We characterize some major algorithmic randomness notions via differentiability of effective functions. (1) We show that a real number z ∈ [0, 1] is computably random if and only if every nondecreasing computable function [0, 1] → R is differentiable at z. (2) A real number z ∈ [0, 1] is weakly 2-random if and only if every almost everywhere differentiable computable function [0, 1] → R is diff...

The traditional concept of smoothness concerns differentiability, and we consider a function to be smooth if it is k-times differentiable. This concept is important in many aspects of numerical analysis, and the quality of reconstructions of signals from certain given observations depend on this information. For instance, a k-times differentiable function on the unit cube [0, 1] can be approxim...

The articles [17], [18], [3], [5], [4], [8], [2], [7], [11], [6], [16], [12], [19], [9], [10], [1], [14], [15], and [13] provide the notation and terminology for this paper. For simplicity, we use the following convention: k, n, m denote elements of N, X denotes a set, s1, s2 denote complex sequences, Y denotes a subset of C, f , f1, f2 denote partial functions from C to C, r denotes a real num...

Let F be a continuous function from an open subset D of a separable Banach space X into a Banach space Y. We show that if there is a dense G8 subset A of D and a Gs subset H of X whose closure has nonempty interior, such that for each a E A and each x E H the directional derivative DxF(a) of F at a in the direction x exists, then F is Giteaux differentiable on a dense G8 subset of D. If X is re...

Suppose an estimator sequence Tn. is asymptotically efficient for a. • smooth functional 1'1;. Then a Hadamard differentia.ble functional rf> applied to Tn is efficient for i f> 0 11':. More generally, Hadamard dif-ferentiability ma.y be used to esta.blish asymptotic efficiency. Three non-trivial applications are discussed: random right censoring, random truncation and estima.ting a. point symm...

The generalized fractional Brownian motion is a Gaussian self-similar process whose increments are not necessarily stationary. It appears in applications as the scaling limit of shot noise with power-law shape function and non-stationary noises variance function. In this paper, we study sample path properties motion, including Hölder continuity, differentiability/non-differentiability, function...

A basic problem in classifying, or searching for similar objects in, a large set of geometric objects is computing similarity between two objects. This has led to extensive work on computing geometric similarity between two objects. This talk discusses some old and some new geometric-similarity algorithms, with an emphasis on transportation distance (also called earth mover’s distance) and Frec...

This paper considers some path properties of real separable Gaussian processes ~ with parameter set an arbitrary interval. The following results are established, among others. At every fixed point the paths of ~ are continuous, or differentiable, with probability zero or one. If ~ is measurable, then with probability one its paths have essentially the same points .~ of continuity and differenti...

In the paper compressible, stationary Navier-Stokes equations are considered. A framework for analysis of such equations is established. In particular, the well-posedness for inhomogeneous boundary value problems of elliptic-hyperbolic type is shown. Analysis is performed for small perturbations of the so-called approximate solutions, i.e., the solutions take form (1.12). The approximate soluti...

In this paper, we study linear fuzzy time-variant optimal control systems using the generalized differentiability concept and we present the general form of optimal controls and states. Some examples are provided to illustrate our results.