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).     

Lax-type isospectral deformations on nilmanifolds

A one-parameter familyg(t) (0 t T) of Riemannian metric on a compact manifold is called an isospectral deformation of a metricg(0) if the Laplace-Beltrami operators associated to the metricsg(t) have the same spectra. Examples of non-trivial isospectral deformations were constructed on solvmanifolds for the first time by C.S. Gordon and E. Wilson on the basis of Kirillov theory. This paper considers the isospectral deformations on nilmanifolds from the dynamical point of view. First, we see for certain isospectral deformations that the associated Hamiltonian systems of geodesic flows are decomposed into a collection of reduced systems which are left invariant as Hamiltonian systems under the deformations. This fact is formulated by the classical Lax equations. Next, by using a quantization procedure, we attempt to obtain Lax equations for the reduced Laplacians from the classical Lax equations. As a result, we show that certain isospectral deformations by Gordon-Wilson are represented by the Lax equations.     

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.     

L ∞ — Decay estimations of the spherical mean value on symmetric spaces

LetM _t[](x) be the spherical mean value operator applied to a function on a symmetric Riemannian space of the non-compact type.L —decay estimations forM _t [](x) as well as for its derivatives with respect to (t, x) are given, provided that belongs to a Banach space with suitable weighted supremum norm. This leads to estimates of the solutions to the wave equation in certain cases in which Huygens' principle is valid.     

On probabilistic commutative spaces

A new characterization is given for the probabilistic commutative Riemannian spaces studied originally (in the compact case) by Roberts and Ursell. Moreover, some earlier basic results are generalized to the non-compact case.     

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.     

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.     

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     

Intrinsic ultracontractivity of Dirichlet Laplacians in non-smooth domains

We give a general criterion for the intrinsic ultracontractivity of Dirichlet Laplacians – _D on domainsD ofR ^d d 3, based on the Lieb's formula. It applies to various classes of domains (e.g. John, Hölder andL ^p-averaging domains) and gives new conditions for intrinsic ultracontractivity in terms of the Minkowski dimension of the boundary D. In particular, isotropic self-similar fractals and domains satisfying a c-covering condition are considered.     

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.     

Infinitesimal obstructions to weakly mixing

Let ^t be the flow (parametrized with respect to arc length) of a smooth unit vector field v on a closed Riemannian manifold M ⁿ , whose orbits are geodesics. Then the (n-1)-plane field normal to v, v, is invariant under d ^t and, for each x M, we define a smooth real function _x (t) : (1 + _i (t)), where the _i(t) are the eigenvalues of AA ^T, A being the matrix (with respect to orthonormal bases) of the non-singular linear map d^2t , restricted to v at the point _x ^-t M ⁿ.Among other things, we prove the Theorem (Theorem II, below). Assume v is also volume preserving and that _x ^' (t) 0 for all x M and real t; then, if _x ^t : M M is weakly missng for some t, it is necessary that vx 0 at all x M.     

Compact elements in the lattice of radicals

Summary A radical R, in the sense of Kurosh--Amitsur, is said to be compact if, given any collection of radicals X such that R ≤;VX, we have R ≤;VX' for some finite subcollection X' of X. A ring A is said to be radical compact if the lower radical on the singleton {A} is compact. This paper explores the relationship between radical compact rings and rings satisfying certain finiteness conditions. Closure properties of the class of all radical compact rings are also investigated.     

Parameter error estimate of near alternation approximation

In practical computation of the discrete best uniform approximation, we usually only get near best (i.e., with the -near alteration property) approximation. We need to estimate the error between the (unknown) best approximation and the achieved approximation. In this paper we estimate the parameter error by means of the generalized strong unicity constants.Visiting scholar from Department of Mathematics, Shanghai University of Science and Technology, Shanghai, P.R. China 201800.     

The Selberg trace formula for SL(3,Z)

In this article the first step toward the generalization of the Selberg trace formula to the case of a rank 2 symmetric space S and a discrete group for which the fundamental region \S goes to infinity nontrivially appears. For S we use the space SL(3,)/SO(3) and for we use SL(3,). The fundamental results are Theorems 9 and 10, in which is calculated the contribution to the matrix trace of the operator K which appears in the right side of the trace formula of the expression h()d^c(), where ^c() is the continuous part of the spectral measure of the quasiregular representation on the space IL₂(\S).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 63, pp. 8–66, 1976.     

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}.     

On the definition of purity of modules

In the paper the coincidence is established of the concepts of purity and bitriangular purity in module theory, introduced in the book of A. P. Mishina and L. A. Skornyakov, Abelian Groups and Modules.Translated from Matematieheskie Zametki, Vol. 11, No. 4, pp. 375–380, April, 1972.In conclusion, the author wishes to thank D. K. Faddeev for interest in this work.     

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).     

The effect of dimension 4 in Aleksandrov's problem of filling a space by polyhedra

This paper deals with filling the hyperbolic space Hⁿ by non-compact polyhedra. In dimensions n <4 the non-compact case is very different from the compact one, which was investigated by A.D. Aleksandrov. For n 4 the compact and non-compact cases are almost similar. This investigation is closely related to deformations of complete and incomplete hyperbolic orbifolds (in the sense of W. Thurston) for which a strong rigidity result is proved-similar to the one for complete hyperbolic manifolds in dimension exceeding two.     

Eine differentialtopologische Berechnung der Totalen Krümmung und Totalen Absolutkrümmung in der sphärischen Differentialgeometrie

The aim of this article is to compute the total (absolute) curvature, i.e. the mean value of the (absolute) Lipschitz-Killing-curvature, of an immersion f: MSⁿ of a compact manifold into the unit sphere in a differential topological manner. Through a generalization of KUIPERs treatment of immersions in Euclidean spaces it can be computed as the mean value of the number of critical points—weighted by (–1)^k (k=Index) resp.not weighted—of certain functions. These functions are the pullback via f of level-functions, which are defined almost everywhere on Sⁿ. Such a level-function is constructed by taking any oriented great circle as a leveling-scale and the orthogonal great (n–1)-spheres as level-surfaces.     

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.