algebra-iv (group theory -ii) - shivaji...

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ALGEBRA-IV (GROUP THEORY -II)

Total marks:100(Theory: 75, Internal Assessment: 25)

5 Periods (4 lectures +1 students’ presentation),

1 Tutorial (per week per student)

(1st, 2nd& 3rd Weeks)

Automorphism, inner automorphism, automorphism groups, automorphism groups of

finite and infinite cyclic groups, applications of factor groups to automorphism groups,

Characteristic subgroups, Commutator subgroup and its properties.

[1]: Chapter 6, Chapter 9 (Theorem 9.4), Exersices1-4 on page168, Exercises 52, 58 on

page Pg 188.

(4th, 5th& 6th Weeks)

Properties of external direct products, the group of units modulo n as an external

direct product, internal direct products, Fundamental Theorem of finite abelian groups.

[1]: Chapter 8, Chapter 9 (Section on internal direct products), Chapter 11.


Group actions, stabilizers and kernels, permutation representation associated with a

given group action, Applications of group actions: Generalized Cayley’s theorem, Index

theorem.


Groups acting on themselves by conjugation, class equation and consequences,

conjugacy in Sn, p-groups, Sylow’s theorems and consequences, Cauchy’s theorem,

Simplicity of An for n ≥ 5, non-simplicity tests.

[2]: Chapter 1 (Section 1.7), Chapter 2 (Section 2.2), Chapter 4 (Section 4.1-4.3, 4.5-4.6).

[1]: Chapter 25.

REFERENCES:

1. Joseph A. Gallian, Contemporary Abstract Algebra (4th Ed.), Narosa Publishing

House, 1999.

2. David S. Dummit and Richard M. Foote, Abstract Algebra (3rd Edition), John Wiley and

Sons (Asia) Pvt. Ltd, Singapore, 2004

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ANALYSIS-IV (METRIC SPACES)

Total marks: 100(Theory: 75, Internal Assessment: 25)



(1st Week)

Metric spaces: definition and examples.

[1] Chapter1, Section 1.2 (1.2.1 to 1.2.6 ).

(2nd Week)

Sequences in metric spaces, Cauchy sequences.

[1] Chapter1, Section 1.3, Section 1.4 (1.4.1 to 1.4.4)

(3rd Week)

Complete Metric Spaces.

[1] Chapter1, Section 1.4 (1.4.5 to 1.4.14 (ii)).

(4th Week)

Open and closed balls, neighbourhood, open set, interior of a set

[1] Chapter2, Section 2.1 (2.1.1 to 2.1.16)

(5th& 6thWeeks)

Limit point of a set, closed set, diameter of a set, Cantor’s Theorem.

[1] Chapter2, Section 2.1 (2.1.17 to 2.1.44)

(7th Week)

Subspaces, dense sets, separable spaces.

[1] Chapter2, Section 2.2, Section 2.3 (2.3.12 to 2.3.16)

(8th Week)

Continuous mappings, sequential criterion and other characterizations of continuity.

[1] Chapter3, Section 3.1

(9th Week)

Uniform continuity

[1] Chapter3, Section3.4 (3.4.1 to 3.4.8)

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(10th Week)

Homeomorphism, Contraction mappings, Banach Fixed point Theorem.

[1] Chapter3, Section 3.5 (3.5.1 to 3.5.7(iv) ), Section 3.7 ( 3.7.1 to 3.7.5)

(11th Week)

Connectedness, connected subsets of R, connectedness and continuous mappings.

[1]Chapter4, Section 4.1 (4.1.1 to 4.1.12)

(12th Week)

Compactness, compactness and boundedness, continuous functions on compact

spaces.

[1] Chapter5, Section 5.1 (5.1.1 to 5.1.6), Section 5.3 (5.3.1 to 5.3.11)

REFERENCES:

[1] SatishShirali&Harikishan L. Vasudeva, Metric Spaces, Springer Verlag London(2006)

(First Indian Reprint 2009)

SUGGESTED READINGS:

[1] S. Kumaresan, Topology of Metric Spaces, Narosa Publishing House, Second Edition

2011.

