Topics of abstract algebra II:
Upon completion of this course, the student should be able to
1. Write mathematical proofs and reason abstractly in exploring properties of groups and rings.
2. Define, construct examples of, and explore properties of groups, including symmetry groups, permutation groups and cyclic groups.
3. Determine subgroups and factor groups of finite groups.
4. Determine, use and apply homomorphism's between groups.
5. Understand definitions, examples, and theorems pertaining to groups.
These outcomes are intended in abstract algebra I.
6. Rings and Ideals.
7. Factor rings
8. Ring homomorphisms.
9. Integral domains
10. Polynomial rings and factorization.
11. Divisiblity in integral domains.
12. Factorization domains. Unique factorization domains including euclidean domains and principal ideal domains
Elementary Number Theory is concerned with exploring properties of integers. The course requires some knowledge in foundations of mathematics.
Many of the problems discussed can be adapted for use by elementary, middle, or secondary school teachers. In recent times, number-theoretical ideas have found important applications, perhaps most notably in the area of computer and network security, and we will mention some of these applications. The course will also emphasize reading and writing proofs; consequently, it will enrich the student’s analytical and problem solving skills.
The course provides an attempt to Use Mathematica Package to help in reinforcing the teaching and learning of some mathematical topics. It also, gives each student his/her enough time and enough training to go through the course according to his/her ability.
Measures of data centers and data variability
Some Vital Statistics Correlation Coefficient
Distribution (frequency) tables; bar graphs and histograms; distribution (frequency )curves, measures of centrality and variability for raw and for grouped data, percentiles; interquartile range, Empirical rule, Chebychev's inequality, coding data and its effect on statistical measures
To identify and solve ordinary differential equation
using different strategies .
This course is a continuation for Calculus I. The course begins with Techniques of integration and it is given as: integration by parts, trigonometric integrals, trigonometric substitutions, partial fractions, rationalizations, half-angle substitution , and improper integrals. Then some applications of definite integrals are given such as: areas between two curves, volumes by washers and cylindrical shells, arc length, and area of a surface. The concept of infinite series is introduced, then tests of convergence are given. Power series, Maclaurin and Taylor expansions are given. Then differentiation and integration of power series are given. Finally, polar coordinates, graphs in polar coordinates, and areas in polar coordinates are given.
1- Classification of PDEs.
2- Some physical models (Heat, Wave and Laplece equations).
3- Second order Linear PDEs. With constant coefficients.
4- Separation of variables.
5- Orthogonal functions.
6- Sturm-Liouville BVP.
1) Revision of some probability distributions:
2) Queuing Theory:
Description of queuing models, the Poisson process, Birth-Death processes, single server queue and some modifications.
3) Reliability Theory:
Failure laws and failure rate, reliability of series and parallel systems.
4) Quality control:
Control charts, acceptance sampling, single sampling plan, other sampling plans.
5) Information theory and Coding:
Uncertainty, information measures and entropies, the first coding theorem discrete channels and the second coding theorem.
1. The completeness property of R.
2. The Archimedean principle in R.
3. Limit of a sequence.
4. Convergent sequences.
5. Monotone and bounded sequences.
6. Cauchy sequences.
7. Subsequences and limit points. Liminf, and limsup.
8. Bolzano-Weierstrass Theorem.
9. Open sets, closed sets, bounded sets and compact sets in R.
10. Limits of real valued functions.
11. Definition of limits by neighborhoods.
12. Definition of limits by sequences.
13. Continuous functions on R.
14. Sequence definition and neighborhood definition of continuity.
15. Boundedness of continous functions on compact intervals.
16. The extreme value theorem.
17. The intermediate value theorem.
18. Uniformly continuous functions.
19. The sequential criterion for uniform continuity.
20. The derivative of functions.
21. Roles Theorem.
22. Mean value theorem.
23. Generalized Mean value theorem.
24. Taylor Theorem with remainder.
25. L’Hospital,s rule.
1. To present Basic properties of complex numbers.
2. To introduce Basic properties of Analytic function.
3. To present Elementary complex functions.
4. To evaluate complex integral.
5. To introduce Laurent series.
6. To present Residue theorem.
7. To use complex integration methods to evaluate Real Improper integrals.
8. To present Basic properties of mappings and conformal mappings.