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https://en.wikipedia.org/wiki/Quillen%E2%80%93Lichtenbaum%20conjecture | In mathematics, the Quillen–Lichtenbaum conjecture is a conjecture relating étale cohomology to algebraic K-theory introduced by , who was inspired by earlier conjectures of . and proved the Quillen–Lichtenbaum conjecture at the prime 2 for some number fields. Voevodsky, using some important results of Markus Rost, has proved the Bloch–Kato conjecture, which implies the Quillen–Lichtenbaum conjecture for all primes.
Statement
The conjecture in Quillen's original form states that if A is a finitely-generated algebra over the integers and l is prime, then there is a spectral sequence analogous to the Atiyah–Hirzebruch spectral sequence, starting at
(which is understood to be 0 if q is odd)
and abutting to
for −p − q > 1 + dim A.
K-theory of the integers
Assuming the Quillen–Lichtenbaum conjecture and the Vandiver conjecture, the K-groups of the integers, Kn(Z), are given by:
0 if n = 0 mod 8 and n > 0, Z if n = 0
Z ⊕ Z/2 if n = 1 mod 8 and n > 1, Z/2 if n = 1.
Z/ck ⊕ Z/2 if n = 2 mod 8
Z/8dk if n = 3 mod 8
0 if n = 4 mod 8
Z if n = 5 mod 8
Z/ck if n = 6 mod 8
Z/4dk if n = 7 mod 8
where ck/dk is the Bernoulli number B2k/k in lowest terms and n is 4k − 1 or 4k − 2 .
References
Algebraic K-theory
Conjectures that have been proved |
https://en.wikipedia.org/wiki/John%20Rhodes%20%28mathematician%29 | John Lewis Rhodes is a mathematician known for work in the theory of semigroups, finite state automata, and algebraic approaches to differential equations.
Education and career
Rhodes was born in Columbus, Ohio, on July 16, 1937, but grew up in Wooster, Ohio, where he founded the Wooster Rocket Society as a teenager. In the fall of 1955, Rhodes entered the Massachusetts Institute of Technology intending to major in physics, but he soon switched to mathematics, earning his B.S. in 1960 and his Ph.D. in 1962. His Ph.D. thesis, co-written with a graduate student from Harvard, Kenneth Krohn, became known as the Prime Decomposition Theorem, or more simply the Krohn–Rhodes Theorem. After a year on an NSF fellowship in Paris, France, he became a member of the Faculty of Mathematics at the University of California, Berkeley, where he spent his entire teaching career.
In the late 1960s Rhodes wrote Applications of Automata Theory and Algebra: Via the Mathematical Theory of Complexity to Biology, Physics, Psychology, Philosophy, and Games, informally known as The Wild Book, which quickly became an underground classic, but remained in typescript until its revision and editing by Chrystopher L. Nehaniv in 2009. The following year Springer published his and Benjamin Steinberg's magnum opus, The q-Theory of Finite Semigroups, both a history of the field and the fruit of eight years' development of finite semigroup theory.
In recent years Rhodes brought semigroups into matroid theory. In 2015 he published, with Pedro V. Silva, the results of his current work in another monograph with Springer, Boolean Representations of Simplicial Complexes and Matroids.
Books and Monographs
John Rhodes and Benjamin Steinberg (2008), The q-theory of finite semigroups. Springer Verlag. .
"The Wild Book", published as Applications of Automata Theory and Algebra via the Mathematical Theory of Complexity to Biology, Physics, Psychology, Philosophy, and Games. John Rhodes; Chrystopher L. Nehaniv (Ed.) (2009), World Scientific.
John Rhodes and Pedro V. Silva (2015), Boolean Representations of Simplicial Complexes and Matroids. Springer Verlag.
See also
Krohn–Rhodes theory
References
External links
Academic homepage
Personal homepage
Springer Monographs download page: q-Theory of Finite Semigroups
Review of Applications of Automata Theory by Attila Egri-Nagy
Springer Monographs download page: Boolean Representations of Simplicial Complexes
1937 births
Living people
20th-century American mathematicians
21st-century American mathematicians
People from Wooster, Ohio
Massachusetts Institute of Technology School of Science alumni
University of California, Berkeley College of Letters and Science faculty
Mathematicians from Ohio |
https://en.wikipedia.org/wiki/Jay%20Kappraff | Jay Kappraff is an American professor of mathematics at the New Jersey Institute of Technology and author.
Biography
Kappraff was trained in engineering, physical sciences and mathematics, earning a B.Ch.E. in chemical engineering at New York Polytechnic in 1958. He went on to be awarded a PhD in applied mathematics in 1974 from the Courant Institute of Mathematical Science, New York University and a M.S. in chemical engineering in 1960 from Iowa State University. He began work for DuPont DeNemours as a chemical engineer from 1961 to 1962 going on to teach mathematics for a brief period before obtaining a position at NASA as an aerospace engineer from 1962 until 1965. He went on to be an instructor of mathematics at the Cooper Union College, New York City from 1968 until 1974. Following this, he joined the New Jersey Institute of Technology, where he currently works. He was a consultant for the Department of Energy in 1976. In 1978 he developed a course in the mathematics of design for computer scientists, mathematicians and architects. In bringing together such an interdisciplinary range of subjects, he began to study what he termed a common language of design and geometry. He has been a lecturer on the relationship between art and science and published a large number or articles on subjects ranging from plasma physics, solar heating, aerospace engineering and fractals. He has also published a number of books on these and related subjects and compiled a series of video lectures on the science of design.
In 1991 his book Connections won a prize for the best book in chemistry, physics, mathematics, astronomy and reference from the Association of American Publishers.
Professional activities
At the NJIT, Kapraff has organized various forums and tuition programs on subjects from Nuclear war and ancient geometry to experimental mathematics. he is a member of the faculty council and chairman of the NJIT Technology and Society Forum committee. He is a member of the Mathematics Association of America and on the editorial board of a new interdisciplinary journal, the International Journal of Biological Systems. He was also guest editor of the journal FORMA for a special issue on the golden mean in 2005.
Selected bibliography
Kappraff, J. "Ancient Harmonic Law". Bridges 2007. (2007)
Kappraff, J. and McClain, E.G. "The Proportions of the Parthenon: A work of musically inspired architecture". Music in Art: International Journal for Music Iconography, Vol. 30/1–2 (Spring–Fall 2005)
Kappraff, J. and Adamson, G.W. Generalized Binet Formulas, Lucas Polynomials, and Cyclic Constants. FORMA vol. 19, No. 4 (2005)
Kappraff, J. and Adamson, G.W. Polygons and Chaos. Journal of Biological Systems and Geometric Theories, Vol. 2 pp 79–94 (Nov. 2004).
Kappraff, J. The Anatomy of a Bud. In Bridges:2004 edited by R. Sarhangi. Winfield,KS:Central Plains books (2004)
Kappraff, J. and Adamson, G.W. The Relationship of the Cotangent Function to Special Relati |
https://en.wikipedia.org/wiki/Lottery%20%28probability%29 | In expected utility theory, a lottery is a discrete distribution of probability on a set of states of nature. The elements of a lottery correspond to the probabilities that each of the states of nature will occur, e.g. (Rain:.70, No Rain:.30). Much of the theoretical analysis of choice under uncertainty involves characterizing the available choices in terms of lotteries.
In economics, individuals are assumed to rank lotteries according to a rational system of preferences, although it is now accepted that people make irrational choices systematically. Behavioral economics studies what happens in markets in which some of the agents display human complications and limitations.
Choice under risk
According to expected utility theory, someone chooses among lotteries by multiplying his subjective estimate of the probabilities of the possible outcomes by a utility attached to each outcome by his personal utility function. Thus, each lottery has an expected utility, a linear combination of the utilities of the outcomes in which weights are the subjective probabilities. It is also founded in the famous example, the St. Petersburg paradox: as Daniel Bernoulli mentioned, the utility function in the lottery could be dependent on the amount of money which he had before the lottery.
For example, let there be three outcomes that might result from a sick person taking either novel drug A or B for his condition: "Cured", "Uncured", and "Dead". Each drug is a lottery. Suppose the probabilities for lottery A are (Cured: .90, Uncured: .00, Dead: .10), and for lottery B are (Cured: .50, Uncured: .50, Dead: .00).
If the person had to choose between lotteries A and B, how would they do it? A theory of choice under risk starts by letting people have preferences on the set of lotteries over the three states of nature—not just A and B, but all other possible lotteries. If preferences over lotteries are complete and transitive, they are called rational. If people follow the axioms of expected utility theory, their preferences over lotteries will follow each lottery's ranking in terms of expected utility. Let the utility values for the sick person be:
Cured: 16 utils
Uncured: 12 utils
Dead: 0 utils
In this case, the expected utility of Lottery A is 14.4 (= .90(16) + .10(12)) and the expected utility of Lottery B is 14 (= .50(16) + .50(12)), so the person would prefer Lottery A. Expected utility theory implies that the same utilities could be used to predict the person's behavior in all possible lotteries. If, for example, he had a choice between lottery A and a new lottery C consisting of (Cured: .80, Uncured: .15 Dead: .05), expected utility theory says he would choose C, because its expected utility is 14.6 (= .80(16) + .15(12) + .05(0)).
The paradox argued by Maurice Allais complicates expected utility in the lottery. In contrast to the former example, let there be outcomes consisting of only losing money. In situation 1, option 1a has a certain loss of $500 |
https://en.wikipedia.org/wiki/Cantellated%205-cubes | In six-dimensional geometry, a cantellated 5-cube is a convex uniform 5-polytope, being a cantellation of the regular 5-cube.
There are 6 unique cantellation for the 5-cube, including truncations. Half of them are more easily constructed from the dual 5-orthoplex
Cantellated 5-cube
Alternate names
Small rhombated penteract (Acronym: sirn) (Jonathan Bowers)
Coordinates
The Cartesian coordinates of the vertices of a cantellated 5-cube having edge length 2 are all permutations of:
Images
Bicantellated 5-cube
In five-dimensional geometry, a bicantellated 5-cube is a uniform 5-polytope.
Alternate names
Bicantellated penteract, bicantellated 5-orthoplex, or bicantellated pentacross
Small birhombated penteractitriacontiditeron (Acronym: sibrant) (Jonathan Bowers)
Coordinates
The Cartesian coordinates of the vertices of a bicantellated 5-cube having edge length 2 are all permutations of:
(0,1,1,2,2)
Images
Cantitruncated 5-cube
Alternate names
Tricantitruncated 5-orthoplex / tricantitruncated pentacross
Great rhombated penteract (girn) (Jonathan Bowers)
Coordinates
The Cartesian coordinates of the vertices of an cantitruncated 5-cube having an edge length of 2 are given by all permutations of coordinates and sign of:
Images
Related polytopes
It is third in a series of cantitruncated hypercubes:
Bicantitruncated 5-cube
Alternate names
Bicantitruncated penteract
Bicantitruncated pentacross
Great birhombated penteractitriacontiditeron (Acronym: gibrant) (Jonathan Bowers)
Coordinates
Cartesian coordinates for the vertices of a bicantitruncated 5-cube, centered at the origin, are all sign and coordinate permutations of
(±3,±3,±2,±1,0)
Images
Related polytopes
These polytopes are from a set of 31 uniform 5-polytopes generated from the regular 5-cube or 5-orthoplex.
References
H.S.M. Coxeter:
H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
Kaleidoscopes: Selected Writings of H.S.M. Coxeter, editied 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]
Norman Johnson Uniform Polytopes, Manuscript (1991)
N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D.
o3o3x3o4x - sirn, o3x3o3x4o - sibrant, o3o3x3x4x - girn, o3x3x3x4o - gibrant
External links
Polytopes of Various Dimensions, Jonathan Bowers
Runcinated uniform polytera (spid), Jonathan Bowers
Multi-dimensional Glossary
5-polytopes |
https://en.wikipedia.org/wiki/Ladislav%20Rieger |
Ladislav Svante Rieger (1916–1963) was a Czechoslovak mathematician who worked in the areas of algebra, mathematical logic, and axiomatic set theory. He is considered to be the founder of mathematical logic in Czechoslovakia, having begun his work around 1957.
Notes
References
Further reading
, especially "3.5 Ladislav Rieger and lattices", pp. 238–250
External links
1916 births
1963 deaths
Czechoslovak mathematicians
Algebraists
Set theorists
Charles University alumni
Academic staff of Charles University
Czechoslovak philosophers |
https://en.wikipedia.org/wiki/Economy-wide%20material%20flow%20accounts | Economy-wide material flow accounts (EW-MFA) is a framework to compile statistics linking flows of materials from natural resources to a national economy. EW-MFA are descriptive statistics, in physical units such as tonnes per year.
EW-MFA is consistent with the principles and system boundaries of the United Nations System of National Accounts (SNA) and follows the residence principle. This means that EW-MFA is also a part of the System of Integrated Environmental and Economic Accounting (SEEA).
Scope
The underlying definition of economy-wide material flow accounts includes statistics on the overall material inputs into national economies, the changes of material stock within the economic system and the material outputs to other economies or to the environment. Statistics on EW-MFA cover all solid, gaseous, and liquid materials, except for water and air. However, water in products is included. EW-MFA includes statistics on material flows crossing the national (geographical) border, i.e. imports and exports.
EW-MFA strives to produce a mass balance of material flows. It systematically categorises material input and output flows crossing the functional border between economy (technosphere, anthroposphere) and environment. Mass balances are defined as "...on the first law of thermodynamics (called the law of conservation of matter), which states that matter (mass, energy) is neither created nor destroyed by any physical process".
Interpreting the statistics
In principle, the statistics will show which countries are dependent on others for natural resources and which are major exporters of natural resources. The statistics also show if a countries production is sustainable, i.e. whether the economy of a country can produce more products using fewer natural resources.
In the European Union between 2000 and 2007, resource productivity increased by almost eight percent. Resource productivity of the EU is expressed by the amount of gross domestic product (GDP) generated per unit of material consumed (Domestic Material Consumption, see below), in other words GDP / DMC in euro per kg. This means that less material was consumed in order to produce the same amount of products in the EU. However, breaking down the components of the index it is seen that both GDP and DMC are increasing, only not equally fast.
History
When the European Council met in Helsinki in December 1999, part of the agenda was dedicated to establishing an understanding of how economies are dependent on the use of natural resources and that these resources are not in limitless supply.
The following year, Eurostat, together with the Wuppertal Institute and national statistical offices in Europe, developed the first statistical guideline for how to compile statistics and subsequent indicators on material flows.
Implementation
There is a link between the System of Integrated Environmental and Economic Accounting (SEEA) and EW-MFA. Statistics are based on the same principles (the re |
https://en.wikipedia.org/wiki/2010%20Chinese%20census | The 2010 Chinese census, officially the Sixth National Population Census of the People's Republic of China (中华人民共和国第六次全国人口普查), was conducted by the National Bureau of Statistics of the People's Republic of China with a zero hour of November 1, 2010.
Census procedure
Census procedure was governed by the Regulations on National Population Census and the Circular of the State Council on the Conduct of the 6th National Population Census. The census cost 700 million RMB.
Results
The main findings of the census were published on April 28, 2011.
Total population
It found the total population of Mainland China to be 1,339,724,852 persons, an increase of 73,899,804 persons from the previous census conducted in 2000. This represented a growth rate of 5.84% over the decade, and an average annual growth rate of 0.57%. The population undercount rate of the census was estimated at 0.12%. The census also listed the population of Hong Kong Special Administrative Region as 7,097,600 persons, the population of Macau Special Administrative Region as 552,300 persons, and the population of Taiwan as 23,162,123 persons.
Population composition and demographics
The census found a total of 401,517,330 family households in Mainland China, with an average of 3.10 persons per household, a decrease of 0.34 persons from the 2000 census. 51.27% of the population is male, and 48.73% is female, giving a male to female ratio of 105.20 men for every 100 women, a decrease from the 2000 figure of 106.74. 49.68% of the population resided in urban areas, and 50.32% resided in rural areas, an increase of 13.46% in the proportion of the urban population. 261,386,075 people had lived in a place different from their household registration for at least six months, with 221,426,652 of these living in a different city from their registration.
According to the 2010 census, males account for 51.27% of China's 1.34 billion people, while females made up 48.73% of the total. The sex ratio (the number of males for each female in a population) at birth was 118.06 boys to every 100 girls (54.14%) in 2010.
16.60% of the population was aged 0–14, 70.14% was aged 15–59, and 13.26% were aged 60 or over. This represented a decrease of 6.29% in the share of the population in the youngest age group, and increases of 3.36% and 2.93% for the 15-59 and 60+ shares, respectively. 91.51% of the population was of the Han Chinese nationality, and 8.49% was of other ethnic groups. The Han population increased by 5.74%, and the population of other groups increased by a combined 6.92%.
Educational attainment
The census found that, in Mainland China, 119,636,790 people had completed higher education, 187,985,979 had completed only senior secondary education, 519,656,445 had completed only junior secondary education, 358,764,003 had completed only primary education, and 54,656,573 were illiterate. Since 2000, out of every 100,000 people, the number with higher education has increased from 3,611 to 8,930, the n |
https://en.wikipedia.org/wiki/Generality%20of%20algebra | In the history of mathematics, the generality of algebra was a phrase used by Augustin-Louis Cauchy to describe a method of argument that was used in the 18th century by mathematicians such as Leonhard Euler and Joseph-Louis Lagrange, particularly in manipulating infinite series. According to Koetsier, the generality of algebra principle assumed, roughly, that the algebraic rules that hold for a certain class of expressions can be extended to hold more generally on a larger class of objects, even if the rules are no longer obviously valid. As a consequence, 18th century mathematicians believed that they could derive meaningful results by applying the usual rules of algebra and calculus that hold for finite expansions even when manipulating infinite expansions.
In works such as Cours d'Analyse, Cauchy rejected the use of "generality of algebra" methods and sought a more rigorous foundation for mathematical analysis.
Example
An example is Euler's derivation of the series
for . He first evaluated the identity
at to obtain
The infinite series on the right hand side of () diverges for all real . But nevertheless integrating this term-by-term gives (), an identity which is known to be true by Fourier analysis.
