Nested Artin strong approximation property

The strong approximation property

Let G be a countable exact discrete group. G has the strong invariant approximation property(SIAP) if and only if C � u(G,S) G = C � � (G) ⊗ S for any Hilbert space H and closed subspace S ⊆ H. We shall use results of Haagerup and Kraus on the approximation property (AP) to investigate some permanence properties of the SIAP for discrete groups. This can be done most efficiently for exact groups. In this paper we describe that the stability properties of the SIAP property pass to semi direct products, and extensions for discrete exact groups.

A fundamental result in the arithmetic of integral quadratic forms is the strong approximation theorem for the spin group Spin of a non-degenerate inde®nite quadratic form in three or more variables. (In practice, one invokes the corresponding version for the kernel O0 of the spinor norm function.) This result is due to the works of Eichler and Kneser [Ei], [Kn] in the 1950s, and has ever since played a central role in the proofs of a number of important theorems for both inde®nite as well as de®nite quadratic forms. On the other hand, it is well-known that such a strong approximation cannot possibly hold for i† the group SO of a inde®nite quadratic form as it is not simply connected, and ii† for the group Spin of a de®nite quadratic form as Spin1 is compact. The purpose of this paper is to show that there exists a version of an ``almost'' strong approximation theorem for de®nite quadratic forms, one forOm and one forO ‡ m m 3† and show that they too have serious applications in the arithmetic theory; e.g., in the asymptotic distributions of classes in a genus and in the representation theory of de®nite forms. It is anticipated that such almost strong approximations would also extend to other classical groups and will be treated in a future article. We give a brief outline of the contents of this paper. Terminology and notations are generally those from [ OM]. For simplicity as well as convenience we shall consider here only positive de®nite integral Z-lattices. A key step in the proof of the strong approximation theorem for O0 is the following result on inde®nite representations with approximation property: Invent. math. 129, 471±487 (1997)

Extending results of Popescu and Brown, the main result of this paper is that excellent henselian R1 and S1 2-dimensional local rings, at least in characteristic zero, have the approximation property of M. Artin.Most of the paper consists of an extension of Néron's desingularization to rings which are R1 and S1; such a theorem was previously known for factorial domains. The main theorem is then deduced from this desingularization theorem using a theorem of Elkik.Because of cohomological obstructions, the desingularization theorem is proved only for quasi-projective varieties. In the previously known case for factorial domains, these obstructions are always zero and the desingularization can be obtained by blowing up subschemes. The more general desingularization of this paper is obtained by blowing up locally free sheaves instead, the obstructions being zero for this case.

Continued Fractions for Algebraic Formal Power Series over a Finite Base Field

The idea to write this paper was the completion of the paper of N. Sankaran (cf.[6]). In fact, in his theorem, he suppose that all automorphisms of the restricted power series rings with respect to the m-adic topology of the maximal ideal of a local ring is anautomorphism by substitution. But this is true for all injective endomorphisms (see our Proposition 1.5.). We study here different classes of endomorphisms of the restricted power series rings with respect to an arbitrary non nilpotent ideal in a commutative ring. To do this, we follow very closely the techniques of M.J. O'Malley concerning the rings of power series, but adapted to the case of the rings of restricted power series (cf.[3],[4]).

Let R be a commutative ring with identity and $${\mathbb {N}}_0$$ be the additive monoid of nonnegative integers. We say that a function $$t : {\mathbb {N}}_0 \times {\mathbb {N}}_0 \rightarrow R$$ is a twist function on R if t satisfies the following three properties for all $$n, m, q \in {\mathbb {N}}_0$$ : (i) $$t(0,q) = 1$$ , (ii) $$t(n,m) = t(m,n)$$ , and (iii) $$t(n,m) \cdot t(n + m, q) = t (n, m + q) \cdot t(m, q)$$ . Let $$R[\![X]\!]$$ (resp., R[X]) be the set of power series (resp., polynomials) with coefficients in R. For $$f = \sum _{n=0}^{\infty } a_nX^n$$ and $$g = \sum _{n=0}^{\infty } b_nX^n \in R[\![X]\!]$$ , let $$f+g = \sum _{n=0}^{\infty } (a_n+b_n)X^n$$ , $$f*_tg = \sum _{n=0}^{\infty }(\sum _{i+j = n}t(i,j)a_ib_j)X^n$$ . Then, $$R^t[\![X]\!]:= (R[\![X]\!], +, *_t)$$ and $$R^t[X] := (R[X], +, *_t)$$ are commutative rings with identity that contain R as a subring. In this paper, we study ring-theoretic properties of $$R^t[\![X]\!]$$ and $$R^t[X]$$ with focus on divisibility properties including UFDs and GCD-domains. We also show how these two rings are related to the usual power series and polynomial rings.

