• Title, Summary, Keyword: n-inner product space

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FUZZY n-INNER PRODUCT SPACE

  • Vijayabalaji, Srinivasan;Thillaigovindan, Natesan
    • Bulletin of the Korean Mathematical Society
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    • v.44 no.3
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    • pp.447-459
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    • 2007
  • The purpose of this paper is to introduce the notion of fuzzy n-inner product space. Ascending family of quasi ${\alpha}$-n-norms corresponding to fuzzy quasi n-norm is introduced and we provide some results on it.

BOHR'S INEQUALITIES IN n-INNER PRODUCT SPACES

  • Cheung, W.S.;Cho, Y.S.;Pecaric, J.;Zhao, D.D.
    • The Pure and Applied Mathematics
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    • v.14 no.2
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    • pp.127-137
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    • 2007
  • The classical Bohr's inequality states that $|z+w|^2{\leq}p|z|^2+q|w|^2$ for all $z,\;w{\in}\mathbb{C}$ and all p, q>1 with $\frac{1}{p}+\frac{1}{q}=1$. In this paper, Bohr's inequality is generalized to the setting of n-inner product spaces for all positive conjugate exponents $p,\;q{\in}\mathbb{R}$. In. In particular, the parallelogram law is recovered and an interesting operator inequality is obtained.

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FUNCTIONAL EQUATIONS ASSOCIATED WITH INNER PRODUCT SPACES

  • Park, Choonkil;Huh, Jae Sung;Min, Won June;Nam, Dong Hoon;Roh, Seung Hyeon
    • Journal of the Chungcheong Mathematical Society
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    • v.21 no.4
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    • pp.455-466
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    • 2008
  • In, [7], Th.M. Rassias proved that the norm defined over a real vector space V is induced by an inner product if and only if for a fixed integer $n{\geq}2$ $$n{\left\|{\frac{1}{n}}{\sum\limits_{i=1}^{n}}x_i{\left\|^2+{\sum\limits_{i=1}^{n}}\right\|}{x_i-{\frac{1}{n}}{\sum\limits_{j=1}^{n}x_j}}\right\|^2}={\sum\limits_{i=1}^{n}}{\parallel}x_i{\parallel}^2$$ holds for all $x_1,{\cdots},x_{n}{\in}V$. Let V,W be real vector spaces. It is shown that if a mapping $f:V{\rightarrow}W$ satisfies $$(0.1){\hspace{10}}nf{\left({\frac{1}{n}}{\sum\limits_{i=1}^{n}}x_i \right)}+{\sum\limits_{i=1}^{n}}f{\left({x_i-{\frac{1}{n}}{\sum\limits_{j=1}^{n}}x_i}\right)}\\{\hspace{140}}={\sum\limits_{i=1}^{n}}f(x_i)$$ for all $x_1$, ${\dots}$, $x_{n}{\in}V$ $$(0.2){\hspace{10}}2f\(\frac{x+y}{2}\)+f\(\frac{x-y}{2} \)+f\(\frac{y}{2}-x\)\\{\hspace{185}}=f(x)+f(y)$$ for all $x,y{\in}V$. Furthermore, we prove the generalized Hyers-Ulam stability of the functional equation (0.2) in real Banach spaces.

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STABILITY OF FUNCTIONAL EQUATIONS ASSOCIATED WITH INNER PRODUCT SPACES: A FIXED POINT APPROACH

  • Park, Choonkil;Hur, Jae Sung;Min, Won June;Nam, Dong Hoon;Roh, Seung Hyeon
    • Korean Journal of Mathematics
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    • v.16 no.3
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    • pp.413-424
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    • 2008
  • In [21], Th.M. Rassias proved that the norm defined over a real vector space V is induced by an inner product if and only if for a fixed integer $n{\geq}2$ $$n{\parallel}\frac{1}{n}\sum\limits_{i=1}^{n}x_i{\parallel}^2+\sum\limits_{i=1}^{n}{\parallel}x_i-\frac{1}{n}\sum\limits_{j=1}^{n}x_j{\parallel}^2=\sum\limits_{i=1}^{n}{\parallel}x_i{\parallel}^2$$ holds for all $x_1,{\dots},x_n{\in}V$. We consider the functional equation $$nf(\frac{1}{n}\sum\limits^n_{i=1}x_i)+\sum\limits_{i=1}^{n}f(x_i-\frac{1}{n}\sum\limits_{j=1}^{n}x_j)=\sum\limits_{i=1}^nf(x_i)$$ Using fixed point methods, we prove the generalized Hyers-Ulam stability of the functional equation $$(1)\;2f(\frac{x+y}{2})+f(\frac{x-y}{2})+f(\frac{y-x}{2})=f(x)+f(y)$$.

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WEIGHTED MOORE-PENROSE INVERSES OF ADJOINTABLE OPERATORS ON INDEFINITE INNER-PRODUCT SPACES

  • Qin, Mengjie;Xu, Qingxiang;Zamani, Ali
    • Journal of the Korean Mathematical Society
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    • v.57 no.3
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    • pp.691-706
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    • 2020
  • Necessary and sufficient conditions are provided under which the weighted Moore-Penrose inverse AMN exists, where A is an adjointable operator between Hilbert C-modules, and the weights M and N are only self-adjoint and invertible. Relationship between weighted Moore-Penrose inverses AMN is clarified when A is fixed, whereas M and N are variable. Perturbation analysis for the weighted Moore-Penrose inverse is also provided.

