Mobius Inversion In Physics

By: Nanxian Chen

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Hardcover | 28 April 2010

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290 Pages

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This book attempts to bridge the gap between the principles of pure mathematics and the applications in physical science. After the Moebius inversion formula had been considered as purely academic, or beyond what was useful in the physics community for more than 150 years, the apparently obscure result in classical mathematics suddenly appears to be connected to a variety of important inverse problems in physical science.This book only requires readers to have some background in elementary calculus and general physics, and the prerequisite knowledge of number theory is not needed. It will be attractive to our multidisciplinary readers interested in the Moebius technique, which is a tiny but important part of the number-theoretic methods. It will inspire many students and researchers in both physics and mathematics. In a practical problem, continuity and discreteness are often correlated, and few textbooks have given attention to this wide and important field as this one.Clearly, this book will be an essential supplement for many existing courses such as mathematical physics, elementary number theory and discrete mathematics.

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Non-Fiction Mathematics Calculus & Mathematical Analysis Functional Analysis & Transforms Science Physics Mathematical Physics Number Theory Applied Mathematics Booktopia Publisher Services World Scientific Publishing

Preface	p. vii
Acknowledgments	p. xi
List of Figures	p. xix
List of Tables	p. xxiii
Basics of Möbius Inversion Formulas	p. 1
Deriving Möbius Series Inversion Formula with an Example in Physics	p. 1
Elementary Concepts of Arithmetic Function	p. 8
Definition of an arithmetic function	p. 8
Dirichlet product between arithmetic functions	p. 9
All reversible functions as a subset of arithmetic functions	p. 11
Multiplicative Functions - a Subgroup of the Group U	p. 12
Group m of multiplicative functions	p. 12
Unit constant function $$$₀ and sum rule of ¿(n)	p. 16
Modified Möbius inversion formulas	p. 18
An alternative Möbius series inversion formula	p. 21
Riemann's ¿(s) and ¿(n)	p. 25
Möbius, Chebyshev and Modulation Transfer Function	p. 28
Witten Index and Möbius Function	p. 32
Cesàro-Möbius Inversion Formula	p. 36
Unification of Eqs. (1.20) and (1.47)	p. 40
Summary	p. 42
Supplement - the Seminal Paper of Möbius	p. 42
Inverse Problems in Boson Systems	p. 47
What is an Inverse Problem?	p. 47
Inverse Blackbody Radiation Problem	p. 49
Bojarski iteration	p. 50
The Möbius inversion for the inverse blackbody radiation	p. 52
Inverse Heat Capacity Problem	p. 53
Historical background	p. 53
Montroll solution	p. 55
The Möbius formula on inverse heat capacity problem	p. 60
General formula for the low temperature limit	p. 61
Temperature dependence of Debye frequency	p. 62
General formula for high temperature limit	p. 63
Some special relations between ¿(s) and ¿(n)	p. 68
Some Inverse Problems Relative to Frequency Spectrum	p. 73
Inverse spontaneous magnetization problem	p. 73
Inverse transmissivity problem	p. 75
Summary	p. 75
Inverse Problems in Fermion Systems	p. 79
The Arithmetic Functions of the Second Kind	p. 79
Definition of an arithmetic function of the second kind	p. 79
Unit function in A₂	p. 81
Inverse of an arithmetic function	p. 81
Möbius Series Inversion Formula of the Second Kind	p. 82
Möbius Inversion and Fourier Deconvolution	p. 83
Solution of Fermi Integral Equation	p. 85
Fermi integral equation	p. 85
Relaxation-time spectra	p. 90
Adsorption integral equation with a Langmuir kernel	p. 91
Generalized Freundlich isotherm	p. 93
Dubinin-Radushkevich isotherm	p. 93
Kernel expression by ¿ - function	p. 95
Möbius and Biorthogonality	p. 96
