• 제목/요약/키워드: convex space

검색결과 408건 처리시간 0.026초

AN EXTENSION OF SCHNEIDER'S CHARACTERIZATION THEOREM FOR ELLIPSOIDS

  • Dong-Soo Kim;Young Ho Kim
    • 대한수학회보
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    • 제60권4호
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    • pp.905-913
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    • 2023
  • Suppose that M is a strictly convex hypersurface in the (n + 1)-dimensional Euclidean space 𝔼n+1 with the origin o in its convex side and with the outward unit normal N. For a fixed point p ∈ M and a positive constant t, we put 𝚽t the hyperplane parallel to the tangent hyperplane 𝚽 at p and passing through the point q = p - tN(p). We consider the region cut from M by the parallel hyperplane 𝚽t, and denote by Ip(t) the (n + 1)-dimensional volume of the convex hull of the region and the origin o. Then Schneider's characterization theorem for ellipsoids states that among centrally symmetric, strictly convex and closed surfaces in the 3-dimensional Euclidean space 𝔼3, the ellipsoids are the only ones satisfying Ip(t) = 𝜙(p)t, where 𝜙 is a function defined on M. Recently, the characterization theorem was extended to centrally symmetric, strictly convex and closed hypersurfaces in 𝔼n+1 satisfying for a constant 𝛽, Ip(t) = 𝜙(p)t𝛽. In this paper, we study the volume Ip(t) of a strictly convex and complete hypersurface in 𝔼n+1 with the origin o in its convex side. As a result, first of all we extend the characterization theorem to strictly convex and closed (not necessarily centrally symmetric) hypersurfaces in 𝔼n+1 satisfying Ip(t) = 𝜙(p)t𝛽. After that we generalize the characterization theorem to strictly convex and complete (not necessarily closed) hypersurfaces in 𝔼n+1 satisfying Ip(t) = 𝜙(p)t𝛽.

A note on convexity on linear vector space

  • Hong, Suk-Kang
    • Journal of the Korean Statistical Society
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    • 제1권1호
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    • pp.18-24
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    • 1973
  • Study on convexity has been improved in many statistical fields, such as linear programming, stochastic inverntory problems and decision theory. In proof of main theorem in Section 3, M. Loeve already proved this theorem with the $r$-th absolute moments on page 160 in [1]. Main consideration is given to prove this theorem using convex theorems with the generalized $t$-th mean when some convex properties hold on a real linear vector space $R_N$, which satisfies all properties of finite dimensional Hilbert space. Throughout this paper $\b{x}_j, \b{y}_j$ where $j = 1,2,......,k,.....,N$, denotes the vectors on $R_N$, and $C_N$ also denotes a subspace of $R_N$.

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APPROXIMATION RESULTS OF A THREE STEP ITERATION METHOD IN BANACH SPACE

  • Omprakash Sahu;Amitabh Banerjee
    • Korean Journal of Mathematics
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    • 제31권3호
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    • pp.269-294
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    • 2023
  • The purpose of this paper is to introduce a new three-step iterative process and show that our iteration scheme is faster than other existing iteration schemes in the literature. We provide a numerical example supported by graphs and tables to validate our proofs. We also prove convergence and stability results for the approximation of fixed points of the contractive-like mapping in the framework of uniformly convex Banach space. In addition, we have established some weak and strong convergence theorems for nonexpansive mappings.

REMARKS ON THE KKM PROPERTY FOR OPEN-VALUED MULTIMAPS ON GENERALIZED CONVEX SPACES

  • KIM HOONJOO;PARK SEHIE
    • 대한수학회지
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    • 제42권1호
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    • pp.101-110
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    • 2005
  • Let (X, D; ${\Gamma}$) be a G-convex space and Y a Hausdorff space. Then $U^K_C$(X, Y) ${\subset}$ KD(X, Y), where $U^K_C$ is an admissible class (dup to Park) and KD denotes the class of multimaps having the KKM property for open-valued multimaps. This new result is used to obtain a KKM type theorem, matching theorems, a fixed point theorem, and a coincidence theorem.

