On the continuity of midpoint hull convex set-valued functions

Summary The concept of hull convexity (midpoint hull convexity) for set-valued functions in vector spaces is examined. This concept, introduced by A. V. Fiacco and J. Kyparisis (Journal of Optimization Theory and Applications,43 (1986), 95–126), is weaker than one of convexity (midpoint convexity).The main result is a sufficient condition for a midpoint hull convex set-valued function to be continuous. This theorem improves a result obtained by K. Nikodem (Bulletin of the Polish Academy of Sciences, Mathematics,34 (1986), 393–399).     

Partly convex programming and zermelo's navigation problems

Mathematical programs, that become convex programs after freezing some variables, are termed partly convex. For such programs we give saddle-point conditions that are both necessary and sufficient that a feasible point be globally optimal. The conditions require cooperation of the feasible point tested for optimality, an assumption implied by lower semicontinuity of the feasible set mapping. The characterizations are simplified if certain point-to-set mappings satisfy a sandwich condition.The tools of parametric optimization and basic point-to-set topology are used in formulating both optimality conditions and numerical methods. In particular, we solve a large class of Zermelo's navigation problems and establish global optimality of the numerical solutions.Research partly supported by NSERC of Canada.     

Measures of inset information on the open domain — I: Inset entropies and information functions of all degrees

This is one in a series of papers studying measures of information in the so-called mixed theory of information (i.e. considering the events as well as their probabilities) on the open domain (i.e. without empty sets and zero probabilities). In this paper we find all-recursive, 3-semisymmetric inset entropies on the open domain. We do this by solving the fundamental equation of inset information of degree () on the open domain.Dedicated to Professor János Aczél on his 60th birthday.     

Axioms forS-pregroups

AnS-pregroup is a subset of a group such that no nontrivial reduced word (a string of length 2 of elements of the set such that the product of no adjacent pair is in the set) represents the identity when placed in the group. To complete the definition we also require anS-pregroup to contain the identity, be closed under inverses and generate the parent group. In [1] John Stallings, coining the term S-pregroups, asked for an internal axiomatic characterization ofS-pregroups. In this paper we describe such a system by modifying the axioms for a pregroup.Presented by J. Mycielski.     

The octahedron equation implies the cube equation: An elementary proof

It has been proved by L. Sweet that the octahedron functional equation implies the cube functional equation in all dimensionsn1. In this note we give an elementary proof of this theorem.     

Applications of the zero–one law to problems connected with Cauchy's functional equation

Summary Marek Kuczma's book, entitled An Introduction To The Theory Of Functional Equations And Inequalities, mentions a certain setV ₀ in several places and presents references as to where this set is discussed in the literature. The main result of this paper is a proof of the fact that the setA _M (V ₀)={xV ₀ f(x)>M} is saturated non-measurable for each additive discontinuous functionf and each real numberM. Other results aboutV ₀ are also presented. Connections between measure and category are stressed. The main tool in our proofs is a certain so-called zero–one law and its topological analogue. In addition it is shown that the zero–one law is equivalent to Smital's lemma.     

Hierarchical decomposition of symmetric discrete systems by matroid and group theories

An algebraic method is proposed for the hierarchical decomposition of large-scale group-symmetric discrete systems into partially ordered subsystems. It aims at extracting substructures and hierarchy for such systems as electrical networks and truss structures.The mathematical problem considered is: given a parametrized family of group invariant structured matricesA, we are to find two constant (=parameter-independent) nonsingular matricesS _r andS _c such thatS _r ^-1 AS _c takes a (common) block-triangular form.The proposed method combines two different decomposition principles developed independently in matroid theory and in group representation theory. The one is the decomposition principle for submodular functions, which has led to the Dulmage—Mendelsohn (DM-) decomposition and further to the combinatorial canonical form (CCF) of layered mixed (LM-) matrices. The other is the full reducibility of group representations, which yields the block-diagonal decomposition of group invariant matrices. The optimality of the proposed method is also discussed.     

Note on the Curvature and Index of an Almost Unitary Contraction Operator

In the recent preprint [1] S. Parrott proves the equality between the Arvesons curvature and the Fredholm index of a pure contraction with finite defect numbers. In the present note one derives a similar formula in the non-pure case.     

Quasigroups satisfying balanced but not Belousov equations are group isotopes

Summary V. D. Belousov (1925–88) made numerous contributions to the study of quasigroups. In particular, his lengthy 1966 paper Balanced identities in quasigroups [4] contains what has been described as a very significant and remarkable theorem [11, pp. 68–69]. Remarkable though it was, this theorem provided only a partial answer to the question as to which balanced equations on quasigroups gave rise to group isotopes. Although not specifically addressed in the paper [12], a characterization of the balanced equations in question may be derived from a generalization of Belousov's Theorem due to E. Falconer. The first author explicitly solved the problem in 1979; however his characterization was of a technical nature and depended on machinery developed over three papers [13].In 1985 Belousov found a characterization which is not only elegant but also lends itself to a simple proof [5]. The purpose of this paper is to provide sufficient background for the non specialist to understand and enjoy what we too would describe as a remarkable theorem.     

Index transforms for multidimensional DFT's and convolutions

Summary Index transforms ofm-dimensional arrays inton-dimensional arrays play a significant role in many fast algorithms of multivariate discrete Fourier transforms (DFT's) and cyclic convolutions. The computation ofm-dimensional long DFT's or convolutions can be transfered to the parallel computation ofn-dimensional short DFT's or convolutions (n>m). In this paper, the nature of index transforms is explored using group-theoretical ideas. We solve the open problems concerning index transforms posed recently by Hekrdla [5, 6].     

