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https://en.wikipedia.org/wiki/Carsten%20Thomassen%20%28mathematician%29
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Carsten Thomassen (born August 22, 1948 in Grindsted) is a Danish mathematician. He has been a Professor of Mathematics at the Technical University of Denmark since 1981, and since 1990 a member of the Royal Danish Academy of Sciences and Letters. His research concerns discrete mathematics and more specifically graph theory.
Thomassen received his Ph.D. in 1976 from the University of Waterloo.
He is editor-in-chief of the Journal of Graph Theory and the Electronic Journal of Combinatorics, and editor of Combinatorica, the Journal of Combinatorial Theory Series B, Discrete Mathematics, and the European Journal of Combinatorics.
He was awarded the Dedicatory Award of the 6th International Conference on the Theory and Applications of Graphs by the Western Michigan University in May 1988, the Lester R. Ford Award by the Mathematical Association of America in 1993, and the Faculty of Mathematics Alumni Achievement Medal by the University of Waterloo in 2005. In 1990 he was an invited speaker (Graphs, random walks and electric networks) at the ICM in Kyōto. He was included on the ISI Web of Knowledge list of the 250 most cited mathematicians.
Selected works
with Bojan Mohar: Graphs on surfaces, Johns Hopkins University Press 2001
5-choosability of planar graphs (see List coloring)
works on Hypohamiltonian graphs
Hamilton connectivity of Tournaments (see Tournament (graph theory)) and of 4-connected planar graphs
his proof of Gr%C3%B6tzsch%27s theorem
See also
List of University of Waterloo people
References
1948 births
Living people
People from Billund Municipality
20th-century Danish mathematicians
Graph theorists
University of Waterloo alumni
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https://en.wikipedia.org/wiki/Soddy%27s%20hexlet
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In geometry, Soddy's hexlet is a chain of six spheres (shown in grey in Figure 1), each of which is tangent to both of its neighbors and also to three mutually tangent given spheres. In Figure 1, the three spheres are the red inner sphere and two spheres (not shown) above and below the plane the centers of the hexlet spheres lie on. In addition, the hexlet spheres are tangent to a fourth sphere (the blue outer sphere in Figure 1), which is not tangent to the three others.
According to a theorem published by Frederick Soddy in 1937, it is always possible to find a hexlet for any choice of mutually tangent spheres A, B and C. Indeed, there is an infinite family of hexlets related by rotation and scaling of the hexlet spheres (Figure 1); in this, Soddy's hexlet is the spherical analog of a Steiner chain of six circles. Consistent with Steiner chains, the centers of the hexlet spheres lie in a single plane, on an ellipse. Soddy's hexlet was also discovered independently in Japan, as shown by Sangaku tablets from 1822 in Kanagawa prefecture.
Definition
Soddy's hexlet is a chain of six spheres, labeled S1–S6, each of which is tangent to three given spheres, A, B and C, that are themselves mutually tangent at three distinct points. (For consistency throughout the article, the hexlet spheres will always be depicted in grey, spheres A and B in green, and sphere C in blue.) The hexlet spheres are also tangent to a fourth fixed sphere D (always shown in red) that is not tangent to the three others, A, B and C.
Each sphere of Soddy's hexlet is also tangent to its neighbors in the chain; for example, sphere S4 is tangent to S3 and S5. The chain is closed, meaning that every sphere in the chain has two tangent neighbors; in particular, the initial and final spheres, S1 and S6, are tangent to one another.
Annular hexlet
The annular Soddy's hexlet is a special case (Figure 2), in which the three mutually tangent spheres consist of a single sphere of radius r (blue) sandwiched between two parallel planes (green) separated by a perpendicular distance 2r. In this case, Soddy's hexlet consists of six spheres of radius r packed like ball bearings around the central sphere and likewise sandwiched. The hexlet spheres are also tangent to a fourth sphere (red), which is not tangent to the other three.
The chain of six spheres can be rotated about the central sphere without affecting their tangencies, showing that there is an infinite family of solutions for this case. As they are rotated, the spheres of the hexlet trace out a torus (a doughnut-shaped surface); in other words, a torus is the envelope of this family of hexlets.
Solution by inversion
The general problem of finding a hexlet for three given mutually tangent spheres A, B and C can be reduced to the annular case using inversion. This geometrical operation always transforms spheres into spheres or into planes, which may be regarded as spheres of infinite radius. A sphere is transformed into
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https://en.wikipedia.org/wiki/P-rep
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In statistical hypothesis testing, p-rep or prep has been proposed as a statistical alternative to the classic p-value. Whereas a p-value is the probability of obtaining a result under the null hypothesis, p-rep purports to compute the probability of replicating an effect. The derivation of p-rep contained significant mathematical errors.
For a while, the Association for Psychological Science recommended that articles submitted to Psychological Science and their other journals report p-rep rather than the classic p-value, but this is no longer the case.
Calculation
Approximation from p
The value of the p-rep (prep) can be approximated based on the p-value (p) as follows:
The above applies for one-tailed distributions.
Criticism
The fact that the p-rep has a one-to-one correspondence with the p-value makes it clear that this new measure brings no additional information beyond that conveyed by the significance of the result. Killeen acknowledges this lack of information, but suggests that p-rep better captures the way naive experimenters conceptualize p-values and statistical hypothesis testing.
Among the criticisms of p-rep is the fact that while it attempts to estimate replicability, it ignores results from other studies which can accurately guide this estimate. For example, an experiment on some unlikely paranormal phenomenon may yield a p-rep of 0.75. Most people would still not conclude the probability of a replication was 75%. Rather, they would conclude it is much closer to 0: Extraordinary claims require extraordinary evidence, and p-rep ignores this. Because of this, p-rep may in fact be harder to interpret than a classical p-value. The fact that p-rep requires assumptions about prior probabilities for it to be valid makes its interpretation complex. Killeen argues that new results should be evaluated in their own right, without the "burden of history", with flat priors: that is what p-rep yields. A more pragmatic estimate of replicability would include prior knowledge, via, for instance, meta-analysis.
Critics have also underscored mathematical errors in the original Killeen paper. For example, the formula relating the effect sizes from two replications of a given experiment erroneously uses one of these random variables as a parameter of the probability distribution of the other while he previously hypothesized these two variables to be independent, criticisms addressed in Killeen's rejoinder.
A further criticism of the p-rep statistic involves the logic of experimentation. The scientific value of replicable data lies in the adequate accounting for previously unmeasured factors (e.g., unmeasured participant variables, experimenter's bias, etc.), The idea that a single study can capture a logical likelihood of such unmeasured factors affecting the outcome, and thus the likelihood of replicability, is a logical fallacy.
References
External links
Statistical tests
Statistical hypothesis testing
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https://en.wikipedia.org/wiki/Composition%20of%20relations
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In the mathematics of binary relations, the composition of relations is the forming of a new binary relation from two given binary relations R and S. In the calculus of relations, the composition of relations is called relative multiplication, and its result is called a relative product. Function composition is the special case of composition of relations where all relations involved are functions.
The word uncle indicates a compound relation: for a person to be an uncle, he must be the brother of a parent. In algebraic logic it is said that the relation of Uncle () is the composition of relations "is a brother of" () and "is a parent of" ().
Beginning with Augustus De Morgan, the traditional form of reasoning by syllogism has been subsumed by relational logical expressions and their composition.
Definition
If and are two binary relations, then
their composition is the relation
In other words, is defined by the rule that says if and only if there is an element such that (that is, and ).
Notational variations
The semicolon as an infix notation for composition of relations dates back to Ernst Schroder's textbook of 1895. Gunther Schmidt has renewed the use of the semicolon, particularly in Relational Mathematics (2011). The use of semicolon coincides with the notation for function composition used (mostly by computer scientists) in category theory, as well as the notation for dynamic conjunction within linguistic dynamic semantics.
A small circle has been used for the infix notation of composition of relations by John M. Howie in his books considering semigroups of relations. However, the small circle is widely used to represent composition of functions which reverses the text sequence from the operation sequence. The small circle was used in the introductory pages of Graphs and Relations until it was dropped in favor of juxtaposition (no infix notation). Juxtaposition is commonly used in algebra to signify multiplication, so too, it can signify relative multiplication.
Further with the circle notation, subscripts may be used. Some authors prefer to write and explicitly when necessary, depending whether the left or the right relation is the first one applied. A further variation encountered in computer science is the Z notation: is used to denote the traditional (right) composition, but ⨾ () denotes left composition.
Mathematical generalizations
Binary relations are morphisms in the category . In Rel the objects are sets, the morphisms are binary relations and the composition of morphisms is exactly composition of relations as defined above. The category Set of sets and functions is a subcategory of where the maps
are functions .
Given a regular category , its category of internal relations has the same objects as , but now the morphisms are given by subobjects in . Formally, these are jointly monic spans between and . Categories of internal relations are allegories. In particular . Given a field (or more
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https://en.wikipedia.org/wiki/Levinson%27s%20inequality
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In mathematics, Levinson's inequality is the following inequality, due to Norman Levinson, involving positive numbers. Let and let be a given function having a third derivative on the range , and such that
for all . Suppose and for . Then
The Ky Fan inequality is the special case of Levinson's inequality, where
References
Scott Lawrence and Daniel Segalman: A generalization of two inequalities involving means, Proceedings of the American Mathematical Society. Vol 35 No. 1, September 1972.
Norman Levinson: Generalization of an inequality of Ky Fan, Journal of Mathematical Analysis and Applications. Vol 8 (1964), 133–134.
Inequalities
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https://en.wikipedia.org/wiki/Census%20in%20Pakistan
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The Census in Pakistan (), is a decennial census and a descriptive count of Pakistan's population on Census Day, and of their dwellings, conducted and supervised by the Pakistan Bureau of Statistics. The 2017 Census in Pakistan marks the first census to take place in Pakistan since 1998. The most recent census was the 2023 Pakistani census.
Overview
A national census is mandated by the Constitution of Pakistan to be held every ten years. After the independence of Pakistan in 1947, the first census took place in 1951 under Finance Minister Sir Malik Ghulam, serving under Prime Minister Liaquat Ali Khan. Since 1951, there have been only 6 nationwide censuses (1961, 1972, 1981, 1998 and 2017). Delays and postponements have often been due to politicization. Pakistan's last completed census took place in 2017. The next national census was scheduled to take place in 2001 and later 2008, and again in 2010, but none of those plans could materialize. There were multiple census counts completed for the latest round in April 2012, but were subsequently thrown out as being "unreliable". A UN led census was to be conducted with staff training and GPS digitisation. As of 2015, the population of Pakistan is estimated at 191.71 million. As of 2016, the population of religious minorities in Pakistan have increased to 3 million. On 25 August 2017, the official results declared Pakistan's population to be 207.74 million.
.
Census
1951
According to 1951 census, the Dominion of Pakistan (both West and East Pakistan) had a population of 75.7 million, in which West Pakistan had a population of 33.7 million and East Pakistan (today Bangladesh) had a population of 42 million. In 1951, minorities constituted 14.4% of the Pakistani population (this includes East Pakistan, today Bangladesh). Breaking down between East and West Pakistan, the population of West Pakistan was 3.44% non-Muslim (1.16 million out of 33.7 million), while East Pakistan (today Bangladesh) was 23.20% non-Muslim (9.744 million out of 42 million). Total non- Muslim population on both sides added up to 10.90 million.
1961
According to the 1961 census, the population of Pakistan was 93 million, with 42.8 million residing in West Pakistan and 50 million residing in East Pakistan. The literacy was 19.2%, in which East Pakistan had a literacy rate of 21.5% while West Pakistan had a literacy rate of 16.9%. Hindus in East Pakistan were 18.4%
1972
The scheduled 1971 census was postponed due to the political crisis of 1970 followed by the India-Pakistan war of 1971 and subsequent loss of East Pakistan. In 1970, the population was 65 million in the East Pakistan(Bangladesh) and 58 million in West Pakistan.
According to the 1972 census, the population of Pakistan was 65.3 million. After 1972, the Census Organization was merged into the Ministry of Interior.
1981
According to the 1981 census, the population of Pakistan was 83.783 million.
1998
2017
2023
Notes
References
External links
Demographics
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https://en.wikipedia.org/wiki/Manihi%20Airport
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Manihi Airport is an airport serving Manihi, an atoll in the Tuamotu archipelago in French Polynesia. It is located 3 km northwest of the village of Paeva.
Airlines and destinations
Statistics
References
External links
Airports in French Polynesia
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https://en.wikipedia.org/wiki/Morphism%20of%20schemes
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In algebraic geometry, a morphism of schemes generalizes a morphism of algebraic varieties just as a scheme generalizes an algebraic variety. It is, by definition, a morphism in the category of schemes.
A morphism of algebraic stacks generalizes a morphism of schemes.
Definition
By definition, a morphism of schemes is just a morphism of locally ringed spaces.
A scheme, by definition, has open affine charts and thus a morphism of schemes can also be described in terms of such charts (compare the definition of morphism of varieties). Let ƒ:X→Y be a morphism of schemes. If x is a point of X, since ƒ is continuous, there are open affine subsets U = Spec A of X containing x and V = Spec B of Y such that ƒ(U) ⊆ V. Then ƒ: U → V is a morphism of affine schemes and thus is induced by some ring homomorphism B → A (cf. #Affine case.) In fact, one can use this description to "define" a morphism of schemes; one says that ƒ:X→Y is a morphism of schemes if it is locally induced by ring homomorphisms between coordinate rings of affine charts.
Note: It would not be desirable to define a morphism of schemes as a morphism of ringed spaces. One trivial reason is that there is an example of a ringed-space morphism between affine schemes that is not induced by a ring homomorphism (for example, a morphism of ringed spaces:
that sends the unique point to s and that comes with .) More conceptually, the definition of a morphism of schemes needs to capture "Zariski-local nature" or localization of rings; this point of view (i.e., a local-ringed space) is essential for a generalization (topos).
Let be a morphism of schemes with . Then, for each point x of X, the homomorphisms on the stalks:
is a local ring homomorphism: i.e., and so induces an injective homomorphism of residue fields
.
(In fact, φ maps th n-th power of a maximal ideal to the n-th power of the maximal ideal and thus induces the map between the (Zariski) cotangent spaces.)
For each scheme X, there is a natural morphism
which is an isomorphism if and only if X is affine; θ is obtained by gluing U → target which come from restrictions to open affine subsets U of X. This fact can also be stated as follows: for any scheme X and a ring A, there is a natural bijection:
(Proof: The map from the right to the left is the required bijection. In short, θ is an adjunction.)
Moreover, this fact (adjoint relation) can be used to characterize an affine scheme: a scheme X is affine if and only if for each scheme S, the natural map
is bijective. (Proof: if the maps are bijective, then and X is isomorphic to by Yoneda's lemma; the converse is clear.)
A morphism as a relative scheme
Fix a scheme S, called a base scheme. Then a morphism is called a scheme over S or an S-scheme; the idea of the terminology is that it is a scheme X together with a map to the base scheme S. For example, a vector bundle E → S over a scheme S is an S-scheme.
An S-morphism from p:X →S to q:Y →S is a morphism ƒ:X →Y of schemes su
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https://en.wikipedia.org/wiki/Central%20Statistics%20Office%20%28India%29
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The Central Statistics Office (CSO) is a governmental agency in India under the Ministry of Statistics and Programme Implementation responsible for co-ordination of statistical activities in India, and evolving and maintaining statistical standards. It has a Graphical Unit. The CSO is located in Delhi. Some portion of Industrial Statistics work pertaining to Annual Survey of industries is carried out in Calcutta. It deals with statistical data of different departments.
Activities
The Central Statistics Office is responsible for co-ordination of statistical activities in the country, and evolving and maintaining statistical standards. Its activities include National Income Accounting; conduct of Annual Survey of Industries, Economic Censuses and its follow up surveys, compilation of Index of Industrial Production, as well as Consumer Price Indices for Urban Non-Manual Employees, Human Development Statistics, Gender Statistics, imparting training in Official Statistics, Five Year Plan work relating to Development of Statistics in the States and Union Territories; dissemination of statistical information, work relating to trade, energy, construction, and environment statistics, revision of National Industrial Classification, etc.
It has two publications :
1. the statistical abstract- InIndia (annual)
2. the monthly abstract of Statistics
Organisation
The CSO is headed by the Director-General who is assisted by Five additional Director-Generals and four Deputy Director-Generals, six Joint Directors, seven special task officers, thirty deputy directors, 48 assistant directors and other supporting staff. The CSO is located in Delhi.
Functions
The Central Statistics Office (CSO) in the Ministry of Statistics and Programme Implementation (MoS & PI) is responsible for the compilation of National Accounts Statistics (NAS). At the State level, State Directorates of Economy and Statistics (DESs) have the responsibility of compiling their State Domestic Product and other aggregates.
It plays an advisory role in statistical matters
It provides national statistics to UN
It has set up a unit to attend to statistical work relating to the five-year plans in collaboration with the planning commission and has expanded training facilities for statistics personnel.
It is also responsible for the compilation and publication of national income statistics.
The CSO through its Industrial Statistical wing conducts the Annual Survey of Industries and publishes the result.
History
The CSO was set up in the cabinet secretariat on 2 May 1951 as a part of the cabinet Secretariat and having co-ordinating and advisory functions. At that time the name of CSO was Central Statistical Institute. In 1954 the CSI merged with CSO and the new name was Central Statistical Organization. Recently, for a third time its name changed and now it called as Central Statistics Office.
Fate
CSO was finally merged with NSSO to make National Statistical Office (NSO) in 2019 under the department
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https://en.wikipedia.org/wiki/Atkin%E2%80%93Lehner%20theory
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In mathematics, Atkin–Lehner theory is part of the theory of modular forms describing when they arise at a given integer level N in such a way that the theory of Hecke operators can be extended to higher levels.
Atkin–Lehner theory is based on the concept of a newform, which is a cusp form 'new' at a given level N, where the levels are the nested congruence subgroups:
of the modular group, with N ordered by divisibility. That is, if M divides N, Γ0(N) is a subgroup of Γ0(M). The oldforms for Γ0(N) are those modular forms f(τ) of level N of the form g(d τ) for modular forms g of level M with M a proper divisor of N, where d divides N/M. The newforms are defined as a vector subspace of the modular forms of level N, complementary to the space spanned by the oldforms, i.e. the orthogonal space with respect to the Petersson inner product.
The Hecke operators, which act on the space of all cusp forms, preserve the subspace of newforms and are self-adjoint and commuting operators (with respect to the Petersson inner product) when restricted to this subspace. Therefore, the algebra of operators on newforms they generate is a finite-dimensional C*-algebra that is commutative; and by the spectral theory of such operators, there exists a basis for the space of newforms consisting of eigenforms for the full Hecke algebra.
Atkin–Lehner involutions
Consider a Hall divisor e of N, which means that not only does e divide N, but also e and N/e are relatively prime (often denoted e||N). If N has s distinct prime divisors, there are 2s Hall divisors of N; for example, if N = 360 = 23⋅32⋅51, the 8 Hall divisors of N are 1, 23, 32, 51, 23⋅32, 23⋅51, 32⋅51, and 23⋅32⋅51.
For each Hall divisor e of N, choose an integral matrix We of the form
with det We = e. These matrices have the following properties:
The elements We normalize Γ0(N): that is, if A is in Γ0(N), then WeAW is in Γ0(N).
The matrix W, which has determinant e2, can be written as eA where A is in Γ0(N). We will be interested in operators on cusp forms coming from the action of We on Γ0(N) by conjugation, under which both the scalar e and the matrix A act trivially. Therefore, the equality W = eA implies that the action of We squares to the identity; for this reason, the resulting operator is called an Atkin–Lehner involution.
If e and f are both Hall divisors of N, then We and Wf commute modulo Γ0(N). Moreover, if we define g to be the Hall divisor g = ef/(e,f)2, their product is equal to Wg modulo Γ0(N).
If we had chosen a different matrix W ′e instead of We, it turns out that We ≡ W ′e modulo Γ0(N), so We and W ′e would determine the same Atkin–Lehner involution.
We can summarize these properties as follows. Consider the subgroup of GL(2,Q) generated by Γ0(N) together with the matrices We; let Γ0(N)+ denote its quotient by positive scalar matrices. Then Γ0(N) is a normal subgroup of Γ0(N)+ of index 2s (where s is the number of distinct prime factors of N); the quotient group is isomorphic to
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https://en.wikipedia.org/wiki/Gabriel%20Kron
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Gabriel Kron (1901 – 1968) was a Hungarian American electrical engineer who promoted the use of methods of linear algebra, multilinear algebra, and differential geometry in the field. His method of system decomposition and solution called Diakoptics is still influential today. Though he published widely, his methods were slow to be assimilated. At Union College a symposium was organized by Schaffer Library on "Gabriel Kron, the Man and His Work", held October 14, 1969. H.H. Happ edited the contributed papers, which were published by Union College Press as Gabriel Kron and Systems Theory.
Early life
Gabriel Kron was born in 1901 in Baia Mare in Transylvania, Hungary. In 1919 he graduated from the gymnasium. By that time Transylvania had been ceded to Romania. Kron's older brother Joseph returned home, which he had left when he was ten years old. Joseph wished for a professional degree, but had no schooling after grade five. Gabriel tutored Joseph, who passed various exams, culminating in the high-school exam in 1920. In December of that year the two left home for the United States. The brothers earned their living in New York City with odd jobs such as dish washer, bus boy, or working machines in garment factories.
