15,073 research outputs found
Interpolation Methods for Binary and Multivalued Logical Quantum Gate Synthesis
A method for synthesizing quantum gates is presented based on interpolation
methods applied to operators in Hilbert space. Starting from the diagonal forms
of specific generating seed operators with non-degenerate eigenvalue spectrum
one obtains for arity-one a complete family of logical operators corresponding
to all the one-argument logical connectives. Scaling-up to n-arity gates is
obtained by using the Kronecker product and unitary transformations. The
quantum version of the Fourier transform of Boolean functions is presented and
a Reed-Muller decomposition for quantum logical gates is derived. The common
control gates can be easily obtained by considering the logical correspondence
between the control logic operator and the binary propositional logic operator.
A new polynomial and exponential formulation of the Toffoli gate is presented.
The method has parallels to quantum gate-T optimization methods using powers of
multilinear operator polynomials. The method is then applied naturally to
alphabets greater than two for multi-valued logical gates used for quantum
Fourier transform, min-max decision circuits and multivalued adders
Tauberian class estimates for vector-valued distributions
We study Tauberian properties of regularizing transforms of vector-valued
tempered distributions, that is, transforms of the form
, where the
kernel is a test function and
. We investigate conditions which
ensure that a distribution that a priori takes values in locally convex space
actually takes values in a narrower Banach space. Our goal is to characterize
spaces of Banach space valued tempered distributions in terms of so-called
class estimates for the transform . Our results
generalize and improve earlier Tauberian theorems of Drozhzhinov and Zav'yalov
[Sb. Math. 194 (2003), 1599-1646]. Special attention is paid to find the
optimal class of kernels for which these Tauberian results hold.Comment: 24 pages. arXiv admin note: substantial text overlap with
arXiv:1012.509
Inverse source problems for degenerate time-fractional PDE
In this paper, we investigate two inverse source problems for degenerate
time-fractional partial differential equation in rectangular domains. The first
problem involves a space-degenerate partial differential equation and the
second one involves a time-degenerate partial differential equation. Solutions
to both problem are expressed in series expansions. For the first problem, we
obtained solutions in the form of Fourier-Legendre series. Convergence and
uniqueness of solutions have been discussed. Solutions to the second problem
are expressed in the form of Fourier-Sine series and they involve a generalized
Mittag- Leffler type function. Moreover, we have established a new estimate for
this generalized Mittag-Leffler type function. The obtained results are
illustrated by providing example solutions using certain given data at the
initial and final time.Comment: 12 pages, 8 figure
Orthogonal structure on a quadratic curve
Orthogonal polynomials on quadratic curves in the plane are studied. These
include orthogonal polynomials on ellipses, parabolas, hyperbolas, and two
lines. For an integral with respect to an appropriate weight function defined
on any quadratic curve, an explicit basis of orthogonal polynomials is
constructed in terms of two families of orthogonal polynomials in one variable.
Convergence of the Fourier orthogonal expansions is also studied in each case.
As an application, we see that the resulting bases can be used to interpolate
functions on the real line with singularities of the form , , or , with exponential convergence
Inverse Harish-Chandra Transform and Difference Operators
We apply a new technique based on double affine Hecke algebras to the
Harish-Chandra theory of spherical zonal functions. The formulas for the
Fourier transforms of the multiplications by the coordinates are obtained as
well as a simple proof of the Harish-Chandra inversion theorem using the Opdam
transform.Comment: AMSTe
Multidimensional Tauberian theorems for vector-valued distributions
We prove several Tauberian theorems for regularizing transforms of vector-valued distributions. The regularizing transform of f is given by the integral transform M-phi(f)(x, y) = (f * phi(y))(x), (x, y) is an element of R-n x R+, with kernel phi(y) (t) = y(-n)phi(t/y). We apply our results to the analysis of asymptotic stability for a class of Cauchy problems, Tauberian theorems for the Laplace transform, the comparison of quasiasymptotics in distribution spaces, and we give a necessary and sufficient condition for the existence of the trace of a distribution on {x(0)} x R-m. In addition, we present a new proof of Littlewood's Tauberian theorem
Cohomological rank functions on abelian varieties
Generalizing the continuous rank function of Barja-Pardini-Stoppino, in this
paper we consider cohomological rank functions of -twisted
(complexes of) coherent sheaves on abelian varieties. They satisfy a natural
transformation formula with respect to the Fourier-Mukai-Poincar\'e transform,
which has several consequences. In many concrete geometric contexts these
functions provide useful invariants. We illustrate this with two different
applications, the first one to GV-subschemes and the second one to
multiplication maps of global sections of ample line bundles on abelian
varieties.Comment: 28 pages, minor changes. Final version to appear on Annales Scient.
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