Circularity of Finite Groups without Fixed Points

Let be a fixed point free group given by the presentation where and are relative prime numbers, t = /s and s = gcd( – 1,), and is the order of modulo . We prove that if (1) = 2, and (2) is embeddable into the multiplicative group of some skew field, then is circular. This means that there is some additive group N on which acts fixed point freely, and |((a)+b)((c)+d)| 2 whenever a,b,c,d N, a0c, are such that (a)+b(c)+d.     

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.     

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.     

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.     

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     

Self-similar random measures are locally scale invariant

Summary In an earlier paper Patzschke and U. Zähle [11] have proved the existence of a fractional tangent measure at the typical point of a self-similar random measure under rather special technical assumptions. In the present paper we remove the most restrictive one. Here we suppose the open set condition for the similarities, a constant positive lower bound for the random contraction ratios, and vanishing on the boundary of the open set with probability 1. The tangent measure isD-scale-invariant, whereD is the similarity dimension of . Moreover, we approximate the tangential distribution by means of and use this in order to prove that the Hausdorff dimension of the tangent measure equalsD. Since the former coincides with the Hausdorff dimension of we obtain an earlier result of Mauldin and Williams [9] as a corollary.     

Compromise optimization of a composite plate with a given probability of realization

A method is proposed for solution of the problem of the compromise optimization of three properties of a composite plate (thermal conductivity, stability, and the probability P^* of design realization), which depend on three initial stochastic data with constant average values, and two variable initial data. The geometry of the domain of plate properties, the curve of optimal Pareto solutions, and the scatter ellipses is determined at four points for a given range of variable parameters. A method of constructing the curves of optimal Pareto solutions for the following assigned probabilities of design realization is proposed and numerically implemented: P^*=0.40, 0.80, and 0.95. The generalized efficiency function ( max, 0 1) of the first two properties decreases from 0.74 to 0.23 as the numerical value of P^* increases from 0.40 to 0.95. A family of isolines = const is plotted for all three properties investigated, and max determined as 0.63.A paper presented at the Tenth International Conference on Mechanics of Composite Materials (Riga, April 1998).Institute of Polymer Mechanics, Riga, Latvia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 33, No. 5, pp. 626–635, September–October, 1997.     

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 .     

Inequalities Between the Radii of Some Spheres That Are Connected with a Convex Surface in Space of Constant Curvature

With a convex surface in space of constant curvature, we associate four numbers (,M,), where is the radius of a largerst sphere freely rolling over the interior side of , is the inradius of , M is the outradius of , and is the radius of a sphere over whose interior may roll freely. Exact inequalities connecting these four numbers are found.     

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     

A Second Descent Problem for Quadratic Forms

Let F be a field of characteristic different from 2. We discuss a new descent problem for quadratic forms, complementing the one studied by Kahn and Laghribi. More precisely, we conjecture that for any quadratic form q over F and any Im(W(F) W(F(q))), there exists a quadratic form W(F) such that dim 2 dim and _F(q), where F(q) is the function field of the projective quadric defined by q = 0. We prove this conjecture for dim 3 and any q, and get partial results for dim {4, 5,6}. We also give other related results.     

Formal Flows on the Non-Archimedean Open Unit Disk

This paper deals with group actions of one-dimensional formal groups defined over the ring of integers in a finite extension of the p-adic field, where the space acted upon is the maximal ideal in the ring of integers of an algebraic closure of the p-adic field. Given a formal group F as above, a formal flow is a series (t,x) satisfying the conditions (0,x)=x and (F(s,t),x)=(s,(t,x)). With this definition, any formal group will act on the disk by left translation, but this paper constructs flows with any specified divisor of fixed points, where a point of the open unit disk is a fixed point of order n if (x–) ⁿ |((t,x)–x). Furthermore, if is an analytic automorphism of the open unit disk with only finitely many periodic points, then there is a flow , an element of the maximal ideal of the ring of constants, and an integer m such that the m-fold iteration of (x) is equal to (,x). All the formal flows constructed here are actions of the additive formal group on the unit disk. Indeed, if the divisor of fixed points of a formal flow is of degree at least two, then the formal group involved must become isomorphic to the additive group when the base is extended to the residue field of the constant ring.     