See also
Principle of permanence
Transfer principle
References
Mathematical analysis
History of calculus |
https://en.wikipedia.org/wiki/Czes%C5%82aw%20Olech | Czesław Olech (22 May 1931 – 1 July 2015) was a Polish mathematician. He was a representative of the Kraków school of mathematics, especially the differential equations school of Tadeusz Ważewski.
Education and career
In 1954 he completed his mathematical studies at the Jagiellonian University in Kraków, obtained his doctorate at the Institute of Mathematical Sciences in 1958, habilitation in 1962, the title of associate professor in 1966, and the title of professor in 1973.
1970–1986: director of The Institute of Mathematics, Polish Academy of Sciences.
1972–1991: director of Stefan Banach International Mathematical Center in Warsaw.
1979–1986: member of the Executive Committee, International Mathematical Union.
1982–1983: president of the Organizing Committee, International Congress of Mathematicians in Warsaw,
1987–1989: president of the Board of Mathematics, Polish Academy of Sciences.
1990–2002: president of the Scientific Council, Institute of Mathematics of the Polish Academy of Sciences.
Czeslaw Olech, often as a visiting professor, was invited by the world's leading mathematical centers in the United States, USSR (later Russia), Canada and many European countries. He cooperated with Solomon Lefschetz, Sergey Nikolsky, Philip Hartman and Roberto Conti, the most distinguished mathematicians involved in the theory of differential equations. Lefschetz highly valued Ważewski's school, and especially the retract method, which Olech applied by developing, among other things, control theory. He supervised nine doctoral dissertations, and reviewed a number of theses and dissertations.
Main fields of research interest
Contributions to ordinary differential equations:
various applications of Tadeusz Ważewski topological method in studying asymptotic behaviour of solutions;
exact estimates of exponential growth of solution of second-order linear differential equations with bounded coefficients;
theorems concerning global asymptotic stability of the autonomous system on the plane with stable Jacobian matrix at each point of the plane, results establishing relation between question of global asymptotic stability of an autonomous system and that of global one-to-oneness of a differentiable map;
contribution to the question whether unicity condition implies convergence of successive approximation to solutions of ordinary differential equations.
Contribution to optimal control theory:
establishing a most general version of the so-called bang-bang principle for linear control problem by detailed study of the integral of set valued map;
existence theorems for optimal control problem with unbounded controls and multidimensional cost functions;
existence of solution of differential inclusions with nonconvex right-hand side;
characterization of controllability of convex processes.
Recognition
Honorary doctorates:
Vilnius University 1989
Jagiellonian University in Kraków 2006
AGH University of Science and Technology in Kraków 2009.
Membership of:
PAN Polis |
https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Tur%C3%A1n%20inequality | In mathematics, the Erdős–Turán inequality bounds the distance between a probability measure on the circle and the Lebesgue measure, in terms of Fourier coefficients. It was proved by Paul Erdős and Pál Turán in 1948.
Let μ be a probability measure on the unit circle R/Z. The Erdős–Turán inequality states that, for any natural number n,
where the supremum is over all arcs A ⊂ R/Z of the unit circle, mes stands for the Lebesgue measure,
are the Fourier coefficients of μ, and C > 0 is a numerical constant.
Application to discrepancy
Let s1, s2, s3 ... ∈ R be a sequence. The Erdős–Turán inequality applied to the measure
yields the following bound for the discrepancy:
This inequality holds for arbitrary natural numbers m,n, and gives a quantitative form of Weyl's criterion for equidistribution.
A multi-dimensional variant of (1) is known as the Erdős–Turán–Koksma inequality.
Notes
Additional references
Inequalities
Turán inequality
Theorems in approximation theory |
https://en.wikipedia.org/wiki/Katatau | Eberti Marques de Toledo (born March 1, 1986) is a Brazilian football player.
Club statistics
References
External links
1986 births
Living people
Brazilian men's footballers
J1 League players
Yokohama FC players
Brazilian expatriate men's footballers
Expatriate men's footballers in Japan
Men's association football forwards |
https://en.wikipedia.org/wiki/Shinya%20Hatta | is a former Japanese football player.
Club statistics
References
External links
1984 births
Living people
Kobe International University alumni
Association football people from Osaka Prefecture
Japanese men's footballers
J2 League players
J3 League players
Japan Football League players
Mito HollyHock players
Kamatamare Sanuki players
FC Ryukyu players
Blaublitz Akita players
Men's association football defenders
People from Takaishi, Osaka |
https://en.wikipedia.org/wiki/Ch%20%28computer%20programming%29 | Ch is a proprietary cross-platform C and C++ interpreter and scripting language environment. It was originally designed by Harry H. Cheng as a scripting language for beginners to learn mathematics, computing, numerical analysis (numeric methods), and programming in C/C++. Ch is now developed and marketed by SoftIntegration, Inc., with multiple versions available, including a freely available student edition and Ch Professional Edition for Raspberry Pi is free for non-commercial use.
Ch can be embedded in C/C++ application programs. It has numerical computing and graphical plotting features. Ch is a combined shell and IDE. Ch shell combines the features of common shell and C language. ChIDE provides quick code navigation and symbolic debugging. It is based on embedded Ch, Scite, and Scintilla.
Ch is written in C and runs on Windows, Linux, macOS, FreeBSD, AIX, Solaris, QNX, and HP-UX. It supports C90 and major C99 features, but it does not support the full set of C++ features. C99 complex number, IEEE-754 floating-point arithmetic, and variable-length array features were supported in Ch before they became part of the C99 standard. An article published by Computer Reseller News (CRN) named Ch as notable among C-based virtual machines for its functionality and the availability of third-party libraries.
Ch has many tool kits that extend its functions. For example, the Ch Mechanism Toolkit is used for design and analysis of commonly used mechanisms such as four-bar linkage, five-bar linkage, six-bar linkage, crank-slider mechanism, and cam-follower system. Ch Control System Toolkit is used for modeling, the design, and analysis of continuous-time or discrete-time linear time-invariant (LTI) control systems. Both tool kits include the source code.
Ch is now used and integrated into the curriculum by many high schools and universities to teach computing and programming in C/C++. Ch has been integrated into free C-STEM Studio, a platform for learning computing, science, technology, engineering, and mathematics (C-STEM) with robotics. C-STEM Studio is developed by the UC Davis Center for Integrated Computing and STEM Education (C-STEM). It offers a curriculum for K-12 students.
Ch supports LEGO Mindstorms NXT and EV3, Arduino, Linkbot, Finch Robot, RoboTalk and Raspberry Pi, Pi Zero, and ARM for robot programming and learning. It can also be embedded into the LabVIEW system design platform and development environment.
Features
Ch supports the 1999 ISO C Standard (C99) and C++ classes. It is a superset of C with C++ classes. Several major features of C99 are supported, such as complex numbers, variable length arrays (VLAs), IEEE-754 floating-point arithmetic, and generic mathematical functions. The specification for wide characters in Addendum 1 for C90 is also supported.
C++ features available in Ch include:
Member functions
Mixed code and declaration
The this -> pointer
Reference type and pass-by-reference
Function-style type conversion |
https://en.wikipedia.org/wiki/Split%20interval | In topology, the split interval, or double arrow space, is a topological space that results from splitting each point in a closed interval into two adjacent points and giving the resulting ordered set the order topology. It satisfies various interesting properties and serves as a useful counterexample in general topology.
Definition
The split interval can be defined as the lexicographic product equipped with the order topology. Equivalently, the space can be constructed by taking the closed interval with its usual order, splitting each point into two adjacent points , and giving the resulting linearly ordered set the order topology. The space is also known as the double arrow space, Alexandrov double arrow space or two arrows space.
The space above is a linearly ordered topological space with two isolated points, and in the lexicographic product. Some authors take as definition the same space without the two isolated points. (In the point splitting description this corresponds to not splitting the endpoints and of the interval.) The resulting space has essentially the same properties.
The double arrow space is a subspace of the lexicographically ordered unit square. If we ignore the isolated points, a base for the double arrow space topology consists of all sets of the form with . (In the point splitting description these are the clopen intervals of the form , which are simultaneously closed intervals and open intervals.) The lower subspace is homeomorphic to the Sorgenfrey line with half-open intervals to the left as a base for the topology, and the upper subspace is homeomorphic to the Sorgenfrey line with half-open intervals to the right as a base, like two parallel arrows going in opposite directions, hence the name.
Properties
The split interval is a zero-dimensional compact Hausdorff space. It is a linearly ordered topological space that is separable but not second countable, hence not metrizable; its metrizable subspaces are all countable.
It is hereditarily Lindelöf, hereditarily separable, and perfectly normal (T6). But the product of the space with itself is not even hereditarily normal (T5), as it contains a copy of the Sorgenfrey plane, which is not normal.
All compact, separable ordered spaces are order-isomorphic to a subset of the split interval.
See also
Notes
References
Arhangel'skii, A.V. and Sklyarenko, E.G.., General Topology II, Springer-Verlag, New York (1996)
Engelking, Ryszard, General Topology, Heldermann Verlag Berlin, 1989.
Topological spaces |
https://en.wikipedia.org/wiki/Heinz%20Lowin | Heinz Lowin (25 December 1938 – 12 October 1987) was a German footballer who plays as a defender.
Career statistics
References
External links
1938 births
1987 deaths
German men's footballers
Men's association football defenders
Germany men's under-21 international footballers
Bundesliga players
VfL Bochum players
Borussia Mönchengladbach players
VVV-Venlo players
German expatriate men's footballers
German expatriate sportspeople in the Netherlands
Expatriate men's footballers in the Netherlands |
https://en.wikipedia.org/wiki/Cyclohedron | In geometry, the cyclohedron is a -dimensional polytope where can be any non-negative integer. It was first introduced as a combinatorial object by Raoul Bott and Clifford Taubes and, for this reason, it is also sometimes called the Bott–Taubes polytope. It was later constructed as a polytope by Martin Markl and by Rodica Simion. Rodica Simion describes this polytope as an associahedron of type B.
The cyclohedron appears in the study of knot invariants.
Construction
Cyclohedra belong to several larger families of polytopes, each providing a general construction. For instance, the cyclohedron belongs to the generalized associahedra that arise from cluster algebra, and to the graph-associahedra, a family of polytopes each corresponding to a graph. In the latter family, the graph corresponding to the -dimensional cyclohedron is a cycle on vertices.
In topological terms, the configuration space of distinct points on the circle is a -dimensional manifold, which can be compactified into a manifold with corners by allowing the points to approach each other. This compactification can be factored as , where is the -dimensional cyclohedron.
Just as the associahedron, the cyclohedron can be recovered by removing some of the facets of the permutohedron.
Properties
The graph made up of the vertices and edges of the -dimensional cyclohedron is the flip graph of the centrally symmetric triangulations of a convex polygon with vertices. When goes to infinity, the asymptotic behavior of the diameter of that graph is given by
.
See also
Associahedron
Permutohedron
Permutoassociahedron
References
Further reading
External links
Polytopes |
https://en.wikipedia.org/wiki/Fulton%E2%80%93MacPherson%20compactification | In geometry, the Fulton–MacPherson compactification of the configuration space of n distinct labeled points in a compact complex manifold is a compact complex manifold that contains the configuration space as an open dense subset and is constructed in a canonical way. The notion was introduced by .
References
Lecture 13: the Fulton–MacPherson compactification by A. Voronov.
Compactification (mathematics) |
https://en.wikipedia.org/wiki/Andr%C3%A9%20Neveu | André Neveu (; born 28 August 1946) is a French physicist working on string theory and quantum field theory who coinvented the Neveu–Schwarz algebra and the Gross–Neveu model.
Biography
Neveu studied in Paris at the École Normale Supérieure (ENS). In 1969 he received his diploma (Thèse de troisième cycle) at University of Paris XI in Orsay with and Claude Bouchiat and in 1971 he completed his doctorate (Doctorat d'État) there.
In 1969 he and his classmate from ENS and Orsay, Joël Scherk, together with John H. Schwarz and David Gross at Princeton University, examined divergences in one-loop diagrams of the bosonic string theory (and discovered the cause of tachyon divergences). From 1971 to 1974 Neveu was at the Laboratory for High Energy Physics of the University of Paris XI where he and Scherk showed that spin-1 excitations of strings could describe Yang–Mills theories. In 1971, Neveu with John Schwarz in Princeton developed, at the same time as Pierre Ramond (1971), the first string theory that also described fermions (called RNS formalism after its three originators). This was an early appearance of the ideas of supersymmetry which were being developed independently at that time by several groups. A few years later, Neveu, working in Princeton with David Gross, developed the Gross–Neveu model. With Roger Dashen and Brosl Hasslacher, he examined, among other things, quantum-field-theoretic models of extended hadrons and semiclassical approximations in quantum field theory which are reflected in the DHN method of the quantization of solitons. From 1972 to 1977 Neveu was at the Institute for Advanced Study while spending half of the time in Orsay. From 1974 to 1983 he was at the Laboratory for Theoretical Physics of the ENS and from 1983 to 1989 in the theory department at CERN. From 1975 he was Maitre de recherche in the CNRS and from 1985 Directeur de recherche. From 1989 he was at the Institute (Laboratory) for Theoretical Physics of the University of Montpellier II (now L2C, Laboratory Charles Coulomb). In 1994/5 he was a visiting professor in the University of California, Berkeley.
In 1973, Neveu received the Paul Langevin Prize of the Société Française de Physique. In 1988 he received the Gentner-Kastler Prize awarded jointly by the Société Française de Physique and the Deutsche Physikalische Gesellschaft (DPG). In 2020 he was awarded the Dirac Medal of the ICTP.
Neveu is married and has three children.
Writings
(On the occasion of the awarding of the Gentner-Kastler Prize)
Notes
External links
Some of my recollections about Joël Scherk
Scientific publications of André Neveu on INSPIRE-HEP
1946 births
Living people
Scientists from Paris
French string theorists
People associated with CERN
Paris-Sud University alumni |
https://en.wikipedia.org/wiki/Aguila%20%28artist%29 | Aguila (born Jan Gielens; January 15, 1937 in Melsbroek, Belgium) is an artist, industrial designer, and founder of the "probability reality", a new art trend in contemporary art. His works consists of paintings, monumental sculpture, monumental installations, conceptual architecture and new concepts on naval design and computer art.
Education
Aguila studied visual and monumental art at the Higher Institute for Arts and Sciences in St Lucas, Brussels under such artists as Jos De Maegd and Maurits Van Saene. During the last year of his art education Aguila came to the conclusion that rather than focus on color, priority should be given to strong compositional elements and the contact the painting had with the environment, extending the composition of the work to the three-dimensional environment.
Aguila discovered industrial design as a student of Professor Elno (1920–1998). He studied industrial design first at L'Ecole Nationale Supérieure des Arts Visuels de la Cambre in Brussels and then at Academy for Industrial Design in Eindhoven, the Netherlands. He later practiced industrial design in Germany.
Early works
Beginning in 1965, Aguila worked as an independent industrial designer for various European industries including Philips – Telecom, Hilson, Bumaco, Brugman, Pirelli Brussels, Verbrughe (furniture), and Munar Antwerp, Belgium. For one of his package designs, Gielens was awarded the Howard Design Award. In the same period he also worked as the Industrial Design advisor at Febelhout, which is the Belgian Federation of Belgian, wood-using industries, Brussels (Belgium). In 1972, with Belgian architect, Bob Van Reeth, he designed his residence in Brussels, Belgium.
In 1974, Aguila was invited to exhibit his paintings at the University of Ghent. The former conservator of the Museum for Modern Art in Brussels, Mrs. Phil Mertens, became interested in Aguila's painting, and recognized him as the leader of a new direction in contemporary art, coined, "Multidimensional Transparent Art".
Aguila later was invited to work as a teacher and lecturer at Cool international school in Brussels 1977–1980, and from 1977–1999 at Universidad Las Colinas in Elche, Spain, where he was time engaged full-time, teaching courses and giving conferences based on his knowledge of painting, conceptual architecture, and landscape art . The painting classes were soon being called "The School of Ilicitis", called after Ilice, the ancient name for Elche. Ilicitis means City of Light. The campus of the University was constructed by Gielens and his architecture students amongst the many palm trees, using very light structures and materials.
Building and art design
The social facet of this cultural artistic movement, is present all over and the aggregation of the multiple facets and disciplines gives birth to a living monumental work of art called "A Creation of the free will", started 1977 and ongoing until at present. This monumental work of art is composed of multi disc |
https://en.wikipedia.org/wiki/Hanan%20Maman | Hanan Maman (; born 28 August 1989) is an Israeli former professional footballer who played as a midfielder. Hanan Maman is the son of former footballer Baruch Maman.
Career statistics
Honours
Hapoel Be'er Sheva
Israeli Premier League: 2017–18
Individual
Israeli Footballer of the Year: 2017–18
External links
Living people
1989 births
Israeli Jews
Israeli men's footballers
Men's association football midfielders
Israel men's international footballers
Hapoel Haifa F.C. players
Hapoel Tel Aviv F.C. players
S.K. Beveren players
Beitar Jerusalem F.C. players
Hapoel Be'er Sheva F.C. players
Liga Leumit players
Israeli Premier League players
Belgian Pro League players
Israeli expatriate men's footballers
Expatriate men's footballers in Belgium
Israeli expatriate sportspeople in Belgium
Footballers from Haifa
Israeli people of Tunisian-Jewish descent
Israeli people of Moroccan-Jewish descent
Israeli Footballer of the Year recipients |
https://en.wikipedia.org/wiki/Mennicke%20symbol | In mathematics, a Mennicke symbol is a map from pairs of elements of a number field to an abelian group satisfying some identities found by . They were named by , who used them in their solution of the congruence subgroup problem.
Definition
Suppose that A is a Dedekind domain and q is a non-zero ideal of A. The set Wq is defined to be the set of pairs (a, b) with a = 1 mod q, b = 0 mod q, such that a and b generate the unit ideal.
A Mennicke symbol on Wq with values in a group C is a function (a, b) → [] from Wq to C such that
[] = 1, [] = [][]
[] = [] if t is in q, [] = [] if t is in A.
There is a universal Mennicke symbol with values in a group Cq such that any Mennicke symbol with values in C can be obtained by composing the universal Mennicke symbol with a unique homomorphism from Cq to C.
