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https://en.wikipedia.org/wiki/Hits%20allowed
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In Baseball statistics, hits allowed (HA) signifies the total number of hits allowed by a pitcher.
See also
Baseball statistics
Pitching statistics
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https://en.wikipedia.org/wiki/Homogeneous%20polynomial
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In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, is a homogeneous polynomial of degree 5, in two variables; the sum of the exponents in each term is always 5. The polynomial is not homogeneous, because the sum of exponents does not match from term to term. The function defined by a homogeneous polynomial is always a homogeneous function.
An algebraic form, or simply form, is a function defined by a homogeneous polynomial. A binary form is a form in two variables. A form is also a function defined on a vector space, which may be expressed as a homogeneous function of the coordinates over any basis.
A polynomial of degree 0 is always homogeneous; it is simply an element of the field or ring of the coefficients, usually called a constant or a scalar. A form of degree 1 is a linear form. A form of degree 2 is a quadratic form. In geometry, the Euclidean distance is the square root of a quadratic form.
Homogeneous polynomials are ubiquitous in mathematics and physics. They play a fundamental role in algebraic geometry, as a projective algebraic variety is defined as the set of the common zeros of a set of homogeneous polynomials.
Properties
A homogeneous polynomial defines a homogeneous function. This means that, if a multivariate polynomial P is homogeneous of degree d, then
for every in any field containing the coefficients of P. Conversely, if the above relation is true for infinitely many then the polynomial is homogeneous of degree d.
In particular, if P is homogeneous then
for every This property is fundamental in the definition of a projective variety.
Any nonzero polynomial may be decomposed, in a unique way, as a sum of homogeneous polynomials of different degrees, which are called the homogeneous components of the polynomial.
Given a polynomial ring over a field (or, more generally, a ring) K, the homogeneous polynomials of degree d form a vector space (or a module), commonly denoted The above unique decomposition means that is the direct sum of the (sum over all nonnegative integers).
The dimension of the vector space (or free module) is the number of different monomials of degree d in n variables (that is the maximal number of nonzero terms in a homogeneous polynomial of degree d in n variables). It is equal to the binomial coefficient
Homogeneous polynomial satisfy Euler's identity for homogeneous functions. That is, if is a homogeneous polynomial of degree in the indeterminates one has, whichever is the commutative ring of the coefficients,
where denotes the formal partial derivative of with respect to
Homogenization
A non-homogeneous polynomial P(x1,...,xn) can be homogenized by introducing an additional variable x0 and defining the homogeneous polynomial sometimes denoted hP:
where d is the degree of P. For example, if
then
A homogenized polynomial can be dehomogenized by setting the additio
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https://en.wikipedia.org/wiki/Discrete%20uniform%20distribution
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In probability theory and statistics, the discrete uniform distribution is a symmetric probability distribution wherein a finite number of values are equally likely to be observed; every one of n values has equal probability 1/n. Another way of saying "discrete uniform distribution" would be "a known, finite number of outcomes equally likely to happen".
A simple example of the discrete uniform distribution is throwing a fair die. The possible values are 1, 2, 3, 4, 5, 6, and each time the die is thrown the probability of a given score is 1/6. If two dice are thrown and their values added, the resulting distribution is no longer uniform because not all sums have equal probability.
Although it is convenient to describe discrete uniform distributions over integers, such as this, one can also consider discrete uniform distributions over any finite set. For instance, a random permutation is a permutation generated uniformly from the permutations of a given length, and a uniform spanning tree is a spanning tree generated uniformly from the spanning trees of a given graph.
The discrete uniform distribution itself is inherently non-parametric. It is convenient, however, to represent its values generally by all integers in an interval [a,b], so that a and b become the main parameters of the distribution (often one simply considers the interval [1,n] with the single parameter n). With these conventions, the cumulative distribution function (CDF) of the discrete uniform distribution can be expressed, for any k ∈ [a,b], as
Estimation of maximum
This example is described by saying that a sample of k observations is obtained from a uniform distribution on the integers , with the problem being to estimate the unknown maximum N. This problem is commonly known as the German tank problem, following the application of maximum estimation to estimates of German tank production during World War II.
The uniformly minimum variance unbiased (UMVU) estimator for the maximum is given by
where m is the sample maximum and k is the sample size, sampling without replacement. This can be seen as a very simple case of maximum spacing estimation.
This has a variance of
so a standard deviation of approximately , the (population) average size of a gap between samples; compare above.
The sample maximum is the maximum likelihood estimator for the population maximum, but, as discussed above, it is biased.
If samples are not numbered but are recognizable or markable, one can instead estimate population size via the capture-recapture method.
Random permutation
See rencontres numbers for an account of the probability distribution of the number of fixed points of a uniformly distributed random permutation.
Properties
The family of uniform distributions over ranges of integers (with one or both bounds unknown) has a finite-dimensional sufficient statistic, namely the triple of the sample maximum, sample minimum, and sample size, but is not an exponential family of distributi
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https://en.wikipedia.org/wiki/Continuous%20uniform%20distribution
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In probability theory and statistics, the continuous uniform distributions or rectangular distributions are a family of symmetric probability distributions. Such a distribution describes an experiment where there is an arbitrary outcome that lies between certain bounds. The bounds are defined by the parameters, and which are the minimum and maximum values. The interval can either be closed (i.e. ) or open (i.e. ). Therefore, the distribution is often abbreviated where stands for uniform distribution. The difference between the bounds defines the interval length; all intervals of the same length on the distribution's support are equally probable. It is the maximum entropy probability distribution for a random variable under no constraint other than that it is contained in the distribution's support.
Definitions
Probability density function
The probability density function of the continuous uniform distribution is:
The values of at the two boundaries and are usually unimportant, because they do not alter the value of over any interval nor of nor of any higher moment. Sometimes they are chosen to be zero, and sometimes chosen to be The latter is appropriate in the context of estimation by the method of maximum likelihood. In the context of Fourier analysis, one may take the value of or to be because then the inverse transform of many integral transforms of this uniform function will yield back the function itself, rather than a function which is equal "almost everywhere", i.e. except on a set of points with zero measure. Also, it is consistent with the sign function, which has no such ambiguity.
Any probability density function integrates to so the probability density function of the continuous uniform distribution is graphically portrayed as a rectangle where is the base length and is the height. As the base length increases, the height (the density at any particular value within the distribution boundaries) decreases.
In terms of mean and variance the probability density function of the continuous uniform distribution is:
Cumulative distribution function
The cumulative distribution function of the continuous uniform distribution is:
Its inverse is:
In terms of mean and variance the cumulative distribution function of the continuous uniform distribution is:
its inverse is:
Example 1. Using the continuous uniform distribution function
For a random variable find
In a graphical representation of the continuous uniform distribution function the area under the curve within the specified bounds, displaying the probability, is a rectangle. For the specific example above, the base would be and the height would be
Example 2. Using the continuous uniform distribution function (conditional)
For a random variable find
The example above is a conditional probability case for the continuous uniform distribution: given that is true, what is the probability that Conditional probability changes the sample space, so a ne
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https://en.wikipedia.org/wiki/R%20v%20Adams
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R v Adams [1996] EWCA Crim 10 and 222, are rulings in the United Kingdom that banned the expression in court of headline (soundbite), standalone Bayesian statistics from the reasoning admissible before a jury in DNA evidence cases, in favour of the calculated average (and maximal) number of matching incidences among the nation's population. The facts involved strong but inconclusive evidence conflicting with the DNA evidence, leading to a retrial.
Facts
A rape victim described her attacker as in his twenties. A suspect, Denis Adams, was arrested and an identity parade was arranged. The woman failed to pick him out, and on being asked if he fitted her description replied in the negative. She had described a man in his twenties and when asked how old Adams looked, she replied about forty. Adams was 37; he had an alibi for the night in question, his girlfriend saying he had spent the night with her. The DNA was the only incriminating evidence heard by the jury, as all the other evidence pointed towards innocence.
Judgment
Use of Bayesian analysis in the court
The DNA profile of the suspect fitted that of evidence left at the scene. The defence argued that the match probability figure put forward by the prosecution (1 in 200 million) was incorrect, and that a figure of 1 in 20 million, or perhaps even 1 in 2 million, was more appropriate. The issue of how the jury should resolve the conflicting evidence was addressed by the defence by a formal statistical method. The jury was instructed in the use of Bayes's theorem by Professor Peter Donnelly of Oxford University. The judge told the jury they could use Bayes's theorem if they wished. Adams was convicted and the case went to appeal. The Appeal Court judges noted that the original trial judge did not direct the jury as to what to do if they did not wish to use Bayes's theorem and ordered a retrial.
At the retrial the defence team again wanted to instruct the new jury in the use of Bayes's theorem (though Prof. Donnelly had doubts about the practicality of the approach). The judge asked that the statistical experts from both sides work together to produce a workable method of implementing Bayes's theorem for use in a courtroom, should the jury wish to use it. A questionnaire was produced which asked a series of questions such as:
"If he were the attacker, what's the chance that she would say he looked nothing like the attacker?"
"If he wasn't the attacker what's the chance that she would say he looked nothing like the attacker?"
These questions were intended to allow the Bayes factors of the various pieces of evidence to be assessed. The questionnaires had boxes where jurors could put their assessments and a formula to enable them to produce the overall odds of guilt or innocence. Adams was convicted once again and again an appeal was made to the Court of Appeal. The appeal was unsuccessful but the Appeal Court ruling was highly critical of the appropriateness of Bayes's theorem in the courtroom
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https://en.wikipedia.org/wiki/School%20Mathematics%20Study%20Group
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The School Mathematics Study Group (SMSG) was an American academic think tank focused on the subject of reform in mathematics education. Directed by Edward G. Begle and financed by the National Science Foundation, the group was created in the wake of the Sputnik crisis in 1958 and tasked with creating and implementing mathematics curricula for primary and secondary education, which it did until its termination in 1977.
The efforts of the SMSG yielded a reform in mathematics education known as New Math which was promulgated in a series of reports, culminating in a series published by Random House called the New Mathematical Library (Vol. 1 is Ivan Niven's Numbers: Rational and Irrational). In the early years, SMSG also produced a set of draft textbooks in typewritten paperback format for elementary, middle and high school students.
Perhaps the most authoritative collection of materials from the School Mathematics Study Group is now housed in the Archives of American Mathematics in the University of Texas at Austin's Center for American History.
See also
Foundations of geometry
Further reading
1958 Letter from Ralph A. Raimi to Fred Quigley concerning the New Math
Whatever Happened to the New Math by Ralph A. Raimi
Some Technical Commentaries on Mathematics Education and History by Ralph A. Raimi
External links
The SMSG Collection at The Center for American History at UT
Archives of American Mathematics at the Center for American History at UT
Mathematics education
Curricula
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https://en.wikipedia.org/wiki/Sequence%20space
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In functional analysis and related areas of mathematics, a sequence space is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it is a function space whose elements are functions from the natural numbers to the field K of real or complex numbers. The set of all such functions is naturally identified with the set of all possible infinite sequences with elements in K, and can be turned into a vector space under the operations of pointwise addition of functions and pointwise scalar multiplication. All sequence spaces are linear subspaces of this space. Sequence spaces are typically equipped with a norm, or at least the structure of a topological vector space.
The most important sequence spaces in analysis are the spaces, consisting of the -power summable sequences, with the p-norm. These are special cases of Lp spaces for the counting measure on the set of natural numbers. Other important classes of sequences like convergent sequences or null sequences form sequence spaces, respectively denoted c and c0, with the sup norm. Any sequence space can also be equipped with the topology of pointwise convergence, under which it becomes a special kind of Fréchet space called FK-space.
Definition
A sequence in a set is just an -valued map whose value at is denoted by instead of the usual parentheses notation
Space of all sequences
Let denote the field either of real or complex numbers. The set of all sequences of elements of is a vector space for componentwise addition
and componentwise scalar multiplication
A sequence space is any linear subspace of
As a topological space, is naturally endowed with the product topology. Under this topology, is Fréchet, meaning that it is a complete, metrizable, locally convex topological vector space (TVS). However, this topology is rather pathological: there are no continuous norms on (and thus the product topology cannot be defined by any norm). Among Fréchet spaces, is minimal in having no continuous norms:
But the product topology is also unavoidable: does not admit a strictly coarser Hausdorff, locally convex topology. For that reason, the study of sequences begins by finding a strict linear subspace of interest, and endowing it with a topology different from the subspace topology.
spaces
For is the subspace of consisting of all sequences satisfying
If then the real-valued function on defined by
defines a norm on In fact, is a complete metric space with respect to this norm, and therefore is a Banach space.
If then is also a Hilbert space when endowed with its canonical inner product, called the , defined for all by
The canonical norm induced by this inner product is the usual -norm, meaning that for all
If then is defined to be the space of all bounded sequences endowed with the norm
is also a Banach space.
If then does not carry a norm, but rather a metric defined by
c, c0 and c00
A is any sequence such
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https://en.wikipedia.org/wiki/Sturmian%20word
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In mathematics, a Sturmian word (Sturmian sequence or billiard sequence), named after Jacques Charles François Sturm, is a certain kind of infinitely long sequence of characters. Such a sequence can be generated by considering a game of English billiards on a square table. The struck ball will successively hit the vertical and horizontal edges labelled 0 and 1 generating a sequence of letters. This sequence is a Sturmian word.
Definition
Sturmian sequences can be defined strictly in terms of their combinatoric properties or geometrically as cutting sequences for lines of irrational slope or codings for irrational rotations. They are traditionally taken to be infinite sequences on the alphabet of the two symbols 0 and 1.
Combinatorial definitions
Sequences of low complexity
For an infinite sequence of symbols w, let σ(n) be the complexity function of w; i.e., σ(n) = the number of distinct contiguous subwords (factors) in w of length n. Then w is Sturmian if σ(n) = n + 1 for all n.
Balanced sequences
A set X of binary strings is called balanced if the Hamming weight of elements of X takes at most two distinct values. That is, for any |s|1 = k or |s|1 = k where |s|1 is the number of 1s in s.
Let w be an infinite sequence of 0s and 1s and let denote the set of all length-n subwords of w. The sequence w is Sturmian if is balanced for all n and w is not eventually periodic.
Geometric definitions
Cutting sequence of irrational
Let w be an infinite sequence of 0s and 1s. The sequence w is Sturmian if for some and some irrational , w is realized as the cutting sequence of the line .
Difference of Beatty sequences
Let w = (wn) be an infinite sequence of 0s and 1s. The sequence w is Sturmian if it is the difference of non-homogeneous Beatty sequences, that is, for some and some irrational
for all or
for all .
Coding of irrational rotation
For , define by . For define the θ-coding of x to be the sequence (xn) where
Let w be an infinite sequence of 0s and 1s. The sequence w is Sturmian if for some and some irrational , w is the θ-coding of x.
Discussion
Example
A famous example of a (standard) Sturmian word is the Fibonacci word; its slope is , where is the golden ratio.
Balanced aperiodic sequences
A set S of finite binary words is balanced if for each n the subset Sn of words of length n has the property that the Hamming weight of the words in Sn takes at most two distinct values. A balanced sequence is one for which the set of factors is balanced. A balanced sequence has at most n+1 distinct factors of length n. An aperiodic sequence is one which does not consist of a finite sequence followed by a finite cycle. An aperiodic sequence has at least n + 1 distinct factors of length n. A sequence is Sturmian if and only if it is balanced and aperiodic.
Slope and intercept
A sequence over {0,1} is a Sturmian word if and only if there exist two real numbers, the slope and the intercept , with irrational, su
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https://en.wikipedia.org/wiki/ZbMATH%20Open
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zbMATH Open, formerly Zentralblatt MATH, is a major reviewing service providing reviews and abstracts for articles in pure and applied mathematics, produced by the Berlin office of FIZ Karlsruhe – Leibniz Institute for Information Infrastructure GmbH. Editors are the European Mathematical Society, FIZ Karlsruhe, and the Heidelberg Academy of Sciences. zbMATH is distributed by Springer Science+Business Media. It uses the Mathematics Subject Classification codes for organising reviews by topic.
History
Mathematicians Richard Courant, Otto Neugebauer, and Harald Bohr, together with the publisher Ferdinand Springer, took the initiative for a new mathematical reviewing journal. Harald Bohr worked in Copenhagen. Courant and Neugebauer were professors at the University of Göttingen. At that time, Göttingen was considered one of the central places for mathematical research, having appointed mathematicians like David Hilbert, Hermann Minkowski, Carl Runge, and Felix Klein, the great organiser of mathematics and physics in Göttingen. His dream of a building for an independent mathematical institute with a spacious and rich reference library was realised four years after his death. The credit for this achievement is particularly due to Richard Courant, who convinced the Rockefeller Foundation to donate a large amount of money for the construction.
The service was founded in 1931, by Otto Neugebauer as Zentralblatt für Mathematik und ihre Grenzgebiete. It contained the bibliographical data of all recently published mathematical articles and book, together with peer reviews done by mathematicians over the world. In the preface to the first volume, the intentions of Zentralblatt are formulated as follows:
Zentralblatt and the Jahrbuch über die Fortschritte der Mathematik had in essence the same agenda, but Zentralblatt published several issues per year. An issue was published as soon as sufficiently many reviews were available, in a frequency of three or four weeks.
In the late 1930s, it began rejecting some Jewish reviewers and a number of reviewers in England and United States resigned in protest. Some of them helped start Mathematical Reviews, a competing publication.
The electronic form was provided under the name INKA-MATH (acronym for Information System Karlsruhe-Database on Mathematics) since at least 1980. The name was later shortened to Zentralblatt MATH.
In addition to the print issue, the services were offered online under the name zbMATH since 1996. Since 2004 older issues were incorporated back to 1826.
The printed issue was discontinued in 2013.
Since January 2021, the access to the database is now open under the name zbMATH Open.
Services
The Zentralblatt MATH abstracting service provides reviews (brief accounts of contents) of current articles, conference papers, books and other publications in mathematics, its applications, and related areas. The reviews are predominantly in English, with occasional entries in German and French. Revi
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https://en.wikipedia.org/wiki/777%20%28number%29
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777 (seven hundred [and] seventy-seven) is the natural number following 776 and preceding 778. The number 777 is significant in numerous religious and political contexts.
In mathematics
777 is an odd, composite, palindromic repdigit. It is also a sphenic number, with 3, 7, and 37 as its prime factors. Its largest prime factor is a concatenation of its smaller two; the only other number below 1000 with this property is 138.
777 is also:
An extravagant number, a lucky number, a polite number, and an amenable number.
A deficient number, since the sum of its divisors is less than 2n.
A congruent number, as it is possible to make a right triangle with a rational number of sides whose area is 777.
An arithmetic number, since the average of its positive divisors is also an integer (152).
A repdigit in senary.
Religious significance
According to the Bible, Lamech, the father of Noah lived for 777 years. Some of the known religious connections to 777 are noted in the sections below.
Judaism
The numbers 3 and 7 both are considered "perfect numbers" under Hebrew tradition.
Christianity
According to the American publication, the Orthodox Study Bible, 777 represents the threefold perfection of the Trinity.
Thelema
777 is also found in the title of the book 777 and Other Qabalistic Writings of Aleister Crowley pertaining to the law of thelema.
Political significance
Afrikaner Weerstandsbeweging
The Afrikaner Resistance Movement (Afrikaner Weerstandsbeweging, AWB), a Boer-nationalist, neo-Nazi, and white supremacist movement in South Africa, used the number 777 as part of their emblem.
The number refers to a triumph of "God's number" 7 over the Devil's number 666. On the AWB flag, the numbers are arranged in a triskelion shape, resembling the Nazi swastika.
Computing
In Unix's chmod, change-access-mode command, the octal value 777 grants all file-access permissions to all user types in a file.
Commercial
Aviation
Boeing, the largest manufacturer of airliners in the United States, released the Boeing 777 (commonly nicknamed the Triple Seven) in June 1995. The 777 family includes the 777-200, 777-200ER, the 777-300, the 777-200LR Worldliner, the 777-300ER, and the 777 Freighter. The -100 wasn't continued in production due to loss of interest. In the 21st century, Boeing has developed what will be in use for some different airlines, called the Boeing 777X. Projects have been delayed because of COVID-19, but will return to normal production.
777 Tower
777 Tower is an office building situated in the US and it was built in 1991.
Gambling and luck
777 is used on most slot machines in the United States to identify a jackpot. As it is considered a lucky number, banknotes with a serial number containing 777 tend to be valued by collectors and numismatists. The US Mint and the Bureau of Engraving and Printing sells uncirculated 777 $1 bills for this reason.
References
Integers
Numerology
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https://en.wikipedia.org/wiki/Nearest%20neighbor
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Nearest neighbor may refer to:
Nearest neighbor search in pattern recognition and in computational geometry
Nearest-neighbor interpolation for interpolating data
Nearest neighbor graph in geometry
Nearest neighbor function in probability theory
Nearest neighbor decoding in coding theory
The k-nearest neighbor algorithm in machine learning, an application of generalized forms of nearest neighbor search and interpolation
The nearest neighbour algorithm for approximately solving the travelling salesman problem
The nearest neighbor method for determining the thermodynamics of nucleic acids
The nearest neighbor method for calculating distances between clusters in hierarchical clustering.
See also
Moore neighborhood
Von Neumann neighborhood
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https://en.wikipedia.org/wiki/Statistical%20discrimination
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Statistical discrimination may refer to:
Statistical discrimination (economics)
Linear discriminant analysis (statistics)
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https://en.wikipedia.org/wiki/Binomial
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Binomial may refer to:
In mathematics
Binomial (polynomial), a polynomial with two terms
Binomial coefficient, numbers appearing in the expansions of powers of binomials
Binomial QMF, a perfect-reconstruction orthogonal wavelet decomposition
Binomial theorem, a theorem about powers of binomials
Binomial type, a property of sequences of polynomials
In probability and statistics
Binomial distribution, a type of probability distribution
Binomial process
Binomial test, a test of significance
In computing science
Binomial heap, a data structure
In linguistics
Binomial pair, a sequence of two or more words or phrases in the same grammatical category, having some semantic relationship and joined by some syntactic device
In biology
Binomial nomenclature, a Latin two-term name for a species, such as Sequoia sempervirens
In finance
Binomial options pricing model, a numerical method for the valuation of options
In politics
Binomial voting system, a voting system used in the parliamentary elections of Chile between 1989 and 2013
See also
List of factorial and binomial topics
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https://en.wikipedia.org/wiki/Cycle%20decomposition
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In mathematics, the term cycle decomposition can mean:
Cycle decomposition (graph theory), a partitioning of the vertices of a graph into subsets, such that the vertices in each subset lie on a cycle
Cycle decomposition (group theory), a useful convention for expressing a permutation in terms of its constituent cycles
In commutative algebra and linear algebra, cyclic decomposition refers to writing a finitely generated module over a principal ideal domain as the direct sum of cyclic modules and one free module.
