On monotone extensions of boundary data

Summary A functionf C (), is called monotone on if for anyx, y the relation x – y ₊ ^s impliesf(x)f(y). Given a domain with a continuous boundary and given any monotone functionf on we are concerned with the existence and regularity ofmonotone extensions i.e., of functionsF which are monotone on all of and agree withf on . In particular, we show that there is no linear mapping that is capable of producing a monotone extension to arbitrarily given monotone boundary data. Three nonlinear methods for constructing monotone extensions are then presented. Two of these constructions, however, have the common drawback that regardless of how smooth the boundary data may be, the resulting extensions will, in general, only be Lipschitz continuous. This leads us to consider a third and more involved monotonicity preserving extension scheme to prove that, when is the unit square [0, 1]² in ², strictly monotone analytic boundary data admit a monotone analytic extension.Research supported by NSF Grant 8922154Research supported by DARPA: AFOSR #90-0323     

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     

A Generalization of the Hausdorff-Young Theorem

Considering mixed-norm sequence spaces lp,q, p, q 1, C. N. Kellogg proved the following theorem: if 1 < p 2 then lp,2 and lp,2 , where 1/p + 1/p = 1. This result extends the Hausdorff-Young Theorem.We introduce here multiple mixed-norm sequence spaces , examine their properties and characterize the multipliers of spaces of the form lp,[s;n],q, with the index s repeated n times. By an interpolation-type argument we prove that (l,[2;n],2, lp,[1;n],1) for 1 < p 2. Using these results we obtain a further generalization of the Hausdorff-Young Theorem: if 1 < p 2 then lp,[2;n] and lp,[2;n] for each n = 0, 1, 2, ¨. The spaces lp,[2;n] decrease and lp,[2;n] increase properly with n for 1 < p < 2 and 1/p + 1/p = 1. We also extend a theorem of J. H. Hedlund on multiplers of Hardy spaces and deduce other results.     

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     

Interpolation theorems for a family of spanning subgraphs

Let G be a graph with order p, size q and component number . For each i between p – and q, let be the family of spanning i-edge subgraphs of G with exactly components. For an integer-valued graphical invariant if H H is an adjacent edge transformation (AET) implies |(H)-(H^')|1 then is said to be continuous with respect to AET. Similarly define the continuity of with respect to simple edge transformation (SET). Let M _j() and m _j() be the invariants defined by . It is proved that both M _p–() and m _p–(;) interpolate over , if is continuous with respect to AET, and that M _j() and m _j() interpolate over , if is continuous with respect to SET. In this way a lot of known interpolation results, including a theorem due to Schuster etc., are generalized.     

Conformally Invariant Regularization and Skeleton Expansions in Gauge Theory

We consider a conformally invariant regularization of an Abelian gauge theory in an Euclidean space of even dimension D 4 and regularized skeleton expansions for vertices and higher Green's functions. We set the respective regularized fields and with the scaling dimensions and into correspondence to the gauge field A and Euclidean current j . We postulate special rules for the limiting transition 0. These rules are different for the transversal and longitudinal components of the field and the current . We show that in the limit 0, there appear conformally invariant fields A and j each of which is transformed by a direct sum of two irreducible representations of the conformal group. Removing the regularization, we obtain a well-defined skeleton theory constructed from conformal two- and three-point correlation functions. We consider skeleton equations on the transversal component of the vertex operator and of the spinor propagator in conformal quantum electrodynamics. For simplicity, we restrict the consideration to an Abelian gauge field A , but generalization to a non-Abelian theory is straightforward.     

Generalized Quasi-Variational Inequalities Without Continuities

Given a nonempty set and two multifunctions , we consider the following generalized quasi-variational inequality problem associated with X, : Find such that . We prove several existence results in which the multifunction is not supposed to have any continuity property. Among others, we extend the results obtained in Ref. 1 for the case (x(X.     

Extrapolation of transformations of random processes perturbed by white noise

We consider the problem of linear mean square optimal estimation of transformation of a stationary random process (t) in observations of process (t) + n(t) for t < – 0, where (t) is white noise uncorrelated with (t). We find least favorable spectral densities f₀() D and minimax (robust) spectral characteristics of an optimal estimator of transformation A for various classesD of densities.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 2, pp. 216–223, February, 1991.     

Almost Fredholm operators in von Neumann algebras

LetA be a von Neumann algebra,J be the ideal of compact operators relative toA and letF ₊ be the left-Fredholm class ofA. We call almost left-Fredholm the class = {A A: if P A is a projection and AP J then P J}. Then and the inclusion is proper unlessA is semifinite and has a non-large center. satisfies all of the algebraic properties ofF ₊ but it is generally not open. IfA is semifinite then A iff there are central projectionsG with G = I such that AG F₊(AG). Let :A A/J. Then the left almost essential spectrum ofA A, , coincides with the set of eigenvalues of (A)     

Das System Totalpolarer und eine Kennzeichnung des Winkels von Hyperebenen in Cayley/Klein-Räumen

In this paper at first we introduce thesystem of total polar subspaces of an arbitrary k-diniensional plane with respect to the absolute (configuration) of an arbitrary n- dimensional CAYLEY/KLEIN space as a generalization of the total polar set of a regular k-plane of . Using the system of total polar sets of the intersection (n–2)-plane of two hyperplanes and we give the followingnew characterization of the angle of . and : for any straight line g with and g=Ø the angle of and is equal to the distance of the two intersection points of g with and .

