Modern mathematical statistics

Dudewicz, E. J. Mishra, S. N. (1988). Modern Mathematical Statistics. New York John Wiley. 

Finally, it should be pointed out once again that obtaining as precise and complete information on a studied chemical or physical system as possible, with a minimal number of experiments and the lowest possible expenses, is the necessary condition for efficient research work. Therefore, application of modern mathematical and statistical methods in designing and analyzing experimental results is a real necessity in all fields and phases of work, starting with purely theoretical considerations of a process, its research and development, all the way to designing equipment and studying optimal operational conditions of a plant. 

The existence of a multitude of new physico-chemical methods of analysis, side by side with the classical methods of chemical analysis, urgently raises the question of finding out rational criteria for comparing the results obtained by various analytical methods. The development and introduction of new analytical methods take place considerably faster than their standardization...It is already evident that an analyst must be as thoroughly familiar with the methods of modern mathematical statistics as the geodesist is with the method of least squares. 

EJ Dudewicz and SN Mishra. Modern Mathematical Statistics. John Wiley Sons, New York, NY, 1988. 

DUDEWICZ, E. J. and MISHRA, S. N. Modern mathematical statistics. New York, Wiley, 1985. 

In summary, it may be stated that for mathematical simulation— depending on the type of utilization—the entire spectrum of modern mathematics will be needed. The most important disciplines are ordinary and partial differential equations, linear algebra and matrix algebra, Boole s algebra, and numerical mathematics as well as probability theory and statistics. 

Devore JL, Berk KN (2011) Modern mathematical statistics with applications. Springer, New York... 

This is a simple analogy of Birkhoff s ergodic theorem for dynamical systems, see A.I. Khinchin, Mathematical Foundation of Statistical Mechanics (Dover, New York 1949) L.E. Reichl, A Modern Course in Statistical Physics (University of Texas Press, Austin, TX 1980) ch. 8. 

In modern crystallography virtually all structure solutions are obtained by direct methods. These procedures are based on the fact that each set of hkl planes in a crystal extends over all atomic sites. The phases of all diffraction maxima must therefore be related in a unique, but not obvious, way. Limited success towards establishing this pattern has been achieved by the use of mathematical inequalities and statistical methods to identify groups of reflections in fixed phase relationship. On incorporating these into multisolution numerical trial-and-error procedures tree structures of sufficient size to solve the complete phase problem can be constructed computationally. Software to solve even macromolecular crystal structures are now available. 

Simplistic and heuristic similarity-based approaches can hardly produce as good predictive models as modern statistical and machine learning methods that are able to assess quantitatively biological or physicochemical properties. QSAR-based virtual screening consists of direct assessment of activity values (numerical or binary) of all compounds in the database followed by selection of hits possessing desirable activity. Mathematical methods used for models preparation can be subdivided into classification and regression approaches. The former decide whether a given compound is active, whereas the latter numerically evaluate the activity values. Classification approaches that assess probability of decisions are called probabilistic. 

The main objective of performing kinetic theory analyzes is to explain physical phenomena that are measurable at the macroscopic level in a gas at- or near equilibrium in terms of the properties of the individual molecules and the intermolecular forces. For instance, one of the original aims of kinetic theory was to explain the experimental form of the ideal gas law from basic principles [65]. The kinetic theory of transport processes determines the transport coefficients (i.e., conductivity, diffusivity, and viscosity) and the mathematical form of the heat, mass and momentum fluxes. Nowadays the kinetic theory of gases originating in statistical mechanics is thus strongly linked with irreversible- or non-equilibrium thermodynamics which is a modern held in thermodynamics describing transport processes in systems that are not in global equilibrium. 

The book is structured to supplement modern texts on kinetics and reaction engineering, not to present an alternative to them. It intentionally concentrates on what is not easily available from other sources. Facets and procedures well covered in standard texts—statistical basis, rates of single-step reactions, experimental reactors, determination of reaction orders, auxiliary experimental techniques (isotopic labeling, spectra, etc.)—are sketched only for ease of reference and to place them in context. Emphasis is on a comprehensive presentation of strategies and streamlined mathematics for network elucidation and modeling suited for industrial practice. 

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