[B. Tech.] BMAS 0101: ENGNEERNG MATHEMATCS Course Objectives: To make the students understand the concepts of differential calculus, differential equations and matrices by giving more emphasis to their applications in engineering. Prerequisites: Credits: 04 Semester L T P: 3 1 0 Differential Calculus: Partial differentiation, Euler s theorem for homogeneous functions, Composite functions, Total derivatives, Expansion of functions of several variables, Asymptotes and Curve Tracing (in Cartesian coordinates), Jacobian and its properties, Extrema of functions of several variables using Lagrange s multipliers. Matrices: Elementary transformations, Rank by echelon form, Consistency and Solution of system of linear equations, Complex matrices, Eigen values and Eigenvectors, Cayley Hamilton theorem & its applications. Ordinary Differential Equations : Solution of Exact and Reducible to exact differential equations. Ordinary Differential Equations : Solution of n th order linear differential equations with constant coefficients, Euler-Cauchy Equations, Simultaneous differential equations, Solution of order differential equations by method of reduction of order, Reduction to normal form, Change of independent variable and Method of variation of parameters, Applications of second order differential equations in electrical circuits and mechanical systems. After studying these topics, the student will be able to Understand partial differentiation and its applications Trace the curves in Cartesian coordinates Find rank of a matrix & its applications in solving systems of linear equations Solve the ordinary differential equations and know their applications in engineering M. Goyal and N. P. Bali, A Text Book of Engineering Mathematics, Laxmi Publication, Delhi, 20. M. K. Jain, S. R. K. yengar and R. K. Jain, Advanced Engineering Mathematics, Narosa Publishing House, New Delhi, 2002. B. S. Grewal, Higher Engineering Mathematics, Khanna Publishers, New Delhi, 20. T. M. Apostol, Calculus, Volume, John Wiley & Sons, nc., USA, 1967. T. M. Apostol, Calculus, Volume, Xerox Corporation, USA, 1969. G. B. Thomas and R. Finney, Calculus & Analytic geometry, Addison Wesley, USA, 1995. W. E. Boyce and R. D. Prima, Elementary Differential Equations, John Wiley & Sons, 2009. 1
[B. Tech.] BMAS 0102: ENGNEERNG MATHEMATCS Course Objectives: To make the students understand the concepts of integral calculus, convergence of infinite series, vectors and partial differential equations by giving more emphasis to their applications in engineering. Credits: 04 L T P: 3 1 0 Semester Convergence of nfinite Series: ntroduction, Limit u n test, Leibnitz test, Comparison and p-series test, Cauchy s root test, Ratio test, Raabe s and Logarithmic tests, De Morgan & Bertrand s test, Special logarithmic test (without proofs). Beta and Gamma Functions: Transformations, Relation between Beta & Gamma functions, Duplication formula (without proof), Applications in solving definite integrals. Multiple ntegrals: Double and Triple integrals, Change of order of integration, Applications to area and volume, Change of variables, Dirichlet integral and its applications. Vector Calculus : ntroduction, Scalar and Vector point functions, Gradient, Divergence and Curl. Vector Calculus : Vector identities, Line, Surface and Volume integrals, Work done by a force, Green s theorem, Gauss' divergence theorem & Stoke's theorem (without proof). Partial Differential Equations (PDEs): ntroduction, order Lagrange's linear PDEs, n th order linear PDEs. After studying these topics, the student will be able to Apply different tests for determining convergence of an infinite series Evaluate double and triple integrals and study their applications Find the gradient of a scalar field and divergence, curl of a vector field Understand various integral theorems related to line, surface and volume integrals Solve partial differential equations of first and higher orders M. Goyal and N. P. Bali, A Text Book of Engineering Mathematics, Laxmi Publication, Delhi, 20. M. K. Jain, S. R. K. yengar and R. K. Jain, Advanced Engineering Mathematics, Narosa Publishing House, Delhi, 2002. H. Kishan, Sure Success in Convergence, Atlantic Publishers & Dist., Delhi, 2005. B. S. Grewal, Higher Engineering Mathematics, Khanna Publishers, New Delhi, 20. T. M. Apostol, Calculus, Volume, John Wiley& Sons, nc., USA, 1967. T. M. Apostol, Calculus, Volume, Xerox Corporation, USA, 1969. G. B. Thomas and R. Finney, Calculus & Analytic Geometry, Addison Wesley, USA, 1995. 2
[B. Tech.] BMAS 0103: ENGNEERNG MATHEMATCS Course Objectives: To make the students understand the concept of Fourier series, Laplace transform, complex analysis and applications of partial differential equations in engineering. Credits: 04 L T P: 3 1 0 Semester Fourier series: Fourier series of period 2, Half range series, Change of interval. Applications of Partial Differential Eqns.: Classification of order PDEs, Method of separation of variables, Applications in solving one and two dimensional wave and heat flow equations, Laplace equation in cartesian and polar coordinates, Transmission line equations. Laplace Transform: Properties of Laplace transform, Laplace transform of derivatives and integrals, Laplace transform of Unit step, Dirac-delta and periodic functions, Properties of inverse Laplace transform, Convolution theorem, Application in solving ordinary differential equations. Complex Analysis: Analytic and Harmonic functions, Line integral in a complex plane, Cauchy s integral theorem and formula, Taylor and Laurent series (without proof), Singularities, Residue at a pole, Residue theorem and its application in evaluation of real integrals (excluding poles on the real axis). After studying these topics, the student will be able to Find Fourier series representation of function of one variable Apply Laplace transforms in solving differential equations Know the concept of analytic function and its applications in engineering M. Goyal and N. P. Bali, A Text Book of Engineering Mathematics, Laxmi Publications, Delhi, 20. M. K. Jain, S. R. K. yengar and R. K. Jain, Advanced Engineering Mathematics, Narosa Publishing House, Delhi, 2002. B. S. Grewal, Higher Engineering Mathematics, Khanna Publishers, New Delhi, 20. R. V. Churchill and J. W. Brown, Complex Variables and Applications, McGraw Hill, New York, 2009. J. M. Howie, Complex Analysis, Springer Verlag, USA, 2004. 3
