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https://en.wikipedia.org/wiki/Inverse
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Inverse or invert may refer to:
Science and mathematics
Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence
Additive inverse (negation), the inverse of a number that, when added to the original number, yields zero
Compositional inverse, a function that "reverses" another function
Inverse element
Inverse function, a function that "reverses" another function
Generalized inverse, a matrix that has some properties of the inverse matrix but not necessarily all of them
Multiplicative inverse (reciprocal), a number which when multiplied by a given number yields the multiplicative identity, 1
Inverse matrix of an Invertible matrix
Other uses
Invert level, the base interior level of a pipe, trench or tunnel
Inverse (website), an online magazine
An outdated term for an LGBT person; see Sexual inversion (sexology)
See also
Inversion (disambiguation)
Inverter (disambiguation)
Opposite (disambiguation)
Reverse (disambiguation)
Complement (disambiguation)
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https://en.wikipedia.org/wiki/Cohomology
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In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology. Some versions of cohomology arise by dualizing the construction of homology. In other words, cochains are functions on the group of chains in homology theory.
From its beginning in topology, this idea became a dominant method in the mathematics of the second half of the twentieth century. From the initial idea of homology as a method of constructing algebraic invariants of topological spaces, the range of applications of homology and cohomology theories has spread throughout geometry and algebra. The terminology tends to hide the fact that cohomology, a contravariant theory, is more natural than homology in many applications. At a basic level, this has to do with functions and pullbacks in geometric situations: given spaces X and Y, and some kind of function F on Y, for any mapping , composition with f gives rise to a function on X. The most important cohomology theories have a product, the cup product, which gives them a ring structure. Because of this feature, cohomology is usually a stronger invariant than homology.
Singular cohomology
Singular cohomology is a powerful invariant in topology, associating a graded-commutative ring with any topological space. Every continuous map f: X → Y determines a homomorphism from the cohomology ring of Y to that of X; this puts strong restrictions on the possible maps from X to Y. Unlike more subtle invariants such as homotopy groups, the cohomology ring tends to be computable in practice for spaces of interest.
For a topological space X, the definition of singular cohomology starts with the singular chain complex:
By definition, the singular homology of X is the homology of this chain complex (the kernel of one homomorphism modulo the image of the previous one). In more detail, Ci is the free abelian group on the set of continuous maps from the standard i-simplex to X (called "singular i-simplices in X"), and ∂i is the i-th boundary homomorphism. The groups Ci are zero for i negative.
Now fix an abelian group A, and replace each group Ci by its dual group and by its dual homomorphism
This has the effect of "reversing all the arrows" of the original complex, leaving a cochain complex
For an integer i, the ith cohomology group of X with coefficients in A is defined to be ker(di)/im(di−1) and denoted by Hi(X, A). The group Hi(X, A) is zero for i negative. The elements of are called singular i-cochains with coefficients in A. (Equivalently, an i-cochain on X can be identified with a function from the set of singular i-simplices in X to A.) Elements of ker(d) and im(d) are called cocycles and coboundaries, respectively, while elements of ker(d)/im(d) = Hi(X, A) are called cohomol
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https://en.wikipedia.org/wiki/Solitary%20wave
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In mathematics and physics, a solitary wave can refer to
The solitary wave (water waves) or wave of translation, as observed by John Scott Russell in 1834, the prototype for a soliton.
A soliton, a generalization of the wave of translation to general systems of partial differential equations
A topological defect, a generalization of the idea of a soliton to any system which is stable against decay due to homotopy theory
zh:孤波
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https://en.wikipedia.org/wiki/Heron%27s%20formula
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In geometry, Heron's formula (or Hero's formula) gives the area of a triangle in terms of the three side lengths , , . If is the semiperimeter of the triangle, the area is,
It is named after first-century engineer Heron of Alexandria (or Hero) who proved it in his work Metrica, though it was probably known centuries earlier.
Example
Let be the triangle with sides , and .
This triangle's semiperimeter is
and so the area is
In this example, the side lengths and area are integers, making it a Heronian triangle. However, Heron's formula works equally well in cases where one or more of the side lengths are not integers.
