Planar Graphs. Euler's Formula. Testing for Planarity. Kuratowski's Theorem. Planar Graphs. When drawing graphs, we usually try to make them look “nice”. Of course, there's no obvious definition of that. But one thing we probably do want if possible: no edges crossing. For example, the drawing on the right is probably “better”.  in a non-planar way originally, but it is actually planar: Like being bipartite or isomorphic, we can't just draw the graph one way and decide it's not planar. There might be another way to draw it so it is planar. Euler's Formula. When we draw a planar graph, it divides the plane up into regions. For example, this graph divides the plane into four regions: three inside and the exterior. While we're counting, on this graph [Math Processing Error]. Plane graphs/planar maps. Planarity is usually considered as a property of graphs, rather than as extra structure. When the structure reading is intended, this is sometimes referred to as a “plane graph” or “planar map”: that is, a graph equipped with an embedding into a genus 0 surface. In this sense, a planar graph is just a graph which is isomorphic to the underlying graph of a planar map.  Related to the Kuratowski and Mac Lane characterizations is the matroid characterization of Hassler Whitney: a graph is planar iff its graphic matroid is dual to another graphic matroid. Related entries. Euler's formula for planar graphs. A graph G is planar if it can be drawn in the plane in such a way that no two edges meet each other except at a vertex to which they are incident. Any such drawing is called a plane drawing of G. For example, the graph K4 is planar, since it can be drawn in the plane without edges crossing. The three plane drawings of Planar Graph Problems And Solutions K4 are: The five Platonic graphs are all planar. On the other hand, the complete bipartite graph K3,3 is not planar, since every drawing of K3,3 contains at least one crossing. why? because K3,3 has a cycle which must appear in any plane drawing. To study planar graphs, we restri. Relational model: Relational algebra, tuple calculus, SQL. Concept of viscosity, shell balances, Bernoulli's equation, mechanical energy balance equation, flow past plane surfaces and through pipes. Look up four in Wiktionary, the free dictionary. Design of irrigation channels and underground pipelines; irrigation scheduling; surface, sprinkler and micro irrigation methods, design and evaluation of irrigation methods; irrigation efficiencies. Prokaryotic and eukaryotic cell structure; planar graph definition in tamil cycle and cell growth control; cell-cell communication; cell signaling and signal transduction; post-translational modifications; protein trafficking; cell death and autophagy; extra-cellular matrix. Interpretation of geological maps. Artisan Books.

Addition reactions to carbon-carbon and carbon-heteroatom N and O multiple bonds. Elimination reactions. Reactive intermediates-carbocations, carbanions, carbenes, nitrenes, arynes and free radicals.

Molecular rearrangements. Synthesis, reactions, mechanisms and selectivity involving the following classes of compounds-alkenes, alkynes, arenes, alcohols, phenols, aldehydes, ketones, carboxylic acids, esters, nitriles, halides, nitro compounds, amines and amides.

Concepts of multistep synthesis-retrosynthetic analysis, strategic disconnections, synthons and synthetic equivalents. Atom economy and green chemistry, umpolung reactivity-formyl and acyl anion equivalents.

Selectivity in organic synthesis-chemo-, regio- and stereoselectivity. Protection and deprotection of functional groups. Concepts of asymmetric synthesis-resolution including enzymatic , desymmetrization and use of chiral auxiliaries, organocatalysis.

Carbon-carbon and carbon-heteroatom bond forming reactions through enolates including boron enolates , enamines and silyl enol ethers. Photochemistry of alkenes, arenes and carbonyl compounds. Photooxidation and photoreduction. Structure, preparation, properties and reactions of furan, pyrrole, thiophene, pyridine, indole, quinoline and isoquinoline.

Structure, properties and reactions of mono-and di-saccharides, physicochemical properties of amino acids, chemical synthesis of peptides, chemical structure determination of peptides and proteins. Structural features of proteins, nucleic acids, lipids, steroids, terpenoids, carotenoids, and alkaloids.

Vector space, basis, linear dependence and independence, matrix algebra, eigenvalues and eigenvectors, rank, solution of linear equations-existence and uniqueness. Mean value theorems, theorems of integral calculus, evaluation of definite and improper integrals, partial derivatives, maxima and minima, multiple integrals, line, surface and volume integrals, Taylor series.

First order equations linear and nonlinear , higher order linear differential equations, Cauchy's and Euler's equations, methods of solution using variation of parameters, complementary function and particular integral, partial differential equations.

Variable separable method, initial and boundary value problems. Vectors in plane and space, vector operations, gradient, divergence and curl, Gauss's, Green's and Stokes' theorems. Analytic functions, Cauchy's integral theorem, Cauchy's integral formula, sequences, series, convergence tests, Taylor and Laurent series, residue theorem.

Mean, median, mode, standard deviation, combinatorial probability, probability distributions, binomial distribution, Poisson distribution, exponential distribution, normal distribution, joint and conditional probability.

