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https://en.wikipedia.org/wiki/Hyperfinite%20set | In nonstandard analysis, a branch of mathematics, a hyperfinite set or *-finite set is a type of internal set. An internal set H of internal cardinality g ∈ *N (the hypernaturals) is hyperfinite if and only if there exists an internal bijection between G = {1,2,3,...,g} and H. Hyperfinite sets share the properties of finite sets: A hyperfinite set has minimal and maximal elements, and a hyperfinite union of a hyperfinite collection of hyperfinite sets may be derived. The sum of the elements of any hyperfinite subset of *R always exists, leading to the possibility of well-defined integration.
Hyperfinite sets can be used to approximate other sets. If a hyperfinite set approximates an interval, it is called a near interval with respect to that interval. Consider a hyperfinite set with a hypernatural n. K is a near interval for [a,b] if k1 = a and kn = b, and if the difference between successive elements of K is infinitesimal. Phrased otherwise, the requirement is that for every r ∈ [a,b] there is a ki ∈ K such that ki ≈ r. This, for example, allows for an approximation to the unit circle, considered as the set for θ in the interval [0,2π].
In general, subsets of hyperfinite sets are not hyperfinite, often because they do not contain the extreme elements of the parent set.
Ultrapower construction
In terms of the ultrapower construction, the hyperreal line *R is defined as the collection of equivalence classes of sequences of real numbers un. Namely, the equivalence class defines a hyperreal, denoted in Goldblatt's notation. Similarly, an arbitrary hyperfinite set in *R is of the form , and is defined by a sequence of finite sets
References
External links
Nonstandard analysis |
https://en.wikipedia.org/wiki/National%20Bureau%20of%20Statistics%2C%20Nigeria | The National Bureau of Statistics oversees and publishes statistics for Nigeria.
Contributing Bureaus
The contributing bureaus are where the National Bureau of Statistics get their information. They include:
National Planning Commission
Economic and Financial Crimes Commission
Federal Ministry of Health
National Population Commission
Nigerian Stock Exchange
Nigerian Embassies and High Commissions
Federal Ministry of Finance
Central Bank of Nigeria
Nigerian National Petroleum Corporation
Nigerian Electricity Regulatory Commission
Some Statistics for Q2, 2009
Telecommunications / Postal Service is accountable for 3.87% of the GDP
Manufacturing is accountable for 3.95% of the GDP
Building & Construction is accountable for 1.94% of the GDP
Crude Petroleum & Natural Gas is accountable for 16.01% of the GDP
Agriculture is accountable for 43% of the GDP
Non-oil Growth was about 8.27%
External links
Government of Nigeria
Nigeria |
https://en.wikipedia.org/wiki/Arthur%20Lee%20Dixon | Arthur Lee Dixon FRS (27 November 1867 — 20 February 1955) was a British mathematician and holder of the Waynflete Professorship of Pure Mathematics at the University of Oxford.
Early life and education
Dixon was born on 27 November 1867 in Pickering, North Riding of Yorkshire to G.T. Dixon, and was the younger brother of Alfred Cardew Dixon. From 1879 to 1885 he studied at Kingswood School, before matriculating at Worcester College, Oxford as a scholar to study mathematics.
Academic career
Dixon became a Fellow of Merton College in 1891, and Waynflete Professor of Pure Mathematics in 1922.
His research was focused on algebra and its application to geometry, elliptic functions and hyperelliptic functions. From 1908 onwards he published a series of papers on algebraic eliminants. He also published a dozen joint papers with W.L. Ferrar on analytic number theory.
Dixon was the last mathematical professor at Oxford to hold a life tenure, and although he was not particularly noted for his mathematical innovations he did publish many papers on analytic number theory and the application of algebra to geometry, elliptic functions and hyperelliptic functions.
He was elected a Fellow of the Royal Society in 1912 and served as president of the London Mathematical Society from 1924 to 1926.
Dixon died on 20 February 1955.
Personal life
In 1902 Dixon married Catherine Rieder. Catherine found the atmosphere in Oxford difficult for her health, and spent a lot of time in Pau to recover. The couple had one child, a daughter, who later married F.J. Baden Fuller; when Catherine died in 1930, Dixon moved in with his daughter and her husband in Sandgate, Kent, where he spent the rest of his life.
References
1867 births
1955 deaths
Fellows of the Royal Society
Fellows of Merton College, Oxford
Alumni of Worcester College, Oxford
People educated at Kingswood School, Bath
Waynflete Professors of Pure Mathematics |
https://en.wikipedia.org/wiki/Wrapped%20distribution | In probability theory and directional statistics, a wrapped probability distribution is a continuous probability distribution that describes data points that lie on a unit n-sphere. In one dimension, a wrapped distribution consists of points on the unit circle. If is a random variate in the interval with probability density function (PDF) , then is a circular variable distributed according to the wrapped distribution and is an angular variable in the interval distributed according to the wrapped distribution .
Any probability density function on the line can be "wrapped" around the circumference of a circle of unit radius. That is, the PDF of the wrapped variable
in some interval of length
is
which is a periodic sum of period . The preferred interval is generally for which .
Theory
In most situations, a process involving circular statistics produces angles () which lie in the interval , and are described by an "unwrapped" probability density function . However, a measurement will yield an angle which lies in some interval of length (for example, 0 to ). In other words, a measurement cannot tell whether the true angle or a wrapped angle , where is some unknown integer, has been measured.
If we wish to calculate the expected value of some function of the measured angle it will be:
.
We can express the integral as a sum of integrals over periods of :
.
Changing the variable of integration to and exchanging the order of integration and summation, we have
where is the PDF of the wrapped distribution and is another unknown integer . The unknown integer introduces an ambiguity into the expected value of , similar to the problem of calculating angular mean. This can be resolved by introducing the parameter , since has an unambiguous relationship to the true angle :
.
Calculating the expected value of a function of will yield unambiguous answers:
.
For this reason, the parameter is preferred over measured angles in circular statistical analysis. This suggests that the wrapped distribution function may itself be expressed as a function of such that:
where is defined such that . This concept can be extended to the multivariate context by an extension of the simple sum to a number of sums that cover all dimensions in the feature space:
where is the th Euclidean basis vector.
Expression in terms of characteristic functions
A fundamental wrapped distribution is the Dirac comb, which is a wrapped Dirac delta function:
.
Using the delta function, a general wrapped distribution can be written
.
Exchanging the order of summation and integration, any wrapped distribution can be written as the convolution of the unwrapped distribution and a Dirac comb:
.
The Dirac comb may also be expressed as a sum of exponentials, so we may write:
.
Again exchanging the order of summation and integration:
.
Using the definition of , the characteristic function of yields a Laurent series about zero for the wrapped distributi |
https://en.wikipedia.org/wiki/Glossary%20of%20module%20theory | Module theory is the branch of mathematics in which modules are studied. This is a glossary of some terms of the subject.
See also: Glossary of linear algebra, Glossary of ring theory, Glossary of representation theory.
A
B
C
D
E
F
G
H
I
J
K
L
M
N
P
Q
R
S
T
U
W
Z
References
Module
Wikipedia glossaries using description lists |
https://en.wikipedia.org/wiki/Louis%20Bourguet | Louis Bourguet (23 April 1678, Nîmes – 31 December 1742, Neuchâtel) was a polymath and correspondent of Leibniz who wrote on archaeology, geology, philosophy, Biblical scholarship and mathematics.
Bourguet entered the College of Zurich in 1688. He became Professor of Philosophy and Mathematics at Neuchâtel in 1731. He tried to integrate Leibnizian philosophy with issues in natural philosophy.
Works
Traité des petrifications, 1742
References
External links
Lettres philosophiques sur la formation de sels et des crystaux (1729, French) - digital facsimile from Linda Hall Library
Traité des petrifications (1742, French) - digital facsimile from Linda Hall Library
1678 births
1742 deaths
French archaeologists
French geologists |
https://en.wikipedia.org/wiki/Luigi%20Ambrosio | Luigi Ambrosio (born 27 January 1963) is a professor at Scuola Normale Superiore in Pisa, Italy. His main fields of research are the calculus of variations and geometric measure theory.
Biography
Ambrosio entered the Scuola Normale Superiore di Pisa in 1981. He obtained his degree under the guidance of Ennio de Giorgi in 1985 at University of Pisa, and the Diploma at Scuola Normale. He obtained his PhD in 1988.
He is currently professor at the Scuola Normale, having taught previously at the University of Rome "Tor Vergata", the University of Pisa, and the University of Pavia. Ambrosio also taught and conducted research at the Massachusetts Institute of Technology, the ETH in Zurich, and the Max Planck Institute for Mathematics in the Sciences in Leipzig.
He is the Managing Editor of the scientific journal Calculus of Variations and Partial Differential Equations, and member of the editorial boards of scientific journals.
Since May 9, 2019 Ambrosio is the director of the Scuola Normale Superiore di Pisa.
Awards
In 1998 Ambrosio won the Caccioppoli Prize of the Italian Mathematical Union. In 2002 he was invited speaker at the International Congress of Mathematicians in Beijing and in 2003 he has been awarded with the Fermat Prize. From 2005 he is a corresponding member of Accademia Nazionale dei Lincei. Ambrosio is listed as an ISI highly cited researcher. In 2018 he was a plenary speaker at the International Congress of Mathematicians in Rio de Janeiro. In 2019 he received the Balzan Prize in Mathematics (Theory of Partial Differential Equations).
Selected publications
References
External links
Site of Caccioppoli Prize
20th-century Italian mathematicians
21st-century Italian mathematicians
Living people
1963 births
People from Alba, Piedmont
Variational analysts
Academic staff of the Scuola Normale Superiore di Pisa |
https://en.wikipedia.org/wiki/Bratteli%20diagram | In mathematics, a Bratteli diagram is a combinatorial structure: a graph composed of vertices labelled by positive integers ("level") and unoriented edges between vertices having levels differing by one. The notion was introduced by Ola Bratteli in 1972 in the theory of operator algebras to describe directed sequences of finite-dimensional algebras: it played an important role in Elliott's classification of AF-algebras and the theory of subfactors. Subsequently Anatoly Vershik associated dynamical systems with infinite paths in such graphs.
Definition
A Bratteli diagram is given by the following objects:
A sequence of sets Vn ('the vertices at level n ') labeled by positive integer set N. In some literature each element v of Vn is accompanied by a positive integer bv > 0.
A sequence of sets En ('the edges from level n to n + 1 ') labeled by N, endowed with maps s: En → Vn and r: En → Vn+1, such that:
For each v in Vn, the number of elements e in En with s(e) = v is finite.
So is the number of e ∈ En−1 with r(e) = v.
When the vertices have markings by positive integers bv, the number av, v ' of the edges with s(e) = v and r(e) = v' for v ∈ Vn and v' ∈ Vn+1 satisfies bv av, v' ≤ bv'.
A customary way to pictorially represent Bratteli diagrams is to align the vertices according to their levels, and put the number bv beside the vertex v, or use that number in place of v, as in
An ordered Bratteli diagram is a Bratteli diagram together with a partial order on En such that for any v ∈ Vn the set { e ∈ En−1 : r(e) = v } is totally ordered. Edges that do not share a common range vertex are incomparable. This partial order allows us to define the set of all maximal edges Emax and the set of all minimal edges Emin. A Bratteli diagram with a unique infinitely long path in Emax and Emin is called essentially simple.
Sequence of finite-dimensional algebras
Any semisimple algebra over the complex numbers C of finite dimension can be expressed as a direct sum ⊕k Mnk(C) of matrix algebras, and the C-algebra homomorphisms between two such algebras up to inner automorphisms on both sides are completely determined by the multiplicity number between 'matrix algebra' components. Thus, an injective homomorphism of ⊕k=1i Mnk(C) into ⊕l=1j Mml(C) may be represented by a collection of positive numbers ak, l satisfying Σ nk ak, l ≤ ml. (The equality holds if and only if the homomorphism is unital; we can allow non-injective homomorphisms by allowing some ak,l to be zero.) This can be illustrated as a bipartite graph having the vertices marked by numbers (nk)k on one hand and the ones marked by (ml)l on the other hand, and having ak, l edges between the vertex nk and the vertex ml.
Thus, when we have a sequence of finite-dimensional semisimple algebras An and injective homomorphisms φn : A<sub>n</sub> → An+1: between them, we obtain a Bratteli diagram by putting
Vn = the set of simple components of An
(each isomorphic to a matrix algebra), marked by the si |
https://en.wikipedia.org/wiki/Semi-membership | In mathematics and theoretical computer science, the semi-membership problem for a set is the problem of deciding which of two possible elements is logically more likely to belong to that set; alternatively, given two elements of which at least one is in the set, to distinguish the member from the non-member.
The semi-membership problem may be significantly easier than the membership problem. For example, consider the set S(x) of finite-length binary strings representing the dyadic rationals less than some fixed real number x. The semi-membership problem for a pair of strings is solved by taking the string representing the smaller dyadic rational, since if exactly one of the strings is an element, it must be the smaller, irrespective of the value of x. However, the language S(x) may not even be a recursive language, since there are uncountably many such x, but only countably many recursive languages.
A function f on ordered pairs (x,y) is a selector for a set S if f(x,y) is equal to either x or y and if f(x,y) is in S whenever at least one of x, y is in S. A set is semi-recursive if it has a recursive selector, and is P-selective or semi-feasible if it is semi-recursive with a polynomial time selector.
Semi-feasible sets have small circuits; they are in the extended low hierarchy; and cannot be NP-complete unless P=NP.
References
Derek Denny-Brown, "Semi-membership algorithms: some recent advances", Technical report, University of Rochester Dept. of Computer Science, 1994
Lane A. Hemaspaandra, Mitsunori Ogihara, "The complexity theory companion", Texts in theoretical computer science, EATCS series, Springer, 2002, , page 294
Lane A. Hemaspaandra, Leen Torenvliet, "Theory of semi-feasible algorithms", Monographs in theoretical computer science, Springer, 2003, , page 1
Ker-I Ko, "Applying techniques of discrete complexity theory to numerical computation" in Ronald V. Book (ed.), "Studies in complexity theory", Research notes in theoretical computer science, Pitman, 1986, , p.40
Computational complexity theory |
https://en.wikipedia.org/wiki/ELSV%20formula | In mathematics, the ELSV formula, named after its four authors , , Michael Shapiro, Alek Vainshtein, is an equality between a Hurwitz number (counting ramified coverings of the sphere) and an integral over the moduli space of stable curves.
Several fundamental results in the intersection theory of moduli spaces of curves can be deduced from the ELSV formula, including the Witten conjecture, the Virasoro constraints, and the -conjecture.
It is generalized by the Gopakumar–Mariño–Vafa formula.
The formula
Define the Hurwitz number
as the number of ramified coverings of the complex projective line (Riemann sphere, that are connected curves of genus g, with n numbered preimages of the point at infinity having multiplicities and m more simple branch points. Here if a covering has a nontrivial automorphism group G it should be counted with weight .
The ELSV formula then reads
Here the notation is as follows:
is a nonnegative integer;
is a positive integer;
are positive integers;
is the number of automorphisms of the n-tuple
is the moduli space of stable curves of genus g with n marked points;
E is the Hodge vector bundle and c(E*) the total Chern class of its dual vector bundle;
ψi is the first Chern class of the cotangent line bundle to the i-th marked point.
The numbers
in the left-hand side have a combinatorial definition and satisfy properties that can be proved combinatorially. Each of these properties translates into a statement on the integrals on the right-hand side of the ELSV formula .
The Hurwitz numbers
The Hurwitz numbers
also have a definition in purely algebraic terms. With K = k1 + ... + kn and m = K + n + 2g − 2, let τ1, ..., τm be transpositions in the symmetric group SK and σ a permutation with n numbered cycles of lengths k1, ..., kn. Then
is a transitive factorization of identity of type (k1, ..., kn) if the product
equals the identity permutation and the group generated by
is transitive.
Definition. is the number of transitive factorizations of identity of type (k1, ..., kn) divided by K!.
Example A. The number is 1/k! times the number of lists of transpositions whose product is a k-cycle. In other words, is 1/k times the number of factorizations of a given k-cycle into a product of k + 2g − 1 transpositions.
The equivalence between the two definitions of Hurwitz numbers (counting ramified coverings of the sphere, or counting transitive factorizations) is established by describing a ramified covering by its monodromy. More precisely: choose a base point on the sphere, number its preimages from 1 to K (this introduces a factor of K!, which explains the division by it), and consider the monodromies of the covering about the branch point. This leads to a transitive factorization.
The integral over the moduli space
The moduli space is a smooth Deligne–Mumford stack of (complex) dimension 3g − 3 + n. (Heuristically this behaves much like complex manifold, except that integrals of characte |
https://en.wikipedia.org/wiki/Adjacency%20algebra | In algebraic graph theory, the adjacency algebra of a graph G is the algebra of polynomials in the adjacency matrix A(G) of the graph. It is an example of a matrix algebra and is the set of the linear combinations of powers of A.
Some other similar mathematical objects are also called "adjacency algebra".
Properties
Properties of the adjacency algebra of G are associated with various spectral, adjacency and connectivity properties of G.
Statement. The number of walks of length d between vertices i and j is equal to the (i, j)-th element of Ad.
Statement. The dimension of the adjacency algebra of a connected graph of diameter d is at least d + 1.
