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Marshall Hall (mathematician)
Marshall Hall Jr. (17 September 1910 – 4 July 1990) was an American mathematician who made significant contributions to group theory and combinatorics. Career Hall studied mathematics at Yale University, graduating in 1932. He studied for a year at Cambridge University under a Henry Fellowship working with G. H. Hardy. He returned to Yale to take his Ph.D. in 1936 under the supervision of Øystein Ore. He worked in Naval Intelligence during World War II, including six months in 1944 at Bletchley Park, the center of British wartime code breaking. In 1946 he took a position at Ohio State University. In 1959 he moved to the California Institute of Technology where, in 1973, he was named the first IBM Professor at Caltech, the first named chair in mathematics. After retiring from Caltech in 1981, he accepted a post at Emory University in 1985. Hall died in 1990 in London on his way to a conference to mark his 80th birthday. Contributions He wrote a number of papers of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
St Louis
St. Louis () is the second-largest city in Missouri, United States. It sits near the confluence of the Mississippi and the Missouri Rivers. In 2020, the city proper had a population of 301,578, while the bi-state metropolitan area, which extends into Illinois, had an estimated population of over 2.8 million, making it the largest metropolitan area in Missouri and the second-largest in Illinois. Before European settlement, the area was a regional center of Native American Mississippian culture. St. Louis was founded on February 14, 1764, by French fur traders Gilbert Antoine de St. Maxent, Pierre Laclède and Auguste Chouteau, who named it for Louis IX of France. In 1764, following France's defeat in the Seven Years' War, the area was ceded to Spain. In 1800, it was retroceded to France, which sold it three years later to the United States as part of the Louisiana Purchase; the city was then the point of embarkation for the Corps of Discovery on the Lewis and Clark Expe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hall's Conjecture
In mathematics, Hall's conjecture is an open question, , on the differences between perfect squares and perfect cubes. It asserts that a perfect square ''y''2 and a perfect cube ''x''3 that are not equal must lie a substantial distance apart. This question arose from consideration of the Mordell equation in the theory of integer points on elliptic curves. The original version of Hall's conjecture, formulated by Marshall Hall, Jr. in 1970, says that there is a positive constant ''C'' such that for any integers ''x'' and ''y'' for which ''y''2 ≠ ''x''3, : , y^2 - x^3, > C\sqrt. Hall suggested that perhaps ''C'' could be taken as 1/5, which was consistent with all the data known at the time the conjecture was proposed. Danilov showed in 1982 that the exponent 1/2 on the right side (that is, the use of , ''x'', 1/2) cannot be replaced by any higher power: for no δ > 0 is there a constant ''C'' such that , ''y''2 - ''x''3, > C, ''x'', 1/2 + δ whenever ''y''2 ≠ ''x''3. In 196 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Burnside's Problem
The Burnside problem asks whether a finitely generated group in which every element has finite order must necessarily be a finite group. It was posed by William Burnside in 1902, making it one of the oldest questions in group theory and was influential in the development of combinatorial group theory. It is known to have a negative answer in general, as Evgeny Golod and Igor Shafarevich provided a counter-example in 1964. The problem has many refinements and variants (see bounded and restricted below) that differ in the additional conditions imposed on the orders of the group elements, some of which are still open questions. Brief history Initial work pointed towards the affirmative answer. For example, if a group ''G'' is finitely generated and the order of each element of ''G'' is a divisor of 4, then ''G'' is finite. Moreover, A. I. Kostrikin was able to prove in 1958 that among the finite groups with a given number of generators and a given prime exponent, there exists ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bletchley Park
Bletchley Park is an English country house and estate in Bletchley, Milton Keynes ( Buckinghamshire) that became the principal centre of Allied code-breaking during the Second World War. The mansion was constructed during the years following 1883 for the financier and politician Sir Herbert Leon in the Victorian Gothic, Tudor, and Dutch Baroque styles, on the site of older buildings of the same name. During World War II, the estate housed the Government Code and Cypher School (GC&CS), which regularly penetrated the secret communications of the Axis Powersmost importantly the German Enigma and Lorenz ciphers. The GC&CS team of codebreakers included Alan Turing, Gordon Welchman, Hugh Alexander, Bill Tutte, and Stuart Milner-Barry. The nature of the work at Bletchley remained secret until many years after the war. According to the official historian of British Intelligence, the "Ultra" intelligence produced at Bletchley shortened the war by two to four years, and without it th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
World War II
World War II or the Second World War, often abbreviated as WWII or WW2, was a world war that lasted from 1939 to 1945. It involved the vast majority of the world's countries—including all of the great powers—forming two opposing military alliances: the Allies and the Axis powers. World War II was a total war that directly involved more than 100 million personnel from more than 30 countries. The major participants in the war threw their entire economic, industrial, and scientific capabilities behind the war effort, blurring the distinction between civilian and military resources. Aircraft played a major role in the conflict, enabling the strategic bombing of population centres and deploying the only two nuclear weapons ever used in war. World War II was by far the deadliest conflict in human history; it resulted in 70 to 85 million fatalities, mostly among civilians. Tens of millions died due to genocides (including the Holocaust), starvation, ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Office Of Naval Intelligence
