Polynomial equation-theorems-Every nth degree equation has exactly n roots-An equation with rational coeffients in conjugate pairs- An equation with real coeffients has imaginary roots in conjugate pairs (Statement only)-problems-Relation between roots and coeffients-Symmetry function of roots-Problems
Unit III
Transformation of an equation - roots with sign changed root-roots multiplied by a given number-reciprocal of a roots-squares of the roots-increasing and decreasing the roots of an equation by a given number-Removing the second term of a given-Transformation in general-Problems.
Unit IV
Sum of the nth power of the roots -Newton's theorem-problems only-Reciprocal equation-Descarte's rule of signs-problems
Unit V
Expansion of sin nq, cos nq and tan nq-Expansion of sin q, cos q-problems-Hyperbolic and inverse Hyperbolic functions-properties-Logarithm of a complex number-Solving the Trigonometric equations- (i) sin q = s ii) cos q = c iii) tan q = t and iv) a cos q+b sin q = c
Text Books
1.
T.K.Manickavasagam Pillai and S.Narayanan: ALGEBERA( VOL I )-Year of Publication 2004
2.
P.R.Vittal: ALGEBRA ANALYTICAL GEOMETRY AND TRIGONOMETRY-Year of Publication 2000
3.
T.K.Manickavasagam Pillai and S.Narayanan: TRIGONOMETRY (VOL I)-Year of Publication 2004.
Reference Books
1.
Burnside and Pantern -ALGEBRA - Macmillan Publishers, UK-1976
2.
N.P. Bali - TRIGONOMETRY -Year of Publication 1994. Krishna Prakasan Mandhir , 9 , Shivaji Road Meerut ( UP )
PAPER-1:ALLIED PHYSICS - I(06UPHA1)
(General Physics, Heat, Optics and Electricity)
UNIT I: Mechanics, Sound
Projectile-range up and down an inclind plane-Impulse-impact-laws of impact-coefficient of restitution-direct impact between two spheres-compound pendulum-theory-determination of acceleration due to gravity.
Production of ultrasonic waves-Piezo electric crystal method-applications-acoustics of buildings-reverberation-reverberation time-absorption coefficient-Sabine's formula.
UNIT II: Properties of Matter
Newton's law of gravitation-determination of gravitational constant-Boy's method-bending of beams- expressions for bending moment-experssion for the depression of the free end of the cantilever-uniform and uniform bending-theory and experiment -torsion-expression for couple per unit twist-torsion pendulum-theory rigidity modulud by static torsion-surface tension-excess pressure inside a curved liquid surface- surface tension and interfacial surface tension drop weight method.
UNIT III: Heat
Specific heats: Determination of Cp and Cv - Van-der-waas's equation -Critical constants and their determination - Expressions for Critical constants - Thermal conductivity of a bad conductor - Lees' disk method -Joul-Thomson effect - Porous plug experiment - Theory - Inversion temperature - Boil temperature - Liquifaction of gases - Hydrogen and Helium - Adiabatic demagnetization.
UNIT IV: Optics
Small angled prism - formation of two thin prisms to produce dispersion without deviation and deviation without dispersion - Constant deviation spectroscope - Interference - Air wedge - Thickness of a wire - Jamins' Interferometer - Rayleigh's Interferometer - Polarisation - Specific rotatory power and its determination
UNIT V: Electricity
Carey Foster's bridge - Theory - Measurement of resistance - Potentiometer - Low range voltmeter and ammeter calibration - Theory of moving coil Ballistic Galvanometer - Determination of current and voltage sensitivities - Comparison of capacities - BH curve - magnetometer method.
BOOKS FOR STUDY AND REFERENCE
1.
Mechanics and mathematical methods- R Murugesand, S Chand & Co.
2.
Properties of matter and acoustics- R Murugesand, S Chand & Co.
3.
Heat and Thermodynamics- Brijlal and Subramaniam, S Chand & Co.
4.
Optics and Spectroscopy- R Murugesand, S Chand & Co.
5.
Electricity and Magnetism- R Murugesand, S Chand & Co.
Reference Books
1.
