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https://en.wikipedia.org/wiki/Cantor%20function
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In mathematics, the Cantor function is an example of a function that is continuous, but not absolutely continuous. It is a notorious counterexample in analysis, because it challenges naive intuitions about continuity, derivative, and measure. Though it is continuous everywhere and has zero derivative almost everywhere, its value still goes from 0 to 1 as its argument reaches from 0 to 1. Thus, in one sense the function seems very much like a constant one which cannot grow, and in another, it does indeed monotonically grow.
It is also called the Cantor ternary function, the Lebesgue function, Lebesgue's singular function, the Cantor–Vitali function, the Devil's staircase, the Cantor staircase function, and the Cantor–Lebesgue function. introduced the Cantor function and mentioned that Scheeffer pointed out that it was a counterexample to an extension of the fundamental theorem of calculus claimed by Harnack. The Cantor function was discussed and popularized by , and .
Definition
To define the Cantor function , let be any number in and obtain by the following steps:
Express in base 3.
If the base-3 representation of contains a 1, replace every digit strictly after the first 1 by 0.
Replace any remaining 2s with 1s.
Interpret the result as a binary number. The result is .
For example:
has the ternary representation 0.02020202... There are no 1s so the next stage is still 0.02020202... This is rewritten as 0.01010101... This is the binary representation of , so .
has the ternary representation 0.01210121... The digits after the first 1 are replaced by 0s to produce 0.01000000... This is not rewritten since it has no 2s. This is the binary representation of , so .
has the ternary representation 0.21102 (or 0.211012222...). The digits after the first 1 are replaced by 0s to produce 0.21. This is rewritten as 0.11. This is the binary representation of , so .
Equivalently, if is the Cantor set on [0,1], then the Cantor function can be defined as
This formula is well-defined, since every member of the Cantor set has a unique base 3 representation that only contains the digits 0 or 2. (For some members of , the ternary expansion is repeating with trailing 2's and there is an alternative non-repeating expansion ending in 1. For example, = 0.13 = 0.02222...3 is a member of the Cantor set). Since and , and is monotonic on , it is clear that also holds for all .
Properties
The Cantor function challenges naive intuitions about continuity and measure; though it is continuous everywhere and has zero derivative almost everywhere, goes from 0 to 1 as goes from 0 to 1, and takes on every value in between. The Cantor function is the most frequently cited example of a real function that is uniformly continuous (precisely, it is Hölder continuous of exponent α = log 2/log 3) but not absolutely continuous. It is constant on intervals of the form (0.x1x2x3...xn022222..., 0.x1x2x3...xn200000...), and every point not in the Cantor set is in
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https://en.wikipedia.org/wiki/Constant%20function
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In mathematics, a constant function is a function whose (output) value is the same for every input value. For example, the function is a constant function because the value of is 4 regardless of the input value (see image).
Basic properties
As a real-valued function of a real-valued argument, a constant function has the general form or just
Example: The function or just is the specific constant function where the output value is The domain of this function is the set of all real numbers R. The codomain of this function is just {2}. The independent variable x does not appear on the right side of the function expression and so its value is "vacuously substituted". Namely and so on. No matter what value of x is input, the output is "2".
Real-world example: A store where every item is sold for the price of 1 dollar.
The graph of the constant function is a horizontal line in the plane that passes through the point
In the context of a polynomial in one variable x, the non-zero constant function is a polynomial of degree 0 and its general form is where is nonzero. This function has no intersection point with the x-axis, that is, it has no root (zero). On the other hand, the polynomial is the identically zero function. It is the (trivial) constant function and every x is a root. Its graph is the x-axis in the plane.
A constant function is an even function, i.e. the graph of a constant function is symmetric with respect to the y-axis.
In the context where it is defined, the derivative of a function is a measure of the rate of change of function values with respect to change in input values. Because a constant function does not change, its derivative is 0. This is often written: . The converse is also true. Namely, if for all real numbers x, then y is a constant function.
Example: Given the constant function The derivative of y is the identically zero function
Other properties
For functions between preordered sets, constant functions are both order-preserving and order-reversing; conversely, if f is both order-preserving and order-reversing, and if the domain of f is a lattice, then f must be constant.
Every constant function whose domain and codomain are the same set X is a left zero of the full transformation monoid on X, which implies that it is also idempotent.
It has zero slope/gradient.
Every constant function between topological spaces is continuous.
A constant function factors through the one-point set, the terminal object in the category of sets. This observation is instrumental for F. William Lawvere's axiomatization of set theory, the Elementary Theory of the Category of Sets (ETCS).
For any non-empty Y, every set X is isomorphic to the set of constant functions in . For any Y and each element x in X, there is a unique function such that for all . Conversely, if a function satisfies for all , is by definition a constant function.
As a corollary, the one-point set is a generator in the category of sets.
Ev
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https://en.wikipedia.org/wiki/Symmedian
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In geometry, symmedians are three particular lines associated with every triangle. They are constructed by taking a median of the triangle (a line connecting a vertex with the midpoint of the opposite side), and reflecting the line over the corresponding angle bisector (the line through the same vertex that divides the angle there in half). The angle formed by the symmedian and the angle bisector has the same measure as the angle between the median and the angle bisector, but it is on the other side of the angle bisector.
The three symmedians meet at a triangle center called the Lemoine point. Ross Honsberger has called its existence "one of the crown jewels of modern geometry".
Isogonality
Many times in geometry, if we take three special lines through the vertices of a triangle, or cevians, then their reflections about the corresponding angle bisectors, called isogonal lines, will also have interesting properties. For instance, if three cevians of a triangle intersect at a point , then their isogonal lines also intersect at a point, called the isogonal conjugate of .
The symmedians illustrate this fact.
In the diagram, the medians (in black) intersect at the centroid .
Because the symmedians (in red) are isogonal to the medians, the symmedians also intersect at a single point, .
This point is called the triangle's symmedian point, or alternatively the Lemoine point or Grebe point.
The dotted lines are the angle bisectors; the symmedians and medians are symmetric about the angle bisectors (hence the name "symmedian.")
Construction of the symmedian
Let be a triangle. Construct a point by intersecting the tangents from and to the circumcircle. Then is the symmedian of .
first proof. Let the reflection of across the angle bisector of meet at . Then:
second proof. Define as the isogonal conjugate of . It is easy to see that the reflection of about the bisector is the line through parallel to . The same is true for , and so, is a parallelogram. is clearly the median, because a parallelogram's diagonals bisect each other, and is its reflection about the bisector.
third proof. Let be the circle with center passing through and , and let be the circumcenter of . Say lines intersect at , respectively. Since , triangles and are similar. Since
we see that is a diameter of and hence passes through . Let be the midpoint of . Since is the midpoint of , the similarity implies that , from which the result follows.
fourth proof. Let be the midpoint of the arc . , so is the angle bisector of . Let be the midpoint of , and It follows that is the Inverse of with respect to the circumcircle. From that, we know that the circumcircle is an Apollonian circle with foci . So is the bisector of angle , and we have achieved our wanted result.
Tetrahedra
The concept of a symmedian point extends to (irregular) tetrahedra. Given a tetrahedron two planes through are isogonal conjugates if they form equal angles with the plane
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https://en.wikipedia.org/wiki/Lemoine%20point
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In geometry, the Lemoine point, Grebe point or symmedian point is the intersection of the three symmedians (medians reflected at the associated angle bisectors) of a triangle.
Ross Honsberger called its existence "one of the crown jewels of modern geometry".
In the Encyclopedia of Triangle Centers the symmedian point appears as the sixth point, X(6). For a non-equilateral triangle, it lies in the open orthocentroidal disk punctured at its own center, and could be any point therein.
The symmedian point of a triangle with side lengths , and has homogeneous trilinear coordinates .
An algebraic way to find the symmedian point is to express the triangle by three linear equations in two unknowns given by the hesse normal forms of the corresponding lines. The solution of this overdetermined system found by the least squares method gives the coordinates of the point. It also solves the optimization problem to find the point with a minimal sum of squared distances from the sides.
The Gergonne point of a triangle is the same as the symmedian point of the triangle's contact triangle.
The symmedian point of a triangle can be constructed in the following way: let the tangent lines of the circumcircle of through and meet at , and analogously define and ; then is the tangential triangle of , and the lines , and intersect at the symmedian point of . It can be shown that these three lines meet at a point using Brianchon's theorem. Line is a symmedian, as can be seen by drawing the circle with center through and .
The French mathematician Émile Lemoine proved the existence of the symmedian point in 1873, and Ernst Wilhelm Grebe published a paper on it in 1847. Simon Antoine Jean L'Huilier had also noted the point in 1809.
For the extension to an irregular tetrahedron see symmedian.
Notes
References
External links
Triangle centers
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https://en.wikipedia.org/wiki/Tessellation
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A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called tiles, with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety of geometries.
A periodic tiling has a repeating pattern. Some special kinds include regular tilings with regular polygonal tiles all of the same shape, and semiregular tilings with regular tiles of more than one shape and with every corner identically arranged. The patterns formed by periodic tilings can be categorized into 17 wallpaper groups. A tiling that lacks a repeating pattern is called "non-periodic". An aperiodic tiling uses a small set of tile shapes that cannot form a repeating pattern (an aperiodic set of prototiles). A tessellation of space, also known as a space filling or honeycomb, can be defined in the geometry of higher dimensions.
A real physical tessellation is a tiling made of materials such as cemented ceramic squares or hexagons. Such tilings may be decorative patterns, or may have functions such as providing durable and water-resistant pavement, floor, or wall coverings. Historically, tessellations were used in Ancient Rome and in Islamic art such as in the Moroccan architecture and decorative geometric tiling of the Alhambra palace. In the twentieth century, the work of M. C. Escher often made use of tessellations, both in ordinary Euclidean geometry and in hyperbolic geometry, for artistic effect. Tessellations are sometimes employed for decorative effect in quilting. Tessellations form a class of patterns in nature, for example in the arrays of hexagonal cells found in honeycombs.
History
Tessellations were used by the Sumerians (about 4000 BC) in building wall decorations formed by patterns of clay tiles.
Decorative mosaic tilings made of small squared blocks called tesserae were widely employed in classical antiquity, sometimes displaying geometric patterns.
In 1619, Johannes Kepler made an early documented study of tessellations. He wrote about regular and semiregular tessellations in his ; he was possibly the first to explore and to explain the hexagonal structures of honeycomb and snowflakes.
Some two hundred years later in 1891, the Russian crystallographer Yevgraf Fyodorov proved that every periodic tiling of the plane features one of seventeen different groups of isometries. Fyodorov's work marked the unofficial beginning of the mathematical study of tessellations. Other prominent contributors include Alexei Vasilievich Shubnikov and Nikolai Belov (1964), and Heinrich Heesch and Otto Kienzle (1963).
Etymology
In Latin, tessella is a small cubical piece of clay, stone, or glass used to make mosaics. The word "tessella" means "small square" (from tessera, square, which in turn is from the Greek word τέσσερα for four). It corresponds to the everyday term tiling, which refers to applications of tessellations, often made of glazed clay.
Overview
Tessellation in two dimension
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https://en.wikipedia.org/wiki/Cunningham%20chain
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In mathematics, a Cunningham chain is a certain sequence of prime numbers. Cunningham chains are named after mathematician A. J. C. Cunningham. They are also called chains of nearly doubled primes.
Definition
A Cunningham chain of the first kind of length n is a sequence of prime numbers (p1, ..., pn) such that pi+1 = 2pi + 1 for all 1 ≤ i < n. (Hence each term of such a chain except the last is a Sophie Germain prime, and each term except the first is a safe prime).
It follows that
or, by setting (the number is not part of the sequence and need not be a prime number), we have
Similarly, a Cunningham chain of the second kind of length n is a sequence of prime numbers (p1, ..., pn) such that pi+1 = 2pi − 1 for all 1 ≤ i < n.
It follows that the general term is
Now, by setting , we have .
Cunningham chains are also sometimes generalized to sequences of prime numbers (p1, ..., pn) such that pi+1 = api + b for all 1 ≤ i ≤ n for fixed coprime integers a and b; the resulting chains are called generalized Cunningham chains.
A Cunningham chain is called complete if it cannot be further extended, i.e., if the previous and the next terms in the chain are not prime numbers.
Examples
Examples of complete Cunningham chains of the first kind include these:
2, 5, 11, 23, 47 (The next number would be 95, but that is not prime.)
3, 7 (The next number would be 15, but that is not prime.)
29, 59 (The next number would be 119 = 7×17, but that is not prime.)
41, 83, 167 (The next number would be 335, but that is not prime.)
89, 179, 359, 719, 1439, 2879 (The next number would be 5759 = 13×443, but that is not prime.)
Examples of complete Cunningham chains of the second kind include these:
2, 3, 5 (The next number would be 9, but that is not prime.)
7, 13 (The next number would be 25, but that is not prime.)
19, 37, 73 (The next number would be 145, but that is not prime.)
31, 61 (The next number would be 121 = 112, but that is not prime.)
Cunningham chains are now considered useful in cryptographic systems since "they provide two concurrent suitable settings for the ElGamal cryptosystem ... [which] can be implemented in any field where the discrete logarithm problem is difficult."
Largest known Cunningham chains
It follows from Dickson's conjecture and the broader Schinzel's hypothesis H, both widely believed to be true, that for every k there are infinitely many Cunningham chains of length k. There are, however, no known direct methods of generating such chains.
There are computing competitions for the longest Cunningham chain or for the one built up of the largest primes, but unlike the breakthrough of Ben J. Green and Terence Tao – the Green–Tao theorem, that there are arithmetic progressions of primes of arbitrary length – there is no general result known on large Cunningham chains to date.
q# denotes the primorial 2 × 3 × 5 × 7 × ... × q.
, the longest known Cunningham chain of either kind is of length 19, discovered by Ja
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https://en.wikipedia.org/wiki/Multiply%20perfect%20number
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In mathematics, a multiply perfect number (also called multiperfect number or pluperfect number) is a generalization of a perfect number.
For a given natural number k, a number n is called (or perfect) if the sum of all positive divisors of n (the divisor function, σ(n)) is equal to kn; a number is thus perfect if and only if it is . A number that is for a certain k is called a multiply perfect number. As of 2014, numbers are known for each value of k up to 11.
It is unknown whether there are any odd multiply perfect numbers other than 1. The first few multiply perfect numbers are:
1, 6, 28, 120, 496, 672, 8128, 30240, 32760, 523776, 2178540, 23569920, 33550336, 45532800, 142990848, 459818240, ... .
Example
The sum of the divisors of 120 is
1 + 2 + 3 + 4 + 5 + 6 + 8 + 10 + 12 + 15 + 20 + 24 + 30 + 40 + 60 + 120 = 360
which is 3 × 120. Therefore 120 is a number.
Smallest known k-perfect numbers
The following table gives an overview of the smallest known numbers for k ≤ 11 :
Properties
It can be proven that:
For a given prime number p, if n is and p does not divide n, then pn is . This implies that an integer n is a number divisible by 2 but not by 4, if and only if n/2 is an odd perfect number, of which none are known.
If 3n is and 3 does not divide n, then n is .
Odd multiply perfect numbers
It is unknown whether there are any odd multiply perfect numbers other than 1. However if an odd number n exists where k > 2, then it must satisfy the following conditions:
The largest prime factor is ≥ 100129
The second largest prime factor is ≥ 1009
The third largest prime factor is ≥ 101
Bounds
In little-o notation, the number of multiply perfect numbers less than x is for all ε > 0.
The number of k-perfect numbers n for n ≤ x is less than , where c and c are constants independent of k.
Under the assumption of the Riemann hypothesis, the following inequality is true for all numbers n, where k > 3
where is Euler's gamma constant. This can be proven using Robin's theorem.
The number of divisors τ(n) of a number n satisfies the inequality
The number of distinct prime factors ω(n) of n satisfies
If the distinct prime factors of n are , then:
Specific values of k
Perfect numbers
A number n with σ(n) = 2n is perfect.
Triperfect numbers
A number n with σ(n) = 3n is triperfect. There are only six known triperfect numbers and these are believed to comprise all such numbers:
120, 672, 523776, 459818240, 1476304896, 51001180160
If there exists an odd perfect number m (a famous open problem) then 2m would be , since σ(2m) = σ(2) σ(m) = 3×2m. An odd triperfect number must be a square number exceeding 1070 and have at least 12 distinct prime factors, the largest exceeding 105.
Variations
Unitary multiply perfect numbers
A similar extension can be made for unitary perfect numbers. A positive integer n is called a unitary multi number if σ*(n) = kn where σ*(n) is the sum of its unitary divisors. (A divisor d of a
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https://en.wikipedia.org/wiki/Abundant%20number
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In number theory, an abundant number or excessive number is a positive integer for which the sum of its proper divisors is greater than the number. The integer 12 is the first abundant number. Its proper divisors are 1, 2, 3, 4 and 6 for a total of 16. The amount by which the sum exceeds the number is the abundance. The number 12 has an abundance of 4, for example.
Definition
A number n for which the sum of divisors σ(n) > 2n, or, equivalently, the sum of proper divisors (or aliquot sum) s(n) > n.
Abundance is the value σ(n) − 2n (or s(n) − n).
Examples
The first 28 abundant numbers are:
12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 102, 104, 108, 112, 114, 120, ... .
For example, the proper divisors of 24 are 1, 2, 3, 4, 6, 8, and 12, whose sum is 36. Because 36 is greater than 24, the number 24 is abundant. Its abundance is 36 − 24 = 12.
Properties
The smallest odd abundant number is 945.
The smallest abundant number not divisible by 2 or by 3 is 5391411025 whose distinct prime factors are 5, 7, 11, 13, 17, 19, 23, and 29 . An algorithm given by Iannucci in 2005 shows how to find the smallest abundant number not divisible by the first k primes. If represents the smallest abundant number not divisible by the first k primes then for all we have
for sufficiently large k.
Every multiple of a perfect number (except the perfect number itself) is abundant. For example, every multiple of 6 greater than 6 is abundant because
Every multiple of an abundant number is abundant. For example, every multiple of 20 (including 20 itself) is abundant because
Consequently, infinitely many even and odd abundant numbers exist.
Furthermore, the set of abundant numbers has a non-zero natural density. Marc Deléglise showed in 1998 that the natural density of the set of abundant numbers and perfect numbers is between 0.2474 and 0.2480.
An abundant number which is not the multiple of an abundant number or perfect number (i.e. all its proper divisors are deficient) is called a primitive abundant number
An abundant number whose abundance is greater than any lower number is called a highly abundant number, and one whose relative abundance (i.e. s(n)/n ) is greater than any lower number is called a superabundant number
Every integer greater than 20161 can be written as the sum of two abundant numbers.
An abundant number which is not a semiperfect number is called a weird number. An abundant number with abundance 1 is called a quasiperfect number, although none have yet been found.
Every abundant number is a multiple of either a perfect number or a primitive abundant number.
Related concepts
Numbers whose sum of proper factors equals the number itself (such as 6 and 28) are called perfect numbers, while numbers whose sum of proper factors is less than the number itself are called deficient numbers. The first known classification of numbers as deficient, perfect or abundant was by Nicomachus in his Introductio A
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https://en.wikipedia.org/wiki/Deficient%20number
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In number theory, a deficient number or defective number is a positive integer for which the sum of divisors of is less than . Equivalently, it is a number for which the sum of proper divisors (or aliquot sum) is less than . For example, the proper divisors of 8 are , and their sum is less than 8, so 8 is deficient.
Denoting by the sum of divisors, the value is called the number's deficiency. In terms of the aliquot sum , the deficiency is .
Examples
The first few deficient numbers are
1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 50, ...
As an example, consider the number 21. Its divisors are 1, 3, 7 and 21, and their sum is 32. Because 32 is less than 42, the number 21 is deficient. Its deficiency is 2 × 21 − 32 = 10.
Properties
Since the aliquot sums of prime numbers equal 1, all prime numbers are deficient. More generally, all odd numbers with one or two distinct prime factors are deficient. It follows that there are infinitely many odd deficient numbers. There are also an infinite number of even deficient numbers as all powers of two have the sum ().
More generally, all prime powers are deficient because their only proper divisors are which sum to , which is at most .
All proper divisors of deficient numbers are deficient. Moreover, all proper divisors of perfect numbers are deficient.
There exists at least one deficient number in the interval for all sufficiently large n.
Related concepts
Closely related to deficient numbers are perfect numbers with σ(n) = 2n, and abundant numbers with σ(n) > 2n.
The natural numbers were first classified as either deficient, perfect or abundant by Nicomachus in his Introductio Arithmetica (circa 100 CE).
See also
Almost perfect number
Amicable number
Sociable number
Superabundant number
References
External links
The Prime Glossary: Deficient number
Arithmetic dynamics
Divisor function
Integer sequences
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https://en.wikipedia.org/wiki/Deficiency
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A deficiency is generally a lack of something. It may also refer to:
A deficient number, in mathematics, a number n for which σ(n) < 2n
Angular deficiency, in geometry, the difference between a sum of angles and the corresponding sum in a Euclidean plane
Deficiency (graph theory), a property describing how far a given graph is from having a perfect matching
Deficiency (medicine), including various types of malnutrition, as well as genetic diseases caused by deficiencies of endogenously produced proteins
A deficiency in construction, an item, or condition that is considered sub-standard, or below minimum expectations
Genetic deletion, in genetics, also called a deficiency
A deficiency judgment, in the law of real estate
A tax deficiency, an amount owed in taxes over and above what has been submitted in payment
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https://en.wikipedia.org/wiki/Cullen%20number
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In mathematics, a Cullen number is a member of the integer sequence (where is a natural number). Cullen numbers were first studied by James Cullen in 1905. The numbers are special cases of Proth numbers.
Properties
In 1976 Christopher Hooley showed that the natural density of positive integers for which Cn is a prime is of the order o(x) for . In that sense, almost all Cullen numbers are composite. Hooley's proof was reworked by Hiromi Suyama to show that it works for any sequence of numbers n·2n + a + b where a and b are integers, and in particular also for Woodall numbers. The only known Cullen primes are those for n equal to:
1, 141, 4713, 5795, 6611, 18496, 32292, 32469, 59656, 90825, 262419, 361275, 481899, 1354828, 6328548, 6679881 .
Still, it is conjectured that there are infinitely many Cullen primes.
A Cullen number Cn is divisible by p = 2n − 1 if p is a prime number of the form 8k − 3; furthermore, it follows from Fermat's little theorem that if p is an odd prime, then p divides Cm(k) for each m(k) = (2k − k)
(p − 1) − k (for k > 0). It has also been shown that the prime number p divides C(p + 1)/2 when the Jacobi symbol (2 | p) is −1, and that p divides C(3p − 1)/2 when the Jacobi symbol (2 | p) is + 1.
It is unknown whether there exists a prime number p such that Cp is also prime.
Cp follows the recurrence relation
.
Generalizations
Sometimes, a generalized Cullen number base b is defined to be a number of the form n·bn + 1, where n + 2 > b; if a prime can be written in this form, it is then called a generalized Cullen prime. Woodall numbers are sometimes called Cullen numbers of the second kind.
As of October 2021, the largest known generalized Cullen prime is 2525532·732525532 + 1. It has 4,705,888 digits and was discovered by Tom Greer, a PrimeGrid participant.
According to Fermat's little theorem, if there is a prime p such that n is divisible by p − 1 and n + 1 is divisible by p (especially, when n = p − 1) and p does not divide b, then bn must be congruent to 1 mod p (since bn is a power of bp − 1 and bp − 1 is congruent to 1 mod p). Thus, n·bn + 1 is divisible by p, so it is not prime. For example, if some n congruent to 2 mod 6 (i.e. 2, 8, 14, 20, 26, 32, ...), n·bn + 1 is prime, then b must be divisible by 3 (except b = 1).
The least n such that n·bn + 1 is prime (with question marks if this term is currently unknown) are
1, 1, 2, 1, 1242, 1, 34, 5, 2, 1, 10, 1, ?, 3, 8, 1, 19650, 1, 6460, 3, 2, 1, 4330, 2, 2805222, 117, 2, 1, ?, 1, 82960, 5, 2, 25, 304, 1, 36, 3, 368, 1, 1806676, 1, 390, 53, 2, 1, ?, 3, ?, 9665, 62, 1, 1341174, 3, ?, 1072, 234, 1, 220, 1, 142, 1295, 8, 3, 16990, 1, 474, 129897, ?, 1, 13948, 1, ?, 3, 2, 1161, 12198, 1, 682156, 5, 350, 1, 1242, 26, 186, 3, 2, 1, 298, 14, 101670, 9, 2, 775, 202, 1, 1374, 63, 2, 1, ...
References
Further reading
.
.
.
.
External links
Chris Caldwell, The Top Twenty: Cullen primes at The Prime Pages.
The Prime Glossary: Cullen number at The Prime Pa
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https://en.wikipedia.org/wiki/Quasiperfect%20number
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In mathematics, a quasiperfect number is a natural number n for which the sum of all its divisors (the divisor function σ(n)) is equal to 2n + 1. Equivalently, n is the sum of its non-trivial divisors (that is, its divisors excluding 1 and n). No quasiperfect numbers have been found so far.
The quasiperfect numbers are the abundant numbers of minimal abundance (which is 1).
Theorems
If a quasiperfect number exists, it must be an odd square number greater than 1035 and have at least seven distinct prime factors.
Related
Numbers do exist where the sum of all the divisors σ(n) is equal to 2n + 2: 20, 104, 464, 650, 1952, 130304, 522752 ... .
Many of these numbers are of the form 2n−1(2n − 3) where 2n − 3 is prime (instead of 2n − 1 with perfect numbers). In addition, numbers exist where the sum of all the divisors σ(n) is equal to 2n − 1, such as the powers of 2.
Betrothed numbers relate to quasiperfect numbers like amicable numbers relate to perfect numbers.
Notes
References
Arithmetic dynamics
Divisor function
Integer sequences
Unsolved problems in mathematics
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https://en.wikipedia.org/wiki/Semiperfect%20number
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In number theory, a semiperfect number or pseudoperfect number is a natural number n that is equal to the sum of all or some of its proper divisors. A semiperfect number that is equal to the sum of all its proper divisors is a perfect number.
The first few semiperfect numbers are: 6, 12, 18, 20, 24, 28, 30, 36, 40, ...
Properties
Every multiple of a semiperfect number is semiperfect. A semiperfect number that is not divisible by any smaller semiperfect number is called primitive.
Every number of the form 2mp for a natural number m and an odd prime number p such that p < 2m+1 is also semiperfect.
In particular, every number of the form 2m(2m+1 − 1) is semiperfect, and indeed perfect if 2m+1 − 1 is a Mersenne prime.
The smallest odd semiperfect number is 945 (see, e.g., Friedman 1993).
A semiperfect number is necessarily either perfect or abundant. An abundant number that is not semiperfect is called a weird number.
With the exception of 2, all primary pseudoperfect numbers are semiperfect.
