On the minimum modulus of a multiple Dirichlet series with monotone coefficients

We establish conditions under which the relation M(x, F) (x, F) m(x, F) holds except for a small set, as ¦x¦ + for an entire function F(z) of several complex variables z (p2) represented by a Dirichlet series, where M(x, F) = sup{¦F(x+iy¦: y ^p}, m(x, F) = inf{¦F(x+iy)¦: y ^p} (x, F) being the maximal term of the Dirichlet series, and x ^p.Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 4, 1997, pp. 21–25.     

Surgery on the Novikov complex

Let M be a closed connected smooth manifold with dim M=n6, and : ₁(M) Z be an epimorphism. Denote by the group ring of ₁(M) and let ^– be its Novikov completion. Let D _* be a free-based finitely generated chain complex over ^– . Assume that D _ii=0 for i1 and in–1 and that D _* has the same simple homotopy type as the Novikov-completed simplicial chain complex of the universal covering M. Let N be an integer. We prove that D _* can be realized, up to the terms of ^– of degree N as the Novikov complex of a Morse map : M S ¹, belonging to . Applications to Arnold's conjectures and to the theory of fibering of M over S ¹ are given.     

Return probabilities for random walk on a half-line

A random walk with reflecting zone on the nonnegative integers is a Markov chain whose transition probabilitiesq(x, y) are those of a random walk (i.e.,q(x, y)=p(y–x)) outside a finite set {0, 1, 2,...,K}, and such that the distributionq(x,·) stochastically dominatesp(·–x) for everyx{0, 1, 2,..., K}. Under mild hypotheses, it is proved that when xp _x>0, the transition probabilities satisfyq ⁿ(x, y)C_xyR^–nn^–3/2 asn, and when xp _x=0,q ⁿ(x, y)C_xyn^–1/2.Supported by National Science Foundation Grant DMS-9307855.     

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.     

A geometric condition for an l.c.K. manifold to be Kähler

We prove that a necessary and sufficient condition for a locally conformal Kähler manifold(M ⁿ , g), n 2, to be Kähler is thatM admits through every one of its points a minimal Kahler submanifold of complex dimensionp2.     

Isotropic random walks in a tree

Summary Let T be an infinite homogeneous tree of order a+1. We study Markov chains {X _n} in T whose transition functions p(x, y)=A[d(x,y)] depend only on the shortest distance between x and y in the graph. The graph T can be represented as a symmetric space of a p-adic matrix group; we prove a series of results using essentially the spherical functions of this symmetric space. Theorem 1. d(X _n,x) n a.s., where >0 if A(0) 1, X ₀=x. Assuming {X _n} is strongly aperiodic, Theorem 2. p ₂(x, y)CRⁿ/n^3/2 for fixed x, y where R=(d) A(d)<1, and if E[d(X₁, X₀)²]<, Theorem 3. R(1–u, x, y) = (1–u)ⁿp_n(x, y)=Ca^–d[exp(–du/)+o_d(1)] as d=d(x,y) uniformly for 0u2. Using Theorem 3, we calculate the Martin boundary Dirichlet kernel of p(x, y) on T, which turns out to be independent of {itA(d)}. We also consider a stepping-stone model of a randomly-mating-and-migrating population on the nodes of T. If initially all individuals are distinct, then in generation n approximately half of the individuals of a given type are within n of a typical one and essentially all are within 2n.This work was partially supported by the National Science Foundation under grant number MCS 75-08098-A01For the academic year 1977–78: Department of Mathematics GN-50, University of Washington, Seattle, Washington 98195 USA     

Asymptote of the eigenvalues of a completely continuous operator

It is proved that if(x) is the majorant of the s-numbers of a completely continuous operator A (i.e.,'(x)- 0, S_n(A) (n)) and if there are found numbers [0, 1] and r₀ > 0 such that r⁰ (r)/(r) will be monotonic in (r₀,), then for some > 0,((x) will be a majorant of the eigenvalues of A.Translated from Matematicheskie Zametki, Vol. 14, No. 4, pp. 487–492, October, 1973.     

On the homogeneous Riemannian structures of type ℐ1⊕ℐ3

We study homogeneous Riemannian structures belonging to the class ₁₃ of the classification given by Tricerri and Vanhecke. The main result is the following: a connected, simply connected Riemannian manifoldM admits a homogeneous structureT of type ₁₃,T₃, if and only ifM is isometric to a hyperbolic space ⁿ .This work was partially supported by M.P.I.     

