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https://en.wikipedia.org/wiki/Mutual%20coherence%20%28linear%20algebra%29 | In linear algebra, the coherence or mutual coherence of a matrix A is defined as the maximum absolute value of the cross-correlations between the columns of A.
Formally, let be the columns of the matrix A, which are assumed to be normalized such that The mutual coherence of A is then defined as
A lower bound is
A deterministic matrix with the mutual coherence almost meeting the lower bound can be constructed by Weil's theorem.
This concept was reintroduced by David Donoho and Michael Elad in the context of sparse representations. A special case of this definition for the two-ortho case appeared earlier in the paper by Donoho and Huo. The mutual coherence has since been used extensively in the field of sparse representations of signals. In particular, it is used as a measure of the ability of suboptimal algorithms such as matching pursuit and basis pursuit to correctly identify the true representation of a sparse signal. Joel Tropp introduced a useful extension of Mutual Coherence, known as the Babel function, which extends the idea of cross-correlation between pairs of columns to the cross-correlation from one column to a set of other columns. The Babel function for two columns is exactly the Mutual coherence, but it also extends the coherence relationship concept in a way that is useful and relevant for any number of columns in the sparse representation matix as well.
See also
Compressed sensing
Restricted isometry property
Babel function
References
Further reading
Mutual coherence
R1magic : R package providing mutual coherence computation.
Signal processing
Matrix theory |
https://en.wikipedia.org/wiki/ICstat | ICstat, "Centro per la Cooperazione Statistica Internazionale - Luigi Bodio" (International Cooperation Center for Statistics) is a non-profit association, based in Rome, created on April 1, 1996. The Association promotes the international cooperation in the field of statistics, economics and law. ICstat co-ordinates technical assistance projects financed by international institutions (EC, World Bank, FAO, MAE etc.) and is particularly involved in several Transition and Developing Countries. The Association supports democratic governance, crisis prevention and recovery, human rights application and monitoring systems, post-conflict political elections and referendum. Moreover ICstat produces studies for scientific and policy purposes.
See also
Sustainable development
Development economics
Economic development
Further reading
CALZARONI M., CAPPIELLO A., DELLA ROCCA G., DI ZIO M., MARTELLI C., PIERACCINI G., TEMBE C., 2005 "O Sector Informal em Moçambique: Resultados do Primeiro Inquérito Nacional (2005) - The Informal Sector in Mozambique: Results of the First National Survey (2005). INE-Mozambique, Maputo", INE Mozambique
Giorgio D'AMORE, 2006 "Mapping existing global systems and initiatives", Water monitoring, FAO UN-Water
Aline PENNISI (Editorial project and co-ordination: 2000). Maritime Transport in the MED Countries, 2000 - Le Transport maritime dans les pays MED, 2000 Eurostat
Aline PENNISI, (Editorial project and co-ordination: 2000) Air Transport in the MED Countries, 1998-2000 - Le Transport aèrien dans les pays MED, 1998-2000, Eurostat
Antonio CAPPIELLO, 2004 "Considerazioni sui recenti censimenti dell'agricolutra, dell'industria e dei servizi in Mozambico. (Considerations on the last Censuses of Agriculture, Industry and Service carried out by INE Mozambique)", Rivista italiana di economia demografia e statistica, volume LVIII n. 3/4.
Antonio CAPPIELLO, 2006 "I currency board come strumento di stabilizzazione economica: come funzionano e dove sono adottati (Currency boards: how they work and where they are adopted). Quaderni di Studi Europei. GIUFFRÉ, VOLUME 1/2006. Centro di Ricerca de "La Sapienza" in Studi europei ed internazionali Eurosapienza.
Antonio CAPPIELLO, 2008 "Estimating Non Observed Economy", Journal of Economic and Social Measurement. Vol. 33 n. 1, 2008 ISSN 0747-9662
External links
ICstat Home page
Eurostat
OECD
Cooperazione Italiana
FAO Statistics
ISTAT
Economics societies
Statistical societies
Population organizations
Non-profit organisations based in Italy
Organisations based in Rome
International economic organizations
International organizations based in Europe |
https://en.wikipedia.org/wiki/Farrell%E2%80%93Jones%20conjecture | In mathematics, the Farrell–Jones conjecture, named after F. Thomas Farrell and Lowell E. Jones, states that certain assembly maps are isomorphisms. These maps are given as certain homomorphisms.
The motivation is the interest in the target of the assembly maps; this may be, for instance, the algebraic K-theory of a group ring
or the L-theory of a group ring
,
where G is some group.
The sources of the assembly maps are equivariant homology theory evaluated on the classifying space of G with respect to the family of virtually cyclic subgroups of G. So assuming the Farrell–Jones conjecture is true, it is possible to restrict computations to virtually cyclic subgroups to get information on complicated objects such as or .
The Baum–Connes conjecture formulates a similar statement, for the topological K-theory of reduced group -algebras .
Formulation
One can find for any ring equivariant homology theories satisfying
respectively
Here denotes the group ring.
The K-theoretic Farrell–Jones conjecture for a group G states that the map induces an isomorphism on homology
Here denotes the classifying space of the group G with respect to the family of virtually cyclic subgroups, i.e. a G-CW-complex whose isotropy groups are virtually cyclic and for any virtually cyclic subgroup of G the fixed point set is contractible.
The L-theoretic Farrell–Jones conjecture is analogous.
Computational aspects
The computation of the algebraic K-groups and the L-groups of a group ring is motivated by obstructions living in those groups (see for example Wall's finiteness obstruction, surgery obstruction, Whitehead torsion). So suppose a group satisfies the Farrell–Jones conjecture for algebraic K-theory. Suppose furthermore we have already found a model for the classifying space for virtually cyclic subgroups:
Choose -pushouts and apply the Mayer-Vietoris sequence to them:
This sequence simplifies to:
This means that if any group satisfies a certain isomorphism conjecture one can compute its algebraic K-theory (L-theory) only by knowing the algebraic K-Theory (L-Theory) of virtually cyclic groups and by knowing a suitable model for .
Why the family of virtually cyclic subgroups ?
One might also try to take for example the family of finite subgroups into account. This family is much easier to handle. Consider the infinite cyclic group . A model for is given by the real line , on which acts freely by translations. Using the properties of equivariant K-theory we get
The Bass-Heller-Swan decomposition gives
Indeed one checks that the assembly map is given by the canonical inclusion.
So it is an isomorphism if and only if , which is the case if is a regular ring. So in this case one can really use the family of finite subgroups. On the other hand this shows that the isomorphism conjecture for algebraic K-Theory and the family of finite subgroups is not true. One has to extend the conjecture to a larger family of subgroups whic |
https://en.wikipedia.org/wiki/Yuki%20Uekusa | is a Japanese football player currently playing for Shimizu S-Pulse.
Club statistics
Updated to 23 February 2018.
References
External links
Profile at Shimizu S-Pulse
1982 births
Living people
Waseda University alumni
People from Ichihara, Chiba
Association football people from Chiba Prefecture
Japanese men's footballers
J1 League players
J2 League players
Kawasaki Frontale players
Montedio Yamagata players
Vissel Kobe players
V-Varen Nagasaki players
Shimizu S-Pulse players
Men's association football goalkeepers |
https://en.wikipedia.org/wiki/Satoshi%20Kukino | is a Japanese football player currently playing for F.C. Machida Zelvia.
Club career statistics
Updated to 23 February 2016.
References
External links
Yokohama FC official
1987 births
Living people
Association football people from Miyazaki Prefecture
Japanese men's footballers
J1 League players
J2 League players
J3 League players
Kawasaki Frontale players
Yokohama FC players
Tochigi SC players
FC Machida Zelvia players
Men's association football forwards
People from Miyazaki (city) |
https://en.wikipedia.org/wiki/Yuji%20Kimura | is a Japanese football player who plays for Kagoshima United FC in the J3 League.
Career statistics
Updated to end of 2018 season.
References
External links
Profile at Mito HollyHock
Profile at Tokushima Vortis
1987 births
Living people
Association football people from Tokyo
Japanese men's footballers
J1 League players
J2 League players
J3 League players
Kawasaki Frontale players
Giravanz Kitakyushu players
Oita Trinita players
Tokushima Vortis players
Roasso Kumamoto players
Mito HollyHock players
Kagoshima United FC players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Kyohei%20Sugiura | is a Japanese football player currently playing for Zweigen Kanazawa.
Career statistics
Updated to end of 2018 season.
References
External links
Profile at Zweigen Kanazawa
Kyohei Sugiura on Instagram
Kawasaki Frontale profile
1989 births
Living people
Japanese men's footballers
J1 League players
J2 League players
Kawasaki Frontale players
Ehime FC players
Vissel Kobe players
Vegalta Sendai players
Zweigen Kanazawa players
Men's association football midfielders
Association football people from Hamamatsu |
https://en.wikipedia.org/wiki/Jun%20Sonoda | is a former Japanese football player who last played for Blaublitz Akita.
Career statistics
Updated to 23 February 2019.
References
External links
Profile at Kawasaki Frontale
Profile at Consadole Sapporo
Profile at Roasso Kumamoto
1989 births
Living people
Japanese men's footballers
J1 League players
J2 League players
J3 League players
Kawasaki Frontale players
FC Machida Zelvia players
Hokkaido Consadole Sapporo players
Roasso Kumamoto players
Fujieda MYFC players
Blaublitz Akita players
Asian Games medalists in football
Footballers at the 2010 Asian Games
Medalists at the 2010 Asian Games
Asian Games gold medalists for Japan
Men's association football defenders
Association football people from Shizuoka (city) |
https://en.wikipedia.org/wiki/Count%20On | Count On is a major mathematics education project in the United Kingdom which was announced by education secretary David Blunkett at the end of 2000. It was the follow-on to Maths Year 2000 which was the UK's contribution to UNICEF's World Mathematical Year.
Count On had two main strands:
The website www.counton.org which won the 2002 BETT prize for best free online learning resource.
"MathFests", which were maths funfairs held around the country, aimed particularly at those who would not normally come into contact with mathematical ideas.
The MathFests were run largely by MatheMagic and the University of York.
The project has now been handed over to the NCETM.
Popularisation of Mathematics
Count On and Maths Year 2000 were some of the first big Popularisation of Mathematics projects. Others are listed below.
International
World Mathematical Year 2000
Statistics 2013
World Maths Day (orig. Australian) - next one is 6 March 2013
Australia
World Maths Day
India
National Mathematics Year
Ireland
Maths Week Ireland
Nigeria
National Mathematics Year
Spain
Matematica Vital
Paul Boron
United Kingdom
Maths Year 2000 Scotland
Maths Cymru (Wales)
United States
Steven Strogatz's blog
References
Mathematics education in the United Kingdom |
https://en.wikipedia.org/wiki/Real%20number | In mathematics, a real number is a number that can be used to measure a continuous one-dimensional quantity such as a distance, duration or temperature. Here, continuous means that pairs of values can have arbitrarily small differences. Every real number can be almost uniquely represented by an infinite decimal expansion.
The real numbers are fundamental in calculus (and more generally in all mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives.
The set of real numbers is denoted or and is sometimes called "the reals".
The adjective real, used in the 17th century by René Descartes, distinguishes real numbers from imaginary numbers such as the square roots of .
The real numbers include the rational numbers, such as the integer and the fraction . The rest of the real numbers are called irrational numbers. Some irrational numbers (as well as all the rationals) are the root of a polynomial with integer coefficients, such as the square root ; these are called algebraic numbers. There are also real numbers which are not, such as ; these are called transcendental numbers.
Real numbers can be thought of as all points on a line called the number line or real line, where the points corresponding to integers () are equally spaced.
Conversely, analytic geometry is the association of points on lines (especially axis lines) to real numbers such that geometric displacements are proportional to differences between corresponding numbers.
The informal descriptions above of the real numbers are not sufficient for ensuring the correctness of proofs of theorems involving real numbers. The realization that a better definition was needed, and the elaboration of such a definition was a major development of 19th-century mathematics and is the foundation of real analysis, the study of real functions and real-valued sequences. A current axiomatic definition is that real numbers form the unique (up to an isomorphism) Dedekind-complete ordered field. Other common definitions of real numbers include equivalence classes of Cauchy sequences (of rational numbers), Dedekind cuts, and infinite decimal representations. All these definitions satisfy the axiomatic definition and are thus equivalent.
Characterizing properties
Real numbers are completely characterized by their fundamental properties that can be summarized by saying that they form an ordered field that is Dedekind complete. Here, "completely characterized" means that there is a unique isomorphism between any two Dedekind complete ordered fields, and thus that their elements have exactly the same properties. This implies that one can manipulate real numbers and compute with them, without knowing how they can be defined; this is what mathematicians and physicists did during several centuries before the first formal definitions were provided in the second half of the 19th century. See Construction of the real numbers for details about these formal defini |
https://en.wikipedia.org/wiki/Vladimir%20E.%20Zakharov | Vladimir Evgen'evich Zakharov (; 1 August 1939 – 20 August 2023) was a Soviet and Russian mathematician and physicist. He was Regents' Professor of mathematics at The University of Arizona, director of the Mathematical Physics Sector at the Lebedev Physical Institute, and was on the committee of the Stefanos Pnevmatikos International Award. Zakharov's research interests covered physical aspects of nonlinear wave theory in plasmas, hydrodynamics, oceanology, geophysics, solid state physics, optics, and general relativity.
Zakharov was awarded the Dirac Medal in 2003 for his contributions to the theory of turbulence, with regard to the exact results and the prediction of inverse cascades, and for "putting the theory of wave turbulence on a firm mathematical ground by finding turbulence spectra as exact solutions and solving the stability problem, and in introducing the notion of inverse and dual cascades in wave turbulence."
Vladimir Zakharov was also a poet. He published several books of poetry in Russia and his works regularly appeared in periodicals, such as Novy Mir, in the 1990s and 2000s. A collection of his poetry in an English translation The Paradise for Clouds was published in the UK in 2009.
Biography
Vladimir Zakharov was born in Kazan, to Evgeniy and Elena Zakharov, an engineer and a schoolteacher. He studied at the Moscow Power Engineering Institute and at the Novosibirsk State University, where he received his specialist degree in physics in 1963 and his Candidate of Sciences degree in 1966, studying under Roald Sagdeev.
Zakharov was married and had three sons. He died in August 2023, at the age of 84.
Academic career
After completing his Candidate of Science degree, Zakharov worked as a researcher at the Budker Institute of Nuclear Physics in Novosibirsk, where in 1971 he completed his Doctor of Sciences degree. In 1974, Zakharov moved to the Landau Institute for Theoretical Physics in Chernogolovka, where he eventually became director. He was elected as a corresponding member of the
Academy of Sciences of the Soviet Union in 1984 and as a full member in 1991. In 1992, Zakharov became a professor of mathematics at the University of Arizona, and in 2004 he became the director of the Mathematical Physics Sector at the Lebedev Physical Institute.
Awards and honors
State Prize for research in Plasma Theory, USSR, 1987
Order of Honors from the State, USSR, 1989
State Prize of Russian Federation for research in Soliton Theory, Russia, 1993
Order for the Service to Russian federation, awarded to 60th anniversary, 1999
Dirac Medal of the Abdus Salam International Center for Theoretical Physics, Trieste, Italy, 2003
Namesake of asteroid 7153 Vladzakharov
Fellow of the American Mathematical Society, 2012
Selected bibliography
S. P. Novikov, S. V. Manakov, L. P. Pitaevskii, V. E. Zakharov, Theory of Solitons: The Inverse Scattering Method, Springer-Verlag (1984),
V. E. Zakharov, What is Integrability?, Springer-Verlag (1991),
V. E |
https://en.wikipedia.org/wiki/Irrational%20number | In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integers. When the ratio of lengths of two line segments is an irrational number, the line segments are also described as being incommensurable, meaning that they share no "measure" in common, that is, there is no length ("the measure"), no matter how short, that could be used to express the lengths of both of the two given segments as integer multiples of itself.
Among irrational numbers are the ratio of a circle's circumference to its diameter, Euler's number e, the golden ratio φ, and the square root of two. In fact, all square roots of natural numbers, other than of perfect squares, are irrational.
