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https://en.wikipedia.org/wiki/Sir%20Cumference
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Sir Cumference is a series of children's educational books about math by Cindy Neuschwander and Wayne Geehan.
The books have been studied for their use in mathematics education.
Characters
Most of the characters of the book are named after math terms, such as Sir Cumference (circumference).
Sir Cumference
Sir Cumference is a knight in the kingdom of Camelot. He has a wife called Lady Di of Ameter and a son named Radius.
Di of Ameter
Di of Ameter is the wife of Sir Cumference. In the first book, she came up with all the different shapes of the table (parallelogram, square, etc.) and in Sir Cumference and the Dragon of Pi, she stayed with Sir Cumference when he turned into a dragon.
Radius
Radius is the son of Di of Ameter and Sir Cumference. He has a friend named Vertex in Sir Cumference and the Sword in the Cone, and plays an important role in both Sir Cumference and the Dragon of Pi and The Sword in the Cone first by turning his father to a dragon and back, and later assisting Vertex in becoming King. He is the focus of Sir Cumference and the Great Knight of Angleland, in which he becomes a knight after rescuing King Lell and his pair of dragons.
Vertex
Vertex is the best friend of Radius. He appears on the first page of Sir Cumference and the Sword in the Cone. He is quoted saying, "I've found out why King Arthur called us all here!" Sir Cumference and Radius agree Vertex should be the heir to the throne.
Series
Currently, there are 11 books in the series:
Sir Cumference and the First Round Table (1997)
Sir Cumference and the Dragon of Pi (1999)
Sir Cumference and the Great Knight of Angleland (2001)
Sir Cumference and the Sword in the Cone (2003)
Sir Cumference and the Isle of Immeter (2006)
Sir Cumference and All the Kings Tens (2009)
Sir Cumference and the Viking's Map (2012)
Sir Cumference and the Off-the-Charts Dessert (2013)
Sir Cumference and the Roundabout Battle (2015)
Sir Cumference and the Fracton Faire (2017)
Sir Cumference Gets Decima's Point (2020)
References
Children's fiction books
Series of children's books
Mathematics fiction books
Series of mathematics books
Children's books based on Arthurian legend
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https://en.wikipedia.org/wiki/Emma%20Darwin%20%28novelist%29
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Emma L. Darwin (born 8 April 1964) is an English historical fiction author, writer of the novels The Mathematics of Love (2006) and A Secret Alchemy (2008) and various short stories. She is the great-great-granddaughter of Charles and Emma Darwin.
Biography
Darwin was born and brought up in London. Her father was Henry Galton Darwin, a lawyer in the Foreign Office, son of Sir Charles Galton Darwin, grandson of Sir George Darwin, and great-grandson of Charles Darwin. Her mother Jane (née Christie), an English teacher, was the younger daughter of John Traill Christie. Darwin has two sisters; Carola and Sophia. Due to the parents' work, the family spent three years commuting between London and Brussels. The family spent many holidays on the Essex/Suffolk border, where much of her novel The Mathematics of Love is set. Darwin has lamented that any reviews of her work inevitably include references to her family background.
She read Drama at the University of Birmingham, and she spent some years in academic publishing. But when she had two small children, she started writing again, and eventually earned an MPhil in Writing at the University of Glamorgan (now the University of South Wales), where her tutor was novelist and poet Christopher Meredith. The novel she wrote for the degree became The Mathematics of Love, which was sold to Headline Review, as the first of a two-book deal. Meanwhile, she had found the form of a research degree so fruitful that she completed a PhD in Creative Writing at Goldsmiths' College in 2010, where her supervisor was Maura Dooley. Darwin now lives with her children in South East London.
The Mathematics of Love was shortlisted for the Commonwealth Writers Best First Book Award for the Europe and South Asia region.
In 2006, her short story Maura's Arm as awarded 3rd place in the Bridport Prize. Previously her story, Closing Time had been longlisted for the 2005 Bridport Prize. She also was highly commended for Nunc Dimittis in the Cadenza Magazine Competition March 2005. Her short story Russian Tea was 2004 Phillip Good Memorial Prize Runner Up, and was included in the 2006 Fish Short Histories Prize anthology.
Publications
The Mathematics of Love London: Headline Review (3 Jul 2006) - paperback published in the UK 8 March 2007 . Published in the US
A Secret Alchemy London: Headline Review 13 Nov 2008
Get Started in Writing Historical Fiction (2016) Teach Yourself
This is not a Book about Charles Darwin: A Writer's Journey through my Family (2019) Holland House Books
References
External links
Author's Website
Author's blog
21st-century English novelists
British women short story writers
English short story writers
English women novelists
Living people
People educated at St Paul's Girls' School
1964 births
Alumni of the University of Birmingham
Alumni of the University of Glamorgan
21st-century British short story writers
21st-century English women writers
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https://en.wikipedia.org/wiki/David%20Kelley%20%28disambiguation%29
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David Kelley (born 1949) is an American philosopher and author.
David Kelley may also refer to:
David C. Kelly, a professor of mathematics
David E. Kelley (born 1956), American television writer and producer
David G. Kelley (born 1928), American politician in the state of California
David H. Kelley (1924–2011), American archaeologist, epigrapher and Mayanist scholar
David M. Kelley (born 1951), American designer and entrepreneur, founder of IDEO
David N. Kelley (born 1959), American attorney and former United States Attorney
David Kelley (poet) (1941–1999), British poet and scholar; co-founder of Black Apollo Press
See also
David Kelly (disambiguation)
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https://en.wikipedia.org/wiki/World%20Maths%20Day
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World Maths Day (World Math Day in American English) is an online international mathematics competition, powered by Mathletics (a learning platform from 3P Learning, the same organisation behind Reading Eggs and Mathseeds). Smaller elements of the wider Mathletics program effectively power the World Maths Day event.
The first World Maths Day started in 2007. Despite these origins, the phrases "World Maths Day" and "World Math Day" are trademarks, and not to be confused with other competitions such as the International Mathematical Olympiad or days such as Pi Day. In 2010, World Maths Day created a Guinness World Record for the Largest Online Maths Competition.
The next World Maths Day will take place on the 8th of March 2023.
Overview
Open to all school-aged students (4 to 18 years old), World Maths Day involves participants playing 20 × 60-second games, with the platform heavily based on "Live Mathletics" found in Mathletics. The contests involve mental maths problems appropriate for each age group, which test the accuracy and speed of the students as they compete against other students across the globe.
The simple but innovative idea of combining the aspects of multi-player online gaming with maths problems has contributed to its popularity around the world. There will be 10 Year group divisions for students to compete in from Kindergarten to Year 9 and above.
An online Hall of Fame will track points throughout the competition with prizes to be awarded to the top students and schools. The Champions Challenge is a new addition to the 2021 competition. Top Year/Grade 9 and above World Maths Day student come together to compete in a knockout tournament. As part of the challenge, students will have their event live streamed, bringing mathematics and Esports together.
History
The inaugural World Maths Day was held on March 13, 2007. 287,000 students from 98 countries answered 38,904,275 questions. The student numbers and the participating countries have steadily increased in the following years.
In 2009, 1.9 million students took part in World Maths Day.
In 2011, World Maths Day sets a Guinness World Record for the Largest Online Maths Competition, with almost 500 million maths questions answered during the event.
In 2012, 3P Learning launched the World Education Games. Over 5.9 Million students from 240 Countries and Territories around the world registered to take part, with World Maths Day being the biggest attraction. In 2013, it was held between 5–7 March and the awards were presented at the Sydney Opera House to the Champions.
In 2015, there were participants from 150 countries. US, UK and Australia all had over 1 million registrations.
The 2019 World Maths Day event was combined with a social media competition, where students around the world were encouraged to dress up in a maths-themed outfit to celebrate maths. Entries included famous mathematicians, an aerial shot of students forming a pi symbol, and human calculators.
In 2022,
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https://en.wikipedia.org/wiki/Rotor%20%28mathematics%29
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A rotor is an object in the geometric algebra (also called Clifford algebra) of a vector space that represents a rotation about the origin. The term originated with William Kingdon Clifford, in showing that the quaternion algebra is just a special case of Hermann Grassmann's "theory of extension" (Ausdehnungslehre). Hestenes defined a rotor to be any element of a geometric algebra that can be written as the product of an even number of unit vectors and satisfies , where is the "reverse" of —that is, the product of the same vectors, but in reverse order.
Definition
In mathematics, a rotor in the geometric algebra of a vector space V is the same thing as an element of the spin group Spin(V). We define this group below.
Let V be a vector space equipped with a positive definite quadratic form q, and let Cl(V) be the geometric algebra associated to V. The algebra Cl(V) is the quotient of the tensor algebra of V by the relations for all . (The tensor product in Cl(V) is what is called the geometric product in geometric algebra and in this article is denoted by .) The Z-grading on the tensor algebra of V descends to a Z/2Z-grading on Cl(V), which we denote by Here, Cleven(V) is generated by even-degree blades and Clodd(V) is generated by odd-degree blades.
There is a unique antiautomorphism of Cl(V) which restricts to the identity on V: this is called the transpose, and the transpose of any multivector a is denoted by . On a blade (i.e., a simple tensor), it simply reverses the order of the factors. The spin group Spin(V) is defined to be the subgroup of Cleven(V) consisting of multivectors R such that That is, it consists of multivectors that can be written as a product of an even number of unit vectors.
Action as rotation on the vector space
Reflections along a vector in geometric algebra may be represented as (minus) sandwiching a multivector M between a non-null vector v perpendicular to the hyperplane of reflection and that vector's inverse v−1:
and are of even grade. Under a rotation generated by the rotor R, a general multivector M will transform double-sidedly as
This action gives a surjective homomorphism presenting Spin(V) as a double cover of SO(V). (See Spin group for more details.)
Restricted alternative formulation
For a Euclidean space, it may be convenient to consider an alternative formulation, and some authors define the operation of reflection as (minus) the sandwiching of a unit (i.e. normalized) multivector:
forming rotors that are automatically normalised:
The derived rotor action is then expressed as a sandwich product with the reverse:
For a reflection for which the associated vector squares to a negative scalar, as may be the case with a pseudo-Euclidean space, such a vector can only be normalized up to the sign of its square, and additional bookkeeping of the sign of the application the rotor becomes necessary. The formulation in terms of the sandwich product with the inverse as above suffers no such shortco
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https://en.wikipedia.org/wiki/Niven%27s%20constant
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In number theory, Niven's constant, named after Ivan Niven, is the largest exponent appearing in the prime factorization of any natural number n "on average". More precisely, if we define H(1) = 1 and H(n) = the largest exponent appearing in the unique prime factorization of a natural number n > 1, then Niven's constant is given by
where ζ is the Riemann zeta function.
In the same paper Niven also proved that
where h(1) = 1, h(n) = the smallest exponent appearing in the unique prime factorization of each natural number n > 1, o is little o notation, and the constant c is given by
and consequently that
References
Further reading
Steven R. Finch, Mathematical Constants (Encyclopedia of Mathematics and its Applications), Cambridge University Press, 2003
External links
Mathematical constants
Number theory
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https://en.wikipedia.org/wiki/String%20metric
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In mathematics and computer science, a string metric (also known as a string similarity metric or string distance function) is a metric that measures distance ("inverse similarity") between two text strings for approximate string matching or comparison and in fuzzy string searching. A requirement for a string metric (e.g. in contrast to string matching) is fulfillment of the triangle inequality. For example, the strings "Sam" and "Samuel" can be considered to be close. A string metric provides a number indicating an algorithm-specific indication of distance.
The most widely known string metric is a rudimentary one called the Levenshtein distance (also known as edit distance). It operates between two input strings, returning a number equivalent to the number of substitutions and deletions needed in order to transform one input string into another. Simplistic string metrics such as Levenshtein distance have expanded to include phonetic, token, grammatical and character-based methods of statistical comparisons.
String metrics are used heavily in information integration and are currently used in areas including fraud detection, fingerprint analysis, plagiarism detection, ontology merging, DNA analysis, RNA analysis, image analysis, evidence-based machine learning, database data deduplication, data mining, incremental search, data integration, malware detection, and semantic knowledge integration.
List of string metrics
Levenshtein distance, or its generalization edit distance
Damerau–Levenshtein distance
Sørensen–Dice coefficient
Block distance or L1 distance or City block distance
Hamming distance
Simple matching coefficient (SMC)
Jaccard similarity or Jaccard coefficient or Tanimoto coefficient
Tversky index
Overlap coefficient
Variational distance
Hellinger distance or Bhattacharyya distance
Information radius (Jensen–Shannon divergence)
Skew divergence
Confusion probability
Tau metric, an approximation of the Kullback–Leibler divergence
Fellegi and Sunters metric (SFS)
Maximal matches
Grammar-based distance
TFIDF distance metric
There also exist functions which measure a dissimilarity between strings, but do not necessarily fulfill the triangle inequality, and as such are not metrics in the mathematical sense. An example of such function is the Jaro–Winkler distance.
Selected string measures examples
References
External links
String Similarity Metrics for Information Integration A fairly complete overview
Carnegie Mellon University open source library
StringMetric project a Scala library of string metrics and phonetic algorithms
Natural project a JavaScript natural language processing library which includes implementations of popular string metrics
Metrics
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https://en.wikipedia.org/wiki/Semi-implicit%20Euler%20method
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In mathematics, the semi-implicit Euler method, also called symplectic Euler, semi-explicit Euler, Euler–Cromer, and Newton–Størmer–Verlet (NSV), is a modification of the Euler method for solving Hamilton's equations, a system of ordinary differential equations that arises in classical mechanics. It is a symplectic integrator and hence it yields better results than the standard Euler method.
Setting
The semi-implicit Euler method can be applied to a pair of differential equations of the form
where f and g are given functions. Here, x and v may be either scalars or vectors. The equations of motion in Hamiltonian mechanics take this form if the Hamiltonian is of the form
The differential equations are to be solved with the initial condition
The method
The semi-implicit Euler method produces an approximate discrete solution by iterating
where Δt is the time step and tn = t0 + nΔt is the time after n steps.
The difference with the standard Euler method is that the semi-implicit Euler method uses vn+1 in the equation for xn+1, while the Euler method uses vn.
Applying the method with negative time step to the computation of from and rearranging leads to the second variant of the semi-implicit Euler method
which has similar properties.
The semi-implicit Euler is a first-order integrator, just as the standard Euler method. This means that it commits a global error of the order of Δt. However, the semi-implicit Euler method is a symplectic integrator, unlike the standard method. As a consequence, the semi-implicit Euler method almost conserves the energy (when the Hamiltonian is time-independent). Often, the energy increases steadily when the standard Euler method is applied, making it far less accurate.
Alternating between the two variants of the semi-implicit Euler method leads in one simplification to the Störmer-Verlet integration and in a slightly different simplification to the leapfrog integration, increasing both the order of the error and the order of preservation of energy.
The stability region of the semi-implicit method was presented by Niiranen although the semi-implicit Euler was misleadingly called symmetric Euler in his paper. The semi-implicit method models the simulated system correctly if the complex roots of the characteristic equation are within the circle shown below. For real roots the stability region extends outside the circle for which the criterion is
As can be seen, the semi-implicit method can simulate correctly both stable systems that have their roots in the left half plane and unstable systems that have their roots in the right half plane. This is clear advantage over forward (standard) Euler and backward Euler. Forward Euler tends to have less damping than the real system when the negative real parts of the roots get near the imaginary axis and backward Euler may show the system be stable even when the roots are in the right half plane.
Example
The motion of a spring satisfying Hooke's law is given by
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https://en.wikipedia.org/wiki/Modular%20multiplicative%20inverse
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In mathematics, particularly in the area of arithmetic, a modular multiplicative inverse of an integer is an integer such that the product is congruent to 1 with respect to the modulus . In the standard notation of modular arithmetic this congruence is written as
which is the shorthand way of writing the statement that divides (evenly) the quantity , or, put another way, the remainder after dividing by the integer is 1. If does have an inverse modulo , then there are an infinite number of solutions of this congruence, which form a congruence class with respect to this modulus. Furthermore, any integer that is congruent to (i.e., in 's congruence class) has any element of 's congruence class as a modular multiplicative inverse. Using the notation of to indicate the congruence class containing , this can be expressed by saying that the modulo multiplicative inverse of the congruence class is the congruence class such that:
where the symbol denotes the multiplication of equivalence classes modulo .
Written in this way, the analogy with the usual concept of a multiplicative inverse in the set of rational or real numbers is clearly represented, replacing the numbers by congruence classes and altering the binary operation appropriately.
As with the analogous operation on the real numbers, a fundamental use of this operation is in solving, when possible, linear congruences of the form
Finding modular multiplicative inverses also has practical applications in the field of cryptography, e.g. public-key cryptography and the RSA algorithm. A benefit for the computer implementation of these applications is that there exists a very fast algorithm (the extended Euclidean algorithm) that can be used for the calculation of modular multiplicative inverses.
Modular arithmetic
For a given positive integer , two integers, and , are said to be congruent modulo if divides their difference. This binary relation is denoted by,
This is an equivalence relation on the set of integers, , and the equivalence classes are called congruence classes modulo or residue classes modulo . Let denote the congruence class containing the integer , then
A linear congruence is a modular congruence of the form
Unlike linear equations over the reals, linear congruences may have zero, one or several solutions. If is a solution of a linear congruence then every element in is also a solution, so, when speaking of the number of solutions of a linear congruence we are referring to the number of different congruence classes that contain solutions.
If is the greatest common divisor of and then the linear congruence has solutions if and only if divides . If divides , then there are exactly solutions.
A modular multiplicative inverse of an integer with respect to the modulus is a solution of the linear congruence
The previous result says that a solution exists if and only if , that is, and must be relatively prime (i.e. coprime). Furthermore, when this
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https://en.wikipedia.org/wiki/Construction%20of%20t-norms
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In mathematics, t-norms are a special kind of binary operations on the real unit interval [0, 1]. Various constructions of t-norms, either by explicit definition or by transformation from previously known functions, provide a plenitude of examples and classes of t-norms. This is important, e.g., for finding counter-examples or supplying t-norms with particular properties for use in engineering applications of fuzzy logic. The main ways of construction of t-norms include using generators, defining parametric classes of t-norms, rotations, or ordinal sums of t-norms.
Relevant background can be found in the article on t-norms.
Generators of t-norms
The method of constructing t-norms by generators consists in using a unary function (generator) to transform some known binary function (most often, addition or multiplication) into a t-norm.
In order to allow using non-bijective generators, which do not have the inverse function, the following notion of pseudo-inverse function is employed:
Let f: [a, b] → [c, d] be a monotone function between two closed subintervals of extended real line. The pseudo-inverse function to f is the function f (−1): [c, d] → [a, b] defined as
Additive generators
The construction of t-norms by additive generators is based on the following theorem:
Let f: [0, 1] → [0, +∞] be a strictly decreasing function such that f(1) = 0 and f(x) + f(y) is in the range of f or equal to f(0+) or +∞ for all x, y in [0, 1]. Then the function T: [0, 1]2 → [0, 1] defined as
T(x, y) = f (-1)(f(x) + f(y))
is a t-norm.
Alternatively, one may avoid using the notion of pseudo-inverse function by having . The corresponding residuum can then be expressed as . And the biresiduum as .
If a t-norm T results from the latter construction by a function f which is right-continuous in 0, then f is called an additive generator of T.
Examples:
The function f(x) = 1 – x for x in [0, 1] is an additive generator of the Łukasiewicz t-norm.
The function f defined as f(x) = –log(x) if 0 < x ≤ 1 and f(0) = +∞ is an additive generator of the product t-norm.
The function f defined as f(x) = 2 – x if 0 ≤ x < 1 and f(1) = 0 is an additive generator of the drastic t-norm.
Basic properties of additive generators are summarized by the following theorem:
Let f: [0, 1] → [0, +∞] be an additive generator of a t-norm T. Then:
T is an Archimedean t-norm.
T is continuous if and only if f is continuous.
T is strictly monotone if and only if f(0) = +∞.
Each element of (0, 1) is a nilpotent element of T if and only if f(0) < +∞.
The multiple of f by a positive constant is also an additive generator of T.
T has no non-trivial idempotents. (Consequently, e.g., the minimum t-norm has no additive generator.)
Multiplicative generators
The isomorphism between addition on [0, +∞] and multiplication on [0, 1] by the logarithm and the exponential function allow two-way transformations between additive and multiplicative generators of a t-norm. If f is an additive gener
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https://en.wikipedia.org/wiki/Weipa%20Airport
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Weipa Airport is an airport in Weipa, Queensland, Australia. The airport is southeast of the town.
Airlines and destinations
Statistics
Weipa Airport was ranked 55th in Australia for the number of revenue passengers served in financial year 2010–2011.
