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https://en.wikipedia.org/wiki/Tetrakis%20square%20tiling
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In geometry, the tetrakis square tiling is a tiling of the Euclidean plane. It is a square tiling with each square divided into four isosceles right triangles from the center point, forming an infinite arrangement of lines. It can also be formed by subdividing each square of a grid into two triangles by a diagonal, with the diagonals alternating in direction, or by overlaying two square grids, one rotated by 45 degrees from the other and scaled by a factor of √2.
Conway, Burgiel, and Goodman-Strauss call it a kisquadrille, represented by a kis operation that adds a center point and triangles to replace the faces of a square tiling (quadrille). It is also called the Union Jack lattice because of the resemblance to the UK flag of the triangles surrounding its degree-8 vertices.
It is labeled V4.8.8 because each isosceles triangle face has two types of vertices: one with 4 triangles, and two with 8 triangles.
As a dual uniform tiling
It is the dual tessellation of the truncated square tiling which has one square and two octagons at each vertex.
Applications
A 5 × 9 portion of the tetrakis square tiling is used to form the board for the Malagasy board game Fanorona. In this game, pieces are placed on the vertices of the tiling, and move along the edges, capturing pieces of the other color until one side has captured all of the other side's pieces. In this game, the degree-4 and degree-8 vertices of the tiling are called respectively weak intersections and strong intersections, a distinction that plays an important role in the strategy of the game. A similar board is also used for the Brazilian game Adugo, and for the game of Hare and Hounds.
The tetrakis square tiling was used for a set of commemorative postage stamps issued by the United States Postal Service in 1997, with an alternating pattern of two different stamps. Compared to the simpler pattern for triangular stamps in which all diagonal perforations are parallel to each other, the tetrakis pattern has the advantage that, when folded along any of its perforations, the other perforations line up with each other, making repeated folding possible.
This tiling also forms the basis for a commonly used "pinwheel", "windmill", and "broken dishes" patterns in quilting.
Symmetry
The symmetry type is:
with the coloring: cmm; a primitive cell is 8 triangles, a fundamental domain 2 triangles (1/2 for each color)
with the dark triangles in black and the light ones in white: p4g; a primitive cell is 8 triangles, a fundamental domain 1 triangle (1/2 each for black and white)
with the edges in black and the interiors in white: p4m; a primitive cell is 2 triangles, a fundamental domain 1/2
The edges of the tetrakis square tiling form a simplicial arrangement of lines, a property it shares with the triangular tiling and the kisrhombille tiling.
These lines form the axes of symmetry of a reflection group (the wallpaper group [4,4], (*442) or p4m), which has the triangles of the tiling as its fundament
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https://en.wikipedia.org/wiki/Theta%20%28disambiguation%29
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Theta is the eighth Greek letter, written Θ (uppercase) or θ (lowercase).
Theta may also refer to:
Science and mathematics
Θ (set theory), the least ordinal α such that there is no surjection from the reals onto α
Theta (gastropod), a genus of sea snails
Theta functions, special functions of several complex variables
Theta meson, a hypothetical meson in quantum physics
Theta representation, a particular representation of the Heisenberg group of quantum mechanics
Theta wave, in biology
Theta*, a pathfinding algorithm in computer science
, a Bachmann–Landau notation in computational complexity theory
The denotation for potential temperature
A common symbol for a variable of the measure of an angle
SARS-CoV-2 Theta variant, one of the variants of SARS-CoV-2, the virus that causes COVID-19
Business
GM Theta platform, an automobile platform of General Motors
Theta Networks a telecommunications software company
Other uses
Theta (finance), in quantitative finance, a first order derivative of an option pricing formula versus time
Theta (musician), a Greek musician
Theta (video game), a 2007 game produced by Kensuke Tanabe
Theta role, in linguistics
Theta, Gauteng, a suburb of Johannesburg, South Africa
An IPA symbol for voiceless dental fricative
Tropical Storm Theta, the record-breaking 29th named storm of the 2020 Atlantic hurricane season
Hyundai Theta engine a four cylinder gasoline engine made by Hyundai.
Kappa Alpha Theta, a North American collegiate sorority
See also
Thetan, in Scientology, the spirit or soul
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https://en.wikipedia.org/wiki/Hirzebruch%E2%80%93Riemann%E2%80%93Roch%20theorem
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In mathematics, the Hirzebruch–Riemann–Roch theorem, named after Friedrich Hirzebruch, Bernhard Riemann, and Gustav Roch, is Hirzebruch's 1954 result generalizing the classical Riemann–Roch theorem on Riemann surfaces to all complex algebraic varieties of higher dimensions. The result paved the way for the Grothendieck–Hirzebruch–Riemann–Roch theorem proved about three years later.
Statement of Hirzebruch–Riemann–Roch theorem
The Hirzebruch–Riemann–Roch theorem applies to any holomorphic vector bundle E on a compact complex manifold X, to calculate the holomorphic Euler characteristic of E in sheaf cohomology, namely the alternating sum
of the dimensions as complex vector spaces, where n is the complex dimension of X.
Hirzebruch's theorem states that χ(X, E) is computable in terms of the Chern classes ck(E) of E, and the Todd classes of the holomorphic tangent bundle of X. These all lie in the cohomology ring of X; by use of the fundamental class (or, in other words, integration over X) we can obtain numbers from classes in The Hirzebruch formula asserts that
where the sum is taken over all relevant j (so 0 ≤ j ≤ n), using the Chern character ch(E) in cohomology. In other words, the products are formed in the cohomology ring of all the 'matching' degrees that add up to 2n. Formulated differently, it gives the equality
where is the Todd class of the tangent bundle of X.
Significant special cases are when E is a complex line bundle, and when X is an algebraic surface (Noether's formula). Weil's Riemann–Roch theorem for vector bundles on curves, and the Riemann–Roch theorem for algebraic surfaces (see below), are included in its scope. The formula also expresses in a precise way the vague notion that the Todd classes are in some sense reciprocals of characteristic classes.
Riemann Roch theorem for curves
For curves, the Hirzebruch–Riemann–Roch theorem is essentially the classical Riemann–Roch theorem. To see this, recall that for each divisor D on a curve there is an invertible sheaf O(D) (which corresponds to a line bundle) such that the linear system of D is more or less the space of sections of O(D). For curves the Todd class is and the Chern character of a sheaf O(D) is just 1+c1(O(D)), so the Hirzebruch–Riemann–Roch theorem states that
(integrated over X).
But h0(O(D)) is just l(D), the dimension of the linear system of D, and by Serre duality h1(O(D)) = h0(O(K − D)) = l(K − D) where K is the canonical divisor. Moreover, c1(O(D)) integrated over X is the degree of D, and c1(T(X)) integrated over X is the Euler class 2 − 2g of the curve X, where g is the genus. So we get the classical Riemann Roch theorem
For vector bundles V, the Chern character is rank(V) + c1(V), so we get Weil's Riemann Roch theorem for vector bundles over curves:
Riemann Roch theorem for surfaces
For surfaces, the Hirzebruch–Riemann–Roch theorem is essentially the Riemann–Roch theorem for surfaces
combined with the Noether formula.
If we want, we
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https://en.wikipedia.org/wiki/Operating%20surplus
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Operating surplus is an accounting concept used in national accounts statistics (such as United Nations System of National Accounts (UNSNA)) and in corporate and government accounts. It is the balancing item of the Generation of Income Account in the UNSNA. It may be used in macro-economics as a proxy for total pre-tax profit income, although entrepreneurial income may provide a better measure of business profits. According to the 2008 SNA, it is the measure of the surplus accruing from production before deducting property income, e.g., land rent and interest.
Operating surplus is a component of value added and GDP. The term "mixed income" is used when operating surplus cannot be distinguished from wage income, for example, in the case of sole proprietorships. Most of operating surplus will normally consist of gross profit income. In principle, it includes the (separately itemised) increase in the value of output inventories held, with or without a valuation adjustment reflecting average prices during the accounting period.
Operating surplus therefore does not necessarily refer to all gross profit income realized in an economy. Profits are also realized from all kinds of property transactions which do not involve new production, such as capital gains, and net profits are often also received from foreign countries or paid to foreign countries. In addition, many profits arising from the use of natural resources, land, and financial assets (in the form of interest income) will not be included.
Derivation of operating surplus in UNSNA
A simple definition of business profit would be "sales less costs", and the accounting derivation of operating surplus is similar (although the SNA concept of entrepreneurial income better matches what is thought of as business profits). Starting off with Gross Output, expenditure on intermediate goods and services are deducted, to arrive at gross value added.
Value added may be stated gross (equal to the net output value, including consumption of fixed capital, i.e. depreciation charges) or net (excluding consumption of fixed capital). The net operating surplus (NOS) is thus the residual balancing item in the product account, obtained as follows:
Gross value added (GV)
less consumption of fixed capital. (CFC)
equals net value added (NV)
less Compensation of employees (CE)
less indirect taxes paid by producers, reduced by producer subsidies received (IT-SU)
equals net operating surplus (NOS)
In simple equations,
NOS=GV - (CE + (IT-SU) + CFC)
or
NOS=NV - (CE + (IT-SU)
Operating surplus can of course also be stated gross (GOS):
GOS=NOS + CFC
In this case, depreciation charges are included.
Exclusions
In UNSNA, "implicit (imputed) rents" on land owned by the enterprise and the "implicit (imputed) interest" chargeable on the use of the enterprise's own funds are excluded from operating surplus.
Operating surplus also excludes property incomes considered to be unrelated to value-adding production.
Mixed
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https://en.wikipedia.org/wiki/FIVB%20Volleyball%20World%20League%20statistics
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This article gives the summarized final standings of each FIVB Volleyball World League tournament, an annual competition involving national men's volleyball teams. The most successful teams, , have been: Brazil, 9 times (1993, 2001, 2003–07, 2009–10) and Italy, 8 times (1990–92, 1994–95, 1997, 1999–2000). The competition has been won 3 times by Russia (2002, 2011, 2013), twice by United States (2008, 2014) and France (2015, 2017) and once by Netherlands (1996), Cuba (1998), Poland (2012) and Serbia (2016).
Summary I
1st - Champions
2nd - Runners-up
3rd - Third place
- – Did not enter / Did not qualify
– Hosts
Summary II
After 2017 World League
Qualifications are not included
Notes
External links
Honours (1990–2016)
Statistics
Volleyball records and statistics
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https://en.wikipedia.org/wiki/National%20Longitudinal%20Surveys
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The National Longitudinal Surveys (NLS) are a set of surveys sponsored by the Bureau of Labor Statistics (BLS) of the U.S. Department of Labor. These surveys have gathered information at multiple points in time on the labor market experiences and other significant life events of several groups of men and women. Each of the NLS samples consists of several thousand individuals, many of whom have been surveyed over several decades.
Surveys
The National Longitudinal Survey of Youth 1997 (NLSY97) began in 1997 with 8,984 men and women born in 1980-84 (ages 12–17 in 1997). Sample members were interviewed annually from 1997 to 2011 and biennially thereafter. The 2015 interview was conducted with 7,103 men and women ages 30–36. Data are available from Round 1 (1997–98) to Round 17 (2015–16).
The National Longitudinal Survey of Youth 1979 (NLSY79) began in 1979 with 12,686 men and women born in 1957-64 (ages 14–22 in 1979). Sample members were interviewed annually from 1979-1994 and biennially thereafter. Oversamples of military and economically disadvantaged, nonblack/non-Hispanic respondents were dropped in 1985 and 1991, leaving a sample size of 9,964. The 2014 interview (Round 26) was conducted with 7,071 men and women ages 49–58.
The NLSY79 Children and Young Adults (NLSCYA) began in 1986 with children born to female NLSY79 respondents. Biennial data collection consists of interviews with the mothers and interviews with the children themselves; from 1994 onward, children turning age 15 and older during the survey year have been administered a Young Adult questionnaire that is similar to the NLSY79 questionnaire. In 2014, 276 children (ages 0–14) and 5,735 young adults (ages 15–42) were interviewed. To date, about 10,500 children have been interviewed in at least one survey round.
The National Longitudinal Surveys of Young Women and Mature Women (NLSW) comprised two separate surveys. The Young Women's survey began in 1968 with 5,159 women born in 1943-53 (ages 14–24 in 1968). Sample members were interviewed 22 times from 1968 to 2003. The final interview in 2003 was conducted with 2,857 women ages 49–59. The Mature Women's survey began in 1967 with 5,083 women born in 1922-37 (ages 30–44 in 1967). Sample members were interviewed 21 times from 1967 to 2003. The final interview in 2003 was conducted with 2,237 women ages 66–80.
The National Longitudinal Surveys of Young Men and Older Men (NLSM) comprised two separate surveys. The Young Men's survey began in 1966 with 5,225 men born in 1941-51 (ages 14–24 in 1966). Sample members were interviewed 12 times from 1966 to 1981. The Older Men's survey began in 1966 with 5,020 men born in 1906-21 (ages 45–59 in 1966). Sample members were interviewed 12 times from 1966 to 1983. A final interview in 1990 was conducted with 2,092 respondents who were 69–83 years old, and 2,206 family members of deceased respondents.
NLSY97
The National Longitudinal Survey of Youth 1997 (NLSY97), the newest survey in the
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https://en.wikipedia.org/wiki/Model%20category
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In mathematics, particularly in homotopy theory, a model category is a category with distinguished classes of morphisms ('arrows') called 'weak equivalences', 'fibrations' and 'cofibrations' satisfying certain axioms relating them. These abstract from the category of topological spaces or of chain complexes (derived category theory). The concept was introduced by .
In recent decades, the language of model categories has been used in some parts of algebraic K-theory and algebraic geometry, where homotopy-theoretic approaches led to deep results.
Motivation
Model categories can provide a natural setting for homotopy theory: the category of topological spaces is a model category, with the homotopy corresponding to the usual theory. Similarly, objects that are thought of as spaces often admit a model category structure, such as the category of simplicial sets.
Another model category is the category of chain complexes of R-modules for a commutative ring R. Homotopy theory in this context is homological algebra. Homology can then be viewed as a type of homotopy, allowing generalizations of homology to other objects, such as groups and R-algebras, one of the first major applications of the theory. Because of the above example regarding homology, the study of closed model categories is sometimes thought of as homotopical algebra.
Formal definition
The definition given initially by Quillen was that of a closed model category, the assumptions of which seemed strong at the time, motivating others to weaken some of the assumptions to define a model category. In practice the distinction has not proven significant and most recent authors (e.g., Mark Hovey and Philip Hirschhorn) work with closed model categories and simply drop the adjective 'closed'.
The definition has been separated to that of a model structure on a category and then further categorical conditions on that category, the necessity of which may seem unmotivated at first but becomes important later. The following definition follows that given by Hovey.
A model structure on a category C consists of three distinguished classes of morphisms (equivalently subcategories): weak equivalences, fibrations, and cofibrations, and two functorial factorizations and subject to the following axioms. A fibration that is also a weak equivalence is called an acyclic (or trivial) fibration and a cofibration that is also a weak equivalence is called an acyclic (or trivial) cofibration (or sometimes called an anodyne morphism).
Axioms
Retracts: if g is a morphism belonging to one of the distinguished classes, and f is a retract of g (as objects in the arrow category , where 2 is the 2-element ordered set), then f belongs to the same distinguished class. Explicitly, the requirement that f is a retract of g means that there exist i, j, r, and s, such that the following diagram commutes:
2 of 3: if f and g are maps in C such that gf is defined and any two of these are weak equivalences then so is the thi
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https://en.wikipedia.org/wiki/Fluent%20%28disambiguation%29
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Fluent is an adjective related to fluency, the ability to communicate in a language quickly and accurately.
Fluent or fluency may also refer to:
Fluent (mathematics), in mathematics, a continuous function
Fluent (artificial intelligence), in artificial intelligence, a condition that varies over time
Fluent, Inc., a company that develops software for computational fluid dynamics
Fluent interface, a software engineering object-oriented construct
Fluent (user interface), introduced in the 2007 Microsoft Office system
Fluent Design System, a design language developed by Microsoft in 2017
Fluentd, open source data collection software
Fluency (handwriting), an aspect of handwriting ability
See also
Fluenz (language learning software), a digital language learning platform
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https://en.wikipedia.org/wiki/Event%20calculus
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The event calculus is a logical language for representing and reasoning about events and their effects first presented by Robert Kowalski and Marek Sergot in 1986. It was extended by Murray Shanahan and Rob Miller in the 1990s. Similar to other languages for reasoning about change, the event calculus represents the effects of actions on fluents. However, events can also be external to the system. In the event calculus, one can specify the value of fluents at some given time points, the events that take place at given time points, and their effects.
Fluents and events
In the event calculus, fluents are reified. This means that they are not formalized by means of predicates but by means of functions. A separate predicate is used to tell which fluents hold at a given time point. For example, means that the box is on the table at time ; in this formula, is a predicate while is a function.
Events are also represented as terms. The effects of events are given using the predicates and . In particular, means that,
if the event represented by the term is executed at time ,
then the fluent will be true after .
The predicate has a similar meaning, with the only difference
being that will be false after .
Domain-independent axioms
Like other languages for representing actions, the event calculus formalizes the correct evolution of the fluent via formulae telling the value of each fluent after an arbitrary action has been performed. The event calculus solves the frame problem in a way that is similar to the successor state axioms of the situation calculus: a fluent is true at time if and only if it has been made true in the past and has not been made false in the meantime.
This formula means that the fluent represented by the term is true at time if:
an event has taken place: ;
this took place in the past: ;
this event has the fluent as an effect: ;
the fluent has not been made false in the meantime:
A similar formula is used to formalize the opposite case in which a fluent is false at a given time. Other formulae are also needed for correctly formalizing fluents before they have been effects of an event. These formulae are similar to the above, but is replaced by .
The predicate, stating that a fluent has been made false during an interval, can be axiomatized, or simply taken as a shorthand, as follows:
Domain-dependent axioms
The axioms above relate the value of the predicates , and , but do not specify which fluents are known to be true and which events actually make fluents true or false. This is done by using a set of domain-dependent axioms. The known values of fluents are stated as simple literals . The effects of events are stated by formulae relating the effects of events with their preconditions. For example, if the event makes the fluent true, but only if is currently true, the corresponding formula in the event calculus is:
The right-hand expression of this equivalence is composed of a disjunction: for ea
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https://en.wikipedia.org/wiki/Tomita%E2%80%93Takesaki%20theory
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In the theory of von Neumann algebras, a part of the mathematical field of functional analysis, Tomita–Takesaki theory is a method for constructing modular automorphisms of von Neumann algebras from the polar decomposition of a certain involution. It is essential for the theory of type III factors, and has led to a good structure theory for these previously intractable objects.
The theory was introduced by , but his work was hard to follow and mostly unpublished, and little notice was taken of it until wrote an account of Tomita's theory.
Modular automorphisms of a state
Suppose that M is a von Neumann algebra acting on a Hilbert space H, and Ω is a cyclic and separating vector of H of norm 1. (Cyclic means that MΩ is dense in H, and separating means that the map from M to MΩ is injective.) We write for the vector state of M, so that H is constructed from using the Gelfand–Naimark–Segal construction. Since Ω is separating, is faithful.
We can define a (not necessarily bounded) antilinear operator S0 on H with dense domain MΩ by setting for all m in M, and similarly we can define a (not necessarily bounded) antilinear operator F0 on H with dense domain M'Ω by setting for m in M′, where M′ is the commutant of M.
These operators are closable, and we denote their closures by S and F = S*. They have polar decompositions
where is an antilinear isometry of H called the modular conjugation and is a positive (hence, self-adjoint) and densely defined operator called the modular operator.
Commutation theorem
The main result of Tomita–Takesaki theory states that:
for all t and that
the commutant of M.
There is a 1-parameter group of modular automorphisms of M associated with the state , defined by .
The modular conjugation operator J and the 1-parameter unitary group satisfy
and
The Connes cocycle
The modular automorphism group of a von Neumann algebra M depends on the choice of state φ. Connes discovered that changing the state does not change the image of the modular automorphism in the outer automorphism group of M. More precisely, given two faithful states φ and ψ of M, we can find unitary elements ut of M for all real t such that
so that the modular automorphisms differ by inner automorphisms, and moreover ut satisfies the 1-cocycle condition
In particular, there is a canonical homomorphism from the additive group of reals to the outer automorphism group of M, that is independent of the choice of faithful state.
KMS states
The term KMS state comes from the Kubo–Martin–Schwinger condition in quantum statistical mechanics.
A KMS state on a von Neumann algebra M with a given 1-parameter group of automorphisms αt is a state fixed by the automorphisms such that for every pair of elements A, B of M there is a bounded continuous function F in the strip , holomorphic in the interior, such that
Takesaki and Winnink showed that any (faithful semi finite normal) state is a KMS state for the 1-parameter group of mod
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https://en.wikipedia.org/wiki/Perspective
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Perspective may refer to:
Vision and mathematics
Perspectivity, the formation of an image in a picture plane of a scene viewed from a fixed point, and its modeling in geometry
Perspective (graphical), representing the effects of visual perspective in graphic arts
Aerial perspective, the effect the atmosphere has on the appearance of an object as it is viewed from a distance
Perspective distortion (photography), the way that viewing a picture from the wrong position gives a perceived distortion
Perspective (geometry), a relation between geometric figures
Vue d'optique or perspective view, a genre of etching popular during the second half of the 18th century and into the 19th.
Entertainment
Perspective (P-Model album), 1982
Perspective (America album), 1984
Perspective (Jason Becker album), 1996
Perspective (Lawson album), 2016
Perspective, a 2010 album by Prague
Perspective (EP), an EP by Tesseract
Perspectives (album), the 2010 album by Australian band House Vs. Hurricane
"Perspectives", a song from the album Sea of Faces by Kutless
Perspective Records, a record label
Perspective (film series), a 2012-2020 film series by B. P. Paquette
Perspective (2019 film), an adult romance drama
Perspective (video game), a puzzle game
Perspectives (TV series), a British arts documentary series
Other
Perspective (pharmacoeconomic), the vantage point from which a pharmacoeconomics analysis is conducted
Point of view (literature), the related experience of the narrator
Point of view (philosophy), in philosophy and psychology, the context for opinions, beliefs and experiences
Perspectives on Political Science, peer-reviewed academic journal
The Perspective, a news and history website
See also
Perspecta, a motion picture sound system
Perspectivism, in philosophy
Point of view (disambiguation)
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https://en.wikipedia.org/wiki/Matplotlib
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Matplotlib is a plotting library for the Python programming language and its numerical mathematics extension NumPy. It provides an object-oriented API for embedding plots into applications using general-purpose GUI toolkits like Tkinter, wxPython, Qt, or GTK. There is also a procedural "pylab" interface based on a state machine (like OpenGL), designed to closely resemble that of MATLAB, though its use is discouraged. SciPy makes use of Matplotlib.
Matplotlib was originally written by John D. Hunter. Since then it has had an active development community and is distributed under a BSD-style license. Michael Droettboom was nominated as matplotlib's lead developer shortly before John Hunter's death in August 2012 and was further joined by Thomas Caswell. Matplotlib is a NumFOCUS fiscally sponsored project.
Comparison with MATLAB
Pyplot is a Matplotlib module that provides a MATLAB-like interface. Matplotlib is designed to be as usable as MATLAB, with the ability to use Python, and the advantage of being free and open-source.
Examples
Toolkits
Several toolkits are available which extend Matplotlib functionality. Some are separate downloads, others ship with the Matplotlib source code but have external dependencies.
Basemap: map plotting with various map projections, coastlines, and political boundaries
Cartopy: a mapping library featuring object-oriented map projection definitions, and arbitrary point, line, polygon and image transformation capabilities. (Matplotlib v1.2 and above)
Excel tools: utilities for exchanging data with Microsoft Excel
GTK tools: interface to the GTK library
Qt interface
Mplot3d: 3-D plots
Natgrid: interface to the natgrid library for gridding irregularly spaced data.
tikzplotlib: export to Pgfplots for smooth integration into LaTeX documents (formerly known as matplotlib2tikz)
Seaborn: provides an API on top of Matplotlib that offers sane choices for plot style and color defaults, defines simple high-level functions for common statistical plot types, and integrates with the functionality provided by Pandas
Related projects
Biggles
Chaco
DISLIN
GNU Octave
gnuplotlib – plotting for numpy with a gnuplot backend
Gnuplot-py
PLplot – Python bindings available
SageMath – uses Matplotlib to draw plots
SciPy (modules plt and gplt)
Plotly – for interactive, online Matplotlib and Python graphs
Bokeh – Python interactive visualization library that targets modern web browsers for presentation
References
External links
Articles with example Python (programming language) code
Free plotting software
Free software programmed in Python
Python (programming language) scientific libraries
Science software that uses GTK
Science software that uses Qt
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https://en.wikipedia.org/wiki/Upper%20set
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In mathematics, an upper set (also called an upward closed set, an upset, or an isotone set in X) of a partially ordered set is a subset with the following property: if s is in S and if x in X is larger than s (that is, if ), then x is in S. In other words, this means that any x element of X that is to some element of S is necessarily also an element of S.
