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https://en.wikipedia.org/wiki/Order%20of%20operations
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In mathematics and computer programming, the order of operations is a collection of rules that reflect conventions about which operations to perform first in order to evaluate a given mathematical expression.
These rules are formalized with a ranking of the operators. The rank of an operator is called its precedence, and an operation with a higher precedence is performed before operations with lower precedence. Calculators generally perform operations with the same precedence from left to right, but some programming languages and calculators adopt different conventions.
For example, multiplication is granted a higher precedence than addition, and it has been this way since the introduction of modern algebraic notation. Thus, in the expression , the multiplication is performed before addition, and the expression has the value , and not . When exponents were introduced in the 16th and 17th centuries, they were given precedence over both addition and multiplication and placed as a superscript to the right of their base. Thus and .
These conventions exist to avoid notational ambiguity while allowing notation to remain brief. Where it is desired to override the precedence conventions, or even simply to emphasize them, parentheses ( ) can be used. For example, forces addition to precede multiplication, while forces addition to precede exponentiation. If multiple pairs of parentheses are required in a mathematical expression (such as in the case of nested parentheses), the parentheses may be replaced by brackets or braces to avoid confusion, as in .
These rules are meaningful only when the usual notation (called infix notation) is used. When functional or Polish notation are used for all operations, the order of operations results from the notation itself.
Internet memes sometimes present ambiguous infix expressions that cause disputes and increase web traffic. Most of these ambiguous expressions involve mixed division and multiplication, where there is no general agreement about the order of operations.
Definition
The order of operations, that is, the order in which the operations in an expression are usually performed, results from a convention adopted throughout mathematics, science, technology and many computer programming languages. It is summarized as:
Parentheses
Exponentiation
Multiplication and Division
Addition and Subtraction
This means that to evaluate an expression, one first evaluates any sub-expression inside parentheses, working inside to outside if there is more than one set. Whether inside parenthesis or not, the operator that is higher in the above list should be applied first.
The commutative and associative laws of addition and multiplication allow adding terms in any order, and multiplying factors in any order—but mixed operations obey the standard order of operations.
In some contexts, it is helpful to replace a division with multiplication by the reciprocal (multiplicative inverse) and a subtraction by addition
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https://en.wikipedia.org/wiki/Packing%20problems
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Packing problems are a class of optimization problems in mathematics that involve attempting to pack objects together into containers. The goal is to either pack a single container as densely as possible or pack all objects using as few containers as possible. Many of these problems can be related to real-life packaging, storage and transportation issues. Each packing problem has a dual covering problem, which asks how many of the same objects are required to completely cover every region of the container, where objects are allowed to overlap.
In a bin packing problem, you are given:
A container, usually a two- or three-dimensional convex region, possibly of infinite size. Multiple containers may be given depending on the problem.
A set of objects, some or all of which must be packed into one or more containers. The set may contain different objects with their sizes specified, or a single object of a fixed dimension that can be used repeatedly.
Usually the packing must be without overlaps between goods and other goods or the container walls. In some variants, the aim is to find the configuration that packs a single container with the maximal packing density. More commonly, the aim is to pack all the objects into as few containers as possible. In some variants the overlapping (of objects with each other and/or with the boundary of the container) is allowed but should be minimized.
Packing in infinite space
Many of these problems, when the container size is increased in all directions, become equivalent to the problem of packing objects as densely as possible in infinite Euclidean space. This problem is relevant to a number of scientific disciplines, and has received significant attention. The Kepler conjecture postulated an optimal solution for packing spheres hundreds of years before it was proven correct by Thomas Callister Hales. Many other shapes have received attention, including ellipsoids, Platonic and Archimedean solids including tetrahedra, tripods (unions of cubes along three positive axis-parallel rays), and unequal-sphere dimers.
Hexagonal packing of circles
These problems are mathematically distinct from the ideas in the circle packing theorem. The related circle packing problem deals with packing circles, possibly of different sizes, on a surface, for instance the plane or a sphere.
The counterparts of a circle in other dimensions can never be packed with complete efficiency in dimensions larger than one (in a one-dimensional universe, the circle analogue is just two points). That is, there will always be unused space if you are only packing circles. The most efficient way of packing circles, hexagonal packing, produces approximately 91% efficiency.
Sphere packings in higher dimensions
In three dimensions, close-packed structures offer the best lattice packing of spheres, and is believed to be the optimal of all packings. With 'simple' sphere packings in three dimensions ('simple' being carefully defined) there are nine pos
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https://en.wikipedia.org/wiki/Order%20topology
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In mathematics, an order topology is a certain topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets.
If X is a totally ordered set, the order topology on X is generated by the subbase of "open rays"
for all a, b in X. Provided X has at least two elements, this is equivalent to saying that the open intervals
together with the above rays form a base for the order topology. The open sets in X are the sets that are a union of (possibly infinitely many) such open intervals and rays.
A topological space X is called orderable or linearly orderable if there exists a total order on its elements such that the order topology induced by that order and the given topology on X coincide. The order topology makes X into a completely normal Hausdorff space.
The standard topologies on R, Q, Z, and N are the order topologies.
Induced order topology
If Y is a subset of X, X a totally ordered set, then Y inherits a total order from X. The set Y therefore has an order topology, the induced order topology. As a subset of X, Y also has a subspace topology. The subspace topology is always at least as fine as the induced order topology, but they are not in general the same.
For example, consider the subset Y = {–1} ∪ {1/n}n∈N in the rationals. Under the subspace topology, the singleton set {–1} is open in Y, but under the induced order topology, any open set containing –1 must contain all but finitely many members of the space.
An example of a subspace of a linearly ordered space whose topology is not an order topology
Though the subspace topology of Y = {–1} ∪ {1/n}n∈N in the section above is shown to be not generated by the induced order on Y, it is nonetheless an order topology on Y; indeed, in the subspace topology every point is isolated (i.e., singleton {y} is open in Y for every y in Y), so the subspace topology is the discrete topology on Y (the topology in which every subset of Y is an open set), and the discrete topology on any set is an order topology. To define a total order on Y that generates the discrete topology on Y, simply modify the induced order on Y by defining -1 to be the greatest element of Y and otherwise keeping the same order for the other points, so that in this new order (call it say <1) we have 1/n <1 –1 for all n ∈ N. Then, in the order topology on Y generated by <1, every point of Y is isolated in Y.
We wish to define here a subset Z of a linearly ordered topological space X such that no total order on Z generates the subspace topology on Z, so that the subspace topology will not be an order topology even though it is the subspace topology of a space whose topology is an order topology.
Let in the real line. The same argument as before shows that the subspace topology on Z is not equal to the induced order topology on Z, but one can show that the subspace topology on Z cannot be equal to any order topology on Z.
An argu
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https://en.wikipedia.org/wiki/Square%20number
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In mathematics, a square number or perfect square is an integer that is the square of an integer; in other words, it is the product of some integer with itself. For example, 9 is a square number, since it equals and can be written as .
The usual notation for the square of a number is not the product , but the equivalent exponentiation , usually pronounced as " squared". The name square number comes from the name of the shape. The unit of area is defined as the area of a unit square (). Hence, a square with side length has area . If a square number is represented by n points, the points can be arranged in rows as a square each side of which has the same number of points as the square root of n; thus, square numbers are a type of figurate numbers (other examples being cube numbers and triangular numbers).
In the real number system, square numbers are non-negative. A non-negative integer is a square number when its square root is again an integer. For example, so 9 is a square number.
A positive integer that has no square divisors except 1 is called square-free.
For a non-negative integer , the th square number is , with being the zeroth one. The concept of square can be extended to some other number systems. If rational numbers are included, then a square is the ratio of two square integers, and, conversely, the ratio of two square integers is a square, for example,
.
Starting with 1, there are square numbers up to and including , where the expression represents the floor of the number .
Examples
The squares smaller than 602 = 3600 are:
02 = 0
12 = 1
22 = 4
32 = 9
42 = 16
52 = 25
62 = 36
72 = 49
82 = 64
92 = 81
102 = 100
112 = 121
122 = 144
132 = 169
142 = 196
152 = 225
162 = 256
172 = 289
182 = 324
192 = 361
202 = 400
212 = 441
222 = 484
232 = 529
242 = 576
252 = 625
262 = 676
272 = 729
282 = 784
292 = 841
302 = 900
312 = 961
322 = 1024
332 = 1089
342 = 1156
352 = 1225
362 = 1296
372 = 1369
382 = 1444
392 = 1521
402 = 1600
412 = 1681
422 = 1764
432 = 1849
442 = 1936
452 = 2025
462 = 2116
472 = 2209
482 = 2304
492 = 2401
502 = 2500
512 = 2601
522 = 2704
532 = 2809
542 = 2916
552 = 3025
562 = 3136
572 = 3249
582 = 3364
592 = 3481
The difference between any perfect square and its predecessor is given by the identity . Equivalently, it is possible to count square numbers by adding together the last square, the last square's root, and the current root, that is, .
Properties
The number m is a square number if and only if one can arrange m points in a square:
The expression for the th square number is . This is also equal to the sum of the first odd numbers as can be seen in the above pictures, where a square results from the previous one by adding an odd number of points (shown in magenta). The formula follows:For example, .
There are several recursive methods for computing square numbers. For example, the th square number can be computed from the previous square by . Alternatively, the th square number can be calculated
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https://en.wikipedia.org/wiki/ZC
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ZC, Zc, or zC may refer to:
ZC, in set theory, a formal system with Zermelo's first five axioms plus the axiom of choice
Zadoff–Chu sequence, in mathematics, a certain complex-valued sequence with the CAZAC property
Zangger Committee, a committee on nuclear proliferation
Zeptocoulomb, another SI unit of electric charge
Zettacoulomb, an SI unit of electric charge
Zimbabwe Cricket, the governing body for cricket in Zimbabwe
Honda D engine, an engine variant produced by Honda Motor Company
Zc(3900), a subatomic particle
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https://en.wikipedia.org/wiki/Polygonal%20number
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In mathematics, a polygonal number is a number represented as dots or pebbles arranged in the shape of a regular polygon. The dots are thought of as alphas (units). These are one type of 2-dimensional figurate numbers.
Definition and examples
The number 10 for example, can be arranged as a triangle (see triangular number):
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But 10 cannot be arranged as a square. The number 9, on the other hand, can be (see square number):
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Some numbers, like 36, can be arranged both as a square and as a triangle (see square triangular number):
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By convention, 1 is the first polygonal number for any number of sides. The rule for enlarging the polygon to the next size is to extend two adjacent arms by one point and to then add the required extra sides between those points. In the following diagrams, each extra layer is shown as in red.
Triangular numbers
Square numbers
Polygons with higher numbers of sides, such as pentagons and hexagons, can also be constructed according to this rule, although the dots will no longer form a perfectly regular lattice like above.
Pentagonal numbers
Hexagonal numbers
Formula
If is the number of sides in a polygon, the formula for the th -gonal number is
or
The th -gonal number is also related to the triangular numbers as follows:
Thus:
For a given -gonal number , one can find by
and one can find by
.
Every hexagonal number is also a triangular number
Applying the formula above:
to the case of 6 sides gives:
but since:
it follows that:
This shows that the th hexagonal number is also the th triangular number . We can find every hexagonal number by simply taking the odd-numbered triangular numbers:
1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, ...
Table of values
The first 6 values in the column "sum of reciprocals", for triangular to octagonal numbers, come from a published solution to the general problem, which also gives a general formula for any number of sides, in terms of the digamma function.
The On-Line Encyclopedia of Integer Sequences eschews terms using Greek prefixes (e.g., "octagonal") in favor of terms using numerals (i.e., "8-gonal").
A property of this table can be expressed by the following identity (see ):
with
Combinations
Some numbers, such as 36 which is both square and triangular, fall into two polygonal sets. The problem of determining, given two such sets, all numbers that belong to both can be solved by reducing the problem to Pell's equation. The simplest example of this is the sequence of square triangular numbers.
The following table summarizes the set of -gonal -gonal numbers for small values of and .
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!
!
! Sequence
! OEIS number
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|4
|3
|1, 36, 1225, 41616, 1413721, 48024900, 16314328
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https://en.wikipedia.org/wiki/Umbral%20calculus
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In mathematics before the 1970s, the term umbral calculus referred to the surprising similarity between seemingly unrelated polynomial equations and certain shadowy techniques used to "prove" them. These techniques were introduced by John Blissard and are sometimes called Blissard's symbolic method. They are often attributed to Édouard Lucas (or James Joseph Sylvester), who used the technique extensively.
Short history
In the 1930s and 1940s, Eric Temple Bell attempted to set the umbral calculus on a rigorous footing.
In the 1970s, Steven Roman, Gian-Carlo Rota, and others developed the umbral calculus by means of linear functionals on spaces of polynomials. Currently, umbral calculus refers to the study of Sheffer sequences, including polynomial sequences of binomial type and Appell sequences, but may encompass systematic correspondence techniques of the calculus of finite differences.
The 19th-century umbral calculus
The method is a notational procedure used for deriving identities involving indexed sequences of numbers by pretending that the indices are exponents. Construed literally, it is absurd, and yet it is successful: identities derived via the umbral calculus can also be properly derived by more complicated methods that can be taken literally without logical difficulty.
An example involves the Bernoulli polynomials. Consider, for example, the ordinary binomial expansion (which contains a binomial coefficient):
and the remarkably similar-looking relation on the Bernoulli polynomials:
Compare also the ordinary derivative
to a very similar-looking relation on the Bernoulli polynomials:
These similarities allow one to construct umbral proofs, which, on the surface, cannot be correct, but seem to work anyway. Thus, for example, by pretending that the subscript n − k is an exponent:
and then differentiating, one gets the desired result:
In the above, the variable b is an "umbra" (Latin for shadow).
See also Faulhaber's formula.
Umbral Taylor series
In differential calculus, the Taylor series of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. That is, a real or complex-valued function f (x) that is infinitely differentiable at can be written as:
Similar relationships were also observed in the theory of finite differences. The umbral version of the Taylor series is given by a similar expression involving the k-th forward differences of a polynomial function f,
where
is the Pochhammer symbol used here for the falling sequential product. A similar relationship holds for the backward differences and rising factorial.
This series is also known as the Newton series or Newton's forward difference expansion.
The analogy to Taylor's expansion is utilized in the calculus of finite differences.
Bell and Riordan
In the 1930s and 1940s, Eric Temple Bell tried unsuccessfully to make this kind of argument logically rigorous. The combinatorialist John Riordan in
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https://en.wikipedia.org/wiki/Linear%20form
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In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers).
If is a vector space over a field , the set of all linear functionals from to is itself a vector space over with addition and scalar multiplication defined pointwise. This space is called the dual space of , or sometimes the algebraic dual space, when a topological dual space is also considered. It is often denoted , or, when the field is understood, ; other notations are also used, such as , or When vectors are represented by column vectors (as is common when a basis is fixed), then linear functionals are represented as row vectors, and their values on specific vectors are given by matrix products (with the row vector on the left).
Examples
The constant zero function, mapping every vector to zero, is trivially a linear functional. Every other linear functional (such as the ones below) is surjective (that is, its range is all of ).
Indexing into a vector: The second element of a three-vector is given by the one-form That is, the second element of is
Mean: The mean element of an -vector is given by the one-form That is,
Sampling: Sampling with a kernel can be considered a one-form, where the one-form is the kernel shifted to the appropriate location.
Net present value of a net cash flow, is given by the one-form where is the discount rate. That is,
Linear functionals in Rn
Suppose that vectors in the real coordinate space are represented as column vectors
For each row vector there is a linear functional defined by
and each linear functional can be expressed in this form.
This can be interpreted as either the matrix product or the dot product of the row vector and the column vector :
Trace of a square matrix
The trace of a square matrix is the sum of all elements on its main diagonal. Matrices can be multiplied by scalars and two matrices of the same dimension can be added together; these operations make a vector space from the set of all matrices. The trace is a linear functional on this space because and for all scalars and all matrices
(Definite) Integration
Linear functionals first appeared in functional analysis, the study of vector spaces of functions. A typical example of a linear functional is integration: the linear transformation defined by the Riemann integral
is a linear functional from the vector space of continuous functions on the interval to the real numbers. The linearity of follows from the standard facts about the integral:
Evaluation
Let denote the vector space of real-valued polynomial functions of degree defined on an interval If then let be the evaluation functional
The mapping is linear since
If are distinct points in then the evaluation functionals form a basis of the dual space of ( proves this last fact using Lagrange interpolation).
Non-example
A function h
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https://en.wikipedia.org/wiki/WMD
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WMD, or wmd, may refer to:
Science and technology
Weapon of mass destruction
Weighted mean in statistics
Wiggle-match-dating or wiggle matching in carbon dating
World Meteorological Day
Transportation
WMD, the National Rail code for Wymondham railway station in Norfolk, UK
WMD, the station code for Westmead railway station in Sydney, Australia
Other uses
West Midlands (county), metropolitan county in England, Chapman code
wmd, the ISO 639-3 code for the Mamaindê language spoken in the Mato Grosso state of Brazil
World Malaria Day
W.M.D., a 2017 film
See also
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https://en.wikipedia.org/wiki/Hate%20Crime%20Statistics%20Act
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The Hate Crime Statistics Act, 28 U.S.C. § 534 (HCSA), passed in 1990 and modified in 2009 by the Matthew Shepard and James Byrd, Jr. Hate Crimes Prevention Act, requires the Attorney General to collect data on crimes committed because of the victim's race, religion, disability, sexual orientation, or ethnicity. The bill was signed into law by George H. W. Bush, and was the first federal statute to "recognize and name gay, lesbian and bisexual people." Since 1992, the Department of Justice through one of its agencies, the FBI, has jointly published an annual report on hate crime statistics.
On November 16, 2020, the FBI released its 2019 Hate Crime Statistics Act (HCSA) report with the total number of reported hate crime incidents rising 2.7% to 7,317 (2019) from 7,120 (2018).
References
1990 in American law
United States federal criminal legislation
Hate crime in the United States
1990 in LGBT history
LGBT law in the United States
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https://en.wikipedia.org/wiki/List%20of%20cities%20and%20towns%20in%20Albania
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This is a list of cities and towns in Albania categorised by municipality, county and population, according to the criteria used by the Institute of Statistics (INSTAT). As of 2014, there were 74 cities classified as urban areas and 2,972 villages as rural areas in Albania. The legislation of Albania provides no official classification on the criteria of how to define a city or urban area. Furthermore, according to the methodology for cities conducted by the Organisation for Economic Co-operation and Development (OECD), five areas, including Tirana, Durrës, Elbasan, Shkodër and Vlorë, can be classified as urban audit cities.
Cities and towns in Albania belong to the following size ranges in terms of the number of population:
One city larger than 250,000: Tirana
Seven cities from 50,000 to 250,000: Durrës, Fier, Elbasan, Kamëz, Korçë, Shkodër and Vlorë
Four cities from 20,000 to 50,000: Berat, Lushnjë, Pogradec and Kavajë
List
Map
Gallery
Notes
References
Cities
Albania
Albania
Albania
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https://en.wikipedia.org/wiki/List%20of%20cities%20and%20largest%20towns%20in%20Bolivia
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According to the National Institute of Statistics of Bolivia (INE), a city is classified as an area where the city limits are identifiable, and its local government is recognized. Bolivia has 1,384 cities. As of 21 November 2012, the date of the most-recent national census, 53 cities have a population of at least 10,000 in Bolivia, as enumerated by the INE. These 53 cities have a population of 6,162,346, accounting for 61.4% of the country's population. The largest city is Santa Cruz de la Sierra, with a population of 1,441,406, a 29.4% increase from the last census date of 5 September 2001. La Guardia had the highest percentage increase, 801.5%, from 2001 to 2012.
From 2001 to 2012, Bolivia had a population increase of 21.1%. Of the 53 cities, 42 had a higher increase than 21.1%, 8 had lower increase and 3 had a small decrease. The three cities that had a negative population growth from 2001 to 2012 are La Paz (−4.1%), Yacuíba (−4.2%), and Santa Ana del Yacuma (−5.4%). With the exception of the Department of La Paz, each department's capital city is the largest city in its respective department. The Department of Santa Cruz has the most cities (18), and Pando and Chuquisaca have the least (1).
List
Notes
A In 2001, Yapacaní did not exist. The INE consolidated 19 cities, which had a 2001 population of 16,572, for the 2012 census to form Yapacaní.
References
Bolivia, List of cities in
Bolivia
Cities
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https://en.wikipedia.org/wiki/List%20of%20cities%20in%20Chile
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This is a list of cities in Chile.
A city is defined by Chile's National Statistics Institute (INE) as an "urban entity" with more than 5,000 inhabitants. This list is based on a June 2005 report by the INE based on the 2002 census which registered 239 cities across the country.
Complete list of cities by region
Largest urban agglomerations
This list includes conurbations, "absorptions" and cities with over 100,000 inhabitants, according to the 2017 census.
See also
List of towns in Chile
Communes of Chile
Notes
References
Further reading
External links
National Statistics Institute (INE)
Cities
Chile, cities in
Chile
Chile
Cities
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https://en.wikipedia.org/wiki/List%20of%20cities%20in%20Mexico
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This is a list of the Top 100 cities in Mexico by fixed population, according to the 2020 Mexican National Census.
According to Mexico's National Institute of Statistics and Geography (INEGI), a locality is "any place settled with one or more dwellings, which may or may not be inhabited, and which is known by a name given by law or tradition". Urban localities are those with more than 2,500 residents, which can be designated as cities, villages or towns according to the laws of each state. The National Urban System, compiled by the National Population Council (CONAPO) in 2018, identifies 401 urban localities in Mexico with more than 15,000 residents as "cities".
This list does not consider the entire population of metropolitan areas and is limited by political boundaries within each municipality or state. Popular notions of city population are not based on official city boundaries, but on metropolitan areas. For population data reflecting common usage, see Metropolitan areas of Mexico.
Top 100 cities by population
Only one state (Tlaxcala) has no cities in the Top 100. Mexico City contains all of the federal entity's area, including rural areas with relatively small populations. All of the map links are of the same scale.
‡ These cities extend beyond the borders of a single municipality.
Distribution
For the Top 100 cities, the following distributions hold as of the 2020 Census.
The total population is 57,930,969, 45.97% of Mexico's total.
The mean city population is 579,310. The median city in population is Villahermosa.
The mean city growth from 2010 to 2020 is 20.77%, compared to a national growth of 12.17%. The median city in population growth is Ixtapaluca.
See also
List of most populous cities in Mexico by decade
List of municipalities in Mexico by population
Metropolitan areas of Mexico
Demographics of Mexico
References
External links
National Population Council (CONAPO) — official website.
National Institute of Statistics and Geography (INEGI) — official website.
Main Cities to Visit in Mexico —
Towns of Mexico — List with all the small villages, towns and cities in Mexico
Mexico
Mexico
Cities
Mexico
Subdivisions of Mexico
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https://en.wikipedia.org/wiki/Replication
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Replication may refer to:
Science
Replication (scientific method), one of the main principles of the scientific method, a.k.a. reproducibility
Replication (statistics), the repetition of a test or complete experiment
Replication crisis
Self-replication, the process in which an entity (a cell, virus, program, etc.) makes a copy of itself
DNA replication or DNA synthesis, the process of copying a double-stranded DNA molecule
Semiconservative replication, mechanism of DNA replication
Viral replication, the process by which viruses produce copies of themselves
Replication (metallography), the use of thin plastic films to duplicate the microstructure of a component
Self-replicating machines
Computing
Replication (computing), the use of redundant resources to improve reliability, fault-tolerance, or performance
Replication (optical media), the manufacture of CDs and DVDs by means other than burning writable discs
See also
Replicator (disambiguation)
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https://en.wikipedia.org/wiki/Equilateral%20polygon
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In geometry, an equilateral polygon is a polygon which has all sides of the same length. Except in the triangle case, an equilateral polygon does not need to also be equiangular (have all angles equal), but if it does then it is a regular polygon. If the number of sides is at least five, an equilateral polygon does not need to be a convex polygon: it could be concave or even self-intersecting.
Examples
All regular polygons and edge-transitive polygons are equilateral. When an equilateral polygon is non-crossing and cyclic (its vertices are on a circle) it must be regular. An equilateral quadrilateral must be convex; this polygon is a rhombus (possibly a square).
A convex equilateral pentagon can be described by two consecutive angles, which together determine the other angles. However, equilateral pentagons, and equilateral polygons with more than five sides, can also be concave, and if concave pentagons are allowed then two angles are no longer sufficient to determine the shape of the pentagon.
A tangential polygon (one that has an incircle tangent to all its sides) is equilateral if and only if the alternate angles are equal (that is, angles 1, 3, 5, ... are equal and angles 2, 4, ... are equal). Thus if the number of sides n is odd, a tangential polygon is equilateral if and only if it is regular.
Measurement
Viviani's theorem generalizes to equilateral polygons: The sum of the perpendicular distances from an interior point to the sides of an equilateral polygon is independent of the location of the interior point.
The principal diagonals of a hexagon each divide the hexagon into quadrilaterals. In any convex equilateral hexagon with common side a, there exists a principal diagonal d1 such that
and a principal diagonal d2 such that
.
