The department offers number of undergraduate and postgraduate courses to other constituent colleges of the university. 

Name of the Program	No of seats	Mode of admission
UG Program
5 year Integrated M.Sc. (Physics)	10	CET
PG Program
M.Sc. (Mathematics)	7	MET
M.Sc. (Statistics)	7+2(Nominated by ICAR)	MET
M.Sc. (Physics)	10	MET

List of Courses for 5 year Integrated M.Sc. (Physics):

Course No	Course Title	Cr. Hr.
Phys. 102	Optics	3+1
Phys. 103	Mechanics	3+1
Phys. 104	Waves and Oscillations	3+0
Phys. 202	Fundamentals of Biophysics	2+1
Phys. 203	Engineering Physics	2+1
Phys. 204	Fundamentals of Quantum Physics	3+0
Phys. 205	Statistical Physics	3+0
Phys. 206	Thermal Physics	3+1
Phys. 207	Electricity and Magnetism-I	3+1
Phys. 208	Electricity and Magnetism-II	3+0
Phys. 301	Introduction to Solid State Physics	3+0
Phys. 303	Electronics-I	3+1
Phys. 305	Nuclear Physics-I	3+0
Phys. 402	Electronics-II	3+0
Phys. 404	Nuclear Physics-II	3+0
Phys. 406	Elements of Modern Physics	2+1
Math. 105	Calculus I	3+0
Math. 213	Differential Equations	3+0
Math. 214	Algebra and Geometry	3+0
Math. 215	Calculus-II	3+0
Math. 216	Vector Calculus	3+0
Math. 302	Number Theory	3+0
Math. 303	Real Analysis	3+0
Math. 304	Fundamentals of numerical analysis	2+1
Math. 305	Linear Algebra	3+0

List of Courses for M.Sc. (Mathematics):

Course No	Course Title	Cr. Hr.
Math.-521	Algebra-I	3+0
Math.-522	Complex Analysis	3+0
Math.-523	Algebra-II	3+0
Math.-524	Ordinary and Partial Differential Equations	3+0
Math.-525	Elements of functional Analysis	2+0
Math.-526	Numerical Methods	2+1
Math.-527	Calculus of Variation and Operations	2+0
Math.-528	Real Analysis	2+0
Math.-529	Principles of Topology	2+0
Math.-530	Transformation Theory	2+0
Math.-531	Measure Theory	2+0
Math.-532	Classical Mechanics	2+0
Math.-535	Operational Methods	2+1

List of Courses for M.Sc. (Statistics):

Course No	Course Title	Cr. Hr.
Stat.-551	Mathematics-I	3+0
Stat.-552	Probability Theory	2+0
Stat.- 553	Statistical Methods	2+1
Stat.-554	Actuarial Statistics	2+0
Stat.-555	Bioinformatics	2+0
Stat.-556	Econometrics	2+0
Stat.-561	Mathematics-II	2+0
Stat.-562	Statistical Inference	2+1
Stat.-563	Design Of Experiments	2+1
Stat.-564	Sampling Techniques	2+0
Stat.-565	Statistical Genetics	2+1
Stat.-566	Statistical Quality Control	2+0
Stat.-567	Optimization Techniques	1+1
Stat.-571	Multivariate Analysis	2+1
Stat.-572	Regression Analysis	1+1
Stat.-573	Statistical Computing	1+1
Stat.-574	Time Series Analysis	1+1
Stat.-575	Demography	2+0
Stat.-576	Statistical Methods For Life Sciences	2+0

List of Courses of M.Sc. (Physics):

Course No	Course Title	Cr. Hr.
Phys.- 501	Classical and Statistical Mechanics	4+0
Phys.-502	Electronics	2+1
Phys.-503	Quantum mechanics	2+1
Phys.-504	Electrodynamics	3+0
Phys.-505	Solid State Physics	3+1
Phys.-506	Nuclear and Particle Physics	3+1
Phys.-507	Nuclear Techniques in Agriculture	2+1
Phys.-508	Material Science	2+1
Phys.-509	Radiation Physics	2+1
Phys.- 515	Modern Physics	2+1
Phys.- 516	Atomic and Molecular Spectroscopy	2+0
Phys.-517/Math.-533	Mathematical Physics-I	3+0
Phys.-518/Math.-534	Mathematical Physics-II	2+1

Course contents for 5 year Integrated M.Sc. (Physics): Phys. 102 Optics                                                                                     3+1                         Sem. II

Electromagnetic nature of light. Definition and properties of wave front. Huygens Principle. Division of amplitude and wavefront. Spatial and temporal coherence, Young’s double slit experiment. Lloyd’s Mirror and Fresnel’s Biprism. Phase change on reflection: Newton’s rings,Michelson interferometer—working, principle and nature of fringes. Interference in thin films, Role of interference in anti-reflection. Multiple beam interference, Fabry-Perot interferometer, nature of fringes. Fresnel’s Assumptions. Fresnel’s Half-Period Zones for Plane Wave. Explanation of Rectilinear Propagation of Light. Theory of a Zone Plate: Multiple Foci of a Zone Plate. Fresnel’s Integral, Fresnel diffraction pattern of a straight edge, a slit and a wire. Fraunhofer diffraction: Single slit. Circular aperture, Resolving Power of a telescope. Double slit. Multiple slits. Diffraction grating. Resolving power of grating. Concept and analytical treatment of unpolarized, plane polarized and elliptically polarized light. Double refraction, Nicol prism, sheet polarisers, retardation plates. Production and analysis of polarized light (quarter and half wave plates). Polarization by reflection, Malus law, Brewster’s angle.   Practicals: Refractive index and dispersive power of the prism using spectrometer. Calibration of prism spectrometer. Newton’s rings. Polarimeter. Diffraction grating. Resolving power of telescope and grating. Ostwald viscometer. Planck’s constant using photovoltaic cell.   Phys. 103 Mechanics                                                                              3+1                         Sem. I

Cartesian and spherical polar co-ordinate systems: area, volume, displacement, velocity and acceleration in these systems, solid angles; Centre of mass system, principle of conservation of momentum about the Centre of mass, Conservation of Energy; Galilean transformation, Galilean Invariance of space & time intervals; Symmetries of space and time; Rigid body motion, Euler’s equation, elementary gyroscope. Fictitious forces. Effect of rotation of earth on ‘g’. Effects of centrifugal and Coriolis forces produced as a result of earth’s rotation. Foucault’s pendulum and its equation of motion. Michelson-Morley Experiment and its outcome. Forces in nature (qualitative). Central forces, Potential energy and force between a point mass and spherical shell, a point mass and solid sphere, gravitational and electrostatic self-energy. Two body problem and concept of reduced mass. Motion of a body under central force; differential equation of the orbit, equation of orbit in inverse-square force field. Kepler’s laws and their derivation, Shape of Galaxy. Elastic collisions in laboratory and center of mass systems; velocities, angles, energies in these systems and their relationships. Rutherford scattering. Practicals: Use of vernier-caliper for volume of different shapes,  screw-gauge, volume of irregular lamina, spherometer, acceleration due to gravity, moment of inertia, co-efficient of friction, compound pendulum, Ketler’s pendulum, elastic Constants of a wire by Searle’s method, Moment of Inertia of a Flywheel, Motion of Spring and calculate (a) Spring constant, (b) g and (c) Modulus of rigidity, Coefficient of Viscosity of water by Capillary Flow Method, coefficient of viscosity of a given liquid by Stoke’s method, one dimensional collision using two hanging spheres of different materials. Melde’s experiment   Phys. 104 Waves and Oscillations                                                          3+0                         Sem. II

Simple harmonic motion, energy of a SHM, Compound Pendulum, Torsional Pendulum, Electrical Oscillations, Transverse Vibrations of a mass on a string, composition of two perpendicular SHM of same period and of period in ratio 1: 2. Decay of free vibrations due to damping, differential equation of motion, types of damping, determination of damping co-efficient; Logarithmic decrement, relaxation time and Q- Factor. Differential equation for forced mechanical and electrical oscillators, Transient and steady state behavior. Displacement and velocity variation with driving force frequency, variation of phase with frequency, resonance. Power supplied to an oscillator and its variation with frequency. Q-value and band width. Q-value as an amplification factor. Stiffness, coupled oscillators, Normal co-ordinates and normal modes of vibration, Inductance coupling of electrical oscillators. Waves in physical media, Wave equation and its solution, Types of waves, particle velocity, acceleration and energy in progressive waves. Longitudinal waves on a rod. Transverse waves on a string, characteristic impedance of a string, Waves in absorbing media. Reflection and Transmission of transverse waves on a string at discontinuity, Reflection and transmission of energy. Reflection and transmission of longitudinal waves at a boundary. Standing wave ratio, Impedance matching, Energy of vibrating string. Wave and group velocity.    Phys. 202 Fundamentals of Biophysics                                                  2+1                         Sem. II

Quantum mechanics- electronic structure of atoms, wave particle duality, wave length of de-Broglie waves, phase and group velocity. Some basic concepts of quantum mechanics. Schrodinger’s wave equations. Particle in a box. Quantum mechanical tunneling.  Ist and IInd law of thermodynamics. Enthalpy. Entropy- statistical and thermodynamic definition of entropy. Helmholtz free energy. Equilibrium thermodynamic. Near-equilibrium thermodynamic. Gibbs free energy, chemical potential. Thermodynamic analysis of membrane transport. Hydration of macromolecules- role of friction, diffusion, sedimentation. The ultracentrifuge. Viscosity. Rotational diffusion. Light scattering. Small angle x-ray scattering. Ultraviolet and visible spectroscopy. Circular dichroism (CD) and optical rotatory dispersion (ORD), fluorescence spectroscopy, infrared spectroscopy, Raman spectroscopy, electron spin resonance and NMR spectroscopy. Light microscopy. Electron optics- transmission electron microscope (TEM), scanning electron microscope (SEM). Preparation of the specimen for electron microscopy. Image reconstruction. Electron diffraction. Tunnelling electron microscope. Atomic force microscope. Crystals and symmetries, crystal systems, point group and space groups. Growth of crystals of biological molecules. X-ray diffraction. Practicals: Refractive index and dispersive power of the prism using spectrometer. Calibration of prism spectrometer. Newton’s rings. Polarimeter. Diffraction grating. Resolving power of telescope and grating. Ostwald viscometer. Planck’s constant using photovoltaic cell. Photospectrometer. Photoelectric effect. Stefan’s constant. Thermal diffusivity in metals.   Phys. 203 Engineering Physics                                                              2+1                         Sem. I

