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https://en.wikipedia.org/wiki/California%20State%20Summer%20School%20for%20Mathematics%20and%20Science
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The California State Summer School for Mathematics and Science (COSMOS) is a summer program for high school students in California for the purpose of preparing them for careers in mathematics and sciences. It is often abbreviated COSMOS, although COSMOS does not contain the correct letters to create an accurate abbreviation. The program is hosted on four different campuses of the University of California, at Davis, Irvine, San Diego, and Santa Cruz.
History
COSMOS was established by the California State Legislature in the summer of 2000 to stimulate the interests of and provide opportunities for talented California high school students. The California State Summer School for Mathematics & Science is modeled after the California State Summer School for the Arts. In the first summer, 292 students enrolled in the program. Each COSMOS campus only holds 150 students, so selection is competitive. It is a great experience in exploring the sciences and a good activity for college applications, especially the University of California application. This program is designed for extremely gifted students who make amazing discoveries in STEM (Science, Technology, Engineering, Mathematics) areas.
References
State evaluation report of the COSMOS program
External links
Official site
Schools in California
Science education in the United States
Schools of mathematics
Summer schools
Science and technology in California
2000 establishments in California
Mathematics summer camps
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https://en.wikipedia.org/wiki/Candidate%20%28disambiguation%29
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A candidate is a person or thing seeking or being considered for some kind of position:
Candidate may also refer to:
Candidate solution, in mathematics
Candidates Tournament, a qualification event for the World Chess Championship
Candidate (degree)
Film
The Candidate (1959 film), an Argentine drama film
The Candidate (1964 film) by Robert Angus, aka Party Girls for the Candidate, aka The Playmates and the Candidate
The Candidate (1972 film) by Michael Ritchie, with Robert Redford
The Candidate (1980 film) by Stefan Aust, Alexander Kluge, Volker Schlöndorff, Alexander von Eschwege, German title: Der Kandidat
The Candidate (1998 film), Taiwanese film by Neil Peng
The Candidate (2008 film) by Kasper Barfoed, Danish title: Kandidaten
Candidate (2013 film) by Jonáš Karásek, Slovak title: Kandidát
The Realm (2018 film) by Rodrigo Sorogoyen, also known as The Candidate, Spanish title: El reino
Television
"The Candidate" (Arrow), an episode of Arrow
The Candidate (TV series), an Afghan TV series supported by the CEPPS agreement
"The Candidate" (Lost), an episode of the sixth and final season of the television drama Lost
"The Candidate" (Frasier), an episode of Frasier
La candidata, a Mexican telenovela
El Candidato (2020), known as The Candidate in English, a Mexican TV series.
Music
Candidate (band), a British rock group
The Candidate (album), a 1979 album by Steve Harley
"Candidate" (David Bowie song)
"Candidate" (Joy Division song)
See also
Candidate of Sciences, a post-graduate scientific degree in many former Eastern Bloc countries
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https://en.wikipedia.org/wiki/John%20Leech%20%28mathematician%29
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John Leech (21 July 1926 in Weybridge, Surrey – 28 September 1992 in Scotland) was a British mathematician working in number theory, geometry and combinatorial group theory. He is best known for his discovery of the Leech lattice in 1965. He also discovered Ta(3) in 1957. Leech was married to Jenifer Haselgrove, a British radio scientist.
References
External links
MacTutor History of Mathematics biography
20th-century British mathematicians
1926 births
1992 deaths
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https://en.wikipedia.org/wiki/Gauss%E2%80%93Manin%20connection
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In mathematics, the Gauss–Manin connection is a connection on a certain vector bundle over a base space S of a family of algebraic varieties . The fibers of the vector bundle are the de Rham cohomology groups of the fibers of the family. It was introduced by for curves S and by in higher dimensions.
Flat sections of the bundle are described by differential equations; the best-known of these is the Picard–Fuchs equation, which arises when the family of varieties is taken to be the family of elliptic curves. In intuitive terms, when the family is locally trivial, cohomology classes can be moved from one fiber in the family to nearby fibers, providing the 'flat section' concept in purely topological terms. The existence of the connection is to be inferred from the flat sections.
Intuition
Consider a smooth morphism of schemes over characteristic 0. If we consider these spaces as complex analytic spaces, then the Ehresmann fibration theorem tells us that each fiber is a smooth manifold and each fiber is diffeomorphic. This tells us that the de-Rham cohomology groups are all isomorphic. We can use this observation to ask what happens when we try to differentiate cohomology classes using vector fields from the base space .
Consider a cohomology class such that where is the inclusion map. Then, if we consider the classes
eventually there will be a relation between them, called the Picard–Fuchs equation. The Gauss–Manin connection is a tool which encodes this information into a connection on the flat vector bundle on constructed from the .
Example
A commonly cited example is the Dwork construction of the Picard–Fuchs equation. Let
be the elliptic curve .
Here, is a free parameter describing the curve; it is an element of the complex projective line (the family of hypersurfaces in dimensions of degree n, defined analogously, has been intensively studied in recent years, in connection with the modularity theorem and its extensions). Thus, the base space of the bundle is taken to be the projective line. For a fixed in the base space, consider an element of the associated de Rham cohomology group
Each such element corresponds to a period of the elliptic curve. The cohomology is two-dimensional. The Gauss–Manin connection corresponds to the second-order differential equation
D-module explanation
In the more abstract setting of D-module theory, the existence of such equations is subsumed in a general discussion of the direct image.
Equations "arising from geometry"
The whole class of Gauss–Manin connections has been used to try to formulate the concept of differential equations that "arise from geometry". In connection with the Grothendieck p-curvature conjecture, Nicholas Katz proved that the class of Gauss–Manin connections with algebraic number coefficients satisfies the conjecture. This result is directly connected with the Siegel G-function concept of transcendental number theory, for meromorphic function solutions. The Bombieri–
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https://en.wikipedia.org/wiki/David%20Corfield
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David Neil Corfield is a British philosopher specializing in philosophy of mathematics and philosophy of psychology. He is Senior Lecturer in Philosophy at the University of Kent.
Education
Corfield studied mathematics at the University of Cambridge, and later earned his MSc and PhD in the philosophy of science and mathematics at King's College London. His doctoral advisor was Donald A. Gillies.
Work
Corfield is the author of Towards a Philosophy of Real Mathematics (2003), in which he argues that the philosophical implications of mathematics did not stop with Kurt Gödel's incompleteness theorems. He has also co-authored a book with Darian Leader about psychology and psychosomatic medicine, Why Do People Get Ill? (2007).
He joined the University of Kent in September 2007 in which he is currently a Senior Lecturer.
He is a member of the informal steering committee of nLab, a wiki-lab for collaborative work on mathematics, physics, and philosophy.
Bibliography
"Assaying Lakatos's Philosophy of Mathematics", Studies in History and Philosophy of Science 28(1), 99–121 (1997).
"Beyond the Methodology of Mathematical Research Programmes", Philosophia Mathematica 6, 272–301 (1998).
"Come the Revolution...", critical notice on The Principles of Mathematics Revisited by Jaakko Hintikka, Philosophical Books 39(3), 150–6 (1998).
"The Importance of Mathematical Conceptualisation", Studies in History and Philosophy of Science 32(3), 507–533 (2001).
"Bayesianism in Mathematics", in Corfield D. and Williamson J. (eds.) (2001), 175–201.
(with J. Williamson), "Bayesianism into the 21st Century", in Corfield D. and Williamson J. (eds.) (2001), 1–16.
Corfield D. and Williamson J. (eds.), Foundations of Bayesianism, Kluwer Applied Logic Series (2001).
"Argumentation and the Mathematical Process", G. Kampis, L. Kvasz & M. Stöltzner (eds.) Appraising Lakatos: Mathematics, Methodology, and the Man, 115–138. Kluwer, Dordrecht (2002).
Review of Conceptual Mathematics by F. W. Lawvere and S. Schanuel and A Primer of Infinitesimal Analysis by J. Bell, Studies in History and Philosophy of Modern Physics, 33B(2), 359–366 (2002).
"From Mathematics to Psychology: Lacan's Missed Encounters" in J. Glynos and Y. Stavrakakis (eds.) Lacan and Science, Karnac Books, 179–206 (2002).
Towards a Philosophy of Real Mathematics, Cambridge University Press (2003).
Review of Opening Skinner's Box by Lauren Slater, The Guardian, 27 March 2004.
Review of
"Categorification as a Heuristic Device", in D. Gillies and C. Cellucci (eds.), Mathematical Reasoning and Heuristics, King's College Publications (2005).
"Some Implications of the Adoption of Category Theory for Philosophy", in Giandomenico Sica (ed.), What is Category Theory?, Polimetrica (2006), 75–94.
(with Darian Leader) Why Do People Get Ill?, Hamish Hamilton (2007).
Modal Homotopy Type Theory: The Prospect of a New Logic for Philosophy, Oxford University Press (2020).
References
External links
Faculty page
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https://en.wikipedia.org/wiki/GD
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GD may refer to:
Arts and entertainment
G-Dragon (born 1988), leader of the South Korean musical group Big Bang
Grateful Dead, an American rock band
Green Day, an American rock band
Geometry Dash, a rhythm-based video game for mobile and PC
Business and economics
Gardner Denver, a US-based manufacturer of industrial equipment
General Dynamics, a US-based defense conglomerate
Good Delivery, a specification for gold and silver bars
Composite Index on the Athens Stock Exchange (stock symbol GD)
Mathematics, science and technology
Biology and medicine
Gaucher's disease, a lipid storage diseases
Gender dysphoria, distress caused by a difference between the sex and gender a person was assigned at birth
Generalized dystonia, a neurological movement disorder
Gestational diabetes, a form of diabetes associated with pregnancy
Graves' disease, an autoimmune thyroid disorder
Gain of deiodinases or Sum activity of peripheral deiodinases, used in diagnosis of thyroid disorders
Grover's disease, another name for Transient acantholytic dermatosis
Video game addiction, also known as gaming disorder
Chemistry
Gadolinium, symbol Gd, a chemical element
Soman, a toxic chemical (NATO designation GD)
Computing
GD Graphics Library, for dynamically manipulating images
GD-ROM, storage media for the Sega Dreamcast
.gd, the country code top-level domain for Grenada
GDScript, the built-in language for the Godot game engine
Mathematics
Gaussian distribution, also called the normal distribution, an important family of continuous probability distributions
Generalized Dirichlet distribution, a probability distribution used in statistics
Gudermannian function, used in map-making
Places
Georgia Dome, a stadium in Atlanta, Georgia and the home of the Atlanta Falcons
Grenada (ISO 3166 country code)
Guangdong, a province of China (Guobiao abbreviation GD)
Other uses
Gangster Disciples, a black street gang in the United States
General Delivery (French: Poste restante), a service where the post office holds mail until the recipient calls for it
Georgian Dream, a political party in the country of Georgia
Goal difference, in sport
Scottish Gaelic language (ISO 639-1 code )
Subaru Impreza (second generation) sedan (ID code: GD)
G-d, a substitution of God used by some religiously observant Jews to refer to YHWH
Toyota GD engine, a straight-4 piston diesel engine developed by Toyota in 2015
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https://en.wikipedia.org/wiki/Refinable%20function
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In mathematics, in the area of wavelet analysis, a refinable function is a function which fulfils some kind of self-similarity. A function is called refinable with respect to the mask if
This condition is called refinement equation, dilation equation or two-scale equation.
Using the convolution (denoted by a star, *) of a function with a discrete mask and the dilation operator one can write more concisely:
It means that one obtains the function, again, if you convolve the function with a discrete mask and then scale it back.
There is a similarity to iterated function systems and de Rham curves.
The operator is linear.
A refinable function is an eigenfunction of that operator.
Its absolute value is not uniquely defined.
That is, if is a refinable function,
then for every the function is refinable, too.
These functions play a fundamental role in wavelet theory as scaling functions.
Properties
Values at integral points
A refinable function is defined only implicitly.
It may also be that there are several functions which are refinable with respect to the same mask.
If shall have finite support
and the function values at integer arguments are wanted,
then the two scale equation becomes a system of simultaneous linear equations.
Let be the minimum index and be the maximum index
of non-zero elements of , then one obtains
Using the discretization operator, call it here, and the transfer matrix of , named , this can be written concisely as
This is again a fixed-point equation. But this one can now be considered as an eigenvector-eigenvalue problem. That is, a finitely supported refinable function exists only (but not necessarily), if has the eigenvalue 1.
Values at dyadic points
From the values at integral points you can derive the values at dyadic points,
i.e. points of the form , with and .
The star denotes the convolution of a discrete filter with a function.
With this step you can compute the values at points of the form .
By replacing iteratedly by you get the values at all finer scales.
Convolution
If is refinable with respect to , and is refinable with respect to , then is refinable with respect to .
Differentiation
If is refinable with respect to ,
and the derivative exists, then is refinable with respect to .
This can be interpreted as a special case of the convolution property, where one of the convolution operands is a derivative of the Dirac impulse.
Integration
If is refinable with respect to , and there is an antiderivative with , then the antiderivative is refinable with respect to mask where the constant must fulfill .
If has bounded support,
then we can interpret integration as convolution with the Heaviside function and apply the convolution law.
Scalar products
Computing the scalar products of two refinable functions and their translates can be broken down to the two above properties.
Let be the translation operator. It holds
where is the adjoint of with respect to convolution, i.e.,
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https://en.wikipedia.org/wiki/Essentially%20surjective%20functor
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In mathematics, specifically in category theory, a functor
is essentially surjective (or dense) if each object of is isomorphic to an object of the form for some object of .
Any functor that is part of an equivalence of categories is essentially surjective. As a partial converse, any full and faithful functor that is essentially surjective is part of an equivalence of categories.
Notes
References
External links
Functors
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https://en.wikipedia.org/wiki/Selman%20Akbulut
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Selman Akbulut (born 1949) is a Turkish mathematician, specializing in research in topology, and geometry. He was a professor at Michigan State University until February 2020.
Career
In 1975 he earned his Ph.D. from the University of California, Berkeley as a student of Robion Kirby. In topology, he has worked on handlebody theory, low-dimensional manifolds, symplectic topology, G2 manifolds. In the topology of real-algebraic sets, he and Henry C. King proved that every compact piecewise-linear manifold is a real-algebraic set; they discovered new topological invariants of real-algebraic sets.
He was a visiting scholar several times at the Institute for Advanced Study (in 1975-76, 1980–81, 2002, and 2005).
On February 14, 2020, Akbulut was removed from his tenured position at MSU by the Board of Trustees, after complaints regarding his teaching attendance and communications with colleagues.
Contributions
He has developed 4-dimensional handlebody techniques, settling conjectures and solving problems about 4-manifolds, such as a conjecture of Christopher Zeeman, the Harer–Kas–Kirby conjecture, a problem of Martin Scharlemann, and problems of Sylvain Cappell and Julius Shaneson.
He constructed an exotic compact 4-manifold (with boundary) from which he discovered "Akbulut corks".
His most recent results concern the 4-dimensional smooth Poincaré conjecture. He has supervised 14 Ph.D students as of 2019. He has more than 100 papers and three books published, and several books edited.
Notes
External links
Akbulut's homepage
Akbulut's papers at ArXiv
Akbulut-King invariants
Real algebraic geometry
Akbulut cork
20th-century Turkish mathematicians
21st-century Turkish mathematicians
Academic scandals
Topologists
University of California, Berkeley alumni
Institute for Advanced Study visiting scholars
Living people
1949 births
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https://en.wikipedia.org/wiki/Surface%20subgroup%20conjecture
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In mathematics, the surface subgroup conjecture of Friedhelm Waldhausen states that the fundamental group of every closed, irreducible 3-manifold with infinite fundamental group has a surface subgroup. By "surface subgroup" we mean the fundamental group of a closed surface not the 2-sphere. This problem is listed as Problem 3.75 in Robion Kirby's problem list.
Assuming the geometrization conjecture, the only open case was that of closed hyperbolic 3-manifolds. A proof of this case was announced in the summer of 2009 by Jeremy Kahn and Vladimir Markovic and outlined in a talk August 4, 2009 at the FRG (Focused Research Group) Conference hosted by the University of Utah. A preprint appeared in the arxiv.org server in October 2009. Their paper was published in the Annals of Mathematics in 2012. In June 2012, Kahn and Markovic were given the Clay Research Awards by the Clay Mathematics Institute at a ceremony in Oxford.
See also
Virtually Haken conjecture
Ehrenpreis conjecture
References
3-manifolds
Conjectures
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https://en.wikipedia.org/wiki/Hamming%20space
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In statistics and coding theory, a Hamming space (named after American mathematician Richard Hamming) is usually the set of all binary strings of length N. It is used in the theory of coding signals and transmission.
More generally, a Hamming space can be defined over any alphabet (set) Q as the set of words of a fixed length N with letters from Q. If Q is a finite field, then a Hamming space over Q is an N-dimensional vector space over Q. In the typical, binary case, the field is thus GF(2) (also denoted by Z2).
In coding theory, if Q has q elements, then any subset C (usually assumed of cardinality at least two) of the N-dimensional Hamming space over Q is called a q-ary code of length N; the elements of C are called codewords. In the case where C is a linear subspace of its Hamming space, it is called a linear code. A typical example of linear code is the Hamming code. Codes defined via a Hamming space necessarily have the same length for every codeword, so they are called block codes when it is necessary to distinguish them from variable-length codes that are defined by unique factorization on a monoid.
The Hamming distance endows a Hamming space with a metric, which is essential in defining basic notions of coding theory such as error detecting and error correcting codes.
Hamming spaces over non-field alphabets have also been considered, especially over finite rings (most notably over Z4) giving rise to modules instead of vector spaces and ring-linear codes (identified with submodules) instead of linear codes. The typical metric used in this case the Lee distance. There exist a Gray isometry between (i.e. GF(22m)) with the Hamming distance and (also denoted as GR(4,m)) with the Lee distance.
References
Coding theory
Linear algebra
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https://en.wikipedia.org/wiki/Tameness%20theorem
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In mathematics, the tameness theorem states that every complete hyperbolic 3-manifold with finitely generated fundamental group is topologically tame, in other words homeomorphic to the interior of a compact 3-manifold.
The tameness theorem was conjectured by . It was proved by and, independently, by Danny Calegari and David Gabai. It is one of the fundamental properties of geometrically infinite hyperbolic 3-manifolds, together with the density theorem for Kleinian groups and the ending lamination theorem.
It also implies the Ahlfors measure conjecture.
History
Topological tameness may be viewed as a property of the ends of the manifold, namely, having a local product structure. An analogous statement is well known in two dimensions, that is, for surfaces. However, as the example of Alexander horned sphere shows, there are wild embeddings among 3-manifolds, so this property is not automatic.
The conjecture was raised in the form of a question by Albert Marden, who proved that any geometrically finite hyperbolic 3-manifold is topologically tame. The conjecture was also called the Marden conjecture or the tame ends conjecture.
There had been steady progress in understanding tameness before the conjecture was resolved. Partial results had been obtained by Thurston, Brock, Bromberg, Canary, Evans, Minsky, Ohshika. An important sufficient condition for tameness in terms of splittings of the fundamental group had been obtained by Bonahon.
The conjecture was proved in 2004 by Ian Agol, and independently, by Danny Calegari and David Gabai. Agol's proof relies on the use of manifolds of pinched negative curvature and on Canary's trick of "diskbusting" that allows to replace a compressible end with an incompressible end, for which the conjecture has already been proved. The Calegari–Gabai proof is centered on the existence of certain closed, non-positively curved surfaces that they call "shrinkwrapped".
See also
Tame topology
References
.
.
.
.
3-manifolds
Conjectures that have been proved
Differential geometry
Hyperbolic geometry
Kleinian groups
Manifolds
Theorems in geometry
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https://en.wikipedia.org/wiki/Philippe%20V%C3%A9ron
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Philippe Véron (2 March 1939 – 7 August 2014) was a French astronomer. He worked at Observatoire de Haute Provence, where he was director from 1985 to 1994.
He studied variability and statistics of quasars, as well as elliptical galaxies. He was married to French astronomer Marie-Paule Véron-Cetty, and together with her compiled and maintained the Veron-Cetty Catalog of Quasars and Active Galactic Nuclei, whose thirteenth edition was published in 2010.
At the time of his death, he was working on the Dictionnaire des Astronomes Français 1850–1950 (Dictionary of French astronomers 1850–1950), which is a biographical encyclopedia. It is unpublished but is available online in PDF form at http://www.obs-hp.fr/dictionnaire.
Asteroid 5260 Philvéron is named after him.