See also
List of airports in Queensland
References
External links
Airport Guide
Airports in Queensland
Weipa Town
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https://en.wikipedia.org/wiki/Squared%20deviations%20from%20the%20mean
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Squared deviations from the mean (SDM) result from squaring deviations. In probability theory and statistics, the definition of variance is either the expected value of the SDM (when considering a theoretical distribution) or its average value (for actual experimental data). Computations for analysis of variance involve the partitioning of a sum of SDM.
Background
An understanding of the computations involved is greatly enhanced by a study of the statistical value
, where is the expected value operator.
For a random variable with mean and variance ,
Therefore,
From the above, the following can be derived:
Sample variance
The sum of squared deviations needed to calculate sample variance (before deciding whether to divide by n or n − 1) is most easily calculated as
From the two derived expectations above the expected value of this sum is
which implies
This effectively proves the use of the divisor n − 1 in the calculation of an unbiased sample estimate of σ2.
Partition — analysis of variance
In the situation where data is available for k different treatment groups having size ni where i varies from 1 to k, then it is assumed that the expected mean of each group is
and the variance of each treatment group is unchanged from the population variance .
Under the Null Hypothesis that the treatments have no effect, then each of the will be zero.
It is now possible to calculate three sums of squares:
Individual
Treatments
Under the null hypothesis that the treatments cause no differences and all the are zero, the expectation simplifies to
Combination
Sums of squared deviations
Under the null hypothesis, the difference of any pair of I, T, and C does not contain any dependency on , only .
total squared deviations aka total sum of squares
treatment squared deviations aka explained sum of squares
residual squared deviations aka residual sum of squares
The constants (n − 1), (k − 1), and (n − k) are normally referred to as the number of degrees of freedom.
Example
In a very simple example, 5 observations arise from two treatments. The first treatment gives three values 1, 2, and 3, and the second treatment gives two values 4, and 6.
Giving
Total squared deviations = 66 − 51.2 = 14.8 with 4 degrees of freedom.
Treatment squared deviations = 62 − 51.2 = 10.8 with 1 degree of freedom.
Residual squared deviations = 66 − 62 = 4 with 3 degrees of freedom.
Two-way analysis of variance
See also
Absolute deviation
Algorithms for calculating variance
Errors and residuals
Least squares
Mean squared error
Residual sum of squares
Root-mean-square deviation
Variance decomposition
References
Statistical deviation and dispersion
Analysis of variance
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https://en.wikipedia.org/wiki/Christophe%20Mandanne
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Christophe Mandanne (born 7 February 1985) is a French professional footballer who plays as a striker.
Career statistics
Honours
Guingamp
Coupe de France: 2013–14
References
External links
1985 births
Living people
French people of Réunion descent
Footballers from Toulouse
French men's footballers
Men's association football forwards
Ligue 1 players
Ligue 2 players
UAE Pro League players
Le Havre AC players
Tours FC players
Dijon FCO players
Stade de Reims players
En Avant Guingamp players
Fujairah FC players
AS Nancy Lorraine players
LB Châteauroux players
French expatriate men's footballers
French expatriate sportspeople in the United Arab Emirates
Expatriate men's footballers in the United Arab Emirates
Black French sportspeople
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https://en.wikipedia.org/wiki/Variations%20in%20published%20cricket%20statistics
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Variations in published cricket statistics have come about because there is no official view of the status of cricket matches played in Great Britain prior to 1895 or in the rest of the world prior to 1947. As a result, historians and statisticians have compiled differing lists of matches that they recognise as (unofficially) first-class. The problem is significant where it touches on some of the sport's first-class records, especially in regards to the playing career of W. G. Grace.
Concept and definition of first-class cricket
The concept of a "first-class standard" was formalised in May 1894 at a meeting of the Marylebone Cricket Club (MCC) committee and the secretaries of the 14 clubs in the official County Championship, which had begun in 1890. As a result, these 14 clubs became officially first-class from 1895 along with MCC, Cambridge University, Oxford University, the main international touring teams and other teams designated as such by MCC (e.g., North v South, Gentlemen v Players, occasional XIs, etc).
First-class cricket was formally defined by the then Imperial Cricket Conference (ICC) in May 1947 as a match of three or more days' duration between two sides of eleven players officially adjudged first-class, with the governing body in each country to decide the status of teams. Significantly, it was stated that the definition does not have retrospective effect.
The absence of any ruling about matches played prior to 1947 (or prior to 1895 in Great Britain) has caused problems for cricket historians and especially statisticians. Matches that are believed to have met the official definitions, assuming they featured teams of the necessary high standard, have been recorded since 1697 (having been in vogue since the 1660s).
It was inevitable that historians and statisticians would seek to apply unofficial first-class status retrospectively, despite the ICC and MCC's directives. The position is that each writer must compile their own list based on personal opinion: as a result, significant differences may be observed in published statistical records, with particular impact on the career records of W. G. Grace, Jack Hobbs and Herbert Sutcliffe. There are also differences in the perceived status of certain matches played by Gloucestershire teams before the county club was formed in 1870, and by Somerset in 1879 and 1881.
One of the problems here is that statisticians have tended not to publish their match lists with their findings: it should, however, be noted that the number of differences is extremely small in terms of the sport's overall statistics.
Development of scoring to 1895
The problem of different versions is as old as cricket scorecards themselves. The earliest known scorecards are dated 1744 but very few were created (or have survived) between 1744 and 1772 when they became habitual.
The main source for scorecards from 1772 until the 1860s is Arthur Haygarth’s Scores & Biographies, which was published in several volumes. H
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https://en.wikipedia.org/wiki/Tangent%20developable
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In the mathematical study of the differential geometry of surfaces, a tangent developable is a particular kind of developable surface obtained from a curve in Euclidean space as the surface swept out by the tangent lines to the curve. Such a surface is also the envelope of the tangent planes to the curve.
Parameterization
Let be a parameterization of a smooth space curve. That is, is a twice-differentiable function with nowhere-vanishing derivative that maps its argument (a real number) to a point in space; the curve is the image of . Then a two-dimensional surface, the tangent developable of , may be parameterized by the map
The original curve forms a boundary of the tangent developable, and is called its directrix or edge of regression. This curve is obtained by first developing the surface into the plane, and then considering the image in the plane of the generators of the ruling on the surface. The envelope of this family of lines is a plane curve whose inverse image under the development is the edge of regression. Intuitively, it is a curve along which the surface needs to be folded during the process of developing into the plane.
Properties
The tangent developable is a developable surface; that is, it is a surface with zero Gaussian curvature. It is one of three fundamental types of developable surface; the other two are the generalized cones (the surface traced out by a one-dimensional family of lines through a fixed point), and the cylinders (surfaces traced out by a one-dimensional family of parallel lines). (The plane is sometimes given as a fourth type, or may be seen as a special case of either of these two types.) Every developable surface in three-dimensional space may be formed by gluing together pieces of these three types; it follows from this that every developable surface is a ruled surface, a union of a one-dimensional family of lines. However, not every ruled surface is developable; the helicoid provides a counterexample.
The tangent developable of a curve containing a point of zero torsion will contain a self-intersection.
History
Tangent developables were first studied by Leonhard Euler in 1772. Until that time, the only known developable surfaces were the generalized cones and the cylinders. Euler showed that tangent developables are developable and that every developable surface is of one of these types.
Notes
References
.
External links
Differential geometry of surfaces
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https://en.wikipedia.org/wiki/ETT
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Ett or ETT may refer to:
Arts
Caspar Ett (1788–1847), German composer and organist
English Touring Theatre
Mathematics
Euler tour technique, in graph theory
Extensional type theory, in logic
Medicine
Endotracheal tube, in respiratory medicine
Epithelioid trophoblastic tumour, a very rare cancer
Ergothioneine transporter, a protein and human gene (SLC22A4)
Exercise Tolerance Test, in cardiology
Military
Embedded Training Teams in Afghanistan
EBR ETT, a French armoured personnel carrier
Other uses
Elementary Teachers of Toronto, a Canadian labour union
English Toy Terrier, a dog breed
Etruscan language, once spoken in Italy (ISO 639-3: ett)
European Transactions on Telecommunications, a scientific journal
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https://en.wikipedia.org/wiki/Hausa%20Sign%20Language
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Hausa Sign Language (HSL) or Maganar Hannu is the indigenous sign language of the Deaf community in northern Nigeria.
Overview
There are no statistics on the number of deaf people in northern Nigeria or in Nigeria in general or on the number of people who use Hausa Sign Language. Estimates as to the number of signers using this language "vary greatly, from 70,000 to five million".
There is no information on the origin of Hausa Sign Language, but it is believed that deaf people have always used HSL for communication. Hausa Sign Language is not taught formally in schools but is handed down from one generation to the next. Deaf children learn it from their parents, from their peers or other members of the deaf community. HSL is constantly enriched whenever deaf people meet, whether informally or in schools, associations or other groups.
Linguistic structure
Hausa Sign Language is a language in its own right with its own lexicon and grammar. It can be analysed linguistically like other spoken and sign languages. The HSL lexicon does, however, include loanwords from spoken Hausa, the surrounding major spoken language. Even though HSL grammar differs from spoken Hausa language grammar, there are influences from the spoken language in some users.
HSL signs are articulated by the hands. Each sign is composed of a number of components that are called the manual parameters, i.e. handshape, orientation, movement, and location. A sign may be articulated by one hand or both. Body posture and movement as well as facial expressions and other non-manual parameters play a role as well. They may be inherent parts of signs but may also be used to express grammatical features, e.g. question markers and emphasis.
References
External links
Website with all signs
Further reading
Hausa SL dictionaries:
Schmaling, Halima C. 2022. Maganar hannu: Harshen bebaye na kasar Hausa. Littafi na tara: Addini. Hamburg: Buske. [Hausa Sign Language, Book 9: Religion.]
Schmaling, Halima C. 2020. Maganar hannu: Harshen bebaye na kasar Hausa. Littafi na farko: Iyali. Hamburg: Buske. [Hausa Sign Language, Book 1: Family.] (2nd rev. and ext. ed.)
Schmaling, Halima C. 2019. Maganar hannu: Harshen bebaye na kasar Hausa. Littafi na takwas: Kayan gida. Hamburg: Buske. [Hausa Sign Language. Book 8: Things in the house.]
Schmaling, Halima C. 2018. Maganar hannu: Harshen bebaye na kasar Hausa. Littafi na bakwai: Ilmi. Hamburg: Buske. [Hausa Sign Language. Book 7: Education.]
Schmaling, Halima C. 2018. Maganar hannu: Harshen bebaye na kasar Hausa. Littafi na shida: Lokaci da yanayi. Hamburg: Buske. [Hausa Sign Language. Book 6: Time and weather.]
Schmaling, Halima C. 2017. Maganar hannu: Harshen bebaye na kasar Hausa. Littafi na biyar: Ayyukan yau da kullum. Hamburg: Buske. [Hausa Sign Language. Book 5: Everyday activities.]
Schmaling, Halima C. 2016. Maganar hannu: Harshen bebaye na kasar Hausa. Littafi na hudu: Kasuwanci da kidaya. Hamburg: Buske.[Hausa Sign Language. Book 4: Comm
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https://en.wikipedia.org/wiki/Roger%20Maddux
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Roger Maddux (born 1948) is an American mathematician specializing in algebraic logic.
He completed his B.A. at Pomona College in 1969, and his Ph.D. in mathematics at the University of California, Berkeley in 1978, where he was one of Alfred Tarski's last students. His career has been at Iowa State University, where he fills a joint appointment in computer science and mathematics.
Maddux is primarily known for his work in relation algebras and cylindric algebras, and as the inventor of relational bases.
Books by Maddux
1990: (with Clifford H. Bergman & Don L. Pigozzi, editors) Algebraic Logic and Universal Algebra in Computer Science, Lecture Notes in Computer Science #425, Springer books
2006: Relation Algebras, vol. 150 in Studies in Logic and the Foundations of Mathematics. Elsevier Science
Notes
External links
Maddux home page at the Iowa State University.
Living people
1948 births
20th-century American mathematicians
21st-century American mathematicians
Pomona College alumni
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https://en.wikipedia.org/wiki/James%20Waddell%20Alexander%20II
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James Waddell Alexander II (September 19, 1888 September 23, 1971) was a mathematician and topologist of the pre-World War II era and part of an influential Princeton topology elite, which included Oswald Veblen, Solomon Lefschetz, and others. He was one of the first members of the Institute for Advanced Study (1933–1951), and also a professor at Princeton University (1920–1951).
Early life, family, and personal life
James was born on September 19, 1888, in Sea Bright, New Jersey. Alexander came from an old, distinguished Princeton family. He was the only child of the American portrait painter John White Alexander and Elizabeth Alexander. His maternal grandfather, James Waddell Alexander, was the president of the Equitable Life Assurance Society. Alexander's affluence and upbringing allowed him to interact with high society in America and elsewhere.
He married Natalia Levitzkaja on January 11, 1918, a Russian woman. Together, they had two children.
They would frequently spend time, until 1937, in the Chamonix area of France, where he would also climb mountains and hills. Alexander was also a noted mountaineer, having succeeded in many major ascents, e.g. in the Swiss Alps and Colorado Rockies. When in Princeton, he liked to climb the university buildings, and always left his office window on the top floor of Fine Hall open so that he could enter by climbing the building.
Education
He graduated from Princeton University with a Bachelor of Science degree in 1910. He received his Masters of Arts degree in 1911 and his doctoral degree in 1915.
Military career
During World War I, Alexander served with tech staff in the Ordnance Department of the United States Army overseas. He retired as a Captain.
Academic career
He was a pioneer in algebraic topology, setting the foundations for Henri Poincaré's ideas on homology theory and furthering it by founding cohomology theory, which developed gradually in the decade after he gave a definition of cochain. For this, in 1928 he was awarded the Bôcher Memorial Prize. He also contributed to the beginnings of knot theory by inventing the Alexander invariant of a knot, which in modern terms is a graded module obtained from the homology of a "cyclic covering" of the knot complement. From this invariant, he defined the first of the polynomial knot invariants.
With Garland Briggs, he also gave a combinatorial description of knot invariance based on certain moves, now (against the history) called the Reidemeister moves; and also a means of computing homological invariants from the knot diagram.
Alexander was an elected member of both the American Philosophical Society and the American Academy of Arts and Sciences.
Towards the end of his life, Alexander became a recluse. He was known as a socialist and his prominence brought him to the attention of McCarthyists. The atmosphere of the McCarthy era pushed him into a greater seclusion. He was not seen in public after 1954, when he appeared to sign a letter support
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https://en.wikipedia.org/wiki/Ragsdale%20conjecture
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The Ragsdale conjecture is a mathematical conjecture that concerns the possible arrangements of real algebraic curves embedded in the projective plane. It was proposed by Virginia Ragsdale in her dissertation in 1906 and was disproved in 1979. It has been called "the oldest and most famous conjecture on the topology of real algebraic curves".
Formulation of the conjecture
Ragsdale's dissertation, "On the Arrangement of the Real Branches of Plane Algebraic Curves," was published by the American Journal of Mathematics in 1906. The dissertation was a treatment of Hilbert's sixteenth problem, which had been proposed by Hilbert in 1900, along with 22 other unsolved problems of the 19th century; it is one of the handful of Hilbert's problems that remains wholly unresolved. Ragsdale formulated a conjecture that provided an upper bound on the number of topological circles of a certain type, along with the basis of evidence.
Conjecture
Ragsdale's main conjecture is as follows.
Assume that an algebraic curve of degree 2k contains p even and n odd ovals. Ragsdale conjectured that
She also posed the inequality
and showed that the inequality could not be further improved. This inequality was later proved by Petrovsky.
Disproving the conjecture
The conjecture was held of very high importance in the field of real algebraic geometry for most of the twentieth century. Later, in 1980, Oleg Viro introduced a technique known as "patchworking algebraic curves" and used to generate a counterexample to the conjecture.
In 1993, Ilia Itenberg produced additional counterexamples to the Ragsdale conjecture, so Viro and Itenberg wrote a paper in 1996 discussing their work on disproving the conjecture using the "patchworking" technique.
The problem of finding a sharp upper bound remains unsolved.
References
Disproved conjectures
Real algebraic geometry
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https://en.wikipedia.org/wiki/Point-finite%20collection
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In mathematics, a collection or family of subsets of a topological space is said to be point-finite if every point of lies in only finitely many members of
A metacompact space is a topological space in which every open cover admits a point-finite open refinement. Every locally finite collection of subsets of a topological space is also point-finite.
A topological space in which every open cover admits a locally finite open refinement is called a paracompact space. Every paracompact space is therefore metacompact.
References
General topology
Families of sets
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https://en.wikipedia.org/wiki/Association%20for%20Women%20in%20Mathematics
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The Association for Women in Mathematics (AWM) is a professional society whose mission is to encourage women and girls to study and to have active careers in the mathematical sciences, and to promote equal opportunity for and the equal treatment of women and girls in the mathematical sciences. The AWM was founded in 1971 and incorporated in the state of Massachusetts. AWM has approximately 5200 members, including over 250 institutional members, such as colleges, universities, institutes, and mathematical societies. It offers numerous programs and workshops to mentor women and girls in the mathematical sciences. Much of AWM's work is supported through federal grants.
History
The Association was founded in 1971 as the Association of Women Mathematicians, but the name was changed almost immediately. As reported in "A Brief History of the Association for Women in Mathematics: The Presidents' Perspectives", by Lenore Blum:
Mary Gray, an early organizer and first president, placed an advertisement in the February 1971 Notices of the AMS, and wrote the first issue of the AWM Newsletter that May. Early goals of the association focused on equal pay for equal work, as well as equal consideration for admission to graduate school and support while there; for faculty appointments at all levels; for promotion and for tenure; for administrative appointments; and for government grants, positions on review and advisory panels and positions in professional organizations. Alice T. Shafer, who succeeded Mary Gray as second president of the AWM, set up an AWM office at Wellesley College. At this point, AWM began to be a recognized established presence in the mathematics scene.
The AWM holds an annual meeting at the Joint Mathematics Meetings. In 2011, during its fortieth-anniversary celebration 40 Years and Counting, the association initiated a biennial research symposium.
The Association for Women in Mathematics Newsletter is the member journal of the organization. The first issue was published in May 1971, a few months after AWM was founded. All regular members of AWM can request that hard copies of the newsletter be sent to them. The newsletter is now open access and anyone can read or download a pdf file of recent or past issues from the AWM website.
Lectures
The AWM sponsors three honorary lecture series.
The Noether Lectures – honor women who "have made fundamental and sustained contributions to the mathematical sciences". Presented in association with the American Mathematical Society, the lecture is given at the annual Joint Mathematics Meetings.
The Falconer Lectures – honor women who "have made distinguished contributions to the mathematical sciences or mathematics education. Presented in association with the Mathematical Association of America, the lecture is given at the annual MathFest.
The Kovalevsky Lectures – honor women who have "made distinguished contributions in applied or computational mathematics". Presented in association wit
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https://en.wikipedia.org/wiki/Limit%20point%20compact
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In mathematics, a topological space is said to be limit point compact or weakly countably compact if every infinite subset of has a limit point in This property generalizes a property of compact spaces. In a metric space, limit point compactness, compactness, and sequential compactness are all equivalent. For general topological spaces, however, these three notions of compactness are not equivalent.
Properties and examples
In a topological space, subsets without limit point are exactly those that are closed and discrete in the subspace topology. So a space is limit point compact if and only if all its closed discrete subsets are finite.
A space is limit point compact if and only if it has an infinite closed discrete subspace. Since any subset of a closed discrete subset of is itself closed in and discrete, this is equivalent to require that has a countably infinite closed discrete subspace.
Some examples of spaces that are not limit point compact: (1) The set of all real numbers with its usual topology, since the integers are an infinite set but do not have a limit point in ; (2) an infinite set with the discrete topology; (3) the countable complement topology on an uncountable set.
Every countably compact space (and hence every compact space) is limit point compact.
For T1 spaces, limit point compactness is equivalent to countable compactness.
An example of limit point compact space that is not countably compact is obtained by "doubling the integers", namely, taking the product where is the set of all integers with the discrete topology and has the indiscrete topology. The space is homeomorphic to the odd-even topology. This space is not T0. It is limit point compact because every nonempty subset has a limit point.
An example of T0 space that is limit point compact and not countably compact is the set of all real numbers, with the right order topology, i.e., the topology generated by all intervals The space is limit point compact because given any point every is a limit point of
For metrizable spaces, compactness, countable compactness, limit point compactness, and sequential compactness are all equivalent.
Closed subspaces of a limit point compact space are limit point compact.
The continuous image of a limit point compact space need not be limit point compact. For example, if with discrete and indiscrete as in the example above, the map given by projection onto the first coordinate is continuous, but is not limit point compact.
