## Interesting Pages

## 2017-05-24

### 1-2017 More on the randomness of randomness.

Some time ago, I published this short article on randomness. I thought I was the only one who had such silly doubts on uch a well-known mathematical concept. And, now, I discover the profoundness of my question. I found this interesting article, on stackexchange. Thanks to the guys who posted the article and the responses. I love the opportunity to learn.

## 2016-10-27

### 12-2016 Men of mathematics -- a book review

Men of Mathematics: The Lives and Achievements of the Great Mathematicians
from Zeno to Poincare, by E. T. Bell

I discovered a great book today `` Men of Mathematics: The Lives and Achievements of the Great Mathematicians from Zeno to Poincare '', by E. T. Bell, a Touchstone Book, published by Simon and Schuster. The book was first published in 1937 by Scottish-born American mathematician and science fiction writer Eric Temple Bell (1883 -- 1960). Besides being a gifted mathematician and a prolific writer, the author was a President of the Mathematical Society of America, Vice-President of the American Mathematical Society and of the American Association for the Advancement of Science. One can expect much from a book written by an author with such impressive credentials. Curious students of mathematics, as well as mature teachers will find this book useful, since it suits both seasoned mathematicians and wannabes.

This is a 590 page book, and is available in paperback edition. The book contains bio-sketches of more than 27 mathematicians, arranged chronologically in 29 chapters. The book starts with Zeno of Elea (ca. 490 -- 430 BC) and ends with George Cantor (1845 -- 1918). Each chapter (except two chapters) is devoted to a single mathematician. Each chapter has an aptly chosen title which gives a hint of what is to follow. Chapters are not chained to any other chapter. The book can therefore be read in any sequence.

Contrary to the impression that the book's title may give, this book is ``NOT intended to be a history of mathematics, or any section of such a history''. It is also not a chronicle of just anecdotes and episodes in the life of these men. In this spirit, the book contains a fair sprinkling of mathematical snippets related to these mathematicians. The dosage of mathematics is just enough to keep the average reader from dozing off.

As mentioned before, the book starts with Zeno of Elea (ca. 490 -- 430 BC).Very little is known about Zeno's life. Zeno of Elea is often remembered for his philosophical problems based on paradoxes. Zeno's arguments are perhaps the first examples of a method of proof called reductio ad absurdum also known as proof by contradiction. The concept of “proof”, so central to all mathematics, seems to have grown from the principles set up by Zeno. It is not surprising that the book starts with a mention of Zeno.

This book gives us detailed images of the persons behind some great mathematical achievements. We can mention a few examples here. This book gives an opportunity to learn about the tug-of-war between Newton and Leibniz. It devotes an entire chapter to each of these two adversaries. Ironically, these two chapters are placed just one after the other. One can also get a glimpse of the miserable life of Abel, or the aristocratic lineage of Fermat

Some people say that this book has a few historical inaccuracies. Complaints like this are common in narrations of such vintage and magnitude. Such blemishes, if any, can be overlooked, considering the huge amount of information compiled by the author.

The lives of many mathematicians are witness to controversies and disputes. Cantor was no exception. He sums up his reactions stoically as ``The essence of mathematics resides in its freedom''. This sentence brings down the curtain on this wonderful book.

It is said that this book has inspired many young people, including the young John Forbes Nash Jr. and Freeman Dyson, to become mathematicians. It can therefore be of immense use to many of my colleagues and students.

* * *

I discovered a great book today `` Men of Mathematics: The Lives and Achievements of the Great Mathematicians from Zeno to Poincare '', by E. T. Bell, a Touchstone Book, published by Simon and Schuster. The book was first published in 1937 by Scottish-born American mathematician and science fiction writer Eric Temple Bell (1883 -- 1960). Besides being a gifted mathematician and a prolific writer, the author was a President of the Mathematical Society of America, Vice-President of the American Mathematical Society and of the American Association for the Advancement of Science. One can expect much from a book written by an author with such impressive credentials. Curious students of mathematics, as well as mature teachers will find this book useful, since it suits both seasoned mathematicians and wannabes.