QUADRATIC MAPPINGS ASSOCIATED WITH INNER PRODUCT SPACES

  • Lee, Sung Jin
    • Korean Journal of Mathematics
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    • v.19 no.1
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    • pp.77-85
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    • 2011
  • In [7], Th.M. Rassias proved that the norm defined over a real vector space V is induced by an inner product if and only if for a fixed integer $n{\geq}2$ $${\sum_{i=1}^{n}}\left\|x_i-{\frac{1}{n}}{\sum_{j=1}^{n}}x_j \right\|^2={\sum_{i=1}^{n}}{\parallel}x_i{\parallel}^2-n\left\|{\frac{1}{n}}{\sum_{i=1}^{n}}x_i \right\|^2$$ holds for all $x_1$, ${\cdots}$, $x_n{\in}V$. Let V, W be real vector spaces. It is shown that if an even mapping $f:V{\rightarrow}W$ satisfies $$(0.1)\;{\sum_{i=1}^{2n}f}\(x_i-{\frac{1}{2n}}{\sum_{j=1}^{2n}}x_j\)={\sum_{i=1}^{2n}}f(x_i)-2nf\({\frac{1}{2n}}{\sum_{i=1}^{2n}}x_i\)$$ for all $x_1$, ${\cdots}$, $x_{2n}{\in}V$, then the even mapping $f:V{\rightarrow}W$ is quadratic. Furthermore, we prove the generalized Hyers-Ulam stability of the quadratic functional equation (0.1) in Banach spaces.

Lp FOURIER-FEYNMAN TRANSFORMS AND CONVOLUTION

  • Ahn, Jae Moon
    • Korean Journal of Mathematics
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    • v.7 no.2
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    • pp.183-198
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    • 1999
  • Let $\mathcal{F}(B)$ be the Fresnel class on an abstract Wiener space (B, H, ${\omega}$) which consists of functionals F of the form : $$F(x)={\int}_H\;{\exp}\{i(h,x)^{\sim}\}df(h),\;x{\in}B$$ where $({\cdot}{\cdot})^{\sim}$ is a stochastic inner product between H and B, and $f$ is in $\mathcal{M}(H)$, the space of all complex-valued countably additive Borel measures on H. We introduce the concepts of an $L_p$ analytic Fourier-Feynman transform ($1{\leq}p{\leq}2$) and a convolution product on $\mathcal{F}(B)$ and verify the existence of the $L_p$ analytic Fourier-Feynman transforms for functionls in $\mathcal{F}(B)$. Moreover, we verify that the Fresnel class $\mathcal{F}(B)$ is closed under the $L_p$ analytic Fourier-Feynman transform and the convolution product, respectively. And we investigate some interesting properties for the $n$-repeated $L_p$ analytic Fourier-Feynman transform on $\mathcal{F}(B)$. Finally, we show that several results in [9] come from our results in Section 3.

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A Study on features and Interpretation of Placeness of Rem Koolhaas' Architecture (렘 콜하스 건축의 장소적 특성과 해석에 관한 연구)

  • Park, Hyung-Jin;Kim, Moon-Duck
    • Korean Institute of Interior Design Journal
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    • v.16 no.2
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    • pp.87-96
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    • 2007
  • This study analyzes the place of modem architecture based on the place theory of C. N, Schultz. For applying Schultz's theory to the modern architecture, It is required to examine the modern cityscape, features of inner space of architecture and features of program. By analyzing the avant-garde architecture of Rem Koolhaas on such basis, the potentiality of placeness of modern architecture could be verified and the alternatives would be searched. It is inferred that the placeness features of Rem Koolhaas' public architecture is under the influence of the interpretation of program based on the humane background rather than the physical aspects of surroundings. The inner space shows the non-linear features, the metaphor of city. The obscurity of physical boundary illustrates the flexible features with ambiguous boundary. Consequently, the inner space expresses the surreal atmosphere that doesn't match the purposes of usage of architecture, the traditional concept. The outer shape is recognized as the by-product from the interpretation of internal program rather than it considered the surrounding context. The outer shape has the relatively simple formative shape and contrasts against the complicated inner space by using the non-physical materials. It is found that Koolhaas' architecture doesn't pursue the features of placeness of traditional concept. However, It is inferred that his architecture has the possibility of placeness by attaching the meaning through the social roles of each architecture. It gives the substantial suggestion to the modern architecture that can't easily acquire the placeness of traditional concept due to the environment of modern city.

EVALUATION OF SOME CONDITIONAL ABSTRACT WIENER INTEGRALS

  • Chung, Dong-Myung;Kang, Soon-Ja
    • Bulletin of the Korean Mathematical Society
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    • v.26 no.2
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    • pp.151-158
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    • 1989
  • Let (H, B, .nu.) be an abstract Wiener space where H is a separable Hilbert space with the inner product <.,.> and the norm vertical bar . vertical bar=.root.<.,.>, which is densely and continuously imbedded into a separable Banach space B with the norm ∥.∥ , and .nu. is a probability measure on the Borel .sigma.-algebra B(B) of B which satisfies (Fig.) where $B^{*}$ is the topological dual of B and (.,.) is the natural dual pairing between B and $B^{*}$. We will regard $B^{*}$.contnd.H.contnd.B in the natural way. Thus we have =(y, x) for all y in $B^{*}$ and x in H. Let $R^{n}$ and C denote the n-dimensional Euclidean space and the complex numbers respectively.ctively.

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