• Title/Summary/Keyword: Quasi norm

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MAX-NORM ERROR ESTIMATES FOR FINITE ELEMENT METHODS FOR NONLINEAR SOBOLEV EQUATIONS

  • CHOU, SO-HSIANG;LI, QIAN
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • v.5 no.2
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    • pp.25-37
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    • 2001
  • We consider the finite element method applied to nonlinear Sobolev equation with smooth data and demonstrate for arbitrary order ($k{\geq}2$) finite element spaces the optimal rate of convergence in $L_{\infty}\;W^{1,{\infty}}({\Omega})$ and $L_{\infty}(L_{\infty}({\Omega}))$ (quasi-optimal for k = 1). In other words, the nonlinear Sobolev equation can be approximated equally well as its linear counterpart. Furthermore, we also obtain superconvergence results in $L_{\infty}(W^{1,{\infty}}({\Omega}))$ for the difference between the approximate solution and the generalized elliptic projection of the exact solution.

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Identification of Manning's Roughness in 1D nonuniform flow (최적화 기법을 이용한 1차원 부등류에서의 매닝조도계수 추정)

  • Lee, Du-Han;Rhee, Dong-Sup;Kim, Myoung-Hwan
    • Proceedings of the Korea Water Resources Association Conference
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    • 2010.05a
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    • pp.679-683
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    • 2010
  • 본 연구에서는 공간적 변수인 조도계수를 기지의 수위값을 이용하여 최적값을 결정하는 방법에 대해서 검토하고자 한다. 최적화 기법에 의한 조도계수는 기지의 수위값과 수치모의에 의한 결과 값의 전체 오차를 최소화하는 값으로 결정된다. 본 연구에서는 3가지 최적화 기법을 이용하였으며 가상 수로에 대해서 적용하였다. 수위계산은 표준축차법에 의해 수행하였으며 사용된 최적화 기법은 quasi-Newton 방법이다. 1차원 모형은 Matlab을 이용하여 표준축자법으로 구성하였으며 BFGS 기법, L-BFGS 기법, Steepest Gradient Descent 기법 등도 Matlab으로 구성하였다. 표준축차법은 조도계수가 입력되면 기지의 수위값과의 2-norm을 계산하도록 구성하였다. 계산 결과에 의하면 세가 기법 모두 20 23회 정도의 반복계산을 수행하고 값이 수렴되었는데, L-BFGS의 경우에는 정확하게 음수의 조도계수로 수렴하였으며, BFGS기법과 Steepest Gradient 기법의 경우에는 양의 값으로 정확하게 수렴하였다.

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OUTER APPROXIMATION METHOD FOR ZEROS OF SUM OF MONOTONE OPERATORS AND FIXED POINT PROBLEMS IN BANACH SPACES

  • Abass, Hammad Anuoluwapo;Mebawondu, Akindele Adebayo;Narain, Ojen Kumar;Kim, Jong Kyu
    • Nonlinear Functional Analysis and Applications
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    • v.26 no.3
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    • pp.451-474
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    • 2021
  • In this paper, we investigate a hybrid algorithm for finding zeros of the sum of maximal monotone operators and Lipschitz continuous monotone operators which is also a common fixed point problem for finite family of relatively quasi-nonexpansive mappings and split feasibility problem in uniformly convex real Banach spaces which are also uniformly smooth. The iterative algorithm employed in this paper is design in such a way that it does not require prior knowledge of operator norm. We prove a strong convergence result for approximating the solutions of the aforementioned problems and give applications of our main result to minimization problem and convexly constrained linear inverse problem.

ITERATIVE ALGORITHMS FOR THE LEAST-SQUARES SYMMETRIC SOLUTION OF AXB = C WITH A SUBMATRIX CONSTRAINT

  • Wang, Minghui;Feng, Yan
    • Journal of applied mathematics & informatics
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    • v.27 no.1_2
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    • pp.1-12
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    • 2009
  • Iterative algorithms are proposed for the least-squares symmetric solution of AXB = E with a submatrix constraint. We characterize the linear mappings from their independent element space to the constrained solution sets, study their properties and use these properties to propose two matrix iterative algorithms that can find the minimum and quasi-minimum norm solution based on the classical LSQR algorithm for solving the unconstrained LS problem. Numerical results are provided that show the efficiency of the proposed methods.