The term lower set (also called a downward closed set, down set, decreasing set, initial segment, or semi-ideal) is defined similarly as being a subset S of X with the property that any element x of X that is to some element of S is necessarily also an element of S.
Definition
Let be a preordered set.
An in (also called an , an , or an set) is a subset that is "closed under going up", in the sense that
for all and all if then
The dual notion is a (also called a , , , , or ), which is a subset that is "closed under going down", in the sense that
for all and all if then
The terms or are sometimes used as synonyms for lower set. This choice of terminology fails to reflect the notion of an ideal of a lattice because a lower set of a lattice is not necessarily a sublattice.
Properties
Every partially ordered set is an upper set of itself.
The intersection and the union of any family of upper sets is again an upper set.
The complement of any upper set is a lower set, and vice versa.
Given a partially ordered set the family of upper sets of ordered with the inclusion relation is a complete lattice, the upper set lattice.
Given an arbitrary subset of a partially ordered set the smallest upper set containing is denoted using an up arrow as (see upper closure and lower closure).
Dually, the smallest lower set containing is denoted using a down arrow as
A lower set is called principal if it is of the form where is an element of
Every lower set of a finite partially ordered set is equal to the smallest lower set containing all maximal elements of
where denotes the set containing the maximal elements of
A directed lower set is called an order ideal.
For partial orders satisfying the descending chain condition, antichains and upper sets are in one-to-one correspondence via the following bijections: map each antichain to its upper closure (see below); conversely, map each upper set to the set of its minimal elements. This correspondence does not hold for more general partial orders; for example the sets of real numbers and are both mapped to the empty antichain.
Upper closure and lower closure
Given an element of a partially ordered set the upper closure or upward closure of denoted by or is defined by
while the lower closure or downward closure of , denoted by or is defined by
The sets and are, respectively, the smallest upper and lower sets containing as an element.
More generally, given a subset define the upper/upward closure and the lower/downward closure of denoted by and respectively, as
and
In this way, and
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https://en.wikipedia.org/wiki/Daniell%20integral
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In mathematics, the Daniell integral is a type of integration that generalizes the concept of more elementary versions such as the Riemann integral to which students are typically first introduced. One of the main difficulties with the traditional formulation of the Lebesgue integral is that it requires the initial development of a workable measure theory before any useful results for the integral can be obtained. However, an alternative approach is available, developed by that does not suffer from this deficiency, and has a few significant advantages over the traditional formulation, especially as the integral is generalized into higher-dimensional spaces and further generalizations such as the Stieltjes integral. The basic idea involves the axiomatization of the integral.
Axioms
We start by choosing a family of bounded real functions (called elementary functions) defined over some set , that satisfies these two axioms:
is a linear space with the usual operations of addition and scalar multiplication.
If a function is in , so is its absolute value .
In addition, every function h in H is assigned a real number , which is called the elementary integral of h, satisfying these three axioms:
Linearity
If h and k are both in H, and and are any two real numbers, then .
Nonnegativity
If for all , then .
Continuity
If is a nonincreasing sequence (i.e. ) of functions in that converges to 0 for all in , then .or (more commonly)If is an increasing sequence (i.e. ) of functions in that converges to h for all in , then .
That is, we define a continuous non-negative linear functional over the space of elementary functions.
These elementary functions and their elementary integrals may be any set of functions and definitions of integrals over these functions which satisfy these axioms. The family of all step functions evidently satisfies the above axioms for elementary functions. Defining the elementary integral of the family of step functions as the (signed) area underneath a step function evidently satisfies the given axioms for an elementary integral. Applying the construction of the Daniell integral described further below using step functions as elementary functions produces a definition of an integral equivalent to the Lebesgue integral. Using the family of all continuous functions as the elementary functions and the traditional Riemann integral as the elementary integral is also possible, however, this will yield an integral that is also equivalent to Lebesgue's definition. Doing the same, but using the Riemann–Stieltjes integral, along with an appropriate function of bounded variation, gives a definition of integral equivalent to the Lebesgue–Stieltjes integral.
Sets of measure zero may be defined in terms of elementary functions as follows. A set which is a subset of is a set of measure zero if for any , there exists a nondecreasing sequence of nonnegative elementary functions in H such that and on .
A set is called a
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https://en.wikipedia.org/wiki/Olav%20Reiers%C3%B8l
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Olav Reiersøl (28 June 1908 – 14 February 2001) was a Norwegian statistician and econometrician, who made several substantial contributions to econometrics and statistics. His works on identifiability and instrumental variables are standard references both in econometrics and statistics, and his work on genetic algebras are frequently cited in genetics.
Reiersøl became interested in the international language Esperanto at a young age, and later in life used it to keep in touch with other mathematicians. He was one of the founders of the Esperanto association Internacia Asocio de Esperantistaj Matematikistoj ("International Association of Esperantist Mathematicians").
He was made a fellow of the Econometric Society in 1952.
References
External links
ET Interviews: Professor Olav Reiersøl on the Econometric Theory page.
1908 births
2001 deaths
University of Oslo alumni
Fellows of the Econometric Society
Norwegian statisticians
Norwegian Esperantists
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https://en.wikipedia.org/wiki/Pandigital%20number
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In mathematics, a pandigital number is an integer that in a given base has among its significant digits each digit used in the base at least once. For example, 1234567890 (one billion two hundred thirty four million five hundred sixty seven thousand eight hundred ninety) is a pandigital number in base 10. The first few pandigital base 10 numbers are given by :
1023456789, 1023456798, 1023456879, 1023456897, 1023456978, 1023456987, 1023457689
The smallest pandigital number in a given base b is an integer of the form
The following table lists the smallest pandigital numbers of a few selected bases.
gives the base 10 values for the first 18 bases.
In a trivial sense, all positive integers are pandigital in unary (or tallying). In binary, all integers are pandigital except for 0 and numbers of the form (the Mersenne numbers). The larger the base, the rarer pandigital numbers become, though one can always find runs of consecutive pandigital numbers with redundant digits by writing all the digits of the base together (but not putting the zero first as the most significant digit) and adding x + 1 zeroes at the end as least significant digits.
Conversely, the smaller the base, the fewer pandigital numbers without redundant digits there are. 2 is the only such pandigital number in base 2, while there are more of these in base 10.
Sometimes, the term is used to refer only to pandigital numbers with no redundant digits. In some cases, a number might be called pandigital even if it doesn't have a zero as a significant digit, for example, 923456781 (these are sometimes referred to as "zeroless pandigital numbers").
No base 10 pandigital number can be a prime number if it doesn't have redundant digits. The sum of the digits 0 to 9 is 45, passing the divisibility rule for both 3 and 9. The first base 10 pandigital prime is 10123457689; lists more.
For different reasons, redundant digits are also required for a pandigital number (in any base except unary) to also be a palindromic number in that base. The smallest pandigital palindromic number in base 10 is 1023456789876543201.
The largest pandigital number without redundant digits to be also a square number is 9814072356 = 990662.
Two of the zeroless pandigital Friedman numbers are: 123456789 = ((86 + 2 × 7)5 − 91) / 34, and 987654321 = (8 × (97 + 6/2)5 + 1) / 34.
A pandigital Friedman number without redundant digits is the square: 2170348569 = 465872 + (0 × 139).
While much of what has been said does not apply to Roman numerals, there are pandigital numbers: MCDXLIV, MCDXLVI, MCDLXIV, MCDLXVI, MDCXLIV, MDCXLVI, MDCLXIV, MDCLXVI. These, listed in , use each of the digits just once, while has pandigital Roman numerals with repeats.
Pandigital numbers are useful in fiction and in advertising. The Social Security number 987-65-4321 is a zeroless pandigital number reserved for use in advertising. Some credit card companies use pandigital numbers with redundant digits as fictitious credit card
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https://en.wikipedia.org/wiki/Doubly%20stochastic%20matrix
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In mathematics, especially in probability and combinatorics, a doubly stochastic matrix
(also called bistochastic matrix) is a square matrix of nonnegative real numbers, each of whose rows and columns sums to 1, i.e.,
Thus, a doubly stochastic matrix is both left stochastic and right stochastic.
Indeed, any matrix that is both left and right stochastic must be square: if every row sums to 1 then the sum of all entries in the matrix must be equal to the number of rows, and since the same holds for columns, the number of rows and columns must be equal.
Birkhoff polytope
The class of doubly stochastic matrices is a convex polytope known as the Birkhoff polytope . Using the matrix entries as Cartesian coordinates, it lies in an -dimensional affine subspace of -dimensional Euclidean space defined by independent linear constraints specifying that the row and column sums all equal 1. (There are constraints rather than because one of these constraints is dependent, as the sum of the row sums must equal the sum of the column sums.) Moreover, the entries are all constrained to be non-negative and less than or equal to 1.
Birkhoff–von Neumann theorem
The Birkhoff–von Neumann theorem (often known simply as Birkhoff's theorem) states that the polytope is the convex hull of the set of permutation matrices, and furthermore that the vertices of are precisely the permutation matrices. In other words, if is a doubly stochastic matrix, then there exist and permutation matrices such that
(Such a decomposition of X is known as a 'convex combination'.) A proof of the theorem based on Hall's marriage theorem is given below.
This representation is known as the Birkhoff–von Neumann decomposition, and may not be unique. It is often described as a real-valued generalization of Kőnig's theorem, where the correspondence is established through adjacency matrices of graphs.
Other properties
The product of two doubly stochastic matrices is doubly stochastic. However, the inverse of a nonsingular doubly stochastic matrix need not be doubly stochastic (indeed, the inverse is doubly stochastic iff it has nonnegative entries).
The stationary distribution of an irreducible aperiodic finite Markov chain is uniform if and only if its transition matrix is doubly stochastic.
Sinkhorn's theorem states that any matrix with strictly positive entries can be made doubly stochastic by pre- and post-multiplication by diagonal matrices.
For , all bistochastic matrices are unistochastic and orthostochastic, but for larger this is not the case.
Van der Waerden's conjecture that the minimum permanent among all doubly stochastic matrices is , achieved by the matrix for which all entries are equal to . Proofs of this conjecture were published in 1980 by B. Gyires and in 1981 by G. P. Egorychev and D. I. Falikman; for this work, Egorychev and Falikman won the Fulkerson Prize in 1982.
Proof of the Birkhoff–von Neumann theorem
Let X be a doubly stochastic matrix. Then
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https://en.wikipedia.org/wiki/Kurosh%20problem
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In mathematics, the Kurosh problem is one general problem, and several more special questions, in ring theory. The general problem is known to have a negative solution, since one of the special cases has been shown to have counterexamples. These matters were brought up by Aleksandr Gennadievich Kurosh as analogues of the Burnside problem in group theory.
Kurosh asked whether there can be a finitely-generated infinite-dimensional algebraic algebra (the problem being to show this cannot happen). A special case is whether or not every nil algebra is locally nilpotent.
For PI-algebras the Kurosh problem has a positive solution.
Golod showed a counterexample to that case, as an application of the Golod–Shafarevich theorem.
The Kurosh problem on group algebras concerns the augmentation ideal I. If I is a nil ideal, is the group algebra locally nilpotent?
There is an important problem which is often referred as the Kurosh's problem on division rings. The problem asks whether there exists an algebraic (over the center) division ring which is not locally finite. This problem has not been solved until now.
References
Vesselin S. Drensky, Edward Formanek (2004), Polynomial Identity Rings, p. 89.
Some open problems in the theory of infinite dimensional algebras (2007). E. Zelmanov.
Ring theory
Unsolved problems in mathematics
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https://en.wikipedia.org/wiki/Pencil%20%28geometry%29
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In geometry, a pencil is a family of geometric objects with a common property, for example the set of lines that pass through a given point in a plane, or the set of circles that pass through two given points in a plane.
Although the definition of a pencil is rather vague, the common characteristic is that the pencil is completely determined by any two of its members. Analogously, a set of geometric objects that are determined by any three of its members is called a bundle. Thus, the set of all lines through a point in three-space is a bundle of lines, any two of which determine a pencil of lines. To emphasize the two-dimensional nature of such a pencil, it is sometimes referred to as a flat pencil.
Any geometric object can be used in a pencil. The common ones are lines, planes, circles, conics, spheres, and general curves. Even points can be used. A pencil of points is the set of all points on a given line. A more common term for this set is a range of points.
Pencil of lines
In a plane, let and be two distinct intersecting lines. For concreteness, suppose that has the equation, and has the equation . Then
,
represents, for suitable scalars and , any line passing through the intersection of = 0 and = 0. This set of lines passing through a common point is called a pencil of lines. The common point of a pencil of lines is called the vertex of the pencil.
In an affine plane with the reflexive variant of parallelism, a set of parallel lines forms an equivalence class called a pencil of parallel lines. This terminology is consistent with the above definition since in the unique projective extension of the affine plane to a projective plane a single point (point at infinity) is added to each line in the pencil of parallel lines, thus making it a pencil in the above sense in the projective plane.
Pencil of planes
A pencil of planes, is the set of planes through a given straight line in three-space, called the axis of the pencil. The pencil is sometimes referred to as a axial-pencil or fan of planes or a sheaf of planes. For example, the meridians of the globe are defined by the pencil of planes on the axis of Earth's rotation.
Two intersecting planes meet in a line in three-space, and so, determine the axis and hence all of the planes in the pencil.
The four-space of quaternions can be seen as an axial pencil of complex planes all sharing the same real line. In fact, quaternions contain a sphere of imaginary units, and a pair of antipodal points on this sphere, together with the real axis, generate a complex plane. The union of all these complex planes constitutes the 4-algebra of quaternions.
Pencil of circles
Any two circles in the plane have a common radical axis, which is the line consisting of all the points that have the same power with respect to the two circles. A pencil of circles (or coaxial system) is the set of all circles in the plane with the same radical axis. To be inclusive, concentric circles are said to have the lin
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https://en.wikipedia.org/wiki/David%20E.%20Rowe
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David E. Rowe (born August 11, 1950) is an American mathematician and historian. He studied mathematics and the history of science at the University of Oklahoma, and took a second doctorate in history at the Graduate Center of the City University of New York. He served as book review editor, managing editor, and editor of the journal Historia Mathematica. In 1992, Rowe was appointed Professor of History of Mathematics and Natural Sciences at the Johannes Gutenberg University in Mainz where he presently teaches. His research has mainly focused on mathematics in Germany, but in recent years he has been concerned with Albert Einstein's general theory of relativity and the broader cultural and political impact of Einstein's ideas. As part of this effort, he and have co-edited a source book entitled Einstein on Politics: His Private Thoughts and Public Stands on Nationalism, Zionism, War, Peace, and the Bomb, published by Princeton University Press in 2007.
Publications
References
External links
Personal Homepage: www.DavidERowe.net
Homepage at the Universität Mainz
Academic staff of Johannes Gutenberg University Mainz
American historians of mathematics
CUNY Graduate Center alumni
University of Oklahoma alumni
Living people
21st-century American historians
21st-century American male writers
1950 births
American male non-fiction writers
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https://en.wikipedia.org/wiki/Behrens%E2%80%93Fisher%20problem
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In statistics, the Behrens–Fisher problem, named after Walter-Ulrich Behrens and Ronald Fisher, is the problem of interval estimation and hypothesis testing concerning the difference between the means of two normally distributed populations when the variances of the two populations are not assumed to be equal, based on two independent samples.
Specification
One difficulty with discussing the Behrens–Fisher problem and proposed solutions, is that there are many different interpretations of what is meant by "the Behrens–Fisher problem". These differences involve not only what is counted as being a relevant solution, but even the basic statement of the context being considered.
Context
Let X1, ..., Xn and Y1, ..., Ym be i.i.d. samples from two populations which both come from the same location–scale family of distributions. The scale parameters are assumed to be unknown and not necessarily equal, and the problem is to assess whether the location parameters can reasonably be treated as equal. Lehmann states that "the Behrens–Fisher problem" is used both for this general form of model when the family of distributions is arbitrary, and for when the restriction to a normal distribution is made. While Lehmann discusses a number of approaches to the more general problem, mainly based on nonparametrics, most other sources appear to use "the Behrens–Fisher problem" to refer only to the case where the distribution is assumed to be normal: most of this article makes this assumption.
Requirements of solutions
Solutions to the Behrens–Fisher problem have been presented that make use of either a classical or a Bayesian inference point of view and either solution would be notionally invalid judged from the other point of view. If consideration is restricted to classical statistical inference only, it is possible to seek solutions to the inference problem that are simple to apply in a practical sense, giving preference to this simplicity over any inaccuracy in the corresponding probability statements. Where exactness of the significance levels of statistical tests is required, there may be an additional requirement that the procedure should make maximum use of the statistical information in the dataset. It is well known that an exact test can be gained by randomly discarding data from the larger dataset until the sample sizes are equal, assembling data in pairs and taking differences, and then using an ordinary t-test to test for the mean-difference being zero: clearly this would not be "optimal" in any sense.
The task of specifying interval estimates for this problem is one where a frequentist approach fails to provide an exact solution, although some approximations are available. Standard Bayesian approaches also fail to provide an answer that can be expressed as straightforward simple formulae, but modern computational methods of Bayesian analysis do allow essentially exact solutions to be found. Thus study of the problem can be used to elucidate the dif
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https://en.wikipedia.org/wiki/Studentization
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In statistics, Studentization, named after William Sealy Gosset, who wrote under the pseudonym Student, is the adjustment consisting of division of a first-degree statistic derived from a sample, by a sample-based estimate of a population standard deviation. The term is also used for the standardisation of a higher-degree statistic by another statistic of the same degree: for example, an estimate of the third central moment would be standardised by dividing by the cube of the sample standard deviation.
A simple example is the process of dividing a sample mean by the sample standard deviation when data arise from a location-scale family. The consequence of "Studentization" is that the complication of treating the probability distribution of the mean, which depends on both the location and scale parameters, has been reduced to considering a distribution which depends only on the location parameter. However, the fact that a sample standard deviation is used, rather than the unknown population standard deviation, complicates the mathematics of finding the probability distribution of a Studentized statistic.
In computational statistics, the idea of using Studentized statistics is of some importance in the development of confidence intervals with improved properties in the context of resampling and, in particular, bootstrapping.
Examples
Studentized range
Studentized residual
See also
Pivotal quantity
References
Statistical ratios
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https://en.wikipedia.org/wiki/Klein%20geometry
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In mathematics, a Klein geometry is a type of geometry motivated by Felix Klein in his influential Erlangen program. More specifically, it is a homogeneous space X together with a transitive action on X by a Lie group G, which acts as the symmetry group of the geometry.
For background and motivation see the article on the Erlangen program.
Formal definition
A Klein geometry is a pair where G is a Lie group and H is a closed Lie subgroup of G such that the (left) coset space G/H is connected. The group G is called the principal group of the geometry and G/H is called the space of the geometry (or, by an abuse of terminology, simply the Klein geometry). The space of a Klein geometry is a smooth manifold of dimension
dim X = dim G − dim H.
There is a natural smooth left action of G on X given by
Clearly, this action is transitive (take ), so that one may then regard X as a homogeneous space for the action of G. The stabilizer of the identity coset is precisely the group H.
Given any connected smooth manifold X and a smooth transitive action by a Lie group G on X, we can construct an associated Klein geometry by fixing a basepoint x0 in X and letting H be the stabilizer subgroup of x0 in G. The group H is necessarily a closed subgroup of G and X is naturally diffeomorphic to G/H.
Two Klein geometries and are geometrically isomorphic if there is a Lie group isomorphism so that . In particular, if φ is conjugation by an element , we see that and are isomorphic. The Klein geometry associated to a homogeneous space X is then unique up to isomorphism (i.e. it is independent of the chosen basepoint x0).
Bundle description
Given a Lie group G and closed subgroup H, there is natural right action of H on G given by right multiplication. This action is both free and proper. The orbits are simply the left cosets of H in G. One concludes that G has the structure of a smooth principal H-bundle over the left coset space G/H:
Types of Klein geometries
Effective geometries
The action of G on need not be effective. The kernel of a Klein geometry is defined to be the kernel of the action of G on X. It is given by
The kernel K may also be described as the core of H in G (i.e. the largest subgroup of H that is normal in G). It is the group generated by all the normal subgroups of G that lie in H.
A Klein geometry is said to be effective if and locally effective if K is discrete. If is a Klein geometry with kernel K, then is an effective Klein geometry canonically associated to .
Geometrically oriented geometries
A Klein geometry is geometrically oriented if G is connected. (This does not imply that G/H is an oriented manifold). If H is connected it follows that G is also connected (this is because G/H is assumed to be connected, and is a fibration).
Given any Klein geometry , there is a geometrically oriented geometry canonically associated to with the same base space G/H. This is the geometry where G0 is the identity component of G. Note
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https://en.wikipedia.org/wiki/Serre%27s%20modularity%20conjecture
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In mathematics, Serre's modularity conjecture, introduced by , states that an odd, irreducible, two-dimensional Galois representation over a finite field arises from a modular form. A stronger version of this conjecture specifies the weight and level of the modular form. The conjecture in the level 1 case was proved by Chandrashekhar Khare in 2005, and a proof of the full conjecture was completed jointly by Khare and Jean-Pierre Wintenberger in 2008.
Formulation
The conjecture concerns the absolute Galois group of the rational number field .
Let be an absolutely irreducible, continuous, two-dimensional representation of over a finite field .
Additionally, assume is odd, meaning the image of complex conjugation has determinant -1.
To any normalized modular eigenform
of level , weight , and some Nebentype character
,
a theorem due to Shimura, Deligne, and Serre-Deligne attaches to a representation
where is the ring of integers in a finite extension of . This representation is characterized by the condition that for all prime numbers , coprime to we have
and
Reducing this representation modulo the maximal ideal of gives a mod representation of .
Serre's conjecture asserts that for any representation as above, there is a modular eigenform such that
.
The level and weight of the conjectural form are explicitly conjectured in Serre's article. In addition, he derives a number of results from this conjecture, among them Fermat's Last Theorem and the now-proven Taniyama–Weil (or Taniyama–Shimura) conjecture, now known as the modularity theorem (although this implies Fermat's Last Theorem, Serre proves it directly from his conjecture).
Optimal level and weight
The strong form of Serre's conjecture describes the level and weight of the modular form.
The optimal level is the Artin conductor of the representation, with the power of removed.
Proof
A proof of the level 1 and small weight cases of the conjecture was obtained in 2004 by Chandrashekhar Khare and Jean-Pierre Wintenberger, and by Luis Dieulefait, independently.
In 2005, Chandrashekhar Khare obtained a proof of the level 1 case of Serre conjecture, and in 2008 a proof of the full conjecture in collaboration with Jean-Pierre Wintenberger.
Notes
References
See also
Wiles's proof of Fermat's Last Theorem
External links
Serre's Modularity Conjecture 50 minute lecture by Ken Ribet given on October 25, 2007 ( slides PDF, other version of slides PDF)
Lectures on Serre's conjectures
Modular forms
Theorems in number theory
Conjectures that have been proved
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https://en.wikipedia.org/wiki/Truncation%20%28disambiguation%29
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Truncation is the term used for limiting the number of digits right of the decimal point by discarding the least significant ones.
Truncation may also refer to:
Mathematics
Truncation (statistics) refers to measurements which have been cut off at some value
Truncation (numerical analysis) refers to truncating an infinite sum by a finite one
Truncation (geometry) is the removal of one or more parts, as for example in truncated cube
Propositional truncation, a type former which truncates a type down to a mere proposition
Computer science
Data truncation, an event that occurs when a file or other data is stored in a location too small to accommodate its entire length
Truncate (SQL), a command in the SQL data manipulation language to quickly remove all data from a table
Biology
Truncate, a leaf shape
Truncated protein, a protein shortened by a mutation which specifically induces premature termination of messenger RNA translation
Other uses
Cheque truncation, the conversion of physical cheques into electronic form for transmission to the paying bank
Clipping (morphology), the word formation process which consists in the reduction of a word to one of its parts
Truncation (archaeology), also 'cut', the removal of archaeological deposits from an archaeological record
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https://en.wikipedia.org/wiki/Forest-fire%20model
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In applied mathematics, a forest-fire model is any of a number of dynamical systems displaying self-organized criticality. Note, however, that according to Pruessner et al. (2002, 2004) the forest-fire model does not behave critically on very large, i.e. physically relevant scales. Early versions go back to Henley (1989) and Drossel and Schwabl (1992). The model is defined as a cellular automaton on a grid with Ld cells. L is the sidelength of the grid and d is its dimension. A cell can be empty, occupied by a tree, or burning. The model of Drossel and Schwabl (1992) is defined by four rules which are executed simultaneously:
A burning cell turns into an empty cell
A tree will burn if at least one neighbor is burning
A tree ignites with probability f even if no neighbor is burning
An empty space fills with a tree with probability p
The controlling parameter of the model is p/f which gives the average number of trees planted between two lightning strikes (see Schenk et al. (1996) and Grassberger (1993)). In order to exhibit a fractal frequency-size distribution of clusters a double separation of time scales is necessary
where Tsmax is the burn time of the largest cluster. The scaling behavior is not simple, however ( Grassberger 1993,2002 and Pruessner et al. 2002,2004).
