Arbitrary torsion classes and almost free abelian groups

Using the set theoretical principle ? for arbitrary large cardinals κ, arbitrary large strongly κ-free abelian groupsA are constructed such that Hom(A, G)={0} for all cotorsion-free groupsG with |G|<κ. This result will be applied to the theory of arbitrary torsion classes for Mod-Z. It allows one, in particular, to prove that the classF of cotorsion-free abelian groups is not cogenerated by aset of abelian groups. This answers a conjecture of Göbel and Wald positively. Furthermore, arbitrary many torsion classes for Mod-Z can be constructed which are not generated or not cogenerated by single abelian groups.     

Cotorsion Theories Cogenerated by a Torsionfree Class

Let R be a right perfect ring, and let (?, 𝒞) be a cotorsion theory in the category of right R-modules ? _R . In this article, it is shown that every right R-module has a superfluous ?-cover if and only if there exists a torsion theory (𝒜, ?) such that (?, 𝒞) is cogenerated by ?. It is also proved that if (𝒜, ?) is a cosplitting torsion theory, then (^⊥?, (^⊥?)^⊥) is a hereditary and complete cotorsion theory, and if (𝒜, ?) is a centrally splitting torsion theory, then (^⊥?, (^⊥?)^⊥) is a hereditary and perfect cotorsion theory.     

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.     

Canonical Dimension Of Orthogonal Groups

We prove Berhuy-Reichstein's conjecture on the canonical dimension of orthogonal groups showing that for any integer n ≥ 1, the canonical dimension of SO_2n+1 and of SO_2n+2 is equal to n(n + 1)/2. More precisely, for a given (2n + 1)-dimensional quadratic form φ defined over an arbitrary field F of characteristic ≠ 2, we establish a certain property of the correspondences on the orthogonal grassmannian X of n-dimensional totally isotropic subspaces of φ, provided that the degree over F of any finite splitting field of φ is divisible by 2ⁿ; this property allows us to prove that the function field of X has the minimal transcendence degree among all generic splitting fields of φ.     

Rings over which all torsion theories cogenerated by simple modules are jansian

Summary It is given a caracterization of (associative) rings (with unit element) over which all torsion theories cogenerated by simple modules are jansian. In the commutative case it is shown that such rings are finite products of local rings.

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Riassunto Vengono caratterizzati gli anelli (associativi con unità) per i quali tutte le teorie di torsione cogenerate da moduli semplici sono Jansiane. Nel caso commutativo si prova che tali anelli sono prodotti finiti di anelli locali.

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This work was written while the second author was visiting Università dell'Aquila, supported by a grant from the Consiglio Nazionale delle Ricerche.     

On Jodeit’s multiplier extension theorems

Let Δ(x) = max {1 - ¦x¦, 0} for all x ∈ ?, and let ξ_[0,1) be the characteristic function of the interval 0 ≤x < 1. Two seminal theorems of M. Jodeit assert that A and ξ_[0,1) act as summability kernels convertingp-multipliers for Fourier series to multipliers forL ^P (?). The summability process corresponding to Δ extendsL ^P (T)-multipliers from ? to ? by linearity over the intervals [n, n + 1],n ∈ ?, when 1 ≤p < ∞, while the summability process corresponding to ξ_[0,1) extends L^P(T)-multipliers by constancy on the intervals [n, n + 1),n ∈ ?, when 1 <p < ∞. We describe how both these results have the following complete generalization: for 1 ≤p < ∞, an arbitrary compactly supported multiplier forL ^P (?) will act as a summability kernel forL ^P (T)-multipliers, transferring maximal estimates from L^P(T) to L^P(?). In particular, specialization of this maximal theorem to Jodeit’s summability kernel ξ_{[0, 1)} provides a quick structural way to recover the fact that the maximal partial sum operator on L^P(?), 1 <p < ∞, inherits strong type (p,p)-boundedness from the Carleson-Hunt Theorem for Fourier series. Another result of Jodeit treats summability kernels lacking compact support, and we show that this aspect of multiplier theory sets up a lively interplay with entire functions of exponential type and sampling methods for band limited distributions.     

