Math with Umair

1. The group of Quaternions is a non-Abelian group of order:

(a) 6
(b) 8
(c) 10
(d) 4

Historical Note: 22

Erhardt Schmidt (1875-1959) was a German Mathematician who studied for his doctoral degree at Gottingen University under David Hilbert, one of the giants of modern mathematics. For most of his life taught at Berlin University where, in addition to making important contributions to many branches of mathematics, he fashioned some of Hilbert's ideas into a general concept, called a Hilbert Space --- a fundamental structure in the study of infinite-dimensional vector spaces. He first described the process that bears his name in a paper on integral equations that he published in 1907.

PROBLEM 40 OF THE AHMES PAPYRUS...

The Ahmes (or Rhind) Papyrus is the source of most of our information about ancient Egyptian Mathematicians. This 5-meter-long papyrus contains 84 short mathematical problems, together with their solutions, and dates from about 1650 B.C.
Problem 40 in this papyrus is the following:

"Divide 100 hekats of barley among five men in arithmetic progression so that the sum of the two smallest is one-seventh the sum of three largest."

YAW , PITCH and ROLL

In aeronautics and astronautics, the orientation of an aircraft or space shuttle relative to an xyz-coordinate system is often described in terms of angles called yaw, pitch and roll. If, for example, an aircraft is flying along the y-axis and the xy-plane defined the horizontal, then the aircraft's angle of rotation about the z-axis is called the yaw, its angle of rotation about the x-axis is called the pitch, and its angle of rotation about the y-axis is called the roll. A combination of yaw, pitch and roll can be achieved by a single rotation about some axis through the origin. This is, in fact, how a space shuttle makes attitude adjustments --- it doesn't perform each rotation separately; it calculates one axis, and rotates about that axis to get the correct orientation. Such rotation maneuvers are used to align an antenna, point the nose toward a celestial object, or position a payload bay for docking.

Biography No.: 33 (W. R. Hamilton)

WILLIAM ROWAN HAMILTON was born on August 3, 1805, in Dublin, Ireland. At three, he was skilled at reading and arithmetic. At five, he read and translated Latin, Greek and Hebrew; at 14, he had mastered 14 languages, including Arabic, Sanskrit, Hindustani, Malay and Bengali.

In 1833, Hamilton provided the first modern treatment of complex numbers. In 1843, he made what he considered his greatest discovery --- the algebra of quaternions. The quaternions represent a natural generalization of the complex numbers with three numbers i , j and k whose squares are -1. With these, rotations in three and four dimensions can be algebraically treated. Of greater significance, however, is the fact that the quaternions are non-commutative under multiplication. This was the first ring to be discovered in which the commutative property does not hold. The essential idea for the quaternions suddenly came to Hamilton after 15 years of fruitless thought!

Today Hamilton's name is attached to several concepts, such as the Hamilton function, which represents the total energy in a physical system; the Hamilton-Jacobi differential equations; and the Cayley-Hamilton Theorem from linear algebra. he also coined the terms vector, scalar and tensor.

In his later years, Hamilton was plagued by alcoholism. He died on September 2, 1865, at the age of 60.

After Isaac Newton, the Greatest Mathematician of the English speaking peoples is William Rowan Hamilton. (Sir Edmund Whittaker - SCIENTIFIC SOCIETY)

Historical Note: 21

Markov chains are named in honor of the Russian Mathematician Andrei Andreyevich Markov, a lover of poetry, who used them to analyze the alternation of vowels and consonants in the poem Eugene Onegin by Pushkin. Markov believed that the only applications of his chains were to the analyze of literary works, so he would be astonished to learn that his discovery is used today in the social sciences, quantum theory and genetics.

Applications of RANK...

The advent of the Internet has stimulated research on finding efficient methods for transmitting large amounts of digital data over communications lines with limited bandwidths. Digital data are commonly stored in matrix form, many techniques for improving transmission speed use the rank of a matrix in some way. Rank plays a role because it measures the "redundancy" in a matrix in the sense that if A is an m x n matrix of rank k, then n - k of the column vectors and m - k of the row vectors can be expressed in terms of k linearly independent column or row vectors. The essential idea in many data compression schemes is to approximate the original data set by a data set with smaller rank conveys nearly the same information, then eliminate redundant vectors in the approximating set to speed up the transmission time.

You know my method. Apply them! (Sherlock Holmes)

For every problem there is a solution which is simple, clean and wrong. (H. L. Mencken)

The paradox of excellence is that it is built upon the foundations of necessary failure. (Matthew Syed)

Biography No. 32 (William Burnside)

WILLIAM BURNSIDE was born on July 2, 1852, in London. After graduating from Cambridge University in 1875. Burnside was appointed lecturer at Cambridge, where he stayed until 1885. He then accepted a position at the Royal Naval College at Greenwich and spent the rest of his career in that post.

Burnside wrote more than 150 research papers in many fields. He is best remembered, however, for his pioneering work in group theory and his classic book "Theory of Groups" which first appeared in 1897. Because of Burnside's emphasis on the abstract approach, many consider him to be the first pure group theorist.

One mark of greatness in a mathematician is the ability to pose important and challenging problems -- problems that open up new areas of research for future generations. Here, Burnside excelled. It was he who first conjectured that a group G of odd order has a series of normal subgroups, G = G0 >= G1 >= G2 >= ... >= Gn = {e} such that G_i / G_(i+1) is Abelian. This extremely important conjecture was finally proved more than 50 years later by Feit and Thompson in 255-page paper. In 1994, Efim Zelmanov received Fields Medal for his work on a variation of one of Burnside's conjecture.

Burnside was elected a Fellow of the Royal Society and awarded two Royal Medals. He served as a president of the council of the London Mathematical Society and received its De Morgan Medal. Burnside died on August 21, 1927.