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.
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
ALLIED PAPER-I
NUMERICAL METHODS.
Paper Code: 06UMCA 01
Max Marks: 100
Unit I
Solution of Algebraic and Transcendental Equations – The Bisection Method – Method of Successive Approximations - The Method of False Position – Newton – Raphson Method – Generalized Newton Method - Problems.
Unit II
Finite Differences – Definition – First Difference – Higher Differences – Difference Tables – Expression of any value of y in terms of the initial value Y0 and the differences – Backward differences – Expression of any value of Y in terms of the initial value Yn and the backward differences of Yn - Central Differences – Properties of the Operator D – Problems.
Unit III
Difference of a Polynomial – The Operator E – Relation between E and D – Relation between D and D – Other Difference Operators - Interpolation – Gregory-Newton’s forward interpolation formula – Gregory-Newton’s backward difference formula – Problems.
Unit IV
Central Differences and Interpolation formulae – Central Different Table - Gauss Forward Interpolation formula – Bessel’s Formula – Everett’s Formula - Problems.
Unit V
Interpolation with Unequal Intervals – Divided Differences – Properties – Newton’s Interpolation formula for unequal intervals – Lagrange’s interpolation formula - Hermite’s interpolation formula – Problems.
Text Books
1.
S.S. SASTRY – INTRODUCTORY METHODS OF NUMERICAL ANALYSIS
2.
Dr. M.K. VENKATARAMAN – NUMERICAL METHODS IN SCIENCE & ENGINEERING.
Reference Books
1.
V. RAJARAMAN – COMPUTER ORIENTED NUMERICAL METHODS.
2.
E. BALAGURUSAMY – NUMERICAL METHODS.
SEMESTER II
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- and -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 forand
and ,simple problems for all the above sections. and
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.Duraipandiyan VECTOR CALCULUS- Year of publication 1984.
4.
P.R.Vittal and V Malini: ECTOR CALCULUS-Year of Publication 1984
ALLIED II - NUMERICAL CALCULUS
PAPER CODE 06UMCA02
Unit I
Numerical differentiation – Newton’s forward difference to compute the derivatives - Newton’s backward difference formula to compute the derivatives – Derivative’s using Stirling’s formula – Bessel’s formula – Newton’s divided difference formula- problems .(12 hour)
Unit II
Numerical Integration – Trapezoidal rule – Simpson’s rule – Simpson’s one third rule – Simpson’s three eight rule – Romberg’s Integration – Newton-Cote’s formula – Gaussian Integers – problems .(12 hour )
Unit III
Difference Equation – definition – linear difference Equation – linear homogeneous equation with constant coefficients – cases when the auxiliary equation has distinct roots, equal roots – imaginary roots –non homogeneous linear difference equation with constant coefficients – useful formula to evaluate the particular in simple cases problem.(12 hour )
Unit IV
Solutions of the linear systems – Direct methods – Matrix inverse method – Gaussian elimination method – Jacobi’s method of iteration – Gauss-siedal method iteration – problems ( 12 hour )
Unit V
Numerical solution of ordinary differential equation- Solution by Taylor’s series –Picard’s method successive approximations – Euler’s method – Runge – Kutta second order method – problems .
Text Books
1.
S.S . SASTRY – INTRODUCTORY METHODS OF NUMRICAL ANALYSIS
2.
Dr. M.K. VENKATARAMAN – NUMERICAL METHODS IN SCIENCE AND ENGINEERING.
3.
3. “COBOL Programming Including MS – COBOL and COBOL – 85”
M.K. Roy, D. Ghosh Dastidar
T.M.H, (Unit – IV & V)
PRACTICAL IN NUMERICAL METHODS AND ANALYSIS
PAPER CODE 06UMCAP01
1. Solutions of algebraic and transcendental Equations
The Bisection method –Method of successive approximation –The method of
False position –Newton-Raphson method – Generalized Newton’s method –
Muller’s method.
2. Interpolation
Newton-Gregory formula for forward and backward differences – Bessel’s
formula and stirling’s formula - Newton’s divided difference formula and
Lagrange’s formula.