Lyapunov exponents and relative entropy for a stochastic flow of diffeomorphisms

Summary The Lyapunov exponents _{1 2}... _d for a stochastic flow of diffeomorphisms of a d-dimensional manifold M (with a strongly recurrent one-point motion) describe the almost-sure limiting exponential growth rates of tangent vectors under the flow. This paper shows how the Lyapunov exponents are related to measure preserving properties of the stochastic flow on M and of the induced stochastic flow on the projective bundle PM. Relative entropy is used to quantify the extent to which a measure fails to be invariant under the flow. The results include the following. If M is compact and if the one-point motion on M is a non-degenerate diffusion with stationary probability measure then ₁+...+ _d 0 with equality if and only if the flow preserves almost surely; if in addition the induced one-point motion on PM satisfies a weak non-degeneracy condition then ₁=...= _d if and only if there is a smooth Riemannian structure on M with respect to which the flow is conformal almost surely.     

A note on the global asymptotic stability of the unreacting state in a simple model for quadratic autocatalysis with linear decay

In this note we consider a reaction-diffusion problem which describes a simple model chemical reaction scheme for quadratic autocatalysis with linear decay. We show that withk>1 (wherek is a parameter measuring the relative strength of the decay step to the autocatalytic step) the dimensionless unreacting state 1, 0 is globally asymptotically stable, with 1+0(t ^–1/2) and 0(t ^–1/2 e ^–(k–1)t) ast-. Here and are the concentrations of the reactant and the autocatalyst respectively, andt is time. The casek<1 has been considered in detail by Merkin et al. [1].     

Semilinear Perturbations of Harmonic Spaces,Liouville Property and a Boundary Value Problem

Let (X,) be a P-harmonic Bauer space and let be a Borel measurable function on X×R satisfying conditions (A) through (D) of Section 2 (e.g., (x,t)=t|t|^–1 where >1). For every Kato family M of potential kernels on X let ^M U(X) denote the set of all real continuous functions on X such that u+K ^M _D (,u)(D) for every open relatively compact subset D of X. We study the existence of a non-trivial function in ^M U(X) which is dominated by a given positive harmonic function on X. If X is a domain of R ^d , is a positive Kato measure on X and L is a second-order differential operator in R ^d , we apply our study to derive a characterization of finite positive measures on the minimal Martin boundary ^M ₁ X for which the boundary value problem Lu=(,u) in X and u= on ^M ₁ X is solvable.     

The classification of compact punctally cohesive Desarguesian projective Klingenberg planes

It is shown that for a compact Desarguesian projective Klingenberg plane P with incidence structure P=#x2119;, , I and neighbour relation , where two distinct points always lie on some line, exactly one of the following holds: P is a non-discrete connected or totally disconnected ordinary projective plane with =id, P is a finite projective plane with =id, P is a finite projective Hjelmslev plane with id, or P is a non-discrete totally disconnected ordinary projective plane with id.Dedicated to H. Salzmann on his 60th birthdayThe author wishes to thank the Natural Sciences and Engineering Research Council of Canada (NSERC) for their financial assistance in the writing of this paper.     

Limits of certain iterates of Post-Widder operators

The paper is a study of the limiting behaviour of the [n t]-th iterates of the well-known Post-Widder operatorsL _{n, x} used in the real inversion of the Laplace transform. It is shown that the limiting operators constitute a semigroup T _t;t0 of class (C ₀) on a family C _,; , >0 of Banach spaces. Applications of the semigroup structure lead to a pointwise saturation theorem forL _{n, x} and a characterization of convex functions inC _, through an inequality involving the action ofL _{n, x}.     

The Dimension Function of Holomorphic Spaces of a Real Submanifold of an Almost Complex Manifold

Let M be a real submanifold of an almost complex manifold and let be the maximal holomorphic subspace, for each x M. We prove that c:M,c(x)=dim H _x is upper-semicontinuous.     