A cluster is defined as a coherent set of cells, all of which have the same state. Cells are coherent if they can reach each other via nearest neighbor relations. In most cases, the von Neumann neighborhood (four adjacent cells) is considered.
The first condition allows large structures to develop, while the second condition keeps trees from popping up alongside a cluster while burning.
In landscape ecology, the forest fire model is used to illustrate the role of the fuel mosaic in the wildfire regime. The importance of the fuel mosaic on wildfire spread is debated. Parsimonious models such as the forest fire model can help to explore the role of the fuel mosaic and its limitations in explaining observed patterns.
References
Henley, C. L. (1989), "Self-organized percolation: a simpler model." Bull. Am. Phys. Soc. 34, 838.
External links
An HTML 5 demo of the forest fire model
Self-organization
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https://en.wikipedia.org/wiki/Trisectrix
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In geometry, a trisectrix is a curve which can be used to trisect an arbitrary angle with ruler and compass and this curve as an additional tool. Such a method falls outside those allowed by compass and straightedge constructions, so they do not contradict the well known theorem which states that an arbitrary angle cannot be trisected with that type of construction. There is a variety of such curves and the methods used to construct an angle trisector differ according to the curve. Examples include:
Limaçon trisectrix (some sources refer to this curve as simply the trisectrix.)
Trisectrix of Maclaurin
Equilateral trefoil (a.k.a. Longchamps' Trisectrix)
Tschirnhausen cubic (a.k.a. Catalan's trisectrix and L'Hôpital's cubic)
Durer's folium
Cubic parabola
Hyperbola with eccentricity 2
Rose curve specified by a sinusoid with angular frequency of one-third.
Parabola
A related concept is a sectrix, which is a curve which can be used to divide an arbitrary angle by any integer. Examples include:
Archimedean Spiral
Quadratrix of Hippias
Sectrix of Maclaurin
Sectrix of Ceva
Sectrix of Delanges
See also
Doubling the cube
Neusis construction
Quadratrix
References
Loy, Jim "Trisection of an Angle", Part VI
"Sectrix curve" at Encyclopédie des Formes Mathématiques Remarquables (In French)
Curves
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https://en.wikipedia.org/wiki/Great%20dirhombicosidodecahedron
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In geometry, the great dirhombicosidodecahedron (or great snub disicosidisdodecahedron) is a nonconvex uniform polyhedron, indexed last as . It has 124 faces (40 triangles, 60 squares, and 24 pentagrams), 240 edges, and 60 vertices.
This is the only non-degenerate uniform polyhedron with more than six faces meeting at a vertex. Each vertex has 4 squares which pass through the vertex central axis (and thus through the centre of the figure), alternating with two triangles and two pentagrams. Another unusual feature is that the faces all occur in coplanar pairs.
This is also the only uniform polyhedron that cannot be made by the Wythoff construction from a spherical triangle. It has a special Wythoff symbol relating it to a spherical quadrilateral. This symbol suggests that it is a sort of snub polyhedron, except that instead of the non-snub faces being surrounded by snub triangles as in most snub polyhedra, they are surrounded by snub squares.
It has been nicknamed "Miller's monster" (after J. C. P. Miller, who with H. S. M. Coxeter and M. S. Longuet-Higgins enumerated the uniform polyhedra in 1954).
Related polyhedra
If the definition of a uniform polyhedron is relaxed to allow any even number of faces adjacent to an edge, then this definition gives rise to one further polyhedron: the great disnub dirhombidodecahedron which has the same vertices and edges but with a different arrangement of triangular faces.
The vertices and edges are also shared with the uniform compounds of 20 octahedra or 20 tetrahemihexahedra. 180 of the 240 edges are shared with the great snub dodecicosidodecahedron.
This polyhedron is related to the nonconvex great rhombicosidodecahedron (quasirhombicosidodecahedron) by a branched cover: there is a function from the great dirhombicosidodecahedron to the quasirhombicosidodecahedron that is 2-to-1 everywhere, except for the vertices.
Cartesian coordinates
Cartesian coordinates for the vertices of a great dirhombicosidodecahedron are all the even permutations of
where τ = (1+)/2 is the golden ratio (sometimes written φ). These vertices result in an edge length of 2.
Gallery
References
Har'El, Z. Uniform Solution for Uniform Polyhedra., Geometriae Dedicata 47, 57-110, 1993. Zvi Har’El, Kaleido software, Images, dual images
Mäder, R. E. Uniform Polyhedra. Mathematica J. 3, 48-57, 1993.
External links
http://www.mathconsult.ch/showroom/unipoly/75.html
http://www.software3d.com/MillersMonster.php
Uniform polyhedra
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https://en.wikipedia.org/wiki/Andrew%20Granville
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Andrew James Granville (born 7 September 1962) is a British mathematician, working in the field of number theory.
He has been a faculty member at the Université de Montréal since 2002. Before moving to Montreal he was a mathematics professor at the University of Georgia (UGA) from 1991 until 2002. He was a section speaker in the 1994 International Congress of Mathematicians together with Carl Pomerance from UGA.
Granville received his Bachelor of Arts (Honours) (1983) and his Certificate of Advanced Studies (Distinction) (1984) from Trinity College, Cambridge University. He received his PhD from Queen's University in 1987 and was inducted into the Royal Society of Canada in 2006.
Granville's work is mainly in number theory, in particular analytic number theory. Along with Carl Pomerance and W. R. (Red) Alford he proved the infinitude of Carmichael numbers in 1994. This proof was based on a conjecture given by Paul Erdős.
Granville won a Lester R. Ford Award in 2007 and again in 2009. In 2008, he won the Chauvenet Prize for expository writing from the Mathematical Association of America for his paper "It is easy to determine whether a given integer is prime". In 2012, he became a fellow of the American Mathematical Society.
Andrew Granville, in collaboration with Jennifer Granville, has written "Prime Suspects: The Anatomy of Integers and Permutations", a graphic novel that investigates key concepts in Mathematics.
References
External links
Professor Granville's Université de Montréal page
Videos of Andrew Granville in the AV-Portal of the German National Library of Science and Technology
1962 births
Living people
20th-century British mathematicians
21st-century British mathematicians
Number theorists
Academic staff of the Université de Montréal
University of Georgia faculty
Queen's University at Kingston alumni
Alumni of Trinity College, Cambridge
Fellows of the Royal Society of Canada
Fellows of the American Mathematical Society
Royal Society Wolfson Research Merit Award holders
British expatriate academics in Canada
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https://en.wikipedia.org/wiki/Fril
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Fril is a programming language for first-order predicate calculus. It includes the semantics of Prolog as a subset, but takes its syntax from the of Logic Programming Associates and adds support for fuzzy sets, support logic, and metaprogramming.
Fril was originally developed by Trevor Martin and Jim Baldwin at the University of Bristol around 1980. In 1986, it was picked up and further developed by Equipu A.I. Research, which later became Fril Systems Ltd. The name Fril was originally an acronym for Fuzzy Relational Inference Language.
Prolog and Fril comparison
Aside from the uncertainty-management features of Fril, there are some minor differences in Fril's implementation of standard Prolog features.
Types
The basic types in Fril are similar to those in Prolog, with one important exception: Prolog's compound data type is the term, with lists defined as nested terms using the . functor; in Fril, the compound type is the list itself, which forms the basis for most constructs. Variables are distinguished by identifiers containing only uppercase letters and underscores (whereas Prolog only requires the first character to be uppercase). As in Prolog, the name _ is reserved to mean "any value", with multiple occurrences of _ replaced by distinct variables.
Syntax
Prolog has a syntax with a typical amount of punctuation, whereas Fril has an extremely simple syntax similar to that of Lisp. A (propositional) clause is a list consisting of a predicate followed by its arguments (if any). Among the types of top-level constructs are rules and direct-mode commands.
Rule
A rule is a list consisting of a conclusion followed by the hypotheses (goals). The general forms look like this:
(fact)
(conclusion goal_1 ... goal_n)
These are equivalent to the respective Prolog constructions:
fact.
conclusion :- goal_1, ..., goal_n.
For example, consider the member predicate in Prolog:
member(E, [E|_]).
member(E, [_|T]) :- member(E, T).
In Fril, this becomes:
((member E (E|_)))
((member E (_|T)) (member E T))
Relation
Some data can be represented in the form of relations. A relation is equivalent to a set of facts with the same predicate name and of constant arity, except that none of the facts can be removed (other than by killing the relation); such a representation consumes less memory internally. A relation is written literally as a list consisting of the predicate name followed by one or more tuples of the relation (all of the arguments of the equivalent fact without the predicate name). A predicate can also be declared a relation by calling the def_rel predicate; this only works if the proposed name does not already exist in the knowledge base. Once a predicate is a relation, anything that would ordinarily add a rule (and does not violate the restrictions of relations) automatically adds a tuple to the relation instead.
Here is an example. The following set of facts:
((my-less-than 2 3))
((my-less-than 8 23))
((my-less-than 42 69))
can be rewri
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https://en.wikipedia.org/wiki/Mathematics%20of%20Sudoku
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Mathematics can be used to study Sudoku puzzles to answer questions such as "How many filled Sudoku grids are there?", "What is the minimal number of clues in a valid puzzle?" and "In what ways can Sudoku grids be symmetric?" through the use of combinatorics and group theory.
The analysis of Sudoku is generally divided between analyzing the properties of unsolved puzzles (such as the minimum possible number of given clues) and analyzing the properties of solved puzzles. Initial analysis was largely focused on enumerating solutions, with results first appearing in 2004.
For classical Sudoku, the number of filled grids is 6,670,903,752,021,072,936,960 (), which reduces to 5,472,730,538 essentially different solutions under the validity preserving transformations. There are 26 possible types of symmetry, but they can only be found in about 0.005% of all filled grids. An ordinary puzzle with a unique solution must have at least 17 clues. There is a solvable puzzle with at most 21 clues for every solved grid. The largest minimal puzzle found so far has 40 clues in the 81 cells.
Similar results are known for variants and smaller grids. No exact results are known for Sudokus larger than the classical 9×9 grid, although there are estimates which are believed to be fairly accurate.
Puzzles
Minimum number of givens
Ordinary Sudokus (proper puzzles) have a unique solution. A minimal Sudoku is a Sudoku from which no clue can be removed leaving it a proper Sudoku. Different minimal Sudokus can have a different number of clues. This section discusses the minimum number of givens for proper puzzles.
Ordinary Sudoku
Many Sudokus have been found with 17 clues, although finding them is not a trivial task. A paper by Gary McGuire, Bastian Tugemann, and Gilles Civario, released on 1 January 2012, explains how it was proved through an exhaustive computer search based on hitting set enumeration that the minimum number of clues in any proper Sudoku is 17.
Symmetrical Sudoku
The fewest clues in a Sudoku with two-way diagonal symmetry (a 180° rotational symmetry) is believed to be 18, and in at least one case such a Sudoku also exhibits automorphism. A Sudoku with 24 clues, dihedral symmetry (a 90° rotational symmetry, which also includes a symmetry on both orthogonal axis, 180° rotational symmetry, and diagonal symmetry) is known to exist, but it is not known if this number of clues is minimal for this class of Sudoku.
Sudokus of other sizes
6×6(2×3) Sudoku: The fewest clues is 8.
8×8(2×4) Sudoku: The fewest clues is 14.
Total number of minimal puzzles
The number of minimal Sudokus (Sudokus in which no clue can be deleted without losing uniqueness of the solution) is not precisely known. However, statistical techniques combined with a generator (), show that there are approximately (with 0.065% relative error):
3.10 × 1037 distinct minimal puzzles,
2.55 × 1025 minimal puzzles that are not pseudo-equivalent (i.e. same arrangement where all instances of one
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https://en.wikipedia.org/wiki/Topological%20K-theory
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In mathematics, topological -theory is a branch of algebraic topology. It was founded to study vector bundles on topological spaces, by means of ideas now recognised as (general) K-theory that were introduced by Alexander Grothendieck. The early work on topological -theory is due to Michael Atiyah and Friedrich Hirzebruch.
Definitions
Let be a compact Hausdorff space and or . Then is defined to be the Grothendieck group of the commutative monoid of isomorphism classes of finite-dimensional -vector bundles over under Whitney sum. Tensor product of bundles gives -theory a commutative ring structure. Without subscripts, usually denotes complex -theory whereas real -theory is sometimes written as . The remaining discussion is focused on complex -theory.
As a first example, note that the -theory of a point is the integers. This is because vector bundles over a point are trivial and thus classified by their rank and the Grothendieck group of the natural numbers is the integers.
There is also a reduced version of -theory, , defined for a compact pointed space (cf. reduced homology). This reduced theory is intuitively modulo trivial bundles. It is defined as the group of stable equivalence classes of bundles. Two bundles and are said to be stably isomorphic if there are trivial bundles and , so that . This equivalence relation results in a group since every vector bundle can be completed to a trivial bundle by summing with its orthogonal complement. Alternatively, can be defined as the kernel of the map induced by the inclusion of the base point into .
-theory forms a multiplicative (generalized) cohomology theory as follows. The short exact sequence of a pair of pointed spaces
extends to a long exact sequence
Let be the -th reduced suspension of a space and then define
Negative indices are chosen so that the coboundary maps increase dimension.
It is often useful to have an unreduced version of these groups, simply by defining:
Here is with a disjoint basepoint labeled '+' adjoined.
Finally, the Bott periodicity theorem as formulated below extends the theories to positive integers.
Properties
(respectively, ) is a contravariant functor from the homotopy category of (pointed) spaces to the category of commutative rings. Thus, for instance, the -theory over contractible spaces is always
The spectrum of -theory is (with the discrete topology on ), i.e. where denotes pointed homotopy classes and is the colimit of the classifying spaces of the unitary groups: Similarly, For real -theory use .
There is a natural ring homomorphism the Chern character, such that is an isomorphism.
The equivalent of the Steenrod operations in -theory are the Adams operations. They can be used to define characteristic classes in topological -theory.
The Splitting principle of topological -theory allows one to reduce statements about arbitrary vector bundles to statements about sums of line bundles.
The Thom isomorphism theorem in topol
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https://en.wikipedia.org/wiki/Ergodic%20sequence
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In mathematics, an ergodic sequence is a certain type of integer sequence, having certain equidistribution properties.
Definition
Let be an infinite, strictly increasing sequence of positive integers. Then, given an integer q, this sequence is said to be ergodic mod q if, for all integers , one has
where
and card is the count (the number of elements) of a set, so that is the number of elements in the sequence A that are less than or equal to t, and
so is the number of elements in the sequence A, less than t, that are equivalent to k modulo q. That is, a sequence is an ergodic sequence if it becomes uniformly distributed mod q as the sequence is taken to infinity.
An equivalent definition is that the sum
vanish for every integer k with .
If a sequence is ergodic for all q, then it is sometimes said to be ergodic for periodic systems.
Examples
The sequence of positive integers is ergodic for all q.
Almost all Bernoulli sequences, that is, sequences associated with a Bernoulli process, are ergodic for all q. That is, let be a probability space of random variables over two letters . Then, given , the random variable is 1 with some probability p and is zero with some probability 1-p; this is the definition of a Bernoulli process. Associated with each is the sequence of integers
Then almost every sequence is ergodic.
See also
Ergodic theory
Ergodic process, for the use of the term in signal processing
Ergodic theory
Integer sequences
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https://en.wikipedia.org/wiki/Tautological%20bundle
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In mathematics, the tautological bundle is a vector bundle occurring over a Grassmannian in a natural tautological way: for a Grassmannian of -dimensional subspaces of , given a point in the Grassmannian corresponding to a -dimensional vector subspace , the fiber over is the subspace itself. In the case of projective space the tautological bundle is known as the tautological line bundle.
The tautological bundle is also called the universal bundle since any vector bundle (over a compact space) is a pullback of the tautological bundle; this is to say a Grassmannian is a classifying space for vector bundles. Because of this, the tautological bundle is important in the study of characteristic classes.
Tautological bundles are constructed both in algebraic topology and in algebraic geometry. In algebraic geometry, the tautological line bundle (as invertible sheaf) is
the dual of the hyperplane bundle or Serre's twisting sheaf . The hyperplane bundle is the line bundle corresponding to the hyperplane (divisor) in . The tautological line bundle and the hyperplane bundle are exactly the two generators of the Picard group of the projective space.
In Michael Atiyah's "K-theory", the tautological line bundle over a complex projective space is called the standard line bundle. The sphere bundle of the standard bundle is usually called the Hopf bundle. (cf. Bott generator.)
More generally, there are also tautological bundles on a projective bundle of a vector bundle as well as a Grassmann bundle.
The older term canonical bundle has dropped out of favour, on the grounds that canonical is heavily overloaded as it is, in mathematical terminology, and (worse) confusion with the canonical class in algebraic geometry could scarcely be avoided.
Intuitive definition
Grassmannians by definition are the parameter spaces for linear subspaces, of a given dimension, in a given vector space . If is a Grassmannian, and is the subspace of corresponding to in , this is already almost the data required for a vector bundle: namely a vector space for each point , varying continuously. All that can stop the definition of the tautological bundle from this indication, is the difficulty that the are going to intersect. Fixing this up is a routine application of the disjoint union device, so that the bundle projection is from a total space made up of identical copies of the , that now do not intersect. With this, we have the bundle.
The projective space case is included. By convention may usefully carry the tautological bundle in the dual space sense. That is, with the dual space, points of carry the vector subspaces of that are their kernels, when considered as (rays of) linear functionals on . If has dimension , the tautological line bundle is one tautological bundle, and the other, just described, is of rank .
Formal definition
Let be the Grassmannian of n-dimensional vector subspaces in as a set it is the set of all n-dimensional vector subspaces of For
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https://en.wikipedia.org/wiki/Chinaman%27s%20chance
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Chinaman's chance is an American idiom which means that a person has little or no chance at success, synonymous with similar idioms of improbability such as a snowball's chance in hell or when pigs fly. Although the origin of the phrase is unclear, it may refer to the historical misfortunes which were suffered by Chinese-American immigrants.
Meaning
The idiom is defined as meaning "no chance at all" in The Columbia Guide to Standard American English. The phrase is sometimes used in extended form as not a Chinaman's chance or not a Chinaman's chance in hell, indicating that the lives and safety of Chinese immigrants were not valued or fair treatment of Chinese immigrants was impossible.
Potential origins
The origin of the phrase is not well documented. In The Chinese looking glass (1967), Dennis Bloodworth asserts the Chinese people have a long association with gambling. He states they believe "it is better to be lucky than clever", concluding the I Ching has encouraged the acceptance of chance and fate: "the philosophy that makes the Chinese the soothsayer's best customer makes him one of the world's outstanding gamblers, too." This in turn resulted in a resilient attitude in response to misfortune and bad luck: "in the course of a long and turbulent history [the Chinese have] sooner or later overthrown by violence every single major dynasty that [have] misruled [them]".
Two potential origins of the phrase have been advanced, depending on the capitalization of the term. Either the phrase is written with the pejorative (capitalized) term "Chinaman", and Chinaman's chance refers to the treatment of Chinese immigrants to America in the 1800s, or the neutral (lower case) "chinaman" is used instead, with chinaman's chance referring to the fragility of fine porcelain.
Chinese immigration
One early potential origin for the phrase is from the California Gold Rush of 1849. The travel time for news of the gold rush to reach China was quite long, and by the time Chinese immigrants arrived to prospect, many of the rich mines were already claimed. These Chinese immigrants who missed out had to work with only those lands which had already been exploited or which were rejected by others, meaning these late-arriving immigrants had a slim chance of success. The historical record, however, indicates that many Chinese combined efforts with each other and did very well in the goldfields, introducing mining techniques then unknown to non-Chinese. Alternatively, in 1920 the phrase was explained to describe the low probability for the Chinese in America to make a fortune at gold mining. Although there were Chinese in the gold mining camps soon after the news broke, "they were extremely unpopular [and] the slightest excuse was sufficient to warrant their being beaten or chased away; consequently they had no chance to get a real foothold" to establish mining rights.
Another potential origin of the phrase Chinaman's chance traces it to the high probability of death
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https://en.wikipedia.org/wiki/Milnor%20conjecture
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In mathematics, the Milnor conjecture was a proposal by of a description of the Milnor K-theory (mod 2) of a general field F with characteristic different from 2, by means of the Galois (or equivalently étale) cohomology of F with coefficients in Z/2Z. It was proved by .
Statement
Let F be a field of characteristic different from 2. Then there is an isomorphism
for all n ≥ 0, where KM denotes the Milnor ring.
About the proof
The proof of this theorem by Vladimir Voevodsky uses several ideas developed by Voevodsky, Alexander Merkurjev, Andrei Suslin, Markus Rost, Fabien Morel, Eric Friedlander, and others, including the newly minted theory of motivic cohomology (a kind of substitute for singular cohomology for algebraic varieties) and the motivic Steenrod algebra.
Generalizations
The analogue of this result for primes other than 2 was known as the Bloch–Kato conjecture. Work of Voevodsky and Markus Rost yielded a complete proof of this conjecture in 2009; the result is now called the norm residue isomorphism theorem.
References
Further reading
K-theory
Conjectures that have been proved
Theorems in algebraic topology
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https://en.wikipedia.org/wiki/Milnor%20K-theory
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In mathematics, Milnor K-theory is an algebraic invariant (denoted for a field ) defined by as an attempt to study higher algebraic K-theory in the special case of fields. It was hoped this would help illuminate the structure for algebraic and give some insight about its relationships with other parts of mathematics, such as Galois cohomology and the Grothendieck–Witt ring of quadratic forms. Before Milnor K-theory was defined, there existed ad-hoc definitions for and . Fortunately, it can be shown Milnor is a part of algebraic , which in general is the easiest part to compute.
Definition
Motivation
After the definition of the Grothendieck group of a commutative ring, it was expected there should be an infinite set of invariants called higher groups, from the fact there exists a short exact sequence
which should have a continuation by a long exact sequence. Note the group on the left is relative . This led to much study and as a first guess for what this theory would look like, Milnor gave a definition for fields. His definition is based upon two calculations of what higher "should" look like in degrees and . Then, if in a later generalization of algebraic was given, if the generators of lived in degree and the relations in degree , then the constructions in degrees and would give the structure for the rest of the ring. Under this assumption, Milnor gave his "ad-hoc" definition. It turns out algebraic in general has a more complex structure, but for fields the Milnor groups are contained in the general algebraic groups after tensoring with , i.e. . It turns out the natural map fails to be injective for a global field pg 96.
Definition
Note for fields the Grothendieck group can be readily computed as since the only finitely generated modules are finite-dimensional vector spaces. Also, Milnor's definition of higher depends upon the canonical isomorphism
(the group of units of ) and observing the calculation of K2 of a field by Hideya Matsumoto, which gave the simple presentation
for a two-sided ideal generated by elements , called Steinberg relations. Milnor took the hypothesis that these were the only relations, hence he gave the following "ad-hoc" definition of Milnor K-theory as
The direct sum of these groups is isomorphic to a tensor algebra over the integers of the multiplicative group modded out by the two-sided ideal generated by:
so
showing his definition is a direct extension of the Steinberg relations.
Properties
Ring structure
The graded module is a graded-commutative ringpg 1-3. If we write
as
then for and we have
From the proof of this property, there are some additional properties which fall out, like for since . Also, if of non-zero fields elements equals , then There's a direct arithmetic application: is a sum of squares if and only if every positive dimensional is nilpotent, which is a powerful statement about the structure of Milnor . In particular, for the fields , with , all of
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https://en.wikipedia.org/wiki/Spin%20connection
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In differential geometry and mathematical physics, a spin connection is a connection on a spinor bundle. It is induced, in a canonical manner, from the affine connection. It can also be regarded as the gauge field generated by local Lorentz transformations. In some canonical formulations of general relativity, a spin connection is defined on spatial slices and can also be regarded as the gauge field generated by local rotations.