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https://en.wikipedia.org/wiki/Bureau%20of%20Transportation%20Statistics
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The Bureau of Transportation Statistics (BTS), part of the United States Department of Transportation, is a government office that compiles, analyzes, and publishes information on the nation's transportation systems across various modes; and strives to improve the DOT's statistical programs through research and the development of guidelines for data collection and analysis. BTS is a principal agency of the U.S. Federal Statistical System.
History
BTS was created in 1992 under the Intermodal Surface Transportation Efficiency Act.
On February 20, 2005, BTS became part of the Research and Innovative Technology Administration (RITA). Through the Fixing America's Surface Transportation (FAST) Act passed on December 4, 2015, BTS and RITA moved to the Office of the Assistant Secretary of Transportation for Research and Technology.
Since 2009, BTS has also maintained a Twitter feed, with regular tweets related to the release of BTS data products and news bulletins concerning transportation trends in the United States. Since 2020, BTS has also maintained a LinkedIn account.
Offices
BTS is divided into seven offices:
Office of Statistical and Economic Analysis
Office of Data Development and Standards
Office of Transportation Analysis
Office of Spatial Analysis and Visualization
Office of Airline Information
Office of Information and Library Sciences
Office of Safety Data and Analysis
Services
Airline Information
BTS' Office of Airline Information is responsible for publishing regular reports—often monthly or quarterly—on airline performance in the United States. Topics include airline financials, origins and destinations, passenger traffic, on-time performance, and mishandled baggage.
Data Catalogs
TranStats is an intermodal transportation collection of downloadable databases. One popular database included in the TranStats collection is the airline on-time performance database, which includes on-time performance of every flight, airline, and airport in the United States. Data.bts.gov is an online dataset collection allowing users to create their own visualizations from selected BTS data.
National Transportation Atlas Database
BTS maintains the National Transportation Atlas Database (NTAD), an open online repository of national-level geographic information systems data and applications related to transportation in the United States.
National Transportation Library
Another BTS product is the National Transportation Library (NTL), an online repository of transportation-centric research, reports, and datasets. Documents, which include products internal and external to the US Department of Transportation, can be accessed through a platform called RosaP.
Other products
COVID-19 and Transportation
Border Crossing Data
Commodity Flow Survey
Freight Analysis Framework
National Census of Ferry Operators
Pocket Guide App
Port Performance Freight Statistics Program
TransBorder Freight Data
Transportation Economic Trends
Transportation Statistics
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https://en.wikipedia.org/wiki/National%20Agricultural%20Statistics%20Service
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The National Agricultural Statistics Service (NASS) is the statistical branch of the U.S. Department of Agriculture and a principal agency of the U.S. Federal Statistical System. NASS has 12 regional offices throughout the United States and Puerto Rico and a headquarters unit in Washington, D.C. NASS conducts hundreds of surveys and issues nearly 500 national reports each year on issues including agricultural production, economics, demographics and the environment. NASS also conducts the United States Census of Agriculture every five years.
History
During the Civil War, USDA collected and distributed crop and livestock statistics to help farmers assess the value of the goods they produced. At that time, commodity buyers usually had more current and detailed market information than did farmers, a circumstance that often prevented farmers from getting a fair price for their goods. Producers in today's marketplace would be similarly handicapped were it not for the information provided by NASS.
The creation of USDA's Crop Reporting Board in 1905 (now called the Agricultural Statistics Board) was another landmark in the development of a nationwide statistical service for agriculture. A USDA reorganization in 1961 led to the creation of the Statistical Reporting Service, known today as National Agricultural Statistics Service (NASS).
The 1997 Appropriations Act shifted the responsibility of conducting the Census of Agriculture from U.S. Census Bureau to USDA. Since then the census has been conducted every five years by NASS. Results from the 2012 Census of Agriculture were released on May 2, 2014.
Surveys and reports
The primary sources of information for NASS reports are farmers, ranchers, livestock feeders, slaughterhouse managers, grain elevator operators and other agribusinesses. NASS relies on these survey respondents to voluntarily supply data for most reports.
NASS surveys are conducted in a variety of ways, including mail surveys, telephone interviews, online response, face-to-face interviews and field observations. Once the information is gathered and interpreted, NASS issues estimates and forecasts for crops and livestock and publishes reports on a variety of topics including production and supplies of food and fiber, prices paid and received by farmers, farm labor and wages, farm income and finances, and agricultural chemical use. NASS's field offices publish local data about many of the same topics.
Importance of NASS data
Producers, farm organizations, agribusinesses, lawmakers and government agencies all rely on the information produced by NASS. For instance:
Statistical information on acreage, production, stocks, prices and value is essential for the smooth operation of federal farm programs.
Agricultural data are indispensable for planning and administering related federal and state programs in such areas as consumer protection, conservation and environmental quality, trade, education and recreation.
NASS data helps to ensure a
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https://en.wikipedia.org/wiki/National%20Center%20for%20Education%20Statistics
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The National Center for Education Statistics (NCES) is the part of the United States Department of Education's Institute of Education Sciences (IES) that collects, analyzes, and publishes statistics on education and public school district finance information in the United States. It also conducts international comparisons of education statistics and provides leadership in developing and promoting the use of standardized terminology and definitions for the collection of those statistics. NCES is a principal agency of the U.S. Federal Statistical System.
History
The functions of NCES have existed in some form since 1867, when Congress passed legislation providing "That there shall be established at the City of Washington, a department of education, for the purpose of collecting such statistics and facts as shall show the condition and progress of education in the several States and Territories, and of diffusing such information respecting the organization and management of schools and school systems, and methods of teaching, as shall aid the people of the United States in the establishment and maintenance of efficient school systems, and otherwise promote the cause of education throughout the country."
Organizational structure
The National Center for Education Statistics fulfills a Congressional mandate to collect, collate, analyze, and report complete statistics on the condition of American education; conduct and publish reports; and review and report on education activities internationally.
The structure and activities of the center consist of the following divisions.
Office of the Commissioner
The Office of the Commissioner sets policy and standards for the center and oversees its operation, thus ensuring that statistical quality and confidentiality are maintained.
Administrative Data Division (ADD)
Administrative Data Division (ADD) oversees planning, design, operations, statistical analysis, reporting, and dissemination of administrative records data at the elementary, secondary, and postsecondary education levels, and on libraries.
Assessment Division (AD)
Assessment Division (AD) creates, designs, develops, implements and reports on the National Assessment of Educational Progress at the national level and coordinates assessment and related data collection activities with the states. The staff also conducts a variety of other related education assessment studies.
Sample Surveys Division (SSD)
Sample Surveys Division (SSD) oversees planning design, operations, statistical analysis reporting, and dissemination of data from sample surveys at all levels of education, including early childhood and adult, and international data, such as High School and Beyond (HS&B). Surveys on vocational and technical education are also included in this division.
Annual Reports and Information Staff (ARIS)
The Annual Reports and Information Staff (ARIS) prepares analyses that synthesize data on a variety of education topics, and disseminates these
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https://en.wikipedia.org/wiki/Rado%27s%20theorem%20%28Ramsey%20theory%29
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Rado's theorem is a theorem from the branch of mathematics known as Ramsey theory. It is named for the German mathematician Richard Rado. It was proved in his thesis, Studien zur Kombinatorik.
Statement
Let be a system of linear equations, where is a matrix with integer entries. This system is said to be -regular if, for every -coloring of the natural numbers 1, 2, 3, ..., the system has a monochromatic solution. A system is regular if it is r-regular for all r ≥ 1.
Rado's theorem states that a system is regular if and only if the matrix A satisfies the columns condition. Let ci denote the i-th column of A. The matrix A satisfies the columns condition provided that there exists a partition C1, C2, ..., Cn of the column indices such that if , then
s1 = 0
for all i ≥ 2, si can be written as a rational linear combination of the cjs in all the Ck with k < i. This means that si is in the linear subspace of Q'm spanned by the set of the cj's.
Special cases
Folkman's theorem, the statement that there exist arbitrarily large sets of integers all of whose nonempty sums are monochromatic, may be seen as a special case of Rado's theorem concerning the regularity of the system of equations
where T ranges over each nonempty subset of the set
Other special cases of Rado's theorem are Schur's theorem and Van der Waerden's theorem. For proving the former apply Rado's theorem to the matrix . For Van der Waerden's theorem with m chosen to be length of the monochromatic arithmetic progression, one can for example consider the following matrix:
Computability
Given a system of linear equations it is a priori unclear how to check computationally that it is regular. Fortunately, Rado's theorem provides a criterion which is testable in finite time. Instead of considering colourings (of infinitely many natural numbers), it must be checked that the given matrix satisfies the columns condition. Since the matrix consists only of finitely many columns, this property can be verified in finite time.
However, the subset sum problem can be reduced to the problem of computing the required partition C1, C2, ..., Cn of columns: Given an input set S for the subset sum problem we can write the elements of S in a matrix of shape 1 × |S|. Then the elements of S corresponding to vectors in the partition C1 sum to zero. The subset sum problem is NP-complete. Hence, verifying that a system of linear equations is regular is also an NP-complete problem.
References
Ramsey theory
Theorems in discrete mathematics
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https://en.wikipedia.org/wiki/David%20Slepian
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David S. Slepian (June 30, 1923 – November 29, 2007) was an American mathematician. He is best known for his work with algebraic coding theory, probability theory, and distributed source coding. He was colleagues with Claude Shannon and Richard Hamming at Bell Labs.
Life and work
Born in Pittsburgh, Pennsylvania, he gained a B.Sc. at University of Michigan before joining the US Army in World War II,
as a sonic deception officer in the Ghost army.
He received his Ph.D. from Harvard University in 1949, writing his dissertation in physics. After post-doctoral work at the
University of Cambridge and University of Sorbonne, he worked at the Mathematics Research Center at Bell Telephone Laboratories, where he pioneered work in algebraic coding theory on group codes, first published in the paper A Class of Binary Signaling Alphabets. Here, he also worked along with other information theory giants such as Claude Shannon and Richard Hamming. He also proved the possibility of singular detection, a perhaps unintuitive result. He is also known for Slepian's lemma in probability theory (1962), and for discovering a fundamental result in
distributed source coding called Slepian–Wolf coding with Jack Keil Wolf (1973).
He later joined the University of Hawaiʻi. His father was Joseph Slepian, also a scientist. His wife is the noted children's author Jan Slepian.
Slepians
Slepian's joint work with H.J. Landau and H.O. Pollak on discrete prolate spheroidal wave functions and sequences (DPSWF, DPSS) eventually led to the naming of the sequences as "Slepians". The naming suggestion was provided by Bob Parker of Scripp's Institute of Oceanography, who suggested that "discrete prolate spheroidal sequences" was a "mouthful".
This work was fundamental to the development of the multitaper, where the discrete form are used as an integral component.
Awards
IEEE Fellow
Fellow of Institute of Mathematical Statistics
Claude E. Shannon Award from the IEEE Information Theory Group 1974, and due to this also the Shannon Lecturer 1974.
National Academy of Engineering elected member 1976
National Academy of Sciences elected member 1977
IEEE Alexander Graham Bell Medal 1981
IEEE Centennial Medal 1984
Society for Industrial and Applied Mathematics’s John von Neumann lecture award 1982
American Academy of Arts and Sciences elected member
References
1923 births
2007 deaths
20th-century American mathematicians
21st-century American mathematicians
Members of the United States National Academy of Sciences
Fellow Members of the IEEE
University of Michigan alumni
Harvard University alumni
University of Paris alumni
University of Hawaiʻi at Mānoa faculty
American information theorists
Scientists at Bell Labs
Scientists from Pittsburgh
IEEE Centennial Medal laureates
United States Army personnel of World War II
United States Army officers
American expatriates in France
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https://en.wikipedia.org/wiki/Bayesian%20experimental%20design
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Bayesian experimental design provides a general probability-theoretical framework from which other theories on experimental design can be derived. It is based on Bayesian inference to interpret the observations/data acquired during the experiment. This allows accounting for both any prior knowledge on the parameters to be determined as well as uncertainties in observations.
The theory of Bayesian experimental design is to a certain extent based on the theory for making optimal decisions under uncertainty. The aim when designing an experiment is to maximize the expected utility of the experiment outcome. The utility is most commonly defined in terms of a measure of the accuracy of the information provided by the experiment (e.g., the Shannon information or the negative of the variance) but may also involve factors such as the financial cost of performing the experiment. What will be the optimal experiment design depends on the particular utility criterion chosen.
Relations to more specialized optimal design theory
Linear theory
If the model is linear, the prior probability density function (PDF) is homogeneous and observational errors are normally distributed, the theory simplifies to the classical optimal experimental design theory.
Approximate normality
In numerous publications on Bayesian experimental design, it is (often implicitly) assumed that all posterior probabilities will be approximately normal. This allows for the expected utility to be calculated using linear theory, averaging over the space of model parameters. Caution must however be taken when applying this method, since approximate normality of all possible posteriors is difficult to verify, even in cases of normal observational errors and uniform prior probability.
Posterior distribution
In many cases, the posterior distribution is not available in closed form and has to be approximated using numerical methods. The most common approach is to use Markov chain Monte Carlo methods to generate samples from the posterior, which can then be used to approximate the expected utility.
Another approach is to use a variational Bayes approximation of the posterior, which can often be calculated in closed form. This approach has the advantage of being computationally more efficient than Monte Carlo methods, but the disadvantage that the approximation might not be very accurate.
Some authors proposed approaches that use the posterior predictive distribution to assess the effect of new measurements on prediction uncertainty, while others suggest maximizing the mutual information between parameters, predictions and potential new experiments.
Mathematical formulation
Given a vector of parameters to determine, a prior probability over those parameters and a likelihood for making observation , given parameter values and an experiment design , the posterior probability can be calculated using Bayes' theorem
where is the marginal probability density in observation space
The expected u
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https://en.wikipedia.org/wiki/Rhombic%20enneacontahedron
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In geometry, a rhombic enneacontahedron (plural: rhombic enneacontahedra) is a polyhedron composed of 90 rhombic faces; with three, five, or six rhombi meeting at each vertex. It has 60 broad rhombi and 30 slim. The rhombic enneacontahedron is a zonohedron with a superficial resemblance to the rhombic triacontahedron.
Construction
It can also be seen as a nonuniform truncated icosahedron with pyramids augmented to the pentagonal and hexagonal faces with heights adjusted until the dihedral angles are zero, and the two pyramid type side edges are equal length. This construction is expressed in the Conway polyhedron notation jtI with join operator j. Without the equal edge constraint, the wide rhombi are kites if limited only by the icosahedral symmetry.
The sixty broad rhombic faces in the rhombic enneacontahedron are identical to those in the rhombic dodecahedron, with diagonals in a ratio of 1 to the square root of 2. The face angles of these rhombi are approximately 70.528° and 109.471°. The thirty slim rhombic faces have face vertex angles of 41.810° and 138.189°; the diagonals are in ratio of 1 to φ2.
It is also called a rhombic enenicontahedron in Lloyd Kahn's Domebook 2.
Close-packing density
The optimal packing fraction of rhombic enneacontahedra is given by
.
It was noticed that this optimal value is obtained in a Bravais lattice by . Since the rhombic enneacontahedron is contained in a rhombic dodecahedron whose
inscribed sphere is identical to its own inscribed sphere, the value of the optimal packing fraction is a corollary of the Kepler conjecture: it can be achieved by putting a rhombicuboctahedron in each cell of the rhombic dodecahedral honeycomb, and it cannot be surpassed, since otherwise the optimal packing density of spheres could be surpassed by putting a sphere in each rhombicuboctahedron of the hypothetical packing which surpasses it.
References
VRML model: George Hart,
George Hart's Conway Generator Try dakD
Domebook2 by Kahn, Lloyd (Editor); Easton, Bob; Calthorpe, Peter; et al., Pacific Domes, Los Gatos, CA (1971), page 102
External links
The Rhombic Enneacontahedron and relations
Rhombic Enneacontahedron
George Hart
A Color-Matching Dissection of the Rhombic Enneacontahedron
Color-Matching Dissection of the Rhombic Enneacontahedron
VRML model
Zonohedra
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https://en.wikipedia.org/wiki/Mapping%20cone%20%28topology%29
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In mathematics, especially homotopy theory, the mapping cone is a construction of topology, analogous to a quotient space. It is also called the homotopy cofiber, and also notated . Its dual, a fibration, is called the mapping fibre. The mapping cone can be understood to be a mapping cylinder , with one end of the cylinder collapsed to a point. Thus, mapping cones are frequently applied in the homotopy theory of pointed spaces.
Definition
Given a map , the mapping cone is defined to be the quotient space of the mapping cylinder with respect to the equivalence relation , . Here denotes the unit interval [0, 1] with its standard topology. Note that some authors (like J. Peter May) use the opposite convention, switching 0 and 1.
Visually, one takes the cone on X (the cylinder with one end (the 0 end) identified to a point), and glues the other end onto Y via the map f (the identification of the 1 end).
Coarsely, one is taking the quotient space by the image of X, so ; this is not precisely correct because of point-set issues, but is the philosophy, and is made precise by such results as the homology of a pair and the notion of an n-connected map.
The above is the definition for a map of unpointed spaces; for a map of pointed spaces (so ), one also identifies all of ; formally, Thus one end and the "seam" are all identified with
Example of circle
If is the circle , the mapping cone can be considered as the quotient space of the disjoint union of Y with the disk formed by identifying each point x on the boundary of to the point in Y.
Consider, for example, the case where Y is the disk , and is the standard inclusion of the circle as the boundary of . Then the mapping cone is homeomorphic to two disks joined on their boundary, which is topologically the sphere .
Double mapping cylinder
The mapping cone is a special case of the double mapping cylinder. This is basically a cylinder joined on one end to a space via a map
and joined on the other end to a space via a map
The mapping cone is the degenerate case of the double mapping cylinder (also known as the homotopy pushout), in which one of is a single point.
Dual construction: the mapping fibre
The dual to the mapping cone is the mapping fibre . Given the pointed map one defines the mapping fiber as
.
Here, I is the unit interval and is a continuous path in the space (the exponential object) . The mapping fiber is sometimes denoted as ; however this conflicts with the same notation for the mapping cylinder.
It is dual to the mapping cone in the sense that the product above is essentially the fibered product or pullback which is dual to the pushout used to construct the mapping cone. In this particular case, the duality is essentially that of currying, in that the mapping cone has the curried form where is simply an alternate notation for the space of all continuous maps from the unit interval to . The two variants are related by an adjoint functor. Observe
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https://en.wikipedia.org/wiki/Robert%20van%20de%20Geijn
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Robert A. van de Geijn is a Professor of Computer Sciences at the University of Texas at Austin. He received his B.S. in Mathematics and Computer Science (1981) from the University of Wisconsin–Madison and his Ph.D. in Applied Mathematics (1987) from the University of Maryland, College Park. His areas of interest include numerical analysis and parallel processing.
Major work
Van de Geijn's has turned toward the theoretical, in particular with his development of the Formal Linear Algebra Method (FLAME). FLAME is an original effort at formalizing the efficient derivation of linear algebra algorithms that are provably correct. This approach benefits from his less theoretical experience; it is designed to ultimately lead to the efficient design and implementation of these algorithms.
He is the principal author of the widely cited book. Using PLAPACK—parallel linear algebra package. Scientific and engineering computation. Cambridge, Mass: MIT Press, 1997.
Personal
Robert van de Geijn was born on August 14, 1962, in the Netherlands. He later moved to the United States, where he enrolled at the University of Wisconsin-Madison in 1978. He is married to a fellow academic, Margaret Myers. They have three children, and now live in a historic house in downtown Pflugerville, Texas.
References
External links
Robert A. van de Geijn
Year of birth missing (living people)
Living people
American computer scientists
Geijn, Robert
Geijn, Robert
University of Texas at Austin faculty
University of Wisconsin–Madison College of Letters and Science alumni
University of Maryland, College Park alumni
People from Pflugerville, Texas
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https://en.wikipedia.org/wiki/Metric%20dimension
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In mathematics, metric dimension may refer to:
Metric dimension (graph theory), the minimum number of vertices of an undirected graph G in a subset S of G such that all other vertices are uniquely determined by their distances to the vertices in S
Minkowski–Bouligand dimension (also called the metric dimension), a way of determining the dimension of a fractal set in a Euclidean space by counting the number of fixed-size boxes needed to cover the set as a function of the box size
Equilateral dimension of a metric space (also called the metric dimension), the maximum number of points at equal distances from each other
Hausdorff dimension, an extended non-negative real number associated with any metric space that generalizes the notion of the dimension of a real vector space
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https://en.wikipedia.org/wiki/Wavefront%20.obj%20file
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OBJ (or .OBJ) is a geometry definition file format first developed by Wavefront Technologies for its Advanced Visualizer animation package. The file format is open and has been adopted by other 3D graphics application vendors.
The OBJ file format is a simple data-format that represents 3D geometry alone — namely, the position of each vertex, the UV position of each texture coordinate vertex, vertex normals, and the faces that make each polygon defined as a list of vertices, and texture vertices. Vertices are stored in a counter-clockwise order by default, making explicit declaration of face normals unnecessary. OBJ coordinates have no units, but OBJ files can contain scale information in a human readable comment line.
File format
Anything following a hash character (#) is a comment.
# this is a comment
An OBJ file may contain vertex data, free-form curve/surface attributes, elements, free-form curve/surface body statements, connectivity between free-form surfaces, grouping and display/render attribute information. The most common elements are geometric vertices, texture coordinates, vertex normals and polygonal faces:
# List of geometric vertices, with (x, y, z, [w]) coordinates, w is optional and defaults to 1.0.
v 0.123 0.234 0.345 1.0
v ...
...
# List of texture coordinates, in (u, [v, w]) coordinates, these will vary between 0 and 1. v, w are optional and default to 0.
vt 0.500 1 [0]
vt ...
...
# List of vertex normals in (x,y,z) form; normals might not be unit vectors.
vn 0.707 0.000 0.707
vn ...
...
# Parameter space vertices in (u, [v, w]) form; free form geometry statement (see below)
vp 0.310000 3.210000 2.100000
vp ...
...
# Polygonal face element (see below)
f 1 2 3
f 3/1 4/2 5/3
f 6/4/1 3/5/3 7/6/5
f 7//1 8//2 9//3
f ...
...
# Line element (see below)
l 5 8 1 2 4 9
Geometric vertex
A vertex is specified via a line starting with the letter v. That is followed by (x,y,z[,w]) coordinates. W is optional and defaults to 1.0. A right-hand coordinate system is used to specify the coordinate locations. Some applications support vertex colors, by putting red, green and blue values after x y and z (this precludes specifying w). The color values range from 0 to 1.
Parameter space vertices
A free-form geometry statement can be specified in a line starting with the string vp. Define points in parameter space of a curve or surface. u only is required for curve points, u and v for surface points and control points of non-rational trimming curves, and u, v and w (weight) for control points of rational trimming curves.
Face elements
Faces are defined using lists of vertex, texture and normal indices in the format vertex_index/texture_index/normal_index for which each index starts at 1 and increases corresponding to the order in which the referenced element was defined. Polygons such as quadrilaterals can be defined by using more than three indices.