3. Numerical Differentiation
Newton Gregory formula for forward and backward differences - Bessel’s
formula and stirling’s formula .
4. Numerical Integration
Trapezoidal rule - Simpson’s one third rule and three eight rule - Bool’e and
Weddle’ rule – Maclaurin’s formula – Gaussian Integration.
5. Numerical Solution
Gaussian elimination method – Jacobi’s method of iteration – Gauss – Seidal
method of iteration solution by Tailor’s series – Picard’s method successive
approximations - Euler’s method – Modified Euler’s method - Runge – Kutta
second order method.
Text 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)
OFFICE AUTOMATION
Paper code : 06UMC04
Unit I
Word 2000 features- text formatting options – Sorting lists, paragraphs and tables – find and replace, create and edit styles – footnotes and endnotes, book marks and cross references.
Unit II
Creating tables – formatting – mail merge features – adding colors and graphics – tool bars – document protection – additional features of word XP.
Unit III
Excel 2000 features – importing and exporting data – working with templates links – report managers – formatting, sorting and filtering data – naming ranges – working tool bars, pivotal tables and pivotal charts, sharing worksheet ;and protection – additional features of Excel XP.
Unit IV
Introduction to MS Power point 2000 – preparing shades and presentation – adding animations – Inserting sounds and movies – additional features of power point XP.
Introduction to MS- Access – creating new database – creating a table – editing a table – entering and editing data into a table – creating a simple repot.
Unit V
Basics of networking – types – LAN, WAN, MAN – Introduction to wed using dial –up network – concept of internet – E-mail – basics, development tools and browsers – surfing the web- communication channels.
Text Books
1.
R. Mansfiebl – working in Microsoft office
2.
C. Futon – Microsoft office 2000 – PHI 2000.
3.
3. “COBOL Programming Including MS – COBOL and COBOL – 85”
M.K. Roy, D. Ghosh Dastidar
T.M.H, (Unit – IV & V)
List of Practicals
1.
Preparation of word document
2.
Preparation of a table using Excel.
3.
Preparation of a slide in Power point.
4.
Creating and editing a table.
5.
Preparation of letters using mail merge.
6.
Creation of simple reports using Ms – Access.
7.
Demonstration of Find, Replace, cut, Pasting texts in a word document.
8.
Creation of version charts ( pie, tire, etc. ).
9.
Creation of animation pictures.
10.
Table formatting.
SEMESTER III
MATHEMATICAL STATISTICS -SYLLABUS
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 distributions, Student ‘t’ distribution – Fisher’s ‘t’ distribution, F distribution – Relationship between them.
Curve Fitting: Fitting of a straight line – Fitting of a second degree parabola – fitting of a power curve – exponential curve.
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.
APPLICATIONS OF INTEGRATION
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.
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.
Reference Books
1.
INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS - K.Sankar Rao
DIFFERENTIAL EQUATION
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
Text Books
1.
Dr.P.R.Vittal-DIFFERENTIAL EQUATIONS & LAPLACE TRANSFORMS Year of Publication 2002 Margham Publications, 24, Rameswaram Road, T.Nagar, Chennai-600017.
2.
S.Sarayanan & others-CALCULUS(vol-III), Year of Publication 2004. Vijay Nicole Imprints Pvt Ltd, # C-&, Nelson Chambers, 115, Nelson Manickam Road, Chennai- 600029.
3.
S.Sankarappan & S.Kalavathi-DIFFERENTIAL EQUATIONS & LAPLACE TRANSFORMATIONS, Year of Publication 2004. Vijay Nicole Imprints Pvt Ltd, # C-&, Nelson Chambers, 115, Nelson Manickam Road, Chennai- 600029.
Reference Books
1.
INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS - K.Sankar Rao
2.
A.Singaravelu-DIFFERENTIAL EQUATIONS & LAPLACE TRANSFORMS, Year of Publication 2002, Meenakshi Publishers, 120, Pushpa Nagar, Medavakkam, Chennai- 600002.
SEMESTER IV
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 III
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.
