Dominions in finitely generated nilpotent groups

In the first part, we prove that the dominion (in the sense of Isbell) of a subgroup of a finitely generated nilpotent group is trivial in the category of all nilpotent groups. In the second part, we show that the dominion of a subgroup of a finitely generated nilpotent group of class two is trivial in the category of all metabelian nilpotent groups.     

Dominions of universal algebras and projective properties

Let A be a universal algebra and H its subalgebra. The dominion of H in A (in a class {ie304-01}) is the set of all elements a ∈ A such that every pair of homomorphisms f, g: A → ∈ {ie304-02} satisfies the following: if f and g coincide on H, then f(a) = g(a). A dominion is a closure operator on a set of subalgebras of a given algebra. The present account treats of closed subalgebras, i.e., those subalgebras H whose dominions coincide with H. We introduce projective properties of quasivarieties which are similar to the projective Beth properties dealt with in nonclassical logics, and provide a characterization of closed algebras in the language of the new properties. It is also proved that in every quasivariety of torsion-free nilpotent groups of class at most 2, a divisible Abelian subgroup H is closed in each group 〈H, a〉 generated by one element modulo H. Translated from Algebra i Logika, Vol. 47, No. 5, pp. 541–557, September–October, 2008.     

On the nonexistence of closed timelike geodesics in flat Lorentz 2-step nilmanifolds

The main purpose of this paper is to prove that there are no closed timelike geodesics in a (compact or noncompact) flat Lorentz 2-step nilmanifold where is a simply connected 2-step nilpotent Lie group with a flat left-invariant Lorentz metric, and a discrete subgroup of acting on by left translations. For this purpose, we shall first show that if is a 2-step nilpotent Lie group endowed with a flat left-invariant Lorentz metric then the restriction of to the center of is degenerate. We shall then determine all 2-step nilpotent Lie groups that can admit a flat left-invariant Lorentz metric. We show that they are trivial central extensions of the three-dimensional Heisenberg Lie group . If is one such group, we prove that no timelike geodesic in can be translated by an element of By the way, we rediscover that the Heisenberg Lie group admits a flat left-invariant Lorentz metric if and only if

     

Strongly closed subgroups of finite groups

This paper gives a complete classification of the finite groups that contain a strongly closed p-subgroup for p any prime.     

Absolutely continuous spectrum of fourth order difference equations

We have studied the absolutely continuous spectrum of a selfadjoint subspace extension generated by a three‐term fourth order difference equation with bounded coefficients using subspace theory. In particular, we have shown that the absolutely continuous spectrum exists outside a certain bounded interval. In addition, we have computed the spectral multiplicity as well as the location of absolutely continuous spectrum of selfadjoint subspace extension under certain asymptotic conditions.     

Absolutely bounded matrices and unconditional convergence

We characterize the so-called absolutely bounded matrices in terms of the (strong) unconditional convergence of their standard decompositions. There is a similar characterization of absolutely compact matrices, and both characterizations are closely related to some natural multiplication operators. Rudiments of the duality theory for the algebra of all absolutely bounded matrices are included.

     

A short note on groups in separably closed valued fields

In this note we show that groups with definable generics in a separably closed valued field K of finite imperfection degree can be embedded into groups definable in the algebraic closure of K.     

Pseudo real closed fields,pseudo p-adically closed fields and NTP2

The main result of this paper is a positive answer to the Conjecture 5.1 of [14] by A. Chernikov, I. Kaplan and P. Simon: If M is a PRC field, then $Th(M)$ is NTP₂ if and only if M is bounded. In the case of PpC fields, we prove that if M is a bounded PpC field, then $Th(M)$ is NTP₂. We also generalize this result to obtain that, if M is a bounded PRC or PpC field with exactly n orders or p-adic valuations respectively, then $Th(M)$ is strong of burden n. This also allows us to explicitly compute the burden of types, and to describe forking.     

Isomorphism of commutative group algebras of closed -local algebraically compact groups

Let be an abelian group and let be a field of 0$">. It is shown via a universal algorithm that if the modified Direct-Factor Problem holds, then the -isomorphism for some group yields provided is a closed -group or a -local algebraically compact group. In particular, this is the case when is closed -primary of arbitrary power, or is -local algebraically compact with cardinality at most and is in cardinality not exceeding . The last claim completely settles a question raised by W. May in Proc. Amer. Math. Soc. (1979) and partially extends our results published in Rend. Sem. Mat. Univ. Padova (1999) and Southeast Asian Bull. Math. (2001).