Like all real numbers, irrational numbers can be expressed in positional notation, notably as a decimal number. In the case of irrational numbers, the decimal expansion does not terminate, nor end with a repeating sequence. For example, the decimal representation of starts with 3.14159, but no finite number of digits can represent exactly, nor does it repeat. Conversely, a decimal expansion that terminates or repeats must be a rational number. These are provable properties of rational numbers and positional number systems and are not used as definitions in mathematics.
Irrational numbers can also be expressed as non-terminating continued fractions and many other ways.
As a consequence of Cantor's proof that the real numbers are uncountable and the rationals countable, it follows that almost all real numbers are irrational.
History
Ancient Greece
The first proof of the existence of irrational numbers is usually attributed to a Pythagorean (possibly Hippasus of Metapontum), who probably discovered them while identifying sides of the pentagram.
The then-current Pythagorean method would have claimed that there must be some sufficiently small, indivisible unit that could fit evenly into one of these lengths as well as the other. Hippasus in the 5th century BC, however, was able to deduce that there was no common unit of measure, and that the assertion of such an existence was a contradiction. He did this by demonstrating that if the hypotenuse of an isosceles right triangle was indeed commensurable with a leg, then one of those lengths measured in that unit of measure must be both odd and even, which is impossible. His reasoning is as follows:
Start with an isosceles right triangle with side lengths of integers a, b, and c. The ratio of the hypotenuse to a leg is represented by c:b.
Assume a, b, and c are in the smallest possible terms (i.e. they have no common factors).
By the Pythagorean theorem: c2 = a2+b2 = b2+b2 = 2b2. (Since the triangle is isosceles, a = b).
Since c2 = 2b2, c2 is divisible by 2, and therefore even.
Since c2 is even, c must be even.
Since c is even, dividing c by 2 yields an in |
https://en.wikipedia.org/wiki/Tomasz%20Mrowka | Tomasz Mrowka (born September 8, 1961) is an American mathematician specializing in differential geometry and gauge theory. He is the Singer Professor of Mathematics and former head of the Department of Mathematics at the Massachusetts Institute of Technology.
Mrowka is the son of Polish mathematician , and is married to MIT mathematics professor Gigliola Staffilani.
Career
A 1983 graduate of the Massachusetts Institute of Technology, he received the PhD from the University of California, Berkeley in 1988 under the direction of Clifford Taubes and Robion Kirby. He joined the MIT mathematics faculty as professor in 1996, following faculty appointments at Stanford University and at the California Institute of Technology (professor 1994–96). At MIT, he was the Simons Professor of Mathematics from 2007–2010. Upon Isadore Singer's retirement in 2010 the name of the chair became the Singer Professor of Mathematics which Mrowka held until 2017. He was named head of the Department of Mathematics in 2014 and held that position for 3 years.
A prior Sloan fellow and Young Presidential Investigator, in 1994 he was an invited speaker at the International Congress of Mathematicians (ICM) in Zurich. In 2007, he received the Oswald Veblen Prize in Geometry from the AMS jointly with Peter Kronheimer, "for their joint contributions to both three- and four-dimensional topology through the development of deep analytical techniques and applications." He was named a Guggenheim Fellow in 2010, and in 2011 he received the Doob Prize with Peter B. Kronheimer for their book Monopoles and Three-Manifolds (Cambridge University Press, 2007). In 2018 he gave a plenary lecture at the ICM in Rio de Janeiro, together with Peter Kronheimer. In 2023 he was awarded the Leroy P. Steele Prize for Seminal Contribution to Research (with Peter Kronheimer).
He became a fellow of the American Academy of Arts & Sciences in 2007, and a member of the National Academy of Sciences in 2015.
Research
Mrowka's work combines analysis, geometry, and topology, specializing in the use of partial differential equations, such as the Yang-Mills equations from particle physics to analyze low-dimensional mathematical objects. Jointly with Robert Gompf, he discovered four-dimensional models of space-time topology.
In joint work with Peter Kronheimer, Mrowka settled many long-standing conjectures, three of which earned them the 2007 Veblen Prize. The award citation mentions three papers that Mrowka and Kronheimer wrote together. The first paper in 1995 deals with Donaldson's polynomial invariants and introduced Kronheimer–Mrowka basic class, which have been used to prove a variety of results about the topology and geometry of 4-manifolds, and partly motivated Witten's introduction of the Seiberg–Witten invariants. The second paper proves the so-called Thom conjecture and was one of the first deep applications of the then brand new Seiberg–Witten equations to four-dimensional topology. In the third |
https://en.wikipedia.org/wiki/Michael%20J.%20Hopkins | Michael Jerome Hopkins (born April 18, 1958) is an American mathematician known for work in algebraic topology.
Life
He received his PhD from Northwestern University in 1984 under the direction of Mark Mahowald, with thesis Stable Decompositions of Certain Loop Spaces. Also in 1984 he also received his D.Phil. from the University of Oxford under the supervision of Ioan James. He has been professor of mathematics at Harvard University since 2005, after fifteen years at the Massachusetts Institute of Technology, a few years of teaching at Princeton University, a one-year position with the University of Chicago, and a visiting lecturer position at Lehigh University.
Work
Hopkins' work concentrates on algebraic topology, especially stable homotopy theory. It can roughly be divided into four parts (while the list of topics below is by no means exhaustive):
The Ravenel conjectures
The Ravenel conjectures very roughly say: complex cobordism (and its variants) see more in the stable homotopy category than you might think. For example, the nilpotence conjecture states that some suspension of some iteration of a map between finite CW-complexes is null-homotopic iff it is zero in complex cobordism. This was proven by Ethan Devinatz, Hopkins and Jeff Smith (published in 1988). The rest of the Ravenel conjectures (except for the telescope conjecture) were proven by Hopkins and Smith soon after (published in 1998). Another result in this spirit proven by Hopkins and Douglas Ravenel is the chromatic convergence theorem, which states that one can recover a finite CW-complex from its localizations with respect to wedges of Morava K-theories.
Hopkins–Miller theorem and topological modular forms
This part of work is about refining a homotopy commutative diagram of ring spectra up to homotopy to a strictly commutative diagram of highly structured ring spectra. The first success of this program was the Hopkins–Miller theorem: It is about the action of the Morava stabilizer group on Lubin–Tate spectra (arising out of the deformation theory of formal group laws) and its refinement to -ring spectra – this allowed to take homotopy fixed points of finite subgroups of the Morava stabilizer groups, which led to higher real K-theories. Together with Paul Goerss, Hopkins later set up a systematic obstruction theory for refinements to -ring spectra. This was later used in the Hopkins–Miller construction of topological modular forms. Subsequent work of Hopkins on this topic includes papers on the question of the orientability of TMF with respect to string cobordism (joint work with Ando, Strickland and Rezk).
The Kervaire invariant problem
On April 21, 2009, Hopkins announced the solution of the Kervaire invariant problem, in joint work with Mike Hill and Douglas Ravenel. This problem is connected to the study of exotic spheres, but got transformed by work of William Browder into a problem in stable homotopy theory. The proof by Hill, Hopkins and Ravenel works purely in |
https://en.wikipedia.org/wiki/Joram%20Lindenstrauss | Joram Lindenstrauss () (October 28, 1936 – April 29, 2012) was an Israeli mathematician working in functional analysis. He was a professor of mathematics at the Einstein Institute of Mathematics.
Biography
Joram Lindenstrauss was born in Tel Aviv. He was the only child of a pair of lawyers who immigrated to Israel from Berlin. He began to study mathematics at the Hebrew University of Jerusalem in 1954 while serving in the army. He became a full-time student in 1956 and received his master's degree in 1959. In 1962 Lindenstrauss earned his Ph.D. from the Hebrew University (dissertation: Extension of Compact Operators, advisors: Aryeh Dvoretzky, Branko Grünbaum). He worked as a postdoc at Yale University and the University of Washington in Seattle from 1962 - 1965. He was appointed senior lecturer at the Hebrew University in 1965, associate professor on 1967 and full professor in 1969. He became the Leon H. and Ada G. Miller Memorial Professor of Mathematics in 1985. He retired in 2005.
Lindenstrauss was married to theoretical computer scientist Naomi Lindenstrauss. Two of their children, Ayelet Lindenstrauss and Fields Medallist Elon Lindenstrauss, are also mathematicians (providing a rare example of father, mother, son and daughter all having papers listed in Mathematical Reviews). Joram was also the cousin of Micha Lindenstrauss.
Research
Lindenstrauss worked in various areas of functional analysis and geometry, particularly Banach space theory, finite- and infinite-dimensional convexity, geometric nonlinear functional analysis and geometric measure theory. He authored more than 100 papers as well as several books in Banach space theory.
Among his results is the Johnson–Lindenstrauss lemma which concerns low-distortion embeddings of points from high-dimensional into low-dimensional Euclidean space. Another of his theorems states that in a Banach space with the Radon–Nikodym property, a closed and bounded set has an extreme point; compactness is not needed.
Awards
In 1981 Lindenstrauss was awarded the Israel Prize, for mathematics. In 1997, Lindenstrauss was the first mathematician from outside Poland to be awarded the Stefan Banach Medal of the Polish Academy of Sciences.
Published works
Classical Banach spaces I (with Lior Tzafriri). Springer-Verlag, 1977.
Classical Banach spaces II (with Lior Tzafriri). Springer-Verlag, 1979.
Banach spaces with a unique unconditional basis, up to permutation (with Jean Bourgain, Peter George Casazza, and Lior Tzafriri). Memoirs of the American Mathematical Society, vol 322. American Mathematical Society, 1985
Geometric nonlinear functional analysis (with Yoav Benyamini). Colloquium publications, 48. American Mathematical Society, 2000.
Handbook of the geometry of Banach spaces (Edited, with William B. Johnson). Elsevier, Vol. 1 (2001), Vol. 2 (2003).
See also
List of Israel Prize recipients
References
1936 births
2012 deaths
20th-century Israeli mathematicians
Functional analysts
Einstein In |
https://en.wikipedia.org/wiki/Hesse%20normal%20form | The Hesse normal form named after Otto Hesse, is an equation used in analytic geometry, and describes a line in or a plane in Euclidean space or a hyperplane in higher dimensions. It is primarily used for calculating distances (see point-plane distance and point-line distance).
It is written in vector notation as
The dot indicates the scalar product or dot product.
Vector points from the origin of the coordinate system, O, to any point P that lies precisely in plane or on line E. The vector represents the unit normal vector of plane or line E. The distance is the shortest distance from the origin O to the plane or line.
Derivation/Calculation from the normal form
Note: For simplicity, the following derivation discusses the 3D case. However, it is also applicable in 2D.
In the normal form,
a plane is given by a normal vector as well as an arbitrary position vector of a point . The direction of is chosen to satisfy the following inequality
By dividing the normal vector by its magnitude , we obtain the unit (or normalized) normal vector
and the above equation can be rewritten as
Substituting
we obtain the Hesse normal form
In this diagram, d is the distance from the origin. Because holds for every point in the plane, it is also true at point Q (the point where the vector from the origin meets the plane E), with , per the definition of the Scalar product
The magnitude of is the shortest distance from the origin to the plane.
References
External links
Analytic geometry |
https://en.wikipedia.org/wiki/Musa%2C%20Azerbaijan | Musa is a village in the municipality of Aşağı Astanlı in the Yardymli Rayon of Azerbaijan. According to Azerbaijan's State Statistics Committee, only eight people lived in the village as of 2014.
References
Populated places in Yardimli District |
https://en.wikipedia.org/wiki/Kirillov%20model | In mathematics, the Kirillov model, studied by , is a realization of a representation of GL2 over a local field on a space of functions on the local field.
If G is the algebraic group GL2 and F is a non-Archimedean local field,
and τ is a fixed nontrivial character of the additive group of F
and π is an irreducible representation of G(F), then the Kirillov model for π is
a representation π on a space of locally constant functions f on F* with compact support in F such that
showed that an irreducible representation of dimension greater than 1 has an essentially unique Kirillov model.
Over a local field, the space of functions with compact support in F* has codimension 0, 1, or 2 in the Kirillov model, depending on whether the irreducible representation is cuspidal, special, or principal.
The Whittaker model can be constructed from the Kirillov model, by defining the image Wξ of a vector ξ of the Kirillov model by
Wξ(g) = π(g)ξ(1)
where π(g) is the image of g in the Kirillov model.
defined the Kirillov model for the general linear group GLn using the mirabolic subgroup. More precisely, a Kirillov model for a representation of the general linear group is an embedding of it in the representation of the mirabolic group induced from a non-degenerate character of the group of upper triangular matrices.
References
Representation theory
Automorphic forms
Langlands program |
https://en.wikipedia.org/wiki/Artinian%20ideal | In abstract algebra, an Artinian ideal, named after Emil Artin, is encountered in ring theory, in particular, with polynomial rings.
Given a polynomial ring R = k[X1, ... Xn] where k is some field, an Artinian ideal is an ideal I in R for which the Krull dimension of the quotient ring R/I is 0. Also, less precisely, one can think of an Artinian ideal as one that has at least each indeterminate in R raised to a power greater than 0 as a generator.
If an ideal is not Artinian, one can take the Artinian closure of it as follows. First, take the least common multiple of the generators of the ideal. Second, add to the generating set of the ideal each indeterminate of the LCM with its power increased by 1 if the power is not 0 to begin with. An example is below.
Examples
Let , and let and . Here, and are Artinian ideals, but is not because in , the indeterminate does not appear alone to a power as a generator.
To take the Artinian closure of , , we find the LCM of the generators of , which is . Then, we add the generators , and to , and reduce. Thus, we have which is Artinian.
References
Commutative algebra
Ring theory |
https://en.wikipedia.org/wiki/Moufang%20polygon | In mathematics, Moufang polygons are a generalization by Jacques Tits of the Moufang planes studied by Ruth Moufang, and are irreducible buildings of rank two that admit the action of root groups.
In a book on the topic, Tits and Richard Weiss classify them all. An earlier theorem, proved independently by Tits and Weiss, showed that a Moufang polygon must be a generalized 3-gon, 4-gon, 6-gon, or 8-gon, so the purpose of the aforementioned book was to analyze these four cases.
Definitions
A generalized n-gon is a bipartite graph of diameter n and girth 2n.
A graph is called thick if all vertices have valence at least 3.
A root of a generalized n-gon is a path of length n.
An apartment of a generalized n-gon is a cycle of length 2n.
The root subgroup of a root is the subgroup of automorphisms of a graph that fix all vertices adjacent to one of the inner vertices of the root.
A Moufang n-gon is a thick generalized n-gon (with n>2) such that the root subgroup of any root acts transitively on the apartments containing the root.
Moufang 3-gons
A Moufang 3-gon can be identified with the incidence graph of a Moufang projective plane. In this identification, the points and lines of the plane correspond to the vertices of the building.
Real forms of Lie groups give rise to examples which are the three main types of Moufang 3-gons. There are four real division algebras: the real numbers, the complex numbers, the quaternions, and the octonions, of dimensions 1,2,4 and 8, respectively. The projective plane over such a division algebra then gives rise to a Moufang 3-gon.
These projective planes correspond to the building attached to SL3(R), SL3(C), a real form of A5 and to a real form of E6, respectively.
In the first diagram the circled nodes represent 1-spaces and 2-spaces in a three-dimensional vector space. In the second diagram the circled nodes represent 1-space and 2-spaces in a 3-dimensional vector space over the quaternions, which in turn represent certain 2-spaces and 4-spaces in a 6-dimensional complex vector space, as expressed by the circled nodes in the A5 diagram.
The fourth case — a form of E6 — is exceptional, and its analogue for Moufang 4-gons is a major feature of Weiss's book.
Going from the real numbers to an arbitrary field, Moufang 3-gons can be divided into three cases as above. The split case in the first diagram exists over any field. The second case extends to all associative, non-commutative division algebras; over the reals these are limited to the algebra of quaternions, which has degree 2 (and dimension 4), but some fields admit central division algebras of other degrees.