Optimality
When an equilateral polygon is inscribed in a Reuleaux polygon, it forms a Reinhardt polygon. Among all convex polygons with the same number of sides, these polygons have the largest possible perimeter for their diameter, the largest possible width for their diameter, and the largest possible width for their perimeter.
References
External links
Equilateral triangle With interactive animation
A Property of Equiangular Polygons: What Is It About? a discussion of Viviani's theorem at Cut-the-knot.
Types of polygons
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https://en.wikipedia.org/wiki/Harmonic%20number
|
In mathematics, the -th harmonic number is the sum of the reciprocals of the first natural numbers:
Starting from , the sequence of harmonic numbers begins:
Harmonic numbers are related to the harmonic mean in that the -th harmonic number is also times the reciprocal of the harmonic mean of the first positive integers.
Harmonic numbers have been studied since antiquity and are important in various branches of number theory. They are sometimes loosely termed harmonic series, are closely related to the Riemann zeta function, and appear in the expressions of various special functions.
The harmonic numbers roughly approximate the natural logarithm function and thus the associated harmonic series grows without limit, albeit slowly. In 1737, Leonhard Euler used the divergence of the harmonic series to provide a new proof of the infinity of prime numbers. His work was extended into the complex plane by Bernhard Riemann in 1859, leading directly to the celebrated Riemann hypothesis about the distribution of prime numbers.
When the value of a large quantity of items has a Zipf's law distribution, the total value of the most-valuable items is proportional to the -th harmonic number. This leads to a variety of surprising conclusions regarding the long tail and the theory of network value.
The Bertrand-Chebyshev theorem implies that, except for the case , the harmonic numbers are never integers.
Identities involving harmonic numbers
By definition, the harmonic numbers satisfy the recurrence relation
The harmonic numbers are connected to the Stirling numbers of the first kind by the relation
The functions
satisfy the property
In particular
is an integral of the logarithmic function.
The harmonic numbers satisfy the series identities
and
These two results are closely analogous to the corresponding integral results
and
Identities involving
There are several infinite summations involving harmonic numbers and powers of :
Calculation
An integral representation given by Euler is
The equality above is straightforward by the simple algebraic identity
Using the substitution , another expression for is
The th harmonic number is about as large as the natural logarithm of . The reason is that the sum is approximated by the integral
whose value is .
The values of the sequence decrease monotonically towards the limit
where is the Euler–Mascheroni constant. The corresponding asymptotic expansion is
where are the Bernoulli numbers.
Generating functions
A generating function for the harmonic numbers is
where ln(z) is the natural logarithm. An exponential generating function is
where Ein(z) is the entire exponential integral. The exponential integral may also be expressed as
where Γ(0, z) is the incomplete gamma function.
Arithmetic properties
The harmonic numbers have several interesting arithmetic properties. It is well-known that is an integer if and only if , a result often attributed to Taeisinger. Indeed, using 2-adic valuation,
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https://en.wikipedia.org/wiki/Repunit
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In recreational mathematics, a repunit is a number like 11, 111, or 1111 that contains only the digit 1 — a more specific type of repdigit. The term stands for "repeated unit" and was coined in 1966 by Albert H. Beiler in his book Recreations in the Theory of Numbers.
A repunit prime is a repunit that is also a prime number. Primes that are repunits in base-2 are Mersenne primes. As of May 2023, the largest known prime number , the largest probable prime R8177207 and the largest elliptic curve primality-proven prime R86453 are all repunits in various bases.
Definition
The base-b repunits are defined as (this b can be either positive or negative)
Thus, the number Rn(b) consists of n copies of the digit 1 in base-b representation. The first two repunits base-b for n = 1 and n = 2 are
In particular, the decimal (base-10) repunits that are often referred to as simply repunits are defined as
Thus, the number Rn = Rn(10) consists of n copies of the digit 1 in base 10 representation. The sequence of repunits base-10 starts with
1, 11, 111, 1111, 11111, 111111, ... .
Similarly, the repunits base-2 are defined as
Thus, the number Rn(2) consists of n copies of the digit 1 in base-2 representation. In fact, the base-2 repunits are the well-known Mersenne numbers Mn = 2n − 1, they start with
1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, 2047, 4095, 8191, 16383, 32767, 65535, ... .
Properties
Any repunit in any base having a composite number of digits is necessarily composite. Only repunits (in any base) having a prime number of digits might be prime. This is a necessary but not sufficient condition. For example,
R35(b) = = 11111 × 1000010000100001000010000100001 = 1111111 × 10000001000000100000010000001,
since 35 = 7 × 5 = 5 × 7. This repunit factorization does not depend on the base-b in which the repunit is expressed.
If p is an odd prime, then every prime q that divides Rp(b) must be either 1 plus a multiple of 2p, or a factor of b − 1. For example, a prime factor of R29 is 62003 = 1 + 2·29·1069. The reason is that the prime p is the smallest exponent greater than 1 such that q divides bp − 1, because p is prime. Therefore, unless q divides b − 1, p divides the Carmichael function of q, which is even and equal to q − 1.
Any positive multiple of the repunit Rn(b) contains at least n nonzero digits in base-b.
Any number x is a two-digit repunit in base x − 1.
The only known numbers that are repunits with at least 3 digits in more than one base simultaneously are 31 (111 in base-5, 11111 in base-2) and 8191 (111 in base-90, 1111111111111 in base-2). The Goormaghtigh conjecture says there are only these two cases.
Using the pigeon-hole principle it can be easily shown that for relatively prime natural numbers n and b, there exists a repunit in base-b that is a multiple of n. To see this consider repunits R1(b),...,Rn(b). Because there are n repunits but only n−1 non-zero residues modulo n there exist two repunits Ri(b) and Rj(b) with 1 ≤ i
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https://en.wikipedia.org/wiki/111%20%28number%29
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111 (One hundred [and] eleven) is the natural number following 110 and preceding 112.
In mathematics
111 is a perfect totient number.
111 is R3 or the second repunit, a number like 11, 111, or 1111 that consists of repeated units, or 1's. It equals 3 × 37, therefore all triplets (numbers like 222 or 777) in base ten are of the form 3n × 37. As a repunit, it also follows that 111 is a palindromic number.
All triplets in all bases are multiples of 111 in that base, therefore the number represented by 111 in a particular base is the only triplet that can ever be prime. 111 is not prime in base ten, but is prime in base two, where 1112 = 710. It is also prime in these other bases up to 128: 3, 5, 6, 8, 12, 14, 15, 17, 20, 21, 24, 27, 33, 38, 41, 50, 54, 57, 59, 62, 66, 69, 71, 75, 77, 78, 80, 89, 90, 99, 101, 105, 110, 111, 117, 119
In base 18, the number 111 is 73 (= 34310) which is the only base where 111 is a perfect power.
The smallest magic square using only 1 and prime numbers has a magic constant of 111:
A six-by-six magic square using the numbers 1 to 36 also has a magic constant of 111:
(The square has this magic constant because 1 + 2 + 3 + ... + 34 + 35 + 36 = 666, and 666 / 6 = 111).
111 is also the magic constant of the n-Queens Problem for n = 6. It is also a nonagonal number.
In base 10, it is a Harshad number as well as a strobogrammatic number.
Nelson
In cricket, the number 111 is sometimes called "a Nelson" after Admiral Nelson, who allegedly only had "One Eye, One Arm, One Leg" near the end of his life. This is in fact inaccurate—Nelson never lost a leg. Alternate meanings include "One Eye, One Arm, One Ambition" and "One Eye, One Arm, One Arsehole".
Particularly in cricket, multiples of 111 are called a double Nelson (222), triple Nelson (333), quadruple Nelson (444; also known as a salamander) and so on.
A score of 111 is considered by some to be unlucky. To combat the supposed bad luck, some watching lift their feet off the ground. Since an umpire cannot sit down and raise his feet, the international umpire David Shepherd had a whole retinue of peculiar mannerisms if the score was ever a Nelson multiple. He would hop, shuffle, or jiggle, particularly if the number of wickets also matched—111/1, 222/2 etc.
In other fields
111 is also:
The atomic number of the element roentgenium (Rg).
The chemical compound 1,1,1-trichloroethane is a chlorinated hydrocarbon that was used as an industrial solvent with a trade name "Solvent 111".
The emergency telephone number in New Zealand; see 111 (emergency telephone number).
NHS 111, a medical helpline in England and Scotland.
It is the lowest positive integer requiring seven syllables to name in English (British and Commonwealth), or six syllables (by dropping the "and") in American English.
Occasionally it is referred to as "eleventy-one", as read in The Fellowship of the Ring by J.R.R. Tolkien.
The first Pacific (4-6-2) locomotive in Great Britain was the Great Western Railway
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https://en.wikipedia.org/wiki/Permanent%20%28mathematics%29
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In linear algebra, the permanent of a square matrix is a function of the matrix similar to the determinant. The permanent, as well as the determinant, is a polynomial in the entries of the matrix. Both are special cases of a more general function of a matrix called the immanant.
Definition
The permanent of an matrix is defined as
The sum here extends over all elements σ of the symmetric group Sn; i.e. over all permutations of the numbers 1, 2, ..., n.
For example,
and
The definition of the permanent of A differs from that of the determinant of A in that the signatures of the permutations are not taken into account.
The permanent of a matrix A is denoted per A, perm A, or Per A, sometimes with parentheses around the argument. Minc uses Per(A) for the permanent of rectangular matrices, and per(A) when A is a square matrix. Muir and Metzler use the notation .
The word, permanent, originated with Cauchy in 1812 as “fonctions symétriques permanentes” for a related type of function, and was used by Muir and Metzler in the modern, more specific, sense.
Properties
If one views the permanent as a map that takes n vectors as arguments, then it is a multilinear map and it is symmetric (meaning that any order of the vectors results in the same permanent). Furthermore, given a square matrix of order n:
perm(A) is invariant under arbitrary permutations of the rows and/or columns of A. This property may be written symbolically as perm(A) = perm(PAQ) for any appropriately sized permutation matrices P and Q,
multiplying any single row or column of A by a scalar s changes perm(A) to s⋅perm(A),
perm(A) is invariant under transposition, that is, perm(A) = perm(AT).
If and are square matrices of order n then, where s and t are subsets of the same size of {1,2,...,n} and are their respective complements in that set.
If is a triangular matrix, i.e. , whenever or, alternatively, whenever , then its permanent (and determinant as well) equals the product of the diagonal entries:
Relation to determinants
Laplace's expansion by minors for computing the determinant along a row, column or diagonal extends to the permanent by ignoring all signs.
For every ,
where is the entry of the ith row and the jth column of B, and is the permanent of the submatrix obtained by removing the ith row and the jth column of B.
For example, expanding along the first column,
while expanding along the last row gives,
On the other hand, the basic multiplicative property of determinants is not valid for permanents. A simple example shows that this is so.
Unlike the determinant, the permanent has no easy geometrical interpretation; it is mainly used in combinatorics, in treating boson Green's functions in quantum field theory, and in determining state probabilities of boson sampling systems. However, it has two graph-theoretic interpretations: as the sum of weights of cycle covers of a directed graph, and as the sum of weights of perfect matchings in a bipartite gra
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https://en.wikipedia.org/wiki/Hilbert%20cube
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In mathematics, the Hilbert cube, named after David Hilbert, is a topological space that provides an instructive example of some ideas in topology. Furthermore, many interesting topological spaces can be embedded in the Hilbert cube; that is, can be viewed as subspaces of the Hilbert cube (see below).
Definition
The Hilbert cube is best defined as the topological product of the intervals for That is, it is a cuboid of countably infinite dimension, where the lengths of the edges in each orthogonal direction form the sequence
The Hilbert cube is homeomorphic to the product of countably infinitely many copies of the unit interval In other words, it is topologically indistinguishable from the unit cube of countably infinite dimension. Some authors use the term "Hilbert cube" to mean this Cartesian product instead of the product of the .
If a point in the Hilbert cube is specified by a sequence with then a homeomorphism to the infinite dimensional unit cube is given by
The Hilbert cube as a metric space
It is sometimes convenient to think of the Hilbert cube as a metric space, indeed as a specific subset of a separable Hilbert space (that is, a Hilbert space with a countably infinite Hilbert basis).
For these purposes, it is best not to think of it as a product of copies of but instead as
as stated above, for topological properties, this makes no difference.
That is, an element of the Hilbert cube is an infinite sequence
that satisfies
Any such sequence belongs to the Hilbert space so the Hilbert cube inherits a metric from there. One can show that the topology induced by the metric is the same as the product topology in the above definition.
Properties
As a product of compact Hausdorff spaces, the Hilbert cube is itself a compact Hausdorff space as a result of the Tychonoff theorem.
The compactness of the Hilbert cube can also be proved without the axiom of choice by constructing a continuous function from the usual Cantor set onto the Hilbert cube.
In no point has a compact neighbourhood (thus, is not locally compact). One might expect that all of the compact subsets of are finite-dimensional.
The Hilbert cube shows that this is not the case.
But the Hilbert cube fails to be a neighbourhood of any point because its side becomes smaller and smaller in each dimension, so that an open ball around of any fixed radius must go outside the cube in some dimension.
Any infinite-dimensional convex compact subset of is homeomorphic to the Hilbert cube. The Hilbert cube is a convex set, whose span is the whole space, but whose interior is empty. This situation is impossible in finite dimensions. The tangent cone to the cube at the zero vector is the whole space.
Every subset of the Hilbert cube inherits from the Hilbert cube the properties of being both metrizable (and therefore T4) and second countable. It is more interesting that the converse also holds: Every second countable T4 space is homeomorphic to a subset of the Hilbert
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https://en.wikipedia.org/wiki/Catalan%20number
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In combinatorial mathematics, the Catalan numbers are a sequence of natural numbers that occur in various counting problems, often involving recursively defined objects. They are named after the French-Belgian mathematician Eugène Charles Catalan.
The nth Catalan number can be expressed directly in terms of the central binomial coefficients by
The first Catalan numbers for n = 0, 1, 2, 3, ... are
1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, ... .
Properties
An alternative expression for Cn is
for
which is equivalent to the expression given above because . This expression shows that Cn is an integer, which is not immediately obvious from the first formula given. This expression forms the basis for a proof of the correctness of the formula.
Another alternative expression is
which can be directly interpreted in terms of the cycle lemma; see below.
The Catalan numbers satisfy the recurrence relations
and
Asymptotically, the Catalan numbers grow as
in the sense that the quotient of the nth Catalan number and the expression on the right tends towards 1 as n approaches infinity.
This can be proved by using the asymptotic growth of the central binomial coefficients, by Stirling's approximation for , or via generating functions.
A more accurate asymptotic analysis shows that the Catalan numbers are approximated by the fourth order approximation
.
The only Catalan numbers Cn that are odd are those for which n = 2k − 1; all others are even. The only prime Catalan numbers are C2 = 2 and C3 = 5.
The Catalan numbers have the integral representations
which immediately yields .
This has a simple probabilistic interpretation. Consider a random walk on the integer line, starting at 0. Let -1 be a "trap" state, such that if the walker arrives at -1, it will remain there. The walker can arrive at the trap state at times 1, 3, 5, 7..., and the number of ways the walker can arrive at the trap state at time is . Since the 1D random walk is recurrent, the probability that the walker eventually arrives at -1 is .
Applications in combinatorics
There are many counting problems in combinatorics whose solution is given by the Catalan numbers. The book Enumerative Combinatorics: Volume 2 by combinatorialist Richard P. Stanley contains a set of exercises which describe 66 different interpretations of the Catalan numbers. Following are some examples, with illustrations of the cases C3 = 5 and C4 = 14.
Cn is the number of Dyck words of length 2n. A Dyck word is a string consisting of n X's and n Y's such that no initial segment of the string has more Y's than X's. For example, the following are the Dyck words up to length 6:
XY
XXYY XYXY
XXXYYY XYXXYY XYXYXY XXYYXY XXYXYY
Re-interpreting the symbol X as an open parenthesis and Y as a close parenthesis, Cn counts the number of expressions containing n pairs of parentheses which are correctly matched:
((())) (()()) (())() ()(()) ()()()
Cn is the
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https://en.wikipedia.org/wiki/Repdigit
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In recreational mathematics, a repdigit or sometimes monodigit is a natural number composed of repeated instances of the same digit in a positional number system (often implicitly decimal). The word is a portmanteau of "repeated" and "digit".
Examples are 11, 666, 4444, and 999999. All repdigits are palindromic numbers and are multiples of repunits. Other well-known repdigits include the repunit primes and in particular the Mersenne primes (which are repdigits when represented in binary).
Repdigits are the representation in base of the number where is the repeated digit and is the number of repetitions. For example, the repdigit 77777 in base 10 is .
A variation of repdigits called Brazilian numbers are numbers that can be written as a repdigit in some base, not allowing the repdigit 11, and not allowing the single-digit numbers (or all numbers will be Brazilian). For example, 27 is a Brazilian number because 27 is the repdigit 33 in base 8, while 9 is not a Brazilian number because its only repdigit representation is 118, not allowed in the definition of Brazilian numbers. The representations of the form 11 are considered trivial and are disallowed in the definition of Brazilian numbers, because all natural numbers n greater than two have the representation 11n − 1. The first twenty Brazilian numbers are
7, 8, 10, 12, 13, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 31, 32, 33, ... .
On some websites (including imageboards like 4chan), it is considered an auspicious event when the sequentially-assigned ID number of a post is a repdigit (or a "GET"), such as 22222222.
History
The concept of a repdigit has been studied under that name since at least 1974, and earlier called them "monodigit numbers". The Brazilian numbers were introduced later, in 1994, in the 9th Iberoamerican Mathematical Olympiad that took place in Fortaleza at Brazil. The first problem in this competition, proposed by Mexico, was as follows:
A number is called "Brazilian" if there exists an integer b such that for which the representation of n in base b is written with all equal digits. Prove that 1994 is Brazilian and that 1993 is not Brazilian.
Primes and repunits
For a repdigit to be prime, it must be a repunit (i.e. the repeating digit is 1) and have a prime number of digits in its base (except trivial single-digit numbers), since, for example, the repdigit 77777 is divisible by 7, in any base > 7. In particular, as Brazilian repunits do not allow the number of digits to be exactly two, Brazilian primes must have an odd prime number of digits. Having an odd prime number of digits is not enough to guarantee that a repunit is prime; for instance, 21 = 1114 = 3 × 7 and 111 = 11110 = 3 × 37 are not prime. In any given base b, every repunit prime in that base with the exception of 11b (if it is prime) is a Brazilian prime. The smallest Brazilian primes are
7 = 1112, 13 = 1113, 31 = 111112 = 1115, 43 = 1116, 73 = 1118, 127 = 11111112, 157 = 11112, ...
While t
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https://en.wikipedia.org/wiki/222%20%28number%29
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222 (two hundred [and] twenty-two) is the natural number following 221 and preceding 223.
In mathematics
It is a decimal repdigit and a strobogrammatic number (meaning that it looks the same turned upside down on a calculator display). It is one of the numbers whose digit sum in decimal is the same as it is in binary.
222 is a noncototient, meaning that it cannot be written in the form n − φ(n) where φ is Euler's totient function counting the number of values that are smaller than n and relatively prime to it.
There are exactly 222 distinct ways of assigning a meet and join operation to a set of ten unlabelled elements in order to give them the structure of a lattice, and exactly 222 different six-edge polysticks.
References
Integers
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https://en.wikipedia.org/wiki/Invertible%20matrix
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In linear algebra, an -by- square matrix is called invertible (also nonsingular, nondegenerate or —rarely used— regular), if there exists an -by- square matrix such thatwhere denotes the -by- identity matrix and the multiplication used is ordinary matrix multiplication. If this is the case, then the matrix is uniquely determined by , and is called the (multiplicative) inverse of , denoted by . Matrix inversion is the process of finding the matrix that satisfies the prior equation for a given invertible matrix .
Over a field, a square matrix that is not invertible is called singular or degenerate. A square matrix with entries in a field is singular if and only if its determinant is zero. Singular matrices are rare in the sense that if a square matrix's entries are randomly selected from any bounded region on the number line or complex plane, the probability that the matrix is singular is 0, that is, it will "almost never" be singular. Non-square matrices, i.e. -by- matrices for which , do not have an inverse. However, in some cases such a matrix may have a left inverse or right inverse. If is -by- and the rank of is equal to , (), then has a left inverse, an -by- matrix such that . If has rank (), then it has a right inverse, an -by- matrix such that .
While the most common case is that of matrices over the real or complex numbers, all these definitions can be given for matrices over any algebraic structure equipped with addition and multiplication (i.e. rings). However, in the case of a ring being commutative, the condition for a square matrix to be invertible is that its determinant is invertible in the ring, which in general is a stricter requirement than it being nonzero. For a noncommutative ring, the usual determinant is not defined. The conditions for existence of left-inverse or right-inverse are more complicated, since a notion of rank does not exist over rings.
The set of invertible matrices together with the operation of matrix multiplication and entries from ring form a group, the general linear group of degree , denoted .
Properties
The invertible matrix theorem
Let be a square -by- matrix over a field (e.g., the field of real numbers). The following statements are equivalent, i.e., they are either all true or all false for any given matrix:
The matrix has a left inverse under matrix multiplication (that is, there exists a such that )
The matrix has a right inverse under matrix multiplication (that is, there exists a such that ).
is invertible, i.e. it has an inverse under matrix multiplication (that is, there exists a such that ).
The linear transformation mapping to has a left inverse under function composition.
The linear transformation mapping to has a right inverse under function composition.
The linear transformation mapping to is invertible, that is, has an inverse under function composition.
is row-equivalent to the -by- identity matrix .
is column-equivalent to the -by- identity
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https://en.wikipedia.org/wiki/Probability%20vector
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In mathematics and statistics, a probability vector or stochastic vector is a vector with non-negative entries that add up to one.
The positions (indices) of a probability vector represent the possible outcomes of a discrete random variable, and the vector gives us the probability mass function of that random variable, which is the standard way of characterizing a discrete probability distribution.
Examples
Here are some examples of probability vectors. The vectors can be either columns or rows.
Geometric interpretation
Writing out the vector components of a vector as
the vector components must sum to one:
Each individual component must have a probability between zero and one:
for all . Therefore, the set of stochastic vectors coincides with the standard -simplex. It is a point if , a segment if , a (filled) triangle if , a (filled) tetrahedron , etc.
Properties
The mean of any probability vector is .
The shortest probability vector has the value as each component of the vector, and has a length of .
The longest probability vector has the value 1 in a single component and 0 in all others, and has a length of 1.
The shortest vector corresponds to maximum uncertainty, the longest to maximum certainty.
The length of a probability vector is equal to ; where is the variance of the elements of the probability vector.
See also
Stochastic matrix
Dirichlet distribution
References
Probability theory
Vectors (mathematics and physics)
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https://en.wikipedia.org/wiki/Stochastic%20matrix
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In mathematics, a stochastic matrix is a square matrix used to describe the transitions of a Markov chain. Each of its entries is a nonnegative real number representing a probability. It is also called a probability matrix, transition matrix, substitution matrix, or Markov matrix. The stochastic matrix was first developed by Andrey Markov at the beginning of the 20th century, and has found use throughout a wide variety of scientific fields, including probability theory, statistics, mathematical finance and linear algebra, as well as computer science and population genetics. There are several different definitions and types of stochastic matrices:
A right stochastic matrix is a real square matrix, with each row summing to 1.
A left stochastic matrix is a real square matrix, with each column summing to 1.
A doubly stochastic matrix is a square matrix of nonnegative real numbers with each row and column summing to 1.
In the same vein, one may define a stochastic vector (also called probability vector) as a vector whose elements are nonnegative real numbers which sum to 1. Thus, each row of a right stochastic matrix (or column of a left stochastic matrix) is a stochastic vector. A common convention in English language mathematics literature is to use row vectors of probabilities and right stochastic matrices rather than column vectors of probabilities and left stochastic matrices; this article follows that convention. In addition, a substochastic matrix is a real square matrix whose row sums are all
History
The stochastic matrix was developed alongside the Markov chain by Andrey Markov, a Russian mathematician and professor at St. Petersburg University who first published on the topic in 1906. His initial intended uses were for linguistic analysis and other mathematical subjects like card shuffling, but both Markov chains and matrices rapidly found use in other fields.
Stochastic matrices were further developed by scholars like Andrey Kolmogorov, who expanded their possibilities by allowing for continuous-time Markov processes. By the 1950s, articles using stochastic matrices had appeared in the fields of econometrics and circuit theory. In the 1960s, stochastic matrices appeared in an even wider variety of scientific works, from behavioral science to geology to residential planning. In addition, much mathematical work was also done through these decades to improve the range of uses and functionality of the stochastic matrix and Markovian processes more generally.
From the 1970s to present, stochastic matrices have found use in almost every field that requires formal analysis, from structural science to medical diagnosis to personnel management. In addition, stochastic matrices have found wide use in land change modeling, usually under the term Markov matrix.
Definition and properties
A stochastic matrix describes a Markov chain over a finite state space with cardinality .