A limit point compact space need not be pseudocompact. An example is given by the same with indiscrete two-point space and the map whose image is not bounded in
A pseudocompact space need not be limit point compact. An example is given by an uncountable set with the cocountable topology.
Every normal pseudocompact space is limit point compact.Proof: Suppose is a normal space that is not limit point compact. There exists a countably infinite closed discrete subset
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https://en.wikipedia.org/wiki/Snub%20%28disambiguation%29
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A snub is a refusal to recognise an acquaintance. It may also refer to:
Snub (geometry), an operation applied to a polyhedron
Lawrence Snub Mosley (1905–1981), American jazz trombonist
Snub Pollard, stage name of Australian-born silent film comedian Harry Fraser (1889–1962)
Snub TV
SNUB, a non-profit organisation aimed at stopping the urban sprawl of Norwich, UK
Supernumerary nipples–uropathies–Becker's nevus syndrome or SNUB syndrome, a medical condition
snub, a nautical term, meaning to suddenly stop a rope that is running out
See also
Snubber, a fluidic, mechanical or electrical device used to suppress transients
Snubbing, a type of heavy well intervention performed on oil and gas wells
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https://en.wikipedia.org/wiki/Subsampling
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Subsampling or sub-sampling may refer to:
Sampling (statistics)
Replication (statistics)
Downsampling in signal processing
Chroma subsampling
Sub-sampling (chemistry)
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https://en.wikipedia.org/wiki/F.%20Yates
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F. Yates is the name of:
Frances Yates (1899–1981), British historian
Frank Yates (1902–1994), pioneer of 20th century statistics
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https://en.wikipedia.org/wiki/Moore%20family
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Moore family may refer to:
Collections of sets that characterize a closure operator, according to mathematician E. H. Moore's theorem in set theory.
The Moore family (Carolinas), a prominent political family of North and South Carolina during the 18th and 19th centuries.
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https://en.wikipedia.org/wiki/Science%20Olympiad%20Foundation
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Science Olympiad Foundation (SOF) is an educational foundation, established in 1998, based in New Delhi, India which promotes science, mathematics, general knowledge, introductory computer education and English language skills among school children in India and many other countries through various Olympiads. However, they are not the official organizer of Olympiads in India.
Olympiads
Every year over 68000 schools from 48 countries register for the 7 Olympiad exams and millions of students appear in them.
Current
Annually, about 5 million students take part in each of the following Olympiad exams:
National Cyber Olympiad (NCO) is a single level exam. It was the second Olympiad conducted by SOF. It has been conducted since 2000. Students from class I-X may participate in the examination.
National Science Olympiad (NSO) is conducted at two levels each year. It was the first Olympiad conducted by SOF. It has been conducted since 1998. Students from class I-XII may participate in the examination.
International Mathematics Olympiad (IMO) is conducted at two levels each year. Students from class I-XII may participate in the examination.
International English Olympiad (IEO) used to be a single level exam each year, but since 2017–2018 it is conducted at two levels. Students from class I-XII can participate in this Olympiad.
International General Knowledge Olympiad (IGKO) is a single level exam. Students from classes I-X may participate in the examination. This Olympiad was introduced in the 2017–2018 academic session.
International Commerce Olympiad (ICO) is a single level exam. Students from classes XI and XII may participate in the examination. It is conducted in partnership with ICSI. Under The Ministry of Corporate Affairs.
SOF International Social Studies Olympiad (ISSO) is a single level exam. Students from classes III-X may participate in the examination.
Former
National Cyber Olympiad Level-2 (NCO) was formerly a two-level exam, but it has been converted to a single level exam since 2017–18.
International Sports Knowledge Olympiad (ISKO) was conducted only once, during the 2016–2017 session. It was a single level exam conducted in partnership with Star Sports in which students of classes 1 to 10 could participate.
Eligibility and pattern
Students from classes 1 through 12 can participate. The exams consist of 35 multiple choice questions of 40 marks for classes I to IV, and 50 multiple choice questions for classes V to XII of 60 marks, to be answered in one hour. Five questions that are part of the 'Achievers' section' carry three marks (and for primary classes excluding 5th, it has only two marks) each whereas the remaining questions carry one mark each. Students are required to mark their answers on an OMR sheet. Results are announced for every student and they include the student's international rank, regional rank (since 2020-21), zonal rank, city rank and school rank and winners are awarded with cash prizes, medals, trophi
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https://en.wikipedia.org/wiki/Channel%20surface
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In geometry and topology, a channel or canal surface is a surface formed as the envelope of a family of spheres whose centers lie on a space curve, its directrix. If the radii of the generating spheres are constant, the canal surface is called a pipe surface. Simple examples are:
right circular cylinder (pipe surface, directrix is a line, the axis of the cylinder)
torus (pipe surface, directrix is a circle),
right circular cone (canal surface, directrix is a line (the axis), radii of the spheres not constant),
surface of revolution (canal surface, directrix is a line),
Canal surfaces play an essential role in descriptive geometry, because in case of an orthographic projection its contour curve can be drawn as the envelope of circles.
In technical area canal surfaces can be used for blending surfaces smoothly.
Envelope of a pencil of implicit surfaces
Given the pencil of implicit surfaces
,
two neighboring surfaces and
intersect in a curve that fulfills the equations
and .
For the limit one gets
.
The last equation is the reason for the following definition.
Let be a 1-parameter pencil of regular implicit surfaces ( being at least twice continuously differentiable). The surface defined by the two equations
is the envelope of the given pencil of surfaces.
Canal surface
Let be a regular space curve and a -function with and . The last condition means that the curvature of the curve is less than that of the corresponding sphere.
The envelope of the 1-parameter pencil of spheres
is called a canal surface and its directrix. If the radii are constant, it is called a pipe surface.
Parametric representation of a canal surface
The envelope condition
of the canal surface above is for any value of the equation of a plane, which is orthogonal to the tangent
of the directrix. Hence the envelope is a collection of circles.
This property is the key for a parametric representation of the canal surface. The center of the circle (for parameter ) has the distance
(see condition above)
from the center of the corresponding sphere and its radius is . Hence
where the vectors and the tangent vector form an orthonormal basis, is a parametric representation of the canal surface.
For one gets the parametric representation of a pipe surface:
Examples
a) The first picture shows a canal surface with
the helix as directrix and
the radius function .
The choice for is the following:
.
b) For the second picture the radius is constant:, i. e. the canal surface is a pipe surface.
c) For the 3. picture the pipe surface b) has parameter .
d) The 4. picture shows a pipe knot. Its directrix is a curve on a torus
e) The 5. picture shows a Dupin cyclide (canal surface).
References
External links
M. Peternell and H. Pottmann: Computing Rational Parametrizations of Canal Surfaces
Surfaces
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https://en.wikipedia.org/wiki/Intercept%20theorem
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The intercept theorem, also known as Thales's theorem, basic proportionality theorem or side splitter theorem is an important theorem in elementary geometry about the ratios of various line segments that are created if two rays with a common starting point are intercepted by a pair of parallels. It is equivalent to the theorem about ratios in similar triangles. It is traditionally attributed to Greek mathematician Thales. It was known to the ancient Babylonians and Egyptians, although its first known proof appears in Euclid's Elements.
Formulation of the theorem
Suppose S is the common starting point of two rays and A, B are the intersections of the first ray with the two parallels, such that B is further away from S than A, and similarly C, D are the intersections of the second ray with the two parallels such that D is further away from S than C. In this configuration the following statements hold:
The ratio of any two segments on the first ray equals the ratio of the according segments on the second ray: , ,
The ratio of the two segments on the same ray starting at S equals the ratio of the segments on the parallels:
The converse of the first statement is true as well, i.e. if the two rays are intercepted by two arbitrary lines and holds then the two intercepting lines are parallel. However, the converse of the second statement is not true (see graphic for a counterexample).
Extensions and conclusions
The first two statements remain true if the two rays get replaced by two lines intersecting in . In this case there are two scenarios with regard to , either it lies between the 2 parallels (X figure) or it does not (V figure). If is not located between the two parallels, the original theorem applies directly. If lies between the two parallels, then a reflection of and at yields V figure with identical measures for which the original theorem now applies. The third statement (converse) however does not remain true for lines.
If there are more than two rays starting at or more than two lines intersecting at , then each parallel contains more than one line segment and the ratio of two line segments on one parallel equals the ratio of the according line segments on the other parallek. For instance if there's a third ray starting at and intersecting the parallels in and , such that is further away from than , then the following equalities holds:
,
For the second equation the converse is true as well, that is if the 3 rays are intercepted by two lines and the ratios of the according line segments on each line are equal, then those 2 lines must be parallel.
Related concepts
Similarity and similar triangles
The intercept theorem is closely related to similarity. It is equivalent to the concept of similar triangles, i.e. it can be used to prove the properties of similar triangles and similar triangles can be used to prove the intercept theorem. By matching identical angles you can always place two similar triangles in one
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https://en.wikipedia.org/wiki/Skew%20lattice
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In abstract algebra, a skew lattice is an algebraic structure that is a non-commutative generalization of a lattice. While the term skew lattice can be used to refer to any non-commutative generalization of a lattice, since 1989 it has been used primarily as follows.
Definition
A skew lattice is a set S equipped with two associative, idempotent binary operations and , called meet and join, that validate the following dual pair of absorption laws
,
.
Given that and are associative and idempotent, these identities are equivalent to validating the following dual pair of statements:
if ,
if .
Historical background
For over 60 years, noncommutative variations of lattices have been studied with differing motivations. For some the motivation has been an interest in the conceptual boundaries of lattice theory; for others it was a search for noncommutative forms of logic and Boolean algebra; and for others it has been the behavior of idempotents in rings. A noncommutative lattice, generally speaking, is an algebra where and are associative, idempotent binary operations connected by absorption identities guaranteeing that in some way dualizes . The precise identities chosen depends upon the underlying motivation, with differing choices producing distinct varieties of algebras.
Pascual Jordan, motivated by questions in quantum logic, initiated a study of noncommutative lattices in his 1949 paper, Über Nichtkommutative Verbände, choosing the absorption identities
He referred to those algebras satisfying them as Schrägverbände. By varying or augmenting these identities, Jordan and others obtained a number of varieties of noncommutative lattices.
Beginning with Jonathan Leech's 1989 paper, Skew lattices in rings, skew lattices as defined above have been the primary objects of study. This was aided by previous results about bands. This was especially the case for many of the basic properties.
Basic properties
Natural partial order and natural quasiorder
In a skew lattice , the natural partial order is defined by if , or dually, . The natural preorder on is given by if or dually . While and agree on lattices, properly refines in the noncommutative case. The induced natural equivalence is defined by if , that is,
and or dually, and . The blocks of the partition are
lattice ordered by if and exist such that . This permits us to draw Hasse diagrams of skew lattices such as the following pair:
E.g., in the diagram on the left above, that and are related is expressed by the dashed
segment. The slanted lines reveal the natural partial order between elements of the distinct -classes. The elements , and form the singleton -classes.
Rectangular Skew Lattices
Skew lattices consisting of a single -class are called rectangular. They are characterized by the equivalent identities: , and . Rectangular skew lattices are isomorphic to skew lattices having the following construction (and conversely): given nonempty
sets and , o
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https://en.wikipedia.org/wiki/Cohomological%20dimension
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In abstract algebra, cohomological dimension is an invariant of a group which measures the homological complexity of its representations. It has important applications in geometric group theory, topology, and algebraic number theory.
Cohomological dimension of a group
As most cohomological invariants, the cohomological dimension involves a choice of a "ring of coefficients" R, with a prominent special case given by R = Z, the ring of integers. Let G be a discrete group, R a non-zero ring with a unit, and RG the group ring. The group G has cohomological dimension less than or equal to n, denoted cdR(G) ≤ n, if the trivial RG-module R has a projective resolution of length n, i.e. there are projective RG-modules P0, ..., Pn and RG-module homomorphisms dk: PkPk − 1 (k = 1, ..., n) and d0: P0R, such that the image of dk coincides with the kernel of dk − 1 for k = 1, ..., n and the kernel of dn is trivial.
Equivalently, the cohomological dimension is less than or equal to n if for an arbitrary RG-module M, the cohomology of G with coefficients in M vanishes in degrees k > n, that is, Hk(G,M) = 0 whenever k > n. The p-cohomological dimension for prime p is similarly defined in terms of the p-torsion groups Hk(G,M){p}.
The smallest n such that the cohomological dimension of G is less than or equal to n is the cohomological dimension of G (with coefficients R), which is denoted .
A free resolution of can be obtained from a free action of the group G on a contractible topological space X. In particular, if X is a contractible CW complex of dimension n with a free action of a discrete group G that permutes the cells, then .
Examples
In the first group of examples, let the ring R of coefficients be .
A free group has cohomological dimension one. As shown by John Stallings (for finitely generated group) and Richard Swan (in full generality), this property characterizes free groups. This result is known as the Stallings–Swan theorem. The Stallings-Swan theorem for a group G says that G is free if and only if every extension by G with abelian kernel is split.
The fundamental group of a compact, connected, orientable Riemann surface other than the sphere has cohomological dimension two.
More generally, the fundamental group of a closed, connected, orientable aspherical manifold of dimension n has cohomological dimension n. In particular, the fundamental group of a closed orientable hyperbolic n-manifold has cohomological dimension n.
Nontrivial finite groups have infinite cohomological dimension over . More generally, the same is true for groups with nontrivial torsion.
Now consider the case of a general ring R.
A group G has cohomological dimension 0 if and only if its group ring RG is semisimple. Thus a finite group has cohomological dimension 0 if and only if its order (or, equivalently, the orders of its elements) is invertible in R.
Generalizing the Stallings–Swan theorem for , Martin Dunwoody proved that a group has cohomological dimensio
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https://en.wikipedia.org/wiki/Trimean
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In statistics the trimean (TM), or Tukey's trimean, is a measure of a probability distribution's location defined as a weighted average of the distribution's median and its two quartiles:
This is equivalent to the average of the median and the midhinge:
The foundations of the trimean were part of Arthur Bowley's teachings, and later popularized by statistician John Tukey in his 1977 book which has given its name to a set of techniques called exploratory data analysis.
Like the median and the midhinge, but unlike the sample mean, it is a statistically resistant L-estimator with a breakdown point of 25%. This beneficial property has been described as follows:
Efficiency
Despite its simplicity, the trimean is a remarkably efficient estimator of population mean. More precisely, for a large data set (over 100 points) from a symmetric population, the average of the 20th, 50th, and 80th percentile is the most efficient 3-point L-estimator, with 88% efficiency. For context, the best single point estimate by L-estimators is the median, with an efficiency of 64% or better (for all n), while using two points (for a large data set of over 100 points from a symmetric population), the most efficient estimate is the 29% midsummary (mean of 29th and 71st percentiles), which has an efficiency of about 81%. Using quartiles, these optimal estimators can be approximated by the midhinge and the trimean. Using further points yield higher efficiency, though it is notable that only three points are needed for very high efficiency.
See also
Truncated mean
Interquartile mean
References
External links
Trimean at MathWorld
Summary statistics
Means
Robust statistics
Exploratory data analysis
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https://en.wikipedia.org/wiki/Ahe%20Airport
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Ahe Airport is an airport on Ahe (Tenukupara), an atoll in French Polynesia .
Airlines and destinations
Statistics
See also
List of airports in French Polynesia
References
External links
Atoll list (in French)
Classification of the French Polynesian atolls by Salvat (1985)
Airports in French Polynesia
Atolls of the Tuamotus
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https://en.wikipedia.org/wiki/Arutua%20Airport
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Arutua Airport is an airport on Arutua atoll in French Polynesia . The airport is 13 km north of the village of Rautini.
Airlines and destinations
Statistics
See also
List of airports in French Polynesia
References
External links
Airports in French Polynesia
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https://en.wikipedia.org/wiki/Makemo%20Airport
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Makemo Airport is an airport on Makemo in French Polynesia. The airport is WNW of the village of Pouheva.
Airlines and destinations
Passenger
Statistics
References
External links
Airports in French Polynesia
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https://en.wikipedia.org/wiki/Rimatara%20Airport
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Rimatara Airport is an airport on Rimatara in French Polynesia . The airport was built in 2006.
The airport was renovated in 2018.
Airlines and destinations
Statistics
References
Airports in French Polynesia
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https://en.wikipedia.org/wiki/Rurutu%20Airport
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Rurutu Airport is an airport on Rurutu in French Polynesia . The airport is located northeast of Moerai. The airport was built in 1977.
Airlines and destinations
Passenger
Statistics
References
External links
Airports in French Polynesia
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https://en.wikipedia.org/wiki/Tubuai%20%E2%80%93%20Mataura%20Airport
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Tubuai – Mataura Airport is an airport on Tubuai in French Polynesia. The airport is southwest of the village of Mataura.
Airlines and destinations
Passenger
Statistics
References
Airports in French Polynesia
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https://en.wikipedia.org/wiki/Zhiming%20Liu%20%28computer%20scientist%29
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Zhiming Liu (, born 10 October 1961, Hebei, China) is a computer scientist. He studied mathematics in Luoyang, Henan in China and obtained his first degree in 1982. He holds a master's degree in Computer Science from the Institute of Software of the Chinese Academy of Sciences (1988), and a PhD degree from the University of Warwick (1991). His PhD thesis was on Fault-Tolerant Programming by Transformations.
After his PhD, Zhiming Liu worked as a guest scientist at the Department of Computer Science, Technical University of Denmark, Lyngby in 1991–1992. Then he returned to the University of Warwick and worked as a postdoctoral research fellow on formal techniques in real-time and fault-tolerant systems till October 1994 when he became a university lecturer in computer science at the University of Leicester (UK). He worked at UNU-IIST during 2002–2013 at UNU-IIST as research fellow and senior research fellow. He joined Birmingham City University (UK) in October 2013 as the Professor of Software Engineering. In 2016, he moved to a new professorial post at Southwest University in Chongqing, China, with funding through the Thousand Talents Program.
Zhiming Liu's main research interest is in the areas of formal methods of computer systems design, including real-time systems, fault-tolerant systems, object-oriented and component-based systems. His research results have been published in mainstream journals and conferences. His joint work with Mathai Joseph work on fault tolerance gives a formal model that defines precisely the notions of fault, error, failure and fault-tolerance, and their relations. It also gives the properties that models of fault-affected programs and fault-tolerant programs in terms of model transformations. They proposed a design process for fault-tolerant systems from requirement specifications and analysis, fault environment identification and analysis, specification of fault-affected design and verification of fault-tolerance for satisfaction of the requirements specification. In collaboration with Zhou Chaochen and Anders Ravn, et al., he also developed a Probabilistic Duration Calculus for system dependability analysis. His recent work with He Jifeng and Xiaoshan Li on the rCOS theory of semantics and refinement of object-oriented and component-based design is being developed into a method with tool support for component-based and model-driven software development.
Zhiming Liu is the founder of International Colloquium on Theoretical Aspects of Computing (ICTAC), the International Symposium on Formal Aspects of Component Software (FACS), and International Symposium on Foundations of Health Information Engineering and systems (FHIES). He has served as a PC chair for a number of conferences and PC members of a number of conferences. He has also edited a number of books.
Zhiming Liu is married to Hong Zhao with two sons, Kim Chang Liu and Edward Tanze Liu.
References
External links
Southwest University home page
UNU-IIST
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https://en.wikipedia.org/wiki/Bassim%20Abbas
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Bassim Abbas Gatea Al-Ogaili (; born 1 July 1982) is an Iraqi former professional footballer who last played for Al-Shorta.
Career statistics
International
Scores and results list Iraq's goal tally first.
Honours
Al-Talaba
Iraqi Premier League: 2001–02
Iraq FA Cup: 2001–02, 2002–03
Iraqi Super Cup: 2002
Umm-Salal
Emir of Qatar Cup: 2007–08
Iraq
West Asian Football Federation Championship: 2002
AFC Asian Cup: 2007
Individual
AFC Asian Cup Best Defender: 2007
Lebanese Premier League Team of the Season: 2006–07
References
External links
1982 births
Living people
Footballers from Baghdad
Expatriate men's footballers in Lebanon
Iraqi expatriate men's footballers
Iraqi men's footballers
Al-Talaba SC players
Iraq men's international footballers
Footballers at the 2004 Summer Olympics
Olympic footballers for Iraq
2004 AFC Asian Cup players
2007 AFC Asian Cup players
2009 FIFA Confederations Cup players
2011 AFC Asian Cup players
AFC Asian Cup-winning players
Expatriate men's footballers in Iran
Expatriate men's footballers in Turkey
Expatriate men's footballers in Qatar
Nejmeh SC players
Umm Salal SC players
Al-Arabi SC (Qatar) players
Diyarbakırspor footballers
Konyaspor footballers
Esteghlal Ahvaz F.C. players
Qatar Stars League players
Amanat Baghdad SC players
Iraqi expatriate sportspeople in Iran
Iraqi expatriate sportspeople in Turkey
Iraqi expatriate sportspeople in Qatar
Men's association football fullbacks
Al-Shorta SC players
Iraqi expatriate sportspeople in Lebanon
Lebanese Premier League players
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https://en.wikipedia.org/wiki/Stable%20normal%20bundle
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In surgery theory, a branch of mathematics, the stable normal bundle of a differentiable manifold is an invariant which encodes the stable normal (dually, tangential) data. There are analogs for generalizations of manifold, notably PL-manifolds and topological manifolds. There is also an analogue in homotopy theory for Poincaré spaces, the Spivak spherical fibration, named after Michael Spivak.
