source stringlengths 31 168 | text stringlengths 51 3k |
|---|---|
https://en.wikipedia.org/wiki/Least-angle%20regression | In statistics, least-angle regression (LARS) is an algorithm for fitting linear regression models to high-dimensional data, developed by Bradley Efron, Trevor Hastie, Iain Johnstone and Robert Tibshirani.
Suppose we expect a response variable to be determined by a linear combination of a subset of potential covariates. Then the LARS algorithm provides a means of producing an estimate of which variables to include, as well as their coefficients.
Instead of giving a vector result, the LARS solution consists of a curve denoting the solution for each value of the L1 norm of the parameter vector. The algorithm is similar to forward stepwise regression, but instead of including variables at each step, the estimated parameters are increased in a direction equiangular to each one's correlations with the residual.
Pros and cons
The advantages of the LARS method are:
It is computationally just as fast as forward selection.
It produces a full piecewise linear solution path, which is useful in cross-validation or similar attempts to tune the model.
If two variables are almost equally correlated with the response, then their coefficients should increase at approximately the same rate. The algorithm thus behaves as intuition would suggest, and also is more stable.
It is easily modified to produce efficient algorithms for other methods producing similar results, like the lasso and forward stagewise regression.
It is effective in contexts where p ≫ n (i.e., when the number of predictors p is significantly greater than the number of points n)
The disadvantages of the LARS method include:
With any amount of noise in the dependent variable and with high dimensional multicollinear independent variables, there is no reason to believe that the selected variables will have a high probability of being the actual underlying causal variables. This problem is not unique to LARS, as it is a general problem with variable selection approaches that seek to find underlying deterministic components. Yet, because LARS is based upon an iterative refitting of the residuals, it appears to be especially sensitive to the effects of noise. This problem is discussed in detail by Weisberg in the discussion section of the Efron et al. (2004) Annals of Statistics article. Weisberg provides an empirical example based upon re-analysis of data originally used to validate LARS that the variable selection appears to have problems with highly correlated variables.
Since almost all high dimensional data in the real world will just by chance exhibit some degree of collinearity across at least some variables, the problem that LARS has with correlated variables may limit its application to high dimensional data.
Algorithm
The basic steps of the Least-angle regression algorithm are:
Start with all coefficients equal to zero.
Find the predictor most correlated with .
Increase the coefficient in the direction of the sign of its correlation with . Take residuals along the way. Stop |
https://en.wikipedia.org/wiki/Ackermann%20ordinal | In mathematics, the Ackermann ordinal is a certain large countable ordinal, named after Wilhelm Ackermann. The term "Ackermann ordinal" is also occasionally used for the small Veblen ordinal, a somewhat larger ordinal.
Unfortunately there is no standard notation for ordinals beyond the Feferman–Schütte ordinal Γ0. Most systems of notation use symbols such as ψ(α), θ(α), ψα(β), some of which are modifications of the Veblen functions to produce countable ordinals even for uncountable arguments, and some of which are "collapsing functions". The last one is an extension of the Veblen functions for more than 2 arguments.
The smaller Ackermann ordinal is the limit of a system of ordinal notations invented by , and is sometimes denoted by or , , or , where Ω is the smallest uncountable ordinal. Ackermann's system of notation is weaker than the system introduced much earlier by , which he seems to have been unaware of.
References
Ordinal numbers |
https://en.wikipedia.org/wiki/Minimal%20prime%20ideal | In mathematics, especially in commutative algebra, certain prime ideals called minimal prime ideals play an important role in understanding rings and modules. The notion of height and Krull's principal ideal theorem use minimal primes.
Definition
A prime ideal P is said to be a minimal prime ideal over an ideal I if it is minimal among all prime ideals containing I. (Note: if I is a prime ideal, then I is the only minimal prime over it.) A prime ideal is said to be a minimal prime ideal if it is a minimal prime ideal over the zero ideal.
A minimal prime ideal over an ideal I in a Noetherian ring R is precisely a minimal associated prime (also called isolated prime) of ; this follows for instance from the primary decomposition of I.
Examples
In a commutative artinian ring, every maximal ideal is a minimal prime ideal.
In an integral domain, the only minimal prime ideal is the zero ideal.
In the ring Z of integers, the minimal prime ideals over a nonzero principal ideal (n) are the principal ideals (p), where p is a prime divisor of n. The only minimal prime ideal over the zero ideal is the zero ideal itself. Similar statements hold for any principal ideal domain.
If I is a p-primary ideal (for example, a symbolic power of p), then p is the unique minimal prime ideal over I.
The ideals and are the minimal prime ideals in since they are the extension of prime ideals for the morphism , contain the zero ideal (which is not prime since , but, neither nor are contained in the zero ideal) and are not contained in any other prime ideal.
In the minimal primes over the ideal are the ideals and .
Let and the images of x, y in A. Then and are the minimal prime ideals of A (and there are no others). Let be the set of zero-divisors in A. Then is in D (since it kills nonzero ) while neither in nor ; so .
Properties
All rings are assumed to be commutative and unital.
Every proper ideal I in a ring has at least one minimal prime ideal above it. The proof of this fact uses Zorn's lemma. Any maximal ideal containing I is prime, and such ideals exist, so the set of prime ideals containing I is non-empty. The intersection of a decreasing chain of prime ideals is prime. Therefore, the set of prime ideals containing I has a minimal element, which is a minimal prime over I.
Emmy Noether showed that in a Noetherian ring, there are only finitely many minimal prime ideals over any given ideal. The fact remains true if "Noetherian" is replaced by the ascending chain conditions on radical ideals.
The radical of any proper ideal I coincides with the intersection of the minimal prime ideals over I. This follows from the fact that every prime ideal contains a minimal prime ideal.
The set of zero divisors of a given ring contains the union of the minimal prime ideals.
Krull's principal ideal theorem says that, in a Noetherian ring, each minimal prime over a principal ideal has height at most one.
Each proper ideal I of a Noetherian ring co |
https://en.wikipedia.org/wiki/Freedom%20of%20religion%20in%20Iraq | According to the most recent government statistics, 97% of the population of Iraq was Muslim in 2010 (60% Shia and 40% Sunni); the constitution states that Islam is the official religion of the country.
In 2023, Iraq was scored 1 out of 4 for religious freedom.
In the same year, it was ranked as the 18th worst place in the world to be a Christian.
Background
In the 2010s, uprisings of the Islamic State (IS), formerly called the Islamic State of Iraq and the Levant (ISIL) or the Islamic State of Iraq and Syria (ISIS), have led to violations of religious freedom in certain parts of Iraq. IS is a Sunni jihadist group that claims religious authority over all Muslims around the world and aspires to bring most of the Muslim-inhabited regions of the world under its political control beginning with Iraq. ISIS follows an extreme anti-Western interpretation of Islam, promotes religious violence and regards those who do not agree with its interpretations as infidels or apostates. Concurrently, IS aims to establish a Salafist-orientated Islamist state in Iraq, Syria and other parts of the Levant.
As ISIL lost territory throughout Iraq in 2016, the armed forces and allied militias restored crosses, and Christians were allowed to return to their homes.
Status of religious freedom
In 2006 The Globe correspondent Khidir Domle stated that the Kurdistan Regional Government (KRG) engaged in discriminatory behaviour against Christians, and that according to Assyrian Christians, the Kurdistan Democratic Party (KDP)-dominated judiciary did so routinely against Assyrians, failing to enforce judgments in their favour. The KRG rejected these accusations.
In 2022, local and international NGOs reported that the government continued to use antiterrorism laws as a pretext for detaining individuals without due process (mostly Sunni Arabs). Yezidis and Christians have also reported verbal and physically abuse from local people; in September 2022, members of the local police and a private security company connected with the Shia militia Kata’ib Hezbollah (KH) threatened to evict 400 internally displaced Christians from the Mariam al-Adra IDP camp in Baghdad.
In 2022, ISIS were still in active in Iraq, carrying out kidnappings and murders; PKK activity is also ongoing.
Recognition
The government recognizes the following religious groups; Muslims, Chaldeans, Assyrians, Assyrian Catholics, Syriac Orthodox, Syriac Catholics, Armenian Apostolic, Armenian Catholics, Roman Catholics, National Protestants, Anglicans, Evangelical Protestant Assyrians, Seventh-day Adventists, Coptic Orthodox, Yezidis, Sabean-Mandeans, and Jews; all recognized religious groups (except Yezidis) have their own personal-status courts which handle marriage, divorce and inheritance issues. Baha’i, Zoroastrian and Kaka’i groups are not allowed to register with the government, although they are recognized in Kurdish areas; Baha'ism is illegal.
Education
Government regulations require three cla |
https://en.wikipedia.org/wiki/Karl%20Ledersteger | Karl Ledersteger (11 November 1900, in Vienna – 24 September 1972, near Vienna) was an important geodesist and geophysicist.
After studies of astronomy, mathematics and geodesy he worked in Germany and later in the National Survey of Austria. Later he set up the scientific department of the Federal Office for Metrology and Survey (BEV), Vienna. In the 1950s he was appointed as a professor of geodesy and astrometry at the Technical University of Vienna. He was head of many research projects, and author of about 200 scientific articles. Still a standard work of astronomical and physical geodesy is his textbook of Erdmessung (Vol. V of the series Handbuch der Vermessungskunde, 871 p.) published 1969.
In 1958/59, Ledersteger was the first geodesist in Central Europe who published on the future fields of satellite geodesy. Other topics of his research were:
Theory of National survey (Landesvermessung) - and practical computation of the ZEN (Zentraleuropäisches Netz, Berlin ~1940) and of parts of the ED50
Theory of equilibrium figures of Earth and planets
Isostasy of the Earth's crust and its effect on geoid determination; a main part was published posthumously by his successor Kurt Bretterbauer
the system of vertical deflections and the definition of reference ellipsoids.
Ledersteger was in intensive contact with the scientific community of whole Europe, US and Russia (e.g. Viktor Ambartsumian, B. Gutenberg, F. Hopfner, W. Heiskanen, M. Kneissl, Sir Harold Jeffreys, Vening Meinesz, H. Moritz, A. Prey, H. H. Schmid, E. Wiechert and S. Zhongolovitch). For almost 20 years he was chair of ÖKIE (Austrian Commission for international Geodesy) and a member of many international commissions and research groups, e.g. in IUGG, DGK and scientific academies of Austria, Germany and Hungary.
Ledersteger received many prizes, several doctorates honoris causa and calls to universities. In Vienna he was asked to begin geodetic lectures immediately after World War II, but his professorship was postponed for 10 years because of his participation in Nazi surveys 1940–1945. He translated several works including a textbook of Magnizki & Browar on theoretical geodesy.
External links, sources and literature
Literature Online, University Innsbruck, 20 titles
Karl Ledersteger 70 years, Harvard
K. Ledersteger: "Astronomische und Physikalische Geodäsie (Erdmessung)", Handbuch der Vermessungskunde, Wilhelm Jordan, Otto Eggert and Max Kneissl ed., Volume V, chapter 4 (Geoid) and 11 (gravity), J.B.Metzler-Verlag, Stuttgart 1968
K. Ledersteger, several lectures and reprints (Austrian & German Univ.libraries), c. 1940 bis 1975
Péter Biró: 100 éve született Karl Ledersteger (K. Ledersteger 100th birthday). Geodézia és Kartográfia, Vol.52/12 (p. 32 ff), Budapest 2000.
1900 births
1972 deaths
Scientists from Vienna
Austrian geodesists
Academic staff of TU Wien |
https://en.wikipedia.org/wiki/Deir%20Ghazaleh | Deir Ghazaleh () is a Palestinian village in the northern West Bank, located nine kilometers northeast of Jenin in the Jenin Governorate. According to the Palestinian Central Bureau of Statistics, Deir Ghazaleh had a population of over 850 inhabitants in mid-year 2006 and 1,129 in 2017, mostly Muslims with a small Christian minority.
History
Ceramics from the Byzantine era have been found here.
Ottoman era
In 1517, Deir Ghazaleh was incorporated into the Ottoman Empire with the rest of Palestine. During the 16th and 17th centuries, it belonged to the Turabay Emirate (1517-1683), which encompassed also the Jezreel Valley, Haifa, Jenin, Beit She'an Valley, northern Jabal Nablus, Bilad al-Ruha/Ramot Menashe, and the northern part of the Sharon plain. By the 1596 tax register it was part of nahiya (subdistrict) of Jinin under the liwa' (district) of Lajjun, with a population of 5 Muslim households. The villagers paid a fixed tax rate of 25% on agricultural products, including wheat, barley, summer crops, olive trees, beehives and/or goats, in addition to occasional revenues; a total of 3,000 akçe.
In 1838, Edward Robinson noted Deir Ghuzal as one of a range of villages round a height, the other villages being named as Beit Qad, Fuku'a, Deir Abu Da'if and Araneh, located in the District of Jenin, also called Haritheh esh-Shemaliyeh.
In 1870 Victor Guérin found it have about fifteen houses, bordered by several antiquated cisterns and silos.
In 1882 the PEF's Survey of Western Palestine found that it resembled Deir Abu Da'if, and that "the ground round it is partly rock, partly arable land."
They further noted a ruin, one mile to the south east of the village, "It is firmly bedded into the earth, which contains fragments of pottery, apparently ancient. The stone seems to have been packed with smaller ones round its base to keep it in position, as found by excavation. The stones are very heavy, and the construction of this monument must have been a considerable labour. It somewhat resembles the vinevard towers existing in other parts of Palestine; but fallen stones sufficient for such a structure were not observed, and there is no reason to suppose it to have ever consisted of more than two courses."
British Mandate era
In the 1922 census of Palestine, conducted by the British Mandate authorities, the village had a population of 134; 120 Muslims and 14 Christians, where the Christians were all Orthodox, increasing slightly in the 1931 census to 186; 169 Muslims and 17 Christians, with 34 houses.
In 1944/5 statistics the population was 270; 240 Muslims and 30 Christians, with a total of 6,588 dunams of land, according to an official land and population survey. Of this, 160 dunams were used for plantations and irrigable land, 4,917 dunams were for cereals, while 6 dunams were built-up (urban) land.
Jordanian era
After the 1948 Arab-Israeli War, Deir Ghazaleh came under Jordanian rule.
In 1961, the population of Deir Ghazzala |
https://en.wikipedia.org/wiki/Filipinos%20in%20Greece | Filipinos in Greece consist of migrants from the Philippines to Greece and their descendants. According to official Greek statistics, there were 5,826 Filipinos in Greece in 1991, which declined to 2,000 by 1996. In reality, there were many more working in the country illegally. The Philippine community have set up a school for their children in downtown Athens.
A large proportion are women (81% ), who generally find employment as domestic workers. The association between Filipinas and domestic work is so strong that a Greek dictionary published in 1998 even defined "Filippineza", a term which literally means Filipina, to be "a domestic worker from the Philippines or a person who performs non-essential auxiliary tasks". Migrants and the Philippine Department of Foreign Affairs protested to the Greek government about the dictionary.
History
Culture
See also
Greek settlement in the Philippines
Philippines–Greece relations
Notes
Sources
. A study of Albanian, Egyptian, and Filipino migrants.
Further reading
Ethnic groups in Greece
Greece |
https://en.wikipedia.org/wiki/Linear%20system%20of%20conics | In algebraic geometry, the conic sections in the projective plane form a linear system of dimension five, as one sees by counting the constants in the degree two equations. The condition to pass through a given point P imposes a single linear condition, so that conics C through P form a linear system of dimension 4. Other types of condition that are of interest include tangency to a given line L.
In the most elementary treatments a linear system appears in the form of equations
with λ and μ unknown scalars, not both zero. Here C and C′ are given conics. Abstractly we can say that this is a projective line in the space of all conics, on which we take
as homogeneous coordinates. Geometrically we notice that any point Q common to C and C′ is also on each of the conics of the linear system. According to Bézout's theorem C and C′ will intersect in four points (if counted correctly). Assuming these are in general position, i.e. four distinct intersections, we get another interpretation of the linear system as the conics passing through the four given points (note that the codimension four here matches the dimension, one, in the five-dimensional space of conics). Note that of these conics, exactly three are degenerate, each consisting of a pair of lines, corresponding to the ways of choosing 2 pairs of points from 4 points (counting via the multinomial coefficient, and accounting for the overcount by a factor of 2 that makes when interested in counting pairs of pairs rather than just selections of size 2).
Applications
A striking application of such a family is in which gives a geometric solution to a quartic equation by considering the pencil of conics through the four roots of the quartic, and identifying the three degenerate conics with the three roots of the resolvent cubic.
Example
For example, given the four points the pencil of conics through them can be parameterized as which are the affine combinations of the equations and corresponding to the parallel vertical lines and horizontal lines; this yields degenerate conics at the standard points of A less elegant but more symmetric parametrization is given by in which case inverting a () interchanges x and y, yielding the following pencil; in all cases the center is at the origin:
hyperbolae opening left and right;
the parallel vertical lines
(intersection point at [1:0:0])
ellipses with a vertical major axis;
a circle (with radius );
ellipses with a horizontal major axis;
the parallel horizontal lines
(intersection point at [0:1:0])
hyperbolae opening up and down,
the diagonal lines
(dividing by and taking the limit as yields )
(intersection point at [0:0:1])
This then loops around to since pencils are a projective line.
In the terminology of , this is a Type I linear system of conics, and is animated in the linked video.
Classification
There are 8 types of linear systems of conics over the complex numbers, depending on intersection multiplicity at the ba |
https://en.wikipedia.org/wiki/UK%20Statistics%20Authority | The UK Statistics Authority (UKSA, ) is a non-ministerial government department of the Government of the United Kingdom responsible for oversight of the Office for National Statistics, maintaining a national code of practice for official statistics, and accrediting statistics that comply with the Code as National Statistics. UKSA was established on 1 April 2008 by the Statistics and Registration Service Act 2007, and is directly accountable to the Parliament of the United Kingdom.
Background
Gordon Brown, then Chancellor of the Exchequer, announced on 28 November 2005, that the government intended to publish plans in early 2006 to legislate to render the Office for National Statistics (ONS) and the statistics it generates independent of government on a model based on the independence of the Monetary Policy Committee of the Bank of England. This was originally a 1997 Labour Party manifesto commitment and was also the policy of the Liberal Democrat and Conservative parties. Such independence was also sought by the Royal Statistical Society and the Statistics Commission. The National Statistician, who is the chief executive of the ONS, would be directly accountable to Parliament through a widely constituted independent governing Statistics Board. The ONS would be a non-ministerial government department so that the staff, including the Director, would remain as civil servants but without being under direct ministerial control. The National Statistician at the time, Karen Dunnell, stated that the legislation would help improve public trust in official statistics although the ONS already acts independently according to its own published guidelines, the National Statistics Code of Practice, which sets out the key principles and standards that official statisticians, including those in other parts of the Government Statistical Service, are expected to follow and uphold.
The details of the plans for independence were considered in Parliament during the 2006/2007 session and resulted in the Statistics and Registration Service Act 2007. In July 2007, Sir Michael Scholar was nominated by the government to be the three-day-a-week non-executive chairman of the Statistics Board which, to re-establish faith in the integrity of government statistics, has statutory responsibility for oversight of UK government statistics and of the Office for National Statistics. It also has a duty to assess all UK government statistics. Following Gordon Brown's later announcement on his 2007 appointment as Prime Minister of new constitutional arrangements for public appointments, Sir Michael also became, on 18 July, the first such nominee to appear before the House of Commons Treasury Committee and to have his nomination subject to confirmation by the House. On 7 February 2008, following the first meeting of the shadow board, it was announced that the body would be known as the UK Statistics Authority.
