In this subsection we consider, briefly, some problems that can arise with certain ways of drawing bar charts and pie charts.

Figure 5 shows what is essentially the same bar chart as Author(s): The Open University

In several different parts of the world, footprints from prehistoric human civilisations have been found preserved in either sand or volcanic ash. From these tracks it is possible to measure the foot length and the length of the stride. These measurements can be used to estimate both the height of the person who made the footprint and also whether the person was walking or running by using the following three formulas:

Author(s): The Open University

Now try the quiz, and see if there are any areas you need to work on.

Author(s): The Open University

Discount can be calculated in the same way as an increase by a percentage. For example, Â£8 with 15% discount means you actually pay

Â Â Â£8 less (15% of Â£8)

Â Â 15% of 8 = Ã— 8 = Author(s): The Open University

Our formal definition of an equivalence relation involves three key properties. A relation that has these three properties partitions the set on which the relation is defined, as we show later in this subsection.

In the last subsection we stated that, for any integer n â‰¥ 2, the set n satisfies the same rules for addition modulo n as the real numbers satisfy for ordinary addition. When it comes to multiplication in Author(s): The Open University

Consider the real exponential function f (x) = ex (that is, f (x) = exp x). We now extend the definition of this function to define a function f(z) = ez whose domain and codomain are .

We expect complex powers
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We will now discuss complex numbers and their properties. We will show how they can be represented as points in the plane and state the Fundamental Theorem of Algebra: that any polynomial equation with complex coefficients has a solution which is a complex number. We will also define the function exp of a complex variable.

Earlier we mentioned several sets of numbers, including Author(s): The Open University

1. By â€˜containsâ€™, we mean that we can find part of the surface that is homeomorphic to a MÃ¶bius band. The edge of the MÃ¶bius band does not need to correspond to an edge at the surface, so that a surface without boundary can be non-orientable (as we shall shortly see).

2. When seeking MÃ¶bius bands in a surface, it can be helpful to look at all possible closed curves on the surface and thicken these into bands.

3. Remember, fro
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1. This theorem applies to all surfaces and not just to surfaces in space.

2. This theorem tells us that the boundary number is a topological invariant for surfaces, i.e. a property that is invariant under homeomorphisms.

3. It follows from the theorem that two surfaces with different boundary numbers cannot be homeomorphic. It does not follow that two surfaces with the same boundary number are homeomorphic â€“
Author(s): The Open University

As we stated in Section 1, our aim is to classify surfaces up to homeomorphism. So it is worthwhile spending a little time examining what sorts of transformations of surfaces are homeomorphisms. We shall restrict the description to surfaces in space, as these are easier to deal with, though the result at the end of this subsection applies to all surfaces.

Recall that a homeomorphism between two topological spaces (such as surfaces in space) is a bijection with the property that b
Author(s): The Open University

Understanding the environment: Thinking styles and models
There is increasing recognition that the reductionist mindset that is currently dominating society, rooted in unlimited economic growth unperceptive to its social and environmental impact, cannot resolve the converging environmental, social and economic crises we now face. The primary aim of this unit is to encourage the shift away from reductionist and human centred thinking towards a holistic and ecological worldview.Author(s): Creator not set

Today I woke up thinking that talking about Identity and Access Control and how your strategy around that affects you (web-) app's architecture without going too deeply into the security lingo that usually comes with it. Here's the 40 minute result.

I start with HTTP's "native" authentication model RFC 2617 and how that's universally bad, with both Basic and Digest authentication having issues Digest being, ironically worse for the overall security strategy. Then I d
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Sir Isaac Newton: Full Biography
Isaac Newton was born December 25, 1642 in Woolsthorpe, Lincolnshire, England, and died March 20, 1727 in London, England. Newton was an English physicist and mathematician, who was also the culminating, figure of the scientific revolution of the 17th century. With discoveries in optics, motion, and mathematics he developed the principles of modern physics. He was the original discoverer of the infinitesimal calculus. Newton's Philosophiae Naturalis Principia Mathematica (Mathematical Principles
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How Did English Evolve?
What is the difference between "a hearty welcome" and "a cordial reception"? In a brief, action-packed history of the English language, Kate Gardoqui explains why these semantically equal phrases evoke such different images.  (05:04)

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Dec 13 - If you are betting on the manufacturing sector creating jobs, you might want to wait till the Markit PMI figures come out Friday - they are not expected to be pretty.
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Results

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SDSU Geological Sciences - Thesis Defense - Shana McCarthy
By: tcarrasc A Geochemical Evaluation of Enhanced In-Situ Bioremediation of Chlorinated Ethenes In Groundwater Shana McCartrhy M.S. Candidate Department of Geological Sciences San Diego State University Advisor Dr. Kathy Thorbjarnarson -------------------------------------------------------------------------------- ABSTRACT Sites impacted with chlorinated solvents present unique technical challenges when compared to most other groundwater contaminants. Chlorinated volatile organic compounds (CV
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Engaging with Detroit
University of Michigan students live, study and work in the Motor City
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