References
External links
Recent publications, form NASA Astrophysics Data System website
20th-century French astronomers
1939 births
2014 deaths
21st-century French astronomers
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https://en.wikipedia.org/wiki/Transfer%20matrix
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In applied mathematics, the transfer matrix is a formulation in terms of a block-Toeplitz matrix of the two-scale equation, which characterizes refinable functions. Refinable functions play an important role in wavelet theory and finite element theory.
For the mask , which is a vector with component indexes from to ,
the transfer matrix of , we call it here, is defined as
More verbosely
The effect of can be expressed in terms of the downsampling operator "":
Properties
See also
Hurwitz determinant
References
(contains proofs of the above properties)
Wavelets
Numerical analysis
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https://en.wikipedia.org/wiki/False%20discovery%20rate
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In statistics, the false discovery rate (FDR) is a method of conceptualizing the rate of type I errors in null hypothesis testing when conducting multiple comparisons. FDR-controlling procedures are designed to control the FDR, which is the expected proportion of "discoveries" (rejected null hypotheses) that are false (incorrect rejections of the null). Equivalently, the FDR is the expected ratio of the number of false positive classifications (false discoveries) to the total number of positive classifications (rejections of the null). The total number of rejections of the null include both the number of false positives (FP) and true positives (TP). Simply put, FDR = FP / (FP + TP). FDR-controlling procedures provide less stringent control of Type I errors compared to family-wise error rate (FWER) controlling procedures (such as the Bonferroni correction), which control the probability of at least one Type I error. Thus, FDR-controlling procedures have greater power, at the cost of increased numbers of Type I errors.
History
Technological motivations
The modern widespread use of the FDR is believed to stem from, and be motivated by, the development in technologies that allowed the collection and analysis of a large number of distinct variables in several individuals (e.g., the expression level of each of 10,000 different genes in 100 different persons). By the late 1980s and 1990s, the development of "high-throughput" sciences, such as genomics, allowed for rapid data acquisition. This, coupled with the growth in computing power, made it possible to seamlessly perform a very high number of statistical tests on a given data set. The technology of microarrays was a prototypical example, as it enabled thousands of genes to be tested simultaneously for differential expression between two biological conditions.
As high-throughput technologies became common, technological and/or financial constraints led researchers to collect datasets with relatively small sample sizes (e.g. few individuals being tested) and large numbers of variables being measured per sample (e.g. thousands of gene expression levels). In these datasets, too few of the measured variables showed statistical significance after classic correction for multiple tests with standard multiple comparison procedures. This created a need within many scientific communities to abandon FWER and unadjusted multiple hypothesis testing for other ways to highlight and rank in publications those variables showing marked effects across individuals or treatments that would otherwise be dismissed as non-significant after standard correction for multiple tests. In response to this, a variety of error rates have been proposed—and become commonly used in publications—that are less conservative than FWER in flagging possibly noteworthy observations. The FDR is useful when researchers are looking for "discoveries" that will give them followup work (E.g.: detecting promising genes for followup studies),
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https://en.wikipedia.org/wiki/Wadge%20hierarchy
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In descriptive set theory, within mathematics, Wadge degrees are levels of complexity for sets of reals. Sets are compared by continuous reductions. The Wadge hierarchy is the structure of Wadge degrees. These concepts are named after William W. Wadge.
Wadge degrees
Suppose and are subsets of Baire space ωω. Then is Wadge reducible to or ≤W if there is a continuous function on ωω with . The Wadge order is the preorder or quasiorder on the subsets of Baire space. Equivalence classes of sets under this preorder are called Wadge degrees, the degree of a set is denoted by []W. The set of Wadge degrees ordered by the Wadge order is called the Wadge hierarchy.
Properties of Wadge degrees include their consistency with measures of complexity stated in terms of definability. For example, if ≤W and is a countable intersection of open sets, then so is . The same works for all levels of the Borel hierarchy and the difference hierarchy. The Wadge hierarchy plays an important role in models of the axiom of determinacy. Further interest in Wadge degrees comes from computer science, where some papers have suggested Wadge degrees are relevant to algorithmic complexity.
Wadge's lemma states that under the axiom of determinacy (AD), for any two subsets of Baire space, ≤W or ≤W ωω\. The assertion that the Wadge lemma holds for sets in Γ is the semilinear ordering principle for Γ or SLO(Γ). Any defines a linear order on the equivalence classes modulo complements. Wadge's lemma can be applied locally to any pointclass Γ, for example the Borel sets, Δ1n sets, Σ1n sets, or Π1n sets. It follows from determinacy of differences of sets in Γ. Since Borel determinacy is proved in ZFC, ZFC implies Wadge's lemma for Borel sets.
Wadge's lemma is similar to the cone lemma from computability theory.
Wadge's lemma via Wadge and Lipschitz games
The Wadge game is a simple infinite game discovered by William Wadge (pronounced "wage"). It is used to investigate the notion of continuous reduction for subsets of Baire space. Wadge had analyzed the structure of the Wadge hierarchy for Baire space with games by 1972, but published these results only much later in his PhD thesis. In the Wadge game , player I and player II each in turn play integers, and the outcome of the game is determined by checking whether the sequences x and y generated by players I and II are contained in the sets A and B, respectively. Player II wins if the outcome is the same for both players, i.e. is in if and only if is in . Player I wins if the outcome is different. Sometimes this is also called the Lipschitz game, and the variant where player II has the option to pass finitely many times is called the Wadge game.
Suppose that the game is determined. If player I has a winning strategy, then this defines a continuous (even Lipschitz) map reducing to the complement of , and if on the other hand player II has a winning strategy then you have a reduction of to . For example, suppose
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https://en.wikipedia.org/wiki/Brauer%27s%20theorem%20on%20forms
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There also is Brauer's theorem on induced characters.
In mathematics, Brauer's theorem, named for Richard Brauer, is a result on the representability of 0 by forms over certain fields in sufficiently many variables.
Statement of Brauer's theorem
Let K be a field such that for every integer r > 0 there exists an integer ψ(r) such that for n ≥ ψ(r) every equation
has a non-trivial (i.e. not all xi are equal to 0) solution in K.
Then, given homogeneous polynomials f1,...,fk of degrees r1,...,rk respectively with coefficients in K, for every set of positive integers r1,...,rk and every non-negative integer l, there exists a number ω(r1,...,rk,l) such that for n ≥ ω(r1,...,rk,l) there exists an l-dimensional affine subspace M of Kn (regarded as a vector space over K) satisfying
An application to the field of p-adic numbers
Letting K be the field of p-adic numbers in the theorem, the equation (*) is satisfied, since , b a natural number, is finite. Choosing k = 1, one obtains the following corollary:
A homogeneous equation f(x1,...,xn) = 0 of degree r in the field of p-adic numbers has a non-trivial solution if n is sufficiently large.
One can show that if n is sufficiently large according to the above corollary, then n is greater than r2. Indeed, Emil Artin conjectured that every homogeneous polynomial of degree r over Qp in more than r2 variables represents 0. This is obviously true for r = 1, and it is well known that the conjecture is true for r = 2 (see, for example, J.-P. Serre, A Course in Arithmetic, Chapter IV, Theorem 6). See quasi-algebraic closure for further context.
In 1950 Demyanov verified the conjecture for r = 3 and p ≠ 3, and in 1952 D. J. Lewis independently proved the case r = 3 for all primes p. But in 1966 Guy Terjanian constructed a homogeneous polynomial of degree 4 over Q2 in 18 variables that has no non-trivial zero. On the other hand, the Ax–Kochen theorem shows that for any fixed degree Artin's conjecture is true for all but finitely many Qp.
Notes
References
Diophantine equations
Theorems in number theory
P-adic numbers
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https://en.wikipedia.org/wiki/Classification%20theorem
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In mathematics, a classification theorem answers the classification problem "What are the objects of a given type, up to some equivalence?". It gives a non-redundant enumeration: each object is equivalent to exactly one class.
A few issues related to classification are the following.
The equivalence problem is "given two objects, determine if they are equivalent".
A complete set of invariants, together with which invariants are solves the classification problem, and is often a step in solving it.
A (together with which invariants are realizable) solves both the classification problem and the equivalence problem.
A canonical form solves the classification problem, and is more data: it not only classifies every class, but provides a distinguished (canonical) element of each class.
There exist many classification theorems in mathematics, as described below.
Geometry
Classification of Euclidean plane isometries
Classification theorems of surfaces
Classification of two-dimensional closed manifolds
Enriques–Kodaira classification of algebraic surfaces (complex dimension two, real dimension four)
Nielsen–Thurston classification which characterizes homeomorphisms of a compact surface
Thurston's eight model geometries, and the geometrization conjecture
Berger classification
Classification of Riemannian symmetric spaces
Classification of 3-dimensional lens spaces
Classification of manifolds
Algebra
Classification of finite simple groups
Classification of Abelian groups
Classification of Finitely generated abelian group
Classification of Rank 3 permutation group
Classification of 2-transitive permutation groups
Artin–Wedderburn theorem — a classification theorem for semisimple rings
Classification of Clifford algebras
Classification of low-dimensional real Lie algebras
Classification of Simple Lie algebras and groups
Classification of simple complex Lie algebras
Classification of simple real Lie algebras
Classification of centerless simple Lie groups
Classification of simple Lie groups
Bianchi classification
ADE classification
Langlands classification
Linear algebra
Finite-dimensional vector spaces (by dimension)
Rank–nullity theorem (by rank and nullity)
Structure theorem for finitely generated modules over a principal ideal domain
Jordan normal form
Sylvester's law of inertia
Analysis
Classification of discontinuities
Complex analysis
Classification of Fatou components
Mathematical physics
Classification of electromagnetic fields
Petrov classification
Segre classification
Wigner's classification
See also
Representation theorem
Comparison theorem
List of manifolds
Mathematical theorems
Mathematical classification systems
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https://en.wikipedia.org/wiki/Convergence%20tests
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In mathematics, convergence tests are methods of testing for the convergence, conditional convergence, absolute convergence, interval of convergence or divergence of an infinite series .
List of tests
Limit of the summand
If the limit of the summand is undefined or nonzero, that is , then the series must diverge. In this sense, the partial sums are Cauchy only if this limit exists and is equal to zero. The test is inconclusive if the limit of the summand is zero. This is also known as the nth-term test, test for divergence, or the divergence test.
Ratio test
This is also known as d'Alembert's criterion.
Suppose that there exists such that
If r < 1, then the series is absolutely convergent. If r > 1, then the series diverges. If r = 1, the ratio test is inconclusive, and the series may converge or diverge.
Root test
This is also known as the nth root test or Cauchy's criterion.
Let
where denotes the limit superior (possibly ; if the limit exists it is the same value).
If r < 1, then the series converges absolutely. If r > 1, then the series diverges. If r = 1, the root test is inconclusive, and the series may converge or diverge.
The root test is stronger than the ratio test: whenever the ratio test determines the convergence or divergence of an infinite series, the root test does too, but not conversely.
Integral test
The series can be compared to an integral to establish convergence or divergence. Let be a non-negative and monotonically decreasing function such that . If
then the series converges. But if the integral diverges, then the series does so as well.
In other words, the series converges if and only if the integral converges.
-series test
A commonly-used corollary of the integral test is the p-series test. Let . Then converges if .
The case of yields the harmonic series, which diverges. The case of is the Basel problem and the series converges to . In general, for , the series is equal to the Riemann zeta function applied to , that is .
Direct comparison test
If the series is an absolutely convergent series and for sufficiently large n , then the series converges absolutely.
Limit comparison test
If , (that is, each element of the two sequences is positive) and the limit exists, is finite and non-zero, then either both series converge or both series diverge.
Cauchy condensation test
Let be a non-negative non-increasing sequence. Then the sum converges if and only if the sum converges. Moreover, if they converge, then holds.
Abel's test
Suppose the following statements are true:
is a convergent series,
is a monotonic sequence, and
is bounded.
Then is also convergent.
Absolute convergence test
Every absolutely convergent series converges.
Alternating series test
Suppose the following statements are true:
are all positive,
and
for every n, .
Then and are convergent series.
This test is also known as the Leibniz criterion.
Dirichlet's test
If is a sequence of real numbers
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https://en.wikipedia.org/wiki/Critical%20line
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Critical line may refer to:
In mathematics, a specific subset of the complex numbers asserted by the Riemann hypothesis to be the locus of all non-trivial zeroes of the Riemann zeta function
Critical line theorem, a mathematical theorem saying that the proportion of nontrivial zeros of the Riemann zeta function lying on the critical line is greater than zero
Critical line (thermodynamics), a higher-dimensional equivalent of a critical point
Critical Line, an art exhibition
Critical line method, a procedure in Portfolio optimization
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https://en.wikipedia.org/wiki/Elementary%20divisors
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In algebra, the elementary divisors of a module over a principal ideal domain (PID) occur in one form of the structure theorem for finitely generated modules over a principal ideal domain.
If is a PID and a finitely generated -module, then M is isomorphic to a finite sum of the form
where the are nonzero primary ideals.
The list of primary ideals is unique up to order (but a given ideal may be present more than once, so the list represents a multiset of primary ideals); the elements are unique only up to associatedness, and are called the elementary divisors. Note that in a PID, the nonzero primary ideals are powers of prime ideals, so the elementary divisors can be written as powers of irreducible elements. The nonnegative integer is called the free rank or Betti number of the module .
The module is determined up to isomorphism by specifying its free rank , and for class of associated irreducible elements and each positive integer the number of times that occurs among the elementary divisors. The elementary divisors can be obtained from the list of invariant factors of the module by decomposing each of them as far as possible into pairwise relatively prime (non-unit) factors, which will be powers of irreducible elements. This decomposition corresponds to maximally decomposing each submodule corresponding to an invariant factor by using the Chinese remainder theorem for R. Conversely, knowing the multiset of elementary divisors, the invariant factors can be found, starting from the final one (which is a multiple of all others), as follows. For each irreducible element such that some power occurs in , take the highest such power, removing it from , and multiply these powers together for all (classes of associated) to give the final invariant factor; as long as is non-empty, repeat to find the invariant factors before it.
See also
Invariant factors
References
Chap.11, p.182.
Chap. III.7, p.153 of
Module theory
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https://en.wikipedia.org/wiki/Fine%20topology
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In mathematics, fine topology can refer to:
Fine topology (potential theory)
The sense opposite to coarse topology, namely:
A term in comparison of topologies which specifies the partial order relation of a topological structure to other one(s)
Final topology
See also
Discrete topology, the most fine topology possible on a given set
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https://en.wikipedia.org/wiki/Geometric%20modeling
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Geometric modeling is a branch of applied mathematics and computational geometry that studies methods and algorithms for the mathematical description of shapes.
The shapes studied in geometric modeling are mostly two- or three-dimensional (solid figures), although many of its tools and principles can be applied to sets of any finite dimension. Today most geometric modeling is done with computers and for computer-based applications. Two-dimensional models are important in computer typography and technical drawing. Three-dimensional models are central to computer-aided design and manufacturing (CAD/CAM), and widely used in many applied technical fields such as civil and mechanical engineering, architecture, geology and medical image processing.
Geometric models are usually distinguished from procedural and object-oriented models, which define the shape implicitly by an opaque algorithm that generates its appearance. They are also contrasted with digital images and volumetric models which represent the shape as a subset of a fine regular partition of space; and with fractal models that give an infinitely recursive definition of the shape. However, these distinctions are often blurred: for instance, a digital image can be interpreted as a collection of colored squares; and geometric shapes such as circles are defined by implicit mathematical equations. Also, a fractal model yields a parametric or implicit model when its recursive definition is truncated to a finite depth.
Notable awards of the area are the John A. Gregory Memorial Award and the Bézier award.
See also
2D geometric modeling
Architectural geometry
Computational conformal geometry
Computational topology
Computer-aided engineering
Computer-aided manufacturing
Digital geometry
Geometric modeling kernel
List of interactive geometry software
Parametric equation
Parametric surface
Solid modeling
Space partitioning
References
Further reading
General textbooks:
This book is out of print and freely available from the author.
For multi-resolution (multiple level of detail) geometric modeling :
Subdivision methods (such as subdivision surfaces):
External links
Geometry and Algorithms for CAD (Lecture Note, TU Darmstadt)
Geometric algorithms
Computer-aided design
Applied geometry
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https://en.wikipedia.org/wiki/Harmonic%20map
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In the mathematical field of differential geometry, a smooth map between Riemannian manifolds is called harmonic if its coordinate representatives satisfy a certain nonlinear partial differential equation. This partial differential equation for a mapping also arises as the Euler-Lagrange equation of a functional called the Dirichlet energy. As such, the theory of harmonic maps contains both the theory of unit-speed geodesics in Riemannian geometry and the theory of harmonic functions.
Informally, the Dirichlet energy of a mapping from a Riemannian manifold to a Riemannian manifold can be thought of as the total amount that stretches in allocating each of its elements to a point of . For instance, an unstretched rubber band and a smooth stone can both be naturally viewed as Riemannian manifolds. Any way of stretching the rubber band over the stone can be viewed as a mapping between these manifolds, and the total tension involved is represented by the Dirichlet energy. Harmonicity of such a mapping means that, given any hypothetical way of physically deforming the given stretch, the tension (when considered as a function of time) has first derivative equal to zero when the deformation begins.
The theory of harmonic maps was initiated in 1964 by James Eells and Joseph Sampson, who showed that in certain geometric contexts, arbitrary maps could be deformed into harmonic maps. Their work was the inspiration for Richard Hamilton's initial work on the Ricci flow. Harmonic maps and the associated harmonic map heat flow, in and of themselves, are among the most widely studied topics in the field of geometric analysis.
The discovery of the "bubbling" of sequences of harmonic maps, due to Jonathan Sacks and Karen Uhlenbeck, has been particularly influential, as their analysis has been adapted to many other geometric contexts. Notably, Uhlenbeck's parallel discovery of bubbling of Yang–Mills fields is important in Simon Donaldson's work on four-dimensional manifolds, and Mikhael Gromov's later discovery of bubbling of pseudoholomorphic curves is significant in applications to symplectic geometry and quantum cohomology. The techniques used by Richard Schoen and Uhlenbeck to study the regularity theory of harmonic maps have likewise been the inspiration for the development of many analytic methods in geometric analysis.
Geometry of mappings between manifolds
Here the geometry of a smooth mapping between Riemannian manifolds is considered via local coordinates and, equivalently, via linear algebra. Such a mapping defines both a first fundamental form and second fundamental form. The Laplacian (also called tension field) is defined via the second fundamental form, and its vanishing is the condition for the map to be harmonic. The definitions extend without modification to the setting of pseudo-Riemannian manifolds.
Local coordinates
Let be an open subset of and let be an open subset of . For each and between 1 and , let be a smooth real-valued f
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https://en.wikipedia.org/wiki/Hilbert%27s%20theorem
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Hilbert's theorem may refer to:
Hilbert's theorem (differential geometry), stating there exists no complete regular surface of constant negative gaussian curvature immersed in
Hilbert's Theorem 90, an important result on cyclic extensions of fields that leads to Kummer theory
Hilbert's basis theorem, in commutative algebra, stating every ideal in the ring of multivariate polynomials over a Noetherian ring is finitely generated
Hilbert's finiteness theorem, in invariant theory, stating that the ring of invariants of a reductive group is finitely generated
Hilbert's irreducibility theorem, in number theory, concerning irreducible polynomials
Hilbert's Nullstellensatz, the basis of algebraic geometry, establishing a fundamental relationship between geometry and algebra
Hilbert's syzygy theorem, a result of commutative algebra in connection with the syzygy problem of invariant theory
See also
List of things named after David Hilbert
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https://en.wikipedia.org/wiki/Apollonius%27s%20theorem
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In geometry, Apollonius's theorem is a theorem relating the length of a median of a triangle to the lengths of its sides.
It states that "the sum of the squares of any two sides of any triangle equals twice the square on half the third side, together with twice the square on the median bisecting the third side".
Specifically, in any triangle if is a median, then
It is a special case of Stewart's theorem. For an isosceles triangle with the median is perpendicular to and the theorem reduces to the Pythagorean theorem for triangle (or triangle ). From the fact that the diagonals of a parallelogram bisect each other, the theorem is equivalent to the parallelogram law.
The theorem is named for the ancient Greek mathematician Apollonius of Perga.
Proof
The theorem can be proved as a special case of Stewart's theorem, or can be proved using vectors (see parallelogram law). The following is an independent proof using the law of cosines.
Let the triangle have sides with a median drawn to side Let be the length of the segments of formed by the median, so is half of Let the angles formed between and be and where includes and includes Then is the supplement of and The law of cosines for and states that
Add the first and third equations to obtain
as required.
