source
stringlengths 31
168
| text
stringlengths 51
3k
|
|---|---|
https://en.wikipedia.org/wiki/Alexander%20Skopin
|
Alexander Ivanovich Skopin (Александр Иванович Скопин) (1927–2003) was a Russian mathematician known for his contributions to abstract algebra.
Biography
Skopin was born on October 22, 1927, in Leningrad, the son of Ivan Alexandrovich Skopin, who was himself also a number theorist and a student of Ivan Matveyevich Vinogradov, and who died in the Siege of Leningrad. After the war, Alexander Skopin studied at Leningrad University, where he was a student of Dmitry Faddeev; From that point to the end of his life, he worked as a researcher at the Steklov Mathematical Institute (where he was scientific secretary from the mid-1960s to the early 1970s) and taught algebra at the St. Petersburg University. He died on September 15, 2003, in St. Petersburg.
Research
Skopin's student work was in abstract algebra, and concerned upper central series of groups and extensions of fields. In the 1970s, Skopin received a second doctorate concerning the application of computer algebra systems to group theory. From that point onward he used computational methods extensively in his research, which focussed on lower central series of Burnside groups. He related this problem to problems in other areas of mathematics including linear algebra and topological sorting of graphs.
References
1927 births
2003 deaths
20th-century Russian mathematicians
Algebraists
|
https://en.wikipedia.org/wiki/Nikolai%20Kochin
|
Nikolai Yevgrafovich Kochin (; 19 May 1901, St Petersburg – 31 December 1944, Moscow) was a Russian and Soviet mathematician specialising in applied mathematics, and especially fluid and gas mechanics.
Biography
Kochin graduated from Petrograd University in 1923. He taught mathematics and mechanics at Leningrad State University from 1924 to 1934.
In 1925 Kochin married Pelageya Polubarinova. They had two daughters.
In 1928 Kochin spent a semester in Göttingen, where he helped Gamow to solve the alpha decay problem through quantum tunneling.
Kochin moved to Moscow 1934. He taught mathematics and mechanics at Moscow State University from 1934 until his death, and was the head of the mechanics section of the Mechanics Institute of the USSR Academy of Sciences from 1939 to 1944.
In 1943 Kochin became ill with sarcoma and died in 1944.
Research interests
Kochin's research was on meteorology, gas dynamics and shock waves in compressible fluids. He gave the solution to the problem of small amplitude waves on the surface of an uncompressed liquid in Towards a Theory of Cauchy-Poisson Waves in 1935.
He also worked on the pitch and roll of ships. In aerodynamics he introduced formulae for aerodynamic force and for the distribution of pressure.
Bibliography
Kochin wrote textbooks on hydromechanics and vector analysis:
Theoretical hydromechanics, by N. E. Kochin, I. A. Kibel, and N. V. Roze. Translated from the fifth Russian ed. by D. Boyanovitch. Edited by J. R. M. Radok. Publisher: New York, Interscience Publishers, [1965,1964]
Articles
Kochin N. On the instability of von Karman vortex streets
Comptes Rendus de l'Académie des Sciences de l'URSS 24: 19-23 1939
External Biographies
Articles
Biography in Dictionary of Scientific Biography (New York 1970-1990).
S. M. Belotserkovskii, The work of N. E. Kochin and some current problems of aerodynamics and hydrodynamics, N E Kochin and the development of mechanics, 'Nauka' (Moscow, 1984), 115-130.
A. A. Dorodnitsyn, N. E. Kochin-principles and system of the scientist's work (Russian), N. E. Kochin and the development of mechanics, 'Nauka' (Moscow, 1984), 8-12.
A. Yu. Ishlinskii, Nikolai Evgrafovich Kochin (Russian), N. E. Kochin and the development of mechanics, 'Nauka' (Moscow, 1984), 5-8.
A. Yu. Ishlinskii, N. E. Kochin and theoretical mechanics, N. E. Kochin and the development of mechanics, 'Nauka' (Moscow, 1984), 169-174.
S. A. Khristianovich, The works of N. E. Kochin (Russian), N. E. Kochin and the development of mechanics, 'Nauka' (Moscow, 1984), 13-19.
Nikolai Evgrafovich Kochin (on the occasion of the ninetieth anniversary of his birth) (Russian), Prikl. Mat. Mekh. 55 (4) (1991), 533-534.
Books
P. Ya. Polubarinova-Kochina, Life and Work of N. Y. Kochin (Leningrad, 1950).
Obituary
Nikolai Evgrafovich Kochin (Russian), Appl. Math. Mech. 9 (1945), 3-12.
References
External links
Picture – from the Russian Academy of Sciences web site.
1901 births
1944 deaths
20th-century Russian mathem
|
https://en.wikipedia.org/wiki/Jos%C3%A9%20Anast%C3%A1cio%20da%20Cunha
|
José Anastácio da Cunha (1744 – January 1, 1787) was a Portuguese mathematician. He is best known for his work on the theory of equations, algebraic analysis, plain and spherical trigonometry, analytical geometry, and differential calculus.
References
External links
1744 births
1787 deaths
18th-century Portuguese mathematicians
People from Lisbon
|
https://en.wikipedia.org/wiki/Edward%20Wegman
|
Edward Wegman is an American statistician and was a professor of statistics at George Mason University until his retirement in 2018. He holds a Ph.D. in mathematical statistics and is a Fellow of the American Statistical Association, a Senior Member of the IEEE, and past chair of the National Research Council Committee on Applied and Theoretical Statistics. In addition to his work in the field of statistical computing, Wegman contributed a report to a Congressional hearing on climate change at the request of Republican Rep. Joe Barton. Wegman's report supported criticisms of the methodology of two specific paleoclimate studies into the temperature record of the past 1000 years, and argued that climate scientists were excessively isolated from the statistical mainstream. Subsequently, significant portions of Wegman's report were found to have been copied without attribution from a variety of sources, including Wikipedia, and a publication based on the report was retracted.
Career
Wegman, a St. Louis, Missouri native, received a B.S. in mathematics from Saint Louis University in 1965. He then went to graduate school at the University of Iowa, where he earned an M.S. in 1967 and a Ph.D. in 1968, both in mathematical statistics. He held a faculty position at the University of North Carolina for ten years. In 1978, Wegman joined the Office of Naval Research, in which he headed the Mathematical Sciences Division. Later, Wegman served as the first program director of the Ultra High Speed Computing basic research program for the Strategic Defense Initiative's Innovative Science and Technology Office. He joined the faculty of George Mason University in 1986 and developed a master's degree program in statistical science. He retired from his position at George Mason University in 2018.
Wegman is credited with coining the phrase "computational statistics" and developing a high-profile research program around the concept that computing resources could transform statistical techniques. He also has been the associate editor of seven academic journals, a member of numerous editorial boards, and the author of more than 160 papers and five books. Wegman is a member of the American Statistical Association, a former president of the International Association for Statistical Computing, and a past chairman of the Committee on Applied and Theoretical Statistics for the United States National Academy of Sciences. Wegman received the 2002 Founders Award from the American Statistical Association for "over thirty years of exceptional service and leadership to the American Statistical Association."
Energy and Commerce hearing & plagiarism
In 2006, Republican Congressman Joe Barton chose Wegman to assist the House Energy and Commerce Committee in its inquiry criticizing the multi-proxy paleoclimate reconstructions which had been dubbed the "hockey stick graph". Wegman produced a report and offered testimony supporting published papers disputing the methodology and
|
https://en.wikipedia.org/wiki/Electronic%20filter%20topology
|
Electronic filter topology defines electronic filter circuits without taking note of the values of the components used but only the manner in which those components are connected.
Filter design characterises filter circuits primarily by their transfer function rather than their topology. Transfer functions may be linear or nonlinear. Common types of linear filter transfer function are; high-pass, low-pass, bandpass, band-reject or notch and all-pass. Once the transfer function for a filter is chosen, the particular topology to implement such a prototype filter can be selected so that, for example, one might choose to design a Butterworth filter using the Sallen–Key topology.
Filter topologies may be divided into passive and active types. Passive topologies are composed exclusively of passive components: resistors, capacitors, and inductors. Active topologies also include active components (such as transistors, op amps, and other integrated circuits) that require power. Further, topologies may be implemented either in unbalanced form or else in balanced form when employed in balanced circuits. Implementations such as electronic mixers and stereo sound may require arrays of identical circuits.
Passive topologies
Passive filters have been long in development and use. Most are built from simple two-port networks called "sections". There is no formal definition of a section except that it must have at least one series component and one shunt component. Sections are invariably connected in a "cascade" or "daisy-chain" topology, consisting of additional copies of the same section or of completely different sections. The rules of series and parallel impedance would combine two sections consisting only of series components or shunt components into a single section.
Some passive filters, consisting of only one or two filter sections, are given special names including the L-section, T-section and Π-section, which are unbalanced filters, and the C-section, H-section and box-section, which are balanced. All are built upon a very simple "ladder" topology (see below). The chart at the bottom of the page shows these various topologies in terms of general constant k filters.
Filters designed using network synthesis usually repeat the simplest form of L-section topology though component values may change in each section. Image designed filters, on the other hand, keep the same basic component values from section to section though the topology may vary and tend to make use of more complex sections.
L-sections are never symmetrical but two L-sections back-to-back form a symmetrical topology and many other sections are symmetrical in form.
Ladder topologies
Ladder topology, often called Cauer topology after Wilhelm Cauer (inventor of the elliptic filter), was in fact first used by George Campbell (inventor of the constant k filter). Campbell published in 1922 but had clearly been using the topology for some time before this. Cauer first picked up on ladders
|
https://en.wikipedia.org/wiki/Dirichlet%20density
|
In mathematics, the Dirichlet density (or analytic density) of a set of primes, named after Peter Gustav Lejeune Dirichlet, is a measure of the size of the set that is easier to use than the natural density.
Definition
If A is a subset of the prime numbers, the Dirichlet density of A
is the limit
if it exists. Note that since as (see Prime zeta function), this is also equal to
This expression is usually the order of the "pole" of
at s = 1, (though in general it is not really a pole as it has non-integral order), at least if this function is a holomorphic function times a (real) power of s−1 near s = 1. For example, if A is the set of all primes, it is the Riemann zeta function which has a pole of order 1 at s = 1, so the set of all primes has Dirichlet density 1.
More generally, one can define the Dirichlet density of a sequence of primes (or prime powers), possibly with repetitions, in the same way.
Properties
If a subset of primes A has a natural density, given by the limit of
(number of elements of A less than N)/(number of primes less than N)
then it also has a Dirichlet density, and the two densities are the same.
However it is usually easier to show that a set of primes has a Dirichlet density, and this is good enough for many purposes. For example, in proving Dirichlet's theorem on arithmetic progressions, it is easy to show that the set of primes
in an arithmetic progression a + nb (for a, b coprime) has Dirichlet density 1/φ(b), which is enough to show that there are an infinite number of such primes, but harder to show that this is the natural density.
Roughly speaking, proving that some set of primes has a non-zero Dirichlet density usually involves showing that certain L-functions do not vanish at the point s = 1, while showing that they have a natural density involves showing that the L-functions have no zeros on the line Re(s) = 1.
In practice, if some "naturally occurring" set of primes has a Dirichlet density, then it also has a natural density, but it is possible to find artificial counterexamples: for example, the set of primes whose first decimal digit is 1 has no natural density, but has Dirichlet density log(2)/log(10).
See also
Natural density
Notes
References
J.-P. Serre, A course in arithmetic, , chapter VI section 4.
Analytic number theory
|
https://en.wikipedia.org/wiki/Limit%20theorem
|
Limit theorem may refer to:
Central limit theorem, in probability theory
Edgeworth's limit theorem, in economics
Plastic limit theorems, in continuum mechanics
Mathematics disambiguation pages
|
https://en.wikipedia.org/wiki/Quantities%20of%20information
|
The mathematical theory of information is based on probability theory and statistics, and measures information with several quantities of information. The choice of logarithmic base in the following formulae determines the unit of information entropy that is used. The most common unit of information is the bit, or more correctly the shannon, based on the binary logarithm. Although "bit" is more frequently used in place of "shannon", its name is not distinguished from the bit as used in data-processing to refer to a binary value or stream regardless of its entropy (information content) Other units include the nat, based on the natural logarithm, and the hartley, based on the base 10 or common logarithm.
In what follows, an expression of the form is considered by convention to be equal to zero whenever is zero. This is justified because for any logarithmic base.
Self-information
Shannon derived a measure of information content called the self-information or "surprisal" of a message :
where is the probability that message is chosen from all possible choices in the message space . The base of the logarithm only affects a scaling factor and, consequently, the units in which the measured information content is expressed. If the logarithm is base 2, the measure of information is expressed in units of shannons or more often simply "bits" (a bit in other contexts is rather defined as a "binary digit", whose average information content is at most 1 shannon).
Information from a source is gained by a recipient only if the recipient did not already have that information to begin with. Messages that convey information over a certain (P=1) event (or one which is known with certainty, for instance, through a back-channel) provide no information, as the above equation indicates. Infrequently occurring messages contain more information than more frequently occurring messages.
It can also be shown that a compound message of two (or more) unrelated messages would have a quantity of information that is the sum of the measures of information of each message individually. That can be derived using this definition by considering a compound message providing information regarding the values of two random variables M and N using a message which is the concatenation of the elementary messages m and n, each of whose information content are given by and respectively. If the messages m and n each depend only on M and N, and the processes M and N are independent, then since (the definition of statistical independence) it is clear from the above definition that .
An example: The weather forecast broadcast is: "Tonight's forecast: Dark. Continued darkness until widely scattered light in the morning." This message contains almost no information. However, a forecast of a snowstorm would certainly contain information since such does not happen every evening. There would be an even greater amount of information in an accurate forecast of snow for a war
|
https://en.wikipedia.org/wiki/Markov%20chain%20mixing%20time
|
In probability theory, the mixing time of a Markov chain is the time until the Markov chain is "close" to its steady state distribution.
More precisely, a fundamental result about Markov chains is that a finite state irreducible aperiodic chain has a unique stationary distribution π and, regardless of the initial state, the time-t distribution of the chain converges to π as t tends to infinity. Mixing time refers to any of several variant formalizations of the idea: how large must t be until the time-t distribution is approximately π? One variant, total variation distance mixing time, is defined as the smallest t such that the total variation distance of probability measures is small:
.
Choosing a different , as long as , can only change the mixing time up to a constant factor (depending on ) and so one often fixes and simply writes .
This is the sense in which proved that the number of riffle shuffles needed to mix an ordinary 52 card deck is 7. Mathematical theory focuses on how mixing times change as a function of the size of the structure underlying the chain. For an -card deck, the number of riffle shuffles needed grows as . The most developed theory concerns randomized algorithms for #P-Complete algorithmic counting problems such as the number of graph colorings of a given vertex graph. Such problems can, for sufficiently large number of colors, be answered using the Markov chain Monte Carlo method and showing that the mixing time grows only as . This example and the shuffling example possess the rapid mixing property, that the mixing time grows at most polynomially fast in (number of states of the chain). Tools for proving rapid mixing include arguments based on conductance and the method of coupling. In broader uses of the Markov chain Monte Carlo method, rigorous justification of simulation results would require a theoretical bound on mixing time, and many interesting practical cases have resisted such theoretical analysis.
See also
Mixing (mathematics) for a formal definition of mixing
References
.
.
.
.
.
Markov processes
|
https://en.wikipedia.org/wiki/Charles%20M.%20Stein
|
Charles Max Stein (March 22, 1920 – November 24, 2016) was an American mathematical statistician and professor of statistics at Stanford University.
He received his Ph.D in 1947 at Columbia University with advisor Abraham Wald. He held faculty positions at Berkeley and the University of Chicago before moving permanently to Stanford in 1953. He is known for Stein's paradox in decision theory, which shows that ordinary least squares estimates can be uniformly improved when many parameters are estimated; for Stein's lemma, giving a formula for the covariance of one random variable with the value of a function of another when the two random variables are jointly normally distributed; and for Stein's method, a way of proving theorems such as the Central Limit Theorem that does not require the variables to be independent and identically distributed. He was a member of the National Academy of Sciences. He died in November 2016 at the age of 96.
Works
Approximate Computation of Expectations, Institute of Mathematical Statistics, Hayward, CA, 1986.
A bound for the error in the normal approximation to the distribution of a sum of dependent random variables, Sixth Berkeley Stanford Symposium, pages 583-602.
Interviews
References
National University of Singapore Program Honoring Prof. Stein
Photograph of Stein
Another photograph
See also
James–Stein estimator
Stein's lemma
Stein's method
Stein's unbiased risk estimate
Stein's loss
Stein discrepancy
1920 births
2016 deaths
American mathematicians
Columbia University alumni
Members of the United States National Academy of Sciences
Stanford University Department of Statistics faculty
University of California, Berkeley faculty
University of Chicago faculty
Mathematical statisticians
People from Brooklyn
|
https://en.wikipedia.org/wiki/Tarski%E2%80%93Grothendieck%20set%20theory
|
Tarski–Grothendieck set theory (TG, named after mathematicians Alfred Tarski and Alexander Grothendieck) is an axiomatic set theory. It is a non-conservative extension of Zermelo–Fraenkel set theory (ZFC) and is distinguished from other axiomatic set theories by the inclusion of Tarski's axiom, which states that for each set there is a Grothendieck universe it belongs to (see below). Tarski's axiom implies the existence of inaccessible cardinals, providing a richer ontology than ZFC. For example, adding this axiom supports category theory.
The Mizar system and Metamath use Tarski–Grothendieck set theory for formal verification of proofs.
Axioms
Tarski–Grothendieck set theory starts with conventional Zermelo–Fraenkel set theory and then adds “Tarski's axiom”. We will use the axioms, definitions, and notation of Mizar to describe it. Mizar's basic objects and processes are fully formal; they are described informally below. First, let us assume that:
Given any set , the singleton exists.
Given any two sets, their unordered and ordered pairs exist.
Given any set of sets, its union exists.
TG includes the following axioms, which are conventional because they are also part of ZFC:
Set axiom: Quantified variables range over sets alone; everything is a set (the same ontology as ZFC).
Axiom of extensionality: Two sets are identical if they have the same members.
Axiom of regularity: No set is a member of itself, and circular chains of membership are impossible.
Axiom schema of replacement: Let the domain of the class function be the set . Then the range of (the values of for all members of ) is also a set.
It is Tarski's axiom that distinguishes TG from other axiomatic set theories. Tarski's axiom also implies the axioms of infinity, choice, and power set. It also implies the existence of inaccessible cardinals, thanks to which the ontology of TG is much richer than that of conventional set theories such as ZFC.
Tarski's axiom (adapted from Tarski 1939). For every set , there exists a set whose members include:
- itself;
- every element of every member of ;
- every subset of every member of ;
- the power set of every member of ;
- every subset of of cardinality less than that of .
More formally:
where “” denotes the power class of x and “” denotes equinumerosity. What Tarski's axiom states (in the vernacular) is that for each set there is a Grothendieck universe it belongs to.
That looks much like a “universal set” for – it not only has as members the powerset of , and all subsets of , it also has the powerset of that powerset and so on – its members are closed under the operations of taking powerset or taking a subset. It's like a “universal set” except that of course it is not a member of itself and is not a set of all sets. That's the guaranteed Grothendieck universe it belongs to. And then any such is itself a member of an even larger “almost universal set” and so on. It's one of the strong cardinality axioms
|
https://en.wikipedia.org/wiki/Paul%20von%20Jank%C3%B3
|
Paul von Jankó (2 June 1856 – 17 March 1919) was a Hungarian pianist, engineer and Idist.
He first studied mathematics and music in Vienna, where he was a pupil of H. Schmitt, J. Krenn and Anton Bruckner. He then moved to Berlin where he during the years 1881 and 1882 studied mathematics at the city's University, and piano with H. Erlich.
Jankó was also a proponent of the international auxiliary language Ido, though he had formerly been an Esperantist. On the 16th of August 1909, Jankó became a member of the Ido-Akademio, the predecessor to the ULI. He was secretary of the Academy from 1912 to 1913. Jankó also created the Ido-Stelo, the symbol of the Ido movement, modelled after the Verda Stelo.
In 1882 Jankó patented the Jankó keyboard, with six rows of keys, drawing upon earlier designs by Conrad Henfling (1708), Johann Rohleder (1791) and William Lunn (1843). From the year 1886 he used this instrument at his own concert journeys. The Norwegian pianist Tekla Nathan Bjerke was a pupil of Jankó, and played many concerts in Norway using this instrument. The Jankó keyboard wasn't used by many people as it was hard for them to relearn new fingering on a strange keyboard.
External links
Obituary in Zeitschrift für Instrumentenbau, Vol. 40, 1919-20
References
1856 births
1919 deaths
Hungarian classical pianists
Hungarian male musicians
Male classical pianists
Hungarian engineers
Hungarian inventors
Idists
19th-century classical pianists
19th-century male musicians
Musicians from Austria-Hungary
Engineers from Austria-Hungary
|
https://en.wikipedia.org/wiki/Ordinal%20notation
|
In mathematical logic and set theory, an ordinal notation is a partial function mapping the set of all finite sequences of symbols, themselves members of a finite alphabet, to a countable set of ordinals. A Gödel numbering is a function mapping the set of well-formed formulae (a finite sequence of symbols on which the ordinal notation function is defined) of some formal language to the natural numbers. This associates each well-formed formula with a unique natural number, called its Gödel number. If a Gödel numbering is fixed, then the subset relation on the ordinals induces an ordering on well-formed formulae which in turn induces a well-ordering on the subset of natural numbers. A recursive ordinal notation must satisfy the following two additional properties:
the subset of natural numbers is a recursive set
the induced well-ordering on the subset of natural numbers is a recursive relation
There are many such schemes of ordinal notations, including schemes by Wilhelm Ackermann, Heinz Bachmann, Wilfried Buchholz, Georg Cantor, Solomon Feferman, Gerhard Jäger, Isles, Pfeiffer, Wolfram Pohlers, Kurt Schütte, Gaisi Takeuti (called ordinal diagrams), Oswald Veblen. Stephen Cole Kleene has a system of notations, called Kleene's O, which includes ordinal notations but it is not as well behaved as the other systems described here.
Usually one proceeds by defining several functions from ordinals to ordinals and representing each such function by a symbol. In many systems, such as Veblen's well known system, the functions are normal functions, that is, they are strictly increasing and continuous in at least one of their arguments, and increasing in other arguments. Another desirable property for such functions is that the value of the function is greater than each of its arguments, so that an ordinal is always being described in terms of smaller ordinals. There are several such desirable properties. Unfortunately, no one system can have all of them since they contradict each other.
A simplified example using a pairing function
As usual, we must start off with a constant symbol for zero, "0", which we may consider to be a function of arity zero. This is necessary because there are no smaller ordinals in terms of which zero can be described. The most obvious next step would be to define a unary function, "S", which takes an ordinal to the smallest ordinal greater than it; in other words, S is the successor function. In combination with zero, successor allows one to name any natural number.
The third function might be defined as one that maps each ordinal to the smallest ordinal that cannot yet be described with the above two functions and previous values of this function. This would map β to ω·β except when β is a fixed point of that function plus a finite number in which case one uses ω·(β+1).
The fourth function would map α to ωω·α except when α is a fixed point of that plus a finite number in which case one uses ωω·(α+1).
ξ-notation
One co
|
https://en.wikipedia.org/wiki/Elizaveta%20Litvinova
|
Elizaveta Fedorovna Litvinova (1845–1919?) was a Russian mathematician and pedagogue. She is the author of over 70 articles about mathematics education.