OBJ files also support free-form geometry which use curves
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https://en.wikipedia.org/wiki/Ap%C3%A9ry%27s%20theorem
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In mathematics, Apéry's theorem is a result in number theory that states the Apéry's constant ζ(3) is irrational. That is, the number
cannot be written as a fraction where p and q are integers. The theorem is named after Roger Apéry.
The special values of the Riemann zeta function at even integers () can be shown in terms of Bernoulli numbers to be irrational, while it remains open whether the function's values are in general rational or not at the odd integers () (though they are conjectured to be irrational).
History
Leonhard Euler proved that if n is a positive integer then
for some rational number . Specifically, writing the infinite series on the left as , he showed
where the are the rational Bernoulli numbers. Once it was proved that is always irrational, this showed that is irrational for all positive integers n.
No such representation in terms of π is known for the so-called zeta constants for odd arguments, the values for positive integers n. It has been conjectured that the ratios of these quantities
are transcendental for every integer .
Because of this, no proof could be found to show that the zeta constants with odd arguments were irrational, even though they were (and still are) all believed to be transcendental. However, in June 1978, Roger Apéry gave a talk titled "Sur l'irrationalité de ζ(3)." During the course of the talk he outlined proofs that and were irrational, the latter using methods simplified from those used to tackle the former rather than relying on the expression in terms of π. Due to the wholly unexpected nature of the proof and Apéry's blasé and very sketchy approach to the subject, many of the mathematicians in the audience dismissed the proof as flawed. However Henri Cohen, Hendrik Lenstra, and Alfred van der Poorten suspected Apéry was on to something and set out to confirm his proof. Two months later they finished verification of Apéry's proof, and on August 18 Cohen delivered a lecture giving full details of the proof. After the lecture Apéry himself took to the podium to explain the source of some of his ideas.
Apéry's proof
Apéry's original proof was based on the well known irrationality criterion from Peter Gustav Lejeune Dirichlet, which states that a number is irrational if there are infinitely many coprime integers p and q such that
for some fixed c, δ > 0.
The starting point for Apéry was the series representation of as
Roughly speaking, Apéry then defined a sequence which converges to about as fast as the above series, specifically
He then defined two more sequences and that, roughly, have the quotient . These sequences were
and
The sequence converges to fast enough to apply the criterion, but unfortunately is not an integer after . Nevertheless, Apéry showed that even after multiplying and by a suitable integer to cure this problem the convergence was still fast enough to guarantee irrationality.
Later proofs
Within a year of Apéry's result an alternative proo
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https://en.wikipedia.org/wiki/Canfield%20%28solitaire%29
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Canfield (US) or Demon (UK) is a patience or solitaire card game with a very low probability of winning. It is an English game first called Demon Patience and described as "the best game for one pack that has yet been invented". It was popularised in the United States in the early 20th century as a result of a story that casino owner Richard A. Canfield had turned it into a gambling game, although it may actually have been Klondike and not Demon that was played at his casino. As a result it became known as Canfield in the United States, while continuing to be called Demon Patience in the United Kingdom and elsewhere. It is closely related to Klondike, and is one of the most popular games of its type.
History
The game is first recorded in 1891 in England by Mary Whitmore Jones as Demon Patience. She describes it as "by far the best game for one pack that has yet been invented," and goes on to say that its "very uncomplimentary name" seems to derive from its ability to frustrate. "Truly a mocking spirit appears to preside over the game, and snatches success from the player often at the last moment, when it seems just within his grasp." Nevertheless when the player does succeed in getting the patience out, "it is a triumph to have conquered the demon."
In Henrietta Stannard's 1895 novel, A Magnificent Young Man, Mrs. Bladenbrook invites the curate to "show me this wonderful new game of yours". He fails to get it out declaring, "Ah, it is no use." Mrs. Bladenbrook asks, "But you are nearly done?" "But I am not quite done," replies the curate, "that is where the demon comes in. It is well called 'Demon Patience'. I have often tried a dozen times to do it, and failed each time when it has seemed just within my grasp. Believe me... it is the one form of Patience which puts all the others into the shade; it is the one form of which one never tires; it is always interesting, always fresh, always tantalizing."
A 1910 publication of Fry's Magazine edited by C.B. Fry confirms that the game is called Demon patience "because the player is so often beaten by the awkward position of a single card which avoids any appearance at the critical period in a perverse manner which at times is quite demoniacal."
Meanwhile, Demon had travelled to America, where the earliest description of it, published in the 1907 Hoyle's Games, confusingly calls it Klondike, actually the name of a quite different game. The author of Hoyle's Games acknowledges that there are several ways of playing the game but only describes what he speculates is "probably the original form". However, it is merely a gambling version of Demon in which "the banker sells a pack of 52 cards for $52, and... agrees to pay $5 for every card the player gets down in the 'top line'".
How Demon came to be called Canfield in the US is unclear, but it is frequently linked to noted gambler Richard A. Canfield, who, in 1894, took over the Clubhouse in Saratoga Springs, New York. Some time after 1900, he encoura
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https://en.wikipedia.org/wiki/Neighbourhood%20system
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In topology and related areas of mathematics, the neighbourhood system, complete system of neighbourhoods, or neighbourhood filter for a point in a topological space is the collection of all neighbourhoods of
Definitions
Neighbourhood of a point or set
An of a point (or subset) in a topological space is any open subset of that contains
A is any subset that contains open neighbourhood of ;
explicitly, is a neighbourhood of in if and only if there exists some open subset with .
Equivalently, a neighborhood of is any set that contains in its topological interior.
Importantly, a "neighbourhood" does have to be an open set; those neighbourhoods that also happen to be open sets are known as "open neighbourhoods."
Similarly, a neighbourhood that is also a closed (respectively, compact, connected, etc.) set is called a (respectively, , , etc.).
There are many other types of neighbourhoods that are used in topology and related fields like functional analysis.
The family of all neighbourhoods having a certain "useful" property often forms a neighbourhood basis, although many times, these neighbourhoods are not necessarily open. Locally compact spaces, for example, are those spaces that, at every point, have a neighbourhood basis consisting entirely of compact sets.
Neighbourhood filter
The neighbourhood system for a point (or non-empty subset) is a filter called the The neighbourhood filter for a point is the same as the neighbourhood filter of the singleton set
Neighbourhood basis
A or (or or ) for a point is a filter base of the neighbourhood filter; this means that it is a subset
such that for all there exists some such that
That is, for any neighbourhood we can find a neighbourhood in the neighbourhood basis that is contained in
Equivalently, is a local basis at if and only if the neighbourhood filter can be recovered from in the sense that the following equality holds:
A family is a neighbourhood basis for if and only if is a cofinal subset of with respect to the partial order (importantly, this partial order is the superset relation and not the subset relation).
Neighbourhood subbasis
A at is a family of subsets of each of which contains such that the collection of all possible finite intersections of elements of forms a neighbourhood basis at
Examples
If has its usual Euclidean topology then the neighborhoods of are all those subsets for which there exists some real number such that For example, all of the following sets are neighborhoods of in :
but none of the following sets are neighborhoods of :
where denotes the rational numbers.
If is an open subset of a topological space then for every is a neighborhood of in
More generally, if is any set and denotes the topological interior of in then is a neighborhood (in ) of every point and moreover, is a neighborhood of any other point.
Said differently, is a neighborhood of a point if and only if
Nei
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https://en.wikipedia.org/wiki/Harmonic%20%28mathematics%29
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In mathematics, a number of concepts employ the word harmonic. The similarity of this terminology to that of music is not accidental: the equations of motion of vibrating strings, drums and columns of air are given by formulas involving Laplacians; the solutions to which are given by eigenvalues corresponding to their modes of vibration. Thus, the term "harmonic" is applied when one is considering functions with sinusoidal variations, or solutions of Laplace's equation and related concepts.
Mathematical terms whose names include "harmonic" include:
Projective harmonic conjugate
Cross-ratio
Harmonic analysis
Harmonic conjugate
Harmonic form
Harmonic function
Harmonic mean
Harmonic mode
Harmonic number
Harmonic series
Alternating harmonic series
Harmonic tremor
Spherical harmonics
Mathematical terminology
Harmonic analysis
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https://en.wikipedia.org/wiki/Proper
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Proper may refer to:
Mathematics
Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact
Proper morphism, in algebraic geometry, an analogue of a proper map for algebraic varieties
Proper transfer function, a transfer function in control theory in which the degree of the numerator does not exceed the degree of the denominator
Proper equilibrium, in game theory, a refinement of the Nash equilibrium
Proper subset
Proper space
Proper complex random variable
Other uses
Proper (liturgy), the part of a Christian liturgy that is specific to the date within the Liturgical Year
Proper frame, such system of reference in which object is stationary (non moving), sometimes also called a co-moving frame
Proper (heraldry), in heraldry, means depicted in natural colors
Proper Records, a UK record label
Proper (album), an album by Into It. Over It. released in 2011
Proper (often capitalized PROPER), a corrected release in response to a previously released online video or movie that contains transcoding or other playback errors
See also
Acceptable (disambiguation)
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https://en.wikipedia.org/wiki/Frobenius%20group
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In mathematics, a Frobenius group is a transitive permutation group on a finite set, such that no non-trivial element
fixes more than one point and some non-trivial element fixes a point.
They are named after F. G. Frobenius.
Structure
Suppose G is a Frobenius group consisting of permutations of a set X. A subgroup H of G fixing a point of X is called a Frobenius complement. The identity element together with all elements not in any conjugate of H form a normal subgroup called the Frobenius kernel K. (This is a theorem due to ; there is still no proof of this theorem that does not use character theory, although see .) The Frobenius group G is the semidirect product of K and H:
.
Both the Frobenius kernel and the Frobenius complement have very restricted structures. proved that the Frobenius kernel K is a nilpotent group. If H has even order then K is abelian. The Frobenius complement H has the property that every subgroup whose order is the product of 2 primes is cyclic; this implies that its Sylow subgroups are cyclic or generalized quaternion groups. Any group such that all Sylow subgroups are cyclic is called a Z-group, and in particular must be a metacyclic group: this means it is the extension of two cyclic groups. If a Frobenius complement H is not solvable then Zassenhaus showed that it
has a normal subgroup of index 1 or 2 that is the product of SL(2,5) and a metacyclic group of order coprime to 30. In particular, if a Frobenius complement coincides with its derived subgroup, then it is isomorphic with SL(2,5). If a Frobenius complement H is solvable then it has a normal metacyclic subgroup such that the quotient is a subgroup of the symmetric group on 4 points. A finite group is a Frobenius complement if and only if it has a faithful, finite-dimensional representation over a finite field in which non-identity group elements correspond to linear transformations without nonzero fixed points.
The Frobenius kernel K is uniquely determined by G as it is the Fitting subgroup, and the Frobenius complement is uniquely determined up to conjugacy by the Schur-Zassenhaus theorem. In particular a finite group G is a Frobenius group in at most one way.
Examples
The smallest example is the symmetric group on 3 points, with 6 elements. The Frobenius kernel K has order 3, and the complement H has order 2.
For every finite field Fq with q (> 2) elements, the group of invertible affine transformations , acting naturally on Fq is a Frobenius group. The preceding example corresponds to the case F3, the field with three elements.
Another example is provided by the subgroup of order 21 of the collineation group of the Fano plane generated by a 3-fold symmetry σ fixing a point and a cyclic permutation τ of all 7 points, satisfying στ = τ2σ. Identifying F8× with the Fano plane, σ can be taken to be the restriction of the Frobenius automorphism σ(x) = x2 of F8 and τ to be multiplication by any element not 0 or 1 (i.e. a generator of the cyclic mu
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https://en.wikipedia.org/wiki/Four-gradient
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In differential geometry, the four-gradient (or 4-gradient) is the four-vector analogue of the gradient from vector calculus.
In special relativity and in quantum mechanics, the four-gradient is used to define the properties and relations between the various physical four-vectors and tensors.
Notation
This article uses the metric signature.
SR and GR are abbreviations for special relativity and general relativity respectively.
indicates the speed of light in vacuum.
is the flat spacetime metric of SR.
There are alternate ways of writing four-vector expressions in physics:
The four-vector style can be used: , which is typically more compact and can use vector notation, (such as the inner product "dot"), always using bold uppercase to represent the four-vector, and bold lowercase to represent 3-space vectors, e.g. . Most of the 3-space vector rules have analogues in four-vector mathematics.
The Ricci calculus style can be used: , which uses tensor index notation and is useful for more complicated expressions, especially those involving tensors with more than one index, such as .
The Latin tensor index ranges in and represents a 3-space vector, e.g. .
The Greek tensor index ranges in and represents a 4-vector, e.g. .
In SR physics, one typically uses a concise blend, e.g. , where represents the temporal component and represents the spatial 3-component.
Tensors in SR are typically 4D -tensors, with upper indices and lower indices, with the 4D indicating 4 dimensions = the number of values each index can take.
The tensor contraction used in the Minkowski metric can go to either side (see Einstein notation):
Definition
The 4-gradient covariant components compactly written in four-vector and Ricci calculus notation are:
The comma in the last part above implies the partial differentiation with respect to 4-position .
The contravariant components are:
Alternative symbols to are and D (although can also signify as the d'Alembert operator).
In GR, one must use the more general metric tensor and the tensor covariant derivative (not to be confused with the vector 3-gradient ).
The covariant derivative incorporates the 4-gradient plus spacetime curvature effects via the Christoffel symbols
The strong equivalence principle can be stated as:
"Any physical law which can be expressed in tensor notation in SR has exactly the same form in a locally inertial frame of a curved spacetime." The 4-gradient commas (,) in SR are simply changed to covariant derivative semi-colons (;) in GR, with the connection between the two using Christoffel symbols. This is known in relativity physics as the "comma to semi-colon rule".
So, for example, if in SR, then in GR.
On a (1,0)-tensor or 4-vector this would be:
On a (2,0)-tensor this would be:
Usage
The 4-gradient is used in a number of different ways in special relativity (SR):
Throughout this article the formulas are all correct for the flat spacetime Minkowski coordinates of
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https://en.wikipedia.org/wiki/Non-perturbative
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In mathematics and physics, a non-perturbative function or process is one that cannot be described by perturbation theory. An example is the function
which does not have a Taylor series at x = 0. Every coefficient of the Taylor expansion around x = 0 is exactly zero, but the function is non-zero if x ≠ 0.
In physics, such functions arise for phenomena which are impossible to understand by perturbation theory, at any finite order. In quantum field theory, 't Hooft–Polyakov monopoles, domain walls, flux tubes, and instantons are examples. A concrete, physical example is given by the Schwinger effect, whereby a strong electric field may spontaneously decay into electron-positron pairs. For not too strong fields, the rate per unit volume of this process is given by,
which cannot be expanded in a Taylor series in the electric charge , or the electric field strength . Here is the mass of an electron and we have used units where .
In theoretical physics, a non-perturbative solution is one that cannot be described in terms of perturbations about some simple background, such as empty space. For this reason, non-perturbative solutions and theories yield insights into areas and subjects that perturbative methods cannot reveal.
See also
Lattice QCD
Soliton
Sphaleron
Instanton
BCFW recursion
Operator product expansion
Conformal bootstrap
Loop quantum gravity
Causal dynamical triangulation
References
Perturbation theory
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https://en.wikipedia.org/wiki/Psi%20function
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Psi function can refer, in mathematics, to
the ordinal collapsing function
the Dedekind psi function
the Chebyshev function
the polygamma function or its special cases
the digamma function
the trigamma function
and in physics to
the quantum mechanical wave function.
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https://en.wikipedia.org/wiki/Hardiness%20zone
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A hardiness zone is a geographic area defined as having a certain average annual minimum temperature, a factor relevant to the survival of many plants. In some systems other statistics are included in the calculations. The original and most widely used system, developed by the United States Department of Agriculture (USDA) as a rough guide for landscaping and gardening, defines 13 zones by long-term average annual extreme minimum temperatures. It has been adapted by and to other countries (such as Canada) in various forms.
Unless otherwise specified, in American contexts "hardiness zone" or simply "zone" usually refers to the USDA scale. For example, a plant may be described as "hardy to zone 10": this means that the plant can withstand a minimum temperature of 30 °F (−1.1 °C) to 40 °F (4.4 °C).
Other hardiness rating schemes have been developed as well, such as the UK Royal Horticultural Society and US Sunset Western Garden Book systems. A heat zone (see below) is instead defined by annual high temperatures; the American Horticultural Society (AHS) heat zones use the average number of days per year when the temperature exceeds .
United States hardiness zones (USDA scale)
The USDA system was originally developed to aid gardeners and landscapers in the United States.
State-by-state maps, along with an electronic system that allows finding the zone for a particular zip code, can be found at the USDA Agricultural Research Service (USDA-ARS) website.
In the United States, most of the warmer zones (zones 9, 10, and 11) are located in the deep southern half of the country and on the southern coastal margins. Higher zones can be found in Hawaii (up to 12) and Puerto Rico (up to 13). The southern middle portion of the mainland and central coastal areas are in the middle zones (zones 8, 7, and 6). The far northern portion on the central interior of the mainland have some of the coldest zones (zones 5, 4, and small area of zone 3) and often have much less consistent range of temperatures in winter due to being more continental, especially further west with higher diurnal temperature variations, and thus the zone map has its limitations in these areas. Lower zones can be found in Alaska (down to 1). The low latitude and often stable weather in Florida, the Gulf Coast, and southern Arizona and California, are responsible for the rarity of episodes of severe cold relative to normal in those areas. The warmest zone in the 48 contiguous states is the Florida Keys (11b) and the coldest is in north-central Minnesota (2b). A couple of locations on the northern coast of Puerto Rico have the warmest hardiness zone in the United States at 13b. Conversely, isolated inland areas of Alaska have the coldest hardiness zone in the United States at 1a.
Definitions
History
The first attempts to create a geographical hardiness zone system were undertaken by two researchers at the Arnold Arboretum in Boston; the first was published in 1927 by Alfred Rehder, and the sec
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https://en.wikipedia.org/wiki/NUTS%20statistical%20regions%20of%20the%20Czech%20Republic
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The Nomenclature of Territorial Units for Statistics (NUTS) is a geocode standard for referencing the subdivisions of the Czech Republic for statistical purposes. The standard is developed and regulated by the European Union. The NUTS standard is instrumental in delivering the European Union's Structural Funds. The NUTS code for the Czech Republic is CZ and a hierarchy of three levels is established by Eurostat. Below these is a further levels of geographic organisation - the local administrative unit (LAU). In the Czech Republic, the LAU 1 is districts and the LAU 2 is municipalities.
Overall
NUTS codes
In the 2003 version, the Vysočina Region was coded CZ061, and the South Moravian Region was coded CZ062.
Local administrative units
Below the NUTS levels, the two LAU (Local Administrative Units) levels are:
The LAU codes of the Czech Republic can be downloaded here: ''
See also
List of Czech regions by Human Development Index
Subdivisions of the Czech Republic
ISO 3166-2 codes of the Czech Republic
FIPS region codes of the Czech Republic
References
Sources
Hierarchical list of the Nomenclature of territorial units for statistics - NUTS and the Statistical regions of Europe
Overview map of EU Countries - NUTS level 1
Correspondence between the NUTS levels and the national administrative units
List of current NUTS codes
Download current NUTS codes (ODS format)
Regions of the Czech Republic, Statoids.com
Districts of the Czech Republic, Statoids.com
External links
NUTS reference of the Czech Republic
NUTS of municipalities
Czech Republic
Nuts
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https://en.wikipedia.org/wiki/Iverson%20bracket
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In mathematics, the Iverson bracket, named after Kenneth E. Iverson, is a notation that generalises the Kronecker delta, which is the Iverson bracket of the statement . It maps any statement to a function of the free variables in that statement. This function is defined to take the value 1 for the values of the variables for which the statement is true, and takes the value 0 otherwise. It is generally denoted by putting the statement inside square brackets:
In other words, the Iverson bracket of a statement is the indicator function of the set of values for which the statement is true.
The Iverson bracket allows using capital-sigma notation without restriction on the summation index. That is, for any property of the integer , one can rewrite the restricted sum in the unrestricted form . With this convention, does not need to be defined for the values of for which the Iverson bracket equals ; that is, a summand must evaluate to 0 regardless of whether is defined.
The notation was originally introduced by Kenneth E. Iverson in his programming language APL, though restricted to single relational operators enclosed in parentheses, while the generalisation to arbitrary statements, notational restriction to square brackets, and applications to summation, was advocated by Donald Knuth to avoid ambiguity in parenthesized logical expressions.
Properties
There is a direct correspondence between arithmetic on Iverson brackets, logic, and set operations. For instance, let A and B be sets and any property of integers; then we have
Examples
The notation allows moving boundary conditions of summations (or integrals) as a separate factor into the summand, freeing up space around the summation operator, but more importantly allowing it to be manipulated algebraically.
Double-counting rule
We mechanically derive a well-known sum manipulation rule using Iverson brackets:
Summation interchange
The well-known rule is likewise easily derived:
Counting
For instance, the Euler phi function that counts the number of positive integers up to n which are coprime to n can be expressed by
Simplification of special cases
Another use of the Iverson bracket is to simplify equations with special cases. For example, the formula
is valid for but is off by for . To get an identity valid for all positive integers (i.e., all values for which is defined), a correction term involving the Iverson bracket may be added:
Common functions
Many common functions, especially those with a natural piecewise definition, may be expressed in terms of the Iverson bracket. The Kronecker delta notation is a specific case of Iverson notation when the condition is equality. That is,
The indicator function, often denoted , or , is an Iverson bracket with set membership as its condition:
The Heaviside step function, sign function, and absolute value function are also easily expressed in this notation:
and
The comparison functions max and min (returning the larger or smaller
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https://en.wikipedia.org/wiki/Iverson%20notation
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Iverson notation can refer to:
APL (programming language)
Iverson bracket, in mathematics
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https://en.wikipedia.org/wiki/Social%20geometry
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Social geometry is a theoretical strategy of sociological explanation, invented by sociologist Donald Black, which uses a multi-dimensional model to explain variations in the behavior of social life. In Black's own use and application of the idea, social geometry is an instance of Pure Sociology.
Variables
While social geometry might entail other elements as well (or instead), Black's own explanation of the model includes five variable aspects: horizontal/morphological (the extent and frequency of interaction among participants), vertical (the unequal distribution of resources), corporate (the degree of organization, or of integration of individuals into organizations), cultural (the amount and frequency of symbolic expressions), and normative (the extent of previously being the target of social control). Black refers to this multi-dimensional amalgam as "social space".
Precursors
Each element of Black's model is arguably an extension of part of something earlier in sociology. For example, vertical space is reminiscent of Marxist concerns, morphological of Émile Durkheim, and cultural perhaps of Pierre Bourdieu. However, several aspects of Black's approach differ from those previous theorists.