BMAO 0001: Operations Research Course Curriculum (Session 2018-2019) [B. Tech.] Course Objectives: To make students understand the concept of quantitative techniques for effective decision making, model formulation and applications in engineering. Credits: 04 L T P: 3 1 0 Linear Programming Problem (LPP): ntroduction, Formulation of LPP, Graphical method, Simplex method, Big- M method, Duality in LPP, Dual simplex method, Degeneracy. Sensitivity Analysis: Sensitivity analysis with respect to change in the Cost coefficient of Objective function and Righthand side of constraints. Non Linear Programming Problem (NLPP): ntroduction and Formulation, Solution of NLPP having more than one inequality constraints using K-T conditions. Transportation Problems: ntroduction, Formulation, Basic feasible solution by North West Corner rule, Row minima method, Column minima method, Least cost method, VAM, Optimality test by MOD method, Degeneracy. Assignment Problem: ntroduction, Hungarian method, Travelling salesman problem. Job Scheduling: ntroduction, Methods, Problems with n jobs & k machines, Problems with 2 jobs & k machines (Graphical method). Game Theory: Saddle point (Pure strategy), Dominance rule, Mixed strategy, n person zero - sum game. Project Management: ntroduction, Rules and Guidelines for drawing network, Dummy activity, Float, Slack, CPM and PERT. Project Crushing: Cost analysis and Crushing the Network, Resource scheduling. Queuing Model: ntroduction, Elements of queuing system, Single-channel and Multi-channel queuing models. Teaching Hours After studying these topics, the student will be able to Construct operation research models from the description of the real systems Learn the mathematical tools to solve various optimization problems Understand the theoretical working of different methods of operation research Text Book: P. K. Gupta and D. S. Hira, Operations Research, S. Chand & Co., Delhi, 2008. (Col.) G.S. Cheema, Operations Research, Laxmi Publications (P) Ltd, 2011. H. A. Taha, Operations Research: An ntroduction, Pearson Education, New Jersey, 2003. 4
[B. Tech.] BMAO 0002: Probability and Applied Statistics Course Objectives: To introduce the concepts of Statistics & probability and illustrate them with engineering applications to support other courses and research. Credits: 04 L T P: 3 1 0 Probability: ntroduction, definitions, axioms, Laws, Conditional & Total probability, Bayes theorem. Random Variables & Distribution functions: Definitions, continuous, discrete and mixed Random Variables, Probability Mass Function (PMF), Probability Density Function (PDF), Cumulative Distribution Function (CDF). Properties of Random Variables: Mean and variance of random variable, Coefficients of variation, Skewness and kurtosis, Moments, Covariance and correlation coefficient. Properties of Distribution Functions. Special Distributions: Uniform, Binomial, Poisson, Exponential, Gamma, Gaussian and Rayleigh distributions. Tests of Hypotheses: ntroduction, Null and Alternate hypotheses, level of significance, types of error, p-value. Small and large sample tests: t- test, F-test, Z-test, χ2 tests. Non Parametric nference: Test of Randomness, Test based on Runs & Sign for one or two sample problems, Median Test, Wilcoxon and Mann-Whitney test, Kolmogorov-Smrinov test for one and two samples. Teaching Hours After studying these topics, the student will be able to apply concept of Statistics and probability in the engineering problems. S.C. Gupta, V.K. Kapoor, Fundamentals of Mathematical Statistics, Sultan Chand Publications, Delhi, 20. V.K. Rohtagi & A.K. Md. E. Saleh, An ntroduction to Probability and Statistics, Wiley, ed., 2008. J.S. Milton & J.C. Arnold, ntroduction to Probability and Statistics, McGraw Higher Ed., 2006. H.J. Larson, ntroduction to Probability Theory and Statistical nference, John Wiley, 1982. S.M. Ross, ntroduction to Probability and Statistics for Engineers and Scientists, Elsevier ndia, New Delhi, 20. 5
[B. Tech.] BMAO 0003: NUMERCAL METHODS Course Objectives: To make the students understand the concept of errors, solution of equations, interpolation, numerical integration & numerical solution of ODEs. Credits: 04 L T P: 3 1 0 Error Analysis: Sources & Types of error, Error in numerical computations, Error in a series approximation. Roots of Equation: Solution of algebraic and transcendental equations by Bisection, iteration, Regula- Falsi and Secant methods. Newton-Raphson method and its order of convergence, Bairstow method for complex roots. nterpolation: Finite differences, Missing term technique, Newton s forward, backward & divided difference formulae, Hermite s interpolation formula. Numerical ntegration: Newton-Cote s quadrature formula, Trapezoidal, Simpson s, Boole s & Weddle s rules, Euler-Maclaurin s formula, Romberg integration. Solution of Simultaneous Linear equations: Gauss elimination method, pivoting, ll-conditioned systems and refinement of solution, LU decomposition and Cholesky methods, Gauss-Seidel iterative method, SOR method. Numerical solution of ODE: Picard s method, Taylorseries method, R-K method of V order, Milne s predictor corrector method, Adams-Bashforth formula. Note: Some problems using software may be given as an assignment. After studying these topics, the students will be able to Understand the propagation of errors in numerical computations Obtain the root(s) of algebraic and transcendental equations by iterative methods Find numerical solution of ordinary differential equations (ODEs) Apply iterative methods to find the solution of system of linear equations nterpolate the data to find the value, not given in the table Use numerical integration to find complex definite integrals M. Goyal, Computer Based Numerical & Statistical Techniques, University Science Press, New Delhi, 2017. S. S. Sastry, ntroductory methods of Numerical Analysis, PH, 2012. Reference Book: M. K. Jain, S. R. K. yengar and R. K. Jain, Numerical Methods for Science & Engineering Computations, New Age nt. Pub., New Delhi, 2012. 6
