corresponding ray paths. We emphasize the anisotrpic case.

Empirical realities for a minimal description risky asset model. the need for fractal features

Abstract: The classical Geometric Brownian motion (GBM) model for the price of a risky asset, from which the huge financial derivatives industry has developed, stipulates that the log returns are iid Gaussian. however, typical log returns data show a distribution with much higher peaks and heavier tails than the Gaussian as well as evidence of strong and persistent dependence. In this paper we describe a simple replacement for GBM, a fractal activity time Geometric Brownian motion (FATGBM) model based on fractal activity time which readily explains these observed features in the data. Consequences of the model are explained, and examples are given to illustrate how the self-similar scaling properties of the activity time check out in practice.

Conditional fourier-feynman transforms and conditional convolution products

Abstract: In this paper we define the concept of a conditionalFourier- Feynman transform and a conditionalconvol product and obtain several interesting relationships between them. In particular we show that the conditionaltransform of the conditionalconvol ution product is the product of conditionaltransforms, and that the conditionalcon- volution product of conditional transforms is the conditional transform of the product of the functionals.

On grams determinant in 2-inner product spaces

Abstract: An analogue of Grams inequality for 2-inner product spaces is given. Further, a number of inequalities involving Grams determinant are stated and proved in terms of 2-inner products.

Regular maps-combinatorial objects relating different fields of mathematics

Abstract: Regular maps and hypermaps are cellular decompositions of closed surfaces exhibiting the highest possible number of symmetries. The five Platonic solids present the most familar examples of regular maps. The gret dodecahedron, a 5-valent pentagonal regular map on the surface of genus 5 discovered by Kepler, is probably the first known non-spherical regular map. Modern history of regular maps goes back at least to Klein (1878) who described in [59] a regular map of type (3, 7) on the orientable surface of genus 3. In its early times, the study of regular maps was closely connected with group theory as one can see in Burnside’s famous monograph [19], and more recently in Coxeter’s and Moser’s book [25] (Chapter 8). The present-time interest in regular maps extends to their connection to Dyck\`s triangle groups, Riemann surfaces, algebraic curves, Galois groups and other areas, Many of these links are nicely surveyed in the recent papers of Jones [55] and Jones and Singerman [54]. The presented survey paper is based on the talk given by the author at the conference “Mathematics in the New Millenium”held in Seoul, October 2000. The idea was, on one hand side, to show the relationship of (regular) maps and hypermaps to the above mentioned fields of mathematics. On the other hand, we wanted to stress some ideas and results that are important for understanding of the nature of these interesting mathematical objects.

Feynman's operational calculi for time dependent noncommuting operators

Abstract: We study Feynman's Operational Calculus for operator- valued functions of time and for measures which are not necessarily probability measures; we also permit the presence of certain un- bounded operators. Further, we relate the disentangling map de- fined within to the solutions of evolution equations and, finally, remark on the application of stability results to the present paper.

Analytic fourier-feynman transform and first variation on abstract wiener space

Abstract: In this paper we express analytic Feynman integral of the first variation of a functional F in terms of analytic Feynman integral of the product of F with a linear factor and obtain an integration by parts formula for the analytic Feynman integral of functionals on abstract Wiener space. We find the Fourier-Feynman transform for the product of functionals in the Fresnel class F(B) with n linear factors.

Integration formulas involving fourier-feynman transforms via a fubini theorem

Abstract: In this paper we use a general Fubini theorem established in [13] to obtain several Feynman integration formulas involving analytic Fourier-Feynman transforms. Included in these formulas is a general Parseval’s relation. 1. Introduction and preliminaries Let C0[0,T ] denote one-parameter Wiener space, that is the space of R-valued continuous functions x(t) on [0,T ] with x(0) = 0. LetM denote the class of all Wiener measurable subsets of C0[0,T ] and let m denote Wiener measure. (C0[0,T ],M,m) is a complete measure space and we denote the Wiener integral of a Wiener integrable functional F by Z