"Fundamentals of Computer Science and Communication Engineering"
Alexis Leon, Mathews Leon,
Vikas Publishing House
New Delhi, 1998
(Unit I & III)
3. "COBOL Programming Including MS - COBOL and COBOL - 85"
M.K. Roy, D. Ghosh Dastidar
T.M.H, (Unit - IV & V)
DIFFERENTIAL CALCULUS PAPERCODE-06UMA02
Unit I
derivatives-definition of derivatives-differentiation techniques-standard formulae-differentiation of implicit functions-successive differentiation-the nth derivative-standard results-leibnitz formula for the nth derivative and applications-meaning of the derivatives-simple problems for all the above sections
Unit II
partial derivatives-definition-successive partial derivatives-function of a function rule-total differential coefficient-implicit functions-homogeneous function and eulers theorem-problems
Unit III
definition-curvature,radius of curvature,circle of curvature and center of curvature in cartesian co-ordinates only-the co ordinates of center of curvature and evolutes-envelopes-definition-method of finding the envelope
Unit IV
asymptotes-definition-methods of finding asymptotes of a plane algebraic curves-special cases-problems-tracing of the curves sinx and cosx-polar co-ordinates-angle between the tangent and the radius vector-slope of the tangent in polar co ordinates-angle of intersection of two curves-pedal equation of a curve
Unit V
radius of curvature in polar co ordinates and p-r co-ordinates-equations of a straight line,a circle and a conic in polar co-ordinates-standard equations only-tracing of the curves cardioid,r=a+bsinq,r=asinq,r=acosq,the leminiscate of bernoulli r2=a2cos2q,the equiangular spiral r=aeqcota and the cycloid x=a(q±sinq),y=a(1±cosq)
Text Books
1.
Dr.P.R.Vittal-DIFFERENTIAL CALCULUS
SEMESTER II
PAPER-II:ALLIED PHYSICS II(06UPHA2)
(Modern Physics, Spectroscopy and Electronics)
UNIT-I: Atomic Physics
Vector Atom Model - Spatial quantization - Spinning electron - Quantum numbers associated with vector atom model - Coupling schemes - LS coupling - jj coupling - Pauli's exclusion principle - Periodic classification of elements - example of electron configuration - Stern - Gerlach experiment
UNIT-II: Nuclear Physcs and Solid State Physics
Nuclear models - liquid drop model - semi empirical mass formula - Merits and demerits - Shell model - evidences - Collective model-Nuclear radiation detectors - Ionisation chamber - Geiger Muller counter - Wilson cloud chamber - Bubble chamber
Particle Accelerators - Synchrocyclotron - Betatron Solid State Physics - Bonding in crystals - ionic bond - covalent bond - metallic bond - molecular bond - hydrogen bond
UNIT - III: Spectroscopy
Molecular spectra - Theory of pure rotational spectrum - origin of vibration - rotation spectrum - electronic spectra - Laser Raman spectroscopy - Resonance spectroscopy - Basic theory of NMR and ESR
UNIT - IV: Basic Electronics
Semiconductor Physics - construction and characteristics of FET, SCR, UJT - Phase shift oscillator working with theory - Multivibrator - astable - monostable - bistable - basic circuits - Operational amplifier as differentiator and integrator.
UNIT - V: Digital Electronics
Binary, octal hexadecimal numbers and their inter conversion - Laws of Boolean algebra - De Morgan's theorems - NAND/NOR as Universal building blocks -Karnaugh mapping - pairs - quads - octets - simplification of Boolean expressions - Half and full adder - Half and full subtractor-
Books for study
1.
Modern Physics, R.Murugesan S Chand & Co Twelfth edition
2.
Digital Principles and applications, Malvino & Leach TMH
3.
Principles of Electronics - V K Mehta S Chand & Co
Books for Reference
1.
Modern Physics J.B. Rajam S Chand & Co
2.
Hand Book of Electronics Gupta & Kumar Pragathi Prakashan
PAPER-III:Allied Physics - Practical(06UPHA3)
1.
Young's modulus - non uniform bending - pin and microscope
2.
Young's modulus - uniform bending - pin and microscope
3.
Torsion pendulum - rigidity modulus
4.
Static torsion- rigidity modulus
5.
surface tension and interfacial surface tension -drop weight method
6.
Specific heat capacity of liquid - method of mixtures - half time radiation correction
7.
Lee's disc - coefficient of thermal conductivity of a bad conductor.
8.
Sonometer - frequency of fork.
9.
Air Wedge - thickness of wire.
10.
Newton's rings - radius of curvature.
11.
Spectrometer - i - d curve
12.
Spectrometer - dispersive power of a grating.
13.
Potentiometer - ammeter calibration
14.
Field along the axis of a coil - BH
15.
Voltage regulator using zener diode.
16.
DeMorgan's theorems using Ics
DIFFERENTIAL EQUATIONS PAPERCODE-06UMA04
Unit I
Differential equations-equations of first order and higher degree-equations solvable for p- solvable for x- solvable for y-clairaut's form-exact differential equations and method of obtaining solution to an exact differential equation-problems
Unit II
Second order differential equations with constant coefficients-particular integrals of eax V where V is of the form x,x2,sin ax,cos ax-problems
Unit III
Second order differential equations with variable coefficients-method of variation of parameters-wronskian-simultaneous differential equations with constant coefficients-problems in all the above sections-total differential equation Pdx+Qdy+Rdz=0-condition for integrability-problems
Unit IV
Formation of partial differential equations by eliminating arbitrary constants and arbitrary functions-non linear differential equations of first order-definition -complete,particular,singular and general integrals-solutions of partial differential equations of standard types-clairauts form,equations reduciable to the clairaut's form-simple problems
Unit V
Lagrange's linear partial differential equations Pp+Qq=R-simple problems only-charpit's method-general method of solving non linear differential equations of first order but of any degree-simple problems only
INTEGRAL CALCULUS AND VECTOR CALCULUS - PAPER CODE 06UMA03
Unit I
Definition of integration-List of standard formulae-Different types of integration-Integration by substation-Integration of rational functions-Integration of irrational functions-Integration by partial fractions-Integral of the type Integration by parts-simple problems for all the above sections
Unit II
Definition-properties of definite integral with problems-Reduction formulae-Bernoulli's formula-Reduction formula for simple problems for all the above sections.
