Mathematics for bsc Computer science, Thesis of Mathematics for Computing

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VIJAYANAGARA

SRI KRISHNADEVARAYA UNIVERSITY,

BELLARY

Scheme And Syllabus

For B.A/B.Sc MATHEMATICS COURSE

(Semester System)

2016 - 17 Batch only

VIJAYANAGARA SRI KRISHNADEVARAYA UNIVERSITY

BELLARY

The syllabus in Mathematics for Six Semesters B.A/B.Sc. Degree Course

1. Following table shows the number of teaching hours, problem solving hours, examination pattern and marks.

Semesters I,II,III,&IV

Semesters V & VI Number of papers in each Semester 2 3 Teaching Hours per paper per Week Teaching^ 4 Hours^ 4 Hours Examination pattern in each paper in each semester

Duration of Examination 3 Hours^ 3 Hours

i)Examination marks a. Maximumb.Minimum for pass^50207028

ii) Internal Assessment marks a. Maximumb.Minimum for pass^25 --^30 --

iii) Total Marks a. Maximumb.Minimum for pass^753010040

2. Internal assessment marks in each paper shall be awarded by the concerned course, teacher based on the two class tests each of one-hour duration conducted during semester.
3. The internal assessment marks awarded shall be carried forward for the repeated examination.
4. The maximum strength of each section for teaching hours is restricted to sixty students.

Paper I: Algebra I

I. Mathematical Logic. Recapitulation of Mathematical Reasoning, Open sentences, compound open sentences, Quantifier, universal Quantifier, Existential quantifier and negation of a quantifier statement. Rule of inference and proofs, Methods of proof. 12 Hrs II. Theory of Equations. Relation between the roots and coefficients of general polynomial equation in one variable, Transformations of equations. Descartes rule of signs. Solution of cubic equation by Cordon’s methods. Biquadratic equation. 15 Hrs III. Matrices. Recapitulation of matrix algebra (Basic concepts), rank of matrix, elementary operations, equivalent matrices, invariance of rank under elementary operations, inverse of a non- singular matrix by elementary operations. System of m-linear equations in n unknowns, matrices associated with linear equation, criterion for existence of non-trivial solution of homogeneous and non-homogeneous system, criterion for uniqueness of solutions. Eigen values and Eigen vectors of square matrix- Cayley-Hamilton theorem - Applications. 25 Hrs

Note: Internal mark: 25

References:

1. F.J. Noronha et al: Introduction to mathematical logic (Bangalore University Publication).
2. Rudraiah et al: College Mathematics Vol-I (Sapna Book House, Bangalore)
3. Schaum's outline of theory and problems of matrices by Frank Ayres (Schaum's Outline Series).
4. TEXTBOOK OF MATRIX ALGEBRA BY SUDDHENDU BISWAS ( PHI Learning Pvt. Ltd. Copyright. )
5. Uspenskey: Theory of equations.
6. C.C. Macduff’s: Theory of equations (John Wiley).

Paper II: CALCULUS-I I. Successive differentiation: nth^ derivatives of the functions: (ax+b)n, log (ax+b), eax, sin(ax+b), cos(ax+b), eaxsin(ax+b), eax^ cos(bx+c), Leibnetz’s theorem and its applications 07Hrs II. Functions of two and three variables: Partial derivatives, Euler’s theorem for homogeneous functions (two variables) and its applications. Total differential, Total derivative and partial derivative of composite functions, Jacobians properties and functional relations. Jacobians of implicit functions. 10Hrs III. Polar co-ordinates: Polar co-ordinates, angle between radius vector and a tangent. Angle of intersections of curves, perpendicular distance drawn from the pole to the tangent and pedal equations (polar and Cartesian) 15Hrs IV. Theory of Plane Curves: Points of inflection, concavity and convexity of curves, derivative of an arc in polar, Cartesian and parametric forms. Radius of curvature of a plane curve in Cartesian, parametric and polar forms, Centre of curvature and Evolutes, Envelops. 20Hrs

NOTE: INTERNAL MARK: 25 References:

1. D.C.Pavate: Modern College Calculus.(Macmillan and Company Limited)
2. Shanti Narayan. : Differential Calculus (S Chand & Company Limited).
3. L.Ben: Calculus Vol-I and II (IBM).
4. Murray.R. Spiegel: Advance Calculus (Schaum's Outline Series).
5. G.K. Ranganath: A text book of Mathematics. (S Chand & Company Limited)
6. Rudraiah et al: College Mathematics Vol-I (Sapna Book House, Bangalore).