Corollary. A connected graph of diameter d has at least d + 1 distinct eigenvalues.
References
Algebraic graph theory |
https://en.wikipedia.org/wiki/Cis%20%28mathematics%29 | is a mathematical notation defined by , where is the cosine function, is the imaginary unit and is the sine function. is the argument of the complex number (angle between line to point and x-axis in polar form). The notation is less commonly used in mathematics than Euler's formula, which offers an even shorter notation for but cis(x) is widely used as a name for this function in software libraries.
Overview
The notation is a shorthand for the combination of functions on the right-hand side of Euler's formula:
where . So,
i.e. "" is an acronym for "".
It connects trigonometric functions with exponential functions in the complex plane via Euler's formula. While the domain of definition is usually , complex values are possible as well:
so the function can be used to extend Euler's formula to a more general complex version.
The function is mostly used as a convenient shorthand notation to simplify some expressions, for example in conjunction with Fourier and Hartley transforms, or when exponential functions shouldn't be used for some reason in math education.
In information technology, the function sees dedicated support in various high-performance math libraries (such as Intel's Math Kernel Library (MKL) or MathCW), available for many compilers, programming languages (including C, C++, Common Lisp, D, Fortran, Haskell, Julia, and Rust), and operating systems (including Windows, Linux, macOS and HP-UX). Depending on the platform the fused operation is about twice as fast as calling the sine and cosine functions individually.
Mathematical identities
Derivative
Integral
Other properties
These follow directly from Euler's formula.
The identities above hold if and are any complex numbers. If and are real, then
History
The notation was first coined by William Rowan Hamilton in Elements of Quaternions (1866) and subsequently used by Irving Stringham (who also called it "sector of x") in works such as Uniplanar Algebra (1893), James Harkness and Frank Morley in their Introduction to the Theory of Analytic Functions (1898), or by George Ashley Campbell (who also referred to it as "cisoidal oscillation") in his works on transmission lines (1901) and Fourier integrals (1928).
In 1942, inspired by the notation, Ralph V. L. Hartley introduced the (for cosine-and-sine) function for the real-valued Hartley kernel, a meanwhile established shortcut in conjunction with Hartley transforms:
In 2016, Reza R. Ahangar, a mathematics professor at TAMUK, defined two hyperbolic function shortcuts as:
Motivation
The notation is sometimes used to emphasize one method of viewing and dealing with a problem over another. The mathematics of trigonometry and exponentials are related but not exactly the same; exponential notation emphasizes the whole, whereas and notations emphasize the parts. This can be rhetorically useful to mathematicians and engineers when discussing this function, and further serve as a mnemonic (for ).
The notation |
https://en.wikipedia.org/wiki/Nuclear%20C%2A-algebra | In the mathematical field of functional analysis, a nuclear C*-algebra is a C*-algebra such that for every C*-algebra the injective and projective C*-cross norms coincides on the algebraic tensor product and the completion of with respect to this norm is a C*-algebra. This property was first studied by under the name "Property T", which is not related to Kazhdan's property T.
Characterizations
Nuclearity admits the following equivalent characterizations:
The identity map, as a completely positive map, approximately factors through matrix algebras. By this equivalence, nuclearity can be considered a noncommutative analogue of the existence of partitions of unity.
The enveloping von Neumann algebra is injective.
It is amenable as a Banach algebra.
(For separable algebras) It is isomorphic to a C*-subalgebra of the Cuntz algebra with the property that there exists a conditional expectation from to .
Examples
The commutative unital C* algebra of (real or complex-valued) continuous functions on a compact Hausdorff space as well as the noncommutative unital algebra of real or complex matrices are nuclear.
See also
References
C*-algebras
Functional analysis
Operator theory
it:C*-algebra#C*-algebra nucleare |
https://en.wikipedia.org/wiki/Anatoly%20Zhigljavsky | Anatoly Aleksandrovich Zhigljavsky (born 19 November 1953) is a professor of Statistics in the School of Mathematics at Cardiff University. He has authored 12 monographs and over 150 papers in refereed journals.
His research interests include stochastic and high-dimensional global optimisation, time series analysis, multivariate data analysis, statistical modeling in market research, probabilistic methods in search and number theory.
He is the Director of the Centre for Optimisation and its Applications, an interdisciplinary centre which encourages joint research and applied projects among members of the Schools of Mathematics, Computer Science and Business and Manufacturing Engineering Centre at Cardiff University. It also encourages increased awareness of the rapidly growing field of optimisation through publications, conferences, joint research and student exchange.
His books include Theory of Global Random Search, Stochastic Global Optimization, Analysis of time series structure: SSA and related techniques, Dynamical Search: Applications of Dynamical Systems in Search and Optimization, and Singular Spectrum Analysis for Time Series. His books received positive comments from reviewers.
Zhigljavsky received an MSc in 1976, a PhD in 1981, and a Habilitation in 1987 from the St. Petersburg State University, Russia.
In 2019 Anatoly Zhigljavsky received the Constantin Caratheodory Prize for his outstanding work and significant contributions in the field of global optimisation.
References
External links
Homepage at Cardiff University
Personal Homepage
British statisticians
1953 births
Living people |
https://en.wikipedia.org/wiki/Minimum%20k-cut | In mathematics, the minimum -cut is a combinatorial optimization problem that requires finding a set of edges whose removal would partition the graph to at least connected components. These edges are referred to as -cut. The goal is to find the minimum-weight -cut. This partitioning can have applications in VLSI design, data-mining, finite elements and communication in parallel computing.
Formal definition
Given an undirected graph with an assignment of weights to the edges and an integer partition into disjoint sets while minimizing
For a fixed , the problem is polynomial time solvable in However, the problem is NP-complete if is part of the input. It is also NP-complete if we specify vertices and ask for the minimum -cut which separates these vertices among each of the sets.
Approximations
Several approximation algorithms exist with an approximation of A simple greedy algorithm that achieves this approximation factor computes a minimum cut in each of the connected components and removes the heaviest one. This algorithm requires a total of max flow computations. Another algorithm achieving the same guarantee uses the Gomory–Hu tree representation of minimum cuts. Constructing the Gomory–Hu tree requires max flow computations, but the algorithm requires an overall max flow computations. Yet, it is easier to analyze the approximation factor of the second algorithm. Moreover, under the small set expansion hypothesis (a conjecture closely related to the unique games conjecture), the problem is NP-hard to approximate to within factor for every constant , meaning that the aforementioned approximation algorithms are essentially tight for large .
A variant of the problem asks for a minimum weight -cut where the output partitions have pre-specified sizes. This problem variant is approximable to within a factor of 3 for any fixed if one restricts the graph to a metric space, meaning a complete graph that satisfies the triangle inequality. More recently, polynomial time approximation schemes (PTAS) were discovered for those problems.
While the minimum -cut problem is W[1]-hard parameterized by , a parameterized approximation scheme can be obtained for this parameter.
See also
Maximum cut
Minimum cut
Notes
References
NP-complete problems
Combinatorial optimization
Computational problems in graph theory
Approximation algorithms |
https://en.wikipedia.org/wiki/Hilbert%E2%80%93Samuel%20function | In commutative algebra the Hilbert–Samuel function, named after David Hilbert and Pierre Samuel, of a nonzero finitely generated module over a commutative Noetherian local ring and a primary ideal of is the map such that, for all ,
where denotes the length over . It is related to the Hilbert function of the associated graded module by the identity
For sufficiently large , it coincides with a polynomial function of degree equal to , often called the Hilbert-Samuel polynomial (or Hilbert polynomial).
Examples
For the ring of formal power series in two variables taken as a module over itself and the ideal generated by the monomials x2 and y3 we have
Degree bounds
Unlike the Hilbert function, the Hilbert–Samuel function is not additive on an exact sequence. However, it is still reasonably close to being additive, as a consequence of the Artin–Rees lemma. We denote by the Hilbert-Samuel polynomial; i.e., it coincides with the Hilbert–Samuel function for large integers.
Proof: Tensoring the given exact sequence with and computing the kernel we get the exact sequence:
which gives us:
.
The third term on the right can be estimated by Artin-Rees. Indeed, by the lemma, for large n and some k,
Thus,
.
This gives the desired degree bound.
Multiplicity
If is a local ring of Krull dimension , with -primary ideal , its Hilbert polynomial has leading term of the form for some integer . This integer is called the multiplicity of the ideal . When is the maximal ideal of , one also says is the multiplicity of the local ring .
The multiplicity of a point of a scheme is defined to be the multiplicity of the corresponding local ring .
See also
j-multiplicity
References
Commutative algebra
Algebraic geometry |
https://en.wikipedia.org/wiki/Primal%20ideal | In mathematics, an element a of a commutative ring A is called (relatively) prime to an ideal Q if whenever ab is an element of Q then b is also an element of Q.
A proper ideal Q of a commutative ring A is said to be primal if the elements that are not prime to it form an ideal.
References
.
Commutative algebra |
https://en.wikipedia.org/wiki/Motorcycle%20testing%20and%20measurement | Motorcycle testing and measurement includes a range of more than two dozen statistics giving the specifications of the motorcycle, and the actual performance, expressed by such things as the output of the engine, and the top speed or acceleration of the motorcycle. Most parameters are uncontroversial and claims made by manufacturers are generally accepted without verification. These might include simple measurements like rake, trail, or wheelbase, or basic features, such as the type of brakes or ignition system.
Other measurements are often doubted or subject to misunderstandings, and the motorcycling press serves as an independent check on sometimes unrealistic sales and marketing claims. Many of these numbers are subject to variable methods of measurements, or disagreement as to the definition of the statistic. The parameters most often in contention for motorcycles are the weight, the engine output (power and torque), and the overall performance, especially acceleration, top speed, and fuel economy. With electric motorcycles and scooters, the range between charges is often a pivotal measurement.
Top speed
Motorcycle speed tests, especially at high speeds, are prone to variation due to human error, limitations in equipment, and atmospheric factors like wind, humidity, and altitude. The published results of two otherwise identical tests could vary depending on whether the result is reported with or without industry standard correction factors calculated to compensate for test conditions. Rounding errors are possible as well when converting to/from miles and kilometers per hour.
Engine power and torque
With power typically being the product of force and speed, a motorcycle's power and torque ratings will be highly indicative of its performance. Reported numbers for power and torque may however vary from one source to another due to inconsistencies in how testing equipment is calibrated, the method of using that equipment, the conditions during the test, and particularly the location that force and speed are being measured at. The power of the engine alone, often called crankshaft power, or power at the crankshaft, will be significantly greater than the power measured at the rear wheel. The amount of power lost due to friction in the transmission (primary drive, gearbox and final drive) depends on the details of the design and construction. Generalizing, a chain drive motorcycle may have some 5-20% less power at the rear wheel than at the crankshaft, while a shaft drive model may lose a little more than that due to greater friction.
While the crankshaft power excludes these transmission losses, still the measurement is often made elsewhere in the drive-train, often at the rear wheel. A correction for the transmission losses is then applied to the measured values to obtain the crankshaft values. For motorcycles, the reported power and torque numbers normally pertain to the crankshaft. In directive 92/61/EEC of 30 June 1992 relating to the ty |
https://en.wikipedia.org/wiki/Rees%20algebra | In commutative algebra, the Rees algebra of an ideal I in a commutative ring R is defined to be The extended Rees algebra of I (which some authors refer to as the Rees algebra of I) is defined asThis construction has special interest in algebraic geometry since the projective scheme defined by the Rees algebra of an ideal in a ring is the blowing-up of the spectrum of the ring along the subscheme defined by the ideal.
Properties
Assume R is Noetherian; then R[It] is also Noetherian. The Krull dimension of the Rees algebra is if I is not contained in any prime ideal P with ; otherwise . The Krull dimension of the extended Rees algebra is .
If are ideals in a Noetherian ring R, then the ring extension is integral if and only if J is a reduction of I.
If I is an ideal in a Noetherian ring R, then the Rees algebra of I is the quotient of the symmetric algebra of I by its torsion submodule.
Relationship with other blow-up algebras
The associated graded ring of I may be defined asIf R is a Noetherian local ring with maximal ideal , then the special fiber ring of I is given byThe Krull dimension of the special fiber ring is called the analytic spread of I.
References
External links
What Is the Rees Algebra of a Module?
Geometry behind Rees algebra (deformation to the normal cone)
Commutative algebra
Algebraic geometry |
https://en.wikipedia.org/wiki/Tony%20Greenfield | Tony Greenfield (26 April 1931 – 19 March 2019) was a British statistical consultant and academic. He was formerly Head of Process Computing and Statistics at the British Iron and Steel Research Association, Sheffield, and Professor of Medical Computing and Statistics at Queen's University, Belfast.
Until he retired, at the age of 80, he was a visiting professor to the Industrial Statistics Research Unit of the University of Newcastle-upon-Tyne and to the Universitat Politècnica de Catalunya.
Greenfield co-authored Design and Analyse your Experiments with Minitab with Andrew Metcalfe and Engineering Statistics with Matlab. His inaugural lecture (1980) at Queen's University is still sold as a booklet. His first book, Research Methods for Postgraduates is highly regarded on both sides of the Atlantic and is now in its third edition, published by Wiley. He has also had a strong hand in The Pocket Statistician, Statistical Practice in Business and Industry and an Encyclopaedia of Statistics in Quality and Reliability. One of his contributions to his local community of Great Hucklow is the editing of a history of lead mining in the area: Lead in the Veins.
Tony was a founding member and Past President of European Network for Business and Industrial Statistics and for many years he was a prominent member of the Royal Statistical Society. He was the first editor of RSS News and of the ENBIS newsletter and magazine. In its first ten years, ENBIS grew to a membership base of around 1500 practitioners spread across more than sixty countries.
Tony was a Chartered Statistician (CStat) and a Chartered Scientist (CSci).
Early life
Tony Greenfield was born in Chapeltown, South Yorkshire on 26 April 1931 to Geoffrey James Greenfield (1900–1978) and Hilda Aynsley (1903–1976).
Tony Greenfield worked in an iron mine when he left Bedford School at the age of 17. He later worked in coal mines, a brass tube factory, and in a copper mine and studied mining engineering at Imperial College London. He received the diploma in journalism from the Regent Street Polytechnic, worked on the Sunday Express and Sunday Mirror before turning to technical journalism for ten years. He was an active member of the Sheffield Junior Chamber of Commerce of which he was chairman of the Business Affairs committee and editor of The Hub, the chamber's monthly magazine. At the 1963 conference in Tel Aviv of Junior Chamber International he was acknowledged as the editor of the best junior chamber magazine in the world.
He moved into the steel industry to write technical reports for Operations Research (OR) scientists. There he found satisfaction in solving production problems, studied OR, mathematics, statistics and computing leading to an external degree from University College London. He moved into steel research and became head of process computing and statistics. Much of his work was in design and analysis of experiments for which he received his PhD. When the laboratories clo |
https://en.wikipedia.org/wiki/ViSta%2C%20The%20Visual%20Statistics%20system | ViSta, the Visual Statistics system is a freeware statistical system developed by Forrest W. Young of the University of North Carolina. ViSta current version maintained by Pedro M. Valero-Mora of the University of Valencia and can be found at . Old versions of ViSta and of the documentation can be found at .
ViSta incorporates a number of special features that are of both theoretical and practical interest: The workmap keeps record of the datasets opened by the user and the subsequent statistical transformations and analysis applied to them. Spreadplots show all the relevant plots for a dataset with a given combination of types of variables. Graphics are the primary way of output in contrast with traditional statistics packages where the textual output is more important.
References
Young, F. W., Valero-Mora, P. M. & Friendly, M. (2006) Visual Statistics: Seeing Data with Interactive Graphics. Wiley
Meissner, W. (2008) Book review of "Visual Statistics: Seeing Data with Interactive Graphics". Psychometrika 73, 1. Springer.
ViSta is mentioned in Michael Friendly's Milestones of Statistical Graphics
External links
This site keeps the last version of ViSta and other information
The original site for ViSta with old versions and documentation
Some Plug-ins to extend the ViSta's analysis options
Current version is 7.9.2.8 (2014, March)
Statistical software |
https://en.wikipedia.org/wiki/The%20Taylor%20Prize%20in%20Mathematics | The Taylor Prize in Mathematics is a cash prize awarded annually to an outstanding graduate student of mathematics, displaying excellence in graduate research and overall accomplishments, at The George Washington University in Washington, DC. The prize is named after James Henry Taylor, a professor of mathematics at GW from 1929 to 1958.
History
James Henry Taylor was a mathematics professor at GW from 1929 to 1958 and then professor emeritus until his death in 1972. Several years after his death the president of the university, Lloyd H. Elliott, along with several of the professors in the Mathematics Department, including Taylor's good friend Fritz Joachim Weyl, decided to create an award in his memory. Money was deposited in an account from which the prize funds would be drawn annually. The interest that builds throughout the year on the account makes up the majority of the annual prize. For example, in 1983 first year graduate student Karma Dajani won the prize, which at the time was $500 but when she won the prize again in 1986 it was worth only $300 because interest had not accumulated for as many years. Thomas J. Carter became the first recipient of the prize in 1977.
About the recipients
The GWU bulletin simply describes the criteria for recipients of the prize as, "awarded to an outstanding mathematics graduate student." Many of the thirty-four winners of the Taylor Prize have gone on to become professors of mathematics at various universities. Most have published essays and books or given lectures on their specific subjects.
