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https://en.wikipedia.org/wiki/Demographics%20of%20Southern%20Norway |
Statistics Norway demographic statistics
The following demographic statistics are from the Statistics Norway, unless otherwise indicated.
Age and sex distribution
Age structure
Norway
(2005 est.)
0–14 years: 19.7% (male 466,243; female 443,075)
15–64 years: 65.6% (male 1,234,384; female 1,486,887)
65 years and over: 14.7% (male 285,389; female 392,331)
Southern Norway
(2009 est.)
0–14 years: 19.8% (male 27,734; female 26,836)
15–64 years: 66.1% (male 92,364; female 89,516)
65 years and over: 14.1% (male 17,311; female 22,831)
Population
257,869 (January 1, 2000)
277,250 (July 1, 2009)
Population growth
19,381 (7.5%)
Population - comparative
slightly larger than Vanuatu and Barbados, but slightly smaller than Iceland and Maldives.
Population growth rate
1.31% (in 2008)
Population growth rate - comparative
slightly larger than Brazil, but slightly smaller than Oregon.
Total fertility rate
1.96 children born/woman (2007)
Language
Literacy
definition: age 15 and over can read and write
total population: 100%
male: NA%
female: NA%
Demographics of Norway |
https://en.wikipedia.org/wiki/Demographics%20of%20Tr%C3%B8ndelag |
Statistics Norway demographic statistics
The following demographic statistics are from the Statistics Norway, unless otherwise indicated.
Age and sex distribution
Age structure
Norway
(2005 est.)
0–14 years: 19.7% (male 466,243; female 443,075)
15–64 years: 65.6% (male 1,234,384; female 1,486,887)
65 years and over: 14.7% (male 285,389; female 392,331)
Trøndelag
(2009 est.)
0–14 years: 19.1% (male 40,746; female 38,777)
15–64 years: 65.7% (male 141,227; female 130,014)
65 years and over: 15.0% (male 27,436; female 35,004)
Population
389,960 (1 January 2000)
418,453 (1 July 2009)
Population growth
28,493 (7.3%)
Population - comparative
slightly larger than Malta, but slightly smaller than Luxembourg.
Population growth rate
1.10% (in 2008)
Population growth rate - comparative
slightly larger than Argentina, but slightly smaller than Ireland.
Total fertility rate
1.90 children born/woman (2007)
Language
Literacy
definition: age 15 and over can read and write
total population: 100%
male: NA%
female: NA%
Demographics of Norway |
https://en.wikipedia.org/wiki/Revolutions%20in%20Mathematics | Revolutions in Mathematics is a 1992 collection of essays in the history and philosophy of mathematics.
Contents
Michael J. Crowe, Ten "laws" concerning patterns of change in the history of mathematics (1975) (15–20);
Herbert Mehrtens, T. S. Kuhn's theories and mathematics: a discussion paper on the "new historiography" of mathematics (1976) (21–41);
Herbert Mehrtens, Appendix (1992): revolutions reconsidered (42–48);
Joseph Dauben, Conceptual revolutions and the history of mathematics: two studies in the growth of knowledge (1984) (49–71);
Joseph Dauben, Appendix (1992): revolutions revisited (72–82);
Paolo Mancosu, Descartes's Géométrie and revolutions in mathematics (83–116);
Emily Grosholz, Was Leibniz a mathematical revolutionary? (117–133);
Giulio Giorello, The "fine structure" of mathematical revolutions: metaphysics, legitimacy, and rigour. The case of the calculus from Newton to Berkeley and Maclaurin (134–168);
Yu Xin Zheng, Non-Euclidean geometry and revolutions in mathematics (169–182);
Luciano Boi, The "revolution" in the geometrical vision of space in the nineteenth century, and the hermeneutical epistemology of mathematics (183–208);
Caroline Dunmore, Meta-level revolutions in mathematics (209–225);
Jeremy Gray, The nineteenth-century revolution in mathematical ontology (226–248);
Herbert Breger, A restoration that failed: Paul Finsler's theory of sets (249–264);
Donald A. Gillies, The Fregean revolution in logic (265–305);
Michael Crowe, Afterword (1992): a revolution in the historiography of mathematics? (306–316).
Reviews
The book was reviewed by Pierre Kerszberg for Mathematical Reviews and by Michael S. Mahoney for American Mathematical Monthly. Mahoney says "The title should have a question mark." He sets the context by referring to paradigm shifts that characterize scientific revolutions as described by Thomas Kuhn in his book The Structure of Scientific Revolutions. According to Michael Crowe in chapter one, revolutions never occur in mathematics. Mahoney explains how mathematics grows upon itself and does not discard earlier gains in understanding with new ones, such as happens in biology, physics, or other sciences. A nuanced version of revolution in mathematics is described by Caroline Dunmore who sees change at the level of "meta-mathematical values of the community that define the telos and methods of the subject, and encapsulate general beliefs about its value." On the other hand, reaction to innovation in mathematics is noted, resulting in "clashes of intellectual and social values".
Editions
Gillies, Donald (1992) Revolutions in Mathematics, Oxford Science Publications, The Clarendon Press, Oxford University Press.
References
Pierre Kerszberg (1994, 2009) Review of Revolutions in Mathematics in Mathematical Reviews.
Michael S. Mahoney (1994) "Review of Revolutions in Mathematics", American Mathematical Monthly 101(3):283–7.
1992 non-fiction books
1992 anthologies
Essay anthologies
Mathematics |
https://en.wikipedia.org/wiki/Biel%20Ribas | Gabriel 'Biel' Ribas Ródenas (born 2 December 1985 in Palma de Mallorca, Balearic Islands) is a Spanish footballer who plays for CF Talavera de la Reina as a goalkeeper.
Career statistics
Honours
UCAM Murcia
Segunda División B: 2015–16
Fuenlabrada
Segunda División B: 2018–19
Spain U17
Meridian Cup: 2003
Spain U19
UEFA European Under-19 Championship: 2004
References
External links
1985 births
Living people
Spanish men's footballers
Footballers from Palma de Mallorca
Men's association football goalkeepers
La Liga players
Segunda División players
Segunda División B players
Tercera División players
Primera Federación players
RCD Espanyol B footballers
RCD Espanyol footballers
Lorca Deportiva CF footballers
UD Salamanca players
CD Atlético Baleares footballers
CD Numancia players
UCAM Murcia CF players
Real Murcia CF players
CF Fuenlabrada footballers
CF Talavera de la Reina players
Spain men's youth international footballers |
https://en.wikipedia.org/wiki/Demographics%20of%20Northern%20Norway |
Statistics Norway demographic statistics
The following demographic statistics are from the Statistics Norway, unless otherwise indicated.
Age and sex distribution
Age structure
Norway
(2005 est.)
0–14 years: 19.7% (male 466,243; female 443,075)
15–64 years: 65.6% (male 1,234,384; female 1,486,887)
65 years and over: 14.7% (male 285,389; female 392,331)
Northern Norway
(2009 est.)
0–14 years: 18.9% (male 44,848; female 42,315)
15–64 years: 65.5% (male 156,476; female 147,465)
65 years and over: 15.6% (male 32,017; female 40,304)
Population
464,328 (January 1, 2000)
464,649 (July 1, 2009)
Population growth
321 (0.06%)
Population - comparative
slightly larger than Malta, but slightly smaller than Luxembourg.
Population growth rate
0.29% (in 2008)
Population growth rate - comparative
slightly larger than United Kingdom, but slightly smaller than Denmark.
Total fertility rate
1.98 children born/woman (2007)
Language
Literacy
definition: age 15 and over can read and write
total population: 100%
male: NA%
female: NA%
Demographics of Norway |
https://en.wikipedia.org/wiki/Endika%20Bordas | Endika Bordas Losada (born 8 March 1982 in Bermeo, Biscay) is a Spanish former professional footballer who played as a midfielder, currently a manager.
Managerial statistics
References
External links
1982 births
Living people
People from Bermeo
Footballers from Biscay
Spanish men's footballers
Men's association football midfielders
La Liga players
Segunda División players
Segunda División B players
Tercera División players
Bermeo FT footballers
CD Basconia footballers
Athletic Bilbao B footballers
Athletic Bilbao footballers
Terrassa FC footballers
CE L'Hospitalet players
Córdoba CF players
UD Salamanca players
SD Amorebieta footballers
Gernika Club footballers
FC Locomotive Tbilisi players
Spain men's youth international footballers
Spanish expatriate men's footballers
Expatriate men's footballers in Georgia (country)
Spanish expatriate sportspeople in Georgia (country)
Spanish football managers
Segunda División B managers
Arenas Club de Getxo managers |
https://en.wikipedia.org/wiki/My%20Science%20Career | The My Science Career website is an Irish online resource for career information in science, technology, engineering and mathematics (STEM).
The website has a famous Irish scientists section, science related articles, a science career glossary and a video interviews section with scientists about their work. “A day in the life” section looks at the everyday working loves of Irish scientists and science broadcasters, ranging from a professor of biochemistry to a marine photographer.
A “Science Ambassadors” section profiles Irish scientists on what it’s like working in various fields and the qualifications they have. The Science Ambassadors range from newly qualified graduates to well established researchers.
MyScienceCareer.ie is an initiative of Ireland’s national integrated awareness programme Discover Science & Engineering (DSE), a government initiative. DSE runs numerous other initiatives, including Science.ie, Science Week Ireland and Discover Primary Science.
External links
Discover Science & Engineering website
My Science Career website
References
Science education in Ireland
Science and technology in the Republic of Ireland |
https://en.wikipedia.org/wiki/Demographics%20of%20Eastern%20Norway |
Statistics Norway demographic statistics
The following demographic statistics are from the Statistics Norway, unless otherwise indicated.
Age and sex distribution
Age structure
Norway
(2005 est.)
0–14 years: 19.7% (male 466,243; female 443,075)
15–64 years: 65.6% (male 1,234,384; female 1,486,887)
65 years and over: 14.7% (male 285,389; female 392,331)
Eastern Norway
(2009 est.)
0–14 years: 18.6% (male 226,334; female 214,775)
15–64 years: 64.5% (male 765,005; female 759,737)
65 years and over: 16.9% (male 150,656; female 201,963)
Population
2,207,164 (January 1, 2000)
2,410,630 (July 1, 2009)
Population growth
203,466 (8.44%)
Population - comparative
slightly larger than Latvia, but slightly smaller than Mongolia and Jamaica.
Population growth rate
1.59% (in 2008)
Population growth rate - comparative
slightly larger than India, but slightly smaller than Turkmenistan.
Total fertility rate
1.85 children born/woman (2007)
Language
Literacy
definition: age 15 and over can read and write
total population: 100%
male: NA%
female: NA%
Demographics of Norway |
https://en.wikipedia.org/wiki/Magdalena%20Municipality%2C%20Beni | Magdalena Municipality is the first municipal section of the Iténez Province in the Beni Department in Bolivia. Its seat is Magdalena.
References
National Institute of Statistics of Bolivia
Municipalities of Beni Department |
https://en.wikipedia.org/wiki/Vladimir%20Korepin | Vladimir E. Korepin (born 1951) is a professor at the C. N. Yang Institute of Theoretical Physics of the Stony Brook University. Korepin made research contributions in several areas of mathematics and physics.
Educational background
Korepin completed his undergraduate study at Saint Petersburg State University, graduating with a diploma in theoretical physics in 1974. In that same year he was employed by the Mathematical Institute of Academy of Sciences. He worked there until 1989, obtaining his PhD in 1977 under the supervision of Ludwig Faddeev. At the same institution he completed his postdoctoral studies.
In 1985, he received a doctor of sciences degree in mathematical physics.
Contributions to physics
Korepin has made contributions to several fields of theoretical physics. Although he is best known for his involvement in condensed matter physics and mathematical physics, he significantly contributed to quantum gravity as well. In recent years, his work has focused on aspects of condensed matter physics relevant for quantum information.
Condensed matter
Among his contributions to condensed matter physics, we mention his studies on low-dimensional quantum gases. In particular, the 1D Hubbard model of strongly correlated fermions, and the 1D Bose gas with delta potential interactions.
In 1979, Korepin presented a solution of the massive Thirring model in one space and one time dimension using the Bethe ansatz. In this work, he provided the exact calculation of the mass spectrum and the scattering matrix.
He studied solitons in the sine-Gordon model. He determined their mass and scattering matrix, both semiclassically and to one loop corrections.
Together with Anatoly Izergin, he discovered the 19-vertex model (sometimes called the Izergin-Korepin model).
In 1993, together with A. R. Its, Izergin and N. A. Slavnov, he calculated space, time and temperature dependent correlation functions in the XX spin chain. The exponential decay in space and time separation of the correlation functions was calculated explicitly.
Quantum gravity
In this field, Korepin has worked on the cancellation of ultra-violet infinities in one loop on mass shell gravity.
Contributions to mathematics
In 1982, Korepin introduced domain wall boundary conditions for the six vertex model, published in Communications in Mathematical Physics. The result plays a role in diverse fields of mathematics such as algebraic combinatorics, alternating sign matrices, domino tiling, Young diagrams and plane partitions. In the same paper the determinant formula was proved for the square of the norm of the Bethe ansatz wave function. It can be represented as a determinant of linearized system of Bethe equations. It can also be represented as a matrix determinant of second derivatives of the Yang action.
The so-called "Quantum Determinant" was discovered in 1981 by A.G. Izergin and V.E. Korepin. It is the center of the Yang–Baxter algebra.
The study of differential equations for |
https://en.wikipedia.org/wiki/Fedosov%20manifold | In mathematics, a Fedosov manifold is a symplectic manifold with a compatible torsion-free connection, that is, a triple (M, ω, ∇), where (M, ω) is a symplectic manifold (that is, is a symplectic form, a non-degenerate closed exterior 2-form, on a -manifold M), and ∇ is a symplectic torsion-free connection on (A connection ∇ is called compatible or symplectic if X ⋅ ω(Y,Z) = ω(∇XY,Z) + ω(Y,∇XZ) for all vector fields X,Y,Z ∈ Γ(TM). In other words, the symplectic form is parallel with respect to the connection, i.e., its covariant derivative vanishes.) Note that every symplectic manifold admits a symplectic torsion-free connection. Cover the manifold with Darboux charts and on each chart define a connection ∇ with Christoffel symbol . Then choose a partition of unity (subordinate to the cover) and glue the local connections together to a global connection which still preserves the symplectic form. The famous result of Boris Vasilievich Fedosov gives a canonical deformation quantization of a Fedosov manifold.
Examples
For example, with the standard symplectic form has the symplectic connection given by the exterior derivative Hence, is a Fedosov manifold.
References
Mathematical physics |
https://en.wikipedia.org/wiki/Deborah%20Loewenberg%20Ball | Deborah Loewenberg Ball is an educational researcher noted for her work in mathematics instruction and the mathematical preparation of teachers. From 2017 to 2018 she served as president of the American Educational Research Association. She served as dean of the School of Education at the University of Michigan from 2005 to 2016, and she currently works as William H. Payne Collegiate Professor of education. Ball directs TeachingWorks, a major project at the University of Michigan to redesign the way that teachers are prepared for practice, and to build materials and tools that will serve the field of teacher education broadly. In a sometimes divisive field,
Ball has a reputation of being respected by both mathematicians and educators. She is also an extremely well respected mentor to junior faculty members and to graduate students.
Education
As an undergraduate at Michigan State University, Ball majored in French and then taught elementary school for seventeen years in East Lansing, Michigan. Ball only started serious study of mathematics when she saw her students struggling in math. In 1988 she received her Ph.D. from Michigan State University from the department of teacher education. Her thesis was titled Knowledge and reasoning in mathematical pedagogy: Examining what prospective teachers bring to teacher education.
Research
Ball's research focuses on improving the effectiveness and quality of mathematical instruction. Much of her work investigates the mathematical knowledge needed for teaching, and she was among the first to suggest that this knowledge is qualitatively different from what is taught in advanced mathematics classes.
Awards and Positions
In 2004, Ball and coauthors David K. Cohen and Stephen W. Raudenbush won the Palmer O. Johnson Award presented by the American Educational Research Association for the best article published in an AERA journal in 2003 for their paper Resources, instruction, and research. In 2007, she was elected member of the National Academy of Education (NAEd). In 2008, she won the Outstanding Contributions to Mathematics Education Award, presented by the Michigan Council of Teachers of Mathematics. In 2009, she won the 19th Louise Hay Award for Outstanding Contributions to Mathematics Education, presented by the Association for Women in Mathematics. In 2012 she became a fellow of the American Mathematical Society. Ball is included in deck 2 of EvenQuads which is a series of playing card decks that feature notable women mathematicians published by the Association of Women in Mathematics.
In 1999, Ball was appointed by U.S. Secretary of Education Richard Riley to serve on the National Commission on Mathematics and Science Teaching for the 21st Century, a committee chaired by Senator John Glenn. From 1999 to 2003, Ball served as chair of the RAND Mathematics Study Panel, whose work culminated in the publication Mathematical Proficiency for All Students: Toward a Strategic Research and Development Program in |
https://en.wikipedia.org/wiki/Subring | In mathematics, a subring of R is a subset of a ring that is itself a ring when binary operations of addition and multiplication on R are restricted to the subset, and which shares the same multiplicative identity as R. For those who define rings without requiring the existence of a multiplicative identity, a subring of R is just a subset of R that is a ring for the operations of R (this does imply it contains the additive identity of R). The latter gives a strictly weaker condition, even for rings that do have a multiplicative identity, so that for instance all ideals become subrings (and they may have a multiplicative identity that differs from the one of R). With definition requiring a multiplicative identity (which is used in this article), the only ideal of R that is a subring of R is R itself.
Definition
A subring of a ring is a subset S of R that preserves the structure of the ring, i.e. a ring with . Equivalently, it is both a subgroup of and a submonoid of .
Examples
The ring and its quotients have no subrings (with multiplicative identity) other than the full ring.