Alternate expressions
Heron's formula can also be written in terms of just the side lengths instead of using the semiperimeter, in several ways,
After expansion, the expression under the square root is a quadratic polynomial of the squared side lengths , , .
The same relation can be expressed using the Cayley–Menger determinant,
History
The formula is credited to Heron (or Hero) of Alexandria ( 60 AD), and a proof can be found in his book Metrica. Mathematical historian Thomas Heath suggested that Archimedes knew the formula over two centuries earlier, and since Metrica is a collection of the mathematical knowledge available in the ancient world, it is possible that the formula predates the reference given in that work.
A formula equivalent to Heron's, namely,
was discovered by the Chinese. It was published in Mathematical Treatise in Nine Sections (Qin Jiushao, 1247).
Proofs
There are many ways to prove Heron's formula, for example using trigonometry as below, or the incenter and one excircle of the triangle, or as a special case of De Gua's theorem (for the particular case of acute triangles), or as a special case of Brahmagupta's formula (for the case of a degenerate cyclic quadrilateral).
Trigonometric proof using the law of cosines
A modern proof, which uses algebra and is quite different from the one provided by Heron, follows.
Let , , be the sides of the triangle and , , the angles opposite those sides.
Applying the law of cosines we get
From this proof, we get the algebraic statement that
The altitude of the triangle on base has length , and it follows
Algebraic proof using the Pythagorean theorem
The following proof is very similar to one given by Raifaizen.
By the Pythagorean theorem we have and according to the figure at the right. Subtracting these yields . This equation allows us to express in terms of the sides of the triangle:
For the height of the triangle we have that . By replacing with the formula given above and applying the difference of squares identity we get
We now apply this result to the formula that calculates the area of a triangle from its height:
Trigonometric proof using the law of cotangents
If is the radius of the incircle of the triangle, then the triangle can be broken into three triangles of equal altitude and bases , , and . Their combined area is
where i
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https://en.wikipedia.org/wiki/Brahmagupta%27s%20formula
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In Euclidean geometry, Brahmagupta's formula, named after the 7th century Indian mathematician, is used to find the area of any cyclic quadrilateral (one that can be inscribed in a circle) given the lengths of the sides. Its generalized version, Bretschneider's formula, can be used with non-cyclic quadrilateral.
Heron's formula can be thought as a special case of the Brahmagupta's formula for triangles.
Formulation
Brahmagupta's formula gives the area of a cyclic quadrilateral whose sides have lengths , , , as
where , the semiperimeter, is defined to be
This formula generalizes Heron's formula for the area of a triangle. A triangle may be regarded as a quadrilateral with one side of length zero. From this perspective, as approaches zero, a cyclic quadrilateral converges into a cyclic triangle (all triangles are cyclic), and Brahmagupta's formula simplifies to Heron's formula.
If the semiperimeter is not used, Brahmagupta's formula is
Another equivalent version is
Proof
Trigonometric proof
Here the notations in the figure to the right are used. The area of the cyclic quadrilateral equals the sum of the areas of and :
But since is a cyclic quadrilateral, . Hence . Therefore,
(using the trigonometric identity).
Solving for common side , in and , the law of cosines gives
Substituting (since angles and are supplementary) and rearranging, we have
Substituting this in the equation for the area,
The right-hand side is of the form and hence can be written as
which, upon rearranging the terms in the square brackets, yields
that can be factored again into
Introducing the semiperimeter yields
Taking the square root, we get
Non-trigonometric proof
An alternative, non-trigonometric proof utilizes two applications of Heron's triangle area formula on similar triangles.
Extension to non-cyclic quadrilaterals
In the case of non-cyclic quadrilaterals, Brahmagupta's formula can be extended by considering the measures of two opposite angles of the quadrilateral:
where is half the sum of any two opposite angles. (The choice of which pair of opposite angles is irrelevant: if the other two angles are taken, half their sum is . Since , we have .) This more general formula is known as Bretschneider's formula.