Complete list of recipients
See also
List of mathematics awards
References
Mathematics awards
George Washington University
1977 establishments in Washington, D.C.
Awards established in 1977 |
https://en.wikipedia.org/wiki/Rigid%20category | In category theory, a branch of mathematics, a rigid category is a monoidal category where every object is rigid, that is, has a dual X* (the internal Hom [X, 1]) and a morphism 1 → X ⊗ X* satisfying natural conditions. The category is called right rigid or left rigid according to whether it has right duals or left duals. They were first defined (following Alexander Grothendieck) by Neantro Saavedra Rivano in his thesis on Tannakian categories.
Definition
There are at least two equivalent definitions of a rigidity.
An object X of a monoidal category is called left rigid if there is an object Y and morphisms and such that both compositions
are identities. A right rigid object is defined similarly.
An inverse is an object X−1 such that both X ⊗ X−1 and X−1 ⊗ X are isomorphic to 1, the identity object of the monoidal category. If an object X has a left (respectively right) inverse X−1 with respect to the tensor product then it is left (respectively right) rigid, and X* = X−1.
The operation of taking duals gives a contravariant functor on a rigid category.
Uses
One important application of rigidity is in the definition of the trace of an endomorphism of a rigid object. The trace can be defined for any pivotal category, i. e. a rigid category such that ( )**, the functor of taking the dual twice repeated, is isomorphic to the identity functor. Then for any right rigid object X, and any other object Y, we may define the isomorphism
and its reciprocal isomorphism
.
Then for any endomorphism , the trace is of f is defined as the composition:
We may continue further and define the dimension of a rigid object to be:
.
Rigidity is also important because of its relation to internal Hom's. If X is a left rigid object, then every internal Hom of the form [X, Z] exists and is isomorphic to Z ⊗ Y. In particular, in a rigid category, all internal Hom's exist.
Alternative terminology
A monoidal category where every object has a left (respectively right) dual is also sometimes called a left (respectively right) autonomous category. A monoidal category where every object has both a left and a right dual is sometimes called an autonomous category. An autonomous category that is also symmetric is called a compact closed category.
Discussion
A monoidal category is a category with a tensor product, precisely the sort of category for which rigidity makes sense.
The category of pure motives is formed by rigidifying the category of effective pure motives.
Notes
References
Monoidal categories |
https://en.wikipedia.org/wiki/Nalini%20Joshi | Nalini Joshi is an Australian mathematician. She is a professor in the School of Mathematics and Statistics at the University of Sydney, the first woman in the School to hold this position, and is a past-president of the Australian Mathematical Society.
Joshi is a member of the School's Applied Mathematics Research Group. Her research concerns integrable systems. She was awarded the Georgina Sweet Australian Laureate Fellowship in 2012. Joshi is also the Vice-President of the International Mathematical Union, and is the first Australian to hold this position.
Early life
Joshi was born and spent her childhood in Burma.
In 2007, she described her experience growing up there:
Education
Joshi attended Fort Street High School and gained her Bachelor of Science with honours in 1980 at the University of Sydney, and her PhD at Princeton University under the supervision of Martin David Kruskal.
Her PhD thesis was entitled The Connection Problem for the First and Second Painlevé Transcendents.
Career
After a postdoctoral fellowship in 1987 and a research fellowship and lectureship (1988–90), both at the Australian National University, Joshi took up a lectureship at the University of New South Wales in Sydney (1990–94) and was promoted to senior lecturer in 1994.
In 1997, she won an Australian Research Council (ARC) senior research fellowship, which she took up at the University of Adelaide, and became an associate professor/reader at that university a year later.
In 2002 she moved to the University of Sydney as Chair of Applied Mathematics; since 2006 she has been director of the Centre for Mathematical Biology, from 2007 to 2009 head of the School of Mathematics and Statistics (associate head since 2010).
In 2015, Joshi co-founded and co-chaired the Science in Australia Gender Equity (SAGE) program, which works to increase retention of women in STEM fields using Athena SWAN principles.
Since 2016, she has served as a member of the SAGE Expert Advisory Group.
Awards and honors
Joshi was elected a fellow of the Australian Academy of Science in March 2008, and has held a number of positions in the Australian Mathematical Society, including its presidency from December 2008 to September 2010.
She was also a board member of the Australian Mathematics Trust (2010–13)
She has been on the National Committee for Mathematical Sciences since January 2010.
In 2012, Joshi became a Georgina Sweet Australian Laureate Fellow, which involves the five-year project, Geometric construction of critical solutions of nonlinear systems.
In 2015, she was the 150th Anniversary Hardy Lecturer, an award by the London Mathematical Society involving an extensive series of lectures throughout the United Kingdom. She is a Fellow of the Royal Society of New South Wales (FRSN). In June 2016, she was appointed an Officer of the Order of Australia.
Joshi was elected vice-president of the International Mathematical Union in July 2018. She was recognised in the October 2019 NSW P |
https://en.wikipedia.org/wiki/Adequate%20equivalence%20relation | In algebraic geometry, a branch of mathematics, an adequate equivalence relation is an equivalence relation on algebraic cycles of smooth projective varieties used to obtain a well-working theory of such cycles, and in particular, well-defined intersection products. Pierre Samuel formalized the concept of an adequate equivalence relation in 1958. Since then it has become central to theory of motives. For every adequate equivalence relation, one may define the category of pure motives with respect to that relation.
Possible (and useful) adequate equivalence relations include rational, algebraic, homological and numerical equivalence. They are called "adequate" because dividing out by the equivalence relation is functorial, i.e. push-forward (with change of codimension) and pull-back of cycles is well-defined. Codimension 1 cycles modulo rational equivalence form the classical group of divisors modulo linear equivalence. All cycles modulo rational equivalence form the Chow ring.
Definition
Let Z*(X) := Z[X] be the free abelian group on the algebraic cycles of X. Then an adequate equivalence relation is a family of equivalence relations, ∼X on Z*(X), one for each smooth projective variety X, satisfying the following three conditions:
(Linearity) The equivalence relation is compatible with addition of cycles.
(Moving lemma) If are cycles on X, then there exists a cycle such that ~X and intersects properly.
(Push-forwards) Let and be cycles such that intersects properly. If ~X 0, then ~Y 0, where is the projection.
The push-forward cycle in the last axiom is often denoted
If is the graph of a function, then this reduces to the push-forward of the function. The generalizations of functions from X to Y to cycles on X × Y are known as correspondences. The last axiom allows us to push forward cycles by a correspondence.
Examples of equivalence relations
The most common equivalence relations, listed from strongest to weakest, are gathered in the following table.
Notes
References
Algebraic geometry
Equivalence (mathematics) |
https://en.wikipedia.org/wiki/Polynomial%20identity%20ring | In ring theory, a branch of mathematics, a ring R is a polynomial identity ring if there is, for some N > 0, an element P ≠ 0 of the free algebra, Z, over the ring of integers in N variables X1, X2, ..., XN such that
for all N-tuples r1, r2, ..., rN taken from R.
Strictly the Xi here are "non-commuting indeterminates", and so "polynomial identity" is a slight abuse of language, since "polynomial" here stands for what is usually called a "non-commutative polynomial". The abbreviation PI-ring is common. More generally, the free algebra over any ring S may be used, and gives the concept of PI-algebra.
If the degree of the polynomial P is defined in the usual way, the polynomial P is called monic if at least one of its terms of highest degree has coefficient equal to 1.
Every commutative ring is a PI-ring, satisfying the polynomial identity XY − YX = 0. Therefore, PI-rings are usually taken as close generalizations of commutative rings. If the ring has characteristic p different from zero then it satisfies the polynomial identity pX = 0. To exclude such examples, sometimes it is defined that PI-rings must satisfy a monic polynomial identity.
Examples
For example, if R is a commutative ring it is a PI-ring: this is true with
The ring of 2 × 2 matrices over a commutative ring satisfies the Hall identity
This identity was used by , but was found earlier by .
A major role is played in the theory by the standard identity sN, of length N, which generalises the example given for commutative rings (N = 2). It derives from the Leibniz formula for determinants
by replacing each product in the summand by the product of the Xi in the order given by the permutation σ. In other words each of the N ! orders is summed, and the coefficient is 1 or −1 according to the signature.
The m × m matrix ring over any commutative ring satisfies a standard identity: the Amitsur–Levitzki theorem states that it satisfies s2m. The degree of this identity is optimal since the matrix ring can not satisfy any monic polynomial of degree less than 2m.
Given a field k of characteristic zero, take R to be the exterior algebra over a countably infinite-dimensional vector space with basis e1, e2, e3, ... Then R is generated by the elements of this basis and
ei ej = − ej ei.
This ring does not satisfy sN for any N and therefore can not be embedded in any matrix ring. In fact sN(e1,e2,...,eN) = N ! e1e2...eN ≠ 0. On the other hand it is a PI-ring since it satisfies [[x, y], z] := xyz − yxz − zxy + zyx = 0. It is enough to check this for monomials in the ei's. Now, a monomial of even degree commutes with every element. Therefore if either x or y is a monomial of even degree [x, y] := xy − yx = 0. If both are of odd degree then [x, y] = xy − yx = 2xy has even degree and therefore commutes with z, i.e. [[x, y], z] = 0.
Properties
Any subring or homomorphic image of a PI-ring is a PI-ring.
A finite direct product of PI-rings is a PI-ring.
A direct p |
https://en.wikipedia.org/wiki/Milner%E2%80%93Rado%20paradox | In set theory, a branch of mathematics, the Milner – Rado paradox, found by , states that every ordinal number less than the successor of some cardinal number can be written as the union of sets where is of order type at most κn for n a positive integer.
Proof
The proof is by transfinite induction. Let be a limit ordinal (the induction is trivial for successor ordinals), and for each , let be a partition of satisfying the requirements of the theorem.
Fix an increasing sequence cofinal in with .
Note .
Define:
Observe that:
and so .
Let be the order type of . As for the order types, clearly .
Noting that the sets form a consecutive sequence of ordinal intervals, and that each is a tail segment of , then:
References
How to prove Milner-Rado Paradox? - Mathematics Stack Exchange
Set theory
Paradoxes |
https://en.wikipedia.org/wiki/Net%20Applications | Net Applications is a web analytics firm. The company is commonly known in the web browser development and technology news communities for its global market share statistics.
History
Since 1999, Net Applications is a source of applications for webmasters and eMarketers. Headquartered in Aliso Viejo, California, Net Applications distributes its services through over 7,000 partners and affiliates.
Criticism
While the statistics released by the company routinely place operating systems sold by Microsoft (Windows) and Apple (Mac OS X) with a high market share in the desktop computer category (through 2013), Vincent Vizzaccaro (EVP – Marketing and Strategic Alliances, Net Applications, 2002–present) has stated that Microsoft and Apple are among the company's clients. The company has also admitted that their statistics are skewed.
See also
Usage share of web browsers
Usage share of operating systems
Web browser
Market share
Browser wars
References
External links
Internet properties established in 1999
Online companies of the United States
Web analytics
1999 establishments in the United States |
https://en.wikipedia.org/wiki/Eric%20Charles%20Milner | Eric Charles Milner, FRSC (May 17, 1928 – July 20, 1997) was a mathematician who worked mainly in combinatorial set theory.
Biography
Born into a South East London working-class family, Milner was sent to a Reading boarding school for the war but, hating it, ran away and roamed the streets of London. Eventually, another school was found for him; Milner attended King's College London starting in 1946, where he competed as a featherweight boxer. He graduated in 1949 as the best mathematics student in his year, and received a master's degree in 1950 under the supervision of Richard Rado and Charles Coulson. Partial deafness prevented him from joining the Navy, and instead, in 1951, he took a position with the Straits Trading Company in Singapore assaying tin. Soon thereafter he joined the mathematics faculty at the University of Malaya in Singapore, where Alexander Oppenheim and Richard K. Guy were already working. In 1958, Milner took a sabbatical at the University of Reading, and in 1961 he took a lecturership there and began his doctoral studies; he obtained a Ph.D. from the University of London in 1963. He joined his former Singapore colleagues Guy and Peter Lancaster as a professor at the University of Calgary in 1967, where he was head of the mathematics department from 1976 to 1980. In 1973, he became a Canadian citizen, and in 1976 he became a fellow of the Royal Society of Canada. In 1974 he was a Plenary Speaker of the International Congress of Mathematicians in Vancouver.
In 1954, while in Singapore, Milner married Esther Stella (Estelle) Lawton, whom he had known as a London student; they had four children who were Paul Milner, Mark Milner, Suzanne Milner, and Simon Milner. Estelle died of cancer in 1975, and in 1979 Milner married Elizabeth Forsyth Borthwick, with whom he had his son Robert Milner.
Research
Milner's interest in set theory was sparked by visits of Paul Erdős to Singapore and by meeting András Hajnal while on sabbatical in Reading.
He generalized Chen Chung Chang's ordinal partition theorem (expressed in the arrow notation for Ramsey theory) ωω→(ωω,3)2 to ωω→(ωω,k)2 for arbitrary finite k. He is also known for the Milner–Rado paradox. He has 15 joint papers with Paul Erdős.
Selected works
References
External links
Eric Milner award
University of Calgary Eric Milner Colloquium
1928 births
1997 deaths
Alumni of King's College London
Set theorists
20th-century Canadian mathematicians
Fellows of the Royal Society of Canada
Academic staff of the University of Calgary
20th-century English mathematicians |
https://en.wikipedia.org/wiki/Jonathan%20Lazare%20Alperin | Jonathan Lazare Alperin (; born 1937) is an American mathematician specializing in the area of algebra known as group theory. He is notable for his work in group theory which has been cited over 500 times according to the Mathematical Reviews. The Alperin–Brauer–Gorenstein theorem is named after him.
Biography
Alperin attended school at Princeton University and wrote his Ph.D. dissertation in 1961 "On a Special Class of Regular p-Groups" under the direction of Graham Higman. He was awarded a Guggenheim Fellowship in 1974. He has several times (1969, 1979, and 1983) been a visiting scholar at the Institute for Advanced Study. In 2012 he became a fellow of the American Mathematical Society.
Alperin was a professor at the University of Chicago. He has published over 60
papers and his work has been cited over 500 times.
He is also known for his conjecture, , a topic of current research in modular representation theory, and for his work on the local control of fusion, , part of local group theory. In , the Alperin–Brauer–Gorenstein theorem was proven, giving the classification of finite simple groups with quasi-dihedral Sylow 2-subgroups.
Selected bibliography
References
External links
Mathematical Reviews author profile
Home page at Chicago
Short biography in the Notices of the AMS
20th-century American mathematicians
21st-century American mathematicians
Group theorists
Institute for Advanced Study visiting scholars
University of Chicago faculty
Princeton University alumni
1937 births
Living people
Fellows of the American Mathematical Society |
https://en.wikipedia.org/wiki/Takahiko%20Sumida | is a former Japanese football player.
Club statistics
References
External links
Profile at Oita Trinita
1991 births
Living people
Association football people from Tottori Prefecture
Japanese men's footballers
J1 League players
J2 League players
J3 League players
Japan Football League players
Oita Trinita players
Gainare Tottori players
Iwate Grulla Morioka players
Men's association football forwards
People from Yonago, Tottori |
https://en.wikipedia.org/wiki/Yudai%20Inoue | is a Japanese football player currently playing for FC Machida Zelvia in the J2 League.
Club statistics
Updated to 1 January 2020.
1Includes Suruga Bank Championship and Promotion Playoffs to J1.
References
External links
Profile at Machida Zelvia
Profile at Oita Trinita
1989 births
Living people
Association football people from Ōita Prefecture
Japanese men's footballers
J1 League players
J2 League players
Oita Trinita players
V-Varen Nagasaki players
FC Machida Zelvia players
Men's association football midfielders
Sportspeople from Ōita (city) |
https://en.wikipedia.org/wiki/2010%20Colo-Colo%20season | The 2010 season was Club Social y Deportivo Colo-Colo's 79th season in the Chilean Primera División. This article shows player statistics and all official matches that the club played during the 2010 season.
Players
Squad information
Squad stats
Players out
Disciplinary records
Players in / out
In
Out
Competitions
Primera División
Standings
Results summary
Results by round
Copa Libertadores
Competitive
Primera División
Copa Libertadores
Group stage
Copa Chile
First round
3 - 3 on points. Curicó won 4 - 3 on penalties
Copa Sudamericana
First round
3 - 3 on global. Universitario Sucre won on away goals
See also
Colo-Colo
References
External links
Official club site
Colo-Colo seasons
Colo-Colo
Colo |
https://en.wikipedia.org/wiki/Differential%20invariant | In mathematics, a differential invariant is an invariant for the action of a Lie group on a space that involves the derivatives of graphs of functions in the space. Differential invariants are fundamental in projective differential geometry, and the curvature is often studied from this point of view. Differential invariants were introduced in special cases by Sophus Lie in the early 1880s and studied by Georges Henri Halphen at the same time. was the first general work on differential invariants, and established the relationship between differential invariants, invariant differential equations, and invariant differential operators.
Differential invariants are contrasted with geometric invariants. Whereas differential invariants can involve a distinguished choice of independent variables (or a parameterization), geometric invariants do not. Élie Cartan's method of moving frames is a refinement that, while less general than Lie's methods of differential invariants, always yields invariants of the geometrical kind.