Construction via embeddings
Given an embedding of a manifold in Euclidean space (provided by the theorem of Hassler Whitney), it has a normal bundle. The embedding is not unique, but for high dimension of the Euclidean space it is unique up to isotopy, thus the (class of the) bundle is unique, and called the stable normal bundle.
This construction works for any Poincaré space X: a finite CW-complex admits a stably unique (up to homotopy) embedding in Euclidean space, via general position, and this embedding yields a spherical fibration over X. For more restricted spaces (notably PL-manifolds and topological manifolds), one gets stronger data.
Details
Two embeddings are isotopic if they are homotopic
through embeddings. Given a manifold or other suitable space X, with two embeddings into Euclidean space these will not in general be isotopic, or even maps into the same space ( need not equal ). However, one can embed these into a larger space by letting the last coordinates be 0:
.
This process of adjoining trivial copies of Euclidean space is called stabilization.
One can thus arrange for any two embeddings into Euclidean space to map into the same Euclidean space (taking ), and, further, if is sufficiently large, these embeddings are isotopic, which is a theorem.
Thus there is a unique stable isotopy class of embedding: it is not a particular embedding (as there are many embeddings), nor an isotopy class (as the target space is not fixed: it is just "a sufficiently large Euclidean space"), but rather a stable isotopy class of maps. The normal bundle associated with this (stable class of) embeddings is then the stable normal bundle.
One can replace this stable isotopy class with an actual isotopy class by fixing the target space, either by using Hilbert space as the target space, or (for a fixed dimension of manifold ) using a fixed sufficiently large, as N depends only on n, not the manifold in question.
More abstractly, rather than stabilizing the embedding, one can take any embedding, and then take a vector bundle direct sum with a sufficient number of trivial line bundles; this corresponds exactly to the normal bundle of the stabilized embedding.
Construction via classifying spaces
An n-manifold M has a tangent bundle, which has a classifying map (up to homotopy)
Composing with the inclusion yields (the homotopy class of a classifying map of) the stable tangent bundle. The normal bundle of an embedding ( large) is an inverse for , such that the Whitney sum is trivial. The homotopy class of the composite
is independent of the choice
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https://en.wikipedia.org/wiki/Reform%20mathematics
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Reform mathematics is an approach to mathematics education, particularly in North America. It is based on principles explained in 1989 by the National Council of Teachers of Mathematics (NCTM). The NCTM document Curriculum and Evaluation Standards for School Mathematics (CESSM) set forth a vision for K–12 (ages 5–18) mathematics education in the United States and Canada. The CESSM recommendations were adopted by many local- and federal-level education agencies during the 1990s. In 2000, the NCTM revised its CESSM with the publication of Principles and Standards for School Mathematics (PSSM). Like those in the first publication, the updated recommendations became the basis for many states' mathematics standards, and the method in textbooks developed by many federally-funded projects. The CESSM de-emphasised manual arithmetic in favor of students developing their own conceptual thinking and problem solving. The PSSM presents a more balanced view, but still has the same emphases.
Mathematics instruction in this style has been labeled standards-based mathematics or reform mathematics.
Principles and standards
Mathematics education reform built up momentum in the early 1980s, as educators reacted to the "new math" of the 1960s and 1970s. The work of Piaget and other developmental psychologists had shifted the focus of mathematics educators from mathematics content to how children best learn mathematics. The National Council of Teachers of Mathematics summarized the state of current research with the publication of Curriculum and Evaluation Standards in 1989 and Principles and Standards for School Mathematics in 2000, bringing definition to the reform movement in North America.
Reform mathematics curricula challenge students to make sense of new mathematical ideas through explorations and projects, often in real-world contexts. Reform texts emphasize written and verbal communication, working in cooperative groups, and making connections between concepts and between representations. In contrast, "traditional" textbooks emphasize procedural mathematics and provide step-by-step examples with skill-building exercises.
Traditional mathematics focuses on teaching algorithms that will lead to the correct answer of a particular problem. Because of this focus on application of algorithms, the student of traditional math must apply the specific method that is being taught. Reform mathematics de-emphasizes this algorithmic dependence. Instead of leading students to find the exact answers to specific problems, reform educators focus students on the overall process which leads to an answer. Students' occasional errors are deemed less important than their understanding of an overall thought process. Research has shown that children make fewer mistakes with calculations and remember algorithms longer when they understand the concepts underlying the methods they use. In general, children in reform classes perform at least as well as children in traditional cl
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https://en.wikipedia.org/wiki/Structure%20theorem%20for%20finitely%20generated%20modules%20over%20a%20principal%20ideal%20domain
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In mathematics, in the field of abstract algebra, the structure theorem for finitely generated modules over a principal ideal domain is a generalization of the fundamental theorem of finitely generated abelian groups and roughly states that finitely generated modules over a principal ideal domain (PID) can be uniquely decomposed in much the same way that integers have a prime factorization. The result provides a simple framework to understand various canonical form results for square matrices over fields.
Statement
When a vector space over a field F has a finite generating set, then one may extract from it a basis consisting of a finite number n of vectors, and the space is therefore isomorphic to Fn. The corresponding statement with the F generalized to a principal ideal domain R is no longer true, since a basis for a finitely generated module over R might not exist. However such a module is still isomorphic to a quotient of some module Rn with n finite (to see this it suffices to construct the morphism that sends the elements of the canonical basis of Rn to the generators of the module, and take the quotient by its kernel.) By changing the choice of generating set, one can in fact describe the module as the quotient of some Rn by a particularly simple submodule, and this is the structure theorem.
The structure theorem for finitely generated modules over a principal ideal domain usually appears in the following two forms.
Invariant factor decomposition
For every finitely generated module over a principal ideal domain , there is a unique decreasing sequence of proper ideals such that is isomorphic to the sum of cyclic modules:
The generators of the ideals are unique up to multiplication by a unit, and are called invariant factors of M. Since the ideals should be proper, these factors must not themselves be invertible (this avoids trivial factors in the sum), and the inclusion of the ideals means one has divisibility . The free part is visible in the part of the decomposition corresponding to factors . Such factors, if any, occur at the end of the sequence.
While the direct sum is uniquely determined by , the isomorphism giving the decomposition itself is not unique in general. For instance if is actually a field, then all occurring ideals must be zero, and one obtains the decomposition of a finite dimensional vector space into a direct sum of one-dimensional subspaces; the number of such factors is fixed, namely the dimension of the space, but there is a lot of freedom for choosing the subspaces themselves (if ).
The nonzero elements, together with the number of which are zero, form a complete set of invariants for the module. Explicitly, this means that any two modules sharing the same set of invariants are necessarily isomorphic.
Some prefer to write the free part of M separately:
where the visible are nonzero, and f is the number of 's in the original sequence which are 0.
Primary decomposition
Every finitely generated modu
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https://en.wikipedia.org/wiki/Jean-Louis%20Calandrini
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Jean-Louis Calandrini (August 30, 1703 – December 29, 1758) was a Genevan scientist. He was a professor of mathematics and philosophy. He was the author of some studies on the aurora borealis, comets, and the effects of lightning, as well as of an important but unpublished work on flat and spherical trigonometry. He also wrote a commentary on the Principia of Isaac Newton (published in Geneva, 1739–42), for which he wrote approximately one hundred footnotes.
He was also known as a botanist. The genus Calandrinia was named after him.
His father was a pastor, also named Jean-Louis, and his mother was Michée Du Pan. He was the grandnephew of Bénédict Calandrini (de) (fr). In 1729, he married Renée Lullin. At the Academy of Geneva, he obtained his thesis in physics (1722). In 1724, Calandrini was named mathematics professor at the same time as Gabriel Cramer, but he first undertook a three-year journey to France and England. He was appointed professor of philosophy from 1734 to 1750. He also played an active role on the political scene of Geneva.
References
Mathematicians from the Republic of Geneva
Politicians from the Republic of Geneva
18th-century botanists from the Republic of Geneva
18th-century mathematicians
Philosophy academics
1703 births
1758 deaths
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https://en.wikipedia.org/wiki/Complete%20set%20of%20invariants
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In mathematics, a complete set of invariants for a classification problem is a collection of maps
(where is the collection of objects being classified, up to some equivalence relation , and the are some sets), such that if and only if for all . In words, such that two objects are equivalent if and only if all invariants are equal.
Symbolically, a complete set of invariants is a collection of maps such that
is injective.
As invariants are, by definition, equal on equivalent objects, equality of invariants is a necessary condition for equivalence; a complete set of invariants is a set such that equality of these is also sufficient for equivalence. In the context of a group action, this may be stated as: invariants are functions of coinvariants (equivalence classes, orbits), and a complete set of invariants characterizes the coinvariants (is a set of defining equations for the coinvariants).
Examples
In the classification of two-dimensional closed manifolds, Euler characteristic (or genus) and orientability are a complete set of invariants.
Jordan normal form of a matrix is a complete invariant for matrices up to conjugation, but eigenvalues (with multiplicities) are not.
Realizability of invariants
A complete set of invariants does not immediately yield a classification theorem: not all combinations of invariants may be realized. Symbolically, one must also determine the image of
References
Mathematical terminology
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https://en.wikipedia.org/wiki/Orientation%20character
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In algebraic topology, a branch of mathematics, an orientation character on a group is a group homomorphism where:
This notion is of particular significance in surgery theory.
Motivation
Given a manifold M, one takes (the fundamental group), and then sends an element of to if and only if the class it represents is orientation-reversing.
This map is trivial if and only if M is orientable.
The orientation character is an algebraic structure on the fundamental group of a manifold, which captures which loops are orientation reversing and which are orientation preserving.
Twisted group algebra
The orientation character defines a twisted involution (*-ring structure) on the group ring , by (i.e., , accordingly as is orientation preserving or reversing). This is denoted .
Examples
In real projective spaces, the orientation character evaluates trivially on loops if the dimension is odd, and assigns -1 to noncontractible loops in even dimension.
Properties
The orientation character is either trivial or has kernel an index 2 subgroup, which determines the map completely.
See also
Characteristic class
Local system
Twisted Poincaré duality
References
External links
Orientation character at the Manifold Atlas
Geometric topology
Group theory
Morphisms
Surgery theory
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https://en.wikipedia.org/wiki/%28%E2%88%922%2C3%2C7%29%20pretzel%20knot
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In geometric topology, a branch of mathematics, the (−2, 3, 7) pretzel knot, sometimes called the Fintushel–Stern knot (after Ron Fintushel and Ronald J. Stern), is an important example of a pretzel knot which exhibits various interesting phenomena under three-dimensional and four-dimensional surgery constructions.
Mathematical properties
The (−2, 3, 7) pretzel knot has 7 exceptional slopes, Dehn surgery slopes which give non-hyperbolic 3-manifolds. Among the enumerated knots, the only other hyperbolic knot with 7 or more is the figure-eight knot, which has 10. All other hyperbolic knots are conjectured to have at most 6 exceptional slopes.
References
Further reading
Kirby, R., (1978). "Problems in low dimensional topology", Proceedings of Symposia in Pure Math., volume 32, 272-312. (see problem 1.77, due to Gordon, for exceptional slopes)
External links
3-manifolds
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https://en.wikipedia.org/wiki/Resolution%20of%20singularities
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In algebraic geometry, the problem of resolution of singularities asks whether every algebraic variety V has a resolution, a non-singular variety W with a proper birational map W→V. For varieties over fields of characteristic 0 this was proved in Hironaka (1964), while for varieties over fields of characteristic p it is an open problem in dimensions at least 4.
Definitions
Originally the problem of resolution of singularities was to find a nonsingular model for the function field of a variety X, in other words a complete non-singular variety X′ with the same function field. In practice it is more convenient to ask for a different condition as follows: a variety X has a resolution of singularities if we can find a non-singular variety X′ and a proper birational map from X′ to X. The condition that the map is proper is needed to exclude trivial solutions, such as taking X′ to be the subvariety of non-singular points of X.
More generally, it is often useful to resolve the singularities of a variety X embedded into a larger variety W. Suppose we have a closed embedding of X into a regular variety W. A strong desingularization of X is given by a proper birational morphism from a regular variety W′ to W subject to some of the following conditions (the exact choice of conditions depends on the author):
The strict transform X′ of X is regular, and transverse to the exceptional locus of the resolution morphism (so in particular it resolves the singularities of X).
The map from the strict transform of X′ to X is an isomorphism away from the singular points of X.
W′ is constructed by repeatedly blowing up regular closed subvarieties of W or more strongly regular subvarieties of X, transverse to the exceptional locus of the previous blowings up.
The construction of W′ is functorial for smooth morphisms to W and embeddings of W into a larger variety. (It cannot be made functorial for all (not necessarily smooth) morphisms in any reasonable way.)
The morphism from X′ to X does not depend on the embedding of X in W. Or in general, the sequence of blowings up is functorial with respect to smooth morphisms.
Hironaka showed that there is a strong desingularization satisfying the first three conditions above whenever X is defined over a field of characteristic 0, and his construction was improved by several authors (see below) so that it satisfies all conditions above.
Resolution of singularities of curves
Every algebraic curve has a unique nonsingular projective model, which means that all resolution methods are essentially the same because they all construct this model. In higher dimensions this is no longer true: varieties can have many different nonsingular projective models.
lists about 20 ways of proving resolution of singularities of curves.
Newton's method
Resolution of singularities of curves was essentially first proved by , who showed the existence of Puiseux series for a curve from which resolution follows easily.
Riemann's method
Ri
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https://en.wikipedia.org/wiki/Superperfect%20group
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In mathematics, in the realm of group theory, a group is said to be superperfect when its first two homology groups are trivial: H1(G, Z) = H2(G, Z) = 0. This is stronger than a perfect group, which is one whose first homology group vanishes. In more classical terms, a superperfect group is one whose abelianization and Schur multiplier both vanish; abelianization equals the first homology, while the Schur multiplier equals the second homology.
Definition
The first homology group of a group is the abelianization of the group itself, since the homology of a group G is the homology of any Eilenberg–MacLane space of type K(G, 1); the fundamental group of a K(G, 1) is G, and the first homology of K(G, 1) is then abelianization of its fundamental group. Thus, if a group is superperfect, then it is perfect.
A finite perfect group is superperfect if and only if it is its own universal central extension (UCE), as the second homology group of a perfect group parametrizes central extensions.
Examples
For example, if G is the fundamental group of a homology sphere, then G is superperfect. The smallest finite, non-trivial superperfect group is the binary icosahedral group (the fundamental group of the Poincaré homology sphere).
The alternating group A5 is perfect but not superperfect: it has a non-trivial central extension, the binary icosahedral group (which is in fact its UCE) is superperfect. More generally, the projective special linear groups PSL(n, q) are simple (hence perfect) except for PSL(2, 2) and PSL(2, 3), but not superperfect, with the special linear groups SL(n,q) as central extensions. This family includes the binary icosahedral group (thought of as SL(2, 5)) as UCE of A5 (thought of as PSL(2, 5)).
Every acyclic group is superperfect, but the converse is not true: the binary icosahedral group is superperfect, but not acyclic.
References
A. Jon Berrick and Jonathan A. Hillman, "Perfect and acyclic subgroups of finitely presentable groups", Journal of the London Mathematical Society (2) 68 (2003), no. 3, 683--698.
Properties of groups
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https://en.wikipedia.org/wiki/Eliran%20Elkayam
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Eliran Elkayam (; born October 30, 1976) is an Israeli football player.
External links
Profile and biography of Eliran Elkayam on Maccabi Haifa's official website
Profile and statistics of Eliran Elkayam on One.co.il
1976 births
Living people
Israeli Jews
Israeli men's footballers
Men's association football defenders
Maccabi Haifa F.C. players
Maccabi Petah Tikva F.C. players
Hapoel Haifa F.C. players
Hapoel Petah Tikva F.C. players
Hapoel Nof HaGalil F.C. players
Bnei Yehuda Tel Aviv F.C. players
Beitar Tel Aviv Bat Yam F.C. players
Maccabi Kafr Kanna F.C. players
Maccabi Ironi Kiryat Ata F.C. players
Liga Leumit players
Israeli Premier League players
Footballers from Kiryat Ata
Israeli people of Moroccan-Jewish descent
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https://en.wikipedia.org/wiki/Undefined%20%28mathematics%29
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In mathematics, the term undefined is often used to refer to an expression which is not assigned an interpretation or a value (such as an indeterminate form, which has the possibility of assuming different values). The term can take on several different meanings depending on the context. For example:
In various branches of mathematics, certain concepts are introduced as primitive notions (e.g., the terms "point", "line" and "plane" in geometry). As these terms are not defined in terms of other concepts, they may be referred to as "undefined terms".
A function is said to be "undefined" at points outside of its domainfor example, the real-valued function is undefined for negative (i.e., it assigns no value to negative arguments).
In algebra, some arithmetic operations may not assign a meaning to certain values of its operands (e.g., division by zero). In which case, the expressions involving such operands are termed "undefined".
In square roots, square roots of any negative number are undefined because you can’t multiply 2 of the same positive nor negative number to get a negative number, like √-4, √-9, √-16 etc. (ex: 6x6=36 and -6x-6=36).
Undefined terms
In ancient times, geometers attempted to define every term. For example, Euclid defined a point as "that which has no part". In modern times, mathematicians recognize that attempting to define every word inevitably leads to circular definitions, and therefore leave some terms (such as "point") undefined (see primitive notion for more).
This more abstract approach allows for fruitful generalizations. In topology, a topological space may be defined as a set of points endowed with certain properties, but in the general setting, the nature of these "points" is left entirely undefined. Likewise, in category theory, a category consists of "objects" and "arrows", which are again primitive, undefined terms. This allows such abstract mathematical theories to be applied to very diverse concrete situations.
In arithmetic
The expression is undefined in arithmetic, as explained in division by zero (the expression is used in calculus to represent an indeterminate form).
Mathematicians have different opinions as to whether should be defined to equal 1, or be left undefined.
Values for which functions are undefined
The set of numbers for which a function is defined is called the domain of the function. If a number is not in the domain of a function, the function is said to be "undefined" for that number. Two common examples are , which is undefined for , and , which is undefined (in the real number system) for negative .
In trigonometry
In trigonometry, for all , the functions and are undefined for all , while the functions and are undefined for all .
In complex analysis
In complex analysis, a point where a holomorphic function is undefined is called a singularity. One distinguishes between removable singularities (i.e., the function can be extended holomorphically to ), poles (i.e., the f
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https://en.wikipedia.org/wiki/Paul%20Glendinning
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Paul Glendinning is a professor of Applied Mathematics, in the School of Mathematics at the University of Manchester who is known for his work on dynamical systems, specifically models of the time-evolution of complex mathematical or physical processes. His main areas of research are bifurcation theory (particularly global bifurcations); synchronization and blowout bifurcations; low-dimensional maps; and quasi-periodically forced systems.
Education
He gained his PhD from King's College, Cambridge in 1985 with a thesis entitled Homoclinic Bifurcations under the supervision of Nigel Weiss.
Career and research
After postdoctoral research at the University of Warwick, he returned to Cambridge, with a Junior Research Fellowship at King's. In 1987 he moved to Gonville and Caius College, Cambridge as Director of Studies in Applied Mathematics. In 1992 he won the Adams Prize. In 1996 he was appointed to a chair at Queen Mary and Westfield College, London and then to a chair at the University of Manchester Institute of Science and Technology (UMIST) in 2000.
In 2004 the Victoria University of Manchester and UMIST merged and he was appointed as head of the School of Mathematics formed by the merger of the Mathematics Departments in the former institutions. His term of office as head of school expired in August 2008.
He was Scientific Director of the International Centre for Mathematical Sciences in Edinburgh from 2016 to 2021. In 2021 he was elected a Fellow of the Royal Society of Edinburgh.
He is on the Editorial Board of the European Journal of Applied Mathematics and the journal Dynamical Systems.
Paul was appointed president of the Institute of Mathematics and its Applications in January 2022
Personal life
Glendinning lives in Marsden, West Yorkshire as of 2012. He is the son of the academic Nigel Glendinning and the writer and broadcaster Victoria Glendinning. His brother is the philosopher Simon Glendinning.
