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.     

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.     

Descendants constructed from matter field in Landau-Ginzburg theories coupled to topological gravity

It is argued that gravitational descendants in the theory of topological gravity coupled to topological Landau-Ginzburg theory (not necessarily conformal) can be constructed from matter fields alone (without metric fields and ghosts). In this sense topological gravity is induced. We discuss the mechanism of this effect (that turns out to be connected with K. Saito's higher residue pairing: Kⁱ(_i(₁),₂)=K⁰(₁,₂)), and demonstrate how it works in a simplest nontrivial example: correlator on a sphere with four marked points. We also discuss some results on k-point correlators on a sphere. From the idea of induced topological gravity it follows that the theory of pure topological gravity (without topological matter) is equivalent to the trivial Landau-Ginzburg theory (with quadratic superpotential).Published in Teoreticheskaya i Matematicheskaya Fizika, Vol. 95, No. 2, pp. 307–316, May, 1993.     

A Selection Theorem for Strongly Regular Multivalued Mappings

We prove the following generalization of a theorem of Ferry concerning selections of strongly regular multivalued maps onto the class of paracompact spaces: Let : X (Z, ) be a map of a paracompact space X into a metric space (Z, ) whose values (x) are complete subspaces of Z and absolute extensors (AE), for every x X. Suppose that can be represented as = , where : X Y is a continuous singlevalued map of X onto some finite-dimensional paracompact space Y and : Y (Z, ) is a strongly regular map. Then for every closed subset A X and every selection r : A Z of the map |A : A Z, there exists an extension : X Z of r such that is a selection of the map . We also prove a local version of this theorem.     

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     

On the equivalence of some rearrangements of the two-parameter Haar system

h ₁, L _p .

---

---

This research was supported in part by MTA-NSF grants INT-8400708 and 8620153.     

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.     

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.     

A condition for nonnegativity of an operator related to the Dirichlet operator

Let L₀ be a positive definite closed linear operator with domain of definition D(L₀) dense in the Hilbert space H; let(, ₁, ₂) be the positive boundary value space of the operator L₀ such that the restriction of L ₀ ^* to ker ₂ is the Friedrichs extension of the operator L₀. We establish a test for nonnegativity of an operator T of the form Ty=L ₀ ^* y+^*(₁–C)y, y D(T)= ker(₂+), where :H and C: are respectively a compact operator and a bounded nonnegative operator.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 32, 1990, pp. 30–33.     

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.     

Net subgroups of Chevalley groups. II. Gauss decomposition

This paper is a continuation of RZhMat 1980, 5A439, where there was introduced the subgroup () of the Chevalley group G(,R) of type over a commutative ring R that corresponds to a net , i.e., to a set =(),, of ideals of R such that ₊ whenever ,,+ . It is proved that if the ring R is semilocal, then () coincides with the group ₀ considered earlier in RZhMat 1976, 10A151; 1977, 10A301; 1978, 6A476. For this purpose there is constructed a decomposition of () into a product of unipotent subgroups and a torus. Analogous results are obtained for sub-radical nets over an arbitrary commutative ring.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 114, pp. 62–76, 1982.In conclusion, the authors would like to thank Z. I. Borevich for his interest in this paper.     

On an application of infinitely divisible distributions to quadrature problems

— - , .     

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 +.     

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.     

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.     

Une variante à la Coxeter du groupe de Steinberg

LetG _n ()be the semi-direct product of the symmetric groupS _n by the Steinberg groupSt _n ()of a ringWe first prove thatG _n ()has a Coxeter-type presentation. The canonical morphism St _n () GL _n ()extends to a group homo G_n() GL _n ()We next determine the kernel of for n = We also give an expression for the generator of the algebraic K group K ₂(Z)of the integers in terms of permutation matrices.     

On Finite Morphisms of Ruled Surfaces

We prove that a surjective morphism :XX between ruled surfaces is finite and it descends to a finite morphism :CC between base curves of X and X. When is restricted to the fibres of X, it has a constant degree, say a, and then deg=a deg. In addition, we have several properties on the inverse image of a minimal section and a fibre of X as well as on the direct images. We also investigate precisely the case when both C and C are elliptic and X is the fibre product C ×_C X especially.     

High order approximation of implicitly defined maps

Summary Approximations for the function implicitly defined by (u)=(u, (u)) are obtained via the iterative scheme _n(u)=(u, _n–1(u)). In this paper the uniform convergence of high order derivatives of _n to the corresponding derivatives of is proved. This result yields a high order approximation theorem for the input-output map generated by a nonlinear control system, using linear combinations of iterated integrals of the control.Lavoro eseguito nell'ambito del G.N.A.F.A. del C.N.R.