NEW TOPICS IN MATHEMATICALPHYSICS RESEARCH -- CONTENTS -- PREFACE -- NONTRIVIAL SYMMETRIES OF dQrevAND EQUILIBRIUM THERMODYNAMICS -- Abstract -- 1 Introduction -- 2 Symmetry and Pfaffian Forms -- 3 Algebraic Structure of Symmetries of w -- 4 A General Framework for Thermodynamics -- 4.1 Pfaffian Form and Fundamental Equation -- 4.2 General Properties of S -- 4.3 Closure Properties of the Domain [16] -- 4.4 m(S) Z(T) -- 4.5 Pfaffian Form and Symmetries -- 5 Existence of a "Local" Nondegenerate Nontrivial Symmetryin Thermodynamics -- 5.1 Transverse Symmetry and Integrating Factors -- 5.2 ˆXL when T is an Independent Variable -- 6 General Nontrivial Symmetry -- 6.1 Metrical Entropy -- 6.2 Generalized Gibbs-Duhem Equation -- 7 Set of Constructive Assumptions -- 8 Quasihomogeneity -- 8.1 A Route to Quasihomogeneity -- 8.2 Gibbs-Duhem Equation Revisited -- 9 Dual Base Symmetry -- 10 Homogeneity in Standard Thermodynamics: the Paradigm -- 10.1 Metrical Entropy S -- 10.2 Corroboration from Gibbs' Approach -- 10.3 Quasihomogeneous Potentials in Standard Thermodynamics -- 10.4 Fundamental Relations, State Equations, and Legendre Transforms -- 10.5 Gibbs-Duhem Equations -- 11 Quasihomogeneous Thermodynamics -- 11.1 Possible Ambiguities -- 11.2 Generalized Gibbs-Duhem Equation -- 11.3 From the Entropy to the Energy Representation -- 11.4 Legendre Transforms -- 11.5 Newtonian Gravity and General Relativity. A Model of Hertel,Narnhofer and Thirring, Chavanis' Studies, and Thermal Geons -- 11.5.1 HNT Model -- 11.5.2 Chavanis' Studies -- 11.5.3 The Thermal Geon -- 11.6 The Black Hole Case -- 12 Conclusions -- References -- SUPERSELECTION RULES INDUCEDBY INFRARED DIVERGENCE -- Abstract -- 1 Introduction -- 2 Induced Superselection Rules -- 2.1 General Considerations -- 2.2 Models -- 3 The Interaction with a Boson Field -- 3.1 The Hamiltonian.

3.2 Coherent States as Reference State -- 3.3 Arbitrary Normal States as Initial State -- 3.4 KMS States as Reference States -- 4 Scattering Processes -- 5 Conclusion -- A Estimates of Operators -- B The van Hove Model -- B.1 The Hamiltonian -- B.2 Evaluation of the Traces -- References -- RIEMANN SURFACES OF SOME STATICDISPERSION RELATION AND PROJECTIVE SPACES -- Abstract -- 1 Introduction -- 2 Analytic Continuation of the S-Matrixto Nonphysical Sheets -- 3 Formulation of the Problem in Projective Spaces -- 4 Conclusion -- Appendix 1 -- Appendix 2 -- References -- THE LIE DERIVATIVE OF SPINOR FIELDS:THEORY AND APPLICATIONS -- Abstract -- Introduction -- 1 Principal Bundles -- 2 Reductive G-structures and their Prolongations -- 3 Gauge-Natural Bundles -- 4 Split Structures on Principal Bundles -- 5 Lie Derivatives on Reductive G-structures -- 6 The G-Killing Condition -- 7 Penrose's Lie Derivative of "Spinor Fields" -- 8 An Application to the Calculus of Variations -- Conclusions -- References -- TOWARDS THE QUANTUM ELECTRODYNAMICSON THE POINCARÉ GROUP -- Abstract -- 1 Introduction -- 2 The Group SU(2) -- 2.1 Harmonic Analysis on the Group SU(2) -- 3 The Group SU(1,1) -- 3.1 Harmonic Analysis on the Group SU(1,1) -- 4 The Group SL(2,C) -- 4.1 Associated Hyperspherical Functions and a Two-dimensional ComplexSphere -- 4.2 Matrix Elements of Principal and Supplementary Seriesof Representations -- 5 Harmonic Analysis on the Group SL(2,C) -- 6 Fields on the Poincaré Group -- 6.1 Harmonic Analysis on SU(2)⊗SU(2) T4 -- 7 Lagrangian Formalism and Field Equations on the Poincar éGroup -- 7.1 Boundary Value Problem -- 8 The Dirac Field -- 8.1 Quantization -- 9 The Maxwell Field -- 9.1 Quantization -- 10 Interacting Fields -- 11 Conclusion -- Acknowledgement -- References -- GENERALIZED LANDEN TRANSFORMATIONFORMULAS FOR JACOBI ELLIPTIC FUNCTIONS -- Abstract.