It is a property of cyclic quadrilaterals (and ultimately of inscribed angles) that opposite angles of a quadrilateral sum to 180°. Consequently, in the case of an inscribed quadrilateral, is 90°, whence the term
giving the basic form of Brahmagupta's formula. It follows from the latter equation that the area of a cyclic quadrilateral is the maximum possible area for any quadrilateral with the given side lengths.
A related formula, which was proved by Coolidge, also gives the area of a general convex quadrilateral. It is
where and are the lengths of the diagonals of the quadrilateral. In a cyclic quadrilateral, according to Ptolemy's theorem, and the formula of Coolidge reduces to Brahmagupta's formula.
Rela
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https://en.wikipedia.org/wiki/Cyclic%20quadrilateral
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In Euclidean geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral whose vertices all lie on a single circle. This circle is called the circumcircle or circumscribed circle, and the vertices are said to be concyclic. The center of the circle and its radius are called the circumcenter and the circumradius respectively. Other names for these quadrilaterals are concyclic quadrilateral and chordal quadrilateral, the latter since the sides of the quadrilateral are chords of the circumcircle. Usually the quadrilateral is assumed to be convex, but there are also crossed cyclic quadrilaterals. The formulas and properties given below are valid in the convex case.
The word cyclic is from the Ancient Greek (kuklos), which means "circle" or "wheel".
All triangles have a circumcircle, but not all quadrilaterals do. An example of a quadrilateral that cannot be cyclic is a non-square rhombus. The section characterizations below states what necessary and sufficient conditions a quadrilateral must satisfy to have a circumcircle.
Special cases
Any square, rectangle, isosceles trapezoid, or antiparallelogram is cyclic. A kite is cyclic if and only if it has two right angles – a right kite. A bicentric quadrilateral is a cyclic quadrilateral that is also tangential and an ex-bicentric quadrilateral is a cyclic quadrilateral that is also ex-tangential. A harmonic quadrilateral is a cyclic quadrilateral in which the product of the lengths of opposite sides are equal.
Characterizations
Circumcenter
A convex quadrilateral is cyclic if and only if the four perpendicular bisectors to the sides are concurrent. This common point is the circumcenter.
Supplementary angles
A convex quadrilateral is cyclic if and only if its opposite angles are supplementary, that is
The direct theorem was Proposition 22 in Book 3 of Euclid's Elements. Equivalently, a convex quadrilateral is cyclic if and only if each exterior angle is equal to the opposite interior angle.
In 1836 Duncan Gregory generalized this result as follows: Given any convex cyclic 2n-gon, then the two sums of alternate interior angles are each equal to (n-1).
Taking the stereographic projection (half-angle tangent) of each angle, this can be re-expressed,
Which implies that
Angles between sides and diagonals
A convex quadrilateral is cyclic if and only if an angle between a side and a diagonal is equal to the angle between the opposite side and the other diagonal. That is, for example,
Pascal Points
Another necessary and sufficient conditions for a convex quadrilateral to be cyclic are: let be the point of intersection of the diagonals, let be the intersection point of the extensions of the sides and , let be a circle whose diameter is the segment, , and let and be Pascal points on sides and formed by the circle .
(1) is a cyclic quadrilateral if and only if points and are collinear with the center , of circle .
(2) is a cyclic quadrilateral if and only if poin
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https://en.wikipedia.org/wiki/Completeness%20%28statistics%29
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In statistics, completeness is a property of a statistic in relation to a parameterised model for a set of observed data.
A complete statistic T is one for which any proposed distribution on the domain of T is predicted by one or more prior distributions on the model parameter space. In other words, the model space is 'rich enough' that every possible distribution of T can be explained by some prior distribution on the model parameter space. In contrast, a sufficient statistic T is one for which any two prior distributions will yield different distributions on T. (This last statement assumes that the model space is identifiable, i.e. that there are no 'duplicate' parameter values. This is a minor point.)