Every practical number that is not a power of two is semiperfect.
The natural density of the set of semiperfect numbers exists.
Primitive semiperfect numbers
A primitive semiperfect number (also called a primitive pseudoperfect number, irreducible semiperfect number or irreducible pseudoperfect number) is a semiperfect number that has no semiperfect proper divisor.
The first few primitive semiperfect numbers are 6, 20, 28, 88, 104, 272, 304, 350, ...
There are infinitely many such numbers. All numbers of the form 2mp, with p a prime between 2m and 2m+1, are primitive semiperfect, but this is not the only form: for example, 770. There are infinitely many odd primitive semiperfect numbers, the smallest being 945, a result of Paul Erdős: there are also infinitely many primitive semiperfect numbers that are not harmonic divisor numbers.
Every semiperfect number is a multiple of a primitive semiperfect number.
See also
Hemiperfect number
Erdős–Nicolas number
Notes
References
Section B2.
External links
Integer sequences
Perfect numbers
de:Vollkommene Zahl#Pseudovollkommene Zahlen
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https://en.wikipedia.org/wiki/Weird%20number
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In number theory, a weird number is a natural number that is abundant but not semiperfect. In other words, the sum of the proper divisors (divisors including 1 but not itself) of the number is greater than the number, but no subset of those divisors sums to the number itself.
Examples
The smallest weird number is 70. Its proper divisors are 1, 2, 5, 7, 10, 14, and 35; these sum to 74, but no subset of these sums to 70. The number 12, for example, is abundant but not weird, because the proper divisors of 12 are 1, 2, 3, 4, and 6, which sum to 16; but 2 + 4 + 6 = 12.
The first few weird numbers are
70, 836, 4030, 5830, 7192, 7912, 9272, 10430, 10570, 10792, 10990, 11410, 11690, 12110, 12530, 12670, 13370, 13510, 13790, 13930, 14770, ... .
Properties
Infinitely many weird numbers exist. For example, 70p is weird for all primes p ≥ 149. In fact, the set of weird numbers has positive asymptotic density.
It is not known if any odd weird numbers exist. If so, they must be greater than 1021.
Sidney Kravitz has shown that for k a positive integer, Q a prime exceeding 2k, and
also prime and greater than 2k, then
is a weird number.
With this formula, he found the large weird number
Primitive weird numbers
A property of weird numbers is that if n is weird, and p is a prime greater than the sum of divisors σ(n), then pn is also weird. This leads to the definition of primitive weird numbers: weird numbers that are not a multiple of other weird numbers . Among the 1765 weird numbers less than one million, there are 24 primitive weird numbers. The construction of Kravitz yields primitive weird numbers, since all weird numbers of the form are primitive, but the existence of infinitely many k and Q which yield a prime R is not guaranteed. It is conjectured that there exist infinitely many primitive weird numbers, and Melfi has shown that the infiniteness of primitive weird numbers is a consequence of Cramér's conjecture.
Primitive weird numbers with as many as 16 prime factors and 14712 digits have been found.
See also
Untouchable number
References
External links
Divisor function
Integer sequences
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https://en.wikipedia.org/wiki/Primeval%20number
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In recreational number theory, a primeval number is a natural number n for which the number of prime numbers which can be obtained by permuting some or all of its digits (in base 10) is larger than the number of primes obtainable in the same way for any smaller natural number. Primeval numbers were first described by Mike Keith.
The first few primeval numbers are
1, 2, 13, 37, 107, 113, 137, 1013, 1037, 1079, 1237, 1367, 1379, 10079, 10123, 10136, 10139, 10237, 10279, 10367, 10379, 12379, 13679, ...
The number of primes that can be obtained from the primeval numbers is
0, 1, 3, 4, 5, 7, 11, 14, 19, 21, 26, 29, 31, 33, 35, 41, 53, 55, 60, 64, 89, 96, 106, ...
The largest number of primes that can be obtained from a primeval number with n digits is
1, 4, 11, 31, 106, 402, 1953, 10542, 64905, 362451, 2970505, ...
The smallest n-digit number to achieve this number of primes is
2, 37, 137, 1379, 13679, 123479, 1234679, 12345679, 102345679, 1123456789, 10123456789, ...
Primeval numbers can be composite. The first is 1037 = 17×61. A Primeval prime is a primeval number which is also a prime number:
2, 13, 37, 107, 113, 137, 1013, 1237, 1367, 10079, 10139, 12379, 13679, 100279, 100379, 123479, 1001237, 1002347, 1003679, 1012379, ...
The following table shows the first seven primeval numbers with the obtainable primes and the number of them.
Base 12
In base 12, the primeval numbers are: (using inverted two and three for ten and eleven, respectively)
1, 2, 13, 15, 57, 115, 117, 125, 135, 157, 1017, 1057, 1157, 1257, 125Ɛ, 157Ɛ, 167Ɛ, ...
The number of primes that can be obtained from the primeval numbers is: (written in base 10)
0, 1, 2, 3, 4, 5, 6, 7, 8, 11, 12, 20, 23, 27, 29, 33, 35, ...
Note that 13, 115 and 135 are composite: 13 = 3×5, 115 = 7×1Ɛ, and 135 = 5×31.
See also
Permutable prime
Truncatable prime
External links
Chris Caldwell, The Prime Glossary: Primeval number at The Prime Pages
Mike Keith, Integers Containing Many Embedded Primes
Base-dependent integer sequences
Prime numbers
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https://en.wikipedia.org/wiki/Woodall%20number
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In number theory, a Woodall number (Wn) is any natural number of the form
for some natural number n. The first few Woodall numbers are:
1, 7, 23, 63, 159, 383, 895, … .
History
Woodall numbers were first studied by Allan J. C. Cunningham and H. J. Woodall in 1917, inspired by James Cullen's earlier study of the similarly defined Cullen numbers.
Woodall primes
Woodall numbers that are also prime numbers are called Woodall primes; the first few exponents n for which the corresponding Woodall numbers Wn are prime are 2, 3, 6, 30, 75, 81, 115, 123, 249, 362, 384, ... ; the Woodall primes themselves begin with 7, 23, 383, 32212254719, ... .
In 1976 Christopher Hooley showed that almost all Cullen numbers are composite. In October 1995, Wilfred Keller published a paper discussing several new Cullen primes and the efforts made to factorise other Cullen and Woodall numbers. Included in that paper is a personal communication to Keller from Hiromi Suyama, asserting that Hooley's method can be reformulated to show that it works for any sequence of numbers , where a and b are integers, and in particular, that almost all Woodall numbers are composite. It is an open problem whether there are infinitely many Woodall primes. , the largest known Woodall prime is 17016602 × 217016602 − 1. It has 5,122,515 digits and was found by Diego Bertolotti in March 2018 in the distributed computing project PrimeGrid.
Restrictions
Starting with W4 = 63 and W5 = 159, every sixth Woodall number is divisible by 3; thus, in order for Wn to be prime, the index n cannot be congruent to 4 or 5 (modulo 6). Also, for a positive integer m, the Woodall number W2m may be prime only if 2m + m is prime. As of January 2019, the only known primes that are both Woodall primes and Mersenne primes are W2 = M3 = 7, and W512 = M521.
Divisibility properties
Like Cullen numbers, Woodall numbers have many divisibility properties. For example, if p is a prime number, then p divides
W(p + 1) / 2 if the Jacobi symbol is +1 and
W(3p − 1) / 2 if the Jacobi symbol is −1.
Generalization
A generalized Woodall number base b is defined to be a number of the form n × bn − 1, where n + 2 > b; if a prime can be written in this form, it is then called a generalized Woodall prime.
The smallest value of n such that n × bn − 1 is prime for b = 1, 2, 3, ... are
3, 2, 1, 1, 8, 1, 2, 1, 10, 2, 2, 1, 2, 1, 2, 167, 2, 1, 12, 1, 2, 2, 29028, 1, 2, 3, 10, 2, 26850, 1, 8, 1, 42, 2, 6, 2, 24, 1, 2, 3, 2, 1, 2, 1, 2, 2, 140, 1, 2, 2, 22, 2, 8, 1, 2064, 2, 468, 6, 2, 1, 362, 1, 2, 2, 6, 3, 26, 1, 2, 3, 20, 1, 2, 1, 28, 2, 38, 5, 3024, 1, 2, 81, 858, 1, 2, 3, 2, 8, 60, 1, 2, 2, 10, 5, 2, 7, 182, 1, 17782, 3, ...
, the largest known generalized Woodall prime with base greater than 2 is 2740879 × 322740879 − 1.
See also
Mersenne prime - Prime numbers of the form 2n − 1.
References
Further reading
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.
.
External links
Chris Caldwell, The Prime Glossary: Woodall number, and The Top Twenty: Woodall, and The
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https://en.wikipedia.org/wiki/Riesel%20number
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In mathematics, a Riesel number is an odd natural number k for which is composite for all natural numbers n . In other words, when k is a Riesel number, all members of the following set are composite:
If the form is instead , then k is a Sierpinski number.
Riesel problem
In 1956, Hans Riesel showed that there are an infinite number of integers k such that is not prime for any integer n. He showed that the number 509203 has this property, as does 509203 plus any positive integer multiple of 11184810. The Riesel problem consists in determining the smallest Riesel number. Because no covering set has been found for any k less than 509203, it is conjectured to be the smallest Riesel number.
To check if there are k < 509203, the Riesel Sieve project (analogous to Seventeen or Bust for Sierpinski numbers) started with 101 candidates k. As of December 2022, 57 of these k had been eliminated by Riesel Sieve, PrimeGrid, or outside persons. The remaining 42 values of k that have yielded only composite numbers for all values of n so far tested are
23669, 31859, 38473, 46663, 67117, 74699, 81041, 107347, 121889, 129007, 143047, 161669, 206231, 215443, 226153, 234343, 245561, 250027, 315929, 319511, 324011, 325123, 327671, 336839, 342847, 344759, 362609, 363343, 364903, 365159, 368411, 371893, 384539, 386801, 397027, 409753, 444637, 470173, 474491, 477583, 485557, 494743.
The most recent elimination was in April 2023, when 97139 × 218397548 − 1 was found to be prime by Ryan Propper. This number is 5,538,219 digits long.
As of April 2023, PrimeGrid has searched the remaining candidates up to n = 14,500,000.
Known Riesel numbers
The sequence of currently known Riesel numbers begins with:
509203, 762701, 777149, 790841, 992077, 1106681, 1247173, 1254341, 1330207, 1330319, 1715053, 1730653, 1730681, 1744117, 1830187, 1976473, 2136283, 2251349, 2313487, 2344211, 2554843, 2924861, ...
Covering set
A number can be shown to be a Riesel number by exhibiting a covering set: a set of prime numbers that will divide any member of the sequence, so called because it is said to "cover" that sequence. The only proven Riesel numbers below one million have covering sets as follows:
has covering set {3, 5, 7, 13, 17, 241}
has covering set {3, 5, 7, 13, 17, 241}
has covering set {3, 5, 7, 13, 19, 37, 73}
has covering set {3, 5, 7, 13, 19, 37, 73}
has covering set {3, 5, 7, 13, 17, 241}.
The smallest n for which k · 2n − 1 is prime
Here is a sequence for k = 1, 2, .... It is defined as follows: is the smallest n ≥ 0 such that is prime, or -1 if no such prime exists.
2, 1, 0, 0, 2, 0, 1, 0, 1, 1, 2, 0, 3, 0, 1, 1, 2, 0, 1, 0, 1, 1, 4, 0, 3, 2, 1, 3, 4, 0, 1, 0, 2, 1, 2, 1, 1, 0, 3, 1, 2, 0, 7, 0, 1, 3, 4, 0, 1, 2, 1, 1, 2, 0, 1, 2, 1, 3, 12, 0, 3, 0, 2, 1, 4, 1, 5, 0, 1, 1, 2, 0, 7, 0, 1, ... . The first unknown n is for that k = 23669.
Related sequences are (not allowing n = 0), for odd ks, see or (not allowing n = 0)
Simultaneously Riesel and Sie
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https://en.wikipedia.org/wiki/Almost%20perfect%20number
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In mathematics, an almost perfect number (sometimes also called slightly defective or least deficient number) is a natural number n such that the sum of all divisors of n (the sum-of-divisors function σ(n)) is equal to 2n − 1, the sum of all proper divisors of n, s(n) = σ(n) − n, then being equal to n − 1. The only known almost perfect numbers are powers of 2 with non-negative exponents . Therefore the only known odd almost perfect number is 20 = 1, and the only known even almost perfect numbers are those of the form 2k for some positive integer k; however, it has not been shown that all almost perfect numbers are of this form. It is known that an odd almost perfect number greater than 1 would have at least six prime factors.
If m is an odd almost perfect number then is a Descartes number. Moreover if a and b are positive odd integers such that and such that and are both primes, then would be an odd weird number.
See also
Perfect number
Quasiperfect number
References
Further reading
External links
Arithmetic dynamics
Divisor function
Integer sequences
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https://en.wikipedia.org/wiki/Hyperperfect%20number
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In number theory, a -hyperperfect number is a natural number for which the equality holds, where is the divisor function (i.e., the sum of all positive divisors of ). A hyperperfect number is a -hyperperfect number for some integer . Hyperperfect numbers generalize perfect numbers, which are 1-hyperperfect.
The first few numbers in the sequence of -hyperperfect numbers are , with the corresponding values of being . The first few -hyperperfect numbers that are not perfect are .
List of hyperperfect numbers
The following table lists the first few -hyperperfect numbers for some values of , together with the sequence number in the On-Line Encyclopedia of Integer Sequences (OEIS) of the sequence of -hyperperfect numbers:
It can be shown that if is an odd integer and and are prime numbers, then is -hyperperfect; Judson S. McCranie has conjectured in 2000 that all -hyperperfect numbers for odd are of this form, but the hypothesis has not been proven so far. Furthermore, it can be proven that if are odd primes and is an integer such that then is -hyperperfect.
It is also possible to show that if and is prime, then for all such that is prime, is -hyperperfect. The following table lists known values of and corresponding values of for which is -hyperperfect:
Hyperdeficiency
The newly introduced mathematical concept of hyperdeficiency is related to the hyperperfect numbers.
Definition (Minoli 2010): For any integer and for integer , define the -hyperdeficiency (or simply the hyperdeficiency) for the number as
A number is said to be -hyperdeficient if
Note that for one gets which is the standard traditional definition of deficiency.
Lemma: A number is -hyperperfect (including ) if and only if the -hyperdeficiency of ,
Lemma: A number is -hyperperfect (including ) if and only if for some , for at least one .
References
Further reading
Articles
.
.
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.
.
.
Books
Daniel Minoli, Voice over MPLS, McGraw-Hill, New York, NY, 2002, (p. 114-134)
External links
MathWorld: Hyperperfect number
A long list of hyperperfect numbers under Data
Divisor function
Integer sequences
Perfect numbers
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https://en.wikipedia.org/wiki/Zermelo%20set%20theory
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Zermelo set theory (sometimes denoted by Z-), as set out in a seminal paper in 1908 by Ernst Zermelo, is the ancestor of modern Zermelo–Fraenkel set theory (ZF) and its extensions, such as von Neumann–Bernays–Gödel set theory (NBG). It bears certain differences from its descendants, which are not always understood, and are frequently misquoted. This article sets out the original axioms, with the original text (translated into English) and original numbering.
The axioms of Zermelo set theory
The axioms of Zermelo set theory are stated for objects, some of which (but not necessarily all) are sets, and the remaining objects are urelements and not sets. Zermelo's language implicitly includes a membership relation ∈, an equality relation = (if it is not included in the underlying logic), and a unary predicate saying whether an object is a set. Later versions of set theory often assume that all objects are sets so there are no urelements and there is no need for the unary predicate.
AXIOM I. Axiom of extensionality (Axiom der Bestimmtheit) "If every element of a set M is also an element of N and vice versa ... then M N. Briefly, every set is determined by its elements."
AXIOM II. Axiom of elementary sets (Axiom der Elementarmengen) "There exists a set, the null set, ∅, that contains no element at all. If a is any object of the domain, there exists a set {a} containing a and only a as an element. If a and b are any two objects of the domain, there always exists a set {a, b} containing as elements a and b but no object x distinct from them both." See Axiom of pairs.
AXIOM III. Axiom of separation (Axiom der Aussonderung) "Whenever the propositional function –(x) is defined for all elements of a set M, M possesses a subset M' containing as elements precisely those elements x of M for which –(x) is true."
AXIOM IV. Axiom of the power set (Axiom der Potenzmenge) "To every set T there corresponds a set T' , the power set of T, that contains as elements precisely all subsets of T ."
AXIOM V. Axiom of the union (Axiom der Vereinigung) "To every set T there corresponds a set ∪T, the union of T, that contains as elements precisely all elements of the elements of T ."
AXIOM VI. Axiom of choice (Axiom der Auswahl) "If T is a set whose elements all are sets that are different from ∅ and mutually disjoint, its union ∪T includes at least one subset S1 having one and only one element in common with each element of T ."
AXIOM VII. Axiom of infinity (Axiom des Unendlichen) "There exists in the domain at least one set Z that contains the null set as an element and is so constituted that to each of its elements a there corresponds a further element of the form {a}, in other words, that with each of its elements a it also contains the corresponding set {a} as element."
Connection with standard set theory
The most widely used and accepted set theory is known as ZFC, which consists of Zermelo–Fraenkel set theory including the axiom of choice (AC). The lin
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https://en.wikipedia.org/wiki/Theoretical%20computer%20science
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Theoretical computer science (TCS) is a subset of general computer science and mathematics that focuses on mathematical aspects of computer science such as the theory of computation, formal language theory, the lambda calculus and type theory.
It is difficult to circumscribe the theoretical areas precisely. The ACM's Special Interest Group on Algorithms and Computation Theory (SIGACT) provides the following description:
History
While logical inference and mathematical proof had existed previously, in 1931 Kurt Gödel proved with his incompleteness theorem that there are fundamental limitations on what statements could be proved or disproved.
Information theory was added to the field with a 1948 mathematical theory of communication by Claude Shannon. In the same decade, Donald Hebb introduced a mathematical model of learning in the brain. With mounting biological data supporting this hypothesis with some modification, the fields of neural networks and parallel distributed processing were established. In 1971, Stephen Cook and, working independently, Leonid Levin, proved that there exist practically relevant problems that are NP-complete – a landmark result in computational complexity theory.
With the development of quantum mechanics in the beginning of the 20th century came the concept that mathematical operations could be performed on an entire particle wavefunction. In other words, one could compute functions on multiple states simultaneously. This led to the concept of a quantum computer in the latter half of the 20th century that took off in the 1990s when Peter Shor showed that such methods could be used to factor large numbers in polynomial time, which, if implemented, would render some modern public key cryptography algorithms like RSA insecure.
Modern theoretical computer science research is based on these basic developments, but includes many other mathematical and interdisciplinary problems that have been posed, as shown below:
Topics
Algorithms
An algorithm is a step-by-step procedure for calculations. Algorithms are used for calculation, data processing, and automated reasoning.
An algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function. Starting from an initial state and initial input (perhaps empty), the instructions describe a computation that, when executed, proceeds through a finite number of well-defined successive states, eventually producing "output" and terminating at a final ending state. The transition from one state to the next is not necessarily deterministic; some algorithms, known as randomized algorithms, incorporate random input.
Automata theory
Automata theory is the study of abstract machines and automata, as well as the computational problems that can be solved using them. It is a theory in theoretical computer science, under discrete mathematics (a section of mathematics and also of computer science). Automata comes from the Greek word αὐτόμα
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https://en.wikipedia.org/wiki/Regularization
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Regularization may refer to:
Regularization (linguistics)
Regularization (mathematics)
Regularization (physics)
Regularization (solid modeling)
Regularization Law, an Israeli law intended to retroactively legalize settlements
See also
Matrix regularization
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https://en.wikipedia.org/wiki/Israel%20Central%20Bureau%20of%20Statistics
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The Israel Central Bureau of Statistics (, HaLishka HaMerkazit LiStatistika; ), abbreviated CBS, is an Israeli government office established in 1949 to carry out research and publish statistical data on all aspects of Israeli life, including population, society, economy, industry, education, and physical infrastructure.
The CBS is headquartered in the Givat Shaul neighborhood of Jerusalem, with another branch in Tel Aviv.
Overview
It is headed by a National Statistician (previously named Government Statistician), who is appointed on the recommendation of the prime minister. Professor Emeritus Danny Pfefferman of Hebrew University has served in that position and as Director of the CBS since 2013. The bureau's annual budget in 2011 was NIS 237 million.
The work of the CBS follows internationally accepted standards which enable comparison of statistical information with other countries. It gathers current, monthly, quarterly and annually data on the national economy (production, consumption, capital formation, labor productivity, savings), the balance of payments and foreign trade, the activity of different economic branches (agriculture, manufacturing, construction, transport, commerce and services, etc.), the price of goods and services, the population, family size, employment, education, health, crime, government services and more. The CBS also conducts a Census of Population and Housing every ten years, as well as periodic and one-time surveys on a variety of subjects.
The work of CBS is overseen by the Public Commission of Statistics. The data is disseminated in a wide variety of publications, among them the Statistical Abstract of Israel. Current and updated statistical information is brought to the public's attention through daily press releases. Government ministries use the data collected by CBS for policymaking, planning and tracking development. The data is also made available to academic research institutions and the general public.
References
External links
CBS's 2018 Israel's population
Israel
Government agencies of Israel
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https://en.wikipedia.org/wiki/Wieferich%20prime
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In number theory, a Wieferich prime is a prime number p such that p2 divides , therefore connecting these primes with Fermat's little theorem, which states that every odd prime p divides . Wieferich primes were first described by Arthur Wieferich in 1909 in works pertaining to Fermat's Last Theorem, at which time both of Fermat's theorems were already well known to mathematicians.
Since then, connections between Wieferich primes and various other topics in mathematics have been discovered, including other types of numbers and primes, such as Mersenne and Fermat numbers, specific types of pseudoprimes and some types of numbers generalized from the original definition of a Wieferich prime. Over time, those connections discovered have extended to cover more properties of certain prime numbers as well as more general subjects such as number fields and the abc conjecture.
, the only known Wieferich primes are 1093 and 3511 .
Equivalent definitions
The stronger version of Fermat's little theorem, which a Wieferich prime satisfies, is usually expressed as a congruence relation . From the definition of the congruence relation on integers, it follows that this property is equivalent to the definition given at the beginning. Thus if a prime p satisfies this congruence, this prime divides the Fermat quotient . The following are two illustrative examples using the primes 11 and 1093:
For p = 11, we get which is 93 and leaves a remainder of 5 after division by 11, hence 11 is not a Wieferich prime. For p = 1093, we get or 485439490310...852893958515 (302 intermediate digits omitted for clarity), which leaves a remainder of 0 after division by 1093 and thus 1093 is a Wieferich prime.
Wieferich primes can be defined by other equivalent congruences. If p is a Wieferich prime, one can multiply both sides of the congruence by 2 to get . Raising both sides of the congruence to the power p shows that a Wieferich prime also satisfies , and hence for all . The converse is also true: for some implies that the multiplicative order of 2 modulo p2 divides gcd, φ, that is, and thus p is a Wieferich prime. This also implies that Wieferich primes can be defined as primes p such that the multiplicative orders of 2 modulo p and modulo p2 coincide: , (By the way, ord10932 = 364, and ord35112 = 1755).
H. S. Vandiver proved that if and only if .
History and search status
In 1902, Meyer proved a theorem about solutions of the congruence ap − 1 ≡ 1 (mod pr). Later in that decade Arthur Wieferich showed specifically that if the first case of Fermat's last theorem has solutions for an odd prime exponent, then that prime must satisfy that congruence for a = 2 and r = 2. In other words, if there exist solutions to xp + yp + zp = 0 in integers x, y, z and p an odd prime with p ∤ xyz, then p satisfies 2p − 1 ≡ 1 (mod p2). In 1913, Bachmann examined the residues of . He asked the question when this residue vanishes and tried to find expressions for answering this question
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https://en.wikipedia.org/wiki/Wilson%20prime
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In number theory, a Wilson prime is a prime number such that divides , where "" denotes the factorial function; compare this with Wilson's theorem, which states that every prime divides . Both are named for 18th-century English mathematician John Wilson; in 1770, Edward Waring credited the theorem to Wilson, although it had been stated centuries earlier by Ibn al-Haytham.
The only known Wilson primes are 5, 13, and 563 . Costa et al. write that "the case is trivial", and credit the observation that 13 is a Wilson prime to . Early work on these numbers included searches by N. G. W. H. Beeger and Emma Lehmer, but 563 was not discovered until the early 1950s, when computer searches could be applied to the problem. If any others exist, they must be greater than 2 × 1013. It has been conjectured that infinitely many Wilson primes exist, and that the number of Wilson primes in an interval is about .
Several computer searches have been done in the hope of finding new Wilson primes.
The Ibercivis distributed computing project includes a search for Wilson primes. Another search was coordinated at the Great Internet Mersenne Prime Search forum.
Generalizations
Wilson primes of order
Wilson's theorem can be expressed in general as for every integer and prime . Generalized Wilson primes of order are the primes such that divides .
It was conjectured that for every natural number , there are infinitely many Wilson primes of order .
The smallest generalized Wilson primes of order are:
Near-Wilson primes
A prime satisfying the congruence with small can be called a near-Wilson prime. Near-Wilson primes with are bona fide Wilson primes. The table on the right lists all such primes with from up to 4.
Wilson numbers
A Wilson number is a natural number such that , where and where the term is positive if and only if has a primitive root and negative otherwise. For every natural number , is divisible by , and the quotients (called generalized Wilson quotients) are listed in . The Wilson numbers are
If a Wilson number is prime, then is a Wilson prime. There are 13 Wilson numbers up to 5.
See also
PrimeGrid
Table of congruences
Wall–Sun–Sun prime
Wieferich prime
Wolstenholme prime
References
Further reading
External links
The Prime Glossary: Wilson prime
Status of the search for Wilson primes
Classes of prime numbers
Factorial and binomial topics
Unsolved problems in number theory
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https://en.wikipedia.org/wiki/Wall%E2%80%93Sun%E2%80%93Sun%20prime
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In number theory, a Wall–Sun–Sun prime or Fibonacci–Wieferich prime is a certain kind of prime number which is conjectured to exist, although none are known.
Definition
Let be a prime number. When each term in the sequence of Fibonacci numbers is reduced modulo , the result is a periodic sequence.
The (minimal) period length of this sequence is called the Pisano period and denoted .
Since , it follows that p divides . A prime p such that p2 divides is called a Wall–Sun–Sun prime.
Equivalent definitions
If denotes the rank of apparition modulo (i.e., is the smallest positive index such that divides ), then a Wall–Sun–Sun prime can be equivalently defined as a prime such that divides .