Every ring has a unique smallest subring, isomorphic to some ring with n a nonnegative integer (see characteristic). The integers correspond to in this statement, since is isomorphic to .
Subring test
The subring test is a theorem that states that for any ring R, a subset S of R is a subring if and only if it is closed under multiplication and subtraction, and contains the multiplicative identity of R.
As an example, the ring Z of integers is a subring of the field of real numbers and also a subring of the ring of polynomials Z[X].
Ring extensions
If S is a subring of a ring R, then equivalently R is said to be a ring extension of S, written as R/S in similar notation to that for field extensions.
Subring generated by a set
Let R be a ring. Any intersection of subrings of R is again a subring of R. Therefore, if X is any subset of R, the intersection of all subrings of R containing X is a subring S of R. S is the smallest subring of R containing X. ("Smallest" means that if T is any other subring of R containing X, then S is contained in T.) S is said to be the subring of R generated by X. If S = R, we may say that the ring R is generated by X.
Relation to ideals
Proper ideals are subrings (without unity) that are closed under both left and right multiplication by elements of R.
If one omits the requirement that rings have a unity element, then subrings need only be non-empty and otherwise conform to the ring structure, and ideals become subrings. Ideals may or may not have their own multiplicative identity (distinct from the identity of the ring):
The ideal I = {(z,0) | z in Z} of the ring Z × Z = {(x,y) | x,y in Z} with componentwise addition and multiplication has the identity (1,0), which is different from the identity (1,1) of the ring. So I is a ring with unity, and a "subring-without-unity", but not a "subring-with-unity" of Z × Z.
The proper ideal |
https://en.wikipedia.org/wiki/Donald%20A.%20Gillies | Donald Angus Gillies (; born 4 May 1944) is a British philosopher and historian of science and mathematics. He is an Emeritus Professor in the Department of Science and Technology Studies at University College London.
Career
After undergraduate studies in mathematics and philosophy at Cambridge, Gillies became a graduate student of Karl Popper and Imre Lakatos (his official PhD supervisor) at the London School of Economics, where he completed a PhD on the foundations of probability.
Gilles is a past President and a current Vice-President of British Society for the Philosophy of Science. From 1982 to 1985 he was an editor of the British Journal for the Philosophy of Science.
Gillies is probably best known for his work on Bayesian confirmation theory, his attempt to simplify and extend Popper’s theory of corroboration. He proposes a novel "principle of explanatory surplus", likening a successful theoretician to a successful entrepreneur. The entrepreneur generates a surplus (of income) over and above his initial investment (the outgoes) to meet the necessary expenses of the enterprise. Similarly, the theoretician generates a surplus (of explanations) over and above his initial investment (of assumptions) to make the necessary explanations of known facts. The size of this surplus is held to be a measure of the confirmation of the theory, but only in qualitative, rather than quantitative, terms.
Gillies has researched the philosophy of science, most particularly the foundations of probability; the philosophy of logic and mathematics; and the interactions of artificial intelligence with some aspects of philosophy, including probability, logic, causality and scientific method. In the philosophy of mathematics, he has developed a method of dealing with very large transfinite cardinals from an Aristotelian point of view.
Books and articles (selection)
Gillies, Donald and Chihara, Charles S. (1988). "An Interchange on the Popper-Miller Argument". Philosophical Studies, Volume 54, pp. 1–8.
Gillies, Donald (1989). "Non-Bayesian Confirmation Theory and the Principle of Explanatory Surplus". The Philosophy of Science Association, PSA 1988, Volume 2, pp. 373–380.
Gillies, Donald ed. (1992). Revolutions in Mathematics. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York.
Gillies, Donald (1996). "Artificial intelligence and scientific method". Oxford: Oxford University Press.
Gillies, Donald (2000). Philosophical Theories of Probability. London: Routledge.
Gillies, Donald (2010). An objective Theory of Probability. London: Routledge.
Gillies, Donald (2011). Frege, Dedekind, and Peano on the Foundations of Arithmetic. London: Routledge.
References
External links
Donald Gillies's personal webpage, University College London
Philosophers of mathematics
Philosophers of science
British historians of mathematics
20th-century English mathematicians
21st-century English mathematicians
Academics of University College London
Li |
https://en.wikipedia.org/wiki/N-curve | We take the functional theoretic algebra C[0, 1] of curves. For each loop γ at 1, and each positive integer n, we define a curve called n-curve. The n-curves are interesting in two ways.
Their f-products, sums and differences give rise to many beautiful curves.
Using the n-curves, we can define a transformation of curves, called n-curving.
Multiplicative inverse of a curve
A curve γ in the functional theoretic algebra C[0, 1], is invertible, i.e.
exists if
If , where , then
The set G of invertible curves is a non-commutative group under multiplication. Also the set H of loops at 1 is an Abelian subgroup of G. If , then the mapping is an inner automorphism of the group G.
We use these concepts to define n-curves and n-curving.
n-curves and their products
If x is a real number and [x] denotes the greatest integer not greater than x, then
If and n is a positive integer, then define a curve by
is also a loop at 1 and we call it an n-curve.
Note that every curve in H is a 1-curve.
Suppose
Then, since .
Example 1: Product of the astroid with the n-curve of the unit circle
Let us take u, the unit circle centered at the origin and α, the astroid.
The n-curve of u is given by,
and the astroid is
The parametric equations of their product are
See the figure.
Since both are loops at 1, so is the product.
Example 2: Product of the unit circle and its n-curve
The unit circle is
and its n-curve is
The parametric equations of their product
are
See the figure.
Example 3: n-Curve of the Rhodonea minus the Rhodonea curve
Let us take the Rhodonea Curve
If denotes the curve,
The parametric equations of are
n-Curving
If , then, as mentioned above, the n-curve . Therefore, the mapping is an inner automorphism of the group G. We extend this map to the whole of C[0, 1], denote it by and call it n-curving with γ.
It can be verified that
This new curve has the same initial and end points as α.
Example 1 of n-curving
Let ρ denote the Rhodonea curve , which is a loop at 1. Its parametric equations are
With the loop ρ we shall n-curve the cosine curve
The curve has the parametric equations
See the figure.
It is a curve that starts at the point (0, 1) and ends at (2π, 1).
Example 2 of n-curving
Let χ denote the Cosine Curve
With another Rhodonea Curve
we shall n-curve the cosine curve.
The rhodonea curve can also be given as
The curve has the parametric equations
See the figure for .
Generalized n-curving
In the FTA C[0, 1] of curves, instead of e we shall take an arbitrary curve , a loop at 1.
This is justified since
Then, for a curve γ in C[0, 1],
and
If , the mapping
given by
is the n-curving. We get the formula
Thus given any two loops and at 1, we get a transformation of curve
given by the above formula.
This we shall call generalized n-curving.
Example 1
Let us take and as the unit circle ``u.’’ and as the cosine curve
Note that
|
https://en.wikipedia.org/wiki/Euclid%27s%20Optics | Optics (), is a work on the geometry of vision written by the Greek mathematician Euclid around 300 BC. The earliest surviving manuscript of Optics is in Greek and dates from the 10th century AD.
The work deals almost entirely with the geometry of vision, with little reference to either the physical or psychological aspects of sight. No Western scientist had previously given such mathematical attention to vision. Euclid's Optics influenced the work of later Greek, Islamic, and Western European Renaissance scientists and artists.
Historical significance
Writers before Euclid had developed theories of vision. However, their works were mostly philosophical in nature and lacked the mathematics that Euclid introduced in his Optics. Efforts by the Greeks prior to Euclid were concerned primarily with the physical dimension of vision. Whereas Plato and Empedocles thought of the visual ray as "luminous and ethereal emanation", Euclid’s treatment of vision in a mathematical way was part of the larger Hellenistic trend to quantify a whole range of scientific fields.
Because Optics contributed a new dimension to the study of vision, it influenced later scientists. In particular, Ptolemy used Euclid's mathematical treatment of vision and his idea of a visual cone in combination with physical theories in Ptolemy's Optics, which has been called "one of the most important works on optics written before Newton". Renaissance artists such as Brunelleschi, Alberti, and Dürer used Euclid's Optics in their own work on linear perspective.
Structure and method
Similar to Euclid's much more famous work on geometry, Elements, Optics begins with a small number of definitions and postulates, which are then used to prove, by deductive reasoning, a body of geometric propositions (theorems in modern terminology) about vision.
The postulates in Optics are:
Let it be assumed
That rectilinear rays proceeding from the eye diverge indefinitely;
That the figure contained by a set of visual rays is a cone of which the vertex is at the eye and the base at the surface of the objects seen;
That those things are seen upon which visual rays fall and those things are not seen upon which visual rays do not fall;
That things seen under a larger angle appear larger, those under a smaller angle appear smaller, and those under equal angles appear equal;
That things seen by higher visual rays appear higher, and things seen by lower visual rays appear lower;
That, similarly, things seen by rays further to the right appear further to the right, and things seen by rays further to the left appear further to the left;
That things seen under more angles are seen more clearly.
The geometric treatment of the subject follows the same methodology as the Elements.
Content
According to Euclid, the eye sees objects that are within its visual cone. The visual cone is made up of straight lines, or visual rays, extending outward from the eye. These visual rays are discrete, but we perce |
https://en.wikipedia.org/wiki/Witness%20%28mathematics%29 | In mathematical logic, a witness is a specific value t to be substituted for variable x of an existential statement of the form ∃x φ(x) such that φ(t) is true.
Examples
For example, a theory T of arithmetic is said to be inconsistent if there exists a proof in T of the formula "0 = 1". The formula I(T), which says that T is inconsistent, is thus an existential formula. A witness for the inconsistency of T is a particular proof of "0 = 1" in T.
Boolos, Burgess, and Jeffrey (2002:81) define the notion of a witness with the example, in which S is an n-place relation on natural numbers, R is an (n+1)-place recursive relation, and ↔ indicates logical equivalence (if and only if):
S(x1, ..., xn) ↔ ∃y R(x1, . . ., xn, y)
"A y such that R holds of the xi may be called a 'witness' to the relation S holding of the xi (provided we understand that when the witness is a number rather than a person, a witness only testifies to what is true)."
In this particular example, the authors defined s to be (positively) recursively semidecidable, or simply semirecursive.
Henkin witnesses
In predicate calculus, a Henkin witness for a sentence in a theory T is a term c such that T proves φ(c) (Hinman 2005:196). The use of such witnesses is a key technique in the proof of Gödel's completeness theorem presented by Leon Henkin in 1949.
Relation to game semantics
The notion of witness leads to the more general idea of game semantics. In the case of sentence the winning strategy for the verifier is to pick a witness for . For more complex formulas involving universal quantifiers, the existence of a winning strategy for the verifier depends on the existence of appropriate Skolem functions. For example, if S denotes then an equisatisfiable statement for S is . The Skolem function f (if it exists) actually codifies a winning strategy for the verifier of S by returning a witness for the existential sub-formula for every choice of x the falsifier might make.
See also
Certificate (complexity), an analogous concept in computational complexity theory
References
George S. Boolos, John P. Burgess, and Richard C. Jeffrey, 2002, Computability and Logic: Fourth Edition, Cambridge University Press, .
Leon Henkin, 1949, "The completeness of the first-order functional calculus", Journal of Symbolic Logic v. 14 n. 3, pp. 159–166.
Peter G. Hinman, 2005, Fundamentals of mathematical logic, A.K. Peters, .
J. Hintikka and G. Sandu, 2009, "Game-Theoretical Semantics" in Keith Allan (ed.) Concise Encyclopedia of Semantics, Elsevier, , pp. 341–343
Logic
Quantifier (logic)
Mathematical logic |
https://en.wikipedia.org/wiki/Erling%20St%C3%B8rmer | Erling Størmer (born 2 November 1937) is a Norwegian mathematician, who has mostly worked with operator algebras.
He was born in Oslo as a son of Leif Størmer. He was a grandson of Carl Størmer and nephew of Per Størmer. He took his doctorate at Columbia University in 1963 with thesis advisor Richard Kadison, and was a professor at the University of Oslo from 1974 to his retirement in 2007.
He is a member of the Norwegian Academy of Science and Letters. In 2012 he became a fellow of the American Mathematical Society.
See also
Jordan operator algebra
References
1937 births
Living people
20th-century Norwegian mathematicians
Columbia University alumni
Norwegian expatriates in the United States
Academic staff of the University of Oslo
Members of the Norwegian Academy of Science and Letters
Fellows of the American Mathematical Society
Scientists from Oslo
Presidents of the Norwegian Mathematical Society |
https://en.wikipedia.org/wiki/Ragni%20Piene | Ragni Piene (born 18 January 1947, Oslo) is a Norwegian mathematician, specializing in algebraic geometry, with particular interest in enumerative results and intersection theory.
Education and career
After a bachelor's degree from the University of Oslo in 1969 and a DEA from Université de Paris in 1970 Piene received a doctorate in mathematics from the Massachusetts Institute of Technology in 1976, advised by Steven Kleiman. Her dissertation was titled Plücker Formulas.
She was appointed professor at the University of Oslo in 1987.
Recognition
She was elected a member of the Norwegian Academy of Science and Letters in 1994,
and in 2012 she became a fellow of the American Mathematical Society and a member of the Academia Europaea. We is also one of the protagonists of the Women of mathematics exhibition.
Service
Since 2003 she has been a member of the executive committee of the International Mathematical Union, and was the chair of the Abel Committee from 2010–2011 to 2013–2014.
References
1947 births
Living people
Algebraic geometers
Massachusetts Institute of Technology School of Science alumni
Norwegian expatriates in the United States
Academic staff of the University of Oslo
Members of the Norwegian Academy of Science and Letters
Members of Academia Europaea
Fellows of the American Mathematical Society
Royal Norwegian Society of Sciences and Letters
20th-century Norwegian mathematicians
Norwegian women mathematicians
20th-century women mathematicians
21st-century women mathematicians
20th-century Norwegian women
20th-century Norwegian people
Presidents of the Norwegian Mathematical Society |
https://en.wikipedia.org/wiki/Relative%20canonical%20model | In the mathematical field of algebraic geometry, the relative canonical model of a singular variety of a mathematical object where
is a particular canonical variety that maps to , which simplifies the structure.
Description
The precise definition is:
If is a resolution define the adjunction sequence to be the sequence of subsheaves if is invertible where is the higher adjunction ideal. Problem. Is finitely generated? If this is true then is called the relative canonical model of , or the canonical blow-up of .
Some basic properties were as follows:
The relative canonical model was independent of the choice of resolution.
Some integer multiple of the canonical divisor of the relative canonical model was Cartier and the number of exceptional components where this agrees with the same multiple of the canonical divisor of Y is also independent of the choice of Y. When it equals the number of components of Y it was called crepant. It was not known whether relative canonical models were Cohen–Macaulay.
Because the relative canonical model is independent of , most authors simplify the terminology, referring to it as the relative canonical model of rather than either the relative canonical model of or the canonical blow-up of . The class of varieties that are relative canonical models have canonical singularities. Since that time in the 1970s other mathematicians solved affirmatively the problem of whether they are Cohen–Macaulay. The minimal model program started by Shigefumi Mori proved that the sheaf in the definition always is finitely generated and therefore that relative canonical models always exist.
References
Algebraic geometry
Birational geometry
Complex manifolds
Dimension |
https://en.wikipedia.org/wiki/Generalized%20chi-squared%20distribution | In probability theory and statistics, the generalized chi-squared distribution (or generalized chi-square distribution) is the distribution of a quadratic form of a multinormal variable (normal vector), or a linear combination of different normal variables and squares of normal variables. Equivalently, it is also a linear sum of independent noncentral chi-square variables and a normal variable. There are several other such generalizations for which the same term is sometimes used; some of them are special cases of the family discussed here, for example the gamma distribution.
Definition
The generalized chi-squared variable may be described in multiple ways. One is to write it as a linear sum of independent noncentral chi-square variables and a normal variable:
Here the parameters are the weights , the degrees of freedom and non-centralities of the constituent chi-squares, and the normal parameters and . Some important special cases of this have all weights of the same sign, or have central chi-squared components, or omit the normal term.
Since a non-central chi-squared variable is a sum of squares of normal variables with different means, the generalized chi-square variable is also defined as a sum of squares of independent normal variables, plus an independent normal variable: that is, a quadratic in normal variables.
Another equivalent way is to formulate it as a quadratic form of a normal vector :
.
Here is a matrix, is a vector, and is a scalar. These, together with the mean and covariance matrix of the normal vector , parameterize the distribution. The parameters of the former expression (in terms of non-central chi-squares, a normal and a constant) can be calculated in terms of the parameters of the latter expression (quadratic form of a normal vector). If (and only if) in this formulation is positive-definite, then all the in the first formulation will have the same sign.
For the most general case, a reduction towards a common standard form can be made by using a representation of the following form:
where D is a diagonal matrix and where x represents a vector of uncorrelated standard normal random variables.
Computing the pdf/cdf/inverse cdf/random numbers
The probability density, cumulative distribution, and inverse cumulative distribution functions of a generalized chi-squared variable do not have simple closed-form expressions. However, numerical algorithms and computer code (Fortran and C, Matlab, R, Python) have been published to evaluate some of these, and to generate random samples.
In the case where , it is possible to obtain an exact expression for the mean and variance of , as shown in the article on quadratic forms.
Applications
The generalized chi-squared is the distribution of statistical estimates in cases where the usual statistical theory does not hold, as in the examples below.
In model fitting and selection
If a predictive model is fitted by least squares, but the residuals have either autocor |
https://en.wikipedia.org/wiki/Mahmoud%20Vahidnia | Mahmoud Vahidnia (; born 1989 in Tehran) is an Iranian philosopher and PhD candidate of philosophy at Shahid Beheshti University.
Life
Vahidnia received his BSc in mathematics from Sharif University of Technology and his MA in philosophy from Shahid Beheshti University.