[B. C. A.] BMAS 01: MATHEMATCS Course Objectives: To make the students understand the concept of algebra, calculus, differential equations and probability. Credits: 04 L T P: 3 1 0 Semester Algebra: ntroduction to matrices, Types, Operations, Minors and Co-factors, Properties of determinant, Adjoint and nverse of a matrix, Cramer s rule, Elementary row operations, Rank of matrix by Echelon form, Solution of a system of simultaneous linear equations. Calculus: Concept of limit, Methods of finding derivatives & integrals (simple problems), Maxima & Minima for the functions of one variable, Expansion of functions of one variable by Taylor & Maclaurin series, ndeterminate forms, L Hospital rule. Ordinary Differential Equations: ntroduction, Variable separable, Linear & Bernoulli forms, n th order linear differential equations with constant coefficients, Complementary function and Particular integral. Probability: ntroduction to probability, Additive and Multiplicative laws of probability (simple problems). After studying these topics, the student will be able to Grasp the applications of matrices and determinant Find rank of a matrix & its applications in solving systems of linear equations Understand the concepts of elementary differentiation and integration Solve the differential equations of first and higher orders Know the basic concepts of probability Text Book: M. Goyal and N. P. Bali, A Text Book of Engineering Mathematics, Laxmi Publications, Delhi, 20. B. S. Grewal, Higher Engineering Mathematics, Khanna Publishers, New Delhi, 20. C. Prasad, Mathematics for Engineers, Prasad Mudranalaya, Allahabad, 1978. Reference Book: M. K. Jain, S. R. K. yengar and R. K. Jain, Advanced Engineering Mathematics, Narosa Publishing House, New Delhi, 2002. E. Kreyszig, Advanced Engineering Mathematics, John Wiley & Sons, Singapore, 2006. 7
[B. C. A.] BMAS 02: Mathematics Course Objectives: To make the students understand the concept of statistics, probability distributions and numerical methods. Credits: 04 L T P: 3 1 0 Semester Statistics: mportance & Limitations of Statistics, Measures of central tendency & Dispersion, Merits and Demerits. Probability Distributions: Binomial, Poisson and Normal distributions, Simple applications. Statistical Methods: Moments, Skewness and Kurtosis by the method of moments, Correlation, Lines of regression, Sampling, Level of significance, Chi-Square test as a goodness of fit and as a test of independence. Numerical Methods: Newton Raphson method and its Order of convergence, Finite differences, Relation between operators, Missing term technique, Newton s interpolation formulae, Newton s divided difference formula, Newton Cote s formula, Trapezoidal and Simpson s rules, Runge Kutta V order formula. After studying these topics, the student will be able to Understand the concepts of statistics Apply numerical methods Use probability distributions G. C. Beri, Business Statistics, TMH, New Delhi, 20. M. Goyal, Computer Based Numerical & Statistical Techniques, Laxmi Pub., New Delhi, 20. S. C. Gupta and V. K. Kapoor, Fundamentals of Statistics, Sultan Chand & Sons, Delhi, 20. P. Gupta, Comprehensive Business Statistics, Laxmi Publications, Delhi, 2011. B. S. Grewal, Higher Engineering Mathematics, Khanna Publishers, Delhi, 20. M. K. Jain, S. R. K. yengar and R. K. Jain, Advanced Engineering Mathematics, Narosa Publishing House, New Delhi, 2002. J. K. Sharma, Business Statistics, Pearson Education, Delhi, 2012. 8
[B. C. A.] BMAS 03: Optimization Methods Course Objectives: To make the students understand the concept of linear programming, transportation and assignment problems, sequencing, network analysis and inventory control. Credits: 04 L T P: 3 1 0 Semester V Linear Programming Problem: Construction of LPP, Feasible and Basic feasible solutions, Optimal solution, Unbounded solution, nfeasible solution, nfinite solutions, Graphical method. Transportation Problem: ntroduction, Basic feasible solution by North West Corner rule, Row Minima method, Column Minima method, Least Cost method, VAM, Optimal solution by MOD method, Degeneracy. Assignment Problem: ntroduction, Hungarian method (Balanced and Unbalanced). Sequencing: ntroduction, Jhonson s rule, Problems with n jobs and k machines, Total elapsed time, Problems with 2 jobs and k machines (Graphical solution). Network Analysis: ntroduction, Rules and guidelines for drawing network, Fulkerson s rule, Dummy activity, CPM and Concept of float, PERT and Concept of slack. nventory Control: ntroduction, Types of inventory, nventory decisions, Economic order quantity, Deterministic inventory problems, EOQ problems with price breaks. After studying these topics, the student will be able to Describe at an intuitive level the process of artificial intelligence and operations research: a real-time cycle of problem understanding, formulation, solution and implementation Formulate simple reasoning, learning and optimization problems, in terms of the representations and methods presented Text Book: P. K. Gupta and D. S. Hira, Operations research, S. Chand & Co. Ltd., Delhi, 20. (Col.) G. S. Cheema, Operations Research, Laxmi Publications Pvt. Ltd, 2011. V. K. Kapoor, Operations research, Sultan Chand & Sons, Delhi, 20. S. D. Sharma, Operations research, Kedar Nath & Ram Nath Publications, Meerut, 2008. H. A. Taha, Operations research: An introduction, Pearson Education, New Jersey, 2003. 9