If the probability of moving from to in one time step is , the st
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https://en.wikipedia.org/wiki/Conjugate%20transpose
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In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an complex matrix is an matrix obtained by transposing and applying complex conjugate on each entry (the complex conjugate of being , for real numbers and ). It is often denoted as or or or (often in physics) .
For real matrices, the conjugate transpose is just the transpose, .
Definition
The conjugate transpose of an matrix is formally defined by
where the subscript denotes the -th entry, for and , and the overbar denotes a scalar complex conjugate.
This definition can also be written as
where denotes the transpose and denotes the matrix with complex conjugated entries.
Other names for the conjugate transpose of a matrix are Hermitian conjugate, adjoint matrix or transjugate. The conjugate transpose of a matrix can be denoted by any of these symbols:
, commonly used in linear algebra
, commonly used in linear algebra
(sometimes pronounced as A dagger), commonly used in quantum mechanics
, although this symbol is more commonly used for the Moore–Penrose pseudoinverse
In some contexts, denotes the matrix with only complex conjugated entries and no transposition.
Example
Suppose we want to calculate the conjugate transpose of the following matrix .
We first transpose the matrix:
Then we conjugate every entry of the matrix:
Basic remarks
A square matrix with entries is called
Hermitian or self-adjoint if ; i.e., .
Skew Hermitian or antihermitian if ; i.e., .
Normal if .
Unitary if , equivalently , equivalently .
Even if is not square, the two matrices and are both Hermitian and in fact positive semi-definite matrices.
The conjugate transpose "adjoint" matrix should not be confused with the adjugate, , which is also sometimes called adjoint.
The conjugate transpose of a matrix with real entries reduces to the transpose of , as the conjugate of a real number is the number itself.
Motivation
The conjugate transpose can be motivated by noting that complex numbers can be usefully represented by real matrices, obeying matrix addition and multiplication:
That is, denoting each complex number by the real matrix of the linear transformation on the Argand diagram (viewed as the real vector space ), affected by complex -multiplication on .
Thus, an matrix of complex numbers could be well represented by a matrix of real numbers. The conjugate transpose, therefore, arises very naturally as the result of simply transposing such a matrix—when viewed back again as an matrix made up of complex numbers.
Properties of the conjugate transpose
for any two matrices and of the same dimensions.
for any complex number and any matrix .
for any matrix and any matrix . Note that the order of the factors is reversed.
for any matrix , i.e. Hermitian transposition is an involution.
If is a square matrix, then where denotes the determinant of .
If is a square matrix, then where denotes the trace of .
is invertible i
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https://en.wikipedia.org/wiki/Complex%20plane
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In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the imaginary numbers.
The complex plane allows for a geometric interpretation of complex numbers. Under addition, they add like vectors. The multiplication of two complex numbers can be expressed more easily in polar coordinates—the magnitude or modulus of the product is the product of the two absolute values, or moduli, and the angle or argument of the product is the sum of the two angles, or arguments. In particular, multiplication by a complex number of modulus 1 acts as a rotation.
The complex plane is sometimes known as the Argand plane or Gauss plane.
Notational conventions
Complex numbers
In complex analysis, the complex numbers are customarily represented by the symbol z, which can be separated into its real (x) and imaginary (y) parts:
for example: z = 4 + 5i, where x and y are real numbers, and i is the imaginary unit. In this customary notation the complex number z corresponds to the point (x, y) in the Cartesian plane.
In the Cartesian plane the point (x, y) can also be represented in polar coordinates as
In the Cartesian plane it may be assumed that the arctangent takes values from −π/2 to π/2 (in radians), and some care must be taken to define the more complete arctangent function for points (x, y) when x ≤ 0. In the complex plane these polar coordinates take the form
where
Here |z| is the absolute value or modulus of the complex number z; θ, the argument of z, is usually taken on the interval ; and the last equality (to |z|eiθ) is taken from Euler's formula. Without the constraint on the range of θ, the argument of z is multi-valued, because the complex exponential function is periodic, with period 2π i. Thus, if θ is one value of arg(z), the other values are given by , where n is any non-zero integer.
While seldom used explicitly, the geometric view of the complex numbers is implicitly based on its structure of a Euclidean vector space of dimension 2, where the inner product of complex numbers and is given by ; then for a complex number its absolute value coincides with its Euclidean norm, and its argument with the angle turning from 1 to .
The theory of contour integration comprises a major part of complex analysis. In this context, the direction of travel around a closed curve is important – reversing the direction in which the curve is traversed multiplies the value of the integral by −1. By convention the positive direction is counterclockwise. For example, the unit circle is traversed in the positive direction when we start at the point z = 1, then travel up and to the left through the point z = i, then down and to the left through −1, then down and to the right through −i, and finally up and to the right to z = 1, where we started.
Almost all of complex analysis is con
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https://en.wikipedia.org/wiki/Domain%20relational%20calculus
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In computer science, domain relational calculus (DRC) is a calculus that was introduced by Michel Lacroix and Alain Pirotte as a declarative database query language for the relational data model.
In DRC, queries have the form:
where each Xi is either a domain variable or constant, and denotes a DRC formula. The result of the query is the set of tuples X1 to Xn that make the DRC formula true.
This language uses the same operators as tuple calculus,
the logical connectives ∧ (and), ∨ (or) and ¬ (not). The existential quantifier (∃) and the universal quantifier (∀) can be used to bind the variables.
Its computational expressiveness is equivalent to that of relational algebra.
Examples
Let (A, B, C) mean (Rank, Name, ID) in the Enterprise relation
and let (D, E, F) mean (Name, DeptName, ID) in the Department relation
All captains of the starship USS Enterprise:
In this example, A, B, C denotes both the result set and a set in the table Enterprise.
Names of Enterprise crew members who are in Stellar Cartography:
In this example, we're only looking for the name, and that's B. The condition F = C is a requirement that describes the intersection of Enterprise crew members AND members of the Stellar Cartography Department.
An alternate representation of the previous example would be:
In this example, the value of the requested F domain is directly placed in the formula and the C domain variable is re-used in the query for the existence of a department, since it already holds a crew member's ID.
See also
Relational calculus
References
External links
DES – An educational tool for working with Domain Relational Calculus and other formal languages
WinRDBI – An educational tool for working with Domain Relational Calculus and other formal languages
Relational model
Logical calculi
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https://en.wikipedia.org/wiki/Kettering%20University
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Kettering University is a private university in Flint, Michigan. It offers bachelor of science and master’s degrees in STEM (science, technology, engineering, and mathematics) and business fields. Kettering University undergraduate students are required to complete at least five co-op terms to graduate. Students gain paid work experience in a variety of industries with Kettering's more than 550 corporate partners, and graduate with professional experiences accompanying their degree.
Kettering University is named after inventor and former head of research for General Motors, Charles F. Kettering. He was a distinguished inventor, researcher, and proponent of cooperative education.
History
Founded as The School of Automotive Trades by Albert Sobey under the direction of the Industrial Fellowship of Flint on October 20, 1919, Kettering University has a long legacy with the automotive industry. The university became known as the Flint Institute of Technology in 1923 before being acquired by General Motors in 1926. It was renamed as the General Motors Institute of Technology and eventually the General Motors Institute in 1932.
Once referred to as the "West Point of the Automobile industry," GMI focused on creating business and industry leaders through the unique co-op model (following the development of this program at the University of Cincinnati in 1907). GMI also pioneered freshman-level manufacturing courses (Production Processes I & II), and automotive degree specialties. A fifth-year thesis requirement was added in 1945, and the Institute was granted the ability to award degrees. The first bachelor's degree was awarded on August 23, 1946.
During the 1950s, the co-op program required applicants to find a GM division to be their sponsor. School and work were mixed in four- or eight-week rotations, dividing the student body into four sections, two (A and B) for the four-week rotations, and two (C and D) for the eight-week rotations. At any given time, when section A was in school, section B was at work, and vice versa. Every four weeks, this situation would be reversed. Sections C and D were scheduled similarly, on an eight-week basis. This resulted in students moving twelve or six times per year during a 48-week school/work year. Because General Motors used the school to train its engineers, tuition was partially subsidized. In June 1979 (the Class of 1984) co-op rotations were expanded to twelve weeks.
Split from GM
After GM reduced operations in Flint, the company and the university separated on July 1, 1982. The name of the institution became "GMI Engineering & Management Institute" and the letters "GMI" were retained to allow easy identification with the old General Motors Institute. The university began charging full tuition as an independent private university. The university kept the cooperative education model, expanding the number of co-op employers for students. The university also began offering graduate programs for both on- and
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https://en.wikipedia.org/wiki/Characteristic%20polynomial
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In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The characteristic polynomial of an endomorphism of a finite-dimensional vector space is the characteristic polynomial of the matrix of that endomorphism over any base (that is, the characteristic polynomial does not depend on the choice of a basis). The characteristic equation, also known as the determinantal equation, is the equation obtained by equating the characteristic polynomial to zero.
In spectral graph theory, the characteristic polynomial of a graph is the characteristic polynomial of its adjacency matrix.
Motivation
In linear algebra, eigenvalues and eigenvectors play a fundamental role, since, given a linear transformation, an eigenvector is a vector whose direction is not changed by the transformation, and the corresponding eigenvalue is the measure of the resulting change of magnitude of the vector.
More precisely, if the transformation is represented by a square matrix an eigenvector and the corresponding eigenvalue must satisfy the equation
or, equivalently,
where is the identity matrix, and
(although the zero vector satisfies this equation for every it is not considered an eigenvector).
It follows that the matrix must be singular, and its determinant
must be zero.
In other words, the eigenvalues of are the roots of
which is a monic polynomial in of degree if is a matrix. This polynomial is the characteristic polynomial of .
Formal definition
Consider an matrix The characteristic polynomial of denoted by is the polynomial defined by
where denotes the identity matrix.
Some authors define the characteristic polynomial to be That polynomial differs from the one defined here by a sign so it makes no difference for properties like having as roots the eigenvalues of ; however the definition above always gives a monic polynomial, whereas the alternative definition is monic only when is even.
Examples
To compute the characteristic polynomial of the matrix
the determinant of the following is computed:
and found to be the characteristic polynomial of
Another example uses hyperbolic functions of a hyperbolic angle φ.
For the matrix take
Its characteristic polynomial is
Properties
The characteristic polynomial of a matrix is monic (its leading coefficient is ) and its degree is The most important fact about the characteristic polynomial was already mentioned in the motivational paragraph: the eigenvalues of are precisely the roots of (this also holds for the minimal polynomial of but its degree may be less than ). All coefficients of the characteristic polynomial are polynomial expressions in the entries of the matrix. In particular its constant coefficient of is the coefficient of is one, and the coefficient of is , where is the trace of (The sign
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https://en.wikipedia.org/wiki/Economics%20and%20Statistics%20Administration
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The Economics and Statistics Administration (ESA) was an agency within the United States Department of Commerce (DOC) that analyzed, disseminated, and reported on national economic and demographic data.
Its three primary missions were the following:
Release and disseminate U.S. National Economic Indicators.
Oversee the missions of the United States Census Bureau (Census) and the Bureau of Economic Analysis (BEA).
Analyze and produce economic reports for the Department of Commerce and the Executive Branch.
References
External links
Economics and Statistics Administration in the Federal Register
List of ESA Reports
Economicindicators.gov, the Administration's official compilation of economic indicators
Census.gov
BEA.gov
United States Department of Commerce agencies
Federal Statistical System of the United States
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https://en.wikipedia.org/wiki/Bureau%20of%20Economic%20Analysis
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The Bureau of Economic Analysis (BEA) of the United States Department of Commerce is a U.S. government agency that provides official macroeconomic and industry statistics, most notably reports about the gross domestic product (GDP) of the United States and its various units—states, cities/towns/townships/villages/counties, and metropolitan areas. They also provide information about personal income, corporate profits, and government spending in their National Income and Product Accounts (NIPAs).
The BEA is one of the principal agencies of the U.S. Federal Statistical System. Its stated mission is to "promote a better understanding of the U.S. economy by providing the most timely, relevant, and accurate economic data in an objective and cost-effective manner".
BEA has about 500 employees and an annual budget of approximately $101 million.
National accounts
BEA's national economic statistics (National Economic Accounts) provide a comprehensive view of U.S. production, consumption, investment, exports and imports, and income and saving. These statistics are best known by summary measures such as gross domestic product (GDP), corporate profits, personal income and spending, and personal saving.
The National Income and Product Accounts (NIPAs) provide information about personal income, corporate profits, government spending, fixed assets, and changes in the net worth of the U.S. Economy.
The accounts also include other approaches and methods of measuring income and spending, such as the gross domestic income (GDI) and gross national income (GNI).
Industry accounts
The industry economic accounts, presented both in an input-output framework and as annual output by each industry, provide a detailed view of the interrelationships between U.S. producers and users and the contribution to production across industries. These accounts are used extensively by policymakers and businesses to understand industry interactions, productivity trends, and the changing structure of the U.S. economy.
There are quarterly and annual reports for "GDP by Industry Accounts", designed for analysis of a specific industry's contribution to overall economic growth and inflation.
Regional Economic Accounts
The regional economic accounts provide information about the geographic distribution of U.S. economic activity and growth. The estimates of gross domestic product (GDP) by state and state and local area personal income (PI), and the accompanying detail, provide a consistent framework for analyzing and comparing individual state and local area economies.
Uses of the regional program estimates
The Federal government uses regional income and product estimates to distribute funds to states:
BEA Regional Income and Product Account Estimates Used to Distribute $406.8 Billion in Federal Funds
FY2016 Federal Funds Distribution Using BEA Regional Income and Product Account Statistics
Federal Uses of BEA Regional Statistics, FY2016
Twenty-six states have set constituti
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https://en.wikipedia.org/wiki/Characteristic%20equation
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Characteristic equation may refer to:
Characteristic equation (calculus), used to solve linear differential equations
Characteristic equation, the equation obtained by equating to zero the characteristic polynomial of a matrix or of a linear mapping
Method of characteristics, a technique for solving partial differential equations
See also
Characteristic (disambiguation)
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https://en.wikipedia.org/wiki/Sheffer%20sequence
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In mathematics, a Sheffer sequence or poweroid is a polynomial sequence, i.e., a sequence of polynomials in which the index of each polynomial equals its degree, satisfying conditions related to the umbral calculus in combinatorics. They are named for Isador M. Sheffer.
Definition
Fix a polynomial sequence (pn). Define a linear operator Q on polynomials in x by
This determines Q on all polynomials. The polynomial sequence pn is a Sheffer sequence if the linear operator Q just defined is shift-equivariant; such a Q is then a delta operator. Here, we define a linear operator Q on polynomials to be shift-equivariant if, whenever f(x) = g(x + a) = Ta g(x) is a "shift" of g(x), then (Qf)(x) = (Qg)(x + a); i.e., Q commutes with every shift operator: TaQ = QTa.
Properties
The set of all Sheffer sequences is a group under the operation of umbral composition of polynomial sequences, defined as follows. Suppose ( pn(x) : n = 0, 1, 2, 3, ... ) and ( qn(x) : n = 0, 1, 2, 3, ... ) are polynomial sequences, given by
Then the umbral composition is the polynomial sequence whose nth term is
(the subscript n appears in pn, since this is the n term of that sequence, but not in q, since this refers to the sequence as a whole rather than one of its terms).
The identity element of this group is the standard monomial basis
Two important subgroups are the group of Appell sequences, which are those sequences for which the operator Q is mere differentiation, and the group of sequences of binomial type, which are those that satisfy the identity
A Sheffer sequence ( pn(x) : n = 0, 1, 2, ... ) is of binomial type if and only if both
and
The group of Appell sequences is abelian; the group of sequences of binomial type is not. The group of Appell sequences is a normal subgroup; the group of sequences of binomial type is not. The group of Sheffer sequences is a semidirect product of the group of Appell sequences and the group of sequences of binomial type. It follows that each coset of the group of Appell sequences contains exactly one sequence of binomial type. Two Sheffer sequences are in the same such coset if and only if the operator Q described above – called the "delta operator" of that sequence – is the same linear operator in both cases. (Generally, a delta operator is a shift-equivariant linear operator on polynomials that reduces degree by one. The term is due to F. Hildebrandt.)
If sn(x) is a Sheffer sequence and pn(x) is the one sequence of binomial type that shares the same delta operator, then
Sometimes the term Sheffer sequence is defined to mean a sequence that bears this relation to some sequence of binomial type. In particular, if ( sn(x) ) is an Appell sequence, then
The sequence of Hermite polynomials, the sequence of Bernoulli polynomials, and the monomials ( xn : n = 0, 1, 2, ... ) are examples of Appell sequences.
A Sheffer sequence pn is characterised by its exponential generating function
where A and B are (formal) power series in t. S
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https://en.wikipedia.org/wiki/Golden%20spiral
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In geometry, a golden spiral is a logarithmic spiral whose growth factor is , the golden ratio. That is, a golden spiral gets wider (or further from its origin) by a factor of for every quarter turn it makes.
Approximations of the golden spiral
There are several comparable spirals that approximate, but do not exactly equal, a golden spiral.
For example, a golden spiral can be approximated by first starting with a rectangle for which the ratio between its length and width is the golden ratio. This rectangle can then be partitioned into a square and a similar rectangle and this rectangle can then be split in the same way. After continuing this process for an arbitrary number of steps, the result will be an almost complete partitioning of the rectangle into squares. The corners of these squares can be connected by quarter-circles. The result, though not a true logarithmic spiral, closely approximates a golden spiral.
Another approximation is a Fibonacci spiral, which is constructed slightly differently. A Fibonacci spiral starts with a rectangle partitioned into 2 squares. In each step, a square the length of the rectangle's longest side is added to the rectangle. Since the ratio between consecutive Fibonacci numbers approaches the golden ratio as the Fibonacci numbers approach infinity, so too does this spiral get more similar to the previous approximation the more squares are added, as illustrated by the image.
Spirals in nature
Approximate logarithmic spirals can occur in nature, for example the arms of spiral galaxies – golden spirals are one special case of these logarithmic spirals, although there is no evidence that there is any general tendency towards this case appearing. Phyllotaxis is connected with the golden ratio because it involves successive leaves or petals being separated by the golden angle; it also results in the emergence of spirals, although again none of them are (necessarily) golden spirals. It is sometimes stated that spiral galaxies and nautilus shells get wider in the pattern of a golden spiral, and hence are related to both and the Fibonacci series.
In truth, many mollusk shells including nautilus shells exhibit logarithmic spiral growth, but at a variety of angles usually distinctly different from that of the golden spiral. This pattern allows the organism to grow without changing shape. Although spiral galaxies have often been modeled as logarithmic spirals, Archimedean spirals, or hyperbolic spirals, their pitch angles vary with distance from the galactic center, unlike logarithmic spirals (for which this angle does not vary), and also at variance with the other mathematical spirals used to model them.
Mathematics
A golden spiral with initial radius 1 is the locus of points of polar coordinates satisfying
where is the Golden Ratio.
The polar equation for a golden spiral is the same as for other logarithmic spirals, but with a special value of the growth factor :
or
with being the base of natural logar
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https://en.wikipedia.org/wiki/Golden%20rectangle
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In geometry, a golden rectangle is a rectangle whose side lengths are in the golden ratio, , which is (the Greek letter phi), where is approximately 1.618.
Golden rectangles exhibit a special form of self-similarity: All rectangles created by adding or removing a square from an end are golden rectangles as well.
Construction
A golden rectangle can be constructed with only a straightedge and compass in four steps:
Drawing a square
Drawing a line from the midpoint of one side of the square to an opposite corner
Using that line as the radius to draw an arc that defines the height of the rectangle
Completing the golden rectangle
A distinctive feature of this shape is that when a square section is added—or removed—the product is another golden rectangle, having the same aspect ratio as the first. Square addition or removal can be repeated infinitely, in which case corresponding corners of the squares form an infinite sequence of points on the golden spiral, the unique logarithmic spiral with this property. Diagonal lines drawn between the first two orders of embedded golden rectangles will define the intersection point of the diagonals of all the embedded golden rectangles; Clifford A. Pickover referred to this point as "the Eye of God".
History
The proportions of the golden rectangle have been observed as early as the Babylonian Tablet of Shamash (c. 888–855 BC), though Mario Livio calls any knowledge of the golden ratio before the Ancient Greeks "doubtful".
According to Livio, since the publication of Luca Pacioli's Divina proportione in 1509, "the Golden Ratio started to become available to artists in theoretical treatises that were not overly mathematical, that they could actually use."
The 1927 Villa Stein designed by Le Corbusier, some of whose architecture utilizes the golden ratio, features dimensions that closely approximate golden rectangles.
Relation to regular polygons and polyhedra
Euclid gives an alternative construction of the golden rectangle using three polygons circumscribed by congruent circles: a regular decagon, hexagon, and pentagon. The respective lengths a, b, and c of the sides of these three polygons satisfy the equation a2 + b2 = c2, so line segments with these lengths form a right triangle (by the converse of the Pythagorean theorem). The ratio of the side length of the hexagon to the decagon is the golden ratio, so this triangle forms half of a golden rectangle.
The convex hull of two opposite edges of a regular icosahedron forms a golden rectangle. The twelve vertices of the icosahedron can be decomposed in this way into three mutually-perpendicular golden rectangles, whose boundaries are linked in the pattern of the Borromean rings.
See also
Golden angle -- Circle with sectors in golden ratio
Notes
References
External links
The Golden Mean and the Physics of Aesthetics
Golden rectangle demonstration With interactive animation
From golden rectangle to golden quadrilaterals Explores some differ
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https://en.wikipedia.org/wiki/Jacques%20Tits
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Jacques Tits () (12 August 1930 – 5 December 2021) was a Belgian-born French mathematician who worked on group theory and incidence geometry. He introduced Tits buildings, the Tits alternative, the Tits group, and the Tits metric.
Life and career
Tits was born in Uccle to Léon Tits, a professor, and Lousia André. Jacques attended the Athénée of Uccle and the Free University of Brussels. His thesis advisor was Paul Libois, and Tits graduated with his doctorate in 1950 with the dissertation Généralisation des groupes projectifs basés sur la notion de transitivité. His academic career includes professorships at the Free University of Brussels (now split into the Université Libre de Bruxelles and the Vrije Universiteit Brussel) (1962–1964), the University of Bonn (1964–1974) and the Collège de France in Paris, until becoming emeritus in 2000. He changed his citizenship to French in 1974 in order to teach at the Collège de France, which at that point required French citizenship. Because Belgian nationality law did not allow dual nationality at the time, he renounced his Belgian citizenship. He has been a member of the French Academy of Sciences since 1979.
Tits was an "honorary" member of the Nicolas Bourbaki group; as such, he helped popularize H.S.M. Coxeter's work, introducing terms such as Coxeter number, Coxeter group, and Coxeter graph.
Honors
Tits received the Wolf Prize in Mathematics in 1993, the Cantor Medal from the Deutsche Mathematiker-Vereinigung (German Mathematical Society) in 1996, and the German distinction "Pour le Mérite". In 2008 he was awarded the Abel Prize, along with John Griggs Thompson, "for their profound achievements in algebra and in particular for shaping modern group theory". He was a member of several Academies of Sciences.
He was a member of the Norwegian Academy of Science and Letters. He became a foreign member of the Royal Netherlands Academy of Arts and Sciences in 1988.
Death
Tits died on 5 December 2021, at the age of 91 in the 13th arrondissement, Paris.
Contributions
He introduced the theory of buildings (sometimes known as Tits buildings), which are combinatorial structures on which groups act, particularly in algebraic group theory (including finite groups, and groups defined over the p-adic numbers). The related theory of (B, N) pairs is a basic tool in the theory of groups of Lie type. Of particular importance is his classification of all irreducible buildings of spherical type and rank at least three, which involved classifying all polar spaces of rank at least three. The existence of these buildings initially depended on the existence of a group of Lie type in each case, but in joint work with Mark Ronan he constructed those of rank at least four independently, yielding the groups directly. In the rank-2 case spherical building are generalized n-gons, and in joint work with Richard Weiss he classified these when they admit a suitable group of symmetries (the so-called Moufang polygons). In col
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https://en.wikipedia.org/wiki/Mathematical%20theory%20%28disambiguation%29
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The term mathematical theory may refer to:
Theory (mathematical logic), a collection of sentences in a formal language.
Mathematical theory, a branch of mathematics
See also
Theory
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https://en.wikipedia.org/wiki/Levinson%20recursion
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Levinson recursion or Levinson–Durbin recursion is a procedure in linear algebra to recursively calculate the solution to an equation involving a Toeplitz matrix. The algorithm runs in time, which is a strong improvement over Gauss–Jordan elimination, which runs in Θ(n3).
The Levinson–Durbin algorithm was proposed first by Norman Levinson in 1947, improved by James Durbin in 1960, and subsequently improved to and then multiplications by W. F. Trench and S. Zohar, respectively.
Other methods to process data include Schur decomposition and Cholesky decomposition. In comparison to these, Levinson recursion (particularly split Levinson recursion) tends to be faster computationally, but more sensitive to computational inaccuracies like round-off errors.
The Bareiss algorithm for Toeplitz matrices (not to be confused with the general Bareiss algorithm) runs about as fast as Levinson recursion, but it uses space, whereas Levinson recursion uses only O(n) space. The Bareiss algorithm, though, is numerically stable, whereas Levinson recursion is at best only weakly stable (i.e. it exhibits numerical stability for well-conditioned linear systems).