Node and mesh analysis, superposition, Thevenin's theorem, Norton's theorem, reciprocity. Sinusoidal steady state analysis: Phasors, complex power, maximum power transfer.

Linear 2-port network parameters, Wye-delta transformation. Fourier series and Fourier transform, sampling theorem and applications. LTI systems: Definition and properties, causality, stability, impulse response, convolution, poles and zeroes, frequency response, group delay, phase delay. Diffusion current, drift current, mobility and resistivity, generation and recombination of carriers, Poisson and continuity equations.

Biasing, ac coupling, small signal analysis, frequency response. Current mirrors and differential amplifiers. Amplifiers, summers, differentiators, integrators, active filters, Schmitt triggers and oscillators. Boolean algebra, minimization of functions using Boolean identities and Karnaugh map, logic gates and their static CMOS implementations, arithmetic circuits, code converters, multiplexers, decoders. Latches and flip-flops, counters, shift-registers, finite state machines, propagation delay, setup and hold time, critical path delay.

Machine instructions and addressing modes, ALU, data-path and control unit, instruction pipelining. Autocorrelation and power spectral density, properties of white noise, filtering of random signals through LTI systems. Amplitude modulation and demodulation, angle modulation and demodulation, spectra of AM and FM, superheterodyne receivers. Differential and integral forms and their interpretation, boundary conditions, wave equation, Poynting vector.

Reflection and refraction, polarization, phase and group velocity, propagation through various media, skin depth. Equations, characteristic impedance, impedance matching, impedance transformation, S-parameters, Smith chart.

Matrix algebra, systems of linear equations, eigen values, eigen vectors. Mean value theorems, theorems of integral calculus, evaluation of definite and improper integrals, partial derivatives, maxima and minima, multiple integrals, Fourier series, vector identities, directional derivatives, line integral, surface integral.

Volume integral, Stokes's theorem, Gauss's theorem, divergence theorem, Green's theorem. First order equations linear and nonlinear , higher order linear differential equations with constant coefficients, method of variation of parameters, Cauchy's equation, Euler's equation, Initial and boundary value problems.

Partial differential equations, method of separation of variables. Analytic functions, Cauchy's integral theorem, Cauchy's integral formula, Taylor series, Laurent series, residue theorem, solution integrals. Sampling theorems, conditional probability, mean, median, mode, standard deviation, random variables, discrete and continuous distributions, Poisson distribution, normal distribution, binomial distribution, correlation analysis, regression analysis.

Ideal voltage and current sources, dependent sources, R, L, C, M elements. Thevenin's, Norton's, superposition and maximum power transfer theorem; transient response of dc and ac networks, sinusoidal steady-state analysis, resonance, two port networks, balanced three phase circuits, star-delta transformation. Complex power and power factor in ac circuits.

Equivalent circuit, phasor diagram, open circuit and short circuit tests, regulation and efficiency. Separately excited, series and shunt, motoring and generating mode of operation and their characteristics, speed control of dc motors. Principle of operation, types, performance, torque-speed characteristics, no-load and blocked-rotor tests, equivalent circuit, starting and speed control.

Cylindrical and salient pole machines, performance and characteristics, regulation and parallel operation of generators, starting of synchronous motors. Heat flow within the earth; gravitational field of the Earth; geomagnetism and paleomagnetism; continental drift; plate tectonics-relationship with earthquakes, volcanism and mountain building; continental and oceanic crust.

Stress, strain and material response; brittle and ductile deformation; nomenclature and classification of folds and faults. Mineralogy-silicate crystal structure and determinative mineralogy of common rock forming minerals. Geomorphic processes and agents; development and evolution of landforms in continental, and oceanic settings; tectonic geomorphology. Forces and mechanism of rock deformation; primary and secondary structures; geometry and genesis of planar, and linear structures bedding, cleavage, schistosity, lineation ; folds, faults, joints, and unconformities; stereographic projection; shear zones.

Thrusts and superposed folding; basement-cover relationship. Interpretation of geological maps. Elements of crystal symmetry, form and twinning; crystallographic projection; crystal chemistry; classification of minerals, physical and optical properties of rock-forming minerals.

Cosmic abundance of elements; meteorites; geochemical evolution of the earth; geochemical cycles; distribution of major, minor, and trace elements in crust and mantle; elements of high temperature and low temperature geochemical thermodynamics. Isotopic evolution of the crust and the mantle, mantle reservoirs; geochemistry of water and water-rock interaction. Classification, forms, textures, and genesis of common igneous rocks; magmatic differentiation; binary and ternary phase diagrams; major and trace elements as monitors of partial melting and magma evolutionary processes.

Mantle plumes, hotspots, and large igneous provinces. Texture, structure, and sedimentary processes; petrology of common sedimentary rocks; sedimentary facies and environments, cyclicities in sedimentary succession; provencance and basin analysis.

Important sedimentary basins of India. Physicochemical conditions of metamorphism and concept of metamorphic facies, grade and baric types; chemographic projections; metamorphism of pelitic, mafic and impure carbonate rocks; role of bulk composition including fluids in metamorphism.

