Ein globales Ergebnis für Bifurkationsaufgaben mit schwach differenzierbaren Operatoren

In this paper we are dealing with positive linear functionals on W-algebras. We introduce the notion of a positive linear functional with ∑-property (see Definition 1.1). It is shown that each positive linear functional on a W-algebra possesses the ∑-property. This fact allows to give a short proof of UHLMANN's conjecture on unitary mixing in the state space of a W-algebra. In proving our main theorem (see Theorem 1.2.) we obtain some results on positive linear functionals and orthoprojections which are useful in other context, too.     

On the optimal coding of one class of vector functions

We consider the problem of coding of parametrically defined vector functions by continuous functionals which act on their coordinate functions. We obtain the optimal method for coding some class of vector functions. In the linear case, we show the relation between this method and the informativeness of functionals with respect to the class of coordinate functions. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 12, pp. 1638–1645, December, 1997.     

Tractability of approximating multivariate linear functionals

We review selected tractability results for approximating linear tensor product functionals defined over reproducing kernel Hilbert spaces. This review is based on Volume II of our book Tractability of Multivariate Problems. In particular, we show that all nontrivial linear tensor product functionals defined over a standard tensor product unweighted Sobolev space suffer the curse of dimensionality and therefore they are intractable. To vanquish the curse of dimensionality we need to consider weighted spaces, in which all groups of variables are monitored by weights. We restrict ourselves to product weights and provide necessary and sufficient conditions on these weights to obtain various kinds of tractability.     

Computation of certain continuous integrals in the space of continuous functions defined on a compact set

We give formulas for computing integrals of functionals that are functions of linear and quadratic functionals with respect to generalized Wiener measure in the sense of J. Kuelbs in the space of continuous functions defined on a compact set.Translated fromMatematicheskie Melody i Fiziko-Mekhanicheskie Polya, Issue 34, 1991, pp. 35–39.     

Nondegeneracy of elliptic problems with periodic nonlinearities in the plane

A class of asymptotically quadratic functionals on Hilbert spaces, called degenerate, is considered and explored. Our results are applied to obtain a extension, in the planar case, of a result published by Solimini in ‘On the solvability of some elliptic partial differential equations with the linear part at resonance’, J. Math. Anal. Appl., 117 (1986), 138-152. Similar extensions had been previously studied in the literature only for domains with particular geometries.     

Informativeness of functionals

We introduce the concept of informativeness of a continuous functional on a metric spaceX with respect to a setMX and a metric x. We pose the problem of finding the most informative functional. For some sets of continuous functions, this problem is solved by reduction to a subset of functionals given by the value of a function at a certain point.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 9, pp. 1156–1163, September, 1994.     

A note on a P ≠ NP result for a restricted class of real machines

For a class of real machines using only linear functionals as computations and equality-tests, we show that the classes P and NP differ.     

Linear differential equations in <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript> with radial derivatives

We obtain a solution of a linear differential equation with a radial derivative. The coefficients and the solution are functionals on L ₂[a, b]. In the same space, we study the properties of solutions of second-order linear homogeneous equations.     

Tukey's linear sensitivity and order statistics

Tukey (1965,Proc. Nat. Acad. Sci. U.S.A.,53, 127–134) introducedlinear sensitivity as a measure of informativeness in a collection of order statistics. Here we study its general properties and discuss how it is related to the best linear unbiased estimator, Fisher information measure, and asymptotic relative efficiency. Also, we obtain explicit and asymptotic expressions for the linear sensitivity of a collection of consecutive order statistics from a location or from a scale family, and discuss its role in the comparison ofL-estimators. We conclude our discussion with examples from uniform, exponential and normal populations.     

A remark on discontinuous translation invariant functionals onL p (G) for certain compact groupsG

We show some criterion for the existence of discontinuous translation invariant linear functionals onC (G) andL _p (G) for a compact groupG.     

Review of Some Functionals of Compound Poisson Processes and Related Stopping Times

The paper reviews recent results of D. Perry, W. Stadje and S. Zacks, on functionals of stopping times and the associated compound Poisson process with lower and upper linear boundaries. In particular, formulae of these functionals are explicitly developed for the total expected discounted cost of discarded service in an M/G/1 queue with restricted accessibility; for the expected total discounted waiting cost in an M/G/1 restricted queue; for the shortage, holding and clearing costs in an inventory system with continuous input; for the risk in sequential estimation and for the transform of the busy period when the upper boundary is random.      