[2] G. F. Simmons, Introduction to Topology and Modern Analysis, Mcgraw-Hill, Edition

2004.

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DIFFERENTIAL EQUATIONS-II (PDE & SYSTEM OF ODE)

Total marks:150 (Theory: 75, Internal Assessment: 25+ Practical: 50)


Practicals(4 periods per week per student)

Section – 1 (1st , 2nd & 3rd Weeks)

Partial Differential Equations – Basic concepts and Definitions, Mathematical Problems.

First-Order Equations: Classification, Construction and Geometrical Interpretation.

Method of Characteristics for obtaining General Solution of Quasi Linear Equations.

Canonical Forms of First-order Linear Equations. Method of Separation of Variables for

solving first – order partial differential equations.

[1]: Chapter 1:1.2, 1.3

[1]: Chapter 2: 2.1-2.7

Section – 2 ( 3-weeks)

Derivation of heat equation, Wave equation and Laplace equation.Classification of

second order linear equations as hyperbolic, parabolic or elliptic. Reduction of second

order Linear Equations to canonical forms.

[1]: Chapter 3: 3.1, 3.2, 3.5, 3.6

[1]: Chapter 4: 4.1-4.5

Section – 3 (3-weeks)

The Cauchy problem, the Cauchy-Kowaleewskaya theorem, Cauchy problem of an

infinite string. Initial Boundary Value Problems, Semi-Infinite String with a fixed end,

Semi-Infinite String with a Free end, Equations with non-homogeneous boundary

conditions, Non-Homogeneous Wave Equation. Method of separation of variables –

Solving the Vibrating String Problem, Solving the Heat Conduction problem

[1]: Chapter 5: 5.1 – 5.5, 5.7

[1]: Chapter 7: 7.1, 7.2, 7.3, 7.5

Section -4 (3-weeks)

Systems of linear differential equations, types of linear systems, differential operators, an

operator method for linear systems with constant coefficients, Basic Theory of linear

systems in normal form, homogeneous linear systems with constant coefficients: Two

Equations in two unknown functions, The method of successive approximations, the

Euler method, the modified Euler method, The Runge- Kutta method.

[2]: Chapter 7: 7.1, 7.3, 7.4,

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[2]:Chapter 8: 8.3, 8.4-A,B,C,D

REFERENCES:

[1]: TynMyint-U and LkenathDebnath, Linear Partial Differential Equations for Scientists

and Engineers, 4th edition, Springer, Indian reprint, 2006

[2]: S. L. Ross, Differential equations, 3rd Edition, John Wiley and Sons, India,2004

Suggested Reading

[1]: Martha L Abell, James P Braselton, Differential equations with MATHEMATICA, 3rd

Edition, Elsevier Academic Press, 2004.

LIST OF PRACTICALS (MODELLING OF FOLLOWING USING

MATLAB/MATHEMATICA/MAPLE)

1. Solution of Cauchy problem for first order PDE.

2. Plotting the characteristics for the first order PDE.

3. Plot the integral surfaces of a given first order PDE with initial data.

4. Solution of wave equation ��

��− ��

��= 0 for any 2 of the following associated

conditions:

(a) �(�, 0) = ∅(�), ��(�, 0) = �(�), � ∈ ℝ, � > 0.

(b) �(�, 0) = ∅(�), ��(�, 0) = �(�), �(0, �) = 0, � ∈ (0,∞), � > 0.

(c) �(�, 0) = ∅(�), ��(�, 0) = �(�), �(0, �) = 0, � ∈ (0,∞), � > 0.

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(d) �(�, 0) = ∅(�), ��(�, 0) = �(�), �(0, �) = 0, �(", �) = 0, 0 < � < ", � > 0.

5. Solution of one-Dimensional heat equation �� = $ � , for a homogeneous rod of

length l.

