Condition numbers of algebraic Riccati equations in the Frobenius norm

Consider the continuous-time algebraic Riccati equation (CARE) and the discrete-time algebraic Riccati equation (DARE) which arise in linear control and system theory. It is known that appropriate assumptions on the coefficient matrices guarantee the existence and uniqueness of Hermitian positive semidefinite stabilizing solutions. In this note, we apply the theory of condition developed by Rice to define condition numbers of the CARE and DARE in the Frobenius norm, and derive explicit expressions of the condition numbers in a uniform manner. Both the complex case and real case are considered, and connections to certain existing condition numbers of the CARE and DARE are discussed.     

On the spectral norm of algebraic numbers

In this paper we continue to study the spectral norms and their completions ([4]) in the case of the algebraic closure $ \overline {\mathbb Q} $ of ? in ?. Let $ \widetilde{\overline{\mathbb{Q}}} $ be the completion of $ \overline {\mathbb Q} $ relative to the spectral norm. We prove that $ \widetilde{\overline{\mathbb{Q}}} $ can be identified with the R‐subalgebra of all symmetric functions of C(G), where C(G) denotes the ?‐Banach algebra of all continuous functions defined on the absolute Galois group G = Gal$ {\overline {\mathbb Q}} / {\mathbb Q} $. We prove that any compact, closed to conjugation subset of ? is the pseudo‐orbit of a suitable element of $ \widetilde{\overline{\mathbb{Q}}} $. We also prove that the topological closure of any algebraic number field in $ \widetilde{\overline{\mathbb{Q}}} $ is of the form $\widetilde{\mathbb{Q}[x]}$ with x in $ \widetilde{\overline{\mathbb{Q}}} $.     

用里特-吴特征集解代数方程直至重数的一个方法

里特—吴特征集提供了用计算机解代数方程的有效方法，但迄今为止，还不能由这一方法给出孤立解的重数．文章给出了孤立解的重数的两个定义，它们是等价的，并且在范德瓦尔登的定义有意义时与后者一致．一个定义是在非标准分析的框架中，另一个则是标准分析的．在证明与范德瓦尔登的定义一致时，非标准分析的定义是本质的．通过再一次在计算机上应用里特—吴方法于由原方程得到的含无穷小参数的代数方程，可以得到原方程的孤立解的重数．文中给出一个例子的计算机计算结果：首先得出有八个解，然后给出它们的重数：其中有两个的重数为六重，另六个为单根．     

Estimates of characteristic numbers of real algebraic varieties

We give some explicit bounds for the number of cobordism classes of real algebraic manifolds of real degree less than d, and for the size of the sum of Betti numbers with Z/2 coefficients for the real form of complex manifolds of complex degree less than d.     

Picard and Lefschetz numbers of real algebraic surfaces

Two Picard numbers and two Lefschetz numbers are defined for a real algebraic surface. They are similar to the Picard number and the Lefschetz number of a complex algebraic surface. For these numbers, some estimates and relations in the form of inequalities are proved.Translated fromMatematicheskie Zametki, Vol. 63, No. 6, pp. 847–852, June, 1998.     

The algebraic theory of the fundamental germ

This paper introduces a notion of fundamental group appropriate for laminations.     

Asymptotic behaviour of Betti numbers of real algebraic surfaces

Let be a nonsingular real algebraic surface of degree m in the complex projective space and its real point set in . In the spirit of the sixteenth Hilbert's problem, one can ask for each degree m about the maximal possible value of the Betti number (i=0 or 1). We show that is asymptotically equivalent to for some real number and prove inequalities and . Received: April 26, 2000     

A characterization of quadrics by intersection numbers

This work is inspired by a paper of Hertel and Pott on maximum non-linear functions (Hertel and Pott, A characterization of a class of maximum non-linear functions. Preprint, 2006). Geometrically, these functions correspond with quasi-quadrics; objects introduced in De Clerck et al. (Australas J Combin 22:151–166, 2000). Hertel and Pott obtain a characterization of some binary quasi-quadrics in affine spaces by their intersection numbers with hyperplanes and spaces of codimension 2. We obtain a similar characterization for quadrics in projective spaces by intersection numbers with low-dimensional spaces. Ferri and Tallini (Rend Mat Appl 11(1): 15–21, 1991) characterized the non-singular quadric Q(4,q) by its intersection numbers with planes and solids. We prove a corollary of this theorem for Q(4,q) and then extend this corollary to all quadrics in PG(n,q),n ≥ 4. The only exceptions occur for q even, where we can have an oval or an ovoid as intersection with our point set in the non-singular part.      