In the fall of 1922 the brothers had saved enough money to enter engineering school at University of Michigan. They continued supporting themselves with casual employment. Gabriel found digging ditches more congenial than dishwashing. He coined the motto: "There are only two occupations compatible with human dignity. One is the study of atomic structure. The other is digging ditches."
In 1925 Gabriel graduated and started on a trip around the world. He planned to walk and hitch hike as much as possible. He ran out of money when he reached Los Angeles, where he worked for the United States Electrical Manufacturing Company. He then transferred to the Robbins and Myers Company in Springfield, Ohio. In 1926 he set out again. From California he took passage on an oil tanker bound for Tahiti. In Sydney, Australia he ran out of money.
After saving 35 pounds from work at the Electricity Metering Manufacturing Company he set out for Northern Australia.
In Fiji he had finished, and buried, Treatise on Differential Equations by Forsyth. In Sydney he searched for a worthy successor, settling on Advanced Vector Analysis with Application to Mathematical Physics by the Australian C.E. Weatherburn. During long hikes in Queensland, Kron saw that vector analysis would be a powerful tool in engineering.
Sea voyages took him to Saigon via Borneo, Manila, and Hong Kong. Hence overland to Cairo and Alexandria by rail, supplemented by many hours of walking. In the spring of 1928 Kron arrived in Romania and stayed with his family till the fall.
After his return Kron was employed as electrical engineer for brief periods with several companies the last of which was Warner Brothers in New York. They closed his department while he was on a continuing highly p
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https://en.wikipedia.org/wiki/Nicolai%20Reshetikhin
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Nicolai Yuryevich Reshetikhin (, born October 10, 1958, in Leningrad, Soviet Union) is a mathematical physicist, currently a professor of mathematics at Tsinghua University, China and a professor of mathematical physics at the University of Amsterdam (Korteweg-de Vries Institute for Mathematics). He is also a professor emeritus at the University of California, Berkeley. His research is in the fields of low-dimensional topology, representation theory, and quantum groups. His major contributions are in the theory of quantum integrable systems, in representation theory of quantum groups
and in quantum topology. He and Vladimir Turaev constructed invariants of 3-manifolds
which are expected to describe quantum Chern-Simons field theory introduced by Edward Witten.
He earned his bachelor's degree and master's degree from Leningrad State University in 1982, and his Ph.D. from the Steklov Mathematical Institute in 1984.
He gave a plenary lecture at the International Congress of Mathematicians in 2010. He was named a Fellow of the American Mathematical Society, in the 2022 class of fellows, "for contributions to the theory of quantum groups, integrable systems, topology, and quantum physics".
See also
Reshetikhin–Turaev invariant
References
External links
1958 births
Living people
20th-century Russian mathematicians
21st-century Russian mathematicians
Mathematicians from Saint Petersburg
Topologists
Academic staff of the University of Amsterdam
University of California, Berkeley faculty
Fellows of the American Mathematical Society
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https://en.wikipedia.org/wiki/Comparable
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Comparable may refer to:
Comparability, in mathematics
Comparative, in grammar, a word that denotes the degree by which an entity has a property greater or less in extent than another
See also
Incomparable (disambiguation)
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https://en.wikipedia.org/wiki/List%20of%20Aston%20Villa%20F.C.%20records%20and%20statistics
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Aston Villa Football Club are an English professional association football club based in Aston, Birmingham, who currently play in the Premier League. The club was founded in 1874 and were founding members of the Football League in 1888, as well as the Premier League in 1992. They are one of the oldest football clubs in England, having won the First Division Championship seven times and the FA Cup seven times. In 1982, the club became one of only six English clubs to win the European Cup.
This list encompasses the honours won by Aston Villa and the records set by the players and the club. The player records section includes details of the club's leading goalscorers and those who have made the most appearances in first-team competitions. Attendance records at Villa Park are also included in the list.
Honours
Aston Villa have won honours both domestically and in European cup competitions. Their most recent domestic honour was a League Cup win in 1996.
European
European Cup:
Winners (1): 1982
European Super Cup:
Winners (1): 1982–83
Intertoto Cup:
Winners (1): 2001
Co-winners (1): 2008
Domestic
League
Football League First Division:
Winners (7): 1894, 1896, 1897, 1899, 1900, 1910, 1981
Runners up (9): 1889, 1903, 1908, 1911, 1913, 1914, 1931, 1933, 1990
Premier League:
Runners up (1): 1993
Football League Second Division:
Winners (2): 1938, 1960
Runners up (2): 1975, 1988
Play-Offs (1): 2019
Football League Third Division:
Winners (1): 1972
Cups
FA Cup*
Winners (7): 1887, 1895, 1897, 1905, 1913, 1920, 1957
Runners up (4): 1892, 1924, 2000, 2015
Football League Cup:
Winners (5): 1961, 1975, 1977, 1994, 1996
Runners up (4): 1963, 1971, 2010, 2020
FA Charity Shield
Winners (1): 1981 (shared)
Runners up (3): 1910, 1957, 1972
Sheriff of London Charity Shield:
Winners (2): 1899, 1901
Runners up (1): 1900
Football League War Cup
Winners (1): 1944
Youth
FA Youth Cup:
Winners (4): 1972, 1980, 2002, 2021
FA Premier League Cup
Winners (1): 2018
HKFC Soccer Sevens
Winners (7): 2002, 2004, 2007, 2008, 2010, 2016, 2023
NextGen Cup:
Winners (1): 2013
Friendly and exhibition
Football World Championship
Winners (3): 1887, 1894, 1900 (shared)
West Bromwich Charity Cup
Winners (1): 1890 (shared)
Bass Charity Vase
Winners (3): 1893, 1894, 2018
Dublin Tournament
Winners (1): 2003
Peace Cup:
Winners (1): 2009
Cup of Traditions
Winners (1): 2017
Queensland Champions Cup
Winners (1): 2022
Al Wahda Challenge Cup
Winners (1): 2022
Orange Trophy
Winners (1): 2023
Player records
Appearances
Youngest first-team player: Jimmy Brown, 15 years 349 days (v. Bolton Wanderers, Division Two, 17 September 1969).
Oldest first-team player: Brad Friedel, 40 years 4 days (v. Liverpool, Premier League, 22 May 2011).
Most appearances
Competitive matches only. Each column contains appearances in the starting eleven, followed by appearances as substitute in brackets.
Other competitions include European Cup, UEFA Cup and Intertoto
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https://en.wikipedia.org/wiki/Prime%20triplet
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In number theory, a prime triplet is a set of three prime numbers in which the smallest and largest of the three differ by 6. In particular, the sets must have the form or . With the exceptions of and , this is the closest possible grouping of three prime numbers, since one of every three sequential odd numbers is a multiple of three, and hence not prime (except for 3 itself).
Examples
The first prime triplets are
(5, 7, 11), (7, 11, 13), (11, 13, 17), (13, 17, 19), (17, 19, 23), (37, 41, 43), (41, 43, 47), (67, 71, 73), (97, 101, 103), (101, 103, 107), (103, 107, 109), (107, 109, 113), (191, 193, 197), (193, 197, 199), (223, 227, 229), (227, 229, 233), (277, 281, 283), (307, 311, 313), (311, 313, 317), (347, 349, 353), (457, 461, 463), (461, 463, 467), (613, 617, 619), (641, 643, 647), (821, 823, 827), (823, 827, 829), (853, 857, 859), (857, 859, 863), (877, 881, 883), (881, 883, 887)
Subpairs of primes
A prime triplet contains a single pair of:
Twin primes: or ;
Cousin primes: or ; and
Sexy primes: .
Higher-order versions
A prime can be a member of up to three prime triplets - for example, 103 is a member of , and . When this happens, the five involved primes form a prime quintuplet.
A prime quadruplet contains two overlapping prime triplets, and .
Conjecture on prime triplets
Similarly to the twin prime conjecture, it is conjectured that there are infinitely many prime triplets. The first known gigantic prime triplet was found in 2008 by Norman Luhn and François Morain. The primes are with . the largest known proven prime triplet contains primes with 20008 digits, namely the primes with .
The Skewes number for the triplet is 87613571, and for the triplet it is 337867.
References
External links
Classes of prime numbers
Unsolved problems in number theory
|
https://en.wikipedia.org/wiki/Rhinos%20Milano
|
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"type": "FeatureCollection",
"features": [
{
"type": "Feature",
"properties": {},
"geometry": {
"type": "Point",
"coordinates": [
9.158057212043788,
45.48146824664756
]
}
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]
}The Rhinos Milano are an American football team based in Milan, Italy. The team was founded in 1976 (first team in Italy) and won the Italian championship in 1981, 1982, 1983, 1990 and 2016. The Rhinos play in the Italian Football League (formerly called Series A) the highest level league in Italy.
History
The team was founded as Pantere Rosa di Piacenza. In 1978 the Rhinos were part of the first game played between two Italian teams where they defeated the Busto Arsizio Frogs.
The 2016 season saw the Rhinos Milano, with Head Coach Chris Ault, winning all of their 13 games played to complete a perfect season.
Seasons
1978 June 25 Rhinos Milano-Frogs Busto Arsizio 34–0 First American Football game between two Italian teams
1979–80 First team able to play at the Nato bases' tournament
1981 Serie A Italian Champions (vs Frogs Gallarate 24–8)
1982 Serie A Italian Champions (vs Frogs Gallarate 11–0)
1983 Serie A Italian Champions (vs Warriors Bologna 20–14)
1984 Serie A 4° place Division North
1985 Serie A 3° place Division East Quarter finals (@ Panthers Parma 14–24)Under 20 Italian Champions
1986 Serie A 2° place Division West Semifinals (@ Angels Pesaro 0–10)
1987 Serie A 2° place Division B Quarter finals (@ Seamen Milano 7–29)
1988 Serie A 1° place Division Center Semifinals (vs Warriors Bologna 13–20)
1989 Serie A 4° place Division A Quarter finals (@ Chiefs Ravenna 13–14)
1990 Serie A Italian Champions (vs Frogs Legnano 33–6)
1991-93 didn't play any game
1994 Serie A 1° place Division B made the (vs Frogs Legnano 27–37)
1995 Serie A 3° place Division B Quarter finals (@ Phoenix Bologna 14–32)
1996 Serie A 5° place Division A
1997 Serie A 3° place Division West Wild Card (@ Bergamo Lions 13–21)
1998 Serie A 3° place Division B
1999-2001 didn't play any game
2002 Serie C (Fivemen) 5° place
2003 Serie C - NWC 1° place NWC BOWL Champions (vs Chargers Novi Ligure 35–12)
2004 Serie C - NWC 1° place Girone A made the NWC BOWL (vs Red Jackets Sarzana 7–22)
2005 Serie B - 4° place Division North Quarter finals (@ Guelfi Firenze 0–25)
2006 Serie A2 - 2° place Division NorthWestern (5° place Sauceda National Ranking) Semifinals (@ Hogs Reggio Emilia 0-42)
2007 Serie A1 - 6° place (record 2–6)
2008 IFL (Italian Football League) - 5° place (record 4–6)
2009 IFL (Italian Football League) - 5° place (record 4–5)
2010 IFL (Italian Football League) - 6° place (record 4–4)
2011 IFL (Italian Football League) - 4° place (record 5–4)
2012 IFL (Italian Football League) - 5° place (record 8–3)
2013 IFL (Italian Football League) - 7° place (record 2–6)
2014 IFL (Italian Football League) - 7° place (record 5–5)
2015 IFL (Italian Football League) - 7° place (re
|
https://en.wikipedia.org/wiki/Sinkov%20statistic
|
Sinkov statistics, also known as log-weight statistics, is a specialized field of statistics that was developed by Abraham Sinkov, while working for the small Signal Intelligence Service organization, the primary mission of which was to compile codes and ciphers for use by the U.S. Army. The mathematics involved include modular arithmetic, a bit of number theory, some linear algebra of two dimensions with matrices, some combinatorics, and a little statistics.
Sinkov did not explain the theoretical underpinnings of his statistics, or characterized its distribution, nor did he give a decision procedure for accepting or rejecting candidate plaintexts on the basis of their S1 scores. The situation becomes more difficult when comparing strings of different lengths because Sinkov does not explain how the distribution of his statistics changes with length, especially when applied to higher-order grams. As for how to accept or reject a candidate plaintext, Sinkov simply said to try all possibilities and to pick the one with the highest S1 value. Although the procedure works for some applications, it is inadequate for applications that require on-line decisions. Furthermore, it is desirable to have a meaningful interpretation of the S1 values.
References
Cryptographic attacks
Computational linguistics
Statistical natural language processing
|
https://en.wikipedia.org/wiki/Minkowski%E2%80%93Steiner%20formula
|
In mathematics, the Minkowski–Steiner formula is a formula relating the surface area and volume of compact subsets of Euclidean space. More precisely, it defines the surface area as the "derivative" of enclosed volume in an appropriate sense.
The Minkowski–Steiner formula is used, together with the Brunn–Minkowski theorem, to prove the isoperimetric inequality. It is named after Hermann Minkowski and Jakob Steiner.
Statement of the Minkowski-Steiner formula
Let , and let be a compact set. Let denote the Lebesgue measure (volume) of . Define the quantity by the Minkowski–Steiner formula
where
denotes the closed ball of radius , and
is the Minkowski sum of and , so that
Remarks
Surface measure
For "sufficiently regular" sets , the quantity does indeed correspond with the -dimensional measure of the boundary of . See Federer (1969) for a full treatment of this problem.
Convex sets
When the set is a convex set, the lim-inf above is a true limit, and one can show that
where the are some continuous functions of (see quermassintegrals) and denotes the measure (volume) of the unit ball in :
where denotes the Gamma function.
Example: volume and surface area of a ball
Taking gives the following well-known formula for the surface area of the sphere of radius , :
where is as above.
References
Calculus of variations
Geometry
Hermann Minkowski
Measure theory
Theorems in measure theory
|
https://en.wikipedia.org/wiki/Brunn%E2%80%93Minkowski%20theorem
|
In mathematics, the Brunn–Minkowski theorem (or Brunn–Minkowski inequality) is an inequality relating the volumes (or more generally Lebesgue measures) of compact subsets of Euclidean space. The original version of the Brunn–Minkowski theorem (Hermann Brunn 1887; Hermann Minkowski 1896) applied to convex sets; the generalization to compact nonconvex sets stated here is due to Lazar Lyusternik (1935).
Statement
Let n ≥ 1 and let μ denote the Lebesgue measure on Rn. Let A and B be two nonempty compact subsets of Rn. Then the following inequality holds:
where A + B denotes the Minkowski sum:
The theorem is also true in the setting where are only assumed to be measurable and non-empty.
Multiplicative version
The multiplicative form of Brunn–Minkowski inequality states that for all .
The Brunn–Minkowski inequality is equivalent to the multiplicative version.
In one direction, use the inequality (exponential is convex), which holds for . In particular, .
Conversely, using the multiplicative form, we find
The right side is maximized at , which gives
.
The Prékopa–Leindler inequality is a functional generalization of this version of Brunn–Minkowski.
On the hypothesis
Measurability
It is possible for to be Lebesgue measurable and to not be; a counter example can be found in "Measure zero sets with non-measurable sum." On the other hand, if are Borel measurable, then is the continuous image of the Borel set , so analytic and thus measurable. See the discussion in Gardner's survey for more on this, as well as ways to avoid measurability hypothesis.
We note that in the case that A and B are compact, so is A + B, being the image of the compact set under the continuous addition map : , so the measurability conditions are easy to verify.
Non-emptiness
The condition that are both non-empty is clearly necessary. This condition is not part of the multiplicative versions of BM stated below.
Proofs
We give two well known proofs of Brunn–Minkowski.
We give a well-known argument that follows a general recipe of arguments in measure theory; namely, it establishes a simple case by direct analysis, uses induction to establish a finitary extension of that special case, and then uses general machinery to obtain the general case as a limit. A discussion of this history of this proof can be found in Theorem 4.1 in Gardner's survey on Brunn–Minkowski.
We prove the version of the Brunn–Minkowski theorem that only requires to be measurable and non-empty.
The case that A and B are axis aligned boxes:
By translation invariance of volumes, it suffices to take . Then . In this special case, the Brunn–Minkowski inequality asserts that . After dividing both sides by , this follows from the AM–GM inequality: .
The case where A and B are both disjoint unions of finitely many such boxes:
We will use induction on the total number of boxes, where the previous calculation establishes the base case of two boxes. First, we observe that there is an
|
https://en.wikipedia.org/wiki/Beta%20wavelet
|
Continuous wavelets of compact support alpha can be built, which are related to the beta distribution. The process is derived from probability distributions using blur derivative. These new wavelets have just one cycle, so they are termed unicycle wavelets. They can be viewed as a soft variety of Haar wavelets whose shape is fine-tuned by two parameters and . Closed-form expressions for beta wavelets and scale functions as well as their spectra are derived. Their importance is due to the Central Limit Theorem by Gnedenko and Kolmogorov applied for compactly supported signals.
Beta distribution
The beta distribution is a continuous probability distribution defined over the interval . It is characterised by a couple of parameters, namely and according to:
.
The normalising factor is ,
where is the generalised factorial function of Euler and is the Beta function.
Gnedenko-Kolmogorov central limit theorem revisited
Let be a probability density of the random variable , i.e.
, and .
Suppose that all variables are independent.
The mean and the variance of a given random variable are, respectively
.
The mean and variance of are therefore and .
The density of the random variable corresponding to the sum is given by the
Central Limit Theorem for distributions of compact support (Gnedenko and Kolmogorov).
Let be distributions such that .
Let , and .
Without loss of generality assume that and .
The random variable holds, as ,
where and
Beta wavelets
Since is unimodal, the wavelet generated by
has only one-cycle (a negative half-cycle and a positive half-cycle).
The main features of beta wavelets of parameters and are:
The parameter is referred to as “cyclic balance”, and is defined as the ratio between the lengths of the causal and non-causal piece of the wavelet. The instant of transition from the first to the second half cycle is given by
The (unimodal) scale function associated with the wavelets is given by
.
A closed-form expression for first-order beta wavelets can easily be derived. Within their support,
Beta wavelet spectrum
The beta wavelet spectrum can be derived in terms of the Kummer hypergeometric function.
Let denote the Fourier transform pair associated with the wavelet.
This spectrum is also denoted by for short. It can be proved by applying properties of the Fourier transform that
where .
Only symmetrical cases have zeroes in the spectrum. A few asymmetric beta wavelets are shown in Fig. Inquisitively, they are parameter-symmetrical in the sense that they hold
Higher derivatives may also generate further beta wavelets. Higher order beta wavelets are defined by
This is henceforth referred to as an -order beta wavelet. They exist for order . After some algebraic handling, their closed-form expression can be found:
Application
Wavelet theory is applicable to several subjects. All wavelet transforms may be considered forms of time-frequency representation for contin
|
https://en.wikipedia.org/wiki/Jung%27s%20theorem
|
In geometry, Jung's theorem is an inequality between the diameter of a set of points in any Euclidean space and the radius of the minimum enclosing ball of that set. It is named after Heinrich Jung, who first studied this inequality in 1901. Algorithms also exist to solve the smallest-circle problem explicitly.
Statement
Consider a compact set
and let
be the diameter of K, that is, the largest Euclidean distance between any two of its points. Jung's theorem states that there exists a closed ball with radius
that contains K. The boundary case of equality is attained by the regular n-simplex.
Jung's theorem in the plane
The most common case of Jung's theorem is in the plane, that is, when n = 2. In this case the theorem states that there exists a circle enclosing all points whose radius satisfies
and this bound is as tight as possible since when K is an equilateral triangle (or its three vertices) one has
General metric spaces
For any bounded set in any metric space, . The first inequality is implied by the triangle inequality for the center of the ball and the two diametral points, and the second inequality follows since a ball of radius centered at any point of will contain all of . Both these inequalities are tight:
In a uniform metric space, that is, a space in which all distances are equal, .
At the other end of the spectrum, in an injective metric space such as the Manhattan distance in the plane, : any two closed balls of radius centered at points of have a non-empty intersection, therefore all such balls have a common intersection, and a radius ball centered at a point of this intersection contains all of .
Versions of Jung's theorem for various non-Euclidean geometries are also known (see e.g. Dekster 1995, 1997).
References
External links
Geometric inequalities
Euclidean geometry
Theorems in geometry
Metric geometry
|
https://en.wikipedia.org/wiki/Harish-Chandra%20Research%20Institute
|
The Harish-Chandra Research Institute (HRI) is an institution dedicated to research in mathematics and theoretical physics, located in Prayagraj, Uttar Pradesh in India. Established in 1975, HRI offers masters and doctoral program in affiliation with the Homi Bhabha National Institute.
HRI has a residential campus in Jhusi town in Prayagraj on the banks of the river Ganga. The institute has over 30 faculty, 50 doctoral students and 25 post-doctoral visiting research fellows and scientists. HRI is funded by the Department of Atomic Energy (DAE) of the Government of India.
