Applied Mathematics: Operations Research Topics

 

- Operations Research Modeling approach

- Linear Programming

- The Simplex Method

- Other Algorithms for Linear Programming

*Dual Simplex Method

*Parametric Linear Programming

*The Upper Bound Technique

*Interior-Point Algorithm

*Goal Programming

- Duality Theory and Sensitivity Analysis

- Transportation and Assignment Problems

- Network Optimization Models

- Dynamic Programming

- Integer Programming

- Non-linear Programming

- Heuristic Programming

- Game Theory

- Decision Analysis

- Queuing Theory

- Inventory Theory

- Simulation Modeling

Applied Mathematics: Numerical Analysis Topics

 

- Errors in Numerical Computations

- Solutions of Equations in One Variable

- Solving Linear Systems of Equations

*Direct Methods

*Iterative Methods

- Solutions of Non-leanear Equations

- Interpolation

- Approximation Theory

- Approximating Eigenvalues

- Numerical Differentiation

- Numerical Integration

- Numerical methods to Ordinary Differential Equations

*Initial value problems

*Systems of Ordinary Differential Equations

*Boundary value problems

- Numerical methods to Partial Differential Equations

- Optimization

Mathematics: Algebra Topics

- Algebra Essentials

*Numbers and Arithmetics

*Mathematical Induction

*sequences and series

*Exponents and Notations

*Radicals and Rational Exponents

*Polynomails

*The binomial Theorem

*Rational Expressions

*Permutations and combinations

- Equations and Inequalities

*Linear Equations

*Complex numbers

*Quadratic Equations

*Other Types of Equations

*Linear Inequalities and Absolute Value Inequalities

- Functions

*Function Notation

*Domain and Range

*Composition of Functions

*Transformation of Functions

*Absolute Value Functions

*Inverses and radical Functions

*Exponential function

*Linear Functions

*Polynomail Functions

*Rational Functions

*Quadratic Functions

*Exponential Functions 

*Logarithmic Functions

*Periodic Functions

*Trigonometric Functions

- System of Equations and Inequalities

*System of Linear Equations

*System of NonLinear Equations and Inequalities

*Partial Fraction

*Matrices

*Determinants

*Solving System of Equations

Mathematics: Linear Algebra Topics

- Linear Algebra

*Systems of Linear equations

*Matrices

*Determinants

*Vector Spaces

*Linear maps between spaces

*Inner product spaces

*Linear Transformations 

*Eigenvalues and Eigenvectors

*Linear and quadratic forms

Computer Science: Data Structures Topics

- Software Engineering Principles

*The Software Process

*Program Design

*Verification of Software Correctness

*C++ Programming Language

- Data Design and Implementation

*Different Views of data

*Data Abstraction

*Object-Oriented Programming

*Constructs for Program Verification

- Abstract Data Type (ADTs)

*Unsorted List and Sorted List

*Stack and Queue

*Lists

- Linked Structure

- Linked Lists

- Programming with Recursion

- Binary Search Trees

- Heaps and Graphs

Change in culture of Software development

Waterfall

• First SDLC Model to be used widely in Software development
• Each phase must be completed before the next phase can begin

Agile

• break the product into small incremental builds. These builds are provided in iterations.
• Every iteration involves cross functional teams working simultaneously on various areas

DevOps

A combination of cultural philosophies, practices, and tools that increases an organization’s ability to deliver applications and services at high velocity.

Software Engineering Principles

Software engineering

The discipline devoted to the design, production and maintenance of computer programs that are developed on time and within cost estimates using tools that help to manage the size and complexing of the resulting software product.

Software process

A standard integrated set of software engineering tools and techniques used on a project or by organization.
Software process activities:

• Problem analysis
• Requirements elicitation
• Requirements definition
• High and low-level design
• Implementation of the design
• Testing and verification
• Delivery
• Operation
• Maintenance

Software engineering tools

• Hardware
• Software
• Idea ware

Software specification

A detailed description of the function, inputs, output and special requirements of a software product, it provides the information needed to design and implement the program
Goals of quality software:

• It works
• It can be modified
• It is reusable
• It is completed on time

Program design tools

• Abstraction
• Information hiding
• Stepwise refinement: Top down – Bottom up – Functional decomposition – round- trip gestalt design
• Visual tools: UML – Class -Responsibilities and collaboration cards (CRC).

Design approaches

• Top down design
• Object oriented design (OOP)

Verification of software correctness

• Testing
• Debugging
• Acceptance test
• Regression testing
• Program verification (fulfills its specification)
• Program validation (fulfills its intended purpose)

Errors

• Specification and design errors
• Compile time errors
• Run time errors

Robustness

• The ability of program to recover following an error
• The ability of a program to continue to operate within its environment.

Designing for correctness

• Assertion: logical preposition that can be true or false.
• Preconditions and post conditions.
• Design review activities: Desk checking – walk through – inspection

Exceptions

An unusual generally unpredictable event detectable by software or hardware, that requires special processing, the event may or may not be erroneous

Program Testing

• Unit testing: test a module of function by itself.
• Functional domain: the set of valid input data for a program or function.

Black-box Testing

• Testing a program or function based on the possible input values.
• Treating the code as a black box.
• Testing based on data coverage.

Clear(White) box testing

• Testing a program or function based on covering all the statements, branches or paths of the code.
• Statement coverage: every statement in the program is executed at least one.
• Branch: a code segment that is not always executed (switch)
• Path: a combination of branches that might be traversed when the program or function is executed.
• Path testing: a testing technique whereby the tester tries to execute all possible paths in the program or function.

Metric-based testing

• Testing based on measurable factors.
• Test plan: a document showing the test cases planned for a program or module, their purposes, input, expected outputs and criteria for success.
• Implementing a test plan: Running the program with the test cases listed in the test plan.
• Test driver: a program that setup the testing environment by declaring and assigning initial values to variables, then calls the sub program to be tested.

Integration Testing

• Testing performed to integrate program modules that have already been independently unit tested.
• Stub: a special function that can be used in top-down testing to stand in for a lower-level function.

References Book: C++ Plus Data Structures, Nell Dale