In 1933 S. Mazur [4] proved the following Theorem 1. Let (X, ·) be a separable real Banach space. Let f be a real-valued convex continuous function defined on an open convex subset Ω ⊂ X. Then there is a subset A ⊂ Ω of the first category such that f is Gateaux differentiable on Ω \ A. The result of Mazur was a starting point for the theory of differentiability of convex functions (cf. the book...

We are interested in the kernel of one-dimensional diffusion equations with continuous coefficients as evaluated by means of explicit discretization schemes of uniform step h > 0 in the limit as h → 0. We consider both semidiscrete triangulations with continuous time and explicit Euler schemes with time step small enough for the method to be stable. We find sharp uniform bounds for the converge...

In this paper, we study the numerical method for solving second-order fuzzy differential equations using Adomian method under strongly generalized differentiability. And, we present an example with initial condition having four different solutions to illustrate the efficiency of the proposed method under strongly generalized differentiability. Keywords—Fuzzy-valued function, Fuzzy initial value...

There are multiple reasons why anisotropic basis functions may be needed or be more appropriate. The most obvious is that if the basis function is to be defined on Rn × T then there is no natural norm on this space that would reflect the unique properties of time. A second reason is that function being interpolated or approximated may incorporate a directional dependence. Thirdly, differentiabi...

In this work we provide a characterization of distinct types (linear and non-linear) maps between Banach spaces in terms the differentiability certain class Lipschitz functions. Our results are stated an abstract bornological non-linear framework. Restricted to linear case, can apply our compact, weakly-compact, limited completely continuous operators. Moreover, yield Gelfand-Phillips recover s...

We introduce Chern degree functions for complexes of coherent sheaves on a polarized surface, which encode information given by stability conditions the boundary $(\alpha, \beta)$-plane. prove that these extend to continuous real valued and we study their differentiability in terms stability. For abelian surfaces, coincide with cohomological rank defined Jiang-Pareschi. illustrate some geometri...

Most of the existing approaches for fuzzy reliability analysis are based on fuzzy probability.  The aim of this paper is to describe fuzzy reliability using fuzzy differential equation. The reliability of a system in real world applications is affected by some uncertain parameters. Fuzzy reliability is a way to present the reliability function uncertainly using fuzzy parameters. In the proposed...

In this paper we prove that if $X $ is a Banach space, then for every lower semi-continuous bounded below function $f, $ there exists a $left(varphi_1, varphi_2right)$-convex function $g, $ with arbitrarily small norm,  such that $f + g $ attains its strong minimum on $X. $ This result extends some of the  well-known varitional principles as that of Ekeland [On the variational principle,  J. Ma...

We prove that the roots of a definable C∞ curve of monic hyperbolic polynomials admit a definable C∞ parameterization, where ‘definable’ refers to any fixed o-minimal structure on (R,+, ·). Moreover, we provide sufficient conditions, in terms of the differentiability of the coefficients and the order of contact of the roots, for the existence of Cp (for p ∈ N) arrangements of the roots in both ...