Unit III
Double and Triple integral -definition-examples-Change of Variables in double integrals-Changing to order of integration-application of double and triple integrals to area in Cartesian and polar coordinates-find the volume of a solid as a double integral and triple integral-problems
Unit IV
Definition of gradient of a Scalar point function-Directional derivative of a vector point function-Unit normal vector-Divergence and Curl of a vector point function-Definitions-solenoidal and irrotational vector-problems.
Unit V
Line integrals-Surface integrals and Volume integrals-Gauss Divergence Theorem-Stoke's Theorem-Green's Theorem (Statements only)-problems.
Text Books
1.
T.K.Manikkavasam & others: CALCULUS (VOL II) Year of publication 2004.
2.
P.R.Vittal: LCULUS - Year publication 2000
3.
P.R.Vittal and V Malini: ECTOR CALCULUS-Year of Publication 1984
Reference Books
1.
K.Viswanathan & S.Selvaraj : VECTOR CALCULUS - Year of publication 1984.
2.
N.P. Bali : CALCULUS: Year of Publication 1994
SEMESTER III
APPLICATIONS OF INTEGRATION
PAPER CODE:06UMA05
Unit I
Laplace transforms-Definition-Standard formula-Elementary theorems with proof-Problems. (12 hours)
Unit II
Inverse Laplace transforms-Standard formulae-Elementary theorems-Problems-Applications to solve Second order differential equations with constant coefficients.
(12 hours)
Unit III
Fourier Series-Definition-Fourier coefficients of period 2 -Even and odd functions-Half Range Series-Problems.
(12 hours)
Unit IV
Introduction-Fourier Integral representation-Fourier Integral Theorems-Statement only-Sine and Cosine integral-Representations-Transformation of elementary functions-Properties of Fourier Transforms-Linearity Property-Change of scale-Shifting property-Simple problems. (12 hours)
Unit V
Beta and Gamma functions-Definition-Properties-Problems-Relation between Beta and Gamma functions-Applications to evaluation of definite integrals. (12 hours)
Text Books
1.
T.K.Manickavasagam Pilli and S.Narayanan -CALCULUS(Vol.III)
2.
P.R.Vittal-CALCULUS.
Reference Books
1.
K.Sankar Rao-INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS.
SUB: Mathematical Statistics Paper Code: 06USTA01
Unit I
Random variables - Discrete and continuous - Distribution Functions - Marginal and Conditional Distributions - Mathematical Expectation - Moment Generating Function - Characteristic function - Chebechev's inequality.
Unit II
Standard distributions - Binomial, Poisson, Rectangular and Normal Distributions.
Unit III
Exact sampling distributions: Chi-Square, Student't', Fisher's't' and F distributions- Relationship between them.
Curve fitting: Fitting of a straight line - fitting of a second degree parabola - Fitting of power curve - Exponential curve.
Reference books:
1.
S.C. Gupta and V.K. Kapoor, Nineth Revised Edition(2001), Fundamentals of mathematical statistics - Sultan chand, New Delhi.
2.
Gupta C.B and Vijay Gupta, (1998), An introduction to Statistical Methods.
STATICS
PAPER CODE:06UMA06
Unit I
Introduction - Force- Definition-Parallelogram law of forces,Triangular law of Forces and converse of triangular law of forces -Lami's theorem - problems - like and unlike parallel forces -problems -moments - definition -Varignon's theorem -problems.
Unit II
Couples -definitions of a Couple -moment of a couple -Theorems -problems -three forces acting on a rigid body - Coplanar forces -General conditions of equilibrium of coplanar forces acting on rigid body - problems.
Unit III
Introduction - Friction - definition - coefficient of friction -Limiting friction - Angle of friction and Cone of friction - Laws of friction - Equilibrium of a particle on a rough inclined plane under any force - problems.
Unit IV
Definition Centre of gravity of uniform bodies like thin rod,thin parallelogram,circular ring and lamina,triangular lamina and trapezium lamina,systems of three uniform rods forming a triangle - Method of Integration for the arc of a circle,sector of a circle - Quadrant of an Ellipse - solid and hollow sphere - solid hollow cone - problems.