BMAS 0201: BUSNESS MATHEMATCS Course Curriculum (Session 2018-2019) [B. B. A.] Course Objectives: To make the students understand the concept of basic algebra, quantitative aptitude, matrices and mathematics of finance. Credits: 04 Semester L T P: 3 1 0 Basic Algebra: ntroduction, Representation of sets, Types & Basic operations on set, Laws of set algebra, Venn diagram, Use of set theory in business, Linear & Quadratic equations in one variable, ntroduction to Permutation and Combination (Simple problems). Quantitative Aptitude: Time & Distance, Time & Work, Boats & Streams, Pipes & Cisterns, Partnership, Percentage, Problems on ages, Problems on trains, Surds, ndices & Logarithms. Mathematics of Finance: Ratio & Proportion, Simple and Compound interest, Profit and Loss, Annuity. Matrices: Definition, Types of matrices, Operations on matrices, Transpose, Adjoint & nverse of matrix, Determinants, Solution of system of equations by Cramer s rule, Use of matrix in business. Teaching Hours After completion of the course, student will be able to Understand the basic concepts of business mathematics Develop basic skills for quantitative application in business situations. nterpret and solve real-life business problems using such concepts as differentiation Understand concepts as matrices and other various mathematical concepts useful in day-to-day scenario D. C. Sancheti and V. K. Kapoor, Business Mathematics, Sultan Chand & Company, Delhi, 20. J. K. Sharma, Business Mathematics, Theory and Applications, Ane Books, Delhi, 2009. P. Gupta and O. P. Chug, Comprehensive Business Mathematics, Laxmi Pub., Delhi, 2008. Q. Zameeruddin, V. K. Khanna and S. K. Bhambhari, Business Mathematics, Vikas Pub., Delhi, 2009. 10
BMAS 0202: BUSNESS STATSTCS Course Curriculum (Session 2018-2019) [B. B. A.] Course Objectives: To make the students understand the concept of statistics and index number. Credits: 04 Semester L T P: 3 1 0 Statistical Data Presentation: Meaning, mportance and Limitations of Statistics, Types of data, Frequency distributions, Diagrammatic & Graphical representation. Statistical Average: Requisite of a good average, Measures of Central tendency & Dispersion, Merits and Demerits. Moments: Computation of moments, Skewness & Kurtosis by the method of moments. ndex Number: ntroduction, Types, Construction of index numbers, Methods of determining index number, Tests, Cost of living index number. Correlation & Regression: ntroduction, Correlation between two variables, Karl Pearson s method, Rank correlation, Lines of regression, Simple applications. Probability: Additive and Multiplicative laws of probability, Binomial & Poisson distributions, Simple applications. Teaching Hours After completion of the course, student will be able to Understand the basic concepts of business statistics nterpret and solve real-life business problems using concepts of statistics Understand and solve problems of regression and correlation Understand and solve problems of probability G. C. Beri, Business Statistics, TMH, New Delhi, 20. S. C. Gupta and V. K. Kapoor, Fundamentals of Mathematical Statistics, Sultan Chand & Sons, Delhi, 20. P. Gupta, Comprehensive Business Statistics, Laxmi Publication, Delhi, 2011. J. K. Sharma, Business Statistics, Pearson Education, Delhi, 2012. R.. Levin and D. Rubin, Statistics for Management, PH, Delhi, 1994. 11
BMAS 0202: BUSNESS STATSTCS Course Curriculum (Session 2018-2019) [B. Com. L. L. B.] Course Objectives: To make the students understand the concepts of statistics and probability. Credits: Semester L T P: 1 1 Statistical Data Presentation: Meaning, mportance and Limitations of Statistics, Types of data, Frequency distributions, Diagrammatic & Graphical representation. Statistical Average: Requisite of a good average, Measures of Central tendency & Dispersion, Merits and Demerits. Moments: Computation of moments, Skewness & Kurtosis by the method of moments. ndex Number: ntroduction, Types, Construction of index numbers, Methods of determining index number, Tests, Cost of living index number. Correlation & Regression: ntroduction, Correlation between two variables, Karl Pearson s method, Rank correlation, Lines of regression, Simple applications. Probability: Additive and Multiplicative laws of probability, Binomial & Poisson distributions, Simple applications. Teaching Hours After completion of the course, student will be able to Understand the basic concepts of business statistics nterpret and solve real-life business problems using concepts of statistics Understand and solve problems of regression and correlation Understand and solve problems of probability G. C. Beri, Business Statistics, TMH, New Delhi, 20. S. C. Gupta and V. K. Kapoor, Fundamentals of Mathematical Statistics, Sultan Chand & Sons, Delhi, 20. P. Gupta, Comprehensive Business Statistics, Laxmi Publications, Delhi, 2011. J. K. Sharma, Business Statistics, Pearson Education, Delhi, 2012. R.. Levin and D. Rubin, Statistics for Management, PH, Delhi, 1994. 12
[B. Com. L. L. B.] BMAS 0202: BUSNESS STATSTCS (PRACTCAL) Course Objectives: To make the students learn how to solve statistical problems using MS-EXCEL and spread sheets. Semester 1. Graphical and Diagrametic representation of statistical data using Bar and Pie diagrams 2. Computation of Mean, Median and Mode 3. Computation of Range, Mean deviation and Standard deviation 1. Computation of raw and central moments 2. Computation of Laspeyre, Paasche and Fisher s index numbers 3. Computation of cost of living index number 1. Fitting of lines of regression 2. Computation of correlation coefficient 3. Computation of probabilities Practical Hours 6 6 6 After completion of the course, student will be able to Understand the basic concepts of Business Statistics through practical approach nterpret and solve real-life business problems using concepts of statistics Understand and solve problems of regression and correlation Understand and solve problems of probability Apply MS-Excel to solve problems of Statistics and Probability G. C. Beri, Business Statistics, TMH, New Delhi, 20. S. C. Gupta and V. K. Kapoor, Fundamentals of Mathematical Statistics, Sultan Chand & Sons, Delhi, 20. P. Gupta, Comprehensive Business Statistics, Laxmi Publications, Delhi, 2011. J. K. Sharma, Business Statistics, Pearson Education, Delhi, 2012. R.. Levin and D. Rubin, Statistics for Management, PH, Delhi, 1994.