Newer algorithms, called asymptotically fast or sometimes superfast Toeplitz algorithms, can solve in for various p (e.g. p = 2, p = 3 ). Levinson recursion remains popular for several reasons; for one, it is relatively easy to understand in comparison; for another, it can be faster than a superfast algorithm for small n (usually n < 256).
Derivation
Background
Matrix equations follow the form
The Levinson–Durbin algorithm may be used for any such equation, as long as M is a known Toeplitz matrix with a nonzero main diagonal. Here is a known vector, and is an unknown vector of numbers xi yet to be determined.
For the sake of this article, êi is a vector made up entirely of zeroes, except for its ith place, which holds the value one. Its length will be implicitly determined by the surrounding context. The term N refers to the width of the matrix above – M is an N×N matrix. Finally, in this article, superscripts refer to an inductive index, whereas subscripts denote indices. For example (and definition), in this article, the matrix Tn is an n×n matrix that copies the upper left n×n block from M – that is, Tnij = Mij.
Tn is also a Toeplitz matrix, meaning that it can be written as
Introductory steps
The algorithm proceeds in two steps. In the first step, two sets of vectors, called the forward and backward vectors, are established. The forward vectors are used to help get the set of backward vectors; then they can be immediately discarded. The backwards vectors are necessary for the second step, where they are used to build the solution desired.
Levinson–Durbin recursion defines the nth "forward vector", denoted , as the vector of length n which satisfies:
The nth "backward vector" is defined similarly; it is the vector of length n which satisfies:
An important simplification can occur whe
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https://en.wikipedia.org/wiki/Set-builder%20notation
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In set theory and its applications to logic, mathematics, and computer science, set-builder notation is a mathematical notation for describing a set by enumerating its elements, or stating the properties that its members must satisfy.
Defining sets by properties is also known as set comprehension, set abstraction or as defining a set's intension.
Sets defined by enumeration
A set can be described directly by enumerating all of its elements between curly brackets, as in the following two examples:
is the set containing the four numbers 3, 7, 15, and 31, and nothing else.
is the set containing , , and , and nothing else (there is no order among the elements of a set).
This is sometimes called the "roster method" for specifying a set.
When it is desired to denote a set that contains elements from a regular sequence, an ellipsis notation may be employed, as shown in the next examples:
is the set of integers between 1 and 100 inclusive.
is the set of natural numbers.
is the set of all integers.
There is no order among the elements of a set (this explains and validates the equality of the last example), but with the ellipses notation, we use an ordered sequence before (or after) the ellipsis as a convenient notational vehicle for explaining which elements are in a set. The first few elements of the sequence are shown, then the ellipses indicate that the simplest interpretation should be applied for continuing the sequence. Should no terminating value appear to the right of the ellipses, then the sequence is considered to be unbounded.
In general, denotes the set of all natural numbers such that . Another notation for is the bracket notation . A subtle special case is , in which is equal to the empty set . Similarly, denotes the set of all for .
In each preceding example, each set is described by enumerating its elements. Not all sets can be described in this way, or if they can, their enumeration may be too long or too complicated to be useful. Therefore, many sets are defined by a property that characterizes their elements. This characterization may be done informally using general prose, as in the following example.
addresses on Pine Street is the set of all addresses on Pine Street.
However, the prose approach may lack accuracy or be ambiguous. Thus, set-builder notation is often used with a predicate characterizing the elements of the set being defined, as described in the following section.
Sets defined by a predicate
Set-builder notation can be used to describe a set that is defined by a predicate, that is, a logical formula that evaluates to true for an element of the set, and false otherwise. In this form, set-builder notation has three parts: a variable, a colon or vertical bar separator, and a predicate. Thus there is a variable on the left of the separator, and a rule on the right of it. These three parts are contained in curly brackets:
or
The vertical bar (or colon) is a separator that can be read as "su
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https://en.wikipedia.org/wiki/Mikoyan-Gurevich%20MiG-23
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The Mikoyan-Gurevich MiG-23 (; NATO reporting name: Flogger) is a variable-geometry fighter aircraft, designed by the Mikoyan-Gurevich design bureau in the Soviet Union. It is a third-generation jet fighter, alongside similar Soviet aircraft such as the Su-17 "Fitter". It was the first Soviet fighter to field a look-down/shoot-down radar, the RP-23 Sapfir, and one of the first to be armed with beyond-visual-range missiles. Production started in 1969 and reached large numbers with over 5,000 aircraft built, making it the most produced variable-sweep wing aircraft in history. The MiG-23 remains in limited service with some export customers.
The basic design was also used as the basis for the Mikoyan MiG-27, a dedicated ground-attack variant. Among many minor changes, the MiG-27 replaced the MiG-23's nose-mounted radar system with an optical panel holding a laser designator and a TV camera.
Development
The MiG-23's predecessor, the MiG-21, was fast and agile, but limited in its operational capabilities by its primitive radar, short range, and limited weapons load (restricted in some aircraft to a pair of short-range R-3/K-13 (AA-2 "Atoll") air-to-air missiles). Work began on a replacement for the MiG-21 in the early 1960s. The new aircraft was required to have better performance and range than the MiG-21, while carrying more capable avionics and weapons including beyond-visual-range (BVR) missiles. A major design consideration was take-off and landing performance. The Soviet Air Force (VVS) demanded the new aircraft have a much shorter take-off run. Low-level speed and handling was also to be improved over the MiG-21. Manoeuvrability was not an urgent requirement. This led Mikoyan to consider two options: lift jets, to provide an additional lift component; and variable-geometry wings, which had been developed by TsAGI for both "clean-sheet" aircraft designs and adaptations of existing designs.
The first option, for an aircraft fitted with lift jets, resulted in the "23-01", also known as the MiG-23PD ( – lift jet), was a tailed delta of similar layout to the smaller MiG-21 but with two lift jets in the fuselage. This first flew on 3 April 1967, but it soon became apparent that this configuration was unsatisfactory, as the lift jets became useless dead weight once airborne. Work on the second strand of development was carried out in parallel by a team led by A.A Andreyev, with MiG directed to build a variable-geometry prototype, the "23-11" in 1965.
The 23-11 featured variable-geometry wings which could be set to angles of 16, 45 and 72 degrees, and it was clearly more promising. The maiden flight of 23–11 took place on 10 June 1967, flown by the famous MiG test pilot Aleksandr Vasilyevich Fedotov (who set the absolute altitude record in 1977 in a Mikoyan-Gurevich MiG-25). Six more flight prototypes and two static-test prototypes were prepared for further flight and system testing. All featured the Tumansky R-27-300 turbojet engine with
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https://en.wikipedia.org/wiki/Geometrization%20conjecture
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In mathematics, Thurston's geometrization conjecture states that each of certain three-dimensional topological spaces has a unique geometric structure that can be associated with it. It is an analogue of the uniformization theorem for two-dimensional surfaces, which states that every simply connected Riemann surface can be given one of three geometries (Euclidean, spherical, or hyperbolic).
In three dimensions, it is not always possible to assign a single geometry to a whole topological space. Instead, the geometrization conjecture states that every closed 3-manifold can be decomposed in a canonical way into pieces that each have one of eight types of geometric structure. The conjecture was proposed by , and implies several other conjectures, such as the Poincaré conjecture and Thurston's elliptization conjecture.
Thurston's hyperbolization theorem implies that Haken manifolds satisfy the geometrization conjecture. Thurston announced a proof in the 1980s and since then several complete proofs have appeared in print.
Grigori Perelman announced a proof of the full geometrization conjecture in 2003 using Ricci flow with surgery in two papers posted at the arxiv.org preprint server. Perelman's papers were studied by several independent groups that produced books and online manuscripts filling in the complete details of his arguments. Verification was essentially complete in time for Perelman to be awarded the 2006 Fields Medal for his work, and in 2010 the Clay Mathematics Institute awarded him its 1 million USD prize for solving the Poincare conjecture, though Perelman declined to accept either award.
The Poincaré conjecture and the spherical space form conjecture are corollaries of the geometrization conjecture, although there are shorter proofs of the former that do not lead to the geometrization conjecture.
The conjecture
A 3-manifold is called closed if it is compact and has no boundary.
Every closed 3-manifold has a prime decomposition: this means it is the connected sum of prime 3-manifolds (this decomposition is essentially unique except for a small problem in the case of non-orientable manifolds). This reduces much of the study of 3-manifolds to the case of prime 3-manifolds: those that cannot be written as a non-trivial connected sum.
Here is a statement of Thurston's conjecture:
Every oriented prime closed 3-manifold can be cut along tori, so that the interior of each of the resulting manifolds has a geometric structure with finite volume.
There are 8 possible geometric structures in 3 dimensions, described in the next section. There is a unique minimal way of cutting an irreducible oriented 3-manifold along tori into pieces that are Seifert manifolds or atoroidal called the JSJ decomposition, which is not quite the same as the decomposition in the geometrization conjecture, because some of the pieces in the JSJ decomposition might not have finite volume geometric structures. (For example, the mapping torus of an Anoso
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https://en.wikipedia.org/wiki/Hsiang%E2%80%93Lawson%27s%20conjecture
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In mathematics, Lawson's conjecture states that the Clifford torus is the only minimally embedded torus in the 3-sphere S3. The conjecture was featured by the Australian Mathematical Society Gazette as part of the Millennium Problems series.
In March 2012, Simon Brendle gave a proof of this conjecture, based on maximum principle techniques.
References
Geometric topology
Theorems in differential geometry
Conjectures that have been proved
Theorems in topology
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https://en.wikipedia.org/wiki/Differential%20form
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In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, especially in geometry, topology and physics.
For instance, the expression is an example of a -form, and can be integrated over an interval contained in the domain of :
Similarly, the expression is a -form that can be integrated over a surface :
The symbol denotes the exterior product, sometimes called the wedge product, of two differential forms. Likewise, a -form represents a volume element that can be integrated over a region of space. In general, a -form is an object that may be integrated over a -dimensional manifold, and is homogeneous of degree in the coordinate differentials
On an -dimensional manifold, the top-dimensional form (-form) is called a volume form.
The differential forms form an alternating algebra. This implies that and This alternating property reflects the orientation of the domain of integration.
The exterior derivative is an operation on differential forms that, given a -form , produces a -form This operation extends the differential of a function (a function can be considered as a -form, and its differential is ) This allows expressing the fundamental theorem of calculus, the divergence theorem, Green's theorem, and Stokes' theorem as special cases of a single general result, the generalized Stokes theorem.
Differential -forms are naturally dual to vector fields on a differentiable manifold, and the pairing between vector fields and -forms is extended to arbitrary differential forms by the interior product. The algebra of differential forms along with the exterior derivative defined on it is preserved by the pullback under smooth functions between two manifolds. This feature allows geometrically invariant information to be moved from one space to another via the pullback, provided that the information is expressed in terms of differential forms. As an example, the change of variables formula for integration becomes a simple statement that an integral is preserved under pullback.
History
Differential forms are part of the field of differential geometry, influenced by linear algebra. Although the notion of a differential is quite old, the initial attempt at an algebraic organization of differential forms is usually credited to Élie Cartan with reference to his 1899 paper. Some aspects of the exterior algebra of differential forms appears in Hermann Grassmann's 1844 work, Die Lineale Ausdehnungslehre, ein neuer Zweig der Mathematik (The Theory of Linear Extension, a New Branch of Mathematics).
Concept
Differential forms provide an approach to multivariable calculus that is independent of coordinates.
Integration and orientation
A differential -form can be integrated over an oriented manifold of dimension . A differential -form can be thought of as measuri
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https://en.wikipedia.org/wiki/Multilinear%20map
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In linear algebra, a multilinear map is a function of several variables that is linear separately in each variable. More precisely, a multilinear map is a function
where () and are vector spaces (or modules over a commutative ring), with the following property: for each , if all of the variables but are held constant, then is a linear function of .
A multilinear map of one variable is a linear map, and of two variables is a bilinear map. More generally, for any nonnegative integer , a multilinear map of k variables is called a k-linear map. If the codomain of a multilinear map is the field of scalars, it is called a multilinear form. Multilinear maps and multilinear forms are fundamental objects of study in multilinear algebra.
If all variables belong to the same space, one can consider symmetric, antisymmetric and alternating k-linear maps. The latter coincide if the underlying ring (or field) has a characteristic different from two, else the former two coincide.
Examples
Any bilinear map is a multilinear map. For example, any inner product on a -vector space is a multilinear map, as is the cross product of vectors in .
The determinant of a matrix is an alternating multilinear function of the columns (or rows) of a square matrix.
If is a Ck function, then the th derivative of at each point in its domain can be viewed as a symmetric -linear function .
Coordinate representation
Let
be a multilinear map between finite-dimensional vector spaces, where has dimension , and has dimension . If we choose a basis for each and a basis for (using bold for vectors), then we can define a collection of scalars by
Then the scalars completely determine the multilinear function . In particular, if
for , then
Example
Let's take a trilinear function
where , and .
A basis for each is Let
where . In other words, the constant is a function value at one of the eight possible triples of basis vectors (since there are two choices for each of the three ), namely:
Each vector can be expressed as a linear combination of the basis vectors
The function value at an arbitrary collection of three vectors can be expressed as
or in expanded form as
Relation to tensor products
There is a natural one-to-one correspondence between multilinear maps
and linear maps
where denotes the tensor product of . The relation between the functions and is given by the formula
Multilinear functions on n×n matrices
One can consider multilinear functions, on an matrix over a commutative ring with identity, as a function of the rows (or equivalently the columns) of the matrix. Let be such a matrix and , be the rows of . Then the multilinear function can be written as
satisfying
If we let represent the th row of the identity matrix, we can express each row as the sum
Using the multilinearity of we rewrite as
Continuing this substitution for each we get, for ,
Therefore, is uniquely determined by how operates on .
Example
In the ca
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https://en.wikipedia.org/wiki/De%20Rham%20cohomology
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In mathematics, de Rham cohomology (named after Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted to computation and the concrete representation of cohomology classes. It is a cohomology theory based on the existence of differential forms with prescribed properties.
On any smooth manifold, every exact form is closed, but the converse may fail to hold. Roughly speaking, this failure is related to the possible existence of "holes" in the manifold, and the de Rham cohomology groups comprise a set of topological invariants of smooth manifolds that precisely quantify this relationship.
Definition
The de Rham complex is the cochain complex of differential forms on some smooth manifold , with the exterior derivative as the differential:
where is the space of smooth functions on , is the space of -forms, and so forth. Forms that are the image of other forms under the exterior derivative, plus the constant function in , are called exact and forms whose exterior derivative is are called closed (see Closed and exact differential forms); the relationship then says that exact forms are closed.
In contrast, closed forms are not necessarily exact. An illustrative case is a circle as a manifold, and the -form corresponding to the derivative of angle from a reference point at its centre, typically written as (described at Closed and exact differential forms). There is no function defined on the whole circle such that is its derivative; the increase of in going once around the circle in the positive direction implies a multivalued function . Removing one point of the circle obviates this, at the same time changing the topology of the manifold.
One prominent example when all closed forms are exact is when the underlying space is contractible to a point, i.e., it is simply connected (no-holes condition). In this case the exterior derivative restricted to closed forms has a local inverse called a homotopy operator. Since it is also nilpotent, it forms a dual chain complex with the arrows reversed compared to the de Rham complex. This is the situation described in the Poincaré lemma.
The idea behind de Rham cohomology is to define equivalence classes of closed forms on a manifold. One classifies two closed forms as cohomologous if they differ by an exact form, that is, if is exact. This classification induces an equivalence relation on the space of closed forms in . One then defines the -th de Rham cohomology group to be the set of equivalence classes, that is, the set of closed forms in modulo the exact forms.
Note that, for any manifold composed of disconnected components, each of which is connected, we have that
This follows from the fact that any smooth function on with zero derivative everywhere is separately constant on each of the connected components of .
De Rham cohomology computed
On
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https://en.wikipedia.org/wiki/Right-hand%20rule
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In mathematics and physics, the right-hand rule is a convention and a mnemonic for deciding the orientation of axes in three-dimensional space. It is a convenient method for determining the direction of the cross product of two vectors.
There are two ways of applying the right hand rule. The first one is conventionally called the Right hand rule or the Flemming's right hand rule. It involves the index finger, the middle finger and the thumb of the right hand. By arranging them as shown in the diagram, the direction of cross product or vector product can be calculated.
The other way, known as Amperes right hand grip rule, right-hand screw rule, coffee-mug rule or the corkscrew-rule involves pointing all fingers of the right hand along the first vector and curling the fingers along the second vector, the direction which the thumb makes is the direction of vector product.
For example, If the curling motion of the fingers represents a movement from the first (x-axis) to the second (y-axis), then the third (z-axis) can point along either thumb in a right handed coordinate system.
Both these rules can be used interchangeably.
The rule can be used to find the direction of the magnetic field, rotation, spirals, electromagnetic fields, mirror images, and enantiomers in mathematics and chemistry.
The sequence is often: index finger along the first vector, then middle finger along the second, then thumb along the third. Two other sequences also work because they preserve the cyclic nature of the cross product (and the underlying Levi-Civita symbol):
Middle finger, thumb, index finger.
Thumb, index finger, middle finger.
Coordinates
For right-handed coordinates, if the thumb of a person's right hand points along the z-axis in the positive direction (third coordinate vector), then the fingers curl from the positive x-axis (first coordinate vector) toward the positive y-axis (second coordinate vector). When viewed at a position along the positive z-axis, the ¼ turn from the positive x- to the positive y-axis is counter-clockwise.
For left-handed coordinates, the above description of the axes is the same, except using the left hand; and the ¼ turn is clockwise.
Interchanging the labels of any two axes reverses the handedness. Reversing the direction of one axis (or three axes) also reverses the handedness. Reversing two axes amounts to a 180° rotation around the remaining axis, also preserving the handedness. These operations can be composed to give repeated changes of handedness. (If the axes do not have a positive or negative direction, then handedness has no meaning.)
Rotations
A rotating body
In mathematics, a rotating body is commonly represented by a pseudovector along the axis of rotation. The length of the vector gives the speed of rotation and the direction of the axis gives the direction of rotation according to the right-hand rule: right fingers curled in the direction of rotation and the right thumb pointing in the positive dir
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https://en.wikipedia.org/wiki/Exterior%20algebra
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In mathematics, the exterior algebra of a vector space is a graded associative algebra
Elements in ∧nV are called -multivectors, and are given by a sum of -blades ("products" of elements of ); it is an abstraction of oriented lengths, areas, volumes and more generally oriented n-volumes for n ≥ 0.
The algebra product ∧ on ∧V is called the exterior product or wedge product; informally, it acts by taking the product of oriented volumes. Every element in ∧V is a sum of v1∧ ... ∧ vk where , and
Such an element is called a k-blade, which in the above figure corresponds to the parallelotope spanned by them.
This product satisfies the alternating property v∧v = 0 for all v ∈ V. This implies it is antisymmetric: u∧v = - v∧u. and more generally any blade flips sign whenever two of its constituent vectors are exchanged, corresponding to a parallelotope of opposite orientation.
The exterior algebra is also called the Grassmann algebra, after Hermann Grassmann.
Formal definitions
There are several equivalent ways to define the exterior algebra of a vector space .
Definition as a quotient of the tensor algebra
Let V be a vector space over a field.
Recall the tensor algebra TV. As a vector space it is spanned by symbols v1⊗ ... ⊗vk for k ≥ 0 and vi ∈ V, and the only relations are those specifying these objects be linear in each variable vi.
∧V inherits the structure of a graded algebra from TV. An element of ∧V may be written (non-uniquely) as a finite sum
v1∧ ... ∧ vk1+ w1∧ ... ∧ wk2 + ... + z1∧ ... ∧ zkn
where each letter is an element of V, and ki, n ≥ 0. It is called a k-vector if all ki = k, in which case its rank is the minimum n for which it can be written in the above form. Rank one k-vectors
v1∧ ... ∧ vk
are also called k-blades.
Definition in terms of explicit basis
Let e1, ..., en be a basis of V.
Motivating examples
The first two examples assume a metric tensor field and an orientation; the third example does not assume either.
Areas in the plane
The Cartesian plane is a real vector space equipped with a basis consisting of a pair of unit vectors
with the orientation and with the metric
Suppose that
are a pair of given vectors in written in components. There is a unique parallelogram having v and w as two of its sides. The area of this parallelogram is given by the standard determinant formula:
Consider now the exterior product of v and w:
where the first step uses the distributive law for the exterior product, and the last uses the fact that the exterior product is an alternating map, and in particular (The fact that the exterior product is an alternating map also forces ) Note that the coefficient in this last expression is precisely the determinant of the matrix . The fact that this may be positive or negative has the intuitive meaning that v and w may be oriented in a counterclockwise or clockwise sense as the vertices of the parallelogram they define. Such an area is called the signed area of the
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https://en.wikipedia.org/wiki/Cotangent%20bundle
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In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold. It may be described also as the dual bundle to the tangent bundle. This may be generalized to categories with more structure than smooth manifolds, such as complex manifolds, or (in the form of cotangent sheaf) algebraic varieties or schemes. In the smooth case, any Riemannian metric or symplectic form gives an isomorphism between the cotangent bundle and the tangent bundle, but they are not in general isomorphic in other categories.
Formal definition via diagonal morphism
There are several equivalent ways to define the cotangent bundle. One way is through a diagonal mapping Δ and germs.
Let M be a smooth manifold and let M×M be the Cartesian product of M with itself. The diagonal mapping Δ sends a point p in M to the point (p,p) of M×M. The image of Δ is called the diagonal. Let be the sheaf of germs of smooth functions on M×M which vanish on the diagonal. Then the quotient sheaf consists of equivalence classes of functions which vanish on the diagonal modulo higher order terms. The cotangent sheaf is defined as the pullback of this sheaf to M:
By Taylor's theorem, this is a locally free sheaf of modules with respect to the sheaf of germs of smooth functions of M. Thus it defines a vector bundle on M: the cotangent bundle.
Smooth sections of the cotangent bundle are called (differential) one-forms.
Contravariance properties
A smooth morphism of manifolds, induces a pullback sheaf on M. There is an induced map of vector bundles .
Examples
The tangent bundle of the vector space is , and the cotangent bundle is , where denotes the dual space of covectors, linear functions .
Given a smooth manifold embedded as a hypersurface represented by the vanishing locus of a function with the condition that the tangent bundle is
where is the directional derivative . By definition, the cotangent bundle in this case is
where Since every covector corresponds to a unique vector for which for an arbitrary
The cotangent bundle as phase space
Since the cotangent bundle X = T*M is a vector bundle, it can be regarded as a manifold in its own right. Because at each point the tangent directions of M can be paired with their dual covectors in the fiber, X possesses a canonical one-form θ called the tautological one-form, discussed below. The exterior derivative of θ is a symplectic 2-form, out of which a non-degenerate volume form can be built for X. For example, as a result X is always an orientable manifold (the tangent bundle TX is an orientable vector bundle). A special set of coordinates can be defined on the cotangent bundle; these are called the canonical coordinates. Because cotangent bundles can be thought of as symplectic manifolds, any real function on the cotangent bundle can be interpreted to be a Hamiltonian; thus the cotangent bundle can be unders
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https://en.wikipedia.org/wiki/Pearson%20correlation%20coefficient
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In statistics, the Pearson correlation coefficient (PCC) is a correlation coefficient that measures linear correlation between two sets of data. It is the ratio between the covariance of two variables and the product of their standard deviations; thus, it is essentially a normalized measurement of the covariance, such that the result always has a value between −1 and 1. As with covariance itself, the measure can only reflect a linear correlation of variables, and ignores many other types of relationships or correlations. As a simple example, one would expect the age and height of a sample of teenagers from a high school to have a Pearson correlation coefficient significantly greater than 0, but less than 1 (as 1 would represent an unrealistically perfect correlation).
Naming and history
It was developed by Karl Pearson from a related idea introduced by Francis Galton in the 1880s, and for which the mathematical formula was derived and published by Auguste Bravais in 1844. The naming of the coefficient is thus an example of Stigler's Law.
Definition
Pearson's correlation coefficient is the covariance of the two variables divided by the product of their standard deviations. The form of the definition involves a "product moment", that is, the mean (the first moment about the origin) of the product of the mean-adjusted random variables; hence the modifier product-moment in the name.
For a population
Pearson's correlation coefficient, when applied to a population, is commonly represented by the Greek letter ρ (rho) and may be referred to as the population correlation coefficient or the population Pearson correlation coefficient. Given a pair of random variables (for example, Height and Weight), the formula for ρ is
where
is the covariance
is the standard deviation of
is the standard deviation of .
The formula for can be expressed in terms of mean and expectation. Since
the formula for can also be written as
where
and are defined as above
is the mean of
is the mean of
is the expectation.
The formula for can be expressed in terms of uncentered moments. Since
the formula for can also be written as
For a sample
Pearson's correlation coefficient, when applied to a sample, is commonly represented by and may be referred to as the sample correlation coefficient or the sample Pearson correlation coefficient. We can obtain a formula for by substituting estimates of the covariances and variances based on a sample into the formula above. Given paired data consisting of pairs, is defined as
where
is sample size
are the individual sample points indexed with i
(the sample mean); and analogously for .