Dia, Para and ferromagnetism-classification. Langevin theory of dia and paramagnetism. Adiabatic demagnetization. Weiss molecular field theory and ferromagnetism. Curie-Weiss law. Wave particle quality, de-Broglie concept, uncertainty principle. Wave function. Time dependent and time independent Schrodinger wave equation, Qualitative explanation of Zeeman effect, Stark effect and Paschan Back effect, Raman spectroscopy. Statement of Bloch’s function. Bands in solids, velocity of Bloch’s electron and effective mass. Distinction between metals, insulators and semiconductors. Intrinsic and extrinsic semiconductors, law of mass action. Determination of energy gap in semiconductors. Donors and acceptor levels. Superconductivity, critical magnetic field. Meissner effect. Isotope effect. Type-I and II superconductors, Josephson’s effect DC and AC, Squids. Introduction to high Tc superconductors. Spontaneous and stimulated emission, Einstein A and B coefficients. Population inversion, He-Ne and Ruby lasers. Ammonia and Ruby masers. Holography-Note. Optical fiber. Physical structure. basic theory. Mode type, input output characteristics of optical fiber and applications. Illumination: laws of illumination, luminous flux, luminous intensity, candle power, brightness.   Practicals:To find the frequency of A.C. supply using an electrical vibrator; To find the low resistance using Carey Foster bridge without calibrating the bridge wire; To determine dielectric constant of material using De Sauty’s bridge; To determine the value of specific charge (e/m) for electrons by helical method; To study the induced e.m.f. as a function of velocity of the magnet; To obtain hysteresis curve (B-H curve) on a C.R.O. and to determine related magnetic quantities; To study the variation of magnetic field with distance along the axis of a current carrying circular coil and to detuning the radius of the coil; To determine the energy band gap in a semiconductor using a p-n Junction diode; To determine the slit width from Fraunhofer diffraction pattern using laser beam; To find the numerical aperture of optical fiber: To set up the fiber optic analog and digital link; To study the phase relationships in L.R. circuit; To study LCR circuit; To study the variations of thermo emf of a copper-constantan thermo-couple with temperature; To find the wave length of light by prism.   Phys. 204 Fundamentals of Quantum Physics                               3+0                       Sem II

Inadequacy of classical Physics: Spectral radiation – Planck’s law. Photoelectric effect – Einstein’s photoelectric equation. Compton’s effect (quantitative) experimental verification. Stability of an atom – Bohr’s atomic theory. Limitations of old quantum theory. Matter Waves: de Broglie’s hypothesis – wavelength of matter waves, properties of matter waves. Phase and group velocities. Davisson and Germer experiment. Uncertainty principle: Heisenberg’s uncertainty principle for position and momentum, energy and time. Fundamental postulates of wave mechanics, eigen function and eigen values, normalization and Born interpretation. Schrodinger wave equation – time dependent and steady state forms, expectation value, particle in a box. step potential, quantum tunneling. Bohr theory of hydrogen atom. Complementary principle of Bohr. Schrodinger equation for hydrogen atom, separation of variables, quantum numbers.

Phys. 205 Statistical Physics                                                          3+0                      Sem-I  

Basics Ideas: probability, distribution of particles, microstate &macrostate, thermodynamic probability, distribution of particles, state of maximum probability and deviation, effect of constraints on the system, equilibrium state of dynamic system. Phase space, division of cells, Maxwell-Boltzmann distribution law, experimental verification of Maxwell-Boltzmann distribution law. Need for quantum mechanics, indistinguishability of particles and its implication. Bose-Einstein Statistics. Fermi-Dirac distribution law, comparison of MB, BE and FD statistics. Application to liquid helium, free electron, Fermi energy and level. Classical theory of radiation, blackbody radiation, Rayleigh-Jean’s law, ultraviolet catastrophe, Planck’s law of radiation, Wein’s displacement laws and Stefan’s law. Statistical interpretation of entropy.   Phys. 206 Thermal Physics                                                              3+1          Sem-II

Concept of temperature, work and heat. Laws of thermodynamics, general relation between  CP and CV, work done during isothermal and adiabatic processes, compressibility and expansion co-efficient. Reversible and Irreversible process with examples. Carnot’s cycle. Refrigerator & coefficient of performance. Thermodynamic scale of temperature.  Concept of Entropy, Clausius theorem. Entropy of a perfect gas and heat death of universe. Thermodynamic Potentials: internal energy, enthalpy, Helmholtz free energy, Gibb’s free energy. Maxwell’s thermodynamic relations, cooling due to adiabatic stretching and compression. Clausius Clapeyron equation, expression of Cp-Cv. Thermodynamic treatment of Joule-Thomson effect and its application for liquification of helium. Production of low temperatures by adiabatic demagnetization, first and second order phase transitions. Negative temperature. Practicals: Stefan’s constant, Searle’s method, Lee-disc method, Joule-Thomson effect, Thermal expansion of solid, Boyle’s law, calibration curve of thermocouple, Newton’s law of cooling, mechanical Equivalent of Heat, J, by Callender and Barne’s constant flow method

Phys. 207 Electricity and Magnetism-I                                     3+1                           Sem I

Introduction to gradient, divergence & curl; their physical significance. Rules for vector derivatives, useful relations involving gradient, divergence & curl. Fundamental theorem for gradients, Gauss’s and Stoke’s theorems. Electric charge and its properties, Coulomb’s law, electric field due to a point charge and continuous charge distributions, field due to electric dipole, electric field lines, flux. Gauss’s law and its applications. Poisson’s and Laplace’s equations. Electric potential due to different charge distribution: wire, ring, disc, spherical sheet, sphere, dipole. Energy for a point and continuous charge distribution. Conductors in the electrostatic field, capacitors, current and current density, drift velocity, expression for current density vector, equation of continuity. Ohm’s Law and expression for electrical conductivity, limitations of Ohm’s law. Equipotential surface method of electrical images. Magnetic fields, magnetic forces, magnetic force on a current carrying wire, torque on a current loop, Biot-Savart law. Field due to infinite wire carrying steady current, field of rings and coils, magnetic field due to a solenoid, force on parallel current carrying wires. Ampere’s circuital law and its applications to infinite hollow cylinder, solenoid and toroid, Hall effect. Practicals: Electromagnetic induction, B-H curve, magnetic field due to solenoid using search coil, permeability of air, low resistance measurement, determination of e/m, charging and discharging of a capacitor, variation of magnetic field with distance. De-Sauty’s bridge.

Phys. 208 Electricity and Magnetism-II                                     3+0                           Sem II

The divergence and curl of electric and magnetic field. Comparison of magnetostatics and electrostatics. Magnetic vector potential and its expression. Faraday’s law of electromagnetic induction, a stationery circuit in a time varying field, a moving conductor in a static magnetic field, a moving circuit in a time varying magnetic field, mutual inductance, reciprocity theorem, self-inductance, energy stored in magnetic field, displacement current, Maxwell’s equations, integral and differential form of Maxwell’s equations. Potential functions, electromagnetic boundary conditions, interface between two loss-less linear media, interface between a dielectric and perfect conductor. Wave equations and their solutions. Time harmonic electromagnetics, source free fields in simple media. Surface current density and change in magnetic field at a current sheet. Field of moving charges, electric field in different frames of references, electric field due to moving charges, electric force in two inertial frames, interaction between moving charges. Flow of electromagnetic power and the Poynting vector, skin depth.   Phys. 301 Introduction to Solid State Physics                         3+0                    Sem. I

Solids: Crystal structure, symmetry operation, Bravais lattices, Lattice with a Basis, unit cell, Miller indices, diamond and NaCl structure. Diffraction: Bragg’s law, experimental methods: Laue method, rotating crystal method, powder method, reciprocal lattices of SC, BCC, FCC Brillouin zones and derivation, structure factor and atomic form factor. Lattice vibration, concepts of phonons, density of modes, quantization of lattice vibrations, Dulong and Petit’s Law, Einstein and Debye theories of specific heat of solids, free electron model, average speed and average kinetic energy, heat capacity.  Band theory: Kronig-Penny model, effective electron mass, metals, insulators and semiconductors. Theory of Dia-, Para-, Ferri- and Ferromagnetic materials, classical Langevin theory of diamagnetism, paramagnetism, Curie-Weiss law. Curie’s law, hysteresis and low temperature production by adiabatic demagnetization.

Phys. 303 Electronics-I                                                                  3+1            Sem-I  

Semiconductor Diodes: P and N type semiconductors. Energy Level Diagram, Fermi level, Conductivity and Mobility, Concept of Drift velocity. PN Junction Fabrication (Simple Idea). Barrier Formation in PN Junction Diode. Static and Dynamic Resistance. Current Flow Mechanism in Forward and Reverse Biased Diode, diode equation. Derivation for Barrier Potential, Barrier Width and Current for Step Junction. Zener diode, light emitting diode, Rectifier Diode: Half-wave Rectifiers, Full-wave Rectifiers, Calculation of Ripple Factor and Rectification Efficiency, L-filter, C-filter, Bipolar Junction transistors: n-p-n and p-n-p Transistors. Characteristics of CB, CE and CC configurations. Current gains α and β. Relations between α and β, transistor as an amplifier, Load line analysis of transistors and Q-point. Transistor Biasing and Stabilization Circuits. Fixed Bias and Voltage Divider Bias. FET: Field Effect Transistor (FET): FET operation, characteristics, difference with BJT. Practicals: PN diode characteristics, Zener diode, CV properties of diode, IV-characteristics of BJT, MOSFET, Op-Amp (723), Op-Amp(741), band-gap of semiconductors, half-wave rectifier, full wave rectifier, Colpitt`s oscillator, Carey Foster’s bridge

Phys. 305 Nuclear Physics-I                                                                       3+0                 Sem-I

General Properties of Nuclei: Constituents of nucleus and their Intrinsic properties, quantitative facts about mass, radii, charge density, binding energy, average binding energy and its variation with mass number, mass defect, angular momentum, parity, magnetic moment, electric moments, nuclear stability, nuclear force and properties, meson theory of nuclear forces. Nuclear Models: Liquid drop model, semi empirical mass formula and significance of its various terms, evidence for nuclear shell structure, nuclear magic numbers, basic assumption of shell model, Radioactivity: law of radioactivity, half-life, mean life, radioactive equilibrium, radioactive series, radioactive dating. Radioactive decay: Alpha decay: basics of α-decay processes, tunnel theory of α emission, Gamow factor, Geiger Nuttall law, α-decay spectroscopy. β-decay: β-, β+, Electron capture, beta energy spectrum, end point energy, neutrino hypothesis, detection of anti-neutrino.  Gamma decay: Gamma rays emission, internal conversion, internal pair conversion, Applications of radioactivity   Phys. 402 Electronics-II                                                                         3+0                        Sem-II 

Feedback in Amplifiers: Effects of Positive and Negative Feedback on Input Impedance, Output Impedance, Gain, Stability, Distortion and Noise, Advantages of Negative feedback, Oscillatory circuits: Fundamentals, classification, feedback. Different types of oscillators: Hartely, Tuned, Colpitts’s, Wien Bridge. Modulation: AM modulation, modulation factor, analysis of AM modulation, Frequency Modulation, Demodulation, AM radio receive. Operational Amplifiers: Characteristics of an Ideal Op-Amp, Open-loop and Closed-loop Gain. Frequency Response. CMRR. Slew Rate.   Phys. 404 Nuclear Physics-II                                                                        3+0                  Sem-II

