• Communications for Statistical Applications and Methods
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• 제15권6호
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• pp.837-842
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• 2008

Students taking linear model courses have difficulty in determining which parametric functions are estimable when the design matrix of a linear model is rank deficient. In this note a special form of estimable functions is presented with a linear combination of some orthogonal estimable functions. Here, the orthogonality means the least squares estimators of the estimable functions are uncorrelated and have the same variance. The number of the orthogonal estimable functions composing the special form is equal to the rank of the design matrix. The orthogonal estimable functions can be easily obtained through the singular value decomposition of the design matrix.

• Structural Engineering and Mechanics
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• 제18권5호
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• pp.565-589
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• 2004

The present paper concerns the macroscopic overall description of rheologic properties for steel wire and synthetic fibre cables under variable loading actions according to non-linear creep and/or relaxation theory. The general constitutive equations of non-linear creep and/or relaxation of tension elements - cables under one-step and the variable stress or strain inputs using the product and two types of additive approximations of the kernel functions are presented in the paper. The derived non-linear constitutive equations describe a non-linear rheologic behaviour of the cables for a variable stress or strain history using the kernel functions determined only by one-step - constant creep or relaxation tests. The developed constitutive equations enable to simulate and to predict in a general way non-linear rheologic behaviour of the cables under an arbitrary loading or straining history. The derived constitutive equations can be used for the various tension structural elements with the non-linear rheologic properties under uniaxial variable stressing or straining.

• Journal of information and communication convergence engineering
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• 제2권3호
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• pp.187-192
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• 2004

A method using two dimension Haar functions approximation for solving the problem of a partial differential equation and estimating the parameters of a non-linear distributed parameter system (DPS) is presented. The applications of orthogonal functions, including Haar functions, and their transforms have been given much attention in system control and communication engineering field since 1970's. The Haar functions set forms a complete set of orthogonal rectangular functions similar in several respects to the Walsh functions. The algorithm adopted in this paper is that of estimating the parameters of non-linear DPS by converting and transforming a partial differential equation into a simple algebraic equation. Two dimension Haar functions approximation method is introduced newly to represent and solve a partial differential equation. The proposed method is supported by numerical examples for demonstration the fast, convenient capabilities of the method.

• ETRI Journal
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• 제15권2호
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• pp.35-51
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• 1993

This paper presents function approximation based on nonparametric estimation. As an estimation model of function approximation, a three layered network composed of input, hidden and output layers is considered. The input and output layers have linear activation units while the hidden layer has nonlinear activation units or kernel functions which have the characteristics of bounds and locality. Using this type of network, a many-to-one function is synthesized over the domain of the input space by a number of kernel functions. In this network, we have to estimate the necessary number of kernel functions as well as the parameters associated with kernel functions. For this purpose, a new method of parameter estimation in which linear learning rule is applied between hidden and output layers while nonlinear (piecewise-linear) learning rule is applied between input and hidden layers, is considered. The linear learning rule updates the output weights between hidden and output layers based on the Linear Minimization of Mean Square Error (LMMSE) sense in the space of kernel functions while the nonlinear learning rule updates the parameters of kernel functions based on the gradient of the actual output of network with respect to the parameters (especially, the shape) of kernel functions. This approach of parameter adaptation provides near optimal values of the parameters associated with kernel functions in the sense of minimizing mean square error. As a result, the suggested nonparametric estimation provides an efficient way of function approximation from the view point of the number of kernel functions as well as learning speed.

• 한국경영과학회지
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• 제11권1호
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• pp.36-43
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• 1986

In this study, the fuzzy programming is extended to handle various types of membership functions by transformation of the complicated fuzzy programming problems into the equivalent crisp linear programming problems with single objective. It is well-known that the fuzzy programming problem with linear membership functions (i.e., ramp type) can be easily transformed into a linear programming problem by introducing one dummy variable to minimize the worst unwanted deviation. However, until recently not many researches have been done to handle various general types of complicated linear membership functions which might be more realistic than ramp-or triangular-type functions. In order to handle these complicated membership functions, the goal dividing concept, which is based on the fuzzy set operation (i. e., intersection and union operations), has been prepared. The linear model obtained using the goal dividing concept is more efficient and single than the previous models [4, 8]. In addition, this result can be easily applied to any nonlinear membership functions by piecewise approximation since the membership function is continuous and monotone increasing or decreasing.

• 대한수학회보
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• 제60권4호
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• pp.985-1001
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• 2023

Recently, Alzer and Choi [2] introduced and studied a set of the four linear Euler sums with parameters. These sums are parametric extensions of Flajolet and Salvy's four kinds of linear Euler sums [9]. In this paper, by using the method of residue computations, we will establish two explicit combined formulas involving two parametric linear Euler sums S⁺⁺_p,q (a, b) and S^+-_p,q (a, b) defined by Alzer and Choi, which can be expressed in terms of a linear combinations of products of trigonometric functions, digamma functions and Hurwitz zeta functions.

• Kyungpook Mathematical Journal
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• 제54권3호
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• pp.365-374
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• 2014

In this paper, we introduce a new subclass of meromorphic functions defined in the punctured unit disc. We derive inclusion relationships, radius problem and some other interesting properties of this class are investigated.

• East Asian mathematical journal
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• 제19권2호
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• pp.207-212
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• 2003

Some new characterization of best approximants from level sets of convex functions in normed linear spaces in terms of norm derivatives are given.

• 대한전기학회:학술대회논문집
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• 대한전기학회 2003년도 학술회의 논문집 정보 및 제어부문 A
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• pp.131-134
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• 2003

In this paper, we consider the problem of finding the stability region in state space for uncertain linear systems with input saturation. For stability analysis, two Lyapunov functions are chosen. One is for the lineal region and the other is for the saturated legion. Piecewise Lyapunov functions are obtained by solving successive linear matrix inequalites(LMIs) relaxations. A sufficient condition for robust stability is derived in the form of stability region of initial conditions. A numerical example shows the effectiveness of the proposed method.

• 한국수학교육학회지시리즈E:수학교육논문집
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• 제24권1호
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• pp.49-65
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• 2010

본 연구는 함수의 개념정의와 개념이미지 형성을 위한 다양한 표상에서 정의역을 고려한 지도-학습을 권고하고자 함에 있다. 중학교 1학년 및 2학년의 수학 교과서에서 함수나 일차함수 개념을 도입하거나 이들 함수의 활용을 학습하는 데 탐구형 문제나 현실적 모델을 가져오고 이 모델로부터 표를 만들기도 하고 함수관계식을 찾게 한 다음 그래프를 그리게 하는 일련의 표상과 표상의 번역과정을 연습한다. 본 연구는 이러한 표상활동에서 일어나는 모델의 적절성과 표현의 적절성을 표상의 동치성과 정의역에 초점을 두어 논한다. 특히 일차함수의 활용에서 교과서에 제시된 휴대폰 요금문제나 저축문제 등에서 일차함수의 식과 그래프로 모델링하는 과정에 나타난 문제점을 비판적으로 검토하고 정의역의 중요성을 지적한다.