Strong Approximation Property Research Articles

The strong approximation property

We introduce and investigate the strong approximation property of Banach spaces which is strictly stronger than the approximation property and at least formally weaker than the weak bounded approximation property. Among others, we show that the weak bounded approximation property is equivalent to a quantitative strengthening of the strong approximation property. Some recent results on the approximation property of Banach spaces and their dual spaces are improved.

Global stability of stochastic high-order neural networks with discrete and distributed delays

High-order neural networks can be considered as an expansion of Hopfield neural networks, and have stronger approximation property, faster convergence rate, greater storage capacity, and higher fault tolerance than lower-order neural networks. In this paper, the global asymptotic stability analysis problem is considered for a class of stochastic high-order neural networks with discrete and distributed time-delays. Based on an Lyapunov–Krasovskii functional and the stochastic stability analysis theory, several sufficient conditions are derived, which guarantee the global asymptotic convergence of the equilibrium point in the mean square. It is shown that the stochastic high-order delayed neural networks under consideration are globally asymptotically stable in the mean square if two linear matrix inequalities (LMIs) are feasible, where the feasibility of LMIs can be readily checked by the Matlab LMI toolbox. It is also shown that the main results in this paper cover some recently published works. A numerical example is given to demonstrate the usefulness of the proposed global stability criteria.

On almost strong approximation for some exceptional groups

If G is a simply connected semisimple group defined over a number field k and ∞ is the set of all infinite places of k , then G has strong approximation with respect to ∞ if and only if the archimedean part of any k -simple component of the adèle group G A is non-compact. Using the affine Bruhat–Tits building, the authors of [W.K. Chan, J. Hsia, On almost strong approximation of algebraic groups, J. Algebra 254 (2002) 441] formulated an almost strong approximation property (ASAP) for groups of compact type, and they proved that ASAP holds for all classical groups of compact type whose Tits indices over k are not 2 A n (d) with d ⩾3. In this paper, we show that ASAP holds for groups of types 3,6 D 4 ,G 2 ,F 4 ,E 7 , or E 8 .

On almost strong approximation for some exceptional groups

If G is a simply connected semisimple group defined over a number field k and ∞ is the set of all infinite places of k, then G has strong approximation with respect to ∞ if and only if the archimedean part of any k-simple component of the adèle group G A is non-compact. Using the affine Bruhat–Tits building, the authors of [W.K. Chan, J. Hsia, On almost strong approximation of algebraic groups, J. Algebra 254 (2002) 441] formulated an almost strong approximation property (ASAP) for groups of compact type, and they proved that ASAP holds for all classical groups of compact type whose Tits indices over k are not 2A n (d) with d⩾3. In this paper, we show that ASAP holds for groups of types 3,6D 4,G 2,F 4,E 7 , or E 8.

Geometrization of Monte-Carlo numerical analysis of an elliptic operator: strong approximation

A one-step scheme is constructed, which, as the Milstein scheme, has the strong approximation property of order 1; in contrast to the Milstein scheme, our scheme does not involve the simulation of iterated Itô integrals of second order. To cite this article: A.B. Cruzeiro et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).

Irreducible Tori in Semisimple Groups

Let G be an absolutely simple simply connected algebraic group over a global field k. In this note, we analyze arithmetic properties of the maximal k-tori of G .We establish density (in the variety of maximal tori) of the set of maximal k-tori which do not contain proper k-subtori (we call such tori k-irreducible). Using the fact that the k-irreducible tori we construct have the weak approximation property, we extend some of our previous results (contained in Publ. Math. Inst. Haute Etudes Sci. 84 (1996), 91-187, § 9) to global function fields. This allows one to establish the congruence subgroup property for the groups of points of G over semi-local subrings of k. Finally, we examine the strong approximation property in the maximal k-tori of G with respect to generalized arithmetic progressions.Irreducible Tori in semisimple groups (Erratum) dx.doi.org/10.1155/S1073792802112074