Nuclear reactions: Types, conserved quantities in nuclear reaction energies of nuclear reaction – Q value and its determination, Exoergic and Endoergic reactions, concept of compound nucleus and direct (pickup and stripping) reactions, Reaction cross-section and its units, Interaction of radiation and charged particle with matter: energy loss formula, stopping power, derivation of Bethe-Bloch formula, straggling, range, interaction of gamma rays with matter. Radiation detection: Gas filled detectors, proportional counter, GM counter, scintillation detector (inorganic only). Accelerators: Linac, Cyclotron, synchrocyclotron, betatron. Elementary particles:  Need of high energy, Fundamental interactions, hadrons, baryon number, lepton number, antiparticle, charge conjugation, isospin, hypercharge, strangeness and Gell-mann Nishijima formula, basic quark model and units in high energy physics.   Phys. 406 Elements of Modern Physics  2+1        Sem II

Special theory of relativity: Galilean transformation, postulates of special theory of relativity. Lorentz transformations, simultaneity principle, Lorentz contraction, time dilation, relativistic transformation of velocity, energy and momentum. Doppler effect, addition of velocities. Variation of mass with velocity, mass-energy equivalence, four vectors. X-ray spectra: production, X-ray diffraction, Bragg’s law, Laue spots, Bragg’s spectrometer, continuous X-ray spectrum, characteristic spectra and absorption of X-ray, Auger effect, Moseley’s law. Lasers: Spontaneous and stimulated emission, Einstein A and B coefficients. Population inversion, laser system, He-Ne , Ruby laser, YAG laser. Superconductivity: critical magnetic field. Meissner effect. Isotope effect. Type-I and II superconductors, Josephson’s effect DC and AC, Squids. BCS theory (qualitative), introduction to high Tc superconductors. Optical fiber: light propagation, different modes in step and graded index fiber, single and multimode fiber, filer loss.   Practicals: Numerical aperture and light attenuation in optical fiber, wavelength of He-Ne laser, verification of logic gates, Hartley oscillator, AM modulation, Study of C.R.O. as display and measuring device, Study of Sine-wave, square wave signals.

Math. 105    Calculus I                                                                        3+0                          Sem. II  Precise definition of limit, continuity, one-sided limit, limits involving infinity, tangents and the derivative at a point, the derivative of a function, extreme values of functions, mean value theorem, monotone functions and the first derivative test, test for concavity. Definite integrals, area between curves, volumes using cross sections and cylindrical shells, arc length and areas of surfaces of revolution. Limits and continuity for functions of several variables, partial derivatives, the chain rule, directional derivatives, gradient vectors, tangent planes, extreme values and saddle points, Lagrange multipliers. Multiple Integrals Double integrals, triple integrals, Jacobian, substitutions in multiple integrals, Green’s theorem, Stoke’s theorem and the divergence’s theorem. Math. 213    Differential Equations                                                     3+0                         Sem. I                         Exact differential equations. First order and higher degree equations solvable for x, y, p. Clairaut’s form. Singular solution as an envelope of general solutions. Geometrical meaning of a differential equation. Orthogonal trajectories. Linear differential equations with constant coefficients. Linear differential equations with variable coefficients- Cauchy and Legendre Equations. Linear differential equations of second order- transformation of the equation by changing the dependent variable/the independent variable, methods of variation of parameters and reduction of order. Simultaneous Differential Equations.  Partial Differential Equations: Origin of first order Partial Differential Equations, Linear Equation of first order, Integral surfaces passing through a given curve, surfaces orthogonal to a given system of surfaces.    Math. 214   Algebra and Geometry                                           3+0                                Sem. II Synthetic division, roots and their multiplicity. Complex roots of real polynomials occur in conjugate pairs with same multiplicity. Relation between roots and co-efficient. Transformation of equations. Descartes’ Rule of Signs. Solution of cubic and bi-quadratic equations: Cardan’s method of solving a cubic, discriminant and nature of roots of real cubic, Descartes’s and Ferrari’s method for a bi-quadratic. Transformation of axes: Shifting of origin and rotation of axes. Sphere: Section of a sphere and a plane, spheres through a given circle, intersection of a line and a sphere, tangent line, tangent plane, angle of intersection of two spheres and condition of orthogonality. Cylinder: Cylinder as a surface generated by a line moving parallel to a fixed line and through a fixed curve, enveloping cylinders.  Cone: Cone with a vertex at the origin, homogeneous equation of second degree in x, y, z.   Math. 215  Calculus-II                                                                 3+0                                  Sem I                                                                                                 Indeterminate forms. Curvature of a curve at a point, radius of curvature of cartesian, parametric, polar curves and implicit functions, evolute and involute, chord of curvature. Multiple points. Asymptotes. Tracing of curves (Cartesian and parametric co-ordinates only).  Integration of hyperbolic and inverse hyperbolic functions. Reduction Formulae. Sequences and series: Bounded and monotonic sequences, convergence of sequences, infinite series and its convergence, comparison tests, absolute convergence. Taylor’s and Maclaurin’s expansions, Taylor’s theorem for functions of two variables.   Math. 216  Vector Calculus                                                        3+0                                   Sem II                                                                                                             Three dimensional coordinate system, vectors, dot product, cross product. Differentiations of vectors, scalar and vector point functions. Gradient, divergence and curl and their physical interpretation. The gradient, maximal and normal property of the gradient, tangent planes. Line integrals: Applications of line integrals, fundamental theorem for line integrals, conservative vector fields, independence of path. Mass and Work. Green’s theorem. Surface integrals, integrals over parametrically defined surfaces, volume integrals. Stoke’s theorem. The Divergence theorem. Math. 302    Number Theory             3+0                              Sem II Divisibility, Greatest common divisor, Euclidean algorithm. The Fundamental theorem of arithmetic. Congruences, residue classes and reduced residue classes. Chinese remainder theorem, Fermat’s little theorem, Wilson’s theorem, Euler’s theorem. Arithmetic functions φ(n), d(n), σ(n), µ(n). Mobius inversion formula. Greatest integer function. Primitive roots and indices.  Math. 303      Real Analysis                                             3+0                                   Sem I Riemann integration: The Riemann Integral and its properties, Integrability of continuous and monotonic functions, mean value theorems of integral calculus. Convergence of improper integrals: Comparison tests, Abel’s and Dirichlet’s tests. Beta and Gamma functions. Sequence and series of functions: Pointwise convergence, Cauchy’s criterion for uniform convergence. Fourier series: Periodic functions, Euler’s formula, Fourier series expansion of piecewise monotonic functions, Fourier series of even and odd functions. Laplace Transformation: Properties of Laplace transform, inverse Laplace transform, Convolution theorem, Laplace transform of derivatives and integrals.

Math 304   Fundamentals of numerical analysis 2+1 Sem II Errors: Relative error, Truncation error, Round-off error. Solution of Non-linear equations: Bisection method, Secant Method, Method of false position, Newton-Raphson Method, fixed-point iteration method. Solution of system of linear equations: Gaussian Elimination method, Gauss-Jordan method, Jacobi’s method, Gauss-Seidel Method. Interpolation: Errors in polynomial interpolation, Finite difference operators, Newton’s forward and backward Formulae, Bessel’s and Striling’s Central difference formulae, Lagrange’s formula and Newton divided difference formula. Numerical Integration: Trapezoidal rule, Simpson’s 1/3 rule, Simpsons 3/8 rule and Weddle’s rule. Solution of ordinary differential equations: Taylor’s series method, Euler’s method, Picard’s approximation method and Runge-Kutta methods.

Practicals: Solution of Non-linear equations, Solution of system of linear equations, Interpolation formulae, Numerical integration, Solution of ordinary differential equations, Application of C programming for solving problems using numerical methods.

Math. 305 Linear Algebra           3+0                             Sem I Vector spaces, basis. Concept of linear independence and examples of different bases. Subspaces of R2, R3. Translation, dilation, rotation, reflection in a point, line and plane. Matrix form of basic geometric transformations. Interpretation of Eigen values and Eigen vectors for such transformations and Eigen spaces as invariant subspaces.  Types of matrices. Rank of  matrix. Elementary transformations. Reduction to normal form. Solutions of linear homogeneous and non-homogeneous equations with number of equations and unknowns upto four. Matrices in diagonal form. Reduction to diagonal form upto matrices of order 3. Computation of matrix inverses using elementary row operations. Course contents for M.Sc. (Mathematics): Math.-521 Algebra-I      3+0 Sem.I Unit I Review of basic concepts of groups with emphasis on exercises, permutation groups, even and odd permutations, conjugacy classes of permutations, alternating groups, simplicity of An, n > 4. Unit II Cayley’s theorem, direct products, fundamental theorem for finite abelian groups, Sylow theorems and their applications, finite Simple groups, groups of order p2 ,pq (p and q primes). Unit III Review of basic concepts of rings, polynomial rings, the ring of Guassian integers, factorization theory in integral domains, divisibility. Unit IV Unique factorization domain (UFD), principal ideal domain (PID), Euclidian domain (ED) and their relationships. Unit V Fields, examples, characteristic of a field, subfield and prime field of a field, field extension, the degree of a field extension, algebraic extensions and transcendental extensions.

Math.-522 Complex Analysis    3+0     Sem. I Unit I Complex plane, geometric representation of complex numbers, joint equation of circle and straight line,  conformal mapping, linear transformation and mapping by another functions. Unit II Bilinear Transformations, Analytic functions, Cauchy-Riemann equations, Harmonic functions and Harmonic conjugates. Unit III  Power series, exponential and trigonometric functions, arg z, log z, az and their continuous branches, Rouche’s Theorem, Maximum Modulus principle. Unit IV Schwarz’ Lemma, Taylor series and Laurent series. Singularities, Cauchy’s residue theorem. Calculus of residues, Zeros and poles of meromorphic functions. Unit V Argument Principle, complex integrals, Cauchy integral formula and evaluation of line integrals by complex integration.

Math.-523 Algebra-II 3+0 Sem.II Unit I Definition and examples of vector spaces (over arbitrary fields), subspaces, direct sum of subspaces, linear dependence and independence, basis and dimensions. Unit II Linear transformations, algebra of linear transformations, linear functions, dual spaces, matrix representation of a linear transformation, rank and nullity of a linear transformation, invariant subspaces.  Unit III Characteristic polynomial and minimal polynomial of a linear transformation, eigen values and eigenvectors of a linear transformation. Unit IV Diagonalization and triangularization of a matrix Jordan canonical forms, bilinear forms, symmetric bilinear forms. Unit V Sylvester’s theorem, quadratic forms, Inner product spaces, Gram-Schmidt orthonormalization process.

