This book provides a brief but accessible introduction to a set of related, mathematical ideas that have proved useful in understanding the brain and behaviour. If you record the eye movements of a group of people watching a riverside scene then some will look at the river, some will look at the barge by the side of the river, some will look at the people on the bridge, and so on, but if a duck takes off then everybody will look at it. How come the brain is so adept at processing such biological objects? In this book it is shown that brains are especially suited to exploiting the geometric properties of such objects. Central to the geometric approach is the concept of a manifold, which extends the idea of a surface to many dimensions. The manifold can be specified by collections of n-dimensional data points or by the paths of a system through state space. Just as tangent planes can be used to analyse the local linear behaviour of points on a surface, so the extension to tangent spaces can be used to investigate the local linear behaviour of manifolds. The majority of the geometric techniques introduced are all about how to do things with tangent spaces. Examples of the geometric approach to neuroscience include the analysis of colour and spatial vision measurements and the control of eye and arm movements. Additional examples are used to extend the applications of the approach and to show that it leads to new techniques for investigating neural systems. An advantage of following a geometric approach is that it is often possible to illustrate the concepts visually and all the descriptions of the examples are complemented by comprehensively captioned diagrams. The book is intended for a reader with an interest in neuroscience who may have been introduced to calculus in the past but is not aware of the many insights obtained by a geometric approach to the brain. Appendices contain brief reviews of the required background knowledge in neuroscience and calculus.
Different animals have different visual systems and so presumably have different ways of seeing. How does the way in which we see depend on the optical, neural and motor components of our visual systems? Originally published in 1993, the mathematical tools needed to answer this question are introduced in this book. Elementary linear algebra is used to describe the transformations of the stimulus that occur in the formation of the optical, neural and motor images in the human visual system. The distinctive feature of the approach is that transformations are specified with enough rigour for readers to be able to set up their own models and generate predictions from them. Underlying the approach of this book is the goal of providing a self-contained source for the derivation of the basic equations of vision science. An introductory section on vector and matrix algebra covers the mathematical techniques which are applied to both sensory and motor aspects of the visual system, and the intervening steps in the mathematical arguments are given in full, in order to make the derivation of the equations easier to follow. A subsidiary goal of this book is to demonstrate the utility of current desktop computer packages which make the application of mathematics very easy. All the numerical results were produced using only a spreadsheet or mathematics package, and example calculations are included in the text.
Since the early 70's, mixed finite elements have been the object of a wide and deep study by the mathematical and engineering communities. The fundamental role of this method for many application fields has been worldwide recognized and its use has been introduced in several commercial codes. An important feature of mixed finite elements is the interplay between theory and application. Discretization spaces for mixed schemes require suitable compatibilities, so that simple minded approximations generally do not work and the design of appropriate stabilizations gives rise to challenging mathematical problems. This volume collects the lecture notes of a C.I.M.E. course held in Summer 2006, when some of the most world recognized experts in the field reviewed the rigorous setting of mixed finite elements and revisited it after more than 30 years of practice. Applications, in this volume, range from traditional ones, like fluid-dynamics or elasticity, to more recent and active fields, like electromagnetism.
Never before had France had a church council so large: almost 1000 churchmen assembled at Bourges on 29 November 1225 to authorize a tax on their incomes in support of the Second Albigensian Crusade. About one third of the participants were representatives sent by corporate bodies, in accordance with a new provision of canon law that insisted, for the first time ever, that there should be no taxation without representation. Basing himself on the rich surviving records, Professor Kay paints a skilful portrait of this council: the political manoeuvering by the papal legate to ensure the tax went through, and his use of this highly public occasion to humiliate members of the University of Paris; and, on the other hand, his failure to win a permanent endowment to support the papal bureaucracy, the bishops' effective protests against the pope's threat to diminish their jurisdiction over monasteries, and a subsequent 'taxpayers' revolt' that challenged the validity of the tax. The book also draws out the importance and implications of what took place, highlighting the council's place at the fountainhead of European representative democracy, the impact of the decisions made on the course of the Albigensian Crusade, the reform of monasticism, and the funding of the papal government which was left to rely on stop-gap expedients, such as the sale of indulgences. In addition, the author suggests that the corpus of texts, newly edited from the original manuscripts and with English translation, could be seen as a model for the revision of the conciliar corpus, most of which still remains based on 18th-century scholarship.
A handsome, compact new edition of the singularly reliable translation of the writings of history's most famous seer-now published with a new afterword on Nostradamus and 2012. The prophecies of Nostradamus have ignited intense fascination throughout the West since they were first published in the sixteenth century. These enigmatic quatrains have gained special meaning for many during our current countdown to the year 2012. The relationship between Nostradamus's prophecies and the Mayan prophecy of 2012 is hotly debated over the Internet and is the focus of today's most popular television documentaries. Now The Essential Nostradamus-the definitive literal translation of his key verses-is redesigned, reset, and repackaged in an affordable and beautifully produced Tarcher Cornerstone Edition. This volume features a brand-new afterword from translator/commentator Richard Smoley, critiquing the Nostradamus legacy in light of the Mayan prophecies about 2012. Smoley is one of today's most respected writers on esotericism, and his new afterword fully (and realistically) assesses the question of Nostradamus's prophecies in relation to 2012, engaging readers who are hungry to learn more and want help sorting fact from fiction. Perfect for any who are curious, and for scholars and students who want to understand the real content of these haunting verses, The Essential Nostradamus is the key sourcebook of its kind.
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