Exponential integrability of sub-Gaussian vectors

Summary In this paper we define two classes of Banach space (B, ·)-valued random vectors called sub-Gaussian vectors and -sub-Gaussian vectors. The main purpose of this paper is to prove the exponential integrability of a sub-Gaussian vectorX, that is, for some >0, in the case whereB=L _p . On the other hand, using the arguments ofX. Fernique and M. Talagrand, we also show that the exponential integrability of a -sub-Gaussian vector in an arbitrary separable Banach space.These two definitions of sub-Gaussian vectors and -sub-Gaussian vectors are not comparable, and neither of these definitions is a necessary condition for the exponential integrability. We shall give illuminating examples.     

On convergence of types and processes in Euclidean space

Let be an Euclidean space; Y _n , Z, U random vectors in ; h _n , g _n affine transformations and let þ be a subgroup of the group G of all the in vertible affine transformations, closed relative to G. Suppose that g_n and where Z is nonsingular. The behaviour of _n = h _n g _n ^–1 as n is discussed first. The results are used then to prove that if for all t(0, ), where h _n þ and Z ₁ is nonsingular and nonsymmetric with respect to þ then H, for all t(0,) and is a continuous homomorphism of the multiplicative group of (0, ) into þ. The explicit forms of the possible are shown.     

Boundary problem with discontinuous translation for two functions which are analytic in domains of different connectivity

In the space L_p(), 1 < p < we consider the problem of finding functions and which are analytic, respectively, in a doubly connected domain with boundary and a simply connected domain with boundary ( and are closed Lyapunov contours) with respect to the boundary condition , where the orientation-preserving translation and the coefficients a, b, c, d admit discontinuities at certain points of . Under specific conditions on the limit values of the derivative of the translation at the points of discontinuity we get necessary and sufficient conditions for the problem to be Noetherian and a formula for calculating its index.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 2, pp. 262–266, February, 1990.     

An application of Shoenfield's absoluteness theorem to the theory of uniform distribution

IfC is a Polish probability space, a Borel set whose sectionsW _x ( have measure one and are decreasing , then we show that the set _x W _x has measure one. We give two proofs of this theorem—one in the language of set theory, the other in the language of probability theory, and we apply the theorem to a question on completely uniformly distributed sequences.Supported by DFG grant Ko 490/7-1.     

The compact support property for solutions to the heat equation with noise

Summary We consider all solutions of a martingale problem associated with the stochastic pde and show thatu(t,·) has compact support for allt0 ifu(0,·) does and if <1. This extends a result of T. Shiga who derived this compact support property for 1/2 and complements a result of C. Mueller who proved this property fails if 1.The author's research was supported by an NSF grant and an NSERC operating grantThe author's research was supported by an NSERC operating grant     

Minimal Surfaces with an Elastic Boundary

LetD:= { C ³ (

Skew-symmetric derivations of a complex Lie algebra

Let be a complex Lie algebra, its underlying real Lie algebra, a real form of and ·, · the euclidean product induced by the real part of an hermitian inner product on . Let aut be the Lie algebra of skew-symmetric derivations of . We give necessary and sufficient conditions to ensure that aut is composed of skew-hermitian derivations. As an application, we study holomorphy in large subgroups of isometries of Lie groups.     

Subexponential distributions and characterizations of related classes

LetL(),0, 0, denote the class of distributionsF satisfying

The Orders of Related Elements of a Finite Field

All finite fields q (q 2, 3, 4, 5, 7, 9, 13, 25, 121) contain a primitive element for which + 1/ is also primitive. All fields of square order q ² (q 3, 5) contain an element of order q + 1 for which + 1/ is a primitive element of the subfield q. These are unconditional versions of general asymptotic results.     

On a Problem of the Theory of Multiply Local Formations

We describe the -closed n-multiply local formations such that the lattice of all -closed n-multiply local formations between and is Boolean.     

Distoriton of area and conditioned Brownian motion

Summary In a simply connected planar domainD the expected lifetime of conditioned Brownian motion may be viewed as a function on the set of hyperbolic geodesics for the domain. We show that each hyperbolic geodesic induces a decomposition ofD into disjoint subregions and that the subregions are obtained in a natural way using Euclidean geometric quantities relating toD. The lifetime associated with on each _j is then shown to be bounded by the product of the diameter of the smallest ball containing _j and the diameter of the largest ball in _j . Because this quantity is never larger than, and in general is much smaller than, the area of the largest ball in _j it leads to finite lifetime estimates in a variety of domains of infinite area.Research of the first author was supported in part by NSF Grant DMS-9100811Research of the second author was supported in part by NSF Grant DMS-9105407     

Towards a model theory for 2-hyponormal operators

We introduce the notion ofweak subnormality, which generalizes subnormality in the sense that for the extension ofT we only require that hold forf ; in this case we call a partially normal extension ofT. After establishing some basic results about weak subnormality (including those dealing with the notion of minimal partially normal extension), we proceed to characterize weak subnormality for weighted shifts and to prove that 2-hyponormal weighted shifts are weakly subnormal. Let { _n } _n=0 be a weight sequence and letW denote the associated unilateral weighted shift on . IfW is 2-hyponormal thenW is weakly subnormal. Moreover, there exists a partially normal extension on such that (i) is hyponormal; (ii) ; and (iii) . In particular, if is strictly increasing then can be obtained as