Unit V
Common catenary - Definition - sag and span - intrinsic,parametric and Cartesian Equations of a catenary - properties -Suspension Bridge - Approximation to the shape of a catenary -problems.
Text Books
1.
P.DURAIPANDIAN - MECHANICS
2.
S.NARAYANN - STATICS
Reference Books
1.
Dr.M.K. VENKATRAMAN - STATICS.
SEMESTER IV
SUB: STATISTIFCAL INFERENCE Paper Code: 06USTA02
Unit I
Concept of Population, sample, statistics-parameter, point estimation-concept of point estimation - Consistency, Unbiasedness, efficiency(Cramer Rao Inequality) and Sufficiency (Rao-Blackwel Theorem)
Unit II
Methods of estimation, Maximum Likelihood, Moments and minimum Chi-square methods, Properties of these estimators - Interval Estimation (concept only)
Unit III
Test of Significance - Standard error - Large Sample test with regard to proportion, mean, difference between means and proportions.
Unit IV
Test of significance - Exact tests based on't' and f distributions with regard to mean, variance and correlation coefficient- Tests based on chi-square distribution.
Unit V
Test of hypothesis - Concept of statistical hypothesis - simple and composite hypothesis - Type I error and Type II errors - Critical region - power of a test - NP Lemma - Simple problems.
Text Books
1.
S.C. Gupta and V.K Kapoor(2001) - Fundamentals of Mathematical Statistics - Sultan Chand, New Delhi
2.
Gupta C.B and Vijay Gupta(1998), An introduction to statistical Methods.
STATISTICS -PRACTICAL SUB CODE-06USTAP01
Unit I
Computation of measures of Location, dispersion (absolute and relative)-Measures of skewness and Kurtosis.
Unit II
Fitting of Binomial, Poisson and Normal distribution and testing of goodness of fit.
Unit III
Curve fitting - Fitting of stline, Second degree parabola-Y = aebx, Y abx, Y = aXb.
Asymptotic and exact tests with regard to mean, variance and correlation coefficient - Independence tests - contingency tables.
DYNAMICS
PAPER CODE:06UMA07
Unit I
Introduction-velocity-definition-resultant velocity-parallelogram law -acceleration-Motion of a particle along a straight line under uniform acceleration- problems-Simple harmonic motion-definition-Equations of S.H.M-Properties of S.H.M-problems-geometric interpretation of S.H.M-composition of two simple harmonic motions.
Unit II
Introduction - Impulse and impulsive force - definitions only -Principle of conservation of linear momentum .Newton's experimental law -Change in K.E is equal to work done-Direct and Oblique impact of two smooth spheres -problems-Impact of a smooth sphere on a fixed plane-problems.
Unit III
Introduction -Projectile,Trajectory,Horizontal, range, velocity of projection and angle of projection -definitions. The path of the projectile is a parabola-Range and time of flight on a horizontal plane-problems-Range and time of flight on a inclined plane-problems.
Unit IV
Definitions of Central orbit, Central force and Areal velocity -Diff. eqn of the central orbit -polar co-ordinates - P-R eqn of the central orbit .Given the central orbit to find the law of force-Given the central law of force to find the orbit-Problems.
Unit V
Standard results of Moments of inertia of standard bodies-parallel axes theorem and perpendicular axes theorem-statement only-motion of a rigid body about a fixed horizontal axis -K.E - Moment of Momentum - Eqn of motion -Compound pendulum-interchangeability-Centre of suspension and centre of oscillation-Simple Equivalent Pendulum.
Text Books
1.
P.DURAIPANDIAN - MECHANICS
2.
2.S.NARAYANAN - DYNAMICS
3.
3. "COBOL Programming Including MS - COBOL and COBOL - 85"
M.K. Roy, D. Ghosh Dastidar
T.M.H, (Unit - IV & V)
Reference Books
1.
Dr.M.K. VENKATARAMAN - DYNAMICS
ENVIRONMENTAL STUDIES
Unit I
The Multidisciplinary nature of environmental studies- definition, scope and importance - Need for public awareness
UNIT-II Natural Resources:
Renewable and Non-renewable resources:
a) Natural resources: Use and over - Exploitation, deforestration, case studies. Timber extraction, mining, dams and their effects on forests and tribal people.
b) water resources: Use and over - Utilization of surface and ground water,
floods, drought, conflicts over water, dams - benefits and problems.
c) Mineral resources: Use and exploitation environmental effects of extracting and using mineral resources, case studies.
d) Food resources: World food problems, changes, caused by agriculture and
overgraing, effects of modern agriculture, fertilizer - pesticide problems, water logging , salinity, case studies.
e) Energy resources: Growing energy needs, renewable and renewable energy sources, use of alternate energy sources. Case studies.
f) Land resources: Land as a resource, land degrading, man induced landslides, soil erosion and desertification.