On the power of standard information for <Emphasis Type="Italic">L</Emphasis><Subscript>∞</Subscript> approximation in the randomized setting

We study approximation of multivariate functions from a general separable reproducing kernel Hilbert space in the randomized setting with the error measured in the L _∞ norm. We consider algorithms that use standard information consisting of function values or general linear information consisting of arbitrary linear functionals. The power of standard or linear information is defined as, roughly speaking, the optimal rate of convergence of algorithms using n function values or linear functionals. We prove under certain assumptions that the power of standard information in the randomized setting is at least equal to the power of linear information in the worst case setting, and that the powers of linear and standard information in the randomized setting differ at most by 1/2. These assumptions are satisfied for spaces with weighted Korobov and Wiener reproducing kernels. For the Wiener case, the parameters in these assumptions are prohibitively large, and therefore we also present less restrictive assumptions and obtain other bounds on the power of standard information. Finally, we study tractability, which means that we want to guarantee that the errors depend at most polynomially on the number of variables and tend to zero polynomially in n ⁻¹ when n function values are used.     

On Multifunctionals in the Musielak-Orlicz Spaces of Multifunctions

We introduce the notions of X_d,s(x),φ-linear multifunctional and X_d,s(x),φ-linear continuous multifunctional. We generalize the theorems on linear bounded functionals in the Musielak-Orlicz space.     

Measurable Functionals and Finitely Absolutely Continuous Measures on Banach Spaces

We consider the structure of orthogonal polynomials in the space L ₂(B, ) for a probability measure on a Banach space B. These polynomials are described in terms of Hilbert–Schmidt kernels on the space of square-integrable linear functionals. We study the properties of functionals of this sort. Certain probability measures are regarded as generalized functionals on the space (B, ).     

Necessity of a Wiener type condition for boundary regularity of quasiminimizers and nonlinear elliptic equations

We use variational methods to obtain a pointwise estimate near a boundary point for quasisubminimizers of the p-energy integral and other integral functionals in doubling metric measure spaces admitting a p-Poincaré inequality. It implies a Wiener type condition necessary for boundary regularity for p-harmonic functions on metric spaces, as well as for (quasi)minimizers of various integral functionals and solutions of nonlinear elliptic equations on R ⁿ .     

Numerical integration based on bivariate quadratic spline quasi-interpolants on bounded domains

In this paper we generate and study new cubature formulas based on spline quasi-interpolants defined as linear combinations of C ¹ bivariate quadratic B-splines on a rectangular domain Ω, endowed with a non-uniform criss-cross triangulation, with discrete linear functionals as coefficients. Such B-splines have their supports contained in Ω and there is no data point outside this domain. Numerical results illustrate the methods.     

Characterization of the D w -Laguerre-Hahn Functionals

We give some characterization theorems for the D w -Laguerre-Hahn linear functionals and we extend the concept of the class of the usual Laguerre-Hahn functionals to the D w -Laguerre-Hahn functionals, recovering the classic results when w tends to zero. Moreover, we show that some transformations carried out on the D w -Laguerre-Hahn linear functionals lead to new D w -Laguerre-Hahn linear functionals. Finally, we analyze the class of the resulting functionals and we give some applications relative to the first associated Charlier, Meixner, Krawtchouk and Hahn orthogonal polynomials.     

Parameter-dependent <Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> Control for Time-varying Delay Polytopic Systems

This paper addresses the robust stabilization and H _∞ control problem for a class of linear polytopic systems with continuously distributed delays. The control objective is to design a robust H _∞ controller that satisfies some exponential stability constraints on the closed-loop poles. Using improved parameter-dependent Lyapunov Krasovskii functionals, new delay-dependent conditions for the robust H _∞ control are established in terms of linear matrix inequalities.     

Minimum asymptotic error of algorithms for solving ODE

We deal with algorithms for solving systems z′(x) = f(x, z(x)), x ε [0, c], z(0) = η where f has r continuous bounded derivatives in [0, c) × ^s. We consider algorithms whose sole dependence on f is through the values of n linear continuous functionals at f. We show that if these functionals are defined by partial derivatives off then, roughly speaking, the error of an algorithm (for a fixed f) cannot converge to zero faster than n^−r as n → +∞. This minimal error is achieved by the Taylor algorithm. If arbitrary linear continuous functionals are allowed, then the error cannot converge to zero faster than n^−(r+1) as n → +∞. This minimal error is achieved by the Taylor-integral algorithm which uses integrals of f.     

-convergence of quadratic functionals with oscillating linear perturbations

We explore the Γ-convergence of quadratic functionals perturbed by linear terms with oscillatory coefficients. In particular, we are interested in understanding the interaction between the oscillations of the quadratic part and the linear term. We also look at this issue from the perspective of the associated equations of optimality as in homogenization.