For a prime p ≠ 2, 5, the rank of apparition is known to divide , where the Legendre symbol has the values
This observation gives rise to an equivalent characterization of Wall–Sun–Sun primes as primes such that divides the Fibonacci number .
A prime is a Wall–Sun–Sun prime if and only if .
A prime is a Wall–Sun–Sun prime if and only if , where is the -th Lucas number.
McIntosh and Roettger establish several equivalent characterizations of Lucas–Wieferich primes. In particular, let ; then the following are equivalent:
Existence
In a study of the Pisano period , Donald Dines Wall determined that there are no Wall–Sun–Sun primes less than . In 1960, he wrote:
It has since been conjectured that there are infinitely many Wall–Sun–Sun primes. No Wall–Sun–Sun primes are known .
In 2007, Richard J. McIntosh and Eric L. Roettger showed that if any exist, they must be > 2.
Dorais and Klyve extended this range to 9.7 without finding such a prime.
In December 2011, another search was started by the PrimeGrid project, however it was suspended in May 2017. In November 2020, PrimeGrid started another project that searches for Wieferich and Wall–Sun–Sun primes simultaneously. The project ended in December 2022, definitely proving that any Wall–Sun–Sun prime must exceed (about ).
History
Wall–Sun–Sun primes are named after Donald Dines Wall, Zhi Hong Sun and Zhi Wei Sun; Z. H. Sun and Z. W. Sun showed in 1992 that if the first case of Fermat's Last Theorem was false for a certain prime p, then p would have to be a Wall–Sun–Sun prime. As a result, prior to Andrew Wiles' proof of Fermat's Last Theorem, the search for Wall–Sun–Sun primes was also the search for a potential counterexample to this centuries-old conjecture.
Generalizations
A tribonacci–Wieferich prime is a prime p satisfying , where h is the least positive integer satisfying [Th,Th+1,Th+2] ≡ [T0, T1, T2] (mod m) and Tn denotes the n-th tribonacci number. No tribonacci–Wieferich prime exists below 1011.
A Pell–Wieferich prime is a prime p satisfying p2 divides Pp−1, when p congruent to 1 or 7 (mod 8), or p2 divides Pp+1, when p congruent to 3 or 5 (mod 8), where Pn denotes the n-th Pell number. For example, 13, 31, and 1546463 are Pell–Wieferich primes, and no others below 109 . In
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https://en.wikipedia.org/wiki/Regular%20prime
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In number theory, a regular prime is a special kind of prime number, defined by Ernst Kummer in 1850 to prove certain cases of Fermat's Last Theorem. Regular primes may be defined via the divisibility of either class numbers or of Bernoulli numbers.
The first few regular odd primes are:
3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 43, 47, 53, 61, 71, 73, 79, 83, 89, 97, 107, 109, 113, 127, 137, 139, 151, 163, 167, 173, 179, 181, 191, 193, 197, 199, ... .
History and motivation
In 1850, Kummer proved that Fermat's Last Theorem is true for a prime exponent p if p is regular. This focused attention on the irregular primes. In 1852, Genocchi was able to prove that the first case of Fermat's Last Theorem is true for an exponent p, if is not an irregular pair. Kummer improved this further in 1857 by showing that for the "first case" of Fermat's Last Theorem (see Sophie Germain's theorem) it is sufficient to establish that either or fails to be an irregular pair.
( is an irregular pair when p is irregular due to a certain condition described below being realized at 2k.)
Kummer found the irregular primes less than 165. In 1963, Lehmer reported results up to 10000 and Selfridge and Pollack announced in 1964 to have completed the table of irregular primes up to 25000. Although the two latter tables did not appear in print, Johnson found that is in fact an irregular pair for and that this is the first and only time this occurs for . It was found in 1993 that the next time this happens is for ; see Wolstenholme prime.
Definition
Class number criterion
An odd prime number p is defined to be regular if it does not divide the class number of the p-th cyclotomic field Q(ζp), where ζp is a primitive p-th root of unity.
The prime number 2 is often considered regular as well.
The class number of the cyclotomic
field is the number of ideals of the ring of integers
Z(ζp) up to equivalence. Two ideals I, J are considered equivalent if there is a nonzero u in Q(ζp) so that . The first few of these class numbers are listed in .
Kummer's criterion
Ernst Kummer showed that an equivalent criterion for regularity is that p does not divide the numerator of any of the Bernoulli numbers Bk for .
Kummer's proof that this is equivalent to the class number definition is strengthened by the Herbrand–Ribet theorem, which states certain consequences of p dividing one of these Bernoulli numbers.
Siegel's conjecture
It has been conjectured that there are infinitely many regular primes. More precisely conjectured that e−1/2, or about 60.65%, of all prime numbers are regular, in the asymptotic sense of natural density. Neither conjecture has been proven to date.
Irregular primes
An odd prime that is not regular is an irregular prime (or Bernoulli irregular or B-irregular to distinguish from other types of irregularity discussed below). The first few irregular primes are:
37, 59, 67, 101, 103, 131, 149, 157, 233, 257, 263, 271, 283, 293, 307, 311, 347, 353, 379, 389,
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https://en.wikipedia.org/wiki/Newman%E2%80%93Shanks%E2%80%93Williams%20prime
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In mathematics, a Newman–Shanks–Williams prime (NSW prime) is a prime number p which can be written in the form
NSW primes were first described by Morris Newman, Daniel Shanks and Hugh C. Williams in 1981 during the study of finite simple groups with square order.
The first few NSW primes are 7, 41, 239, 9369319, 63018038201, … , corresponding to the indices 3, 5, 7, 19, 29, … .
The sequence S alluded to in the formula can be described by the following recurrence relation:
The first few terms of the sequence are 1, 1, 3, 7, 17, 41, 99, … . Each term in this sequence is half the corresponding term in the sequence of companion Pell numbers. These numbers also appear in the continued fraction convergents to .
Further reading
External links
The Prime Glossary: NSW number
Classes of prime numbers
Unsolved problems in mathematics
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https://en.wikipedia.org/wiki/Finite%20group
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In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving transformations. Important examples of finite groups include cyclic groups and permutation groups.
The study of finite groups has been an integral part of group theory since it arose in the 19th century. One major area of study has been classification: the classification of finite simple groups (those with no nontrivial normal subgroup) was completed in 2004.
History
During the twentieth century, mathematicians investigated some aspects of the theory of finite groups in great depth, especially the local theory of finite groups and the theory of solvable and nilpotent groups. As a consequence, the complete classification of finite simple groups was achieved, meaning that all those simple groups from which all finite groups can be built are now known.
During the second half of the twentieth century, mathematicians such as Chevalley and Steinberg also increased our understanding of finite analogs of classical groups, and other related groups. One such family of groups is the family of general linear groups over finite fields.
Finite groups often occur when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving transformations. The theory of Lie groups, which may be viewed as dealing with "continuous symmetry", is strongly influenced by the associated Weyl groups. These are finite groups generated by reflections which act on a finite-dimensional Euclidean space. The properties of finite groups can thus play a role in subjects such as theoretical physics and chemistry.
Examples
Permutation groups
The symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself. Since there are n! (n factorial) possible permutations of a set of n symbols, it follows that the order (the number of elements) of the symmetric group Sn is n!.
Cyclic groups
A cyclic group Zn is a group all of whose elements are powers of a particular element a where , the identity. A typical realization of this group is as the complex roots of unity. Sending a to a primitive root of unity gives an isomorphism between the two. This can be done with any finite cyclic group.
Finite abelian groups
An abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order (the axiom of commutativity). They are named after Niels Henrik Abel.
An arbitrary finite abelian group is isomorphic to a direct sum of finite cyclic groups of prime power order, and these orders are uniquely determine
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https://en.wikipedia.org/wiki/Office%20for%20National%20Statistics
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The Office for National Statistics (ONS; ) is the executive office of the UK Statistics Authority, a non-ministerial department which reports directly to the UK Parliament.
Overview
The ONS is responsible for the collection and publication of statistics related to the economy, population and society of the UK; responsibility for some areas of statistics in Scotland, Northern Ireland and Wales is devolved to the devolved governments for those areas. The ONS functions as the executive office of the National Statistician, who is also the UK Statistics Authority's Chief Executive and principal statistical adviser to the UK's National Statistics Institute, and the 'Head Office' of the Government Statistical Service (GSS). Its main office is in Newport near the United Kingdom Intellectual Property Office and Tredegar House, but another significant office is in Titchfield in Hampshire, and a small office is in London. ONS co-ordinates data collection with the respective bodies in Northern Ireland and Scotland, namely NISRA and NRS.
History
The ONS was formed on 1 April 1996 by the merger of the Central Statistical Office (CSO) and the Office of Population Censuses and Surveys (OPCS). Following the Statistics and Registration Service Act 2007, the United Kingdom Statistics Authority became a non-ministerial department on 1 April 2008.
Purpose and scope
ONS produces and publishes a wide range of the information about the United Kingdom that can be used for social and economic policy-making as well as painting a portrait of the country as its population evolves over time. This is often produced in ways that make comparison with other societies and economies possible. Much of the data on which policy-makers depend is produced by ONS through a combination of a decennial population census, samples and surveys and analysis of data generated by businesses and organisations such as the National Health Service and the register of births, marriages and deaths. Its publications, and analyses by other users based on its published data, are reported and discussed daily in the media as the basis for the public understanding of the country in which they live.
Applications of data
The reliance on some of these data by government (both local and national) makes ONS material central to debates about the determination of priorities, the allocation of resources and for decisions on interest rates or borrowing. The complexity and degree and speed of change in the society, combined with the challenge of measuring some of these (e.g. in relation to longevity, migration or illness patterns or fine movements in inflation or other aspects of national accounts) give rise to periodic debates about some of its indicators and portrayals. Many of these rely on sources which are outside ONS, while some of its own sources need to be supplemented, for example between censuses, by updated but less rigorously obtained information from other sources. Consequently, unexpected or inco
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https://en.wikipedia.org/wiki/Palestinian%20Central%20Bureau%20of%20Statistics
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The Palestinian Central Bureau of Statistics (PCBS; ) is the official
statistical institution of the State of Palestine. Its main task is to provide credible statistical figures at the national and international levels. It is a state institution that provides service to the governmental, non – governmental and private sectors in addition to research institutions and universities. It is established as an independent statistical bureau. The PCBS publishes the Statistical Yearbook of Palestine and the Jerusalem Statistical Yearbook annually.
The head office of the agency is in Ein Munjed Quarter, Ramallah.
Activities
Besides general statistics, such as the Retail Price Index, the PCBS also carries out special projects. It conducted the first Palestinian census in 1997, although Israel prevented the national census team from surveying the population in East Jerusalem. In 2007, the second census was carried out. In the 2007 census, a limited census was carried out in East Jerusalem.
Also, the PCBS provided the 2003 "Survey on the Impact of separation Wall on the Location Where it Passed Through".
The PCBS publishes the Statistical Yearbook of Palestine and the Jerusalem Statistical Yearbook annually.
Offices
The PCBS has its main office in the Balu'a area of Ramallah. In October 2001, the building was raided by the Israeli Defence Forces. The soldiers confiscated hard drives and vandalized a number of the offices. In March and April 2002, its Fieldwork section in downtown Ramallah was raided four times; soldiers searched the apartments.
Presidents
Dr. Hasan Abu Libdeh (1993–2005)
Dr. Luay Shabeneh (2005–2010)
Ola Awad (2011–present)
References
External links
Palestinian Central Bureau of Statistics
Palestinian Central Bureau of Statistics
Statistical Yearbook of Palestine 2013, December 2013. (File size: 14.1 MB)
Government of the State of Palestine
PCBS
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https://en.wikipedia.org/wiki/Isadore%20Singer
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Isadore Manuel Singer (May 3, 1924 – February 11, 2021) was an American mathematician. He was an Emeritus Institute Professor in the Department of Mathematics at the Massachusetts Institute of Technology and a Professor Emeritus of Mathematics at the University of California, Berkeley.
Singer is noted for his work with Michael Atiyah, proving the Atiyah–Singer index theorem in 1962, which paved the way for new interactions between pure mathematics and theoretical physics. In early 1980s, while a professor at Berkeley, Singer co-founded the Mathematical Sciences Research Institute (MSRI) with Shiing-Shen Chern and Calvin Moore.
Biography
Early life and education
Singer was born on May 3, 1924, in Detroit, Michigan, to Polish Jewish immigrants. His father Simon was employed as a printer and only spoke Yiddish, and his mother, Freda (Rosemaity), worked as a seamstress. Singer learned English swiftly and subsequently taught it to the rest of his family. Isadore was born with a prominent hemangioma birthmark under his right eye.
Singer studied physics at the University of Michigan, graduating in 1944 after just two-and-a-half years so that he could join the military. He was stationed in the US Army in the Philippines, where he was a radar officer. During the daytime, he operated a communications school for the Philippine Army. He undertook correspondence courses in mathematics at night in order to satisfy the prerequisites for relativity and quantum mechanics. Upon his return from military service, Singer studied mathematics for one year at the University of Chicago. Although he initially intended to go back to physics, his interest in math was piqued, and he continued with the subject, earning an M.S. in Mathematics in 1948 and a Ph.D. in Mathematics in 1950 under the supervision of Irving Segal.
Career
Singer held a postdoctoral fellowship as a CLE Moore instructor at the Massachusetts Institute of Technology in 1950. After appointments at the University of California, Los Angeles, Columbia University, and Princeton University, he returned to MIT as a professor in 1956 and was appointed as the Norbert Wiener Professor from 1970 to 1979. In 1979, he moved to the University of California, Berkeley as Miller Professor. He returned to MIT in 1983 as the first John D. MacArthur Professor, before being appointed as an Institute Professor in 1987.
Singer was chair of the Committee of Science & Public Policy of the United States National Academy of Sciences, a member of the White House Science Council (1982–88), and on the Governing Board of the United States National Research Council (1995–99). He was one of the founders of the independent non-profit Mathematical Sciences Research Institute, based in Berkeley, California.
Singer died on February 11, 2021, at his home in Boxborough, Massachusetts. He was 96.
Research
Partnering with British-Lebanese mathematician Michael Atiyah, Singer created a linkage between the fields of analysis, especially dif
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https://en.wikipedia.org/wiki/Financial%20engineering
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Financial engineering is a multidisciplinary field involving financial theory, methods of engineering, tools of mathematics and the practice of programming. It has also been defined as the application of technical methods, especially from mathematical finance and computational finance, in the practice of finance.
Financial engineering plays a key role in a bank's customer-driven derivatives business
— delivering bespoke OTC-contracts and "exotics", and implementing various structured products —
which encompasses quantitative modelling, quantitative programming and risk managing financial products in compliance with the regulations and Basel capital/liquidity requirements.
An older use of the term "financial engineering" that is less common today is aggressive restructuring of corporate balance sheets.
Mathematical finance is the application of mathematics to finance. Computational finance and mathematical finance are both subfields of financial engineering. Computational finance is a field in computer science and deals with the data and algorithms that arise in financial modeling.
Discipline
Financial engineering draws on tools from applied mathematics, computer science, statistics and economic theory.
In the broadest sense, anyone who uses technical tools in finance could be called a financial engineer, for example any computer programmer in a bank or any statistician in a government economic bureau. However, most practitioners restrict the term to someone educated in the full range of tools of modern finance and whose work is informed by financial theory. It is sometimes restricted even further, to cover only those originating new financial products and strategies.
Despite its name, financial engineering does not belong to any of the fields in traditional professional engineering even though many financial engineers have studied engineering beforehand and many universities offering a postgraduate degree in this field require applicants to have a background in engineering as well. In the United States, the Accreditation Board for Engineering and Technology (ABET) does not accredit financial engineering degrees. In the United States, financial engineering programs are accredited by the International Association of Quantitative Finance.
Quantitative analyst ("Quant") is a broad term that covers any person who uses math for practical purposes, including financial engineers. Quant is often taken to mean "financial quant", in which case it is similar to financial engineer. The difference is that it is possible to be a theoretical quant, or a quant in only one specialized niche in finance, while "financial engineer" usually implies a practitioner with broad expertise.
"Rocket scientist" (aerospace engineer) is an older term, first coined in the development of rockets in WWII (Wernher von Braun), and later, the NASA space program; it was adapted by the first generation of financial quants who arrived on Wall Street in the late 1970s and early
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https://en.wikipedia.org/wiki/Atiyah%E2%80%93Singer%20index%20theorem
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In differential geometry, the Atiyah–Singer index theorem, proved by Michael Atiyah and Isadore Singer (1963), states that for an elliptic differential operator on a compact manifold, the analytical index (related to the dimension of the space of solutions) is equal to the topological index (defined in terms of some topological data). It includes many other theorems, such as the Chern–Gauss–Bonnet theorem and Riemann–Roch theorem, as special cases, and has applications to theoretical physics.
History
The index problem for elliptic differential operators was posed by Israel Gel'fand. He noticed the homotopy invariance of the index, and asked for a formula for it by means of topological invariants. Some of the motivating examples included the Riemann–Roch theorem and its generalization the Hirzebruch–Riemann–Roch theorem, and the Hirzebruch signature theorem. Friedrich Hirzebruch and Armand Borel had proved the integrality of the  genus of a spin manifold, and Atiyah suggested that this integrality could be explained if it were the index of the Dirac operator (which was rediscovered by Atiyah and Singer in 1961).
The Atiyah–Singer theorem was announced in 1963. The proof sketched in this announcement was never published by them, though it appears in Palais's book. It appears also in the "Séminaire Cartan-Schwartz 1963/64" that was held in Paris simultaneously with the seminar led by Richard Palais at Princeton University. The last talk in Paris was by Atiyah on manifolds with boundary. Their first published proof replaced the cobordism theory of the first proof with K-theory, and they used this to give proofs of various generalizations in another sequence of papers.
1965: Sergey P. Novikov published his results on the topological invariance of the rational Pontryagin classes on smooth manifolds.
Robion Kirby and Laurent C. Siebenmann's results, combined with René Thom's paper proved the existence of rational Pontryagin classes on topological manifolds. The rational Pontryagin classes are essential ingredients of the index theorem on smooth and topological manifolds.
1969: Michael Atiyah defines abstract elliptic operators on arbitrary metric spaces. Abstract elliptic operators became protagonists in Kasparov's theory and Connes's noncommutative differential geometry.
1971: Isadore Singer proposes a comprehensive program for future extensions of index theory.
1972: Gennadi G. Kasparov publishes his work on the realization of K-homology by abstract elliptic operators.
1973: Atiyah, Raoul Bott, and Vijay Patodi gave a new proof of the index theorem using the heat equation, described in a paper by Melrose.
1977: Dennis Sullivan establishes his theorem on the existence and uniqueness of Lipschitz and quasiconformal structures on topological manifolds of dimension different from 4.
1983: Ezra Getzler motivated by ideas of Edward Witten and Luis Alvarez-Gaume, gave a short proof of the local index theorem for operators that are locally Dirac op
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https://en.wikipedia.org/wiki/Baby%20monster%20group
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In the area of modern algebra known as group theory, the baby monster group B (or, more simply, the baby monster) is a sporadic simple group of order
241313567211131719233147
= 4154781481226426191177580544000000
= 4,154,781,481,226,426,191,177,580,544,000,000
≈ 4.
B is one of the 26 sporadic groups and has the second highest order of these, with the highest order being that of the monster group. The double cover of the baby monster is the centralizer of an element of order 2 in the monster group. The outer automorphism group is trivial and the Schur multiplier has order 2.
History
The existence of this group was suggested by Bernd Fischer in unpublished work from the early 1970s during his investigation of {3,4}-transposition groups: groups generated by a class of transpositions such that the product of any two elements has order at most 4. He investigated its properties and computed its character table. The first construction of the baby monster was later realized as a permutation group on 13 571 955 000 points using a computer by Jeffrey Leon and Charles Sims. Robert Griess later found a computer-free construction using the fact that its double cover is contained in the monster group. The name "baby monster" was suggested by John Horton Conway.
Representations
In characteristic 0, the 4371-dimensional representation of the baby monster does not have a nontrivial invariant algebra structure analogous to the Griess algebra, but showed that it does have such an invariant algebra structure if it is reduced modulo 2.
The smallest faithful matrix representation of the Baby Monster is of size 4370 over the finite field of order 2.
constructed a vertex operator algebra acted on by the baby monster.
Generalized monstrous moonshine
Conway and Norton suggested in their 1979 paper that monstrous moonshine is not limited to the monster, but that similar phenomena may be found for other groups. Larissa Queen and others subsequently found that one can construct the expansions of many Hauptmoduln from simple combinations of dimensions of sporadic groups. For the Baby monster B or F2, the relevant McKay–Thompson series is where one can set the constant term .
and η(τ) is the Dedekind eta function.
Maximal subgroups
found the 30 conjugacy classes of maximal subgroups of B as follows:
2.2E6(2):2 This is the centralizer of an involution, and is the subgroup fixing a point of the smallest permutation representation on 13 571 955 000 points.
21+22.Co2
Fi23
29+16.S8(2)
Th
(22 × F4(2)):2
22+10+20.(M22:2 × S3)
[230].L5(2)
S3 × Fi22:2
[235].(S5 × L3(2))
HN:2
O8+(3):S4
31+8.21+6.U4(2).2
(32:D8 × U4(3).2.2).2
5:4 × HS:2
S4 × 2F4(2)
[311].(S4 × 2S4)
S5 × M22:2
(S6 × L3(4):2).2
53.L3(5)
51+4.21+4.A5.4
(S6 × S6).4
52:4S4 × S5
L2(49).23
L2(31)
M11
L3(3)
L2(17):2
L2(11):2
47:23
References
External links
MathWorld: Baby monster group
Atlas of Finite Group Representations: Baby Monster group
Sporadic groups
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https://en.wikipedia.org/wiki/Sporadic%20group
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In mathematics, a sporadic group is one of the 26 exceptional groups found in the classification of finite simple groups.
A simple group is a group G that does not have any normal subgroups except for the trivial group and G itself. The classification theorem states that the list of finite simple groups consists of 18 countably infinite families plus 26 exceptions that do not follow such a systematic pattern. These 26 exceptions are the sporadic groups. They are also known as the sporadic simple groups, or the sporadic finite groups. Because it is not strictly a group of Lie type, the Tits group is sometimes regarded as a sporadic group, in which case there would be 27 sporadic groups.
The monster group, or friendly giant, is the largest of the sporadic groups, and all but six of the other sporadic groups are subquotients of it.
Names
Five of the sporadic groups were discovered by Mathieu in the 1860s and the other 21 were found between 1965 and 1975. Several of these groups were predicted to exist before they were constructed. Most of the groups are named after the mathematician(s) who first predicted their existence. The full list is:
Mathieu groups M11 (M11), M12 (M12), M22 (M22), M23 (M23), M24 (M24)
Janko groups J1 (J1), J2 or HJ (J2), J3 or HJM (J3), J4 (J4)
Conway groups Co1 (Co1), Co2 (), Co3 (Co3)
Fischer groups Fi22 (Fi22), Fi23 (Fi23), Fi24′ or F3+ (Fi24)
Higman–Sims group HS
McLaughlin group McL
Held group He or F7+ or F7
Rudvalis group Ru
Suzuki group Suz or F3−
O'Nan group O'N (ON)
Harada–Norton group HN or F5+ or F5
Lyons group Ly
Thompson group Th or F3|3 or F3
Baby Monster group B or F2+ or F2
Fischer–Griess Monster group M or F1
Various constructions for these groups were first compiled in , including character tables, individual conjugacy classes and lists of maximal subgroup, as well as Schur multipliers and orders of their outer automorphisms. These are also listed online at , updated with their group presentations and semi-presentations. The degrees of minimal faithful representation or Brauer characters over fields of characteristic p ≥ 0 for all sporadic groups have also been calculated, and for some of their covering groups. These are detailed in .
An exception found in the classification of sporadic groups within finite simple groups is the Tits group T, that is sometimes also considered as being sporadic — it is almost but not strictly a group of Lie type — which is why in some sources the number of sporadic groups is given as 27, instead of 26. In some other sources, the Tits group is regarded as neither sporadic nor of Lie type. The Tits group is the of the infinite family of commutator groups ; thus by definition not sporadic. For these finite simple groups coincide with the groups of Lie type also known as Ree groups of type 2F4.
The earliest use of the term sporadic group may be where he comments about the Mathieu groups: "These apparently sporadic simple groups would probably repay a clo
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https://en.wikipedia.org/wiki/Bimonster%20group
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In mathematics, the bimonster is a group that is the wreath product of the monster group M with Z2:
The Bimonster is also a quotient of the Coxeter group corresponding to the Dynkin diagram Y555, a Y-shaped graph with 16 nodes:
John H. Conway conjectured that a presentation of the bimonster could be given by adding a certain extra relation to the presentation defined by the Y555 diagram; this was proved in 1990 by A. A. Ivanov a mathematician not the famous painter and Simon P. Norton.
See also
Triality - simple Lie group D4, Y111
Affine E_6 Y222
References
.
.
.
.
.
.
External links
(Note: incorrectly named here as [36,6,6])
Group theory
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https://en.wikipedia.org/wiki/Ethnomathematics
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In mathematics education, ethnomathematics is the study of the relationship between mathematics and culture. Often associated with "cultures without written expression", it may also be defined as "the mathematics which is practised among identifiable cultural groups". It refers to a broad cluster of ideas ranging from distinct numerical and mathematical systems to multicultural mathematics education. The goal of ethnomathematics is to contribute both to the understanding of culture and the understanding of mathematics, and mainly to lead to an appreciation of the connections between the two.
Development and meaning
The term "ethnomathematics" was introduced by the Brazilian educator and mathematician Ubiratan D'Ambrosio in 1977 during a presentation for the American Association for the Advancement of Science. Since D'Ambrosio put forth the term, people - D'Ambrosio included - have struggled with its meaning ("An etymological abuse leads me to use the words, respectively, ethno and mathema for their categories of analysis and tics from (from techne)".).
The following is a sampling of some of the definitions of ethnomathematics proposed between 1985 and 2006:
"The mathematics which is practiced among identifiable cultural groups such as national-tribe societies, labour groups, children of certain age brackets and professional classes".
"The mathematics implicit in each practice".
"The study of mathematical ideas of a non-literate culture".
"The codification which allows a cultural group to describe, manage and understand reality".
"Mathematics…is conceived as a cultural product which has developed as a result of various activities".
"The study and presentation of mathematical ideas of traditional peoples".
"Any form of cultural knowledge or social activity characteristic of a social group and/or cultural group that can be recognized by other groups such as Western anthropologists, but not necessarily by the group of origin, as mathematical knowledge or mathematical activity".
"The mathematics of cultural practice".
"The investigation of the traditions, practices and mathematical concepts of a subordinated social group".
"I have been using the word ethnomathematics as modes, styles, and techniques (tics) of explanation, of understanding, and of coping with the natural and cultural environment (mathema) in distinct cultural systems (ethnos)".