References
20th-century British mathematicians
21st-century British mathematicians
Living people
Academics of the University of Manchester
Academics of Queen Mary University of London
Alumni of King's College, Cambridge
Fellows of Gonville and Caius College, Cambridge
Year of birth missing (living people)
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https://en.wikipedia.org/wiki/Polynomial%20greatest%20common%20divisor
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In algebra, the greatest common divisor (frequently abbreviated as GCD) of two polynomials is a polynomial, of the highest possible degree, that is a factor of both the two original polynomials. This concept is analogous to the greatest common divisor of two integers.
In the important case of univariate polynomials over a field the polynomial GCD may be computed, like for the integer GCD, by the Euclidean algorithm using long division. The polynomial GCD is defined only up to the multiplication by an invertible constant.
The similarity between the integer GCD and the polynomial GCD allows extending to univariate polynomials all the properties that may be deduced from the Euclidean algorithm and Euclidean division. Moreover, the polynomial GCD has specific properties that make it a fundamental notion in various areas of algebra. Typically, the roots of the GCD of two polynomials are the common roots of the two polynomials, and this provides information on the roots without computing them. For example, the multiple roots of a polynomial are the roots of the GCD of the polynomial and its derivative, and further GCD computations allow computing the square-free factorization of the polynomial, which provides polynomials whose roots are the roots of a given multiplicity of the original polynomial.
The greatest common divisor may be defined and exists, more generally, for multivariate polynomials over a field or the ring of integers, and also over a unique factorization domain. There exist algorithms to compute them as soon as one has a GCD algorithm in the ring of coefficients. These algorithms proceed by a recursion on the number of variables to reduce the problem to a variant of the Euclidean algorithm. They are a fundamental tool in computer algebra, because computer algebra systems use them systematically to simplify fractions. Conversely, most of the modern theory of polynomial GCD has been developed to satisfy the need for efficiency of computer algebra systems.
General definition
Let and be polynomials with coefficients in an integral domain , typically a field or the integers.
A greatest common divisor of and is a polynomial that divides and , and such that every common divisor of and also divides . Every pair of polynomials (not both zero) has a GCD if and only if is a unique factorization domain.
If is a field and and are not both zero, a polynomial is a greatest common divisor if and only if it divides both and , and it has the greatest degree among the polynomials having this property. If , the GCD is 0. However, some authors consider that it is not defined in this case.
The greatest common divisor of and is usually denoted "".
The greatest common divisor is not unique: if is a GCD of and , then the polynomial is another GCD if and only if there is an invertible element of such that
and
.
In other words, the GCD is unique up to the multiplication by an invertible constant.
In the case of the integers, this
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https://en.wikipedia.org/wiki/Split%20graph
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In graph theory, a branch of mathematics, a split graph is a graph in which the vertices can be partitioned into a clique and an independent set. Split graphs were first studied by , and independently introduced by .
A split graph may have more than one partition into a clique and an independent set; for instance, the path is a split graph, the vertices of which can be partitioned in three different ways:
the clique and the independent set
the clique and the independent set
the clique and the independent set
Split graphs can be characterized in terms of their forbidden induced subgraphs: a graph is split if and only if no induced subgraph is a cycle on four or five vertices, or a pair of disjoint edges (the complement of a 4-cycle).
Relation to other graph families
From the definition, split graphs are clearly closed under complementation. Another characterization of split graphs involves complementation: they are chordal graphs the complements of which are also chordal. Just as chordal graphs are the intersection graphs of subtrees of trees, split graphs are the intersection graphs of distinct substars of star graphs. Almost all chordal graphs are split graphs; that is, in the limit as n goes to infinity, the fraction of n-vertex chordal graphs that are split approaches one.
Because chordal graphs are perfect, so are the split graphs. The double split graphs, a family of graphs derived from split graphs by doubling every vertex (so the clique comes to induce an antimatching and the independent set comes to induce a matching), figure prominently as one of five basic classes of perfect graphs from which all others can be formed in the proof by of the Strong Perfect Graph Theorem.
If a graph is both a split graph and an interval graph, then its complement is both a split graph and a comparability graph, and vice versa. The split comparability graphs, and therefore also the split interval graphs, can be characterized in terms of a set of three forbidden induced subgraphs. The split cographs are exactly the threshold graphs. The split permutation graphs are exactly the interval graphs that have interval graph complements;
these are the permutation graphs of skew-merged permutations. Split graphs have cochromatic number 2.
Algorithmic problems
Let be a split graph, partitioned into a clique and an independent set . Then every maximal clique in a split graph is either itself, or the neighborhood of a vertex in . Thus, it is easy to identify the maximum clique, and complementarily the maximum independent set in a split graph. In any split graph, one of the following three possibilities must be true:
There exists a vertex in such that is complete. In this case, is a maximum clique and is a maximum independent set.
There exists a vertex in such that is independent. In this case, is a maximum independent set and is a maximum clique.
is a maximal clique and is a maximal independent set. In this case, has a unique partition
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https://en.wikipedia.org/wiki/1%20%2B%201%20%2B%201%20%2B%201%20%2B%20%E2%8B%AF
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In mathematics, , also written , , or simply , is a divergent series, meaning that its sequence of partial sums does not converge to a limit in the real numbers. The sequence 1 can be thought of as a geometric series with the common ratio 1. Unlike other geometric series with rational ratio (except −1), it converges in neither the real numbers nor in the -adic numbers for some . In the context of the extended real number line
since its sequence of partial sums increases monotonically without bound.
Where the sum of occurs in physical applications, it may sometimes be interpreted by zeta function regularization, as the value at of the Riemann zeta function:
The two formulas given above are not valid at zero however, but the analytic continuation is.
Using this one gets (given that ),
where the power series expansion for about follows because has a simple pole of residue one there. In this sense .
Emilio Elizalde presents a comment from others about the series:
See also
Grandi's series
1 − 2 + 3 − 4 + · · ·
1 + 2 + 3 + 4 + · · ·
1 + 2 + 4 + 8 + · · ·
1 − 2 + 4 − 8 + ⋯
1 − 1 + 2 − 6 + 24 − 120 + · · ·
Harmonic series
Notes
External links
Arithmetic series
Divergent series
Geometric series
1 (number)
Mathematical paradoxes
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https://en.wikipedia.org/wiki/Lubrication%20theory
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In fluid dynamics, lubrication theory describes the flow of fluids (liquids or gases) in a geometry in which one dimension is significantly smaller than the others. An example is the flow above air hockey tables, where the thickness of the air layer beneath the puck is much smaller than the dimensions of the puck itself.
Internal flows are those where the fluid is fully bounded. Internal flow lubrication theory has many industrial applications because of its role in the design of fluid bearings. Here a key goal of lubrication theory is to determine the pressure distribution in the fluid volume, and hence the forces on the bearing components. The working fluid in this case is often termed a lubricant.
Free film lubrication theory is concerned with the case in which one of the surfaces containing the fluid is a free surface. In that case, the position of the free surface is itself unknown, and one goal of lubrication theory is then to determine this. Examples include the flow of a viscous fluid over an inclined plane or over topography. Surface tension may be significant, or even dominant. Issues of wetting and dewetting then arise. For very thin films (thickness less than one micrometre), additional intermolecular forces, such as Van der Waals forces or disjoining forces, may become significant.
Theoretical basis
Mathematically, lubrication theory can be seen as exploiting the disparity between two length scales. The first is the characteristic film thickness, , and the second is a characteristic substrate length scale . The key requirement for lubrication theory is that the ratio is small, that is, .
The Navier–Stokes equations (or Stokes equations, when fluid inertia may be neglected) are expanded in this small parameter, and the leading-order equations are then
where and are coordinates in the direction of the substrate and perpendicular to it respectively. Here is the fluid pressure, and is the fluid velocity component parallel to the substrate; is the fluid viscosity. The equations show, for example, that pressure variations across the gap are small, and that those along the gap are proportional to the fluid viscosity. A more general formulation of the lubrication approximation would include a third dimension, and the resulting differential equation is known as the Reynolds equation.
Further details can be found in the literature or in the textbooks given in the bibliography.
Applications
An important application area is lubrication of machinery components such as fluid bearings and mechanical seals. Coating is another major application area including the preparation of thin films, printing, painting and adhesives.
Biological applications have included studies of red blood cells in narrow capillaries and of liquid flow in the lung and eye.
Notes
References
Aksel, N.; Schörner M. (2018), Films over topography: from creeping flow to linear stability, theory, and experiments, a review, Acta Mech. 229, 1453–1482
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https://en.wikipedia.org/wiki/Mathematics%20and%20Computing%20College
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Mathematics and Computing Colleges were introduced in England in 2002 and Northern Ireland in 2006 as part of the Government's Specialist Schools programme which was designed to raise standards in secondary education. Specialist schools focus on their chosen specialism but must also meet the requirements of the National Curriculum and deliver a broad and balanced education to all their pupils. Mathematics and Computing Colleges must focus on mathematics and either computing or ICT.
Colleges are expected to disseminate good practice and share resources with other schools and the wider community. They often develop active partnerships with local organisations and their feeder primary schools. They also work with local businesses to promote the use of mathematics and computing outside of school.
In 2007 there were 222 schools in England which were designated as specialist Mathematics and Computing Colleges. A further 21 schools were designated in combined specialisms which included mathematics and computing, and 15 had a second specialism in Mathematics and Computing.
The Specialist Schools programme ended in 2011. Since then, schools in England have to either become an academy or apply through the Dedicated Schools Grant if they wish to become a Mathematics and Computing College. As of 2021 there are few Mathematics and Computing Colleges left in the United Kingdom.
References
External links
Vision for Mathematics and Computing Colleges, The Standards Site
Specialist Schools Programme
Computer science education in the United Kingdom
Mathematics education in the United Kingdom
2002 introductions
Specialist schools programme
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https://en.wikipedia.org/wiki/Hartman%E2%80%93Grobman%20theorem
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In mathematics, in the study of dynamical systems, the Hartman–Grobman theorem or linearisation theorem is a theorem about the local behaviour of dynamical systems in the neighbourhood of a hyperbolic equilibrium point. It asserts that linearisation—a natural simplification of the system—is effective in predicting qualitative patterns of behaviour. The theorem owes its name to Philip Hartman and David M. Grobman.
The theorem states that the behaviour of a dynamical system in a domain near a hyperbolic equilibrium point is qualitatively the same as the behaviour of its linearization near this equilibrium point, where hyperbolicity means that no eigenvalue of the linearization has real part equal to zero. Therefore, when dealing with such dynamical systems one can use the simpler linearization of the system to analyse its behaviour around equilibria.
Main theorem
Consider a system evolving in time with state that satisfies the differential equation for some smooth map . Now suppose the map has a hyperbolic equilibrium state : that is, and the Jacobian matrix of at state has no eigenvalue with real part equal to zero. Then there exists a neighbourhood of the equilibrium and a homeomorphism ,
such that and such that in the neighbourhood the flow of is topologically conjugate by the continuous map to the flow of its linearisation .
Even for infinitely differentiable maps , the homeomorphism need not to be smooth, nor even locally Lipschitz. However, it turns out to be Hölder continuous, with an exponent depending on the constant of hyperbolicity of .
The Hartman–Grobman theorem has been extended to infinite-dimensional Banach spaces, non-autonomous systems (potentially stochastic), and to cater for the topological differences that occur when there are eigenvalues with zero or near-zero real-part.
Example
The algebra necessary for this example is easily carried out by a web service that computes normal form coordinate transforms of systems of differential equations, autonomous or non-autonomous, deterministic or stochastic.
Consider the 2D system in variables evolving according to the pair of coupled differential equations
By direct computation it can be seen that the only equilibrium of this system lies at the origin, that is . The coordinate transform, where , given by
is a smooth map between the original and new coordinates, at least near the equilibrium at the origin. In the new coordinates the dynamical system transforms to its linearisation
That is, a distorted version of the linearisation gives the original dynamics in some finite neighbourhood.
See also
Linear approximation
Stable manifold theorem
References
Further reading
External links
Theorems in analysis
Theorems in dynamical systems
Approximations
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https://en.wikipedia.org/wiki/Disjoint%20union%20of%20graphs
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In graph theory, a branch of mathematics, the disjoint union of graphs is an operation that combines two or more graphs to form a larger graph.
It is analogous to the disjoint union of sets, and is constructed by making the vertex set of the result be the disjoint union of the vertex sets of the given graphs, and by making the edge set of the result be the disjoint union of the edge sets of the given graphs. Any disjoint union of two or more nonempty graphs is necessarily disconnected.
Notation
The disjoint union is also called the graph sum, and may be represented either by a plus sign or a circled plus sign: If and are two graphs, then or denotes their disjoint union.
Related graph classes
Certain special classes of graphs may be represented using disjoint union operations. In particular:
The forests are the disjoint unions of trees.
The cluster graphs are the disjoint unions of complete graphs.
The 2-regular graphs are the disjoint unions of cycle graphs.
More generally, every graph is the disjoint union of connected graphs, its connected components.
The cographs are the graphs that can be constructed from single-vertex graphs by a combination of disjoint union and complement operations.
References
Graph operations
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https://en.wikipedia.org/wiki/1%20%E2%88%92%202%20%2B%204%20%E2%88%92%208%20%2B%20%E2%8B%AF
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In mathematics, is the infinite series whose terms are the successive powers of two with alternating signs. As a geometric series, it is characterized by its first term, 1, and its common ratio, −2.
As a series of real numbers it diverges, so in the usual sense it has no sum. In a much broader sense, the series is associated with another value besides ∞, namely , which is the limit of the series using the 2-adic metric.
Historical arguments
Gottfried Leibniz considered the divergent alternating series as early as 1673. He argued that by subtracting either on the left or on the right, one could produce either positive or negative infinity, and therefore both answers are wrong and the whole should be finite:
Now normally nature chooses the middle if neither of the two is permitted, or rather if it cannot be determined which of the two is permitted, and the whole is equal to a finite quantity
Leibniz did not quite assert that the series had a sum, but he did infer an association with following Mercator's method. The attitude that a series could equal some finite quantity without actually adding up to it as a sum would be commonplace in the 18th century, although no distinction is made in modern mathematics.
After Christian Wolff read Leibniz's treatment of Grandi's series in mid-1712, Wolff was so pleased with the solution that he sought to extend the arithmetic mean method to more divergent series such as . Briefly, if one expresses a partial sum of this series as a function of the penultimate term, one obtains either or . The mean of these values is , and assuming that at infinity yields as the value of the series. Leibniz's intuition prevented him from straining his solution this far, and he wrote back that Wolff's idea was interesting but invalid for several reasons. The arithmetic means of neighboring partial sums do not converge to any particular value, and for all finite cases one has , not . Generally, the terms of a summable series should decrease to zero; even could be expressed as a limit of such series. Leibniz counsels Wolff to reconsider so that he "might produce something worthy of science and himself."
Modern methods
Geometric series
Any summation method possessing the properties of regularity, linearity, and stability will sum a geometric series
In this case a = 1 and r = −2, so the sum is .
Euler summation
In his 1755 Institutiones, Leonhard Euler effectively took what is now called the Euler transform of , arriving at the convergent series . Since the latter sums to , Euler concluded that . His ideas on infinite series do not quite follow the modern approach; today one says that is Euler summable and that its Euler sum is .
The Euler transform begins with the sequence of positive terms:
a0 = 1,
a1 = 2,
a2 = 4,
a3 = 8,...
The sequence of forward differences is then
Δa0 = a1 − a0 = 2 − 1 = 1,
Δa1 = a2 − a1 = 4 − 2 = 2,
Δa2 = a3 − a2 = 8 − 4 = 4,
Δa3 = a4 − a3 = 16 − 8 = 8,...
which is just the same sequence. He
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https://en.wikipedia.org/wiki/Adjoint%20bundle
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In mathematics, an adjoint bundle is a vector bundle naturally associated to any principal bundle. The fibers of the adjoint bundle carry a Lie algebra structure making the adjoint bundle into a (nonassociative) algebra bundle. Adjoint bundles have important applications in the theory of connections as well as in gauge theory.
Formal definition
Let G be a Lie group with Lie algebra , and let P be a principal G-bundle over a smooth manifold M. Let
be the (left) adjoint representation of G. The adjoint bundle of P is the associated bundle
The adjoint bundle is also commonly denoted by . Explicitly, elements of the adjoint bundle are equivalence classes of pairs [p, X] for p ∈ P and X ∈ such that
for all g ∈ G. Since the structure group of the adjoint bundle consists of Lie algebra automorphisms, the fibers naturally carry a Lie algebra structure making the adjoint bundle into a bundle of Lie algebras over M.
Restriction to a closed subgroup
Let G be any Lie group with Lie algebra , and let H be a closed subgroup of G.
Via the (left) adjoint representation of G on , G becomes a topological transformation group of .
By restricting the adjoint representation of G to the subgroup H,
also H acts as a topological transformation group on . For every h in H, is a Lie algebra automorphism.
Since H is a closed subgroup of the Lie group G, the homogeneous space M=G/H is the base space of a principal bundle with total space G and structure group H. So the existence of H-valued transition functions is assured, where is an open covering for M, and the transition functions form a cocycle of transition function on M.
The associated fibre bundle is a bundle of Lie algebras, with typical fibre , and a continuous mapping induces on each fibre the Lie bracket.
Properties
Differential forms on M with values in are in one-to-one correspondence with horizontal, G-equivariant Lie algebra-valued forms on P. A prime example is the curvature of any connection on P which may be regarded as a 2-form on M with values in .
The space of sections of the adjoint bundle is naturally an (infinite-dimensional) Lie algebra. It may be regarded as the Lie algebra of the infinite-dimensional Lie group of gauge transformations of P which can be thought of as sections of the bundle where conj is the action of G on itself by (left) conjugation.
If is the frame bundle of a vector bundle , then has fibre the general linear group (either real or complex, depending on ) where . This structure group has Lie algebra consisting of all matrices , and these can be thought of as the endomorphisms of the vector bundle . Indeed there is a natural isomorphism .
Notes
References
. As PDF
Lie algebras
Vector bundles
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https://en.wikipedia.org/wiki/1/2%20%E2%88%92%201/4%20%2B%201/8%20%E2%88%92%201/16%20%2B%20%E2%8B%AF
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In mathematics, the infinite series
is a simple example of an alternating series that converges absolutely.
It is a geometric series whose first term is and whose common ratio is −, so its sum is
Hackenbush and the surreals
A slight rearrangement of the series reads
The series has the form of a positive integer plus a series containing every negative power of two with either a positive or negative sign, so it can be translated into the infinite blue-red Hackenbush string that represents the surreal number :
LRRLRLR... = .
A slightly simpler Hackenbush string eliminates the repeated R:
LRLRLRL... = .
In terms of the Hackenbush game structure, this equation means that the board depicted on the right has a value of 0; whichever player moves second has a winning strategy.
Related series
The statement that is absolutely convergent means that the series is convergent. In fact, the latter series converges to 1, and it proves that one of the binary expansions of 1 is 0.111....
Pairing up the terms of the series results in another geometric series with the same sum, . This series is one of the first to be summed in the history of mathematics; it was used by Archimedes circa 250–200 BC.
The Euler transform of the divergent series is . Therefore, even though the former series does not have a sum in the usual sense, it is Euler summable to .
Notes
References
Geometric series
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https://en.wikipedia.org/wiki/Masakazu%20Tashiro
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Masakazu Tashiro (, born June 26, 1988) is a Japanese football player who currently plays for the Iwate Grulla Morioka in J3 League.
Club statistics
Updated to 1 March 2019.
References
External links
Profile at V-Varen Nagasaki
Profile at Yokohama F. Marinos
1988 births
Living people
Association football people from Tokyo
Japanese men's footballers
J1 League players
J2 League players
J3 League players
Japan Football League players
Yokohama F. Marinos players
FC Machida Zelvia players
JEF United Chiba players
Montedio Yamagata players
V-Varen Nagasaki players
Yokohama FC players
Omiya Ardija players
Iwate Grulla Morioka players
Men's association football defenders
People from Meguro
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https://en.wikipedia.org/wiki/Yota%20Akimoto
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is a Japanese footballer. He currently plays for Ehime FC on loan from Shonan Bellmare.
Career statistics
Club
Updated to 18 January 2019.
1Includes Japanese Super Cup.
References
External links
Profile at Shonan Bellmare
Profile at FC Tokyo
1987 births
Living people
Association football people from Tokyo
Japanese men's footballers
J1 League players
J2 League players
Yokohama F. Marinos players
Ehime FC players
Shonan Bellmare players
FC Tokyo players
FC Machida Zelvia players
Men's association football goalkeepers
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https://en.wikipedia.org/wiki/Robin%20boundary%20condition
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In mathematics, the Robin boundary condition (; properly ), or third type boundary condition, is a type of boundary condition, named after Victor Gustave Robin (1855–1897). When imposed on an ordinary or a partial differential equation, it is a specification of a linear combination of the values of a function and the values of its derivative on the boundary of the domain. Other equivalent names in use are Fourier-type condition and radiation condition.
