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Part I The General Theory of Relativity and Some of Its Applications |
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Chapter 1 The Special Theory of Relativity |
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1 | (18) |
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1.1 Conflict between Newtonian mechanics and Maxwell's theory of electromagnetism |
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1 | (2) |
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1.2 The experiments of Michelson and Morley |
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3 | (1) |
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4 | (1) |
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1.4 The notion of proper time, time dilation and length contraction |
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4 | (1) |
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5 | (1) |
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1.6 The equations of mechanics in special relativity |
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6 | (1) |
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1.7 Mass, velocity, momentum and energy in special relativity, Einstein's derivation of the energy mass relation E = mc2 |
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7 | (1) |
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1.8 Four vectors and tensors in special relativity and their Lorentz transformation laws |
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8 | (2) |
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1.9 The general from of the Lorentz group consisting of boosts and rotations |
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10 | (1) |
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1.10 The Poincare group consisting of Lorentz tranformations with space-time translations |
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11 | (1) |
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1.11 Irreducible representations of the Poincare group with applications to Wigner's particle classfication theory |
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12 | (1) |
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1.12 Lorentz transformations of the electromagnetic field |
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13 | (2) |
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1.13 Relative velocity in inspecial relativity |
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15 | (1) |
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1.14 Fluid dynamics in special relativity |
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16 | (1) |
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1.15 Plasma physics and magnetohydrodynamics in special relativity |
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16 | (1) |
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1.16 Particle moving in a constant magnetic field in special relativity |
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17 | (2) |
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Chapter 2 The General Theory of Relativity |
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19 | (26) |
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2.1 Drawbacks with the special theory of relativity |
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19 | (1) |
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2.2 The principle of equivalence |
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19 | (1) |
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2.3 Why gravitational field is not a force? |
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20 | (1) |
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2.4 Four vectors and tensors in the general theory of relativity |
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21 | (1) |
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2.5 Basics of Riemannian geometry |
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22 | (15) |
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2.6 The energy-momentum tensor of matter in a background curved metric |
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37 | (1) |
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2.7 Maxwell's equations in a background curved metric |
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38 | (1) |
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2.8 The energy-momentum tensor of the electromagnetic field in a background curved metric |
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39 | (1) |
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2.9 The Einstein field equations of gravitation (i) In the absence of matter and radiation, (ii) In the presence of matter and radiation |
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40 | (1) |
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2.10 Proof of the consistency of the Einstein field equations with the fluid dynamical equations based on the Bianchi identity for the Einstein tensor |
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41 | (1) |
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2.11 The weak field limit of Einstein's field equations is Newton's inverse square law of gravitation |
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42 | (1) |
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2.12 The post-Newtonian equations of celestial mechanics, gravitation and hydrodynamics |
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42 | (3) |
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Chapter 3 Engineering Applications of General Relativity |
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45 | (62) |
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3.1 Applications of general relativity to global positioning systems |
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45 | (3) |
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3.2 General relativistic corrections to the Klein-Gordon wave propagation |
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48 | (1) |
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3.3 Calculating the effect of general relativity on the motion of a plasma with applications to estimation of the metric from the radiation field produced by the plasma in motion |
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49 | (1) |
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50 | (1) |
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3.5 Quantum theory of fields |
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51 | (15) |
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3.6 Energy-momentum tensor of matter with viscous and thermal corrections |
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66 | (3) |
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3.7 Energy-momentum tensor of the electromagnetic field in a background curved space-time |
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69 | (1) |
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3.8 Relativistic Fermi fluid in a gravitational field |
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70 | (1) |
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3.9 The post-Newtonian approximation |
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71 | (4) |
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3.10 Energy-Momentum tensor of matter with viscous and thermal corrections |
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75 | (4) |
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3.11 Energy-momentum tensor of the electromagnetic field in a background curved spacetime |
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79 | (1) |
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3.12 Relativistic Fermi fluid in a gravitational field. The Dirac equation in a gravitational field has the form |
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80 | (1) |
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3.13 The post-Newtonian approximation |
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81 | (4) |
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3.14 The BCS theory of superconductivity |
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85 | (2) |
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3.15 Quantum scattering theory in the presence of a gravitational field |
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87 | (2) |
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3.16 Maxwell's equations in the Schwarzchild space-time |
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89 | (2) |
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3.17 Some more problems in general relativity |
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91 | (10) |
