Monthly Archives: February 2017

A Legal Tangle in Rome !!

Ancient  Roman mathematical works were utilitarian. Here is a Roman inheritance problem:

A dying Roman, knowing his wife was pregnant, left a will saying that if she had a son, he would inherit two-thirds of the estate and the widow one third; but if she had a daughter, the daughter would get one third and the widow two-thirds.

Soon after his death, his widow had twins — a boy and a girl. This is a possibility that the will maker had not foreseen. What division of the estate keeps as close as possible to the terms of the will?

(Ref: The Moscow Puzzles by Boris A. Kordemsky; Dover Publications).

Regards,

Nalin Pithwa

Donald Knuth, mathematician and computer scientist

Donald Knuth, American mathematician, computer scientist (born 1938):

(based on his brief biography presented in “Discrete Mathematics and its Applications by Kenneth H. Rosen):

Knuth grew up in Milwaukee, where his father taught book-keeping at a Lutheran high school and owned a small printing business. He was an excellent student, earning academic achievement awards. He applied his intelligence in unconditional ways, winning a contest, when he was in the eighth grade by finding over 4500 words that could be formed from the letters in “Ziegler’s Giant Bar.” This won a television set for his school and a candy bar for everyone in his class.

Knuth had a difficult time choosing physics over music as his major at the Case Institute of Technology. He then switched from physics to mathematics, and in 1960 he received his bachelor of science degree simultaneously receiving a master of science degree by a special award of the faculty who considered his work outstanding. At Case, he managed the basketball team and applied his talents by constructing a formula for the value of each player. This novel approach was covered by Newsweek and by Walter Cronkite on the CBS television network. Knuth began graduate work at the California Institute of Technology in 1960 and received his Ph.D. there in 1963. During this time, he worked as a consultant, writing compilers for different computers.

Knuth joined the staff of the California Institute of Technology in 1963, where he remained until 1968, when he took a job as a full time professor of Stanford University. He retired as Professor Emeritus in 1992 to concentrate on writing. He is especially interested in updating and completing new volumes of his series, The Art of Computer Programming, a work that has had a profound influence on the development of computer science, which he began writing as a graduate student in 1962, focusing on compilers. In common jargon, “Knuth” referring to The Art of Computer Programming has come to mean the reference that answers all questions about such topics as data structures and algorithms.

Knuth is the founder of the modern study of computational complexity. He has made fundamental contributions to the subject of compilers. His dissatisfaction with mathematics typography sparked him to invent the widely used TeX and Metafont systems. TeX has become a standard language for computer typography. Two of the many awards Knuth has received are the 1974 Turing Award and the 1979 National Medal of Technology, awarded to him by President Carter.

Knuth has written for a wide range of professional journals in computer science and in mathematics. However, his first publication in 1952, when he was a college freshman, was a parody of the metric system called “The Potrzebie Systems of Weight and Measures,” which appeared in MAD magazine and has been in reprint several times. He is a church organist, as his father was. He is also a composer of music for the organ. Knuth believes that writing computer programs can be an aesthetic experience, much like writing poetry or composing music.

Knuth pays USD 2.56 for the first person to find each error in his books and USD 0.32 for significant suggestionss, if you send him a letter with an error (you will need to use regular mail because he has given up reading e-mail), he will eventually inform you whether you were the first person to tell him about this error. Be prepared for a long wait, because he receives an overwhelming amount of mail. (The author Prof Kenneth H. Rosen received a letter years after sending an error report to Knuth, nothing that this report arrived several months after the first report of this error.)

With regards to Prof. Donald Knuth,

Nalin Pithwa

 

 

Derivatives of different orders — Leibniz’ Rule: IITJEE Maths training

Let a function y=f(x) be differentiable on some interval [a,b]. Generally speaking, the values of the derivative f^{'}(x) depend on x, which is to say that the derivative f^{'}(x) is also a function of x. Differentiating this function, we obtain the so-called second derivative of the function f(x).

The derivative of a first derivative is called a derivative of the second order or the second derivative of the original function and is denoted by the symbol y^{''} or f^{''}(x): y^{''}=(y^{'})^{'} = f^{''}(x)

For example, if y=x^{5}, then y^{'}=5x^{4} and y^{''} = (5x^{4})^{'}=20x^{3}

The derivative of the second derivative is called a derivative of the third order or the third derivative and is denoted by y^{'''} or f^{'''}(x).

Generally, a derivative of the nth order of a function f(x) is called the derivative (first order) of the derivative of the (n-1)th order and is denoted by the symbol y^{(n)} or f^{(n)}(x):

y^{(n)} = (y^{(n-1)})^{'}=f^{(n)}(x)

(Note: the order of the derivative is taken in parentheses so as to avoid confusion with the exponent of a power.)

Derivatives of the fourth, fifth and higher orders are also denoted by Roman numerals: y^{IV}, y^{V}, y^{VI}, \ldots. Here, the order of the derivative may be written without brackets. For instance, if y=x^{5}, y^{'}=5x^{4}, y^{''}=20x^{3}, y^{'''}=60x^{2}, y^{IV}=y^{(4)}=120x, y^{V}=y^{(5)}=100, y^{(6)}=y^{(7)}=\ldots = 0

Example 1:

Given a function y=e^{kx}, where k is a constant, find the expression of its derivative of any order n.

Solution 1:

y^{'}=ke^{kx}, y^{''}=k^{2}e^{kx}, y^{(n)}=k^{n}e^{kx}

Example 2:

y=\sin{x}. Find y^{(n)}.

