A unified Witten–Reshetikhin–Turaev invariant for integral homology spheres

We construct an invariant J _M of integral homology spheres M with values in a completion of the polynomial ring ℤ[q] such that the evaluation at each root of unity ζ gives the the SU(2) Witten–Reshetikhin–Turaev invariant τ_ζ(M) of M at ζ. Thus J _M unifies all the SU(2) Witten–Reshetikhin–Turaev invariants of M. It also follows that τ_ζ(M) as a function on ζ behaves like an “analytic function” defined on the set of roots of unity.     

Ideals in Morita Rings and Morita Semigroups

We characterize the lattice of all ideals of a Morita ring （semigroup） when the corresponding pair of rings （semigroups） in the Morita context are Morita equivalent s-unital （like-unitv） rings （semigroups）.     

Polynomials with high multiplicity at unity and Tarry’s problem

The existence of a polynomial with integer coefficients of relatively small length and sufficiently large multiplicity at unity is established. The proof of the corresponding statement is based on an estimate of the number of solutions of Tarry’s system. Translated fromMatematicheskie Zametki, Vol. 65, No. 6, pp. 810–815, June, 1999.     

Centers of Mal’tsev and alternative algebras

Let ϕ be an associative commutative ring with unity containing 1/6. Let A and B be a free Mal’tsev and a free alternative ϕ-algebras on a set of k≥6 free generators, respectively. We construct nonzero homogeneous elements of degree 7 belonging to an annihilatorAnnA of A, and nonzero homogeneous elements of degree 7 belonging to the center Z(B) of B. It is shown that a nilpotent Mal’tsev algebra of index 8 on a set of 6 generators has no faithful representation. Supported by RFFR grant No. 96-01-01511, and by the Program “Universities of Russia: Fundamental Research.” Translated fromAlgebra i Logika, Vol. 38, No. 5, pp. 613–635, September–October, 1999.     

Ionization of Atoms by Intense Laser Pulses

The process of ionization of a hydrogen atom by a short infrared laser pulse is studied in the regime of very large pulse intensity, in the dipole approximation. Let A denote the integral of the electric field of the pulse over time at the location of the atomic nucleus. It is shown that, in the limit where |A| → ∞, the ionization probability approaches unity and the electron is ejected into a cone opening in the direction of −A and of arbitrarily small opening angle. Asymptotics of various physical quantities in |A|⁻¹ is studied carefully. Our results are in qualitative agreement with experimental data reported in Eckle et al. (Science 322, 1525–1529; 2008, Nature (physics) 4, 565–570 2008).     

Completion of Linearly Ordered Groups

We give examples of linearly ordered groups that are not embeddable in divisible orderable. In the first example, the group does not embed in any divisible group with strictly isolated unity. In the second example, the group in question is an O*-group, and in the third, it is a group with a central system of convex subgroups. To my teacher A. I. Kokorin Supported by RFBR grant Nos. 96-01-00358, 99-01-00335, and 03-01-00320. __________ Translated from Algebra i Logika, Vol. 44, No. 6, pp. 664–681, November–December, 2005.     

Multiplicative relations with conjugate algebraic numbers

We study what algebraic numbers can be represented by a product of algebraic numbers conjugate over a fixed number field K in fixed integer powers. The problem is nontrivial if the sum of these integer powers is equal to zero. The norm of such a number over K must be a root of unity. We show that there are infinitely many algebraic numbers whose norm over K is a root of unity and which cannot be represented by such a product. Conversely, every algebraic number can be expressed by every sufficiently long product in algebraic numbers conjugate over K. We also construct nonsymmetric algebraic numbers, i.e., algebraic numbers such that no elements of the corresponding Galois group acting on the full set of their conjugates form a Latin square. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 7, pp. 890–900, July, 2007.     

Global Classification of Two-Component Approximately Integrable Evolution Equations

We globally classify two-component evolution equations, with homogeneous diagonal linear part, admitting infinitely many approximate symmetries. Important ingredients are the symbolic calculus of Gel’fand and Dikiĭ, the Skolem–Mahler–Lech theorem, an algorithm of Smyth, and results on diophantine equations in roots of unity obtained by Beukers.      