We continue the first and second authors’ study of q-commutative power series rings and Laurent series rings , specializing to the case in which the commutation parameters qij are all roots of unity. In this setting, R is a PI algebra, and we can apply results of De Concini, Kac, and Procesi to show that L is an Azumaya algebra whose degree can be inferred from the qij. Our main results establish an exact criterion (dependent on the qij) for determining when the centers of L and R are commutative Laurent series and commutative power series rings, respectively. In the event this criterion is satisfied, it follows that L is a unique factorization ring in the sense of Chatters and Jordan, and it further follows, by results of Dumas, Launois, Lenagan, and Rigal, that R is a unique factorization ring.

This paper deals with algebraic power series over a commutative semiringA. A characteristization result states that a power series is algebraic if and only if it is the behavior of a proper pushdown automaton.

In this chapter we prepare the ground for the proof of the Jung-Abhyankar theorem in section 2 and the study of quasiordinary power series in section 4 of chapter V. We assume that the reader is acquainted with the notion of power series over a field; in section 1, for the convenience of the reader, we give some background, introduce convergent power series over the real and complex numbers in section 2, and prove Weierstraß division and preparation theorem in section 3. The category of formal (resp. analytic) algebras—these are homomorphic images of formal (resp. convergent) rings of power series—is treated in section 4. The last section considers, in particular, the completion of an analytic algebra; these results are needed to prove the convergent case of the Jung-Abhyankar theorem.

Among commutative rings, the polynomial rings in a finite number of indeterminates enjoy important special properties and are frequently used in applications. As they are also of paramount importance in Algebraic Geometry, polynomial rings have been intensively studied. On the other hand, rings of formal power series have been extensively used in “algebroid geometry” and have many properties which are parallel to those of polynomial rings. In the first section of this chapter we shall define formal power series rings and we shall show that the main properties of polynomial rings which have been derived in previous chapters (see, in particular, Vol. I, Ch. I, §§ 16–18) hold also for formal power series rings. In the later sections of this chapter we shall give deeper properties of polynomial rings and, whenever possible, the parallel properties of power series rings.

Let D be a principal ideal domain (PID), I be an ideal of D, and X be an indeterminate over D. Let [D;I][X] be the subring of the power series ring consisting of all power series in such that for all large i. By definition, the polynomial ring and the power series ring are special cases of when and I = D, respectively. In this article, we investigate the ring in the case I is a nonzero proper ideal of D. We prove that R is a two-dimensional non-Noetherian ring. For each maximal ideal P of D, it is shown that is a height-one prime ideal of R. The set of units of R is given and the spectrum of R is also described. Unlike the power series ring the ring R is not a unique factorization domain (UFD). Furthermore, when I is a nonzero prime ideal, R does not satisfy both ACCP and the atomic property. In obtaining results on R, we introduce and sometimes use results on the ring RS , where I = dD with and Closely related to R, the ring RS is shown to be a Noetherian UFD with Krull dimension at most two. Moreover, RS has Krull dimension two exactly when I is not contained in the Jacobson radical of D; otherwise RS is a PID.

High-order neural networks can be considered as an expansion of Hopfield neural networks, and have stronger approximation property and faster convergence rate. However, in practice high order network is seldom to be used to solve combinatorial optimization problem. In this paper crossbar switch problem, which is an NP-complete problem, is used as an example to demonstrate how to use high order discrete Hopfield neural network to solve engineering optimization problems. The construction method of energy function and the neural computing algorithm are presented. It is also discussed the method how to speed the convergence and escape from local minima. Experimental results show that high order network has a quick convergence speed, and outperforms the traditional discrete Hopfield network.KeywordsHopfield networkconstraint satisfactioncrossbar switch problem

In the field of formal power series over a finite field, we prove a result which enables us to construct explicit examples of$U_{m}$-numbers by using continued fraction expansions of algebraic formal power series of degree$m>1$.

Let P = P(z,d/dz) be a linear ordinary differential operator with holomorphic coefficients in a neighbourgood X of the origin 0 in ℂ. Let Pu = f be an inhomogeneous differential equation in the space O {0} or in O {0}, where 0 {0} is the ring of convergent power series and O {0} is the ring of formal power series at the origin 0.