That is - solve the IBVP:

�� = $�, 0 < � < ", � > 0,

�(0, �) = 0, �(", �) = 0, � ≥ 0,

�(�, 0) = &(�), 0 ≤ � ≤ "

6. Solving systems of ordinary differential equations.

7. Approximating solution to Initial Value Problems using any of the following approximate

methods:

(a) The Euler Method

(b) The Modified Euler Method.

(c) The Runge-Kutta Method.

Comparison between exact and approximate results for any representative differential

equation.

8. Draw the following sequence of functions on given the interval and

discuss the pointwise convergence:

(i) ( ) for Rnnf x x x= ∈ , (ii) ( ) for Rn

x

nf x x= ∈ ,

(iii) 2

( ) for Rn

x nx

nf x x

+= ∈ , (iv)

sin( ) for R

nx n

nf x x

+= ∈

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(v) ( ) for R, 0n

x

x nf x x x

+= ∈ ≥ , (vi)

2 21( ) for Rn

nx

n xf x x

+= ∈ ,

(Vii) 1

( ) for R, 0n

nx

nxf x x x

+= ∈ ≥ ,

(viii) 1

( ) for R, 0n

nn

x

xf x x x

+= ∈ ≥

9. Discuss the uniform convergence of sequence of functions above.

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PROBABILITY AND STATISTICS




Use of Calculators is allowed

1st Week

Sample space, Probability axioms, Real random variables (discrete and continuous).

2nd Week

Cumulative distribution function, Probability mass/density functions, Mathematical

expectation.

3rd Week

Moments, Moment generating function, Characteristic function.

4th Week

Discrete distributions: uniform, binomial, Poisson, Geometric, Negative Binomial

distributions.

5th Week

Continuous distributions: Uniform, Normal, Exponential, Gamma distributions

[1]Chapter 1 (Section 1.1, .3, 1.5-1.9)

[2]Chapter 5 (Section 5.1-5.5,5.7), Chapter 6 (Sections 6.2-6.3,6.5-6.6)

6th Week

Joint cumulative distribution Function and its properties, Joint probability density functions

– marginal and conditional distributions

7th Week

Expectation of a function of two random variables, Conditional expectations, Independent

random variables, Covariance and correlation coefficient.

8th Week

Bivariate normal distribution, Joint moment generating function.

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9th Week

Linear regression for two variables, The rank correlation coefficient.

[1]Chapter 2 (Section 2.1, 2.3-2.5)

[2]Chapter 4 (Exercise 4.47), Chapter 6 (Sections 6.7), Chapter 14 (Section 14.1,

14.2), Chapter 16 (Section 16.7)

10thWeek

Chebyshev’s inequality, statement and interpretation of (weak) law of large numbers and

strong law of large numbers.

11thWeek

Central Limit Theorem for independent and identically distributed random variables with

finite variance.

12thWeek

Markov Chains, Chapman – Kolmogorov Equations, Classification of states.

[2] Chapter 4 (Section 4.4)

[3] Chapter 2 (Sections 2.7), Chapter 4 (Section 4.1-4.3)

REFERENCES:

1. Robert V. Hogg, Joseph W. Mc Kean and Allen T. Craig. Introduction of Mathematical Statistics, Pearson Education, Asia, 2007

2. Irvin Miller and Marylees Miller, John E. Freund’s Mathematical Statistics with Applications (7thEdn), Pearson Education, Asia, 2006.

3. Sheldon Ross, Introduction to Probability Models (9th Edition), Academic Press, Indian Reprint, 2007

PRACTICALS /LAB WORK TO BE PERFORMED ON A COMPUTER

USING SPSS/EXCEL/Mathematica etc.