Paper IV: Advanced Calculus

I. Differentiability : Rolle’s Theorem, Lagrange’s and Cauchy’s mean value theorem. Taylor theorem with Lagrange’s form of the remainder. Taylors and Maclaurins series problems on transcendental functions. Indeterminate forms, L’Hospital’s rules. 15 Hrs II. Integral Calculus: Reduction formulae for the function: sinnx, cosnx, tannx, cotnx, secnx, coscnx, sinmxcosmx Application of integration to find area bounded by the curve, surface area, length of an arc & volumes of solids of revolution for standard curves in Cartesian and polar forms. 12 Hrs III. Line and multiple integrals: Definition of line integrals, basic properties. Examples on evaluation of the integrals. Definition of double integral: evaluation of double integrals (i) under given limits (ii) in regions bounded by given curve. Change of variables. Surface areas as double integrals, definition of a triple integral and evaluation of volume as triple integrals. 15 Hrs IV. Gamma and Beta functions : Gamma and Beta functions, connection between two functions, application to evaluation of integrals. 10 Hrs Note: Internal Marks: 25

References:

1. D.C.Pavate: Modern College Calculus. (Macmillan and Company Limited).
2. Shanti Narayan: Integral calculus (S Chand & Company Limited).
3. Murry R.Spigel: Advanced calculus (Schaum's Outline Series).
4. Rudraiah et al: College Mathematics Vol-I (Sapna Book House, Bangalore).
5. Sokoilnikoff I S: Advanced Calculus (McGraw Hill).

Paper V: Algebra III

I. Rings, Integral domains, fields : Rings, types of rings, Properties of rings, rings of integer modulo n-sub rings, Ideals-Principal and maximal ideals in a commutative ring-examples and standard properties. Homomorphism and Isomorphism, properties of homomorphism. Quotient rings, Integral domains- fields- properties following the definition- field is an integral domain- finite integral domain is a field. 27 Hrs II. Linear Algebra : Vector spaces, examples including Rn^ and Cn. Properties of vector spaces: subspaces. Criteria for a subset to be a subspace. Linear combination concepts of linearly independent and dependent subsets. Basis and dimension of a vector space and standard results related to a basis. Examples illustrating concept and result (with emphasis on R3). Linear transformations: Properties of linear transformations, matrix of a linear transformation, change of basis, range and kernel of a linear transformation, rank-nullity theorem. 25 Hrs

Note: Internal Marks-

References:

1. Hertein.I.N: Topic in Algebra (Wiley Student Edition)
2. Fraleigh J.B: A first course in abstract Algebra (PEARSON Education)
3. Lipschiz S: Linear Algebra (Schaum's Outline Series)
4. Shepherd G.C: Vector spaces of finite Dimension (Oliver and Boyd)
5. N. Jacobson: Basic Algebra Vol I & II, (Dover publications.)