References
Erratum
. Errata
Group theory
Algebraic K-theory |
https://en.wikipedia.org/wiki/James%20Demmel | James Weldon Demmel Jr. (born October 19, 1955) is an American mathematician and computer scientist, the Dr. Richard Carl Dehmel Distinguished Professor of Mathematics and Computer Science at the University of California, Berkeley.
In 1999, Demmel was elected a member of the National Academy of Engineering for contributions to numerical linear algebra and scientific computing.
Biography
Born in Pittsburgh, Demmel did his undergraduate studies at the California Institute of Technology, graduating in 1975 with a B.S. in mathematics. He earned his Ph.D. in computer science in 1983 from UC Berkeley, under the supervision of William Kahan; his dissertation was entitled A Numerical Analyst's Jordan Canonical Form. After holding a faculty position at New York University for six years, he moved to Berkeley in 1990.
Academic works
Demmel is known for his work on LAPACK, a software library for numerical linear algebra and more generally for research in numerical algorithms combining mathematical rigor with high performance implementation. Prometheus, a parallel multigrid finite element solver written by Demmel, Mark Adams, and Robert Taylor, won the Carl Benz Award at Supercomputing 1999 and the Gordon Bell Prize for Adams and his coworkers at Supercomputing 2004.
Honors and awards
Demmel was elected as a member of the National Academy of Engineering in 1999, a fellow of the Association for Computing Machinery in 1999, a fellow of the IEEE in 2001, a fellow of SIAM in 2009, and a member of the United States National Academy of Sciences in 2011. Demmel was one of two scientists honored in 1986 with the Leslie Fox Prize for Numerical Analysis. In 1993, Demmel won the J.H. Wilkinson Prize in Numerical Analysis and Scientific Computing, and in 2010, he was the winner of the IEEE's Sidney Fernbach Award "for computational science leadership in creating adaptive, innovative, high-performance linear algebra software". In 2012 he became a fellow of the American Mathematical Society. He received the IEEE Computer Society Charles Babbage Award in 2013.
Personal life
Demmel is married to Katherine Yelick, who is also an ACM Fellow and professor of electrical engineering and computer science at UC Berkeley, and Associate Lab Director for Computing Sciences at Lawrence Berkeley National Laboratory.
References
External links
Home page at UC Berkeley
Living people
20th-century American mathematicians
21st-century American mathematicians
American computer scientists
Numerical analysts
California Institute of Technology alumni
University of California, Berkeley alumni
New York University faculty
Fellows of the Association for Computing Machinery
Fellows of the American Mathematical Society
Fellow Members of the IEEE
Members of the United States National Academy of Sciences
Members of the United States National Academy of Engineering
UC Berkeley College of Engineering faculty
1955 births
People from Pittsburgh |
https://en.wikipedia.org/wiki/Vogel%20plane | In mathematics, the Vogel plane is a method of parameterizing simple Lie algebras by eigenvalues α, β, γ of the Casimir operator on the symmetric square of the Lie algebra, which gives a point (α: β: γ) of P2/S3, the projective plane P2 divided out by the symmetric group S3 of permutations of coordinates. It was introduced by , and is related by some observations made by . generalized Vogel's work to higher symmetric powers.
The point of the projective plane (modulo permutations) corresponding to a simple complex Lie algebra is given by three eigenvalues α, β, γ of the Casimir operator acting on spaces A, B, C, where the symmetric square of the Lie algebra (usually) decomposes as a sum of the complex numbers and 3 irreducible spaces A, B, C.
See also
E7½
References
Lie groups
Lie algebras |
https://en.wikipedia.org/wiki/Annals%20of%20Functional%20Analysis | The Annals of Functional Analysis is a peer-reviewed mathematics journal founded by Professor Mohammad Sal Moslehian and published by the Tusi Mathematical Research Group in cooperation with Springer (Birkhäuser). The journal was established in 2009 and covers functional analysis and operator theory and related topics.
Abstracting and indexing
The journal is abstracted and indexed in Scopus, Science Citation Index Expanded, Mathematical Reviews, and Zentralblatt MATH. The journal is included in the prestigious Reference List Journal of MathSciNet published by the American Mathematical Society. Two other related journals are Banach Journal of Mathematical Analysis and Advances in Operator Theory.
References
External links
Mathematics journals
Academic journals established in 2010
English-language journals
Quarterly journals |
https://en.wikipedia.org/wiki/Football%20records%20and%20statistics%20in%20Indonesia | This page details football records in Indonesia.
National team
Individual
Most appearances: 111, Abdul Kadir (1965 - 1979) (105 in official FIFA matches)
Most goals: 70, Abdul Kadir (1965 - 1979) (68 in official FIFA matches)
Best scoring percentage (players with 20+ goals): 63% (70 goals in 111 games), Abdul Kadir (1965 - 1979)
Scorelines
Biggest win: 12 goals margin
12 - 0 ( v. Philippines, 22 September 1972)
13 - 1 ( v. Philippines, 23 December 2002)
Biggest home win: 13 - 1, v. Philippines (23 December 2002)
Biggest away win: 8 - 0, v. New Zealand (4 October 1975)
Biggest win in neutral ground: 12 - 0, v. Philippines (Seoul, 22 September 1972)
Biggest defeat: 0 - 10, v. Bahrain (29 February 2012)
Biggest home defeat: 1 - 7, v. Uruguay (8 October 2010)
Biggest away defeat: 0 - 10, v. Bahrain (29 February 2012)
Biggest defeat in neutral ground: 0 - 6, v. Hungary (Reims, 5 June 1938)
Highest scoring: 14 goals, Indonesia 13 - 1 Philippines (23 December 2002)
Top-Tier Professional League
Records in this section refer to Liga Indonesia Premier Division from its founding in 1994 until 2007 (when it was still the top-tier league), to the Indonesia Super League from 2008 to the present day, and to the Indonesian Premier League during the dualism era.
Titles
Most League titles: 4, Persipura Jayapura (2005, 2008–09, 2010–11, 2013)
Top-flight Appearances
Most appearances: 26 seasons, Persib Bandung, Persija Jakarta
Most consecutive seasons in top-flight: 26 seasons, Persib Bandung, Persija Jakarta (1994 to 2022/23)
Individual
Most career goals: 344, Cristian Gonzáles (2003 to 2018)
Most goals in a season: 37, Sylvano Comvalius Dominique (2017)
Most top goalscorer awards: 4, Cristian Gonzáles (2005, 2006, 2007-08, 2008-09)
Most goals in a game: 6, Ilham Jayakesuma (Persita Tangerang 10 - 1 Persikab, 28 April 2002)
Most hat-tricks: 11, Cristian Gonzáles (2003 to 2010-11)
Scorelines
Biggest home win: Persita Tangerang 10-1 Persikab Bandung (28 April 2002)
Highest scoring: 12 goals
Pelita Jaya 10-2 Persijatim Jakarta Timur (14 June 1995)
Sriwijaya 10-2 Persegres Gresik United (5 November 2017)
Indonesia Super League – since 2008-09 season
Titles
Most titles: 3, Persipura Jayapura (2008–09, 2010–11, 2013)
Top-flight appearances
Most seasons in top flight overall: 13 seasons, Arema, Madura United, Persib Bandung, Persija Jakarta
Most consecutive seasons in top flight: 13 seasons, Arema, Madura United, Persib Bandung, Persija Jakarta
Fewest seasons in top flight overall: 1 season, Kalteng Putra, Persema Malang, Persiba Bantul, PSAP Sigli
Wins
Most wins overall: 165, Persipura Jayapura
Most wins in a season: 25
Persipura Jayapura (2008–09, 2013)
Sriwijaya (2011–12)
Fewest wins in a season: 2
Persijap Jepara (2014)
Persiba Bantul (2014)
Persegres Gresik United (2017)
Persiraja Banda Aceh (2021–22)
Most consecutive wins: 9, Persipura Jayapura (15 March 2009 – 20 May 2009)
Most consecutive home wins: 15, Arema (27 January 2010 – 27 March |
https://en.wikipedia.org/wiki/2011%E2%80%9312%20Burton%20Albion%20F.C.%20season | The 2011–12 season was Burton Albion's third consecutive season in League Two.
League table
Squad statistics
Appearances and goals
|-
|colspan="14"|Players featured for Burton but left before the end of the season:
|-
|colspan="14"|Players played for Burton on loan who have returned to their parent club:
|}
Top scorers
Disciplinary record
Club
Coaching and Medical Staff
Last updated 17 September 2012.
Source:
Includes staff registered with club on 5 May 2012.
Managerial change
Following Burton's victory over Northampton Town on 26 December 2011, the club then went 14 consecutive games without a victory. This prompted chairman Ben Robinson to sack Paul Peschisolido on 17 March 2012. Gary Rowett and Kevin Poole were put in temporary charge until a new manager could be found. Rowett was subsequently put in charge of the club on a permanent basis on 11 May 2012 in time for the new season.
Players
As of 5 May 2012.
Source: Burton Albion, Soccerbase
Ordered by position then squad number.
Appearances (starts and substitute appearances) and goals include those in competitive matches in The Football League, The Football Conference, FA Cup, League Cup, Football League Trophy, FA Trophy and Conference League Cup.
1Player/Goalkeeping coach. Oldest registered player in The Football League.
2Club Captain.
3Undisclosed fee reported by the Burton Mail to be £20K.
4Appearances include previous spell with club in 2010–11
Kit
|
|
|
Burton's away kit was retained from the previous season, as was the Mr. Cropper sponsorship brand. TAG Leisure continue to manufacture the club's matchday and training attire. The new home kit was unveiled on 15 July before the pre-season friendly with Derby County. Following 16 years of plain yellow shirts, it marks a return to the traditional black and yellow stripes that had been worn by the club from its foundation through to the mid-1990s. The kit will be used for all club competitions and will remain in use until the end of the 2012–13 league season.
Other Information
Results
Pre-Season Friendlies
League Two
League Two Results summary
Results by round
FA Cup
League Cup
Football League Trophy
Transfers
Awards
Source: Burton AlbionLast updated 18 May 2012.
References
Burton Albion
Burton Albion F.C. seasons |
https://en.wikipedia.org/wiki/Kiran%20Kedlaya | Kiran Sridhara Kedlaya (; born July 1974) is an Indian American mathematician. He currently is a Professor of Mathematics and the Stefan E. Warschawski Chair in Mathematics at the University of California, San Diego.
Biography
Kiran Kedlaya was born into a Tulu Brahmin family. At age 16, Kedlaya won a gold medal at the International Mathematics Olympiad, and would later win a silver and another gold medal. While an undergraduate student at Harvard, he was a three-time Putnam Fellow in 1993, 1994, and 1995. A 1996 article by The Harvard Crimson described him as "the best college-age student in math in the United States".
Kedlaya was runner-up for the 1995 Morgan Prize, for a paper in which he substantially improved on results of László Babai and Vera Sós (1985) on the size of the largest product-free subset of a finite group of order n.
He gave an invited talk at the International Congress of Mathematicians in 2010, on the topic of "Number Theory".
In 2012 he became a fellow of the American Mathematical Society.
Game shows
He was also a contestant on the game show Jeopardy! in 2011, winning one episode.
Selected works
p-adic Differential Equations, Cambridge Studies in Advanced Mathematics, Band 125, Cambridge University Press 2010
with David Savitt, Dinesh Thakur, Matt Baker, Brian Conrad, Samit Dasgupta, Jeremy Teitelbaum p-adic Geometry, Lectures from the 2007 Arizona Winter School, American Mathematical Society 2008
with Bjorn Poonen, Ravi Vakil The William Lowell Putnam Mathematical Competition 1985-2000: Problems, Solutions and Commentary, Mathematical Association of America, 2002
References
External links
Kiran Kedlaya's website
1974 births
Living people
Algebraic geometers
Indian number theorists
American people of Indian descent
People from Silver Spring, Maryland
Harvard University alumni
Princeton University alumni
Massachusetts Institute of Technology School of Science alumni
Massachusetts Institute of Technology faculty
University of California, San Diego faculty
Fellows of the American Mathematical Society
International Mathematical Olympiad participants
21st-century Indian mathematicians
Putnam Fellows
Recipients of the Presidential Early Career Award for Scientists and Engineers |
https://en.wikipedia.org/wiki/Non-uniform%20random%20variate%20generation | Non-uniform random variate generation or pseudo-random number sampling is the numerical practice of generating pseudo-random numbers (PRN) that follow a given probability distribution.
Methods are typically based on the availability of a uniformly distributed PRN generator. Computational algorithms are then used to manipulate a single random variate, X, or often several such variates, into a new random variate Y such that these values have the required distribution.
The first methods were developed for Monte-Carlo simulations in the Manhattan project, published by John von Neumann in the early 1950s.
Finite discrete distributions
For a discrete probability distribution with a finite number n of indices at which the probability mass function f takes non-zero values, the basic sampling algorithm is straightforward. The interval [0, 1) is divided in n intervals [0, f(1)), [f(1), f(1) + f(2)), ... The width of interval i equals the probability f(i).
One draws a uniformly distributed pseudo-random number X, and searches for the index i of the corresponding interval. The so determined i will have the distribution f(i).
Formalizing this idea becomes easier by using the cumulative distribution function
It is convenient to set F(0) = 0. The n intervals are then simply [F(0), F(1)), [F(1), F(2)), ..., [F(n − 1), F(n)). The main computational task is then to determine i for which F(i − 1) ≤ X < F(i).
This can be done by different algorithms:
Linear search, computational time linear in n.
Binary search, computational time goes with log n.
Indexed search, also called the cutpoint method.
Alias method, computational time is constant, using some pre-computed tables.
There are other methods that cost constant time.
Continuous distributions
Generic methods for generating independent samples:
Rejection sampling for arbitrary density functions
Inverse transform sampling for distributions whose CDF is known
Ratio of uniforms, combining a change of variables and rejection sampling
Slice sampling
Ziggurat algorithm, for monotonically decreasing density functions as well as symmetric unimodal distributions
Convolution random number generator, not a sampling method in itself: it describes the use of arithmetics on top of one or more existing sampling methods to generate more involved distributions.
Generic methods for generating correlated samples (often necessary for unusually-shaped or high-dimensional distributions):
Markov chain Monte Carlo, the general principle
Metropolis–Hastings algorithm
Gibbs sampling
Slice sampling
Reversible-jump Markov chain Monte Carlo, when the number of dimensions is not fixed (e.g. when estimating a mixture model and simultaneously estimating the number of mixture components)
Particle filters, when the observed data is connected in a Markov chain and should be processed sequentially
For generating a normal distribution:
Box–Muller transform
Marsaglia polar method
For generating a Poisson distribution:
S |
https://en.wikipedia.org/wiki/D.%20S.%20Malik | Davender S. Malik is an Indian American mathematician and professor of mathematics and computer science at Creighton University.
Education
Malik attended the University of Delhi in New Delhi, India, receiving his bachelor's and master's degrees in mathematics, where he won the Prof. Ram Behari Gold Medal in 1980 for his high marks. Then at the University of Waterloo in Ontario, Canada, he received a master's degree in pure mathematics. In the United States, Malik went to Ohio University, earning an M.S. in computer science, and a Ph.D. in mathematics in 1985, writing his dissertation on "A Study of Q-Hypercyclic Rings."
Career
In 1985, Malik joined the faculty of Creighton University, teaching in the mathematics department. In 2013 he became the first holder of the Frederick H. and Anna K. Scheerer Endowed Chair in Mathematics. His research has focused on ring theory, abstract algebra, information science, and fuzzy mathematics, including fuzzy automata theory, fuzzy logic, and applications of fuzzy set theory in other disciplines.
In the academic community, Malik has been a member of the American Mathematical Society and Phi Kappa Phi. Within his community, co-created a Creighton program in which faculty help area high school students pursue scientific research, to be published in their own student journal.
Malik has published more than 45 papers and 18 books. He has created a computer science line of textbooks that includes extensive and complete programming examples, exercises, and case studies throughout using programming languages such as C++ and Java.
Books
The books he has written include:
Programming
C++ Programming: From Problem Analysis to Program Design (1st ed., 2002; 8th ed. 2017)
C++ Programming: Program Design Including Data Structures (1st ed., 2002; 8th ed. 2017)
Data Structures Using C++ (1st ed., 2003; 2nd ed. 2010)
Data Structures Using Java (2003)
Java programming: From Problem Analysis to Program Design (1st ed., 2003; 5th ed. 2012)
Java programming: Program Design including Data structures (2006)
Java programming: Guided Learning With Early Objects (2009)
Introduction to C++ Programming, Brief Edition (2009)
Mathematics
Fundamentals of Abstract Algebra (1997)
Fuzzy Commutative Algebra (1998)
Fuzzy Discrete Structures (2000)
Fuzzy Mathematics in Medicine (2000)
Fuzzy Automata and Languages: Theory and Applications (2002)
Fuzzy Semigroups (2003)
Application of Fuzzy Logic to Social Choice Theory (2015)
References
External links
Faculty webpage at Creighton University
20th-century American mathematicians
21st-century American mathematicians
Algebraists
American textbook writers
American male writers of Indian descent
Creighton University faculty
Indian emigrants to the United States
20th-century Indian mathematicians
1958 births
Living people
Mathematics educators
Ohio University alumni
Science and technology studies scholars
Delhi University alumni
University of Waterloo alumni
American male non- |
https://en.wikipedia.org/wiki/Walter%20Edwin%20Arnoldi | Walter Edwin Arnoldi (December 14, 1917 – October 5, 1995) was an American engineer mainly known for the Arnoldi iteration, an eigenvalue algorithm used in numerical linear algebra. His main research interests included modelling vibrations, acoustics, aerodynamics of aircraft propeller, and oxygen reclamation problems of space science. His 1951 paper The principle of minimized iterations in the solution of the eigenvalue problem is one of the most cited papers in numerical linear algebra.
Life and career
Born in New York City, Arnoldi earned a degree in mechanical engineering from the Stevens Institute of Technology in 1937 before achieving a Master of Science degree from Harvard University. He worked at the Hamilton Standard Division of the United Technologies Corporation from 1939 until his retirement in 1977; known as United Aircraft for much of his career. He was married to Flora (von Weiler) Arnoldi with whom he had two sons, Douglas and Carl. He lived in West Hartford, Connecticut since 1950. He died in West Hartford on October 5, 1995.