BMAS 0251: COMMERCAL STATSTCS Course Curriculum (Session 2018-2019) [B. Com. (H)] Course Objectives: To make the students understand the concept of statistics, index numbers and mathematics of finance. Credits: 04 Semester L T P: 3 1 0 Content Data Collection & Representation: Meaning, mportance, Scope and Limitations of statistics, Types of data, Frequency distribution, Diagrammatic and Graphical representation. Mathematics of Finance: Percentage, Ratio and Proportion, Simple & Compound interest, Annuity & its types. Statistical Average: Requisite of a good average, Measures of Central tendency & Dispersion, Merits and Demerits. Moments: Computation of moments, Skewness & Kurtosis by the method of moments. Correlation & Regression: ntroduction, Correlation between two variables, Karl Pearson s method, Rank correlation, Lines of regression, Simple applications. ndex Number: ntroduction, Types, Constructions of index number, Methods, Tests, Cost of living index number. Teaching Hours After completion of the course, student will be able to Understand the basic concepts of commercial statistics nterpret and solve real-life business problem using the concepts of statistics Understand and solve problems of measures of central tendency & dispersion, regression, correlation etc. G. C. Beri, Business Statistics, TMH, New Delhi, 20. S. C. Gupta and V. K. Kapoor, Fundamentals of Mathematical Statistics, Sultan Chand & Sons, Delhi, 20. P. Gupta, Comprehensive Business Statistics, Laxmi Publications, Delhi, 2011. R.. Levin and D. Rubin, Statistics for Management, PH, New Delhi, 1994. J. K. Sharma, Business Statistics, Pearson Education, Delhi, 2012.
BMAS 0251: COMMERCAL STATSTCS Course Curriculum (Session 2018-2019) [B. A. (H) Economics] Course Objectives: To make the students understand the concept of statistics, index numbers and mathematics of finance. Credits: 04 Semester L T P: 3 1 0 Content Data Collection & Representation: Meaning, mportance, Scope and Limitations of statistics, Types of data, Frequency distribution, Diagrammatic and Graphical representation. Mathematics of Finance: Percentage, Ratio and Proportion, Simple & Compound interest, Annuity & its types. Statistical Average: Requisite of a good average, Measures of Central tendency & Dispersion, Merits and Demerits. Moments: Computation of moments, Skewness & Kurtosis by the method of moments. Correlation & Regression: ntroduction, Correlation between two variables, Karl Pearson s method, Rank correlation, Lines of regression, Simple applications. ndex Number: ntroduction, Types, Constructions of index number, Methods, Tests, Cost of living index number. Teaching Hours After completion of the course, student will be able to Understand the basic concepts of commercial statistics nterpret and solve real-life business problem using the concepts of statistics Understand and solve problems of measures of central tendency & dispersion, regression, correlation etc. G. C. Beri, Business Statistics, TMH, New Delhi, 20. S. C. Gupta and V. K. Kapoor, Fundamentals of Mathematical Statistics, Sultan Chand & Sons, Delhi, 20. P. Gupta, Comprehensive Business Statistics, Laxmi Publications, Delhi, 2011. R.. Levin and D. Rubin, Statistics for Management, PH, New Delhi, 1994. J. K. Sharma, Business Statistics, Pearson Education, Delhi, 2012.