Paper VII: Real and Complex Analysis

IV. Real Analysis: Reimann Integration: Recapitulation of real number system, postulates and their consequences, inequalities and absolute values, lower and upper bounds. The upper and lower sums, necessary and sufficient conditions for integrability. Algebra of integrable functions. Integrability of continous and monotonic functions. Fundamental theorem of calculus, change of variables. Integration by parts. The first and second mean value theorems of integral calculus. 17 Hrs II Complex Analysis: Recapitulation of complex numbers and complex plane, conjugate and modulus of a complex number. The polar form, geometrical representation, Euler’s formula eiө=CiSө. Function of complex variable: Limits, continuity and differentiability. Analytic functions, Cauchy-Reimann equations in Cartesian and polar forms. Sufficient conditions for analyticity (in Cartesian form). Real and imaginary parts of analytic functions which are harmonic. Construction of analytic function given real and imaginary parts. Some standard transformation: Conformal transformation, special conformal transformation. The complex line integral: examples and properties (definitions of the concepts like Neighborhood of a point, closed contour etc. at appropriate places should be mentioned.) Cauchy integral theorem (statement) and its consequences. The Cauchy’s integral formulae for the function and its derivatives, applications to the evaluation of simple line integrals. 35Hrs

Note: Internal Marks-25.

References:

1. S C Malik: Mathematical Analysis (New Age International Pvt Ltd).
2. Sharma and Vasistha: Real Analysis (Krishna Prakashan Mandir, Meerut).
3. Churchill R V : Introduction to complex variables and applications (Mcgraw Hill)
4. Murry R Spiegel : Complex Variables (Schaum’s Outline series)
5. Choudhary B: The elements of complex analysis (New Age International Pvt Ltd).
6. L.V. Alhfors : Complex Analysis: An Introduction to The Theory of Analytic Functions of One Complex Variable (Mcgraw Hill)

Paper VIII: Special Functions and PDE-I I. Special Functions: Legendre’s differential equation, Legendre polynomials Pn(x) as a solution, Rodrigue’s formula, generating polynomials theorem, orthogonal property and basic relation. Recurrence relations. Bessel differential equation, Bessel function Jn(x) as a solution – generation formulae, integral formula for Jn(x), orthogonal property, recurrence relations, basic relation problems there on. Laguerre’s differential equations, Laguerre polynomials Ln(x) as a solution, generating function, orthogonal property, recurrence relations, basic relation problems there on. Hermite’s differential equations, Hermite polynomials Hn(x) as a solution, generating function, orthogonal property, recurrence relations, basic relation problems there on. 32 Hrs II Partial Differential Equations (PDE-I): Formation of Partial Differential Equations, Lagrange’s linear equations Pp+Qq=R, Standard types of first order linear Partial Differential Equations and equations reducible to standard form, Charpit’s method. Standard type of Non-linear PDE of first kind. 20 Hrs

Note: Internal Marks- References:

1. Ayres F : Differential Equations (Schaum’s Outline Series)
2. Stophenson.G: An introduction to Partial Differential Equations(ELBS)
3. B.S Grewal: Higher Engineering Mathematics (Khanna Publishers).
4. M.D Raisinghania: Advanced Differential equations (S.Chand & co)
5. Ian N. Sneddan: Elements of Partial Differential Equations, McGraw Hill.

Paper X: Applied Mathematics

I. Vector Analysis : Scalar field, Quotient of a scalar field, geometrical meaning, Directional Derivatives, Vector field, Divergence and curl of a vector field. Solenoidal and irrigational fields. Expression for , div f and curl f (Cartesian co-ordinates), vector identities, Greens theorem in the plane with proof and its applications, Gauss divergence theorem (Statement only), Stoke’s theorem (Statement only), examples based on them. 22Hrs II Calculus of Variation: Introduction, Functionals, Euler’s equations, solutions of Euler’s equation. Geodesics. Isoperimetric problems, several dependent variables, functionals involving higher order derivatives. 15Hrs III Partial Differential Equation (PDE-II): Solution of second order linear partial differential equations in two variables with constant Coefficients by finding complimentary function and particular integral, canonical forms for parabolic, elliptic and hyperbolic equations, solution by separation of variables. Solutions of one- dimensional heat and wave equations and two dimensional Laplace equation by the method of separation of variables. 15Hrs Note: Internal Marks- References:

1. Murry. R. Spiegel: Vector analysis (Schaum’s Outline series).
2. Spain.B: Vector analysis.( D. Van Nostrand Company, London, 1967)
3. Stophenson.G: An Introduction to Partial Differential Equations (ELBS).
4. Ian N. Sneddan: Elements of Partial Differential Equations, McGraw Hill.
5. M.D Raisinghania: Advanced Differential Equations (S.Chand & Co).
6. B.S Grewal: Higher Engineering Mathematics (Khanna Publishers).
7. K.Shankar Rao: Introduction to Partial Differential Equations (PHI Pvt Ltd, New Delhi)

Paper XII: Trigonometry and Complex analysis I. Trigonometry: Expression of sine and cosines using De-Moiver’s theorem. Series of sines and cosines. Hyperbolic functions. Logarithm of complex number (simple example), summation of trigonometric series (simple problems) 12 hrs II. Topological Spaces : Definition of a topology and examples: Types of topologies: Discrete, indiscrete and co-finite topology (or finite complement topology). Open and closed sets. Simple examples, elementary concepts closure and closure properties, neighborhoods, limit points and derived sets, interior, exterior and boundary of a set. Bases and sub bases: Definition, base for a topology, properties of base for a topology. Characterization of a topological space in terms of base. III. Relative topology: Definition, Elementary properties and examples. Separation axioms, T 1 -Spaces and T 2 - Spaces (Definition and simple properties) 30 Hrs IV. Fuzzy Sets: The vocabulary of Fuzzy logic- Boolean sets-operators-Fuzzy sets-Fuzzy Quantifiers-Fuzzy set operators-operations on Fuzzy sets-illustrations-Applications. 10 Hrs Note: Internal Marks- References:

1. Churchill R V : Introduction to complex variables and applications (Mcgraw Hill)
2. Murry R Spiegel : Complex Variables (Schaum’s Outline series)
3. Choudhary B: The elements of complex analysis (New Age International Pvt Ltd).
4. L.V. Alhfors : Complex Analysis: An Introduction to The Theory of Analytic Functions of One Complex Variable (Mcgraw Hill)
5. E. Sampath Kumar and K.S. Amur: Introduction to Modern Algebra and Topology.
6. General Topology by Seymour Lipschutz (Schum’s Outline series).
7. Klir and Yaan: Introduction to fuzzy set theory (PHI Publication).

OPTIONAL PAPERS FOR FIFTH SEMESTER Paper XI (A) Title: GRAPH THEORY- I Introduction: Graphs, finite and null graphs. Connectedness and component, degree of vertex, minimum and maximum degree, ∑deg vi= 2q. The number of vertices of odd degree is even. Isomorphism, complete graph, line graph, total graph. Sub graphs, spanning and induced sub graphs, walk, trial, path, cycle, the shortest path problems, bipartite graph characterization of bipartite graph in terms of its cycles. 22Hrs Eulerian and Hamiltonian graphs : Introduction the Kenigsberg bridge (New name as kalingrad) problem and travelling salesman problem, Characterization of Eulerian graphs and properties of Hamiltonian graphs some applications graphs in electronic network., Cut vertex, bridge, block, tree, spanning tree, rooted and binary trees, forest. Some properties of trees. 15Hrs Connectivity : Vertex and edge connectivity. Some external problems, Mengers theorems (statement), Properties of n-Connected graphs with respect to vertices and edges, Matrix representation: Incidence, adjacency, power of adjacency matrix, edge sequence in adjacency matrix, circuit matrix, some applications 15Hrs

Note: Internal marks: 30

References:

1. Robin J Wilson: Introduction to Graph theory Longman (London), UK.
2. Narsing Deo : Graph theory and applications (PHI), India.
3. Frank. Harrary : Graph Theory, Narosa Publications, India.
4. V.K.Balakrishnan: Graph Theory, (Schum’s Outline Series).

Paper XI (B): DISCRETE MATHEMATICS-I

Sets and propositions-Cardinality. Mathematical induction. Principle of Inclusion and exclusion. Computability and formal languages- Ordered sets. Languages, Phrase structure grammers. Types of grammers and languages.

Permutation, Combinations and discrete probability. Relations and Functions: Binary relations. Equivalence relations and partitions. Partial order relations and lattices. Chains and anti-chains. Functions and the pigeonhole principle.