1. Calculation of

(i) Arithmetic Mean, geometric Mean, harmonic Mean

(ii) Variance

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2. Fitting of

(i) Binomial Distribution

(ii) Poisson Distribution

(iii) Negative Binomial Distribution

(iv) Normal Distribution

3. Calculation of

(i) Correlation coefficients

(ii) Rank correlation

4. Fitting of polynomials

5. Regression curves

6. Illustrations of Central limit theorem.

7. Draw the following surfaces and find level curves at the given heights:

(i) 2 2; 1, 6, 9( , ) 10 y z z zf x y x − = = == − ,

(ii) 2 2; 1, 6, 9( , ) y z z zf x y x + = = == , (iii) 3 ; 1, 6( , ) y z zf x y x − = == ,

(iv)2

2 ; 1, 5, 84

( , )y

z z zf x y x + = = ==

(v)2 2; 0, 1, 3, 5( , ) 4 y z z z zf x y x + = = = == ,

(vi) ; 6, 4, 2, 0, 2, 4, 6( , ) 2 z z z z z z zf x y x y = − = − = − = = = == − − .

8. Draw the following surfaces and discuss whether limit exits or not as

( , )x y approaches to the given points. Find the limit, if it exists:

(i) ; ( , ) (0,0) and ( , ) (1,3)( , )x y

x y x yx y

f x y+

→ →−

= ,

(ii) 2 2; ( , ) (0,0) and ( , ) (2,1),( , )

x yx y x y

x yf x y

−→ →

+=

(iii) ( ) ; ( , ) (1,1) and ( , ) (1,0)( , ) xyx y e x y x yf x y + → →= ,

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(iv) ; ( , ) (0,0) and ( , ) (1,0)( , ) xye x y x yf x y → →= ,

(v)

2

2 2; ( , ) (0,0)( , )

x yx y

x yf x y

−→

+= ,

(vi) 2

2 2; ( , ) (0,0)( , )

x yx y

x yf x y

+→

+= ,

(vii) 2 2

2 2; ( , ) (0,0) and ( , ) (2,1)( , )

x yx y x y

x yf x y

−→ →

+= ,

(viii) 2

; ( , ) (0,0) and ( , ) (1, 1).( , )x y

x y x yx y

f x y−

→ → −+

=

9. Draw the tangent plane to the following surfaces at the given point:

(i) 2 2 at (3,1, 10)( , ) x yf x y += , (ii) 2 2 at (2,2,2)( , ) 10 x yf x y − −= ,

(iii) 2 2 2 =9 at (3,0,0)x y z+ + ,

(iv) 4

arctan at (1, 3, ) and (2,2, )3

z xππ

= ,

(v) 2log | | at ( 3, 2,0)z x y= + − − .

10. Use an incremental approximation to estimate the following functions at

the given point and compare it with calculated value:

(i) 4 42 at (1.01, 2.03)( , ) 3x yf x y += , (ii) 5 32 at (0.98, 1.03)( , ) x yf x y −= ,

(iii) ( , ) at (1.01, 0.98)xyf x y e= ,

(iv) 2 2

( , ) at (1.01, 0.98)x yf x y e= .

11. Find critical points and identify relative maxima, relative minima or

saddle points to the following surfaces, if it exist:

(i) 2 2yz x += , (ii) 2 2

xz y −= , (iii) 2 21 yz x −= − , (iv) 2 4yz x= .

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12. Draw the following regions D and check whether these regions are of

Type I or Type II:

(i) { }( , ) | 0 2,1 xD x y x y e= ≤ ≤ ≤ ≤ ,

(ii) { }2( , ) | log 2,1D x y y x y e= ≤ ≤ ≤ ≤ ,

(iii) { }( , ) | 0 1, 1D x y x x y= ≤ ≤ ≤ ≤ ,

(iv) The region D is bounded by 2 2y x= − and the line ,y x=

(v) { }3( , ) | 0 1, 1D x y x x y= ≤ ≤ ≤ ≤ ,

(vi) { }3( , ) | 0 ,0 1D x y x y y= ≤ ≤ ≤ ≤ ,

(vii) ( , ) | 0 ,cos sin4

D x y x x y xπ

= ≤ ≤ ≤ ≤

.

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ALGEBRA- V (RINGS AND LINEAR ALGEBRA -II)

Total marks: 100(Theory: 75, Internal Assessment: 25)



(1st,2nd , 3rd& 4th Weeks) Polynomial rings over commutative rings, division algorithm and consequences, principal

ideal domains, factorization of polynomials, reducibility tests, irreducibility tests,

Eisenstein criterion, unique factorization in Z[x].