Definition
Robin boundary conditions are a weighted combination of Dirichlet boundary conditions and Neumann boundary conditions. This contrasts to mixed boundary conditions, which are boundary conditions of different types specified on different subsets of the boundary. Robin boundary conditions are also called impedance boundary conditions, from their application in electromagnetic problems, or convective boundary conditions, from their application in heat transfer problems (Hahn, 2012).
If Ω is the domain on which the given equation is to be solved and ∂Ω denotes its boundary, the Robin boundary condition is:
for some non-zero constants a and b and a given function g defined on ∂Ω. Here, u is the unknown solution defined on Ω and denotes the normal derivative at the boundary. More generally, a and b are allowed to be (given) functions, rather than constants.
In one dimension, if, for example, Ω = [0,1], the Robin boundary condition becomes the conditions:
Notice the change of sign in front of the term involving a derivative: that is because the normal to [0,1] at 0 points in the negative direction, while at 1 it points in the positive direction.
Application
Robin boundary conditions are commonly used in solving Sturm–Liouville problems which appear in many contexts in science and engineering.
In addition, the Robin boundary condition is a general form of the insulating boundary condition for convection–diffusion equations. Here, the convective and diffusive fluxes at the boundary sum to zero:
where D is the diffusive constant, u is the convective velocity at the boundary and c is the concentration. The second term is a result of Fick's law of diffusion.
References
Bibliography
Gustafson, K. and T. Abe, (1998a). The third boundary condition – was it Robin's?, The Mathematical Intelligencer, 20, #1, 63–71.
Gustafson, K. and T. Abe, (1998b). (Victor) Gustave Robin: 1855–1897, The Mathematical Intelligencer, 20, #2, 47–53.
Boundary conditions
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https://en.wikipedia.org/wiki/Schr%C3%B6der%E2%80%93Bernstein%20theorems%20for%20operator%20algebras
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The Schröder–Bernstein theorem from set theory has analogs in the context operator algebras. This article discusses such operator-algebraic results.
For von Neumann algebras
Suppose M is a von Neumann algebra and E, F are projections in M. Let ~ denote the Murray-von Neumann equivalence relation on M. Define a partial order « on the family of projections by E « F if E ~ F' ≤ F. In other words, E « F if there exists a partial isometry U ∈ M such that U*U = E and UU* ≤ F.
For closed subspaces M and N where projections PM and PN, onto M and N respectively, are elements of M, M « N if PM « PN.
The Schröder–Bernstein theorem states that if M « N and N « M, then M ~ N.
A proof, one that is similar to a set-theoretic argument, can be sketched as follows. Colloquially, N « M means that N can be isometrically embedded in M. So
where N0 is an isometric copy of N in M. By assumption, it is also true that, N, therefore N0, contains an isometric copy M1 of M. Therefore, one can write
By induction,
It is clear that
Let
So
and
Notice
The theorem now follows from the countable additivity of ~.
Representations of C*-algebras
There is also an analog of Schröder–Bernstein for representations of C*-algebras. If A is a C*-algebra, a representation of A is a *-homomorphism φ from A into L(H), the bounded operators on some Hilbert space H.
If there exists a projection P in L(H) where P φ(a) = φ(a) P for every a in A, then a subrepresentation σ of φ can be defined in a natural way: σ(a) is φ(a) restricted to the range of P. So φ then can be expressed as a direct sum of two subrepresentations φ = φ' ⊕ σ.
Two representations φ1 and φ2, on H1 and H2 respectively, are said to be unitarily equivalent if there exists a unitary operator U: H2 → H1 such that φ1(a)U = Uφ2(a), for every a.
In this setting, the Schröder–Bernstein theorem reads:
If two representations ρ and σ, on Hilbert spaces H and G respectively, are each unitarily equivalent to a subrepresentation of the other, then they are unitarily equivalent.
A proof that resembles the previous argument can be outlined. The assumption implies that there exist surjective partial isometries from H to G and from G to H. Fix two such partial isometries for the argument. One has
In turn,
By induction,
and
Now each additional summand in the direct sum expression is obtained using one of the two fixed partial isometries, so
This proves the theorem.
See also
Schröder–Bernstein theorem for measurable spaces
Schröder–Bernstein property
References
B. Blackadar, Operator Algebras, Springer, 2006.
C*-algebras
Functional analysis
Operator theory
Von Neumann algebras
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https://en.wikipedia.org/wiki/Circular%20mean
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In mathematics and statistics, a circular mean or angular mean is a mean designed for angles and similar cyclic quantities, such as times of day, and fractional parts of real numbers.
This is necessary since most of the usual means may not be appropriate on angle-like quantities. For example, the arithmetic mean of 0° and 360° is 180°, which is misleading because 360° equals 0° modulo a full cycle. As another example, the "average time" between 11 PM and 1 AM is either midnight or noon, depending on whether the two times are part of a single night or part of a single calendar day.
The circular mean is one of the simplest examples of directional statistics and of statistics of non-Euclidean spaces.
This computation produces a different result than the arithmetic mean, with the difference being greater when the angles are widely distributed. For example, the arithmetic mean of the three angles 0°, 0°, and 90° is (0° + 0° + 90°) / 3 = 30°, but the vector mean is arctan(1/2) = 26.565°. Moreover, with the arithmetic mean the circular variance is only defined ±180°.
Definition
Since the arithmetic mean is not always appropriate for angles, the following method can be used to obtain both a mean value and measure for the variance of the angles:
Convert all angles to corresponding points on the unit circle, e.g., to . That is, convert polar coordinates to Cartesian coordinates. Then compute the arithmetic mean of these points. The resulting point will lie within the unit disk but generally not on the unit circle. Convert that point back to polar coordinates. The angle is a reasonable mean of the input angles. The resulting radius will be 1 if all angles are equal. If the angles are uniformly distributed on the circle, then the resulting radius will be 0, and there is no circular mean. (In fact, it is impossible to define a continuous mean operation on the circle.) In other words, the radius measures the concentration of the angles.
Given the angles a common formula of the mean using the atan2 variant of the arctangent function is
Using complex arithmetic
An equivalent definition can be formulated using complex numbers:
.
In order to match the above derivation using arithmetic means of points, the sums would have to be divided by . However, the scaling does not matter for and , thus it can be omitted.
This may be more succinctly stated by realizing that directional data are in fact vectors of unit length. In the case of one-dimensional data, these data points can be represented conveniently as complex numbers of unit magnitude , where is the measured angle. The mean resultant vector for the sample is then:
The sample mean angle is then the argument of the mean resultant:
The length of the sample mean resultant vector is:
and will have a value between 0 and 1. Thus the sample mean resultant vector can be represented as:
Similar calculations are also used to define the circular variance.
Properties
The circular mean,
maximizes the likeliho
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https://en.wikipedia.org/wiki/Modes%20of%20convergence
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In mathematics, there are many senses in which a sequence or a series is said to be convergent. This article describes various modes (senses or species) of convergence in the settings where they are defined. For a list of modes of convergence, see Modes of convergence (annotated index)
Note that each of the following objects is a special case of the types preceding it: sets, topological spaces, uniform spaces, TAGs (topological abelian groups), normed spaces, Euclidean spaces, and the real/complex numbers. Also, note that any metric space is a uniform space.
Elements of a topological space
Convergence can be defined in terms of sequences in first-countable spaces. Nets are a generalization of sequences that are useful in spaces which are not first countable. Filters further generalize the concept of convergence.
In metric spaces, one can define Cauchy sequences. Cauchy nets and filters are generalizations to uniform spaces. Even more generally, Cauchy spaces are spaces in which Cauchy filters may be defined. Convergence implies "Cauchy-convergence", and Cauchy-convergence, together with the existence of a convergent subsequence implies convergence. The concept of completeness of metric spaces, and its generalizations is defined in terms of Cauchy sequences.
Series of elements in a topological abelian group
In a topological abelian group, convergence of a series is defined as convergence of the sequence of partial sums. An important concept when considering series is unconditional convergence, which guarantees that the limit of the series is invariant under permutations of the summands.
In a normed vector space, one can define absolute convergence as convergence of the series of norms (). Absolute convergence implies Cauchy convergence of the sequence of partial sums (by the triangle inequality), which in turn implies absolute-convergence of some grouping (not reordering). The sequence of partial sums obtained by grouping is a subsequence of the partial sums of the original series. The norm convergence of absolutely convergent series is an equivalent condition for a normed linear space to be Banach (i.e.: complete).
Absolute convergence and convergence together imply unconditional convergence, but unconditional convergence does not imply absolute convergence in general, even if the space is Banach, although the implication holds in .
Convergence of sequence of functions on a topological space
The most basic type of convergence for a sequence of functions (in particular, it does not assume any topological structure on the domain of the functions) is pointwise convergence. It is defined as convergence of the sequence of values of the functions at every point. If the functions take their values in a uniform space, then one can define pointwise Cauchy convergence, uniform convergence, and uniform Cauchy convergence of the sequence.
Pointwise convergence implies pointwise Cauchy-convergence, and the converse holds if the space in which the fu
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https://en.wikipedia.org/wiki/%C3%8Ele%20des%20Pins%20Airport
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Île des Pins Airport () is an airport on Île des Pins in New Caledonia .
Airlines and destinations
Statistics
References
Airports in New Caledonia
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https://en.wikipedia.org/wiki/Ouanaham%20Airport
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Ouanaham Airport is an airport serving Lifou, Lifou Island, New Caledonia.
Airlines and destinations
Statistics
References
Lifou
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https://en.wikipedia.org/wiki/Mar%C3%A9%20Airport
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Maré Airport is an airport in Maré, New Caledonia.
Airlines and destinations
Statistics
References
Airports in New Caledonia
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https://en.wikipedia.org/wiki/Moscow%20Institute%20of%20Electronics%20and%20Mathematics
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Moscow Institute of Electronics and Mathematics, MIEM (; also occasionally referred to as Moscow Institute of Electronic Engineering) — a Russian higher educational institution in the field of electronics, computer engineering, and applied mathematics.
History
The institute was founded by the joint decree of the Communist Party Central Committee and the USSR government of 21 April 1962 as the Moscow Institute of Electronic Machine Building () from the Moscow Evening Machine Building Institute ( (founded in 1929). It was designed to educate personnel for the technologically advanced enterprises of the USSR's military industry. The institute changed over to the current name in 1993, retaining the same abbreviation.
In 2011, the institute was incorporated into the National Research University Higher School of Economics. In December 2014, the institute moved to a new building located in the northwestern suburb of Moscow, Strogino, from its previous location at 3 Tryokhsvyatitelskiy lane in central Moscow.
Faculties
As of 2015, the institute has 3 departments:
Faculty of Electronic Engineering;
Faculty of Computer Engineering;
Faculty of Applied Mathematics.
References
External links
Official Site (in Russian)
See also
National Research University of Electronic Technology, another Russian technical institute founded in 1960s within the scope of the Soviet microelectronics program
Universities and colleges established in 1962
Universities and institutes established in the Soviet Union
1962 establishments in the Soviet Union
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https://en.wikipedia.org/wiki/MIEM
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MIEM may refer to:
Member of the Institute of Emergency Management
Moscow Institute of Electronics and Mathematics
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https://en.wikipedia.org/wiki/Fullbrook%20School
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Fullbrook School is a secondary school and sixth form in north west Surrey, England. The school has held Specialist Science, Technology, Mathematics and Computing College status since 2002. The school gained Grant Maintained status in the mid-1990s and was then given foundation status in 1999. In 2011, the school became an academy. Its main catchment areas are Byfleet, West Byfleet and New Haw with some pupils coming from Addlestone, Woking, Goldsworth Park and Sheerwater. The school has around 1550 students and there are about 250 students in the school's Sixth Form. In January 2017, Mrs. A Turner retired as head of Fullbrook School, and was succeeded by Mrs. K Moore. In 2022, Mr. A McKenzie was appointed as head of the school, coinciding with his lead as principal at King's College, Guildford.
History
Fullbrook was first established on its present site in 1954, when West Byfleet County Secondary School was divided into two as numbers at that school, due to post-war expansion, reached 747 in September 1953. Pupils living to the north of the line from Sheerwater Road, the Basingstoke Canal and then along the railway to West Weybridge (now Byfleet and New Haw) transferred to Fullbrook County Secondary School.
Above the main entrance is a stone plaque that was loaned to the school from the London County Council (LCC) during the building programme in the early 1950s. The Sheerwater Estate was being built to provide over spill housing for Londoners. The school there, now Bishop David Brown, had not yet been built and so the plaque came to the just completed Fullbrook School. The Headteacher and Governors used it to inspire the school badge. The plaque features an eagle and a squirrel. It was decided that as an eagle was often used on badges, to use choose the squirrel as it was different. The Festival of Britain was held at that time and celebrated contemporary design. The original Fullbrook building is in a typical 1950s style and perhaps the plaque was placed over the main entrance to decorate an otherwise plain architectural design. The plaque design is very much in the Art Deco style and it is possible that it came from a London building destroyed during The Blitz.
Mr. W. H. Bean, the school's first headmaster retired in 1968.
In 1976, West Byfleet County Secondary School closed and its pupils and teachers joined Fullbrook, merged to create the mixed school we are familiar with today.
Houses
The houses of the school were originally named after British Royal Houses - Tudor, Stuart, Hanover and Windsor and then later famous battles - Alamein, Blenheim, Agincourt, Hastings, Trafalgar and Waterloo and were formed from an entire school vote on a list of 50 house name ideas which were themed around the idea of Maths and Computing (the school's Specialist Subjects). The six that won were Newton (yellow), Cyber (blue), Enigma (red), Fibonacci (purple), Galileo (orange) and Matrix (green). In 2017, another vote was held, updating the house names wi
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https://en.wikipedia.org/wiki/Nth-term%20test
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In mathematics, the nth-term test for divergence is a simple test for the divergence of an infinite series:If or if the limit does not exist, then diverges.Many authors do not name this test or give it a shorter name.
When testing if a series converges or diverges, this test is often checked first due to its ease of use.
In the case of p-adic analysis the term test is a necessary and sufficient condition for convergence due to the non-archimedean triangle inequality.
Usage
Unlike stronger convergence tests, the term test cannot prove by itself that a series converges. In particular, the converse to the test is not true; instead all one can say is:If then may or may not converge. In other words, if the test is inconclusive.The harmonic series is a classic example of a divergent series whose terms limit to zero. The more general class of p-series,
exemplifies the possible results of the test:
If p ≤ 0, then the term test identifies the series as divergent.
If 0 < p ≤ 1, then the term test is inconclusive, but the series is divergent by the integral test for convergence.
If 1 < p, then the term test is inconclusive, but the series is convergent, again by the integral test for convergence.
Proofs
The test is typically proven in contrapositive form:If converges, then
Limit manipulation
If sn are the partial sums of the series, then the assumption that the series
converges means that
for some number L. Then
Cauchy's criterion
The assumption that the series converges means that it passes Cauchy's convergence test: for every there is a number N such that
holds for all n > N and p ≥ 1. Setting p = 1 recovers the definition of the statement
Scope
The simplest version of the term test applies to infinite series of real numbers. The above two proofs, by invoking the Cauchy criterion or the linearity of the limit, also work in any other normed vector space (or any (additively written) abelian group).
Notes
References
Convergence tests
Articles containing proofs
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https://en.wikipedia.org/wiki/Scoring%20algorithm
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Scoring algorithm, also known as Fisher's scoring, is a form of Newton's method used in statistics to solve maximum likelihood equations numerically, named after Ronald Fisher.
Sketch of derivation
Let be random variables, independent and identically distributed with twice differentiable p.d.f. , and we wish to calculate the maximum likelihood estimator (M.L.E.) of . First, suppose we have a starting point for our algorithm , and consider a Taylor expansion of the score function, , about :
where
is the observed information matrix at . Now, setting , using that and rearranging gives us:
We therefore use the algorithm
and under certain regularity conditions, it can be shown that .
Fisher scoring
In practice, is usually replaced by , the Fisher information, thus giving us the Fisher Scoring Algorithm:
..
Under some regularity conditions, if is a consistent estimator, then (the correction after a single step) is 'optimal' in the sense that its error distribution is asymptotically identical to that of the true max-likelihood estimate.
See also
Score (statistics)
Score test
Fisher information
References
Further reading
Maximum likelihood estimation
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https://en.wikipedia.org/wiki/Graph%20enumeration
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In combinatorics, an area of mathematics, graph enumeration describes a class of combinatorial enumeration problems in which one must count undirected or directed graphs of certain types, typically as a function of the number of vertices of the graph. These problems may be solved either exactly (as an algebraic enumeration problem) or asymptotically.
The pioneers in this area of mathematics were George Pólya, Arthur Cayley and J. Howard Redfield.
Labeled vs unlabeled problems
In some graphical enumeration problems, the vertices of the graph are considered to be labeled in such a way as to be distinguishable from each other, while in other problems any permutation of the vertices is considered to form the same graph, so the vertices are considered identical or unlabeled. In general, labeled problems tend to be easier. As with combinatorial enumeration more generally, the Pólya enumeration theorem is an important tool for reducing unlabeled problems to labeled ones: each unlabeled class is considered as a symmetry class of labeled objects.
Exact enumeration formulas
Some important results in this area include the following.
The number of labeled n-vertex simple undirected graphs is 2n(n −1)/2.
The number of labeled n-vertex simple directed graphs is 2n(n −1).
The number Cn of connected labeled n-vertex undirected graphs satisfies the recurrence relation
from which one may easily calculate, for n = 1, 2, 3, ..., that the values for Cn are
1, 1, 4, 38, 728, 26704, 1866256, ...
The number of labeled n-vertex free trees is nn−2 (Cayley's formula).
The number of unlabeled n-vertex caterpillars is
Graph database
Various research groups have provided searchable database that lists graphs with certain properties of a small sizes. For example
The House of Graphs
Small Graph Database
References
Enumerative combinatorics
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https://en.wikipedia.org/wiki/Arthur%20Mattuck
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Arthur Paul Mattuck (June 11, 1930 – October 8, 2021) was an emeritus professor of mathematics at the Massachusetts Institute of Technology. He may be best known for his 1998 book, Introduction to Analysis () and his differential equations video lectures featured on MIT's OpenCourseWare.
Mattuck was a student of Emil Artin at Princeton University, where he completed his PhD in 1954.
Recognition
In 2012 he became a fellow of the American Mathematical Society.
Personal life
From 1959 to 1977 Mattuck was married to chemist Joan Berkowitz. Mattuck is quoted extensively in Sylvia Nasar's biography of John Nash, A Beautiful Mind.
He was the brother of the physicist Richard Mattuck.
He died on October 8, 2021, at age of 91. He was survived by his daughter Rosemary and her partner Jeffrey Broadman, and three nephews (Allan, Robin, and Martin).
References
External links
Initial lecture to 18.03 Differential equations, by Prof. Arthur Mattuck-- demonstrating Prof. Mattuck's ability to jump his students into quick learning
Prof. Arthur Mattuck Home Page
"The Unofficial 18.02/18.03 Quote Book"
1930 births
2021 deaths
Academics from Brooklyn
American textbook writers
American male non-fiction writers
20th-century American mathematicians
21st-century American mathematicians
Swarthmore College alumni
Massachusetts Institute of Technology School of Science faculty
Fellows of the American Mathematical Society
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https://en.wikipedia.org/wiki/Observed%20information
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In statistics, the observed information, or observed Fisher information, is the negative of the second derivative (the Hessian matrix) of the "log-likelihood" (the logarithm of the likelihood function). It is a sample-based version of the Fisher information.
Definition
Suppose we observe random variables , independent and identically distributed with density f(X; θ), where θ is a (possibly unknown) vector. Then the log-likelihood of the parameters given the data is
.
We define the observed information matrix at as
Since the inverse of the information matrix is the asymptotic covariance matrix of the corresponding maximum-likelihood estimator, the observed information is often evaluated at the maximum-likelihood estimate for the purpose of significance testing or confidence-interval construction. The invariance property of maximum-likelihood estimators allows the observed information matrix to be evaluated before being inverted.
Alternative definition
Andrew Gelman, David Dunson and Donald Rubin define observed information instead in terms of the parameters' posterior probability, :
Fisher information
The Fisher information is the expected value of the observed information given a single observation distributed according to the hypothetical model with parameter :
.
Comparison with the expected information
The comparison between the observed information and the expected information remains an active and ongoing area of research and debate. Efron and Hinkley provided a frequentist justification for preferring the observed information to the expected information when employing normal approximations to the distribution of the maximum-likelihood estimator in one-parameter families in the presence of an ancillary statistic that affects the precision of the MLE. Lindsay and Li showed that the observed information matrix gives the minimum mean squared error as an approximation of the true information if an error term of is ignored. In Lindsay and Li's case, the expected information matrix still requires evaluation at the obtained ML estimates, introducing randomness.