References
20th-century American mathematicians
Numerical linear algebra
1917 births
1995 deaths
American mechanical engineers
Engineers from New York (state)
Stevens Institute of Technology alumni
Harvard University alumni
20th-century American engineers
People from West Hartford, Connecticut |
https://en.wikipedia.org/wiki/Cantic%207-cube | In seven-dimensional geometry, a cantic 7-cube or truncated 7-demicube as a uniform 7-polytope, being a truncation of the 7-demicube.
A uniform 7-polytope is vertex-transitive and constructed from uniform 6-polytope facets, and can be represented a coxeter diagram with ringed nodes representing active mirrors. A demihypercube is an alternation of a hypercube.
Its 3-dimensional analogue would be a truncated tetrahedron (truncated 3-demicube), and Coxeter diagram or as a cantic cube.
Alternate names
Truncated demihepteract
Truncated hemihepteract (thesa) (Jonathan Bowers)
Cartesian coordinates
The Cartesian coordinates for the 1344 vertices of a truncated 7-demicube centered at the origin and edge length 6 are coordinate permutations:
(±1,±1,±3,±3,±3,±3,±3)
with an odd number of plus signs.
Images
It can be visualized as a 2-dimensional orthogonal projections, for example the a D7 Coxeter plane, containing 12-gonal symmetry. Most visualizations in symmetric projections will contain overlapping vertices, so the colors of the vertices are changed based on how many vertices are at each projective position, here shown with red color for no overlaps.
Related polytopes
There are 95 uniform polytopes with D6 symmetry, 63 are shared by the B6 symmetry, and 32 are unique:
Notes
References
H.S.M. Coxeter:
H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
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]
Norman Johnson Uniform Polytopes, Manuscript (1991)
N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D.
External links
Polytopes of Various Dimensions
Multi-dimensional Glossary
7-polytopes |
https://en.wikipedia.org/wiki/Rectified%209-simplexes | In nine-dimensional geometry, a rectified 9-simplex is a convex uniform 9-polytope, being a rectification of the regular 9-simplex.
These polytopes are part of a family of 271 uniform 9-polytopes with A9 symmetry.
There are unique 4 degrees of rectifications. Vertices of the rectified 9-simplex are located at the edge-centers of the 9-simplex. Vertices of the birectified 9-simplex are located in the triangular face centers of the 9-simplex. Vertices of the trirectified 9-simplex are located in the tetrahedral cell centers of the 9-simplex. Vertices of the quadrirectified 9-simplex are located in the 5-cell centers of the 9-simplex.
Rectified 9-simplex
The rectified 9-simplex is the vertex figure of the 10-demicube.
Alternate names
Rectified decayotton (reday) (Jonathan Bowers)
Coordinates
The Cartesian coordinates of the vertices of the rectified 9-simplex can be most simply positioned in 10-space as permutations of (0,0,0,0,0,0,0,0,1,1). This construction is based on facets of the rectified 10-orthoplex.
Images
Birectified 9-simplex
This polytope is the vertex figure for the 162 honeycomb. Its 120 vertices represent the kissing number of the related hyperbolic 9-dimensional sphere packing.
Alternate names
Birectified decayotton (breday) (Jonathan Bowers)
Coordinates
The Cartesian coordinates of the vertices of the birectified 9-simplex can be most simply positioned in 10-space as permutations of (0,0,0,0,0,0,0,1,1,1). This construction is based on facets of the birectified 10-orthoplex.
Images
Trirectified 9-simplex
Alternate names
Trirectified decayotton (treday) (Jonathan Bowers)
Coordinates
The Cartesian coordinates of the vertices of the trirectified 9-simplex can be most simply positioned in 10-space as permutations of (0,0,0,0,0,0,1,1,1,1). This construction is based on facets of the trirectified 10-orthoplex.
Images
Quadrirectified 9-simplex
Alternate names
Quadrirectified decayotton
Icosayotton (icoy) (Jonathan Bowers)
Coordinates
The Cartesian coordinates of the vertices of the quadrirectified 9-simplex can be most simply positioned in 10-space as permutations of (0,0,0,0,0,1,1,1,1,1). This construction is based on facets of the quadrirectified 10-orthoplex.
Images
Notes
References
H.S.M. Coxeter:
H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
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]
Norman Johnson Uniform Polytopes, Manuscript (1991)
N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966)
o3x3o3o3o3o3o3o3o - reday, o3o3x3o |
https://en.wikipedia.org/wiki/Ridit%20scoring | In statistics, ridit scoring is a statistical method used to analyze ordered qualitative measurements.
The tools of ridit analysis were developed and first applied by Bross, who coined the term "ridit" by analogy with other statistical transformations such as probit and logit. A ridit describes how the distribution of the dependent variable in row i of a contingency table compares relative to an identified distribution (e.g., the marginal distribution of the dependent variable).
Calculation of ridit scores
Choosing a reference data set
Since ridit scoring is used to compare two or more sets of ordered qualitative data, one set is designated as a reference against which other sets can be compared. In econometric studies, for example, the ridit scores measuring taste survey answers of a competing or historically important product are often used as the reference data set against which taste surveys of new products are compared. Absent a convenient reference data set, an accumulation of pooled data from several sets or even an artificial or hypothetical set can be used.
Determining the probability function
After a reference data set has been chosen, the reference data set must be converted to a probability function. To do this, let x1, x2,..., xn denote the ordered categories of the preference scale. For each j, xj represents a choice or judgment. Then, let the probability function p be defined with respect to the reference data set as
Determining ridits
The ridit scores, or simply ridits, of the reference data set are then easily calculated as
Each of the categories of the reference data set are then associated with a ridit score.
More formally, for each , the value wj is the ridit score of the choice xj.
Interpretation and examples
Intuitively, ridit scores can be understood as a modified notion of percentile ranks. For any j, if xj has a low (close to 0) ridit score, one can conclude that
is very small, which is to say that very few respondents have chosen a category "lower" than xj.
Applications
Ridit scoring has found use primarily in the health sciences (including nursing and epidemiology) and econometric preference studies.
A mathematical approach
Besides having intuitive appeal, the derivation for ridit scoring can be arrived at with mathematically rigorous methods as well. Brockett and Levine presented a derivation of the above ridit score equations based on several intuitively uncontroversial mathematical postulates.
Notes
R statistical computing package for Ridit Analysis: https://cran.r-project.org/package=Ridit
Further reading
Econometric modeling
Categorical data |
https://en.wikipedia.org/wiki/Abel%27s%20sum%20formula | Abel's sum formula may refer to:
Abel's summation formula, a formula used in number theory to compute series
Summation by parts, a transformation of the summation of products of sequences into other summations |
https://en.wikipedia.org/wiki/Terry%20Lyons%20%28mathematician%29 | Terence John Lyons FLSW is a British mathematician, specialising in stochastic analysis. Lyons, previously the Wallis Professor of Mathematics, is a fellow of St Anne's College, Oxford and a Faculty Fellow at The Alan Turing Institute. He was the director of the Oxford-Man Institute from 2011 to 2015 and the president of the London Mathematical Society from 2013 to 2015. His mathematical contributions have been to probability, harmonic analysis, the numerical analysis of stochastic differential equations, and quantitative finance. In particular he developed what is now known as the theory of rough paths. Together with Patrick Kidger he proved a universal approximation theorem for neural networks of arbitrary depth.
Education
Lyons obtained his B.A. at Trinity College, Cambridge and his D.Phil at the University of Oxford.
Career
Lyons has held positions at UCLA, Imperial College London, the University of Edinburgh and since 2000 has been Wallis Professor of Mathematics at the University of Oxford. He was the Director of the Oxford-Man Institute at the University of Oxford from 15 June 2011 to 15 December 2015. He also held a number of visiting positions in Europe and North America.
Together with Zhongmin Qian he wrote the monograph System Control and Rough Paths, and together with Michael J. Caruana and Thierry Lévy the book Differential Equations Driven by Rough Paths.
Honours and awards
In 1985 he was awarded the Rollo Davidson Prize. In 1986 he was awarded the Whitehead Prize of the London Mathematical Society. In 2000 he was awarded the Pólya Prize of the London Mathematical Society.
He was elected a fellow of the Royal Society of Edinburgh in 1988, and elected a Fellow of the Royal Society in 2002; he was made a fellow of the Institute of Mathematical Statistics, in 2005 and a fellow of the Learned Society of Wales in 2011. In 2013, he was elected president of the London Mathematical Society.
In 2007 he was awarded a Doctor Honoris Causa from the University of Toulouse, he was made an Honorary Fellow of Aberystwyth University in 2010 and Cardiff University in 2012. In 2017 he was awarded an honorary Doctor of Mathematics from the University of Waterloo.
References
Living people
20th-century British mathematicians
21st-century British mathematicians
Probability theorists
Wallis Professors of Mathematics
Fellows of St Anne's College, Oxford
Academics of the University of Edinburgh
Fellows of the Royal Society
1953 births
Fellows of the Learned Society of Wales |
https://en.wikipedia.org/wiki/Boris%20Rozovsky | Boris Rozovsky is Ford Foundation Professor of Applied Mathematics at Brown University. His research is in stochastic analysis, particularly the study of stochastic partial differential equations.
Rozovsky started his studies in art school, but switched to mathematics; he earned a master's degree in 1968 and a Ph.D. in 1973 from Moscow State University. He moved to the U.S. in 1988; after teaching for fourteen years at the University of Southern California, he joined the Brown University faculty in 2006.
Since 1997, he is a Fellow of the Institute of Mathematical Statistics.
In 1997, he was awarded the Peter-the-Great Medal.
In 2003 he was awarded the Kolmogorov Medal.
References
Living people
20th-century American mathematicians
21st-century American mathematicians
Probability theorists
Brown University faculty
1945 births |
https://en.wikipedia.org/wiki/Alfr%C3%A9d%20R%C3%A9nyi%20Institute%20of%20Mathematics | The Alfréd Rényi Institute of Mathematics () is the research institute in mathematics of the Hungarian Academy of Sciences. It was created in 1950 by Alfréd Rényi, who directed it until his death. Since its creation, the institute has been the center of mathematical research in Hungary. It received the title Centre of Excellence of the European Union (2001). The current director is András Stipsicz. The institute publishes the research journal Studia Scientiarum Mathematicarum Hungarica.
Research divisions and research groups
Algebra (head: Mátyás Domokos)
Algebraic geometry and differential topology (head: András Némethi)
Algebraic Logic (head: Hajnal Andréka)
Analysis (head: András Kroó)
Combinatorics and discrete mathematics (head: Ervin Győri)
Geometry (head: Gábor Fejes Tóth)
Number theory (head: János Pintz)
Probability & statistics (head: Péter Major)
Set theory and general topology (head: Lajos Soukup)
Cryptology (head: Gábor Tardos)
Financial Mathematics (Momentum research group of the Hungarian Academy of Sciences, head: Miklós Rásonyi)
Groups and Graphs (Momentum research group of the Hungarian Academy of Sciences, European Research Council research group, head: Miklós Abért)
Limits of Structures (Momentum research group of the Hungarian Academy of Sciences, European Research Council research group, head: Balázs Szegedy)
Low Dimensional Topology (Momentum research group of the Hungarian Academy of Sciences, European Research Council research group, head: András Stipsicz)
Regularity (European Research Council research group, head: Endre Szemerédi)
Discrete and Convex Geometry (European Research Council research group, head: Imre Bárány)
Didactics (head: Péter Juhász)
Name
The institute's name originally was Applied Mathematics Institute of the HAS () then Mathematical Research Institute of the HAS (). It obtained its current name on 1 July 1999 after Alfréd Rényi, the eminent mathematician who founded the institute and was its director for 20 years.
Some of the notable researchers
Imre Bárány, combinatorialist, geometer, member of the Hungarian Academy of Sciences
Imre Csiszár, information theorist, Shannon Award, Dobrushin Prize, member of the Hungarian Academy of Sciences
Zoltán Füredi, combinatorialist, member of the Hungarian Academy of Sciences
Tibor Gallai, combinatorialist
András Hajnal, set theorist, member of the Hungarian Academy of Sciences
István Juhász, working in set theoretical topology, member of the Hungarian Academy of Sciences
Imre Lakatos, philosopher of mathematics and science
Péter Major, probability theorist, member of the Hungarian Academy of Sciences
Katalin Marton, information theorist, Shannon Award
Péter Pál Pálfy, algebra, member of the Hungarian Academy of Sciences
János Pintz, number theorist, AMS Cole Prize, member of the Hungarian Academy of Sciences
Lajos Pósa, combinatorialist, educator
László Pyber, algebra, member of the Hungarian Academy of Sciences
Imre Z. Ruzsa, n |
https://en.wikipedia.org/wiki/Kummer%27s%20congruence | In mathematics, Kummer's congruences are some congruences involving Bernoulli numbers, found by .
used Kummer's congruences to define the p-adic zeta function.
Statement
The simplest form of Kummer's congruence states that
where p is a prime, h and k are positive even integers not divisible by p−1 and the numbers Bh are Bernoulli numbers.
More generally if h and k are positive even integers not divisible by p − 1, then
whenever
where φ(pa+1) is the Euler totient function, evaluated at pa+1 and a is a non negative integer. At a = 0, the expression takes the simpler form, as seen above.
The two sides of the Kummer congruence are essentially values of the p-adic zeta function, and the Kummer congruences imply that the p-adic zeta function for negative integers is continuous, so can be extended by continuity to all p-adic integers.
See also
Von Staudt–Clausen theorem, another congruence involving Bernoulli numbers
References
Theorems in number theory
Modular arithmetic |
https://en.wikipedia.org/wiki/Lothar%20Geisler | Lothar Geisler (8 December 1936 – 28 April 2019) was a German footballer.
Career
Statistics
1 1960–61 and 1962–63 include the German football championship playoffs.
References
External links
1936 births
Bundesliga players
German men's footballers
VfL Bochum players
Borussia Dortmund players
Borussia Dortmund II players
Men's association football defenders
Men's association football midfielders
2019 deaths
Footballers from Dortmund
West German men's footballers
Sportspeople from the Province of Westphalia |
https://en.wikipedia.org/wiki/Judith%20D.%20Sally | Judith D. Sally (born Judith Donovan; March 23, 1937, in Manhattan, New York) is a Professor Emeritus of Mathematics at Northwestern University. Her research is in commutative algebra, particularly in the study of Noetherian local rings and graded rings.
Life and education
Judith Donovan was born to Dr. and Mrs. Edward J. Donovan in Manhattan, New York in 1937. She finished high school at the Convent of Sacred Heart in New York and pursued her undergraduate studies at Barnard College, earning her bachelor's degree in 1958. After graduating from Barnard, she began graduate studies in mathematics at Brandeis University in Waltham, Massachusetts. At Brandeis, she met Paul J. Sally, Jr, who was in the doctoral program in mathematics at Brandeis. Judith and Paul were married in November 1959, while Paul was still in graduate school. In 1960, Judith Sally was awarded a master's degree in mathematics from Brandeis. Judith and Paul had three sons, David, Stephen, and Paul III, while Paul was completing his dissertation and consequently, Judith postponed her doctoral studies. Paul completed his Ph.D. at Brandeis in 1965 and joined the faculty at the University of Chicago that same year.
In 1968, Judith entered the doctoral program in mathematics at Chicago.
In 1971, Judith Sally was awarded her Ph.D. in mathematics from University of Chicago. Her thesis "Regular Overrings of Regular Local Rings" was supervised by Irving Kaplansky.
Career
After completion of her doctoral studies, Sally spent 1971–1972 in a postdoctoral position at Rutgers University in New Brunswick, New Jersey. Sally joined the faculty at Northwestern University in 1972. In 1977, she received a Sloan Fellowship. She received a Bunting Fellowship at the Mary Ingraham Institute at Radcliffe College for the 1981-1982 academic year. Sally was awarded a National Science Foundation Visiting Professorship for Women for the 1988–1989 academic year, during which time she visited Purdue University in West Lafayette, Indiana. At Northwestern she won the College of Arts and Sciences Teaching Award. In 1995, she was invited to give the Association for Women in Mathematics Noether Lecture, an honor "for fundamental and sustained contributions to the mathematical sciences". She wrote a research monograph Number of generators of ideals in rings that was published by Marcel Dekker in 1978. She has published several books on mathematics education with her husband, Paul Sally.
Selected publications
References
External links
Judith D. Sally's Author Profile on MathSciNet
Judith D. Sally's Profile on zbMATH
1937 births
Living people
University of Chicago alumni
Barnard College alumni
American women mathematicians
Northwestern University faculty
20th-century American mathematicians
21st-century American mathematicians
Place of birth missing (living people)
20th-century women mathematicians
21st-century women mathematicians
20th-century American women
21st-century American women |
https://en.wikipedia.org/wiki/Stoyan%20Predev | Stoyan Predev (; born 19 August 1993, in Sofia) is a Bulgarian footballer who plays as a defender for Dunav Ruse.
Career
On 1 July 2018, Predev signed with Kariana.
Career statistics
Club
References
External links
1993 births
Living people
Footballers from Sofia
Bulgarian men's footballers
Men's association football defenders
First Professional Football League (Bulgaria) players
Second Professional Football League (Bulgaria) players
PFC Slavia Sofia players
FC Vitosha Bistritsa players
FC Sportist Svoge players
PFC Septemvri Sofia players
FC Pirin Razlog players
FC Lokomotiv 1929 Sofia players
FC Kariana Erden players
FC Montana players
FC Dunav Ruse players |
https://en.wikipedia.org/wiki/David%20L.%20Fulton | David L. Fulton is a private collector of Cremonese instruments.
Born in 1944, he grew up in Eugene, Oregon, playing the violin from an early age. He studied mathematics at the University of Chicago, and was concertmaster of the University of Chicago Orchestra while he was there.
Fulton performed professionally with the Hartford Symphony Orchestra as a violinist. In 1970 he founded the Department of Computer Science at Bowling Green State University, serving as Professor and Chairman for 10 years. While still at Bowling Green, he co-founded Fox Software, which ultimately gained international recognition for its database management application, FoxPro. Following the sale of Fox Software to Microsoft in 1992, Dr. Fulton served as Microsoft’s Vice President for Database Products until his retirement in 1994.
Fulton has produced several documentary films about violins and music. The first was Homage (2008), which won the 2009 Juno award as "Classical Album of the Year: Solo or Chamber Ensemble". The film features violinist James Ehnes performing on fourteen instruments from Fulton's collection.
