A generalization of a theorem on biseparating maps

In J. Math. Anal. Appl. 12 (1995) 258–265, Araujo et al. proved that for any linear biseparating map  from C(X) onto C(Y), where X and Y are completely regular, there exist ω in C(Y) and an homeomorphism h from the realcompactification vX of X onto vY, such that

The compact version of this result was proved before by Jarosz in Bull. Canad. Math. Soc. 33 (1990) 139–144. In Contemp. Math., Vol. 253, 2000, pp. 125–144, Henriksen and Smith asked to what extent the result above can be generalized to a larger class of algebras. In the present paper, we give an answer to that question as follows. Let A and B be uniformly closed Φ-algebras. We first prove that every order bounded linear biseparating map from A onto B is automatically a weighted isomorphism, that is, there exist ω in B and a lattice and algebra isomorphism ψ between A and B such that

(a)=ωψ(a) for all aA.

We then assume that every universally σ-complete projection band in A is essentially one-dimensional. Under this extra condition and according to a result from Mem. Amer. Math. Soc. 143 (2000) 679 by Abramovich and Kitover, any linear biseparating map from A onto B is automatically order bounded and, by the above, a weighted isomorphism. It turns out that, indeed, the latter result is a generalization of the aforementioned theorem by Araujo et al. since we also prove that every universally σ-complete projection band in the uniformly closed Φ-algebra C(X) is essentially one-dimensional.     

A generalization of a minimax theorem of Fan via a theorem of the alternative

A minimization problem for a functional on a convex subsetC of a normed linear space is considered. Under certain hypotheses, optimality in a certain subset ofC implies the validity of first-order necessary optimality conditions for the problem inC. The result is applied to a problem in optimal periodic control of neutral functional differential equations.This work was partially supported by a grant from Deutsche Forschungsgemeinschaft and by AFOSR under Grant No. AFOSR-84-0398.     

A matrix version of the Ky Fan generalization of the kantorovich inequality

Ky Fan extended some inequalities for Hermitian positive definite matrices to sums of such matrics. These and similar results are further generalized by showing that the inequalities are maintained with orthogonal matrices substituted for vectors.     

Applications of a theorem by Ky Fan in the theory of graph energy

The energy of a graph G is equal to the sum of the absolute values of the eigenvalues of G, which in turn is equal to the sum of the singular values of the adjacency matrix of G. Let X, Y, and Z be matrices, such that X+Y=Z. The Ky Fan theorem establishes an inequality between the sum of the singular values of Z and the sum of the sum of the singular values of X and Y. This theorem is applied in the theory of graph energy, resulting in several new inequalities, as well as new proofs of some earlier known inequalities.     

A generalization of simplicial elimination orderings

We consider the class of graphs where every induced subgraph possesses a vertex whose neighborhood has no P₄ and no 2K₂. We prove that Berge's Strong Perfect Graph Conjecture holds for such graphs. The class generalizes several well-known families of perfect graphs, such as triangulated graphs and bipartite graphs. Testing membership in this class and computing the maximum clique size for a graph in this class is not hard, but finding an optimal coloring is NP-hard. We give a polynomial-time algorithm for optimally coloring the vertices of such a graph when it is perfect. © 1996 John Wiley & Sons, Inc.     

一般化凸空间上的Ky Fan型重合点定理及其应用

我们应用已有的一般化凸空间上的KKM型定理得到K y Fan型重合点定理,然后作为应用给出截口定理和择一性问题.我们的主要结果对文[1-5]中的相应的结果进行了改进和一般化.     

A Generalized Ky Fan Minimax Inequality on Finite-Dimensional Spaces

Journal of Optimization Theory and Applications - An existence result for a generalized inequality over a possible unbounded domain in a finite-dimensional space is established. The proof technique...     

A homology transfer for a class of simplicial maps

We construct a homology transfer _*H_*(B) H_*(E) for a certain class of proper simplicial maps pE B. Roughly, the important hypothesis is that there is a compact space F, so that all fibres F_x>=p^–1(x), xB, are quotient spaces of F, in a certain locally controlled manner. The composition p_** H_*(B)H_*(E)H_*(B) is multiplication by the Euler characteristic of F.     

Minimax inequalities of Ky Fan

In this paper, we found a new result by relaxing the condition of [1, Theorem 3]. As its direct consequence, we have obtained some new minimax inequalities of Ky Fan and minimax theorems.     

A generalization of a theorem of Pringsheim

We consider noncommutative continued fractions of the form b₀ + a₁(b₁ + a₂(b₂ + a₃(…)⁻¹ c₃)⁻¹ c₂)⁻¹ c₁, (1) where a_n, b_n and c_n are elements of some Banach algebra B and b_n⁻¹ exists. Such expressions play an important role in the numerical investigation of various problems in theoretical physics and in applied mathematics, but up to now their convergence was not studied in the general case. In this paper we prove a theorem which is an extension of a wellknown theorem of Pringsheim and, in particular, guarantees the convergence of (1) under the following hypotheses:

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. As an application, we give a generalization of a theorem of van Vleck. The paper closes with an extensive bibliography.     

A generalization of a theorem of Foxby

In this paper, it is proved that a commutative noetherian local ring admitting a finitely generated module of finite projective and injective dimensions with respect to a semidualizing module is Gorenstein. This result recovers a celebrated theorem of Foxby.      

A generalization of a theorem of Neumaier

In this paper, we show that for given positive integer p, the number of connected biregular graphs with exactly three distinct eigenvalues, and with smallest eigenvalue at least \(-p\), is finite.