Contents -- Preface -- Acknowledgments -- List of Figures -- List of Tables -- 1. Basics of Mobius Inversion Formulas -- 1.1 Deriving Mobius Series Inversion Formula with an Example in Physics -- 1.2 Elementary Concepts of Arithmetic Function -- 1.2.1 Definition of an arithmetic function -- 1.2.2 Dirichlet product between arithmetic functions -- 1.2.3 All reversible functions as a subset of arithmetic functions -- 1.3 Multiplicative Functions - a Subgroup of the Group U -- 1.3.1 Group M of multiplicative functions -- 1.3.2 Unit constant function 0 and sum rule of (n) -- 1.3.3 Modified Mobius inversion formulas -- 1.3.4 An alternative Mobius series inversion formula -- 1.3.4.1 An alternative Mobius series inversion -- 1.3.4.2 Madelung constant in a linear ionic chain -- 1.3.4.3 An inverse problem for intrinsic semiconductors -- 1.4 Riemann's (s) and (n) -- 1.5 Mobius, Chebyshev and Modulation Transfer Function -- 1.6 Witten Index and M obius Function -- 1.7 Cesaro-Mobius Inversion Formula -- 1.8 Unification of Eqs. (1.20) and (1.47) -- 1.9 Summary -- 1.10 Supplement - the Seminal Paper of Mobius -- 2. Inverse Problems in Boson Systems -- 2.1 What is an Inverse Problem? -- 2.2 Inverse Blackbody Radiation Problem -- 2.2.1 Bojarski iteration -- 2.2.2 The Mobius inversion for the inverse blackbody radiation -- 2.3 Inverse Heat Capacity Problem -- 2.3.1 Historical background -- 2.3.2 Montroll solution -- 2.3.3 The Mobius formula on inverse heat capacity problem -- 2.3.4 General formula for the low temperature limit -- 2.3.5 Temperature dependence of Debye frequency -- 2.3.6 General formula for high temperature limit -- 2.3.7 Some special relations between (s) and (n) -- 2.4 Some Inverse Problems Relative to Frequency Spectrum -- 2.4.1 Inverse spontaneous magnetization problem -- 2.4.2 Inverse transmissivity problem -- 2.5 Summary.

3. Inverse Problems in Fermion Systems -- 3.1 The Arithmetic Functions of the Second Kind -- 3.1.1 Definition of an arithmetic function of the second kind -- 3.1.2 Unit function in A2 -- 3.1.3 Inverse of an arithmetic function -- 3.2 Mobius Series Inversion Formula of the Second Kind -- 3.3 Mobius Inversion and Fourier Deconvolution -- 3.4 Solution of Fermi Integral Equation -- 3.4.1 Fermi integral equation -- 3.4.2 Relaxation-time spectra -- 3.4.3 Adsorption integral equation with a Langmuir kernel -- 3.4.4 Generalized Freundlich isotherm -- 3.4.5 Dubinin-Radushkevich isotherm -- 3.4.6 Kernel expression by - function -- 3.5 Mobius and Biorthogonality -- 3.5.1 Chebyshev formulation -- 3.5.2 From orthogonality to biorthogonality -- 3.5.3 Multiplicative dual orthogonality and square wave representation -- 3.5.4 Multiplicative biorthogonal representation for saw waves -- 3.6 Construction of Additive Biorthogonality -- 3.6.1 Basic theorem on additively orthogonal expansion -- 3.6.2 Derivative biorthogonality from even square waves -- 3.6.3 Derivative set from triangular wave -- 3.6.4 Another derivative set by saw wave -- 3.6.5 Biorthogonal modulation in communication -- 3.7 Cesaro Inversion Formula of the Second Kind -- 3.8 Summary -- 4. Arithmetic Fourier Transform -- 4.1 Concept of Arithmetic Fourier Transform -- 4.2 Fundamental Theorem of AFT (Wintner) -- 4.2.1 Statement of the Wintner theorem -- 4.2.2 Proof of Eq. (4.2) -- 4.2.3 Proof of Eq. (4.3) -- 4.3 The Improvement of Wintner Algorithm by Reed -- 4.3.1 Two other modified Mobius inverse formulas -- 4.3.2 Reed's expression -- 4.4 Fundamental Theorem of AFT (Bruns) -- 4.4.1 Proof of Eq. (4.41) -- 4.4.2 The relationship between a(n) -- b(n) and B(2n -- ) -- 4.5 Uniformly Sampling in AFT based on Ramanujan Sum -- 4.5.1 What is the Ramanujan sum rule? -- 4.5.2 Proof of Ramanujan sum rule.