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Model order reduction for Campbell diagram analysis of shaft-disc-blade system in 3D finite elements

  • Phuor, Ty;Yoon, GilHo
    • Structural Engineering and Mechanics
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    • v.81 no.4
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    • pp.411-428
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    • 2022
  • This paper presents the Campbell diagram analysis of the rotordynamic system using the full order model (FOM) and the reduced order model (ROM) techniques to determine the critical speeds, identify the stability and reduce the computational time. Due to the spin-speed-dependent matrices (e.g., centrifugal stiffening matrix), several model order reduction (MOR) techniques may be considered, such as the modal superposition (MS) method and the Krylov subspace-based MOR techniques (e.g., Ritz vector (RV), quasi-static Ritz vector (QSRV), multifrequency quasi-static Ritz vector (MQSRV), multifrequency/ multi-spin-speed quasi-static Ritz vector (MMQSRV) and the combined Ritz vector & modal superposition (RV+MS) methods). The proposed MMQSRV method in this study is extended from the MQSRV method by incorporating the rotational-speed-dependent stiffness matrices into the Krylov subspace during the MOR process. Thus, the objective of this note is to respond to the question of whether to use the MS method or the Krylov subspace-based MOR technique in establishing the Campbell diagram of the shaft-disc-blade assembly systems in three-dimensional (3D) finite element analysis (FEA). The Campbell diagrams produced by the FOM and various MOR methods are presented and discussed thoroughly by computing the norm of relative errors (ER). It is found that the RV and the MS methods are dominant at low and high rotating speeds, respectively. More precisely, as the spinning velocity becomes large, the calculated ER produced by the RV method is significantly increased; in contrast, the ER produced by the MS method is smaller and more consistent. From a computational point of view, the MORs have substantially reduced the time computing considerably compared to the FOM. Additionally, the verification of the 3D FE rotordynamic model is also provided and found to be in close agreement with the existing solutions.

SOLVABILITY AND BOUNDEDNESS FOR GENERAL VARIATIONAL INEQUALITY PROBLEMS

  • Luo, Gui-Mei
    • Bulletin of the Korean Mathematical Society
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    • v.50 no.2
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    • pp.589-599
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    • 2013
  • In this paper, we propose a sufficient condition for the existence of solutions to general variational inequality problems (GVI(K, F, $g$)). The condition is also necessary when F is a $g-P^M_*$ function. We also investigate the boundedness of the solution set of (GVI(K, F, $g$)). Furthermore, we show that when F is norm-coercive, the general complementarity problems (GCP(K, F, $g$)) has a nonempty compact solution set. Finally, we establish some existence theorems for (GNCP(K, F, $g$)).

Error estimation for 2-D crack analysis by utilizing an enriched natural element method

  • Cho, J.R.
    • Structural Engineering and Mechanics
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    • v.76 no.4
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    • pp.505-512
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    • 2020
  • This paper presents an error estimation technique for 2-D crack analysis by an enriched natural element (more exactly, enriched Petrov-Galerkin NEM). A bare solution was approximated by PG-NEM using Laplace interpolation functions. Meanwhile, an accurate quasi-exact solution was obtained by a combined use of enriched PG-NEM and the global patch recovery. The Laplace interpolation functions are enriched with the near-tip singular fields, and the approximate solution obtained by enriched PG-NEM was enhanced by the global patch recovery. The quantitative error amount is measured in terms of the energy norm, and the accuracy (i.e., the effective index) of the proposed method was evaluated using the errors which obtained by FEM using a very fine mesh. The error distribution was investigated by calculating the local element-wise errors, from which it has been found that the relative high errors occurs in the vicinity of crack tip. The differences between the enriched and non-enriched PG-NEMs have been investigated from the effective index, the error distribution, and the convergence rate. From the comparison, it has been justified that the enriched PG-NEM provides much more accurate error information than the non-enriched PG-NEM.

