144 research outputs found
Microscopic Computational Model of a Superfluid
A finite one-dimensional microscopic model of a superfulid is presented. The
model consists of interacting Bose particles with an additional impurity
particle confined to a ring. Both semiclassical and exact quantum calculations
reveal dissipationless motion of impurity with increased effective mass due to
its interaction with the excitations of Bose fluid. It is shown that both the
excitation spectrum of Bose fluid and the excitation spectrum of impurity can
be analyzed using the structure of the ground state of the system
{\sigma}-Galois theory of linear difference equations
We develop a Galois theory for systems of linear difference equations with an
action of an endomorphism {\sigma}. This provides a technique to test whether
solutions of such systems satisfy {\sigma}-polynomial equations and, if yes,
then characterize those. We also show how to apply our work to study
isomonodromic difference equations and difference algebraic properties of
meromorphic functions
Isomonodromic differential equations and differential categories
We study isomonodromicity of systems of parameterized linear differential
equations and related conjugacy properties of linear differential algebraic
groups by means of differential categories. We prove that isomonodromicity is
equivalent to isomonodromicity with respect to each parameter separately under
a filtered-linearly closed assumption on the field of functions of parameters.
Our result implies that one does not need to solve any non-linear differential
equations to test isomonodromicity anymore. This result cannot be further
strengthened by weakening the requirement on the parameters as we show by
giving a counterexample. Also, we show that isomonodromicity is equivalent to
conjugacy to constants of the associated parameterized differential Galois
group, extending a result of P. Cassidy and M. Singer, which we also prove
categorically. We illustrate our main results by a series of examples, using,
in particular, a relation between Gauss-Manin connection and parameterized
differential Galois groups.Comment: 31 page
On Computation of Kolchin Characteristic Sets: Ordinary and Partial Cases
In this paper we study the problem of computing a Kolchin characteristic set
of a radical differential ideal. The central part of the article is the
presentation of algorithms solving this problem in two principal cases: for
ordinary differential polynomials and in the partial differential case. Our
computations are mainly performed with respect to orderly rankings. We also
discuss the usefulness of regular and characteristic decompositions of radical
differential ideals. In the partial differential case we give an algorithm for
computing characteristic sets in the special case of radical differential
ideals satisfying the property of consistency. For this class of ideals we show
how to deal with arbitrary differential rankings.Comment: Minor change
Triviality of differential Galois cohomologies of linear differential algebraic groups
We show that the triviality of the differential Galois cohomologies over a
partial differential field K of a linear differential algebraic group is
equivalent to K being algebraically, Picard-Vessiot, and linearly
differentially closed. This former is also known to be equivalent to the
uniqueness up to an isomorphism of a Picard-Vessiot extension of a linear
differential equation with parameters
On bounds for the effective differential Nullstellensatz
Understanding bounds for the effective differential Nullstellensatz is a
central problem in differential algebraic geometry. Recently, several bounds
have been obtained using Dicksonian and antichains sequences (with a given
growth rate). In the present paper, we make these bounds more explicit and,
therefore, more applicable to understanding the computational complexity of the
problem, which is essential to designing more efficient algorithms
Corrigendum: 'On bounds for the effective differential Nullstellensatz, arXiv:1508.07508'
We correct a small gap found in the authors' paper 'On bounds for the
effective differential Nullstellensatz' (J Algebra 449:1-21, 2016). This gap is
due to an inequality that does not generally hold. However, under one
additional assumption, it does hold. In this note, we provide a detailed proof
of this. We then point out that this assumption is satisfied in all instances
in which the inequality was used
Galois theory of difference equations with periodic parameters
We develop a Galois theory for systems of linear difference equations with
periodic parameters, for which we also introduce linear difference algebraic
groups. We then apply this to constructively test if solutions of linear
q-difference equations, with complex q, not a root of unity, satisfy any
polynomial q'-difference equations with q' being a root of unity.Comment: 42 page
A Method of the Study of the Cauchy Problem for a Singularly Perturbed Linear Inhomogeneous Differential Equation
We construct a sequence that converges to a solution of the Cauchy problem
for a singularly perturbed linear inhomogeneous differential equation of an
arbitrary order. This sequence is also an asymptotic sequence in the following
sense: the deviation (in the norm of the space of continuous functions) of its
th element from the solution of the problem is proportional to the th
power of the parameter of perturbation. This sequence can be used for
justification of asymptotics obtained by the method of boundary functions.Comment: 12 page
Effective difference elimination and Nullstellensatz
We prove effective Nullstellensatz and elimination theorems for difference
equations in sequence rings. More precisely, we compute an explicit function of
geometric quantities associated to a system of difference equations (and these
geometric quantities may themselves be bounded by a function of the number of
variables, the order of the equations, and the degrees of the equations) so
that for any system of difference equations in variables and , if these equations have
any nontrivial consequences in the variables, then such a
consequence may be seen algebraically considering transforms up to the order of
our bound. Specializing to the case of , we obtain an effective method
to test whether a given system of difference equations is consistent
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