He won a gold medal at the Iranian Mathematical Olympiad in 2007. He is also a winner of the silver medal in national computer Olympiad in Iran. Although some media reported he was the "international mathematics Olympiad winner", he didn't compete at any international mathematics Olympiads.
Criticism of Iranian Supreme Leader
Vahidnia got significant media attention when he was a student in the Department of Mathematics at Sharif University of Technology for his face-to-face criticism of Iran's supreme leader on October 28, 2009, during a meeting between Ali Khamenei and students. During this meeting, Khamenei was challenged by Vahidnia in what was called "an unusual encounter". The event was being broadcast by Iranian state-run TV. That made the authorities stop airing the programme.
Citations
On October 28, 2009, during the annual meeting of Tehran intellectual elites with the Supreme Leader, in Tehran University, Vahidinia spoke for 20 minutes without interruptions, critiquing the status of ignorant idol in a golden cage of the Supreme Ayatollah Khamenei. Some time before, Khamenei had announced that "Contestation of the 12 June vote is the worst crime possible". The Iranian state-run TV stopped the broadcasting, but the audience's cellphones managed to record the full speech. Mahmoud asked: "I want to ask you something: why does nobody in this country dare to criticize you? Do you think that you never make mistakes? Isn't this ignorance? You have been changed into a kind of inaccessible idol that nobody can criticize. I don't understand why everybody is forbidden to criticize your choices.".
Structure of the speech
Vahidnia classified his criticism in four parts:
State-run TV: IRIB (Islamic Republic of Iran Broadcasting) for trying to show a reverse image of what is happening in Iran after June 12, 2009, election and destroying the figures that people trust. He brought up that since the head of IRIB is selected by the Supreme Leader, Ali Khamenei, Khamenei is either unaware of what is happening in an organization under his control or he has direct control and is responsible for their programs.
Freedom of speech: intelligence-based atmosphere ruling the media and press and brought up the issues that critical newspapers have been facing. He asked for an end to closure of press offices and demanded freedom of the press even when they criticize the supreme leader.
Supreme leader is criticize-able: Lack of openness in society so that people and intellectuals could freely criticize the supreme leader since the supreme leader, like anyone else, is prone to making mistakes. In his speech, Vahidnia mentioned that the ones around the supreme leader are making an idol.
Organiza |
https://en.wikipedia.org/wiki/Svetlana%20Kuznetsova%20career%20statistics | This is a list of the main career statistics of professional tennis player Svetlana Kuznetsova. Since her professional debut in 2000, she has won 18 singles titles and 16 doubles titles on the WTA Tour. Some of her major titles include two Grand Slam titles in singles (2004 US Open and 2009 French Open) and two in doubles (Australian Open in 2005 and 2012). During her career, she has made 62 top 10 wins, including seven wins over world No. 1 in that moment and earned more than $25M prize money. In singles, she reached career-highest ranking of place 2, while in doubles she is former world No. 3. Playing for Russia at the Billie Jean King Cup, she won three titles, in 2004, 2007 and 2008.
Performance timelines
Only main-draw results in WTA Tour, Grand Slam tournaments, Fed Cup/Billie Jean King Cup and Olympic Games are included in win–loss records.
Singles
Current through 2021 Wimbledon Championships.
Doubles
Current through the suspension of the 2020 WTA Tour.
Grand Slam finals
Singles: 4 (2 titles, 2 runner-ups)
Doubles: 7 (2 titles, 5 runner-ups)
Tier I / Premier Mandatory & Premier 5 finals
Singles: 13 (2 titles, 11 runners-up)
Doubles: 6 (4 titles, 2 runner-ups)
WTA career finals
Singles: 42 (18 titles, 24 runners–up)
Doubles: 31 (16 titles, 15 runners–up)
Junior Grand Slam finals
Singles: 2 (2 runners-up)
Doubles: 4 (1 title, 3 runners-up)
WTA Tour career earnings
Current as of 15 November 2021
Fed Cup/Billie Jean King Cup participation
This table is current through the 2015 Fed Cup
Singles (21–11)
Doubles (6–2)
Career Grand Slam tournament seedings
Record against other players
Record against top 10 players
Kuznetsova's record against players who have been ranked in the top 10. Active players are in boldface.
No. 1 wins
Top 10 wins
See also
List of Grand Slam Women's Singles champions
WTA Tour records
Notes
References
External links
Tennis career statistics |
https://en.wikipedia.org/wiki/Qaiser%20Mushtaq | Qaiser Mushtaq (born 28 February 1954), (D.Phil.(Oxon), ASA, KIA), is a Pakistani mathematician and academic who has made numerous contributions in the field of Group theory and Semigroup. He has been vice-chancellor of The Islamia University Bahawalpur from December 2014 to December 2018. Mushtaq is one of the leading mathematicians and educationists in Pakistan. Through his research and writings, he has exercised a profound influence on mathematics in Pakistan. Mushtaq is an honorary full professor at the Mathematics Division of the Institute for Basic Research, Florida, US.
His research contributions in the fields of group theory and LA-semigroup theory have won him recognition at both national and international levels. In Graham Higman's words, "he has laid the foundation of coset diagrams for the modular group", to study the actions of groups on various spaces and projective lines over Galois fields. This work has been cited in the Encyclopedia of Design Theory.
Biography
Qaiser Mushtaq was born in Sheikhupura, Pakistan to Pir Mushtaq Ali and Begum Saghira Akhter, and belongs to the Qureshi family of Gujranwala. He is a descendant of Shah Jamal Nuri. Mushtaq married Aileen Qaiser, a senior journalist educated from the National University of Singapore and Wolfson College, Oxford. They have two daughters, Shayyan Qaiser and Zara Qaiser.
He received primary education from the Convent of Jesus and Mary, Sialkot, and secondary education from Government Pilot Secondary School, Sialkot. Mushtaq studied for a certain period at Murray College till his family moved to Rawalpindi, where he studied at Gordon College. He did his MSc and M.Phil. from Quaid-i-Azam University, Islamabad. He then joined Bahauddin Zakariya University, Multan, as a lecturer for a period of one year before returning to Quaid-i-Azam University in 1979. Later, in 1980, he received the Royal Scholarship to do his D.Phil. at Wolfson College, Oxford. He was a doctoral student of Graham Higman and was awarded a doctorate in 1983 for a thesis entitled Coset Diagrams for the Modular Group. In 1990 he was at the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy, as a visiting mathematician. He also worked as an associate professor at the Universiti Brunei Darussalam from 1993 to 1999, after which he returned to Pakistan.
Mushtaq is a tenured professor at Quaid-i-Azam University, and a former syndicate member, Quaid-i-Azam University, Islamabad. He is also an honorary full professor at the Mathematics Division, Institute for Basic Research, Florida, US.
He served as vice chancellor of The Islamia University of Bahawalpur from 19 December 2014 to December 2018.
Research in mathematics
He has parametrised actions of the modular group on projective lines over Galois fields. This method has proven to be so effective and rewarding, that its wide uses can be seen in Combinatorial Group Theory, Algebraic Number Theory, and Theory of Group Graphs. His graphical techn |
https://en.wikipedia.org/wiki/Bernstein%20set | In mathematics, a Bernstein set is a subset of the real line that meets every uncountable closed subset of the real line but that contains none of them.
A Bernstein set partitions the real line into two pieces in a peculiar way: every measurable set of positive measure meets both the Bernstein set and its complement, as does every set with the property of Baire that is not a meagre set.
References
Descriptive set theory
Sets of real numbers |
https://en.wikipedia.org/wiki/Intelligent%20Home%20Control | Intelligent House Concept is a building automation system using a star configured topology with wires to each device. Originally made by LK, but now owned by Schneider Electric and sold as "IHC Intelligent House Concept".
The system is made up of a central controller and up to 8 input modules and 16 output modules. Each input module can have 16 digital (on/off) inputs and each output module 8 digital (on/off) outputs, resulting in a total of 128 input and 128 outputs per controller.
Module control protocol
The central controller has one point-to-point data communication wire connected to each module. The protocol between the central controller and the modules uses a 5V pulse width encoding as follows:
A header that is 4100µs high and 300µs low
One pulse per I/O port, i.e. 16 pulses for input modules and 8 pulses for output modules
One addition parity pulse; an even number of pulses is 0 parity and odd number pulses is 1 parity
The pulse width is 600µs
A 0 (input or output off, or even parity) is encoded as 300µs high and 300µs low
A 1 (input or output on, or odd parity) is encoded as 150µs high and 450µs low
The above signal constantly repeats.
References
Smart Home Automation for Better, Smarter Living
Building automation
Home automation |
https://en.wikipedia.org/wiki/Frederick%20Higginbottom | Frederick James Higginbottom (21 October 1859 – 12 May 1943) was a British journalist and newspaper editor.
The son of a mathematics tutor, Higginbottom was born in Accrington, Lancashire. He began his career as a journalist with the Southport Daily News at the age of fifteen, and became the editor of the Southport Visiter just five years later. Though a small paper, it provided him with an opportunity to demonstrate his skills, and he was hired by the Press Association in 1881 to serve as their Dublin correspondent in 1882.
Higginbottom moved to London in 1892, where he served briefly as a correspondent for an Irish newspaper before founding the London Press Exchange, which provided news and advertising for the provincial press. He also started working for the Pall Mall Gazette as their parliamentary correspondent. In 1900, he left for a position with the Daily Chronicle, but returned to the Pall Mall Gazette soon afterward.
In 1909, Higginbottom was named editor of the Pall Mall Gazette by its owner, William Waldorf Astor. As editor, Higginbottom proved capable but unimaginative. He did little to change the paper's position on the issues of the day, nor did he succeed in restoring the Gazette to profitability. After three years as editor, Astor replaced him with J. L. Garvin and Higginbottom returned to his position as parliamentary correspondent. He continued with the Pall Mall Gazette (apart from a brief period as director of press intelligence for the Ministry of National Service in 1917-18) until 1919, when he moved to the Daily Chronicle. He worked for the Chronicle until his retirement in 1930.
Works
The Vivid Life: A Journalist's Career (London: Simpkin Marshall Limited, 1934)
References
Further reading
1859 births
1943 deaths
British male journalists
British newspaper editors |
https://en.wikipedia.org/wiki/Semilinear | Semilinear or semi-linear (literally, "half linear") may refer to:
Mathematics
Antilinear map, also called a "semilinear map"
Semilinear order
Semilinear map
Semilinear set
Semilinearity (operator theory)
Semilinear equation, a type of differential equation which is linear in the highest order derivative(s) of the unknown function
Various forms of "mild" nonlinearity are referred to as "semilinear"
Other
Semilinear response, physics
Artificial neuron, also called a "semi-linear unit"
Semi-linear resolution
A mixture of linear and nonlinear gameplay in video games may be referred to as "semi-linear gameplay" |
https://en.wikipedia.org/wiki/Semilinear%20map | In linear algebra, particularly projective geometry, a semilinear map between vector spaces V and W over a field K is a function that is a linear map "up to a twist", hence semi-linear, where "twist" means "field automorphism of K". Explicitly, it is a function that is:
additive with respect to vector addition:
there exists a field automorphism θ of K such that , where is the image of the scalar under the automorphism. If such an automorphism exists and T is nonzero, it is unique, and T is called θ-semilinear.
Where the domain and codomain are the same space (i.e. ), it may be termed a semilinear transformation. The invertible semilinear transforms of a given vector space V (for all choices of field automorphism) form a group, called the general semilinear group and denoted by analogy with and extending the general linear group. The special case where the field is the complex numbers and the automorphism is complex conjugation, a semilinear map is called an antilinear map.
Similar notation (replacing Latin characters with Greek) are used for semilinear analogs of more restricted linear transform; formally, the semidirect product of a linear group with the Galois group of field automorphism. For example, PΣU is used for the semilinear analogs of the projective special unitary group PSU. Note however, that it is only recently noticed that these generalized semilinear groups are not well-defined, as pointed out in – isomorphic classical groups G and H (subgroups of SL) may have non-isomorphic semilinear extensions. At the level of semidirect products, this corresponds to different actions of the Galois group on a given abstract group, a semidirect product depending on two groups and an action. If the extension is non-unique, there are exactly two semilinear extensions; for example, symplectic groups have a unique semilinear extension, while has two extensions if n is even and q is odd, and likewise for PSU.
Definition
A map for vector spaces and over fields and respectively is -semilinear, or simply semilinear, if there exists a field homomorphism such that for all , in and in it holds that
A given embedding of a field in allows us to identify with a subfield of , making a -semilinear map a K-linear map under this identification. However, a map that is -semilinear for a distinct embedding will not be K-linear with respect to the original identification , unless is identically zero.
More generally, a map between a right -module and a left -module is -semilinear if there exists a ring antihomomorphism such that for all , in and in it holds that
The term semilinear applies for any combination of left and right modules with suitable adjustment of the above expressions, with being a homomorphism as needed.
The pair is referred to as a dimorphism.
Related
Transpose
Let be a ring isomorphism, a right -module and a right -module, and a -semilinear map. Define the transpose of as the mapping tha |
https://en.wikipedia.org/wiki/Rank%20factorization | In mathematics, given a field , nonnegative integers , and a matrix , a rank decomposition or rank factorization of is a factorization of of the form , where and , where is the rank of .
Existence
Every finite-dimensional matrix has a rank decomposition: Let be an matrix whose column rank is . Therefore, there are linearly independent columns in ; equivalently, the dimension of the column space of is . Let be any basis for the column space of and place them as column vectors to form the matrix . Therefore, every column vector of is a linear combination of the columns of . To be precise, if is an matrix with as the -th column, then
where 's are the scalar coefficients of in terms of the basis . This implies that , where is the -th element of .
Non-uniqueness
If is a rank factorization, taking and
gives another rank factorization for any invertible matrix of compatible dimensions.
Conversely, if are two rank factorizations of , then there exists an invertible matrix such that and .
Construction
Rank factorization from reduced row echelon forms
In practice, we can construct one specific rank factorization as follows: we can compute , the reduced row echelon form of . Then is obtained by removing from all non-pivot columns (which can be determined by looking for columns in which do not contain a pivot), and is obtained by eliminating any all-zero rows of .
Note: For a full-rank square matrix (i.e. when ), this procedure will yield the trivial result and (the identity matrix).
Example
Consider the matrix
is in reduced echelon form.
Then is obtained by removing the third column of , the only one which is not a pivot column, and by getting rid of the last row of zeroes from , so
It is straightforward to check that
Proof
Let be an permutation matrix such that in block partitioned form, where the columns of are the pivot columns of . Every column of is a linear combination of the columns of , so there is a matrix such that , where the columns of contain the coefficients of each of those linear combinations. So , being the identity matrix. We will show now that .
Transforming into its reduced row echelon form amounts to left-multiplying by a matrix which is a product of elementary matrices, so , where . We then can write , which allows us to identify , i.e. the nonzero rows of the reduced echelon form, with the same permutation on the columns as we did for . We thus have , and since is invertible this implies , and the proof is complete.
Singular value decomposition
If then one can also construct a full-rank factorization of via a singular value decomposition
Since is a full-column-rank matrix and is a full-row-rank matrix, we can take and .
Consequences
rank(A) = rank(AT)
An immediate consequence of rank factorization is that the rank of is equal to the rank of its transpose . Since the columns of are the rows of , the column rank of equals its row rank.
Proof: To see |
https://en.wikipedia.org/wiki/PFC%20Litex%20Lovech%20in%20European%20football |
Total statistics
Statistics by country
Statistics by competition
UEFA Champions League / European Cup
UEFA Europa League / UEFA Cup
References
European cup history
Bulgarian football clubs in international competitions |
https://en.wikipedia.org/wiki/%C3%81d%C3%A1m%20Balajti | Ádám Balajti (born 7 March 1991) is a Hungarian professional footballer who plays for Vasas II.
Career statistics
Honours
FIFA U-20 World Cup:
Third place: 2009
References
External links
1991 births
Footballers from Eger
Living people
Hungarian men's footballers
Hungary men's youth international footballers
Hungary men's under-21 international footballers
Men's association football forwards
Diósgyőri VTK players
Debreceni VSC players
Újpest FC players
MTK Budapest FC players
Mezőkövesdi SE footballers
Szolnoki MÁV FC footballers
Vác FC players
Vasas SC players
Tiszakécske FC footballers
Nemzeti Bajnokság I players
Nemzeti Bajnokság II players
Nemzeti Bajnokság III players |
https://en.wikipedia.org/wiki/Vera%20T.%20S%C3%B3s | Vera Turán Sós (11 September 1930 – 22 March 2023) was a Hungarian mathematician who specialized in number theory and combinatorics. She was a student and close collaborator of both Paul Erdős and Alfréd Rényi. She also collaborated frequently with her husband Pál Turán, an analyst, number theorist, and combinatorist. Until 1987, she worked at the Department of Analysis at the Eötvös Loránd University, Budapest. Afterwards, she was employed by the Alfréd Rényi Institute of Mathematics. She was elected a corresponding member (1985) and member (1990) of the Hungarian Academy of Sciences. In 1997, Sós was awarded the Széchenyi Prize.
One of her contributions is the Kővári–Sós–Turán theorem concerning the maximum possible number of edges in a bipartite graph that does not contain certain complete subgraphs. Another is the following so-called friendship theorem proved with Paul Erdős and Alfréd Rényi: if, in a finite graph, any two vertices have exactly one common neighbor, then some vertex is joined to all others. In number theory, Sós proved the three-gap theorem, conjectured by Hugo Steinhaus and proved independently by Stanisław Świerczkowski.
