The geometry of a deformation of the standard addition on the integral lattice

Let $\mathfrak A_n$ be the subset of the standard integer lattice $\mathbb Z^n$, $\mathfrak A_n\subset\mathbb Z^n$ which is defined by the condition $\mathfrak A_n=((a_1,...,a_n)\in\mathbb Z^n | a_i\not\equiv a_j\mod n, \forall i,j\in {1,... n})$. It is clear that the standard addition on the lattice $\mathbb Z^n$ does not induce the group structure on the set $\mathfrak A_n$ since the componentwise sum of some two vectors may contain components which are equal modulo $n$. Our aim is to find a new associative multiplication on the lattice $\mathbb Z^n$ such that the induced multiplication on the set $\mathfrak A_n$ gives it the group structure. In this paper the group structure on the subset $\mathfrak A_n$ of the integer lattice $\mathbb Z^n$ is studied by means of the constructions of a deformation of a group multiplication. The geometric realization of this group in the enveloping space and its generators and relations between them are found. We begin with the main constructions and the results we need for them

On factorization and solution of multidimensional linear partial differential equations

We describe a method of obtaining closed-form complete solutions of certain second-order linear partial differential equations with more than two independent variables. This method generalizes the classical method of Laplace transformations of second-order hyperbolic equations in the plane and is based on an idea given by Ulisse Dini in 1902.Comment: 11 pages, Plain LaTeX; Submitted to Proceedings of Waterloo Workshop on Computer Algebra devoted to the 60th birthday Of S.A.Abramov. The second version includes grant acknowledgements and minor changes in a couple of place

Generalized Laplace transformations and integration of hyperbolic systems of linear partial differential equations

We give a new procedure for generalized factorization and construction of the complete solution of strictly hyperbolic linear partial differential equations or strictly hyperbolic systems of such equations in the plane. This procedure generalizes the classical theory of Laplace transformations of second-order equations in the plane.Comment: LaTeX, 17 pages, Submitted to ISSAC 2005, Beijing, China, July 24--27 200

On rational definite summation

We present a partial proof of van Hoeij-Abramov conjecture about the algorithmic possibility of computation of finite sums of rational functions. The theoretical results proved in this paper provide an algorithm for computation of a large class of sums $S(n) = \sum_{k=0}^{n-1}R(k,n)$.Comment: LaTeX 2.09, 7 pages, submitted to "Programming & Computer Software

Classical differential geometry and integrability of systems of hydrodynamic type

Remarkable parallelism between the theory of integrable systems of first-order quasilinear PDE and some old results in projective and affine differential geometry of conjugate nets, Laplace equations, their Bianchi-Baecklund transformations is exposed. These results were recently applied by I.M.Krichever and B.A.Dubrovin to prove integrability of some models in topological field theories. Within the geometric framework we derive some new integrable (in a sense to be discussed) generalizations describing N-wave resonant interactions.Comment: 12 pages. To be published in: Proc. NATO ARW "Applications of analytic and geometric methods to nonlinear differential equations, 14-19 July 1992, Exeter, UK

Factorization of linear partial differential operators and Darboux integrability of nonlinear PDEs

Using a new definition of generalized divisors we prove that the lattice of such divisors for a given linear partial differential operator is modular and obtain analogues of the well-known theorems of the Loewy-Ore theory of factorization of linear ordinary differential operators. Possible applications to factorized Groebner bases computations in the commutative and non-commutative cases are discussed, an application to finding criterions of Darboux integrability of nonlinear PDEs is given.Comment: LaTeX 2.09, acmconf.sty (included in the tar file), 8 pages. Presented at the Poster session of ISSAC'98 (Rostock, Germany

New Error Tolerant Method to Search Long Repeats in Symbol Sequences

A new method to identify all sufficiently long repeating substrings in one or several symbol sequences is proposed. The method is based on a specific gauge applied to symbol sequences that guarantees identification of the repeating substrings. It allows the matching of substrings to contain a given level of errors. The gauge is based on the development of a heavily sparse dictionary of repeats, thus drastically accelerating the search procedure. Some genomic applications illustrate the method. This paper is the extended and detailed version of the presentation at the third International Conference on Algorithms for Computational Biology to be held at Trujillo, Spain, June 21-22, 2016.Comment: 13 pages, 4 figure

The Moutard transformation: an algebraic formalism via pseudodifferential operators and applications

We consider the Moutard transformation which is a two-dimensional version of the well-known Darboux transformation. We give an algebraic interpretation of the Moutard transformation as a conjugation in an appropriate ring and the corresponding version of the algebro-geometric formalism for two-dimensional Schroedinger operators. An application to some problems of the spectral theory of two-dimensional Schroedinger operators and to the $(2+1)$-dimensional Novikov--Veselov equation is sketched.Comment: 15 pages, 2 figure

Classical Mechanical Systems with one-and-a-half Degrees of Freedom and Vlasov Kinetic Equation

We consider non-stationary dynamical systems with one-and-a-half degrees of freedom. We are interested in algorithmic construction of rich classes of Hamilton's equations with the Hamiltonian H=p^2/2+V(x,t) which are Liouville integrable. For this purpose we use the method of hydrodynamic reductions of the corresponding one-dimensional Vlasov kinetic equation. Also we present several examples of such systems with first integrals with non-polynomial dependencies w.r.t. to momentum. The constructed in this paper classes of potential functions {$V(x,t)$} which give integrable systems with one-and-a-half degrees of freedom are parameterized by arbitrary number of constants.Comment: 32 pages, standard LaTeX, in the second version: misprints corrected, Section "Multi-Time Generalization. Explicit Solutions" adde

On Local Description of Two-Dimensional Geodesic Flows with a Polynomial First Integral

In this paper we construct multiparametric families of two dimensional metrics with polynomial first integral. Such integrable geodesic flows are described by solutions of some semi-Hamiltonian hydrodynamic type system. We find infinitely many conservation laws and commuting flows for this system. This procedure allows us to present infinitely many particular metrics by the generalized hodograph method.Comment: 21 page