Stochastics, Control and Robotics

Description

This book discusses various problems in stochastic Processes, Control Theory, Electromagnetics, Classical and Quantum Field Theory & Quantum Stochastics. The problems are chosen to motivate the interested reader to learn more about these subjects from other standard sources. Stochastic Process theory is applied to the study of differential equations of mechanics subject to external noise. Some issues in general relativity like Geodesic motion, field theory in curved space time etc. are discussed via isolated problems. The more recent quantum stochastic process theory as formulated by R.L. Hudson and K. R. Parathasarathy is discussed. This provides a non commutative operator theoretic version of stochastic process theory. V.P. Belavkin's approach to quantum filtering based on non demolition measurements and Hudson Parathasarathy calculus has been discussed in detail. Quantum versions of the simple exclusion model in Markov process theory have been included. 3D Robots carring a current density interacting with an external Klein- Gordon or Electromagnetic field has been given some attention. The readers will after going through this book, be ready to carry out independent research in classical and quantum field theory and stochastic processes as applied to practical problems. Cover Half Title Title Page Copyright Page Brief Contents Table of Contents Preface 1. Classical Robotics and Quantum Stochastics [1] A Problem in Robotics with Current Carrying Links [2] Quantum Filtering [3] Evan’s Hudson Flow [4] Quantum Version of a Gauss-Markov Process [5] Problems on the δ-Function [6] Conditional Expectation in Quantum Filtering [7] K.R. Parthasarathy Quantum Markov Processes [8] Quantum Stochastic Lyapunov Theory [9] Let P be a Solution to the Martingale Problem Π(x, a, b), i.e. [10] Problem from Revuz and Yor [11] Gough and Kostler Paper on Quantum Filtering Some Remarks [12] Quantization of Master. Slave Robot Motion Using Evans-Hudson Flows [13] Problem From KRP QSC [14] Moment Generating Function for Certain Quantum Random Variables [15] Belavkin Filtering Contd. [16] Modelling Noise in Quantum Mechanical Systems [17] Belavkin has Derived a Quantum Stochastic Differential Equation (qsde) for X̂(t|t) [18] Waveguide Modes [19] Estimating the Initial State from Measurements of an Observable after the State Passes through a Sequence of Quantum Channels [20] Quantum Poisson Random Variable [21] Quantum Relative Entropy Evolution [22] Gauge Group Invariance of a Wave Eqn. [23] Problem [24] Quantum Oscillator Perturbed by an Electromagnetic Field [25] Quantization of Mechanical Systems [26] Quantum Filtering [27] Perspective Projection by Camera Attached to a Robot [28] Proof of The Positivity of Quantum Relative Entropy [29] Entropy of a Quantum System [30] Dirac Eqn. Based Temperature Estimation of Black-Body Temperature Field [31] Quantum Image Processing on a C∞ – Manifold [32] Belavkin Contd. (Quantum Non-Linear Filtering) [33] Remarks on Theorems from V. Kac. Infinite Dimensional Lic Algebra. P.87 [34] Quantum Mutual Information [35] Scattering Theory Applied to Quantum Gate Design [36] Quantization of the Simple Exclusion Model [37] Quantum Version of Nearest Neighbour Interactions [38] Remarks and Comments on “Collected Papers of S.R.S Varadhan” Vol. 4, Particle System and Their Large Deviations [39] Quantum String Theory [40] Quantum Shannon Theory Contd. [41] Basics of Cq-Information Theory [42] Typical Sequences and Error Probability in Classical and Quantum Coding [43] Quantum Image Processing Some Basic Problems 2. Electromagnetus and Related Partial Differential Equation [1] Computation of the Perturbe Characteristics Frequencies in a Cavity Resonator with in Homogeneous Dielectric and Permittivity [2] Numerical Methods for pde [3] Tracking of Moving Targets Using a Camera Attached to the Tip of a 2 = Link Robot [4] Problem [5] n-Dimensional Helmholtz Green’s Funtion [6] Edge Diffusion [7] Waves in Metamaterials [8] Stroock Partial Differential Equations for Probabilities φ ≃ Function, u ≃ Sobolev Distribution 3. Radon and Group Theoretic Transforms with Robotics Applications [1] Modern Trends in Signal Processing by Harish Parthasarathy, NSIT [2] Radon-Transform Based Image Processing [3] Image Processing Using Invariants of the Permutation Group [4] Radon Transform of Rotated and Translated Image Field [5] Estimating the Rotation Applied to the 3-D Robot Links from Electromagnetic Field Pattern [6] Kinetic Energy of a Two Link Robot [7] Vector Potential Generated by a 2-Link Robot [8] Group Associated with Robot Motion [9] Radon Transform [10] Campbelt Baker-Hausdorff Formula [11] Problem on Rigid Body Motion [12] Remarks on Theorems from V. Kac. Infinite Dimensional Lic