• Journal of the Korean Mathematical Society
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• v.41 no.4
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• pp.717-735
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• 2004

A problem raised by Selfridge and solved by Pomerance asks to find the pairs (a, b) of natural numbers for which $2^a\;-\;2^b$ divides $n^a\;-\;n^b$ for all integers n. Vajaitu and one of the authors have obtained a generalization which concerns elements ${\alpha}_1,\;{\cdots},\;{{\alpha}_{\kappa}}\;and\;{\beta}$ in the ring of integers A of a number field for which ${\Sigma{\kappa}{i=1}}{\alpha}_i{\beta}^{{\alpha}i}\;divides\;{\Sigma{\kappa}{i=1}}{\alpha}_i{z^{{\alpha}i}}\;for\;any\;z\;{\in}\;A$. Here we obtain a further generalization, proving the corresponding finiteness results in a multidimensional setting.

• Journal for History of Mathematics
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• v.15 no.1
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• pp.83-92
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• 2002

Gator Szego was one of the most brilliant Mathematicians. Mathematical science owes him several fundamental contributions in such fields as theory of functions of a complex variables, conformal mapping, Fourier series, theory of orthogonal polynomials, and many others. He wrote the famous Polya-Szego Problems and Theorem in Analysis which is the two volume of concentrated mathematical beauty. In this paper, we mention Szego's life, Szego's work, and Szego reproducing kernel.

• Journal of the Korean Mathematical Society
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• v.40 no.3
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• pp.341-371
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• 2003

This article discusses in detail how the study of proper holomorphic rational mappings between balls in different dimensions relates to positivity conditions and to isometric imbedding of holomorphic bundles. The first chapter discusses rational proper mappings between balls; the second chapter discusses seven distinct positivity conditions for real-valued polynomials in several complex variables; the third chapter reveals how these issues relate to an isometric imbedding theorem for holomorphic vector bundles proved by the author and Catlin.

• Steel and Composite Structures
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• v.39 no.3
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• pp.291-306
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• 2021

Based on Reissner's mixed variational theorem (RMVT), the authors develop a semi-analytical finite element (FE) method for a three-dimensional (3D) bending analysis of nonhomogeneous orthotropic, complete and incomplete toroidal shells subjected to uniformly-distributed loads. In this formulation, the toroidal shell is divided into several finite annular prisms (FAPs) with quadrilateral cross-sections, where trigonometric functions and serendipity polynomials are used to interpolate the circumferential direction and meridian-radial surface variations in the primary field variables of each individual prism, respectively. The material properties of the toroidal shell are considered to be nonhomogeneous orthotropic over the meridianradial surface, such that homogeneous isotropic toroidal shells, laminated cross-ply toroidal shells, and single- and bi-directional functionally graded toroidal shells can be included as special cases in this work. Implementation of the current FAP methods shows that their solutions converge rapidly, and the convergent FAP solutions closely agree with the 3D elasticity solutions available in the literature.

• Proceedings of the KIEE Conference
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• 2003.07d
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• pp.2519-2521
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• 2003

In this paper, we propose Self-Organizing Fuzzy Polynomial Neural Networks(SOFPNN) architecture for optimal model identification and discuss a comprehensive design methodology supporting its development. It is shown that this network exhibits a dynamic structure as the number of its layers as well as the number of nodes in each layer of the SOFPNN are not predetermined (as this is the case in a popular topology of a multilayer perceptron). As the form of the conclusion part of the rules, especially the regression polynomial uses several types of high-order polynomials such as linear, quadratic, and modified quadratic. As the premise part of the rules, both triangular and Gaussian-like membership function are studied and the number of the premise input variables used in the rules depends on that of the inputs of its node in each layer. We introduce two kinds of SOFPNN architectures, that is, the basic and modified one with both the generic and the advanced type. The superiority and effectiveness of the proposed SOFPNN architecture is demonstrated through nonlinear function numerical example.

• Journal of the Korea Institute of Military Science and Technology
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• v.19 no.2
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• pp.227-235
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• 2016

Heavy-weight high speed underwater vehicle(HSUV) is launched from the submerged mother ship. For the safety point of view, it is important to confirm whether the HSUV would touch the launching mother ship. In this paper, the hydrodynamic force and moment were modeled by the polynomials of motion variables and the simple lift and drag acting on a plate and cylinder which consist of the HSUV's several parts. The mother ship was assumed as the Rankine half body to consider the flow field near the moving ship. Such hydrodynamic force and moment were included in the 6 DOF equations of motion of the HSUV and the dynamic simulations for the various conditions of the HSUV until the propeller activation were performed. Developed simulation program is expected to reduce the number of expensive sea trial test to develop safety logic of the HSUV at the initial firing stage.