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
Environmental Pollution
Definition Causes, effects and control measures of :-
Air pollution
Water pollution
Soil pollution
Marine pollution
Noise pollution
thermal pollution
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 VI
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 VII
Unit 5: 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 VIII
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)
FINANCIAL MATHEMATICS
Unit I
Probability – Probabilities and events – conditional probability – Random variables and expected values – Convergence 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 – rRate of return – continuation 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
Valuingby 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 analsis of Risk – Neutral priced Call options – r\Rates of Return – single period and Geometric Brownian Motion – Simple Problems
Text Book
Sheldon M Ross, AN ELEMENTARY INTRODUCTION TO MATHEMATICAL FINANCE ii Edition, Cambridge University press – 2005.
Reference Books:
S.M. Ross, A FIRST COURSE IN PROBABILITY – 6th Edition, Englewood cliffs Prentice Hall
J. Cox and M. Rubinstein – OPTIONS MARKETS – Englewood cliffs Prencice Hall
J. E. Ingersill – THEORY OF FINANCIAL DECISION MAKING, Lanjarn, MD Rowerman of Little fields
Unit VIII
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)
STATISTICS –PRACTICAL
Unit I
Probability – Probabilities and events – conditional probability – Random variables and expected values – Convergence 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 – rRate of return – continuation 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
Valuingby 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 analsis of Risk – Neutral priced Call options – r\Rates of Return – single period and Geometric Brownian Motion – Simple Problems
Text Book
Sheldon M Ross, AN ELEMENTARY INTRODUCTION TO MATHEMATICAL FINANCE ii Edition, Cambridge University press – 2005.
Reference Books:
S.M. Ross, A FIRST COURSE IN PROBABILITY – 6th Edition, Englewood cliffs Prentice Hall
J. Cox and M. Rubinstein – OPTIONS MARKETS – Englewood cliffs Prencice Hall
J. E. Ingersill – THEORY OF FINANCIAL DECISION MAKING, Lanjarn, MD Rowerman of Little fields
Unit VIII
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)
PROGRAMMING IN C WITH PRACTICAL
Unit I
Introduction- Basic structure of C – programs – character set – keywords and identifiers – constants – variables – data types – declaration of variables – Assigning values to variables – Defining symbolic constants, operators and expressions. (12 hours).
Unit II
Reading and writing a character – Formatted input & output – IF- IF ELSE – ELSE IF ladder – Switch statement - ? Operator – GO TO statement – WHILE – DO – FOR statement. (12 hours)
Unit III
Array – introducing one dimensional & two dimensional arrays – initializing two dimensional arrays- Handing of character string . (12 hours)
Unit IV
User defined functions – Form of C functions - Return values & their types – Calling a function – Three categories functions – Structures and unions – introduction – Structure definition – giving values to members – structure initialization – unions.
(12 hours)
Unit V
Pointers – introduction – understanding pointers accessing the address of a variables – declaring & initialization pointers.
File management – Introduction – defining, opening and closing a file – I/O operation files. (12 hours)
Text Books
1.
E.Balagurusamy – PROGRAMMING IN ANSI C , Tata McGraw hill publications company .Ltd.Ed 2.1. -1998.
STATISTIFCAL INFERENCE
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
SEMESTER V
ALGEBRAIC STRUCTURE
Unit I
Groups – Axioms of a group – Definition – Examples – Addition modulo n – Multiplication modulo n – Klien’s four group – Symmetric group – matrix group – Simple properties – Lemma – Problems – Cyclic groups – Definition and Examples – Order of an element – properties – problems – Permutation groups – examples – Alternating groups (12hrs)
Homomorphism – Definition – Examples – Properties – Kernel of a Homomorphism – properties – Isomorphism – Definition – properties – Fundamental theorem on Homomorphism – Theorems on cyclic groups – Problems – Isomorphism theorems – Automorphism – Definition – Properties – Cayley’s theorem on permutation group. (12 hrs)
Unit IV
Rings – Definition – Examples – Ring of real Quaternion – Properties – Special classes of Rings – Zero divisor – Integral domain – definition – properties – Unit – Division ring – Field – Definition – Examples – Properties – Ideals – Definition – Examples – Properties – Quotient Rings – Principal Ideal – Maximal Ideal, Prime Ideal – Definition – Properties – The characteristic of an integral domain – Definition – Properties. (12 hrs)
Unit V
Definition – Properties – Greatest common divisor – Definition – Properties – Associates – Definition and properties – Irreducible element – Definition – Properties – relatively prime elements – Definition – Properties – Ideal generated by a prime element is a maximal ideal – The Unique factorization theorem – Euclidean ring – Gaussian integers. (12 hrs)
Text Books
1.