Put another way: assume that we have an identifiable model space parameterised by , and a statistic (which is effectively just a function of one or more i.i.d. random variables drawn from the model). Then consider the map which takes each distribution on model parameter to its induced distribution on statistic . The statistic is said to be complete when is surjective, and sufficient when is injective.
Definition
Consider a random variable X whose probability distribution belongs to a parametric model Pθ parametrized by θ.
Say T is a statistic; that is, the composition of a measurable function with a random sample X1,...,Xn.
The statistic T is said to be complete for the distribution of X if, for every measurable function g,:
The statistic T is said to be boundedly complete for the distribution of X if this implication holds for every measurable function g that is also bounded.
Example 1: Bernoulli model
The Bernoulli model admits a complete statistic. Let X be a random sample of size n such that each Xi has the same Bernoulli distribution with parameter p. Let T be the number of 1s observed in the sample, i.e. . T is a statistic of X which has a binomial distribution with parameters (n,p). If the parameter space for p is (0,1), then T is a complete statistic. To see this, note that
Observe also that neither p nor 1 − p can be 0. Hence if and only if:
On denoting p/(1 − p) by r, one gets:
First, observe that the range of r is the positive reals. Also, E(g(T)) is a polynomial in r and, therefore, can only be identical to 0 if all coefficients are 0, that is, g(t) = 0 for all t.
It is important to notice that the result that all coefficients must be 0 was obtained because of the range of r. Had the parameter space been finite and with a number of elements less than or equal to n, it might be possible to solve the linear equations in g(t) obtained by substituting the values of r and get solutions different from 0. For example, if n = 1 and the parameter space is {0.5}, a single observation and a single parameter value, T is not complete. Observe that, with the definition:
then, E(g(T)) = 0 although g(t) is not 0 for t = 0 nor for t = 1.
Relation to sufficient statistics
For some parametric families, a complete sufficient stat
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https://en.wikipedia.org/wiki/Moment-generating%20function
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In probability theory and statistics, the moment-generating function of a real-valued random variable is an alternative specification of its probability distribution. Thus, it provides the basis of an alternative route to analytical results compared with working directly with probability density functions or cumulative distribution functions. There are particularly simple results for the moment-generating functions of distributions defined by the weighted sums of random variables. However, not all random variables have moment-generating functions.
As its name implies, the moment-generating function can be used to compute a distribution’s moments: the nth moment about 0 is the nth derivative of the moment-generating function, evaluated at 0.
In addition to real-valued distributions (univariate distributions), moment-generating functions can be defined for vector- or matrix-valued random variables, and can even be extended to more general cases.
The moment-generating function of a real-valued distribution does not always exist, unlike the characteristic function. There are relations between the behavior of the moment-generating function of a distribution and properties of the distribution, such as the existence of moments.
Definition
Let be a random variable with CDF . The moment generating function (mgf) of (or ), denoted by , is
provided this expectation exists for in some neighborhood of 0. That is, there is an such that for all in , exists. If the expectation does not exist in a neighborhood of 0, we say that the moment generating function does not exist.
In other words, the moment-generating function of X is the expectation of the random variable . More generally, when , an -dimensional random vector, and is a fixed vector, one uses instead of :
always exists and is equal to 1. However, a key problem with moment-generating functions is that moments and the moment-generating function may not exist, as the integrals need not converge absolutely. By contrast, the characteristic function or Fourier transform always exists (because it is the integral of a bounded function on a space of finite measure), and for some purposes may be used instead.
The moment-generating function is so named because it can be used to find the moments of the distribution. The series expansion of is
Hence
where is the th moment. Differentiating times with respect to and setting , we obtain the th moment about the origin, ;
see Calculations of moments below.
If is a continuous random variable, the following relation between its moment-generating function and the two-sided Laplace transform of its probability density function holds:
since the PDF's two-sided Laplace transform is given as
and the moment-generating function's definition expands (by the law of the unconscious statistician) to
This is consistent with the characteristic function of being a Wick rotation of when the moment generating function exists, as the characteris
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https://en.wikipedia.org/wiki/Orthonormality
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In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal unit vectors. A unit vector means that the vector has a length of 1, which is also known as normalized. Orthogonal means that the vectors are all perpendicular to each other. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of unit length. An orthonormal set which forms a basis is called an orthonormal basis.