Effect Assessment of Safety Education in Laboratories of Department of Dental Technology Applied of Theory of Planned Behavior (TPB) (TPB 이론을 적용한 치기공학 전공 대학생들의 실습실 안전교육 효과 평가)

  • Park, Jong-Hee;Lee, Myung-Sun
    • The Journal of Korean Society for School & Community Health Education
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    • v.9 no.2
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    • pp.15-31
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    • 2008
  • Objective: This study is a quasi-experimental study by nonequivalent control group pretest-posttest design, performed in order to identify whether the program developed on the basis of Ajzen(1988, 1991)'s theory of planned behavior (TPB) for predicting the safety behavior of students of department of dental technology affects intention toward safety behavior and general behavior. Therefore, the purpose of this study is to identify whether the safety education in laboratories in which knowledge is reinforced on the basis of the TPB theory affects safety behavior. Methodology: The theoretical framework of this study was established on the basis of the TPB by Ajzen(1988, 1991). This study was performed between May 20 and December 5, 2006, and the subjects included 101 in the experimental group and 98 in the control group, a total of 199. The experimental group was educated through six 50-minute sessions (twice a week) for three weeks. The experimental group was pretested, educated, and then asked to answer the pro-questionnaire after finishing the sixth session, but the control group was free from intervention. The research tool in this study was developed by the authors with reference to the process of tool development used in Schifter and Ajzen (1985) based on Ajzen and Fishbein(1980)'s tool development guide. Results: The data were analyzed through SPSS 14.0 version. The reliability test of attitude, subjective norm, perceived norm, intention, and behavior toward safety behavior as TPB variables was calculated of Cronbach's alpha coefficient, and general characteristics and the homogeneity test of the experimental and the control groups were analyzed through t test and $X^2$ test. According to the results of the analysis, the experiment group may be higher than the control group after the safety education in the scores of attitude, the subjective norm, the perceived norm, and the intention toward safety behavior as TPB independent variables showed no significant difference, but there was significant difference in the behavior and the knowledge. Meanwhile, according to the results of the pretest and posttest, eleven items in the experimental group and two items in the control group showed significant difference in the behaviors, and twelve in the former and four in the latter were significantly different in the knowledge. Conclusion: The safety education program developed on the basis of TPB was identified to enhance safety behavior and knowledge. Therefore, the safety education program used in this study for students of department of dental technology may be utilized as database for safety education in the department of dental technology.

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Systematic Design Method of Fuzzy Logic Controllers by Using Fuzzy Control Cell (퍼지제어 셀을 이용한 퍼지논리제어기의 조직적인 설계방법)

  • 남세규;김종식;유완석
    • Transactions of the Korean Society of Mechanical Engineers
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    • v.16 no.7
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    • pp.1234-1243
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    • 1992
  • A systematic procedure to design fuzzy PID controllers is developed in this paper. The concept of local fuzzy control cell is proposed by introducing both an adequate global control rule and membership functions to simplify a fuzzy logic controller. Fuzzy decision is made by using algebraic product and parallel firing arithematic mean, and a defuzzification strategy is adopted for improving the computational efficiency based on nonfuzzy micro-processor. A direct method, transforming the typical output of quasi-linear fuzzy operator to the digital compensator of PID form, is also proposed. Finally, the proposed algorithm is applied to an DC-servo motor. It is found that this algorithm is systematic and robust through computer simulations and implementation of controller using Intel 8097 micro-processor.

On the Bibliographies of Chinese Historical Books - Classifying and cataloguing system of six historical bibliographies - (중국의 사지서목에 대하여 -육사예문$\cdot$경적지의 분류 및 편목체재 비교를 중심으로-)

