Pansystems clustering analysis of complex systems

In this paper, by use of equivalence operators δ_i and semi-equivalence operators ε_i we study the clustering problems of complex systems, present δ_i(1,3) disconnection principle,dual transformation principle and large-scale systems decomposition principle for analizing and operating complex systems, discuss interconnectivity and disconnectivity of complex systems in detail and present some related theorems. Finally, we discuss the levels of systems according to pansystems, clustering approach proposed in this paper.     

Nonequivalence transformation of λ‐matrix eigenproblems and model embedding approach to model tuning

In this paper we present some theory for a non‐equivalence transformation of matrix eigenvalues for λ‐matrix polynomials. Application of this transformation to eigenvalue embedding for model tuning on a realistic industry problem is illustrated. The new approach has several advantages such as flexibility, efficiency, and structure‐preservation. A numerical experiment on a pseudosimulation data set from The Boeing Company is reported. Copyright © 2001 John Wiley & Sons, Ltd.     

Quantifier elimination on some pseudo-algebraically closed valued fields

In this article, some theories of pseudo-algebraically closed non-trivially valued fields are shown to admit quantifier elimination in the language obtained by adjoining to the language of rings the function symbols for splitting coefficients, the function symbols for relative p-coordinate functions, and the division predicate for a valuation.     

等价度量下的直径型填充测度

本文研究了在等价度量下,直径型填充测度之间的关系,证明了对任意紧度量空间(X,ρ),Pρ,g和Pcρ,g等价当且仅当纲函数g满足加倍条件.所得结论揭示了填充测度与加倍条件之间的关系.     

A note on complex conference matrices

Let ${\mathbb{F}}_q$ be the Galois field of order $q=p^m$, p a prime number and m a positive integer. We prove in this article that for any nontrivial multiplicative character ϰ of ${\mathbb{F}}_q^{⁎}$ and for any $b\in {\mathbb{F}}_q^{⁎}$ we have$\sum _{a\in {\mathbb{F}}_q^{⁎}}\varkappa (a)\overline{\varkappa (a+b)}=-1$. Whenever q is odd and ϰ is the Legendre symbol this formula reduces to the well-known Jacobsthal's formula. A complex conference matrix is a square matrix of order n with zero diagonal and unimodular complex numbers elsewhere such that $C^{⁎}C=(n-1)I$. Paley used finite fields with odd orders $q=p^m$, p prime and the real Legendre symbol to construct real symmetric conference matrices of orders $q+1$ whenever $q\equiv 1(\mathrm{mod}4)$ and real skew-symmetric conference matrices of orders $q+1$ whenever $q\equiv -1(\mathrm{mod}4)$. In this article we extend Paley construction to the complex setting. We extend Jacobsthal's formula to all other nontrivial characters to produce a complex symmetric conference matrix of order $q+1$ whenever $q\ge 4$ is any prime power as well as a complex skew-symmetric conference matrix of order $q+1$ whenever q is any odd prime power. These matrices were constructed very recently in connection with harmonic Grassmannian codes, by use of finite fields and the character table of their additive characters. We propose here a new proof of their construction by use of the above generalized formula similarly as was done by Paley in the real case. We also classify, up to equivalence, the complex conference matrices constructed with some nontrivial characters. In particular, we prove that the complex conference matrix constructed with any nontrivial multiplicative character ϰ and that one constructed with ${\varkappa }^{p^k}$ for any integer $k=1,...m-1$ are permutation equivalent. Moreover, we determine the spectrum of any complex conference matrix obtained from this construction.     

Relative plus constructions

Let $h$ be a connective homology theory. We construct a functorial relative plus construction as a Bousfield localization functor in the category of maps of spaces. It allows us to associate to a pair $(X,H)$, consisting of a connected space $X$ and an $h$-perfect normal subgroup $H$ of the fundamental group ${\pi }_1\left(X\right)$, an $h$-acyclic map $X\to X_H^{+h}$ inducing the quotient by $H$ on the fundamental group. We show that this map is terminal among the $h$-acyclic maps that kill a subgroup of $H$. When $h$ is an ordinary homology theory with coefficients in a commutative ring with unit $R$, this provides a functorial and well-defined counterpart to a construction by cell attachment introduced by Broto, Levi, and Oliver in the spirit of Quillen’s plus construction. We also clarify the necessity to use a strongly $R$-perfect group $H$ in characteristic zero.