Intuitive overview
The construction of orthogonality of vectors is motivated by a desire to extend the intuitive notion of perpendicular vectors to higher-dimensional spaces. In the Cartesian plane, two vectors are said to be perpendicular if the angle between them is 90° (i.e. if they form a right angle). This definition can be formalized in Cartesian space by defining the dot product and specifying that two vectors in the plane are orthogonal if their dot product is zero.
Similarly, the construction of the norm of a vector is motivated by a desire to extend the intuitive notion of the length of a vector to higher-dimensional spaces. In Cartesian space, the norm of a vector is the square root of the vector dotted with itself. That is,
Many important results in linear algebra deal with collections of two or more orthogonal vectors. But often, it is easier to deal with vectors of unit length. That is, it often simplifies things to only consider vectors whose norm equals 1. The notion of restricting orthogonal pairs of vectors to only those of unit length is important enough to be given a special name. Two vectors which are orthogonal and of length 1 are said to be orthonormal.
Simple example
What does a pair of orthonormal vectors in 2-D Euclidean space look like?
Let u = (x1, y1) and v = (x2, y2).
Consider the restrictions on x1, x2, y1, y2 required to make u and v form an orthonormal pair.
From the orthogonality restriction, u • v = 0.
From the unit length restriction on u, ||u|| = 1.
From the unit length restriction on v, ||v|| = 1.
Expanding these terms gives 3 equations:
Converting from Cartesian to polar coordinates, and considering Equation and Equation immediately gives the result r1 = r2 = 1. In other words, requiring the vectors be of unit length restricts the vectors to lie on the unit circle.
After substitution, Equation becomes . Rearranging gives . Using a trigonometric identity to convert the cotangent term gives
It is clear that in the plane, orthonormal vectors are simply radii of the unit circle whose difference in angles equals 90°.
Definition
Let be an inner-product space. A set of vectors
is called orthonormal if and only if
where is the Kronecker delta and is the inner product defined over .
Significance
Orthonormal sets are not especially significant on their own. However, they display certain features that make them fundamental in exploring the notion of diagonalizability of certain operators on vector spaces.
Properties
Orthonormal sets
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https://en.wikipedia.org/wiki/Main%20diagonal
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In linear algebra, the main diagonal (sometimes principal diagonal, primary diagonal, leading diagonal, major diagonal, or good diagonal) of a matrix is the list of entries where . All off-diagonal elements are zero in a diagonal matrix. The following four matrices have their main diagonals indicated by red ones:
Antidiagonal
The antidiagonal (sometimes counter diagonal, secondary diagonal, trailing diagonal, minor diagonal, off diagonal, or bad diagonal) of an order square matrix is the collection of entries such that for all . That is, it runs from the top right corner to the bottom left corner.
See also
Trace
References
Matrices
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https://en.wikipedia.org/wiki/Orthonormal%20basis
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In mathematics, particularly linear algebra, an orthonormal basis for an inner product space V with finite dimension is a basis for whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, the standard basis for a Euclidean space is an orthonormal basis, where the relevant inner product is the dot product of vectors. The image of the standard basis under a rotation or reflection (or any orthogonal transformation) is also orthonormal, and every orthonormal basis for arises in this fashion.
For a general inner product space an orthonormal basis can be used to define normalized orthogonal coordinates on Under these coordinates, the inner product becomes a dot product of vectors. Thus the presence of an orthonormal basis reduces the study of a finite-dimensional inner product space to the study of under dot product. Every finite-dimensional inner product space has an orthonormal basis, which may be obtained from an arbitrary basis using the Gram–Schmidt process.
In functional analysis, the concept of an orthonormal basis can be generalized to arbitrary (infinite-dimensional) inner product spaces. Given a pre-Hilbert space an orthonormal basis for is an orthonormal set of vectors with the property that every vector in can be written as an infinite linear combination of the vectors in the basis. In this case, the orthonormal basis is sometimes called a Hilbert basis for Note that an orthonormal basis in this sense is not generally a Hamel basis, since infinite linear combinations are required. Specifically, the linear span of the basis must be dense in but it may not be the entire space.