"What is the difference between ethnomathematics and the general practice of creating a mathematical model of a cultural phenomenon (e.g., the "mathematical anthropology" of Paul Kay [1971] and others)? The essential issue is the relation between intentionality and epistemological status. A single drop of water issuing from a watering can, for example, can be modeled mathematically, but we would not attribute knowledge of that mathematics to the average gardener. Estimating the increase in seeds required for an increased garden plot, on the other hand, would qualify".
"N.C. Ghosh included Ethnomathematics in
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https://en.wikipedia.org/wiki/Radical%20center
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The term radical center can refer to:
Radical centrism, a political movement
a mathematical construct: also called the power center (geometry)
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https://en.wikipedia.org/wiki/Modular%20group
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In mathematics, the modular group is the projective special linear group of matrices with integer coefficients and determinant 1. The matrices and are identified. The modular group acts on the upper-half of the complex plane by fractional linear transformations, and the name "modular group" comes from the relation to moduli spaces and not from modular arithmetic.
Definition
The modular group is the group of linear fractional transformations of the upper half of the complex plane, which have the form
where , , , are integers, and . The group operation is function composition.
This group of transformations is isomorphic to the projective special linear group , which is the quotient of the 2-dimensional special linear group over the integers by its center . In other words, consists of all matrices
where , , , are integers, , and pairs of matrices and are considered to be identical. The group operation is the usual multiplication of matrices.
Some authors define the modular group to be , and still others define the modular group to be the larger group .
Some mathematical relations require the consideration of the group of matrices with determinant plus or minus one. ( is a subgroup of this group.) Similarly, is the quotient group . A matrix with unit determinant is a symplectic matrix, and thus , the symplectic group of matrices.
Finding elements
To find an explicit matrix in , begin with two coprime integers , and solve the determinant equation(Notice the determinant equation forces to be coprime since otherwise there would be a factor such that , , hencewould have no integer solutions.) For example, if then the determinant equation readsthen taking and gives , henceis a matrix. Then, using the projection, these matrices define elements in .
Number-theoretic properties
The unit determinant of
implies that the fractions , , , are all irreducible, that is having no common factors (provided the denominators are non-zero, of course). More generally, if is an irreducible fraction, then
is also irreducible (again, provided the denominator be non-zero). Any pair of irreducible fractions can be connected in this way; that is, for any pair and of irreducible fractions, there exist elements
such that
Elements of the modular group provide a symmetry on the two-dimensional lattice. Let and be two complex numbers whose ratio is not real. Then the set of points
is a lattice of parallelograms on the plane. A different pair of vectors and will generate exactly the same lattice if and only if
for some matrix in . It is for this reason that doubly periodic functions, such as elliptic functions, possess a modular group symmetry.
The action of the modular group on the rational numbers can most easily be understood by envisioning a square grid, with grid point corresponding to the fraction (see Euclid's orchard). An irreducible fraction is one that is visible from the origin; the action of the modular group on a fra
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https://en.wikipedia.org/wiki/Domain%20theory
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Domain theory is a branch of mathematics that studies special kinds of partially ordered sets (posets) commonly called domains. Consequently, domain theory can be considered as a branch of order theory. The field has major applications in computer science, where it is used to specify denotational semantics, especially for functional programming languages. Domain theory formalizes the intuitive ideas of approximation and convergence in a very general way and is closely related to topology.
Motivation and intuition
The primary motivation for the study of domains, which was initiated by Dana Scott in the late 1960s, was the search for a denotational semantics of the lambda calculus. In this formalism, one considers "functions" specified by certain terms in the language. In a purely syntactic way, one can go from simple functions to functions that take other functions as their input arguments. Using again just the syntactic transformations available in this formalism, one can obtain so-called fixed-point combinators (the best-known of which is the Y combinator); these, by definition, have the property that f(Y(f)) = Y(f) for all functions f.
To formulate such a denotational semantics, one might first try to construct a model for the lambda calculus, in which a genuine (total) function is associated with each lambda term. Such a model would formalize a link between the lambda calculus as a purely syntactic system and the lambda calculus as a notational system for manipulating concrete mathematical functions. The combinator calculus is such a model. However, the elements of the combinator calculus are functions from functions to functions; in order for the elements of a model of the lambda calculus to be of arbitrary domain and range, they could not be true functions, only partial functions.
Scott got around this difficulty by formalizing a notion of "partial" or "incomplete" information to represent computations that have not yet returned a result. This was modeled by considering, for each domain of computation (e.g. the natural numbers), an additional element that represents an undefined output, i.e. the "result" of a computation that never ends. In addition, the domain of computation is equipped with an ordering relation, in which the "undefined result" is the least element.
The important step to finding a model for the lambda calculus is to consider only those functions (on such a partially ordered set) that are guaranteed to have least fixed points. The set of these functions, together with an appropriate ordering, is again a "domain" in the sense of the theory. But the restriction to a subset of all available functions has another great benefit: it is possible to obtain domains that contain their own function spaces, i.e. one gets functions that can be applied to themselves.
Beside these desirable properties, domain theory also allows for an appealing intuitive interpretation. As mentioned above, the domains of computation are always partia
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https://en.wikipedia.org/wiki/Kirkland%20Lake
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Kirkland Lake is a town and municipality in Timiskaming District of Northeastern Ontario. The 2016 population, according to Statistics Canada, was 7,981.
The community name was based on a nearby lake which in turn was named after Winnifred Kirkland, a secretary of the Ontario Department of Mines in Toronto. The lake was named by surveyor Louis Rorke in 1907. Miss Kirkland never visited the town, and the lake that bore her name did not exist as it dried because of mine tailings, but recently due to floodings of the mineshafts has come back up to about half of its initial deepness. The community comprises Kirkland Lake (Teck Township), as well as Swastika, Chaput Hughes, Bernhardt, and Morrisette Township.
Kirkland Lake was built on gold, but is also notable for its hockey players. As well, hockey broadcaster Foster Hewitt called Kirkland Lake "the town that made the NHL." The town celebrated this via Hockey Heritage North, which was renamed Heritage North.
Until January 1, 1972, the town was known as Township of Teck. A by-law was introduced, on July 20, 1971, to change the municipality's name to Town of Kirkland Lake, effective January 1, 1972.
History
Gold in the Kirkland Lake region was originally reported in the late 1800s by Chief Ignace Tonené of the Temagami First Nation. In fact, he staked a claim near the north arm of Larder Lake but stated it was jumped (stolen). No action was taken on his report. Chief Tonene Lake was named in his honour. Chief Ignace Tonené would go on to help form Beaverhouse First Nation.
Later, Tom Price discovered a boulder containing gold on a visit to the Kirkland Lake area in 1906.
In 1911, important claims were made along the Main Break. John Hunton staked claims on February 18, 1911, which were incorporated as the Hunton Gold Mines Ltd. in April 1914, eventually becoming part of the Amalgamated Kirkland. Stephen Orr filed claims on February 22, 1911, the basis for the Teck-Hughes Mine and the Orr Gold Mines Ltd, which was incorporated in June 1913. George Minaker staked claims on February 23, 1911, part of which he sold to (Sir) Harry Oakes in September 1912, becoming part of the Lake Shore Mine. John Reamsbottom filed claims on April 18, 1911, which became part of the Teck-Hughes Mine.
It was at Teck-Hughes mine that miners and engineers developed Teck cable for sturdy electrical transmission. That type of cable is now used on electrical projects around the world.
C.A. McKane staked claims on April 20, 1911, which became the Kirkland Lake Gold Mine. A. Maracle staked claims on June 5, 1911, which became part of the Townsite claims. Melville McDougall staked claims on June 27, 1911, which he transferred to Oakes on September 6, 1912, and became the part of the Lake Shore Mine. Jack Matchett staked a claim on July 7, 1911, later acquired by Oakes, which became part of the Townsite Mine. On July 10, 1911, Dave Elliott staked claims which became the Macassa Mine. "Swift" Burnside staked claims on July
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https://en.wikipedia.org/wiki/Successor%20function
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In mathematics, the successor function or successor operation sends a natural number to the next one. The successor function is denoted by S, so S(n) = n +1. For example, S(1) = 2 and S(2) = 3. The successor function is one of the basic components used to build a primitive recursive function.
Successor operations are also known as zeration in the context of a zeroth hyperoperation: H0(a, b) = 1 + b. In this context, the extension of zeration is addition, which is defined as repeated succession.
Overview
The successor function is part of the formal language used to state the Peano axioms, which formalise the structure of the natural numbers. In this formalisation, the successor function is a primitive operation on the natural numbers, in terms of which the standard natural numbers and addition is defined. For example, 1 is defined to be S(0), and addition on natural numbers is defined recursively by:
{|
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| m + 0 || = m,
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| m + S(n) || = S(m + n).
|}
This can be used to compute the addition of any two natural numbers. For example, 5 + 2 = 5 + S(1) = S(5 + 1) = S(5 + S(0)) = S(S(5 + 0)) = S(S(5)) = S(6) = 7.
Several constructions of the natural numbers within set theory have been proposed. For example, John von Neumann constructs the number 0 as the empty set {}, and the successor of n, S(n), as the set n ∪ {n}. The axiom of infinity then guarantees the existence of a set that contains 0 and is closed with respect to S. The smallest such set is denoted by N, and its members are called natural numbers.
The successor function is the level-0 foundation of the infinite Grzegorczyk hierarchy of hyperoperations, used to build addition, multiplication, exponentiation, tetration, etc. It was studied in 1986 in an investigation involving generalization of the pattern for hyperoperations.
It is also one of the primitive functions used in the characterization of computability by recursive functions.
See also
Successor ordinal
Successor cardinal
Increment and decrement operators
Sequence
References
Mathematical logic
Arithmetic
Logic in computer science
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https://en.wikipedia.org/wiki/Hopf%20algebra
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In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously an (unital associative) algebra and a (counital coassociative) coalgebra, with these structures' compatibility making it a bialgebra, and that moreover is equipped with an antiautomorphism satisfying a certain property. The representation theory of a Hopf algebra is particularly nice, since the existence of compatible comultiplication, counit, and antipode allows for the construction of tensor products of representations, trivial representations, and dual representations.
Hopf algebras occur naturally in algebraic topology, where they originated and are related to the H-space concept, in group scheme theory, in group theory (via the concept of a group ring), and in numerous other places, making them probably the most familiar type of bialgebra. Hopf algebras are also studied in their own right, with much work on specific classes of examples on the one hand and classification problems on the other. They have diverse applications ranging from condensed-matter physics and quantum field theory to string theory and LHC phenomenology.
Formal definition
Formally, a Hopf algebra is an (associative and coassociative) bialgebra H over a field K together with a K-linear map S: H → H (called the antipode) such that the following diagram commutes:
Here Δ is the comultiplication of the bialgebra, ∇ its multiplication, η its unit and ε its counit. In the sumless Sweedler notation, this property can also be expressed as
As for algebras, one can replace the underlying field K with a commutative ring R in the above definition.
The definition of Hopf algebra is self-dual (as reflected in the symmetry of the above diagram), so if one can define a dual of H (which is always possible if H is finite-dimensional), then it is automatically a Hopf algebra.
Structure constants
Fixing a basis for the underlying vector space, one may define the algebra in terms of structure constants for multiplication:
for co-multiplication:
and the antipode:
Associativity then requires that
while co-associativity requires that
The connecting axiom requires that
Properties of the antipode
The antipode S is sometimes required to have a K-linear inverse, which is automatic in the finite-dimensional case, or if H is commutative or cocommutative (or more generally quasitriangular).
In general, S is an antihomomorphism, so S2 is a homomorphism, which is therefore an automorphism if S was invertible (as may be required).
If S2 = idH, then the Hopf algebra is said to be involutive (and the underlying algebra with involution is a *-algebra). If H is finite-dimensional semisimple over a field of characteristic zero, commutative, or cocommutative, then it is involutive.
If a bialgebra B admits an antipode S, then S is unique ("a bialgebra admits at most 1 Hopf algebra structure"). Thus, the antipode does not pose any extra structure which we can choose: Being a Hopf algebra is a property of
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https://en.wikipedia.org/wiki/Lucas%E2%80%93Lehmer%20primality%20test
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In mathematics, the Lucas–Lehmer test (LLT) is a primality test for Mersenne numbers. The test was originally developed by Édouard Lucas in 1878 and subsequently proved by Derrick Henry Lehmer in 1930.
The test
The Lucas–Lehmer test works as follows. Let Mp = 2p − 1 be the Mersenne number to test with p an odd prime. The primality of p can be efficiently checked with a simple algorithm like trial division since p is exponentially smaller than Mp. Define a sequence for all i ≥ 0 by
The first few terms of this sequence are 4, 14, 194, 37634, ... .
Then Mp is prime if and only if
The number sp − 2 mod Mp is called the Lucas–Lehmer residue of p. (Some authors equivalently set s1 = 4 and test sp−1 mod Mp). In pseudocode, the test might be written as
// Determine if Mp = 2p − 1 is prime for p > 2
Lucas–Lehmer(p)
var s = 4
var M = 2p − 1
repeat p − 2 times:
s = ((s × s) − 2) mod M
if s == 0 return PRIME else return COMPOSITE
Performing the mod M at each iteration ensures that all intermediate results are at most p bits (otherwise the number of bits would double each iteration). The same strategy is used in modular exponentiation.
Alternate starting values
Starting values s0 other than 4 are possible, for instance 10, 52, and others . The Lucas-Lehmer residue calculated with these alternative starting values will still be zero if Mp is a Mersenne prime. However, the terms of the sequence will be different and a non-zero Lucas-Lehmer residue for non-prime Mp will have a different numerical value from the non-zero value calculated when s0 = 4.
It is also possible to use the starting value (2 mod Mp)(3 mod Mp)−1, usually denoted by 2/3 for short. This starting value equals (2p + 1) /3, the Wagstaff number with exponent p.
Starting values like 4, 10, and 2/3 are universal, that is, they are valid for all (or nearly all) p. There are infinitely many additional universal starting values. However, some other starting values are only valid for a subset of all possible p, for example s0 = 3 can be used if p = 3 (mod 4). This starting value was often used where suitable in the era of hand computation, including by Lucas in proving M127 prime.
The first few terms of the sequence are 3, 7, 47, ... .
Sign of penultimate term
If sp−2 = 0 mod Mp then the penultimate term is sp−3 = ± 2(p+1)/2 mod Mp. The sign of this penultimate term is called the Lehmer symbol ϵ(s0, p).
In 2000 S.Y. Gebre-Egziabher proved that for the starting value 2/3 and for p ≠ 5 the sign is:
That is, ϵ(2/3, p) = +1 iff p = 1 (mod 4) and p ≠ 5.
The same author also proved that the Lehmer symbols for starting values 4 and 10 when p is not 2 or 5 are related by:
That is, ϵ(4, p) × ϵ(10, p) = 1 iff p = 5 or 7 (mod 8) and p ≠ 2, 5.
OEIS sequence shows ϵ(4, p) for each Mersenne prime Mp.
Time complexity
In the algorithm as written above, there are two expensive operations during each iteration: the multiplication s × s, and the mod M operation. The mod
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https://en.wikipedia.org/wiki/Graph%20%28discrete%20mathematics%29
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In discrete mathematics, and more specifically in graph theory, a graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense "related". The objects correspond to mathematical abstractions called vertices (also called nodes or points) and each of the related pairs of vertices is called an edge (also called link or line). Typically, a graph is depicted in diagrammatic form as a set of dots or circles for the vertices, joined by lines or curves for the edges. Graphs are one of the objects of study in discrete mathematics.
The edges may be directed or undirected. For example, if the vertices represent people at a party, and there is an edge between two people if they shake hands, then this graph is undirected because any person A can shake hands with a person B only if B also shakes hands with A. In contrast, if an edge from a person A to a person B means that A owes money to B, then this graph is directed, because owing money is not necessarily reciprocated.
Graphs are the basic subject studied by graph theory. The word "graph" was first used in this sense by J. J. Sylvester in 1878 due to a direct relation between mathematics and chemical structure (what he called a chemico-graphical image).
Definitions
Definitions in graph theory vary. The following are some of the more basic ways of defining graphs and related mathematical structures.
Graph
A graph (sometimes called an undirected graph to distinguish it from a directed graph, or a simple graph to distinguish it from a multigraph) is a pair , where is a set whose elements are called vertices (singular: vertex), and is a set of paired vertices, whose elements are called edges (sometimes links or lines).
The vertices and of an edge are called the endpoints of the edge. The edge is said to join and and to be incident on and . A vertex may belong to no edge, in which case it is not joined to any other vertex.
A multigraph is a generalization that allows multiple edges to have the same pair of endpoints. In some texts, multigraphs are simply called graphs.
Sometimes, graphs are allowed to contain loops, which are edges that join a vertex to itself. To allow loops, the pairs of vertices in must be allowed to have the same node twice. Such generalized graphs are called graphs with loops or simply graphs when it is clear from the context that loops are allowed.
Generally, the set of vertices is supposed to be finite; this implies that the set of edges is also finite. Infinite graphs are sometimes considered, but are more often viewed as a special kind of binary relation, as most results on finite graphs do not extend to the infinite case, or need a rather different proof.
An empty graph is a graph that has an empty set of vertices (and thus an empty set of edges). The order of a graph is its number of vertices . The size of a graph is its number of edges . However, in some contexts, such as for expressing the computational complexity of
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https://en.wikipedia.org/wiki/Graph%20drawing
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Graph drawing is an area of mathematics and computer science combining methods from geometric graph theory and information visualization to derive two-dimensional depictions of graphs arising from applications such as social network analysis, cartography, linguistics, and bioinformatics.
A drawing of a graph or network diagram is a pictorial representation of the vertices and edges of a graph. This drawing should not be confused with the graph itself: very different layouts can correspond to the same graph. In the abstract, all that matters is which pairs of vertices are connected by edges. In the concrete, however, the arrangement of these vertices and edges within a drawing affects its understandability, usability, fabrication cost, and aesthetics. The problem gets worse if the graph changes over time by adding and deleting edges (dynamic graph drawing) and the goal is to preserve the user's mental map.
Graphical conventions
Graphs are frequently drawn as node–link diagrams in which the vertices are represented as disks, boxes, or textual labels and the edges are represented as line segments, polylines, or curves in the Euclidean plane. Node–link diagrams can be traced back to the 14th-16th century works of Pseudo-Lull which were published under the name of Ramon Llull, a 13th century polymath. Pseudo-Lull drew diagrams of this type for complete graphs in order to analyze all pairwise combinations among sets of metaphysical concepts.
In the case of directed graphs, arrowheads form a commonly used graphical convention to show their orientation; however, user studies have shown that other conventions such as tapering provide this information more effectively. Upward planar drawing uses the convention that every edge is oriented from a lower vertex to a higher vertex, making arrowheads unnecessary.
Alternative conventions to node–link diagrams include adjacency representations such as circle packings, in which vertices are represented by disjoint regions in the plane and edges are represented by adjacencies between regions; intersection representations in which vertices are represented by non-disjoint geometric objects and edges are represented by their intersections; visibility representations in which vertices are represented by regions in the plane and edges are represented by regions that have an unobstructed line of sight to each other; confluent drawings, in which edges are represented as smooth curves within mathematical train tracks; fabrics, in which nodes are represented as horizontal lines and edges as vertical lines; and visualizations of the adjacency matrix of the graph.
Quality measures
Many different quality measures have been defined for graph drawings, in an attempt to find objective means of evaluating their aesthetics and usability. In addition to guiding the choice between different layout methods for the same graph, some layout methods attempt to directly optimize these measures.
The crossing number of a drawing is t
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https://en.wikipedia.org/wiki/Extendible%20cardinal
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In mathematics, extendible cardinals are large cardinals introduced by , who was partly motivated by reflection principles. Intuitively, such a cardinal represents a point beyond which initial pieces of the universe of sets start to look similar, in the sense that each is elementarily embeddable into a later one.
Definition
For every ordinal η, a cardinal κ is called η-extendible if for some ordinal λ there is a nontrivial elementary embedding j of Vκ+η into Vλ, where κ is the critical point of j, and as usual Vα denotes the αth level of the von Neumann hierarchy. A cardinal κ is called an extendible cardinal if it is η-extendible for every nonzero ordinal η (Kanamori 2003).
Properties
For a cardinal , say that a logic is -compact if for every set of -sentences, if every subset of or cardinality has a model, then has a model. (The usual compactness theorem shows -compactness of first-order logic.) Let be the infinitary logic for second-order set theory, permitting infinitary conjunctions and disjunctions of length . is extendible iff is -compact.
Variants and relation to other cardinals
A cardinal κ is called η-C(n)-extendible if there is an elementary embedding j witnessing that κ is η-extendible (that is, j is elementary from Vκ+η to some Vλ with critical point κ) such that furthermore, Vj(κ) is Σn-correct in V. That is, for every Σn formula φ, φ holds in Vj(κ) if and only if φ holds in V. A cardinal κ is said to be C(n)-extendible if it is η-C(n)-extendible for every ordinal η. Every extendible cardinal is C(1)-extendible, but for n≥1, the least C(n)-extendible cardinal is never C(n+1)-extendible (Bagaria 2011).
Vopěnka's principle implies the existence of extendible cardinals; in fact, Vopěnka's principle (for definable classes) is equivalent to the existence of C(n)-extendible cardinals for all n (Bagaria 2011). All extendible cardinals are supercompact cardinals (Kanamori 2003).
See also
List of large cardinal properties
Reinhardt cardinal
References
Large cardinals
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https://en.wikipedia.org/wiki/Isoperimetric%20inequality
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In mathematics, the isoperimetric inequality is a geometric inequality involving the perimeter of a set and its volume. In -dimensional space the inequality lower bounds the surface area or perimeter of a set by its volume ,
,
where is a unit sphere. The equality holds only when is a sphere in .
On a plane, i.e. when , the isoperimetric inequality relates the square of the circumference of a closed curve and the area of a plane region it encloses. Isoperimetric literally means "having the same perimeter". Specifically in , the isoperimetric inequality states, for the length L of a closed curve and the area A of the planar region that it encloses, that
and that equality holds if and only if the curve is a circle.
The isoperimetric problem is to determine a plane figure of the largest possible area whose boundary has a specified length. The closely related Dido's problem asks for a region of the maximal area bounded by a straight line and a curvilinear arc whose endpoints belong to that line. It is named after Dido, the legendary founder and first queen of Carthage. The solution to the isoperimetric problem is given by a circle and was known already in Ancient Greece. However, the first mathematically rigorous proof of this fact was obtained only in the 19th century. Since then, many other proofs have been found.
The isoperimetric problem has been extended in multiple ways, for example, to curves on surfaces and to regions in higher-dimensional spaces. Perhaps the most familiar physical manifestation of the 3-dimensional isoperimetric inequality is the shape of a drop of water. Namely, a drop will typically assume a symmetric round shape. Since the amount of water in a drop is fixed, surface tension forces the drop into a shape which minimizes the surface area of the drop, namely a round sphere.
The isoperimetric problem in the plane
The classical isoperimetric problem dates back to antiquity. The problem can be stated as follows: Among all closed curves in the plane of fixed perimeter, which curve (if any) maximizes the area of its enclosed region? This question can be shown to be equivalent to the following problem: Among all closed curves in the plane enclosing a fixed area, which curve (if any) minimizes the perimeter?
This problem is conceptually related to the principle of least action in physics, in that it can be restated: what is the principle of action which encloses the greatest area, with the greatest economy of effort? The 15th-century philosopher and scientist, Cardinal Nicholas of Cusa, considered rotational action, the process by which a circle is generated, to be the most direct reflection, in the realm of sensory impressions, of the process by which the universe is created. German astronomer and astrologer Johannes Kepler invoked the isoperimetric principle in discussing the morphology of the solar system, in Mysterium Cosmographicum (The Sacred Mystery of the Cosmos, 1596).
Although the circle appears to be an obv
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https://en.wikipedia.org/wiki/Power%20center%20%28geometry%29
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In geometry, the power center of three circles, also called the radical center, is the intersection point of the three radical axes of the pairs of circles. If the radical center lies outside of all three circles, then it is the center of the unique circle (the radical circle) that intersects the three given circles orthogonally; the construction of this orthogonal circle corresponds to Monge's problem. This is a special case of the three conics theorem.
The three radical axes meet in a single point, the radical center, for the following reason. The radical axis of a pair of circles is defined as the set of points that have equal power with respect to both circles. For example, for every point on the radical axis of circles 1 and 2, the powers to each circle are equal: . Similarly, for every point on the radical axis of circles 2 and 3, the powers must be equal, . Therefore, at the intersection point of these two lines, all three powers must be equal, . Since this implies that , this point must also lie on the radical axis of circles 1 and 3. Hence, all three radical axes pass through the same point, the radical center.
The radical center has several applications in geometry. It has an important role in a solution to Apollonius' problem published by Joseph Diaz Gergonne in 1814. In the power diagram of a system of circles, all of the vertices of the diagram are located at radical centers of triples of circles. The Spieker center of a triangle is the radical center of its excircles. Several types of radical circles have been defined as well, such as the radical circle of the Lucas circles.
Notes
Further reading
External links
Radical Center at Cut-the-Knot
Radical Axis and Center at Cut-the-Knot
Elementary geometry
Geometric centers
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https://en.wikipedia.org/wiki/Covering%20lemma
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In the foundations of mathematics, a covering lemma is used to prove that the non-existence of certain large cardinals leads to the existence of a canonical inner model, called the core model, that is, in a sense, maximal and approximates the structure of the von Neumann universe V. A covering lemma asserts that under some particular anti-large cardinal assumption, the core model exists and is maximal in a sense that depends on the chosen large cardinal. The first such result was proved by Ronald Jensen for the constructible universe assuming 0# does not exist, which is now known as Jensen's covering theorem.
Example
For example, if there is no inner model for a measurable cardinal, then the Dodd–Jensen core model, KDJ is the core model and satisfies the covering property, that is for every uncountable set x of ordinals, there is y such that y ⊃ x, y has the same cardinality as x, and y ∈ KDJ. (If 0# does not exist, then KDJ = L.)
Versions
If the core model K exists (and has no Woodin cardinals), then
If K has no ω1-Erdős cardinals, then for a particular countable (in K) and definable in K sequence of functions from ordinals to ordinals, every set of ordinals closed under these functions is a union of a countable number of sets in K. If L=K, these are simply the primitive recursive functions.
If K has no measurable cardinals, then for every uncountable set x of ordinals, there is y ∈ K such that x ⊂ y and |x| = |y|.
If K has only one measurable cardinal κ, then for every uncountable set x of ordinals, there is y ∈ K[C] such that x ⊂ y and |x| = |y|. Here C is either empty or Prikry generic over K (so it has order type ω and is cofinal in κ) and unique except up to a finite initial segment.