Thermobarometry and metamorphic P-T-t paths, and their tectonic significance. Diversity of life through time, mass extinctions-causes and effects; taphonomy-processes of fossilization. Morphology and functional morphology of invertebrates bivalves, brachiopods, gastropods, echinoids, ammonites ; microfossils foraminifera, ostracoda, conodonts, bryozoa ; vertebrate paleonology equus, probicidea, human.

Paleobotany plant, spores, pollens. Fossils and paleoenvironments. Principles of stratigraphy and concepts of correlation; lithostratigraphy, biostratigraphy, and chronostratigraphy. Principles of sequence stratigraphy and applications. Stratigraphy of peninsular and extra-peninsular India.

Boundary problems in Indian stratigraphy. Coal and petroleum geology; marine mineral resources. Prospecting and exploration of economic mineral deposits-sampling, ore reserve estimation, geostatistics, mining methods. Ore dressing and mineral economics. Distribution of mineral, fossil, and nuclear fuel deposits in India.

Plate motions, driving mechanisms, plate boundaries, supercontinent cycles. Physicomechanical properties of rocks and soils; rock index tests; rock failure criteria Mohr-Coulomb, Griffith, and Hoek-Brown criteria ; shear strength of rock discontinuities; rock mass classifications RMR and Q Systems ; in-situ stresses. Rocks as construction materials; geological factors in the construction of engineering structures including dams, tunnels, and excavation sites.

Analysis of slope stability. Groundwater flow and exploration, well hydraulics and water quality. Energy sources and radiation principles, atmospheric absorption, interaction of energy with earth's surface, aerial-photo interpretation, multispectral remote sensing in visible, infrared, thermal IR and microwave regions. Digital processing of satellite images. GIS-basic concepts, raster and vector mode operations. The earth as a planet; different motions of the earth; gravity field of the earth, Clairaut's theorem, size and shape of earth; geomagnetic field, paleomagnetism; geothermics and heat flow; seismology and interior of the earth; variation of density.

Velocity, pressure, temperature, electrical and magnetic properties of the earth. Elements of elasticity theory-stress and strain tensors, generalized Hooke's law; body and surface waves; rotational, dilatational, irrotational, and equivolumnal waves. Reflection and refraction of elastic waves; inhomogeneous and evanescent waves, and bounded waves; Eikonal equation and ray theory; earthquakes causes and measurements, magnitude and intensity, focal mechanisms; earthquake quantification.

Source characteristics, seismotectonics, and seismic hazards; digital seismographs, earthquake statistics, wave propagation in elastic media, quantifying earthquake source from seismological data.

Elements of seismic tomography. Scalar and vector potential fields; Laplace, Maxwell, and Helmholtz equations for solution of different types of boundary value problems in Cartesian, cylindrical and spherical polar coordinates; Green's theorem; image theory. Integral equations in potential and time-varying field theory.

Absolute and relative gravity measurements; gravimeters; land, airborne, shipborne, and bore-hole gravity surveys; tensorial What Are Planar Graphs Used For gravity sensors and surveys; various corrections for gravity data reduction-free air, Bouguer, and isostatic anomalies.

Density estimates of rocks; regional and residual gravity separation; principle of equivalent stratum; data enhancement techniques, upward and downward continuation; derivative maps, wavelength filtering; preparation and analysis of gravity maps. Gravity anomalies and their interpretation-anomalies due to geometrical and irregular shaped bodies, depth rules, calculation of mass.

Elements of Earth's magnetic field, units of measurement, magnetic susceptibility of rocks and measurements, magnetometers, and magnetic gradiometers, land, airborne and marine magnetic, and magnetic gradiometer surveys. Various corrections applied to magnetic data, IGRF, reduction to Pole transformation, Poisson's relation of gravity, and magnetic potential field, preparation of magnetic maps, upward and downward continuation.

Magnetic anomalies due to geometrical and irregular shaped bodies; image processing concepts in processing of magnetic anomaly maps; depth rules; interpretation of processed magnetic anomaly data; derivative.

Analytic signal and Euler depth solutions. Applications of gravity and magnetic methods for mineral, and oil exploration. Conduction of electricity through rocks, electrical conductivities of metals, non-metals, rock forming minerals and different rocks, concepts of DC resistivity measurement and depth of investigation; apparent resistivity and apparent chargeability.

Concept of negative apparent resistivity and negative apparent chargeability; theory of reciprocity, sounding and profiling, various electrode arrangements, application of linear filter theory, sounding curves over multi-layered earth. Dar-Zarrouk parameters, reduction of layers, triangle of anisotropy, interpretation of resistivity field data, principles of equivalence and suppression, self-potential method and its origin; electrical resistivity tomography ERT ; induced polarization.

