Unit - 1.
Group, order of a group, finite and infinite groups, sub-groups, Right and Left Cosets of sub-group, Lagrange's theorem and its decuctions order of an element index of a sub-group in group. Normal Sub-group and quotient groups. Homomophisms, Isomorphisms, and.
Unit - 2.
Automorphisms; Kernal of a homomorphism. Fundamental theorem of homomorphism Cyclic group. Permutation group- Cycle and transposition even and old permutation.
Unit - 3.
Definition and example of rings. Properties of rings .Sub-rings. Zero divisors Indegral Domain. Characteristics or ring Homomorphism of rings, Quotient ring.
Unit - 4.
Ideals Principal ideal and principal ideal ring, Maximal and prime ideal and corresponding Quotient rings. Quotient field of an intergral domain. Polynomials, degree of- polymomials; F(X)
Unit - 5.
As an intergral domain, Division algouthem for polymomials, greatest Uommon divisor of two polynomials reducible and irredutible polynomials, unique theorem for polynomials unique factorisation theorem for polynomials roots of and degree polyno- mials equation.
Note :- All units Carry Equal Marks.
Books Recommended :
Name of Book Author Publisher.
1.
Algebraic
Structures
: Dr. A.R. Singal,
REstogi & Co. 1985.
2. Elements of
Modern Algebra : B.S. Vatssa
I.N. Herstein
3. Topics in
Algebra
: Fathar Wols.
Willey-Eastern 1983.
4. Modern Algebra
(2nd Ed.) : Surjitsingh
Vikas Publihouse
Quzi, Zameruddin Ltd. (2nd Ed. 1975)
5. University
Algebra
: N.S. Gopal Krishna
1. Number Systems :
The real
field to be developed by ordered set approach. Equivalence of this approach
and Dedikind's approach. Extended real number
system. The complex number system Euclidean spaces.
2. Basic Topology :
Finite,
countable and uncountable sets Metric spaces, Neighbour hoods in metric
spaces Limit points of a set, Open, closed bounded,
Compact, connected and convex sub-sets of metric spaces.
3. Sequences and Series :
Convergent
Sequences, Sub-sequences Cauchy Sequences, upper and lower limits Special
Sequences and Series Series of non-nagative
terms The number Root and ratio tests.
4. Power Series with real
(Complex) terms interval (circle) of convergence and radius of a power series
Summation by parts absolute
convergence. Addition and multiplication of series Rearrangements.
5. Limits and continuity :
Limits
and continuity for functions from a metric space into another metric space
continuity of a composite function, structureal properties of
continuous function from a metric space into Rk. continuity and compactness.
Continuity and connectedness Discontinuities
Monotonity function. infinite limits and limits at infinity. Differentiation :
Derivatives of a real function continuity and
differentiability structural properties of the class of differentiable
function, Mean value theorems. Continuity of derivatives L'Hospital's
Rules Derivatives of higher order Taylor's theorems Differentiation of vector
values function on (a,b). The course is roughly covered
chapters 1,2,3,4,5 of the book entitled "Principles of Mathematical
Analysis" by walter Rudin McGraw Hill
(International
student Edition) 3rd Edition.
Note : All units
carry equal marks.
1. Riemann -
Stieltje's integral :
Riemann integral and
stieltje's integral Properties of Reimann and stieltje's integrals
Intergration and differentitation integration of Vector values
functions Rectifiable curves.
2. Sequences and series of function :
Sequences of function.
Limit of a sequence of function Uniform convergence. Tests for uniform
convergence, uniform convergence and contindity .
Uniform convergence and differentiation.
3. Some Special function :
Power series, the
expontial function Logarithmic function Trigonometric function.
4. Functions of several variables :
Linear transformation.
Linear oprators on a finite dimensional vector space. Invertible linear
operators, Differentiation partial derivatives
Derivatives of Higher order- Diffrentiation of integral.