The second, Violin Masters: Two Gentlemen of Cremona, (2010), narrated by Alfred Molina and featuring renowned violinists James Ehnes, Joshua Bell, Midori, Itzhak Perlman among others, examines the history and modern use of Stradivari and Guarneri del Gesù violins. Violin Masters won a 2012 Emmy in the "Documentary - Historical" category.
The most recent film, Transcendence: A Meeting of Greats, (2014), documents the sessions at which the Miró Quartet recorded Schubert's great String Quartet No. 15, in G Major, D. 887. This film was nominated for two 2014 Emmy awards in the Special Event Coverage category, winning Best Director in that category.
In January 2022 Fulton published a book about his violin collection entitled The Fulton Collection - A Guided Tour. This deluxe 1st edition volume is a large-format, linen-bound hardcover containing museum-quality photos, archival material, and first-hand recollections. The Fulton collection may be seen and heard on the book's associated website which features detailed high-definition video of the instruments being played and where the book may be ordered: / Collection Book Website
Notable Instruments
ViolinsStradivari La Pucelle 1709
Stradivari "General Kyd, Perlman" 1714
Stradivari Marsick 1715
Stradivari "Baron d'Assignies" 1713
Stradivari "Alba, Herzog, Coronation" 1719
Stradivari "Sassoon" 1733
Stradivari "Baron Knoop, Bevan" 1715
Guarneri del Gesù "King Joseph" 1737
Guarneri del Gesù "Stern, Panette, Balâtre, Alard" 1737
Guarneri del Gesù "Lord Wilton" 1742
Guarneri del Gesù "Haddock" 1734
Guarneri del Gesù "d'Egville" 1735
Guarneri del Gesù "Kemp, Emperor" 1738
Guarneri del Gesù "Carrodus" 1743
Pietro Guarneri, of Mantua "Shapiro" 1698
Carlo Bergonzi "Kreisler, Perlman" 1735(?)
Giovanni Battista Guadagnini, Turin 1778
Violas
Andrea Guarneri "Conte Vitale" 1676
Gasparo da Salò "Krasner, Kelley" c |
https://en.wikipedia.org/wiki/Stephen%20F.%20Barker | Stephen Francis Barker (January 11, 1927 – December 16, 2019) was an American Philosopher of Mathematics, a Professor Emeritus of Philosophy in the Department of Philosophy, Krieger School of Arts and Sciences at Johns Hopkins University. He was also a faculty member at the University of Southern California, the University of Virginia and Ohio State University.
He was born in Ann Arbor, Michigan. In 1948, he served in the Navy for a year. He then proceeded in 1949 to earn a bachelor's degree from Swarthmore College. He got his master's degree as well as ph.D in Philosophy from Harvard University in 1951 and 1954 respectively. While at Harvard, he won the Bechtel Prize in 1951 for his essay, "A Study of Phenomenalism".
Later, he became an instructor in the University of Southern California in 1954–55. He became an Assistant Professor in the University of Virginia and then was made an Associate Professor in the same university from 1956 to 1961. He became a professor at Ohio State University from 1961 to1964. He reached the peak of his career when he arrived at Johns Hopkins as a professor in 1964 where he was named Professor Emeritus upon his retirement in 2002.
Barker was not just a Professor or a Professor Emeritus, he was also a Sheldon Traveling Fellow in England in 1952–53. Harvard awarded him the George Santayana Fellowship for the academic year 1955–56. In addition, he became a Guggenheim Fellow in 1964–65.
Barker was married to Evelyn Barker who was also a Philosopher and died in 2003. Baker himself eventually died at Roland Park Place at 92 years of age after living an exceptionally brilliant life.
Books
Barker is the author of:
Induction and hypothesis: a study of the logic of confirmation (Cornell University Press, 1957). This study of theories of informal reasoning is structured in four parts: an investigation of the problem of induction, a rejection of explanations based on overriding premises (such as the uniformity of nature) as a form of begging the question, an overview of positivist approaches to the problem, and finally a resolution to the problem based on theories of John George Kemeny involving the selection of the most likely hypothesis to fit a set of observations.
Philosophy of mathematics (Prentice-Hall, 1964). Part of a series of books (edited by Elizabeth and Monroe Beardsley) overviewing the main areas of philosophy, this book describes the main problems in the philosophy of mathematics and evaluates their proposed solutions. Its five chapters concern Euclidean and non-Euclidean geometry, and literalist and non-literalist views on the meaning of numbers.
The elements of logic (McGraw Hill, 1965)
Thomas Reid critical interpretations (with Tom L. Beauchamp, Philosophical monographs, 1976)
In addition, he edited John Wisdom's Proof and explanation: the Virginia lectures (University Press of America, 1991), co-edited The Legacy of logical positivism; studies in the philosophy of science with Peter Achinstein (Joh |
https://en.wikipedia.org/wiki/FRAX | FRAX (Fracture Risk Assessment Tool) is a diagnostic tool used to evaluate the 10-year probability of bone fracture risk. It was developed by the University of Sheffield. FRAX integrates clinical risk factors and bone mineral density at the femoral neck to calculate the 10-year probability of hip fracture and the 10-year probability of a major osteoporotic fracture (clinical spine, forearm, hip or shoulder fracture). The models used to develop the FRAX diagnostic tool were derived from studying patient populations in North America, Europe, Latin America, Asia and Australia.
Components
The parameters included in a FRAX assessment are:
Country
Age
Sex
Weight
Height
Previous fracture
Hip fracture in the subject's mother or father
Smoking
Glucocorticoid treatment
Rheumatoid arthritis
Disease strongly associated with osteoporosis
Alcohol intake of 3 or more standard drinks per day
Bone mineral density (BMD) of the femoral neck
Trabecular bone score (optional)
Availability and usage
FRAX is freely accessible online, and commercially available as a desktop application, in paper-form as a FRAX Pad, as an iPhone application, and as an Android application. The tool is compatible with 58 models for 53 countries, and is available in 28 languages.
FRAX is incorporated into many national guidelines around the world, including those of Belgium, Canada, Japan, Netherlands, Poland, Sweden, Switzerland, UK (NOGG), and US (NOF). FRAX assessments are intended to provide guidance for determining access to treatment in healthcare systems.
Adjustments
Glucocorticoid use is included FRAX as a dichotomous variable, whereas the increased risk for fractures seen with glucocorticoid use is dependent on glucocorticoid dose and duration of use. Several methods have been proposed how to adjust FRAX accordingly.
Though known to be a risk factor for fractures, Type 2 Diabetes is not included as such in FRAX. Some clinicians choose rheumatoid arthritis as an equivalent risk factor instead.
FRAX was developed and most commonly used to assess fracture risk for previously untreated individuals, though some have suggested is can also be used in those treated in the past or even on current treatment for osteoporosis.
References
External links
Alternate FRAX calculator that doesn't use Java
Medical diagnosis
Medical terminology
Nosology |
https://en.wikipedia.org/wiki/Patrick%20Billingsley | Patrick Paul Billingsley (May 3, 1925 – April 22, 2011) was an American mathematician and stage and screen actor, noted for his books in advanced probability theory and statistics. He was born and raised in Sioux Falls, South Dakota, and graduated from the United States Naval Academy in 1946.
Academic career
After earning a Ph.D. in mathematics at Princeton University in 1955, he was attached to the NSA until his discharge from the Navy in 1957. In 1958 he became a professor of mathematics and statistics at the University of Chicago, where he served as chair of the Department of Statistics from 1980 to 1983, and retired in 1994. In 1964–65 he was a Fulbright Fellow and visiting professor at the University of Copenhagen. In 1971–72 he was a Guggenheim Fellow and visiting professor at the University of Cambridge (Peterhouse). From 1976 to 1979 he edited the Annals of Probability. In 1983 he was president of the Institute of Mathematical Statistics. He was given the Lester R. Ford Award for his article "Prime Numbers and Brownian Motion." He was elected a Fellow of the American Academy of Arts and Sciences in 1986.
He starred in a number of plays at Court Theatre and Body Politic Theatre in Chicago and appeared in at least nine films.
In Young Men and Fire, fellow University of Chicago professor Norman Maclean wrote about Billingsley that "he is a distinguished statistician and one of the best amateur actors I have ever seen".
Books
Statistical Inference for Markov Processes (1961)
Ergodic Theory and Information (1965)
Convergence of Probability Measures (1st Edition 1968, 2nd Edition 1999)
The Elements of Statistical Inference (with David L. Huntsberger, 1986)
Probability and Measure (1st Edition 1976, 2nd Edition 1986, 3rd Edition, 1995, Anniversary Edition 2012 )
Stage plays
Three Magic Keys, Taliesin (1964)
The Pirates of Penzance, Pirate (1966)
Read Me a Story (1966)
Clue of the Circus Clowns, Circus Master (1968)
Finian's Rainbow, Buzz Collins (1968)
Beadle-Levi Show (parody of The Homecoming) (1968)
Guys and Dolls, Arvide Abernathy (1969)
We Bombed in New Haven (Court Theatre, 1970)
Victorian Children (1970)
Vaudeville Show, singer (1970)
The Threepenny Opera, street singer (1970)
Four Plays of Fantasy and the Unusual (1970)
Moulin Rouge (1971)
Oh, What a Lovely War! (1973)
Midsummer Night's Dream, Theseus (Court Theatre, 1973)
The Caretaker, Aston (Court Theatre, 1973)
The Father, The Captain (1974)
Murder in the Cathedral, First Knight (1974)
Twelfth Night, Feste (Court Theatre, 1974)
The Same Room, Tom Ferris (1975)
Dracula, Dr. Seward (1975)
Much Ado about Nothing, Balthazar and Friar Frances (Court Theatre, 1975)
Exits and Entrances (1976)
Trifles, Sheriff Peters (1976)
The Lover, Richard-Max (Court Theatre, 1977)
The Tempest, Alonzo (Court Theatre, 1977)
She Stoops to Conquer, Mr. Hardcastle (Court Theatre, 1978)
Measure for Measure, The Duke (Court Theatre, 1979)
Mrs. Warren's Profession, R |
https://en.wikipedia.org/wiki/Beatrice%20Mabel%20Cave-Browne-Cave | Beatrice Mabel Cave-Browne-Cave, MBE AFRAeS (30 May 1874 – 9 July 1947) was an English mathematician who undertook pioneering work in the mathematics of aeronautics.
Birth and education
Beatrice Cave-Browne-Cave was the daughter of Sir Thomas Cave-Browne-Cave and Blanche Matilda Mary Ann (née Milton). She was one of six siblings. The family surname (Cave-Browne-Cave) came from a variety of historical circumstances but she and her younger sister Frances tended to use the single surname Cave professionally. Cave was educated at home in Streatham and entered Girton College, Cambridge with Frances in 1895. In 1898, she completed a degree in the mathematical tripos, earning second-class honors. The following year, Cave passed part II of the mathematical tripos with third-class honors.
Career
Cave spent eleven years teaching mathematics to girls at a high school in Clapham in south-west London and doing computing work at home.
In the years just before the First World War, Cave worked under Professor Karl Pearson in the Galton Laboratory at University College, London. In 1903, she was among six researchers, including her sister Frances, that collaborated on a large child development study led by Pearson. They worked unpaid until the Worshipful Company of Drapers provided a grant that paid them a stipend in 1904. Pearson hoped to establish evidence of the inheritance of attributes by collecting physical and mental data from 4000 children and their parents, which included some of Cave's high school students. She assisted in the collection and processing of data as well as related computations. Cave published in Biometrika and also conducted statistical analyses for the Treasury and Board of Trade. Cave started working full-time as a computer at the Galton Laboratory in 1913, in which time she co-authored two papers published in Biometrika, including Numerical Illustrations of the Variate Difference Method. Cave also created correlation tables in 1917 based on a series of mice breeding experiments by Raphael Weldon, a colleague of Pearson's at University College. Her correlation tables included tables showing amount of pigment, connecting old and new process of determining amount of pigmentation, mother and son pigmentation percentages, grandparents and offspring, and father and son amount of pigment in mice.
In 1916, Cave began working for the government on airplane design. She carried out original research for the government on the mathematics of aeronautics which remained classified under the Official Secrets Act for fifty years. She examined the effects of loads on different areas of planes during flight, and her research helped to improve aircraft stability and propeller efficiency. Some of her works are held in UCL archives which include correspondence from her time at the Galton Laboratory for work on bomb trajectories, terminal velocities, timber tests, and detonators, for the Admiralty Air Department and Ministry of Munitions.
Cave was elect |
https://en.wikipedia.org/wiki/Valentiner%20group | In mathematics, the Valentiner group is the perfect triple cover of the alternating group on 6 points, and is a group of order 1080. It was found by in the form of an action of A6 on the complex projective plane, and was studied further by .
All perfect alternating groups have perfect double covers. In most cases this is the universal central extension. The two exceptions are A6 (whose perfect triple cover is the Valentiner group) and A7, whose universal central extensions have centers of order 6.
Representations
The alternating group A6 acts on the complex projective plane, and showed that the group acts on the 6 conics of Gerbaldi's theorem. This gives a homomorphism to PGL3(C), and the lift of this to the triple cover GL3(C) is the Valentiner group. This embedding can be defined over the field generated by the 15th roots of unity.
The product of the Valentiner group with a group of order 2 is a 3-dimensional complex reflection group of order 2160 generated by 45 complex reflections of order 2. The invariants form a polynomial algebra with generators of degrees 6, 12, and 30.
The Valentiner group has complex irreducible faithful group representations of dimension 3, 3, 3, 3, 6, 6, 9, 9, 15, 15.
The Valentiner group can be represented as the monomial symmetries of the hexacode, the 3-dimensional subspace of F spanned by (001111), (111100), and (0101ω), where the elements of the finite field F4 are 0, 1, ω, .
The group PGL3(F4) acts on the 2-dimensional projective plane over F4 and acts transitively on its hyperovals (sets of 6 points such that no three are on a line). The subgroup fixing a hyperoval is a copy of the alternating group A6. The lift of this to the triple cover GL3(F4) of PGL3(F4) is the Valentiner group.
described the representations of the Valentiner group as a Galois group, and gave an order 3 differential equation with the Valentiner group as its differential Galois group.
References
Finite groups |
https://en.wikipedia.org/wiki/Gerbaldi%27s%20theorem | In linear algebra and projective geometry, Gerbaldi's theorem, proved by , states that one can find six pairwise apolar linearly independent nondegenerate ternary quadratic forms. These are permuted by the Valentiner group.
References
Quadratic forms
Theorems in linear algebra
Theorems in projective geometry |
https://en.wikipedia.org/wiki/Cyclotruncated%205-simplex%20honeycomb | In five-dimensional Euclidean geometry, the cyclotruncated 5-simplex honeycomb or cyclotruncated hexateric honeycomb is a space-filling tessellation (or honeycomb). It is composed of 5-simplex, truncated 5-simplex, and bitruncated 5-simplex facets in a ratio of 1:1:1.
Structure
Its vertex figure is an elongated 5-cell antiprism, two parallel 5-cells in dual configurations, connected by 10 tetrahedral pyramids (elongated 5-cells) from the cell of one side to a point on the other. The vertex figure has 8 vertices and 12 5-cells.
It can be constructed as six sets of parallel hyperplanes that divide space. The hyperplane intersections generate cyclotruncated 5-cell honeycomb divisions on each hyperplane.
Related polytopes and honeycombs
See also
Regular and uniform honeycombs in 5-space:
5-cubic honeycomb
5-demicubic honeycomb
5-simplex honeycomb
Omnitruncated 5-simplex honeycomb
Notes
References
Norman Johnson Uniform Polytopes, Manuscript (1991)
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] (1.9 Uniform space-fillings)
(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
Honeycombs (geometry)
6-polytopes |
https://en.wikipedia.org/wiki/Omnitruncated%205-simplex%20honeycomb | In five-dimensional Euclidean geometry, the omnitruncated 5-simplex honeycomb or omnitruncated hexateric honeycomb is a space-filling tessellation (or honeycomb). It is composed entirely of omnitruncated 5-simplex facets.
The facets of all omnitruncated simplectic honeycombs are called permutahedra and can be positioned in n+1 space with integral coordinates, permutations of the whole numbers (0,1,..,n).
A5* lattice
The A lattice (also called A) is the union of six A5 lattices, and is the dual vertex arrangement to the omnitruncated 5-simplex honeycomb, and therefore the Voronoi cell of this lattice is an omnitruncated 5-simplex.
∪
∪
∪
∪
∪
= dual of
Related polytopes and honeycombs
Projection by folding
The omnitruncated 5-simplex honeycomb can be projected into the 3-dimensional omnitruncated cubic honeycomb by a geometric folding operation that maps two pairs of mirrors into each other, sharing the same 3-space vertex arrangement:
See also
Regular and uniform honeycombs in 5-space:
5-cube honeycomb
5-demicube honeycomb
5-simplex honeycomb
Notes
References
Norman Johnson Uniform Polytopes, Manuscript (1991)
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] (1.9 Uniform space-fillings)
(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
Honeycombs (geometry)
6-polytopes |
https://en.wikipedia.org/wiki/Omnitruncated%20simplectic%20honeycomb | In geometry an omnitruncated simplectic honeycomb or omnitruncated n-simplex honeycomb is an n-dimensional uniform tessellation, based on the symmetry of the affine Coxeter group. Each is composed of omnitruncated simplex facets. The vertex figure for each is an irregular n-simplex.
The facets of an omnitruncated simplectic honeycomb are called permutahedra and can be positioned in n+1 space with integral coordinates, permutations of the whole numbers (0,1,..,n).
Projection by folding
The (2n-1)-simplex honeycombs can be projected into the n-dimensional omnitruncated hypercubic honeycomb by a geometric folding operation that maps two pairs of mirrors into each other, sharing the same vertex arrangement:
See also
Hypercubic honeycomb
Alternated hypercubic honeycomb
Quarter hypercubic honeycomb
Simplectic honeycomb
Truncated simplectic honeycomb
References
George Olshevsky, Uniform Panoploid Tetracombs, Manuscript (2006) (Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs)
Branko Grünbaum, Uniform tilings of 3-space. Geombinatorics 4(1994), 49 - 56.