Rearranging gives us this formula for :
where are defined as above.
This formula suggests a convenient single-pass algorithm for calculating sample correlations, though depending on the numbers involved, it can sometimes be numerically unstable.
Rearranging again gives us this formula for :
where are defined as above.
An equ
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https://en.wikipedia.org/wiki/Standard%20score
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In statistics, the standard score is the number of standard deviations by which the value of a raw score (i.e., an observed value or data point) is above or below the mean value of what is being observed or measured. Raw scores above the mean have positive standard scores, while those below the mean have negative standard scores.
It is calculated by subtracting the population mean from an individual raw score and then dividing the difference by the population standard deviation. This process of converting a raw score into a standard score is called standardizing or normalizing (however, "normalizing" can refer to many types of ratios; see Normalization for more).
Standard scores are most commonly called z-scores; the two terms may be used interchangeably, as they are in this article. Other equivalent terms in use include z-value, z-statistic, normal score, standardized variable and pull in high energy physics.
Computing a z-score requires knowledge of the mean and standard deviation of the complete population to which a data point belongs; if one only has a sample of observations from the population, then the analogous computation using the sample mean and sample standard deviation yields the t-statistic.
Calculation
If the population mean and population standard deviation are known, a raw score
x is converted into a standard score by
where:
μ is the mean of the population,
σ is the standard deviation of the population.
The absolute value of z represents the distance between that raw score x and the population mean in units of the standard deviation. z is negative when the raw score is below the mean, positive when above.
Calculating z using this formula requires use of the population mean and the population standard deviation, not the sample mean or sample deviation. However, knowing the true mean and standard deviation of a population is often an unrealistic expectation, except in cases such as standardized testing, where the entire population is measured.
When the population mean and the population standard deviation are unknown, the standard score may be estimated by using the sample mean and sample standard deviation as estimates of the population values.
In these cases, the z-score is given by
where:
is the mean of the sample,
S is the standard deviation of the sample.
Though it should always be stated, the distinction between use of the population and sample statistics often is not made. In either case, the numerator and denominator of the equations have the same units of measure so that the units cancel out through division and z is left as a dimensionless quantity.
Applications
Z-test
The z-score is often used in the z-test in standardized testing – the analog of the Student's t-test for a population whose parameters are known, rather than estimated. As it is very unusual to know the entire population, the t-test is much more widely used.
Prediction intervals
The standard score can be used in the calculation o
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https://en.wikipedia.org/wiki/Mensuration
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Mensuration may refer to:
Measurement
Theory of measurement
Mensuration (mathematics), a branch of mathematics that deals with measurement of various parameters of geometric figures and many more
Forest mensuration, a branch of forestry that deals with measurements of forest stand
Mensural notation of music
Mensuration canon, a musical composition wherein the main melody is accompanied by one or more imitations of that melody in other voices
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https://en.wikipedia.org/wiki/Euler%27s%20four-square%20identity
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In mathematics, Euler's four-square identity says that the product of two numbers, each of which is a sum of four squares, is itself a sum of four squares.
Algebraic identity
For any pair of quadruples from a commutative ring, the following expressions are equal:
Euler wrote about this identity in a letter dated May 4, 1748 to Goldbach (but he used a different sign convention from the above). It can be verified with elementary algebra.
The identity was used by Lagrange to prove his four square theorem. More specifically, it implies that it is sufficient to prove the theorem for prime numbers, after which the more general theorem follows. The sign convention used above corresponds to the signs obtained by multiplying two quaternions. Other sign conventions can be obtained by changing any to , and/or any to .
If the and are real numbers, the identity expresses the fact that the absolute value of the product of two quaternions is equal to the product of their absolute values, in the same way that the Brahmagupta–Fibonacci two-square identity does for complex numbers. This property is the definitive feature of composition algebras.
Hurwitz's theorem states that an identity of form,
where the are bilinear functions of the and is possible only for n = 1, 2, 4, or 8.
Proof of the identity using quaternions
Comment: The proof of Euler's four-square identity is by simple algebraic evaluation. Quaternions derive from the four-square identity, which can be written as the product of two inner products of 4-dimensional vectors, yielding again an inner product of 4-dimensional vectors: . This defines the quaternion multiplication rule , which simply reflects Euler's identity, and some mathematics of quaternions. Quaternions are, so to say, the "square root" of the four-square identity. But let the proof go on:
Let and be a pair of quaternions. Their quaternion conjugates are and . Then
and
The product of these two is , where is a real number, so it can commute with the quaternion , yielding
No parentheses are necessary above, because quaternions associate. The conjugate of a product is equal to the commuted product of the conjugates of the product's factors, so
where is the Hamilton product of and :
Then
If where is the scalar part and is the vector part, then so
So,
Pfister's identity
Pfister found another square identity for any even power:
If the are just rational functions of one set of variables, so that each has a denominator, then it is possible for all .
Thus, another four-square identity is as follows:
where and are given by
Incidentally, the following identity is also true:
See also
Brahmagupta–Fibonacci identity (sums of two squares)
Degen's eight-square identity
Pfister's sixteen-square identity
Latin square
References
External links
A Collection of Algebraic Identities
Lettre CXV from Euler to Goldbach
Elementary algebra
Elementary number theory
Mathematical identities
Squares in number theory
Leon
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https://en.wikipedia.org/wiki/Edmund%20Landau
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Edmund Georg Hermann Landau (14 February 1877 – 19 February 1938) was a German mathematician who worked in the fields of number theory and complex analysis.
Biography
Edmund Landau was born to a Jewish family in Berlin. His father was Leopold Landau, a gynecologist, and his mother was Johanna Jacoby. Landau studied mathematics at the University of Berlin, receiving his doctorate in 1899 and his habilitation (the post-doctoral qualification required to teach in German universities) in 1901. His doctoral thesis was 14 pages long.
In 1895, his paper on scoring chess tournaments is the earliest use of eigenvector centrality.
Landau taught at the University of Berlin from 1899 to 1909, after which he held a chair at the University of Göttingen. He married Marianne Ehrlich, the daughter of the Nobel Prize-winning biologist Paul Ehrlich, in 1905.
At the 1912 International Congress of Mathematicians Landau listed four problems in number theory about primes that he said were particularly hard using current mathematical methods. They remain unsolved to this day and are now known as Landau's problems.
During the 1920s, Landau was instrumental in establishing the Mathematics Institute at the nascent Hebrew University of Jerusalem. Intent on eventually settling in Jerusalem, he taught himself Hebrew and delivered a lecture entitled Solved and unsolved problems in elementary number theory in Hebrew on 2 April 1925 during the university's groundbreaking ceremonies. He negotiated with the university's president, Judah Magnes, regarding a position at the university and the building that was to house the Mathematics Institute.
Landau and his family emigrated to Mandatory Palestine in 1927 and he began teaching at the Hebrew University. The family had difficulty adjusting to the primitive living standards then available in Jerusalem. In addition, Landau became a pawn in a struggle for control of the university between Magnes and Chaim Weizmann and Albert Einstein. Magnes suggested that Landau be appointed Rector of the university, but Einstein and Weizmann supported Selig Brodetsky. Landau was disgusted by the dispute and decided to return to Göttingen, remaining there until he was forced out by the Nazi regime after the Machtergreifung in 1933, in a boycott organized by Oswald Teichmüller. Thereafter, he lectured only outside Germany. He moved to Berlin in 1934, where he died in early 1938 of natural causes.
In 1903, Landau gave a much simpler proof than was then known of the prime number theorem and later presented the first systematic treatment of analytic number theory in the Handbuch der Lehre von der Verteilung der Primzahlen (the "Handbuch"). He also made important contributions to complex analysis.
G. H. Hardy and Hans Heilbronn wrote that "No one was ever more passionately devoted to mathematics than Landau".
Works
Handbuch der Lehre von der Verteilung der Primzahlen, Taubner, Leipzig, 1909.
Darstellung und Begründung einiger neuerer Ergebnisse
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https://en.wikipedia.org/wiki/Multivalued%20function
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In mathematics, a multivalued function is a set-valued function with additional properties depending on context. The terms multifunction and many-valued function are sometimes also used.
A multivalued function of sets f : X → Y is a subset
Write f(x) for the set of those y ∈ Y with (x,y) ∈ Γf. If f is an ordinary function, it is a multivalued function by taking its graph
They are called single-valued functions to distinguish them.
Motivation
The term multivalued function originated in complex analysis, from analytic continuation. It often occurs that one knows the value of a complex analytic function in some neighbourhood of a point . This is the case for functions defined by the implicit function theorem or by a Taylor series around . In such a situation, one may extend the domain of the single-valued function along curves in the complex plane starting at . In doing so, one finds that the value of the extended function at a point depends on the chosen curve from to ; since none of the new values is more natural than the others, all of them are incorporated into a multivalued function.
For example, let be the usual square root function on positive real numbers. One may extend its domain to a neighbourhood of in the complex plane, and then further along curves starting at , so that the values along a given curve vary continuously from . Extending to negative real numbers, one gets two opposite values for the square root—for example for —depending on whether the domain has been extended through the upper or the lower half of the complex plane. This phenomenon is very frequent, occurring for th roots, logarithms, and inverse trigonometric functions.
To define a single-valued function from a complex multivalued function, one may distinguish one of the multiple values as the principal value, producing a single-valued function on the whole plane which is discontinuous along certain boundary curves. Alternatively, dealing with the multivalued function allows having something that is everywhere continuous, at the cost of possible value changes when one follows a closed path (monodromy). These problems are resolved in the theory of Riemann surfaces: to consider a multivalued function as an ordinary function without discarding any values, one multiplies the domain into a many-layered covering space, a manifold which is the Riemann surface associated to .
Inverses of functions
If f : X → Y is an ordinary function, then its inverse the multivalued function
defined as Γf, viewed as a subset of X × Y. When f is a differentiable function between manifolds, the inverse function theorem gives conditions for this to be single-valued locally in X.
For example, the complex logarithm log(z) is the multivalued inverse of the exponential function ez : C → C×, with graph
It is not single valued, given a single w with w = log(z), we have
Given any holomorphic function on an open subset of the complex plane C, its analytic continuation is always
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https://en.wikipedia.org/wiki/Linnik%27s%20theorem
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Linnik's theorem in analytic number theory answers a natural question after Dirichlet's theorem on arithmetic progressions. It asserts that there exist positive c and L such that, if we denote p(a,d) the least prime in the arithmetic progression
where n runs through the positive integers and a and d are any given positive coprime integers with 1 ≤ a ≤ d − 1, then:
The theorem is named after Yuri Vladimirovich Linnik, who proved it in 1944. Although Linnik's proof showed c and L to be effectively computable, he provided no numerical values for them.
It follows from Zsigmondy's theorem that p(1,d) ≤ 2d − 1, for all d ≥ 3. It is known that p(1,p) ≤ Lp, for all primes p ≥ 5, as Lp is congruent to 1 modulo p for all prime numbers p, where Lp denotes the p-th Lucas number. Just like Mersenne numbers, Lucas numbers with prime indices have divisors of the form 2kp+1.
Properties
It is known that L ≤ 2 for almost all integers d.
On the generalized Riemann hypothesis it can be shown that
where is the totient function,
and the stronger bound
has been also proved.
It is also conjectured that:
Bounds for L
The constant L is called Linnik's constant and the following table shows the progress that has been made on determining its size.
Moreover, in Heath-Brown's result the constant c is effectively computable.
Notes
Theorems in analytic number theory
Theorems about prime numbers
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https://en.wikipedia.org/wiki/Harald%20Cram%C3%A9r
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Harald Cramér (; 25 September 1893 – 5 October 1985) was a Swedish mathematician, actuary, and statistician, specializing in mathematical statistics and probabilistic number theory. John Kingman described him as "one of the giants of statistical theory".
Biography
Early life
Harald Cramér was born in Stockholm, Sweden on 25 September 1893. Cramér remained close to Stockholm for most of his life. He entered the University of Stockholm as an undergraduate in 1912, where he studied mathematics and chemistry. During this period, he was a research assistant under the famous chemist, Hans von Euler-Chelpin, with whom he published his first five articles from 1913 to 1914. Following his lab experience, he began to focus solely on mathematics. He eventually began his work on his doctoral studies in mathematics which were supervised by Marcel Riesz at the University of Stockholm. Also influenced by G. H. Hardy, Cramér's research led to a PhD in 1917 for his thesis "On a class of Dirichlet series".
Academic professional career
Following his PhD, he served as an Assistant Professor of Mathematics at Stockholm University from 1917 to 1929. Early on, Cramér was highly involved in analytic number theory. He also made some important statistical contributions to the distribution of primes and twin primes. His most famous paper on this subject is entitled "On the order of magnitude of the difference between consecutive prime numbers", which provided a rigorous account of the constructive role in which probability applied to number theory and included an estimate for prime gaps that became known as Cramér's conjecture.
In the late 1920s, Cramér became interested in the field of probability, which at the time was not an accepted branch of mathematics. Cramér knew that a radical change was needed in this field, and in a paper in 1926 said, "The probability concept should be introduced by a purely mathematical definition, from which its fundamental properties and the classical theorems are deduced by purely mathematical operations." Cramér took an interest in the rigorous mathematical formulation of probability in the work of French and Russian mathematicians such as Kolmogorov, Lévy, Bernstein, and Khinchin in the early 1930s. Cramér also made significant development to the revolution in probability theory.
Cramér later wrote his careful study of the field in his Cambridge publication Random variables and probability distributions which appeared in 1937 (with a 2nd edition in 1962 and a 3rd edition in 1970). Shortly after World War II, Cramér went on to publish the influential Mathematical Methods of Statistics in 1946. This text was one that "showed the way in which statistical practice depended on a body of rigorous mathematical analysis as well as Fisherian intuition." His 1955 book Elements of Probability Theory and Some of its Applications introduces probability theory at a more elementary level than Mathematical Methods of Statistics.
In 1929, Cramér was
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https://en.wikipedia.org/wiki/Parametric%20statistics
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Parametric statistics is a branch of statistics which assumes that sample data comes from a population that can be adequately modeled by a probability distribution that has a fixed set of parameters. Conversely a non-parametric model does not assume an explicit (finite-parametric) mathematical form for the distribution when modeling the data. However, it may make some assumptions about that distribution, such as continuity or symmetry.
Most well-known statistical methods are parametric. Regarding nonparametric (and semiparametric) models, Sir David Cox has said, "These typically involve fewer assumptions of structure and distributional form but usually contain strong assumptions about independencies".
Example
The normal family of distributions all have the same general shape and are parameterized by mean and standard deviation. That means that if the mean and standard deviation are known and if the distribution is normal, the probability of any future observation lying in a given range is known.
Suppose that we have a sample of 99 test scores with a mean of 100 and a standard deviation of 1. If we assume all 99 test scores are random observations from a normal distribution, then we predict there is a 1% chance that the 100th test score will be higher than 102.33 (that is, the mean plus 2.33 standard deviations), assuming that the 100th test score comes from the same distribution as the others. Parametric statistical methods are used to compute the 2.33 value above, given 99 independent observations from the same normal distribution.
A non-parametric estimate of the same thing is the maximum of the first 99 scores. We don't need to assume anything about the distribution of test scores to reason that before we gave the test it was equally likely that the highest score would be any of the first 100. Thus there is a 1% chance that the 100th score is higher than any of the 99 that preceded it.
History
Parametric statistics was mentioned by R. A. Fisher in his work Statistical Methods for Research Workers in 1925, which created the foundation for modern statistics.
See also
Aggregated distribution
All models are wrong
Inverse problem
Parametric model
Sufficient statistic
References
Statistical inference
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https://en.wikipedia.org/wiki/Nonparametric%20statistics
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Nonparametric statistics is a type of statistical analysis that does not rely on the assumption of a specific underlying distribution (such as the normal distribution), or any other specific assumptions about the population parameters (such as mean and variance). This is in contrast to parametric statistics, which make such assumptions about the population. In nonparametric statistics, a distribution may not be specified at all, or a distribution may be specified but its parameters, such as the mean and variance, are not assumed to have a known value or distribution in advance. In some cases, parameters may be generated from the data, such as the median. Nonparametric statistics can be used for descriptive statistics or statistical inference. Nonparametric tests are often used when the assumptions of parametric tests are evidently violated.
Definitions
The term "nonparametric statistics" has been defined imprecisely in the following two ways, among others:
Applications and purpose
Non-parametric methods are widely used for studying populations that have a ranked order (such as movie reviews receiving one to four "stars"). The use of non-parametric methods may be necessary when data have a ranking but no clear numerical interpretation, such as when assessing preferences. In terms of levels of measurement, non-parametric methods result in ordinal data.
As non-parametric methods make fewer assumptions, their applicability is much more general than the corresponding parametric methods. In particular, they may be applied in situations where less is known about the application in question. Also, due to the reliance on fewer assumptions, non-parametric methods are more robust.
Non-parametric methods are sometimes considered simpler to use and more robust than parametric methods, even when the assumptions of parametric methods are justified. This is due to their more general nature, which may make them less susceptible to misuse and misunderstanding. Non-parametric methods can be considered a conservative choice, as they will work even when their assumptions are not met, whereas parametric methods can produce misleading results when their assumptions are violated.
The wider applicability and increased robustness of non-parametric tests comes at a cost: in cases where a parametric test would be appropriate, non-parametric tests have less statistical power. In other words, a larger sample size can be required to draw conclusions with the same degree of confidence.
Non-parametric models
Non-parametric models differ from parametric models in that the model structure is not specified a priori but is instead determined from data. The term non-parametric is not meant to imply that such models completely lack parameters but that the number and nature of the parameters are flexible and not fixed in advance.
A histogram is a simple nonparametric estimate of a probability distribution.
Kernel density estimation is another method to estimate a probabilit
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https://en.wikipedia.org/wiki/Hypatia%20%28disambiguation%29
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Hypatia (c. 370–415), was a Greek scholar and philosopher who was considered the first notable woman in mathematics.
Hypatia may also refer to:
Fiction
Hypatia (novel) by Charles Kingsley
Hypatia, a character based on Hypatia of Alexandria in the series The Heirs of Alexandria by Mercedes Lackey, Eric Flint and Dave Freer
Hypatia, a character based on Hypatia of Alexandria in the novel Baudolino by Umberto Eco
Hypatia, the main character in the 2009 film Agora, played by Rachel Weisz
Dr. Alexandria Hypatia, a character named for Hypatia of Alexandria in the video game Dishonored 2
The scientific research ship Hypatia in the novel Illuminae by Amie Kaufman and Jay Kristoff
Philosophy
Hypatia, a 1720 work by John Toland
Hypatia: A Journal of Feminist Philosophy, a scholarly publication for research in feminism and philosophy
Hypatia transracialism controversy, a dispute that began in April 2017 about an article in Hypatia
Astronomy
Hypatia (planet) or Iota Draconis b, an exoplanet
Hypatia (crater), a feature of the Moon
238 Hypatia, a C-type main belt asteroid
Hypatia (stone), a series of small stones, claimed by some to derive from the core of the hypothetical LDG comet
Other uses
Hypatia (moth), a genus of moths in the subfamily Arctiinae
Hypatia Sans an Adobe font created in 2002
Hyapatia Lee, actress
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https://en.wikipedia.org/wiki/Wallace%E2%80%93Bolyai%E2%80%93Gerwien%20theorem
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In geometry, the Wallace–Bolyai–Gerwien theorem, named after William Wallace, Farkas Bolyai and P. Gerwien, is a theorem related to dissections of polygons. It answers the question when one polygon can be formed from another by cutting it into a finite number of pieces and recomposing these by translations and rotations. The Wallace–Bolyai–Gerwien theorem states that this can be done if and only if two polygons have the same area.
Wallace had proven the same result already in 1807.
According to other sources, Bolyai and Gerwien had independently proved the theorem in 1833 and 1835, respectively.
Formulation
There are several ways in which this theorem may be formulated. The most common version uses the concept of "equidecomposability" of polygons: two polygons are equidecomposable if they can be split into finitely many triangles that only differ by some isometry (in fact only by a combination of a translation and a rotation). In this case the Wallace–Bolyai–Gerwien theorem states that two polygons are equidecomposable if and only if they have the same area.
Another formulation is in terms of scissors congruence: two polygons are scissors-congruent if they can be decomposed into finitely many polygons that are pairwise congruent. Scissors-congruence is an equivalence relation. In this case the Wallace–Bolyai–Gerwien theorem states that the equivalence classes of this relation contain precisely those polygons that have the same area.
Proof sketch
The theorem can be understood in a few steps. Firstly, every polygon can be cut into triangles. There are a few methods for this. For convex polygons one can cut off each vertex in turn, while for concave polygons this requires more care. A general approach that works for non-simple polygons as well would be to choose a line not parallel to any of the sides of the polygon and draw a line parallel to this one through each of the vertices of the polygon. This will divide the polygon into triangles and trapezoids, which in turn can be converted into triangles.
Secondly, each of these triangles can be transformed into a right triangle and subsequently into a rectangle with one side of length 1. Alternatively, a triangle can be transformed into one such rectangle by first turning it into a parallelogram and then turning this into such a rectangle. By doing this for each triangle, the polygon can be decomposed into a rectangle with unit width and height equal to its area.
Since this can be done for any two polygons, a "common subdivision" of the rectangle in between proves the theorem. That is, cutting the common rectangle (of size 1 by its area) according to both polygons will be an intermediate between both polygons.
Notes about the proof
First of all, this proof requires an intermediate polygon. In the formulation of the theorem using scissors-congruence, the use of this intermediate can be reformulated by using the fact that scissor-congruences are transitive. Since both the first polygon and the
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https://en.wikipedia.org/wiki/Chain%20%28algebraic%20topology%29
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In algebraic topology, a -chain
is a formal linear combination of the -cells in a cell complex. In simplicial complexes (respectively, cubical complexes), -chains are combinations of -simplices (respectively, -cubes), but not necessarily connected. Chains are used in homology; the elements of a homology group are equivalence classes of chains.
Definition
For a simplicial complex , the group of -chains of is given by:
where are singular -simplices of . that any element in not necessary to be a connected simplicial complex.
Integration on chains
Integration is defined on chains by taking the linear combination of integrals over the simplices in the chain with coefficients (which are typically integers).
The set of all k-chains forms a group and the sequence of these groups is called a chain complex.
Boundary operator on chains
The boundary of a chain is the linear combination of boundaries of the simplices in the chain. The boundary of a k-chain is a (k−1)-chain. Note that the boundary of a simplex is not a simplex, but a chain with coefficients 1 or −1 – thus chains are the closure of simplices under the boundary operator.
Example 1: The boundary of a path is the formal difference of its endpoints: it is a telescoping sum. To illustrate, if the 1-chain is a path from point to point , where
,
and
are its constituent 1-simplices, then
Example 2: The boundary of the triangle is a formal sum of its edges with signs arranged to make the traversal of the boundary counterclockwise.
A chain is called a cycle when its boundary is zero. A chain that is the boundary of another chain is called a boundary. Boundaries are cycles,
so chains form a chain complex, whose homology groups (cycles modulo boundaries) are called simplicial homology groups.
Example 3: The plane punctured at the origin has nontrivial 1-homology group since the unit circle is a cycle, but not a boundary.
In differential geometry, the duality between the boundary operator on chains and the exterior derivative is expressed by the general Stokes' theorem.
References
Algebraic topology
Integration on manifolds
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https://en.wikipedia.org/wiki/Peter%20Armitage%20%28statistician%29
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Peter Armitage CBE (born 15 June 1924) is a statistician specialising in medical statistics.
Peter Armitage attended Huddersfield College and went on to read mathematics at Trinity College, Cambridge. Armitage belonged to the generation of mathematicians who came to maturity in the Second World War. He joined the weapons procurement agency, the Ministry of Supply where he worked on statistical problems with George Barnard.
After the war he resumed his studies and then worked as a statistician for the Medical Research Council from 1947 to 1961. From 1961 to 1976, he was Professor of Medical Statistics at the London School of Hygiene and Tropical Medicine where he succeeded Austin Bradford Hill. His main work there was on sequential analysis. He moved to Oxford as Professor of Biomathematics and became Professor of Applied Statistics and head of the new Department of Statistics, retiring in 1990. He was president of the Royal Statistical Society in 1982–4. He was president of the International Society for Clinical Biostatistics in 1990–1991. He is editor-in-chief of the Encyclopedia of Biostatistics. He lives in Wallingford, Oxfordshire.