Solution 2:

y^{'}=\cos{x}=\sin(x+\frac{\pi}{2})

y^{''}=-\sin{x}=\sin(x+2\frac{\pi}{2})

y^{'''}=-\cos{x}=\sin(x+3\frac{\pi}{2})

y^{IV}=\sin{x}=\sin(x+4\frac{\pi}{2})

\vdots

y^{(n)}=\sin(x+n\frac{\pi}{2})

In similar fashion, we can also derive the formulae for the derivatives of any order of certain other elementary functions. You can find yourself the formulae for derivatives of the n^{th} order of the functions y=x^{k}, y=\cos{x}, y=\ln (x).

Let us derive a formula called the Leibniz rule that will enable us to calculate the n^{th} derivative of the product of two functions u(x), v(x). To obtain this formula, let us first find several derivatives and then establish the general rule for finding the derivative of any order:

y=uv

y^{'}=u^{'}v+uv^{'}

y^{''}=u^{''}v+u^{'}v^{'}+u^{'}v^{'}+uv^{''}=u^{''}v+2u^{'}v^{'}+uv^{''}

y^{'''}=u^{'''}v+u^{''}v^{'}+2u^{''}v^{'}+2u^{'}v^{''}+u^{'}v^{''}+uv^{'''}

which in turn equals u^{'''}v+3u^{''}v^{'}+3u^{'}v^{''}+uv^{'''}

y^{IV}=u^{IV}v+4u^{'''}v+6u^{''}v^{''}+4u^{'}v^{'''}+uv^{IV}

The rule for forming derivatives holds for the derivative of any order and obviously consists in the following:

The expression (u+v)^{n} is expanded by the binomial theorem, and in the expansion obtained the exponents of the powers of and are replaced by indices that are the orders of the derivatives, and the zero  powers (u^{0}=v^{0}=1) in the end terms of the expansion are replaced by the function themselves (that is, “derivatives of zero order”):

y^{(n)}=(uv)^{(n)}=u^{(n)}v+nu^{(n-1)}v^{'}+\frac{n(n-1)}{1.2}u^{(n-2)}v^{''}+\ldots + uv^{(n)}

This is the Leibniz Rule.

A rigorous proof of this formula may be carried out by the method of complete mathematical induction (in other words, to prove that if this formula holds for the nth order, it will also hold for the order n+1).

Example:

y=e^{ax}x^{2}. Find the derivative y^{(n)}.

Solution:

u=e^{ax}, v=x^{2}

u^{'}=ae^{ax}, v^{'}=2x

u^{''}=a^{2}e^{ax}, v^{''}=2

\vdots, \vdots

u^{(n)}=a^{n}e^{ax}, v^{'''}=v^{IV}=\ldots=0

y^{(n)}=a^{n}e^{ax}x^{2}+na^{n-1}e^{ax}.2x+\frac{n(n-1)}{1.2}a^{n-2}e^{ax}.2, or

y^{(n)}=e^{ax}[a^{n}x^{2}+2na^{n-1}x+n(n-1)a^{n-2}]

More calculus in the pipeline…there is no limit to it 🙂

Nalin Pithwa

 

 

False Analogy from Moscow :-)

False Analogy:

Scientific discoveries are sometimes made by using analogy. It certain features of two objects are similar, perhaps other features are also similar. Analogy, however, is only a tool for good guesses. The guesses have to be tested.

Analogy has its place in math also, but alas, so does false analogy:

“By how much is 40 larger than 32?”

“By 8.”

“By how much is 32 smaller than 40?”

“By 8.”

“By what percentage is 40 larger than 32?”

“By 25 percent.”

“By what percentage is 32 smaller than 40?”

“By 25 percent.”

But, it is only 20 percent smaller.

(A) Suppose your monthly income increases 30 percent. By what percentage does your purchasing power increase?

(B) Suppose that your monthly income does not change, but prices go down 30 percent. By what percentage does your purchasing power increase?

(C) When a secondhand bookstore holds a sale with a 10 percent reduction in prices, it  makes an 8 percent profit on each book sold. What was the profit before the sale?

(D) If a metal worker reduces the time per part by p percent, how much does he increase his productivity?

Cheers,

Nalin Pithwa

Reference:

The Moscow Puzzles by Boris A. Kordemsky, Dover Publications.

PS: One of the best quote about  analogies in math was made by Stefan Banach, a famous Polish mathematician, with whom Stanislaw Ulam had collaborated. The quote is as follows:

One can imagine that the ultimate mathematician is one who can see analogies between analogies. A mathematician is a person who can find analogies between theorems; a better mathematician is one who can see analogies between proofs and the best mathematician can notice analogies between theories.

Edmund Landau mathematician

Edmund Landau (1877-1938)

Edmund Landau, the son of a Berlin gyanaecologist attended high school and university in Berlin. He received his doctorate in 1899, under the direction of Frobenius. Landau first taught at the University of Berlin and then moved to Gottingen, where he was a full professor until the Nazis forced him to stop teaching. Landau’s main contributions to mathematics were in the field of analytic number theory. In particular, he established several important results concerning the distribution of primes. He authored a three volume exposition on number theory as well as other books on number theory and (mathematical) analysis.

In particular, I like his classic, Foundations of Analysis, AMS Chelsea Publishing:

http://www.amazon.in/Foundations-Analysis-Third-Chelsea-Publishing/dp/082182693X/ref=sr_1_15?s=books&ie=UTF8&qid=1486127483&sr=1-15&keywords=Edmund+landau

-Nalin Pithwa