弱代数性自反与强代数性自反

Algebraic reflexivity introduced by Hadwin is related to linear interpolation. In this paper, the concepts of weakly algebraic reflexivity and strongly algebraic reflexivity which are also related to linear interpolation are introduced. Some properties of them are obtained and some relations between them revealed.     

Coherent algebraic sheaves in real algebraic geometry

Summary In the following paper we give examples of vector bundlesF→X defined on a regular, compact, connected affine varietyX such thatF has no algebraic structure.

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Riassunto Nel presente lavoro si danno esempi di fibrati vettorialiF→X, definiti su varietà affini, regolari, compatte, connesseX che non ammettono struttura algebrica.

     

On algebraic approximations of certain algebraic numbers

Using hypergeometric functions and the Thue-Siegel method we give an effective improvement of Liouville's approximation theorem. As an application, we derive effective upper bounds for the solutions (X,Y) of the two-parametric family of quartic Thue inequalities

|BX4−AX3Y−6BX2Y2+AXY3+BY4|?N     

On the question of the algebraic independence of algebraic powers of algebraic numbers

We obtain results showing that transcendental numbers of the form , wherea0, 1, is irrational, anda and are algebraic numbers, cannot be expressed algebraically in terms of two of the numbers. The proof is carried out by A. O. Gel'fond's method.Translated from Matematicheskie Zametki, Vol. 11, No. 6, pp. 635–644, June, 1972.     

Affine algebraic sets relative to an algebraic theory

For an arbitrary algebraic theory T and a prescribed T-algebraL, the geometrical category AfAlgSet(L) of affine algebraic sets over the affine lineL is build up. It is proved to be dually equivalent to the category FcAlg(L) of functional T-algebras overL. The Galois theory ofL is defined and a Galois duality established. These geometrical categories are characterized up to an equivalence of categories.     

Matroids algebraic overF(t) are algebraic overF

In his thesis [3] M. J. Piff conjectured that a matroid, which is algebraic over a fieldFit) witht transcendent overF, must be algebraic overF. Two proofs have appeared, one by Shameeva [5] and another one by the author [2], but both are unsatisfactory. In this paper I will settle conjecture by applying a theorem of Seidenberg.     

Simplifying method for algebraic approximation of certain algebraic numbers

A new simple method for approximating certain algebraic numbers is developed. By applying this method, an effective upper bound is derived for the integral solutions of the quartic Thue equation with two parameters $tx^4 - 4sx^3 y - 6tx^2 y^2 + 4sxy^3 + ty^4 = N$ , where s > 32t ³. As an application, Ljunggren’s equation is solved in an elementary way.     

Rational and algebraic approximations of algebraic numbers and their application

Effective rational and algebraic approximations of a large class of algebraic numbers are obtained by Thue-Siegel’s method. As an application of this result, it is proved that: if D>0 is not a square, and ε =x ₀ denotes the fundamental solution ofx ²?Dy ²=?1, thenx ²+1=Dy ⁴ is solvable if and only ify ₀=A ², where A is an integer. Moreover, if ε>64, thenx ²+1=Dy ⁴ has at most one positive integral solution (x, y).     

Ordered algebraic structures

Book Reviews

Ordered algebraic structuresJorge Martinez (ed.): Proceedings of the Caribbean Mathematics Foundation Conference on Ordered Algebraic Structures, Curaçao, August 1988, Mathematics and Its Applications, vol. 55, Kluwer Academic Publishers, Dordrecht, Boston, London, 1989, xx + 278 pp., ISBN 0-7923-0489-6, US$ 104.     

Applied algebraic dynamics

A review of the book Applied Algebraic Dynamics (De Gruyter Expositions in Mathematics 49, (2009)), written by Vladimir Anashin and Andrei Khrennikov, is presented.     

Adaptive algebraic smoothers

This paper will present a new method of adaptively constructing block iterative methods based on Local Sensitivity Analysis (LSA). The method can be used in the context of geometric and algebraic multigrid methods for constructing smoothers, and in the context of Krylov methods for constructing block preconditioners. It is suitable for both constant and variable coefficient problems. Furthermore, the method can be applied to systems arising from both scalar and coupled system partial differential equations (PDEs), as well as linear systems that do not arise from PDEs. The simplicity of the method will allow it to be easily incorporated into existing multigrid and Krylov solvers while providing a powerful tool for adaptively constructing methods tuned to a problem.     

Difference algebraic subgroups of commutative algebraic groups over finite fields

We study the question of which torsion subgroups of commutative algebraic groups over finite fields are contained in modular difference algebraic groups for some choice of a field automorphism. We show that if G is a simple commutative algebraic group over a finite field of characteristic p, ? is a prime different from p, and for some difference closed field (?, σ) the ?-primary torsion of G(?) is contained in a modular group definable in (?, σ), then it is contained in a group of the form {x∈G(?) :σ(x) =[a](x) } with a∈ℕ\p ^ℕ. We show that no such modular group can be found for many G of interest. In the cases that such equations may be found, we recover an effective version of a theorem of Boxall. Received: 28 May 1998 / Revised version: 20 December 1998