Ein Störungssatz für abgeschlossene,lineare Operatoren

Zusammenfassung In vorliegender Note wird ein Satz von Kato [7] über die Störung eines abgeschlossenen, normal auflösbaren OperatorsT mit endlichem Null-defekt (T) durch einen streng singulären Operator verallgemeinert. Zu diesem Zweck wird für jedes 0 mit Hilfe des Kuratowskischen Nichtkompaktheitsmaßes eine KlasseC von beschränkten, linearen Operatoren eingeführt, welche sowohl die streng singulären Operatoren als auch die OperatorenS mit S enthält.Das erzielte Resultat steht in engem Zusammenhang mit den Untersuchungen von Gol'denteinn, Gohberg und Markus [5] und von Gol'denteienn und Markus [6].     

A network simplex method

Simple combinatorial modifications are given which ensure finiteness in the primal simplex method for the transshipment problem and the upper-bounded primal simplex method for the minimum cost flow problem. The modifications involve keeping strongly feasible bases. An efficient algorithm is given for converting any feasible basis into a strongly feasible basis. Strong feasibility is preserved by a rule for choosing the leaving basic variable at each simplex iteration. The method presented is closely related to a new perturbation technique and to previously known degeneracy modifications for shortest path problems and maximum flow problems.The author holds a National Research Council of Canada Post-Doctorate Fellowship.     

A note on treating a second order cone program as a special case of a semidefinite program

It is well known that a vector is in a second order cone if and only if its arrow matrix is positive semidefinite. But much less well-known is about the relation between a second order cone program (SOCP) and its corresponding semidefinite program (SDP). The correspondence between the dual problem of SOCP and SDP is quite direct and the correspondence between the primal problems is much more complicated. Given a SDP primal optimal solution which is not necessarily arrow-shaped, we can construct a SOCP primal optimal solution. The mapping from the primal optimal solution of SDP to the primal optimal solution of SOCP can be shown to be unique. Conversely, given a SOCP primal optimal solution, we can construct a SDP primal optimal solution which is not an arrow matrix. Indeed, in general no primal optimal solutions of the SOCP-related SDP can be an arrow matrix.Mathematics Subject Classification (2000): 20E28, 20G40, 20C20     

Einschließungsaussagen für gewöhnliche Differentialoperatoren

Summary For differential operatorsM of second order (as defined in (1.1)) we describe a method to prove Range-Domain implications—Mu–u and an algorithm to construct these functions , , , . This method has been especially developed for application to non-inverse-positive differential operators. For example, for non-negativea ₂ and for given functions = we require =C ₀[0, 1] C ₂([0, 1]–T) whereT is some finite set), (M) (t)(t), (t[0, 1]–T) and certain additional conditions for eachtT. Such Range-Domain implications can be used to obtain a numerical error estimation for the solution of a boundary value problemMu=r; further, we use them to guarantee the existence of a solution of nonlinear boundary value problems between the bounds- and .     

On a Particular Class of Minihypers and Its Applications. I. The Result for General q

This article is the first in a series of three articles that discuss a particular class of minihypers and its applications. Proving that for small and < N, a {v _{+ 1}, v ; N, q}-minihyper consists of a sum of -spaces, we show that the excess points of an s-cover with excess of PG(N, q), (s + 1)|(N + 1), form a sum of s-spaces, and that no maximal partial s-spreads with deficiency of PG(N, q), (s + 1)|(N + 1), exist. The case q square will be studied in greater detail in [7] and further applications of these classification results on this class of minihypers will be published in [8].     

Critical sets in parametric optimization

We deal with one-parameter families of optimization problems in finite dimensions. The constraints are both of equality and inequality type. The concept of a generalized critical point (g.c. point) is introduced. In particular, every local minimum, Kuhn-Tucker point, and point of Fritz John type is a g.c. point. Under fairly weak (even generic) conditions we study the set consisting of all g.c. points. Due to the parameter, the set is pieced together from one-dimensional manifolds. The points of can be divided into five (characteristic) types. The subset of nondegenerate critical points (first type) is open and dense in (nondegenerate means: strict complementarity, nondegeneracy of the corresponding quadratic form and linear independence of the gradients of binding constraints). A nondegenerate critical point is completely characterized by means of four indices. The change of these indices along is presented. Finally, the Kuhn-Tucker subset of is studied in more detail, in particular in connection with the (failure of the) Mangasarian-Fromowitz constraint qualification.     

