Bases,spanning sets,and the axiom of choice

Two theorems are proved: First that the statement “there exists a field F such that for every vector space over F, every generating set contains a basis” implies the axiom of choice. This generalizes theorems of Halpern, Blass, and Keremedis. Secondly, we prove that the assertion that every vector space over ?₂ has a basis implies that every well‐ordered collection of two‐element sets has a choice function. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

OnQ-order andR-order of convergence

We give sufficient conditions for a sequence to have theQ-order and/or theR-order of convergence greater than one. If an additional condition is satisfied, then the sequence has an exactQ-order of convergence. We show that our results are sharp and we compare them with older results.This work was supported in part by the National Science Foundation under Grant No. DMS-85-03365. The author wishes to thank J. E. Dennis and R. A. Tapia for helpful comments, and the referee for pointing out a number of typographical and mathematical errors in the original version of this paper.     

Dynamical Running Mass of Quark in the Dyson-Schwinger Equation Approach

Based on the Dyson-Schwinger equations of QCD in the "rainbow" approximation, the fully dressed quarkpropagator Sf(p) is investigated, and then an algebraic parametrization form of the propagator is obtained as a solutionof the equations. The dressed quark amplitudes Af and Bf built up the fully dressed quark propagator and the dynamicalrunning masses Mf defined by Af and Bf for light quarks u, d and s are calculated, respectively. Using the predictedrunning masses Mf, quark condensates <0|q(0)q(0)|0> = -(0.255 GeV)a for u, d quarks, and <0|s s|0> = 0.8<0|q(0)q(0)]0)for s quark, and experimental pion decay constant fπ = 0.093 GeV, the masses of Goldstone bosons K, π, and η are alsoevaluated. The numerical results show that the masses of quarks are dependent on their momentum p2. The fully dressedquark amplitudes Af and Bf have correct behaviors which can be used for many purposes in our future researches onnonperturbative QCD.     

Restorations of punctured languages and similarity of languages

Punctured languages are languages whose words are partial words in the sense that the letters at some positions are unknown. We investigate to which extent restoration of punctured languages is possible if the number of unknown positions or the proportion of unknown positions per word, respectively, is bounded, and we study their relationships for different boundings. The considered restoration classes coincide with similarity classes according to some kind of similarity for languages. Thus all results we can also formulate in the language of similarity. We show some hierarchies of similarity classes for each class from the Chomsky hierarchy and prove the existence of linear languages which are not δ ‐similar to any regular language for any δ < ½. For δ ≥ ½ this is unknown but it could only be possible in the case of non‐slender linear languages. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Assessment of thermal expansion coefficient of methylsilsesquioxane thin film

For the methylsilsesquioxane film whose optical birefringence is almost zero, it was recently reported that its vertical thermal expansion coefficient (CTE) was approximately one order of magnitude larger than the lateral CTE. Though the birefringence is not an absolute predictor of anisotropic behavior, the discrepancy in both the CTEs was so remarkable that it was essential to investigate whether the anisotropy was intrinsic property or not. If the effect of Poisson's ratio is considered in the calculation of the vertical CTE and when elastic modulus measured by surface acoustic wave spectroscopy is used in the assessment of the lateral CTE, both the CTEs are coincident with each other. Therefore, it can be concluded that the discrepancy in the CTEs can be attributed to a higher in‐plane polymer chain orientation but it can also arise from the misleadingly assumed modulus and Poisson's ratio. © 2006 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 44: 3109–3120, 2006     

On disjoint configurations in infinite graphs

For a graph A and a positive integer n, let nA denote the union of n disjoint copies of A; similarly, the union of ?₀ disjoint copies of A is referred to as ?₀A. It is shown that there exist (connected) graphs A and G such that nA is a minor of G for all n??, but ?₀A is not a minor of G. This supplements previous examples showing that analogous statements are true if, instead of minors, isomorphic embeddings or topological minors are considered. The construction of A and G is based on the fact that there exist (infinite) graphs G₁, G₂,… such that G_i is not a minor of G_j for all i ≠ j. In contrast to previous examples concerning isomorphic embeddings and topological minors, the graphs A and G presented here are not locally finite. The following conjecture is suggested: for each locally finite connected graph A and each graph G, if nA is a minor of G for all n ? ?, then ?₀A is a minor of G, too. If true, this would be a far‐reaching generalization of a classical result of R. Halin on families of disjoint one‐way infinite paths in graphs. © 2002 Wiley Periodicals, Inc. J Graph Theory 39: 222–229, 2002; DOI 10.1002/jgt.10016     

Optimality conditions and duality theory for minimizing sums of the largest eigenvalues of symmetric matrices

The sum of the largest eigenvalues of a symmetric matrix is a nonsmooth convex function of the matrix elements. Max characterizations for this sum are established, giving a concise characterization of the subdifferential in terms of a dual matrix. This leads to a very useful characterization of the generalized gradient of the following convex composite function: the sum of the largest eigenvalues of a smooth symmetric matrix-valued function of a set of real parameters. The dual matrix provides the information required to either verify first-order optimality conditions at a point or to generate a descent direction for the eigenvalue sum from that point, splitting a multiple eigenvalue if necessary. Connections with the classical literature on sums of eigenvalues and eigenvalue perturbation theory are discussed. Sums of the largest eigenvalues in the absolute value sense are also addressed.This paper is dedicated to Phil Wolfe on the occasion of his 65th birthday.The work of this author was supported by the National Science Foundation under grants CCR-8802408 and CCR-9101640.The work of this author was supported in part during a visit to Argonne National Laboratory by the Applied Mathematical Sciences subprogram of the Office of Energy Research of the U.S. Department of Energy under contract W-31-109-Eng-38, and in part during a visit to the Courant Institute by the U.S. Department of Energy under Contract DEFG0288ER25053.     

因子分析在企业竞争力评价中的应用

根据因子分析原理,建立了企业竞争力的规模因子、效益因子和成长因子得分模型。通过现实竞争力和潜在竞争力分析,实现了企业竞争力和可持续发展的评价。     

ON SOME INTERPRETATIONS OF CLASSICAL LOGIC

In distinction from the well-known double-negation embeddings of the classical logic we consider some variants of single-negation embeddings and describe some classes of superintuitionistic first-order predicate logics in which the classical first-order calculus is interpretable in such a way. Also we find the minimal extensions of Heyting's logic in which the classical predicate logic can be embedded by means of these translations.