Role of an individual in conservation of natural.
Equitable use of resources for sustainable lifestyles.
Unit 3: Ecosystems:
Concept of Ecosystem.
Structure and function of ecosystem.
Producers, consumers and decomposers.
Energy flow in the ecosystem.
Ecological succession.
Food chains, food webs and ecological pyramids.
Introduction, types, characteristics features, structure and function of the following gecosystem.
a) Forest ecosystem
b) Grassland ecosystem
c) Desert ecosystem
d) Aquatic ecosystem(ponds, steams, lakes, rivers, oceans, estuaries)
Unit 4: Biodiversity and its conservation
introduction - definition: genetic, species and ecosystem diversity.
Biogeogtaphically classification of India.
Value of biodiversity: consumptive use, productive use,, social, ethical, aesthetic ad option values.
Biodiversithy at global, national and local levels.
India as a mega - diversity nation.
Hot-spots of biodiversity
Threats to biodiversity: habitat loss, poaching of wildlife, man wildlife conflicts.
Endangered and endemic species of India.
Conservation of biodiversity: In-situ and ex-situ conservastion of biodiversity.
Unit V
Table Handling: OCURS Clause And Subscripting - Assigning Values To Tale Elements - Multidimensional Tables - PERFORM Verb And Table Handling - Indexed Tables And Indexing - SET Verb - SEARCH Verb - OCCURS DEPENDING Clause - Sorting A Table - Index Data Items. Sequential Files: File Characteristic - File - control Entries For Sequential Files - File Description - Fixed Length Records - Statements For Magnetic - Tape Files - I - O - CONTROL Paragraph - Implementation Differences. Sorting And Merging Of Files : The Simple Sort Verb - File Updation - Variations of Updation - Simple MERGE Verb - INPUT And OUTPUT PROCEDURE In SORT Statement - An Example of SORT Statement with INPUT\OUTPUT PROCEDURES - MERGE Verb With OUTPUT PROCEDURES - SAME SORT AREA Clause - MEMORY SIZE Clause. Direct Access Files Relative Files - Indexed Sequential Files.
Unit I
The Multidisciplinary nature of environmental studies- definition, scope and importance – Need for public awareness
UNIT-II Natural Resources:
Renewable and Non-renewable resources:
a) Natural resources: Use and over – Exploitation, deforestration, case studies. Timber extraction, mining, dams and their effects on forests and tribal people.
b) water resources: Use and over – Utilization of surface and ground water,
floods, drought, conflicts over water, dams – benefits and problems.
c) Mineral resources: Use and exploitation environmental effects of extracting and using mineral resources, case studies.
d) Food resources: World food problems, changes, caused by agriculture and
overgraing, effects of modern agriculture, fertilizer – pesticide problems, water logging , salinity, case studies.
e) Energy resources: Growing energy needs, renewable and renewable energy sources, use of alternate energy sources. Case studies.
f) Land resources: Land as a resource, land degrading, man induced landslides, soil erosion and desertification.
Role of an individual in conservation of natural.
Equitable use of resources for sustainable lifestyles.
Unit 3: Ecosystems:
Concept of Ecosystem.
Structure and function of ecosystem.
Producers, consumers and decomposers.
Energy flow in the ecosystem.
Ecological succession.
Food chains, food webs and ecological pyramids.
Introduction, types, characteristics features, structure and function of the following gecosystem.
a) Forest ecosystem
b) Grassland ecosystem
c) Desert ecosystem
d) Aquatic ecosystem(ponds, steams, lakes, rivers, oceans, estuaries)
Unit 4: Biodiversity and its conservation
introduction - definition: genetic, species and ecosystem diversity.
Biogeogtaphically classification of India
Value of biodiversity: consumptive use, productive use,, social, ethical, aesthetic ad option values.
Biodiversithy at global, national and local levels.
India as a mega - diversity nation
Hot-spots of biodiversity
Threats to biodiversity: habitat loss, poaching of wildlife, man wildlife conflicts.
Endangered and endemic species of India
Conservation of biodiversity: In-situ and ex-situ conservastion of biodiversity.
Unit 5: Environmental Pollution
Definition Causes, effects and control measures of :-
a) Air pollution
b) Water pollution
c) Soil pollution
d) Marine pollution
e) Noise pollution
f) thermal pollution
g) Nuclear hazards
* Solid waste management:
Role of an individual in prevention of pollution.
Solid waste management: Causes, effect and control measures of urban and industrial wastes.
Pollution - case studies.
Disaster management : floods, earthquake, cyclone and landslides.
Unit - 6. Social issues and the Environment
From Unsustainable to sustainable development
Urban problems relate to energy.
Water conservation, rain water hharvesting watershed management
Resettlement and rehabilitation of people; its problems and concerns, Case studies,
Environmental ethics: Issues and possible solutions.