This is a 590 page book, and is available in paperback edition. The book contains bio-sketches of more than 27 mathematicians, arranged chronologically in 29 chapters. The book starts with Zeno of Elea (ca. 490 -- 430 BC) and ends with George Cantor (1845 -- 1918). Each chapter (except two chapters) is devoted to a single mathematician. Each chapter has an aptly chosen title which gives a hint of what is to follow. Chapters are not chained to any other chapter. The book can therefore be read in any sequence.

Contrary to the impression that the book's title may give, this book is ``NOT intended to be a history of mathematics, or any section of such a history''. It is also not a chronicle of just anecdotes and episodes in the life of these men. In this spirit, the book contains a fair sprinkling of mathematical snippets related to these mathematicians. The dosage of mathematics is just enough to keep the average reader from dozing off.

As mentioned before, the book starts with Zeno of Elea (ca. 490 -- 430 BC).Very little is known about Zeno's life. Zeno of Elea is often remembered for his philosophical problems based on paradoxes. Zeno's arguments are perhaps the first examples of a method of proof called reductio ad absurdum also known as proof by contradiction. The concept of “proof”, so central to all mathematics, seems to have grown from the principles set up by Zeno. It is not surprising that the book starts with a mention of Zeno.

This book gives us detailed images of the persons behind some great mathematical achievements. We can mention a few examples here. This book gives an opportunity to learn about the tug-of-war between Newton and Leibniz. It devotes an entire chapter to each of these two adversaries. Ironically, these two chapters are placed just one after the other. One can also get a glimpse of the miserable life of Abel, or the aristocratic lineage of Fermat

Some people say that this book has a few historical inaccuracies. Complaints like this are common in narrations of such vintage and magnitude. Such blemishes, if any, can be overlooked, considering the huge amount of information compiled by the author.

The lives of many mathematicians are witness to controversies and disputes. Cantor was no exception. He sums up his reactions stoically as ``The essence of mathematics resides in its freedom''. This sentence brings down the curtain on this wonderful book.

It is said that this book has inspired many young people, including the young John Forbes Nash Jr. and Freeman Dyson, to become mathematicians. It can therefore be of immense use to many of my colleagues and students.

* * *

## 2016-05-07

### 11-2016 Enjoy this Major Technology Breakthrough !

Announcing the new Built-in Orderly Organized Knowledge-device (BOOK). The
BOOK is a revolutionary breakthrough in technology: No wires, no electric
circuits, no batteries, nothing to be connected or switched on. It's so
easy to use even a child can operate it. Just lift its cover!

Compact and portable, it can be used anywhere -- even sitting in an armchair by the fire -- yet it is powerful enough to hold as much information as a CD-ROM disc. Here's how it works...

Each BOOK is constructed of sequentially numbered sheets of paper (recyclable), each capable of holding thousands of bits of information. These pages are locked together with a custom-fit device called a binder which keeps the sheets in their correct sequence. Opaque Paper Technology (OPT) allows manufacturers to use both sides of the sheet, doubling the information density and cutting costs in half. Experts are divided on the prospects for further increases in information density; for now BOOKs with more information simply use more pages. This makes them thicker and harder to carry, and has drawn some criticism from the mobile computing crowd.

Each sheet is scanned optically, registering information directly into your brain. A flick of the finger takes you to the next sheet. The BOOK may be taken up at any time and used by merely opening it. The BOOK never crashes and never needs rebooting, though like other display devices it can become unusable if dropped overboard. The "browse" feature allows you to move instantly to any sheet, and move forward or backward as you wish. Many come with an "index" feature, which pinpoints the exact location of any selected information for instant retrieval.

An optional "BOOKmark" accessory allows you to open the BOOK to the exact place you left it in a previous session -- even if the BOOK has been closed. BOOKmarks fit universal design standards; thus, a single BOOKmark can be used in BOOKs by various manufacturers. Conversely, numerous BOOKmarks can be used in a single BOOK if the user wants to store numerous views at once. The number is limited only by the number of pages in the BOOK.