UKSA was established on 1 April 2008 by the Statistics and Registration Se |
https://en.wikipedia.org/wiki/Robin%20Plackett | Robin L. Plackett (3 September 1920 – 23 June 2009) was a statistician best known for his contributions to the history of statistics and to experimental design, most notably the Plackett–Burman designs.
Early life and education
Plackett attended Liverpool Collegiate School from 1932 to 1939. He then attended Clare College, Cambridge, where he graduated in 1942.
Early career
During World War II, Plackett was requested to work for the Ministry of Supply, in SR17 which was a statistical branch. He began to develop a methodology for applying statistical knowledge, and would pass it down to new recruits.
First scientific paper
In 1946, he would publish his first paper which was written jointly with Peter Burman in an journal called Biometrika. The paper, titled "The design of optimum multifactorial experiments", introduced Plackett–Burman experimental designs.
Academic career
In 1947, he became a lecturer at Liverpool University. He would also publish research on the history of statistics. Then, in 1962, he took a short post for the Professor of Statistics at King's College, Durham before the college merged with Newcastle University in 1963.
He was the first professor of statistics at Newcastle University and held the post until his retirement in 1983. In 1987 the Royal Statistical Society awarded him the Guy Medal in Gold, having awarded him both the bronze and silver medals earlier in his career.
He authored several books on statistics, including Principles of Regression Analysis (1960), The Analysis of Categorical Data (1974) and An Introduction to the Interpretation of Quantal Responses in Biology (1979, with P. S. Hewlett).
Personal life
Plackett was said to have had a keen interest in climbing. He met his wife, Carol whom they have been married for 65 years. He also had three children: Adam, Jane and Martin.
References
Further reading
Academics of Newcastle University
English statisticians
1920 births
Fellows of the American Statistical Association
2009 deaths
Alumni of Clare College, Cambridge |
https://en.wikipedia.org/wiki/Coxeter%27s%20loxodromic%20sequence%20of%20tangent%20circles | In geometry, Coxeter's loxodromic sequence of tangent circles is an infinite sequence of circles arranged so that any four consecutive circles in the sequence are pairwise mutually tangent. This means that each circle in the sequence is tangent to the three circles that precede it and also to the three circles that follow it.
Properties
The radii of the circles in the sequence form a geometric progression with ratio
where is the golden ratio. This ratio and its reciprocal satisfy the equation
and so any four consecutive circles in the sequence meet the conditions of Descartes' theorem.
The centres of the circles in the sequence lie on a logarithmic spiral. Viewed from the centre of the spiral, the angle between the centres of successive circles is
The angle between consecutive triples of centers is
the same as one of the angles of the Kepler triangle, a right triangle whose construction also involves the square root of the golden ratio.
History and related constructions
The construction is named after geometer H. S. M. Coxeter, who generalised the two-dimensional case to sequences of spheres and hyperspheres in higher dimensions. It can be interpreted as a degenerate special case of the Doyle spiral.
See also
Apollonian gasket
References
External links
Circle packing
Golden ratio |
https://en.wikipedia.org/wiki/Hydra%20game | In mathematics, specifically in graph theory and number theory, a hydra game is a single-player iterative mathematical game played on a mathematical tree called a hydra where, usually, the goal is to cut off the hydra's "heads" while the hydra simultaneously expands itself. Hydra games can be used to generate large numbers or infinite ordinals or prove the strength of certain mathematical theories.
Unlike their combinatorial counterparts like TREE and SCG, no search is required to compute these fast-growing function values – one must simply keep applying the transformation rule to the tree until the game says to stop.
Introduction
A simple hydra game can be defined as follows:
A hydra is a finite rooted tree, which is a connected graph with no cycles and a specific node designated as the root of the tree. In a rooted tree, each node has a single parent (with the exception of the root, which has no parent) and a set of children, as opposed to an unrooted tree where there is no parent-child relationship and we simply refer to edges between nodes.
The player selects a leaf node from the tree and a natural number during each turn. A leaf node can be defined as a node with no children, or a node of degree 1 which is not .
Remove the leaf node . Let be 's parent. If , return to stage 2. Otherwise if , let be the parent of . Then create leaf nodes as children of such that the new nodes would appear after any existing children of during a post-order traversal (visually, these new nodes would appear to the right side of any existing children). Then return to stage 2.
Even though the hydra may grow by an unbounded number of leaves at each turn, the game will eventually end in finitely many steps: if is the greatest distance between the root and the leaf, and the number of leaves at this distance, induction on can be used to demonstrate that the player will always kill the hydra. If , removing the leaves can never cause the hydra to grow, so the player wins after turns. For general , we consider two kinds of moves: those that involve a leaf at a distance less than from the root, and those that involve a leaf at a distance of exactly . Since moves of the first kind are also identical to moves in a game with depth , the induction hypothesis tells us that after finitely many such moves, the player will have no choice but to choose a leaf at depth . No move introduces new nodes at this depth, so this entire process can only repeat up to times, after which there are no more leaves at depth and the game now has depth (at most) . Invoking the induction hypothesis again, we find that the player must eventually win overall.
While this shows that the player will win eventually, it can take a very long time. As an example, consider the following algorithm. Pick the rightmost leaf and set the first time, the second time, and so on, always increasing by one. If a hydra has a single -length branch, then for , the hydra is killed in a single st |
https://en.wikipedia.org/wiki/Steven%20Kleiman | Steven Lawrence Kleiman (born March 31, 1942) is an American mathematician.
Professional career
Kleiman is a professor emeritus of mathematics at the Massachusetts Institute of Technology. Born in Boston, he did his undergraduate studies at MIT. He received his Ph.D. from Harvard University in 1965, after studying there with Oscar Zariski and David Mumford, and joined the MIT faculty in 1969. Kleiman held the prestigious NATO Postdoctoral Fellowship (1966-1967), Sloan Fellowship (1968), and Guggenheim Fellowship (1979).
Contributions
Kleiman is known for his work in algebraic geometry and commutative algebra. He has made seminal contributions in motivic cohomology, moduli theory, intersection theory and enumerative geometry. A 2002 study of 891 academic collaborations in enumerative geometry and intersection theory covered by Mathematical Reviews found that he was not only the most prolific author in those areas, but also the one with the most collaborative ties, and the most central author of the field in terms of closeness centrality; the study's authors proposed to name the collaboration graph of the field in his honor.
Awards and honors
In 1989 the University of Copenhagen awarded him an honorary doctorate and in May 2002 the Norwegian Academy of Science and Letters hosted a conference in honor of his 60th birthday and elected him as a foreign member. In 1992 Kleiman was elected foreign member of the Royal Danish Academy of Sciences and Letters.
In 2012 he became a fellow of the American Mathematical Society. He was an invited speaker at the International Congress of Mathematics at Nice in 1970.
Selected publications
.
.
.
.
.
.
.
See also
Cone of curves (Kleiman-Mori cone)
Kleiman's theorem
References
External links
1942 births
Living people
20th-century American mathematicians
21st-century American mathematicians
Algebraic geometers
Massachusetts Institute of Technology School of Science faculty
Harvard University alumni
Members of the Norwegian Academy of Science and Letters
Fellows of the American Mathematical Society
Massachusetts Institute of Technology alumni
Mathematicians from Massachusetts |
https://en.wikipedia.org/wiki/Isoparametric%20manifold | In Riemannian geometry, an isoparametric manifold is a type of (immersed) submanifold of Euclidean space whose normal bundle is flat and whose principal curvatures are constant along any parallel normal vector field. The set of isoparametric manifolds is stable under the mean curvature flow.
Examples
A straight line in the plane is an obvious example of isoparametric manifold. Any affine subspace of the Euclidean n-dimensional space is also an example since the principal curvatures of any shape operator are zero.
Another simplest example of an isoparametric manifold is a sphere in Euclidean space.
Another example is as follows. Suppose that G is a Lie group and G/H is a symmetric space with canonical decomposition
of the Lie algebra g of G into a direct sum (orthogonal with respect to the Killing form) of the Lie algebra h or H with a complementary subspace p. Then a principal orbit of the adjoint representation of H on p is an isoparametric manifold in p. Non principal orbits are examples of the so-called submanifolds with principal constant curvatures. Actually, by Thorbergsson's theorem any complete, full and irreducible isoparametric submanifold of codimension > 2 is an orbit of a s-representation, i.e. an H-orbit as above where the symmetric space G/H has no flat factor.
The theory of isoparametric submanifolds is deeply related to the theory of holonomy groups. Actually, any isoparametric submanifold is foliated by the holonomy tubes of a submanifold with constant principal curvatures i.e. a focal submanifold. The paper "Submanifolds with constant principal curvatures and normal holonomy groups" is a very good introduction to such theory. For more detailed explanations about holonomy tubes and focalizations see the book Submanifolds and Holonomy.
References
See also
Isoparametric function
Riemannian geometry
Manifolds |
https://en.wikipedia.org/wiki/Divina%20proportione | Divina proportione (15th century Italian for Divine proportion), later also called De divina proportione (converting the Italian title into a Latin one) is a book on mathematics written by Luca Pacioli and illustrated by Leonardo da Vinci, completed by February 9th, 1498 in Milan and first printed in 1509. Its subject was mathematical proportions (the title refers to the golden ratio) and their applications to geometry, to visual art through perspective, and to architecture. The clarity of the written material and Leonardo's excellent diagrams helped the book to achieve an impact beyond mathematical circles, popularizing contemporary geometric concepts and images.
Some of its content was plagiarised from an earlier book by Piero della Francesca, De quinque corporibus regularibus.
Contents of the book
The book consists of three separate manuscripts, which Pacioli worked on between 1496 and 1498. He credits Fibonacci as the main source for the mathematics he presents.
Compendio divina proportione
The first part, Compendio divina proportione (Compendium on the Divine Proportion), studies the golden ratio from a mathematical perspective (following the relevant work of Euclid), giving mystical and religious meanings to this ratio, in seventy-one chapters. Pacioli points out that golden rectangles can be inscribed by an icosahedron, and in the fifth chapter, gives five reasons why the golden ratio should be referred to as the "Divine Proportion":
Its value represents divine simplicity.
Its definition invokes three lengths, symbolizing the Holy Trinity.
Its irrationality represents God's incomprehensibility.
Its self-similarity recalls God's omnipresence and invariability.
Its relation to the dodecahedron, which represents the quintessence
It also contains a discourse on the regular and semiregular polyhedra, as well as a discussion of the use of geometric perspective by painters such as Piero della Francesca, Melozzo da Forlì and Marco Palmezzano.
Trattato dell'architettura
The second part, Trattato dell'architettura (Treatise on Architecture), discusses the ideas of Vitruvius (from his De architectura) on the application of mathematics to architecture in twenty chapters. The text compares the proportions of the human body to those of artificial structures, with examples from classical Greco-Roman architecture.
Libellus in tres partiales divisus
The third part, Libellus in tres partiales divisus (Book divided into three parts), is a translation into Italian of Piero della Francesca's Latin book De quinque corporibus regularibus [On [the] Five Regular Solids]. It does not credit della Francesca for this material, and in 1550 Giorgio Vasari wrote a biography of della Francesca, in which he accused Pacioli of plagiarism and claimed that he stole della Francesca's work on perspective, on arithmetic and on geometry. Because della Francesca's book had been lost, these accusations remained unsubstantiated until the 19th century, when a copy of della Fr |
https://en.wikipedia.org/wiki/Greg%20Kuperberg | Greg Kuperberg (born July 4, 1967) is a Polish-born American mathematician known for his contributions to geometric topology, quantum algebra, and combinatorics. Kuperberg is a professor of mathematics at the University of California, Davis.
Biography
Kuperberg is the son of two mathematicians, Krystyna Kuperberg and Włodzimierz Kuperberg. He was born in Poland in 1967, but his family emigrated to Sweden in 1969 due to the 1968 Polish political crisis. In 1972, Kuperberg's family moved to the United States, eventually settling in Auburn, Alabama.
Kuperberg wrote three computer games for the IBM Personal Computer in 1982 and 1983 (which were published by Orion Software): Paratrooper, PC-Man and J-Bird. (video game clones of Sabotage, Pac-Man and Q*bert, respectively)
He enrolled at Harvard University in 1983 and received a bachelor's degree in 1987. He was ranked Top 10 in the 1986 William Lowell Putnam Mathematical Competition. Upon leaving Harvard, Kuperberg studied at the University of California, Berkeley under Andrew Casson, receiving a Ph.D. in geometric topology and quantum algebra in 1991. From 1991 until 1992, Kuperberg was a NSF postdoctoral fellow and adjunct assistant professor at Berkeley, and from 1992 to 1995 held a Dickson Instructorship at the University of Chicago. From 1995 through 1996, Kuperberg was Gibbs Assistant Professor at Yale University after which he joined the mathematics faculty at the University of California, Davis. In 2012 he became a fellow of the American Mathematical Society.
Kuperberg is married to physicist Rena Zieve, who is a professor of physics at UC Davis.
Selected publications
Kuperberg has over fifty publications, including two in the Annals of Mathematics.
with Krystyna Kuperberg:
References
External links
Greg Kuperberg, faculty page at UC-Davis
Bits from my personal collection - the original IBM PC and Orion Software
1967 births
Polish mathematicians
20th-century American mathematicians
21st-century American mathematicians
Auburn High School (Alabama) alumni
Harvard University alumni
Living people
People from Auburn, Alabama
Polish emigrants to the United States
Topologists
University of California, Berkeley alumni
University of California, Davis faculty
Fellows of the American Mathematical Society
American video game programmers |
https://en.wikipedia.org/wiki/Liouville%27s%20formula | In mathematics, Liouville's formula, also known as the Abel-Jacobi-Liouville Identity, is an equation that expresses the determinant of a square-matrix solution of a first-order system of homogeneous linear differential equations in terms of the sum of the diagonal coefficients of the system. The formula is named after the French mathematician Joseph Liouville. Jacobi's formula provides another representation of the same mathematical relationship.
Liouville's formula is a generalization of Abel's identity and can be used to prove it. Since Liouville's formula relates the different linearly independent solutions of the system of differential equations, it can help to find one solution from the other(s), see the example application below.
Statement of Liouville's formula
Consider the -dimensional first-order homogeneous linear differential equation
on an interval of the real line, where for denotes a square matrix of dimension with real or complex entries. Let denote a matrix-valued solution on , meaning that is the so-called fundamental matrix, a square matrix of dimension with real or complex entries and the derivative satisfies
Let
denote the trace of , the sum of its diagonal entries. If the trace of is a continuous function, then the determinant of satisfies
for all and in .
Example application
This example illustrates how Liouville's formula can help to find the general solution of a first-order system of homogeneous linear differential equations. Consider
on the open interval . Assume that the easy solution
is already found. Let
denote another solution, then
is a square-matrix-valued solution of the above differential equation. Since the trace of is zero for all , Liouville's formula implies that the determinant
is actually a constant independent of . Writing down the first component of the differential equation for , we obtain using () that
Therefore, by integration, we see that
involving the natural logarithm and the constant of integration . Solving equation () for and substituting for gives
which is the general solution for . With the special choice and we recover the easy solution we started with, the choice and yields a linearly independent solution. Therefore,
is a so-called fundamental solution of the system.
Proof of Liouville's formula
We omit the argument for brevity. By the Leibniz formula for determinants, the derivative of the determinant of can be calculated by differentiating one row at a time and taking the sum, i.e.
Since the matrix-valued solution satisfies the equation , we have for every entry of the matrix
or for the entire row
When we subtract from the -th row the linear combination
of all the other rows, then the value of the determinant remains unchanged, hence
for every } by the linearity of the determinant with respect to every row. Hence
by () and the definition of the trace. It remains to show that this representation of the derivative implies Liouville's formula.
|
https://en.wikipedia.org/wiki/Disjunction%20property%20of%20Wallman | In mathematics, especially in order theory, a partially ordered set with a unique minimal element 0 has the disjunction property of Wallman when for every pair (a, b) of elements of the poset, either b ≤ a or there exists an element c ≤ b such that c ≠ 0 and c has no nontrivial common predecessor with a. That is, in the latter case, the only x with x ≤ a and x ≤ c is x = 0.
A version of this property for lattices was introduced by , in a paper showing that the homology theory of a topological space could be defined in terms of its distributive lattice of closed sets. He observed that the inclusion order on the closed sets of a T1 space has the disjunction property. The generalization to partial orders was introduced by .
References
.
.
Order theory |
https://en.wikipedia.org/wiki/Goodman%20and%20Kruskal%27s%20lambda | In probability theory and statistics, Goodman & Kruskal's lambda () is a measure of proportional reduction in error in cross tabulation analysis. For any sample with a nominal independent variable and dependent variable (or ones that can be treated nominally), it indicates the extent to which the modal categories and frequencies for each value of the independent variable differ from the overall modal category and frequency, i.e., for all values of the independent variable together. is defined by the equation
where
is the overall non-modal frequency, and
is the sum of the non-modal frequencies for each value of the independent variable.
Values for lambda range from zero (no association between independent and dependent variables) to one (perfect association).
Weaknesses
Although Goodman and Kruskal's lambda is a simple way to assess the association between variables, it yields a value of 0 (no association) whenever two variables are in accord—that is, when the modal category is the same for all values of the independent variable, even if the modal frequencies or percentages vary. As an example, consider the table below, which describes a fictitious sample of 350 individuals, categorized by relationship status and blood pressure. Assume that the relationship status is the independent variable, the blood pressure is the dependent variable, i.e., the question asked is "can the blood pressure be predicted better if the relationship status is known?"
For this sample,
The reason is that the predicted nominal blood pressure is actually "Normal" in both columns (both upper numbers are higher than the corresponding lower number). Thus, considering the relationship status will not change the prediction that people have a normal blood pressure, even though the data indicate that being married increases the probability of high blood pressure.
If the question is changed, e.g. by asking "What is the predicted relationship status based on blood pressure?," will have a non-zero value.
That is:
See also
Proportional reduction in loss
References
Goodman, L.A., Kruskal, W.H. (1954) "Measures of association for cross classifications". Part I. Journal of the American Statistical Association, 49, 732–764.
Goodman, L.A., Kruskal, W.H. (1959) "Measures of Association for Cross Classifications. II: Further Discussion and References". Journal of the American Statistical Association, 52, 123–163.
Goodman, L.A., Kruskal, W.H. (1963) "Measures of Association for Cross Classifications III: Approximate Sampling Theory", Journal of the American Statistical Association, 58, 310–364.
Statistical ratios
Summary statistics for contingency tables |
https://en.wikipedia.org/wiki/Sanur%2C%20Jenin | Sanur (, also spelled Sanour) is a Palestinian village located southwest of Jenin, in the Jenin Governorate of the State of Palestine. According to the Palestinian Central Bureau of Statistics, Sanur had a population of 4,067 in 2007 and 5,036 in 2017. During the late Ottoman era, Sanur served as a fortified village of the Jarrar family and played a key role in limiting the centralized power of the Ottoman sultanate, the Ottoman governors of Damascus and Acre and the Ottoman-aligned Tuqan family of Nablus from exerting direct authority over the rural highlands of Jabal Nablus (modern-day northern West Bank).
History
An old cistern is found by the mosque. Cisterns are also carved into rock on the steep slopes, as are tombs.
