Generalized Variation of Mappings with Applications to Composition Operators and Multifunctions

We study (set-valued) mappings of bounded -variation defined on the compact interval I and taking values in metric or normed linear spaces X. We prove a new structural theorem for these mappings and extend Medvedev's criterion from real valued functions onto mappings with values in a reflexive Banach space, which permits us to establish an explicit integral formula for the -variation of a metric space valued mapping. We show that the linear span GV (I;X) of the set of all mappings of bounded -variation is automatically a Banach algebra provided X is a Banach algebra. If h:I× X Y is a given mapping and the composition operator is defined by (f)(t)=h(t,f(t)), where tI and f:I X, we show that :GV (I;X) GV (I;Y) is Lipschitzian if and only if h(t,x)=h₀(t)+h₁(t)x, tI, xX. This result is further extended to multivalued composition operators with values compact convex sets. We prove that any (not necessarily convex valued) multifunction of bounded -variation with respect to the Hausdorff metric, whose graph is compact, admits regular selections of bounded -variation.     

Best Φ- and N Φ-approximants in Orlicz spaces of vector valued functions

Summary Let (, A, P) be a probability space and E be a Banach space. We study the approximation of an E-valued random variable X, which is an element of the Orlicz space L(, A, P; E), by a function YL, which is measurable with respect to a sub--field of A and takes values in a closed convex subset of E. Two types of approximation are considered: (X – Y) dP=inf, and N(X–Y)=inf with the Orlicz space norm N. We give conditions for the existence of best approximants. If E is reflexive, we obtain martingale type convergence theorems for best approximants and discuss the continuity of the operator X best approximant of X.This paper is a part of the authors doctoral thesis, written under the guidance of D. Landers     

Elliptic genera,involutions, and homogeneous spin manifolds

We study the normalized elliptic genera (X)=(X)/ ^k/2 for 4k-dimensional homogeneous spin manifolds X and show that they are constant as modular functions. The basic tool is a reduction formula relating (X) to that of the self-intersection of the fixed point set of an involution on X. When (X) is a constant it equals the signature of X. We derive a general formula for sign(G/H), GH compact Lie groups, and determine its value in some cases by making use of the theory of involutions in compact Lie groups.Dedicated to Jacques Tits on the occasion of his sixtieth birthday     

Über die Minimalflächen des isotropen Raumes,welche zugleich Affinminimalflächen sind

One determines all the minimal surfaces of the isotropic space, which simultaneously are affinminimal surfaces. A characteristic property of those surfaces is that the isotropic spherical imagines of the asymptotic lines of form two orthogonal pencils of circles. There are three types of such surfaces : first the well known right helicoid _I , second an interesting transcendental surface _II , and third the isotropic analogy _III of the minimal surface ofEnneper. The surfaces permit cinematic generations. Especially _II and _III can be generated byClifford screws in a certain indefinite quasielliptic space.In the isotropic space conjugate to the surfaces are isotropic minimal surfaces ^* with plane lines of curvature. There are also three types of such surfaces: _I ^* is a logarithmic surface of revolution, _II ^* is an interesting transcendental surface, and _III ^* is again the isotropic minimal surface ofEnnerper.     

Compact groups on compact projective planes

LetP=(P, L) be a compact projective plane with 0P< and let be a compact connected subgroup of Aut(P). If dim dimE – dimP, whereE is the elliptic motion group of the corresponding classical plane, then E or is isomorphic to a point stabilizerE ₀ inE, cf. [31]. Here we consider the case E ₀. It is shown that the action of on the point spaceP is equivalent to the classical action ofE ₀. For dimP {8, 16} the planeP is uniquely determined by a 2-dimensional subplane with SO₂ Aut().Für H. Reiner Salzmann zum 65. Geburtstag     

Generalized ornstein-uhlenbeck process having a characteristic operator with polynomial coefficients

Summary Let be a weighted Schwartz's space of rapidly decreasing functions, the dual space and (t) a perturbed diffusion operator with polynomial coefficients from into itself. It is proven that (t) generates the Kolmogorov evolution operator from into itself via stochastic method. As applications, we construct a unique solution of a Langevin's equation on : whereW(t) is a Brownian motion and ^*(t) is the adjoint of (t) and show a central limit theorem for interacting multiplicative diffusions.     