5. Complex valued function on Sub-sets
of a complex plane :
Limit diffrentiatiability
and analyticity of complex function Cauchy-Riemann equations Harmonic
function. Conjugate harmonic function Replace
analyticity of elementary functions.
(1)
The course is roughly covered by chapters 7 (omit 7.28 to 7.33) Chapters 8
(omit 8.8 to 8.22 Chapters 9 (omit 9.9 and 9.24 to 9.38) of the book Entitled.
(2)
"Principles of Mathematical Analysis" by walter Rudin, McGraw-Hill
(International student Edition)- 3rd Edition and Chapter 2,8 of the book entitled
(3) Complex variables and applications by R.V. Churchil.
Book Recommended :
Name of the Book Author Publisher.
1. Principles of
Mathematical Analysis : T.M. Apostol
2. Pure
Mathematics
: G.H.
Hardy Cambridge Uni.
Press
3. Advance
calculus
: H.K.Nicherson D.C. Spener &
East-
West (1968) N.E.
4. Principles of real
analysis
: S.C. Malik
Viley Easterin Limited
New Delhi 1982.
5. Methods of Real
analysis
: R.R. Goldberg Oxford
publication
Bombay (1979)
Note : All units carry equal marks.
Unit - 1. Method of
plane Statics :
Equilibrium
of a particle, Equilibrium of a system of particles. Work and potential
energy.
Unit - 2. Application of plane statics
:
Mass
centers and centers of gravity friction, flexible cables.
Unit - 3 Plane Kinematics :
Kinematics
of particles. Motion of rigid body parallel to a fixed plane.
Unit - 4 Motion of Rigid Body :
Moment
of Inertia, Kinetic energy and angular moment Rotation of a rigid body about a
fixed axis, general motion of a rigid body paralled to a fixed plane stability of equilibrium.
Unit - 5. Plane Impulsive Motion :
General
Theory of plane impulsive motion collisions.
Note : All units carry equal marks.
The course is roughly covered by :Principles of Mechanics" by Synge and
Griffith McGraw-Hill (3rd Edition) 1959.
Unit - 1. Linear
Programming :
General
LP problem. Basic feasible solutions Simplex method Degeneracy revised Simplex
method Duality Sensitivity analysis and parametric LP. Integer Programming.
Unit - 2. Transportation Problem :
Triantular basis.
Finding a basic feasible solution Testing for optimality changing the basis
Degeneracy unbalanced problem, transportation with transshipment Caterer problem. Assignment
problem.
Unit - 3. Flow and pontential in
Networks :
Graphs
Minimum path problem Spenning tree of minimum lenght. Problem of minimum
potentia differences Scheduling of sequential activities Maximum flow problem and duality in it.
Unit - 4. Nonlinear Convex Programming
:
Legragian
Function saddle point Kuhntucker Theory Quadratic Programming.
Dyamic Programming : Minimum
path problem Singal additive constaint with additively separable return,
single additive constraint with multiplicative separable.
Unit - 5. Theory of Games :
Matrix
(retangular games) minimal theorem saddle point strategies and pay - off
theorem of matrix games, Graphical solution notion of dominance. Rectangular game as an LP Problem.
Note : All units carry equal Marks.
The syllabus is roughly covered by "Optimization Methods in O.R. and
system Analysis" by K.V.Mital (Wiley Eastern) Chapter 3,4,5,6,7 and 9.
(a) Lattices and Boolean Algebra :
Unit - 1. Lattices :
Partially
ordered sets (POSETS), Hasse diagram, Lattices as posets Lattices as algebraic
systems The equivalence of two definitions of a lattice Direct product of two lattices Order
isomorphism of two posets. Isomorphic lattices Completelattices
Distributive lattices complemented lattices. Boolean algebra. Examples of
Boolean algebra. Boolean algebra of logic circuits and switches. Direct product of two Boolean algebra.
Homomorphism.