Math.-524  Ordinary and Partial Differential Equations 3+0 Sem.II Unit I Existence and Uniqueness of solution of first order equations. Boundary value problems and Strum-Liouville theory. ODE in more than 2-variables.  Unit II Series solution technique for the solution of linear ordinary differential equations of variable coefficients. Frobenius Method.  Unit III Bessel equation, Bessel and Neumann functions, Legendre equation, Legendre function, Legendre associated functions and their properties.  Unit IV Hermite, Laguerre equations and respective special functions. Orthogonal property of special functions, Partial differential equations of first order. Partial differential equations of higher order with constant coefficients.  Unit V Partial differential equations of second order and their classification. variable separable technique for the solution of heat equation, wave equation, Laplace equation.

Math.-525 Elements of Functional Analysis  2+0          Sem. II Unit I Revision of vector spaces, inner product spaces and normed spaces. Unit II Banach Spaces with examples of LP ( [a,b] ) and C ( [a,b] ), Hahn Banach theorem, open mapping theorem, closed graph theorem. Unit III Baire Category theorem, Banach Steinhauns theorem (Uniform boundedness principle), Boundedness and continuity of linear transformation. Unit IV Dual Spaces. Hilbert space, orthonormal basis, Bessel’s inequality, Riesz Fischer theorem, Parseval’s identity. Unit V Bounded Linear functionals; projections, Riesz Representation theorem, adjoint operators, self adjoint, normal, Unitary and isometric operators.

Math.-526 Numerical Methods 2+1 Sem. I Unit I Computational errors, absolute and relative errors, difference operators, divided differences, interpolating polynomials using finite differences. Unit II Hermite interpolation, piecewise and spline interpolation, bivariate interpolation. Numerical solution of algebraic and transcendental equations by bisection, secant and Newton-Raphson Methods. Unit III Solution of polynomial equations by Birge-Vieta, Bairstow’s and Graffe’s root squaring methods.  Unit IV Numerical Differentiation, Numerical Integration: General formulae, Trapezoidal rule, Simpson’s 1/3 and 3/8 rule, Romberg integration, Newton-Cotes formulae, Gaussian integration.  Unit V Solution of Ordinary Differential Equations: Taylor’s series, Picard method of Successive approximations, Euler’s method, Modified Euler’s method, Runge-Kutta Method – 2nd and 4th order, Predictor-Corrector methods, Milne-Simpson’s method, Adam’s – Bashforth method, Finite difference method for boundary value problems. Practical Computational errors, absolute and relative errors, difference operators, divided differences, interpolating polynomials using finite differences, Hermite interpolation, piecewise and spline interpolation, bivariate interpolation, numerical solution of algebraic and transcendental equations by bisection, secant and Newton-Raphson methods, solution of polynomial equations by Birge-Vieta, Bairstow’s and Graffe’s root squaring methods, numerical differentiation, numerical integration: general formulae, trapezoidal rule, Simpson’s 1/3 and 3/8 rule, Romberg integration, Newton-Cotes formulae, Gaussian integration, solution of ordinary differential equations: Taylor’s series, Picard method of successive approximations, Euler’s method, modified Euler’s method, Runge-Kutta method – 2nd and 4th order, predictor-corrector methods, Milne-Simpson’s method, Adam’s – Bashforth method, finite difference method for boundary value problems.

Math.-527 Calculus of Variation and Operations 2+0 Sem.I Unit I Functional and their properties, Motivating problems of Calculus of variations, Shortest distance, minimum surface of revolution. Unit II Brachistochrone problem, Isoperimetric problems, Geodesics, Fundamental lemma of Calculus of Variations. Unit III Euler’s equation for one dependent function  and its generalization to (i) n dependent functions, (ii) higher order derivatives. Unit IV Variational problems with moving boundaries, Variation under constraints. Unit V Variational methods of Rayleigh-Ritz and Galerkin.

Math.-528 Real Analysis     2+0 Sem.I Unit I Basic Topology: Finite, countable and uncountable sets, Metric spaces, Perfect, compact and connected sets.  Unit II Sequences and series: Convergent sequences, Subsequences, Cauchy sequences (in metric spaces). Unit III Absolute convergence. Continuity: Limits of functions (in metric spaces), Continuous and monotonic functions, Continuity and compactness, Continuity and connectedness. Unit IV The Riemann-Stieltjes integral: Definition and properties of the Riemann-Stieltjes integral, Rectifiable curves.   Unit V Sequences and series of functions: Pointwise and Uniform convergence, Uniform convergence and continuity, Uniform convergence and integration, Uniform convergence and differentiation, Stone Weierstrass Theorem.

Math.-529 Principles of Topology 2+0 Sem.I Unit I Connected sets, topological spaces, bases for a topology, the order topology, the product topology on X × Y, the subspace topology. Unit II Closed sets and limit points, continuous functions, the product topology, the metric topology, the quotient topology. Unit III Compact spaces, compact space of the real line, limit point compactness, local compactness, nets, the  countability axioms. Unit IV Nets ,the  countability axioms, The separation axioms, normal spaces Unit V The Urysohn lemma, the Urysohnmetrization theorem, the Tietze extension theorem, the Tychonoff  theorem.

Math.-530 Transformation Theory 2+0 Sem. II Unit I Laplace transformation, inverse Laplace transformation, applications. inverse Laplace transformation, Fourier transformation, convolution theorem . Unit II Parseval’s Identity, applications of transforms to boundary value problems, Z- transforms, definition of integral equations and their classifications. Unit III Eigen values and Eigen functions, special kinds of kernel, Convolution integral, inner or scalar product of two functions, reduction to a system of algebraic equations. Unit IV Fredholm alternative, Fredholm theorem, Fredholm alternative theorem, an approximate method, method of successive approximations. Unit V Iterative scheme for Fredholm and Volterrra integral equations of the second kind, conditions of Uniform convergence and Uniqueness of series solution.

Math.-531 Measure Theory                                                                      2+0         Sem.I Unit I Lebesgue outer measure, measurable sets, Lebesgue measure, regularity of the Lebesgue measure, measurable functions. Unit II Borel and Lebesgue measurability, a non-measurable set, Littlewood’s three principles.  Unit III The Lebesgue integral: Integration of non-negative measurable functions, the general Lebesgue integral, monotone convergence theorem and dominated convergence theorem. Unit IV Integration of series, relationship with Riemann integral. Unit V Positive Borel measures: vector spaces, the Riesz representation theorem (statement only).

Math.-532 Classical Mechanics 2+0             Sem.II Unit I Variation under constraints, Variational methods of Rayleigh-Ritz and Galerkin. Lagrangian Mechanics: Generalized coordinates, Constraints, Holonomic and non-holonomic systems, Scleronomic and Rheonomic systems. Unit II  Generalized velocity, potential and force, Velocity dependent Potentials and Dissipation function. Unit III Lagrangian Mechanics: Hamilton’s principle, Principle of Least action, Cyclic coordinates, Conjugate momentum, Conservation theorems. Hamiltonian Mechanics: Legendre’s transformation, Hamilton’s equations, Routhian. Unit IV Poisson Bracket, Jacobi identity for Poisson bracket, Poission theorem, Canonical Transformation. Unit V Hamilton-Jacobi equations, Method of Separation of variables, Action – Angle variables, Lagrange Bracket.

Math.-535 Operational Methods 2+1                Sem.II Unit I Linear Programming and examples, Convex Sets, Hyperplane, Open and Closed half-spaces, Feasible, Basic Feasible and Optimal Solutions, Extreme Point & graphical methods, Simplex method, Charnes-M method, Two phase method, Determination of Optimal solutions, unrestricted variables. Unit II Duality theory, Dual linear Programming Problems, fundamental properties of dual Problems, Complementary slackness, Unbounded solution in Primal. Dual Simplex Algorithm, Sensitivity analysis.  Unit III Cost minimization Transportation Problems, Balanced and unbalanced Transportation problems, U-V method, Paradox in Transportation problem. Unit IV Time Minimization Transportation Problems, Cost Minimization and Time Minimization Assignment problems, Hungarian Method and its convergence.  Unit V Integer Programming problems: Pure and Mixed Integer Programming problems, Gomary’s algorithm, Branch & Bound Technique, Travelling salesman problem.

Practical Linear programming and examples, convex sets, hyperplane, open and closed half-spaces, feasible, basic feasible and optimal solutions,  extreme point & graphical methods, simplex method, Charnes-m method, two phase method, determination of optimal solutions, unrestricted variables, duality theory, dual linear programming problems, fundamental properties of dual problems, complementary slackness, unbounded solution in primal, dual simplex algorithm, sensitivity analysis, cost minimization transportation problems, balanced and unbalanced transportation problems, u-v method, paradox in transportation problem, time minimization transportation problems, cost minimization and time minimization assignment problems, Hungarian method and its convergence, integer programming problems: pure and mixed integer programming problems, Gomary’s algorithm, branch & bound technique, travelling salesman problem. Course contents for M.Sc. (Statistics): Stat.-551 Mathematics-I 3+0 Sem. I Unit I Calculus: Limit and continuity, differentiation of functions, successive differentiation, partial differentiation, mean value theorems, Taylor and Maclaurin’s series. Application of derivatives, L’hospital’s rule. Unit II Real Analysis: Convergence and divergence of infinite series, use of comparison tests -D’Alembert’s Ratio – test, Cauchy’s nth root test, Raabe’s test, Kummer’s test, Gauss test. Absolute and conditional convergence. Riemann integration, concept of Lebesgue integration, power series, Fourier, Laplace and Laplace -Steiltjes’ transformation, multiple integrals.Integration of rational, irrational and trigonometric functions. Application of integration. Unit III Differential equation: Differential equations of first order, linear differential equations of higher order with constant coefficient. Unit IV Numerical Analysis: Simple interpolation, Divided differences, Numerical differentiation and integration.

Stat.-552 Probability Theory   2+0 Sem. I Unit I Basic concepts of probability. Elements of measure theory: class of sets, field, sigma field, minimal sigma field, Borel sigma field in R, measure- probability measure. Axiomatic approach to probability. Properties of probability based on axiomatic definition. Addition and multiplication theorems. Conditional probability and independence of events. Bayes theorem. Unit II Random variables: definition of random variable, discrete and continuous, functions of random variables. Probability mass function and Probability density function, Distribution function and its properties. Notion of bivariate random variables, bivariate distribution function and its properties. Joint, marginal and conditional distributions. Independence of random variables. Transformation of random variables (two dimensional case only). Mathematical expectation: Mathematical expectation of functions of a random variable. Raw and central moments and their relation, covariance, skewness and kurtosis. Addition and multiplication theorems of expectation. Definition of moment generating function, cumulating generating function, probability generating function and statements of their properties. Unit III Conditional expectation and conditional variance. Characteristic function and its properties. Inversion and Uniqueness theorems. Chebyshev,  Markov, Cauchy-Schwartz, Sequence of random variables and modes of convergence (convergence in distribution in probability, almost surely, and quadratic mean) and their interrelations.  Unit IV Laws of large numbers: WLLN, Bernoulli and Kintchin’s WLLN. Kolmogorov inequality, Kolmogorov‘s  SLLNs. Central Limit theorems: Demoviere- Laplace CLT, Lindberg – Levy CLT and simple applications.

