Galaxies, stars and planets
This free course is a general introduction to galaxies, stars and planets, including scale of the universe from the very large to the very small; orbits and gravity; the Solar System; the Sun and other stars; galaxies and the composition of astronomical objects. First published on Tue, 22 Mar 2016 as Author(s): Creator not set

1.2 Taxa and relationships

Until the mid-20th century, inferences about evolutionary relationships between species were generally based upon as wide a range of evidence as could be mustered. Evolutionary systematics is the name given to this eclectic approach, because of its explicit focus on evolutionary conclusions. The disparate nature of the evidence used (ranging from the taxonomic attributes and geographical distribution of living organisms to the stratigraphical distribution of fossils) meant that there w
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7.1 Introduction

So far, we have concentrated on the electronic and spatial structures of chemical substances, but we have not said much about chemical reactions. Now we turn to the question of why chemical reactions happen. To remind you of the basic ideas, we shall concentrate on one particular reaction which occurs in the modern motor car.

Table 2 shows typical percentages of the main constituents of the exhaust gas that emerges from a modern car engine. The two most dangerous pollutants are carbon
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2.2 Wave packets and scattering in one dimension

Figure 6 shows the scattering of a wave packet, incident from the left, on a target represented by a potential energy function of the form

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Introduction

In this unit we shall consider two physical phenomena of fundamental importance: scattering and tunnelling. Each will be treated using both a stationary-state approach and a wave-packet approach.

We can consider two approaches to describing the state of a system in wave mechanics. In cases where the probability distributions are independent of time, a stationary-state approach can be used. In other cases, where probabilities are time-dependent and motion is r
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5 The Devonian Period

Before going any further, click on 'View document' below and read pages 76–77 from Douglas Palmer's Atlas of the Prehistoric World. Do not worry too much about all the different names of fish groups in this, the ‘Age of Fishes’.

3.2.1 Fats

Fats, also known as lipids, are important components of living tissues, and are used by the body for making cell membranes and for storing energy. Fats come in a variety of different biochemical types, which may be obtained from the diet or can be synthesised within the body. Many cells of the body can convert certain types of fat into others, but by preference, fats will be obtained from the diet, if available. The fatty acids that cannot be synthesised by the body and therefore must
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2.11 Using a calculator for statistical calculations

Table 3 shows all the values for each step in the process of calculating a standard deviation, so that you can see what the operations encapsulated by Equation 7 actually entail, but you will probably be relieved to hear that it is not usually necessary to carry out such detailed calculations. Scientific and graphics calculators (or computer sp
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2.10 The distribution of repeated measurements

As noted in the previous section, if the same quantity is measured repeatedly, the results will generally be scattered across a range of values. This is perhaps best illustrated using a real example. Table 2 shows 10 measurements of a quantity called the ‘unit cell constant’ for an industrial catalyst used in the refining of petrol; this is an important quantity which determines how well the catalyst works, and can be measured by X-ray diffraction techniques. Notice that the cell constant
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Introduction

This course will introduce you to a number of ways of representing data graphically and of summarising data numerically. You will learn the uses for pie charts, bar charts, histograms and scatterplots. You will also be introduced to various ways of summarising data and methods for assessing location and dispersion.

This OpenLearn course is an adapted extract from the Open University course Author(s): The Open University

Introduction

This course is concerned with two main topics. In Section 1, you will learn about another kind of graphical display, the boxplot. Boxplots are particularly useful for assessing quickly the location, dispersion, and symmetry or skewness of a set of data, and for making comparisons of these features in two or more data sets. The other topic, is that of dealing with data presented in tabular form. You are, no doubt, familiar with such tables: they are common in the media and in reports an
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Copyright © 2016 The Open University

Introduction

This unit provides an overview of the processes involved in developing models. It starts by explaining how to specify the purpose of the model and moves on to look at aspects involved in creating models, such as simplifying problems, choosing variables and parameters, formulating relationships and finding solutions. You will also look at interpreting results and evaluating models.

This unit, the third in a series of five, builds on the ideas introduced and developed in Modelling poll
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Examples 1 and 2

Example 1

Example 1
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3.1 Introduction

In this section we shall develop the correspondence between C and R × R by obtaining a geometric representation of elements of C and operations on C. We shall define the polar form of a complex number and the modulus and argument of a complex number. We shall see that not only does
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1 Overview

A fundamental concept in mathematics is that of a function.

Consider, for example, the function f defined by

This is an example of a real function, because it associates with a given real number x the real number 2x2 − 1: it maps real numbers to real n
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Learning outcomes

After studying this unit you should be able to:

  • understand how the wave and diffusion partial differential equations can be used to model certain systems;

  • determine appropriate simple boundary and initial conditions for such models;

  • find families of solutions for the wave equation, damped wave equation, diffusion equation and similar homogeneous linear second-order partial differential equations, subject to simple boundary conditions, using the meth
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1.6 Do, talk and record triad

The do–talk–record triad (DTR) is a description of what is likely to take place in collaborative mathematics classrooms. It is concerned with observable events, and with the learner rather than the teacher, though many teaching insights flow from it. Although the order of the triad suggests that it should be followed in a particular sequence, this is not necessarily the case. Sometimes talking comes before doing or recording before talking. It also takes time for a learner to move
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Introduction

This unit is an adapted extract from the Open University course Complex analysis (M337)

This unit is devoted solely to complex numbers.

In Section 1, we define complex numbers and show you how to manipulate them, stressing the similarities with the manipulation of real numbers.

Section 2 is devoted to the geometric representation of complex numbers. You will find that
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1.3.9 Summary

There is a lot of information available on maths and statistics via the internet. Try the activities below to start exploring what is available.

Activity

1.3.3 Books and electronic books

Books are a good source of information. The publishing process (where a book is checked by an editor before publishing, and often reviewed by another author) means that books are reliable sources of information, although they may need to be evaluated for bias. A growing number of books can be found online.

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