Time and frequency domain IP measurements; interpretation and applications of SP, resistivity and IP data sets for ground-water exploration, mineral exploration, environmental and engineering applications. Geoelectromagnetic spectrum; Biot Savart's law; Maxwell's equation, Helmholtz equation, basic concept of EM induction in the earth, skin-depth, elliptic polarization, in-phase and quadrature components, phasor diagrams.

Response function and response parameters; ground and airborne methods, measurements in different source-receiver configurations; Earth's natural electromagnetic methods-tellurics, geomagnetic depth sounding and magnetotellurics. Electromagnetic profiling and sounding, time domain EM method; EM scale modelling, processing of EM data and interpretation; ground penetrating radar GPR methods; effect of conducting overburden; geological applications including groundwater.

Mineral environmental and hydrocarbon exploration. Elastic properties of earth materials; reflection, refraction, and CDP surveys; land and marine seismic sources, generation, and propagation of elastic waves, velocity-depth models, geophones, hydrophones, digital recording systems, digital formats. Field layouts, seismic noise, and noise profile analysis, optimum geophone grouping, noise cancellation by shot and geophone arrays, 2D, 3D, and 4D seismic data acquisition, processing and interpretation; CDP stacking charts, binning, filtering.

Static and dynamic corrections, digital seismic data processing, seismic deconvolution and migration methods, attribute analysis, bright and dim spots, seismic stratigraphy, high resolution seismics, VSP, AVO, multi.

Component seismics and seismic interferometry. Reservoir geophysics-rock physics and petrophysics. Geophysical survey design. Sampling theorem, Nyquist frequency, aliasing, Fourier series, periodic waveform, Fourier and Hilbert transform, Z-transform and wavelet transform; power spectrum, delta function, auto correlation, cross correlation, convolution, deconvolution.

Principles of digital filters, windows, poles, and zeros. Principles and techniques of geophysical well-logging, SP, resistivity, induction, gamma ray, neutron, density, sonic, temperature, dip meter, caliper, nuclear magnetic resonance-longitudinal and transverse relaxation, CPMG sequence. Porosity characterization, cement bond logging, micro-logs. Pulsed neutron devices and spectroscopy; multi-array and triaxial induction devices; quantitative evaluation of formations from well logs; logging while drilling; high angle and horizontal wells; clay quantification; lithology and porosity estimation.

Saturation and permeability estimation; application of bore hole geophysics in ground water, mineral, and oil exploration. Prospecting and assaying of mineral radioactive and non-radioactive deposits, half-life, decay constant, radioactive equilibrium, GM counter, scintillation detector, semiconductor devices, application of radiometric for exploration.

Assaying and radioactive waste disposal. Basic concepts of forward and inverse problems, ill-posedness of inverse problems, condition number, non-uniqueness and stability of solutions; L1, L2, and Lp norms, overdetermined, underdetermined, and mixed determined inverse problems.

Quasi-linear and non-linear methods including Tikhonov's regularization method, singular value decomposition, Backus-Gilbert method, simulated annealing, genetic algorithms, swarm intelligence, machine learning, and artificial neural networks. Statistics of misfit and likelihood, Bayesian construction of posterior probabilities, sparsity promoting L1 optimization. Ambiguity and uncertainty in geophysical interpretation.

Matrix algebra, systems of linear equations, consistency and rank, eigen values and eigen vectors. Mean value theorems, theorems of integral calculus, partial derivatives, maxima and minima, multiple integrals, Fourier series, vector identities, line, surface and volume integrals, Stokes, Gauss and Green's theorems.

First order equation linear and nonlinear , second order linear differential equations with constant coefficients, method of variation of parameters, Cauchy's and Euler's equations, initial and boundary value problems. Solution of partial differential equations: variable separable method.

Analytic functions, Cauchy's integral theorem and integral formula, Taylor's and Laurent's series, residue theorem, solution of integrals. Sampling theorems, conditional probability, mean, median, mode, standard deviation and variance; random variables: Discrete and continuous distributions: Normal, Poisson and binomial distributions. Matrix inversion, solutions of non-linear algebraic equations, iterative methods for solving differential equations, numerical integration, regression and correlation analysis.

Coulomb's law, electric field intensity, electric flux density, Gauss's law, divergence, electric field and potential due to point, line, plane and spherical charge distributions, effect of dielectric medium, capacitance of simple configurations. Voltage and current sources: Independent, dependent, ideal and practical; V-I relationships of resistor, inductor, mutual inductance and capacitor; transient analysis of RLC circuits with DC excitation.

Kirchhoff's laws, mesh and nodal analysis, superposition, Thevenin, Norton, maximum power transfer and reciprocity theorems. Peak-, average- and RMS values of ac quantities; apparent-, active- and reactive powers; phasor analysis, impedance and admittance; series and parallel resonance, locus diagrams, realization of basic filters with R, L and C elements. One-port and two-port networks, driving point impedance and admittance, open-, and short circuit parameters.