Elliptic curves and their torsion subgroups over number fields of type (2, 2, ..., 2)

Suppose thatE: y ² =x(x + M) (x + N) is an elliptic curve, whereM N are rational numbers (#0, ±1), and are relatively prime. LetK be a number field of type (2,...,2) with degree 2′. For arbitrary n, the structure of the torsion subgroup E(K) _tors of theK-rational points (Mordell group) ofE is completely determined here. Explicitly given are the classification, criteria and parameterization, as well as the groups E(K) _tors themselves. The order of E( K)_tors is also proved to be a power of 2 for anyn. Besides, for any elliptic curveE over any number field F, it is shown that E( L)_tors = E( F) _tors holds for almost all extensionsL/F of degree p(a prime number). These results have remarkably developed the recent results by Kwon about torsion subgroups over quadratic fields.     

Laurent-Padé approximants to four kinds of Chebyshev polynomial expansions. Part II: Clenshaw-Lord type approximants

Laurent-Padé (Chebyshev) rational approximantsP _m (w, w ⁻¹)/Q _n (w, w ⁻¹) of Clenshaw-Lord type [2,1] are defined, such that the Laurent series ofP _m /Q _n matches that of a given functionf(w, w ⁻¹) up to terms of orderw ^±(m+n) , based only on knowledge of the Laurent series coefficients off up to terms inw ^±(m+n) . This contrasts with the Maehly-type approximants [4,5] defined and computed in part I of this paper [6], where the Laurent series ofP _m matches that ofQ _n f up to terms of orderw ^±(m+n ), but based on knowledge of the series coefficients off up to terms inw ^±(m+2n). The Clenshaw-Lord method is here extended to be applicable to Chebyshev polynomials of the 1st, 2nd, 3rd and 4th kinds and corresponding rational approximants and Laurent series, and efficient systems of linear equations for the determination of the Padé-Chebyshev coefficients are obtained in each case. Using the Laurent approach of Gragg and Johnson [4], approximations are obtainable for allm≥0,n≥0. Numerical results are obtained for all four kinds of Chebyshev polynomials and Padé-Chebyshev approximants. Remarkably similar results of formidable accuracy are obtained by both Maehly-type and Clenshaw-Lord type methods, thus validating the use of either.     

On polynomial approximations to f a (Z)(z-a)−1 with complex a and some applications to certain non-hermitian matrices

We are concerned with the problem 1 $$\mathop {min}\limits_{p \in P_n } \mathop {max}\limits_{z \in [ - 1,1]} |w(z)(f_a (z) - p(z))|,a \in C/[ - 1,1],n = 0 \cdots $$ of best polynomial approximation of degree n to f_a(z)=(z?a)^?1 on the unit interval. Here P_n denotes the class of complex polynomials of degree at most n, and ω belongs to a certain classical family of weight functions. For real a the solution of this approximation problem is known. In this paper, we obtain the best approximations for purely imaginary a. For general a, close approximations to the optimal polynomials are derived by solving the approximation problem expli citly for a certain subclass of P_n. We then use these polynomials to devise an iterative method for the solution of linear systems Ax=b with coefficient matrices of the form A=cI+dT where T=T^H and c, d ∈C. Finally, as a further appication of our results, we derive bounds for the decay rates of the inverses of banded matrices A=cI+dT.     

Cubical 4-Polytopes with Few Vertices

A cubical polytope is a convex polytope all of whose facets are conbinatorial cubes. A d-polytope Pis called almost simple if, in the graph of P, each vertex of Pis d-valent of (d+ 1)-valent. It is known that, for d> 4, all but one cubical d-polytopes with up to 2^d+1vertices are almost simple, which provides a complete enumeration of all the cubical d-polytopes with up to 2^d+1vertices. We show that this result is also true for d=4.     

On Homogeneity of Radicals of Semigroup Algebras

We answer a known question showing that not all radicals of semigroup Q algebras of Q₀ -complete commutative, cancellative and torsion-free semigroups are homogeneous.     

The lower order of functions of the classB

The class of functions Φ(z, t) defined for z∈ Cⁿ and t ≥0 such that the functions Φ(z, ¦w¦), w∈C, are plurisubharmonic in Cⁿ⁺¹ is called the classD. A typical example of functions of the classB are functions of the form \(\ln M_g (z,t) = \mathop {\ln \sup |}\limits_{|w| = t} g(z,w)|\) where g(z, w), z∈Cⁿ, w∈C, is an entire function in Cⁿ⁺¹. In this note it is proved under certain restrictions on the function Φ(z, t)εB that its lower order relative to the variable t is the same for all z∈Cⁿ except, possibly, for the points z of a set of zero Γ capacity. See [5].