Climate change, global warming, c=[acid rain, ozone layer depletion, nuclear accidens and holocaust. Case studies.
Wasteland reclamation.
Consumerism and waste products.
Environment Production Act.
Air (Prevention of Control of Pollution) Act
Water (Prevention of Control of Pollution) Act
Wildlife Protection Act
Forest Conservation Act
Issues involved in enformcement of environment legislation Public awareness.
Unit 7: Human population and the Environment
Population growth, variation among nations.
Population explosion - Family Welfare Programme
Environment and hukan health
Human Rights
Value Education
HIV/AIDS
Women and Child Welfare
Role of Information Technology in Environment and human health
Case Studies.
Unit 8; Field work
Visit to local area to documents environmental assets - river/forest/grassland/hill/mountain
Study of common plants, insects, birds.
Study of simple ecosystems - ponds, river, hill slopes, ect.(Field work eual to 5 lecture hours)
Subject: FINANCIAL MATHEMATICS PAPER CODE: 06UMA08 Class: II B.Sc(Mathematics) Sem:Even
Unit I
Probability - Probabilities and Events - Conditional Probability - Random Variables and Expected values - Covariance and Correlation - Continuous Random Variables - Normal Random Variables - Properties of Normal Random Variables - The Central Limit Theorem - Simple Problems.
Unit II
Geometric Brownian Motion - G.B.M as a limit of simple models - Brownian Motion - Simple problems - Interest rates - Present value analysis - Rate of return - Continuous of varying interest rates - An example of option pricing - other examples of pricing via arbitrage.
Unit III
The Arbitrage theorem - the multiperiod Binomial model - Proof of the Arbitrage theorem - Black Scholes formula - properties of the Black-Scholes option cost - Derivation of Black Scholes formula - simple problems.
Unit IV
Additional results on options - Call options on Dividend paying Securities - Pricing American put options - Adding Jumps to Geometric Brownian Motion - Estimating the Volatility Parameter - Simple problems.
Unit V
Valuing by Expected Utility - Limitation of Arbitrage pricing - Valuing Investments by Expected utility - The portfolio selection problem - Value at risk and conditional value at risk - The capital assets pricing model - Mean variance analysis of Risk - Neutral priced call options - Rates of Return - Single period and Geometric Brownian Motion - Simple problems.
Text Books
1.
Sheldon M.Ross - AN ELEMENTARY INTRODUCTION TO MATHEMATICAL FINANCE second edition - Cambridge University Press - 2005.
Reference Books
1.
S.M. Ross, A FIRST COURSE IN PROBABILITY - 6th Edition - Englewood Cliffs Prentice Hall - NJ 2002,
SEMESTER V
III B.Sc Mathematics Semester - V Subject Code: BXA
Unit I
Sub groups - A counting principle, Normal sub groups and Quotient groups - Homomorphisms (Section 2.4 to 2.7. Omit Application I and II, Corollary).
Unit II
Automorphisms - Cayley's theorem - Permutation groups (Section 2.8, 2.9 and 2.10).
Unit III
Definition and examples of Rings - some special classes of rings - Homomorphism - More ideals and quotient rings (Section 3.1 to 3.5).
Unit IV
The field of Quotients of an Integral domain - Euclidean rings - A particular Euclidean ring (Ring of Gaussian integers) (Sections 3.6, 3.7 and 3.8).
Unit V
Polynomial Rings, Polynomials over the rational field - Polynomial rings over commutative rings (Section 3.9 to 3.11).
Content & Treatment as in "Topics in Algebra" by I.N.Herstein.( for 4 units)
Reference Books
1.
A Text book of Modern Algebra by Dr.R.Balakrishnan & Dr.N.Ramabadran (Vikas)
2.
Modern Algebra by Santiago M.L, Tata Mc-Graw Hill Publications (2003) and Arul publications (1988).
3.
A First Course in Abstract Algebra by John B. Fraleign.
4.
Modern Algebra by A.R.Vasishta, Krishna Prakasan Mandir.
COMPLEX ANALYSIS I subject code- BXC
Unit I
Functions of a complex variable - Mapping - Limits ,Theorems on limits continuity
Derivative ,Differentiation formulae - C-R equations Cartesian and polar - sufficient condition - Analytic functions, Harmonic functions
Unit II
Linear functions - The functions Z^2,Z^n,Z^1/2,1/Z - The linear transformation and properties - Cross ratio - some general Properties of linear transformation
Unit III
The tranformation w = expZ - W = sinZ, W = cosZ, W = 1/2(Z+1/Z), W = sinhZ,
W = coshZ - Conformal mapping - some basic problems
Cauchy integral formula - Derivative of analyticfunctions - Morearas theorem - Cauchy Inequality - Maximum modulus principle - Lioville's theorem - Fundamental theorem of algebra
Text Books
1.
Complex variables and application by R.V. Churchill and J.W.Brown; McGraw hill
Reference Books
1.
