Tag Archives: cauchy

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.

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So many Integrals – I

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So many Integrals – I

We all know that, area is  the basis of integration theory, just as counting is basis of the real number system. So, we can say:

An integral is a mathematical operator that can be interpreted as an area under curve.

But, in mathematics we have various flavors of integrals named after their discoverers. Since the topic is a bit long, I have divided it into two posts. In this and next post I will write their general form and then will briefly discuss them.

Cauchy Integral

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Newton, Leibniz and Cauchy (left to right)

This was rigorous formulation of Newton’s & Leibniz’s idea of integration, in 1826 by French mathematician, Baron Augustin-Louis Cauchy.

Let f be a positive continuous function defined on an interval [a, b],\quad a, b being real numbers. Let P : a = x_0 < x_1 < x_2<\ldots < x_n = b, n being an integer, be a partition of the interval [a, b] and form the sum

S_p = \sum_{i=1}^n (x_i - x_{i-1}) f(t_i)

where t_i \in [x_{i-1} , x_i]f be such that f(t_i) = \text{Minimum} \{ f(x) : x \in [x_{i-1}, x_{i}]\}

By adding more points to the partition P, we can get a new partition, say P', which we call a ‘refinement’ of P and then form the sum S_{P'}.  It is trivial to see that S_P \leq S_{P'} \leq \text{Area bounded between x-axis and function}f

Since, f is continuous (and positive), then S_P becomes closer and closer to a unique real number, say kf, as we take more and more refined partitions in such a way that |P| := \text{Maximum} \{x_i - x_{i-1}, 1 \leq i \leq n\} becomes closer to zero. Such a limit will be independent of the partitions. The number k is the area bounded by function and x-axis and we call it the Cauchy integral of f over a  to b. Symbolically, \int_{a}^{b} f(x) dx (read as “integral of f(x)dx from a to b”).


 

Riemann Integral

Riemann_3

Riemann

Cauchy’s definition of integral can readily be extended to a bounded function with finitely many discontinuities. Thus, Cauchy integral does not require either the assumption of continuity or any analytical expression of f to prove that the sum S_p indeed converges to a unique real number.

In 1851, a German mathematician, Georg Friedrich Bernhard Riemann gave a more general definition of integral.

Let [a,b] be a closed interval in \mathbb{R}. A finite, ordered set of points P :\{ a = x_0 < x_1 < x_2<\ldots < x_n = b\}, n being an integer, be a partition of the interval [a, b]. Let, I_j denote the interval [x_{j-1}, x_j], j= 1,2,3,\ldots , n. The symbol \Delta_j denotes the length of I_j. The mesh of P, denoted by m(P), is defined to be max\Delta_j.

Now, let f be a function defined on interval [a,b]. If, for each j, s_j is an element of I_j, then we define:

S_P = \sum_{j=1}^n f(s_j) \Delta_j

Further, we say that S_P tend to a limit k as m(P) tends to 0 if, for any \epsilon > 0, there is a \delta >0 such that, if P is any partition of [a,b] with m(P) < \delta, then |S_P - k| < \epsilon for every choice of s_j \in I_j.

Now, if S_P tends to a finite limit as m(P) tends to zero, the value of the limit is called Riemann integral of f over [a,b] and is denoted by \int_{a}^{b} f(x) dx


 

Darboux Integral

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Darboux

In 1875, a French mathematician, Jean Gaston Darboux  gave his way of looking at the Riemann integral, defining upper and lower sums and defining a function to be integrable if the difference between the upper and lower sums tends to zero as the mesh size gets smaller.

Let f be a bounded function defined on an interval [a, b],\quad a, b being real numbers. Let P : a = x_0 < x_1 < x_2<\ldots < x_n = b, n being an integer, be a partition of the interval [a, b] and form the sum

S_P = \sum_{i=1}^n (x_i - x_{i-1}) f(t_i), \quad \overline{S}_P =\sum_{i=1}^n (x_i - x_{i-1}) f(s_i)

where t_i,s_i \in [x_{i-1} , x_i] be such that

f(t_i) = \text{sup} \{ f(x) : x \in [x_{i-1}, x_{i}]\},

f(s_i) = \text{inf} \{ f(x) : x \in [x_{i-1}, x_{i}]\}

The sums S_P and \overline{S}_P represent the areas and  S_P \leq \text{Area bounded by curve} \leq \overline{S}_P. Moreover, if P' is a refinement of P, then

S_p \leq S_{P'} \leq \text{Area bounded by curve} \leq \overline{S}_{P'} \leq \overline{S}_{P}

Using the boundedness of f, one can show that S_P, \overline{S}_P converge as the partition get’s finer and finer, that is |P| := \text{Maximum}\{x_i - x_{i-1}, 1 \leq i \leq n\} \rightarrow 0, to some real numbers, say k_1, k_2 respectively. Then:

k_l \leq \text{Area bounnded by the curve} \leq k_2

If k_l = k_2 , then we have \int_{a}^{b} f(x) dx = k_l = k_2.


There are two more flavours of integrals which I will discuss in next post. (namely, Stieltjes Integral and Lebesgue Integral)