Composition algebras of the second kind

The concept of a composition algebra of the second kind is introduced. We prove that such algebras are non-degenerate monocomposition algebras without unity. A big number of these algebras in any finite dimension are constructed, as well as two algebras in a countable dimension. The constructed algebras each contains a non-isotropic idempotent e² = e. We describe all orthogonally non-isomorphic composition algebras of the second kind in the following forms: (1) a two-dimensional algebra (which has turned out to be unique); (2) three-dimensional algebras in the constructed series. For every algebra A, the group Ortaut A of orthogonal automorphisms is specified. __________ Translated from Algebra i Logika, Vol. 46, No. 4, pp. 428–447, July–August, 2007.     

On Galois cohomology and realizability of 2-groups as Galois groups II

In [Michailov I.M., On Galois cohomology and realizability of 2-groups as Galois groups, Cent. Eur. J. Math., 2011, 9(2), 403–419] we calculated the obstructions to the realizability as Galois groups of 14 non-abelian groups of order 2 ⁿ , n ≥ 4, having a cyclic subgroup of order 2 ⁿ⁻², over fields containing a primitive 2 ⁿ⁻³th root of unity. In the present paper we obtain necessary and sufficient conditions for the realizability of the remaining 8 groups that are not direct products of smaller groups.     

Generalized Coherent States for the q-Oscillator Associated with Discrete q-Hermite Polynomials

We continue studying generalized coherent states of the Barut-Girardello type for oscillator-like systems related to a given set of orthogonal polynomials. In this paper we construct a family of coherent states associated with discrete q-Hermite polynomials of the II-type and prove the overcompleteness of this family by constructing the measure in the unity decomposition for this family of coherent states. Bibliography: 49 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 308, 2004, pp. 48–66.     

Primitive algebraic relations of skew derivations of prime rings

In [1], the question was posed as to whether or not all algebraic relations of skew derivations of prime rings follow from primitive algebraic relations. Here we argue to obtain a negative answer to a milder question, and namely, an example is constructed in which a pointed Hopf algebra H (generated as an algebra with unity by its relatively primitive elements) acts trivially on the generalized centroid C of a prime ring R, but not all algebraic relations of skew derivations (corresponding to relatively primitive elements in H) follow from primitive algebraic ones. The R in the counterexample is a free associative C-algebra. Supported by ISF grant No. RPS300 and by RFFR grant No. 95-01-01356a. Translated from Algebra i Logika, Vol. 36, No. 4, pp. 407–421, July–August, 1997.     

Volume-preserving Fields and Reeb Fields on 3-manifolds

Volume-preserving field X on a 3-manifold is the one that satisfies LxΩ = 0 for some volume Ω. The Reeb vector field of a contact form is of volume-preserving, but not conversely. On the basis of Geiges-Gonzalo＇s parallelization results, we obtain a volume-preserving sphere, which is a triple of everywhere linearly independent vector fields such that all their linear combinations with constant coefficients are volume-preserving fields. From many aspects, we discuss the distinction between volume-preserving fields and Reeb-like fields. We establish a duality between volume-preserving fields and h-closed 2-forms to understand such distinction. We also give two kinds of non-Reeb-like but volume-preserving vector fields to display such distinction.     

Computing ECT-B-splines recursively

ECT-spline curves for sequences of multiple knots are generated from different local ECT-systems via connection matrices. Under appropriate assumptions there is a basis of the space of ECT-splines consisting of functions having minimal compact supports, normalized to form a nonnegative partition of unity. The basic functions can be defined by generalized divided differences [24]. This definition reduces to the classical one in case of a Schoenberg space. Under suitable assumptions it leads to a recursive method for computing the ECT-B-splines that reduces to the de Boor–Mansion–Cox recursion in case of ordinary polynomial splines and to Lyche's recursion in case of Tchebycheff splines. For sequences of simple knots and connection matrices that are nonsingular, lower triangular and totally positive the spline weights are identified as Neville–Aitken weights of certain generalized interpolation problems. For multiple knots they are limits of Neville–Aitken weights. In many cases the spline weights can be computed easily by recurrence. Our approach covers the case of Bézier-ECT-splines as well. They are defined by different local ECT-systems on knot intervals of a finite partition of a compact interval [a,b] connected at inner knots all of multiplicities zero by full connection matrices A ^[i] that are nonsingular, lower triangular and totally positive. In case of ordinary polynomials of order n they reduce to the classical Bézier polynomials. We also present a recursive algorithm of de Boor type computing ECT-spline curves pointwise. Examples of polynomial and rational B-splines constructed from given knot sequences and given connection matrices are added. For some of them we give explicit formulas of the spline weights, for others we display the B-splines or the B-spline curves. *Supported in part by INTAS 03-51-6637.     

