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B.Sc.PART-III EXAM, 2011 3 4 SANT GADGE BABA AMRAVATI UNIVERSITY PROSPECTUS Unit-II : Ring Thoery : Ring Homomorphism

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B.Sc.PART-III EXAM, 2011 3 4 SANT GADGE BABA AMRAVATI UNIVERSITY PROSPECTUS Unit-II : Ring Thoery : Ring Homomorphism B.Sc. Final Prospectus No.2011123 PUBLISHED BY Exam., 2011 Dineshkumar Joshi Registrar Sant Gadge Baba ∫…∆i… M……b˜M…‰ §……§…… +®…Æ˙…¥…i…“ ¥…t…{…“`ˆ Amravati University Amravati - 444602 SANT GADGE BABA AMRAVATI UNIVERSITY ¥…Y……x… ¥…t…∂……J…… (FACULTY OF SCIENCE) +¶™……∫…GÚ ®…EÚ… ¥…Y……x… ∫x……i…EÚ +xi™… {…Æ˙“I……, 2011 ( j…¥…π…‘™… +¶™……∫…GÚ®…) PROSPECTUS OF B.Sc. Final Examination, 2011 (Three Year Degree Course) 2010 Visit us at www.sgbau.ac.in "™…… +¶™……∫…GÚ ®…E‰Úi…“±… (Prospectus) EÚ…‰h…i……Ω˛“ ¶……M… ∫…∆i… M……b˜M…‰ §……§…… +®…Æ˙…¥…i…“ ¥…t…{…“`ˆ…S™…… {…⁄¥……«x…÷®…i…“ ∂…¥……™… EÚ…‰h……∫…Ω˛“ {…÷x…®…÷« p˘i… EÚÆ˙i…… ™…‰h……Æ˙ x……Ω˛“.' (Price-Rs. 13/-) ”No part of this prospectus can be reprinted or published without specific permission of Sant Gadge Baba Amravati University.” SANT GADGE BABA AMRAVATI UNIVERSITY Ordinance No. 19 : Admission of Candidates to Degrees. SPECIAL NOTE FOR INFORMATION OF THE STUDENTS Ordinance No. 109 : Recording of a change of name of a University (1) Notwithstanding anything to the contrary, it is notified for general student in the records of the University. information and guidance of all concerned that a person, who has passed Ordinance No. 138 : For improvement of Division/Grade. the qualifying examination and is eligible for admission only to the Ordinance No.19/2001 : An Ordinance for Central Assessment Programme, corresponding next higher examination as an ex-student or an external Scheme of Evaluation and Moderation of candidate, shall be examined in accordance with the syllabus of such next answerbooks and preparation of results of the higher examination in force at the time of such examination in such subjects examinations, conducted by the University, papers or combination of papers in which students from University Ordinance 2001. Departments or Colleges are to be examined by the University. (2) Be it known to all the students desirous to take examination/s for Dineshkumar Joshi Registrar which this prospectus has been prescribed should, if found necessary for Sant Gadge Baba any other information regarding examinations etc., refer the University Amravati University. Ordinance Booklet the various conditions/provisions pertaining to examination as prescribed in the following Ordinances. PATTERN OF QUESTION PAPER ON THE UNIT SYSTEM Ordinance No. 1 : Enrolment of Students. Ordinance No. 2 : Admission of Students The pattern of question paper as per unit system will be boradly based Ordinance No. 4 : National cadet corps on the following pattern. Ordinance No. 6 : Examinations in General (relevent extracts) (1) Syllabus has been divided into units equal to the number of question to Ordinance No. 18/2001 : An Ordinance to provide grace marks for be answered in the paper. On each unit there will be a question either a long answer type or a short answer type. passing in a Head of passing and Inprovement of Division (Higher Class) and getting (2) Number of question will be in accordance with the unit prescribed in the syllabi for each paper i.e. there will be one question on each unit. Distinction in the subject and condonation of (3) For every question long answer type or short answer type there will be defficiency of marks in a subject in all the an alternative choice from the same unit. However, there will be no faculties prescribed by the Statute NO.18, internal choice in a question. Ordinance 2001. (4) Division of marks between long answer and short answer type question Ordinance No. 9 : Conduct of Examinations (relevent extracts) will be in the ratio of 40 and 60. Ordinance No. 10 : Providing for Exemptions and Compartments (5) Each short answer type question shall Contain 4 to 8 short sub question with no internal choice. 2 SANT GADGE BABA AMRAVAI UNIVERSITY PROSPECTUS SYLLABUS FOR References : 1. MATHEMATICS 1. R.R.Goldberg, Methods of Real Analysis, Oxford, IBH publishing Co., New (W.E.F. THE SESSION 2005-06) Delhi, 1970. PAPER-VII 2. T.M.Karade, J.N.Salunke, K.S.Adhau, M.S.Bendre : Lectures on Analysis; ANALYSIS Sonu-Nilu Publication Nagpur. Unit-I : Real Analysis : Series of arbitary terms, convergence, divergence 3. Walter Rudin : Principles of Mathematical Analysis. International students and oscillation. Abel’s and Dirichlet’s tests. Multiplication of edition (Third Edition) series. Double series. 4. T.M.Apostol, Mathematical Analysis, Narosa Publishing House, New Delhi, Partial derivative and differentiability of real-valued functions of 1985. two variables. Schwarz and Young’s theorem. Implicit function 5. S.Lang, Undergraduate Analysis, Springer-Verlag, New York, 1983. theorem. 