Complex analysis by P.Duraipandiyan.
2.
Theory and problems of complex variable by L.muuray,R.spiegal; schaum outline series.
SUBJECT: NUMERICAL ANALYSIS CODE: BXD
Unit I
Solutions of Algebric and transcendental equations - Bisection method - the iteration method - the method of false position - Newton -Rapshon method - Generalised Newton's Method - muller's method
Unit II
Interpolation: Finite differences - forward differences - backward differences - central differences - symbolic relations - detection of errors by use of differences tables - differences of polynomial - Newton formula for interpolation - Gauss'central difference formula - Stirling's formula - Bessels formula - Evertt's formula
Unit III
Numerical Differentiation: Numerical Differentiation - maxima and minima of tabulated function -Numerical Integration : Trapezoidal rule- Simposon's 1/3 rule, 3/8 rule - Booles's and Weddle's rule
Unit IV
Solutions of linear Systems - Direct methods- Matrix inversion method - Gaussion elimination method - modification of the gauss method to compute the inverse - Solutions of linear systems - Iterative methods.
Unit V
Numerical solution of ordinary differential Equations - Solutions by Taylo's Series - Picard's method of successive approximations - Euler's method - Modified Euler's method - Runge-Kutta Method.
Text Books
1.
Introduction methods of Numerical Analysis by S.S.Sastry (1990)
Subject: OPERATIONS RESEARCH - I PAPER CODE:BXE
Unit I
Definition and scope of OR - phase and limitations of OR - definition and standard of LPP - canonical form of LPP - mathematical formulation and graphical method - simplex algorithm - simple problems.
Unit II
Algorithm of Charnes method of penalties - problems in Charnes method - Algorithm of Two-Phase simplex method and problems - Duality - Dual Simplex Method.
Assignment Problem - Mathematical Formulation - Balanced Assignment problem - Maximization case in A.P - Unbalanced A.P - Maximization case.
Unit V
Sequencing Problem - n jobs to operated on 2 machines - n jobs to be operated on 3 machines - n jobs to be operated on m machines - graphical method.
Reference Books
1.
OPERATIONS RESEARCH by H.A. Taha.
2.
OPERATIONS RESEARCH by P.K. Gupta, Man Mohan and Kanti Swarup.
3.
OPERATIONS RESEARCH by V.K. Kapoor.
4.
OPERATIONS RESEARCH by P.K. Gupta and D.S. Hira.
5.
OPERATIONS RESEARCH by S.D. Sharma.
Paper X REAL ANALYSIS- I Paper Code : BXB
Unit I
Equivalence, countability, Real numbers, Least upper bounds - Definition of sequence and subsequence - Limit of a sequence - convergent sequences - Divergent sequences. ( 1.5 to 1.7 and 2.1 to 2.4 )
Unit II
Bounded sequences - Monotone sequences - Operation on convergent sequences - operations on divergent sequences - Limit superior and limit inferior - Cauchy sequences. ( 2.5 to 2.10)
Unit III
Series of real numbers - Convergence and Divergence - Series with non-negative term - Alternating series - conditional convergence and absolute convergence - Test for absolute convergence - series whose terms form non increasing sequence - summation by parts ( 3.1 to 3.4 & 3.6 to 3.7 )
Unit IV
Limit of a function on the real line - Metric spaces - Limit in metric spaces - Functions continuous at a point on the real line - Reformulation (4.1 to 4.3 and 5.1 to 5.2)
Unit V
Functions continuous on Metric spaces - open sets - closed sets - discontinuous functions on R'.
Text Books
1.
Content and treatment as in 'METHOD OF REAL ANALYSIS" by Richard R. Gold Berg, Oxford & IBH.
SEMESTER VI
III B.Sc Mathematics Semester - VI Subject Code: BXF
Unit I
Vector spaces - Elementary basic concepts - Homomorphism - Subspaces - Quotient spaces - internal direct sum - external direct sum. (Section 4.1)
Unit II
Linear independence - dimension of a vector space - Bases - dimension of Quotient spaces. (Section 4.2)
Linear Transformation - The Algebra of Linear transformation, characteristic roots - matrices - canonical forms - Triangular form.(Sections 6.1 to 6.4)
Unit V
Matrix - Rank of a matrix - Consistency of simultaneous linear equation - solution of simultaneous equation using matrices - Cayley-Hamilton theorem with proof - simple application, Eigen values & Eigen vectors - Diagonalization of matrices
Content & Treatment as in "Topics in Algebra" by I.N.Herstein. (for 4 units)
Reference Books
1.
A Text book of Modern Algebra by Dr.R.Balakrishnan & Dr.N.Ramabadran (Vikas
2.
Modern Algebra by Santiago M.L.
3.
Modern Algebra by A.R.Vasishta, Krishna Prakasan Mandir.