However, when the construction of confidence intervals is of primary focus, there are reported findings that the expected information outperforms the observed counterpart. Yuan and Spall showed that the expected information outperforms the observed counterpart for confidence-interval constructions of scalar parameters in the mean squared error sense. This finding was later generalized to multiparameter cases, although the claim had been weakened to the expected information matrix performing at least as well as the observed information matrix.
See also
Fisher information matrix
Fisher information metric
References
Information theory
Estimation theory
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https://en.wikipedia.org/wiki/Reiss
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Reiss may refer to:
Reiss (surname)
Reiss (brand), fashion brand
Reiss, Scotland
Reiss relation in mathematics
Reiss (ship), an historic steam tug -- see Northeastern Maritime Historical Foundation
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https://en.wikipedia.org/wiki/Affine%20manifold
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In differential geometry, an affine manifold is a differentiable manifold equipped with a flat, torsion-free connection.
Equivalently, it is a manifold that is (if connected) covered by an open subset of , with monodromy acting by affine transformations. This equivalence is an easy corollary of Cartan–Ambrose–Hicks theorem.
Equivalently, it is a manifold equipped with an atlas—called the affine structure—such that all transition functions between charts are affine transformations (that is, have constant Jacobian matrix); two atlases are equivalent if the manifold admits an atlas subjugated to both, with transitions from both atlases to a smaller atlas being affine. A manifold having a distinguished affine structure is called an affine manifold and the charts which are affinely related to those of the affine structure are called affine charts. In each affine coordinate domain the coordinate vector fields form a parallelisation of that domain, so there is an associated connection on each domain. These locally defined connections are the same on overlapping parts, so there is a unique connection associated with an affine structure. Note there is a link between linear connection (also called affine connection) and a web.
Formal definition
An affine manifold is a real manifold with charts such that for all where denotes the Lie group of affine transformations. In fancier words it is a (G,X)-manifold where and is the group of affine transformations.
An affine manifold is called complete if its universal covering is homeomorphic to .
In the case of a compact affine manifold , let be the fundamental group of and be its universal cover. One can show that each -dimensional affine manifold comes with a developing map , and a homomorphism , such that is an immersion and equivariant with respect to .
A fundamental group of a compact complete flat affine manifold is called an affine crystallographic group. Classification of affine crystallographic groups is a difficult problem, far from being solved. The Riemannian crystallographic groups (also known as Bieberbach groups) were classified by Ludwig Bieberbach, answering a question posed by David Hilbert. In his work on Hilbert's 18-th problem, Bieberbach proved that any Riemannian crystallographic group contains an abelian subgroup of finite index.
Important longstanding conjectures
Geometry of affine manifolds is essentially a network of longstanding conjectures; most of them proven in low dimension and some other special cases.
The most important of them are:
Markus conjecture (1962) stating that a compact affine manifold is complete if and only if it has parallel volume. Known in dimension 2.
Auslander conjecture (1964) stating that any affine crystallographic group contains a polycyclic subgroup of finite index. Known in dimensions up to 6, and when the holonomy of the flat connection preserves a Lorentz metric. Since every virtually polycyclic crystallographic group preserves a volume
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https://en.wikipedia.org/wiki/F1%20in%20Schools
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F1 in Schools is an international STEM (science, technology, engineering, mathematics) competition for school children (aged 11–19), in which groups of 3–6 students have to design and manufacture a miniature car out of the official F1 Model Block using CAD/CAM design tools. The cars are powered by CO2 cartridges and are attached to a track by a nylon wire. They are timed from the moment they are launched to when they pass the finish line by a computer.
The cars have to follow extensive regulations, in a similar fashion to Formula 1 (e.g. the wheels of the car must be in contact with the track at all times). The cars are raced on a 20m long track with two lanes, to allow two cars to be raced simultaneously.
Software called F1 Virtual Wind Tunnel was designed specifically for the challenge.
The competition is currently operational in over 40 countries. The competition was first introduced in the UK in 1999. The competition's aim is to introduce younger people to engineering in a more fun environment. The competition is held annually, with Regional and National Finals. The overall winners of the National Finals are invited to compete at the World Finals, which are held at a different location each year, usually held in conjunction with a Formula One Grand Prix. In the UK competition there are 3 classes of entry: Professional Class aimed at 11- to 19-year-olds; Development Class aimed at 11- to 19-year-olds in their first year; and Entry Class aimed at 11- to 14-year-olds.
, the F1 in Schools World Champions are Recoil Racing from Marie-Therese-Gymnasium Erlangen, in Germany.
The F1 in Schools World Record was set in 2016 by the Australian team Infinitude and is 0.916 seconds.
After safety issues concerning the use of extended canister chambers coupled with the Launch Energy Recovery System (LERS), the controversial device was banned globally from the 2017 World Finals season onwards, after being innovated in 2014 by Colossus F1.
Denford Ltd. unveiled a new track and timing system that debuted at the 2017 World Finals. All components are now manufactured in-house, resulting in a lower overall cost in comparison to the Pitsco produced track that it succeeds. The track's launching mechanism has had numerous reliability issues since its introduction.
In 2018, the competition's logo was updated to incorporate Formula One's updated logo. Consequently, the Bernie Ecclestone World Champions trophy was replaced, with the new World Champions trophy incorporating the new logo and the car of the 2017 World Champions, Hyperdrive.
The 2020 F1 in Schools World Finals has been postponed twice due to the effects of the COVID-19 pandemic. The World Finals 2020/21 was held as a virtual event in the UK in June 2021 with 43 competing teams.
Aspects of the competition
Specifications judging
Specifications judging is a detailed inspection process where the race car is assessed for compliance with the F1 in Schools Technical
Regulations. Scrutineering is condu
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https://en.wikipedia.org/wiki/Content%20%28measure%20theory%29
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In mathematics, in particular in measure theory, a content is a real-valued function defined on a collection of subsets such that
That is, a content is a generalization of a measure: while the latter must be countably additive, the former must only be finitely additive.
In many important applications the is chosen to be a ring of sets or to be at least a semiring of sets in which case some additional properties can be deduced which are described below. For this reason some authors prefer to define contents only for the case of semirings or even rings.
If a content is additionally σ-additive it is called a pre-measure and if furthermore is a σ-algebra, the content is called a measure. Therefore every (real-valued) measure is a content, but not vice versa. Contents give a good notion of integrating bounded functions on a space but can behave badly when integrating unbounded functions, while measures give a good notion of integrating unbounded functions.
Examples
A classical example is to define a content on all half open intervals by setting their content to the length of the intervals, that is, One can further show that this content is actually σ-additive and thus defines a pre-measure on the semiring of all half-open intervals. This can be used to construct the Lebesgue measure for the real number line using Carathéodory's extension theorem. For further details on the general construction see article on Lebesgue measure.
An example of a content that is not a measure on a σ-algebra is the content on all subsets of the positive integers that has value on any integer and is infinite on any infinite subset.
An example of a content on the positive integers that is always finite but is not a measure can be given as follows. Take a positive linear functional on the bounded sequences that is 0 if the sequence has only a finite number of nonzero elements and takes value 1 on the sequence so the functional in some sense gives an "average value" of any bounded sequence. (Such a functional cannot be constructed explicitly, but exists by the Hahn–Banach theorem.) Then the content of a set of positive integers is the average value of the sequence that is 1 on this set and 0 elsewhere. Informally, one can think of the content of a subset of integers as the "chance" that a randomly chosen integer lies in this subset (though this is not compatible with the usual definitions of chance in probability theory, which assume countable additivity).
Properties
Frequently contents are defined on collections of sets that satisfy further constraints. In this case additional properties can be deduced that fail to hold in general for contents defined on any collections of sets.
On semirings
If forms a Semiring of sets then the following statements can be deduced:
Every content is monotone that is,
Every content is subadditive that is,
for such that
On rings
If furthermore is a Ring of sets one gets additionally:
Subtractivity: for sat
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https://en.wikipedia.org/wiki/Khintchine%20inequality
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In mathematics, the Khintchine inequality, named after Aleksandr Khinchin and spelled in multiple ways in the Latin alphabet, is a theorem from probability, and is also frequently used in analysis. Heuristically, it says that if we pick complex numbers , and add them together each multiplied by a random sign , then the expected value of the sum's modulus, or the modulus it will be closest to on average, will be not too far off from .
Statement
Let be i.i.d. random variables
with for ,
i.e., a sequence with Rademacher distribution. Let and let . Then
for some constants depending only on (see Expected value for notation). The sharp values of the constants were found by Haagerup (Ref. 2; see Ref. 3 for a simpler proof). It is a simple matter to see that when , and when .
Haagerup found that
where and is the Gamma function.
One may note in particular that matches exactly the moments of a normal distribution.
Uses in analysis
The uses of this inequality are not limited to applications in probability theory. One example of its use in analysis is the following: if we let be a linear operator between two Lp spaces and , , with bounded norm , then one can use Khintchine's inequality to show that
for some constant depending only on and .
Generalizations
For the case of Rademacher random variables, Pawel Hitczenko showed that the sharpest version is:
where , and and are universal constants independent of .
Here we assume that the are non-negative and non-increasing.
See also
Marcinkiewicz–Zygmund inequality
Burkholder-Davis-Gundy inequality
References
Thomas H. Wolff, "Lectures on Harmonic Analysis". American Mathematical Society, University Lecture Series vol. 29, 2003.
Uffe Haagerup, "The best constants in the Khintchine inequality", Studia Math. 70 (1981), no. 3, 231–283 (1982).
Fedor Nazarov and Anatoliy Podkorytov, "Ball, Haagerup, and distribution functions", Complex analysis, operators, and related topics, 247–267, Oper. Theory Adv. Appl., 113, Birkhäuser, Basel, 2000.
Theorems in analysis
Probabilistic inequalities
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https://en.wikipedia.org/wiki/Sample%20mean%20and%20covariance
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The sample mean (sample average) or empirical mean (empirical average), and the sample covariance or empirical covariance are statistics computed from a sample of data on one or more random variables.
The sample mean is the average value (or mean value) of a sample of numbers taken from a larger population of numbers, where "population" indicates not number of people but the entirety of relevant data, whether collected or not. A sample of 40 companies' sales from the Fortune 500 might be used for convenience instead of looking at the population, all 500 companies' sales. The sample mean is used as an estimator for the population mean, the average value in the entire population, where the estimate is more likely to be close to the population mean if the sample is large and representative. The reliability of the sample mean is estimated using the standard error, which in turn is calculated using the variance of the sample. If the sample is random, the standard error falls with the size of the sample and the sample mean's distribution approaches the normal distribution as the sample size increases.
The term "sample mean" can also be used to refer to a vector of average values when the statistician is looking at the values of several variables in the sample, e.g. the sales, profits, and employees of a sample of Fortune 500 companies. In this case, there is not just a sample variance for each variable but a sample variance-covariance matrix (or simply covariance matrix) showing also the relationship between each pair of variables. This would be a 3×3 matrix when 3 variables are being considered. The sample covariance is useful in judging the reliability of the sample means as estimators and is also useful as an estimate of the population covariance matrix.
Due to their ease of calculation and other desirable characteristics, the sample mean and sample covariance are widely used in statistics to represent the location and dispersion of the distribution of values in the sample, and to estimate the values for the population.
Definition of the sample mean
The sample mean is the average of the values of a variable in a sample, which is the sum of those values divided by the number of values. Using mathematical notation, if a sample of N observations on variable X is taken from the population, the sample mean is:
Under this definition, if the sample (1, 4, 1) is taken from the population (1,1,3,4,0,2,1,0), then the sample mean is , as compared to the population mean of . Even if a sample is random, it is rarely perfectly representative, and other samples would have other sample means even if the samples were all from the same population. The sample (2, 1, 0), for example, would have a sample mean of 1.
If the statistician is interested in K variables rather than one, each observation having a value for each of those K variables, the overall sample mean consists of K sample means for individual variables. Let be the ith independently drawn observat
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https://en.wikipedia.org/wiki/Hiroki%20Iikura
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Hiroki Iikura (, born June 1, 1986, in Aomori, Japan) is a Japanese professional footballer who plays as a goalkeeper club for Yokohama F. Marinos.
Club statistics
.
Honours
Vissel Kobe
Emperor's Cup: 2019
Japanese Super Cup: 2020
References
External links
Profile at Yokohama F-Marinos
1986 births
Living people
Association football people from Aomori Prefecture
Japanese men's footballers
J1 League players
Japan Football League players
Yokohama F. Marinos players
Roasso Kumamoto players
Vissel Kobe players
Men's association football goalkeepers
People from Aomori (city)
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https://en.wikipedia.org/wiki/Nicholas%20Higham
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Nicholas John Higham FRS (born 25 December 1961 in Salford) is a British numerical analyst. He is Royal Society Research Professor and Richardson Professor of Applied Mathematics in the Department of Mathematics at the University of Manchester.
Education and career
Higham was educated at Eccles Grammar School, Eccles College, and the University of Manchester, from which he gained his B.Sc. in mathematics (1982), M.Sc. in Numerical Analysis and Computing (1983), and PhD in Numerical Analysis (1985). His PhD thesis was supervised by George Hall. He was appointed lecturer in mathematics at the University of Manchester in 1985, where he has been Richardson Professor Professor of Applied Mathematics since 1998.
In 1988–1989 he was Visiting Assistant Professor of Computer Science at Cornell University, Ithaca, New York.
Research
Higham is best known for his work on the accuracy and stability of numerical algorithms. He has more than 140 refereed publications on topics such as rounding error analysis, linear systems, least squares problems, matrix functions and nonlinear matrix equations, matrix nearness problems, condition number estimation, and generalized eigenvalue problems. He has contributed software to LAPACK and the NAG library, and has contributed code included in the MATLAB distribution.
Higham's books include Functions of Matrices: Theory and Computation, (2008),Accuracy and Stability of Numerical Algorithms, Handbook of Writing for the Mathematical Sciences, and MATLAB Guide, co-authored with his brother Desmond Higham. He is Editor of the Princeton Companion to Applied Mathematics and a contributor to the Penguin Dictionary of Mathematics. His books have been translated into Chinese, Japanese and Korean.
Awards and honours
Higham's honours include the Alston S. Householder Award VI, 1987 (for the best PhD thesis in numerical algebra 1984–1987), the 1988 Leslie Fox Prize for Numerical Analysis, a 1999 Junior Whitehead Prize from the London Mathematical Society, a 2020 IMA Gold Medal, the 2019 Naylor Prize and Lectureship by the London Mathematical Society, the 2021 George Pólya Prize for Mathematical Exposition by the Society for Industrial and Applied Mathematics (SIAM), and the 2022 Hans Schneider Prize in Linear Algebra. Higham held a prestigious Royal Society Wolfson Research Merit Award (2003–2008). He was elected as a Fellow of the Royal Society in 2007 and as a ACM Fellow in 2020. In 2008 he was awarded the Fröhlich Prize in recognition of 'his leading contributions to numerical linear algebra and numerical stability analysis'. He was elected a Member of Academia Europaea in 2016. In 2022 he became Fellow of the Royal Academy of Engineering.
Higham is a Fellow of the Institute of Mathematics and Its Applications, a Fellow of the Institute of Engineering and Technology, and a Fellow of the Society for Industrial and Applied Mathematics. He is also a Fellow of the Alan Turing Institute.
Professional service
Higham served as p
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https://en.wikipedia.org/wiki/Dyserth
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Dyserth () is a village, community and electoral ward in Denbighshire, Wales. Its population at the 2011 United Kingdom census was 2,269 and was estimated by the Office for National Statistics as 2,271 in 2019. It lies within the historic county boundaries of Flintshire. Features include quarrying remains, waterfalls and the mountain Moel Hiraddug. Its railway line, once part of the London and North Western Railway, finally closed in 1973 and is now a footpath.
Overview
Dyserth is mentioned in the Domesday book of 1086, listed in the Hundred of Ati's Cross and within Cheshire:
Dyserth also had a nearby castle, which suffered at the hands of Llywelyn ap Gruffudd; destroyed after a six-week siege in 1263. The remains of the castle were quarried away during World War I.
The oldest industry in the village and surrounding area is mining, with lead, copper and limestone just some of the minerals being mined locally in the past. These quarries are still visible and form a major part of the village's geography, though mining ceased when Dyserth Quarry closed in 1981.
Traditionally, there has been a strong Welsh language speaking community in the village and until recent times many families and village folk knew, or knew of, each other. This is typical of a rural community whose life often centred on its many churches and chapels. Many of the village's families have their roots in agriculture, with many notable farms in or around Dyserth, including Hottia, Bryn Cnewyllyn and Ty Newydd.
Places of worship
The Parish Church of St Bridget and St Cwyfan, of the Church in Wales (the Wales based churches of the Anglican Communion), is a Grade II* listed building. The church is dedicated to Saint Brigid of Kildare, and includes the name of the Celtic monk Saint Cwyfan, believed to have founded the original place of worship near the Dyserth Waterfall during the 6th century. The church is notable for a Jesse Window dating from the 16th century.
Dyserth Chapel, in Dyserth High Street, built in 1927, also has stained glass. It houses the English-speaking Horeb United Reformed Church.
Railway
The Dyserth branch line was opened by the London and North Western Railway in 1869 to tap limestone quarries and a lead mine. A passenger, parcels and goods service was introduced in 1905 to serve local needs and an expanding holiday industry. The company designed and built a single carriage, steam-powered Motor Train for such lines, with the Dyserth Branch using the first example. The passenger service was a success before the First World War. Services were doubled and an additional unit provided for the motor trains. After the war the motor trains were replaced by locomotive-propelled push-pull trains. Road competition and the 1926 General Strike ate into profits, leading the London, Midland and Scottish Railway to withdraw the passenger service in 1930.
The line remained open for minerals, parcels and general goods until the end of November 1951, when parcels and gene
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https://en.wikipedia.org/wiki/Toronto%20Raptors%20all-time%20roster
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The following is a list of players, both past and current, who appeared on the roster for the Toronto Raptors NBA franchise.
Players
Note: Statistics are correct through the end of the season.
A to B
-
|align="left" bgcolor="#CCFFCC"|x || align="center"|F/C || align="left"|Memphis || align="center"|2 || align="center"|– || 128 || 2,865 || 801 || 132 || 1,172 || 22.4 || 6.3 || 1.0 || 9.2 || align=center|
|-
|align="left"| ||align="center"|F/C || align="left"|Baylor || align="center"|2 || align="center"|– || 36 || 403 || 92 || 15 || 135 || 11.2 || 2.6 || 0.4 || 3.8 || align=center|
|-
|align="left"| || align="center"|G || align="left"|Arizona || align="center"|1 || align="center"| || 12 || 52 || 7 || 1 || 11 || 4.3 || 0.6 || 0.1 || 0.9 || align=center|
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|align="left"| || align="center"|F/C || align="left"| HTV || align="center"|1 || align="center"| || 24 || 265 || 61 || 8 || 114 || 11.0 || 2.5 || 0.3 || 4.8 || align=center|
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|align="left"| || align="center"|C || align="left"|Florida State || align="center"|2 || align="center"|– || 26 || 181 || 61 || 5 || 39 || 7.0 || 2.3 || 0.2 || 1.5 || align=center|
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|align="left"| || align="center"|G || align="left"|Fresno State || align="center"|2 || align="center"| || 127 || 3,697 || 386 || 706 || 1,502 || 29.1 || 3.0 || 5.6 || 11.8 || align=center|
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|align="left"| || align="center"|C || align="left"| Virtus Bologna || align="center"|1 || align="center"| || 11 || 150 || 34 || 7 || 56 || 13.6 || 3.1 || 0.6 || 5.1 || align=center|
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|align="left"| || align="center"|G/F || align="left"|Michigan State || align="center"|2 || align="center"|– || 82 || 1,956 || 182 || 129 || 856 || 23.9 || 2.2 || 1.6 || 10.4 || align=center|
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|align="left"| || align="center"|G/F || align="left"|Georgia || align="center"|1 || align="center"| || 49 || 1,564 || 186 || 149 || 606 || 31.9 || 3.8 || 3.0 || 12.4 || align=center|
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|align="left" bgcolor="#CCFFCC"|x || align="center"|F || align="left"|Indiana || align="center"|6 || align="center"|– || 368 || 10,446 || 1,578 || 560 || 4,268 || 28.4 || 4.3 || 1.5 || 11.6 || align=center|
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|align="left"| || align="center"|C || align="left"|BYU || align="center"|2 || align="center"|– || 111 || 1,337 || 329 || 31 || 317 || 12.0 || 3.0 || 0.3 || 2.9 || align=center|
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|align="left"| || align="center"|F || align="left"|Illinois || align="center"|1 || align="center"| || 30 || 246 || 50 || 12 || 29 || 8.2 || 1.7 || 0.4 || 1.0 || align=center|
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|align="left"| || align="center"|G || align="left"|FIU || align="center"|1 || align="center"| || 17 || 96 || 12 || 21 || 30 || 5.6 || 0.7 || 1.2 || 1.8 || align=center|
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|align="left"| || align="center"|G || align="left"|Texas || align="center"|1 || align="center"| || 10 || 82 || 4 || 10 || 21 || 8.2 || 0.4 || 1.0 || 2.1 || align=center|
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|align="left"| || align="center"|G || align="left"|Ohio State || align="center"|1 || align="center"| || 1 || 2 || 0 || 0 || 0 || 2.0 || 0.0 || 0.0 || 0.0 || align=center|
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https://en.wikipedia.org/wiki/Excellent%20ring
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In commutative algebra, a quasi-excellent ring is a Noetherian commutative ring that behaves well with respect to the operation of completion, and is called an excellent ring if it is also universally catenary. Excellent rings are one answer to the problem of finding a natural class of "well-behaved" rings containing most of the rings that occur in number theory and algebraic geometry. At one time it seemed that the class of Noetherian rings might be an answer to this problem, but Masayoshi Nagata and others found several strange counterexamples showing that in general Noetherian rings need not be well-behaved: for example, a normal Noetherian local ring need not be analytically normal.