You can also make personal notes next to BOOK text entries with an optional programming tool, the Portable Erasable Nib Cryptic Intercommunication Stylus (PENCILS).

Portable, durable, and affordable, the BOOK is being hailed as the entertainment wave of the future. The BOOK's appeal seems so certain that thousands of content creators have committed to the platform.

Look for a flood of new titles soon.

(by Alan S. Zaben?)

Found at : http://www.infiltec.com/j-book.htm

Compact and portable, it can be used anywhere -- even sitting in an armchair by the fire -- yet it is powerful enough to hold as much information as a CD-ROM disc. Here's how it works...

Each BOOK is constructed of sequentially numbered sheets of paper (recyclable), each capable of holding thousands of bits of information. These pages are locked together with a custom-fit device called a binder which keeps the sheets in their correct sequence. Opaque Paper Technology (OPT) allows manufacturers to use both sides of the sheet, doubling the information density and cutting costs in half. Experts are divided on the prospects for further increases in information density; for now BOOKs with more information simply use more pages. This makes them thicker and harder to carry, and has drawn some criticism from the mobile computing crowd.

Each sheet is scanned optically, registering information directly into your brain. A flick of the finger takes you to the next sheet. The BOOK may be taken up at any time and used by merely opening it. The BOOK never crashes and never needs rebooting, though like other display devices it can become unusable if dropped overboard. The "browse" feature allows you to move instantly to any sheet, and move forward or backward as you wish. Many come with an "index" feature, which pinpoints the exact location of any selected information for instant retrieval.

An optional "BOOKmark" accessory allows you to open the BOOK to the exact place you left it in a previous session -- even if the BOOK has been closed. BOOKmarks fit universal design standards; thus, a single BOOKmark can be used in BOOKs by various manufacturers. Conversely, numerous BOOKmarks can be used in a single BOOK if the user wants to store numerous views at once. The number is limited only by the number of pages in the BOOK.

You can also make personal notes next to BOOK text entries with an optional programming tool, the Portable Erasable Nib Cryptic Intercommunication Stylus (PENCILS).

Portable, durable, and affordable, the BOOK is being hailed as the entertainment wave of the future. The BOOK's appeal seems so certain that thousands of content creators have committed to the platform.

Look for a flood of new titles soon.

(by Alan S. Zaben?)

Found at : http://www.infiltec.com/j-book.htm

## 2016-03-20

### 10-2016 Another incredible story of a woman mathematician

Christine Ladd-Franklin

Did a PhD dissertation: "On the Algebra of Logic" at The Johns Hopkins University 1926 (1882) United States

Her dissertation was completed in 1882, however, the school did not award her with a Ph.D until 1926. She waited 44 years to actually get her PhD, and died a mere 4 years later.

Source : Mathematics Genealogy Project http://genealogy.math.ndsu.nodak.edu/id.php?id=41944

Biography: Biography of Christine Ladd Franklin

Case study of the prejudices she suffered: Christine Ladd’s life story is a casebook study of the prejudices that women, who wished to enter academia suffered in the nineteenth and early twentieth centuries.

Did a PhD dissertation: "On the Algebra of Logic" at The Johns Hopkins University 1926 (1882) United States

Her dissertation was completed in 1882, however, the school did not award her with a Ph.D until 1926. She waited 44 years to actually get her PhD, and died a mere 4 years later.

Source : Mathematics Genealogy Project http://genealogy.math.ndsu.nodak.edu/id.php?id=41944

Biography: Biography of Christine Ladd Franklin

Case study of the prejudices she suffered: Christine Ladd’s life story is a casebook study of the prejudices that women, who wished to enter academia suffered in the nineteenth and early twentieth centuries.

## 2016-03-14

### 9-2016 Do not join the army.

René Eugène Gâteaux (5 May 1889 – 3 October 1914), was a French mathematician. He is known for the Gâteaux derivative. The Gâteaux derivative is a generalization of the concept of directional derivative in differential calculus. It is defined for functions between locally convex topological vector spaces such as Banach spaces.

Gâteaux was killed during World War I. Gâteaux was a fairly well-decorated soldier and was recalled for service in the war. He was killed during a retreat early in the conflict in France.