Norman Johnson Uniform Polytopes, Manuscript (1991)
Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition,
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] (1.9 Uniform space-fillings)
(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
Honeycombs (geometry)
Polytopes
Truncated tilings |
https://en.wikipedia.org/wiki/P-adic%20distribution | In mathematics, a p-adic distribution is an analogue of ordinary distributions (i.e. generalized functions) that takes values in a ring of p-adic numbers.
Definition
If X is a topological space, a distribution on X with values in an abelian group G is a finitely additive function from the compact open subsets of X to G. Equivalently, if we define the space of test functions to be the locally constant and compactly supported integer-valued functions, then a distribution is an additive map from test functions to G. This is formally similar to the usual definition of distributions, which are continuous linear maps from a space of test functions on a manifold to the real numbers.
p-adic measures
A p-adic measure is a special case of a p-adic distribution, analogous to a measure on a measurable space. A p-adic distribution taking values in a normed space is called a p-adic measure if the values on compact open subsets are bounded.
References
Number theory
p-adic numbers |
https://en.wikipedia.org/wiki/Separation%20relation | In mathematics, a separation relation is a formal way to arrange a set of objects in an unoriented circle. It is defined as a quaternary relation satisfying certain axioms, which is interpreted as asserting that a and c separate b from d.
Whereas a linear order endows a set with a positive end and a negative end, a separation relation forgets not only which end is which, but also where the ends are located. In this way it is a final, further weakening of the concepts of a betweenness relation and a cyclic order. There is nothing else that can be forgotten: up to the relevant sense of interdefinability, these three relations are the only nontrivial reducts of the ordered set of rational numbers.
Application
The separation may be used in showing the real projective plane is a complete space. The separation relation was described with axioms in 1898 by Giovanni Vailati.
=
=
⇒ ¬
∨ ∨
∧ ⇒ .
The relation of separation of points was written AC//BD by H. S. M. Coxeter in his textbook The Real Projective Plane. The axiom of continuity used is "Every monotonic sequence of points has a limit." The separation relation is used to provide definitions:
{An} is monotonic ≡ ∀ n > 1
M is a limit ≡ (∀ n > 2 ) ∧ (∀ P ⇒ ∃ n ).
References
Order theory |
https://en.wikipedia.org/wiki/Lorentz%20surface | In mathematics, a Lorentz surface is a two-dimensional oriented smooth manifold with a conformal equivalence class of Lorentzian metrics. It is the analogue of a Riemann surface in indefinite signature.
Further reading
Conformal geometry
Surfaces
Hendrik Lorentz |
https://en.wikipedia.org/wiki/2011%E2%80%9312%20PFC%20CSKA%20Sofia%20season | The 2011–12 season was PFC CSKA Sofia's 64th consecutive season in A Group. This article shows player statistics and all matches (official and friendly) that the club will play during the 2011–12 season.
Players
Squad stats
Appearances for competitive matches only
|-
|colspan="14"|Players sold or loaned out after the start of the season:
|}
As of 23 May 2012
Players in/out
Summer transfers
In:
Out:
Winter transfers
In:
Out:
Pre-season and friendlies
Pre-season
On-season (autumn)
Mid-season
On-season (spring)
Competitions
A Group
Table
Results summary
Results by round
Fixtures and results
Bulgarian Cup
Bulgarian Super Cup
By winning in the 2010–11 Bulgarian Cup, CSKA Sofia will play against the 2010–11 Bulgarian champions Litex Lovech for the Supercup.
Europa League
By winning in the 2010–11 Bulgarian Cup, CSKA Sofia qualified for the Europa League. They started in the play-off round.
Play-off round
UEFA Club Rankings
This is the current UEFA Club Rankings, including season 2010–11.
See also
PFC CSKA Sofia
References
External links
CSKA Official Site
CSKA Fan Page with up-to-date information
Bulgarian A Professional Football Group
UEFA Profile
PFC CSKA Sofia seasons
Cska Sofia
CSKA Sofia |
https://en.wikipedia.org/wiki/Motivic%20L-function | In mathematics, motivic L-functions are a generalization of Hasse–Weil L-functions to general motives over global fields. The local L-factor at a finite place v is similarly given by the characteristic polynomial of a Frobenius element at v acting on the v-inertial invariants of the v-adic realization of the motive. For infinite places, Jean-Pierre Serre gave a recipe in for the so-called Gamma factors in terms of the Hodge realization of the motive. It is conjectured that, like other L-functions, that each motivic L-function can be analytically continued to a meromorphic function on the entire complex plane and satisfies a functional equation relating the L-function L(s, M) of a motive M to , where M∨ is the dual of the motive M.
Examples
Basic examples include Artin L-functions and Hasse–Weil L-functions. It is also known , for example, that a motive can be attached to a newform (i.e. a primitive cusp form), hence their L-functions are motivic.
Conjectures
Several conjectures exist concerning motivic L-functions. It is believed that motivic L-functions should all arise as automorphic L-functions, and hence should be part of the Selberg class. There are also conjectures concerning the values of these L-functions at integers generalizing those known for the Riemann zeta function, such as Deligne's conjecture on special values of L-functions, the Beilinson conjecture, and the Bloch–Kato conjecture (on special values of L-functions).
Notes
References
alternate URL
Zeta and L-functions
Algebraic geometry |
https://en.wikipedia.org/wiki/Iwasawa%20algebra | In mathematics, the Iwasawa algebra Λ(G) of a profinite group G is a variation of the group ring of G with p-adic coefficients that take the topology of G into account. More precisely, Λ(G) is the inverse limit of the group rings Zp(G/H) as H runs through the open normal subgroups of G. Commutative Iwasawa algebras were introduced by in his study of Zp extensions in Iwasawa theory, and non-commutative Iwasawa algebras of compact p-adic analytic groups were introduced by .
Iwasawa algebra of the p-adic integers
In the special case when the profinite group G is isomorphic to the additive group of the ring of p-adic integers Zp, the Iwasawa algebra Λ(G) is isomorphic to the ring of the formal power series Zp[[T]] in one variable over Zp. The isomorphism is given by identifying 1 + T with a topological generator of G. This ring is a 2-dimensional complete Noetherian regular local ring, and in particular a unique factorization domain.
It follows from the Weierstrass preparation theorem for formal power series over a complete local ring that the prime ideals of this ring are as follows:
Height 0: the zero ideal.
Height 1: the ideal (p), and the ideals generated by irreducible distinguished polynomials (polynomials with leading coefficient 1 and all other coefficients divisible by p).
Height 2: the maximal ideal (p,T).
Finitely generated modules
The rank of a finitely generated module is the number of times the module Zp[[T]] occurs in it. This is well-defined and is additive for short exact sequences of finitely-generated modules. The rank of a finitely generated module is zero if and only if the module is a torsion module, which happens if and only if the support has dimension at most 1.
Many of the modules over this algebra that occur in Iwasawa theory are finitely generated torsion modules. The structure of such modules can be described as follows. A quasi-isomorphism of modules is a homomorphism whose kernel and cokernel are both finite groups, in other words modules with support either empty or the height 2 prime ideal. For any finitely generated torsion module there is a quasi-isomorphism to a finite sum of modules of the form Zp[[T]]/(fn) where f
is a generator of a height 1 prime ideal. Moreover, the number of times any module Zp[[T]]/(f) occurs in the module is well defined and independent of the composition series. The torsion module therefore has a characteristic power series, a formal power series given by the product of the power series fn, that is uniquely defined up to multiplication by a unit. The ideal generated by the characteristic power series is called the characteristic ideal of the Iwasawa module. More generally, any generator of the characteristic ideal is called a characteristic power series.
The μ-invariant of a finitely-generated torsion module is the number of times the module Zp[[T]]/(p) occurs in it. This invariant is additive on short exact sequences of finitely generated torsion modules (though it is not addit |
https://en.wikipedia.org/wiki/6-simplex%20honeycomb | In six-dimensional Euclidean geometry, the 6-simplex honeycomb is a space-filling tessellation (or honeycomb). The tessellation fills space by 6-simplex, rectified 6-simplex, and birectified 6-simplex facets. These facet types occur in proportions of 1:1:1 respectively in the whole honeycomb.
A6 lattice
This vertex arrangement is called the A6 lattice or 6-simplex lattice. The 42 vertices of the expanded 6-simplex vertex figure represent the 42 roots of the Coxeter group. It is the 6-dimensional case of a simplectic honeycomb. Around each vertex figure are 126 facets: 7+7 6-simplex, 21+21 rectified 6-simplex, 35+35 birectified 6-simplex, with the count distribution from the 8th row of Pascal's triangle.
The A lattice (also called A) is the union of seven A6 lattices, and has the vertex arrangement of the dual to the omnitruncated 6-simplex honeycomb, and therefore the Voronoi cell of this lattice is the omnitruncated 6-simplex.
∪
∪
∪
∪
∪
∪
= dual of
Related polytopes and honeycombs
Projection by folding
The 6-simplex honeycomb can be projected into the 3-dimensional cubic honeycomb by a geometric folding operation that maps two pairs of mirrors into each other, sharing the same vertex arrangement:
See also
Regular and uniform honeycombs in 6-space:
6-cubic honeycomb
6-demicubic honeycomb
Truncated 6-simplex honeycomb
Omnitruncated 6-simplex honeycomb
222 honeycomb
Notes
References
Norman Johnson Uniform Polytopes, Manuscript (1991)
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] (1.9 Uniform space-fillings)
(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
Honeycombs (geometry)
7-polytopes |
https://en.wikipedia.org/wiki/Wolff-Michael%20Roth | Wolff-Michael Roth (born June 28, 1953, Heidelberg) is a learning scientist at the University of Victoria conducting research on how people across the life span know and learn mathematics and science. He has contributed to numerous fields of research: learning science in learning communities, coteaching, authentic school science education, cultural-historical activity theory, social studies of science, gesture studies, qualitative research methods, embodied cognition, situated cognition, and the role of language in learning science and mathematics.
Career
Roth received a master's degree of physics from the University of Würzburg and completed a doctorate in the College of Science and Technology at the University of Southern Mississippi (Hattiesburg, Mississippi) with concentrations in cognition, statistics, and physical chemistry. He began to establish himself as a researcher while teaching at Appleby College (Oakville, Ontario). There, he did the research for what became one of the first research articles on the social construction of knowledge in science classrooms: "The social construction of scientific concepts or The concept map as conscription device and tool for social thinking in high school science." There he also conducted the work that would lead to Authentic School Science, in which he provides evidence for how high school students learn science when provided with opportunities to frame their own research questions, which they then answer by designing and conducting experiments. They write up their results, which they have to be able to defend within their learning community. In 1992, he joined Simon Fraser University (Burnaby, British Columbia), where he was mainly responsible for teaching statistics in the Faculty of Education. In his research, he initially focused on learning in science classrooms, but soon expanded his work to learning mathematics and science among future teachers, research scientists, and designers of computer software. In 1997, he was appointed Lansdowne Professor of Applied Cognitive Science at the University of Victoria. There he further expanded his research on the learning of mathematics and science, for example, on the use of graphs in scientific research and in technical professions.
Main Contributions
Although he worked within a neo-Piagetian (information processing oriented) paradigm during his doctoral research, using statistical methods, his subsequent work was initially based in school science classrooms and later extended to mathematics and science in fish hatcheries, environmental activism, field ecology, scientific laboratories, dental practice, water technicians, construction sites, and in local communities.
Graphing as Social Practice
Psychologist tend to theorize graphing, as all other forms of representing activity, as a faculty of the mind. Based on his ethnographic studies of mathematics among scientists, Roth proposes to view graphing as a social practice that humans learn in relation wit |
https://en.wikipedia.org/wiki/Teichm%C3%BCller%20character | In number theory, the Teichmüller character ω (at a prime p) is a character of (Z/qZ)×, where if is odd and if , taking values in the roots of unity of the p-adic integers. It was introduced by Oswald Teichmüller. Identifying the roots of unity in the p-adic integers with the corresponding ones in the complex numbers, ω can be considered as a usual Dirichlet character of conductor q. More generally, given a complete discrete valuation ring O whose residue field k is perfect of characteristic p, there is a unique multiplicative section of the natural surjection . The image of an element under this map is called its Teichmüller representative. The restriction of ω to k× is called the Teichmüller character.
Definition
If x is a p-adic integer, then is the unique solution of that is congruent to x mod p. It can also be defined by
The multiplicative group of p-adic units is a product of the finite group of roots of unity and a group isomorphic to the p-adic integers. The finite group is cyclic of order p – 1 or 2, as p is odd or even, respectively, and so it is isomorphic to (Z/qZ)×. The Teichmüller character gives a canonical isomorphism between these two groups.
A detailed exposition of the construction of Teichmüller representatives for the p-adic integers, by means of Hensel lifting, is given in the article on Witt vectors, where they provide an important role in providing a ring structure.
See also
Witt vector
References
Section 4.3 of
Class field theory |
https://en.wikipedia.org/wiki/Hopf%20lemma | In mathematics, the Hopf lemma, named after Eberhard Hopf, states that if a continuous real-valued function in a domain in Euclidean space with sufficiently smooth boundary is harmonic in the interior and the value of the function at a point on the boundary is greater than the values at nearby points inside the domain, then the derivative of the function in the direction of the outward pointing normal is strictly positive. The lemma is an important tool in the proof of the maximum principle and in the theory of partial differential equations. The Hopf lemma has been generalized to describe the behavior of the solution to an elliptic problem as it approaches a point on the boundary where its maximum is attained.
In the special case of the Laplacian, the Hopf lemma had been discovered by Stanisław Zaremba in 1910. In the more general setting for elliptic equations, it was found independently by Hopf and Olga Oleinik in 1952, although Oleinik's work is not as widely known as Hopf's in Western countries. There are also extensions which allow domains with corners.
Statement for harmonic functions
Let Ω be a bounded domain in Rn with smooth boundary. Let f be a real-valued function continuous on the closure of Ω and harmonic on Ω. If x is a boundary point such that f(x) > f(y) for all y in Ω sufficiently close to x, then the (one-sided) directional derivative of f in the direction of the outward pointing normal to the boundary at x is strictly positive.
Proof for harmonic functions
Subtracting a constant, it can be assumed that f(x) = 0 and f is strictly negative at interior points near x. Since the boundary of Ω is smooth there is a small ball contained in Ω the closure of which is tangent to the boundary at x and intersects the boundary only at x. It is then sufficient to check the result with Ω replaced by this ball. Scaling and translating, it is enough to check the result for the unit ball in Rn, assuming f(x) is zero for some unit vector x and f(y) < 0 if |y| < 1.
By Harnack's inequality applied to −f
for r < 1. Hence
Hence the directional derivative at x is bounded below by the strictly positive constant on the right hand side.
General discussion
Consider a second order, uniformly elliptic operator of the form
Here is an open, bounded subset of .
The Weak Maximum Principle states that a solution of the equation in attains its maximum value on the closure at some point on the boundary . Let be such a point, then necessarily
where denotes the outer normal derivative. This is simply a consequence of the fact that must be nondecreasing as approach . The Hopf Lemma strengthens this observation by proving that, under mild assumptions on and , we have
A precise statement of the Lemma is as follows. Suppose that is a bounded region in and let be the operator described above. Let be of class and satisfy the differential inequality
Let be given so that .
If (i) is at , and (ii) , then either is a constant, or , where |
https://en.wikipedia.org/wiki/Rectified%208-cubes | In eight-dimensional geometry, a rectified 8-cube is a convex uniform 8-polytope, being a rectification of the regular 8-cube.
There are unique 8 degrees of rectifications, the zeroth being the 8-cube, and the 7th and last being the 8-orthoplex. Vertices of the rectified 8-cube are located at the edge-centers of the 8-cube. Vertices of the birectified 8-cube are located in the square face centers of the 8-cube. Vertices of the trirectified 8-cube are located in the 7-cube cell centers of the 8-cube.
Rectified 8-cube
Alternate names
rectified octeract
Images
Birectified 8-cube
Alternate names
Birectified octeract
Rectified 8-demicube
Images
Trirectified 8-cube
Alternate names
trirectified octeract
Images
Notes
References
H.S.M. Coxeter:
H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
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]
Norman Johnson Uniform Polytopes, Manuscript (1991)
N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D.
o3o3o3o3o3o3x4o, o3o3o3o3o3x3o4o, o3o3o3o3x3o3o4o
External links
Polytopes of Various Dimensions
Multi-dimensional Glossary
8-polytopes |
https://en.wikipedia.org/wiki/Eisenstein%E2%80%93Kronecker%20number | In mathematics, Eisenstein–Kronecker numbers are an analogue for imaginary quadratic fields of generalized Bernoulli numbers. They are defined in terms of classical Eisenstein–Kronecker series, which were studied by Kenichi Bannai and Shinichi Kobayashi using the Poincaré bundle.
Eisenstein–Kronecker numbers are algebraic and satisfy congruences that can be used in the construction of two-variable p-adic L-functions. They are related to critical L-values of Hecke characters.
Definition
When is the area of the fundamental domain of divided by , where is a lattice in :
when
where and is the complex conjugate of .
References
Number theory |
https://en.wikipedia.org/wiki/Kronecker%20coefficient | In mathematics, Kronecker coefficients gλμν describe the decomposition of the tensor product (= Kronecker product) of two irreducible representations of a symmetric group into irreducible representations. They play an important role algebraic combinatorics and geometric complexity theory. They were introduced by Murnaghan in 1938.
Definition
Given a partition λ of n, write Vλ for the Specht module associated to λ. Then the Kronecker coefficients gλμν are given by the rule
One can interpret this on the level of symmetric functions, giving a formula for the Kronecker product of two Schur polynomials:
This is to be compared with Littlewood–Richardson coefficients, where one instead considers the induced representation
and the corresponding operation of symmetric functions is the usual product. Also note that the Littlewood–Richardson coefficients are the analogue of the Kronecker coefficients for representations of GLn, i.e. if we write Wλ for the irreducible representation corresponding to λ (where λ has at most n parts), one gets that
Properties
showed that computing Kronecker coefficients is #P-hard and contained in GapP. A recent work by shows that deciding whether a given Kronecker coefficient is non-zero is NP-hard. This recent interest in computational complexity of these coefficients arises from its relevance in the Geometric Complexity Theory program.
