Large transversals to small families of unit disks

Summary It is proved that if the nonempty intersection of bounded closed convex sets AnB is contained in (A + F)U(B+F) and one of the following holds true: (i) the space X is less-than-three dimensional, (ii) AUB is convex, (iii) F is a one-point set, then AnBCA+F or AnBCB+F (Theorems 2 and 3). Moreover, under some hypotheses the characterization of A and B such that AnB is a summand of AUB is given (Theorem 3).     

Martingale Hardy spaces,BMO and VMO spaces with nonlinearly ordered stochastic basis

VMO. , ⁺, «» VMO . « » .     

On affine scaling algorithms for nonconvex quadratic programming

We investigate the use of interior algorithms, especially the affine-scaling algorithm, to solve nonconvex — indefinite or negative definite — quadratic programming (QP) problems. Although the nonconvex QP with a polytope constraint is a hard problem, we show that the problem with an ellipsoidal constraint is easy. When the hard QP is solved by successively solving the easy QP, the sequence of points monotonically converge to a feasible point satisfying both the first and the second order optimality conditions.Research supported in part by NSF Grant DDM-8922636 and the College Summer Grant, College of Business Administration, The University of Iowa.     

Borel fibrations and G-spaces

Considering discrete groups G only, we present an elementary proof of the familiar equivalence of the category of G-spaces (with maps equivariant up to homotopy) and the category of Borel fibrations over BG.     

Orthonormal systems satisfying an inequality of S. M. Nikol'skii

N- (p, q) (1 pN-, L _p - L _q -. , , , L L _q - , , .     

Elastic properties of honeycomb sandwich plates with nontraditional cell forms

A method for correcting the elastic properties of honeycomb sandwich plates by changing the cell shape is described. Compact and convenient analytical expressions (1) are proposed for calculating the reduced elastic parameters of the honeycomb sandwich plates. The acceptable range of changes in the size of the honeycombs is restricted by the cell elongation region [0.5;2] and the region of variation of the angle between cell walls [/6; 5/6]. The results of the calculation experiment showed that the traditional cell shape (=/3, =1) is not effective as an optimum structure (Table 1).Institute of Transport Systems and Technologies (Transmag), National Academy of Sciences of Ukraine, Dnepropetrovsk, Ukraine. Translated from Mekhanika Kompozitnykh Materialov, Vol. 33, No. 2, pp. 187–191, March–April, 1997.     

A new class of cutting planes for the symmetric travelling salesman problem

A comprehensive class of cutting planes for the symmetric travelling salesman problem (TSP) is proposed which contains the known comb inequalities, the path inequalities and the 3-star constraints as special cases. Its relation to the clique tree inequalities is discussed. The cutting planes are shown to be valid for a relaxed version of the TSP, the travelling salesman problem on a road network, and—under certain conditions—to define facets of the polyhedron associated with this problem.     

Certain nonlocal problems for two-dimensional equations of motion of Oldroyd fluids

The solvability of a boundary-value problem on the semi-axis t0 is studied for two-dimensional equations of motion of Oldroyd fluids (1), and with trivial problem data a proof is given of the existence of a solution which is periodic with respect to t and has the period . This solution has an absolute term which is also periodic with respect to t and has the period . Substantiation is given for the principle of linearization (first Liapunov method) in the theory of the exponential stability of solutions at t.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 189, pp. 101–121, 1991.     

Best approximation and best unilateral approximation of the kernel of a biharmonic equation and optimal renewal of the values of operators

For the classB _p , 0 < 1, 1p , of 2-periodic functions of the form f(t)=u(,t), whereu (,t) is a biharmonic function in the unit disk, we obtain the exact values of the best approximation and best unilateral approximation of the kernel K(t) of the convolution f= K *g, gl, with respect to the metric of L₁. We also consider the problem of renewal of the values of the convolution operator by using the information about the values of the boundary functions.Translated from Ukrainskii Matematicheskii Zhurnal, Vol.47, No. 11, pp. 1549–1557, November, 1995.     

Boolean Term Orders and the Root System Bn

A Boolean term order is a total order on subsets of [n] ={1,..., n} such that for all [n], , and for all with ( ) = . Boolean term orders arise in several different areas of mathematics, including Gröbner basis theory for the exterior algebra, and comparative probability.The main result of this paper is that Boolean term orders correspond to one-element extensions of the oriented matroid M(B_n), where B_n is the root system {e_i : 1 i n} {e_i ± e_j : 1 i < j n}. This establishes Boolean term orders in the framework of the Baues problem, in the sense of (Reiner, 1998). We also define a notion of coherence for a Boolean term order, and a flip relation between different term orders. Other results include examples of noncoherent term orders, including an example exhibiting flip deficiency, and enumeration of Boolean term orders for small values of n.     