If we go on to Hilbert spaces, a non-orthonormal set of vectors having the same linear span as an orthonormal basis may not be a basis at all. For instance, any square-integrable function on the interval can be expressed (almost everywhere) as an infinite sum of Legendre polynomials (an orthonormal basis), but not necessarily as an infinite sum of the monomials
A different generalisation is to pseudo-inner product spaces, finite-dimensional vector spaces equipped with a non-degenerate symmetric bilinear form known as the metric tensor. In such a basis, the metric takes the form with positive ones and negative ones.
Examples
For , the set of vectors is called the standard basis and forms an orthonormal basis of with respect to the standard dot product. Note that both the standard basis and standard dot product rely on viewing as the Cartesian product
Proof: A straightforward computation shows that the inner products of these vectors equals zero, and that each of their magnitudes equals one, This means that is an orthonormal set. All vectors can be expressed as a sum of the basis vectors scaled so spans and hence must be a basis. It may also be shown that the standard basis rotated about an axis through the origin or reflected in a plane through the origin also forms an orthon
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https://en.wikipedia.org/wiki/Negligible%20set
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In mathematics, a negligible set is a set that is small enough that it can be ignored for some purpose.
As common examples, finite sets can be ignored when studying the limit of a sequence, and null sets can be ignored when studying the integral of a measurable function.
Negligible sets define several useful concepts that can be applied in various situations, such as truth almost everywhere.
In order for these to work, it is generally only necessary that the negligible sets form an ideal; that is, that the empty set be negligible, the union of two negligible sets be negligible, and any subset of a negligible set be negligible.
For some purposes, we also need this ideal to be a sigma-ideal, so that countable unions of negligible sets are also negligible.
If and are both ideals of subsets of the same set , then one may speak of -negligible and -negligible subsets.
The opposite of a negligible set is a generic property, which has various forms.
Examples
Let X be the set N of natural numbers, and let a subset of N be negligible if it is finite.
Then the negligible sets form an ideal.
This idea can be applied to any infinite set; but if applied to a finite set, every subset will be negligible, which is not a very useful notion.
Or let X be an uncountable set, and let a subset of X be negligible if it is countable.
Then the negligible sets form a sigma-ideal.
Let X be a measurable space equipped with a measure m, and let a subset of X be negligible if it is m-null.
Then the negligible sets form a sigma-ideal.
Every sigma-ideal on X can be recovered in this way by placing a suitable measure on X, although the measure may be rather pathological.
Let X be the set R of real numbers, and let a subset A of R be negligible if for each ε > 0, there exists a finite or countable collection I1, I2, … of (possibly overlapping) intervals satisfying:
and
This is a special case of the preceding example, using Lebesgue measure, but described in elementary terms.
Let X be a topological space, and let a subset be negligible if it is of first category, that is, if it is a countable union of nowhere-dense sets (where a set is nowhere-dense if it is not dense in any open set).
Then the negligible sets form a sigma-ideal.
Let X be a directed set, and let a subset of X be negligible if it has an upper bound.
Then the negligible sets form an ideal.
The first example is a special case of this using the usual ordering of N.
In a coarse structure, the controlled sets are negligible.
Derived concepts
Let X be a set, and let I be an ideal of negligible subsets of X.
If p is a proposition about the elements of X, then p is true almost everywhere if the set of points where p is true is the complement of a negligible set.
That is, p may not always be true, but it's false so rarely that this can be ignored for the purposes at hand.
If f and g are functions from X to the same space Y, then f and g are equivalent if they are equal almost everywhere.
To make the in
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https://en.wikipedia.org/wiki/Riemannian%20geometry
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Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as smooth manifolds with a Riemannian metric (an inner product on the tangent space at each point that varies smoothly from point to point). This gives, in particular, local notions of angle, length of curves, surface area and volume. From those, some other global quantities can be derived by integrating local contributions.
