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...

Acylisothiocyanate (1) was allowed to react with benzylidene malononitrile producing oxazine derivative (2). Also, compound (1) was reacted with sodium azide followed by refluxing with sodium ethoxide affording thiazinotetrazole derivative (4). The reaction of acetylacetone with compound (1) gave pyridine derivative (5) by intramolecular cycloaddition while it was reacted with N-methyl aniline ...

Analysis of the roles of the short chain fatty acid receptor, free fatty acid 3 receptor (FFA3), has been severely limited by the low potency of its endogenous ligands, the crossover of function of these on the closely related free fatty acid 2 receptor, and a dearth of FFA3-selective synthetic ligands. From a series of hexahydroquinolone-3-carboxamides, we demonstrate that 4-(furan-2-yl)-2-met...

2-Methylpyridinum p-toluenesulfonate ([2-MPy][p-TSA]) as a novel room temperature ionic liquidwas synthesized and evaluated as a recoverable catalyst for the one-pot synthesis ofhexahydroquinoline-3-carboxamide derivatives by four-component reaction of arylaldehydes,dimedone, acetoacetanilide and ammonium acetate in high to excellent yield in ethanol at 50 oC.The [2-MPy]...

This paper is a review of the authors’ results on the Dynamical Systems Method (DSM) for solving operator equation (*) F (u)= f . It is assumed that (*) is solvable. The novel feature of the results is the minimal assumption on the smoothness of F . It is assumed that F is continuously Fréchet differentiable, but no smoothness assumptions on F ′(u) are imposed. The DSM for solving equation (*) ...

In this paper, firstly, in $E_1^3$, we defined normal Fermi-Walker derivative and applied for the adapted frame. Normal Fermi-Walker parallelism, normal non-rotating frame, and Darboux vector expressions of normal Fermi-Walker derivative by normal Fermi-Walker derivative are given for adapted frame. Being conditions of normal Fermi-Walker derivative and normal non-rotating frame are examined fo...

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 ...