The third case involves "alternative" division algebras (which satisfy a weakened form of the associative law), and a theorem of Richard Bruck and Erwin Kleinfeld shows that these are Cayley-Dickson algebras. This concludes the discussion of Moufang 3-gons.
Moufang 4-gons
Moufang 4-gons are also called Moufang quadrangles.
The classification of Mouf |
https://en.wikipedia.org/wiki/Infinity%20and%20the%20Mind | Infinity and the Mind: The Science and Philosophy of the Infinite is a popular mathematics book by American mathematician, computer scientist, and science fiction writer Rudy Rucker.
Synopsis
The book contains accessible popular expositions on the mathematical theory of infinity, and a number of related topics. These include Gödel's incompleteness theorems and their relationship to concepts of artificial intelligence and the human mind, as well as the conceivability of some unconventional cosmological models. The material is approached from a variety of viewpoints, some more conventionally mathematical and others being nearly mystical. There is a brief account of the author's personal contact with Kurt Gödel.
An appendix contains one of the few popular expositions on set theory research on what are known as "strong axioms of infinity."
Reception
Dave Langford reviewed Infinity and the Mind for White Dwarf #41, and stated that "a must for anyone who enjoyed Hofstadter's Godel, Escher, Bach or the works of Martin Gardner."
Infinity and the Mind was reviewed by the New Yorker, which asserted that "Rudy Rucker's Infinity and the Mind is a terrific study with real mathematical depth." Martin Gardner described the book as "Informal, amusing, witty, profound... In an extraordinary burst of creative energy, Rudy Rucker has managed to bring together every aspect of mathematical infinity.... A dizzying glimpse into that boundless region of blinding light where the mysteries of transcendence shatter the clarity of logic, set theory, proof theory, and contemporary physics."
References
External links
Infinity and the Mind at Princeton University Press
1982 non-fiction books
Books by Rudy Rucker
Mathematics books
Birkhäuser books |
https://en.wikipedia.org/wiki/Whittaker%20model | In representation theory, a branch of mathematics, the Whittaker model is a realization of a representation of a reductive algebraic group such as GL2 over a finite or local or global field on a space of functions on the group. It is named after E. T. Whittaker even though he never worked in this area, because pointed out that for the group SL2(R) some of the functions involved in the representation are Whittaker functions.
Irreducible representations without a Whittaker model are sometimes called "degenerate", and those with a Whittaker model are sometimes called "generic". The representation θ10 of the symplectic group Sp4 is the simplest example of a degenerate representation.
Whittaker models for GL2
If G is the algebraic group GL2 and F is a local field, and is a fixed non-trivial character of the additive group of F and is an irreducible representation of a general linear group G(F), then the Whittaker model for is a representation on a space of functions ƒ on G(F) satisfying
used Whittaker models to assign L-functions to admissible representations of GL2.
Whittaker models for GLn
Let be the general linear group , a smooth complex valued non-trivial additive character of and the subgroup of consisting of unipotent upper triangular matrices. A non-degenerate character on is of the form
for ∈ and non-zero ∈ . If is a smooth representation of , a Whittaker functional is a continuous linear functional on such that for all ∈ , ∈ . Multiplicity one states that, for unitary irreducible, the space of Whittaker functionals has dimension at most equal to one.
Whittaker models for reductive groups
If G is a split reductive group and U is the unipotent radical of a Borel subgroup B, then a Whittaker model for a representation is an embedding of it into the induced (Gelfand–Graev) representation Ind(), where is a non-degenerate character of U, such as the sum of the characters corresponding to simple roots.
See also
Gelfand–Graev representation, roughly the sum of Whittaker models over a finite field.
Kirillov model
References
J. A. Shalika, The multiplicity one theorem for , The Annals of Mathematics, 2nd. Ser., Vol. 100, No. 2 (1974), 171-193.
Further reading
Representation theory
Automorphic forms
Langlands program
E. T. Whittaker |
https://en.wikipedia.org/wiki/Aleksandre%20Guruli | Alexander Guruli (; born 9 November 1985) is a professional Georgian football midfielder who plays for US Saint-Omer.
Personal life
He is the son of Gija Guruli.
Career statistics
International
Statistics accurate as of match played 14 November 2012
International goals
References
External links
1985 births
Living people
Sportspeople from Batumi
Men's footballers from Georgia (country)
Men's association football midfielders
Georgia (country) men's under-21 international footballers
Georgia (country) men's international footballers
Expatriate men's footballers from Georgia (country)
Expatriate men's footballers in France
Expatriate men's footballers in Ukraine
Expatriate men's footballers in Belarus
Expatriate men's footballers in Azerbaijan
Expatriate sportspeople from Georgia (country) in Ukraine
Expatriate sportspeople from Georgia (country) in Azerbaijan
Ukrainian Premier League players
Azerbaijan Premier League players
Olympique Lyonnais players
US Boulogne players
Lyon La Duchère players
US Lesquin players
FC Karpaty Lviv players
FC Dila Gori players
FC Dinamo Batumi players
FC Zestafoni players
FC Shakhtyor Soligorsk players
FC Samtredia players
Shuvalan FK players
FC Shukura Kobuleti players
Olympique Grande-Synthe players
US Saint-Omer players |
https://en.wikipedia.org/wiki/Catalog%20of%20articles%20in%20probability%20theory | This page lists articles related to probability theory. In particular, it lists many articles corresponding to specific probability distributions. Such articles are marked here by a code of the form (X:Y), which refers to number of random variables involved and the type of the distribution. For example (2:DC) indicates a distribution with two random variables, discrete or continuous. Other codes are just abbreviations for topics. The list of codes can be found in the table of contents.
Core probability: selected topics
Probability theory
Basic notions (bsc)
Random variable
Continuous probability distribution / (1:C)
Cumulative distribution function / (1:DCR)
Discrete probability distribution / (1:D)
Independent and identically-distributed random variables / (FS:BDCR)
Joint probability distribution / (F:DC)
Marginal distribution / (2F:DC)
Probability density function / (1:C)
Probability distribution / (1:DCRG)
Probability distribution function
Probability mass function / (1:D)
Sample space
Instructive examples (paradoxes) (iex)
Berkson's paradox / (2:B)
Bertrand's box paradox / (F:B)
Borel–Kolmogorov paradox / cnd (2:CM)
Boy or Girl paradox / (2:B)
Exchange paradox / (2:D)
Intransitive dice
Monty Hall problem / (F:B)
Necktie paradox
Simpson's paradox
Sleeping Beauty problem
St. Petersburg paradox / mnt (1:D)
Three Prisoners problem
Two envelopes problem
Moments (mnt)
Expected value / (12:DCR)
Canonical correlation / (F:R)
Carleman's condition / anl (1:R)
Central moment / (1:R)
Coefficient of variation / (1:R)
Correlation / (2:R)
Correlation function / (U:R)
Covariance / (2F:R) (1:G)
Covariance function / (U:R)
Covariance matrix / (F:R)
Cumulant / (12F:DCR)
Factorial moment / (1:R)
Factorial moment generating function / anl (1:R)
Fano factor
Geometric standard deviation / (1:R)
Hamburger moment problem / anl (1:R)
Hausdorff moment problem / anl (1:R)
Isserlis Gaussian moment theorem / Gau
Jensen's inequality / (1:DCR)
Kurtosis / (1:CR)
Law of the unconscious statistician / (1:DCR)
Moment / (12FU:CRG)
Law of total covariance / (F:R)
Law of total cumulance / (F:R)
Law of total expectation / (F:DR)
Law of total variance / (F:R)
Logmoment generating function
Marcinkiewicz–Zygmund inequality / inq
Method of moments / lmt (L:R)
Moment problem / anl (1:R)
Moment-generating function / anl (1F:R)
Second moment method / (1FL:DR)
Skewness / (1:R)
St. Petersburg paradox / iex (1:D)
Standard deviation / (1:DCR)
Standardized moment / (1:R)
Stieltjes moment problem / anl (1:R)
Trigonometric moment problem / anl (1:R)
Uncorrelated / (2:R)
Variance / (12F:DCR)
Variance-to-mean ratio / (1:R)
Inequalities (inq)
Chebyshev's inequality / (1:R)
An inequality on location and scale parameters / (1:R)
Azuma's inequality / (F:BR)
Bennett's inequality / (F:R)
Bernstein inequalities / (F:R)
Bhatia–Davis inequality
Chernoff bound / (F:B)
Doob's martingale inequality / (FU:R)
Dudley's theorem / |
https://en.wikipedia.org/wiki/Infinite%20broom | In topology, a branch of mathematics, the infinite broom is a subset of the Euclidean plane that is used as an example distinguishing various notions of connectedness. The closed infinite broom is the closure of the infinite broom, and is also referred to as the broom space.
Definition
The infinite broom is the subset of the Euclidean plane that consists of all closed line segments joining the origin to the point as n varies over all positive integers, together with the interval (½, 1] on the x-axis.
The closed infinite broom is then the infinite broom together with the interval (0, ½] on the x-axis. In other words, it consists of all closed line segments joining the origin to the point or to the point .
Properties
Both the infinite broom and its closure are connected, as every open set in the plane which contains the segment on the x-axis must intersect slanted segments. Neither are locally connected. Despite the closed infinite broom being arc connected, the standard infinite broom is not path connected.
The interval [0,1] on the x-axis is a deformation retract of the closed infinite broom, but it is not a strong deformation retract.
See also
Comb space
Integer broom topology
List of topologies
References
Topological spaces
Trees (topology) |
https://en.wikipedia.org/wiki/Reuven%20Rubinstein | Reuven Rubinstein (1938-2012)() was an Israeli scientist known for his contributions to Monte Carlo simulation, applied probability, stochastic modeling and stochastic optimization, having authored more than one hundred papers and six books.
During his career, Rubinstein has made fundamental and important contributions in these fields and has advanced the theory and application of adaptive importance sampling, rare event simulation, stochastic optimization, sensitivity analysis of simulation-based models, the splitting method, and counting problems concerning NP-complete problems.
He is well known as the founder of several breakthrough methods, such as
the score function method, stochastic counterpart method and cross-entropy method, which have numerous applications in combinatorial optimization and simulation.
His citation index is in the top 5% among his colleagues in operations research and management sciences. His 1981 book "Simulation and the Monte Carlo Method", Wiley (second edition 2008 and third edition 2017, with Dirk Kroese) alone has over 10,000 citations.
He has held visiting positions in various research institutes, including Columbia University, Harvard University, Stanford University, IBM and Bell Laboratories. He has given invited and plenary lectures in many international conferences around the globe.
In 2010 Prof. Rubinstein won the INFORMS Simulation Society highest prize
- the Lifetime Professional Achievement Award (LPAA), which recognizes scholars who have made fundamental contributions to the field of simulation that persist over most of a professional career.
In 2011 Reuven Rubinstein won from the Operations Research Society of Israel (ORSIS) the Lifetime Professional Award (LPA), which recognizes scholars who have made fundamental contributions to the field of operations research over most of a professional career and constitutes ORSIS's highest award.
Publications
Rubinstein R.Y., "Simulation and the Monte Carlo Method", Wiley, 1981.
Rubinstein, R.Y. and A. Shapiro, "Discrete Event Systems: Sensitivity Analysis and Stochastic Optimization", Wiley, 1993.
Rubinstein, R.Y., "The cross-entropy method for combinatorial and continuous Optimization", Methodology and Computing in Applied Probability, 2, 127—190, 1999.
Rubinstein R.Y., "Randomized algorithms with splitting: Why the classic randomized algorithms do not work and how to make them work", Methodology and Computing in Applied Probability, 2009.
Rubinstein R.Y. and D.P. Kroese, "The Cross-Entropy Method: a Unified Approach to Combinatorial Optimization, Monte-Carlo Simulation and Machine Learning", Springer, 2004.
Rubinstein R.Y. and D.P. Kroese, "Simulation and the Monte Carlo Method", Second Edition, Wiley, 2008.
Rubinstein R.Y. and D.P. Kroese, "Simulation and the Monte Carlo Method", Third Edition, Wiley, 2017.
References
Israeli Jews
Israeli mathematicians
Israeli operations researchers
Columbia University staff
Harvard University staff
Stanford Unive |
https://en.wikipedia.org/wiki/Shunsuke%20Oyama | is a former Japanese football player.
Club statistics
Updated to 23 February 2016.
References
External links
1986 births
Living people
Japanese men's footballers
J1 League players
J2 League players
J3 League players
Urawa Red Diamonds players
Ehime FC players
Shonan Bellmare players
Kataller Toyama players
Men's association football midfielders
Association football people from Saitama (city) |
https://en.wikipedia.org/wiki/Der-Tsai%20Lee | Der-Tsai Lee (aka. D. T. Lee) is a Taiwanese computer scientist, known for his work in computational geometry. For many years he was a professor at Northwestern University. He has been a distinguished research fellow of the Institute for Information Science at the Academia Sinica in Taipei, Taiwan since 1998. From 1998 to 2008, he was director of this institute. He was the 14th President of National Chung Hsing University from August 1, 2011.
Lee received a B.S. degree in electrical engineering from
National Taiwan University in 1971, an M.S. from the University of Illinois at Urbana-Champaign in 1976, and a Ph.D. from UIUC under the supervision of Franco Preparata in 1978. After holding a faculty position at Northwestern University for 20 years, he moved to the Academia Sinica in 1998. He also holds faculty positions at National Taiwan University, National Taiwan University of Science and Technology, and National Chiao Tung University. He is a Fellow of the IEEE and the ACM. He was elected as the Academician of Academia Sinica, Taiwan in 2004. He also won the Humboldt Research Award in 2007 and elected as the member of The Academy of Sciences for the Developing World (also known as Third World Academy of Sciences) (TWAS) in 2008. In 2010, he became the Humboldt Ambassador Scientist. He has published near 200 research papers, and an ISI highly cited researcher. He is editor in chief of the International Journal of Computational Geometry and Applications.
He was awarded the German-Taiwanese Friendship Medal by Michael Zickerick, the Director General of the German Institute Taipei, in May 2014.
References
Researchers in geometric algorithms
National Taiwan University alumni
American people of Taiwanese descent
Grainger College of Engineering alumni
Northwestern University faculty
Fellows of the Association for Computing Machinery
Fellow Members of the IEEE
Living people
1949 births
Taiwanese computer scientists
Members of Academia Sinica
Place of birth missing (living people)
Presidents of universities and colleges in Taiwan
Academic staff of the National Chung Hsing University |
https://en.wikipedia.org/wiki/Napkin%20folding%20problem | The napkin folding problem is a problem in geometry and the mathematics of paper folding that explores whether folding a square or a rectangular napkin can increase its perimeter. The problem is known under several names, including the Margulis napkin problem, suggesting it is due to Grigory Margulis, and the Arnold's rouble problem referring to Vladimir Arnold and the folding of a Russian ruble bank note. Some versions of the problem were solved by Robert J. Lang, Svetlana Krat, Alexey S. Tarasov, and Ivan Yaschenko. One form of the problem remains open.
Formulations
There are several way to define the notion of folding, giving different interpretations. By convention, the napkin is always a unit square.
Folding along a straight line
Considering the folding as a reflection along a line that reflects all the layers of the napkin, the perimeter is always non-increasing, thus never exceeding 4.
By considering more general foldings that possibly reflect only a single layer of the napkin (in this case, each folding is a reflection of a connected component of folded napkin on one side of a straight line), it is still open if a sequence of these foldings can increase the perimeter. In other words, it is still unknown if there exists a solution that can be folded using some combination of mountain folds, valley folds, reverse folds, and/or sink folds (with all folds in the latter two cases being formed along a single line). Also unknown, of course, is whether such a fold would be possible using the more-restrictive pureland origami.
Folding without stretching
One can ask for a realizable construction within the constraints of rigid origami where the napkin is never stretched whilst being folded. In 2004 A. Tarasov showed that such constructions can indeed be obtained. This can be considered a complete solution to the original problem.
Where only the result matters
One can ask whether there exists a folded planar napkin (without regard as to how it was folded into that shape).