First, they emphasized a largely unidimensional model: Marx, for example, emphasized solely economic status (and derivatives of it, from base to superstructure) while Durkheim and Weber de-emphasized economic differentiation.
Second, by including multiple dimensions, Black's model allows for consideration of each variable while holding others constant. That is, the theoretical propositions hold under a condition of ceteris paribus, a probabilistic approach characteristic of science generally and contrary to the general cleavage of sociology between purported determinists and those who are anti-scientific. (Later versions of Black's work, such as "The Elementary Forms of Social Control", utilize multiple dimensions in a different way - as simultaneous dimensions, to generate a typology of social settings and conflict management patterns.)
Further, the inclusion of these variables within the same model allows for the possibility of both interaction effects between variables as well as correlation between them, with any one variable being used to explain any other. Black himself uses each of the dimensions to explain variation in normative behavior, but relational or cultural behavior might also be jointly accountable by the other dimensions.
Most significantly, Black's Social Geometry entails an epistemological departure from reliance on individualistic explanations, teleology, and even individuals as such. That is, it is an instance of pure sociology, and thus uses a different logic and language than any precursors from whose work Black's ideas may said to be extended or derived.
Measurements
The model allows for several different kinds of measurement along these dimensions. First, location: For example, any case (individual, group, etc.) can b
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https://en.wikipedia.org/wiki/Unipotent
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In mathematics, a unipotent element r of a ring R is one such that r − 1 is a nilpotent element; in other words, (r − 1)n is zero for some n.
In particular, a square matrix M is a unipotent matrix if and only if its characteristic polynomial P(t) is a power of t − 1. Thus all the eigenvalues of a unipotent matrix are 1.
The term quasi-unipotent means that some power is unipotent, for example for a diagonalizable matrix with eigenvalues that are all roots of unity.
In the theory of algebraic groups, a group element is unipotent if it acts unipotently in a certain natural group representation. A unipotent affine algebraic group is then a group with all elements unipotent.
Definition
Definition with matrices
Consider the group of upper-triangular matrices with 's along the diagonal, so they are the group of matrices
Then, a unipotent group can be defined as a subgroup of some . Using scheme theory the group can be defined as the group scheme
and an affine group scheme is unipotent if it is a closed group scheme of this scheme.
Definition with ring theory
An element x of an affine algebraic group is unipotent when its associated right translation operator, rx, on the affine coordinate ring A[G] of G is locally unipotent as an element of the ring of linear endomorphism of A[G]. (Locally unipotent means that its restriction to any finite-dimensional stable subspace of A[G] is unipotent in the usual ring-theoretic sense.)
An affine algebraic group is called unipotent if all its elements are unipotent. Any unipotent algebraic group is isomorphic to a closed subgroup of the group of upper triangular matrices with diagonal entries 1, and conversely any such subgroup is unipotent. In particular any unipotent group is a nilpotent group, though the converse is not true (counterexample: the diagonal matrices of GLn(k)).
For example, the standard representation of on with standard basis has the fixed vector .
Definition with representation theory
If a unipotent group acts on an affine variety, all its orbits are closed, and if it acts linearly on a finite-dimensional vector space then it has a non-zero fixed vector. In fact, the latter property characterizes unipotent groups. In particular, this implies there are no non-trivial semisimple representations.
Examples
Un
Of course, the group of matrices is unipotent. Using the lower central series
where
and
there are associated unipotent groups. For example, on , the central series are the matrix groups
, , , and
given some induced examples of unipotent groups.
Gan
The additive group is a unipotent group through the embedding
Notice the matrix multiplication gives
hence this is a group embedding. More generally, there is an embedding from the map
Using scheme theory, is given by the functor
where
Kernel of the Frobenius
Consider the functor on the subcategory , there is the subfunctor where
so it is given by the kernel of the Frobenius endomorphism.
Classification of u
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https://en.wikipedia.org/wiki/NUTS%20statistical%20regions%20of%20Ireland
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Ireland uses the Nomenclature of Territorial Units for Statistics (NUTS) geocode standard for referencing country subdivisions for statistical purposes. The standard is developed and regulated by the European Union. The NUTS standard is instrumental in delivering European Structural and Investment Funds. The NUTS code for Ireland is IE and a hierarchy of three levels is established by Eurostat. A further level of geographic organisation, the local administrative unit (LAU), in Ireland is the local electoral area.
Overview
NUTS levels 1, 2 and 3
The most recent revision of NUTS regions was made in 2016 and took effect in 2018. The eligibility of regions for funding under the European Regional Development Fund and the European Social Fund Plus was revised in 2021. NUTS 2 Regions may be classified as less developed regions, transition regions, or more developed regions.
Demographic statistics by NUTS 3 region
Local administrative units
The local administrative units in Ireland are the local electoral areas. These are subdivisions of local government areas used for local elections. In counties outside Dublin and in the cities and counties, they also form the basis of municipal districts within local authorities.
Regional Assemblies
Each of the three NUTS 2 regions has a Regional Assembly. These are divided into strategic planning areas, which correspond to the NUTS 3 regions. Prior to 2014, the eight NUTS 3 regions had Regional Authorities. The 2014 act abolished these and transferred their functions to the Regional Assemblies. Assembly members are nominated by constituent local authorities from among their elected councillors.
See also
ISO 3166-2 codes of Ireland
FIPS region codes of Ireland
List of Irish regions by Human Development Index
Local government in the Republic of Ireland
Sources
LAU codes
Hierarchical list of the Nomenclature of territorial units for statistics - NUTS and the Statistical regions of Europe
Overview map of EU Countries - NUTS level 1
NUTS Maps 2016
Correspondence between the NUTS levels and the national administrative units
References
Ireland
1 NUTS
Ireland
Ireland
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https://en.wikipedia.org/wiki/%C3%89mile%20L%C3%A9onard%20Mathieu
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Émile Léonard Mathieu (; 15 May 1835, in Metz – 19 October 1890, in Nancy) was a French mathematician. He is known for his work in group theory and mathematical physics. He has given his name to the Mathieu functions, Mathieu groups and Mathieu transformation. He authored a treatise of mathematical physics in 6 volumes.
Volume 1 is an exposition of the techniques to solve the differential equations of mathematical physics,
and contains an account of the applications of Mathieu functions to electrostatics.
Volume 2 deals with capillarity. Volumes 3 and 4 deal with electrostatics and magnetostatics.
Volume 5 deals with electrodynamics, and volume 6 with elasticity. The asteroid 27947 Emilemathieu was named in his honour.
Early Life
Émile Mathieu was born into a family of minor civil servants. His father, Nicolas Mathieu, was a cashier at the Tax Office of the city. His mother, Amélie Antoinette Aubertin, was from Metz while her brother, Pierre Aubertin, the uncle of our mathematician, had attended the École Polytechnique, was a colonel of artillery and the director of a foundry which made cannons. Émile Mathieu was brought up in Metz, and he attended school at the Lycée de Metz in that town. He excelled at school, first in classical studies showing remarkable abilities in Latin and Greek compositions. However, once he had met mathematics when he was in his teenage years, it became the only subject which he wanted to pursue. We mentioned above that his uncle Pierre Aubertin had studied at the École Polytechnique and he advised Émile on preparing himself for the entrance examinations which he took successfully in 1854.
Books by Émile Mathieu
Traité de physique mathématique (6 vols.) (Gauthier-Villars, 1873-1890)
Dynamique Analytique (Gauthier-Villars, 1878)
References
External links
19th-century French mathematicians
1835 births
1890 deaths
École Polytechnique alumni
Group theorists
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https://en.wikipedia.org/wiki/Mathieu%20function
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In mathematics, Mathieu functions, sometimes called angular Mathieu functions, are solutions of Mathieu's differential equation
where are real-valued parameters. Since we may add to to change the sign of , it is a usual convention to set .
They were first introduced by Émile Léonard Mathieu, who encountered them while studying vibrating elliptical drumheads. They have applications in many fields of the physical sciences, such as optics, quantum mechanics, and general relativity. They tend to occur in problems involving periodic motion, or in the analysis of partial differential equation (PDE) boundary value problems possessing elliptic symmetry.
Definition
Mathieu functions
In some usages, Mathieu function refers to solutions of the Mathieu differential equation for arbitrary values of and . When no confusion can arise, other authors use the term to refer specifically to - or -periodic solutions, which exist only for special values of and . More precisely, for given (real) such periodic solutions exist for an infinite number of values of , called characteristic numbers, conventionally indexed as two separate sequences and , for . The corresponding functions are denoted and , respectively. They are sometimes also referred to as cosine-elliptic and sine-elliptic, or Mathieu functions of the first kind.
As a result of assuming that is real, both the characteristic numbers and associated functions are real-valued.
and can be further classified by parity and periodicity (both with respect to ), as follows:
{| class="wikitable"
! Function !! Parity !! Period
|-
|
| even
|
|-
|
| even
|
|-
|
| odd
|
|-
|
| odd
|
|}
The indexing with the integer , besides serving to arrange the characteristic numbers in ascending order, is convenient in that and become proportional to and as . With being an integer, this gives rise to the classification of and as Mathieu functions (of the first kind) of integral order. For general and , solutions besides these can be defined, including Mathieu functions of fractional order as well as non-periodic solutions.
Modified Mathieu functions
Closely related are the modified Mathieu functions, also known as radial Mathieu functions, which are solutions of Mathieu's modified differential equation
which can be related to the original Mathieu equation by taking . Accordingly, the modified Mathieu functions of the first kind of integral order, denoted by and , are defined from
These functions are real-valued when is real.
Normalization
A common normalization, which will be adopted throughout this article, is to demand
as well as require and as .
Stability
The Mathieu equation has two parameters. For almost all choices of parameter, by Floquet theory (see next section), any solution either converges to zero or diverges to infinity.
Parametrize Mathieu equation as , where . The regions of stability and instability are separated by curves
Floquet theory
Many properties of the Mat
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https://en.wikipedia.org/wiki/Zassenhaus%20group
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In mathematics, a Zassenhaus group, named after Hans Zassenhaus, is a certain sort of doubly transitive permutation group very closely related to rank-1 groups of Lie type.
Definition
A Zassenhaus group is a permutation group G on a finite set X with the following three properties:
G is doubly transitive.
Non-trivial elements of G fix at most two points.
G has no regular normal subgroup. ("Regular" means that non-trivial elements do not fix any points of X; compare free action.)
The degree of a Zassenhaus group is the number of elements of X.
Some authors omit the third condition that G has no regular normal subgroup. This
condition is put in to eliminate some "degenerate" cases. The extra examples one gets by omitting it are either Frobenius groups or certain groups of degree 2p and order
2p(2p − 1)p for a prime p, that are generated by all semilinear mappings and Galois automorphisms of a field of order 2p.
Examples
We let q = pf be a power of a prime p, and write Fq for the finite field of order q. Suzuki proved that any Zassenhaus group is of one of the following four types:
The projective special linear group PSL2(Fq) for q > 3 odd, acting on the q + 1 points of the projective line. It has order (q + 1)q(q − 1)/2.
The projective general linear group PGL2(Fq) for q > 3. It has order (q + 1)q(q − 1).
A certain group containing PSL2(Fq) with index 2, for q an odd square. It has order (q + 1)q(q − 1).
The Suzuki group Suz(Fq) for q a power of 2 that is at least 8 and not a square. The order is (q2 + 1)q2(q − 1)
The degree of these groups is q + 1 in the first three cases, q2 + 1 in the last case.
Further reading
Finite Groups III (Grundlehren Der Mathematischen Wissenschaften Series, Vol 243) by B. Huppert, N. Blackburn,
Permutation groups
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https://en.wikipedia.org/wiki/Cauchy%20boundary%20condition
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In mathematics, a Cauchy () boundary condition augments an ordinary differential equation or a partial differential equation with conditions that the solution must satisfy on the boundary; ideally so as to ensure that a unique solution exists. A Cauchy boundary condition specifies both the function value and normal derivative on the boundary of the domain. This corresponds to imposing both a Dirichlet and a Neumann boundary condition. It is named after the prolific 19th-century French mathematical analyst Augustin-Louis Cauchy.
Second-order ordinary differential equations
Cauchy boundary conditions are simple and common in second-order ordinary differential equations,
where, in order to ensure that a unique solution exists, one may specify the value of the function and the value of the derivative at a given point , i.e.,
and
where is a boundary or initial point. Since the parameter is usually time, Cauchy conditions can also be called initial value conditions or initial value data or simply Cauchy data. An example of such a situation is Newton's laws of motion, where the acceleration depends on position , velocity , and the time ; here, Cauchy data corresponds to knowing the initial position and velocity.
Partial differential equations
For partial differential equations, Cauchy boundary conditions specify both the function and the normal derivative on the boundary. To make things simple and concrete, consider a second-order differential equation in the plane
where is the unknown solution, denotes derivative of with respect to etc. The functions specify the problem.
We now seek a that satisfies the partial differential equation in a domain , which is a subset of the plane, and such that the Cauchy boundary conditions
hold for all boundary points . Here is the derivative in the direction of the normal to the boundary. The functions and are the Cauchy data.
Notice the difference between a Cauchy boundary condition and a Robin boundary condition. In the former, we specify both the function and the normal derivative. In the latter, we specify a weighted average of the two.
We would like boundary conditions to ensure that exactly one (unique) solution exists, but for second-order partial differential equations, it is not as simple to guarantee existence and uniqueness as it is for ordinary differential equations. Cauchy data are most immediately relevant for hyperbolic problems (for example, the wave equation) on open domains (for example, the half plane).
See also
Dirichlet boundary condition
Mixed boundary condition
Neumann boundary condition
Robin boundary condition
References
Boundary conditions
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https://en.wikipedia.org/wiki/Cauchy%20problem
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A Cauchy problem in mathematics asks for the solution of a partial differential equation that satisfies certain conditions that are given on a hypersurface in the domain. A Cauchy problem can be an initial value problem or a boundary value problem (for this case see also Cauchy boundary condition). It is named after Augustin-Louis Cauchy.
Formal statement
For a partial differential equation defined on Rn+1 and a smooth manifold S ⊂ Rn+1 of dimension n (S is called the Cauchy surface), the Cauchy problem consists of finding the unknown functions of the differential equation with respect to the independent variables that satisfies
subject to the condition, for some value ,
where are given functions defined on the surface (collectively known as the Cauchy data of the problem). The derivative of order zero means that the function itself is specified.
Cauchy–Kowalevski theorem
The Cauchy–Kowalevski theorem states that If all the functions are analytic in some neighborhood of the point , and if all the functions are analytic in some neighborhood of the point , then the Cauchy problem has a unique analytic solution in some neighborhood of the point .
See also
Cauchy boundary condition
Cauchy horizon
References
External links
Cauchy problem at MathWorld.
Partial differential equations
Mathematical problems
Boundary value problems
de:Anfangswertproblem#Partielle Differentialgleichungen
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https://en.wikipedia.org/wiki/FK-space
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In functional analysis and related areas of mathematics a FK-space or Fréchet coordinate space is a sequence space equipped with a topological structure such that it becomes a Fréchet space. FK-spaces with a normable topology are called BK-spaces.
There exists only one topology to turn a sequence space into a Fréchet space, namely the topology of pointwise convergence. Thus the name coordinate space because a sequence in an FK-space converges if and only if it converges for each coordinate.
FK-spaces are examples of topological vector spaces. They are important in summability theory.
Definition
A FK-space is a sequence space , that is a linear subspace of vector space of all complex valued sequences, equipped with the topology of pointwise convergence.
We write the elements of as
with .
Then sequence in converges to some point if it converges pointwise for each That is
if for all
Examples
The sequence space of all complex valued sequences is trivially an FK-space.
Properties
Given an FK-space and with the topology of pointwise convergence the inclusion map
is a continuous function.
FK-space constructions
Given a countable family of FK-spaces with a countable family of seminorms, we define
and
Then is again an FK-space.
See also
− FK-spaces with a normable topology
References
F-spaces
Fréchet spaces
Topological vector spaces
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https://en.wikipedia.org/wiki/BK-space
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In functional analysis and related areas of mathematics, a BK-space or Banach coordinate space is a sequence space endowed with a suitable norm to turn it into a Banach space. All BK-spaces are normable FK-spaces.
Examples
The space of convergent sequences the space of vanishing sequences and the space of bounded sequences under the supremum norm
The space of absolutely p-summable sequences with and the norm
See also
References
Banach spaces
Topological vector spaces
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https://en.wikipedia.org/wiki/NUTS%20statistical%20regions%20of%20the%20Netherlands
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In the NUTS (Nomenclature of Territorial Units for Statistics) codes of the Netherlands (NL), the three levels are:
NUTS codes
Local administrative units
Below the NUTS levels, the two LAU (Local Administrative Units) levels are:
The LAU codes of the Netherlands can be downloaded here:
See also
Subdivisions of the Netherlands
ISO 3166-2 codes of the Netherlands
FIPS region codes of the Netherlands
References
External links
Hierarchical list of the Nomenclature of territorial units for statistics - NUTS and the Statistical regions of Europe
Overview map of EU Countries - NUTS level 1
NEDERLAND - NUTS level 2
NEDERLAND - NUTS level 3
Correspondence between the NUTS levels and the national administrative units
List of current NUTS codes
Download current NUTS codes (ODS format)
Provinces of Netherlands, Statoids.com
Netherlands
Nuts
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https://en.wikipedia.org/wiki/List%20of%20French%20Open%20men%27s%20doubles%20champions
|
Champions
French Championships
French Open
Statistics
Multiple champions
Champions by country
If the doubles partners are from the same country then that country gets two titles instead of one, while if they are from different countries then each country will get one title apiece.
Notes
References
See also
French Open other competitions
List of French Open men's singles champions
List of French Open women's singles champions
List of French Open women's doubles champions
List of French Open mixed doubles champions
Grand Slam men's doubles
List of Australian Open men's doubles champions
List of Wimbledon gentlemen's doubles champions
List of US Open men's doubles champions
List of Grand Slam men's doubles champions
Mens
Lists of male tennis players
French Open champions
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https://en.wikipedia.org/wiki/Relativistic%20Breit%E2%80%93Wigner%20distribution
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The relativistic Breit–Wigner distribution (after the 1936 nuclear resonance formula of Gregory Breit and Eugene Wigner) is a continuous probability distribution with the following probability density function,
where is a constant of proportionality, equal to
with
(This equation is written using natural units, .)
It is most often used to model resonances (unstable particles) in high-energy physics. In this case, is the center-of-mass energy that produces the resonance, is the mass of the resonance, and Γ is the resonance width (or decay width), related to its mean lifetime according to . (With units included, the formula is .)
Usage
The probability of producing the resonance at a given energy is proportional to , so that a plot of the production rate of the unstable particle as a function of energy traces out the shape of the relativistic Breit–Wigner distribution. Note that for values of off the maximum at such that , (hence for ), the distribution has attenuated to half its maximum value, which justifies the name for Γ, width at half-maximum.
In the limit of vanishing width, Γ → 0, the particle becomes stable as the Lorentzian distribution sharpens infinitely to .
In general, Γ can also be a function of ; this dependence is typically only important when Γ is not small compared to and the phase space-dependence of the width needs to be taken into account. (For example, in the decay of the rho meson into a pair of pions.) The factor of 2 that multiplies Γ2 should also be replaced with 2
(or 4/2, etc.) when the resonance is wide.
The form of the relativistic Breit–Wigner distribution arises from the propagator of an unstable particle, which has a denominator of the form . (Here, 2 is the square of the four-momentum carried by that particle in the tree Feynman diagram involved.) The propagator in its rest frame then is proportional to the quantum-mechanical amplitude for the decay utilized to reconstruct that resonance,
The resulting probability distribution is proportional to the absolute square of the amplitude, so then the above relativistic Breit–Wigner distribution for the probability density function.
The form of this distribution is similar to the amplitude of the solution to the classical equation of motion for a driven harmonic oscillator damped and driven by a sinusoidal external force. It has the standard resonance form of the Lorentz, or Cauchy distribution, but involves relativistic variables = 2, here = 2. The distribution is the solution of the differential equation for the amplitude squared w.r.t. the energy energy (frequency), in such a classical forced oscillator,
with
Gaussian broadening
In experiment, the incident beam that produces resonance always has some spread of energy around a central value. Usually, that is a Gaussian/normal distribution. The resulting resonance shape in this case is given by the convolution of the Breit–Wigner and the Gaussian distribution,
This function can be
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https://en.wikipedia.org/wiki/Somer%E2%80%93Lucas%20pseudoprime
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In mathematics, in particular number theory, an odd composite number N is a Somer–Lucas d-pseudoprime (with given d ≥ 1) if there exists a nondegenerate Lucas sequence with the discriminant such that and the rank appearance of N in the sequence U(P, Q) is
where is the Jacobi symbol.
Applications
Unlike the standard Lucas pseudoprimes, there is no known efficient primality test using the Lucas d-pseudoprimes. Hence they are not generally used for computation.
See also
Lawrence Somer, in his 1985 thesis, also defined the Somer d-pseudoprimes. They are described in brief on page 117 of Ribenbaum 1996.
References
Pseudoprimes
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https://en.wikipedia.org/wiki/Absorbing%20set
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In functional analysis and related areas of mathematics an absorbing set in a vector space is a set which can be "inflated" or "scaled up" to eventually always include any given point of the vector space.
Alternative terms are radial or absorbent set.
Every neighborhood of the origin in every topological vector space is an absorbing subset.
Definition
Notation for scalars
Suppose that is a vector space over the field of real numbers or complex numbers and for any let
denote the open ball (respectively, the closed ball) of radius in centered at
Define the product of a set of scalars with a set of vectors as and define the product of with a single vector as
Preliminaries
Balanced core and balanced hull
A subset of is said to be if for all and all scalars satisfying this condition may be written more succinctly as and it holds if and only if
Given a set the smallest balanced set containing denoted by is called the of while the largest balanced set contained within denoted by is called the of
These sets are given by the formulas
and
(these formulas show that the balanced hull and the balanced core always exist and are unique).
A set is balanced if and only if it is equal to its balanced hull () or to its balanced core (), in which case all three of these sets are equal:
If is any scalar then
while if is non-zero or if then also
One set absorbing another
If and are subsets of then is said to if it satisfies any of the following equivalent conditions:
Definition: There exists a real such that for every scalar satisfying Or stated more succinctly, for some
If the scalar field is then intuitively, " absorbs " means that if is perpetually "scaled up" or "inflated" (referring to as ) then (for all positive sufficiently large), all will contain and similarly, must also eventually contain for all negative sufficiently large in magnitude.
This definition depends on the underlying scalar field's canonical norm (that is, on the absolute value ), which thus ties this definition to the usual Euclidean topology on the scalar field. Consequently, the definition of an absorbing set (given below) is also tied to this topology.
There exists a real such that for every non-zero scalar satisfying Or stated more succinctly, for some
Because this union is equal to where is the closed ball with the origin removed, this condition may be restated as: for some
The non-strict inequality can be replaced with the strict inequality which is the next characterization.