• The Transactions of the Korean Institute of Electrical Engineers D
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• v.53 no.3
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• pp.135-144
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• 2004

In this paper, we propose competitive fuzzy polynomial neurons-based advanced Self-Organizing Neural Networks(SONN) architecture for optimal model identification and discuss a comprehensive design methodology supporting its development. The proposed SONN dwells on the ideas of fuzzy rule-based computing and neural networks. And it consists of layers with activation nodes based on fuzzy inference rules and regression polynomial. Each activation node is presented as Fuzzy Polynomial Neuron(FPN) which includes either the simplified or regression polynomial fuzzy inference rules. As the form of the conclusion part of the rules, especially the regression polynomial uses several types of high-order polynomials such as linear, quadratic, and modified quadratic. As the premise part of the rules, both triangular and Gaussian-like membership (unction are studied and the number of the premise input variables used in the rules depends on that of the inputs of its node in each layer. We introduce two kinds of SONN architectures, that is, the basic and modified one with both the generic and the advanced type. Here the basic and modified architecture depend on the number of input variables and the order of polynomial in each layer. The number of the layers and the nodes in each layer of the SONN are not predetermined, unlike in the case of the popular multi-layer perceptron structure, but these are generated in a dynamic way. The superiority and effectiveness of the Proposed SONN architecture is demonstrated through two representative numerical examples.

• Advances in aircraft and spacecraft science
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• v.5 no.3
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• pp.363-383
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• 2018

One-dimensional (1D) models of incompressible flows, can be of interest for many applications in which fast resolution times are demanded, such as fluid-structure interaction of flows in compliant pipes and hemodynamics. This work proposes a higher-order 1D theory for the flow-field analysis of incompressible, laminar, and viscous fluids in rigid pipes. This methodology is developed in the domain of the Carrera Unified Formulation (CUF), which was first employed in structural mechanics. In the framework of 1D modelling, CUF allows to express the primary variables (i.e., velocity and pressure fields in the case of incompressible flows) as arbitrary expansions of the generalized unknowns, which are functions of the 1D computational domain coordinate. As a consequence, the governing equations can be expressed in terms of fundamental nuclei, which are invariant of the theory approximation order. Several numerical examples are considered for validating this novel methodology, including simple Poiseuille flows in circular pipes and more complex velocity/pressure profiles of Stokes fluids into non-conventional computational domains. The attention is mainly focused on the use of hierarchical McLaurin polynomials as well as piece-wise nonlocal Lagrange expansions of the generalized unknowns across the pipe section. The preliminary results show the great advantages in terms of computational costs of the proposed method. Furthermore, they provide enough confidence for future extensions to more complex fluid-dynamics problems and fluid-structure interaction analysis.

• The Transactions of the Korean Institute of Electrical Engineers D
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• v.53 no.8
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• pp.561-570
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• 2004

In this paper, new architectures and comprehensive design methodologies of Genetic Algorithms(GAs) based Genetically optimized Neurofuzzy Networks(GoNFN) are introduced, and a series of numeric experiments are carried out. The proposed GoNFN is based on the rule-based Neurofuzzy Networks(NFN) with the extended structure of the premise and the consequence parts of fuzzy rules being formed within the networks. The premise part of the fuzzy rules are designed by using space partitioning in terms of fuzzy sets defined in individual variables. In the consequence part of the fuzzy rules, three different forms of the regression polynomials such as constant, linear and quadratic are taken into consideration. The structure and parameters of the proposed GoNFN are optimized by GAs. GAs being a global optimization technique determines optimal parameters in a vast search space. But it cannot effectively avoid a large amount of time-consuming iteration because GAs finds optimal parameters by using a given space. To alleviate the problems, the dynamic search-based GAs is introduced to lead to rapidly optimal convergence over a limited region or a boundary condition. In a nutshell, the objective of this study is to develop a general design methodology o GAs-based GoNFN modeling, come up a logic-based structure of such model and propose a comprehensive evolutionary development environment in which the optimization of the model can be efficiently carried out both at the structural as well as parametric level for overall optimization by utilizing the separate or consecutive tuning technology. To evaluate the performance of the proposed GoNFN, the models are experimented with the use of several representative numerical examples.