Robert J. Lang showed in 1997 that several classical origami constructions give rise to an easy solution. In fact, Lang showed that the perimeter can be made as large as desired by making the construction more complicated, while still resulting in a flat folded solution. However his constructions are not necessarily rigid origami because of their use of sink folds and related forms. Although no stretching is needed in sink and unsink folds, it is often (though not always) necessary to curve facets and/or sweep one or more creases continuously through the paper in intermediate steps before obtaining a flat result. Whether a general rigidly foldable solution exists based on sink folds is an open problem.
In 1998, I. Yaschenko constructed a 3D folding with projection onto a plane which has a bigger perimeter. This indicated to mathematicians that there was probably a flat folded solution to the problem.
The same conclusion was made by Svetlana Krat. Her approach is diffe |
https://en.wikipedia.org/wiki/Miodrag%20Petkovi%C4%87 | Miodrag S. Petković (born 10 February 1948 in Niš, Serbia in the former Yugoslavia) is a mathematician and computer scientist. In 1991 he became a full professor of mathematics at the Faculty of Electronic Engineering, University of Niš in Serbia.
Biography
Petković specializes in the theory of iterative processes for solving nonlinear equations and Interval mathematics. He wrote 270 academic papers (153 in Clarivate Analytics' SCI journals) and 28 books,
including four monographs Iterative Methods for Simultaneous Inclusion of Polynomial Zeros (Springer-Verlag 1989), Complex Interval Arithmetic and Its Applications (Wiley-VCH 1998), Point Estimation of Root Finding Methods (Springer-Verlag 2008), and Multipoint Methods for Solving Nonlinear Equations (Elsevier 2013). Petković papers were cited 1193 times with Hirsch index h=20, while Elsevier's Reference Manager Mendeley displays 1565 citations and h=21.
He was visiting professor at the University of Oldenburg from 1989 to 2001, the Louis Pasteur University, Strasbourg (France, 1992), the University of Tsukuba (Japan, 2001), and a scientific researcher/invited lecturer at Columbia University, Harvard University, and at the universities of Freiburg, Zurich (ETH), Oldenburg, Berlin (Humboldt University), London, Sofia, Kiel, Tokyo, Tsukuba, Nagoya and Vienna. He took part at 60 conferences and congresses, and he was the invited lecturer on two world's congresses in 1992 and 1996, and several international conferences. He was a co-organizer of the international conference at the University of Kiel (Germany) 1998.
Petković is an Associate Editor in Journal of Computational and Applied Mathematics and Applied Mathematics and Computation and a member of editorial board of Reliable Computing, Journal of Applied Mathematics, Journal of Mathematics and Computing Systems, Journal of Complex Analysis, Mathematical Aeterna, and Novi Sad J. Math.
Petković is a member of the Serbian Scientific Society, New York Academy of Science, American Mathematical Society, GAMM, and was a member of the Serbian National Council of Science from 2010 to 2015.
Publications
Publications include:
(a collection of 110 problems in algebra, geometry, and combinatorics based on the rules of the chess game – part of this book can be read on Google books)
(preview readable on Google books)
References
External links
Personal website, list of Web of Science citation
ResearchGate website, list of publications
ResearchGate website, list of Web of Science citation
Academic profile on Mendeley.com
University of Niš alumni
Academic staff of the University of Niš
Serbian mathematicians
Recreational mathematicians
Mathematics popularizers
1948 births
Living people |
https://en.wikipedia.org/wiki/Patrick%20Pircher | Patrick Pircher (born 7 April 1982, in Bregenz) is an Austrian footballer playing for FC Dornbirn.
National team statistics
Honours
Austrian Football Bundesliga winner: 2002–03
Austrian Cup winner: 2002–03
References
External links
1982 births
Living people
Austrian men's footballers
Austria men's international footballers
Austria men's under-21 international footballers
Schwarz-Weiß Bregenz players
FK Austria Wien players
FC Admira Wacker Mödling players
SC Rheindorf Altach players
FC Augsburg players
FC Juniors OÖ players
Austrian Football Bundesliga players
2. Bundesliga players
Austrian expatriate men's footballers
Expatriate men's footballers in Germany
Men's association football defenders
Sportspeople from Bregenz
Footballers from Vorarlberg |
https://en.wikipedia.org/wiki/Dunford%E2%80%93Schwartz%20theorem | In mathematics, particularly functional analysis, the Dunford–Schwartz theorem, named after Nelson Dunford and Jacob T. Schwartz, states that the averages of powers of certain norm-bounded operators on L1 converge in a suitable sense.
Statement of the theorem
The statement is no longer true when the boundedness condition is relaxed to even .
Notes
Theorems in functional analysis |
https://en.wikipedia.org/wiki/Jurgis%20Jurgelis | Jurgis Jurgelis (born 9 August 1942 in Šiauliai, Generalbezirk Litauen, Reichskommissariat Ostland) is a mathematics teacher, politician, and signatory of the 1990 Act of the Re-Establishment of the State of Lithuania.
References
1942 births
Living people
People from Šiauliai
20th-century Lithuanian politicians
Members of the Seimas |
https://en.wikipedia.org/wiki/Hill%20tetrahedron | In geometry, the Hill tetrahedra are a family of space-filling tetrahedra. They were discovered in 1896 by M. J. M. Hill, a professor of mathematics at the University College London, who showed that they are scissor-congruent to a cube.
Construction
For every , let
be three unit vectors with angle between every two of them.
Define the Hill tetrahedron as follows:
A special case is the tetrahedron having all sides right triangles, two with sides and two with sides . Ludwig Schläfli studied as a special case of the orthoscheme, and H. S. M. Coxeter called it the characteristic tetrahedron of the cubic spacefilling.
Properties
A cube can be tiled with six copies of .
Every can be dissected into three polytopes which can be reassembled into a prism.
Generalizations
In 1951 Hugo Hadwiger found the following n-dimensional generalization of Hill tetrahedra:
where vectors satisfy for all , and where . Hadwiger showed that all such simplices are scissor congruent to a hypercube.
References
M. J. M. Hill, Determination of the volumes of certain species of tetrahedra without employment of the method of limits, Proc. London Math. Soc., 27 (1895–1896), 39–53.
H. Hadwiger, Hillsche Hypertetraeder, Gazeta Matemática (Lisboa), 12 (No. 50, 1951), 47–48.
H.S.M. Coxeter, Frieze patterns, Acta Arithmetica 18 (1971), 297–310.
E. Hertel, Zwei Kennzeichnungen der Hillschen Tetraeder, J. Geom. 71 (2001), no. 1–2, 68–77.
Greg N. Frederickson, Dissections: Plane and Fancy, Cambridge University Press, 2003.
N.J.A. Sloane, V.A. Vaishampayan, Generalizations of Schobi’s Tetrahedral Dissection, .
External links
Three piece dissection of a Hill tetrahedron into a triangular prism
Space-Filling Tetrahedra
Polyhedra
Space-filling polyhedra |
https://en.wikipedia.org/wiki/Togliatti%20surface | In algebraic geometry, a Togliatti surface is a nodal surface of degree five with 31 nodes. The first examples were constructed by . proved that 31 is the maximum possible number of nodes for a surface of this degree, showing this example to be optimal.
See also
Barth surface
Endrass surface
Sarti surface
List of algebraic surfaces
References
.
.
External links
Algebraic surfaces
Complex surfaces |
https://en.wikipedia.org/wiki/J.%20C.%20P.%20Miller | Jeffrey Charles Percy Miller (31 August 1906 – 24 April 1981) was an English mathematician and computing pioneer. He worked in number theory and on geometry, particularly polyhedra, where Miller's monster refers to the great dirhombicosidodecahedron.
He was an early member of the Computing Laboratory of the University of Cambridge. He contributed in computation to the construction and documentation of mathematical tables, and by the proposal of certain algorithms. Miller's recurrence algorithm is mentioned in the Handbook of Mathematical Functions.
As the reference says, this technique was subsequently much developed and applied, and was enunciated rather casually by Miller in a 1952 book of tables of Bessel functions.
In volume 2 of The Art of Computer Programming, Donald Knuth attributes to Miller a basic technique on formal power series, for recursive evaluation of coefficients of powers or more general functions.
In the theory of stellation of polyhedra, he made some influential suggestions to H. S. M. Coxeter. These became known as Miller's rules. The 1938 book on the fifty-nine icosahedra resulted, written by Coxeter and Patrick du Val. In the 1930s, Coxeter and Miller found 12 new uniform polyhedra, a step in the process of their complete classification in the 1950s. Miller also made an early investigation into what is now known as the Rule 90 cellular automaton.
Dr Miller was married to Germaine Miller (née Gough) in 1934 and had three children (David, Alison and Jane). Germaine died in Cambridge in her 100th year in March 2010 and is buried at St Andrew's Church, Chesterton, Cambridge.
Notes
Further reading
Doron Zeilberger,The J. C. P. Miller recurrence for exponentiating a polynomial, and its q-analog, Journal of Difference Equations and Applications, Volume 1, Issue 1 1995, pages 57 – 60.
1906 births
1981 deaths
English computer scientists
Cellular automatists
20th-century English mathematicians |
https://en.wikipedia.org/wiki/Kim%20Yoo-jin%20%28footballer%2C%20born%201983%29 | Kim Yoo-Jin (born June 19, 1983) is a South Korean retired football defender.
His previous clubs include K-League side Suwon Bluewings, Busan I'Park and J2 League side Yokohama.
Club statistics
References
External links
1983 births
Living people
South Korean men's footballers
Men's association football defenders
Suwon Samsung Bluewings players
Asan Mugunghwa FC players
Busan IPark players
Sagan Tosu players
Yokohama FC players
Liaoning F.C. players
Kim Yoo-jin
K League 1 players
J2 League players
Chinese Super League players
South Korean expatriate men's footballers
South Korean expatriate sportspeople in Japan
South Korean expatriate sportspeople in China
South Korean expatriate sportspeople in Thailand
Expatriate men's footballers in Japan
Expatriate men's footballers in China
Expatriate men's footballers in Thailand
Footballers from Busan |
https://en.wikipedia.org/wiki/Inoue%E2%80%93Hirzebruch%20surface | In mathematics, a Inoue–Hirzebruch surface is a complex surface with no meromorphic functions introduced by . They have Kodaira dimension κ = −∞, and are non-algebraic surfaces of class VII with positive second Betti number. studied some higher-dimensional analogues.
See also
List of algebraic surfaces
References
Complex surfaces |
https://en.wikipedia.org/wiki/Humbert%20surface | In algebraic geometry, a Humbert surface, studied by , is a surface in the moduli space of principally polarized abelian surfaces consisting of the surfaces with a symmetric endomorphism of some fixed discriminant.
References
Humbert, G., Sur les fonctionnes abéliennes singulières. I, II, III. J. Math. Pures Appl. serie 5, t. V, 233–350 (1899); t. VI, 279–386 (1900); t. VII, 97–123 (1901)
Algebraic surfaces
Complex surfaces |
https://en.wikipedia.org/wiki/Daniel%20Hershkowitz | Daniel Hershkowitz (; born 2 January 1953 in Haifa, Israel) is an Israeli politician, mathematician, and Orthodox rabbi. Since 2018, he has headed the . He is professor emeritus of mathematics at the Technion, and is also rabbi of the Ahuza neighborhood in Haifa. He was president of Bar-Ilan University from 2013-17.
Early life
Hershkowitz was born in Haifa. His parents were Holocaust survivors from Hungary, and his father was wounded in the 1948 Arab-Israeli War.
Hershkowitz studied at a religious high school, and graduated at age 16. He served for five years in the Intelligence Corps of the Israel Defense Forces, reaching the rank of Major.
Hershkowitz earned his BSc in mathematics in 1973, MSc in 1976, and DSc in 1982, all from the Technion – Israel Institute of Technology. His yeshiva studies were conducted at Mercaz HaRav; he received his Semikha (ordination) in 1995 from Rabbis She'ar Yashuv Cohen, Shlomo Chelouche, and Nehemyah Roth, as well as an additional ordination "Rabbi of the City" from the Chief Rabbinate of Israel (2001).
Academia
He has published over 80 mathematics articles in academic journals. He was President of the International Linear Algebra Society (2002-2008), and was previously a Professor of Mathematics at the University of Wisconsin–Madison. In 1982, he was awarded the Landau Research Prize in Mathematics; in 1990, the New England Academic Award for Excellence in Research; in 1990, the Technion's Award for Excellence in Teaching; and in 1991, the Henri Gutwirth Award for Promotion of Research.
Political career
In 2009, he was elected to the Knesset as the leader of the Jewish Home, and was appointed Minister of Science and Technology after joining Benjamin Netanyahu's government. He did not contest the 2013 elections, and subsequently left the Knesset. Since September 2018, he is the Head of the Civil Service Commission under the office of the Prime Minister of Israel.
Bar-Ilan University
He was president of Bar-Ilan University from 2013 to 2017, succeeding Moshe Kaveh and followed by Arie Zaban.
References
External links
Prof. Daniel Hershkowitz, MK, Israeli Ministry of Foreign Affairs
Daniel Hershkowitz's homepage at the Technion Mathematics Department
Rabbinic homepage
Biography, borhatorah.org
1953 births
Living people
Government ministers of Israel
Ministers of Science of Israel
Algebraists
Israeli mathematicians
Israeli Orthodox rabbis
Mercaz HaRav alumni
Jewish scientists
The Jewish Home leaders
Members of the 18th Knesset (2009–2013)
Politicians from Haifa
Technion – Israel Institute of Technology alumni
Academic staff of Technion – Israel Institute of Technology
University of Wisconsin–Madison faculty
Academic staff of Bar-Ilan University
Presidents of universities in Israel
Israeli people of Hungarian-Jewish descent
Electronic Journal of Linear Algebra editors
Rabbinic members of the Knesset |
https://en.wikipedia.org/wiki/Vandermonde%20polynomial | In algebra, the Vandermonde polynomial of an ordered set of n variables , named after Alexandre-Théophile Vandermonde, is the polynomial:
(Some sources use the opposite order , which changes the sign times: thus in some dimensions the two formulas agree in sign, while in others they have opposite signs.)
It is also called the Vandermonde determinant, as it is the determinant of the Vandermonde matrix.
The value depends on the order of the terms: it is an alternating polynomial, not a symmetric polynomial.
Alternating
The defining property of the Vandermonde polynomial is that it is alternating in the entries, meaning that permuting the by an odd permutation changes the sign, while permuting them by an even permutation does not change the value of the polynomial – in fact, it is the basic alternating polynomial, as will be made precise below.
It thus depends on the order, and is zero if two entries are equal – this also follows from the formula, but is also consequence of being alternating: if two variables are equal, then switching them both does not change the value and inverts the value, yielding and thus (assuming the characteristic is not 2, otherwise being alternating is equivalent to being symmetric).
Conversely, the Vandermonde polynomial is a factor of every alternating polynomial: as shown above, an alternating polynomial vanishes if any two variables are equal, and thus must have as a factor for all .
Alternating polynomials
Thus, the Vandermonde polynomial (together with the symmetric polynomials) generates the alternating polynomials.
Discriminant
Its square is widely called the discriminant, though some sources call the Vandermonde polynomial itself the discriminant.
The discriminant (the square of the Vandermonde polynomial: ) does not depend on the order of terms, as , and is thus an invariant of the unordered set of points.
If one adjoins the Vandermonde polynomial to the ring of symmetric polynomials in n variables , one obtains the quadratic extension , which is the ring of alternating polynomials.
Vandermonde polynomial of a polynomial
Given a polynomial, the Vandermonde polynomial of its roots is defined over the splitting field; for a non-monic polynomial, with leading coefficient a, one may define the Vandermonde polynomial as
(multiplying with a leading term) to accord with the discriminant.
Generalizations
Over arbitrary rings, one instead uses a different polynomial to generate the alternating polynomials – see (Romagny, 2005).
The Vandermonde determinant is a very special case of the Weyl denominator formula applied to the trivial representation of the special unitary group .