History
The institute was founded as the Mehta Research Institute of Mathematics and Mathematical Physics in 1975, with an endowment from the B.S. Mehta Trust, Calcutta. The institute was initially managed by Badri Nath Prasad and following his death in January 1966 by S.R. Sinha, both from the Allahabad University. The first official director of the institute was Prabhu Lal Bhatnagar in 1975 when it became truly operational. He was followed by S.R. Sinha again.
On 29 November 1975 B. Devadas Acharya joined the Mehta Research Institute (MRI) as its first postdoctoral fellow and on 1 January 1980 was appointed as the first assistant professor of mathematics at MRI. During his research work between 1975 and 1984, he gave many talks on graph theory and its applications in computing. In one of his talks to international audiences, he envisioned a computing engine based on matrices which would be much more powerful.
Sharadchandra Shankar Shrikhande joined the institute as its director in January 1983. The institute was facing financial difficulties, and Shrikhande sought DAE support for the institute. Following the recommendations of the DAE review committee, the Government of Uttar Pradesh committed to provide a campus for HRI, while the DAE committed to provide full funding for all operational expenses.
In January 1990, the institute was granted about in Jhusi town of Prayagraj district and H.S. Mani took over as Director. The institute moved to its present campus in 1996. Since then, the institute has grown in facilities, scope of research as well as number of faculty and students.
In October 2000, the institute was renamed in honor of renowned Indian mathematician Harish-Chandra.
Ravi S. Kulkarni succeeded Mani as the director in August 2001 and was followed by Amitava Raychaudhuri in July 2005. Jayanta Kumar Bhattacharjee followed in May 2011.
Research activities
The HRI Mathematics research group has four teams with focus on Algebra, Analysis, Geometry & Topology and Number Theory. The HRI Physics research group consists of teams focused on Astrophysics, Condensed Matter Physics, High Energy Physics, String Theory and Quantum Information & Computation. Prominent HRI faculty members in the area of String Theory include Ashoke Sen and Rajesh Gopakumar. HRI faculty member in the area of Quantum Information and Computation includes Arun K. Pati, Aditi Sen De and Ujjawal S
|
https://en.wikipedia.org/wiki/Lambda%20function
|
Lambda function may refer to:
Mathematics
Dirichlet lambda function, λ(s) = (1 – 2−s)ζ(s) where ζ is the Riemann zeta function
Liouville function, λ(n) = (–1)Ω(n)
Von Mangoldt function, Λ(n) = log p if n is a positive power of the prime p
Modular lambda function, λ(τ), a highly symmetric holomorphic function on the complex upper half-plane
Carmichael function, λ(n), in number theory and group theory
Computing
Lambda calculus, in computer science
Lambda function (computer programming), or lambda abstraction
AWS Lambda, a form of serverless computing
See also
Lambda point, of fluid helium
|
https://en.wikipedia.org/wiki/Metacyclic%20group
|
In group theory, a metacyclic group is an extension of a cyclic group by a cyclic group. That is, it is a group G for which there is a short exact sequence
where H and K are cyclic. Equivalently, a metacyclic group is a group G having a cyclic normal subgroup N, such that the quotient G/N is also cyclic.
Properties
Metacyclic groups are both supersolvable and metabelian.
Examples
Any cyclic group is metacyclic.
The direct product or semidirect product of two cyclic groups is metacyclic. These include the dihedral groups and the quasidihedral groups.
The dicyclic groups are metacyclic. (Note that a dicyclic group is not necessarily a semidirect product of two cyclic groups.)
Every finite group of squarefree order is metacyclic.
More generally every Z-group is metacyclic. A Z-group is a group whose Sylow subgroups are cyclic.
References
Properties of groups
Solvable groups
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https://en.wikipedia.org/wiki/Brauer%E2%80%93Siegel%20theorem
|
In mathematics, the Brauer–Siegel theorem, named after Richard Brauer and Carl Ludwig Siegel, is an asymptotic result on the behaviour of algebraic number fields, obtained by Richard Brauer and Carl Ludwig Siegel. It attempts to generalise the results known on the class numbers of imaginary quadratic fields, to a more general sequence of number fields
In all cases other than the rational field Q and imaginary quadratic fields, the regulator Ri of Ki must be taken into account, because Ki then has units of infinite order by Dirichlet's unit theorem. The quantitative hypothesis of the standard Brauer–Siegel theorem is that if Di is the discriminant of Ki, then
Assuming that, and the algebraic hypothesis that Ki is a Galois extension of Q, the conclusion is that
where hi is the class number of Ki. If one assumes that all the degrees are bounded above by a uniform constant
N, then one may drop the assumption of normality - this is what is actually proved in Brauer's paper.
This result is ineffective, as indeed was the result on quadratic fields on which it built. Effective results in the same direction were initiated in work of Harold Stark from the early 1970s.
References
Richard Brauer, On the Zeta-Function of Algebraic Number Fields, American Journal of Mathematics 69 (1947), 243–250.
Analytic number theory
Theorems in algebraic number theory
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https://en.wikipedia.org/wiki/Hypercycle%20%28geometry%29
|
In hyperbolic geometry, a hypercycle, hypercircle or equidistant curve is a curve whose points have the same orthogonal distance from a given straight line (its axis).
Given a straight line and a point not on , one can construct a hypercycle by taking all points on the same side of as , with perpendicular distance to equal to that of . The line is called the axis, center, or base line of the hypercycle. The lines perpendicular to , which are also perpendicular to the hypercycle, are called the normals of the hypercycle. The segments of the normals between and the hypercycle are called the radii. Their common length is called the distance or radius of the hypercycle.
The hypercycles through a given point that share a tangent through that point converge towards a horocycle as their distances go towards infinity.
Properties similar to those of Euclidean lines
Hypercycles in hyperbolic geometry have some properties similar to those of lines in Euclidean geometry:
In a plane, given a line and a point not on it, there is only one hypercycle of that of the given line (compare with Playfair's axiom for Euclidean geometry).
No three points of a hypercycle are on a circle.
A hypercycle is symmetrical to each line perpendicular to it. (Reflecting a hypercycle in a line perpendicular to the hypercycle results in the same hypercycle.)
Properties similar to those of Euclidean circles
Hypercycles in hyperbolic geometry have some properties similar to those of circles in Euclidean geometry:
A line perpendicular to a chord of a hypercycle at its midpoint is a radius and it bisects the arc subtended by the chord.
Let AB be the chord and M its middle point.
By symmetry the line R through M perpendicular to AB must be orthogonal to the axis L.
Therefore R is a radius.
Also by symmetry, R will bisect the arc AB.
The axis and distance of a hypercycle are uniquely determined.
Let us assume that a hypercycle C has two different axes L1 and L2.
Using the previous property twice with different chords we can determine two distinct radii R1 and R2. R1 and R2 will then have to be perpendicular to both L1 and L2, giving us a rectangle. This is a contradiction because the rectangle is an impossible figure in hyperbolic geometry.
Two hypercycles have equal distances if and only if they are congruent.
If they have equal distance, we just need to bring the axes to coincide by a rigid motion and also all the radii will coincide; since the distance is the same, also the points of the two hypercycles will coincide.
Vice versa, if they are congruent the distance must be the same by the previous property.
A straight line cuts a hypercycle in at most two points.
Let the line K cut the hypercycle C in two points A and B. As before, we can construct the radius R of C through the middle point M of AB. Note that K is ultraparallel to the axis L because they have the common perpendicular R. Also, two ultraparallel lines have minimum distance at the common perp
|
https://en.wikipedia.org/wiki/Hypercycle
|
Hypercycle may refer to:
Hypercycle (chemistry), a kind of reaction network prominent in a theory of the self-organization of matter
Hypercycle (geometry), a curve in hyperbolic space whose points have the same orthogonal distance from a given straight line
|
https://en.wikipedia.org/wiki/Mathematical%20Correspondent
|
The Mathematical Correspondent was the first American "specialized scientific journal" and the first American mathematics journal, established in 1804, under the editorial guidance of George Baron. The journal published an essay by Robert Adrian which was the first to introduce Diophantine analysis in the United States. In 1807, Adrian, a main contributor to the journal, became editor for one year.
References
Publications established in 1804
Mathematics journals
Defunct journals of the United States
Publications with year of disestablishment missing
|
https://en.wikipedia.org/wiki/List%20of%20Atlas%20launches%20%282000%E2%80%932009%29
|
Notable missions
Mars Reconnaissance Orbiter (MRO)
New Horizons
Lunar Reconnaissance Orbiter (LRO)
Launch statistics
Rocket configurations
Launch sites
Launch outcomes
Launch history
Photo gallery
References
Atlas
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https://en.wikipedia.org/wiki/Bayesian%20multivariate%20linear%20regression
|
In statistics, Bayesian multivariate linear regression is a
Bayesian approach to multivariate linear regression, i.e. linear regression where the predicted outcome is a vector of correlated random variables rather than a single scalar random variable. A more general treatment of this approach can be found in the article MMSE estimator.
Details
Consider a regression problem where the dependent variable to be predicted is not a single real-valued scalar but an m-length vector of correlated real numbers. As in the standard regression setup, there are n observations, where each observation i consists of k−1 explanatory variables, grouped into a vector of length k (where a dummy variable with a value of 1 has been added to allow for an intercept coefficient). This can be viewed as a set of m related regression problems for each observation i:
where the set of errors are all correlated. Equivalently, it can be viewed as a single regression problem where the outcome is a row vector and the regression coefficient vectors are stacked next to each other, as follows:
The coefficient matrix B is a matrix where the coefficient vectors for each regression problem are stacked horizontally:
The noise vector for each observation i is jointly normal, so that the outcomes for a given observation are correlated:
We can write the entire regression problem in matrix form as:
where Y and E are matrices. The design matrix X is an matrix with the observations stacked vertically, as in the standard linear regression setup:
The classical, frequentists linear least squares solution is to simply estimate the matrix of regression coefficients using the Moore-Penrose pseudoinverse:
To obtain the Bayesian solution, we need to specify the conditional likelihood and then find the appropriate conjugate prior. As with the univariate case of linear Bayesian regression, we will find that we can specify a natural conditional conjugate prior (which is scale dependent).
Let us write our conditional likelihood as
writing the error in terms of and yields
We seek a natural conjugate prior—a joint density which is of the same functional form as the likelihood. Since the likelihood is quadratic in , we re-write the likelihood so it is normal in (the deviation from classical sample estimate).
Using the same technique as with Bayesian linear regression, we decompose the exponential term using a matrix-form of the sum-of-squares technique. Here, however, we will also need to use the Matrix Differential Calculus (Kronecker product and vectorization transformations).
First, let us apply sum-of-squares to obtain new expression for the likelihood:
We would like to develop a conditional form for the priors:
where is an inverse-Wishart distribution
and is some form of normal distribution in the matrix . This is accomplished using the vectorization transformation, which converts the likelihood from a function of the matrices to a function of the vectors .
Writ
|
https://en.wikipedia.org/wiki/Continuous%20group%20action
|
In topology, a continuous group action on a topological space X is a group action of a topological group G that is continuous: i.e.,
is a continuous map. Together with the group action, X is called a G-space.
If is a continuous group homomorphism of topological groups and if X is a G-space, then H can act on X by restriction: , making X a H-space. Often f is either an inclusion or a quotient map. In particular, any topological space may be thought of as a G-space via (and G would act trivially.)
Two basic operations are that of taking the space of points fixed by a subgroup H and that of forming a quotient by H. We write for the set of all x in X such that . For example, if we write for the set of continuous maps from a G-space X to another G-space Y, then, with the action ,
consists of f such that ; i.e., f is an equivariant map. We write . Note, for example, for a G-space X and a closed subgroup H, .
References
See also
Lie group action
Group actions (mathematics)
Topological groups
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https://en.wikipedia.org/wiki/Conjugate%20%28square%20roots%29
|
In mathematics, the conjugate of an expression of the form is provided that does not appear in and . One says also that the two expressions are conjugate.
In particular, the two solutions of a quadratic equation are conjugate, as per the in the quadratic formula .
Complex conjugation is the special case where the square root is the imaginary unit.
Properties
As
and
the sum and the product of conjugate expressions do not involve the square root anymore.
This property is used for removing a square root from a denominator, by multiplying the numerator and the denominator of a fraction by the conjugate of the denominator (see Rationalisation). An example of this usage is:
Hence:
A corollary property is that the subtraction:
leaves only a term containing the root.
See also
Conjugate element (field theory), the generalization to the roots of a polynomial of any degree
Elementary algebra
|
https://en.wikipedia.org/wiki/Nested%20intervals
|
In mathematics, a sequence of nested intervals can be intuitively understood as an ordered collection of intervals on the real number line with natural numbers as an index. In order for a sequence of intervals to be considered nested intervals, two conditions have to be met:
Every interval in the sequence is contained in the previous one ( is always a subset of ).
The length of the intervals get arbitrarily small (meaning the length falls below every possible threshold after a certain index ).
In other words, the left bound of the interval can only increase (), and the right bound can only decrease ().
Historically - long before anyone defined nested intervals in a textbook - people implicitly constructed such nestings for concrete calculation purposes. For example, the ancient Babylonians discovered a method for computing square roots of numbers. In contrast, the famed Archimedes constructed sequences of polygons, that inscribed and surcumscribed a unit circle, in order to get a lower and upper bound for the circles circumference - which is the circle number Pi ().
The central question to be posed is the nature of the intersection over all the natural numbers, or, put differently, the set of numbers, that are found in every Interval (thus, for all ). In modern mathematics, nested intervals are used as a construction method for the real numbers (in order to complete the field of rational numbers).
Historic motivation
As stated in the introduction, historic users of mathematics discovered the nesting of intervals and closely related algorithms as methods for specific calculations. Some variations and modern interpretations of these ancient techniques will be introduced here:
Computation of square roots
One intuitive algorithm is so easy to understand, that it could well be found by engaged high school students. When trying to find the square root of a number , one can be certain that , which gives the first interval , in which has to be found. If one knows the next higher perfect square , one can get an even better candidate for the first interval: .
The other intervals can now be defined recursively by looking at the sequence of midpoints . Given the interval is already known (starting at ), one can define
To put this into words, one can compare the midpoint of to in order to determine whether the midpoint is smaller or larger than . If the midpoint is smaller, one can set it as the lower bound of the next interval , and if the midpoint is larger, one can set it as the upper bound of the next interval. This guarantees that . With this construction the intervals are nested and their length get halved in every step of the recursion. Therefore, it is possible to get lower and upper bounds for with arbitrarily good precision (given enough computational time).
One can also compute , when . In this case , and the algorithm can be used by setting and calculating the reciprocal after the desired level of precision has been acqu
|
https://en.wikipedia.org/wiki/Cone%20%28category%20theory%29
|
In category theory, a branch of mathematics, the cone of a functor is an abstract notion used to define the limit of that functor. Cones make other appearances in category theory as well.
Definition
Let F : J → C be a diagram in C. Formally, a diagram is nothing more than a functor from J to C. The change in terminology reflects the fact that we think of F as indexing a family of objects and morphisms in C. The category J is thought of as an "index category". One should consider this in analogy with the concept of an indexed family of objects in set theory. The primary difference is that here we have morphisms as well. Thus, for example, when J is a discrete category, it corresponds most closely to the idea of an indexed family in set theory. Another common and more interesting example takes J to be a span. J can also be taken to be the empty category, leading to the simplest cones.
Let N be an object of C. A cone from N to F is a family of morphisms
for each object X of J, such that for every morphism f : X → Y in J the following diagram commutes:
The (usually infinite) collection of all these triangles can
be (partially) depicted in the shape of a cone with the apex N. The cone ψ is sometimes said to have vertex N and base F.
One can also define the dual notion of a cone from F to N (also called a co-cone) by reversing all the arrows above. Explicitly, a co-cone from F to N is a family of morphisms
for each object X of J, such that for every morphism f : X → Y in J the following diagram commutes:
Equivalent formulations
At first glance cones seem to be slightly abnormal constructions in category theory. They are maps from an object to a functor (or vice versa). In keeping with the spirit of category theory we would like to define them as morphisms or objects in some suitable category. In fact, we can do both.
Let J be a small category and let CJ be the category of diagrams of type J in C (this is nothing more than a functor category). Define the diagonal functor Δ : C → CJ as follows: Δ(N) : J → C is the constant functor to N for all N in C.
If F is a diagram of type J in C, the following statements are equivalent:
ψ is a cone from N to F
ψ is a natural transformation from Δ(N) to F
(N, ψ) is an object in the comma category (Δ ↓ F)
The dual statements are also equivalent:
ψ is a co-cone from F to N
ψ is a natural transformation from F to Δ(N)
(N, ψ) is an object in the comma category (F ↓ Δ)
These statements can all be verified by a straightforward application of the definitions. Thinking of cones as natural transformations we see that they are just morphisms in CJ with source (or target) a constant functor.
Category of cones
By the above, we can define the category of cones to F as the comma category (Δ ↓ F). Morphisms of cones are then just morphisms in this category. This equivalence is rooted in the observation that a natural map between constant functors Δ(N), Δ(M) corresponds to a morphism between N and M. In this s
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https://en.wikipedia.org/wiki/Dorian%20M.%20Goldfeld
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Dorian Morris Goldfeld (born January 21, 1947) is an American mathematician working in analytic number theory and automorphic forms at Columbia University.
Professional career
Goldfeld received his B.S. degree in 1967 from Columbia University. His doctoral dissertation, entitled "Some Methods of Averaging in the Analytical Theory of Numbers", was completed under the supervision of Patrick X. Gallagher in 1969, also at Columbia. He has held positions at the University of California at Berkeley (Miller Fellow, 1969–1971), Hebrew University (1971–1972), Tel Aviv University (1972–1973), Institute for Advanced Study (1973–1974), in Italy (1974–1976), at MIT (1976–1982), University of Texas at Austin (1983–1985) and Harvard (1982–1985). Since 1985, he has been a professor at Columbia University.
He is a member of the editorial board of Acta Arithmetica and of The Ramanujan Journal. On January 1, 2018 he became the Editor-in-Chief of the Journal of Number Theory.
He is a co-founder and board member of Veridify Security, formerly SecureRF, a corporation that has developed the world's first linear-based security solutions.
Goldfeld advised several doctoral students including M. Ram Murty. In 1986, he brought Shou-Wu Zhang to the United States to study at Columbia.
Research interests
Goldfeld's research interests include various topics in number theory. In his thesis, he proved a version of Artin's conjecture on primitive roots on the average without the use of the Riemann Hypothesis.
In 1976, Goldfeld provided an ingredient for the effective solution of Gauss' class number problem for imaginary quadratic fields. Specifically, he proved an effective lower bound for the class number of an imaginary quadratic field assuming the existence of an elliptic curve whose L-function had a zero of order at least 3 at . (Such a curve was found soon after by Gross and Zagier). This effective lower bound then allows the determination of all imaginary fields with a given class number after a finite number of computations.
His work on the Birch and Swinnerton-Dyer conjecture includes the proof of an estimate for a partial Euler product associated to an elliptic curve, bounds for the
order of the Tate–Shafarevich group.
Together with his collaborators, Dorian Goldfeld has introduced the theory of multiple Dirichlet series, objects that extend the fundamental Dirichlet series in one variable.
He has also made contributions to the understanding of Siegel zeroes, to the ABC conjecture, to modular forms on , and to cryptography (Arithmetica cipher, Anshel–Anshel–Goldfeld key exchange).
Together with his wife, Dr. Iris Anshel, and father-in-law, Dr. Michael Anshel, both mathematicians, Dorian Goldfeld founded the field of braid group cryptography.
Awards and honors
In 1987 he received the Frank Nelson Cole Prize in Number Theory, one of the prizes in Number Theory, for his solution of Gauss' class number problem for imaginary quadratic fields. He has also held the
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https://en.wikipedia.org/wiki/Tacnode
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In classical algebraic geometry, a tacnode (also called a point of osculation or double cusp) is a kind of singular point of a curve. It is defined as a point where two (or more) osculating circles to the curve at that point are tangent. This means that two branches of the curve have ordinary tangency at the double point.
The canonical example is
A tacnode of an arbitrary curve may then be defined from this example, as a point of self-tangency locally diffeomorphic to the point at the origin of this curve. Another example of a tacnode is given by the links curve shown in the figure, with equation
More general background
Consider a smooth real-valued function of two variables, say where and are real numbers. So is a function from the plane to the line. The space of all such smooth functions is acted upon by the group of diffeomorphisms of the plane and the diffeomorphisms of the line, i.e. diffeomorphic changes of coordinate in both the source and the target. This action splits the whole function space up into equivalence classes, i.e. orbits of the group action.
One such family of equivalence classes is denoted by where is a non-negative integer. This notation was introduced by V. I. Arnold. A function is said to be of type if it lies in the orbit of i.e. there exists a diffeomorphic change of coordinate in source and target which takes into one of these forms. These simple forms are said to give normal forms for the type -singularities.
A curve with equation will have a tacnode, say at the origin, if and only if has a type -singularity at the origin.
Notice that a node corresponds to a type -singularity. A tacnode corresponds to a type -singularity. In fact each type -singularity, where is an integer, corresponds to a curve with self-intersection. As increases, the order of self-intersection increases: transverse crossing, ordinary tangency, etc.
The type -singularities are of no interest over the real numbers: they all give an isolated point. Over the complex numbers, type -singularities and type -singularities are equivalent: gives the required diffeomorphism of the normal forms.
