Program Outcomes (B.Sc.)

PO1. Disciplinary Knowledge: Capable of demonstrating comprehensive knowledge and understanding of one or more other disciplines that form a part of an undergraduate program of study.

PO 2. Critical Thinking: Take informed actions after identifying the assumptions that frame our thinking and actions, checking out the degree to which these assumptions are accurate and valid, and looking at our ideas and decisions (intellectual, organizational, and personal) from different perspectives.

PO3: Communication Skills: Ability to express thoughts and ideas effectively in writing and orally;communicate with others using appropriate media; and present complex information in a clear and concise manner to different groups.

PO 4. Social Interaction: Ability to work effectively and respectfully with diverse teams; facilitate cooperative or coordinated effort on the part of a group and act together as a group or a team in the interests of a common cause.

PO 5. Moral and Ethical Awareness: Ability to embrace moral/ ethical values in conducting one’s life, possess knowledge of the values and beliefs of multiple cultures and a global perspective; and capability to effectively engage in a multicultural society and interact respectfully with diverse groups.

PO 6: Environment and Sustainability: Understand the issues of environmental contexts and sustainable development.

PO 7: Information and Digital Literacy: Capability to use ICT in a variety of learning situations. Demonstrate ability to access, evaluate and use a variety of relevant information sources; and use appropriate software for analysis of data.

PO 8: Research –related skills: A sense of inquiry and capability for asking relevant/ appropriate questions, synthesizing and articulating; Ability to recognize cause- and- effect relationships, define problems, formulate hypotheses,interpret and draw conclusions from data, ability to plan, execute and report the results of an experiment or investigation. Ability to apply one’s learning to real life situations.

PO 9: Employability: After completion of the program students become employable on the basis of their qualification, acquired knowledge and skills.

Program Specific Outcomes

PSO-1. The students will love mathematics and learn how to appreciate its internal aesthetic beauty.

PSO-2. The students will be aware of the interlink between the topics included in the syllabus as well as the different fields of sciences,

economics etc.

PSO-3. They will be able to answer the basic concept-based questions.

PSO-4. The students will learn the art of mathematical calculation/computation as well as analytical concepts and abstraction.

PSO-5. They will get motivated to pursue this rich subject in future.

PSO-6. They will be able to think logically to come to a conclusion.

PSO-7. The students will learn the art of original and creative thinking

Learning Outcomes of Core Course-5

On successful completion of this course, students will be able to:

1. know the ∈-δ approach of limit of a function and sequential criterion of limit.

2. understand algebra of limits and find different important limits.

3. get the concept of continuity, its sequential criterion; and be familiar with different continuous functions.

4. know neighbourhood property of continuous functions.

5. understand and apply results like boundedness property, Bolano’s theorem on continuity, intermediate value theorem, etc.

6. identify different types of discontinuities.

7. deal with continuity of monotonic functions and inverse functions.

8. get ideas of uniform continuity, Lipschitz functions, and their relations.

9. get the concept of differentiability of a function, understand algebra of differentiable functions and meaning of sign of derivative.

10. understand and apply Darboux theorem, Rolle’s theorem, Lagrange’s mean value theorem, Cauchy’s mean value theorem, Taylor’s theorem

11. expand different functions as infinite series.

12. apply L’ Hospital’s rule.

13. determine principles of local maximum/minimum of functions using derivatives and apply these principles in different problems.

Learning Outcomes of Core Course- 6

1. At the end of course , students will be able to know about a new branch of Algebra. This course provides an axiomatic description of an abstract vector space with different types of examples, concepts of basis, different ways to form basis according to own choice and concept of subspaces (with Geometrical interpretation).

2. Another important topic of this course is linear transformation. Students will be able to construct different types of mappings on vector spaces with special property linearity. This course makes them understand the concept of nullity, rank and relation between them (according to the basic property of a mapping such as injectivity, surjectivity, bijectivity).

3. Students will be surprised to see the relation between matrices and linear transformations, the change of matrix presentation of a linear transformation with the change of base of vector spaces and related theorems.

4. At last the course will provide the concept of eigenvalue, eigenvector, Cayley-Hamilton theorem and their application in Algebra.

Learning Outcomes of Core Course-7

Unit:1 (Ordinary Differential Equations)

1. Students are able to distinguish between linear, nonlinear, partial and ordinary differential equations.

2. Classifies the differential equations with respect to their order and linearity.

3. Expresses the basic existence theorem for higher- order linear differential equations.

4. Able to use the rules of different integrating factors in various non-exact equations.

5. Able to understand the concept of Linear dependent and independent solutions (Wronskian ) .

6. To inculcate knowledge on solving of first and second order differential equations with algebraic manipulations.

7. Applies the method of undetermined coefficients to solve the non-homogeneous linear differential equations with constant coefficients. Uses the method "variations of parameters" to find to solution of higher-order linear differential equations with variable coefficients.