M.L.Santiago – Modern Algebra – Year of publication 1994- Tata Mc-graw hill,New Delhi
Reference Books
1.
A.R.Vasistha- A First course in Modern Algebra – Year of publication 1983,Krishna Prakasan Mandir,9,Sivaji Road,Meerut(UP)
2.
I.N.Herstein – Topics in Algebra- 2nd edition,Year of Publication 1975,John Wiley, New York
3.
K.Viswanatha Naik – Modern Algebra- Year of publication 1988,Emerald publication,135,Anna Salai,Chennai -600 002
4.
Dr. R. Balakrishnan and Dr. N. Ramabadran – A text book of Modern Algebra- Year of publication 1994. Vikas Publishing house, New Delhi
DISCRETE MATHEMATICS
Unit I
Connectives: Negation, conjunction, disjunction, WFF, Tautologies, equivalence & Duality – Normal forms: DNF, CNF, PDNF, PCNF, - Theory of inference calculus validity using truth table – predicate calculus: Predicates, statement function, variables & Quantifiers – inference theory of predicate calculus: valid formulae & Equivalences. (12 hrs)
Unit II
Relations & ordering: Relations, properties of binary relation in a set – Functions: Definition & Introduction, composition of functions, inverse function, binary and composition of functions, inverse function, binary and n-array operations, hashing functions – Natural numbers: Peano axioms & mathematical induction, cardinality. (12 hrs)
Unit III
Algebraic systems: Definition & examples, semigroups and monoids – definition and examples – homomorphism of semigroups * monoids, sub semigroups & sub monoids – polish expression and their compilation – polish rotation, conversion of infix to polish – group codes: the communication model and basic notations of error correction – generation of codes by using parity checks – error recovery in group codes. (12 hrs)
Unit IV
Latices as partially ordered sets: Definition & examples – some properties of lattices – sub lattices – direct product and homomorphism – Boolean algebra : Definition & examples – sub algebra – direct product and homomorphism – Boolean forms & free Boolean algebra – values of Boolean expression & Boolean function. (12 hrs)
Unit V
Grammar & language: Discussion of grammar – formal definition of language – finite state machines: introductory sequential circuit, equivalence of finite state machines – finite state acceptors & regular grammars. (12 hrs)
Reference Books
1.
J.P. Trembley , R. Manohar – Discrete Mathematical Structures with Application to Computer Science –Year of Publication 2001-Tata Mcgraw hill- New Delhi.
2.
Prof. V.Sunderesan,K.S.Ganapathy Subramaniam, K.Ganesan – Discrete mathematics – Year of Publication 2000- Tata Mcgraw hill –New Delhi.
3.
L.Lovarz, J.Pelikan, K.Vexztergombi – Discrete Mathematics – Year of Publication 2002-Springer International edition.