MORAL OF THE STORY: Do not join the army.

Gâteaux was killed during World War I. Gâteaux was a fairly well-decorated soldier and was recalled for service in the war. He was killed during a retreat early in the conflict in France.

MORAL OF THE STORY: Do not join the army.

### 8-2016 Topologists should not go swimming

Pavel Urysohn (1898-1924) was a Russian mathematician of Jewish origin who is best known for his contributions in dimension theory, and for developing Urysohn's Metrization Theorem and Urysohn's Lemma, both of which are fundamental results in topology.

Urysohn drowned while swimming in the sea, on vacation in France off the coast of Brittany, France, near Batz-sur-Mer, and is buried there. He was accompanied by his colleague and friend, Pavel Alexandrov (another famous mathematician). It is believed that Alexandrov was incredibly distressed by this tragedy, and deeply regretted his inability to save his friend.

MORAL OF THE STORY : Topologists should not go swimming.

Urysohn drowned while swimming in the sea, on vacation in France off the coast of Brittany, France, near Batz-sur-Mer, and is buried there. He was accompanied by his colleague and friend, Pavel Alexandrov (another famous mathematician). It is believed that Alexandrov was incredibly distressed by this tragedy, and deeply regretted his inability to save his friend.

MORAL OF THE STORY : Topologists should not go swimming.

### 7-2016 Algebraists are also human

Yutaka Taniyama (12 November 1927 – 17 November 1958) was a Japanese mathematician known for the Taniyama–Shimura-Weil conjecture. This conjecture inspired Andrew Wiles to work for a number of years in secrecy on it, and to prove enough of it to prove Fermat's Last Theorem (FLT). The correct proof of FLT was published in May 1995. Owing to the pioneering contribution of Wiles and the efforts of a number of mathematicians the Taniyama–Shimura-Weil conjecture was finally proven in 1999.

On 17 November 1958, Taniyama committed suicide. His enigmatic suicide note, which shows symptoms of stress and mental depression, mentions tiredness and a loss of confidence in his future. About a month later, Misako Suzuki, the woman whom he was planning to marry, also committed suicide, leaving a note reading: "We promised each other that no matter where we went, we would never be separated. Now that he is gone, I must go too in order to join him."

MORAL OF THE STORY : Algebraists are also human.

On 17 November 1958, Taniyama committed suicide. His enigmatic suicide note, which shows symptoms of stress and mental depression, mentions tiredness and a loss of confidence in his future. About a month later, Misako Suzuki, the woman whom he was planning to marry, also committed suicide, leaving a note reading: "We promised each other that no matter where we went, we would never be separated. Now that he is gone, I must go too in order to join him."

MORAL OF THE STORY : Algebraists are also human.

## 2016-03-13

### 6-2016 Taxis are no good for game theorists

John Forbes Nash, Jr. (June 13, 1928 – May 23, 2015) was an American mathematician who made fundamental contributions to game theory, differential geometry, and the study of partial differential equations. Nash's work has provided insight into the factors that govern chance and decision making inside complex systems found in daily life

Serving as a Senior Research Mathematician at Princeton University during the latter part of his life, he shared the 1994 Nobel Memorial Prize in Economic Sciences with game theorists Reinhard Selten and John Harsanyi. In 2015, he was awarded the Abel Prize for his work on nonlinear partial differential equations. One would have expected him to be a leading contender, perhaps even a virtual certainty, for a 1962 Fields' Medal, but mental illness destroyed his career, long before those decisions were made.

Nash’s research into game theory and his long struggle with paranoid schizophrenia became well known to the general public because of the Academy Award-winning motion picture A Beautiful Mind (2001), which was based on Sylvia Nasar’s 1998 biography of the same name. The film opened in the United States cinemas on December 21, 2001. It went on to gross over $313 million worldwide and win four Academy Awards, for Best Picture, Best Director, Best Adapted Screenplay and Best Supporting Actress. It was also nominated for Best Actor, Best Film Editing, Best Makeup, and Best Original Score.