Crack Analysis of Piezoelectric Material Considering Bounded Uncertain Material Properties

  • Kim, Tae-Uk;Shin, Jeong-Woo
    • International Journal of Aeronautical and Space Sciences
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    • 제4권2호
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    • pp.9-16
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    • 2003
  • Piezoelectric materials are widely used to construct smart or adaptive structures. Although extensive efforts have been devoted to the analysis of piezoelectric materials in recent years, most researches have been conducted by assuming that the material properties are fixed and have no uncertainties. Intrinsically, material properties have a certain amount of scatter and such uncertainties can affect the performance of component. In this paper, the convex modeling is used to consider such uncertainties in calculating the crack extension force of piezoelectric material and the results are compared with the one obtained via the Monte Carlo simulation. Numerical results show that crack extension forces increase when uncertainties considered, which indicates that such uncertainties should not be ignored for reliable lifetime prediction. Also, the results obtained by the convex modeling and the Monte Carlo simulation show good agreement, which demonstrates the effectiveness of the convex modeling.

CONVERGENCE OF VISCOSITY APPROXIMATIONS TO FIXED POINTS OF NONEXPANSIVE NONSELF-MAPPINGS IN BANACH SPACES

  • Jung, Jong-Soo
    • East Asian mathematical journal
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    • 제24권1호
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    • pp.81-95
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    • 2008
  • Let E be a uniformly convex Banach space with a uniformly $G{\hat{a}}teaux$ differentiable norm, C a nonempty closed convex subset of E, and $T\;:\;C\;{\rightarrow}\;E$ a nonexpansive mapping satisfying the weak inwardness condition. Assume that every weakly compact convex subset of E has the fixed point property. For $f\;:\;C\;{\rightarrow}\;C$ a contraction and $t\;{\in}\;(0,\;1)$, let $x_t$ be a unique fixed point of a contraction $T_t\;:\;C\;{\rightarrow}\;E$, defined by $T_tx\;=\;tf(x)\;+\;(1\;-\;t)Tx$, $x\;{\in}\;C$. It is proved that if {$x_t$} is bounded, then $x_t$ converges to a fixed point of T, which is the unique solution of certain variational inequality. Moreover, the strong convergence of other implicit and explicit iterative schemes involving the sunny nonexpansive retraction is also given in a reflexive and strictly convex Banach space with a uniformly $G{\hat{a}}teaux$ differentiable norm.

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APPLICATIONS OF RESULTS ON ABSTRACT CONVEX SPACES TO TOPOLOGICAL ORDERED SPACES

  • Kim, Hoonjoo
    • 대한수학회보
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    • 제50권1호
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    • pp.305-320
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    • 2013
  • Topological semilattices with path-connected intervals are special abstract convex spaces. In this paper, we obtain generalized KKM type theorems and their analytic formulations, maximal element theorems and collectively fixed point theorems on abstract convex spaces. We also apply them to topological semilattices with path-connected intervals, and obtain generalized forms of the results of Horvath and Ciscar, Luo, and Al-Homidan et al..

COINCIDENCE THEOREMS ON A PRODUCT OF GENERALIZED CONVEX SPACES AND APPLICATIONS TO EQUILIBRIA

  • Park, Se-Hie;Kim, Hoon-Joo
    • 대한수학회지
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    • 제36권4호
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    • pp.813-828
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    • 1999
  • In this paper, we give a Peleg type KKM theorem on G-convex spaces and using this, we obtain a coincidence theorem. First, these results are applied to a whole intersection property, a section property, and an analytic alternative for multimaps. Secondly, these are used to proved existence theorems of equilibrium points in qualitative games with preference correspondences and in n-person games with constraint and preference correspondences for non-paracompact wetting of commodity spaces.

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GENERALIZED MINIMAX THEOREMS IN GENERALIZED CONVEX SPACES

  • Kim, Hoon-Joo
    • 호남수학학술지
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    • 제31권4호
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    • pp.559-578
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    • 2009
  • In this work, we obtain intersection theorem, analytic alternative and von Neumann type minimax theorem in G-convex spaces. We also generalize Ky Fan minimax inequality to acyclic versions in G-convex spaces. The result is applied to formulate acyclic versions of other minimax results, a theorem of systems of inequalities and analytic alternative.

𝛿-CONVEX STRUCTURE ON RECTANGULAR METRIC SPACES CONCERNING KANNAN-TYPE CONTRACTION AND REICH-TYPE CONTRACTION

  • Sharma, Dileep Kumar;Tiwari, Jayesh
    • 한국수학교육학회지시리즈B:순수및응용수학
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    • 제29권4호
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    • pp.293-306
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    • 2022
  • In the present paper, we introduce the notation of 𝛿-convex rectangular metric spaces with the help of convex structure. We investigate fixed point results concerning Kannan-type contraction and Reich-type contraction in such spaces. We also propound an ingenious example in reference of given new notion.