| 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 |
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