4.
A Text book of Matrices by M.D.Raisingania, S.Chand & Co.Ltd.
5.
A Text book of Matrices by R.S.Agarwal, S.Chand & Co. Ltd.
COPMLEX ANALYSIS II subject code- BXH
Unit I
Sequence
Limit of sequence
Convergence of sequence
Series and its convergence
Taylors series
Laurents series
Absolute convergence
Unit II
Uniform convergence
Power series
Integration of of power series
Differentiation of power series (term by term)
Zeroes of analytic function
Unit III
Residues
Residues at a pole
Cauchy's residue theorem
Unit IV
Evaluation of
1 . Improper real integrals
2. Improper integrals involving sines and cosines
3. Definite integrals involving sines and cosines
4 . Integration through a branch cut
Unit V
Singular points
Different kinds of singular points
Number of zeroes of meromorphic functions
Argument principle
Rouches theorem and its applications
Text Books
1.
Complex variables and application by R.V. Churchill and J.W.Brown; McGraw hill
Reference Books
1.
Complex analysis by P.Duraipandiyan.
2.
Theory and problems of complex variable by L.muuray,R.spiegal; schaum outline series.
SUBJECT : OPERATIONS RESEARCH - II
CODE:
Unit I
Inventory models: Types of Inventory models. Deterministic
i) uniform rate of demand, infinite rate of production and no shortage
ii) Uniform rate of demand, finite rate of replacement and no shortage
iii) Uniform rate of demand, instantaneous production with shortage
iv) Uniform rate of demand, instantaneous production with shortage and fixed time
Unit II
Probabilistic models: News paper boy problem - discrete and continuous type cases - inventory models with one price break.
Unit III
Queuing Theory: General concept and definition - classification of queues - Poisson process - properties of poisson process. Models: M/M/1: /FCFS
M/M/I:n/FCFS
Unit IV
Network Analysis: Drawing network diagram- Critical Path Method- Concept of Slack and Float on network - PERT - Algorithm for PERT -Difference of PERT and CPM
Unit V
Resource Analysis in Network Scheduling - Project cost - crashing cost - Time cost optimization Algorithm - Resource allocation and Scheduling.
Text Books
1.
Operations Research by P.K.Gupta and Manmohan
Paper XV REAL ANALYSIS - II Paper Code : BXG
Unit I
More about open sets - connected sets - Bounded set and totally bounded sets - complete metric spaces ( 6.1 to 6.4)
Unit II
Compact metric spaces - continuous functions on compact metric spaces - continuity of the inverse function - uniform continuity ( 6.5 to 6.8)
Unit III
Sets of measure zero - Definition of the Riemann integral - Existence of the Riemann integral - Properties of the Riemann integral - Derivatives. (7.1 to 7.5)
Unit IV
Rolle's theorem - the law of mean - Fundamental theorem of calculus - Improper integral - improper integral continued. ( 7.6 to 7.10)
Unit V
Pointwise convergence of sequence of functions - uniform convergence of sequence of functions - consequences of uniform convergence - convergence and uniform convergence of series of functions. ( 9.1 to 9.4 )
Content and treatment as in 'METHOD OF REAL ANALYSIS" by Richard R. Gold Berg, Oxford & IBH.
Paper XVII PROGRAMMING IN C Paper Code : BXI
(Practical)
Unit I
Basic structure of C programme - Character set - constants - key words and identifiers - variables - data types - declaration of variables - assigning values to variables - defining symbolic constants.
Unit II
Arithmetic operators - relational operators - logical operators - assignment operators - increment and decrement operators - conditional operators - special operators - arithmetic expressions - evaluation of expressions - type conversions in expressions.
Unit III
Reading and writing character - formatted input and output - decision making with if statement - the if . else statements - nested if .. Else statement - the switch statement - the goto statement - while statement - go statement - for statement jumps in looks.
Unit IV
One, two dimensional arrays - initating 2 dimensional array - multi dimensional arrays - declaring and initializing string variables - reading string terminal - writing strings on the screen - arithemetic operations on characters.
Unit V
User defined functions - their need - multiplication programme - the form of C functions - Return values and their types - calling functions - category of functions - no arguments and no return values.
Paper XVII PROGRAMMING IN C Paper Code : BXI
(Practical)
Program List
1.
Program to find the sum of n numbers
2.
Program to find the largest of 3 numbers
3.
Program to solve a quadratic equations.
4.
Program to find the simple and compound interest.
5.
Program to read an integer and determine whether it is prime or not.
6.
Program to generate Fibonacci series.
7.
Program to find the numbers in ascending and descending order.
8.
Program to find the mean and standard deviation.
9.
Program for matrix addition, subtraction and multiplication.
10.
Program to read an integer N and find the sum of
1 + 1/1! + ½! +.......
Integration by parts-simple problems for all the above sections
simple problems for all the above sections.