Geometric families of constant reductions and the Skolem property

Let F | K F|K be a function field in one variable and V \mathcal V be a family of independent valuations of the constant field K . K. Given v ∈ V , v\in \mathcal V , a valuation prolongation v \mathrm v to F F is called a constant reduction if the residue fields F v | K v F\mathrm v |Kv again form a function field of one variable. Suppose t ∈ F t\in F is a non-constant function, and for each v ∈ V v\in \mathcal V let V t V_{t} be the set of all prolongations of the Gauß valuation v t v_{t} on K ( t ) K(t) to F . F. The union of the sets V t V_{t} over all v ∈ V v\in \mathcal V is denoted by V t . \mathbfit {V}_{t}. The aim of this paper is to study families of constant reductions V \mathbfit {V} of F F prolonging the valuations of V \mathcal V and the criterion for them to be principal, that is to be sets of the type V t . \mathbfit {V}_{t}. The main result we prove is that if either V \mathcal V is finite and each v ∈ V v\in \mathcal V has rational rank one and residue field algebraic over a finite field, or if V \mathcal V is any set of non-archimedean valuations of a global field K K satisfying the strong approximation property, then each geometric family of constant reductions V \mathbfit {V} prolonging V \mathcal V is principal. We also relate this result to the Skolem property for the existence of V \mathcal V -integral points on varieties over K , K, and Rumely’s existence theorem. As an application we give a birational characterization of arithmetic surfaces X / S \mathcal X /S in terms of the generic points of the closed fibre. The characterization we give implies the existence of finite morphisms to P S 1 . \mathbb P ^{1}_{S}.

A stable range for quadratic forms over commutative rings

A commutative ring A has quadratic stable range 1 (qsr( A) = 1) if each primitive binary quadratic form over A represents a unit. It is shown that qsr( A) = 1 implies that every primitive quadratic form over A represents a unit, A has stable range 1 and finitely generated constant rank projectives over A are free. A classification of quadratic forms is provided over Bezout domains with characteristic other than 2, quadratic stable range 1, and a strong approximation property for a certain subset of their maximum spectrum. These domains include rings of holomorphic functions on connected noncompact Riemann surfaces. Examples of localizations of rings of algebraic integers are provided to show that the classical concept of stable range does not behave well in either direction under finite integral extensions and that qsr( A) = 1 does not descend from such extensions.

Strong Approximation Property for Baer Orderings on ∗-Fields

Let ( D, ∗) be a ∗-field with [ D: Z( D)] being finite. Our main objective is to show that the space of all Baer orderings (resp. weak ∗-orderings) of ( D, ∗) satisfies the strong approximation property iff every Baer ordering of ( D, ∗) is in fact a weak ∗-ordering. This shows that the notions of Baer orderings and weak ∗-orderings are respectively the "correct" analogues for semiorderings and orderings. We also intro-duce the concept of Baer formally real ∗-fields and Baer preorderings. We prove that a ∗-field admits a Baer ordering iff it is Baer formally real. In addition, some new results on weak ∗-orderings are also discussed.

Quadratic forms and Linkage of Quaternion Algebras

AbstractA field F satisfies n-linkage on a subset of Ḟ if whenever the quaternion algebrasare equal in Br(F) there exist z ∈ Ḟ withfor i = 1, 2, . . ., n. This linkage of quaternion algebras is examined and its relationship to the torsion freeness of I2(F) and to the strong approximation property is investigated.

Approximation of Arbitrary Distributed Lag Structures by a Modified Polynomial Lag: An Extension

Abstract Since a uniform approximation of distributed lag models by representatives of some subfamily of the family of all possible lag structures guarantees neither a reasonable standard error of forecast in large samples nor a good approximation of the true long-run response and mean lag, it is shown that the modified polynomial lags fulfill a stronger approximation property. A possible simplification of the estimation procedure for these models is mentioned.

Extremal and monogenic additive set functions

The extreme points of the convex set of all additive set functions on a field, which coincide on a subfield are characterized by a simple approximation property. It is proved that a stronger approximation property is characteristic for a so-called monogenic additive set function on a field, which can be generated uniquely by an additive set function on a subfield. Finally it is shown that a simple decomposition property must hold if the convex set above has a finite number of extreme points.