A major unsolved problem in representation theory and combinatorics is to give a combinatorial description of the Kronecker coefficients. It has been open since 1938, when Murnaghan asked for such a combinatorial description. A combinatorial description would also imply that the problem is # P-complete in light of the above result.
The Kronecker coefficients can be computed as
where is the character value of the irreducible representation corresponding to partition on a permutation .
The Kronecker coefficients also appear in the generalized Cauchy identity
See also
Littlewood–Richardson coefficient
References
Algebraic combinatorics
Representation theory
Symmetric functions |
https://en.wikipedia.org/wiki/1511%20in%20science | The year 1511 in science and technology included a number of events, some of which are listed here.
Cartography
A form of the Bonne projection is used by Sylvano.
Mathematics
Charles de Bovelles publishes Géométrie en françoys, the first scientific work printed in French.
Births
September 29(?) – Michael Servetus, Aragonese polymath (died 1553)
October 22 – Erasmus Reinhold, German astronomer and mathematician (born 1511)
Deaths
Matthias Ringmann, German geographer (born 1482)
References
16th century in science
1510s in science |
https://en.wikipedia.org/wiki/Brjuno%20number | In mathematics, a Brjuno number (sometimes spelled Bruno or Bryuno) is a special type of irrational number named for Russian mathematician Alexander Bruno, who introduced them in .
Formal definition
An irrational number is called a Brjuno number when the infinite sum
converges to a finite number.
Here:
is the denominator of the th convergent of the continued fraction expansion of .
is a Brjuno function
Importance
The Brjuno numbers are important in the one–dimensional analytic small divisors problems. Bruno improved the diophantine condition in Siegel's Theorem, showed that germs of holomorphic functions with linear part are linearizable if is a Brjuno number. showed in 1987 that this condition is also necessary, and for quadratic polynomials is necessary and sufficient.
Properties
Intuitively, these numbers do not have many large "jumps" in the sequence of convergents, in which the denominator of the ()th convergent is exponentially larger than that of the th convergent. Thus, in contrast to the Liouville numbers, they do not have unusually accurate diophantine approximations by rational numbers.
Brjuno function
Brjuno sum
The Brjuno sum or Brjuno function is
where:
is the denominator of the th convergent of the continued fraction expansion of .
Real variant
The real Brjuno function is defined for irrational numbers
and satisfies
for all irrational between 0 and 1.
Yoccoz's variant
Yoccoz's variant of the Brjuno sum defined as follows:
where:
is irrational real number:
is the fractional part of
is the fractional part of
This sum converges if and only if the Brjuno sum does, and in fact their difference is bounded by a universal constant.
See also
Transcendental number
References
Notes
Dynamical systems
Number theory |
https://en.wikipedia.org/wiki/Palm%20calculus | In the study of stochastic processes, Palm calculus, named after Swedish teletrafficist Conny Palm, is the study of the relationship between probabilities conditioned on a specified event and time-average probabilities. A Palm probability or Palm expectation, often denoted or , is a probability or expectation conditioned on a specified event occurring at time 0.
Little's formula
A simple example of a formula from Palm calculus is Little's law , which states that the time-average number of users (L) in a system is equal to the product of the rate () at which users arrive and the Palm-average waiting time (W) that a user spends in the system. That is, the average W gives equal weight to the waiting time of all customers, rather than being the time-average of "the waiting times of the customers currently in the system".
Feller's paradox
An important example of the use of Palm probabilities is Feller's paradox, often associated with the analysis of an M/G/1 queue. This states that the (time-)average time between the previous and next points in a point process is greater than the expected interval between points. The latter is the Palm expectation of the former, conditioning on the event that a point occurs at the time of the observation. This paradox occurs because large intervals are given greater weight in the time average than small intervals.
References
Palm, C. (1943) "Intensitätsschwankungen im Fernsprechverkehr" Ericsson Techniks, No. 44
Queueing theory
Stochastic calculus
Telecommunication theory |
https://en.wikipedia.org/wiki/Arithmetic%20surface | In mathematics, an arithmetic surface over a Dedekind domain R with fraction field is a geometric object having one conventional dimension, and one other dimension provided by the infinitude of the primes. When R is the ring of integers Z, this intuition depends on the prime ideal spectrum Spec(Z) being seen as analogous to a line. Arithmetic surfaces arise naturally in diophantine geometry, when an algebraic curve defined over K is thought of as having reductions over the fields R/P, where P is a prime ideal of R, for almost all P; and are helpful in specifying what should happen about the process of reducing to R/P when the most naive way fails to make sense.
Such an object can be defined more formally as an R-scheme with a non-singular, connected projective curve for a generic fiber and unions of curves (possibly reducible, singular, non-reduced ) over the appropriate residue field for special fibers.
Formal definition
In more detail, an arithmetic surface (over the Dedekind domain ) is a scheme with a morphism with the following properties: is integral, normal, excellent, flat and of finite type over and the generic fiber is a non-singular, connected projective curve over and for other in ,
is a union of curves over .
Over a Dedekind scheme
In even more generality, arithmetic surfaces can be defined over Dedekind schemes, a typical example of which is the spectrum of the ring of integers of a number field (which is the case above). An arithmetic surface is then a regular fibered surface over a Dedekind scheme of dimension one. This generalisation is useful, for example, it allows for base curves which are smooth and projective over finite fields, which is important in positive characteristic.
What makes them "arithmetic"?
Arithmetic surfaces over Dedekind domains are the arithmetic analogue of fibered surfaces over algebraic curves. Arithmetic surfaces arise primarily in the context of number theory. In fact, given a curve over a number field , there exists an arithmetic surface over the ring of integers whose generic fiber is isomorphic to . In higher dimensions one may also consider arithmetic schemes.
Properties
Dimension
Arithmetic surfaces have dimension 2 and relative dimension 1 over their base.
Divisors
We can develop a theory of Weil divisors on arithmetic surfaces since every local ring of dimension one is regular. This is briefly stated as "arithmetic surfaces are regular in codimension one." The theory is developed in Hartshorne's Algebraic Geometry, for example.
Examples
Projective line
The projective line over Dedekind domain is a smooth, proper arithmetic surface over . The fiber over any maximal ideal is the projective line over the field
Regular minimal models
Néron models for elliptic curves, initially defined over a global field, are examples of this construction, and are much studied examples of arithmetic surfaces. There are strong analogies with elliptic fibrations.
Intersection theory
Given |
https://en.wikipedia.org/wiki/Cantellated%206-cubes | In six-dimensional geometry, a cantellated 6-cube is a convex uniform 6-polytope, being a cantellation of the regular 6-cube.
There are 8 cantellations for the 6-cube, including truncations. Half of them are more easily constructed from the dual 5-orthoplex.
Cantellated 6-cube
Alternate names
Cantellated hexeract
Small rhombated hexeract (acronym: srox) (Jonathan Bowers)
Images
Bicantellated 6-cube
Alternate names
Bicantellated hexeract
Small birhombated hexeract (acronym: saborx) (Jonathan Bowers)
Images
Cantitruncated 6-cube
Alternate names
Cantitruncated hexeract
Great rhombihexeract (acronym: grox) (Jonathan Bowers)
Images
It is fourth in a series of cantitruncated hypercubes:
Bicantitruncated 6-cube
Alternate names
Bicantitruncated hexeract
Great birhombihexeract (acronym: gaborx) (Jonathan Bowers)
Images
Related polytopes
These polytopes are part of a set of 63 uniform 6-polytopes generated from the B6 Coxeter plane, including the regular 6-cube or 6-orthoplex.
Notes
References
H.S.M. Coxeter:
H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
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]
Norman Johnson Uniform Polytopes, Manuscript (1991)
N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D.
o3o3o3x3o4x - srox, o3o3x3o3x4o - saborx, o3o3o3x3x4x - grox, o3o3x3x3x4o - gaborx
External links
Polytopes of Various Dimensions
Multi-dimensional Glossary
6-polytopes |
https://en.wikipedia.org/wiki/File%20dynamics | The term file dynamics is the motion of many particles in a narrow channel.
In science: in chemistry, physics, mathematics and related fields, file dynamics (sometimes called, single file dynamics) is the diffusion of N (N → ∞) identical Brownian hard spheres in a quasi-one-dimensional channel of length L (L → ∞), such that the spheres do not jump one on top of the other, and the average particle's density is approximately fixed. The most famous statistical properties of this process is that the mean squared displacement (MSD) of a particle in the file follows, , and its probability density function (PDF) is Gaussian in position with a variance MSD.
Results in files that generalize the basic file include:
In files with a density law that is not fixed, but decays as a power law with an exponent a with the distance from the origin, the particle in the origin has a MSD that scales like, , with a Gaussian PDF.
When, in addition, the particles' diffusion coefficients are distributed like a power law with exponent γ (around the origin), the MSD follows, , with a Gaussian PDF.
In anomalous files that are renewal, namely, when all particles attempt a jump together, yet, with jumping times taken from a distribution that decays as a power law with an exponent, −1 − α, the MSD scales like the MSD of the corresponding normal file, in the power of α.
In anomalous files of independent particles, the MSD is very slow and scales like, . Even more exciting, the particles form clusters in such files, defining a dynamical phase transition. This depends on the anomaly power α: the percentage of particles in clusters ξ follows, .
Other generalizations include: when the particles can bypass each other with a constant probability upon encounter, an enhanced diffusion is seen. When the particles interact with the channel, a slower diffusion is observed. Files in embedded in two-dimensions show similar characteristics of files in one dimension.
Generalizations of the basic file are important since these models represent reality much more accurately than the basic file. Indeed, file dynamics are used in modeling numerous microscopic processes: the diffusion within biological and synthetic pores and porous material, the diffusion along 1D objects, such as in biological roads, the dynamics of a monomer in a polymer, etc.
Mathematical formulation
Simple files
In simple Brownian files, , the joint probability density function (PDF) for all the particles in file, obeys a normal diffusion equation:
In , is the set of particles' positions at time and is the set of the particles' initial positions at the initial time (set to zero). Equation (1) is solved with the appropriate boundary conditions, which reflect the hard-sphere nature of the file:
and with the appropriate initial condition:
In a simple file, the initial density is fixed, namely,, where is a parameter that represents a microscopic length. The PDFs' coordinates must |
https://en.wikipedia.org/wiki/Georg%20Scheffers | Georg Scheffers (21 November 1866 – 12 August 1945) was a German mathematician specializing in differential geometry.
Life
Scheffers was born on 21 November 1866 in the village of Altendorf near Holzminden (today incorporated into Holzminden). Scheffers began his university career at the University of Leipzig where he studied with Felix Klein and Sophus Lie. Scheffers was a coauthor with Lie for three of the earliest expressions of Lie theory:
Lectures on Differential equations with known Infinitesimal transformations (1893),
Lectures on Continuous groups (1893), and
Geometry of Contact Transformations (1896).
All three are now available online through archive.org.
In 1896 Scheffers became docent at the Technical University of Darmstadt, where he was raised to professor in 1900. From 1907 to 1935, when he retired, Scheffers was a professor at the Technical University of Berlin.
In 1901–1902 he published a famous two-volume textbook entitled Anwendung der Differential- und Integralrechnung auf die Geometrie (application of differential and integral calculus to geometry). The first volume subtitled Einführung in die Theorie der Curven in der Ebene und in Raum was published in 1901 and dealt with curves. The second volume subtitled Einführung in die Theorie der Flächen (introduction to the theory of surfaces) was published in 1902. A second edition was published in 1910 (volume 2, 1913), and a third edition in 1922.
In 1907 Scheffers published the first two volumes of his extensive revision and rewriting of Georg Bohlmann's 1897–1899 revision of Harnack's 1884 German translation of Serret's famous two-volume Cours de calcul différentiel et intégral, which was first published by Gauthier-Villars in 1868. In 1909 Scheffers published the third and last volume of his rewriting of Bohlman's version of Serret's two-volume work. For a new edition, Scheffers added an appendix with 46 pages of historical notes for the first and second volumes.
Another very successful book was prepared for students of science and technology: Lehrbuch der Mathematik (textbook of mathematics). It provided an introduction to analytic geometry as well as calculus of derivatives and integrals. In 1958 this book was republished for the fourteenth time.
Scheffers is known for an article on special transcendental curves (including W-curves) which appeared in the Enzyklopädie der mathematischen Wissenschaften in 1903: "Besondere transzendenten Kurven" (special transcendental curves). He wrote on translation surfaces for Acta Mathematica in 1904: "Das Abel'sche und das Lie'sche Theorem über Translationsflächen" (the theorem of Abel and Lie on translation surfaces).
Other books written by Scheffers are Lehrbuch der darstellenden Geometrie (textbook on descriptive geometry) (1919), Allerhand aus der zeichnenden Geometrie (1930), and Wie findet und zeichnet man Gradnetze von Land- und Sternkarten? (1934).
Georg Scheffers died 12 August 1945, in Berlin.
Hypercomplex numbers
|
https://en.wikipedia.org/wiki/2008%E2%80%9309%20Crystal%20Palace%20F.C.%20season | The 2008–09 season was Crystal Palace Football Club's 4th consecutive season in the Championship, after their play-off defeat in the previous campaign.
Statistics
Last updated on 10 January 2010.
|}
Club
Management
League table
Matches
Preseason
Errea South West Challenge Cup
Group phase
Crystal Palace finished second in their group with three points and thus qualified for the semifinal stage. However, with the squad heavily depleted through injury, Palace opted to give their semifinal place to third-placed team Torquay United
Football League Championship
Football League Cup
Round 1
Round 2
FA Cup
Round 3
Round 4
End-of-season awards
References
Notes
External links
Crystal Palace F.C. official website
Crystal Palace F.C. on Soccerbase
Crystal Palace F.C. Reserve site
Crystal Palace F.C. seasons
Crystal Palace |
https://en.wikipedia.org/wiki/Hall%20word | In mathematics, in the areas of group theory and combinatorics, Hall words provide a unique monoid factorisation of the free monoid. They are also totally ordered, and thus provide a total order on the monoid. This is analogous to the better-known case of Lyndon words; in fact, the Lyndon words are a special case, and almost all properties possessed by Lyndon words carry over to Hall words. Hall words are in one-to-one correspondence with Hall trees. These are binary trees; taken together, they form the Hall set. This set is a particular totally ordered subset of a free non-associative algebra, that is, a free magma. In this form, the Hall trees provide a basis for free Lie algebras, and can be used to perform the commutations required by the Poincaré–Birkhoff–Witt theorem used in the construction of a universal enveloping algebra. As such, this generalizes the same process when done with the Lyndon words. Hall trees can also be used to give a total order to the elements of a group, via the commutator collecting process, which is a special case of the general construction given below. It can be shown that Lazard sets coincide with Hall sets.
The historical development runs in reverse order from the above description. The commutator collecting process was described first, in 1934, by Philip Hall and explored in 1937 by Wilhelm Magnus. Hall sets were introduced by Marshall Hall based on work of Philip Hall on groups.
Subsequently, Wilhelm Magnus showed that they arise as the graded Lie algebra associated with the filtration on a free group given by the lower central series. This correspondence was motivated by commutator identities in group theory due to Philip Hall and Ernst Witt.
Hall set
The Hall set is a totally ordered subset of a free non-associative algebra, that is, a free magma. Let be a set of generators, and let be the free magma over . The free magma is simply the set of non-associative strings in the letters of , with parenthesis retained to show grouping. Parenthesis may be written with square brackets, so that elements of the free magma may be viewed as formal commutators. Equivalently, the free magma is the set of all binary trees with leaves marked by elements of .
The Hall set can be constructed recursively (in increasing order) as follows:
The elements of are given an arbitrary total order.
The Hall set contains the generators:
A formal commutator if and only if and and and:
Either (and thus ),
Or with and and .
The commutators can be ordered arbitrarily, provided that always holds.
The construction and notation used below are nearly identical to that used in the commutator collecting process, and so can be directly compared; the weights are the string-lengths. The difference is that no mention of groups is required. These definitions all coincide with that of X. Viennot.
Note that some authors reverse the order of the inequality. Note also that one of the conditions, the , may feel "backwards"; this |
https://en.wikipedia.org/wiki/Onsager%20Medal | The Onsager Medal (Onsagermedaljen) is a scholastic presentation awarded to researchers in one or more subject areas of chemistry, physics or mathematics. The medal is awarded in memory of Lars Onsager, who received Nobel Prize in Chemistry in 1968. The medal, designed by Harald Wårvik, commemorates the efforts of a single individual as chosen by the Onsager committee at the Norwegian University of Science and Technology (NTNU).
The professorship awardee is expected to spend 3–6 months working at NTNU. The lectureship awardee will give a lecture at the university.
Professorship
1993 George Stell, State University of New York, Stony Brook (statistical physics)
1994 Vladisav Borisovic Lazarev, Kurnakov Institute of General and Inorganic Chemistry, Moscow (chemistry)
1995 F. W. Gehring, University of Michigan (mathematics)
1996 J. M. J. van Leeuwen, Leiden University(statistical physics)
1997 Dick Bedeaux, Leiden University (physical chemistry)
1998 V. S. Varadarajan, University of California, Los Angeles (mathematics)
1999 Arieh Iserles, University of Cambridge (mathematics)
2000 V. Havin, St Petersburg (mathematics)
2001 David A. Brant, University of California, Irvine (chemistry)
2002 John S. Newman, Department of Chemical Engineering, University of California (chemical engineering)
2003 Miguel Rubí Capaceti, Facultat de Fisica, Departament Fisica Fonamental, Universitat de Barcelona, Spain (chemistry)
2004 George Batrouni, University of Nice Sophia Antipolis, France (physics)
2005 Alexander Volberg, Michigan State University (mathematics)
2006 John R. Klauder, Departments of Physics and Mathematics, University of Florida (physics and mathematics)
2007 Matthieu H. Ernst, Institute for Theoretical Physics, Utrecht University, The Netherlands (physics)
2008 Peter S. Riseborough, Department of Physics, Temple University, Philadelphia, USA (physics)
2009 Gerrit Ernst-Wilhelm Bauer, Kavli Institute of NanoScience, Delft University of Technology, The Netherlands (physics)
2010 Elisabeth Bouchaud, Head of the Division of Physics and Chemistry of Surfaces and Interfaces at CEA-SACLAY, Gif-sur-Yvette, France (physics)
2011 George W. Scherer, W. L. Knapp Professor of Civil & Environmental Engineering, Princeton University
2012 Richard Spontak, Department of Materials Science & Engineering, and Department of Chemical & Biomolecular Engineering, North Carolina State University
2013 Reinout Quispel, La Trobe University, Melbourne, Australia
2014 Xiang-Yu Zhou, Chinese Academy of Sciences, Beijing, China
2015 Matthias Eschrig, Royal Holloway, University of London, UK
2016 Jan Vermant, ETH Zürich, Switzerland
2017 Jian-Min Zuo, University of Illinois at Urbana-Champaign, United States.
2018 Eero Saksman, University of Helsinki, Finland
2019 Daan Frenkel, University of Cambridge
2020 Lorenz T. Biegler, Carnegie Mellon University, United States
Lectureship
1993 M. E. Fisher, University of Maryland (statistical physics)
1994 B |
https://en.wikipedia.org/wiki/Cyclic%20category | In mathematics, the cyclic category or cycle category or category of cycles is a category of finite cyclically ordered sets and degree-1 maps between them. It was introduced by .
