Page 142 - FULL FINAL HANDBOOK 20232024

Faculty of Science Handbook, Academic Session 2023/2024

functions. Primitive roots. Assessment:
Continuous Assessment: 40%
Assessment: Final Examination: 60%
Continuous Assessment: 40%
Final Examination: 60%
SIM3008 GROUP THEORY
SIM3004 ADVANCED LINEAR ALGEBRA The three isomorphism theorems. Cyclic groups. Direct
product of groups. Introduction to the three Sylow’s
Inner product spaces, the Cauchy-Schwarz inequality, Theorems. Classification of groups up to order 8. Finitely
the Gram-Schmidt orthogonalization process, generated abelian groups. Permutation groups.
orthogonal complements, orthogonal projections.
Adjoint operators, normal operators, self-adjoint Assessment:
operators, unitary operators, positive definite operators. Continuous Assessment: 40%
Bilinear forms, congruence, rank, Sylvester’s law of Final Examination: 60%
inertia, classification of symmetric bilinear forms, real
quadratic forms. The Schur triangularization theorem,
the primary decomposition theorem, the Jordan
canonical form. SIM3009 DIFFERENTIAL GEOMETRY
Assessment: Vector algebra on Euclidean space. Lines and planes.
Continuous Assessment: 40% Change of coordinates. Differential geometry of curves.
Final Examination: 60% Frenet Equations. Local theory of surfaces in Euclidean
space. First and second fundamental forms. Gaussian
curvatures and mean curvatures. Geodesics. Gauss-
Bonnet Theorem.
SIM3005 MATRIX THEORY
Assessment:
Rank and nullity of matrices, Sylvester’s law inequality, Continuous Assessment: 40%
the Frobenius inner product, the Gram-Schmidt process, Final Examination: 60%
the continuity argument. Rank and full rank
decompositions, LU and QR decompositions, spectral
decompositions, singular value decompositions, polar
decompositions, Cholesky decompositions. Generalized SIM3010 TOPOLOGY
inverses, Moore-Penrose inverses, the best
approximation solutions, least squares solutions. Topological Spaces. Continuity, connectedness and
Kronecker products of matrices, permutations, matrix compactness. Separation axioms and countability.
functions of Kronecker products, Schmidt rank and Metric spaces. Product spaces.
decompositions.
Assessment:
Assessment: Continuous Assessment: 40%
Continuous Assessment: 40% Final Examination: 60%
Final Examination: 60%
SIM3011 COMPLEX ANALYSIS
SIM3006 ALGEBRA II
Infinite series expansions: convergence and divergence
This is a second course in abstract algebra and will cover and region of convergence. Taylor and Laurent
more advanced topics on groups and rings. Topics on theorems. Classification of isolated singularities. Zeroes
groups include the isomorphism theorems, various and poles. Calculus of residues; calculation of definite
subgroups such as the centre and commutator integrals. Residue Theory. Evaluation of certain
subgroups, finitely generated abelian groups, integrals. Argument principle, Rouche’s Theorem.
automorphism groups, permutation groups, and p- Maximum modulus principle. Conformal mappings.
groups.
Assessment:
For rings, the focus is on commutative rings. Topics on Continuous Assessment: 40%
rings include the maximal and prime ideals, polynomial Final Examination: 60%
rings, irreducible polynomials and the unique
factorization theorem.
Assessment: SIM3012 REAL ANALYSIS
Continuous Assessment: 40%
Final Examination: 60% Infinite series, convergence. Tests of convergence.
Absolute and conditional convergence. Rearrangement
of series. Pointwise and uniform convergence.
Properties of uniform convergence. Superior limit and
SIM3007 RING THEORY inferior limit. Power series, radius of convergence.
Taylor series. Riemann integral. Integrable functions.
This course includes both commutative and non- Properties of the Riemann integral. Integration in relation
commutative rings. Topics that will be discussed include to differentiation. Differentiation of integrals. Improper
subrings, subfields and ideals; internal direct sum and integrals. Sequences and series of functions.
external direct product; nil ideals, nilpotent ideals;
modules and submodules; prime ideals, maximal ideals; Assessment:
prime radical and Jacobson radical; semiprime and Continuous Assessment: 40%
semiprimitive rings; rings with chain conditions; group Final Examination: 60%
rings.
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