Life and career
Vera Sós was the daughter of a school teacher. As an adolescent, Sós attended the Abonyi Street Jewish high school in Budapest and graduated in 1948. She was later introduced to Alfréd Rényi and Paul Erdős, with whom she later collaborated, by her teacher Tibor Gallai. (Together, she and Erdős wrote thirty papers.) Sós considered Gallai to be the person who discovered her gift for mathematics. Sós was also one of three girls in Gallai's class to became mathematicians. Sós later attended Eötvös Loránd University. There, she studied as a mathematics and physics major and graduated in 1952. Although she was still a student, Sós taught at Eötvös University in 1950. At the age of twenty, Sós attended a Mathematical Congress in Budapest, Hungary and attended a summer internship. Sós met her husband and collaborator Paul Turán in college. They married in 1952. The two had two children in 1953 and 1960, Gyorgy and Thomas Turán. Turán died in September 1976.
In 1965, Sós began the weekly Hajnal–Sós seminar at the Mathematical Institute of the Hungarian Academy for Science with András Hajnal. The seminar is considered to be a "forum for new results in combinatorics." This weekly seminar continues to this day.
Throughout her years working in mathematics, Sós had been honored with many awards as a result of her work. One of the many awards includes the Széchenyi Prize, which she received in 1997. The Széchenyi Prize is an award given to those who have greatly contributed to the academic life of Hungary.
Awards
Member of Academia Europaea, 2013
Széchenyi Prize, 1997
Academic Award, 1983
Cross of the Hungarian Order of Merit, 2002
Tibor Szele Medal, 1974
Selected publications
Notes
References
MTI Ki Kicsoda [Who's Who] 2009, Magyar Távirati Iroda Zrt., Budapest, 2008, pp. 1130.,
A Magyar Tu |
https://en.wikipedia.org/wiki/Sagitta%20%28geometry%29 | In geometry, the sagitta (sometimes abbreviated as sag) of a circular arc is the distance from the midpoint of the arc to the midpoint of its chord. It is used extensively in architecture when calculating the arc necessary to span a certain height and distance and also in optics where it is used to find the depth of a spherical mirror or lens. The name comes directly from Latin sagitta, meaning an "arrow".
Formulas
In the following equations, denotes the sagitta (the depth or height of the arc), equals the radius of the circle, and the length of the chord spanning the base of the arc. As and are two sides of a right triangle with as the hypotenuse, the Pythagorean theorem gives us
This may be rearranged to give any of the other three:
The sagitta may also be calculated from the versine function, for an arc that spans an angle of , and coincides with the versine for unit circles
Approximation
When the sagitta is small in comparison to the radius, it may be approximated by the formula
Alternatively, if the sagitta is small and the sagitta, radius, and chord length are known, they may be used to estimate the arc length by the formula
where is the length of the arc; this formula was known to the Chinese mathematician Shen Kuo, and a more accurate formula also involving the sagitta was developed two centuries later by Guo Shoujing.
Applications
Architects, engineers, and contractors use these equations to create "flattened" arcs that are used in curved walls, arched ceilings, bridges, and numerous other applications.
The sagitta also has uses in physics where it is used, along with chord length, to calculate the radius of curvature of an accelerated particle. This is used especially in bubble chamber experiments where it is used to determine the momenta of decay particles. Likewise historically the sagitta is also utilised as a parameter in the calculation of moving bodies in a centripetal system. This method is utilised in Newton's Principia.
See also
Circular segment
Versine
References
External links
Calculating the Sagitta of an Arc
Architectural terminology
Geometric measurement |
https://en.wikipedia.org/wiki/Goran%20Arnaut | Goran Arnaut (Serbian Cyrillic: Горан Арнаут; born 27 August 1979) is a former Serbian footballer who played mainly as a central midfielder.
Statistics
References
External links
HLSZ profile
PrvaLiga profile
1979 births
Living people
People from Bačka Topola
Footballers from North Bačka District
Serbian men's footballers
Men's association football midfielders
OFK Beograd players
FK Partizan players
FK Radnički 1923 players
FK Čukarički players
FK Teleoptik players
NK Primorje players
Slovenian PrvaLiga players
Shamakhi FK players
Serbian expatriate men's footballers
Expatriate men's footballers in Slovenia
Serbian expatriate sportspeople in Slovenia
Expatriate men's footballers in Azerbaijan
Serbian expatriate sportspeople in Azerbaijan
Expatriate men's footballers in Hungary
Serbian expatriate sportspeople in Hungary
FK Partizan non-playing staff
Serbia and Montenegro expatriate sportspeople in Slovenia
Serbia and Montenegro men's footballers
Serbia and Montenegro expatriate men's footballers |
https://en.wikipedia.org/wiki/Tensor%20product%20model%20transformation | In mathematics, the tensor product (TP) model transformation was proposed by Baranyi and Yam as key concept for higher-order singular value decomposition of functions. It transforms a function (which can be given via closed formulas or neural networks, fuzzy logic, etc.) into TP function form if such a transformation is possible. If an exact transformation is not possible, then the method determines a TP function that is an approximation of the given function. Hence, the TP model transformation can provide a trade-off between approximation accuracy and complexity.
A free MATLAB implementation of the TP model transformation can be downloaded at or an old version of the toolbox is available at MATLAB Central . A key underpinning of the transformation is the higher-order singular value decomposition.
Besides being a transformation of functions, the TP model transformation is also a new concept in qLPV based control which plays a central role in the providing a valuable means of bridging between identification and polytopic systems theories. The TP model transformation is uniquely effective in manipulating the convex hull of polytopic forms, and, as a result has revealed and proved the fact that convex hull manipulation is a necessary and crucial step in achieving optimal solutions and decreasing conservativeness in modern LMI based control theory. Thus, although it is a transformation in a mathematical sense, it has established a conceptually new direction in control theory and has laid the ground for further new approaches towards optimality. Further details on the control theoretical aspects of the TP model transformation can be found here: TP model transformation in control theory.
The TP model transformation motivated the definition of the "HOSVD canonical form of TP functions", on which further information can be found here. It has been proved that the TP model transformation is capable of numerically reconstructing this HOSVD based canonical form. Thus, the TP model transformation can be viewed as a numerical method to compute the HOSVD of functions, which provides exact results if the given function has a TP function structure and approximative results otherwise.
The TP model transformation has recently been extended in order to derive various types of convex TP functions and to manipulate them. This feature has led to new optimization approaches in qLPV system analysis and design, as described at TP model transformation in control theory.
Definitions
Finite element TP function A given function , where , is a TP function if it has the structure:
that is, using compact tensor notation (using the tensor product operation of ):
where core tensor is constructed from , and row vector contains continuous univariate weighting functions . The function is the -th weighting function defined on the -th dimension, and is the -the element of vector . Finite element means that is bounded for all . For qLPV modelling and control a |
https://en.wikipedia.org/wiki/Lieb%E2%80%93Liniger%20model | The Lieb–Liniger model describes a gas of particles moving in one dimension and satisfying Bose–Einstein statistics.
Introduction
A model of a gas of particles moving in one dimension and satisfying Bose–Einstein statistics was introduced in 1963 in order to study whether the available approximate theories of such gases, specifically Bogoliubov's theory, would conform to the actual properties of the model gas. The model is based on a well defined Schrödinger Hamiltonian for particles interacting with each other via a two-body potential, and all the eigenfunctions and eigenvalues of this Hamiltonian can, in principle, be calculated exactly. Sometimes it is called one dimensional Bose gas with delta interaction. It also can be considered as quantum non-linear Schrödinger equation.
The ground state as well as the low-lying excited states were computed and found to be in agreement with Bogoliubov's theory when the potential is small, except for the fact that there are actually two types of elementary excitations instead of one, as predicted by Bogoliubov's and other theories.
The model seemed to be only of academic interest until, with the sophisticated experimental techniques developed in the first decade of the 21st century, it became possible to produce this kind of gas using real atoms as particles.
Definition and solution of the model
There are boson particles with coordinates on the line , with periodic boundary conditions. Thus, a state of the N-body system must be described by a wave function that remains unchanged under permutation of any two particles (permutation symmetry), i.e., for all and satisfies for all . The Hamiltonian, in appropriate units, is
where is the Dirac delta function, i.e., the interaction is a contact interaction. The constant denotes its strength. The delta function gives rise to a boundary condition when two coordinates, say and are equal; this condition is that as , the derivative satisfies . The hard core limit is known as the Tonks–Girardeau gas.
Schrödinger's time independent equation, is solved by explicit construction of . Since is symmetric it is completely determined by its values in the simplex , defined by the condition that . In this region one looks for a of the form considered by H.A. Bethe in 1931 in the context of magnetic spin systems—the Bethe ansatz. That is, for certain real numbers , to be determined,
where the sum is over all permutations, , of the integers , and maps to . The coefficients , as well as the 's are determined by the condition , and this leads to
Dorlas (1993) proved that all eigenfunctions of are of this form.
These equations determine in terms of the 's, which, in turn, are determined by the periodic boundary conditions. These lead to equations:
where are integers when is odd and, when is even, they take values . For the ground state the 's satisfy
The first kind of elementary excitation consists in choosing as before, but i |
https://en.wikipedia.org/wiki/Group%20testing | In statistics and combinatorial mathematics, group testing is any procedure that breaks up the task of identifying certain objects into tests on groups of items, rather than on individual ones. First studied by Robert Dorfman in 1943, group testing is a relatively new field of applied mathematics that can be applied to a wide range of practical applications and is an active area of research today.
A familiar example of group testing involves a string of light bulbs connected in series, where exactly one of the bulbs is known to be broken. The objective is to find the broken bulb using the smallest number of tests (where a test is when some of the bulbs are connected to a power supply). A simple approach is to test each bulb individually. However, when there are a large number of bulbs it would be much more efficient to pool the bulbs into groups. For example, by connecting the first half of the bulbs at once, it can be determined which half the broken bulb is in, ruling out half of the bulbs in just one test.
Schemes for carrying out group testing can be simple or complex and the tests involved at each stage may be different. Schemes in which the tests for the next stage depend on the results of the previous stages are called adaptive procedures, while schemes designed so that all the tests are known beforehand are called non-adaptive procedures. The structure of the scheme of the tests involved in a non-adaptive procedure is known as a pooling design.
Group testing has many applications, including statistics, biology, computer science, medicine, engineering and cyber security. Modern interest in these testing schemes has been rekindled by the Human Genome Project.
Basic description and terms
Unlike many areas of mathematics, the origins of group testing can be traced back to a single report written by a single person: Robert Dorfman. The motivation arose during the Second World War when the United States Public Health Service and the Selective service embarked upon a large-scale project to weed out all syphilitic men called up for induction. Testing an individual for syphilis involves drawing a blood sample from them and then analysing the sample to determine the presence or absence of syphilis. At the time, performing this test was expensive, and testing every soldier individually would have been very expensive and inefficient.
Supposing there are soldiers, this method of testing leads to separate tests. If a large proportion of the people are infected then this method would be reasonable. However, in the more likely case that only a very small proportion of the men are infected, a much more efficient testing scheme can be achieved. The feasibility of a more effective testing scheme hinges on the following property: the soldiers can be pooled into groups, and in each group the blood samples can be combined together. The combined sample can then be tested to check if at least one soldier in the group has syphilis. This is the central ide |
https://en.wikipedia.org/wiki/Pedro%20Baquero | Pedro Jesús López Baquero (born 2 October 1980 in Huelva, Andalusia) is a Spanish retired footballer who played as a central defender.
Club statistics
Honours
Recreativo
Segunda División: 2005–06
Rayo Vallecano
Segunda División B: 2007–08
Cádiz
Segunda División B: 2011–12
References
External links
1980 births
Living people
Spanish men's footballers
Footballers from Huelva
Men's association football defenders
Segunda División players
Segunda División B players
Tercera División players
Atlético Onubense players
Gimnàstic de Tarragona footballers
Recreativo de Huelva players
Rayo Vallecano players
Lorca Deportiva CF footballers
Pontevedra CF footballers
Cádiz CF players
Real Oviedo players
Lleida Esportiu footballers
CD San Roque de Lepe footballers
Cypriot First Division players
Doxa Katokopias FC players
Spanish expatriate men's footballers
Expatriate men's footballers in Cyprus
Spanish expatriate sportspeople in Cyprus |
https://en.wikipedia.org/wiki/Robert%20Sorgenfrey | Robert Henry Sorgenfrey (August 14, 1915 – January 7, 1996) was an American mathematician and Professor Emeritus of Mathematics at the University of California, Los Angeles. The Sorgenfrey line and the Sorgenfrey plane are named after him; the Sorgenfrey line was the first example of a normal topological space whose product with itself is not normal.
References
1915 births
1996 deaths
20th-century American mathematicians |
https://en.wikipedia.org/wiki/Jonathan%20Mart%C3%ADn%20%28footballer%29 | Jonathan Martín Carabias (born 6 March 1981 in Salamanca, Region of León) is a Spanish former footballer who plays as a defender.
Career statistics
Club
References
External links
1981 births
Living people
Footballers from Salamanca
Spanish men's footballers
Men's association football defenders
La Liga players
Segunda División players
Segunda División B players
Tercera División players
Real Valladolid Promesas players
Real Valladolid players
Cultural y Deportiva Leonesa players
Racing de Ferrol footballers
CD Guijuelo footballers |
https://en.wikipedia.org/wiki/Kasner | Kasner is a surname. Notable people with the surname include:
Angela Merkel, née Kasner (born 1954), German Chancellor
Edward Kasner (1878–1955), American mathematician, Tutor on Mathematics in the Columbia University Mathematics Department
Marliese Kasner (born Marliese Miller) (born 1982), Canadian curler from Canwood, Saskatchewan
Stephen Kasner (1970–2019), painter, illustrator, musician, photographer, graphic artist, and magician
See also
Kasner metric, an exact solution to Einstein's theory of general relativity
Kasner polygon of a polygon is the polygon whose vertices are the midpoints of the edges of
Kasner's dwarf burrowing skink (Scelotes kasneri) is a species of skink in the family Scincidae
Kassner
German-language surnames |
https://en.wikipedia.org/wiki/Renato%20Gonz%C3%A1lez | Renato Patricio González De La Hoz (born 19 February 1990) is a Chilean footballer that currently plays for Deportes Valdivia of the Segunda División Profesional de Chile.
Career statistics
International goals
Honours
Club
San Marcos de Arica
Primera B: 2012
References
External links
1990 births
Living people
Footballers from Santiago
Chilean men's footballers
Chilean expatriate men's footballers
Chile men's international footballers
Club Deportivo Palestino footballers
Associação Atlética Ponte Preta players
C.D. Universidad de Concepción footballers
C.D. Cobresal footballers
San Marcos de Arica footballers
Club Universidad de Chile footballers
C.D. Antofagasta footballers
Santiago Morning footballers
Deportes Puerto Montt footballers
Deportes Recoleta footballers
Deportes Valdivia footballers
Chilean Primera División players
Campeonato Brasileiro Série B players
Primera B de Chile players
Segunda División Profesional de Chile players
Chilean expatriate sportspeople in Brazil
Expatriate men's footballers in Brazil
Men's association football midfielders |
https://en.wikipedia.org/wiki/Radical%20symbol | In mathematics, the radical symbol, radical sign, root symbol, radix, or surd is a symbol for the square root or higher-order root of a number. The square root of a number is written as
while the th root of is written as
It is also used for other meanings in more advanced mathematics, such as the radical of an ideal.
In linguistics, the symbol is used to denote a root word.
Principal square root
Each positive real number has two square roots, one positive and the other negative. The square root symbol refers to the principal square root, which is the positive one. The two square roots of a negative number are both imaginary numbers, and the square root symbol refers to the principal square root, the one with a positive imaginary part. For the definition of the principal square root of other complex numbers, see Square root#Principal square root of a complex number.
Origin
The origin of the root symbol √ is largely speculative. Some sources imply that the symbol was first used by Arab mathematicians. One of those mathematicians was Abū al-Hasan ibn Alī al-Qalasādī (1421–1486). Legend has it that it was taken from the Arabic letter "" (ǧīm), which is the first letter in the Arabic word "" (jadhir, meaning "root"). However, Leonhard Euler believed it originated from the letter "r", the first letter of the Latin word "radix" (meaning "root"), referring to the same mathematical operation.
The symbol was first seen in print without the vinculum (the horizontal "bar" over the numbers inside the radical symbol) in the year 1525 in Die Coss by Christoff Rudolff, a German mathematician. In 1637 Descartes was the first to unite the German radical sign √ with the vinculum to create the radical symbol in common use today.
Encoding
The Unicode and HTML character codes for the radical symbols are:
However, these characters differ in appearance from most mathematical typesetting by omitting the overline connected to the radical symbol, which surrounds the argument of the square root function. The OpenType math table allows adding this overline following the radical symbol.
Legacy encodings of the square root character U+221A include:
0xC3 in Mac OS Roman and Mac OS Cyrillic
0xFB (+) in Code page 437 and Code page 866 (but not Code page 850) on DOS and the Windows console
0xD6 in the Symbol font encoding
02-69 (7-bit 0x2265, SJIS 0x81E3, EUC 0xA2E5) in Japanese JIS X 0208
01-78 (EUC/UHC 0xA1EE) in Korean Wansung code
01-44 (EUC 0xA1CC) in Mainland Chinese GB 2312 or GBK
Traditional Chinese: 0xA1D4 in Big5 or 1-2235 (kuten 01-02-21, EUC 0xA2B5 or 0x8EA1A2B5) in CNS 11643
The Symbol font displays the character without any vinculum whatsoever; the overline may be a separate character at 0x60. The JIS, Wansung and CNS 11643 code charts include a short overline attached to the radical symbol, whereas the GB 2312 and GB 18030 charts do not.
Additionally a "Radical Symbol Bottom" (U+23B7, ⎷) is available in the Miscellaneous Technical block. This |
https://en.wikipedia.org/wiki/Maximal%20common%20divisor | In abstract algebra, particularly ring theory, maximal common divisors are an abstraction of the number theory concept of greatest common divisor (GCD). This definition is slightly more general than GCDs, and may exist in rings in which GCDs do not. Halter-Koch (1998) provides the following definition.
is a maximal common divisor of a subset, , if the following criteria are met:
for all
Suppose , and for all . Then .