There exists a real such that for every non-zero scalar satisfying Or stated more succinctly, for some
Here is the open ball with the origin removed and
If is a balanced set then this list can be extended to include:
There exists a non-zero scalar such that
If then the requirement may be dropped.
There exists a non-zero scalar such that
If (a necessary condition for to be an absorbing set, or t
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https://en.wikipedia.org/wiki/Geometry%20of%20Love
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Geometry of Love is the fifteenth studio album by French electronic musician and composer Jean-Michel Jarre, released by Warner Music in October 2003.
This album has more in common with the preceding Sessions 2000 album than releases prior, but the style here is still more electronica than jazz. The music was to be lounge music, played in the background or in the chill-out area of a club. The album was commissioned by Jean-Roch, as a soundtrack for his 'VIP Room' nightclub in France. The CD was initially meant to come out in only 2000 copies. However, it was later released as a generally available CD. The physical CD was a long time out of print (available only in digital download format), but in 2018 remastered reissue was released on CD again.
The album cover is a pixelated and turned counter-clockwise photo of the pubis of Isabelle Adjani, Jarre's girlfriend at the time.
The track "Velvet Road" is a remake of the unreleased composition "Children of Space" created by Jarre for the "Rendez-Vous in Space" concert in Okinawa, in 2001. Some of the sounds in Geometry of Love were used earlier on Interior Music released in 2001. Several tracks from Geometry of Love were included on Jarre's 2006 compilation release Sublime Mix.
Track listing
Equipment
Roland XP-80
Eminent 310U
ARP 2600
Minimoog
Korg KARMA
Novation Digital Music Systems Supernova II
microKORG
Roland JP-8000
Korg Mini Pops 7
Digisequencer
E-mu Systems XL7
Roland HandSonic
EMS Synthi AKS
EMS VCS 3
RMI Harmonic Synthesizer
Pro Tools
References
Notes
External links
Geometry of Love at Discogs
2003 albums
Jean-Michel Jarre albums
Lounge music albums
Electronic albums by French artists
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https://en.wikipedia.org/wiki/Local%20diffeomorphism
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In mathematics, more specifically differential topology, a local diffeomorphism is intuitively a map between Smooth manifolds that preserves the local differentiable structure. The formal definition of a local diffeomorphism is given below.
Formal definition
Let and be differentiable manifolds. A function
is a local diffeomorphism, if for each point there exists an open set containing such that is open in and
is a diffeomorphism.
A local diffeomorphism is a special case of an immersion where the image of under locally has the differentiable structure of a submanifold of Then and may have a lower dimension than
Characterizations
A map is a local diffeomorphism if and only if it is a smooth immersion (smooth local embedding) and an open map.
The inverse function theorem implies that a smooth map is a local diffeomorphism if and only if the derivative is a linear isomorphism for all points This implies that and must have the same dimension.
A map between two connected manifolds of equal dimension () is a local diffeomorphism if and only if it is a smooth immersion (smooth local embedding), or equivalently, if and only if it is a smooth submersion.
This is because every smooth immersion is a locally injective function while invariance of domain guarantees that any continuous injective function between manifolds of equal dimensions is necessarily an open map.
Discussion
For instance, even though all manifolds look locally the same (as for some ) in the topological sense, it is natural to ask whether their differentiable structures behave in the same manner locally. For example, one can impose two different differentiable structures on that make into a differentiable manifold, but both structures are not locally diffeomorphic (see below). Although local diffeomorphisms preserve differentiable structure locally, one must be able to "patch up" these (local) diffeomorphisms to ensure that the domain is the entire (smooth) manifold. For example, there can be no global diffeomorphism from the 2-sphere to Euclidean 2-space although they do indeed have the same local differentiable structure. This is because all local diffeomorphisms are continuous, the continuous image of a compact space is compact, the sphere is compact whereas Euclidean 2-space is not.
Properties
If a local diffeomorphism between two manifolds exists then their dimensions must be equal.
Every local diffeomorphism is also a local homeomorphism and therefore a locally injective open map.
A local diffeomorphism has constant rank of
Examples
A diffeomorphism is a bijective local diffeomorphism.
A smooth covering map is a local diffeomorphism such that every point in the target has a neighborhood that is by the map.
Local flow diffeomorphisms
See also
References
.
Theory of continuous functions
Diffeomorphisms
Functions and mappings
Inverse functions
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https://en.wikipedia.org/wiki/194%20%28number%29
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194 (one hundred [and] ninety-four) is the natural number following 193 and preceding 195.
In mathematics
194 is the smallest Markov number that is neither a Fibonacci number nor a Pell number
194 is the smallest number written as the sum of three squares in five ways
194 is the number of irreducible representations of the Monster group
194!! - 1 is prime
See also
194 (disambiguation)
References
Integers
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https://en.wikipedia.org/wiki/Bi-directional%20delay%20line
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In mathematics, a bi-directional delay line is a numerical analysis technique used in computer simulation for solving ordinary differential equations by converting them to hyperbolic equations. In this way an explicit solution scheme is obtained with highly robust numerical properties. It was introduced by Auslander in 1968.
It originates from simulation of hydraulic pipelines where wave propagation was studied. It was then found that it could be used as an efficient numerical technique for numerically insulating different parts of a simulation model in each times step. It is used in the HOPSAN simulation package (Krus et al. 1990).
It is also known as the Transmission Line Modelling (TLM) from an independent development by Johns and O'Brian 1980. This is also extended to partial differential equations.
References
D.M. Auslander, "Distributed System Simulation with Bilateral Delay Line Models", Journal of Basic Engineering, Trans. ASME p195-p200. June 1968.
P. B. Johns and M.O'Brien. "Use of the transmission line modelling (t.l.m) method to solve nonlinear lumped networks", The Radio Electron and Engineer. 1980.
P Krus, A Jansson, J-O Palmberg, K Weddfeldt. "Distributed Simulation of Hydromechanical Systems". Presented at Third Bath International Fluid Power Workshop, Bath, UK 1990.
Numerical differential equations
Numerical analysis
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https://en.wikipedia.org/wiki/Polytree
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In mathematics, and more specifically in graph theory, a polytree (also called directed tree, oriented tree or singly connected network) is a directed acyclic graph whose underlying undirected graph is a tree. In other words, if we replace its directed edges with undirected edges, we obtain an undirected graph that is both connected and acyclic.
A polyforest (or directed forest or oriented forest) is a directed acyclic graph whose underlying undirected graph is a forest. In other words, if we replace its directed edges with undirected edges, we obtain an undirected graph that is acyclic.
A polytree is an example of an oriented graph.
The term polytree was coined in 1987 by Rebane and Pearl.
Related structures
An arborescence is a directed rooted tree, i.e. a directed acyclic graph in which there exists a single source node that has a unique path to every other node. Every arborescence is a polytree, but not every polytree is an arborescence.
A multitree is a directed acyclic graph in which the subgraph reachable from any node forms a tree. Every polytree is a multitree.
The reachability relationship among the nodes of a polytree forms a partial order that has order dimension at most three. If the order dimension is three, there must exist a subset of seven elements , , and such that, for either or , with these six inequalities defining the polytree structure on these seven elements.
A fence or zigzag poset is a special case of a polytree in which the underlying tree is a path and the edges have orientations that alternate along the path. The reachability ordering in a polytree has also been called a generalized fence.
Enumeration
The number of distinct polytrees on unlabeled nodes, for , is
Sumner's conjecture
Sumner's conjecture, named after David Sumner, states that tournaments are universal graphs for polytrees, in the sense that every tournament with vertices contains every polytree with vertices as a subgraph. Although it remains unsolved, it has been proven for all sufficiently large values of .
Applications
Polytrees have been used as a graphical model for probabilistic reasoning. If a Bayesian network has the structure of a polytree, then belief propagation may be used to perform inference efficiently on it.
The contour tree of a real-valued function on a vector space is a polytree that describes the level sets of the function. The nodes of the contour tree are the level sets that pass through a critical point of the function and the edges describe contiguous sets of level sets without a critical point. The orientation of an edge is determined by the comparison between the function values on the corresponding two level sets.
See also
Glossary of graph theory
Notes
References
.
.
.
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.
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Trees (graph theory)
Directed acyclic graphs
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https://en.wikipedia.org/wiki/Federigo%20Enriques
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Abramo Giulio Umberto Federigo Enriques (5 January 1871 – 14 June 1946) was an Italian mathematician, now known principally as the first to give a classification of algebraic surfaces in birational geometry, and other contributions in algebraic geometry.
Biography
Enriques was born in Livorno, and brought up in Pisa, in a Sephardi Jewish family of Portuguese descent. His younger brother was zoologist Paolo Enriques who was also the father of Enzo Enriques Agnoletti and Anna Maria Enriques Agnoletti. He became a student of Guido Castelnuovo (who later became his brother-in-law after marrying his sister Elbina), and became an important member of the Italian school of algebraic geometry. He also worked on differential geometry. He collaborated with Castelnuovo, Corrado Segre and Francesco Severi. He had positions at the University of Bologna, and then the University of Rome La Sapienza. In 1931 he sworn allegiance to fascism, and in 1933 he became a member of the PNF. Despite this, he lost his position in 1938, when the Fascist government enacted the "leggi razziali" (racial laws), which in particular banned Jews from holding professorships in Universities.
The Enriques classification, of complex algebraic surfaces up to birational equivalence, was into five main classes, and was background to further work until Kunihiko Kodaira reconsidered the matter in the 1950s. The largest class, in some sense, was that of surfaces of general type: those for which the consideration of differential forms provides linear systems that are large enough to make all the geometry visible. The work of the Italian school had provided enough insight to recognise the other main birational classes. Rational surfaces and more generally ruled surfaces (these include quadrics and cubic surfaces in projective 3-space) have the simplest geometry. Quartic surfaces in 3-spaces are now classified (when non-singular) as cases of K3 surfaces; the classical approach was to look at the Kummer surfaces, which are singular at 16 points. Abelian surfaces give rise to Kummer surfaces as quotients. There remains the class of elliptic surfaces, which are fiber bundles over a curve with elliptic curves as fiber, having a finite number of modifications (so there is a bundle that is locally trivial actually over a curve less some points). The question of classification is to show that any surface, lying in projective space of any dimension, is in the birational sense (after blowing up and blowing down of some curves, that is) accounted for by the models already mentioned.
No more than other work in the Italian school would the proofs by Enriques now be counted as complete and rigorous. Not enough was known about some of the technical issues: the geometers worked by a mixture of inspired guesswork and close familiarity with examples. Oscar Zariski started to work in the 1930s on a more refined theory of birational mappings, incorporating commutative algebra methods. He also began work on t
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https://en.wikipedia.org/wiki/Hieronymus%20Georg%20Zeuthen
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Hieronymus Georg Zeuthen (15 February 1839 – 6 January 1920) was a Danish mathematician. He is known for work on the enumerative geometry of conic sections, algebraic surfaces, and history of mathematics.
Biography
Zeuthen was born in Grimstrup near Varde where his father was a minister. In 1849, his father moved to a church in Sorø where Zeuthen began his secondary schooling. In 1857 he entered the University of Copenhagen to study mathematics and graduated with a master's degree in 1862. Following this he earned a scholarship to study abroad, and decided to visit Paris where he studied geometry with Michel Chasles.
After returning to Copenhagen, Zeuthen submitted his doctoral dissertation on a new method to determine the characteristics of conic systems in 1865. Enumerative geometry remained his focus up until 1875.
In 1871 he was appointed as an extraordinary professor at the University of Copenhagen, as well as becoming an editor of Matematisk Tidsskrift, a position he held for 18 years. For 39 years he served as secretary of the Royal Danish Academy of Sciences and Letters, during which he also lectured at the Polytechnic Institute. In 1886, he was promoted to ordinary professor at the University of Copenhagen, where he twice served as rector.
After 1875 Zeuthen began to make contributions in other areas such as mechanics and algebraic geometry, as well as being recognised as an expert on the history of medieval and Greek mathematics. He wrote 40 papers and books on the history of mathematics, which covered many topics and several periods. He was an invited speaker at the International Congress of Mathematicians in 1897 at Zurich, in 1904 at Heidelberg, and in 1908 at Rome.
See also
Zeuthen–Segre invariant
Ingeborg Hammer-Jensen, notable student and historian of science
Publications
Abriß einer elementar-geometrischen Kegelschnittlehre. Teubner 1882.
Die Lehre von den Kegelschnitten im Altertum. Kopenhagen 1886 (Danish version 1885 in Forh.Vid.Selskab).
Geschichte der Mathematik im Altertum und Mittelalter. Kopenhagen 1896 (Danish version 1893 publ. by Verlag A.F.Hoest).
Histoire des Mathématiques dans l'Antiquité et le Moyen Age. Paris, Gauthier-Villars, 1902.
Geschichte der Mathematik im XVI. und XVII. Jahrhundert. Teubner 1903, and as Heft 17 of Abhandlungen zur Geschichte der mathematischen Wissenschaften (ed. Moritz Cantor). The Danish version was published 1903 in Copenhagen.
Die Mathematik im Altertum und im Mittelalter. Kopenhagen 1912.
Lehrbuch der abzählenden Methoden der Geometrie. Teubner 1914.
Hvorledes Mathematiken i tiden fra Platon til Euklid blev rationel Videnskab. Avec un résumé en francais. Forh.Dansk Vid.Selskab 1917, pp.199-369.
References
External links
1839 births
1920 deaths
People from Esbjerg Municipality
Danish mathematicians
Members of the Royal Danish Academy of Sciences and Letters
Members of the French Academy of Sciences
Danish historians of mathematics
University of Copenhagen alumni
Aca
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https://en.wikipedia.org/wiki/Naimark%27s%20problem
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Naimark's problem is a question in functional analysis asked by . It asks whether every C*-algebra that has only one irreducible -representation up to unitary equivalence is isomorphic to the -algebra of compact operators on some (not necessarily separable) Hilbert space.
The problem has been solved in the affirmative for special cases (specifically for separable and Type-I C*-algebras). used the -Principle to construct a C*-algebra with generators that serves as a counterexample to Naimark's Problem. More precisely, they showed that the existence of a counterexample generated by elements is independent of the axioms of Zermelo–Fraenkel set theory and the Axiom of Choice ().
Whether Naimark's problem itself is independent of remains unknown.
See also
List of statements undecidable in
Gelfand–Naimark Theorem
References
Conjectures
C*-algebras
Independence results
Unsolved problems in mathematics
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https://en.wikipedia.org/wiki/Barrelled%20space
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In functional analysis and related areas of mathematics, a barrelled space (also written barreled space) is a topological vector space (TVS) for which every barrelled set in the space is a neighbourhood for the zero vector.
A barrelled set or a barrel in a topological vector space is a set that is convex, balanced, absorbing, and closed.
Barrelled spaces are studied because a form of the Banach–Steinhaus theorem still holds for them.
Barrelled spaces were introduced by .
Barrels
A convex and balanced subset of a real or complex vector space is called a and it is said to be , , or .
A or a in a topological vector space (TVS) is a subset that is a closed absorbing disk; that is, a barrel is a convex, balanced, closed, and absorbing subset.
Every barrel must contain the origin. If and if is any subset of then is a convex, balanced, and absorbing set of if and only if this is all true of in for every -dimensional vector subspace thus if then the requirement that a barrel be a closed subset of is the only defining property that does not depend on (or lower)-dimensional vector subspaces of
If is any TVS then every closed convex and balanced neighborhood of the origin is necessarily a barrel in (because every neighborhood of the origin is necessarily an absorbing subset). In fact, every locally convex topological vector space has a neighborhood basis at its origin consisting entirely of barrels. However, in general, there exist barrels that are not neighborhoods of the origin; "barrelled spaces" are exactly those TVSs in which every barrel is necessarily a neighborhood of the origin. Every finite dimensional topological vector space is a barrelled space so examples of barrels that are not neighborhoods of the origin can only be found in infinite dimensional spaces.
Examples of barrels and non-barrels
The closure of any convex, balanced, and absorbing subset is a barrel. This is because the closure of any convex (respectively, any balanced, any absorbing) subset has this same property.
A family of examples: Suppose that is equal to (if considered as a complex vector space) or equal to (if considered as a real vector space). Regardless of whether is a real or complex vector space, every barrel in is necessarily a neighborhood of the origin (so is an example of a barrelled space). Let be any function and for every angle let denote the closed line segment from the origin to the point Let Then is always an absorbing subset of (a real vector space) but it is an absorbing subset of (a complex vector space) if and only if it is a neighborhood of the origin. Moreover, is a balanced subset of if and only if for every (if this is the case then and are completely determined by 's values on ) but is a balanced subset of if and only it is an open or closed ball centered at the origin (of radius ). In particular, barrels in are exactly those closed balls centered at the origin with radius in If then is a closed
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https://en.wikipedia.org/wiki/Darboux%20basis
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A Darboux basis may refer to:
A Darboux basis of a symplectic vector space
In differential geometry, a Darboux frame on a surface
A Darboux tangent in the dovetail joint
Mathematics disambiguation pages
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https://en.wikipedia.org/wiki/Balanced%20set
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In linear algebra and related areas of mathematics a balanced set, circled set or disk in a vector space (over a field with an absolute value function ) is a set such that for all scalars satisfying
The balanced hull or balanced envelope of a set is the smallest balanced set containing
The balanced core of a set is the largest balanced set contained in
Balanced sets are ubiquitous in functional analysis because every neighborhood of the origin in every topological vector space (TVS) contains a balanced neighborhood of the origin and every convex neighborhood of the origin contains a balanced convex neighborhood of the origin (even if the TVS is not locally convex). This neighborhood can also be chosen to be an open set or, alternatively, a closed set.
Definition
Let be a vector space over the field of real or complex numbers.
Notation
If is a set, is a scalar, and then let and and for any let
denote, respectively, the open ball and the closed ball of radius in the scalar field centered at where and
Every balanced subset of the field is of the form or for some
Balanced set
A subset of is called a or balanced if it satisfies any of the following equivalent conditions:
Definition: for all and all scalars satisfying
for all scalars satisfying
where
For every
is a (if ) or (if ) dimensional vector subspace of
If then the above equality becomes which is exactly the previous condition for a set to be balanced. Thus, is balanced if and only if for every is a balanced set (according to any of the previous defining conditions).
For every 1-dimensional vector subspace of is a balanced set (according to any defining condition other than this one).
For every there exists some such that or
If is a convex set then this list may be extended to include:
for all scalars satisfying
If then this list may be extended to include:
is symmetric (meaning ) and
Balanced hull
The of a subset of denoted by is defined in any of the following equivalent ways:
Definition: is the smallest (with respect to ) balanced subset of containing
is the intersection of all balanced sets containing
Balanced core
The of a subset of denoted by is defined in any of the following equivalent ways:
Definition: is the largest (with respect to ) balanced subset of
is the union of all balanced subsets of
if while if
Examples
The empty set is a balanced set. As is any vector subspace of any (real or complex) vector space. In particular, is always a balanced set.
Any non-empty set that does not contain the origin is not balanced and furthermore, the balanced core of such a set will equal the empty set.
Normed and topological vectors spaces
The open and closed balls centered at the origin in a normed vector space are balanced sets. If is a seminorm (or norm) on a vector space then for any constant the set is balanced.
If is any subset and then is a balanced set.
In particular, if is
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https://en.wikipedia.org/wiki/Absolutely%20convex%20set
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In mathematics, a subset C of a real or complex vector space is said to be absolutely convex or disked if it is convex and balanced (some people use the term "circled" instead of "balanced"), in which case it is called a disk.
The disked hull or the absolute convex hull of a set is the intersection of all disks containing that set.
Definition
A subset of a real or complex vector space is called a and is said to be , , and if any of the following equivalent conditions is satisfied:
is a convex and balanced set.
for any scalars and if then
for all scalars and if then
for any scalars and if then
for any scalars if then
The smallest convex (respectively, balanced) subset of containing a given set is called the convex hull (respectively, the balanced hull) of that set and is denoted by (respectively, ).
Similarly, the , the , and the of a set is defined to be the smallest disk (with respect to subset inclusion) containing
The disked hull of will be denoted by or and it is equal to each of the following sets:
which is the convex hull of the balanced hull of ; thus,
In general, is possible, even in finite dimensional vector spaces.
the intersection of all disks containing
Sufficient conditions
The intersection of arbitrarily many absolutely convex sets is again absolutely convex; however, unions of absolutely convex sets need not be absolutely convex anymore.
If is a disk in then is absorbing in if and only if
Properties
If is an absorbing disk in a vector space then there exists an absorbing disk in such that
If is a disk and and are scalars then and
The absolutely convex hull of a bounded set in a locally convex topological vector space is again bounded.
If is a bounded disk in a TVS and if is a sequence in then the partial sums are Cauchy, where for all In particular, if in addition is a sequentially complete subset of then this series converges in to some point of
The convex balanced hull of contains both the convex hull of and the balanced hull of Furthermore, it contains the balanced hull of the convex hull of thus
where the example below shows that this inclusion might be strict.
However, for any subsets if then which implies
Examples
Although the convex balanced hull of is necessarily equal to the balanced hull of the convex hull of
For an example where let be the real vector space and let
Then is a strict subset of that is not even convex;
in particular, this example also shows that the balanced hull of a convex set is necessarily convex.
The set is equal to the closed and filled square in with vertices and (this is because the balanced set must contain both and where since is also convex, it must consequently contain the solid square which for this particular example happens to also be balanced so that ). However, is equal to the horizontal closed line segment between the two points in so that is instead a closed "hour glas
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https://en.wikipedia.org/wiki/Cross-entropy
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In information theory, the cross-entropy between two probability distributions and over the same underlying set of events measures the average number of bits needed to identify an event drawn from the set if a coding scheme used for the set is optimized for an estimated probability distribution , rather than the true distribution .
Definition
The cross-entropy of the distribution relative to a distribution over a given set is defined as follows:
,
where is the expected value operator with respect to the distribution .
The definition may be formulated using the Kullback–Leibler divergence , divergence of from (also known as the relative entropy of with respect to ).
where is the entropy of .
For discrete probability distributions and with the same support this means
The situation for continuous distributions is analogous. We have to assume that and are absolutely continuous with respect to some reference measure (usually is a Lebesgue measure on a Borel σ-algebra). Let and be probability density functions of and with respect to . Then
and therefore
NB: The notation is also used for a different concept, the joint entropy of and .
Motivation
In information theory, the Kraft–McMillan theorem establishes that any directly decodable coding scheme for coding a message to identify one value out of a set of possibilities can be seen as representing an implicit probability distribution over , where is the length of the code for in bits. Therefore, cross-entropy can be interpreted as the expected message-length per datum when a wrong distribution is assumed while the data actually follows a distribution . That is why the expectation is taken over the true probability distribution and not . Indeed the expected message-length under the true distribution is
Estimation
There are many situations where cross-entropy needs to be measured but the distribution of is unknown. An example is language modeling, where a model is created based on a training set , and then its cross-entropy is measured on a test set to assess how accurate the model is in predicting the test data. In this example, is the true distribution of words in any corpus, and is the distribution of words as predicted by the model. Since the true distribution is unknown, cross-entropy cannot be directly calculated. In these cases, an estimate of cross-entropy is calculated using the following formula:
where is the size of the test set, and is the probability of event estimated from the training set. In other words, is the probability estimate of the model that the i-th word of the text is . The sum is averaged over the words of the test. This is a Monte Carlo estimate of the true cross-entropy, where the test set is treated as samples from .
