• Title/Summary/Keyword: G. D. Birkhoff

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ON UNIVERSAL FUNCTIONS

  • Aron, Richard;Markose, Dinesh
    • Journal of the Korean Mathematical Society
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    • v.41 no.1
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    • pp.65-76
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    • 2004
  • An entire function $f\;{\in}\;H(\mathbb{C})$ is called universal with respect to translations if for any $g\;{\in}\;H(\mathbb{C}),\;R\;>\;0,\;and\;{\epsilon}\;>\;0$, there is $n\;{\in}\;{\mathbb{N}}$ such that $$\mid$f(z\;+\;n)\;-\;g(z)$\mid$\;<\;{\epsilon}$ whenever $$\mid$z$\mid$\;{\leq}\;R$. Similarly, it is universal with respect to differentiation if for any g, R, and $\epsilon$, there is n such that $$\mid$f^{(n)}(z)\;-\;g(z)$\mid$\;<\;{\epsilon}\;for\;$\mid$z$\mid$\;{\leq}\;R$. In this note, we review G. MacLane's proof of the existence of universal functions with respect to differentiation, and we give a simplified proof of G. D. Birkhoff's theorem showing the existence of universal functions with respect to translation. We also discuss Godefroy and Shapiro's extension of these results to convolution operators as well as some new, related results and problems.

A Study for Design based on Measure and Information as BIRKHOFF's Aesthetic Category (디자인을 위한 비르크호프-G. D. Birkhoff의 미학적 카테고리로서의 척도와 인포메이션에 대한 고찰)

  • Choi, Sung-Woon;Kim, Kyung-Bae
    • Archives of design research
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    • v.17 no.4
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    • pp.107-118
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    • 2004
  • Nowadays Industrial Design appears to us as provided products in the market through only User Interface and Technical Improvement. Consumer purchasing products intend to grant the meaning for the product by mobilizing his specific feeling and experience the moment he contact products the first time. As metaphorical character shown in the product is named to the Product Language, Industrial designer shall constantly proceed the study on the exact understanding or processing method. As Objects or products using by us imply aesthetical value, it appears to be different according to the diverse condition of products be different according to the diverse condition of products to be used. In case objectivity is unsatisfactory, aesthetic character shown in the product requires new instrumental system. Product analysis applied aesthetical measure of BIRKHOFF can be applied as tool for inherent metaphorical character of product and product development. With this, Design operation shall be much easier. So Application of aesthetical measure is not only a system but a tool giving help to carry out more smooth design process.

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Comparison of early tertiary mathematics in USA and Korea (미국과 한국의 초기 고등수학 발전과정 비교연구)

  • Lee, Sang-Gu;Seol, Han-Guk;Ham, Yoon-Mee
    • Communications of Mathematical Education
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    • v.23 no.4
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    • pp.977-998
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    • 2009
  • In this article, we give a comparative study on the last 300 years of USA and Korean tertiary mathematics. The first mathematics classes in United States were offered before July, 1638, but the real founding of tertiary mathematics courses was in 1640 when Henry Dunster assumed the duties of the presidency at Harvard. President Dunster read arithmetics and geometry on Mondays and Tuesdays to the third year students during the first three quarters, and astronomy in the last quarter. So tertiary mathematics education in United States began at Harvard which is the oldest college in USA. After 230 years since then, Benjamin Peirce in 1870 made a major and first American contribution to mathematics and got an attention from European mathematicians. Major change on the role of Harvard mathematics from teaching to research made by G.D. Birkhoff when he joined as an assistant professor in 1912. Tertiary mathematics education in Korea started long before Chosun Dynasty. But it was given to only small number of government actuarial officers. Modern mathematics education of tertiary level in Korea was given at Sungkyunkwan, Ewha, Paichai, and Soongsil. But all college level education opportunity, particularly in mathematics, was taken over by colonial government after 1920. And some technical and normal schools offered some tertiary mathematics courses. There was no college mathematics department in Korea until 1945. After the World War II, the first college mathematics department was established, and Rimhak Ree in 1949 made a major and first Korean contribution to modern mathematics, and later found Ree group. He got an attention from western mathematicians for the first time as a Korean. It can be compared with Benjamin Peirce's contribution for USA.