Definition
The simplest case is for differential invariants for one independent variable x and one dependent variable y. Let G be a Lie group acting on R2. Then G also acts, locally, on the space of all graphs of the form y = ƒ(x). Roughly speaking, a k-th order differential invariant is a function
depending on y and its first k derivatives with respect to x, that is invariant under the action of the group.
The group can act on the higher-order derivatives in a nontrivial manner that requires computing the prolongation of the group action. The action of G on the first derivative, for instance, is such that the chain rule continues to hold: if
then
Similar considerations apply for the computation of higher prolongations. This method of computing the prolongation is impractical, however, and it is much simpler to work infinitesimally at the level of Lie algebras and the Lie derivative along the G action.
More generally, differential invariants can be considered for mappings from any smooth manifold X into another smooth manifold Y for a Lie group acting on the Cartesian product X×Y. The graph of a mapping X → Y is a submanifold of X×Y that is everywhere transverse to the fibers over X. The group G acts, locally, on the space of such graphs, and induces an action on the k-th prolongation Y(k) consisting of graphs passing through each point modulo the relation of k-th order contact. A differential invariant is a function on Y(k) that is invariant under the prolongation of the group action.
Applications
Solving equivalence problems
Differential invariants can be applied to the study of systems of partial differential equations: seeking similarity solutions that are invariant under the action of a particular group can reduce the dimension of the problem (i.e. yield a "reduced system").
Noether's theorem implies the existence of differential invariants corresponding to every differentiable symmetry of a variational problem.
F |
https://en.wikipedia.org/wiki/Boolean%20matrix | In mathematics, a Boolean matrix is a matrix with entries from a Boolean algebra. When the two-element Boolean algebra is used, the Boolean matrix is called a logical matrix. (In some contexts, particularly computer science, the term "Boolean matrix" implies this restriction.)
Let U be a non-trivial Boolean algebra (i.e. with at least two elements). Intersection, union, complementation, and containment of elements is expressed in U. Let V be the collection of n × n matrices that have entries taken from U. Complementation of such a matrix is obtained by complementing each element. The intersection or union of two such matrices is obtained by applying the operation to entries of each pair of elements to obtain the corresponding matrix intersection or union. A matrix is contained in another if each entry of the first is contained in the corresponding entry of the second.
The product of two Boolean matrices is expressed as follows:
According to one author, "Matrices over an arbitrary Boolean algebra β satisfy most of the properties over β0 = {0, 1}. The reason is that any Boolean algebra is a sub-Boolean algebra of for some set S, and we have an isomorphism from n × n matrices over "
References
R. Duncan Luce (1952) "A Note on Boolean Matrices", Proceedings of the American Mathematical Society 3: 382–8, Jstor link
Jacques Riguet (1954) "Sur l'extension du calcul des relations binaires au calcul des matrices à éléments dans une algèbre de Boole", Comptes Rendus 238: 2382–2385
Further reading
Stan Gudder & Frédéric Latrémolière (2009) "Boolean inner-product spaces and Boolean matrices", Linear Algebra and Its Applications 431: 274–96
D.E. Rutherford (1963) "Inverses of Boolean matrices", Proceedings of the Glasgow Mathematical Association 6: 49–63
T.S. Blythe (1967) "Eigenvectors of Boolean Matrices", Proceedings of the Royal Society of Edinburgh 67: 196–204
Steven Kirkland & Norman J. Pullman (1993) "Linear Operators Preserving Invariants of Non-binary Boolean Matrices", Linear and Multilinear Algebra 33: 295–300
Kyung-Kae Kang, Seok-Zun Song & Young-Bae Jung (2011) "Linear Preservers of Regular Matrices over General Boolean Algebras", Bulletin of the Malaysian Mathematical Sciences Society, second series, 34(1): 113–25
Matrices
Boolean algebra |
https://en.wikipedia.org/wiki/Shunsuke%20Iwanuma | is a Japanese football player currently playing for Nagano Parceiro.
Club statistics
Updated to 23 February 2017.
References
External links
Profile at Nagano Parceiro
Profile at Consadole Sapporo
1988 births
Living people
Association football people from Gunma Prefecture
Japanese men's footballers
J1 League players
J2 League players
J3 League players
Hokkaido Consadole Sapporo players
Matsumoto Yamaga FC players
Kyoto Sanga FC players
AC Nagano Parceiro players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Adolf%20Bestelmeyer | Adolf (Christoph Wilhelm) Bestelmeyer (21 December 1875 – 21 November 1957) was a German experimental physicist.
Life and work
Bestelmeyer studied mathematics and physics at the Technical University of Berlin, the Technical University of Munich and the University of Munich. After his promotion, he worked in 1904 as an assistant at the University of Göttingen. In World War I he was active in torpedo research, and afterwards he was professor of physics at the University of Greifswald from 1917 to 1921. He then served until the end of World War II as a laboratory manager in various companies (like Askania), especially in the area of torpedo construction.
In 1907, Bestelmeyer questioned the accuracy of the measurements by Walter Kaufmann regarding the speed dependence of the electromagnetic mass. Bestelmeyer used a velocity filter for his own experiments on cathode rays, and this method was later also used by Alfred Bucherer. While Bucherer saw the results of his experiments as a confirmation of special relativity, his methods were criticized by Bestelmeyer, thus a polemical dispute between these two researchers arose. It took years until those problems could be resolved, and the results of further experiments confirmed the predictions of special relativity. Which are now again under suspicion.
Bestelmeyer is also known for developing a magnetic detonator for torpedoes in 1917, for which he was awarded the Iron Cross 1st Class (although there was no time for testing this device in World War I anymore).
See also
Kaufmann–Bucherer–Neumann experiments
References
External links
1875 births
1957 deaths
20th-century German physicists
Academic staff of the University of Greifswald
Experimental physicists |
https://en.wikipedia.org/wiki/The%20Mathematical%20Gazette | The Mathematical Gazette is a triannual peer-reviewed academic journal published by Cambridge University Press on behalf of the Mathematical Association. It covers mathematics education with a focus on the 15–20 years age range.
The journal was established in 1894 by Edward Mann Langley as the successor to the Reports of the Association for the Improvement of Geometrical Teaching. William John Greenstreet was its editor-in-chief for more than thirty years (1897–1930). Since 2000, the editor is Gerry Leversha.
Editors-in-chief
The following persons are or have been editor-in-chief:
Abstracting and indexing
The journal is abstracted and indexed in EBSCO databases, Emerging Sources Citation Index, Scopus, and zbMATH Open.
References
External links
Mathematics education journals
Mathematics education in the United Kingdom
1894 establishments in England
Cambridge University Press academic journals
Academic journals established in 1894
English-language journals
Triannual journals |
https://en.wikipedia.org/wiki/Kunen%27s%20inconsistency%20theorem | In set theory, a branch of mathematics, Kunen's inconsistency theorem, proved by , shows that several plausible large cardinal axioms are inconsistent with the axiom of choice.
Some consequences of Kunen's theorem (or its proof) are:
There is no non-trivial elementary embedding of the universe V into itself. In other words, there is no Reinhardt cardinal.
If j is an elementary embedding of the universe V into an inner model M, and λ is the smallest fixed point of j above the critical point κ of j, then M does not contain the set j "λ (the image of j restricted to λ).
There is no ω-huge cardinal.
There is no non-trivial elementary embedding of Vλ+2 into itself.
It is not known if Kunen's theorem still holds in ZF (ZFC without the axiom of choice), though showed that there is no definable elementary embedding from V into V. That is there is no formula J in the language of set theory such that for some parameter p∈V for all sets x∈V and y∈V:
Kunen used Morse–Kelley set theory in his proof. If the proof is re-written to use ZFC, then one must add the assumption that replacement holds for formulas involving j. Otherwise one could not even show that j "λ exists as a set. The forbidden set j "λ is crucial to the proof. The proof first shows that it cannot be in M. The other parts of the theorem are derived from that.
It is possible to have models of set theory that have elementary embeddings into themselves, at least if one assumes some mild large cardinal axioms. For example, if 0# exists then there is an elementary embedding from the constructible universe L into itself. This does not contradict Kunen's theorem because if 0# exists then L cannot be the whole universe of sets.
See also
Rank-into-rank
References
Large cardinals |
https://en.wikipedia.org/wiki/Tapering%20%28mathematics%29 | In mathematics, physics, and theoretical computer graphics, tapering is a kind of shape deformation. Just as an affine transformation, such as scaling or shearing, is a first-order model of shape deformation, tapering is a higher order deformation just as twisting and bending. Tapering can be thought of as non-constant scaling by a given tapering function. The resultant deformations can be linear or nonlinear.
To create a nonlinear taper, instead of scaling in x and y for all z with constants as in:
let a and b be functions of z so that:
An example of a linear taper is , and a quadratic taper .
As another example, if the parametric equation of a cube were given by ƒ(t) = (x(t), y(t), z(t)), a nonlinear taper could be applied so that the cube's volume slowly decreases (or tapers) as the function moves in the positive z direction. For the given cube, an example of a nonlinear taper along z would be if, for instance, the function T(z) = 1/(a + bt) were applied to the cube's equation such that ƒ(t) = (T(z)x(t), T(z)y(t), T(z)z(t)), for some real constants a and b.
See also
3D projection
References
External links
, Computer Graphics Notes. University of Toronto. (See: Tapering).
, 3D Transformations. Brown University. (See: Nonlinear deformations).
, ScienceWorld article on Tapering in Image Synthesis.
Linear algebra
Functions and mappings |
https://en.wikipedia.org/wiki/Junya%20Osaki | is a Japanese football player currently playing for Tochigi SC.
Club statistics
Updated to end of 2018 season.
1Includes FIFA Club World Cup and Promotion Playoffs to J1.
National Team Career
Last update: 29 January 2010
Appearances in major competitions
References
External links
Profile at Renofa Yamaguchi
1991 births
Living people
Association football people from Toyama Prefecture
Japanese men's footballers
J1 League players
J2 League players
Sanfrecce Hiroshima players
Tokushima Vortis players
Renofa Yamaguchi FC players
Tochigi SC players
Men's association football midfielders
People from Toyama (city) |
https://en.wikipedia.org/wiki/Steve%20Shnider | Steve Shnider is a retired professor of mathematics at Bar Ilan University.
He received a PhD in Mathematics from Harvard University in 1972, under Shlomo Sternberg.
His main interests are in the differential geometry of fiber bundles; algebraic methods in the theory of deformation of geometric structures; symplectic geometry; supersymmetry; operads; and Hopf algebras.
He retired in 2014.
Book on operads
A 2002 book of Markl, Shnider and Stasheff Operads in algebra, topology, and physics was the first book to provide a systematic treatment of operad theory, an area of mathematics that came to prominence in 1990s and found many applications in algebraic topology, category theory, graph cohomology, representation theory, algebraic geometry, combinatorics, knot theory, moduli spaces, and other areas. The book was the subject of a Featured Review in Mathematical Reviews by Alexander A. Voronov which stated, in particular: "The first book whose main goal is the theory of operads per se ... a book such as this one has been long awaited by a wide scientific readership, including mathematicians and theoretical physicists ... a great piece of mathematical literature and will be helpful to anyone who needs to use operads, from graduate students to mature mathematicians and physicists."
Bibliography
According to Mathematical Reviews, Shnider's work has been cited over 300 times by over 300 authors by 2010.
Books
Selected papers
.
References
21st-century Israeli mathematicians
Living people
Geometers
Harvard University alumni
Year of birth missing (living people) |
https://en.wikipedia.org/wiki/Tsubasa%20Oya | is a former Japanese football player who last played for Gainare Tottori.
Career
After a long career, Oya retired in February 2020 to become a school coach for Vissel Kobe.
Club statistics
Updated to 23 February 2020.
References
External links
1986 births
Living people
Kansai University alumni
Association football people from Shimane Prefecture
Japanese men's footballers
J1 League players
J2 League players
Vissel Kobe players
Fagiano Okayama players
Omiya Ardija players
Tokushima Vortis players
Men's association football defenders |
https://en.wikipedia.org/wiki/Federal%20Statistical%20Office%20%28Switzerland%29 | The Federal Statistical Office (FSO) is a Federal agency of the Swiss Confederation. It is the statistics office of Switzerland, situated in Neuchâtel and attached to the Federal Department of Home Affairs.
The Federal Statistical Office is the national service provider and competence centre for statistical observations in areas of national, social, economic and environmental importance. The FSO is the main producer of statistics in the country and runs the Swiss Statistics data pool. It provides information on all subject areas covered by official statistics.
The office is closely linked to the national statistics scene as well as to partners in the worlds of science, business and politics. It works closely with Eurostat, the Statistics Office of the European Union, in order to provide information that is also comparable at an international level.
The key principles upheld by the office throughout its statistical activities are data protection, scientific reliability, impartiality, topicality and service orientation.
History
With the founding of the Swiss Federal State in 1848, statistics gained in importance at the national level. In 1849, statistics became the task of the Department of Home Affairs under Federal Councillor Stefano Franscini, who conducted the first population census in the newly founded federal state in 1850. In 1860, the Federal Statistics Bureau (the present Federal Statistical Office) was founded in Bern, where it was located until 1998. Since 1998 all sections of the FSO have been centrally located in one building in Neuchâtel.
In the year of the FSO's foundation, a federal act was passed on the population census to be conducted every ten years. Ten years later the law was extended. In 1870, Parliament approved a brief law confined to organisational issues about "official statistical surveys in Switzerland". In 1992 this was replaced with the more up-to-date Federal Statistics Act. The new Federal Constitution of 1999 included for the first time an article (Art. 65) regarding statistics. In 2002 the Charter of Swiss Public Statistics was approved. One of the aims of the Charter is to establish universal principles that are based upon international standards but that also take particularities of the Swiss statistical system into account. The bilateral cooperation agreement between Switzerland and the European Union in the area of statistics came into force in 2007.
The Statistical Yearbook of Switzerland was first published in 1891 and has since then been published without interruption by the FSO. Since 1987, the FSO has been making important statistical information available online in electronic form, and in 1996 this service was extended and the STATINF database and website were added.
Mission
The FSO produces and publishes key statistical information on the current situation and development of the nation and society, of the economy and the environment. It completes these with comprehensive analyses, it creates |
https://en.wikipedia.org/wiki/Akihito%20Kusunose | is a former Japanese football player.
Club statistics
References
External links
Profile at Vissel Kobe
1986 births
Living people
Ryutsu Keizai University alumni
Association football people from Kōchi Prefecture
Japanese men's footballers
J1 League players
J2 League players
Vissel Kobe players
Matsumoto Yamaga FC players
Men's association football midfielders
People from Kōchi, Kōchi |
https://en.wikipedia.org/wiki/Bill%20Whelton | William Whelton (born August 28, 1959) is an American former professional ice hockey player who played two games in the National Hockey League. He played for the Winnipeg Jets.
Career statistics
Regular season and playoffs
External links
1959 births
Living people
American men's ice hockey defensemen
Boston University Terriers men's ice hockey players
Brunico SG players
ECH Chur players
Ice hockey players from Massachusetts
Lahti Pelicans players
Sherbrooke Jets players
Sportspeople from Everett, Massachusetts
Ice hockey people from Middlesex County, Massachusetts
Tulsa Oilers (1964–1984) players
Winnipeg Jets (1979–1996) draft picks
Winnipeg Jets (1979–1996) players |
https://en.wikipedia.org/wiki/Szymanski%27s%20conjecture | In mathematics, Szymanski's conjecture, named after , states that every permutation on the n-dimensional doubly directed hypercube graph can be routed with edge-disjoint paths. That is, if the permutation σ matches each vertex v to another vertex σ(v), then for each v there exists a path in the hypercube graph from v to σ(v) such that no two paths for two different vertices u and v use the same edge in the same direction.
Through computer experiments it has been verified that the conjecture is true for n ≤ 4 . Although the conjecture remains open for n ≥ 5, in this case there exist permutations that require the use of paths that are not shortest paths in order to be routed .
References
.
.
.
Conjectures
Unsolved problems in graph theory
Network topology |
https://en.wikipedia.org/wiki/The%20Problem%20%28album%29 | The Problem is an album by Wu-Tang Clan DJ Mathematics, released on June 28, 2005, by Nature Sounds. Most tracks have cameos by members of the Wu-Tang Clan. The Problem is Mathematics' second solo album.
Track listing
"Intro"
"C What I C" (T-Slugz and Eyeslow)
"Strawberries & Cream" (Allah Real, Inspectah Deck, The RZA and Ghostface Killah)
"Can I Rise" (HahFlamez)
"John – 3:16" (Method Man and Panama P.I.)
"Winta Sno" (Eyeslow, L.S., and Ali Vegas)
"Two Shots Of Henny" (Buddah Bless, Angie Neil, HahFlamez, Eyeslow, Panama P.I. and Allah Real)
"Bullet Scar" (T-Slugz)
"Real Nillaz" (Ghostface Killah, Buddah Bless, Eyeslow, and Raekwon)
"Coach Talk" (Bald Head)
"Rush" (Method Man and GZA)
"U.S.A." (Ghostface Killah, Masta Killa, Todd, Panama P.I., Eyeslow and HahFlamez)
"Tommy" (Allah Real, Eyeslow, Angie Neil and Bald Head)
"Break That" (Ol Dirty Bastard, Masta Killa and U-God)
"Spot Lite" (Method Man, U-God, Inspectah Deck and Cappadonna) [bonus track]
Mathematics (producer) albums
2005 albums
Nature Sounds albums
Albums produced by Mathematics |
https://en.wikipedia.org/wiki/Dushnik%E2%80%93Miller%20theorem | In mathematics, the Dushnik–Miller theorem is a result in order theory stating that every infinite linear order has a non-identity order embedding into itself. It is named for Ben Dushnik and E. W. Miller, who published this theorem for countable linear orders in 1940. More strongly, they showed that in the countable case there exists an order embedding into a proper subset of the given order; however, they provided examples showing that this strengthening does not always hold for uncountable orders.
