• Journal of Educational Research in Mathematics
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• v.12 no.2
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• pp.229-246
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• 2002

In this paper, we start with the question "what is algebraic thinking\ulcorner". The problem is that the algebraic thinking is not exactly defined. We consider algebraic thinking from the various perspectives. But in the discussion relating to the definition of algebraic thinking, we verify that there is the algebraic notations in the core of algebraic thinking. So we device algebraic notations into the six categories, and investigate these examples from the school mathematics. In order to investigate this relation of algebraic thinking and algebraic notations, we present 'the algebraic thinking process analysis model' from Frege' idea. In this model, there are three components of algebraic notations which interplays; sense, expression, denotapion. Thus many difficulties of algebraic thinking can be explained by this model. We suppose that the difficulty in the algebraic thinking may be caused by the discord of these three components. And through the transformation of conceptual frame, we can explain the dynamics of algebraic thinking. Also, we present examples which show these difficulties and dynamics of algebraic thinking. As a result of these analysis, we conclude that algebraic thinking can be explained through the semiotic aspects of algebraic notations.

• Journal of Mechanical Science and Technology
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• v.16 no.11
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• pp.1522-1539
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• 2002

In the present study, an explicit algebraic stress model is shown to be the exact tensor representation of algebraic stress model by directly solving a set of algebraic equations without resort to tensor representation theory. This repeals the constraints on the Reynolds stress, which are based on the principle of material frame indifference and positive semi-definiteness. An a priori test of the explicit algebraic stress model is carried out by using the DNS database for a fully developed channel flow at Rer = 135. It is confirmed that two-point correlation function between the velocity fluctuation and the Laplacians of the pressure-gradient i s anisotropic and asymmetric in the wall-normal direction. Thus, a novel composite algebraic Reynolds stress model is proposed and applied to the channel flow calculation, which incorporates non-local effect in the algebraic framework to predict near-wall behavior correctly.

• Journal of the Korean Mathematical Society
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• v.56 no.2
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• pp.539-558
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• 2019

An ideal of a commutative ring is called a d-ideal if it contains the annihilator of the annihilator of each of its elements. Denote by DId(A) the lattice of d-ideals of a ring A. We prove that, as in the case of f-rings, DId(A) is an algebraic frame. Call a ring homomorphism "compatible" if it maps equally annihilated elements in its domain to equally annihilated elements in the codomain. Denote by $SdRng_c$ the category whose objects are rings in which the sum of two d-ideals is a d-ideal, and whose morphisms are compatible ring homomorphisms. We show that $DId:\;SdRng_c{\rightarrow}CohFrm$ is a functor (CohFrm is the category of coherent frames with coherent maps), and we construct a natural transformation $RId{\rightarrow}DId$, in a most natural way, where RId is the functor that sends a ring to its frame of radical ideals. We prove that a ring A is a Baer ring if and only if it belongs to the category $SdRng_c$ and DId(A) is isomorphic to the frame of ideals of the Boolean algebra of idempotents of A. We end by showing that the category $SdRng_c$ has finite products.

• Journal of the Korea institute for structural maintenance and inspection
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• v.9 no.3
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• pp.139-150
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• 2005

Finite element method is applied to the determinations of the two eigenvalues(the elastic critical load and the natural frequence of lateral vibrations) of single story-3 equal bay rectangular frame. The analysis parameters are taper parameter ${\alpha}$ for column, and beam span to column height ratio, ${\beta}$ and second moment area ratio of beam to column, ${\Upsilon}$. Support condition at the column base and sway condition at the column top are also considered in the stability analysis of frame. The changes in the coefficient of eigenvalue are represented by algebraic function of analysis parameter. The coefficients estimated by the proposed algebraic function show good agreement with those determined by finite element method, which suggest the design aid role of the proposed function. By increasing the column axial forces step by step, the corresponding frequencies are also determined, which makes one examine or confirm the relationship suggested by other studies.

• Journal of the Korean Mathematical Society
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• v.54 no.1
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• pp.215-225
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• 2017

In this paper, we study some equivalence relations between continuous frames in a Hilbert space ${\mathcal{H}}$. In particular, we seek two necessary and sufficient conditions under which two continuous frames are near. Moreover, we investigate a distance between continuous frames in order to acquire the closest and nearest tight continuous frame to a given continuous frame. Finally, we implement these results for shearlet and wavelet frames in two examples.

• Transactions of the Korean Society of Mechanical Engineers A
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• v.33 no.11
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• pp.1341-1346
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• 2009

In this paper, the technique to reinforce the durability performance of structure using the sensitivity information for the frame structure is applied. The fatigue life calculation for the frame structure is performed from the quasi-static and transient analysis and the characteristics of two methods are compared for the fatigue analysis. Then the reinforcement technique is applied. First, some design variables related to the locations of fatigue failure is selected. Then sensitivities of fatigue life at fracture points with respect to the variation of design variables are calculated and the vector composed of gaps between the target life and initial life cycles is calculated. If the number of fatigue fracture points is same as the number of design variables, the variations of the design variables are calculated from the linear algebraic equation. If not, the variations of the design variables are calculated from the optimization formulation with the constraints.

• Journal of Welding and Joining
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• v.16 no.3
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• pp.141-147
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• 1998

Design sensitivity analysis of a vehicle system is an essential tool for design optimization and trade-off studies. Most optimization algorithms require the derivatives of cost and constraint function with respect to design in order to calculate the next improved design. This paper presents an efficient algorithm application for the design sensitivity analysis, using the direct differentiation method. A mounting area of suspension that welded on chassis frame is analyzed to show the validity and the efficiency of the proposed method. A mounting area of suspension that welded on chassis frame is analyzed to show the validity and the efficiency of the proposed method.

• Proceedings of the Computational Structural Engineering Institute Conference
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• 2003.10a
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• pp.243-250
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• 2003

A method for the determination of effective length factors of the framed columns with sinusoidally tapered sections is proposed. In the study, the stability analysis of the single story-two equal bay, frame with tapered columns is performed first by finite element method. The changes of the critical load coefficients of frames are reprersented by algebraic equations of the analysis parameters. The effective length factor formula is expressed in terms of proposed algebraic equation. The effective length factors for the prismatic columns (α=0.0) estimated by the proposed method coincide fairly well with those determined by the analytical method.

• Journal of the Korean Statistical Society
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• v.11 no.2
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• pp.118-130
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• 1982

Random walks as specia Markov stochastic processes have received particular attention in recent years. Not only the applicability of the theory already developed but also its extension within the frame work of probability measures on algebraic-topological structures such as semigroups, groups and linear spaces became a new challenge for research work in the field. At the same time new insights into classical problems were obtained which in various cases lead to a more efficient presentation of the subject. Consequently the teaching of random walks at all levels should profit from the recent development.

• Journal of the Korean Mathematical Society
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• v.48 no.5
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• pp.899-912
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• 2011

We introduce ${\varphi}$-frames in $L^2$(G), as a generalization of a-frames defined in [8], where G is a locally compact Abelian group and ${\varphi}$ is a topological automorphism on G. We give a characterization of ${\varphi}$-frames with regard to usual frames in $L^2$(G) and show that ${\varphi}$-frames share several useful properties with frames. We define the associated ${\varphi}$-analysis and ${\varphi}$-preframe operators, with which we obtain criteria for a sequence to be a ${\varphi}$-frame or a ${\varphi}$-Bessel sequence. We also define ${\varphi}$-Riesz bases in $L^2$(G) and establish equivalent conditions for a sequence in $L^2$(G) to be a ${\varphi}$-Riesz basis.