  • Kang Soon-Ae
    • Journal of the Korean Society for Library and Information Science
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    • v.24
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    • pp.289-332
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    • 1993
  • In china, six bibliographies of offical historical books are evaluated at the most important things among the systematically-editing bibliographies. These bibliographies would be usful to study the orign of classical sciences and their development, bibliographic research of Chinese classics, bibliographic judgement on genuine books, titles, authors, volumes. They could be refered to research into graving, correcting, and existence of ancient books. therefore, these bibliographies would be applied to estimation the phase of scientific and cultural development. The study of these bibliographies has been not yet made in Korea. This thesis lays its importance on the background of their appearance, their classification norms, organizing system of their catalogue, and comparison between their difference. 1. Editing and compiling of Chilyak (칠약) by Liu Chin (유흠) and official histories played an important role of entering an apperance of historical book's bibliographies. Chilyak has been lost. However, its classification and compiling system of classical books would be traced by Hansoyemunji(한서예문지) of which basic system is similar to Chilyak. It classified books according to their scientific characteristic. If a few books didn't have their own categories, they were combined by the circles parallel to the books' characteristic. With the books classified under the same scientific characteristic, they were again divided into the scientific schools or structures. It also arranged the same kinds of books according to the chronology. The some books wi th duplicate subjects were classified multiplely by their duplicate subject. 2. Ssu-ma Chon's (사마천) The Historical Records (Saki, 사기) and Pan Ku's (반고) The History of the Former Han Dynasty (Hanso, 한서) has also took effects on appearance of historical books' bibliographies. Covering overall history, Saki was structured by the five parts: The basic annals(본기), the chronological tables (표), the documents (서), the hereditary houses (세가), biographies (열전). The basic annals dealt with kings and courts' affairs according to the chronology. The chronological tables was the records of the annals. The documents described overall the social and cultural systems. The hereditary houses recorded courts' meritorious officials and public figures. The biographies showed exemplars of seventy peoples selected by their social status. Pan Ku(반구)'s The History of the Former Han Dynasty(한서) deserved to be called the prototype for the offical histories after Saki's (사기; The Historical Records) apperance. Although it modelled on Saki, it had set up its own cataloguing system. It was organized by four parts; the basic annals (본기), the chronological tables (표), treatises(지), biographies (열전). The documents in the Hanso(한서) was converted into treatises(지). The hereditary houses and biographies were merged. For the first time, the treatise with The Yemunji could operate function for historical bibliographies. 3. There were six historical bibliographies: Hansoyemunji(한서예문지), Susokyongjeokji (수서경적지), Kudangsokyongjeokji(구당서경적지), Shindangsoyemunji (신당서예문지), Songsayemunji (송사예문지), Myongsayemunji (명사예문지). 1) Modelling on Liu Chin's Chilyak except Chipryak(집략), Hansoyemunji divided the characteristic of the books and documents into six parts: Yukrye(육예), Cheja(제자), Shibu(시부), Pyongsoh(병서), Susul(수술), Pangki(방기). Under six parts, there were thirty eight orders in Hansoyemunji. To its own classification, Hansoyemunji applied the Chilyak's theory of classification that the books or documents were managed according to characteristic of sciences, the difference of schools, the organization of sentences. However the overlapped subjects were deleted and unified into one. The books included into an unsuitable subject were corrected and converted into another. The Hansoyemunji consisted of main preface (Taesoh 대서), minor preface (Sosoh 소서) , the general preface (Chongso 총서). It also recorded the introduction of books and documents, the origin of sciences, the outline of subjects, and the establishment of orders. The books classified by the subject had title, author, and volumes. They were rearranged by titles and the chronological publication year. Sometimes author was the first access point to catalogue the books. If it was necessary for the books to take footnotes, detail notes were formed. The Volume number written consecutively to order and subject could clarify the quantity of books. 