Iteration of closed geodesics in stationary Lorentzian manifolds

Following the lines of Bott in (Commun Pure Appl Math 9:171–206, 1956), we study the Morse index of the iterates of a closed geodesic in stationary Lorentzian manifolds, or, more generally, of a closed Lorentzian geodesic that admits a timelike periodic Jacobi field. Given one such closed geodesic γ, we prove the existence of a locally constant integer valued map Λ_γ on the unit circle with the property that the Morse index of the iterated γ ^N is equal, up to a correction term ε_γ∈{0,1}, to the sum of the values of Λ_γ at the N-th roots of unity. The discontinuities of Λ_γ occur at a finite number of points of the unit circle, that are special eigenvalues of the linearized Poincaré map of γ. We discuss some applications of the theory.     

Quadratic forms for a 1-form on an isolated complete intersection singularity

We consider a holomorphic 1-form ω with an isolated zero on an isolated complete intersection singularity (V,0). We construct quadratic forms on an algebra of functions and on a module of differential forms associated to the pair (V,ω). They generalize the Eisenbud–Levine–Khimshiashvili quadratic form defined for a smooth V. Partially supported by the DFG-programme 'Global methods in complex geometry' (Eb 102/4–3) grants RFBR–04–01–00762, NSh–1972.2003.1.     

Quasi-projective reduction of toric varieties

We define a quasi–projective reduction of a complex algebraic variety X to be a regular map from X to a quasi–projective variety that is universal with respect to regular maps from X to quasi–projective varieties. A toric quasi–projective reduction is the analogous notion in the category of toric varieties. For a given toric variety X we first construct a toric quasi–projective reduction. Then we show that X has a quasi–projective reduction if and only if its toric quasi–projective reduction is surjective. We apply this result to characterize when the action of a subtorus on a quasi–projective toric variety admits a categorical quotient in the category of quasi–projective varieties. Received October 29, 1998; in final form December 28, 1998     

Special finite-dimensional monocomposition algebras

Special monocomposition algebras introduced in [1] are studied. Using their properties, we prove that any nondegenerate monocomposition algebra ,dim ≥ 3, with unity contains no proper ideal of dimension ≤8. This implies that if 3≤dim ≤ 9, then is a central simple algebra. Translated fromAlgebra i Logika, Vol. 35, No. 2, pp. 125–144, March–April, 1996.     

A unified quantum <Emphasis Type="Italic">SO</Emphasis>(3) invariant for rational homology 3-spheres

Given a rational homology 3-sphere M with |H ₁(M,ℤ)|=b and a link L inside M, colored by odd numbers, we construct a unified invariant I _M,L belonging to a modification of the Habiro ring where b is inverted. Our unified invariant dominates the whole set of the SO(3) Witten–Reshetikhin–Turaev invariants of the pair (M,L). If b=1 and L=∅, I _M coincides with Habiro’s invariant of integral homology 3-spheres. For b>1, the unified invariant defined by the third author is determined by I _M . Important applications are the new Ohtsuki series (perturbative expansions of I _M ) dominating quantum SO(3) invariants at roots of unity whose order is not a power of a prime. These series are not known to be determined by the LMO invariant.     

Topological Structure of Non-wandering Set of a Graph Map

Let G be a graph （i.e., a finite one-dimensional polyhedron） and f ： G → G be a continuous map. In this paper, we show that every isolated recurrent point of f is an isolated non-wandering point; every accumulation point of the set of non-wandering points of f with infinite orbit is a two-order accumulation point of the set of recurrent points of f; the derived set of an ω-limit set of f is equal to the derived set of an the set of recurrent points of f; and the two-order derived set of non-wandering set of f is equal to the two-order derived set of the set of recurrent points of f.