Algebra. P.87 [13] Theta a Function From Group Theory P 252 (V.Kac) [14] Iwasava Decomposition 4. Stochastic Filtering and Control, Interacting Particles [1] Bernoulli Filters [2] A Problem in Large Deviation Theory [3] Large Deviations in Image Trajectory on a Screen Taken by a Robot Camera in Motion [4] P406 Revuz and Yor [5] Problem from “Continuous Martingales and Brownian Motion” by Revuz and Yor [6] Problem from Revuz and Yor [7] Collected Papers of Varadhan-Interacting Particle System, Vol. 4. Hamiltonian System and Hydrodynamical Equations [8] (Ph.D. Problem of Rohit) For Rohit and Dr. Vijayant. Skorohod Optional Stochastic Control. Optimal Stochastic Control of Master and Slave Robot [9] English-Chapter-1 [10] Markov Processes-Some Application [11] Comments and Proofs on “Particle System” Collected Papers of S.R.S. Varadhan Vol. 4 [12] Interacting Diffusions [13] This is Independent of s. It is Reasonable to Expert that at Equilibrium [14] Wong Zakai Filtering Equation [15] Stochastic Control of Finance Market [16] Wong-Zokai Equation [17] Filtering Theory with State and Measurement Noises Having Correlation 5. Classical and Quantum Robotics [1] Mathematical Pre-Requisites for Robotics [2] Belavkin Filtering Contd. [3] Dirac Eqn., for Rigid Body Robot (Single Link) [4] Quantum Robot Dynamics [5] Hamiltonian Based Quantum Robot Tracking [6] Robot Interacting with a Klein-Gordon Field 6. Large Deviations, Classical and Quantum General Relativity with GPS Application [1] Intuitive Derivation of the Gartner Ellis LDP [2] LDP in Robotic Vision [3] LMS Algorithm for Non-Linear Disturbance Observer in a Two-Link Robot System [4] Disturbance Observer Design in Robotics [5] Disturbance Observer Continuation [6] Results in the LMS’ Iteration [7] Notation for ψ(t) in the LMS Disturbance Observer [8] Approximate Convergence Analysis [9] Statistical Model for Disturbance Estimation Error Based on LMS [10] Statistics of LMS Coefficients for the Disturbance Observer [11] Ḇ(t) ∈ ℝd d-Vector Valued Standard Brownian Motion [12] General Relativistic Correction to GPS [13] Thiemann Modern QGTR [14] Generalization of General Relativistic String Theory [15] Symmetric Space-Times in General Relativity and Cosmology [16] Selected Topics in General Relativity [17] Perturbations in the Tetrad Caused by Metric Perturbation [18] Geodesic Equation in Tetra Formalism [19] Quantum General Relativistic Scattering [20] Derive Dirac Equation in Curved Space-Time Using the Newman Penrose Formalism: Special-Relativity [21] Proofs of Identities from S. Chandra Sekhar, “The Mathematical Theory of Blackholes”. Cartain’s Second Equation of Structure [22] Mathematical Preliminaries for Cartan’s Eqn. of Structure [23] Reflection and Refraction at a Metamaterial Surface [24] Maxwell’s Equations in an Unhomogeneous Medium with Background Gravitation Taken into Account [25] Relativistic Kinematics* Motions of a Rigid Body in a Gravitational Field a General Relativistic Calculation [26] Random Fluctuation in the Gravitational Field Produced by Random Fluctuation in the Energy-Momentum Tensor of the Matter Plus Radiation Field [27] Special Relativistic Plasma Physics 7. Quantum Signal Processing [1] Quantum Scattering of a Rigid Body [2] Optimal Control of a Field [3] Example from Quantum Mechanics [4] Example from Quantum Field Theory [5] Quantum Signal Processing (Contd.) [6] em Field Theory Based Image Processing [7] Derivation of the Kallianpur-Striebel Formula [8] Slave Dynamics with Parametric Uncertainties Using Adaptive Controllers [9] Belavkin’s Theory of Quantum Non-Linear Filtering [10] A Problem Related to the Generator of Brownian Motion [11] A Problem Related to Poisson Random Fields [12] Relativistic Kinematics of Rigid Bodies [13] Quantum Robotics Disturbance Observer [14] Quantum Field Theory in the Presence of Stochastic Disturbance [15] Simultaneous Tracking, Parametric Uncertainties Estimation and Disturbance Observer in a Single Robot [16] Quantum Filtering to Signal Estimation and Image Processing [17] Image Transmission Through Quantum Channels [18] Entanglement Theory in Quantum Mechanics [19] Some Other Remarks Related to The Above Problem [20] Image Processing from the Em Field View Point [21] CAR [22] An Identity Concerning Positive Definite Functions on a Group [23] Example of Construction of a Spherical Function [24] Spherical Functions in Image Processing [25] CAR from CCR (KRP Quantum Stochastic Calculus) [26] Spherical Function (Helgason Vol. II) [27] Clarifications of Problems Related to the I to measure of Brownian motion [28] Problem from Revuz and Yor [29] Group Theory and Robotics [30] Suppose ∇2 f(x) = 0, x ∈ D [31] Entropy Evolution in Sudarshan-Lindblad Equation [32] Problem on Rigid Body Motion [33] Problem from Revuz and Yor