Single phase transformer: Equivalent circuit, phasor diagram, open circuit and short circuit tests, regulation and efficiency. Three phase induction motors: Principle of operation, types, performance, torque-speed characteristics, no-load and blocked rotor tests, equivalent circuit, starting and speed control; types of losses and efficiency calculations of electric machines.

Periodic, aperiodic and impulse signals; Laplace, Fourier and z-transforms; transfer function, frequency response of first and second order linear time invariant systems, impulse response of systems; convolution, correlation. Feedback principles, signal flow graphs, transient response, steady-state-errors, Bode plot, phase and gain margins, Routh and Nyquist criteria, root loci, design of lead, lag and lead-lag compensators, state-space representation of systems.

Time-delay systems; mechanical, hydraulic and pneumatic system components, synchro pair, servo and stepper motors, servo valves; on-off, P, PI, PID, cascade, feedforward, and ratio controllers, tuning of PID controllers and sizing of control valves. Characteristics of ideal and practical operational amplifiers; applications of opamps: Adder, subtractor, integrator, differentiator, difference amplifier, instrumentation amplifier, precision rectifier, active filters, oscillators, signal generators.

Voltage controlled oscillators and phase locked loop, sources and effects of noise and interference in electronic circuits. Combinational logic circuits, minimization of boolean functions. Arithmetic circuits, comparators, Schmitt trigger, multi-vibrators, sequential circuits, flipflops, shift registers, timers and counters; sample-and-hold circuit, multiplexer.

Analog-to-digital successive approximation, integrating, flash and sigma-delta and digital-to-analog converters weighted R, R-2R ladder and current steering logic. SI units, standards R,L,C, voltage, current and frequency , systematic and random errors in measurement, expression of uncertainty - accuracy and precision, propagation of errors, linear and weighted regression. Digital multimeter; oscilloscope, shielding and grounding.

Resistive-, capacitive-, inductive-, piezoelectric-, Hall effect sensors and associated signal conditioning circuits; transducers for industrial instrumentation: Displacement linear and angular , velocity, acceleration, force, torque, vibration, shock. Conductivity and viscosity measurement.

Amplitude- and frequency modulation and demodulation; Shannon's sampling theorem, pulse code modulation; frequency and time division multiplexing, amplitude-, phase-, frequency-, quadrature amplitude, pulse shift keying for digital modulation. Optical sources and detectors: LED, laser, photo-diode, light dependent resistor, square law detectors and their characteristics; interferometer: Applications in metrology; basics of fiber optic sensing.

UV-VIS spectro photometers, mass spectrometer. Volume and surface area; vector calculus: Gradient, divergence, and curl, line integrals and surface integrals, Green's theorem, Stokes' theorem, and Gauss divergence theorem.

Finite dimensional vector spaces over real or complex fields; linear transformations and their matrix representations, rank and nullity; systems of linear equations, characteristic polynomial, eigen values and eigen vectors, diagonalization. Minimal polynomial, Cayley-Hamilton theorem, finite dimensional inner product spaces, Gram-Schmidt orthonormalization process, symmetric, skew-symmetric, Hermitian, skew-Hermitian, normal, orthogonal and unitary matrices.

Diagonalization by a unitary matrix, Jordan canonical form; bilinear and quadratic forms. Metric spaces, connectedness, compactness, completeness; sequences and series of functions, uniform convergence, Ascoli-Arzela theorem; weier strass approximation theorem; contraction mapping principle, power series. Functions of a complex variable: Continuity, differentiability, analytic functions, harmonic functions; Complex integration: Cauchy's integral theorem and formula; Liouville's theorem, maximum modulus principle, Morera's theorem; zeros and singularities.

Power series, radius of convergence, Taylor's series and Laurent's series; residue theorem and applications for evaluating real integrals; Rouche's theorem, argument principle, Schwarz lemma; conformal mappings, Mobius transformations.

First order ordinary differential equations, existence and uniqueness theorems for initial value problems, linear ordinary differential equations of higher order with constant coefficients. Second order linear ordinary differential equations with variable coefficients; Cauchy-Euler equation, method of Laplace transforms for solving ordinary differential equations, series solutions power series, Frobenius method.

Legendre and Bessel functions and their orthogonal properties; systems of linear first order ordinary differential equations, Sturm's oscillation and separation theorems, Sturm-Liouville eigen value problems. Planar autonomous systems of ordinary differential equations: Stability of stationary points for linear systems with constant coefficients, linearized stability, Lyapunov functions.

Groups, subgroups, normal subgroups, quotient groups, homomorphisms, automorphisms; cyclic groups, permutation groups, group action, Sylow's theorems and their applications; rings, ideals, prime and maximal ideals, quotient rings.

Unique factorization domains, principle ideal domains, euclidean domains, polynomial rings, Eisenstein's irreducibility criterion; fields, finite fields, field extensions, algebraic extensions, algebraically closed fields. Normed linear spaces, banach spaces, Hahn-Banach theorem, open mapping and closed graph theorems, principle of uniform boundedness; inner-product spaces, Hilbert spaces, orthonormal bases, projection theorem, Riesz representation theorem.