Early life and education
Born in 1845 in czarist Russia as Elizaveta Fedorovna Ivashkina, she completed her early education at a women's high school in Saint Petersburg. In 1866 Elizaveta married Viktor Litvinov, which, unlike Vladimir Kovalevsky (Sofia Kovalevskaya's husband), would not allow her to travel to Europe to study at the universities there. Thus, Litvinova started to study with Strannoliubskii, who had also privately tutored Kovalevskaya.
In 1872, as soon as her husband died, Litvinova went to Zürich and enrolled at a polytechnic institute. In 1873 the Russian czar decreed all Russian women studying in Zürich had to return to Russia or face the consequences. Litvinova was one of the few to ignore the decree and she remained to continue her studies, earning her baccalaureate in Zürich in 1876 and her doctoral degree in 1878 from the University of Bern.
Career and later life
When Litvinova returned to Russia, she was denied university appointments because she had defied the 1873 recall. She taught at a women's high school and supplemented her meager income by writing biographies of more famous mathematicians such as Kovalevskaya and Aristotle. After retiring, it is believed that Litvinova died during the Russian Revolution in 1919.
Bibliography
A. H. Koblitz, Sofia Vasilevna Kovalevskaia in
External links
"Elizaveta Litvinova", Biographies of Women Mathematicians, Agnes Scott College
1845 births
1919 deaths
Russian mathematicians
Russian women mathematicians
Women mathematicians
|
https://en.wikipedia.org/wiki/Dmitrii%20Sintsov
|
Dmitrii Matveevich Sintsov (21 November 1867, in Vyatka – 28 January 1946) was a Russian mathematician known for his work in the theory of conic sections and non-holonomic geometry.
He took a leading role in the development of mathematics at the University of Kharkiv, serving as chairman of the Kharkov Mathematical Society for forty years, from 1906 until his death at the age of 78.
See also
Aleksandr Lyapunov
Bibliography
External links
1867 births
1946 deaths
People from Kirov, Kirov Oblast
People from Vyatsky Uyezd
Mathematicians from the Russian Empire
Soviet mathematicians
Academic staff of the National University of Kharkiv
First convocation members of the Verkhovna Rada of the Ukrainian Soviet Socialist Republic
|
https://en.wikipedia.org/wiki/Feliks%20Bara%C5%84ski
|
Feliks Barański (1915-2006) was a Polish mathematician and an active member of the so-called Lwów School of Mathematics. Born May 1915 in Lwów, Austria-Hungary (modern Lviv, Ukraine), he joined the circle of young, talented mathematicians formed around Stefan Banach and Hugo Steinhaus. During the period of German occupation of his hometown he made his living as a lice feeder in the institute of Rudolf Weigl. Expelled from Lwow after the war, he settled in Kraków, where he joined the local Kraków University of Technology. He was also admitted into the Polish Mathematics Society.
Lwów School of Mathematics
1915 births
2006 deaths
|
https://en.wikipedia.org/wiki/Jan%20Rajewski
|
Jan Rajewski (14 May 1857 – 30/31 December 1906) was a professor of the University of Lviv. He was a mathematician.
External links
Mathematics at Lviv University
Mathematicians from Austria-Hungary
1857 births
1906 deaths
Burials at Lychakiv Cemetery
|
https://en.wikipedia.org/wiki/Chebyshev%20rational%20functions
|
In mathematics, the Chebyshev rational functions are a sequence of functions which are both rational and orthogonal. They are named after Pafnuty Chebyshev. A rational Chebyshev function of degree is defined as:
where is a Chebyshev polynomial of the first kind.
Properties
Many properties can be derived from the properties of the Chebyshev polynomials of the first kind. Other properties are unique to the functions themselves.
Recursion
Differential equations
Orthogonality
Defining:
The orthogonality of the Chebyshev rational functions may be written:
where for and for ; is the Kronecker delta function.
Expansion of an arbitrary function
For an arbitrary function the orthogonality relationship can be used to expand :
where
Particular values
Partial fraction expansion
References
Rational functions
|
https://en.wikipedia.org/wiki/Math%20A
|
Mathematics courses named Math A, Maths A, and similar are found in:
Mathematics education in New York: Math A, Math A/B, Math B
Mathematics education in Australia: Maths A, Maths B, Maths C
Mathematics disambiguation pages
|
https://en.wikipedia.org/wiki/Pefkos
|
{
"type": "FeatureCollection",
"features": [
{
"type": "Feature",
"properties": {},
"geometry": {
"type": "Point",
"coordinates": [
28.057022,
36.065752
]
}
}
]
}Pefkos or Pefki, Greek: Πεύκος (Πεύκοι), is a well known beach resort located on eastern coast of Rhodes, just southwest of Lindos, and from the capital city Rhodes. The island of Rhodes is the largest of the Dodecanese islands, on the eastern Aegean Sea, just a few miles from the coast of the Asia Minor. Pefkos was once known as a fishermen's hamlet located along the coastal road that connects the villages of Lindos and Lardos. Originally Pefkos was mainly used as a summer temporary residence for those who lived further inland but grew crops such as grapes, olives, tomatoes, figs and corn. They couldn't return home daily due to the heat and distance, so had small very basic houses in Pefkos. Visiting Pefkos by day will leave one with the impression of a quiet and relaxed holiday resort; however when the lights come on the resort is bustling with warm, friendly activity.
The main Pefkos beach (Lee beach) is a pure sand beach and is a Blue Flag status awarded beach for 2008. The beach is busy with tourists during the day, currently there are 3 restaurants on the waterfront. The beach is quiet, with no motorised watersports.
Pefkos, until recently was under developed, but is now swiftly evolving in an important tourist resort, mostly offering self-catering holiday accommodation (villas and apartments) while its commercial center has many amenities and a large range of restaurants, tavernas and bars. The resort is set against a hill which is surrounded by Pine trees from which the town gets its name. It is largely visited by Scandinavians, British, Germans, Polish and Austrians.
References
Photos
External links
Pefkos page at discover-rhodes.com
Populated places in Rhodes
|
https://en.wikipedia.org/wiki/Joint%20Academic%20Coding%20System
|
The Joint Academic Coding System (JACS) system was used by the Higher Education Statistics Agency (HESA) and the Universities and Colleges Admissions Service (UCAS) in the United Kingdom to classify academic subjects. It was replaced by the Higher Education Classification of Subjects (HECoS) and the Common Aggregation Hierarchy (CAH) for the 2019/20 academic year.
A JACS code for a single subject consists of a letter and three numbers. The letter represents the broad subject classification, e.g. F for physical sciences. The first number represents the principal subject area, e.g. F3 for physics, and subsequent numbers represent further details, similar to the Dewey Decimal System. The principal subject of physics, for example, is broken into 19 detailed subjects, represented by a letter plus three numbers: e.g., F300 represents physics, F330 environmental physics, and F331 atmospheric physics.
History
HESA and UCAS used to operate two different (though similar) subject coding systems - HESAcode and Standard Classification of Academic Subjects (SCAS) respectively. In 1996 a joint project was launched to bring these two systems together to create a unified structure. A project team was established with two people from each of the two organizations. The project team became known as JACS since this was an acronym of their names (Jonathan Waller and Andy Youell from HESA, Clive Sillence and Sara Goodwins from UCAS).
The first operational version (v1.7) of the Joint Academic Coding System (retaining the JACS acronym) was published in 1999 and became operational in UCAS and HESA systems for the year 2002/03.
An update exercise took place in 2005 and JACS 2 was introduced for the academic year 2007/08. JACS 3 was introduced for the 2012/13 year.
Codes
The letter codes assigned to the subject areas and the letter + number codes assigned to the principal subjects in JACS 3 are:
Y codes (combined studies) are only used at the Course level in the HESA database and are not used to describe individual modules.
JACS Codes in the UCAS system
Course codes in the UCAS system are assigned by course providers and do not necessarily correspond to the JACS codes of the course subject. UCAS course codes are four characters in length but, unlike JACS codes, may consist of any combination of letters and numbers in any order. However, historically UCAS created course codes from the JACS subject code, and many institutions continue to do this, which can lead to confusion between the two concepts.
Where a course involves more than one subject, UCAS historically created the course code based on an aggregation of the JACS codes. For courses that are split 50:50 between two subjects, a code with two letters and two numbers is used, which combines the principal subject codes that would be used for the two subjects if studied as individual degrees.
Example
Consider the BSc course Mathematics and Physics:
The principal subject code for Mathematics was G1, and the princi
|
https://en.wikipedia.org/wiki/Transport%20of%20structure
|
In mathematics, particularly in universal algebra and category theory, transport of structure refers to the process whereby a mathematical object acquires a new structure and its canonical definitions, as a result of being isomorphic to (or otherwise identified with) another object with a pre-existing structure. Definitions by transport of structure are regarded as canonical.
Since mathematical structures are often defined in reference to an underlying space, many examples of transport of structure involve spaces and mappings between them. For example, if and are vector spaces with being an inner product on , such that there is an isomorphism from to , then one can define an inner product on by the following rule:
Although the equation makes sense even when is not an isomorphism, it only defines an inner product on when is, since otherwise it will cause to be degenerate. The idea is that allows one to consider and as "the same" vector space, and by following this analogy, then one can transport an inner product from one space to the other.
A more elaborated example comes from differential topology, in which the notion of smooth manifold is involved: if is such a manifold, and if is any topological space which is homeomorphic to , then one can consider as a smooth manifold as well. That is, given a homeomorphism , one can define coordinate charts on by "pulling back" coordinate charts on through . Recall that a coordinate chart on is an open set together with an injective map
for some natural number ; to get such a chart on , one uses the following rules:
and .
Furthermore, it is required that the charts cover (the fact that the transported charts cover follows immediately from the fact that is a bijection). Since is a smooth manifold, if U and V, with their maps and , are two charts on , then the composition, the "transition map"
(a self-map of )
is smooth. To verify this for the transported charts on , notice that
,
and therefore
, and
.
Thus the transition map for and is the same as that for and , hence smooth. That is, is a smooth manifold via transport of structure. This is a special case of transport of structures in general.
The second example also illustrates why "transport of structure" is not always desirable. Namely, one can take to be the plane, and to be an infinite one-sided cone. By "flattening" the cone, a homeomorphism of and can be obtained, and therefore the structure of a smooth manifold on , but the cone is not "naturally" a smooth manifold. That is, one can consider as a subspace of 3-space, in which context it is not smooth at the cone point.
A more surprising example is that of exotic spheres, discovered by Milnor, which states that there are exactly 28 smooth manifolds which are homeomorphic (but by definition not diffeomorphic) to , the 7-dimensional sphere in 8-space. Thus, transport of structure is most productive when there exists a canonical isomorphism between t
|
https://en.wikipedia.org/wiki/David%20Gabai
|
David Gabai is an American mathematician and the Hughes-Rogers Professor of Mathematics at Princeton University. Focused on low-dimensional topology and hyperbolic geometry, he is a leading researcher in those subjects.
Biography
David Gabai received his B.S. in mathematics from MIT in 1976 and his Ph.D. in mathematics from Princeton University in 1980. Gabai completed his doctoral dissertation, titled "Foliations and genera of links", under the supervision of William Thurston.
After positions at Harvard and University of Pennsylvania, Gabai spent most of the period of 1986–2001 at Caltech, and has been at Princeton since 2001. Gabai was the Chair of the Department of Mathematics at Princeton University from 2012 to 2019.
Honours and awards
In 2004, David Gabai was awarded the Oswald Veblen Prize in Geometry, given every three years by the American Mathematical Society.
He was an invited speaker in the International Congress of Mathematicians 2010, Hyderabad on the topic of topology.
In 2011, he was elected to the United States National Academy of Sciences. In 2012, he became a fellow of the American Mathematical Society.
Gabai was elected as a member of the American Academy of Arts and Sciences in 2014.
Work
David Gabai has played a key role in the field of topology of 3-manifolds in the last three decades. Some of the foundational results he and his collaborators have proved are as follows: Existence of taut foliation in 3-manifolds, Property R Conjecture, foundation of essential laminations, Seifert fiber space conjecture, rigidity of homotopy hyperbolic 3-manifolds, weak hyperbolization for 3-manifolds with genuine lamination, Smale conjecture for hyperbolic 3-manifolds, Marden's Tameness Conjecture, Weeks manifold being the minimum volume closed hyperbolic 3-manifold.
Selected works
Foliations and the topology of 3-manifolds; I: J. Differential Geom. 18 (1983), no. 3, 445–503; II: J. Differential Geom. 26 (1987), no. 3, 461–478; III: J. Differential Geom. 26 (1987), no. 3, 479–536.
with U. Oertel: Essential laminations in 3-manifolds, Ann. of Math. (2) 130 (1989), no. 1, 41–73.
Convergence groups are Fuchsian groups, Ann. of Math. (2) 136 (1992), no. 3, 447–510.
with G. R. Meyerhoff, N. Thurston: Homotopy hyperbolic 3-manifolds are hyperbolic, Ann. of Math. (2) 157 (2003), no. 2, 335–431.
with D. Calegari: Shrinkwrapping and the taming of hyperbolic 3-manifolds, J. Amer. Math. Soc. 19 (2006), no. 2, 385–446.
with G. R. Meyerhoff, P. Milley: Minimum volume cusped hyperbolic three-manifolds, J. Amer. Math. Soc. 22 (2009), no. 4, 1157–1215.
References
External links
1954 births
Living people
Topologists
Princeton University faculty
Members of the United States National Academy of Sciences
Clay Research Award recipients
Fellows of the American Mathematical Society
Geometers
|
https://en.wikipedia.org/wiki/Schouten%20tensor
|
In Riemannian geometry the Schouten tensor is a second-order tensor introduced by Jan Arnoldus Schouten defined for by:
where Ric is the Ricci tensor (defined by contracting the first and third indices of the Riemann tensor), R is the scalar curvature, g is the Riemannian metric, is the trace of P and n is the dimension of the manifold.
The Weyl tensor equals the Riemann curvature tensor minus the Kulkarni–Nomizu product of the Schouten tensor with the metric. In an index notation
The Schouten tensor often appears in conformal geometry because of its relatively simple conformal transformation law
where
Further reading
Arthur L. Besse, Einstein Manifolds. Springer-Verlag, 2007. See Ch.1 §J "Conformal Changes of Riemannian Metrics."
Spyros Alexakis, The Decomposition of Global Conformal Invariants. Princeton University Press, 2012. Ch.2, noting in a footnote that the Schouten tensor is a "trace-adjusted Ricci tensor" and may be considered as "essentially the Ricci tensor."
Wolfgang Kuhnel and Hans-Bert Rademacher, "Conformal diffeomorphisms preserving the Ricci tensor", Proc. Amer. Math. Soc. 123 (1995), no. 9, 2841–2848. Online eprint (pdf).
T. Bailey, M.G. Eastwood and A.R. Gover, "Thomas's Structure Bundle for Conformal, Projective and Related Structures", Rocky Mountain Journal of Mathematics, vol. 24, Number 4, 1191-1217.
See also
Weyl–Schouten theorem
Cotton tensor
Curvature tensors
Riemannian geometry
Tensors in general relativity
|
https://en.wikipedia.org/wiki/Weyl%E2%80%93Schouten%20theorem
|
In the mathematical field of differential geometry, the existence of isothermal coordinates for a (pseudo-)Riemannian metric is often of interest. In the case of a metric on a two-dimensional space, the existence of isothermal coordinates is unconditional. For higher-dimensional spaces, the Weyl–Schouten theorem (named after Hermann Weyl and Jan Arnoldus Schouten) characterizes the existence of isothermal coordinates by certain equations to be satisfied by the Riemann curvature tensor of the metric.
Existence of isothermal coordinates is also called conformal flatness, although some authors refer to it instead as local conformal flatness; for those authors, conformal flatness refers to a more restrictive condition.
Theorem
In terms of the Riemann curvature tensor, the Ricci tensor, and the scalar curvature, the Weyl tensor of a pseudo-Riemannian metric of dimension is given by
The Schouten tensor is defined via the Ricci and scalar curvatures by
As can be calculated by the Bianchi identities, these satisfy the relation that
The Weyl–Schouten theorem says that for any pseudo-Riemannian manifold of dimension :
If then the manifold is conformally flat if and only if its Weyl tensor is zero.
If then the manifold is conformally flat if and only if its Schouten tensor is a Codazzi tensor.
As known prior to the work of Weyl and Schouten, in the case , every manifold is conformally flat. In all cases, the theorem and its proof are entirely local, so the topology of the manifold is irrelevant.
There are varying conventions for the meaning of conformal flatness; the meaning as taken here is sometimes instead called local conformal flatness.
Sketch of proof
The only if direction is a direct computation based on how the Weyl and Schouten tensors are modified by a conformal change of metric. The if direction requires more work.
Consider the following equation for a 1-form :
Let denote the tensor on the right-hand side. The Frobenius theorem states that the above equation is locally solvable if and only if
is symmetric in and for any 1-form . A direct cancellation of terms shows that this is the case if and only if
for any 1-form . If then the left-hand side is zero since the Weyl tensor of any three-dimensional metric is zero; the right-hand side is zero whenever the Schouten tensor is a Codazzi tensor. If then the left-hand side is zero whenever the Weyl tensor is zero; the right-hand side is also then zero due to the identity given above which relates the Weyl tensor to the Schouten tensor.
As such, under the given curvature and dimension conditions, there always exists a locally defined 1-form solving the given equation. From the symmetry of the right-hand side, it follows that must be a closed form. The Poincaré lemma then implies that there is a real-valued function with . Due to the formula for the Ricci curvature under a conformal change of metric, the (locally defined) pseudo-Riemannian metric is Ricci-flat. If then every R
|
https://en.wikipedia.org/wiki/Graeffe%27s%20method
|
In mathematics, Graeffe's method or Dandelin–Lobachesky–Graeffe method is an algorithm for finding all of the roots of a polynomial. It was developed independently by Germinal Pierre Dandelin in 1826 and Lobachevsky in 1834. In 1837 Karl Heinrich Gräffe also discovered the principal idea of the method. The method separates the roots of a polynomial by squaring them repeatedly. This squaring of the roots is done implicitly, that is, only working on the coefficients of the polynomial. Finally, Viète's formulas are used in order to approximate the roots.
Dandelin–Graeffe iteration
Let be a polynomial of degree
Then
Let be the polynomial which has the squares as its roots,
Then we can write:
can now be computed by algebraic operations on the coefficients of the polynomial alone. Let:
then the coefficients are related by
Graeffe observed that if one separates into its odd and even parts:
then one obtains a simplified algebraic expression for :
This expression involves the squaring of two polynomials of only half the degree, and is therefore used in most implementations of the method.
Iterating this procedure several times separates the roots with respect to their magnitudes. Repeating k times gives a polynomial of degree :
with roots
If the magnitudes of the roots of the original polynomial were separated by some factor , that is, , then the roots of the k-th iterate are separated by a fast growing factor
.
Classical Graeffe's method
Next the Vieta relations are used
If the roots are sufficiently separated, say by a factor , , then the iterated powers of the roots are separated by the factor , which quickly becomes very big.
The coefficients of the iterated polynomial can then be approximated by their leading term,
and so on,
implying
Finally, logarithms are used in order to find the absolute values of the roots of the original polynomial. These magnitudes alone are already useful to generate meaningful starting points for other root-finding methods.
To also obtain the angle of these roots, a multitude of methods has been proposed, the most simple one being to successively compute the square root of a (possibly complex) root of , m ranging from k to 1, and testing which of the two sign variants is a root of . Before continuing to the roots of , it might be necessary to numerically improve the accuracy of the root approximations for , for instance by Newton's method.
Graeffe's method works best for polynomials with simple real roots, though it can be adapted for polynomials with complex roots and coefficients, and roots with higher multiplicity. For instance, it has been observed that for a root with multiplicity d,
the fractions
tend to
for . This allows to estimate the multiplicity structure of the set of roots.
From a numerical point of view, this method is problematic since the coefficients of the iterated polynomials span very quickly many orders of magnitude, which implies serious numerical errors. O
|
https://en.wikipedia.org/wiki/Cevian
|
In geometry, a cevian is a line segment which joins a vertex of a triangle to a point on the opposite side of the triangle. Medians and angle bisectors are special cases of cevians. The name "cevian" comes from the Italian mathematician Giovanni Ceva, who proved a well-known theorem about cevians which also bears his name.
Length
Stewart's theorem
The length of a cevian can be determined by Stewart's theorem: in the diagram, the cevian length is given by the formula
Less commonly, this is also represented (with some rearrangement) by the following mnemonic:
Median
If the cevian happens to be a median (thus bisecting a side), its length can be determined from the formula
or
since
Hence in this case
Angle bisector
If the cevian happens to be an angle bisector, its length obeys the formulas
and
and
where the semiperimeter
The side of length is divided in the proportion .
Altitude
If the cevian happens to be an altitude and thus perpendicular to a side, its length obeys the formulas
and
where the semiperimeter
Ratio properties
There are various properties of the ratios of lengths formed by three cevians all passing through the same arbitrary interior point: Referring to the diagram at right,
The first property is known as Ceva's theorem. The last two properties are equivalent because summing the two equations gives the identity .
Splitter
A splitter of a triangle is a cevian that bisects the perimeter. The three splitters concur at the Nagel point of the triangle.
Area bisectors
Three of the area bisectors of a triangle are its medians, which connect the vertices to the opposite side midpoints. Thus a uniform-density triangle would in principle balance on a razor supporting any of the medians.
Angle trisectors
If from each vertex of a triangle two cevians are drawn so as to trisect the angle (divide it into three equal angles), then the six cevians intersect in pairs to form an equilateral triangle, called the Morley triangle.
Area of inner triangle formed by cevians
Routh's theorem determines the ratio of the area of a given triangle to that of a triangle formed by the pairwise intersections of three cevians, one from each vertex.
See also
Mass point geometry
Menelaus' theorem
Notes
References
Ross Honsberger (1995). Episodes in Nineteenth and Twentieth Century Euclidean Geometry, pages 13 and 137. Mathematical Association of America.
Vladimir Karapetoff (1929). "Some properties of correlative vertex lines in a plane triangle." American Mathematical Monthly 36: 476–479.
Indika Shameera Amarasinghe (2011). “A New Theorem on any Right-angled Cevian Triangle.” Journal of the World Federation of National Mathematics Competitions, Vol 24 (02), pp. 29–37.
Straight lines defined for a triangle
|
https://en.wikipedia.org/wiki/DFFITS
|
In statistics, DFFIT and DFFITS ("difference in fit(s)") are diagnostics meant to show how influential a point is in a linear regression, first proposed in 1980.
DFFIT is the change in the predicted value for a point, obtained when that point is left out of the regression:
where and are the prediction for point i with and without point i included in the regression.
DFFITS is the Studentized DFFIT, where Studentization is achieved by dividing by the estimated standard deviation of the fit at that point:
where is the standard error estimated without the point in question, and is the leverage for the point.
DFFITS also equals the products of the externally Studentized residual () and the leverage factor ():
Thus, for low leverage points, DFFITS is expected to be small, whereas as the leverage goes to 1 the distribution of the DFFITS value widens infinitely.
For a perfectly balanced experimental design (such as a factorial design or balanced partial factorial design), the leverage for each point is p/n, the number of parameters divided by the number of points. This means that the DFFITS values will be distributed (in the Gaussian case) as times a t variate. Therefore, the authors suggest investigating those points with DFFITS greater than .
Although the raw values resulting from the equations are different, Cook's distance and DFFITS are conceptually identical and there is a closed-form formula to convert one value to the other.
Development
Previously when assessing a dataset before running a linear regression, the possibility of outliers would be assessed using histograms and scatterplots. Both methods of assessing data points were subjective and there was little way of knowing how much leverage each potential outlier had on the results data. This led to a variety of quantitative measures, including DFFIT, DFBETA.
References
Regression diagnostics
|
https://en.wikipedia.org/wiki/Steuart%20Campbell
|
Steuart Campbell (born in ) is a British writer who lives in Edinburgh.