Divisibility in integral domains, irreducibles, primes, unique factorization domains,

Euclidean domains.

[1]: Chapter 16, Chapter 17, Chapter 18.

(5th, 6th, 7th& 8th Weeks) Dual spaces, dual basis, double dual, transpose of a linear transformationand its matrix in

the dual basis, annihilators, Eigenspaces of a linearoperator, diagonalizability, invariant

subspaces and Cayley-Hamiltontheorem, the minimal polynomial for a linear operator.

[2]: Chapter 2 (Section 2.6), Chapter 5 (Sections 5.1-5.2, 5.4), Chapter 7(Section 7.3).

(9th, 10th, 11th& 12th Weeks) Inner product spaces and norms, Gram-Schmidt orthogonalisation process, orthogonal

complements, Bessel’s inequality, the adjoint of a linear operator, Least Squares

Approximation, minimal solutions to systems of linear equations, Normal and self-adjoint

operators, Orthogonal projections and Spectral theorem.

[2]: Chapter 6 (Sections 6.1-6.4, 6.6).

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REFERENCES: 1. Joseph A. Gallian, Contemporary Abstract Algebra (4th Ed.), Narosa Publishing

House, 1999.

2. Stephen H. Friedberg, Arnold J. Insel, Lawrence E. Spence, Linear Algebra

(4th Edition), Prentice-Hall of India Pvt. Ltd., New Delhi, 2004.

SUGGESTED READING: (Linear Algebra)

1. S Lang, Introduction to Linear Algebra (2nd edition), Springer, 2005

2. Gilbert Strang, Linear Algebra and its Applications, Thomson, 2007

3. S. Kumaresan, Linear Algebra- A Geometric Approach, Prentice Hall of India,

1999.

4. Kenneth Hoffman, Ray Alden Kunze, Linear Algebra 2nd Ed., Prentice-Hall Of

India Pvt. Limited, 1971

(Ring theory and group theory)

1. John B.Fraleigh, A first course in Abstract Algebra, 7th Edition, Pearson

Education India, 2003.

2. Herstein, Topics in Algebra (2nd edition), John Wiley & Sons, 2006

3. Michael Artin, Algebra (2nd edition), Pearson Prentice Hall, 2011

4. Robinson, Derek John Scott., An introduction to abstract algebra, Hindustan

book agency, 2010.

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ANALYSIS-V (COMPLEX ANALYSIS )

Total marks: 150(Theory: 75, Practical: 50, Internal Assessment: 25)


Practical (4 periods per week per student),

(1st& 2nd Week): Limits, Limits involving the point at infinity, continuity. Properties of complex numbers, regions in the complex plane, functions of complex variable, mappings. Derivatives, differentiation formulas, Cauchy-Riemann equations, sufficient conditions for differentiability. [1]: Chapter 1 (Section 11), Chapter 2 (Section 12, 13) Chapter 2 (Sections 15, 16, 17,

18, 19, 20, 21, 22) (3rd, 4th & 5th Week): Analytic functions, examples of analytic functions, exponential function, Logarithmic function, trigonometric function, derivatives of functions, definite integrals of functions. [1]: Chapter 2 (Sections 24, 25), Chapter 3 (Sections 29, 30, 34),Chapter 4 (Section 37,

38) (6th Week): Contours, Contour integrals and its examples, upper bounds for moduli of contour integrals. [1]: Chapter 4 (Section 39, 40, 41, 43) (7th Week): Antiderivatives, proof of antiderivative theorem, Cauchy-Goursat theorem, Cauchy integral formula. [1]: Chapter 4 (Sections 44, 45, 46, 50) (8th Week): An extension of Cauchy integral formula, consequences of Cauchy integral formula, Liouville’s theorem and the fundamental theorem of algebra. [1]: Chapter 4 (Sections 51, 52, 53) (9th Week): Convergence of sequences and series, Taylor series and its examples. [1]: Chapter 5 (Sections 55, 56, 57, 58, 59)

