"natural theology > notes > 2012 > notes 16 September 2012"

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Acemoglu, Daron, and James Robinson, Why Nations Fail: The Origins of Power, Prosperity and Poverty, Crown Business 2012 "Some time ago a little-known Scottish philosopher wrote a book on what makes nations succeed and what makes them fail. The Wealth of Nations is still being read today. With the same perspicacity and with the same broad historical perspective, Daron Acemoglu and James Robinson have retackled this same question for our own times. Two centuries from now our great-great- . . . -great grandchildren will be, similarly, reading Why Nations Fail." —George Akerlof, Nobel laureate in economics, 2001
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Casti, John L, Five Golden Rules: Great Theories of 20th-Century Mathematics - and Why They Matter, John Wiley and Sons 1996 Preface: '[this book] is intended to tell the general reader about mathematics by showcasing five of the finest achievements of the mathematician's art in this [20th] century.' p ix. Treats the Minimax theorem (game theory), the Brouwer Fixed-Point theorem (topology), Morse's theorem (singularity theory), the Halting theorem (theory of computation) and the Simplex method (optimisation theory).
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Debreu, Gerard, Theory of Value: An Axiomatic Analysis of Economic Equilibrium, Yale University Press 1972 Amazon customer review: 'This is not an easy book. The mathematics are very rigorous, but everything is well defined, and it is self-contained. However, it pays to read this short book. If you want to understand the foundations of the modern economic analysis, this the place to look.' A Customer
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Heath, Thomas Little, Thirteen Books of Euclid's Elements (volume 1, I-II), Dover 1956 'This is the definitive edition of one of the very greatest classics of all time - the full Euclid, not an abridgement. Utilizing the text established by Heiberg, Sir Thomas Heath encompasses almost 2500 years of mathematical and historical study upon Euclid.'
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Heath, Thomas Little, Thirteen Books of Euclid's Elements (volume 2, III-IX), Dover 1956 'This is the definitive edition of one of the very greatest classics of all time - the full Euclid, not an abridgement. Utilizing the text established by Heiberg, Sir Thomas Heath encompasses almost 2500 years of mathematical and historical study upon Euclid.'
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Heath, Thomas Little, Thirteen Books of Euclid's Elements (volume 3, X-XIII), Dover 1956 'This is the definitive edition of one of the very greatest classics of all time - the full Euclid, not an abridgement. Utilizing the text established by Heiberg, Sir Thomas Heath encompasses almost 2500 years of mathematical and historical study upon Euclid.'
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Marr, David, Patrick White: A Life, Knopf 1992 Editorial review from Library Journal : 'From Library Journal An admirably readable biography of the Nobel Prize-winning author of Voss , The Tree of Man , and many other books, this work is full of detail on White's family and prosperous background, the events and people in his life, his writing habits, his religious beliefs, his cantankerousness and temper, his causes and doubts, his attraction to the theater, and much more. White helped Marr gain access to people and material, even authorizing him to collect his letters, "the backbone of this book." Marr deals intelligently with important issues (among them, White's rootedness in and dissatisfaction with Australia, his sense of himself as an outsider, his relation to his mother, and, in particular his homosexuality, which White considered central to his novelistic and theatrical ability), avoiding psychoanalytical speculations and other intrusions. White reviewed the book shortly before he died, finding it "so painful he often found himself reading through tears. He did not ask Marr to change a line."' Richard Kuczkowski Copyright 1992 Reed Business Information, Inc.
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Newton, Isaac, Philosophiae Naturalis Principia Mathematica , Harvard University Press 1972 One of the most important contributions to human knowledge. First translated from the Latin by Andrew Motte in 1729,
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Newton, Isaac, Principia volume II The System of the World, University of California Press 1966 back

Newton, Isaac, and Julia Budenz, I. Bernard Cohen, Anne Whitman (Translators), The Principia : Mathematical Principles of Natural Philosophy, University of California Press 1999 This completely new translation, the first in 270 years, is based on the third (1726) edition, the final revised version approved by Newton; it includes extracts from the earlier editions, corrects errors found in earlier versions, and replaces archaic English with contemporary prose and up-to-date mathematical forms. ... The illuminating Guide to the Principia by I. Bernard Cohen, along with his and Anne Whitman's translation, will make this preeminent work truly accessible for today's scientists, scholars, and students.
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Stewart, Ian, Why Beauty is Truth: A History of Symmetry, Basic Books/Perseus 2007 Jacket: ' ... Symmetry has been a key idea for artists, architects and musicians for centuries but within mathematics it remained, until very recently ,an arcane pursuit. In the twentieth century, however, symmetry emerged as central to the most fundamental ideas in physics and cosmology. Why beauty is truth tells its history, from ancient Babylon to twenty-first century physics.'
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White, Patrick, and David Marr (Editor), Patrick White Letters, University Of Chicago Press 1996 Amazon book description: '"Letters are the devil, and I always hope that any I have written have been destroyed."—Patrick White Patrick White spent his whole life writing letters. He wanted them all burnt, but thousands survive to reveal him as one of the greatest letter-writers of his time. Patrick White: Letters is an unexpected and final volume of prose by Australia's most acclaimed novelist. Only a few scraps of White's letters have been published before. From the aftermath of the First World War until his death in 1990, letters poured from White's pen: they are shrewd, funny, dramatic, pigheaded, camp, and above all, hauntingly beautiful. He wrote novels to sway a hostile world, but letters were for friends.'
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Links

