Monthly Archives: November 2015

Nov25

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

Darboux.jpg

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)

Nov19

Gauss Reborn

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Carl Friedrich Gauss wrote a beautiful textbook in Latin for (Algebraic) Number Theory titled “Disquisitiones Arithmeticae” (when he was 21 years old).  In 21st century, Manjul Bhargava is extending and simplifying the ideas of Gauss. His PhD thesis (at age of 26 years) generalized Gauss’s classical law for composition of binary quadratic forms to many other situations. For a glimpse of his work, refer his article: The Factorial Function and Generalizations

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Manjul Bhargava (left); Carl Friedrich Gauss (right)

Like other great mathematicians, he lives mathematics, for example read: Numbers, toys and music: A conversation with Manjul Bhargava

Last year (at age of 40 years) he was awarded the most prestigious award on our planet for young mathematicians, “Fields Medal” for :

 developing powerful new methods in the geometry of numbers, which he applied to count rings of small rank and to bound the average rank of elliptic curves.

You can read his beautiful lecture on his contributions to mathematics at: Answers on a donut – the Fields medal lecture of Manjul Bhargava

 

Nov15

Puzzle : Free cab ride

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Today, intra-city cab services are provided with aid of mobile apps. So to promote usage of their apps they have following condition and offer:

If you install the app, you need to register your app with your mobile number and you can’t register with your mobile number more than once. You suggest the app to others and those people who use your reference code in registration, get a free ride. Once the people using your reference code avail their first free ride, you get a complementary free ride (corresponding to each person). So, every time a person avail a free ride using your reference code, you get a free ride!

Now my problem for you is:

There are n people (each having only one mobile number) acquainted to each other and want to exploit this offer of free rides. What is the maximum number of free rides this group of n people can enjoy?

You can post you solution in comments, I will post my solution later. Happy puzzling.

Nov5

Metamathematics

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I like to predict things going to happen each day based upon some (il)logical reasoning. Also, earlier this year I wrote a blog post: “Inquisitive Mathematical Thinking“.

Today, inspired by various “wrong” interpretation of “Principle of Mathematical Induction” which leads to various absurd results, I came up with an idea of giving a “flawed” proof of (i.e. I will deliberately mimic what I call “rape of Mathematics”):

Every day is good day.

Let me start with following induction Hypothesis:
P(1): Today is a good day.
This is trivially true (just like the statement “the sun rises in east”).

Now, let’s assume the truth of following statement:
P(k): k^{th} day is good day.

Now, what remains to prove is that P(k) \Rightarrow P(k+1), where:
P(k+1): (k+1)^{th} day is good day.

Since, our past actions determine our future results (metaphysical truth), if today is good day then I will be able to prepare for tomorrow’s challenges and hence my tomorrow will be good. Thus proving the inductive step.

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The above proof has lot of logical flaws like: ” How do you define a day to be good?” and “Implication is based on a metaphysical truth”.

Bottom line: Life not as simple as Mathematics!