Career
Campbell trained as an architect and worked as one until the mid-1970s. He then gained a degree in mathematics and science from the Open University (BA, 1983).
Campbell is the Secretary/Treasurer of the Edinburgh Secular Society.
Writings
The Loch Ness Monster: The Evidence. 1986 The Aquarian Press (Thorsons Publishing Group) Wellingborough: ; Revised ed. 1991 Aberdeen University Press (Macmillan Pergamon Publishing Corporation) Aberdeen: ; 1996 Birlinn Ltd, Edinburgh: ; 1997 (without subtitle); Prometheus Books, Amhurst: ; 2002 Birlinn Ltd, Edinburgh 1997: ). Argues against the existence of the Loch Ness Monster by analysis of the purported evidence.
The UFO Mystery Solved 1994 Explicit Books, Edinburgh: . A critical examination of UFO reports and their explanation in terms of meteorological and astronomical phenomena;
The Rise and Fall of Jesus with a foreword by Prof. James Thrower 1996 Explicit Books, Edinburgh: ; 2009 Revised and updated ed. WPS (WritersPrintShop): ; 2019 Revised 3rd ed. Tectum Verlag (Nomos Publishing Company), Marburg: (print), (ePDF). Exploration of the origins of Christianity, asserting that Jesus wanted to be crucified.
Chinook Crash (The crash of RAF Chinook helicopter ZD576 on the Mull of Kintyre) 2004 Pen & Sword Aviation (Pen & Sword Books Ltd, Barnsley) (print) (ebook). An examination of and an explanation for the fatal crash on 2 June 1994.
References
External links
1937 births
Alumni of the Open University
British sceptics
Critics of cryptozoology
Living people
Writers from Birmingham, West Midlands
UFO skeptics
|
https://en.wikipedia.org/wiki/Tunnell%27s%20theorem
|
In number theory, Tunnell's theorem gives a partial resolution to the congruent number problem, and under the Birch and Swinnerton-Dyer conjecture, a full resolution.
Congruent number problem
The congruent number problem asks which positive integers can be the area of a right triangle with all three sides rational. Tunnell's theorem relates this to the number of integral solutions of a few fairly simple Diophantine equations.
Theorem
For a given square-free integer n, define
Tunnell's theorem states that supposing n is a congruent number, if n is odd then 2An = Bn and if n is even then 2Cn = Dn. Conversely, if the Birch and Swinnerton-Dyer conjecture holds true for elliptic curves of the form , these equalities are sufficient to conclude that n is a congruent number.
History
The theorem is named for Jerrold B. Tunnell, a number theorist at Rutgers University, who proved it in .
Importance
The importance of Tunnell's theorem is that the criterion it gives is testable by a finite calculation. For instance, for a given , the numbers can be calculated by exhaustively searching through in the range .
See also
Birch and Swinnerton-Dyer conjecture
Congruent number
References
Theorems in number theory
Diophantine equations
|
https://en.wikipedia.org/wiki/Producer%27s%20risk
|
Producer's risk is the probability that a good product will be rejected as a bad product by the consumer.
When the acceptance reliability level (ARL) is pi0, we can define the producer's risk as:
P(Test is Failed|pi0)
It calculates the probability of loss from (1) rejecting a batch which, in fact, should have been accepted, or (2) accepting a batch that, in fact, will be rejected by the customer.
See also
consumer's risk
Quality control
References
Production economics
|
https://en.wikipedia.org/wiki/Softmax%20function
|
The softmax function, also known as softargmax or normalized exponential function, converts a vector of real numbers into a probability distribution of possible outcomes. It is a generalization of the logistic function to multiple dimensions, and used in multinomial logistic regression. The softmax function is often used as the last activation function of a neural network to normalize the output of a network to a probability distribution over predicted output classes, based on Luce's choice axiom.
Definition
The softmax function takes as input a vector of real numbers, and normalizes it into a probability distribution consisting of probabilities proportional to the exponentials of the input numbers. That is, prior to applying softmax, some vector components could be negative, or greater than one; and might not sum to 1; but after applying softmax, each component will be in the interval , and the components will add up to 1, so that they can be interpreted as probabilities. Furthermore, the larger input components will correspond to larger probabilities.
The standard (unit) softmax function where is defined by the formula
In words, it applies the standard exponential function to each element of the input vector and normalizes these values by dividing by the sum of all these exponentials. The normalization ensures that the sum of the components of the output vector is 1. The term "softmax" derives from the amplifying effects of the exponential on any maxima in the input vector. For example, the standard softmax of is approximately , which amounts to assigning almost all of the total unit weight in the result to the position of the vector's maximal element (of 8).
In general, instead of a different base can be used. If , smaller input components will result in larger output probabilities, and decreasing the value of will create probability distributions that are more concentrated around the positions of the smallest input values. Conversely, as above, if larger input components will result in larger output probabilities, and increasing the value of will create probability distributions that are more concentrated around the positions of the largest input values. Writing or (for real ) yields the expressions:
In some fields, the base is fixed, corresponding to a fixed scale, while in others the parameter is varied.
Interpretations
Smooth arg max
The name "softmax" is misleading. The function is not a smooth maximum (that is, a smooth approximation to the maximum function), but is rather a smooth approximation to the arg max function: the function whose value is the index of a vector's largest element. In fact, the term "softmax" is also used for the closely related LogSumExp function, which is a smooth maximum. For this reason, some prefer the more accurate term "softargmax", but the term "softmax" is conventional in machine learning. This section uses the term "softargmax" to emphasize this interpretation.
Formally, i
|
https://en.wikipedia.org/wiki/Scott%E2%80%93Potter%20set%20theory
|
An approach to the foundations of mathematics that is of relatively recent origin, Scott–Potter set theory is a collection of nested axiomatic set theories set out by the philosopher Michael Potter, building on earlier work by the mathematician Dana Scott and the philosopher George Boolos.
Potter (1990, 2004) clarified and simplified the approach of Scott (1974), and showed how the resulting axiomatic set theory can do what is expected of such theory, namely grounding the cardinal and ordinal numbers, Peano arithmetic and the other usual number systems, and the theory of relations.
ZU etc.
Preliminaries
This section and the next follow Part I of Potter (2004) closely. The background logic is first-order logic with identity. The ontology includes urelements as well as sets, which makes it clear that there can be sets of entities defined by first-order theories not based on sets. The urelements are not essential in that other mathematical structures can be defined as sets, and it is permissible for the set of urelements to be empty.
Some terminology peculiar to Potter's set theory:
ι is a definite description operator and binds a variable. (In Potter's notation the iota symbol is inverted.)
The predicate U holds for all urelements (non-collections).
ιxΦ(x) exists iff (∃!x)Φ(x). (Potter uses Φ and other upper-case Greek letters to represent formulas.)
{x : Φ(x)} is an abbreviation for ιy(not U(y) and (∀x)(x ∈ y ⇔ Φ(x))).
a is a collection if {x : x∈a} exists. (All sets are collections, but not all collections are sets.)
The accumulation of a, acc(a), is the set {x : x is an urelement or ∃b∈a (x∈b or x⊂b)}.
If ∀v∈V(v = acc(V∩v)) then V is a history.
A level is the accumulation of a history.
An initial level has no other levels as members.
A limit level is a level that is neither the initial level nor the level above any other level.
A set is a subcollection of some level.
The birthday of set a, denoted V(a), is the lowest level V such that a⊂V.
Axioms
The following three axioms define the theory ZU.
Creation: ∀V∃V' (V∈V' ).
Remark: There is no highest level, hence there are infinitely many levels. This axiom establishes the ontology of levels.
Separation: An axiom schema. For any first-order formula Φ(x) with (bound) variables ranging over the level V, the collection {x∈V : Φ(x)} is also a set. (See Axiom schema of separation.)
Remark: Given the levels established by Creation, this schema establishes the existence of sets and how to form them. It tells us that a level is a set, and all subsets, definable via first-order logic, of levels are also sets. This schema can be seen as an extension of the background logic.
Infinity: There exists at least one limit level. (See Axiom of infinity.)
Remark: Among the sets Separation allows, at least one is infinite. This axiom is primarily mathematical, as there is no need for the actual infinite in other human contexts, the human sensory order being necessarily finite. For mathemat
|
https://en.wikipedia.org/wiki/Titu%20Andreescu
|
Titu Andreescu (born August 19, 1956) is an associate professor of mathematics at the University of Texas at Dallas. He is firmly involved in mathematics contests and olympiads, having been the Director of American Mathematics Competitions (as appointed by the Mathematical Association of America), Director of the Mathematical Olympiad Program, Head Coach of the United States International Mathematical Olympiad Team, and Chairman of the United States of America Mathematical Olympiad. He has also authored a large number of books on the topic of problem solving and olympiad-style mathematics.
Biography
Andreescu was born in the Romanian city of Timișoara in 1956. From an early age, an interest in higher-level mathematics was encouraged by his father and by his uncle Andrew, who was a retired university professor. As a high school student, he excelled in mathematics, and in 1973, 1974, and 1975 won the Romanian national problem solving contests organized by the journal Gazeta Matematică. After graduating with a B.S. degree from the University of Timișoara, Andreescu was appointed a professor of mathematics at the Constantin Diaconovici Loga school of Mathematics and Physics. Between the years 1981–1989 he also worked as the editor-in-chief of the Periodical Revista mathematică din Timișoara. In 1990, as the Eastern Bloc began to collapse, Andreescu emigrated to the United States, where he first taught at the Illinois Mathematics and Science Academy. He earned a Ph.D. degree from the University of Timișoara in 2003.
Mathematics coaching and contests
During the 1980s, Andreescu served as a coach for the Romanian International Mathematical Olympiad (IMO) team and in 1983 was presented with the national award of "Distinguished Professor". In 1984 he was appointed as "Counselor of the Romanian Ministry of Education". After emigrating to the United States, Andreescu became involved in coaching the American IMO team. His most notable success came in 1994 when the American team obtained a perfect score at the Hong Kong International Mathematical Olympiad, the first time that any team had achieved this. In recognition of this achievement, he was awarded a Certificate of Appreciation from the Mathematical Association of America for "outstanding service" as coach of the United States Mathematical Olympiad Program.
AwesomeMath Program
In 2006, Andreescu established a math camp for bright and motivated middle and high school mathematicians. The first AwesomeMath summer program was very effective, with noted professors serving as instructors, and mentors and assistants who had performed well at Olympiads. The program has now been expanded to include locations at the University of Puget Sound (formerly at the University of California, Berkeley) and Cornell University, as well as the original location at the University of Texas at Dallas. The program continues throughout the year as the related AwesomeMath Year-round program, or AMY.
Metroplex Math Circle
In 20
|
https://en.wikipedia.org/wiki/G%C3%BCrkan%20Sermeter
|
Gürkan Sermeter (born 14 February 1974) is a Swiss former footballer who last played for AC Bellinzona in the Swiss Challenge League.
External links
Statistics at T-Online.de
AC Bellinzona profile
1974 births
Swiss men's footballers
Living people
FC Aarau players
Grasshopper Club Zürich players
BSC Young Boys players
FC Luzern players
AC Bellinzona players
Swiss Super League players
Swiss Challenge League players
Men's association football midfielders
|
https://en.wikipedia.org/wiki/Algorithmic%20Number%20Theory%20Symposium
|
Algorithmic Number Theory Symposium (ANTS) is a biennial academic conference, first held in Cornell in 1994, constituting an international forum for the presentation of new research in computational number theory. They are devoted to algorithmic aspects of number theory, including elementary number theory, algebraic number theory, analytic number theory, geometry of numbers, arithmetic geometry, finite fields, and cryptography.
Selfridge Prize
In honour of the many contributions of John Selfridge to mathematics, the Number Theory Foundation has established a prize to be awarded to those individuals who have authored the best paper accepted for presentation at ANTS. The prize, called the Selfridge Prize, is awarded every two years in an even numbered year. The prize winner(s) receive a cash award and a sculpture.
The prize winners and their papers selected by the ANTS Program Committee are:
2006 – ANTS VII – Werner Bley and Robert Boltje – Computation of locally free class groups.
2008 – ANTS VIII – Juliana Belding, Reinier Bröker, Andreas Enge and Kristin Lauter – Computing hilbert class polynomials.
2010 – ANTS IX – John Voight – Computing automorphic forms on Shimura curves over fields with arbitrary class number.
2012 – ANTS X – Andrew Sutherland – On the evaluation of modular polynomials.
2014 – ANTS XI – Tom Fisher – Minimal models for 6-coverings of elliptic curves.
2016 – ANTS XII – Jan Steffen Müller and Michael Stoll – Computing canonical heights on elliptic curves in quasi-linear time.
2018 – ANTS XIII – Michael Musty, Sam Schiavone, Jeroen Sijsling and John Voight – A database of Belyĭ maps.
2020 – ANTS XIV – Jonathan Love and Dan Boneh – Supersingular curves with small non-integer endomorphisms.
2022 – ANTS XV – Harald Helfgott and Lola Thompson – Summing mu(n): a faster elementary algorithm.
Proceedings
Prior to ANTS X, the refereed Proceedings of ANTS were published in the Springer Lecture Notes in Computer Science (LNCS). The proceedings of ANTS X, ANTS XIII, and ANTS XIV were published in the Mathematical Sciences Publishers Open Book Series (OBS). The proceedings of ANTS XI and ANTS XII were published as a special issue of the LMS Journal of Computation and Mathematics (JCM). The proceedings for ANTS XV will be published in Research in Number Theory.
Conferences
1994: ANTS I – Cornell University (Ithaca, NY, USA) – LNCS 877
1996: ANTS II – Universite Bordeaux 1 (Talence, FR) – LNCS 1122
1998: ANTS III – Reed College (Portland, OR, USA) – LNCS 1423
2000: ANTS IV – Universiteit Leiden (Leiden, NL) – LNCS 1838
2002: ANTS V – University of Sydney (Sydney, AU) – LNCS 2369
2004: ANTS VI – University of Vermont (Burlington, VT, USA) – LNCS 3076
2006: ANTS VII – Technische Universität Berlin (Berlin, DE) – LNCS 4076
2008: ANTS VIII – Banff Centre (Banff, AB, CA) – LNCS 5011
2010: ANTS IX – INRIA (Nancy, FR) – LNCS 6197
2012: ANTS X – University of California, San Diego (San Diego, CA, USA) – OBS 1
2014:
|
https://en.wikipedia.org/wiki/List%20of%20Sheffield%20Wednesday%20F.C.%20records%20and%20statistics
|
These are Sheffield Wednesday F.C. records. They cover all competitive matches dating back to the team's first appearance in the FA Cup in 1880.
Record Games
Seasonal records
Record Runs
All records relate to league games only
Players
General
Transfers
Appearances and goals
Highest Average attendance in a season
Honours
Top tier
Lower tier
Local
References
Club records (last accessed 28 July 2006)
Appearances (last accessed 28 July 2006)
Goalscorers (last accessed 28 July 2006)
Honours (last archived 15 October 2012)
Club Records (last accessed 1 September 2016)
Record Signing (last accessed 1 September 2016)
Records
Sheffield Wednesday
|
https://en.wikipedia.org/wiki/Modal%20%CE%BC-calculus
|
In theoretical computer science, the modal μ-calculus (Lμ, Lμ, sometimes just μ-calculus, although this can have a more general meaning) is an extension of propositional modal logic (with many modalities) by adding the least fixed point operator μ and the greatest fixed point operator ν, thus a fixed-point logic.
The (propositional, modal) μ-calculus originates with Dana Scott and Jaco de Bakker, and was further developed by Dexter Kozen into the version most used nowadays. It is used to describe properties of labelled transition systems and for verifying these properties. Many temporal logics can be encoded in the μ-calculus, including CTL* and its widely used fragments—linear temporal logic and computational tree logic.
An algebraic view is to see it as an algebra of monotonic functions over a complete lattice, with operators consisting of functional composition plus the least and greatest fixed point operators; from this viewpoint, the modal μ-calculus is over the lattice of a power set algebra. The game semantics of μ-calculus is related to two-player games with perfect information, particularly infinite parity games.
Syntax
Let P (propositions) and A (actions) be two finite sets of symbols, and let Var be a countably infinite set of variables. The set of formulas of (propositional, modal) μ-calculus is defined as follows:
each proposition and each variable is a formula;
if and are formulas, then is a formula;
if is a formula, then is a formula;
if is a formula and is an action, then is a formula; (pronounced either: box or after necessarily )
if is a formula and a variable, then is a formula, provided that every free occurrence of in occurs positively, i.e. within the scope of an even number of negations.
(The notions of free and bound variables are as usual, where is the only binding operator.)
Given the above definitions, we can enrich the syntax with:
meaning
(pronounced either: diamond or after possibly ) meaning
means , where means substituting for in all free occurrences of in .
The first two formulas are the familiar ones from the classical propositional calculus and respectively the minimal multimodal logic K.
The notation (and its dual) are inspired from the lambda calculus; the intent is to denote the least (and respectively greatest) fixed point of the expression where the "minimization" (and respectively "maximization") are in the variable , much like in lambda calculus is a function with formula in bound variable ; see the denotational semantics below for details.
Denotational semantics
Models of (propositional) μ-calculus are given as labelled transition systems where:
is a set of states;
maps to each label a binary relation on ;
, maps each proposition to the set of states where the proposition is true.
Given a labelled transition system and an interpretation of the variables of the -calculus, , is the function defined by the following rules:
;
;
;
;
;
,
|
https://en.wikipedia.org/wiki/Polyharmonic%20spline
|
In applied mathematics, polyharmonic splines are used for function approximation and data interpolation. They are very useful for interpolating and fitting scattered data in many dimensions. Special cases include thin plate splines and natural cubic splines in one dimension.
Definition
A polyharmonic spline is a linear combination of polyharmonic radial basis functions (RBFs) denoted by plus a polynomial term:
where
( denotes matrix transpose, meaning is a column vector) is a real-valued vector of independent variables,
are vectors of the same size as (often called centers) that the curve or surface must interpolate,
are the weights of the RBFs,
are the weights of the polynomial.
The polynomial with the coefficients improves fitting accuracy for polyharmonic smoothing splines and also improves extrapolation away from the centers See figure below for comparison of splines with polynomial term and without polynomial term.
The polyharmonic RBFs are of the form:
Other values of the exponent are not useful (such as ), because a solution of the interpolation problem might not exist. To avoid problems at (since ), the polyharmonic RBFs with the natural logarithm might be implemented as:
or, more simply adding a continuity extension in
The weights and are determined such that the function interpolates given points (for ) and fulfills the orthogonality conditions
All together, these constraints are equivalent to the symmetric linear system of equations
where
In order for this system of equations to have a unique solution, must be full rank. is full rank for very mild conditions on the input data. For example, in two dimensions, three centers forming a non-degenerate triangle ensure that is full rank, and in three dimensions, four centers forming a non-degenerate tetrahedron ensure that B is full rank. As explained later, the linear transformation resulting from the restriction of the domain of the linear transformation to the null space of is positive definite. This means that if is full rank, the system of equations () always has a unique solution and it can be solved using a linear solver specialised for symmetric matrices. The computed weights allow evaluation of the spline for any using equation (). Many practical details of implementing and using polyharmonic splines are explained in Fasshauer. In Iske polyharmonic splines are treated as special cases of other multiresolution methods in scattered data modelling.
Discussion
The main advantage of polyharmonic spline interpolation is that usually very good interpolation results are obtained for scattered data without performing any "tuning", so automatic interpolation is feasible. This is not the case for other radial basis functions. For example, the Gaussian function needs to be tuned, so that is selected according to the underlying grid of the independent variables. If this grid is non-uniform, a proper selection of to achieve a good interpolat
|
https://en.wikipedia.org/wiki/Hugo%20Morales
|
Hugo Alberto Morales (born 30 July 1974 in Buenos Aires) is an Argentine retired footballer who played as a midfielder.
External links
Argentine League statistics
1974 births
Living people
Footballers from Buenos Aires
Argentine men's footballers
Men's association football midfielders
Argentine Primera División players
Club Atlético Huracán footballers
Club Atlético Independiente footballers
Club Atlético Lanús footballers
Talleres de Córdoba footballers
La Liga players
Segunda División players
CD Tenerife players
Categoría Primera A players
Atlético Nacional footballers
Millonarios F.C. players
Club Deportivo Universidad Católica footballers
Argentina men's youth international footballers
Argentina men's under-20 international footballers
Argentina men's international footballers
Footballers at the 1996 Summer Olympics
Olympic footballers for Argentina
Olympic silver medalists for Argentina
Olympic medalists in football
Argentine expatriate men's footballers
Argentine expatriate sportspeople in Spain
Expatriate men's footballers in Spain
Expatriate men's footballers in Colombia
Expatriate men's footballers in Chile
Medalists at the 1996 Summer Olympics
|
https://en.wikipedia.org/wiki/Well-known%20text%20representation%20of%20geometry
|
Well-known text (WKT) is a text markup language for representing vector geometry objects. A binary equivalent, known as well-known binary (WKB), is used to transfer and store the same information in a more compact form convenient for computer processing but that is not human-readable. The formats were originally defined by the Open Geospatial Consortium (OGC) and described in their Simple Feature Access. The current standard definition is in the ISO/IEC 13249-3:2016 standard.
Geometric objects
WKT can represent the following distinct geometric objects:
Point, MultiPoint
LineString, MultiLineString
Polygon, MultiPolygon, Triangle
PolyhedralSurface
TIN (Triangulated irregular network)
GeometryCollection
Coordinates for geometries may be 2D (x, y), 3D (x, y, z), 4D (x, y, z, m) with an m value that is part of a linear referencing system or 2D with an m value (x, y, m). Three-dimensional geometries are designated by a "Z" after the geometry type and geometries with a linear referencing system have an "M" after the geometry type. Empty geometries that contain no coordinates can be specified by using the symbol EMPTY after the type name.
WKT geometries are used throughout OGC specifications and are present in applications that implement these specifications. For example, PostGIS contains functions that can convert geometries to and from a WKT representation, making them human readable.
The OGC standard definition requires a polygon to be topologically closed. It also states that if the exterior linear ring of a polygon is defined in a counterclockwise direction, then it will be seen from the "top". Any interior linear rings should be defined in opposite fashion compared to the exterior ring, in this case, clockwise.
The following are some other examples of geometric WKT strings: (Note: Each item below is an individual geometry.)
GEOMETRYCOLLECTION(POINT(4 6),LINESTRING(4 6,7 10))
POINT ZM (1 1 5 60)
POINT M (1 1 80)
POINT EMPTY
MULTIPOLYGON EMPTY
TRIANGLE((0 0 0,0 1 0,1 1 0,0 0 0))
TIN (((0 0 0, 0 0 1, 0 1 0, 0 0 0)), ((0 0 0, 0 1 0, 1 1 0, 0 0 0)))
POLYHEDRALSURFACE Z ( PATCHES
((0 0 0, 0 1 0, 1 1 0, 1 0 0, 0 0 0)),
((0 0 0, 0 1 0, 0 1 1, 0 0 1, 0 0 0)),
((0 0 0, 1 0 0, 1 0 1, 0 0 1, 0 0 0)),
((1 1 1, 1 0 1, 0 0 1, 0 1 1, 1 1 1)),
((1 1 1, 1 0 1, 1 0 0, 1 1 0, 1 1 1)),
((1 1 1, 1 1 0, 0 1 0, 0 1 1, 1 1 1))
)
Well-known binary
Well-known binary (WKB) representations are typically shown in hexadecimal strings.
The first byte indicates the byte order for the data:
00 : big endian
01 : little endian
The next 4 bytes are a 32-bit unsigned integer for the geometry type, as described below:
Each data type has a unique data structure, such as the number of points or linear rings, followed by coordinates in 64-bit double numbers.
