Mobius Inversion in Physics (Tsinghua Report and Review in Physics)

Contents......Page 14
Preface......Page 8
Acknowledgments......Page 12
List of Figures......Page 19
List of Tables......Page 23
1.1 Deriving Mobius Series Inversion Formula with an Example in Physics......Page 25
1.2.1 Definition of an arithmetic function......Page 32
1.2.2 Dirichlet product between arithmetic functions......Page 33
1.2.3 All reversible functions as a subset of arithmetic functions......Page 35
1.3.1 Group M of multiplicative functions......Page 36
1.3.2 Unit constant function  0 and sum rule of  (n)......Page 40
1.3.3 Modified Mobius inversion formulas......Page 42
1.3.4.1 An alternative Mobius series inversion......Page 45
1.3.4.2 Madelung constant in a linear ionic chain......Page 46
1.3.4.3 An inverse problem for intrinsic semiconductors......Page 48
1.4 Riemann\'s  (s) and  (n)......Page 49
1.5 Mobius, Chebyshev and Modulation Transfer Function......Page 52
1.6 Witten Index and M obius Function......Page 56
1.7 Cesaro–Mobius Inversion Formula......Page 60
1.8 Unification of Eqs. (1.20) and (1.47)......Page 64
1.10 Supplement  – the Seminal Paper of Mobius......Page 66
2.1 What is an Inverse Problem?......Page 71
2.2 Inverse Blackbody Radiation Problem......Page 73
2.2.1 Bojarski iteration......Page 74
2.2.2 The Mobius inversion for the inverse blackbody radiation......Page 76
2.3.1 Historical background......Page 77
2.3.2 Montroll solution......Page 79
2.3.3 The Mobius formula on inverse heat capacity problem......Page 84
2.3.4 General formula for the low temperature limit......Page 85
2.3.5 Temperature dependence of Debye frequency......Page 86
2.3.6 General formula for high temperature limit......Page 87
2.3.7 Some special relations between  (s) and  (n)......Page 92
2.4.1 Inverse spontaneous magnetization problem......Page 97
2.5 Summary......Page 99
3.1.1 Definition of an arithmetic function of the second kind......Page 103
3.1.3 Inverse of an arithmetic function......Page 105
3.2 Mobius Series Inversion Formula of the Second Kind......Page 106
3.3 Mobius Inversion and Fourier Deconvolution......Page 107
3.4.1 Fermi integral equation......Page 109
3.4.2 Relaxation-time spectra......Page 114
3.4.3 Adsorption integral equation with a Langmuir kernel......Page 115
3.4.5 Dubinin–Radushkevich isotherm......Page 117
3.4.6 Kernel expression by  – function......Page 119
3.5.1 Chebyshev formulation......Page 120
3.5.2 From orthogonality to biorthogonality......Page 123
3.5.3 Multiplicative dual orthogonality and square wave representation......Page 126
3.5.4 Multiplicative biorthogonal representation for saw waves......Page 127
3.6 Construction of Additive Biorthogonality......Page 128
3.6.1 Basic theorem on additively orthogonal expansion......Page 129
3.6.2 Derivative biorthogonality from even square waves......Page 131
3.6.3 Derivative set from triangular wave......Page 136
3.6.4 Another derivative set by saw wave......Page 138
3.6.5 Biorthogonal modulation in communication......Page 140
3.7 Cesaro Inversion Formula of the Second Kind......Page 143
3.8 Summary......Page 146
4.1 Concept of Arithmetic Fourier Transform......Page 149
4.2.2 Proof of Eq. (4.2)......Page 151
4.2.3 Proof of Eq. (4.3)......Page 153
4.3.1 Two other modified Mobius inverse formulas......Page 154
4.3.2 Reed\'s expression......Page 156
4.4 Fundamental Theorem of AFT (Bruns)......Page 159
4.4.1 Proof of Eq. (4.41)......Page 160
4.4.2 The relationship between a(n); b(n) and B(2n;  )......Page 161
4.5.1 What is the Ramanujan sum rule?......Page 164
4.5.2 Proof of Ramanujan sum rule......Page 165
4.5.3 Uniformly sampling AFT (USAFT)......Page 166
4.5.3.1 Example for N = 4......Page 167
4.5.3.3 N = 4[1; 2; 3; ..., t]......Page 168
4.5.4 Note on application of generalized function......Page 169
4.6 Summary......Page 170
5.1 Concept of Low Dimensional Structures......Page 173
5.2 Linear Atomic Chains......Page 174
5.3 Simple Example in a Square Lattice......Page 176
5.4.2 Unit elements, associates with reducible and irreducible integers in G......Page 178
5.4.3 Unique factorization theorem in G......Page 179
5.4.4 Criteria for reducibility......Page 180
5.4.5 Procedure for factorization into irreducibles......Page 182
5.4.6 Sum rule of Mobius functions and Mobius inverse formula......Page 183
5.4.7 Coordination numbers in 2D square lattice......Page 184
5.4.8 Application to the 2D arithmetic Fourier transform......Page 188
5.4.9.1 The derivation of Bruns version for 2D AFT......Page 192
5.4.9.2 Example for 2D AFT computation......Page 196
5.5.1 Definition of Eisenstein integers......Page 197
5.5.2 Norm and associates of an Eisenstein integer......Page 198
5.5.3 Reducibility of an Eisenstein integer......Page 199
5.5.4 Factorization procedure of an arbitrary Eisenstein integer......Page 200
5.5.5 Mobius inverse formula on Eisenstein integers......Page 201
5.5.6.2 Calculated results of elastic constants......Page 203
5.5.7 Coordination number in a hexagonal lattice......Page 205
5.6 Summary......Page 206
6.1 A Brief Historical Review......Page 207
6.2.1 CGE solution......Page 208
6.3 Mobius Inversion for a General 3D Lattice......Page 210
6.4.1 Inversion formula for a fcc lattice......Page 213
6.4.2 Inversion formula in a bcc structure......Page 216
6.4.3 Inversion formula for the cross potentials in a L12 structure......Page 219
6.5 Atomistic Analysis of the Field-Ion Microscopy Image of Fe3Al......Page 221
6.6.1 Expression based on a cubic crystal cell......Page 225
6.6.2 Expression based on a unit cell......Page 227
6.7 The Stability and Phase Transition in NaCl......Page 229
6.8 Inversion of Stretching Curve......Page 233
6.9 Lattice Inversion Technique for Embedded Atom Method......Page 235
6.10.1 Interface between two matched rectangular lattices......Page 239
6.10.2 Metal/MgO interface......Page 241
6.10.3 Matal/SiC interface......Page 251
6.11 Summary......Page 254
A.1 TOSET......Page 257
A.2 POSET......Page 258
A.3 Interval and Chain......Page 259
A.4 Local Finite POSET......Page 260
A.5 M obius Function on Locally Finite POSET......Page 261
A.5.1 Example......Page 262
A.6.1 Mobius inverse formula A......Page 263
A.6.2 Mobius inverse formula B......Page 264
A.7 Principle of Inclusion and Exclusion......Page 265
A.8 Cluster Expansion Method......Page 269
Epilogue......Page 275
Bibliography......Page 277
Index......Page 287