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Ground of the revolutionary change in early 20C American Mathematics (20세기 초 미국수학계의 혁명적변화의 바탕)

  • Lee, Sang-Gu;Hwang, Suk-Geun;Cheon, Gi-Sang
    • Journal for History of Mathematics
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    • v.20 no.3
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    • pp.127-146
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    • 2007
  • From 1876 to 1883, British mathematician James Joseph Sylvester worked as the founding head of Mathematics Department at the Johns Hopkins University which has been known as America's first school of mathematical research. Sylvester established the American Journal of Mathematics, the first sustained mathematics research journal in the United States. It is natural that we think this is the most exciting and important period in American mathematics. But we found out that the International Congress of Mathematicians held at the World's Columbian Exposition in Chicago, August 21-26, 1893 was the real turning point in American's dedication to mathematical research. The University of Chicago was founded in 1890 by the American Baptist Education Society and John D. Rockefeller. The founding head of mathematics department Eliakim Hastings Moore was the one who produced many excellent American mathematics Ph.D's in early stage. Many of Moore's students contributed to build up real American mathematics research power in early 20 century The University also has a well-deserved reputation as the "teacher of teachers". Beginning with Sylvester, we analyze what E.H. Moore had done as a teacher and a head of the new department that produced many mathematical talents such as L.E. Dickson(1896), H. Slaught(1898), O. Veblen(1903), R.L. Moore(1905), G.D. Birkhoff(1907), T.H. Hilderbrants(1910), E.W. Chittenden(1912) who made the history of American mathematics. In this article, we study how Moore's vision, new system and new way of teaching influenced American mathematical society at early stage of the top class mathematical research. and the meaning that early University of Chicago case gave.

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Contribution of Oswald Veblen to AMS and its meaning in Korea (Oswald Veblen이 미국수학계에 미친 영향과 한국에서의 의미)

  • Lee, Sang-Gu;Ham, Yoon-Mee
    • Journal for History of Mathematics
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    • v.22 no.2
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    • pp.27-52
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    • 2009
  • This article discusses the contributions of the leader Oswald Veblen, who was the president of AMS during 1923-1924. In 2006, Korea ranked 12th in SCIE publications in mathematics, more than doubling its publications in less than 10 years, a successful model for a country with relatively short history of modern mathematical research. Now there are 192 four-year universities in Korea. Some 42 of these universities have Ph.D. granting graduate programs in mathematics and/or mathematical education in Korea. Rapid growth is observed over a broad spectrum including a phenomenal performance surge in International Mathematical Olympiad. Western mathematics was first introduced in Korea in the 17th century, but real significant mathematical contributions by Korean mathematicians in modern mathematics were not much known yet to the world. Surprisingly there is no Korean mathematician who could be found in MaC Tutor History Birthplace Map. We are at the time, to have a clear vision and leadership for the 21st century. Even with the above achievement, Korean mathematical community has had obstacles in funding. Many people thinks that mathematical research can be done without funding rather unlike other science subjects, even though they agree fundamental mathematical research is very important. We found that the experience of early American mathematical community can help us to give a vision and role model for Korean mathematical community. When we read the AMS Notice article 'The Vision, Insight, and Influence of Oswald Veblen' by Steve Batterson, it answers many of our questions on the development of American mathematics in early 20th century. We would like to share the story and analyze its meaning for the development of Korean Mathematics of 21st century.