The Office of Naval Intelligence (ONI) is the military intelligence agency of the United States Navy. Established in 1882 primarily to advance the Navy's modernization efforts, it is the oldest member of the U.S. Intelligence Community and serves as the nation's premier source of maritime intelligence. Since the First World War, ONI's mission has broadened to include real-time reporting on the developments and activities of foreign navies; protecting maritime resources and interests; monitoring and countering transnational maritime threats; providing technical, operational, and tactical support to the U.S. Navy and its partners; and surveying the global maritime environment. ONI employs over 3,000 military and civilian personnel worldwide and is headquartered at the National Maritime Intelligence Center in Suitland, Maryland. History Despite playing an active and decisive role in the American Civil War, in the following years the U.S. Navy fell into precipitous decline. A lack o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Howard Engstrom
Howard T. Engstrom (1902 – 1962) was a Yale University mathematics professor and headed research operations at the United States Navy's Communication Supplementary Activities CSAW during World War II. Along with William Norris and others he founded Engineering Research Associates in 1946. He was one of the co-creators of the Univac computer, and served as deputy director of the National Security Agency. Education Engstrom graduated with a bachelor's degree in chemical engineering from Northeastern University in 1925, and received a master's from the University of Maine in 1922, where he also worked as a mathematics instructor. He was announced as an Instructor of mathematics at Yale in 1927, received his PhD from Yale in 1929, and was promoted to associate professor in 1941. Engstrom "was a national research fellow at the California Institute of Technology in 1930 and an international research fellow at Göttingen, Germany, in 1931". He was also a "fellow of the Institute of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics and from evolutionary biology to computer science. Combinatorics is well known for the breadth of the problems it tackles. Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, as well as in its many application areas. Many combinatorial questions have historically been considered in isolation, giving an ''ad hoc'' solution to a problem arising in some mathematical context. In the later twentieth century, however, powerful and general theoretical methods were developed, making combinatorics into an independent branch of mathematics in its own right. One of the oldest and most accessible parts of combinatorics is gra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field (mathematics), fields, and vector spaces, can all be seen as groups endowed with additional operation (mathematics), operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right. Various physical systems, such as crystals and the hydrogen atom, and Standard Model, three of the four known fundamental forces in the universe, may be modelled by symmetry groups. Thus group theory and the closely related representation theory have many important applications in physics, chemistry, and materials science. Group theory is also ce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History One of the earliest known mathematicians were Thales of Miletus (c. 624–c.546 BC); he has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed. He is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem. The number of known mathematicians grew when Pythagoras of Samos (c. 582–c. 507 BC) established the Pythagorean School, whose doctrine it was that mathematics ruled the universe and whose motto was "All is number". It was the Pythagoreans who coined the term "mathematics", and with whom the study of mathematics for its own sake begins. The first woman mathematician recorded by history was Hypati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Planar Ternary Ring
In mathematics, an algebraic structure (R,T) consisting of a non-empty set R and a ternary mapping T \colon R^3 \to R \, may be called a ternary system. A planar ternary ring (PTR) or ternary field is special type of ternary system used by Marshall Hall to construct projective planes by means of coordinates. A planar ternary ring is not a ring in the traditional sense, but any field gives a planar ternary ring where the operation T is defined by T(a,b,c) = ab + c. Thus, we can think of a planar ternary ring as a generalization of a field where the ternary operation takes the place of both addition and multiplication. In effect, in computer architecture, this ternary operation is known, e.g., as the multiply–accumulate operation (MAC). There is wide variation in the terminology. Planar ternary rings or ternary fields as defined here have been called by other names in the literature, and the term "planar ternary ring" can mean a variant of the system defined here. The term "ter ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
A-group
The A-Group culture was an ancient culture that flourished between the First and Second Cataracts of the Nile in Nubia. It lasted from 3800 BC to 3100 BC. Overview In 1907, the Egyptologist George A. Reisner first discovered artifacts belonging to the A-Group culture. Early hubs of this civilization included Kubaniyya in the north and Buhen in the south, with Aswan, Sayala, Toshka and Qustul in between. The A-Group population have been described as ethnically “very similar” to the pre-dynastic Egyptians in physical characteristics. The A-Group makers maintained commercial ties with the Ancient Egyptians. They traded commodities like incense, ebony and ivory, which were gathered from the southern riverine area. They also bartered carnelian from the Western Desert as well as gold mined from the Eastern Desert in exchange for Egyptian products, olive oil and other items from the Mediterranean basin. Excavation Findings The A-Group makers left behind a number of cemeter ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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