Unit - 2. Atoms of a
Boolean Algebra :
A
(x) (the set of all the atoms of a Boolean algebra less than or equal to x)
and its properties Isomorphism of a finite Boolean algebra and (P(S),C). Order of a finite Boolean algebra as 2".
Boolean function / expressions. Minterms, Maxterms, representation of a Boolean expression as a sum of products canonical form and
as product of sum canonocal form Karnaugh map. Minimization of a Boolean expression by Cube array method and by Karnaugh
method.
Unit - 3. Graph Theory :
Graphs,
Basic Definition Undirected Directed, Mixed Weighted graphs incidence
and degree Isomorphism. subgraphs , Walks Paths Circuits Connected disconnected graph Strong weak and unilateral
components Euler graphs operation on graphs Hamiltonion paths Trees Binary and M-ary tress To. count trees spanning trees.
Unit - 4.
Cut
sets Connectivity and speparability 1- isomorphism, 2- isomorphism planar
graphs and their different representations.
Detection of planarity Geometric and combinatorial duals vector space
associate with graph circuit and cut-set subspces Orthogonal
and spaces.
Unit - 5.
Incidence matrix adjacency matrix of a path matrix and their relationship. Chromatic number, chromatic partitioning, coverings. Acylic digraphs and Decylization.
Reference Book :
1. "Discrete
Mathematical Structure with Application to computer Science" by Trembley
I.P and Manohar R.
2. "Discrete Mathematics structure
with Application to computer science"
by Trenbley R.S. Hamming and E.A.
Feigebaum.
3. "Discrete Mathematics structure
with application to computer science" by B. Kolman and R.C. Bubsy.
4. "Graph Theory with a application
to Engg. And computer science" by Narshigh Deo
5. "Graph Theory" by Harary F.
6. "Graph theory and its
Application" by Harris.
7. "Boolean Algebra abd its
Application" by Whitestitt J.E.
8. "Boolean Algebra-Abstract and
Concrere" by A.P. Bowran.
Unit - 1. Metric
Spaces :
Definition
of a metric space and example, continuous function open and closed spheres and
their properties, neighbourhood of a point and its properties open sets and its properties limit
of a sequence, definition of closed sets and its properties.
Unit - 2 Metric Space
(cont.) and Topological Spaces :
Product
spaces, subspaces, equivalence of metric spaces, definition of a topological
space equivalence of metric spaces, definition of a topological space metrizable space and
neighbourhood of a point.
Unit - 3. Topological
Space : (Cont.)
Hausdorff
space, definition of closure interior boundary of a set and their
properties functions continuity and Homeorplisms, subspace topology and product topology
Unit - 4. Connectedness
:.
Definitions
of connected and disconected spaces connectedness on the real line and
applications of connectedness, components and local connectedness Locally - connected at a
point.
Unit - 5. Compactness
:
Definition
of a compact space subsets of a reline product of compact spaces compact
metric spaces Bolzano Weierstrass property.
Note : The course is roughly covered by.
Chapter : 2,3 : (Omit : Nbhd. space
Definition. 3.4 page 91 to end of thm 3.8 page 94 Omit : Closure -space-Defi.. (4.8 page 98 to end of 4.13 page 99).
Chapter :
4 : (Omit thm. 4.6 page
128 to end of thm 4.8 page 133 Omit Artical-6,7,8 page 138 up to page 161)
Chapter :
5 : (Omit Defn.
6.1 page 187 to end of thm 6.5 page 189 omit article-7 193 up to page 210)
of the book entitled "Introduction to Topology" by B. Mendelson CBS
publication 1983.
Reference Book :
1.
"Introduction to
Topology"
: B.M. J. Mansfield (CBCS Pub.)
2. "Introduction to Topology
and : By G.F. Simmons
(McGraw-Hill) Modern
Analysis"
3. "Topology - A First
course"
: by J.R. Munkres (Prentice-Hal of India).