Einige Erläuterungen zum Khacian-Verfahren

Zusammenfassung 1979 veröffentlichte L.G. Khacian einen neuartigen Algorithmus zur Lösung der Linearen Optimierungsaufgabe. Da diese Arbeit viel Aufsehen erregte, teilweise falsch interpretiert wurde und an manchen Stellen offenbar nur in verkürzter Fassung vorliegt, wird das Verfahren im folgenden anhand der vollständigen Originalarbeit dargestellt. Gegenüber der Originalarbeit sind einige zusätzliche Beweise aufgenommen worden.

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Summary In 1979 L.G. Khacian has published a new algorithm for the solution of the Linear Programming Problem. This paper has been discussed in many places, sometimes it was misinterpreted seriously. It is the aim of the following paper to give a careful introduction into Khacian's method by using the original complete paper. Some additional proofs have been inserted.

     

Book Reviews

Theory of Operator Algebras I by Maramichi Takesaki. Springer Verlag New York Inc., 1979

An Introduction to Stochastic Processes by D. Kannan, North Holland, New York, 1979, xii + 296 pp., $24.50.     

On the Divergence of Collocation Solutions in Smooth Piecewise Polynomial Spaces for Volterra Integral Equations

In this paper we present a characterization of those smooth piecewise polynomial collocation spaces that lead to divergent collocation solutions for Volterra integral equations of the second kind. The key to these results is an equivalence result between such collocation solutions and collocation solutions in slightly smoother spaces for initial-value problems for ordinary differential equations. For the latter problems Mülthei (1979/1980) established a complete divergence (and convergence) theory. Our analysis can be extended to furnish divergence results for smooth collocation solutions to Volterra integral equations of the first kind. AMS subject classification (2000) 65R20, 65L20, 65L60.Received May 2004. Accepted September 2004. Communicated by Tom Lyche.Hermann Brunner: This research was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC).     

Canonical recursive functions and operations

A series of properties of canonical recursive functions and operations are established, allowing the possibility of extrapolating to these functions and operations the method proposed by R. L. Goodstein for constructing equational calculi.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 88, pp. 218–235, 1979.     

OSCILLATION THEOREMS FOR DELAY AND NEUTRAL DELAY DIFFERENTIAL EQUATIONS

Oscillation criteria for the delay differential equationx′(t)+p(t)x(t-т(t))=0,where p, т are non-negative real-valued continuous functions are investigated inthe case when the numbers k=integral from n=t-т(t) to t(p(s)ds),L=integral from n=t-т(t) to t(p(s)ds)satisfy 0≤k<1/e and 1/e≤L<1. The present result improves almost allresults of the literature concerning it. Furthermore, it is established that allsolutions of the odd-order neutral delay differential equation(x(t)-px(t-т))~(n)+Q(t)x(t-σ)=0,where 0≤p<1,т,σ∈(0,∞)and Q(t)≥0,are oscillatory ifintegral from n=t-σ to t ((t-s)~(n-1)Q(s)ds>(1/e)(1-p)(n-1))!.This result generalizes a theorem of Gopalsamy et. al (Czech. Math. J., 42(1992),313-323)and also extends a very well-known result of Ladas (ApplicableAnalysls, 9(1979),93-98).     

A remark on convergence in the limit theorem for the distribution of random functionals

Estimates are established which characterize the rate of convergence in a global limit theorem for random functionals.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 85, pp. 197–206, 1979.The author is grateful to L. N. Dovbysh for assistance and to I. A. Ibragimov for valuable consultations.     

Book review

Applied Functional Analysis, by Jean-Pierre Aubin, Wiley-Intersciece 1979. Reviewed by Roy B. Leipnik     

Some results on compact Kähler surfaces with non-positive bisectional curvature

In 1961 L. Frankel conjectured that any compact Kähler manifold of positive bisectional curvature is biholomorphic to a projective space. This was solved by S. Mori in 1979 and Siu and Yau in 1980. N. Mok in 1988 gave a full solution to the generalized Frankel’s conjecture and proposed a dual problem. The results in this paper are attempts to give modified solutions to this problem in complex dimension two.     

Reconstruction of the error distribution in the linear regression scheme

Under a certain condition, the characterization of the additive type of an error distribution by the distribution of a maximal invariant is proved in the linear regression scheme. With an added condition on the design matrix, the possibility of reconstructing the error distribution itself is proved.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 87, pp. 36–40, 1979.The author is grateful to L. B. Klebanov for his interest in this article.     

Short Book Reviews

Problems in Linear Algebra, by I. V. Proskuryakov. Mir Publishers, Moscow, 1978= 453 pp.