In reverse mathematics, the Dushnik–Miller theorem for countable linear orders has the same strength as the arithmetical comprehension axiom (ACA0), one of the "big five" subsystems of second-order arithmetic. This result is closely related to the fact that (as Louise Hay and Joseph Rosenstein proved) there exist computable linear orders with no computable non-identity self-embedding.
See also
Cantor's isomorphism theorem
Laver's theorem
References
Order theory |
https://en.wikipedia.org/wiki/Robert%20Williams%20%28geometer%29 | Robert Edward Williams (born 1942) is an American designer, mathematician, and architect. He is noted for books on the geometry of natural structure, the discovery of a new space-filling polyhedron, the development of theoretical principles of Catenatic Geometry, and the invention of the Ars-Vivant Wild-life Protector System for repopulating the Western Mojave Desert in California, USA with desert tortoises.
Biography—life, theories, and work
Robert Williams was born in Cincinnati, Ohio, the son of Robert Finley Williams and Edna Rita Brotherton. His father was the oldest member of the Williams Brothers, a quartet of musical entertainers, who appeared on recordings, radio, and television, from the late 1930s to the present.
Williams's work was originally inspired by the design principles in natural structure systems promoted by R. Buckminster Fuller. He was introduced to the work of Fuller by designer Peter Pearce in 1963. He finished graduate studies in structural design at Southern Illinois University in 1967, where Fuller was University Professor. While at SIU, he invented a system of clustering dome structures by using small circle Catenatic Geometry principles rather than great circles, or geodesics, as Fuller had designed into geodesic dome structures. From his research with naturally packed cell systems (biological cells, soap bubble packings, and metal crystallites) he also discovered a new space-filling polyhedron, the β-tetrakaidecahedron, the faces of which closely approximate the actual distribution of the kinds of faces found in experimental samples of cell geometry in natural systems.
Williams met astronomer, Albert George Wilson at the Rand Corporation in 1966. Wilson invited him to conduct research at the McDonnell-Douglas Corporation Advanced Research Laboratories (DARL) in Huntington Beach, California, USA. After graduate studies, he joined Dr. Wilson in September 1967 and continued his research into general structure principles in natural systems. He was the geometry and structure consultant to NASA engineer, Charles A. Willits, on the initiatory work in the development of large scale structure systems for space stations. The first of four editions of his structural geometry research was published by DARL in 1969, with the title:Handbook of Structure. His paper in the journal Science proposed that his discovery of the β-tetrakaidecahedron is the most reasonable alternative to Lord Kelvin's α-tetrakaidecahedron.
As an organizer and presenter at the First International Conference on Hierarchical Structures sponsored by DARL in 1968, Williams was an early proponent advocating the discipline of Hierarchical Structure to be a legitimate area of scientific research.
In the spring of 1970, Williams became a visiting lecturer in Design at Southern Illinois University. A year later he returned to California, and started the design company Mandala Design Associates. In 1972, Eudaemon Press published Natural Structure: Toward a Form |
https://en.wikipedia.org/wiki/Stokes%27%20theorem | Stokes' theorem, also known as the Kelvin–Stokes theorem after Lord Kelvin and George Stokes, the fundamental theorem for curls or simply the curl theorem, is a theorem in vector calculus on . Given a vector field, the theorem relates the integral of the curl of the vector field over some surface, to the line integral of the vector field around the boundary of the surface. The classical theorem of Stokes can be stated in one sentence: The line integral of a vector field over a loop is equal to the flux of its curl through the enclosed surface. It is illustrated in the figure, where the direction of positive circulation of the bounding contour , and the direction of positive flux through the surface , are related by a right-hand-rule. For the right hand the fingers circulate along and the thumb is directed along .
Stokes' theorem is a special case of the generalized Stokes theorem. In particular, a vector field on can be considered as a 1-form in which case its curl is its exterior derivative, a 2-form.
Theorem
Let be a smooth oriented surface in with boundary . If a vector field is defined and has continuous first order partial derivatives in a region containing , then
More explicitly, the equality says that
The main challenge in a precise statement of Stokes' theorem is in defining the notion of a boundary. Surfaces such as the Koch snowflake, for example, are well-known not to exhibit a Riemann-integrable boundary, and the notion of surface measure in Lebesgue theory cannot be defined for a non-Lipschitz surface. One (advanced) technique is to pass to a weak formulation and then apply the machinery of geometric measure theory; for that approach see the coarea formula. In this article, we instead use a more elementary definition, based on the fact that a boundary can be discerned for full-dimensional subsets of .
A more detailed statement will be given for subsequent discussions.
Let be a piecewise smooth Jordan plane curve. The Jordan curve theorem implies that divides into two components, a compact one and another that is non-compact. Let denote the compact part; then is bounded by . It now suffices to transfer this notion of boundary along a continuous map to our surface in . But we already have such a map: the parametrization of .
Suppose is piecewise smooth at the neighborhood of , with . If is the space curve defined by then we call the boundary of , written .
With the above notation, if is any smooth vector field on , then
Here, the "" represents the dot product in .
Proof
The proof of the theorem consists of 4 steps. We assume Green's theorem, so what is of concern is how to boil down the three-dimensional complicated problem (Stokes' theorem) to a two-dimensional rudimentary problem (Green's theorem). When proving this theorem, mathematicians normally deduce it as a special case of a more general result, which is stated in terms of differential forms, and proved using more sophisticated machinery. Whi |
https://en.wikipedia.org/wiki/Vladislav%20Kadyrov | Vladislav Azizovich Kadyrov (; ; born 16 November 1970) is an Azerbaijani professional football coach and a former player. He also holds Russian citizenship.
Managerial statistics
Honours
1987 FIFA U-16 World Championship champion for USSR.
USSR Federation Cup finalist: 1988.
Russian Second League top scorer: 1994 (Zone East, 34 goals).
External links
Career summary by KLISF
1970 births
Living people
Russian men's footballers
Soviet men's footballers
Azerbaijani men's footballers
Azerbaijani football managers
Azerbaijan men's international footballers
Azerbaijani expatriate men's footballers
FC Arsenal Tula players
Footballers from Baku
Mughan FK managers
Men's association football midfielders
FC Lokomotiv Nizhny Novgorod players
Neftçi PFK players
Russian people of Chechen descent
Azerbaijani people of Chechen descent
Ravan Baku FK managers
FC Spartak Ryazan players
FC Akademiya Tolyatti players
FC Portovik-Energiya Kholmsk players |
https://en.wikipedia.org/wiki/Sliced%20inverse%20regression | Sliced inverse regression (or SIR) is a tool for dimensionality reduction in the field of multivariate statistics.
In statistics, regression analysis is a method of studying the relationship between a response variable y and its input variable , which is a p-dimensional vector. There are several approaches in the category of regression. For example, parametric methods include multiple linear regression, and non-parametric methods include local smoothing.
As the number of observations needed to use local smoothing methods scales exponentially with high-dimensional data (as p grows), reducing the number of dimensions can make the operation computable. Dimensionality reduction aims to achieve this by showing only the most important dimension of the data. SIR uses the inverse regression curve, , to perform a weighted principal component analysis.
Model
Given a response variable and a (random) vector of explanatory variables, SIR is based on the model
where are unknown projection vectors, is an unknown number smaller than , is an unknown function on as it only depends on arguments, and is a random variable representing error with and a finite variance of . The model describes an ideal solution, where depends on only through a dimensional subspace; i.e., one can reduce the dimension of the explanatory variables from to a smaller number without losing any information.
An equivalent version of is: the conditional distribution of given depends on only through the dimensional random vector . It is assumed that this reduced vector is as informative as the original in explaining .
The unknown are called the effective dimension reducing directions (EDR-directions). The space that is spanned by these vectors is denoted by the effective dimension reducing space (EDR-space).
Relevant linear algebra background
Given , then , the set of all linear combinations of these vectors is called a linear subspace and is therefore a vector space. The equation says that vectors span , but the vectors that span space are not unique.
The dimension of is equal to the maximum number of linearly independent vectors in . A set of linear independent vectors of makes up a basis of . The dimension of a vector space is unique, but the basis itself is not. Several bases can span the same space. Dependent vectors can still span a space, but the linear combinations of the latter are only suitable to a set of vectors lying on a straight line.
Inverse regression
Computing the inverse regression curve (IR) means instead of looking for
, which is a curve in
it is actually
, which is also a curve in , but consisting of one-dimensional regressions.
The center of the inverse regression curve is located at . Therefore, the centered inverse regression curve is
which is a dimensional curve in .
Inverse regression versus dimension reduction
The centered inverse regression curve lies on a -dimensional subspace spanned by . This is a connection between the |
https://en.wikipedia.org/wiki/List%20of%20Northampton%20Town%20F.C.%20records%20and%20statistics | This page details Northampton Town Football Club records.
Honours
League
English 2nd Tier
Runners-up: 1964–65
English 3rd Tier
Champions: 1962–63
Runners-up: 1927–28, 1949–50
English 4th Tier
Champions: 1986–87, 2015–16
Runners-up: 1975–76, 2005–06
Promoted: 1960–61, 1999–2000, 2022–23
Promoted as Play-off Winners: 1996–97, 2019–20
Southern Football League
Champions: 1908–09
Runners-up: 1910–11
Cups
Charity Shield
Runners-up: 1909
FA Cup:
Best: Round 5 1969–70
Football League Cup:
Best: Quarter-final 1964–65, 1966–67
Football League Trophy:
Best: Area Semi-final 1996–97, 2003–04
Team records
Record win:
11-1 v Southend United (H), Southern League 1909/10
Record defeat:
0-11 v Southampton (A), Southern League, 1901/02
Most points gained in a season:
99 (1986/87, Football League Fourth Division), (2015/16, Football League Two)
Fewest points gained in a season:
19 (1906/07, Southern League)
Most goals scored in a season:
109: 1952-53 (Football League Third Division South) and 1962-63 (Football League Third Division)
Appearances
Most appearances
All-time most appearances (Does not include wartime appearances)
Current players in bold.
Most appearances – 552 by Tommy Fowler (1946–1961)
Most league appearances – 521 by Tommy Fowler (1946–1961)
Most appearances in the first tier (Premier League and predecessors) – 42 by Joe Kiernan
Most appearances in the second tier (Championship and predecessors) – 111 by Terry Branston
Most appearances in the third tier (League One and predecessors) – 138 by Peter Gleasure
Most appearances in the fourth tier (League Two and predecessors) – 521 by Tommy Fowler
Most FA Cup appearances – 31 by Tommy Fowler
Most League Cup appearances – 25 by Joe Kiernan and Peter Gleasure
Most appearances in a single season – 58 by John Frain (45 in FL, 3 PO, 5 FAC, 2 FLC, 3 AWS – 1997–98)
Youngest and oldest appearances
Longest Spell at club – 15 years by Tommy Fowler (1946–1961)
Youngest first-team player – 15 years 336 days by Josh Tomlinson (v Brighton & Hove Albion U23's, 3 November 2021)
Oldest first-team player – 43 years 42 days by Lloyd Davies (v Exeter City, 3 April 1920)
Goalscorers
Top goalscorers
Top 20 all-time top goalscorers (Does not include wartime appearances.)
Current players in bold.
Most goals – 143 by Jack English
Most league goals – 135 by Jack English
Most goals in the first tier (Premier League and predecessors) – 9 by Bobby Brown
Most goals in the second tier (Championship and predecessors) – 33 by Don Martin
Most goals in the third tier (League One and predecessors) – 50 by Cliff Holton
Most goals in the fourth tier (League Two and predecessors) – 135 by Jack English
Most FA Cup goals – 12 by Albert Dawes
Most League Cup goals – 11 by Don Martin
Top goalscorers in individual matches and seasons
Most goals scored in a single season – 39 by Cliff Holton (1961–62)
Most goals scored in a season in the first tier (Premier League and predecessors) – 9 by Bobby Brown (1965–66) |
https://en.wikipedia.org/wiki/George%20James%20Lidstone | George James Lidstone FIA FSA FRSE (1870–1952) was a British actuary who made several contributions to the field of statistics. He is known for Lidstone smoothing and Lidstone series. He served as President of the Faculty of Actuaries from 1924 to 1926.
Life
He was born in London on 11 December 1870, the youngest of five children of Eliza Munnings and her husband, William Thompson Lidstone from Devon.
He was educated at Birkbeck School in Clapton. He qualified as an actuary in 1891 and began working for the Alliance Assurance Company in 1893. He was promoted rapidly and ended as Secretary of The Equitable Life Assurance Society in 1905.
In 1913 he moved to Edinburgh in Scotland as Manager and Actuary of the Scottish Widows Fund. In 1918 he was elected a Fellow of the Royal Society of Edinburgh. His proposers were Sir Edmund Taylor Whittaker, George MacRitchie Low, John Horne and Cargill Gilston Knott.
The University of Edinburgh granted him an honorary doctorate (LLD) in 1925.
From 1929 his health failed and he stepped down from most active roles. He also went totally blind. He died in Edinburgh on 12 May 1952.
Family
His wife, Florence Mary Gay, died in 1942. They lived at Hermiston House, west of Edinburgh.
After his wife's death he moved to 23 Wester Coates Avenue, in Edinburgh's West End.
References
Notes
1870 births
1952 deaths
British actuaries
Fellows of the Royal Society of Edinburgh
People from London |
https://en.wikipedia.org/wiki/Roger%20Kortko | Roger Kortko (born February 1, 1963) is a former professional ice hockey player who played 79 games in the National Hockey League. He played with the New York Islanders.
Career statistics
References
1963 births
Canadian ice hockey centres
New York Islanders draft picks
New York Islanders players
Living people
People from Rosthern, Saskatchewan
Ice hockey people from Saskatchewan
Saskatoon Blades players
Springfield Indians players
Binghamton Whalers players
Tilburg Trappers players
Indianapolis Checkers (CHL) players
EV Füssen players |
https://en.wikipedia.org/wiki/Mike%20Posavad | Mike Posavad (born January 3, 1964) is a Canadian former professional ice hockey player who played eight games in the National Hockey League for the St. Louis Blues.
Career statistics
External links
1964 births
Canadian ice hockey defencemen
Living people
Peoria Rivermen (IHL) players
Peterborough Petes (ice hockey) players
Salt Lake Golden Eagles (CHL) players
Ice hockey people from Brantford
St. Louis Blues draft picks
St. Louis Blues players |
https://en.wikipedia.org/wiki/Septic%20equation | In algebra, a septic equation is an equation of the form
where .
A septic function is a function of the form
where . In other words, it is a polynomial of degree seven. If , then f is a sextic function (), quintic function (), etc.
The equation may be obtained from the function by setting .
The coefficients may be either integers, rational numbers, real numbers, complex numbers or, more generally, members of any field.
Because they have an odd degree, septic functions appear similar to quintic or cubic function when graphed, except they may possess additional local maxima and local minima (up to three maxima and three minima). The derivative of a septic function is a sextic function.
Solvable septics
Some seventh degree equations can be solved by factorizing into radicals, but other septics cannot. Évariste Galois developed techniques for determining whether a given equation could be solved by radicals which gave rise to the field of Galois theory. To give an example of an irreducible but solvable septic, one can generalize the solvable de Moivre quintic to get,
,
where the auxiliary equation is
.
This means that the septic is obtained by eliminating and between , and .
It follows that the septic's seven roots are given by
where is any of the 7 seventh roots of unity. The Galois group of this septic is the maximal solvable group of order 42. This is easily generalized to any other degrees , not necessarily prime.
Another solvable family is,
whose members appear in Kluner's Database of Number Fields. Its discriminant is
The Galois group of these septics is the dihedral group of order 14.
The general septic equation can be solved with the alternating or symmetric Galois groups or . Such equations require hyperelliptic functions and associated theta functions of genus 3 for their solution. However, these equations were not studied specifically by the nineteenth-century mathematicians studying the solutions of algebraic equations, because the sextic equations' solutions were already at the limits of their computational abilities without computers.
Septics are the lowest order equations for which it is not obvious that their solutions may be obtained by superimposing continuous functions of two variables. Hilbert's 13th problem was the conjecture this was not possible in the general case for seventh-degree equations. Vladimir Arnold solved this in 1957, demonstrating that this was always possible. However, Arnold himself considered the genuine Hilbert problem to be whether for septics their solutions may be obtained by superimposing algebraic functions of two variables (the problem still being open).
Galois groups
Septic equations solvable by radicals have a Galois group which is either the cyclic group of order 7, or the dihedral group of order 14 or a metacyclic group of order 21 or 42.
