Information-Theoretic Descriptors of Molecular States and Electronic Communications between Reactants

The classical (modulus/probability) and nonclassical (phase/current) components of molecular states are reexamined and their information contributions are summarized. The state and information continuity relations are discussed and a nonclassical character of the resultant gradient information source is emphasized. The states of noninteracting and interacting subsystems in the model donor-acceptor reactive system are compared and configurations of the mutually-closed and -open equidensity orbitals are tackled. The density matrices for subsystems in reactive complexes are used to describe the entangled molecular fragments and electron communications in donor-acceptor systems which determine the entropic multiplicity and composition of chemical bonds between reactants.

Approximation Properties of an Extended Family of the Szász–Mirakjan Beta-Type Operators

Approximation and some other basic properties of various linear and nonlinear operators are potentially useful in many different areas of researches in the mathematical, physical, and engineering sciences. Motivated essentially by this aspect of approximation theory, our present study systematically investigates the approximation and other associated properties of a class of the Szász-Mirakjan-type operators, which are introduced here by using an extension of the familiar Beta function. We propose to establish moments of these extended Szász-Mirakjan Beta-type operators and estimate various convergence results with the help of the second modulus of smoothness and the classical modulus of continuity. We also investigate convergence via functions which belong to the Lipschitz class. Finally, we prove a Voronovskaja-type approximation theorem for the extended Szász-Mirakjan Beta-type operators.

A New Modulus of Smoothness for Uniform Approximation

The estimates of best approximation using classical modulus of smoothness is not uniform. Also we sometimes need to improve the degree of best approximation near the end points. Thus we need to improve this classical modulus of smoothness. Here we define a new modulus of smoothness to achieve uniform estimates of best approximation and an improvement of a degree of such version of best approximation. Our modulus of smoothness is for k-monotone functions. Estimates for using our modulus of smoothness are introduced. Applications for these estimates are also introduced

A genuine family of Bernstein-Durrmeyer type operators based on Polya basis functions

In this paper, we construct a genuine family of Bernstein-Durrmeyer type operators based on Polya basis functions. We establish some moment estimates and the direct results which include global approximation theorem in terms of classical modulus of continuity, local approximation theorem in terms of the second order Ditizian-Totik modulus of smoothness, Voronovskaya-type asymptotic theorem and a quantitative estimate of the same type. Lastly, we study the approximation of functions having a derivative of bounded variation.

Control Conditions for the Second–Class Crack of Concrete Based on Different Modulus Theory

Many engineering materials exhibit different elastic properties under the extension and compression. Among these materials, the concrete is a typical material which has different extension modulus and compression modulus. Based on different modulus theory, mechanical analysis was performed in this paper for reinforced concrete beams, stress formula was deduced, and the corresponding formula was applied to concrete structures with second-class crack control, which has improved the original crack control conditions in the code for design of concrete structures in China based on the classical modulus theory.