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(10th Week): Laurent series and its examples, absolute and uniform convergence of power series, uniqueness of series representations of power series. [1]: Chapter 5 (Sections 60, 62, 63, 66) (11th Week): Isolated singular points, residues, Cauchy’s residue theorem, residue at infinity. [1]: Chapter 6 (Sections 68, 69, 70, 71) (12th Week): Types of isolated singular points, residues at poles and its examples, definite integrals involving sines and cosines. [1]: Chapter 6 (Sections 72, 73, 74), Chapter 7 (Section 85). REFERENCES:

1. James Ward Brown and Ruel V. Churchill, Complex Variables and Applications

(Eighth Edition), McGraw – Hill International Edition, 2009.

SUGGESTED READING:

1. Joseph Bak and Donald J. Newman, Complex analysis (2nd Edition),

Undergraduate Texts in Mathematics, Springer-Verlag New York, Inc., New

York, 1997.

LAB WORK TO BE PERFORMED ON A COMPUTER

(MODELING OF THE FOLLOWING PROBLEMS USING MATLAB/ MATHEMATICA/

MAPLE ETC.)

1. Declaring a complex number and graphical representation.

e.g. Z1 =3 + 4i, Z2 = 4 – 7i

2. Program to discuss the algebra of complex numbers.

e.g., if Z1 =3 + 4i, Z2 = 4 – 7i, then find Z1 + Z2, Z1 - Z2, Z1 * Z2,and Z1 / Z2

3. To find conjugate, modulus and phase angle of an array of complex numbers.

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e.g., Z = [ 2+ 3i 4-2i 6+11i 2-5i]

4. To compute the integral over a straight line path between the two specified end points.

e. g., ∫C

dz sin Z , where C is the straight line path from -1+ i to 2 - i.

5. To perform contour integration.

e.g., (i) ∫ +−C

2 1)dz2Z(Z , where C is the Contour given by x = y2 +1; 22 ≤≤− y .

(ii) ∫ ++C

23 1)dz2Z(Z , where C is the contour given by 122 =+ yx , which can be

parameterized by x = cos (t), y = sin (t) for π20 ≤≤ y .

6. To plot the complex functions and analyze the graph .

e.g., (i) f(z) = Z

(ii) f(z)=Z3

(iii) f(z) = (Z4-1)1/4

(iv) ( ) ,f z z= ( )f z iz= , 2( ) ,f z z= ( ) zf z e= etc.

7. To perform the Taylor series expansion of a given function f(z) around a given point z.

The number of terms that should be used in the Taylor series expansion is given for each

function. Hence plot the magnitude of the function and magnitude of its Taylors series

expansion.

e.g., (i) f(z) = exp(z) around z = 0, n =40.

(ii) f(z)=exp(z2) around z = 0, n = 160.

8. To determines how many terms should be used in the Taylor series expansion of a

given function f(z) around z = 0 for a specific value of z to get a percentage error of less

than 5 %.

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e.g., For f(z) = exp(z) around z =0, execute and determine the number of

necessary terms to get a percentage error of less than 5 % for the following values

of z:

(i) z = 30 + 30 i

(ii) z = 10 +10 3 i

9. To perform Laurents series expansion of a given function f(z) around a given point z.

e.g., (i) f(z)= (sin z -1)/z4 around z = 0

(ii) f(z) = cot (z)/z4 around z = 0.

10. To compute the poles and corresponding residues of complex functions.

e.g.,3

z +1f(z) =

- 2z + 2z

12. To perform Conformal Mapping and Bilinear Transformations.

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DIFFERENTIAL EQUATIONS-III (TRANSFORMS & BOUNDARY

VALUE PROBLEMS)



Practicals( 4 periods per week per student)

(1st, 2nd& 3rd Weeks)

Introduction: power series solution methods, series solutions near ordinary point, series

solution about regular singular point. Special functions: Bessel’s equation and function,

Legendre’s equation and function.