1 Introduction -- 2 Generalized Landen Formulas -- 2.1 "dn" Landen Formulas -- 2.2 "cn" Landen Formulas -- 2.3 "sn" Landen Formulas -- 3 Alternative Forms for Landen Transformations -- 3.1 Relation between the Transformed Modulus Parameters m˜i and m -- 3.2 Alternative Forms of Landen Transformation Formulas -- 4 Landen Transformations for Products of JacobiElliptic Functions -- 4.1 Any Integer p -- 4.2 Odd Integer Case -- 4.3 Even Integer Case -- 5 Gauss Transformation Formulas -- 6 Landen Transformation Formulas and Cyclic Identities -- 7 Conclusion -- Acknowledgment -- References -- ON NON-ORTHOGONALSIGNAL REPRESENTATION -- Abstract -- 1 Introduction -- 2 Frames -- 2.1 Frames in Finite Dimension -- 2.1.1 Building Orthogonal Projectors -- 3 Recursive Biorthogonalisation Approach for ConstructingOrthogonal Projectors -- 3.1 Forward Biorthogonalisation -- 3.2 Backward Biorthogonalisation -- 4 Atoms Selection -- 4.1 Optimised Orthogonal Matching Pursuit Approach -- 4.2 Backward Optimised Orthogonal Matching Pursuit Approach -- 4.3 Numerical Example -- 5 Using Redundancy for Embedding Encrypted Information -- 5.1 Numerical Examples -- 6 Compressing Redundancy -- 6.1 Indices Selection -- 6.2 Reducing Lagrange Multipliers -- 6.3 Data Expansion and Interpolation -- 6.4 Numerical Example -- 7 Conclusions -- Acknowledgements -- A Appendix -- References -- FICTITIOUS CHARGES AS A UNIVERSAL TOOLIN THE PROBLEM OF LATTICE SUMMATIONOF COULOMB AND MULTIPOLE SERIES -- Abstract -- 1 Introduction -- 2 Direct Summation Problems of Coulomb Series -- 3 Specifications of Triclinic Geometry -- 4 Shell-wise Summation and Periodic Boundary Conditions -- 5 Translation-Invariant Bulk Potentials -- 6 General Topological Effects in Infinite Crystals -- 7 Lorentz Field -- 8 Topological Potential as a Lorentz-Tensor Effect -- 9 Slice-wise Summation and Other Limiting Events.

10 Practical Efficiency of Shell-wise Summation -- 11 Jellium Model and Dipolar Sums -- 12 Conclusion -- A Characteristic Triclinic Surface Integral -- B Coulomb Interactions of Cubes Charged Uniformly -- References -- SEPARATION OF VARIABLES AND EXACTSOLUTION OF THE DIRAC EQUATIONSIN SOME COSMOLOGICAL BACKGROUNDS -- Abstract -- 1 Introduction -- 2 Hamilton-Jacobi Equation -- 3 Klein-Gordon Equation -- 4 Dirac Equation -- 5 Algebraic Method of Separation of Variables -- 6 Exact Solution in Some Cosmological Spaces -- 7 RelativisticWave Equations in a Spatially Open Universe -- 7.1 Solution of the Hamilton-Jacobi Equation -- 7.2 Solution of the Klein Gordon Equation -- 7.3 Solution of the Dirac Equation -- 8 Conclusions and Final Remarks -- Acknowledgments -- References -- O(3) NONLINEAR SIGMA MODEL, HOPF MAP AND THE KNOTINVARIANT -- 1 INTRODUCTION -- 2 TOPOLOGICAL LINEAR DEFECTS IN THE NONLINEAR SIGMA MODEL -- 2.1 Soliton in the NSLM -- 2.2 Knot in the NSLM -- 3 TOPOLOGICAL POINT DEFECTS IN THE NLSM -- 4 CONCLUSION -- ACKNOWLEDGMENT -- REFERENCES -- HERMITIAN MODIFICATIONS OF TOEPLITZLINEAR FUNCTIONALSAND ORTHOGONAL POLYNOMIALS -- Abstract -- 1 Introduction -- 2 Quasi-definite Hermitian modified functionals -- 2.1 Lemma -- 2.2 Corollary -- 2.3 Definition ([1]) -- 2.4 Proposition -- 2.5 Theorem -- 2.6 Remarks -- 2.7 Theorem -- 2.8 Proof -- 2.9 Corollary -- 2.10 Lemma -- 2.11 Theorem -- 2.12 Corollary -- 3 Examples. -- Acknowledgements -- References -- INDEX.