Spectral theorem for compact self-adjoint operators. Systems of linear equations: Direct methods Gaussian elimination, LU decomposition, Cholesky factorization , iterative methods Gauss-Seidel and Jacobi and their convergence for diagonally dominant coefficient matrices. Numerical solutions of nonlinear equations: Bisection method, secant method, Newton-Raphson method, fixed point iteration; interpolation: Lagrange and Newton forms of interpolating polynomial, error in polynomial interpolation of a function.

Numerical differentiation and error, numerical integration: Trapezoidal and Simpson rules, Newton-Cotes integration formulas, composite rules, mathematical errors involved in numerical integration formulae.

Numerical solution of initial value problems for ordinary differential equations: Methods of Euler, Runge-Kutta method of order 2.

Method of characteristics for first order linear and quasilinear partial differential equations; second order partial differential equations in two independent variables: Classification and canonical forms.

Second order partial differential equations in two independent variables: Method of separation of variables for Laplace equation in Cartesian and polar coordinates, heat and wave equations in one space variable.

Wave equation: Cauchy problem and D' Alembert formula, domains of dependence and influence, non-homogeneous wave equation; heat equation: Cauchy problem; Laplace and Fourier transform methods.

Basic concepts of topology, bases, subbases, subspace topology, order topology, product topology, quotient topology, metric topology, connectedness, compactness, countability and separation axioms, Urysohn's Lemma.

Linear programming models, convex sets, extreme points; basic feasible solution, graphical method, simplex method, two phase methods, revised simplex method ; infeasible and unbounded linear programming models, alternate optima; duality theory. Modified distribution method; solving assignment problems, hungarian method. Functions of single variable, limit, continuity, and differentiability, mean value theorems, indeterminate forms; evaluation of definite and improper integrals; double and triple integrals; partial derivatives, total derivative.

Taylor series in one and two variables , maxima and minima, Fourier series; gradient, divergence, and curl, vector identities, directional derivatives, line, surface and volume integrals, applications of Gauss, Stokes, and Green's theorems. First order equations linear and nonlinear ; higher order linear differential equations with constant coefficients; Euler-Cauchy equation; initial and boundary value problems; Laplace transforms; solutions of heat, wave and Laplace's equations.

Analytic functions; Cauchy-Riemann equations; Cauchy's integral theorem and integral formula; Taylor and Laurent series. Definitions of probability, sampling theorems, conditional probability; mean, median, mode, and standard deviation; random variables, binomial, Poisson, and normal distributions.

Numerical solutions of linear and non-linear algebraic equations; integration by trapezoidal and Simpson's rules; single and multi-step methods for differential equations. Free-body diagrams and equilibrium; friction and its applications including rolling friction, belt-pulley, brakes, clutches, screw jack, wedge, vehicles, etc; trusses and frames; virtual work; kinematics and dynamics of rigid bodies in plane motion.

Impulse and momentum linear and angular , and energy formulations; Lagrange's equation. Stress and strain, elastic constants, Poisson's ratio; Mohr's circle for plane stress and plane strain; thin cylinders; shear force and bending moment diagrams; bending and shear stresses; concept of shear centre; deflection of beams.

Torsion of circular shafts; Euler's theory of columns; energy methods; thermal stresses; strain gauges and rosettes; testing of materials with universal testing machine; testing of hardness and impact strength. Displacement, velocity, and acceleration analysis of plane mechanisms; dynamic analysis of linkages; cams; gears and gear trains; flywheels and governors; balancing of reciprocating and rotating masses; gyroscope.

Free and forced vibration of single degree of freedom systems, effect of damping; vibration isolation; resonance; critical speeds of shafts. Design for static and dynamic loading; failure theories; fatigue strength and the S-N diagram; principles of the design of machine elements such as bolted, riveted and welded joints; shafts, gears, rolling and sliding contact bearings.

Brakes and clutches, springs. Fluid properties; fluid statics, forces on submerged bodies, stability of floating bodies; control-volume analysis of mass, momentum, and energy; fluid acceleration; differential equations of continuity and momentum; Bernoulli's equation.

Dimensional analysis; viscous flow of incompressible fluids, boundary layer, elementary turbulent flow, flow through pipes, head losses in pipes, bends, and fittings; basics of compressible fluid flow.

Modes of heat transfer; one dimensional heat conduction, resistance concept and electrical analogy, heat transfer through fins; unsteady heat conduction, lumped parameter system, Heisler's charts; thermal Planar Graph boundary layer.

Dimensionless parameters in free and forced convective heat transfer, heat transfer correlations for flow over flat plates and through pipes, effect of turbulence; heat exchanger performance, LMTD and NTU methods; radiative heat transfer.

Stefan-Boltzmann law, Wien's displacement law, black and grey surfaces, view factors, radiation network analysis. Thermodynamic systems and processes; properties of pure substances, behaviour of ideal and real gases; zeroth and first laws of thermodynamics, calculation of work and heat in various processes; second law of thermodynamics.