References
Abstract algebra |
https://en.wikipedia.org/wiki/George%20W.%20Lilley | George W. Lilley (February 9, 1850 – June 8, 1904) was an American academic, professor of mathematics, and the first president of two American universities, today known as South Dakota State University and Washington State University.
Early life and education
George W. Lilley was born February 9, 1850, in Kewanee, Henry County, Illinois, the son of William Lilley, a native of England, and Harriet Huntley Lilley, a native of Canada. Pioneers in Henry County, George's parents owned two farms including one with a substantial coal mining operation. They had six children.
George Lilley began his college studies at Knox College in Galesburg, Illinois, earning a bachelor's degree about 1873 or 1874. In 1886, Knox College also granted him an honorary degree. He continued his studies at the University of Michigan in Ann Arbor and at Illinois Wesleyan University in Bloomington, Illinois. The latter institution awarded him both Ph.D. and LL.D. degrees.
Career
Lilley began his career in business in Corning, Iowa, from 1878 to 1880. In early 1884 he was selected to be the first president of Dakota Agricultural College, today South Dakota State University in Brookings. Under his leadership, the first building was constructed and enrollment grew from 35 to 252 students with varying educational preparation. While he resigned from the presidency in 1886, he is credited with laying out many ideas and plans that eventually were realized. He then served four years as professor of mathematics there.
On May 1, 1891, Lilley was appointed as the first president of the Washington Agricultural College and School of Science for a one-year term. His responsibilities included serving as director of college's agricultural experiment station. The town of Pullman in a fertile agricultural region of southeast Washington called the Palouse had just been chosen as the location for Washington's new land-grant institution, today Washington State University. Again he led a fledgling college through construction of its first small building, the hiring of five faculty members and the opening of classes. That occurred on January 12, 1892, with 59 students attending, 13 in collegiate courses and 46 in preparatory courses. Lilley himself served as professor of mathematics and elementary physics, while his brother-in-law Charles E. Munn was appointed professor of veterinary science.
He served the Washington institution through the end of 1892 when the Board of Regents chose John W. Heston as the institution's second president. Lilley is remembered as "a genial Midwesterner" who was popular with the students.
From Pullman, Lilley moved to Portland, Oregon, where he served from 1894 to 1896 as principal of the Park School, a Portland public school established in 1878. Then, in 1897, he became a professor of mathematics at the University of Oregon in Eugene, Oregon, a position he held until his death in 1904. He was the author of several books on algebra and was a contributor to Ameri |
https://en.wikipedia.org/wiki/Ministry%20of%20Statistics | Ministry of Statistics may refer to:
Ministry of Statistics (Pakistan), Pakistan
Ministry of Statistics & Analysis (Minstat), Republic of Belarus
Ministry of Statistics and Programme Implementation (India)
Statistical Committee of Armenia |
https://en.wikipedia.org/wiki/Special%20Data%20Dissemination%20Standard | Special Data Dissemination Standard (SDDS) is an International Monetary Fund standard to guide member countries in the dissemination of national statistics to the public.
It was established in April 1996.
Members
There are currently 65 members.
Argentina
Armenia (November 7, 2003; 54th member)
Australia
Austria
Belarus
Belgium
Brazil
Bulgaria
Canada
Chile
China
Colombia
Costa Rica
Croatia
Czech Republic
Denmark
Ecuador
Egypt
El Salvador
Estonia
Finland
France
Germany
Greece
Hong Kong
Hungary
Iceland
India
Indonesia
Ireland
Israel
Italy
Japan
Kazakhstan
Korea
Kyrgyz Republic
Latvia
Lithuania
Luxembourg
Malaysia
Mauritius
Mexico
Moldova
Morocco
Netherlands
Norway
Peru
Philippines
Poland
Portugal
Romania
Russian Federation
Saudi Arabia
Senegal
Singapore
Slovak Republic
Slovenia
South Africa
Spain
Sri Lanka
Sweden
Switzerland
Thailand
Tunisia
Turkey
Ukraine
United Kingdom
United States
Uruguay
West Bank and Gaza ("Palestine" as of April 19, 2012; 71st member)
References
External links
SDDS - Overview, IMF
Subscribing Countries, IMF
International Monetary Fund
1996 introductions |
https://en.wikipedia.org/wiki/Statistics%20education | Statistics education is the practice of teaching and learning of statistics, along with the associated scholarly research.
Statistics is both a formal science and a practical theory of scientific inquiry, and both aspects are considered in statistics education. Education in statistics has similar concerns as does education in other mathematical sciences, like logic, mathematics, and computer science. At the same time, statistics is concerned with evidence-based reasoning, particularly with the analysis of data. Therefore, education in statistics has strong similarities to education in empirical disciplines like psychology and chemistry, in which education is closely tied to "hands-on" experimentation.
Mathematicians and statisticians often work in a department of mathematical sciences (particularly at colleges and small universities). Statistics courses have been sometimes taught by non-statisticians, against the recommendations of some professional organizations of statisticians and of mathematicians.
Statistics education research is an emerging field that grew out of different disciplines and is currently establishing itself as a unique field that is devoted to the improvement of teaching and learning statistics at all educational levels.
Goals of statistics education
Statistics educators have cognitive and noncognitive goals for students. For example, former American Statistical Association (ASA) President Katherine Wallman defined statistical literacy as including the cognitive abilities of understanding and critically evaluating statistical results as well as appreciating the contributions statistical thinking can make.
Cognitive goals
In the text rising from the 2008 joint conference of the International Commission on Mathematical Instruction and the International Association of Statistics Educators, editors Carmen Batanero, Gail Burrill, and Chris Reading (Universidad de Granada, Spain, Michigan State University, USA, and University of New England, Australia, respectively) note worldwide trends in curricula which reflect data-oriented goals. In particular, educators currently seek to have students: "design investigations; formulate research questions; collect data using observations, surveys, and experiments; describe and compare data sets; and propose and justify conclusions and predictions based on data." The authors note the importance of developing statistical thinking and reasoning in addition to statistical knowledge.
Despite the fact that cognitive goals for statistics education increasingly focus on statistical literacy, statistical reasoning, and statistical thinking rather than on skills, computations and procedures alone, there is no agreement about what these terms mean or how to assess these outcomes. A first attempt to define and distinguish between these three terms appears in the ARTIST website which was created by Garfield, delMas and Chance and has since been included in several publications.
Brief definitions |
https://en.wikipedia.org/wiki/Van%20der%20Corput%20lemma%20%28harmonic%20analysis%29 | In mathematics, in the field of harmonic analysis,
the van der Corput lemma is an estimate for oscillatory integrals
named after the Dutch mathematician J. G. van der Corput.
The following result is stated by E. Stein:
Suppose that a real-valued function is smooth in an open interval ,
and that for all .
Assume that either , or that
and is monotone for .
Then there is a constant , which does not depend on ,
such that
for any .
Sublevel set estimates
The van der Corput lemma is closely related to the sublevel set estimates,
which give the upper bound on the measure of the set
where a function takes values not larger than .
Suppose that a real-valued function is smooth
on a finite or infinite interval ,
and that for all .
There is a constant , which does not depend on ,
such that
for any
the measure of the sublevel set
is bounded by .
References
Inequalities
Harmonic analysis
Fourier analysis |
https://en.wikipedia.org/wiki/Negative%20multinomial%20distribution | In probability theory and statistics, the negative multinomial distribution is a generalization of the negative binomial distribution (NB(x0, p)) to more than two outcomes.
As with the univariate negative binomial distribution, if the parameter is a positive integer, the negative multinomial distribution has an urn model interpretation. Suppose we have an experiment that generates m+1≥2 possible outcomes, {X0,...,Xm}, each occurring with non-negative probabilities {p0,...,pm} respectively. If sampling proceeded until n observations were made, then {X0,...,Xm} would have been multinomially distributed. However, if the experiment is stopped once X0 reaches the predetermined value x0 (assuming x0 is a positive integer), then the distribution of the m-tuple {X1,...,Xm} is negative multinomial. These variables are not multinomially distributed because their sum X1+...+Xm is not fixed, being a draw from a negative binomial distribution.
Properties
Marginal distributions
If m-dimensional x is partitioned as follows
and accordingly
and let
The marginal distribution of is . That is the marginal distribution is also negative multinomial with the removed and the remaining p'''s properly scaled so as to add to one.
The univariate marginal is said to have a negative binomial distribution.
Conditional distributions
The conditional distribution of given is . That is,
Independent sums
If and If are independent, then
. Similarly and conversely, it is easy to see from the characteristic function that the negative multinomial is infinitely divisible.
Aggregation
If
then, if the random variables with subscripts i and j'' are dropped from the vector and replaced by their sum,
This aggregation property may be used to derive the marginal distribution of mentioned above.
Correlation matrix
The entries of the correlation matrix are
Parameter estimation
Method of Moments
If we let the mean vector of the negative multinomial be
and covariance matrix
then it is easy to show through properties of determinants that . From this, it can be shown that
and
Substituting sample moments yields the method of moments estimates
and
Related distributions
Negative binomial distribution
Multinomial distribution
Inverted Dirichlet distribution, a conjugate prior for the negative multinomial
Dirichlet negative multinomial distribution
References
Waller LA and Zelterman D. (1997). Log-linear modeling with the negative multi-
nomial distribution. Biometrics 53: 971–82.
Further reading
Factorial and binomial topics
Multivariate discrete distributions |
https://en.wikipedia.org/wiki/Oscillatory%20integral%20operator | In mathematics, in the field of harmonic analysis, an oscillatory integral operator is an integral operator of the form
where the function S(x,y) is called the phase of the operator and the function a(x,y) is called the symbol of the operator. λ is a parameter. One often considers S(x,y) to be real-valued and smooth, and a(x,y) smooth and compactly supported. Usually one is interested in the behavior of Tλ for large values of λ.
Oscillatory integral operators often appear in many fields of mathematics (analysis, partial differential equations, integral geometry, number theory) and in physics. Properties of oscillatory integral operators have been studied by Elias Stein and his school.
Hörmander's theorem
The following bound on the L2 → L2 action of oscillatory integral operators (or L2 → L2 operator norm) was obtained by Lars Hörmander in his paper on Fourier integral operators:
Assume that x,y ∈ Rn, n ≥ 1. Let S(x,y) be real-valued and smooth, and let a(x,y) be smooth and compactly supported. If everywhere on the support of a(x,y), then there is a constant C such that Tλ, which is initially defined on smooth functions, extends to a continuous operator from L2(Rn) to L2(Rn), with the norm bounded by , for every λ ≥ 1:
References
Microlocal analysis
Harmonic analysis
Singular integrals
Fourier analysis
Integral transforms
Inequalities |
https://en.wikipedia.org/wiki/Veselin%20Stoykov | Veselin Stoykov (; born 27 August 1986) is a Bulgarian footballer, who plays as a striker for Strumska Slava.
Club statistics
As of 1 August 2011
References
1986 births
Living people
Bulgarian men's footballers
OFC Pirin Blagoevgrad players
PFC Pirin Blagoevgrad players
PFC Vidima-Rakovski Sevlievo players
FC Caspiy players
FC Septemvri Simitli players
First Professional Football League (Bulgaria) players
Second Professional Football League (Bulgaria) players
Men's association football forwards
Footballers from Blagoevgrad |
https://en.wikipedia.org/wiki/Endre%20S%C3%BCli | Endre Süli (also, Endre Suli or Endre Šili) is a mathematician. He is Professor of Numerical Analysis in the Mathematical Institute, University of Oxford, Fellow and Tutor in Mathematics at Worcester College, Oxford and Adjunct Fellow of Linacre College, Oxford. He was educated at the University of Belgrade and, as a British Council Visiting Student, at the University of Reading and St Catherine's College, Oxford. His research is concerned with the mathematical analysis of numerical algorithms for nonlinear partial differential equations.
Biography
Süli is a Foreign Member of the Serbian Academy of Sciences and Arts (2009), Fellow of the European Academy of Sciences (FEurASc, 2010), Fellow of the Society for Industrial and Applied Mathematics (FSIAM, 2016), a Member of the Academia Europaea (MAE, 2020), and a Fellow of the Royal Society (FRS, 2021). He was an invited speaker at the International Congress of Mathematicians in Madrid in 2006 and was Chair of the Society for the Foundations of Computational Mathematics
(2002–2005). Other honours include: Fellow of the Institute of Mathematics and its Applications (FIMA, 2007), Charlemagne Distinguished Lecture (2011), IMA Service Award (2011), Professor Hospitus Universitatis Carolinae Pragensis, Charles University in Prague (2012–), Distinguished Visiting Chair Professor Shanghai Jiao Tong University (2013), President, SIAM United Kingdom and Republic of Ireland Section (2013–2015), London Mathematical Society/New Zealand Mathematical Society Forder Lectureship (2015), Aziz Lecture (2015), BIMOS Distinguished Lecture (2016), John von Neumann Lecture (2016), Sibe Mardešić Lecture (2018), London Mathematical Society Naylor Prize and Lectureship (2021). Since 2005 Süli has been co-Editor-in-Chief of the IMA Journal of Numerical Analysis published by Oxford University Press. He is a member of the Scientific Advisory Board of the Berlin Mathematics Research Center MATH+ and the Board of the Doctoral School for Mathematical and Physical Sciences for Advanced Materials and Technologies of the Scuola Superiore Meridionale at the University of Naples, and was a member of the Scientific Steering Committee of the Isaac Newton Institute for Mathematical Sciences at the University of Cambridge (2011–2014), the Scientific Advisory Board of the Berlin Mathematical School (2016–2018), the Scientific Council of Société de Mathématiques Appliquées et Industrielles (SMAI) (2014–2020), the Scientific Committee of the Mathematisches Forschungsinstitut Oberwolfach (Mathematical Research Institute of Oberwolfach) (2013–2021), and the Scientific Advisory Board of the Archimedes Center for Modeling, Analysis and Computation at the University of Crete (2010–2014). Between 2014 and 2022 he served as the Delegate for Mathematics to the Board of Delegates of Oxford University Press.
He grew up in Subotica and is a recipient of the Pro Urbe Prize of the City of Subotica (2021). He is the father of Sterija Award-winning Serbi |
https://en.wikipedia.org/wiki/Franco%20Dolci | Pablo Franco Dolci (born 1 January 1984) is an Argentine professional football midfielder who plays for Deportivo Maipú in the Torneo Federal A.
External links
Argentine Primera statistics at Fútbol XXI
Living people
1984 births
Argentine men's footballers
Argentine expatriate men's footballers
Men's association football wingers
Argentine Primera División players
Primera Nacional players
Torneo Federal A players
Ligue 1 players
Ligue 2 players
OGC Nice players
SC Bastia players
Chacarita Juniors footballers
Newell's Old Boys footballers
Ferro Carril Oeste footballers
Sportivo Belgrano footballers
Independiente Rivadavia footballers
Deportivo Maipú players
Argentine expatriate sportspeople in France
Expatriate men's footballers in France
Footballers from Córdoba Province, Argentina |
https://en.wikipedia.org/wiki/Paul%20Tseng | Paul Tseng () was a Chinese-American (Hakka Taiwanese) and Canadian applied mathematician and a professor at the Department of Mathematics at the University of Washington, in Seattle, Washington. Tseng was recognized by his peers to be one of the leading optimization researchers of his generation. On August 13, 2009, Paul Tseng went missing while kayaking in the Jinsha River in the Yunnan province of China and is presumed dead.
Biography
Tseng was born September 21, 1959, in Hsinchu, Taiwan. In December 1970, Tseng's family moved to Vancouver, British Columbia. Tseng received his B.Sc. from Queen's University in 1981 and his Ph.D. from Massachusetts Institute of Technology in 1986. In 1990 Tseng moved to the University of Washington's Department of Mathematics. Tseng has conducted research primarily in continuous optimization and secondarily in discrete optimization and distributed computation.
Research
Tseng made many contributions to mathematical optimization, publishing many articles and helping to develop quality software that has been widely used.
He published over 120 papers in optimization and had close collaborations with several colleagues, including Dimitri Bertsekas and Zhi-Quan Tom Luo.
Tseng's research subjects include:
Efficient algorithms for structured convex programs and network flow problems,
Complexity analysis of interior point methods for linear programming,
Parallel and distributed computing,
Error bounds and convergence analysis of iterative algorithms for optimization problems and variational inequalities,
Interior point methods and semidefinite relaxations for hard quadratic and matrix optimization problems, and
Applications of large scale optimization techniques in signal processing and machine learning.
In his research, Tseng gave a new proof for the sharpest complexity result for path-following interior-point methods for linear programming. Furthermore, together with Tom Luo, he resolved a long-standing open question on the convergence of matrix splitting algorithms for linear complementarity problems and affine variational inequalities. Tseng was the first to establish the convergence of the affine scaling algorithm for linear programming in the presence of degeneracy.
Tseng has coauthored (with his Ph.D. advisor, Dimitri Bertsekas) a publicly available network optimization program, called RELAX, which has been widely used in industry and academia for research purposes. This software has been used by statisticians like Paul R. Rosenbaum and Donald Rubin in their work on propensity score matching. Tseng's software for matching has similarly been used in nonparametric statistics to implement exact tests. Tseng has also developed a program called ERELAXG, for network optimization problems with gains. In 2010 conferences in his honor were held at the University of Washington and at Fudan University in Shanghai. Tseng's personal web page can be accessed in the exact state it was at the time of his disappearance |
https://en.wikipedia.org/wiki/Michael%20Saks%20%28mathematician%29 | Michael Ezra Saks is an American mathematician. He is currently the Department Chair of the Mathematics Department at Rutgers University (2017–) and from 2006 until 2010 was director of the Mathematics Graduate Program at Rutgers University. Saks received his Ph.D. from the Massachusetts Institute of Technology in 1980 after completing his dissertation titled Duality Properties of Finite Set Systems under his advisor Daniel J. Kleitman.