See also
References
External links
David B. Surowski: Advanced High-School Mathematics. p. 27
Euclidean geometry
Articles containing proofs
Theorems about triangles
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https://en.wikipedia.org/wiki/List%20of%20Grand%20Slam%20boys%27%20singles%20champions
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List of Boys' Singles Junior Grand Slam tournaments tennis champions.
Champions by year
Statistics
Most Grand Slam singles titles
Note: when a tie, the person to reach the mark first is listed first.
Grand Slam singles titles by country (since 1973)
Grand Slam achievements
Grand Slam
Players who held all four Grand Slam titles simultaneously (in a calendar year).
Career Grand Slam
Players who won all four Grand Slam titles over the course of their careers.
The event at which the Career Grand Slam was achieved is indicated in bold.
Multiple titles in a season
Three titles in a single season
Note: players who won 4 titles in a season are not included here.
Two titles in a single season
Note: players who won 3+ titles in a season are not included here.
Australian—French:
1952 Ken Rosewall
1961 John Newcombe
1962 John Newcombe (2)
1968 Phil Dent
1997 Daniel ElsnerAustralian—Wimbledon:
1989 Nicklas Kulti
1991 Thomas Enqvist
Australian—U.S.:
1995 Nicolas Kiefer
2000 Andy RoddickFrench—Wimbledon:
1958 Butch Buchholz
1963 Nicky Kalogeropoulos
1966 Vladimir Korotkov
1976 Heinz Günthardt
1978 Ivan Lendl
1979 Ramesh Krishnan
2018 Tseng Chun-hsin
French—U.S.:
1990 Andrea Gaudenzi
2002 Richard GasquetWimbledon—U.S.:
1973 Billy Martin
1974 Billy Martin (2)
1977 Van Winitsky
1982 Pat Cash
2008 Grigor Dimitrov
2012 Filip Peliwo
Surface Slam
Players who won Grand Slam titles on clay, grass and hard courts in a calendar year.
Channel Slam
Players who won the French Open-Wimbledon double.
See also
List of Grand Slam girls' singles champions
List of Grand Slam boys' doubles champions
List of Grand Slam girls' doubles champions
External links
International Tennis Federation - List of past winners
TennisProGuru.com - How successful are winners of Junior Grand Slams in Pro Tennis
Boys
boys
BOys
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https://en.wikipedia.org/wiki/Australian%20Mathematical%20Society
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The Australian Mathematical Society (AustMS) was founded in 1956 and is the national society of the mathematics profession in Australia.
One of the Society's listed purposes is to promote the cause of mathematics in the community by representing the interests of the profession to government. The Society also publishes three mathematical journals. In December 2020, Ole Warnaar moved from President-Elect to President, succeeding Jacqui Ramagge, who was elected in 2018.
Society awards
The Australian Mathematical Society Medal
The George Szekeres Medal
The Gavin Brown Prize
The Mahler Lectureship
The B.H. Neumann Prize
Society journals
The society publishes three journals through Cambridge University Press:
Journal of the Australian Mathematical Society
ANZIAM Journal (formerly Series B, Applied Mathematics)
Bulletin of the Australian Mathematical Society
ANZIAM
ANZIAM (Australia and New Zealand Industrial and Applied Mathematics) is a division of The Australian Mathematical Society (AustMS). Members are interested in applied mathematical research, mathematical applications in industry and business, and mathematics education at tertiary level.
The ANZIAM meeting is held annually at a different location in Australia or New Zealand. The 2020 ANZIAM meeting was held in the Hunter Valley, NSW.
ANZIAM awards three medals to members on the basis of research achievements, activities enhancing applied or industrial mathematics, and contributions to ANZIAM: the J.H. Michell Medal (for early-career awardees), the Ernie Tuck Medal (mid-career), and the ANZIAM Medal. In addition, each year the best student presentation is awarded the T.M. Cherry prize. As a tongue in cheek response, each year the student body also awards the best non-student talk a Cherry Ripe chocolate bar.
Special Interest Groups
ANZIAM has a number of special interest groups, based on specific research themes within applied mathematics: the Computational Mathematics Group, the Engineering Mathematics Group, Mathsport (concerned with the application of mathematics and computation to sport), the Mathematics in Industry Study Group, SIGMAOPT (concerning optimisation), and the Mathematical Biology Group. Each Special Interest Group runs an annual or biennial workshop or conference.
See also
American Mathematical Society
References
Resources
1956 establishments in Australia
Learned societies of Australia
Mathematical societies
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https://en.wikipedia.org/wiki/Nils%20Lid%20Hjort
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Nils Lid Hjort (born 12 January 1953) is a Norwegian statistician, who has been a professor of mathematical statistics at the University of Oslo since 1991. Hjort's research themes are varied, with particularly noteworthy contributions in the fields of Bayesian probability (Beta processes for use in non- and semi-parametric models, particularly within survival analysis and event history analysis, but also with links to Indian buffet processes in machine learning), density estimation and nonparametric regression (local likelihood methodology), model selection (focused information criteria and model averaging), confidence distributions, and change detection. He has also worked with spatial statistics, statistics of remote sensing, pattern recognition, etc.
An article on frequentist model averaging, with co-author Gerda Claeskens, was selected as Fast Breaking Paper in the field of mathematics by the Essential Science Indicators in 2005. This and a companion paper, both published in Journal of the American Statistical Association in 2003, introduced focused information criteria, along with a clear large-sample analysis of subset and post-selection estimators.
Hjort has been a core member of the Centre of Excellence Centre for Ecological and Evolutionary Synthesis, on the scientific advisory board of the Centre for Innovation Statistics for Innovation, and has also been involved with the Centre for Biostatistical Modelling in the Medical Sciences, all within the University of Oslo.
Hjort is an elected member of the Norwegian Academy of Science and Letters since 1999, the Royal Norwegian Society of Sciences and Letters since 2016, and was the third recipient of the Sverdrup Prize, awarded by the Norwegian Statistical Association in 2013. He was also the first recipient of the Ludwig von Drake Award.
He has also served on the editorial boards on various journals dedicated to the methodology and application of statistical research, including the Scandinavian Journal of Statistics, Journal of the Royal Statistical Society, Series B, and the Annals of Statistics, and has been on the programme committees of numerous international conferences. He has led the 2014–2019 Research Council of Norway funded project FocuStat: Focus Driven Statistical Inference With Complex Data at the University of Oslo, and is co-leading the 2022–2023 project Stability and Change at the Centre for Advanced Study at the Norwegian Academy of Science and Letters.
Over the years, Hjort has supervised or co-supervised about 40 Master's degree students and about 15 PhDs. Among these are Steffen Grønneberg and Céline Cunen, both winners of the Sverdrup Young Researcher Prize, Martin Jullum and Ingrid Dæhlen, both winners of the Norwegian Computing Centre Master's Prize.
On the applied side, Hjort demonstrated in 1994 that there is a small but Olympically significant difference between inner-lane and outer-lane starts for 500 m speedskating races, after a systematic analysis of w
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https://en.wikipedia.org/wiki/Deterrence%20%28penology%29
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Deterrence in relation to criminal offending is the idea or theory that the threat of punishment will deter people from committing crime and reduce the probability and/or level of offending in society. It is one of five objectives that punishment is thought to achieve; the other four objectives are denunciation, incapacitation (for the protection of society), retribution and rehabilitation.
Criminal deterrence theory has two possible applications: the first is that punishments imposed on individual offenders will deter or prevent that particular offender from committing further crimes; the second is that public knowledge that certain offences will be punished has a generalised deterrent effect which prevents others from committing crimes.
Two different aspects of punishment may have an impact on deterrence, the first being the certainty of punishment, by increasing the likelihood of apprehension and punishment, this may have a deterrent effect. The second relates to the severity of punishment; how severe the punishment is for a particular crime may influence behavior if the potential offender concludes that the punishment is so severe, it is not worth the risk of getting caught.
An underlying principle of deterrence is that it is utilitarian or forward-looking. As with rehabilitation, it is designed to change behaviour in the future rather than simply provide retribution or punishment for current or past behaviour.
Categories
There are two main goals of deterrence theory.
Individual deterrence is the aim of punishment to discourage the offender from criminal acts in the future. The belief is that when punished, offenders recognise the unpleasant consequences of their actions on themselves and will change their behaviour accordingly.
General deterrence is the intention to deter the general public from committing crime by punishing those who do offend. When an offender is punished by, for example, being sent to prison, a clear message is sent to the rest of society that behaviour of this sort will result in an unpleasant response from the criminal justice system. Most people do not want to end up in prison and so they are deterred from committing crimes that might be punished that way.
Underlying assumptions
A key assumption underlying deterrence theory is that offenders weigh up the pros and cons of a certain course of action and make rational choices. Known as rational choice theory, it assumes the following:
People are able to freely choose their actions and behaviour (as opposed to their offending being driven by socio-economic factors such as unemployment, poverty, limited education and/or addiction).
The offender is capable of assessing the likelihood of getting caught.
The offender knows the likely punishment that will be received.
The offender is able to calculate whether the pain or severity of the likely punishment outweighs the gain or benefit of getting away with the crime.
Other assumptions relate to the concept of marg
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https://en.wikipedia.org/wiki/P%C3%B3lya%20conjecture
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In number theory, the Pólya conjecture (or Pólya's conjecture) stated that "most" (i.e., 50% or more) of the natural numbers less than any given number have an odd number of prime factors. The conjecture was set forth by the Hungarian mathematician George Pólya in 1919, and proved false in 1958 by C. Brian Haselgrove. Though mathematicians typically refer to this statement as the Pólya conjecture, Pólya never actually conjectured that the statement was true; rather, he showed that the truth of the statement would imply the Riemann hypothesis. For this reason, it is more accurately called "Pólya's problem".
The size of the smallest counterexample is often used to demonstrate the fact that a conjecture can be true for many cases and still fail to hold in general, providing an illustration of the strong law of small numbers.
Statement
The Pólya conjecture states that for any n > 1, if the natural numbers less than or equal to n (excluding 0) are partitioned into those with an odd number of prime factors and those with an even number of prime factors, then the former set has at least as many members as the latter set. Repeated prime factors are counted repeatedly; for instance, we say that 18 = 2 × 3 × 3 has an odd number of prime factors, while 60 = 2 × 2 × 3 × 5 has an even number of prime factors.
Equivalently, it can be stated in terms of the summatory Liouville function, with the conjecture being that
for all n > 1. Here, λ(k) = (−1)Ω(k) is positive if the number of prime factors of the integer k is even, and is negative if it is odd. The big Omega function counts the total number of prime factors of an integer.
Disproof
The Pólya conjecture was disproved by C. Brian Haselgrove in 1958. He showed that the conjecture has a counterexample, which he estimated to be around 1.845 × 10361.
An explicit counterexample, of n = 906,180,359 was given by R. Sherman Lehman in 1960; the smallest counterexample is n = 906,150,257, found by Minoru Tanaka in 1980.
The conjecture fails to hold for most values of n in the region of 906,150,257 ≤ n ≤ 906,488,079. In this region, the summatory Liouville function reaches a maximum value of 829 at n = 906,316,571.
References
External links
Disproved conjectures
Conjectures about prime numbers
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https://en.wikipedia.org/wiki/Green%27s%20matrix
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In mathematics, and in particular ordinary differential equations, a Green's matrix helps to determine a particular solution to a first-order inhomogeneous linear system of ODEs. The concept is named after George Green.
For instance, consider where is a vector and is an matrix function of , which is continuous for , where is some interval.
Now let be linearly independent solutions to the homogeneous equation and arrange them in columns to form a fundamental matrix:
Now is an matrix solution of .
This fundamental matrix will provide the homogeneous solution, and if added to a particular solution will give the general solution to the inhomogeneous equation.
Let be the general solution. Now,
This implies or where is an arbitrary constant vector.
Now the general solution is
The first term is the homogeneous solution and the second term is the particular solution.
Now define the Green's matrix
The particular solution can now be written
External links
An example of solving an inhomogeneous system of linear ODEs and finding a Green's matrix from www.exampleproblems.com.
Ordinary differential equations
Matrices
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https://en.wikipedia.org/wiki/Infinite%20group
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In group theory, an area of mathematics, an infinite group is a group whose underlying set contains an infinite number of elements. In other words, it is a group of infinite order.
Examples
(Z, +), the group of integers with addition is infinite
Non-discrete Lie groups are infinite. For example, (R, +), the group of real numbers with addition is an infinite group
The general linear group of order n > 0 over an infinite field is infinite
See also
Finite group
Infinite group theory
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https://en.wikipedia.org/wiki/Map%20coloring
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In cartography, map coloring is the act of choosing colors as a form of map symbol to be used on a map. In mathematics, map coloring is the act of assigning colors to features of a map such that no two adjacent features have the same color using the minimum number of colors.
Cartography
Color is a very useful attribute to depict different features on a map. Typical uses of color include displaying different political divisions, different elevations, or different kinds of roads. A choropleth map is a thematic map in which areas are colored differently to show the measurement of a statistical variable being displayed on the map. The choropleth map provides an easy way to visualize how a measurement varies across a geographic area or it shows the level of variability within a region. In addition to choropleth maps, a cartographer should strive to depict colors effectively on any kind of map.
Displaying the data in different hues can greatly affect the understanding or feel of the map. In many cultures, certain colors have connotations. These connotations lie under a field of study called color symbolism. For example, coloring a certain nation a color that has a negative connotation in their culture could be counterproductive. Likewise, using assumed skin colors to show racial or ethnic patterns will likely cause offense. It is not possible to always predict the color connotations of every map reader or to avoid negative connotations, but it is helpful to be aware of common color connotations in order to make a map as appealing and understandable as possible.
Cartographers may also choose to pick hues that are associated with what they are mapping. For example, when mapping precipitation, they may choose to use shades of blue or for a map of wildfires they may use yellows, reds, and oranges. Carefully choosing colors ensures that the map is intuitive and easy to read. This process is referred to as feature association. Also, the cartographer must take into account that many people have impaired color vision, and colors must be used that are easily distinguishable by such readers.
A general rule is that most people can differentiate only between 5-8 different shades of one color. Rather than more than 8 shades of a color, it is best to use multiple colors. Most GIS programs provide users with carefully curated color schemes to choose from, thus making the process of selecting colors easier.
Colors can also be used to produce three-dimensional effects from two-dimensional maps, either by explicit color-coding of the two images intended for different eyes, or by using the characteristics of the human visual system to make the map look three-dimensional.
Mathematics
In mathematics there is a close link between map coloring and graph coloring, since every map showing different areas has a corresponding graph. By far the most famous result in this area is the four color theorem, which states that any planar map can be colored with at most four
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https://en.wikipedia.org/wiki/Characteristic%20multiplier
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In mathematics, and particularly ordinary differential equations, a characteristic multiplier is an eigenvalue of a monodromy matrix. The logarithm of a characteristic multiplier is also known as characteristic exponent. They appear in Floquet theory of periodic differential operators and in the Frobenius method.
See also
Multiplier (disambiguation)
References
External links
Examples of finding characteristic multipliers of systems of ODEs from www.exampleproblems.com.
Ordinary differential equations
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https://en.wikipedia.org/wiki/Monodromy%20matrix
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In mathematics, and particularly ordinary differential equations (ODEs), a monodromy matrix is the fundamental matrix of a system of ODEs evaluated at the period of the coefficients of the system. It is used for the analysis of periodic solutions of ODEs in Floquet theory.
See also
Floquet theory
Monodromy
Riemann–Hilbert problem
References
Ordinary differential equations
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https://en.wikipedia.org/wiki/182%20%28number%29
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182 (one hundred [and] eighty-two) is the natural number following 181 and preceding 183.
In mathematics
182 is an even number
182 is a composite number, as it is a positive integer with a positive divisor other than one or itself
182 is a deficient number, as the sum of its proper divisors, 154, is less than 182
182 is a member of the Mian–Chowla sequence: 1, 2, 4, 8, 13, 21, 31, 45, 66, 81, 97, 123, 148, 182
182 is a nontotient number, as there is no integer with exactly 182 coprimes below it
182 is an odious number
182 is a pronic number, oblong number or heteromecic number, a number which is the product of two consecutive integers (13 × 14)
182 is a repdigit in the D'ni numeral system (77), and in base 9 (222)
182 is a sphenic number, the product of three prime factors
182 is a square-free number
182 is an Ulam number
Divisors of 182: 1, 2, 7, 13, 14, 26, 91, 182
In astronomy
182 Elsa is a S-type main belt asteroid
OGLE-TR-182 is a star in the constellation Carina
In the military
JDS Ise (DDH-182), a Hyūga-class helicopter destroyer of the Japan Maritime Self-Defense Force
The United States Air Force 182d Airlift Wing unit at Greater Peoria Regional Airport, Peoria, Illinois
was a United States Navy troop transport during World War II
was a United States Navy yacht during World War I
was a United States Navy Alamosa-class cargo ship during World War II
was a United States Navy during World War II
was a United States Navy during World War II
was a United States Navy following World War I
182nd Fighter Squadron, Texas Air National Guard unit of the Texas Air National Guard
182nd Infantry Regiment, now known as the 182nd Cavalry Squadron (RSTA), is the oldest combat regiment in the United States Army
182nd Battalion, Canadian Expeditionary Force during World War I
In music
Blink-182, an American pop punk band
Blink-182 (album), their 2003 eponymous album
In transportation
Alfa Romeo 182 Formula One car
182nd–183rd Streets station on the IND Concourse Line () of the New York City Subway
London Buses route 182
Bücker Bü 182 was a single-seat advanced trainer plane in Germany
Cessna 182 marketed under the name Skylane, is a four-seat, single-engine, light airplane
Flight 182
Pacific Southwest Airlines Flight 182 collided with a Cessna skyhawk in San Diego on September 25, 1978
Air India Flight 182, exploded by a terrorist bombing off the coast of Ireland on June 23, 1985
DDG-182 Mirai, a fictional ship in the anime Zipang
In other fields
182 is also:
The year AD 182 or 182 BC
The atomic number of an element temporarily called Unoctbium
The Laird o Logie is child ballad #182
The human gene GPR182 (or G protein-coupled receptor 182)
The Star of Bombay is a cabochon-cut star sapphire originating from Sri Lanka
The band Blink 182 used the number randomly to distinguish themselves from another band named Blink
See also
List of highways numbered 182
United Nations Security Council Resolution
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https://en.wikipedia.org/wiki/The%20Algebra%20of%20Infinite%20Justice
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The Algebra of Infinite Justice (2001) is a collection of essays written by Booker Prize winner Arundhati Roy. The book discusses a wide range of issues including political euphoria in India over its successful nuclear bomb tests, the effect of public works projects on the environment, the influence of foreign multinational companies on policy in poorer countries, and the "war on terror". Some of the essays in the collection were republished later, along with later writing, in her book My Seditious Heart.
The official introduction to the book by Penguin India states :
A few weeks after India detonated a thermonuclear device in 1998, Arundhati Roy wrote ‘The End of Imagination’. The essay attracted worldwide attention as the voice of a brilliant Indian writer speaking out with clarity and conscience against nuclear weapons. Over the next three and a half years, she wrote a series of political essays on a diverse range of momentous subjects: from the illusory benefits of big dams, to the downside of corporate globalization and the US Government’s war against terror.
Essays
The end of Imagination
This is the name of the first essay in the 2001 book. It was later used as the title of a comprehensive collection of Roy's essays in 2016
The greater common good
Essay concerning the controversial Sardar Sarovar Dam project in India's Narmada Valley.
Power politics
This essay examines Indian dam construction and challenges the idea that only "experts" can influence economic policy. It explores the human costs of the privatization of India’s power supply and the construction of monumental dams in India. This is the second essay in the original 2001 book. There is also a 2002 book of Roy's essays with this title Power Politics.
The ladies have feelings so...
The Algebra of Infinite Justice
War is peace
The world doesn't have to choose between the Taliban and the US government. All the beauty of the world—literature, music, art—lies between these two fundamentalist poles.
Democracy Who’s She When She’s at Home
This essay examines the horrific communal violence in Gujarat
War talk Summer Games with Nuclear Bombs’
When India and Pakistan conducted their nuclear tests in 1998 hypocrisy of Western nuclear powers, implicitly racist, denunciation of the tests. Roy explores the double standard while she finds nuclear weapons unspeakable. Her final sentence is: Why do we tolerate these men who use nuclear weapons to blackmail the entire human race?