The spin connection occurs in two common forms: the Levi-Civita spin connection, when it is derived from the Levi-Civita connection, and the affine spin connection, when it is obtained from the affine connection. The difference between the two of these is that the Levi-Civita connection is by definition the unique torsion-free connection, whereas the affine connection (and so the affine spin connection) may contain torsion.
Definition
Let be the local Lorentz frame fields or vierbein (also known as a tetrad), which is a set of orthonormal space time vector fields that diagonalize the metric tensor
where is the spacetime metric and is the Minkowski metric. Here, Latin letters denote the local Lorentz frame indices; Greek indices denote general coordinate indices. This simply expresses that , when written in terms of the basis , is locally flat. The Greek vierbein indices can be raised or lowered by the metric, i.e. or . The Latin or "Lorentzian" vierbein indices can be raised or lowered by or respectively. For example, and
The torsion-free spin connection is given by
where are the Christoffel symbols. This definition should be taken as defining the torsion-free spin connection, since, by convention, the Christoffel symbols are derived from the Levi-Civita connection, which is the unique metric compatible, torsion-free connection on a Riemannian Manifold. In general, there is no restriction: the spin connection may also contain torsion.
Note that using the gravitational covariant derivative of the contravariant vector . The spin connection may be written purely in terms of the vierbein field as
which by definition is anti-symmetric in its internal indices .
The spin connection defines a covariant derivative on generalized tensors. For example, its action on is
Cartan's structure equations
In the Cartan formalism, the spin connection is used to define both torsion and curvature. These are easiest to read by working with differential forms, as this hides some of the profusion of indexes. The equations presented here are effectively a restatement of those that can be found in the article on the connection form and the curvature form. The primary difference is that these retain the indexes on the vierbein, instead of completely hiding them. More narrowly, the Cartan formalism is to be interpreted in its historical setting, as a generalization of the idea of an affine connection to a homogeneous space; it is not yet as general as the idea of a principal connection on a fiber bundle. It serves as a suit
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https://en.wikipedia.org/wiki/Orbifold%20notation
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In geometry, orbifold notation (or orbifold signature) is a system, invented by the mathematician William Thurston and promoted by John Conway, for representing types of symmetry groups in two-dimensional spaces of constant curvature. The advantage of the notation is that it describes these groups in a way which indicates many of the groups' properties: in particular, it follows William Thurston in describing the orbifold obtained by taking the quotient of Euclidean space by the group under consideration.
Groups representable in this notation include the point groups on the sphere (), the frieze groups and wallpaper groups of the Euclidean plane (), and their analogues on the hyperbolic plane ().
Definition of the notation
The following types of Euclidean transformation can occur in a group described by orbifold notation:
reflection through a line (or plane)
translation by a vector
rotation of finite order around a point
infinite rotation around a line in 3-space
glide-reflection, i.e. reflection followed by translation.
All translations which occur are assumed to form a discrete subgroup of the group symmetries being described.
Each group is denoted in orbifold notation by a finite string made up from the following symbols:
positive integers
the infinity symbol,
the asterisk, *
the symbol o (a solid circle in older documents), which is called a wonder and also a handle because it topologically represents a torus (1-handle) closed surface. Patterns repeat by two translation.
the symbol (an open circle in older documents), which is called a miracle and represents a topological crosscap where a pattern repeats as a mirror image without crossing a mirror line.
A string written in boldface represents a group of symmetries of Euclidean 3-space. A string not written in boldface represents a group of symmetries of the Euclidean plane, which is assumed to contain two independent translations.
Each symbol corresponds to a distinct transformation:
an integer n to the left of an asterisk indicates a rotation of order n around a gyration point
the asterisk, * indicates a reflection
an integer n to the right of an asterisk indicates a transformation of order 2n which rotates around a kaleidoscopic point and reflects through a line (or plane)
an indicates a glide reflection
the symbol indicates infinite rotational symmetry around a line; it can only occur for bold face groups. By abuse of language, we might say that such a group is a subgroup of symmetries of the Euclidean plane with only one independent translation. The frieze groups occur in this way.
the exceptional symbol o indicates that there are precisely two linearly independent translations.
Good orbifolds
An orbifold symbol is called good if it is not one of the following: p, pq, *p, *pq, for p, q ≥ 2, and p ≠ q.
Chirality and achirality
An object is chiral if its symmetry group contains no reflections; otherwise it is called achiral. The corresponding orbifold i
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https://en.wikipedia.org/wiki/Dissection%20problem
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In geometry, a dissection problem is the problem of partitioning a geometric figure (such as a polytope or ball) into smaller pieces that may be rearranged into a new figure of equal content. In this context, the partitioning is called simply a dissection (of one polytope into another). It is usually required that the dissection use only a finite number of pieces. Additionally, to avoid set-theoretic issues related to the Banach–Tarski paradox and Tarski's circle-squaring problem, the pieces are typically required to be well-behaved. For instance, they may be restricted to being the closures of disjoint open sets.
The Bolyai–Gerwien theorem states that any polygon may be dissected into any other polygon of the same area, using interior-disjoint polygonal pieces. It is not true, however, that any polyhedron has a dissection into any other polyhedron of the same volume using polyhedral pieces (see Dehn invariant). This process is possible, however, for any two honeycombs (such as cube) in three dimension and any two zonohedra of equal volume (in any dimension).
A partition into triangles of equal area is called an equidissection. Most polygons cannot be equidissected, and those that can often have restrictions on the possible numbers of triangles. For example, Monsky's theorem states that there is no odd equidissection of a square.
See also
Dissection puzzle
Hilbert's third problem
Hinged dissection
References
External links
David Eppstein, Dissection Tiling.
Discrete geometry
Euclidean geometry
Geometric dissection
Polygons
Polyhedra
Polytopes
Mathematical problems
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https://en.wikipedia.org/wiki/Dissection%20%28disambiguation%29
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Dissection is the dismembering of the body of a deceased animal or plant to study its anatomical structure.
Dissection may also refer to:
The dissection problem in geometry
Dissection (medical), a tear in a blood vessel
Dissection (band), a Swedish extreme metal band
Dissection (album), a 1997 Crimson Thorn album
Dissected plateau, a plateau area
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https://en.wikipedia.org/wiki/List%20of%20combinatorial%20computational%20geometry%20topics
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List of combinatorial computational geometry topics enumerates the topics of computational geometry that states problems in terms of geometric objects as discrete entities and hence the methods of their solution are mostly theories and algorithms of combinatorial character.
See List of numerical computational geometry topics for another flavor of computational geometry that deals with geometric objects as continuous entities and applies methods and algorithms of nature characteristic to numerical analysis.
Construction/representation
Boolean operations on polygons
Convex hull
Hyperplane arrangement
Polygon decomposition
Polygon triangulation
Minimal convex decomposition
Minimal convex cover problem (NP-hard)
Minimal rectangular decomposition
Tessellation problems
Shape dissection problems
Straight skeleton
Stabbing line problem
Triangulation
Delaunay triangulation
Point-set triangulation
Polygon triangulation
Voronoi diagram
Extremal shapes
Minimum bounding box (Smallest enclosing box, Smallest bounding box)
2-D case: Smallest bounding rectangle (Smallest enclosing rectangle)
There are two common variants of this problem.
In many areas of computer graphics, the bounding box (often abbreviated to bbox) is understood to be the smallest box delimited by sides parallel to coordinate axes which encloses the objects in question.
In other applications, such as packaging, the problem is to find the smallest box the object (or objects) may fit in ("packaged"). Here the box may assume an arbitrary orientation with respect to the "packaged" objects.
Smallest bounding sphere (Smallest enclosing sphere)
2-D case: Smallest bounding circle
Largest empty rectangle (Maximum empty rectangle)
Largest empty sphere
2-D case: Maximum empty circle (largest empty circle)
Interaction/search
Collision detection
Line segment intersection
Point location
Point in polygon
Polygon intersection
Range searching
Orthogonal range searching
Simplex range searching
Ray casting (not to be confused with ray tracing of computer graphics)
Slab method
Proximity problems
Closest pair of points
Closest point problem
Diameter of a point set
Delaunay triangulation
Voronoi diagram
Visibility
Visibility (geometry)
Art gallery problem (The museum problem)
Visibility graph
Watchman route problem
Computer graphics applications:
Hidden surface determination
Hidden line removal
Ray casting (not to be confused with ray tracing of computer graphics)
Other
Happy ending problem
Ham sandwich problem
shape assembly problems
shape matching problems
Klee's measure problem
Problems on isothetic polygons and isothetic polyhedra
Orthogonal convex hull
Path planning
Paths among obstacles
Shortest path in a polygon
Polygon containment
Robust geometric computation addresses two main issues: fixed-precision representation of real numbers in computers and possible geometrical degeneracy (mathematics) of input data
Computational geometry
Computationa
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https://en.wikipedia.org/wiki/List%20of%20numerical%20computational%20geometry%20topics
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List of numerical computational geometry topics enumerates the topics of computational geometry that deals with geometric objects as continuous entities and applies methods and algorithms of nature characteristic to numerical analysis. This area is also called "machine geometry", computer-aided geometric design, and geometric modelling.
See List of combinatorial computational geometry topics for another flavor of computational geometry that states problems in terms of geometric objects as discrete entities and hence the methods of their solution are mostly theories and algorithms of combinatorial character.
Curves
In the list of curves topics, the following ones are fundamental to geometric modelling.
Parametric curve
Bézier curve
Spline
Hermite spline
Beta spline
B-spline
Higher-order spline
NURBS
Contour line
Surfaces
Bézier surface
Isosurface
Parametric surface
Other
Level-set method
Computational topology
Mathematics-related lists
Geometric algorithms
Geometry
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https://en.wikipedia.org/wiki/Cabri%20Geometry
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Cabri Geometry is a commercial interactive geometry software produced by the French company Cabrilog for teaching and learning geometry and trigonometry. It was designed with ease-of-use in mind. The program allows the user to animate geometric figures, proving a significant advantage over those drawn on a blackboard. Relationships between points on a geometric object may easily be demonstrated, which can be useful in the learning process. There are also graphing and display functions which allow exploration of the connections between geometry and algebra. The program can be run under Windows or the Mac OS.
See also
Interactive geometry software – alternatives to Cabri Geometry
References
External links
Cabri Geometry
Cabri belongs to the Inter2Geo European project aiming at interoperability between interactive geometry software.
Interactive geometry software
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https://en.wikipedia.org/wiki/David%20Salsburg
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David S. Salsburg (born 1931) is an author. His 2002 book The Lady Tasting Tea, subtitled How Statistics Revolutionized Science in the Twentieth Century, provides a layman's overview of important developments in the field of statistics in the late 19th and early 20th century, particularly in the areas of experiment design, the study of random distributions, and the careers of major researchers in the field such as Ronald Fisher, Karl Pearson, and Jerzy Neyman.
Salsburg is a retired pharmaceutical company statistician (having been a senior research fellow at Pfizer's central research department until 1995) who has taught at Harvard, Yale, Connecticut College, the University of Connecticut, the University of Pennsylvania, Rhode Island College, and Trinity College and has been a Fellow of the American Statistical Association since 1978. Salsburg was also the first statistician hired by Pfizer. In 1994, Salsburg was awarded the Career Achievement Award of the Biostatistics Section of the Pharmaceutical Research and Manufacturers of America, given annually for "significant contributions to the advancement of biostatistics in the pharmaceutical industry". The Mathematical Association of America characterised him as follows
"Salsburg believes that the public is not fully aware of the degree to which recent developments in statistics impact the way we perceive the world. He correctly points out that the twentieth century saw the fading of a deterministic outlook and the rise of a statistical/probabilistic way of looking at the world. This ongoing revolution is not only in the physical sciences, it also touches the social sciences and even the humanities. Though profound, it is a quiet revolution that has been unnoticed by many."
Salsburg's most recent book, Errors, Blunders and Lies: How to Tell the Difference was published in 2017.
Publications
1. Understanding Randomness: Exercises for Statisticians (Lecture Notes in Statistics) (1983)
2. Statistics for Toxicologists (Drug and Chemical Toxicology) (1986)
3. The Use of Restricted Significance Tests in Clinical Trials (Statistics for Biology and Health) (1992)
4. The Lady Tasting Tea: How Statistics Revolutionized Science in the Twentieth Century (2002)
5. When the Band has Ceased to Play, American Presidents after Leaving Office (2013)
6. Jonah in the Garden of Eden: a statistical investigation of the Hebrew Bible (2013)
7. Love Feeds Among the Lilies (2013)
8. Errors, Blunders and Lies: How to Tell the Difference (ASA-CRC Series on Statistical Reasoning in Science and Society) (2017)
References
External links
Photograph of David Salsburg, from near the bottom of this web page
American statisticians
1931 births
Living people
Harvard University staff
University of Pennsylvania staff
20th-century American non-fiction writers
21st-century American non-fiction writers
Yale University staff
21st-century American historians
21st-century American male writers
20th-century American mathematicians
Fellow
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https://en.wikipedia.org/wiki/Separatrix%20%28mathematics%29
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In mathematics, a separatrix is the boundary separating two modes of behaviour in a differential equation.
Example: simple pendulum
Consider the differential equation describing the motion of a simple pendulum:
where denotes the length of the pendulum, the gravitational acceleration and the angle between the pendulum and vertically downwards. In this system there is a conserved quantity H (the Hamiltonian), which is given by
With this defined, one can plot a curve of constant H in the phase space of system. The phase space is a graph with along the horizontal axis and on the vertical axis – see the thumbnail to the right. The type of resulting curve depends upon the value of H.
If then no curve exists (because must be imaginary).
If then the curve will be a simple closed curve which is nearly circular for small H and becomes "eye" shaped when H approaches the upper bound. These curves correspond to the pendulum swinging periodically from side to side.
If then the curve is open, and this corresponds to the pendulum forever swinging through complete circles.
In this system the separatrix is the curve that corresponds to . It separates — hence the name — the phase space into two distinct areas, each with a distinct type of motion. The region inside the separatrix has all those phase space curves which correspond to the pendulum oscillating back and forth, whereas the region outside the separatrix has all the phase space curves which correspond to the pendulum continuously turning through vertical planar circles.
Example: FitzHugh–Nagumo model
In the FitzHugh–Nagumo model, when the linear nullcline pierces the cubic nullcline at the left, middle, and right branch once each, the system has a separatrix. Trajectories to the left of the separatrix converge to the left stable equilibrium, and similarly for the right. The separatrix itself is the stable manifold for the saddle point in the middle. Details are found in the page.
The separatrix is clearly visible by numerically solving for trajectories backwards in time. Since when solving for the trajectories forwards in time, trajectories diverge from the separatrix, when solving backwards in time, trajectories converge to the separatrix.
References
Logan, J. David, Applied Mathematics, 3rd Ed., 2006, John Wiley and Sons, Hoboken, NJ, pg. 65.
External links
Separatrix from MathWorld.
Dynamical systems
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https://en.wikipedia.org/wiki/Dihedral%20symmetry%20in%20three%20dimensions
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In geometry, dihedral symmetry in three dimensions is one of three infinite sequences of point groups in three dimensions which have a symmetry group that as an abstract group is a dihedral group Dihn (for n ≥ 2).
Types
There are 3 types of dihedral symmetry in three dimensions, each shown below in 3 notations: Schönflies notation, Coxeter notation, and orbifold notation.
Chiral
Dn, [n,2]+, (22n) of order 2n – dihedral symmetry or para-n-gonal group (abstract group: Dihn).
Achiral
Dnh, [n,2], (*22n) of order 4n – prismatic symmetry or full ortho-n-gonal group (abstract group: Dihn × Z2).
Dnd (or Dnv), [2n,2+], (2*n) of order 4n – antiprismatic symmetry or full gyro-n-gonal group (abstract group: Dih2n).
For a given n, all three have n-fold rotational symmetry about one axis (rotation by an angle of 360°/n does not change the object), and 2-fold rotational symmetry about a perpendicular axis, hence about n of those. For n = ∞, they correspond to three Frieze groups. Schönflies notation is used, with Coxeter notation in brackets, and orbifold notation in parentheses. The term horizontal (h) is used with respect to a vertical axis of rotation.
In 2D, the symmetry group Dn includes reflections in lines. When the 2D plane is embedded horizontally in a 3D space, such a reflection can either be viewed as the restriction to that plane of a reflection through a vertical plane, or as the restriction to the plane of a rotation about the reflection line, by 180°. In 3D, the two operations are distinguished: the group Dn contains rotations only, not reflections. The other group is pyramidal symmetry Cnv of the same order, 2n.
With reflection symmetry in a plane perpendicular to the n-fold rotation axis, we have Dnh, [n], (*22n).
Dnd (or Dnv), [2n,2+], (2*n) has vertical mirror planes between the horizontal rotation axes, not through them. As a result, the vertical axis is a 2n-fold rotoreflection axis.
Dnh is the symmetry group for a regular n-sided prism and also for a regular n-sided bipyramid. Dnd is the symmetry group for a regular n-sided antiprism, and also for a regular n-sided trapezohedron. Dn is the symmetry group of a partially rotated prism.
n = 1 is not included because the three symmetries are equal to other ones:
D1 and C2: group of order 2 with a single 180° rotation.
D1h and C2v: group of order 4 with a reflection in a plane and a 180° rotation about a line in that plane.
D1d and C2h: group of order 4 with a reflection in a plane and a 180° rotation about a line perpendicular to that plane.
For n = 2 there is not one main axis and two additional axes, but there are three equivalent ones.
D2, [2,2]+, (222) of order 4 is one of the three symmetry group types with the Klein four-group as abstract group. It has three perpendicular 2-fold rotation axes. It is the symmetry group of a cuboid with an S written on two opposite faces, in the same orientation.
D2h, [2,2], (*222) of order 8 is the symmetry group of a cuboid.
D2d, [4,2+], (2*2
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https://en.wikipedia.org/wiki/The%20Lady%20Tasting%20Tea
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The Lady Tasting Tea: How Statistics Revolutionized Science in the Twentieth Century () is a book by David Salsburg about the history of modern statistics and the role it played in the development of science and industry.
The title comes from the "lady tasting tea", an example from the famous book, The Design of Experiments, by Ronald A. Fisher. Regarding Fisher's example, the statistician Debabrata Basu wrote that "the famous case of the 'lady tasting tea'" was "one of the two supporting pillars [...] of the randomization analysis of experimental data".
Summary
The book discusses the statistical revolution which took place in the twentieth century, where science shifted from a deterministic view (Clockwork universe) to a perspective concerned primarily with probabilities and distributions and parameters. Salsburg does this through a collection of stories about the people who were fundamental in the change, starting with men like R.A. Fisher and Karl Pearson. He discusses at length how many of these people had their own philosophy of statistics, and in particular their own understanding of statistical significance. Throughout, he introduces in a very nontechnical fashion a variety of statistical ideas and methods, such as maximum likelihood estimation and bootstrapping.
Reception
The book was generally well-received, receiving coverage in a variety of medical and statistical journals. Reviewers from the medical field enjoyed Salsburg's coverage of Fisher's opposition to early research on the health effects of tobacco. Critics disagreed with certain opinions that Salsburg voiced, like his barebones portrayal of Bayesian statistics and his seeming disdain for pure mathematics. Nevertheless, almost all reviewers appreciated the interesting read and recommended the book to people in their field as well as a general audience.
List of scholars mentioned
The book discusses a wide variety of statisticians, mathematicians, as well as other scientists and scholars. This is a list of those mentioned, broken down into groups of chapters.
Chapters 1-9
Chapters 10-19
Chapters 20-29
References
External links
Publisher's web page
2002 non-fiction books
Statistics books
History of probability and statistics
Henry Holt and Company books
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https://en.wikipedia.org/wiki/Interprime
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In mathematics, an interprime is the average of two consecutive odd primes. For example, 9 is an interprime because it is the average of 7 and 11. The first interprimes are:
4, 6, 9, 12, 15, 18, 21, 26, 30, 34, 39, 42, 45, 50, 56, 60, 64, 69, 72, 76, 81, 86, 93, 99, ...
Interprimes cannot be prime themselves (otherwise the primes would not have been consecutive).
There are infinitely many primes and therefore also infinitely many interprimes. The largest known interprime may be the 388342-digit n = 2996863034895 · 21290000, where n + 1 is the largest known twin prime.
See also
Prime gap
Twin primes
Cousin prime
Sexy prime
Balanced prime – a prime number with equal-sized prime gaps above and below it
References
Integer sequences
Prime numbers
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https://en.wikipedia.org/wiki/Odious%20number
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In number theory, an odious number is a positive integer that has an odd number of 1s in its binary expansion. Non-negative integers that are not odious are called evil numbers.
In computer science, an odious number is said to have odd parity.
Examples
The first odious numbers are:
Properties
If denotes the th odious number (with ), then for all , .
Every positive integer has an odious multiple that is at most . The numbers for which this bound is tight are exactly the Mersenne numbers with even exponents, the numbers of the form , such as 3, 15, 63, etc. For these numbers, the smallest odious multiple is exactly .
Related sequences
The odious numbers give the positions of the nonzero values in the Thue–Morse sequence. Every power of two is odious, because its binary expansion has only one nonzero bit. Except for 3, every Mersenne prime is odious, because its binary expansion consists of an odd prime number of consecutive nonzero bits.
Non-negative integers that are not odious are called evil numbers. The partition of the non-negative integers into the odious and evil numbers is the unique partition of these numbers into two sets that have equal multisets of pairwise sums.
References
External links
Integer sequences
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https://en.wikipedia.org/wiki/Cyclic%20symmetry%20in%20three%20dimensions
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In three dimensional geometry, there are four infinite series of point groups in three dimensions (n≥1) with n-fold rotational or reflectional symmetry about one axis (by an angle of 360°/n) that does not change the object.
They are the finite symmetry groups on a cone. For n = ∞ they correspond to four frieze groups. Schönflies notation is used. The terms horizontal (h) and vertical (v) imply the existence and direction of reflections with respect to a vertical axis of symmetry. Also shown are Coxeter notation in brackets, and, in parentheses, orbifold notation.
Types
Chiral
Cn, [n]+, (nn) of order n - n-fold rotational symmetry - acro-n-gonal group (abstract group Zn); for n=1: no symmetry (trivial group)
AchiralCnh, [n+,2], (n*) of order 2n - prismatic symmetry or ortho-n-gonal group (abstract group Zn × Dih1); for n=1 this is denoted by Cs (1*) and called reflection symmetry, also bilateral symmetry. It has reflection symmetry with respect to a plane perpendicular to the n-fold rotation axis.Cnv, [n], (*nn) of order 2n - pyramidal symmetry or full acro-n-gonal group (abstract group Dihn); in biology C2v is called biradial symmetry. For n=1 we have again Cs (1*). It has vertical mirror planes. This is the symmetry group for a regular n-sided pyramid.S2n, [2+,2n+], (n×) of order 2n - gyro-n-gonal group (not to be confused with symmetric groups, for which the same notation is used; abstract group Z2n); It has a 2n-fold rotoreflection axis, also called 2n-fold improper rotation axis, i.e., the symmetry group contains a combination of a reflection in the horizontal plane and a rotation by an angle 180°/n. Thus, like Dnd, it contains a number of improper rotations without containing the corresponding rotations.
for n=1 we have S2 (1×), also denoted by Ci; this is inversion symmetry.
C2h, [2,2+] (2*) and C2v, [2], (*22) of order 4 are two of the three 3D symmetry group types with the Klein four-group as abstract group. C2v applies e.g. for a rectangular tile with its top side different from its bottom side.
Frieze groups
In the limit these four groups represent Euclidean plane frieze groups as C∞, C∞h, C∞v, and S∞. Rotations become translations in the limit. Portions of the infinite plane can also be cut and connected into an infinite cylinder.
Examples
See also
Dihedral symmetry in three dimensions
References
On Quaternions and Octonions, 2003, John Horton Conway and Derek A. Smith
The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass,
Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
N.W. Johnson: Geometries and Transformations, (2018) Chapter 11: Finite symmetry groups, 11.5 Spherical Coxeter groups
Symmetry
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https://en.wikipedia.org/wiki/Square%20root%20of%20a%20matrix
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In mathematics, the square root of a matrix extends the notion of square root from numbers to matrices. A matrix is said to be a square root of if the matrix product is equal to .
Some authors use the name square root or the notation only for the specific case when is positive semidefinite, to denote the unique matrix that is positive semidefinite and such that (for real-valued matrices, where is the transpose of ).
Less frequently, the name square root may be used for any factorization of a positive semidefinite matrix as , as in the Cholesky factorization, even if . This distinct meaning is discussed in .
Examples
In general, a matrix can have several square roots. In particular, if then as well.
The 2×2 identity matrix has infinitely many square roots. They are given by
and
where are any numbers (real or complex) such that .