Stat.- 553 Statistical Methods  2+1 Sem. I Unit I Descriptive statistics: probability distributions: Discrete probability distributions ~Bernoulli, Binomial, Poisson, Negative-binomial, Geometric and Hyper Geometric, Uniform, multinomial ~ Properties of these distributions and real life examples. Continuous probability distributions ~ rectangular, exponential, Cauchy, normal, gamma, beta of two kinds, Weibull, lognormal, logistic, Pareto. Properties of these distributions. Probability distributions of functions of random variables.  Unit II Concepts of compound, truncated and mixture distributions (definitions and examples). Sampling distributions of sample mean and sample variance from Normal population, central and non–central chi-Square, t and F distributions, their properties and inter relationships. Unit III Concepts of random vectors, moments and their distributions.  Bivariate  Normal distribution – marginal and conditional distributions. Distribution of quadratic forms. Cochran theorem. Correlation, rank correlation, correlation ratio and intra-class correlation. Regression analysis, partial and multiple correlation and regression.  Unit IV Sampling distribution of correlation coefficient, regression coefficient. Categorical data analysis, Association between attributes. Variance Stabilizing Transformations. Unit V Order statistics, distribution of r-th order statistics, joint distribution of several order statistics and their functions, marginal distributions of order statistics. Practical Fitting of discrete distributions and test for goodness of fit, fitting of continuous distributions and test for goodness of fit, fitting of truncated distribution, computation of simple, multiple and partial correlation coefficient, correlation ratio and intra-class correlation, regression coefficients and regression equations, fitting of Pearsonian curves, analysis of association between attributes, categorical data and log-linear models.

Stat.-554 Actuarial Statistics  2+0 Sem. I Unit I Insurance and utility theory, models for individual claims and their sums, survival function, curtate future lifetime, force of mortality. Unit II Life table and its relation with survival function, examples, assumptions for fractional ages, some analytical laws of mortality, select and ultimate tables. Unit III Multiple life functions, joint life and last survivor status, insurance and annuity benefits through multiple life functions evaluation for special mortality laws. Multiple decrement models, deterministic and random survivorship groups, associated single decrement tables, central rates of multiple decrement, net single premiums and their numerical evaluations.  Unit IV Distribution of aggregate claims, compound Poisson distribution and its applications. Unit V Principles of compound interest: Nominal and effective rates of interest and discount, force of interest and discount, compound interest, accumulation factor, continuous compounding. Unit VI Insurance payable at the moment of death and at the end of the year of death-level benefit insurance, endowment insurance, deferred insurance and varying benefit insurance, recursions, commutation functions. Unit VII Life annuities: Single payment, continuous life annuities, discrete life annuities, life annuities with monthly payments, commutation functions, varying annuities, recursions, complete annuities-immediate and apportionable annuities-due. Unit VIII Net premiums: Continuous and discrete premiums, true monthly payment premiums, apportionable premiums, commutation functions, accumulation type benefits. Payment premiums, apportionable premiums, commutation functions, accumulation type benefits. Net premium reserves: Continuous and discrete net premium reserve, reserves on a semi-continuous basis, reserves based on true monthly premiums, reserves on an apportionable or discounted continuous basis, reserves at fractional durations, allocations of loss to policy years, recursive formulas and differential equations for reserves, commutation functions. Unit IX Some practical considerations: Premiums that include expenses-general expenses types of expenses, per policy expenses. Claim amount distributions, approximating the individual model, stop-loss insurance.

Stat.-555 Bioinformatics       2+0 Sem. I Unit I Basic Biology: Cell, genes, gene structures, gene expression and regulation, Molecular tools, nucleotides, nucleic acids, markers, proteins and enzymes, bioenergetics, single nucleotide polymorphism, expressed sequence tag. Structural and functional genomics: Organization and structure of genomes, genome mapping, assembling of physical maps, strategies and techniques for genome sequencing and analysis. Unit II Computing techniques: OS and Programming Languages – Linux, perl, bioperl, python, biopython, cgi, My SQL, php My Admin; Coding for browsing biological databases on web, parsing & annotation of genomic sequences; Database designing; Computer networks – Internet, World wide web, Web browsers– EMB net, NCBI; Databases on public domain pertaining to Nucleic acid sequences, protein sequences, SNPs, etc.; Searching sequence databases, Structural databases. Unit III Statistical Techniques: MANOVA, Cluster analysis, Discriminant analysis, Principal component analysis, Principal coordinate analysis, Multidimensional scaling; Multiple regression analysis; Likelihood approach in estimation and testing; Resampling techniques – Bootstrapping and Jack-knifing; Hidden Markov Models; Bayesian estimation and Gibbs sampling;  Unit IV Tools for Bioinformatics: DNA Sequence Analysis – Features of DNA sequence analysis, Approaches to EST analysis; Pairwise alignment techniques: Comparing two sequences, PAM and BLOSUM, Global alignment (The Needleman and Wunsch algorithm), Local Alignment (The Smith-Waterman algorithm), Dynamic programming, Pairwise database searching; Sequence analysis– BLAST and other related tools, Multiple alignment and database search using motif models, Clustal W, Phylogeny; Databases on SNPs; EM algorithm and other methods to discover common motifs in biosequences; Gene prediction based on Neural Networks, Genetic algorithms, Computational analysis of protein sequence, structure and function; Design and Analysis of microarray/RNA seq experiments.

Stat.-556 Econometrics 2+0 Sem. I Unit I Representation of Economic phenomenon, relationship among economic variables, linear and non-linear economic models, single equation general linear regression model, basic assumptions, Ordinary least squares method of estimation for simple and multiple regression models; summary statistics correlation matrix, co-efficient of multiple determination, standard errors of estimated parameters, tests of significance and confidence interval estimation. BLUE properties of Least Squares estimates. Chow test, test of improvement of fit through additional regressors. Maximum likelihood estimation. Unit II Heteroscedasticity, Auto-correlation, Durbin Watson test, Multi- collinearity. Stochastic regressors, Errors in variables, Use of instrumental variables in regression analysis. Dummy Variables. Distributed Lag models: Koyck’s Geometric Lag scheme, Adaptive Expectation and Partial Adjustment Mode, Rational Expectation Models and test for rationality. Unit III Simultaneous equation model: Basic rationale, Consequences of simultaneous relations, Identification problem, Conditions of Identification, Indirect Least Squares, Two-stage least squares, K-class estimators, Limited Information and Full Information Maximum Likelihood Methods, three stage least squares, Generalized least squares, Recursive models, SURE Models. Mixed Estimation Methods, use of instrumental variables, pooling of cross-section and time series data, Principal Component Methods. Unit IV Problem and Construction of index numbers and their tests; fixed and chain based index numbers; Construction of cost of living index number.  Unit V Demand analysis – Demand and Supply Curves; Determination of demand curves from market data. Engel’s Law and the Engel’s Curves, Income distribution and method of its estimation, Pareto’s Curve, Income inequality measures.

Stat.-561 Mathematics-II       2+0 Sem. II Unit I Linear Algebra: Group, ring, field and vector spaces, Sub-spaces, basis, Gram Schmidt’s orthogonalization, Galois field – Fermat’s theorem and primitive elements. Linear transformations. Graph theory: Concepts and applications. Unit II Matrix Algebra: Basic terminology, linear independence and dependence of vectors. Row and column spaces, Echelon form. Determinants, Trace of matrices rank and inverse of matrices. Special matrices – idempotent, symmetric, orthogonal. Eigen values and eigen vectors, Spectral decomposition of matrices. Unit III Unitary, Similar, Hadamard, Circulant, Helmert’s matrices. Kronecker and Hadamard product of matrices, Kronecker sum of matrices. Sub- matrices and partitioned matrices, Permutation matrices, full rank factorization, Grammian root of a symmetric matrix. Solutions of linear equations, Equations having many solutions. Unit IV Generalized inverses, Moore-Penrose inverse, Applications of g-inverse. Inverse and Generalized inverse of partitioned matrices, Differentiation and integration of vectors and matrices, Quadratic forms.

Stat.-562 Statistical Inference      2+1 Sem. II Unit I Concepts of point estimation: unbiasedness, consistency, efficiency and sufficiency. Statement of Neyman’s Factorization theorem with applications. MVUE, Rao-Blackwell theorem, completeness, Lehmann- Scheffe theorem. Fisher information, Cramer-Rao lower bound and its applications. Unit II Moments, minimum chi-square, least square and maximum likelihood methods of estimation and their properties. Interval estimation-Confidence level, shortest length CI. CI for the parameters of Normal, Exponential, Binomial and Poisson distributions. Unit III Fundamentals of hypothesis testing-statistical hypothesis, statistical test, critical region, types of errors, test function, randomized and non- randomized tests, level of significance, power function, most powerful tests: Neyman-Pearson fundamental lemma, MLR families and UMP tests for one parameter exponential families. Concepts of consistency, unbiasedness and invariance of tests. Likelihood Ratio tests, asymptotic properties of LR tests with applications (including homogeneity of means and variances).Relation between confidence interval estimation and testing of hypothesis. Unit IV Sequential Probability ratio test, Properties of SPRT. Termination property of SPRT, SPRT for Binomial, Poisson, Normal and Exponential distributions. Concepts of loss, risk and decision functions, admissible and optimal decision functions, estimation and testing viewed as decision problems, conjugate families, Bayes and Minimax decision functions with applications to estimation with quadratic loss. Unit V Non-parametric tests: Sign test, Wilcoxon signed rank test, Runs test for randomness, Kolmogorov – Smirnov test for goodness of fit, Median test and Wilcoxon-Mann-Whitney U-test. Chi-square test for goodness of fit and test for independence of attributes. Spearman’s rank correlation and Kendall’s Tau tests for independence. Practical Methods of estimation – maximum likelihood, minimum t2 and moments, confidence interval estimation, mp and ump tests, large sample tests, non-parametric tests, sequential probability ratio test, decision functions.

Stat.-563 Design Of Experiments      2+1 Sem. II Unit I Elements of linear estimation, Gauss Markoff Theorem, relationship between BLUEs and linear zero-functions. Aitken’s transformation, test of hypothesis, Analysis of Variance, partitioning of degrees of freedom. Unit II Orthogonality, contrasts, mutually orthogonal contrasts, analysis of covariance; Basic principles of design of experiments, Uniformity trials, size and shape of plots and blocks, Randomization procedure. Unit III Basic designs – completely randomized design, randomized complete block design and Latin square design; Construction of orthogonal Latin squares, mutually orthogonal Latin squares (MOLS), Youden square designs, Graeco Latin squares. Unit IV Balanced incomplete block (BIB) designs – general properties and analysis without and with recovery of intra block information, construction of BIB designs. Partially balanced incomplete block designs with two associate classes – properties, analysis and construction, Lattice designs, alpha designs, cyclic designs, augmented designs. Unit V Factorial experiments, confounding in symmetrical factorial experiments (2nand 3nseries), partial and total confounding, asymmetrical factorials. Unit VI Cross-over designs. Missing plot technique; Split plot and Strip plot design; Groups of experiments. Sampling in field experiments. Practical Determination of size and shape of plots and blocks from uniformity trials data, analysis of data generated from completely randomized design, randomized complete block design, latin square design, Youden square design, analysis of data generated from a bib design, lattice design, pbib designs; 2n, 3n factorial experiments without and with confounding, split and strip plot designs, repeated measurement design, missing plot techniques, analysis of covariance, analysis of groups of experiments, analysis of clinical trial experiments.