Reception
Mithu C Banerji , in a review in The Observer (2002) stated: Roy's writing reflects her fiction, and meanders between polemic and sentiment. Yet whether she is talking about the 'death of my world' or about 'one country's terrorist being another's freedom fighter', she is always passionately intense.S. Prasannarajan of India Today said: ...marvel at the italicised banality of her text, its remoteness from the context. This is the rebel without a context, and no textual exaggeration, assist
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https://en.wikipedia.org/wiki/Holomorphic%20separability
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In mathematics in complex analysis, the concept of holomorphic separability is a measure of the richness of the set of holomorphic functions on a complex manifold or complex-analytic space.
Formal definition
A complex manifold or complex space is said to be holomorphically separable, if whenever x ≠ y are two points in , there exists a holomorphic function , such that f(x) ≠ f(y).
Often one says the holomorphic functions separate points.
Usage and examples
All complex manifolds that can be mapped injectively into some are holomorphically separable, in particular, all domains in and all Stein manifolds.
A holomorphically separable complex manifold is not compact unless it is discrete and finite.
The condition is part of the definition of a Stein manifold.
References
Complex analysis
Several complex variables
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https://en.wikipedia.org/wiki/Vanishing
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Vanishing may refer to:
Entertainment
Vanishing, a type of magical effect.
Mathematics
The mathematical concept, see root of a function
Music
A song from the A Perfect Circle album Thirteenth Step
A song from Mariah Carey (album)
A song by Bryan Adams from Waking Up the Neighbours
A song by Barenaked Ladies from Barenaked Ladies Are Me
Art and literature
A Void, 1969 French novel, also translated under the titles A Vanishing and Vanish'd
Vanishing (2022 film), a French-South Korean film
The Vanishing (disambiguation), various films and novels
See also
Vanish (disambiguation)
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https://en.wikipedia.org/wiki/187%20%28number%29
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187 (one hundred [and] eighty-seven) is the natural number following 186 and preceding 188.
In mathematics
There are 187 ways of forming a sum of positive integers that adds to 11, counting two sums as equivalent when they are cyclic permutations of each other. There are also 187 unordered triples of 5-bit binary numbers whose bitwise exclusive or is zero.
Per Miller's rules, the triakis tetrahedron produces 187 distinct stellations. It is the smallest Catalan solid, dual to the truncated tetrahedron, which only has 9 distinct stellations.
In other fields
There are 187 chapters in the Hebrew Torah.
See also
187 (disambiguation)
References
Integers
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https://en.wikipedia.org/wiki/Sergey%20Krasnikov
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Serguei Vladilenovich Krasnikov (; 1961) is a Russian physicist.
Life
Krasnikov obtained a doctorate (CSc.) in physics and mathematics from Saint Petersburg University. He is currently based at Pulkovo Observatory in St. Petersburg, Russia.
Krasnikov’s work is focused on theoretical physics, specifically on the development of the Krasnikov tube and its applications in causality, closed timelike curves, and hyperfast travel.
In 2001, Krasnikov worked at Starlab, in a joint NASA/USAF-funded project to assess the viability of time travel under realistic physical conditions.
In 2002 he attended the 11th UK Conference on the Foundations of Physics hosted by the Faculty of Philosophy, University of Oxford at which he delivered the paper "Time machine (1988-2001)".
See also
Alcubierre drive
Wormhole
References
External links
Homepage at the Alexander Friedmann Laboratory for Theoretical Physics
Curriculum Vitae at the Friedmann Laboratory website
TEDxBrussels 2009 Serguei Krasnikov on time travel
1961 births
Living people
Russian physicists
Quantum physicists
Time travel
Russian scientists
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https://en.wikipedia.org/wiki/Brahmagupta%20matrix
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In mathematics, the following matrix was given by Indian mathematician Brahmagupta:
It satisfies
Powers of the matrix are defined by
The and are called Brahmagupta polynomials. The Brahmagupta matrices can be extended to negative integers:
See also
Brahmagupta's identity
Brahmagupta's function
References
External links
Eric Weisstein. Brahmagupta Matrix, MathWorld, 1999.
Brahmagupta
Matrices
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https://en.wikipedia.org/wiki/Horseshoe%20lemma
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In homological algebra, the horseshoe lemma, also called the simultaneous resolution theorem, is a statement relating resolutions of two objects and to resolutions of
extensions of by . It says that if an object is an extension of by , then a resolution of can be built up inductively with the nth item in the resolution equal to the coproduct of the nth items in the resolutions of and . The name of the lemma comes from the shape of the diagram illustrating the lemma's hypothesis.
Formal statement
Let be an abelian category with enough projectives. If
is a diagram in such that the column is exact and the
rows are projective resolutions of and respectively, then
it can be completed to a commutative diagram
where all columns are exact, the middle row is a projective resolution
of , and for all n. If is an
abelian category with enough injectives, the dual statement also holds.
The lemma can be proved inductively. At each stage of the induction, the properties of projective objects are used to define maps in a projective resolution of . Then the snake lemma is invoked to show that the simultaneous resolution constructed so far has exact rows.
See also
Nine lemma
References
Homological algebra
Lemmas in category theory
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https://en.wikipedia.org/wiki/University%20of%20Campinas%20Institute%20of%20Computing
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The Institute of Computing (), formerly the Department of Computer Science at the Institute of Mathematics, Statistics and Computer Science, is the main unit of education and research in computer science at the State University of Campinas (Unicamp). The institute is located at the Zeferino Vaz campus, in the district of Barão Geraldo in Campinas, São Paulo, Brazil.
History
The origins of the Institute traces back to 1969 when Unicamp created a baccalaureate in Computer Science. The first one of its kind in Brazil, it served as a model for many computing courses in other universities in the country. In the same year, the Department of Computer Science (DCC) was established at the Institute of Mathematics, Statistics and Computer Science (IMECC). In March 1996, the department was separated from IMECC and became a full institute, the 20th academic unit of Unicamp. The reorganization was completed formally when its first dean came to office in the next year (March 1997).
Courses
The institute offers two undergraduate courses: a baccalaureate in Computer Science (evening period) and another in Computer Engineering (in partnership with the School of Electric and Computer Engineering). The institute offers also graduate programs at the level of master's and doctorate in Computer Science. These courses have received top evaluations from the ministry of education, and attract students from many Latin America countries. The institute also offers many post-graduate specialization and continued education directed mainly towards the qualification and specialization of information technology professionals.
Departments
The IC is organized into three departments:
DSC - Department of Computer Systems
DSI - Department of Information Systems
DTC - Department of Computer Theory
See also
University of Campinas
School of Electrical and Computer Engineering (Campinas)
Jorge Stolfi
References
External links
IC Home Page
FEEC
Unicamp
University of Campinas
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https://en.wikipedia.org/wiki/188%20%28number%29
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188 (one hundred [and] eighty-eight) is the natural number following 187 and preceding 189.
In mathematics
There are 188 different four-element semigroups, and 188 ways a chess queen can move from one corner of a board to the opposite corner by a path that always moves closer to its goal. The sides and diagonals of a regular dodecagon form 188 equilateral triangles.
In other fields
The number 188 figures prominently in the film The Parallel Street (1962) by German experimental film director . The opening frame of the film is just an image of this number.
See also
The year AD 188 or 188 BC
List of highways numbered 188
References
Integers
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https://en.wikipedia.org/wiki/Lipkovo
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Lipkovo (, ) is a village in North Macedonia. It is the seat of Lipkovo Municipality.
History
According to the statistics of the Bulgarian ethnographer Vasil Kanchov from 1900, 490 inhabitants lived in Lipkovo, 250 Muslim Albanians and 240 Bulgarian Exarchists.
Lipkovo was a central strategic village during the 2001 armed conflict between the Albanian National Liberation Army and the Macedonian Army. Today, it has a dam which supplies water and electricity to the Kumanovo region.
Demographics
As of the 2021 census, Lipkovo had 2,138 residents with the following ethnic composition:
Albanians 2,104
Persons for whom data are taken from administrative sources 32
Others 2
According to the 2002 census, the town had a total of 2644 inhabitants. Ethnic groups in the village include:
Albanians 2631
Macedonians 2
Others 11
Sister Towns
Mustafakemalpaşa, the main town of Bursa Province in the Marmara region of Turkey.
References
External links
Municipal flag of Lipkovo
Villages in Lipkovo Municipality
Albanian communities in North Macedonia
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https://en.wikipedia.org/wiki/Diagonal%20morphism
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In category theory, a branch of mathematics, for any object in any category where the product exists, there exists the diagonal morphism
satisfying
for
where is the canonical projection morphism to the -th component. The existence of this morphism is a consequence of the universal property that characterizes the product (up to isomorphism). The restriction to binary products here is for ease of notation; diagonal morphisms exist similarly for arbitrary products. The image of a diagonal morphism in the category of sets, as a subset of the Cartesian product, is a relation on the domain, namely equality.
For concrete categories, the diagonal morphism can be simply described by its action on elements of the object . Namely, , the ordered pair formed from . The reason for the name is that the image of such a diagonal morphism is diagonal (whenever it makes sense), for example the image of the diagonal morphism on the real line is given by the line that is the graph of the equation . The diagonal morphism into the infinite product may provide an injection into the space of sequences valued in ; each element maps to the constant sequence at that element. However, most notions of sequence spaces have convergence restrictions that the image of the diagonal map will fail to satisfy.
See also
Diagonal functor
Diagonal embedding
References
Morphisms
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https://en.wikipedia.org/wiki/Diagonal%20functor
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In category theory, a branch of mathematics, the diagonal functor is given by , which maps objects as well as morphisms. This functor can be employed to give a succinct alternate description of the product of objects within the category : a product is a universal arrow from to . The arrow comprises the projection maps.
More generally, given a small index category , one may construct the functor category , the objects of which are called diagrams. For each object in , there is a constant diagram that maps every object in to and every morphism in to . The diagonal functor assigns to each object of the diagram , and to each morphism in the natural transformation in (given for every object of by ). Thus, for example, in the case that is a discrete category with two objects, the diagonal functor is recovered.
Diagonal functors provide a way to define limits and colimits of diagrams. Given a diagram , a natural transformation (for some object of ) is called a cone for . These cones and their factorizations correspond precisely to the objects and morphisms of the comma category , and a limit of is a terminal object in , i.e., a universal arrow . Dually, a colimit of is an initial object in the comma category , i.e., a universal arrow .
If every functor from to has a limit (which will be the case if is complete), then the operation of taking limits is itself a functor from to . The limit functor is the right-adjoint of the diagonal functor. Similarly, the colimit functor (which exists if the category is cocomplete) is the left-adjoint of the diagonal functor. For example, the diagonal functor described above is the left-adjoint of the binary product functor and the right-adjoint of the binary coproduct functor.
See also
Diagram (category theory)
Cone (category theory)
Diagonal morphism
References
Category theory
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https://en.wikipedia.org/wiki/Jacobson%20ring
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In algebra, a Hilbert ring or a Jacobson ring is a ring such that every prime ideal is an intersection of primitive ideals. For commutative rings primitive ideals are the same as maximal ideals so in this case a Jacobson ring is one in which every prime ideal is an intersection of maximal ideals.
Jacobson rings were introduced independently by , who named them after Nathan Jacobson because of their relation to Jacobson radicals, and by , who named them Hilbert rings after David Hilbert because of their relation to Hilbert's Nullstellensatz.
Jacobson rings and the Nullstellensatz
Hilbert's Nullstellensatz of algebraic geometry is a special case of the statement that the polynomial ring in finitely many variables over a field is a Hilbert ring. A general form of the Nullstellensatz states that if R is a Jacobson ring, then so is any finitely generated R-algebra S. Moreover, the pullback of any maximal ideal J of S is a maximal ideal I of R, and S/J is a finite extension of the field R/I.
In particular a morphism of finite type of Jacobson rings induces a morphism of the maximal spectrums of the rings. This explains why for algebraic varieties over fields it is often sufficient to work with the maximal ideals rather than with all prime ideals, as was done before the introduction of schemes. For more general rings such as local rings, it is no longer true that morphisms of rings induce morphisms of the maximal spectra, and the use of prime ideals rather than maximal ideals gives a cleaner theory.
Examples
Any field is a Jacobson ring.
Any principal ideal domain or Dedekind domain with Jacobson radical zero is a Jacobson ring. In principal ideal domains and Dedekind domains, the nonzero prime ideals are already maximal, so the only thing to check is if the zero ideal is an intersection of maximal ideals. Asking for the Jacobson radical to be zero guarantees this. In principal ideal domains and Dedekind domains, the Jacobson radical vanishes if and only if there are infinitely many prime ideals.
Any finitely generated algebra over a Jacobson ring is a Jacobson ring. In particular, any finitely generated algebra over a field or the integers, such as the coordinate ring of any affine algebraic set, is a Jacobson ring.
A local ring has exactly one maximal ideal, so it is a Jacobson ring exactly when that maximal ideal is the only prime ideal. Thus any commutative local ring with Krull dimension zero is Jacobson, but if the Krull dimension is 1 or more, the ring cannot be Jacobson.
showed that any countably generated algebra over an uncountable field is a Jacobson ring.
Tate algebras over non-archimedean fields are Jacobson rings.
A commutative ring R is a Jacobson ring if and only if R[x], the ring of polynomials over R, is a Jacobson ring.
Characterizations
The following conditions on a commutative ring R are equivalent:
R is a Jacobson ring
Every prime ideal of R is an intersection of maximal ideals.
Every radical ideal is an intersection o
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https://en.wikipedia.org/wiki/Australian%20Mathematics%20Competition
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The Australian Mathematics Competition is a mathematics competition run by the Australian Maths Trust for students from year 3 up to year 12 in Australia, and their equivalent grades in other countries.
Since its inception in 1976 in the Australian Capital Territory, the participation numbers have increased to around 600,000, with around 100,000 being from outside Australia, making it the world's largest mathematics competition.
History
The forerunner of the competition, first held in 1976, was open to students within the Australian Capital Territory, and attracted 1,200 entries. In 1976 and 1977 the outstanding entrants were awarded the Burroughs medal. In 1978, the competition became a nationwide event, and became known as the Australian Mathematics Competition for the Wales awards with 60,000 students from Australia and New Zealand participating. In 1983 the medals were renamed the Westpac awards following a change to the name of the title sponsor Westpac. Other sponsors since the inception of the competition have been the Canberra Mathematical Association and the University of Canberra (previously known as the Canberra College of Advanced Education).
The competition has since spread to countries such as New Zealand, Singapore, Fiji, Tonga, Taiwan, China and Malaysia, which submit thousands of entries each. A French translation of the paper has been available since the current competition was established in 1978, with Chinese translation being made available to Hong Kong (Traditional Chinese Characters) and Taiwan (Traditional Chinese Characters) students in 2000. Large print and braille versions are also available.
In 2004, the competition was expanded to allow two more divisions, one for year five and six students, and another for year three and four students.
In 2005, students from 38 different countries entered the competition.
Format
The competition paper consists of twenty-five multiple-choice questions and five integer questions, which are ordered in increasing difficulty. Students record their personal details and mark their answers by pencil on a carbon-mark answer sheet, which is marked by computer in the Australian Maths Trust offices. Since 2016, an online option has been available to schools. The online competition has the same content as the paper version and results from both options are assessed together, with options being jumbled between computers (to hinder attempts on cheating). However, the paper is undertaken on a browser with a onetime pin with focus monitors (accessible by the AMT and supervisors). There are five divisions in total: Senior (for years 11 and 12), Intermediate (for years 9 and 10), Junior (for years 7 and 8), Upper Primary (for years 5 and 6) and Middle Primary (for years 3 and 4).
Students are allowed 75 minutes (60 minutes for the two primary papers) to read and answer the questions. Calculators are not permitted for secondary-level entrants, but geometrical aids such as rulers, compasses, pro
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https://en.wikipedia.org/wiki/Alain%20Desrosi%C3%A8res
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Alain Desrosières (18 April 1940 – 15 February 2013) was a statistician, sociologist and historian of science in France, well known for his work in the history of statistics He is the author of The Politics of Large Numbers: A History of Statistical Reasoning, published in 1993, translated into several languages, including English in 1998, and subsequently reviewed in the LRB in 2000. This described the origins of statistics as technical machinery for administration in the 19th and 20th centuries, including the attempts to measure human and economic development. The text is an account of the statistics and their use in abstracting features of society to better measure and understand them, with particular aims.
His major technical work on the socio-professional categorisation scheme used in French official statistics was updated in five editions over more than fifteen years. Further collected papers were published in two volumes as The Statistical Argument in 2008, and a final collection published posthumously in 2014 as Prouver et Gouverner. His major contribution was to frame public statistics as constructed reality, with categories created to describe society, but tracked carefully using these definitions. Thus bridging the opposing views that they are either objective facts or political propaganda due to his unusual combination of sociological study and statistical training.
References
External links
Obituary, Le Monde
Obituary, Libération
Special issue, Statistique et Société, 2018 (in memoriam)
French statisticians
Historians of science
Sociologists of science
French sociologists
2013 deaths
1940 births
French male non-fiction writers
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https://en.wikipedia.org/wiki/Abstract%20algebraic%20logic
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In mathematical logic, abstract algebraic logic is the study of the algebraization of deductive systems
arising as an abstraction of the well-known Lindenbaum–Tarski algebra, and how the resulting algebras are related to logical systems.
History
The archetypal association of this kind, one fundamental to the historical origins of algebraic logic and lying at the heart of all subsequently developed subtheories, is the association between the class of Boolean algebras and classical propositional calculus. This association was discovered by George Boole in the 1850s, and then further developed and refined by others, especially C. S. Peirce and Ernst Schröder, from the 1870s to the 1890s. This work culminated in Lindenbaum–Tarski algebras, devised by Alfred Tarski and his student Adolf Lindenbaum in the 1930s. Later, Tarski and his American students (whose ranks include Don Pigozzi) went on to discover cylindric algebra, whose representable instances algebraize all of classical first-order logic, and revived relation algebra, whose models include all well-known axiomatic set theories.
Classical algebraic logic, which comprises all work in algebraic logic until about 1960, studied the properties of specific classes of algebras used to "algebraize" specific logical systems of particular interest to specific logical investigations. Generally, the algebra associated with a logical system was found to be a type of lattice, possibly enriched with one or more unary operations other than lattice complementation.
Abstract algebraic logic is a modern subarea of algebraic logic that emerged in Poland during the 1950s and 60s with the work of Helena Rasiowa, Roman Sikorski, Jerzy Łoś, and Roman Suszko (to name but a few). It reached maturity in the 1980s with the seminal publications of the Polish logician Janusz Czelakowski, the Dutch logician Wim Blok and the American logician Don Pigozzi. The focus of abstract algebraic logic shifted from the study of specific classes of algebras associated with specific logical systems (the focus of classical algebraic logic), to the study of:
Classes of algebras associated with classes of logical systems whose members all satisfy certain abstract logical properties;
The process by which a class of algebras becomes the "algebraic counterpart" of a given logical system;
The relation between metalogical properties satisfied by a class of logical systems, and the corresponding algebraic properties satisfied by their algebraic counterparts.
The passage from classical algebraic logic to abstract algebraic logic may be compared to the passage from "modern" or abstract algebra (i.e., the study of groups, rings, modules, fields, etc.) to universal algebra (the study of classes of algebras of arbitrary similarity types (algebraic signatures) satisfying specific abstract properties).
The two main motivations for the development of abstract algebraic logic are closely connected to (1) and (3) above. With respect to (1), a criti
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https://en.wikipedia.org/wiki/Kazuya%20Kato
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is a Japanese mathematician who works at the University of Chicago and specializes in number theory and arithmetic geometry.
Early life and education
Kazuya Kato grew up in the prefecture of Wakayama in Japan. He attended college at the University of Tokyo, from which he also obtained his master's degree in 1975, and his PhD in 1980.
Career
Kato was a professor at Tokyo University, Tokyo Institute of Technology and Kyoto University. He joined the faculty of the University of Chicago in 2009.
A special volume of Documenta Mathematica was published in honor of his 50th birthday, together with research papers written by leading number theorists and former students it contains Kato's song on Prime Numbers.
Research
Kato's first work was in the higher-dimensional generalisations of local class field theory using algebraic K-theory. His theory was then extended to higher global class field theory in which several of his papers were written jointly with Shuji Saito.
He contributed to various other areas such as p-adic Hodge theory, logarithmic geometry (he was one of its creators together with Jean-Marc Fontaine and Luc Illusie), comparison conjectures, special values of zeta functions including applications to the Birch-Swinnerton-Dyer conjecture, the Bloch-Kato conjecture on Tamagawa numbers, and Iwasawa theory.
Awards and honors
In 2005, Kato received the Imperial Prize of the Japan Academy for "Research on Arithmetic Geometry".
Books
Kato has published several books in Japanese, of which some have already been translated into English.
He wrote a book on Fermat's Last Theorem and is also the coauthor of two volumes of the trilogy on Number Theory, which have been translated into English.
