A NOTE ON THAT KUBOTA＇S AD－INTEGRAL IS MORE GENERAL THAN BURKILLS AP－INTEGRAL

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Asymptotic expansions forL- andR-statistics under alternatives

In this paper we establish asymptotic expansions (a.e.) under alternatives for the distribution functions of sums of independent identically distributed random variables (i.i.d.r.v.'s.), linear combinations of order statistics, and one-sample rank statistics (L- and R-statistics). The general Lemma from [V. E. Bening,Bull. Moscow State Univ., Ser. 15, 2 36–44 (1994)] is applied to these statistics. Section 1 contains the statement of the theorem, in Sec. 2 the theorems is proved; its proof involves four auxiliary lemmas, also contained in Sec. 2. Finally Sec. 3 contains the proofs of these lemmas. Proceedings of the Seminar on Stability Problems for Stochastic Models, Moscow, 1993.     

A constructive proof of Ky Fan's generalization of Tucker's lemma

We present a proof of Ky Fan's combinatorial lemma on labellings of triangulated spheres that differs from earlier proofs in that it is constructive. We slightly generalize the hypotheses of Fan's lemma to allow for triangulations of Sⁿ that contain a flag of hemispheres. As a consequence, we can obtain a constructive proof of Tucker's lemma that holds for a more general class of triangulations than the usual version.     

A constructive Schwarz reflection principle

We prove a constructive version of the Schwarz reflection principle. Our proof techniques are in line with Bishop's development of constructive analysis. The principle we prove enables us to reflect analytic functions in the real line, given that the imaginary part of the function converges to zero near the real line in a uniform fashion. This form of convergence to zero is classically equivalent to pointwise convergence, but may be a stronger condition from the constructivist point of view.

     

Center manifolds of infinite dimensions II: Proofs of the main results

In this paper, the part II of a bipartite work, we present the proof of Theorem 1 which was stated in part I, i.e. the precursor of the present paper. This proof establishes the existence of a smooth center manifold, i.e. of a smooth decoupling function for the abstract evolution system considered in part I. In an appendix, the proofs of some auxiliary lemmas are presented, some of which were stated in part I, while the others are related to the present proof of Theorem 1.     

内点同伦方法求解更一般非凸集上的不动点问题(英文)

In this paper,we are mainly devoted to solving fixed point problems in more general nonconvex sets via an interior point homotopy method.Under suitable conditions,a constructive proof is given to prove the existence of fixed points,which can lead to an implementable globally convergent algorithm.     

完全覆盖和加标分割在分析中的应用

通过对数学分析中众多定理的证法的分析,说明利用文中的两个引理可使一些定理的证法既直接又易于接受,使黎曼可积的一个充要条件的证明简捷,文章最后给出了罗尔中值定理的一种简单的新证法.     

Glivenko sequent classes in the light of structural proof theory

In 1968, Orevkov presented proofs of conservativity of classical over intuitionistic and minimal predicate logic with equality for seven classes of sequents, what are known as Glivenko classes. The proofs of these results, important in the literature on the constructive content of classical theories, have remained somehow cryptic. In this paper, direct proofs for more general extensions are given for each class by exploiting the structural properties of G3 sequent calculi; for five of the seven classes the results are strengthened to height-preserving statements, and it is further shown that the constructive and minimal proofs are identical in structure to the classical proof from which they are obtained.     

On Roussel–Rubio-type lemmas and their consequences

Roussel and Rubio proved a lemma which is essential in the proof of the Strong Perfect Graph Theorem. We give a new short proof of the main case of this lemma. In this note, we also give a short proof of Hayward’s decomposition theorem for weakly chordal graphs, relying on a Roussel–Rubio-type lemma. We recall how Roussel–Rubio-type lemmas yield very short proofs of the existence of even pairs in weakly chordal graphs and Meyniel graphs.     

On reid’s 3-simplicial matroid theorem

In this paper we prove the following result of Ralph Reid (which was never published nor completely proved). Theorem. Let M be a matroid coordinatizable (representable) over a prime field F. Then there is a 3-simplicial matroid M′ over F which is a series extension of M. The proof we give is different from the original proof of Reid which uses techniques of algebraic topology. Our proof is constructive and uses elementary matrix operations.     

Balancedness of Sequencing Games with Multiple Parallel Machines

We provide simple constructive proofs of balancedness of classes of m-PS (m-Parallel Sequencing) games, which arise from sequencing situations with m parallel machines. This includes the setting that is studied by Calleja et al. (2001) and Calleja et al. (2002), who provided a complex constructive proof and a simple non-constructive proof of balancedness of a restricted class of 2-PS games, respectively. Furthermore, we provide a counterexample to illustrate that our balancedness results cannot be extended to a general setting.     