See also
Acnode
Cusp or Spinode
Crunode
References
Further reading
External links
Curves
Singularity theory
Algebraic curves
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https://en.wikipedia.org/wiki/Locally%20simply%20connected%20space
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In mathematics, a locally simply connected space is a topological space that admits a basis of simply connected sets. Every locally simply connected space is also locally path-connected and locally connected.
The circle is an example of a locally simply connected space which is not simply connected. The Hawaiian earring is a space which is neither locally simply connected nor simply connected. The cone on the Hawaiian earring is contractible and therefore simply connected, but still not locally simply connected.
All topological manifolds and CW complexes are locally simply connected. In fact, these satisfy the much stronger property of being locally contractible.
A strictly weaker condition is that of being semi-locally simply connected. Both locally simply connected spaces and simply connected spaces are semi-locally simply connected, but neither converse holds.
References
Properties of topological spaces
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https://en.wikipedia.org/wiki/Public%20library%20ratings
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There are several national systems for assessing, evaluating, or otherwise rating the quality of public libraries.
United States
Basic library statistics (not rankings) were initially maintained by the National Center for Educational Statistics; that body continues to collect data for academic libraries, but administration of the Public Libraries Survey and the State Library Agencies Survey was transferred to the Institute of Museum and Library Services (IMLS) in October 2007. IMLS continues to conduct public library surveys as well as distribute historical data from surveys back to 1988.
The Library Data Archives includes longitudinal data sets.
HAPLR and subsequent debate
The system that would become Hennen's American Public Library Ratings (HAPLR) was first published in the January 1999 issue of American Libraries prepared by Thomas J. Hennen Jr., Director of Waukesha County Federated Library System in Wisconsin. Libraries were ranked on 15 measures with comparisons in broad population categories. HAPLR was updated annually through 2010 and was the focus of widespread professional debate in the field of librarianship.
Oregon State Librarian Jim Scheppke noted that the statistics that HAPLR relies on are misleading because they rely too much on output measures, such as circulation, funding, etc. and not on input measures, such as open hours and patron satisfaction. "To give HAPLR some credit, collectively, the libraries in the top half of the list are definitely better than the libraries in the bottom half, but when it gets down to individual cases, which is what HAPLR claims to be able to do, it doesn't work."
In contrast, Library Journal editor, John N. Berry, noted: "Unfortunately, when you or your library receives any kind of honor, it stimulates the flow of competitive hormones in your professional colleagues. This jealousy rears its ugly head in many ways. We've suffered endless tutorials on the defects in Hennen's rankings. So what? They work!"
Keith Curry Lance and Marti Cox, both of the Library Research Service, took issue with HAPLR reasoning backwards from statistics to conclusion, point out the redundancy of HAPLR's statistical categories, and question its arbitrary system of weighting criteria.
Hennen responded, saying Lance and Cox seem to suggest "that the job of comparing libraries cannot be done, so I am at fault for having tried. Somehow, unique among American public or private institutions, libraries are just too varied and too local to be compared. Yet despite these assertions, the authors urge individuals to use the NCES Public Library Peer Comparison tool (nces.ed.gov/surveys/libraries/publicpeer/) to do this impossible task."
A 2006 Library School Student Writing Award article questioned HAPLR's weighting of factors, and its failure to account for local factors (such as a library's mission) in measuring a library's success, the index's failure to measure computer and Internet usage, and its lack of focus of on
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https://en.wikipedia.org/wiki/Hyperconnected%20space
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In the mathematical field of topology, a hyperconnected space or irreducible space is a topological space X that cannot be written as the union of two proper closed sets (whether disjoint or non-disjoint). The name irreducible space is preferred in algebraic geometry.
For a topological space X the following conditions are equivalent:
No two nonempty open sets are disjoint.
X cannot be written as the union of two proper closed sets.
Every nonempty open set is dense in X.
The interior of every proper closed set is empty.
Every subset is dense or nowhere dense in X.
No two points can be separated by disjoint neighbourhoods.
A space which satisfies any one of these conditions is called hyperconnected or irreducible. Due to the condition about neighborhoods of distinct points being in a sense the opposite of the Hausdorff property, some authors call such spaces anti-Hausdorff.
An irreducible set is a subset of a topological space for which the subspace topology is irreducible. Some authors do not consider the empty set to be irreducible (even though it vacuously satisfies the above conditions).
Examples
Two examples of hyperconnected spaces from point set topology are the cofinite topology on any infinite set and the right order topology on .
In algebraic geometry, taking the spectrum of a ring whose reduced ring is an integral domain is an irreducible topological space—applying the lattice theorem to the nilradical, which is within every prime, to show the spectrum of the quotient map is a homeomorphism, this reduces to the irreducibility of the spectrum of an integral domain. For example, the schemes , are irreducible since in both cases the polynomials defining the ideal are irreducible polynomials (meaning they have no non-trivial factorization). A non-example is given by the normal crossing divisorsince the underlying space is the union of the affine planes , , and . Another non-example is given by the schemewhere is an irreducible degree 4 homogeneous polynomial. This is the union of the two genus 3 curves (by the genus–degree formula)
Hyperconnectedness vs. connectedness
Every hyperconnected space is both connected and locally connected (though not necessarily path-connected or locally path-connected).
Note that in the definition of hyper-connectedness, the closed sets don't have to be disjoint. This is in contrast to the definition of connectedness, in which the open sets are disjoint.
For example, the space of real numbers with the standard topology is connected but not hyperconnected. This is because it cannot be written as a union of two disjoint open sets, but it can be written as a union of two (non-disjoint) closed sets.
Properties
The nonempty open subsets of a hyperconnected space are "large" in the sense that each one is dense in X and any pair of them intersects. Thus, a hyperconnected space cannot be Hausdorff unless it contains only a single point.
Every hyperconnected space is both connected and locally connec
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https://en.wikipedia.org/wiki/1979%20Machchhu%20dam%20failure
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{
"type": "FeatureCollection",
"features": [ {
"type": "Feature",
"properties": { "marker-symbol": "dam", "marker-size": "small", "title": "Machhu dam" },
"geometry": { "type": "Point", "coordinates": [70.865833, 22.763889] }
},
{
"type": "Feature",
"properties": { "marker-symbol": "town", "marker-size": "small", "title": "Morbi" },
"geometry": { "type": "Point", "coordinates": [70.83, 22.82] }
},
]
}
The Machchhu dam failure or Morbi disaster is a dam-related flood disaster which occurred on 11 August 1979. The Machchu-2 dam, situated on the Machchhu river, failed, sending a wall of water through the town of Morbi (now in the Morbi district) of Gujarat, India. Estimates of the number of people killed vary greatly ranging from 1,800 to 25,000 people.
The Machchu II dam
The first dam on the Machchhu river, named Machchhu I, was built in 1959, having a catchment area of . The Machchhu II dam was constructed downstream of Machchhu I in 1972, and has a catchment area of .
It was an earthfill dam. The dam was meant to serve an irrigation scheme. Considering the long history of drought in Saurashtra region, the primary consideration at the time of design was water supply, not flood control. It consisted of a masonry spillway of consisting 18 sluice gates across the river section and long earthen embankments on both sides. The spillway capacity provided for . The embankments were of and of length on left and right side respectively. The embankments had a 6.1 m top width, with upstream and downstream slopes 1:3 (V:H) and 1:2 respectively; and a clay core extending through alluvium to bedrock. The upstream face consisted of 61 cm small gravel and a 61 cm hand packed rip-rap. The dam stood above the river bed and its overflow section was long. The reservoir had a storage capacity of .
Failure
The failure was caused by excessive rain and massive flooding leading to the disintegration of the earthen walls of the four kilometre long Machchhu-2 dam. The actual observed flow following the intense rainfall reached 16,307 m3/s, thrice what the dam was designed for, resulting in its collapse. of the left and of the right embankment of the dam collapsed. Within 20 minutes the floods of height inundated the low-lying areas of Morbi industrial town located 5 km below the dam.
Around 3.30 pm the tremendous swirling flow of water struck Morbi. Water level rose to within the next 15 minutes and some low lying areas of city were under of water for the next 6 hours.
The Morbi dam failure was listed as the worst dam burst in the Guinness Book of Records (before the death toll of the 1975 Banqiao Dam failure was declassified in 2005). The book No One Had A Tongue To Speak by Tom Wooten and Utpal Sandesara debunks the official claims that the dam failure was an act of God and points to structural and communication failures that led to and exacerbated the disaster. There was great economic loss. The flood damaged farmland, leading t
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https://en.wikipedia.org/wiki/%C3%89tienne%20Ghys
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Étienne Ghys (born 29 December 1954) is a French mathematician. His research focuses mainly on geometry and dynamical systems, though his mathematical interests are broad. He also expresses much interest in the historical development of mathematical ideas, especially the contributions of Henri Poincaré.
Alumnus of the École normale supérieure de Saint-Cloud, Ghys obtained his PhD in 1979 from the University of Lille with thesis Sur les actions localement libres du groupe affine written under the direction of Gilbert Hector.
He is currently a CNRS directeur de recherche at the École normale supérieure in Lyon. He is also editor-in-chief of the Publications Mathématiques de l'IHÉS and a member of the French Academy of Sciences.
Ghys was an invited speaker at the International Congress of Mathematicians (ICM) in Kyoto in 1990, and a plenary speaker at the ICM in Madrid in 2006. In 2015, he was awarded the inaugural Clay Award for Dissemination of Mathematical Knowledge.
He co-authored the computer graphics mathematical movie Dimensions: A walk through mathematics!. His doctoral students include Serge Cantat.
External links
« Notice sur les travaux scientifiques d'Étienne Ghys », an overview of his mathematical interests and results, written for his entry at the French Academy of Sciences.
References
1954 births
Living people
20th-century French mathematicians
French geometers
Members of the French Academy of Sciences
ENS Fontenay-Saint-Cloud-Lyon alumni
Dynamical systems theorists
Members of Academia Europaea
University of Lille Nord de France alumni
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https://en.wikipedia.org/wiki/Steinmetz%20solid
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In geometry, a Steinmetz solid is the solid body obtained as the intersection of two or three cylinders of equal radius at right angles. Each of the curves of the intersection of two cylinders is an ellipse.
The intersection of two cylinders is called a bicylinder. Topologically, it is equivalent to a square hosohedron. The intersection of three cylinders is called a tricylinder. A bisected bicylinder is called a vault, and a cloister vault in architecture has this shape.
Steinmetz solids are named after mathematician Charles Proteus Steinmetz, who solved the problem of determining the volume of the intersection. However, the same problem had been solved earlier, by Archimedes in the ancient Greek world, Zu Chongzhi in ancient China, and Piero della Francesca in the early Italian Renaissance. They appear prominently in the sculptures of Frank Smullin.
Bicylinder
A bicylinder generated by two cylinders with radius has the
volume
and the
surface area
.
The upper half of a bicylinder is the square case of a domical vault, a dome-shaped solid based on any convex polygon whose cross-sections are similar copies of the polygon, and analogous formulas calculating the volume and surface area of a domical vault as a rational multiple of the volume and surface area of its enclosing prism hold more generally. In China, the bicylinder is known as Mou he fang gai, literally "two square umbrella"; it was described by the third-century mathematician Liu Hui.
Proof of the volume formula
For deriving the volume formula it is convenient to use the common idea for calculating the volume of a sphere: collecting thin cylindric slices. In this case the thin slices are square cuboids (see diagram). This leads to
.
It is well known that the relations of the volumes of a right circular cone, one half of a sphere and a right circular cylinder with same radii and heights are 1 : 2 : 3. For one half of a bicylinder a similar statement is true:
The relations of the volumes of the inscribed square pyramid (), the half bicylinder () and the surrounding squared cuboid () are 1 : 2 : 3.
Using Multivariable Calculus
Consider the equations of the cylinders:
The volume will be given by:
With the limits of integration:
Substituting, we have:
Proof of the area formula
The surface area consists of two red and two blue cylindrical biangles. One red biangle is cut into halves by the y-z-plane and developed into the plane such that half circle (intersection with the y-z-plane) is developed onto the positive -axis and the development of the biangle is bounded upwards by the sine arc . Hence the area of this development is
and the total surface area is:
.
Alternate proof of the volume formula
Deriving the volume of a bicylinder (white) can be done by packing it in a cube (red). A plane (parallel with the cylinders' axes) intersecting the bicylinder forms a square and its intersection with the cube is a larger square. The difference between the areas of the two square
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https://en.wikipedia.org/wiki/AJM
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AJM may refer to:
Distinguished Young Women, formerly known as America's Junior Miss
Air Jamaica (ICAO airline designator AJM)
Abrasive jet machining
American Journal of Mathematics
Association des Juristes Maliennes, an association of women jurists in Mali
Australian Jazz Museum
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https://en.wikipedia.org/wiki/W.%20R.%20%28Red%29%20Alford
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William Robert "Red" Alford, Ph.D, J.D. (July 21, 1937 – May 29, 2003) was an American mathematician and lawyer, who was best known for his work in the fields of topology and number theory.
Personal life
Alford was born in Canton, Mississippi, to parents Clayton and Pennington Alford. After graduating high school, Alford became a member of the United States Air Force, and earned his Bachelor of Science in mathematics and physics from The Citadel (1959). Subsequently, he earned his Ph.D in mathematics from Tulane University (1963), and his J.D. from the University of Georgia School of Law (1976). After earning his J.D., he practiced law in Athens, Georgia, before returning to the mathematics faculty at the University of Georgia. He retired in 2002, and passed the next year after suffering from a brain tumor.
Mathematics
Alford's dissertation at Tulane was entitled: Some Wild Embeddings of the One and Two Dimensional Spheres in the Three Sphere
In 1994, in a paper with Andrew Granville and Carl Pomerance, he proved the infinitude of Carmichael numbers based on a conjecture given by Paul Erdős.
MathSciNet credits Alford with eleven publications, of which two were in the prestigious Annals of Mathematics-the Carmichael numbers paper, and a 1970 paper in knot theory.
External links
1937 births
2003 deaths
Deaths from brain cancer in the United States
20th-century American mathematicians
21st-century American mathematicians
Number theorists
University of Georgia faculty
University of Georgia alumni
United States Air Force airmen
The Citadel, The Military College of South Carolina alumni
Georgia (U.S. state) lawyers
Mathematicians from Mississippi
Mathematicians from Georgia (U.S. state)
20th-century American lawyers
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https://en.wikipedia.org/wiki/Pierre%20de%20Fermat
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Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he is recognized for his discovery of an original method of finding the greatest and the smallest ordinates of curved lines, which is analogous to that of differential calculus, then unknown, and his research into number theory. He made notable contributions to analytic geometry, probability, and optics. He is best known for his Fermat's principle for light propagation and his Fermat's Last Theorem in number theory, which he described in a note at the margin of a copy of Diophantus' Arithmetica. He was also a lawyer at the Parlement of Toulouse, France.
Biography
Fermat was born in 1607 in Beaumont-de-Lomagne, France—the late 15th-century mansion where Fermat was born is now a museum. He was from Gascony, where his father, Dominique Fermat, was a wealthy leather merchant and served three one-year terms as one of the four consuls of Beaumont-de-Lomagne. His mother was Claire de Long. Pierre had one brother and two sisters and was almost certainly brought up in the town of his birth.
He attended the University of Orléans from 1623 and received a bachelor in civil law in 1626, before moving to Bordeaux. In Bordeaux, he began his first serious mathematical researches, and in 1629 he gave a copy of his restoration of Apollonius's De Locis Planis to one of the mathematicians there. Certainly, in Bordeaux he was in contact with Beaugrand and during this time he produced important work on maxima and minima which he gave to Étienne d'Espagnet who clearly shared mathematical interests with Fermat. There he became much influenced by the work of François Viète.
In 1630, he bought the office of a councilor at the Parlement de Toulouse, one of the High Courts of Judicature in France, and was sworn in by the Grand Chambre in May 1631. He held this office for the rest of his life. Fermat thereby became entitled to change his name from Pierre Fermat to Pierre de Fermat. On 1 June 1631, Fermat married Louise de Long, a fourth cousin of his mother Claire de Fermat (née de Long). The Fermats had eight children, five of whom survived to adulthood: Clément-Samuel, Jean, Claire, Catherine, and Louise.
Fluent in six languages (French, Latin, Occitan, classical Greek, Italian and Spanish), Fermat was praised for his written verse in several languages and his advice was eagerly sought regarding the emendation of Greek texts. He communicated most of his work in letters to friends, often with little or no proof of his theorems. In some of these letters to his friends, he explored many of the fundamental ideas of calculus before Newton or Leibniz. Fermat was a trained lawyer making mathematics more of a hobby than a profession. Nevertheless, he made important contributions to analytical geometry, probability, number theory and calculus.
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https://en.wikipedia.org/wiki/Orbit%20portrait
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In mathematics, an orbit portrait is a combinatorial tool used in complex dynamics for understanding the behavior of one-complex dimensional quadratic maps.
In simple words one can say that it is :
a list of external angles for which rays land on points of that orbit
graph showing above list
Definition
Given a quadratic map
from the complex plane to itself
and a repelling or parabolic periodic orbit of , so that (where subscripts are taken 1 + modulo ), let be the set of angles whose corresponding external rays land at .
Then the set is called the orbit portrait of the periodic orbit .
All of the sets must have the same number of elements, which is called the valence of the portrait.
Examples
Parabolic or repelling orbit portrait
valence 2
valence 3
Valence is 3 so rays land on each orbit point.
For complex quadratic polynomial with c= -0.03111+0.79111*i portrait of parabolic period 3 orbit is :
Rays for above angles land on points of that orbit . Parameter c is a center of period 9 hyperbolic component of Mandelbrot set.
For parabolic julia set c = -1.125 + 0.21650635094611*i. It is a root point between period 2 and period 6 components of Mandelbrot set. Orbit portrait of period 2 orbit with valence 3 is :
valence 4
Formal orbit portraits
Every orbit portrait has the following properties:
Each is a finite subset of
The doubling map on the circle gives a bijection from to and preserves cyclic order of the angles.
All of the angles in all of the sets are periodic under the doubling map of the circle, and all of the angles have the same exact period. This period must be a multiple of , so the period is of the form , where is called the recurrent ray period.
The sets are pairwise unlinked, which is to say that given any pair of them, there are two disjoint intervals of where each interval contains one of the sets.
Any collection of subsets of the circle which satisfy these four properties above is called a formal orbit portrait. It is a theorem of John Milnor that every formal orbit portrait is realized by the actual orbit portrait of a periodic orbit of some quadratic one-complex-dimensional map. Orbit portraits contain dynamical information about how external rays and their landing points map in the plane, but formal orbit portraits are no more than combinatorial objects. Milnor's theorem states that, in truth, there is no distinction between the two.
Trivial orbit portraits
Orbit portrait where all of the sets have only a single element are called trivial, except for orbit portrait . An alternative definition is that an orbit portrait is nontrivial if it is maximal, which in this case means that there is no orbit portrait that strictly contains it (i.e. there does not exist an orbit portrait such that ). It is easy to see that every trivial formal orbit portrait is realized as the orbit portrait of some orbit of the map , since every external ray of this map lands, and they all land at distinct point
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https://en.wikipedia.org/wiki/Beckman%E2%80%93Quarles%20theorem
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In geometry, the Beckman–Quarles theorem states that if a transformation of the Euclidean plane or a higher-dimensional Euclidean space preserves unit distances, then it preserves all Euclidean distances. Equivalently, every homomorphism from the unit distance graph of the plane to itself must be an isometry of the plane. The theorem is named after Frank S. Beckman and Donald A. Quarles Jr., who published this result in 1953; it was later rediscovered by other authors and re-proved in multiple ways. Analogous theorems for rational subsets of Euclidean spaces, or for non-Euclidean geometry, are also known.
Statement and proof idea
Formally, the result is as follows. Let be a function or multivalued function from a -dimensional Euclidean space to itself, and suppose that, for every pair of points and that are at unit distance from each other, every pair of images and are also at unit distance from each other. Then must be an isometry: it is a one-to-one function that preserves distances between all pairs of
One way of rephrasing the Beckman–Quarles theorem involves graph homomorphisms, mappings between undirected graphs that take vertices to vertices and edges to edges. For the unit distance graph whose vertices are all of the points in the plane, with an edge between any two points at unit distance, a homomorphism from this graph to itself is the same thing as a unit-distance-preserving transformation of the plane. Thus, the Beckman–Quarles theorem states that the only homomorphisms from this graph to itself are the obvious ones coming from isometries of the For this graph, all homomorphisms are symmetries of the graph, the defining property of a class of graphs called
As well as the original proofs of Beckman and Quarles of the theorem, and the proofs in later papers rediscovering the several alternative proofs have been If is the set of distances preserved by a then it follows from the triangle inequality that certain comparisons of other distances with members of are preserved Therefore, if can be shown to be a dense set, then all distances must be preserved. The main idea of several proofs of the Beckman–Quarles theorem is to use the structural rigidity of certain unit distance graphs, such as the graph of a regular simplex, to show that a mapping that preserves unit distances must preserve enough other distances to form a
Counterexamples for other spaces
Beckman and Quarles observe that the theorem is not true for the real line (one-dimensional Euclidean space). As an example, consider the function that returns if is an integer and returns otherwise. This function obeys the preconditions of the theorem: it preserves unit distances. However, it does not preserve the distances between integers and
Beckman and Quarles provide another counterexample showing that their theorem cannot be generalized to an infinite-dimensional space, the Hilbert space of square-summable sequences of real numbers. "Square-summable" means tha
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https://en.wikipedia.org/wiki/Dispersion%20point
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In topology, a dispersion point or explosion point is a point in a topological space the removal of which leaves the space highly disconnected.