Stat.-564 Sampling Techniques    2+1 Sem. I Unit I Sample survey vs complete enumeration, probability sampling, sample space, sampling design, sampling strategy; Determination of sample size; Confidence-interval; Simple random sampling, Estimation of population proportion, Stratified random sampling, Proportional allocation and optimal allocation, Inverse sampling. Unit II Ratio, Product and regression methods of estimation, Cluster sampling, Systematic sampling, Multistage sampling with equal probability, Separate and combined ratio estimator, Double sampling, Successive sampling –two occasions. Unbiased ratio type estimators Unit III Non-sampling errors – sources and classification, Non-response in surveys, Randomized response techniques, Response errors/Measurement error – interpenetrating sub-sampling. Unit IV PPS Sampling with and without replacement, Cumulative method and Lahiri’s method of selection, Horvitz-Thompson estimator, Ordered and unordered estimators, Sampling strategies due to Midzuno-Sen and Rao- Hartley-Cochran. Inclusion probability proportional to size sampling. Practical Determination of sample size and selection of sample, simple random sampling, inverse sampling, stratified random sampling, cluster sampling, systematic sampling, ratio and regression methods of estimation, double sampling, multi-stage sampling, imputation methods; randomized response techniques; sampling with varying probabilities.

Stat.-565 Statistical Genetics    2+1 Sem. II Unit I Physical basis of inheritance. Analysis of segregation, detection and estimation of linkage for qualitative characters. Amount of information about linkage, combined estimation, disturbed segregation. Unit II Gene and genotypic frequencies, Random mating and Hardy -Weinberg law, Application and extension of the equilibrium law, Fisher’s fundamental theorem of natural selection. Disequilibrium due to linkage for two pairs of genes, sex-linked genes, Theory of path coefficients. Unit III Concepts of inbreeding, Regular system of inbreeding. Forces affecting gene frequency – selection, mutation and migration, equilibrium between forces in large populations, Random genetic drift, Effect of finite population size. Unit IV Polygenic system for quantitative characters, concepts of breeding value and dominance deviation. Genetic variance and its partitioning, Effect of inbreeding on quantitative characters, Multiple allelism in continuous variation, Sex-linked genes, Maternal effects – estimation of their contribution. Unit V Correlations between relatives, Heritability, Repeatability and Genetic correlation. Response due to selection, Selection index and its applications in plants and animals’ improvement programmes, Correlated response to selection. Unit VI Restricted selection index. Variance component approach and linear regression approach for the analysis of GE interactions. Measurement of stability and adaptability for genotypes. Concepts of general and specific combining ability. Diallel and partial diallel crosses – construction and analysis. Practical Test for the single factor segregation ratios, homogeneity of the families with regard to single factor segregation, detection and estimation of linkage parameter by different procedures, estimation of genotypic and gene frequency from a given data, Hardy-Weinberg law; estimation of changes in gene frequency due to systematic forces, inbreeding coefficient, genetic components of variation, heritability and repeatability coefficient, genetic correlation coefficient, examination of effect of linkage, epistasis and inbreeding on mean and variance of metric traits, mating designs; construction of selection index including phenotypic index, restricted selection index, correlated response to selection.

Stat.-566   Statistical Quality Control  2+0 Sem. II Unit I Introduction to Statistical Quality Control; Control Charts for Variables – Mean, Standard deviation and Range charts; Statistical basis; Rational subgroups. Unit II Control charts for attributes- ‘np’, ‘p’ and ‘c’ charts.   Unit III Fundamental concepts of acceptance, sampling plans, single, double and sequential sampling plans for attributes inspection. Unit IV Sampling inspection tables for selection of single and double sampling plans.

Stat.-567 Optimization Techniques    1+1 Sem. II Unit I Classification of optimization problems, Classical optimization techniques: single variable optimization, multivariable optimization techniques with no constraints, multivariable optimization techniques with equality constraints, multivariable optimization techniques with inequality constraints. UNIT II Linear programming: simplex method, duality, sensitivity analysis, Karmarkar’s method, transportation problem. Unit III Nonlinear programming Unconstrained optimization techniques: direct search methods such as random search, grid search, Hooke and Jeeves’ method, Powel’s method. Descent methods such as gradient method, steepest descent method, conjugate gradient method, Newton’s method, Marquardt method. Unit IV Quadratic programming, integer linear programming, integer nonlinear programming, geometric programming, dynamic programming, stochastic programming, multiobjective optimization, optimal control theory, genetic algorithms, simulated annealing, neural network based optimization Practical Problems based on classical optimization techniques, optimization techniques with constraints, minimization problems using numerical methods, linear programming (lp) problems through graphical method, simplex method, simplex two-phase method, primal and dual method, sensitivity analysis for lp problem, lp problem using Karmarkar’s method, problems based on quadratic programming, integer programming, dynamic programming, stochastic programming, problems based on pontryagin’s maximum principle, problems based on multi objective optimization.

Stat.-571 Multivariate Analysis 2+1 Sem. I Unit I Concept of random vector, its expectation and Variance-Covariance matrix. Marginal and joint distributions. Conditional distributions and Independence of random vectors. Multinomial distribution. Multivariate Normal distribution, marginal and conditional distributions. Sample mean vector and its distribution. Maximum likelihood estimates of mean vector and dispersion matrix. Tests of hypothesis about mean vector. Unit II Wishart distribution and its simple properties. Hotelling’s T2 and Mahalanobis D2 statistics. Null distribution of Hotelling’s T2. Rao’s U statistics and its distribution. Wilks’ λ criterion and its properties. Concepts of discriminant analysis, computation of linear discriminant function, classification between k ( ≥2) multivariate normal populations based on LDF and MahalanobisD2. Unit III Principal Component Analysis, factor analysis. Canonical variables and canonical correlations. Cluster analysis: similarities and dissimilarities of qualitative and quantitative characteristics, Hierarchical clustering. Single, Complete and Average linkage methods. K-means cluster analysis. Unit IV Path analysis and computation of path coefficients, introduction to multidimensional scaling, some theoretical results, similarities, metric and non-metric scaling methods. Practical Maximum likelihood estimates of mean-vector and dispersion matrix, testing of hypothesis on mean vectors of multivariate normal populations, cluster analysis, discriminant function, canonical correlation, principal component analysis, factor analysis, multivariate analysis of variance and covariance, multidimensional scaling.

Stat.-572 Regression Analysis     1+1 Sem. I Unit I Simple and Multiple linear regressions: Least squares fit, Properties and examples. Polynomial regression: Use of orthogonal polynomials. Unit II Assumptions of regression; diagnostics and transformations; residual analysis ~ Studentized residuals, applications of residuals in detecting outliers, identification of influential observations. Lack of fit, Pure error. Test of normality, test of linearity, Testing homoscedasticity and normality of errors, Durbin-Watson test. Test of goodness of fit for the model evaluation and validation. Concept of multi-collinearity Unit III Weighted least squares method: Properties, and examples. Box-Cox family of transformations. Use of dummy variables, Over fitting and under fitting of model, Selection of variables: Forward selection, Backward elimination. Stepwise and Stagewise regressions. Unit IV Introduction to non-linear models, nonlinear estimation: Least squares for nonlinear models. Practical Multiple regression fitting with three and four independent variables, estimation of residuals, their applications in outlier detection, distribution of residuals, test of homoscedasticity, and normality, box-cox transformation, restricted estimation of parameters in the model, hypothesis testing, step wise regression analysis; least median of squares norm, orthogonal polynomial fitting.

Stat.-573 Statistical Computing     1+1 Sem. I Unit I Introduction to statistical packages and computing: data types and structures, Use of Software packages like, SAS, SPSS or “R: The R Project for Statistical Computing”. Data analysis principles and practice, Summarization and tabulation of data, Exploratory data analysis; Graphical representation of data. Statistical Distributions: Fitting and testing the goodness of fit of discrete and continuous probability distributions; Unit II ANOVA, regression and categorical data methods; model formulation, fitting, diagnostics and validation; Matrix computations in linear models. Analysis of discrete data. Multiple comparisons, Contrast analysis. Unit III Numerical linear algebra, numerical optimization, graphical techniques, numerical approximations, Time Series Analysis. Unit IV Analysis of mixed models; Estimation of variance components, Analysis of Covariance, Fitting of non-linear model, Discriminant function; Principal component analysis. techniques in the analysis of survival data and longitudinal studies, Approaches to handling missing data, and meta- analysis.

Practical Data management, graphical representation of data, descriptive statistics, general linear models ~ fitting and analysis of residuals, outlier detection, fitting and testing the goodness of fit of probability distributions, testing the hypothesis for one sample t-test, two sample t– test, paired t-test, test for large samples – Chi-squares test, f test, one way analysis of variance, contrast and its testing, pairwise  comparisons; mixed effect models, estimation of variance components, categorical data analysis, dissimilarity measures, similarity measures, analysis of discrete data, analysis of binary data, numerical algorithms, spatial modeling, cohort studies, clinical trials, analysis of survival data, handling missing data, analysis of time series data – fitting of arima models.   Stat.-574 Time Series Analysis      1+1 Sem. I Unit I Components of a time-series. Autocorrelation and Partial autocorrelation functions, Correlogram and periodogram analysis. Unit II Linear stationary models: Autoregressive, moving average and Mixed processes. Linear non-stationary models: Autoregressive integrated moving average processes. Unit III Forecasting: Minimum mean square forecasts and their properties, Calculating and updating forecasts. Unit IV Model identification: Objectives, Techniques, and Initial estimates. Model estimation: Likelihood function, Sum of squares function, Least squares estimates. Seasonal models. Intervention analysis models and Outlier detection. Practical Time series analysis, autocorrelations, correlogram and periodogram, linear stationary model, linear non-stationary model, model identification and model estimation intervention analysis and outlier detection

Stat.-575 Demography 2+0 Sem. I Unit I Introduction to vital statistics, crude and standard mortality and morbidityrates, Estimation of mortality, Measures of fertility and mortality, period and cohortmeasures. Unit II Life tables and their applications, methods of construction of abridged lifetables,Increment-DecrementLifeTables. Unit III Stationary and stable populations, Migration and immigration. Application of stable population theory to estimate vital rates, migration and its estimation. Demographic relations in non stable populations. Measurement of population growth, Lotka’s model (deterministic) and intrinsic rate of growth, Measures of mortality and morbidity period. Unit IV Principle of biological assays, parallel lineand slope ratio assays, choice of doses and efficiency in assays quantal responses, probit and logit transformations, epidemiological models.