The transport of nuclear contamination in fractured porous media

Abstract: The objects of this paper are to formulate a model for the transport of a chain of radioactive waste products in a fractured porous medium, to devise an eective and ecient numerical method for approximating the solution of the model, and to demonstrate the convergence of the numerical method. The formulation begins from a model in an unfractured (single porosity) medium, passes through a double porosity model in a fractured medium, and ends with a modified single porosity model that takes the relevant time scales of the flow and the nuclear decay. numerical method for approximating the solution of the model, and demonstrate the convergence of the numerical method. The model is intended to be a reasonably accurate description of the transport and dispersion of nuclear contamination through a granitic medium having densely spaced fractures that are the result of shrinkage; geologic faults are not covered herein. Many nuclear waste facilities are expected to be located in such granitic media. We begin with a standard model for the transport and dispersion of a nuclear chain in an unfractured, single-porosity porous medium (23) and then introduce a dual porosity model derivable by homogenization. By considering the size of the matrix blocks and the time scale of the problem, we reduce the dual porosity model to a so-called "limit model",

Weak convergence to common fixed points of countable nonexpansive mappings and its applications

Abstract: In this paper, we introduce an iteration generated by countable nonexpansive mappings and prove a weak convergence theorem which is connected with the feasibility problem. This result is used to solve the problem of finding a solution of the countable convex inequality system and the problem of finding a common fixed point for a commuting countable family of nonexpansive mappings.

A recent progress in algorithmic analysis of fifo queues with markovian arrival steams

Abstract: This paper summarizes recent development of analytical and algorithmical results for stationary FIFO queues with multiple Markovian arrival streams, where service time distributions are general and they may differ for different arrival streams. While this kind of queues naturally arises in considering queues with a superposition of independent phase-type arrivals, the conventional approach based on the queue length dynamics (i.e., M/G/1 pradigm) is not applicable to this kind of queues. On the contrary, the workload process has a Markovian property, so that it is analytically tractable. This paper first reviews the results for the stationary distributions of the amount of work-in-system, actual waiting time and sojourn time, all of which were obtained in the last six years by the author. Further this paper shows an alternative approach, recently developed by the author, to analyze the joint queue length distribution based on the waiting time distribution. An emphasis is placed on how to construct a numerically feasible recursion to compute the stationary queue length mass function.

High dimension prufer domains of integer-valued polynomials

Abstract: Let V be any valuation domain and let E be a subset of the quotient field K of V. We study the ring of integer-valued polynomials on E, that is, Int(E, V)={fK[X]｜f(E)⊆V}. We show that, if E is precompact, then Int(E, V) has many properties similar to those of the classical ring Int(Z).In particular, Int(E, V) is dense in the ring of continuous functions C(E, V); each finitely generated ideal of Int(E, V) may be generated by two elements; and finally, Int(E, V) is a Prufer domain.

Well-posedness for the benjamin equations

Abstract: We consider the time local well-posedness of the Ben- jamin equation. Like the result due to Kenig-Ponce-Vega (10), (12), we show that the initial value problem is time locally well posed in the Sobolev space H s (R) for s > i3=4.

Complete system of finite order for cr mappings between real analytic hypersurfaces of degenerate levi form

Abstract: We prove that the germ of a CR mapping f between real analytic real hypersurfaces has a holomorphic extension and satisfies a complete system of finite order if the source is of finite type in the sense of Bloom-Graham and the target is k-nondegenerate under cer- tain generic assumptions on f.

Reidemeister zeta function for group extensions

Abstract: In this paper, we study the rationality of the Reidemeister zeta function of an endomorphism of a group extension. As an application, we give sufficient conditions for the rationality of the Reidemeister and the Nielsen zeta functions of selfmaps on an exponential solvmanifold or an infra-nilmanifold or the coset space of a compact connected Lie group by a finite subgroup.