One-parametric bottleneck transportation problems

We discuss the bottleneck transportation problem with one nonlinear parameter in the bottleneck objective function. A finite sequence of feasible basic solutions which are optimal in subsequent closed parameter-intervals is generated using a primal method for the nonparametric subproblems. The best among three primal codes for solving these subproblems is selected on extensive computational comparisons. We discuss computational experience with the sequential method for the case of linear and quadratic dependence on one parameter. Observed computational behaviour is O((n ·m)), with 2.     

Increase of Dimension by Homogenization

The paper deals with the homogenization of a Neumann's problem in a thin periodic weakly connected domain of R ³. The domain _n is composed of a large number n of disjoint periodic connected components linked by a periodic lattice _n of very thin bridges. According to the distribution and to the size of the linking bridges, the limit problem as n tends to infinity is either a 4d Neumann's problem or a 4d nonlocal problem. The additional term corresponding to the increase of dimension is due to the connection effect of the bridges.     

Über eine Verallgemeinerung der Nicoletti-Aufgabe für Funktional-Differentialgleichungen mit voreilendem Argument

We consider a functional differential equation (1) (t)=F(t,) fort[0,+) together with a generalized Nicoletti condition (2)H()=. The functionF: [0,+)×C ⁰[0,+)B is given (whereB denotes the Banach space) and the value ofF (t, ) may depend on the values of (t) fort[0,+);H: C ⁰[0,+)B is a given linear operator and B. Under suitable assumptions we show that when the solution :[0,+)B satisfies a certain growth condition, then there exists exactly one solution of the problem (1), (2).     

The Laplacian with Robin Boundary Conditions on Arbitrary Domains

Using a capacity approach, we prove in this article that it is always possible to define a realization of the Laplacian on L ²() with generalized Robin boundary conditions where is an arbitrary open subset of R ⁿ and is a Borel measure on the boundary of . This operator generates a sub-Markovian C ₀-semigroup on L ²(). If d=d where is a strictly positive bounded Borel measurable function defined on the boundary and the (n–1)-dimensional Hausdorff measure on , we show that the semigroup generated by the Laplacian with Robin boundary conditions has always Gaussian estimates with modified exponents. We also obtain that the spectrum of the Laplacian with Robin boundary conditions in L ^p () is independent of p[1,). Our approach constitutes an alternative way to Daners who considers the (n–1)-dimensional Hausdorff measure on the boundary. In particular, it allows us to construct a conterexample disproving Daners' closability conjecture.     

Imbedding problem with non-Abelian kernel of order p4

This paper considers the imbedding problem for numerical fields and p-groups with nonabelian kernel of order p⁴, two generators and , and defining relations =1, ^p=1, [,,]=1, and [,,]= 1. For p=2 and almost always for odd p, the Hasse principle is valid, and the problem is solvable if and only if all related local problems are solvable. Counterexamples in which the Hasse principle is not valid are constructed for some exceptional cases.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 175, pp. 46–62, 1989.     

Connected and alternating vectors: Polyhedra and algorithms

Given a graphG = (V, E), leta ^S, S L, be the edge set incidence vectors of its nontrivial connected subgraphs.The extreme points of = {x R ^E: a^sx |V(S)| - |S|, S L} are shown to be integer 0/± 1 and characterized. They are the alternating vectorsb ^k, k K, ofG. WhenG is a tree, the extreme points ofB 0,b ^kx 1,k K} are shown to be the connected vectors ofG together with the origin. For the four LP's associated with andA, good algorithms are given and total dual integrality of andA proven.On leave from Swiss Federal Institute of Technology, Zurich.     

Stability of the extreme point set of a polyhedron

This paper is focused on the stability properties of the extreme point set of a polyhedron. We consider a polyhedral setX(A,b) which is defined by a linear system of equality and inequality constraintsAxb, where the matrixA and the right-hand sideb are subject to perturbations. The extreme point setE(X(A,b)) of the polyhedronX(A,b) defines a multivalued map :(A,b)E(X(A,b)). In the paper, characterization of continuity and Lipschitz continuity of the map is obtained. Boundedness of the setX(A,b) is not assumed It is shown that lower Lipschitz continuity is equivalent to the lower semicontinuity of the map and to the Robinson and Mangasarian-Fromovitz constraint qualifications. Upper Lipschitz continuity is proved to be equivalent to the upper semicontinuity of the map . It appears that the upper semicontinuity of the map implies the lower semicontinuity of this map. Some examples of using the conditions obtained are provided.The author wishes to thank Dr. N. M. Novikova, Dr. S. K. Zavriev, and anonymous referees for their helpful comments and advice. The research described in this publication was made possible in part by Grant NJCU100 from the International Science Foundation and Russian Government, and by the Euler Grant, Deutsche Mathematiker Vereinigung.     