Ceramic remains (sherds) have been found here, dating from the Middle Bronze Age IIB, Iron Age I and IA II, Persian, Hellenistic, early and late Roman, Byzantine, early Muslim and Medieval eras.
Ottoman era
Sanur, like the rest of Palestine, was incorporated into the Ottoman Empire in 1517, and in 1596 it appeared in Ottoman tax registers as being in the nahiya (subdistrcit) of Jabal Sami, part of Nablus Sanjak. It had a population of 23 households and five bachelors, all Muslims. The villagers paid a fixed tax rate of 33.3% agricultural products, including on wheat, barley, summer crops, olive trees, goats and/or beehives, in addition to occasional revenues; a total of 5,200 akçe.
Modern Sanur was founded by a branch of the Jarrar family that migrated to the site from Jaba', during the late Ottoman era. Sanur served as the Jarrar family's throne village, from where they controlled many of the villages in the region of Jenin, Lajjun, the Jezreel Valley (Marj Ibn Amer) and Nazareth. In 1785, under the leadership of Sheikh Yusuf al-Jarrar, a formidable fortress was built in the village, which guarded access to Nablus from the north. Part of the fortress's walls had been built earlier by Sheikh Yusuf's father, Muhammad Zabin. The fortress, along with their large peasant militia, solidified the Jarrar family's military strength. Unlike the other roughly two dozen throne villages in Palestine's central highlands, Sanur was completely encircled by fortified walls.
In the mid and late 18th century, the Arab sheikh Zahir al-Umar emerged as the autonomous ruler of the Galilee and the coastal town of Acre, which he fortified. From Acre, Zahir extended his control southward into Jarrar territory. The Jarrar family entered into a coalition with the Beni Sakhr tribe, but failed to prevent Zahir from taking over Nazareth and the Jezreel Valley in 1735. Zahir pursued the Jarrar family's forces into Jabal Nablus but once he reached Sanur, he realized he would not be able to overcome its fortress. The Jarrars' successful resistance against Zahir rendered the region of Jabal Nablus to be largely outside Zahir's control. Sanur marked the limit of Zahir's influence and continued to limit the control of successive rulers of Acre. In |
https://en.wikipedia.org/wiki/Rauch%20comparison%20theorem | In Riemannian geometry, the Rauch comparison theorem, named after Harry Rauch, who proved it in 1951, is a fundamental result which relates the sectional curvature of a Riemannian manifold to the rate at which geodesics spread apart. Intuitively, it states that for positive curvature, geodesics tend to converge, while for negative curvature, geodesics tend to spread.
The statement of the theorem involves two Riemannian manifolds, and allows to compare the infinitesimal rate at which geodesics spread apart in the two manifolds, provided that their curvature can be compared. Most of the time, one of the two manifolds is a "comparison model", generally a manifold with constant curvature , and the second one is the manifold under study : a bound (either lower or upper) on its sectional curvature is then needed in order to apply Rauch comparison theorem.
Statement
Let be Riemannian manifolds, on which are drawn unit speed geodesic segments and . Assume that has no conjugate points along , and let be two normal Jacobi fields along and such that :
and
.
If the sectional curvature of every 2-plane containing is less or equal than the sectional curvature of every 2-plane containing , then for all .
Conditions of the theorem
The theorem is formulated using Jacobi fields to measure the variation in geodesics. As the tangential part of a Jacobi field is independent of the geometry of the manifold, the theorem focuses on normal Jacobi fields, i.e. Jacobi fields which are orthogonal to the speed vector of the geodesic for all time . Up to reparametrization, every variation of geodesics induces a normal Jacobi field.
Jacobi fields are requested to vanish at time because the theorem measures the infinitesimal divergence (or convergence) of a family of geodesics issued from the same point , and such a family induces a Jacobi field vanishing at .
Analog theorems
Under very similar conditions, it is also possible to compare the Hessian of the distance function to a given point. It is also possible to compare the Laplacian of this function (which is the trace of the Hessian), with some additional condition on one of the two manifolds: it is then enough to have an inequality on the Ricci curvature (which is the trace of the curvature tensor).
See also
Toponogov's theorem
References
do Carmo, M.P. Riemannian Geometry, Birkhäuser, 1992.
Lee, J. M., Riemannian Manifolds: An Introduction to Curvature, Springer, 1997.
Theorems in Riemannian geometry |
https://en.wikipedia.org/wiki/Verusco | Verusco Technologies is a company based in Palmerston North, New Zealand. They supply video analysis software and statistics to rugby union teams, including the Super Rugby franchise the Blue Bulls, most South African professional teams and other teams in France, Japan, and Australia.
While involved with analysis of the 2007 Rugby World Cup, they discovered that the New Zealand national rugby union team, the "All Blacks", made 57 tackles to Frances' 269, and they had 66 percent possession and 60 percent territory. The playing time, or the time the ball was in play, was the longest of any game Verusco had ever recorded.
The company has analysed 1,500 games since 2000.
References
External links
Rugby union in New Zealand
Sports software
Software companies of New Zealand |
https://en.wikipedia.org/wiki/Henry%20Wallman | Henry "Hank" Wallman (1915–1992) was an American mathematician, known for his work in lattice theory, dimension theory, topology, and electronic circuit design.
A native of Brooklyn and a 1933 graduate of Brooklyn College, Wallman received his Ph.D. in mathematics from Princeton University in 1937, under the supervision of Solomon Lefschetz and became a faculty member at the Massachusetts Institute of Technology, where he was associated with the Radiation Laboratory. During World War II he did classified work at MIT, possibly involving radar. In 1948, he left MIT to become a professor of electrotechnics at the Chalmers University of Technology in Gothenburg, Sweden, which awarded him the Chalmers medal in 1980 and where he eventually retired. In 1950 he was elected as a foreign member to the Swedish Royal Academy. He was elected a member of the Royal Swedish Academy of Engineering Sciences in 1960 and of the Royal Swedish Academy of Engineering Sciences in 1970.
The disjunction property of Wallman is named after Wallman, as is the Wallman compactification, and he co-authored an important monograph on dimension theory with Witold Hurewicz. Wallman was also a radio enthusiast, and in the postwar period co-authored a book comprehensively documenting what was known at the time about vacuum tube amplification technology, including new developments such as showing how the central limit theorem could be used to describe the rise time of cascaded circuits. At Chalmers, Wallman helped build the Electronic Differential Analyser, an early example of an analog computer, and performed pioneering research in biomedical engineering combining video displays with X-ray imaging.
References
1915 births
1992 deaths
Lattice theorists
Topologists
American biomedical engineers
20th-century American mathematicians
Swedish mathematicians
Princeton University alumni
Massachusetts Institute of Technology faculty
Academic staff of the Chalmers University of Technology
Members of the Royal Swedish Academy of Sciences
Members of the Royal Swedish Academy of Engineering Sciences
Brooklyn College alumni |
https://en.wikipedia.org/wiki/Richard%20Davis%20Anderson | Richard Davis Anderson, Sr. (February 17, 1922 – March 4, 2008) was an American mathematician known internationally for his work in infinite-dimensional topology. Much of his early work focused on proofs surrounding Hilbert space and Hilbert cubes.
Life
Richard Anderson and his twin brother, John, were born February 17, 1922, in Hamden, Connecticut. He received a bachelor's degree in mathematics from the University of Minnesota in 1941, after just two years of study. He went on to graduate school at the University of Texas at Austin, where he studied under R. L. Moore. His graduate work was interrupted by World War II. Two days after the Japanese attack on Pearl Harbor, he enlisted in the United States Navy. During his term in the U. S. Navy, he served on the USS Rocky Mount. After returning from the war, he finished his doctoral work at the University of Texas and went on to teach mathematics at the University of Pennsylvania, where he went through the ranks of instructor, assistant professor, and associate professor (from 1951 to 1956). During this time he also spent two years (the academic years 1951–1952 and 1955–1956) at the Institute for Advanced Study in Princeton, New Jersey. He then accepted a post at Louisiana State University, where he became the university's first Boyd Professor of mathematics. Boyd Professor is Louisiana State University's highest professor rank.
Accomplishments
Served as vice president of the American Mathematical Society in 1972 and 1973.
Served as president of the Mathematical Association of America in 1981 and 1982.
Served as chair of the Council of Scientific Society Presidents in 1984.
Received the Award for Distinguished Service to Mathematics from the Mathematical Association of America in 1978.
Received the Bolzano Medal from the Czechoslovakian Academy of Sciences in 1981.
Invited lectures at conferences and colloquia in many places in the US and in cities of 21 other countries.
Invited Speaker at the ICM in 1970 in Nice
References
Further reading
Interview with Anderson and reminiscences from his colleagues.
1922 births
2008 deaths
People from Hamden, Connecticut
University of Texas at Austin College of Natural Sciences alumni
United States Navy personnel of World War II
United States Navy sailors
20th-century American mathematicians
21st-century American mathematicians
University of Minnesota College of Science and Engineering alumni
University of Pennsylvania faculty
Topologists
Presidents of the Mathematical Association of America
Mathematicians from Connecticut
American twins
Functional analysts |
https://en.wikipedia.org/wiki/Domain%20decomposition%20methods | In mathematics, numerical analysis, and numerical partial differential equations, domain decomposition methods solve a boundary value problem by splitting it into smaller boundary value problems on subdomains and iterating to coordinate the solution between adjacent subdomains. A coarse problem with one or few unknowns per subdomain is used to further coordinate the solution between the subdomains globally. The problems on the subdomains are independent, which makes domain decomposition methods suitable for parallel computing. Domain decomposition methods are typically used as preconditioners for Krylov space iterative methods, such as the conjugate gradient method, GMRES, and LOBPCG.
In overlapping domain decomposition methods, the subdomains overlap by more than the interface. Overlapping domain decomposition methods include the Schwarz alternating method and the additive Schwarz method. Many domain decomposition methods can be written and analyzed as a special case of the abstract additive Schwarz method.
In non-overlapping methods, the subdomains intersect only on their interface. In primal methods, such as Balancing domain decomposition and BDDC, the continuity of the solution across subdomain interface is enforced by representing the value of the solution on all neighboring subdomains by the same unknown. In dual methods, such as FETI, the continuity of the solution across the subdomain interface is enforced by Lagrange multipliers. The FETI-DP method is hybrid between a dual and a primal method.
Non-overlapping domain decomposition methods are also called iterative substructuring methods.
Mortar methods are discretization methods for partial differential equations, which use separate discretization on nonoverlapping subdomains. The meshes on the subdomains do not match on the interface, and the equality of the solution is enforced by Lagrange multipliers, judiciously chosen to preserve the accuracy of the solution. In the engineering practice in the finite element method, continuity of solutions between non-matching subdomains is implemented by multiple-point constraints.
Finite element simulations of moderate size models require solving linear systems with millions of unknowns. Several hours per time step is an average sequential run time, therefore, parallel computing is a necessity. Domain decomposition methods embody large potential for a parallelization of the finite element methods, and serve a basis for distributed, parallel computations.
Example 1: 1D Linear BVP
The exact solution is:
Subdivide the domain into two subdomains, one from and another from . In the left subdomain define the interpolating function and in the right define . At the interface between these two subdomains the following interface conditions shall be imposed:
Let the interpolating functions be defined as:
Where is the nth cardinal function of the chebyshev polynomials of the first kind with input argument y.
If N=4 then the following approximation |
https://en.wikipedia.org/wiki/PTD | PTD may refer to:
Law
Permanent total disability, in insurance law
Pre-trial diversion, in criminal justice
Protected trust deed, in Scottish bankruptcy law
Science, technology and mathematics
Pathfinder Technology Demonstrator, ongoing NASA missions to test miniaturized satellites
Peak–trough difference, of an oscillating curve
Poloidal–toroidal decomposition, in vector calculus
Pre-term delivery, a human birth under 37 weeks gestation
PTD-DBM, a synthetic peptide to reverse hair loss
Pre-Thread Data file, in Microsoft's .NET Framework
Other uses
Participatory technology development, in international agriculture programs
or the Dominican Workers' Party
Potsdam Municipal Airport, New York, US (FAA LID:PTD)
Post travel depression, a mood disorder
Prevention through design, in occupational health and safety |
https://en.wikipedia.org/wiki/Timeline%20of%20abelian%20varieties | This is a timeline of the theory of abelian varieties in algebraic geometry, including elliptic curves.
Early history
c. 1000 Al-Karaji writes on congruent numbers
Seventeenth century
Fermat studies descent for elliptic curves
1643 Fermat poses an elliptic curve Diophantine equation
1670 Fermat's son published his Diophantus with notes
Eighteenth century
1718 Giulio Carlo Fagnano dei Toschi, studies the rectification of the lemniscate, addition results for elliptic integrals.
1736 Leonhard Euler writes on the pendulum equation without the small-angle approximation.
1738 Euler writes on curves of genus 1 considered by Fermat and Frenicle
1750 Euler writes on elliptic integrals
23 December 1751 – 27 January 1752: Birth of the theory of elliptic functions, according to later remarks of Jacobi, as Euler writes on Fagnano's work.
1775 John Landen publishes Landen's transformation, an isogeny formula.
1786 Adrien-Marie Legendre begins to write on elliptic integrals
1797 Carl Friedrich Gauss discovers double periodicity of the lemniscate function
1799 Gauss finds the connection of the length of a lemniscate and a case of the arithmetic-geometric mean, giving a numerical method for a complete elliptic integral.
Nineteenth century
1826 Niels Henrik Abel, Abel-Jacobi map
1827 Inversion of elliptic integrals independently by Abel and Carl Gustav Jacob Jacobi
1829 Jacobi, Fundamenta nova theoriae functionum ellipticarum, introduces four theta functions of one variable
1835 Jacobi points out the use of the group law for diophantine geometry, in De usu Theoriae Integralium Ellipticorum et Integralium Abelianorum in Analysi Diophantea
1836-7 Friedrich Julius Richelot, the Richelot isogeny.
1847 Adolph Göpel gives the equation of the Kummer surface
1851 Johann Georg Rosenhain writes a prize essay on the inversion problem in genus 2.
c. 1850 Thomas Weddle - Weddle surface
1856 Weierstrass elliptic functions
1857 Bernhard Riemann lays the foundations for further work on abelian varieties in dimension > 1, introducing the Riemann bilinear relations and Riemann theta function.
1865 Carl Johannes Thomae, Theorie der ultraelliptischen Funktionen und Integrale erster und zweiter Ordnung
1866 Alfred Clebsch and Paul Gordan, Theorie der Abel'schen Functionen
1869 Karl Weierstrass proves an abelian function satisfies an algebraic addition theorem
1879, Charles Auguste Briot, Théorie des fonctions abéliennes
1880 In a letter to Richard Dedekind, Leopold Kronecker describes his Jugendtraum, to use complex multiplication theory to generate abelian extensions of imaginary quadratic fields
1884 Sofia Kovalevskaya writes on the reduction of abelian functions to elliptic functions
1888 Friedrich Schottky finds a non-trivial condition on the theta constants for curves of genus , launching the Schottky problem.
1891 Appell–Humbert theorem of Paul Émile Appell and Georges Humbert, classifies the holomorphic line bundles on an abelian surface by co |
https://en.wikipedia.org/wiki/CRR | CRR may refer to:
Capital Requirements Regulation, a European regulation on prudential requirements for credit institutions and investment firms
Coefficient of residuals resistance, (in Statistics) a random measurement on residuals in piecewise regression analysis
Convergence rate of residuals, (in Statistics) an alternative term with the same meanings as the coefficient of residuals resistance
Corrour railway station
Cross River Rail
Reserve requirement or cash reserve ratio
Binomial options pricing model or Cox Ross Rubinstein option pricing model
Clinchfield Railroad
Cat Righting Reflex, The intrinsic ability for cats to land on their feet by correcting their orientation while falling
Carolina Algonquian language (ISO 639-3 language code)
The Center For Reproductive Rights
The Current Run Rate (Cricket)
Curia Regis roll
Cross Region Replication, used to copy objects across Amazon S3 buckets in different AWS Regions
Cost Revenue Ratio, also known as efficiency ratio |
https://en.wikipedia.org/wiki/Giorgos%20Markopoulos | Giorgos Markopoulos (; born 1951) is a Greek poet. He read Economics and Statistics at the University of Piraeus (then Higher School for Industrial Studies).
He belongs to the so-called Genia tou 70, which is a literary term referring to Greek authors who began publishing their work during the 1970s, especially towards the end of the Greek military junta of 1967-1974 and at the first years of the Metapolitefsi.
He was awarded the State Prize for Poetry in 1999 for his collection, Μη σκεπάζεις το ποτάμι, which was also nominated for the European Union Prize in 2000. His work has been translated into English, French, Italian and Polish.
Selected poetry
Έβδομη Συμφωνία (Seventh Symphony), 1968
Η θλίψις του προαστίου (The Sadness of the Suburbs), 1976
Οι πυροτεχνουργοί (The Bomb Squad), 1979
Ποιήματα 1968–1987 (Poems 1968–1987), 1992
Mη σκεπάζεις το ποτάμι (Don't Cover the River), 1998
Notes
External links
His entry for the 2001 Frankfurt Book Fair (Greek)
His page at the website of the Hellenic Authors' Society (Greek)
His page at Kastaniotis publishers
1951 births
Living people
Modern Greek poets
20th-century Greek poets
People from Messini |
https://en.wikipedia.org/wiki/Von%20Neumann%27s%20inequality | In operator theory, von Neumann's inequality, due to John von Neumann, states that, for a fixed contraction T, the polynomial functional calculus map is itself a contraction.
Formal statement
For a contraction T acting on a Hilbert space and a polynomial p, then the norm of p(T) is bounded by the supremum of |p(z)| for z in the unit disk."
Proof
The inequality can be proved by considering the unitary dilation of T, for which the inequality is obvious.
Generalizations
This inequality is a specific case of Matsaev's conjecture. That is that for any polynomial P and contraction T on
where S is the right-shift operator. The von Neumann inequality proves it true for and for and it is true by straightforward calculation.
S.W. Drury has shown in 2011 that the conjecture fails in the general case.
References
Operator theory
Inequalities
John von Neumann |
https://en.wikipedia.org/wiki/Schwarz%20alternating%20method | In mathematics, the Schwarz alternating method or alternating process is an iterative method introduced in 1869–1870 by Hermann Schwarz in the theory of conformal mapping. Given two overlapping regions in the complex plane in each of which the Dirichlet problem could be solved, Schwarz described an iterative method for solving the Dirichlet problem in their union, provided their intersection was suitably well behaved. This was one of several constructive techniques of conformal mapping developed by Schwarz as a contribution to the problem of uniformization, posed by Riemann in the 1850s and first resolved rigorously by Koebe and Poincaré in 1907. It furnished a scheme for uniformizing the union of two regions knowing how to uniformize each of them separately, provided their intersection was topologically a disk or an annulus. From 1870 onwards Carl Neumann also contributed to this theory.
In the 1950s Schwarz's method was generalized in the theory of partial differential equations to an iterative method for finding the solution of an elliptic boundary value problem on a domain which is the union of two overlapping subdomains. It involves solving the boundary value problem on each of the two subdomains in turn, taking always the last values of the approximate solution as the next boundary conditions. It is used in numerical analysis, under the name multiplicative Schwarz method (in opposition to additive Schwarz method) as a domain decomposition method.
History
It was first formulated by H. A. Schwarz and served as a theoretical tool: its convergence for general second order elliptic partial differential equations was first proved much later, in 1951, by Solomon Mikhlin.
