41,306 research outputs found
Discrete Fourier analysis with lattices on planar domains
A discrete Fourier analysis associated with translation lattices is developed
recently by the authors. It permits two lattices, one determining the integral
domain and the other determining the family of exponential functions. Possible
choices of lattices are discussed in the case of lattices that tile \RR^2 and
several new results on cubature and interpolation by trigonometric, as well as
algebraic, polynomials are obtained
Discrete Fourier Analysis and Chebyshev Polynomials with Group
The discrete Fourier analysis on the
-- triangle is deduced from the
corresponding results on the regular hexagon by considering functions invariant
under the group , which leads to the definition of four families
generalized Chebyshev polynomials. The study of these polynomials leads to a
Sturm-Liouville eigenvalue problem that contains two parameters, whose
solutions are analogues of the Jacobi polynomials. Under a concept of
-degree and by introducing a new ordering among monomials, these polynomials
are shown to share properties of the ordinary orthogonal polynomials. In
particular, their common zeros generate cubature rules of Gauss type
Discrete Fourier analysis, Cubature and Interpolation on a Hexagon and a Triangle
Several problems of trigonometric approximation on a hexagon and a triangle
are studied using the discrete Fourier transform and orthogonal polynomials of
two variables. A discrete Fourier analysis on the regular hexagon is developed
in detail, from which the analysis on the triangle is deduced. The results
include cubature formulas and interpolation on these domains. In particular, a
trigonometric Lagrange interpolation on a triangle is shown to satisfy an
explicit compact formula, which is equivalent to the polynomial interpolation
on a planer region bounded by Steiner's hypocycloid. The Lebesgue constant of
the interpolation is shown to be in the order of . Furthermore, a
Gauss cubature is established on the hypocycloid.Comment: 29 page
Pairing Properties of Symmetric Nuclear Matter in Relativistic Mean Field Theory
The properties of pairing correlations in symmetric nuclear matter are
studied in the relativistic mean field (RMF) theory with the effective
interaction PK1. Considering well-known problem that the pairing gap at Fermi
surface calculated with RMF effective interactions are three times larger than
that with Gogny force, an effective factor in the particle-particle channel is
introduced. For the RMF calculation with PK1, an effective factor 0.76 give a
maximum pairing gap 3.2 MeV at Fermi momentum 0.9 fm, which are
consistent with the result with Gogny force.Comment: 14 pages, 6 figures
Variational quantum simulation of general processes
Variational quantum algorithms have been proposed to solve static and dynamic
problems of closed many-body quantum systems. Here we investigate variational
quantum simulation of three general types of tasks---generalised time evolution
with a non-Hermitian Hamiltonian, linear algebra problems, and open quantum
system dynamics. The algorithm for generalised time evolution provides a
unified framework for variational quantum simulation. In particular, we show
its application in solving linear systems of equations and matrix-vector
multiplications by converting these algebraic problems into generalised time
evolution. Meanwhile, assuming a tensor product structure of the matrices, we
also propose another variational approach for these two tasks by combining
variational real and imaginary time evolution. Finally, we introduce
variational quantum simulation for open system dynamics. We variationally
implement the stochastic Schr\"odinger equation, which consists of dissipative
evolution and stochastic jump processes. We numerically test the algorithm with
a six-qubit 2D transverse field Ising model under dissipation.Comment: 18 page
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