A Niche Width Model of Optimal Specialization

Niche width theory, a part of organizational ecology, predicts whether specialist or generalist forms of organizations have higher fitness, in a continually changing environment. To this end, niche width theory uses a mathematical model borrowed from biology. In this paper, we first loosen the specialist-generalist dichotomy, so that we can predict the optimal degree of specialization. Second, we generalize the model to a larger class of environmental conditions, on the basis of the model's underlying assumptions. Third, we criticize the way the biological model is treated in sociological theory. Two of the model's dimensions seem to be confused, i.e., that of trait and environment; the predicted optimal specialization is a property of individual organizations, not of populations; and, the distinction between fine and coarse grained environments is superfluous.     

Almost-periodic solutions for a second order abstract Cauchy problem

Summary Let M be a number field of degree m with ring of integers \bZ_M. Let F\in\bZ_M[X,Y] be a form of degree n such that F(X,1) has distinct roots. Let\break G\in\bZ[X,Y] be an arbitrary polynomial of degree k. Assuming that k\le n-2m\pl 1 if all roots of F^{(i)}(X,1) (1\le i\le n) are complex and k\le n-4m\pl 1 otherwise, we provide an efficient algorithm for finding all solutions X,Y\in\bZ_M, \max\b(\overline{|X|},\overline{|Y|}\,\b)\ki C of the inequality \overline{\b|F(X,Y)\b|\!}\,\le c \cdot \overline{\b|G(X,Y)\b|\!}\,. We provide numerical examples with m=3 and C=10^{100}.     

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.     

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.     

On the Mechanism for Nonlinear Representations of the Lorentz Group Arising in Quantum Field Theory

A mechanism is identified that leads to the correct law for the relativistic Wigner function transformation with respect to the Lorentz group as long as the corresponding relativistic wave functions have special transformation properties.     

A note on sequence-covering mappings

Summary Let f: X &rarr; Y be a mapping. f is called a sequence-covering mapping if in case S is a convergent sequence containing its limit point in Y then there is a compact subset K of X such that f(K) = S. It is shown that each quotient and compact mapping of a metric space is sequence-covering.     

Stellare Abänderungen und Schälbarkeit von Komplexen und Polytopen

Brugesser and Mani proved that the boundary-complex of a convex polytope can be shelled. This result lead to McMullen's proof of the Upper-bound-conjecture. We show that the shellability of complexes has a close connection to the theory of stellar operations. Several results on special shelling procedures and on non-shellable complexes are obtained.     

Hugoniót–Maslov Chains for Singular Vortical Solutions to Quasilinear Hyperbolic Systems and Typhoon Trajectory

According to Maslov, many 2D quasilinear systems of PDE possess only three algebras of singular solutions with properties of structural self-similarity and stability. They are the algebras of shock waves, narrow solitons, and square-root point singularities (solitary vortices). Their propagation is described by infinite chains of ODE (the Hugoniót–Maslov chains). We consider the Hugoniót-Maslov chain for the square-root point singularities of the shallow water equations. We discuss different related mathematical questions (in particular, unexpected integrability effects) as well as their possible application to the problem of typhoon dynamics.     

Diagonal Bases in Orlik-Solomon Type Algebras

To encode an important property of the no broken circuit bases of the Orlik- Solomon-Terao algebras, Szenes has introduced a particular type of bases, the so called diagonal basis. We prove that this definition extends naturally to a large class of algebras, the so called - algebras. Our definitions make also use of an iterative residue formula based on the matroidal operation of contraction. This formula can be seen as the combinatorial analogue of an iterative residue formula introduced by Szenes. As an application we deduce nice formulas to express a pure element in a diagonal basis.AMS Subject Classification: 52C35, 05B35, 14F40.     

Representation of text strings in binary computers

The optimal internal representation of text strings in binary computers is discussed. The average storage space required can be reduced if shift signals are used. Under certain conditions a text, composed of appr. 100 different symbols, can be economically stored using 32 different 5-bit code signals, including a set of special shift signals which specify the significance of the 5-bit configurations in much the same way as upper and lower case signals remove the ambiguity of 5-bit teletype signals.