A list of his publications and collaborations may be found at DBLP.
In 2016 he became a Fellow of the Association for Computing Machinery.
Research
Saks' research in computational complexity theory, combinatorics, and graph theory has contributed to the study of lower bounds in order theory, randomized computation, and space–time tradeoff.
In 1984, Saks and Jeff Kahn showed that there exist a tight information-theoretical lower bound for sorting under partially ordered information up to a multiplicative constant.
In the first super-linear lower bound for the noisy broadcast problem was proved. In a noisy broadcast model, processors are assigned a local input bit . Each processor may perform a noisy broadcast to all other processors where the received bits may be independently flipped with a fixed probability. The problem is for processor to determine for some function . Saks et al. showed that an existing protocol by Gallager was indeed optimal by a reduction from a generalized noisy decision tree and produced a lower bound on the depth of the tree that learns the input.
In 2003, P. Beame, Saks, X. Sun, and E. Vee published the first time–space lower bound trade-off for randomized computation of decision problems was proved.
Positions
Saks holds positions in the following journal editorial boards:
SIAM Journal on Computing, Associate Editor
Combinatorica, Editorial Board member
Journal of Graph Theory, Editorial Board member
Discrete Applied Mathematics, Editorial Board member
Selected publications
References
External links
Living people
Rutgers University faculty
Gödel Prize laureates
Massachusetts Institute of Technology School of Science alumni
Theoretical computer scientists
Fellows of the Association for Computing Machinery
Year of birth missing (living people)
Combinatorialists |
https://en.wikipedia.org/wiki/A.%20Edward%20Nussbaum | Adolf Edward Nussbaum (10 January 1925 – 31 October 2009) was a German-born American theoretical mathematician who was a professor of mathematics at Washington University in St. Louis for nearly 40 years. He worked with others in 20th-century theoretical physics and mathematics such as J. Robert Oppenheimer and John von Neumann, and was acquainted with Albert Einstein.
Early years
Nussbaum was born to a Jewish family in Rheydt, a borough of the German city Mönchengladbach in northwestern Germany, in 1925. The youngest of three children, he was a Holocaust survivor and was orphaned after the Nazi takeover of Germany.
Both his father, Karl Nussbaum, a wounded veteran of World War I during which he had been awarded the Iron Cross, and his mother, Franziska, was murdered at Auschwitz. His brother, Erwin Nussbaum, was also captured and killed. Nussbaum and his sister, Lieselotte, were separated and sent on a Kindertransport to Belgium in 1939.
When Belgium was invaded by Germany, Nussbaum escaped to southern France, then under the Vichy regime. He lived there at an orphanage known as Château de la Hille. He began his teaching career there, while still a teenager, teaching mathematics to the younger children.
After being captured twice, and jailed once by the Nazis, he escaped on foot to Switzerland, where he attended the University of Zurich, studying both mathematics and physics. In 1947, he was sponsored by relatives in New Jersey to emigrate to the United States.
Career
Shortly after emigrating to the United States, he studied mathematics at Brooklyn College before transferring to Columbia University in New York where he earned his Master of Arts degree in 1950 and his Ph.D. in 1957.
While writing his thesis for Columbia, he worked in the academic year 1952–1953 at the Institute for Advanced Study in Princeton with John von Neumann, a mathematician who used Hilbert spaces in his development of the mathematical basis of quantum mechanics. Hilbert spaces eventually became Nussbaum's area of expertise and he wrote several papers with von Neumann on this topic. During this period, Nussbaum also became acquainted with Albert Einstein, another of the original group at the Institute for Advanced Study.
Nussbaum's thesis was accepted with no revisions and he received his doctorate shortly thereafter.
In the meantime he had worked at the University of Connecticut in Storrs, where he co-authored papers with Allen Devinatz, and at the Rensselaer Polytechnic Institute in Troy, New York. He followed Devinatz to St. Louis to teach at Washington University in 1958.
In 1962, he was a visiting scholar at the Institute for Advanced Studies working with Robert Oppenheimer; in 1967–68 he was a visiting scholar at Stanford University in Palo Alto, California.
He joined Washington University's mathematics faculty as an assistant professor in 1958. He became a full professor in 1966 and taught until 1995, when he was named an emeritus professor.
Personal li |
https://en.wikipedia.org/wiki/Equational | Equational may refer to:
Equative, a construction in linguistics
something pertaining to equations, in mathematics
something pertaining to equality, in logic
See also
Equation (disambiguation)
Equality (disambiguation) |
https://en.wikipedia.org/wiki/Ravi%20Sood | Ravi Sood (born July 5, 1976) is a Canadian financier and venture capitalist. Sood was raised in Waterloo, Ontario and resides in Hong Kong. He received a bachelor's degree in mathematics from the University of Waterloo. He is the co-founder and former CEO of Navina Asset Management and its predecessor company Lawrence Asset Management, which at its peak controlled over $800 million in assets globally. On August 6, 2010, Sood sold Navina to Aston Hill Financial. and moved on to focus on other endeavors.
Sood has also founded several businesses operating in emerging markets, including Buchanan Renewables, Feronia Inc., Jade Power Trust, and Galane Gold.
Sood became well known in Canada as a regular TV personality and frequent guest host of the Business News Network's evening news programme "Squeeze Play". He is best known for commenting in the media on the income trust sector, global markets, natural resources and agriculture.
References
1976 births
Businesspeople from Waterloo, Ontario
Canadian financial businesspeople
Canadian people of Indian descent
Living people
University of Waterloo alumni |
https://en.wikipedia.org/wiki/Fr%C3%A9chet%20mean | In mathematics and statistics, the Fréchet mean is a generalization of centroids to metric spaces, giving a single representative point or central tendency for a cluster of points. It is named after Maurice Fréchet. Karcher mean is the renaming of the Riemannian Center of Mass construction developed by Karsten Grove and Hermann Karcher. On the real numbers, the arithmetic mean, median, geometric mean, and harmonic mean can all be interpreted as Fréchet means for different distance functions.
Definition
Let (M, d) be a complete metric space. Let x1, x2, …, xN be points in M. For any point p in M, define the Fréchet variance to be the sum of squared distances from p to the xi:
The Karcher means are then those points, m of M, which locally minimise Ψ:
If there is an m of M that globally minimises Ψ, then it is Fréchet mean.
Sometimes, the xi are assigned weights wi. Then, the Fréchet variance is calculated as a weighted sum,
Examples of Fréchet means
Arithmetic mean and median
For real numbers, the arithmetic mean is a Fréchet mean, using the usual Euclidean distance as the distance function.
The median is also a Fréchet mean, if the definition of the function Ψ is generalized to the non-quadratic
where , and the Euclidean distance is the distance function d. In higher-dimensional spaces, this becomes the geometric median.
Geometric mean
On the positive real numbers, the (hyperbolic) distance function can be defined. The geometric mean is the corresponding Fréchet mean. Indeed is then an isometry from the euclidean space to this "hyperbolic" space and must respect the Fréchet mean: the Fréchet mean of the is the image by of the Fréchet mean (in the Euclidean sense) of the , i.e. it must be:
.
Harmonic mean
On the positive real numbers, the metric (distance function):
can be defined. The harmonic mean is the corresponding Fréchet mean.
Power means
Given a non-zero real number , the power mean can be obtained as a Fréchet mean by introducing the metric
f-mean
Given an invertible and continuous function , the f-mean can be defined as the Fréchet mean obtained by using the metric:
This is sometimes called the generalised f-mean or quasi-arithmetic mean.
Weighted means
The general definition of the Fréchet mean that includes the possibility of weighting observations can be used to derive weighted versions for all of the above types of means.
See also
Circular mean
Fréchet distance
M-estimator
Geometric median
References
Means |
https://en.wikipedia.org/wiki/OpenCTM | OpenCTM is a 3D geometry technology for storing triangle-based
meshes in a compact format.
Overview
OpenCTM is an open source project that has evolved around a technology for compressing 3D triangle meshes. The technology is divided into three different parts:
An open, binary file format.
An open source software library for reading and writing OpenCTM files.
A software tool set for converting and viewing OpenCTM files.
The triangle mesh data structure that OpenCTM uses is directly compatible with high performance graphics rendering pipelines, such as OpenGL, which makes is suitable for interactive computer graphics applications. Other application types, such as CAD/CAM tools, usually need to convert the mesh data into a custom mesh format for more efficient data handling.
OpenCTM uses a triangle index array to represent the triangle connectivity information, and several arrays for representing vertex data (vertex coordinates, normals, UV coordinates and custom vertex attributes).
File format
The file format, which is binary, uses 32-bit little endian format for all integer fields, and 32-bit binary IEEE 754 format for all floating point fields (also little endian).
The file begins with a special integer identifier, 0x4D54434F, which, if interpreted as four ASCII characters, forms the string “OCTM”. Following the identifier is an integer value that specifies the file format version (the latest official file format version is 5).
The rest of the file, which is described in the file format specification, contains the triangle mesh information. This includes a compressed triangle index array and compressed vertex arrays (one array for each vertex attribute).
Due to the many steps of data processing that are required for implementing the compression, interacting with the file format directly is usually more complex than interacting with other, uncompressed triangle mesh file formats (for instance STL and PLY).
Compression
The compression is based on lossless entropy reduction, by means of various
differentiation operations, followed by lossless entropy coding using the
LZMA compression library.
See also
List of file formats
Lossless data compression
References
External links
The OpenCTM website
CAD file formats
3D graphics software
2009 software
Software using the zlib license |
https://en.wikipedia.org/wiki/Almost%20simple%20group | In mathematics, a group is said to be almost simple if it contains a non-abelian simple group and is contained within the automorphism group of that simple group – that is, if it fits between a (non-abelian) simple group and its automorphism group. In symbols, a group A is almost simple if there is a (non-abelian) simple group S such that
Examples
Trivially, non-abelian simple groups and the full group of automorphisms are almost simple, but proper examples exist, meaning almost simple groups that are neither simple nor the full automorphism group.
For or the symmetric group is the automorphism group of the simple alternating group so is almost simple in this trivial sense.
For there is a proper example, as sits properly between the simple and due to the exceptional outer automorphism of Two other groups, the Mathieu group and the projective general linear group also sit properly between and
Properties
The full automorphism group of a non-abelian simple group is a complete group (the conjugation map is an isomorphism to the automorphism group), but proper subgroups of the full automorphism group need not be complete.
Structure
By the Schreier conjecture, now generally accepted as a corollary of the classification of finite simple groups, the outer automorphism group of a finite simple group is a solvable group. Thus a finite almost simple group is an extension of a solvable group by a simple group.
See also
Quasisimple group
Semisimple group
Notes
External links
Almost simple group at the Group Properties wiki
Properties of groups |
https://en.wikipedia.org/wiki/2008%E2%80%9309%20Club%20Nacional%20de%20Football%20season |
Squad
As of August 2, 2009
Transfers
In
Out
Statistics
Appearances and goals
Last updated on August 2, 2009.
|}
Note
* = Players who left the club mid-season
Top scorers
Includes all competitive matches
Disciplinary record
Note
* = Players who left the club mid-season
Captains
Penalties Awarded
International players
The following is a list of all squad members who have played for their national sides during the 2008–09 season. Players in bold were in the starting XI for their national side.
14 October 2008
20 January 2009
22 January 2009
24 January 2009
28 January 2009
31 January 2009
2 February 2009
4 February 2009
6 February 2009
8 February 2009
2 April 2009
6 June 2009
11 June 2009
Starting 11
Overall
{|class="wikitable"
|-
|Games played || 50 (35 Primera División Uruguaya, 5 Liguilla Pre-Libertadores, 10 Copa Libertadores)
|-
|Games won || 25 (20 Primera División Uruguaya, 1 Liguilla Pre-Libertadores, 4 Copa Libertadores)
|-
|Games drawn || 14 (9 Primera División Uruguaya, 1 Liguilla Pre-Libertadores, 4 Copa Libertadores)
|-
|Games lost || 11 (6 Primera División Uruguaya, 3 Liguilla Pre-Libertadores, 2 Copa Libertadores)
|-
|Goals scored || 83
|-
|Goals conceded || 58
|-
|Goal difference || +25
|-
|Yellow cards || 115
|-
|Red cards || 15
|-
|Worst discipline || Santiago García (6 , 3 )
|-
|Best result || 3-0 (A) v Nacional - Copa Libertadores 2009.02.134-1 (A) v Racing - Primera División Uruguaya 2009.03.143-0 (H) v River Plate - Copa Libertadores 2009.03.193-0 (N) v Defensor Sporting - Primera División Uruguaya 2009.07.08
|-
|Worst result || 0-3 (H) v Central Español - Primera División Uruguaya 2009.06.13
|-
|Most appearances || Santiago García (42 appearances)
|-
|Top scorer || Santiago García (13 goals)
|-
Club
Coaching staff
Friendlies
Copa Bimbo
Semi-finals
Final
Primera División Uruguaya
Apertura's table
The apertura's winner qualifies for the semifinal of the Primera División
Matches
Final
Clausura's table
Villa Española was relegated due to financial issues after the Apertura.
The clausura's winner qualifies for the semifinal of the Primera División
Matches
Aggregate table
The aggregate's winners qualify to Primera División Uruguaya final
The aggregate's top-six qualify to Liguilla Pre-Libertadores
Results by round
Relegation table
As of June 18, 2009.
The three clubs with the lowest average of points over the last two seasons are relegated.
Semi-finals
First leg
Second leg
Third legNacional won 5–2 on points.Final
First leg
Second legNacional won 6–0 on points.Copa Libertadores
Group stage
Round of 16
First leg
Second legSan Luis withdrew from the tournament over the H1N1 flu outbreak in MexicoQuarterfinals
First leg
Second legNacional 1–1 Palmeiras on aggregate. Nacional won on away goals.Semi-finals
First leg
Second legEstudiantes won 3–1 on aggregate.Liguilla Pre-Libertadores
The Liguilla Pre-Libertadores' champion qualify to Copa Libertadores 2010 group stage |
https://en.wikipedia.org/wiki/Poly-Weibull%20distribution | In probability theory and statistics, the poly-Weibull distribution is a continuous probability distribution. The distribution is defined to be that of a random variable defined to be the smallest of a number of statistically independent random variables having non-identical Weibull distributions.
References
Preprint
Continuous distributions
Survival analysis |
https://en.wikipedia.org/wiki/Ranked%20lists%20of%20Chilean%20regions | This article includes several ranked indicators for Chile's regions.
By area
Chilean regions by area.
Sources: "División Político Administrativa y Censal 2007", National Statistics Office, 2007 (Chile area data); CIA's The World Factbook (country area comparison).
Note: It does not include the Chilean Antarctic Territory, annexed to the Magallanes Region and totalling .
Population
By population
Chilean regions by population as of June 30, 2015.
Sources: National Statistics Office's September 2014 projections (Chile's population), accessed on 28 June 2015; UNDESA's World Population Prospects: The 2012 Revision, September 2013, accessed on 28 June 2015 (country comparison).
By urban and rural population
Chilean regions by their urban and rural population as of June 30, 2010.
Sources: National Statistics Office (Chile's population).
By population density
Chilean regions by population density as of 2010.
Sources: National Statistics Office (Chile area data, Chile's population); Wikipedia's List of countries and dependencies by population density (country comparison).
Note: It does not include the internationally unrecognized Chilean Antarctic Territory, annexed to the Magallanes and Antártica Chilena Region and totalling .
By number of households and household size
Chilean regions by number of households and household size in 2013.
Source: Casen Survey 2013, Ministry of Social Development of Chile.
Note: Data exclude live-in domestic workers and their family.
By indigenous population
Chilean regions by persons self-identifying as belonging to one of Chile's indigenous groups in 2013.
Source: Ministry of Social Development of Chile's 2013 Casen Survey.
a Easter Island—where the majority of the Rapanui people live—was not included in the survey.
By foreign nationals
Chilean regions by number of foreign nationals and country/area of citizenship in 2017.
Source: Ministry of Social Development of Chile's 2017 Casen Survey.
Economy
By regional GDP (PPP)
Chilean regions by their 2014 regional gross domestic product at purchasing power parity in billions of 2014 international dollars.
Sources: Central Bank of Chile (Chile's 2014 Regional GDP in current prices), accessed on 9 April 2016. OECD's OECD.Stat (Chile's 2014 PPP conversion factor for GDP (375.432372)), accessed on 9 April 2016. World Bank's World Development Indicators (2014 GDP (PPP) for world countries), accessed on 9 April 2016.
Notes: The aggregate Regional GDP is less than the National GDP because it does not include extra-regio GDP, VAT taxes and import duties.
By regional GDP (PPP) per capita
Chilean regions by their 2014 regional gross domestic product per capita at purchasing power parity in 2014 international dollars.
Sources: Central Bank of Chile (Chile's 2014 Regional GDP in current prices), accessed on 9 April 2016. National Statistics Office of Chile (Chile's 2014 national and regional population), accessed on 9 April 2016. OECD's OECD.Stat (Chile's 2014 |
https://en.wikipedia.org/wiki/Director%20circle | In geometry, the director circle of an ellipse or hyperbola (also called the orthoptic circle or Fermat–Apollonius circle) is a circle consisting of all points where two perpendicular tangent lines to the ellipse or hyperbola cross each other.
Properties
The director circle of an ellipse circumscribes the minimum bounding box of the ellipse. It has the same center as the ellipse, with radius , where and are the semi-major axis and semi-minor axis of the ellipse. Additionally, it has the property that, when viewed from any point on the circle, the ellipse spans a right angle.
The director circle of a hyperbola has radius , and so, may not exist in the Euclidean plane, but could be a circle with imaginary radius in the complex plane.
