Posts tagged with finite

A history of 20th-century mathematics in ~500 words, with example pictures:

The inheritance of Cantor’s Set Theory allowed the 20th century to create the domain of “Functional Analysis”.

This comes about as an extension of the classical Differential and Integral Calculus in which one considers not merely a particular function (like the exponential function or a trigonometric function), but the operations and transformations which can be performed on all functions of a certain type.

The creation of a “new” theory of integration, by Émile Borel and above all Henri Lebesgue, at the beginning of the 20th century, followed by the invention of normed spaces by Maurice Fréchet, Norbert Wiener and especially Stefan Banach, yielded new tools for construction and proof in mathematics.

The theory is seductive by its generality, its simplicity and its harmony, and it is capable of resolving difficult problems with elegance. The price to pay is that it usually makes use of non-constructive methods (the Hahn-Banach theorem, Baire’s theorem and its consequences), which enable one to prove the existence of a mathematical object, but without giving an effective construction.

[I]n 1955 … [Nancy] … was the golden age of French mathematics, where, in the orbit of Bourbaki and impelled above all by Henri Cartan, Laurent Schwartz and Jean-Pierre Serre, mathematicians attacked the most difficult problems of geometry, group theory and topology.

$\dpi{200} \bg_white \begin{matrix} ^1_4 \Box ^2_3 \ \overset{\mathbf{V}} \to \ ^4_1 \Box ^3_2 \ \overset{\mathbf{R}} \to \ ^1_2 \Box^4_3 \ \overset{\mathbf{V}} \to \ ^2_1 \Box ^3_4 \\ (\mathbf{VRV} = \mathbf{R}^{-1} = \mathbf{R}^3 ) \end{matrix}$
New tools appeared: sheaf theory and homological algebra … which were admirable for their generality and flexibility.

The apples of the garden of the Hesperides were the famous conjectures stated by André Weil in 1954: these conjectures appeared as a combinatorial problem … of a discouraging generality….

The fascinating aspect of these conjectures is that they … fuse … opposites: “discrete” and “continuous”, or “finite” and “infinite”.

Methods invented in topology to keep track of invariants under the continuous deformation of geometric objects, must be employed to enumerate a finite number of configurations.

Like Moses, André Weil caught sight of the Promised Land, but unlike Moses, he was unable to cross the Red Sea on dry land, nor did he have an adequate vessel. André Weil … was not … ignorant of these techniques …. But [he] was suspicious of “big machinery” ….

Homological algebra, conceived as a general tool reaching beyond all special cases, was invented by Cartan and Eilenberg (… in 1956). This book is a very precise exposition, but limited to the theory of modules over rings and the associated functors “Ext” and “Tor”. It was already a vast synthesis of known methods and results, but sheaves do not enter into this picture. Sheaves … were created together with their homology, but the homology theory is constructed in an ad hoc manner….

In the autumn of 1950, Eilenberg …undertook with Cartan to [axiomatise] sheaf homology; yet the construction … preserves its initial ad hoc character.

When Serre introduced sheaves into algebraic geometry, in 1953, the seemingly pathological nature of the “Zariski topology” forced him into some very indirect constructions.

………

Count.

2¹⁰⁰

Readers of isomorphismes, you might enjoy powers of two tumblr.

2100 = 1,267,650,600,228,229,401,496,703,205,376 — one nonillion, two hundred sixty-seven octillion, six hundred fifty septillion, six hundred sextillion, two hundred twenty-eight quintillion, two hundred twenty-nine quadrillion, four hundred one trillion, four hundred ninety-six billion, seven hundred three million, two hundred five thousand, three hundred seventy-six (31 digits, 320 characters)

I think I’ve been subscribed since the 30’s. Never a letdown. And of course it’s only going to get more exciting.

This is an amazing fact that comes up in so many applications.  It’s used in the valuation of companies, solution of equations, ……… any time you want to convert an infinite stream into something finite.

f is a proper fraction. (0 < f < 1)

$\large \dpi{200} \bg_white f^0+f^1+f^2+f^3+f^4+f^5+f^6+ \ldots = {1 \over 1-f}$

Or, in fancy notation:

$\large \dpi{200} \bg_white \sum_{i=0}^\infty f^{\,i} = {1 \over 1-f} , \quad 0

Or, in C++:

long big = 9999999999;float frac = .70;double total = 0;for ( i = 0; i < big; i++){  total += frac∗∗i;  }cout << total;                 # in this case, prints 1 / .3 = 10/3cout << total - 1/(1-frac)     # prints 0 for any value of frac

Isn’t it strange that adding together an infinite number of things can give you a finite answer?  The ancient philosopher Zeno thought that he could disprove reality through the following thought experiment

1. An arrow fired at a tree first covers half the distance to the tree.
2. Then it covers half the remaining distance to the tree.
3. Then it covers half the remaining distance to the tree.
4. Etc….so it only ever covers less than all the distance to the tree!  Because it just keeps adding halves of halves of halves of ….
5. So, since we see it hit the tree, but logically it cannot hit the tree, logically reality must be false!  (Motion is impossible, and we observe motion, so our observations are impossible.)

But calculus proves that:

$\large \dpi{200} \bg_white {1 \over 2} + {1 \over 4} +{1 \over 8} + {1 \over 16 } + {1 \over 32} + {1 \over 64} + {1 \over 128} + {1 \over 256} + \ldots = {^1\! / \!_2 \over 1- {1 \over 2} } = 1$

Take that, Zeno!