Relation to maximum likelihood
In classification problems we want to estimate the probability of different outcomes. Let the estimated probability of outcome be with to-be-optimized parameters and let t
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https://en.wikipedia.org/wiki/Hall%20subgroup
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In mathematics, specifically group theory, a Hall subgroup of a finite group G is a subgroup whose order is coprime to its index. They were introduced by the group theorist .
Definitions
A Hall divisor (also called a unitary divisor) of an integer n is a divisor d of n such that
d and n/d are coprime. The easiest way to find the Hall divisors is to write the prime power factorization of the number in question and take any subset of the factors. For example, to find the Hall divisors of 60, its prime power factorization is 22 × 3 × 5, so one takes any product of 3, 22 = 4, and 5. Thus, the Hall divisors of 60 are 1, 3, 4, 5, 12, 15, 20, and 60.
A Hall subgroup of G is a subgroup whose order is a Hall divisor of the order of G. In other words, it is a subgroup whose order is coprime to its index.
If π is a set of primes, then a Hall π-subgroup is a subgroup whose order is a product of primes in π, and whose index is not divisible by any primes in π.
Examples
Any Sylow subgroup of a group is a Hall subgroup.
The alternating group A4 of order 12 is solvable but has no subgroups of order 6 even though 6 divides 12, showing that Hall's theorem (see below) cannot be extended to all divisors of the order of a solvable group.
If G = A5, the only simple group of order 60, then 15 and 20 are Hall divisors of the order of G, but G has no subgroups of these orders.
The simple group of order 168 has two different conjugacy classes of Hall subgroups of order 24 (though they are connected by an outer automorphism of G).
The simple group of order 660 has two Hall subgroups of order 12 that are not even isomorphic (and so certainly not conjugate, even under an outer automorphism). The normalizer of a Sylow of order 4 is isomorphic to the alternating group A4 of order 12, while the normalizer of a subgroup of order 2 or 3 is isomorphic to the dihedral group of order 12.
Hall's theorem
proved that if G is a finite solvable group and π
is any set of primes, then G has a Hall π-subgroup, and any two Hall are conjugate. Moreover, any subgroup whose order is
a product of primes in π is contained in some Hall . This result can be thought of as a generalization of Sylow's Theorem to Hall subgroups, but the examples above show that such a generalization is false when the group is not solvable.
The existence of Hall subgroups can be proved by induction on the order of G, using the fact that every finite solvable group has a normal elementary abelian subgroup. More precisely, fix a minimal normal subgroup A, which is either a or a as G is . By induction there is a subgroup H of G containing A such that H/A is a Hall of G/A. If A is a then H is a Hall of G. On the other hand, if A is a , then by the Schur–Zassenhaus theorem A has a complement in H, which is a Hall of G.
A converse to Hall's theorem
Any finite group that has a Hall for every set of primes π is solvable. This is a generalization of Burnside's theorem that any group whose order is of th
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https://en.wikipedia.org/wiki/Tomahawk%20%28geometry%29
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The tomahawk is a tool in geometry for angle trisection, the problem of splitting an angle into three equal parts. The boundaries of its shape include a semicircle and two line segments, arranged in a way that resembles a tomahawk, a Native American axe. The same tool has also been called the shoemaker's knife, but that name is more commonly used in geometry to refer to a different shape, the arbelos (a curvilinear triangle bounded by three mutually tangent semicircles).
Description
The basic shape of a tomahawk consists of a semicircle (the "blade" of the tomahawk), with a line segment the length of the radius extending along the same line as the diameter of the semicircle (the tip of which is the "spike" of the tomahawk), and with another line segment of arbitrary length (the "handle" of the tomahawk) perpendicular to the diameter. In order to make it into a physical tool, its handle and spike may be thickened, as long as the line segment along the handle continues to be part of the boundary of the shape. Unlike a related trisection using a carpenter's square, the other side of the thickened handle does not need to be made parallel to this line segment.
In some sources a full circle rather than a semicircle is used, or the tomahawk is also thickened along the diameter of its semicircle, but these modifications make no difference to the action of the tomahawk as a trisector.
Trisection
To use the tomahawk to trisect an angle, it is placed with its handle line touching the apex of the angle, with the blade inside the angle, tangent to one of the two rays forming the angle, and with the spike touching the other ray of the angle. One of the two trisecting lines then lies on the handle segment, and the other passes through the center point of the semicircle. If the angle to be trisected is too sharp relative to the length of the tomahawk's handle, it may not be possible to fit the tomahawk into the angle in this way, but this difficulty may be worked around by repeatedly doubling the angle until it is large enough for the tomahawk to trisect it, and then repeatedly bisecting the trisected angle the same number of times as the original angle was doubled.
If the apex of the angle is labeled , the point of tangency of the blade is , the center of the semicircle is , the top of the handle is , and the spike is , then triangles and are both right triangles with a shared base and equal height, so they are congruent triangles. Because the sides and of triangle are respectively a tangent and a radius of the semicircle, they are at right angles to each other and is also a right triangle; it has the same hypotenuse as and the same side lengths , so again it is congruent to the other two triangles, showing that the three angles formed at the apex are equal.
Although the tomahawk may itself be constructed using a compass and straightedge, and may be used to trisect an angle, it does not contradict Pierre Wantzel's 1837 theorem that arbitrary angl
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https://en.wikipedia.org/wiki/Legendre%20sieve
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In mathematics, the Legendre sieve, named after Adrien-Marie Legendre, is the simplest method in modern sieve theory. It applies the concept of the Sieve of Eratosthenes to find upper or lower bounds on the number of primes within a given set of integers. Because it is a simple extension of Eratosthenes' idea, it is sometimes called the Legendre–Eratosthenes sieve.
Legendre's identity
The central idea of the method is expressed by the following identity, sometimes called the Legendre identity:
where A is a set of integers, P is a product of distinct primes, is the Möbius function, and is the set of integers in A divisible by d, and S(A, P) is defined to be:
i.e. S(A, P) is the count of numbers in A with no factors common with P.
Note that in the most typical case, A is all integers less than or equal to some real number X, P is the product of all primes less than or equal to some integer z < X, and then the Legendre identity becomes:
(where denotes the floor function). In this example the fact that the Legendre identity is derived from the Sieve of Eratosthenes is clear: the first term is the number of integers below X, the second term removes the multiples of all primes, the third term adds back the multiples of two primes (which were miscounted by being "crossed out twice") but also adds back the multiples of three primes once too many, and so on until all (where denotes the number of primes below z) combinations of primes have been covered.
Once S(A, P) has been calculated for this special case, it can be used to bound using the expression
which follows immediately from the definition of S(A, P).
Limitations
The Legendre sieve has a problem with fractional parts of terms accumulating into a large error, which means the sieve only gives very weak bounds in most cases. For this reason it is almost never used in practice, having been superseded by other techniques such as the Brun sieve and Selberg sieve. However, since these more powerful sieves are extensions of the basic ideas of the Legendre sieve, it is useful to first understand how this sieve works.
References
Sieve theory
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https://en.wikipedia.org/wiki/Passive%20optical%20network
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A passive optical network (PON) is a fiber-optic telecommunications technology for delivering broadband network access to end-customers. Its architecture implements a point-to-multipoint topology in which a single optical fiber serves multiple endpoints by using unpowered (passive) fiber optic splitters to divide the fiber bandwidth among the endpoints. Passive optical networks are often referred to as the last mile between an Internet service provider (ISP) and its customers. Many fiber ISPs prefer this technology.
Components and characteristics
A passive optical network consists of an optical line terminal (OLT) at the service provider's central office (hub), passive (non-power-consuming) optical splitters, and a number of optical network units (ONUs) or optical network terminals (ONTs), which are near end users. A PON reduces the amount of fiber and central office equipment required compared with point-to-point architectures. A passive optical network is a form of fiber-optic access network.
In most cases, downstream signals are broadcast to all premises sharing multiple fibers. Encryption can prevent eavesdropping.
Upstream signals are combined using a multiple access protocol, usually time-division multiple access (TDMA).
History
Passive optical networks were first proposed by British Telecommunications in 1987.
Two major standard groups, the Institute of Electrical and Electronics Engineers (IEEE) and the Telecommunication Standardization Sector of the International Telecommunication Union (ITU-T), develop standards along with a number of other industry organizations.
The Society of Cable Telecommunications Engineers (SCTE) also specified radio frequency over glass for carrying signals over a passive optical network.
FSAN and ITU
Starting in 1995, work on fiber to the home architectures was done by the Full Service Access Network (FSAN) working group, formed by major telecommunications service providers and system vendors. The International Telecommunication Union (ITU) did further work, and standardized on two generations of PON. The older ITU-T G.983 standard was based on Asynchronous Transfer Mode (ATM), and has therefore been referred to as APON (ATM PON). Further improvements to the original APON standard – as well as the gradual falling out of favor of ATM as a protocol – led to the full, final version of ITU-T G.983 being referred to more often as broadband PON, or BPON. A typical APON/BPON provides 622 megabits per second (Mbit/s) (OC-12) of downstream bandwidth and 155 Mbit/s (OC-3) of upstream traffic, although the standard accommodates higher rates.
The ITU-T G.984 Gigabit-capable Passive Optical Networks (GPON, G-PON) standard represented an increase, compared to BPON, in both the total bandwidth and bandwidth efficiency through the use of larger, variable-length packets. Again, the standards permit several choices of bit rate, but the industry has converged on 2.488 gigabits per second (Gbit/s) of downstream bandwidth
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https://en.wikipedia.org/wiki/Steinberg%20group
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In mathematics, Steinberg group means either of two distinct, though related, constructions of the mathematician Robert Steinberg:
Steinberg group (K-theory) St(A) in algebraic K-theory.
Steinberg group (Lie theory) is a 'twisted' group of Lie type, in particular one of the groups of type 3D4 or 2E6.
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https://en.wikipedia.org/wiki/Ree%20group
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In mathematics, a Ree group is a group of Lie type over a finite field constructed by from an exceptional automorphism of a Dynkin diagram that reverses the direction of the multiple bonds, generalizing the Suzuki groups found by Suzuki using a different method. They were the last of the infinite families of finite simple groups to be discovered.
Unlike the Steinberg groups, the Ree groups are not given by the points of a connected reductive algebraic group defined over a finite field; in other words, there is no "Ree algebraic group" related to the Ree groups in the same way that (say) unitary groups are related to Steinberg groups. However, there are some exotic pseudo-reductive algebraic groups over non-perfect fields whose construction is related to the construction of Ree groups, as they use the same exotic automorphisms of Dynkin diagrams that change root lengths.
defined Ree groups over infinite fields of characteristics 2 and 3. and introduced Ree groups of infinite-dimensional Kac–Moody algebras.
Construction
If is a Dynkin diagram, Chevalley constructed split algebraic groups corresponding to , in particular giving groups with values in a field . These groups have the following automorphisms:
Any endomorphism of the field induces an endomorphism of the group
Any automorphism of the Dynkin diagram induces an automorphism of the group .
The Steinberg and Chevalley groups can be constructed as fixed points of an endomorphism of X(F) for the algebraic closure of a field. For the Chevalley groups, the automorphism is the Frobenius endomorphism of , while for the Steinberg groups the automorphism is the Frobenius endomorphism times an automorphism of the Dynkin diagram.
Over fields of characteristic 2 the groups and and over fields of characteristic 3 the groups have an endomorphism whose square is the endomorphism associated to the Frobenius endomorphism of the field . Roughly speaking, this endomorphism comes from the order 2 automorphism of the Dynkin diagram where one ignores the lengths of the roots.
Suppose that the field has an endomorphism whose square is the Frobenius endomorphism: . Then the Ree group is defined to be the group of elements of such that . If the field is perfect then and are automorphisms, and the Ree group is the group of fixed points of the involution of .
In the case when is a finite field of order (with p = 2 or 3) there is an endomorphism with square the Frobenius exactly when k = 2n + 1 is odd, in which case it is unique. So this gives the finite Ree groups as subgroups of B2(22n+1), F4(22n+1), and G2(32n+1) fixed by an involution.
Chevalley groups, Steinberg group, and Ree groups
The relation between Chevalley groups, Steinberg group, and Ree groups is roughly as follows. Given a Dynkin diagram X, Chevalley constructed a group scheme over the integers whose values over finite fields are the Chevalley groups. In general one can take the fixed points of an endomorphism o
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https://en.wikipedia.org/wiki/Suzuki%20group
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In the mathematical discipline known as group theory, the phrase Suzuki group refers to:
The Suzuki sporadic group, Suz or Sz is a sporadic simple group of order 213 · 37 · 52 · 7 · 11 · 13 = 448,345,497,600 discovered by Suzuki in 1969
One of an infinite family of Suzuki groups of Lie type discovered by Suzuki
Group theory
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https://en.wikipedia.org/wiki/Alternating%20sign%20matrix
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In mathematics, an alternating sign matrix is a square matrix of 0s, 1s, and −1s such that the sum of each row and column is 1 and the nonzero entries in each row and column alternate in sign. These matrices generalize permutation matrices and arise naturally when using Dodgson condensation to compute a determinant. They are also closely related to the six-vertex model with domain wall boundary conditions from statistical mechanics. They were first defined by William Mills, David Robbins, and Howard Rumsey in the former context.
Examples
A permutation matrix is an alternating sign matrix, and an alternating sign matrix is a permutation matrix if and only if no entry equals .
An example of an alternating sign matrix that is not a permutation matrix is
Alternating sign matrix theorem
The alternating sign matrix theorem states that the number of alternating sign matrices is
The first few terms in this sequence for n = 0, 1, 2, 3, … are
1, 1, 2, 7, 42, 429, 7436, 218348, … .
This theorem was first proved by Doron Zeilberger in 1992. In 1995, Greg Kuperberg gave a short proof based on the Yang–Baxter equation for the six-vertex model with domain-wall boundary conditions, that uses a determinant calculation due to Anatoli Izergin. In 2005, a third proof was given by Ilse Fischer using what is called the operator method.
Razumov–Stroganov problem
In 2001, A. Razumov and Y. Stroganov conjectured a connection between O(1) loop model, fully packed loop model (FPL) and ASMs.
This conjecture was proved in 2010 by Cantini and Sportiello.
References
Further reading
Bressoud, David M., Proofs and Confirmations, MAA Spectrum, Mathematical Associations of America, Washington, D.C., 1999.
Bressoud, David M. and Propp, James, How the alternating sign matrix conjecture was solved, Notices of the American Mathematical Society, 46 (1999), 637–646.
Mills, William H., Robbins, David P., and Rumsey, Howard Jr., Proof of the Macdonald conjecture, Inventiones Mathematicae, 66 (1982), 73–87.
Mills, William H., Robbins, David P., and Rumsey, Howard Jr., Alternating sign matrices and descending plane partitions, Journal of Combinatorial Theory, Series A, 34 (1983), 340–359.
Propp, James, The many faces of alternating-sign matrices, Discrete Mathematics and Theoretical Computer Science, Special issue on Discrete Models: Combinatorics, Computation, and Geometry (July 2001).
Razumov, A. V., Stroganov Yu. G., Combinatorial nature of ground state vector of O(1) loop model, Theor. Math. Phys., 138 (2004), 333–337.
Razumov, A. V., Stroganov Yu. G., O(1) loop model with different boundary conditions and symmetry classes of alternating-sign matrices], Theor. Math. Phys., 142 (2005), 237–243,
Robbins, David P., The story of , The Mathematical Intelligencer, 13 (2), 12–19 (1991), .
Zeilberger, Doron, Proof of the refined alternating sign matrix conjecture, New York Journal of Mathematics 2 (1996), 59–68.
External links
Alternating sign matrix entry in MathWorld
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https://en.wikipedia.org/wiki/Dodgson%20condensation
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In mathematics, Dodgson condensation or method of contractants is a method of computing the determinants of square matrices. It is named for its inventor, Charles Lutwidge Dodgson (better known by his pseudonym, as Lewis Carroll, the popular author), who discovered it in 1866. The method in the case of an n × n matrix is to construct an (n − 1) × (n − 1) matrix, an (n − 2) × (n − 2), and so on, finishing with a 1 × 1 matrix, which has one entry, the determinant of the original matrix.
General method
This algorithm can be described in the following four steps:
Let A be the given n × n matrix. Arrange A so that no zeros occur in its interior. An explicit definition of interior would be all ai,j with . One can do this using any operation that one could normally perform without changing the value of the determinant, such as adding a multiple of one row to another.
Create an (n − 1) × (n − 1) matrix B, consisting of the determinants of every 2 × 2 submatrix of A. Explicitly, we write
Using this (n − 1) × (n − 1) matrix, perform step 2 to obtain an (n − 2) × (n − 2) matrix C. Divide each term in C by the corresponding term in the interior of A so .
Let A = B, and B = C. Repeat step 3 as necessary until the 1 × 1 matrix is found; its only entry is the determinant.
Examples
Without zeros
One wishes to find
All of the interior elements are non-zero, so there is no need to re-arrange the matrix.
We make a matrix of its 2 × 2 submatrices.
We then find another matrix of determinants:
We must then divide each element by the corresponding element of our original matrix. The interior of the original matrix is
, so after dividing we get
.
The process must be repeated to arrive at a 1 × 1 matrix.
Dividing by the interior of the 3 × 3 matrix, which is just −5, gives and −8 is indeed the determinant of the original matrix.
With zeros
Simply writing out the matrices:
Here we run into trouble. If we continue the process, we will eventually be dividing by 0. We can perform four row exchanges on the initial matrix to preserve the determinant and repeat the process, with most of the determinants precalculated:
Hence, we arrive at a determinant of 36.
Desnanot–Jacobi identity and proof of correctness of the condensation algorithm
The proof that the condensation method computes the determinant of the matrix if no divisions by zero are encountered is based on an identity known as the Desnanot–Jacobi identity (1841) or, more generally, the Sylvester determinant identity (1851).
Let be a square matrix, and for each , denote by the matrix that results from by deleting the -th row and the -th column. Similarly, for
, denote by the matrix that results from by deleting the -th and -th rows and the -th and -th columns.
Desnanot–Jacobi identity
Proof of the correctness of Dodgson condensation
Rewrite the identity as
Now note that by induction it follows that when applying the Dodgson condensation procedure to a square matrix of order , the matrix in the
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https://en.wikipedia.org/wiki/Convergent%20series
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In mathematics, a series is the sum of the terms of an infinite sequence of numbers. More precisely, an infinite sequence defines a series that is denoted
The th partial sum is the sum of the first terms of the sequence; that is,
A series is convergent (or converges) if the sequence of its partial sums tends to a limit; that means that, when adding one after the other in the order given by the indices, one gets partial sums that become closer and closer to a given number. More precisely, a series converges, if there exists a number such that for every arbitrarily small positive number , there is a (sufficiently large) integer such that for all ,
If the series is convergent, the (necessarily unique) number is called the sum of the series.
The same notation
is used for the series, and, if it is convergent, to its sum. This convention is similar to that which is used for addition: denotes the operation of adding and as well as the result of this addition, which is called the sum of and .
Any series that is not convergent is said to be divergent or to diverge.
Examples of convergent and divergent series
The reciprocals of the positive integers produce a divergent series (harmonic series):
Alternating the signs of the reciprocals of positive integers produces a convergent series (alternating harmonic series):
The reciprocals of prime numbers produce a divergent series (so the set of primes is "large"; see divergence of the sum of the reciprocals of the primes):
The reciprocals of triangular numbers produce a convergent series:
The reciprocals of factorials produce a convergent series (see e):
The reciprocals of square numbers produce a convergent series (the Basel problem):
The reciprocals of powers of 2 produce a convergent series (so the set of powers of 2 is "small"):
The reciprocals of powers of any n>1 produce a convergent series:
Alternating the signs of reciprocals of powers of 2 also produces a convergent series:
Alternating the signs of reciprocals of powers of any n>1 produces a convergent series:
The reciprocals of Fibonacci numbers produce a convergent series (see ψ):
Convergence tests
There are a number of methods of determining whether a series converges or diverges.
Comparison test. The terms of the sequence are compared to those of another sequence . If,
for all n, , and converges, then so does
However,
if, for all n, , and diverges, then so does
Ratio test. Assume that for all n, is not zero. Suppose that there exists such that
If r < 1, then the series is absolutely convergent. If then the series diverges. If the ratio test is inconclusive, and the series may converge or diverge.
Root test or nth root test. Suppose that the terms of the sequence in question are non-negative. Define r as follows:
where "lim sup" denotes the limit superior (possibly ∞; if the limit exists it is the same value).
If r < 1, then the series converges. If then the series diver
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https://en.wikipedia.org/wiki/Equianharmonic
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In mathematics, and in particular the study of Weierstrass elliptic functions, the equianharmonic case occurs when the Weierstrass invariants satisfy g2 = 0 and g3 = 1.
This page follows the terminology of Abramowitz and Stegun; see also the lemniscatic case. (These are special examples of complex multiplication.)
In the equianharmonic case, the minimal half period ω2 is real and equal to
where is the Gamma function. The half period is
Here the period lattice is a real multiple of the Eisenstein integers.
The constants e1, e2 and e3 are given by
The case g2 = 0, g3 = a may be handled by a scaling transformation.
Modular forms
Elliptic curves
Elliptic functions
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https://en.wikipedia.org/wiki/List%20of%20finite%20simple%20groups
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In mathematics, the classification of finite simple groups states that every finite simple group is cyclic, or alternating, or in one of 16 families of groups of Lie type, or one of 26 sporadic groups.
The list below gives all finite simple groups, together with their order, the size of the Schur multiplier, the size of the outer automorphism group, usually some small representations, and lists of all duplicates.
Summary
The following table is a complete list of the 18 families of finite simple groups and the 26 sporadic simple groups, along with their orders. Any non-simple members of each family are listed, as well as any members duplicated within a family or between families. (In removing duplicates it is useful to note that no two finite simple groups have the same order, except that the group A8 = A3(2) and A2(4) both have order 20160, and that the group Bn(q) has the same order as Cn(q) for q odd, n > 2. The smallest of the latter pairs of groups are B3(3) and C3(3) which both have order 4585351680.)
There is an unfortunate conflict between the notations for the alternating groups An and the groups of Lie type An(q). Some authors use various different fonts for An to distinguish them. In particular,
in this article we make the distinction by setting the alternating groups An in Roman font and the Lie-type groups An(q) in italic.
In what follows, n is a positive integer, and q is a positive power of a prime number p, with the restrictions noted. The notation (a,b) represents the greatest common divisor of the integers a and b.