More specifically, if X is a connected topological space containing the point p and at least two other points, p is a dispersion point for X if and only if is totally disconnected (every subspace is disconnected, or, equivalently, every connected component is a single point). If X is connected and is totally separated (for each two points x and y there exists a clopen set containing x and not containing y) then p is an explosion point. A space can have at most one dispersion point or explosion point. Every totally separated space is totally disconnected, so every explosion point is a dispersion point.
The Knaster–Kuratowski fan has a dispersion point; any space with the particular point topology has an explosion point.
If p is an explosion point for a space X, then the totally separated space is said to be pulverized.
References
. (Note that this source uses hereditarily disconnected and totally disconnected for the concepts referred to here respectively as totally disconnected and totally separated.)
Topology
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https://en.wikipedia.org/wiki/5-orthoplex
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In five-dimensional geometry, a 5-orthoplex, or 5-cross polytope, is a five-dimensional polytope with 10 vertices, 40 edges, 80 triangle faces, 80 tetrahedron cells, 32 5-cell 4-faces.
It has two constructed forms, the first being regular with Schläfli symbol {33,4}, and the second with alternately labeled (checkerboarded) facets, with Schläfli symbol {3,3,31,1} or Coxeter symbol 211.
It is a part of an infinite family of polytopes, called cross-polytopes or orthoplexes. The dual polytope is the 5-hypercube or 5-cube.
Alternate names
pentacross, derived from combining the family name cross polytope with pente for five (dimensions) in Greek.
Triacontaditeron (or triacontakaiditeron) - as a 32-facetted 5-polytope (polyteron).
As a configuration
This configuration matrix represents the 5-orthoplex. The rows and columns correspond to vertices, edges, faces, cells and 4-faces. The diagonal numbers say how many of each element occur in the whole 5-orthoplex. The nondiagonal numbers say how many of the column's element occur in or at the row's element.
Cartesian coordinates
Cartesian coordinates for the vertices of a 5-orthoplex, centered at the origin are
(±1,0,0,0,0), (0,±1,0,0,0), (0,0,±1,0,0), (0,0,0,±1,0), (0,0,0,0,±1)
Construction
There are three Coxeter groups associated with the 5-orthoplex, one regular, dual of the penteract with the C5 or [4,3,3,3] Coxeter group, and a lower symmetry with two copies of 5-cell facets, alternating, with the D5 or [32,1,1] Coxeter group, and the final one as a dual 5-orthotope, called a 5-fusil which can have a variety of subsymmetries.
Other images
Related polytopes and honeycombs
This polytope is one of 31 uniform 5-polytopes generated from the B5 Coxeter plane, including the regular 5-cube and 5-orthoplex.
References
H.S.M. Coxeter:
H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
(Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
(Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
Norman Johnson Uniform Polytopes, Manuscript (1991)
N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966)
External links
Polytopes of Various Dimensions
Multi-dimensional Glossary
5-polytopes
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https://en.wikipedia.org/wiki/Iran%20Bioinformatics%20Center
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Iran Bioinformatics Center (IBC) is the only academic center in Iran working on Bioinformatics. Although there are some independent research groups such as Bioinformatics and Biomathematics Unit in Mazandaran University of Medical Sciences working on Bioinformatics but IBC is a part of Institute of Biochemistry and Biophysics (IBB) in Tehran University. IBC offers a Ph.D. program for Bioinformatics.
See also
Institute of Biochemistry and Biophysics
Tehran University
Bioinformatics and Biomathematics Unit
Bioinformatics
External links
Iran Bioinformatics Center
Institute of Biophysics and Biochemistry
Bioinformatics Center of Institute of Bio-IT
University of Tehran
Bioinformatics organizations
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https://en.wikipedia.org/wiki/Division%20No.%204%2C%20Newfoundland%20and%20Labrador
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Census Division No. 4 is a Statistics Canada statistical division that comprises the areas of the province of Newfoundland and Labrador called St. George's. It covers a land area of 7087.65 km² and had a population of 20,387 at the 2016 census.
Towns
Cape St. George
Gallants
Kippens
Lourdes
Port au Port East
Port au Port West-Aguathuna-Felix Cove
St. George's
Stephenville
Stephenville Crossing
Unorganized subdivisions
Subdivision A (including Codroy, Cape Anguille, Doyles, South Branch)
Subdivision B (including Highlands, Jeffrey’s, Robinsons)
Subdivision C (including St. Teresa, Flat Bay, Barachois Brook)
Subdivision D (including Fox Island River)
Subdivision E (including Mainland)
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Division No. 4 had a population of living in of its total private dwellings, a change of from its 2016 population of . With a land area of , it had a population density of in 2021.
References
Sources
004
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https://en.wikipedia.org/wiki/One-sided
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One-sided may refer to:
Biased
One-sided argument, a logical fallacy
In calculus, one-sided limit, either of the two limits of a function of a real variable as approaches a specified point
One-sided (algebra)
One-sided overhand bend, simple method of joining two cords or threads together
One-sided test, a statistical test
See also
Unilateralism
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https://en.wikipedia.org/wiki/Dudley%20E.%20Littlewood
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Dudley Ernest Littlewood (7 September 1903, London –
6 October 1979, Llandudno) was a British mathematician known for his work in group representation theory.
He read mathematics at Trinity College, Cambridge, where his tutor was John Edensor Littlewood (they were not related). He was a lecturer at University College, Swansea from 1928 to 1947, and in 1948 took up the chair of mathematics at University College of North Wales, Bangor, retiring in 1970.
He worked on invariant theory and group representation theory, especially of the symmetric group, often in collaboration with Archibald Read Richardson of Swansea. They introduced the immanant of a matrix, studied Schur functions and developed the Littlewood–Richardson rule for their multiplication. Littlewood was also interested in the application of representation theory to quantum mechanics.
Selected publications
The theory of group characters and matrix representations of groups. 1940; 2nd edition Oxford 1950.
The skeleton key of mathematics: a simple account of complex algebraic theories, Hutchison & Company, London, 1949; 2nd edition Harper & Brothers, New York, 1960; 2002 Dover pbk reprint; 2013 Dover ebook edition
A university algebra. (1950); 2nd edition 1961.
See also
Hall–Littlewood polynomials
Plethysm
Restricted representation
References
External links
1903 births
1976 deaths
20th-century British mathematicians
Group theorists
Alumni of Trinity College, Cambridge
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https://en.wikipedia.org/wiki/Archibald%20Read%20Richardson
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Archibald Read Richardson FRS (21 August 1881 – 4 November 1954) was a British mathematician known for his work in algebra.
Career
Richardson collaborated with Dudley E. Littlewood on invariants and group representation theory. They introduced the immanant of a matrix, studied Schur functions and developed the Littlewood–Richardson rule for their multiplication.
Awards and honours
Richardson was elected a Fellow of the Royal Society on 21 March 1946.
See also
Quasideterminant
References
1881 births
1954 deaths
20th-century British mathematicians
Algebraists
Fellows of the Royal Society
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https://en.wikipedia.org/wiki/George%20Lilley
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George Lilley may refer to:
George L. Lilley (1859–1909), United States Representative and Governor of Connecticut
George W. Lilley (1850–1904), American academic, professor of mathematics and university president
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https://en.wikipedia.org/wiki/Hall%20algebra
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In mathematics, the Hall algebra is an associative algebra with a basis corresponding to isomorphism classes of finite abelian p-groups. It was first discussed by but forgotten until it was rediscovered by , both of whom published no more than brief summaries of their work. The Hall polynomials are the structure constants of the Hall algebra. The Hall algebra plays an important role in the theory of Masaki Kashiwara and George Lusztig regarding canonical bases in quantum groups. generalized Hall algebras to more general categories, such as the category of representations of a quiver.
Construction
A finite abelian p-group M is a direct sum of cyclic p-power components where
is a partition of called the type of M. Let be the number of subgroups N of M such that N has type and the quotient M/N has type . Hall proved that the functions g are polynomial functions of p with integer coefficients. Thus we may replace p with an indeterminate q, which results in the Hall polynomials
Hall next constructs an associative ring over , now called the Hall algebra. This ring has a basis consisting of the symbols and the structure constants of the multiplication in this basis are given by the Hall polynomials:
It turns out that H is a commutative ring, freely generated by the elements corresponding to the elementary p-groups. The linear map from H to the algebra of symmetric functions defined on the generators by the formula
(where en is the nth elementary symmetric function) uniquely extends to a ring homomorphism and the images of the basis elements may be interpreted via the Hall–Littlewood symmetric functions. Specializing q to 0, these symmetric functions become Schur functions, which are thus closely connected with the theory of Hall polynomials.
References
George Lusztig, Quivers, perverse sheaves, and quantized enveloping algebras, Journal of the American Mathematical Society 4 (1991), no. 2, 365–421.
Algebras
Invariant theory
Symmetric functions
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https://en.wikipedia.org/wiki/Division%20No.%204%2C%20Subdivision%20B%2C%20Newfoundland%20and%20Labrador
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Division No. 4, Subd. B is an unorganized subdivision on St. George's Bay on the island of Newfoundland in Newfoundland and Labrador, Canada. It is in Division No. 4.
According to the 2016 Statistics Canada Census:
Population: 1174
% Change (2011 to 2016): -9.6%
Dwellings: 948
Area: 1847.38 km2
Density: 0.6 people/km2
Division No. 4, Subd. B includes the unincorporated communities of
Cartyville
Heatherton
Highlands
Jeffrey's
Loch Leven
McKay's
Robinsons
St. Fintan's
St. David's
References
Newfoundland and Labrador subdivisions
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https://en.wikipedia.org/wiki/James%20Munkres
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James Raymond Munkres (born August 18, 1930) is a Professor Emeritus of mathematics at MIT and the author of several texts in the area of topology, including Topology (an undergraduate-level text), Analysis on Manifolds, Elements of Algebraic Topology, and Elementary Differential Topology. He is also the author of Elementary Linear Algebra.
Munkres completed his undergraduate education at Nebraska Wesleyan University and received his Ph.D. from the University of Michigan in 1956; his advisor was Edwin E. Moise. Earlier in his career he taught at the University of Michigan and at Princeton University.
Among Munkres' contributions to mathematics is the development of what is sometimes called the Munkres assignment algorithm. A significant contribution in topology is his obstruction theory for the smoothing of homeomorphisms. These developments establish a connection between the John Milnor groups of differentiable structures on spheres and the smoothing methods of classical analysis.
He was elected to the 2018 class of fellows of the American Mathematical Society.
Textbooks
References
External links
20th-century American mathematicians
21st-century American mathematicians
American textbook writers
American male non-fiction writers
Topologists
Nebraska Wesleyan University alumni
University of Michigan alumni
University of Michigan faculty
Princeton University faculty
Massachusetts Institute of Technology School of Science faculty
Living people
1930 births
Fellows of the American Mathematical Society
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https://en.wikipedia.org/wiki/Monotone%20likelihood%20ratio
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A monotonic likelihood ratio in distributions and
The ratio of the density functions above is monotone in the parameter , so satisfies the monotone likelihood ratio property.
In statistics, the monotone likelihood ratio property is a property of the ratio of two probability density functions (PDFs). Formally, distributions ƒ(x) and g(x) bear the property if
that is, if the ratio is nondecreasing in the argument .
If the functions are first-differentiable, the property may sometimes be stated
For two distributions that satisfy the definition with respect to some argument x, we say they "have the MLRP in x." For a family of distributions that all satisfy the definition with respect to some statistic T(X), we say they "have the MLR in T(X)."
Intuition
The MLRP is used to represent a data-generating process that enjoys a straightforward relationship between the magnitude of some observed variable and the distribution it draws from. If satisfies the MLRP with respect to , the higher the observed value , the more likely it was drawn from distribution rather than . As usual for monotonic relationships, the likelihood ratio's monotonicity comes in handy in statistics, particularly when using maximum-likelihood estimation. Also, distribution families with MLR have a number of well-behaved stochastic properties, such as first-order stochastic dominance and increasing hazard ratios. Unfortunately, as is also usual, the strength of this assumption comes at the price of realism. Many processes in the world do not exhibit a monotonic correspondence between input and output.
Example: Working hard or slacking off
Suppose you are working on a project, and you can either work hard or slack off. Call your choice of effort and the quality of the resulting project . If the MLRP holds for the distribution of q conditional on your effort , the higher the quality the more likely you worked hard. Conversely, the lower the quality the more likely you slacked off.
Choose effort where H means high, L means low
Observe drawn from . By Bayes' law with a uniform prior,
Suppose satisfies the MLRP. Rearranging, the probability the worker worked hard is
which, thanks to the MLRP, is monotonically increasing in (because is decreasing in ). Hence if some employer is doing a "performance review" he can infer his employee's behavior from the merits of his work.
Families of distributions satisfying MLR
Statistical models often assume that data are generated by a distribution from some family of distributions and seek to determine that distribution. This task is simplified if the family has the monotone likelihood ratio property (MLRP).
A family of density functions indexed by a parameter taking values in an ordered set is said to have a monotone likelihood ratio (MLR) in the statistic if for any ,
is a non-decreasing function of .
Then we say the family of distributions "has MLR in ".
List of families
Hypothesis testing
If the family of random
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https://en.wikipedia.org/wiki/Division%20No.%204%2C%20Subdivision%20C%2C%20Newfoundland%20and%20Labrador
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Division No. 4, Subd. C is an unorganized subdivision on St. George's Bay on the island of Newfoundland in Newfoundland and Labrador, Canada. It is in Division No. 4.
According to the 2016 Statistics Canada Census:
Population: 747
% Change (2011 to 2016): 2.6
Dwellings: 490
Area: 2378.34 km2
Density: 0.3 people/km2
Division No. 4, Subd. C includes the unincorporated communities of
Barachois Brook
Flat Bay
Mattis Point
St. Teresa
References
Newfoundland and Labrador subdivisions
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https://en.wikipedia.org/wiki/Narrowing
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Narrowing may refer to:
Narrowing (computer science), a type of algorithm for solving equations between symbolic expressions
Narrowing of algebraic value sets, a method for the elimination of values from a solution set which are inconsistent with the equations being solved
Narrowing (historical linguistics), a type of semantic change
Collisional narrowing of a spectral line due to collisions of the emitting species
Motional narrowing of a resonant frequency due to the inhomogeneity of the system averaging out over time
Perceptual narrowing, a process in brain development
Q-based narrowing, a concept in pragmatics
Stenosis, the narrowing of a blood vessel or other tubular organ
See also
Narrow (disambiguation)
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https://en.wikipedia.org/wiki/Primon%20gas
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In mathematical physics, the primon gas or Riemann gas discovered by Bernard Julia is a model illustrating correspondences between number theory and methods in quantum field theory, statistical mechanics and dynamical systems such as the Lee-Yang theorem. It is a quantum field theory of a set of non-interacting particles, the primons; it is called a gas or a free model because the particles are non-interacting. The idea of the primon gas was independently discovered by Donald Spector. Later works by Ioannis Bakas and Mark Bowick, and Spector explored the connection of such systems to string theory.
The model
State space
Consider a Hilbert space H with an orthonormal basis of states labelled by the prime numbers p. Second quantization gives a new Hilbert space K, the bosonic Fock space on H, where states describe collections of primes - which we can call primons if we think of them as analogous to particles in quantum field theory. This Fock space has an orthonormal basis given by finite multisets of primes. In other words, to specify one of these basis elements we can list the number of primons for each prime :
where the total is finite. Since any positive natural number has a unique factorization into primes:
we can also denote the basis elements of the Fock space as simply where
In short, the Fock space for primons has an orthonormal basis given by the positive natural numbers, but we think of each such number as a collection of primons: its prime factors, counted with multiplicity.
Identifying the Hamiltonian via the Koopman operator
Given the state , we may use the Koopman operator to lift dynamics from the space of states to the space of observables:
where is an algorithm for integer factorisation, analogous to the discrete logarithm, and is the successor function. Thus, we have:
A precise motivation for defining the Koopman operator is that it represents a global linearisation of , which views linear combinations of eigenstates as
integer partitions. In fact, the reader may easily check that the successor function is
not a linear function:
Hence, is canonical.
Energies
If we take a simple quantum Hamiltonian H to have eigenvalues proportional to log p, that is,
with
we are naturally led to
Statistics of the phase-space dimension
Let's suppose we would like to know the average time, suitably-normalised, that the Riemann gas spends in a particular subspace. How might this frequency be related to the dimension of this subspace?
If we characterize distinct linear subspaces as Erdős-Kac data which have the form of sparse binary vectors, using the Erdős-Kac theorem we may actually demonstrate that this frequency depends upon nothing more than the dimension of the subspace. In fact, if counts the number of unique prime divisors of then the Erdős-Kac law tells us that for large :
has the standard normal distribution.
What is even more remarkable is that although the Erdős-Kac theorem has the form of
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https://en.wikipedia.org/wiki/Parry%E2%80%93Daniels%20map
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In mathematics, the Parry–Daniels map is a function studied in the context of dynamical systems. Typical questions concern the existence of an invariant or ergodic measure for the map.
It is named after the English mathematician Bill Parry and the British statistician Henry Daniels, who independently studied the map in papers published in 1962.
Definition
Given an integer n ≥ 1, let Σ denote the n-dimensional simplex in Rn+1 given by
Let π be a permutation such that
Then the Parry–Daniels map
is defined by
References
Dynamical systems
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https://en.wikipedia.org/wiki/Hilbert%20dimension
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In mathematics the term Hilbert dimension may refer to:
Hilbert space dimension
Hilbert dimension in ring theory, see Hilbert's basis theorem
See also
Hilbert series and Hilbert polynomial
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https://en.wikipedia.org/wiki/Hilbert%20algebra
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In mathematics, Hilbert algebras and left Hilbert algebras occur in the theory of von Neumann algebras in:
Tomita–Takesaki theory#Left Hilbert algebras
Von Neumann algebras
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https://en.wikipedia.org/wiki/List%20of%20Tottenham%20Hotspur%20F.C.%20records%20and%20statistics
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Tottenham Hotspur are an English association football club based in Tottenham, London. They are among the most successful clubs in English football, with 26 league and cup victories.
Club records
Record wins
Record win: 13–2 v Crewe Alexandra, FA Cup, 3 February 1960
Record league victory: 9–0 v Bristol Rovers, Division 2, 22 October 1977
Record Premier League victory: 9–1 v Wigan Athletic, 22 November 2009
Most league goals scored: 10–4 v Everton, 11 October 1958.
Record cup victory: 13–2 v Crewe Alexandra, FA Cup, 3 February 1960
Record home win: 13–2 v Crewe Alexandra, FA Cup, 3 February 1960
Record UEFA Cup win: 9–0 v Keflavík (Iceland) 28 September 1971 (aggregate 15–1, including 1–6 win away on 14 September 1971)
Record away wins:
7–0 v Tranmere Rovers, FA Cup, 4 January 2019
6–0 v Drogheda United, UEFA Cup, 14 September 1983
6–0 v Oldham Athletic, Football League Cup, 23 September 2004
7–1 v Hull City, Premier League, 21 May 2017.
Record defeats
Record defeat: 0–8 v 1. FC Köln, UEFA Intertoto Cup, 22 July 1995
Record Champions League defeat: 2–7 v Bayern Munich, 1 October 2019
Most league goals conceded: 2–8 v Derby County, Division 1, 16 October 1976
Record league defeat: 0–7 v Liverpool, Division 1, 2 September 1978
Record Premier League defeat:
1–7 v Newcastle United, 28 December 1996
0–6 v Sheffield United, 2 March 1993
0–6 v Manchester City, 24 November 2013
Record cup defeat: 1–6 v Newcastle United, FA Cup, 23 December 1999
Record home defeat: 0–6
v Sunderland, Football League First Division, 19 December 1914
v Arsenal, Football League First Division, 6 March 1935
Record away defeat: 0–8 v 1. FC Köln, UEFA Intertoto Cup, 22 July 1995
Additional records
Record attendance: 85,512 v Bayer Leverkusen, Champions League, 2 November 2016 (at Wembley)
Most league points (under 2 for a win system): 70, Division 2, 1919–20
Most league points (under 3 for a win system): 86, Premier League, 2016–17
Most league goals: 115, Division 1, 1960–61
Most goals in total: 280 Harry Kane, 2011–2023
Most league goals in total: 220 Jimmy Greaves, 1961–70
Most goals in a season: 49 Clive Allen, 1986–87
Youngest goalscorer: Alfie Devine, 16 years, 163 days against Marine (A), 10 January 2021
Most league appearances: 655 Steve Perryman, 1969–1986
Most appearances: 854 Steve Perryman, 1969–1986
Youngest first team player: Alfie Devine, 16 years, 163 days against Marine (A), 10 January 2021
Youngest first team player in a European game: Dane Scarlett, 16 years, 247 days against Ludogorets Razgrad (H), 26 November 2020
Oldest first team player: Brad Friedel, 42 years, 176 days against Newcastle United (H), 10 November 2013
Transfer record (received): £100M from Bayern Munich for Harry Kane 12 August 2023
Transfer record (paid): £60 million to Everton for Richarlison, July 2022
London derbies best attendances
– Arsenal:
Tottenham 1–0 Arsenal, 83,222, 10 February 2018, Premier League, New Wembley
Tottenham 3–1 Arsenal, 77,893, 14 April 1991, FA Cup
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https://en.wikipedia.org/wiki/Ronald%20Gillespie
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Ronald James Gillespie, (August 21, 1924 – February 26, 2021) was a British chemist specializing in the field of molecular geometry, who arrived in Canada after accepting an offer that included his own laboratory with new equipment, which post-World War II Britain could not provide. He was responsible for establishing inorganic chemistry education in Canada.
