(1) For all sets X1, X2 and for all sets a1, a2 holds [:X1 7−→ a1, X2 7−→ a2 :] = [:X1, X2 :] 7−→ 〈a1, a2〉. Let I be a set. One can check that 0I is function yielding. Next we state two propositions: (2) For all functions f , g holds x(g · f ) = g ·x f . (3) For all functions f , g, h holds x( f · [:g, h :]) = x f · [:h, g :]. Let f be a function yielding function. Observe that x f is function ...

The molecular-replacement method works well with good models and simple unit cells, but often fails with more difficult problems. Experience with likelihood in other areas of crystallography suggests that it would improve performance significantly. For molecular replacement, the form of the required likelihood function depends on whether there is ambiguity in the relative phases of the contribu...

A likelihood function based on the multivariate probability distribution of all observed structure-factor amplitudes from a single isomorphous replacement with anomalous scattering experiment has been derived and implemented for use in substructure refinement and phasing as well as macromolecular model refinement. Efficient calculation of a multidimensional integration required for function eva...

In this paper, we introduce and investigate two new subclasses of the functions class $ Sigma $ of bi-univalent functions defined in the open unit disk, which are associated with the Aghalary-Ebadian-Wang operator. We  estimate the coefficients $|a_{2} |$ and  $|a_{3} |$ for functions in these new subclasses. Several  consequences of the result are also pointed out.

In the present paper, we introduce and investigate some properties of two subclasses $ Lambda_{n}( lambda , beta ) $ and $ Lambda_{n}^{+}( lambda , beta ) $;  meromorphic and starlike  functions of order $ beta $. In particular, several inclusion relations, coefficient estimates, distortion theorems and covering theorems are proven here for each of these function classes.

A theoretical investigation is performed for studying the flow of heat transfer characteristics of Sakiadis and Blasius flow of magnetohydrodynamic Maxwell fluid with thermal radiation. We also incorporated the Cattaneo-Chirstov heat flux model to control the heat transfer phenomena. Numerical solutions are carried out by using Shooting technique. The effects of various governing parameters on ...

class AbstractList { //@ model public int size; //@ in isEmpty; //@ model public boolean isEmpty; //@ public represents isEmpty <(size == 0); //@ public invariant size >= 0; //@ assignable isEmpty; //@ ensures isEmpty; abstract public void clear(); //@ assignable size; // but isEmpty is not assignable //@ ensures size <= \old(size); //@ ensures \old(size) > 0 ==> size > 0; abstract public void ...

In the field of Geometric Function Theory, one can not deny the importance of analytic and univalent functions. The characteristics of these functions including their taylor series expansion, their coefficients in these representations as well as their associated functional inequalities have always attracted the researchers. In particular, Fekete-Szegö inequality is one of such vastly studied a...

The main goal of the current research is to investigate numerical computation Ag/Al2O3 nanofluid over a Riga plate with injection/suction. energy equation formulated using Cattaneo–Christov heat flux, non-linear thermal radiation, and sink/source. leading equations are non-dimensionalized by employing suitable transformations, results achieved MATLAB bvp4c technique. fluctuations fluid flow tra...

The impact of heat-absorbing viscoelastic nanofluidic flow along with a convectively heated porous Riga plate Cattaneo-Christov double flux was analytically investigated. Buongiorno model nanofluid implemented the diversity Brownian motion and thermophoresis. Making use transformations; PDE systems are altered into an ODE system. We homotopy analysis method to solve these analytically. reaction...