References
20th-century Japanese mathematicians
21st-century Japanese mathematicians
Number theorists
1952 births
Living people
People from Wakayama Prefecture
University of Tokyo alumni
Academic staff of Kyoto University
Academic staff of the University of Tokyo
Academic staff of Tokyo Institute of Technology
University of Chicago faculty
Laureates of the Imperial Prize
Fellows of the American Academy of Arts and Sciences
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https://en.wikipedia.org/wiki/Leyland%20number
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In number theory, a Leyland number is a number of the form
where x and y are integers greater than 1. They are named after the mathematician Paul Leyland. The first few Leyland numbers are
8, 17, 32, 54, 57, 100, 145, 177, 320, 368, 512, 593, 945, 1124 .
The requirement that x and y both be greater than 1 is important, since without it every positive integer would be a Leyland number of the form x1 + 1x. Also, because of the commutative property of addition, the condition x ≥ y is usually added to avoid double-covering the set of Leyland numbers (so we have 1 < y ≤ x).
Leyland primes
A Leyland prime is a Leyland number that is also a prime. The first such primes are:
17, 593, 32993, 2097593, 8589935681, 59604644783353249, 523347633027360537213687137, 43143988327398957279342419750374600193, ...
corresponding to
32+23, 92+29, 152+215, 212+221, 332+233, 245+524, 563+356, 3215+1532.
One can also fix the value of y and consider the sequence of x values that gives Leyland primes, for example x2 + 2x is prime for x = 3, 9, 15, 21, 33, 2007, 2127, 3759, ... ().
By November 2012, the largest Leyland number that had been proven to be prime was 51226753 + 67535122 with 25050 digits. From January 2011 to April 2011, it was the largest prime whose primality was proved by elliptic curve primality proving. In December 2012, this was improved by proving the primality of the two numbers 311063 + 633110 (5596 digits) and 86562929 + 29298656 (30008 digits), the latter of which surpassed the previous record. There are many larger known probable primes such as 3147389 + 9314738, but it is hard to prove primality of large Leyland numbers. Paul Leyland writes on his website: "More recently still, it was realized that numbers of this form are ideal test cases for general purpose primality proving programs. They have a simple algebraic description but no obvious cyclotomic properties which special purpose algorithms can exploit."
There is a project called XYYXF to factor composite Leyland numbers.
Leyland number of the second kind
A Leyland number of the second kind is a number of the form
where x and y are integers greater than 1. The first such numbers are:
0, 1, 7, 17, 28, 79, 118, 192, 399, 431, 513, 924, 1844, 1927, 2800, 3952, 6049, 7849, 8023, 13983, 16188, 18954, 32543, 58049, 61318, 61440, 65280, 130783, 162287, 175816, 255583, 261820, ...
A Leyland prime of the second kind is a Leyland number of the second kind that is also prime. The first few such primes are:
7, 17, 79, 431, 58049, 130783, 162287, 523927, 2486784401, 6102977801, 8375575711, 13055867207, 83695120256591, 375700268413577, 2251799813682647, ...
For the probable primes, see Henri Lifchitz & Renaud Lifchitz, PRP Top Records search.
References
External links
Eponymous numbers in mathematics
Integer sequences
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https://en.wikipedia.org/wiki/Family-wise%20error%20rate
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In statistics, family-wise error rate (FWER) is the probability of making one or more false discoveries, or type I errors when performing multiple hypotheses tests.
Familywise and Experimentwise Error Rates
John Tukey developed in 1953 the concept of a familywise error rate as the probability of making a Type I error among a specified group, or "family," of tests. Ryan (1959) proposed the related concept of an experimentwise error rate, which is the probability of making a Type I error in a given experiment. Hence, an experimentwise error rate is a familywise error rate for all of the tests that are conducted within an experiment.
As Ryan (1959, Footnote 3) explained, an experiment may contain two or more families of multiple comparisons, each of which relates to a particular statistical inference and each of which has its own separate familywise error rate. Hence, familywise error rates are usually based on theoretically informative collections of multiple comparisons. In contrast, an experimentwise error rate may be based on a co-incidental collection of comparisons that refer to a diverse range of separate inferences. Consequently, some have argued that it may not be useful to control the experimentwise error rate. Indeed, Tukey was against the idea of experimentwise error rates (Tukey, 1956, personal communication, in Ryan, 1962, p. 302). More recently, Rubin (2021) criticised the automatic consideration of experimentwise error rates, arguing that “in many cases, the joint studywise [experimentwise] hypothesis has no relevance to researchers’ specific research questions, because its constituent hypotheses refer to comparisons and variables that have no theoretical or practical basis for joint consideration.”
Background
Within the statistical framework, there are several definitions for the term "family":
Hochberg & Tamhane (1987) defined "family" as "any collection of inferences for which it is meaningful to take into account some combined measure of error".
According to Cox (1982), a set of inferences should be regarded a family:
To take into account the selection effect due to data dredging
To ensure simultaneous correctness of a set of inferences as to guarantee a correct overall decision
To summarize, a family could best be defined by the potential selective inference that is being faced: A family is the smallest set of items of inference in an analysis, interchangeable about their meaning for the goal of research, from which selection of results for action, presentation or highlighting could be made (Yoav Benjamini).
Classification of multiple hypothesis tests
Definition
The FWER is the probability of making at least one type I error in the family,
or equivalently,
Thus, by assuring , the probability of making one or more type I errors in the family is controlled at level .
A procedure controls the FWER in the weak sense if the FWER control at level is guaranteed only when all null hypotheses are true (i.e. when , meaning
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https://en.wikipedia.org/wiki/Singular%20cardinals%20hypothesis
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In set theory, the singular cardinals hypothesis (SCH) arose from the question of whether the least cardinal number for which the generalized continuum hypothesis (GCH) might fail could be a singular cardinal.
According to Mitchell (1992), the singular cardinals hypothesis is:
If κ is any singular strong limit cardinal, then 2κ = κ+.
Here, κ+ denotes the successor cardinal of κ.
Since SCH is a consequence of GCH, which is known to be consistent with ZFC, SCH is consistent with ZFC. The negation of SCH has also been shown to be consistent with ZFC, if one assumes the existence of a sufficiently large cardinal number. In fact, by results of Moti Gitik, ZFC + ¬SCH is equiconsistent with ZFC + the existence of a measurable cardinal κ of Mitchell order κ++.
Another form of the SCH is the following statement:
2cf(κ) < κ implies κcf(κ) = κ+,
where cf denotes the cofinality function. Note that κcf(κ)= 2κ for all singular strong limit cardinals κ. The second formulation of SCH is strictly stronger than the first version, since the first one only mentions strong limits. From a model in which the first version of SCH fails at ℵω and GCH holds above ℵω+2, we can construct a model in which the first version of SCH holds but the second version of SCH fails, by adding ℵω Cohen subsets to ℵn for some n.
Jack Silver proved that if κ is singular with uncountable cofinality and 2λ = λ+ for all infinite cardinals λ < κ, then 2κ = κ+. Silver's original proof used generic ultrapowers. The following important fact follows from Silver's theorem: if the singular cardinals hypothesis holds for all singular cardinals of countable cofinality, then it holds for all singular cardinals. In particular, then, if is the least counterexample to the singular cardinals hypothesis, then .
The negation of the singular cardinals hypothesis is intimately related to violating the GCH at a measurable cardinal. A well-known result of Dana Scott is that if the GCH holds below a measurable cardinal on a set of measure one—i.e., there is normal -complete ultrafilter D on such that , then . Starting with a supercompact cardinal, Silver was able to produce a model of set theory in which is measurable and in which . Then, by applying Prikry forcing to the measurable , one gets a model of set theory in which is a strong limit cardinal of countable cofinality and in which —a violation of the SCH. Gitik, building on work of Woodin, was able to replace the supercompact in Silver's proof with measurable of Mitchell order . That established an upper bound for the consistency strength of the failure of the SCH. Gitik again, using results of inner model theory, was able to show that a measurable cardinal of Mitchell order is also the lower bound for the consistency strength of the failure of SCH.
A wide variety of propositions imply SCH. As was noted above, GCH implies SCH. On the other hand, the proper forcing axiom, which implies and hence is incompatible with GCH also imp
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https://en.wikipedia.org/wiki/SETAR%20%28model%29
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In statistics, Self-Exciting Threshold AutoRegressive (SETAR) models are typically applied to time series data as an extension of autoregressive models, in order to allow for higher degree of flexibility in model parameters through a regime switching behaviour.
Given a time series of data xt, the SETAR model is a tool for understanding and, perhaps, predicting future values in this series, assuming that the behaviour of the series changes once the series enters a different regime. The switch from one regime to another depends on the past values of the x series (hence the Self-Exciting portion of the name).
The model consists of k autoregressive (AR) parts, each for a different regime. The model is usually referred to as the SETAR(k, p) model where k is the number of threshold, there are k+1 number of regime in the model, and p is the order of the autoregressive part (since those can differ between regimes, the p portion is sometimes dropped and models are denoted simply as SETAR(k).
Definition
Autoregressive Models
Consider a simple AR(p) model for a time series yt
where:
for i=1,2,...,p are autoregressive coefficients, assumed to be constant over time;
stands for white-noise error term with constant variance.
written in a following vector form:
where:
is a row vector of variables;
is the vector of parameters :;
stands for white-noise error term with constant variance.
SETAR as an Extension of the Autoregressive Model
SETAR models were introduced by Howell Tong in 1977 and more fully developed in the seminal paper (Tong and Lim, 1980). They can be thought of in terms of extension of autoregressive models, allowing for changes in the model parameters according to the value of weakly exogenous threshold variable zt, assumed to be past values of y, e.g. yt-d, where d is the delay parameter, triggering the changes.
Defined in this way, SETAR model can be presented as follows:
if
where:
is a column vector of variables;
are k+1 non-trivial thresholds dividing the domain of zt into k different regimes.
The SETAR model is a special case of Tong's general threshold autoregressive models (Tong and Lim, 1980, p. 248). The latter allows the threshold variable to be very flexible, such as an exogenous time series in the open-loop threshold autoregressive system (Tong and Lim, 1980, p. 249), a Markov chain in the Markov-chain driven threshold autoregressive model (Tong and Lim, 1980, p. 285), which is now also known as the Markov switching model.
For a comprehensive review of developments over the 30 years
since the birth of the model, see Tong (2011).
Basic Structure
In each of the k regimes, the AR(p) process is governed by a different set of p variables :. In such setting, a change of the regime (because the past values of the series yt-d surpassed the threshold) causes a different set of coefficients : to govern the process y.
See also
Logistic Smooth-Transmission Model
References
Hansen, B.E. (1997). Inference in TAR Models, St
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https://en.wikipedia.org/wiki/Perplexity
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In information theory, perplexity is a measurement of how well a probability distribution or probability model predicts a sample. It may be used to compare probability models. A low perplexity indicates the probability distribution is good at predicting the sample. Perplexity was originally introduced in 1977 in the context of speech recognition by Frederick Jelinek, Robert Leroy Mercer, Lalit R. Bahl, and James K. Baker.
Perplexity of a Probability Distribution
The perplexity PP of a discrete probability distribution p is a concept widely used in information theory, machine learning, and statistical modeling. It is defined as
where H(p) is the entropy (in bits) of the distribution, and x ranges over the events. The base of the logarithm need not be 2: The perplexity is independent of the base, provided that the entropy and the exponentiation use the same base. In some contexts, this measure is also referred to as the (order-1 true) diversity.
Perplexity of a random variable X may be defined as the perplexity of the distribution over its possible values x. It can be thought of as a measure of uncertainty or "surprise" related to the outcomes.
In the special case where p models a fair k-sided die (a uniform distribution over k discrete events), its perplexity is k. A random variable with perplexity k has the same uncertainty as a fair k-sided die. One is said to be "k-ways perplexed" about the value of the random variable. Unless it is a fair k-sided die, more than k values may be possible, but the overall uncertainty is not greater because some values may have a probability greater than 1/k.
Perplexity is sometimes used as a measure of the difficulty of a prediction problem. However, it's not always an accurate representation. For example, if you have two choices, one with probability 0.9, your chances of a correct guess using the optimal strategy are 90 percent. Yet, the perplexity is 2−0.9 log2 0.9 - 0.1 log2 0.1= 1.38. The inverse of the perplexity, 1/1.38 = 0.72, does not correspond to the 0.9 probability.
The perplexity is the exponentiation of the entropy, a more straightforward quantity. Entropy measures the expected or "average" number of bits required to encode the outcome of the random variable using an optimal variable-length code. It can also be regarded as the expected information gain from learning the outcome of the random variable, providing insight into the uncertainty and complexity of the underlying probability distribution.
Perplexity of a probability model
A model of an unknown probability distribution p, may be proposed based on a training sample that was drawn from p. Given a proposed probability model q, one may evaluate q by asking how well it predicts a separate test sample x1, x2, ..., xN also drawn from p. The perplexity of the model q is defined as
where is customarily 2. Better models q of the unknown distribution p will tend to assign higher probabilities q(xi) to the test events. Thus, they have low
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https://en.wikipedia.org/wiki/Crystal%20ball%20%28disambiguation%29
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A crystal ball is a scrying or fortune telling orb object
Crystal Ball may also refer to:
Crystal Ball (detector), a hermetic particle detector
Crystal Ball function, a probability density function
Crystal Ball (G.I. Joe), a fictional villain in the G.I. Joe universe, member of Cobra
Sabato's Crystal Ball, a web site analyzing and predicting national political races
"The Crystal Ball" (fairy tale), a German fairy tale collected by the Brothers Grimm
The Crystal Ball (film), a 1943 film starring Ray Milland
The Crystal Ball (painting), a 1902 painting by John William Waterhouse
Crystal Ball, a "lifeline" in the Who Wants to Be a Millionaire? franchise
Music
Albums
Crystal Ball (box set), a 1998 box set by Prince
Crystal Ball (EP), a 2019 EP by Purplebeck
Crystal Ball (Styx album), a 1976 album by Styx
Crystal Ball (unreleased album), album by Prince, recorded in 1986
Songs
"Crystal Ball" (Keane song), a 2006 song by Keane
"Crystal Ball" (Styx song), the 1976 album's title track
"Crystal Ball", a song by Pink from the album Funhouse
"Crystal Ball", a song by State Champs from the album Living Proof
"Crystal Ball", a song by Timeflies from the album After Hours
"Crystal Ball", a 1998 (recorded in 1986) Prince song from the box set of the same name.
See also
Cristóbal (disambiguation), Spanish equivalent of "Christopher"
"Crystal Baller", a 2003 song by Third Eye Blind
Krystal Ball, news anchor and former MSNBC co-host
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https://en.wikipedia.org/wiki/David%20George%20Kendall
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David George Kendall FRS (15 January 1918 – 23 October 2007) was an English statistician and mathematician, known for his work on probability, statistical shape analysis, ley lines and queueing theory. He spent most of his academic life in the University of Oxford (1946–1962) and the University of Cambridge (1962–1985). He worked with M. S. Bartlett during World War II, and visited Princeton University after the war.
Life and career
David George Kendall was born on 15 January 1918 in Ripon, West Riding of Yorkshire, and attended Ripon Grammar School before attending Queen's College, Oxford, graduating in 1939.
He worked on rocketry at the Ministry of Supply's Projectile Development Establishment during the World War II, before moving to Magdalen College, Oxford, in 1946.
In 1962 he was appointed the first Professor of Mathematical Statistics in the Statistical Laboratory, University of Cambridge; in which post he remained until his retirement in 1985. He was elected to a professorial fellowship at Churchill College, and he was a founding trustee of the Rollo Davidson Trust. In 1986, he was awarded an Honorary Degree (Doctor of Science) by the University of Bath.
Kendall was an expert in probability and data analysis, and pioneered statistical shape analysis including the study of ley lines. He defined Kendall's notation for queueing theory.
The Royal Statistical Society awarded him the Guy Medal in Silver in 1955, followed in 1981 by the Guy Medal in Gold. In 1980 the London Mathematical Society awarded Kendall their Senior Whitehead Prize, and in 1989 their De Morgan Medal. He was elected a fellow of the Royal Society in 1964.
Kendall also played a key role in founding the Bernoulli Society in 1975, and was its initial president.
He was married to Diana Fletcher from 1952 until his death. They had two sons and four daughters, including Wilfrid Kendall, professor in the Department of Statistics at the University of Warwick, journalist Bridget Kendall MBE, Felicity Kendall Hickman, Judy Kendall, reader and poet at University of Salford, George Kendall, and Harriet Strudwick, the Antipodean sibling.
Selected bibliography
References
External links
The Papers of Professor David Kendall held at Churchill Archives Centre
Royal Society citation
1918 births
2007 deaths
People from Ripon
20th-century English mathematicians
21st-century English mathematicians
English statisticians
Queueing theorists
Alumni of The Queen's College, Oxford
Fellows of Churchill College, Cambridge
Fellows of Magdalen College, Oxford
Fellows of the Royal Society
Probability theorists
Cambridge mathematicians
People educated at Ripon Grammar School
Professors of the University of Cambridge
Mathematical statisticians
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https://en.wikipedia.org/wiki/Bombieri%E2%80%93Vinogradov%20theorem
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In mathematics, the Bombieri–Vinogradov theorem (sometimes simply called Bombieri's theorem) is a major result of analytic number theory, obtained in the mid-1960s, concerning the distribution of primes in arithmetic progressions, averaged over a range of moduli. The first result of this kind was obtained by Mark Barban in 1961 and the Bombieri–Vinogradov theorem is a refinement of Barban's result. The Bombieri–Vinogradov theorem is named after Enrico Bombieri and A. I. Vinogradov, who published on a related topic, the density hypothesis, in 1965.
This result is a major application of the large sieve method, which developed rapidly in the early 1960s, from its beginnings in work of Yuri Linnik two decades earlier. Besides Bombieri, Klaus Roth was working in this area. In the late 1960s and early 1970s, many of the key ingredients and estimates were simplified by Patrick X. Gallagher.
Statement of the Bombieri–Vinogradov theorem
Let and be any two positive real numbers with
Then
Here is the Euler totient function, which is the number of summands for the modulus q, and
where denotes the von Mangoldt function.
A verbal description of this result is that it addresses the error term in the prime number theorem for arithmetic progressions, averaged over the moduli q up to Q. For a certain range of Q, which are around if we neglect logarithmic factors, the error averaged is nearly as small as . This is not obvious, and without the averaging is about of the strength of the Generalized Riemann Hypothesis (GRH).
See also
Elliott–Halberstam conjecture (a generalization of Bombieri–Vinogradov)
Vinogradov's theorem (named after Ivan Matveyevich Vinogradov)
Notes
External links
The Bombieri-Vinogradov Theorem, R.C. Vaughan's Lecture note.
Sieve theory
Theorems in analytic number theory
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https://en.wikipedia.org/wiki/Vinogradov%27s%20theorem
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In number theory, Vinogradov's theorem is a result which implies that any sufficiently large odd integer can be written as a sum of three prime numbers. It is a weaker form of Goldbach's weak conjecture, which would imply the existence of such a representation for all odd integers greater than five. It is named after Ivan Matveyevich Vinogradov, who proved it in the 1930s. Hardy and Littlewood had shown earlier that this result followed from the generalized Riemann hypothesis, and Vinogradov was able to remove this assumption. The full statement of Vinogradov's theorem gives asymptotic bounds on the number of representations of an odd integer as a sum of three primes. The notion of "sufficiently large" was ill-defined in Vinogradov's original work, but in 2002 it was shown that 101346 is sufficiently large. Additionally numbers up to 1020 had been checked via brute force methods, thus only a finite number of cases to check remained before the odd Goldbach conjecture would be proven or disproven. In 2013, Harald Helfgott proved Goldbach's weak conjecture for all cases.
Statement of Vinogradov's theorem
Let A be a positive real number. Then
where
using the von Mangoldt function , and
A consequence
If N is odd, then G(N) is roughly 1, hence for all sufficiently large N. By showing that the contribution made to r(N) by proper prime powers is , one sees that
This means in particular that any sufficiently large odd integer can be written as a sum of three primes, thus showing Goldbach's weak conjecture for all but finitely many cases.
Strategy of proof
The proof of the theorem follows the Hardy–Littlewood circle method. Define the exponential sum
.
Then we have
,
where denotes the number of representations restricted to prime powers . Hence
.
If is a rational number , then can be given by the distribution of prime numbers in residue classes modulo . Hence, using the Siegel–Walfisz theorem we can compute the contribution of the above integral in small neighbourhoods of rational points with small denominator. The set of real numbers close to such rational points is usually referred to as the major arcs, the complement forms the minor arcs. It turns out that these intervals dominate the integral, hence to prove the theorem one has to give an upper bound for for contained in the minor arcs. This estimate is the most difficult part of the proof.
