The metric anomaly of analytic torsion at the boundary of an even dimensional cone

The formula for analytic torsion of a cone in even dimensions is composed of three terms. The first two terms are well understood and given by an algebraic combination of the Betti numbers and the analytic torsion of the cone base. The third “singular” contribution is an intricate spectral invariant of the cone base. We identify the third term as the metric anomaly of the analytic torsion coming from the non-product structure of the cone at its regular boundary. Hereby we filter out the actual contribution of the concial singularity and identify the analytic torsion of an even-dimensional cone purely in terms of the Betti numbers and the analytic torsion of the cone base.     

Minimal curves of constant torsion

The Griffiths formalism is applied to find constant torsion curves which are extremal for arclength with respect to variations preserving torsion, fixing the endpoints and the binormals at the endpoints. The critical curves are elastic rods of constant torsion, which are shown to not realize certain boundary conditions.

     

TORSION PRERADICALS—A TOOL FOR ANALYSING RINGS

Abstract

This paper surveys a selection of results in the literature on torsion preradicals; these are left exact preradical functors on the category of unital right modules over an associative ring with identity. Various well known classes of rings such as semisimple, artinian, perfect and strongly prime are characterized in terms of torsion preradicals. A classification of prime rings using torsion preradicals is also exhibited. Rings all of whose torsion preradicals are radicals and rings whose torsion preradicals commute, are investigated. An application of the latter condition to Jacobson's Conjecture is presented.     

Geometric equivalence of nilpotent groups

Nilpotent, torsion free groups are considered. Sufficient conditions are presented for a nilpotent, torsion free group to be geometrically equivalent to its Mal’tsev completion. Also some results are achieved in describing the classes of geometric equivalence of class 2 nilpotent, torsion free groups with center of small rank. Bibliography: 15 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 330, 2006, pp. 259–270.     

$ L^2 $-torsion of Hyperbolic Manifolds of Finite Volume

Suppose is a compact connected odd-dimensional manifold with boundary, whose interior M comes with a complete hyperbolic metric of finite volume. We will show that the -topological torsion of and the -analytic torsion of the Riemannian manifold M are equal. In particular, the -topological torsion of is proportional to the hyperbolic volume of M, with a constant of proportionality which depends only on the dimension and which is known to be nonzero in odd dimensions [HS]. In dimension 3 this proves the conjecture [Lü2, Conjecture 2.3] or [LLü, Conjecture 7.7] which gives a complete calculation of the -topological torsion of compact -acyclic 3-manifolds which admit a geometric JSJT-decomposition.?In an appendix we give a counterexample to an extension of the Cheeger-Müller theorem to manifolds with boundary: if the metric is not a product near the boundary, in general analytic and topological torsion are not equal, even if the Euler characteristic of the boundary vanishes. Submitted: March 1998, revised: July 1998.     

Superposition of large butt-end and coaxial torsional and axial shear deformations of homogeneous and fiber-reinforced thick-wall cylinders

A mathematical model of superposition of large butt-end and coaxial torsional and axial shear deformations of homogeneous and fiber-reinforced thick-wall cylinders is constructed. The macroscopic stresses of the reinforced material are additively determined by matrix stresses and by tensile or constrained compression stresses in the reinforcing fibers. The model is based on the numerical solution of two boundary-value problems, one of which corresponds to the butt-end torsion and the other to the coaxial torsion and axial shear. The boundary-value problem on joint deformations is solved with the use of the displacement field determined from the solution to the boundary-value problem on butt-end torsion. The results obtained by applying this method to homogeneous and axially-radially reinforced thick-wall cylinders subjected to butt-end torsion with subsequent coaxial torsion and axial shear are presented. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 43, No. 4, pp. 465–492, July–August, 2007.     

Approximating absolute Galois groups

In this paper we identify a class of profinite groups (totally torsion free groups) that includes all separable Galois groups of fields containing an algebraically closed subfield, and demonstrate that it can be realized as an inverse limit of torsion free virtually finitely generated abelian (tfvfga) profinite groups. We show by examples that the condition is quite restrictive. In particular, semidirect products of torsion free abelian groups are rarely totally torsion free. The result is of importance for K-theoretic applications, since descent problems for tfvfga groups are relatively manageable.     

A refinement of the Ray–Singer torsion

We propose a refinement of the Ray–Singer torsion, which can be viewed as an analytic counterpart of the refined combinatorial torsion introduced by Turaev. Given a closed, oriented manifold of odd dimension with fundamental group Γ, the refined torsion is a complex valued, holomorphic function defined for representations of Γ which are close to the space of unitary representations. When the representation is unitary the absolute value of the refined torsion is equal to the Ray–Singer torsion, while its phase is determined by the η-invariant. As an application we extend and improve a result of Farber about the relationship between the absolute torsion of Farber–Turaev and the η-invariant. To cite this article: M. Braverman, T. Kappeler, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

Torsion free extensions and perfect localizations

This note may be viewed as a continuation of the ideas developed in [13]. We start by reviewing some principles concering bimodule localization, the fundamentals of which the reader is assumed to be familiar with. In particular, the main properties of perfect localization and their relation to extensions are recollected. Next, we introduce the idea of a torsion free extension and prove its main features, especially in relation to torsion theories. We conclude by generalizing a theorem of L. Rowen's [9] concerning rings which are torsion free over their center to arbitrary torsion free extensions. His results are thus shown to be of torsion theoretic nature.     

Lattice Properties of Torsion Theories

The main purpose of this brief note is to show that the sets of all hereditary torsion theories which are noetherian, strongly noetherian, or of finite type, respectively, form a sublattice of the lattice of all hereditary torsion theories for the category R-mod.     