[B. Sc. (H) Bio Tech.] BMAS 0501: BOSTATSTCS Course Objectives: To make the students understand the concept of biostatistics, probability, calculus and algebra. Credits: 04 Semester L T P: 3 1 0 Content Logarithms, ntroduction and simple problems on Differentiation, ntegration (excluding trigonometric functions), Scalar and Vector quantities, Types of vectors, Addition and Subtraction of vectors, Scalar and Vector product of two vectors, Types of matrices, Operations on matrices (addition, subtraction and multiplication). ntroduction to Biostatistics, Data collection, Tabulation and Classification of data, Frequency distributions, Diagrammatical & Graphical representation of data, Measures of Central tendency and Dispersion, ntroduction to Probability (simple problems). Correlation between two variables, Karl Pearson s formula for finding correlation coefficient, Rank correlation, Regression lines, Fitting of straight line & second degree parabola by the method of least squares, Population and sample, Testing of hypothesis, Level of significance, t-test, Chi-square test as a goodness of fit. Teaching Hours After completion of the course, student will be able to Attain a basic proficiency in quantitative skills, understand and critically assess data collection and its representation Get the knowledge of averages, measures of dispersion to interpret the data Understand the correlation between the two variables and concept of regression lines Test the hypothesis and apply t and Chi square tests P. Banerjee, ntroduction to Biostatistics, S. Chand & Co., Delhi, 2006. G. C. Beri, Business Statistics, TMH, New Delhi, 20. H. Kishan, Differential Calculus, Atlantic Pub. and Distributors, Delhi, 2008. H. Kishan, ntegral Calculus, Atlantic Pub. and Distributors, Delhi, 2005. S. C. Gupta and V. K. Kapoor, Fundamentals of Mathematical Statistics, Sultan Chand & Sons, Delhi, 20. B. K. Mahajan, Methods in Biostatistics, Jaypee Publications, New Delhi, 2010. 16
[M. Sc. Bio Tech.] MMAS 0501: ADVANCED BOSTATSTCS Course Objectives: To make the students understand the advanced concepts of biostatistics, algebra and differential equations. Credits: 04 Semester L T P: 3 1 0 Content ntroduction to Vector algebra, Scalar & Vector triple products, Collinear and Coplanar vectors, Determinant and its properties, Adjoint and nverse of a matrix (simple problems), Formation of ordinary differential equations (ODEs), Solution of ODE of order and degree (Variable separable and Linear forms). ntroduction to Biostatistics, Revision of measures of central tendency and dispersion, Computation of moments, Skewness and Kurtosis by the method of moments, ntroduction to probability, Additive and multiplicative laws, Conditional probability. Method of least squares for fitting of exponential curves, Sampling, Testing of hypothesis, Type and type errors, Level of Significance, Degree of freedom, Students t-test, F test, Chi-square test as a goodness of fit and as a test of independence, ANOVA (one way classification). Teaching Hours After completion of the course, student will be able to Recognize and give example of different type of data arising in public health and clinical studies nterpret difference in data, select appropriate test for comparing populations. Test the hypothesis and apply t, F and Chi-square tests Understand one way classification of analysis of variance P. Banerjee, ntroduction to Biostatistics, S. Chand & Co., Delhi, 2006. G. C. Beri, Business Statistics, TMH, New Delhi, 20. H. Kishan, Differential Equations, Atlantic Publishers and Distributors, Delhi, 2008. S. C. Gupta and V. K. Kapoor, Fundamentals of Mathematical Statistics, Sultan Chand & Sons, Delhi, 20. B. K. Mahajan, Methods in Biostatistics, Jaypee Brothers Pub., New Delhi, 2010. 17
[B. Sc. (H) Chemistry] BMAS 0502: ALGEBRA & CALCULUS Course Objectives: To make the students understand the concepts of algebra and calculus by giving more emphasis to their applications in the field of chemistry. Prerequisites: Credits: 03 Semester L T P: 3 0 0 Matrices: ntroduction, nverse by elementary transformations, Rank by Echelon form, Solution of system of linear equations by elementary transformations and Cramer s rule, Linear dependence & independence of vectors, Complex matrices, Eigen values and Eigen vectors. Differential Calculus: Successive differentiation, calculation of n th derivative, Leibnitz theorem, Partial differentiation, Euler s theorem, Composite functions, Total derivative, Jacobian and its properties, Expansion of functions of one and two variables. ntegral Calculus: ntegration of rational & irrational functions, Beta & Gamma functions, Double and Triple integrals, Change of order, Change of variables. Vector Calculus: Gradient, Divergence and Curl, Green s theorem, Gauss' divergence theorem & Stoke's theorem (without proof). After studying these topics, the student will be able to Understand partial differentiation and its applications Find rank of a matrix & its applications in solving systems of linear equations Evaluate double, triple integrals and study their applications Find the gradient of a scalar point function and divergence, curl of a vector field M. Goyal and N. P. Bali, A Text Book of Engineering Mathematics, Laxmi Publication, Delhi, 20. M. K. Jain, S. R. K. yengar and R. K. Jain, Advanced Engineering Mathematics, Narosa Publishing House, New Delhi, 2002. W. E. Boyce and R. D. Prima, Elementary Differential Equations, John Wiley & Sons, 2009. G. B. Thomas and R. Finney, Calculus & Analytic geometry, Addison Wesley, USA, 1995. 18
[B. Sc. (H) Chemistry] BMAS 0503: ADVANCED MATHEMATCS Course Objectives: To make the students understand the concepts of ordinary differential equations, statistics and numerical methods by giving more emphasis to their applications in chemistry. Credits: 03 L T P: 3 0 0 Semester Ordinary Differential Equations (ODEs): ntroduction, Formation, Solution of ODEs of order & degree, Solution of n th order linear differential equations with constant coefficients, Euler-Cauchy Equations, Simultaneous differential equations. Numerical Methods: Errors & its types, teration & Newton Raphson method, Finite differences, Missing term technique, nterpolation by Newton s forward, backward and divided difference formulae, Numerical integration by trapezoidal and Simpson s rules, Numerical solution of order ODE by R-K V order method. Statistics: Measures of central tendency & Dispersion, Correlation and Regression, Fitting of straight line by method of least squares, Binomial & Poisson distributions, Statistical hypotheses, Level of significance, Chi square test as a test of goodness of fit. After studying these topics, the student will be able to Solve the ordinary differential equations and know their applications in chemistry Apply numerical techniques for numerical differentiation and integration Understand the probability distributions Test the hypothesis by Chi-square test M. Goyal and N. P. Bali, A Text Book of Engineering Mathematics, Laxmi Publication, Delhi, 20. S. C. Gupta and V. K. Kapoor, Fundamentals of Statistics, Sultan Chand & Sons, Delhi, 20. M. K. Jain, S. R. K. yengar and R. K. Jain, Advanced Engineering Mathematics, Narosa Publishing House, Delhi, 2002. B. S. Grewal, Higher Engineering Mathematics, Khanna Publishers, New Delhi, 20. G. C. Beri, Business Statistics, TMH, New Delhi, 20. 19