Inequalities: Theory of Majorization and its Applications. by A. W. Marshall and I. Olkin. Academic Press, 1979, 569 pp.

Problems in Linear Algebra, by I. V. Proskuryakov. Mir Publishers, Moscow, 1978. 453 pp. Inequalities: Theory of Majorization and its Applications. by A. W. Marshall and I. Olkin. Academic Press, 1979, 569 pp. Introduction to Numerical Analysis. by F. Stummel and K. Hainer. Scottish Academic Press (in United States, Columbia University Press), 1980. 282 pp., soft cover.

Introduction to Numerical Analysis. by F. Stummel and K. Hainer. Scottish Academic Press (in the United States, Columbia University Press), 1980. 282 pp., soft cover.     

Book Reviews

Matrix Methods For Engineers and Scientists,by S. Barnett.McGraw-Hill Book Company, London, 1979. 185 pp

Permanents, by Henry Minc. Encylopedia of Mathematics and its Applications, Vol.6. Addison-Wesley, Reading, 1978.     

An estimate of the Hecke L-functions on the critical line

One obtains, under certain restrictions, an estimate for the Hecke L -functions on the critical line, similar to the estimate for the Riemann zeta-function.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 91, pp. 40–51, 1979.In conclusion, I would like to express my gratitude to A. I. Vinogradov for his constant interest and help in the writing of this paper.     

Multivariate semi-Weibull distributions

Some multivariate semi-Weibull (denoted by MSW) distributions including the Marshall–Olkin multivariate semi-Weibull (denoted by MO-MSW) one are introduced. They are more general than the multivariate Weibull distributions proposed by Lee [L. Lee, Multivariate distributions having Weibull properties, J. Multivariate Anal. 9 (1979) 267–277]. The Marshall–Olkin multivariate semi-Pareto (denoted by MO-MSP) distribution is also defined. Two characterization theorems for the homogeneous MSW are proved. The multivariate minima domain of partial attraction of MSW is studied, and the interrelationships between MO-MSP and MSW are examined. The MSW distribution possesses the minima-semi-stability and minima-infinite divisibility properties.     

Scientific contributions of Leo Khachiyan (a short overview)

On April 29, 2005, Leo Khachiyan passed away with tragic suddenness in the middle of his scientific career. He became famous for his polynomial time algorithm for linear programming (LP)[L. Khachiyan, A polynomial algorithm in linear programming, Soviet Math. Doklady 20 (1) (1979) 191-194; L. Khachiyan, Polynomial algorithms for linear programming, USSR Comp. Math. and Math. Phys. 20 (2) (1980) 51-68]. However, he contributed fundamentally to several other areas, as well. In this introductory paper, we would like to recall briefly his main contributions, and provide a complete (up to our best knowledge) list of his publications.     

A note on Bailey's Lemma

Combining the qa-binomial theorem in the version of J. Cigler (Monatsh. Math. 88, 87–105 (1979)) with the ε-technique of L. J. Rogers (G. E. Andrews, Math. Chronicle, 11, 1–15, 1982) a short operator proof of a significant special case of Bailey's Lemma is given.     

A controlled-topology proof of the product structure theorem

The controlled end and h-cobordism theorems (Ends of maps I, 1979) are used to give quick proofs of the Top/PL and PL/DIFF product structure theorems.     

Representation of integers by positive ternary quadratic forms (a new modification of the discrete ergodic method)

One gives a detailed presentation of a new modification of the discrete ergodic method outlined in A. V. Malyshev's note (this Zapiski,50, 179–186 (1975)). One gives new proofs for the asymptotic formulas obtained in Chap VI of A. V. Malyshev's monography (Tr. Mat. Inst. Akad. Nauk SSSR,65 (1962). One obtains estimates for the remainder terms of this formulas under the assumption of some hypotheses about the zeros of Dirichlet's L -functions.Translated from Zapiski Nauchnykh Seminarog Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 82, pp. 33–87, 1979.     

A note on bi-criteria transportation problems

The present note reveals a drawback in an algorithm ofAneja/Nair [1979] for the determination of efficient solutions in a bi-criteria transportation problem with an additional time objective function and develops a modified procedure that overcomes this weakness.

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Zusammenfassung In der vorliegenden Arbeit wird ein Fehlschluß im Algorithmus vonAneja/Nair [1979] zum Auffinden aller effizienten Lösungen eines Transportproblems mit zwei Summen- und einer Engpaßzielfunktion aufgezeigt. Ferner wird ein Verfahren angegeben, in dem diese Unstimmigkeiten nicht mehr auftreten.