4.6 The Classification Theorem

In this subsection we state the Classification Theorem for surfaces, which classifies a surface in terms of its boundary number β, its orientability number ω and its Euler characteristic χ, each of which is a topological invariant – it is preserved under homeomorphisms.

Let us remind ourselves of these three numbers.

  • A surface may or may not have a boundary, and, if it does, then the boundary has finitely many disjoint pieces. The nu
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7.2 Understanding the limits of rationality

An important first step in making more effective decisions is to understand the limits of human rationality. Because of these limits we have developed formal processes for reasoning: statistics; probability theory; modelling methods; and so on. We have also developed technologies such as computers to support us in processing information. These are certainly useful, but it is always important to remember they are used by humans and can be easily subverted. For example:

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Counting to Ten in Chinese (1-10)
Learn how to count to 10 in Chinese.  (03:03)
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Java4585: Getting Started with Search Engines
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The purpose of this module is to get you started with Search Engines
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7.342 G-Protein Coupled Receptors: Vision and Disease (MIT)
How do we communicate with the outside world? How are our senses of vision, smell, taste and pain controlled at the cellular and molecular levels? What causes medical conditions like allergies, hypertension, depression, obesity and various central nervous system disorders? G-protein coupled receptors (GPCRs) provide a major part of the answer to all of these questions. GPCRs constitute the largest family of cell-surface receptors and in humans are encoded by more than 1,000 genes. GPCRs convert
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Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative C

Lesson 04 - One Minute Luxembourgish
In lesson 04 of One Minute Luxembourgish you will learn how to say that you don't understand something. Remember - even a few phrases of a language can help you make friends and enjoy travel more. Find out more about One Minute Languages at our website - http://www.oneminutelanguages.com. One Minute Luxembourgish is brought to you by the Radio Lingua Network and is ©Copyright 2008.Author(s): No creator set

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3.1 Social work values and legal values

Social work practice is founded on and informed by a value base; however, this value base is uncertain and changing (Shardlow, 1998). It is important that practitioners are able to reflect on their values and prejudices and consider the implications of these for practice. The next activity requires you to think about this before going on to look in more detail at what is meant by social work values.

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Técnicas de trabajo cooperativo y aprendizaje cooperativo en ambientes virtuales
Questo modulo introduce e discute metodologie e tecniche per promuovere l'apprendimento cooperativo e collaborativo in ambienti virtuali.,This module introduces and further discusses methodologies and tecniques for promoting cooperative and collaborative learning in virtual learning environments.,Este modulo afronta el tema de la sociedad del conocimiento y de la inteligencia conectiva en la cual los individuos no deberàn “saber todo” sino que deberàn disponer sobretodo de competencias met
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Earth had a climate long before we showed up and started noticing it and it's influenced by a whole series of cycles that have been churning along for hundreds of millions of years. In most cases those cycles will continue long after we're gone. A look at the history of climate change on Earth can give us some much needed perspective on our current climate dilemma because the surprising truth is, what we're experiencing now is different than anything this planet has encountered before. So, let's
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6 Concluding thoughts

We seem to have come a long way and covered a great deal of ground since I approached this subject by explaining that a mechanism must exist to help us focus on one sound out of many. That clearly is one function of attention, but attention seems to have other functions too. The results of visual search experiments show that attention is a vital factor in joining together the features that make up an object, and the experiences of brain-damaged patients suggest that this feature-assembly role
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1.3.7 Models of adjustment

Here we have talked about changes of place as having a particular impact on an individual's sense of well-being or self-esteem. Relocation and separation from familiar places just like separation from loved ones can be experienced as a form of loss which can have devastating effects for some people. Some authors have seen changes in self-esteem as the key to understanding how people cope with change. For example, Hopson and Adams (1976) suggest that any transition, whatever triggers it, sets
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Health and environment
To be able to understand the importance of the environment for our health, we need to know a little about the interdependence between environment and humankind. This free course, Health and environment, will look at interactions between plants, animals and the physical and chemical environment, as well as considering ways in which humans have altered, and are altering this environment. Author(s): Creator not set

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Except for third party materials and otherwise stated (see http://www.open.ac.uk/conditions terms and conditions), this content is made available under a http://creativecommons.org/licenses/by-nc-sa/2

1.5 Ways of working and contributing

The physical design of most primary schools reflects the expectation that teachers work in classrooms with large numbers of children. In fact, given their large classes, most schools feel quite crowded. The employment of teaching assistants has doubled the number of adults working in some classrooms and, as Schlapp and Davidson (2001) note, this has sometimes led to problems in teaching assistants finding work spaces they can call their own. Often they give support while working alongside tea
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3.1 Conclusion

In this unit you have considered a range of responses and feelings that services users may experience during the transition into residential care, and have identified strategies that can be used to support them with this move. Passing on comprehensive information about the service user to care providers will help them to respond more effectively to the service user's needs. Being able to provide relatives and service users with information about possible placements and negotiating with provid
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Say what you see
Try describing this picture.  If you are just beginning, try finding nouns to name the different items you see in the picture?  How about some adjectives?  Do you know the […]

3.1 Needing help

Some people will cope with parenting without the need for additional support, some will need some help at some time, while for others parenting may be a task which appears to overwhelm them. Some individuals are affected by circumstances which make them more likely to require support or assistance. In this section we explore these issues in more detail.

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CAAD Education and Post-Graduate Opportunities (At Mimar Sinan University)
This paper addresses new design teaching strategies at an important and traditional university in Istanbul, founded as the Academy of Fine Arts 110 years ago. It will include a short review of design education before the Academy changed into a university, and a description of the present situation with regard to computers. Nearly two years ago, CAAD education was introduced as an elective subject. The students show great interest in CAD; most Turkish architects now work with computers and CAAD g
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Frink - Horse and Rider DP183064

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Dover Street and Piccadilly, Mayfair, London. Elisabeth Frink sculpture "Horse and Rider" from south east. Photographed by Chris Redgrave in 2015.
© Historic England


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The Growing Stage, The Children’s Theatre of New Jersey, will present the world premiere of "And Then Came Tango" on Friday, March 6th. Inspired by a true story, the play revolves around the case of two male penguins at the Central Park Zoo who established a bond to hatch an orphaned egg. Written by Emily Freeman, and featuring choreography by Jillian Petrie and original music by Paul Marbach, "And Then Came Tango" is a celebration of tolerance and the eternal instinct that guides one generati
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Jeremy Bowen presents a look back at the creation of the state of Israel, examining the the events leading up to the Israeli war of Independence in 1949, its...
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