On Vector Valued Measure Spaces of Bounded Φ-variation containing copies of ℓ∞

Given a Young function , we study the existence of copies of c ₀ and in cabv (,X) and in cabsv (,X), the countably additive, -continuous, and X-valued measure spaces of bounded -variation and bounded -semivariation, respectively.     

Positivity,Trotter Products,and Blow-up

We give a new approach to blow-up results for nonlinear evolution equations of the form u(t) =Au(t) +(u(t)) on ordered Banach spaces, using the Lie--Trotter product formula for obtaining lower and upper bounds for the solution.     

Lie algebras satisfying identities of degree 5

Let be an associative ring with unity, containing 1/6.We prove that every prime Lie -algebra satisfying the identity [(yx)(zx)]x = 0is embedded as a subring of a special form in a three-dimensional simple Lie algebra over some field A. It follows that there exists no central simple Lie algebra which is not three-dimensional and the cube of every inner derivation in which is a derivation. It is proved that if a semiprime Lie algebra over a field satisfies an arbitrary identity of degree 5 (not following from the anticommutativity and Jacobi identities), then it also satisfies the standard identity of degree 5. Essentially used in the proof is the notion of antiderivation. In passing we show that every prime Lie algebra having a nonzero antiderivation satisfies the standard identity of degree 5. Translated fromAlgebra i Logika, Vol. 34, No. 6, pp. 681-705, November-December, 1995.Supported by RFFR grant No. 94-01-00381-a.     

On the Convergence of Backward Differentiation Formulas for Stiff Initial Value Problems

The influence of a time-dependent transformation to a numerical method is studied. Thus convergence results of backward differentiation formulas applied to the non-autonomous stiff system y = A(t)y + (t) are given. The approach is based on a special decomposition of the companion matrix.This revised version was published online in October 2005 with corrections to the Cover Date.     

Das kinematische Bild einer Regelfläche des einfach isotropen Raumes

TheClifford parallelism of the three-dimensional isotropic space J ₃ ⁽¹⁾ induces thekinematic mapping of an element of surface in J ₃ ⁽¹⁾ onto a pair of points in a fixed plane ₀ . By identifying a regular surface with the manifold of its osculating elements of surface thekinematic image of is defined and equivalent to an area preserving transformation of the plane ₀ . In this paper we will examine the connections between the isotropic invariants of a skew ruled surface and its kinematic image. Since the striction line y of exactly consists of the singular points of the kinematic images of the osculating planes of y are considered in addition. In this way we obtain a correspondence between the theory of ruled surfaces and the elementary geometry of two pairs of plane curves with a common middle curve.     

On the Representation of Approximate Subdifferentials for a Class of Generalized Convex Functions

Given a family of real-valued functions defined in a normed vector space X, we study a class of -convex functions having a simpler representation for the --subdifferential. The case =X^* with X being a Banach space (the Fenchel case) is particularly analysed, and we find that the sublinear lower semicontinuous functions satisfy the simpler representation with respect to X^*. As a side result, we provide various new subdifferential-type charaterizations of positively homogeneous functions among those which are lower semicontinuous and convex. In addition, we also discuss that family related to the the so-called prox-bounded functions. In this more general framework our simpler representation may give rise to a new notion of enlargement of the subdifferential.Mathematics Subject Classifications (2000) 47H05, 46B99, 47H17.This work is based on research material supported in part by CONICYT-Chile through FONDECYT 101-0116 and FONDAP-Matemáticas Aplicadas II.     

On the existence of a cyclic vector for some families of operators

Under certain restrictions, it is proved that a family of self-adjoint commuting operatorsA=(A ) _where is a nuclear space, possesses a cyclic vector iff there exists a Hubert spaceH of full operator-valued measureE, where is the space dual to andE is the joint resolution of the identity of the familyA.Published in Ukrainskii Matematicheskii Zhurnal, Vol.45, No. 10, pp. 1362–1370, October, 1993.     