In particular if is any Pythagorean triple—that is, any set of positive integers such that , then
is a square root matrix of which is symmetric and has rational entries.
Thus
Minus identity has a square root, for example:
which can be used to represent the imaginary unit and hence all complex numbers using 2×2 real matrices, see Matrix representation of complex numbers.
Just as with the real numbers, a real matrix may fail to have a real square root, but have a square root with complex-valued entries.
Some matrices have no square root. An example is the matrix
While the square root of a nonnegative integer is either again an integer or an irrational number, in contrast an integer matrix can have a square root whose entries are rational, yet non-integral, as in examples above.
Positive semidefinite matrices
A symmetric real n × n matrix is called positive semidefinite if for all (here denotes the transpose, changing a column vector into a row vector).
A square real matrix is positive semidefinite if and only if for some matrix .
There can be many different such matrices .
A positive semidefinite matrix can also have many matrices such that .
However, always has precisely one square root that is positive semidefinite (and hence symmetric).
In particular, since is required to be symmetric, , so the two conditions or are equivalent.
For complex-valued matrices, the conjugate transpose is used instead and positive semidefinite matrices are Hermitian, meaning .
This unique matrix is called the principal, non-negative, or positive square root (the latter in the case of positive definite matrices).
The principal square root of a real positive semidefinite matrix is real.
The principal square root of a positive definite matrix is positive definite; more generally, the rank of the principal square root of is the same as the rank of .
The operation of taking the principal square root is continuous on this set of matrices. These properties are consequences of the holomorphic functional calculus applied to matrices.
The existence and uniqueness of the principal square root can be de
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https://en.wikipedia.org/wiki/Zvi%20Hecker
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Zvi Hecker (; 31 May 1931 – 24 September 2023) was a Polish-born Israeli architect. His work is known for its emphasis on geometry and asymmetry.
Biography
Zvi Hecker was born as Tadeusz Hecker in Kraków, Poland. He grew up in Poland and Samarkand. He began his education in architecture at the Cracow University of Technology. He immigrated to Israel in 1950. There he studied architecture at the Technion - Israel Institute of Technology, graduating in 1955. At the Technion, Eldar Sharon was a classmate, and Alfred Neumann was their professor. Between 1955 and 1957, he studied painting at the Avni Institute of Art and Design, before beginning his career as an architect. Between 1957 and 1959, Hecker served in the Combat Engineering Corps of the Israel Defense Forces.
Hecker died on 24 September 2023, at the age of 92.
Architectural career
After his military service, he founded a firm with Eldar Sharon (until 1964) and Alfred Neumann (until 1966). The physical and economic conditions in Israel at the time, allowed them to complete a fair number of works in a relatively brief period of time, which brought international attention. Their joint works include the Mediterranean Sea Club in Achzib (1960–1961), Dubiner House (1963), the Chaim Laskov Officer Training School (1963–1967) Bahad 1, the main officer training school of the Israel Defense Forces, just later the synagogue (1969–1971) at the same academy, and the Bat Yam city hall (1963–1969). Their designs shared aspects in common with the metabolist movement, borrowing metaphoric shapes from nature for use in planning morphological structures. The modularity of these works, such as the Dubiner House, provided an architectural precedent for the Habitat 67 project by Moshe Safdie.
Hecker resided in Berlin and Tel Aviv. He was involved in planning projects for the German Jewish community as well as other international projects.
Academic career
Hecker taught in Canada, the United States, Israel, and Austria at the Université Laval, University of Texas at Arlington, Washington University in St. Louis, Iowa State University, Technion – Israel Institute of Technology, and the University of Applied Arts Vienna. Zvi Hecker wrote about his work periodically, co-authoring books with Sir Peter Cook, John Hejduk, and others.
Projects
The early projects of Zvi Hecker, designed in partnership with Sharon and Neumann, have architectural qualities that were developed later in his career. The officer school (Bahad 1) was built to give a respectable living environment to soldiers in the Negev desert, and special emphasis was given to the large spaces between the structures, in order to form a micro-environment there, separating the people inside from the harsh desert outskirts. Raw concrete was chosen because it did not require constant maintenance and renovation in light of the strong sandy winds. About the school, Hecker said: "The location of the base has a special relevance to the vision of David Ben-Guri
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https://en.wikipedia.org/wiki/MIMA
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MIMA may refer to:
Member of the Institute of Mathematics and its Applications
MiMA (building), an apartment building whose name means Middle of Manhattan, New York City, United States
Middlesbrough Institute of Modern Art, art gallery in Middlesbrough, England
Modern Improvisational Music Association, a public charity in New Jersey, United States
Multicultural and Indigenous Media Awards, former name of NSW Premier's Multicultural Communication Awards, Australia
Millennium Iconoclast Museum of Art, museum in Brussels
See also
Mima (disambiguation)
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https://en.wikipedia.org/wiki/Logarithm%20of%20a%20matrix
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In mathematics, a logarithm of a matrix is another matrix such that the matrix exponential of the latter matrix equals the original matrix. It is thus a generalization of the scalar logarithm and in some sense an inverse function of the matrix exponential. Not all matrices have a logarithm and those matrices that do have a logarithm may have more than one logarithm. The study of logarithms of matrices leads to Lie theory since when a matrix has a logarithm then it is in an element of a Lie group and the logarithm is the corresponding element of the vector space of the Lie algebra.
Definition
The exponential of a matrix A is defined by
.
Given a matrix B, another matrix A is said to be a matrix logarithm of .
Because the exponential function is not bijective for complex numbers (e.g. ), numbers can have multiple complex logarithms, and as a consequence of this, some matrices may have more than one logarithm, as explained below. If the matrix logarithm of exists and is unique, then it is written as in which case
Power series expression
If B is sufficiently close to the identity matrix, then a logarithm of B may be computed by means of the following power series:
.
Specifically, if , then the preceding series converges and .
Example: Logarithm of rotations in the plane
The rotations in the plane give a simple example. A rotation of angle α around the origin is represented by the 2×2-matrix
For any integer n, the matrix
is a logarithm of A.
⇔
where
…
qed.
Thus, the matrix A has infinitely many logarithms. This corresponds to the fact that the rotation angle is only determined up to multiples of 2π.
In the language of Lie theory, the rotation matrices A are elements of the Lie group SO(2). The corresponding logarithms B are elements of the Lie algebra so(2), which consists of all skew-symmetric matrices. The matrix
is a generator of the Lie algebra so(2).
Existence
The question of whether a matrix has a logarithm has the easiest answer when considered in the complex setting. A complex matrix has a logarithm if and only if it is invertible. The logarithm is not unique, but if a matrix has no negative real eigenvalues, then there is a unique logarithm that has eigenvalues all lying in the strip . This logarithm is known as the principal logarithm.
The answer is more involved in the real setting. A real matrix has a real logarithm if and only if it is invertible and each Jordan block belonging to a negative eigenvalue occurs an even number of times. If an invertible real matrix does not satisfy the condition with the Jordan blocks, then it has only non-real logarithms. This can already be seen in the scalar case: no branch of the logarithm can be real at -1. The existence of real matrix logarithms of real 2×2 matrices is considered in a later section.
Properties
If A and B are both positive-definite matrices, then
Suppose that A and B commute, meaning that AB = BA. Then
if and only if , where is an eigenvalue of and is t
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https://en.wikipedia.org/wiki/Infinity%20%28disambiguation%29
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Infinity (symbol: ) is a mathematical concept that is involved in almost all branches of mathematics, and used in many scientific and non-scientific areas.
Infinity or infinities may also refer to:
Infinity (philosophy), a related philosophical and metaphysical concept
Mathematics
Infinity symbol
Aleph number, symbols for representing different kinds of mathematical infinity
Axiom of infinity
Actual infinity
Buildings
The Infinity, a highrise condo in San Francisco, California, US
Infinity Tower (Dubai), former name of the Cayan Tower skyscraper in Dubai, UAE
Infinity Tower (Brisbane), a skyscraper in Australia
Tower Infinity, a skyscraper in Korea
Technology
BT Infinity, a broadband service in the United Kingdom provided by BT Retail
GTS Infinity, a celebrity Millennium-Class cruise ship
Infinity Firearms, a brand name of Strayer Voight Inc, manufacturer of M1911-styled pistols
Infinity Engine, a game engine used in several popular computer role-playing games
U-Turn Infinity, a German paraglider design
Organizations
Infinity Broadcasting Corporation, now known as CBS Radio, one of the largest radio corporations in the United States
Infinity Systems, a manufacturer of loudspeakers
Infinity Power Chutes, an American aircraft manufacturer
Arts and entertainment
Infinity (1996 film), a biographical film starring Matthew Broderick as physicist Richard Feynman
Infinity (2023 film), an Indian Tamil-language film
Games
Infinity (role-playing game), a tabletop role-playing game
Infinity (wargame), a science fiction 28mm miniature skirmish game
Infinity (video game series), a series of visual novel games produced by KID
Never 7: The End of Infinity, which was originally released as Infinity
Disney Infinity, a toys-to-life video game series
Disney Infinity (video game), the first game in the series
Print media
Infinity Science Fiction, a science fiction magazine
Infinity (comic book), a crossover comic book published by Marvel Comics in 2013
Infinity Comics, a digital comics lineup exclusive to Marvel Unlimited which feature the vertical scroll format
Infinity, Inc., a team of superheroes appearing in comic books published by DC Comics
Star Wars Infinities, a 2002-2004 Star Wars comic book published by Dark Horse Comics
The Infinities, a 2009 novel by John Banville
Music
Infinity Recordings, a British record label
Infinity Records, a short-lived American record label from the 1970s
Infinity (band), a Eurodance band from Norway
Infinity (producer) (born 1983), American record producer
Albums
Infinity (Charice album), 2011
Infinity (Crematory album)
Infinity (Deep Obsession album), 1999
Infinity (Devin Townsend album), 1998
Infinity (End of Green album), the debut album of German alternative metal band End Of Green
Infinity (f.i.r. album)
Infinity (Guru Josh album), the debut album by English acid house musician Guru Josh
Infinity (Jesu album), 2009
Infinity (John Coltrane album), 1972
Infinity (Journ
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https://en.wikipedia.org/wiki/Circles%20of%20Apollonius
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The circles of Apollonius are any of several sets of circles associated with Apollonius of Perga, a renowned Greek geometer. Most of these circles are found in planar Euclidean geometry, but analogs have been defined on other surfaces; for example, counterparts on the surface of a sphere can be defined through stereographic projection.
The main uses of this term are fivefold:
Apollonius showed that a circle can be defined as the set of points in a plane that have a specified ratio of distances to two fixed points, known as foci. This Apollonian circle is the basis of the Apollonius pursuit problem. It is a particular case of the first family described in #2.
The Apollonian circles are two families of mutually orthogonal circles. The first family consists of the circles with all possible distance ratios to two fixed foci (the same circles as in #1), whereas the second family consists of all possible circles that pass through both foci. These circles form the basis of bipolar coordinates.
The circles of Apollonius of a triangle are three circles, each of which passes through one vertex of the triangle and maintains a constant ratio of distances to the other two. The isodynamic points and Lemoine line of a triangle can be solved using these circles of Apollonius.
Apollonius' problem is to construct circles that are simultaneously tangent to three specified circles. The solutions to this problem are sometimes called the circles of Apollonius.
The Apollonian gasket—one of the first fractals ever described—is a set of mutually tangent circles, formed by solving Apollonius' problem iteratively.
Apollonius' definition of a circle
A circle is usually defined as the set of points P at a given distance r (the circle's radius) from a given point (the circle's center). However, there are other, equivalent definitions of a circle. Apollonius discovered that a circle could be defined as the set of points P that have a given ratio of distances k = to two given points (labeled A and B in Figure 1). These two points are sometimes called the foci.
Proof using vectors in Euclidean spaces
Let d, d be non-equal positive real numbers.
Let C be the internal division point of AB in the ratio d : d and D the external division point of AB in the same ratio, d : d.
Then,
Therefore, the point P is on the circle which has the diameter CD.
Proof using the angle bisector theorem
First consider the point on the line segment between and , satisfying the ratio. By the definition
and from the angle bisector theorem the angles and are equal.
Next take the other point on the extended line that satisfies the ratio. So
Also take some other point anywhere on the extended line . Also by the Angle bisector theorem the line bisects the exterior angle . Hence, and are equal and . Hence by Thales's theorem lies on the circle which has as a diameter.
Apollonius pursuit problem
The Apollonius pursuit problem is one of finding whether a ship leaving fr
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https://en.wikipedia.org/wiki/GCD%20domain
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In mathematics, a GCD domain is an integral domain R with the property that any two elements have a greatest common divisor (GCD); i.e., there is a unique minimal principal ideal containing the ideal generated by two given elements. Equivalently, any two elements of R have a least common multiple (LCM).
A GCD domain generalizes a unique factorization domain (UFD) to a non-Noetherian setting in the following sense: an integral domain is a UFD if and only if it is a GCD domain satisfying the ascending chain condition on principal ideals (and in particular if it is Noetherian).
GCD domains appear in the following chain of class inclusions:
Properties
Every irreducible element of a GCD domain is prime. A GCD domain is integrally closed, and every nonzero element is primal. In other words, every GCD domain is a Schreier domain.
For every pair of elements x, y of a GCD domain R, a GCD d of x and y and an LCM m of x and y can be chosen such that , or stated differently, if x and y are nonzero elements and d is any GCD d of x and y, then xy/d is an LCM of x and y, and vice versa. It follows that the operations of GCD and LCM make the quotient R/~ into a distributive lattice, where "~" denotes the equivalence relation of being associate elements. The equivalence between the existence of GCDs and the existence of LCMs is not a corollary of the similar result on complete lattices, as the quotient R/~ need not be a complete lattice for a GCD domain R.
If R is a GCD domain, then the polynomial ring R[X1,...,Xn] is also a GCD domain.
R is a GCD domain if and only if finite intersections of its principal ideals are principal. In particular, , where is the LCM of and .
For a polynomial in X over a GCD domain, one can define its content as the GCD of all its coefficients. Then the content of a product of polynomials is the product of their contents, as expressed by Gauss's lemma, which is valid over GCD domains.
Examples
A unique factorization domain is a GCD domain. Among the GCD domains, the unique factorization domains are precisely those that are also atomic domains (which means that at least one factorization into irreducible elements exists for any nonzero nonunit).
A Bézout domain (i.e., an integral domain where every finitely generated ideal is principal) is a GCD domain. Unlike principal ideal domains (where every ideal is principal), a Bézout domain need not be a unique factorization domain; for instance the ring of entire functions is a non-atomic Bézout domain, and there are many other examples. An integral domain is a Prüfer GCD domain if and only if it is a Bézout domain.
If R is a non-atomic GCD domain, then R[X] is an example of a GCD domain that is neither a unique factorization domain (since it is non-atomic) nor a Bézout domain (since X and a non-invertible and non-zero element a of R generate an ideal not containing 1, but 1 is nevertheless a GCD of X and a); more generally any ring R[X1,...,Xn] has these properties.
A commutative
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https://en.wikipedia.org/wiki/Euler%20class
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In mathematics, specifically in algebraic topology, the Euler class is a characteristic class of oriented, real vector bundles. Like other characteristic classes, it measures how "twisted" the vector bundle is. In the case of the tangent bundle of a smooth manifold, it generalizes the classical notion of Euler characteristic. It is named after Leonhard Euler because of this.
Throughout this article is an oriented, real vector bundle of rank over a base space .
Formal definition
The Euler class is an element of the integral cohomology group
constructed as follows. An orientation of amounts to a continuous choice of generator of the cohomology
of each fiber relative to the complement of zero. From the Thom isomorphism, this induces an orientation class
in the cohomology of relative to the complement of the zero section . The inclusions
where includes into as the zero section, induce maps
The Euler class e(E) is the image of u under the composition of these maps.
Properties
The Euler class satisfies these properties, which are axioms of a characteristic class:
Functoriality: If is another oriented, real vector bundle and is continuous and covered by an orientation-preserving map , then . In particular, .
Whitney sum formula: If is another oriented, real vector bundle, then the Euler class of their direct sum is given by
Normalization: If possesses a nowhere-zero section, then .
Orientation: If is with the opposite orientation, then .
Note that "Normalization" is a distinguishing feature of the Euler class. The Euler class obstructs the existence of a non-vanishing section in the sense that if then has no non-vanishing section.
Also unlike other characteristic classes, it is concentrated in a degree which depends on the rank of the bundle: . By contrast, the Stiefel Whitney classes live in independent of the rank of . This reflects the fact that the Euler class is unstable, as discussed below.
Vanishing locus of generic section
The Euler class corresponds to the vanishing locus of a section of in the following way. Suppose that is an oriented smooth manifold of dimension . Let be a smooth section that transversely intersects the zero section. Let be the zero locus of . Then is a codimension submanifold of which represents a homology class and is the Poincaré dual of .
Self-intersection
For example, if is a compact submanifold, then the Euler class of the normal bundle of in is naturally identified with the self-intersection of in .
Relations to other invariants
In the special case when the bundle E in question is the tangent bundle of a compact, oriented, r-dimensional manifold, the Euler class is an element of the top cohomology of the manifold, which is naturally identified with the integers by evaluating cohomology classes on the fundamental homology class. Under this identification, the Euler class of the tangent bundle equals the Euler characteristic of the manifold. In the language of chara
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https://en.wikipedia.org/wiki/Robion%20Kirby
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Robion Cromwell Kirby (born February 25, 1938) is a Professor of Mathematics at the University of California, Berkeley who specializes in low-dimensional topology. Together with Laurent C. Siebenmann he developed the Kirby–Siebenmann invariant for classifying the piecewise linear structures on a topological manifold. He also proved the fundamental result on the Kirby calculus, a method for describing 3-manifolds and smooth 4-manifolds by surgery on framed links. Along with his significant mathematical contributions, he has over 50 doctoral students and is the editor of an influential problem list.
He received his Ph.D. from the University of Chicago in 1965, with thesis "Smoothing Locally Flat Imbeddings" written under the direction of . He soon became an assistant professor at UCLA. While there he developed his "torus trick" which enabled him to solve, in dimensions greater than four (with additional joint work with Siebenmann), four of John Milnor's seven most important problems in geometric topology.
In 1971, he was awarded the Oswald Veblen Prize in Geometry by the American Mathematical Society.
In 1995 he became the first mathematician to receive the NAS Award for Scientific Reviewing from the National Academy of Sciences for his problem list in low-dimensional topology. He was elected to the National Academy of Sciences in 2001. In 2012 he became a fellow of the American Mathematical Society.
Kirby is also the President of Mathematical Sciences Publishers, a small non-profit academic publishing house that focuses on mathematics and engineering journals.
Books
References
External links
Kirby's home page.
Biographical notes from the Proceedings of the Kirbyfest in honour of his 60th birthday in 1998.
Video Lectures by Kirby at Edinburgh
1938 births
Living people
Mathematicians from Illinois
Scientists from Chicago
20th-century American mathematicians
21st-century American mathematicians
Topologists
University of Chicago alumni
University of California, Los Angeles faculty
University of California, Berkeley College of Letters and Science faculty
Fellows of the American Mathematical Society
Members of the United States National Academy of Sciences
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https://en.wikipedia.org/wiki/Klein%20quadric
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In mathematics, the lines of a 3-dimensional projective space, S, can be viewed as points of a 5-dimensional projective space, T. In that 5-space, the points that represent each line in S lie on a quadric, Q known as the Klein quadric.
If the underlying vector space of S is the 4-dimensional vector space V, then T has as the underlying vector space the 6-dimensional exterior square Λ2V of V. The line coordinates obtained this way are known as Plücker coordinates.
These Plücker coordinates satisfy the quadratic relation
defining Q, where
are the coordinates of the line spanned by the two vectors u and v.
The 3-space, S, can be reconstructed again from the quadric, Q: the planes contained in Q fall into two equivalence classes, where planes in the same class meet in a point, and planes in different classes meet in a line or in the empty set. Let these classes be and . The geometry of S is retrieved as follows:
The points of S are the planes in C.
The lines of S are the points of Q.
The planes of S are the planes in C’.
The fact that the geometries of S and Q are isomorphic can be explained by the isomorphism of the Dynkin diagrams A3 and D3.
References
Albrecht Beutelspacher & Ute Rosenbaum (1998) Projective Geometry : from foundations to applications, page 169, Cambridge University Press
Arthur Cayley (1873) "On the superlines of a quadric surface in five-dimensional space", Collected Mathematical Papers 9: 79–83.
Felix Klein (1870) "Zur Theorie der Liniencomplexe des ersten und zweiten Grades", Mathematische Annalen 2: 198
Oswald Veblen & John Wesley Young (1910) Projective Geometry, volume 1, Interpretation of line coordinates as point coordinates in S5, page 331, Ginn and Company.
.
Projective geometry
Quadrics
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https://en.wikipedia.org/wiki/Plane%20partition
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In mathematics and especially in combinatorics, a plane partition is a two-dimensional array of nonnegative integers (with positive integer indices i and j) that is nonincreasing in both indices. This means that
and for all i and j.
Moreover, only finitely many of the may be nonzero. Plane partitions are a generalization of partitions of an integer.
A plane partition may be represented visually by the placement of a stack of unit cubes above the point (i, j) in the plane, giving a three-dimensional solid as shown in the picture. The image has matrix form
Plane partitions are also often described by the positions of the unit cubes. From this point of view, a plane partition can be defined as a finite subset of positive integer lattice points (i, j, k) in , such that if (r, s, t) lies in and if satisfies , , and , then (i, j, k) also lies in .
The sum of a plane partition is
The sum describes the number of cubes of which the plane partition consists. Much interest in plane partitions concerns the enumeration of plane partitions in various classes. The number of plane partitions with sum n is denoted by PL(n). For example, there are six plane partitions with sum 3
so PL(3) = 6.
Plane partitions may be classified by how symmetric they are. Many symmetric classes of plane partitions are enumerated by simple product formulas.
Generating function of plane partitions
The generating function for PL(n) is
.
It is sometimes referred to as the MacMahon function, as it was discovered by Percy A. MacMahon.
This formula may be viewed as the 2-dimensional analogue of Euler's product formula for the number of integer partitions of n. There is no analogous formula known for partitions in higher dimensions (i.e., for solid partitions). The asymptotics for plane partitions were first calculated by E. M. Wright. One obtains, for large , that
Evaluating numerically yields
Plane partitions in a box
Around 1896, MacMahon set up the generating function of plane partitions that are subsets of the box in his first paper on plane partitions. The formula is given by
A proof of this formula can be found in the book Combinatory Analysis written by MacMahon. MacMahon also mentions the generating functions of plane partitions. The formula for the generating function can be written in an alternative way, which is given by
Multiplying each component by , and setting q = 1 in the formulas above yields that the total number of plane partitions that fit in the box is equal to the following product formula:
The planar case (when t = 1) yields the binomial coefficients:
The general solution is
Special plane partitions
Special plane partitions include symmetric, cyclic and self-complementary plane partitions, and combinations of these properties.
In the subsequent sections, the enumeration of special sub-classes of plane partitions inside a box are considered.
These articles use the notation for the number of such plane partitions, where , , an
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https://en.wikipedia.org/wiki/Quantum%20calculus
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Quantum calculus, sometimes called calculus without limits, is equivalent to traditional infinitesimal calculus without the notion of limits. It defines "q-calculus" and "h-calculus", where h ostensibly stands for Planck's constant while q stands for quantum. The two parameters are related by the formula
where is the reduced Planck constant.
Differentiation
In the q-calculus and h-calculus, differentials of functions are defined as
and
respectively. Derivatives of functions are then defined as fractions by the q-derivative
and by
In the limit, as h goes to 0, or equivalently as q goes to 1, these expressions take on the form of the derivative of classical calculus.
Integration
q-integral
A function F(x) is a q-antiderivative of f(x) if DqF(x) = f(x). The q-antiderivative (or q-integral) is denoted by and an expression for F(x) can be found from the formula
which is called the Jackson integral of f(x). For , the series converges to a function F(x) on an interval (0,A] if |f(x)xα| is bounded on the interval for some .
The q-integral is a Riemann–Stieltjes integral with respect to a step function having infinitely many points of increase at the points qj, with the jump at the point qj being qj. If we call this step function gq(t) then dgq(t) = dqt.
h-integral
A function F(x) is an h-antiderivative of f(x) if DhF(x) = f(x). The h-antiderivative (or h-integral) is denoted by . If a and b differ by an integer multiple of h then the definite integral is given by a Riemann sum of f(x) on the interval partitioned into subintervals of width h.