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Herrn Prof. Dr.Oswald Giering zum 60. Geburtstag gewidmet     

On flag-transitive affine planes of orderq 3

Let be a translation plane of orderq ³,q an odd prime power, whose kern GF(q). Letl be the line at infinity of . LetG be a solvable collineation group of in the linear translation complement, which acts transitively onl , and letH be a maximal normal cyclic subgroup ofG. Then the restriction ofH onl acts semiregularly onl and {1, 2, 3, 6}, where is the restriction ofG onl (ifq –1(mod 3), then {1, 2}). Ifq {3, 5} and {1, 2}, then is determined completely, using a computer.     

Testing Extreme Value Models

We consider the full statistical families of extreme value distributions and generalized Pareto distributions , where , and denote the shape, scale and location parameters, respectively. We consider the testing problems against and = 0 against 0, where and are treated as nuisance parameters. Showing local asymptotic normality (LAN), we derive asymptotic envelope power functions for test sequences and establish tests which attain these upper bounds. The finite sample size behavior is studied by simulations.     

An open mapping theorem for measures

LetX andY be Hausdorff spaces and denote byM (X) andM (Y) the corresponding spaces of finite and non-negative Borel measures, endowed with the weak topology. A Borel map :XY induces the map :M (X)M (Y). We give necessary and sufficient conditions for to be open. In case of being a surjection between Suslin spaces, is open if and only if is.     

On regularities of the distribution of special sequences

Ifp2 is an integer, then every nonnegative integerk is represented by an expression of the form with integersa _i (k), 0a _i (k)p–1,i=0.1,...,s. The radical-inverse function to the basep, _p (k), is defined by . The sequence is uniformly distributed modulo 1 (it may be called a one-dimensional Halton sequence). In the casep=2 it is the van der Corput sequence. The set of all numbers (0, 1] such that the local discrepancy is bounded inn is determined.     

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 .     

Optimal conditioning in the convex class of rank two updates

Davidon's new quasi-Newton optimization algorithm selects the new inverse Hessian approximation at each step to be the optimally conditioned member of a certain one-parameter class of rank two updates to the last inverse Hessian approximationH. In this paper we show that virtually the same goals of conditioning can be achieved while restricting to the convex class of updates, which are bounded by the popular DFP and BFGS updates. This suggests the computational testing of alternatives to the optimal conditioning strategy.This research supported by NSF grant 73-03413, contract P04361 of the National Bureau of Economic Research, Cambridge, Massachusetts, and a National Science Foundation Graduate Fellowship, forms a portion of the author's doctoral thesis at Cornell University directed by Professor J.E. Dennis.     

On the exactness of a theorem of Mazur and Orlicz

For a preassigned unbounded sequence {S_n} of complex numbers, and preassigned complex numbers z₁ and z₂z₁ we construct: 1) regular matrices A=a_nk and B=b_nk such that the same bounded sequences are summable by these matrices and that , and ; 2) regular matrices A⁽¹⁾)=a _nk ⁽¹⁾ and B⁽¹⁾=b _nk ⁽¹⁾ such that B⁽¹⁾ A⁽¹⁾, and, . Our results show that the well known theorem of MazurOrlicz on the bounded consistency of two regular matrices, one of which is boundedly stronger than the other, is exact.Translated from Matematicheskie Zametki, Vol. 11, No. 4, pp. 431–436, April, 1972.     

Deterministic approximations of probability inequalities

A simple general framework for derivingexplicit deterministic approximations of probability inequalities of the formP(a) is presented. These approximations are based on limited parametric information about the involved random variables (such as their mean, variance, range or upper bound values). First the case of a single random variable is analysed, followed by the cases of independent and dependent summands . As examples of possible applications, a stochastic extension of the knapsack problem and the stochastic linear programming problem with separate chance-constraints are investigated: we provide approximate deterministic surrogates for these problems.

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Zusammenfassung Es wird ein Rahmen zur Ableitung expliziter deterministischer Approximation für Wahrscheinlichkeitsungleichungen der FormP(a) angegeben. Diese Approximationen basieren auf begrenzter parametrischer Information über die beteiligten Zufallsvariablen (wie ihr Erwartungswert, Varianz, Wertebereich oder obere Schranken). Zuerst wird der Fail einer Zufallsvariablen analysiert, sodann werden Summen von unabhängigen Summanden betrachtet. Als Beispiele für mögliche Anwendungen wird eine stochastische Erweiterung des Rucksack-problems untersucht sowie stochastische lineare Programme mit separablen Wahrscheinlichkeitsrestriktionen. Für diese Probleme werden näherungsweise deterministische Ersatzprobleme angegeben.

     

Polynomial Filtrations and Lannes'' T Functor

This paper defines and studies the polynomial filtration [p_k ] of the shift functor : F , where F is the category of functors between F-vector spaces over a finite field F. The functors [p_k ] correspond to a system of functors (p_k T):U U, related to Lannes'T-functor on the category U of unstable modules over the Steenrod algebra. The main results concern the behaviour of the quotients ~ _s:=~/[p_s-1~ filtrations by ~_s-nilpotent functors are introduced and it is shown that the full subcategory of _s-nilpotent functors is thick.