Cyclic groups, Zp
Simplicity: Simple for p a prime number.
Order: p
Schur multiplier: Trivial.
Outer automorphism group: Cyclic of order p − 1.
Other names: Z/pZ, Cp
Remarks: These are the only simple groups that are not perfect.
Alternating groups, An, n > 4
Simplicity: Solvable for n < 5, otherwise simple.
Order: n!/2 when n > 1.
Schur multiplier: 2 for n = 5 or n > 7, 6 for n = 6 or 7; see Covering groups of the alternating and symmetric groups
Outer automorphism group: In general 2. Exceptions: for n = 1, n = 2, it is trivial, and for n = 6, it has order 4 (elementary abelian).
Other names: Altn.
Isomorphisms: A1 and A2 are trivial. A3 is cyclic of order 3. A4 is isomorphic to A1(3) (solvable). A5 is isomorphic to A1(4) and to A1(5). A6 is isomorphic to A1(9) and to the derived group B2(2)′. A8 is isomorphic to A3(2).
Remarks: An index 2 subgroup of the symmetric group of permutations of n points when n > 1.
Groups of Lie type
Notation: n is a positive integer, q > 1 is a power of a prime number p, and is the order of some underlying finite field. The order of the outer automorphism group is written as d⋅f⋅g, where d is the order of the group of "diagonal automorphisms", f is the order of the (cyclic) group of "field automorphisms" (generated by a Frobenius automorphism), and g is the order of the group of "graph automorphisms" (coming from automorphisms of the Dynkin diagram). The ou
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https://en.wikipedia.org/wiki/Sublinear%20function
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In linear algebra, a sublinear function (or functional as is more often used in functional analysis), also called a quasi-seminorm or a Banach functional, on a vector space is a real-valued function with only some of the properties of a seminorm. Unlike seminorms, a sublinear function does not have to be nonnegative-valued and also does not have to be absolutely homogeneous. Seminorms are themselves abstractions of the more well known notion of norms, where a seminorm has all the defining properties of a norm that it is not required to map non-zero vectors to non-zero values.
In functional analysis the name Banach functional is sometimes used, reflecting that they are most commonly used when applying a general formulation of the Hahn–Banach theorem.
The notion of a sublinear function was introduced by Stefan Banach when he proved his version of the Hahn-Banach theorem.
There is also a different notion in computer science, described below, that also goes by the name "sublinear function."
Definitions
Let be a vector space over a field where is either the real numbers or complex numbers
A real-valued function on is called a (or a if ), and also sometimes called a or a , if it has these two properties:
Positive homogeneity/Nonnegative homogeneity: for all real and all
This condition holds if and only if for all positive real and all
Subadditivity/Triangle inequality: for all
This subadditivity condition requires to be real-valued.
A function is called or if for all although some authors define to instead mean that whenever these definitions are not equivalent.
It is a if for all
Every subadditive symmetric function is necessarily nonnegative.
A sublinear function on a real vector space is symmetric if and only if it is a seminorm.
A sublinear function on a real or complex vector space is a seminorm if and only if it is a balanced function or equivalently, if and only if for every unit length scalar (satisfying ) and every
The set of all sublinear functions on denoted by can be partially ordered by declaring if and only if for all
A sublinear function is called if it is a minimal element of under this order.
A sublinear function is minimal if and only if it is a real linear functional.
Examples and sufficient conditions
Every norm, seminorm, and real linear functional is a sublinear function.
The identity function on is an example of a sublinear function (in fact, it is even a linear functional) that is neither positive nor a seminorm; the same is true of this map's negation
More generally, for any real the map
is a sublinear function on and moreover, every sublinear function is of this form; specifically, if and then and
If and are sublinear functions on a real vector space then so is the map More generally, if is any non-empty collection of sublinear functionals on a real vector space and if for all then is a sublinear functional on
A function which is subadd
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https://en.wikipedia.org/wiki/B%E2%82%80
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B0, that is "B subscript zero", is also generally used in Magnetic Resonance Imaging to denote the net magnetization vector. Although in physics and mathematics the notation to represent a physical quantity can be arbitrary, it is generally accepted in the literature, such as the International Society for Magnetic Resonance in Medicine that B0 represents net magnetization. This is particularly prominent in areas of science where magnetic fields are important such as spectroscopy. By convention, B0 is interpreted as a vector quantity pointing the z-direction, with subsequent x and y cartesian axes oriented with the right hand rule.
B0 is also the symbol often used to denote the reference magnetization in which equations with electromagnetic fields are normalized.
References
Magnetic resonance imaging
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https://en.wikipedia.org/wiki/Johannes%20Werner
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Johann(es) Werner (; February 14, 1468 – May 1522) was a German mathematician. He was born in Nuremberg, Germany, where he became a parish priest. His primary work was in astronomy, mathematics, and geography, although he was also considered a skilled instrument maker.
Mathematics
His mathematical works were in the areas of spherical trigonometry, as well as conic sections. He published an original work on conic sections in 1522 and is one of several mathematicians sometimes credited with the invention of prosthaphaeresis, which simplifies tedious computations by the use of trigonometric formulas, sometimes called Werner's formulas.
Astronomy
In 1500 he observed a comet, and kept observations of its movements from June 1 until the 24th.
This work further developed the suggestion of Regiomontanus that the occurrences of eclipses and cometary orbits could be used to find longitude, giving a practical approach for this method by means of the cross-staff. (The approach did not actually solve the problem as the instrument was not sufficiently accurate.)
His trepidations method to describe precession of the equinoxes was posthumously challenged in 1524 by Nicolaus Copernicus in The Letter against Werner.
Geography
He is most noted for his work, , published in Nuremberg in 1514, a translation of Claudius Ptolemy's Geography. In it, he refined and promoted the Werner map projection, a cordiform (heart-shape) projection map that had been developed by Johannes Stabius (Stab) of Vienna around 1500. This projection would be used for world maps and some continental maps through the 16th century and into the 17th century. It was used by Mercator, Oronce Fine, and Ortelius in the late 16th century for maps of Asia and Africa. By the 18th century, it was replaced by the Bonne projection for continental maps. The Werner projection is only used today for instructional purposes and as a novelty.
In this work, Werner also proposed an astronomical method to determine longitude, by measuring the position of the moon relative to the background stars. The idea was later discussed in detail by Petrus Apianus in his (Landshut 1524) and became known as the lunar distance method.
Meteorology
Many consider Werner as a pioneer of modern meteorology and weather forecasting.
Between 1513 and 1520, Johann Werner made the first regular observations of the weather conditions in Germany.
Notable publications
, Nürnberg 1514
, Nürnberg, Petrejus 1522
, Leibzig, B.G. Teubner 1907 [written early 16th century].
, 1546
Honours
The crater Werner on the Moon is named after him.
Some of the trigonometric identities used in prosthaphaeresis, an early method for rapid computation of products, were named Werner formulas in honor of Werner's role in development of the algorithm.
See also
History of longitude
References
External links
Johann Werner
Werner Map Projection
Bonne Map Projection
Cordiform Map Projection
1468 births
1528 deaths
Scientists from Nuremberg
German
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https://en.wikipedia.org/wiki/143%20%28number%29
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143 (one hundred [and] forty-three) is the natural number following 142 and preceding 144.
In mathematics
143 is the sum of seven consecutive primes (11 + 13 + 17 + 19 + 23 + 29 + 31). But this number is never the sum of an integer and its base 10 digits, making it a self number. It is also the product of a twin prime pair (11 × 13).
Every positive integer is the sum of at most 143 seventh powers (see Waring's problem).
143 is the difference in the first exception to the pattern shown below:
.
In the military
Vickers Type 143 was a British single-seat fighter biplane in 1929
United States Air Force 143d Airlift Wing airlift unit at Quonset Point, Rhode Island
was a United States Navy during World War II
was a United States Navy during World War II
was a United States Navy during World War II
was a United States Navy in World War II
was a United States Navy patrol boat
was a United States Navy during the Cuban Missile Crisis
was a United States Navy during World War I
In transportation
London Buses route 143 is a Transport for London contracted bus route in London
Air Canada Flight 143, landed at Gimli, Manitoba Air Force Base after gliding after running out of fuel on July 23, 1983
Philippine Airlines Flight 143 exploded prior to takeoff on May 11, 1990, at Manila Airport
Bristol Type 143 was a British twin-engined monoplane aircraft of the Bristol Aeroplane Company
British Rail Class 143 diesel multiple unit, part of the Pacer family of trains introduced in 1985
East 143rd Street–St. Mary's Street station on the IRT Pelham Line of the New York City Subway
143rd Street station on Metra's SouthWest Service in Orland Park, Illinois
In media
Musicians Ray J and Bobby Brackins wrote the song "143"
On Mister Rogers' Neighborhood: "Transformations", 143 is used to mean "I love you". 1 meaning I for 1 letter, 4 meaning Love for the 4 letters, and 3 meaning You for the 3 letters. Reportedly, Fred Rogers maintained his weight at exactly for the last thirty years of his life, and associated the number with the phrase "I love you".
Jake Shimabukuro released the song "143" based on his experience in high school when 143 was sent on a pager to indicate "I Love You".
Sal Governale of The Howard Stern Show had a long running saga on the show about his wife who had an emotional friend. He discovered the severity of their relationship when he read their text messages and emails which included "143", shorthand for "I love you".
"Case 143", song by Stray Kids.
In popular culture
143. A popular pager number to communicate "I love you" (based on the number of letters in each of the three words)
In other fields
143 is also:
The year AD 143 or 143 BC
143 AH is a year in the Islamic calendar that corresponds to 760 – 761 CE
143 Adria is a large main belt asteroid
143 Records label of producer David Foster, a sub-label of Atlantic Records
Psalm 143
Sonnet 143 by William Shakespeare
Slovenia ranks #143 in world populatio
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https://en.wikipedia.org/wiki/Ennio%20de%20Giorgi
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Ennio De Giorgi (8 February 1928 – 25 October 1996) was an Italian mathematician who worked on partial differential equations and the foundations of mathematics.
Mathematical work
De Giorgi's first work was in geometric measure theory, on the topic of the sets of finite perimeters which he called in 1958 as Caccioppoli sets, after his mentor and friend. His definition applied some important analytic tools and the De Giorgi's theorem for the sets established a new tool for set theory as well as his own works. This achievement not only brought Ennio immediate recognition but displayed his ability to attack problems using completely new and effective methods which, though conceived before, can be used with greater precision as shown in his research works.
He solved Bernstein's problem about minimal surfaces for 8 dimensions in 1969 with Enrico Bombieri and Enrico Giusti, for which Bombieri won the Fields Medal in 1974.
His earliest work was on the aim to develop a regularity theory for minimal hypersurfaces, changing how we view the advanced theory of minimal surfaces and calculus of variations forever. The proof required De Giorgi to develop his own version of geometric measure theory along with a related key compactness theorem. With these results, he was able to conclude that a minimal hypersurface is analytic outside a closed subset of codimension at least two. He also established regularity theory for all minimal surfaces in a similar manner.
He solved 19th Hilbert problem on the regularity of solutions of elliptic partial differential equations. Before his results, mathematicians were not able to venture beyond second order nonlinear elliptic equations in two variables. In a major breakthrough, De Giorgi proved that solutions of uniformly elliptic second order equations of divergence form, with only measurable coefficients, were Hölder continuous. His proof was proved in 1956/57 in parallel with John Nash's, who was also working on and solved Hilbert's problem. His results were the first to be published, and it was anticipated that either mathematician would win the 1958 Fields Medal, but it was not to be. Nevertheless, de Giorgi's work opened up the field of nonlinear elliptic partial differential equations in higher dimensions which paved a new period for all of mathematical analysis.
Almost all of his work relates to partial differential equations, minimal surfaces and calculus of variations; these notify the early triumphs of the then-unestablished field of geometric analysis. The work of Karen Uhlenbeck, Shing-Tung Yau and many others have taken inspiration from De Giorgi which have been and continue to be extended and rebuilt in powerful and effective mannerisms.
De Giorgi's conjecture for boundary reaction terms in dimension ≤ 5 was solved by Alessio Figalli and Joaquim Serra, which was one of the results mentioned in Figalli's 2018 Fields Medal lecture given by Luis Caffarelli.
His work on minimal surfaces, partial differential
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https://en.wikipedia.org/wiki/Contractible%20space
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In mathematics, a topological space X is contractible if the identity map on X is null-homotopic, i.e. if it is homotopic to some constant map. Intuitively, a contractible space is one that can be continuously shrunk to a point within that space.
Properties
A contractible space is precisely one with the homotopy type of a point. It follows that all the homotopy groups of a contractible space are trivial. Therefore any space with a nontrivial homotopy group cannot be contractible. Similarly, since singular homology is a homotopy invariant, the reduced homology groups of a contractible space are all trivial.
For a topological space X the following are all equivalent:
X is contractible (i.e. the identity map is null-homotopic).
X is homotopy equivalent to a one-point space.
X deformation retracts onto a point. (However, there exist contractible spaces which do not strongly deformation retract to a point.)
For any path-connected space Y, any two maps f,g: Y → X are homotopic.
For any space Y, any map f: Y → X is null-homotopic.
The cone on a space X is always contractible. Therefore any space can be embedded in a contractible one (which also illustrates that subspaces of contractible spaces need not be contractible).
Furthermore, X is contractible if and only if there exists a retraction from the cone of X to X.
Every contractible space is path connected and simply connected. Moreover, since all the higher homotopy groups vanish, every contractible space is n-connected for all n ≥ 0.
Locally contractible spaces
A topological space X is locally contractible at a point x if for every neighborhood U of x there is a neighborhood V of x contained in U such that the inclusion of V is nulhomotopic in U. A space is locally contractible if it is locally contractible at every point. This definition is occasionally referred to as the "geometric topologist's locally contractible," though is the most common usage of the term. In Hatcher's standard Algebraic Topology text, this definition is referred to as "weakly locally contractible," though that term has other uses.
If every point has a local base of contractible neighborhoods, then we say that X is strongly locally contractible. Contractible spaces are not necessarily locally contractible nor vice versa. For example, the comb space is contractible but not locally contractible (if it were, it would be locally connected which it is not). Locally contractible spaces are locally n-connected for all n ≥ 0. In particular, they are locally simply connected, locally path connected, and locally connected. The circle is (strongly) locally contractible but not contractible.
Strong local contractibility is a strictly stronger property than local contractibility; the counterexamples are sophisticated, the first being given by Borsuk and Mazurkiewicz in their paper Sur les rétractes absolus indécomposables, C.R.. Acad. Sci. Paris 199 (1934), 110-112).
There is some disagreement about which definition is the "s
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https://en.wikipedia.org/wiki/Chi-square
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The term chi-square, chi-squared, or has various uses in statistics:
chi-square distribution, a continuous probability distribution
chi-square test, name given to some tests using chi-square distribution
chi-square target models, a mathematical model used in radar cross-section
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https://en.wikipedia.org/wiki/Inverse%20semigroup
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In group theory, an inverse semigroup (occasionally called an inversion semigroup) S is a semigroup in which every element x in S has a unique inverse y in S in the sense that and , i.e. a regular semigroup in which every element has a unique inverse. Inverse semigroups appear in a range of contexts; for example, they can be employed in the study of partial symmetries.
(The convention followed in this article will be that of writing a function on the right of its argument, e.g. x f rather than f(x), and
composing functions from left to right—a convention often observed in semigroup theory.)
Origins
Inverse semigroups were introduced independently by Viktor Vladimirovich Wagner in the Soviet Union in 1952, and by Gordon Preston in the United Kingdom in 1954. Both authors arrived at inverse semigroups via the study of partial bijections of a set: a partial transformation α of a set X is a function from A to B, where A and B are subsets of X. Let α and β be partial transformations of a set X; α and β can be composed (from left to right) on the largest domain upon which it "makes sense" to compose them:
where α−1 denotes the preimage under α. Partial transformations had already been studied in the context of pseudogroups. It was Wagner, however, who was the first to observe that the composition of partial transformations is a special case of the composition of binary relations. He recognised also that the domain of composition of two partial transformations may be the empty set, so he introduced an empty transformation to take account of this. With the addition of this empty transformation, the composition of partial transformations of a set becomes an everywhere-defined associative binary operation. Under this composition, the collection of all partial one-one transformations of a set X forms an inverse semigroup, called the symmetric inverse semigroup (or monoid) on X, with inverse the functional inverse defined from image to domain (equivalently, the converse relation). This is the "archetypal" inverse semigroup, in the same way that a symmetric group is the archetypal group. For example, just as every group can be embedded in a symmetric group, every inverse semigroup can be embedded in a symmetric inverse semigroup (see below).
The basics
The inverse of an element x of an inverse semigroup S is usually written x−1. Inverses in an inverse semigroup have many of the same properties as inverses in a group, for example, . In an inverse monoid, xx−1 and x−1x are not necessarily equal to the identity, but they are both idempotent. An inverse monoid S in which , for all x in S (a unipotent inverse monoid), is, of course, a group.
There are a number of equivalent characterisations of an inverse semigroup S:
Every element of S has a unique inverse, in the above sense.
Every element of S has at least one inverse (S is a regular semigroup) and idempotents commute (that is, the idempotents of S form a semilattice).
Every -class and every -cl
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https://en.wikipedia.org/wiki/Superabundant%20number
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In mathematics, a superabundant number is a certain kind of natural number. A natural number is called superabundant precisely when, for all :
where denotes the sum-of-divisors function (i.e., the sum of all positive divisors of , including itself). The first few superabundant numbers are . For example, the number 5 is not a superabundant number because for , and 5, the sigma is , and .
Superabundant numbers were defined by . Unknown to Alaoglu and Erdős, about 30 pages of Ramanujan's 1915 paper "Highly Composite Numbers" were suppressed. Those pages were finally published in The Ramanujan Journal 1 (1997), 119–153. In section 59 of that paper, Ramanujan defines generalized highly composite numbers, which include the superabundant numbers.
Properties
proved that if n is superabundant, then there exist a k and a1, a2, ..., ak such that
where pi is the i-th prime number, and
That is, they proved that if n is superabundant, the prime decomposition of n has non-increasing exponents (the exponent of a larger prime is never more than that a smaller prime) and that all primes up to are factors of n. Then in particular any superabundant number is an even integer, and it is a multiple of the k-th primorial
In fact, the last exponent ak is equal to 1 except when n is 4 or 36.
Superabundant numbers are closely related to highly composite numbers. Not all superabundant numbers are highly composite numbers. In fact, only 449 superabundant and highly composite numbers are the same . For instance, 7560 is highly composite but not superabundant. Conversely, 1163962800 is superabundant but not highly composite.
Alaoglu and Erdős observed that all superabundant numbers are highly abundant.
Not all superabundant numbers are Harshad numbers. The first exception is the 105th superabundant number, 149602080797769600. The digit sum is 81, but 81 does not divide evenly into this superabundant number.
Superabundant numbers are also of interest in connection with the Riemann hypothesis, and with Robin's theorem that the Riemann hypothesis is equivalent to the statement that
for all n greater than the largest known exception, the superabundant number 5040. If this inequality has a larger counterexample, proving the Riemann hypothesis to be false, the smallest such counterexample must be a superabundant number .
Not all superabundant numbers are colossally abundant.
Extension
The generalized -super abundant numbers are those such that for all , where is the sum of the -th powers of the divisors of .
1-super abundant numbers are superabundant numbers. 0-super abundant numbers are highly composite numbers.
For example, generalized 2-super abundant numbers are 1, 2, 4, 6, 12, 24, 48, 60, 120, 240, ...
References
.
.
.
External links
MathWorld: Superabundant number
Divisor function
Integer sequences
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https://en.wikipedia.org/wiki/Colossally%20abundant%20number
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In number theory, a colossally abundant number (sometimes abbreviated as CA) is a natural number that, in a particular, rigorous sense, has many divisors. Particularly, it is defined by a ratio between the sum of an integer's divisors and that integer raised to a power higher than one. For any such exponent, whichever integer has the highest ratio is a colossally abundant number. It is a stronger restriction than that of a superabundant number, but not strictly stronger than that of an abundant number.
Formally, a number is said to be colossally abundant if there is an such that for all ,
where denotes the sum-of-divisors function.
The first 15 colossally abundant numbers, 2, 6, 12, 60, 120, 360, 2520, 5040, 55440, 720720, 1441440, 4324320, 21621600, 367567200, 6983776800 are also the first 15 superior highly composite numbers, but neither set is a subset of the other.
History
Colossally abundant numbers were first studied by Ramanujan and his findings were intended to be included in his 1915 paper on highly composite numbers. Unfortunately, the publisher of the journal to which Ramanujan submitted his work, the London Mathematical Society, was in financial difficulties at the time and Ramanujan agreed to remove aspects of the work to reduce the cost of printing. His findings were mostly conditional on the Riemann hypothesis and with this assumption he found upper and lower bounds for the size of colossally abundant numbers and proved that what would come to be known as Robin's inequality (see below) holds for all sufficiently large values of n.
The class of numbers was reconsidered in a slightly stronger form in a 1944 paper of Leonidas Alaoglu and Paul Erdős in which they tried to extend Ramanujan's results.
Properties
Colossally abundant numbers are one of several classes of integers that try to capture the notion of having many divisors. For a positive integer n, the sum-of-divisors function σ(n) gives the sum of all those numbers that divide n, including 1 and n itself. Paul Bachmann showed that on average, σ(n) is around πn / 6. Grönwall's theorem, meanwhile, says that the maximal order of σ(n) is ever so slightly larger, specifically there is an increasing sequence of integers n such that for these integers σ(n) is roughly the same size as eγn log(log(n)), where γ is the Euler–Mascheroni constant. Hence colossally abundant numbers capture the notion of having many divisors by requiring them to maximise, for some ε > 0, the value of the function
over all values of n. Bachmann and Grönwall's results ensure that for every ε > 0 this function has a maximum and that as ε tends to zero these maxima will increase. Thus there are infinitely many colossally abundant numbers, although they are rather sparse, with only 22 of them less than 1018.
Just like with superior highly composite numbers, an effective construction of the set of all colossally abundant numbers is given by the following monotonic mapping from the positive real
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https://en.wikipedia.org/wiki/Highly%20abundant%20number
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In number theory, a highly abundant number is a natural number with the property that the sum of its divisors (including itself) is greater than the sum of the divisors of any smaller natural number.
Highly abundant numbers and several similar classes of numbers were first introduced by , and early work on the subject was done by . Alaoglu and Erdős tabulated all highly abundant numbers up to 104, and showed that the number of highly abundant numbers less than any is at least proportional to .
Formal definition and examples
Formally, a natural number n is called highly abundant if and only if for all natural numbers m < n,
where σ denotes the sum-of-divisors function. The first few highly abundant numbers are
1, 2, 3, 4, 6, 8, 10, 12, 16, 18, 20, 24, 30, 36, 42, 48, 60, ... .
