Separation of variables in two-dimensional wave equations with potential

In the present paper, we solve the problem of separation of variables in the wave equationu _tt –u _xx +V(x)u=0 and give a complete classification of potentialsV(x) forwhich equations of this type admit nontrivial separation of variables. Furthermore, we construct all coordinate systems in which this separation is possible.Published in Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 10, pp. 1343–1361, October, 1994.     

Enumeration of nilpotent loops via cohomology

The isomorphism problem for centrally nilpotent loops can be tackled by methods of cohomology. We develop tools based on cohomology and linear algebra that either lend themselves to direct count of the isomorphism classes (notably in the case of nilpotent loops of order 2q, q a prime), or lead to efficient classification computer programs. This allows us to enumerate all nilpotent loops of order less than 24.     

Classification of bernstein algebras of type (3, n - 3)

A classification of Bernstein algebras in dimensions n ? 4 has been made by Holgate in [2], however that article contains no classification up to isomorphism, the problem is solved by Lyubich in [4] when K = R or C, and by Cortes [1] in the general case. Also Lyubich has given in [5] a classification of the regular nonexceptional Bernstein algebra of type (3,n?3) and a classification but not up to isomorphism of nonregular nonexceptional Bernstein algebras of type (3,n ? 3) when K = C. The aim of this paper it to characterize, up to isomorphism, Bernstein algebras of type(2, n ? 2) and nonexceptional of type(3, n ?3) over a infinite commutative field K whose characteristic is different from 2.     

Integrable Many-Body Systems via the Inosemtsev Limit

The Inozemtsev limit (IL), or the scaling limit, is known as a procedure applied to the elliptic Calogero-Moser model. It is a combination of the trigonometric limit, infinite shifts of particle coordinates, and coupling-constant rescalings. This results in an interaction of the exponential type. We show that the IL applied to the sl(N,) elliptic Euler-Calogero-Moser model and to the elliptic Gaudin model produces new Toda-like systems of N interacting particles endowed with additional degrees of freedom corresponding to a coadjoint orbit of sl(n,). The limits corresponding to the complete degeneration of the orbital degrees of freedom lead to recovering only the known periodic and nonperiodic Toda systems. We classify the systems appearing in the IL in the sl(3,) case. This classification is represented on a two-dimensional plane of parameters describing infinite shifts of particle coordinates. This space is subdivided into symmetric domains. In this picture, a mixture of the Toda and trigonometric Calogero-Sutherland potentials emerges on lower-dimensional domain walls. Because of obvious symmetries, this classification can be generalized to an arbitrary number of particles. We also apply the IL to the sl(2,) elliptic Gaudin model on a two-punctured elliptic curve and discuss the main properties of its possible limits. The limits of Lax matrices are also considered.     

The Ring of Modules with Endo-permutation Source

In this paper we consider certain subalgebras of the Green algebra (representation algebra) of a finite group G. One such algebra is spanned by the isomorphism classes of all indecomposable modules whose source is an endo-permutation module. This algebra can be embedded into a finite direct product of Laurent polynomial rings in finitely many variables over a field. Another such algebra is spanned by the isomorphism classes of all irreducibly generated modules. When G is p-solvable then this algebra is finite-dimensional and split semisimple.R. Boltje was supported by the NSF, DMS-0200592 and 0128969. B. Külshammer was supported by the DAAD.     

Gaudin magnet and off-mass-shell Bethe wave functions

The general Bethe equation (or off-shell Bethe ansatz equation) is proved for the Gaudin magnet for a broad class of simple complex Lie algebras. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 113, No. 1, pp. 13–28, October, 1997.     

On a new form of Bethe ansatz equations and separation of variables in the $$\mathfrak{s}\mathfrak{l}_3 $$ Gaudin model

A new form of Bethe ansatz equations is introduced. A version of a separation of variables for the quantum Gaudin model is presented. Dedicated to V.I. Arnold on the occasion of his 70th birthday     

Motives of Uniruled 3-Folds

We construct an elliptic analogue of Sklyanins separation of variables for the sl(2) Gaudin system, using an adaptation of Drinfelds Radon transformations.     

Factor orbit equivalence of compact group extensions and classification of finite extensions of ergodic automorphisms

In §§1–5, we classifyn-point extensions of ergodic automorphisms up to factor orbit-equivalence (which is the natural analogue of factor isomorphism). This classification is in terms of conjugacy classes of subgroups of the symmetric group onn points, and parallels D. Rudolph’s classification ofn-point extensions of Bernoulli shifts up to factor isomorphism. In §6, we give another proof of A. Fieldsteel’s theorem on factor orbit-equivalence of compact group extensions.     