6. D.Somasundaram & B.Choudhari, A first Course in Mathematical Analysis, Fourier series. Fourier expansion of piecewise monotonic Narosa Publishing House, New Delhi, 1997. functions. 7. Shanti Narayan, A Course of Mathematical Analysis, S.Chand & Co. New Unit-II : Riemann integral. Integrability of continuous and monotonic Delhi. functions. The fundamental theorem of integral calculus. Mean 8. P.K.Jain & S.K.Kaushik, An introduction to Real Analysis, S.Chand & Co. value theorems of integral calculus. New Delhi, 2000. Improper integrals and their convergence, comparison tests, 9. R.V.Churchill and J.W.Brown, Complex Variables and Applications, 5th Abel’s and Dirichelet’s tests. Edition, McGraw Hill, New York, 1990. Unit-III : Complex Analysis : Stereographic projection. 10. Mark J. Ablowitz & A.S.Fokas, Complex Variables : Introduction and Application, Cambridge University Press, South Asian Edition, 1998. Continuity and differentiability of complex functions. Analytic functions. Cauchy-Riemann equations. Harmonic functions. 11. Shanti Narayan, Theory of Functions of a Complex Variable, S.Chand & Elementary functions. Mapping by elementary functions. Co. New Delhi. 12. E.T.Copson, Metric Spaces, Cambridge University Press, 1968. Mobius transformations. Fixed points. Cross ratio. Inverse points and critical points. Conformal mappings. 13. P.K.Jain & K.Ahmed, Metric Spaces, Narosa Publishing House, New Delhi, 1996. Unit-IV : Metric Spaces : Countable and uncountable sets. Definition & examples of metric spaces. Neighbourhoods. Limit points, interior 14. G.F.Simmons, Introduction to Topology and Modern Analysis, McGraw points. Open and closed sets. Closure, interior & boundary Hill, 1963. points. Sub-space of a metric space. Cauchy sequences. ***** Completeness. Cantor’s intersection theorem. Baire Category PAPER-VIII theorem. ABSTRACT ALGEBRA Unit-V : Compactness. Connectedness. Limit of functions. Continuous Unit-I : Group-Automorphisms, inner automorphism. Automorphism functions. Uniform continuous functions. Continuity and groups and their computations. Conjugacy relation. Normaliser. compactness. Continuity and connectedness. Counting principle and the class equations of a finite group. Center for group of prime-order. B.Sc.PART-III EXAM, 2011 3 4 SANT GADGE BABA AMRAVATI UNIVERSITY PROSPECTUS Unit-II : Ring thoery : Ring homomorphism. Ideals and Quotient Rings. 8) S.Kumaresan. Linear Algebra. A Geometric Approach. Prentice-Hall of India, Field of Quotients of an Integral Domain. Euclidean Rings. 2000. Polynomial Rings. Polynomials over the Rational Field. The 9) Vivek Sahai and Vikas Bist, Algebra, Norosa Publishing House, 1997. Eisenstein Criterion. 10) I.S.Luther and I.B.S.Passi, Algebra, Vol.-I-Groups, Vol.-II-Rings. Narosa Unit-III : Vector Spaces : Definition and examples of vector spaces. Publishing House (Vol.I-1996, Vol.-II - 1999) Subspaces. Sum and direct sum of subspaces. Linear span. Linear dependence, independence and their basic properties. Basis. 11) D.S.Malik, J.N.Mordeson, and M.K.Sen, Fundamentals of Abstract Algebra, Finite dimensional vector spaces. Existence theorem for bases. McGraw-Hill International Edition, 1997. Invariance of the number of elements of a basis set. Dimension. 12) T.M.Karade, J.N.Salunke, K.S.Adhau, M.S.Bendre : Lectures on Unit-IV : Linear Transformations : Linear transformations and their “ABSTRACT ALGEBRA” Sonu-Nilu Publication.2nd Publication. representation as matrices. The Algebra of linear transformations. ***** The rank nullity theorem. Change of basis. Dual space. Bidual PAPER-IX space and natural isomorphism. Adjoint of a linear transformation. MATHEMATICAL MODELLING Eigenvalues and eigenvectors of a linear transformation. Unit-I : The Process of applied mathematics. Setting of First-order Unit-V : Inner Product Spaces : Inner product spaces - Cauchy-Schwarz differential equations - Qualitative Solutions Sketching. inquality. Orthogonal vectors. Orthogonal Complements. Unit-II : Difference and Differential Equation growth models - single- Orthonormal sets and bases. Bessel’s inequality for finite species population models. Population growth- An age structure dimensional spaces. Gram-Schmidt Orthogonalization process. model. The spread of Technological innovation. Modules, submodules. Quotient modules. Homomorphism and Unit-III : Higher order linear models - A model for the detection of diabetes. isomorphism theorems. Combat modes. Traffic models - Car-following models. Equillibrium References : speed distributions. 1) I.N.Herstein, Topics in Algebra, Wiley Eastern Ltd. New Delhi, 1975. Unit-IV : Non-linear population growth models. Prey-Predator models. Epidemic growth models. Models from political Science - 2) N.Jacobson, Basic Algebra, Vols. I & II W.H.Freeman. 1980 (also published Proportional representation - cumulative voting, comparison by Hindustan Publishing Company) voting. 3) Shanti Narayan, A Text Book of Modern Abstract Algebra. S.Chand & Co.
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