If K has no inaccessible limit of measurable cardinals and no proper class of measurable cardinals, then there is a maximal and unique (except for a finite set of ordinals) set C (called a system of indiscernibles) for K such that for every sequence S in K of measure one sets consisting of one set for each measurable cardinal, C minus ∪S is finite. Note that every κ \ C is either finite or Prikry generic for K at κ except for members of C below a measurable cardinal below κ. For every uncountable set x of ordinals, there is y ∈ K[C] such that x ⊂ y and |x| = |y|.
For every uncountable set x of ordinals, there is a set C of indiscernibles for total extenders on K such that there is y ∈ K[C] and x ⊂ y and |x| = |y|.
K computes the successors of singular and weakly compact cardinals correctly (Weak Covering Property). Moreover, if |κ| > ω1, then cofinality((κ+)K) ≥ |κ|.
Extenders and indiscernibles
For core models without overlapping total extenders, the systems of indiscernibles are well understood. Although (if K has an inaccessible limit of measurable cardinals), the system may depend on the set to be covered, it is well-determined and unique in a weaker sense. One application of the covering is counting the number of (sequences of) indisce
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https://en.wikipedia.org/wiki/Reverse%20mathematics
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Reverse mathematics is a program in mathematical logic that seeks to determine which axioms are required to prove theorems of mathematics. Its defining method can briefly be described as "going backwards from the theorems to the axioms", in contrast to the ordinary mathematical practice of deriving theorems from axioms. It can be conceptualized as sculpting out necessary conditions from sufficient ones.
The reverse mathematics program was foreshadowed by results in set theory such as the classical theorem that the axiom of choice and Zorn's lemma are equivalent over ZF set theory. The goal of reverse mathematics, however, is to study possible axioms of ordinary theorems of mathematics rather than possible axioms for set theory.
Reverse mathematics is usually carried out using subsystems of second-order arithmetic, where many of its definitions and methods are inspired by previous work in constructive analysis and proof theory. The use of second-order arithmetic also allows many techniques from recursion theory to be employed; many results in reverse mathematics have corresponding results in computable analysis. In higher-order reverse mathematics, the focus is on subsystems of higher-order arithmetic, and the associated richer language.
The program was founded by and brought forward by Steve Simpson. A standard reference for the subject is , while an introduction for non-specialists is . An introduction to higher-order reverse mathematics, and also the founding paper, is .
General principles
In reverse mathematics, one starts with a framework language and a base theory—a core axiom system—that is too weak to prove most of the theorems one might be interested in, but still powerful enough to develop the definitions necessary to state these theorems. For example, to study the theorem “Every bounded sequence of real numbers has a supremum” it is necessary to use a base system that can speak of real numbers and sequences of real numbers.
For each theorem that can be stated in the base system but is not provable in the base system, the goal is to determine the particular axiom system (stronger than the base system) that is necessary to prove that theorem. To show that a system S is required to prove a theorem T, two proofs are required. The first proof shows T is provable from S; this is an ordinary mathematical proof along with a justification that it can be carried out in the system S. The second proof, known as a reversal, shows that T itself implies S; this proof is carried out in the base system. The reversal establishes that no axiom system S′ that extends the base system can be weaker than S while still proving T.
Use of second-order arithmetic
Most reverse mathematics research focuses on subsystems of second-order arithmetic. The body of research in reverse mathematics has established that weak subsystems of second-order arithmetic suffice to formalize almost all undergraduate-level mathematics. In second-order arithmetic, all o
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https://en.wikipedia.org/wiki/Finitely%20generated%20module
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In mathematics, a finitely generated module is a module that has a finite generating set. A finitely generated module over a ring R may also be called a finite R-module, finite over R, or a module of finite type.
Related concepts include finitely cogenerated modules, finitely presented modules, finitely related modules and coherent modules all of which are defined below. Over a Noetherian ring the concepts of finitely generated, finitely presented and coherent modules coincide.
A finitely generated module over a field is simply a finite-dimensional vector space, and a finitely generated module over the integers is simply a finitely generated abelian group.
Definition
The left R-module M is finitely generated if there exist a1, a2, ..., an in M such that for any x in M, there exist r1, r2, ..., rn in R with x = r1a1 + r2a2 + ... + rnan.
The set {a1, a2, ..., an} is referred to as a generating set of M in this case. A finite generating set need not be a basis, since it need not be linearly independent over R. What is true is: M is finitely generated if and only if there is a surjective R-linear map:
for some n (M is a quotient of a free module of finite rank).
If a set S generates a module that is finitely generated, then there is a finite generating set that is included in S, since only finitely many elements in S are needed to express any finite generating set, and these finitely many elements form a generating set. However, it may occur that S does not contain any finite generating set of minimal cardinality. For example the set of the prime numbers is a generating set of viewed as -module, and a generating set formed from prime numbers has at least two elements, while the singleton is also a generating set.
In the case where the module M is a vector space over a field R, and the generating set is linearly independent, n is well-defined and is referred to as the dimension of M (well-defined means that any linearly independent generating set has n elements: this is the dimension theorem for vector spaces).
Any module is the union of the directed set of its finitely generated submodules.
A module M is finitely generated if and only if any increasing chain Mi of submodules with union M stabilizes: i.e., there is some i such that Mi = M. This fact with Zorn's lemma implies that every nonzero finitely generated module admits maximal submodules. If any increasing chain of submodules stabilizes (i.e., any submodule is finitely generated), then the module M is called a Noetherian module.
Examples
If a module is generated by one element, it is called a cyclic module.
Let R be an integral domain with K its field of fractions. Then every finitely generated R-submodule I of K is a fractional ideal: that is, there is some nonzero r in R such that rI is contained in R. Indeed, one can take r to be the product of the denominators of the generators of I. If R is Noetherian, then every fractional ideal arises in this way.
Finitely generated modu
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https://en.wikipedia.org/wiki/Free%20module
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In mathematics, a free module is a module that has a basis, that is, a generating set consisting of linearly independent elements. Every vector space is a free module, but, if the ring of the coefficients is not a division ring (not a field in the commutative case), then there exist non-free modules.
Given any set and ring , there is a free -module with basis , which is called the free module on or module of formal -linear combinations of the elements of .
A free abelian group is precisely a free module over the ring of integers.
Definition
For a ring and an -module , the set is a basis for if:
is a generating set for ; that is to say, every element of is a finite sum of elements of multiplied by coefficients in ; and
is linearly independent if for every of distinct elements, implies that (where is the zero element of and is the zero element of ).
A free module is a module with a basis.
An immediate consequence of the second half of the definition is that the coefficients in the first half are unique for each element of M.
If has invariant basis number, then by definition any two bases have the same cardinality. For example, nonzero commutative rings have invariant basis number. The cardinality of any (and therefore every) basis is called the rank of the free module . If this cardinality is finite, the free module is said to be free of finite rank, or free of rank if the rank is known to be .
Examples
Let R be a ring.
R is a free module of rank one over itself (either as a left or right module); any unit element is a basis.
More generally, If R is commutative, a nonzero ideal I of R is free if and only if it is a principal ideal generated by a nonzerodivisor, with a generator being a basis.
Over a principal ideal domain (e.g., ), a submodule of a free module is free.
If R is commutative, the polynomial ring in indeterminate X is a free module with a possible basis 1, X, X2, ....
Let be a polynomial ring over a commutative ring A, f a monic polynomial of degree d there, and the image of t in B. Then B contains A as a subring and is free as an A-module with a basis .
For any non-negative integer n, , the cartesian product of n copies of R as a left R-module, is free. If R has invariant basis number, then its rank is n.
A direct sum of free modules is free, while an infinite cartesian product of free modules is generally not free (cf. the Baer–Specker group).
A finitely generated module over a commutative local ring is free if and only if it is faithfully flat. Also, Kaplansky's theorem states a projective module over a (possibly non-commutative) local ring is free.
Sometimes, whether a module is free or not is undecidable in the set-theoretic sense. A famous example is the Whitehead problem, which asks whether a Whitehead group is free or not. As it turns out, the problem is independent of ZFC.
Formal linear combinations
Given a set and ring , there is a free -module that has as a basis: namely, the direct sum o
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https://en.wikipedia.org/wiki/Mathematics%20education
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In contemporary education, mathematics education—known in Europe as the didactics or pedagogy of mathematics—is the practice of teaching, learning, and carrying out scholarly research into the transfer of mathematical knowledge.
Although research into mathematics education is primarily concerned with the tools, methods, and approaches that facilitate practice or the study of practice, it also covers an extensive field of study encompassing a variety of different concepts, theories and methods. National and international organisations regularly hold conferences and publish literature in order to improve mathematics education.
History
Ancient
Elementary mathematics were a core part of education in many ancient civilisations, including ancient Egypt, ancient Babylonia, ancient Greece, ancient Rome, and Vedic India. In most cases, formal education was only available to male children with sufficiently high status, wealth, or caste. The oldest known mathematics textbook is the Rhind papyrus, dated from circa 1650 BCE.
Pythagorean theorem
Historians of Mesopotamia have confirmed that use of the Pythagorean rule dates back to the Old Babylonian Empire (20th–16th centuries BC) and that it was being taught in scribal schools over one thousand years before the birth of Pythagoras.
In Plato's division of the liberal arts into the trivium and the quadrivium, the quadrivium included the mathematical fields of arithmetic and geometry. This structure was continued in the structure of classical education that was developed in medieval Europe. The teaching of geometry was almost universally based on Euclid's Elements. Apprentices to trades such as masons, merchants, and moneylenders could expect to learn such practical mathematics as was relevant to their profession.
Medieval and early modern
In the Middle Ages, the academic status of mathematics declined, because it was strongly associated with trade and commerce, and considered somewhat un-Christian. Although it continued to be taught in European universities, it was seen as subservient to the study of natural, metaphysical, and moral philosophy. The first modern arithmetic curriculum (starting with addition, then subtraction, multiplication, and division) arose at reckoning schools in Italy in the 1300s. Spreading along trade routes, these methods were designed to be used in commerce. They contrasted with Platonic math taught at universities, which was more philosophical and concerned numbers as concepts rather than calculating methods. They also contrasted with mathematical methods learned by artisan apprentices, which were specific to the tasks and tools at hand. For example, the division of a board into thirds can be accomplished with a piece of string, instead of measuring the length and using the arithmetic operation of division.
The first mathematics textbooks to be written in English and French were published by Robert Recorde, beginning with The Grounde of Artes in 1543. However, there are ma
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https://en.wikipedia.org/wiki/Brahmagupta%E2%80%93Fibonacci%20identity
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In algebra, the Brahmagupta–Fibonacci identity expresses the product of two sums of two squares as a sum of two squares in two different ways. Hence the set of all sums of two squares is closed under multiplication. Specifically, the identity says
For example,
The identity is also known as the Diophantus identity, as it was first proved by Diophantus of Alexandria. It is a special case of Euler's four-square identity, and also of Lagrange's identity.
Brahmagupta proved and used a more general Brahmagupta identity, stating
This shows that, for any fixed A, the set of all numbers of the form x2 + Ay2 is closed under multiplication.
These identities hold for all integers, as well as all rational numbers; more generally, they are true in any commutative ring. All four forms of the identity can be verified by expanding each side of the equation. Also, (2) can be obtained from (1), or (1) from (2), by changing b to −b, and likewise with (3) and (4).
History
The identity first appeared in Diophantus' Arithmetica (III, 19), of the third century A.D.
It was rediscovered by Brahmagupta (598–668), an Indian mathematician and astronomer, who generalized it to Brahmagupta's identity, and used it in his study of what is now called Pell's equation. His Brahmasphutasiddhanta was translated from Sanskrit into Arabic by Mohammad al-Fazari, and was subsequently translated into Latin in 1126. The identity was introduced in western Europe in 1225 by Fibonacci, in The Book of Squares, and, therefore, the identity has been often attributed to him.
Related identities
Analogous identities are Euler's four-square related to quaternions, and Degen's eight-square derived from the octonions which has connections to Bott periodicity. There is also Pfister's sixteen-square identity, though it is no longer bilinear.
These identities are strongly related with Hurwitz's classification of composition algebras.
The Brahmagupta–Fibonacci identity is a special form of Lagrange's identity, which is itself a special form of Binet–Cauchy identity, in turn a special form of the Cauchy–Binet formula for matrix determinants.
Multiplication of complex numbers
If a, b, c, and d are real numbers, the Brahmagupta–Fibonacci identity is equivalent to the multiplicativity property for absolute values of complex numbers:
This can be seen as follows: expanding the right side and squaring both sides, the multiplication property is equivalent to
and by the definition of absolute value this is in turn equivalent to
An equivalent calculation in the case that the variables a, b, c, and d are rational numbers shows the identity may be interpreted as the statement that the norm in the field Q(i) is multiplicative: the norm is given by
and the multiplicativity calculation is the same as the preceding one.
Application to Pell's equation
In its original context, Brahmagupta applied his discovery of this identity to the solution of Pell's equation x2 − Ay2 = 1. Using the identity in
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https://en.wikipedia.org/wiki/Ramsey%20cardinal
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In mathematics, a Ramsey cardinal is a certain kind of large cardinal number introduced by and named after Frank P. Ramsey, whose theorem establishes that ω enjoys a certain property that Ramsey cardinals generalize to the uncountable case.
Let [κ]<ω denote the set of all finite subsets of κ. A cardinal number κ is called Ramsey if, for every function
f: [κ]<ω → {0, 1}
there is a set A of cardinality κ that is homogeneous for f. That is, for every n, the function f is constant on the subsets of cardinality n from A. A cardinal κ is called ineffably Ramsey if A can be chosen to be a stationary subset of κ. A cardinal κ is called virtually Ramsey if for every function
f: [κ]<ω → {0, 1}
there is C, a closed and unbounded subset of κ, so that for every λ in C of uncountable cofinality, there is an unbounded subset of λ that is homogenous for f; slightly weaker is the notion of almost Ramsey where homogenous sets for f are required of order type λ, for every λ < κ.
The existence of any of these kinds of Ramsey cardinal is sufficient to prove the existence of 0#, or indeed that every set with rank less than κ has a sharp.
Every measurable cardinal is a Ramsey cardinal, and every Ramsey cardinal is a Rowbottom cardinal.
A property intermediate in strength between Ramseyness and measurability is existence of a κ-complete normal non-principal ideal I on κ such that for every and for every function
f: [κ]<ω → {0, 1}
there is a set B ⊂ A not in I that is homogeneous for f. This is strictly stronger than κ being ineffably Ramsey.
The existence of a Ramsey cardinal implies the existence of 0# and this in turn implies the falsity of the Axiom of Constructibility of Kurt Gödel.
References
Large cardinals
Ramsey theory
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https://en.wikipedia.org/wiki/Erd%C5%91s%20cardinal
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In mathematics, an Erdős cardinal, also called a partition cardinal is a certain kind of large cardinal number introduced by .
A cardinal κ is called α-Erdős if for every function there is a set of order type that is homogeneous for . In the notation of the partition calculus, κ is α-Erdős if
.
The existence of zero sharp implies that the constructible universe satisfies "for every countable ordinal , there is an -Erdős cardinal". In fact, for every indiscernible satisfies "for every ordinal , there is an -Erdős cardinal in " (the Levy collapse to make countable).
However, the existence of an -Erdős cardinal implies existence of zero sharp. If is the satisfaction relation for (using ordinal parameters), then the existence of zero sharp is equivalent to there being an -Erdős ordinal with respect to . Thus, the existence of zero sharp implies that the axiom of constructibility is false.
If κ is -Erdős, then it is -Erdős in every transitive model satisfying " is countable."
See also
List of large cardinal properties
References
Large cardinals
Cardinal
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https://en.wikipedia.org/wiki/Subtle%20cardinal
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In mathematics, subtle cardinals and ethereal cardinals are closely related kinds of large cardinal number.
A cardinal κ is called subtle if for every closed and unbounded C ⊂ κ and for every sequence A of length κ for which element number δ (for an arbitrary δ), Aδ ⊂ δ, there exist α, β, belonging to C, with α < β, such that Aα = Aβ ∩ α.
A cardinal κ is called ethereal if for every closed and unbounded C ⊂ κ and for every sequence A of length κ for which element number δ (for an arbitrary δ), Aδ ⊂ δ and Aδ has the same cardinal as δ, there exist α, β, belonging to C, with α < β, such that card(α) = card(Aβ ∩ Aα).
Subtle cardinals were introduced by . Ethereal cardinals were introduced by . Any subtle cardinal is ethereal, and any strongly inaccessible ethereal cardinal is subtle.
Theorem
There is a subtle cardinal ≤ κ if and only if every transitive set S of cardinality κ contains x and y such that x is a proper subset of y and x ≠ Ø and x ≠ {Ø}. An infinite ordinal κ is subtle if and only if for every λ < κ, every transitive set S of cardinality κ includes a chain (under inclusion) of order type λ.
See also
List of large cardinal properties
References
Large cardinals
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https://en.wikipedia.org/wiki/Alan%20Sokal
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Alan David Sokal (; born January 24, 1955) is an American professor of mathematics at University College London and professor emeritus of physics at New York University. He works in statistical mechanics and combinatorics. He is a critic of postmodernism, and caused the Sokal affair in 1996 when his deliberately nonsensical paper was published by Duke University Press's Social Text. He also co-authored a paper criticizing the critical positivity ratio concept in positive psychology.
Academic career
Sokal received his BA from Harvard College in 1976 and his PhD from Princeton University in 1981. He was advised by Arthur Wightman. In the summers of 1986, 1987, and 1988, Sokal taught mathematics at the National Autonomous University of Nicaragua, when the Sandinistas were heading the elected government.
Research interests
Sokal's research lies in mathematical physics and combinatorics. In particular, he studies the interplay between these fields based on questions arising in statistical mechanics and quantum field theory. This includes work on the chromatic polynomial and the Tutte polynomial, which appear both in algebraic graph theory and in the study of phase transitions in statistical mechanics. His interests include computational physics and algorithms, such as Markov chain Monte Carlo algorithms for problems in statistical physics. He also co-authored a book on quantum triviality.
In 2013, Sokal co-authored a paper with Nicholas Brown and Harris Friedman, rejecting the Losada Line, a concept popular in positive psychology. Named after its proposer, Marcial Losada, it refers to a critical range for an individual's ratio of positive to negative emotions, outside of which the individual will tend to have poorer life and occupational outcomes. This concept of a critical positivity ratio was highly cited and popularised by psychologists such as Barbara Fredrickson. The trio's paper, published in American Psychologist, contended that the ratio was based on faulty mathematical reasoning and therefore invalid.
Sokal affair
In 1996, Sokal was curious to see whether the then-non-peer-reviewed postmodern cultural studies journal Social Text (published by Duke University Press) would publish a submission which "flattered the editors' ideological preconceptions". Sokal submitted a grand-sounding but completely nonsensical paper titled "Transgressing the Boundaries: Toward a Transformative Hermeneutics of Quantum Gravity."
After holding the article back from earlier issues because of Sokal's refusal to consider revisions, the staff published it in the "Science Wars" issue as a relevant contribution. Soon thereafter, Sokal then revealed that the article was a hoax in the journal Lingua Franca, arguing that the left and social science would be better served by intellectual underpinnings based on reason. The affair was front-page news in The New York Times on May 18, 1996. Sokal responded to leftist and postmodernist criticism of the deception by asser
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https://en.wikipedia.org/wiki/Urysohn%20and%20completely%20Hausdorff%20spaces
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In topology, a discipline within mathematics, an Urysohn space, or T2½ space, is a topological space in which any two distinct points can be separated by closed neighborhoods. A completely Hausdorff space, or functionally Hausdorff space, is a topological space in which any two distinct points can be separated by a continuous function. These conditions are separation axioms that are somewhat stronger than the more familiar Hausdorff axiom T2.
Definitions
Suppose that X is a topological space. Let x and y be points in X.
We say that x and y can be separated by closed neighborhoods if there exists a closed neighborhood U of x and a closed neighborhood V of y such that U and V are disjoint (U ∩ V = ∅). (Note that a "closed neighborhood of x" is a closed set that contains an open set containing x.)
We say that x and y can be separated by a function if there exists a continuous function f : X → [0,1] (the unit interval) with f(x) = 0 and f(y) = 1.
A Urysohn space, also called a T2½ space, is a space in which any two distinct points can be separated by closed neighborhoods.
A completely Hausdorff space, or functionally Hausdorff space, is a space in which any two distinct points can be separated by a continuous function.
Naming conventions
The study of separation axioms is notorious for conflicts with naming conventions used. The definitions used in this article are those given by Willard (1970) and are the more modern definitions. Steen and Seebach (1970) and various other authors reverse the definition of completely Hausdorff spaces and Urysohn spaces. Readers of textbooks in topology must be sure to check the definitions used by the author. See History of the separation axioms for more on this issue.
Relation to other separation axioms
Any two points which can be separated by a function can be separated by closed neighborhoods. If they can be separated by closed neighborhoods then clearly they can be separated by neighborhoods. It follows that every completely Hausdorff space is Urysohn and every Urysohn space is Hausdorff.
One can also show that every regular Hausdorff space is Urysohn and every Tychonoff space (=completely regular Hausdorff space) is completely Hausdorff. In summary we have the following implications:
One can find counterexamples showing that none of these implications reverse.
Examples
The cocountable extension topology is the topology on the real line generated by the union of the usual Euclidean topology and the cocountable topology. Sets are open in this topology if and only if they are of the form U \ A where U is open in the Euclidean topology and A is countable. This space is completely Hausdorff and Urysohn, but not regular (and thus not Tychonoff).
There exist spaces which are Hausdorff but not Urysohn, and spaces which are Urysohn but not completely Hausdorff or regular Hausdorff. Examples are non trivial; for details see Steen and Seebach.
Notes
References
Stephen Willard, General Topology, Addison-We
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https://en.wikipedia.org/wiki/Fischer%20group
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In the area of modern algebra known as group theory, the Fischer groups are the three sporadic simple groups Fi22, Fi23 and Fi24 introduced by .
3-transposition groups
The Fischer groups are named after Bernd Fischer who discovered them while investigating 3-transposition groups.
These are groups G with the following properties:
G is generated by a conjugacy class of elements of order 2, called 'Fischer transpositions' or 3-transpositions.
The product of any two distinct transpositions has order 2 or 3.
The typical example of a 3-transposition group is a symmetric group,
where the Fischer transpositions are genuinely transpositions. The symmetric group Sn can be generated by transpositions: (12), (23), ..., .
Fischer was able to classify 3-transposition groups that satisfy certain extra technical conditions. The groups he found fell mostly into several infinite classes (besides symmetric groups: certain classes of symplectic, unitary, and orthogonal groups), but he also found 3 very large new groups. These groups are usually referred to as Fi22, Fi23 and Fi24. The first two of these are simple groups, and the third contains the simple group Fi24′ of index 2.
A starting point for the Fischer groups is the unitary group PSU6(2), which could be thought of as a group Fi21 in the series of Fischer groups, of order . Actually it is the double cover 2.PSU6(2) that becomes a subgroup of the new group. This is the stabilizer of one vertex in a graph of 3510 (= 2⋅33⋅5⋅13). These vertices are identified as conjugate 3-transpositions in the symmetry group Fi22 of the graph.
The Fischer groups are named by analogy with the large Mathieu groups. In Fi22 a maximal set of 3-transpositions all commuting with one another has size 22 and is called a basic set. There are 1024 3-transpositions, called anabasic that do not commute with any in the particular basic set. Any one of other 2464, called hexadic, commutes with 6 basic ones. The sets of 6 form an S(3,6,22) Steiner system, whose symmetry group is M22. A basic set generates an abelian group of order 210, which extends in Fi22 to a subgroup 210:M22.
The next Fischer group comes by regarding 2.Fi22 as a one-point stabilizer for a graph of 31671 (= 34⋅17⋅23) vertices, and treating these vertices as the 3-transpositions in a group Fi23. The 3-transpositions come in basic sets of 23, 7 of which commute with a given outside 3-transposition.
Next one takes Fi23 and treats it as a one-point stabilizer for a graph of 306936 (= 23⋅33⋅72⋅29) vertices to make a group Fi24. The 3-transpositions come in basic sets of 24, eight of which commute with a given outside 3-transposition. The group Fi24 is not simple, but its derived subgroup has index 2 and is a sporadic simple group.
Notation
There is no uniformly accepted notation for these groups. Some authors use F in place of Fi (F22, for example).
Fischer's notation for them was M(22), M(23) and M(24)′, which emphasised their close relationship with the three
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https://en.wikipedia.org/wiki/Decagon
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In geometry, a decagon (from the Greek δέκα déka and γωνία gonía, "ten angles") is a ten-sided polygon or 10-gon. The total sum of the interior angles of a simple decagon is 1440°.
Regular decagon
A regular decagon has all sides of equal length and each internal angle will always be equal to 144°. Its Schläfli symbol is {10} and can also be constructed as a truncated pentagon, t{5}, a quasiregular decagon alternating two types of edges.
Side length
The picture shows a regular decagon with side length and radius of the circumscribed circle.
The triangle has two equally long legs with length and a base with length
The circle around with radius intersects in a point (not designated in the picture).
Now the triangle is a isosceles triangle with vertex and with base angles .
Therefore . So and hence is also a isosceles triangle with vertex . The length of its legs is , so the length of is .
The isosceles triangles and have equal angles of 36° at the vertex, and so they are similar, hence:
Multiplication with the denominators leads to the quadratic equation:
This equation for the side length has one positive solution:
So the regular decagon can be constructed with ruler and compass.
Further conclusions
and the base height of (i.e. the length of ) is and the triangle has the area: .
Area
The area of a regular decagon of side length a is given by:
In terms of the apothem r (see also inscribed figure), the area is:
In terms of the circumradius R, the area is:
An alternative formula is where d is the distance between parallel sides, or the height when the decagon stands on one side as base, or the diameter of the decagon's inscribed circle.
By simple trigonometry,
and it can be written algebraically as
Sides
A regular decagon has 10 sides and is equilateral. It has 35 diagonals
Construction
As 10 = 2 × 5, a power of two times a Fermat prime, it follows that a regular decagon is constructible using compass and straightedge, or by an edge-bisection of a regular pentagon.
An alternative (but similar) method is as follows:
Construct a pentagon in a circle by one of the methods shown in constructing a pentagon.
Extend a line from each vertex of the pentagon through the center of the circle to the opposite side of that same circle. Where each line cuts the circle is a vertex of the decagon. In other words, the image of a regular pentagon under a point reflection with respect of its center is a concentric congruent pentagon, and the two pentagons have in total the vertices of a concentric regular decagon.
The five corners of the pentagon constitute alternate corners of the decagon. Join these points to the adjacent new points to form the decagon.
Nonconvex regular decagon
The length ratio of two inequal edges of a golden triangle is the golden ratio, denoted by , or its multiplicative inverse:
So we can get the properties of a regular decagonal star, through a tiling by golden triangles that fills this star pol
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https://en.wikipedia.org/wiki/Sabi%20%28Korea%29
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|
https://en.wikipedia.org/wiki/Pick%27s%20theorem
|
In geometry, Pick's theorem provides a formula for the area of a simple polygon with integer vertex coordinates, in terms of the number of integer points within it and on its boundary. The result was first described by Georg Alexander Pick in 1899. It was popularized in English by Hugo Steinhaus in the 1950 edition of his book Mathematical Snapshots. It has multiple proofs, and can be generalized to formulas for certain kinds of non-simple polygons.