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3.18 Neural networks for learning the expansion of our universe |
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101 | (1) |
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3.19 Quantum stochastic differential equations in general relativity |
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102 | (5) |
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Chapter 4 Some Basic Problems in Electromagnetics Related to General Relativity (gtr) |
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107 | (145) |
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4.1 Em waves and quantum communication |
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107 | (1) |
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4.2 Cavity resonator antennas with current source in a gravitational field |
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108 | (2) |
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110 | (2) |
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4.4 Restricted quantum gravity in one spatial dimension and one time dimension |
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112 | (1) |
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4.5 Quantum theory of fields |
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113 | (13) |
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4.6 Energy-momentum tensor of matter with viscous and thermal corrections |
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126 | (3) |
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4.7 Energy-momentum tensor of the electromagnetic field in a background curved spacetime |
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129 | (1) |
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4.8 Relativistic Fermi fluid in a gravitational field |
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130 | (1) |
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4.9 The post-Newtonian approximation |
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130 | (5) |
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4.10 The BCS theory of superconductivity |
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135 | (2) |
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4.11 Quantum scattering theory in the presence of a gravitational field |
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137 | (1) |
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4.12 Maxwell's equations in the Schwarzchild spacetime |
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138 | (3) |
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4.13 Some more problems in general relativity |
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141 | (111) |
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Chapter 5 Basic Problems in Algebra, Geometry and Differential Equations |
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252 | (21) |
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5.1 Algebra, Triangle geometry, Integration and basic probability |
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165 | (4) |
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169 | (1) |
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5.3 Brownian motion simulation |
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169 | (1) |
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170 | (1) |
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170 | (1) |
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5.6 Hamiltonian mechanics from Lagrangians |
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170 | (1) |
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5.7 Rate of a chemical reaction |
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171 | (1) |
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5.8 Linearization of the Navier-Stokes Fluid equations with gravitational self interaction |
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171 | (1) |
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5.9 Wave equations in mechanics |
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172 | (1) |
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5.10 Surface of revolution |
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172 | (1) |
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5.11 1-D Schrodinger equation |
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172 | (1) |
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5.12 Lagrange's triangle in mechanics |
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173 | (1) |
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174 | (1) |
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5.14 Blurring of 3-D objects in random motion |
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175 | (1) |
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5.15 Commutators of products of matrices |
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176 | (1) |
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5.16 Path of a light ray in an medium having inhomogeneous refractive index |
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176 | (1) |
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176 | (1) |
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177 | (1) |
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5.19 Jacobian formula for multiple integrals |
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178 | (1) |
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5.20 Existence of only five regular polyhedra in nature |
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179 | (1) |
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5.21 Definition of the derivative and its properties |
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180 | (1) |
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5.22 Pattern recognition using group representations |
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181 | (6) |
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5.23 Using characters of group representations to estimate the group transformation element |
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187 | (1) |
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5.24 Explicit formulas for the induced representation for semidirect products of finite groups |
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188 | (1) |
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5.25 Applications of the Extended Kalman filter and the Recursive Least Squares Algorithm to System Identification Problems using Neural Networks |
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189 | (26) |
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5.26 Application of neural networks to the gravitational metric estimation problem |
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215 | (1) |
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5.27 Problems in quantum scattering theory |
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216 | (1) |
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217 | (1) |
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5.29 Estimating the metric parameters from geodesic measurements |
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217 | (1) |
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5.30 Perturbations to the band structure of semiconductors |
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218 | (1) |
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5.31 Scattering into cones for Schrodinger Hamiltonians |
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218 | (1) |
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5.32 Study projects involving conventional field theory in curved background metrics |
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219 | (5) |
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5.33 Intuitive explanation of an invariance principle in scattering theory |
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224 | (1) |
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5.34 Scattering theory for the Dirac Hamiltonian in curved space-time |
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225 | (1) |
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5.35 Derivation of the approximate Schrodinger Hamiltonian for a particle in curved spacetime with corrections upto fourth order in the space derivatives |
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226 | (1) |
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5.36 Quantum scattering theory in the presence of time dependent Hamiltonians arising in general relativity |
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226 | (2) |