The Galois group (of order 168) is formed by the permutations of the 7 vertex labels which preserve the 7 "lines" in the Fano plane. S |
https://en.wikipedia.org/wiki/Constructive%20nonstandard%20analysis | In mathematics, constructive nonstandard analysis is a version of Abraham Robinson's nonstandard analysis, developed by Moerdijk (1995), Palmgren (1998), Ruokolainen (2004). Ruokolainen wrote:
The possibility of constructivization of nonstandard analysis was studied by Palmgren (1997, 1998, 2001). The model of constructive nonstandard analysis studied there is an extension of Moerdijk’s (1995) model for constructive nonstandard arithmetic.
See also
Smooth infinitesimal analysis
John Lane Bell
References
Ieke Moerdijk, A model for intuitionistic nonstandard arithmetic, Annals of Pure and Applied Logic, vol. 73 (1995), pp. 37–51.
"Abstract: This paper provides an explicit description of a model for intuitionistic nonstandard arithmetic, which can be formalized in a constructive metatheory without the axiom of choice."
Erik Palmgren, Developments in Constructive Nonstandard Analysis, Bulletin of Symbolic Logic Volume 4, Number 3 (1998), 233–272.
"Abstract: We develop a constructive version of nonstandard analysis, extending Bishop's constructive analysis with infinitesimal methods. ..."
Juha Ruokolainen 2004, Constructive Nonstandard Analysis Without Actual Infinity
Constructivism (mathematics)
Nonstandard analysis |
https://en.wikipedia.org/wiki/Wilf%20Edwards | Wilfred James Edwards (12 August 1905 – 1976) was a footballer who played in the Football League for Crewe Alexandra and Lincoln City. He was born in Fenton, England.
Career statistics
References
1905 births
1976 deaths
People from Fenton, Staffordshire
English men's footballers
Men's association football outside forwards
Crewe Alexandra F.C. players
Stoke City F.C. players
Stafford Rangers F.C. players
Burton Town F.C. players
Loughborough Corinthians F.C. players
Lincoln City F.C. players
English Football League players |
https://en.wikipedia.org/wiki/William%20Wood%20%28footballer%2C%20born%201900%29 | William Wood (born 1900, date of death unknown) was a footballer who played in the Football League for Aberdare Athletic and Stoke. He was born in Stoke-on-Trent, England.
Career statistics
References
English men's footballers
Aberdare Athletic F.C. players
Stoke City F.C. players
English Football League players
1900 births
Year of death missing
Men's association football outside forwards
Footballers from Stoke-on-Trent |
https://en.wikipedia.org/wiki/Takumi%20Yamada | is a Japanese football player who plays for Montedio Yamagata.
Career statistics
Updated to 26 July 2022.
References
External links
Profile at Montedio Yamagata
1989 births
Living people
Association football people from Tokyo
Japanese men's footballers
J1 League players
J2 League players
Montedio Yamagata players
Men's association football defenders |
https://en.wikipedia.org/wiki/Amitai%20Regev | Amitai Regev (born December 7, 1940) is an Israeli mathematician, known for his work in ring theory.
He is the Herman P. Taubman Professor of Mathematics at the Weizmann Institute of Science. He received his doctorate from the Hebrew University of Jerusalem in 1972, under the direction of Shimshon Amitsur.
Regev has made significant contributions to the theory of polynomial identity rings (PI rings). In particular, he proved Regev's theorem that the tensor product of two PI rings is again a PI ring. He developed so-called "Regev theory" that connects PI rings to representations of the symmetric group, and hence to Young tableaux. He has made seminal contributions to the asymptotic enumeration of Young tableaux and tableaux of hook shape, and together with William Beckner proved the Macdonald-Selberg conjecture for the infinite Lie algebras of type B, C, and D.
References
External links
Amitai Regev's homepage
1940 births
Living people
Jewish scientists
20th-century Israeli mathematicians
21st-century Israeli mathematicians |
https://en.wikipedia.org/wiki/Sadratnamala | Sadratnamala is an astronomical-mathematical treatise in Sanskrit written by Sankara Varman, an astronomer-mathematician of the Kerala school of mathematics, in 1819. Even though the book has been written at a time when western mathematics and astronomy had been introduced in India, it is composed purely in the traditional style followed by the mathematicians of the Kerala school. Sankara Varman has also written a detailed commentary on the book in Malayalam.
Sadratnamala is one of the books cited in C. M. Whish's paper on the achievements of the Kerala school of mathematics. This paper published in the Transactions of the Royal Asiatic Society of Great Britain and Ireland in 1834, was the first ever attempt to bring the accomplishments of Keralese mathematicians to the attention of Western mathematical scholarship.
Whish wrote in his paper thus: "The author of Sadratnamalah is SANCARA VARMA, the younger brother of the present Raja of Cadattanada near Tellicherry, a very intelligent man and acute mathematician. This work, which is a complete system of Hindu astronomy, is comprehended in two hundred and eleven verses of different measures, and abounds with fluxional forms and series, to be found in no work of foreign or other Indian countries."
Synopsis of the book
The book contains 212 verses divided into six chapters, called prakarana-s.
Chapter 1: Gives the names of numerals; defines the eight operations of addition, subtraction, multiplication, division, squaring, extracting square root, cubing, and extracting cube root.
Chapter 2: Lists the different measures, namely, the measures of time, angles, lunar days, planets and stars, almanacs, length, grain weight, money and the directions.
Chapter 3: Defines the rule of three and syllabic enumeration; explains methods for the computation of the elements of the almanac, namely, mean and true sun, moon and planets, lunar day, yoga and karana; gives methods for determining the time elapsed after sunrise and after sunset.
Chapter 4: Deals with arcs and sines and its application in astronomical measurements and computations.
Chapter 5: Deals with computations relating to the shadow, eclipse, vyatipata, retrograde motion of the planets and apses of the moon.
Chapter 6: Explains the necessity of periodic revision of astronomical constants; gives a full description of parahita-karana.
Sankara Varman (1774–1839)
Sankara Varman, author of Sadratnamala, was born as a younger prince in the principality of Katathanad in the North Malabar in Kerala. To the local people he was known as Appu Thampuran. The date of birth of Sankara Varman is still uncertain. There are some strong arguments in favour of the year 1774 CE. Sankara Varman died in 1839 CE.
References
Kerala school of astronomy and mathematics
Hindu astronomy
Astronomy books
Indian mathematics
Indian astronomy texts |
https://en.wikipedia.org/wiki/Irakli%20Vashakidze | Irakli Vashakidze (born 13 March 1976) is a Georgian former professional football player.
Career statistics
Achievements
Dinamo Tbilisi
Umaglesi Liga: 1998–99, 2002–03
Georgian Cup: 2003
Georgian Super Cup: 1999
Torpedo Kutaisi
Umaglesi Liga: 2000–01
Georgian Cup: 2000–01
References
External links
1976 births
Living people
Men's footballers from Georgia (country)
Georgia (country) men's international footballers
Men's association football defenders
FC Dinamo Tbilisi players
Aris Thessaloniki F.C. players
Turan Tovuz players
Gabala SC players
Expatriate men's footballers from Georgia (country)
Expatriate men's footballers in Greece
Expatriate men's footballers in Azerbaijan
Expatriate sportspeople from Georgia (country) in Azerbaijan
Footballers from Tbilisi |
https://en.wikipedia.org/wiki/Tornike%20Aptsiauri | Tornike Aptsiauri (; born 29 November 1979 in Tbilisi) is a Georgian professional football player.
Career statistics
Achievements
Dinamo-2 Tbilisi
Pirveli Liga
Winner (1): 1998–99 Regionuli Liga (East, B zone)
Olimpi Rustavi
Umaglesi Liga
Winner (1): 2006–07
Zestafoni
Umaglesi Liga
Winner (2): 2010–11, 2011–12
Georgian Cup
Runner-Up (1): 2011–12
Georgian Super Cup
Winner (2): 2011, 2012
References
External links
1979 births
Living people
Men's footballers from Georgia (country)
Footballers from Tbilisi
Expatriate men's footballers from Georgia (country)
Expatriate men's footballers in Azerbaijan
Georgia (country) men's international footballers
FC Locomotive Tbilisi players
FC Tbilisi players
FC Merani Tbilisi players
FC Dinamo Tbilisi players
FC Borjomi players
FC Metalurgi Rustavi players
Gabala SC players
Expatriate sportspeople from Georgia (country) in Azerbaijan
FC Zestafoni players
FC Sioni Bolnisi players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Giorgi%20Gabidauri | Giorgi Gabidauri (; born 6 December 1979) is a Georgian professional football player.
Career statistics
References
External links
1979 births
Living people
Jews from Georgia (country)
Emigrants from Georgia (country) to Israel
Men's footballers from Georgia (country)
FC Dinamo Tbilisi players
Anorthosis Famagusta FC players
Panachaiki F.C. players
FC Shakhter Karagandy players
Georgia (country) men's international footballers
Gabala SC players
Hapoel Bnei Lod F.C. players
Maccabi Umm al-Fahm F.C. players
Liga Leumit players
Cypriot First Division players
Expatriate men's footballers from Georgia (country)
Expatriate men's footballers in Cyprus
Expatriate men's footballers in Greece
Expatriate men's footballers in Kazakhstan
Expatriate men's footballers in Azerbaijan
Expatriate sportspeople from Georgia (country) in Cyprus
Expatriate sportspeople from Georgia (country) in Greece
Expatriate sportspeople from Georgia (country) in Kazakhstan
Expatriate sportspeople from Georgia (country) in Azerbaijan
Men's association football midfielders |
https://en.wikipedia.org/wiki/Herbert%20Busemann | Herbert Busemann (12 May 1905 – 3 February 1994) was a German-American mathematician specializing in convex and differential geometry. He is the author of Busemann's theorem in Euclidean geometry and geometric tomography. He was a member of the Royal Danish Academy and a winner of the Lobachevsky Medal (1985), the first American mathematician to receive it. He was also a Fulbright scholar in New Zealand in 1952.
Biography
Herbert Busemann was born in Berlin to a well-to-do family. His father, Alfred Busemann, was a director of Krupp, where Busemann also worked for several years. He studied at University of Munich, Paris, and Rome. He defended his dissertation in University of Göttingen in 1931, where his advisor was Richard Courant. He remained in Göttingen as an assistant until 1933, when he escaped Nazi Germany to Copenhagen (he had a Jewish grandfather). He worked at the University of Copenhagen until 1936, when he left to the United States. There, he got married in 1939 and naturalized in 1943. He had temporary positions at the Institute for Advanced Study, Johns Hopkins University, Illinois Institute of Technology, Smith College, and eventually became a professor in 1947 at University of Southern California. He advanced to a distinguished professor in 1964, and continued working at USC until his retirement in 1970. Over the course of his work at USC, he supervised over 10 Ph.D. students.
He is the author of six monographs, two of which were translated into Russian.
He received Lobachevsky Medal in 1985 for his book The geometry of geodesics.
Busemann was also an active mathematical citizen. At different times, he was the president of the California chapter of Mathematical Association of America, and a member of the council of the American Mathematical Society.
Busemann was also an accomplished linguist; he was able to read and speak in French, German, Spanish, Italian, Russian, and Danish. He could also read Arabic, Latin, Greek and Swedish. He translated a number of papers and monograph, most notably from Russian, a rare language at the time. He was also an accomplished artist and had several public exhibitions of his Hard-edge paintings]. He died in Santa Ynez, California on February 3, 1994, at the age of 88.
Busemann's Selected papers are now available in two volumes (908 and 842 pages), with introductory biographical material and commentaries on his work, and published by edited by Athanase Papadopoulos, Springer Verlag, 2018.
Books
Herbert Busemann, Selected Works, (Athanase Papadopoulos, ed.) Volume I, , XXXII, 908 p., Springer International Publishing, 2018.
Herbert Busemann, Selected Works, (Athanase Papadopoulos, ed.) Volume II, , XXXV, 842 p., Springer International Publishing, 2018.
Introduction to algebraic manifolds, Princeton University Press, 1939.
with Paul J. Kelly: Projective geometry and projective metrics, Academic Press, 1953, Dover 2006.
Convex Surfaces, Interscience 1958, Dover, 2008.
Geo |
https://en.wikipedia.org/wiki/Richard%20S.%20Varga | Richard Steven Varga (October 9, 1928 - February 25, 2022) was an American mathematician who specialized in numerical analysis and linear algebra. He was an Emeritus University Professor of Mathematical Sciences at Kent State University and an adjunct Professor at Case Western Reserve University. Varga was known for his contributions to many areas of mathematics, including matrix analysis, complex analysis, approximation theory, and scientific computation. He was the author of the classic textbook Matrix Iterative Analysis. Varga served as the Editor-in-Chief of the journal Electronic Transactions on Numerical Analysis (ETNA).
Birth and education
Richard Varga was born in Cleveland, Ohio of Hungarian-born parents in 1928. He obtained a bachelor's degree in mathematics from Case Institute of Technology (present Case Western Reserve University) in 1950. Varga was a member of the collegiate wrestling team of Case.
Following the advice of Professor Max Morris at Case, Varga joined Harvard University for the master's degree and obtained an A.M. in mathematics. Continuing his doctoral work at Harvard under the supervision of Joseph L. Walsh, Varga worked on the theory of rational approximation of complex analytic functions. Varga received his Ph.D. degree in 1954 with a dissertation Properties of a Special Set of Entire Functions and their Respective Partial Sums.
While at Harvard, Varga also studied with Garrett Birkhoff, who later came to collaborate with Varga in research both on iterative methods for differential equations and on positive matrices (and positive operators on partially ordered vector spaces).
Career
From 1954 until 1960, Varga worked for Bettis Atomic Power Laboratory in Pittsburgh. In 1960 he returned to Case Institute of Technology as a professor of mathematics and remained there for the next nine years. He then moved to Kent State University as University Professor of mathematics. At Kent Varga has held numerous academic positions, including director (1980–1988) and research director (1988–2006) of the Institute for Computational Mathematics. His work includes numerical analysis—particularly iterative methods in numerical linear algebra, matrix theory, and differential equations—complex approximation theory, particularly Padé approximation (often with Edward B. Saff, Jr.)—and analytic number theory, including high-precision calculations related to the Riemann hypothesis. He is also known for advocating experimentation in mathematics, and for writing a monograph surveying his contributions on scientific computing to resolve open problems and conjectures.
Awards and honors
In 2012 he became a fellow of the American Mathematical Society.
See also
Matrix splitting
References
External links
1928 births
2022 deaths
Numerical analysts
Approximation theorists
Mathematical analysts
Number theorists
20th-century American mathematicians
21st-century American mathematicians
Kent State University faculty
Case Western Reserve Universi |
https://en.wikipedia.org/wiki/Noncommutative%20algebraic%20geometry | Noncommutative algebraic geometry is a branch of mathematics, and more specifically a direction in noncommutative geometry, that studies the geometric properties of formal duals of non-commutative algebraic objects such as rings as well as geometric objects derived from them (e.g. by gluing along localizations or taking noncommutative stack quotients).
For example, noncommutative algebraic geometry is supposed to extend a notion of an algebraic scheme by suitable gluing of spectra of noncommutative rings; depending on how literally and how generally this aim (and a notion of spectrum) is understood in noncommutative setting, this has been achieved in various level of success. The noncommutative ring generalizes here a commutative ring of regular functions on a commutative scheme. Functions on usual spaces in the traditional (commutative) algebraic geometry have a product defined by pointwise multiplication; as the values of these functions commute, the functions also commute: a times b equals b times a. It is remarkable that viewing noncommutative associative algebras as algebras of functions on "noncommutative" would-be space is a far-reaching geometric intuition, though it formally looks like a fallacy.
Much of the motivation for noncommutative geometry, and in particular for the noncommutative algebraic geometry, is from physics; especially from quantum physics, where the algebras of observables are indeed viewed as noncommutative analogues of functions, hence having the ability to observe their geometric aspects is desirable.
One of the values of the field is that it also provides new techniques to study objects in commutative algebraic geometry such as Brauer groups.
The methods of noncommutative algebraic geometry are analogs of the methods of commutative algebraic geometry, but frequently the foundations are different. Local behavior in commutative algebraic geometry is captured by commutative algebra and especially the study of local rings. These do not have a ring-theoretic analogue in the noncommutative setting; though in a categorical setup one can talk about stacks of local categories of quasicoherent sheaves over noncommutative spectra. Global properties such as those arising from homological algebra and K-theory more frequently carry over to the noncommutative setting.
History
Classical approach: the issue of non-commutative localization
Commutative algebraic geometry begins by constructing the spectrum of a ring. The points of the algebraic variety (or more generally, scheme) are the prime ideals of the ring, and the functions on the algebraic variety are the elements of the ring. A noncommutative ring, however, may not have any proper non-zero two-sided prime ideals. For instance, this is true of the Weyl algebra of polynomial differential operators on affine space: The Weyl algebra is a simple ring. Therefore, one can for instance attempt to replace a prime spectrum by a primitive spectrum: there are also the theory of non |
https://en.wikipedia.org/wiki/Partition%20cardinal | In mathematics, a partition cardinal is either:
An Erdős cardinal; or
A Strong partition cardinal. |
https://en.wikipedia.org/wiki/Olho%20d%27%C3%81gua | Olho d'Água is a municipality in the state of Paraíba in the Northeast Region of Brazil.
Demographics
According to the 2020 IBGE statistics, 6,462 people live in this municipality.
Attractions
In the Recanto Ecológico Rio da Prata wildlife park, the Olho D'Água river gets flooded every seven years, submerging the local hiking trails. This has become an attractive venue for divers.