8. Create and analyses mathematical models using first order differential equations to solve application problems such as circuits, mixture problems, population modelling, orthogonal trajectories, and slope fields.

9. Create and analyses mathematical models using higher order differential equations to solve application problems such as harmonic oscillator and circuits.

10. Communicate mathematical applications, concepts, computations, and results with classmates and colleagues in the fields of science, technology, engineering, and mathematics.

Unit: 2 (Multivariate Calculus I)

1. Analyse real world scenarios to: recognize when vectors, geometry of space, multivariate and vector functions, partial differentiation, and multiple integration are appropriate, formulate and model these scenarios (using technology, if appropriate) in order to find solutions using multiple approaches, judge if the results are reasonable, and then interpret these results.

2. Recognize the underlying mathematical concepts of vectors, geometry of space, multivariate and vector functions, partial differentiation, and multiple integration.

Learning Outcomes of GE-3

1. At the end of the course students can get a precise knowledge about Integral Calculus , Numerical Methods and Linear Programming .In Integral Calculus they will learn definite integral , improper integral , double integral , volume integral , surface integral and their applications.

2. In Numerical Methods they will be familiar with different numerical operators , different types of interpolation , numerical integration , solution of numerical equation.

3. In Linear Programming they will know use of linear programming in real life problem, formulation of L.P.P. and solving them using different methods like graphical method , simplex method, dual method.

Learning Outcomes of SEC-A

What the students learn from the course of C Programming Language

1. C is robust language and has rich set of built-in functions (Library Functions) , data types and operators which can be used to write any

complex program.

2. Program written in C are efficient due to availability of several data types and operators.

3. C has the capabilities of an assembly language (low level features) with the feature of high level language so it is well suited for writing both system software and application software.

4. C is highly portable language i.e. code written in one machine can be moved to other which is very important and powerful feature.

5. Another very important feature of C programming Language is that it allows users to define User Defined Functions according to their need.

6. Single list of instructions within main() functions are known as monolithic program – i.e. program containing a large single list of

instructions. These types of programs are very difficult to understand, debug, test and maintain. So to avoid these difficulties we use user defined functions.

7. User defined functions in C programming has following advantages:

8. Reduction in Program Size: Since any sequence of statements which are repeatedly used in a program can be combined together to form a user defined functions. And this functions can be called as many times as required. This avoids writing of same code again and again reducing program size.

9. Reducing Complexity of Program: Complex program can be decomposed into small sub-programs or user defined functions.

10. Easy to Debug and Maintain : During debugging it is very easy to locate and isolate faulty functions. It is also easy to maintain program that uses user defined functions.

11. Readability of Program: Since while using user defined function, a complex problem is divided in to different sub-programs with clear

objective and interface which makes easy to understand the logic behind the program.

12. Code Reusability: Once user defined function is implemented it can be called or used as many times as required which reduces code

repeatability and increases code reusability.

Learning Outcomes of Core Course-11

On successful completion of the course students will be able to:

1. To understand the basic principles of probability including the laws for unions, intersections, and complementation, Bayes theorem and use these principles in problem solving situations. Apply the basic probability rules, including additive and multiplicative laws, independent and mutually exclusive events in probability models.

2. Understand the concept of Sigma-algebra and concept of probability space based on sigma algebra.

3. Understand the definitions of discrete, continuous, and joint random variables, compute the mean,variance and covariance of random variables, know the definition of mass (density) function and distribution function of a random variable and be able to find one from the other, and be able to find the marginal mass (density) function and distribution functions from the joint mass (density) function and distribution function.

4. Identify an appropriate probability distribution for a given discrete or continuous random variable and use its properties to calculate probabilities.

5. Calculate probabilities for joint distributions including marginal and conditional probabilities.

6. Calculate the moments and formulates the Moment Generating Function, Characteristic Function.

7. Calculate statistics such as the mean and variance of common probability distributions. Summarize the roles of different distributions in practical situations and illustrate with examples the nature of limiting distributions for large samples .

8. To develop a questionnaire, organize a sample survey by implementing different sampling techniques and predict population characteristics.

9. Explain the desirable properties of estimators. Calculate and interpret maximum likelihood estimates and their confidence intervals.

10. Explain the role of probability in hypothesis testing and describe issues related to interpreting statistical significance.