On May 23, 2015, John and his wife Alicia Nash were killed in a car crash while riding in a taxi on the New Jersey Turnpike near Monroe Township, New Jersey, USA. They were on their way home after a visit to Norway, where Nash had received the Abel Prize.

MORAL OF THE STORY: Game theorists should not hire taxis.

Serving as a Senior Research Mathematician at Princeton University during the latter part of his life, he shared the 1994 Nobel Memorial Prize in Economic Sciences with game theorists Reinhard Selten and John Harsanyi. In 2015, he was awarded the Abel Prize for his work on nonlinear partial differential equations. One would have expected him to be a leading contender, perhaps even a virtual certainty, for a 1962 Fields' Medal, but mental illness destroyed his career, long before those decisions were made.

Nash’s research into game theory and his long struggle with paranoid schizophrenia became well known to the general public because of the Academy Award-winning motion picture A Beautiful Mind (2001), which was based on Sylvia Nasar’s 1998 biography of the same name. The film opened in the United States cinemas on December 21, 2001. It went on to gross over $313 million worldwide and win four Academy Awards, for Best Picture, Best Director, Best Adapted Screenplay and Best Supporting Actress. It was also nominated for Best Actor, Best Film Editing, Best Makeup, and Best Original Score.

On May 23, 2015, John and his wife Alicia Nash were killed in a car crash while riding in a taxi on the New Jersey Turnpike near Monroe Township, New Jersey, USA. They were on their way home after a visit to Norway, where Nash had received the Abel Prize.

MORAL OF THE STORY: Game theorists should not hire taxis.

### 5-2016 Do not depend on your wife for food

Kurt Friedrich Gödel (April 28, 1906 – January 14, 1978) was an Austrian, and later American, logician, mathematician, and philosopher. Considered with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel made an immense impact upon scientific and philosophical thinking in the 20th century, a time when others such as Bertrand Russell, A. N. Whitehead, and David Hilbert were pioneering the use of logic and set theory to understand the foundations of mathematics. Gödel published his two incompleteness theorems in 1931 when he was 25 years old, one year after finishing his doctorate at the University of Vienna.

He sadly succumbed to crippling paranoia later in life. In his sixties, he became convinced that his food was being poisoned, and would only trust the cooking of his wife Adele. When she was hospitalized for six months in 1977, Gödel refused to eat, and subsequently died of starvation.

MORAL OF THE STORY : If you are a logician, do not depend on your wife for food

He sadly succumbed to crippling paranoia later in life. In his sixties, he became convinced that his food was being poisoned, and would only trust the cooking of his wife Adele. When she was hospitalized for six months in 1977, Gödel refused to eat, and subsequently died of starvation.

MORAL OF THE STORY : If you are a logician, do not depend on your wife for food

## 2016-03-09

### 4-2016 Might is not always right

Archimedes of Syracuse (287 BC – 212 BC) was an Ancient Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. He is generally considered the greatest mathematician of antiquity and one of the greatest of all time.

Archimedes was contemplating a mathematical diagram when the city of Syracuse was captured. A Roman soldier commanded him to come and meet General Marcellus but he declined, saying that he had to finish working on the problem. The soldier was enraged by this, and killed Archimedes with his sword. Plutarch also gives a lesser-known account of the death of Archimedes which suggests that he may have been killed while attempting to surrender to a Roman soldier. According to this story, Archimedes was carrying mathematical instruments, and was killed because the soldier thought that they were lethal weapons.

MORAL OF THE STORY: If an armed brute is talking to you, hide all the mathematical instruments you have.

Archimedes was contemplating a mathematical diagram when the city of Syracuse was captured. A Roman soldier commanded him to come and meet General Marcellus but he declined, saying that he had to finish working on the problem. The soldier was enraged by this, and killed Archimedes with his sword. Plutarch also gives a lesser-known account of the death of Archimedes which suggests that he may have been killed while attempting to surrender to a Roman soldier. According to this story, Archimedes was carrying mathematical instruments, and was killed because the soldier thought that they were lethal weapons.

MORAL OF THE STORY: If an armed brute is talking to you, hide all the mathematical instruments you have.

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