Riemannian geometry originated with the vision of Bernhard Riemann expressed in his inaugural lecture "Ueber die Hypothesen, welche der Geometrie zu Grunde liegen" ("On the Hypotheses on which Geometry is Based"). It is a very broad and abstract generalization of the differential geometry of surfaces in R3. Development of Riemannian geometry resulted in synthesis of diverse results concerning the geometry of surfaces and the behavior of geodesics on them, with techniques that can be applied to the study of differentiable manifolds of higher dimensions. It enabled the formulation of Einstein's general theory of relativity, made profound impact on group theory and representation theory, as well as analysis, and spurred the development of algebraic and differential topology.
Introduction
Riemannian geometry was first put forward in generality by Bernhard Riemann in the 19th century. It deals with a broad range of geometries whose metric properties vary from point to point, including the standard types of non-Euclidean geometry.
Every smooth manifold admits a Riemannian metric, which often helps to solve problems of differential topology. It also serves as an entry level for the more complicated structure of pseudo-Riemannian manifolds, which (in four dimensions) are the main objects of the theory of general relativity. Other generalizations of Riemannian geometry include Finsler geometry.
There exists a close analogy of differential geometry with the mathematical structure of defects in regular crystals. Dislocations and disclinations produce torsions and curvature.
The following articles provide some useful introductory material:
Metric tensor
Riemannian manifold
Levi-Civita connection
Curvature
Riemann curvature tensor
List of differential geometry topics
Glossary of Riemannian and metric geometry
Classical theorems
What follows is an incomplete list of the most classical theorems in Riemannian geometry. The choice is made depending on its importance and elegance of formulation. Most of the results can be found in the classic monograph by Jeff Cheeger and D. Ebin (see below).
The formulations given are far from being very exact or the most general. This list is oriented to those who already know the basic definitions and want to know what these definitions are about.
General theorems
Gauss–Bonnet theorem The integral of the Gauss curvature on a compact 2-dimensional Riemannian manifold is equal to 2πχ(M) where χ(M) denotes the Euler characteristic of M. This theorem has a generalization to any compact even-
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https://en.wikipedia.org/wiki/Jacobian%20matrix%20and%20determinant
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In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variables as input as the number of vector components of its output, its determinant is referred to as the Jacobian determinant. Both the matrix and (if applicable) the determinant are often referred to simply as the Jacobian in literature.
Example
Suppose is a function such that each of its first-order partial derivatives exist on . This function takes a point as input and produces the vector as output. Then the Jacobian matrix of is defined to be an matrix, denoted by , whose th entry is , or explicitly
where is the transpose (row vector) of the gradient of the -th component.
The Jacobian matrix, whose entries are functions of , is denoted in various ways; common notations include , , , and . Some authors define the Jacobian as the transpose of the form given above.
The Jacobian matrix represents the differential of at every point where is differentiable. In detail, if is a displacement vector represented by a column matrix, the matrix product is another displacement vector, that is the best linear approximation of the change of in a neighborhood of , if is differentiable at . This means that the function that maps to is the best linear approximation of for all points close to . The linear map is known as the derivative or the differential of at .
When , the Jacobian matrix is square, so its determinant is a well-defined function of , known as the Jacobian determinant of . It carries important information about the local behavior of . In particular, the function has a differentiable inverse function in a neighborhood of a point if and only if the Jacobian determinant is nonzero at (see Jacobian conjecture for a related problem of global invertibility). The Jacobian determinant also appears when changing the variables in multiple integrals (see substitution rule for multiple variables).
When , that is when is a scalar-valued function, the Jacobian matrix reduces to the row vector ; this row vector of all first-order partial derivatives of is the transpose of the gradient of , i.e.
. Specializing further, when , that is when is a scalar-valued function of a single variable, the Jacobian matrix has a single entry; this entry is the derivative of the function .
These concepts are named after the mathematician Carl Gustav Jacob Jacobi (1804–1851).