11. One area that needs to be strengthened in response to the career climate is student preparation in statistics and data science. The Higher Education recently listed the growth of data science programs as a key trend in higher education. However, they also noted that data science programs are being added without careful attention to what a data science curriculum should look like. Moreover, because data and statistics play an important role in all disciplines, undergraduate curriculum in statistics and data science may be embedded within different disciplinary contexts. As such, there is a need for a set of comprehensive learning outcomes to help guide data learning across the disciplines.

Learning Outcomes of Core Course-12

1. Group is a part of Abstract Algebra which includes a solid introduction to the traditional topics of mathematics. There is a connection between abstract algebra , number theory and geometry. Students have been given the definition of groups , various examples of groups and large number of group properties in the previous classes. Cyclic groups, Group Homomorphism and Isomorphism concepts are known to the students.

2. In this course, Automorphism ,which is an isomorphism of a group onto itself, is introduced. The set of all automorphisms on a group onto itself forms a group under composition of mappings and is called the Automorphism group. The students will enjoy the beauty of automorphism groups of finite and infinite cyclic groups

3. Symmetry itself is a vast subject, significant in art and nature. Mathematics lies at its roots. External Direct Product and Internal Direct Product of groups are introduced. The students will be able to get a better insight to the fact that the Universe is an enormous Direct Product Representation of Symmetry groups . The Properties of Finite Abelian groups will help them to study the Sylows Theorems

4. The concepts and methodologies are used by applied mathematician. Computer scientists, physicists and chemists.

Learning Outcomes of DSE-A1 (Bio-mathematics)

On successful completion of this course, students will learn the following.

1. There is a synergistic relationship between mathematics and biology, and Biomathematics is the most exciting modern application of mathematics.

2. Many biological processes can be represented in terms of mathematics, mathematical models can be formulated, and analysis of these models can be used for gaining better insight and help deepen our understanding about these biological processes.

3. Nothing is permanent except change; and differential equations and difference equations can serve as excellent tools for the study of change in the physical world.

4. The concepts stability and bifurcations can be used in many real-life situations.

5. Formulation and analysis of single-species continuous-time models like Malthus model, Verhulst logistic model, Gompertz model are very useful from both mathematical and practical point of view.

6. Allee effect and problems related to harvesting can be modeled and analyzed mathematically.

7. Many continuous-time predator-prey models (including the famous LotkaVolterra model) , chemostat models (including Michaelis-Menten kinetics) can be formulated and analyzed to help understanding the dynamics of those systems.

8. The concept of functional responses of Holling type can be used to construct realistic predator-prey models.

9. Formulation and analysis of epidemiological models (including KermackMcKendrick model and many models of types SI, SIR, SIS, SIC, etc.) can help in understanding the disease dynamics in more sophisticated way.

10. Formulation and analysis of many discrete-type single-species model, predator-prey model, host-parasitoid models (like Nicholson-Bailey model) can be very useful in real-life situations.

11. Bio-mathematics has stimulated revolutionary developments in the theory of differential equations and difference equations and these equations are serving as indispensable tools in the field of theoretical biology

Learning Outcomes ( DSE-B2: Boolean Algebra and Automata Theory)

1. Lattice theory is an essential preliminary to the full understanding of logic, set theory, probability, functional analysis, projective geometry, the decomposition theorems of abstract algebra, and many other branches of mathematics.

2. The prime feature of lattice theory is its versatility. It connects many areas. Algebra, analysis, topology, logic, computer science, combinatorics, linear algebra, geometry, category theory, probability.

3. One of the most important practical applications and also one of the oldest applications of modern algebra, especially lattice theory, is the use of Boolean algebras in modelling and simplifying switching or relay circuits.

4. Logic networks and automata are facets of digital systems. The change of the design of logic networks from skills and art into a scientific discipline was possible by the development of the underlying mathematical theory called the Switching Theory. The fundamentals of this theory come from the attempts towards an algebraic description of laws of thoughts presented in the works by George J. Boole and the works on logic by Augustus De Morgan.

5. Along with the development of modern science and technology, the Automata Theory has become the important foundations of theory and application. Theoretical computer science is divided into three key areas: automata theory, computability theory, and complexity theory. The goal is to ascertain the power and limits of computation.

6. In order to study these aspects, it is necessary to define precisely what constitutes a model of computation as well as what constitutes a computational problem. This is the purpose of automata theory.

7. The computational models are automata, while the computational problems are formulated as formal languages.

8. The Church-Turing thesis conjectures that no model of computation that is physically realizable is more powerful than the Turing Machine. In other words, the Church-Turing thesis conjectures that any problem that can be solved via computational means, can be solved by a Turing Machine.