LINEAR PROGRAMMING
Unit I
Introduction – Definition of O.R – Scope, phases & limitation of O.R – Linear programming problem – Mathematical formulation – Characteristic of a LPP – Matrix form of LPP – Graphical method – Definition of bounded, unbounded & optimal solutions – procedure of solving LPP by graphical method – problems – Simplex technique – Definition of basic, non-basic variables – basic solutions – slack variables and optimal solution, simplex procedure of solving LPP – Problems. (12 hrs)
Unit II
Introduction – Big-M method – definition of Big-M method, surplus variables & artificial variables – procedure of solving an LPP by Big-M method – Pseudo optimal solution – Problems – Two phase simplex method – Procedure of solving an LPP by two-phase method – problems. (12 hrs)
Unit III
Introduction – Balanced & unbalanced T.P, feasible solution – basic feasible solution – Optimum solution – degeneracy in a T.P – Mathematical formulations – North-West corner rule – Vogell’s approximation method (unit penalty method) Method of matrix minima (Least cost method) – problems algorithm of optimality test (Modi method) – Problems. (12 hrs)
Unit IV
Introduction – Definition of Assignment problem, balanced & unbalanced assignment problem – restriction on assignment problem – Mathematical formulation – formulation & solution of an assignment problem (Hungarian method) – degeneracy in an assignment problem. (12 hrs)
Unit V
Introduction – definition – Basic assumption – n jobs to be operated on two machines - n jobs to be operated on three machines – n-jobs to be operated on m-machines – problems – two jobs to be operated on m machines (graphical method) – problems. (12 hrs)
Reference Books
1.
P.K.Gupta, Man Mohan and Kanti Swarup – Operations Research,ninth edition, Year of Publication 2001, Sultan Chand and sons, New Delhi
REAL ANALYSIS
Unit I
Field of Real numbers – Axioms – Bounded sets – lub & glb – Completeness axioms – Archimedean property – Definition of Rational numbers – Monotone sequences – Theorems on nested intervals – Dedikind’s cut property – square roots – absolute value. (12 hrs)
Unit II
Bounded Sequences – Null sequence – Convergent sequence – Subsequence – Bolzano-Weirestrass Theorem – Cauchy’s criterion for convergence – Limit superior and limit inferior of a bounded sequence.(12 hrs)
Unit III
Infinite Series – Convergence – Divergence – Oscillatory series – Examples – partial sums – series of positive terms – Examples – Absolute Convergence – Examples – Tests for Convergence – Comparison test – Ratio test – Cauchy’s root test – Condensation test – only statements of the above tests – problems. (12 hrs)
Unit IV
Intervals – Closed sets – open sets – Neighborhoods – finite and infinite sets – Heine-Borel theorem – Limit of a function at a point – Deleted neighborhoods – Limits and continuity – Characterization of limits – Algebra of limits. (12 hrs)
Unit V
Continuity of a function at a point – Algebra of continuity - examples – one sided continuity – composition – continuous function on an interval – Intermediate value theorem – Continuous function on closed interval – Monotonic continuous functions – Inverse function theorems – Uniform continuity. (12 hrs)
Reference Books
1.
Sreling K.Barberian – A first course in Real Analysis – Year of publication 2004-Springer (India)Private limited, New Delhi
VISUAL BASIC
Unit I
Introduction – Data Access – developing for the internet , new control, VB’S control set building controls in VB,IDE and VB – Development environment, Event driving programming, working with objects and controls – Tool box, VB modules, Event driven code , designing a form.(12 hours)
Unit II
Designing user interface- visual elements of VB- menus, toolbars an tab strips Activex an other controls – status bars on animation and times events, Aligning controls, setting focus and tab order: Right mouse button support working with printer , common dialog, Derivers, fiolers and files. Adding graphic and multimedia.(12 hours)
Unit III
Connecting a data base : Buliding a data base project – ODBC-DAO,ADO,OLEDB,DB controls building reports, data environment designer . (12 hours)
Unit IV
Building internet application: Internet basics with VB, HTML basics, IIS and Active server pages, WEB class designer. (12 hours)
Unit V
IIS object model – building DHTML applications – DHTML page designer, Building the interface. (12 hours)
Reference Books
1.
Corel - VISUAL BASIC 6.0 – The complete reference Tata Mc Graw Hill Publication company, New Delhi -2002
SEMESTER VI
COMPLEX ANALYSIS
Unit I
Functions of a complex variable – limit of a function at a point – therems on limits – continuity – derivatives – Cauchy-Riemann equations – necessary and sufficient conditions – analytic function – examples – harmonic function – properties – to find an analytic part is given.
