Science, Maths & Technology

### How you can be more secure than the President

To keep your digital world protected, you don't need to build a border wall - just follow some good security housekeeping advice

Science, Maths & Technology

### Safe computing

Tracking the security threats in the connected world - and how you can protect yourself from them

Science, Maths & Technology

### Modelling events in time

This free course develops ideas about probability and random processes. Sections 1 and 2 introduce the fundamental ideas of random processes through a series of examples. Section 3 describes a model that is appropriate for events occurring ‘at random’ in such a way that their rate of occurrence remains constant. Section 4 derives the main results from Section 3. Section 5 introduces the multivariate Poisson process in which each event may be just one of several different types of event. Section 6 introduces the non-homogeneous Poisson process in which events occur at a rate that varies with time.

Science, Maths & Technology

### Modelling events in time

This free course develops ideas about probability and random processes. Sections 1 and 2 introduce the fundamental ideas of random processes through a series of examples. Section 3 describes a model that is appropriate for events occurring ‘at random’ in such a way that their rate of occurrence remains constant. Section 4 derives the main results from Section 3. Section 5 introduces the multivariate Poisson process in which each event may be just one of several different types of event. Section 6 introduces the non-homogeneous Poisson process in which events occur at a rate that varies with time.

Science, Maths & Technology

### Kinematics of fluids

This free course examines the basic kinematics of two-dimensional fluid flows. Section 1 introduces the diﬀerential equations for pathlines and streamlines. Section 2 introduces a scalar ﬁeld, called the stream function, which for an incompressible ﬂuid provides an alternative method of modelling the ﬂow and ﬁnding the streamlines. Sections 2 and 3 derive the stream functions for several simple two-dimensional ﬂow types (the uniform ﬂow, source, doublet and vortex), and suitable combinations of these are used to model more complicated ﬂows. Section 4 introduces the idea of diﬀerentiation following the motion, and Euler’s equation is developed in Section 5.

Science, Maths & Technology

### Kinematics of fluids

This free course examines the basic kinematics of two-dimensional fluid flows. Section 1 introduces the diﬀerential equations for pathlines and streamlines. Section 2 introduces a scalar ﬁeld, called the stream function, which for an incompressible ﬂuid provides an alternative method of modelling the ﬂow and ﬁnding the streamlines. Sections 2 and 3 derive the stream functions for several simple two-dimensional ﬂow types (the uniform ﬂow, source, doublet and vortex), and suitable combinations of these are used to model more complicated ﬂows. Section 4 introduces the idea of diﬀerentiation following the motion, and Euler’s equation is developed in Section 5.

Science, Maths & Technology

### Linear programming – the basic ideas

This free course examines the formulation and solution of small linear programming problems. Section 1 deals with the formulation of linear programming models, describing how mathematical models of suitable real-world problems can be constructed. Section 2 looks at graphical representations of two-dimensional models, considers some theoretical implications and examines the graphical solution of such models. Section 3 introduces the simplex method for solving linear programming models and Section 4 uses matrix notation to formalize the simplex method.

Science, Maths & Technology

### Linear programming – the basic ideas

This free course examines the formulation and solution of small linear programming problems. Section 1 deals with the formulation of linear programming models, describing how mathematical models of suitable real-world problems can be constructed. Section 2 looks at graphical representations of two-dimensional models, considers some theoretical implications and examines the graphical solution of such models. Section 3 introduces the simplex method for solving linear programming models and Section 4 uses matrix notation to formalize the simplex method.

Science, Maths & Technology

### Introduction to the calculus of variations

This free course concerns the calculus of variations. Section 1 introduces some key ingredients by solving a seemingly simple problem – finding the shortest distance between two points in a plane. The section also introduces the notions of a functional and of a stationary path. Section 2 describes basic problems that can be formulated in terms of functionals. Section 3 looks at partial and total derivatives. Section 4 contains a derivation of the Euler-Lagrange equation. In Section 5 the Euler-Lagrange equation is used to solve some of the earlier problems, as well as one arising from a new topic, Fermat’s principle.

Science, Maths & Technology

### Introduction to the calculus of variations

This free course concerns the calculus of variations. Section 1 introduces some key ingredients by solving a seemingly simple problem – finding the shortest distance between two points in a plane. The section also introduces the notions of a functional and of a stationary path. Section 2 describes basic problems that can be formulated in terms of functionals. Section 3 looks at partial and total derivatives. Section 4 contains a derivation of the Euler-Lagrange equation. In Section 5 the Euler-Lagrange equation is used to solve some of the earlier problems, as well as one arising from a new topic, Fermat’s principle.

Science, Maths & Technology

### Point estimation

This free course looks at point estimation, that is, the estimation of the value of the parameter of a statistical model by a single number, a point estimate for the parameter. Section 1 develops some aspects of maximum likelihood estimation. In particular, you will find out how to obtain the maximum likelihood estimator of an unknown parameter, using calculus. You will need to do lots of differentiation in this section. Section 2 introduces a number of important properties of point estimation.

Science, Maths & Technology

### Point estimation

This free course looks at point estimation, that is, the estimation of the value of the parameter of a statistical model by a single number, a point estimate for the parameter. Section 1 develops some aspects of maximum likelihood estimation. In particular, you will find out how to obtain the maximum likelihood estimator of an unknown parameter, using calculus. You will need to do lots of differentiation in this section. Section 2 introduces a number of important properties of point estimation.