9. To this day, the Church-Turing thesis remains an open conjecture. For this reason, the notion of an algorithm is equated with a Turing Machine.

10. In a nut shell, this course helps the readers have an idea - how set theory, relations and mathematical logic (the topics of Philosophy of Mathematics) are used to construct an abstract theoretical machine which was developed by the mathematicians much earlier than programming language and computers.

Learning Outcomes ( DSE-B3: LPP)

After the completion of this course students will be able to:

1) mathematically formulate an applied word problem involving revenue, costs, and

constraints as a linear program.

2) apply the simplex algorithm to solve a linear programming problem.

3) utilize computer software to solve a linear programming problem.

4) solve a linear programming problem using either the M-Method or the Two-Phase

Simplex Method.

5) describe the Dual Theorem and its consequences.

6) use duality to analyze changes to a linear programming problem’s optimal solution.

Learning Outcomes ( DSE-A: GRAPH THEORY) For General

Graph Theory is mainly the study of relationships. Given a set of nodes and connections (edges), which can denote anything from city layouts to computer data, graph theory provides a helpful tool to quantify & simplify the many moving parts of dynamic systems. Studying graphs through a framework provides answers to many arrangement, networking, optimization, matching and operational problems. Graphs can be used to model many types of relations and processes in physical, biological, social and information systems, and has a wide range of useful applications ,such as,

1. Ranking/ordering hyperlinks in search engines;

2. GPS/Google maps to find the shortest path home;

3. Study of molecules and atoms in chemistry;

4. DNA sequencing;

5. Computer network security;

Learning Outcomes of Core Course-13

1. Metric Space is a part of Functional Analysis which is an abstract branch of mathematics that originated from classical analysis. It plays an increasing role in applied sciences as well as in mathematics itself. Metric space play a role similar to that of the real line R in calculus

2. A metric space is a set X with a metric d (distance function on X) satisfying certain axioms. The concept of open sets, closed sets, interior point, limit point, bounded sets, diameters of sets are included in the course. Convergent sequence are very important concepts in real analysis as well as in Metric spaces.

3. The learners understand the ideas of completeness properties in R and in a general metric spaces. Cauchy’s criterion of convergence of sequences play an important role in R. Every convergent sequence is Cauchy and bounded, but the converse is not true in a metric space. R is a complete metric space. Q is not.

4. They study the Cantors Intersection Theorem which gives a necessary and sufficient condition for completeness of a metric space. Continuity and Uniform continuity of functions between two metric spaces are studied essentially in analogous ways as is done for the real valued functions on R. Open cover and compactness are introduced and certain theorems follow from these concepts. Idea of Connectedness is also given with examples from R

5. By studying the Contraction theorem and Banach Fixed point Theorem students will be able to understand the iteration methods for solving system of linear algebraic equations and yield sufficient conditions for convergence and error bounds.

6. After successful completion of the course, students will be able to understand the concept of stereographic projection, different kinds of complex valued functions and their limit, continuity , differentiability and analyticity. They will know a new transformation namely Mobius Transformation, its different properties and applications. They will also understand Cauchy theorem and Cauchy integral formulas and application of these to evaluate complex contour integrals. At last students will know about complete concept of Power series of complex variable.

Learning Outcomes of Core Course-14

1. Numerical analysis is one of part of mathematics. To deal with a physical problem one often tries to construct a mathematical model. These models in general lead to a differential equation or difference equation which cannot be solved analytically in very few situations one can get analytic solution . Therefore one has to adopt approximate methods or numerical methods .these methods are based on series expansions or they may be purely numerical leading to the estimation of the unknown at specific points in its interval of definition by simple arithmetic means.

2. Approximation Theory and Numerical Analysis are closely related areas of mathematics. Approximation Theory lies in the crossroads of pure and applied mathematics. It includes a wide spectrum of areas ranging from abstract problems in real, complex, and functional analysis to direct applications in engineering and industry. Therefore, Approximation Theory employs a great variety of methods, which originate in analysis, operator theory, harmonic analysis, quantum calculus, algorithms, probability theory, and further areas of mathematics.

3. Mathematical problems arising from scientific applications present a wide variety of difficulties that prevent us from solving them exactly. This has led to an equally wide variety of techniques for computing approximations to quantities occurring in such problems in order to obtain approximate solutions.

4. Determining the condition, or sensitivity, of a problem is an important task in the error analysis of an algorithm designed to solve the problem, but it does not provide sufficient information to determine whether an algorithm will yield an accurate approximate solution.