Uniform boundedness of <Emphasis Type="Italic">p</Emphasis>-primary torsion of abelian schemes

Let k be a field finitely generated over ℚ and p a prime. The torsion conjecture (resp. p-primary torsion conjecture) for abelian varieties over k predicts that the k-rational torsion (resp. the p-primary k-rational torsion) of a d-dimensional abelian variety A over k should be bounded only in terms of k and d. These conjectures are only known for d=1. The p-primary case was proved by Y. Manin, in 1969; the general case was completed by L. Merel, in 1996, after a series of contributions by B. Mazur, S. Kamienny and others. Due to the fact that moduli of elliptic curves are 1-dimensional, the d=1 case of the torsion conjecture (resp. p-primary torsion conjecture) is closely related to the following. For any k-curve S and elliptic scheme E→S, the k-rational torsion (resp. the p-primary k-rational torsion) is uniformly bounded in the fibres E _s , s∈S(k). In this paper, we extend this result in the p-primary case to arbitrary abelian schemes over curves.     

Radical and Torsion Theory for Acts

After recapitulating the rudiments of the Kurosh-Amitsur radical theory of S-acts, hereditary radicals are discussed. The hereditary Hoehnke radical assignments r which designate a Rees congruence r(A) to each S-act A, are the hereditary Kurosh-Amitsur radical assignments. Then the corresponding radical as well as semisimple classes are characterized. Equivalence classes of injective S-acts determine hereditary torsion assignments t, these are just the hereditary Hoehnke radicals, but the congruence t(A) of A need not be a Rees congruence. Torsion and torsionfree classes are characterized; several hereditary torsion assignments may determine the same torsion class which is always a radical class closed under taking subacts. Examples show that a hereditary torsion assignment need not be a hereditary radical.     

Intermediately fully invariant subgroups of abelian groups

Describing intermediately fully invariant subgroups of divisible and torsion groups, we show that the intermediately fully invariant subgroups are direct summands in a completely decomposable group whose every homogeneous component is decomposable. For torsion groups, we find out when all their fully invariant subgroups are intermediately fully invariant; and for torsion-free groups, this question comes down to the reduced case. Also, in a torsion group that is the sum of cyclic subgroups, its subgroup is shown to be intermediately inert if and only if it is commensurable with some intermediately fully invariant subgroup.     

Macpherson's chern classes of singular algebraic varieties

Sacerdote [Sa] has shown that the non-Abelian free groups satisfy precisely the same universal-existential sentences Th(F₂)??? in a first-order language Lo appropriate for group theory. It is shown that in every model of Th(F₂)??? the maximal Abelian subgroups are elementarily equivalent to locally cyclic groups (necessarily nontrivial and torsion free). Two classes of groups are interpolated between the non-Abelian locally free groups and Remeslennikov’s ?-free groups. These classes are the almost locally free groups and the quasi-locally free groups. In particular, the almost locally free groups are the models of Th(F₂)??? while the quasi-locally free groups are the ?-free groups with maximal Abelian subgroups elementarily equivalent to locally cyclic groups (necessarily nontrivial and torsion free). Two principal open questions at opposite ends of a spectrum are: (1.) Is every finitely generated almost locally free group free? (2.) Is every quasi-locally free group almost locally free? Examples abound of finitely generated quasi-locally free groups containing nontrivial torsion in their Abelianizations. The question of whether or not almost locally free groups have torsion free Abelianization is related to a bound in a free group on the number of factors needed to express certain elements of the derived group as a product of commutators     

Additive Groups of Local Near-Rings

The object of this note is to obtain information about the additive groups of local near-rings. It will be shown that the additive group of a torsion local near-ring is a bounded p-group. The torsion Abelian groups which are additive groups of local near-rings will be described completely. A method will be given to construct groups of almost any prime power order, which are not additive groups of local near-rings.     

Equivalence of higher torsion invariants

We show that the smooth torsion of bundles of manifolds constructed by Dwyer, Weiss, and Williams satisfies the axioms for higher torsion developed by Igusa. As a consequence we obtain that the smooth Dwyer–Weiss–Williams torsion is proportional to the higher torsion of Igusa and Klein.     

Structural changes during torsion of polyethylene

X-ray structural studies on the changes in oriented polyethylene fibers during torsion at large and small angles showed that the crystallites undergo shear deformation, and the angle x between the axes of the macromolecules and the upper (lower) faces of the crystallites deviates appreciably from a right angle. As a result of torsion of the fibers, an axial texture of the fibrill is formed, inside which the crystallites are skewed, while the c axes of the crystallites are inclined to the texture axis. The nature of the more specific changes in the structure depends on the ratio between the temperatures of the preliminary orientational drawing of the samples, and the subsequent torsion.The V. I. Lenin Tadzhik State University, Dushanbe, Institute of Macromolecular Compounds, Academy of Sciences of the USSR, Leningrad. Translated from Mekhanika Polimerov, No. 1, pp. 160–162, January–February, 1974.     

High rank elliptic curves with prescribed torsion group over quadratic fields

There are 26 possibilities for the torsion groups of elliptic curves defined over quadratic number fields. We present examples of high rank elliptic curves with a given torsion group which set the current rank records for most of the torsion groups. In particular, we show that for each possible torsion group, except maybe for \(\mathbb {Z}/15\mathbb {Z}\) , there exists an elliptic curve over some quadratic field with this torsion group and with rank \(\ge 2\) .     

边裂纹柱扭转的强奇性积分方程解法*

本文利用单裂纹扭转的位错型解答,使用有限部积分的概念和方法,最后将含有单根水平裂纹的柱体扭转问题归为解一个强奇性积分方程,并为其建立了数值求解方法,文末作了若干数值例子的计算,结果令人满意.