关于实Hilbert环

通过引进“强实Hilbert环”这一概念,本文证明了,一个环A是强实Hilbert环,当且仅当多项式环A[X]是实Hilbert环,当且仅当A[X]的每个实极大理想在A上的局限是实极大的,从而文献[1]中两个主要结果被否定.此外,本文还研究了所谓的“严格的实Hilbert环”,这类环对于半代数零点定理等方面的探讨更具应用意义.     

关于严实Hilbert环

在本文中，严实Ｈｉｌｂｅｒｔ环得到了更进一步的刻划．本文的主要结果是：一个环Ａ是严实Ｈｉｌｂｅｒｔ环，当且仅当多项式环Ａ［Ｘ］的每个实极大理想在Ａ上的局限是Ａ的一个极大理想，当且仅当Ａ是实Ｈｉｌｂｅｒｔ环，且Ａ［Ｘ］的每个实极大理想是极大的．     

关于严实Hilbert环

在本文中，严实Ｈｉｌｂｅｒｔ环得到了更进一步的刻划．本文的主要结果是：一个环Ａ是严实Ｈｉｌｂｅｒｔ环，当且仅当多项式环Ａ［Ｘ］的每个实极大理想在Ａ上的局限是Ａ的一个极大理想，当且仅当Ａ是实Ｈｉｌｂｅｒｔ环，且Ａ［Ｘ］的每个实极大理想是极大的．     

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     

Cohomological non-rigidity of generalized real Bott manifolds of height 2

We investigate the following problem: When do two generalized real Bott manifolds of height 2 have isomorphic cohomology rings with ?/2 coefficients and also when are they diffeomorphic? It turns out that in general cohomology rings with ?/2 coefficients do not distinguish those manifolds up to diffeomorphism. This gives a negative answer to the cohomological rigidity problem for real toric manifolds posed earlier by Y. Kamishima and the present author. We also prove that generalized real Bott manifolds of height 2 are diffeomorphic if they are homotopy equivalent.     

Approximate real inversion formulas of the gaussian convolution

We shall give very and surprisingly simple approximate real inversion formulas of the Gaussian convolution (the Weierstrass transform) for the first-order Sobolev Hilbert space on the whole real line by using best approximations and the theory of reproducing kernels and by using a good connection with the Tikhonov regularization.     

The ascending chain condition for real ideas

If A is a ring satisfying the ascending chain condition for real ideals, then this condition is also satisfied by the polynomial ring A[X]. However, an example is given to illustrate that the condition need not to hold in the power series ring A[[X]]. It is also shown that if every real prime ideal is the real ideal generated by finitely many elements, then the ring satisfies the ascending chain condition for real ideals. So, the analogues of Hilbert basis theorem and Cohen's theorem hold for real ideals.     

A multidimensional real Schwarz Lemma for the Hilbert metric

We present a real multidimensional version of the Schwarz Lemma on a bounded convex domain D of ℝ ⁿ endowed with the Hilbert metric. We provide as an application an extension of a Birkhoff’s Theorem on mappings contracting the Hilbert metric.     

On formally real modules

The aim of this paper is to develop the study of modules in real algebra. By introducing two kinds of reality for modules over commutative rings, we establish some results on semireal modules and real modules. Our results involve real radicals of submodules and orderings on modules.     

Approximation of the Hilbert Transform on the real line using Hermite zeros

The authors study the Hilbert Transform on the real line. They introduce some polynomial approximations and some algorithms for its numerical evaluation. Error estimates in uniform norm are given.

     

Computing Stark units for totally real cubic fields

A method for computing provably accurate values of partial zeta functions is used to numerically confirm the rank one abelian Stark Conjecture for some totally real cubic fields of discriminant less than 50000. The results of these computations are used to provide explicit Hilbert class fields and some ray class fields for the cubic extensions.

     

On linear relations for L-values over real quadratic fields

In this paper, we give a method to construct a classical modular form from a Hilbert modular form. By applying this method, we can get linear formulas which relate the Fourier coefficients of the Hilbert and classical modular forms. The paper focuses on the Hilbert modular forms over real quadratic fields. We will state a construction of relations between the special values of L-functions, especially at 0, and arithmetic functions. We will also give a relation between the sum of squares functions with underlying fields \(\mathbb {Q}(\sqrt{D})\) and \(\mathbb {Q}\).     

Theta functions of indefinite quadratic forms over real number fields

We define theta functions attached to indefinite quadratic forms over real number fields and prove that these theta functions are Hilbert modular forms by regarding them as specializations of symplectic theta functions. The eighth root of unity which arises under modular transformations is determined explicitly.

     

On representations of real Jacobi groups

We consider a category of continuous Hilbert space representations and a category of smooth Fr’echet representations,of a real Jacobi group G.By Mackey’s theory,they are respectively equivalent to certain categories of representations of a real reductive group L.Within these categories,we show that the two functors that take smooth vectors for G and for L are consistent with each other.By using Casselman-Wallach’s theory of smooth representations of real reductive groups,we define matrix coefficients for distributional vectors of certain representations of G.We also formulate Gelfand-Kazhdan criteria for real Jacobi groups which could be used to prove multiplicity one theorems for Fourier-Jacobi models.     

Pointwise bounds of orthogonal expansions on the real line via weighted Hilbert transforms

We study pointwise bounds of orthogonal expansions on the real line for a class of exponential weights of smooth polynomial decay at infinity. As a consequence of our main results, we establish pointwise bounds for weighted Hilbert transforms which are of independent interest. Mathematics subject classifications (2000)  41A10, 42C05     

A converse to the Eidelheit theorem in real Hilbert spaces

We establish the following converse to the Eidelheit theorem: an unbounded closed and convex set of a real Hilbert space may be separated by a closed hyperplane from every other disjoint closed and convex set, if and only if it has a finite codimension and a non-empty interior with respect to its affine hull.     

Quadratic forms over formally real fields with eight square classes

Complete classification of formally real fields with 8 square classes with respect to the behaviour of quadratic forms is given. Two fields F and K are equivalent with respect to quadratic forms if the quadratic form schemes of the two fields are isomorphic or in other words, if the Witt rings W(F) and W(K) are isomorphic. It is shown here that for formally real fields with 8 square classes there are exactly seven possible quadratic form schemes and for each of the seven schemes a formally real field with 8 square classes and the given scheme is constructed.     

Existence and uniqueness results for mild solutions of random impulsive abstract neutral partial differential equation over real axis

In this paper, we discuss the existence and uniqueness of mild solutions of random impulsive abstract neutral partial differential equations in a real separable Hilbert space. The results are obtained by using Leray-Schauder Alternative and Banach Contraction Principle. Finally an example is given to illustrate our problem.     

Densities for some real quadratic fields with infinite Hilbert 2-class field towers

Let K be a real quadratic field with 2-class rank equal to 4 or 5 and 4-class rank equal to 3. This paper computes density information for such fields to have infinite Hilbert 2-class field towers.