Extensions of distance preserving mappings in euclidean and hyperbolic geometry

Suppose that X is a real inner product space of (finite or infinite) dimension at least 2. A distance preserving mapping , where is a (finite or infinite) subset of a finite-dimensional subspace of X, can be extended to an isometry of X. This holds true for euclidean as well as for hyperbolic geometry. To both geometries there exist examples of non-extentable distance preserving , where S is not contained in a finite-dimensional subspace of X.     

Strong topologies on vector-valued function spaces

Let be a real Banach space and let E be an ideal of L ⁰ over a -finite measure space (, , ). Let (X) be the space of all strongly -measurable functions f: X such that the scalar function , defined by , belongs to E. The paper deals with strong topologies on E(X). In particular, the strong topology the order continuous dual of E(X)) is examined. We generalize earlier results of [PC] and [FPS] concerning the strong topologies.     

Subdifferentials with respect to dualities

LetX andW be two sets and: ¯R^X ¯R^W a duality (i.e., a mapping such that for all and all index setsI). We introduce and study the subdifferential of a function at a pointx _o X, with respect to. We also consider the particular cases when is a (Fenchel-Moreau) conjugation, or a -duality, or a -duality, in the sense of [8].     

Continuity of Stochastic Convolutions

Let B be a Brownian motion, and let be the space of all continuous periodic functions f with period 1. It is shown that the set of all f such that the stochastic convolution does not have a modification with bounded trajectories, and consequently does not have a continuous modification, is of the second Baire category.     

Some remarks on the product of two C α-compact subsets

For a cardinal , we say that a subset B of a space X is C -compact in X if for every continuous function is a compact subset of . If B is a C-compact subset of a space X, then (B, X) denotes the degree of C -compactness of B in X. A space X is called -pseudocompact if X is C -compact into itself. For each cardinal , we give an example of an -pseudocompact space X such that X × X is not pseudocompact: this answers a question posed by T. Retta in Some cardinal generalizations of pseudocompactness Czechoslovak Math. J. 43 (1993), 385–390. The boundedness of the product of two bounded subsets is studied in some particular cases. A version of the classical Glicksberg's Theorem on the pseudocompactness of the product of two spaces is given in the context of boundedness. This theorem is applied to several particular cases.     

An existence theorem for generalized quasi-variational inequalities

In this paper, we deal with the following generalized quasi-variational inequality problem: given a closed convex subsetX ⁿ , a multifunction :X 2 ⁿ and a multifunction :X 2 ^X , find a point ( ) X × ⁿ such that We prove an existence theorem in which, in particular, the multifunction is not supposed to be upper semicontinuous.     

On the Lipschitz Constant for a Nonisometric Bi-Lipschitz Transformation of a Cantor Set

A bi-Lipschitz continuous mapping of a space X is a bijection such that , where . We write if f is a Lipschitz (bi-Lipschitz) mapping of X into itself and denote by the set of all bi-Lipschitz mappings of X that are not isometry. Thus, if and blip . For X we consider a standard Cantor set K on the real line (with standard metric). The main result of this paper is formulated as follows: where Bibliography: 2 titles.     

QUADRATIC ESTIMATORS OF QUADRATIC FUNCTIONS WITH PARAMETERS IN NORMAL LINEAR MODELS

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Rates of Decay and h-Processes for One Dimensional Diffusions Conditioned on Non-Absorption

Let (X _t ) be a one dimensional diffusion corresponding to the operator , starting from x>0 and T ₀ be the hitting time of 0. Consider the family of positive solutions of the equation with (0, ), where . We show that the distribution of the h-process induced by any such is , for a suitable sequence of stopping times (S _M : M0) related to which converges to with M. We also give analytical conditions for , where is the smallest point of increase of the spectral measure associated to .     

Obstructions to the Section Problem in a Fibration with a Weak Formal Base

For a class of Serre fibratations with a weak formal base X (or with a degenerated -algebra structure on the integral cohomology H*(X)), obstructions are defined by means of spherical twisting cochains of . In particular, for a given section on n-skeleton of X, the problem of avoiding the (n+1)th obstruction to the existence of a section on X ⁿ⁺¹ reduces to solving a system of linear equations with respect to cohomology elements of the groups Homotopy classification theorems for sections as well as for weak formal maps are given, too.     

A New Plethysm Formula for Symmetric Functions

This paper gives a new formula for the plethysm of power-sum symmetric functions and Schur symmetric functions with one part. The form of the main result is that for b,

An Extremal Limit Theorem for the Argmax Process of Brownian Motion Minus a Parabolic Drift

We study the extremal behavior of the stationary processes and , on increasing intervals [0,T], as , where V(t) is the location of the maximum of standard two-sided Brownian motion minus a parabolic drift. The result can be applied to the asymptotic behavior of the -risk of several nonparametric maximum likelihood estimators.     