If we assume the Generalized Riemann Hypothesis, the argument used for the major arcs can be extended to the minor arcs. This was done by Hardy and Littlewood in 1923. In 1937 Vinogradov gave an unconditional upper bound for . His argument began with a simple sieve identity, the resulting terms were then rearranged in a complicated way to obtain some cancellation. In 1977 R. C. Vaughan found a much simpler argument, based on what later became known as Vaughan's identity. He proved that if , then
.
Using the Siegel–Walfisz theorem we can deal with up to arbitrary powers of , using Dirichlet's approxim
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https://en.wikipedia.org/wiki/Admissible%20ordinal
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In set theory, an ordinal number α is an admissible ordinal if Lα is an admissible set (that is, a transitive model of Kripke–Platek set theory); in other words, α is admissible when α is a limit ordinal and Lα ⊧ Σ0-collection. The term was coined by Richard Platek in 1966.
The first two admissible ordinals are ω and (the least nonrecursive ordinal, also called the Church–Kleene ordinal). Any regular uncountable cardinal is an admissible ordinal.
By a theorem of Sacks, the countable admissible ordinals are exactly those constructed in a manner similar to the Church–Kleene ordinal, but for Turing machines with oracles. One sometimes writes for the -th ordinal that is either admissible or a limit of admissibles; an ordinal that is both is called recursively inaccessible. There exists a theory of large ordinals in this manner that is highly parallel to that of (small) large cardinals (one can define recursively Mahlo ordinals, for example). But all these ordinals are still countable. Therefore, admissible ordinals seem to be the recursive analogue of regular cardinal numbers.
Notice that α is an admissible ordinal if and only if α is a limit ordinal and there does not exist a γ < α for which there is a Σ1(Lα) mapping from γ onto α. is an admissible ordinal iff there is a standard model of KP whose set of ordinals is , in fact this may be take as the definition of admissibility. The th admissible ordinal is sometimes denoted by p.174 or .
The Friedman-Jensen-Sacks theorem states that countable is admissible iff there exists some such that is the least ordinal not recursive in .
See also
α-recursion theory
Large countable ordinals
Constructible universe
Regular cardinal
References
Ordinal numbers
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https://en.wikipedia.org/wiki/Coefficient%20matrix
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In linear algebra, a coefficient matrix is a matrix consisting of the coefficients of the variables in a set of linear equations. The matrix is used in solving systems of linear equations.
Coefficient matrix
In general, a system with linear equations and unknowns can be written as
where are the unknowns and the numbers are the coefficients of the system. The coefficient matrix is the matrix with the coefficient as the th entry:
Then the above set of equations can be expressed more succinctly as
where is the coefficient matrix and is the column vector of constant terms.
Relation of its properties to properties of the equation system
By the Rouché–Capelli theorem, the system of equations is inconsistent, meaning it has no solutions, if the rank of the augmented matrix (the coefficient matrix augmented with an additional column consisting of the vector ) is greater than the rank of the coefficient matrix. If, on the other hand, the ranks of these two matrices are equal, the system must have at least one solution. The solution is unique if and only if the rank equals the number of variables. Otherwise the general solution has free parameters; hence in such a case there are an infinitude of solutions, which can be found by imposing arbitrary values on of the variables and solving the resulting system for its unique solution; different choices of which variables to fix, and different fixed values of them, give different system solutions.
Dynamic equations
A first-order matrix difference equation with constant term can be written as
where is and and are . This system converges to its steady-state level of if and only if the absolute values of all eigenvalues of are less than 1.
A first-order matrix differential equation with constant term can be written as
This system is stable if and only if all eigenvalues of have negative real parts.
References
Linear algebra
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https://en.wikipedia.org/wiki/Archibald%20Henderson%20%28professor%29
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Archibald Henderson (July 17, 1877 – December 6, 1963) was an American professor of mathematics who wrote on a variety of subjects, including drama and history. He is well known for his friendship with George Bernard Shaw.
Early life
He was born at Salisbury, North Carolina, was educated at the University of North Carolina (A.B., 1898; Ph.D., 1902), where he was a member of Sigma Nu Fraternity, and studied more at Chicago, Cambridge, and Berlin universities, and at the Sorbonne (Paris). After 1899 he taught at the University of North Carolina, becoming professor of pure mathematics in 1908.
Bernard Shaw
In 1903 in Chicago Henderson saw the first performance in the United States of Bernard Shaw's play You Never Can Tell. Henderson became so enthusiastic about the playwright and his personality that he determined to write Shaw's biography. After some communication between Shaw and Henderson, Henderson arrived in London in 1907 on the very same train carrying Mark Twain who was en route to Oxford to receive an honorary degree. Having established his relation with Shaw, Henderson went on to write three different versions of Shaw's biography covering Shaw's entire career up to the playwright's death in 1950, including several other miscellaneous works about Shaw. The Libraries at the University of North Carolina hold about 380 of Henderson's own writings on various topics, including an invaluable collection of 75 scrap books devoted to articles about Shaw.
Works
His publications include:
Lines on the Cubic Surface (1911)
Interpreters of Life and the Modern Spirit (1911)
Mark Twain (1911)
George Bernard Shaw: His Life and Works (1911)
George Bernard Shaw: Playboy and Prophet (1932)
George Bernard Shaw: Man of the Century (1956)
George Bernard Shaw: Table Talk of G.B.S. (1926)
George Bernard Shaw: Is Bernard Shaw a Dramatist? (1929)
Forerunners of the Republic (1913)
The Life and Times of Richard Henderson (1913)
European Dramatists (1913)
The Changing Drama (1914)
References
"Archibald Henderson", Dictionary of North Carolina Biography, edited by William S. Powell. University of North Carolina Press, 1979–1996.
External links
American biographers
American male biographers
American non-fiction writers
University of Chicago alumni
University of Paris alumni
1877 births
1963 deaths
University of North Carolina at Chapel Hill faculty
People from Salisbury, North Carolina
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https://en.wikipedia.org/wiki/Routh%27s%20theorem
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In geometry, Routh's theorem determines the ratio of areas between a given triangle and a triangle formed by the pairwise intersections of three cevians. The theorem states that if in triangle points , , and lie on segments , , and , then writing , , and , the signed area of the triangle formed by the cevians , , and is
where is the area of the triangle .
This theorem was given by Edward John Routh on page 82 of his Treatise on Analytical Statics with Numerous Examples in 1896. The particular case has become popularized as the one-seventh area triangle. The case implies that the three medians are concurrent (through the centroid).
Proof
Suppose that the area of triangle is 1. For triangle and line using Menelaus's theorem, We could obtain:
Then
So the area of triangle is:
Similarly, we could know: and
Thus the area of triangle is:
Citations
The citation commonly given for Routh's theorem is Routh's Treatise on Analytical Statics with Numerous Examples, Volume 1, Chap. IV, in the second edition of 1896
p. 82, possibly because that edition has been easier to hand. However, Routh gave the theorem already in the first edition of 1891, Volume 1, Chap. IV, p. 89. Although there is a change in pagination between the editions, the wording of the relevant footnote remained the same.
Routh concludes his extended footnote with a caveat:
"The author has not met with these expressions for the areas of two triangles that often occur. He has therefore placed them here in order that the argument in the text may be more easily understood."
Presumably, Routh felt those circumstances had not changed in the five years between editions. On the other hand, the title of Routh's book had been used earlier by Isaac Todhunter; both had been coached by William Hopkins.
Although Routh published the theorem in his book, that is not the first published statement. It is stated and proved as rider (vii) on page 33 of Solutions of the Cambridge Senate-house Problems and Riders for the Year 1878, i.e., the mathematical tripos of that year, and the link is https://archive.org/details/solutionscambri00glaigoog. It is stated that the author of the problems with roman numerals is Glaisher.
Routh was a famous Mathematical Tripos coach when his book came out and was surely familiar with the content of the 1878 tripos examination. Thus, his statement The author has not met with these expressions for the areas of two triangles that often occur. is puzzling.
Problems in this spirit have a long history in recreational mathematics and mathematical paedagogy, perhaps one of the oldest instances of being the determination of the proportions of the fourteen regions of the Stomachion board. With Routh's Cambridge in mind, the one-seventh-area triangle, associated in some accounts with Richard Feynman, shows up, for example, as Question 100, p. 80, in Euclid's Elements of Geometry (Fifth School Edition), by Robert Potts (1805--1885,) of Trinity College, published
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https://en.wikipedia.org/wiki/Quasi-finite%20field
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In mathematics, a quasi-finite field is a generalisation of a finite field. Standard local class field theory usually deals with complete valued fields whose residue field is finite (i.e. non-archimedean local fields), but the theory applies equally well when the residue field is only assumed quasi-finite.
Formal definition
A quasi-finite field is a perfect field K together with an isomorphism of topological groups
where Ks is an algebraic closure of K (necessarily separable because K is perfect). The field extension Ks/K is infinite, and the Galois group is accordingly given the Krull topology. The group is the profinite completion of integers with respect to its subgroups of finite index.
This definition is equivalent to saying that K has a unique (necessarily cyclic) extension Kn of degree n for each integer n ≥ 1, and that the union of these extensions is equal to Ks. Moreover, as part of the structure of the quasi-finite field, there is a generator Fn for each Gal(Kn/K), and the generators must be coherent, in the sense that if n divides m, the restriction of Fm to Kn is equal to Fn.
Examples
The most basic example, which motivates the definition, is the finite field K = GF(q). It has a unique cyclic extension of degree n, namely Kn = GF(qn). The union of the Kn is the algebraic closure Ks. We take Fn to be the Frobenius element; that is, Fn(x) = xq.
Another example is K = C((T)), the ring of formal Laurent series in T over the field C of complex numbers. (These are simply formal power series in which we also allow finitely many terms of negative degree.) Then K has a unique cyclic extension
of degree n for each n ≥ 1, whose union is an algebraic closure of K called the field of Puiseux series, and that a generator of Gal(Kn/K) is given by
This construction works if C is replaced by any algebraically closed field C of characteristic zero.
Notes
References
Class field theory
Field (mathematics)
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https://en.wikipedia.org/wiki/Encyclopedia%20of%20Mathematics
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The Encyclopedia of Mathematics (also EOM and formerly Encyclopaedia of Mathematics) is a large reference work in mathematics.
Overview
The 2002 version contains more than 8,000 entries covering most areas of mathematics at a graduate level, and the presentation is technical in nature. The encyclopedia is edited by Michiel Hazewinkel and was published by Kluwer Academic Publishers until 2003, when Kluwer became part of Springer. The CD-ROM contains animations and three-dimensional objects.
The encyclopedia has been translated from the Soviet Matematicheskaya entsiklopediya (1977) originally edited by Ivan Matveevich Vinogradov and extended with comments and three supplements adding several thousand articles.
Until November 29, 2011, a static version of the encyclopedia could be browsed online free of charge online. This URL now redirects to the new wiki incarnation of the EOM.
Encyclopedia of Mathematics wiki
A new dynamic version of the encyclopedia is now available as a public wiki online. This new wiki is a collaboration between Springer and the European Mathematical Society. This new version of the encyclopedia includes the entire contents of the previous online version, but all entries can now be publicly updated to include the newest advancements in mathematics. All entries will be monitored for content accuracy by members of an editorial board selected by the European Mathematical Society.
Versions
Vinogradov, I. M. (Ed.), Matematicheskaya entsiklopediya, Moscow, Sov. Entsiklopediya, 1977.
Hazewinkel, M. (Ed.), Encyclopaedia of Mathematics (set), Kluwer, 1994 ().
Hazewinkel, M. (Ed.), Encyclopaedia of Mathematics, Vol. 1 (A–B), Kluwer, 1987 ().
Hazewinkel, M. (Ed.), Encyclopaedia of Mathematics, Vol. 2 (C), Kluwer, 1988 ().
Hazewinkel, M. (Ed.), Encyclopaedia of Mathematics, Vol. 3 (D–Fey), Kluwer, 1989 ().
Hazewinkel, M. (Ed.), Encyclopaedia of Mathematics, Vol. 4 (Fib–H), Kluwer, 1989 ().
Hazewinkel, M. (Ed.), Encyclopaedia of Mathematics, Vol. 5 (I–Lit), Kluwer, 1990 ().
Hazewinkel, M. (Ed.), Encyclopaedia of Mathematics, Vol. 6 (Lob–Opt), Kluwer, 1990 ().
Hazewinkel, M. (Ed.), Encyclopaedia of Mathematics, Vol. 7 (Orb–Ray), Kluwer, 1991 ().
Hazewinkel, M. (Ed.), Encyclopaedia of Mathematics, Vol. 8 (Rea–Sti), Kluwer, 1992 ().
Hazewinkel, M. (Ed.), Encyclopaedia of Mathematics, Vol. 9 (Sto–Zyg), Kluwer, 1993 ().
Hazewinkel, M. (Ed.), Encyclopaedia of Mathematics, Vol. 10 (Index), Kluwer, 1994 ().
Hazewinkel, M. (Ed.), Encyclopaedia of Mathematics, Supplement I, Kluwer, 1997 ().
Hazewinkel, M. (Ed.), Encyclopaedia of Mathematics, Supplement II, Kluwer, 2000 ().
Hazewinkel, M. (Ed.), Encyclopaedia of Mathematics, Supplement III, Kluwer, 2002 ().
Hazewinkel, M. (Ed.), Encyclopaedia of Mathematics on CD-ROM, Kluwer, 1998 ().
Encyclopedia of Mathematics, public wiki monitored by an editorial board under the management of the European Mathematical Society.
See also
List of online encyclopedias
References
External links
Germa
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https://en.wikipedia.org/wiki/Cameron%20Gordon
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Cameron Gordon may refer to:
Cameron Gordon (mathematician), professor of mathematics at the University of Texas, Austin
Cam Gordon, Green Party councillor for Minneapolis, Minnesota
Cameron Gordon (American football) (born 1991), American football linebacker
See also
Gordon Cameron (disambiguation)
Cam Gordon (rugby union), Australian rugby union player whose full name is George Campbell Gordon
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https://en.wikipedia.org/wiki/Iterated%20monodromy%20group
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In geometric group theory and dynamical systems the iterated monodromy group of a covering map is a group describing the monodromy action of the fundamental group on all iterations of the covering. A single covering map between spaces is therefore used to create a tower of coverings, by placing the covering over itself repeatedly. In terms of the Galois theory of covering spaces, this construction on spaces is expected to correspond to a construction on groups. The iterated monodromy group provides this construction, and it is applied to encode the combinatorics and symbolic dynamics of the covering, and provide examples of self-similar groups.
Definition
The iterated monodromy group of f is the following quotient group:
where :
is a covering of a path-connected and locally path-connected topological space X by its subset ,
is the fundamental group of X and
is the monodromy action for f.
is the monodromy action of the iteration of f, .
Action
The iterated monodromy group acts by automorphism on the rooted tree of preimages
where a vertex is connected by an edge with .
Examples
Iterated monodromy groups of rational functions
Let :
f be a complex rational function
be the union of forward orbits of its critical points (the post-critical set).
If is finite (or has a finite set of accumulation points), then the iterated monodromy group of f is the iterated monodromy group of the covering , where is the Riemann sphere.
Iterated monodromy groups of rational functions usually have exotic properties from the point of view of classical group theory. Most of them are infinitely presented, many have intermediate growth.
IMG of polynomials
The Basilica group is the iterated monodromy group of the polynomial
See also
Growth rate (group theory)
Amenable group
Complex dynamics
Julia set
References
Volodymyr Nekrashevych, Self-Similar Groups, Mathematical Surveys and Monographs Vol. 117, Amer. Math. Soc., Providence, RI, 2005; .
Kevin M. Pilgrim, Combinations of Complex Dynamical Systems, Springer-Verlag, Berlin, 2003; .
External links
arXiv.org - Iterated Monodromy Group - preprints about the Iterated Monodromy Group.
Laurent Bartholdi's page - Movies illustrating the Dehn twists about a Julia set.
mathworld.wolfram.com - The Monodromy Group page.
Geometric group theory
Homotopy theory
Complex analysis
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https://en.wikipedia.org/wiki/Trevor%20Wooley
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Trevor Dion Wooley FRS (born 17 September 1964) is a British mathematician and currently Professor of Mathematics at Purdue University. His fields of interest include analytic number theory, Diophantine equations and Diophantine problems, harmonic analysis,
the Hardy-Littlewood circle method, and the theory and applications of exponential sums. He has made significant breakthroughs on Waring's problem, for which he was awarded the Salem Prize in 1998.
He received his bachelor's degree in 1987 from the University of Cambridge and his PhD, supervised by Robert Charles Vaughan, in 1990 from the University of London. In 2007, he was elected Fellow of the Royal Society.
Awards and honours
Alfred P. Sloan Research Fellow, 1993–1995
Salem Prize, 1998
Invited speaker, International Congress of Mathematicians, Beijing 2002
Elected Fellow of the Royal Society, 2007.
Fröhlich Prize, 2012.
Fellow of the American Mathematical Society, 2012.
Invited speaker, International Congress of Mathematicians, Seoul 2014
Selected publications
References
External links
Living people
20th-century British mathematicians
21st-century British mathematicians
Purdue University faculty
Fellows of the Royal Society
Fellows of the American Mathematical Society
Alumni of Imperial College London
1964 births
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https://en.wikipedia.org/wiki/List%20of%20national%20parks%20of%20Venezuela
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The national parks of Venezuela are protected areas in Venezuela covering a wide range of habitats. In 2007 there were 43 national parks, covering 21.76% of Venezuela's territory.
Statistics
Every Venezuela state has one or more national parks.
5 national parks - Lara, Amazonas
4 national parks - Falcón, Mérida, Miranda, Portuguesa, and Táchira.
3 national parks - Apure, Sucre, and Trujillo.
2 national parks - Barinas, Bolívar, Carabobo, Distrito Capital, Guárico, Nueva Esparta, Yaracuy, and Zulia.
1 national park - Anzoátegui, Aragua, Cojedes, Delta Amacuro, Federal Dependencies, Monagas, and Vargas.
18 national parks are over 1000 km2; 15 over 2000 km2; 5 over 5000 km2 and 3 over 10,000 km2. The largest parks, in the Guayana Region, are Parima Tapirapecó National Park (39,000 km2) and Canaima National Park (30,000 km2).
List of national parks
See also
List of national parks
Venezuelan bolívar banknotes
References
External links
Instituto Nacional de Parques de Venezuela
Ministerio del Poder Popular para el Ambiente (Ministry of Environment)
National Parks in Venezuela
Venezuela
National parks
National parks
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https://en.wikipedia.org/wiki/Fibonacci%27s%20identity
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Fibonacci's identity may refer either to:
the Brahmagupta–Fibonacci identity in algebra, showing that the set of all sums of two squares is closed under multiplication
the Cassini and Catalan identities on Fibonacci numbers
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https://en.wikipedia.org/wiki/Sean%20Hood
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Sean Hood (born August 13, 1966) is an American screenwriter and film director.
Early life
Hood graduated from Brown University, with a double major in pure mathematics and studio art, and then spent several years working in Hollywood as a set dresser, prop assistant and art director working with filmmakers as diverse as James Cameron, David Fincher and David Lynch. He continued his studies at the USC School of Cinematic Arts, graduating in 1997 with an MFA in production. His student short film, "The Shy and the Naked" won a grant from the Sloan Foundation for the positive portrayal of science.
Career
Screenwriting
Hood sold his first spec screenplay to MTV Films in 2000, and went on to sign a deal with Dimension Films, which included rewrites on Halloween: Resurrection and Cursed. He went on to work on Conan the Barbarian for producer Avi Lerner, and Hercules: The Legend Begins. Most recently, he penned an screenplay adaptation of the novel Rolling in The Deep, which will be produced by J. Todd Harris and directed by Mary Lambert (director). In 2011, Hood was hired to write the script for the fifth Rambo film, titled Rambo: Last Stand, however, Hood's script was put on hold in early 2012. In 2014, Hood's Rambo: Last Stand script was abandoned in favor of a new script by Sylvester Stallone.
Television
He wrote the episode Echoes for the NBC horror anthology series Fear Itself. Similarly, he contributed to the Showtime horror anthology series Masters of Horror by penning the episode Sick Girl.
Hood also worked on Sick for The CW. and The Dorm for MTV. In 2022, he adapted Warren Ellis' comic FreakAngels into an animated series for Crunchyroll.