Brownian charges around loops

Summary Let (X _t ⁿ ) be a Poisson sequence of independent Brownian motions in ^d ,d3; Let be a compact oriented submanifold of d, of dimensiond–2 and volume ; let t be the sum of the windings of (X _s ⁿ , 0st) around ; then t/t converges in law towards a Cauchy variable of parameter /2. A similar result is valid when the winding is replaced by the integral of a harmonic 1-form in ^d .     

Die Weingarten-Regelflächen

Referring to articles of BELTRAMI (1865), DINI (1866) and CHARIAR (1978), but using a completely different approach, we determine allruled surfaces in Euclidean space ³, which are (at leastlocally) WEINGARTEN —-surfaces under theminimal assumption C². Theskew ruled WEINGARTEN —surfaces can be characterized by havingconstant invariants d O (parameter of distribution), k (skewness of distribution) and (striction angle); theirfunctional (WEINGARTEN-)relation between the mean curvature H and the Gaussian curvature K of is of the form H= (-K)^1/4 + (-K)^3/4 with arbitrary real constants ,. These facts allow various geometric interpretations.

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Herrn Prof. Dr. Oswald Giering zum 60. Geburtstag gewidmet     

Local invariants of differential equations

We consider an analytic system X=(X) in the neighborhood of the fixed point X=0. Depending on the characteristic numbers of the matrix (/x)₀, we define the integer d 0 as the dimension of the normal form or as the multiplicity of the resonance. We show that a system with d=1, subject to certain additional assumptions, has a finite number of invariants relative to reversible formal changes of variablesx = (Y). All these invariants are the coefficients of some normal form. We touch upon questions concerning invariants of relatively smooth and continuous substitutions.Translated from Matematicheskie Zametki, Vol. 14, No. 4, pp. 499–507, October, 1973.     

Drachennetze mit geradlinigen Hauptdiagonalen

A C^s-net of curves N (s1) [3] in a regular C^s-2-surface Eⁿ (n2) is called a C^s-kite- net [4] if N and the net N₁ of its angular bisecting curves form a pair of diagonal nets [1] in such a way that each mesh of N-curves possessing two N₁-diagonals shows, with respect to one of these (calledmain diagonal), the same symmetry of angles and lengths as a rectilinear kite in E². Referring to the fact that the main diagonals of any C^s-kite-net N (s2) are geodesics in [5], we ask in this paper for all C^s-kite-nets and, more generally, C^s-D-nets [5] (s1) withstraight main diagonals. This leads, among other results, to a characterization of the skew ruled surfaces in Eⁿ (n3) with constant parameter of distribution and the constant striction /2.

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Herrn Professor Dr. WERNER BURAU zum 70. Geburtstag gewidmet     

Finitely generated relatively universal varieties of Heyting algebras

Given distinct varieties and of the same type, we say that is relatively -universal if there exists an embedding :K from a universal categoryK such that for every pairA, B ofK-objects, a homomorphismf:A B has the formf=_g for someK-morphismg:A B if and only if Im(f) . Finitely generated relatively -universal varieties of Heyting algebras are described for the variety of Boolean algebras, the variety generated by a three element chain, and for the variety generated by the four element Boolean algebra with an added greatest element.Dedicated to the memory of Alan DayPresented by J. Sichler.The support of the NSERC is gratefully acknowledged.     

On the Growth of the Maximum of the Modulus of an Entire Function on a Sequence

Let M _f(r) and _f(r) be, respectively, the maximum of the modulus and the maximum term of an entire function f and let be a continuously differentiable function convex on (–, +) and such that x = o((x)) as x +. We establish that, in order that the equality be true for any entire function f, it is necessary and sufficient that ln (x) = o((x)) as x +.     

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.     

A characterization of semimartingales on nuclear spaces

Summary This work is devoted to prove the following fact: Suppose that is a nuclear space whose dual is nuclear under the strong topology. IfX is a weakly adapted mapping with values in such that for any,X'() has a modification which is a semimartingale then there exists a unique projective system of Hubert space-valued semimartingales indexed by the Hilbert-Schmidt neighbourhood base of the dual space whose projective limit isX.In the last part we study in detail a semimartingale defined as the convolution of a distribution by a random Dirac measure whose support is determined by the trajectories of a real-valued semimartingale.