The algorithm
The original problem considered by Schwarz was a Dirichlet problem (with the Laplace's equation) on a domain consisting of a circle and a partially overlapping square. To solve the Dirichlet problem on one of the two subdomains (the square or the circle), the value of the solution must be known on the border: since a part of the border is contained in the other subdomain, the Dirichlet problem must be solved jointly on the two subdomains. An iterative algorithm is introduced:
Make a first guess of the solution on the circle's boundary part that is contained in the square
Solve the Dirichlet problem on the circle
Use the solution in (2) to approximate the solution on the square's boundary
Solve the Dirichlet problem on the square
Use the solution in (4) to approximate the solution on the circle's boundary, then go to step (2).
At convergence, the solution on the overlap is the same when computed on the square or on the circle.
Optimized Schwarz methods
The convergence speed depends on the size of the overlap between the subdomains, and on the transmission conditions (boundary conditions used in the interface between the subdomains). It is possible to increase the convergence speed of the Schwarz methods by choosing adapted transmission conditions: theses |
https://en.wikipedia.org/wiki/Balancing%20domain%20decomposition%20method | In numerical analysis, the balancing domain decomposition method (BDD) is an iterative method to find the solution of a symmetric positive definite system of linear algebraic equations arising from the finite element method. In each iteration, it combines the solution of local problems on non-overlapping subdomains with a coarse problem created from the subdomain nullspaces. BDD requires only solution of subdomain problems rather than access to the matrices of those problems, so it is applicable to situations where only the solution operators are available, such as in oil reservoir simulation by mixed finite elements. In its original formulation, BDD performs well only for 2nd order problems, such elasticity in 2D and 3D. For 4th order problems, such as plate bending, it needs to be modified by adding to the coarse problem special basis functions that enforce continuity of the solution at subdomain corners, which makes it however more expensive. The BDDC method uses the same corner basis functions as, but in an additive rather than multiplicative fashion. The dual counterpart to BDD is FETI, which enforces the equality of the solution between the subdomain by Lagrange multipliers. The base versions of BDD and FETI are not mathematically equivalent, though a special version of FETI designed to be robust for hard problems has the same eigenvalues and thus essentially the same performance as BDD.
The operator of the system solved by BDD is the same as obtained by eliminating the unknowns in the interiors of the subdomain, thus reducing the problem to the Schur complement on the subdomain interface. Since the BDD preconditioner involves the solution of Neumann problems on all subdomain, it is a member of the Neumann–Neumann class of methods, so named because they solve a Neumann problem on both sides of the interface between subdomains.
In the simplest case, the coarse space of BDD consists of functions constant on each subdomain and averaged on the interfaces. More generally, on each subdomain, the coarse space needs to only contain the nullspace of the problem as a subspace.
References
External links
BDD reference implementation at mgnet.org
Domain Decomposition – Theory, publications, methods, algorithms.
Domain decomposition methods |
https://en.wikipedia.org/wiki/Neumann%E2%80%93Neumann%20methods | In mathematics, Neumann–Neumann methods are domain decomposition preconditioners named so because they solve a Neumann problem on each subdomain on both sides of the interface between the subdomains. Just like all domain decomposition methods, so that the number of iterations does not grow with the number of subdomains, Neumann–Neumann methods require the solution of a coarse problem to provide global communication. The balancing domain decomposition is a Neumann–Neumann method with a special kind of coarse problem.
More specifically, consider a domain Ω, on which we wish to solve the Poisson equation
for some function f. Split the domain into two non-overlapping subdomains Ω1 and Ω2 with common boundary Γ and let u1 and u2 be the values of u in each subdomain. At the interface between the two subdomains, the two solutions must satisfy the matching conditions
where is the unit normal vector to Γ in each subdomain.
An iterative method with iterations k=0,1,... for the approximation of each ui (i=1,2) that satisfies the matching conditions is to first solve the Dirichlet problems
for some function λ(k) on Γ, where λ(0) is any inexpensive initial guess. We then solve the two Neumann problems
We then obtain the next iterate by setting
for some parameters ω, θ1 and θ2.
This procedure can be viewed as a Richardson iteration for the iterative solution of the equations arising from the Schur complement method.
This continuous iteration can be discretized by the finite element method and then solved—in parallel—on a computer. The extension to more subdomains is straightforward, but using this method as stated as a preconditioner for the Schur complement system is not scalable with the number of subdomains; hence the need for a global coarse solve.
See also
Neumann–Dirichlet method
References
Domain decomposition methods |
https://en.wikipedia.org/wiki/Neumann%E2%80%93Dirichlet%20method | In mathematics, the Neumann–Dirichlet method is a domain decomposition preconditioner which involves solving Neumann boundary value problem on one subdomain and Dirichlet boundary value problem on another, adjacent across the interface between the subdomains. On a problem with many subdomains organized in a rectangular mesh, the subdomains are assigned Neumann or Dirichlet problems in a checkerboard fashion.
See also
Neumann–Neumann method
References
Domain decomposition methods |
https://en.wikipedia.org/wiki/John%20Frith%20%28rugby%20league%29 | John Frith (born 24 April 1985 in Roma, Queensland) is an Australian professional rugby league footballer for the North Queensland Cowboys in the National Rugby League (NRL) competition.
Statistics
Club career
References
External links
Player Details at cowboys.com.au
1985 births
Living people
Australian rugby league players
North Queensland Cowboys players
People from Roma, Queensland
Rugby league players from Queensland
Rugby league props |
https://en.wikipedia.org/wiki/Formalism%20%28philosophy%20of%20mathematics%29 | In the philosophy of mathematics, formalism is the view that holds that statements of mathematics and logic can be considered to be statements about the consequences of the manipulation of strings (alphanumeric sequences of symbols, usually as equations) using established manipulation rules. A central idea of formalism "is that mathematics is not a body of propositions representing an abstract sector of reality, but is much more akin to a game, bringing with it no more commitment to an ontology of objects or properties than ludo or chess." According to formalism, the truths expressed in logic and mathematics are not about numbers, sets, or triangles or any other coextensive subject matter — in fact, they aren't "about" anything at all. Rather, mathematical statements are syntactic forms whose shapes and locations have no meaning unless they are given an interpretation (or semantics). In contrast to mathematical realism, logicism, or intuitionism, formalism's contours are less defined due to broad approaches that can be categorized as formalist.
Along with realism and intuitionism, formalism is one of the main theories in the philosophy of mathematics that developed in the late nineteenth and early twentieth century. Among formalists, David Hilbert was the most prominent advocate.
Early formalism
The early mathematical formalists attempted "to block, avoid, or sidestep (in some way) any ontological commitment to a problematic realm of abstract objects." German mathematicians Eduard Heine and Carl Johannes Thomae are considered early advocates of mathematical formalism. Heine and Thomae's formalism can be found in Gottlob Frege's criticisms in The Foundations of Arithmetic.
According to Alan Weir, the formalism of Heine and Thomae that Frege attacks can be "describe[d] as term formalism or game formalism." Term formalism is the view that mathematical expressions refer to symbols, not numbers. Heine expressed this view as follows: "When it comes to definition, I take a purely formal position, in that I call certain tangible signs numbers, so that the existence of these numbers is not in question."
Thomae is characterized as a game formalist who claimed that "[f]or the formalist, arithmetic is a game with signs which are called empty. That means that they have no other content (in the calculating game) than they are assigned by their behaviour with respect to certain rules of combination (rules of the game)."
Frege provides three criticisms of Heine and Thomae's formalism: "that [formalism] cannot account for the application of mathematics; that it confuses formal theory with metatheory; [and] that it can give no coherent explanation of the concept of an infinite sequence." Frege's criticism of Heine's formalism is that his formalism cannot account for infinite sequences. Dummett argues that more developed accounts of formalism than Heine's account could avoid Frege's objections by claiming they are concerned with abstract symbols rather t |
https://en.wikipedia.org/wiki/Manolis%20Xexakis | Manolis Xexakis () (born 1948 in Rethymnon, Crete) is a Greek poet and prose writer. He studied physics and mathematics at the University of Thessaloniki. He has worked as a journalist, teacher, and also in advertising.
Poetry
Ασκήσεις Μαθηματικών (Math Exercises), 1980
Πλόες ερωτικοί (Erotic Sea Ways), 1980
Κάτοπτρα μελαγχολικού λόγου (Mirrors of Melancholic Word), 1987
Prose
Ο θάνατος του ιππικού (The Death of the Cavalry), 1977
Πού κούκος; Πού άνεμος; (Where the Cuckoo? Where the Wind?), 1987
Σονάτα κομπολογιών (A String of Beads Sonata), 2000
External links
His page at the website of the Hellenic Authors' Society (Greek)
Notes
Cretan poets
1948 births
Living people
People from Rethymno |
https://en.wikipedia.org/wiki/Timeline%20of%20ancient%20Greek%20mathematicians | This is a timeline of mathematicians in ancient Greece.
Timeline
Historians traditionally place the beginning of Greek mathematics proper to the age of Thales of Miletus (ca. 624–548 BC), which is indicated by the at 600 BC. The at 300 BC indicates the approximate year in which Euclid's Elements was first published. The at 300 AD passes through Pappus of Alexandria (), who was one of the last great Greek mathematicians of late antiquity. Note that the solid thick is at year zero, which is a year that does not exist in the Anno Domini (AD) calendar year system
The mathematician Heliodorus of Larissa is not listed due to the uncertainty of when he lived, which was possibly during the 3rd century AD, after Ptolemy.
Overview of the most important mathematicians and discoveries
Of these mathematicians, those whose work stands out include:
Thales of Miletus () is the first known individual to use deductive reasoning applied to geometry, by deriving four corollaries to Thales' theorem. He is the first known individual to whom a mathematical discovery has been attributed.
Pythagoras () was credited with many mathematical and scientific discoveries, including the Pythagorean theorem, Pythagorean tuning, the five regular solids, the Theory of Proportions, the sphericity of the Earth, and the identity of the morning and evening stars as the planet Venus.
Theaetetus () Proved that there are exactly five regular convex polyhedra (it is emphasized that it was, in particular, proved that there does not exist any regular convex polyhedra other than these five). This fact led these five solids, now called the Platonic solids, to play a prominent role in the philosophy of Plato (and consequently, also influenced later Western Philosophy) who associated each of the four classical elements with a regular solid: earth with the cube, air with the octahedron, water with the icosahedron, and fire with the tetrahedron (of the fifth Platonic solid, the dodecahedron, Plato obscurely remarked, "...the god used [it] for arranging the constellations on the whole heaven"). The last book (Book XIII) of the Euclid's Elements, which is probably derived from the work of Theaetetus, is devoted to constructing the Platonic solids and describing their properties; Andreas Speiser has advocated the view that the construction of the 5 regular solids is the chief goal of the deductive system canonized in the Elements. Astronomer Johannes Kepler proposed a model of the Solar System in which the five solids were set inside one another and separated by a series of inscribed and circumscribed spheres.
Eudoxus of Cnidus () is considered by some to be the greatest of classical Greek mathematicians, and in all antiquity second only to Archimedes. Book V of Euclid's Elements is though to be largely due to Eudoxus.
Aristarchus of Samos () presented the first known heliocentric model that placed the Sun at the center of the known universe with the Earth revolving around it. |
https://en.wikipedia.org/wiki/Biruaca%20Municipality | The Biruaca Municipality is one of the seven municipalities (municipios) that makes up the Venezuelan state of Apure and, according to the 2011 census by the National Institute of Statistics of Venezuela, the municipality has a population of 54,323. The town of Biruaca is the municipal seat of the Biruaca Municipality.
Demographics
The Biruaca Municipality, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, has a population of 54,233 (up from 45,180 in 2000). This amounts to 11.4% of the state's population. The municipality's population density is .
Government
The mayor of the Biruaca Municipality is Daniel Blanco, re-elected November 23, 2008 with 38% of the vote. The municipality contains one parish; Urbana Biruaca.
Climate
Minimum monthly temperature in the Biruaca municipality lies between 21.2 and 23.9 °C, while maximum temperature is between 29.2 and 35.3 °C. Total annual precipitation fluctuates between 1336 and 1820 mm. Most rain falls between June and August, while the driest quarter comprises January to March.
References
External links
biruaca-apure.gob.ve
Municipalities of Apure |
https://en.wikipedia.org/wiki/Skew%20normal%20distribution | In probability theory and statistics, the skew normal distribution is a continuous probability distribution that generalises the normal distribution to allow for non-zero skewness.
Definition
Let denote the standard normal probability density function
with the cumulative distribution function given by
,
where "erf" is the error function. Then the probability density function (pdf) of the skew-normal distribution with parameter is given by
This distribution was first introduced by O'Hagan and Leonard (1976). Alternative forms to this distribution, with the corresponding quantile function, have been given by Ashour and Abdel-Hamid and by Mudholkar and Hutson.
A stochastic process that underpins the distribution was described by Andel, Netuka and Zvara (1984). Both the distribution and its stochastic process underpinnings were consequences of the symmetry argument developed in Chan and Tong (1986), which applies to multivariate cases beyond normality, e.g. skew multivariate t distribution and others. The distribution is a particular case of a general class of distributions with probability density functions of the form where is any PDF symmetric about zero and is any CDF whose PDF is symmetric about zero.
To add location and scale parameters to this, one makes the usual transform . One can verify that the normal distribution is recovered when , and that the absolute value of the skewness increases as the absolute value of increases. The distribution is right skewed if and is left skewed if . The probability density function with location , scale , and parameter becomes
The skewness () of the distribution is limited to slightly less than the interval .
As has been shown, the mode (maximum) of the distribution is unique. For general there is no analytic expression for , but a quite accurate (numerical) approximation is:
where and
Estimation
Maximum likelihood estimates for , , and can be computed numerically, but no closed-form expression for the estimates is available unless . In contrast, the method of moments has a closed-form expression since the skewness equation can be inverted with
where and the sign of is the same as the sign of . Consequently, , , and where and are the mean and standard deviation. As long as the sample skewness is not too large, these formulas provide method of moments estimates , , and based on a sample's , , and .
The maximum (theoretical) skewness is obtained by setting in the skewness equation, giving . However it is possible that the sample skewness is larger, and then cannot be determined from these equations. When using the method of moments in an automatic fashion, for example to give starting values for maximum likelihood iteration, one should therefore let (for example) .
Concern has been expressed about the impact of skew normal methods on the reliability of inferences based upon them.
Related distributions
The exponentially modified normal distribution is another 3-para |
https://en.wikipedia.org/wiki/Mu%C3%B1oz%20Municipality | The Muñoz Municipality is one of the seven municipalities (municipios) that makes up the Venezuelan state of Apure and, according to the 2011 census by the National Institute of Statistics of Venezuela, the municipality has a population of 27,542. The town of Bruzual is the municipal seat of the Muñoz Municipality. The municipality is named for Major General , leader of a division of independence forces during the Venezuelan War of Independence.
Demographics
The Muñoz Municipality, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, has a population of 30,197 (up from 26,141 in 2000). This amounts to 6.4% of the state's population. The municipality's population density is .
Government
The mayor of the Muñoz Municipality is Ramón de Jesús Bona Arraíz, re-elected November 23, 2008 with 58% of the vote. The municipality is divided into five parishes; Urbana Bruzual, Mantecal, Quintero, Rincón Hondo, and San Vicente.
References
External links
munoz-apure.gob.ve
Municipalities of Apure |
https://en.wikipedia.org/wiki/P%C3%A1ez%20Municipality%2C%20Apure | Páez is one of the seven municipalities (municipios) that makes up the Venezuelan state of Apure and, according to the 2011 census by the National Institute of Statistics of Venezuela, the municipality has a population of 100,125. The town of Guasdualito is the municipal seat of the Páez Municipality.
Name
The municipality is one of several in Venezuela named "Páez Municipality", for independence hero José Antonio Páez.
Demographics
The Páez Municipality, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, has a population of 111,059 (up from 93,010 in 2000). This amounts to 23.4% of the state's population. The municipality's population density is .
Government
The mayor of the Páez Municipality is José del Carmen Alvarado, re-elected November 23, 2008 with 76% of the vote. The municipality is divided into five parishes; Urbana Guasdualito, Aramendi, El Amparo, San Camilo, and Urdaneta.
References
Municipalities of Apure |
https://en.wikipedia.org/wiki/Pedro%20Camejo%20Municipality | The Pedro Camejo Municipality is one of the seven municipalities (municipios) that makes up the Venezuelan state of Apure and, according to the 2011 census by the National Institute of Statistics of Venezuela, the municipality has a population of 28,966. The town of San Juan de Payara is the municipal seat of the Pedro Camejo Municipality. The municipality is named after Afro-Venezuelan independence hero Pedro Camejo.
Demographics
The Pedro Camejo Municipality, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, has a population of 31,187 (up from 26,407 in 2000). This amounts to 6.5% of the state's population. The municipality's population density is .
Government
The mayor of the Pedro Camejo Municipality is Pedro Danilo Leal, re-elected November 23, 2008 with 56% of the vote. The municipality is divided into three parishes; Urbana San Juan de Payara, Codazzi, and Cunaviche.
References
External links
pedrocamejo-apure.gob.ve
Municipalities of Apure |
https://en.wikipedia.org/wiki/R%C3%B3mulo%20Gallegos%20Municipality%2C%20Apure | The Rómulo Gallegos Municipality is one of the seven municipalities (municipios) that makes up the Venezuelan state of Apure and, according to the 2011 census by the National Institute of Statistics of Venezuela, the municipality has a population of 24,418. The town of Elorza is the shire town of the Rómulo Gallegos Municipality.
Name
The municipality is one of a number in Venezuela named "Rómulo Gallegos Municipality", in honour of the writer Rómulo Gallegos.
Demographics
The Rómulo Gallegos Municipality, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, has a population of 23,839 (up from 20,331 in 2000). This amounts to 5% of the state's population. The municipality's population density is .
Government
The mayor of the Rómulo Gallegos Municipality is Jesus Leopoldo Estrada Moreno, re-elected November 23, 2008 with 48% of the vote. The municipality is divided into two parishes; Urbana Elorza and La Trinidad.
References
External links
romulogallegos-apure.gob.ve
Municipalities of Apure |
https://en.wikipedia.org/wiki/Abstract%20additive%20Schwarz%20method | In mathematics, the abstract additive Schwarz method, named after Hermann Schwarz, is an abstract version of the additive Schwarz method for boundary value problems on partial differential equations, formulated only in terms of linear algebra without reference to domains, subdomains, etc. Many if not all domain decomposition methods can be cast as abstract additive Schwarz method, which is often the first and most convenient approach to their analysis.
References
Domain decomposition methods |
https://en.wikipedia.org/wiki/Cox%E2%80%93Zucker%20machine | In arithmetic geometry, the Cox–Zucker machine is an algorithm created by David A. Cox and Steven Zucker. This algorithm determines whether a given set of sections provides a basis (up to torsion) for the Mordell–Weil group of an elliptic surface E → S, where S is isomorphic to the projective line.
The algorithm was first published in the 1979 article "Intersection numbers of sections of elliptic surfaces" by Cox and Zucker and was later named the "Cox–Zucker machine" by Charles Schwartz in 1984.
Name origin
The name sounds similar to the obscenity "". This was a deliberate choice by Cox and Zucker, who conceived of the idea of coauthoring a paper in 1970, while first-year graduate students at Princeton University, for the express purpose of enabling this joke. They followed through on it five years later, as members of the faculty at Rutgers, the State University of New Jersey.
