c o l o r e d l i g h t r e f l e c t i o n s p r o d u c e d b y h i s m o b i l e s . H i s s h o r t J a m e s E . D a v i s , " F i g u r e S t u d y , " p e n c i l o n p a p e r , 1 9 3 4 . f i l m s , a v e r a g i n g a b o u t t e n m i n u t e s i n l e n g t h , w e r e o f t e n a c c o m p a n i e d b y o r i g i n a l m u s i c , s o m e t i m e s c o m p o s e d a n d a r r a n g...

In this work, we augment the approximate density functional method SCC-DFTB (DFTB3) with the chemical-potential equalization (CPE) approach in order to improve the performance for molecular electronic polarizabilities. The CPE method, originally implemented for the NDDO type of methods by Giese and York, has been shown to significantly emend minimal basis methods with respect to the response pr...

In this paper, we apply the concept of Caputo’s H-differentiability, constructed based on the generalized Hukuhara difference, to solve the fuzzy fractional differential equation (FFDE) with uncertainty. This is in contrast to conventional solutions that either require a quantity of fractional derivatives of unknown solution at the initial point (Riemann–Liouville) or a solution with increasing...

A module $M$ is called $emph{H}$-cofinitely supplemented if for every cofinite submodule $E$ (i.e. $M/E$ is finitely generated) of $M$ there exists a direct summand $D$ of $M$ such that $M = E + X$ holds if and only if $M = D + X$, for every submodule $X$ of $M$. In this paper we study factors, direct summands and direct sums of $emph{H}$-cofinitely supplemented modules. Let $M$ be an $emph{H}...

Let M be a right R-module. We call M Rad-H-supplemented iffor each Y M there exists a direct summand D of M such that(Y + D)/D (Rad(M) + D)/D and (Y + D)/Y (Rad(M) + Y )/Y .It is shown that:(1) Let M = M1M2, where M1 is a fully invariant submodule of M.If M is Rad-H-supplemented, thenM1 andM2 are Rad-H-supplemented.(2) Let M = M1 M2 be a duo module and Rad--supplemented. IfM1 is radical M2-...