Tag Archives: euler

Prime Consequences

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Most of us are aware of the following consequence of Fundamental Theorem of Arithmetic:

There are infinitely many prime numbers.

The classic proof by Euclid is easy to follow. But I wanted to share the following two analytic equivalents (infinite series and infinite products) of the above purely arithmetical statement:

  • \displaystyle{\sum_{p}\frac{1}{p}}   diverges.

For proof, refer to this discussion: https://math.stackexchange.com/q/361308/214604

  • \displaystyle{\sum_{n=1}^\infty \frac{1}{n^{s}} = \prod_p\left(1-\frac{1}{p^s}\right)^{-1}}, where s is any complex number with \text{Re}(s)>1.

The outline of proof,   when s is a real number, has been discussed here: http://mathworld.wolfram.com/EulerProduct.html

What is Topology?

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A couple of years ago, I was introduced to topology via proof of Euler’s Polyhedron formula given in the book “What is Mathematics?” by Richard Courant and Herbert Robbins. Then I got attracted towards topology by reading the book “Euler’s gem – the polyhedron formula and the birth of topology” by David S. Richeson. But now after doing a semester course on “introduction to topology” I have realized that all this was a lie. These books were not presenting the real picture of subject, they were presenting just the motivational pictures. For example, this is my favourite video about introduction to topology by Tadashi Tokieda (though it doesn’t give the true picture):

Few months ago I read the book “The Poincaré Conjecture” by Donal O’Shea and it gave an honest picture of algebraic topology. But, then I realized that half of my textbook on topology is about point-set topology (while other half was about algebraic topology). This part of topology has no torus or Möbius strip (checkout this photo) but rather dry set theoretic arguments. So I decided to dig deeper into what really Topology is all about? Is is just a fancy graph theory (in 1736, both Topology and graph theory started from Euler’s Polyhedron formula) or it’s a new form of Geometry which we study using set theory, algebra and analysis.

The subject of topology itself consists of several different branches, such as:

  • Point-Set topology
  • Algebraic topology
  • Differential topology
  • Geometric topology

Point-set topology grew out of analysis, following Cauchy’s contribution to the foundations of analysis and in particular trigonometric representation of a function (Fourier series). In 1872, Georg Cantor desired a more solid foundation for standard operations (addition, etc.) performed on the real numbers. To this end, he defined a Cauchy sequence of rational numbers. He creates a bijection between the number line and the possible limits of sequence of rational numbers. He took the converse, that “the geometry of the straight line is complete,” as an axiom (note that thinking of points on the real line as limits of sequence of rational numbers is “for clarity” and not essential to what he is doing). Then Cantor proved following theorem:

If there is an equation of form \displaystyle{0=C_0+C_1+\ldots +C_n+\ldots} where C_0 = \frac{d_0}{2} and C_n = c_n\sin{(nx)} +d_n\cos{(nx)} for all values of x except those which correspond to points in the interval (0,2\pi) give a point set P of the \nuth kind, where \nu signifies any large number, then d_0=1, c_n=d_n=0

This theorem lead to definition of point set to be a finite or infinite set of points. This in turn lead to definition of cluster point, derived set, …. and whole of introductory course in topology. Modern mathematics tends to view the term “point-set” as synonymous with “open set.” Here I would like to quote James Munkres (from point-set topology part of my textbook):

A problem of fundamental importance in topology is to find conditions on a topological space that will guarantee that it is metrizable…. Although the metrization problem is an important problem in topology, the study of metric spaces as such does not properly belong to topology as much as it does to analysis.

Now, what is generally publicised to be “the topology” is actually the algebraic topology. This aspect of topology is indeed beautiful. It lead to concepts like fundamental groups which are inseparable from modern topology. In 1895, Henri Poincaré topologized Euler’s proof of Polyhedron formula leading to what we call today Euler’s Characteristic. This marked the beginning of what we today call algebraic topology.

For long time, differential geometry and algebraic topology remained the centre of attraction for geometers.But, in 1956, John Milnor discovered that there were distinct different differentiable structures (even I don’t know what it actually means!) on seven sphere. His arguments brought together topology and analysis in an unexpected way, and doing so initiated the field of differential topology.

Geometric topology has borrowed enormously from the rest of algebraic topology it has returned very scant interest on this “borrowed” capital. It is however full of problems with some of the simplest, in formulation, as yet unsolved. Knot Theory (or in general low-dimensional topology) is one of the most active area of research of this branch of topology. Here I would like to quote R.J. Daverman and R.B. Sher:

Geometric Topology focuses on matters arising in special spaces such as manifolds, simplicial complexes, and absolute neighbourhood retracts. Fundamental issues driving the subject involve the search for topological characterizations of the more important objects and for topological classification within key classes.
Some key contributions to this branch of topology came from Stephen Smale (1960s), William Thurston (1970s), Michael Freedman (1982), Simon Donaldson (1983), Lowell Edwin Jones (1993), F. Thomas Farrel (1993), … and the story continues.