As Cox explained in a memorial tribute to Zucker in Notices of the American Mathematical Society in 2021: "A few weeks after we met, we realized that we had to write a joint paper because the combination of our last names, in the usual alphabetical order, is remarkably obscene."
See also
Cox ring
References
Complex manifolds
Birational geometry
Algebraic surfaces
Mathematical humor
English profanity |
https://en.wikipedia.org/wiki/Pseudoscientific%20metrology | Some approaches in the branch of historic metrology are highly speculative and can be qualified as pseudoscience.
Origins
In 1637, John Greaves, professor of geometry at Gresham College, made his first of several studies in Egypt and Italy, making numerous measurements of buildings and monuments, including the Great Pyramid. These activities fuelled many centuries of interest in metrology of the ancient cultures by the likes of Isaac Newton and the French Academy.
Charles Piazzi Smyth
John Taylor, in his 1859 book The Great Pyramid: Why Was It Built? & Who Built It?, claimed that the Great Pyramid was planned and the building supervised by the biblical Noah, and that it was "built to make a record of the measure of the Earth".
A paper presented to the Royal Academy on the topic was rejected.
Taylor's theories were, however, the inspiration for the deeply religious archaeologist Charles Piazzi Smyth to go to Egypt to study and measure the pyramid, subsequently publishing his book Our Inheritance in the Great Pyramid (1864), claiming that the measurements he obtained from the Great Pyramid of Giza indicated a unit of length, the pyramid inch, equivalent to 1.001 British inches, that could have been the standard of measurement by the pyramid's architects. From this he extrapolated a number of other measurements, including the pyramid pint, the sacred cubit, and the pyramid scale of temperature.
Smyth claimed—and presumably believed—that the inch was a God-given measure handed down through the centuries from the 'Time of Israel', and that the architects of the pyramid could only have been directed by the hand of God. To support this Smyth said that, in measuring the pyramid, he found the number of inches in the perimeter of the base equalled 1000 times the number of days in a year, and found a numeric relationship between the height of the pyramid in inches to the distance from Earth to the Sun, measured in statute miles.
Smyth used this as an argument against the introduction of the metre in Britain, which he considered a product of the minds of atheistic French radicals.
The grand scheme
By the time measurements of Mesopotamia were discovered, by doing various exercises of mathematics on the definitions of the major ancient measurement systems, various people (Jean-Adolphe Decourdemanche in 1909, August Oxé in 1942) came to the conclusion that the relationship between them was well planned.
Livio C. Stecchini claims in his A History of Measures:
The relation among the units of length can be explained by the ratio 15:16:17:18 among the four fundamental feet and cubits. Before I arrived at this discovery, Decourdemanche and Oxé discovered that the cubes of those units are related according to the conventional specific gravities of oil, water, wheat and barley.
Stecchini makes claims that imply that the Egyptian measures of length, originating from at least the 3rd millennium BC, were directly derived from the circumference of the earth with |
https://en.wikipedia.org/wiki/Francisco%20de%20Miranda%20Municipality%2C%20Anzo%C3%A1tegui | Francisco de Miranda is one of the 21 municipalities (municipios) that makes up the eastern Venezuelan state of Anzoátegui and, according to the 2011 census by the National Institute of Statistics of Venezuela, the municipality has a population of 43,173. The town of Pariaguán is the shire town of the Francisco de Miranda Municipality.
Name
The municipality is one of several in Venezuela named "Francisco de Miranda Municipality" in honour of Venezuelan independence hero Francisco de Miranda.
Demographics
The Francisco de Miranda Municipality, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, has a population of 42,357 (up from 36,970 in 2000). This amounts to 2.9% of the state's population. The municipality's population density is .
Government
Tomás Bello, elected on 23 November 2008 with 62% of the vote. He replaced Lorenzo Emilio Rondon shortly after the elections. The municipality is divided into five parishes; Capital Francisco de Miranda, Atapirire, Boca del Pao, El Pao, and Múcura (created 27 June 1995).
See also
Miranda (disambiguation)
References
External links
franciscodemiranda-anzoategui.gob.ve
Municipalities of Anzoategui |
https://en.wikipedia.org/wiki/Bjorn%20Poonen | Bjorn Mikhail Poonen (born July 27, 1968, in Boston, Massachusetts) is a mathematician, four-time Putnam Competition winner, and a Distinguished Professor in Science in the Department of Mathematics at the Massachusetts Institute of Technology.
His research is primarily in arithmetic geometry, but he has occasionally published in other subjects such as probability and computer science.
He has edited two books.
He is the founding managing editor of the journal Algebra & Number Theory, and serves also on the editorial boards of Involve: A Journal of Mathematics and the A K Peters Research Notes in Mathematics book series.
Education
Poonen is a 1985 alumnus of Winchester High School in Winchester, Massachusetts. In 1989, Poonen graduated from Harvard University with an A.B. in Mathematics and Physics, summa cum laude. He then studied under Kenneth Alan Ribet at the University of California, Berkeley, completing a PhD there in 1994.
Academic positions
Poonen held postdoctoral positions at Mathematical Sciences Research Institute and Princeton University and served on the faculty of the University of California, Berkeley from 1997 to 2008, before moving to MIT.
He has also held visiting positions at the Isaac Newton Institute (1998 and 2005), the Université Paris-Sud (2001), Harvard (2007), and MIT (2007).
Major honors and awards
Joseph L. Doob Prize, 2023
Fellow of the American Mathematical Society, 2012.
American Academy of Arts and Sciences: elected in 2012
Chauvenet Prize: the 2011 winner, for his article "Undecidability in number theory"
Miller Research Professorship – University of California Berkeley.
David and Lucile Packard Fellowship
Sloan Research Fellowship
William Lowell Putnam Mathematical Competition: winner in 1985, 1986, 1987, and 1988 (the only other four-time winners since 1938 are Don Coppersmith, Arthur Rubin, Ravi D. Vakil, Gabriel Carroll, Reid W. Barton, Daniel Kane and Brian R. Lawrence).
International Mathematical Olympiad: silver medalist in 1985.
American High School Mathematics Examination: only participant (out of 380,000) to receive a perfect score in 1985.
Trivia
Poonen co-authored a paper entitled "How to spread rumors fast".
References
External links
Personal webpage
1968 births
Living people
20th-century American mathematicians
21st-century American mathematicians
Harvard University alumni
University of California, Berkeley alumni
Academic staff of Paris-Sud University
Putnam Fellows
People from Winchester, Massachusetts
Fellows of the American Academy of Arts and Sciences
Fellows of the American Mathematical Society
International Mathematical Olympiad participants
Sloan Research Fellows
Simons Investigator
Winchester High School (Massachusetts) alumni
Number theorists
Arithmetic geometers
American people of Ojibwe descent |
https://en.wikipedia.org/wiki/Chris%20Heyde | Christopher Charles Heyde AM (20 April 1939, in Sydney – 6 March 2008, in Canberra) was a prominent Australian statistician who did leading research in probability, stochastic processes and statistics.
Heyde was a professor at Columbia University, the University of Melbourne, CSIRO, University of Manchester, University of Sheffield, Michigan State University, and The Australian National University, Canberra.
In 2008, Heyde died of metastatic melanoma.
Honours
1972 – Member of the International Statistical Institute
1973 – Fellow of the Institute of Mathematical Statistics
1977 – Fellow of the Australian Academy of Science (FAA)
1981 – Honorary Life Member of the Statistical Society of Australia Inc. (SSAI)
1988 – Awarded the Statistical Society of Australia's Pitman Medal and served as President of the Society
1994 – Shared the Australian Academy of Science's Hannan Medal with Peter Hall.
1995 – Thomas Ranken Lyle Medal of the Australian Academy of Science.
1998 – D.Sc. honoris causa, University of Sydney
2003 – Fellow of the Academy of the Social Sciences in Australia (FASSA)
2003 – Appointed Member of the Order of Australia (AM) "for service to mathematics, particularly through research into statistics and probability, and to the advancement of learning in these disciplines".
Offices held
Vice President of the International Statistical Institute
President of the Bernoulli Society
President of the Statistical Society of Australia (1979–1981)
Vice President of the Australian Mathematical Society
Editor of the Australian Journal of Statistics
Editor of Stochastic Processes and Their Applications (1983–1989)
Editor-in-chief of Journal of Applied Probability (1990–2008)
Editor-in-chief of Advances in Applied Probability (1990–2008).
References
External links
Memoir full transcription from Historical Records of Australian Science, vol.20, no.1, 2009
1939 births
2008 deaths
Probability theorists
Australian statisticians
Academic staff of the Australian National University
Members of the Order of Australia
Fellows of the Australian Academy of Science
Fellows of the Academy of the Social Sciences in Australia
People educated at Barker College
Michigan State University faculty
Academics of the University of Sheffield
Mathematical statisticians
People from Sydney |
https://en.wikipedia.org/wiki/Hjelmslev%27s%20theorem | In geometry, Hjelmslev's theorem, named after Johannes Hjelmslev, is the statement that if points P, Q, R... on a line are isometrically mapped to points P´, Q´, R´... of another line in the same plane, then the midpoints of the segments PP´, QQ´, RR´... also lie on a line.
The proof is easy if one assumes the classification of plane isometries. If the given isometry is odd, in which case it is necessarily either a reflection in a line or a glide-reflection (the product of three reflections in a line and two perpendiculars to it), then the statement is true of any points in the plane whatsoever: the midpoint of PP´ lies upon the axis of the (glide-)reflection for any P. If the isometry is even, compose it with reflection in line PQR to obtain an odd isometry with the same effect on P, Q, R... and apply the previous remark.
The importance of the theorem lies in the fact that it has a different proof that does not presuppose the parallel postulate and is therefore valid in non-Euclidean geometry as well. By its help, the mapping that maps every point P of the plane to the midpoint of the segment P´P´´, where P´and P´´ are the images of P under a rotation (in either sense) by a given acute angle about a given center, is seen to be a collineation mapping the whole hyperbolic plane in a 1-1 way onto the inside of a disk, thus providing a good intuitive notion of the linear structure of the hyperbolic plane. In fact, this is called the Hjelmslev transformation.
References
.
External links
Hjelmslev's Theorem by Jay Warendorff, the Wolfram Demonstrations Project.
Hjelmslev's Theorem from cut-the-knot
Theorems in plane geometry |
https://en.wikipedia.org/wiki/SCORUS | An acronym for "Standing Committee of Regional and Urban Statistics", SCORUS is a sub-committee of the International Association for Official Statistics (IAOS) which is a section of the International Statistical Institute. The sub-committee has specific responsibility for regional and urban statistics and research. Its members form a dedicated international network of persons interested in regional and urban statistical issues.
External links
http://www.scorus.org
References
Statistical organizations
Official statistics |
https://en.wikipedia.org/wiki/Girardot%20Municipality%2C%20Cojedes | The Girardot Municipality is one of the nine municipalities (municipios) that makes up the Venezuelan state of Cojedes and, according to the 2011 census by the National Institute of Statistics of Venezuela, the municipality has a population of 11,906. The town of El Baúl is the municipal seat of the Girardot Municipality.
Demographics
The Girardot Municipality, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, has a population of 11,579 (up from 10,387 in 2000). This amounts to 3.8% of the state's population. The municipality's population density is .
Government
The mayor of the Girardot Municipality is Gustavo Ramón Guillén Brizuela, re-elected on October 31, 2004, with 30% of the vote. The municipality is divided into two parishes; El Baúl and Sucre.
References
Municipalities of Cojedes (state) |
https://en.wikipedia.org/wiki/Lima%20Blanco%20Municipality | The Lima Blanco Municipality is one of the nine municipalities (municipios) that make up the Venezuelan state of Cojedes and, according to the 2011 census by the National Institute of Statistics of Venezuela, the municipality has a population of 9,454. The town of Macapo is the shire town of the Lima Blanco Municipality.
Demographics
The Lima Blanco Municipality, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, had a population of 9,161 (up from 7,965 in 2000). This amounted to 3.1% of the state's population. The municipality's population density is .
Government
The mayor of the Lima Blanco Municipality is José Orangel Sánchez Fernández, elected on October 31, 2004, with 36% of the vote. He replaced Moraima Machado shortly after the elections. The municipality is divided into two parishes; Macapo and La Aguadita (previous to December 30, 1994, the Lima Blanco Municipality contained only a single parish).
References
Municipalities of Cojedes (state) |
https://en.wikipedia.org/wiki/Ricaurte%20Municipality | The Ricaurte Municipality is one of the nine municipalities (municipios) that makes up the Venezuelan state of Cojedes and, according to the 2011 census by the National Institute of Statistics of Venezuela, the municipality has a population of 12,657. The town of Libertad is the municipal seat of the Ricaurte Municipality. The municipality is named for Venezuelan independence hero Antonio Ricaurte.
Demographics
The Ricaurte Municipality, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, has a population of 12,340 (up from 10,883 in 2000). This amounts to 4.1% of the state's population. The municipality's population density is .
Government
The mayor of the Ricaurte Municipality is Ramón A Peralta Ruíz, elected on October 31, 2004, with 54% of the vote. He replaced Violeta Montoya shortly after the elections. The municipality is divided into two parishes; Libertad de Cojedes and El Amparo.
References
Municipalities of Cojedes (state) |
https://en.wikipedia.org/wiki/B%C3%A1lint%20T%C3%B3th | Bálint Tόth (born 1955, Cluj/Kolozsvár/Klausenburg) is a Hungarian mathematician whose work concerns probability theory, stochastic process and probabilistic aspects of mathematical physics. He obtained PhD in 1988 from the Hungarian Academy of Sciences, worked as senior researcher at the Institute of Mathematics of the HAS and as professor of mathematics at TU Budapest. He holds the Chair of Probability at the University of Bristol and is a research professor at the Alfréd Rényi Institute of Mathematics, Budapest.
He has worked on microscopic models of Brownian motion, quantum spin systems, limit theorems for random walks with long memory, and non-conventional stochastic processes, hydrodynamic limits, etc. In particular, Tόth contributed to the theory of self-interacting motions, that is, motions that are "reinforced", "self-avoiding" or "self-repellent". In collaboration with Wendelin Werner he constructed the random geometric object later called the Brownian web.
Tóth was an invited speaker of the International Congress of Mathematicians ICM-2018 (Rio de Janeiro), and of the European Congress of Mathematics ECM-2000 (Barcelona). He was plenary speaker at three Conferences on Stochastic Processes and their Applications SPA-2005 (Santa Barbara), SPA-2014 (Buenos Aires), and IMS Medallion Lecturer at SPA-2022 (Wuhan).
He is a member of Academia Europaea and a corresponding member of the Hungarian Academy of Sciences.
Tóth was Editor-in-Chief of the journals Electronic Journal of Probability (2009-2011) and Annals of Applied Probability (2016-2018). Currently he is co-Editor-in-Chief (jointly with Fabio Toninelli) of Probability Theory and Related Fields.
References
External links
Arxiv papers
Personal homepage
20th-century Hungarian mathematicians
21st-century Hungarian mathematicians
Living people
1955 births
Probability Theory and Related Fields editors |
https://en.wikipedia.org/wiki/R%C3%B3mulo%20Gallegos%20Municipality%2C%20Cojedes | The Rómulo Gallegos Municipality is one of the nine municipalities (municipios) that makes up the Venezuelan state of Cojedes and, according to the 2011 census by the National Institute of Statistics of Venezuela, the municipality has a population of 18,297. The town of Las Vegas is the municipal seat of the Rómulo Gallegos Municipality. The municipality is one of a number in Venezuela named "Rómulo Gallegos Municipality", in honour of the writer Rómulo Gallegos.
Demographics
The Rómulo Gallegos Municipality, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, has a population of 15,141 (up from 12,815 in 2000). This amounts to 5% of the state's population. The municipality's population density is .
Government
The mayor of the Rómulo Gallegos Municipality is Jesus Francisco Oviedo Guerra, elected on October 31, 2004, with 51% of the vote. He replaced Alberto Molina shortly after the elections. The municipality is divided into one parish; Rómulo Gallegos.
References
Municipalities of Cojedes (state) |
https://en.wikipedia.org/wiki/Ezequiel%20Zamora%20Municipality%2C%20Cojedes | The San Carlos Municipality is one of the nine municipalities (municipios) that makes up the Venezuelan state of Cojedes and, according to the 2011 census by the National Institute of Statistics of Venezuela, the municipality has a population of 106,760. The city of San Carlos is the municipal seat of the San Carlos Municipality.
History
Father Capuchino Fray Pedro de Berja founded the city of San Carlos de Austria on April 27, 1678.
Demographics
The San Carlos Municipality, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, has a population of 97,860 (up from 85,402 in 2000). This amounts to 32.6% of the state's population. The municipality's population density is .
Government
The mayor of the San Carlos Municipality is José Jesús Betancourt Sanoja, re-elected on October 31, 2004, with 52% of the vote. The municipality is divided into three parishes; “San Carlos de Austria, Juan Angel Bravo, and Manuel Manrique.
Sites of interest
Religious buildings
Catedral de San Carlos
Iglesia Santo Domingo
Iglesia San Juan
Squares and parks
Plaza Bolívar
References
External links
Information on the municipalities of Cojedes
Flag & Coat of Arms of San Carlos
Municipalities of Cojedes (state) |
https://en.wikipedia.org/wiki/Tinaco%20Municipality | The Tinaco Municipality is one of the nine municipalities (municipios) that makes up the Venezuelan state of Cojedes and, according to the 2011 census by the National Institute of Statistics of Venezuela, the municipality has a population of 32,564. The town of Tinaco is the municipal seat of the Tinaco Municipality.
Demographics
The Tinaco Municipality, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, has a population of 31,371 (up from 27,203 in 2000). This amounts to 10.4% of the state's population. The municipality's population density is .
Government
The mayor of the Tinaco Municipality is Teodoro Venancio Bolívar Caballero, elected on October 31, 2004, with 65% of the vote. He replaced Enrique Centeno shortly after the elections. The municipality is divided into one parishes; General en Jefe José Laurencio Silva (the parish's name was officially changed on December 30, 1994, it was formerly known as Tinaco Parish).
For the 2008 mayor's elections there are three major candidates these are:
Enrique Centeno
Francisco Ojeda
Celida Veloz
References
Municipalities of Cojedes (state) |
https://en.wikipedia.org/wiki/Discrete%20Fourier%20transform%20over%20a%20ring | In mathematics, the discrete Fourier transform over a ring generalizes the discrete Fourier transform (DFT), of a function whose values are commonly complex numbers, over an arbitrary ring.
Definition
Let be any ring, let be an integer, and let be a principal nth root of unity, defined by:
The discrete Fourier transform maps an n-tuple of elements of to another n-tuple of elements of according to the following formula:
By convention, the tuple is said to be in the time domain and the index is called time. The tuple is said to be in the frequency domain and the index is called frequency. The tuple is also called the spectrum of . This terminology derives from the applications of Fourier transforms in signal processing.
If is an integral domain (which includes fields), it is sufficient to choose as a primitive nth root of unity, which replaces the condition () by:
for
Another simple condition applies in the case where n is a power of two: () may be replaced by .
Inverse
The inverse of the discrete Fourier transform is given as:
where is the multiplicative inverse of in (if this inverse does not exist, the DFT cannot be inverted).
