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...

In this paper we consider how to use the convolution method to construct approximations, which consist of fuzzy numbers sequences with good properties, for a general fuzzy number. It shows that this convolution method can generate differentiable approximations in finite steps for fuzzy numbers which have finite nondifferentiable points. In the previous work, this convolution method only can be ...

It is known that the variational inequality problem (VIP) can be converted to a differentiable unconstrained optimization problem via a merit function first considered by Peng and later studied further by Yamashita, Taji and Fukushima. This merit function, called the D-gap function, though is differentiable, is not twice differentiable and its generalized Hessian with existing definitions is ve...

On utilizing the spectral representation of selfadjoint operators in Hilbert spaces, some error bounds in approximating $n$-time differentiable functions of selfadjoint operators in Hilbert Spaces via a Taylor's type expansion are given.

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

Any spectral function can be written as a composition of a symmetric function f : IR 7→ IR and the eigenvalue function λ(·) : S 7→ IR, often denoted by (f ◦ λ), where S is the subspace of n×n symmetric matrices. In this paper, we present some nonsmooth analysis for such spectral functions. Our main results are: (a) (f ◦ λ) is directionally differentiable if f is semidifferentiable; (b) (f ◦ λ) ...

While conventional computational differentiation based on the forward or reverse modes allows highly accurate computation of derivatives, there are situations where these modes fail to produce the values of derivatives, although the underlying function is differentiable. Typical examples of this phenomenon are connected to the occurrence of branch points in coding as in IF-ELSE structures as we...

Owing to the importance of non-differentiable point in a function for economy, engineering and theoretical analysis, this paper brings forward a novel methodology to figure out the non-differentiable point of function which is based on adjusting the models for global extremum and local extremum of particle swarm optimazation (PSO). The algorithm takes the difference between left and right diffe...

‎the purpose of this paper is to introduce a new mapping for a finite‎ ‎family of accretive operators and introduce an iterative algorithm‎ ‎for finding a common zero of a finite family of accretive operators‎ ‎in a real reflexive strictly convex banach space which has a‎ ‎uniformly g^ateaux differentiable norm and admits the duality‎ ‎mapping $j_{varphi}$‎, ‎where $varphi$ is a gauge function ...

n this paper, we consider continuity properties(especially, regularity, also viewed as an approximation property) for $%mathcal{p}_{0}(x)$-valued set multifunctions ($x$ being a linear,topological space), in order to obtain egoroff and lusin type theorems forset multifunctions in the vietoris hypertopology. some mathematicalapplications are established and several physical implications of thema...