• Title/Summary/Keyword: Hilbert

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A metric characterization of Hilbert spaces

  • Mok, Jin-Sik
    • Bulletin of the Korean Mathematical Society
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    • v.33 no.1
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    • pp.35-38
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    • 1996
  • The aim of this paper is to present a characterization of Hilbert spaces in terms of the lengths of four sides and two diagonals of a parallelogram.

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Some New Hilbert Type Inequalities

  • Salem, Shaban Raslan
    • Kyungpook Mathematical Journal
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    • v.46 no.1
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    • pp.19-29
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    • 2006
  • In this paper, we obtain some inequalities similar to Hilbert type. Some new inequalities are also given.

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AN EXISTENCE OF LINEAR SYSTEMS WITH GIVEN TRANSFER FUNCTION

  • Yang, Meehyea
    • Bulletin of the Korean Mathematical Society
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    • v.30 no.1
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    • pp.99-107
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    • 1993
  • A vector space K with scalar product <.,.> is called a Krein space if it can be decomposed as a northogonal sum of a Hilbert space and an anti-space of a Hilbert space. The space K induces a Hilbert space $K_{J}$ in the inner product <.,.> $K_{J}$=<.,.>K, where $J^{2}$=I. the eigenspaces of J are denoted by $K^{+}$$_{J}$, which is a Hilbert space and $K^{-}$$_{J}$, which is an anti-space of a Hilbert space. Then the Krein space K is the orthogonal sum of $K^{+}$$_{J}$ and $K^{-}$$_{J}$. Such a decomposition of K is called a fundamental decomposition. In general, fundamental decompositions are not unique. The norm of the Hilbert space depends on the choice of a fundamental decomposion, but such norms are equivalent. The topology generated by these norms is called the strong or Mackey topology of K. It is used to define all topological notions on the Krein space K with respect to this topology. The Pontryagin index of a Krein space is the dimension of the antispace of a Hilbert space in any such decomposition. the dimension does not depend on the choice of orthogonal decomposition. A Krein space is called a Pontryagin space if it has finite Pontryagin index.dex.yagin index.dex.

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The privacy protection algorithm of ciphertext nearest neighbor query based on the single Hilbert curve

  • Tan, Delin;Wang, Huajun
    • KSII Transactions on Internet and Information Systems (TIIS)
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    • v.16 no.9
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    • pp.3087-3103
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    • 2022
  • Nearest neighbor query in location-based services has become a popular application. Aiming at the shortcomings of the privacy protection algorithms of traditional ciphertext nearest neighbor query having the high system overhead because of the usage of the double Hilbert curves and having the inaccurate query results in some special circumstances, a privacy protection algorithm of ciphertext nearest neighbor query which is based on the single Hilbert curve has been proposed. This algorithm uses a single Hilbert curve to transform the two-dimensional coordinates of the points of interest into Hilbert values, and then encrypts them by the order preserving encryption scheme to obtain the one-dimensional ciphertext data which can be compared in numerical size. Then stores the points of interest as elements composed of index value and the ciphertext of the other information about the points of interest on the server-side database. When the user needs to use the nearest neighbor query, firstly calls the approximate nearest neighbor query algorithm proposed in this paper to query on the server-side database, and then obtains the approximate nearest neighbor query results. After that, the accurate nearest neighbor query result can be obtained by calling the precision processing algorithm proposed in this paper. The experimental results show that this privacy protection algorithm of ciphertext nearest neighbor query which is based on the single Hilbert curve is not only feasible, but also optimizes the system overhead and the accuracy of ciphertext nearest neighbor query result.

GENERALIZED JENSEN'S EQUATIONS IN A HILBERT MODULE

  • An, Jong Su;Lee, Jung Rye;Park, Choonkil
    • Korean Journal of Mathematics
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    • v.15 no.2
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    • pp.135-148
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    • 2007
  • We prove the stability of generalized Jensen's equations in a Hilbert module over a unital $C^*$-algebra. This is applied to show the stability of a projection, a unitary operator, a self-adjoint operator, a normal operator, and an invertible operator in a Hilbert module over a unital $C^*$-algebra.

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THE GENERALIZED INVERSES A(1,2)T,S OF THE ADJOINTABLE OPERATORS ON THE HILBERT C^*-MODULES

  • Xu, Qingxiang;Zhang, Xiaobo
    • Journal of the Korean Mathematical Society
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    • v.47 no.2
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    • pp.363-372
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    • 2010
  • In this paper, we introduce and study the generalized inverse $A^{(1,2)}_{T,S}$ with the prescribed range T and null space S of an adjointable operator A from one Hilbert $C^*$-module to another, and get some analogous results known for finite matrices over the complex field or associated rings, and the Hilbert space operators.

A Central Limit Theorem for the Linear Process in a Hilbert Space under Negative Association

  • Ko, Mi-Hwa
    • Communications for Statistical Applications and Methods
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    • v.16 no.4
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    • pp.687-696
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    • 2009
  • We prove a central limit theorem for the negatively associated random variables in a Hilbert space and extend this result to the linear process generated by negatively associated random variables in a Hilbert space. Our result implies an extension of the central limit theorem for the linear process in a real space under negative association to a simplest case of infinite dimensional Hilbert space.

CHARACTERIZATION OF THE HILBERT BALL BY ITS AUTOMORPHISMS

  • Kim, Kang-Tae;Ma, Daowei
    • Journal of the Korean Mathematical Society
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    • v.40 no.3
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    • pp.503-516
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    • 2003
  • We show in this paper that every domain in a separable Hilbert space, say H, which has a $C^2$ smooth strongly pseudoconvex boundary point at which an automorphism orbit accumulates is biholomorphic to the unit ball of H. This is the complete generalization of the Wong-Rosay theorem to a separable Hilbert space of infinite dimension. Our work here is an improvement from the preceding work of Kim/Krantz [10] and subsequent improvement of Byun/Gaussier/Kim [3] in the infinite dimensions.