References
Basic career information is in the entry in
Who's Who 2005
There are recollections in
Peter Armitage "Purposes, methods, philosophies", Significance Volume 1 Issue 4 Page 170 - December 2004
External links
A brief biography at wiley.co.uk (publisher of the Encyclopedia of Biostatics)
There is a photograph at the 'Peter Armitage on the Portraits of Statisticians' page
Academics of the London School of Hygiene and Tropical Medicine
Alumni of Trinity College, Cambridge
Commanders of the Order of the British Empire
English statisticians
Fellows of St Peter's College, Oxford
Living people
People educated at Huddersfield New College
Presidents of the Royal Statistical Society
1924 births
Biostatisticians
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https://en.wikipedia.org/wiki/Grigori%20Perelman
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Grigori Yakovlevich Perelman (; born 13 June 1966) is a Russian mathematician who is known for his contributions to the fields of geometric analysis, Riemannian geometry, and geometric topology. In 2005, Perelman abruptly quit his research job at the Steklov Institute of Mathematics, and in 2006 stated that he had quit professional mathematics, due to feeling disappointed over the ethical standards in the field. He lives in seclusion in Saint Petersburg, and has not accepted offers for interviews since 2006.
In the 1990s, partly in collaboration with Yuri Burago, Mikhael Gromov, and Anton Petrunin, he made contributions to the study of Alexandrov spaces. In 1994, he proved the soul conjecture in Riemannian geometry, which had been an open problem for the previous 20 years. In 2002 and 2003, he developed new techniques in the analysis of Ricci flow, and proved the Poincaré conjecture and Thurston's geometrization conjecture, the former of which had been a famous open problem in mathematics for the past century. The full details of Perelman's work were filled in and explained by various authors over the following several years.
In August 2006, Perelman was offered the Fields Medal for "his contributions to geometry and his revolutionary insights into the analytical and geometric structure of the Ricci flow", but he declined the award, stating: "I'm not interested in money or fame; I don't want to be on display like an animal in a zoo." On 22 December 2006, the scientific journal Science recognized Perelman's proof of the Poincaré conjecture as the scientific "Breakthrough of the Year", the first such recognition in the area of mathematics.
On 18 March 2010, it was announced that he had met the criteria to receive the first Clay Millennium Prize for resolution of the Poincaré conjecture. On 1 July 2010, he rejected the prize of one million dollars, saying that he considered the decision of the board of the Clay Institute to be unfair, in that his contribution to solving the Poincaré conjecture was no greater than that of Richard S. Hamilton, the mathematician who pioneered the Ricci flow partly with the aim of attacking the conjecture. He had previously rejected the prestigious prize of the European Mathematical Society in 1996.
Early life and education
Grigori Yakovlevich Perelman was born in Leningrad, Soviet Union (now Saint Petersburg, Russia) on 13 June 1966, to Jewish parents, Yakov (who now lives in Israel) and Lyubov (who still lives in Saint Petersburg with Grigori). Grigori's mother Lyubov gave up graduate work in mathematics to raise him. Grigori's mathematical talent became apparent at the age of ten, and his mother enrolled him in Sergei Rukshin's after-school mathematics training program.
His mathematical education continued at the Leningrad Secondary School 239, a specialized school with advanced mathematics and physics programs. Grigori excelled in all subjects except physical education. In 1982, as a member of the Soviet Union
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https://en.wikipedia.org/wiki/Closeness
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Closeness may refer to:
closeness (mathematics)
closeness (graph theory), the shortest path between one vertex and another vertex
the personal distance between two people in proxemics
Social connectedness
Closeness (album), a 1976 album by Charlie Haden
Closeness (film), a 2017 Russian film
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https://en.wikipedia.org/wiki/Vienna%20Circle
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The Vienna Circle () of logical empiricism was a group of elite philosophers and scientists drawn from the natural and social sciences, logic and mathematics who met regularly from 1924 to 1936 at the University of Vienna, chaired by Moritz Schlick. The Vienna Circle had a profound influence on 20th-century philosophy, especially philosophy of science and analytic philosophy.
The philosophical position of the Vienna Circle was called logical empiricism (German: logischer Empirismus), logical positivism or neopositivism. It was influenced by Ernst Mach, David Hilbert, French conventionalism (Henri Poincaré and Pierre Duhem), Gottlob Frege, Bertrand Russell, Ludwig Wittgenstein and Albert Einstein. The Vienna Circle was pluralistic and committed to the ideals of the Enlightenment. It was unified by the aim of making philosophy scientific with the help of modern logic. Main topics were foundational debates in the natural and social sciences, logic and mathematics; the modernization of empiricism by modern logic; the search for an empiricist criterion of meaning; the critique of metaphysics and the unification of the sciences in the unity of science.
The Vienna Circle appeared in public with the publication of various book series – Schriften zur wissenschaftlichen Weltauffassung (Monographs on the Scientific World-Conception), Einheitswissenschaft (Unified Science) and the journal Erkenntnis – and the organization of international conferences in Prague; Königsberg (today known as Kaliningrad); Paris; Copenhagen; Cambridge, UK, and Cambridge, Massachusetts. Its public profile was provided by the Ernst Mach Society (German: Verein Ernst Mach) through which members of the Vienna Circle sought to popularize their ideas in the context of programmes for popular education in Vienna.
During the era of Austrofascism and after the annexation of Austria by Nazi Germany most members of the Vienna Circle were forced to emigrate. The murder of Schlick in 1936 by former student Johann Nelböck put an end to the Vienna Circle in Austria.
History of the Vienna Circle
The history and development of the Vienna Circle shows various stages:
The "First Vienna Circle" (1907–1912)
The pre-history of the Vienna Circle began with meetings on the philosophy of science and epistemology from 1908 on, promoted by Philipp Frank, Hans Hahn and Otto Neurath.
Hans Hahn, the oldest of the three (1879–1934), was a mathematician. He received his degree in mathematics in 1902. Afterwards he studied under the direction of Ludwig Boltzmann in Vienna and David Hilbert, Felix Klein and Hermann Minkowski in Göttingen. In 1905 he received the Habilitation in mathematics. He taught at Innsbruck (1905–1906) and Vienna (from 1909).
Otto Neurath (1882–1945) studied mathematics, political economy, and history in Vienna and Berlin. From 1907 to 1914 he taught in Vienna at the Neue Wiener Handelsakademie (Viennese Commercial Academy). Neurath married Olga, Hahn's sister, in 1911.
Philipp Fran
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https://en.wikipedia.org/wiki/Percentile%20rank
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In statistics, the percentile rank (PR) of a given score is the percentage of scores in its frequency distribution that are less than that score.
Formulation
Its mathematical formula is
where CF—the cumulative frequency—is the count of all scores less than or equal to the score of interest, F is the frequency for the score of interest, and N is the number of scores in the distribution. Alternatively, if CF is the count of all scores less than the score of interest, then
Example
The figure illustrates the percentile rank computation and shows how the 0.5 × F term in the formula ensures that the percentile rank reflects a percentage of scores less than the specified score. For example, for the 10 scores shown in the figure, 60% of them are below a score of 4 (five less than 4 and half of the two equal to 4) and 95% are below 7 (nine less than 7 and half of the one equal to 7). Occasionally the percentile rank of a score is mistakenly defined as the percentage of scores lower than or equal to it, but that would require a different computation, one with the 0.5 × F term deleted. Typically percentile ranks are only computed for scores in the distribution but, as the figure illustrates, percentile ranks can also be computed for scores whose frequency is zero. For example, 90% of the scores are less than 6 (nine less than 6, none equal 6).
Usage
In educational measurement, a range of percentile ranks, often appearing on a score report, shows the range within which the test taker's "true" percentile rank probably occurs. The "true" value refers to the rank the test taker would obtain if there were no random errors involved in the testing process.
Percentile ranks are commonly used to clarify the interpretation of scores on standardized tests. For the test theory, the percentile rank of a raw score is interpreted as the percentage of examinees in the norm group who scored below the score of interest.
Caveats
Percentile ranks are not on an equal-interval scale; that is, the difference between any two scores is not the same as between any other two scores whose difference in percentile ranks is the same. For example, is not the same distance as because of the bell-curve shape of the distribution. Some percentile ranks are closer to some than others. Percentile rank 30 is closer on the bell curve to 40 than it is to 20. If the distribution is normally distributed, the percentile rank can be inferred from the standard score.
See also
Quantile
Percentile
References
Summary statistics
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https://en.wikipedia.org/wiki/1675%20in%20literature
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This article contains information about the literary events and publications of 1675.
Events
November 11 – Gottfried Leibniz's notebooks record a breakthrough in his work on calculus.
New books
Prose
Joshua Barnes – Gerania; a New Discovery of a Little Sort of People, anciently discoursed of, called Pygmies
John Barret – Fifty Queries Seriously Propounded to those that Question or Deny Infants Right to Baptism
Friderich Martens – Spitzbergische oder Groenlandische Reise-Beschreibung, gethan im Jahre 1671
Edward Phillips – Theatrum poetarum
A Satire Against Separatists, variously attributed to Abraham Cowley or Peter Hausted
Philipp Jakob Spener – Pia Desideria
Marie-Catherine de Villedieu – Les Désordres de l’amour
John Wilkins – Of the Principle and Duties of Natural Religion
Miguel de Molinos
Guía espiritual
Breve tratado de la comunión cotidiana
Denis Vairasse – The History of the Sevarites or Sevarambi
Drama
John Crowne
Calisto, or the Chaste Nymph (masque)
Country Wit
John Dryden – Aureng-zebe
Thomas Duffet – Psyche Debauch'd
Sir Francis Fane – Love in the Dark
Nathaniel Lee –
Nero, Emperor of Rome
Sophonisba
Thomas Otway – Alcibiades
Henry Nevil Payne – The Siege of Constantinople
Thomas Shadwell – The Libertine
William Wycherley – The Country Wife
Poetry
John Wilmot, 2nd Earl of Rochester – A Satire Against Mankind (published 1679)
Births
February 26 (baptized) – Abel Evans, English clergyman, academic and poet (died 1737)
September 2 – William Somervile, English poet (died 1742)
October 11 – Samuel Clarke, English philosopher and cleric (died 1729)
Deaths
April 8 – Veit Erbermann, German theologian and controversialist (born 1597)
September – Heinrich Müller, German devotional writer (born 1631)
September 12 – Girolamo Graziani, Italian poet (born 1604)
September 23 – Valentin Conrart, co-founder of French Academie (born 1603)
November 11 – Thomas Willis, English physician and natural philosopher (born 1621)
December 6 – John Lightfoot, English scholar and cleric (born 1602)
References
Years of the 17th century in literature
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https://en.wikipedia.org/wiki/1647%20in%20literature
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This article contains information about the literary events and publications of 1647.
Events
Summer – Thomas Hobbes gives up his work as mathematics tutor to the future Charles II of England because of a serious illness.
October 6 – London authorities raid the Salisbury Court Theatre, breaking up an illicit performance of Beaumont and Fletcher's A King and No King.
unknown date – Plagiarist Robert Baron publishes his Deorum Dona, a masque, and Gripus and Hegio, a pastoral, which draws heavily on the poems of Edmund Waller and John Webster's The Duchess of Malfi. The masque claims to have been performed before "Flaminius and Clorinda, King and Queen of Cyprus, at their regal palace in Nicosia," a fantasy with no relation to the actual history of Cyprus.
New books
Prose
René Descartes – Les Principes de la philosophie (French version of original Latin)
Antonio Enríquez Gómez – El siglo pitagórico. La vida de don Gregorio Guadaña
Baltasar Gracián – Oráculo manual y arte de prudencia
Johannes Hevelius – Selenographia
John Lilburne – Rash Oaths
John Lilly – Christian Astrology
Thomas May – The History of the Parliament of England
Adam Olearius – Beschreibung der muscowitischen und persischen Reise (Description of [his] Muscovite and Persian Journey)
María de Zayas y Sotomayor – Desengaños amorosos. Parte segunda del sarao y entretenimiento honesto
Drama
"Beaumont and Fletcher" – Comedies and Tragedies... (first folio collection of the plays of John Fletcher and his various collaborators)
Antony Brewer – The Country Girl
Marchamont Nedham (Mercurius Pragmaticus) – The Levellers Levelled, or the Independents' Conspiracy to Root Out Monarchy
Jean Rotrou
Don Bertrand de Cabrère
Venceslas
Samuel Sheppard – The Committee-Man Curried
Johann Rist – Das friedewünschende Teutschland (Peace-loving Germany)
Lope de Vega – Parte XXV de comedias
Poetry
Abraham Cowley – The Mistress
Henry More – Philosophical Poems
Johan van Heemskerk – Batavische Arcadia
Births
April 1 – John Wilmot, 2nd Earl of Rochester, English poet (died 1680)
August 12 – Johann Heinrich Acker, German religious historian writing in Latin (died 1719)
November 18 – Pierre Bayle French encyclopedist (died 1706)
Unknown dates
Henry Aldrich, English theologian and philosopher (died 1710)
Petter Dass, Norwegian poet (died 1707)
Glückel of Hameln, German diarist (died 1727)
Deaths
January 29 – Francis Meres, English miscellanist and cleric (born 1565)
April – Ephraim Pagit, English writer on comparative religion (born c. 1575)
May 21 – Pieter Corneliszoon Hooft, Dutch historian, poet and dramatist (born 1581)
June 12 – Thomas Farnaby, English classicist and cleric (born c. 1575)
References
Years of the 17th century in literature
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https://en.wikipedia.org/wiki/Accumulation%20point
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In mathematics, a limit point, accumulation point, or cluster point of a set in a topological space is a point that can be "approximated" by points of in the sense that every neighbourhood of with respect to the topology on also contains a point of other than itself. A limit point of a set does not itself have to be an element of
There is also a closely related concept for sequences. A cluster point or accumulation point of a sequence in a topological space is a point such that, for every neighbourhood of there are infinitely many natural numbers such that This definition of a cluster or accumulation point of a sequence generalizes to nets and filters.
The similarly named notion of a (respectively, a limit point of a filter, a limit point of a net) by definition refers to a point that the sequence converges to (respectively, the filter converges to, the net converges to). Importantly, although "limit point of a set" is synonymous with "cluster/accumulation point of a set", this is not true for sequences (nor nets or filters). That is, the term "limit point of a sequence" is synonymous with "cluster/accumulation point of a sequence".
The limit points of a set should not be confused with adherent points (also called ) for which every neighbourhood of contains some point of . Unlike for limit points, an adherent point of may have a neighbourhood not containing points other than itself. A limit point can be characterized as an adherent point that is not an isolated point.
Limit points of a set should also not be confused with boundary points. For example, is a boundary point (but not a limit point) of the set in with standard topology. However, is a limit point (though not a boundary point) of interval in with standard topology (for a less trivial example of a limit point, see the first caption).
This concept profitably generalizes the notion of a limit and is the underpinning of concepts such as closed set and topological closure. Indeed, a set is closed if and only if it contains all of its limit points, and the topological closure operation can be thought of as an operation that enriches a set by uniting it with its limit points.
Definition
Accumulation points of a set
Let be a subset of a topological space
A point in is a limit point or cluster point or if every neighbourhood of contains at least one point of different from itself.
It does not make a difference if we restrict the condition to open neighbourhoods only. It is often convenient to use the "open neighbourhood" form of the definition to show that a point is a limit point and to use the "general neighbourhood" form of the definition to derive facts from a known limit point.
If is a space (such as a metric space), then is a limit point of if and only if every neighbourhood of contains infinitely many points of In fact, spaces are characterized by this property.
If is a Fréchet–Urysohn space (which all metric spaces and first-
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https://en.wikipedia.org/wiki/Logistic%20regression
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In statistics, the logistic model (or logit model) is a statistical model that models the probability of an event taking place by having the log-odds for the event be a linear combination of one or more independent variables. In regression analysis, logistic regression (or logit regression) is estimating the parameters of a logistic model (the coefficients in the linear combination). Formally, in binary logistic regression there is a single binary dependent variable, coded by an indicator variable, where the two values are labeled "0" and "1", while the independent variables can each be a binary variable (two classes, coded by an indicator variable) or a continuous variable (any real value). The corresponding probability of the value labeled "1" can vary between 0 (certainly the value "0") and 1 (certainly the value "1"), hence the labeling; the function that converts log-odds to probability is the logistic function, hence the name. The unit of measurement for the log-odds scale is called a logit, from logistic unit, hence the alternative names. See and for formal mathematics, and for a worked example.
Binary variables are widely used in statistics to model the probability of a certain class or event taking place, such as the probability of a team winning, of a patient being healthy, etc. (see ), and the logistic model has been the most commonly used model for binary regression since about 1970. Binary variables can be generalized to categorical variables when there are more than two possible values (e.g. whether an image is of a cat, dog, lion, etc.), and the binary logistic regression generalized to multinomial logistic regression. If the multiple categories are ordered, one can use the ordinal logistic regression (for example the proportional odds ordinal logistic model). See for further extensions. The logistic regression model itself simply models probability of output in terms of input and does not perform statistical classification (it is not a classifier), though it can be used to make a classifier, for instance by choosing a cutoff value and classifying inputs with probability greater than the cutoff as one class, below the cutoff as the other; this is a common way to make a binary classifier.
Analogous linear models for binary variables with a different sigmoid function instead of the logistic function (to convert the linear combination to a probability) can also be used, most notably the probit model; see . The defining characteristic of the logistic model is that increasing one of the independent variables multiplicatively scales the odds of the given outcome at a constant rate, with each independent variable having its own parameter; for a binary dependent variable this generalizes the odds ratio. More abstractly, the logistic function is the natural parameter for the Bernoulli distribution, and in this sense is the "simplest" way to convert a real number to a probability. In particular, it maximizes entropy (minimizes added in
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https://en.wikipedia.org/wiki/Unit%20cell
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In geometry, biology, mineralogy and solid state physics, a unit cell is a repeating unit formed by the vectors spanning the points of a lattice. Despite its suggestive name, the unit cell (unlike a unit vector, for example) does not necessarily have unit size, or even a particular size at all. Rather, the primitive cell is the closest analogy to a unit vector, since it has a determined size for a given lattice and is the basic building block from which larger cells are constructed.
The concept is used particularly in describing crystal structure in two and three dimensions, though it makes sense in all dimensions. A lattice can be characterized by the geometry of its unit cell, which is a section of the tiling (a parallelogram or parallelepiped) that generates the whole tiling using only translations.
There are two special cases of the unit cell: the primitive cell and the conventional cell. The primitive cell is a unit cell corresponding to a single lattice point, it is the smallest possible unit cell. In some cases, the full symmetry of a crystal structure is not obvious from the primitive cell, in which cases a conventional cell may be used. A conventional cell (which may or may not be primitive) is a unit cell with the full symmetry of the lattice and may include more than one lattice point. The conventional unit cells are parallelotopes in n dimensions.
Primitive cell
A primitive cell is a unit cell that contains exactly one lattice point. For unit cells generally, lattice points that are shared by cells are counted as of the lattice points contained in each of those cells; so for example a primitive unit cell in three dimensions which has lattice points only at its eight vertices is considered to contain of each of them. An alternative conceptualization is to consistently pick only one of the lattice points to belong to the given unit cell (so the other lattice points belong to adjacent unit cells).
The primitive translation vectors , , span a lattice cell of smallest volume for a particular three-dimensional lattice, and are used to define a crystal translation vector
where , , are integers, translation by which leaves the lattice invariant. That is, for a point in the lattice , the arrangement of points appears the same from as from .
Since the primitive cell is defined by the primitive axes (vectors) , , , the volume of the primitive cell is given by the parallelepiped from the above axes as
Usually, primitive cells in two and three dimensions are chosen to take the shape parallelograms and parallelepipeds, with an atom at each corner of the cell. This choice of primitive cell is not unique, but volume of primitive cells will always be given by the expression above.
Wigner–Seitz cell
In addition to the parallelepiped primitive cells, for every Bravais lattice there is another kind of primitive cell called the Wigner–Seitz cell. In the Wigner–Seitz cell, the lattice point is at the center of the cell, and
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https://en.wikipedia.org/wiki/Yates%27s%20correction%20for%20continuity
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In statistics, Yates's correction for continuity (or Yates's chi-squared test) is used in certain situations when testing for independence in a contingency table. It aims
at correcting the error introduced by assuming that the discrete probabilities of frequencies in the table can be approximated by a continuous distribution (chi-squared). In some cases, Yates's correction may adjust too far, and so its current use is limited.
Correction for approximation error
Using the chi-squared distribution to interpret Pearson's chi-squared statistic requires one to assume that the discrete probability of observed binomial frequencies in the table can be approximated by the continuous chi-squared distribution. This assumption is not quite correct, and introduces some error.
To reduce the error in approximation, Frank Yates, an English statistician, suggested a correction for continuity that adjusts the formula for Pearson's chi-squared test by subtracting 0.5 from the difference between each observed value and its expected value in a 2 × 2 contingency table. This reduces the chi-squared value obtained and thus increases its p-value.
The effect of Yates's correction is to prevent overestimation of statistical significance for small data. This formula is chiefly used when at least one cell of the table has an expected count smaller than 5. Unfortunately, Yates's correction may tend to overcorrect. This can result in an overly conservative result that fails to reject the null hypothesis when it should (a type II error). So it is suggested that Yates's correction is unnecessary even with quite low sample sizes, such as:
The following is Yates's corrected version of Pearson's chi-squared statistics:
where:
Oi = an observed frequency
Ei = an expected (theoretical) frequency, asserted by the null hypothesis
N = number of distinct events
2 × 2 table
As a short-cut, for a 2 × 2 table with the following entries:
In some cases, this is better.
See also
Continuity correction
Wilson score interval with continuity correction
References
Statistical tests for contingency tables
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https://en.wikipedia.org/wiki/Meagre%20set
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In the mathematical field of general topology, a meagre set (also called a meager set or a set of first category) is a subset of a topological space that is small or negligible in a precise sense detailed below. A set that is not meagre is called nonmeagre, or of the second category. See below for definitions of other related terms.
The meagre subsets of a fixed space form a σ-ideal of subsets; that is, any subset of a meagre set is meagre, and the union of countably many meagre sets is meagre.
Meagre sets play an important role in the formulation of the notion of Baire space and of the Baire category theorem, which is used in the proof of several fundamental results of functional analysis.
Definitions
Throughout, will be a topological space.
The definition of meagre set uses the notion of a nowhere dense subset of that is, a subset of whose closure has empty interior. See the corresponding article for more details.
A subset of is called a of or of the in if it is a countable union of nowhere dense subsets of . Otherwise, the subset is called a of or of the in The qualifier "in " can be omitted if the ambient space is fixed and understood from context.
A topological space is called (respectively, ) if it is a meagre (respectively, nonmeagre) subset of itself.
A subset of is called in or in if its complement is meagre in . (This use of the prefix "co" is consistent with its use in other terms such as "cofinite".)
A subset is comeagre in if and only if it is equal to a countable intersection of sets, each of whose interior is dense in
Remarks on terminology
The notions of nonmeagre and comeagre should not be confused. If the space is meagre, every subset is both meagre and comeagre, and there are no nonmeagre sets. If the space is nonmeager, no set is at the same time meagre and comeager, every comeagre set is nonmeagre, and there can be nonmeagre sets that are not comeagre, that is, with nonmeagre complement. See the Examples section below.
As an additional point of terminology, if a subset of a topological space is given the subspace topology induced from , one can talk about it being a meagre space, namely being a meagre subset of itself (when considered as a topological space in its own right). In this case can also be called a meagre subspace of , meaning a meagre space when given the subspace topology. Importantly, this is not the same as being meagre in the whole space . (See the Properties and Examples sections below for the relationship between the two.) Similarly, a nonmeagre subspace will be a set that is nonmeagre in itself, which is not the same as being nonmeagre in the whole space. Be aware however that in the context of topological vector spaces some authors may use the phrase "meagre/nonmeagre subspace" to mean a vector subspace that is a meagre/nonmeagre set relative to the whole space.
The terms first category and second category were the original ones used by René Baire i
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https://en.wikipedia.org/wiki/Finite%20mathematics
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In mathematics education, Finite Mathematics is a syllabus in college and university mathematics that is independent of calculus. A course in precalculus may be a prerequisite for Finite Mathematics.
Contents of the course include an eclectic selection of topics often applied in social science and business, such as finite probability spaces, matrix multiplication, Markov processes, finite graphs, or mathematical models. These topics were used in Finite Mathematics courses at Dartmouth College as developed by John G. Kemeny, Gerald L. Thompson, and J. Laurie Snell and published by Prentice-Hall. Other publishers followed with their own topics. With the arrival of software to facilitate computations, teaching and usage shifted from a broad-spectrum Finite Mathematics with paper and pen, into development and usage of software.