     

Absolutely continuous spectrum for one-dimensional Schrödinger operators with slowly decaying potentials: Some optimal results

The absolutely continuous spectrum of one-dimensional Schrödinger operators is proved to be stable under perturbation by potentials satisfying mild decay conditions. In particular, the absolutely continuous spectra of free and periodic Schrödinger operators are preserved under all perturbations satisfying This result is optimal in the power scale. Slightly more general classes of perturbing potentials are also treated. A general criterion for stability of the absolutely continuous spectrum of one-dimensional Schrödinger operators is established. In all cases analyzed, the main term of the asymptotic behavior of the generalized eigenfunctions is shown to have WKB form for almost all energies. The proofs rely on maximal function and norm estimates, and on almost everywhere convergence results for certain multilinear integral operators.

     

Lorentz Geometry of 2-Step Nilpotent Lie Groups

We study the geometry of 2-step nilpotent Lie groups endowed with left-invariant Lorentz metrics. After integrating explicitly the geodesic equations, we discuss the problem of the existence of translated geodesics in those groups. A good part of the paper focuses on the existence of closed timelike geodesics in compact Lorentz 2-step nilmanifolds. Other related results, corollaries, and examples are also presented.     

Absolutely continuous spectrum for random Schrödinger operators on the Bethe strip

The Bethe strip of width m is the cartesian product $\mathbb {B}\times \lbrace 1,\ldots ,m\rbrace$, where $\mathbb {B}$ is the Bethe lattice (Cayley tree). We prove that Anderson models on the Bethe strip have “extended states” for small disorder. More precisely, we consider Anderson‐like Hamiltonians $H_\lambda =\frac{1}{2} \Delta \otimes 1 + 1 \otimes A\,+\,\lambda \mathcal {V}$ on a Bethe strip with connectivity K ≥ 2, where A is an m × m symmetric matrix, $\mathcal {V}$ is a random matrix potential, and λ is the disorder parameter. Given any closed interval $I\subset \big (\!-\!\sqrt{K}+a_{{\rm max}},\sqrt{K}+a_{\rm {min}}\big )$, where a_min and a_max are the smallest and largest eigenvalues of the matrix A, we prove that for λ small the random Schrödinger operator H_λ has purely absolutely continuous spectrum in I with probability one and its integrated density of states is continuously differentiable on the interval I.     

Finite groups with 2-nilpotent subgroups of even index

Finite groups with 2-nilpotent subgroups of even index are described. Translated fromMatematicheskie Zametki, Vol. 61, No. 5, pp. 700–705, May, 1997. Translated by A. I. Shtern     

Isomorphisms,definable relations,and Scott families of class 2 nilpotent groups

The paper deals in questions on the complexity of isomorphisms and relations on universes of structures, on the number and properties of numberings in various hierarchies of sets, and on the existence of connections between semantic and syntactic properties of the structures and relations for the class of nilpotent groups of class two. Supported by the Council for Grants (under RF President) and State Aid of Fundamental Science Schools via project NSh-4413.2006.1. __________ Translated from Algebra i Logika, Vol. 46, No. 4, pp. 514–524, July–August, 2007.     

Limits of commutative triangular systems on simply connected step 2-nilpotent Lie groups

We prove that on simply connected step 2-nilpotent Lie groupsG any limit of a commutative infinitesimals triangular system of probability measures which are either all symmetric or supported by some discrete subgroupH is infinitely divisible onG resp.H.     

Description of finite nilpotent groups of the 2nd degree with a prime odd period

We describe the mentioned groups, assigning values to centralizers of the generating elements and defining a group operation on corteges of elements of the Galois field with a prime odd characteristic. On the indicated groups we define odules over the Galois field and describe these odules. We prove that the considered groups are those with unique extraction of root.     

Conjugate Points on 2-Step Nilpotent Groups

Let N be a simply connected 2-step nilpotent Lie group equipped with a left-invariant metric. We consider the characterizations of Jacobi fields and conjugate points along geodesics emanating from the identity element in N. We obtain a partial result for N and the complete result for N with a one-dimensional center.     

On the dynamics of certain actions of free groups on closed real analytic manifolds

Let M be a closed connected real analytic manifold; let be a free group on two generators. The set of analytic actions of on M endowed with Taken‚s topology contains a nonempty open subset whose corresponding actions share three properties: (a) they have every orbit dense, (b) they leave invariant no geometric structure on M, (c) any homeomorphism conjugating two of them is analytic. Received: October 3, 2001     

Closed Curves in E 2 and S 2 Whose Total Squared Curvatures and Almost Minimal

The total squared curvature of a closed plane curve is minimized by a covering of a circle, if the rotation number of the curve is not zero. We measure the closeness of a closed plane curve to a circle by the difference of its total squared curvature from the minimum. A similar result is obtained for a simple closed curve in the 2-sphere.