Directing
He was one of the founding members of Filmmakers Alliance and often collaborates creatively with FA's president, Jacques Thelemaque. His most recent short film is Melancholy Baby.
Blogging
Hood wrote the screenwriting blog Genre Hacks from 2008 until 2017.
Teaching at USC
He is currently an adjunct professor at the USC School of Cinematic Arts. He teaches the courses "Writing the Feature Script" and "Advanced Motion Picture Script Analysis,"and "Creating The Short Film."
Credits
Film
Television
References
External links
Genre Hacks
Scripts & Scribes Interview with Sean Hood
1966 births
Living people
20th-century American screenwriters
21st-century American screenwriters
USC School of Cinematic Arts alumni
Brown University alumni
American television writers
American educators
American male screenwriters
American male television writers
Screenwriting instructors
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https://en.wikipedia.org/wiki/Glaisher%27s%20theorem
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In number theory, Glaisher's theorem is an identity useful to the study of integer partitions. Proved in 1883 by James Whitbread Lee Glaisher, it states that the number of partitions of an integer into parts not divisible by is equal to the number of partitions in which no part is repeated or more times. This generalizes a result established in 1748 by Leonhard Euler for the case .
Statement
It states that the number of partitions of an integer into parts not divisible by is equal to the number of partitions in which no part is repeated d or more times, which can be written formally as partitions of the form where and .
When this becomes the special case known as Euler's theorem, that the number of partitions of into distinct parts is equal to the number of partitions of into odd parts.
In the following examples, we use the multiplicity notation of partitions. For example, is a notation for the partition 1 + 1 + 1 + 1 + 2 + 3 + 3.
Example for d=2 (Euler's theorem case)
Among the 15 partitions of the number 7, there are 5, shown in bold below, that contain only odd parts (i.e. only odd numbers):
If we count now the partitions of 7 with distinct parts (i.e. where no number is repeated), we also obtain 5:
The partitions in bold in the first and second case are not the same, and it is not obvious why their number is the same.
Example for d=3
Among the 11 partitions of the number 6, there are 7, shown in bold below, that contain only parts not divisible by 3:
And if we count the partitions of 6 with no part that repeats more than 2 times, we also obtain 7:
Proof
A proof of the theorem can be obtained with generating functions. If we note the number of partitions with no parts divisible by d and the number of partitions with no parts repeated more than d-1 times, then the theorem means that for all n . The uniqueness of ordinary generating functions implies that instead of proving that for all n, it suffices to prove that the generating functions of and are equal, i.e. that .
Each generating function can be rewritten as infinite products (with a method similar to the infinite product of the partition function) :
(i.e. the product of terms where n is not divisible by d).
If we expand the infinite product for :
we see that each term in the numerator cancels with the corresponding multiple of d in the denominator. What remains after canceling all the numerator terms is exactly the infinite product for .
Hence the generating functions for and are equal.
Rogers-Ramanujan identities
If instead of counting the number of partitions with distinct parts we count the number of partitions with parts differing by at least 2, a further generalization is possible. It was first discovered by Leonard James Rogers in 1894, and then independently by Ramanujan in 1913 and Schur in 1917, in what are now known as the Rogers-Ramanujan identities. It states that:
1) The number of partitions whose parts differ by at least 2 is equal to
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https://en.wikipedia.org/wiki/Crofton%20formula
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In mathematics, the Crofton formula, named after Morgan Crofton (1826–1915), is a classic result of integral geometry relating the length of a curve to the expected number of times a "random" line intersects it.
Statement
Suppose is a rectifiable plane curve. Given an oriented line ℓ, let (ℓ) be the number of points at which and ℓ intersect. We can parametrize the general line ℓ by the direction in which it points and its signed distance from the origin. The Crofton formula expresses the arc length of the curve in terms of an integral over the space of all oriented lines:
The differential form
is invariant under rigid motions of , so it is a natural integration measure for speaking of an "average" number of intersections. It is usually called the kinematic measure.
The right-hand side in the Crofton formula is sometimes called the Favard length.
In general, the space of oriented lines in is the tangent bundle of , and we can similarly define a kinematic measure on it, which is also invariant under rigid motions of . Then for any rectifiable surface of codimension 1, we have where
Proof sketch
Both sides of the Crofton formula are additive over concatenation of curves, so it suffices to prove the formula for a single line segment. Since the right-hand side does not depend on the positioning of the line segment, it must equal some function of the segment's length. Because, again, the formula is additive over concatenation of line segments, the integral must be a constant times the length of the line segment. It remains only to determine the factor of 1/4; this is easily done by computing both sides when γ is the unit circle.
The proof for the generalized version proceeds exactly as above.
Poincare’s formula for intersecting curves
Let be the Euclidean group on the plane. It can be parametrized as , such that each defines some : rotate by counterclockwise around the origin, then translate by . Then is invariant under action of on itself, thus we obtained a kinematic measure on .
Given rectifiable simple (no self-intersection) curves in the plane, then The proof is done similarly as above. First note that both sides of the formula are additive in , thus the formula is correct with an undetermined multiplicative constant. Then explicitly calculate this constant, using the simplest possible case: two circles of radius 1.
Other forms
The space of oriented lines is a double cover of the space of unoriented lines. The Crofton formula is often stated in terms of the corresponding density in the latter space, in which the numerical factor is not 1/4 but 1/2. Since a convex curve intersects almost every line either twice or not at all, the unoriented Crofton formula for convex curves can be stated without numerical factors: the measure of the set of straight lines which intersect a convex curve is equal to its length.
The same formula (with the same multiplicative constants) apply for hyperbolic spaces and spherical spaces, when the
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https://en.wikipedia.org/wiki/List%20of%20formulae%20involving%20%CF%80
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The following is a list of significant formulae involving the mathematical constant . Many of these formulae can be found in the article Pi, or the article Approximations of .
Euclidean geometry
where is the circumference of a circle, is the diameter, and is the radius. More generally,
where and are, respectively, the perimeter and the width of any curve of constant width.
where is the area of a circle. More generally,
where is the area enclosed by an ellipse with semi-major axis and semi-minor axis .
where is the area between the witch of Agnesi and its asymptotic line; is the radius of the defining circle.
where is the area of a squircle with minor radius , is the gamma function and is the arithmetic–geometric mean.
where is the area of an epicycloid with the smaller circle of radius and the larger circle of radius (), assuming the initial point lies on the larger circle.
where is the area of a rose with angular frequency () and amplitude .
where is the perimeter of the lemniscate of Bernoulli with focal distance .
where is the volume of a sphere and is the radius.
where is the surface area of a sphere and is the radius.
where is the hypervolume of a 3-sphere and is the radius.
where is the surface volume of a 3-sphere and is the radius.
Regular convex polygons
Sum of internal angles of a regular convex polygon with sides:
Area of a regular convex polygon with sides and side length :
Inradius of a regular convex polygon with sides and side length :
Circumradius of a regular convex polygon with sides and side length :
Physics
The cosmological constant:
Heisenberg's uncertainty principle:
Einstein's field equation of general relativity:
Coulomb's law for the electric force in vacuum:
Magnetic permeability of free space:
Approximate period of a simple pendulum with small amplitude:
Exact period of a simple pendulum with amplitude ( is the arithmetic–geometric mean):
Kepler's third law of planetary motion:
The buckling formula:
A puzzle involving "colliding billiard balls":
is the number of collisions made (in ideal conditions, perfectly elastic with no friction) by an object of mass m initially at rest between a fixed wall and another object of mass b2Nm, when struck by the other object. (This gives the digits of π in base b up to N digits past the radix point.)
Formulae yielding
Integrals
(integrating two halves to obtain the area of the unit circle)
(see also Cauchy distribution)
(see Gaussian integral).
(when the path of integration winds once counterclockwise around 0. See also Cauchy's integral formula).
(see also Proof that 22/7 exceeds ).
(where is the arithmetic–geometric mean; see also elliptic integral)
Note that with symmetric integrands , formulas of the form can also be translated to formulas .
Efficient infinite series
(see also Double factorial)
(see Chudnovsky algorithm)
(see Srinivasa Ramanujan, Ramanujan–Sato series)
The following are
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https://en.wikipedia.org/wiki/Morgan%20Crofton
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Morgan Crofton (1826, Dublin, Ireland – 1915, Brighton, England) was an Irish mathematician who contributed to the field of geometric probability theory. He also worked with James Joseph Sylvester and contributed an article on probability to the 9th edition of the Encyclopædia Britannica. Crofton's formula is named in his honour.
Early life
Morgan Crofton was born into a wealthy Anglo-Irish family. His father, the Reverend William Crofton, Rector of Skreen, County Sligo, was the younger brother of Sir Malby Crofton, 2nd Baronet of Longford House. He was also the cousin of Lord Edward Crofton, Baron Crofton of the Mote. Despite being born into an aristocratic, Anglican family, Crofton joined to the Roman Catholic Church in the 1850s in part due to an interest in Cardinal John Henry Newman.
This led to his resignation at Queen's College, Galway and transference to various Catholic colleges.
He married twice: firstly on 31 August 1857 Julia Agnes Cecilia, daughter of J B Kernan (died 1902) and secondly Katherine, daughter of Holland Taylor of Manchester.
Career
He was Professor of Mathematics at the Royal Military Academy, Woolwich and Professor of Natural Philosophy at Queen's University of Ireland. He was elected a Fellow of the Royal Society in June, 1868.
References
External links
MacTutor biography of Crofton
1826 births
1915 deaths
20th-century Irish mathematicians
Converts to Roman Catholicism from Anglicanism
Fellows of the Royal Society
Morgan
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https://en.wikipedia.org/wiki/Sobolev%20inequality
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In mathematics, there is in mathematical analysis a class of Sobolev inequalities, relating norms including those of Sobolev spaces. These are used to prove the Sobolev embedding theorem, giving inclusions between certain Sobolev spaces, and the Rellich–Kondrachov theorem showing that under slightly stronger conditions some Sobolev spaces are compactly embedded in others. They are named after Sergei Lvovich Sobolev.
Sobolev embedding theorem
Let denote the Sobolev space consisting of all real-valued functions on whose first weak derivatives are functions in . Here is a non-negative integer and . The first part of the Sobolev embedding theorem states that if , and are two real numbers such that
then
and the embedding is continuous. In the special case of and , Sobolev embedding gives
where is the Sobolev conjugate of , given by
This special case of the Sobolev embedding is a direct consequence of the Gagliardo–Nirenberg–Sobolev inequality. The result should be interpreted as saying that if a function in has one derivative in , then itself has improved local behavior, meaning that it belongs to the space where . (Note that , so that .) Thus, any local singularities in must be more mild than for a typical function in .
The second part of the Sobolev embedding theorem applies to embeddings in Hölder spaces . If and
with then one has the embedding
This part of the Sobolev embedding is a direct consequence of Morrey's inequality. Intuitively, this inclusion expresses the fact that the existence of sufficiently many weak derivatives implies some continuity of the classical derivatives. If then for every .
In particular, as long as , the embedding criterion will hold with and some positive value of . That is, for a function on , if has derivatives in and , then will be continuous (and actually Hölder continuous with some positive exponent ).
Generalizations
The Sobolev embedding theorem holds for Sobolev spaces on other suitable domains . In particular (; ), both parts of the Sobolev embedding hold when
is a bounded open set in with Lipschitz boundary (or whose boundary satisfies the cone condition; )
is a compact Riemannian manifold
is a compact Riemannian manifold with boundary and the boundary is Lipschitz (meaning that the boundary can be locally represented as a graph of a Lipschitz continuous function).
is a complete Riemannian manifold with injectivity radius and bounded sectional curvature.
If is a bounded open set in with continuous boundary, then is compactly embedded in ().
Kondrachov embedding theorem
On a compact manifold with boundary, the Kondrachov embedding theorem states that if andthen the Sobolev embedding
is completely continuous (compact). Note that the condition is just as in the first part of the Sobolev embedding theorem, with the equality replaced by an inequality, thus requiring a more regular space .
Gagliardo–Nirenberg–Sobolev inequality
Assume that is a continuously
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https://en.wikipedia.org/wiki/Sequential%20space
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In topology and related fields of mathematics, a sequential space is a topological space whose topology can be completely characterized by its convergent/divergent sequences. They can be thought of as spaces that satisfy a very weak axiom of countability, and all first-countable spaces (especially metric spaces) are sequential.
In any topological space if a convergent sequence is contained in a closed set then the limit of that sequence must be contained in as well. This property is known as sequential closure. Sequential spaces are precisely those topological spaces for which sequentially closed sets are in fact closed. (These definitions can also be rephrased in terms of sequentially open sets; see below.) Said differently, any topology can be described in terms of nets (also known as Moore–Smith sequences), but those sequences may be "too long" (indexed by too large an ordinal) to compress into a sequence. Sequential spaces are those topological spaces for which nets of countable length (i.e., sequences) suffice to describe the topology.
Any topology can be refined (that is, made finer) to a sequential topology, called the sequential coreflection of
The related concepts of Fréchet–Urysohn spaces, -sequential spaces, and -sequential spaces are also defined in terms of how a space's topology interacts with sequences, but have subtly different properties.
Sequential spaces and -sequential spaces were introduced by S. P. Franklin.
History
Although spaces satisfying such properties had implicitly been studied for several years, the first formal definition is due to S. P. Franklin in 1965. Franklin wanted to determine "the classes of topological spaces that can be specified completely by the knowledge of their convergent sequences", and began by investigating the first-countable spaces, for which it was already known that sequences sufficed. Franklin then arrived at the modern definition by abstracting the necessary properties of first-countable spaces.
Preliminary definitions
Let be a set and let be a sequence in ; that is, a family of elements of , indexed by the natural numbers. In this article, means that each element in the sequence is an element of and, if is a map, then For any index the tail of starting at is the sequence A sequence is eventually in if some tail of satisfies
Let be a topology on and a sequence therein. The sequence converges to a point written (when context allows, ), if, for every neighborhood of eventually is in is then called a limit point of
A function between topological spaces is sequentially continuous if implies
Sequential closure/interior
Let be a topological space and let be a subset. The topological closure (resp. topological interior) of in is denoted by (resp. ).
The sequential closure of in is the setwhich defines a map, the sequential closure operator, on the power set of If necessary for clarity, this set may also be written or
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https://en.wikipedia.org/wiki/Jenny%20Harrison
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Jenny Harrison is a professor of mathematics at the University of California, Berkeley.
Education and career
Harrison grew up in Tuscaloosa, Alabama. On graduating from the University of Alabama, she won a Marshall Scholarship which she used to fund her graduate studies at the University of Warwick. She completed her doctorate there in 1975, supervised by Christopher Zeeman. Hassler Whitney was her postdoctoral adviser at the Institute for Advanced Study, and she was also one of the Miller Research Fellows at Berkeley. She was on the tenured faculty at the University of Oxford (Somerville College) from 1978 to 1981, before returning to Berkeley as an assistant professor.
In 1986, after being denied tenure at Berkeley, Harrison filed a lawsuit based on gender discrimination. Stephen Smale and Robion Kirby were the most vocal opponents to her receiving tenure during the case, while Morris Hirsch and James Yorke were her most vocal supporters. The 1993 settlement led to a new review of her work by a panel of seven mathematicians and science faculty who unanimously recommended tenure as a full professor.
Research contributions
Harrison specializes in geometric analysis and areas in the intersection of algebra, geometry, and geometric measure theory. She introduced and developed with collaborators a theory of generalized functions called differential chains that unifies an infinitesimal calculus with the classical theory of the smooth continuum, a long outstanding problem. The infinitesimals are constructive and arise from methods of standard analysis, as opposed to the nonstandard analysis of Abraham Robinson. The methods apply equally well to domains such as soap films, fractals, charged particles, and Whitney stratified spaces, placing them on the same footing as smooth submanifolds in the resulting calculus. The results include optimal generalizations and simplifications of the theorems of Stokes, Gauss and Green. She has pioneered applications of differential chains to the calculus of variations, physics, and continuum mechanics. Her solution to Plateau's problem is the first proof of existence of a solution to a universal Plateau's problem for finitely many boundary curves, taking into account all soap films arising in nature, including nonorientable films with triple junctions, as well as solutions of Jesse Douglas, Herbert Federer and Wendell Fleming. Recently, she and Harrison Pugh have announced existence and soap film regularity of a solution to a universal Plateau's problem for codimension one surfaces using Hausdorff measure to define area.
As a graduate student at the University of Warwick, where Zeeman introduced her to Plateau's problem. She found a counterexample to the Seifert conjecture at Oxford. In a Berkeley seminar in 1983 she proposed the existence of a general theory linking these together, and the theory of differential chains began to evolve. Jenny Harrison and Harrison Pugh proved that the topological vect
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https://en.wikipedia.org/wiki/Joint%20Policy%20Board%20for%20Mathematics
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The Joint Policy Board for Mathematics (JPBM) consists of the American Mathematical Society, the American Statistical Association, the Mathematical Association of America, and the Society for Industrial and Applied Mathematics.
The Board has nearly 55,000 mathematicians and scientists who are members of the four organizations.
Each April, the JPBM celebrates Mathematics and Statistics Awareness Month (previously, the month was called Mathematics Awareness Month) to increase public understanding of and appreciation for mathematics and statistics. The event was renamed by the JPBM in 2017. To simplify coordination efforts, the JPBM also decided in 2017 that there will no longer be an annual assigned theme for the month. This celebration of mathematics, and now mathematics and statistics, began as Mathematics Awareness Week in 1986.
JPBM Communications Award
Each January at the Joint Mathematics Meeting the JPBM gives its Communications Award to a journalist or other communicator for bringing accurate mathematical information to non-mathematical audiences.
JPBM Communications Award winners
2023: Grant Sanderson and Jordan Ellenberg
2022: Talithia Williams
2021: John Bailer, Richard Campbell, Rosemary Pennington, and Erica Klarreich
2020: Christopher Budd and James Tanton
2019: Margot Lee Shetterly
2018: Vi Hart and Matt Parker
2017: Siobhan Roberts, for Expository and Popular Books, and Arthur T. Benjamin, for Public Outreach
2016: Simon Singh, for Expository and Popular Books, and the National Museum of Mathematics, for Public Outreach
2015: Nate Silver
2014: Danica McKellar
2013: John Allen Paulos
2012: Dana Mackenzie
2011: Nicolas Falacci and Cheryl Heuton
2010: Marcus du Sautoy
2009: George Csicsery
2008: Carl Bialik
2007: Steven H. Strogatz
2006: Roger Penrose
2005: Barry Arthur Cipra
2003: Robert Osserman
2002: Helaman Ferguson and Claire Ferguson
2001: Keith J. Devlin
2000: Sylvia Nasar
1999: Ian Stewart
1998: Constance Reid
1997: Philip J. Davis
1996: Gina Kolata
1994: Martin Gardner
1993: Joel Schneider
1991: Ivars Peterson
1990: Hugh Whitemore
1988: James Gleick
External links
JPBM
Mathematics and Statistics Awareness Month
MAA: JPBM Communications Award
Mathematical societies
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https://en.wikipedia.org/wiki/NVSS
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NVSS may refer to:
National Vital Statistics System, a U.S. government vital statistics system
NRAO VLA Sky Survey, an astronomical survey of the northern hemisphere
NVSS designation, names like NVSS 2146+82 for objects catalogued by the survey
North View Secondary School, a former school in Yishun, Singapore
North Vista Secondary School, a school in Sengkang, Singapore
N. V. S. S. Prabhakar, a politician from Telangana, India
Santo-Pekoa International Airport, ICAO code NVSS
Nechako Valley Secondary School, in School District 91 Nechako Lakes in Canada
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https://en.wikipedia.org/wiki/Buzen%27s%20algorithm
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In queueing theory, a discipline within the mathematical theory of probability, Buzen's algorithm (or convolution algorithm) is an algorithm for calculating the normalization constant G(N) in the Gordon–Newell theorem. This method was first proposed by Jeffrey P. Buzen in his 1971 PhD dissertation and subsequently published in a refereed journal in 1973. Computing G(N) is required to compute the stationary probability distribution of a closed queueing network.
Performing a naïve computation of the normalizing constant requires enumeration of all states. For a closed network with N circulating customers and M service facilities, G(N) is the sum of individual terms, with each term consisting of M factors raised to powers whose sum is N. Buzen's algorithm computes G(N) using only NM multiplications and NM additions. This dramatic improvement opened the door to applying the Gordon-Newell theorem to models of real world computer systems as well as flexible manufacturing systems and other cases where bottlenecks and queues can form within networks of inter-connected service facilities. The values of G(1), G(2) ... G(N -1), which can be used to calculate other important quantities of interest, are computed as by-products of the algorithm.