Power engineering: Air and gas compressors; vapour and gas power cycles, concepts of regeneration and reheat. IC engines: Air-standard Otto, diesel, and dual cycles. Refrigeration and air-conditioning: Vapour and gas refrigeration and heat pump cycles; properties of moist air, psychrometric chart, basic psychrometric processes.

Turbomachinery: Impulse and reaction principles, velocity diagrams, pelton-wheel, Francis and Kaplan turbines; steam and gas turbines. Structure and properties of engineering materials, phase diagrams, heat treatment, stress-strain diagrams for engineering materials. Different types of castings, design of patterns, moulds and cores; solidification and cooling; riser and gating design.

Plastic deformation and yield criteria; fundamentals of hot and cold working processes; load estimation for bulk forging, rolling, extrusion, drawing and sheet shearing, deep drawing, bending metal forming processes; principles of powder metallurgy. Principles of welding, brazing, soldering, and adhesive bonding.

Mechanics of machining; basic machine tools; single and multi-point cutting tools, tool geometry and materials, tool life and wear; economics of machining; principles of non-traditional machining processes; principles of work holding, jigs, and fixtures.

Limits, fits, and tolerances; linear and angular measurements; comparators; interferometry; form and finish measurement; alignment and testing methods; tolerance analysis in manufacturing and assembly; concepts of coordinate-measuring machine CMM.

Forecasting models, aggregate production planning, scheduling, materials requirement planning; lean manufacturing. Deterministic models; safety stock inventory control systems. Limit, continuity, and differentiability; partial derivatives; maxima and minima; sequences and series; test for convergence; Fourier series.

Gradient; divergence and curl; line; surface and volume integrals; Stokes, Gauss, and Green's theorems. Definitions of probability and sampling theorems, conditional probability, mean, median, mode, and standard deviation; random variables; Poisson, normal, and binomial distributions; analysis of experimental data; linear least squares method. First law-energy conservation, second law-entropy; enthalpy, Gibbs and Helmholtz free energy; Maxwell's relations; chemical potential; applications to metallurgical systems, solutions, ideal and regular solutions; Gibbs phase rule, phase equilibrium.

Binary phase diagram and lever rule, free-energy vs. Single electrode potential, electrochemical cells, Nernst equation, potential-pH diagrams. Concept of viscosity, shell balances, Bernoulli's equation, mechanical energy balance equation, flow past plane surfaces and through pipes.

Conduction, Fourier's law, 1-D steady state conduction. Heat transfer coefficient relations for forced convection. Black body radiation, Stefan-Boltzmann law, Kirchhoff's law. Diffusion and Fick's laws, mass transfer coefficients. Buckingham Pi theorem, significance of dimensionless numbers. First order reactions, reaction rate constant, Arrhenius relation, heterogeneous reactions, oxidation kinetics.

Material and heat balance in blast furnace; structure and properties of slags, and molten salts-basicity of slags, sulphide, and phosphate capacity of slags; production of metallurgical coke.

Basic oxygen furnace, process dynamics, oxidation reactions, electric arc furnace. Ladle process-deoxidation, argon stirring, desulphurization, inclusion shape control, principles of degassing methods; basics of stainless steel manufacturing. Fluid flow in the tundish and mould, heat transfer in the mould, segregation, inclusion control. Ionic, covalent, metallic, and secondary bonding in materials, crystal structure of solids-metals and alloys, ionic and covalent solids, and polymers.

Bragg's law, optical metallography, principles of SEM imaging. Point, line, and surface defects; coherent, semi-coherent, and incoherent interfaces.

Diffusion equation, steady state, and error function solutions; examples-homogenenization and carburization; Kirkendall effect; uphill diffusion; atomic models for interstitial and substitutional diffusion; pipe diffusion and grain boundary diffusion.

Driving force, homogeneous and heterogeneous nucleation, growth kinetics. Precipitation, spinoidal decomposition, ordering, massive transformation, discontinuous precipitation, eutectoid transformation, diffusionless transformations; precipitate coarsening, Gibbs-Thomson effect.

Edge, screw, and mixed dislocations, source and multiplication of dislocations, stress fields around dislocations; partial dislocations, dislocation interactions, and reactions. Cyclic stress strain behaviour-low and high cycle fatigue, crack growth. Mould design involving feeding, gating, and risering, casting practices, casting defects.

Metal forming-fundamentals of metal forming processes of rolling, forging, extrusion, wire drawing, and sheet metal forming, defects in forming. Principles of soldering, brazing, and welding, welding metallurgy, defects in welded joints in steels and aluminium alloys.

Dye-penetrant, ultrasonic, radiography, eddy current, acoustic emission and magnetic particle inspection methods. Matrices and determinants; inverse and rank of matrix; systems of linear equations; eigen values and eigen vectors.

Measures of central tendency and dispersion; hypothesis testing; binomial, Poisson, exponential, and normal distributions; correlation and regression analysis. Solutions of linear algebraic equations; interpolation; integration of trapezoidal and Simpson's rule; single and multi-step methods for differential equations.