Stability of Frobenius pull-backs of tangent bundles and generic injectivity of Gauss maps in positive characteristic

For a smooth projective variety X of dimension n in a projective space defined over an algebraically closed field k, the Gauss mapis a morphism from X to the Grassmannian of n-plans in sending to the embedded tangent space .The purpose of this paper is to prove the generic injectivity of Gauss mapsin positive characteristic for two cases; (1) weighted complete intersectionsof dimension of general type; (2) surfaces or 3-folds with -semistable tangent bundles; based on a criterion of Kaji by looking atthe stability of Frobenius pull-backs of their tangent bundles. The first result implies that a conjecture of Kleiman--Piene is true in case X is of general type of dimension . The second result is a generalization of the injectivity for curves.     

On a joint generalization of Cauchy's and d'Alembert's functional equations

LetX be a locally compact abelian group and let a compact groupG act onX. We find solutionsf: X inL(X) of the following functional equation:

A classification of martingale Hardy spaces associated with rearrangement-invariant function spaces

Let X be a rearrangement-invariant Banach function space over a complete probability space , and denote by the Hardy space consisting of all martingales such that . We prove that implies for any filtration if and only if Doobs inequality holds in X, where denotes the martingale defined by , n = 0, 1, 2, ..., and a.s.Received: 1 August 2000     

Invariance principles for random walks on hypergroups on ℝ2 and ℕ

Let (X _n:n) be i.i.d. with finite variance and values in a hypergroupK:=₊ or and _j=1 ⁿ X _j be the randomized sum of these random variables. It is shown that the processes converge in distribution to a Gaussian process in the caseK=₊, that the processes converge towards a Bessel process on ₊ in the case of polynomial growth of the hypergroupK=₊ or , and that in the case of exponential growth converges towards a Brownian motion asn.     

An Optimal Way of Moving a Sequence of Points onto a Curve in Two Dimensions

Let (t), 0 t T, be a smooth curve and let _i , i = 1, 2, , n, be a sequence of points in two dimensions. An algorithm is given that calculates the parameters t_i, i = 1, 2, , n, that minimize the function max{ _i – (t_i) ₂ : i = 1, 2, , n } subject to the constraints 0 t₁ t₂ t_n T. Further, the final value of the objective function is best lexicographically, when the distances _i – (t_i)₂, i = 1, 2, , n, are sorted into decreasing order. The algorithm finds the global solution to this calculation. Usually the magnitude of the total work is only about n when the number of data points is large. The efficiency comes from techniques that use bounds on the final values of the parameters to split the original problem into calculations that have fewer variables. The splitting techniques are analysed, the algorithm is described, and some numerical results are presented and discussed.     

Robust tests in group sequential analysis: One- and two-sided hypotheses in the linear model

Consider the linear modelY=X+E in the usual matrix notation where the errors are independent and identically distributed. We develop robust tests for a large class of one- and two-sided hypotheses about when the data are obtained and tests are carried out according to a group sequential design. To illustrate the nature of the main results, let and be anM- and the least squares estimator of respectively which are asymptotically normal about with covariance matrices ²(X ^t X)^–1 and ²(X ^t X)^–1 respectively. Let the Wald-type statistics based on and be denoted byRW andW respectively. It is shown thatRW andW have the same asymptotic null distributions; here the limit is taken with the number of groups fixed but the numbers of observations in the groups increase proportionately. Our main result is that the asymptotic Pitman efficiency ofRW relative toW is (²/²). Thus, the asymptotic efficiency-robustness properties of relative to translate to asymptotic power-robustness ofRW relative toW. Clearly, this is an attractive result since we already have a large literature which shows that is efficiency-robust compared to . The results of a simulation study show that with realistic sample sizes,RW is likely to have almost as much power asW for normal errors, and substantially more power if the errors have long tails. The simulation results also illustrate the advantages of group sequential designs compared to a fixed sample design, in terms of sample size requirements to achieve a specified power.     

Kolmogorov equation and large-time behaviour for fractional brownian motion driven linear sde’s

We consider a stochastic process X _t ^x which solves an equation

A Constructive Proof of the Generalized Gelfand Isomorphism

Using an analog of the classical Frobenius recursion, we define the notion of a Frobenius -homomorphism. For , this is an ordinary ring homomorphism. We give a constructive proof of the following theorem. Let X be a compact Hausdorff space, the th symmetric power of X, and the algebra of continuous complex-valued functions on X with the sup-norm; then the evaluation map defined by the formula identifies the space with the space of all Frobenius -homomorphisms of the algebra into with the weak topology.