Mathematical Analysis of Problems in the Natural Sciences

by Zorich, Vladimir; Gould, Gerald

• ISBN13: 9783642148125
• ISBN10: 3642148123
• Format: Hardcover
• Copyright: 2010-11-30
• Publisher: Springer Nature

Summary

Based on a two-semester course aimed at illustrating various interactions of "pure mathematics" with other sciences, such as hydrodynamics, thermodynamics, statistical physics and information theory, this text unifies three general topics of analysis and physics, which are as follows: the dimensional analysis of physical quantities, which contains various applications including Kolmogorov's model for turbulence; functions of very large number of variables and the principle of concentration along with the non-linear law of large numbers, the geometric meaning of the Gauss and Maxwell distributions, and the Kotelnikov-Shannon theorem; and, finally, classical thermodynamics and contact geometry, which covers two main principles of thermodynamics in the language of differential forms, contact distributions, the Frobenius theorem and the Carnot-Caratheodory metric. It includes problems, historical remarks, and Zorich's popular article, "Mathematics as language and method."

Author Biography

Vladimir A. Zorich is a distinguished Professor of Mathematics at the University of Moscow who solved the problem of global homeomorphism for space quasiconformal mappings and provided its far-reaching generalizations. In Mathematical Analysis of Problems in the Natural Sciences he uses a lively and accessible style to unify three topics of analysis and physics, which are as follows: the dimensional analysis of physical quantities, which contains various applications including Kolmogorov's model for turbulence; functions of very large numbers of variables and the principle of concentration along with the non-linear law of large numbers, the geometric meaning of the Gauss and Maxwell distributions, and the Kotelnikov-Shannon theorem; and, finally, classical thermodynamics and contact geometry, which covers two main principles of thermodynamics in the language of differential forms, contact distributions, the Frobenius theorem and the Carnot-Caratheodory metric.