Formula
Suppose that a polygon has integer coordinates for all of its vertices. Let be the number of integer points interior to the polygon, and let be the number of integer points on its boundary (including both vertices and points along the sides). Then the area of this polygon is:
The example shown has interior points and boundary points, so its area is square units.
Proofs
Via Euler's formula
One proof of this theorem involves subdividing the polygon into triangles with three integer vertices and no other integer points. One can then prove that each subdivided triangle has area exactly . Therefore, the area of the whole polygon equals half the number of triangles in the subdivision. After relating area to the number of triangles in this way, the proof concludes by using Euler's polyhedral formula to relate the number of triangles to the number of grid points in the polygon.
The first part of this proof shows that a triangle with three integer vertices and no other integer points has area exactly , as Pick's formula states. The proof uses the fact that all triangles tile the plane, with adjacent triangles rotated by 180° from each other around their shared edge. For tilings by a triangle with three integer vertices and no other integer points, each point of the integer grid is a vertex of six tiles. Because the number of triangles per grid point (six) is twice the number of grid points per triangle (three), the triangles are twice as dense in the plane as the grid points. Any scaled region of the plane contains twice as many triangles (in the limit as the scale factor goes to infinity) as the number of grid points it contains. Therefore, each triangle has area , as needed for the proof. A different proof that these triangles have area is based on the use of Minkowski's theorem on lattice points in symmetric convex sets.
This already proves Pick's formula for a polygon that is one of these special triangles. Any other polygon can be subdivided into special triangles: add non-crossing line segments within the polygon between pairs of grid points until no more line segments can be added. The only polygons that cannot be subdivided in this way are the special triangles considered above; therefore, only special triangles can appear in the resulting subdivision. Because each special triangle has area , a polygon of area will be subdivided into special triangles.
The subdivision of the polygon into triangles forms a planar graph, and Euler's formula gives an equation that applies to the
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https://en.wikipedia.org/wiki/Annihilator%20%28ring%20theory%29
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In mathematics, the annihilator of a subset of a module over a ring is the ideal formed by the elements of the ring that give always zero when multiplied by each element of .
Over an integral domain, a module that has a nonzero annihilator is a torsion module, and a finitely generated torsion module has a nonzero annihilator.
The above definition applies also in the case noncommutative rings, where the left annihilator of a left module is a left ideal, and the right-annihilator, of a right module is a right ideal.
Definitions
Let R be a ring, and let M be a left R-module. Choose a non-empty subset S of M. The annihilator of S, denoted AnnR(S), is the set of all elements r in R such that, for all s in S, . In set notation,
for all
It is the set of all elements of R that "annihilate" S (the elements for which S is a torsion set). Subsets of right modules may be used as well, after the modification of "" in the definition.
The annihilator of a single element x is usually written AnnR(x) instead of AnnR({x}). If the ring R can be understood from the context, the subscript R can be omitted.
Since R is a module over itself, S may be taken to be a subset of R itself, and since R is both a right and a left R module, the notation must be modified slightly to indicate the left or right side. Usually and or some similar subscript scheme are used to distinguish the left and right annihilators, if necessary.
If M is an R-module and , then M is called a faithful module.
Properties
If S is a subset of a left R module M, then Ann(S) is a left ideal of R.
If S is a submodule of M, then AnnR(S) is even a two-sided ideal: (ac)s = a(cs) = 0, since cs is another element of S.
If S is a subset of M and N is the submodule of M generated by S, then in general AnnR(N) is a subset of AnnR(S), but they are not necessarily equal. If R is commutative, then the equality holds.
M may be also viewed as an R/AnnR(M)-module using the action . Incidentally, it is not always possible to make an R module into an R/I module this way, but if the ideal I is a subset of the annihilator of M, then this action is well-defined. Considered as an R/AnnR(M)-module, M is automatically a faithful module.
For commutative rings
Throughout this section, let be a commutative ring and a finitely generated (for short, finite) -module.
Relation to support
Recall that the support of a module is defined as
Then, when the module is finitely generated, there is the relation
,
where is the set of prime ideals containing the subset.
Short exact sequences
Given a short exact sequence of modules,
the support property
together with the relation with the annihilator implies
More specifically, we have the relations
If the sequence splits then the inequality on the left is always an equality. In fact this holds for arbitrary direct sums of modules, as
Quotient modules and annihilators
Given an ideal and let be a finite module, then there is the relation
on the support.
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https://en.wikipedia.org/wiki/Solid%20of%20revolution
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In geometry, a solid of revolution is a solid figure obtained by rotating a plane figure around some straight line (the axis of revolution), which may not intersect the generatrix (except at its boundary). The surface created by this revolution and which bounds the solid is the surface of revolution.
Assuming that the curve does not cross the axis, the solid's volume is equal to the length of the circle described by the figure's centroid multiplied by the figure's area (Pappus's second centroid theorem).
A representative disc is a three-dimensional volume element of a solid of revolution. The element is created by rotating a line segment (of length ) around some axis (located units away), so that a cylindrical volume of units is enclosed.
Finding the volume
Two common methods for finding the volume of a solid of revolution are the disc method and the shell method of integration. To apply these methods, it is easiest to draw the graph in question; identify the area that is to be revolved about the axis of revolution; determine the volume of either a disc-shaped slice of the solid, with thickness , or a cylindrical shell of width ; and then find the limiting sum of these volumes as approaches 0, a value which may be found by evaluating a suitable integral. A more rigorous justification can be given by attempting to evaluate a triple integral in cylindrical coordinates with two different orders of integration.
Disc method
The disc method is used when the slice that was drawn is perpendicular to the axis of revolution; i.e. when integrating parallel to the axis of revolution.
The volume of the solid formed by rotating the area between the curves of and and the lines and about the -axis is given by
If (e.g. revolving an area between the curve and the -axis), this reduces to:
The method can be visualized by considering a thin horizontal rectangle at between on top and on the bottom, and revolving it about the -axis; it forms a ring (or disc in the case that ), with outer radius and inner radius . The area of a ring is , where is the outer radius (in this case ), and is the inner radius (in this case ). The volume of each infinitesimal disc is therefore . The limit of the Riemann sum of the volumes of the discs between and becomes integral (1).
Assuming the applicability of Fubini's theorem and the multivariate change of variables formula, the disk method may be derived in a straightforward manner by (denoting the solid as D):
Cylinder method
The cylinder method is used when the slice that was drawn is parallel to the axis of revolution; i.e. when integrating perpendicular to the axis of revolution.
The volume of the solid formed by rotating the area between the curves of and and the lines and about the -axis is given by
If (e.g. revolving an area between curve and -axis), this reduces to:
The method can be visualized by considering a thin vertical rectangle at with height , and revolving it about the -axis; it form
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https://en.wikipedia.org/wiki/Disk%20%28mathematics%29
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In geometry, a disk (also spelled disc) is the region in a plane bounded by a circle. A disk is said to be closed if it contains the circle that constitutes its boundary, and open if it does not.
For a radius, , an open disk is usually denoted as and a closed disk is . However in the field of topology the closed disk is usually denoted as while the open disk is .
Formulas
In Cartesian coordinates, the open disk of center and radius R is given by the formula:
while the closed disk of the same center and radius is given by:
The area of a closed or open disk of radius R is πR2 (see area of a disk).
Properties
The disk has circular symmetry.
The open disk and the closed disk are not topologically equivalent (that is, they are not homeomorphic), as they have different topological properties from each other. For instance, every closed disk is compact whereas every open disk is not compact. However from the viewpoint of algebraic topology they share many properties: both of them are contractible and so are homotopy equivalent to a single point. This implies that their fundamental groups are trivial, and all homology groups are trivial except the 0th one, which is isomorphic to Z. The Euler characteristic of a point (and therefore also that of a closed or open disk) is 1.
Every continuous map from the closed disk to itself has at least one fixed point (we don't require the map to be bijective or even surjective); this is the case n=2 of the Brouwer fixed point theorem. The statement is false for the open disk:
Consider for example the function
which maps every point of the open unit disk to another point on the open unit disk to the right of the given one. But for the closed unit disk it fixes every point on the half circle
As a statistical distribution
A uniform distribution on a unit circular disk is occasionally encountered in statistics. It most commonly occurs in operations research in the mathematics of urban planning, where it may be used to model a population within a city. Other uses may take advantage of the fact that it is a distribution for which it is easy to compute the probability that a given set of linear inequalities will be satisfied. (Gaussian distributions in the plane require numerical quadrature.)
"An ingenious argument via elementary functions" shows the mean Euclidean distance between two points in the disk to be , while direct integration in polar coordinates shows the mean squared distance to be .
If we are given an arbitrary location at a distance from the center of the disk, it is also of interest to determine the average distance from points in the distribution to this location and the average square of such distances. The latter value can be computed directly as .
Average distance to an arbitrary internal point
To find we need to look separately at the cases in which the location is internal or external, i.e. in which , and we find that in both cases the result can only be expressed in terms of complete
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https://en.wikipedia.org/wiki/Stationary%20process
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In mathematics and statistics, a stationary process (or a strict/strictly stationary process or strong/strongly stationary process) is a stochastic process whose unconditional joint probability distribution does not change when shifted in time. Consequently, parameters such as mean and variance also do not change over time. If you draw a line through the middle of a stationary process then it should be flat; it may have 'seasonal' cycles around the trend line, but overall it does not trend up nor down.
Since stationarity is an assumption underlying many statistical procedures used in time series analysis, non-stationary data are often transformed to become stationary. The most common cause of violation of stationarity is a trend in the mean, which can be due either to the presence of a unit root or of a deterministic trend. In the former case of a unit root, stochastic shocks have permanent effects, and the process is not mean-reverting. In the latter case of a deterministic trend, the process is called a trend-stationary process, and stochastic shocks have only transitory effects after which the variable tends toward a deterministically evolving (non-constant) mean.
A trend stationary process is not strictly stationary, but can easily be transformed into a stationary process by removing the underlying trend, which is solely a function of time. Similarly, processes with one or more unit roots can be made stationary through differencing. An important type of non-stationary process that does not include a trend-like behavior is a cyclostationary process, which is a stochastic process that varies cyclically with time.
For many applications strict-sense stationarity is too restrictive. Other forms of stationarity such as wide-sense stationarity or N-th-order stationarity are then employed. The definitions for different kinds of stationarity are not consistent among different authors (see Other terminology).
Strict-sense stationarity
Definition
Formally, let be a stochastic process and let represent the cumulative distribution function of the unconditional (i.e., with no reference to any particular starting value) joint distribution of at times . Then, is said to be strictly stationary, strongly stationary or strict-sense stationary if
Since does not affect , is independent of time.
Examples
White noise is the simplest example of a stationary process.
An example of a discrete-time stationary process where the sample space is also discrete (so that the random variable may take one of N possible values) is a Bernoulli scheme. Other examples of a discrete-time stationary process with continuous sample space include some autoregressive and moving average processes which are both subsets of the autoregressive moving average model. Models with a non-trivial autoregressive component may be either stationary or non-stationary, depending on the parameter values, and important non-stationary special cases are where unit roots exist in the model.
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https://en.wikipedia.org/wiki/Discretization
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In applied mathematics, discretization is the process of transferring continuous functions, models, variables, and equations into discrete counterparts. This process is usually carried out as a first step toward making them suitable for numerical evaluation and implementation on digital computers. Dichotomization is the special case of discretization in which the number of discrete classes is 2, which can approximate a continuous variable as a binary variable (creating a dichotomy for modeling purposes, as in binary classification).
Discretization is also related to discrete mathematics, and is an important component of granular computing. In this context, discretization may also refer to modification of variable or category granularity, as when multiple discrete variables are aggregated or multiple discrete categories fused.
Whenever continuous data is discretized, there is always some amount of discretization error. The goal is to reduce the amount to a level considered negligible for the modeling purposes at hand.
The terms discretization and quantization often have the same denotation but not always identical connotations. (Specifically, the two terms share a semantic field.) The same is true of discretization error and quantization error.
Mathematical methods relating to discretization include the Euler–Maruyama method and the zero-order hold.
Discretization of linear state space models
Discretization is also concerned with the transformation of continuous differential equations into discrete difference equations, suitable for numerical computing.
The following continuous-time state space model
where v and w are continuous zero-mean white noise sources with power spectral densities
can be discretized, assuming zero-order hold for the input u and continuous integration for the noise v, to
with covariances
where
, if is nonsingular
and is the sample time, although is the transposed matrix of . The equation for the discretized measurement noise is a consequence of the continuous measurement noise being defined with a power spectral density.
A clever trick to compute Ad and Bd in one step is by utilizing the following property:
Where and are the discretized state-space matrices.
Discretization of process noise
Numerical evaluation of is a bit trickier due to the matrix exponential integral. It can, however, be computed by first constructing a matrix, and computing the exponential of it
The discretized process noise is then evaluated by multiplying the transpose of the lower-right partition of G with the upper-right partition of G:
Derivation
Starting with the continuous model
we know that the matrix exponential is
and by premultiplying the model we get
which we recognize as
and by integrating..
which is an analytical solution to the continuous model.
Now we want to discretise the above expression. We assume that u is constant during each timestep.
We recognize the bracketed expression as , and the second term
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https://en.wikipedia.org/wiki/Kernel%20%28set%20theory%29
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In set theory, the kernel of a function (or equivalence kernel) may be taken to be either
the equivalence relation on the function's domain that roughly expresses the idea of "equivalent as far as the function can tell", or
the corresponding partition of the domain.
An unrelated notion is that of the kernel of a non-empty family of sets which by definition is the intersection of all its elements:
This definition is used in the theory of filters to classify them as being free or principal.
Definition
For the formal definition, let be a function between two sets.
Elements are equivalent if and are equal, that is, are the same element of
The kernel of is the equivalence relation thus defined.
The is
The kernel of is also sometimes denoted by The kernel of the empty set, is typically left undefined.
A family is called and is said to have if its is not empty.
A family is said to be if it is not fixed; that is, if its kernel is the empty set.
Quotients
Like any equivalence relation, the kernel can be modded out to form a quotient set, and the quotient set is the partition:
This quotient set is called the coimage of the function and denoted (or a variation).
The coimage is naturally isomorphic (in the set-theoretic sense of a bijection) to the image, specifically, the equivalence class of in (which is an element of ) corresponds to in (which is an element of ).
As a subset of the square
Like any binary relation, the kernel of a function may be thought of as a subset of the Cartesian product
In this guise, the kernel may be denoted (or a variation) and may be defined symbolically as
The study of the properties of this subset can shed light on
Algebraic structures
If and are algebraic structures of some fixed type (such as groups, rings, or vector spaces), and if the function is a homomorphism, then is a congruence relation (that is an equivalence relation that is compatible with the algebraic structure), and the coimage of is a quotient of
The bijection between the coimage and the image of is an isomorphism in the algebraic sense; this is the most general form of the first isomorphism theorem.
In topology
If is a continuous function between two topological spaces then the topological properties of can shed light on the spaces and
For example, if is a Hausdorff space then must be a closed set.
Conversely, if is a Hausdorff space and is a closed set, then the coimage of if given the quotient space topology, must also be a Hausdorff space.
A space is compact if and only if the kernel of every family of closed subsets having the finite intersection property (FIP) is non-empty; said differently, a space is compact if and only if every family of closed subsets with F.I.P. is fixed.
See also
References
Bibliography
Abstract algebra
Basic concepts in set theory
Set theory
Topology
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https://en.wikipedia.org/wiki/Differentiable%20function
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In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its domain. A differentiable function is smooth (the function is locally well approximated as a linear function at each interior point) and does not contain any break, angle, or cusp.
If is an interior point in the domain of a function , then is said to be differentiable at if the derivative exists. In other words, the graph of has a non-vertical tangent line at the point . is said to be differentiable on if it is differentiable at every point of . is said to be continuously differentiable if its derivative is also a continuous function over the domain of the function . Generally speaking, is said to be of class if its first derivatives exist and are continuous over the domain of the function .
For a multivariable function, as shown here, the differentiability of it is something more than the existence of the partial derivatives of it.
Differentiability of real functions of one variable
A function , defined on an open set , is said to be differentiable at if the derivative
exists. This implies that the function is continuous at .
This function is said to be differentiable on if it is differentiable at every point of . In this case, the derivative of is thus a function from into
A continuous function is not necessarily differentiable, but a differentiable function is necessarily continuous (at every point where it is differentiable) as being shown below (in the section Differentiability and continuity). A function is said to be continuously differentiable if its derivative is also a continuous function; there exist functions that are differentiable but not continuously differentiable (an example is given in the section Differentiability classes).
Differentiability and continuity
If is differentiable at a point , then must also be continuous at . In particular, any differentiable function must be continuous at every point in its domain. The converse does not hold: a continuous function need not be differentiable. For example, a function with a bend, cusp, or vertical tangent may be continuous, but fails to be differentiable at the location of the anomaly.
Most functions that occur in practice have derivatives at all points or at almost every point. However, a result of Stefan Banach states that the set of functions that have a derivative at some point is a meagre set in the space of all continuous functions. Informally, this means that differentiable functions are very atypical among continuous functions. The first known example of a function that is continuous everywhere but differentiable nowhere is the Weierstrass function.
Differentiability classes
A function is said to be if the derivative exists and is itself a continuous function. Although the derivative of a diffe
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https://en.wikipedia.org/wiki/Row%20echelon%20form
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In linear algebra, a matrix is in row echelon form if it can be obtained as the result of Gaussian elimination. In particular, every matrix can be put in row echelon form by a succession of elementary row operations. The term echelon comes from the French "échelon" ("level" or step of a ladder), and refers to the fact that the nonzero entries of a matrix in row echelon form look like an inverted staircase.
For square matrices, an upper triangular matrix with nonzero entries on the diagonal is in row echelon form, and a matrix in row echelon form is (weakly) upper triangular. Thus, the row echelon form can be viewed as a generalization of upper triangular form for rectangular matrices.
A matrix is in reduced row echelon form if it is in row echelon form, the first nonzero entry of each row is equal to and the ones above it within the same column equal . The reduced row echelon form of a matrix is unique and does not depend on the sequence of elementary row operations used to obtain it. The variant of Gaussian elimination that transforms a matrix to reduced row echelon form is sometimes called Gauss–Jordan elimination.
A matrix is in column echelon form if its transpose is in row echelon form. So, all properties of column echelon forms can be easily deduced from the corresponding properties of row echelon forms. Therefore, only row echelon forms are considered in the remainder of the article.
(General) row echelon form
A matrix is in row echelon form if
All rows having only zero entries are at the bottom.
The leading entry (that is, the left-most nonzero entry) of every nonzero row, called the pivot, is on the right of the leading entry of every row above.
Some texts add the condition that the leading coefficient must be 1 while others require this only in reduced row echelon form.
These two conditions imply that all entries in a column below a leading coefficient are zeros.
The following is an example of a matrix in row echelon form, but not in reduced row echelon form (see below):
Many properties of matrices may be easily deduced from their row echelon form, such as the rank and the kernel.
Reduced row echelon form
A matrix is in reduced row echelon form (also called row canonical form) if it satisfies the following conditions:
It is in row echelon form.
The leading entry in each nonzero row is (called a leading one).
Each column containing a leading has zeros in all its other entries.
For a matrix in row echelon form, the last condition is equivalent to
Each column containing a leading has zeros in all entries above the leading .
While a matrix may have several echelon forms, its reduced echelon form is unique.
Given a matrix in reduced row echelon form, if one permutes the columns in order to have the leading of the th row in the th column, one gets a matrix of the form
where is an identity matrix, is a matrix with the same number of rows as , and the two 's are zero matrices of appropriate size. However, since
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https://en.wikipedia.org/wiki/Rank%E2%80%93nullity%20theorem
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The rank–nullity theorem is a theorem in linear algebra, which asserts:
the number of columns of a matrix is the sum of the rank of and the nullity of ; and
the dimension of the domain of a linear transformation is the sum of the rank of (the dimension of the image of ) and the nullity of (the dimension of the kernel of ).
It follows that for linear transformations of vector spaces of equal finite dimension, either injectivity or surjectivity implies bijectivity.
Stating the theorem
Linear transformations
Let be a linear transformation between two vector spaces where 's domain is finite dimensional. Then
where is the rank of (the dimension of its image) and is the nullity of (the dimension of its kernel). In other words,
This theorem can be refined via the splitting lemma to be a statement about an isomorphism of spaces, not just dimensions. Explicitly, since induces an isomorphism from to the existence of a basis for that extends any given basis of implies, via the splitting lemma, that Taking dimensions, the rank–nullity theorem follows.
Matrices
Linear maps can be represented with matrices. More precisely, an matrix represents a linear map where is the underlying field. So, the dimension of the domain of is , the number of columns of , and the rank–nullity theorem for an matrix is
Proofs
Here we provide two proofs. The first operates in the general case, using linear maps. The second proof looks at the homogeneous system where is a with rank and shows explicitly that there exists a set of linearly independent solutions that span the null space of .
While the theorem requires that the domain of the linear map be finite-dimensional, there is no such assumption on the codomain. This means that there are linear maps not given by matrices for which the theorem applies. Despite this, the first proof is not actually more general than the second: since the image of the linear map is finite-dimensional, we can represent the map from its domain to its image by a matrix, prove the theorem for that matrix, then compose with the inclusion of the image into the full codomain.
First proof
Let be vector spaces over some field and defined as in the statement of the theorem with .
As is a subspace, there exists a basis for it. Suppose and let
be such a basis.
We may now, by the Steinitz exchange lemma, extend with linearly independent vectors to form a full basis of .
Let
such that
is a basis for .
From this, we know that
We now claim that is a basis for .
The above equality already states that is a generating set for ; it remains to be shown that it is also linearly independent to conclude that it is a basis.
Suppose is not linearly independent, and let
for some .
Thus, owing to the linearity of , it follows that
This is a contradiction to being a basis, unless all are equal to zero. This shows that is linearly independent, and more specifically that it is a basis for .
To summarize, we have
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https://en.wikipedia.org/wiki/Riemann%E2%80%93Stieltjes%20integral
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In mathematics, the Riemann–Stieltjes integral is a generalization of the Riemann integral, named after Bernhard Riemann and Thomas Joannes Stieltjes. The definition of this integral was first published in 1894 by Stieltjes. It serves as an instructive and useful precursor of the Lebesgue integral, and an invaluable tool in unifying equivalent forms of statistical theorems that apply to discrete and continuous probability.
Formal definition
The Riemann–Stieltjes integral of a real-valued function of a real variable on the interval with respect to another real-to-real function is denoted by
Its definition uses a sequence of partitions of the interval
The integral, then, is defined to be the limit, as the mesh (the length of the longest subinterval) of the partitions approaches , of the approximating sum
where is in the -th subinterval . The two functions and are respectively called the integrand and the integrator. Typically is taken to be monotone (or at least of bounded variation) and right-semicontinuous (however this last is essentially convention). We specifically do not require to be continuous, which allows for integrals that have point mass terms.
The "limit" is here understood to be a number A (the value of the Riemann–Stieltjes integral) such that for every ε > 0, there exists δ > 0 such that for every partition P with norm(P) < δ, and for every choice of points ci in [xi, xi+1],
Properties
The Riemann–Stieltjes integral admits integration by parts in the form
and the existence of either integral implies the existence of the other.
On the other hand, a classical result shows that the integral is well-defined if f is α-Hölder continuous and g is β-Hölder continuous with .
If is bounded on , increases monotonically, and is Riemann integrable, then the Riemann–Stieltjes integral is related to the Riemann integral by
For a step function
where , if is continuous at , then
Application to probability theory
If g is the cumulative probability distribution function of a random variable X that has a probability density function with respect to Lebesgue measure, and f is any function for which the expected value is finite, then the probability density function of X is the derivative of g and we have
But this formula does not work if X does not have a probability density function with respect to Lebesgue measure. In particular, it does not work if the distribution of X is discrete (i.e., all of the probability is accounted for by point-masses), and even if the cumulative distribution function g is continuous, it does not work if g fails to be absolutely continuous (again, the Cantor function may serve as an example of this failure). But the identity
holds if g is any cumulative probability distribution function on the real line, no matter how ill-behaved. In particular, no matter how ill-behaved the cumulative distribution function g of a random variable X, if the moment E(Xn) exists, then it is equal to
Applicat
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https://en.wikipedia.org/wiki/Disc%20integration
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Disc integration, also known in integral calculus as the disc method, is a method for calculating the volume of a solid of revolution of a solid-state material when integrating along an axis "parallel" to the axis of revolution. This method models the resulting three-dimensional shape as a stack of an infinite number of discs of varying radius and infinitesimal thickness. It is also possible to use the same principles with rings instead of discs (the "washer method") to obtain hollow solids of revolutions. This is in contrast to shell integration, which integrates along an axis perpendicular to the axis of revolution.
Definition
Function of
If the function to be revolved is a function of , the following integral represents the volume of the solid of revolution:
where is the distance between the function and the axis of rotation. This works only if the axis of rotation is horizontal (example: or some other constant).
Function of
If the function to be revolved is a function of , the following integral will obtain the volume of the solid of revolution:
where is the distance between the function and the axis of rotation. This works only if the axis of rotation is vertical (example: or some other constant).
Washer method
To obtain a hollow solid of revolution (the “washer method”), the procedure would be to take the volume of the inner solid of revolution and subtract it from the volume of the outer solid of revolution. This can be calculated in a single integral similar to the following:
where is the function that is farthest from the axis of rotation and is the function that is closest to the axis of rotation. For example, the next figure shows the rotation along the -axis of the red "leaf" enclosed between the square-root and quadratic curves:
The volume of this solid is:
One should take caution not to evaluate the square of the difference of the two functions, but to evaluate the difference of the squares of the two functions.
(This formula only works for revolutions about the -axis.)
To rotate about any horizontal axis, simply subtract from that axis from each formula. If is the value of a horizontal axis, then the volume equals
For example, to rotate the region between and along the axis , one would integrate as follows:
The bounds of integration are the zeros of the first equation minus the second. Note that when integrating along an axis other than the , the graph of the function that is farthest from the axis of rotation may not be that obvious. In the previous example, even though the graph of is, with respect to the x-axis, further up than the graph of , with respect to the axis of rotation the function is the inner function: its graph is closer to or the equation of the axis of rotation in the example.
The same idea can be applied to both the -axis and any other vertical axis. One simply must solve each equation for before one inserts them into the integration formula.