He was educated at the University of London obtaining a B.Sc. in 1945, a Ph.D. in 1949 and a D.Sc. in 1957. He was assistant lecturer and then lecturer in the Department of Chemistry at University College London in England from 1950 to 1958.
He moved to McMaster University, Hamilton, Ontario, Canada, in 1958, dying on February 26, 2021, at the age of ninety-six in the nearby town of Dundas, Ontario. He was elected as a Fellow of the Royal Society of Canada in 1965, a Fellow of the Royal Society of London in 1977, and made a member of the Order of Canada in 2007.
Gillespie did extensive work on expanding the idea of the Valence Shell Electron Pair Repulsion (VSEPR) model of Molecular Geometry, which he developed with Ronald Nyholm (and thus is also known as the Gillespie-Nyholm theory), and setting the rules for assigning numbers. He has written several books on this VSEPR topic in chemistry. With other workers he developed LCP theory, (ligand close packing theory), which for some molecules allows geometry to be predicted on the basis of ligand-ligand repulsions. Gillespie has also done extensive work on interpreting the covalent radius of fluorine. The covalent radius of most atoms is found by taking half the length of a single bond between two similar atoms in a neutral molecule. Calculating the covalent radius for fluorine is more difficult because of its high electronegativity compared to its small atomic radius size. Gillespie's work on the bond length of fluorine focuses on theoretically determining the covalent radius of fluorine by examining its covalent radius when it is attached to several different atoms.
Publications
Chemical Bonding and Molecular Geometry: From Lewis to Electron Densities (Topics in Inorganic Chemistry) by Ronald J. Gillespie and Paul L. A. Popelier
Atoms, Molecules and Reactions: An Introduction to Chemistry by Ronald J. Gillespie
Chemistry by Ronald J. Gillespie, David Humphreys, Colin Baird, and E. A. Robinson
''Atoms, Molecules and Reactions: An Introduction to Chemistry with D.A. Humphreys, E.A. Robinson and D.R. Eaton, Prentice Hall, 1994
References
1924 births
2021 deaths
British chemists
Inorganic chemists
Members of the Order of Canada
Academic staff of McMaster University
Alumni of the University of London
Academics of University College London
Fellows of the Royal Society of Canada
Fellows of the Royal Society
British emigrants to Canada
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https://en.wikipedia.org/wiki/Noise%20margin
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In electrical engineering, noise margin is the maximum voltage amplitude of extraneous signal that can be algebraically added to the noise-free worst-case input level without causing the output voltage to deviate from the allowable logic voltage level. It is commonly used in at least two contexts as follows:
In communications system engineering, noise margin is the ratio by which the signal exceeds the minimum acceptable amount. It is normally measured in decibels.
In a digital circuit, the noise margin is the amount by which the signal exceeds the threshold for a proper '0' (logic low) or '1' (logic high). For example, a digital circuit might be designed to swing between 0.0 and 1.2 volts, with anything below 0.2 volts considered a '0', and anything above 1.0 volts considered a '1'. Then the noise margin for a '0' would be the amount that a signal is below 0.2 volts, and the noise margin for a '1' would be the amount by which a signal exceeds 1.0 volt. In this case noise margins are measured as an absolute voltage, not a ratio. Noise margins for CMOS chips are usually much greater than those for TTL because the VOH min is closer to the power supply voltage and VOL max is closer to zero.
Real digital inverters do not instantaneously switch from a logic high (1) to a logic low (0), there is some capacitance. While an inverter is transitioning from a logic high to low, there is an undefined region where the voltage cannot be considered high or low. This is considered a noise margin. There are two noise margins to consider: Noise margin high (NMH) and noise margin low (NML). NMH is the amount of voltage between an inverter transitioning from a logic high (1) to a logic low (0) and vice versa for NML. The equations are as follows: NMH ≡ VOH - VIH and NML ≡ VIL - VOL. Typically, in a CMOS inverter VOH will equal VDD and VOL will equal the ground potential, as mentioned above.
VIH is defined as the highest input voltage at which the slope of the voltage transfer characteristic (VTC) is equal to -1, where the VTC is the plot of all valid output voltages vs. input voltages. Similarly, VIL is defined as the lowest input voltage where slope of the VTC is equal to -1.
In practice, noise margins are the amount of noise, that a logic circuit can withstand.
Noise margins are generally defined so that positive values ensure proper operation, and negative margins result in compromised operation, or outright failure.
See also
Digital circuit
Signal integrity
Substrate coupling
ITU G.992.1
signal-to-noise ratio
signal
References
External links
DMT, a DSL monitoring and downstream noise margin tweaking program.
MIT, PDF of a PowerPoint Presentation on for Digital Noise Margin.
Electronic engineering
Electronic design
Electronic design automation
Integrated circuits
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https://en.wikipedia.org/wiki/Spouge%27s%20approximation
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In mathematics, Spouge's approximation is a formula for computing an approximation of the gamma function. It was named after John L. Spouge, who defined the formula in a 1994 paper. The formula is a modification of Stirling's approximation, and has the form
where a is an arbitrary positive integer and the coefficients are given by
Spouge has proved that, if Re(z) > 0 and a > 2, the relative error in discarding εa(z) is bounded by
The formula is similar to the Lanczos approximation, but has some distinct features. Whereas the Lanczos formula exhibits faster convergence, Spouge's coefficients are much easier to calculate and the error can be set arbitrarily low. The formula is therefore feasible for arbitrary-precision evaluation of the gamma function. However, special care must be taken to use sufficient precision when computing the sum due to the large size of the coefficients ck, as well as their alternating sign. For example, for a = 49, one must compute the sum using about 65 decimal digits of precision in order to obtain the promised 40 decimal digits of accuracy.
See also
Stirling's approximation
Lanczos approximation
References
Gamma and related functions
Computer arithmetic algorithms
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https://en.wikipedia.org/wiki/1994%20in%20heavy%20metal%20music
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This is a timeline documenting the events of heavy metal music in the year 1994.
Newly formed bands
20 Dead Flower Children
36 Crazyfists
Abaddon Incarnate
Abscess
Abstrakt Algebra
The Abyss
Agoraphobic Nosebleed
Amen
Antaeus
Aurora Borealis
Behexen
Blindside
Blut Aus Nord
Coalesce
Colour Haze
Corrupted
Craft
Creed
Dark Fortress
Deadguy
Deströyer 666
DGM
Disillusion
Dismal Euphony
Disturbed (as Brawl)
Dødheimsgard
Draconian
Edguy
Ektomorf
Empyrium
Epoch Of Unlight
Eternal Tears of Sorrow
Evoken
Flowing Tears
Gluecifer
Gov't Mule
Graveworm
Gravity Kills
Guano Apes
HammerFall
Hatebreed
Heavenly
Hed PE
The Hellacopters
Hollenthon
Horde
Iron Monkey
Kalisia
Kampfar
Knut
Kobong
Kronos
Lacuna Coil
Lake of Tears
Lamb of God (as Burn the Priest)
Limbonic Art
Limp Bizkit
Lost Horizon
Lux Occulta
Martyr
Melechesh
Mob Rules
Moonspell
Muse
Mushroomhead
Naer Mataron
Nailbomb
Necrophagist
Negură Bunget
Neuraxis
Nightingale
Nile
Nothingface
October Tide
Officium Triste
O.N.A.
Page and Plant
Portal
Primordial
Quo Vadis
Ragnarok
Rammstein
Scarve
Sevendust
Siebenbürgen
Six Feet Under
Skylark
Slash's Snakepit
Spiritual Beggars
Static-X
Strapping Young Lad
Stone Sour
Suidakra
Symphony X
System of a Down
Tenacious D
Theatres des Vampires
Throes of Dawn
Thyrane
Trail of Tears
Ulver
Vintersorg
Virgin Black
Windir
Albums
Accept – Death Row
Acid Bath – When the Kite String Pops
Acid King – Acid King (EP)
Alice Cooper – The Last Temptation
Alice in Chains – Jar of Flies
Aggressor – Of Long Duration Anguish
The Almighty – Crank
Altar - Youth Against Christ
Amorphis – Tales from the Thousand Lakes
Ancient – Svartalvheim
Annihilator – King of the Kill
Arcturus – Constellation (EP)
Asphyx - Asphyx
At the Gates - Terminal Spirit Disease
Bang Tango – Love After Death
Bathory – Requiem
Behemoth – And the Forests Dream Eternally (EP)
Benediction - The Grotesque / Ashen Epitaph (EP)
Bestial Warlust - Vengeance War Till Death
Biohazard – State of the World Address
Black Sabbath – Cross Purposes
The Black Crowes – Amorica
Body Count – Born Dead
Bolt Thrower – ...For Victory
Bon Jovi – Cross Road
Bruce Dickinson – Balls to Picasso
Brutality - When the Sky Turns Black
Brutal Truth - Need to Control
Burzum – Hvis lyset tar oss
Cannibal Corpse – The Bleeding
Cathedral – Cosmic Requiem (EP)
Cathedral – In Memorium (EP)
Cathedral – Statik Majik (EP)
Cemetary – Black Vanity
Cinderella – Still Climbing
Converge – Halo in a Haystack
Corrosion of Conformity – Deliverance
Cradle of Filth – The Principle of Evil Made Flesh
Cryptopsy – Blasphemy Made Flesh
The Cult – The Cult
Dangerous Toys – Pissed
Danzig – Danzig 4
Dark Funeral – Dark Funeral (EP)
Darkthrone – Transilvanian Hunger
Deliverance – River Disturbance
Destruction – Destruction (EP)
Desultory - Bitterness
Dio – Strange Highways (US release)
DGeneration – DGeneration
Downset. – downset.
Dream Theater – Awake
Edge of Sani
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https://en.wikipedia.org/wiki/P-matrix
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In mathematics, a -matrix is a complex square matrix with every principal minor is positive. A closely related class is that of -matrices, which are the closure of the class of -matrices, with every principal minor 0.
Spectra of -matrices
By a theorem of Kellogg, the eigenvalues of - and - matrices are bounded away from a wedge about the negative real axis as follows:
If are the eigenvalues of an -dimensional -matrix, where , then
If , , are the eigenvalues of an -dimensional -matrix, then
Remarks
The class of nonsingular M-matrices is a subset of the class of -matrices. More precisely, all matrices that are both -matrices and Z-matrices are nonsingular -matrices. The class of sufficient matrices is another generalization of -matrices.
The linear complementarity problem has a unique solution for every vector if and only if is a -matrix. This implies that if is a -matrix, then is a -matrix.
If the Jacobian of a function is a -matrix, then the function is injective on any rectangular region of .
A related class of interest, particularly with reference to stability, is that of -matrices, sometimes also referred to as -matrices. A matrix is a -matrix if and only if is a -matrix (similarly for -matrices). Since , the eigenvalues of these matrices are bounded away from the positive real axis.
See also
Hurwitz matrix
Linear complementarity problem
M-matrix
Q-matrix
Z-matrix
Perron–Frobenius theorem
Notes
References
David Gale and Hukukane Nikaido, The Jacobian matrix and global univalence of mappings, Math. Ann. 159:81-93 (1965)
Li Fang, On the Spectra of - and -Matrices, Linear Algebra and its Applications 119:1-25 (1989)
R. B. Kellogg, On complex eigenvalues of and matrices, Numer. Math. 19:170-175 (1972)
Matrix theory
Matrices
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https://en.wikipedia.org/wiki/1993%20National%20Scout%20Jamboree
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The 1993 National Scout Jamboree was the 13th national Scout jamboree of the Boy Scouts of America and was held from August 4-10, 1993, at Fort A.P. Hill, Virginia.
Statistics
This event was attended by 34,449 scouts.
List of sub-camps
The 1993 National Scout Jamboree was divided into four regional encampments which consisted of a total of 19 sub-camps. Each subcamp consisted of approximately 1300 participants each dispersed among 30-40 troops. Each troop occupied a campsite with dimensions of approximately X 90 feet. Each subcamp had a special patch depicting a historical flag.
Central region
Subcamp 1: Green Mountain
Subcamp 2: Rhode Island
Subcamp 3: Guilford Courthouse
Subcamp 4: French Fleur-de-lis
Western region
Subcamp 5: Union Jack
Subcamp 6: Grand Union
Subcamp 7: Fremont
Subcamp 8: Sons of Liberty
Subcamp 9: Gadsden
Southern region
Subcamp 15: Navy Jack
Subcamp 16: Serapis
Subcamp 17: Fort Moultrie
Subcamp 18: Lions & Castles
Subcamp 19: Commodore Perry
Northeast region
Subcamp 10: Bunker Hill
Subcamp 11: Bennington
Subcamp 12: Washington Cruisers
Subcamp 13: Phila, Light Horse
Subcamp 14: Taunton
Program
Jamboree attendees were able to participate in a number of activities. Singer Lee Greenwood and performance group Up With People performed at the opening ceremony, and singer Louise Mandrell performed at the closing ceremony. A list of the main activities is given below.
Action centers
"Action Alley"
Air-Rifle
Archery
"Bikathalon"
"Buckskin Games"
"Confidence Course"
Motocross
"Patrol Challenge"
Pioneering
Trap Shooting
Rappelling
Remote centers
Conservation
Fishing - More than 20,000 bass, channel catfish, bluegill and other fish were stocked in Fishhook Lake.
Aquatics
"Raft Encounter"
Racing Shell Run
Canoe Sprint
"Kayak Fun"
Canoe Slalom
"Discover Scuba"
"Snorkel Search"
Exhibits and displays
National Exhibits
Merit Badge Midway
Arts and Science Expo
Brownsea Island Camp
Daily Stage Shows
Amateur Radio Station
Order of the Arrow Jamboree Rendezvous
The Order of the Arrow Jamboree Rendezvous was held on the evening of Monday, August 9.
Severe weather
A major rainstorm occurred on Friday, August 6 which caused localized flooding throughout Fort A. P. Hill and necessitated the cancellation of all Jamboree activities for the afternoon. This storm deposited over of rain on the jamboree site in a 13-hour period.
Newspaper
A daily newspaper entitled Jamboree Today was distributed to all jamboree participants to inform them of events at the jamboree.
References
1993 in Virginia
1993
August 1993 events in the United States
Boy Scouts of America
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https://en.wikipedia.org/wiki/Hyperfinite
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Hyperfinite may refer to:
Hyperfinite set, a type of internal set in non-standard analysis
Hyperfinite von Neumann algebra, also called amenable von Neumann algebras
Hyperfinite type II factor, a unique von Neumann algebra that is a factor of type II and also hyperfinite
Hyper-finite field, an uncountable field similar in many ways to finite fields
Uniformly hyperfinite algebra, a C*-algebra that can be written as the closure of an increasing union of finite-dimensional full matrix algebras
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https://en.wikipedia.org/wiki/Jim%20Warren%20%28computer%20specialist%29
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Jim Warren (July 20, 1936 – November 24, 2021) was an American mathematics and computing educator, computer professional, entrepreneur, editor, publisher and continuing sometime activist.
Early career
From 1957 to 1967, Warren was a mathematics teacher at secondary-school level, and professor at college and university levels, with his last full-time academic position being Chair of the Mathematics Department at the College of Notre Dame, Belmont, a small liberal arts college in Belmont, California. He later taught computer courses at Stanford University, San Jose State University and San Francisco State University.
He had his first full-time teaching contract, for an annual salary of , when he was 20 years old and had completed only three years of college. In the ensuing decade, he was also a National Science Foundation Guest Lecturer, was the founder and Director of Summer Mathematics Institutes at Our Lady of the Lake University in San Antonio, Texas, and earned national recognition for innovative weekly enrichment programs he created for secondary school students, and for in-service programs for elementary and secondary school teachers, all without cost, as Chair of the Alamo District [South Texas] Council of Teachers of Mathematics (1960–1962).
In the late 1960s, Warren was involved in the radical, utopian, alternative, hippie Midpeninsula Free University, including serving pro-bono as its elected General Secretary for three terms. In that time, he created and edited its irregular magazine, which he titled The Free You.
Computing
From 1968 through the mid-1970s Warren worked as a freelance minicomputer programmer and computer consultant, operating under the name, Frelan Associates (for "free land"), creating assembler-level real-time data-acquisition and process-control programs for biomedical research at Stanford Medical Center, and control programs for various high-tech companies around Silicon Valley. In those years, he also chaired the Association for Computing Machinery's regional chapters of SIGPLAN, SIGMICRO and the San Francisco Peninsula ACM.
In 1977, Warren co-founded the West Coast Computer Faire which, for a half-dozen years, was the largest public microcomputer convention in the world. He was its self-titled "Faire Chaircreature," organizing eight conventions. In 1983, he sold the Faire to Prentice-Hall, "for 100% down; nothin' to pay".
To promote the Computer Faires and circulate news and gossip about the then-infant microcomputer industry, he founded and edited the first free tabloid newspaper about microcomputing, the irregular Silicon Gulch Gazette (SGG), published from issue #0 in February 1977, through issue #43, in January 1986, with one issue named Business Systems Journal.
Beginning in 1978, Warren created and published the Intelligent Machines Journal (IMJ, which is also Pig Latin for "Jim"), the first subscription news periodical about microcomputing, published as a tabloid newspaper, with Tom Williams as its
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https://en.wikipedia.org/wiki/Invariant%20measure
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In mathematics, an invariant measure is a measure that is preserved by some function. The function may be a geometric transformation. For examples, circular angle is invariant under rotation, hyperbolic angle is invariant under squeeze mapping, and a difference of slopes is invariant under shear mapping.
Ergodic theory is the study of invariant measures in dynamical systems. The Krylov–Bogolyubov theorem proves the existence of invariant measures under certain conditions on the function and space under consideration.
Definition
Let be a measurable space and let be a measurable function from to itself. A measure on is said to be invariant under if, for every measurable set in
In terms of the pushforward measure, this states that
The collection of measures (usually probability measures) on that are invariant under is sometimes denoted The collection of ergodic measures, is a subset of Moreover, any convex combination of two invariant measures is also invariant, so is a convex set; consists precisely of the extreme points of
In the case of a dynamical system where is a measurable space as before, is a monoid and is the flow map, a measure on is said to be an invariant measure if it is an invariant measure for each map Explicitly, is invariant if and only if
Put another way, is an invariant measure for a sequence of random variables (perhaps a Markov chain or the solution to a stochastic differential equation) if, whenever the initial condition is distributed according to so is for any later time
When the dynamical system can be described by a transfer operator, then the invariant measure is an eigenvector of the operator, corresponding to an eigenvalue of this being the largest eigenvalue as given by the Frobenius-Perron theorem.
Examples
Consider the real line with its usual Borel σ-algebra; fix and consider the translation map given by: Then one-dimensional Lebesgue measure is an invariant measure for
More generally, on -dimensional Euclidean space with its usual Borel σ-algebra, -dimensional Lebesgue measure is an invariant measure for any isometry of Euclidean space, that is, a map that can be written as for some orthogonal matrix and a vector
The invariant measure in the first example is unique up to trivial renormalization with a constant factor. This does not have to be necessarily the case: Consider a set consisting of just two points and the identity map which leaves each point fixed. Then any probability measure is invariant. Note that trivially has a decomposition into -invariant components and
Area measure in the Euclidean plane is invariant under the special linear group of the real matrices of determinant
Every locally compact group has a Haar measure that is invariant under the group action.
See also
References
John von Neumann (1999) Invariant measures, American Mathematical Society
Dynamical systems
Measures (measure theory)
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https://en.wikipedia.org/wiki/Krylov%E2%80%93Bogolyubov%20theorem
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In mathematics, the Krylov–Bogolyubov theorem (also known as the existence of invariant measures theorem) may refer to either of the two related fundamental theorems within the theory of dynamical systems. The theorems guarantee the existence of invariant measures for certain "nice" maps defined on "nice" spaces and were named after Russian-Ukrainian mathematicians and theoretical physicists Nikolay Krylov and Nikolay Bogolyubov who proved the theorems.
Formulation of the theorems
Invariant measures for a single map
Theorem (Krylov–Bogolyubov). Let (X, T) be a compact, metrizable topological space and F : X → X a continuous map. Then F admits an invariant Borel probability measure.
That is, if Borel(X) denotes the Borel σ-algebra generated by the collection T of open subsets of X, then there exists a probability measure μ : Borel(X) → [0, 1] such that for any subset A ∈ Borel(X),
In terms of the push forward, this states that
Invariant measures for a Markov process
Let X be a Polish space and let be the transition probabilities for a time-homogeneous Markov semigroup on X, i.e.