Textbooks
1957: Kemeny, Thompson, Snell, Introduction to Finite Mathematics, (2nd edition 1966) Prentice-Hall
1959: Hazelton Mirkil & Kemeny, Thompson, Snell, Finite Mathematical Structures, Prentice-Hall
1962: Arthur Schliefer Jr. & Kemeny, Thompson, Snell, Finite Mathematics with Business Applications, Prentice-Hall
1969: Marvin Marcus, A Survey of Finite Mathematics, Houghton-Mifflin
1970: Guillermo Owen, Mathematics for Social and Management Sciences, Finite Mathematics, W. B. Saunders
1970: Irving Allen Dodes, Finite Mathematics: A Liberal Arts Approach, McGraw-Hill
1971: A.W. Goodman & J. S. Ratti, Finite Mathematics with Applications, Macmillan
1971: J. Conrad Crown & Marvin L. Bittinger, Finite Mathematics: a modeling approach, (2nd edition 1981) Addison-Wesley
1977: Robert F. Brown & Brenda W. Brown, Applied Finite Mathematics, Wadsworth Publishing
1980: L.J. Goldstein, David I. Schneider, Martha Siegel, Finite Mathematics and Applications, (7th edition 2001) Prentice-Hall
1981: John J. Costello, Spenser O. Gowdy, Agnes M. Rash, Finite Mathematics with Applications, Harcourt, Brace, Jovanovich
1982: James Radlow, Understanding Finite Mathematics, PWS Publishers
1984: Daniel Gallin, Finite Mathematics, Scott Foresman
1984: Gary G. Gilbert & Donald O. Koehler, Applied Finite Mathematics, McGraw-Hill
1984: Frank S. Budnick, Finite Mathematics with Applications in Management and the Social Sciences, McGraw Hill
2011: Rupinder Sekhon, Applied Finite Mathematics, Open Textbook Library
2015: Chris P. Tsokos & Rebecca D. Wooton, The Joy of Finite Mathematics, Academic Press
See also
Discrete mathematics
Finite geometry
Finite group, Finite ring, Finite field
Finite topological space
References
Mathematics education
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https://en.wikipedia.org/wiki/Duality
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Duality may refer to:
Mathematics
Duality (mathematics), a mathematical concept
Dual (category theory), a formalization of mathematical duality
Duality (optimization)
Duality (order theory), a concept regarding binary relations
Duality (projective geometry), general principle of projective geometry
Duality principle (Boolean algebra), the extension of order-theoretic duality to Boolean algebras
S-duality (homotopy theory)
Philosophy, logic, and psychology
Dualistic cosmology, a twofold division in several spiritual and religious worldviews
Dualism (philosophy of mind), where the body and mind are considered to be irreducibly distinct
De Morgan's laws, specifically the ability to generate the dual of any logical expression
Complementary duality of Carl Jung's functions and types in Socionics
Duality (CoPs), refers to the notion of a duality in a Community of Practice
Science
Electrical and mechanical
Duality (electrical circuits), regarding isomorphism of electrical circuits
Duality (mechanical engineering), regarding isomorphism of some mechanical laws
Physics
AdS/CFT correspondence (anti de Sitter/conformal field theory correspondence), sometimes called the Maldacena duality
Dual resonance model
Duality (electricity and magnetism)
Englert–Greenberger duality relation
Holographic duality
Kramers–Wannier duality
Mirror symmetry (string theory)
Montonen–Olive duality
Mysterious duality
String duality, a class of symmetries
S-duality
T-duality
U-duality
Wave–particle duality
Music
Albums
Duality (Peter Leitch and John Hicks album), 1994
Duality (Lisa Gerrard and Pieter Bourke album), 1998
Duality (Set It Off album), 2014
Duality (Ra album), 2005
Duality (mixtape), 2012
Duality (Luna Li album), 2022
Duality, a 2009 album by Darker Half
Duality, a 2020 album by Duke Dumont
Duality, a 2018 album by Big Scoob
Songs
"Duality", a 2014 single from pop rock/punk rock band Set It Off
"Duality" (song), a 2004 single & Grammy-nominated song by metal band Slipknot
"Duality", a 2007 single from alternative rock/punk band Bayside
Other
Duality (film), a 2001 Star Wars fan film by Dave Macomber and Mark Thomas
Duality, a large format audio mixing console by Solid State Logic
Dual (grammatical number), grammatical number that some languages use in addition to singular and plural
See also
Double (disambiguation)
Dual (disambiguation)
Duality principle (disambiguation)
List of dualities: philosophy, mathematics, physics and engineering
Nondualism (philosophy)
Triality (mathematics)
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https://en.wikipedia.org/wiki/Wilson%27s%20theorem
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In algebra and number theory, Wilson's theorem states that a natural number n > 1 is a prime number if and only if the product of all the positive integers less than n is one less than a multiple of n. That is (using the notations of modular arithmetic), the factorial satisfies
exactly when n is a prime number. In other words, any number n is a prime number if, and only if, (n − 1)! + 1 is divisible by n.
History
This theorem was stated by Ibn al-Haytham (c. 1000 AD), and, in the 18th century, by the English mathematician John Wilson. Edward Waring announced the theorem in 1770, although neither he nor his student Wilson could prove it. Lagrange gave the first proof in 1771. There is evidence that Leibniz was also aware of the result a century earlier, but he never published it.
Example
For each of the values of n from 2 to 30, the following table shows the number (n − 1)! and the remainder when (n − 1)! is divided by n. (In the notation of modular arithmetic, the remainder when m is divided by n is written m mod n.)
The background color is blue for prime values of n, gold for composite values.
Proofs
The proofs (for prime moduli) below use the fact that the residue classes modulo a prime number are a field—see the article prime field for more details. Lagrange's theorem, which states that in any field a polynomial of degree n has at most n roots, is needed for all the proofs.
Composite modulus
If n is composite it is divisible by some prime number q, where . Because divides , let for some integer . Suppose for the sake of contradiction that were congruent to where n is composite. Then (n-1)! would also be congruent to −1 (mod q) as implies that for some integer which shows (n-1)! being congruent to -1 (mod q). But (n − 1)! ≡ 0 (mod q) by the fact that q is a term in (n-1)! making (n-1)! a multiple of q. A contradiction is now reached.
In fact, more is true. With the sole exception of 4, where 3! = 6 ≡ 2 (mod 4), if n is composite then (n − 1)! is congruent to 0 (mod n). The proof is divided into two cases: First, if n can be factored as the product of two unequal numbers, , where 2 ≤ a < b ≤ n − 2, then both a and b will appear in the product and (n − 1)! will be divisible by n. If n has no such factorization, then it must be the square of some prime q, q > 2. But then 2q < q2 = n, both q and 2q will be factors of (n − 1)!, and again n divides (n − 1)!.
Prime modulus
Elementary proof
The result is trivial when , so assume p is an odd prime, . Since the residue classes (mod p) are a field, every non-zero a has a unique multiplicative inverse, a−1. Lagrange's theorem implies that the only values of a for which are (because the congruence can have at most two roots (mod p)). Therefore, with the exception of ±1, the factors of can be arranged in disjoint pairs such that product of each pair is congruent to 1 modulo p. This proves Wilson's theorem.
For example, for , one has
Proof using Fermat's little theorem
Again, t
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https://en.wikipedia.org/wiki/Stochastic%20calculus
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Stochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes. This field was created and started by the Japanese mathematician Kiyosi Itô during World War II.
The best-known stochastic process to which stochastic calculus is applied is the Wiener process (named in honor of Norbert Wiener), which is used for modeling Brownian motion as described by Louis Bachelier in 1900 and by Albert Einstein in 1905 and other physical diffusion processes in space of particles subject to random forces. Since the 1970s, the Wiener process has been widely applied in financial mathematics and economics to model the evolution in time of stock prices and bond interest rates.
The main flavours of stochastic calculus are the Itô calculus and its variational relative the Malliavin calculus. For technical reasons the Itô integral is the most useful for general classes of processes, but the related Stratonovich integral is frequently useful in problem formulation (particularly in engineering disciplines). The Stratonovich integral can readily be expressed in terms of the Itô integral, and vice versa. The main benefit of the Stratonovich integral is that it obeys the usual chain rule and therefore does not require Itô's lemma. This enables problems to be expressed in a coordinate system invariant form, which is invaluable when developing stochastic calculus on manifolds other than Rn.
The dominated convergence theorem does not hold for the Stratonovich integral; consequently it is very difficult to prove results without re-expressing the integrals in Itô form.
Itô integral
The Itô integral is central to the study of stochastic calculus. The integral is defined for a semimartingale X and locally bounded predictable process H.
Stratonovich integral
The Stratonovich integral or Fisk–Stratonovich integral of a semimartingale against another semimartingale Y can be defined in terms of the Itô integral as
where [X, Y]tc denotes the quadratic covariation of the continuous parts of X
and Y. The alternative notation
is also used to denote the Stratonovich integral.
Applications
An important application of stochastic calculus is in mathematical finance, in which asset prices are often assumed to follow stochastic differential equations. For example, the Black–Scholes model prices options as if they follow a geometric Brownian motion, illustrating the opportunities and risks from applying stochastic calculus.
Stochastic integrals
Besides the classical Itô and Fisk–Stratonovich integrals, many different notion of stochastic integrals exist such as the Hitsuda–Skorokhod integral, the Marcus integral, the Ogawa integral and more.
See also
Itô calculus
Itô's lemma
Stratonovich integral
Semimartingale
Wiener process
References
Fima C Klebaner, 2012, Introduction to Stochastic Calculus with Application (3rd E
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https://en.wikipedia.org/wiki/Viterbi%20algorithm
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The Viterbi algorithm is a dynamic programming algorithm for obtaining the maximum a posteriori probability estimate of the most likely sequence of hidden states—called the Viterbi path—that results in a sequence of observed events, especially in the context of Markov information sources and hidden Markov models (HMM).
The algorithm has found universal application in decoding the convolutional codes used in both CDMA and GSM digital cellular, dial-up modems, satellite, deep-space communications, and 802.11 wireless LANs. It is now also commonly used in speech recognition, speech synthesis, diarization, keyword spotting, computational linguistics, and bioinformatics. For example, in speech-to-text (speech recognition), the acoustic signal is treated as the observed sequence of events, and a string of text is considered to be the "hidden cause" of the acoustic signal. The Viterbi algorithm finds the most likely string of text given the acoustic signal.
History
The Viterbi algorithm is named after Andrew Viterbi, who proposed it in 1967 as a decoding algorithm for convolutional codes over noisy digital communication links. It has, however, a history of multiple invention, with at least seven independent discoveries, including those by Viterbi, Needleman and Wunsch, and Wagner and Fischer. It was introduced to Natural Language Processing as a method of part-of-speech tagging as early as 1987.
Viterbi path and Viterbi algorithm have become standard terms for the application of dynamic programming algorithms to maximization problems involving probabilities.
For example, in statistical parsing a dynamic programming algorithm can be used to discover the single most likely context-free derivation (parse) of a string, which is commonly called the "Viterbi parse". Another application is in target tracking, where the track is computed that assigns a maximum likelihood to a sequence of observations.
Extensions
A generalization of the Viterbi algorithm, termed the max-sum algorithm (or max-product algorithm) can be used to find the most likely assignment of all or some subset of latent variables in a large number of graphical models, e.g. Bayesian networks, Markov random fields and conditional random fields. The latent variables need, in general, to be connected in a way somewhat similar to a hidden Markov model (HMM), with a limited number of connections between variables and some type of linear structure among the variables. The general algorithm involves message passing and is substantially similar to the belief propagation algorithm (which is the generalization of the forward-backward algorithm).
With the algorithm called iterative Viterbi decoding one can find the subsequence of an observation that matches best (on average) to a given hidden Markov model. This algorithm is proposed by Qi Wang et al. to deal with turbo code. Iterative Viterbi decoding works by iteratively invoking a modified Viterbi algorithm, reestimating the score for a filler u
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https://en.wikipedia.org/wiki/Bernoulli%20polynomials
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In mathematics, the Bernoulli polynomials, named after Jacob Bernoulli, combine the Bernoulli numbers and binomial coefficients. They are used for series expansion of functions, and with the Euler–MacLaurin formula.
These polynomials occur in the study of many special functions and, in particular, the Riemann zeta function and the Hurwitz zeta function. They are an Appell sequence (i.e. a Sheffer sequence for the ordinary derivative operator). For the Bernoulli polynomials, the number of crossings of the x-axis in the unit interval does not go up with the degree. In the limit of large degree, they approach, when appropriately scaled, the sine and cosine functions.
A similar set of polynomials, based on a generating function, is the family of Euler polynomials.
Representations
The Bernoulli polynomials Bn can be defined by a generating function. They also admit a variety of derived representations.
Generating functions
The generating function for the Bernoulli polynomials is
The generating function for the Euler polynomials is
Explicit formula
for n ≥ 0, where Bk are the Bernoulli numbers, and Ek are the Euler numbers.
Representation by a differential operator
The Bernoulli polynomials are also given by
where D = d/dx is differentiation with respect to x and the fraction is expanded as a formal power series. It follows that
cf. integrals below. By the same token, the Euler polynomials are given by
Representation by an integral operator
The Bernoulli polynomials are also the unique polynomials determined by
The integral transform
on polynomials f, simply amounts to
This can be used to produce the inversion formulae below.
Another explicit formula
An explicit formula for the Bernoulli polynomials is given by
That is similar to the series expression for the Hurwitz zeta function in the complex plane. Indeed, there is the relationship
where ζ(s, q) is the Hurwitz zeta function. The latter generalizes the Bernoulli polynomials, allowing for non-integer values of n.
The inner sum may be understood to be the nth forward difference of xm; that is,
where Δ is the forward difference operator. Thus, one may write
This formula may be derived from an identity appearing above as follows. Since the forward difference operator Δ equals
where D is differentiation with respect to x, we have, from the Mercator series,
As long as this operates on an mth-degree polynomial such as xm, one may let n go from 0 only up to m.
An integral representation for the Bernoulli polynomials is given by the Nörlund–Rice integral, which follows from the expression as a finite difference.
An explicit formula for the Euler polynomials is given by
The above follows analogously, using the fact that
Sums of pth powers
Using either the above integral representation of or the identity , we have
(assuming 00 = 1).
The Bernoulli and Euler numbers
The Bernoulli numbers are given by
This definition gives for .
An alternate convention defines the Bernoull
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https://en.wikipedia.org/wiki/Additive%20inverse
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In mathematics, the additive inverse of a number (sometimes called the opposite of ) is the number that, when added to , yields zero. The operation taking a number to its additive inverse is known as sign change or negation. For a real number, it reverses its sign: the additive inverse (opposite number) of a positive number is negative, and the additive inverse of a negative number is positive. Zero is the additive inverse of itself.
The additive inverse of is denoted by unary minus: (see also below). For example, the additive inverse of 7 is −7, because , and the additive inverse of −0.3 is 0.3, because .
Similarly, the additive inverse of is which can be simplified to . The additive inverse of is , because .
The additive inverse is defined as its inverse element under the binary operation of addition (see also below), which allows a broad generalization to mathematical objects other than numbers. As for any inverse operation, double additive inverse has no net effect: .
Common examples
For a number (and more generally in any ring), the additive inverse can be calculated using multiplication by −1; that is, . Examples of rings of numbers are integers, rational numbers, real numbers, and complex numbers.
Relation to subtraction
Additive inverse is closely related to subtraction, which can be viewed as an addition of the opposite:
.
Conversely, additive inverse can be thought of as subtraction from zero:
.
Hence, unary minus sign notation can be seen as a shorthand for subtraction (with the "0" symbol omitted), although in a correct typography, there should be no space after unary "−".
Other properties
In addition to the identities listed above, negation has the following algebraic properties:
, it is an Involution operation
notably,
Formal definition
The notation + is usually reserved for commutative binary operations (operations where for all , ). If such an operation admits an identity element (such that for all ), then this element is unique (). For a given , if there exists such that , then is called an additive inverse of .
If + is associative, i.e., for all , , , then an additive inverse is unique. To see this, let and each be additive inverses of ; then
.
For example, since addition of real numbers is associative, each real number has a unique additive inverse.
Other examples
All the following examples are in fact abelian groups:
Complex numbers: . On the complex plane, this operation rotates a complex number 180 degrees around the origin (see the image above).
Addition of real- and complex-valued functions: here, the additive inverse of a function is the function defined by , for all , such that , the zero function ( for all ).
More generally, what precedes applies to all functions with values in an abelian group ('zero' meaning then the identity element of this group):
Sequences, matrices and nets are also special kinds of functions.
In a vector space, the additive inverse is often called
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https://en.wikipedia.org/wiki/Ruud%20Janssen
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Ruud Janssen (born July 29, 1959, in Tilburg) is a Dutch Fluxus and mail artist currently living in Breda in the Netherlands.
Life and Work
Janssen studied physics and mathematics before he became active with mail art in 1980, doing several international mail art projects. From 1994 till 2001 he has conducted interviews with Fluxus and mail artists. These interviews have been published in booklets and on the internet since 1996. In later years, Janssen focused on acrylic painting and individual correspondences.
Janssen was selected to publish an essay as one of eleven contemporary "New Fluxus" artists who are seen to 'inhabit the site of Fluxus, developing and interpreting the Fluxus tradition in a new way.' in a special double issue of the journal Visible Language on Fluxus.
In 1994, Janssen began a series of mail-interviews. He interviewed Fluxus and mail art personalities by using all the communication-forms that were available (fax, e-mail, envelope, personal meeting, telephone). The question traveled by one form and the answer could be sent back in another form. This concept resulted in a series of mail-interviews. These interviews were published in booklet-form and the complete texts are also available online. In 2008, a selection of the mail-interviews were published in book form.
In 1997 Janssen had a solo exhibition of his mail art work and interviews in Guy Bleus' E-Mail Art Archives (Provincial Center for Visual Arts, now Z33) in Hasselt, Belgium.
TAM-Publications
Janssen publishes articles, magazines and booklets with his TAM-Publications and participates in international mail art projects, collaborations and exhibitions.
International Union of Mail-Artists
He founded IUOMA (International Union of Mail-Artists) in 1988 and is also the curator of the TAM-Rubberstamp Archive, the result of a Mail Art collection that has been accumulated by him from 1983 till now. The archive contains art prints, original rubber stamps, magazines and literature.
Fluxus Heidelberg Center
In 2003 Janssen and Litsa Spathi founded the Fluxus Heidelberg Center for which they are building up a collection of Fluxus material and where they also publish their own works.
Publications
References
External links
Official site of Ruud Janssen, IUOMA and TAM-Publications
Fluxus Heidelberg Center website
1959 births
Living people
Fluxus
Dutch male bloggers
People from Tilburg
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https://en.wikipedia.org/wiki/JSJ%20decomposition
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In mathematics, the JSJ decomposition, also known as the toral decomposition, is a topological construct given by the following theorem:
Irreducible orientable closed (i.e., compact and without boundary) 3-manifolds have a unique (up to isotopy) minimal collection of disjointly embedded incompressible tori such that each component of the 3-manifold obtained by cutting along the tori is either atoroidal or Seifert-fibered.
The acronym JSJ is for William Jaco, Peter Shalen, and Klaus Johannson. The first two worked together, and the third worked independently.
The characteristic submanifold
An alternative version of the JSJ decomposition states:
A closed irreducible orientable 3-manifold M has a submanifold Σ that is a Seifert manifold (possibly disconnected and with boundary) whose complement is atoroidal (and possibly disconnected).
The submanifold Σ with the smallest number of boundary tori is called the characteristic submanifold of M; it is unique (up to isotopy). Cutting the manifold along the tori bounding the characteristic submanifold is also sometimes called a JSJ decomposition, though it may have more tori than the standard JSJ decomposition.
The boundary of the characteristic submanifold Σ is a union of tori that are almost the same as the tori appearing in the JSJ decomposition. However there is a subtle difference: if one of the tori in the JSJ decomposition is "non-separating", then the boundary of the characteristic submanifold has two parallel copies of it (and the region between them is a Seifert manifold isomorphic to the product of a torus and a unit interval).
The set of tori bounding the characteristic submanifold can be characterised as the unique (up to isotopy) minimal collection of disjointly embedded incompressible tori such that closure of each component of the 3-manifold obtained by cutting along the tori is either atoroidal or Seifert-fibered.
The JSJ decomposition is not quite the same as the decomposition in the geometrization conjecture, because some of the pieces in the JSJ decomposition might not have finite volume geometric structures. For example, the mapping torus of an Anosov map of a torus has a finite volume sol structure, but its JSJ decomposition cuts it open along one torus to produce a product of a torus and a unit interval, and the interior of this has no finite volume geometric structure.
See also
Geometrization conjecture
Manifold decomposition
Satellite knot
References
.
Jaco, William; Shalen, Peter B. Seifert fibered spaces in 3-manifolds. Geometric topology (Proc. Georgia Topology Conf., Athens, Ga., 1977), pp. 91–99, Academic Press, New York-London, 1979.
Jaco, William; Shalen, Peter B. A new decomposition theorem for irreducible sufficiently-large 3-manifolds. Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 2, pp. 71–84, Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978.
Johannson, Klaus, Homotopy equ
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https://en.wikipedia.org/wiki/Eigenfunction
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In mathematics, an eigenfunction of a linear operator D defined on some function space is any non-zero function in that space that, when acted upon by D, is only multiplied by some scaling factor called an eigenvalue. As an equation, this condition can be written as
for some scalar eigenvalue The solutions to this equation may also be subject to boundary conditions that limit the allowable eigenvalues and eigenfunctions.
An eigenfunction is a type of eigenvector.
Eigenfunctions
In general, an eigenvector of a linear operator D defined on some vector space is a nonzero vector in the domain of D that, when D acts upon it, is simply scaled by some scalar value called an eigenvalue. In the special case where D is defined on a function space, the eigenvectors are referred to as eigenfunctions. That is, a function f is an eigenfunction of D if it satisfies the equation
where λ is a scalar. The solutions to Equation may also be subject to boundary conditions. Because of the boundary conditions, the possible values of λ are generally limited, for example to a discrete set λ1, λ2, … or to a continuous set over some range. The set of all possible eigenvalues of D is sometimes called its spectrum, which may be discrete, continuous, or a combination of both.
Each value of λ corresponds to one or more eigenfunctions. If multiple linearly independent eigenfunctions have the same eigenvalue, the eigenvalue is said to be degenerate and the maximum number of linearly independent eigenfunctions associated with the same eigenvalue is the eigenvalue's degree of degeneracy or geometric multiplicity.
Derivative example
A widely used class of linear operators acting on infinite dimensional spaces are differential operators on the space C∞ of infinitely differentiable real or complex functions of a real or complex argument t. For example, consider the derivative operator with eigenvalue equation
This differential equation can be solved by multiplying both sides by and integrating. Its solution, the exponential function
is the eigenfunction of the derivative operator, where f0 is a parameter that depends on the boundary conditions. Note that in this case the eigenfunction is itself a function of its associated eigenvalue λ, which can take any real or complex value. In particular, note that for λ = 0 the eigenfunction f(t) is a constant.
Suppose in the example that f(t) is subject to the boundary conditions f(0) = 1 and . We then find that
where λ = 2 is the only eigenvalue of the differential equation that also satisfies the boundary condition.
Link to eigenvalues and eigenvectors of matrices
Eigenfunctions can be expressed as column vectors and linear operators can be expressed as matrices, although they may have infinite dimensions. As a result, many of the concepts related to eigenvectors of matrices carry over to the study of eigenfunctions.
Define the inner product in the function space on which D is defined as
integrated over some range of interes
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https://en.wikipedia.org/wiki/Suslin%27s%20problem
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In mathematics, Suslin's problem is a question about totally ordered sets posed by and published posthumously.
It has been shown to be independent of the standard axiomatic system of set theory known as ZFC; showed that the statement can neither be proven nor disproven from those axioms, assuming ZF is consistent.
(Suslin is also sometimes written with the French transliteration as , from the Cyrillic .)
Formulation
Suslin's problem asks: Given a non-empty totally ordered set R with the four properties
R does not have a least nor a greatest element;
the order on R is dense (between any two distinct elements there is another);
the order on R is complete, in the sense that every non-empty bounded subset has a supremum and an infimum; and
every collection of mutually disjoint non-empty open intervals in R is countable (this is the countable chain condition for the order topology of R),
is R necessarily order-isomorphic to the real line R?
If the requirement for the countable chain condition is replaced with the requirement that R contains a countable dense subset (i.e., R is a separable space), then the answer is indeed yes: any such set R is necessarily order-isomorphic to R (proved by Cantor).
The condition for a topological space that every collection of non-empty disjoint open sets is at most countable is called the Suslin property.
Implications
Any totally ordered set that is not isomorphic to R but satisfies properties 1–4 is known as a Suslin line. The Suslin hypothesis says that there are no Suslin lines: that every countable-chain-condition dense complete linear order without endpoints is isomorphic to the real line. An equivalent statement is that every tree of height ω1 either has a branch of length ω1 or an antichain of cardinality .
The generalized Suslin hypothesis says that for every infinite regular cardinal κ every tree of height κ either has a branch of length κ or an antichain of cardinality κ. The existence of Suslin lines is equivalent to the existence of Suslin trees and to Suslin algebras.
The Suslin hypothesis is independent of ZFC.
and independently used forcing methods to construct models of ZFC in which Suslin lines exist. Jensen later proved that Suslin lines exist if the diamond principle, a consequence of the axiom of constructibility V = L, is assumed. (Jensen's result was a surprise, as it had previously been conjectured that V = L implies that no Suslin lines exist, on the grounds that V = L implies that there are "few" sets.) On the other hand, used forcing to construct a model of ZFC without Suslin lines; more precisely, they showed that Martin's axiom plus the negation of the continuum hypothesis implies the Suslin hypothesis.
The Suslin hypothesis is also independent of both the generalized continuum hypothesis (proved by Ronald Jensen) and of the negation of the continuum hypothesis. It is not known whether the generalized Suslin hypothesis is consistent with the generalized continuum hypothesis
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https://en.wikipedia.org/wiki/Logical%20biconditional
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In logic and mathematics, the logical biconditional, also known as material biconditional or equivalence or biimplication or bientailment, is the logical connective used to conjoin two statements and to form the statement " if and only if " (often abbreviated as " iff "), where is known as the antecedent, and the consequent.
