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Introduction

This unit is an adapted extract from the course Pure mathematics (M208)

The idea of vectors and conics may be new to you. In this unit we look at some of the ways that we represent points, lines and planes in mathematics.

In Section 1 we revise coordinate geometry in two-dimensional Euclidean space,
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The Limits of Acceptable Biological Variation in Elite Athletes: Should Sex Ambiguity Be Treated Dif
By: WentzMR Dr. J. Michael Bostwick, Professor of Psychiatry at Mayo Clinic in Rochester, MN, discusses his article appearing in the June 2012 issue of Mayo Clinic Proceedings, where he explores sex, gender, and other genetic traits in elite athletes. Available at: http://www.mayoclinicproceedings.org/article/S0025-6196(12)00439-9/fulltext
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4.2.3 ATM adaptation layer

The basic function of the ATM adaptation layer is to convert the user data supplied by higher layers into 48-byte blocks of data. The ATM adaptation layer is divided into two sub-layers – the convergence sub-layer, and the segmentation and re-assembly sub-layer. The convergence sub-layer provides services to higher layers through a set of protocols, but I do not need to describe these here. The segmentation and re-assembly sub-layer separates the messages from the conve
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2.2.1 Soil pH

pH (a measure of acidity or alkalinity) is an important environmental factor, particularly in soils. Soil is derived partly from accumulated decaying vegetation and partly from broken up fragments of the underlying rocks. Soil pH is determined by both these components and also by the water that fills the spaces between solid soil particles.

How might you expect the pH of soil overlying limestone (or chalk, which is a particular form of limestone) to compare with that of soil overlying s
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3.8 Summary of Part B

In Part B you learned more about the ECHR and the procedures of the ECtHR and how protocols have been used to ensure that the ECHR remains a living instrument. Part B also explored the new challenges created by the rapid expansion of HCPs at the end of the last century and the proposals for reform of the ECtHR.


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6.2 The supremacy of EU law

Whenever there is a conflict between the provisions of EU law and the provisions of the domestic (national) law of a member state, then EU law will prevail. This is a principle which was developed by the ECJ as the relationship between domestic and EU law is not clarified by treaty provisions. This is an important principle, as it ensures the proper functioning of the EU. If an EU member state had the power to annul EU law by adopting new domestic (national) law which was in conflict with the
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2.4 The European Court of Human Rights

Common law and the court hierarchy, statutory interpretation and judicial precedent are all peculiar to the domestic English law. The European Court of Human Rights operates in a different way. The rights in the European Convention on Human Rights are stated in general terms and are interpreted according to international legal principles. For example, Article 31(1) of the Vienna Convention on the Law of Treaties states:

<
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2.2 The Convention itself

The ECHR is essentially a charter of rights. Any charter of rights represents a consensus, a negotiated agreement between the drafters. Every state intending to adopt a charter will have its own vision and aims, and the drafters have to find a way of accommodating these visions and aims. This often results in the creation of provisions that are a compromise and are drafted in the widest possible terms. The ECHR is drafted in such a way. It is a vaguely worded aspirational charter inten
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2.1 History

The Council of Europe was set up in 1949. It is an intergovernmental organisation (based in Strasbourg, France) set up to protect human rights, promote cultural diversity and to combat social problems such as intolerance. Its creation was seen as a way of achieving a European approach to the protection of certain individual rights. Although presented now as historical events, the horrors of what had taken place in the Second World War were then fresh in the minds of the governments and
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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
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3.19 Multiplication with negative numbers

Now that you have rules for addition and subtraction of negative numbers, think about multiplication and division.

Example 27

Describe each of the following in terms of the number line and the value of Thomas's piggy bank:

  • (a) the mul
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3.15.1 Subtraction on the number line

Now what about subtraction? You can think of subtraction as undoing addition: adding 3 to 8 gets you 11, and so subtracting 3 from the answer, 11, gets you back to 8. Therefore, in terms of the number line, subtracting 3 from 11 means starting at 11 and moving 3 units to the left.

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3.5.1 Try some yourself

Activity 30

Carry out the following calculations, without your calculator.

  • (a) 3 × (60 + 70).

  • (b) (3 × 60) + 70.

  • (c) (70 − 60) ÷ 5.


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3.4 Did I make a rough estimate to act as a check?

When using a calculator many people have ‘blind
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Try some yourself

Question 1

Which of these triangles are similar?

2.7 Rotational symmetry

There is another kind of symmetry which is often used in designs. It can be seen, for instance, in a car wheel trim.

Look at the trim on the left. It does not have line symmetry but
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5 Approaches to problem solving

You should not expect always to be able to read a problem and then just write down the answer. When you are faced with a written mathematical question or problem to solve, read it carefully. It is important that you get to grips with the question in two ways: first, that you absorb the information given; and second, that you find out what the question is really asking. Your solution will link the two. This method can be summarised by the following questions.

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Mediating Change: Culture and Climate Change
Every generation faces challenges that previous generations could scarcely imagine. Twenty years ago, few people were talking about climate change, now it's one of the most hotly-contested areas in politics. How do artists, writers, musicians and broadcasters respond when a new subject appears that is as large and significant as this? What kind of novels, plays, paintings, sculptures, movies and music begin to emerge? ‘Mediating Change’ is a four-part series, chaired by BBC Radio 4’s Que
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Studying mammals: Food for thought
Who were our ancestors? How are apes and humans related? And where does the extinct Homo erectus fit into the puzzle? In this free course, Studying mammals: Food for thought, we will examine culture, tool use and social structure in both apes and humans to gain an understanding of where we come from and why we behave as we do. This is the tenth course in the Studying mammals series. Author(s): Creator not set

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