For example, the geometry POINT(2.0 4.0) is represented as: 000000000140000000000000004010000000000000, where:
1-byte integer 00 or 0: big endian
4-byte integer 00000001 or 1: POINT (2D)
8-by
|
https://en.wikipedia.org/wiki/Sonny%27s%20Blues
|
"Sonny's Blues" is a 1957 short story written by James Baldwin, originally published in Partisan Review. The story contains the recollections of a black algebra teacher in 1950s Harlem as he reacts to his brother Sonny's drug addiction, arrest, and recovery. Baldwin republished the work in the 1965 short story collection Going to Meet the Man.
Plot
"Sonny's Blues" is a story written in the first-person singular narrative style. Much of the story is told through a series of flashbacks as memory and family history are revealed to be central drivers of the trauma and alienation experienced by Sonny and the Narrator.
The story opens with the unnamed narrator reading about a heroin bust resulting in the arrest of a man named Sonny, his brother. The narrator goes about his day as an algebra teacher at a high school in Harlem, but begins to ponder Sonny's fate and worry about the boys in his class. After school, he meets a friend of Sonny, who laments Sonny will struggle with addiction even after his detox and release.
After the narrator's daughter Grace dies of polio, he finally decides to reach out to Sonny. The narrator remembers leaving for the war, leaving Sonny with his wife Isabel and her parents. Sonny decides to play the piano, and his passion is obsessive. Once Isabel's parents find out that Sonny has not been attending school, he leaves their house, drops out of school, and joins the Navy.
Sonny returns from the war. Their relationship sours, as the narrator intermittently fights with Sonny.
Back in the present, the narrator reveals that Grace's death has caused him to reflect on his role as an older brother, surmising that his absence impaired Sonny's personal growth. The narrator resolves to reconcile with Sonny.
While Isabel takes her children to see their grandparents, the narrator contemplates searching Sonny's room. He changes his mind, however, when he sees Sonny in a revival meeting in the street below his apartment, where a woman sings with a tambourine alongside her brother and sister, and enraptures the audience.
Some time later, Sonny invites the narrator to watch him play in Greenwich Village. The narrator begrudgingly agrees to go. Sonny explains his heroin addiction in vague analogies. The woman's performance reminded him of the rush he got using heroin, equating it to a need to feel in control. The narrator asks Sonny if he has to feel like that to play. Sonny answers that some people do. The narrator then asks Sonny if it is worth killing himself just to try to escape suffering. Sonny replies that he will not die faster than anyone else trying not to suffer. Sonny reveals that the reason he wanted to leave Harlem was to escape the drugs.
The brothers go to the jazz club in Greenwich Village. The narrator realizes how revered Sonny is there as he hears him play. In the beginning, Sonny falters, as he has not played for over a year, but his playing eventually proves to be brilliant and he wins over the narrator and
|
https://en.wikipedia.org/wiki/Division%20No.%201%2C%20Saskatchewan
|
Division No. 1 is one of eighteen census divisions in the province of Saskatchewan, Canada, as defined by Statistics Canada. It is located in the southeast corner of the province, bordering Manitoba and North Dakota. The most populous community in this division is Estevan.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Division No. 1 had a population of living in of its total private dwellings, a change of from its 2016 population of . With a land area of , it had a population density of in 2021.
Census subdivisions
The following census subdivisions (municipalities or municipal equivalents) are located within Saskatchewan's Division No. 1.
Cities
Estevan
Towns
Alameda
Arcola
Bienfait
Carlyle
Carnduff
Lampman
Oxbow
Redvers
Stoughton
Wawota
Villages
Alida
Carievale
Fairlight
Forget
Frobisher
Gainsborough
Glen Ewen
Heward
Kennedy
Kenosee Lake
Kisbey
Manor
Maryfield
North Portal
Roche Percee
Storthoaks
Rural municipalities
RM No. 1 Argyle
RM No. 2 Mount Pleasant
RM No. 3 Enniskillen
RM No. 4 Coalfields
RM No. 5 Estevan
RM No. 31 Storthoaks
RM No. 32 Reciprocity
RM No. 33 Moose Creek
RM No. 34 Browning
RM No. 35 Benson
RM No. 61 Antler
RM No. 63 Moose Mountain
RM No. 64 Brock
RM No. 65 Tecumseh
RM No. 91 Maryfield
RM No. 92 Walpole
RM No. 93 Wawken
RM No. 94 Hazelwood
RM No. 95 Golden West
Indian reserves
Ocean Man First Nation
Ocean Man 69
Ocean Man 69A
Ocean Man 69B
Ocean Man 69C
Ocean Man 69E
Ocean Man 69F
Ocean Man 69G
Ocean Man 69H
Ocean Man 69I
Pheasant Rump Nakota First Nation
Pheasant Rump 68
White Bear First Nation
White Bear 70
See also
List of census divisions of Saskatchewan
List of communities in Saskatchewan
Notes
References
01
|
https://en.wikipedia.org/wiki/Division%20No.%203%2C%20Saskatchewan
|
Division No. 3 is one of eighteen census divisions in the province of Saskatchewan, Canada, as defined by Statistics Canada. It is located in the south-southwestern part of the province, adjacent to the border with Montana, United States. The most populous community in this division is Assiniboia.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Division No. 3 had a population of living in of its total private dwellings, a change of from its 2016 population of . With a land area of , it had a population density of in 2021.
Census subdivisions
The following census subdivisions (municipalities or municipal equivalents) are located within Saskatchewan's Division No. 3.
Towns
Assiniboia
Coronach
Gravelbourg
Mossbank
Ponteix
Rockglen
Willow Bunch
Villages
Hazenmore
Kincaid
Limerick
Mankota
Neville
Vanguard
Wood Mountain
Rural municipalities
RM No. 11 Hart Butte
RM No. 12 Poplar Valley
RM No. 42 Willow Bunch
RM No. 43 Old Post
RM No. 44 Waverley
RM No. 45 Mankota
RM No. 46 Glen McPherson
RM No. 71 Excel
RM No. 72 Lake of the Rivers
RM No. 73 Stonehenge
RM No. 74 Wood River
RM No. 75 Pinto Creek
RM No. 76 Auvergne
RM No. 101 Terrell
RM No. 102 Lake Johnston
RM No. 103 Sutton
RM No. 104 Gravelbourg
RM No. 105 Glen Bain
RM No. 106 Whiska Creek
Indian reserves
Cowessess First Nation
Cowessess 73
Sturgeon Lake First Nation
Sturgeon Lake 101C
Wood Mountain Lakota First Nation
Wood Mountain 160
Other communities
Aneroid
Bateman
Congress
Crane Valley
Ferland
Fife Lake
Flintoft
Glentworth
Killdeer
Lafleche
Mazenod
Mccord
Melaval
Meyronne
Ormiston
Pambrun
Scout Lake
Spring Valley
St. Victor
Verwood
Viceroy
Woodrow
See also
List of census divisions of Saskatchewan
List of communities in Saskatchewan
Notes
References
Division No. 3, Saskatchewan Statistics Canada
03
|
https://en.wikipedia.org/wiki/Division%20No.%204%2C%20Saskatchewan
|
Division No. 4 is one of eighteen census divisions in the province of Saskatchewan, Canada, as defined by Statistics Canada. It is located in the southwest corner of the province, bordering Alberta to the west and Montana, United States to the south. The most populous community in this division is Maple Creek.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Division No. 4 had a population of living in of its total private dwellings, a change of from its 2016 population of . With a land area of , it had a population density of in 2021.
Census subdivisions
The following census subdivisions (municipalities or municipal equivalents) are located within Saskatchewan's Division No. 4.
Towns
Eastend
Maple Creek
Shaunavon
Villages
Bracken
Cadillac
Carmichael
Climax
Consul
Frontier
Neville
Val Marie
Rural municipalities
RM No. 17 Val Marie
RM No. 18 Lone Tree
RM No. 19 Frontier
RM No. 49 White Valley
RM No. 51 Reno
RM No. 77 Wise Creek
RM No. 78 Grassy Creek
RM No. 79 Arlington
RM No. 107 Lac Pelletier
RM No. 108 Bone Creek
RM No. 109 Carmichael
RM No. 110 Piapot
RM No. 111 Maple Creek
Other communities
Special service areas
Admiral
Organized hamlets
Darlings Beach
Hamlets
Orkney
Piapot
Simmie
Unincorporated communities
Battle Creek
Beaver Valley
Belanger
Blumenort
Canuck
Carnagh
Claydon
Cross
Cummings
Divide
Dollard
East Fairwell
Edgell
Fort Walsh, National historic site
Garden Head
Govenlock
Hatton
Hillandale
Illerbrun
Instow
Kealey Springs
Klintonel
Lac Pelletier
Loomis
Masefield
Merryflat
Nashlyn
Neighbour
Neuhoffnung
Olga
Oxarat
Palisade
Rangeview
Ravenscrage
Robsart
Rosefield
Scotsguard
Senate
Sidewood
Skull Creek
South Fork
Staynor Hall
Vidora
West Plains
Willow Creek
Indian reserves
Nekaneet Cree Nation
Nekaneet Reserve
Little Pine First Nation
Little Pine 116
Carry the Kettle Nakoda Nation
Carry the Kettle 76-7
Carry the Kettle 76-69
Carry the Kettle 76-70
Carry the Kettle 76-71
Carry the Kettle 76-72
Carry the Kettle 76-73
Carry the Kettle 76-74
Carry the Kettle 76-75
Carry the Kettle 76-76
Carry the Kettle 76-77
Carry the Kettle 76-78
Carry the Kettle 76-79
Carry the Kettle 76-80
Carry the Kettle 76-81
Carry the Kettle 76-82
Carry the Kettle 76-84
See also
List of census divisions of Saskatchewan
List of communities in Saskatchewan
Notes
References
Division No. 4, Saskatchewan Statistics Canada
04
|
https://en.wikipedia.org/wiki/James%20Ax
|
James Burton Ax (10 January 1937 – 11 June 2006) was an American mathematician who made groundbreaking contributions in algebra and number theory using model theory. He shared, with Simon B. Kochen, the seventh Frank Nelson Cole Prize in Number Theory, which was awarded for a series of three joint papers on Diophantine problems.
Education and career
James Ax graduated from Stuyvesant High School in New York City and then the Brooklyn Polytechnic University. He earned his Ph.D. from the University of California, Berkeley in 1961 under the direction of Gerhard Hochschild, with a dissertation on The Intersection of Norm Groups. After a year at Stanford University, he joined the mathematics faculty at Cornell University. He spent the academic year 1965–1966 at Harvard University on a Guggenheim Fellowship. In 1969, he moved from Cornell to the mathematics department at Stony Brook University and remained on the faculty until 1977, when he retired from his academic career. In 1970 he was an Invited Speaker at the ICM in Nice with talk Transcendence and differential algebraic geometry. In the 1970s, he worked on the fundamentals of physics, including an axiomatization of space-time and the group theoretical properties of the axioms of quantum mechanics.
In the 1980s, he and Berkeley classmate Jim Simons founded a quantitative finance firm, Axcom Trading Advisors, which was later acquired by Renaissance Technologies and renamed the Medallion Fund. The latter fund was named after the Cole Prize won by James Ax and the Veblen Prize won by James Simons.
In the early 1990s, Ax retired from his financial career and went to San Diego, California, where he studied further on the foundations of quantum mechanics and also attended, at the University of California, San Diego, courses on playwriting and screenwriting. (In 2005 he completed a thriller screenplay entitled Bots.)
The Ax Library in the Department of Mathematics at the University of California, San Diego houses his mathematical books.
Personal
Ax is the father of American cosmologist Brian Keating and Kevin B. Keating (b. 1967), who is the president of the Kevin and Masha Keating Family Foundation. After Ax and his first wife divorced, she remarried a man named Keating, and young Brian and his older brother Kevin took the stepfather's name. Brian Keating explained (in 2020) that he and his father were not close during his childhood; his father often joked that 'I don't really care about kids until they learn algebra.'
Selected publications
See also
Leopoldt's conjecture
Schanuel's conjecture
References
External links
James B. Ax Library - at UCSD.
1937 births
2006 deaths
Stuyvesant High School alumni
Polytechnic Institute of New York University alumni
University of California, Berkeley alumni
Stanford University Department of Mathematics faculty
Cornell University faculty
Stony Brook University faculty
Model theorists
Mathematicians from New York (state)
Number theorists
|
https://en.wikipedia.org/wiki/Cryptomorphism
|
In mathematics, two objects, especially systems of axioms or semantics for them, are called cryptomorphic if they are equivalent but not obviously equivalent. In particular, two definitions or axiomatizations of the same object are "cryptomorphic" if it is not obvious that they define the same object. Examples of cryptomorphic definitions abound in matroid theory and others can be found elsewhere, e.g., in group theory the definition of a group by a single operation of division, which is not obviously equivalent to the usual three "operations" of identity element, inverse, and multiplication.
This word is a play on the many morphisms in mathematics, but "cryptomorphism" is only very distantly related to "isomorphism", "homomorphism", or "morphisms". The equivalence may in a cryptomorphism, if it is not actual identity, be informal, or may be formalized in terms of a bijection or equivalence of categories between the mathematical objects defined by the two cryptomorphic axiom systems.
Etymology
The word was coined by Garrett Birkhoff before 1967, for use in the third edition of his book Lattice Theory. Birkhoff did not give it a formal definition, though others working in the field have made some attempts since.
Use in matroid theory
Its informal sense was popularized (and greatly expanded in scope) by Gian-Carlo Rota in the context of matroid theory: there are dozens of equivalent axiomatic approaches to matroids, but two different systems of axioms often look very different.
In his 1997 book Indiscrete Thoughts, Rota describes the situation as follows:
Though there are many cryptomorphic concepts in mathematics outside of matroid theory and universal algebra, the word has not caught on among mathematicians generally. It is, however, in fairly wide use among researchers in matroid theory.
See also
Combinatorial class, an equivalence among combinatorial enumeration problems hinting at the existence of a cryptomorphism
References
Birkhoff, G.: Lattice Theory, 3rd edition. American Mathematical Society Colloquium Publications, Vol. XXV. 1967.
Crapo, H. and Rota, G.-C.: On the foundations of combinatorial theory: Combinatorial geometries. M.I.T. Press, Cambridge, Mass. 1970.
Elkins, James: Chapter Cryptomorphs in Why Are Our Pictures Puzzles?: On the Modern Origins of Pictorial Complexity, 1999
Rota, G.-C.: Indiscrete Thoughts, Birkhäuser Boston, Inc., Boston, MA. 1997.
White, N., editor: Theory of Matroids, Encyclopedia of Mathematics and its Applications, 26. Cambridge University Press, Cambridge. 1986.
Mathematical terminology
Matroid theory
|
https://en.wikipedia.org/wiki/Thomas%20Harriot%20College%20of%20Arts%20and%20Sciences
|
The Thomas Harriot College of Arts and Sciences is the liberal arts college at East Carolina University. Its Departments comprise courses of study in mathematics, the natural sciences, the social sciences, and the humanities.
In 1941, the Board of Trustees approved an undergraduate degree program in liberal arts disciplines for students wanting to pursue a non-teaching degree. When East Carolina College was elevated to university status in 1967, the School of Arts and Sciences became the College of Arts and Sciences, the home of the liberal arts. The school is named for Thomas Harriot, a cartographer, historian, and surveyor who took part in Sir Walter Raleigh's second expedition to Virginia.
Organization
The Departments of the College are:
Anthropology
Biology
Chemistry
Economics
English
Foreign Languages and Literatures
Geography
Geology
History
Mathematics
Philosophy
Physics
Political Science
Psychology
Sociology
Urban and Regional Planning
There are interdisciplinary programs in:
Asian studies
African and African-American studies
Classical studies
Coastal and marine studies
The Great Books
Institute for Historical and Cultural Research (IHCR)
International studies
Medieval and Renaissance studies
North Carolinian studies
Religious studies
Russian studies
Security studies
Women's studies
External links
Official website
East Carolina University divisions
Liberal arts colleges at universities in the United States
|
https://en.wikipedia.org/wiki/Division%20No.%205%2C%20Saskatchewan
|
Division No. 5 is one of eighteen census divisions in the province of Saskatchewan, Canada, as defined by Statistics Canada. It is located in the east-southeastern part of the province, bordering Manitoba. The most populous community in this division is Melville.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Division No. 5 had a population of living in of its total private dwellings, a change of from its 2016 population of . With a land area of , it had a population density of in 2021.
Census subdivisions
The following census subdivisions (municipalities or municipal equivalents) are located within Saskatchewan's Division No. 5.
Cities
Melville
Towns
Bredenbury
Broadview
Churchbridge
Esterhazy
Fleming
Grenfell
Kipling
Langenburg
Lemberg
Moosomin
Rocanville
Saltcoats
Wapella
Whitewood
Wolseley
Villages
Atwater
Bangor
Dubuc
Duff
Fenwood
Gerald
Glenavon
Goodeve
Grayson
Killaly
MacNutt
Neudorf
Spy Hill
Stockholm
Tantallon
Waldron
Welwyn
Windthorst
Yarbo
Resort villages
Bird's Point
Melville Beach
West End
Rural municipalities
RM No. 121 Moosomin
RM No. 122 Martin
RM No. 123 Silverwood
RM No. 124 Kingsley
RM No. 125 Chester
RM No. 151 Rocanville
RM No. 152 Spy Hill
RM No. 153 Willowdale
RM No. 154 Elcapo
RM No. 155 Wolseley
RM No. 181 Langenburg
RM No. 183 Fertile Belt
RM No. 184 Grayson
RM No. 185 McLeod
RM No. 211 Churchbridge
RM No. 213 Saltcoats
RM No. 214 Cana
RM No. 215 Stanley
Indian reserves
Cowessess First Nation
Cowessess 73
Kahkewistahaw First Nation
Kahkewistahaw 72
Ochapowace Nation
Ochapowace 71
Ochapowace 71-7
Ochapowace 71-10
Ochapowace 71-18
Ochapowace 71-26
Ochapowace 71-44
Ochapowace 71-51
Ochapowace 71-54
Ochapowace 71-70
Sakimay First Nation
Little Bone 74B
Sakimay 74
Shesheep 74A
See also
List of census divisions of Saskatchewan
List of communities in Saskatchewan
Notes
References
Division No. 5, Saskatchewan Statistics Canada
05
|
https://en.wikipedia.org/wiki/Division%20No.%206%2C%20Saskatchewan
|
Division No. 6 is one of eighteen census divisions in the province of Saskatchewan, Canada, as defined by Statistics Canada. It is located in the south-central part of the province. The most populous community in this division is Regina, the provincial capital.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Division No. 6 had a population of living in of its total private dwellings, a change of from its 2016 population of . With a land area of , it had a population density of in 2021.
Census subdivisions
The following census subdivisions (municipalities or municipal equivalents) are located within Saskatchewan's Division No. 6.
Cities
Regina
Towns
Balcarres
Balgonie
Cupar
Fort Qu'Appelle
Francis
Grand Coulee
Indian Head
Lumsden
Pilot Butte
Qu'Appelle
Regina Beach
Rouleau
Sintaluta
Southey
Strasbourg
White City
Villages
Abernethy
Belle Plaine
Bethune
Briercrest
Buena Vista
Bulyea
Chamberlain
Craven
Dilke
Disley
Drinkwater
Dysart
Earl Grey
Edenwold
Findlater
Holdfast
Kendal
Lebret
Lipton
Markinch
McLean
Montmartre
Odessa
Pense
Sedley
Silton
Vibank
Wilcox
Resort villages
Alice Beach
B-Say-Tah
Fort San
Glen Harbour
Grandview Beach
Island View
Kannata Valley
Katepwa
Lumsden Beach
North Grove
Pelican Pointe
Saskatchewan Beach
Sunset Cove
Wee Too Beach
Rural municipalities
RM No. 126 Montmartre
RM No. 127 Francis
RM No. 128 Lajord
RM No. 129 Bratt's Lake
RM No. 130 Redburn
RM No. 156 Indian Head
RM No. 157 South Qu'Appelle
RM No. 158 Edenwold
RM No. 159 Sherwood
RM No. 160 Pense
RM No. 186 Abernethy
RM No. 187 North Qu'Appelle
RM No. 189 Lumsden
RM No. 190 Dufferin
RM No. 216 Tullymet
RM No. 217 Lipton
RM No. 218 Cupar
RM No. 219 Longlaketon
RM No. 220 McKillop
RM No. 221 Sarnia
Indian reserves
Treaty Four Reserve Grounds 77 (shared by 33 First Nations)
Carry the Kettle Nakoda First Nation
Assiniboine 76
Little Black Bear First Nation
Little Black Bear 84
Muscowpetung First Nation
Muscowpetung 80
Okanese First Nation
Okanese 82
Pasqua First Nation
Pasqua 79
Peepeekisis Cree Nation
Peepeekisis 81
Piapot Cree Nation
Piapot 75
Standing Buffalo Dakota Nation
Standing Buffalo 78
Star Blanket Cree Nation
Atim Ka-mihkosit Reserve
Star Blanket 83
Star Blanket 83C
Wa-pii-moos-toosis 83A
See also
List of census divisions of Saskatchewan
List of communities in Saskatchewan
Notes
References
Division No. 6, Saskatchewan Statistics Canada
06
|
https://en.wikipedia.org/wiki/Division%20No.%207%2C%20Saskatchewan
|
Division No. 7 is one of eighteen census divisions in the province of Saskatchewan, Canada, as defined by Statistics Canada. It is located in the south-central part of the province. The most populous community in this division is Moose Jaw.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Division No. 7 had a population of living in of its total private dwellings, a change of from its 2016 population of . With a land area of , it had a population density of in 2021.
Census subdivisions
The following census subdivisions (municipalities or municipal equivalents) are located within Saskatchewan's Division No. 7.
Cities
Moose Jaw
Towns
Central Butte
Craik
Herbert
Morse
Villages
Aylesbury
Beechy
Brownlee
Caronport
Chaplin
Coderre
Ernfold
Eyebrow
Hodgeville
Keeler
Lucky Lake
Marquis
Mortlach
Riverhurst
Rush Lake
Shamrock
Tugaske
Tuxford
Waldeck
Resort villages
Beaver Flat
Coteau Beach
Mistusinne
South Lake
Sun Valley
Rural municipalities
RM No. 131 Baildon
RM No. 132 Hillsborough
RM No. 133 Rodgers
RM No. 134 Shamrock
RM No. 135 Lawtonia
RM No. 136 Coulee
RM No. 161 Moose Jaw
RM No. 162 Caron
RM No. 163 Wheatlands
RM No. 164 Chaplin
RM No. 165 Morse
RM No. 166 Excelsior
RM No. 191 Marquis
RM No. 193 Eyebrow
RM No. 194 Enfield
RM No. 222 Craik
RM No. 223 Huron
RM No. 224 Maple Bush
RM No. 225 Canaan
RM No. 226 Victory
RM No. 255 Coteau
RM No. 256 King George
Other communities
Hamlets
Bateman
Birsay
Bushell Park
Caron
Courval
Crestwynd
Demaine
Dunblane
Gouldtown
Main Centre
Neidpath
Parkbeg
Prairie View
Riverhurst
Rush Lake
Shamrock
Tugaske
Tuxford
See also
List of census divisions of Saskatchewan
List of communities in Saskatchewan
Notes
References
Division No. 7, Saskatchewan Statistics Canada
07
|
https://en.wikipedia.org/wiki/Division%20No.%208%2C%20Saskatchewan
|
Division No. 8 is one of eighteen census divisions in the province of Saskatchewan, Canada, as defined by Statistics Canada. It is located in the west-southwestern part of the province, bordering Alberta. The most populous community in this division is Swift Current.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Division No. 8 had a population of living in of its total private dwellings, a change of from its 2016 population of . With a land area of , it had a population density of in 2021.
