• Bulletin of the Korean Mathematical Society
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• v.40 no.3
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• pp.509-520
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• 2003

Let p be a prime number, $Q_{p}$ the field of p-adic numbers, K a finite extension of $Q_{p}$, $\bar{K}}$ a fixed algebraic closure of K and $C_{p}$ the completion of K with respect to the p-adic valuation. Let E be a closed subfield of $C_{p}$, containing K. Given elements $t_1$...,$t_{r}$ $\in$ E for which the field K($t_1$...,$t_{r}$) is dense in E, we construct integral bases of E over K.

• Journal of the Chungcheong Mathematical Society
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• v.22 no.4
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• pp.717-725
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• 2009

We show that to every real polynomial of degree n, there corresponds a certain basis for the space of polynomials of degree less than or equal to (n-1). As an application, we give a new proof for the existence and uniqueness of the partial fraction decomposition of a rational function.

• Journal of applied mathematics & informatics
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• v.9 no.1
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• pp.261-275
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• 2002

Besides asymptotic method, the method of orthogonal polynomials has been used to obtain the solution of the Fredholm integral equation. The principal (singular) part of the kerne1 which corresponds to the selected domain of parameter variation is isolated. The unknown and known functions are expanded in a Chebyshev polynomial and an infinite a1gebraic system is obtained.

• Kyungpook Mathematical Journal
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• v.28 no.1
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• pp.39-44
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• 1988

In 1940, B. H. Neumann, working with a system more general than a near-field, proved that the additive group of such a system (and of a near-field) is commutative. The algebraic structure he used is known as a Neumann system (N-system). Here, the prime N-systems are classified and for each possible characteristic, examples of N-systems which are neither near-fields nor rings are given. It is also shown that a necessary condition for the set of all odd polynomials over GF(p) to be an N-system is that p is a Fermat prime.

• Coupled systems mechanics
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• v.9 no.6
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• pp.563-573
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• 2020

Dynamic responses of a laminated composite cantilever beam under a harmonic are investigated in this study. The governing equations of problem are derived by using the Lagrange procedure. The Timoshenko beam theory is considered and the Ritz method is implemented in the solution of the problem. The algebraic polynomials are used with the trivial functions for the Ritz method. In the solution of dynamic problem, the Newmark average acceleration method is used in the time history. In the numerical examples, the effects of load parameter, the fiber orientation angles and stacking sequence of laminas on the dynamic responses of the laminated beam are investigated.

• Korean Journal of Mathematics
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• v.30 no.2
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• pp.187-198
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• 2022

In this paper, we introduce a new concept in number theory called ζ-factors associated with a positive integer n. Applications of ζ-factors are in the arrangement of the defining polynomials in cyclic n-roots algebraic system and are thoroughly investigated. More precisely, ζ-factors arise in the proofs of vanishing theorems in regard to associated prime factors of the system. Exact computations through concrete examples of positive dimensions for n = 16, 18 support the results.

• Transactions of the Korean Society for Noise and Vibration Engineering
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• v.15 no.4 s.97
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• pp.457-464
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• 2005

A three-dimensional (3-D) method of analysis is presented for determining the free vibration frequencies and mode shapes of thick, complete (not truncated) conical shells of revolution, Unlike conventional shell theories, which are mathematically two-dimensional (2-D). the present method is based upon the 3-D dynamic equations of elasticity. Displacement components $u_{r},\;u_{z},\;and\;u_{\theta}$ in the radial, axial, and circumferential directions, respectively, are taken to be sinusoidal in time, periodic in , and algebraic polynomials in the r and z directions. Potential (strain) and kinetic energies of the conical shells are formulated, the Ritz method is used to solve the eigenvalue problem, thus yielding upper bound values of the frequencies by minimizing the frequencies. As the degree of the polynomials is increased, frequencies converge to the exact values. Convergence to four-digit exactitude is demonstrated for the first five frequencies of theconical shells. Novel numerical results are presented for thick, complete conical shells of revolution based upon the 3-D theory. Comparisons are also made between the frequencies from the present 3-D Ritz method and a 2-D thin shell theory.

• Transactions of the Korean Society for Noise and Vibration Engineering
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• v.16 no.10 s.115
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• pp.1024-1035
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• 2006

A three-dimensional (3-D) method of analysis is presented for determining the free vibration frequencies and mode shapes of thick, complete (circumferentially closed), circular rings with an elliptical or circular cross-section. Displacement components $u_r,\;u_\theta\;and\;u_z$ in the radial, circumferential, and axial directions, respectively, are taken to be periodic in ${\theta}$ and in time, and algebraic polynomials in the r and z directions. Potential (strain) and kinetic energies of the circular rings are formulated, and upper bound values of the frequencies are obtained by minimizing the frequencies. As the degree of the polynomials is increased, frequencies converge to the exact values. Convergence to four-digit exactitude is demonstrated for the first five frequencies of the rings. Novel numerical results are presented for the circular rings having an elliptical cross-section based upon 3-D theory. Comparisons are also made between the frequencies from the present 3-D Ritz method and ones obtained from thin and thick ring theories, experiments, and another 3-D method.

• Structural Engineering and Mechanics
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• v.85 no.5
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• pp.621-633
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• 2023

The major objective of this paper is to study the receding contact problem between two functional graded layers under a flat indenter. The gravity is assumed negligible, and the shear moduli of both layers are assumed to vary exponentially along the thickness direction. In the absence of body forces, the problem is reduced to a system of Fredholm singular integral equations with the contact pressure and contact size as unknowns via Fourier integral transform, which is transformed into an algebraic one by the Gauss-Chebyshev quadratures and polynomials of both the first and second kinds. Then, an iterative speediest descending algorithm is proposed to numerically solve the system of algebraic equations. Both semi-analytical and finite element method, FEM solutions for the presented problem validate each other. To improve the accuracy of the numerical result of FEM, a graded FEM solution is performed to simulate the FGM mechanical characteristics. The results reveal the potential links between the contact stress/size and the indenter size, the thickness, as well as some other material properties of FGM.

• Journal of the Korea Institute of Information Security & Cryptology
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• v.33 no.3
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• pp.375-381
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• 2023

Fully homomorphic encryption enables algebraic operations on encrypted data, and recently, methods for approximating non-algebraic operations such as the maximum function have been studied. However, precise approximation of max-pooling operations for four or more numbers have not been researched yet. In this study, we propose a precise max-pooling approximation method using the composition of approximate polynomials of the maximum function and theoretically analyze its precision. Experimental results show that the proposed approximate max-pooling has a small amortized runtime of less than 1ms and high precision that matches the theoretical analysis.