Matrix formulation
Since the discrete Fourier transform is a linear operator, it can be described by matrix multiplication. In matrix notation, the discrete Fourier transform is expressed as follows:
The matrix for this transformation is called the DFT matrix.
Similarly, the matrix notation for the inverse Fourier transform is
Polynomial formulation
Sometimes it is convenient to identify an -tuple with a formal polynomial
By writing out the summation in the definition of the discrete Fourier transform (), we obtain:
This means that is just the value of the polynomial for , i.e.,
The Fourier transform can therefore be seen to relate the coefficients and the values of a polynomial: the coefficients are in the time-domain, and the values are in the frequency domain. Here, of course, it is important that the polynomial is evaluated at the th roots of unity, which are exactly the powers of .
Similarly, the definition of the inverse Fourier transform () can be written:
With
this means that
We can summarize this as follows: if the values of are the coefficients of , then the values of are the coefficients of , up to a scalar factor and reordering.
Special cases
Complex numbers
If is the field of complex numbers, then the th roots of unity can be visualized as points on the unit circle of the complex plane. In this case, one usually takes
which yields the usual formula for the complex discrete Fourier transform:
Over the complex numbers, it is often customary to normalize the formulas for the DFT and inverse DFT by using the scalar factor in both formulas, rather than in the formula for the DFT and in the formula for the inverse DFT. With this normalization, the DFT matrix is then unitary.
Note that does not make sense in an arbitrary field.
Finite fields
If is a finite |
https://en.wikipedia.org/wiki/John%20Stephen%20Roy%20Chisholm | J. S. R. (Roy) Chisholm (26 November 1926 – 10 August 2015) was an English mathematical physicist. He was Professor Emeritus of Applied Mathematics at the University of Kent in Canterbury, where he worked from its founding in 1965 until 1994. Before that he held positions at the University of Glasgow (1951-1954) and Cardiff (1954-1962) following which he was appointed Dublin University Professor of Natural Philosophy at Trinity College Dublin (1962-1966). He held BA (1948) and PhD (1952) degrees from Cambridge.
Chisholm developed a method for rational approximations of two variable functions generalising Padé approximant.
Roy Chisholm initiated the first International Conference on Clifford Algebras and Their Applications in Mathematical Physics (ICCA) conference in 1985 which took place at the University of Kent in Canterbury, United Kingdom, and which has happened triennially since then.
He was married to Monty Chisholm, author of a book on English literary figure wife Lucy Clifford and mathematician husband William Kingdon Clifford.
Books
An Introduction to Statistical Mechanics, with A.H.de Borde, Pergamon Press (1958)
Mathematical Methods in Physics, with Rosa Morris, North-Holland (1964)
Vectors in Three-Dimensional Space (Cambridge University Press,1978)
Clifford Algebras and Their Applications in Mathematical Physics, co-edited with A.K.Common (Reidel 1986)
Clifford Analysis and its Applications, co-eds F.Brackx & V.Soucek (Kluwer 2001)
Changing Stations, a Campus Drama (Moat Sole 2014)
References
1926 births
2015 deaths
English mathematicians
Academics of the University of Kent
Alumni of Christ's College, Cambridge |
https://en.wikipedia.org/wiki/Crespo%20Municipality | The Crespo Municipality is one of the nine municipalities (municipios) that makes up the Venezuelan state of Lara and, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, the municipality has a population of 49,813. The town of Duaca is the shire town of the Crespo Municipality.
Demographics
The Crespo Municipality, according to a 2011 population census by the National Institute of Statistics of Venezuela, has a population of 49,958 (up from 43,407 in 2000). This amounts to 2.8% of the state's population. The municipality's population density is .
Government
The mayor of the Crespo Municipality is Miguel Valecillos Paul, re-elected on October 31, 2004, with 83% of the vote. The municipality is divided into two parishes; Fréitez and José María Blanco
.
See also
Duaca
Lara
Municipalities of Venezuela
References
External links
crespo-lara.gob.ve
Municipalities of Lara (state) |
https://en.wikipedia.org/wiki/Iribarren%20Municipality | The Iribarren Municipality is one of the nine municipalities (municipios) that make up the Venezuelan state of Lara and, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, the municipality has a population of 1,027,022. The City of Barquisimeto is the shire town of the Iribarren Municipality.
Demographics
The Iribarren Municipality, according to a 2011 population census by the National Institute of Statistics of Venezuela, has a population of 996.230 (up from 915,634 in 2000). This amounts to 56% of the state's population. The municipality's population density is .
Government
The mayor of the Iribarren Municipality is Alfredo Ramos, elected on December 8, 2013. The municipality is divided into 10 parishes; Catedral, Concepción, El Cují, Juan de Villegas, Santa Rosa, Tamaca, Unión, Aguedo Felipe Alvarado, Buena Vista, and Juárez.
References
External links
iribarren-lara.gob.ve
Nuevo Sistema Ferroviario de Venezuela
Video of Barquisimeto and its surroundings
Municipalities of Lara (state) |
https://en.wikipedia.org/wiki/Jim%C3%A9nez%20Municipality%2C%20Lara | The Jiménez Municipality is one of the nine municipalities (municipios) that makes up the Venezuelan state of Lara and, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, the municipality has a population of 97,524. The town of Quíbor is the shire town of the Jiménez Municipality.
Demographics
The Jiménez Municipality, according to a 2011 population estimate by the National Institute of Statistics of Venezuela, has a population of 100,997 (up from 97,524 in 2007). This amounts to 5.6% of the state's population. The municipality's population density is .
Government
The mayor of the Jiménez Municipality is Luis Alberto Plaza Paz, elected on October 31, 2004, with 73% of the vote. He replaced Manuel Diaz shortly after the elections. The municipality is divided into eight parishes; Juan Bautista Rodríguez, Cuara, Diego de Lozada, Paraíso de San José, San Miguel, Tintorero, José Bernardo Dorante, and Coronel.
See also
Quíbor
Lara
Municipalities of Venezuela
References
External links
jimenez-lara.gob.ve
Municipalities of Lara (state) |
https://en.wikipedia.org/wiki/Mor%C3%A1n%20Municipality | The Morán Municipality is one of the nine municipalities (municipios) that makes up the Venezuelan state of Lara and, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, the municipality has a population of 128,674. The town of El Tocuyo is the shire town of the Morán Municipality.
History
The town of El Tocuyo was founded by Juan de Carvajal in 1545 on the banks of the Tocuyo River and it was the administrative capital of Venezuela from 1546 to 1548. Its original name was Nuestra Señora de la Pura y Limpia Concepción del Tocuyo.
Geography
The surrounding area has good soil and an ideal climate for agriculture, dry and warm with plenty of water available from the Tocuyo River. The area has been occupied for over 20,000 years. When the Spanish arrived they found the "Gayones" Indians, who inhabited this valley, sowing corn and other agricultural products as cotton and yucca. After the Spanish came, sugar cane was, for centuries, the biggest crop; but since 1980 vegetables such as tomatoes, onions, chiles, and potatoes are taking its place.
Demographics
The Morán Municipality, according to a 2011 population estimate by the National Institute of Statistics of Venezuela, has a population of 123,880 (up from 115,166 in 2000). This amounts to 7.2% of the state's population. The municipality's population density is .
Government
The mayor of the Morán Municipality is Pedro Emilio Alastre López, re-elected on October 31, 2004, with 61% of the vote. The municipality is divided into eight parishes; Bolívar, Anzoátegui, Guarico, Hilario Luna y Luna, Humocaro Alto, Humocaro Bajo, La Candelaria, and Morán
.
See also
El Tocuyo
Lara
Municipalities of Venezuela
References
External links
moran-lara.gob.ve
"Historia de Venezuela para nosotros: El Tocuyo" Fundación Empresas Polar
https://web.archive.org/web/20160303220628/http://www.eltocuyo.8m.com/
Municipalities of Lara (state) |
https://en.wikipedia.org/wiki/Torres%20Municipality | The Torres Municipality is one of the eight municipalities (municipios) that makes up the Venezuelan state of Lara and, according to a 2011 population estimate by the National Institute of Statistics of Venezuela, the municipality has a population of 185.275. The town of Carora is the shire town of the Torres Municipality.
History
The town of Carora was founded twice. The first time was in 1569 by Juan de Tejo, but it was evacuated because of attacks by local natives. It was founded again in 1572 by Juan de Salmanca. Carora has one of the most beautiful and well-conserved colonial zones in Venezuela. This is visible in the streets and the colonial houses of this part of Carora. One of the interesting characteristics of the colonial zone is that most of the houses are still occupied by descendants of the original owners.
The cathedral of San Juan Bautista (Saint John the Baptist) was constructed at the beginning of 1600, with a very simple facade. In the inner part it is decorated with wood pillars and forged iron lights, in addition to a gold and wooden altar. Other sites of interest in Carora are: The José Zubillaga Perera Library, the birthplace of the Venezuelan hero Juan Jacinto Lara, and Lara House, the Fine arts Center. Also the Chapel of El Calvario, an example of the colonial baroque architecture. Also in the historic zone is the well-known Torres Club. Founded in 1898, this is in a colonial house. It has a restaurant and a "Healthcare center", which is only for members, but can usually be visited by tourists.
Economy
Agricultural potential: The main commercial activity of the region is cattle ranching, specifically dairy farming. Other agriculture products of the region are sugar cane and grapes. The former has the greater area under cultivation. In the case of the grapes cultivated in the Valley of Altagracia, these represents 86% of the crop cultivated in the Lara state. This activity is tied to the wine industry. That has acquired a national reputation with some of best wines of the country.
There is a possibility of extending the Center-West railway network to Carora, which would contribute to consolidating the industrial zone.
Commerce in Carora, has a high proportion of small and larger retailers, and an institute for the control of this sector, the ACIC, has been created.
Touristic potential: The city includes two different types of landscapes, one dry with Xerophile vegetation, and another with subhumid characteristics. The latter, the Carora's historic zone and the Cerro Saroche national park, has a great touristic potential.
Demographics
The Torres Municipality, according to a 2011 population estimate by the National Institute of Statistics of Venezuela, has a population of 185,275. This amounts to 10.4% of the state's population. The municipality's population density is .
Government
The municipality is divided into 17 parishes; Trinidad Samuel, Antonio Díaz, Camacaro, Castañeda, Cecilio Zubillaga, Chi |
https://en.wikipedia.org/wiki/Urdaneta%20Municipality%2C%20Lara | Urdaneta is one of the nine municipalities (municipios) that make up the Venezuelan state of Lara and, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, the municipality has a population of 63,213. The town of Siquisique is the municipal seat of the Urdaneta Municipality. The municipality is one of several in Venezuela named "Urdaneta Municipality" in honour of Venezuelan independence hero Rafael Urdaneta.
Demographics
The Urdaneta Municipality, according to a 2011 population census by the National Institute of Statistics of Venezuela, has a population of 61,381 (up from 56,063 in 2000). This amounts to 3.5% of the state's population. The municipality's population density is .
Government
The mayor of the Urdaneta Municipality is Luis Gonzaga Ladino, elected on October 31, 2004, with 64% of the vote. He replaced Willians Ereu shortly after the elections. The municipality is divided into four parishes; Siquisique, Moroturo, San Miguel, and Xaguas.
References
External links
urdaneta-lara.gob.ve
Municipalities of Lara (state) |
https://en.wikipedia.org/wiki/Bentley%E2%80%93Ottmann%20algorithm | In computational geometry, the Bentley–Ottmann algorithm is a sweep line algorithm for listing all crossings in a set of line segments, i.e. it finds the intersection points (or, simply, intersections) of line segments. It extends the Shamos–Hoey algorithm, a similar previous algorithm for testing whether or not a set of line segments has any crossings. For an input consisting of line segments with crossings (or intersections), the Bentley–Ottmann algorithm takes time . In cases where , this is an improvement on a naïve algorithm that tests every pair of segments, which takes .
The algorithm was initially developed by ; it is described in more detail in the textbooks , , and . Although asymptotically faster algorithms are now known by and , the Bentley–Ottmann algorithm remains a practical choice due to its simplicity and low memory requirements.
Overall strategy
The main idea of the Bentley–Ottmann algorithm is to use a sweep line approach, in which a vertical line L moves from left to right (or, e.g., from top to bottom) across the plane, intersecting the input line segments in sequence as it moves. The algorithm is described most easily in its general position, meaning:
No two line segment endpoints or crossings have the same x-coordinate
No line segment endpoint lies upon another line segment
No three line segments intersect at a single point.
In such a case, L will always intersect the input line segments in a set of points whose vertical ordering changes only at a finite set of discrete events. Specifically, a discrete event can either be associated with an endpoint (left or right) of a line-segment or intersection point of two line-segments. Thus, the continuous motion of L can be broken down into a finite sequence of steps, and simulated by an algorithm that runs in a finite amount of time.
There are two types of events that may happen during the course of this simulation. When L sweeps across an endpoint of a line segment s, the intersection of L with s is added to or removed from the vertically ordered set of intersection points. These events are easy to predict, as the endpoints are known already from the input to the algorithm. The remaining events occur when L sweeps across a crossing between (or intersection of) two line segments s and t. These events may also be predicted from the fact that, just prior to the event, the points of intersection of L with s and t are adjacent in the vertical ordering of the intersection points.
The Bentley–Ottmann algorithm itself maintains data structures representing the current vertical ordering of the intersection points of the sweep line with the input line segments, and a collection of potential future events formed by adjacent pairs of intersection points. It processes each event in turn, updating its data structures to represent the new set of intersection points.
Data structures
In order to efficiently maintain the intersection points of the sweep line L with the input line segment |
https://en.wikipedia.org/wiki/Frequency%20partition%20of%20a%20graph | In graph theory, a discipline within mathematics, the frequency partition of a graph (simple graph) is a partition of its vertices grouped by their degree. For example, the degree sequence of the left-hand graph below is (3, 3, 3, 2, 2, 1) and its frequency partition is 6 = 3 + 2 + 1. This indicates that it has 3 vertices with some degree, 2 vertices with some other degree, and 1 vertex with a third degree. The degree sequence of the bipartite graph in the middle below is (3, 2, 2, 2, 2, 2, 1, 1, 1) and its frequency partition is 9 = 5 + 3 + 1. The degree sequence of the right-hand graph below is (3, 3, 3, 3, 3, 3, 2) and its frequency partition is 7 = 6 + 1.
In general, there are many non-isomorphic graphs with a given frequency partition. A graph and its complement have the same frequency partition. For any partition p = f1 + f2 + ... + fk of an integer p > 1, other than p = 1 + 1 + 1 + ... + 1, there is at least one (connected) simple graph having this partition as its frequency partition.
Frequency partitions of various graph families are completely identified; frequency partitions of many families of graphs are not identified.
Frequency partitions of Eulerian graphs
For a frequency partition p = f1 + f2 + ... + fk of an integer p > 1, its graphic degree sequence is denoted as ((d1)f1,(d2)f2, (d3)f3, ..., (dk) fk) where degrees di's are different and fi ≥ fj for i < j.
Bhat-Nayak et al. (1979) showed that a partition of p with k parts, k ≤ integral part of is a frequency partition of a Eulerian graph and conversely.
Frequency partition of trees, Hamiltonian graphs, tournaments and hypegraphs
The frequency partitions of families of graphs such as trees, Hamiltonian graphs directed graphs and tournaments and to k-uniform hypergraphs. have been characterized.
Unsolved problems in frequency partitions
The frequency partitions of the following families of graphs have not yet been characterized:
Line graphs
Bipartite graphs
References
External section
Graph theory |
https://en.wikipedia.org/wiki/Bivariate | Bivariate may refer to:
Mathematics
Bivariate function, a function of two variables
Bivariate polynomial, a polynomial of two indeterminates
Statistics
Bivariate data, that shows the relationship between two variables
Bivariate analysis, statistical analysis of two variables
Bivariate distribution, a joint probability distribution for two variables
Other
Bivariate map, a single map that displays two variables
See also
Two-dimensional curve
Multivariate (disambiguation) |
https://en.wikipedia.org/wiki/Blitz%2B%2B | Blitz++ is a high-performance vector mathematics library written in C++. This library is intended for use in scientific applications that might otherwise be implemented with Fortran or MATLAB.
Blitz++ utilizes advanced C++ template metaprogramming techniques, including expression templates, to provide speed-optimized mathematical operations on sequences of data without sacrificing the natural syntax provided by other mathematical programming systems. Indeed, it has been recognized as a pioneer in the area of C++ template metaprogramming.
References
External links
C++ numerical libraries
Free software programmed in C++ |
https://en.wikipedia.org/wiki/P%C3%ADritu%20Municipality%2C%20Anzo%C3%A1tegui | The Píritu Municipality is one of the 21 municipalities (municipios) that makes up the eastern Venezuelan state of Anzoátegui and, according to the 2011 census by the National Institute of Statistics of Venezuela, the municipality has a population of 23,248. The town of Píritu, Anzoátegui is the shire town of the Píritu Municipality.
History
The Píritu Municipality separated in 1993 from the Fernando de Peñalver Municipality.
Demographics
The Píritu Municipality, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, has a population of 24,532 (up from 19,834 in 2000). This amounts to 1.7% of the state's population. The municipality's population density is .
Government
The mayor of the Píritu Municipality is Rita Jimenez, elected on November 23, 2008, with 47% of the vote. She replaced Antonio Barrios shortly after the elections. The municipality is divided into two parishes; Capital Píritu and San Francisco (previous to June 27, 1995, the Píritu Municipality contained only a single parish)
.
See also
Municipalities of Venezuela
References
External links
piritu-anzoategui.gob.ve
Municipalities of Anzoategui |
https://en.wikipedia.org/wiki/Wiener%20sausage | In the mathematical field of probability, the Wiener sausage is a neighborhood of the trace of a Brownian motion up to a time t, given by taking all points within a fixed distance of Brownian motion. It can be visualized as a sausage of fixed radius whose centerline is Brownian motion. The Wiener sausage was named after Norbert Wiener by because of its relation to the Wiener process; the name is also a pun on Vienna sausage, as "Wiener" is German for "Viennese".
The Wiener sausage is one of the simplest non-Markovian functionals of Brownian motion. Its applications include stochastic phenomena including heat conduction. It was first described by , and it was used by to explain results of a Bose–Einstein condensate, with proofs published by .
Definitions
The Wiener sausage Wδ(t) of radius δ and length t is the set-valued random variable on Brownian paths b (in some Euclidean space) defined by
is the set of points within a distance δ of some point b(x) of the path b with 0≤x≤t.
Volume of the Wiener sausage
There has been a lot of work on the behavior of the volume (Lebesgue measure) |Wδ(t)| of the Wiener sausage as it becomes thin (δ→0); by rescaling, this is essentially equivalent to studying the volume as the sausage becomes long (t→∞).
showed that in 3 dimensions the expected value of the volume of the sausage is
In dimension d at least 3 the volume of the Wiener sausage is asymptotic to
as t tends to infinity. In dimensions 1 and 2 this formula gets replaced by and respectively. , a student of Spitzer, proved similar results for generalizations of Wiener sausages with cross sections given by more general compact sets than balls.
References
Especially chapter 22.
(Reprint of 1964 edition)
An advanced monograph covering the Wiener sausage.
Mathematical physics
Statistical mechanics
Wiener process |
https://en.wikipedia.org/wiki/Santa%20Ana%20Municipality%2C%20Anzo%C3%A1tegui | The Santa Ana Municipality is one of the 21 municipalities (municipios) that makes up the eastern Venezuelan Anzoátegui State and, according to the 2011 census by the National Institute of Statistics of Venezuela, the municipality has a population of 9,636. The town of Santa Ana is the shire town of the Santa Ana Municipality.