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5.37 Band structure of a semiconductor altered by a massive gravitational field |
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228 | (1) |
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5.38 Design of quantum gates using quantum physical systems in a gravitational field |
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229 | (1) |
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5.39 Quantum phase estimation |
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230 | (1) |
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5.40 Noisy Schrodinger equations, pure and mixed states |
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231 | (1) |
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5.41 Constructions using ruler and compass |
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232 | (1) |
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5.42 Application of the Jordan canonical form for matrices in general relativity |
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232 | (1) |
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5.43 Application of the Jordan canonical form in solving fluid dynamical equations when the velocity field is a small perturbation of a constant velocity field |
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233 | (1) |
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5.44 The Jordan canonical form |
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233 | (1) |
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5.45 Some topics in scattering theory in L2(Rn) |
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234 | (2) |
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5.46 MATLAB problems on applications of linear algebra to signal processing |
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236 | (2) |
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5.47 Applications of the RLS lattice algorithms to general relativity |
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238 | (1) |
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5.48 Knill-Laflamme theorem on quantum coding theory, a different proof |
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239 | (2) |
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5.49 Ashtekar's quantization of gravity |
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241 | (4) |
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5.50 Example of an error correcting quantum code from quantum mechanics |
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245 | (1) |
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5.51 An application of the Jordan canonical form to noisy quantum theory |
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246 | (1) |
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5.52 An algorithm for computing the Jordan canonical form |
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246 | (1) |
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5.53 Rotating blackhole analysis using the tetrad formalism |
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247 | (1) |
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5.54 Maxwell's equations in the rotating blackhole metric |
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247 | (1) |
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5.55 Some notions on operators in an infinite/finite dimensional Hilbert space |
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248 | (2) |
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5.56 Some versions of the quantum Boltzmann equation |
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250 | (23) |
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Part II Quantum Mechanics |
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1 The De-Broglie Duality of particle and wave properties of matter |
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273 | (1) |
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2 Bohr's correspondence principle |
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273 | (1) |
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3 Bohr-Sommerfeld's quantization rules |
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274 | (1) |
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4 The principle of superposition of wave functions and its application to the Young double slit diffraction experiment |
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275 | (1) |
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5 Schrodinger's wave mechanics and Heisenberg's matrix mechanics |
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276 | (2) |
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6 Dirac's replacement of the Poisson bracket by the quantum Lie bracket |
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278 | (1) |
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7 Duality between the Schrodinger and Heisenberg mechanics based on Dirac's idea |
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278 | (2) |
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8 Quantum dynamics in Dirac's interaction picture |
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280 | (1) |
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9 The Pauli equation: Incorporating spin in the Schrodinger wave equation in the presence of a magnetic field |
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281 | (1) |
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281 | (1) |
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11a The spectrum of the Hydrogen atom |
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282 | (2) |
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11b The spectrum of particle in a 3 -- D box |
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284 | (1) |
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11c The spectrum of a quantum harmonic oscillator |
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285 | (1) |
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12 Time independent perturbation theory |
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285 | (2) |
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13 Time dependent perturbation theory |
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287 | (1) |
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14 The full Dyson series for the evolution operator of a quantum system in the presence of a time varying potential |
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288 | (1) |
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15 The transition probabilities in the presence of a stochastically time varying potential |
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288 | (1) |
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16 Basics of quantum gates and their realization using perturbed quantum systems |
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289 | (1) |
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17 Bounded and unbounded linear operators in a Hilbert space |
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290 | (1) |
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18 The spectral theorem for compact normal and bounded and unbounded self-adjoint operators in a Hilbert space |
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291 | (2) |
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19 The general theory of Events, states and observables in the quantum theory |
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293 | (3) |
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20 The evolution of the density operator in the absence of noise |
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296 | (1) |
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21 The Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) equation for noisy quantum systems |
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296 | (1) |
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22 Distinguishable and indistinguishable particles |
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296 | (1) |
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23 The relationship between spin and statistics |
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296 | (1) |
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24(a) Tensor products of Hilbert spaces |
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296 | (1) |
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24(b) Symmetric and antisymmetric tensor products of Hilbert spaces, the Fock spaces |
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297 | (2) |
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24(c) Coherent/exponential vectors in the Fock spaces |
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299 | (1) |