Nowadays, notations to represent equivalence include .
is logically equivalent to both and , and the XNOR (exclusive nor) boolean operator, which means "both or neither".
Semantically, the only case where a logical biconditional is different from a material conditional is the case where the hypothesis (antecedent) is false but the conclusion (consequent) is true. In this case, the result is true for the conditional, but false for the biconditional.
In the conceptual interpretation, means "All 's are 's and all 's are 's". In other words, the sets and coincide: they are identical. However, this does not mean that and need to have the same meaning (e.g., could be "equiangular trilateral" and could be "equilateral triangle"). When phrased as a sentence, the antecedent is the subject and the consequent is the predicate of a universal affirmative proposition (e.g., in the phrase "all men are mortal", "men" is the subject and "mortal" is the predicate).
In the propositional interpretation, means that implies and implies ; in other words, the propositions are logically equivalent, in the sense that both are either jointly true or jointly false. Again, this does not mean that they need to have the same meaning, as could be "the triangle ABC has two equal sides" and could be "the triangle ABC has two equal angles". In general, the antecedent is the premise, or the cause, and the consequent is the consequence. When an implication is translated by a hypothetical (or conditional) judgment, the antecedent is called the hypothesis (or the condition) and the consequent is called the thesis.
A common way of demonstrating a biconditional of the form is to demonstrate that and separately (due to its equivalence to the conjunction of the two converse conditionals). Yet another way of demonstrating the same biconditional is by demonstrating that and .
When both members of the biconditional are propositions, it can be separated into two conditionals, of which one is called a theorem and the other its reciprocal. Thus whenever a theorem and its reciprocal are true, we have a biconditional. A simple theorem gives rise to an implication, whose antecedent is the hypothesis and whose consequent is the thesis of the theorem.
It is often said that the hypothesis is the sufficient condition of the thesis, and that the thesis is the necessary condition of the hypothesis. That is, it is sufficient that the hypothesis be true for the thesis to be true, while it is necessary that the thesis be true if the hypothesis were true. When a theorem and its reciprocal are true, its hypothesis is said to be the necessary and sufficient
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https://en.wikipedia.org/wiki/Truncation
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In mathematics and computer science, truncation is limiting the number of digits right of the decimal point.
Truncation and floor function
Truncation of positive real numbers can be done using the floor function. Given a number to be truncated and , the number of elements to be kept behind the decimal point, the truncated value of x is
However, for negative numbers truncation does not round in the same direction as the floor function: truncation always rounds toward zero, the floor function rounds towards negative infinity. For a given number , the function ceil is used instead
.
In some cases is written as . See Notation of floor and ceiling functions.
Causes of truncation
With computers, truncation can occur when a decimal number is typecast as an integer; it is truncated to zero decimal digits because integers cannot store non-integer real numbers.
In algebra
An analogue of truncation can be applied to polynomials. In this case, the truncation of a polynomial P to degree n can be defined as the sum of all terms of P of degree n or less. Polynomial truncations arise in the study of Taylor polynomials, for example.
See also
Arithmetic precision
Quantization (signal processing)
Precision (computer science)
Truncation (statistics)
References
External links
Wall paper applet that visualizes errors due to finite precision
Numerical analysis
ja:端数処理
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https://en.wikipedia.org/wiki/Propositional%20function
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In propositional calculus, a propositional function or a predicate is a sentence expressed in a way that would assume the value of true or false, except that within the sentence there is a variable (x) that is not defined or specified (thus being a free variable), which leaves the statement undetermined. The sentence may contain several such variables (e.g. n variables, in which case the function takes n arguments).
Overview
As a mathematical function, A(x) or A(x, x, ..., x), the propositional function is abstracted from predicates or propositional forms. As an example, consider the predicate scheme, "x is hot". The substitution of any entity for x will produce a specific proposition that can be described as either true or false, even though "x is hot" on its own has no value as either a true or false statement. However, when a value is assigned to x , such as lava, the function then has the value true; while one assigns to x a value like ice, the function then has the value false.
Propositional functions are useful in set theory for the formation of sets. For example, in 1903 Bertrand Russell wrote in The Principles of Mathematics (page 106):
"...it has become necessary to take propositional function as a primitive notion.
Later Russell examined the problem of whether propositional functions were predicative or not, and he proposed two theories to try to get at this question: the zig-zag theory and the ramified theory of types.
A Propositional Function, or a predicate, in a variable x is an open formula p(x) involving x that becomes a proposition when one gives x a definite value from the set of values it can take.
According to Clarence Lewis, "A proposition is any expression which is either true or false; a propositional function is an expression, containing one or more variables, which becomes a proposition when each of the variables is replaced by some one of its values from a discourse domain of individuals." Lewis used the notion of propositional functions to introduce relations, for example, a propositional function of n variables is a relation of arity n. The case of n = 2 corresponds to binary relations, of which there are homogeneous relations (both variables from the same set) and heterogeneous relations.
See also
Propositional formula
Boolean-valued function
Formula (logic)
Sentence (logic)
Truth function
Open sentence
References
Functions and mappings
Mathematical relations
Concepts in logic
Predicate logic
Logical expressions
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https://en.wikipedia.org/wiki/Delta%20operator
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In mathematics, a delta operator is a shift-equivariant linear operator on the vector space of polynomials in a variable over a field that reduces degrees by one.
To say that is shift-equivariant means that if , then
In other words, if is a "shift" of , then is also a shift of , and has the same "shifting vector" .
To say that an operator reduces degree by one means that if is a polynomial of degree , then is either a polynomial of degree , or, in case , is 0.
Sometimes a delta operator is defined to be a shift-equivariant linear transformation on polynomials in that maps to a nonzero constant. Seemingly weaker than the definition given above, this latter characterization can be shown to be equivalent to the stated definition when has characteristic zero, since shift-equivariance is a fairly strong condition.
Examples
The forward difference operator
is a delta operator.
Differentiation with respect to x, written as D, is also a delta operator.
Any operator of the form
(where Dn(ƒ) = ƒ(n) is the nth derivative) with is a delta operator. It can be shown that all delta operators can be written in this form. For example, the difference operator given above can be expanded as
The generalized derivative of time scale calculus which unifies the forward difference operator with the derivative of standard calculus is a delta operator.
In computer science and cybernetics, the term "discrete-time delta operator" (δ) is generally taken to mean a difference operator
the Euler approximation of the usual derivative with a discrete sample time . The delta-formulation obtains a significant number of numerical advantages compared to the shift-operator at fast sampling.
Basic polynomials
Every delta operator has a unique sequence of "basic polynomials", a polynomial sequence defined by three conditions:
Such a sequence of basic polynomials is always of binomial type, and it can be shown that no other sequences of binomial type exist. If the first two conditions above are dropped, then the third condition says this polynomial sequence is a Sheffer sequence—a more general concept.
See also
Pincherle derivative
Shift operator
Umbral calculus
References
External links
Linear algebra
Polynomials
Finite differences
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https://en.wikipedia.org/wiki/Monge%20array
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In mathematics applied to computer science, Monge arrays, or Monge matrices, are mathematical objects named for their discoverer, the French mathematician Gaspard Monge.
An m-by-n matrix is said to be a Monge array if, for all such that
one obtains
So for any two rows and two columns of a Monge array (a 2 × 2 sub-matrix) the four elements at the intersection points have the property that the sum of the upper-left and lower right elements (across the main diagonal) is less than or equal to the sum of the lower-left and upper-right elements (across the antidiagonal).
This matrix is a Monge array:
For example, take the intersection of rows 2 and 4 with columns 1 and 5.
The four elements are:
17 + 7 = 24
23 + 11 = 34
The sum of the upper-left and lower right elements is less than or equal to the sum of the lower-left and upper-right elements.
Properties
The above definition is equivalent to the statement
A matrix is a Monge array if and only if for all and .
Any subarray produced by selecting certain rows and columns from an original Monge array will itself be a Monge array.
Any linear combination with non-negative coefficients of Monge arrays is itself a Monge array.
One interesting property of Monge arrays is that if you mark with a circle the leftmost minimum of each row, you will discover that your circles march downward to the right; that is to say, if , then for all . Symmetrically, if you mark the uppermost minimum of each column, your circles will march rightwards and downwards. The row and column maxima march in the opposite direction: upwards to the right and downwards to the left.
The notion of weak Monge arrays has been proposed; a weak Monge array is a square n-by-n matrix which satisfies the Monge property only for all .
Every Monge array is totally monotone, meaning that its row minima occur in a nondecreasing sequence of columns, and that the same property is true for every subarray. This property allows the row minima to be found quickly by using the SMAWK algorithm.
Monge matrix is just another name for submodular function of two discrete variables. Precisely, A is a Monge matrix if and only if A[i,j] is a submodular function of variables i,j.
Applications
A square Monge matrix which is also symmetric about its main diagonal is called a Supnick matrix (after Fred Supnick); this kind of matrix has applications to the traveling salesman problem (namely, that the problem admits of easy solutions when the distance matrix can be written as a Supnick matrix). Any linear combination of Supnick matrices is itself a Supnick matrix.
References
Theoretical computer science
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https://en.wikipedia.org/wiki/Horizontal
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Horizontal may refer to:
Horizontal plane, in astronomy, geography, geometry and other sciences and contexts
Horizontal coordinate system, in astronomy
Horizontalism, in monetary circuit theory
Horizontalism, in sociology
Horizontal market, in microeconomics
Horizontal (album), a 1968 album by the Bee Gees
"Horizontal" (song)" is a 1968 song by the Bee Gees
See also
Horizontal and vertical
Horizontal fissure (disambiguation), anatomical features
Horizontal bar, an apparatus used by male gymnasts in artistic gymnastics
Vertical (disambiguation)
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https://en.wikipedia.org/wiki/Difference
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Difference commonly refers to:
Difference (philosophy), the set of properties by which items are distinguished
Difference (mathematics), the result of a subtraction
Difference, The Difference, Differences or Differently may also refer to:
Music
Difference (album), by Dreamtale, 2005
Differently (album), by Cassie Davis, 2009
"Differently" (song), by Cassie Davis, 2009
The Difference (album), Pendleton, 2008
"The Difference" (The Wallflowers song), 1997
"Difference", a song by Benjamin Clementine from the 2022 album And I Have Been
"The Difference", a song by Westlife from the 2009 album Where We Are
"The Difference", a song by Nick Jonas from the 2016 album Last Year Was Complicated
"The Difference", a song by Meek Mill featuring Quavo, from the 2016 mixtape DC4
"The Difference", a song by Matchbox Twenty from the 2002 album More Than You Think You Are
"The Difference", a 2020 song by Flume featuring Toro y Moi
"The Difference", a 2022 song by Ni/Co which represented Alabama in the American Song Contest
"Differences" (song), by Ginuwine, 2001
Science and mathematics
Difference (mathematics), the result of a subtraction
Difference equation, a type of recurrence relation
Differencing, in statistics, an operation on time-series data
Data differencing, in computer science
Set difference, the result of removing the elements of a set from another set
Other uses
Difference (heraldry), or cadency, a way of distinguishing similar coats of arms
Difference (philosophy), a key concept of philosophy
Differences (journal), a journal of feminist cultural studies
"Differences", a Series D episode of the television series QI (2006)
See also
Different (disambiguation)
Differential (disambiguation)
Distinction (disambiguation)
Deference, submitting to one's superior
Différance, a French term coined by Jacques Derrida
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https://en.wikipedia.org/wiki/Fractional%20calculus
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Fractional calculus is a branch of mathematical analysis that studies the several different possibilities of defining real number powers or complex number powers of the differentiation operator
and of the integration operator
and developing a calculus for such operators generalizing the classical one.
In this context, the term powers refers to iterative application of a linear operator to a function , that is, repeatedly composing with itself, as in
For example, one may ask for a meaningful interpretation of
as an analogue of the functional square root for the differentiation operator, that is, an expression for some linear operator that, when applied twice to any function, will have the same effect as differentiation. More generally, one can look at the question of defining a linear operator
for every real number in such a way that, when takes an integer value , it coincides with the usual -fold differentiation if , and with the -th power of when .
One of the motivations behind the introduction and study of these sorts of extensions of the differentiation operator is that the sets of operator powers defined in this way are continuous semigroups with parameter , of which the original discrete semigroup of for integer is a denumerable subgroup: since continuous semigroups have a well developed mathematical theory, they can be applied to other branches of mathematics.
Fractional differential equations, also known as extraordinary differential equations, are a generalization of differential equations through the application of fractional calculus.
Historical notes
In applied mathematics and mathematical analysis, a fractional derivative is a derivative of any arbitrary order, real or complex. Its first appearance is in a letter written to Guillaume de l'Hôpital by Gottfried Wilhelm Leibniz in 1695. Around the same time, Leibniz wrote to one of the Bernoulli brothers describing the similarity between the binomial theorem and the Leibniz rule for the fractional derivative of a product of two functions. Fractional calculus was introduced in one of Niels Henrik Abel's early papers where all the elements can be found: the idea of fractional-order integration and differentiation, the mutually inverse relationship between them, the understanding that fractional-order differentiation and integration can be considered as the same generalized operation, and even the unified notation for differentiation and integration of arbitrary real order.
Independently, the foundations of the subject were laid by Liouville in a paper from 1832.
The autodidact Oliver Heaviside introduced the practical use of fractional differential operators in electrical transmission line analysis circa 1890. The theory and applications of fractional calculus expanded greatly over the 19th and 20th centuries, and numerous contributors have given different definitions for fractional derivatives and integrals.
Nature of the fractional derivative
The -th derivative
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https://en.wikipedia.org/wiki/Multiplicative%20inverse
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In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x−1, is a number which when multiplied by x yields the multiplicative identity, 1. The multiplicative inverse of a fraction a/b is b/a. For the multiplicative inverse of a real number, divide 1 by the number. For example, the reciprocal of 5 is one fifth (1/5 or 0.2), and the reciprocal of 0.25 is 1 divided by 0.25, or 4. The reciprocal function, the function f(x) that maps x to 1/x, is one of the simplest examples of a function which is its own inverse (an involution).
Multiplying by a number is the same as dividing by its reciprocal and vice versa. For example, multiplication by 4/5 (or 0.8) will give the same result as division by 5/4 (or 1.25). Therefore, multiplication by a number followed by multiplication by its reciprocal yields the original number (since the product of the number and its reciprocal is 1).
The term reciprocal was in common use at least as far back as the third edition of Encyclopædia Britannica (1797) to describe two numbers whose product is 1; geometrical quantities in inverse proportion are described as in a 1570 translation of Euclid's Elements.
In the phrase multiplicative inverse, the qualifier multiplicative is often omitted and then tacitly understood (in contrast to the additive inverse). Multiplicative inverses can be defined over many mathematical domains as well as numbers. In these cases it can happen that ; then "inverse" typically implies that an element is both a left and right inverse.
The notation f −1 is sometimes also used for the inverse function of the function f, which is for most functions not equal to the multiplicative inverse. For example, the multiplicative inverse is the cosecant of x, and not the inverse sine of x denoted by or . The terminology difference reciprocal versus inverse is not sufficient to make this distinction, since many authors prefer the opposite naming convention, probably for historical reasons (for example in French, the inverse function is preferably called the bijection réciproque).
Examples and counterexamples
In the real numbers, zero does not have a reciprocal (division by zero is undefined) because no real number multiplied by 0 produces 1 (the product of any number with zero is zero). With the exception of zero, reciprocals of every real number are real, reciprocals of every rational number are rational, and reciprocals of every complex number are complex. The property that every element other than zero has a multiplicative inverse is part of the definition of a field, of which these are all examples. On the other hand, no integer other than 1 and −1 has an integer reciprocal, and so the integers are not a field.
In modular arithmetic, the modular multiplicative inverse of a is also defined: it is the number x such that . This multiplicative inverse exists if and only if a and n are coprime. For example, the inverse of 3 modulo 11 is 4 because . The extended Euclidean al
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https://en.wikipedia.org/wiki/Falling%20and%20rising%20factorials
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In mathematics, the falling factorial (sometimes called the descending factorial, falling sequential product, or lower factorial) is defined as the polynomial
The rising factorial (sometimes called the Pochhammer function, Pochhammer polynomial, ascending factorial, rising sequential product, or upper factorial) is defined as
The value of each is taken to be 1 (an empty product) when These symbols are collectively called factorial powers.
The Pochhammer symbol, introduced by Leo August Pochhammer, is the notation , where is a non-negative integer. It may represent either the rising or the falling factorial, with different articles and authors using different conventions. Pochhammer himself actually used with yet another meaning, namely to denote the binomial coefficient
In this article, the symbol is used to represent the falling factorial, and the symbol is used for the rising factorial. These conventions are used in combinatorics,
although Knuth's underline and overline notations and are increasingly popular.
In the theory of special functions (in particular the hypergeometric function) and in the standard reference work Abramowitz and Stegun, the Pochhammer symbol is used to represent the rising factorial.
When is a positive integer, gives the number of -permutations (sequences of distinct elements) from an -element set, or equivalently the number of injective functions from a set of size to a set of size . The rising factorial gives the number of partitions of an -element set into ordered sequences (possibly empty).
Examples and combinatorial interpretation
The first few falling factorials are as follows:
The first few rising factorials are as follows:
The coefficients that appear in the expansions are Stirling numbers of the first kind (see below).
When the variable is a positive integer, the number is equal to the number of -permutations from a set of items, that is, the number of ways of choosing an ordered list of length consisting of distinct elements drawn from a collection of size . For example, is the number of different podiums—assignments of gold, silver, and bronze medals—possible in an eight-person race. In this context, other notations like , , , or are also sometimes used. On the other hand, is "the number of ways to arrange flags on flagpoles",
where all flags must be used and each flagpole can have any number of flags. Equivalently, this is the number of ways to partition a set of size (the flags) into distinguishable parts (the poles), with a linear order on the elements assigned to each part (the order of the flags on a given pole).
Properties
The rising and falling factorials are simply related to one another:
Falling and rising factorials of integers are directly related to the ordinary factorial:
Rising factorials of half integers are directly related to the double factorial:
The falling and rising factorials can be used to express a binomial coefficient:
Thus many identities o
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https://en.wikipedia.org/wiki/Lorentz%20group
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In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch physicist Hendrik Lorentz.
For example, the following laws, equations, and theories respect Lorentz symmetry:
The kinematical laws of special relativity
Maxwell's field equations in the theory of electromagnetism
The Dirac equation in the theory of the electron
The Standard Model of particle physics
The Lorentz group expresses the fundamental symmetry of space and time of all known fundamental laws of nature. In small enough regions of spacetime where gravitational variances are negligible, physical laws are Lorentz invariant in the same manner as special relativity.
Basic properties
The Lorentz group is a subgroup of the Poincaré group—the group of all isometries of Minkowski spacetime. Lorentz transformations are, precisely, isometries that leave the origin fixed. Thus, the Lorentz group is the isotropy subgroup with respect to the origin of the isometry group of Minkowski spacetime. For this reason, the Lorentz group is sometimes called the homogeneous Lorentz group while the Poincaré group is sometimes called the inhomogeneous Lorentz group. Lorentz transformations are examples of linear transformations; general isometries of Minkowski spacetime are affine transformations.
Mathematically, the Lorentz group may be described as the indefinite orthogonal group O(1,3), the matrix Lie group that preserves the quadratic form
on (The vector space equipped with this quadratic form is sometimes written ). This quadratic form is, when put on matrix form (see classical orthogonal group), interpreted in physics as the metric tensor of Minkowski spacetime.
The Lorentz group is a six-dimensional noncompact non-abelian real Lie group that is not connected. The four connected components are not simply connected. The identity component (i.e., the component containing the identity element) of the Lorentz group is itself a group, and is often called the restricted Lorentz group, and is denoted . The restricted Lorentz group consists of those Lorentz transformations that preserve both the orientation of space and the direction of time. Its fundamental group has order 2, and its universal cover, the indefinite spin group , is isomorphic to both the special linear group and to the symplectic group . These isomorphisms allow the Lorentz group to act on a large number of mathematical structures important to physics, most notably spinors. Thus, in relativistic quantum mechanics and in quantum field theory, it is very common to call the Lorentz group, with the understanding that is a specific representation (the vector representation) of it.
A recurrent representation of the action of the Lorentz group on Minkowski space uses biquaternions, which form a composition algebra. The isometry property of Lorentz transformati
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https://en.wikipedia.org/wiki/Fock%20space
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The Fock space is an algebraic construction used in quantum mechanics to construct the quantum states space of a variable or unknown number of identical particles from a single particle Hilbert space . It is named after V. A. Fock who first introduced it in his 1932 paper "Konfigurationsraum und zweite Quantelung" ("Configuration space and second quantization").
Informally, a Fock space is the sum of a set of Hilbert spaces representing zero particle states, one particle states, two particle states, and so on. If the identical particles are bosons, the -particle states are vectors in a symmetrized tensor product of single-particle Hilbert spaces . If the identical particles are fermions, the -particle states are vectors in an antisymmetrized tensor product of single-particle Hilbert spaces (see symmetric algebra and exterior algebra respectively). A general state in Fock space is a linear combination of -particle states, one for each .
Technically, the Fock space is (the Hilbert space completion of) the direct sum of the symmetric or antisymmetric tensors in the tensor powers of a single-particle Hilbert space ,
Here is the operator which symmetrizes or antisymmetrizes a tensor, depending on whether the Hilbert space describes particles obeying bosonic or fermionic statistics, and the overline represents the completion of the space. The bosonic (resp. fermionic) Fock space can alternatively be constructed as (the Hilbert space completion of) the symmetric tensors (resp. alternating tensors ). For every basis for there is a natural basis of the Fock space, the Fock states.
Definition
The Fock space is the (Hilbert) direct sum of tensor products of copies of a single-particle Hilbert space
Here , the complex scalars, consists of the states corresponding to no particles, the states of one particle, the states of two identical particles etc.
A general state in is given by
where
is a vector of length 1 called the vacuum state and is a complex coefficient,
is a state in the single particle Hilbert space and is a complex coefficient,
, and is a complex coefficient, etc.
The convergence of this infinite sum is important if is to be a Hilbert space. Technically we require to be the Hilbert space completion of the algebraic direct sum. It consists of all infinite tuples such that the norm, defined by the inner product is finite
where the particle norm is defined by
i.e., the restriction of the norm on the tensor product
For two general states
and
the inner product on is then defined as
where we use the inner products on each of the -particle Hilbert spaces. Note that, in particular the particle subspaces are orthogonal for different .
Product states, indistinguishable particles, and a useful basis for Fock space
A product state of the Fock space is a state of the form
which describes a collection of particles, one of which has quantum state , another and so on up to the th particle, where each is any state from
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https://en.wikipedia.org/wiki/Seismic%20hazard
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A seismic hazard is the probability that an earthquake will occur in a given geographic area, within a given window of time, and with ground motion intensity exceeding a given threshold. With a hazard thus estimated, risk can be assessed and included in such areas as building codes for standard buildings, designing larger buildings and infrastructure projects, land use planning and determining insurance rates. The seismic hazard studies also may generate two standard measures of anticipated ground motion, both confusingly abbreviated MCE; the simpler probabilistic Maximum Considered Earthquake (or Event ), used in standard building codes, and the more detailed and deterministic Maximum Credible Earthquake incorporated in the design of larger buildings and civil infrastructure like dams or bridges. It is important to clarify which MCE is being discussed.
Calculations for determining seismic hazard were first formulated by C. Allin Cornell in 1968 and, depending on their level of importance and use, can be quite complex.
The regional geology and seismology setting is first examined for sources and patterns of earthquake occurrence, both in depth and at the surface from seismometer records; secondly, the impacts from these sources are assessed relative to local geologic rock and soil types, slope angle and groundwater conditions. Zones of similar potential earthquake shaking are thus determined and drawn on maps. The well known San Andreas Fault is illustrated as a long narrow elliptical zone of greater potential motion, like many areas along continental margins associated with the Pacific ring of fire. Zones of higher seismicity in the continental interior may be the site for intraplate earthquakes) and tend to be drawn as broad areas, based on historic records, like the 1812 New Madrid earthquake, since specific causative faults are generally not identified as earthquake sources.
Each zone is given properties associated with source potential: how many earthquakes per year, the maximum size of earthquakes (maximum magnitude), etc. Finally, the calculations require formulae that give the required hazard indicators for a given earthquake size and distance. For example, some districts prefer to use peak acceleration, others use peak velocity, and more sophisticated uses require response spectral ordinates.
The computer program then integrates over all the zones and produces probability curves for the key ground motion parameter. The final result gives a 'chance' of exceeding a given value over a specified amount of time. Standard building codes for homeowners might be concerned with a 1 in 500 years chance, while nuclear plants look at the 10,000 year time frame. A longer-term seismic history can be obtained through paleoseismology. The results may be in the form of a ground response spectrum for use in seismic analysis.
More elaborate variations on the theme also look at the soil conditions. Higher ground motions are likely to be
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