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.     

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.     

On Some Sets of Group Functions

Let G be a group, let A be an Abelian group, and let n be an integer such that n –1. In the paper, the sets _n (G,A) of functions from G into A of degree not greater than n are studied. In essence, these sets were introduced by Logachev, Sal'nikov, and Yashchenko. We describe all cases in which any function from G into A is of bounded (or not necessarily bounded) finite degree. Moreover, it is shown that if G is finite, then the study of the set _n (G,A) is reduced to that of the set n(G/O ^p (G),A _p ) for primes p dividing G/G. Here O ^p (G) stands for the p-coradical of the group G, A _p for the p-component of A, and G for the commutator subgroup of G.     

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.     

Normality preserving multiplication operators

We show that a multiplication operator (T)=ATB is normality preserving if and only if it is hyponormality preserving, if and only if it is either of the formA=fg,B=h f, orA=D,B=D* for someC andD* D=I. Also we show that is (semi-) Fredholmness prserving if and only ifA andB are (semi-) Fredholm operators.Supported by the Science Foundation of Zhejiang Province and NSF.     

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.     

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.     

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 the parallel factorization of matrices over principal ideal rings

We describe all the factorizations A=BC (up to associates) of a matrix A over a commutative principal ideal domain parallel to the factorization D^A= of its canonical diagonal form D^A ( and are diagonal matrices), that is, the factorizations such that the matrices B and C are equivalent to the matrices and respectively.Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 4, 1997, pp. 96–100.     

Topological transitivity of cylindrical cascades

The existence is proved of a topologically transitive (t.t.) homeomorphism U of the space W = × Z of the formU (, z)=(T,, z+f ()) ( , z Z), where is a complete separable metric space, T is a t.t. homeomorphism of onto itself, Z is a separable banach space, andf is a continuous map: z. For the special case W = S¹×R, T=+ ( is incommensurable with 2) the existence is proved of t.t. homeomorphisms (1) of two types: 1) with zero measure of the set of transitive points, 2) with zero measure of the set of intransitive points. An example is presented of a continuous functionf: S¹R for which the corresponding homeomorphism (1) is t.t. for all incommensurable with 2.Translated from Matematicheskie Zametki, Vol. 14, No. 3, pp. 441–452, September, 1973.The author thanks D. V. Anosov for advice and interest in the work.     

Anticommutative Algebras Satisfying Standard Identities of Degree Four

We define an anticommutative -algebra A(D,a) whose multiplication generalizes the concept of a Jacobi bracket in the form (4). It is proved that A(D,a) is a J-algebra and that it satisfies a standard identity of degree four. A subclass of algebras A(D,a) over which is connected with some class of 3-Lie algebras is distinguished. We establish a criterion of being simple for factor algebras of non-Lie algebras in , given a 1-dimensional annihilator, and then use it to construct examples of simple infinite-dimensional (of dimension p³-1) non-Lie J-algebras over a field satisfying standard identities of degree 4, if the characteristic p of is zero (for p > 2). Also, the criterion of algebras belonging to is given.     

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.     

Total oscillation diminishing property for scalar conservation laws

Summary. We prove a BV estimate for scalar conservation laws that generalizes the classical Total Variation Diminishing property. In fact, for any Lipschitz continuous monotone :, we have that |(u)|_TV() is nonincreasing in time. We call this property Total Oscillation Diminishing because it is in contradiction with the oscillations observed recently in some numerical computations based on TVD schemes. We also show that semi-discrete Total Variation Diminishing finite volume schemes are TOD and that the fully discrete Godunov scheme is TOD.Mathematics Subject Classification (2000): 35L65, 35K55, 65M20     

A Favard theorem for orthogonal rational functions on the unit circle

We consider forn=0, 1,... the nested spaces _n of rational functions of degreen at most with given poles . Given a finite measure supported on the unit circle, we associate with it a nested orthogonal basis of rational functions ₀,..., _n for _n ,n=0, 1,.... These _n satisfy a recurrence relation that generalizes the recurrence for Szeg polynomials.In this paper we shall prove a Favard type theorem which says that if one has a sequence of rational functions _{n n} which are generated by such a recurrence, then there will be a measure supported on the unit circle to which they are orthogonal. We shall give a sufficient condition for the uniqueness of this measure.     

A class of nonseparable dynamic programming problems

A solution procedure is proposed for a class of deterministic sequential decision problems whose objective functions are of the form f _n(x _n)+(g _n(x _n)) where is differentiable and either concave or convex. The procedure calls for the collaboration between dynamic programming and c-programming, and is demonstrated in our treatment of a minimum variance type problem.     

Dualities for the Domany–Kinzel Model

We study the Domany–Kinzel model, which is a class of discrete-time Markov processes in one-dimension with two parameters (p ₁,p ₂)[0,1]². When p ₁= and p ₂=(2– ²) with (,)[0,1]², the process can be identified with the mixed site-bond oriented percolation model on a square lattice with probabilities of a site being open and of a bond being open. This paper treats dualities for the Domany–Kinzel model _t ^A and the DKdual _t ^A starting from A. We prove that , as long as one of A,B is finite and p ₂p ₁.     

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.     

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.     

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.