Example
The derivative of the function (for some positive integer ) in the classical calculus is . The corresponding expressions in q-calculus and h-calculus are
with the q-bracket
and
respectively. The expression is then the q-calculus analogue of the simple power rule for
positive integral powers. In this sense, the function is still nice in the q-calculus, but rather ugly in the h-calculus – the h-calculus analog of is instead the falling factorial,
One may proceed further and develop, for example, equivalent notions of Taylor expansion, et cetera, and even arrive at q-calculus analogues for all of the usual functions one would want to have, such as an analogue for the sine function whose q-derivative is the appropriate analogue for the cosine.
History
The h-calculus is just the calculus of finite differences, which had been studied by George Boole and others, and has proven useful in a number of fields, among them combinatorics and fluid mechanics. The q-calculus, while dating in a sense back to Leonhard Euler and Carl Gustav Jacobi, is only recently beginning to see more usefulness in quantum mechanics, having an intimate connection with commutativity relations and Lie algebra.
See also
Noncommutative geometry
Quantum differential calculus
Time scale calculus
q-analog
Basic hypergeometric series
Quantum dilogarithm
Further reading
George Gasper, Mizan Rahman, Basic Hyper
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https://en.wikipedia.org/wiki/Quasi-algebraically%20closed%20field
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In mathematics, a field F is called quasi-algebraically closed (or C1) if every non-constant homogeneous polynomial P over F has a non-trivial zero provided the number of its variables is more than its degree. The idea of quasi-algebraically closed fields was investigated by C. C. Tsen, a student of Emmy Noether, in a 1936 paper ; and later by Serge Lang in his 1951 Princeton University dissertation and in his 1952 paper . The idea itself is attributed to Lang's advisor Emil Artin.
Formally, if P is a non-constant homogeneous polynomial in variables
X1, ..., XN,
and of degree d satisfying
d < N
then it has a non-trivial zero over F; that is, for some xi in F, not all 0, we have
P(x1, ..., xN) = 0.
In geometric language, the hypersurface defined by P, in projective space of degree , then has a point over F.
Examples
Any algebraically closed field is quasi-algebraically closed. In fact, any homogeneous polynomial in at least two variables over an algebraically closed field has a non-trivial zero.
Any finite field is quasi-algebraically closed by the Chevalley–Warning theorem.
Algebraic function fields of dimension 1 over algebraically closed fields are quasi-algebraically closed by Tsen's theorem.
The maximal unramified extension of a complete field with a discrete valuation and a perfect residue field is quasi-algebraically closed.
A complete field with a discrete valuation and an algebraically closed residue field is quasi-algebraically closed by a result of Lang.
A pseudo algebraically closed field of characteristic zero is quasi-algebraically closed.
Properties
Any algebraic extension of a quasi-algebraically closed field is quasi-algebraically closed.
The Brauer group of a finite extension of a quasi-algebraically closed field is trivial.
A quasi-algebraically closed field has cohomological dimension at most 1.
Ck fields
Quasi-algebraically closed fields are also called C1. A Ck field, more generally, is one for which any homogeneous polynomial of degree d in N variables has a non-trivial zero, provided
dk < N,
for k ≥ 1. The condition was first introduced and studied by Lang. If a field is Ci then so is a finite extension. The C0 fields are precisely the algebraically closed fields.
Lang and Nagata proved that if a field is Ck, then any extension of transcendence degree n is Ck+n. The smallest k such that K is a Ck field ( if no such number exists), is called the diophantine dimension dd(K) of K.
C1 fields
Every finite field is C1.
C2 fields
Properties
Suppose that the field k is C2.
Any skew field D finite over k as centre has the property that the reduced norm D∗ → k∗ is surjective.
Every quadratic form in 5 or more variables over k is isotropic.
Artin's conjecture
Artin conjectured that p-adic fields were C2, but
Guy Terjanian found p-adic counterexamples for all p. The Ax–Kochen theorem applied methods from model theory to show that Artin's conjecture was true for Qp with p large enough (depending
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https://en.wikipedia.org/wiki/Chevalley%E2%80%93Warning%20theorem
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In number theory, the Chevalley–Warning theorem implies that certain polynomial equations in sufficiently many variables over a finite field have solutions. It was proved by and a slightly weaker form of the theorem, known as Chevalley's theorem, was proved by . Chevalley's theorem implied Artin's and Dickson's conjecture that finite fields are quasi-algebraically closed fields .
Statement of the theorems
Let be a finite field and be a set of polynomials such that the number of variables satisfies
where is the total degree of . The theorems are statements about the solutions of the following system of polynomial equations
The Chevalley–Warning theorem states that the number of common solutions is divisible by the characteristic of . Or in other words, the cardinality of the vanishing set of is modulo .
The Chevalley theorem states that if the system has the trivial solution , that is, if the polynomials have no constant terms, then the system also has a non-trivial solution .
Chevalley's theorem is an immediate consequence of the Chevalley–Warning theorem since is at least 2.
Both theorems are best possible in the sense that, given any , the list has total degree and only the trivial solution. Alternatively, using just one polynomial, we can take f1 to be the degree n polynomial given by the norm of x1a1 + ... + xnan where the elements a form a basis of the finite field of order pn.
Warning proved another theorem, known as Warning's second theorem, which states that if the system of polynomial equations has the trivial solution, then it has at least solutions where is the size of the finite field and . Chevalley's theorem also follows directly from this.
Proof of Warning's theorem
Remark: If then
so the sum over of any polynomial in of degree less than also vanishes.
The total number of common solutions modulo of is equal to
because each term is 1 for a solution and 0 otherwise.
If the sum of the degrees of the polynomials is less than n then this vanishes by the remark above.
Artin's conjecture
It is a consequence of Chevalley's theorem that finite fields are quasi-algebraically closed. This had been conjectured by Emil Artin in 1935. The motivation behind Artin's conjecture was his observation that quasi-algebraically closed fields have trivial Brauer group, together with the fact that finite fields have trivial Brauer group by Wedderburn's theorem.
The Ax–Katz theorem
The Ax–Katz theorem, named after James Ax and Nicholas Katz, determines more accurately a power of the cardinality of dividing the number of solutions; here, if is the largest of the , then the exponent can be taken as the ceiling function of
The Ax–Katz result has an interpretation in étale cohomology as a divisibility result for the (reciprocals of) the zeroes and poles of the local zeta-function. Namely, the same power of divides each of these algebraic integers.
See also
Combinatorial Nullstellensatz
References
External
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https://en.wikipedia.org/wiki/Noetherian%20topological%20space
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In mathematics, a Noetherian topological space, named for Emmy Noether, is a topological space in which closed subsets satisfy the descending chain condition. Equivalently, we could say that the open subsets satisfy the ascending chain condition, since they are the complements of the closed subsets. The Noetherian property of a topological space can also be seen as a strong compactness condition, namely that every open subset of such a space is compact, and in fact it is equivalent to the seemingly stronger statement that every subset is compact.
Definition
A topological space is called Noetherian if it satisfies the descending chain condition for closed subsets: for any sequence
of closed subsets of , there is an integer such that
Properties
A topological space is Noetherian if and only if every subspace of is compact (i.e., is hereditarily compact), and if and only if every open subset of is compact.
Every subspace of a Noetherian space is Noetherian.
The continuous image of a Noetherian space is Noetherian.
A finite union of Noetherian subspaces of a topological space is Noetherian.
Every Hausdorff Noetherian space is finite with the discrete topology.
Proof: Every subset of X is compact in a Hausdorff space, hence closed. So X has the discrete topology, and being compact, it must be finite.
Every Noetherian space X has a finite number of irreducible components. If the irreducible components are , then , and none of the components is contained in the union of the other components.
From algebraic geometry
Many examples of Noetherian topological spaces come from algebraic geometry, where for the Zariski topology an irreducible set has the intuitive property that any closed proper subset has smaller dimension. Since dimension can only 'jump down' a finite number of times, and algebraic sets are made up of finite unions of irreducible sets, descending chains of Zariski closed sets must eventually be constant.
A more algebraic way to see this is that the associated ideals defining algebraic sets must satisfy the ascending chain condition. That follows because the rings of algebraic geometry, in the classical sense, are Noetherian rings. This class of examples therefore also explains the name.
If R is a commutative Noetherian ring, then Spec(R), the prime spectrum of R, is a Noetherian topological space. More generally, a Noetherian scheme is a Noetherian topological space. The converse does not hold, since there are non-Noetherian rings with only one prime ideal, so that Spec(R) consists of exactly one point and therefore is a Noetherian space.
Example
The space (affine -space over a field ) under the Zariski topology is an example of a Noetherian topological space. By properties of the ideal of a subset of , we know that if
is a descending chain of Zariski-closed subsets, then
is an ascending chain of ideals of Since is a Noetherian ring, there exists an integer such that
Since is the closure of Y f
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https://en.wikipedia.org/wiki/Concordance%20%28genetics%29
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In genetics, concordance is the probability that a pair of individuals will both have a certain characteristic (phenotypic trait) given that one of the pair has the characteristic. Concordance can be measured with concordance rates, reflecting the odds of one person having the trait if the other does. Important clinical examples include the chance of offspring having a certain disease if the mother has it, if the father has it, or if both parents have it. Concordance among siblings is similarly of interest: what are the odds of a subsequent offspring having the disease if an older child does? In research, concordance is often discussed in the context of both members of a pair of twins. Twins are concordant when both have or both lack a given trait. The ideal example of concordance is that of identical twins, because the genome is the same, an equivalence that helps in discovering causation via deconfounding, regarding genetic effects versus epigenetic and environmental effects (nature versus nurture).
In contrast, discordance occurs when a similar trait is not shared by the persons. Studies of twins have shown that genetic traits of monozygotic twins are fully concordant, whereas in dizygotic twins, half of genetic traits are concordant, while the other half are discordant. Discordant rates that are higher than concordant rates express the influence of the environment on twin traits.
Studies
A twin study compares the concordance rate of identical twins to that of fraternal twins. This can help suggest whether a disease or a certain trait has a genetic cause. Controversial uses of twin data have looked at concordance rates for homosexuality and intelligence. Other studies have involved looking at the genetic and environmental factors that can lead to increased LDL in women twins.
Because identical twins are genetically virtually identical, it follows that a genetic pattern carried by one would very likely also be carried by the other. If a characteristic identified in one twin is caused by a certain gene, then it would also very likely be present in the other twin. Thus, the concordance rate of a given characteristic helps suggest whether or to what extent a characteristic is related to genetics.
There are several problems with this assumption:
A given genetic pattern may not have 100% penetrance, in which case it may have different phenotypic consequences in genetically identical individuals;
Developmental and environmental conditions may be different for genetically identical individuals. If developmental and environmental conditions contribute to the development of the disease or other characteristic, there can be differences in the outcome of genetically identical individuals;
The logic is further complicated if the characteristic is polygenic, i.e., caused by differences in more than one gene.
Epigenetic effects can alter the genetic expressions in twins through varied factors. The expression of the epigenetic effect is typically weak
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https://en.wikipedia.org/wiki/Greater%20Manchester%20Built-up%20Area
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The Greater Manchester Built-up Area is an area of land defined by the Office for National Statistics (ONS), consisting of the large conurbation that encompasses the urban element of the city of Manchester and the metropolitan area that forms much of Greater Manchester in North West England. According to the United Kingdom Census 2011, the Greater Manchester Built-up Area has a population of 2,553,379 making it the second most populous conurbation in the United Kingdom after the Greater London Built-up Area. This was an increase of 14% from the population recorded at the United Kingdom Census 2001 of 2,240,230, when it was known as the Greater Manchester Urban Area.
The Greater Manchester Built-up Area is not conterminous with Greater Manchester, a metropolitan county of the same name (and, until 1974, part of the county of Lancashire) for it excludes settlements such as Wigan and Marple from Greater Manchester, but includes hinterland settlements which lie outside its statutory boundaries, such as Wilmslow in Cheshire, Glossop in Derbyshire, Whitworth in Lancashire and Newton-le-Willows in Merseyside.
Constituent parts
The largest settlements (in descending order of population) within the Greater Manchester Built-up Area are Manchester, Bolton, Stockport, Oldham, Rochdale, Salford, and Bury. These settlements are not coterminous with the Metropolitan Boroughs of the same name, and the ONS takes some of its settlement boundaries within the conurbation from the contiguous urban core of pre-Local Government Act 1972 local government districts. This means that the GMUA bears a much closer resemblance to the earlier "SELNEC" area than to the Greater Manchester Metropolitan County. Unlike most urban areas which expanded outwards around a central core of employment the Greater Manchester Urban Area was formed from the inward expansion of several large manufacturing towns towards a centralised marketplace for the trading of goods and raw materials.
Wigan is separate from the Greater Manchester Built-up Area, although eastern parts of the wider Metropolitan Borough of Wigan, such as Leigh, Tyldesley and Atherton are included. Wigan and Ince-in-Makerfield are included in a separate Wigan Urban Area, which includes places outside Greater Manchester. The gap is formed between Ince-in-Makerfield (of Wigan Urban Area) and Hindley (Greater Manchester Urban Area).
Other built-up areas, including New Mills and the rest of High Peak are narrowly avoided as is Ashton-in-Makerfield (which is included in the Liverpool Urban Area) and much of the parish of Saddleworth.
Settlements
According to the ONS definitions the Greater Manchester Built-up Area consists of the following settlements:
Notes:
See also
Geography of Greater Manchester
List of Greater Manchester settlements by population
References
External links
Map showing makeup of the conurbation according to the ONS (PDF)
Geography of Greater Manchester
Urban areas of England
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https://en.wikipedia.org/wiki/Ancestral%20graph
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In statistics and Markov modeling, an ancestral graph is a type of mixed graph to provide a graphical representation for the result of marginalizing one or more vertices in a graphical model that takes the form of a directed acyclic graph.
Definition
Ancestral graphs are mixed graphs used with three kinds of edges: directed edges, drawn as an arrow from one vertex to another, bidirected edges, which have an arrowhead at both ends, and undirected edges, which have no arrowheads. It is required to satisfy some additional constraints:
If there is an edge from a vertex u to another vertex v, with an arrowhead at v (that is, either an edge directed from u to v or a bidirected edge), then there does not exist a path from v to u consisting of undirected edges and/or directed edges oriented consistently with the path.
If a vertex v is an endpoint of an undirected edge, then it is not also the endpoint of an edge with an arrowhead at v.
Applications
Ancestral graphs are used to depict conditional independence relations between variables in Markov models.
References
Extensions and generalizations of graphs
Graphical models
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https://en.wikipedia.org/wiki/M-separation
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In statistics, m-separation is a measure of disconnectedness in ancestral graphs and a generalization of d-separation for directed acyclic graphs. It is the opposite of m-connectedness.
Suppose G is an ancestral graph. For given source and target nodes s and t and a set Z of nodes in G\{s, t}, m-connectedness can be defined as follows. Consider a path from s to t. An intermediate node on the path is called a collider if both edges on the path touching it are directed toward the node. The path is said to m-connect the nodes s and t, given Z, if and only if:
every non-collider on the path is outside Z, and
for each collider c on the path, either c is in Z or there is a directed path from c to an element of Z.
If s and t cannot be m-connected by any path satisfying the above conditions, then the nodes are said to be m-separated.
The definition can be extended to node sets S and T. Specifically, S and T are m-connected if each node in S can be m-connected to any node in T, and are m-separated otherwise.
References
Drton, Mathias and Thomas Richardson. Iterative Conditional Fitting for Gaussian Ancestral Graph Models. Technical Report 437, December 2003.
See also
d-separation
Graphical models
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https://en.wikipedia.org/wiki/Adragon%20De%20Mello
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Adragon De Mello (born October 8, 1976) graduated from the University of California, Santa Cruz with a degree in computational mathematics in 1988, at age 11. At the time, he was the youngest college graduate in U.S. history, a record broken in 1994 by Michael Kearney. His early achievements may have been more due to endless hard work than to inherent intellectual capabilities.
Father's beliefs
Adragon was the only child of Cathy Gunn and Agustin Eastwood De Mello (1929–2003). His father planned an ideal life for a "boy genius" before Adragon was born; it included not only graduating from college early, but also getting a doctorate in physics by age 12, winning the Nobel Prize in Physics by age 16, being elected a senator by age 20 (US senators must be at least 30 years old), becoming president of the United States by age 26 (the minimum age set by the US Constitution is 35), then head of a world government by age 30, and chairman of an intergalactic government after that. Since his father had set the goal that his son would become a Nobel Prize winner by age 16, he obsessively pushed his son in mathematics and other academic subjects from an early age. For example, when doing math homework, his father insisted that he solve an equation five times, even when he got the correct answer on the first attempt.
His father also sought publicity for his son. In 1987, while at university, Adragon and his father were interviewed by Morley Safer on 60 Minutes II. They also appeared on 48 Hours and The Tonight Show. During these interviews, Adragon would repeat the goals his father had chosen, saying he wanted to get a Ph.D. in physics and win a Nobel Prize by age 16 or 17.
When his father enrolled him in Popper-Keizer, a school for gifted children, standardized tests Adragon took suggested he was around the 85th percentile for students his age, where most students enrolled in such schools were in the 95th percentile. His father removed him from the school for gifted students "after tests showed the boy was less gifted than his father believed".
Schools
In 1981, Adragon joined Mensa. He has also been a member of Intertel, another organization for people with high intelligence.
After attending seven different elementary schools in the space of just three years, Adragon enrolled in Cabrillo College for two years starting in 1984. After that, he transferred to UC–Santa Cruz. While he did graduate from university in 1988, some of his math teachers later claimed that his grades were borderline.
Adragon was accepted into a graduate program at the Florida Institute of Technology, but did not enroll.
Teenage years
After graduating from university and being legally removed from his father's custody, he opted to enroll in Sunnyvale Junior High School (now Sunnyvale Middle School) under the assumed name of James Gunn – James after the fictional spy, James Bond, with his mother's last name. He took all of the classes except math, and played in Little
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https://en.wikipedia.org/wiki/Breusch%E2%80%93Pagan%20test
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In statistics, the Breusch–Pagan test, developed in 1979 by Trevor Breusch and Adrian Pagan, is used to test for heteroskedasticity in a linear regression model. It was independently suggested with some extension by R. Dennis Cook and Sanford Weisberg in 1983 (Cook–Weisberg test). Derived from the Lagrange multiplier test principle, it tests whether the variance of the errors from a regression is dependent on the values of the independent variables. In that case, heteroskedasticity is present.
Suppose that we estimate the regression model
and obtain from this fitted model a set of values for , the residuals. Ordinary least squares constrains these so that their mean is 0 and so, given the assumption that their variance does not depend on the independent variables, an estimate of this variance can be obtained from the average of the squared values of the residuals. If the assumption is not held to be true, a simple model might be that the variance is linearly related to independent variables. Such a model can be examined by regressing the squared residuals on the independent variables, using an auxiliary regression equation of the form
This is the basis of the Breusch–Pagan test. It is a chi-squared test: the test statistic is distributed nχ2 with k degrees of freedom. If the test statistic has a p-value below an appropriate threshold (e.g. p < 0.05) then the null hypothesis of homoskedasticity is rejected and heteroskedasticity assumed.
If the Breusch–Pagan test shows that there is conditional heteroskedasticity, one could either use weighted least squares (if the source of heteroskedasticity is known) or use heteroscedasticity-consistent standard errors.
Procedure
Under the classical assumptions, ordinary least squares is the best linear unbiased estimator (BLUE), i.e., it is unbiased and efficient. It remains unbiased under heteroskedasticity, but efficiency is lost. Before deciding upon an estimation method, one may conduct the Breusch–Pagan test to examine the presence of heteroskedasticity. The Breusch–Pagan test is based on models of the type for the variances of the observations where explain the difference in the variances. The null hypothesis is equivalent to the parameter restrictions:
The following Lagrange multiplier (LM) yields the test statistic for the Breusch–Pagan test:
This test can be implemented via the following three-step procedure:
Step 1: Apply OLS in the model
Step 2: Compute the regression residuals, , square them, and divide by the Maximum Likelihood estimate of the error variance from the Step 1 regression, to obtain what Breusch and Pagan call :
Step 2: Estimate the auxiliary regression
where the z terms will typically but not necessarily be the same as the original covariates x.
Step 3: The LM test statistic is then half of the explained sum of squares from the auxiliary regression in Step 2:
where TSS is the sum of squared deviations of the from their mean of 1, and SSR is the sum of squared
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https://en.wikipedia.org/wiki/Carus%20Mathematical%20Monographs
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The Carus Mathematical Monographs is a monograph series published by the Mathematical Association of America. Books in this series are intended to appeal to a wide range of readers in mathematics and science.
Scope and audience
While the books are intended to cover nontrivial material, the emphasis is on exposition and clear communication rather than novel results and a systematic Bourbaki-style presentation. The webpage for the series states:
The exposition of mathematical subjects that the monographs contain are set forth in a manner comprehensible not only to teachers and students specializing in mathematics, but also to scientific workers in other fields. More generally, the monographs are intended for the wide circle of thoughtful people familiar with basic graduate or advanced undergraduate mathematics encountered in the study of mathematics itself or in the context of related disciplines who wish to extend their knowledge without prolonged and critical study of the mathematical journals and treatises.
Many of the books in the series have become classics in the genre of general mathematical exposition.
Series listing
Calculus of Variations, by G. A. Bliss (out of print)
Analytic Functions of a Complex Variable, by D. R. Curtiss (out of print)
Mathematical Statistics, by H. L. Rietz (out of print)
Projective Geometry, by J. W. Young (out of print)
A History of Mathematics in America before 1900, by D. E. Smith and Jekuthiel Ginsburg (out of print)
Fourier Series and Orthogonal Polynomials, by Dunham Jackson (out of print)
Vectors and Matrices, by C. C. MacDuffee (out of print)
Rings and Ideals, by N. H. McCoy (out of print)
The Theory of Algebraic Numbers, second edition, by Harry Pollard and Harold G. Diamond
The Arithmetic Theory of Quadratic Forms, by B. W. Jones (out of print)
Irrational Numbers, by Ivan Niven
Statistical Independence in Probability, Analysis and Number Theory, by Mark Kac
A Primer of Real Functions, third edition, by Ralph P. Boas, Jr.
Combinatorial Mathematics, by Herbert John Ryser
Noncommutative Rings, by I. N. Herstein (out of print)
Dedekind Sums, by Hans Rademacher and Emil Grosswald (out of print)
The Schwarz Function and its Applications, by Philip J. Davis
Celestial Mechanics, by Harry Pollard (out of print)
Field Theory and its Classical Problems, by Charles Robert Hadlock
The Generalized Riemann Integral, by Robert M. McLeod (out of print)
From Error-Correcting Codes through Sphere Packings to Simple Groups, by Thomas M. Thompson
Random Walks and Electric Networks, by Peter G. Doyle and J. Laurie Snell
Complex Analysis: The Geometric Viewpoint, by Steven G. Krantz
Knot Theory, by Charles Livingston
Algebra and Tiling: Homomorphisms in the Service of Geometry, by Sherman K. Stein and Sándor Szabó
The Sensual (Quadratic) Form, by John H. Conway assisted by Francis Y. C. Fung, 1997,
A Panorama of Harmonic Analysis, by Steven G. Krantz, 1999,
Inequalities from Complex Analysis, by John P. D'Angelo, 2002,
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https://en.wikipedia.org/wiki/Lefschetz%20zeta%20function
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In mathematics, the Lefschetz zeta-function is a tool used in topological periodic and fixed point theory, and dynamical systems. Given a continuous map , the zeta-function is defined as the formal series
where is the Lefschetz number of the -th iterate of . This zeta-function is of note in topological periodic point theory because it is a single invariant containing information about all iterates of .
Examples
The identity map on has Lefschetz zeta function
where is the Euler characteristic of , i.e., the Lefschetz number of the identity map.
For a less trivial example, let be the unit circle, and let be reflection in the x-axis, that is, . Then has Lefschetz number 2, while is the identity map, which has Lefschetz number 0. Likewise, all odd iterates have Lefschetz number 2, while all even iterates have Lefschetz number 0. Therefore, the zeta function of is
Formula
If f is a continuous map on a compact manifold X of dimension n (or more generally any compact polyhedron), the zeta function is given by the formula
Thus it is a rational function. The polynomials occurring in the numerator and denominator are essentially the characteristic polynomials of the map induced by f on the various homology spaces.
Connections
This generating function is essentially an algebraic form of the Artin–Mazur zeta function, which gives geometric information about the fixed and periodic points of f.
See also
Lefschetz fixed-point theorem
Artin–Mazur zeta function
Ruelle zeta function
References
Zeta and L-functions
Dynamical systems
Fixed points (mathematics)
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https://en.wikipedia.org/wiki/Scalar%20boson
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A scalar boson is a boson whose spin equals zero. A boson is a particle whose wave function is symmetric under particle exchange and therefore follows Bose–Einstein statistics. The spin–statistics theorem implies that all bosons have an integer-valued spin. Scalar bosons are the subset of bosons with zero-valued spin.