Census subdivisions
The following census subdivisions (municipalities or municipal equivalents) are located within Saskatchewan's Division No. 8.
Cities
Swift Current
Towns
Burstall
Cabri
Eatonia
Elrose
Eston
Gull Lake
Kyle
Leader
Villages
Abbey
Fox Valley
Golden Prairie
Hazlet
Lancer
Mendham
Pennant
Prelate
Richmound
Sceptre
Shackleton
Stewart Valley
Success
Tompkins
Webb
Rural municipalities
RM No. 137 Swift Current
RM No. 138 Webb
RM No. 139 Gull Lake
RM No. 141 Big Stick
RM No. 142 Enterprise
RM No. 167 Saskatchewan Landing
RM No. 168 Riverside
RM No. 169 Pittville
RM No. 171 Fox Valley
RM No. 228 Lacadena
RM No. 229 Miry Creek
RM No. 230 Clinworth
RM No. 231 Happyland
RM No. 232 Deer Forks
RM No. 257 Monet
RM No. 259 Snipe Lake
RM No. 260 Newcombe
RM No. 261 Chesterfield
Indian reserves
Carry the Kettle Nakoda Nation
Carry the Kettle 76-33
Carry the Kettle 76-37
Carry the Kettle 76-38
Unincorporated communities
Hamlets
Laporte
Organized hamlets
White Bear
Wymark
Special service areas
Mantario
Unincorporated communities
Abbey Colony
Aikins
Beverley
Cantuar
Chipperfield
Cuthbert
Duncairn
Dunelm
Estuary
Eyre
Forgan
Gascoigne
Glidden
Greenan
Gunnworth
Hak
High Point
Horsham
Hughton
Inglebright
Isham
Java
Lacadena
Leinan
Lemsford
Lille
Linacre
Madison
Matador
Mondou
Nadeauville
Penkill
Plato
Player
Portreeve
Rhineland
Richlea
Roadene
Rosengart
Roseray
Saltburn
Sanctuary
Schantzenfeld
Schoenfeld
Schoenwiese
Shackleton
Snipe Lake
Springfeld
Surprise
Tuberose
Tunstall
Tyner
Verlo
Wartime
Wheatland Colony
Witley
Wyatt
See also
List of census divisions of Saskatchewan
List of communities in Saskatchewan
Notes
References
Division No. 8, Saskatchewan Statistics Canada
08
|
https://en.wikipedia.org/wiki/Division%20No.%209%2C%20Saskatchewan
|
Division No. 9, Canada, is one of the eighteen census divisions within the province of Saskatchewan, as defined by Statistics Canada. It is located in the eastern part of the province, bordering Manitoba. The most populous community in this division is Yorkton.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Division No. 9 had a population of living in of its total private dwellings, a change of from its 2016 population of . With a land area of , it had a population density of in 2021.
Census subdivisions
The following census subdivisions (municipalities or municipal equivalents) are located within Saskatchewan's Division No. 9.
Cities
Yorkton
Towns
Canora
Kamsack
Norquay
Preeceville
Springside
Sturgis
Villages
Arran
Buchanan
Calder
Ebenezer
Endeavour
Hyas
Invermay
Lintlaw
Pelly
Rama
Rhein
Sheho
Stenen
Theodore
Togo
Rural municipalities
RM No. 241 Calder
RM No. 243 Wallace
RM No. 244 Orkney
RM No. 245 Garry
RM No. 271 Cote
RM No. 273 Sliding Hills
RM No. 274 Good Lake
RM No. 275 Insinger
RM No. 301 St. Philips
RM No. 303 Keys
RM No. 304 Buchanan
RM No. 305 Invermay
RM No. 331 Livingston
RM No. 333 Clayton
RM No. 334 Preeceville
RM No. 335 Hazel Dell
Indian reserves
Cote First Nation
Cote 64
Keeseekoose First Nation
Keeseekoose 66
Keeseekoose 66A
Keeseekoose 66-CA-04
Keeseekoose 66-CA-05
Keeseekoose 66-CA-06
Keeseekoose 66-KE-04
Keeseekoose 66-KE-05
The Key First Nation
The Key 65
See also
List of census divisions of Saskatchewan
List of communities in Saskatchewan
Notes
References
Division No. 9, Saskatchewan Statistics Canada
09
|
https://en.wikipedia.org/wiki/Division%20No.%2010%2C%20Saskatchewan
|
Division No. 10 is one of eighteen census divisions in the province of Saskatchewan, Canada, as defined by Statistics Canada. It is located in the east-central part of the province. The most populous community in this division is Wynyard.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Division No. 10 had a population of living in of its total private dwellings, a change of from its 2016 population of . With a land area of , it had a population density of in 2021.
Census subdivisions
The following census subdivisions (municipalities or municipal equivalents) are located within Saskatchewan's Division No. 10.
Cities
none
Towns
Foam Lake
Ituna
Leroy
Raymore
Wadena
Watson
Wynyard
Villages
Elfros
Hubbard
Jansen
Kelliher
Leross
Lestock
Margo
Punnichy
Quill Lake
Quinton
Semans
Resort villages
Chorney Beach
Leslie Beach
Rural municipalities
RM No. 246 Ituna Bon Accord
RM No. 247 Kellross
RM No. 248 Touchwood
RM No. 276 Foam Lake
RM No. 277 Emerald
RM No. 279 Mount Hope
RM No. 307 Elfros
RM No. 308 Big Quill
RM No. 309 Prairie Rose
RM No. 336 Sasman
RM No. 337 Lakeview
RM No. 338 Lakeside
RM No. 339 Leroy
Source: Statistics Canada 2002 2001 Community Profiles
Indian reserves
Beardy's and Okemasis 96 and 97A
Day Star 87
Fishing Lake 89
Fishing Lake 89A
Gordon 86
Muskowekwan 85
Muskowekwan 85-1
Muskowekwan 85-10
Muskowekwan 85-12
Muskowekwan 85-15
Muskowekwan 85-17
Muskowekwan 85-22
Muskowekwan 85-23
Muskowekwan 85-24
Muskowekwan 85-26
Muskowekwan 85-27
Muskowekwan 85-28
Muskowekwan 85-29
Muskowekwan 85-2A
Muskowekwan 85-31
Muskowekwan 85-33
Muskowekwan 85-8
Poorman 88
Source: Statistics Canada 2002 2001 Community Profiles.
See also
List of census divisions of Saskatchewan
List of communities in Saskatchewan
Notes
References
Division No. 10, Saskatchewan Statistics Canada
10
|
https://en.wikipedia.org/wiki/Division%20No.%2011%2C%20Saskatchewan
|
Division No. 11 is one of eighteen census divisions in the province of Saskatchewan, Canada, as defined by Statistics Canada. It is located in the central part of the province and includes the largest city in the province, Saskatoon.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Division No. 11 had a population of living in of its total private dwellings, a change of from its 2016 population of . With a land area of , it had a population density of in 2021.
Census subdivisions
The following census subdivisions (municipalities or municipal equivalents) are located within Saskatchewan's Division No. 11.
Cities
Martensville
Saskatoon
Warman
Towns
Allan
Colonsay
Dalmeny
Davidson
Dundurn
Govan
Hanley
Imperial
Langham
Lanigan
Nokomis
Osler
Outlook
Watrous
Villages
Bladworth
Bradwell
Broderick
Clavet
Drake
Duval
Elbow
Glenside
Kenaston
Liberty
Loreburn
Hawarden
Meacham
Plunkett
Simpson
Strongfield
Viscount
Young
Zelma
Resort villages
Etters Beach
Manitou Beach
Shields
Thode
Rural municipalities
RM No. 250 Last Mountain Valley
RM No. 251 Big Arm
RM No. 252 Arm River
RM No. 253 Willner
RM No. 254 Loreburn
RM No. 280 Wreford
RM No. 281 Wood Creek
RM No. 282 McCraney
RM No. 283 Rosedale
RM No. 284 Rudy
RM No. 310 Usborne
RM No. 312 Morris
RM No. 313 Lost River
RM No. 314 Dundurn
RM No. 340 Wolverine
RM No. 341 Viscount
RM No. 342 Colonsay
RM No. 343 Blucher
RM No. 344 Corman Park
Indian reserves
Whitecap 94
See also
List of census divisions of Saskatchewan
List of communities in Saskatchewan
Notes
References
Division No. 11, Saskatchewan Statistics Canada
11
|
https://en.wikipedia.org/wiki/Division%20No.%2012%2C%20Saskatchewan
|
Division No. 12 is one of the eighteen census divisions in the province of Saskatchewan, Canada, as defined by Statistics Canada. It is located in the west-central part of the province. The most populous community in this division is Battleford.
Demographics
In the 2021 Canadian census conducted by Statistics Canada, Division No. 12 had a population of living in of its total private dwellings, a change of from its 2016 population of . With a land area of , it had a population density of in 2021.
Census subdivisions
The following census subdivisions (municipalities or municipal equivalents) are located within Saskatchewan's Division No. 12.
Cities
none
Towns
Battleford
Biggar
Delisle
Rosetown
Zealandia
Villages
Asquith
Conquest
Dinsmore
Harris
Kinley
Macrorie
Milden
Perdue
Tessier
Vanscoy
Wiseton
Rural municipalities
RM No. 285 Fertile Valley
RM No. 286 Milden
RM No. 287 St. Andrews
RM No. 288 Pleasant Valley
RM No. 315 Montrose
RM No. 316 Harris
RM No. 317 Marriott
RM No. 318 Mountain View
RM No. 345 Vanscoy
RM No. 346 Perdue
RM No. 347 Biggar
RM No. 376 Eagle Creek
RM No. 377 Glenside
RM No. 378 Rosemount
RM No. 408 Prairie
RM No. 438 Battle River
Indian reserves
Grizzly Bear's Head 110 and Lean Man 111
Mosquito 109
Red Pheasant 108
Sweet Grass 113
Sweet Grass 113-M16
See also
List of census divisions of Saskatchewan
List of communities in Saskatchewan
Notes
References
Division No. 12, Saskatchewan Statistics Canada
12
|
https://en.wikipedia.org/wiki/Division%20No.%2013%2C%20Saskatchewan
|
Division No. 13 is one of eighteen census divisions in the province of Saskatchewan, Canada, as defined by Statistics Canada. It is located in the western part of the province, bordering Alberta. The most populous community in this division is Kindersley.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Division No. 13 had a population of living in of its total private dwellings, a change of from its 2016 population of . With a land area of , it had a population density of in 2021.
Census subdivisions
The following census subdivisions (municipalities or municipal equivalents) are located within Saskatchewan's Division No. 13.
Cities
none
Towns
Cut Knife
Kerrobert
Kindersley
Luseland
Macklin
Scott
Unity
Wilkie
Villages
Brock
Coleville
Denzil
Dodsland
Flaxcombe
Landis
Major
Marengo
Marsden
Neilburg
Netherhill
Plenty
Ruthilda
Senlac
Smiley
Tramping Lake
Rural municipalities
RM No. 290 Kindersley
RM No. 292 Milton
RM No. 319 Winslow
RM No. 320 Oakdale
RM No. 321 Prairiedale
RM No. 322 Antelope Park
RM No. 349 Grandview
RM No. 350 Mariposa
RM No. 351 Progress
RM No. 352 Heart's Hill
RM No. 379 Reford
RM No. 380 Tramping Lake
RM No. 381 Grass Lake
RM No. 382 Eye Hill
RM No. 409 Buffalo
RM No. 410 Round Valley
RM No. 411 Senlac
RM No. 439 Cut Knife
RM No. 440 Hillsdale
RM No. 442 Manitou Lake
Indian reserves
Indian Reserve - Little Pine 116
Indian Reserve - Poundmaker 114
See also
List of census divisions of Saskatchewan
List of communities in Saskatchewan
Notes
References
Division No. 13, Saskatchewan Statistics Canada
13
|
https://en.wikipedia.org/wiki/Division%20No.%2014%2C%20Saskatchewan
|
Division No. 14 is one of eighteen census divisions in the province of Saskatchewan, Canada, as defined by Statistics Canada. It is located on the northern portion of Southeast Saskatchewan, bordering Manitoba. The most populous community in this division is the city of Melfort. Other important communities are the towns of Nipawin and Tisdale.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Division No. 14 had a population of living in of its total private dwellings, a change of from its 2016 population of . With a land area of , it had a population density of in 2021.
Census subdivisions
The following census subdivisions (municipalities or municipal equivalents) are located within Saskatchewan's Division No. 14.
Cities
Melfort
Towns
Arborfield
Carrot River
Choiceland
Hudson Bay
Kelvington
Naicam
Nipawin
Porcupine Plain
Rose Valley
Star City
Tisdale
Villages
Archerwill
Aylsham
Bjorkdale
Codette
Fosston
Love
Mistatim
Pleasantdale
Ridgedale
Smeaton
Spalding
Valparaiso
Weekes
White Fox
Zenon Park
Resort villages
Tobin Lake
Rural municipalities
RM No. 366 Kelvington
RM No. 367 Ponass Lake
RM No. 368 Spalding
RM No. 394 Hudson Bay
RM No. 395 Porcupine
RM No. 397 Barrier Valley
RM No. 398 Pleasantdale
RM No. 426 Bjorkdale
RM No. 427 Tisdale
RM No. 428 Star City
RM No. 456 Arborfield
RM No. 457 Connaught
RM No. 458 Willow Creek
RM No. 486 Moose Range
RM No. 487 Nipawin
RM No. 488 Torch River
Indian reserves
Carrot River 29A
Kinistin 91
Opaskwayak Cree Nation 27A (formerly Carrot River 27A)
Red Earth 29
Shoal Lake 28A
Yellow Quill 90
See also
List of census divisions of Saskatchewan
List of communities in Saskatchewan
References
Division No. 14, Saskatchewan Statistics Canada
14
|
https://en.wikipedia.org/wiki/Division%20No.%2015%2C%20Saskatchewan
|
Division No. 15 is one of eighteen census divisions in the province of Saskatchewan, Canada, as defined by Statistics Canada. It is located in the north-central part of the province. The most populous community in this division is Prince Albert.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Division No. 15 had a population of living in of its total private dwellings, a change of from its 2016 population of . With a land area of , it had a population density of in 2021.
Census subdivisions
The following census subdivisions (municipalities or municipal equivalents) are located within Saskatchewan's Division No. 15.
Cities
Humboldt
Prince Albert
Melfort
Towns
Aberdeen
Birch Hills
Bruno
Cudworth
Duck Lake
Hague
Kinistino
Rosthern
St. Brieux
Vonda
Wakaw
Waldheim
Villages
Albertville
Alvena
Annaheim
Beatty
Christopher Lake
Englefeld
Hepburn
Laird
Lake Lenore
Meath Park
Middle Lake
Muenster
Paddockwood
Pilger
Prud'Homme
St. Benedict
St. Gregor
St. Louis
Weirdale
Weldon
Resort villages
Candle Lake
Wakaw Lake
Rural municipalities
RM No. 369 St. Peter
RM No. 370 Humboldt
RM No. 371 Bayne
RM No. 372 Grant
RM No. 373 Aberdeen
RM No. 399 Lake Lenore
RM No. 400 Three Lakes
RM No. 401 Hoodoo
RM No. 402 Fish Creek
RM No. 403 Rosthern
RM No. 404 Laird
RM No. 429 Flett's Springs
RM No. 430 Invergordon
RM No. 431 St. Louis
RM No. 459 Kinistino
RM No. 460 Birch Hills
RM No. 461 Prince Albert
RM No. 463 Duck Lake
RM No. 490 Garden River
RM No. 491 Buckland
RM No. 520 Paddockwood
RM No. 521 Lakeland
Reserves
Beardy's 97 and Okemasis 96
Beardy's and Okemasis 96 & 97-B
Chief Joseph Custer
Cumberland 100A
James Smith 100
Little Red River 106C
Montreal Lake 106B
Muskoday Reserve
One Arrow 95
One Arrow 95-1A
One Arrow 95-1C
One Arrow 95-1D
Wahpaton 94A
See also
List of census divisions of Saskatchewan
List of communities in Saskatchewan
References
Division No. 15, Saskatchewan Statistics Canada
15
|
https://en.wikipedia.org/wiki/Division%20No.%2016%2C%20Saskatchewan
|
Division No. 16 is one of eighteen census divisions in the province of Saskatchewan, Canada, as defined by Statistics Canada. It is located in the north-central part of the province. The most populous community in this division is North Battleford.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Division No. 16 had a population of living in of its total private dwellings, a change of from its 2016 population of . With a land area of , it had a population density of in 2021.
Census subdivisions
The following census subdivisions (municipalities or municipal equivalents) are located within Saskatchewan's Division No. 16.
Cities
North Battleford
Towns
Big River
Blaine Lake
Hafford
Radisson
Shellbrook
Spiritwood
Villages
Borden
Canwood
Debden
Denholm
Krydor
Leask
Leoville
Marcelin
Maymont
Medstead
Parkside
Richard
Ruddell
Shell Lake
Speers
Resort villages
Big Shell
Echo Bay
Pebble Baye
Rural municipalities
RM No. 405 Great Bend
RM No. 406 Mayfield
RM No. 434 Blaine Lake
RM No. 435 Redberry
RM No. 436 Douglas
RM No. 437 North Battleford
RM No. 464 Leask
RM No. 466 Meeting Lake
RM No. 467 Round Hill
RM No. 493 Shellbrook
RM No. 494 Canwood
RM No. 496 Spiritwood
RM No. 497 Medstead
RM No. 555 Big River
Crown colonies
North Battleford Crown Colony
Unorganized areas
Prince Albert National Park
Indian reserves
Indian Reserve --Ahtahkakoop 104
Indian Reserve --Big River 118
Indian Reserve --Chitek Lake 191
Indian Reserve --Little Red River 106D
Indian Reserve --Lucky Man
Indian Reserve --Mistawasis 103
Indian Reserve --Muskeg Lake 102B
Indian Reserve --Muskeg Lake 102D
Indian Reserve --Muskeg Lake 102E
Indian Reserve --Muskeg Lake 102F
Indian Reserve --Muskeg Lake 102G
Indian Reserve --Muskeg Lake Cree Nation 102
Indian Reserve --Pelican Lake 191A
Indian Reserve --Pelican Lake 191B
Indian Reserve --Saulteaux 159A
Indian Reserve --Sturgeon Lake 101
Indian Reserve --Sweet Grass 113-L6
Indian Reserve --Witchekan Lake 117
Indian Reserve --Witchekan Lake 117D
See also
List of census divisions of Saskatchewan
List of communities in Saskatchewan
References
Division No. 16, Saskatchewan Statistics Canada
16
|
https://en.wikipedia.org/wiki/Division%20No.%2017%2C%20Saskatchewan
|
Division No. 17 is one of eighteen census divisions in the province of Saskatchewan, Canada, as defined by Statistics Canada. It is located in the west-northwest part of the province, bordering Alberta. The most populous community in this division is the interprovincial city of Lloydminster. Another important population centre is the town of Meadow Lake.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Division No. 17 had a population of living in of its total private dwellings, a change of from its 2016 population of . With a land area of , it had a population density of in 2021.
Census subdivisions
The following census subdivisions (municipalities or municipal equivalents) are located within Saskatchewan's Division No. 17.
Cities
Lloydminster
Meadow Lake
Towns
Lashburn
Maidstone
Marshall
St. Walburg
Turtleford
Villages
Dorintosh
Edam
Glaslyn
Goodsoil
Loon Lake
Makwa
Meota
Mervin
Paradise Hill
Paynton
Pierceland
Waseca
Resort villages
Aquadeo
Cochin
Greig Lake
Kivimaa-Moonlight Bay
Metinota
Rural municipalities
RM No. 468 Meota
RM No. 469 Turtle River
RM No. 470 Paynton
RM No. 471 Eldon
RM No. 472 Wilton
RM No. 498 Parkdale
RM No. 499 Mervin
RM No. 501 Frenchman Butte
RM No. 502 Brittania
RM No. 561 Loon Lake
RM No. 588 Meadow Lake
RM No. 622 Beaver River
Indian reserves
Big Island Lake Cree Nation
Eagles Lake 165C
Flying Dust First Nation 105
Makaoo 120
Makwa Lake 129
Makwa Lake 129A
Makwa Lake 129B
Makwa Lake 129C
Meadow Lake 105A
Min-A-He-Quo-Sis 116C
Ministikwan 161
Ministikwan 161A
Moosomin 112B
Saulteaux 159
Seekaskootch 119
Thunderchild First Nation 115B
Thunderchild First Nation 115C
Thunderchild First Nation 115D
Waterhen 130
See also
List of census divisions of Saskatchewan
List of communities in Saskatchewan
References
Division No. 17, Saskatchewan Statistics Canada
17
|
https://en.wikipedia.org/wiki/Division%20No.%2018%2C%20Saskatchewan
|
Division No. 18, Saskatchewan, Canada, is one of the eighteen Statistics Canada census divisions within the province, occupying the northern half of the province. The census division is coextensive with the Northern Saskatchewan Administration District (NSAD).
The census division is the largest in the province terms of area at , representing 46 per cent of the province's entire area of .
The most populous communities in the census division are La Ronge and La Loche with populations of 2,743 and 2,611 respectively.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Division No. 18 had a population of living in of its total private dwellings, a change of from its 2016 population of . With a land area of , it had a population density of in 2021.
Census subdivisions
Division No. 18 has 58 census subdivisions, of which 24 are municipalities (including a portion of the City of Flin Flon, a city bisected by the Saskatchewan-Manitoba border, 2 northern towns, 11 northern villages and 10 northern hamlets), 32 are First Nations communities (31 Indian reserves and an Indian settlement), an unincorporated northern settlement and the unorganized balance of Division No. 18. All municipalities within the census division, except for the Northern Hamlet of Black Point, are recognized as census subdivisions.
Cities
Northern towns
Northern villages
Northern hamlets
Indian settlements
Indian reserves
Unincorporated communities
A northern settlement is an unincorporated community in the Northern Saskatchewan Administration District, and its administration is regulated by The Northern Municipalities Act. Saskatchewan has 11 northern settlements. One northern settlement, Missinipe, is recognized as a census subdivision by Statistics Canada.
See also
List of census divisions of Saskatchewan
List of communities in Saskatchewan
References
18
|
https://en.wikipedia.org/wiki/Ren%C3%A9%20Gateaux
|
René Eugène Gateaux (; 5 May 1889 – 3 October 1914) was a French mathematician. He is principally known for the Gateaux derivative, used in the calculus of variations and in the theory of optimal control. He died in combat during World War I. Paul Lévy produced a posthumous edition of his works, extending them considerably, in his Leçons d'analyse fonctionnelle of 1922.
Life
Early years
Gateaux was born on at Vitry-le-François, Marne, 222 years after another mathematician, Abraham de Moivre, was born there (de Moivre, being of Huguenot ancestry, fled to London after the Edict of Fontainebleau of 1685). His father had a small saddlery and upholstery business, and his mother was a seamstress. He was schooled at Reims, and in 1907 entered the École normale supérieure (ENS) on the rue d'Ulm. He was well regarded as one of the most promising mathematicians among his peers. During his time at ENS, Gateaux converted to Roman Catholicism.
Schoolteacher
In 1910, he sat the mathematics examination (being placed 11th of 16 in his year, a somewhat unimpressive result perhaps due to his being so young, according to the ENS's deputy head Émile Borel). He became a teacher at the lycée in Bar-le-Duc, Meuse in 1912, having completed his two years' military service (the first as a private soldier, and the second as a sub-lieutenant, as was required by a 1905 law concerning the service of students from some Grandes Écoles).
