83 research outputs found
A note on Verhulst's logistic equation and related logistic maps
We consider the Verhulst logistic equation and a couple of forms of the
corresponding logistic maps. For the case of the logistic equation we show that
using the general Riccati solution only changes the initial conditions of the
equation. Next, we consider two forms of corresponding logistic maps reporting
the following results. For the map x_{n+1} = rx_n(1 - x_n) we propose a new way
to write the solution for r = -2 which allows better precision of the iterative
terms, while for the map x_{n+1}-x_n = rx_n(1 - x_{n+1}) we show that it
behaves identically to the logistic equation from the standpoint of the general
Riccati solution, which is also provided herein for any value of the parameter
r.Comment: 6 pages, 3 figures, 7 references with title
On convergence towards a self-similar solution for a nonlinear wave equation - a case study
We consider the problem of asymptotic stability of a self-similar attractor
for a simple semilinear radial wave equation which arises in the study of the
Yang-Mills equations in 5+1 dimensions. Our analysis consists of two steps. In
the first step we determine the spectrum of linearized perturbations about the
attractor using a method of continued fractions. In the second step we
demonstrate numerically that the resulting eigensystem provides an accurate
description of the dynamics of convergence towards the attractor.Comment: 9 pages, 5 figure
Bohl-Perron type stability theorems for linear difference equations with infinite delay
Relation between two properties of linear difference equations with infinite
delay is investigated: (i) exponential stability, (ii) \l^p-input
\l^q-state stability (sometimes is called Perron's property). The latter
means that solutions of the non-homogeneous equation with zero initial data
belong to \l^q when non-homogeneous terms are in \l^p. It is assumed that
at each moment the prehistory (the sequence of preceding states) belongs to
some weighted \l^r-space with an exponentially fading weight (the phase
space). Our main result states that (i) (ii) whenever and a certain boundedness condition on coefficients is
fulfilled. This condition is sharp and ensures that, to some extent,
exponential and \l^p-input \l^q-state stabilities does not depend on the
choice of a phase space and parameters and , respectively. \l^1-input
\l^\infty-state stability corresponds to uniform stability. We provide some
evidence that similar criteria should not be expected for non-fading memory
spaces.Comment: To be published in Journal of Difference Equations and Application
Multiple-scale analysis of discrete nonlinear partial difference equations: the reduction of the lattice potential KdV
We consider multiple lattices and functions defined on them. We introduce
slow varying conditions for functions defined on the lattice and express the
variation of a function in terms of an asymptotic expansion with respect to the
slow varying lattices.
We use these results to perform the multiple--scale reduction of the lattice
potential Korteweg--de Vries equation.Comment: 17 pages. 1 figur
Moment inversion problem for piecewise D-finite functions
We consider the problem of exact reconstruction of univariate functions with
jump discontinuities at unknown positions from their moments. These functions
are assumed to satisfy an a priori unknown linear homogeneous differential
equation with polynomial coefficients on each continuity interval. Therefore,
they may be specified by a finite amount of information. This reconstruction
problem has practical importance in Signal Processing and other applications.
It is somewhat of a ``folklore'' that the sequence of the moments of such
``piecewise D-finite''functions satisfies a linear recurrence relation of
bounded order and degree. We derive this recurrence relation explicitly. It
turns out that the coefficients of the differential operator which annihilates
every piece of the function, as well as the locations of the discontinuities,
appear in this recurrence in a precisely controlled manner. This leads to the
formulation of a generic algorithm for reconstructing a piecewise D-finite
function from its moments. We investigate the conditions for solvability of the
resulting linear systems in the general case, as well as analyze a few
particular examples. We provide results of numerical simulations for several
types of signals, which test the sensitivity of the proposed algorithm to
noise
On the universality of the Discrete Nonlinear Schroedinger Equation
We address the universal applicability of the discrete nonlinear Schroedinger
equation. By employing an original but general top-down/bottom-up procedure
based on symmetry analysis to the case of optical lattices, we derive the most
widely applicable and the simplest possible model, revealing that the discrete
nonlinear Schroedinger equation is ``universally'' fit to describe light
propagation even in discrete tensorial nonlinear systems and in the presence of
nonparaxial and vectorial effects.Comment: 6 Pages, to appear in Phys. Rev.
Quadratic fermionic interactions yield effective Hamiltonians for adiabatic quantum computing
Polynomially-large ground-state energy gaps are rare in many-body quantum
systems, but useful for adiabatic quantum computing. We show analytically that
the gap is generically polynomially-large for quadratic fermionic Hamiltonians.
We then prove that adiabatic quantum computing can realize the ground states of
Hamiltonians with certain random interactions, as well as the ground states of
one, two, and three-dimensional fermionic interaction lattices, in polynomial
time. Finally, we use the Jordan-Wigner transformation and a related
transformation for spin-3/2 particles to show that our results can be restated
using spin operators in a surprisingly simple manner. A direct consequence is
that the one-dimensional cluster state can be found in polynomial time using
adiabatic quantum computing.Comment: 14 page
Towards the Formalization of Fractional Calculus in Higher-Order Logic
Fractional calculus is a generalization of classical theories of integration
and differentiation to arbitrary order (i.e., real or complex numbers). In the
last two decades, this new mathematical modeling approach has been widely used
to analyze a wide class of physical systems in various fields of science and
engineering. In this paper, we describe an ongoing project which aims at
formalizing the basic theories of fractional calculus in the HOL Light theorem
prover. Mainly, we present the motivation and application of such formalization
efforts, a roadmap to achieve our goals, current status of the project and
future milestones.Comment: 9 page
Consistency Conditions for Fundamentally Discrete Theories
The dynamics of physical theories is usually described by differential
equations. Difference equations then appear mainly as an approximation which
can be used for a numerical analysis. As such, they have to fulfill certain
conditions to ensure that the numerical solutions can reliably be used as
approximations to solutions of the differential equation. There are, however,
also systems where a difference equation is deemed to be fundamental, mainly in
the context of quantum gravity. Since difference equations in general are
harder to solve analytically than differential equations, it can be helpful to
introduce an approximating differential equation as a continuum approximation.
In this paper implications of this change in view point are analyzed to derive
the conditions that the difference equation should satisfy. The difference
equation in such a situation cannot be chosen freely but must be derived from a
fundamental theory. Thus, the conditions for a discrete formulation can be
translated into conditions for acceptable quantizations. In the main example,
loop quantum cosmology, we show that the conditions are restrictive and serve
as a selection criterion among possible quantization choices.Comment: 33 page
Discrete Determinants and the Gel'fand-Yaglom formula
I present a partly pedagogic discussion of the Gel'fand-Yaglom formula for
the functional determinant of a one-dimensional, second order difference
operator, in the simplest settings. The formula is a textbook one in discrete
Sturm-Liouville theory and orthogonal polynomials. A two by two matrix approach
is developed and applied to Robin boundary conditions. Euler-Rayleigh sums of
eigenvalues are computed. A delta potential is introduced as a simple,
non-trivial example and extended, in an appendix, to the general case. The
continuum limit is considered in a non--rigorous way and a rough comparison
with zeta regularised values is made. Vacuum energies are also considered in
the free case. Chebyshev polynomials act as free propagators and their
properties are developed using the two-matrix formulation, which has some
advantages and appears to be novel. A trace formula, rather than a determinant
one, is derived for the Gel'fand-Yaglom function.Comment: 29 pages. Submitted version. Typos corrected and adjustments made.
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