Math Is Fun Forum

Hi ganesh,

M#223 is the same as M#220

Nutted it out. Here's my answer:

The discrepancy between intuition and mathematics stems from several unrealistic features in the mathematical description of the game. Given the formal statement of the game, the player wishing only to maximise money should indeed pay any price at all to play. This conclusion is difficult to accept primarily because of bias towards risk-aversion, and the widespread, correct judgement that money in reality does not possess a linear utility. In fact, money usually possesses diminishing utility, and is sometimes modelled as proportional to the base-10 logarithm of wealth rather than to wealth itself. Applying this principle, the amount of utility derived from the game can be calculated as that given by precisely $4. However, the paradox reemerges for a variant of the game which has infinite utility: the pot starts at $100 and is multiplied by 10 to the power of 2 to the power of the number of heads for each additional heads. A like intuition might possibly suggest that this second game cannot, as a matter of certainty, be worth more than $10 septillion, and is virtually certain to be worth closer to $100,000; in any case, here too intuition does not consider it to be of infinite value.
The reason for this further discrepancy is the widespread and, again, correct judgement that there exists some huge amount of wealth after which absolutely no more utility is possible from gaining money. Suppose, for example, this amount is $75 trillion, approximately world GDP. For the original game, this brings the monetary value down to $47.06 floored, and the utility-adjusted value down to $3.99 floored. For the variant game, this brings the monetary value down to $9,375,012,502,550 (about $9.375 trillion), and the utility-adjusted value down to $54,247.86 floored. Clearly, the utility-and-maximum-wealth-adjusted values can provide reasonable answers given real human preferences.

#35. Henri Poincaré

Jules Henri Poincaré (29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and a philosopher of science. He is often described as a polymath, and in mathematics as The Last Universalist by Eric Temple Bell, since he excelled in all fields of the discipline as it existed during his lifetime.

As a mathematician and physicist, he made many original fundamental contributions to pure and applied mathematics, mathematical physics, and celestial mechanics. He was responsible for formulating the Poincaré conjecture, which was one of the most famous unsolved problems in mathematics until it was solved in 2002–2003. In his research on the three-body problem, Poincaré became the first person to discover a chaotic deterministic system which laid the foundations of modern chaos theory. He is also considered to be one of the founders of the field of topology.

Poincaré made clear the importance of paying attention to the invariance of laws of physics under different transformations, and was the first to present the Lorentz transformations in their modern symmetrical form. Poincaré discovered the remaining relativistic velocity transformations and recorded them in a letter to Dutch physicist Hendrik Lorentz (1853–1928) in 1905. Thus he obtained perfect invariance of all of Maxwell's equations, an important step in the formulation of the theory of special relativity.

The Poincaré group used in physics and mathematics was named after him.

Poincaré made many contributions to different fields of pure and applied mathematics such as: celestial mechanics, fluid mechanics, optics, electricity, telegraphy, capillarity, elasticity, thermodynamics, potential theory, quantum theory, theory of relativity and physical cosmology.

He was also a populariser of mathematics and physics and wrote several books for the lay public.

Among the specific topics he contributed to are the following:

algebraic topology
the theory of analytic functions of several complex variables
the theory of abelian functions
algebraic geometry
Poincaré was responsible for formulating one of the most famous problems in mathematics, the Poincaré conjecture, proven in 2003 by Grigori Perelman.
Poincaré recurrence theorem
hyperbolic geometry
number theory
the three-body problem
the theory of diophantine equations
the theory of electromagnetism
the special theory of relativity
In an 1894 paper, he introduced the concept of the fundamental group.
In the field of differential equations Poincaré has given many results that are critical for the qualitative theory of differential equations, for example the Poincaré sphere and the Poincaré map.
Poincaré on "everybody's belief" in the Normal Law of Errors
Published an influential paper providing a novel mathematical argument in support of quantum mechanics.