At the same time as he took the post at Bar-le-Duc, he started to work on his thesis about functional analysis, following the work of Vito Volterra and Jacques Hadamard, and its applications to potential theory. Even though it is unknown why Gateaux chose this subject, he may have been encouraged by Hadamard himself, who had just completed a course on the subject at the Collège de France. Among others, in 1911 Paul Lévy had undertaken a brilliant thesis on this type of question, and in 1912 Joseph Pérès, an alumnus of the ENS in the year before Gateaux, had left for Rome to work under Volterra.
Student in Rome
In 1913, Gateaux asked for, and was awarded, a bursary from the David Weill Foundation to go with him to Rome. Before leaving, he sent a letter to Borel and Volterra, on the subjects they had proposed he work on in Rome. Within it was the theme of integration of real functions in infinite-dimensional space.
He stayed in Rome from October 1913, following Volterra's course and working hard. He published numerous notes in the Rendiconti dell'Accademia dei Lincei, and presented a seminar at the University of Rome. He returned to France in June 1914, intending to return in the September after being awarded a Commercy bursary for another year.
Death in combat
Gateaux was caught off-guard by mobilisation and the August 1914 declaration of war. He was sent to Toul as a lieutenant in the 269th Infantry Regiment with responsibility for the 2nd Machine Guns section. Having helped defend Nancy in the Battle of Grand Couronné, his regiment
|
https://en.wikipedia.org/wiki/De%20La%20Salle%20University%20College%20of%20Science
|
The College of Science (COS) of De La Salle University was originally part of the College of Arts and Sciences. In 1982, the departments of Biology, Chemistry, Mathematics and Physics separated to form the College of Science while the liberal arts departments formed the College of Liberal Arts. Although the College of Science is the youngest and the second smallest college formed in the university, its contribution to the academe and to the country has had tremendous impact to scientific research development and nation-building. And thus in recognition of their efforts, all of the Science and Mathematics programs of the college have been granted the recognition of Center of Excellence in the Philippines by the Commission on Higher Education.
Academic departments of COS
Biology
The Department of Biology focuses itself in laboratory work, lectures, research, and field trips. The Department boasts of its advanced facilities, located in the either St. Joseph Building and in STRC Building. It is a Center for Excellence in Biology by CHED.
Chemistry
The Department of Chemistry has been awarded as a center of excellence by the CHED. It has since been teaming up with the various departments of the university to provide Engineering and Liberal Arts students with the important facts of Chemistry. The Department also encourages the use of computers in Chemical analyses. The Department has three programs, Bachelor of Science in Chemistry, Bachelor of Science in Chemistry Minor in Business and Bachelor of Science in Biochemistry.
Mathematics
Established in 1946, the Department of Mathematics trains its students in logical thought, critical analysis, imagination, and problem solving. It also handles graduate and undergraduate courses in the College of Business and Economics, Liberal Arts and Education. It was awarded as a center of excellence by the CHED.
Physics
The Department of Physics was recognized by the CHED as a Center of Excellence. The Department caters to a variety of interests in physics: solid-state physics and material science, medical instrumentation, laser physics, instrumentation physics, quantum field theory, and physics education.
Degree offerings
Undergraduate programs
BS in Biochemistry
BS in Biology
BS in Chemistry
BS in Chemistry minor in Business Studies
BS in Chemistry major in Food Science
BS in Human Biology
BS in Mathematics with specialization in Business Applications
BS in Mathematics with specialization in Computer Applications
BS in Statistics major in Actuarial Science
BS in Physics minor in Economics
BS in Physics minor in Finance
BS in Physics with specialization in Materials Science
BS in Physics with specialization in Medical Instrumentation
BS in Premed Physics
Graduate programs
Doctor of Philosophy in Biology
Doctor of Philosophy in Chemistry
Doctor of Philosophy in Mathematics (Regular and Straight Program)
Doctor of Philosophy in Physics (Regular and Straight Program)
Master of Science in Biology
Master of Scie
|
https://en.wikipedia.org/wiki/Fleming%E2%80%93Viot%20process
|
In probability theory, a Fleming–Viot process (F–V process) is a member of a particular subset of probability measure-valued Markov processes on compact metric spaces, as defined in the 1979 paper by Wendell Helms Fleming and Michel Viot. Such processes are martingales and diffusions.
The Fleming–Viot processes have proved to be important to the development of a mathematical basis for the theories behind allele drift.
They are generalisations of the Wright–Fisher process and arise as infinite population limits of suitably rescaled variants of Moran processes.
See also
Coalescent theory
Voter model
References
Fleming, W. H., Michel Viot, M. (1979) "Some measure-valued Markov processes in population genetics theory" (PDF format) Indiana University Mathematics Journal, 28 (5), 817–843.
Ferrari, Pablo A.; Mari, Nevena "Quasi stationary distributions and Fleming Viot processes" , Lecture presentation
Markov processes
Statistical genetics
Martingale theory
|
https://en.wikipedia.org/wiki/Cox%20process
|
In probability theory, a Cox process, also known as a doubly stochastic Poisson process is a point process which is a generalization of a Poisson process where the intensity that varies across the underlying mathematical space (often space or time) is itself a stochastic process. The process is named after the statistician David Cox, who first published the model in 1955.
Cox processes are used to generate simulations of spike trains (the sequence of action potentials generated by a neuron), and also in financial mathematics where they produce a "useful framework for modeling prices of financial instruments in which credit risk is a significant factor."
Definition
Let be a random measure.
A random measure is called a Cox process directed by , if is a Poisson process with intensity measure .
Here, is the conditional distribution of , given .
Laplace transform
If is a Cox process directed by , then has the Laplace transform
for any positive, measurable function .
See also
Poisson hidden Markov model
Doubly stochastic model
Inhomogeneous Poisson process, where λ(t) is restricted to a deterministic function
Ross's conjecture
Gaussian process
Mixed Poisson process
References
Notes
Bibliography
Cox, D. R. and Isham, V. Point Processes, London: Chapman & Hall, 1980
Donald L. Snyder and Michael I. Miller Random Point Processes in Time and Space Springer-Verlag, 1991 (New York) (Berlin)
Poisson point processes
|
https://en.wikipedia.org/wiki/List%20of%20people%20from%20Milan
|
The following is a list of people from Milan.
Scientists
Mathematics
Eugenio Calabi (1923–2023)
Marco Abate (born 1962)
Maria Gaetana Agnesi (1718–1799), the world's first woman to write a mathematics handbook and the first woman appointed as a mathematics professor at a university, wrote the first book discussing both differential and integral calculus
Enzo Tonti (1935–2021)
Fencing theorists
Camillo Agrippa (1535–1595), is considered to be one of the greatest fencing theorists of all time
Fine arts
Architects
Donato Felice d'Allio (16771761), Rococo style, worked in Austria
Sculptors
Carlo Abate (18591941)
Painters
Filippo Abbiati (16401715)
Mario Acerbi (painter) (18871982)
Angelo Achini (18501930)
Franz Adam (18151886)
Luigi Ademollo (17641849)
Carlo Paolo Agazzi (18701922)
Federico Agnelli (16261702), engraver
Claudio Detto (born 1950), Italian contemporary art painter
Giorgio Salmoiraghi (1936–2022)
Guglielmo Stella (1828–1888), painter and writer
Photographers
Gabriele Basilico (19442013)
Fabio Ponzio (born 1959)
Oliviero Toscani (born 1942)
Writers and historians
Ottavio Codogno (1570/74–1630), author of a guidebook to the postal services of early 17th-century Europe
Musicians
Arrangers
Pino Presti (born 1943) Italian bassist, arranger, composer, conductor and record producer
Pianists
Marcello Abbado (19262020)
Maria Teresa Agnesi Pinottini (17201795), harpsichordist
Singers
Iris Adami Corradetti (19041998)
Alessandro Mahmoud (born 1992), singer-songwriter, Italian representative at the Eurovision Song Contest in and
Orchestral conductors
Claudio Abbado (19332014)
Roberto Abbado (born 1954)
Riccardo Chailly
Pop rock artists
Manuel Agnelli (born 1966), alternative rock, member of the band Afterhours
Ghigo Agosti (born 1936), also comedy rock
Adriano Celentano (born 1938)
Cristina Scabbia (born 1972), singer of Lacuna Coil
Politicians
Agnese Visconti (13631391), consort of Francesco I Gonzaga Lord of Mantua
Vittorio Agnoletto (born 1958), (Communist Refoundation Party), member of the European Parliament
Silvio Berlusconi (1936-2023), Italian politician who served as Prime Minister of Italy in four governments
Bettino Craxi (19342000), Italian politician, leader of the Italian Socialist Party from 1976 to 1993 and Prime Minister of Italy from 1983 to 1987
Mario Monti (born 1943), Italian economist who served as the Prime Minister of Italy from 2011 to 2013
Catholic religious
Alberto Ablondi (19242010)
Ferdinando d'Adda (16501719), cardinal of San Clemente, San Pietro in Vincoli, Santa Balbina and Albano, archbishop of Amasya and apostolic nuncio to Great Britain
Aicone (died 918), archbishop of Milan
Media
Actors/Actresses of Film, Theatre and TV
Diego Abatantuono (born 1955)
Cele Abba (19061992)
Marta Abba (19001988)
Marco Lui (1975), mime and comedian
TV and radio presenter
Lucilla Agosti (born 1978), TV and radio presenter and actress
Athletes
Footballers
Camillo Achilli (1921–1998), football pl
|
https://en.wikipedia.org/wiki/Yellow%20jersey%20statistics
|
Since the first Tour de France in 1903, there have been 2,205 stages, up to and including the final stage of the 2021 Tour de France. Since 1919, the race leader following each stage has been awarded the yellow jersey ().
Although the leader of the classification after a stage gets a yellow jersey, he is not considered the winner of the yellow jersey, only the wearer. Only after the final stage, the wearer of the yellow jersey is considered the winner of the yellow jersey, and thereby the winner of the Tour de France.
In this article first-place-classifications before 1919 are also counted as if a yellow jersey was awarded. There have been more yellow jerseys given than there were stages: In 1914, 1929, and 1931, there were multiple cyclists with the same leading time, and the 1988 Tour de France had a "prelude", an extra stage for a select group of cyclists. As of 2021 a total of 2,208 yellow jerseys have been awarded in the Tour de France to 295 riders.
Individual records
In previous tours, sometimes a stage was broken in two (or three). On such occasions, only the cyclist leading at the end of the day is counted. The "Jerseys" column lists the number of days that the cyclist wore the yellow jersey; the "Tour wins" column gives the number of times the cyclist won the general classification. The next four columns indicate the number of times the rider won the points classification, the King of the Mountains classification, and the young rider competition, and the years in which the yellow jersey was worn, with bold years indicating an overall Tour win. For example: Eddy Merckx has spent 96 days in the yellow jersey, won the general classification five times, won the points classification three times, and won the mountains classification twice, but never won the young rider classification. He wore the yellow jersey in the Tours of 1969, 1970, 1971, 1972, 1974 (which he all won) and 1975 (which he did not win). Three cyclists (Jean Robic in 1947, Charly Gaul in 1958 and Jan Janssen in 1968) have won the Tour de France with only two yellow jerseys in their career.
Until the results of Lance Armstrong were annulled for cheating in 2012, he was ranked second in this list, leading the Tour for 83 stages from 1999 to 2005. Alberto Contador was stripped of the yellow jersey and 6 days of wearing it in 2010 Tour de France because he tested positive for doping.
Fabian Cancellara is, as of 2022, the rider with the most yellow jerseys for someone who has not won the Tour with twenty-nine days in yellow.
This table is updated to the last stage of the 2023 Tour de France (i.e. the stage is included).
Number of wearers per year
The largest number of riders wearing the yellow jersey in any year is 8. The smallest is 1.
Notes
Per country
The yellow jersey has been awarded to 25 countries since 1903. In the table below, "Jerseys" indicates the number of yellow jerseys that were given to cyclists of each country. "Tour wins" stands for the number of to
|
https://en.wikipedia.org/wiki/Normal%20curve%20equivalent
|
In educational statistics, a normal curve equivalent (NCE), developed for the United States Department of Education by the RMC Research Corporation, is a way of normalizing scores received on a test into a 0-100 scale similar to a percentile rank, but preserving the valuable equal-interval properties of a z-score.
It is defined as:
70770 + /qnorm(.99) × z
or, approximately
50 + 21.063 × z,
where z is the standard score or "z-score", i.e. z is how many standard deviations above the mean the raw score is (z is negative if the raw score is below the mean). The reason for the choice of the number 21.06 is to bring about the following result: If the scores are normally distributed (i.e. they follow the "bell-shaped curve") then
the normal equivalent score is 99 if the percentile rank of the raw score is 99;
the normal equivalent score is 50 if the percentile rank of the raw score is 50;
the normal equivalent score is 1 if the percentile rank of the raw score is 1.
This relationship between normal equivalent scores and percentile ranks does not hold at values other than 1, 50, and 99. It also fails to hold in general if scores are not normally distributed.
The number 21.06 was chosen because
It is desired that a score of 99 correspond to the 99th percentile;
The 99th percentile in a normal distribution is 2.3263 standard deviations above the mean;
99 is 49 more than 50—thus 49 points above the mean;
49/2.3263 = 21.06.
Normal curve equivalents are on an equal-interval scale. This is advantageous compared to percentile rank scales, which suffer from the problem that the difference between any two scores is not the same as that between any other two scores (see below or percentile rank for more information).
The major advantage of NCEs over percentile ranks is that NCEs can be legitimately averaged.
Caution
Careful consideration is required when computing effect sizes using NCEs. NCEs differ from other scores, such as raw and scaled scores, in the magnitude of the effect sizes. Comparison of NCEs typically results in smaller effect sizes, and using the typical ranges for other effect sizes may result in interpretation errors.
Excel formula for conversion from Percentile to NCE:
=21.06*NORMSINV(PR/100)+50, where PR is the percentile value.
Excel formula for conversion from NCE to Percentile:
=100*NORMSDIST((NCE-50)/21.06), where NCE is the Normal Curve Equivalent (NCE) value
References
External links
Norm Scale Calculator (Utility for the Transformation and Visualization of Norm Scores)
Scholastic Testing Service, a glossary of terms related to the bell or normal curve.
UCLA stats: How should I analyze percentile rank data describing how to convert percentile ranks to NCEs with Stata.
Statistics of education
|
https://en.wikipedia.org/wiki/Transport%20in%20Winnipeg
|
Transport in Winnipeg involves various transportation systems, including both private and public services, and modes of transport in the capital city of Manitoba.
According to Statistics Canada, in 2011, the dominant form of travel in Winnipeg was by car as a driver (69%), followed by commute trips using public transit (15%), as a car passenger (7%), walking (6%), bicycle (2%), and other modes (1%).
In the province of Manitoba, transportation is the largest contributor to greenhouse gas emissions, representing almost half of the personal emissions for households. As such, the City of Winnipeg government aims for its residents to ultimately adopt sustainable transport methods—i.e., walking, cycling, and public transit—as their preferred choice of transportation.
Transportation structures within the city are the responsibility of the Winnipeg government's Public Works Department. More generally, transportation in Manitoba is regulated by The Driver and Vehicles Act and The Highway Traffic Act. Moreover, insurance is mandatory in the province, and is made available via Manitoba Public Insurance and Autopac brokers.
Pre-incorporation
For thousands of years, the region's Indigenous peoples used various networks of rivers across what is now known as the province of Manitoba.
Situated at the confluence of the Red and the Assiniboine rivers in what is now downtown Winnipeg, The Forks became an early meeting place for the purpose of trade and would prove to be the most important location for European and First Nations trade in Manitoba. The common method of transportation on these waterways during this time were often birch bark canoes generally used by the Indigenous peoples, while European traders would tend to use York boats.
Overland transport in the 19th century was often by ox-drawn Red River cart, which could be built and maintained using only locally obtained material.
Winnipeg was incorporation as a city on 8 November 1873, and has since continued to grow and expand, along with its transportation needs and its inventory of structures.
Roads and expressways
As the City is situated at the confluence of the Red and the Assiniboine rivers, it was necessary for Winnipeg in its early years to construct various bridges, allowing the city to grow and enabling those on opposite sides of the rivers to be united.
First constructing bridges in order to cross the Red and the Assiniboine rivers, the early growth of Winnipeg saw the need for additional structures to be built; either to either go over (overpasses) or go under (underpasses) railroad tracks and/or roadways. As the street network developed and expanded, the City built various other structures (culverts) in order for the creeks within Winnipeg (e.g. Bunn's Creek, Omand's Creek, Sturgeon Creek, etc.) to flow under the newly-constructed streets.
Today, the City of Winnipeg's Public Works Department is responsible for street and sidewalk maintenance and for managing structures within the ci
|
https://en.wikipedia.org/wiki/Rectified%20120-cell
|
In geometry, a rectified 120-cell is a uniform 4-polytope formed as the rectification of the regular 120-cell.
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as tC120.
There are four rectifications of the 120-cell, including the zeroth, the 120-cell itself. The birectified 120-cell is more easily seen as a rectified 600-cell, and the trirectified 120-cell is the same as the dual 600-cell.
Rectified 120-cell
In geometry, the rectified 120-cell or rectified hecatonicosachoron is a convex uniform 4-polytope composed of 600 regular tetrahedra and 120 icosidodecahedra cells. Its vertex figure is a triangular prism, with three icosidodecahedra and two tetrahedra meeting at each vertex.
Alternative names:
Rectified 120-cell (Norman Johnson)
Rectified hecatonicosichoron / rectified dodecacontachoron / rectified polydodecahedron
Icosidodecahedral hexacosihecatonicosachoron
Rahi (Jonathan Bowers: for rectified hecatonicosachoron)
Ambohecatonicosachoron (Neil Sloane & John Horton Conway)
Projections
Related polytopes
Notes
References
Kaleidoscopes: Selected Writings of H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
(Paper 22) H.S.M. Coxeter, Regular and Semi-Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
(Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
J.H. Conway and M.J.T. Guy: Four-Dimensional Archimedean Polytopes, Proceedings of the Colloquium on Convexity at Copenhagen, page 38 und 39, 1965
N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
External links
rectified 120-cell Marco Möller's Archimedean polytopes in R4 (German)
Four-dimensional Archimedean Polytopes, Marco Möller, 2004 PhD dissertation
H4 uniform polytopes with coordinates: r{5,3,3}
4-polytopes
|
https://en.wikipedia.org/wiki/Eastern%20Military%20Academy
|
Eastern Military Academy (EMA) was a high school military academy founded in 1944 in Connecticut, United States, by Roland R. Robinson, a former mathematics teacher at Peekskill Military Academy (now also defunct), and his brother-in-law, Carleton Witham. The relationship with the local town was poor from the start, and in 1948 the school moved to Cold Spring Hills on Long Island, New York, until the school closed in 1979.
History
At its new location, the school was based in one of the largest mansions ever constructed in the United States, Oheka Castle, built by Otto Kahn, a multimillionaire. Following Kahn's death in 1934, his heirs had little interest in the estate, and the town of Huntington briefly used it as a retirement home for municipal employees.
EMA was organized for most of its existence as a battalion, with a band company, troop (using horses stabled a few miles away), two infantry companies of high school and junior high school students, a company of children sixth grade and below, a company of day students, i.e. students who did not board in the school, battalion staff of two to four members, and a four-member color guard. For several years of very high enrollment, the school organized as a regiment.
Robinson and Witham died within six weeks of one another in the summer of 1968, leaving the school in the hands of Alice Robinson, who was Robinson's widow and Witham's sister. According to an article in Newsday on September 30, 1968, she was then the first woman ever to head a military academy in the USA. In 1970 she sold the school to three investors, without notifying the longtime dean and new Headmaster, Leopold Hedbavny, or alumni. These investors immediately took out a three million dollar bank mortgage, although they had paid Mrs. Robinson only $50,000 plus stock in their new corporation, and stock in several other of the ultimately nine schools they bought. All nine schools were eventually closed following the taking out of large mortgages.
In 1955 the Army granted EMA status as an honor Junior ROTC unit. From 1951 to 1968 students ninth grade and up were taken to a military training base, Camp Smith, for a week in May for riflery training and practice. This continued at a base in New Jersey until 1975, the same year the school lost its honor rating. In 1977 the Army struck it from the rolls of recognized Junior ROTC units, and removed all military supplied equipment, mostly M-1 Garand rifles (prior to 1955 cadets used Springfield M1903 rifles). In 1979, enrollment was down to just ninety from a high of over 350. The school closed in 1979 after a fire had been set in the dormitory floors.
In the late 1960s, as the school was declining, two groups of disgruntled staff broke away and founded the General Douglas MacArthur Military Academy and the Marine Military Academy.
For a few years EMA had an armored unit based on some surplus Armored Personnel Carriers. Operating these when gasoline prices began rising beca
|
https://en.wikipedia.org/wiki/Geodesics%20as%20Hamiltonian%20flows
|
In mathematics, the geodesic equations are second-order non-linear differential equations, and are commonly presented in the form of Euler–Lagrange equations of motion. However, they can also be presented as a set of coupled first-order equations, in the form of Hamilton's equations. This latter formulation is developed in this article.
Overview
It is frequently said that geodesics are "straight lines in curved space". By using the Hamilton–Jacobi approach to the geodesic equation, this statement can be given a very intuitive meaning: geodesics describe the motions of particles that are not experiencing any forces. In flat space, it is well known that a particle moving in a straight line will continue to move in a straight line if it experiences no external forces; this is Newton's first law. The Hamiltonian describing such motion is well known to be with p being the momentum. It is the conservation of momentum that leads to the straight motion of a particle. On a curved surface, exactly the same ideas are at play, except that, in order to measure distances correctly, one must use the Riemannian metric. To measure momenta correctly, one must use the inverse of the metric. The motion of a free particle on a curved surface still has exactly the same form as above, i.e. consisting entirely of a kinetic term. The resulting motion is still, in a sense, a "straight line", which is why it is sometimes said that geodesics are "straight lines in curved space". This idea is developed in greater detail below.
Geodesics as an application of the principle of least action
Given a (pseudo-)Riemannian manifold M, a geodesic may be defined as the curve that results from the application of the principle of least action. A differential equation describing their shape may be derived, using variational principles, by minimizing (or finding the extremum) of the energy of a curve. Given a smooth curve
that maps an interval I of the real number line to the manifold M, one writes the energy
where is the tangent vector to the curve at point .
Here, is the metric tensor on the manifold M.
Using the energy given above as the action, one may choose to solve either the Euler–Lagrange equations or the Hamilton–Jacobi equations. Both methods give the geodesic equation as the solution; however, the Hamilton–Jacobi equations provide greater insight into the structure of the manifold, as shown below. In terms of the local coordinates on M, the (Euler–Lagrange) geodesic equation is
where the xa(t) are the coordinates of the curve γ(t), are the Christoffel symbols, and repeated indices imply the use of the summation convention.
Hamiltonian approach to the geodesic equations
Geodesics can be understood to be the Hamiltonian flows of a special Hamiltonian vector field defined on the cotangent space of the manifold. The Hamiltonian is constructed from the metric on the manifold, and is thus a quadratic form consisting entirely of the kinetic term.
The geodesic equa
|
https://en.wikipedia.org/wiki/Thomas%20Jech
|
Thomas J. Jech (, ; born January 29, 1944, in Prague) is a mathematician specializing in set theory who was at Penn State for more than 25 years.
Life
He was educated at Charles University (his advisor was Petr Vopěnka) and from 2000 is at the Institute of Mathematics of the Academy of Sciences of the Czech Republic.
Work
Jech's research also includes mathematical logic, algebra, analysis, topology, and measure theory.
Jech gave the first published proof of the consistency of the existence of a Suslin line.
With Karel Prikry, he introduced the notion of precipitous ideal. He gave several models where the axiom of choice failed, for example one with ω1 measurable. The concept of a Jech–Kunen tree is named after him and Kenneth Kunen.