Circle orders,N-gon orders and the crossing number

Let ={P ₁,...,P _m } be a family of sets. A partial order P(, <) on is naturally defined by the condition P _i <P _j iff P _i is contained in P _j . When the elements of are disks (i.e. circles together with their interiors), P(, <) is called a circle order; if the elements of are n-polygons, P(, <) is called an n-gon order. In this paper we study circle orders and n-gon orders. The crossing number of a partial order introduced in [5] is studied here. We show that for every n, there are partial orders with crossing number n. We prove next that the crossing number of circle orders is at most 2 and that the crossing number of n-gon orders is at most 2n. We then produce for every n4 partial orders of dimension n which are not circle orders. Also for every n>3, we prove that there are partial orders of dimension 2n+2 which are not n-gon orders. Finally, we prove that every partial order of dimension 2n is an n-gon order.This research was supported under Natural Sciences and Engineering Research Council of Canada (NSERC Canada) grant numbers A2507 and A0977.     

Hauptregelflächen einer Verallgemeinerten Regelfläche mit Zentralregelfläche

We consider generalized ruled surfaces in euclidean n-space ⁿ with k-dimensional generators and central ruled surface of dimension k–m+1 (O < m < k). Every orthogonal trajectoryy of the generators of defines a principal ruled surface _y with generators totally orthogonal to the generators of . In each generator of _y there exists an ellipsoid — called the indicatrix of the distribution parameters — which is defined by the distribution parameters of the tangent spaces to or _y. Formulars will be given for the distribution parameters of and _y .

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Herrn Prof. Dr. H.R. Müller zum 70. Geburtstag     

A Theorem of Liouville Type for p-Harmonic Morphisms

In this article we prove a Liouville type theorem for p-harmonic morphisms. We show that if : MNis a p-harmonic morphism (p2) from a complete noncompact Riemannian manifold Mof nonnegative Ricci curvature into a Riemannian manifold Nof nonpositive scalar curvature such that the p-energy E _p (), or (2p–2)-energy E _2p–2() is finite, then is constant.     

On a zero-crossing probability

Let {X(t), 0E{exp (–sX(t))}=exp (–t(s)), where (s)=(1–(s)), is the intensity of the Poisson process, and (s) is the Laplace transform of the distribution of nonnegative jumps. Consider the zero-crossing probability =P{X(t)–t=0 for some t,0<t<}. We show that =() where is the largest nonnegative root of the equation (s)=s. It is conjectured that this result holds more generally for any stochastic process with stationary independent increments and with sample paths that are nondecreasing step functions vanishing at 0.     

Contingent Derivatives of Implicit (Multi-) Functions and Stationary Points

For an implicit multifunction (p) defined by the generally nonsmooth equation F(x,p)=0, contingent derivative formulas are derived, being similar to the formula =–F _x ^–1 F _p in the standard implicit function theorem for smooth F and . This will be applied to the projection X(p)={xy: (x,y)(p)} of the solution set (p) of the system F(x,y,p)=0 onto the x-space. In particular settings, X(p) may be interpreted as stationary solution sets. We discuss in detail the situation in which X(p) arises from the Karush–Kuhn–Tucker system of a nonlinear program.     

Convergence des solutions formelles de certaines équations fonctionnelles

Résumé Soitq un nombre algébrique de module 1, qui ne soit pas une racine de l'unité, etP [X, Y ₀,Y ₁] un polynôme non nul. Dans cet article, nous montrons que toute solution de l'équation fonctionnelleP(z, (z), (qz))=0, qui est une série formelle (z) dansQ[[z]], a un rayon de convergence non nul.

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Summary Letq Q be an algebraic number of modulus one that is not a root of unity. LetP Q[X, Y ₀,Y ₁] be a non zero polynomial. In this paper, we show that every formal power series,(z) Q[[z]], solution of the functional equationP(z), (z), (qz)) = 0 has a non zero radius of convergence.