After defending his doctoral thesis on the study of singular points of the system of differential equations, Poincaré wrote a series of memoirs under the title "On curves defined by differential equations" (1881–1882). In these articles, he built a new branch of mathematics, called "qualitative theory of differential equations". Poincaré showed that even if the differential equation can not be solved in terms of known functions, yet from the very form of the equation, a wealth of information about the properties and behavior of the solutions can be found. In particular, Poincaré investigated the nature of the trajectories of the integral curves in the plane, gave a classification of singular points (saddle, focus, center, node), introduced the concept of a limit cycle and the loop index, and showed that the number of limit cycles is always finite, except for some special cases. Poincaré also developed a general theory of integral invariants and solutions of the variational equations. For the finite-difference equations, he created a new direction – the asymptotic analysis of the solutions. He applied all these achievements to study practical problems of mathematical physics and celestial mechanics, and the methods used were the basis of its topological works.

#34. Ettore Majorana

Ettore Majorana (born on 5 August 1906 – probably died after 1959) was an Italian theoretical physicist who worked on neutrino masses. He disappeared suddenly under mysterious circumstances while going by ship from Palermo to Naples. The Majorana equation and Majorana fermions are named after him. In 2006, the Majorana Prize was established in his memory.

"At Fermi's urging, Majorana left Italy early in 1933 on a grant from the National Research Council. In Leipzig, Germany, he met Werner Heisenberg. In letters he subsequently wrote to Heisenberg, Majorana revealed that he had found in him, not only a scientific colleague, but a warm personal friend." Majorana also travelled to Copenhagen, where he worked with Niels Bohr, another Nobel Prize winner, and a friend and mentor of Heisenberg.

The Nazis had come to power in Germany as Majorana arrived there. He studied with Werner Heisenberg in Leipzig, and worked on a theory of the nucleus (published in German in 1933) which, in its treatment of exchange forces, represented a further development of Heisenberg's theory of the nucleus. Majorana's last-published paper, in 1937, this time in Italian, was an elaboration of a symmetrical theory of electrons and positrons.

"In the fall of 1933, Majorana returned to Rome in poor health, having developed acute gastritis in Germany and apparently suffering from nervous exhaustion. Put on a strict diet, he grew reclusive and became harsh in his dealings with his family. To his mother, with whom he had previously shared a warm relationship, he had written from Germany that he would not accompany her on their customary summer vacation by the sea. Appearing at the institute less frequently, he soon was scarcely leaving his home; the promising young physicist had become a hermit. For nearly four years he shut himself off from friends and stopped publishing."

During these years, in which he published few articles, Majorana wrote many small works on several topics, from geophysics, to electrical engineering, from mathematics to relativity. These unpublished papers, preserved in Domus Galileiana in Pisa, recently have been edited by Erasmo Recami and Salvatore Esposito.

He became a full professor of theoretical physics at the University of Naples in 1937, without needing to take an examination because of his "high fame of singular expertise reached in the field of theoretical physics", independently of the competition rules.

Majorana did prescient theoretical work on neutrino masses, a currently active subject of research. He also worked on an idea that mass may exert a small shielding effect on gravitational waves, which did not gain much traction.

Hi (:

Suppose your brain is scanned; is it correct to one-box or two-box? The probability of the prediction going wrong is close to zero in this example (although I think the problem persists whenever the probability is below 49.5%). The predictor chooses, based on the prediction, what is in the box (for the sake of argument, say the prediction occurs before the boxes are presented to you). A million dollars if it predicts you will take one box, and nothing if it predicts you will take both.

Shall I explain the intuitions of those arriving at either answer to make clear what is so difficult?

That's true, it is worth at least $2. But the question is: What is the maximum entry fee (if any) that you should be willing to pay to play this game? Or: As the coin tosser, how high do I need to set the entry fee to make this game worthwhile (if it ever can be)? Or: What is the value of this game?

Hey!

Edit: Wait that below was for 4.4cm x 2.6cm x 1cm hahaha. No wonder it's so tiny It's out by a factor of 1,000,000

#32. Genghis Khan

Genghis Khan (c. 1162 – August 18 1227, born Temüjin, was the founder and Great Khan (emperor) of the Mongol Empire, which became the largest contiguous empire in history after his death.