Bibliography
Lectures in set theory, with particular emphasis on the method of forcing, Springer-Verlag Lecture Notes in Mathematics 217 (1971) ()
The axiom of choice, North-Holland 1973 (Dover paperback edition )
(with K. Hrbáček) Introduction to set theory, Marcel Dekker, 3rd edition 1999 ()
Multiple forcing, Cambridge University Press 1986 ()
Set Theory: The Third Millennium Edition, revised and expanded, 2006, Springer Science & Business Media, . 1st ed. 1978; 2nd (corrected) ed. 1997
References
External links
Home page, with a copy at Penn state.
1944 births
Living people
20th-century Czech mathematicians
21st-century Czech mathematicians
Set theorists
Czechoslovak mathematicians
Charles University alumni
|
https://en.wikipedia.org/wiki/Mark%20Pinsky
|
Mark A. Pinsky (15 July 1940 – 8 December 2016) was Professor of Mathematics at Northwestern University. His research areas included probability theory, mathematical analysis, Fourier Analysis and wavelets. Pinsky earned his Ph.D at Massachusetts Institute of Technology (MIT).
His published works include 125 research papers and ten books, including several conference proceedings and textbooks. His 2002 book, Introduction to Fourier Analysis and Wavelets, has been translated into Spanish.
Biography
Pinsky was at Northwestern beginning in 1968, following a two-year postdoctoral position at Stanford. He completed the Ph.D. at Massachusetts Institute of Technology in 1966, under the direction of Henry McKean and became Full Professor in 1976. He was married to the artist Joanna Pinsky since 1963; they have three children, Seth, Jonathan and Lea, and four grandchildren, Nathan, Jason, Justin and Jasper.
Academic memberships and services
Pinsky was a member of the American Mathematical Society (AMS), a fellow of the Institute of Mathematical Statistics, Mathematical Association of America, and has provided services for Mathematical Sciences Research Institute (MSRI), most recently as Consulting Editor for the AMS. He served on the Executive Committee of MSRI for the period 1996–2000.
Pinsky was an invited speaker at the meeting to honor Stanley Zietz in Philadelphia at University of the Sciences in Philadelphia, on 20 March 2008.
Pinsky was a Fellow of the Institute of Mathematical Statistics and member of the Editorial Board of Journal of Theoretical Probability.
Mathematical works
His early work was directed toward generalizations of the central limit theorem, known as random evolution, on which he wrote a monograph in 1991. At the same time he became interested in differential equations with noise, computing the Lyapunov exponents of various stochastic differential equations. His many interests include classical harmonic analysis and stochastic Riemannian geometry. The Pinsky phenomenon, a term coined by J.P. Kahane, has become a popular topic for research in harmonic analysis.
Pinsky was coordinator of the twenty-ninth Midwest Probability Colloquium, held at Northwestern University in October 2007.
In 2008, the Department of Mathematics at Northwestern University received a private donation from Mark and Joanna Pinsky to endow an annual lecture series.
Selected publications
Introduction to Fourier Analysis and Wavelets (Brooks/Cole Series in Advanced Mathematics), 2002,
Fourier series of radial functions in several variables
Pointwise Fourier inversion and related eigenfunction expansions
Eigenfunction expansions with general boundary conditions
Pointwise Fourier Inversion-A Wave Equation Approach
A generalized Kolmogorov for the Hilbert transform
See list of publication with pdfs.
External links
Home page at Northwestern
References
Northwestern University faculty
1940 births
2016 deaths
Stanford University staff
20th-cent
|
https://en.wikipedia.org/wiki/Faug%C3%A8re%27s%20F4%20and%20F5%20algorithms
|
In computer algebra, the Faugère F4 algorithm, by Jean-Charles Faugère, computes the Gröbner basis of an ideal of a multivariate polynomial ring. The algorithm uses the same mathematical principles as the Buchberger algorithm, but computes many normal forms in one go by forming a generally sparse matrix and using fast linear algebra to do the reductions in parallel.
The Faugère F5 algorithm first calculates the Gröbner basis of a pair of generator polynomials of the ideal. Then it uses this basis to reduce the size of the initial matrices of generators for the next larger basis:
If Gprev is an already computed Gröbner basis (f2, …, fm) and we want to compute a Gröbner basis of (f1) + Gprev then we will construct matrices whose rows are m f1 such that m is a monomial not divisible by the leading term of an element of Gprev.
This strategy allows the algorithm to apply two new criteria based on what Faugère calls signatures of polynomials. Thanks to these criteria, the algorithm can compute Gröbner bases for a large class of interesting polynomial systems, called regular sequences, without ever simplifying a single polynomial to zero—the most time-consuming operation in algorithms that compute Gröbner bases. It is also very effective for a large number of non-regular sequences.
Implementations
The Faugère F4 algorithm is implemented
in FGb, Faugère's own implementation, which includes interfaces for using it from C/C++ or Maple,
in Maple computer algebra system, as the option method=fgb of function Groebner[gbasis]
in the Magma computer algebra system,
in the SageMath computer algebra system,
Study versions of the Faugère F5 algorithm is implemented in
the SINGULAR computer algebra system;
the SageMath computer algebra system.
in SymPy Python package.
Applications
The previously intractable "cyclic 10" problem was solved by F5, as were a number of systems related to cryptography; for example HFE and C*.
References
Till Stegers Faugère's F5 Algorithm Revisited (alternative link). Diplom-Mathematiker Thesis, advisor Johannes Buchmann, Technische Universität Darmstadt, September 2005 (revised April 27, 2007). Many references, including links to available implementations.
External links
Faugère's home page (includes pdf reprints of additional papers)
An introduction to the F4 algorithm.
Computer algebra
|
https://en.wikipedia.org/wiki/Raghunath%20Dhondo%20Karve
|
Raghunath Dhondo Karve (14 January 1882 – 14 October 1953) was a professor of mathematics and a social reformer from Maharashtra, India. He was a pioneer in initiating family planning and birth control for masses in Mumbai in 1921.
Born in a Chitpavan Brahmin family, Raghunath was the eldest son of Bharat Ratna Maharshi Dhondo Keshav Karve. His mother Radhabai died during childbirth in 1891, when he was nine. He was born in Murud. He studied at New English School, Pune. He stood first in a matriculation examination conducted in 1899. He went to Fergusson College, Pune where he obtained a Bachelor of Arts degree in 1904. Karve started his professional career as a professor of mathematics at Wilson College in Mumbai. However, when he started publicly expressing his views about family planning, population control, and women's right to experience sexual/sensual pleasure as much as men, the conservative Christian administrators of the college asked him to resign from the professorship. He then devoted himself to the above causes.
On his own initiative, Karve started the very first birth control clinic in India in 1921, the same year when the first birth control clinic opened in London.
Books authored
‘Santatiniyaman Aachar ani Vichar’ (Family planning: Thoughts and Action) in 1923
‘Guptrogapasun Bachav’ & ‘Aadhunik Kaamashastra’
In 1927 he published ‘Samajswasthya’; a monthly on social health, and continued it till death (14 October 1953). Through this monthly, he tried to educated people about sex education. He hardly could meet his own needs, yet he never ceased from continuing the monthly
‘Adhunik Kamashastra’ (1934)
‘Adhunik Aharshastra’ (1938)
‘Vaishya Vyavasay’ (1940), which had a scientific approach.
Some of his other light themed books were ‘Parischya Ghari’ (1946) and ’13 Goshti’ (1940)
Samaj Swasthya
Karve published a Marathi magazine Samaj Swasthya (समाजस्वास्थ्य) starting from July 1927 until 1953. In it, he continually discussed issues of society's well-being through population control through use of contraceptives so as prevent unwanted pregnancies and induced abortions. He promoted responsible parenting by men, gender equality, and women's empowerment and right to experience sexual/sensual pleasure. As an illustration of some of Karve's radical thoughts, he expressed the thought that so long as childbirth and venereal diseases are prevented, women could engage in promiscuity—even perhaps with male prostitutes—for the sake of variety in sexual pleasure, if they so desire, without, in fact, harming their husbands. All the issues of this magazine are now available at www.radhonkarve.com
Wife's support
Karve's wife, Malati, supported his cause though it brought them social ostracism besides his loss of his professorial career. She shared the couple's financial responsibility, and the two chose to remain childless.
Apart from his wife, he had support of Dr. Ambedkar, "Wrangler Paranjape", Riyastkar Sardesai, and Mama Varekar
|
https://en.wikipedia.org/wiki/Bais%20%28Rajput%20clan%29
|
The Bais () is a Rajput clan from India.
History
Their wealth caused Donald Butter, a visiting doctor who wrote Outlines of the Topography and Statistics of the Southern Districts of Oudh, and of the Cantonment of Sultanpur-Oudh, to describe the Bais Rajput in the 1830s as the "best dressed and housed people of the southern Oudh".
The Bais Rajputs were known for well-building.
See also
Baiswara
Rajput clans
References
Rajput clans of Uttar Pradesh
|
https://en.wikipedia.org/wiki/Transferable%20belief%20model
|
The transferable belief model (TBM) is an elaboration on the Dempster–Shafer theory (DST), which is a mathematical model used to evaluate the probability that a given proposition is true from other propositions that are assigned probabilities. It was developed by Philippe Smets who proposed his approach as a response to Zadeh’s example against Dempster's rule of combination. In contrast to the original DST the TBM propagates the open-world assumption that relaxes the assumption that all possible outcomes are known. Under the open world assumption Dempster's rule of combination is adapted such that there is no normalization. The underlying idea is that the probability mass pertaining to the empty set is taken to indicate an unexpected outcome, e.g. the belief in a hypothesis outside the frame of discernment. This adaptation violates the probabilistic character of the original DST and also Bayesian inference. Therefore, the authors substituted notation such as probability masses and probability update with terms such as degrees of belief and transfer giving rise to the name of the method: The transferable belief model.
Zadeh’s example in TBM context
Lofti Zadeh describes an information fusion problem. A patient has an illness that can be caused by three different factors A, B or C. Doctor 1 says that the patient's illness is very likely to be caused by A (very likely, meaning probability p = 0.95), but B is also possible but not likely (p = 0.05). Doctor 2 says that the cause is very likely C (p = 0.95), but B is also possible but not likely (p = 0.05). How is one to make one's own opinion from this?
Bayesian updating the first opinion with the second (or the other way round) implies certainty that the cause is B. Dempster's rule of combination lead to the same result. This can be seen as paradoxical, since although the two doctors point at different causes, A and C, they both agree that B is not likely. (For this reason the standard Bayesian approach is to adopt Cromwell's rule and avoid the use of 0 or 1 as probabilities.)
Formal definition
The TBM describes beliefs at two levels:
a credal level where beliefs are entertained and quantified by belief functions,
a pignistic level where beliefs can be used to make decisions and are quantified by probability functions.
Credal level
According to the DST, a probability mass function is defined such that:
with
where the power set contains all possible subsets of the frame of discernment . In contrast to the DST the mass allocated to the empty set is not required to be zero, and hence generally holds true. The underlying idea is that the frame of discernment is not necessarily exhaustive, and thus belief allocated to a proposition , is in fact allocated to where is the set of unknown outcomes. Consequently, the combination rule underlying the TBM corresponds to Dempster's rule of combination, except the normalization that grants . Hence, in the TBM any two independent functions and ar
|
https://en.wikipedia.org/wiki/Sommerfeld%20radiation%20condition
|
In applied mathematics, and theoretical physics the Sommerfeld radiation condition is a concept from theory of differential equations and scattering theory used for choosing a particular solution to the Helmholtz equation. It was introduced by Arnold Sommerfeld in 1912
and is closely related to the limiting absorption principle (1905) and the limiting amplitude principle (1948).
The boundary condition established by the principle essentially chooses a solution of some wave equations which only radiates outwards from known sources. It, instead, of allowing arbitrary inbound waves from the infinity propagating in instead detracts from them.
The theorem most underpinned by the condition only holds true in three spatial dimensions. In two it breaks down because wave motion doesn't retain its power as one over radius squared. On the other hand, in spatial dimensions four and above, power in wave motion falls off much faster in distance.
Formulation
Arnold Sommerfeld defined the condition of radiation for a scalar field satisfying the Helmholtz equation as
"the sources must be sources, not sinks of energy. The energy which is radiated from the sources must scatter to infinity; no energy may be radiated from infinity into ... the field."
Mathematically, consider the inhomogeneous Helmholtz equation
where is the dimension of the space, is a given function with compact support representing a bounded source of energy, and is a constant, called the wavenumber. A solution to this equation is called radiating if it satisfies the Sommerfeld radiation condition
uniformly in all directions
(above, is the imaginary unit and is the Euclidean norm). Here, it is assumed that the time-harmonic field is If the time-harmonic field is instead one should replace with in the Sommerfeld radiation condition.
The Sommerfeld radiation condition is used to solve uniquely the Helmholtz equation. For example, consider the problem of radiation due to a point source in three dimensions, so the function in the Helmholtz equation is where is the Dirac delta function. This problem has an infinite number of solutions, for example, any function of the form
where is a constant, and
Of all these solutions, only satisfies the Sommerfeld radiation condition and corresponds to a field radiating from The other solutions are unphysical . For example, can be interpreted as energy coming from infinity and sinking at
See also
Limiting absorption principle
Limiting amplitude principle
Nonradiation condition
References
"Eighty years of Sommerfeld’s radiation condition", Steven H. Schot, Historia Mathematica 19, #4 (November 1992), pp. 385–401, .
External links
Radiation
Boundary conditions
|
https://en.wikipedia.org/wiki/List%20of%20scientific%20priority%20disputes
|
This is a list of priority disputes in science and science-related fields (such as mathematics).
Mathematics
Rule for solving cubic equations: Niccolò Tartaglia, Gerolamo Cardano
Leibniz–Newton calculus controversy: Isaac Newton, Gottfried Leibniz
Physics
Mechanical equivalent of heat: James Prescott Joule, Julius von Mayer
Radio waves: James Clerk Maxwell, Oliver Lodge, Heinrich Hertz, David Edward Hughes
Special relativity priority dispute: Albert Einstein, Henri Poincaré, Hendrik Lorentz
General relativity priority dispute: Albert Einstein, David Hilbert
Chandrasekhar limit: Subrahmanyan Chandrasekhar, Edmund Clifton Stoner, Wilhelm Anderson
Eightfold Way: Murray Gell-Mann, Yuval Ne'eman
Accelerating expansion of the universe: High-Z Supernova Search Team, Supernova Cosmology Project.
Astronomy
Controversy over the discovery of Haumea: José Luis Ortiz Moreno et al., Michael E. Brown et al.
Sunspots: Galileo, Christoph Scheiner
Geoheliocentric system: Tycho Brahe, Nicolaus Raimarus Ursus
Galilean moons: Galileo, Simon Marius
Prediction of Neptune: Urbain Le Verrier, John Couch Adams
Chemistry
Oxygen: Joseph Priestley, Carl Wilhelm Scheele, Antoine Laurent Lavoisier
Periodic table: Dmitri Mendeleev, Lothar Meyer
Biology and medicine
Evolution: Charles Darwin, Alfred Russel Wallace, Patrick Matthew
Opiate receptor: Candace Pert, Solomon H. Snyder
DNA structure: Francis Crick, James D. Watson, Rosalind Franklin, Erwin Chargaff, Oswald Avery
Lymphatic system: Olof Rudbeck, Thomas Bartholin
Blood transfusion: Richard Lower, Henry Oldenburg, Jean-Baptiste Denis
Life cycle of malarial parasite: Giovanni Battista Grassi, Ronald Ross
Magnetic resonance imaging (MRI): Paul Lauterbur, Peter Mansfield, Raymond Vahan Damadian, and others (see 2003 Nobel Prize in Physiology or Medicine)
HIV: Robert Gallo, Luc Montagnier (see 2008 Nobel Prize in Physiology or Medicine)
Teaching a deaf-mute person to speak: John Wallis, William Holder
Technology
Watch balance spring: Robert Hooke, Christiaan Huygens
Light bulb: Joseph Swan, Thomas Edison
Elisha Gray and Alexander Bell telephone controversy: Johann Philipp Reis, Antonio Meucci, Alexander Graham Bell, Elisha Gray
Incandescent light bulb: Thomas Edison, Joseph Swan
Radio: Oliver Lodge, Jagadish Chandra Bose, Reginald Fessenden, Guglielmo Marconi, Roberto Landell de Moura, Alexander Popov, Nikola Tesla(see invention of radio)
Electronic television: Philo T. Farnsworth, Vladimir Zworykin(see history of television)
Claims to the first powered flight: Shivkar Bapuji Talpade in the Marutsakhā (1895), Clément Ader in the Avion III (1897), Gustave Whitehead in his No's. 21 and 22 aeroplanes (1901–1903), Richard Pearse in his monoplane (1903–1904), Samuel Pierpont Langley's Aerodrome A (1903), Karl Jatho in Jatho biplane (1903), The Wright brothers in the Wright Flyer (1903), Alberto Santos-Dumont in the 14 Bis (1906)
Notes
See also
List of examples of Stigler's law
Nobel Prize controversies
List
|
https://en.wikipedia.org/wiki/Necklace%20%28disambiguation%29
|
A necklace is an article of jewelry worn around the neck.
Necklace may also refer to:
Necklace (combinatorics) or fixed necklace, a concept in combinatorial mathematics
"The Necklace", a short story by Guy de Maupassant
"The Necklace (Dynasty)", a 1981 episode of the TV series Dynasty
Necklace (horse)
See also
Necklace of Harmonia, a fabled object in Greek mythology
Necklace splitting problem, another application in combinatorics
The Affair of the Necklace (disambiguation)
Antoine's necklace, in topology
Necklacing, a form of execution
Necklace Nebula, nebula located in the constellation Sagitta
|
https://en.wikipedia.org/wiki/P%C3%B3lya%20Prize
|
Pólya Prize may refer to:
George Pólya Prize, awarded by the Society for Industrial and Applied Mathematics (SIAM)
Pólya Prize (LMS), awarded by the London Mathematical Society
See also
George Pólya Award, awarded by the Mathematical Association of America
|
https://en.wikipedia.org/wiki/George%20P%C3%B3lya%20Prize
|
The Society for Industrial and Applied Mathematics (SIAM) has three prizes named after George Pólya: the George Pólya Prize for Mathematical Exposition, established in 2013; the George Pólya Prize in Applied Combinatorics, established in 1969, and first awarded in 1971; and the George Pólya Prize in Mathematics, established in 1992, to complement the exposition and applied combinatorics prizes.
Frank Harary and William T. Tutte donated money to establish the original 1969 prize in combinatorics. Currently, funding for the three SIAM prizes is provided by the estate of Stella Pólya, the wife of George Pólya.
Combinatorics Winners
1971 Ronald L. Graham, Klaus Leeb, B. L. Rothschild, A. W. Hales, and R. I. Jewett
1975 Richard P. Stanley, Endre Szemerédi, and Richard M. Wilson
1979 László Lovász
1983 Anders Björner and Paul Seymour
1987 Andrew Yao
1992 Gil Kalai and Saharon Shelah
1994 Gregory Chudnovsky and Harry Kesten
1996 Jeff Kahn and David Reimer
1998 Percy Deift, Xin Zhou, and Peter Sarnak
2000 Noga Alon
2002 Craig Tracy and Harold Widom
2004 Neil Robertson and Paul Seymour
2006 Gregory F. Lawler, Oded Schramm, Wendelin Werner
2008 Van H. Vu
2010 Emmanuel Candès and Terence Tao
2012 Vojtěch Rödl and Mathias Schacht
2014 Adam Marcus, Daniel Spielman and Nikhil Srivastava
2016 Jozsef Balogh, Robert Morris, and Wojciech Samotij, David Saxton and Andrew Thomason
2018 No Award Given
2021 Assefaw Gebremedhin, Fredrik Manne, Alex Pothen
2022 Antti Kupiainen, Rémi Rhodes, Vincent Vargas
(List of winners from Pólya Prize page at SIAM website.)
See also
List of mathematics awards
References
Awards established in 1969
Awards of the Society for Industrial and Applied Mathematics
|
https://en.wikipedia.org/wiki/Double%20counting%20%28fallacy%29
|
Double counting is a fallacy in reasoning.
An example of double counting is shown starting with the question: What is the probability of seeing at least one 5 when throwing a pair of dice? An erroneous argument goes as follows: The first die shows a 5 with probability 1/6, and the second die shows a 5 with probability 1/6; therefore, the probability of seeing a 5 on at least one of the dice is 1/6 + 1/6 = 1/3 = 12/36. However, the correct answer is 11/36, because the erroneous argument has double-counted the event where both dice show 5s.
Double counting can be generalized as the fallacy in which, when counting events or occurrences in probability or in other areas, a solution counts events two or more times, resulting in an erroneous number of events or occurrences which is higher than the true result. This results in the calculated sum of probabilities for all possible outcomes to be higher than 100%, which is impossible.
In mathematical terms, the previous example calculated the probability of P(A or B) as P(A)+P(B). However, by the inclusion-exclusion principle, P(A or B) = P(A) + P(B) - P(A and B), one compensates for double counting by subtracting those objects which were double counted.
Another example is made in the joke where a man explains to his boss why he has to be an hour late to work every day:
8760 (365*24) hours compose one year.
He needs 8 hours sleep daily (365*8) 2920 hours leaving 5840 hours.
He uses an hour and 30 minutes per meal, (1.5*365) or 547.5 hours, leaving 5250.5.
He needs 20 minutes a day to bathe, 109.5 leaving 5183.
Weekends use 2 days a week, 52 weeks, 2496, leaving 2687.
Vacation uses two weeks, 336 hours, leaving 2361.
The company celebrates 5 holidays a year, 120, leaving 2231.
He commutes to work 1 hour each way, 2 hours a day, 5 days a week, 50 weeks a year, 500, leaving 1731.
The work week is 8 hours a day, 5 days a week, 50 weeks a year, 2000 hours, leaving him short by 269 hours, or roughly 1 hour of each work day.
All of the numbers are correct, but the man is counting them incorrectly. Sleeping, bathing and eating are also parts of the weekends, holidays and vacation times that are being included, making these hours double counted.
Further reading
Stephen Campbell, Flaws and Fallacies in Statistical Thinking (2012), in series Dover Books on Mathematics, Courier Corporation,
Informal fallacies
Misuse of statistics
|
https://en.wikipedia.org/wiki/Don%20Berry%20%28statistician%29
|
Donald Arthur Berry (born May 26, 1940) is an American statistician and a practitioner and proponent of Bayesian statistics in medical science. He was the chairman of the Department of Biostatistics and Applied Mathematics at the University of Texas M. D. Anderson Cancer Center from 1999-2010, where he played a role in the use of Bayesian methods to develop innovative, adaptive clinical trials. He is best known for the development of statistical theory relating to the design of clinical trials. He is a fellow of the American Statistical Association and the Institute of Mathematical Sciences. He founded Berry Consultants, a statistical consulting group, with Scott Berry in 2000.
Biography
Berry was born in Southbridge, Massachusetts, in 1940, and obtained an A.B. in mathematics from Dartmouth College, before moving to Yale University where he received an M.A. and Ph.D. in statistics. Berry initially "flunked out" of his undergraduate education at Dartmouth and joined the army, being stationed in Panama, but at the request of his Dean he returned to Dartmouth to complete his undergraduate education in mathematics.
References
External links
1940 births
Living people
American statisticians
Bayesian statisticians
Yale Graduate School of Arts and Sciences alumni
University of Minnesota faculty
Duke University faculty
University of Texas faculty
Fellows of the American Statistical Association
Dartmouth College alumni
People from Southbridge, Massachusetts
|
https://en.wikipedia.org/wiki/Rectified%2024-cell
|
In geometry, the rectified 24-cell or rectified icositetrachoron is a uniform 4-dimensional polytope (or uniform 4-polytope), which is bounded by 48 cells: 24 cubes, and 24 cuboctahedra. It can be obtained by rectification of the 24-cell, reducing its octahedral cells to cubes and cuboctahedra.