See also
List of municipalities in Paraíba
References
Municipalities in Paraíba |
https://en.wikipedia.org/wiki/Irau%C3%A7uba | Irauçuba is a municipality in the state of Ceará in the Northeast region of Brazil.
According to the resident population estimate made by the Brazilian Institute of Geography and Statistics IBGE in 2020, its population was 24,305 inhabitants.
Geography
The climate is tropical. Summer has much less rainfall than the winter. The climate classification is according to Köppen and Geiger. The average temperature is 26.3 °C in Irauçuba. Average annual rainfall of 629 semi-arid hot mm. Tropical with average rainfall of 539 mm, with rainfall concentrated 12 January to April 13. Irauçuba has one of medium pluviometrias smaller state. The city is beating too desertification key Irauçuba already have been quite committed to desertification.
See also
List of municipalities in Ceará
References
Municipalities in Ceará |
https://en.wikipedia.org/wiki/Uruburetama | Uruburetama is a municipality in the state of Ceará in the Northeast region of Brazil. Its population, according to the census of the Brazilian Institute of Geography and Statistics, IBGE, 2010, is 19,765 people.
Toponymy
"Uruburetama" is a word derived from the Tupi-Guaraní language which means "land of the crows," by combining the terms uru'bu (crow, vulture, buzzard) and retama (land).
History
The municipality is located in the former 'sesmaria' (colonial land allotment) granted to the Captain General Bento Coelho de Morais on November 19, 1720.
These lands were donated to the Father Estevão Velho Cabral de Melo for priestly heritage by Manuel Pereira Pinto, a lieutenant colonel, who received the inheritance of the Captain General Bento Coelho de Morais, his father in law.
In 1761 came the first time, the toponym "Sítio Arraial" (Ranch Hamlet), in a document that Father Estevão reverted the land to its donors, reserving for himself only a quarter of a league.
In 1878, the Fathers João Francisco Dias Nogueira and José Tomaz Albuquerque concluded the current main church thanks to donations from the people.
On 1 August 1890, Decree 34 the settlement was elevated to town with the toponym of St. John of Arraial. But the following year, by court term, the council was dissolved and attached to the municipalities of San Francisco (now Itapajé) and Itapipoca.
On July 28, 1899, through Law 526, the village was restored with the name of St. John Uruburetama.
The village was elevated to city status under the name of Arraial in 1931, under the State Decree 262 of 28 July 1931. However, the name was replaced by Uruburetama in 1938. At the time, the council was made up of districts: Uruburetama (headquarters), Curu (now São Luís do Curu), Natavidade (Cemoaba), Riachuelo (now Umirim) and Tururu, all independent today.
Currently the municipality consists of the districts Uruburetama, Santa Luzia, Itacolomy, Retiro, Severino, Canto Escuro, Bananal and Tamboatá.
Geographical
It is located in the coastal region of the state, with a distance of about 111 km, in line straight from the state capital Fortaleza.
Cities surrounding Uruburetama
The diagram below show the cities 40 km near Uruburetama.
See also
List of municipalities in Ceará
References
Municipalities in Ceará |
https://en.wikipedia.org/wiki/So%20Wai%20Chuen | So Wai Chuen (born 26 March 1988) is a former Hong Kong professional footballer. He played as a centre-back.
Career statistics
International
Hong Kong U-23
As of 15 November 2010
Hong Kong
As of 28 July 2011
Notes and references
1988 births
Living people
Hong Kong men's footballers
Sun Hei SC players
Hong Kong First Division League players
Hong Kong Premier League players
Eastern Sports Club footballers
Hong Kong Pegasus FC players
Hong Kong men's international footballers
Footballers at the 2010 Asian Games
Men's association football central defenders
Asian Games competitors for Hong Kong |
https://en.wikipedia.org/wiki/Sotan%20Tanabe | is a Japanese footballer who plays for Avispa Fukuoka as a right winger.
Career statistics
Club
Last updated in the end of 2018 season
1 Includes Emperor's Cup and Copa del Rey.
2Includes J.League Cup.
3Includes AFC Champions League.
Honours
Club
FC Tokyo
J.League Division 2 (1) : 2011
Emperor's Cup (1) : 2011
J.League Cup (1) : 2009
Suruga Bank Championship (1) : 2010
See also
List of Japanese footballers playing in Europe
References
External links
FC Tokyo official profile
1990 births
Living people
Association football people from Tokyo
Japanese men's footballers
J1 League players
J2 League players
J3 League players
Segunda División players
FC Tokyo players
FC Tokyo U-23 players
CE Sabadell FC footballers
Avispa Fukuoka players
Japanese expatriate men's footballers
Japanese expatriate sportspeople in Spain
Expatriate men's footballers in Spain
Men's association football midfielders |
https://en.wikipedia.org/wiki/Argentia%20Beach | Argentia Beach is a summer village in Alberta, Canada. It is located on the northern shore of Pigeon Lake.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, the Summer Village of Argentia Beach had a population of 39 living in 25 of its 101 total private dwellings, a change of from its 2016 population of 27. With a land area of , it had a population density of in 2021.
In the 2016 Census of Population conducted by Statistics Canada, the Summer Village of Argentia Beach had a population of 27 living in 17 of its 100 total private dwellings, an increase of from its 2011 population of 15. With a land area of , it had a population density of in 2016.
See also
List of communities in Alberta
List of summer villages in Alberta
List of resort villages in Saskatchewan
References
External links
1967 establishments in Alberta
Summer villages in Alberta |
https://en.wikipedia.org/wiki/Ieke%20Moerdijk | Izak (Ieke) Moerdijk (; born 23 January 1958) is a Dutch mathematician, currently working at Utrecht University, who in 2012 won the Spinoza prize.
Education and career
Moerdijk studied mathematics, philosophy and general linguistics at the University of Amsterdam. He obtained his PhD cum laude in 1985 at the same institution. His thesis was entitled Topics in intuitionism and topos theory and was written under the supervision of Anne Sjerp Troelstra.
After that, he worked as postdoctoral researcher at the University of Chicago and Cambridge. From 1988 to 2011 he was professor at Utrecht University. After working at the Mathematical Institute of the Radboud University Nijmegen for a few years, he returned to Utrecht University in 2016.
In 2000 Moerdijk was an invited speaker to the 3rd European Congress of Mathematics. He was elected member of the Royal Netherlands Academy of Arts and Sciences in 2006 and of the Academia Europaea in 2014.
Moerdijk received the 2011 Descartes-Huygens prize for his contribution to French–Dutch scientific collaborations from the Académie des Sciences in Paris. In 2012 he received the Spinoza prize from the Netherlands Organisation for Scientific Research.
Research
Moerdijk's research interests lie in the fields of category theory, algebraic and differential topology, and their applications to mathematical logic.
Moerdijk is seen, together with André Joyal, as one of the founders of algebraic set theory. In 1992 he wrote, together with Saunders Mac Lane, a book on topos theory that became the standard reference on the subject: Sheaves in geometry and logic. A first introduction to topos theory. In 1995 he made pioneering contributions to constructive non-standard analysis, of which he is one of the founders.
Moerdijk's research has also covered topics in differential geometry; in particular, he wrote in 2003 an influential monograph on foliations and Lie groupoids. Recently Moerdijk pursues, among other topics, research on the theory of operads, on the logic structure of quantum information theory, and on dendroidal sets.
Moerdijk has written more than a hundred publications and is the author of several influential books. He supervised 19 PhD students as of 2021.
Selected books
Joyal, André; Moerdijk, Ieke (1995) Algebraic set theory. London Mathematical Society Lecture Note Series, 220. Cambridge University Press, Cambridge.
Mac Lane, Saunders; Moerdijk, Ieke (1994) Sheaves in geometry and logic. A first introduction to topos theory. Corrected reprint of the 1992 edition. Universitext. Springer-Verlag, New York, 1994.
Moerdijk, Ieke.; Mrčun, Janez (2003) Introduction to foliations and Lie groupoids. Cambridge Studies in Advanced Mathematics, 91. Cambridge University Press, Cambridge.
Moerdijk, Ieke; Reyes, Gonzalo E. (1991) Models for smooth infinitesimal analysis. Springer-Verlag, New York.
Moerdijk, Ieke, Classifying spaces and classifying topoi, Lecture Notes in Mathematics 1616, Springer 1995. vi |
https://en.wikipedia.org/wiki/Kyohei%20Noborizato | is a Japanese footballer who plays as a left back for Kawasaki Frontale.
Career statistics
Club
Honours
Japan
Asian Games (1) : 2010
Club
Kawasaki Frontale
J1 League (4) : 2017, 2018, 2020, 2021
J.League Cup (1) : 2019
Japanese Super Cup (1) : 2019
Individual
J.League Best XI (1) : 2020
References
External links
Profile at Kawasaki Frontale
1990 births
Living people
Association football people from Osaka Prefecture
People from Higashiōsaka
Japanese men's footballers
J1 League players
Kawasaki Frontale players
Asian Games medalists in football
Footballers at the 2010 Asian Games
Asian Games gold medalists for Japan
Men's association football defenders
Medalists at the 2010 Asian Games |
https://en.wikipedia.org/wiki/Edwards%20curve | In mathematics, the Edwards curves are a family of elliptic curves studied by Harold Edwards in 2007. The concept of elliptic curves over finite fields is widely used in elliptic curve cryptography. Applications of Edwards curves to cryptography were developed by Daniel J. Bernstein and Tanja Lange: they pointed out several advantages of the Edwards form in comparison to the more well known Weierstrass form.
Definition
The equation of an Edwards curve over a field K which does not
have characteristic 2 is:
for some scalar .
Also the following form with parameters c and d is called an Edwards curve:
where c, d ∈ K with cd(1 − c4·d) ≠ 0.
Every Edwards curve is birationally equivalent to an elliptic curve in Montgomery form, and thus admits an algebraic group law once one chooses a point to serve as a neutral element. If K is finite, then a sizeable fraction of all elliptic curves over K can be written as Edwards curves.
Often elliptic curves in Edwards form are defined having c=1, without loss of generality. In the following sections, it is assumed that c=1.
The group law
(See also Weierstrass curve group law)
Every Edwards curve over field K with characteristic not equal to 2 with is birationally equivalent to an elliptic curve over the same field: , where and the point is mapped to the infinity O. This birational mapping induces a group on any Edwards curve.
Edwards addition law
On any elliptic curve the sum of two points is given by a rational expression of the coordinates of the points, although in general one may need to use several formulas to cover all possible pairs. For the Edwards curve, taking the neutral element to be the point (0, 1), the sum of the points and is given by the formula
The opposite of any point is . The point has order 2, and the points have order 4. In particular, an Edwards curve always has a point of order 4 with coordinates in K.
If d is not a square in K and , then there are no exceptional points: the denominators and are always nonzero. Therefore, the Edwards addition law is complete when d is not a square in K. This means that the formulas work for all pairs of input points on the Edwards curve with no exceptions for doubling, no exception for the neutral element, no exception for negatives, etc. In other words, it is defined for all pairs of input points on the Edwards curve over K and the result gives the sum of the input points.
If d is a square in K, then the same operation can have exceptional points, i.e. there can be pairs of points such that one of the denominators becomes zero: either or .
One of the attractive feature of the Edwards Addition law is that it is strongly unified i.e. it can also be used to double a point, simplifying protection against side-channel attack. The addition formula above is faster than other unified formulas and has the strong property of completeness
Example of addition law :
Consider the elliptic curve in the Edwards form with d=2
and the po |
https://en.wikipedia.org/wiki/Johan%20Lindbom | Johan Stig Mikael Lindbom (born July 8, 1971) is a former Swedish professional ice hockey player who played 38 games in the National Hockey League (NHL) with the New York Rangers.
Career statistics
External links
1971 births
Living people
Hartford Wolf Pack players
HV71 players
IF Troja/Ljungby players
New York Rangers draft picks
New York Rangers players
People from Alvesta Municipality
Swedish ice hockey right wingers
Sportspeople from Kronoberg County
HV71 coaches
Swedish Hockey League coaches |
https://en.wikipedia.org/wiki/Amir%20Mohebi | Amir Mohebi, (born 24 February 1981) is an Iranian footballer who currently plays for Pas Hamedan in Iran's Premier Football League.
Club career
Club career statistics
Assist Goals
International career
In 2009, Mohebi was summoned by the Iranian Football Federation to participate in a friendly match against Iceland.
References
Iranian men's footballers
Iran men's international footballers
F.C. Aboomoslem players
Tractor S.C. players
Living people
1981 births
PAS Tehran F.C. players
Men's association football midfielders
Place of birth missing (living people)
Persepolis F.C. non-playing staff |
https://en.wikipedia.org/wiki/Algebra%20Colloquium | Algebra Colloquium is a journal founded in 1994. It was initially published by Springer-Verlag Hong Kong Ltd. In 2005, from volume 12 onwards, publishing rights were taken over by World Scientific. The company now publishes the journal quarterly.
The journal is jointly edited by the Chinese Academy of Sciences and Soochow University. The journal mainly covers the field of pure and applied algebra.
According to the Journal Citation Reports, the journal has a 2020 impact factor of 0.429.
Abstracting and indexing
Science Citation Index Expanded
Research Alert
CompuMath Citation Index
MathSciNet
Mathematical Reviews
Zentralblatt MATH
AJ VINITI (Russian)
Chinese Science Citation Index
Chinese Math Abstract
References
External links
AC Journal Website
World Scientific academic journals
Mathematics journals
Academic journals established in 1994
English-language journals |
https://en.wikipedia.org/wiki/List%20of%20Atlas%20launches%20%282010%E2%80%932019%29 |
Notable missions
USA-212
Juno
Mars Science Laboratory/Curiosity
MAVEN
OSIRIS-REx
Launch statistics
Rocket configurations
Launch sites
Launch outcomes
Launch history
See also
List of Thor and Delta launches (2010-2019)
References
Atlas |
https://en.wikipedia.org/wiki/Truth%20table | A truth table is a mathematical table used in logic—specifically in connection with Boolean algebra, boolean functions, and propositional calculus—which sets out the functional values of logical expressions on each of their functional arguments, that is, for each combination of values taken by their logical variables. In particular, truth tables can be used to show whether a propositional expression is true for all legitimate input values, that is, logically valid.
A truth table has one column for each input variable (for example, A and B), and one final column showing all of the possible results of the logical operation that the table represents (for example, A XOR B). Each row of the truth table contains one possible configuration of the input variables (for instance, A=true, B=false), and the result of the operation for those values.
A truth table is a structured representation that presents all possible combinations of truth values for the input variables of a Boolean function and their corresponding output values. A function f from A to F is a special relation, a subset of A×F, which simply means that f can be listed as a list of input-output pairs. Clearly, for the Boolean functions, the output belongs to a binary set, i.e. F = {0, 1}. For an n-ary Boolean function, the inputs come from a domain that is itself a Cartesian product of binary sets corresponding to the input Boolean variables. For example for a binary function, f(A, B), the domain of f is A×B, which can be listed as: A×B = {(A = 0, B = 0), (A = 0, B = 1), (A = 1, B = 0), (A = 1, B = 1)}. Each element in the domain represents a combination of input values for the variables A and B. These combinations now can be combined with the output of the function corresponding to that combination, thus forming the set of input-output pairs as a special relation that is subset of A×F. For a relation to be a function, the special requirement is that each element of the domain of the function must be mapped to one and only one member of the codomain. Thus, the function f itself can be listed as: f = {((0, 0), f0), ((0, 1), f1), ((1, 0), f2), ((1, 1), f3)}, where f0, f1, f2, and f3 are each Boolean, 0 or 1, values as members of the codomain {0, 1}, as the outputs corresponding to the member of the domain, respectively. Rather than a list (set) given above, the truth table then presents these input-output pairs in a tabular format, in which each row corresponds to a member of the domain paired with its corresponding output value, 0 or 1. Of course, for the Boolean functions, we do not have to list all the members of the domain with their images in the codomain; we can simply list the mappings that map the member to “1”, because all the others will have to be mapped to “0” automatically (that leads us to the minterms idea).
Ludwig Wittgenstein is generally credited with inventing and popularizing the truth table in his Tractatus Logico-Philosophicus, which was completed in 1918 and published |
https://en.wikipedia.org/wiki/Curling%20at%20the%202010%20Winter%20Olympics%20%E2%80%93%20Statistics | This is a statistical synopsis of event of Curling at the 2010 Winter Olympics.
6 of the 80 curlers at the 1998 Nagano Olympics are competing in Vancouver (plus a seventh who serves as a coach).
12 of the 100 curlers at the 2002 Salt Lake City Olympics are in Vancouver.
22 of the 100 curlers at the 2006 Turin Olympics are back in Vancouver.
Percentages
In curling, each player is graded on their shots on a scale of zero to four. Their cumulative point total is then marked as a percentage out of the total points possible. This score is just for statistical purposes, and has nothing to do with the outcome of the game.
Men's tournament
Percentages by draw.
Leads
1 Normally throws second
Seconds
1Normally throws third rocks
2Includes percentages from playing lead
3Includes percentages from playing third
Thirds
1 Normally throws last rocks
2 Normally throws second rocks
3 Includes games played as fourth
4 Includes games played as second
Skips
1 Normally throws third
2 Includes games played as third
Team totals
Women's tournament
Percentages by draw.