Stat.-576 Statistical Methods For Life Sciences 2+0 Sem. I Unit I Proportions and counts, contingency tables, logistic regression models, Poisson regression and log-linear models, models for polytomous data and generalized linear models. Unit II Computing techniques, numerical methods, simulation and general implementation of biostatistical analysis techniques with emphasis on data applications. Analysis of survival time data using parametric and non- parametric models, hypothesis testing, and methods for analyzing censored (partially observed) data with covariates. Topics include marginal estimation of a survival function, estimation of a generalized multivariate linear regression model (allowing missing covariates and/or outcomes). Unit III Proportional Hazard model: Methods of estimation, estimation of survival functions, time-dependent covariates, estimation of a multiplicative intensity model (such as Cox proportional hazards model) and estimation of causal parameters assuming marginal structural models. Unit IV General theory for developing locally efficient estimators of the parameters of interest in censored data models. Rank tests with censored data. Computing techniques, numerical methods, simulation and general implementation of bio-statistical analysis techniques with emphasis on data applications. Unit V Newton, scoring, and EM algorithms for maximization; smoothing methods; bootstrapping; trees and neural networks; clustering; isotonic regression; Markov chain Monte Carlo methods.

Course contents for M.Sc. (Physics): Phys.-501 Classical and Statistical Mechanics 4+0 Sem-I Unit I Mechanics of a system of particles, constraints of motion, generalized coordinates, D’Alembert’s Principle and Lagrange’s velocity-dependent forces and the dissipation function, Lagrange’s equations of motion, Calculation of variations, Hamilton’s Principle and Euler-Lagrange’s differential equations of motion, Advantages of variational principle formulation, Conservation theorems and symmetry properties. Legendre Transformation, Hamilton’s equations of motion, Cyclic-co-ordinates, Hamilton’s equations from variational principle, Principle of least action.  Unit II Hamilton’s equations of motion, Canonical transformation, Poisson’s brackets and its relations, Infinitesimal contact transformation, Hamilton Jacobi equations for principal and characteristic functions, Harmonic oscillator problem, Action angle variables. First integrals, Equivalent one-dimensional problem and classification of orbits, differential equation for the orbit, Kepler’s problem, Scattering in a central force field, Euler angles, Coriolis force. Independent co-ordinates of rigid body, orthogonal transformations, Motion of a rigid body, Euler angles and Euler’s theorem, Rate of change of a vector, infinitesimal rotation, Inertia tensor and principal axis transformation. Unit III Macroscopic and microscopic states, contact between statistics and thermodynamics, classical ideal gas, Gibbs paradox and its solution. Elements of ensemble theory: phase space and Liouville’s theorem, the microcanonical ensemble theory and its application to ideal gas of monatomic particles, equipartition and virial theorems. Canonical ensemble: Equilibrium between a system and a heat reservoir, a system in the canonical ensemble and, physical significance of various statistical quantities, partition function, classical ideal gas in canonical ensemble theory, energy fluctuations, a system of harmonic oscillators as canonical ensemble, statistics of paramagnetism, thermodynamics of magnetic systems and negative temperatures. Unit IV The grand canonical ensemble: Equilibrium between a system and a particle-energy reservoir and significance of statistical quantities. Classical ideal gas in grand canonical ensemble theory. Density and energy fluctuations. Elements of Quantum Statistics: Quantum states and phase space, quantum statistics of various ensembles. An ideal gas in quantum mechanical ensembles, statistics of occupation numbers. Basic concepts and thermodynamic behaviour of an ideal Bose gas, Bose-Einstein condensation, Ideal Fermi Systems: thermodynamic behaviour of an ideal fermi gas, discussion of heat capacity of a free-electron gas at low temperatures.                                        Phys.-502 Electronics 2+1 Sem-I Unit I High pass RC circuits: response to step, pulse, square and exponential wave forms, application as differentiator. Low pass RC circuit: response to step, pulse, square and exponential waveforms, application as integrator. Multivibrators: bistable, monostable and astable: operation and triggering and frequency of oscillation. Diode as clipper, series clippers, transistor clippers Unit II Concept of DC positive logic, negative logic systems, Boolean algebra, logic gates, de Morgans theorem, Data processing circuits: Multiplexers, Demultiplexers, Encoders, Decoders, Parity generators. Half-adder, full-adder, subtractor, flip flops: SR, JK and Master Slave JK, D type and T-type. Shift registers, counters and their types and applications. Unit III Analogue to digital converters, digital to analogue converter, digital display, seven segment display. Operational amplifiers: emitter coupled differential amplifier, characteristics, common mode rejection ratio. Basics operational amplifier applications: scale, adder, summer etc. Non-linear analog system: logarithmic, anti-log amplifier, multiplier and integrator. Unit IV  Microprocessor: Fundamentals of microprocessors, buffer registers, bus organized computers, SAP-I, Microprocessor (μP) 8085 architecture, memory interfacing, interfacing I/O devices. SAP-2 and SAP-3: architecture, instructions, programming in assembly language. Generation and detection of SSB waves. Practical Verification of all logic gates, verification of half adder and full adder, subtractor. To determine the frequency of astable multivibrator, analogue to digital conversion (a/d) digital to analogue conversion (d/a), k/e ratio measurement,  To measure voltage, gain of op-amps (741), To study zener diode characteristics, band filters, signal shaper, design and study of various flip flops circuits (rs, d, jk, t) , design and study of various counter circuits (up, down, ring, mod-n) 

Phys.-503 Quantum mechanics 3+0 Sem-II Unit I Basic concepts and quantum kinematics: Dirac’s bra and ket notations; Linear vector space; inner product; operators and properties of operators; Eigenkets of an observable; eigenkets as basekets, Matrix representations; Change of basis; unitary transformation; Postulates of quantum mechanics; Matrix approach to quantum mechanics; Quantum dynamics. Unit II Theory of angular momentum: Schrodinger equation for spherically symmetric potential, orbital angular momentum operator, Eigen values and eigen vectors of L2 and LZ.Rotations and angular momentum commutation relations. Spin angular momentum. Eigenvalues and eigenvectors of J2 and JZ. Pauli spin matrices and spin wave functions, raising and lowering operators, matrix formulation of general angular momenta. Addition of angular momenta, C.G. coefficients. Unit III Approximation methods: Non-degenerate and degenerate time independent perturbation theories with application to Stark effect. Variational method with application to ground states of harmonic oscillator, hydrogen atom and other simple cases. First order time dependent perturbation theory, constant and harmonic perturbations. Fermi’s golden rule and its application to radiative transitions in atoms.  WKB approximation. Unit IV Scattering theory: Scattering cross-section and scattering amplitude, partial wave analysis. Green’s function in scattering. Born approximation and its application to square well potential and screened coulomb potential. Optical theorem, Scattering of identical particles: symmetric and anti-symmetric wave functions. Heitler-London theory of hydrogen molecule, helium atom. Unit V Relativistic quantum mechanics: Klein Gordon equation, Dirac equation and its plane wave solution, significance of negative energy states. Existence of spin angular momentum of Dirac particles. Electron in electromagnetic fields. Dirac equation for a particle in central field.

Phys.-504 Electrodynamics 3+0 Sem-I Unit I Electrostatics and Magnetostatics: Gauss’s law and its applications, Laplace and Poisson’s equations, Electrostatic potential energy. The differential equations of magnetostatics, Vector potential, Magnetic field of a localized current distribution, Force and torque on a localized current distribution. Electrostatics of Dielectrics: Static fields in material media. Polarization vector, macroscopic equations. Molecular polarizability and electric susceptibility. Clasusius-Mossetti relation. Models of Molecular Polarizability. Energy of charges in dielectric media. Multipole expansion of the scalar potential of a charge distribution. Dipole moment, quadrupole moment. Multipole expansion of the energy of a charge distribution in an external field.  Unit II Boundary value problems: Uniqueness Theorem. Dirichlet of Neumann Boundary conditions, Green’s Theorem, Formal solution of Electrostatic Boundary value problem with Green function. Method of images with examples. Magnetostatics Boundary value problems. Time varying fields and Maxwell equations: Faraday’s Law of induction. displacement current. Maxwell equations. Scalar and vector potentials. Gauge transformation, Lorentz and Coulomb gauges,General Expression for the electromagnetic field energy, conservation of energy, Poynting’s Theorem. Conservation of momentum. Unit III Electromagnetic Waves: Wave equation, plane waves in free space and isotropic dielectrics, polarization, energy transmitted by a plane wave, Poynting’s theorem for a complex vector field, waves in conducting media. skin depth. E.M. waves in rare field plasma and their propagation in ionosphere. Reflection and Refraction of EM waves at plane interface, Frensel’s amplitude relations. Reflection and transmission coefficients. Polarization by reflection. Brewster’s angle, Total internal reflection. EM wave guides. TE and TM waves, Rectangular wave guides. Energy flow and attenuation in wave guides. Cavity resonator. Unit IV Radiation from Localized Time Varying Sources: Solutions of the inhomogeneous wave equation in the absence of boundaries. Fields and Radiation of a localized oscillating source. Electric dipole and electric quadrupole fields, centre fed linear antenna. Radiation From Accelerated Charges: Lienard-Wiechert Potentials, Field of a charge in arbitrary motion and uniform motion, Radiated power from an accelerated charge at low velocities-Larmor-Power formula. Radiation from a charged particle with collinear velocity and acceleration. Radiation from a charged particle in a circular orbit, Radiation from an ultra- relativistic particle, Bremsstrahlung and Cerenkov radiation. Radiation reaction. Line-width and level shift of an oscillator. Unit V Special Theory of Relativity: Lorentz transformation as orthogonal transformation in 4-dimension, relativistic equation of motion, applications of energy momentum conservation, Disintegration of a particle, C.M. System and reaction thresholds. Covariant Formulation of Electrodynamics in Vacuum: Four vectors in Electrodynamics, 4-current density, 4-potential, covariant continuity equation, wave equation, covariance of Maxwell equations. Electromagnetic field tensor, transformation of EM fields. Invariants of the EM fields. Energy momentum tensor of the EM fields and the conservation laws.