[1] Chapter 8: 8.1-8.3

[2]Chapter 8: 8.6,8.9


Sturm Theory: Self-Adjoint equation of the second order, Abel’s formula, Sturm

separation and comparison theorems, method of Separation of variables: The Laplace

and beam equations, non-homogeneous equation.

[3] Chapter 11: 11.8

[2] Chapter 7: 7.7, 7.8

(7th , 8th& 9th Weeks)

Boundary Value Problem: Introduction, maximum and minimum Principal, Uniqueness

and continuity theorem, Dirichlet problem for a circle, Neumann Problem for a circle.

[2] Chapter 9: 9.1-9.4, 9.6

(10th,11th& 12th Weeks)

Integral transform-introduction, Fourier transforms, properties of Fourier transforms,

convolution Theorem of Fourier transforms, Laplace transforms, properties of Laplace

transforms, convolution theorem of Laplace transforms

[2] Chapter 12: 12.1-12.4, 12.8-12.10

REFERENCE :

1.C.H. Edwards and D.E. Penny, Differential Equations and Boundary value Problems Computing

and Modelling, Pearson Education India, 2005.

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2.TynMyint-U, LokenathDebnath, Linear Partial Differential Equations for Scientists and

Engineers, 4th edition, Springer, Indian reprint, 2006.

3. S.L. Ross, Differential Equations, 3rd edition, John Wiley and Sons, India, 2004.

Suggested Reading

1. Martha L Abell, James P Braselton, Differential equations with MATHEMATICA, 3rd Edition,

Elsevier Academic Press, 2004.

LIST OF PRACTICALS FOR DE-III

(MODELLING OF FOLLOWING USING

MATLAB/MATHEMATICA/MAPLE)

1. Plotting of [0,1]. Legendre polynomial for n=1 to 5 in the interval Verifying

graphically that all roots of Pn(x) lie in the interval [0,1].

2. Automatic computation of coefficients in the series solution near ordinary points.

3. Plotting of the Bessel’s function of first kind of order 0 to 3.

4. Automating the Frobeniusseries method.

5. Use of Laplace transforms to plot the solutions of

a. Massspring systems with and without external forces and draw

inferences.

b. LCR circuits with applied voltage. Plot the graph of current and charge

w.r.t. time and draw inferences.

6. Find Fourier series of different functions. Plot the graphs for n = 1-6. Draw

inferences for the solutions as n tends to infinity.

7. Finding and plotting Laplace transforms of various functions and solving a

differential equation using Laplace transform.

8. Finding and plotting Fourier transforms of various functions, and solving any

representative partial differential equation using Fourier transform.

9. Finding and plotting the convolution of 2 functions and verify the Convolution

theorem of the Fourier Transform/Laplace Transform.

10. Solve the Laplace equation describing the steady state temperature distribution in

a thin rectangular slab, the problem being written as:

∇�� = 0, 0 < � < *, 0 < + < ,,

�(�, 0) = &(�), 0 ≤ � ≤ *,

�(�, ,) = 0, 0 ≤ � ≤ *,

�(0, +) = 0, �(*, +) = 0,

for prescribed values of a and b, and given function f(x).

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CALCULUS OF VARIATIONS AND LINEAR PROGRAMMING




(1st&2nd Week) Functionals, Some simple variational problems, The variation of a functional, A necessary condition for an extremum, The simplest variational problem, Euler’s equation, A simple variable end point problem. [1]: Chapter 1 (Sections 1, 3, 4 and 6). (3rd&4th Week) Introduction to linear programming problem, Graphical method of solution, Basic feasible solutions, Linear programming and Convexity. [2]: Chapter 2 (Section 2.2), Chapter 3 (Sections 3.1, 3.2 and 3.9). (5th& 6th Week) Introduction to the simplex method, Theory of the simplex method, Optimality and Unboundedness. [2]: Chapter 3 (Sections 3.3 and 3.4). (7th& 8th Week) The simplex tableau and examples, Artificial variables. [2]: Chapter 3 (Sections 3.5 and 3.6). (9th&10th Week) Introduction to duality, Formulation of the dual problem, Primal‐dual relationship, The duality theorem, The complementary slackness theorem. [2]: Chapter 4 (Sections 4.1, 4.2, 4.4 and 4.5). (11th&12th Week) Transportation problem and its mathematical formulation, Northwest‐corner method, Least-cost method and Vogel approximation method for determination of starting basic solution, Algorithm for solving transportation problem, Assignment problem and its mathematical formulation, Hungarian method for solving assignment problem. [3]: Chapter 5 (Sections 5.1, 5.3 and 5.4)