See also
Capelli polynomial (ref)
References
The fundamental theorem of alternating functions, by Matthieu Romagny, September 15, 2005
Polynomials
Symmetric functions |
https://en.wikipedia.org/wiki/Alternating%20polynomial | In algebra, an alternating polynomial is a polynomial such that if one switches any two of the variables, the polynomial changes sign:
Equivalently, if one permutes the variables, the polynomial changes in value by the sign of the permutation:
More generally, a polynomial is said to be alternating in if it changes sign if one switches any two of the , leaving the fixed.
Relation to symmetric polynomials
Products of symmetric and alternating polynomials (in the same variables ) behave thus:
the product of two symmetric polynomials is symmetric,
the product of a symmetric polynomial and an alternating polynomial is alternating, and
the product of two alternating polynomials is symmetric.
This is exactly the addition table for parity, with "symmetric" corresponding to "even" and "alternating" corresponding to "odd". Thus, the direct sum of the spaces of symmetric and alternating polynomials forms a superalgebra (a -graded algebra), where the symmetric polynomials are the even part, and the alternating polynomials are the odd part.
This grading is unrelated to the grading of polynomials by degree.
In particular, alternating polynomials form a module over the algebra of symmetric polynomials (the odd part of a superalgebra is a module over the even part); in fact it is a free module of rank 1, with the Vandermonde polynomial in n variables as generator.
If the characteristic of the coefficient ring is 2, there is no difference between the two concepts: the alternating polynomials are precisely the symmetric polynomials.
Vandermonde polynomial
The basic alternating polynomial is the Vandermonde polynomial:
This is clearly alternating, as switching two variables changes the sign of one term and does not change the others.
The alternating polynomials are exactly the Vandermonde polynomial times a symmetric polynomial: where is symmetric.
This is because:
is a factor of every alternating polynomial: is a factor of every alternating polynomial, as if , the polynomial is zero (since switching them does not change the polynomial, we get
so is a factor), and thus is a factor.
an alternating polynomial times a symmetric polynomial is an alternating polynomial; thus all multiples of are alternating polynomials
Conversely, the ratio of two alternating polynomials is a symmetric function, possibly rational (not necessarily a polynomial), though the ratio of an alternating polynomial over the Vandermonde polynomial is a polynomial.
Schur polynomials are defined in this way, as an alternating polynomial divided by the Vandermonde polynomial.
Ring structure
Thus, denoting the ring of symmetric polynomials by Λn, the ring of symmetric and alternating polynomials is , or more precisely , where is a symmetric polynomial, the discriminant.
That is, the ring of symmetric and alternating polynomials is a quadratic extension of the ring of symmetric polynomials, where one has adjoined a square root of the discriminant.
Alternatively, it is:
|
https://en.wikipedia.org/wiki/Donizete%20Oliveira | Donizete Francisco de Oliveira (born 21 February 1968), sometimes known as just Donizete, is a Brazilian former professional footballer who played as a midfielder.
Career statistics
Club
International
References
External links
1968 births
Living people
Brazilian men's footballers
Footballers from Bauru
Men's association football midfielders
Brazil men's international footballers
Campeonato Brasileiro Série A players
J1 League players
Fluminense FC players
Grêmio Foot-Ball Porto Alegrense players
Clube Atlético Bragantino players
São Paulo FC players
Cruzeiro Esporte Clube players
Esporte Clube Vitória players
Sporting Cristal footballers
Urawa Red Diamonds players
CR Vasco da Gama players
Brazilian expatriate men's footballers
Brazilian expatriate sportspeople in Japan
Expatriate men's footballers in Japan |
https://en.wikipedia.org/wiki/Schl%C3%A4fli%20orthoscheme | In geometry, a Schläfli orthoscheme is a type of simplex. The orthoscheme is the generalization of the right triangle to simplex figures of any number of dimensions. Orthoschemes are defined by a sequence of edges that are mutually orthogonal. They were introduced by Ludwig Schläfli, who called them orthoschemes and studied their volume in Euclidean, hyperbolic, and spherical geometries. H. S. M. Coxeter later named them after Schläfli. As right triangles provide the basis for trigonometry, orthoschemes form the basis of a trigonometry of n dimensions, as developed by Schoute who called it polygonometry. J.-P. Sydler and Børge Jessen studied orthoschemes extensively in connection with Hilbert's third problem.
Orthoschemes, also called path-simplices in the applied mathematics literature, are a special case of a more general class of simplices studied by Fiedler, and later rediscovered by Coxeter. These simplices are the convex hulls of trees in which all edges are mutually perpendicular. In an orthoscheme, the underlying tree is a path.
In three dimensions, an orthoscheme is also called a birectangular tetrahedron (because its path makes two right angles at vertices each having two right angles) or a quadrirectangular tetrahedron (because it contains four right angles).
Properties
All 2-faces are right triangles.
All facets of a d-dimensional orthoscheme are (d − 1)-dimensional orthoschemes.
The dihedral angles that are disjoint from edges of the path have acute angles; the remaining dihedral angles are all right angles.
The midpoint of the longest edge is the center of the circumscribed sphere.
The case when is a generalized Hill tetrahedron.
Every hypercube in d-dimensional space can be dissected into d! congruent orthoschemes. A similar dissection into the same number of orthoschemes applies more generally to every hyperrectangle but in this case the orthoschemes may not be congruent.
Every regular polytope can be dissected radially into g congruent orthoschemes that meet at its center, where g is the order of the regular polytope's symmetry group.
In 3- and 4-dimensional Euclidean space, every convex polytope is scissor congruent to an orthoscheme.
Every orthoscheme can be trisected into three smaller orthoschemes.
In 3-dimensional hyperbolic and spherical spaces, the volume of orthoschemes can be expressed in terms of the Lobachevsky function, or in terms of dilogarithms.
Dissection into orthoschemes
Hugo Hadwiger conjectured in 1956 that every simplex can be dissected into finitely many orthoschemes. The conjecture has been proven in spaces of five or fewer dimensions, but remains unsolved in higher dimensions.
Hadwiger's conjecture implies that every convex polytope can be dissected into orthoschemes.
Characteristic simplex of the general regular polytope
Coxeter identifies various orthoschemes as the characteristic simplexes of the polytopes they generate by reflections. The characteristic simplex is the |
https://en.wikipedia.org/wiki/Monostatic%20polytope | In geometry, a monostatic polytope (or unistable polyhedron) is a d-polytope which "can stand on only one face". They were described in 1969 by J. H. Conway, M. Goldberg, R. K. Guy and K. C. Knowlton. The monostatic polytope in 3-space constructed independently by Guy and Knowlton has 19 faces. In 2012, Andras Bezdek discovered an 18-face solution, and in 2014, Alex Reshetov published a 14-face object.
Definition
A polytope is called monostatic if, when filled homogeneously, it is stable on only one facet. Alternatively, a polytope is monostatic if its centroid (the center of mass) has an orthogonal projection in the interior of only one facet.
Properties
No convex polygon in the plane is monostatic. This was shown by V. Arnold via reduction to the four-vertex theorem.
There are no monostatic simplices in dimension up to 8. In dimension 3 this is due to Conway. In dimension up to 6 this is due to R. J. M. Dawson. Dimensions 7 and 8 were ruled out by R. J. M. Dawson, W. Finbow, and P. Mak.
(R. J. M. Dawson) There exist monostatic simplices in dimension 10 and up.
(Lángi) There are monostatic polytopes in dimension 3 whose shapes are arbitrarily close to a sphere.
(Lángi) There are monostatic polytopes in dimension 3 with k-fold rotational symmetry for an arbitrary positive integer k.
See also
Gömböc
Roly-poly toy
References
J. H. Conway, M. Goldberg and R. K. Guy, Problem 66-12, SIAM Review 11 (1969), 78–82.
K. C. Knowlton, A unistable polyhedron with only 19 faces, Bell Telephone Laboratories MM 69-1371-3 (Jan. 3, 1969).
H. Croft, K. Falconer, and R. K. Guy, Problem B12 in Unsolved Problems in Geometry, New York: Springer-Verlag, p. 61, 1991.
R. J. M. Dawson, Monostatic simplexes. Amer. Math. Monthly 92 (1985), no. 8, 541–546.
R. J. M. Dawson, W. Finbow, P. Mak, Monostatic simplexes. II. Geom. Dedicata 70 (1998), 209–219.
R. J. M. Dawson, W. Finbow, Monostatic simplexes. III. Geom. Dedicata 84 (2001), 101–113.
Z. Lángi, A solution to some problems of Conway and Guy on monostable polyhedra, Bull. Lond. Math. Soc. 54 (2022), no. 2, 501–516.
Igor Pak, Lectures on Discrete and Polyhedral Geometry, Section 9.
A. Reshetov, A unistable polyhedron with 14 faces. Int. J. Comput. Geom. Appl. 24 (2014), 39–60.
External links
YouTube: The uni-stable polyhedron
Wolfram Demonstrations Project: Bezdek's Unistable Polyhedron With 18 Faces
Polyhedra |
https://en.wikipedia.org/wiki/Pentagon | In geometry, a pentagon (from the Greek πέντε pente meaning five and γωνία gonia meaning angle) is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°.
A pentagon may be simple or self-intersecting. A self-intersecting regular pentagon (or star pentagon) is called a pentagram.
Regular pentagons
A regular pentagon has Schläfli symbol {5} and interior angles of 108°.
A regular pentagon has five lines of reflectional symmetry, and rotational symmetry of order 5 (through 72°, 144°, 216° and 288°). The diagonals of a convex regular pentagon are in the golden ratio to its sides. Given its side length its height (distance from one side to the opposite vertex), width (distance between two farthest separated points, which equals the diagonal length ) and circumradius are given by:
The area of a convex regular pentagon with side length is given by
If the circumradius of a regular pentagon is given, its edge length is found by the expression
and its area is
since the area of the circumscribed circle is the regular pentagon fills approximately 0.7568 of its circumscribed circle.
Derivation of the area formula
The area of any regular polygon is:
where P is the perimeter of the polygon, and r is the inradius (equivalently the apothem). Substituting the regular pentagon's values for P and r gives the formula
with side length t.
Inradius
Similar to every regular convex polygon, the regular convex pentagon has an inscribed circle. The apothem, which is the radius r of the inscribed circle, of a regular pentagon is related to the side length t by
Chords from the circumscribed circle to the vertices
Like every regular convex polygon, the regular convex pentagon has a circumscribed circle. For a regular pentagon with successive vertices A, B, C, D, E, if P is any point on the circumcircle between points B and C, then PA + PD = PB + PC + PE.
Point in plane
For an arbitrary point in the plane of a regular pentagon with circumradius , whose distances to the centroid of the regular pentagon and its five vertices are and
respectively, we have
If are the distances from the vertices of a regular pentagon to any point on its circumcircle, then
Geometrical constructions
The regular pentagon is constructible with compass and straightedge, as 5 is a Fermat prime. A variety of methods are known for constructing a regular pentagon. Some are discussed below.
Richmond's method
One method to construct a regular pentagon in a given circle is described by Richmond and further discussed in Cromwell's Polyhedra.
The top panel shows the construction used in Richmond's method to create the side of the inscribed pentagon. The circle defining the pentagon has unit radius. Its center is located at point C and a midpoint M is marked halfway along its radius. This point is joined to the periphery vertically above the center at point D. Angle CMD is bisected, and the bisector intersects the vertical axis at poin |
https://en.wikipedia.org/wiki/Squaregraph | In graph theory, a branch of mathematics, a squaregraph is a type of undirected graph that can be drawn in the plane in such a way that every bounded face is a quadrilateral and every vertex with three or fewer neighbors is incident to an unbounded face.
Related graph classes
The squaregraphs include as special cases trees, grid graphs, gear graphs, and the graphs of polyominos.
As well as being planar graphs, squaregraphs are median graphs, meaning that for every three vertices u, v, and w there is a unique median vertex m(u,v,w) that lies on shortest paths between each pair of the three vertices. As with median graphs more generally, squaregraphs are also partial cubes: their vertices can be labeled with binary strings such that the Hamming distance between strings is equal to the shortest path distance between vertices.
The graph obtained from a squaregraph by making a vertex for each zone (an equivalence class of parallel edges of quadrilaterals) and an edge for each two zones that meet in a quadrilateral is a circle graph determined by a triangle-free chord diagram of the unit disk.
Characterization
Squaregraphs may be characterized in several ways other than via their planar embeddings:
They are the median graphs that do not contain as an induced subgraph any member of an infinite family of forbidden graphs. These forbidden graphs are the cube (the simplex graph of K3), the Cartesian product of an edge and a claw K1,3 (the simplex graph of a claw), and the graphs formed from a gear graph by adding one more vertex connected to the hub of the wheel (the simplex graph of the disjoint union of a cycle with an isolated vertex).
They are the graphs that are connected and bipartite, such that (if an arbitrary vertex r is picked as a root) every vertex has at most two neighbors closer to r, and such that at every vertex v, the link of v (a graph with a vertex for each edge incident to v and an edge for each 4-cycle containing v) is either a cycle of length greater than three or a disjoint union of paths.
They are the dual graphs of arrangements of lines in the hyperbolic plane that do not have three mutually-crossing lines.
Algorithms
The characterization of squaregraphs in terms of distance from a root and links of vertices can be used together with breadth first search as part of a linear time algorithm for testing whether a given graph is a squaregraph, without any need to use the more complex linear-time algorithms for planarity testing of arbitrary graphs.
Several algorithmic problems on squaregraphs may be computed more efficiently than in more general planar or median graphs; for instance, and present linear time algorithms for computing the diameter of squaregraphs, and for finding a vertex minimizing the maximum distance to all other vertices.
Notes
References
.
.
.
.
Graph families
Planar graphs
Bipartite graphs |
https://en.wikipedia.org/wiki/Leverage%20%28statistics%29 | In statistics and in particular in regression analysis, leverage is a measure of how far away the independent variable values of an observation are from those of the other observations. High-leverage points, if any, are outliers with respect to the independent variables. That is, high-leverage points have no neighboring points in space, where is the number of independent variables in a regression model. This makes the fitted model likely to pass close to a high leverage observation. Hence high-leverage points have the potential to cause large changes in the parameter estimates when they are deleted i.e., to be influential points. Although an influential point will typically have high leverage, a high leverage point is not necessarily an influential point. The leverage is typically defined as the diagonal elements of the hat matrix.
Definition and interpretations
Consider the linear regression model , . That is, , where, is the design matrix whose rows correspond to the observations and whose columns correspond to the independent or explanatory variables. The leverage score for the independent observation is given as:
, the diagonal element of the ortho-projection matrix (a.k.a hat matrix) .
Thus the leverage score can be viewed as the 'weighted' distance between to the mean of 's (see its relation with Mahalanobis distance). It can also be interpreted as the degree by which the measured (dependent) value (i.e., ) influences the fitted (predicted) value (i.e., ): mathematically,
.
Hence, the leverage score is also known as the observation self-sensitivity or self-influence. Using the fact that (i.e., the prediction is ortho-projection of onto range space of ) in the above expression, we get . Note that this leverage depends on the values of the explanatory variables of all observations but not on any of the values of the dependent variables .
Properties
The leverage is a number between 0 and 1, Proof: Note that is idempotent matrix () and symmetric (). Thus, by using the fact that , we have . Since we know that , we have .
Sum of leverages is equal to the number of parameters in (including the intercept). Proof: .
Determination of outliers in X using leverages
Large leverage corresponds to an that is extreme. A common rule is to identify whose leverage value is more than 2 times larger than the mean leverage (see property 2 above). That is, if , shall be considered an outlier. Some statisticians prefer the threshold of instead of .
Relation to Mahalanobis distance
Leverage is closely related to the Mahalanobis distance (proof). Specifically, for some matrix , the squared Mahalanobis distance of (where is row of ) from the vector of mean of length , is , where is the estimated covariance matrix of 's. This is related to the leverage of the hat matrix of after appending a column vector of 1's to it. The relationship between the two is:
This relationship enables us to decompose leverage into m |
https://en.wikipedia.org/wiki/List%20of%20mathematics%20education%20journals | This is a list of notable academic journals in the field of mathematics education.