Theorem (Krylov–Bogolyubov). If there exists a point for which the family of probability measures { Pt(x, ·) | t > 0 } is uniformly tight and the semigroup (Pt) satisfies the Feller property, then there exists at least one invariant measure for (Pt), i.e. a probability measure μ on X such that
See also
For the 1st theorem: Ya. G. Sinai (Ed.) (1997): Dynamical Systems II. Ergodic Theory with Applications to Dynamical Systems and Statistical Mechanics. Berlin, New York: Springer-Verlag. . (Section 1).
For the 2nd theorem: G. Da Prato and J. Zabczyk (1996): Ergodicity for Infinite Dimensional Systems. Cambridge Univ. Press. . (Section 3).
Notes
Ergodic theory
Theorems in dynamical systems
Probability theorems
Random dynamical systems
Theorems in measure theory
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https://en.wikipedia.org/wiki/Dickman%20function
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In analytic number theory, the Dickman function or Dickman–de Bruijn function ρ is a special function used to estimate the proportion of smooth numbers up to a given bound.
It was first studied by actuary Karl Dickman, who defined it in his only mathematical publication, which is not easily available, and later studied by the Dutch mathematician Nicolaas Govert de Bruijn.
Definition
The Dickman–de Bruijn function is a continuous function that satisfies the delay differential equation
with initial conditions for 0 ≤ u ≤ 1.
Properties
Dickman proved that, when is fixed, we have
where is the number of y-smooth (or y-friable) integers below x.
Ramaswami later gave a rigorous proof that for fixed a, was asymptotic to , with the error bound
in big O notation.
Applications
The main purpose of the Dickman–de Bruijn function is to estimate the frequency of smooth numbers at a given size. This can be used to optimize various number-theoretical algorithms such as P-1 factoring and can be useful of its own right.
It can be shown using that
which is related to the estimate below.
The Golomb–Dickman constant has an alternate definition in terms of the Dickman–de Bruijn function.
Estimation
A first approximation might be A better estimate is
where Ei is the exponential integral and ξ is the positive root of
A simple upper bound is
Computation
For each interval [n − 1, n] with n an integer, there is an analytic function such that . For 0 ≤ u ≤ 1, . For 1 ≤ u ≤ 2, . For 2 ≤ u ≤ 3,
with Li2 the dilogarithm. Other can be calculated using infinite series.
An alternate method is computing lower and upper bounds with the trapezoidal rule; a mesh of progressively finer sizes allows for arbitrary accuracy. For high precision calculations (hundreds of digits), a recursive series expansion about the midpoints of the intervals is superior.
Extension
Friedlander defines a two-dimensional analog of . This function is used to estimate a function similar to de Bruijn's, but counting the number of y-smooth integers with at most one prime factor greater than z. Then
See also
Buchstab function, a function used similarly to estimate the number of rough numbers, whose convergence to is controlled by the Dickman function
Golomb–Dickman constant
References
Further reading
Analytic number theory
Special functions
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https://en.wikipedia.org/wiki/BIT%20predicate
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In mathematics and computer science, the BIT predicate, sometimes is a predicate that tests whether the bit of the (starting from the least significant digit) when is written as a binary number. Its mathematical applications include modeling the membership relation of hereditarily finite sets, and defining the adjacency relation of the Rado graph. In computer science, it is used for efficient representations of set data structures using bit vectors, in defining the private information retrieval problem from communication complexity, and in descriptive complexity theory to formulate logical descriptions of complexity classes.
History
The BIT predicate was first introduced in 1937 by Wilhelm Ackermann to define the Ackermann coding, which encodes hereditarily finite sets as The BIT predicate can be used to perform membership tests for the encoded sets: is true if and only if the set encoded is a member of the set encoded
Ackermann denoted the predicate using a Fraktur font to distinguish it from the notation that he used for set membership (short for an element in German). The notation and the name "the BIT predicate", come from the work of Ronald Fagin and Neil Immerman, who applied this predicate in computational complexity theory as a way to encode and decode information in the late 1980s and early
Description and implementation
The binary representation of a number is an expression for as a sum of distinct powers of two,
where each bit in this expression is either 0 or 1. It is commonly written in binary notation as just the sequence of these bits, . Given this expansion for , the BIT predicate is defined to equal . It can be calculated from the formula
where is the floor function and mod is the modulo function.
The BIT predicate is a primitive recursive function. As a binary relation (producing true and false values rather than 1 and 0 respectively), the BIT predicate is asymmetric: there do not exist two numbers and for which both and are true.
In programming languages such as C, C++, Java, or Python that provide a and a the BIT predicate can be implemented by the expression
(i>>j)&1. The subexpression i>>j shifts the bits in the binary representation of so that is shifted to and the masks off the remaining bits, leaving only the bit in As with the modular arithmetic formula above, the value of the expression is respectively as the value of is true or false.
Applications
Set data structures
For a set represented as a bit array, the BIT predicate can be used to test set membership. For instance, subsets of the non-negative integers may be represented by a bit array with a one in when is a member of the subset, and a zero in that position when it is not a member. When such a bit array is interpreted as a binary number, the set for distinct is represented as the binary number . If is a set, represented in this way, and is a number that may or may not be an element of , then returns a nonzero valu
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https://en.wikipedia.org/wiki/Map%20algebra
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Map algebra is an algebra for manipulating geographic data, primarily fields. Developed by Dr. Dana Tomlin and others in the late 1970s, it is a set of primitive operations in a geographic information system (GIS) which allows one or more raster layers ("maps") of similar dimensions to produce a new raster layer (map) using mathematical or other operations such as addition, subtraction etc.
History
Prior to the advent of GIS, the overlay principle had developed as a method of literally superimposing different thematic maps (typically an isarithmic map or a chorochromatic map) drawn on transparent film (e.g., cellulose acetate) to see the interactions and find locations with specific combinations of characteristics. The technique was largely developed by landscape architects and city planners, starting with Warren Manning and further refined and popularized by Jaqueline Tyrwhitt, Ian McHarg and others during the 1950s and 1960s.
In the mid-1970s, landscape architecture student C. Dana Tomlin developed some of the first tools for overlay analysis in raster as part of the IMGRID project at the Harvard Laboratory for Computer Graphics and Spatial Analysis, which he eventually transformed into the Map Analysis Package (MAP), a popular raster GIS during the 1980s. While a graduate student at Yale University, Tomlin and Joseph K. Berry re-conceptualized these tools as a mathematical model, which by 1983 they were calling "map algebra." This effort was part of Tomlin's development of cartographic modeling, a technique for using these raster operations to implement the manual overlay procedures of McHarg. Although the basic operations were defined in his 1983 PhD dissertation, Tomlin had refined the principles of map algebra and cartographic modeling into their current form by 1990. Although the term cartographic modeling has not gained as wide an acceptance as synonyms such as suitability analysis, suitability modeling and multi-criteria decision making, "map algebra" became a core part of GIS. Because Tomlin released the source code to MAP, its algorithms were implemented (with varying degrees of modification) as the analysis toolkit of almost every raster GIS software package starting in the 1980s, including GRASS, IDRISI (now TerrSet), and the GRID module of ARC/INFO (later incorporated into the Spatial Analyst module of ArcGIS).
This widespread implementation further led to the development of many extensions to map algebra, following efforts to extend the raster data model, such as adding new functionality for analyzing spatiotemporal and three-dimensional grids.
Map algebra operations
Like other algebraic structures, map algebra consists of a set of objects (the domain) and a set of operations that manipulate those objects with closure (i.e., the result of an operation is itself in the domain, not something completely different). In this case, the domain is the set of all possible "maps," which are generally implemented as raster gri
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https://en.wikipedia.org/wiki/Schwartz%20kernel%20theorem
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In mathematics, the Schwartz kernel theorem is a foundational result in the theory of generalized functions, published by Laurent Schwartz in 1952. It states, in broad terms, that the generalized functions introduced by Schwartz (Schwartz distributions) have a two-variable theory that includes all reasonable bilinear forms on the space of test functions. The space itself consists of smooth functions of compact support.
Statement of the theorem
Let and be open sets in .
Every distribution defines a
continuous linear map such that
for every .
Conversely, for every such continuous linear map
there exists one and only one distribution such that () holds.
The distribution is the kernel of the map .
Note
Given a distribution one can always write the linear map K informally as
so that
.
Integral kernels
The traditional kernel functions of two variables of the theory of integral operators having been expanded in scope to include their generalized function analogues, which are allowed to be more singular in a serious way, a large class of operators from to its dual space of distributions can be constructed. The point of the theorem is to assert that the extended class of operators can be characterised abstractly, as containing all operators subject to a minimum continuity condition. A bilinear form on arises by pairing the image distribution with a test function.
A simple example is that the natural embedding of the test function space into - sending every test function into the corresponding distribution - corresponds to the delta distribution
concentrated at the diagonal of the underlined Euclidean space, in terms of the Dirac delta function . While this is at most an observation, it shows how the distribution theory adds to the scope. Integral operators are not so 'singular'; another way to put it is that for a continuous kernel, only compact operators are created on a space such as the continuous functions on . The operator is far from compact, and its kernel is intuitively speaking approximated by functions on with a spike along the diagonal and vanishing elsewhere.
This result implies that the formation of distributions has a major property of 'closure' within the traditional domain of functional analysis. It was interpreted (comment of Jean Dieudonné) as a strong verification of the suitability of the Schwartz theory of distributions to mathematical analysis more widely seen. In his Éléments d'analyse volume 7, p. 3 he notes that the theorem includes differential operators on the same footing as integral operators, and concludes that it is perhaps the most important modern result of functional analysis. He goes on immediately to qualify that statement, saying that the setting is too 'vast' for differential operators, because of the property of monotonicity with respect to the support of a function, which is evident for differentiation. Even monotonicity with respect to singular support is not characteristic of the
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https://en.wikipedia.org/wiki/Division%20No.%204%2C%20Subdivision%20D%2C%20Newfoundland%20and%20Labrador
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Division No. 4, Subd. D is an unorganized subdivision on the island of Newfoundland in Newfoundland and Labrador, Canada. It is in Division No. 4.
According to the 2016 Statistics Canada Census:
Population: 860
% Change (2011 to 2016): +3.6
Dwellings: 646
Area: 1,149.70 km2
Density: 0.7 people/km2
Division No. 4, Subd. D includes the unincorporated communities of
Fox Island River
Point au Mal
References
Newfoundland and Labrador subdivisions
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https://en.wikipedia.org/wiki/Bessel%20process
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In mathematics, a Bessel process, named after Friedrich Bessel, is a type of stochastic process.
Formal definition
The Bessel process of order n is the real-valued process X given (when n ≥ 2) by
where ||·|| denotes the Euclidean norm in Rn and W is an n-dimensional Wiener process (Brownian motion).
For any n, the n-dimensional Bessel process is the solution to the stochastic differential equation (SDE)
where W is a 1-dimensional Wiener process (Brownian motion). Note that this SDE makes sense for any real parameter (although the drift term is singular at zero).
Notation
A notation for the Bessel process of dimension started at zero is .
In specific dimensions
For n ≥ 2, the n-dimensional Wiener process started at the origin is transient from its starting point: with probability one, i.e., Xt > 0 for all t > 0. It is, however, neighbourhood-recurrent for n = 2, meaning that with probability 1, for any r > 0, there are arbitrarily large t with Xt < r; on the other hand, it is truly transient for n > 2, meaning that Xt ≥ r for all t sufficiently large.
For n ≤ 0, the Bessel process is usually started at points other than 0, since the drift to 0 is so strong that the process becomes stuck at 0 as soon as it hits 0.
Relationship with Brownian motion
0- and 2-dimensional Bessel processes are related to local times of Brownian motion via the Ray–Knight theorems.
The law of a Brownian motion near x-extrema is the law of a 3-dimensional Bessel process (theorem of Tanaka).
References
Williams D. (1979) Diffusions, Markov Processes and Martingales, Volume 1 : Foundations. Wiley. .
Stochastic processes
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https://en.wikipedia.org/wiki/Honest%20leftmost%20branch
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In set theory, an honest leftmost branch of a tree T on ω × γ is a branch (maximal chain) ƒ ∈ [T] such that for each branch g ∈ [T], one has ∀ n ∈ ω : ƒ(n) ≤ g(n). Here, [T] denotes the set of branches of maximal length of T, ω is the smallest infinite ordinal (represented by the natural numbers N), and γ is some other ordinal.
See also
scale (computing)
Suslin set
References
Akihiro Kanamori, The Higher Infinite, Perspectives in Mathematical Logic, Springer, Berlin, 1997.
Yiannis N. Moschovakis, Descriptive set theory, North-Holland, Amsterdam, 1980.
Trees (set theory)
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https://en.wikipedia.org/wiki/Diagram%20%28category%20theory%29
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In category theory, a branch of mathematics, a diagram is the categorical analogue of an indexed family in set theory. The primary difference is that in the categorical setting one has morphisms that also need indexing. An indexed family of sets is a collection of sets, indexed by a fixed set; equivalently, a function from a fixed index set to the class of sets. A diagram is a collection of objects and morphisms, indexed by a fixed category; equivalently, a functor from a fixed index category to some category.
The universal functor of a diagram is the diagonal functor; its right adjoint is the limit of the diagram and its left adjoint is the colimit. The natural transformation from the diagonal functor to some arbitrary diagram is called a cone.
Definition
Formally, a diagram of type J in a category C is a (covariant) functor
The category J is called the index category or the scheme of the diagram D; the functor is sometimes called a J-shaped diagram. The actual objects and morphisms in J are largely irrelevant; only the way in which they are interrelated matters. The diagram D is thought of as indexing a collection of objects and morphisms in C patterned on J.
Although, technically, there is no difference between an individual diagram and a functor or between a scheme and a category, the change in terminology reflects a change in perspective, just as in the set theoretic case: one fixes the index category, and allows the functor (and, secondarily, the target category) to vary.
One is most often interested in the case where the scheme J is a small or even finite category. A diagram is said to be small or finite whenever J is.
A morphism of diagrams of type J in a category C is a natural transformation between functors. One can then interpret the category of diagrams of type J in C as the functor category CJ, and a diagram is then an object in this category.
Examples
Given any object A in C, one has the constant diagram, which is the diagram that maps all objects in J to A, and all morphisms of J to the identity morphism on A. Notationally, one often uses an underbar to denote the constant diagram: thus, for any object in C, one has the constant diagram .
If J is a (small) discrete category, then a diagram of type J is essentially just an indexed family of objects in C (indexed by J). When used in the construction of the limit, the result is the product; for the colimit, one gets the coproduct. So, for example, when J is the discrete category with two objects, the resulting limit is just the binary product.
If J = −1 ← 0 → +1, then a diagram of type J (A ← B → C) is a span, and its colimit is a pushout. If one were to "forget" that the diagram had object B and the two arrows B → A, B → C, the resulting diagram would simply be the discrete category with the two objects A and C, and the colimit would simply be the binary coproduct. Thus, this example shows an important way in which the idea of the diagram generalizes that of the index
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https://en.wikipedia.org/wiki/Suslin%20cardinal
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In mathematics, a cardinal λ < Θ is a Suslin cardinal if there exists a set P ⊂ 2ω such that P is λ-Suslin but P is not λ'-Suslin for any λ' < λ. It is named after the Russian mathematician
Mikhail Yakovlevich Suslin (1894–1919).
See also
Suslin representation
Suslin line
AD+
References
Howard Becker, The restriction of a Borel equivalence relation to a sparse set, Arch. Math. Logic 42, 335–347 (2003),
Cardinal numbers
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https://en.wikipedia.org/wiki/PCB%20congener%20list
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This is a complete list of polychlorinated biphenyl (PCB) congeners.
PCB congener list
Explanation of PCB "descriptors"
Congener descriptors give a shorthand notation for geometry and substituent positions. The twelve congeners that display all four of the descriptors are referred to as being "dioxin-like", referring both to their toxicity and structural features which make them similar to 2,3,7,8-tetrachlorodibenzo-p-dioxin (2378-TCDD). Individual congeners are identified by the number and position of the chlorine atoms around the biphenyl rings.
CP0 / CP1
These 68 coplanar congeners fall into one of two groups. The first group of 20 congeners consists of those with chlorine substitution at none of the ortho positions on the biphenyl backbone and are referred to as CP0 or non-ortho congeners. The second group of 48 congeners includes those with chlorine substitution at only one of the ortho positions and are referred to as CP1 or mono-ortho congeners.
4CL
These 169 congeners have a total of four or more chlorine substituents, regardless of position.
PP
These 54 congeners have both para positions chlorinated.
2M
These 140 congeners have two or more of the meta positions chlorinated.
References
External links
U.S. Environmental Protection Agency: PCB Congeners
Organochlorides
Biphenyls
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https://en.wikipedia.org/wiki/Szilassi%20polyhedron
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In geometry, the Szilassi polyhedron is a nonconvex polyhedron, topologically a torus, with seven hexagonal faces.
Coloring and symmetry
The 14 vertices and 21 edges of the Szilassi polyhedron form an embedding of the Heawood graph onto the surface of a torus.
Each face of this polyhedron shares an edge with each other face. As a result, it requires seven colours to colour all adjacent faces. This example shows that, on surfaces topologically equivalent to a torus, some subdivisions require seven colors, providing the lower bound for the seven colour theorem. The other half of the theorem states that all toroidal subdivisions can be colored with seven or fewer colors.
The Szilassi polyhedron has an axis of 180-degree symmetry. This symmetry swaps three pairs of congruent faces, leaving one unpaired hexagon that has the same rotational symmetry as the polyhedron.
Complete face adjacency
The tetrahedron and the Szilassi polyhedron are the only two known polyhedra in which each face shares an edge with each other face.
If a polyhedron with f faces is embedded onto a surface with h holes, in such a way that each face shares an edge with each other face, it follows by some manipulation of the Euler characteristic that
This equation is satisfied for the tetrahedron with h = 0 and f = 4, and for the Szilassi polyhedron with h = 1 and f = 7.
The next possible solution, h = 6 and f = 12, would correspond to a polyhedron with 44 vertices and 66 edges. However, it is not known whether such a polyhedron can be realized geometrically without self-crossings (rather than as an abstract polytope). More generally this equation can be satisfied precisely when f is congruent to 0, 3, 4, or 7 modulo 12.
History
The Szilassi polyhedron is named after Hungarian mathematician Lajos Szilassi, who discovered it in 1977. The dual to the Szilassi polyhedron, the Császár polyhedron, was discovered earlier by ; it has seven vertices, 21 edges connecting every pair of vertices, and 14 triangular faces. Like the Szilassi polyhedron, the Császár polyhedron has the topology of a torus.
References
External links
.
.
Szilassi Polyhedron – Papercraft model at CutOutFoldUp.com
Nonconvex polyhedra
Toroidal polyhedra
Unsolved problems in mathematics
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https://en.wikipedia.org/wiki/Suslin%20representation
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In mathematics, a Suslin representation of a set of reals (more precisely, elements of Baire space) is a tree whose projection is that set of reals. More generally, a subset A of κω is λ-Suslin if there is a tree T on κ × λ such that A = p[T].
By a tree on κ × λ we mean here a subset T of the union of κi × λi for all i ∈ N (or i < ω in set-theoretical notation).
Here, p[T] = { f | ∃g : (f,g) ∈ [T] } is the projection of T,
where [T] = { (f, g ) | ∀n ∈ ω : (f(n), g(n)) ∈ T } is the set of branches through T.
Since [T] is a closed set for the product topology on κω × λω where κ and λ are equipped with the discrete topology (and all closed sets in κω × λω come in this way from some tree on κ × λ), λ-Suslin subsets of κω are projections of closed subsets in κω × λω.
When one talks of Suslin sets without specifying the space, then one usually means Suslin subsets of R, which descriptive set theorists usually take to be the set ωω.
See also
Suslin cardinal
Suslin operation
External links
R. Ketchersid, The strength of an ω1-dense ideal on ω1 under CH, 2004.
Set theory
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https://en.wikipedia.org/wiki/Dan-Virgil%20Voiculescu
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Dan-Virgil Voiculescu (born 14 June 1949) is a Romanian professor of mathematics at the University of California, Berkeley. He has worked in single operator theory, operator K-theory and von Neumann algebras. More recently, he developed free probability theory.
Education and career
Voiculescu studied at the University of Bucharest, receiving his PhD in 1977 under the direction of Ciprian Foias. He was an assistant at the University of Bucharest (1972–1973), a researcher at the Institute of Mathematics of the Romanian Academy (1973–1975), and a researcher at INCREST (1975–1986). He came to Berkeley in 1986 for the International Congress of Mathematicians, and stayed on as visiting professor. Voiculescu was appointed professor at Berkeley in 1987.
Awards and honors
He received the 2004 NAS Award in Mathematics from the National Academy of Sciences (NAS) for “the theory of free probability, in particular, using random matrices and a new concept of entropy to solve several hitherto intractable problems in von Neumann algebras.”
Voiculescu was elected to the National Academy of Sciences in 2006. In 2012 he became a fellow of the American Mathematical Society.