Demographics
The Santa Ana Municipality, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, has a population of 10,919 (up from 9,675 in 2000). This amounts to 0.7% of the state's population. The municipality's population density is .
Government
The mayor of the Santa Ana Municipality is Gerson Martínez, re-elected November 23, 2008 with 49% of the vote. The municipality is divided into two parishes; Capital Santa Ana and Pueblo Nuevo.
See also
Santa Ana
Anzoátegui
Municipalities of Venezuela
References
External links
santaana-anzoategui.gob.ve
Municipalities of Anzoategui |
https://en.wikipedia.org/wiki/Bol%C3%ADvar%20Municipality%2C%20Barinas | Bolívar Municipality is one of the 12 municipalities (municipios) that makes up the Venezuelan state of Barinas and, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, the municipality has a population of 47,878. The town of Barinitas is the shire town of the Bolívar Municipality.
Name
The municipality is one of several in Venezuela named "Bolívar Municipality" in honour of Venezuelan independence hero Simón Bolívar.
Demographics
The Bolívar Municipality, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, has a population of 47,878 (up from 40,863 in 2000). This amounts to 6.3% of the state's population. The municipality's population density is .
Government
The mayor of the Bolívar Municipality is Iván Dario Maldonado, elected on October 31, 2004, with 49% of the vote. He replaced Alberto Melean shortly after the elections. The municipality is divided into three parishes; Barinitas, Altamira, and Calderas.
References
External links
Municipalities of Barinas (state) |
https://en.wikipedia.org/wiki/Cruz%20Paredes%20Municipality | The Cruz Paredes Municipality is one of the 12 municipalities (municipios) that makes up the Venezuelan state of Barinas and, according to the 2011 census by the National Institute of Statistics of Venezuela, the municipality has a population of 26,042. The town of Barrancas is the municipal seat of the Cruz Paredes Municipality.
Demographics
The Cruz Paredes Municipality, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, has a population of 25,178 (up from 21,197 in 2000). This amounts to 3.3% of the state's population. The municipality's population density is .
Government
The mayor of the Cruz Paredes Municipality is María Guadalupe Fernández Acuña, elected on October 31, 2004, with 42% of the vote. She replaced Carlos Ramirez shortly after the elections. The municipality is divided into three parishes; Barrancas, El Socorro, and Masparrito.
References
External links
cruzparedes-barinas.gob.ve
Municipalities of Barinas (state) |
https://en.wikipedia.org/wiki/Ezequiel%20Zamora%20Municipality%2C%20Barinas | The Ezequiel Zamora Municipality is one of the 12 municipalities (municipios) that makes up the Venezuelan state of Barinas and, according to the 2011 census by the National Institute of Statistics of Venezuela, the municipality has a population of 53,580. The town of Santa Bárbara is the shire town of the Ezequiel Zamora Municipality. The municipality is named for the 19th century Venezuelan soldier Ezequiel Zamora.
Demographics
The Ezequiel Zamora Municipality, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, has a population of 51,436 (up from 43,367 in 2000). This amounts to 6.8% of the state's population. The municipality's population density is .
Government
The mayor of the Ezequiel Zamora Municipality is Levid Emilio Méndez, re-elected on October 31, 2004, with 38% of the vote. The municipality is divided into four parishes; Santa Bárbara, José Ignacio del Pumar, Pedro Briceño Méndez, and Ramón Ignacio Méndez.
References
External links
ezequielzamora-barinas.gob.ve
Municipalities of Barinas (state) |
https://en.wikipedia.org/wiki/Iwasawa%20group |
In mathematics, a group is called an Iwasawa group, M-group or modular group if its lattice of subgroups is modular. Alternatively, a group G is called an Iwasawa group when every subgroup of G is permutable in G .
proved that a p-group G is an Iwasawa group if and only if one of the following cases happens:
G is a Dedekind group, or
G contains an abelian normal subgroup N such that the quotient group G/N is a cyclic group and if q denotes a generator of G/N, then for all n ∈ N, q−1nq = n1+ps where s ≥ 1 in general, but s ≥ 2 for p=2.
In , Iwasawa's proof was deemed to have essential gaps, which were filled by Franco Napolitani and Zvonimir Janko. has provided an alternative proof along different lines in his textbook. As part of Schmidt's proof, he proves that a finite p-group is a modular group if and only if every subgroup is permutable, by .
Every subgroup of a finite p-group is subnormal, and those finite groups in which subnormality and permutability coincide are called PT-groups. In other words, a finite p-group is an Iwasawa group if and only if it is a PT-group.
Examples
The Iwasawa group of order 16 is isomorphic to the modular maximal-cyclic group of order 16.
See also
Modular law for groups
Further reading
Both finite and infinite M-groups are presented in textbook form in . Modern study includes .
References
Finite groups
Properties of groups |
https://en.wikipedia.org/wiki/Obispos%20Municipality | The Obispos Municipality is one of the 12 municipalities (municipios) that makes up the Venezuelan state of Barinas and, according to the 2011 census by the National Institute of Statistics of Venezuela, the municipality has a population of 37,493. The town of Obispos is the municipal seat of the Obispos Municipality.
Demographics
The Obispos Municipality, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, has a population of 31,694 (up from 26,595 in 2000). This amounts to 4.2% of the state's population. The municipality's population density is .
Government
The mayor of the Obispos Municipality is Luis Manuel Zambrano Volcan, re-elected on October 31, 2004, with 54% of the vote. The municipality is divided into four parishes; Obispos, El Real, La Luz, and Los Guasimitos.
References
Municipalities of Barinas (state) |
https://en.wikipedia.org/wiki/Maximal%20function | Maximal functions appear in many forms in harmonic analysis (an area of mathematics). One of the most important of these is the Hardy–Littlewood maximal function. They play an important role in understanding, for example, the differentiability properties of functions, singular integrals and partial differential equations. They often provide a deeper and more simplified approach to understanding problems in these areas than other methods.
The Hardy–Littlewood maximal function
In their original paper, G.H. Hardy and J.E. Littlewood explained their maximal inequality in the language of cricket averages. Given a function f defined on Rn, the uncentred Hardy–Littlewood maximal function Mf of f is defined as
at each x in Rn. Here, the supremum is taken over balls B in Rn which contain the point x and |B| denotes the measure of B (in this case a multiple of the radius of the ball raised to the power n). One can also study the centred maximal function, where the supremum is taken just over balls B which have centre x. In practice there is little difference between the two.
Basic properties
The following statements are central to the utility of the Hardy–Littlewood maximal operator.
(a) For f ∈ Lp(Rn) (1 ≤ p ≤ ∞), Mf is finite almost everywhere.
(b) If f ∈ L1(Rn), then there exists a c such that, for all α > 0,
(c) If f ∈ Lp(Rn) (1 < p ≤ ∞), then Mf ∈ Lp(Rn) and
where A depends only on p and c.
Properties (b) is called a weak-type bound of Mf. For an integrable function, it corresponds to the elementary Markov inequality; however, Mf is never integrable, unless f = 0 almost everywhere, so that the proof of the weak bound (b) for Mf requires a less elementary argument from geometric measure theory, such as the Vitali covering lemma. Property (c) says the operator M is bounded on Lp(Rn); it is clearly true when p = ∞, since we cannot take an average of a bounded function and obtain a value larger than the largest value of the function. Property (c) for all other values of p can then be deduced from these two facts by an interpolation argument.
It is worth noting (c) does not hold for p = 1. This can be easily proved by calculating Mχ, where χ is the characteristic function of the unit ball centred at the origin.
Applications
The Hardy–Littlewood maximal operator appears in many places but some of its most notable uses are in the proofs of the Lebesgue differentiation theorem and Fatou's theorem and in the theory of singular integral operators.
Non-tangential maximal functions
The non-tangential maximal function takes a function F defined on the upper-half plane
and produces a function F* defined on Rn via the expression
Observe that for a fixed x, the set is a cone in with vertex at (x,0) and axis perpendicular to the boundary of Rn. Thus, the non-tangential maximal operator simply takes the supremum of the function F over a cone with vertex at the boundary of Rn.
Approximations of the identity
One particularly important form of functions |
https://en.wikipedia.org/wiki/Muckenhoupt%20weights | In mathematics, the class of Muckenhoupt weights consists of those weights for which the Hardy–Littlewood maximal operator is bounded on . Specifically, we consider functions on and their associated maximal functions defined as
where is the ball in with radius and center at . Let , we wish to characterise the functions for which we have a bound
where depends only on and . This was first done by Benjamin Muckenhoupt.
Definition
For a fixed , we say that a weight belongs to if is locally integrable and there is a constant such that, for all balls in , we have
where is the Lebesgue measure of , and is a real number such that: .
We say belongs to if there exists some such that
for all and all balls .
Equivalent characterizations
This following result is a fundamental result in the study of Muckenhoupt weights.
Theorem. A weight is in if and only if any one of the following hold.
(a) The Hardy–Littlewood maximal function is bounded on , that is
for some which only depends on and the constant in the above definition.
(b) There is a constant such that for any locally integrable function on , and all balls :
where:
Equivalently:
Theorem. Let , then if and only if both of the following hold:
This equivalence can be verified by using Jensen's Inequality.
Reverse Hölder inequalities and
The main tool in the proof of the above equivalence is the following result. The following statements are equivalent
for some .
There exist such that for all balls and subsets , implies .
There exist and (both depending on ) such that for all balls we have:
We call the inequality in the third formulation a reverse Hölder inequality as the reverse inequality follows for any non-negative function directly from Hölder's inequality. If any of the three equivalent conditions above hold we say belongs to .
Weights and BMO
The definition of an weight and the reverse Hölder inequality indicate that such a weight cannot degenerate or grow too quickly. This property can be phrased equivalently in terms of how much the logarithm of the weight oscillates:
(a) If then (i.e. has bounded mean oscillation).
(b) If , then for sufficiently small , we have for some .
This equivalence can be established by using the exponential characterization of weights above, Jensen's inequality, and the John–Nirenberg inequality.
Note that the smallness assumption on in part (b) is necessary for the result to be true, as , but:
is not in any .
Further properties
Here we list a few miscellaneous properties about weights, some of which can be verified from using the definitions, others are nontrivial results:
If , then defines a doubling measure: for any ball , if is the ball of twice the radius, then where is a constant depending on .
If , then there is such that .
If , then there is and weights such that .
Boundedness of singular integrals
It is not only the Hardy–Littlewood maximal operator that is bounded on these weight |
https://en.wikipedia.org/wiki/Pedraza%20Municipality | The Pedraza Municipality is one of the 12 municipalities (municipios) that makes up the Venezuelan state of Barinas and, according to the 2011 census by the National Institute of Statistics of Venezuela, the municipality has a population of 65,390. The town of Ciudad Bolivia is the shire town of the Pedraza Municipality.
Demographics
The Pedraza Municipality, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, has a population of 62,847 (up from 52,411 in 2000). This amounts to 8.3% of the state's population. The municipality's population density is .
Government
The mayor of the Pedraza Municipality is Frenchy Tomas Díaz Rodríguez, re-elected on October 31, 2004, with 69% of the vote. The municipality is divided into four parishes; Ciudad Bolivia, Ignacio Briceño, José Félix Ribas, Páez.
See also
Barinas
Municipalities of Venezuela
References
External links
pedraza-barinas.gob.ve
Municipalities of Barinas (state) |
https://en.wikipedia.org/wiki/Rojas%20Municipality | The Rojas Municipality is one of the 12 municipalities (municipios) that makes up the Venezuelan state of Barinas and, according to the 2011 census by the National Institute of Statistics of Venezuela, the municipality has a population of 40,126. The town of Libertad is the shire town of the Rojas Municipality.
Demographics
The Rojas Municipality, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, has a population of 40,264 (up from 34,132 in 2000). This amounts to 5.3% of the state's population. The municipality's population density is .
Government
The mayor of the Rojas Municipality is Orlando Leonel Gómez Mendoza, re-elected on October 31, 2004, with 48% of the vote. The municipality is divided into four parishes; Libertad, Dolores, Palacios Fajardo, and Santa Rosa.
See also
Libertad
Barinas
Municipalities of Venezuela
References
External links
rojas-barinas.gob.ve
Municipalities of Barinas (state) |
https://en.wikipedia.org/wiki/Sosa%20Municipality | The Sosa Municipality is one of the 12 municipalities (municipios) that makes up the Venezuelan state of Barinas and, according to the 2011 census by the National Institute of Statistics of Venezuela, the municipality has a population of 24,142. The town of Ciudad de Nutrias is the municipal seat of the Sosa Municipality.
Demographics
The Sosa Municipality, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, has a population of 27,445 (up from 21,960 in 2000). This amounts to 3.6% of the state's population. The municipality's population density is .
Government
The mayor of the Sosa Municipality is Francisco Ramon Ramírez, re-elected on October 31, 2004, with 41% of the vote. The municipality is divided into four parishes; Ciudad de Nutrias, El Regalo, Puerto de Nutrias, and Santa Catalina.
References
Municipalities of Barinas (state) |
https://en.wikipedia.org/wiki/Andr%C3%A9s%20Eloy%20Blanco%20Municipality%2C%20Barinas | The Andrés Eloy Blanco Municipality is one of the 12 municipalities (municipios) that makes up the Venezuelan state of Barinas and, according to the 2011 census by the National Institute of Statistics of Venezuela, the municipality has a population of 16,144. The town of El Cantón is the shire town of the Andrés Eloy Blanco Municipality. The municipality is one of several in Venezuela named "Andrés Eloy Blanco Municipality" in honour of Venezuelan poet Andrés Eloy Blanco.
In 2008, it established a sister-relationship with Dane County, Wisconsin -- http://www.dane-andres.org/
Demographics
The Andrés Eloy Blanco Municipality, according to a 2017 population estimate by the National Institute of Statistics of Venezuela, has a population of 18,816 (up from 15,847 in 2000). This amounts to 2.5% of the state's population. The municipality's population density is .
Government
The mayor of the Andrés Eloy Blanco Municipality is Zulay Martínez Fuentes, re-elected on October 31, 2004, with 69% of the vote. The municipality is divided into three parishes; El Cantón, Santa Cruz de Guacas, and Puerto Vivas.
References
External links
andreseloyblanco-barinas.gob.ve]
Municipalities of Barinas (state) |
https://en.wikipedia.org/wiki/Antonio%20Jos%C3%A9%20de%20Sucre%20Municipality | Antonio José de Sucre is one of the 12 municipalities (municipios) that makes up the Venezuelan state of Barinas and, according to the 2011 census by the National Institute of Statistics of Venezuela, the municipality has a population of 81,665. The town of Socopó is the shire town of the Antonio José de Sucre Municipality. The municipality is one of several in Venezuela named in honour of Venezuelan independence hero Antonio José de Sucre (the others include only his surname, as "Sucre Municipality").
Demographics
The Antonio José de Sucre Municipality, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, has a population of 74,968 (up from 63,947 in 2000). This amounts to 9.9% of the state's population. The municipality's population density is .
Government
The mayor of the Antonio José de Sucre Municipality is Salvador Guerrero, elected on October 31, 2004, with 46% of the vote. He replaced Trino Hurtado shortly after the elections. The municipality is divided into three parishes; Ticoporo, Andrés Bello, and Nicolás Pulido.
References
External links
antoniojosedesucre-barinas.gob.ve
Municipalities of Barinas (state) |
https://en.wikipedia.org/wiki/Arakelov%20theory | In mathematics, Arakelov theory (or Arakelov geometry) is an approach to Diophantine geometry, named for Suren Arakelov. It is used to study Diophantine equations in higher dimensions.
Background
The main motivation behind Arakelov geometry is the fact there is a correspondence between prime ideals and finite places , but there also exists a place at infinity , given by the Archimedean valuation, which doesn't have a corresponding prime ideal. Arakelov geometry gives a technique for compactifying into a complete space which has a prime lying at infinity. Arakelov's original construction studies one such theory, where a definition of divisors is constructor for a scheme of relative dimension 1 over such that it extends to a Riemann surface for every valuation at infinity. In addition, he equips these Riemann surfaces with Hermitian metrics on holomorphic vector bundles over X(C), the complex points of . This extra Hermitian structure is applied as a substitute for the failure of the scheme Spec(Z) to be a complete variety.
Note that other techniques exist for constructing a complete space extending , which is the basis of F1 geometry.
Original definition of divisors
Let be a field, its ring of integers, and a genus curve over with a non-singular model , called an arithmetic surface. Also, we let be an inclusion of fields (which is supposed to represent a place at infinity). Also, we will let be the associated Riemann surface from the base change to . Using this data, we can define a c-divisor as a formal linear combination where is an irreducible closed subset of of codimension 1, , and , and the sum represents the sum over every real embedding of and over one embedding for each pair of complex embeddings . The set of c-divisors forms a group .
Results
defined an intersection theory on the arithmetic surfaces attached to smooth projective curves over number fields, with the aim of proving certain results, known in the case of function fields,
in the case of number fields. extended Arakelov's work by establishing results such as a Riemann-Roch theorem, a Noether formula, a Hodge index theorem and the nonnegativity of the self-intersection of the dualizing sheaf in this context.
Arakelov theory was used by Paul Vojta (1991) to give a new proof of the Mordell conjecture, and by in his proof of Serge Lang's generalization of the Mordell conjecture.
developed a more general framework to define the intersection pairing defined on an arithmetic surface over the spectrum of a ring of integers by Arakelov.
Arakelov's theory was generalized by Henri Gillet and Christophe Soulé to higher dimensions. That is, Gillet and Soulé defined an intersection pairing on an arithmetic variety. One of the main results of Gillet and Soulé is the arithmetic Riemann–Roch theorem of , an extension of the Grothendieck–Riemann–Roch theorem to arithmetic varieties.
For this one defines arithmetic Chow groups CHp(X) of an arithmetic variety X, an |
https://en.wikipedia.org/wiki/Arismendi%20Municipality%2C%20Barinas | The Arismendi Municipality is one of the 12 municipalities (municipios) that makes up the Venezuelan state of Barinas and, according to the 2011 census by the National Institute of Statistics of Venezuela, the municipality has a population of 23,727. The town of Arismendi is the municipal seat of the Arismendi Municipality.
Demographics
The Arismendi Municipality, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, has a population of 20,305 (up from 18,941 in 2000). This amounts to 2.7% of the state's population. The municipality's population density is .
Government
The mayor of the Arismendi Municipality is Ramón Frías, elected on October 31, 2004, with 61% of the vote. He replaced Noheli Abreu shortly after the elections. The municipality is divided into four parishes; Arismendi, Guadarrama, La Unión, and San Antonio.
References
Municipalities of Barinas (state) |
https://en.wikipedia.org/wiki/Chemical%20bonding%20model | A chemical bonding model is a theoretical model used to explain atomic bonding structure, molecular geometry, properties, and reactivity of physical matter. This can refer to:
VSEPR theory, a model of molecular geometry.
Valence bond theory, which describes molecular electronic structure with localized bonds and lone pairs.
Molecular orbital theory, which describes molecular electronic structure with delocalized molecular orbitals.
Crystal field theory, an electrostatic model for transition metal complexes.
Ligand field theory, the application of molecular orbital theory to transition metal complexes.