Problem setup
Consider a closed queueing network with M service facilities and N circulating customers. Assume that the service time for a customer at service facility i is given by an exponentially distributed random variable with parameter μi and that, after completing service at service facility i, a customer will proceed next to service facility j with probability pij.
Let be the steady state probability that the number of customers at service facility i is equal to ni for i = 1, 2, ... , M . It follows from the Gordon–Newell theorem that
....
This result is usually written more compactly as
The values of Xi are determined by solving
G(N) is a normalizing constant chosen so that the sum of all values of is equal to 1. Buzen's algorithm represents the first efficient procedure for computing G(N).
Algorithm description
The individual terms that must be added together to compute G(N) all have the following form:
.... . Note that this set of terms can be partitioned into two groups. The first group comprises all terms for which the exponent of is greater than or equal to 1. This implies that raised to the power 1 can be factored out of each of these terms.
After factoring out , a surprising result emerges: the modified terms in the first group are identical to the terms used to compute the normalizing constant for the same network with one customer removed. Thus, the sum of the terms in the first group can be written as “XM times G(N -1)”. This insight provides the foundation for the development of the algorithm.
Next consider the second group. The exponent of XM for every term in this group is zero. As a result, service facility M effectively disappears from all term
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https://en.wikipedia.org/wiki/Pseudorandom%20permutation
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In cryptography, a pseudorandom permutation (PRP) is a function that cannot be distinguished from a random permutation (that is, a permutation selected at random with uniform probability, from the family of all permutations on the function's domain) with practical effort.
Definition
Let F be a mapping . F is a PRP if and only if
For any , is a bijection from to , where .
For any , there is an "efficient" algorithm to evaluate for any ,.
For all probabilistic polynomial-time distinguishers : , where is chosen uniformly at random and is chosen uniformly at random from the set of permutations on n-bit strings.
A pseudorandom permutation family is a collection of pseudorandom permutations, where a specific permutation may be chosen using a key.
The model of block ciphers
The idealized abstraction of a (keyed) block cipher is a truly random permutation on the mappings between plaintext and ciphertext. If a distinguishing algorithm exists that achieves significant advantage with less effort than specified by the block cipher's security parameter (this usually means the effort required should be about the same as a brute force search through the cipher's key space), then the cipher is considered broken at least in a certificational sense, even if such a break doesn't immediately lead to a practical security failure.
Modern ciphers are expected to have super pseudorandomness.
That is, the cipher should be indistinguishable from a randomly chosen permutation on the same message space, even if the adversary has black-box access to the forward and inverse directions of the cipher.
Connections with pseudorandom function
Michael Luby and Charles Rackoff showed that a "strong" pseudorandom permutation can be built from a pseudorandom function using a Luby–Rackoff construction which is built using a Feistel cipher.
Related concepts
Unpredictable permutation
An unpredictable permutation (UP) Fk is a permutation whose values cannot be predicted by a fast randomized algorithm. Unpredictable permutations may be used as a cryptographic primitive, a building block for cryptographic systems with more complex properties.
An adversary for an unpredictable permutation is defined to be an algorithm that is given access to an oracle for both forward and inverse permutation operations. The adversary is given a challenge input k and is asked to predict the value of Fk. It is allowed to make a series of queries to the oracle to help it make this prediction, but is not allowed to query the value of k itself.
A randomized algorithm for generating permutations generates an unpredictable permutation if its outputs are permutations on a set of items (described by length-n binary strings) that cannot be predicted with accuracy significantly better than random by an adversary that makes a polynomial (in n) number of queries to the oracle prior to the challenge round, whose running time is polynomial in n, and whose error probability is less than 1/2 for all insta
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https://en.wikipedia.org/wiki/Fatou%E2%80%93Bieberbach%20domain
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In mathematics, a Fatou–Bieberbach domain is a proper subdomain of , biholomorphically equivalent to . That is, an open set is called a Fatou–Bieberbach domain if there exists a bijective holomorphic function whose inverse function is holomorphic. It is well-known that the inverse can not be polynomial.
History
As a consequence of the Riemann mapping theorem, there are no Fatou–Bieberbach domains in the case n = 1.
Pierre Fatou and Ludwig Bieberbach first explored such domains in higher dimensions in the 1920s, hence the name given to them later. Since the 1980s, Fatou–Bieberbach domains have again become the subject of mathematical research.
References
Fatou, Pierre: "Sur les fonctions méromorphes de deux variables. Sur certains fonctions uniformes de deux variables." C.R. Paris 175 (1922)
Bieberbach, Ludwig: "Beispiel zweier ganzer Funktionen zweier komplexer Variablen, welche eine schlichte volumtreue Abbildung des auf einen Teil seiner selbst vermitteln". Preussische Akademie der Wissenschaften. Sitzungsberichte (1933)
Rosay, J.-P. and Rudin, W: "Holomorphic maps from to ". Trans. Amer. Math. Soc. 310 (1988)
Several complex variables
Inverse functions
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https://en.wikipedia.org/wiki/Sphere%20theorem
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In Riemannian geometry, the sphere theorem, also known as the quarter-pinched sphere theorem, strongly restricts the topology of manifolds admitting metrics with a particular curvature bound. The precise statement of the theorem is as follows. If M is a complete, simply-connected, n-dimensional Riemannian manifold with sectional curvature taking values in the interval then M is homeomorphic to the n-sphere. (To be precise, we mean the sectional curvature of every tangent 2-plane at each point must lie in .) Another way of stating the result is that if M is not homeomorphic to the sphere, then it is impossible to put a metric on M with quarter-pinched curvature.
Note that the conclusion is false if the sectional curvatures are allowed to take values in the closed interval . The standard counterexample is complex projective space with the Fubini–Study metric; sectional curvatures of this metric take on values between 1 and 4, with endpoints included. Other counterexamples may be found among the rank one symmetric spaces.
Differentiable sphere theorem
The original proof of the sphere theorem did not conclude that M was necessarily diffeomorphic to the n-sphere. This complication is because spheres in higher dimensions admit smooth structures that are not diffeomorphic. (For more information, see the article on exotic spheres.) However, in 2007 Simon Brendle and Richard Schoen utilized Ricci flow to prove that with the above hypotheses, M is necessarily diffeomorphic to the n-sphere with its standard smooth structure. Moreover, the proof of Brendle and Schoen only uses the weaker assumption of pointwise rather than global pinching. This result is known as the differentiable sphere theorem.
History of the sphere theorem
Heinz Hopf conjectured that a simply connected manifold with pinched sectional curvature is a sphere. In 1951, Harry Rauch showed that a simply connected manifold with curvature in [3/4,1] is homeomorphic to a sphere. In 1960, Marcel Berger and Wilhelm Klingenberg proved the topological version of the sphere theorem with the optimal pinching constant.
References
Riemannian geometry
Theorems in topology
Theorems in Riemannian geometry
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https://en.wikipedia.org/wiki/Ernst%20Meissel
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Daniel Friedrich Ernst Meissel (31 July 1826, Eberswalde, Brandenburg Province – 11 March 1895, Kiel) was a German astronomer who contributed to various aspects of number theory.
See also
Meissel–Lehmer algorithm
Meissel–Mertens constant
External links
1826 births
1895 deaths
19th-century German astronomers
19th-century German mathematicians
Number theorists
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https://en.wikipedia.org/wiki/183%20%28number%29
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183 (one hundred [and] eighty-three) is the natural number following 182 and preceding 184.
In mathematics
183 is a perfect totient number, a number that is equal to the sum of its iterated totients
Because , it is the number of points in a projective plane over the finite field . 183 is the fourth element of a divisibility sequence in which the th number can be computed as
for a transcendental number . This sequence counts the number of trees of height in which each node can have at most two children.
There are 183 different semiorders on four labeled elements.
See also
The year AD 183 or 183 BC
List of highways numbered 183
References
Integers
183
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https://en.wikipedia.org/wiki/Reinhardt%20cardinal
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In set theory, a branch of mathematics, a Reinhardt cardinal is a kind of large cardinal. Reinhardt cardinals are considered under ZF (Zermelo–Fraenkel set theory without the Axiom of Choice), because they are inconsistent with ZFC (ZF with the Axiom of Choice). They were suggested by American mathematician William Nelson Reinhardt (1939–1998).
Definition
A Reinhardt cardinal is the critical point of a non-trivial elementary embedding of into itself.
This definition refers explicitly to the proper class . In standard ZF, classes are of the form for some set and formula . But it was shown in
that no such class is an elementary embedding . So Reinhardt cardinals are inconsistent with this notion of class.
There are other formulations of Reinhardt cardinals which are not known to be inconsistent. One is to add a new function symbol to the language of ZF, together with axioms stating that is an elementary embedding of , and Separation and Collection axioms for all formulas involving . Another is to use a class theory such as NBG or KM, which admit classes which need not be definable in the sense above.
Kunen's inconsistency theorem
proved his inconsistency theorem, showing that the existence of an elementary embedding contradicts NBG with the axiom of choice (and ZFC extended by ). His proof uses the axiom of choice, and it is still an open question as to whether such an embedding is consistent with NBG without the axiom of choice (or with ZF plus the extra symbol and its attendant axioms).
Kunen's theorem is not simply a consequence of ,
as it is a consequence of NBG, and hence does not require the assumption that is a definable class.
Also, assuming exists, then there is an elementary embedding of a transitive model of ZFC (in fact Goedel's constructible universe ) into itself. But such embeddings are not classes of .
Stronger axioms
There are some variations of Reinhardt cardinals, forming a hierarchy of hypotheses asserting the existence of elementary embeddings .
A super Reinhardt cardinal is such that for every ordinal , there is an elementary embedding with and having critical point .
J3: There is a nontrivial elementary embedding
J2: There is a nontrivial elementary embedding and DC holds, where is the least fixed-point above the critical point.
J1: For every ordinal , there is an elementary embedding with and having critical point .
Each of J1 and J2 immediately imply J3. A cardinal as in J1 is known as a super Reinhardt cardinal.
Berkeley cardinals are stronger large cardinals suggested by Woodin.
See also
List of large cardinal properties
References
Citations
External links
Large cardinals
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https://en.wikipedia.org/wiki/Base%20flow
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The term base flow may refer to:
Baseflow in hydrology
Base flow (random dynamical systems) in the study of random dynamical systems in mathematics
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https://en.wikipedia.org/wiki/Block%20matrix%20pseudoinverse
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In mathematics, a block matrix pseudoinverse is a formula for the pseudoinverse of a partitioned matrix. This is useful for decomposing or approximating many algorithms updating parameters in signal processing, which are based on the least squares method.
Derivation
Consider a column-wise partitioned matrix:
If the above matrix is full column rank, the Moore–Penrose inverse matrices of it and its transpose are
This computation of the pseudoinverse requires (n + p)-square matrix inversion and does not take advantage of the block form.
To reduce computational costs to n- and p-square matrix inversions and to introduce parallelism, treating the blocks separately, one derives
where orthogonal projection matrices are defined by
The above formulas are not necessarily valid if does not have full rank – for example, if , then
Application to least squares problems
Given the same matrices as above, we consider the following least squares problems, which
appear as multiple objective optimizations or constrained problems in signal processing.
Eventually, we can implement a parallel algorithm for least squares based on the following results.
Column-wise partitioning in over-determined least squares
Suppose a solution solves an over-determined system:
Using the block matrix pseudoinverse, we have
Therefore, we have a decomposed solution:
Row-wise partitioning in under-determined least squares
Suppose a solution solves an under-determined system:
The minimum-norm solution is given by
Using the block matrix pseudoinverse, we have
Comments on matrix inversion
Instead of , we need to calculate directly or indirectly
In a dense and small system, we can use singular value decomposition, QR decomposition, or Cholesky decomposition to replace the matrix inversions with numerical routines. In a large system, we may employ iterative methods such as Krylov subspace methods.
Considering parallel algorithms, we can compute and in parallel. Then, we finish to compute and also in parallel.
See also
References
External links
The Matrix Reference Manual by Mike Brookes
Linear Algebra Glossary by John Burkardt
The Matrix Cookbook by Kaare Brandt Petersen
Lecture 8: Least-norm solutions of undetermined equations by Stephen P. Boyd
Numerical linear algebra
Matrix theory
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https://en.wikipedia.org/wiki/The%20Book%20of%20Squares
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The Book of Squares, (Liber Quadratorum in the original Latin) is a book on algebra by Leonardo Fibonacci, published in 1225. It was dedicated to Frederick II, Holy Roman Emperor.
The Liber quadratorum has been passed down by a single 15th-century manuscript, the so-called ms. E 75 Sup. of the Biblioteca Ambrosiana (Milan, Italy), ff. 19r-39v. During the 19th century, the work has been published for the first time in a printed edition by Baldassarre Boncompagni Ludovisi, prince of Piombino.
Appearing in the book is Fibonacci's identity, establishing that the set of all sums of two squares is closed under multiplication. The book anticipated the works of later mathematicians such as Fermat and Euler. The book examines several topics in number theory, among them an inductive method for finding Pythagorean triples based on the sequence of odd integers, the fact that the sum of the first odd integers is , and the solution to the congruum problem.
Notes
Further reading
B. Boncompagni Ludovisi, Opuscoli di Leonardo Pisano secondo un codice della Biblioteca Ambrosiana di Milano contrassegnato E.75. Parte Superiore, in Id., Scritti di Leonardo Pisano matematico del secolo decimoterzo, vol. II, Roma 1862, pp. 253–283
P. Ver Eecke, Léonard de Pise. Le livre des nombres carrés. Traduit pour la première fois du Latin Médiéval en Français, Paris, Blanchard-Desclée - Bruges 1952.
G. Arrighi, La fortuna di Leonardo Pisano alla corte di Federico II, in Dante e la cultura sveva. Atti del Convegno di Studi, Melfi, 2-5 novembre 1969, Firenze 1970, pp. 17–31.
E. Picutti, Il Libro dei quadrati di Leonardo Pisano e i problemi di analisi indeterminata nel Codice Palatino 557 della Biblioteca Nazionale di Firenze, in «Physis. Rivista Internazionale di Storia della Scienza» XXI, 1979, pp. 195–339.
L.E. Sigler, Leonardo Pisano Fibonacci, the book of squares. An annotated translation into modern English, Boston 1987.
M. Moyon, Algèbre & Practica geometriæ en Occident médiéval latin: Abū Bakr, Fibonacci et Jean de Murs, in Pluralité de l’algèbre à la Renaissance, a cura di S. Rommevaux, M. Spiesser, M.R. Massa Esteve, Paris 2012, pp. 33–65.
External links
Fibonacci and Square Numbers at
1225 books
13th-century Latin books
Mathematics books
Squares in number theory
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https://en.wikipedia.org/wiki/Mikl%C3%B3s%20Schweitzer%20Competition
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The Miklós Schweitzer Competition (Schweitzer Miklós Matematikai Emlékverseny in Hungarian) is an annual Hungarian mathematics competition for university undergraduates, established in 1949.
It is named after Miklós Schweitzer (1 February 1923 – 28 January 1945), a young Hungarian mathematician who died under the Siege of Budapest in the Second World War.
The Schweitzer contest is uniquely high-level among mathematics competitions. The problems, written by prominent Hungarian mathematicians, are challenging and require in-depth knowledge of the fields represented. The competition is open-book and competitors are allowed ten days to come up with solutions.
The problems on the competition can be classified roughly in the following categories:
1. Algebra
2. Combinatorics
3. Theory of Functions
4. Geometry
5. Measure Theory
6. Number Theory
7. Operators
8. Probability Theory
9. Sequences and Series
10. Topology
11. Set Theory
Recently a similar competition has been started in France.
References
Contests in higher mathematics (Hungary, 1949–1961). In memoriam, Miklós Schweitzer. (G. Szasz, L. Geher, I. Kovacs, L. Pinter, eds), Akadémiai Kiadó, Budapest, 1968 260 pp.
Miklós Schweitzer Competition Problems in recent years
Problems of the Miklós Schweitzer Memorial Competition at http://artofproblemsolving.com/
Mathematics competitions
Recurring events established in 1949
Student events
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https://en.wikipedia.org/wiki/Enneper%20surface
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In differential geometry and algebraic geometry, the Enneper surface is a self-intersecting surface that can be described parametrically by:
It was introduced by Alfred Enneper in 1864 in connection with minimal surface theory.
The Weierstrass–Enneper parameterization is very simple, , and the real parametric form can easily be calculated from it. The surface is conjugate to itself.
Implicitization methods of algebraic geometry can be used to find out that the points in the Enneper surface given above satisfy the degree-9 polynomial equation
Dually, the tangent plane at the point with given parameters is where
Its coefficients satisfy the implicit degree-6 polynomial equation
The Jacobian, Gaussian curvature and mean curvature are
The total curvature is . Osserman proved that a complete minimal surface in with total curvature is either the catenoid or the Enneper surface.
Another property is that all bicubical minimal Bézier surfaces are, up to an affine transformation, pieces of the surface.
It can be generalized to higher order rotational symmetries by using the Weierstrass–Enneper parameterization for integer k>1. It can also be generalized to higher dimensions; Enneper-like surfaces are known to exist in for n up to 7.
See also
for higher order algebraic Enneper surfaces.
References
External links
Mathematical Sciences Research Institute - The Generalized Enneper's Surfaces (Archived)
Harvey Mudd College - Enneper's Surface (Archived)
Algebraic surfaces
Minimal surfaces
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https://en.wikipedia.org/wiki/Axiom%20of%20choice%20%28disambiguation%29
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Axiom of choice is an axiom of set theory.
Axiom of choice may also refer to:
Axiom of Choice (band), a world music group of Iranian émigrés
See also
Axiom of countable choice
Axiom of dependent choice
Axiom of global choice
Axiom of non-choice
Axiom of finite choice
Luce's choice axiom
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https://en.wikipedia.org/wiki/Topology%20%28journal%29
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Topology was a peer-reviewed mathematical journal covering topology and geometry. It was established in 1962 and was published by Elsevier. The last issue of Topology appeared in 2009.
Pricing dispute
On 10 August 2006, after months of unsuccessful negotiations with Elsevier about the price policy of library subscriptions, the entire editorial board of the journal handed in their resignation, effective 31 December 2006. Subsequently, two more issues appeared in 2007 with papers that had been accepted before the resignation of the editors. In early January the former editors instructed Elsevier to remove their names from the website of the journal, but Elsevier refused to comply, justifying their decision by saying that the editorial board should remain on the journal until all of the papers accepted during its tenure had been published.
In 2007 the former editors of Topology announced the launch of the Journal of Topology, published by Oxford University Press on behalf of the London Mathematical Society at a significantly lower price. Its first issue appeared in January 2008.
References
External links
Journals declaring independence
Mathematics journals
English-language journals
Academic journals established in 1962
Elsevier academic journals
Publications disestablished in 2009
Bimonthly journals
Defunct journals
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https://en.wikipedia.org/wiki/Easton%27s%20theorem
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In set theory, Easton's theorem is a result on the possible cardinal numbers of powersets. (extending a result of Robert M. Solovay) showed via forcing that the only constraints on permissible values for 2κ when κ is a regular cardinal are
(where cf(α) is the cofinality of α) and
Statement
If G is a class function whose domain consists of ordinals and whose range consists of ordinals such that
G is non-decreasing,
the cofinality of is greater than for each α in the domain of G, and
is regular for each α in the domain of G,
then there is a model of ZFC such that
for each in the domain of G.
The proof of Easton's theorem uses forcing with a proper class of forcing conditions over a model satisfying the generalized continuum hypothesis.
The first two conditions in the theorem are necessary. Condition 1 is a well known property of cardinality, while condition 2 follows from König's theorem.
In Easton's model the powersets of singular cardinals have the smallest possible cardinality compatible with the conditions that 2κ has cofinality greater than κ and is a non-decreasing function of κ.
No extension to singular cardinals
proved that a singular cardinal of uncountable cofinality cannot be the smallest cardinal for which the generalized continuum hypothesis fails. This shows that Easton's theorem cannot be extended to the class of all cardinals. The program of PCF theory gives results on the possible values of for singular cardinals . PCF theory shows that the values of the continuum function on singular cardinals are strongly influenced by the values on smaller cardinals, whereas Easton's theorem shows that the values of the continuum function on regular cardinals are only weakly influenced by the values on smaller cardinals.
See also
Singular cardinal hypothesis
Aleph number
Beth number
References
Set theory
Theorems in the foundations of mathematics
Cardinal numbers
Forcing (mathematics)
Independence results
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https://en.wikipedia.org/wiki/Realizer
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Realizer may refer to:
For its use in mathematics see Order dimension
CA-Realizer, the programming language similar to Visual Basic created by Computer Associates
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