Minerals, rocks, and their origin, classification, ore genesis; structural geology. Methods of access to deposits; underground drivages; drilling methods and machines; explosives and energetics, blasting devices, blast design practices; rock-tool interaction applicable to mechanical cutting systems and their selection. Levels and levelling, theodolite, tacheometry, triangulation; contouring; errors and adjustments; correlation; underground surveying; curves; photogrammetry; EDM, total station, GPS, basics of GIS and remote sensing.

Equivalent force systems; equations of equilibrium; two dimensional frames and trusses; free body diagrams; friction forces; particle kinematics and dynamics; beam analysis. Geotechnical properties of rocks; rock mass classification; instrumentation and in-situ stress measurement techniques; theories of rock failure; ground vibrations; stress distribution around mine openings; subsidence; slope stability.

Strata control and monitoring plan. Surface mining: Layout, development, loading, transportation and mechanization, continuous surface mining systems; highwall mining; underground coal mining: Bord and pillar systems, room and pillar mining, longwall mining, thick seam mining methods. Underground metal mining: Open, supported, and caved stoping methods, stope mechanization, ore handling systems.

Generation and transmission of mechanical, hydraulic and pneumatic power; materials handling: wire ropes, haulages, conveyors, face and development machinery, hoisting systems, pumps; comminution methods and machinery.

Air, water, and soil pollution: Standards of quality, causes and dispersion of contamination, and control; noise pollution and control; land reclamation; EIA. Underground atmosphere; heat load sources and thermal environment; air cooling; mechanics of airflow, distribution, natural and mechanical ventilation; mine fans and their usage; auxiliary ventilation; ventilation survey and planning; ventilation networks.

Mine gases, methane drainage; underground hazards from fires, explosions, dust and inundation; rescue apparatus and practices; safety management plan; accident data analysis; assessment; mine lighting; mine legislation; occupational health and safety.

Mineral resource classification; discounted cash flow analysis; mine valuation; mineral taxation. Sampling methods, practices, and interpretation; reserve estimation techniques: Basics of geostatistics and quality control; optimization of facility location; mine planning and its components, determination of mine size and mine life. Concepts of reliability; reliability of simple systems; maintainability and availability; linear programming, transportation, and assignment problems; network analysis; inventory models; queuing theory; decision trees.

Functions of single variable, limit, continuity, and differentiability, Taylor series, mean value theorems, evaluation of definite and improper integrals, partial derivatives, total derivative, maxima and minima, gradient, divergence, and curl. Svetovid is portrayed as having four heads Voices of Democracy.

The Economist. ISSN November Archived PDF from the original on Protein Structure and Function. New Science Press. Oligomers containing two, three, four, five, six or even more subunits are known as dimers, trimers, tetramers, pentamers, hexamers, and so on. Exoplanets and Alien Solar Systems. New Earth Labs. The Solar System. Our Planet Earth Publishing. M4 is a globular star cluster near Antares in Scorpius.

The Milky Way. Harvard University Press. Academic Press. An Introductory Text to Bioengineering. World Scientific. The mammalian heart consists of four chambers, Biology: a search for order in complexity.

Zondervan Pub. Except for the flies, mosquitoes, and some others, insects with wings have four wings. California Master Gardener Handbook, 2nd Edition. Pratiyogita Darpan. National Geographic Books. Feeding Dairy Cows. Department of Agriculture. The cow's Planar Graph Perfect Matching Is In Nc stomach is divided into four compartments.

Great unsolved mysteries of science. Beryllium Chemistry and Processing. ASM International. Plasma is one of the four fundamental states of matter, the others being solid, liquid, and gas. Relativity and the Nature of Spacetime.

College Physics: Reasoning and Relationships. We have referred to the four fundamental forces in nature, The Probabilistic Method. The quote beginning "Almost from prehistoric times Aristotle's Four Causes. Peter Lang. The OODA loop consists of four steps. Good Housekeeping.

Designing for Print. Discovering Computers, Essentials. Certified Examiner Study Guide. A byte also contains two 4-bit nibbles Popular Mechanics.

Hearst Magazines. The Everything Kids' Basketball Book: The all-time greats, legendary teams, today's superstars - and tips on playing like a pro.

Cambridge University Press. But one confused re-spelling is fower for 'four'. Mythic Astrology: Archetypal Powers in the Horoscope. The Key to the Universe. Health Research Books. The 4th Tarot Card is called "The Emperor.

Video Games. Tetra means "four" in Greek. The court will not issue a writ unless at least four justices approve of it. This is called the rule of four.

From Music to Mathematics: Exploring the Connections. JHU Press. Princeton University Press. Uploaded by Dipesh. Date uploaded Dec 16, Did you find this document useful? Is this content inappropriate? Report this Document. Description: List of MAC. Flag for inappropriate content. Download now. List of Amc. Related titles.

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