<i>Advanced Mathematical Methods in Science and Engineering </i>

Hayek, Sabih I.; Pierucci, Mauro

doi:10.1121/1.1428546

Cited by 5 publications

(6 citation statements)

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“…This method is appropriate for obtaining asymptotic value of complex integrals of the form γ F (z)e iT G(z) dz, where T > 0 is a large number and γ is a path for the integration. Assuming F (z) is a slowly varying function over γ and G(z) is an analytic function, one can argue that the major contribution to the integral comes from the point(s) z 0 at which G(z) has a minimum, whence [66] …”

Section: Exponential Estimatementioning

confidence: 99%

Accuracy versus run time in an adiabatic quantum search

2010

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Adiabatic quantum algorithms are characterized by their run time and accuracy. The relation between the two is essential for quantifying adiabatic algorithmic performance yet is often poorly understood. We study the dynamics of a continuous time, adiabatic quantum search algorithm and find rigorous results relating the accuracy and the run time. Proceeding with estimates, we show that under fairly general circumstances the adiabatic algorithmic error exhibits a behavior with two discernible regimes: The error decreases exponentially for short times and then decreases polynomially for longer times. We show that the well-known quadratic speedup over classical search is associated only with the exponential error regime. We illustrate the results through examples of evolution paths derived by minimization of the adiabatic error. We also discuss specific strategies for controlling the adiabatic error and run time.

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Section: Exponential Estimatementioning

confidence: 99%

Accuracy versus run time in an adiabatic quantum search

2010

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“…The inhomogeneous biharmonic equation arises in areas of continuum mechanics, including linear elasticity theory and the solution of Stokes flows (cf. [24,32,42]). Most important applications of the theory of functions of one complex variable were obtained in the plane theory of elasticity and in the approximate theory of plates subject to normal loading (cf.…”

Section: Preliminaries and Main Resultsmentioning

confidence: 99%

Schwarz-Type Lemma, Landau-Type Theorem, and Lipschitz-Type Space of Solutions to Inhomogeneous Biharmonic Equations

Chen

Wang

2018

J Geom Anal

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The purpose of this paper is to study the properties of the solutions to the inhomogeneous biharmonic equations: ∆(∆f ) = g, where g : D → C is a continuous function and D denotes the closure of the unit disk D in the complex plane C. In fact, we establish the following properties for those solutions: Firstly, we establish the Schwarz type lemma. Secondly, by using the obtained results, we get a Landau type theorem. Thirdly, we discuss their Lipschitz type property.2000 Mathematics Subject Classification. Primary: 31A30; Secondary: 31A05.

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“…where the constant s-wave spherical harmonic, Y d , can be obtained simply by keeping in mind that in d dimensions we have that [19]:…”

Section: A Theoretical Formulationmentioning

confidence: 99%

Confinement of two-body systems and calculations in d dimensions

Garrido

Jensen

2019

Phys. Rev. Research

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A continuous transition for a system moving in a three-dimensional (3D) space to moving in a lower-dimensional space, 2D or 1D, can be made by means of an external squeezing potential. A squeeze along one direction gives rise to a 3D to 2D transition, whereas a simultaneous squeeze along two directions produces a 3D to 1D transition, without going through an intermediate 2D configuration. In the same way, for a system moving in a 2D space, a squeezing potential along one direction produces a 2D to 1D transition. In this work we investigate the equivalence between this kind of confinement procedure and calculations without an external field, but where the dimension d is taken as a parameter that changes continuously from d = 3 to d = 1. The practical case of an external harmonic oscillator squeezing potential acting on a two-body system is investigated in details. For the three transitions considered, 3D → 2D, 2D → 1D, and 3D → 1D, a universal connection between the harmonic oscillator parameter and the dimension d is found. This relation is well established for infinitely large 3D scattering lengths of the two-body potential for 3D → 2D and 3D → 1D transitions, and for infinitely large 2D scattering length for the 2D → 1D case. For finite scattering lengths size corrections must be applied. The traditional wave functions for external squeezing potentials are shown to be uniquely related with the wave functions for specific non-integer dimension parameters, d.

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Advanced Mathematical Methods in Science and Engineering

Abstract: These reviews of books and other forms of information express the opinions of the individual reviewers and are not necessarily endorsed by the Editorial Board of this Journal.

Cited by 5 publications

References 0 publications

Accuracy versus run time in an adiabatic quantum search

Accuracy versus run time in an adiabatic quantum search

Schwarz-Type Lemma, Landau-Type Theorem, and Lipschitz-Type Space of Solutions to Inhomogeneous Biharmonic Equations

Confinement of two-body systems and calculations in d dimensions

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