The name scalar boson arises from quantum field theory, which demands that fields of spin-zero particles transform like a scalar under Lorentz transformation (i.e. are Lorentz invariant).
A pseudoscalar boson is a scalar boson that has odd parity, whereas "regular" scalar bosons have even parity.
Examples
Scalar
The only fundamental scalar boson in the Standard Model of particle physics is the Higgs boson, the existence of which was confirmed on 14 March 2013 at the Large Hadron Collider by CMS and ATLAS. As a result of this confirmation, the 2013 Nobel Prize in physics was awarded to Peter Higgs and François Englert.
Various known composite particles are scalar bosons, e.g. the alpha particle and scalar mesons.
The φ4-theory or quartic interaction is a popular "toy model" quantum field theory that uses scalar bosonic fields, used in many introductory quantum textbooks to introduce basic concepts in field theory.
Pseudoscalar
There are no fundamental pseudoscalars in the Standard Model, but there are pseudoscalar mesons, like the pion.
See also
Scalar field theory
Klein–Gordon equation
Vector boson
Higgs boson
References
Bosons
Quantum field theory
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https://en.wikipedia.org/wiki/Hemicontinuity
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In mathematics, the notion of the continuity of functions is not immediately extensible to set-valued functions between two sets A and B.
The dual concepts of upper hemicontinuity and lower hemicontinuity facilitate such an extension.
A set-valued function that has both properties is said to be continuous in an analogy to the property of the same name for single-valued functions.
Roughly speaking, a function is upper hemicontinuous if when (1) a convergent sequence of points in the domain maps to a sequence of sets in the range which (2) contain another convergent sequence, then the image of the limiting point in the domain must contain the limit of the sequence in the range.
Lower hemicontinuity essentially reverses this, saying if a sequence in the domain converges, given a point in the range of the limit, then you can find a sub-sequence whose image contains a convergent sequence to the given point.
Upper hemicontinuity
A set-valued function is said to be upper hemicontinuous at the point if, for any open with , there exists a neighbourhood of such that for all is a subset of
Sequential characterization
For a set-valued function with closed values, if is upper hemicontinuous at then for all sequences in and all sequences such that
if and then
If B is compact, the converse is also true.
Closed graph theorem
The graph of a set-valued function is the set defined by
If is an upper hemicontinuous set-valued function with closed domain (that is, the set of points where is not the empty set is closed) and closed values (i.e. is closed for all ), then is closed.
If is compact, then the converse is also true.
Lower hemicontinuity
A set-valued function is said to be lower hemicontinuous at the point
if for any open set intersecting there exists a neighbourhood of such that intersects for all (Here means nonempty intersection ).
Sequential characterization
is lower hemicontinuous at if and only if for every sequence in such that in and all there exists a subsequence of and also a sequence such that and for every
Open graph theorem
A set-valued function have if the set
is open in for every If values are all open sets in then is said to have .
If has an open graph then has open upper and lower sections and if has open lower sections then it is lower hemicontinuous.
The open graph theorem says that if is a set-valued function with convex values and open upper sections, then has an open graph in if and only if is lower hemicontinuous.
Properties
Set-theoretic, algebraic and topological operations on set-valued functions (like union, composition, sum, convex hull, closure)
usually preserve the type of continuity. But this should be taken with appropriate care since, for example, there exists a pair of lower hemicontinuous set-valued functions whose intersection is not lower hemicontinuous.
This can be fixed upon strengthening continuity properties: if one of those low
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https://en.wikipedia.org/wiki/Saturated%20measure
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In mathematics, a measure is said to be saturated if every locally measurable set is also measurable. A set , not necessarily measurable, is said to be a if for every measurable set of finite measure, is measurable. -finite measures and measures arising as the restriction of outer measures are saturated.
References
Measures (measure theory)
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https://en.wikipedia.org/wiki/Stable%20homotopy%20theory
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In mathematics, stable homotopy theory is the part of homotopy theory (and thus algebraic topology) concerned with all structure and phenomena that remain after sufficiently many applications of the suspension functor. A founding result was the Freudenthal suspension theorem, which states that given any pointed space , the homotopy groups stabilize for sufficiently large. In particular, the homotopy groups of spheres stabilize for . For example,
In the two examples above all the maps between homotopy groups are applications of the suspension functor. The first example is a standard corollary of the Hurewicz theorem, that . In the second example the Hopf map, , is mapped to its suspension , which generates .
One of the most important problems in stable homotopy theory is the computation of stable homotopy groups of spheres. According to Freudenthal's theorem, in the stable range the homotopy groups of spheres depend not on the specific dimensions of the spheres in the domain and target, but on the difference in those dimensions. With this in mind the k-th stable stem is
.
This is an abelian group for all k. It is a theorem of Jean-Pierre Serre that these groups are finite for . In fact, composition makes into a graded ring. A theorem of Goro Nishida states that all elements of positive grading in this ring are nilpotent. Thus the only prime ideals are the primes in . So the structure of is quite complicated.
In the modern treatment of stable homotopy theory, spaces are typically replaced by spectra. Following this line of thought, an entire stable homotopy category can be created. This category has many nice properties that are not present in the (unstable) homotopy category of spaces, following from the fact that the suspension functor becomes invertible. For example, the notion of cofibration sequence and fibration sequence are equivalent.
See also
Adams filtration
Adams spectral sequence
Chromatic homotopy theory
Equivariant stable homotopy theory
Nilpotence theorem
References
Homotopy theory
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https://en.wikipedia.org/wiki/Tokyo%20International%20Film%20Festival
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The is a film festival established in 1985. The event was held biennially from 1985 to 1991 and annually thereafter. According to FIAPF statistics, it is one of Asia's competitive film festivals, is considered to be the second largest film festival in Asia behind the Shanghai International Film Festival, and the only Japanese festival accredited by the FIAPF.
The awards handed out during the festival have changed throughout its existence, but the Tokyo Grand Prix, handed to the best film, has stayed as the top award. Other awards that have been given regularly include the Special Jury Award and awards for best actor, best actress and best director.
In recent years, the festival's main events have been held over one week in late October, at the Roppongi Hills development. Events include open-air screenings, voice-over screenings, and appearances by actors, as well as seminars and symposiums related to the film market.
List of festivals and award winners
Other awards
Best Screenplay Award
2017 - Euthanizer, Teemu Nikki
2018 - Amanda, Mikhael Hers and Maud Ameline
2019 - A Beloved Wife, Shin Adachi
Best Artistic Contribution Award
2012 - Pankaj Kumar, Ship of Theseus
2014 - Ispytanie, Aleksandr Kott
2015 - Family Film, Olmo Omerzu
2017 - The Looming Storm
2018 - The White Crow
2019 - Chaogtu with Sarula
2021 - Crane Lantern
2022 - Peacock Lament, Sanjeewa Pushpakumara
Audience Award
2013 - Red Family, Lee Ju-hyoung
2014 - Pale Moon, Daihachi Yoshida
2015 - God Willing, Edoardo Falcone
2016 - Die Beautiful, Jun Lana
2017 - Tremble All You Want, Akiko Ooku
2018 - Another World, Junji Sakamoto
2019 - Only the Animals, Dominik Moll
2020 - Hold Me Back, Akiko Ooku (sole award given)
2021 - Just Remembering, Daigo Matsui
2022 - By the Window, Rikiya Imaizumi
Asian Future Best Film Award
2014 - Bedone marz بدون مرز, Amirhossein Asgari امیرحسین عسگری
2015 - Pimpaka Towira, The Island Funeral
2016 - Birdshot
2017 - Passage of Life
2018 - A First Farewell
2019 - Summer Knight
2021 - World, Northern Hemisphere
2022 - Butterflies Live Only One Day, Mohammadreza Vatandoust
Japanese Cinema Splash Best Picture Award
2014 - 100 Yen Love, Masaharu Take
2015 - Ken and Kazu, Hiroshi Shoji
2017 - Of Love & Law, Hikaru Toda
2016 - Poolsideman, Hirobumi Watanabe
2018 - Lying to Mom, Katsumi Nojiri
2019 - i -Documentary of the Journalist-, Tatsuya Mori
Tokyo Gemstone Award
2017 - Mayu Matsuoka, Shizuka Ishibashi, Adeline D'Hermy, Daphne Low
2018 - Lien Binh Phat, Karelle Tremblay, Mai Kiryu, Nijiro Murakami
2019 - Josefine Frida Pettersen, Sairi Ito, Riru Yoshina, Yui Sakuma
References
External links
Official site of the Tokyo International Film Festival
English site of the Tokyo International Film Festival
Tokyo International Film Festival at the Internet Movie Database
Film festivals in Tokyo
Film festivals established in 1985
October events
1985 establishments in Japan
Annual events in Japan
Autumn events in Japan
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https://en.wikipedia.org/wiki/European%20Cup%20and%20UEFA%20Champions%20League%20records%20and%20statistics
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This page details statistics of the European Cup and Champions League. Unless noted, these statistics concern all seasons since the inception of the European Cup in the 1955–56 season, and renamed since 1992 as the UEFA Champions League. This does not include the qualifying rounds of the UEFA Champions League, unless otherwise noted.
General performances
By club
A total of 23 clubs have won the tournament since its 1955 inception, with Real Madrid being the only team to win it fourteen times, including the first five. Only three other clubs have reached ten or more finals: AC Milan, Bayern Munich and Liverpool. A total of thirteen clubs have won the tournament multiple times: the four forementioned clubs, along with Benfica, Inter Milan, Ajax, Nottingham Forest, Juventus, Manchester United, Porto, Barcelona and Chelsea. A total of nineteen clubs have reached the final without ever managing to win the tournament.
Clubs from ten countries have provided tournament winners. Spanish clubs have been the most successful, winning nineteen titles. England is second with fifteen and Italy is third with twelve, while the other multiple-time winners are Germany with eight, the Netherlands with six, and Portugal with four. The only other countries to provide a tournament winner are Scotland, Romania, Yugoslavia, and France. Greece, Belgium and Sweden have all provided losing finalists.
By nation
Overall team records
In this ranking two points are awarded for a win, 1 for a draw and 0 for a loss. As per statistical convention in football, matches decided in extra time are counted as wins and losses, while matches decided by penalty shoot-outs are counted as draws. Teams are ranked by total points, then by goal difference, then by goals scored. Only the top 25 are listed (includes qualifying rounds).
Number of participating clubs of the Champions League era (from 1992–present)
A total of 149 clubs from 34 national associations have played in or qualified for the Champions League group stage. Season in bold represents teams qualified for the knockout phase that season. Between 1999–2000 and 2002–03, qualification is considered from the second group stage. Starting from the 2024–25 season with the introduction of a league phase, the top eight are considered to be qualified as well as the eight play-off winners.
European Cup group stage participants
(only one season was played in this format)
1991–92:
Anderlecht
Barcelona
Benfica
Dynamo Kyiv
Panathinaikos
Red Star Belgrade
Sampdoria
Sampdoria is the only side to have played in 1991–92 European Cup group stage, but to have not played in the Champions League group stage.
Sparta Prague
Goals
Most goals scored in a matchday: 63 (matchday 1 of the first group stage, 2000–01 season).
Most goals scored in a season: 449 (2000–01 season).
Host of the finals
The city that has hosted the final the most times is London, doing so on seven occasions. Of these, five have been played at the or
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https://en.wikipedia.org/wiki/Matthew%20Stephens%20%28statistician%29
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Matthew Stephens (born 1970) is a Bayesian statistician and professor in the departments of human genetics and statistics at the University of Chicago. He is known for the Li and Stephens model as an efficient coalescent.
Education
Stephens has a PhD from Magdalen College, Oxford University where his advisor was Brian D. Ripley. He then went on to work with Peter Donnelly as a postdoctoral researcher.
Career
Stephens conducted postdoctoral research with Peter Donnelly at the University of Oxford. It was there that he developed the Structure computer program, along with Jonathan Pritchard, whic is used for determining population structure and estimating individual admixture. He then went on to develop the influential Li and Stephens model as an efficient model for linkage disequilibrium.
Awards
Stephens was awarded the Guy Medal (bronze) in 2006. He was elected a Fellow of the Royal Society in 2023.
Notes
1970 births
British statisticians
Population geneticists
Statistical geneticists
Living people
20th-century British mathematicians
21st-century British mathematicians
Genetic epidemiologists
Fellows of the Royal Society
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https://en.wikipedia.org/wiki/Wythoff%20construction
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In geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling. It is often referred to as Wythoff's kaleidoscopic construction.
Construction process
The method is based on the idea of tiling a sphere, with spherical triangles – see Schwarz triangles. This construction arranges three mirrors at the sides of a triangle, like in a kaleidoscope. However, different from a kaleidoscope, the mirrors are not parallel, but intersect at a single point. They therefore enclose a spherical triangle on the surface of any sphere centered on that point and repeated reflections produce a multitude of copies of the triangle. If the angles of the spherical triangle are chosen appropriately, the triangles will tile the sphere, one or more times.
If one places a vertex at a suitable point inside the spherical triangle enclosed by the mirrors, it is possible to ensure that the reflections of that point produce a uniform polyhedron. For a spherical triangle ABC we have four possibilities which will produce a uniform polyhedron:
A vertex is placed at the point A. This produces a polyhedron with Wythoff symbol a|b c, where a equals π divided by the angle of the triangle at A, and similarly for b and c.
A vertex is placed at a point on line AB so that it bisects the angle at C. This produces a polyhedron with Wythoff symbol a b|c.
A vertex is placed so that it is on the incenter of ABC. This produces a polyhedron with Wythoff symbol a b c|.
The vertex is at a point such that, when it is rotated around any of the triangle's corners by twice the angle at that point, it is displaced by the same distance for every angle. Only even-numbered reflections of the original vertex are used. The polyhedron has the Wythoff symbol |a b c.
The process in general also applies for higher-dimensional regular polytopes, including the 4-dimensional uniform 4-polytopes.
Non-Wythoffian constructions
Uniform polytopes that cannot be created through a Wythoff mirror construction are called non-Wythoffian. They generally can be derived from Wythoffian forms either by alternation (deletion of alternate vertices) or by insertion of alternating layers of partial figures. Both of these types of figures will contain rotational symmetry. Sometimes snub forms are considered Wythoffian, even though they can only be constructed by the alternation of omnitruncated forms.
See also
Wythoff symbol - a symbol for the Wythoff construction of uniform polyhedra and uniform tilings.
Coxeter–Dynkin diagram - a generalized symbol for the Wythoff construction of uniform polytopes and honeycombs.
References
Coxeter Regular Polytopes, Third edition, (1973), Dover edition, (Chapter V: The Kaleidoscope, Section: 5.7 Wythoff's construction)
Coxeter The Beauty of Geometry: Twelve Essays, Dover Publications, 1999, (Chapter 3: Wythoff's Construction for Uniform Polytopes)
Har'El, Z. Uniform Solution for Uniform
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https://en.wikipedia.org/wiki/Cram%C3%A9r%E2%80%93Wold%20theorem
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In mathematics, the Cramér–Wold theorem in measure theory states that a Borel probability measure on is uniquely determined by the totality of its one-dimensional projections. It is used as a method for proving joint convergence results. The theorem is named after Harald Cramér and Herman Ole Andreas Wold.
Let
and
be random vectors of dimension k. Then converges in distribution to if and only if:
for each , that is, if every fixed linear combination of the coordinates of converges in distribution to the correspondent linear combination of coordinates of .
If takes values in , then the statement is also true with .
Footnotes
References
External links
Project Euclid: "When is a probability measure determined by infinitely many projections?"
Theorems in measure theory
Probability theorems
Convergence (mathematics)
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https://en.wikipedia.org/wiki/Wedge%20%28geometry%29
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In solid geometry, a wedge is a polyhedron defined by two triangles and three trapezoid faces. A wedge has five faces, nine edges, and six vertices.
A wedge is a subclass of the prismatoids with the base and opposite ridge in two parallel planes.
A wedge can also be classified as a digonal cupola.
Comparisons:
A wedge is a parallelepiped where a face has collapsed into a line.
A quadrilaterally-based pyramid is a wedge in which one of the edges between two trapezoid faces has collapsed into a point.
Volume
For a rectangle based wedge, the volume is
where the base rectangle is a by b, c is the apex edge length parallel to a, and h the height from the base rectangle to the apex edge.
Examples
Wedges can be created from decomposition of other polyhedra. For instance, the dodecahedron can be divided into a central cube with 6 wedges covering the cube faces. The orientations of the wedges are such that the triangle and trapezoid faces can connect and form a regular pentagon.
A triangular prism is a special case wedge with the two triangle faces being translationally congruent.
Two obtuse wedges can be formed by bisecting a regular tetrahedron on a plane parallel to two opposite edges.
References
Harris, J. W., & Stocker, H. "Wedge". §4.5.2 in Handbook of Mathematics and Computational Science. New York: Springer, p. 102, 1998.
External links
Polyhedra
Prismatoid polyhedra
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https://en.wikipedia.org/wiki/Medial
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Medial may refer to:
Mathematics
Medial magma, a mathematical identity in algebra
Geometry
Medial axis, in geometry the set of all points having more than one closest point on an object's boundary
Medial graph, another graph that represents the adjacencies between edges in the faces of a plane graph
Medial triangle, the triangle whose vertices lie at the midpoints of an enclosing triangle's sides
Polyhedra:
Medial deltoidal hexecontahedron
Medial disdyakis triacontahedron
Medial hexagonal hexecontahedron
Medial icosacronic hexecontahedron
Medial inverted pentagonal hexecontahedron
Medial pentagonal hexecontahedron
Medial rhombic triacontahedron
Linguistics
A medial sound or letter is one that is found in the middle of a larger unit (like a word)
Syllable medial, a segment located between the onset and the rime of a syllable
In the older literature, a term for the voiced stops (like b, d, g)
Medial or second person demonstrative, a demonstrative indicating things near the addressee
Anatomy
Medial (anatomy), term of location meaning 'towards the centre'
Medial ligament (disambiguation), term used for various ligaments toward the midline of the human body
Medial rotation, rotation toward the centre of the body
See also
Medial border (disambiguation)
Medial plantar (disambiguation)
Medial wall (disambiguation)
Median (disambiguation)
Medial capitals or CamelCase, use of capital letters in the middle of a compound word or abbreviation
Mid vowel, a vowel sound pronounced with the tongue midway between open and closed vowel positions
Medial s <ſ>, a form of the letter s written in the middle of a word
Human anatomical terms § Standard terms
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https://en.wikipedia.org/wiki/Filtered%20algebra
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In mathematics, a filtered algebra is a generalization of the notion of a graded algebra. Examples appear in many branches of mathematics, especially in homological algebra and representation theory.
A filtered algebra over the field is an algebra over that has an increasing sequence of subspaces of such that
and that is compatible with the multiplication in the following sense:
Associated graded algebra
In general there is the following construction that produces a graded algebra out of a filtered algebra.
If is a filtered algebra then the associated graded algebra is defined as follows:
The multiplication is well-defined and endows with the structure of a graded algebra, with gradation Furthermore if is associative then so is . Also if is unital, such that the unit lies in , then will be unital as well.
As algebras and are distinct (with the exception of the trivial case that is graded) but as vector spaces they are isomorphic. (One can prove by induction that is isomorphic to as vector spaces).
Examples
Any graded algebra graded by , for example , has a filtration given by .
An example of a filtered algebra is the Clifford algebra of a vector space endowed with a quadratic form The associated graded algebra is , the exterior algebra of
The symmetric algebra on the dual of an affine space is a filtered algebra of polynomials; on a vector space, one instead obtains a graded algebra.
The universal enveloping algebra of a Lie algebra is also naturally filtered. The PBW theorem states that the associated graded algebra is simply .
Scalar differential operators on a manifold form a filtered algebra where the filtration is given by the degree of differential operators. The associated graded algebra is the commutative algebra of smooth functions on the cotangent bundle which are polynomial along the fibers of the projection .
The group algebra of a group with a length function is a filtered algebra.
See also
Filtration (mathematics)
Length function
References
Algebras
Homological algebra
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https://en.wikipedia.org/wiki/Cairo%20pentagonal%20tiling
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In geometry, a Cairo pentagonal tiling is a tessellation of the Euclidean plane by congruent convex pentagons, formed by overlaying two tessellations of the plane by hexagons and named for its use as a paving design in Cairo. It is also called MacMahon's net after Percy Alexander MacMahon, who depicted it in his 1921 publication New Mathematical Pastimes. John Horton Conway called it a 4-fold pentille.
Infinitely many different pentagons can form this pattern, belonging to two of the 15 families of convex pentagons that can tile the plane. Their tilings have varying symmetries; all are face-symmetric. One particular form of the tiling, dual to the snub square tiling, has tiles with the minimum possible perimeter among all pentagonal tilings. Another, overlaying two flattened tilings by regular hexagons, is the form used in Cairo and has the property that every edge is collinear with infinitely many other edges.
In architecture, beyond Cairo, the Cairo tiling has been used in Mughal architecture in 18th-century India, in the early 20th-century Laeiszhalle in Germany, and in many modern buildings and installations. It has also been studied as a crystal structure and appears in the art of M. C. Escher.
Structure and classification
The union of all edges of a Cairo tiling is the same as the union of two tilings of the plane by hexagons. Each hexagon of one tiling surrounds two vertices of the other tiling, and is divided by the hexagons of the other tiling into four of the pentagons in the Cairo tiling. Infinitely many different pentagons can form Cairo tilings, all with the same pattern of adjacencies between tiles and with the same decomposition into hexagons, but with varying edge lengths, angles, and symmetries. The pentagons that form these tilings can be grouped into two different infinite families, drawn from the 15 families of convex pentagons that can tile the plane, and the five families of pentagon found by Karl Reinhardt in 1918 that can tile the plane isohedrally (all tiles symmetric to each other).
One of these two families consists of pentagons that have two non-adjacent right angles, with a pair of sides of equal length meeting at each of these right angles. Any pentagon meeting these requirements tiles the plane by copies that, at the chosen right angled corners, are rotated by a right angle with respect to each other. At the pentagon sides that are not adjacent to one of these two right angles, two tiles meet, rotated by a 180° angle with respect to each other. The result is an isohedral tiling, meaning that any pentagon in the tiling can be transformed into any other pentagon by a symmetry of the tiling. These pentagons and their tiling are often listed as "type 4" in the listing of types of pentagon that can tile. For any type 4 Cairo tiling, twelve of the same tiles can also cover the surface of a cube, with one tile folded across each cube edge and three right angles of tiles meeting at each cube vertex, to form the same co
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https://en.wikipedia.org/wiki/Rational%20point
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In number theory and algebraic geometry, a rational point of an algebraic variety is a point whose coordinates belong to a given field. If the field is not mentioned, the field of rational numbers is generally understood. If the field is the field of real numbers, a rational point is more commonly called a real point.
Understanding rational points is a central goal of number theory and Diophantine geometry. For example, Fermat's Last Theorem may be restated as: for , the Fermat curve of equation has no other rational points than , , and, if is even, and .
Definition
Given a field , and an algebraically closed extension of , an affine variety over is the set of common zeros in of a collection of polynomials with coefficients in :
These common zeros are called the points of .
A -rational point (or -point) of is a point of that belongs to , that is, a sequence of elements of such that for all . The set of -rational points of is often denoted .
Sometimes, when the field is understood, or when is the field of rational numbers, one says "rational point" instead of "-rational point".
For example, the rational points of the unit circle of equation
are the pairs of rational numbers
where is a Pythagorean triple.
The concept also makes sense in more general settings. A projective variety in projective space over a field can be defined by a collection of homogeneous polynomial equations in variables A -point of written is given by a sequence of elements of , not all zero, with the understanding that multiplying all of by the same nonzero element of gives the same point in projective space. Then a -point of means a -point of at which the given polynomials vanish.
More generally, let be a scheme over a field . This means that a morphism of schemes is given. Then a -point of means a section of this morphism, that is, a morphism such that the composition is the identity on . This agrees with the previous definitions when is an affine or projective variety (viewed as a scheme over ).
When is a variety over an algebraically closed field , much of the structure of is determined by its set of -rational points. For a general field , however, gives only partial information about . In particular, for a variety over a field and any field extension of , also determines the set of -rational points of , meaning the set of solutions of the equations defining with values in .
Example: Let be the conic curve in the affine plane over the real numbers Then the set of real points is empty, because the square of any real number is nonnegative. On the other hand, in the terminology of algebraic geometry, the algebraic variety over is not empty, because the set of complex points is not empty.