An Elementary Approach to the Twin Primes Problem

Hardy-Littlewood [4] conjectured an asymptotic formula for the number of prime pairs (twin primes) (p, p+2d) with p+2dy, where d N is fixed and y . Up to now, no one has been able to prove this conjecture, but employing Hardy-Littlewoods circle method, Lavrik [5] showed that in a certain sense this formula holds true for almost-all dy/2.In the present paper, we use a completely different method to prove Lavriks almost-all result. Our method is based on an elementary approach developed by Pan Chengdong [7] to the twin primes problem. By a slight modification of our method, we get a corresponding almost-all result for the binary Goldbach problem. From this, according to [3], we derive Vinogradovs [8] well-known Three-Primes-Theorem.     

On the dispersion spectrum

A measure for the denseness of sequences (an) mod 1, irrational, is the dispersion constantD() introduced byH. Niederreiter. In this paper the smallest accumulation point ₁ of the set of theD() is determined and all those are explicitely given for whichD () < ₁ holds.     

Eigenvalues of convex processes and convergence properties of differential inclusions

In this paper, we study (real) eigenvalues and eigenvectors of convex processes, and provide conditions for the existence of eigenvectors in a given convex coneK ⁿ . It is established that the maximal eigenvalue ofG(·) inK is expressed by (whereK ⁰ is the polar cone ofK) provided that the minimum is attained in intK ⁰. This result is applied to study the asymptotic behaviour of certain differential inclusions{G(x(t)). We extend some known results for the von Neumann-Gale model to our more general framework. We prove that ifx ₀ is the unique eigenvector corresponding to the maximal eigenvalue ₀ ofG(·) inK, then the nonexistence of solutions of a certain special trigonometric form is necessary and sufficient for every viable solutionx(·) to satisfy^- ₀ ^t x(t)cx ₀ ast for somec0. Our method is to study the family of convex conesW =cl{v–x :xK,vG(x) where is any real number. We characterize the maximal eigenvalue ₀ as the minimal for whichW can be separated fromK.The research was supported in part by a grant from the ministry of science and the Maagara special project for the absorption of new immigrants in the Department of Mathematics at Technion.     

Adjungierte Funktoren in der Kategorie der p-Bornologischen Räume

In the category p b of p-convex vector spaces and linear maps preserving bounded sets a p-bornological topology will be introduced on the tensor product of two spaces, likewise on the spaces of morphisms Hom(E,F). Thus one gets a pair of adjoint functors from p to p , p being the category of p-bornological spaces and continuous linear maps, and the topologies being introduced will be characterized by extreme properties with respect to the adjoint transformations.

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Dieser Arbeit liegt ein Teil der Dissertation des Autors, Kiel 1967, zugrunde.     

Algorithm for a sum of reciprocals

An algorithm is described for the approximate calculation of a collection of sums of the form _k= _j–1 ⁿ c_j/(_j+_k), 1kn, where 0<_j. The working time of the algorithm is 0(n(t+ log n)(t+log n)) if _k calculated to within 2^–t; here the function (l) denotes the time of multiplication of twoZ-bit numbers.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 137, pp. 3–6, 1984.In conclusion, the author expresses thanks to A. O. Slisenko and Yu. A. Kuznetsov, who drew the attention of the author to the problem considered.     

Complete problems withL-samplable distributions

The average case complexity classes P, L-samplable and NL, L-samplable are defined. We show that Deterministic Bounded Halting is complete for P, L-samplable and that Graph Reachability is complete for NL-samplable, both problems with a universal logspace samplable distribution.     

A class of non-Moufang Bol loops isomorphic to all their loop isotopes

For any two primes, , such that< and divides–1, it is shown that there exists a non-Moufang Bol loop of order ² which is isomorphic to each of its loop isotopes.