127,082 research outputs found
Existence of meromorphic solutions of first order difference equations
It is shown that if It is shown that if
\begin{equation}\label{abstract_eq}
f(z+1)^n=R(z,f),\tag{\dag}
\end{equation} where is rational in with meromorphic
coefficients and , has an admissible meromorphic solution,
then either satisfies a difference linear or Riccati equation with
meromorphic coefficients, or \eqref{abstract_eq} can be transformed into one in
a list of ten equations with certain meromorphic or algebroid coefficients. In
particular, if \eqref{abstract_eq}, where the assumption has
been discarded, has rational coefficients and a transcendental meromorphic
solution of hyper-order , then either satisfies a difference linear
or Riccati equation with rational coefficients, or \eqref{abstract_eq} can be
transformed into one in a list of five equations which consists of four
difference Fermat equations and one equation which is a special case of the
symmetric QRT map. Solutions to all of these equations are presented in terms
of Weierstrass or Jacobi elliptic functions, or in terms of meromorphic
functions which are solutions to a difference Riccati equation. This provides a
natural difference analogue of Steinmetz' generalization of Malmquist's
theorem.Comment: 25 pages, some errors are correcte
Linear difference equations, frieze patterns and combinatorial Gale transform
We study the space of linear difference equations with periodic coefficients
and (anti)periodic solutions. We show that this space is isomorphic to the
space of tame frieze patterns and closely related to the moduli space of
configurations of points in the projective space. We define the notion of
combinatorial Gale transform which is a duality between periodic difference
equations of different orders. We describe periodic rational maps generalizing
the classical Gauss map
Finite-order meromorphic solutions and the discrete Painleve equations
Let w(z) be a finite-order meromorphic solution of the second-order
difference equation w(z+1)+w(z-1) = R(z,w(z)) (eqn 1) where R(z,w(z)) is
rational in w(z) and meromorphic in z. Then either w(z) satisfies a difference
linear or Riccati equation or else equation (1) can be transformed to one of a
list of canonical difference equations. This list consists of all known
difference Painleve equation of the form (1), together with their autonomous
versions. This suggests that the existence of finite-order meromorphic
solutions is a good detector of integrable difference equations.Comment: 34 page
Extensions of differential representations of SL(2) and tori
Linear differential algebraic groups (LDAGs) measure differential algebraic
dependencies among solutions of linear differential and difference equations
with parameters, for which LDAGs are Galois groups. The differential
representation theory is a key to developing algorithms computing these groups.
In the rational representation theory of algebraic groups, one starts with
SL(2) and tori to develop the rest of the theory. In this paper, we give an
explicit description of differential representations of tori and differential
extensions of irreducible representation of SL(2). In these extensions, the two
irreducible representations can be non-isomorphic. This is in contrast to
differential representations of tori, which turn out to be direct sums of
isotypic representations.Comment: 21 pages; few misprints corrected; Lemma 4.6 adde
Discrete Nonlinear Planar Systems and Applications to Biological Population Models
We study planar systems of difference equations and applications to biological models of species populations. Central to the analysis of this study is the idea of folding - the method of transforming systems of difference equations into higher order scalar difference equations. Two classes of second order equations are studied: quadratic fractional and exponential.
We investigate the boundedness and persistence of solutions, the global stability of the positive fixed point and the occurrence of periodic solutions of the quadratic rational equations. These results are applied to a class of linear/rational systems that can be transformed into a quadratic fractional equation via folding. These results apply to systems with negative parameters, instances not commonly considered in previous studies. We also identify ranges of parameter values that provide sufficient conditions on existence of chaotic and multiple stable orbits of different periods for the planar system.
We study a second order exponential difference equation with time varying parameters and obtain sufficient conditions for boundedness of solutions and global convergence to zero. For the autonomous case, we show occurrence of multistable periodic and nonperiodic orbits. For the case where parameters are periodic, we show that the nature of the solutions differs qualitatively depending on whether the period of the parameters is even or odd.
The above results are applied to biological models of populations. We investigate a broad class of planar systems that arise in the study of stage-structured single species populations. In biological contexts, these results include conditions on extinction or survival of the species in some balanced form, and possible occurrence of complex and chaotic behavior. Special rational (Beverton-Holt) and exponential (Ricker) cases are considered to explore the role of inter-stage competition, restocking strategies, as well as seasonal fluctuations in the vital rates
Converging to Gosper's Algorithm
Given two polynomials, we find a convergence property of the GCD of the
rising factorial and the falling factorial. Based on this property, we present
a unified approach to computing the universal denominators as given by Gosper's
algorithm and Abramov's algorithm for finding rational solutions to linear
difference equations with polynomial coefficients.Comment: 13 page
- …