2) Refering to Classfication System by Seven Norms (칠분법) and Classification System by Four Norms(사분법), Susokyongjeokji(수서경적지) had accomplished the classification by four norms. In fact, its classification largely imitated Wanhyosoh(완효서)'s Chilrok(칠록), Susokyongjeokji's system of classification consisted of four parts-Kyung(경), Sa(사), Cha(자), Chip(칩). The four parts were divided into 40 orders. Its appendix was again divided into two parts, Buddihism and Taiosm. Under the two parts there were fifteen orders. Totally Susokyongjeokji was made of six parts and fifty five orders. In comparison with Hansoyemunji(한서예문지), it clearly showed the conception of Kyung, Sa, Cha, Chip. Especially it deserved to be paid attention that Hansoyemunji laied history off Chunchu(춘추) and removed history to Sabu(사부). However Chabu(사부) put many contrary subjects such as Cheja(제자), Kiye(기예), Sulsu(술수), Sosol(소설) into the same boundary, which committed errors insufficient theoretical basis. Anothor demerit of Susokyongjeokji was that it dealt with Taiosm scriptures and Buddism scriptures at the appendix because they were considered as quasi-religion. Its compilation of bibliographical facts consisted of main preface(Taesoh 대서), minor preface(Sosoh 소서), general preface (Chongsoh 총서), postscript (Husoh 후서). Its bibliological facts mainly focused on the titles. Its recorded authors' birth date and their position. It wrote the lost and existence of books consecutive to total number of books, which revealed total of the lost books in Su Dynasty. 3) Modelling on the basis of Kokumsorok(고분서록) and Naewaekyongrok(내외경록), Kudangsokyongjeokji(구당서경적지) had four parts and fourty five orders. It was estimated as the important role of establishing basic frame of classification by four norms in classification theory's history. However it had also its own limit. Editing and compling orders of Kudangsokyongjeokji had been not progressively changed. Its orders imitated by and large Susokyongjeokji. In Its system of organizing catalogue, with its minor preface and general preface deleting, Kudangsokyongjeokji by titles after orders sometimes broke out confusion because of unclear boundaries between orders. 4) Shindangsoyemunji(신당서예문지), adding 28,469 books to Kudangsokyongjeokji, recorded 82,384 books which were divided by four parts and fourty four orders. In comparison with Kudangkyongjeokj, Sindangsoyemunji corrected unclear order's norm. It merged the analogical norms four orders (for instance, Kohun 고훈 and Sohakryu 소학류) and seperated the different norms four orders (for example, Hyokyong 효경 and Noneuhryu 논어류, Chamwi 참위 and Kyonghaeryu 경해류, Pyonryon 편년 and Wisaryu 위사류). Recording kings' behaviors and speeches (Kikochuryu 기거주류) in the historical parts induced the concept of specfication category. For the first time, part of Chipbu (집부) set up the order of classification norm for historical and literatural books and documents (Munsaryu 문사류). Its editing and compiling had been more simplified than Kudangsokyongjeokji. Introduction was written at first part of bibliographies. Appendants except bibliographic items such subject, author, title, volume number, total were omitted. 5) Songsayemunji(송사예문지) were edited in the basis of combining Puksong(북송) and Namsong(남송), depending on Sabukuksayemunji(사부국사예문지). Generally Songsayemunji had lost a lot of bibliographical facts of many books. They were duplicated and wrongly classified books because it committed an error of the incorrectly annalistic editing. Particularly Namsong showed more open these defaults. Songsayemunji didin't include the books published since the king Youngchong(영종). Its system of classification was more better controlled. Chamwiryu(참위류) in the part of Kyongbu(경부) was omitted. In the part of history(Sabu 사부), recordings of kings' behaviors and speeches more merged in the annals. Historical abstract documents (Sachoryu 사초류) were seperately arranged. In the part of Chabu(자부), Myongdangkyongmaekryu(명당경맥류) and Euisulryu(의술류) were combined. Ohangryu(오행류) were laied off Shikuryu(시구류). In the part of Chipbu(집부), historical and literatural books (Munsaryu 문사류) were independentely arranged. There were the renamed orders; from Wisa(위사) to Paesa(패사), Chapsa (잡사) to Pyolsa(열사), Chapchonki(잡전기) to Chonki(전기), Ryusoh(류서) to Ryusa(류서). Introduction had only main preface. The books of each subject catalogued by title, the volume number, and author and arranged mainly by authors. Annotations were written consecutively after title and the volume number. In the afternote the number of not-treated books were revealed. Difference from Singdangsohyemunji(신당서예문지) were that the concept and boundary of orders became more clearer. It also wrote the number of books consecutive to main subject. 6) Modelling on Chonkyongdangsomok (경당서목), Myongsayemunji(명사예문지) was compiled in the basis of books and documents published in the Ming Danasty. In classification system, Myongsayemunji partly merged and the seperated some orders for it. It also deleted and renamed some of orders. In case of necessity, combining of orders' norm was occured particulary in the part of Sabu(사부) and Chabu(자부). Therefore these merging of orders norm didn't offer sufficient theretical background. For example, such demerits were seen in the case that historical books edited by annals were combined with offical historical ones which were differently compiled and edited from the former. In the part of Chabu(자부), it broke out another confusion that Pubga(법가), Meongga(명가), Mukga(묵가), Chonghweongka's(종횡가) thoughts were classified in the Chapka(잡가). Scriptures of Taiosim and Buddhism were seperated from each other. There were some deleted books such as Mokrokryu(목록류), Paesaryu(패사류) in the part of history (Sabu 사부) and Chosaryu(초사류) in the part of Chipbu(집부). The some in the each orders had been renamed. Imitating compiling system of Songsayemunji(송사예문지), with reffering to its differ-ence, Myongsayemunji(명사예문지) wrote the review and the change of the books by author. The number of not-treated books didn't appear at the total. It also deleted the total following main subject.

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