C
College Mathematics Journal
E
Educational Studies in Mathematics
F
For the Learning of Mathematics
I
International Journal of Science and Mathematics Education
Investigations in Mathematics Learning
J
Journal for Research in Mathematics Education
Journal of Mathematics Teacher Education
M
The Mathematics Educator
The Mathematics Enthusiast
Mathematics Teacher
Mathematics Teaching
P
Philosophy of Mathematics Education Journal
Problems, Resources, and Issues in Mathematics Undergraduate Studies
S
Science & Education
T
Teaching Mathematics and Its Applications
See also
List of probability journals
List of statistics journals
List of mathematics journals
External links
Online list of some journals in mathematics education
Mathematics education |
https://en.wikipedia.org/wiki/Barth%20surface |
In algebraic geometry, a Barth surface is one of the complex nodal surfaces in 3 dimensions with large numbers of double points found by . Two examples are the Barth sextic of degree 6 with 65 double points, and the Barth decic of degree 10 with 345 double points.
For degree 6 surfaces in P3, showed that 65 is the maximum number of double points possible.
The Barth sextic is a counterexample to an incorrect claim by Francesco Severi in 1946 that 52 is the maximum number of double points possible.
Informal accounting of the 65 ordinary double points of the Barth Sextic
The Barth Sextic may be visualized in three dimensions as featuring 50 finite and 15 infinite ordinary double points (nodes).
Referring to the figure, the 50 finite ordinary double points are arrayed as the vertices of 20 roughly tetrahedral shapes oriented such that the bases of these four-sided "outward pointing" shapes form the triangular faces of a regular icosidodecahedron. To these 30 icosidodecahedral vertices are added the summit vertices of the 20 tetrahedral shapes. These 20 points themselves are the vertices of a concentric regular dodecahedron circumscribed about the inner icosidodecahedron. Together, these are the 50 finite ordinary double points of the figure.
The 15 remaining ordinary double points at infinity correspond to the 15 lines that pass through the opposite vertices of the inscribed icosidodecahedron, all 15 of which also intersect in the center of the figure. .
See also
Endrass surface
Sarti surface
Togliatti surface
List of algebraic surfaces
References
.
.
.
External links
Algebraic surfaces
Complex surfaces |
https://en.wikipedia.org/wiki/Real%20RAM | In computing, especially computational geometry, a real RAM (random-access machine) is a mathematical model of a computer that can compute with exact real numbers instead of the binary fixed point or floating point numbers used by most actual computers. The real RAM was formulated by Michael Ian Shamos in his 1978 Ph.D. dissertation.
Model
The "RAM" part of the real RAM model name stands for "random-access machine". This is a model of computing that resembles a simplified version of a standard computer architecture. It consists of a stored program, a computer memory unit consisting of an array of cells, and a central processing unit with a bounded number of registers. Each memory cell or register can store a real number. Under the control of the program, the real RAM can transfer real numbers between memory and registers, and perform arithmetic operations on the values stored in the registers.
The allowed operations typically include addition, subtraction, multiplication, and division, as well as comparisons, but not modulus or rounding to integers. The reason for avoiding integer rounding and modulus operations is that allowing these operations could give the real RAM unreasonable amounts of computational power, enabling it to solve PSPACE-complete problems in polynomial time.
When analyzing algorithms for the real RAM, each allowed operation is typically assumed to take constant time.
Implementation
Software libraries such as LEDA have been developed which allow programmers to write computer programs that work as if they were running on a real RAM.
These libraries represent real values using data structures which allow them to perform arithmetic and comparisons with the same results as a real RAM would produce. For example, In LEDA, real numbers are represented using the leda_real datatype, which supports k-th roots for any natural number k, rational operators, and comparison operators. The time analysis of the underlying real RAM algorithm using these real datatypes
can be interpreted as counting the number of library calls needed by a given algorithm.
Related models
The real RAM closely resembles the later Blum–Shub–Smale machine. However, the real RAM is typically used for the analysis of concrete algorithms in computational geometry, while the Blum–Shub–Smale machine instead forms the basis for extensions of the theory of NP-completeness to real-number computation.
An alternative to the real RAM is the word RAM, in which both the inputs to a problem and the values stored in memory and registers are assumed to be integers with a fixed number of bits. The word RAM model can perform some operations more quickly than the real RAM; for instance, it allows fast integer sorting algorithms, while sorting on the real RAM must be done with slower comparison sorting algorithms. However, some computational geometry problems have inputs or outputs that cannot be represented exactly using integer coordinates; see for instance the Perles configurati |
https://en.wikipedia.org/wiki/Computing%20the%20permanent | In linear algebra, the computation of the permanent of a matrix is a problem that is thought to be more difficult than the computation of the determinant of a matrix despite the apparent similarity of the definitions.
The permanent is defined similarly to the determinant, as a sum of products of sets of matrix entries that lie in distinct rows and columns. However, where the determinant weights each of these products with a ±1 sign based on the parity of the set, the permanent weights them all with a +1 sign.
While the determinant can be computed in polynomial time by Gaussian elimination, it is generally believed that the permanent cannot be computed in polynomial time. In computational complexity theory, a theorem of Valiant states that computing permanents is #P-hard, and even #P-complete for matrices in which all entries are 0 or 1 . This puts the computation of the permanent in a class of problems believed to be even more difficult to compute than NP. It is known that computing the permanent is impossible for logspace-uniform ACC0 circuits.
The development of both exact and approximate algorithms for computing the permanent of a matrix is an active area of research.
Definition and naive algorithm
The permanent of an n-by-n matrix A = (ai,j) is defined as
The sum here extends over all elements σ of the symmetric group Sn, i.e. over all permutations of the numbers 1, 2, ..., n. This formula differs from the corresponding formula for the determinant only in that, in the determinant, each product is multiplied by the sign of the permutation σ while in this formula each product is unsigned. The formula may be directly translated into an algorithm that naively expands the formula, summing over all permutations and within the sum multiplying out each matrix entry. This requires n! n arithmetic operations.
Ryser formula
The best known general exact algorithm is due to .
Ryser’s method is based on an inclusion–exclusion formula that can be given as follows: Let be obtained from A by deleting k columns, let be the product of the row-sums of , and let be the sum of the values of over all possible . Then
It may be rewritten in terms of the matrix entries as follows
Ryser's formula can be evaluated using arithmetic operations, or by processing the sets in Gray code order.
Balasubramanian–Bax–Franklin–Glynn formula
Another formula that appears to be as fast as Ryser's (or perhaps even twice as fast) is to be found in the two Ph.D. theses; see , ; also
. The methods to find the formula are quite different, being related to the combinatorics of the Muir algebra, and to finite difference theory respectively. Another way, connected with invariant theory is via the polarization identity for a symmetric tensor . The formula generalizes to infinitely many others, as found by all these authors, although it is not clear if they are any faster than the basic one. See .
The simplest known formula of this type (when the characteristic of the |
https://en.wikipedia.org/wiki/Pequea%20Valley%20High%20School | Pequea Valley High School is the only secondary school in the Pequea Valley School District. It is located in Kinzers, Lancaster County, Pennsylvania, United States.
Statistics
Attendance at Pequea Valley Senior High School during the 2005–2006 school year was 92.91%, compared with the 87.97% scored in the prior year. Students were 57.1% proficient in math and 73.4% proficient in reading.
In 2012, the school's varsity soccer team won the state title.
References
Public high schools in Pennsylvania
Schools in Lancaster County, Pennsylvania |
https://en.wikipedia.org/wiki/Geometriae%20Dedicata | Geometriae Dedicata is a mathematical journal, founded in 1972, concentrating on geometry and its relationship to topology, group theory and the theory of dynamical systems. It was created on the initiative of Hans Freudenthal in Utrecht, the Netherlands. It is published by Springer Netherlands. The Editor-in-Chief is Richard Alan Wentworth.
References
External links
Springer site
Mathematics journals
Springer Science+Business Media academic journals |
https://en.wikipedia.org/wiki/Clique%20complex | Clique complexes, independence complexes, flag complexes, Whitney complexes and conformal hypergraphs are closely related mathematical objects in graph theory and geometric topology that each describe the cliques (complete subgraphs) of an undirected graph.
Clique complex
The clique complex of an undirected graph is an abstract simplicial complex (that is, a family of finite sets closed under the operation of taking subsets), formed by the sets of vertices in the cliques of . Any subset of a clique is itself a clique, so this family of sets meets the requirement of an abstract simplicial complex that every subset of a set in the family should also be in the family.
The clique complex can also be viewed as a topological space in which each clique of vertices is represented by a simplex of dimension . The 1-skeleton of (also known as the underlying graph of the complex) is an undirected graph with a vertex for every 1-element set in the family and an edge for every 2-element set in the family; it is isomorphic to .
Negative example
Every clique complex is an abstract simplicial complex, but the opposite is not true. For example, consider the abstract simplicial complex over with maximal sets If it were the of some graph , then had to have the edges so should also contain the clique
Independence complex
The independence complex of an undirected graph is an abstract simplicial complex formed by the sets of vertices in the independent sets of . The clique complex of is equivalent to the independence complex of the complement graph of .
Flag complex
A flag complex is an abstract simplicial complex with an additional property called "2-determined": for every subset S of vertices, if every pair of vertices in S is in the complex, then S itself is in the complex too.
Every clique complex is a flag complex: if every pair of vertices in S is a clique of size 2, then there is an edge between them, so S is a clique.
Every flag complex is a clique complex: given a flag complex, define a graph G on the set of all vertices, where two vertices u,v are adjacent in G iff {u,v} is in the complex (this graph is called the 1-skeleton of the complex). By definition of a flag complex, every set of vertices that are pairwise-connected, is in the complex. Therefore, the flag complex is equal to the clique complex on G.
Thus, flag complexes and clique complexes are essentially the same thing. However, in many cases it is convenient to define a flag complex directly from some data other than a graph, rather than indirectly as the clique complex of a graph derived from that data.
Mikhail Gromov defined the no-Δ condition to be the condition of being a flag complex.
Whitney complex
Clique complexes are also known as Whitney complexes, after Hassler Whitney. A Whitney triangulation or clean triangulation of a two-dimensional manifold is an embedding of a graph onto the manifold in such a way that every face is a triangle and every triangle |
https://en.wikipedia.org/wiki/Simplex%20graph | In graph theory, a branch of mathematics, the simplex graph of an undirected graph is itself a graph, with one node for each clique (a set of mutually adjacent vertices) in . Two nodes of are linked by an edge whenever the corresponding two cliques differ in the presence or absence of a single vertex.
The empty set is included as one of the cliques of that are used to form the clique graph, as is every set of one vertex and every set of two adjacent vertices. Therefore, the simplex graph contains within it a subdivision of itself. The simplex graph of a complete graph is a hypercube graph, and the simplex graph of a cycle graph of length four or more is a gear graph. The simplex graph of the complement graph of a path graph is a Fibonacci cube.
The complete subgraphs of can be given the structure of a median algebra: the median of three cliques , , and is formed by the vertices that belong to a majority of the three cliques. Any two vertices belonging to this median set must both belong to at least one of , , or , and therefore must be linked by an edge, so the median of three cliques is itself a clique. The simplex graph is the median graph corresponding to this median algebra structure. When is the complement graph of a bipartite graph, the cliques of can be given a stronger structure as a distributive lattice, and in this case the simplex graph is the graph of the lattice. As is true for median graphs more generally, every simplex graph is itself bipartite.
The simplex graph has one vertex for every simplex in the clique complex of , and two vertices are linked by an edge when one of the two corresponding simplexes is a facet of the other. Thus, the objects (vertices in the simplex graph, simplexes in ) and relations between objects (edges in the simplex graph, inclusion relations between simplexes in ) are in one-to-one correspondence between and .
Simplex graphs were introduced by , who observed that a simplex graph has no cubes if and only if the underlying graph is triangle-free, and showed that the chromatic number of the underlying graph equals the minimum number such that the simplex graph can be isometrically embedded into a Cartesian product of trees. As a consequence of the existence of triangle-free graphs with high chromatic number, they showed that there exist two-dimensional topological median algebras that cannot be embedded into products of finitely many real trees. also use simplex graphs as part of their proof that testing whether a graph is triangle-free or whether it is a median graph may be performed equally quickly.
Notes
References
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Graph operations
Graph families
Bipartite graphs |
https://en.wikipedia.org/wiki/Operator%20space | In functional analysis, a discipline within mathematics, an operator space is a normed vector space (not necessarily a Banach space) "given together with an isometric embedding into the space B(H) of all bounded operators on a Hilbert space H.". The appropriate morphisms between operator spaces are completely bounded maps.
Equivalent formulations
Equivalently, an operator space is a subspace of a C*-algebra.
Category of operator spaces
The category of operator spaces includes operator systems and operator algebras. For operator systems, in addition to an induced matrix norm of an operator space, one also has an induced matrix order. For operator algebras, there is still the additional ring structure.
See also
Gilles Pisier
Operator system
References
Banach spaces
Operator theory |
https://en.wikipedia.org/wiki/B%C3%B8rge%20Jessen | Børge Christian Jessen (19 June 1907 – 20 March 1993) was a Danish mathematician best known for his work in analysis, specifically on the Riemann zeta function, and in geometry, specifically on Hilbert's third problem.
Early years
Jessen was born on 19 June 1907 in Copenhagen to Hans Jessen and Christine Jessen (née Larsen). He attended Skt. Jørgens Gymnasium, where he was taught by the Hungarian mathematician Julius Pal during his first year. In 1925, Jessen graduated from the gymnasium and enrolled at the University of Copenhagen. During his time at the university he got to know Harald Bohr, then a leading figure in Danish mathematics. In 1928, Bohr established a collaboration with Jessen, which would last until Bohr's death in 1951.
After receiving his master's degree in the spring of 1929, Jessen embarked on a stay abroad. Supported by the Carlsberg Foundation, he spent the fall of 1929 at the University of Szeged, where he met Frigyes Riesz, Alfréd Haar, and Lipót Fejér. He then spent the winter semester of 1929–30 at the University of Göttingen, where he attended lectures by David Hilbert and Edmund Landau while working on his PhD thesis. On 1 May 1930 Jessen defended his thesis in Copenhagen. He later elaborated the thesis into an article that was published in Acta Mathematica in 1934. The same year, he was appointed as a docent at The Royal Veterinary and Agricultural University in Denmark.
In 1931, Jessen married Ellen Pedersen (1903–1979), cand. mag. in mathematics and the daughter of Peder Oluf Pedersen. Jessen continued to travel frequently in the early 1930s, visiting Paris, Cambridge, England, the Institute for Advanced Study, Yale and Harvard University in America.
Career
Jessen was a professor of descriptive geometry at the Technical University of Denmark from 1935 till 1942, when he moved back to the University of Copenhagen where he was professor from 1942 to 1977 when he retired. He was the president of the Carlsberg Foundation in 1955-1963 and one of the founders of the Hans Christian Ørsted Institute. He was the Secretary of the Interim Executive Committee of the International Mathematical Union (1950–1952), and in September 1951 he officially declared the founding of the Union, with its first domicile in Copenhagen. He was also active in the Danish Mathematical Society. After his death, the society named an award in his honor (Børge Jessen Diploma Award).
See also
Jessen's icosahedron
Jessen–Wintner theorem
References
External links
Bernard Bru and Salah Eid "Jessen’s theorem and Lévy’s lemma" in JEHPS June 2009
A short biography
1907 births
1993 deaths
Scientists from Copenhagen
Danish mathematicians
20th-century Danish mathematicians
University of Copenhagen alumni
Academic staff of the University of Copenhagen
Mathematical analysts
Geometers
Institute for Advanced Study visiting scholars |
https://en.wikipedia.org/wiki/Educational%20Studies%20in%20Mathematics | Educational Studies in Mathematics is a peer-reviewed academic journal covering mathematics education. It was established by Hans Freudenthal in 1968. The journal is published by Springer Science+Business Media and the editors-in-chief are Susanne Prediger (Technical University of Dortmund) and David Wagner (University of New Brunswick). According to the Journal Citation Reports, the journal has a 2020 impact factor of 2.402.