4.5.3 Uniformly sampling AFT (USAFT) -- 4.5.3.1 Example for N = 4 -- 4.5.3.2 Example for N = 8 -- 4.5.3.3 N = 4[1 -- 2 -- 3 -- ..., t] -- 4.5.4 Note on application of generalized function -- 4.6 Summary -- 5. Inverse Lattice Problems in Low Dimensions -- 5.1 Concept of Low Dimensional Structures -- 5.2 Linear Atomic Chains -- 5.3 Simple Example in a Square Lattice -- 5.4 Arithmetic Functions on Gaussian Integers -- 5.4.1 Gaussian integers -- 5.4.2 Unit elements, associates with reducible and irreducible integers in G -- 5.4.3 Unique factorization theorem in G -- 5.4.4 Criteria for reducibility -- 5.4.5 Procedure for factorization into irreducibles -- 5.4.6 Sum rule of Mobius functions and Mobius inverse formula -- 5.4.7 Coordination numbers in 2D square lattice -- 5.4.8 Application to the 2D arithmetic Fourier transform -- 5.4.9 Bruns version of 2D AFT and VLSI architecture -- 5.4.9.1 The derivation of Bruns version for 2D AFT -- 5.4.9.2 Example for 2D AFT computation -- 5.5 2D Hexagonal Lattice and Eisenstein Integers -- 5.5.1 Definition of Eisenstein integers -- 5.5.2 Norm and associates of an Eisenstein integer -- 5.5.3 Reducibility of an Eisenstein integer -- 5.5.4 Factorization procedure of an arbitrary Eisenstein integer -- 5.5.5 Mobius inverse formula on Eisenstein integers -- 5.5.6 Application to monolayer graphite -- 5.5.6.1 Calculation of pair potential -- 5.5.6.2 Calculated results of elastic constants -- 5.5.7 Coordination number in a hexagonal lattice -- 5.6 Summary -- 6. Inverse Lattice Problems -- 6.1 A Brief Historical Review -- 6.2 3D Lattice Inversion Problem -- 6.2.1 CGE solution -- 6.2.2 Bazant iteration -- 6.3 Mobius Inversion for a General 3D Lattice -- 6.4 Inversion Formulas for some Common Lattice Structures -- 6.4.1 Inversion formula for a fcc lattice -- 6.4.2 Inversion formula in a bcc structure.

6.4.3 Inversion formula for the cross potentials in a L12 structure -- 6.5 Atomistic Analysis of the Field-Ion Microscopy Image of Fe3Al -- 6.6 Interaction between Unlike Atoms in B1 and B3 structures -- 6.6.1 Expression based on a cubic crystal cell -- 6.6.2 Expression based on a unit cell -- 6.7 The Stability and Phase Transition in NaCl -- 6.8 Inversion of Stretching Curve -- 6.9 Lattice Inversion Technique for Embedded Atom Method -- 6.10 Interatomic Potentials between Atoms across Interface -- 6.10.1 Interface between two matched rectangular lattices -- 6.10.2 Metal/MgO interface -- 6.10.3 Matal/SiC interface -- 6.11 Summary -- Appendix : Mobius Inverse Formula on a Partially Ordered Set -- A.1 TOSET -- A.2 POSET -- A.3 Interval and Chain -- A.4 Local Finite POSET -- A.5 M obius Function on Locally Finite POSET -- A.5.1 Example -- A.5.2 Example -- A.6 Mobius Inverse Formula on Locally Finite POSET -- A.6.1 Mobius inverse formula A -- A.6.2 Mobius inverse formula B -- A.7 Principle of Inclusion and Exclusion -- A.8 Cluster Expansion Method -- Epilogue -- Bibliography -- Index.