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25 Creation, Conservation and Annihilation Operators in the Boson Fock Space |
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300 | (1) |
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26 The general theory of quantum stochastic processes in the sense of Hudson and Parthasarathy |
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300 | (1) |
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27 The quantum Ito formula of Hudson and Parthasarathy |
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301 | (1) |
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28 The general theory of quantum stochastic differential equations |
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301 | (1) |
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29 The Hudson-Parthasarathy noisy Schrodinger equation and the derivation of the GKSL equation from its partial trace |
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301 | (1) |
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30 The Feynman path integral for solving the Schrodinger equation |
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301 | (1) |
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31 Comparison between the Hamiltonian (Schrodinger-Heisenberg) and Lagrangian (path integral) approaches to quantum mechanics |
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302 | (1) |
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32 The quantum theory of fields |
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303 | (24) |
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33 Dirac's wave equation in a gravitational field |
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327 | (1) |
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34 Canonical quantization of the gravitational field |
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327 | (2) |
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35 The scattering matrix for the interaction between photons, electrons, positrons and gravitons |
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329 | (1) |
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36 Atom interacting with a Laser |
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329 | (4) |
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37 The classical and quantum Boltzmann equations |
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333 | (2) |
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38 Bands in a semiconductor |
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335 | (2) |
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39 The Hartree-Fock apporoximate method for computing the wave functions of a many electron atom |
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337 | (1) |
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40 The Born-Oppenheimer approximate method for computing the wave functions of electrons and nuclei in a lattice |
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338 | (2) |
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41 The performance of quantum gates in the presence of classical and quantum noise |
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340 | (1) |
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42 Design of quantum gates by applying a time varying electromagnetic field on atoms and oscillators |
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341 | (1) |
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43 Solution of Dirac's equation in the Coulomb potential |
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342 | (1) |
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44 Dirac's equation in general radial potentials |
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342 | (6) |
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45 The Schrodinger equation in an electromangetic field described as a quantum stochastic process |
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348 | (1) |
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46 Dirac's equation in an electromagnetic field described as a quantum stochastic process |
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348 | (2) |
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47 General Scattering theory, the Moller and wave operators, the scattering matrix, the Lippman-Schwinger equation for the scattering matrix, Born scattering |
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350 | (2) |
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48 Design of quantum gates using time dependent scattering theory |
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352 | (1) |
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49 Evans-Hudson flows and its application to the quantization of the fluid dynamical equations in noise |
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353 | (2) |
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50 Classical non-linear filtering |
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355 | (1) |
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51 Derivation of the extended Kalman filter (EKF) as an approximation to the Kushner filter |
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356 | (1) |
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52 Belavkin's theory of non-demolition measurements and quantum filtering in coherent states based on the Hudson-Parthasarathy Boson Fock space theory of quantum noise, The quantum Kallianpur-Striebel formula |
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357 | (4) |
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53 Classical control of a stochastic dynamical system by error feedback based on a state observer derived from the EKF |
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361 | (1) |
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54 Quantum control using error feedback based on Belavkin quantum filters for the quantum state observer |
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361 | (2) |
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55 Lyapunov's stability theory with application to classical and quantum dynamical systems |
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363 | (1) |
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56 Imprimitivity systems as a description of covariant observables under a group action |
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363 | (2) |
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57 Schwinger's analysis of the interaction between the electron and a quantum electromagnetic field |
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365 | (1) |
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366 | (1) |
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59 Quantum error correcting codes |
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367 | (8) |
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60 Quantum hypothesis testing |
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375 | (3) |
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61 The Sudarshan-Lindblad equation for observables in an open quantum system |
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378 | (1) |
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62 The Yang-Mills field and its quantization using path integrals |
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379 | (2) |
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63 A general remark on path integral computations for gauge invariant actions |
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381 | (2) |
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64 Calculation of the normalized spherical harmonics |
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383 | (3) |
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65 Volterra systems in quantum mechanics |
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386 | (5) |
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66a RLS lattice algorithms for quantum observable estimation |
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391 | (1) |
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66b Quantum scattering theory, the wave operators and the scattering matrix |
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392 | (3) |
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67 Quantum systems driven by Stroock-Varadhan martingales |
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395 | (258) |
| Appendix |
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653 | (2) |
| References |
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655 | (13) |
| Index |
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668 | |
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