See also
Solid of revolution
Shell integ
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https://en.wikipedia.org/wiki/Monoidal%20category
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In mathematics, a monoidal category (or tensor category) is a category equipped with a bifunctor
that is associative up to a natural isomorphism, and an object I that is both a left and right identity for ⊗, again up to a natural isomorphism. The associated natural isomorphisms are subject to certain coherence conditions, which ensure that all the relevant diagrams commute.
The ordinary tensor product makes vector spaces, abelian groups, R-modules, or R-algebras into monoidal categories. Monoidal categories can be seen as a generalization of these and other examples. Every (small) monoidal category may also be viewed as a "categorification" of an underlying monoid, namely the monoid whose elements are the isomorphism classes of the category's objects and whose binary operation is given by the category's tensor product.
A rather different application, of which monoidal categories can be considered an abstraction, is that of a system of data types closed under a type constructor that takes two types and builds an aggregate type; the types are the objects and is the aggregate constructor. The associativity up to isomorphism is then a way of expressing that different ways of aggregating the same data—such as and —store the same information even though the aggregate values need not be the same. The aggregate type may be analogous to the operation of addition (type sum) or of multiplication (type product). For type product, the identity object is the unit , so there is only one inhabitant of the type, and that is why a product with it is always isomorphic to the other operand. For type sum, the identity object is the void type, which stores no information and it is impossible to address an inhabitant. The concept of monoidal category does not presume that values of such aggregate types can be taken apart; on the contrary, it provides a framework that unifies classical and quantum information theory.
In category theory, monoidal categories can be used to define the concept of a monoid object and an associated action on the objects of the category. They are also used in the definition of an enriched category.
Monoidal categories have numerous applications outside of category theory proper. They are used to define models for the multiplicative fragment of intuitionistic linear logic. They also form the mathematical foundation for the topological order in condensed matter physics. Braided monoidal categories have applications in quantum information, quantum field theory, and string theory.
Formal definition
A monoidal category is a category equipped with a monoidal structure. A monoidal structure consists of the following:
a bifunctor called the monoidal product, or tensor product,
an object called the monoidal unit, unit object, or identity object,
three natural isomorphisms subject to certain coherence conditions expressing the fact that the tensor operation:
is associative: there is a natural (in each of three arguments , , ) isomorphism ,
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https://en.wikipedia.org/wiki/Shell%20integration
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Shell integration (the shell method in integral calculus) is a method for calculating the volume of a solid of revolution, when integrating along an axis perpendicular to the axis of revolution. This is in contrast to disc integration which integrates along the axis parallel to the axis of revolution.
Definition
The shell method goes as follows: Consider a volume in three dimensions obtained by rotating a cross-section in the -plane around the -axis. Suppose the cross-section is defined by the graph of the positive function on the interval . Then the formula for the volume will be:
If the function is of the coordinate and the axis of rotation is the -axis then the formula becomes:
If the function is rotating around the line then the formula becomes:
and for rotations around it becomes
The formula is derived by computing the double integral in polar coordinates.
Derivation of the formula
Example
Consider the volume, depicted below, whose cross section on the interval [1, 2] is defined by:
In the case of disc integration we would need to solve for given and because the volume is hollow in the middle we would find two functions, one that defined the inner solid and one that defined the outer solid. After integrating these two functions with the disk method we would subtract them to yield the desired volume.
With the shell method all we need is the following formula:
By expanding the polynomial the integral becomes very simple. In the end we find the volume is cubic units.
See also
Solid of revolution
Disc integration
References
Frank Ayres, Elliott Mendelson. Schaum's Outlines: Calculus. McGraw-Hill Professional 2008, . pp. 244–248 ()
Integral calculus
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https://en.wikipedia.org/wiki/Squeeze%20theorem
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In calculus, the squeeze theorem (also known as the sandwich theorem, among other names) is a theorem regarding the limit of a function that is trapped between two other functions.
The squeeze theorem is used in calculus and mathematical analysis, typically to confirm the limit of a function via comparison with two other functions whose limits are known. It was first used geometrically by the mathematicians Archimedes and Eudoxus in an effort to compute , and was formulated in modern terms by Carl Friedrich Gauss.
Statement
The squeeze theorem is formally stated as follows.
The functions and are said to be lower and upper bounds (respectively) of .
Here, is not required to lie in the interior of . Indeed, if is an endpoint of , then the above limits are left- or right-hand limits.
A similar statement holds for infinite intervals: for example, if , then the conclusion holds, taking the limits as .
This theorem is also valid for sequences. Let be two sequences converging to , and a sequence. If we have , then also converges to .
Proof
According to the above hypotheses we have, taking the limit inferior and superior:
so all the inequalities are indeed equalities, and the thesis immediately follows.
A direct proof, using the -definition of limit, would be to prove that for all real there exists a real such that for all with we have Symbolically,
As
means that
and
means that
then we have
We can choose . Then, if , combining () and (), we have
which completes the proof. Q.E.D
The proof for sequences is very similar, using the -definition of the limit of a sequence.
Examples
First example
The limit
cannot be determined through the limit law
because
does not exist.
However, by the definition of the sine function,
It follows that
Since , by the squeeze theorem, must also be 0.
Second example
Probably the best-known examples of finding a limit by squeezing are the proofs of the equalities
The first limit follows by means of the squeeze theorem from the fact that
for close enough to 0. The correctness of which for positive can be seen by simple geometric reasoning (see drawing) that can be extended to negative as well. The second limit follows from the squeeze theorem and the fact that
for close enough to 0. This can be derived by replacing in the earlier fact by and squaring the resulting inequality.
These two limits are used in proofs of the fact that the derivative of the sine function is the cosine function. That fact is relied on in other proofs of derivatives of trigonometric functions.
Third example
It is possible to show that
by squeezing, as follows.
In the illustration at right, the area of the smaller of the two shaded sectors of the circle is
since the radius is and the arc on the unit circle has length . Similarly, the area of the larger of the two shaded sectors is
What is squeezed between them is the triangle whose base is the vertical segment whose endpoints are the two dots
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https://en.wikipedia.org/wiki/Saharon%20Shelah
|
Saharon Shelah (; born July 3, 1945) is an Israeli mathematician. He is a professor of mathematics at the Hebrew University of Jerusalem and Rutgers University in New Jersey.
Biography
Shelah was born in Jerusalem on July 3, 1945. He is the son of the Israeli poet and political activist Yonatan Ratosh. He received his PhD for his work on stable theories in 1969 from the Hebrew University.
Shelah is married to Yael, and has three children. His brother, magistrate judge Hamman Shelah was murdered along with his wife and daughter by an Egyptian soldier in the Ras Burqa massacre in 1985.
Shelah planned to be a scientist while at primary school, but initially was attracted to physics and biology, not mathematics. Later he found mathematical beauty in studying geometry: He said, "But when I reached the ninth grade I began studying geometry and my eyes opened to that beauty—a system of demonstration and theorems based on a very small number of axioms which impressed me and captivated me." At the age of 15, he decided to become a mathematician, a choice cemented after reading Abraham Halevy Fraenkel's book An Introduction to Mathematics.
He received a B.Sc. from Tel Aviv University in 1964, served in the Israel Defense Forces Army between 1964 and 1967, and obtained a M.Sc. from the Hebrew University (under the direction of Haim Gaifman) in 1967. He then worked as a teaching assistant at the Institute of Mathematics of the Hebrew University of Jerusalem while completing a Ph.D. there under the supervision of Michael Oser Rabin, on a study of stable theories.
Shelah was a lecturer at Princeton University during 1969–70, and then worked as an assistant professor at the University of California, Los Angeles during 1970–71. He became a professor at Hebrew University in 1974, a position he continues to hold.
He has been a visiting professor at the following universities: the University of Wisconsin (1977–78), the University of California, Berkeley (1978 and 1982), the University of Michigan (1984–85), at Simon Fraser University, Burnaby, British Columbia (1985), and Rutgers University, New Jersey (1985). He has been a distinguished visiting professor at Rutgers University since 1986.
Academic career
Shelah's personal webpage, lists 1123 published and accepted mathematical papers, as well as more than 100 preprints and papers in preparation, including joint papers with 288 co-authors; the American Mathematical Society's database MathSciNet lists 1147 published books and journal articles with 266 coauthors. His main interests lie in mathematical logic, model theory in particular, and in axiomatic set theory.
In model theory, he developed classification theory, which led him to a solution of Morley's problem. In set theory, he discovered the notion of proper forcing, an important tool in iterated forcing arguments. With PCF theory, he showed that in spite of the undecidability of the most basic questions of cardinal arithmetic (such as the continuum hyp
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https://en.wikipedia.org/wiki/Moore%E2%80%93Penrose%20inverse
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In mathematics, and in particular linear algebra, the Moore–Penrose inverse of a matrix is the most widely known generalization of the inverse matrix. It was independently described by E. H. Moore in 1920, Arne Bjerhammar in 1951, and Roger Penrose in 1955. Earlier, Erik Ivar Fredholm had introduced the concept of a pseudoinverse of integral operators in 1903. When referring to a matrix, the term pseudoinverse, without further specification, is often used to indicate the Moore–Penrose inverse. The term generalized inverse is sometimes used as a synonym for pseudoinverse.
A common use of the pseudoinverse is to compute a "best fit" (least squares) solution to a system of linear equations that lacks a solution (see below under § Applications).
Another use is to find the minimum (Euclidean) norm solution to a system of linear equations with multiple solutions. The pseudoinverse facilitates the statement and proof of results in linear algebra.
The pseudoinverse is defined and unique for all matrices whose entries are real or complex numbers. It can be computed using the singular value decomposition. In the special case where is a normal matrix (for example, a Hermitian matrix), the pseudoinverse annihilates the kernel of and acts as a traditional inverse of on the subspace orthogonal to the kernel.
Notation
In the following discussion, the following conventions are adopted.
will denote one of the fields of real or complex numbers, denoted , , respectively. The vector space of matrices over is denoted by .
For , the transpose is denoted and the Hermitian transpose (also called conjugate transpose) is denoted . If , then .
For , (standing for "range") denotes the column space (image) of (the space spanned by the column vectors of ) and denotes the kernel (null space) of .
For any positive integer , the identity matrix is denoted .
Definition
For , a pseudoinverse of is defined as a matrix satisfying all of the following four criteria, known as the Moore–Penrose conditions:
need not be the general identity matrix, but it maps all column vectors of to themselves:
acts like a weak inverse:
is Hermitian:
is also Hermitian:
These conditions are equivalent to being the projection onto the support of , and being the projection onto the image of . The pseudoinverse exists for any matrix . If furthermore is full rank, that is, its rank is , then has a particularly simple algebraic expression.
In particular, when has linearly independent columns (equivalently, is injective, and thus is invertible), can be computed asThis particular pseudoinverse is a left inverse, that is, . If on the other hand has linearly independent rows (equivalently, is surjective, and thus is invertible), can be computed asThis is a right inverse, as .
In the more general case, the pseudoinverse can be expressed leveraging the singular value decomposition. Any matrix can be decomposed as for some isometries and diagonal positive rea
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https://en.wikipedia.org/wiki/Kuratowski%20closure%20axioms
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In topology and related branches of mathematics, the Kuratowski closure axioms are a set of axioms that can be used to define a topological structure on a set. They are equivalent to the more commonly used open set definition. They were first formalized by Kazimierz Kuratowski, and the idea was further studied by mathematicians such as Wacław Sierpiński and António Monteiro, among others.
A similar set of axioms can be used to define a topological structure using only the dual notion of interior operator.
Definition
Kuratowski closure operators and weakenings
Let be an arbitrary set and its power set. A Kuratowski closure operator is a unary operation with the following properties:
A consequence of preserving binary unions is the following condition:
In fact if we rewrite the equality in [K4] as an inclusion, giving the weaker axiom [K4''] (subadditivity):
then it is easy to see that axioms [K4'] and [K4''] together are equivalent to [K4] (see the next-to-last paragraph of Proof 2 below).
includes a fifth (optional) axiom requiring that singleton sets should be stable under closure: for all , . He refers to topological spaces which satisfy all five axioms as T1-spaces in contrast to the more general spaces which only satisfy the four listed axioms. Indeed, these spaces correspond exactly to the topological T1-spaces via the usual correspondence (see below).
If requirement [K3] is omitted, then the axioms define a Čech closure operator. If [K1] is omitted instead, then an operator satisfying [K2], [K3] and [K4'] is said to be a Moore closure operator. A pair is called Kuratowski, Čech or Moore closure space depending on the axioms satisfied by .
Alternative axiomatizations
The four Kuratowski closure axioms can be replaced by a single condition, given by Pervin:
Axioms [K1]–[K4] can be derived as a consequence of this requirement:
Choose . Then , or . This immediately implies [K1].
Choose an arbitrary and . Then, applying axiom [K1], , implying [K2].
Choose and an arbitrary . Then, applying axiom [K1], , which is [K3].
Choose arbitrary . Applying axioms [K1]–[K3], one derives [K4].
Alternatively, had proposed a weaker axiom that only entails [K2]–[K4]:
Requirement [K1] is independent of [M] : indeed, if , the operator defined by the constant assignment satisfies [M] but does not preserve the empty set, since . Notice that, by definition, any operator satisfying [M] is a Moore closure operator.
A more symmetric alternative to [M] was also proven by M. O. Botelho and M. H. Teixeira to imply axioms [K2]–[K4]:
Analogous structures
Interior, exterior and boundary operators
A dual notion to Kuratowski closure operators is that of Kuratowski interior operator, which is a map satisfying the following similar requirements:
For these operators, one can reach conclusions that are completely analogous to what was inferred for Kuratowski closures. For example, all Kuratowski interior operators are isotonic, i.e. they satisfy
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https://en.wikipedia.org/wiki/Prime-counting%20function
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In mathematics, the prime-counting function is the function counting the number of prime numbers less than or equal to some real number x. It is denoted by (x) (unrelated to the number ).
Growth rate
Of great interest in number theory is the growth rate of the prime-counting function. It was conjectured in the end of the 18th century by Gauss and by Legendre to be approximately
where log is the natural logarithm, in the sense that
This statement is the prime number theorem. An equivalent statement is
where li is the logarithmic integral function. The prime number theorem was first proved in 1896 by Jacques Hadamard and by Charles de la Vallée Poussin independently, using properties of the Riemann zeta function introduced by Riemann in 1859. Proofs of the prime number theorem not using the zeta function or complex analysis were found around 1948 by Atle Selberg and by Paul Erdős (for the most part independently).
More precise estimates
In 1899, de la Vallée Poussin proved that
for some positive constant . Here, is the big notation.
More precise estimates of are now known. For example, in 2002, Kevin Ford proved that
Mossinghoff and Trudgian proved an explicit upper bound for the difference between and :
for .
For values of that are not unreasonably large, is greater than . However, is known to change sign infinitely many times. For a discussion of this, see Skewes' number.
Exact form
For let when is a prime number, and otherwise. Bernhard Riemann, in his work On the Number of Primes Less Than a Given Magnitude, proved that is equal to
where
is the Möbius function, is the logarithmic integral function, ρ indexes every zero of the Riemann zeta function, and is not evaluated with a branch cut but instead considered as where is the exponential integral. If the trivial zeros are collected and the sum is taken only over the non-trivial zeros ρ of the Riemann zeta function, then may be approximated by
The Riemann hypothesis suggests that every such non-trivial zero lies along .
Table of (x), x / log x, and li(x)
The table shows how the three functions (x), x / log x and li(x) compare at powers of 10. See also, and
{| class="wikitable" style="text-align: right"
! x
! (x)
!
!
!
!
|-
| 10
| 4
| 0
| 2
| 2.500
| -8.57%
|-
| 102
| 25
| 3
| 5
| 4.000
|13.14%
|-
| 103
| 168
| 23
| 10
| 5.952
|13.83%
|-
| 104
| 1,229
| 143
| 17
| 8.137
|11.66%
|-
| 105
| 9,592
| 906
| 38
| 10.425
|9.45%
|-
| 106
| 78,498
| 6,116
| 130
| 12.739
|7.79%
|-
| 107
| 664,579
| 44,158
| 339
| 15.047
|6.64%
|-
| 108
| 5,761,455
| 332,774
| 754
| 17.357
|5.78%
|-
| 109
| 50,847,534
| 2,592,592
| 1,701
| 19.667
|5.10%
|-
| 1010
| 455,052,511
| 20,758,029
| 3,104
| 21.975
|4.56%
|-
| 1011
| 4,118,054,813
| 169,923,159
| 11,588
| 24.283
|4.13%
|-
| 1012
| 37,607,912,018
| 1,416,705,193
| 38,263
| 26.590
|3.77%
|-
| 1013
| 346,065,536,839
| 11,992,858,452
| 108,971
| 28.896
|3.47%
|-
| 1014
| 3,204,941,750,802
| 102,838,308,636
| 314,890
| 31.202
|3.21
|
https://en.wikipedia.org/wiki/Algebra%20homomorphism
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In mathematics, an algebra homomorphism is a homomorphism between two algebras. More precisely, if A and B are algebras over a field (or a ring) K, it is a function such that, for all k in K and x, y in A, one has
The first two conditions say that F is a K-linear map, and the last condition says that F preserves the algebra multiplication. So, if the algebras are associative, F is a , and, if the algebras are rings and F preserves the identity, it is a ring homomorphism.
If F admits an inverse homomorphism, or equivalently if it is bijective, F is said to be an isomorphism between A and B.
Unital algebra homomorphisms
If A and B are two unital algebras, then an algebra homomorphism is said to be unital if it maps the unity of A to the unity of B. Often the words "algebra homomorphism" are actually used to mean "unital algebra homomorphism", in which case non-unital algebra homomorphisms are excluded.
A unital algebra homomorphism is a (unital) ring homomorphism.
Examples
Every ring is a Z-algebra since there always exists a unique homomorphism . See for the explanation.
Any homomorphism of commutative rings gives S the structure of a commutative -algebra. Conversely, if S is a commutative R-algebra, the map is a homomorphism of commutative rings. It is straightforward to deduce that the overcategory of the commutative rings over R is the same as the category of commutative R-algebras.
If A is a subalgebra of B, then for every invertible b in B the function that takes every a in A to b−1 a b is an algebra homomorphism (in case , this is called an inner automorphism of B). If A is also simple and B is a central simple algebra, then every homomorphism from A to B is given in this way by some b in B; this is the Skolem–Noether theorem.
See also
Morphism
Spectrum of a ring
Augmentation (algebra)
References
Algebras
Ring theory
Morphisms
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https://en.wikipedia.org/wiki/F-distribution
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In probability theory and statistics, the F-distribution or F-ratio, also known as Snedecor's F distribution or the Fisher–Snedecor distribution (after Ronald Fisher and George W. Snedecor), is a continuous probability distribution that arises frequently as the null distribution of a test statistic, most notably in the analysis of variance (ANOVA) and other F-tests.
Definition
The F-distribution with d1 and d2 degrees of freedom is the distribution of
where and are independent random variables with chi-square distributions with respective degrees of freedom and .
It can be shown to follow that the probability density function (pdf) for X is given by
for real x > 0. Here is the beta function. In many applications, the parameters d1 and d2 are positive integers, but the distribution is well-defined for positive real values of these parameters.
The cumulative distribution function is
where I is the regularized incomplete beta function.
The expectation, variance, and other details about the F(d1, d2) are given in the sidebox; for d2 > 8, the excess kurtosis is
The k-th moment of an F(d1, d2) distribution exists and is finite only when 2k < d2 and it is equal to
The F-distribution is a particular parametrization of the beta prime distribution, which is also called the beta distribution of the second kind.
The characteristic function is listed incorrectly in many standard references (e.g.,). The correct expression is
where U(a, b, z) is the confluent hypergeometric function of the second kind.
Characterization
A random variate of the F-distribution with parameters and arises as the ratio of two appropriately scaled chi-squared variates:
where
and have chi-squared distributions with and degrees of freedom respectively, and
and are independent.
In instances where the F-distribution is used, for example in the analysis of variance, independence of and might be demonstrated by applying Cochran's theorem.
Equivalently, the random variable of the F-distribution may also be written
where and , is the sum of squares of random variables from normal distribution and is the sum of squares of random variables from normal distribution .
In a frequentist context, a scaled F-distribution therefore gives the probability , with the F-distribution itself, without any scaling, applying where is being taken equal to . This is the context in which the F-distribution most generally appears in F-tests: where the null hypothesis is that two independent normal variances are equal, and the observed sums of some appropriately selected squares are then examined to see whether their ratio is significantly incompatible with this null hypothesis.
The quantity has the same distribution in Bayesian statistics, if an uninformative rescaling-invariant Jeffreys prior is taken for the prior probabilities of and . In this context, a scaled F-distribution thus gives the posterior probability , where the observed sums and are now taken as
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https://en.wikipedia.org/wiki/Ian%20Stewart%20%28mathematician%29
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Ian Nicholas Stewart (born 24 September 1945) is a British mathematician and a popular-science and science-fiction writer. He is Emeritus Professor of Mathematics at the University of Warwick, England.
Education and early life
Stewart was born in 1945 in Folkestone, England. While in the sixth form at Harvey Grammar School in Folkestone he came to the attention of the mathematics teacher. The teacher had Stewart sit mock A-level examinations without any preparation along with the upper-sixth students; Stewart was placed first in the examination. He was awarded a scholarship to study at the University of Cambridge as an undergraduate student of Churchill College, Cambridge, where he studied the Mathematical Tripos and obtained a first-class Bachelor of Arts degree in mathematics in 1966. Stewart then went to the University of Warwick where his PhD on Lie algebras was supervised by Brian Hartley and completed in 1969.
Career and research
After his PhD, Stewart was offered an academic position at Warwick. He is well known for his popular expositions of mathematics and his contributions to catastrophe theory.
While at Warwick, Stewart edited the mathematical magazine Manifold. He also wrote a column called "Mathematical Recreations" for Scientific American magazine from 1991 to 2001. This followed the work of past columnists like Martin Gardner, Douglas Hofstadter, and A. K. Dewdney. Altogether, he wrote 96 columns for Scientific American, which were later reprinted in the books "Math Hysteria", "How to Cut a Cake: And Other Mathematical Conundrums" and "Cows in the Maze".
Stewart has held visiting academic positions in Germany (1974), New Zealand (1976), and the US (University of Connecticut 1977–78, University of Houston 1983–84).
Stewart has published more than 140 scientific papers, including a series of influential papers co-authored with Jim Collins on coupled oscillators and the symmetry of animal gaits.
Stewart has collaborated with Jack Cohen and Terry Pratchett on four popular science books based on Pratchett's Discworld. In 1999 Terry Pratchett made both Jack Cohen and Professor Ian Stewart "Honorary Wizards of the Unseen University" at the same ceremony at which the University of Warwick gave Terry Pratchett an honorary degree.
In March 2014 Ian Stewart's iPad app, Incredible Numbers by Professor Ian Stewart, launched in the App Store. The app was produced in partnership with Profile Books and Touch Press.
Mathematics and popular science
Manifold, mathematical magazine published at the University of Warwick (1960s)
Nut-crackers: Puzzles and Games to Boggle the Mind (Piccolo Books) with John Jaworski, 1971.
Concepts of Modern Mathematics (1975)
Oh! Catastrophe (1982, in French)
Does God Play Dice? The New Mathematics of Chaos (1989)
Game, Set and Math (1991)
Fearful Symmetry (1992)
Another Fine Math You've Got Me Into (1992)
The Collapse of Chaos: Discovering Simplicity in a Complex World, with Jack Cohen (1995)
Nature's Nu
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https://en.wikipedia.org/wiki/Sociable%20number
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In mathematics, sociable numbers are numbers whose aliquot sums form a periodic sequence. They are generalizations of the concepts of perfect numbers and amicable numbers. The first two sociable sequences, or sociable chains, were discovered and named by the Belgian mathematician Paul Poulet in 1918. In a sociable sequence, each number is the sum of the proper divisors of the preceding number, i.e., the sum excludes the preceding number itself. For the sequence to be sociable, the sequence must be cyclic and return to its starting point.
The period of the sequence, or order of the set of sociable numbers, is the number of numbers in this cycle.
If the period of the sequence is 1, the number is a sociable number of order 1, or a perfect number—for example, the proper divisors of 6 are 1, 2, and 3, whose sum is again 6. A pair of amicable numbers is a set of sociable numbers of order 2. There are no known sociable numbers of order 3, and searches for them have been made up to as of 1970.
It is an open question whether all numbers end up at either a sociable number or at a prime (and hence 1), or, equivalently, whether there exist numbers whose aliquot sequence never terminates, and hence grows without bound.
Example
As an example, the number 1,264,460 is a sociable number whose cyclic aliquot sequence has a period of 4:
The sum of the proper divisors of () is
1 + 2 + 4 + 5 + 10 + 17 + 20 + 34 + 68 + 85 + 170 + 340 + 3719 + 7438 + 14876 + 18595 + 37190 + 63223 + 74380 + 126446 + 252892 + 316115 + 632230 = 1547860,
the sum of the proper divisors of () is
1 + 2 + 4 + 5 + 10 + 20 + 193 + 386 + 401 + 772 + 802 + 965 + 1604 + 1930 + 2005 + 3860 + 4010 + 8020 + 77393 + 154786 + 309572 + 386965 + 773930 = 1727636,
the sum of the proper divisors of () is
1 + 2 + 4 + 521 + 829 + 1042 + 1658 + 2084 + 3316 + 431909 + 863818 = 1305184, and
the sum of the proper divisors of () is
1 + 2 + 4 + 8 + 16 + 32 + 40787 + 81574 + 163148 + 326296 + 652592 = 1264460.
List of known sociable numbers
The following categorizes all known sociable numbers by the length of the corresponding aliquot sequence:
It is conjectured that if n is congruent to 3 modulo 4 then there is no such sequence with length n.
The 5-cycle sequence is: 12496, 14288, 15472, 14536, 14264
The only known 28-cycle is: 14316, 19116, 31704, 47616, 83328, 177792, 295488, 629072, 589786, 294896, 358336, 418904, 366556, 274924, 275444, 243760, 376736, 381028, 285778, 152990, 122410, 97946, 48976, 45946, 22976, 22744, 19916, 17716 . It was discovered by Ben Orlin.
These two sequences provide the only sociable numbers below 1 million (other than the perfect and amicable numbers).
Searching for sociable numbers
The aliquot sequence can be represented as a directed graph, , for a given integer , where denotes the
sum of the proper divisors of .
Cycles in represent sociable numbers within the interval . Two special cases are loops that represent perfect numbers and cycles of length two t
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https://en.wikipedia.org/wiki/Rate%20%28mathematics%29
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In mathematics, a rate is the quotient of two quantities in different units of measurement, often represented as a fraction. If the divisor (or fraction denominator) in the rate is equal to one expressed as a single unit, and if it is assumed that this quantity can be changed systematically (i.e., is an independent variable), then the dividend (the fraction numerator) of the rate expresses the corresponding rate of change in the other (dependent) variable.
Temporal rate is a common type of rate ("per unit of time"), such as speed, heart rate, and flux.