Chebyshev formulation	p. 96
From orthogonality to biorthogonality	p. 99
Multiplicative dual orthogonality and square wave representation	p. 102
Multiplicative biorthogonal representation for saw waves	p. 103
Construction of Additive Biorthogonality	p. 104
Basic theorem on additively orthogonal expansion	p. 105
Derivative biorthogonality from even square waves	p. 107
Derivative set from triangular wave	p. 112
Another derivative set by saw wave	p. 114
Biorthogonal modulation in communication	p. 116
Cesàro Inversion Formula of the Second Kind	p. 119
Summary	p. 122
Arithmetic Fourier Transform	p. 125
Concept of Arithmetic Fourier Transform	p. 125
Fundamental Theorem of AFT (Wintner)	p. 127
Statement of the Wintner theorem	p. 127
Proof of Eq. (4.2)	p. 127
Proof of Eq. (4.3)	p. 129
The Improvement of Wintner Algorithm by Reed	p. 130
Two other modified Möbius inverse formulas	p. 130
Reed's expression	p. 132
Fundamental Theorem of AFT (Bruns)	p. 135
Proof of Eq. (4.41)	p. 136
The relationship between a(n),b(n) and B(2n,¿)	p. 137
Uniformly Sampling in AFT based on Ramanujan Sum	p. 140
What is the Ramanujan sum rule?	p. 140
Proof of Ramanujan sum rule	p. 141
Uniformly sampling AFT (USAFT)	p. 142
Note on application of generalized function	p. 145
Summary	p. 146
Inverse Lattice Problems in Low Dimensions	p. 149
Concept of Low Dimensional Structures	p. 149
Linear Atomic Chains	p. 150
Simple Example in a Square Lattice	p. 152
Arithmetic Functions on Gaussian Integers	p. 154
Gaussian integers	p. 154
Unit elements, associates with reducible and irreducible integers in G	p. 154
Unique factorization theorem in G	p. 155
Criteria for reducibility	p. 156
Procedure for factorization into irreducibles	p. 158
Sum rule of Möbius functions and Möbius inverse formula	p. 159
Coordination numbers in 2D square lattice	p. 160
Application to the 2D arithmetic Fourier transform	p. 164
Bruns version of 2D AFT and VLSI architecture	p. 168
2D Hexagonal Lattice and Eisenstein Integers	p. 173
Definition of Eisenstein integers	p. 173
Norm and associates of an Eisenstein integer	p. 174
Reducibility of an Eisenstein integer	p. 175
Factorization procedure of an arbitrary Eisenstein integer	p. 176
Möbius inverse formula on Eisenstein integers	p. 177
Application to monolayer graphite	p. 179
Coordination number in a hexagonal lattice	p. 181
Summary	p. 182
Inverse Lattice Problems	p. 183
A Brief Historical Review	p. 183
3D Lattice Inversion Problem	p. 184
CGE solution	p. 184
Bazant iteration	p. 186
Möbius Inversion for a General 3D Lattice	p. 186
Inversion Formulas for some Common Lattice Structures	p. 189
Inversion formula for a fcc lattice	p. 189
Inversion formula in a bcc structure	p. 192
Inversion formula for the cross potentials in a L1₂ structure	p. 195
Atomistic Analysis of the Field-Ion Microscopy Image of Fe₃Al	p. 197
Interaction between Unlike Atoms in B₁ and B₃ structures	p. 201
Expression based on a cubic crystal cell	p. 201
Expression based on a unit cell	p. 203
The Stability and Phase Transition in NaCl	p. 205
Inversion of Stretching Curve	p. 209
Lattice Inversion Technique for Embedded Atom Method	p. 211
Interatomic Potentials between Atoms across Interface	p. 215
Interface between two matched rectangular lattices	p. 215
Metal/MgO interface	p. 217
Matal/SiC interface	p. 227
Summary	p. 230
Appendix: Möbius Inverse Formula on a Partially Ordered Set	p. 233
TOSET	p. 233
POSET	p. 234
Interval and Chain	p. 235
Local Finite POSET	p. 236
Möbius Function on Locally Finite POSET	p. 237
Example	p. 238
Example	p. 239
Möbius Inverse Formula on Locally Finite POSET	p. 239
Möbius inverse formula A	p. 239
Möbius inverse formula B	p. 240
Principle of Inclusion and Exclusion	p. 241
Cluster Expansion Method	p. 245
Epilogue	p. 251
Bibliography	p. 253
Index	p. 263
Table of Contents provided by Ingram. All Rights Reserved.

Mobius Inversion In Physics

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