References
External links
Berkeley page
Notes on Free probability aspects of random matrices
Dan-Virgil Voiculescu: visionary operator algebraist and creator of free probability theory
Romanian emigrants to the United States
Members of the United States National Academy of Sciences
20th-century Romanian mathematicians
20th-century American mathematicians
21st-century American mathematicians
Mathematical analysts
Probability theorists
University of California, Berkeley faculty
Romanian academics
University of Bucharest alumni
Scientists from Bucharest
Fellows of the American Mathematical Society
Living people
1949 births
21st-century Romanian mathematicians
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https://en.wikipedia.org/wiki/Pereira%20%28footballer%2C%20born%201960%29
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Luiz Carlos Pereira (born 6 March 1960 in São Paulo), nicknamed "The Spanish Goose", is a retired Brazilian football player.
Club statistics
Honours
Individual Honors
J. League Most Valuable Player: 1994
J. League Best Eleven: 1993, 1994
Japanese Footballer of the Year: 1994
Team Honors
J1 League: 1993, 1994
References
External links
CBF BID
1960 births
Living people
Brazilian men's footballers
Brazilian expatriate men's footballers
Expatriate men's footballers in Japan
J1 League players
Japan Football League (1992–1998) players
Guarani FC players
Tokyo Verdy players
Hokkaido Consadole Sapporo players
Footballers from São Paulo
Men's association football defenders
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https://en.wikipedia.org/wiki/Alexander%20Soifer
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Alexander Soifer is a Russian-born American mathematician and mathematics author. His works include over 400 articles and 13 books.
Soifer obtained his Ph.D. in 1973 and has been a professor of mathematics at the University of Colorado since 1979. He was visiting fellow at Princeton University from 2002 to 2004, and again in 2006–2007. Soifer also teaches courses on art history and European cinema. His publications include 13 books and over 400 articles.
Every spring, Soifer, along with other mathematician colleagues, sponsors the Colorado Mathematical Olympiad (CMO) at the University of Colorado Colorado Springs. Soifer compiles and writes most of the problems for the contest. The CMO was founded by Soifer on April 18, 1983.
For the Olympiad's 30th anniversary, the university produced a film about it. In May 2018, in recognition of 35 years of leadership, the judges and winners decided in 2018 to rename the Colorado Mathematical Olympiad to the Soifer Mathematical Olympiad.
In 1991 Soifer founded the research quarterly Geombinatorics, and publishes it with the Geombinatorics editorial board.
In July 2006 at the University of Cambridge, Soifer was presented with the Paul Erdős Award by the World Federation of National Mathematics Competitions.
Soifer was the President of the World Federation of National Mathematics Competitions from 2012 to 2018. His Erdős number is 1.
Selected books
The Scholar and the State: In Search of Van der Waerden Springer, New York, 2015 (publisher: Birkhauser-Springer, Basel)
Mathematics as Problem Solving Center for Excellence in Mathematical Education, Colorado Springs, 1987
How does one cut a triangle? Center for Excellence in Mathematical Education, Colorado Springs, 1990
Colorado Mathematical Olympiad: The First Ten Years and Further Explorations Center for Excellence in Mathematical Education, Colorado Springs, 1991
Geometric Etudes in Combinatorial Mathematics Center for Excellence in Mathematical Education, Colorado Springs, 1994
The Mathematical Coloring Book Springer, New York 2009
Mathematics as Problem Solving 2nd ed. Springer, New York 2009
How Does One Cut a Triangle? 2nd ed. Springer, New York 2009
The Colorado Mathematical Olympiad and Further Explorations: From the Mountains of Colorado to the Peaks of Mathematics Springer, New York 2011
The Colorado Mathematical Olympiad; The Third Decade and Further Explorations: From the Mountains of Colorado to the Peaks of Mathematics Springer, New York 2017
Geometric Etudes in Combinatorial Mathematics 2nd ed. Springer, New York 2010
Ramsey Theory: Yesterday, Today, and Tomorrow, (editor and contributor) Birkhäuser-Springer 2011
Life and Fate: In Search of Van der Waerden, appeared in November 2008 in Russian. The expanded English edition, The Scholar and the State: In Search of Van der Waerden, was published by Birkhäuser-Springer in 2017.
Geombinatorics
Geombinatorics is a quarterly scientific journal of mathematics. It was established by
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https://en.wikipedia.org/wiki/Folk%20mathematics
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Folk mathematics may refer to:
The mathematical folklore that circulates among mathematicians
The informal mathematics used in everyday life
See also
Folk theorem (disambiguation)
Numerals in Koro Language -language of Indigenous People by N. C. Ghosh. Science and culture, 82(5-6) 189-193, 2016
Folk Mathematics : Concepts & Definition - An Out Line by N.C.Ghosh, Rabindra Bharati Patrika Vol. XII, No. 2, 2009
Folklore Study. LOKDARPAN - Journal of the Dept. of Folklore by N.C.Ghosh, Kalyani University. Vol. 3, No. 2, 2007
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https://en.wikipedia.org/wiki/GRAPE
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GRAPE, or GRAphics Programming Environment is a software development environment for mathematical visualization, especially differential geometry and continuum mechanics. In 1994, it won the European Academic Software Award.
The term graphical refers to the applications; the programming itself is mostly based on C. GRAPE was developed by the University of Bonn in Germany and is available for free for non-commercial purposes. It has not been developed actively since 1998.
qfix Grape
Another graphical programming environment called GRAPE is developed by qfix and the University of Ulm. Here, it is used as a graphical tool for developing object oriented programs for controlling autonomous mobile robots. After arranging graphical program entities to receive the desired flow chart, the graphical program can be translated to source code (e.g. C++). A modular interface makes the environment easy to extend, so additional classes can be integrated or different flowchart-to-code translator or compilers can be used.
References
External links
official homepage (University of Bonn)
Homepage qfix robotics
qfix Grape
Data visualization software
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https://en.wikipedia.org/wiki/United%20States%20cities%20by%20crime%20rate%20%28100%2C000%E2%80%93250%2C000%29
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The following table is based on Federal Bureau of Investigation Uniform Crime Reports statistics.
The population numbers are based on U.S. Census estimates for the year end. The number of murders includes nonnegligent manslaughter. This list is based on the reporting agency. In most cases the city and the reporting agency are identical. However, in some cases such as Charlotte, Honolulu and Las Vegas, the reporting agency as more than one city.
Murder is the only statistic that all agencies are required to report. Consequently, some agencies do not report all the crimes. If components are missing the total is adjusted to "0."
Note about population
Data are voluntarily submitted by each jurisdiction and some jurisdictions do not appear in the table because they either did not submit data or they did not meet deadlines.
According to the FBI website has this disclaimer on population estimates:
For the 2008 population estimates used in this table, the FBI computed individual rates of growth from one year to the next for every city/town and county using 2000 decennial population counts and 2001 through 2007 population estimates from the U.S. Census Bureau. Each agency’s rates of growth were averaged; that average was then applied and added to its 2007 Census population estimate to derive the agency’s 2008 population estimate.
2012 Calendar Year Ratios of Crime Per 100,000 Population
Rates are based on cases per 100,000 for all of calendar 2011.
Criticism of ranking crime data
The FBI web site recommends against using its data for ranking because these rankings lead to simplistic and/or incomplete analyses that often create misleading perceptions adversely affecting cities and counties, along with their residents. The FBI web site also recommends against using its data to judge how effective law enforcement agencies are, since there are many factors that influence crime rates other than law enforcement.
In November 2007, the executive board of the American Society of Criminology (ASC) went further than the FBI itself, and approved a resolution opposing not only the use of the ratings to judge police departments, but also opposing any development of city crime rankings from FBI Uniform Crime Reports (UCRs) at all. The resolution opposed these rankings on the grounds that they "fail to account for the many conditions affecting crime rates" and "divert attention from the individual and community characteristics that elevate crime in all cities", though it did not provide sources or further elaborate on these claims. The resolution states the rankings "represent an irresponsible misuse of the data and do groundless harm to many communities" and "work against a key goal of our society, which is a better understanding of crime-related issues by both scientists and the public".
The U.S. Conference of Mayors passed a similar statement, which also committed the Conference to working with the FBI and the U.S. Department of Justice "to educate repo
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https://en.wikipedia.org/wiki/United%20States%20cities%20by%20crime%20rate%20%2860%2C000%E2%80%93100%2C000%29
|
The following table is based on Federal Bureau of Investigation Uniform Crime Reports statistics.
The population numbers are based on U.S. Census estimates for the year end. The number of murders includes nonnegligent manslaughter. This list is based on the reporting agency. In most cases the city and the reporting agency are identical. However, in some cases such as Charlotte, Honolulu and Las Vegas, the reporting agency as more than one city.
Murder is the only statistic that all agencies are required to report. Consequently, some agencies do not report all the crimes. If components are missing the total is adjusted to "0."
Note about population
Data are voluntarily submitted by each jurisdiction and some jurisdictions do not appear in the table because they either did not submit data or they did not meet deadlines.
According to the FBI website has this disclaimer on population estimates:
For the 2008 population estimates used in this table, the FBI computed individual rates of growth from one year to the next for every city/town and county using 2000 decennial population counts and 2001 through 2007 population estimates from the U.S. Census Bureau. Each agency’s rates of growth were averaged; that average was then applied and added to its 2007 Census population estimate to derive the agency’s 2008 population estimate.
2014 Calendar Year Ratios of Crime Per 100,000 Population
Criticism of ranking crime data
The FBI web site recommends against using its data for ranking because these rankings lead to simplistic and/or incomplete analyses that often create misleading perceptions adversely affecting cities and counties, along with their residents. The FBI web site also recommends against using its data to judge how effective law enforcement agencies are, since there are many factors that influence crime rates other than law enforcement.
In November 2007, the executive board of the American Society of Criminology (ASC) went further than the FBI itself, and approved a resolution opposing not only the use of the ratings to judge police departments, but also opposing any development of city crime rankings from FBI Uniform Crime Reports (UCRs) at all. The resolution opposed these rankings on the grounds that they "fail to account for the many conditions affecting crime rates" and "divert attention from the individual and community characteristics that elevate crime in all cities", though it did not provide sources or further elaborate on these claims. The resolution states the rankings "represent an irresponsible misuse of the data and do groundless harm to many communities" and "work against a key goal of our society, which is a better understanding of crime-related issues by both scientists and the public".
The U.S. Conference of Mayors passed a similar statement, which also committed the Conference to working with the FBI and the U.S. Department of Justice "to educate reporters, elected officials, and citizens on what the (UCR) data
|
https://en.wikipedia.org/wiki/United%20States%20cities%20by%20crime%20rate%20%2840%2C000%E2%80%9360%2C000%29
|
The following table is based on Federal Bureau of Investigation Uniform Crime Reports statistics.
The population numbers are based on U.S. Census estimates for the year end. The number of murders includes nonnegligent manslaughter. This list is based on the reporting agency. In most cases the city and the reporting agency are identical. However, in some cases such as Charlotte, Honolulu and Las Vegas, the reporting agency as more than one city.
Murder is the only statistic that all agencies are required to report. Consequently, some agencies particularly in Illinois do not report all the crimes. If components are missing the total is adjusted to "0."
Note about population
Data is voluntarily submitted by each jurisdiction and some jurisdictions do not appear in the table because they either did not submit data or it did not meet deadlines.
According to the FBI website has this disclaimer on population estimates:
For the 2007 population estimates used in this table, the FBI computed individual rates of growth from one year to the next for every city/town and county using 2000 decennial population counts and 2001 through 2006 population estimates from the U.S. Census Bureau. Each agency's rates of growth were averaged; that average was then applied and added to its 2006 Census population estimate to derive the agency's 2007 population estimate
2010 Calendar Year Ratios of Crime Per 100,000 Population
Criticism of ranking crime data
The FBI web site recommends against using its data for ranking because these rankings lead to simplistic and/or incomplete analyses that often create misleading perceptions adversely affecting cities and counties, along with their residents. The FBI web site also recommends against using its data to judge how effective law enforcement agencies are, since there are many factors that influence crime rates other than law enforcement.
In November 2007, the executive board of the American Society of Criminology (ASC) went further than the FBI itself, and approved a resolution opposing not only the use of the ratings to judge police departments, but also opposing any development of city crime rankings from FBI Uniform Crime Reports (UCRs) at all. The resolution opposed these rankings on the grounds that they "fail to account for the many conditions affecting crime rates" and "divert attention from the individual and community characteristics that elevate crime in all cities", though it did not provide sources or further elaborate on these claims. The resolution states the rankings "represent an irresponsible misuse of the data and do groundless harm to many communities" and "work against a key goal of our society, which is a better understanding of crime-related issues by both scientists and the public".
The U.S. Conference of Mayors passed a similar statement, which also committed the Conference to working with the FBI and the U.S. Department of Justice "to educate reporters, elected officials, and citizens on
|
https://en.wikipedia.org/wiki/Charles%20E.%20M.%20Pearce
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Charles Edward Miller Pearce (29 March 1940 – 8 June 2012) was a New Zealand/Australian mathematician.
At the time of his death on 8 June 2012 he was the (Sir Thomas) Elder Professor of Mathematics at the University of Adelaide.
Early life
Pearce was born in Wellington. His early schooling was in Wellington and he was dux of Hutt Valley High School in 1957. He earned his Bachelor of Science (a double major in Applied and Pure Mathematics and a further double major in Physics and Mathematical Physics) and in 1962 he earned a Masters of Science with first class honours in Mathematics, all from Victoria University of Wellington. The bachelor's degree was from the University of New Zealand, as the constituent colleges of UNZ, of which Victoria University College was one of four, had proliferated into four autonomous Universities by the time Pearce completed his master's degree.
New Zealand origins
Pearce always remained proud of his New Zealand origins. Being descended from Maori people, he claimed his New Zealand ancestry was longer than almost all his peers from New Zealand.
Pearce is descended from Alexander Gray, one of just five Scots who settled in New Zealand as part of the original and largely strong interest in Maoritanga and claimed ancestral connection to three waka (canoes) in the heke (migration): Aotea, Kurahaupo and Takatimu. His principal tribal connection was with the Ngati Ruanui, based in the southern Taranaki.
Life and career
In 1963 Pearce left New Zealand for doctoral study at the Australian National University (ANU) in Canberra, under the supervision of Pat Moran. Thereafter followed short stints (1 to 3 years) as lecturer at: ANU; University of Queensland (visiting Professor); Université de Rennes 1, France; and University of Sheffield (1966–68). He was appointed to the University of Adelaide in 1968 and remained there for the ensuing years, having been promoted to senior lecturer in 1971, Reader in 1982 and professor in 2003. He was a leading figure in the Department of Applied Mathematics there, being appointed in 2005 to the Elder Chair of Mathematics.
While at ANU, he met and married Frances (née O'Connor), and they brought up their two daughters, Emma and Ann, in Adelaide. Charles died in a motor vehicle accident near Fox Glacier on the NZ South Island on 8 June 2012.
Mathematical work
He is known for probabilistic and statistical modelling. Pearce published prolifically in the area of probabilistic and statistical modelling and analysis, with strong contributions being made in both theory and practice. His book with Dragomir addresses the fine points of the Hermite–Hadamard inequality and is published by Kluwer Academic Press. His applied interests included queuing theory, road traffic, telecommunications, and urban planning. With former student Bill Henderson, who followed him from Sheffield to Adelaide, he helped establish the successful Teletraffic Centre in the University of Adelaide. Publications are numero
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https://en.wikipedia.org/wiki/Local%20Langlands%20conjectures
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In mathematics, the local Langlands conjectures, introduced by , are part of the Langlands program. They describe a correspondence between the complex representations of a reductive algebraic group G over a local field F, and representations of the Langlands group of F into the L-group of G. This correspondence is not a bijection in general. The conjectures can be thought of as a generalization of local class field theory from abelian Galois groups to non-abelian Galois groups.
Local Langlands conjectures for GL1
The local Langlands conjectures for GL1(K) follow from (and are essentially equivalent to) local class field theory. More precisely the Artin map gives an isomorphism from the group GL1(K)= K* to the abelianization of the Weil group. In particular irreducible smooth representations of GL1(K) are 1-dimensional as the group is abelian, so can be identified with homomorphisms of the Weil group to GL1(C). This gives the Langlands correspondence between homomorphisms of the Weil group to GL1(C) and
irreducible smooth representations of GL1(K).
Representations of the Weil group
Representations of the Weil group do not quite correspond to irreducible smooth representations of general linear groups. To get a bijection, one has to slightly modify the notion of a representation of the Weil group, to something called a Weil–Deligne representation. This consists of a representation of the Weil group on a vector space V together with a nilpotent endomorphism N of V such that wNw−1=||w||N, or equivalently a representation of the Weil–Deligne group. In addition the representation of the Weil group should have an open kernel, and should be (Frobenius) semisimple.
For every Frobenius semisimple complex n-dimensional Weil–Deligne representation ρ of the Weil group of F there is an L-function L(s,ρ) and a local ε-factor ε(s,ρ,ψ) (depending on a character ψ of F).
Representations of GLn(F)
The representations of GLn(F) appearing in the local Langlands correspondence are smooth irreducible complex representations.
"Smooth" means that every vector is fixed by some open subgroup.
"Irreducible" means that the representation is nonzero and has no subrepresentations other than 0 and itself.
Smooth irreducible complex representations are automatically admissible.
The Bernstein–Zelevinsky classification reduces the classification of irreducible smooth representations to cuspidal representations.
For every irreducible admissible complex representation π there is an L-function L(s,π) and a local ε-factor ε(s,π,ψ) (depending on a character ψ of F). More generally, if there are two irreducible admissible representations π and π' of general linear groups there are local Rankin–Selberg convolution L-functions L(s,π×π') and ε-factors ε(s,π×π',ψ).
described the irreducible admissible representations of general linear groups over local fields.
Local Langlands conjectures for GL2
The local Langlands conjecture for GL2 of a local field says that there
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https://en.wikipedia.org/wiki/Nesmith%20Ankeny
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Nesmith Cornett Ankeny (1927, Walla Walla, Washington – 4 August 1993, Seattle) was an American mathematician specialising in number theory.
After Army service, he studied at Stanford University and obtained his Ph.D. at Princeton University in 1950 under the supervision of Emil Artin. He was a Fellow at Princeton and the Institute for Advanced Study, then assistant professor at Johns Hopkins University from 1952 to 1955, when he joined MIT. He was a Guggenheim fellow in 1958; he became a full professor in 1964 and retired in 1992.
His research was mainly in analytic number theory, on consequences of the generalized Riemann hypothesis.
He was also interested in game theory and gaming: he wrote a book on mathematical analysis of poker strategies, especially bluffing.
See also
Ankeny–Artin–Chowla congruence
Works
N.C. Ankeny, Poker strategy, Basic Books (1981), .
References
External links
1927 births
1993 deaths
20th-century American mathematicians
Number theorists
Princeton University alumni
Institute for Advanced Study visiting scholars
Stanford University alumni
Princeton University fellows
Johns Hopkins University faculty
Massachusetts Institute of Technology faculty
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https://en.wikipedia.org/wiki/Normal%20coordinates
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In differential geometry, normal coordinates at a point p in a differentiable manifold equipped with a symmetric affine connection are a local coordinate system in a neighborhood of p obtained by applying the exponential map to the tangent space at p. In a normal coordinate system, the Christoffel symbols of the connection vanish at the point p, thus often simplifying local calculations. In normal coordinates associated to the Levi-Civita connection of a Riemannian manifold, one can additionally arrange that the metric tensor is the Kronecker delta at the point p, and that the first partial derivatives of the metric at p vanish.
A basic result of differential geometry states that normal coordinates at a point always exist on a manifold with a symmetric affine connection. In such coordinates the covariant derivative reduces to a partial derivative (at p only), and the geodesics through p are locally linear functions of t (the affine parameter). This idea was implemented in a fundamental way by Albert Einstein in the general theory of relativity: the equivalence principle uses normal coordinates via inertial frames. Normal coordinates always exist for the Levi-Civita connection of a Riemannian or Pseudo-Riemannian manifold. By contrast, in general there is no way to define normal coordinates for Finsler manifolds in a way that the exponential map are twice-differentiable .
Geodesic normal coordinates
Geodesic normal coordinates are local coordinates on a manifold with an affine connection defined by means of the exponential map
and an isomorphism
given by any basis of the tangent space at the fixed basepoint . If the additional structure of a Riemannian metric is imposed, then the basis defined by E may be required in addition to be orthonormal, and the resulting coordinate system is then known as a Riemannian normal coordinate system.
Normal coordinates exist on a normal neighborhood of a point p in M. A normal neighborhood U is an open subset of M such that there is a proper neighborhood V of the origin in the tangent space TpM, and expp acts as a diffeomorphism between U and V. On a normal neighborhood U of p in M, the chart is given by:
The isomorphism E, and therefore the chart, is in no way unique.
A convex normal neighborhood U is a normal neighborhood of every p in U. The existence of these sort of open neighborhoods (they form a topological basis) has been established by J.H.C. Whitehead for symmetric affine connections.
Properties
The properties of normal coordinates often simplify computations. In the following, assume that is a normal neighborhood centered at a point in and are normal coordinates on .
Let be some vector from with components in local coordinates, and be the geodesic with and . Then in normal coordinates, as long as it is in . Thus radial paths in normal coordinates are exactly the geodesics through .
The coordinates of the point are
In Riemannian normal coordinates at a point th
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