Phys.-505 Solid State Physics 3+1 Sem-II Unit I  Reciprocal lattice, Phonon momentum, inelastic scattering by phonons, phonon heat capacity, density of states, Einstein and Debye’s model, anharmonic crystal interactions, Thermal conductivity, Thermal resistivity, Umklapp process. Unit II   One-electron approximation, Bloch theorem, Kronig-Penney model, Nearly free electron theory, zone schemes, Tight binding theory, construction of fermi surfaces, calculations of energy bands, De Haas-van-Alphen effect, Cyclotron resonance, Magnetoresistance. Unit III Quantum theories of dia, para and ferromagnetism, Weiss molecular field theory of ferromagnetism, Ferrimagnetism and antiferromagnetism, Heisenberg exchange model. Spin waves and magnons. Unit IV  Superconductivity: experimental survey, basic phenomenology, BCS pairing mechanism and nature of BCS ground state, Flux quantization, Vortex state of a Type II superconductors, Josephson effect, Tunnelling experiments, High Tc superconductors. Unit V Many electron crystal Hamiltonian, Hall effect, Hartree-Fock approximation, Debye-Waller Factor, Electrostatic screening (Thomas-Fermi and random phase approximations), Quantum Hall effect.  Practical Susceptibility of liquids and solids, hysteresis loss of materials, electron spin resonance, four probe methods, Hall effect measurements, dispersion relations with lattice dynamics kit, measurement of dielectric constant, microwave experiment, curie temperature of ferrimagnetics, To measure cut-off frequency of monoatomic lattice, study the phenomena of magnetoresistance, measurement of velocity of ultrasonic sound waves through given liquid media and calculation of its adiabatic compressibility.

Phys.-506 Nuclear and Particle Physics 3+1 Sem-II Unit I Nuclear mass, size determination, angular momentum, spin, parity, iso-spin and moments of nuclei, binding energy, nuclear magneton. Deuteron binding energy, dipole and quadrupole moments of the deuteron, central and tensor forces. Evidence for saturation property of nuclear force, Neutron-proton scattering, exchange character, spin dependence (ortho and para-hydrogen), charge independence and charge symmetry.  Proton- proton scattering (qualitative idea only). Magic numbers, nuclear shell model and applications, collective model. Unit II Nuclear cross-section, interaction of heavy charged particles, light charge particles and gamma rays, interaction of neutrons. Basics statistics and treatment of experimental data. Detector response, energy resolution, efficiency, dead time. Gaseous ionization detectors, drift and mobility, choice of fill gas, multiwire proportional chamber, the drift chamber, properties of drift gases. Unit III Scintillation detectors: general characteristics, organic scintillators, inorganic scintillators, temperature dependence. Particle classification, Yukawa theory, fundamental forces, parity, charge conjugation and time reversal.  Spin and parity determination of pions and strange particles, isospin in pion-nucleon system, Gell-Mann Nishijimascheme,Daltiz Plot. Quarks model, Gelmann Okubo formula, w-Φ mixing. Unit IV Symmetries, elementary ideas of SU(2) and SU(3) symmetry groups and hadron classification. Fermi theory of beta decay,Cabibbo mixing, CP violation, CPT theorem, introduction to the standard model. Klein-Gordan equation, concept of particle and anti-particle, Dirac equation, V-A theory of weak interactions, Higgs mechanism. Practical To measure energy resolution using Gamma ray spectroscop, verification of ynverse square law, To determine operating voltage of GM counter, To measure attenuation coefficient using aluminium plates of varying thickness, To determine dead time and efficiency of GM counter using beta radiation, To measure half-life of beta sources, Counting of statistics, end-point energy of beta sources, To check randomcity of GM counter, To measure cross-section for p-p interactions in bubble chamber, To measure cross-section K+-d interaction in bubble chamber.

Phys.-507 Nuclear Techniques in Agriculture 2+1 Sem-II Unit I Natural, artificial and induced radioactivity, units of radioactivity, interactions of nuclear radiations with matter Unit II Detection and measurement of nuclear radiations, GM counter, solid and liquid scintillation Counters, characteristics of alpha, beta and gamma radiations, nuclear technique.  Unit III Crop improvement, adaptability of fertilizers by the plants using tracer techniques, interaction of neutrons with matter and biological materials, nuclear techniques for determining moisture content of soils.  Unit IV Determination of nitrogen and protein contents of seeds using nuclear technique, seed oil measurement using NMR technique, water dating, application of Mossbauer spectroscopy in agriculture, radiation hormesis, radiation induced phenomena: food irradiation and radiation protection, radioactive waste disposals. Practical Half-life determination, neutron activation-based experiments, attenuation studies of beta particle in plant leaves, To determine isotope uptake by plants,  To study statistical fluctuations in measurement of counts from beta source, To determine the thickness of leaves.

Phys.-508 Material Science                                      2+1 Sem-I Unit I Crystal Structure: Solids: Amorphous and Crystalline Materials. Lattice Translation Vectors. Lattice with a Basis – Central and Non-Central Elements. Unit Cell. Miller Indices. Reciprocal Lattice. Types of Lattices. Brillouin Zones. Diffraction of X-rays by Crystals. Bragg’s Law. Atomic and Geometrical Factor, Crystal defects. Unit II Polymers: Basic concepts & definitions, monomer & functionality, oligomer, polymer, repeating unites, degree of polymerization, molecular weight & molecular weight distribution. Classification of polymers thermoplastic/ thermoset, addition/ condensation, natural /synthetic, crystalline/amorphous, step-growth /chain growth, branched/ crosslinked, Classification of polymers based on end-use. Unit III Phase diagrams: the phase rule, single-component single system, and binary phase diagrams. Microstructural changes during cooling. The lever rule. Nucleation and growth. Nucleation kinetics. The glass transition, recovery, recrystallization, and grain growth. Unit IV Dielectric Properties of Materials: Polarization. Local Electric Field at an Atom. Depolarization Field. Electric Susceptibility. Polarizability. Clausius Mosotti Equation. Superconductivity: Experimental Results. Critical Temperature. Critical magnetic field. Meissner effect. Type I and type II Superconductors, London’s Equation and Penetration Depth. Practical Cooling curve of binary alloys, study and demonstrate the mechanical behaviour of elastomers, electrical conductivity of ionic solids, dielectric constant and dielectric loss measurement of organic liquids, To measure susceptibility of materials, To measure hysteresis loop of selected materials , Four probe methods, Hall effect measurements, dispersion relations with lattice dynamics kit

Phys.-509 Radiation Physics     2+1 Sem-II Unit I Types of radiations, radioactivity, half-life, radioactive series, alpha decay – quantum mechanical tunnelling (qualitative). Beta decay – continuous beta ray spectrum – neutrino hypothesis. Fermi’s theory of beta decay. Detection of neutrino – non-conservation of parity in beta decay. Gamma decay Internal conversion and electron capture (qualitative only). Unit II Interactions of heavy charged particle and electrons with matter – specific energy loss, stopping power, radiative mode of energy loss, electron range and transmission curves, radiation length. Interaction of gamma rays with matter – Elastic scattering, photoelectric effect, Compton scattering, Klein-Nishina formula (qualitative) and pair production processes, cross section, gamma ray attenuation, linear and mass absorption coefficients.  Unit III  Radiation detectors – Gas filled counters – general features – ionization chamber, proportional counter and GM counter. Van de Graaf accelerator, cyclotron, linac, synchronous accelerator, colliding beams. Unit IV Interaction of neutrons: general properties, energy classification, elastic and inelastic scattering, nuclear reaction, neutron activation and induced activity, radioisotope production, Nuclear fission. Radiation quantities and units – radiation exposure, absorbed dose, equivalent dose and effective dose.  Practicals: Half-life determination, Verification of inverse square law, attenuation studies of beta particle in plant leaves, To determine isotope uptake by plants, To study statistical fluctuations in measurement of counts from beta source, To measure half-life of beta sources, counting of statistics, end-point energy of beta sources, To check randomcity of GM counter, energy resolution of a scintillation spectrometer

 

Phys.-515 Modern Physics 2+1 Sem-I Unit I Blackbody spectra, UV catastrophe, Planck’s hypothesis, Compton effect (qualitative only), pair production, DE Broglie waves, phase and group velocity, Quantum theory of hydrogen atom. Unit II Special theory of relativity, time dilation, Doppler effect, length contraction, twin paradox, massless state, relativistic momentum, mass and energy equivalence, four vector and space-time diagram. Unit III Magnetic properties of materials. Orbital and spin magnetic moments of an electron. Bohr magnetron, Classical theory of dia, para and ferromagnetism. Specific heat, heat capacity, Debye and Einstein models for specific heat of solids. Unit IV Optical fibre, light propagation in step and graded index fibre, fibre losses. Optical and laser source, holography and applications. Practical Determination of magnetic susceptibility, dielectric constant measurement, hysteresis loss measurement of various materials, e/m ratio measurement of electron, michelson interferometer, characteristics of photocell, variation of magnetic field (ghee’s method), verification of inverse square law for intensity, photoelectric effect and measurement of planck’s constant

Phys.-516 Atomic and Molecular Spectroscopy 2+0 Sem-II Unit I Vector model of atom, quantum states of an electron in an atom, electron spin, spectrum of helium and alkali atom, relativisticcorrections for energy levels of hydrogen atom, hyperfine structure and isotopic shift, spectral terms for equivalent electrons, width of spectral lines, LS & JJ couplings.  Unit II Zeeman and Anomalous Zeeman effect, Paschen-Bach &Stark effects. Electromagnetic spectrum, Electron spin resonance. Nuclear magnetic resonance, chemical shift.  Unit III Frank-Condon principle. Electronic spectra, rotational and vibrational Raman spectra, selection rules, Mossbauer spectroscopy.

Phys.-517/Math.-533 Mathematical Physics-I 3+0 Sem. I Unit I Laplace transform, its properties and its applications for the solution of differential equations. Fourier series, Fourier integral and its applications Unit II Series solution technique for the solution of linear ordinary differential equations of variable coefficients, Frobenius Method  Unit III Bessel equation, properties of Bessel functions, Legendre equation, Legendre associated functions and their properties. Hermite, Laguerre and respective special functions. Unit IV Complex integrals, Cauchy’s integral theorem, Cauchy’s integral formula and evaluation of line integral by complex integration, Taylor’s and Laurant’s series. Unit V Integration by methods of residuals, Contour integration. Complex analytic function, potential theory and applications to two-dimensional fluid flow.

Phys.-518/Math.-534 Mathematical  Physics-II   2+1 Sem-II Unit I Dimensional analysis. Vector algebra and vector calculus. Linear algebra, matrices, Cayley-Hamilton Theorem. Eigen values and eigen vectors. Unit II Linear ordinary differential equations of second & higher orders, methods of finding complementary functions and particular integrals, method of variation of parameters, Cauchy’s and Legendre’s linear equations, Green’s function as a solution technique for non-homogeneous ordinary differential equations. Unit III Partial differential equations of first order, variable separable technique for the solution of heat equation, wave equation, Laplace equation. Unit IV Numerical solution of algebraic and transcendental equations by bisection, secant and Newton-Raphson, Difference operators, divided differences, interpolating polynomials using finite differences, Numerical Integration: General formulae, Trapezoidal rule, Simpson’s 1/3 and 3/8 rule, Solution of first order differential equation using Runge-Kutta method. Finite difference methods. Unit V Introductory group theory, Transformation of co-ordinates. Indicial and summation conventions, covariant and contravariant vectors, invariant tensors. Practical Vector algebra and vector calculus, eigen values and eigen vectors, linear ordinary differential equations of second & higher orders, heat equation, wave equation, Laplace equation, solution of first order differential equation, numerical solution of algebraic and transcendental equations, numerical integration, finite difference methods