Chemical bonding |
https://en.wikipedia.org/wiki/List%20of%20Chicago%20Blackhawks%20statistics%20and%20records | This article is a list of statistics and records relating to the Chicago Blackhawks. The Chicago Blackhawks are a professional ice hockey team that joined the National Hockey League (NHL) in 1926 as one of the Original Six. The Blackhawks, who were known as the Black Hawks 1926 to 1986, has won the Stanley Cup six times in their 87-year history. This list encompasses the major honours won by the Blackhawks, records set by the team, their managers and their players, and details the team's NHL performances.
Team honors and achievements
The Chicago Blackhawks have won the Stanley Cup, the highest team honor in the National Hockey League, on six occasions. They also won the Clarence S. Campbell Bowl six times as the Western Conference champions, most recently in 2014-15. The Blackhawks won the Prince of Wales Trophy in 1966–67 as the regular season champions, then again in 1969–70 as the East Division champions. In 1990–91 and again in 2012-13 the team won the Presidents' Trophy for accumulating the most points in the regular season.
Stanley Cup
1933–34
1937–38
1960–61
2009–10
2012–13
2014–15
Prince of Wales Trophy
1966–67
1969–70
Clarence S. Campbell Bowl
1970–71
1971–72
1972–73
1991–92
2009–10
2012–13
2014–15
Presidents' Trophy
1990–91
2012–13
Player records
Appearances
Points
Most goals in a season: 58, by Bobby Hull in 1968–69.
Most assists in a season: 87, by Denis Savard in 1981–82 and 1987–88.
Most points in a season: 131, by Denis Savard in 1987–88.
Top goal scorers
Goalies
All Time
Wins – Tony Esposito – 418
Losses – Tony Esposito – 302
Shutouts – Tony Esposito – 74
Games – Tony Esposito – 873
Minutes – Tony Esposito – 51,839:00
GAA – Charlie Gardiner – 2.02 (Min 100 games)
Save % – Corey Crawford – .917 (Min 100 games)
Single season
Wins – Ed Belfour – 43 (1990–1)
Losses – Al Rollins – 47 (1953–4)
Shutouts – Tony Esposito – 15 (1969–70)
Games – Ed Belfour – 74 (1990–1)
Minutes – Tony Esposito – 4,219:00
GAA – Charlie Gardiner – 1.64 (1933–4) min 20 games
Save % – Jeff Hackett – .927 (1996–7) min 20 games
Penalty minutes
Most penalty minutes in a season: 408, by Mike Peluso in 1991–92.
Team records
Goals
Most goals scored in a season: 351 in 80 games, 1985–86.
Fewest goals scored in a season:
33 in 44 games, 1928–29.
133 in 70 games, 1953–54.
Most goals against in a season: 363 in 80 games, 1981–82.
Fewest goals against in a season:
77 in 44 games, 2012–13
78 in 48 games, 1930–31.
164 in 78 games, 1973–74.
189 in 82 games, 2014–15.
Points
Most points in a season:
107 in 78 games, 1970–71
107 in 78 games, 1971–72
112 in 82 games, 2009–10
Fewest points in a season:
17 in 44 games, 1927–28
31 in 70 games, 1953–54
Games
Record scores
Record win: 12–0 against the Philadelphia Flyers at The Spectrum on January 30, 1969.
Record defeat: 0–12 against the Detroit Red Wings at Joe Lewis Arena on December 4, 1987.
Consecutive results
Record consecutive wins: 12 (from December 29, 2015 to present).
Rec |
https://en.wikipedia.org/wiki/Caron%C3%AD%20Municipality | The Caroní Municipality is one of the 11 municipalities (municipios) that makes up the Venezuelan state of Bolívar and, according to the 2011 census by the National Institute of Statistics of Venezuela, the municipality has a population of 704,585. The city of Guayana City is the shire town of the Caroní Municipality.
City Hall Foundation
The historical background of this municipality dates back to the foundation in 1724 of the mission of Purísima Concepción de Nuestra Señora del Caroní. Later, in 1817, General Manuel Piar established a command and won the Battle of San Félix against the royalist General La Torre.
In 1819, the Congress of Angostura decreed the division of the missions' territory into four districts, in Bajo Caroní the municipalities of San Félix, Caruachi, Murucuri, Caroní and San Miguel were included; this in accordance with the ordinance of 1841 was transferred to the Puerto de Tablas.
In 1864, the Sovereign State of Guayana divided the territory into four departments: Ciudad Bolívar, Upata, Alto and Baixo Orinoco; San Félix is not mentioned. At the end of the 19th century, the name of San Félix reappears and appears as a foreign municipality in the District of Piar. In the 1950s, the Companhia Mineira do Orinoco, at the confluence of the Orinoco and Caroní, established the infrastructure to explore the iron deposits of Cerro Bolívar.
In 1952 Puerto Ordaz is founded. In 1959, the Macagua I Dam came into operation. In 1960, the CVG was created. In 1961 Matanzas, Puerto Ordaz and San Félix were merged under the name Santo Tomé de Guayana (today Ciudad Guayana).
The creation of the District of Caroní took place on June 29, 1961, with the reform of the Law on Political Territorial Division of the state of Bolívar, with the capital of San Félix de Guayana and the populated centers of Puerto Ordaz, Matanzas, Castillito, Caruachi, La Ceiba and Alta Vista.
On December 29, 1960, Rómulo Betancourt, decreed by Organic Statute, the development of Guayana and created the Venezuelan Corporation of Guayana as the governing body. The Corporation chose July 2, 1961 to commemorate the anniversary of Ciudad Guayana, the date on which the first stone was laid.
In 1979 the District capital was changed (San Félix de Guayana by Santo Tomé de Guayana) and on June 25, 1986, the name was replaced by Ciudad Guayana, through the reform of the Territorial Political Division Law.
The Organic Law of the Municipal Regime of June 15, 1989 created the Municipality of Caroní, with its capital in San Félix, administered by various powers, with the City Hall being the local executive power and the City Council the local legislative power.
Demographics
The Caroní Municipality, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, has a population of 777,283 (up from 678,179 in 2000). This amounts to 50.6% of the state's population. The municipality's population density is .
Languages
Most of the population spe |
https://en.wikipedia.org/wiki/Cede%C3%B1o%20Municipality%2C%20Bol%C3%ADvar | The Cedeño Municipality is one of the 11 municipalities (municipios) that makes up the Venezuelan state of Bolívar. According to the 2011 census by the National Institute of Statistics of Venezuela, the municipality has a population of 67,000. The town of Caicara del Orinoco is the municipal seat of the Cedeño Municipality.
Demographics
The Cedeño Municipality, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, has a population of 88,296 (up from 69,664 in 2000). This amounts to 5.8% of the state's population. The municipality's population density is .
Government
The mayor of the Cedeño Municipality is Luis Guillermo Troconiz Marquez, elected on October 31, 2004, with 25% of the vote. He replaced Ismael Ortuño shortly after the elections. The municipality is divided into five (six if you count the Capital Cedeño section) parishes; Altagracia, Ascensión Farreras, Guaniamo, La Urbana, and Pijiguaos.
See also
Caicara del Orinoco
Bolívar
Municipalities of Venezuela
References
External links
cedeno-bolivar.gob.ve
Municipalities of Bolívar (state) |
https://en.wikipedia.org/wiki/El%20Callao%20Municipality | The El Callao Municipality is one of the 11 municipalities (municipios) that makes up the Venezuelan state of Bolívar and, according to the 2011 census by the National Institute of Statistics of Venezuela, the municipality has a population of 21,769. The town of El Callao is the shire town of the El Callao Municipality.
History
The town of El Callao, between the 18th and 19th centuries, before the arrival of the Spanish and the colony, was inhabited by the indigenous Maquiritare. Then some explorers settled in this territory looking for escaped slaves. These African slaves were in the vicinity of the Yuruari River and used this hiding place, if they were found they were returned to their work. But many of them stayed there because it was a perfect place to build a town, they killed and enslaved indigenous people from the area, and founded the town of El Callao. Over time, events would come that would make El Callao a population with several languages derived from the speech of African slaves in this region, mixed with that used by the English and French in their invasions of Spanish territories, and a few Creoles who speak Spanish. , in an almost entirely Spanish-speaking country. Large gold deposits were found in Yuruari, which made El Callao develop rapidly and led to the penetration of English and Brazilian miners. Recently the national government signed several treaties with various countries around the world to exploit it.
In 1854, the first miners settled in the area of El Callao, dedicated to the exploitation of gold-bearing quartz in the area of Ejidos de la Nueva Providencia de Caratal, on the banks of the Yuruari River. These miners were mostly immigrants from the Caribbean Antilles: San Marteen, Santa Lucía, San Cristóbal, Trinidad among others, generating a varied cultural conglomerate, both idiomatic and gastronomic, making this area of Guayana particularly rich and special culturally. because in this area English, Papiamento and other languages were spoken.
The municipality becomes official or is created on December 7, 1991. Along with the creation of the municipality came the coat of arms and the flag. The municipality clearly demonstrates its passion for football, by dedicating a barracks to it (specifically the right one), in its honor, and they also dedicate spaces to its main economy, gold, it is the left barracks, and in the lower one see the entire municipality.
Hydrography
The municipal hydrography is recognized mainly by the Yuruari River, by the veins of pure gold in its interior.
Geology
The El Callao region within the hydrographic basin of the Yuruari River dates from the Precambrian, mainly from the Archean and the Cenozoic. The volcanic rocks metamorphosed into the green schist facies, received the name of El Callao formation because they were found in this locality. In the District there are between 250 and 300 veins of gold. In the early 1970s, 88 veins were being exploited, of which 68 were in national minin |
https://en.wikipedia.org/wiki/Doubly%20stochastic%20model | In statistics, a doubly stochastic model is a type of model that can arise in many contexts, but in particular in modelling time-series and stochastic processes.
The basic idea for a doubly stochastic model is that an observed random variable is modelled in two stages. In one stage, the distribution of the observed outcome is represented in a fairly standard way using one or more parameters. At a second stage, some of these parameters (often only one) are treated as being themselves random variables. In a univariate context this is essentially the same as the well-known concept of compounded distributions. For the more general case of doubly stochastic models, there is the idea that many values in a time-series or stochastic model are simultaneously affected by the underlying parameters, either by using a single parameter affecting many outcome variates, or by treating the underlying parameter as a time-series or stochastic process in its own right.
The basic idea here is essentially similar to that broadly used in latent variable models except that here the quantities playing the role of latent variables usually have an underlying dependence structure related to the time-series or spatial context.
An example of a doubly stochastic model is the following. The observed values in a point process might be modelled as a Poisson process in which the rate (the relevant underlying parameter) is treated as being the exponential of a Gaussian process.
See also
Cox process
References
Further reading
Latent variable models
Hidden stochastic models |
https://en.wikipedia.org/wiki/Municipio%20Gran%20Sabana | The Municipio Gran Sabana is one of the 11 municipalities (municipios) that makes up the Venezuelan state of Bolívar and, according to the 2011 census by the National Institute of Statistics of Venezuela, the municipality has a population of 28,450. The town of Santa Elena de Uairén is the shire town of the Gran Sabana Municipality.
Demographics
The Gran Sabana Municipality, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, has a population of 34,095 (up from 27,027 in 2000). This amounts to 2.2% of the state's population. The municipality's population density is . As of 2011, the Gran Sabana Municipality has amongst the highest proportion of indigenous people in the country, roughly 77.67%, the majority of which being Pemon.
Government
The mayor of the Gran Sabana Municipality is Emilio González, elected in the 2017 Venezuelan municipal elections. He replaced Manuel Vallés shortly after the elections. The municipality is divided into one (two if you count the Capital Gran Sabana section) parish (Ikabarú).
See also
Gran Sabana
Santa Elena de Uairén
Bolívar
Angel Falls
Canaima National Park
Municipalities of Venezuela
References
External links
gransabana-bolivar.gob.ve
Gran Sabana |
https://en.wikipedia.org/wiki/Padre%20Pedro%20Chien%20Municipality | The Padre Pedro Chien Municipality is one of the 11 municipalities (municipios) that makes up the Venezuelan state of Bolívar and, according to the 2011 census by the National Institute of Statistics of Venezuela, the municipality has a population of 15,488. The town of El Palmar is the shire town of the Padre Pedro Chien Municipality.
Demographics
The Padre Pedro Chien Municipality, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, has a population of 14,749 (up from 12,810 in 2000). This amounts to 1% of the state's population. The municipality's population density is .
Government
The mayor of the Padre Pedro Chien Municipality is Franklin Gonzalez, re-elected on October 31, 2004, with 76% of the vote. The municipality is divided into one parish (Capital Padre Pedro Chien).
See also
El Palmar
Bolívar
Municipalities of Venezuela
References
External links
padrepedrochien-bolivar.gob.ve
Municipalities of Bolívar (state) |
https://en.wikipedia.org/wiki/Piar%20Municipality%2C%20Bol%C3%ADvar | The Piar Municipality is one of the 11 municipalities (municipios) that makes up the Venezuelan state of Bolívar and, according to the 2011 census by the National Institute of Statistics of Venezuela, the municipality has a population of 98,274. The town of Upata is the shire town of the Piar Municipality.
History
The creation of this municipality took place between 1960 and 1990. The municipality was named after the important independence hero, Manuel Piar.
Demographics
The Piar Municipality, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, has a population of 107,643 (up from 94,274 in 2000). This amounts to 7% of the state's population. The municipality's population density is .
Government
The mayor of the Piar Municipality is Francisco Contreras, elected on October 31, 2004, with 44% of the vote. He replaced Americo De Grazia shortly after the elections. The municipality is divided into two (three if you count the Capital Piar section) parishes; Andrés Eloy Blanco and Pedro Cova.
Weather
The climate is tropical savannah in the north and center, alternating with tropical rainforest in the south and in the highlands of Imataca. Average temperatures in the plains of 26 degrees, averages of 25 to 24 degrees in the areas located above 300 meters above sea level, up to averages of 22 to 23 degrees in the fringes of tepuis and plateaus that are located on the border with the Gran Municipality of savannah Minimums around 21 degrees average, with maximums of 34 degrees in its areas of greater insolation and with a more pronounced dry season, located in the Center and North of the municipality.
Parish Organization
The municipality has 3 parishes
Andrés Eloy Blanco (El Pao, this parish is the smallest in the municipality, located in the North and Northwest of the municipality). Its approximate population is 4 thousand inhabitants. The population of El Pao was home for more than 50 years to the second largest iron mining company in Venezuela, where a high-quality mine was produced or theorized from Cerro Florero, at an average of 2 million tons per year, the first explored by the North American company. American Iron Mining Company, later nationalized and transferred its assets to the public company of the Corporação Venezuelana de Guayana Ferrominera del Orinoco. This population center emerged as a typical encampment of mining technicians, currently the authorities are trying to redefine its role as a residential area, small business, tourism, education, small craft offices and food processing. Around them are populated by agricultural producers in the rural areas of El Retumbo, El Trical, El Arrozal, Cerro Azul, Los Morrocoyes, Mina Abajo, Pao Viejo, El Corozo, Cunaviche, Las Adjuntas and Los Jabillos.
Capital section (Upata), covering a third of the municipal territory, is located to the north and center of the territory up to the area of the Oronata Carichapo rivers, which separates it from |
https://en.wikipedia.org/wiki/Sucre%20Municipality%2C%20Bol%C3%ADvar | Sucre is one of the 11 municipalities (municipios) that makes up the Venezuelan state of Bolívar and, according to the 2011 census by the National Institute of Statistics of Venezuela, the municipality has a population of 20,359. The town of Maripa is the shire town of the Sucre Municipality.
Name
The municipality is one of several in Venezuela named "Sucre Municipality" in honour of Venezuelan independence hero Antonio José de Sucre.
Demographics
The Sucre Municipality, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, has a population of 27,218 (up from 20,108 in 2000). This amounts to 1.8% of the state's population. The municipality's population density is .
Government
The mayor of the Sucre Municipality is Juan Carlos Figarella Díaz, re-elected on October 31, 2004, with 47% of the vote. The municipality is divided into four (five if you count the Capital Sucre section) parishes; Aripao, Guarataro, Las Majadas, and Moitaco.
References
Municipalities of Bolívar (state) |
https://en.wikipedia.org/wiki/Topological%20dynamics | In mathematics, topological dynamics is a branch of the theory of dynamical systems in which qualitative, asymptotic properties of dynamical systems are studied from the viewpoint of general topology.
Scope
The central object of study in topological dynamics is a topological dynamical system, i.e. a topological space, together with a continuous transformation, a continuous flow, or more generally, a semigroup of continuous transformations of that space. The origins of topological dynamics lie in the study of asymptotic properties of trajectories of systems of autonomous ordinary differential equations, in particular, the behavior of limit sets and various manifestations of "repetitiveness" of the motion, such as periodic trajectories, recurrence and minimality, stability, non-wandering points. George Birkhoff is considered to be the founder of the field. A structure theorem for minimal distal flows proved by Hillel Furstenberg in the early 1960s inspired much work on classification of minimal flows. A lot of research in the 1970s and 1980s was devoted to topological dynamics of one-dimensional maps, in particular, piecewise linear self-maps of the interval and the circle.
Unlike the theory of smooth dynamical systems, where the main object of study is a smooth manifold with a diffeomorphism or a smooth flow, phase spaces considered in topological dynamics are general metric spaces (usually, compact). This necessitates development of entirely different techniques but allows an extra degree of flexibility even in the smooth setting, because invariant subsets of a manifold are frequently very complicated topologically (cf limit cycle, strange attractor); additionally, shift spaces arising via symbolic representations can be considered on an equal footing with more geometric actions. Topological dynamics has intimate connections with ergodic theory of dynamical systems, and many fundamental concepts of the latter have topological analogues (cf Kolmogorov–Sinai entropy and topological entropy).
See also
Poincaré–Bendixson theorem
Symbolic dynamics
Topological conjugacy
References
Robert Ellis, Lectures on topological dynamics. W. A. Benjamin, Inc., New York 1969
Walter Gottschalk, Gustav Hedlund, Topological dynamics. American Mathematical Society Colloquium Publications, Vol. 36. American Mathematical Society, Providence, R. I., 1955
J. de Vries, Elements of topological dynamics. Mathematics and its Applications, 257. Kluwer Academic Publishers Group, Dordrecht, 1993
Ethan Akin, The General Topology of Dynamical Systems, AMS Bookstore, 2010,
J. de Vries, Topological Dynamical Systems: An Introduction to the Dynamics of Continuous Mappings, De Gruyter Studies in Mathematics, 59, De Gruyter, Berlin, 2014,
Jian Li and Xiang Dong Ye, Recent development of chaos theory in topological dynamics, Acta Mathematica Sinica, English Series, 2016, Volume 32, Issue 1, pp. 83–114. |
https://en.wikipedia.org/wiki/Monomial%20group | In mathematics, in the area of algebra studying the character theory of finite groups, an M-group or monomial group is a finite group whose complex irreducible characters are all monomial, that is, induced from characters of degree 1 .
In this section only finite groups are considered. A monomial group is solvable by , presented in textbook in and . Every supersolvable group and every solvable A-group is a monomial group. Factor groups of monomial groups are monomial, but subgroups need not be, since every finite solvable group can be embedded in a monomial group, as shown by and in textbook form in .
The symmetric group is an example of a monomial group that is neither supersolvable nor an A-group. The special linear group is the smallest finite group that is not monomial: since the abelianization of this group has order three, its irreducible characters of degree two are not monomial.
References
Finite groups
Properties of groups |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.