Some asymptotic results related to the law of iterated logarithm for Brownian motion

For 0<<1, let . The questions addressed in this paper are motivated by a result due to Strassen: almost surely, lim sup _t U ((t))=1–exp{–4(^–1)–1}. We show that Strassen's result is closely related to a large deviations principle for the family of random variablesU (t), t>0. Also, when =1,U (t)0 almost surely and we obtain some bounds on the rate of convergence. Finally, we prove an analogous limit theorem for discounted averages of the form as 0, whereD is a suitable discount function. These results also hold for symmetric random walks.     

Scalar-quadratic stabilizability of the Petersen counterexample via a linear static controller

A new property called scalar-quadratic is presented for establishing the stabilizability of linear time-varyring uncertain systems. It is applied to a well-known linear time-varying system _OL which contains two uncertainties ₁(t) and ₂(t). Using the Lyapunov functionsV(x)=x ^T Px, whereP is a constant postitive-definite symmetric matrix, previous authors have shown that _OL is stabilizable by linear static controllers when the time-varying uncertainties are bounded by a normalized bound satisfying < 0.8. We extend the bound to < 1.0 by using the more general Lyapunov functions satisfying the scalar-quadratic propertyV(ax)=a ² V(x), aR, xR ₀ ² .Our proof uses a hexagon as a closed, convex hypersuface to construct a scalar-quadratic Lyapunov function, so that the Lyapunov time derivative satisfies the quadratic convergence condition , >0, for the closed-loop system _CL formed from _OL and a stabilizing linear static controller. The critical condition in the proof of the quaratic convergence ondition is the satisfaction of the inequality , where _max is a normalization bound for ₁(t) and ₂(t) and wheree ₁ ande ₂ are parameters for the controller. For the controller parametrized bye ₁=8 ande ₂=20, this inequality reduces to _max < 2.2096. This result, in particular, establishes that the Petersen counterexample is stabilitzable by the linear static controller withe ₁=8 ande ₂=20. Furthermore, it establishes the amazing result that _OL is stabilizable by a linear static controlle on any compact subset of the constant uncertainaty controllability space defined by ₁>0 and ₂>0.     

A representation theorem for weak automorphisms of a universal algebra

Given a group G and a descending chainG ₀,G ₁,...,G _n, of normal subgroups ofG, we prove that there exists a universal algebra , such that the chain ...W_n( )...W₁( }) W₀( )W( ) is isomorphic to the chain ...G _n ...G ₁G ₀G, where W( ) is the group of weak automorphisms of , and W_n( ) is the group of weak automorphisms of that leaves alln-ary operations fixed.We also prove that there are an infinite number of non-isomorphic algebras that satisfy the above.These results are a generalization of those proved by J. Sichler, in the special case when G=G₀, and G₁=G₂=...=G_n=....Presented by J. Mycielski.This paper comprises part of the author's doctoral dissertation at the University of Notre Dame in 1983. The author wishes to express her deep gratitude to Professor Abraham Goetz for suggesting this problem, for being extremely generous with his time and experience, and for giving her his constant encouragement. The author also thanks the reviewer for his helpful comments.     

Impossibility of a Certain Cyclotomic Equation with Applicationsto Difference Sets

Under certain conditions, we show the nonexistence ofan element in the p-th cyclotomicfield over , that satisfies . As applications, we establish the nonexistence ofsome difference sets and affine difference sets.     

Some Generalizations of the Notion of Length

Two numerical characteristics of a nonrectifiable arc generalizing the notion of length are introduced. Geometrically, this notion can naturally be generalized as the least upper bound of the sums , where are the lengths of segments of a polygonal line inscribed in the curve and is a given function. On the other hand, the length of is the norm of the functional in the space ; its norms in other spaces can be considered as analytical generalizations of length. In this paper, we establish conditions under which the generalized geometric rectifiability of a curve implies its generalized analytic rectifiability.     

Krümmungskennzeichnung der Ketten einer allgemeinen Geometrie Σ(ℜ,\mathfrak{L})

Lie theory renders a concept of curvature for the geometries (,) introduced by W. Benz [1]. Among regular curves in the chains of (, ) are characterized by =0.     

Estimates of the Kolmogorov Widths of Classes of Analytic Functions Representable by Cauchy-Type Integrals. I

In the Banach space of functions analytic in a Jordan domain , we establish order estimates for the Kolmogorov widths of certain classes of functions that can be represented in by Cauchy-type integrals along the rectifiable curve = and can be analytically continued to or to .     

On the Growth of the Denominators of Convergents

Let log . We prove that there exist non-denumerably many pairwise not equivalent irrational numbers such that and where qn() denotes the denominator of the nth convergent of .