To fully understand mathematics, the essential rules of logic for the formulation of mathematical proofs have to be studied. The various types of mathematical proofs are discussed, mainly using examples in elementary set theory and number systems.
In this course, students are introduced to the basic operations that are performed on sets. In so doing the concepts of relations, functions and binary operations are examined. Next we examine natural numbers and the derived principle of mathematical induction. We discuss permutations, combinations and sequences that are defined recursively with respect to natural numbers. The properties of the integers such as divisibility, greatest common divisor and Euclidean algorithm are examined. We study the field axioms in relation to rational numbers. We solve linear and non-linear inequalities involving the real numbers.
In a broad sense, calculus is the mathematical study of change consisting of differential calculus (concerning rates of change and slopes of curves) and integral calculus (concerning accumulation of quantities and the area under/between curves).
The course continues the study of the fundamental concepts of the differential and integral calculus of a single variable. The used approach is intuitive and informal but nevertheless prepares students to understand the rigorous proofs at a subsequent level. While there is a considerable overlap with the ideas encountered in Calculus A, a greater emphasis is put on the fundamental ideas involved in differentiation and integration. The main results are derived and illustrated although often heuristic proofs are used.
This course will treat the limits, continuity and differentiability of a function of a single variable from a more rigorous point of view. Double integrals are introduced and methods for their evaluation are considered.
This course exposes students to rigorous mathematical definitions
of limits of sequences of numbers and functions, classical
results about continuity and series of numbers and their proofs.
A major emphasis is placed on the proper use of definitions for
the rigorous proof of theorems. The following topics will be
covered: The real number system, topological properties of real
numbers, sequences, continuity and uniform continuity.
This course is an introduction to complex analysis, or the theory of the analytic functions of a complex variable. Such functions, beautiful on their own, are immediately useful in Physics, Engineering, and Signal Processing. However, the concepts introduced in complex analysis are nowadays also ubiquitous in number theory and combinatorics.
Students will calculate line integrals, and the techniques they will learn in this course will help them get past many of the seeming “dead ends” they ran up against in calculus. Indeed, most of the definite integrals they will learn to evaluate cannot be solved except through techniques from complex analysis.
Numerical analysis is the study of the computational methods used to solve problems involving continuous variables. It uses the ability of modern computers to perform billions of arithmetic operations per second, to find approximate solutions to otherwise intractable problems.
For example, students will survey some of the basic problems and methods needed to simulate the solutions of ordinary differential equations. Making sure that such a simulation is accurate but also can be performed fast is one of the main tasks in numerical analysis, e.g., when trying to predict the path of a tropical storm.
In this course, students will write programs to solve equations, approximate functions, integrate functions, and solve initial value problems. Thus this course will concentrate on the analytical side of numerical analysis, leaving direct and iterative methods in linear algebra to the course Linear Algebra 2.
This course is a survey of the major ideas of inference,
experimental design and statistical methods. The course may be
viewed as consisting of three closely connected parts. In the
first section, students are introduced to the basics of the
statistical packages Minitab / R and their use in descriptive
statistics. Emphasis is placed on the use of real data and both
summary statistical measures and graphical descriptive devices
for continuous and discrete data are discussed.
In the second section, we discuss the frequentist theory of inference, including point estimation, confidence intervals and hypothesis testing. Section three is devoted to various statistical methods. The major ones are regression models and the use of ANOVA in designed experiments. Several of the important basic designs are discussed. We also discuss methods for the analysis of discrete data, such as in contingency tables, and non-parametric procedures.
Ordinary differential equations (ODEs) arise in many contexts of mathematics and science (social as well as natural). Mathematical descriptions of change use differentials and derivatives. Various differentials, derivatives, and functions become related to each other via equations, and thus a differential equation is a result that describes dynamically changing phenomena, evolution, and variation. Often, quantities are defined as the rate of change of other quantities (for example, derivatives of displacement with respect to time), or gradients of quantities, which is how they enter differential equations.
- Composition (language), in literature and rhetoric,
producing a work in spoken tradition and written discourse, to
include visuals and digital space
- Composition (music), an original piece of music and its creation
- Composition (visual arts), the plan, placement or arrangement of the elements of art in a work
- Composition (dance), practice and teaching of choreography
- Function composition (mathematics), an operation that takes functions and gives a single function as the result
- Composition of relations (mathematics), an operation that takes relations and gives a single relation as the result
- Composition (combinatorics), a way of writing a positive integer as a sum of positive integers
- Composition algebra (mathematics), an algebra over a field with composing norm
- Binary function or law of composition