[M.Sc. Chemistry] MMAS 0502: MATHEMATCS FOR CHEMSTS Course Objectives: To make the students understand the concepts of algebra, calculus and statistics by giving more emphasis to their applications in the field of chemistry. Prerequisites: Credits: 02 Semester L T P: 2 0 0 Algebra: ntroduction to matrices, Determinant, Adjoint and nverse of a matrix, Elementary operations, Rank of a matrix by Echelon form, Solution of system of linear equations by rank test and Cramer s rule, Revision of vector products, Point functions, Gradient, Divergence and Curl. Calculus: Differentiation and integration of standard functions, Product, quotient and chain rules for differentiation, Extrema of functions of one variable, ntegration by substitution, by parts and by partial fraction, Definite integral and its properties. Simple applications. Statistics: Measures of central tendency and dispersion, Correlation and Regression, Fitting of straight line by method of least squares. ntroduction to probability, Binomial and Poisson distributions. After studying these topics, the student will be able to Understand differentiation and integration Find rank of a matrix & its applications in solving systems of linear equations Calculate the measures of central tendency and dispersion Find the gradient of a scalar point function and divergence, curl of a vector field M. Goyal and N. P. Bali, A Text Book of Engineering Mathematics, Laxmi Pub., Delhi, 20. Mathematics books for Class X & X, NCERT Publications. P. Gupta, Comprehensive Mathematics (for Class X & X), Laxmi Pub. (P) Ltd. Delhi. S. C. Gupta and V. K. Kapoor, Fundamentals of Statistics, Sultan Chand & Sons, Delhi, 20. 10 10 10 20
[Ph.D. Course Work] PAM 9002: RELATVTY AND COSMOLOGY Course Objectives: To make the students understand the concept of general relativity and cosmology by giving more emphasis to their applications. Credits: 04 Semester L T P: 4 0 0 Special Theory of Relativity: nertial and non-inertial frames, Special and General Galilean transformations, Lorentz transformation and its geometrical interpretation, Transformation formula for mass, density, momentum, energy and force, Minkowski-space, Relativistic equation of motion, Four vectors and tensors in Minkowski space, Lagrangian and Hamiltonian formulation of Relativistic Mechanics. General Relativity: Principles of equivalence and general covariance, Mach s Principle, Einstein s field equations, Energy momentum tensors, Gravitational equations, Vectors and tensors, Experimental tests of general relativity, Alternatives theories of gravitations, Cosmological solutions in Brans-Dicke Theory, Kaluza s five dimensional theory, Cosmological models, Singularity in cosmological models. Cosmology: Static cosmological models, Newtonian cosmology, Einstein universe, Expanding universe, Friedmann models, Cosmological models with non-zero cosmological term, The early universe, The inflationary universe, Primordial black holes, Dark energy and dark matter, Observational constraints on cosmological parameters, Standard cosmology. After studying these topics, the student will be able to Understand the basic principles of cosmology. Know the significance the Einstein s theories of special and general relativity. Deal with the cosmological models S. R. Roy & Raj Bali, Theory of Relativity, Jaipur Publishing House, 2008. J. V. Narlikar, An ntroduction to Cosmology, Cambridge University Press, 2002. S. Weinberg, Cosmology, Oxford University Press, 2008. S. K. Srivastava, General Relativity and Cosmology, PH Pvt. Ltd., 2008. 21
[Ph.D. Course Work] PAM 9003: PARTAL DFFERENTAL EQNS: METHODS & APPLCATONS Course Objectives: To make the students understand the various analytic and semi analytic approaches to solve linear and nonlinear partial differential equations with given initial and boundary conditions by giving more emphasis to their applications in the real world. Credits: 04 L T P: 4 0 0 Semester Order Linear Partial Differential Equations (PDEs): Solution by Adomian decomposition method (ADM), The noise terms phenomenon. Solution by the modified decomposition method (MDM) and the variational iteration method (VM). Method of characteristics, Solution of systems of linear PDEs by ADM and VM. Heat Flow and Wave Equations by ADM and VM: Solution of homogeneous and inhomogeneous one dimensional heat and wave equations by ADM and VM. Solution of higher dimensional heat flow & wave equations by ADM. Solution of Laplace s equation with Dirichlet, Neumann and Robin Boundary conditions by ADM. Non linear Partial Differential Equations: Calculation of Adomian polynomials, Solution of nonlinear PDEs by ADM, MDM and VM, Solution of non-linear PDEs systems, Nonlinear advection problem, Goursat problem, Klein Gordon, Sine Gordon, Burger s, Telegraph, Schrodinger, Korteweg devries (KdV) equations by ADM and VM. After studying these topics, the student will be able to Understand various methods for solving Partial differential equations Apply the semi analytic methods in real world problems Explore new applications of the aforesaid methods A. M. Wazwaz, Partial Differential Equations and Solitary Wave Theory, Springer, 2009. A. M. Wazwaz, Partial Differential Equations: Methods and Applications, Balkema Publishers, 2002. G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Academic Publishers, 1994. L. Debnath, Non Linear Partial Differential Equations for Scientists and Engineers, Birkhauser, Springer, 2012. L. C. Evans, Partial Differential Equations, American Mathematical Society, 1998. 22