The intersection of complete geodesics on <Emphasis Type="Italic">S</Emphasis><Superscript>2</Superscript> with positive curvature

Let M be a Riemannian manifold. A complete geodesic on M means that :(-,+)M is a normalized geodesic. In this paper, we prove that on (S²,g) with positive curvature, any two complete geodesics must intersect an infinite number of times, and a complete geodesic must self-intersect an infinite number of times. Mathematics Subject Classification (2000) 53C40 (53C22)     

Jordan axioms for C*-algebras

A complex Banach spaceA which is also an associative algebra provided with a conjugate linear vector space involution * satisfying (a ²)^*=(a ^*)², aa ^* a=a³ and ab+ba2ab for alla, b inA is shown to be a C^*-algebra. The assumptions onA can be expressed in terms of the Jordan algebra obtained by symmetrization of the product ofA and are satisfied by any C^*-algebra. Thus we obtain a purely Jordan characterization of C^*-algebras.     

Smoothability of the Conformal Boundary of a Lorentz Surface Implies 'Global Smoothability'

In 1985, Kulkarni defined the conformal boundary of a simply connected and time-oriented Lorentzian surface . He also introduced a notion of 'smoothability' of this boundary, depending only on local properties of . In this paper we show that smoothability of is in fact a global property of . In doing so, we classify Lorentzian surfaces with smoothable boundaries up to conformal homeomorphism. To be specific, suppose that the Minkowski plane E ² ₁ is the x,y-plane with metric dxdy. Our main theorem states that if is smoothable then is conformally homeomorphic to the interior U of a Jordan curve in E ² ₁ that is locally the graph of a continuous function over either the x-axis or the y-axis at each point of U.     

The approach to normality of the distance-neighbour function when used to describe relative turbulent dispersion

Batchelor [1] suggested that the Distance-Neighbour Function when used to describe the relative turbulent dispersion of a cloud of marked fluid, whose diameter is well within the length scale range of the universal inertial sub-range of turbulence, would become of Gaussian form. This paper determines that a necessary condition for the Gaussian form to apply is that the non-dimensional time of (/v)^1/2 t 300, following the release of the marked fluid with an initial diameter <(v ³/t)^1/4, be attained.

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Zusammenfassung Batchelor [1] hat die Behauptung aufgestellt, dass die Abstands-Nachbar-Funktion (räumliche Korrelation) die Gauss'sche Form annimmt, wen man sie zur Beschreibung der relativen turbulenten Dispersion einer Wolke von markierter Flüssigkeit benützt, falls der Durchmesser der Wolke nicht über die Längenskala des universellen Inertial-Unterbereichs der Turbulenz hinausgeht. In dieser Arbeit wird gezeigt, dass eine notwendige Bedingung der Anwendbarkeit der Gauss'schen Form darin besteht, dass die dimensionsloze Zeit (/v)^1/2 t 300 nach dem Ablösen einer markierten Flüssigkeits-wolke mit dem Anfangsdurchmesser <(v ³/t)^1/4 erreicht wird.

     

Analytic discs in pseudoconvex submanifolds of of higher codimension

We prove that a small analytic disc A attached to a pseudoconvex submanifold M of and which shares a conormal with M at some boundary point, is in fact contained in M. The proof uses an argument of reduction to a hypersurface by a symplectic complex transformation. The result is classical in case codM=1 (cf. e.g. [4]); it is a consequence of Hopfs Lemma applied to a plurisubharmonic defining function of M as in [5]. It was already generalized to the higher codimension in [8] but under the additional assumption, in the present paper, that A has an analytic lift A* in T*X attached to T*_MX (i.e. A has defect 1 in the terminology of [6], [7]).     

A chord-stretching map of a convex loop is an isometry

Let ⁿ be n-dimensional Euclidean space, and let : [0, L] ⁿ and : [0, L] ⁿ be closed rectifiable arcs in ⁿ of the same total length L which are parametrized via their arc length. is said to be a chord-stretched version of if for each 0s tL, |(t)–(s)| |(t)–(s)|. is said to be convex if is simple and if ([0, L]) is the frontier of some plane convex set. Individual work by Professors G. Choquet and G. T. Sallee demonstrated that if were simple then there existed a convex chord-stretched version of . This result led Professor Yang Lu to conjecture that if were convex and were a chord-stretched version of then and would be congruent, i.e. any chord-stretching map of a convex arc is an isometry. Professor Yang Lu has proved this conjecture in the case where and are C ² curves. In this paper we prove the conjecture in general.