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A Destructive Method in the Connection of the Algorithm and Design in the Digital media - Centered on the Rapid Prototyping Systems of Product Design - (디지털미디어 환경(環境)에서 디자인 특성(特性)에 관한 연구(硏究) - 실내제품(室內製品) 디자인을 중심으로 -)

  • Kim Seok-Hwa
    • Journal of Science of Art and Design
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    • v.5
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    • pp.87-129
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    • 2003
  • The purpose of this thesis is to propose a new concept of design of the 21st century, on the basis of the study on the general signification of the structures and the signs of industrial product design, by examining the difference between modern and post-modern design, which is expected to lead the users to different design practice and interpretation of it. The starting point of this study is the different styles and patterns of 'Gestalt' in the post-modern design of the late 20th century from modern design - the factor of determination in industrial product design. That is to say, unlike functional and rational styles of modern product design, the late 20th century is based upon the pluralism characterized by complexity, synthetic and decorativeness. So far, most of the previous studies on design seem to have excluded visual aspects and usability, focused only on effective communication of design phenomena. These partial studies on design, blinded by phenomenal aspects, have resulted in failure to discover a principle of fundamental system. However, design varies according to the times; and the transformation of design is reflected in Design Pragnanz to constitute a new text of design. Therefore, it can be argued that Design Pragnanz serves as an essential factor under influence of the significance of text. In this thesis, therefore, I delve into analysis of the 20th century product design, in the light of Gestalt theory and Design Pragnanz, which have been functioning as the principle of the past design. For this study, I attempted to discover the fundamental elements in modern and post-modern designs, and to examine the formal structure of product design, the users' aesthetic preference and its semantics, from the integrative viewpoint. Also, with reference to history and theory of design my emphasis is more on fundamental visual phenomena than on structural analysis or process of visualization in product design, in order to examine the formal properties of modern and post-modern designs. Firstly, In Chapter 1, 'Issues and Background of the Study', I investigated the Gestalt theory and Design Pragnanz, on the premise of formal distinction between modern and post-modern designs. These theories are founded upon the discussion on visual perception of Gestalt in Germany in 1910's, in pursuit of the principle of perception centered around visual perception of human beings. In Chapter 2, I dealt with functionalism of modern design, as an advance preparation for the further study on the product design of the late 20th century. First of all, in Chapter 2-1, I examined the tendency of modern design focused on functionalism, which can be exemplified by the famous statement 'Form follows function'. Excluding all unessential elements in design - for example, decoration, this tendency has attained the position of the international style based on the spirit of Bauhause - universality and regularity - in search of geometric order, standardization and rationalization. In Chapter 2-2, I investigated the anthropological viewpoint that modern design started representing culture in a symbolic way including overall aspects of the society - politics, economics and ethics, and its criticism on functionalist design that aesthetic value is missing in exchange of excessive simplicity in style. Moreover, I examined the pluralist phenomena in post-modern design such as kitsch, eclecticism, reactionism, hi-tech and digital design, breaking away from functionalist purism of modern design. In Chapter 3, I analyzed Gestalt Pragnanz in design in a practical way, against the background of design trends. To begin with, I selected mass product design among those for the 20th century products as a target of analysis, highlighting representative styles in each category of the products. For this analysis, I adopted the theory of J. M Lehnhardt, who gradated in percentage the aesthetic and semantic levels of Pragnantz in design expression, and that of J. K. Grutter, who expressed it in a formula of M = O : C. I also employed eight units of dichotomies, according to the G. D. Birkhoff's aesthetic criteria, for the purpose of scientific classification of the degree of order and complexity in design; and I analyzed phenomenal aspects of design form represented in each unit. For Chapter 4, I executed a questionnaire about semiological phenomena of Design Pragnanz with 28 units of antonymous adjectives, based upon the research in the previous chapter. Then, I analyzed the process of signification of Design Pragnanz, founded on this research. Furthermore, the interpretation of the analysis served as an explanation to preference, through systematic analysis of Gestalt and Design Pragnanz in product design of the late 20th century. In Chapter 5, I determined the position of Design Pragnanz by integrating the analyses of Gestalt and Pragnanz in modern and post-modern designs In this process, 1 revealed the difference of each Design Pragnanz in formal respect, in order to suggest a vision of the future as a result, which will provide systemic and structural stimulation to current design.

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