Brouwer fixed point theorem - Wikipedia Brouwer fixed point theorem - Wikipedia, the free encyclopedia 'Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after Luitzen Brouwer. It states that for any continuous function f with certain properties there is a point x0 such that f(x0) = x0. The simplest form of Brouwer's theorem is for continuous functions f from a disk D to itself. A more general form is for continuous functions from a convex compact subset K of Euclidean space to itself. back

Eugene Wigner The Unreasonable Effectiveness of Mathematics in the Natural Sciences 'The first point is that the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and that there is no rational explanation for it. Second, it is just this uncanny usefulness of mathematical concepts that raises the question of the uniqueness of our physical theories.' back

Galois theory - Wikipedia Galois theory - Wikipedia, the free encyclopedia 'In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory. Using Galois theory, certain problems in field theory can be reduced to group theory, which is in some sense simpler and better understood. Originally Galois used permutation groups to describe how the various roots of a given polynomial equation are related to each other. The modern approach to Galois theory, developed by Richard Dedekind, Leopold Kronecker and Emil Artin, among others, involves studying automorphisms of field extensions.' back

Homeomorphism - Wikipedia Homeomorphism - Wikipedia, the free encyclopedia 'In the mathematical field of topology, a homeomorphism or topological isomorphism (from the Greek words (homoios) = similar and (morph) = shape = form (. . . ) is a bicontinuous function between two topological spaces. Homeomorphisms are the isomorphisms in the category of topological spaces — that is, they are the mappings which preserve all the topological properties of a given space. Two spaces with a homeomorphism between them are called homeomorphic, and from a topological viewpoint they are the same.' back

Perron-Frobenius theorem - Wikipedia Perron-Frobenius theorem - Wikipedia, the free encyclopedia 'In linear algebra, the Perron–Frobenius theorem, proved by Oskar Perron (1907) and Georg Frobenius (1912), asserts that a real square matrix with positive entries has a unique largest real eigenvalue and that the corresponding eigenvector has strictly positive components, and also asserts a similar statement for certain classes of nonnegative matrices. This theorem has important applications to probability theory (ergodicity of Markov chains); to the theory of dynamical systems (subshifts of finite type); to economics (Leontief's input-output model); to demography (Leslie population age distribution model)[2] to mathematical background of the Internet search engines and even to ranking of football teams.' back

Schauder fixed point theorem - Wikipedia Schauder fixed point theorem - Wikipedia, the free encyclopedia 'The Schauder fixed point theorem is an extension of the Brouwer fixed point theorem to topological vector spaces, which may be of infinite dimension. It asserts that if K is a convex subset of a topological vector space V and T is a continuous mapping of K into itself so that T(K) is contained in a compact subset of K , then T has a fixed point.' back

Sperner's lemma - Wikipedia Sperner's lemma - Wikipediam the free encyclopedia 'In mathematics, Sperner's lemma is a combinatorial analog of the Brouwer fixed point theorem, which follows from it. Sperner's lemma states that every Sperner coloring (described below) of a triangulation of an n-dimensional simplex contains a cell colored with a complete set of colors. The initial result of this kind was proved by Emanuel Sperner, in relation with proofs of invariance of domain. Sperner colorings have been used for effective computation of fixed points, in root-finding algorithms, and are applied in fair division (cake cutting) algorithms. Unfortunately it is now believed to be an intractable computational problem to find a Brouwer fixed point or equivalently a Sperner coloring even in the plane, in the general case. The problem is PPAD-Complete, a complexity class invented by Papadimitriou.' back

Wiles's proof of Fermat's Last Theorem - Wikipedia Wiles's proof of Fermat's Last Theorem - Wikipedia, the free encyclopedia 'Wiles's proof of Fermat's Last Theorem is a proof of the modularity theorem for semistable elliptic curves released by Andrew Wiles, which, together with Ribet's theorem, provides a proof for Fermat's Last Theorem. Wiles first announced his proof in June 1993 in a version that was soon recognized as having a serious gap. The widely accepted version of the proof was released by Andrew Wiles in September 1994, and published in 1995.' back

vol VII: Notes

2012

Notes

Sunday 16 September 2012

Monday 17 September 2012

Tuesday 18 September 2012

Wednesday 19 September 2012

Thursday 20 September 2012

Friday 21 September 2012

Saturday 22 September 2012

Further reading

Books

Links