Unit II
Bilinear transformations – definition – properties – invariance of cross ratio – fixed points – problems – special bilinear transformations - problems
Unit III
Simply – connected domain – Cauchy’s fundamental theorem – proof using Goursat’s fundamental theorem for multiply connected domains – Cauchy’s integral formula & Cauchy’s formula for the first derivative – Morera’s theorem.
Unit IV
Cauchy’s inequality – Liouville’s theorem – Fundamental theorem of Algebra –Maximum modules theorem – Taylor’s series - problems
Unit V
Singularities – types of singularities – isolated singularly – removable singularity – pole – essential singularity – determination of the nature of singularity - residue – definition – calculation of residues – Cauchy’s Residue theorem – Contour integration around a unit circle ( Statement only ) – integration of functions with poles on the real axis.
Reference Books
1.
For Units I: Ruel V Churchill & James ward Brown – COMPLEX VARIABLES AND APPLICATIONS (IV – edition) - Year of Publication 1986.Mcgraw Hill international Book Company, New York.
2.
Unit II , III , IV & V – P. Duraipandian & Laxmi Duraipandian , D.Muhilan – COMPLEX ANALYSIS – Year of Publication 1988.Emerald Publishers,135, Anna Salai, Chennai – 600002.
GRAPH THEORY
Unit I
Introduction – Definition and examples – Degrees – Definition – Theorem 1 and corollary – Theorem 2 and problems – subgraphs – Definitions – Theorem 1 – Operations on Graphs – Definition – Theorem-1.
K.R.Parthasarathy – BASICS OF GRAPH THREORY – Year of Publication 2001 –TMH Publishing company Ltd, NewDelhi.
2.
S. Kumaravelu & Suseela Kumaravelu – GRAPH THEORY – Year of Publication 1996 – SKV Printers.
3.
A.Chandran – A FIRST COURSE IN GRAPH THEORY – Year of Publication 1997 – Macmillan Publishers, Chennai.
JAVA AND INTERNET WITH PRACTICAL
Unit I
Java history – java features – How Java differs from C and C ++, compiling and running a simple Java program. Constants – variables – Data types – declaration of variables – giving values to variables – type casting getting values of variables – Arithmetic operators – Relational operators – Logical operators – Assignment operators – increment and decrement operators – Conditional operators – expressions.
Unit II
If Statements – conditional operators – WHILE statement – DO statement – FOR statement – nesting of FOR loops – examples – Defining a class – Creating objects – constructors – method overloading – Static members – simple programs.
Packages – Creating packages – Accessing and using packages – simple programs. Exception handling i.e, managing errors and exceptions – try, catch, multiple catch, finally statements – simple programs.
Unit V
Introduction to Internet – types of browsers – Browsing through the WEB – creating E-mail id – Connecting to the dial- up net work – Inter connecting LAN and Internet using proxy server – ORL –FTP -Basic concepts.
Introduction – markup languages – HTTP – An introduction – introducing to HTML –Basic tags, Images, Links – Text Formatting tags, ordered, Unordered and definition list.
Reference Books
1.
Neil Ranelall – THE INTERNET – Second edition, Prentice Hall India, New Delhi 1996.
2.
Patrick Naughton and Herbert Schedelt - THE COMPLETE REFERENCE – JAVA2, Tata McGraw Hill Publishing Co. Ltd., New Delhi, Third edition 2000.
LINEAR ALGEBRA
Unit I
Vector Space – definition and simple properties – example – subspaces – quotient spaces definition – sums & direct sums – definition – Linear dependence and Linear independence of vectors – definition – problems – linear span L(S).
Unit II
Basis and Dimension – definition – properties – theorems – Homomorphism – definition – Isomorphism – theorems – kernel of a homomorphism – simple theorems – Rank of a homomorphism – theorems – problems.
The algebra of Linear transformations – definition – theorems – minimal polynomial – Invertible and Singular Transformations – examples – Rank of a linear transformations – theorems – problems – Eigen values and Eigen vectors – definition – Trace and Transpose – definition – properties – theorems.
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