References:

[1] Nicholas Scoville (Ursinus College), “Georg Cantor at the Dawn of Point-Set Topology,” Convergence (May 2012), doi:10.4169/loci003861

[2] André Weil, “Riemann, Betti and the Birth of Topology.” Archive for History of Exact Sciences 20, no. 2 (1979): 91–96. doi:10.1007/bf00327626.

[3] Johnson, Dale M. “The Problem of the Invariance of Dimension in the Growth of Modern Topology, Part I.” Archive for History of Exact Sciences 20, no. 2 (1979): 97–188. doi:10.1007/bf00327627.

[4] Johnson, Dale M. “The Problem of the Invariance of Dimension in the Growth of Modern Topology, Part II.” Archive for History of Exact Sciences 25, no. 2–3 (December 1981): 85–266. doi:10.1007/bf02116242.

[5] Lefschetz, Solomon. “The Early Development of Algebraic Topology.” Boletim Da Sociedade Brasileira de Matemática 1, no. 1 (January 1970): 1–48. doi:10.1007/bf02628194.

Celebrity Mathematicians

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In my opinion, currency notes are one of the biggest motivation to learn arithmetical operations (like addition, multiplication,…). In fact, most of our elementary school problems are about buying a particular quantity of something.

Historically, there had been currencies notes featuring great mathematicians like Carl Friedrich Gauss, Leonard Euler and Rene Descartes. But, today there are no currencies featuring mathematicians. The database of currency notes featuring mathematicians is available here: http://web.olivet.edu/~hathaway/math_money.html

Since honouring people by featuring them on currency notes is politically challenging, government rather issues special postage stamps. The database of stamps featuring mathematicians is available here:  http://jeff560.tripod.com/stamps.html

Apart from illustrating various mathematical concepts (like graphs, metric system, binomial theorem… ) on stamps, India Post has issued stamps to honour mathematicians like Damodar Dharmananda Kosambi , Srinivasa Ramanujan Iyengar and Bertrand Russell.

 

Integration & Summation

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A few months ago I wrote a series of blog posts on “rigorous”  definitions of integration [Part 1, Part 2]. Last week I identified an interesting flaw in my “imagination” of integration in terms of “limiting summation” and it lead to an interesting investigation.

The Paradox

While defining integration as area under curve, we consider rectangles of equal width and let that width approach zero. Hence I used to imagine integration as summation of individual heights, since width approaches zero in limiting case. It was just like extending summation over integers to summation over real numbers.

integration (1)

My Thought Process..

But as per my above imagination, since width of line segment is zero,  I am considering rectangles of zero width. Then each rectangle is of zero area (I proved it recently). So the area under curve will be zero! Paradox!

I realized that, just like ancient greeks, I am using very bad imagination of limiting process!

The Insight

But, as it turns out, my imagination is NOT completely wrong.  I googled and stumbled upon this stack exchange post:

There is the answer by Jonathan to this question which captures my imagination:

The idea is that since \int_0^n f(x)dx can be approximated by the Riemann sum, thus \displaystyle{\sum_{i=0}^n f(i) = \int_{0}^n f(x)dx + \text{higher order corrections}}

The generalization of above idea gives us the Euler–Maclaurin formula 

\displaystyle{\sum_{i=m+1}^n f(i) = \int^n_m f(x)\,dx + B_1 \left(f(n) - f(m)\right) + \sum_{k=1}^p\frac{B_{2k}}{(2k)!}\left(f^{(2k - 1)}(n) - f^{(2k - 1)}(m)\right) + R}

where m,n,p are natural numbers, f (x) is a real valued continuous function, B_k are the Bernoulli numbers and R is an error term which is normally small for suitable values of p (depends on n, m, p and f).

Proof of above formula is by principle of mathematical induction. For more details, see this beautiful paper: Apostol, T. M.. (1999). An Elementary View of Euler’s Summation Formula. The American Mathematical Monthly, 106(5), 409–418. http://doi.org/10.2307/2589145 

Maze

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Solving mazes is a great time-pass, if you “too” have nothing to do just try to solve this maze made by me (very easy one, can be solved within few seconds)

New Doc 8_1

Very easy example of maze

Since I am writing about mazes, there must be something mathematical about them. Indeed “solving a maze” is a NP-complete problem [Non-deterministic Polynomial-time solvable problem], one of most popular class of problems in Theoretical Computer Science. But, no surprise here, since lot of puzzles like Minesweeper, sudoku etc. too are NP-complete.

Leonhard Euler was one of the first to analyze plane mazes mathematically, and in doing so made the first significant contributions to the branch of mathematics known as topology.

Mazes containing no loops are known as “standard”, or “perfect” mazes, and are equivalent to a tree in graph theory. Thus many maze solving algorithms are closely related to graph theory. Intuitively, if one pulled and stretched out the paths in the maze in the proper way, the result could be made to resemble a tree (dead ends become the leaves of the tree.).

To make my maze I used “adding walls algorithm” i.e. I laid a set of obstructions within an open area.  Can you tell whether my maze above is “perfect” one??

As it turns out, there are various maze generation as well as maze solving algorithms. A detailed discussion of these can be found at: http://www.astrolog.org/labyrnth/algrithm.htm

Also, there is an interesting article on Maze solving using image analysis: http://www.crisluengo.net/index.php/archives/277