Leads
Seconds
1 Normally throws third
2Includes games played third
Thirds
1 Normally throws last rocks
2 Normally throws second rocks
3 Includes games played as fourth
4 Includes games played as second
Fourth
1 Normally throws third rocks
2 Includes games played as third
Team totals
Team statistics
Men's tournament
Women's tournament
Key
PF/G = Average number of points per game
PA/G = Average number of points against per game
EF/G = Average number of ends won per game
EA/G = Average number of ends lost per game
PF/E = Average number of points per end
PA/E = Average number of post against per end
BE/G = Average number of blank ends per game
HE% = Percentage of ends with last rock advantage where a team scores at least two points
SE% = Percentage of ends where a team without last rock advantage scores at least one point
FE% = Percentage of ends where a team without last rock advantage forces their opposition to just one point
Statistics |
https://en.wikipedia.org/wiki/Ceyuan%20haijing | Ceyuan haijing () is a treatise on solving geometry problems with the algebra of Tian yuan shu written by the mathematician Li Zhi in 1248 in the time of the Mongol Empire. It is a collection of 692 formula and 170 problems, all derived from the same master diagram of a round town inscribed in a right triangle and a square. They often involve two people who walk on straight lines until they can see each other, meet or reach a tree or pagoda in a certain spot. It is an algebraic geometry book, the purpose of book is to study intricated geometrical relations by algebra.
Majority of the geometry problems are solved by polynomial equations, which are represented using a method called tian yuan shu, "coefficient array method" or literally "method of the celestial unknown". Li Zhi is the earliest extant source of this method, though it was known before him in some form. It is a positional system of rod numerals to represent polynomial equations.
Ceyuan haijing was first introduced to the west by the British Protestant Christian missionary to China, Alexander Wylie in his book Notes on Chinese Literature, 1902. He wrote:
This treatise consists of 12 volumes.
Volume 1
Diagram of a Round Town
The monography begins with a master diagram called the Diagram of Round Town(圆城图式). It shows a circle inscribed in a right angle triangle and four horizontal lines, four vertical lines.
TLQ, the large right angle triangle, with horizontal line LQ, vertical line TQ and hypotenuse TL
C: Center of circle:
NCS: A vertical line through C, intersect the circle and line LQ at N(南north side of city wall), intersects south side of circle at S(南).
NCSR, Extension of line NCS to intersect hypotenuse TL at R(日)
WCE: a horizontal line passing center C, intersects circle and line TQ at W(西, west side of city wall) and circle at E (东, east side of city wall).
WCEB:extension of line WCE to intersect hypotenuse at B(川)
KSYV: a horizontal tangent at S, intersects line TQ at K(坤), hypotenuse TL at Y(月).
HEMV: vertical tangent of circle at point E, intersects line LQ at H, hypotenuse at M(山, mountain)
HSYY, KSYV, HNQ, QSK form a square, with inscribed circle C.
Line YS, vertical line from Y intersects line LQ at S(泉, spring)
Line BJ, vertical line from point B, intersects line LQ at J(夕, night)
RD, a horizontal line from R, intersects line TQ at D(旦, day)
The North, South, East and West direction in Li Zhi's diagram are opposite to our present convention.
Triangles and their sides
There are a total of fifteen right angle triangles formed by the intersection between triangle TLQ, the four horizontal lines, and four vertical lines.
The names of these right angle triangles and their sides are summarized in the following table
In problems from Vol 2 to Vol 12, the names of these triangles are used in very terse terms. For instance
"明差","MING difference" refers to the "difference between the vertical side and horizontal side of MING triangle.
"叀差","ZHUANG difference" refers |
https://en.wikipedia.org/wiki/Yigu%20yanduan | Yigu yanduan (益古演段 Old Mathematics in Expanded Sections) is a 13th-century mathematical work by Yuan dynasty mathematician Li Zhi.
Overview
Yigu yanduan was based on Northern Song mathematician Jiang Zhou's (蒋周) Yigu Ji (益古集 Collection of Old Mathematics) which is not extant. However, from fragments quoted in Yang Hui's work The Complete Algorithms of Acreage (田亩比类算法大全), this lost mathematical treatise Yigu Ji was about solving area problems with geometry.
Li Zhi used the examples of Yigu Ji to introduce the art of Tian yuan shu to newcomers to this field. Although Li Zhi's previous monograph Ceyuan haijing also used Tian yuan shu, it is harder to understand than Yigu yanduan.
Yigu yanduan was later collected into Siku Quanshu.
Yigu yanduan consists of three volumes with 64 problems solved using Tian yuan sh] in parallel with the geometrical method. Li Zhi intended to introduce students to the art of Tian yuan shu through ancient geometry. Yigu yanduan together with Ceyuan haijing are considered major contributions to Tian yuan shu by Li Zhi. These two works are also considered as the earliest extant documents on Tian yuans shu.
All the 64 problems followed more or less the same format, starting with a question (问), followed by an answer (答曰), a diagram, then an algorithm (术), in which Li Zhi explained step by step how to set up algebra equation with Tian yuan shu, then followed by geometrical interpretation (Tiao duan shu). The order of arrangement of Tian yuan shu equation in Yigu yanduan is the reverse of that in Ceyuan haijing, i.e., here with the constant term at top, followed by first order tian yuan, second order tian yuan, third order tian yuan etc. This later arrangement conformed with contemporary convention of algebra equation( for instance, Qin Jiushao's Mathematical Treatise in Nine Sections), and later became a norm.
Yigu yanduan was first introduced to the English readers by the British Protestant Christian missionary to China, Alexander Wylie who wrote:
In 1913 Van Hée translated all 64 problems in Yigu yanduan into French.
Volume I
Problem 1 to 22, all about the mathematics of a circle embedded in a square.
Example: problem 8
There is a square field, with a circular pool in the middle, given that the land is 13.75 mu, and the sum of the circumferences of the square field and the circular pool equals to 300 steps, what is the circumferences of the square and circle respective ?
Anwwer: The circumference of the square is 240 steps, the circumference of the circle is 60 steps.
Method: set up tian yuan one (celestial element 1) as the diameter of the circle, x
TAI
multiply it by 3 to get the circumference of the circle 3x (pi ~~3)
TAI
subtract this from the sum of circumferences to obtain the circumference of the square
TAI
The square of it equals to 16 times the area of the square
TAI
Again set up tian yuan 1 as the diameter of circle, square it up and multiplied by 12 to get
16 times the are |
https://en.wikipedia.org/wiki/Communications%20in%20Contemporary%20Mathematics | Communications in Contemporary Mathematics (CCM) is a journal published by World Scientific since 1999. It covers research in the fields such as applied mathematics, dynamical systems, mathematical physics, and topology.
Abstracting and indexing
The journal is indexed in Zentralblatt MATH, Mathematical Reviews, ISI Alerting Services, CompuMath Citation Index, Current Contents/Physical, Chemical and Earth Sciences, and the Science Citation Index.
References
Academic journals established in 1999
Mathematics journals
World Scientific academic journals
English-language journals |
https://en.wikipedia.org/wiki/Cosmos%20%28journal%29 | COSMOS is the scientific journal of the Singapore National Academy of Science. It is published twice annually by World Scientific and covers interdisciplinary research in Science and Mathematics.
See also
Cosmos. Problems of Biological Sciences
Cosmos magazine
References
Academic journals established in 2005
World Scientific academic journals
English-language journals
Multidisciplinary scientific journals |
https://en.wikipedia.org/wiki/Feature%20detection%20%28nervous%20system%29 | Feature detection is a process by which the nervous system sorts or filters complex natural stimuli in order to extract behaviorally relevant cues that have a high probability of being associated with important objects or organisms in their environment, as opposed to irrelevant background or noise.
Feature detectors are individual neurons—or groups of neurons—in the brain which code for perceptually significant stimuli. Early in the sensory pathway feature detectors tend to have simple properties; later they become more and more complex as the features to which they respond become more and more specific.
For example, simple cells in the visual cortex of the domestic cat (Felis catus), respond to edges—a feature which is more likely to occur in objects and organisms in the environment. By contrast, the background of a natural visual environment tends to be noisy—emphasizing high spatial frequencies but lacking in extended edges. Responding selectively to an extended edge—either a bright line on a dark background, or the reverse—highlights objects that are near or very large. Edge detectors are useful to a cat, because edges do not occur often in the background "noise" of the visual environment, which is of little consequence to the animal.
History
Early in the history of sensory neurobiology, physiologists favored the idea that the nervous system detected specific features of stimuli, rather than faithfully copying the sensory world onto a sensory map in the brain. For example, they favored the idea that the visual system detects specific features of the visual world. This view contrasted with the metaphor that the retina acts like a camera and the brain acts like film that preserves all elements without making assumptions about what is important in the environment. It wasn't until the late 1950s that the feature detector hypothesis fully developed, and over the last fifty years, it has been the driving force behind most work on sensory systems.
Horace B. Barlow was one of the first investigators to use the concept of the feature detector to relate the receptive field of a neuron to a specific animal behavior. In 1953, H.B. Barlow's electrophysiological recordings from excised retina of the frog provided the first evidence for the presence of an inhibitory surround in the receptive field of a frog's retinal ganglion cell. In reference to "on-off" ganglion cells—which respond to both the transition from light to dark and the transition from dark to light—and also had very restricted receptive fields of visual angle (about the size of a fly at the distance that the frog could strike), Barlow stated, "It is difficult to avoid the conclusion that the 'on-off' units are matched to the stimulus and act as fly detectors". In the same year, Stephen Kuffler published in vivo evidence for an excitatory center, inhibitory surround architecture in the ganglion cells of the mammalian retina which further supported Barlow's suggestion that on-off units can |
https://en.wikipedia.org/wiki/Trivial%20cylinder | In geometry and topology, trivial cylinders are certain pseudoholomorphic curves appearing in certain cylindrical manifolds.
In Floer homology and its variants, chain complexes or differential graded algebras are generated by certain combinations of closed orbits of vector fields. In symplectic Floer homology, one considers the Hamiltonian vector field of a Hamiltonian function on a symplectic manifold; in symplectic field theory, contact homology, and their variants, one considers the Reeb vector field associated to a contact form, or more generally a stable Hamiltonian structure.
The differentials all count some flavor of pseudoholomorphic curves in a manifold with a cylindrical almost-complex structure whose ends at negative infinity are the given collection of closed orbits. For instance, in symplectic Floer homology, one considers the product of the mapping torus of a symplectomorphism with the real numbers; in symplectic field theory, one considers the symplectization of a contact manifold.
The product of a given embedded closed orbit with R is always a pseudoholomorphic curve, and such a curve is called a trivial cylinder. Trivial cylinders do not generally contribute to the aforementioned differentials, but they may appear as components of more complicated curves which do.
Algebraic geometry
Complex manifolds
Symplectic topology |
https://en.wikipedia.org/wiki/2005%E2%80%9306%20Norwich%20City%20F.C.%20season | The 2005–06 season was Norwich City's first year back in the Football League Championship after being relegated from the Premier League in the previous season. This article shows statistics and lists all matches that Norwich City played in the season.
Season overview
Final league table
Matches
League
August
September
October
November
December
January
February
March
April
League Cup
FA Cup
Transfers
Summer
In
Out
Winter
In
Out
Players
First team squad
Squad at end of season.
Left club during season
Board and staff members
Board members
Coaching staff
Notes
References
Norwich City F.C. seasons
Norwich City |
https://en.wikipedia.org/wiki/GJMS%20operator | In the mathematical field of differential geometry, the GJMS operators are a family of differential operators, that are defined on a Riemannian manifold. In an appropriate sense, they depend only on the conformal structure of the manifold. The GJMS operators generalize the Paneitz operator and the conformal Laplacian. The initials GJMS are for its discoverers Graham, Jenne, Mason & Sparling (1992).
Properly, the GJMS operator on a conformal manifold of dimension n is a conformally invariant operator between the line bundle of conformal densities of weight for k a positive integer
The operators have leading symbol given by a power of the Laplace–Beltrami operator, and have lower order correction terms that ensure conformal invariance.
The original construction of the GJMS operators used the ambient construction of Charles Fefferman and Robin Graham. A conformal density defines, in a natural way, a function on the null cone in the ambient space. The GJMS operator is defined by taking density ƒ of the appropriate weight and extending it arbitrarily to a function F off the null cone so that it still retains the same homogeneity. The function ΔkF, where Δ is the ambient Laplace–Beltrami operator, is then homogeneous of degree , and its restriction to the null cone does not depend on how the original function ƒ was extended to begin with, and so is independent of choices. The GJMS operator also represents the obstruction term to a formal asymptotic solution of the Cauchy problem for extending a weight function off the null cone in the ambient space to a harmonic function in the full ambient space.
The most important GJMS operators are the critical GJMS operators. In even dimension n, these are the operators Ln/2 that take a true function on the manifold and produce a multiple of the volume form.
References
.
Conformal geometry
Differential operators |
https://en.wikipedia.org/wiki/Quantum%20KZ%20equations | In mathematical physics, the quantum KZ equations or quantum Knizhnik–Zamolodchikov equations or qKZ equations are the analogue for quantum affine algebras of the Knizhnik–Zamolodchikov equations for affine Kac–Moody algebras. They are a consistent system of difference equations satisfied by the N-point functions, the vacuum expectations of products of primary fields. In the limit as the deformation parameter q approaches 1, the N-point functions of the quantum affine algebra tend to those of the affine Kac–Moody algebra and the difference equations become partial differential equations. The quantum KZ equations have been used to study exactly solved models in quantum statistical mechanics.
See also
Quantum affine algebras
Yang–Baxter equation
Quantum group
Affine Hecke algebra
Kac–Moody algebra
Two-dimensional conformal field theory
References
Mathematical physics
Conformal field theory
Quantum groups
Equations of physics |
https://en.wikipedia.org/wiki/2010%20Judgments%20of%20the%20Supreme%20Court%20of%20the%20United%20Kingdom | This is a list of the judgments given by the Supreme Court of the United Kingdom in 2010 and statistics associated thereupon. Since the Supreme Court began its work on 1 October 2009, this year was its first full year of operation. In total, 58 cases were heard in 2010.
The table lists judgments made by the court and the opinions of the judges in each case. Judges are treated as having concurred in another's judgment when they either formally attach themselves to the judgment of another or speak only to acknowledge their concurrence with one or more judges. Any judgment which reaches a conclusion which differs from the majority on one or more major points of the appeal has been treated as dissent.
Because every judge in the court is entitled to hand down a judgment, it is not uncommon for 'factions' to be formed who reach the same conclusion in different ways, or for all members of the court to reach the same conclusion in different ways. The table does not reflect this.
Table key
2010 Judgments
Notes
External links
Supreme Court decided cases, 2010
Supreme Court of the United Kingdom cases
Supreme Court of the United Kingdom |
https://en.wikipedia.org/wiki/Edis%20Seliminski | Edis Seliminski ( (born 6 September 1990) is a Bulgarian football striker who currently plays for Botev Lukovit.
References
External links
Profile at Akademik Sofia
PFL.bg statistics
https://web.archive.org/web/20120927134023/http://cdn.football24.bg/cache/220x0/assets/media/players/3071edis.jpg
1990 births
Living people
Bulgarian men's footballers
Bulgarian people of Turkish descent
PFC Akademik Sofia players
First Professional Football League (Bulgaria) players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Fractals%20%28journal%29 | Fractals is a peer-reviewed scientific journal devoted to explaining complex phenomena using fractal geometry and scaling. It is published by World Scientific and has explored diverse topics from turbulence and colloidal aggregation to stock markets.
Abstracting and indexing
The journal is indexed and abstracted in:
According to the Journal Citation Reports, the journal has a 2018 impact factor of 1.629.
References
External links
Mathematics journals
English-language journals
World Scientific academic journals
Quarterly journals
Academic journals established in 1993 |
https://en.wikipedia.org/wiki/Infinite%20Dimensional%20Analysis%2C%20Quantum%20Probability%20and%20Related%20Topics | Infinite Dimensional Analysis, Quantum Probability and Related Topics is a quarterly peer-reviewed scientific journal published since 1998 by World Scientific. It covers the development of infinite dimensional analysis, quantum probability, and their applications to classical probability and other areas of physics.
Abstracting and indexing
The journal is abstracted and indexed in CompuMath Citation Index, Current Contents/Physical, Chemical & Earth Sciences, Mathematical Reviews, Science Citation Index, Scopus, and Zentralblatt MATH. According to the Journal Citation Reports, the journal has a 2020 impact factor of 0.793.
References
External links
Mathematics journals
Academic journals established in 1988
World Scientific academic journals
English-language journals
Quarterly journals |
https://en.wikipedia.org/wiki/International%20Journal%20of%20Algebra%20and%20Computation | The International Journal of Algebra and Computation is published by World Scientific, and contains articles on general mathematics, as well as:
Combinatorial group theory and semigroup theory
Universal algebra
Algorithmic and computational problems in algebra
Theory of automata
Formal language theory
Theory of computation
Theoretical computer science
According to the Journal Citation Reports, the journal has a 2020 impact factor of 0.719.
Abstracting and indexing
The journal is indexed in:
ISI Alerting Services
CompuMath Citation Index
Science Citation Index
Current Contents/Physical, Chemical and Earth Sciences
Mathematical Reviews
INSPEC
Zentralblatt MATH
Computer Abstracts
Mathematics journals
Academic journals established in 1991
World Scientific academic journals
English-language journals |
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