PAM 9004: nventory Modeling Course Curriculum (Session 2018-2019) [Ph.D. Course Work] Course Objectives: To make the students understand the various inventory models with deterministic & probabilistic demands, price breaks and different demand functions. Credits: 04 Semester L T P: 4 0 0 Basics of nventory Models (M): Necessity of M, nventory costs, Basic definitions, Classification of M. nventory Models with Deterministic Demand: (1) Demand rate uniform, production rate infinite, (2) Demand rate non uniform & production rate infinite, (3) Demand rate uniform & production rate finite, (4) Demand rate uniform, production rate infinite & shortages allowed, (5) Demand rate uniform, production rate finite& shortages allowed. nventory Models with Probabilistic Demand: (1) nstantaneous demand, setup cost zero, stock level discrete & lead time zero (2) nstantaneous demand, setup cost zero, stock level continuous& lead time zero (3) Continuous demand, setup cost zero, stock level discrete& lead time zero (4) Continuous demand, setup cost zero, stock level continuous& lead time zero (5) Continuous demand, setup cost zero, stock level discrete with lead time. nventory Models with Price Breaks: Purchase inventory models with one and two price breaks nventory Models with different demand functions: nventory models with price and time dependent demand, nventory models with price, time& stock dependent demand, nventory models with seasonal demand. After studying these topics, the students will be able to Understand the inventory models with uniform and non-uniform demand rate Know the inventory models with instantaneous/ continuous demand with discrete/ continuous stock level and lead time zero. Minimize the total inventory holding costs and ordering costs P. K. Gupta & D. S. Hira, Operations Research, S. Chand Publication, 20. Sven Axsäter, nventory Control, Springer, 2000. Reference Book: E. L. Porteus, Foundations of Stochastic nventory Theory, Stanford University Press, 2002. 23
[Ph.D. Course Work] PAM 9005: Computer Skills, Tensor and Riemannian Geometry Course Objectives: To make the students understand the concept of tensors and Riemannian geometry and know the software packages by giving more emphasis to their applications. Credits: 04 Semester L T P: 4 0 0 Computer Skills : ntroduction to Latex, MATLAB and MAPLE. Tensor and Riemannian Geometry : Tensor calculus, n-dimensional space V n, Superscript and subscript, Transformation of coordinates, Transformation law of tensor, Product of two tensors, Contraction, Trace of a tensor, Quotient law, Metric tensor and Riemann space. Computer Skills : ntroduction to Excel, Chart, Functions. Power point presentation, ntroduction to C language. Tensor and Riemannian Geometry : Associated and Reciprocal or conjugate tensor, Symmetric and anti symmetric tensor, Tensor density, Levi-Civita tensor, Christoffel symbols, Law of transformation of Christoffel symbols, Covariant differentiation, Riemannian Affine connection, Covariant derivative of a vector density, Riemannian metric, Geodesics, Null geodesics, Tensor form of gradient, divergence, Laplacian and curl. Tensor and Riemannian Geometry : ntrinsic derivative, Riemannian and normal coordinates, Gaussian coordinates, Parallel transport, Geodesics as auto parallel curves, Parallel propagation, Riemann curvature tensor R i ljk, Covariant curvature tensor, Symmetric properties of R i ljk, Covariant curvature tensor R hljk, Number of independent components of R hljk, Ricci tensor, Bianchi identities, Conformal curvature tensor, Algebraic classification of the conformal curvature tensor, Conformal invariance, Geodesic deviation, Lie derivative. After studying these topics, the students will be able to Use software packages in research Understand covariant differentiation, Bianchi identities and their applications Teaching Hours S. R. Roy and Raj Bali, Theory of Relativity, Jaipur Publishing House, 2008. S. K. Srivastava, General Relativity and Cosmology, PH Pvt. Ltd., 2008. Leslie Lamport, A Document preparation System: Latex, Addison-Wesley Professional, 1994. Y. K. Singh and B. B. Chaudhary, MATLAB Programming, PH, 2007. Reference Book: J. V. Narlikar, Cosmology, Cambridge University Press, 2002. 24
25 Course Curriculum (Session 2018-2019) [B. B. A. (FB) & (H)] BFBC 0002: QUANTTATVE ASPECTS OF BUSNESS ntroduction: The course will enable the students in terms of understanding the quantitative aspects related to business thereby enhancing their skills. Objective: Studying this subject would improve the mathematical abilities and statistical skills of the students and help them in understanding related concepts. Prerequisites: Credits: 04 Semester L T P: 3 2 0 Matrices: Addition, Subtraction, Multiplication of matrices, nverse of matrices, Systems of linear equations and its applications. Differentiation of functions of single variable (excluding trigonometric functions), Breakeven analysis, Simple problems of maxima and minima. Role of Statistics in Business, Classification & presentation of data with the help of MS-Excel. Application of measures of central tendency Mean, Mode, Median, and measures of dispersion Range, Quartile Deviation, Standard deviation in business decision making. Correlation Karl Pearson s Correlation Rank Correlation. Regression- Fitting regression equations, Uses of Regression in Business Problems. Time Series- Components of time Series Measurement of Trend Semi Average method Moving Average method Method of Least Squares Measurement of Seasonal Variations Simple Average Method Ratio to Moving Average Method. ndex Numbers Un weighted ndex Numbers, weighted ndex Numbers: Laspeyre s, Paasche, Fisher, Bowley s and Marshall Edgeworth s method, Cost of Living ndex Number Test on index numbers. MS-excel based application to address the issues of Time series. Reference Books/ Text Books/ Cases: Sharma, J. K. (2010), Business Mathematics, New Delhi: Ane Books (P) Ltd. Bajpai, N. (2001). Business Statistics. New Delhi: Pearson Education (P) Ltd. Levin, R.. (1997), Statistics for Management. New Delhi: Pearson Education (P) Ltd. Beri, G. C. (2010). Business Statistics. New Delhi: Tata McGraw Hill Publishing Company Ltd. Gupta, S. C. (2006). Fundamentals of Statistics. New Delhi: Himalaya Publishing House ntended Outcomes: After completion of the course, student will be able to: Understand the basic concepts of business statistics. nterpret and solve real-life business problem using the concepts of statistics. Understand the business & industry problems along with working out their solutions. Understand the business & industry problems along with working out their solutions. Understand various mathematical concepts useful in day-to-day scenario.