More generally, for a scheme over a commutative ring and any commutative -algebra , the set of -points of means the set of morphisms over . The scheme is determined up to isomorphism by the functor ; this is the philoso
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https://en.wikipedia.org/wiki/Weil%20cohomology%20theory
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In algebraic geometry, a Weil cohomology or Weil cohomology theory is a cohomology satisfying certain axioms concerning the interplay of algebraic cycles and cohomology groups. The name is in honor of André Weil. Any Weil cohomology theory factors uniquely through the category of Chow motives, but the category of Chow motives itself is not a Weil cohomology theory, since it is not an abelian category.
Definition
Fix a base field k of arbitrary characteristic and a "coefficient field" K of characteristic zero. A Weil cohomology theory is a contravariant functor
satisfying the axioms below. For each smooth projective algebraic variety X of dimension n over k, then the graded K-algebra
is required to satisfy the following:
is a finite-dimensional K-vector space for each integer i.
for each i < 0 or i > 2n.
is isomorphic to K (the so-called orientation map).
Poincaré duality: there is a perfect pairing
There is a canonical Künneth isomorphism
For each integer r, there is a cycle map defined on the group of algebraic cycles of codimension r on X,
satisfying certain compatibility conditions with respect to functoriality of H and the Künneth isomorphism. If X is a point, the cycle map is required to be the inclusion Z ⊂ K.
Weak Lefschetz axiom: For any smooth hyperplane section j: W ⊂ X (i.e. W = X ∩ H, H some hyperplane in the ambient projective space), the maps
are isomorphisms for and injections for
Hard Lefschetz axiom: Let W be a hyperplane section and be its image under the cycle class map. The Lefschetz operator is defined as
where the dot denotes the product in the algebra Then
is an isomorphism for i = 1, ..., n.
Examples
There are four so-called classical Weil cohomology theories:
singular (= Betti) cohomology, regarding varieties over C as topological spaces using their analytic topology (see GAGA),
de Rham cohomology over a base field of characteristic zero: over C defined by differential forms and in general by means of the complex of Kähler differentials (see algebraic de Rham cohomology),
-adic cohomology for varieties over fields of characteristic different from ,
crystalline cohomology.
The proofs of the axioms for Betti cohomology and de Rham cohomology are comparatively easy and classical. For -adic cohomology, for example, most of the above properties are deep theorems.
The vanishing of Betti cohomology groups exceeding twice the dimension is clear from the fact that a (complex) manifold of complex dimension n has real dimension 2n, so these higher cohomology groups vanish (for example by comparing them to simplicial (co)homology).
The de Rham cycle map also has a down-to-earth explanation: Given a subvariety Y of complex codimension r in a complete variety X of complex dimension n, the real dimension of Y is 2n−2r, so one can integrate any differential (2n−2r)-form along Y to produce a complex number. This induces a linear functional . By Poincaré duality, to give such a funct
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https://en.wikipedia.org/wiki/Immigration%20to%20France
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According to the French National Institute of Statistics INSEE, the 2021 census counted nearly 7 million immigrants (foreign-born people) in France, representing 10.3% of the total population. This is a decrease from INSEE statistics in 2018 in which there were 9 million immigrants (foreign-born people) in France, which at the time represented 14% of the country's total population.
The area with the largest proportion of immigrants is the Parisian urban area (Greater Paris), where almost 40% of immigrants lived in 2012. Other regions with important immigrant populations are Rhône-Alpes (Lyon) and Provence-Alpes-Côte d'Azur (Marseille).
The Paris region is a magnet for immigrants, hosting one of the largest concentrations of immigrants in Europe. As of 2006, about 45% of people (6 million) living in the region were either immigrant (25%) or born to at least one immigrant parent (20%).
Among the 802,000 newborns in metropolitan France in 2010, 27.3% had one or both parents foreign-born, and about one quarter (23.9%) had one parent or both born outside of Europe. Including grandparents, about 22% of newborns in France between 2006 and 2008 had at least one foreign-born grandparent (9% born in another European country, 8% born in Maghreb and 2% born in another region of the world).
In 2014, the National Institute of Statistics (INSEE) published a study reporting that the number of Spanish, Portuguese, and Italian immigrants in France between 2009 and 2012 has doubled. This increase resulting from the financial crisis that hit several European countries in that period, has pushed up the number of Europeans settled in France.
Statistics on Spanish immigrants in France show a growth of 107 percent between 2009 and 2012, i.e. in this period went from 5,300 to 11,000 people.
Of the total of 229,000 new foreigners coming to France in 2012, nearly 8% were Portuguese, British 5%, Spanish 5%, Italians 4%, Germans 4%, Romanians 3%, and Belgians 3%.
By 2022, the total number of new foreigners coming to France rose above 320,000 for the first time, with nearly a majority coming from Africa. A significant increase in students, family reunification and labor migration occurred under the presidency of Emmanuel Macron.
History
France's population dynamics began to change in the middle of the 19th century, as France joined the Industrial Revolution. The pace of industrial growth attracted millions of European immigrants over the next century, with especially large numbers arriving from Poland, Belgium, Portugal, Italy, and Spain. In the wake of the First World War, in which France suffered six million casualties, significant numbers of workers from French colonies came. By 1930, the Paris region alone had a North African Muslim population of 70,000.
1945–1974
Right after the Second World War, immigration to France significantly increased. During the period of reconstruction, France lacked labor, and as a result, the French government was eager to recruit im
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https://en.wikipedia.org/wiki/Fluent%20calculus
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The fluent calculus is a formalism for expressing dynamical domains in first-order logic. It is a variant of the situation calculus; the main difference is that situations are considered representations of states. A binary function symbol is used to concatenate the terms that represent facts that hold in a situation. For example, that the box is on the table in the situation is represented by the formula . The frame problem is solved by asserting that the situation after the execution of an action is identical to the one before but for the conditions changed by the action. For example, the action of moving the box from the table to the floor is formalized as:
This formula states that the state after the move is added the term and removed the term . Axioms specifying that is commutative and non-idempotent are necessary for such axioms to work.
See also
Fluent (artificial intelligence)
Frame problem
Situation calculus
Event calculus
References
M. Thielscher (1998). Introduction to the fluent calculus. Electronic Transactions on Artificial Intelligence, 2(3–4):179–192.
M. Thielscher (2005). Reasoning Robots - The Art and Science of Programming Robotic Agents. Volume 33 of Applied Logic Series. Springer, Dordrecht.
Logical calculi
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https://en.wikipedia.org/wiki/Blattner%27s%20conjecture
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In mathematics, Blattner's conjecture or Blattner's formula is a description of the discrete series representations of a general semisimple group G in terms of their restricted representations to a maximal compact subgroup K (their so-called K-types). It is named after Robert James Blattner, despite not being formulated as a conjecture by him.
Statement
Blattner's formula says that if a discrete series representation with infinitesimal character λ is restricted to a maximal compact subgroup K, then the representation of K with highest weight μ occurs with multiplicity
where
Q is the number of ways a vector can be written as a sum of non-compact positive roots
WK is the Weyl group of K
ρc is half the sum of the compact roots
ρn is half the sum of the non-compact roots
ε is the sign character of WK.
Blattner's formula is what one gets by formally restricting the Harish-Chandra character formula for a discrete series representation to the maximal torus of a maximal compact group. The problem in proving the Blattner formula is that this only gives the character on the regular elements of the maximal torus, and one also needs to control its behavior on the singular elements. For non-discrete irreducible representations the formal restriction of Harish-Chandra's character formula need not give the decomposition under the maximal compact subgroup: for example, for the principal series representations of SL2 the character is identically zero on the non-singular elements of the maximal compact subgroup, but the representation is not zero on this subgroup. In this case the character is a distribution on the maximal compact subgroup with support on the singular elements.
History
Harish-Chandra orally attributed the conjecture to Robert James Blattner as a question Blattner raised, not a conjecture made by Blattner. Blattner did not publish it in any form. It first appeared in print in , where it was first referred to as "Blattner's Conjecture," despite the results of that paper having been obtained without knowledge of Blattner's question and notwithstanding Blattner's not having made such a conjecture. mentioned a special case of it slightly earlier.
Schmid (1972) proved Blattner's formula in some special cases.
showed that Blattner's formula gave an upper bound for the multiplicities of K-representations, proved Blattner's conjecture for groups whose symmetric space is Hermitian, and proved Blattner's conjecture for linear semisimple groups. Blattner's conjecture (formula) was also proved by by infinitesimal methods which were totally new and completely different from those of Hecht and Schmid (1975). Part of the impetus for Enright’s paper (1979) came from several sources: from , , . In Enright (1979) multiplicity formulae are given for the so-called mock-discrete series representations also. used his ideas to obtain results on the construction and classification of irreducible Harish-Chandra modules of any real semisimple Lie algebra.
Re
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https://en.wikipedia.org/wiki/Naum%20Sekulovski
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Naum Sekulovski (born 14 May 1982) is an Australian soccer player who plays for Preston Lions in the NPL 2 Victoria competition.
A League career statistics
(Correct as of 21 March 2010)
Honours
Perth Glory FC
Best Clubman: 2010
References
External links
Perth Glory profile
Oz Football profile
1982 births
Living people
Soccer players from Melbourne
Sekulovski|Naum
Oakleigh Cannons FC players
Falcons 2000 SC players
Wollongong Wolves FC players
Parramatta Power SC players
Preston Lions FC players
Perth Glory FC players
Persema Malang players
Hume City FC players
Goulburn Valley Suns FC players
A-League Men players
National Premier Leagues players
Indonesian Premier League players
Victorian Institute of Sport alumni
Men's association football defenders
Australian men's soccer players
Australian expatriate men's soccer players
Expatriate men's footballers in Indonesia
Australian expatriate sportspeople in Indonesia
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https://en.wikipedia.org/wiki/Solvable%20Lie%20algebra
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In mathematics, a Lie algebra is solvable if its derived series terminates in the zero subalgebra. The derived Lie algebra of the Lie algebra is the subalgebra of , denoted
that consists of all linear combinations of Lie brackets of pairs of elements of . The derived series is the sequence of subalgebras
If the derived series eventually arrives at the zero subalgebra, then the Lie algebra is called solvable. The derived series for Lie algebras is analogous to the derived series for commutator subgroups in group theory, and solvable Lie algebras are analogs of solvable groups.
Any nilpotent Lie algebra is a fortiori solvable but the converse is not true. The solvable Lie algebras and the semisimple Lie algebras form two large and generally complementary classes, as is shown by the Levi decomposition. The solvable Lie algebras are precisely those that can be obtained from semidirect products, starting from 0 and adding one dimension at a time.
A maximal solvable subalgebra is called a Borel subalgebra. The largest solvable ideal of a Lie algebra is called the radical.
Characterizations
Let be a finite-dimensional Lie algebra over a field of characteristic . The following are equivalent.
(i) is solvable.
(ii) , the adjoint representation of , is solvable.
(iii) There is a finite sequence of ideals of :
(iv) is nilpotent.
(v) For -dimensional, there is a finite sequence of subalgebras of :
with each an ideal in . A sequence of this type is called an elementary sequence.
(vi) There is a finite sequence of subalgebras of ,
such that is an ideal in and is abelian.
(vii) The Killing form of satisfies for all in and in . This is Cartan's criterion for solvability.
Properties
Lie's Theorem states that if is a finite-dimensional vector space over an algebraically closed field of characteristic zero, and is a solvable Lie algebra, and if is a representation of over , then there exists a simultaneous eigenvector of the endomorphisms for all elements .
Every Lie subalgebra and quotient of a solvable Lie algebra are solvable.
Given a Lie algebra and an ideal in it,
is solvable if and only if both and are solvable.
The analogous statement is true for nilpotent Lie algebras provided is contained in the center. Thus, an extension of a solvable algebra by a solvable algebra is solvable, while a central extension of a nilpotent algebra by a nilpotent algebra is nilpotent.
A solvable nonzero Lie algebra has a nonzero abelian ideal, the last nonzero term in the derived series.
If are solvable ideals, then so is . Consequently, if is finite-dimensional, then there is a unique solvable ideal containing all solvable ideals in . This ideal is the radical of .
A solvable Lie algebra has a unique largest nilpotent ideal , called the nilradical, the set of all such that is nilpotent. If is any derivation of , then .
Completely solvable Lie algebras
A Lie algebra is called completely solvable or split solvable if it has an
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https://en.wikipedia.org/wiki/Final%20stellation%20of%20the%20icosahedron
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In geometry, the complete or final stellation of the icosahedron is the outermost stellation of the icosahedron, and is "complete" and "final" because it includes all of the cells in the icosahedron's stellation diagram. That is, every three intersecting face planes of the icosahedral core intersect either on a vertex of this polyhedron, or inside of it.
This polyhedron is the seventeenth stellation of the icosahedron, and given as Wenninger model index 42.
As a geometrical figure, it has two interpretations, described below:
As an irregular star (self-intersecting) polyhedron with 20 identical self-intersecting enneagrammic faces, 90 edges, 60 vertices.
As a simple polyhedron with 180 triangular faces (60 isosceles, 120 scalene), 270 edges, and 92 vertices. This interpretation is useful for polyhedron model building.
Johannes Kepler researched stellations that create regular star polyhedra (the Kepler-Poinsot polyhedra) in 1619, but the complete icosahedron, with irregular faces, was first studied in 1900 by Max Brückner.
History
1619: In Harmonices Mundi, Johannes Kepler first applied the stellation process, recognizing the small stellated dodecahedron and great stellated dodecahedron as regular polyhedra.
1809: Louis Poinsot rediscovered Kepler's polyhedra and two more, the great icosahedron and great dodecahedron as regular star polyhedra, now called the Kepler–Poinsot polyhedra.
1812: Augustin-Louis Cauchy made a further enumeration of star polyhedra, proving there are only 4 regular star polyhedra.
1900: Max Brückner extended the stellation theory beyond regular forms, and identified ten stellations of the icosahedron, including the complete stellation.
1924: A.H. Wheeler in 1924 published a list of 20 stellation forms (22 including reflective copies), also including the complete stellation.
1938: In their 1938 book The Fifty Nine Icosahedra, H. S. M. Coxeter, P. Du Val, H. T. Flather and J. F. Petrie stated a set of stellation rules for the regular icosahedron and gave a systematic enumeration of the fifty-nine stellations which conform to those rules. The complete stellation is referenced as the eighth in the book.
1974: In Wenninger's 1974 book Polyhedron Models, the final stellation of the icosahedron is included as the 17th model of stellated icosahedra with index number W42.
1995: Andrew Hume named it in his Netlib polyhedral database as the echidnahedron (the echidna, or spiny anteater is a small mammal that is covered with coarse hair and spines and which curls up in a ball to protect itself).
Interpretations
As a stellation
The stellation of a polyhedron extends the faces of a polyhedron into infinite planes and generates a new polyhedron that is bounded by these planes as faces and the intersections of these planes as edges. The Fifty Nine Icosahedra enumerates the stellations of the regular icosahedron, according to a set of rules put forward by J. C. P. Miller, including the complete stellation. The Du Val symb
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https://en.wikipedia.org/wiki/Weil%20reciprocity%20law
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In mathematics, the Weil reciprocity law is a result of André Weil holding in the function field K(C) of an algebraic curve C over an algebraically closed field K. Given functions f and g in K(C), i.e. rational functions on C, then
f((g)) = g((f))
where the notation has this meaning: (h) is the divisor of the function h, or in other words the formal sum of its zeroes and poles counted with multiplicity; and a function applied to a formal sum means the product (with multiplicities, poles counting as a negative multiplicity) of the values of the function at the points of the divisor. With this definition there must be the side-condition, that the divisors of f and g have disjoint support (which can be removed).
In the case of the projective line, this can be proved by manipulations with the resultant of polynomials.
To remove the condition of disjoint support, for each point P on C a local symbol
(f, g)P
is defined, in such a way that the statement given is equivalent to saying that the product over all P of the local symbols is 1. When f and g both take the values 0 or ∞ at P, the definition is essentially in limiting or removable singularity terms, by considering (up to sign)
fagb
with a and b such that the function has neither a zero nor a pole at P. This is achieved by taking a to be the multiplicity of g at P, and −b the multiplicity of f at P. The definition is then
(f, g)P = (−1)ab fagb.
See for example Jean-Pierre Serre, Groupes algébriques et corps de classes, pp. 44–46, for this as a special case of a theory on mapping algebraic curves into commutative groups.
There is a generalisation of Serge Lang to abelian varieties (Lang, Abelian Varieties).
References
André Weil, Oeuvres Scientifiques I, p. 291 (in Lettre à Artin, a 1942 letter to Artin, explaining the 1940 Comptes Rendus note Sur les fonctions algébriques à corps de constantes finis)
for a proof in the Riemann surface case
Algebraic curves
Theorems in algebraic geometry
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https://en.wikipedia.org/wiki/Joseph%20Wedderburn
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Joseph Henry Maclagan Wedderburn FRSE FRS (2 February 1882 – 9 October 1948) was a Scottish mathematician, who taught at Princeton University for most of his career. A significant algebraist, he proved that a finite division algebra is a field, and part of the Artin–Wedderburn theorem on simple algebras. He also worked on group theory and matrix algebra.
His younger brother was the lawyer Ernest Wedderburn.
Life
Joseph Wedderburn was the tenth of fourteen children of Alexander Wedderburn of Pearsie, a physician, and Anne Ogilvie. He was educated at Forfar Academy then in 1895 his parents sent Joseph and his younger brother Ernest to live in Edinburgh with their paternal uncle, J R Maclagan Wedderburn, allowing them to attend George Watson's College. This house was at 3 Glencairn Crescent in the West End of the city.
In 1898 Joseph entered the University of Edinburgh. In 1903, he published his first three papers, worked as an assistant in the Physical Laboratory of the University, obtained an MA degree with First Class Honours in mathematics, and was elected a Fellow of the Royal Society of Edinburgh, upon the proposal of George Chrystal, James Gordon MacGregor, Cargill Gilston Knott and William Peddie. Aged only 21 he remains one of the youngest Fellows ever.
He then studied briefly at the University of Leipzig and the University of Berlin, where he met the algebraists Frobenius and Schur. A Carnegie Scholarship allowed him to spend the 1904–1905 academic year at the University of Chicago where he worked with Oswald Veblen, E. H. Moore, and most importantly, Leonard Dickson, who was to become the most important American algebraist of his day.
Returning to Scotland in 1905, Wedderburn worked for four years at the University of Edinburgh as an assistant to George Chrystal, who supervised his D.Sc, awarded in 1908 for a thesis titled On Hypercomplex Numbers. He gained a PhD in algebra from the University of Edinburgh in 1908. From 1906 to 1908, Wedderburn edited the Proceedings of the Edinburgh Mathematical Society. In 1909, he returned to the United States to become a Preceptor in Mathematics at Princeton University; his colleagues included Luther P. Eisenhart, Oswald Veblen, Gilbert Ames Bliss, and George Birkhoff.
Upon the outbreak of the First World War, Wedderburn enlisted in the British Army as a private. He was the first person at Princeton to volunteer for that war, and had the longest war service of anyone on the staff. He served with the Seaforth Highlanders in France, as Lieutenant (1914), then as Captain of the 10th Battalion (1915–18). While a Captain in the Fourth Field Survey Battalion of the Royal Engineers in France, he devised sound-ranging equipment to locate enemy artillery.
He returned to Princeton after the war, becoming Associate Professor in 1921 and editing the Annals of Mathematics until 1928. While at Princeton, he supervised only three PhDs, one of them being Nathan Jacobson. In his later years, Wedderburn became
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https://en.wikipedia.org/wiki/Tonelli%27s%20theorem
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In mathematics, Tonelli's theorem may refer to
Tonelli's theorem in measure theory, a successor of Fubini's theorem
Tonelli's theorem in functional analysis, a fundamental result on the weak lower semicontinuity of nonlinear functionals on Lp spaces
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https://en.wikipedia.org/wiki/Penrose%20graphical%20notation
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In mathematics and physics, Penrose graphical notation or tensor diagram notation is a (usually handwritten) visual depiction of multilinear functions or tensors proposed by Roger Penrose in 1971. A diagram in the notation consists of several shapes linked together by lines.
The notation widely appears in modern quantum theory, particularly in matrix product states and quantum circuits. In particular, Categorical quantum mechanics which includes ZX-calculus is a fully comprehensive reformulation of quantum theory in terms of Penrose diagrams, and is now widely used in quantum industry.
The notation has been studied extensively by Predrag Cvitanović, who used it, along with Feynman's diagrams and other related notations in developing "birdtracks", a group-theoretical diagram to classify the classical Lie groups. Penrose's notation has also been generalized using representation theory to spin networks in physics, and with the presence of matrix groups to trace diagrams in linear algebra.
Interpretations
Multilinear algebra
In the language of multilinear algebra, each shape represents a multilinear function. The lines attached to shapes represent the inputs or outputs of a function, and attaching shapes together in some way is essentially the composition of functions.
Tensors
In the language of tensor algebra, a particular tensor is associated with a particular shape with many lines projecting upwards and downwards, corresponding to abstract upper and lower indices of tensors respectively. Connecting lines between two shapes corresponds to contraction of indices. One advantage of this notation is that one does not have to invent new letters for new indices. This notation is also explicitly basis-independent.
Matrices
Each shape represents a matrix, and tensor multiplication is done horizontally, and matrix multiplication is done vertically.
Representation of special tensors
Metric tensor
The metric tensor is represented by a U-shaped loop or an upside-down U-shaped loop, depending on the type of tensor that is used.
Levi-Civita tensor
The Levi-Civita antisymmetric tensor is represented by a thick horizontal bar with sticks pointing downwards or upwards, depending on the type of tensor that is used.
Structure constant
The structure constants () of a Lie algebra are represented by a small triangle with one line pointing upwards and two lines pointing downwards.
Tensor operations
Contraction of indices
Contraction of indices is represented by joining the index lines together.
Symmetrization
Symmetrization of indices is represented by a thick zig-zag or wavy bar crossing the index lines horizontally.
Antisymmetrization
Antisymmetrization of indices is represented by a thick straight line crossing the index lines horizontally.
Determinant
The determinant is formed by applying antisymmetrization to the indices.
Covariant derivative
The covariant derivative () is represented by a circle around the tensor(s) to be diff
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https://en.wikipedia.org/wiki/Idempotent%20matrix
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In linear algebra, an idempotent matrix is a matrix which, when multiplied by itself, yields itself. That is, the matrix is idempotent if and only if . For this product to be defined, must necessarily be a square matrix. Viewed this way, idempotent matrices are idempotent elements of matrix rings.
Example
Examples of idempotent matrices are:
Examples of idempotent matrices are:
Real 2 × 2 case
If a matrix is idempotent, then
implying so or
implying so or
Thus, a necessary condition for a matrix to be idempotent is that either it is diagonal or its trace equals 1.
For idempotent diagonal matrices, and must be either 1 or 0.
If , the matrix will be idempotent provided so a satisfies the quadratic equation
or
which is a circle with center (1/2, 0) and radius 1/2. In terms of an angle θ,
is idempotent.
However, is not a necessary condition: any matrix
with is idempotent.
Properties
Singularity and regularity
The only non-singular idempotent matrix is the identity matrix; that is, if a non-identity matrix is idempotent, its number of independent rows (and columns) is less than its number of rows (and columns).
This can be seen from writing , assuming that has full rank (is non-singular), and pre-multiplying by to obtain .
When an idempotent matrix is subtracted from the identity matrix, the result is also idempotent. This holds since
If a matrix is idempotent then for all positive integers n, . This can be shown using proof by induction. Clearly we have the result for , as . Suppose that . Then, , since is idempotent. Hence by the principle of induction, the result follows.
Eigenvalues
An idempotent matrix is always diagonalizable. Its eigenvalues are either 0 or 1: if is a non-zero eigenvector of some idempotent matrix and its associated eigenvalue, then which implies This further implies that the determinant of an idempotent matrix is always 0 or 1. As stated above, if the determinant is equal to one, the matrix is invertible and is therefore the identity matrix.
Trace
The trace of an idempotent matrix — the sum of the elements on its main diagonal — equals the rank of the matrix and thus is always an integer. This provides an easy way of computing the rank, or alternatively an easy way of determining the trace of a matrix whose elements are not specifically known (which is helpful in statistics, for example, in establishing the degree of bias in using a sample variance as an estimate of a population variance).
Relationships between idempotent matrices
In regression analysis, the matrix is known to produce the residuals from the regression of the vector of dependent variables on the matrix of covariates . (See the section on Applications.) Now, let be a matrix formed from a subset of the columns of , and let . It is easy to show that both and are idempotent, but a somewhat surprising fact is that . This is because , or in other words, the residuals from the regression of the columns of
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