We prove in the general setting of lower semicontinuous functions on Banach spaces the relation between the Rockafellar directional derivative and the mixed lower limit of the lower Dini derivatives. As a byproduct we derive the famous inclusions of tangent cones of closed sets in Banach spaces. The results are established using as principal tool multidirectional mean value inequalities [Aussel...

We define the notions of unilateral metric derivatives and “metric derived numbers” in analogy with Dini derivatives (also referred to as “derived numbers”) and establish their basic properties. We also prove that the set of points where a path with values in a metric space with continuous metric derivative is not “metrically differentiable” (in a certain strong sense) is σsymmetrically porous ...

Predictor-corrector (PC) methods for the numerical solution of stiff ODEs can be extended to include the second derivative of the solution. In this paper, we consider second derivative PC methods with the three-step second derivative Adams-Bashforth as predictor and two-step second derivative Adams-Moulton as corrector which both methods have order six. Implementation of the proposed PC method ...

The present study aims at indicating the existence and uniqueness result of system in extended colombeau algebra. The Caputo fractional derivative is used for solving the system of ODEs. In addition, Riesz fractional derivative of  Colombeau generalized algebra is considered. The purpose of introducing Riesz fractional derivative is regularizing it in Colombeau sense. We also give a solution to...

We complete the study of Manin triples of real simple Lie algebras. In the Part 2 of the article we classify the Manin triples (g(R), W, g(R) ⊕ g(R)) (case 2 of the doubles) up to weak and gauge equivalence. First we recall the main definitions of the Part 1. Definition 1. Let g 1 , g 2 , d be Lie algebras over a field K and let Q be a symmetric nondegenerate bilinear form on d. A triple (g 1 ,...

Gravity and magnetic measurements are common remote sensing strategies to obtain the property change of observed targets. Nowadays, characteristic value derivatives gravity anomalies is commonly used detect source horizontal edge. We found that coordinates different-order not directly corresponding edge source, which varies with depth size source. The spatial derivative (SDD) method was develop...

Abstract We consider a linear system that consists of wave equation on horizontal hypersurface and parabolic in the half space below. The model describes longitudinal elastic waves organic monolayers at water–air interface, which is an experimental setup relevant for understanding propagation biological membranes. study scaling regime where length scale much larger than vertical provide rigorou...

a green and efficient one-pot, four-component synthesis of 6-amino-4-aryl-3-methyl-2,4-dihydropyrano[2,3- c ]pyrazo-carbonitrile derivatives catalyzed by borax in water is described. this method provides several advantages such as environmental friendliness, shorter reaction time, excellent yields, and simple workup procedure. the in vitro cytotoxic activity of the synthesized test compounds wa...

a mixture of acetaminophen, diphenhydramine hydrochloride and pseudoephedrine hydrochloride is used for the symptomatic treatment of common cold. in this study, a derivative spectrophotometric method based on zero-crossing technique was proposed for simultaneous determination of acetaminophen, diphenhydramine hydrochloride and pseudoephedrine hydrochloride. determination of these drugs was perf...

We develop a non-anticipative calculus for functionals of a continuous semimartingale, using an extension of the Ito formula to path-dependent functionals which possess certain directional derivatives. The construction is based on a pathwise derivative, introduced by B Dupire, for functionals on the space of right-continuous functions with left limits. We show that this functional derivative ad...