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as tC24.
It can also be considered a cantellated 16-cell with the lower symmetries B4 = [3,3,4]. B4 would lead to a bicoloring of the cuboctahedral cells into 8 and 16 each. It is also called a runcicantellated demitesseract in a D4 symmetry, giving 3 colors of cells, 8 for each.
Construction
The rectified 24-cell can be derived from the 24-cell by the process of rectification: the 24-cell is truncated at the midpoints. The vertices become cubes, while the octahedra become cuboctahedra.
Cartesian coordinates
A rectified 24-cell having an edge length of has vertices given by all permutations and sign permutations of the following Cartesian coordinates:
(0,1,1,2) [4!/2!×23 = 96 vertices]
The dual configuration with edge length 2 has all coordinate and sign permutations of:
(0,2,2,2) [4×23 = 32 vertices]
(1,1,1,3) [4×24 = 64 vertices]
Images
Symmetry constructions
There are three different symmetry constructions of this polytope. The lowest construction can be doubled into by adding a mirror that maps the bifurcating nodes onto each other. can be mapped up to symmetry by adding two mirror that map all three end nodes together.
The vertex figure is a triangular prism, containing two cubes and three cuboctahedra. The three symmetries can be seen with 3 colored cuboctahedra in the lowest construction, and two colors (1:2 ratio) in , and all identical cuboctahedra in .
Alternate names
Rectified 24-cell, Cantellated 16-cell (Norman Johnson)
Rectified icositetrachoron (Acronym rico) (George Olshevsky, Jonathan Bowers)
Cantellated hexadecachoron
Disicositetrachoron
Amboicositetrachoron (Neil Sloane & John Horton Conway)
Related polytopes
The convex hull of the rectified 24-cell and its dual (assuming that they are congruent) is a nonuniform polychoron composed of 192 cells: 48 cubes, 144 square antiprisms, and 192 vertices. Its vertex figure is a triangular bifrustum.
Related uniform polytopes
The rectified 24-cell can also be derived as a cantellated 16-cell:
Citations
References
T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 26. pp. 409: Hemicubes: 1n1)
Norman Johnson Uniform Polytopes, Manuscript (1991)
N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966)
4-polytopes
|
https://en.wikipedia.org/wiki/Analysis%20of%20rhythmic%20variance
|
In statistics, analysis of rhythmic variance (ANORVA) is a method for detecting rhythms in biological time series, published by Peter Celec (Biol Res. 2004, 37(4 Suppl A):777–82). It is a procedure for detecting cyclic variations in biological time series and quantification of their probability. ANORVA is based on the premise that the variance in groups of data from rhythmic variables is low when a time distance of one period exists between the data entries.
References
Analysis of rhythmic variance--ANORVA. A new simple method for detecting rhythms in biological time series.
Analysis of Rhythmic Variance
Analysis of variance
Time series
Biostatistics
|
https://en.wikipedia.org/wiki/LU%20decomposition
|
In numerical analysis and linear algebra, lower–upper (LU) decomposition or factorization factors a matrix as the product of a lower triangular matrix and an upper triangular matrix (see matrix decomposition). The product sometimes includes a permutation matrix as well. LU decomposition can be viewed as the matrix form of Gaussian elimination. Computers usually solve square systems of linear equations using LU decomposition, and it is also a key step when inverting a matrix or computing the determinant of a matrix. The LU decomposition was introduced by the Polish astronomer Tadeusz Banachiewicz in 1938. To quote: "It appears that Gauss and Doolittle applied the method
[of elimination] only to symmetric equations. More recent authors, for example, Aitken, Banachiewicz, Dwyer, and Crout … have emphasized the use of the method, or variations of it, in connection with non-symmetric problems … Banachiewicz … saw the point … that the basic problem is really one of matrix factorization, or “decomposition” as he called it."
It's also referred to as LR decomposition (factors into left and right triangular matrices).
Definitions
Let A be a square matrix. An LU factorization refers to the factorization of A, with proper row and/or column orderings or permutations, into two factors – a lower triangular matrix L and an upper triangular matrix U:
In the lower triangular matrix all elements above the diagonal are zero, in the upper triangular matrix, all the elements below the diagonal are zero. For example, for a 3 × 3 matrix A, its LU decomposition looks like this:
Without a proper ordering or permutations in the matrix, the factorization may fail to materialize. For example, it is easy to verify (by expanding the matrix multiplication) that . If , then at least one of and has to be zero, which implies that either L or U is singular. This is impossible if A is nonsingular (invertible). This is a procedural problem. It can be removed by simply reordering the rows of A so that the first element of the permuted matrix is nonzero. The same problem in subsequent factorization steps can be removed the same way; see the basic procedure below.
LU factorization with partial pivoting
It turns out that a proper permutation in rows (or columns) is sufficient for LU factorization. LU factorization with partial pivoting (LUP) refers often to LU factorization with row permutations only:
where L and U are again lower and upper triangular matrices, and P is a permutation matrix, which, when left-multiplied to A, reorders the rows of A. It turns out that all square matrices can be factorized in this form, and the factorization is numerically stable in practice. This makes LUP decomposition a useful technique in practice.
LU factorization with full pivoting
An LU factorization with full pivoting involves both row and column permutations:
where L, U and P are defined as before, and Q is a permutation matrix that reorders the columns of A.
Lower-diagonal-upper (L
|
https://en.wikipedia.org/wiki/Heteroclinic%20network
|
In mathematics, a heteroclinic network is an invariant set in the phase space of a dynamical system. It can be thought of loosely as the union of more than one heteroclinic cycle. Heteroclinic networks arise naturally in a number of different types of applications, including fluid dynamics and populations dynamics.
The dynamics of trajectories near to heteroclinic networks is intermittent: trajectories spend a long time performing one type of behaviour (often, close to equilibrium), before switching rapidly to another type of behaviour. This type of intermittent switching behaviour has led to several different groups of researchers using them as a way to model and understand various type of neural dynamics.
References
Dynamical systems
|
https://en.wikipedia.org/wiki/John%20H.%20Smith%20%28mathematician%29
|
John Howard Smith is an American mathematician and retired professor of mathematics at Boston College. He received his Ph.D. from the Massachusetts Institute of Technology in 1963, under the supervision of Kenkichi Iwasawa.
In voting theory, he is known for the Smith set, the smallest nonempty set of candidates such that, in every pairwise matchup (two-candidate election/runoff) between a member and a non-member, the member is the winner by majority rule, and for the Smith criterion, a property of certain election systems in which the winner is guaranteed to belong to the Smith set. He has also made contributions to spectral graph theory and additive number theory.
References
Year of birth missing (living people)
Living people
20th-century American mathematicians
21st-century American mathematicians
Voting theorists
Graph theorists
Number theorists
Massachusetts Institute of Technology School of Science alumni
Boston College faculty
|
https://en.wikipedia.org/wiki/Galbraith%20plot
|
In statistics, a Galbraith plot (also known as Galbraith's radial plot or just radial plot) is one way of displaying several estimates of the same quantity that have different standard errors.
It can be used to examine heterogeneity in a meta-analysis, as an alternative or supplement to a forest plot.
A Galbraith plot is produced by first calculating the standardized estimates or z-statistics by dividing each estimate by its standard error (SE). The Galbraith plot is then a scatter plot of each z-statistic (vertical axis) against 1/SE (horizontal axis). Larger studies (with smaller SE and larger 1/SE) will be observed to aggregate away from the origin.
See also
Plot
Funnel plot
References
External links
Galbraith plots are available within the metafor package in R, along with various other diagnostic and summary plots.
MIX 2.0 Software to perform meta-analysis and create Galbraith plots in Excel.
RadialPlotter Java application for fission track, luminescence and other radial plots from P. Vermeesch.
RadialPlotter() function within the R package 'numOSL' from Peng Jun for statistical age models analysis in optically stimulated luminescence dating.
plot_RadialPlot() function within the R package 'Luminescence' to produce Galbraith plots.
Further reading
Galbraith, R.F., 1990. The radial plot: Graphical assessment of spread in ages. International Journal of Radiation Applications and Instrumentation. Part D. Nuclear Tracks and Radiation Measurements, 17 (3), pp. 207–214. .
Galbraith, R. & Green, P., 1990. Estimating the component ages in a finite mixture. International Journal of Radiation Applications and Instrumentation. Part D. Nuclear Tracks and Radiation Measurements, 17 (3), pp. 197–206.
Galbraith, R.F. & Laslett, G.M., 1993. Statistical models for mixed fission track ages. Nuclear Tracks And Radiation Measurements, 21 (4), pp. 459–470.
Galbraith, R.F., 1994. Some Applications of Radial Plots. Journal of the American Statistical Association, 89 (428), pp. 1232–1242.
Galbraith, R.F., 2010. On plotting OSL equivalent doses. Ancient TL, 28 (1), pp. 1–10. PDF on Ancient TL website
Galbraith, R.F. & Roberts, R.G., 2012. Statistical aspects of equivalent dose and error calculation and display in OSL dating: An overview and some recommendations. Quaternary Geochronology, 11, pp. 1–27.
Statistical charts and diagrams
Meta-analysis
|
https://en.wikipedia.org/wiki/West%20York%20Area%20High%20School
|
West York Area High School is a high school located in York, York County in south central Pennsylvania. According to the National Center for Education Statistics, the school reported an enrollment of 871 students in grades 9 through 12 in the 2018–2019 school year.
Part of the West York Area School District, the school serves children in grades nine through twelve. It operates on a two-semester, six term year with block scheduling of four-period days, each period lasting 80 minutes. Students may attend the local technical school or specialized center on a part-time basis. Career internships and diversified occupations programs are also available for seniors.
Campus
In 2009, the West York Area School Board began the process of upgrading the facilities at the school, beginning with the opening of a new gym in October 2014. The development was completed in 2016 when a new two-story wing was added that includes eight additional classrooms and a large group instruction/meeting room. The existing science labs were renovated and three new labs were added. A new choral room was provided, along with a new guidance suite and the school auditorium was equipped with upgraded stage rigging, lights and sound systems.
Extracurricular activities
West York Area High School students have access to a wide variety of clubs, activities, and an extensive sports program.
Sport teams
The school uses navy blue and white as spirit colors, and its mascot is a Bulldog. The school is a member of PIAA AAA athletics class. In 2008, the school's football team won their Division with a 9-1 regular season record. They went on to become the first team from YAIAA to win an AA or AAA district championship, when they won the District III Championship in Hershey. The Bulldogs' 2008 season came to an end, though, on December 5 at Altoona's Mansion Park Stadium, where the Bulldogs fell 49–21 to defending state champions, Thomas Jefferson High School. West York's final record was 13–2, the best in school history.
Boys
Baseball
Basketball
Cross Country
Football
Golf
Lacrosse
Soccer
Swimming and Diving
Tennis
Track and Field
Volleyball
Wrestling
Girls
Basketball
Cross Country
Field Hockey
Golf
Lacrosse
Soccer (Fall)
Softball
Swimming and Diving
Tennis
Track and Field
Volleyball
Music
The school produces a musical each year; recent productions have included Crazy for You in 2012, Brigadoon in 2011 and Thoroughly Modern Millie in 2010.
The school's marching band has been directed by Rod Meckley since 2012. The band show each year is themed; recent themes have included the music of Ray Charles in 2012-13 and Elvis Presley]] in 2011–12. Previous themes have included American Tapestry, Island Pulse and the songs of Billy Joel.
The school also maintains a Jazz Ensemble (also directed by Rod Meckley). This group performs a few times a year at the high school as well as at the York County: Distinguished Young Women competition.
References
External links
School District website
|
https://en.wikipedia.org/wiki/Proper%20forcing%20axiom
|
In the mathematical field of set theory, the proper forcing axiom (PFA) is a significant strengthening of Martin's axiom, where forcings with the countable chain condition (ccc) are replaced by proper forcings.
Statement
A forcing or partially ordered set P is proper if for all regular uncountable cardinals , forcing with P preserves stationary subsets of .
The proper forcing axiom asserts that if P is proper and Dα is a dense subset of P for each α<ω1, then there is a filter G P such that Dα ∩ G is nonempty for all α<ω1.
The class of proper forcings, to which PFA can be applied, is rather large. For example, standard arguments show that if P is ccc or ω-closed, then P is proper. If P is a countable support iteration of proper forcings, then P is proper. Crucially, all proper forcings preserve .
Consequences
PFA directly implies its version for ccc forcings, Martin's axiom. In cardinal arithmetic, PFA implies . PFA implies any two -dense subsets of R are isomorphic, any two Aronszajn trees are club-isomorphic, and every automorphism of the Boolean algebra /fin is trivial. PFA implies that the Singular Cardinals Hypothesis holds. An especially notable consequence proved by John R. Steel is that the axiom of determinacy holds in L(R), the smallest inner model containing the real numbers. Another consequence is the failure of square principles and hence existence of inner models with many Woodin cardinals.
Consistency strength
If there is a supercompact cardinal, then there is a model of set theory in which PFA holds. The proof uses the fact that proper forcings are preserved under countable support iteration, and the fact that if is supercompact, then there exists a Laver function for .
It is not yet known how much large cardinal strength comes from PFA.
Other forcing axioms
The bounded proper forcing axiom (BPFA) is a weaker variant of PFA which instead of arbitrary dense subsets applies only to maximal antichains of size ω1. Martin's maximum is the strongest possible version of a forcing axiom.
Forcing axioms are viable candidates for extending the axioms of set theory as an alternative to large cardinal axioms.
The Fundamental Theorem of Proper Forcing
The Fundamental Theorem of Proper Forcing, due to Shelah, states that any countable support iteration of proper forcings is itself proper. This follows from the Proper Iteration Lemma, which states that whenever is a countable support forcing iteration based on and is a countable elementary substructure of for a sufficiently large regular cardinal , and and and is -generic and forces "," then there exists such that is -generic and the restriction of to equals and forces the restriction of to to be stronger or equal to .
This version of the Proper Iteration Lemma, in which the name is not assumed to be in , is due to Schlindwein.
The Proper Iteration Lemma is proved by a fairly straightforward induction on , and the Fundamental Theorem of Proper Forcing fo
|
https://en.wikipedia.org/wiki/JAMA%20%28numerical%20linear%20algebra%20library%29
|
JAMA is a software library for performing numerical linear algebra tasks created at National Institute of Standards and Technology in 1998 similar in functionality to LAPACK.
Functionality
The main capabilities provided by JAMA are:
Eigensystem solving
LU decomposition
Singular value decomposition
QR decomposition
Cholesky decomposition
Versions exist for both C++ and the Java programming language. The C++ version uses the Template Numerical Toolkit for lower-level operations. The Java version provides the lower-level operations itself.
History
As work of US governmental organization the algorithm and source code have been released to the public domain around 1998. JAMA has had little development since the year 2000, with only the occasional bug fix being released. The project's webpage contains the following statement, "(JAMA) is no longer actively developed to keep track of evolving usage patterns in the Java language, nor to further improve the API. We will, however, fix outright errors in the code." The last bug fix was released November 2012, with the previous one being released in 2005.
Usage Example
Example of Singular Value Decomposition (SVD):
SingularValueDecomposition s = matA.svd();
Matrix U = s.getU();
Matrix S = s.getS();
Matrix V = s.getV();
Example of matrix multiplication:
Matrix result = A.times(B);
See also
List of numerical libraries
References
External links
JAMA/C++ download and documentation page at NIST
JAMA/Java homepage at NIST
Numerical software
Public-domain software with source code
C++ numerical libraries
|
https://en.wikipedia.org/wiki/Central%20composite%20design
|
In statistics, a central composite design is an experimental design, useful in response surface methodology, for building a second order (quadratic) model for the response variable without needing to use a complete three-level factorial experiment.
After the designed experiment is performed, linear regression is used, sometimes iteratively, to obtain results. Coded variables are often used when constructing this design.
Implementation
The design consists of three distinct sets of experimental runs:
A factorial (perhaps fractional) design in the factors studied, each having two levels;
A set of center points, experimental runs whose values of each factor are the medians of the values used in the factorial portion. This point is often replicated in order to improve the precision of the experiment;
A set of axial points, experimental runs identical to the centre points except for one factor, which will take on values both below and above the median of the two factorial levels, and typically both outside their range. All factors are varied in this way.
Design matrix
The design matrix for a central composite design experiment involving k factors is derived from a matrix, d, containing the following three different parts corresponding to the three types of experimental runs:
The matrix F obtained from the factorial experiment. The factor levels are scaled so that its entries are coded as +1 and −1.
The matrix C from the center points, denoted in coded variables as (0,0,0,...,0), where there are k zeros.
A matrix E from the axial points, with 2k rows. Each factor is sequentially placed at ±α and all other factors are at zero. The value of α is determined by the designer; while arbitrary, some values may give the design desirable properties. This part would look like:
Then d is the vertical concatenation:
The design matrix X used in linear regression is the horizontal concatenation of a column of 1s (intercept), d, and all elementwise products of a pair of columns of d:
where d(i) represents the ith column in d.
Choosing α
There are many different methods to select a useful value of α. Let F be the number of points due to the factorial design and T = 2k + n, the number of additional points, where n is the number of central points in the design. Common values are as follows (Myers, 1971):
Orthogonal design:: , where ;
Rotatable design: α = F1/4 (the design implemented by MATLAB’s ccdesign function).
Application of central composite designs for optimization
Statistical approaches such as Response Surface Methodology can be employed to maximize the production of a special substance by optimization of operational factors. In contrast to conventional methods, the interaction among process variables can be determined by statistical techniques. For instance, in a study, a central composite design was employed to investigate the effect of critical parameters of organosolv pretreatment of rice straw including temperature, time, and ethanol c
|
https://en.wikipedia.org/wiki/Kai%20Michalke
|
Kai Michalke (born 5 April 1976) is German former professional footballer who played as a forward or left winger.
Career statistics
Honours
Germany U16
UEFA European Under-16 Football Championship: 1992
References
External links
Living people
1976 births
Footballers from Bochum
German men's footballers
Men's association football forwards
Germany men's under-21 international footballers
Germany men's youth international footballers
Bundesliga players
2. Bundesliga players
Eredivisie players
VfL Bochum players
Hertha BSC players
1. FC Nürnberg players
Alemannia Aachen players
MSV Duisburg players
Heracles Almelo players
SG Wattenscheid 09 players
German expatriate men's footballers
German expatriate sportspeople in the Netherlands
Expatriate men's footballers in the Netherlands
West German men's footballers
|
https://en.wikipedia.org/wiki/359%20%28number%29
|
359 (three hundred [and] fifty-nine) is the natural number following 358 and preceding 360. 359 is the 72nd prime number.
In mathematics
359 is a Sophie Germain prime: (also a Sophie Germain prime).
It is also a safe prime, because subtracting 1 and halving it gives another prime number (179, itself also safe).
Since the reversal of its digits gives 953, which is prime, it is also an emirp.
359 is an Eisenstein prime with no imaginary part and a Chen prime.
It is a strictly non-palindromic number.
In other fields
According to the author Douglas Adams, 359 is the funniest three-digit number.
Integers
|
https://en.wikipedia.org/wiki/Numerical%20cognition
|
Numerical cognition is a subdiscipline of cognitive science that studies the cognitive, developmental and neural bases of numbers and mathematics. As with many cognitive science endeavors, this is a highly interdisciplinary topic, and includes researchers in cognitive psychology, developmental psychology, neuroscience and cognitive linguistics. This discipline, although it may interact with questions in the philosophy of mathematics, is primarily concerned with empirical questions.
Topics included in the domain of numerical cognition include:
How do non-human animals process numerosity?
How do infants acquire an understanding of numbers (and how much is inborn)?
How do humans associate linguistic symbols with numerical quantities?
How do these capacities underlie our ability to perform complex calculations?
What are the neural bases of these abilities, both in humans and in non-humans?
What metaphorical capacities and processes allow us to extend our numerical understanding into complex domains such as the concept of infinity, the infinitesimal or the concept of the limit in calculus?
Heuristics in numerical cognition
Comparative studies
A variety of research has demonstrated that non-human animals, including rats, lions and various species of primates have an approximate sense of number (referred to as "numerosity"). For example, when a rat is trained to press a bar 8 or 16 times to receive a food reward, the number of bar presses will approximate a Gaussian or Normal distribution with peak around 8 or 16 bar presses. When rats are more hungry, their bar-pressing behavior is more rapid, so by showing that the peak number of bar presses is the same for either well-fed or hungry rats, it is possible to disentangle time and number of bar presses. In addition, in a few species the parallel individuation system has been shown, for example in the case of guppies which successfully discriminated between 1 and 4 other individuals.
Similarly, researchers have set up hidden speakers in the African savannah to test natural (untrained) behavior in lions. These speakers can play a number of lion calls, from 1 to 5. If a single lioness hears, for example, three calls from unknown lions, she will leave, while if she is with four of her sisters, they will go and explore. This suggests that not only can lions tell when they are "outnumbered" but that they can do this on the basis of signals from different sensory modalities, suggesting that numerosity is a multisensory concept.
Developmental studies
Developmental psychology studies have shown that human infants, like non-human animals, have an approximate sense of number. For example, in one study, infants were repeatedly presented with arrays of (in one block) 16 dots. Careful controls were in place to eliminate information from "non-numerical" parameters such as total surface area, luminance, circumference, and so on. After the infants had been presented with many displays containing 16 items, they habi
|
https://en.wikipedia.org/wiki/Folded%20normal%20distribution
|
The folded normal distribution is a probability distribution related to the normal distribution. Given a normally distributed random variable X with mean μ and variance σ2, the random variable Y = |X| has a folded normal distribution. Such a case may be encountered if only the magnitude of some variable is recorded, but not its sign. The distribution is called "folded" because probability mass to the left of x = 0 is folded over by taking the absolute value. In the physics of heat conduction, the folded normal distribution is a fundamental solution of the heat equation on the half space; it corresponds to having a perfect insulator on a hyperplane through the origin.
Definitions
Density
The probability density function (PDF) is given by
for x ≥ 0, and 0 everywhere else. An alternative formulation is given by
,
where cosh is the cosine Hyperbolic function. It follows that the cumulative distribution function (CDF) is given by:
for x ≥ 0, where erf() is the error function. This expression reduces to the CDF of the half-normal distribution when μ = 0.
The mean of the folded distribution is then
or
where is the normal cumulative distribution function:
The variance then is expressed easily in terms of the mean:
Both the mean (μ) and variance (σ2) of X in the original normal distribution can be interpreted as the location and scale parameters of Y in the folded distribution.
Properties
Mode
The mode of the distribution is the value of for which the density is maximised. In order to find this value, we take the first derivative of the density with respect to and set it equal to zero. Unfortunately, there is no closed form. We can, however, write the derivative in a better way and end up with a non-linear equation
.
Tsagris et al. (2014) saw from numerical investigation that when , the maximum is met when , and when becomes greater than , the maximum approaches . This is of course something to be expected, since, in this case, the folded normal converges to the normal distribution. In order to avoid any trouble with negative variances, the exponentiation of the parameter is suggested. Alternatively, you can add a constraint, such as if the optimiser goes for a negative variance the value of the log-likelihood is NA or something very small.
Characteristic function and other related functions
The characteristic function is given by
.
The moment generating function is given by
.
The cumulant generating function is given by
.
The Laplace transformation is given by
.
The Fourier transform is given by
.
Related distributions
When , the distribution of is a half-normal distribution.
The random variable has a noncentral chi-squared distribution with 1 degree of freedom and noncentrality equal to .
The folded normal distribution can also be seen as the limit of the folded non-standardized t distribution as the degrees of freedom go to infinity.
There is a bivariate version developed by Psarakis and Panaretos (20
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.