A general module theoretic framework for vector M-Padé and matrix rational interpolation

A general module theoretic framework is used to solve several classical interpolation problems and generalizations thereof in a unified way. These problems are divided into two main families. The first family contains the classical linearized Padé, Padé-Hermite and M-Padé problems and the generalization to the vector M-Padé problem. The second family consists of the Padé problem, the scalar, vector and matrix rational interpolation problems. The solution method is straightforward, recursive and efficient. It can follow any path in the solution table even if this solution table is nonnormal (nonperfect). Reordering of the interpolation data is not required.     

Maximum monoreflections

It is shown that, in a category with a specified class of monics and under some mild hypothesis,there is a monoreflection maximum among those whose reflection maps lie in . Thus, for example, any variety, and most SP-classes in a variety, have both amaximum monoreflection and amaximum essential reflection (which might be the same, but frequently aren't, and which might be the identity functor, but frequently aren't). And, for example, under some mild hypotheses, beneath each completion lies a maximum monoreflection, so that, for example, any category of rings has amaximum functorial ring of quotients.     

Packing directed circuits fractionally

LetG be a digraph, and letk1, such that no fractional packing of directed circuits ofG has value >k, when every vertex is given capacity 1. We prove there is a set ofO (k logk logk) vertices meeting all directed circuits ofG.     

Staircase kernels for orthogonald-polytopes

LetS be a finite union of boxes inR ^d . Forx inS, defineA _x ={yx is clearly visible fromy via staircase paths inS}, and let KerS denote the staircase kernel ofS. Then KerS={A _x x is a point of local nonconvexity ofS}. A similar result holds with clearly visible replaced by visible and points of local nonconvexity ofS replaced by boundary points ofS.Supported in part by NSF grant DMS-9207019.     

Weyl's theorem for operator matrices

Weyl's theorem holds for an operator when the complement in the spectrum of the Weyl spectrum coincides with the isolated points of the spectrum which are eigenvalues of finite multiplicity. By comparison Browder's theorem holds for an operator when the complement in the spectrum of the Weyl spectrum coincides with Riesz points. Weyl's theorem and Browder's theorem are liable to fail for 2×2 operator matrices. In this paper we explore how Weyl's theorem and Browder's theorem survive for 2×2 operator matrices on the Hilbert space.Supported in part by BSRI-97-1420 and KOSEF 94-0701-02-01-3.     

Chapter 1 Introduction: Extensions and new developments in DEA

The extensions, new developments and new interpretations for DEA covered in this paper include: (1) new measures of efficiency, (2) new models and (3) new ways of implementing established models with new results and interpretations presented that include treatments of congestion, returns-to-scale and mix and technical inefficiencies and measures of efficiency that can be used to reflect all pertinent properties. Previously used models, such as those used to identify allocative inefficiencies, are extended by means of assurance region approaches which are less demanding in their information requirements and underlying assumptions. New opportunities for research are identified in each section of this chapter. Sources of further developments and possible sources for further help are also suggested with references supplied to other papers that appear in this volume and which are summarily described in this introductory chapter.     

Construction of Pseudorandom Binary Sequences Using Additive Characters

In earlier papers the authors studied finite pseudorandom binary sequences, and they constructed sequences with strong pseudorandom properties. In these earlier constructions multiplicative characters were used. In this paper a new construction is presented which utilizes properties of additive characters. These new sequences can be computed fast, they are well-distributed relative to arithmetic progressions and their correlations of small order are small, but the price paid for the fast computation is that the correlations of large order can be large.     

On the convergence of the multivariate “homogeneous”qd-algorithm

The convergence of columns in the univariateqd-algorithm to reciprocals of polar singularities of meromorphic functions has often proved to be very useful. A multivariateqd-algorithm was discovered in 1982 for the construction of the so-called homogeneous Padé approximants.In the first section we repeat the univariate convergence results. In the second section we summarize the homogeneous multivariateqd-algorithm. In the third section a multivariate convergence result is proved by combining results from the previous sections. This convergence result is compared with another theorem for the general order multivariateqdg-algorithm. The main difference lies in the fact that the homogeneous form detects the polar singularities pointwise while the general form detects them curvewise.     

Stability of axial motions of nonlinearly elastic loops

We establish the stability of axial motions (steady motions along the lengthwise direction) of nonlinearly elastic loops of string. A key observation here is that a linear combination of the total energy and the total circulation of the string, both of which are conserved quantities, yields an appropriate Liapunov function. From our previous work [5], we know that there are uncountably many shapes corresponding to a given axial speed. Accordingly, we establish orbitai stability (modulo this collection of relative equilibria). For a well-defined class of soft materials, there is an upper bound on the axial speed sufficient for stability; stiff materials are shown to be orbitally stable at any axial speed.     

Bergman-Vekua theory in differential form

We develop Bergman-Vekua integral operator theory on the basis of a new kind of series suggested by polynomial operators (classP _j operators); in a sense, this is an analog of the Weierstrass approach as opposed to the Cauchy-Riemann approach in classical complex analysis, which also has advantages over the original form in boundary value problems and other applications.Dedicated to the memory of my friend, Peter Henrici