Editors-in-chief
The following persons are or have been editors-in-chief:
1968-1977: Hans Freudenthal
1978-1989: Alan Bishop
1990-1995: Willibald Dörfler
1996-2000: Kenneth Ruthven
2001-2005: Anna Sierpińska
2006-2008: Tommy Dreyfus
2009-2013: Norma Presmeg
2014-2018: Merrilyn Goos
2019-2020: Arthur Bakker
2021-2022: Arthur Bakker and David Wagner
2022-present: Susanne Prediger and David Wagner
See also
List of mathematics education journals
References
External links
Academic journals established in 1968
Mathematics education journals
Springer Science+Business Media academic journals
English-language journals |
https://en.wikipedia.org/wiki/Yu%20Shimasaki | is a former Japanese football player.
Club statistics
References
External links
1985 births
Living people
Juntendo University alumni
Association football people from Osaka Prefecture
People from Ibaraki, Osaka
Japanese men's footballers
J2 League players
Sagan Tosu players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Koya%20Shimizu | is a Japanese football player. He plays for Briobecca Urayasu.
Club statistics
References
External links
1982 births
Living people
Kokushikan University alumni
Association football people from Tokyo
Japanese men's footballers
J2 League players
Japan Football League players
Vegalta Sendai players
Sagan Tosu players
Tokyo Verdy players
Briobecca Urayasu players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Toyoki%20Hasegawa | was a Japanese football player he is currently assistant manager Japan Football League club Verspah Oita.
Hasegawa previously played for Sagan Tosu in the J2 League.
Club statistics
References
External links
1986 births
Living people
Association football people from Kumamoto Prefecture
Japanese men's footballers
J2 League players
Japan Football League players
Sagan Tosu players
Verspah Oita players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Sho%20Shimoji | is a Japanese football player for Thai League 2 club Samut Prakan City.
Club statistics
References
External links
1985 births
Living people
Aoyama Gakuin University alumni
Association football people from Okinawa Prefecture
Japanese men's footballers
J2 League players
Sagan Tosu players
Sportivo Luqueño players
Clube Atlético Linense players
Sho Shimoji
Sho Shimoji
Japanese expatriate men's footballers
Japanese expatriate sportspeople in Paraguay
Expatriate men's footballers in Paraguay
Expatriate men's footballers in Brazil
Expatriate men's footballers in Thailand
Men's association football midfielders |
https://en.wikipedia.org/wiki/Yusuke%20Yada | is a former Japanese football player.
Club statistics
References
External links
1983 births
Living people
Hosei University alumni
Association football people from Saitama Prefecture
Japanese men's footballers
J2 League players
Japan Football League players
Kataller Toyama players
Sagan Tosu players
Men's association football defenders |
https://en.wikipedia.org/wiki/Yuki%20Kuriyama | is a Japanese former footballer.
Club statistics
References
External links
1988 births
Living people
Japanese men's footballers
J2 League players
Japan Football League players
Sagan Tosu players
Matsumoto Yamaga FC players
Kagoshima United FC players
Men's association football midfielders
Association football people from Kumamoto |
https://en.wikipedia.org/wiki/Hirokazu%20Hasegawa | is a former Japanese football player.
Club statistics
References
External links
1986 births
Living people
Hiroshima University of Economics alumni
Japanese men's footballers
J2 League players
Sagan Tosu players
Oita Trinita players
Men's association football midfielders
Association football people from Hiroshima |
https://en.wikipedia.org/wiki/Influence%20of%20nonstandard%20analysis | Abraham Robinson's theory of nonstandard analysis has been applied in a number of fields.
Probability theory
"Radically elementary probability theory" of Edward Nelson combines the discrete and the continuous theory through the infinitesimal approach. The model-theoretical approach of nonstandard analysis together with Loeb measure theory allows one to define Brownian motion as a hyperfinite random walk, obviating the need for cumbersome measure-theoretic developments. Jerome Keisler used this classical approach of nonstandard analysis to characterize general stochastic processes as hyperfinite ones.
Economics
Economists have used nonstandard analysis to model markets with large numbers of agents (see Robert M. Anderson (economist)).
Education
An article by Michèle Artigue concerns the teaching of analysis. Artigue devotes a section, "The non standard analysis and its weak impact on education" on page 172, to non-standard analysis. She writes:
The non-standard analysis revival and its weak impact on education. The publication in 1966 of Robinson's book NSA constituted in some sense a rehabilitation of infinitesimals which had fallen into disrepute [...] [Robinson's proposal] was met with suspicion, even hostility, by many mathematicians [...] Nevertheless, despite the obscurity of this first work, NSA developed rapidly [...] The attempts at simplification were often conducted with the aim of producing an elementary way of teaching NSA. This was the case with the work of Keisler and Henle-Kleinberg [...]
Artigue continues specifically with reference to the calculus textbook:
[Keisler's work] served as a reference text for a teaching experiment in the first year in university in the Chicago area in 1973-74. Sullivan used 2 questionnaires to evaluate the effects of the course, one for teachers, the other for students. The 11 teachers involved gave a very positive opinion of the experience. The student questionnaire revealed no significant difference in technical performance [...] but showed that those following the NSA course were better able to interpret the sense of the mathematical formalism of calculus [...] The appearance of the 2nd book of Keisler led to a virulent criticism by Bishop, accusing Keisler of seeking [...] to convince students that mathematics is only "an esoteric and meaningless exercise in technique", detached from any reality. These criticisms were in opposition to the declarations of the partisans of NSA who affirmed with great passion its simplicity and intuitive character. [...] However, it is necessary to emphasize the weak impact of NSA on contemporary education. The small number of reported instances of this approach are often accompanied with passionate advocacy, but this rarely rises above the level of personal conviction.
Authors of books on hyperreals
Sergio Albeverio
Robert M. Anderson (economist)
Leif Arkeryd
Nigel Cutland
Martin Davis
Jens Erik Fenstad
Robert Goldblatt
Reuben Hersh
Raphael Høegh-Krohn
Karel H |
https://en.wikipedia.org/wiki/Hyperbolic%20law%20of%20cosines | In hyperbolic geometry, the "law of cosines" is a pair of theorems relating the sides and angles of triangles on a hyperbolic plane, analogous to the planar law of cosines from plane trigonometry, or the spherical law of cosines in spherical trigonometry. It can also be related to the relativistic velocity addition formula.
History
Describing relations of hyperbolic geometry, Franz Taurinus showed in 1826 that the spherical law of cosines can be related to spheres of imaginary radius, thus he arrived at the hyperbolic law of cosines in the form:
which was also shown by Nikolai Lobachevsky (1830):
Ferdinand Minding gave it in relation to surfaces of constant negative curvature:
as did Delfino Codazzi in 1857:
The relation to relativity using rapidity was shown by Arnold Sommerfeld in 1909 and Vladimir Varićak in 1910.
Hyperbolic laws of cosines
Take a hyperbolic plane whose Gaussian curvature is . Given a hyperbolic triangle with angles and side lengths , , and , the following two rules hold. The first is an analogue of Euclidean law of cosines, expressing the length of one side in terms of the other two and the angle between the latter:
The second law has no Euclidean analogue, since it expresses the fact that lengths of sides of a hyperbolic triangle are determined by the interior angles:
Houzel indicates that the hyperbolic law of cosines implies the angle of parallelism in the case of an ideal hyperbolic triangle:
Hyperbolic law of Haversines
In cases where is small, and being solved for, the numerical precision of the standard form of the hyperbolic law of cosines will drop due to rounding errors, for exactly the same reason it does in the Spherical law of cosines. The hyperbolic version of the law of haversines can prove useful in this case:
Relativistic velocity addition via hyperbolic law of cosines
Setting in (), and by using hyperbolic identities in terms of the hyperbolic tangent, the hyperbolic law of cosines can be written:
In comparison, the velocity addition formulas of special relativity for the x and y-directions as well as under an arbitrary angle , where is the relative velocity between two inertial frames, the velocity of another object or frame, and the speed of light, is given by
It turns out that this result corresponds to the hyperbolic law of cosines - by identifying with relativistic rapidities the equations in () assume the form:
See also
Hyperbolic law of sines
Hyperbolic triangle trigonometry
History of Lorentz transformations
References
Bibliography
External links
Non Euclidean Geometry, Math Wiki at TU Berlin
Velocity Compositions and Rapidity, at MathPages
Hyperbolic geometry
Special relativity
es:Teorema del coseno#Geometría hiperbólica
fr:Théorème d'Al-Kashi#Géométrie hyperbolique
pl:Twierdzenie cosinusów#Wzory cosinusów w geometriach nieeuklidesowych |
https://en.wikipedia.org/wiki/Jun%20Suzuki%20%28footballer%2C%20born%201989%29 | is a Japanese football player currently playing for Lithuanian side Sūduva.
Career statistics
Updated to end of 2018 season.
National team career statistics
Appearances in major competitions
References
External links
Profile at Oita Trinita
1989 births
Living people
Association football people from Fukuoka (city)
Japanese men's footballers
J1 League players
J2 League players
J3 League players
Avispa Fukuoka players
Tokyo Verdy players
Oita Trinita players
Fujieda MYFC players
Men's association football midfielders
Japanese expatriate men's footballers
Japanese expatriate sportspeople in Lithuania
Expatriate men's footballers in Lithuania |
https://en.wikipedia.org/wiki/Kyohei%20Oyama | is a former Japanese football player.
He had experience for Japan at U18, U19 and U20 levels, but was released by Avispa Fukuoka at the end of the 2010 season.
Club statistics
References
External links
1989 births
Living people
Association football people from Fukuoka Prefecture
Japanese men's footballers
J2 League players
Avispa Fukuoka players
Renofa Yamaguchi FC players
Men's association football forwards |
https://en.wikipedia.org/wiki/Yasuomi%20Kugisaki | is a former Japanese football player.
Club statistics
References
External links
1982 births
Living people
Tokai University alumni
Association football people from Miyazaki Prefecture
Japanese men's footballers
J1 League players
J2 League players
Japan Football League players
Avispa Fukuoka players
Minebea Mitsumi FC players
Men's association football defenders |
https://en.wikipedia.org/wiki/Shingo%20Honda | is a former Japanese football player. He last played for Honda FC. Honda previously played for Avispa Fukuoka in the J2 League.
Club statistics
Updated to 1 March 2018.
References
External links
1987 births
Living people
Association football people from Kumamoto Prefecture
Japanese men's footballers
J1 League players
J2 League players
J3 League players
Japan Football League players
Avispa Fukuoka players
Japan Soccer College players
Matsumoto Yamaga FC players
Zweigen Kanazawa players
Honda FC players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Shilpa%20Rao | Shilpa Rao (born Apeksha Rao; 11 April 1984) is an Indian singer born and raised in Jamshedpur. She completed her Post Graduation in Applied Statistics from St. Xavier's College, Mumbai, before working as a jingle singer for three years. During her college days, composer Mithoon offered her to record the song "Tose Naina" from Anwar (2007), making her Bollywood career debut, and the song went on to become one of the most popular songs of 2007.
Rao rose to widespread prominence with the release of "Woh Ajnabi" from The Train (2007) and "Khuda Jaane" from Bachna Ae Haseeno (2008), the latter of which earned her a nomination for the Filmfare Award for Best Female Playback Singer. The following year, she collaborated with Ilaiyaraaja for Paa (2009), where she performed the song "Mudi Mudi Ittefaq Se" which earned her a second Filmfare nomination in the same category. In 2012, Rao teamed up with A. R. Rahman for the song "Ishq Shava" for Yash Chopra's swan song Jab Tak Hai Jaan, followed by Pritam's "Malang" from Dhoom 3 (2013) and Vishal–Shekhar's "Meherbaan" from Bang Bang! (2014). Rao's collaborations with Amit Trivedi were also acclaimed with songs such as "Manmarziyaan" from Lootera (2013) receiving particular praise. She was the final Indian singer to perform in Coke Studio Pakistan with the song "Paar Chanaa De" (2016) and received praise for singing the song "Aaj Jaane Ki Zid Na Karo" from the deluxe edition of the Ae Dil Hai Mushkil soundtrack (2016).
Rao is particularly known in the media for trying new styles in her songs and singing for different genres. Rao, who considers her father as her biggest inspiration in the music career, has supported charitable organisations for a number of causes.
Early life
Born on 11 April 1984 in Jamshedpur, Rao was initially named as Apeksha Rao but later changed to Shilpa Rao. According to her, she relates to the name Shilpa more, since the name has "to do with art". She started singing while she was a kid, tutored and instructed by her father, S Venkat Rao, who holds a degree in music. He taught Rao to understand the "nuances" of different ragas: "His method of teaching was casual and at the same time very effective because it was designed to pique my interest". For her education, Rao went to Little Flower School and Loyola School, Jamshedpur, where she was part of the choir group in school. In 1997, she visited to Mumbai along with her family, to do a postgraduate diploma in statistics from the University of Mumbai.
Rao was motivated to be a singer upon meeting with Hariharan, at the age of 13, and commenced training under Ustad Ghulam Mustafa Khan, insisted by Hariharan. Initially, she found herself struggling in "finding references" and meeting music composers since she was not "so active" in social networking media during the time. About her early days in the city, Rao said: "Jamshedpur is home for me but Mumbai with the people, the place, the pace everything has made a more patient person and a m |
https://en.wikipedia.org/wiki/Billie%20Jean%20King%20career%20statistics | This article shows the main career statistics of former tennis player Billie Jean King.
Grand Slam finals
Singles: 18 (12 titles, 6 runners-up)
Doubles: 29 (16 titles, 13 runners-up)
Mixed doubles: 18 (11 titles, 7 runners-up)
By winning the 1968 Australian Championships title, King became the 7th player to complete the mixed doubles career Grand Slam.
Grand Slam tournament timelines
Singles
Note: The Australian Open was held twice in 1977, in January and December.
See also
Singles performance timelines for all female tennis players who reached at least one Grand Slam or Olympic final
Women's doubles
Note: The Australian Open was held twice in 1977, in January and December.
Mixed doubles
Note: The Australian Open was held twice in 1977, in January and December.
Grand Slam singles records
Australian Championships/Open
King's overall win–loss record at the Australian Championships/Open was 16–4 .800 in 5 years (1965, 1968, 1969, 1982, 1983). (Her win total does not include any first round byes.)
King was 1–1 in finals, 2–1 in semifinals, and 3–1 in quarterfinals.
King was 5–1 in three set matches, 11–3 in two set matches, and 1–0 in deuce third sets, i.e., sets that were tied 5–5 before being resolved.
King was seeded all 5 years she entered the tournament.
Seeded #1 overall in 1969 (losing finalist), 1968 (champion).
Seeded #2 foreign in 1965 (semifinalist).
Seeded #7 overall in 1983 (lost 2nd round).
Seeded #9 overall in 1982 (quarterfinalist).
King was 6–3 .667 against seeded players and 10–1 .909 against unseeded players.
Versus #1 seeds (domestic, foreign, or overall), King was 0–1 (Margaret Court (1965)).
Versus #2 seeds (domestic, foreign, or overall), King was 0–2 (Chris Evert 1982, Margaret Court (1969)).
Versus #3 seeds (domestic, foreign, or overall), King was 2–0 (Ann Haydon-Jones (1969), Judy Tegart-Dalton (1968)).
Versus #4 seeds (domestic, foreign, or overall), King was 1–0 (Robyn Ebbern (1965)).
Versus #6 seeds (domestic, foreign, or overall), King was 1–0 (Karen Krantzcke (1969)).
Versus #7 seeds (domestic, foreign, or overall), King was 2–0 (Barbara Potter (1982), Margaret Court (1968)).
Against her major rivals at the Australian Championships/Open, King was 1–0 versus Kerry Melville Reid, 1–0 versus Judy Tegart-Dalton, 1–0 versus Evonne Goolagong Cawley, 1–0 versus Ann Haydon-Jones, 1–2 versus Margaret Court, and 0–1 versus Chris Evert.
French Championships/Open
King's overall win–loss record at the French Championships/Open was 22–6 .786 in 7 years (1967–1970, 1972, 1980, 1982). (Her win total does not include any first round byes but does include one walkover.)