The Problem of the Classification of the Nilpotent Class 2 Torsion Free Groups up to Geometric Equivalence

In this article, we consider the problem of classification of the nilpotent class 2 finitely generated torsion free groups up to geometric equivalence. By a very easy technique it is proved that this problem is equivalent to the problem of classification of the complete in the Maltsev sense nilpotent torsion free finite rank groups up to isomorphism. This result leads to better understanding of the complexity of the problem of the classification of the quasi-varieties of the nilpotent class 2 groups. It is well known that the variety of the nilpotent class s groups is Noetherian for every s ∈ ?. So the problem of the classification of the quasi-varieties generated even by a single nilpotent class 2 finitely generated torsion free group is equivalent to the problem of classification of the complete in the Maltsev sense nilpotent torsion free finite rank groups up to isomorphism.     

The Steiner quadruple systems of order 16

The Steiner quadruple systems of order 16 are classified up to isomorphism by means of an exhaustive computer search. The number of isomorphism classes of such designs is 1,054,163. Properties of the designs—including the orders of the automorphism groups and the structures of the derived Steiner triple systems of order 15—are tabulated. A consistency check based on double counting is carried out to gain confidence in the correctness of the classification.     

Affine Mendelsohn triple systems and the Eisenstein integers

We define a Mendelsohn triple system (MTS) of order coprime with 3, and having multiplication affine over an abelian group, to be affine, nonramified. By exhibiting a one‐to‐one correspondence between isomorphism classes of affine MTS and those of modules over the Eisenstein integers, we solve the isomorphism problem for affine, nonramified MTS and enumerate these isomorphism classes (extending the work of Donovan, Griggs, McCourt, Opr?al, and Stanovský). As a consequence, all entropic MTSs of order coprime with 3 and distributive MTS of order coprime with 3 are classified. Partial results on the isomorphism problem for affine MTS with order divisible by 3 are given, and a complete classification is conjectured. We also prove that for any affine MTS, the qualities of being nonramified, pure, and self‐orthogonal are equivalent.     

Forms of pointed Hopf algebras

Using descent theory, we study Hopf algebra forms of pointed Hopf algebras. It turns out that the set of isomorphism classes of such forms are in one-to-one correspondence to other known invariants, for example the set of isomorphism classes of Galois extensions with a certain group F, or the set of isometry classes of m-ary quadratic forms. Our theory leads to a classification of all Hopf algebras over a field of characteristic zero that become pointed after a base extension, in dimension p, p ² and p ³, with p odd. Received: 22 November 1998     

Counting isomorphism classes of pointed hyperelliptic curves of genus 4 over finite fields with odd characteristic

This paper is devoted to computing the number of isomorphism classes of pointed hyperelliptic curves over finite fields. We deal with the genus-4 case and the finite fields are of odd characteristic. The number of isomorphism classes is computed. This number can be represented as a polynomial in q of degree 7, where q is the order of the finite field. The results have applications in the classification problems and in the hyperelliptic curve cryptosystems.     

On spherical representations of quivers and generalized complexes

For any quiverQ we consider spherical representationsV ofQ such that the isomorphism class ofV is a spherical variety. We suggest an approach for classifying such representations for anyQ and obtain a classification forQ being an equioriented Dynkin diagramA _n. In particular, all complexes are spherical representations. We introduce a category of representations that we call generalized complexes and classify spherical generalized complexes. For the quivers that we call crumbly we prove that any spherical generalized complex has a polynomial algebra of covariants on the closure of its isomorphism class.Partially supported by INTAS-OPEN grant 97-1570 and RFFI grant 98-01-00598.     

Classification of three-dimensional solvable <Emphasis Type="Italic">p</Emphasis>-adic Leibniz algebras

The present paper is devoted to the study of low dimensional Leibniz algebras over the field of p-adic numbers. The classification up to isomorphism of three-dimensional Lie algebras over the integer p-adic numbers is already known [8]. Here, we extend this classification to solvable Lie and non-Lie Leibniz algebras over the field of p-adic numbers.     

On the use of Lie-Bäcklund operators in quantum mechanics

This paper uses Lie-Bäcklund operators to study the connection between classical and quantum-mechanical invariants and their relations to symmetry and to separation of variables. The problem of an isomorphic correspondence between classical and quantum mechanics is studied concretely and constructively. For functions at most quadratic in the momenta an isomorphism is possible which agrees with Weyl's transform and which takes invariants into invariants. It is not possible to extend the isomorphism indefinitely. The requirement that an invariant goes into an invariant may necessitate variants of Weyl's transform. This is illustrated for the case of cubic invariants. Finally, the case of a specific value of energy is considered.     

On the classification of toric Fano 4-folds

The biregular classification of smoothd-dimensional toric Fano varieties is equivalent to the classification of special simplicial polyhedraP in ℝ ^d , the so-called Fano polyhedra, up to an isomorphism of the standard lattice . In this paper, we explain the complete biregular classification of all 4-dimensional smooth toric Fano varieties. The main result states that there exist exactly 123 different types of toric Fano 4-folds somorphism. Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory Vol. 56. Algebraic Geometry-9, 1998.