Two Proofs on Max-Min-Max Principle of Jerk Control in Time-Optimal Rectilinear Motion

This paper provides two proofs to the well-known principle of max-min-max jerk control in time-optimal jerk-controlled rectilinear motion, whose proof seems to be missing in the literature. One proof is constructive in that it computes and finds out the solution; the other is a disproof in that it shows all other control schemes lead to longer motion time. The ideas employed in the proofs may be used to solve the same kind of problems in a more general setting.     

Eliminating disjunctions by disjunction elimination

Completeness and other forms of Zorn’s Lemma are sometimes invoked for semantic proofs of conservation in relatively elementary mathematical contexts in which the corresponding syntactical conservation would suffice. We now show how a fairly general syntactical conservation theorem that covers plenty of the semantic approaches follows from an utmost versatile criterion for conservation due to Scott.To this end we work with multi-conclusion entailment relations as extending single-conclusion entailment relations. In a nutshell, the additional axioms with disjunctions in positive position can be eliminated by reducing them to the corresponding disjunction elimination rules, which turn out provable in a wealth of mathematical instances. In deduction terms this means to fold up branchings of proof trees by way of properties of the relevant mathematical structures.Applications include syntactical counterparts of the theorems or lemmas known under the names of Artin–Schreier, Krull–Lindenbaum and Szpilrajn, as well as of the spatiality of coherent locales. Related work has been done before on individual instances, e.g. in locale theory, dynamical algebra, formal topology and proof analysis.     

A new variant of the Schwarz–Pick–Ahlfors Lemma

We prove a “general shrinking lemma” that resembles the Schwarz–Pick–Ahlfors Lemma and its many generalizations, but differs in applying to maps of a finite disk into a disk, rather than requiring the domain of the map to be complete. The conclusion is that distances to the origin are all shrunk, and by a limiting procedure we can recover the original Ahlfors Lemma, that all distances are shrunk. The method of proof is also different in that it relates the shrinking of the Schwarz–Pick–Ahlfors-type lemmas to the comparison theorems of Riemannian geometry. Received: 26 May 1998 / Revised version: 4 May 1999     

A note on subset selection for matrices

In an earlier paper the authors examined the problem of selecting rows of a matrix so that the resulting matrix is as “non-singular” as possible. However, the proof of the key result in that paper is not constructive. In this note we give a constructive proof for that result. In addition, we examine a case where as non-singular as possible means maximizing a determinant and provide a new bound and a constructive proof for this case also.     

Erratum to: Special primitive pairs in finite groups

During the review of the article “Special primitive pairs in finite groups”, an error was found in the proof of one of the preliminary lemmas. This note corrects this error and the author wishes to thank George Glauberman for pointing it out.     

A correct proof of the McMorris-Powers’ theorem on the consensus of phylogenies

McMorris and Powers proved an Arrow-type theorem on phylogenies given as collections of quartets. There is an error in one of the main lemmas used to prove this theorem. However, this lemma (and thereby the theorem) is still true, and we provide a corrected proof.     

构造弹性力学位移函数的机械化算法*

本文证明线性算子方程组Au=f的一般解为u=Cv+e,其中v满足方程组Dv=g,D是对角矩阵。以Hilbert零点定理的构造性证明为基础,给出了C,D,e的机械化求示。用此方法可以给出各种弹性力学位移函数的机械化算法。     

条件数学期望与随机变量独立性的一个充要条件

随机试验的独立性、随机事件的独立性、随机变量的独立性均是概率统计中的重要概念,不少学者都在这些方面有所讨论.本文作者就二维离散形随机向量(ξ,η)中两个分量ξ与η的相互独立性展开讨论.先是证明了三个引理,其中引理1在一般概率论教科书中均有介绍,但为使读者方便,作者也作了证明.利用三个引理,作者找到随机变量独立性的一个充要条件.     

Rigid Rings and Makar-Limanov Techniques

A ring is rigid if it admits no nonzero locally nilpotent derivation. Although a “generic” ring should be rigid, it is not trivial to show that a ring is rigid. We provide several examples of rigid rings and we outline two general strategies to help determine if a ring is rigid, which we call “parametrization techniques.” and “filtration techniques.” We provide many tools and lemmas which may be useful in other situations. Also, we point out some pitfalls to beware when using these techniques. Finally, we give some reasonably simple rings for which the question of rigidity remains unsettled.