Global Optimality Conditions in Maximizing a Convex Quadratic Function under Convex Quadratic Constraints

For the problem of maximizing a convex quadratic function under convex quadratic constraints, we derive conditions characterizing a globally optimal solution. The method consists in exploiting the global optimality conditions, expressed in terms of -subdifferentials of convex functions and -normal directions, to convex sets. By specializing the problem of maximizing a convex function over a convex set, we find explicit conditions for optimality.     

Unbounded convex mappings of the ball in

In this paper, we study univalent holomorphic mappings of the unit ball in that have the property that the image contains a line for some , . We show that under certain rather reasonable conditions, up to composition with a holomorphic automorphism of the ball, the mapping is an extension of the strip mapping in the plane to higher dimensions.

     

Generalized Variation of Mappings with Applications to Composition Operators and Multifunctions

We study (set-valued) mappings of bounded -variation defined on the compact interval I and taking values in metric or normed linear spaces X. We prove a new structural theorem for these mappings and extend Medvedev's criterion from real valued functions onto mappings with values in a reflexive Banach space, which permits us to establish an explicit integral formula for the -variation of a metric space valued mapping. We show that the linear span GV (I;X) of the set of all mappings of bounded -variation is automatically a Banach algebra provided X is a Banach algebra. If h:I× X Y is a given mapping and the composition operator is defined by (f)(t)=h(t,f(t)), where tI and f:I X, we show that :GV (I;X) GV (I;Y) is Lipschitzian if and only if h(t,x)=h₀(t)+h₁(t)x, tI, xX. This result is further extended to multivalued composition operators with values compact convex sets. We prove that any (not necessarily convex valued) multifunction of bounded -variation with respect to the Hausdorff metric, whose graph is compact, admits regular selections of bounded -variation.     

Remarks on Estimation Problem for Stationary Processes in Continuous Time

The problem of estimating of the law (in the space of the paths) and the common marginal distribution for a strictly stationary ergodic process X is discussed. We show, in particular, that:(1) The empirical measure

Necessary and sufficient conditions for the lipschitzian invertibility of nonlinear difference operators in the spacesl p (ℤ, ℝ) with 1≤p≤∞with 1≤p≤∞

Necessary and sufficient conditions for the Lipschitzian invertibility of the difference map

On the ε-entropy of classes of holomorphic functions

Suppose thatB _R ^d is a ball of radiusR in ℂ ^d and σ is the standard measure on the unit sphere in ℂ ^d . ForR>1, 1≤p≤∞, and for the natural numbersl, d, byH _R ⁰ (l, p, d) we denote the class of functionsf holomorphic inB _R ^d and such that in the homogeneous polynomial expansion of the firstl summands the zero and radial derivatives of orderl belong to the closed unit ball of the Hardy spaceH ^p (B _R ^d ). In this paper an asymptotic formula for the ε-entropy of the classH _R ⁰ (l, p, d) in the spacesL ^p (σ), 1≤p<∞, and is obtained. Translated fromMatematicheskie Zametki, Vol. 68, No. 2, pp. 286–293, August, 2000.     

Probability Density Function Estimation Using Gamma Kernels

We consider estimating density functions which have support on [0, ) using some gamma probability densities as kernels to replace the fixed and symmetric kernel used in the standard kernel density estimator. The gamma kernels are non-negative and have naturally varying shape. The gamma kernel estimators are free of boundary bias, non-negative and achieve the optimal rate of convergence for the mean integrated squared error. The variance of the gamma kernel estimators at a distance x away from the origin is O(n ^–4/5 x ^–1/2) indicating a smaller variance as x increases. Finite sample comparisons with other boundary bias free kernel estimators are made via simulation to evaluate the performance of the gamma kernel estimators.     

Testing for Unit Roots in a Nearly Nonstationary Spatial Autoregressive Process

The limiting distribution of the normalized periodogram ordinate is used to test for unit roots in the first-order autoregressive model _st= _s-1,t+_s,t-1- _s-1,t-1+_st. Moreover, for the sequence _n = e ^c/n , _n = e ^d/n of local Pitman-type alternatives, the limiting distribution of the normalized periodogram ordinate is shown to be a linear combination of two independent chi-square random variables whose coefficients depend on c and d. This result is used to tabulate the asymptotic power of a test for various values of c and d. A comparison is made between the periodogram test and a spatial domain test.     

On Nonlinear Integro-differential Operators in Generalized Orlicz–Sobolev Spaces

A nonlinear integral operator T of the form (Tf)(s)=∫_G K(t, f (σ(s, t))) dμ(t), for sG, is defined and investigated in the measure space (G, Σ, μ), where f and K are vector-valued functions with values in normed linear spaces E and F, respectively. The results are applied to the case of integro-differential operators in generalized Orlicz–Sobolev spaces. There are studied problems of existence, embeddings, and approximation by means of T.     

Extremal Solutions for Quasilinear Elliptic Inclusions in All of ℝN with State-Dependent Subdifferentials

This paper deals with quasilinear elliptic differential inclusions defined in all of ^N and governed in general by a nonpotential quasilinear elliptic operator of the Leray–Lions type and a multivalued term in form of a (nonmonotone) state-dependent subdifferential. We prove the existence of entire extremal solutions within a sector of an ordered pair of appropriately defined upper and lower solutions without imposing any condition at infinity. Therefore, standard variational methods cannot be applied here. Furthermore, due to the unboundedness of the domain and due to lack of monotonicity of the operators involved, no comparison results are available such that the problem under consideration becomes even more difficult.