In fact, often rate is a synonym of rhythm or frequency, a count per second (i.e., hertz); e.g., radio frequencies or sample rates.
In describing the units of a rate, the word "per" is used to separate the units of the two measurements used to calculate the rate; for example, a heart rate is expressed as "beats per minute".
Rates that have a non-time divisor or denominator include exchange rates, literacy rates, and electric field (in volts per meter).
A rate defined using two numbers of the same units will result in a dimensionless quantity, also known as ratio or simply as a rate (such as tax rates) or counts (such as literacy rate). Dimensionless rates can be expressed as a percentage (for example, the global literacy rate in 1998 was 80%), fraction, or multiple.
Properties and examples
Rates and ratios often vary with time, location, particular element (or subset) of a set of objects, etc. Thus they are often mathematical functions.
A rate (or ratio) may often be thought of as an output-input ratio, benefit-cost ratio, all considered in the broad sense. For example, miles per hour in transportation is the output (or benefit) in terms of miles of travel, which one gets from spending an hour (a cost in time) of traveling (at this velocity).
A set of sequential indices may be used to enumerate elements (or subsets) of a set of ratios under study. For example, in finance, one could define I by assigning consecutive integers to companies, to political subdivisions (such as states), to different investments, etc. The reason for using indices I is so a set of ratios (i=0, N) can be used in an equation to calculate a function of the rates such as an average of a set of ratios. For example, the average velocity found from the set of v I 's mentioned above. Finding averages may involve using weighted averages and possibly using the harmonic mean.
A ratio r=a/b has both a numerator "a" and a denominator "b". The value of a and b may be a real number or integer. The inverse of a ratio r is 1/r = b/a. A rate may be equivalently expressed as an inverse of its value if the ratio of its units is also inverse. For example, 5 miles (mi) per kilowatt-hour (kWh) corresponds to 1/5 kWh/mi (or 200 Wh/mi).
Rates are relevant to many aspects of everyday life. For example:
How fast are you driving? The speed of the car (often expressed in miles per hour) is a rate. What interest does your savings account p
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https://en.wikipedia.org/wiki/Media%20access%20unit
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A media access unit (MAU), also known as a multistation access unit (MAU or MSAU), is a device to attach multiple network stations in a star topology as a Token Ring network, internally wired to connect the stations into a logical ring (generally passive i.e. non-switched and unmanaged; however managed Token Ring MAUs do exist in the form of CAUs, or Controlled Access Units).
Passive Token Ring was an active IBM networking product in the 1997 time-frame, after which it was rapidly displaced by switched networking.
Advantages and disadvantages
Passive networking without power
The majority of IBM-implemented (actual) passive Token Ring MAUs operated without the requirement of power; instead the passive MAU used a series of relays that adjusted themselves as data is passed through: this is also why Token Ring generally used relays to terminate disconnected or failed ports. The power-less IBM 8228 Multistation Access Unit requires a special 'Setup Aid' tool to re-align the relays after the unit has been moved which causes them to be in incorrect states: this is accomplished by a 9v battery sending a charge to snap the relays back in a proper state. The advantages of having a MAU operate without power is that they can be placed in areas without outlets, the disadvantage is that they must be primed each time the internal relays experience excessive force. The IBM 8226 MAU, while containing a power jack, primarily uses this for the LEDs: relays are still used inside the unit but do not require priming.
Bandwidth
In theory, this networking technology supported large geographic areas (with a total ring circumference of several kilometers). But with the bandwidth shared by all stations, in practice separate networks spanning smaller areas were joined using bridges. This bridged network technology was soon displaced by high-bandwidth switched networks.
Fault tolerance
Multistation Access Units contain relays to short out non-operating stations. Multiple MAUs can be connected into a larger ring through their ring in/ring out connectors.
An MAU is also called a "ring in a box". The loop that used to make up the ring of the token Ring is now integrated into this device. In Token Ring, when a link is broken in the ring, the entire network goes down; however with an MAU, the broken circuit is closed within 1ms; allowing stations on the ring to have their cords unplugged without disabling the entire network.
See also
Lobe Attachment Module
References
External links
Networking hardware
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https://en.wikipedia.org/wiki/Primitive%20element
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In mathematics, the term primitive element can mean:
Primitive root modulo n, in number theory
Primitive element (field theory), an element that generates a given field extension
Primitive element (finite field), an element that generates the multiplicative group of a finite field
Primitive element (lattice), an element in a lattice that is not a positive integer multiple of another element in the lattice
Primitive element (coalgebra), an element X on which the comultiplication Δ has the value Δ(X) = X⊗1 + 1⊗X
Primitive element (free group), an element of a free generating set
Primitive element (Lie algebra), a Borel-weight vector
See also
Primitive element theorem
Primitive root (disambiguation)
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https://en.wikipedia.org/wiki/Hereditarily%20finite%20set
|
In mathematics and set theory, hereditarily finite sets are defined as finite sets whose elements are all hereditarily finite sets. In other words, the set itself is finite, and all of its elements are finite sets, recursively all the way down to the empty set.
Formal definition
A recursive definition of well-founded hereditarily finite sets is as follows:
Base case: The empty set is a hereditarily finite set.
Recursion rule: If a1,...,ak are hereditarily finite, then so is {a1,...,ak}.
and only sets that can be built by a finite number of applications of these two rules are hereditarily finite.
The set is an example for such a hereditarily finite set and so is the empty set .
On the other hand, the sets or are examples of finite sets that are not hereditarily finite. For example, the first cannot be hereditarily finite since it contains at least one infinite set as an element, when .
Discussion
The class of hereditarily finite sets is denoted by , meaning that the cardinality of each member is smaller than . (Analogously, the class of hereditarily countable sets is denoted by .)
It can also be denoted by , which denotes the th stage of the von Neumann universe.
The class is countable.
Ackermann coding
In 1937, Wilhelm Ackermann introduced an encoding of hereditarily finite sets as natural numbers.
It is defined by a function that maps each hereditarily finite set to a natural number, given by the following recursive definition:
For example, the empty set contains no members, and is therefore mapped to an empty sum, that is, the number zero. On the other hand, a set with distinct members is mapped to .
The inverse of , which maps natural numbers back to sets, is
where BIT denotes the BIT predicate.
The Ackermann coding can be used to construct a model of finitary set theory in the natural numbers. More precisely, (where is the converse relation of BIT, swapping its two arguments) models Zermelo–Fraenkel set theory without the axiom of infinity. Here, each natural number models a set, and the BIT relation models the membership relation between sets.
Representation
This class of sets is naturally ranked by the number of bracket pairs necessary to represent the sets:
(i.e. , the Neumann ordinal "0")
(i.e. or , the Neumann ordinal "1")
and then also (i.e. , the Neumann ordinal "2"),
, as well as ,
... sets represented with bracket pairs, e.g. . There are six such sets
... sets represented with bracket pairs, e.g. . There are twelve such sets
... sets represented with bracket pairs, e.g. or (i.e. , the Neumann ordinal "3")
... etc.
In this way, the number of sets with bracket pairs is
Axiomatizations
Theories of finite sets
The set also represents the first von Neumann ordinal number, denoted .
And indeed all finite von Neumann ordinals are in and thus the class of sets representing the natural numbers, i.e it includes each element in the standard model of natural numbers.
Robinson arithmetic can alr
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https://en.wikipedia.org/wiki/Quotient
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In arithmetic, a quotient (from 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics. It has two definitions: either the integer part of a division (in the case of Euclidean division), or as a fraction or a ratio (in the case of a general division). For example, when dividing 20 (the dividend) by 3 (the divisor), the quotient is 6 (with a remainder of 2) in the first sense, and (a repeating decimal) in the second sense.
In metrology (International System of Quantities and the International System of Units), "quotient" refers to the general case with respect to the units of measurement of physical quantities.
Ratios is the special case for dimensionless quotients of two quantities of the same kind.
Quotients with a non-trivial dimension and compound units, especially when the divisor is a duration (e.g., "per second"), are known as rates.
For example, density (mass divided by volume, in units of kg/m3) is said to be a "quotient", whereas mass fraction (mass divided by mass, in kg/kg or in percent) is a "ratio".
Specific quantities are intensive quantities resulting from the quotient of a physical quantity by mass, volume, or other measures of the system "size".
Notation
The quotient is most frequently encountered as two numbers, or two variables, divided by a horizontal line. The words "dividend" and "divisor" refer to each individual part, while the word "quotient" refers to the whole.
Integer part definition
The quotient is also less commonly defined as the greatest whole number of times a divisor may be subtracted from a dividend—before making the remainder negative. For example, the divisor 3 may be subtracted up to 6 times from the dividend 20, before the remainder becomes negative:
20 − 3 − 3 − 3 − 3 − 3 − 3 ≥ 0,
while
20 − 3 − 3 − 3 − 3 − 3 − 3 − 3 < 0.
In this sense, a quotient is the integer part of the ratio of two numbers.
Quotient of two integers
A rational number can be defined as the quotient of two integers (as long as the denominator is non-zero).
A more detailed definition goes as follows:
A real number r is rational, if and only if it can be expressed as a quotient of two integers with a nonzero denominator. A real number that is not rational is irrational.
Or more formally:
Given a real number r, r is rational if and only if there exists integers a and b such that and .
The existence of irrational numbers—numbers that are not a quotient of two integers—was first discovered in geometry, in such things as the ratio of the diagonal to the side in a square.
More general quotients
Outside of arithmetic, many branches of mathematics have borrowed the word "quotient" to describe structures built by breaking larger structures into pieces. Given a set with an equivalence relation defined on it, a "quotient set" may be created which contains those equivalence classes as elements. A quotient group may be formed by breaking a g
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https://en.wikipedia.org/wiki/Fluctuation%20theorem
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The fluctuation theorem (FT), which originated from statistical mechanics, deals with the relative probability that the entropy of a system which is currently away from thermodynamic equilibrium (i.e., maximum entropy) will increase or decrease over a given amount of time. While the second law of thermodynamics predicts that the entropy of an isolated system should tend to increase until it reaches equilibrium, it became apparent after the discovery of statistical mechanics that the second law is only a statistical one, suggesting that there should always be some nonzero probability that the entropy of an isolated system might spontaneously decrease; the fluctuation theorem precisely quantifies this probability.
Statement
Roughly, the fluctuation theorem relates to the probability distribution of the time-averaged irreversible entropy production, denoted . The theorem states that, in systems away from equilibrium over a finite time t, the ratio between the probability that takes on a value A and the probability that it takes the opposite value, −A, will be exponential in At.
In other words, for a finite non-equilibrium system in a finite time, the FT gives a precise mathematical expression for the probability that entropy will flow in a direction opposite to that dictated by the second law of thermodynamics.
Mathematically, the FT is expressed as:
This means that as the time or system size increases (since is extensive), the probability of observing an entropy production opposite to that dictated by the second law of thermodynamics decreases exponentially. The FT is one of the few expressions in non-equilibrium statistical mechanics that is valid far from equilibrium.
Note that the FT does not state that the second law of thermodynamics is wrong or invalid. The second law of thermodynamics is a statement about macroscopic systems. The FT is more general. It can be applied to both microscopic and macroscopic systems. When applied to macroscopic systems, the FT is equivalent to the Second Law of Thermodynamics.
History
The FT was first proposed and tested using computer simulations, by Denis Evans, E.G.D. Cohen and Gary Morriss in 1993. The first derivation was given by Evans and Debra Searles in 1994. Since then, much mathematical and computational work has been done to show that the FT applies to a variety of statistical ensembles. The first laboratory experiment that verified the validity of the FT was carried out in 2002. In this experiment, a plastic bead was pulled through a solution by a laser. Fluctuations in the velocity were recorded that were opposite to what the second law of thermodynamics would dictate for macroscopic systems. In 2020, observations at high spatial and spectral resolution of the solar photosphere have shown that solar turbulent convection satisfies the symmetries predicted by the fluctuation relation at a local level.
Second law inequality
A simple consequence of the fluctuation theorem given above is that if w
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https://en.wikipedia.org/wiki/Regular%20polygon
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In Euclidean geometry, a regular polygon is a polygon that is direct equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either convex, star or skew. In the limit, a sequence of regular polygons with an increasing number of sides approximates a circle, if the perimeter or area is fixed, or a regular apeirogon (effectively a straight line), if the edge length is fixed.
General properties
These properties apply to all regular polygons, whether convex or star.
A regular n-sided polygon has rotational symmetry of order n.
All vertices of a regular polygon lie on a common circle (the circumscribed circle); i.e., they are concyclic points. That is, a regular polygon is a cyclic polygon.
Together with the property of equal-length sides, this implies that every regular polygon also has an inscribed circle or incircle that is tangent to every side at the midpoint. Thus a regular polygon is a tangential polygon.
A regular n-sided polygon can be constructed with compass and straightedge if and only if the odd prime factors of n are distinct Fermat primes. See constructible polygon.
A regular n-sided polygon can be constructed with origami if and only if for some , where each distinct is a Pierpont prime.
Symmetry
The symmetry group of an n-sided regular polygon is dihedral group Dn (of order 2n): D2, D3, D4, ... It consists of the rotations in Cn, together with reflection symmetry in n axes that pass through the center. If n is even then half of these axes pass through two opposite vertices, and the other half through the midpoint of opposite sides. If n is odd then all axes pass through a vertex and the midpoint of the opposite side.
Regular convex polygons
All regular simple polygons (a simple polygon is one that does not intersect itself anywhere) are convex. Those having the same number of sides are also similar.
An n-sided convex regular polygon is denoted by its Schläfli symbol {n}. For n < 3, we have two degenerate cases:
Monogon {1} Degenerate in ordinary space. (Most authorities do not regard the monogon as a true polygon, partly because of this, and also because the formulae below do not work, and its structure is not that of any abstract polygon.)
Digon {2}; a "double line segment" Degenerate in ordinary space. (Some authorities do not regard the digon as a true polygon because of this.)
In certain contexts all the polygons considered will be regular. In such circumstances it is customary to drop the prefix regular. For instance, all the faces of uniform polyhedra must be regular and the faces will be described simply as triangle, square, pentagon, etc.
Angles
For a regular convex n-gon, each interior angle has a measure of:
degrees;
radians; or
full turns,
and each exterior angle (i.e., supplementary to the interior angle) has a measure of degrees, with the sum of the exterior angles equal to 360 degrees or 2π radians or one full turn.
As n approaches inf
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https://en.wikipedia.org/wiki/Michael%20Spivak
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Michael David Spivak (May 25, 1940October 1, 2020) was an American mathematician specializing in differential geometry, an expositor of mathematics, and the founder of Publish-or-Perish Press. Spivak was the author of the five-volume A Comprehensive Introduction to Differential Geometry.
Biography
Spivak was born in Queens, New York. He received his Bachelor of Arts (A.B.) from Harvard University in 1960, and in 1964 he received his Ph.D. from Princeton University under the supervision of John Milnor, with thesis On Spaces Satisfying Poincaré Duality. In 1985, Spivak received the Leroy P. Steele Prize.
Spivak lectured on elementary physics. Spivak's book, Physics for Mathematicians: Mechanics I (published December 6, 2010), contains the material that these lectures stemmed from and more. Spivak was also the designer of the MathTime Professional 2 fonts (which are widely used in academic publishing) and the creator of Science International.
Writing
His five-volume A Comprehensive Introduction to Differential Geometry (Publish or Perish Inc., 1970; 2nd ed., 1979; 3rd ed., 1999, revised 2005) is among his most influential and celebrated works. The distinctive pedagogical aim of the work, as stated in its preface, was to elucidate for graduate students the often obscure relationship between classical differential geometry—geometrically intuitive but imprecise—and its modern counterpart, replete with precise but unintuitive algebraic definitions. On several occasions, most prominently in Volume 2, Spivak "translates" the classical language that Gauss or Riemann would be familiar with to the abstract language that a modern differential geometer might use. The Leroy P. Steele Prize was awarded to Spivak in 1985 for his authorship of the work.
Spivak also authored several well-known undergraduate textbooks. Among them, his textbook Calculus (W. A. Benjamin Inc., 1967; Publish or Perish, 4th ed., 2008) takes a rigorous and theoretical approach to introductory calculus and includes proofs of many theorems taken on faith in most other introductory textbooks. Spivak acknowledged in the preface of the second edition that the work is arguably an introduction to mathematical analysis rather than a calculus textbook. Another of his well-known textbooks is Calculus on Manifolds (W. A. Benjamin Inc., 1965; Addison-Wesley, revised edition, 1968), a concise (146 pages) but rigorous and modern treatment of multivariable calculus accessible to advanced undergraduates.
Spivak also wrote The Joy of TeX: A Gourmet Guide to Typesetting with the AMS-TeX Macro Package and The Hitchhiker's Guide to Calculus. The book Morse Theory by John Milnor was based on lecture notes by Spivak and Robert Wells (as mentioned on the cover page of the booklet).
Spivak pronouns
Spivak used a set of English gender-neutral pronouns in his book The Joy of TeX, which are often referred to as Spivak pronouns. (Spivak stated that he did not originate these pronouns.)
Bibliography
Ca
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https://en.wikipedia.org/wiki/Nowhere%20continuous%20function
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In mathematics, a nowhere continuous function, also called an everywhere discontinuous function, is a function that is not continuous at any point of its domain. If is a function from real numbers to real numbers, then is nowhere continuous if for each point there is some such that for every we can find a point such that and . Therefore, no matter how close we get to any fixed point, there are even closer points at which the function takes not-nearby values.
More general definitions of this kind of function can be obtained, by replacing the absolute value by the distance function in a metric space, or by using the definition of continuity in a topological space.
Examples
Dirichlet function
One example of such a function is the indicator function of the rational numbers, also known as the Dirichlet function. This function is denoted as and has domain and codomain both equal to the real numbers. By definition, is equal to if is a rational number and it is if otherwise.
More generally, if is any subset of a topological space such that both and the complement of are dense in then the real-valued function which takes the value on and on the complement of will be nowhere continuous. Functions of this type were originally investigated by Peter Gustav Lejeune Dirichlet.
Non-trivial additive functions
A function is called an if it satisfies Cauchy's functional equation:
For example, every map of form where is some constant, is additive (in fact, it is linear and continuous). Furthermore, every linear map is of this form (by taking ).
Although every linear map is additive, not all additive maps are linear. An additive map is linear if and only if there exists a point at which it is continuous, in which case it is continuous everywhere. Consequently, every non-linear additive function is discontinuous at every point of its domain.
Nevertheless, the restriction of any additive function to any real scalar multiple of the rational numbers is continuous; explicitly, this means that for every real the restriction to the set is a continuous function.
Thus if is a non-linear additive function then for every point is discontinuous at but is also contained in some dense subset on which 's restriction is continuous (specifically, take if and take if ).
Discontinuous linear maps
A linear map between two topological vector spaces, such as normed spaces for example, is continuous (everywhere) if and only if there exists a point at which it is continuous, in which case it is even uniformly continuous. Consequently, every linear map is either continuous everywhere or else continuous nowhere.
Every linear functional is a linear map and on every infinite-dimensional normed space, there exists some discontinuous linear functional.
Other functions
The Conway base 13 function is discontinuous at every point.
Hyperreal characterisation
A real function is nowhere continuous if its natural hyperreal extension has th
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https://en.wikipedia.org/wiki/Free%20object
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In mathematics, the idea of a free object is one of the basic concepts of abstract algebra. Informally, a free object over a set A can be thought of as being a "generic" algebraic structure over A: the only equations that hold between elements of the free object are those that follow from the defining axioms of the algebraic structure. Examples include free groups, tensor algebras, or free lattices.
The concept is a part of universal algebra, in the sense that it relates to all types of algebraic structure (with finitary operations). It also has a formulation in terms of category theory, although this is in yet more abstract terms.
Definition
Free objects are the direct generalization to categories of the notion of basis in a vector space. A linear function between vector spaces is entirely determined by its values on a basis of the vector space The following definition translates this to any category.
A concrete category is a category that is equipped with a faithful functor to Set, the category of sets. Let be a concrete category with a faithful functor . Let be a set (that is, an object in Set), which will be the basis of the free object to be defined. A free object on is a pair consisting of an object in and an injection (called the canonical injection), that satisfies the following universal property:
For any object in and any map between sets , there exists a unique morphism in such that . That is, the following diagram commutes:
If free objects exist in , the universal property implies every map between two sets induces a unique morphism between the free objects built on them, and this defines a functor . It follows that, if free objects exist in , the functor , called the free functor is a left adjoint to the forgetful functor ; that is, there is a bijection
Examples
The creation of free objects proceeds in two steps. For algebras that conform to the associative law, the first step is to consider the collection of all possible words formed from an alphabet. Then one imposes a set of equivalence relations upon the words, where the relations are the defining relations of the algebraic object at hand. The free object then consists of the set of equivalence classes.
Consider, for example, the construction of the free group in two generators. One starts with an alphabet consisting of the five letters . In the first step, there is not yet any assigned meaning to the "letters" or ; these will be given later, in the second step. Thus, one could equally well start with the alphabet in five letters that is . In this example, the set of all words or strings will include strings such as aebecede and abdc, and so on, of arbitrary finite length, with the letters arranged in every possible order.
In the next step, one imposes a set of equivalence relations. The equivalence relations for a group are that of multiplication by the identity, , and the multiplication of inverses: . Applying these relations to the strings above, one obt
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https://en.wikipedia.org/wiki/Ultrametric%20space
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In mathematics, an ultrametric space is a metric space in which the triangle inequality is strengthened to . Sometimes the associated metric is also called a non-Archimedean metric or super-metric.
Formal definition
An ultrametric on a set is a real-valued function
(where denote the real numbers), such that for all :
;
(symmetry);
;
if then ;
} (strong triangle inequality or ultrametric inequality).
An ultrametric space is a pair consisting of a set together with an ultrametric on , which is called the space's associated distance function (also called a metric).
If satisfies all of the conditions except possibly condition 4 then is called an ultrapseudometric on . An ultrapseudometric space is a pair consisting of a set and an ultrapseudometric on .
In the case when is an Abelian group (written additively) and is generated by a length function (so that ), the last property can be made stronger using the Krull sharpening to:
with equality if .
We want to prove that if , then the equality occurs if . Without loss of generality, let us assume that . This implies that . But we can also compute . Now, the value of cannot be , for if that is the case, we have contrary to the initial assumption. Thus, , and . Using the initial inequality, we have and therefore .
Properties
From the above definition, one can conclude several typical properties of ultrametrics. For example, for all , at least one of the three equalities or or holds. That is, every triple of points in the space forms an isosceles triangle, so the whole space is an isosceles set.
Defining the (open) ball of radius centred at as , we have the following properties:
Every point inside a ball is its center, i.e. if then .
Intersecting balls are contained in each other, i.e. if is non-empty then either or .
All balls of strictly positive radius are both open and closed sets in the induced topology. That is, open balls are also closed, and closed balls (replace with ) are also open.
The set of all open balls with radius and center in a closed ball of radius forms a partition of the latter, and the mutual distance of two distinct open balls is (greater or) equal to .
Proving these statements is an instructive exercise. All directly derive from the ultrametric triangle inequality. Note that, by the second statement, a ball may have several center points that have non-zero distance. The intuition behind such seemingly strange effects is that, due to the strong triangle inequality, distances in ultrametrics do not add up.
Examples
The discrete metric is an ultrametric.
The p-adic numbers form a complete ultrametric space.
Consider the set of words of arbitrary length (finite or infinite), Σ*, over some alphabet Σ. Define the distance between two different words to be 2−n, where n is the first place at which the words differ. The resulting metric is an ultrametric.
The set of words with glued ends of the length n over some alphabet Σ is an
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https://en.wikipedia.org/wiki/Isolated%20singularity
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In complex analysis, a branch of mathematics, an isolated singularity is one that has no other singularities close to it. In other words, a complex number z0 is an isolated singularity of a function f if there exists an open disk D centered at z0 such that f is holomorphic on D \ {z0}, that is, on the set obtained from D by taking z0 out.
Formally, and within the general scope of general topology, an isolated singularity of a holomorphic function is any isolated point of the boundary of the domain . In other words, if is an open subset of , and is a holomorphic function, then is an isolated singularity of .
Every singularity of a meromorphic function on an open subset is isolated, but isolation of singularities alone is not sufficient to guarantee a function is meromorphic. Many important tools of complex analysis such as Laurent series and the residue theorem require that all relevant singularities of the function be isolated.
There are three types of isolated singularities: removable singularities, poles and essential singularities.
Examples
The function has 0 as an isolated singularity.
The cosecant function has every integer as an isolated singularity.
Nonisolated singularities
Other than isolated singularities, complex functions of one variable may exhibit other singular behavior. Namely, two kinds of nonisolated singularities exist:
Cluster points, i.e. limit points of isolated singularities: if they are all poles, despite admitting Laurent series expansions on each of them, no such expansion is possible at its limit.
Natural boundaries, i.e. any non-isolated set (e.g. a curve) around which functions cannot be analytically continued (or outside them if they are closed curves in the Riemann sphere).
Examples
The function is meromorphic on , with simple poles at , for every . Since , every punctured disk centered at has an infinite number of singularities within, so no Laurent expansion is available for around , which is in fact a cluster point of its poles.
The function has a singularity at 0 which is not isolated, since there are additional singularities at the reciprocal of every integer, which are located arbitrarily close to 0 (though the singularities at these reciprocals are themselves isolated).
The function defined via the Maclaurin series converges inside the open unit disk centred at and has the unit circle as its natural boundary.
External links
Ahlfors, L., Complex Analysis, 3 ed. (McGraw-Hill, 1979).
Rudin, W., Real and Complex Analysis, 3 ed. (McGraw-Hill, 1986).
Complex analysis
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https://en.wikipedia.org/wiki/Self-healing%20ring
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A self-healing ring, or SHR, is a telecommunications term for loop network topology, a common configuration in telecommunications transmission systems. Like roadway and water distribution systems, a loop or ring is used to provide redundancy. SDH, SONET and WDM systems are often configured in self-healing rings.
Description
The system consists of a ring of bidirectional links between a set of stations, typically using optical fiber communications.
In normal use, traffic is dispatched in the direction of the shortest path towards its destination.
In the event of the loss of a link, or of an entire station, the two nearest surviving stations "loop back" their ends of the ring. In this way, traffic can still travel to all surviving parts of the ring, even if it has to travel "the long way round".
A second break in the ring may divide it into two sub-rings, but in such a case each sub-ring will remain functional.
Advantages
Self-healing rings offer high levels of resilience at low cost, since it is often geographically easy to take multiple paths across the landscape and link them up into a ring with very little extra fiber length.
Recent submarine communications cables are typically built in pairs to function as a self-healing ring. Very high resilience systems are typically built on interconnected meshes of self-healing rings.
Another example of a self-healing ring network technology is the FDDI local-area network. Resilient Packet Ring is a new technology for packet-switched self-healing ring networks.
See also
Redundancy (engineering)
Telecommunications equipment
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