The class of excellent rings was defined by Alexander Grothendieck (1965) as a candidate for such a class of well-behaved rings. Quasi-excellent rings are conjectured to be the base rings for which the problem of resolution of singularities can be solved; showed this in characteristic 0, but the positive characteristic case is (as of 2016) still a major open problem. Essentially all Noetherian rings that occur naturally in algebraic geometry or number theory are excellent; in fact it is quite hard to construct examples of Noetherian rings that are not excellent.
Definitions
The definition of excellent rings is quite involved, so we recall the definitions of the technical conditions it satisfies. Although it seems like a long list of conditions, most rings in practice are excellent, such as fields, polynomial rings, complete Noetherian rings, Dedekind domains over characteristic 0 (such as ), and quotient and localization rings of these rings.
Recalled definitions
A ring containing a field is called geometrically regular over if for any finite extension of the ring is regular.
A homomorphism of rings from is called regular if it is flat and for every the fiber is geometrically regular over the residue field of .
A ring is called a G-ring (or Grothendieck ring) if it is Noetherian and its formal fibers are geometrically regular; this means that for any , the map from the local ring to its completion is regular in the sense above.
Finally, a ring is J-2 if any finite type -algebra is J-1, meaning the regular subscheme is open.
Definition of (quasi-)excellence
A ring is called quasi-excellent if it is a G-ring and J-2 ring. It is called excellentpg 214 if it is quasi-excellent and universally catenary. In practice almost all Noetherian rings are universally catenary, so there is little difference between excellent and quasi-excellent rings.
A scheme is called excellent or quasi-excellent if it has a cover by open affine subschemes with the same property, which implies that every open affine subscheme has this property.
Properties
Because an excellent ring is a G-ring, it is Noetherian by definition. Because it is universally catenary, every maximal chain of prime ideals has the same length. This is useful for studying the dimension theory
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https://en.wikipedia.org/wiki/Development%20%28differential%20geometry%29
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In classical differential geometry, development refers to the simple idea of rolling one smooth surface over another in Euclidean space. For example, the tangent plane to a surface (such as the sphere or the cylinder) at a point can be rolled around the surface to obtain the tangent plane at other points.
Properties
The tangential contact between the surfaces being rolled over one another provides a relation between points on the two surfaces. If this relation is (perhaps only in a local sense) a bijection between the surfaces, then the two surfaces are said to be developable on each other or developments of each other. Differently put, the correspondence provides an isometry, locally, between the two surfaces.
In particular, if one of the surfaces is a plane, then the other is called a developable surface: thus a developable surface is one which is locally isometric to a plane. The cylinder is developable, but the sphere is not.
Flat connections
Development can be generalized further using flat connections. From this point of view, rolling the tangent plane over a surface defines an affine connection on the surface (it provides an example of parallel transport along a curve), and a developable surface is one for which this connection is flat.
More generally any flat Cartan connection on a manifold defines a development of that manifold onto the model space. Perhaps the most famous example is the development of conformally flat n-manifolds, in which the model-space is the n-sphere. The development of a conformally flat manifold is a conformal local diffeomorphism from the universal cover of the manifold to the n-sphere.
Undevelopable surfaces
The class of double-curved surfaces (undevelopable surfaces) contains objects that cannot be simply unfolded (developed). Such surfaces can be developed only approximately with some distortions of linear surface elements (see the Stretched grid method)
See also
Developable surface
Ruled surface
References
Differential geometry
Connection (mathematics)
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https://en.wikipedia.org/wiki/Anscombe%27s%20quartet
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Anscombe's quartet comprises four data sets that have nearly identical simple descriptive statistics, yet have very different distributions and appear very different when graphed. Each dataset consists of eleven (x, y) points. They were constructed in 1973 by the statistician Francis Anscombe to demonstrate both the importance of graphing data when analyzing it, and the effect of outliers and other influential observations on statistical properties. He described the article as being intended to counter the impression among statisticians that "numerical calculations are exact, but graphs are rough".
Data
For all four datasets:
The first scatter plot (top left) appears to be a simple linear relationship, corresponding to two correlated variables, where y could be modelled as gaussian with mean linearly dependent on x.
For the second graph (top right), while a relationship between the two variables is obvious, it is not linear, and the Pearson correlation coefficient is not relevant. A more general regression and the corresponding coefficient of determination would be more appropriate.
In the third graph (bottom left), the modelled relationship is linear, but should have a different regression line (a robust regression would have been called for). The calculated regression is offset by the one outlier, which exerts enough influence to lower the correlation coefficient from 1 to 0.816.
Finally, the fourth graph (bottom right) shows an example when one high-leverage point is enough to produce a high correlation coefficient, even though the other data points do not indicate any relationship between the variables.
The quartet is still often used to illustrate the importance of looking at a set of data graphically before starting to analyze according to a particular type of relationship, and the inadequacy of basic statistic properties for describing realistic datasets.
The datasets are as follows. The x values are the same for the first three datasets.
It is not known how Anscombe created his datasets. Since its publication, several methods to generate similar data sets with identical statistics and dissimilar graphics have been developed.
One of these, the Datasaurus Dozen, consists of points tracing out the outline of a dinosaur, plus twelve other data sets that have the same summary statistics.
See also
Exploratory data analysis
Goodness of fit
Regression validation
Simpson's paradox
Statistical model validation
References
External links
Department of Physics, University of Toronto
Dynamic Applet made in GeoGebra showing the data & statistics and also allowing the points to be dragged (Set 5).
Animated examples from Autodesk called the "Datasaurus Dozen".
Documentation for the datasets in R.
Misuse of statistics
Statistical charts and diagrams
Statistical data sets
1973 introductions
1973 in science
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https://en.wikipedia.org/wiki/Richardson%20Professor%20of%20Applied%20Mathematics
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The Richardson Chair of Applied Mathematics is an endowed professorial position in the School of Mathematics, University of Manchester, England. The chair was founded by an endowment of £3,600 from one John Richardson, in 1890. The endowment was originally used to support the Richardson Lectureship in Mathematics.
One holder of the Richardson Lectureship was John Edensor Littlewood (1907-1910). The position lapsed in 1918, but was resurrected as a lectureship in Pure Mathematics between 1935 and 1944. There was then a further hiatus until the establishment of the Richardson Chair of Applied Mathematics in 1998. The current holder (since 1998) is Nicholas Higham.
A complete list of Richardson Lecturers and Professors is as follows:
F. T. Swanwick (1891-1907) Lecturer in Mathematics
J. E. Littlewood (1907-1910) Lecturer in Mathematics
H. R. Hasse (1910-1912) Lecturer in Mathematics
W. D. Evans (1912-1918) Lecturer in Mathematics
W. N. Bailey (1935-1944) Lecturer in Pure Mathematics
N. J. Higham (1998- ) Professor of Applied Mathematics
The School of Mathematics has three other endowed chairs, the others being the Beyer Chair, the Fielden Chair of Pure Mathematics and the Sir Horace Lamb Chair.
References
Professorships in mathematics
Professorships at the University of Manchester
Mathematics education in the United Kingdom
1890 establishments in England
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https://en.wikipedia.org/wiki/Josef%20Paldus
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Josef Paldus, (born November 25, 1935 in Bzí, Czech Republic, died January 15, 2023 in Kitchener, Canada) was a Distinguished Professor Emeritus of Applied Mathematics at the University of Waterloo, Ontario, Canada.
Josef Paldus became associate professor at the University of Waterloo after emigration to Canada from (former) Czechoslovakia in 1968. In 1975 he was promoted to full professor at this university, and he retired in 2001.
Paldus' research was mainly in the field of quantum chemistry and especially in the mathematical aspects of it. He is known for his collaborative work with Jiří Čížek on coupled cluster theory. Paldus and Čížek adapted the many-body coupled cluster method to many-electron systems, thus making it a viable method in the study of the electronic correlation that occurs in atoms and molecules.
Other well-known work by Paldus is the Unitary Group Approach. This approach regards the computation of Hamiltonian matrix elements over N-electron spin eigenstates that appear in electronic correlation problems.
Josef Paldus has (co)authored over 330 scientific papers.
Paldus possessed several doctoral degrees: In 1961 he received a PhD in Physical Chemistry at the Czechoslovak Academy of Sciences. In 1995 he became DrSc at the Charles University in Prague. In June 2006 he became Dr.h.c. at the Comenius University in Bratislava and in June 2008 he was awarded the honorary degree Docteur Honoris Causa by the Université Louis Pasteur in Strasbourg, France.
Other honors received by Paldus are inter alia:
1981 Corresponding Member of the European Academy of Sciences, Arts and Letters, Paris.
1983 Fellow of the Royal Society of Canada.
1984 Member of the International Academy of Quantum Molecular Science.
1985 Member of the New York Academy of Sciences.
1990 Member of the Board of Directors, International Society for Theoretical Chemical Physics.
1992 J. Heyrovský Gold Medal of the Czechoslovak Academy of Sciences.
1994 Gold Medal of the Faculty of Mathematics and Physics, Comenius University, Bratislava, Slovakia.
1995 Honorary Member of The Learned Society of the Czech Republic.
2002 Fellow of the Fields Institute for Research in Mathematical Sciences.
2005 Gold Medal of Charles University.
2007 Honorary Medal De Scientia et Humanitate Optime Meritis of the Academy of Sciences of the Czech Republic, Prague, Czech Republic.
2010 Honorary Fellowship of the European Society of Computational Methods in Sciences and Engineering.
2013 Fellow of the American Institute of Physics.
See also
List of University of Waterloo people
References
External links
Josef Paldus
Josef Paldus Department of Applied Mathematics, University of Waterloo homepage
1935 births
Living people
Fellows of the Royal Society of Canada
Members of the International Academy of Quantum Molecular Science
Theoretical chemists
Academic staff of the University of Waterloo
Computational chemists
Fellows of the American Physical Society
Charles University alumni
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https://en.wikipedia.org/wiki/Catenary%20ring
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In mathematics, a commutative ring R is catenary if for any pair of prime ideals p, q, any two strictly increasing chains
p = p0 ⊂ p1 ⊂ ... ⊂ pn = q
of prime ideals are contained in maximal strictly increasing chains from p to q of the same (finite) length. In a geometric situation, in which the dimension of an algebraic variety attached to a prime ideal will decrease as the prime ideal becomes bigger, the length of such a chain n is usually the difference in dimensions.
A ring is called universally catenary if all finitely generated algebras over it are catenary rings.
The word 'catenary' is derived from the Latin word catena, which means "chain".
There is the following chain of inclusions.
Dimension formula
Suppose that A is a Noetherian domain and B is a domain containing A that is finitely generated over A. If P is a prime ideal of B and p its intersection with A, then
The dimension formula for universally catenary rings says that equality holds if A is universally catenary. Here κ(P) is the residue field of P and tr.deg. means the transcendence degree (of quotient fields). In fact, when A is not universally catenary, but , then equality also holds.
Examples
Almost all Noetherian rings that appear in algebraic geometry are universally catenary.
In particular the following rings are universally catenary:
Complete Noetherian local rings
Dedekind domains (and fields)
Cohen-Macaulay rings (and regular local rings)
Any localization of a universally catenary ring
Any finitely generated algebra over a universally catenary ring.
A ring that is catenary but not universally catenary
It is delicate to construct examples of Noetherian rings that are not universally catenary. The first example was found by , who found a 2-dimensional Noetherian local domain that is catenary but not universally catenary.
Nagata's example is as follows. Choose a field k and a formal power series z=Σi>0aixi in the ring S of formal power series in x over k such that z and x are algebraically independent.
Define z1 = z and zi+1=zi/x–ai.
Let R be the (non-Noetherian) ring generated by x and all the elements zi.
Let m be the ideal (x), and let n be the ideal generated by x–1 and all the elements zi. These are both maximal ideals of R, with residue fields isomorphic to k. The local ring Rm is a regular local ring of dimension 1 (the proof of this uses the fact that z and x are algebraically independent) and the local ring Rn is a regular Noetherian local ring of dimension 2.
Let B be the localization of R with respect to all elements not in either m or n. Then B is a 2-dimensional Noetherian semi-local ring with 2 maximal ideals, mB (of height 1) and nB (of height 2).
Let I be the Jacobson radical of B, and let A = k+I. The ring A is a local domain of dimension 2 with maximal ideal I, so is catenary because all 2-dimensional local domains are catenary. The ring A is Noetherian because B is Noetherian and is a finite A-module. However A is not universally cat
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https://en.wikipedia.org/wiki/Claiton%20%28footballer%2C%20born%201978%29
|
Claiton Alberto Fontoura dos Santos, (born 25 January 1978) better known as Claiton, is a Brazilian football manager and former player who played as a midfielder.
Club statistics
Honours
Internacional
Rio Grande do Sul State Championship: 1997, 2002, 2003
Vitória
Bahia State Championship: 2000
Bahia
Bahia State Championship: 2001
Nordeste Cup: 2001
Santos
Brazilian League: 2004
Flamengo
Guanabara Cup: 2007
Rio de Janeiro State Championship: 2007
External links
furacao
CBF
zerpzero.pt
globoesporte
sambafoot
Living people
1978 births
Footballers from Porto Alegre
Brazilian men's footballers
Brazilian football managers
Men's association football midfielders
Brazilian expatriate men's footballers
Expatriate men's footballers in Japan
Expatriate men's footballers in Switzerland
Campeonato Brasileiro Série A players
Swiss Super League players
J1 League players
J2 League players
Sport Club Internacional players
Esporte Clube Bahia players
Esporte Clube Vitória players
Servette FC players
Santos FC players
Nagoya Grampus players
Botafogo de Futebol e Regatas players
CR Flamengo footballers
Hokkaido Consadole Sapporo players
Club Athletico Paranaense players
Esporte Clube Pelotas players
Esporte Clube Novo Hamburgo players
Esporte Clube Passo Fundo players
Alecrim Futebol Clube players
Clube Esportivo Aimoré managers
Sport Club São Paulo managers
Esporte Clube Cruzeiro managers
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https://en.wikipedia.org/wiki/Random%20measure
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In probability theory, a random measure is a measure-valued random element. Random measures are for example used in the theory of random processes, where they form many important point processes such as Poisson point processes and Cox processes.
Definition
Random measures can be defined as transition kernels or as random elements. Both definitions are equivalent. For the definitions, let be a separable complete metric space and let be its Borel -algebra. (The most common example of a separable complete metric space is )
As a transition kernel
A random measure is a (a.s.) locally finite transition kernel from a (abstract) probability space to .
Being a transition kernel means that
For any fixed , the mapping
is measurable from to
For every fixed , the mapping
is a measure on
Being locally finite means that the measures
satisfy for all bounded measurable sets
and for all except some -null set
In the context of stochastic processes there is the related concept of a stochastic kernel, probability kernel, Markov kernel.
As a random element
Define
and the subset of locally finite measures by
For all bounded measurable , define the mappings
from to . Let be the -algebra induced by the mappings on and the -algebra induced by the mappings on . Note that .
A random measure is a random element from to that almost surely takes values in
Basic related concepts
Intensity measure
For a random measure , the measure satisfying
for every positive measurable function is called the intensity measure of . The intensity measure exists for every random measure and is a s-finite measure.
Supporting measure
For a random measure , the measure satisfying
for all positive measurable functions is called the supporting measure of . The supporting measure exists for all random measures and can be chosen to be finite.
Laplace transform
For a random measure , the Laplace transform is defined as
for every positive measurable function .
Basic properties
Measurability of integrals
For a random measure , the integrals
and
for positive -measurable are measurable, so they are random variables.
Uniqueness
The distribution of a random measure is uniquely determined by the distributions of
for all continuous functions with compact support on . For a fixed semiring that generates in the sense that , the distribution of a random measure is also uniquely determined by the integral over all positive simple -measurable functions .
Decomposition
A measure generally might be decomposed as:
Here is a diffuse measure without atoms, while is a purely atomic measure.
Random counting measure
A random measure of the form:
where is the Dirac measure, and are random variables, is called a point process or random counting measure. This random measure describes the set of N particles, whose locations are given by the (generally vector valued) random variables . The diffuse component is null for a counting measure.
In the formal not
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https://en.wikipedia.org/wiki/Oberwolfach
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Oberwolfach () is a town in the district of Ortenau in Baden-Württemberg, Germany. It is the site of the Oberwolfach Research Institute for Mathematics, or Mathematisches Forschungsinstitut Oberwolfach.
Geography
Geographical situation
The town of Oberwolfach lies between 270 and 948 meters above sea level in the central Schwarzwald (Black Forest) on the river Wolf, a tributary of the Kinzig.
Neighbouring localities
The district is neighboured by Bad Peterstal-Griesbach to the north, Bad Rippoldsau-Schapbach in Landkreis Freudenstadt to the east, by the towns of Wolfach and Hausach to the south, and by Oberharmersbach to the west.
References
External links
Gemeinde Oberwolfach: Official Homepage (in German)
Oberwolfach Mineral Museum
Ortenaukreis
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https://en.wikipedia.org/wiki/Martin%20Kreuzer
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Martin Kreuzer (born 15 July 1962 in Ihrlerstein) is a German mathematics professor and chess player who holds the chess titles of International Correspondence Chess Grandmaster and FIDE Master.
Kreuzer did his graduate studies in mathematics at the University of Regensburg, located on the Danube River in Bavaria. After spending one year in the United States as a foreign exchange student at Brandeis University in Boston, he finished his diploma in Mathematics in Regensburg in 1986. Next came a post-doctoral fellowship at Queen's University in Kingston, Ontario, Canada, from 1989 to 1991, working in algebraic geometry with Professor Anthony Geramita. He then returned to Germany, worked as a scientific assistant at the University of Regensburg and gained his habilitation in Mathematics in 1997. After substituting for the chair of algebraic geometry at the University of Bayreuth in 2000–2001 and for the chair of algebra at Technical University of Dortmund from 2002 to 2007, he moved to Passau where he holds the chair of symbolic computation at the University of Passau. His main research interests are computer algebra, cryptography, computational commutative algebra, algebraic geometry, and their industrial applications.
Kreuzer's chess skills have earned him the FIDE Master title for over-the-board play. During his short stay in Canada, he finished fourth at the 1990 Open Canadian Chess Championship at Edmundston. Further notable over-the-board results include the prize for the best player without Elo number in the Elo rating system at the 1987 Open "Chess for Peace" in London and a fourth place at the first Novotel Open in Genova, Italy in 1997.
He gained the title of International Correspondence Chess Grandmaster (GMC) 1994, from his result in the 1988–95 Von Massow Memorial tournament. Kreuzer played board six in the finals, on the German team which shared the gold medal at the 11th Correspondence Chess Olympiad, 1992–1999. He was a member the German team which won the 12th Correspondence Chess Olympiad, 1998–2004, and a member of the German team which won the 13th Correspondence Chess Olympiad, 2004–2009.
A selection of his games can be found at chessbase.com.
Writings
Computational Commutative Algebra I, by Martin Kreuzer and Lorenzo Robbiano, Heidelberg, Springer-Verlag 2000, .
Computational Commutative Algebra II, by Martin Kreuzer and Lorenzo Robbiano, Heidelberg, Springer-Verlag 2005, .
References
External links
1962 births
Living people
20th-century German mathematicians
21st-century German mathematicians
German male writers
German chess players
Correspondence chess grandmasters
Chess FIDE Masters
Brandeis University alumni
Queen's University at Kingston alumni
Academic staff of the University of Passau
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