Rings with Zassenhaus Families of Ideals

Let R be a ring with identity. We call a family ? of left ideals of R a Zassenhaus family if the only additive endomorphisms of R that leave all members of ? invariant are the left multiplications by elements of R. Moreover, if R is torsion-free and there is some left R-module M such that R ? M ? R?_?? and End _?(M) = R we call R a “Zassenhaus ring”. It is well known that all Zassenhaus rings have Zassenhaus families. We will give examples to show that the converse does not hold even for torsion-free rings of finite rank.     

Sums of squares of regular functions on real algebraic varieties

Let be an affine algebraic variety over (or any other real closed field ). We ask when it is true that every positive semidefinite (psd) polynomial function on is a sum of squares (sos). We show that for the answer is always negative if has a real point. Also, if is a smooth non-rational curve all of whose points at infinity are real, the answer is again negative. The same holds if is a smooth surface with only real divisors at infinity. The ``compact' case is harder. We completely settle the case of smooth curves of genus : If such a curve has a complex point at infinity, then every psd function is sos, provided the field is archimedean. If is not archimedean, there are counter-examples of genus .

     

Vector bundles over real algebraic varieties

The object of this paper is to study continuous vector bundles, over real algebraic varieties, admitting an algebraic structure. For large classes of real varieties, we obtain explicit information concerning the Grothendieck group of algebraic vector bundles. We show that in many cases this group is small compared to the corresponding group of continuous vector bundles. These results are used elsewhere to study the geometry of real algebraic varieties.Dedicated to Professor Alexander Grothendieck on the occasion of his 60th birthdaySupported by the NSF Grant DMS-8602672.     

Real zeros of the zero-dimensional parametric piecewise algebraic variety

The piecewise algebraic variety is the set of all common zeros of multivariate splines. We show that solving a parametric piecewise algebraic variety amounts to solve a finite number of parametric polynomial systems containing strict inequalities. With the regular decomposition of semi-algebraic systems and the partial cylindrical algebraic decomposition method, we give a method to compute the supremum of the number of torsion-free real zeros of a given zero-dimensional parametric piecewise algebraic variety, and to get distributions of the number of real zeros in every n-dimensional cell when the number reaches the supremum. This method also produces corresponding necessary and sufficient conditions for reaching the supremum and its distributions. We also present an algorithm to produce a necessary and sufficient condition for a given zero-dimensional parametric piecewise algebraic variety to have a given number of distinct torsion-free real zeros in every n-cell in the n-complex. This work was supported by National Natural Science Foundation of China (Grant Nos. 10271022, 60373093, 60533060), the Natural Science Foundation of Zhejiang Province (Grant No. Y7080068) and the Foundation of Department of Education of Zhejiang Province (Grant Nos. 20070628 and Y200802999)     

Linear continuous division for exterior and interior products

We consider the complex

>> {\Lambda\sb 0(M;E)} @>{\partial\sb \omega}>> ... ...>>{\dots} @>{\partial\sb\omega}>> {\Lambda\sb m(M;E)}, \end{CD}\end{displaymath}">

where is a finite-dimensional vector bundle over a suitable differential manifold , denotes the space of all smooth or real analytic or holomorphic sections of the -exterior product of and for . We give sufficient and necessary conditions for the above complex to be exact and, in smooth and holomorphic cases, we give sufficient conditions for its splitting, i.e., for existence of linear continuous right inverse operators for .

Analogous results are obtained whenever is replaced by a suitable closed subset or are replaced by the interior product operators , for a given section of the dual bundle .

     

On the Krull Dimension of Rings of Transfer Functions

In this article, we prove that the Krull dimension of several commonly used classes of transfer functions of infinite dimensional linear control systems is infinite. On the other hand, we also show that the weak Krull dimension of the Hardy algebra , the disk algebra and the Wiener algebra is equal to 1. A. Sasane is supported by the Nuffield Grant NAL/32420.