An Intuitionistic Axiomatisation of Real Closed Fields

We give an intuitionistic axiomatisation of real closed fields which has the constructive reals as a model. The main result is that this axiomatisation together with just the decidability of the order relation gives the classical theory of real closed fields. To establish this we rely on the quantifier elimination theorem for real closed fields due to Tarski, and a conservation theorem of classical logic over intuitionistic logic for geometric theories.     

Gabriel filters in real closed rings

Real closed rings arise in semi-algebraic geometry and topology as well as in the investigation of partially ordered rings. It is shown that localizations of real closed rings with respect to Gabriel filters, or more generally: multiplicative filters, are again real closed. Thus, real closedness is preserved under a large number of important ring theoretic constructions. For a few particularly simple cases the multiplicative filters are classified and the localizations are determined. Received: August 26, 1996     

正则的正实部函数的导数估计

讨论了正则的正实部函数的导数估计问题,利用正实部函数的性质,得到三阶导数、四阶导数的准确估计式.     

Classification of three-dimensional zeropotent algebras over the real number field

A nonassociative algebra is defined to be zeropotent if the square of any element is zero. In this paper, we give a complete classification of three-dimensional zeropotent algebras over the real number field up to isomorphism. By restricting the result to the subclass of Lie algebras, we can obtain a classification of three-dimensional real Lie algebras, which is in accordance with the Bianchi classification. Moreover, three-dimensional zeropotent algebras over a real closed field are classified in the same manner as those over the real number field.     

具有连续对合运算的实Banach*代数的Jordan结构

本文讨论了实Banach*代数的Jordan结构．主要结果:第一部分指出映射到 *-半单实Banach*代数上的Jordan*同态是连续的,且其核空间是闭*理想;由映射到交换实Banach*代数上的Jordan*同态诱导的因子代数也是交换的．第二部分介绍了两个不同的锥,并讨论了他们间的关系．另外,我们得到了关于实Banach*代数*- 根基的一个新的刻画．本文是Satish Shirali的工作的实化．     

Convex subrings of partially ordered rings

The relationship between a poring and a convex subring has been studied most successfully for porings with bounded inversion. The best conceivable results are known for real closed rings. The present paper focuses on the connections between the prime spectra and between the real spectra of a poring and a convex subring. Examples show that for arbitrary porings one may not expect a very close relationship. But with the assumption of bounded inversion or spectral compatibility the results are similar to those for real closed rings (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Local C~k──boundary Regularity of Integral Solution Operators of the ──equations on Open Set with Piecewise C~1──boundary

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Metrizability of Spaces of ℝ-places of Function Fields of Transcendence Degree 1 Over Real Closed Fields

In this article we discuss the following question “When do different orderings of the rational function field R(X) (where R is a real closed field) induce the same ?-place?” We use this to show that if R contains a dense real closed subfield R′, then the spaces of ?-places of R(X) and R′(X) are homeomorphic. For the function field K = R(X), we prove that its space M(K) of ?-places is metrizible if and only if R contains a countable dense subfield. Moreover, we show that this condition is neccessary for the metrizability of M(F) for any function field F of transcendence degree 1 over R.     

Maximal Points of Stable and Related Polynomials

We prove that any polynomial having all its roots in a closed half-plane, whose boundary contains the origin, has either one or two maximal points, and only one if it has at least one root in the open half-plane. This result concerns stable polynomials as well as polynomials having only real roots, including real orthogonal polynomials.     

实Clifford分析中正则函数列的几条性质

在解析函数列的收敛性定理的基础上 ,定义了实 Clifford分析中正则函数列的一致有界、内闭一致有界及内闭一致收敛等概念 ,并讨论了正则函数列的几条性质 .     

Regular versus continuous rational maps

We prove that a continuous map from a compact nonsingular real algebraic variety X into the unit 2-sphere can be approximated by regular maps if and only if it is homotopic to a continuous map which is regular in the complement of a Zariski closed subvariety A of X of codimension at least 3. The assumption on the codimension of A is essential.     

加强P除环上自共轭矩阵的几个定理

证明了如果Ｏ是加强Ｐ－除环,则为实封闭域．利用该结果还讨论了加强Ｐ一除环上自共轭矩阵的正定性．     

Second order homological obstructions on real algebraic manifolds

Let Y be a compact nonsingular real algebraic set whose homology classes (over Z/2) are represented by Zariski closed subsets. It is well known that every smooth map from a compact smooth manifold to Y is unoriented bordant to a regular map. In this paper, we show how to construct smooth maps from compact nonsingular real algebraic sets to Y not homotopic to any regular map starting from a nonzero homology class of Y of positive degree. We use these maps to obtain obstructions to the existence of local algebraic tubular neighborhoods of algebraic submanifolds of Rn and to study some algebro-homological properties of rational real algebraic manifolds.     

实Clifford分析中超正则函数列和函数空间的性质

定义了实Clifford分析中超正则函数列的一致有界、内闭一致有界及内闭一致收敛等概念,并讨论了超正则函数列及超正则函数空间的几条性质.     

The solutions to some operator equations with corresponding operators not necessarily having closed ranges

In this paper, we study the solvability of the operator equations in the general setting of infinite-dimensional Hilbert space with corresponding operators no necessarily having closed range. We get the necessary and sufficient conditions for the existences of the solutions and obtain formulae in each case for the general selfadjoint, positive and real positive solutions to these operator systems.     

Refinement monoids with weak comparability and applications to regular rings and -algebras

We prove a cancellation theorem for simple refinement monoids satisfying the weak comparability condition, first introduced by K.C. O'Meara in the context of von Neumann regular rings. This result is then applied to von Neumann regular rings and -algebras of real rank zero via the monoid of isomorphism classes of finitely generated projective modules.

     

Closed irregularity sets of linear differential systems with a parameter multiplying the derivative

For any closed subset M of the real line that does not contain zero, we construct a linear differential system with bounded piecewise continuous coefficient matrix A(·) such that the corresponding system with coefficient matrix μA(·) linearly depending on a real parameter μ is Lyapunov irregular for all μ in M and Lyapunov regular for all other parameter values.     

Idéaux fermés dans des intersections de classes non-quasi-analytiques

The author proves that any ideal generated by a finite number of real analytic functions in an intersection of non-quasi-analytic classes is closed for the natural Frechet topology associated to such a space. She also establishes a closedness criterion when the generators are less regular. To cite this article: E. Croix, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Some classes of functions of a linear closed operator

A linear closed densely defined operator and some domain Ω lying in the regular set of the operator and containing the negative real semiaxis of the real line are specified in a Banach space. We assume that power estimates for the norm of the resolvent operator are known at zero and infinity. We use the Cauchy integral formula to introduce operator functions generated by scalar functions that are analytic in a certain domain not containing the origin and containing the complement of Ω and satisfy power estimates for their absolute values at zero and infinity. We study some properties of operator functions, which were studied by the authors earlier for the case of an operator whose inverse is bounded; in particular, we study the multiplicative property.     

On Sign Conditions Over Real Multivariate Polynomials

We present a new probabilistic algorithm to find a finite set of points intersecting the closure of each connected component of the realization of every sign condition over a family of real polynomials defining regular hypersurfaces that intersect transversally. This enables us to show a probabilistic procedure to list all feasible sign conditions over the polynomials. In addition, we extend these results to the case of closed sign conditions over an arbitrary family of real multivariate polynomials. The complexity bounds for these procedures improve the known ones.