Generalization
More generally, for any collection of points , weights , and constant , one can define a circle as the locus of points such that
The director circle of an ellipse is a special case of this more general construction with two points and at the foci of the ellipse, weights , and equal to the square of the major axis of the ellipse. The Apollonius circle, the locus of points such that the ratio of distances of to two foci and is a fixed constant , is another special case, with , , and .
Related constructions
In the case of a parabola the director circle degenerates to a straight line, the directrix of the parabola.
Notes
References
Conic sections
Circles |
https://en.wikipedia.org/wiki/Sinc%20numerical%20methods | In numerical analysis and applied mathematics, sinc numerical methods are numerical techniques for finding approximate solutions of partial differential equations and integral equations based on the translates of sinc function and Cardinal function C(f,h) which is an expansion of f defined by
where the step size h>0 and where the sinc function is defined by
Sinc approximation methods excel for problems whose solutions may have singularities, or infinite domains, or boundary layers.
The truncated Sinc expansion of f is defined by the following series:
.
Sinc numerical methods cover
function approximation,
approximation of derivatives,
approximate definite and indefinite integration,
approximate solution of initial and boundary value ordinary differential equation (ODE) problems,
approximation and inversion of Fourier and Laplace transforms,
approximation of Hilbert transforms,
approximation of definite and indefinite convolution,
approximate solution of partial differential equations,
approximate solution of integral equations,
construction of conformal maps.
Indeed, Sinc are ubiquitous for approximating every operation of calculus
In the standard setup of the sinc numerical methods, the errors (in big O notation) are known to be with some c>0, where n is the number of nodes or bases used in the methods. However, Sugihara has recently found that the errors in the Sinc numerical methods based on double exponential transformation are with some k>0, in a setup that is also meaningful both theoretically and practically and are found to be best possible in a certain mathematical sense.
Reading
References
Numerical analysis |
https://en.wikipedia.org/wiki/Pseudoisotopy%20theorem | In mathematics, the pseudoisotopy theorem is a theorem of Jean Cerf's which refers to the connectivity of a group of diffeomorphisms of a manifold.
Statement
Given a differentiable manifold M (with or without boundary), a pseudo-isotopy diffeomorphism of M is a diffeomorphism of M × [0, 1] which restricts to the identity on .
Given a pseudo-isotopy diffeomorphism, its restriction to is a diffeomorphism of M. We say g is pseudo-isotopic to the identity. One should think of a pseudo-isotopy as something that is almost an isotopy—the obstruction to ƒ being an isotopy of g to the identity is whether or not ƒ preserves the level-sets for .
Cerf's theorem states that, provided M is simply-connected and dim(M) ≥ 5, the group of pseudo-isotopy diffeomorphisms of M is connected. Equivalently, a diffeomorphism of M is isotopic to the identity if and only if it is pseudo-isotopic to the identity.
Relation to Cerf theory
The starting point of the proof is to think of the height function as a 1-parameter family of smooth functions on M by considering the function . One then applies Cerf theory.
References
Theorems in differential topology
Singularity theory |
https://en.wikipedia.org/wiki/Healthcare%20in%20Costa%20Rica | Costa Rica provides universal health care to its citizens and permanent residents. Both the private and public health care systems in Costa Rica are continually being upgraded. Statistics from the World Health Organization (WHO) frequently place Costa Rica in the top country rankings in the world for long life expectancy. WHO's 2000 survey ranked Costa Rica as having the 36th best health care system, placing it one spot above the United States at the time. In addition, the UN (United Nations) has ranked Costa Rica’s public health system within the top 20 worldwide and the number 1 in Latin America.
The Human Rights Measurement Initiative finds that Costa Rica is fulfilling 94.7% of what it should be fulfilling for the right to health based on its level of income. When looking at the right to health with respect to children, Costa Rica achieves 97.7% of what is expected based on its current income. In regards to the right to health amongst the adult population, the country achieves 94.8% of what is expected based on the nation's level of income. Costa Rica falls into the "fair" category when evaluating the right to reproductive health because the nation is fulfilling 91.7% of what the nation is expected to achieve based on the resources (income) it has available.
Costs
In the early 1970s, the Ministry of Health was heavily subsidized by foreign aid. By 1977, health programs had been so successful that US Aid for International Development in the sector was ended, as the country was found "too healthy" to continue being a recipient. In 2001, total spending on health care was 7% of GDP, with 3/4 of that being the public sector. Currently, costs tend to be much less than doctor, hospital, and prescription costs in the U.S. The country spends one tenth as much per capita on health care as the United States, focusing on preventive care.
Public
The Costa Rican Social Security Fund or Caja Costarricense de Seguro Social (as it is known in Spanish) is in charge of most of the nation's public health sector. Its role in public health (as the administrator of health institutions) is key in Costa Rica, playing an important part in the state's national health policy making. Worker and employer contribution are mandated by law, under the principle of solidarity. Workers need to be cover by a "poliza de riesgo del trabajo" an insurance policy that complements the health care provided by the "Caja" for injuries related to labor risks.
Caja services are guaranteed to all residents, including the uninsured. In 1989 this was expanded to include undocumented immigrants as well, which constituted up to 8% of the population at the time. The percent of residents with health insurance increased gradually, as the program was originally only intended for urban workers. It was not until 1961 that universal mandatory coverage became a stated goal. After reaching a peak of 92% coverage in 1990, rates have remained around 88%. 12% of the insured are low-income resident |
https://en.wikipedia.org/wiki/Contiguity%20%28probability%20theory%29 | In probability theory, two sequences of probability measures are said to be contiguous if asymptotically they share the same support. Thus the notion of contiguity extends the concept of absolute continuity to the sequences of measures.
The concept was originally introduced by as part of his foundational contribution to the development of asymptotic theory in mathematical statistics. He is best known for the general concepts of local asymptotic normality and contiguity.
Definition
Let be a sequence of measurable spaces, each equipped with two measures Pn and Qn.
We say that Qn is contiguous with respect to Pn (denoted ) if for every sequence An of measurable sets, implies .
The sequences Pn and Qn are said to be mutually contiguous or bi-contiguous (denoted ) if both Qn is contiguous with respect to Pn and Pn is contiguous with respect to Qn.
The notion of contiguity is closely related to that of absolute continuity. We say that a measure Q is absolutely continuous with respect to P (denoted ) if for any measurable set A, implies . That is, Q is absolutely continuous with respect to P if the support of Q is a subset of the support of P, except in cases where this is false, including, e.g., a measure that concentrates on an open set, because its support is a closed set and it assigns measure zero to the boundary, and so another measure may concentrate on the boundary and thus have support contained within the support of the first measure, but they will be mutually singular. In summary, this previous sentence's statement of absolute continuity is false. The contiguity property replaces this requirement with an asymptotic one: Qn is contiguous with respect to Pn if the "limiting support" of Qn is a subset of the limiting support of Pn. By the aforementioned logic, this statement is also false.
It is possible however that each of the measures Qn be absolutely continuous with respect to Pn, while the sequence Qn not being contiguous with respect to Pn.
The fundamental Radon–Nikodym theorem for absolutely continuous measures states that if Q is absolutely continuous with respect to P, then Q has density with respect to P, denoted as , such that for any measurable set A
which is interpreted as being able to "reconstruct" the measure Q from knowing the measure P and the derivative ƒ. A similar result exists for contiguous sequences of measures, and is given by the Le Cam's third lemma.
Properties
For the case for all n it applies .
It is possible that is true for all n without .
Le Cam's first lemma
For two sequences of measures on measurable spaces the following statements are equivalent:
for any statistics .
where and are random variables on .
Applications
Econometrics
See also
Asymptotic theory (statistics)
Contiguity (disambiguation)
Probability space
Notes
References
Additional literature
Roussas, George G. (1972), Contiguity of Probability Measures: Some Applications in Statistics, CUP, .
Scott, D.J. (1982) C |
https://en.wikipedia.org/wiki/Dini%27s%20surface | In geometry, Dini's surface is a surface with constant negative curvature that can be created by twisting a pseudosphere. It is named after Ulisse Dini and described by the following parametric equations:
Another description is a generalized helicoid constructed from the tractrix.
See also
Breather surface
References
Surfaces of constant negative curvature |
https://en.wikipedia.org/wiki/Beno%C3%AEt%20Poulain | Benoît Guy Jean Poulain (born 27 July 1987) is a French professional footballer who plays as a centre-back.
Career
On 14 July 2022, Poulain returned to Nîmes on a one-season deal.
Career statistics
Honours
Club Brugge
Belgian Super Cup: 2016, 2018
Belgian Pro League: 2016, 2018
References
External links
1987 births
Living people
French men's footballers
Men's association football midfielders
Nîmes Olympique players
K.V. Kortrijk players
Club Brugge KV players
Kayserispor footballers
K.A.S. Eupen players
Ligue 2 players
Championnat National players
Belgian Pro League players
Süper Lig players
French expatriate men's footballers
French expatriate sportspeople in Belgium
Expatriate men's footballers in Belgium
French expatriate sportspeople in Turkey
Expatriate men's footballers in Turkey |
https://en.wikipedia.org/wiki/Kakeya | Kakeya may refer to:
Kakeya, Shimane town
Sōichi Kakeya, mathematician
Kakeya set in mathematics |
https://en.wikipedia.org/wiki/Gauthier%20Pinaud | Gauthier Pinaud (born 8 January 1988) is a French professional footballer who played most recently for Orléans as a right-back.
Career statistics
Notes
External links
Gauthier Pinaud foot-national.com Profile
1988 births
Living people
French men's footballers
Ligue 2 players
Championnat National players
Championnat National 2 players
Championnat National 3 players
LB Châteauroux players
RC Strasbourg Alsace players
US Orléans players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Phil%20Scarf | Philip A. Scarf is Professor of Management Mathematics at Cardiff Business School, Cardiff University. He was formerly Professor of Applied Statistics at Salford Business School, University of Salford. A statistician, his interests are in modeling in sport, maintenance and reliability, and corrosion engineering. He advised the Press Association and the Football Association on the development of the Actim Index: the "official player rating system of the Premier League and Championship".
Early life
He obtained his BSc from the University of Sheffield and PhD from the University of Manchester. His thesis concerned the statistical modelling of metallic corrosion, and the application of extreme value theory.
Academic and research career
He was appointed as Professor of Applied Statistics by Salford in 2008.
He is the lead editor of the IMA Journal of Management Mathematics.
References
External links
Phil Scarf's profile at Salford Business School
Living people
Academics of the University of Salford
Alumni of the University of Manchester
Alumni of the University of Sheffield
British statisticians
Year of birth missing (living people)
Academics of Cardiff Business School
Academics of Cardiff University |
https://en.wikipedia.org/wiki/Root%20%28mathematics%29%20%28disambiguation%29 | {{safesubst:#invoke:RfD||INTDABLINK of redirects from incomplete disambiguation|month = October
|day = 14
|year = 2023
|time = 06:45
|timestamp = 20231014064523
|content=#REDIRECT Root (disambiguation)
}} |
https://en.wikipedia.org/wiki/Uncertainty%20theory | Uncertainty theory is a branch of mathematics based on normality, monotonicity, self-duality, countable subadditivity, and product measure axioms.
Mathematical measures of the likelihood of an event being true include probability theory, capacity, fuzzy logic, possibility, and credibility, as well as uncertainty.
Four axioms
Axiom 1. (Normality Axiom) .
Axiom 2. (Self-Duality Axiom) .
Axiom 3. (Countable Subadditivity Axiom) For every countable sequence of events , we have
.
Axiom 4. (Product Measure Axiom) Let be uncertainty spaces for . Then the product uncertain measure is an uncertain measure on the product σ-algebra satisfying
.
Principle. (Maximum Uncertainty Principle) For any event, if there are multiple reasonable values that an uncertain measure may take, then the value as close to 0.5 as possible is assigned to the event.
Uncertain variables
An uncertain variable is a measurable function ξ from an uncertainty space to the set of real numbers, i.e., for any Borel set B of real numbers, the set
is an event.
Uncertainty distribution
Uncertainty distribution is inducted to describe uncertain variables.
Definition: The uncertainty distribution of an uncertain variable ξ is defined by .
Theorem (Peng and Iwamura, Sufficient and Necessary Condition for Uncertainty Distribution): A function is an uncertain distribution if and only if it is an increasing function except and .
Independence
Definition: The uncertain variables are said to be independent if
for any Borel sets of real numbers.
Theorem 1: The uncertain variables are independent if
for any Borel sets of real numbers.
Theorem 2: Let be independent uncertain variables, and measurable functions. Then are independent uncertain variables.
Theorem 3: Let be uncertainty distributions of independent uncertain variables respectively, and the joint uncertainty distribution of uncertain vector . If are independent, then we have
for any real numbers .
Operational law
Theorem: Let be independent uncertain variables, and a measurable function. Then is an uncertain variable such that
where are Borel sets, and means for any.
Expected Value
Definition: Let be an uncertain variable. Then the expected value of is defined by
provided that at least one of the two integrals is finite.
Theorem 1: Let be an uncertain variable with uncertainty distribution . If the expected value exists, then
Theorem 2: Let be an uncertain variable with regular uncertainty distribution . If the expected value exists, then
Theorem 3: Let and be independent uncertain variables with finite expected values. Then for any real numbers and , we have
Variance
Definition: Let be an uncertain variable with finite expected value . Then the variance of is defined by
Theorem: If be an uncertain variable with finite expected value, and are real numbers, then
Critical value
Definition: Let be an uncertain variable, and . Then
is called the α-optimistic value to , and
|
https://en.wikipedia.org/wiki/Baba%20Malick | Baba Malick N'Diaye (born 3 September 1983 in Cordina) is a Senegalese born naturalized Qatari footballer, who currently plays as a goalkeeper.
Club career statistics
Statistics accurate as of 10 April 2023
1Includes Emir of Qatar Cup.
2Includes Sheikh Jassem Cup.
3Includes AFC Champions League.
International career
He is also player of the Qatar national football team and was nominated among the Asian Football Confederation Players of the Year 2009.
External links
References
1983 births
Living people
Qatari men's footballers
Qatar men's international footballers
Umm Salal SC players
Al-Duhail SC players
Al-Shamal SC players
Al Kharaitiyat SC players
Men's association football goalkeepers
Qatar Stars League players
Qatari Second Division players
Senegalese emigrants to Qatar
Naturalised citizens of Qatar
Qatari people of Senegalese descent |
https://en.wikipedia.org/wiki/Khaled%20Al%20Zakiba | Khaled Al Zakiba is Qatari footballer who is a defender. He is a member of the Qatar national football team.
Club career statistics
Statistics accurate as of 21 August 2011
1Includes Emir of Qatar Cup.
2Includes Sheikh Jassem Cup.
3Includes AFC Champions League.
References
External links
Player profile - QSL.com.qa
Player profile - fifa.com
Living people
Al-Shamal SC players
Qatar SC players
Al-Rayyan SC players
1986 births
Umm Salal SC players
Muaither SC players
Al Bidda SC players
Qatari men's footballers
Qatar men's international footballers
Qatar Stars League players
Qatari Second Division players
Men's association football defenders |
https://en.wikipedia.org/wiki/Victor%20Bangert | Victor Bangert (born 28 November 1950) is Professor of Mathematics at the Mathematisches Institut in Freiburg, Germany. His main interests are differential geometry and dynamical systems theory. He specialises in the theory of closed geodesics, wherein one of his significant results, combined with another one due to John Franks, implies that every Riemannian 2-sphere possesses infinitely many closed geodesics. He also made important contributions to Aubry–Mather theory.
He obtained his Ph.D. from Universität Dortmund in 1977 under the supervision of Rolf Wilhelm Walter, with the thesis Konvexität in riemannschen Mannigfaltigkeiten.
He served in the editorial board of "manuscripta mathematica" from 1996 to 2017.
Bangert was an invited speaker at the 1994 International Congress of Mathematicians in Zürich.
Selected publications
References
External links
Personal webpage
Oberwolfach photos of Victor Bangert
20th-century German mathematicians
21st-century German mathematicians
1950 births
Living people
Geometers
Dynamical systems theorists
Scientists from Osnabrück |
https://en.wikipedia.org/wiki/John%20Alderson%20%28footballer%29 | John Henry Alderson (5 March 1910 – 1986) was a footballer who played in the Football League for Darlington. He was born in England.
Career statistics
References
English men's footballers
Darlington F.C. players
Stoke City F.C. players
English Football League players
Men's association football outside forwards
Bishop Auckland F.C. players
1910 births
1986 deaths
People from County Durham (district)
Footballers from County Durham |
https://en.wikipedia.org/wiki/Supersonic%20airfoils | A supersonic airfoil is a cross-section geometry designed to generate lift efficiently at supersonic speeds. The need for such a design arises when an aircraft is required to operate consistently in the supersonic flight regime.
Supersonic airfoils generally have a thin section formed of either angled planes or opposed arcs (called "double wedge airfoils" and "biconvex airfoils" respectively), with very sharp leading and trailing edges. The sharp edges prevent the formation of a detached bow shock in front of the airfoil as it moves through the air. This shape is in contrast to subsonic airfoils, which often have rounded leading edges to reduce flow separation over a wide range of angle of attack. A rounded edge would behave as a blunt body in supersonic flight and thus would form a bow shock, which greatly increases wave drag. The airfoils' thickness, camber, and angle of attack are varied to achieve a design that will cause a slight deviation in the direction of the surrounding airflow.
Lift and drag
At supersonic conditions, aircraft drag is originated due to:
Skin-friction drag due to shearing.
