• Title, Summary, Keyword: 2-inner product

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ON GRAMS DETERMINANT IN 2-INNER PRODUCT SPACES

  • Cho, Y.J.;Matic, M.;Pecaric, J.
    • Journal of the Korean Mathematical Society
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    • v.38 no.6
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    • pp.1125-1156
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    • 2001
  • An analogue of Grams inequality for 2-inner product spaces is given. Further, a number of inequalities involving Grams determinant are stated and proved in terms of 2-inner products.

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FUZZY SEMI-INNER-PRODUCT SPACE

  • Cho, Eui-Whan;Kim, Young-Key;Shin, Chae-Seob
    • The Pure and Applied Mathematics
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    • v.2 no.2
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    • pp.163-172
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    • 1995
  • G.Lumer [8] introduced the concept of semi-product space. H.M.El-Hamouly [7] introduced the concept of fuzzy inner product spaces. In this paper, we defined fuzzy semi-inner-product space and investigated some properties of fuzzy semi product space.

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CHARACTERIZATIONS OF AN INNER PRODUCT SPACE BY GRAPHS

  • Lin, C.S.
    • The Pure and Applied Mathematics
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    • v.16 no.4
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    • pp.359-367
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    • 2009
  • The graph of the parallelogram law is well known, which gives rise to the characterization of an inner product space among normed linear spaces [6]. In this paper we will sketch graphs of its deformations according to our previous paper [7, Theorem 3.1 and 3.2]; each one of which characterizes an inner product space among normed linear spaces. Consequently, the graphs of some classical characterizations of an inner product space follow easily.

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On Bessel's and Grüss Inequalities for Orthonormal Families in 2-Inner Product Spaces and Applications

  • Dragomir, Sever Silverstru;Cho, Yeol-Je;Kim, Seong-Sik;Kim, Young-Ho
    • Kyungpook Mathematical Journal
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    • v.48 no.2
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    • pp.207-222
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    • 2008
  • A new counterpart of Bessel's inequality for orthonormal families in real or complex 2-inner product spaces is obtained. Applications for some Gr$\"{u}$ss inequality for determinantal integral inequalities are also provided.

An Analysis of the Vector and Inner Product Concepts in Geometry and Vector Curriculum ('기하와 벡터' 교육과정의 벡터와 내적 개념 분석)

  • Shin, BoMi
    • Journal of the Korean School Mathematics Society
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    • v.16 no.4
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    • pp.841-862
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    • 2013
  • This study analyzed issues in the mathematics curriculum concerning the cognitive development of the vector and inner product concepts in the light of Tall's and Watson's research(Tall, 2004a; Tall, 2004b; Watson et al., 2003; Watson, 2002). Some suggestions in teaching the vector and inner product concepts were elaborated in the terms of these analyses. First, the position vector needs to be represented by an arrow on the coordinate system in order to introduce the component form of a vector represented by a directed line segment. Second, proofs of the vector operation law should be carried out by symbolic manipulations based on the algebraic concept of a vector in the symbolic world. Third, it is appropriate that the inner product is defined as $\vec{a}{\cdot}\vec{b}=a_1b_1+a_2b_2$ (when, $\vec{a}=(a_1,a_2)$, $\vec{b}=(b_1,b_2)$) when it comes to considering the meaning of the inner product relevant to vector space in the formal world. Cognitive growth of concepts of the vector and inner product can be properly induced through revising explanation methods about the concepts in the curriculum in the basis of the above suggestions.

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SOME NEW RESULTS RELATED TO BESSEL AND GRUSS INEQUALITIES IN 2-INNER PRODUCT SPACES AND APPLICATIONS

  • DRAGOMIR S.S.;CHO, Y.J.;KIM, S.S.
    • Bulletin of the Korean Mathematical Society
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    • v.42 no.3
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    • pp.591-608
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    • 2005
  • Some new reverses of Bessel's inequality for orthonormal families in real or complex 2-inner product spaces are pointed out. Applications for some Gruss type inequalities and for determinantal integral inequalities are given as well.

Connecting the Inner and Outer Product of Vectors Based on the History of Mathematics (수학사에 기초한 벡터의 내적과 외적의 연결)

  • Oh, Taek-Keun
    • Journal of Educational Research in Mathematics
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    • v.25 no.2
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    • pp.177-188
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    • 2015
  • In this paper, I investigated the historical development process for the product of two vectors in the plane and space, and draw implications for educational guidance to internal and external product of vectors based on it. The results of the historical analysis show that efforts to define the product of the two line segments having different direction in the plane justified the rules of complex algebraic calculations with its length of the product of their lengths and its direction of the sum of their directions. Also, the efforts to define the product of the two line segments having different direction in three dimensional space led to the introduction of quaternion. In addition, It is founded that the inner product and outer product of vectors was derived from the real part and vector part of multiplication of two quaternions. Based on these results, I claimed that we should review the current deployment method of making inner product and outer product as multiplications that are not related to each other, and suggested one approach for connecting the inner and outer product.