5. Is it always reasonable to assume that any approximate solution is the exact solution to a nearby problem? Unfortunately, it is not. It is possible that an algorithm that yields an accurate approximation for given data may be unreasonably sensitive to perturbations in that data. This leads to the concept of a stable algorithm: an algorithm applied to a given problem with given data x is said to be stable if it computes an approximate solution that is the exact solution to the same problem with data xˆ, where xˆ is a small perturbation of x.

6. It can be shown that if a problem is well-conditioned, and if we have a stable algorithm for solving it, then the computed solution can be considered accurate, in the sense that the relative error in the computed solution is small. On the other hand, a stable algorithm applied to an ill-conditioned problem cannot be expected to produce an accurate solution.

7. The calculus of finite differences is the study of changes in the dependent variable y=f(x) with respect to the finite changes in the independent variable in finite difference we study many operators such as forward difference operators, backward difference operators, central difference operators, shift operators, averaging operator etc. We also study the interpolations for equal interval and unequal intervals .interpolation is also defined as the technique of estimating the function for any intermediate value of interpolations is the art of reading between the lines of a table. Interpolation can be think of a technique of achieving the most likely estimate of a certain quantity under certain specific assumptions.

8. Central difference interpolation formulae are used for interpolating the functional value near the middle of a given set of data.

9. Just like differential equations, the difference equations play a important role in dealing with the problems of economics, social and other science, especially in mathematical models corresponding to a given physical problem. Hence it is essential to study the difference equations. Numerical differentiation is the process of evaluating the derivative of a function at a point when the exact form of function is not .For this one can obtain the suitable polynomial which is the best fit for the given data by the means of suitable interpolation formula and then known but a set of values of that function is known.

10. The problem of numerical integration is solved by approximating the integrand by a polynomial with the help of an interpolation formula and then integrating the expansion between the desired limits.

11. Methods for finding the numerical solution of first order differential equations having numerical coefficient with given initial conditions to any desired degree of accuracy. The solution is obtained step by step through a series of equal intervals in the independent variable.

12. Solution of algebraic and transcendental equations is also possible in numerical technique by bisection method, Regula Falsi method, Newton Raphson method, Iteration method etc.

Learning Outcomes of DSE-A2 ( DIFFERENTIAL GEOMETRY)

1. Gauss, Riemann and Christoffel contributed largely to the development of Differential Geometry where they introduced the concept of Tensors. The chief aim of Tensor calculus is the investigation of relations which remain valid when we change from one coordinate system to any other coordinate system. It is invaluable in its application to Differential Geometry and most branches of theoretical Physics. Metric tensor, Riemannian space and Einstein space is studied.

2. The students learn about the space curves. curvature, torsion and Serret-Frenet formula. Osculating circles, and spheres. Existence of space curves. Evolutes and involutes of curves are explained in detail. First and second Fundamental forms. Principal and Gaussian curvatures. Lines of curvature, Euler’s theorem. Rodrigue’s formula, Conjugate and asymptotic lines are studied.

3. Idea of Developables on plane and Surfaces is given. Geodesics equations. Nature of geodesics on a surface of revolution. Clairaut’s theorem, Normal property of geodesics. Torsion of a geodesic. Geodesic curvature. Gauss-Bonnet theorem are also studied.

Learning Outcomes of DSE-B2 (Point Set Topology)

1. This course is designed to provide the students an intense foundation in fundamental concepts of point set topology.

2. After completing the course, the students will be able to understand whether a given family of subsets is a topology or not, relationship between base and subbase of a topology.

3. They can work with basic problems (proofs, construction of examples, counter-examples, or argue that a claim is false) in the topology of R, topology of Metric Spaces, Hausdorff spaces.

4. They will be surprised to see how various basic concepts are generalised from metric spaces to topological spaces . Further they shall become familiar with seperability, completeness, connectedness, compactness of metric space and topological space .

Learning Outcomes of DSE-B (ADVANCED CALCULUS) for GENERAL

Upon completion of this course, the students will be able to

1. Understand the Concept of Point-wise and Uniform convergence of sequence of functions and series of functions with special reference of Power Series.

2. Determine the Radius of convergence of Power Series. Statement of properties of continuity of sum function power series. Term by term integration and Term by term differentiation of Power Series. Statements of Abel’s Theorems on Power Series.

3. Understand the Dirichlet’s conditions of convergence and statement of the theorem on convergence of Fourier Sine and Cosine series.

4. Understand the Laplace Transform and its application to ODE and initial value problems.