The wave drag due to the thickness (or volume) or zero-lift wave drag
Drag due to lift
Therefore, the Drag coefficient on a supersonic airfoil is described by the following expression:
CD= CD,friction+ CD,thickness+ CD,lift
Experimental data allow us to reduce this expression to:
CD= CD,O + KCL2
Where CDO is the sum of C(D,friction) and C D,thickness, and k for supersonic flow is a function of the Mach number. The skin-friction component is derived from the presence of a viscous boundary layer which is infinitely close to the surface of the aircraft body. At the boundary wall, the normal component of velocity is zero; therefore an infinitesimal area exists where there is no slip. The zero-lift wave drag component can be obtained based on the supersonic area rule which tells us that the wave-drag of an aircraft in a steady supersonic flow is identical to the average of a series of equivalent bodies of revolution. The bodies of revolution are defined by the cuts through the aircraft made by the tangent to the fore Mach cone from a distant point of the aircraft at an azimuthal angle. This average is over all azimuthal angles. The drag due-to lift component is calculated using lift-analysis programs. The wing design and the lift-analysis programs are separate lifting-surfaces methods that solve the direct or inverse problem of design and lift analysis.
Supersonic wing design
Years of research and experience with the unusual conditions of supersonic flow have led to some interesting conclusions about airfoil design. Considering a rectangular wing, the pressure at a point P with coordinates (x,y) on the wing is defined only by the pressure disturbances originated at points within the upstream Mach cone emanating from point P. As result, the wing tips modify the flow within their own rearward Mach cones. The remaining area of the wing does not suffe |
https://en.wikipedia.org/wiki/Reach%20%28mathematics%29 | Let X be a subset of Rn. Then the reach of X is defined as
Examples
Shapes that have reach infinity include
a single point,
a straight line,
a full square, and
any convex set.
The graph of ƒ(x) = |x| has reach zero.
A circle of radius r has reach r.
References
Geometric measurement
Real analysis
Topology |
https://en.wikipedia.org/wiki/Projection%20pursuit%20regression | In statistics, projection pursuit regression (PPR) is a statistical model developed by Jerome H. Friedman and Werner Stuetzle which is an extension of additive models. This model adapts the additive models in that it first projects the data matrix of explanatory variables in the optimal direction before applying smoothing functions to these explanatory variables.
Model overview
The model consists of linear combinations of ridge functions: non-linear transformations of linear combinations of the explanatory variables. The basic model takes the form
where xi is a 1 × p row of the design matrix containing the explanatory variables for example i, yi is a 1 × 1 prediction, {βj} is a collection of r vectors (each a unit vector of length p) which contain the unknown parameters, {fj} is a collection of r initially unknown smooth functions that map from ℝ → ℝ, and r is a hyperparameter. Good values for r can be determined through cross-validation or a forward stage-wise strategy which stops when the model fit cannot be significantly improved. As r approaches infinity and with an appropriate set of functions {fj}, the PPR model is a universal estimator, as it can approximate any continuous function in ℝp.
Model estimation
For a given set of data , the goal is to minimize the error function
over the functions and vectors . No method exists for solving over all variables at once, but it can be solved via alternating optimization. First, consider each pair individually: Let all other parameters be fixed, and find a "residual", the variance of the output not accounted for by those other parameters, given by
The task of minimizing the error function now reduces to solving
for each j in turn. Typically new pairs are added to the model in a forward stage-wise fashion.
Aside: Previously-fitted pairs can be readjusted after new fit-pairs are determined by an algorithm known as backfitting, which entails reconsidering a previous pair, recalculating the residual given how other pairs have changed, refitting to account for that new information, and then cycling through all fit-pairs this way until parameters converge. This process typically results in a model that performs better with fewer fit-pairs, though it takes longer to train, and it is usually possible to achieve the same performance by skipping backfitting and simply adding more fits to the model (increasing r).
Solving the simplified error function to determine an pair can be done with alternating optimization, where first a random is used to project in to 1D space, and then the optimal is found to describe the relationship between that projection and the residuals via your favorite scatter plot regression method. Then if is held constant, assuming is once differentiable, the optimal updated weights can be found via the Gauss-Newton method—a quasi-Newton method in which the part of the Hessian involving the second derivative is discarded. To derive this, first Taylor expand , then plug th |
https://en.wikipedia.org/wiki/Backfitting%20algorithm | In statistics, the backfitting algorithm is a simple iterative procedure used to fit a generalized additive model. It was introduced in 1985 by Leo Breiman and Jerome Friedman along with generalized additive models. In most cases, the backfitting algorithm is equivalent to the Gauss–Seidel method, an algorithm used for solving a certain linear system of equations.
Algorithm
Additive models are a class of non-parametric regression models of the form:
where each is a variable in our -dimensional predictor , and is our outcome variable. represents our inherent error, which is assumed to have mean zero. The represent unspecified smooth functions of a single . Given the flexibility in the , we typically do not have a unique solution: is left unidentifiable as one can add any constants to any of the and subtract this value from . It is common to rectify this by constraining
for all
leaving
necessarily.
The backfitting algorithm is then:
Initialize ,
Do until converge:
For each predictor j:
(a) (backfitting step)
(b) (mean centering of estimated function)
where is our smoothing operator. This is typically chosen to be a cubic spline smoother but can be any other appropriate fitting operation, such as:
local polynomial regression
kernel smoothing methods
more complex operators, such as surface smoothers for second and higher-order interactions
In theory, step (b) in the algorithm is not needed as the function estimates are constrained to sum to zero. However, due to numerical issues this might become a problem in practice.
Motivation
If we consider the problem of minimizing the expected squared error:
There exists a unique solution by the theory of projections given by:
for i = 1, 2, ..., p.
This gives the matrix interpretation:
where . In this context we can imagine a smoother matrix, , which approximates our and gives an estimate, , of
or in abbreviated form
An exact solution of this is infeasible to calculate for large np, so the iterative technique of backfitting is used. We take initial guesses and update each in turn to be the smoothed fit for the residuals of all the others:
Looking at the abbreviated form it is easy to see the backfitting algorithm as equivalent to the Gauss–Seidel method for linear smoothing operators S.
Explicit derivation for two dimensions
Following, we can formulate the backfitting algorithm explicitly for the two dimensional case. We have:
If we denote as the estimate of in the ith updating step, the backfitting steps are
By induction we get
and
If we set then we get
Where we have solved for by directly plugging out from .
We have convergence if . In this case, letting :
We can check this is a solution to the problem, i.e. that and converge to and correspondingly, by plugging these expressions into the original equations.
Issues
The choice of when to stop the algorithm is arbitra |
https://en.wikipedia.org/wiki/Saint-Norbert%2C%20Quebec | Saint-Norbert is a parish municipality in D'Autray Regional County Municipality the Lanaudière region of Quebec, Canada.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Saint-Norbert had a population of living in of its total private dwellings, a change of from its 2016 population of . With a land area of , it had a population density of in 2021.
Population trend:
Population in 2011: 1059 (2006 to 2011 population change: -0.7%)
Population in 2006: 1067
Population in 2001: 1046
Population in 1996: 1070
Population in 1991: 971
Mother tongue:
English as first language: 0%
French as first language: 100%
English and French as first language: 0%
Other as first language: 0%
Education
Commission scolaire des Samares operates francophone public schools, including:
École Sainte-Anne
The Sir Wilfrid Laurier School Board operates anglophone public schools, including:
Joliette Elementary School in Saint-Charles-Borromée
Joliette High School in Joliette
See also
List of parish municipalities in Quebec
References
External links
Saint-Norbert - MRC d'Autray
Incorporated places in Lanaudière
Parish municipalities in Quebec |
https://en.wikipedia.org/wiki/%F0%9D%95%8A | is the blackboard bold letter S. It can refer to:
The sphere spectrum
The -dimensional sphere
The algebra of sedenions |
https://en.wikipedia.org/wiki/Kolmogorov%27s%20three-series%20theorem | In probability theory, Kolmogorov's Three-Series Theorem, named after Andrey Kolmogorov, gives a criterion for the almost sure convergence of an infinite series of random variables in terms of the convergence of three different series involving properties of their probability distributions. Kolmogorov's three-series theorem, combined with Kronecker's lemma, can be used to give a relatively easy proof of the Strong Law of Large Numbers.
Statement of the theorem
Let be independent random variables. The random series converges almost surely in if the following conditions hold for some , and only if the following conditions hold for any :
Proof
Sufficiency of conditions ("if")
Condition (i) and Borel–Cantelli give that for large, almost surely. Hence converges if and only if converges. Conditions (ii)-(iii) and Kolmogorov's Two-Series Theorem give the almost sure convergence of .
Necessity of conditions ("only if")
Suppose that converges almost surely.
Without condition (i), by Borel–Cantelli there would exist some such that for infinitely many , almost surely. But then the series would diverge. Therefore, we must have condition (i).
We see that condition (iii) implies condition (ii): Kolmogorov's two-series theorem along with condition (i) applied to the case gives the convergence of . So given the convergence of , we have converges, so condition (ii) is implied.
Thus, it only remains to demonstrate the necessity of condition (iii), and we will have obtained the full result. It is equivalent to check condition (iii) for the series
where for each , and are IID—that is, to employ the assumption that , since is a sequence of random variables bounded by 2, converging almost surely, and with . So we wish to check that if converges, then converges as well. This is a special case of a more general result from martingale theory with summands equal to the increments of a martingale sequence and the same conditions (; the series of the variances is converging; and the summands are bounded).
Example
As an illustration of the theorem, consider the example of the harmonic series with random signs:
Here, "" means that each term is taken with a random sign that is either or with respective probabilities , and all random signs are chosen independently. Let in the theorem denote a random variable that takes the values and with equal probabilities. With the summands of the first two series are identically zero and var(Yn)=. The conditions of the theorem are then satisfied, so it follows that the harmonic series with random signs converges almost surely. On the other hand, the analogous series of (for example) square root reciprocals with random signs, namely
diverges almost surely, since condition (3) in the theorem is not satisfied for any A. Note that this is different from the behavior of the analogous series with alternating signs, , which does converge.
Notes
Mathematical series
Probability theorems |
https://en.wikipedia.org/wiki/2009%E2%80%9310%20Sporting%20CP%20season | The 2009–10 season was Sporting Clube de Portugal's 76th season in the top flight, the Liga Sagres. This article shows player statistics and all matches (official and friendly) that the club have and will play during the 2009–10 season.
Squad
First team squad
As of 1 January 2010.
Youth Squad
(a) According to LPFP
UEFA Europa League squad
As of 14 September 2009.
List B Players included in UEFA
Transfers
In
Summer
Winter
UEFA Europa League
Group D
Results by round
Round of 32
|}
Round of 16
Carlsberg Cup
Third round
Group B
Knock-out stage
Semi-finals
Competitive
Liga Sagres
Kickoff times are in UTC+0.
UEFA Champions League
Play-off round
Fiorentina 3–3 Sporting CP on aggregate. Fiorentina won on away goals.
UEFA Europa League
Kickoff times are in CET.
Group Stage
Round of 32
Sporting CP won 4–2 on aggregate.
Round of 16
Atlético Madrid won 2–2 on aggregate from the away goals rule.
Pre-season and friendlies
References
Portuguese football clubs 2009–10 season
2010 |
https://en.wikipedia.org/wiki/1902%E2%80%9303%20Primera%20Fuerza%20season | The 1902–03 Primera Fuerza season was the first season of the amateur era in Méxican Football. Statistics of Primera Fuerza in season 1902–03. The tournament started 9 October 1902 and ended 1 February 1903.
Overview
It was contested by 5 teams, and Orizaba won the championship. Teams played each other once and the winner was determined by the points accumulated at the end of the last game.
Matches
On 20 September 1902 the calendar for the tournament was presented. The tournament was supposed to begin 19 October 1902 and end 28 December 1902 in order to not interfere with the tournament of cricket. The meeting took place in the offices of British Club. The games that were going to be played in Ciudad de México would be played in Campo del Reforma while Orizaba and Pachuca would play their respective games in their own cities.
League standings
Results
Top goalscorers
Players sorted first by goals scored, then by last name.
References
Mexico - List of final tables (RSSSF)
See also
1902-03
Mex
1902–03 in Mexican football |
https://en.wikipedia.org/wiki/1903%E2%80%9304%20Primera%20Fuerza%20season | Statistics of Primera División de México in season 1903–04.
Overview
1903-04 was the first season when the Mexican championship was played under a normal league system. Defending champions Orizaba only managed to place fourth. The new champions were Mexico Cricket Club San Pedro de los Pinos, who from 1894 to 1903 had played under the name Mexican National Cricket Club. This club was founded in the small town San Pedro de los Pinos, which now lies in the outskirts of Mexico City. The line-up of Mexico's second national champions, which was mainly British, included the three brothers Bruce, "Jack" and Walter Willy.
League standings
Top goalscorers
Players sorted first by goals scored, then by last name.
References
Mexico - List of final tables (RSSSF)
1903-04
Mex
1903–04 in Mexican football |
https://en.wikipedia.org/wiki/1904%E2%80%9305%20Primera%20Fuerza%20season | Statistics of Primera Fuerza in season 1904–05.
Overview
Pachuca won the Mexican national championship for the first time in 1905, if only because their goal average (goal ratio) was better than that of British Club (Mexico City). A Mexican triumph this was not, however, as all of Pachuca's players were British, many of whom had gained experience playing in the English leagues. The champions' line-up included the three brothers "Stan", "Charly" and "Jack" Dawe, as well as outside right "Willie" Rule and wing half-back "Jack" Rabling. Goalkeeper Charles Quickmore was a priest and minister of the Protestant mine churches in Mexico. Orizaba, Mexican champions in 1903, could no longer play in the league as the travels involved would have been too long and would have taken too much time.
League standings
Top goalscorers
Players sorted first by goals scored, then by last name.
References
Mexico - List of final tables (RSSSF)
1904-05
Mex
1904–05 in Mexican football |
https://en.wikipedia.org/wiki/1905%E2%80%9306%20Primera%20Fuerza%20season | Statistics of Primera División de México in season 1905–06.
Overview
In Mexico, the 1905–06 season saw the first club from the capital - Reforma Athletic Club- win the Mexican championship. This club, though dominated by Brits, did also allow native players. It was remarkable that in 20 league matches there were 16 where at last one or both sides failed to score. Champions Reforma had two excellent full-backs in the brothers Robert and Charles Blackmore, while most of their goals were not scored by a forward but by right half-back Charles Butlin. Mexico Cricket Club, which had changed name to San Pedro Golf Club, were runners-up.
Reforma 1905-06 Champion Squad
England
Thomas R. Phillips(C)
Morton S. Turner
Robert J. Blackmore
Charles Blackmore
Charles M. Butlin
Ebenezer Johnson
Charles D. Gibson
Robert Locke
Thomas R. Phillips
Paul M. Bennett
Thomas R. Phillips (Manager)
Mexico
Julio Lacaud
Vicente Etchegaray
France
Edward Bourchier
League standings
Top goalscorers
Players sorted first by goals scored, then by last name.
References
Mexico - List of final tables (RSSSF)
1905-06
Mex
1905–06 in Mexican football |
https://en.wikipedia.org/wiki/1906%E2%80%9307%20Primera%20Fuerza%20season | Statistics of Primera Fuerza in season 1906–07.
Overview
It was contested by 5 teams, and Reforma won the championship.
League standings
Top goalscorers
Players sorted first by goals scored, then by last name.
References
Mexico - List of final tables (RSSSF)
1906-07
Mex
1906–07 in Mexican football |
https://en.wikipedia.org/wiki/1907%E2%80%9308%20Primera%20Fuerza%20season | Statistics of Primera Fuerza in season 1907–08.
Overview
It was contested by four teams, and the British Club won the championship. It was the first Mexican championship title for British Football Club (Mexico City), whose players were almost exclusively British and where player-trainer Percy Clifford, centre half-back "Jack" Caldwall and the Hogg brothers were the most prominent characters.
British Club 1907–08 Champion squad
John Easton
Alexander Dewar
Pierce Mennill
Percy Clifford
Bryan White
John Johnson
John Hogg
Douglas Watson
Stephen Crowder
George Ratcliff
John Caldwall
Horace Hogg
League standings
Top goalscorers
Players sorted first by goals scored, then by last name.
References
Mexico - List of final tables (RSSSF)
1907-08
Mex
1907–08 in Mexican football |
https://en.wikipedia.org/wiki/1908%E2%80%9309%20Primera%20Fuerza%20season | Statistics of Primera Fuerza in season 1908–09.
Overview
In the 1907–08 tournament, Puebla withdrew from the Primera Fuerza due to the long distances they would have had to travel. Reforma went on to win the championship.
League standings
Top goalscorers
Players sorted first by goals scored, then by last name.
References
Mexico – List of final tables (RSSSF)
1908-09
Mex
1908–09 in Mexican football |
https://en.wikipedia.org/wiki/1909%E2%80%9310%20Primera%20Fuerza%20season | Statistics of Primera Fuerza in season 1909–10.
Overview
In three fourths of all matches in Mexico’s 1909/10 championship season, one (if not both) sides failed to score. Reforma Athletic Club went unbeaten again and successfully defended their title. Only two players remained of the side which had won in 1909, Charles Butlin and Robert Blackmore. The latter, a win half-back, even finished top scorer. English trainer "Tom" Phillips had built a new team and led them right on to victory.
League standings
Top goalscorers
Players sorted first by goals scored, then by last name.
References
Mexico - List of final tables (RSSSF)
1909-10
Mex
1909–10 in Mexican football |
https://en.wikipedia.org/wiki/1910%E2%80%9311%20Primera%20Fuerza%20season | Statistics of Primera Fuerza in season 1910–11.
Overview
It was contested by 3 teams, and Reforma won the championship.
League standings
Top goalscorers
Players sorted first by goals scored, then by last name.
References
Mexico - List of final tables (RSSSF)
1910-11
Mex
1910–11 in Mexican football |
https://en.wikipedia.org/wiki/1911%E2%80%9312%20Primera%20Fuerza%20season | Statistics of Primera Fuerza in season 1911–12.
Overview
It was contested by 3 teams, and Reforma won the championship.
League standings
Top goalscorers
Players sorted first by goals scored, then by last name.
References
Mexico - List of final tables (RSSSF)
1911-12
Mex
1911–12 in Mexican football |
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