Integral gruss inequality for mappings with values in hilbert spaces and applications

Abstract: In this paper we prove a version of Gruss integral inequality for mappings with values in Hilbert spaces. Some applications for convex functions defined on Hibert space are also given.

A note on uniqueness and stability for the inverse conductivity problem with one measurement

Abstract: We consider the inverse conductivity problem to identify the unknown conductivity as well as the domain D. We show hat, unlike the case when is known, even a two or three dimensional ball may not be identified uniquely if the conductivity constant is not known. We find a necessary and sufficient condition on the Cauchy data (u│∂Ω, g) for the uniqueness in identification of and D. We also discuss on failure of stability.

White noise approach to Feynman integrals

Abstract: The trajectory of a classical dynamics is determined by the least action principle. As soon as we come to quantum dynamics, we have to consider all possible trajectories which are proposed to be a sum of the classical trajectory and Brownian fluctuation. Thus, the action involves the square of the derivative B(t) (white noise) of a Brownian motion B(t). The square is a typical example of a generalized white noise functional. The Feynman propagator should therefore be an average of a certain generalized white noise functional. This idea can be applied to a large class of dynamics with various kinds of Lagrangians.

A fubini theorem for analytic feynman integrals with applications

Abstract: In this paper we establish a Fubini theorem for various analytic Wiener and Feynman integrals. We then proceed to obtain several integration formulas as corollaries.

A generalization of the krasnoselskii-petryshyn compression and expansion theorem: an essential map approach

Abstract: This paper introduces the notions of essential and inessential map for countably -set contractive maps These ideas enable us to establish a Krasnoselskii-Petryshyn compression and expansion theorem in a cone for countably -set contractive maps

A note on approximate similarity

Abstract: This paper answers some old questions about approximate similarity and raises new ones. We provide positive evidence and a technique for finding negative evidence on the question of whether approximate similarity is the equivalence relation generated by approximate equivalence and similarity.

Functional integration, kontsevich integral and formal integration

Abstract: This paper is an exposition of the relationship between Witten's functional integral and Vassiliev invariants.

Penalized approach and analysis of an optimal shape control problem for the stationary navier-stokes equations

Abstract: This paper is concerned with an optimal shape control problem for the stationary Navier-Stokes system. A two-dimensional channel flow ofan incompressible, viscous fluid is examined to deter- mine the shape ofa bump on a part ofthe boundary that minimizes the viscous drag. By introducing an artificial compressibility term to relax the incompressibility constraints, we take the penalty method. The existence ofoptimal solutions f or the penalized problem will be shown. Next, by employing Lagrange multipliers method and the ma- terial derivatives, we derive the shape gradient for the minimization problem ofthe shape f which represents the viscous drag.

Hadamard and dragomir-agarwal inequalities, higher-order convexity and the euler formula

Abstract: We obtain bounds relating to Euler's formula for the case of a function with higher-order convexity properties. These are used to derive estimates of the error involved in the use of the trapezoidal formula for integrating such a function.

Feynman integrals, diffusion processes and quantum symplectic two-forms

Abstract: This is an introduction to a stochastic version of E. Cartan's symplectic mechanics. A class of time-symmetric ("Bern- stein") diusion processes is used to deform stochastically the ex- terior derivative of the Poincare-Cartan one-form on the extended phase space. The resulting symplectic two-form is shown to contain the (a.e.) dynamical laws of the diusions. This can be regarded as a geometrization of Feynman's path integral approach to quantum theory; when Planck's constant reduce to zero, we recover Cartan's mechanics. The underlying strategy is the one of "Euclidean Quan- tum Mechanics".

Generalizations of the nash equilibrium theorem on generalized convex spaces

Abstract: Generalized forms of the von neumann-Sion type minimax theorem, the Fan-Ma intersection theorem, the Fan-a type analytic alternative, and the Nash-Ma equilibrium theorem hold for generalized convex spaces without having any linear structure.