King was 1–0 in finals, 1–1 in semifinals, and 2–4 in quarterfinals. She failed to reach the quarterfinals only once, in 1982 when she lost to Lucia Romanov in the third round.
King was 3–3 in three set matches, 19–3 in two set matches, and 1–0 in deuce third sets, i.e., sets that were tied 5–5 before being resolved.
King was seeded all 7 y |
https://en.wikipedia.org/wiki/Half%20Moon%20Lake%2C%20Alberta | Half Moon Lake is a hamlet in Alberta, Canada within Strathcona County. It is also recognized as a designated place by Statistics Canada under the name of Half Moon Estates. The community is located on the shores of Half Moon Lake, just north of Highway 629, approximately southeast of Sherwood Park.
The hamlet was founded in the late 1950s when the land north of the lake was subdivided into residential lots, with the subdivision of the south side following soon after.
Demographics
The population of Half Moon Lake according to the 2022 municipal census conducted by Strathcona County is 187, a decrease from its 2018 municipal census population count of 214.
In the 2021 Census of Population conducted by Statistics Canada, Half Moon Lake had a population of 87 living in 33 of its 35 total private dwellings, a change of from its 2016 population of 223. With a land area of , it had a population density of in 2021.
As a designated place in the 2016 Census of Population conducted by Statistics Canada, Half Moon Lake had a population of 218 living in 88 of its 103 total private dwellings, a change of from its 2011 population of 250. With a land area of , it had a population density of in 2016.
Lake
Half Moon Lake is a crescent-shaped body of water that is approximately in length, in width and a maximum of in depth.
Although the lake is surrounded by private land, visitors to Strathcona County will find the commercially run Half Moon Lake Resort at the south end of the lake, which provides access to the lake. The resort, open during the summer months, has campsites, a developed beach, and boat launch.
See also
List of communities in Alberta
List of designated places in Alberta
List of hamlets in Alberta
References
Hamlets in Alberta
Strathcona County
Designated places in Alberta |
https://en.wikipedia.org/wiki/Lorents%20Lorentsen | Lorents Lorentsen (born 5 March 1947) is a Norwegian civil servant.
A candidatus oeconomices by education, he was hired in Statistics Norway in 1979, and was promoted to head of research in 1987. He left in 1992 to become deputy under-secretary of State in the Ministry of Finance and Customs. He served as acting permanent under-secretary of State of the Ministry of Finance from 2002 to 2003, before being appointed as head of the Environment Directorate in the Organisation for Economic Co-operation and Development in 2003.
References
1947 births
Living people
Norwegian civil servants
Norwegian economists
OECD officials |
https://en.wikipedia.org/wiki/Y%C5%8Dsuke%20Nakata | is a former Japanese football player.
Club statistics
References
External links
1981 births
Living people
Komazawa University alumni
Association football people from Iwate Prefecture
Japanese men's footballers
J2 League players
Vegalta Sendai players
Yokohama FC players
Iwate Grulla Morioka players
Men's association football defenders
Universiade medalists in football
FISU World University Games gold medalists for Japan |
https://en.wikipedia.org/wiki/Kakutani%27s%20theorem%20%28geometry%29 | Kakutani's theorem is a result in geometry named after Shizuo Kakutani. It states that every convex body in 3-dimensional space has a circumscribed cube, i.e. a cube all of whose faces touch the body. The result was further generalized by Yamabe and Yujobô to higher dimensions, and by Floyd to other circumscribed parallelepipeds.
References
.
.
.
Theorems in convex geometry |
https://en.wikipedia.org/wiki/Shota%20Suzuki%20%28footballer%2C%20born%201984%29 | is a former Japanese football player.
Club statistics
References
External links
1984 births
Living people
Association football people from Kanagawa Prefecture
Japanese men's footballers
J1 League players
J2 League players
Omiya Ardija players
Kashiwa Reysol players
Shonan Bellmare players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Takahiro%20Yamaguchi | is a former Japanese footballer who last played for Oita Trinita.
Career statistics
Updated to 2 February 2018.
1Includes J1 Promotion Playoffs and J2/J3 Promotion-Relegation Playoffs.
References
External links
Profile at Oita Trinita
1984 births
Living people
Waseda University alumni
Association football people from Tokyo
Japanese men's footballers
J1 League players
J2 League players
J3 League players
Shonan Bellmare players
V-Varen Nagasaki players
Oita Trinita players
Men's association football defenders |
https://en.wikipedia.org/wiki/Nobutaka%20Suzuki | is a former Japanese football player.
Club statistics
References
External links
1983 births
Living people
Association football people from Saitama Prefecture
Japanese men's footballers
Japanese expatriate men's footballers
Expatriate men's footballers in Germany
J1 League players
J2 League players
Shonan Bellmare players
Gainare Tottori players
Japanese expatriate sportspeople in Germany
People from Ageo, Saitama
Men's association football defenders |
https://en.wikipedia.org/wiki/Ryota%20Nagata | is a Japanese football player currently playing for Kamatamare Sanuki.
Club statistics
Updated to 23 February 2018.
References
External links
Profile at Kamatamare Sanuki
1985 births
Living people
Ritsumeikan University alumni
Japanese men's footballers
J1 League players
J2 League players
J3 League players
Shonan Bellmare players
Sagan Tosu players
Montedio Yamagata players
Thespakusatsu Gunma players
Kamatamare Sanuki players
Men's association football midfielders
Association football people from Kyoto |
https://en.wikipedia.org/wiki/Yuya%20Nakamura%20%28footballer%2C%20born%201986%29 | is a Japanese football player who plays for Aventura Kawaguchi.
Club statistics
Updated to 23 February 2018.
References
External links
Profile at Machida Zelvia
1986 births
Living people
Association football people from Saitama Prefecture
Japanese men's footballers
J1 League players
J2 League players
J3 League players
Urawa Red Diamonds players
Shonan Bellmare players
FC Machida Zelvia players
Tochigi City FC players
Men's association football forwards |
https://en.wikipedia.org/wiki/Satoru%20Hayashi | is a Japanese football player. He plays for Arterivo Wakayama.
He previously played for Shonan Bellmare.
Club statistics
References
External links
1988 births
Living people
People from Zama, Kanagawa
Association football people from Kanagawa Prefecture
Japanese men's footballers
J1 League players
J2 League players
Japan Football League players
Shonan Bellmare players
Gainare Tottori players
Zweigen Kanazawa players
FC Osaka players
Nara Club players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Shoma%20Kamata | is a Japanese footballer who plays for Kataller Toyama.
Club statistics
Updated to 27 December 2021.
Honours
Blaublitz Akita
J3 League (1): 2020
References
External links
Profile at Shimizu S-Pulse
1989 births
Living people
Association football people from Kanagawa Prefecture
Japanese men's footballers
J1 League players
J2 League players
J3 League players
Blaublitz Akita players
Shonan Bellmare players
JEF United Chiba players
Fagiano Okayama players
Shimizu S-Pulse players
Fukushima United FC players
Kataller Toyama players
Asian Games medalists in football
Footballers at the 2010 Asian Games
Asian Games gold medalists for Japan
Medalists at the 2010 Asian Games
Men's association football defenders |
https://en.wikipedia.org/wiki/Daisuke%20Kikuchi | is a Japanese footballer who plays for FC Gifu in the J3 League.
Career statistics
Updated to end of 2018 season.
1Includes Japanese Super Cup and FIFA Club World Cup.
National Team career
As of 6 October 2010
Appearances in major competitions
References
External links
Profile at Urawa Red Diamonds
1991 births
Living people
Association football people from Kanagawa Prefecture
Japanese men's footballers
J1 League players
J2 League players
J3 League players
Shonan Bellmare players
Thespakusatsu Gunma players
Urawa Red Diamonds players
Kashiwa Reysol players
Avispa Fukuoka players
Tochigi SC players
FC Gifu players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Kim%20Yeong-gi | Kim Yeong-gi (; born 24 January 1985 in Himeji, Hyōgo, Japan) is a South Korean footballer.
Career statistics
Updated to 23 February 2017.
References
External links
Profile at Nagano Parceiro
1985 births
Living people
Momoyama Gakuin University alumni
Association football people from Hyōgo Prefecture
South Korean men's footballers
J1 League players
J2 League players
J3 League players
Shonan Bellmare players
Oita Trinita players
Avispa Fukuoka players
AC Nagano Parceiro players
South Korean expatriate men's footballers
Expatriate men's footballers in Japan
South Korean expatriate sportspeople in Japan
Men's association football goalkeepers
Zainichi Korean men's footballers |
https://en.wikipedia.org/wiki/Bangladesh%20Bureau%20of%20Statistics | The Bangladesh Bureau of Statistics (BBS) is the centralized official bureau in Bangladesh for collecting statistics on demographics, the economy, and other facts about the country and disseminating the information.
History
Although independent statistical programs had existed in the country before, they were often incomplete or produced inaccurate results, which led the Government of Bangladesh establishing an official bureau in August 1974, by merging four of the previous larger statistical agencies, the Bureau of Statistics, the Bureau of Agriculture Statistics, the Agriculture Census Commission and the Population Census Commission.
In July 1975, the Statistics and Informatics Division was created under the Planning Ministry (Bangladesh) and tasked to oversee the BBS. Between 2002 and 2012, the division remained abolished but was later reinstated.
The Bangladesh Bureau of Statistics is headquartered in Dhaka. As of 2019, it has 8 Divisional statistical offices, 64 District statistical offices and 489 Upazila/Thana offices.
References
External links
Bangladesh Bureau of Statistics, "Census Reports: Population Census-2001", 2001. The 1991 census figures can be seen compared to the 2001 census.
National statistical services
Demographics of Bangladesh
Government agencies of Bangladesh
Research institutes in Bangladesh
1974 establishments in Bangladesh
Organisations based in Dhaka |
https://en.wikipedia.org/wiki/Kensaku%20Abe | is a former Japanese football player.
Club statistics
References
External links
1980 births
Living people
University of Tsukuba alumni
Association football people from Kanagawa Prefecture
Japanese men's footballers
J1 League players
J2 League players
Ventforet Kofu players
Vissel Kobe players
Men's association football goalkeepers |
https://en.wikipedia.org/wiki/Michitaka%20Akimoto | is a Japanese football player.
Club statistics
Honour
Thai Honda FC
Thai Division 1 League Champion; 2016
References
External links
Michitaka Akimoto on Instagram
1982 births
Living people
Hosei University alumni
Association football people from Shizuoka Prefecture
Japanese men's footballers
J1 League players
J2 League players
Kyoto Sanga FC players
Ventforet Kofu players
Kataller Toyama players
Men's association football defenders |
https://en.wikipedia.org/wiki/Takafumi%20Mikuriya | is a former Japanese football player who is currently a football referee.
Club statistics
References
External links
1984 births
Living people
Osaka University of Health and Sport Sciences alumni
Association football people from Nagasaki Prefecture
Japanese men's footballers
J1 League players
J2 League players
Ventforet Kofu players
Thespakusatsu Gunma players
Kataller Toyama players
Men's association football defenders
Japanese football referees |
https://en.wikipedia.org/wiki/DigiListan | DigiListan is a Swedish radio programme in SR P3 airing the top singles sold electronically to computers, mobile phones and other kinds of media players in Sweden (downloaded music). The statistics are created using Nielsen SoundScan.
The programme was first aired in January 2007.
See also
Sverigetopplistan—Swedish national music chart
External links
DigiListan
Listen to all lists on Spotify
Swedish radio programs |
https://en.wikipedia.org/wiki/Kazunari%20Hosaka | is a Japanese former professional footballer who played as a midfielder.
Club career statistics
.
References
External links
1983 births
Living people
Tokyo Gakugei University alumni
Association football people from Tokyo Metropolis
People from Fuchū, Tokyo
Japanese men's footballers
J1 League players
J2 League players
Ventforet Kofu players
Fagiano Okayama players
Men's association football midfielders
FISU World University Games gold medalists for Japan
Universiade medalists in football |
https://en.wikipedia.org/wiki/Daiki%20Tamori | is a Japanese football manager and former player.
Club career statistics
Updated to 31 December 2018.
References
External links
Profile at FC Gifu
1983 births
Living people
Hosei University alumni
Association football people from Hiroshima Prefecture
Japanese men's footballers
J1 League players
J2 League players
Ventforet Kofu players
Ehime FC players
Kyoto Sanga FC players
FC Gifu players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Yohei%20Onishi | is a former Japanese soccer player.
Club statistics
References
External links
1982 births
Living people
Hannan University alumni
Association football people from Okayama Prefecture
Japanese men's footballers
J1 League players
J2 League players
J3 League players
Ventforet Kofu players
Kataller Toyama players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Jun%20Uruno | is a former Japanese football player.
Club statistics
Honours
Air Force Central
Thai Division 1 League: 2013
References
External links
1979 births
Living people
Association football people from Saitama Prefecture
Japanese men's footballers
J1 League players
J2 League players
Japan Football League (1992–1998) players
Japan Football League players
Honda FC players
Ventforet Kofu players
Roasso Kumamoto players
Jun Uruno
Jun Uruno
Expatriate men's footballers in Thailand
Japanese expatriate sportspeople in Thailand
Men's association football forwards |
https://en.wikipedia.org/wiki/Masafumi%20Maeda | is a former Japanese football player.
Club statistics
References
External links
1983 births
Living people
Kansai University alumni
Association football people from Shiga Prefecture
Japanese men's footballers
J1 League players
J2 League players
Gamba Osaka players
Ventforet Kofu players
Thespakusatsu Gunma players
Men's association football midfielders
Universiade medalists in football
FISU World University Games gold medalists for Japan |
https://en.wikipedia.org/wiki/Elementary%20calculus | Elementary calculus may refer to:
The elementary aspects of differential and integral calculus;
Elementary Calculus: An Infinitesimal Approach, a textbook by Jerome Keisler. |
https://en.wikipedia.org/wiki/1991%20Bangladeshi%20census | In 1991, the Bangladesh Bureau of Statistics, conducted a national census in Bangladesh. They recorded data from all of the districts and upazilas and main cities in Bangladesh including statistical data on population size, households, sex and age distribution, marital status, economically active population, literacy and educational attainment, religion, number of children etc. According to the census, Hindus were 10.5 per cent of the population, down from 12.1 per cent as of 1981.
Bangladesh have a population of 106,314,992 as per 1991 census report. Majority of 93,886,769 reported that they were Muslims, 11,184,337 reported as Hindus, 616,626 as Buddhists, 350,839 as Christians and 276,418 as others.
See also
Demographics of Bangladesh
2001 Census of Bangladesh
2011 Census of Bangladesh
2022 Census of Bangladesh
References
External links
Bangladesh Bureau of Statistics, "Census Reports: Population Census-2001", 2001. The 1991 census figures can be seen compared to the 2001 census.
Censuses in Bangladesh
Bangladesh
Census |
https://en.wikipedia.org/wiki/2001%20Bangladeshi%20census | In 2001, the Bangladesh Bureau of Statistics conducted a national census in Bangladesh, ten years after the 1991 census. They recorded data from all of the districts, upazilas, and main cities in Bangladesh including statistical data on population size, households, sex and age distribution, marital status, economically active population, literacy and educational attainment, religion, number of children, etc. According to the adjusted 2001 census figures, Bangladesh's population stood at 129.3 million (an initial count put it at 124.4 million; an adjustment for the standard rate of undercounting then boosted the figure). According to the census, Hindus were 9.2 per cent of the population, down from 10.5 per cent as of 1991.
The census data were collected from January 23 to 27, 2001. The 2001 census was the first in Bangladesh to use optical mark recognition (OMR) technology.
Bangladesh have a population of 124,355,263 as per 2001 census report. Majority of 111,397,444 reported that they were Muslims, 11,614,781 reported as Hindus, 771,002 as Buddhists, 385,501 as Christians and 186,532 as others.
See also
Demographics of Bangladesh
1991 Census of Bangladesh
2011 Census of Bangladesh
2022 Census of Bangladesh
References
External links
Censuses in Bangladesh
Census
Bangladesh |
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