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PRACTICAL/LAB WORK TO BE PERFORMED ON A COMPUTER:

(MODELLING OF THE FOLLOWING PROBLEMS USING EXCEL

SOLVER/LINGO/MATHEMATICA, ETC.)

(i) Formulating and solving linear programming models on a spreadsheet using excel solver.

[2]: Appendix E and Chapter 3 (Examples 3.10.1 and 3.10.2).

[4]: Chapter 3 (Section 3.5 with Exercises 3.5-2 to 3.5-5)

(ii) Finding solution by solving its dual using excel solver and giving an interpretation of the

dual.

[2]: Chapter 4 (Examples 4.3.1 and 4.4.2)

(iii) Using the excel solver table to find allowable range for each objective function coefficient,

and the allowable range for each right-hand side.

[4]: Chapter 6 (Exercises 6.8-1 to 6.8-5).

(iv) Formulating and solving transportation and assignment models on a spreadsheet using

solver.

[4]: Chapter 8 (CASE 8.1: Shipping Wood to Market, CASE 8.3: Project Pickings).

From the Metric space paper, exercises similar to those given below:

1. Calculate d(x,y) for the following metrics

(i) X=R, d(x,y)=Ix-yI, (ii) X=R3, d(x,y)= (∑(xi-yi)

2)1/2

x: 0, 1, π, e x: (0,1,-1), (1,2,π), (2,-3,5)

y: 1, 2, ½, √2 y: (1, 2, .5), ( e,2,4), (-2,-3,5)

(iii) X=C[0,1], d(f,g)= sup If(x)-g(x)I

f(x): x2 , sin x, tan x

g(x): x , IxI, cos x

2. Draw open balls of the above metrics with centre and radius of your choice.

3. Find the fixed points for the following functions

f(x)=x2 , g(x)= sin x, h(x)= cos x in X=[-1, 1],

f(x,y)= (sin x, cos y), g(x,y) = (x2 , y2 ) in X= { (x,y): x2+y2≤1},

of 91

under the Euclidean metrics on R and R2 respectively.

4. Determine the compactness and connectedness by drawing sets in R2.

REFERENCES:

[1]. I. M. Gelfand and S. V. Fomin, Calculus of Variations, Dover Publications, Inc., New

York, 2000.

[2]. Paul R. Thie and Gerard E. Keough, An Introduction to Linear Programming and

Game Theory, Third Edition, John Wiley & Sons, Inc., Hoboken, New Jersey, 2008.

[3]. Hamdy A. Taha, Operations Research: An Introduction, Ninth Edition, Prentice Hall,

2011.

[4]. Frederick S. Hillier and Gerald J. Lieberman, Introduction to Operations Research,

Ninth Edition, McGraw-Hill, Inc., New York, 2010.

SUGGESTED READING:

[1].R. Weinstock, Calculus of Variations, Dover Publications, Inc. New York, 1974.

[2].M. L. Krasnov, G. I. Makarenko and A. I. Kiselev, Problems and Exercises in the

Calculus of Variations, Mir Publishers, Moscow, 1975.

[3].Mokhtar S. Bazaraa, John J. Jarvis and Hanif D, Sherali, Linear Programming and

Network Flows, Fourth Edition, John Wiley & Sons, Inc., Hoboken, New Jersey,

2010.

[4].G. Hadley, Linear Programming, Narosa Publishing House, New Delhi, 2002.

algebra-iv (group theory -ii) - shivaji...

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