For instance, 5 is not highly abundant because σ(5) = 5+1 = 6 is smaller than σ(4) = 4 + 2 + 1 = 7, while 8 is highly abundant because σ(8) = 8 + 4 + 2 + 1 = 15 is larger than all previous values of σ.
The only odd highly abundant numbers are 1 and 3.
Relations with other sets of numbers
Although the first eight factorials are highly abundant, not all factorials are highly abundant. For example,
σ(9!) = σ(362880) = 1481040,
but there is a smaller number with larger sum of divisors,
σ(360360) = 1572480,
so 9! is not highly abundant.
Alaoglu and Erdős noted that all superabundant numbers are highly abundant, and asked whether there are infinitely many highly abundant numbers that are not superabundant. This question was answered affirmatively by .
Despite the terminology, not all highly abundant numbers are abundant numbers. In particular, none of the first seven highly abundant numbers (1, 2, 3, 4, 6, 8, and 10) is abundant. Along with 16, the ninth highly abundant number, these are the only highly abundant numbers that are not abundant.
7200 is the largest powerful number that is also highly abundant: all larger highly abundant numbers have a prime factor that divides them only once. Therefore, 7200 is also the largest highly abundant number with an odd sum of divisors.
Notes
References
Divisor function
Integer sequences
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https://en.wikipedia.org/wiki/Superior%20highly%20composite%20number
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In number theory, a superior highly composite number is a natural number which, in a particular rigorous sense, has many divisors. Particularly, it is defined by a ratio between the number of divisors an integer has and that integer raised to some positive power. For any possible exponent, whichever integer has the highest ratio is a superior highly composite number. It is a stronger restriction than that of a highly composite number, which is defined as having more divisors than any smaller positive integer.
The first 10 superior highly composite numbers and their factorization are listed.
For a superior highly composite number there exists a positive real number such that for all natural numbers smaller than we have
and for all natural numbers larger than we have
where , the divisor function, denotes the number of divisors of . The term was coined by Ramanujan (1915).
For example, the number with the most divisors per square root of the number itself is 12; this can be demonstrated using some highly composites near 12.
120 is another superior highly composite number because it has the highest ratio of divisors to itself raised to the .4 power.
The first 15 superior highly composite numbers, 2, 6, 12, 60, 120, 360, 2520, 5040, 55440, 720720, 1441440, 4324320, 21621600, 367567200, 6983776800 are also the first 15 colossally abundant numbers, which meet a similar condition based on the sum-of-divisors function rather than the number of divisors. Neither set, however, is a subset of the other.
Properties
All superior highly composite numbers are highly composite. This is easy to prove: if there is some number k that has the same number of divisors as n but is less than n itself (i.e. , but ), then for all positive ε, so if a number "n" is not highly composite, it cannot be superior highly composite.
An effective construction of the set of all superior highly composite numbers is given by the following monotonic mapping from the positive real numbers. Let
for any prime number p and positive real x. Then
is a superior highly composite number.
Note that the product need not be computed indefinitely, because if then , so the product to calculate can be terminated once .
Also note that in the definition of , is analogous to in the implicit definition of a superior highly composite number.
Moreover, for each superior highly composite number exists a half-open interval such that .
This representation implies that there exist an infinite sequence of such that for the n-th superior highly composite number holds
The first are 2, 3, 2, 5, 2, 3, 7, ... . In other words, the quotient of two successive superior highly composite numbers is a prime number.
Superior highly composite radices
The first few superior highly composite numbers have often been used as radices, due to their high divisibility for their size. For example:
Binary (base 2)
Senary (base 6)
Duodecimal (base 12)
Sexagesimal (base 60)
Bigger SHCNs can be
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https://en.wikipedia.org/wiki/Abu%20Ja%27far%20al-Khazin
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Abu Jafar Muhammad ibn Husayn Khazin (; 900–971), also called Al-Khazin, was an Iranian Muslim astronomer and mathematician from Khorasan. He worked on both astronomy and number theory.
Al-Khazin was one of the scientists brought to the court in Ray, Iran by the ruler of the Buyid dynasty, Adhad ad-Dowleh, who ruled from 949 to 983. In 959/960, Khazin was required by the vizier of Ray, who was appointed by ad-Dowleh, to measure the obliquity of the ecliptic.
One of Al-Khazin's works ("Tables of the Disks of the Astrolabe") was described by his successors as the best work in the field and they make many references to it. The work describes some astronomical instruments, in particular an astrolabe fitted with plates inscribed with tables, and a commentary on the use of these. A copy of this instrument was made, but it vanished in Germany during World War II. A photograph of this copy was taken and examined by the historian David King in 1980.
Al-Khazin also wrote a commentary on the Roman polymath Ptolemy's Almagest in which he gives 19 propositions relating to statements by Ptolemy, and proposed a different model of the cosmos.
References
Sources
Further reading
(PDF version)
900 births
971 deaths
10th-century Iranian mathematicians
10th-century Iranian astronomers
Astronomers of the medieval Islamic world
Scholars under the Buyid dynasty
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https://en.wikipedia.org/wiki/Minimum%20mean%20square%20error
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In statistics and signal processing, a minimum mean square error (MMSE) estimator is an estimation method which minimizes the mean square error (MSE), which is a common measure of estimator quality, of the fitted values of a dependent variable. In the Bayesian setting, the term MMSE more specifically refers to estimation with quadratic loss function. In such case, the MMSE estimator is given by the posterior mean of the parameter to be estimated. Since the posterior mean is cumbersome to calculate, the form of the MMSE estimator is usually constrained to be within a certain class of functions. Linear MMSE estimators are a popular choice since they are easy to use, easy to calculate, and very versatile. It has given rise to many popular estimators such as the Wiener–Kolmogorov filter and Kalman filter.
Motivation
The term MMSE more specifically refers to estimation in a Bayesian setting with quadratic cost function. The basic idea behind the Bayesian approach to estimation stems from practical situations where we often have some prior information about the parameter to be estimated. For instance, we may have prior information about the range that the parameter can assume; or we may have an old estimate of the parameter that we want to modify when a new observation is made available; or the statistics of an actual random signal such as speech. This is in contrast to the non-Bayesian approach like minimum-variance unbiased estimator (MVUE) where absolutely nothing is assumed to be known about the parameter in advance and which does not account for such situations. In the Bayesian approach, such prior information is captured by the prior probability density function of the parameters; and based directly on Bayes theorem, it allows us to make better posterior estimates as more observations become available. Thus unlike non-Bayesian approach where parameters of interest are assumed to be deterministic, but unknown constants, the Bayesian estimator seeks to estimate a parameter that is itself a random variable. Furthermore, Bayesian estimation can also deal with situations where the sequence of observations are not necessarily independent. Thus Bayesian estimation provides yet another alternative to the MVUE. This is useful when the MVUE does not exist or cannot be found.
Definition
Let be a hidden random vector variable, and let be a known random vector variable (the measurement or observation), both of them not necessarily of the same dimension. An estimator of is any function of the measurement . The estimation error vector is given by and its mean squared error (MSE) is given by the trace of error covariance matrix
where the expectation is taken over conditioned on . When is a scalar variable, the MSE expression simplifies to . Note that MSE can equivalently be defined in other ways, since
The MMSE estimator is then defined as the estimator achieving minimal MSE:
Properties
When the means and variances are finite, the MMSE estimat
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https://en.wikipedia.org/wiki/Complex%20convexity
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Complex convexity is a general term in complex geometry.
Definition
A set in is called if its intersection with any complex line is contractible.
Background
In complex geometry and analysis, the notion of convexity and its generalizations play an important role in understanding function behavior. Examples of classes of functions with a rich structure are, in addition to the convex functions, the subharmonic functions and the plurisubharmonic functions.
Geometrically, these classes of functions correspond to convex domains and pseudoconvex domains, but there are also other types of domains, for instance lineally convex domains which can be generalized using convex analysis.
A great deal is already known about these domains, but there remain some fascinating, unsolved problems. This theme is mainly theoretical, but there are computational aspects of the domains studied, and these computational aspects are certainly worthy of further study.
References
Complex analysis
Convex analysis
External links
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https://en.wikipedia.org/wiki/Parabolic%20cylinder%20function
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In mathematics, the parabolic cylinder functions are special functions defined as solutions to the differential equation
This equation is found when the technique of separation of variables is used on Laplace's equation when expressed in parabolic cylindrical coordinates.
The above equation may be brought into two distinct forms (A) and (B) by completing the square and rescaling , called H. F. Weber's equations:
and
If is a solution, then so are
If is a solution of equation (), then is a solution of (), and, by symmetry,
are also solutions of ().
Solutions
There are independent even and odd solutions of the form (). These are given by (following the notation of Abramowitz and Stegun (1965)):
and
where is the confluent hypergeometric function.
Other pairs of independent solutions may be formed from linear combinations of the above solutions. One such pair is based upon their behavior at infinity:
where
The function approaches zero for large values of and , while diverges for large values of positive real .
and
For half-integer values of a, these (that is, U and V) can be re-expressed in terms of Hermite polynomials; alternatively, they can also be expressed in terms of Bessel functions.
The functions U and V can also be related to the functions (a notation dating back to Whittaker (1902)) that are themselves sometimes called parabolic cylinder functions:
Function was introduced by Whittaker and Watson as a solution of eq.~() with bounded at . It can be expressed in terms of confluent hypergeometric functions as
Power series for this function have been obtained by Abadir (1993).
References
Special hypergeometric functions
Special functions
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https://en.wikipedia.org/wiki/Laplace%E2%80%93Beltrami%20operator
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In differential geometry, the Laplace–Beltrami operator is a generalization of the Laplace operator to functions defined on submanifolds in Euclidean space and, even more generally, on Riemannian and pseudo-Riemannian manifolds. It is named after Pierre-Simon Laplace and Eugenio Beltrami.
For any twice-differentiable real-valued function f defined on Euclidean space Rn, the Laplace operator (also known as the Laplacian) takes f to the divergence of its gradient vector field, which is the sum of the n pure second derivatives of f with respect to each vector of an orthonormal basis for Rn. Like the Laplacian, the Laplace–Beltrami operator is defined as the divergence of the gradient, and is a linear operator taking functions into functions. The operator can be extended to operate on tensors as the divergence of the covariant derivative. Alternatively, the operator can be generalized to operate on differential forms using the divergence and exterior derivative. The resulting operator is called the Laplace–de Rham operator (named after Georges de Rham).
Details
The Laplace–Beltrami operator, like the Laplacian, is the (Riemannian) divergence of the (Riemannian) gradient:
An explicit formula in local coordinates is possible.
Suppose first that M is an oriented Riemannian manifold. The orientation allows one to specify a definite volume form on M, given in an oriented coordinate system xi by
where is the absolute value of the determinant of the metric tensor, and the dxi are the 1-forms forming the dual frame to the frame
of the tangent bundle and is the wedge product.
The divergence of a vector field on the manifold is then defined as the scalar function with the property
where LX is the Lie derivative along the vector field X. In local coordinates, one obtains
where here and below the Einstein notation is implied, so that the repeated index i is summed over.
The gradient of a scalar function ƒ is the vector field grad f that may be defined through the inner product on the manifold, as
for all vectors vx anchored at point x in the tangent space TxM of the manifold at point x. Here, dƒ is the exterior derivative of the function ƒ; it is a 1-form taking argument vx. In local coordinates, one has
where gij are the components of the inverse of the metric tensor, so that with δik the Kronecker delta.
Combining the definitions of the gradient and divergence, the formula for the Laplace–Beltrami operator applied to a scalar function ƒ is, in local coordinates
If M is not oriented, then the above calculation carries through exactly as presented, except that the volume form must instead be replaced by a volume element (a density rather than a form). Neither the gradient nor the divergence actually depends on the choice of orientation, and so the Laplace–Beltrami operator itself does not depend on this additional structure.
Formal self-adjointness
The exterior derivative and are formal adjoints, in the sense that for a compactly supp
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https://en.wikipedia.org/wiki/Tensor%20density
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In differential geometry, a tensor density or relative tensor is a generalization of the tensor field concept. A tensor density transforms as a tensor field when passing from one coordinate system to another (see tensor field), except that it is additionally multiplied or weighted by a power W of the Jacobian determinant of the coordinate transition function or its absolute value. A tensor density with a single index is called a vector density. A distinction is made among (authentic) tensor densities, pseudotensor densities, even tensor densities and odd tensor densities. Sometimes tensor densities with a negative weight W are called tensor capacity. A tensor density can also be regarded as a section of the tensor product of a tensor bundle with a density bundle.
Motivation
In physics and related fields, it is often useful to work with the components of an algebraic object rather than the object itself. An example would be decomposing a vector into a sum of basis vectors weighted by some coefficients such as
where is a vector in 3-dimensional Euclidean space, are the usual standard basis vectors in Euclidean space. This is usually necessary for computational purposes, and can often be insightful when algebraic objects represent complex abstractions but their components have concrete interpretations. However, with this identification, one has to be careful to track changes of the underlying basis in which the quantity is expanded; it may in the course of a computation become expedient to change the basis while the vector remains fixed in physical space. More generally, if an algebraic object represents a geometric object, but is expressed in terms of a particular basis, then it is necessary to, when the basis is changed, also change the representation. Physicists will often call this representation of a geometric object a tensor if it transforms under a sequence of linear maps given a linear change of basis (although confusingly others call the underlying geometric object which hasn't changed under the coordinate transformation a "tensor", a convention this article strictly avoids). In general there are representations which transform in arbitrary ways depending on how the geometric invariant is reconstructed from the representation. In certain special cases it is convenient to use representations which transform almost like tensors, but with an additional, nonlinear factor in the transformation. A prototypical example is a matrix representing the cross product (area of spanned parallelogram) on The representation is given by in the standard basis by
If we now try to express this same expression in a basis other than the standard basis, then the components of the vectors will change, say according to where is some 2 by 2 matrix of real numbers. Given that the area of the spanned parallelogram is a geometric invariant, it cannot have changed under the change of basis, and so the new representation of this matrix must be:
wh
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https://en.wikipedia.org/wiki/George%20Batchelor
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George Keith Batchelor FRS (8 March 1920 – 30 March 2000) was an Australian applied mathematician and fluid dynamicist.
He was for many years a Professor of Applied Mathematics in the University of Cambridge, and was founding head of the Department of Applied Mathematics and Theoretical Physics (DAMTP). In 1956 he founded the influential Journal of Fluid Mechanics which he edited for some forty years. Prior to Cambridge he studied at Melbourne High School and University of Melbourne.
As an applied mathematician (and for some years at Cambridge a co-worker with Sir Geoffrey Taylor in the field of turbulent flow), he was a keen advocate of the need for physical understanding and sound experimental basis.
His An Introduction to Fluid Dynamics (CUP, 1967) is still considered a classic of the subject, and has been re-issued in the Cambridge Mathematical Library series, following strong current demand. Unusual for an 'elementary' textbook of that era, it presented a treatment in which the properties of a real viscous fluid were fully emphasised. He was elected a Foreign Honorary Member of the American Academy of Arts and Sciences in 1959.
The Batchelor Prize award, is named in his honour and is awarded every four years at the meeting of the International Congress on Theoretical and Applied Mechanics.
References
External links
An Introduction to Fluid Dynamics by G. K. Batchelor at Cambridge Mathematical Library.
Obituaries for George Batchelor (with portraits) at the Department of Applied Mathematics and Theoretical Physics (DAMTP) of the University of Cambridge website
Obituary by Julian Hunt
Video recording of the K. Moffatt's lecture on life and work of George Batchelor
1920 births
2000 deaths
Academics of the University of Cambridge
Alumni of the University of Cambridge
20th-century Australian mathematicians
Fellows of the American Academy of Arts and Sciences
Fellows of the Royal Society
Foreign associates of the National Academy of Sciences
Fluid dynamicists
Cambridge mathematicians
People educated at Melbourne High School
Royal Medal winners
Australian textbook writers
Mathematicians from Melbourne
Journal of Fluid Mechanics editors
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https://en.wikipedia.org/wiki/Polar%20set
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In functional and convex analysis, and related disciplines of mathematics, the polar set is a special convex set associated to any subset of a vector space lying in the dual space
The bipolar of a subset is the polar of but lies in (not ).
Definitions
There are at least three competing definitions of the polar of a set, originating in projective geometry and convex analysis.
In each case, the definition describes a duality between certain subsets of a pairing of vector spaces over the real or complex numbers ( and are often topological vector spaces (TVSs)).
If is a vector space over the field then unless indicated otherwise, will usually, but not always, be some vector space of linear functionals on and the dual pairing will be the bilinear () defined by
If is a topological vector space then the space will usually, but not always, be the continuous dual space of in which case the dual pairing will again be the evaluation map.
Denote the closed ball of radius centered at the origin in the underlying scalar field of by
Functional analytic definition
Absolute polar
Suppose that is a pairing.
The polar or absolute polar of a subset of is the set:
where denotes the image of the set under the map defined by
If denotes the convex balanced hull of which by definition is the smallest convex and balanced subset of that contains then
This is an affine shift of the geometric definition;
it has the useful characterization that the functional-analytic polar of the unit ball (in ) is precisely the unit ball (in ).
The prepolar or absolute prepolar of a subset of is the set:
Very often, the prepolar of a subset of is also called the polar or absolute polar of and denoted by ;
in practice, this reuse of notation and of the word "polar" rarely causes any issues (such as ambiguity) and many authors do not even use the word "prepolar".
The bipolar of a subset of often denoted by is the set ;
that is,
Real polar
The real polar of a subset of is the set:
and the real prepolar of a subset of is the set:
As with the absolute prepolar, the real prepolar is usually called the real polar and is also denoted by
It's important to note that some authors (e.g. [Schaefer 1999]) define "polar" to mean "real polar" (rather than "absolute polar", as is done in this article) and use the notation for it (rather than the notation that is used in this article and in [Narici 2011]).
The real bipolar of a subset of sometimes denoted by is the set ;
it is equal to the -closure of the convex hull of
For a subset of is convex, -closed, and contains
In general, it is possible that but equality will hold if is balanced.
Furthermore, where denotes the balanced hull of
Competing definitions
The definition of the "polar" of a set is not universally agreed upon.
Although this article defined "polar" to mean "absolute polar", some authors define "polar" to mean "real polar" and other autho
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https://en.wikipedia.org/wiki/African%20Mathematical%20Union
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The African Mathematical Union or Union Mathematique Africaine is an African organization dedicated to the development of mathematics in Africa. It was founded in 1976 in Rabat, Morocco, during the first Pan-African Congress of Mathematicians with Henri Hogbe Nlend as its first President. Another key figure in its early years was George Saitoti, later a prominent Kenyan politician.
Mission
The mission of the African Mathematical Union is twofold:
To coordinate and promote the quality of teaching, research and outreach activities in all areas of activities in all areas of mathematics throughout Africa.
To advance mathematical research and education towards the economic, social and cultural development of the continent.
Commissions
The Union has five Commissions:
AMU-CAWM. Commission on Women in Mathematics in Africa, led by Marie Françoise Ouedraogo since 2009.
AMU-CMEA. Commission on Mathematics Education in Africa.
AMU-CHMA. Commission on the History of Mathematics in Africa.
AMU-CRIMS. Commission for Research and Innovations.
AMU-PAMOC. Pan African Mathematics Olympiads Commission.
Commission on Women in Mathematics
The Commission on Women in Mathematics (AMUCAWM) published a report on women with a doctorate in mathematics.
The Commission on Women in Mathematics (AMUCAWM) was created in 1986. At the AMUCWMA's 2012 conference in Ouagadougou, a panel on the state on women in mathematics in Africa recommended the creation of an association for African female mathematicians. The AWUCWMA held another conference soon after in July 2013 in Cape Town, where the African Women in Mathematics Association was formed.
Journal
Since 1978 the Union has published the journal Afrika Matematica (), has been edited by Daouda Sangare until 2009.
As of 2010, the journal is edited by Jacek Banasiak of the University of KwaZulu-Natal, Durban, South Africa.
The deputy editors of the journal are:
Moussa Ouattara, University of Ouagadougou, Ouagadougou, Burkina Faso
Daniel Makinde, Cape Peninsula University of Technology, Cape Town, South Africa.
The members of the advisory committee are:
Aderemi Kuku, Grambling State University, Grambling, USA
Laurent Lafforgue, Institut des Hautes Études Scientifiques, Bures-sur-Yvette, France
Ari Laptev, Imperial College, London, UK
Claudio Procesi, University of Rome 'La Sapienza', Rome, Italy
Michel Waldschmidt, Pierre and Marie Curie University, Paris, France
References
External links
The official homepage since 2009
Unofficial homepage
link to submit papers
Mathematical societies
Organizations based in Rabat
Scientific organizations established in 1976
1976 establishments in Morocco
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https://en.wikipedia.org/wiki/Comprehensive%20School%20Mathematics%20Program
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Comprehensive School Mathematics Program (CSMP) stands for both the name of a curriculum and the name of the project that was responsible for developing curriculum materials in the United States.
Two major curricula were developed as part of the overall CSMP project: the Comprehensive School Mathematics Program (CSMP), a K–6 mathematics program for regular classroom instruction, and the Elements of Mathematics (EM) program, a grades 7–12 mathematics program for gifted students. EM treats traditional topics rigorously and in-depth, and was the only curriculum that strictly adhered to Goals for School Mathematics: The Report of the Cambridge Conference on School Mathematics (1963). As a result, it includes much of the content generally required for an undergraduate mathematics major. These two curricula are unrelated to one another, but certain members of the CSMP staff contributed to the development of both projects. Additionally, some staff of the Elements of Mathematics were also involved with the Secondary School Mathematics Curriculum Improvement Study program being. What follows is a description of the K–6 program that was designed for a general, heterogeneous audience.
The CSMP project was established in 1966, under the direction of Burt Kaufman, who remained director until 1979, succeeded by Clare Heidema. It was originally affiliated with Southern Illinois University in Carbondale, Illinois. After a year of planning, CSMP was incorporated into the Central Midwest Regional Educational Laboratory (later CEMREL, Inc.), one of the national educational laboratories funded at that time by the U.S. Office of Education. In 1984, the project moved to Mid-continental Research for Learning (McREL) Institute's Comprehensive School Reform program, who supported the program until 2003. Heidema remained director to its conclusion. In 1984, it was implemented in 150 school districts in 42 states and about 55,000 students.
Overview
The CSMP project employs four non-verbal languages for the purpose of posing problems and representing mathematical concepts: the Papy Minicomputer (mental computation), Arrows (relations), Strings (classification), and Calculators (patterns). It was designed to teach mathematics as a problem-solving activity rather than simply teaching arithmetic skills, and uses the Socratic method, guiding students to figure out concepts on their own rather than directly lecturing or demonstrating the material. The curriculum uses a spiral structure and philosophy, providing students chances to learn materials at different times and rates. By giving students repeated exposure to a variety of content – even if all students may not initially fully understand – students may experience, assimilate, apply, and react to a variety of mathematical experiences, learning to master different concepts over time, at their own paces, rather than being presented with a single topic to study until mastered.
The curriculum introduced many basic concepts
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