He came to power by uniting many of the nomadic tribes of Northeast Asia. After founding the Mongol Empire and being proclaimed "Genghis Khan", he started the Mongol invasions that resulted in the conquest of most of Eurasia. These included raids or invasions of the Qara Khitai, Caucasus, Khwarezmid Empire, Western Xia and Jin dynasties. These campaigns were often accompanied by wholesale massacres of the civilian populations – especially in the Khwarezmian and Xia controlled lands. By the end of his life, the Mongol Empire occupied a substantial portion of Central Asia and China.

Before Genghis Khan died, he assigned Ögedei Khan as his successor and split his empire into khanates among his sons and grandsons. He died in 1227 after defeating the Western Xia. He was buried in an unmarked grave somewhere in Mongolia at an unknown location. His descendants extended the Mongol Empire across most of Eurasia by conquering or creating vassal states out of all of modern-day China, Korea, the Caucasus, Central Asia, and substantial portions of modern Eastern Europe, Russia, and Southwest Asia. Many of these invasions repeated the earlier large-scale slaughters of local populations. As a result, Genghis Khan and his empire have a fearsome reputation in local histories.

Beyond his military accomplishments, Genghis Khan also advanced the Mongol Empire in other ways. He decreed the adoption of the Uyghur script as the Mongol Empire's writing system. He also practiced meritocracy and encouraged religious tolerance in the Mongol Empire while unifying the nomadic tribes of Northeast Asia. Present-day Mongolians regard him as the founding father of Mongolia.

Although known for the brutality of his campaigns and considered by many to have been a genocidal ruler, Genghis Khan is also credited with bringing the Silk Road under one cohesive political environment. This brought communication and trade from Northeast Asia into Muslim Southwest Asia and Christian Europe, thus expanding the horizons of all three cultural areas.

Does anyone yet have more to contribute to resolving this matter? I am afraid I have never quite gotten over it

#29. Ludwig van Beethoven

Ludwig van Beethoven (baptised 17 December 1770 – 26 March 1827) was a German composer. A crucial figure in the transition between the Classical and Romantic eras in Western art music, he remains one of the most famous and influential of all composers. His best-known compositions include 9 symphonies, 5 piano concertos, 1 violin concerto, 32 piano sonatas, 16 string quartets, his great Mass the Missa solemnis and an opera, Fidelio.

Born in Bonn, then the capital of the Electorate of Cologne and part of the Holy Roman Empire, Beethoven displayed his musical talents at an early age and was taught by his father Johann van Beethoven and by composer and conductor Christian Gottlob Neefe. At the age of 21 he moved to Vienna, where he began studying composition with Joseph Haydn, and gained a reputation as a virtuoso pianist. He lived in Vienna until his death. By his late 20s his hearing began to deteriorate, and by the last decade of his life he was almost totally deaf. In 1811 he gave up conducting and performing in public but continued to compose; many of his most admired works come from these last 15 years of his life.

Beethoven is acknowledged as one of the giants of classical music; he is occasionally referred to as one of the "three Bs" (along with Bach and Brahms) who epitomise that tradition. He was also a pivotal figure in the transition from the 18th century musical classicism to 19th century romanticism, and his influence on subsequent generations of composers was profound. His music features twice on the Voyager Golden Record, a phonograph record containing a broad sample of the images, common sounds, languages, and music of Earth, sent into outer space with the two Voyager probes.

Beethoven composed in several musical genres and for a variety of instrument combinations. His works for symphony orchestra include nine symphonies (the Ninth Symphony includes a chorus), and about a dozen pieces of "occasional" music. He wrote seven concerti for one or more soloists and orchestra, as well as four shorter works that include soloists accompanied by orchestra. His only opera is Fidelio; other vocal works with orchestral accompaniment include two masses and a number of shorter works.

His large body of compositions for piano includes 32 piano sonatas and numerous shorter pieces, including arrangements of some of his other works. Works with piano accompaniment include 10 violin sonatas, 5 cello sonatas, and a sonata for French horn, as well as numerous lieder.

Beethoven also wrote a significant quantity of chamber music. In addition to 16 string quartets, he wrote five works for string quintet, seven for piano trio, five for string trio, and more than a dozen works for various combinations of wind instruments.