Definition
The cyclic category Λ has one object Λn for each natural number n = 0, 1, 2, ...
The morphisms from Λm to Λn are represented by increasing functions f from the integers to the integers, such that f(x+m+1) = f(x)+n+1, where two functions f and g represent the same morphism when their difference is divisible by n+1.
Informally, the morphisms from Λm to Λn can be thought of as maps of (oriented)
necklaces with m+1 and n+1 beads. More precisely, the morphisms can be identified with homotopy classes of degree 1 increasing maps from S1 to itself that map the subgroup Z/(m+1)Z to Z/(n+1)Z.
Properties
The number of morphisms from Λm to Λn is (m+n+1)!/m!n!.
The cyclic category is self dual.
The classifying space BΛ of the cyclic category is a classifying space BS1of the circle group S1.
Cyclic sets
A cyclic set is a contravariant functor from the cyclic category to sets. More generally a cyclic object in a category C is a contravariant functor from the cyclic category to C.
See also
Cyclic homology
Simplex category
References
External links
Cycle category in nLab
Categories in category theory
Homology theory |
https://en.wikipedia.org/wiki/Langlands%E2%80%93Shahidi%20method | In mathematics, the Langlands–Shahidi method provides the means to define automorphic L-functions in many cases that arise with connected reductive groups over a number field. This includes Rankin–Selberg products for cuspidal automorphic representations of general linear groups. The method develops the theory of the local coefficient, which links to the global theory via Eisenstein series. The resulting L-functions satisfy a number of analytic properties, including an important functional equation.
The local coefficient
The setting is in the generality of a connected quasi-split reductive group G, together with a Levi subgroup M, defined over a local field F. For example, if G = Gl is a classical group of rank l, its maximal Levi subgroups are of the form GL(m) × Gn, where Gn is a classical group of rank n and of the same type as Gl, l = m + n. F. Shahidi develops the theory of the local coefficient for irreducible generic representations of M(F). The local coefficient is defined by means of the uniqueness property of Whittaker models paired with the theory of intertwining operators for representations obtained by parabolic induction from generic representations.
The global intertwining operator appearing in the functional equation of Langlands' theory of Eisenstein series can be decomposed as a product of local intertwining operators. When M is a maximal Levi subgroup, local coefficients arise from Fourier coefficients of appropriately chosen Eisenstein series and satisfy a crude functional equation involving a product of partial L-functions.
Local factors and functional equation
An induction step refines the crude functional equation of a globally generic cuspidal automorphic representation to individual functional equations of partial L-functions and γ-factors:
The details are technical: s a complex variable, S a finite set of places (of the underlying global field) with unramified for v outside of S, and is the adjoint action of M on the complex Lie algebra of a specific subgroup of the Langlands dual group of G. When G is the special linear group SL(2), and M = T is the maximal torus of diagonal matrices, then π is a Größencharakter and the corresponding γ-factors are the local factors of Tate's thesis.
The γ-factors are uniquely characterized by their role in the functional equation and a list of local properties, including multiplicativity with respect to parabolic induction. They satisfy a relationship involving Artin L-functions and Artin root numbers when v gives an archimedean local field or when v is non-archimedean and is a constituent of an unramified principal series representation of M(F). Local L-functions and root numbers ε are then defined at every place, including , by means of Langlands classification for p-adic groups. The functional equation takes the form
where and are the completed global L-function and root number.
Examples of automorphic L-functions
, the Rankin–Selberg L-function of cuspidal automorph |
https://en.wikipedia.org/wiki/Rectified%207-cubes | In seven-dimensional geometry, a rectified 7-cube is a convex uniform 7-polytope, being a rectification of the regular 7-cube.
There are unique 7 degrees of rectifications, the zeroth being the 7-cube, and the 6th and last being the 7-cube. Vertices of the rectified 7-cube are located at the edge-centers of the 7-ocube. Vertices of the birectified 7-cube are located in the square face centers of the 7-cube. Vertices of the trirectified 7-cube are located in the cube cell centers of the 7-cube.
Rectified 7-cube
Alternate names
rectified hepteract (Acronym rasa) (Jonathan Bowers)
Images
Cartesian coordinates
Cartesian coordinates for the vertices of a rectified 7-cube, centered at the origin, edge length are all permutations of:
(±1,±1,±1,±1,±1,±1,0)
Birectified 7-cube
Alternate names
Birectified hepteract (Acronym bersa) (Jonathan Bowers)
Images
Cartesian coordinates
Cartesian coordinates for the vertices of a birectified 7-cube, centered at the origin, edge length are all permutations of:
(±1,±1,±1,±1,±1,0,0)
Trirectified 7-cube
Alternate names
Trirectified hepteract
Trirectified 7-orthoplex
Trirectified heptacross (Acronym sez) (Jonathan Bowers)
Images
Cartesian coordinates
Cartesian coordinates for the vertices of a trirectified 7-cube, centered at the origin, edge length are all permutations of:
(±1,±1,±1,±1,0,0,0)
Related polytopes
Notes
References
H.S.M. Coxeter:
H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
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]
Norman Johnson Uniform Polytopes, Manuscript (1991)
N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D.
o3o3o3x3o3o4o - sez, o3o3o3o3x3o4o - bersa, o3o3o3o3o3x4o - rasa
External links
Polytopes of Various Dimensions
Multi-dimensional Glossary
7-polytopes |
https://en.wikipedia.org/wiki/Tsubasa%20Oshima | is a former Japanese football defender. He played in the J2 League for Fagiano Okayama in 2009.
Club statistics
References
External links
Matsumoto Yamaga FC
1983 births
Living people
University of Tsukuba alumni
Association football people from Saitama Prefecture
Japanese men's footballers
J1 League players
J2 League players
Japan Football League players
Ventforet Kofu players
Fagiano Okayama players
Matsumoto Yamaga FC players
Kamatamare Sanuki players
Men's association football defenders |
https://en.wikipedia.org/wiki/Keiichi%20Misawa | is a former Japanese football player.
Club statistics
References
External links
1988 births
Living people
Association football people from Nagano Prefecture
Japanese men's footballers
J1 League players
J2 League players
Vissel Kobe players
Thespakusatsu Gunma players
Men's association football defenders |
https://en.wikipedia.org/wiki/Ryo%20Kanazawa | is a former Japanese football player.
Club statistics
References
External links
JEF United Chiba
1988 births
Living people
Association football people from Kyoto Prefecture
Japanese men's footballers
J1 League players
J2 League players
JEF United Chiba players
Men's association football forwards |
https://en.wikipedia.org/wiki/Kenneth%20Davidson | Kenneth or Kenny Davidson may refer to:
Kenneth Davidson (mathematician), professor of pure mathematics at the University of Waterloo
Walter Davidson (Canadian politician) (Kenneth Walter Davidson, born 1937), former political figure in British Columbia, Canada
Kenneth Davidson (cricketer) (1905–1954), English cricketer
Kenneth S. M. Davidson, mechanical engineering professor
Kenny Davidson (American football) (born 1967), former American football defensive end
Kenny Davidson (Scottish footballer) (born 1952)
W. K. Davidson, known as Kenny, American restaurateur and Illinois politician |
https://en.wikipedia.org/wiki/Kang%20Hyun-su | Kang Hyun-su (born June 16, 1984) is a North Korean football player.
Club statistics
References
External links
Entry at j-league.or.jp
1984 births
Living people
Momoyama Gakuin University alumni
Association football people from Osaka
North Korean men's footballers
J2 League players
Japan Football League players
Kyoto Sanga FC players
Kataller Toyama players
SP Kyoto FC players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Chebyshev%27s%20bias | In number theory, Chebyshev's bias is the phenomenon that most of the time, there are more primes of the form 4k + 3 than of the form 4k + 1, up to the same limit. This phenomenon was first observed by Russian mathematician Pafnuty Chebyshev in 1853.
Description
Let (x; n, m) denote the number of primes of the form nk + m up to x. By the prime number theorem (extended to arithmetic progression),
That is, half of the primes are of the form 4k + 1, and half of the form 4k + 3. A reasonable guess would be that (x; 4, 1) > (x; 4, 3) and (x; 4, 1) < (x; 4, 3) each also occur 50% of the time. This, however, is not supported by numerical evidence — in fact, (x; 4, 3) > (x; 4, 1) occurs much more frequently. For example, this inequality holds for all primes x < 26833 except 5, 17, 41 and 461, for which (x; 4, 1) = (x; 4, 3). The first x such that (x; 4, 1) > (x; 4, 3) is 26861, that is, (x; 4, 3) ≥ (x; 4, 1) for all x < 26861.
In general, if 0 < a, b < n are integers, gcd(a, n) = gcd(b, n) = 1, a is a quadratic residue mod n, b is a quadratic nonresidue mod n, then (x; n, b) > (x; n, a) occurs more often than not. This has been proved only by assuming strong forms of the Riemann hypothesis. The stronger conjecture of Knapowski and Turán, that the density of the numbers x for which (x; 4, 3) > (x; 4, 1) holds is 1 (that is, it holds for almost all x), turned out to be false. They, however, do have a logarithmic density, which is approximately 0.9959....
Generalizations
This is for k = −4 to find the smallest prime p such that (where is the Kronecker symbol), however, for a given nonzero integer k (not only k = −4), we can also find the smallest prime p satisfying this condition. By the prime number theorem, for every nonzero integer k, there are infinitely many primes p satisfying this condition.
For positive integers k = 1, 2, 3, ..., the smallest primes p are
2, 11100143, 61981, 3, 2082927221, 5, 2, 11100143, 2, 3, 577, 61463, 2083, 11, 2, 3, 2, 11100121, 5, 2082927199, 1217, 3, 2, 5, 2, 17, 61981, 3, 719, 7, 2, 11100143, 2, 3, 23, 5, 11, 31, 2, 3, 2, 13, 17, 7, 2082927199, 3, 2, 61463, 2, 11100121, 7, 3, 17, 5, 2, 11, 2, 3, 31, 7, 5, 41, 2, 3, ... ( is a subsequence, for k = 1, 5, 8, 12, 13, 17, 21, 24, 28, 29, 33, 37, 40, 41, 44, 53, 56, 57, 60, 61, ... )
For negative integers k = −1, −2, −3, ..., the smallest primes p are
2, 3, 608981813029, 26861, 7, 5, 2, 3, 2, 11, 5, 608981813017, 19, 3, 2, 26861, 2, 643, 11, 3, 11, 31, 2, 5, 2, 3, 608981813029, 48731, 5, 13, 2, 3, 2, 7, 11, 5, 199, 3, 2, 11, 2, 29, 53, 3, 109, 41, 2, 608981813017, 2, 3, 13, 17, 23, 5, 2, 3, 2, 1019, 5, 263, 11, 3, 2, 26861, ... ( is a subsequence, for k = −3, −4, −7, −8, −11, −15, −19, −20, −23, −24, −31, −35, −39, −40, −43, −47, −51, −52, −55, −56, −59, ... )
For every (positive or negative) nonsquare integer k, there are more primes p with than with (up to the same limit) more often than not.
Extension to higher power residue
Let m and n be integers such that m |
https://en.wikipedia.org/wiki/Central%20limit%20theorem%20for%20directional%20statistics | In probability theory, the central limit theorem states conditions under which the average of a sufficiently large number of independent random variables, each with finite mean and variance, will be approximately normally distributed.
Directional statistics is the subdiscipline of statistics that deals with directions (unit vectors in Rn), axes (lines through the origin in Rn) or rotations in Rn. The means and variances of directional quantities are all finite, so that the central limit theorem may be applied to the particular case of directional statistics.
This article will deal only with unit vectors in 2-dimensional space (R2) but the method described can be extended to the general case.
The central limit theorem
A sample of angles are measured, and since they are indefinite to within a factor of , the complex definite quantity is used as the random variate. The probability distribution from which the sample is drawn may be characterized by its moments, which may be expressed in Cartesian and polar form:
It follows that:
Sample moments for N trials are:
where
The vector [] may be used as a representation of the sample mean and may be taken as a 2-dimensional random variate. The bivariate central limit theorem states that the joint probability distribution for and in the limit of a large number of samples is given by:
where is the bivariate normal distribution and is the covariance matrix for the circular distribution:
Note that the bivariate normal distribution is defined over the entire plane, while the mean is confined to be in the unit ball (on or inside the unit circle). This means that the integral of the limiting (bivariate normal) distribution over the unit ball will not be equal to unity, but rather approach unity as N approaches infinity.
It is desired to state the limiting bivariate distribution in terms of the moments of the distribution.
Covariance matrix in terms of moments
Using multiple angle trigonometric identities
It follows that:
The covariance matrix is now expressed in terms of the moments of the circular distribution.
The central limit theorem may also be expressed in terms of the polar components of the mean. If is the probability of finding the mean in area element , then that probability may also be written .
References
Directional statistics
Asymptotic theory (statistics) |
https://en.wikipedia.org/wiki/Truncated%207-cubes | In seven-dimensional geometry, a truncated 7-cube is a convex uniform 7-polytope, being a truncation of the regular 7-cube.
There are 6 truncations for the 7-cube. Vertices of the truncated 7-cube are located as pairs on the edge of the 7-cube. Vertices of the bitruncated 7-cube are located on the square faces of the 7-cube. Vertices of the tritruncated 7-cube are located inside the cubic cells of the 7-cube. The final three truncations are best expressed relative to the 7-orthoplex.
Truncated 7-cube
Alternate names
Truncated hepteract (Jonathan Bowers)
Coordinates
Cartesian coordinates for the vertices of a truncated 7-cube, centered at the origin, are all sign and coordinate permutations of
(1,1+√2,1+√2,1+√2,1+√2,1+√2,1+√2)
Images
Related polytopes
The truncated 7-cube, is sixth in a sequence of truncated hypercubes:
Bitruncated 7-cube
Alternate names
Bitruncated hepteract (Jonathan Bowers)
Coordinates
Cartesian coordinates for the vertices of a bitruncated 7-cube, centered at the origin, are all sign and coordinate permutations of
(±2,±2,±2,±2,±2,±1,0)
Images
Related polytopes
The bitruncated 7-cube is fifth in a sequence of bitruncated hypercubes:
Tritruncated 7-cube
Alternate names
Tritruncated hepteract (Jonathan Bowers)
Coordinates
Cartesian coordinates for the vertices of a tritruncated 7-cube, centered at the origin, are all sign and coordinate permutations of
(±2,±2,±2,±2,±1,0,0)
Images
Notes
References
H.S.M. Coxeter:
H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
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]
Norman Johnson Uniform Polytopes, Manuscript (1991)
N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D.
o3o3o3o3o3x4x - taz, o3o3o3o3x3x4o - botaz, o3o3o3x3x3o4o - totaz
External links
Polytopes of Various Dimensions
Multi-dimensional Glossary
7-polytopes |
https://en.wikipedia.org/wiki/Truncated%208-cubes | In eight-dimensional geometry, a truncated 8-cube is a convex uniform 8-polytope, being a truncation of the regular 8-cube.
There are unique 7 degrees of truncation for the 8-cube. Vertices of the truncation 8-cube are located as pairs on the edge of the 8-cube. Vertices of the bitruncated 8-cube are located on the square faces of the 8-cube. Vertices of the tritruncated 7-cube are located inside the cubic cells of the 8-cube. The final truncations are best expressed relative to the 8-orthoplex.
Truncated 8-cube
Alternate names
Truncated octeract (acronym tocto) (Jonathan Bowers)
Coordinates
Cartesian coordinates for the vertices of a truncated 8-cube, centered at the origin, are all 224 vertices are sign (4) and coordinate (56) permutations of
(±2,±2,±2,±2,±2,±2,±1,0)
Images
Related polytopes
The truncated 8-cube, is seventh in a sequence of truncated hypercubes:
Bitruncated 8-cube
Alternate names
Bitruncated octeract (acronym bato) (Jonathan Bowers)
Coordinates
Cartesian coordinates for the vertices of a truncated 8-cube, centered at the origin, are all the sign coordinate permutations of
(±2,±2,±2,±2,±2,±1,0,0)
Images
Related polytopes
The bitruncated 8-cube is sixth in a sequence of bitruncated hypercubes:
Tritruncated 8-cube
Alternate names
Tritruncated octeract (acronym tato) (Jonathan Bowers)
Coordinates
Cartesian coordinates for the vertices of a truncated 8-cube, centered at the origin, are all the sign coordinate permutations of
(±2,±2,±2,±2,±1,0,0,0)
Images
Quadritruncated 8-cube
Alternate names
Quadritruncated octeract (acronym oke) (Jonathan Bowers)
Coordinates
Cartesian coordinates for the vertices of a bitruncated 8-orthoplex, centered at the origin, are all sign and coordinate permutations of
(±2,±2,±2,±2,±1,0,0,0)
Images
Related polytopes
Notes
References
H.S.M. Coxeter:
H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
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]
Norman Johnson Uniform Polytopes, Manuscript (1991)
N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D.
o3o3o3o3o3o3x4x – tocto, o3o3o3o3o3x3x4o – bato, o3o3o3o3x3x3o4o – tato, o3o3o3x3x3o3o4o – oke
External links
Polytopes of Various Dimensions
Multi-dimensional Glossary
8-polytopes |
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2007 |
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Kawasaki Frontale seasons |
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Yokohama F. Marinos
Yokohama F. Marinos seasons |
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