Table of Contents

Analysis of Dimensions of Physical Quantities
A few introductory words	p. 3
Elements of the theory	p. 5
Dimension of a physical quantity (preliminaries)	p. 5
Measurement, unit of measurement, measuring process	p. 5
Basic and derived units	p. 5
Dependent and independent units	p. 6
A formula for the dimension of a physical quantity	p. 6
Change of the numerical values of a physical quantity under a change of the sizes of the basic units	p. 6
Postulate of the invariance of the ratio of the values of physical quantities with the same name	p. 7
Function of dimension and a formula for the dimension of a physical quantity in a given basis	p. 7
Fundamental theorem of dimension theory	p. 9
The ¿-Theorem	p. 9
Principle of similarity	p. 10
Examples of applications	p. 11
Period of rotation of a body in a circular orbit (laws of similarity)	p. 11
The gravitional constant. Kepler's third law and the degree exponent in Newton's law of universal gravitation	p. 12
Period of oscillation of a heavy pendulum (inclusion of g)	p. 13
Outflow of volume and mass in a waterfall	p. 14
Drag force for the motion of a ball in a non-viscous medium	p. 14
Drag force for the motion of a ball in a viscous medium	p. 15
Exercises	p. 16
Concluding remarks	p. 18
Further applications: hydrodynamics and turbulence	p. 23
Equations of hydrodynamics (general information)	p. 23
Loss of stability of the flow and comments on bifurcations in dynamical systems	p. 25
Turbulence (initial ideas)	p. 26
The Kolmogorov model	p. 27
The multiscale property of turbulent motions	p. 27
Developed turbulence in the inertia interval	p. 28
Specific energy	p. 28
Reynolds number of motions of a given scale	p. 29
The Kolmogorov-Obukhov law	p. 29
Inner scale of turbulence	p. 30
Energy spectrum of turbulent pulsations	p. 30
Turbulent mixing and dispersion of particles	p. 31
Multidimensional Geometry and Functions of a Very Large Number of Variables
Introduction	p. 35
Some examples of functions of very many variables in natural science and technology	p. 37
Digital sampling of a signal (CIM-code-impulse modulation)	p. 37
The linear device and its mathematical description (convolution)	p. 37
Fourier-reciprocal (spectral) description of a linear device	p. 38
Functions and devices with a compactly supported spectrum	p. 39
The ideal filter and its instrumental function	p. 39
The sampling theorem (Kotel'nikov-Shannon formula)	p. 40
Code-impulse modulation of a signal (CIM)	p. 41
Transmission capacity of an ideal communication channel	p. 42
Evaluation of the dimension of a TV signal	p. 42
Some other areas of multiparameter phenomena and spaces of large dimension	p. 42
Molecular theory of matter	p. 43
Phase space in classical Hamiltonian mechanics	p. 43
The Gibbs thermodynamic ensembles	p. 43
Probability theory	p. 44
Concentration principle and its applications	p. 45
The ball and sphere in Euclidean space Rn with n ” 1	p. 45
Concentration of the volume of a ball as n → ∞	p. 45
Thermodynamic limit	p. 45
Concentration of the area of a sphere	p. 46
Isoperimetric inequality and almost constancy of a function on a sphere of very large dimension	p. 48
Some remarks	p. 50
The various averages	p. 50
The multidimensional cube and the principle of concentration	p. 51
Principle of concentration, thermodynamics, ergodicity	p. 52
Principle of concentration and limiting distributions	p. 53
Communication in the presence of noise	p. 55
Discrete recording of a continuous signal - concretization	p. 55
Energy and mean power of a signal	p. 55
Quantization by levels	p. 56
Ideal multilevel communication channel	p. 57
Noise (white noise)	p. 57
Transmission capacity of a communication channel with noise	p. 58
Rough estimate of the transmission capacity of a communication channel with noise	p. 58
Geometry of signal and noise	p. 58
Shannon's theorem	p. 60
Discussion of Shannon's theorem, examples and supplementary remarks	p. 62
Shannon's commentary	p. 62
Weak signal in a large amount of noise	p. 62
Redundancy of language	p. 63
Precise measurements in a crude piece of apparatus	p. 63
Shannon-Fano code	p. 64
Statistical characteristics of an optimal code	p. 64
Encoding and decoding - ¿-entropy and ¿-capacity	p. 65
Mathematical model of a channel with noise	p. 67
Simplest model and formulating the problem	p. 67
Information and entropy (preliminary considerations)	p. 68
Conditional entropy and information	p. 70
Interpretation of loss of information in a channel with noise	p. 72
Calculating the transmission capacity of an abstract communication channel	p. 74
Classical Thermodynamics and Contact Geometry
Introduction	p. 79
Classical thermodynamics (basic ideas)	p. 81
The two principles of thermodynamics	p. 81
Energy and the perpetual motion machine	p. 81
Perpetual motion machine of the second kind and entropy	p. 81
The two principles of theormodynamics in a mathematical setting	p. 83
Differential form of heat exchange	p. 84
The two principles of thermodynamics in the language of differential forms	p. 85
Thermodynamics without heat	p. 87
Adiabatic transition and Carathéodory's axiom	p. 88
Thermodynamics and contact geometry	p. 91
Contact distributions	p. 91
Adiabatic process and contact distribution	p. 91
Formalization	p. 91
Integrability of distributions	p. 92
The Frobenius theorem	p. 92
Integrability, connectibility, controllability	p. 93
Carnot-Carathéodory metric	p. 94
The Gibbs contact form	p. 95
Concluding remarks	p. 96
Thermodynamics classical and statistical	p. 99
Kinetic theories	p. 99
Molecules and pressure	p. 99
The Maxwell distribution	p. 99
Entropy according to Boltzmann	p. 100
The Gibbs ensemble and the "thermodynamicization" of mechanics	p. 101
Ergodicity	p. 102
Paradoxes, problems, difficulties	p. 104
Quantum statistical thermodynamics (a few words)	p. 106
Calculation of states and the conditional extremum	p. 106
Clarifying remarks and additional comments	p. 108
References	p. 115
Appendix
Mathematics as language and method	p. 127
Unforeseen epilogue	p. 135
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