Graphs and Planar Graphs: Basic terminology, Multi-Graphs. Weighted graphs. Paths and circuits. Hamiltonian paths and circuits. Travelling salesman problem. Planar Graphs. Trees: Trees, Routed trees, Binary search trees. Spanning trees and cut sets. Transport Networks Finite state Machines: Equivalent machines. Finite state machines as Language Recognizers. Recurrence relations: First order relations, second order linear homogeneous relations, Third and higher order linear homogeneous relations, linear non-Homogeneous relations of second and higher order. 52Hrs

Note: Internal marks: 30 References:

1. Liu C.L: Elements of discrete Mathematics (McGraw Hill).
2. Trambley J.P. and Manohar P: Discrete Mathematical Structures with Application to computer Science (TMH).
3. Narsingh Deo: Graph Theory with Application to Engineering and Computer Science (PHI).
4. Kolamn B. and Busy R.C: Discrete Mathematical Structures for Computer Science (PHI).

Paper XI (D): MECHANICS-I Dynamics of a Particle and System of Particles : Conservation Principle. Mechanics of particle Conservation of linear momentum, angular momentum and Energy. Mechanics and system of particles- Conservation of linear momentum, angular momentum and Energy. Tangential and normal components of velocity and acceleration. Constrained motion of a particle under gravity along, inside and outside of a circle and a cycloid. Radial and transverse compounds of velocity and acceleration. Motion of a particle in a central force field, determination of orbit from central forces and vice-versa, Kepler’s Laws of Planetary Motion. 27 Hrs

Dynamics of Rigid Bodies : Centre of mass of a rigid body, static equilibrium of rigid body, rotation of rigid body about a fixed axes. Moment of Inertia. Laminar motion of a rigid body, body rolling down an inclined plane. Angular momentum of a rigid body. Product of inertia, moment of inertia of a rigid body, about an arbitary axes, momental ellipsoid. D’Alembert’s Principle, General equation of motion of a rigid body, motion of centre of inertia, motion relavative to centre of inertia. 25 Hrs

Note: Internal marks: 30

References:

1. S.L.Gupta, V.Kumar and H.V.Sharma: Classical Mechanics, Pragati Prakashan, Meerut.
2. F.Chorlton: Textbook of Dynamics, CBS Publishers, New Delhi.
3. Murray R Spiegal: Theoretical Mechanics, Schaum Series.
4. S.L.Loney: An elementary treatise on the dynamics of a particle and of rigid bodies, Cambridge University Press, 1958.
5. Grant R.Fowles: Analytical Mechanics, Holt, Rinehart and Winston Inc.

Paper XI (E): MATHEMATICAL MODELLING-I

Introduction The technique of mathematical modeling, Characteristics of mathematical models, Limitations of mathematical modeling.

Mathematical Modelling through Ordinary Differential Equations : Linear Growth and decay models: Single Sprecies population models, Population growth, effecta of immigration and emigration on populations size, spread of scientific and technological innovation, radioactive decay, diffusion, diffusion of medicine in the blood stream.

Higher Order Linear Models : A model for the detection of diabetes, modeling in dynamics, vibration of a mass on a spring free and undamped, damped forced motion, electric circuit problem

Modelling of Epidemics : A simple epidemic model, a susceptible-infected-susceptible (SIS) model, simple epidence model with carriers and removal model for arm race, combat model, traffic model.

52Hrs Note: Internal marks: 30

References:

1. Differential Equation Models, Eds. Martin Braun, C.S.Colman, D.A.Drew, Springer Verlag, 1982.
2. Discrete & System models, W.R.Lucas, F.S.Roberts, R.M.Thrall, Springer Verlag, 1982.
3. Life Science Models, H.M.Roberts and M.Thompson, Springer Verlag, 1982.
4. Models in Applied Mathematics, Springer Verlag, 1982.
5. Mathematical Modelling, J.N.Kapur, Wiley Eastern, 1988.