Separation of the Schrödinger operator with an operator potential in the Hilbert spaces

Mohamed, Adel S.; Atia, H.A.

doi:10.1080/0036810410001712790

Cited by 10 publications

(7 citation statements)

References 6 publications

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“…This knowledge can be used to solve the partial differential equation explicitly. Following [63], the solution of (D.1) is unique. The rest easily follows from showing that the provided density function satisfies the partial differential equation and its boundary conditions.…”

Section: Proof Of Corollarymentioning

confidence: 99%

Ensuring privacy with constrained additive noise by minimizing Fisher information

Farokhi

2019

Automatica

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The problem of preserving the privacy of individual entries of a database when responding to linear or nonlinear queries with constrained additive noise is considered. For privacy protection, the response to the query is systematically corrupted with an additive random noise whose support is a subset or equal to a pre-defined constraint set. A measure of privacy using the inverse of the trace of the Fisher information matrix is developed. The Cramér-Rao bound relates the variance of any estimator of the database entries to the introduced privacy measure. The probability density that minimizes the trace of the Fisher information (as a proxy for maximizing the measure of privacy) is computed. An extension to dynamic problems is also presented. Finally, the results are compared to the differential privacy methodology.

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Section: Proof Of Corollarymentioning

confidence: 99%

Ensuring privacy with constrained additive noise by minimizing Fisher information

Farokhi

2019

Automatica

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“…More fundamental results of separation for differential operators were obtained by Everitt and Giertz [3,4]. A number of results concerning the property referred to the separation of differential operators was discussed by Biomatov [5], Otelbaev [6], Zettle [7] and Mohamed et al [8][9][10][11][12][13]. The separation for the differential operators with the matrix potential was first studied by Bergbaev [14].…”

Section: Introductionmentioning

confidence: 99%

Inequalities and Separation for a Biharmonic Laplace-Beltrami Differential Operator in a Hilbert Space Associated with the Existence and Uniqueness Theorem

Zayed¹

2016

JPDE

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“…More fundamental results of separation of differential operator were obtained by Everitt and Giertz [8,9]. A number of results concerning the property referred to the separation of differential operators was discussed by Biomatov [2], Otelbaev [16], Zettle [20] and Mohamed et al [10][11][12][13][14][15]. The separation for the differential operators with the matrix potentials was first studied by Bergbaev [1].…”

Section: Introductionmentioning

confidence: 99%

Separation for the biharmonic differential operator in the Hilbert space associated with the existence and uniqueness theorem

Zayed

2008

Journal of Mathematical Analysis and Applications

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In this paper, we have studied the separation for the following biharmonic differential operator:is the space of all bounded linear operators on the Hilbert space H 1 and u is the biharmonic differential operator, while u = n i=1 ∂ 2 u ∂x 2 i is the Laplace operator in R n . Moreover, we have studied the existence and uniqueness of the solution of the biharmonic differential equationin the Hilbert space H , where f (x) ∈ H .

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Separation of the Schrödinger operator with an operator potential in the Hilbert spaces

Cited by 10 publications

References 6 publications

Ensuring privacy with constrained additive noise by minimizing Fisher information

Ensuring privacy with constrained additive noise by minimizing Fisher information

Inequalities and Separation for a Biharmonic Laplace-Beltrami Differential Operator in a Hilbert Space Associated with the Existence and Uniqueness Theorem

Separation for the biharmonic differential operator in the Hilbert space associated with the existence and uniqueness theorem

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