Jacobian matrix
The Jacobian of a vector-valued function in several variables generalizes the gradient of a scalar-valued function in several variables, which in turn generalizes the derivative of a scalar-valued function of a single variable. In other words, the Jacobian matrix of a scalar-valued function in several variables is (the transpose of) its gradient and the gradient of a scalar-valued function of a single variable is its derivativ
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https://en.wikipedia.org/wiki/Einstein%20notation
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In mathematics, especially the usage of linear algebra in mathematical physics, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies summation over a set of indexed terms in a formula, thus achieving brevity. As part of mathematics it is a notational subset of Ricci calculus; however, it is often used in physics applications that do not distinguish between tangent and cotangent spaces. It was introduced to physics by Albert Einstein in 1916.
Introduction
Statement of convention
According to this convention, when an index variable appears twice in a single term and is not otherwise defined (see Free and bound variables), it implies summation of that term over all the values of the index. So where the indices can range over the set ,
is simplified by the convention to:
The upper indices are not exponents but are indices of coordinates, coefficients or basis vectors. That is, in this context should be understood as the second component of rather than the square of (this can occasionally lead to ambiguity). The upper index position in is because, typically, an index occurs once in an upper (superscript) and once in a lower (subscript) position in a term (see below). Typically, would be equivalent to the traditional .
In general relativity, a common convention is that
the Greek alphabet is used for space and time components, where indices take on values 0, 1, 2, or 3 (frequently used letters are ),
the Latin alphabet is used for spatial components only, where indices take on values 1, 2, or 3 (frequently used letters are ),
In general, indices can range over any indexing set, including an infinite set. This should not be confused with a typographically similar convention used to distinguish between tensor index notation and the closely related but distinct basis-independent abstract index notation.
An index that is summed over is a summation index, in this case "". It is also called a dummy index since any symbol can replace "" without changing the meaning of the expression (provided that it does not collide with other index symbols in the same term).
An index that is not summed over is a free index and should appear only once per term. If such an index does appear, it usually also appears in every other term in an equation. An example of a free index is the "" in the equation , which is equivalent to the equation .
Application
Einstein notation can be applied in slightly different ways. Typically, each index occurs once in an upper (superscript) and once in a lower (subscript) position in a term; however, the convention can be applied more generally to any repeated indices within a term. When dealing with covariant and contravariant vectors, where the position of an index also indicates the type of vector, the first case usually applies; a covariant vector can only be contracted with a contravariant vector, corresponding to summation of the p
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https://en.wikipedia.org/wiki/Metric%20tensor
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In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows defining distances and angles there. More precisely, a metric tensor at a point of is a bilinear form defined on the tangent space at (that is, a bilinear function that maps pairs of tangent vectors to real numbers), and a metric tensor on consists of a metric tensor at each point of that varies smoothly with .
A metric tensor is positive-definite if for every nonzero vector . A manifold equipped with a positive-definite metric tensor is known as a Riemannian manifold. Such a metric tensor can be thought of as specifying infinitesimal distance on the manifold. On a Riemannian manifold , the length of a smooth curve between two points and can be defined by integration, and the distance between and can be defined as the infimum of the lengths of all such curves; this makes a metric space. Conversely, the metric tensor itself is the derivative of the distance function (taken in a suitable manner).
While the notion of a metric tensor was known in some sense to mathematicians such as Gauss from the early 19th century, it was not until the early 20th century that its properties as a tensor were understood by, in particular, Gregorio Ricci-Curbastro and Tullio Levi-Civita, who first codified the notion of a tensor. The metric tensor is an example of a tensor field.
The components of a metric tensor in a coordinate basis take on the form of a symmetric matrix whose entries transform covariantly under changes to the coordinate system. Thus a metric tensor is a covariant symmetric tensor. From the coordinate-independent point of view, a metric tensor field is defined to be a nondegenerate symmetric bilinear form on each tangent space that varies smoothly from point to point.
Introduction
Carl Friedrich Gauss in his 1827 Disquisitiones generales circa superficies curvas (General investigations of curved surfaces) considered a surface parametrically, with the Cartesian coordinates , , and of points on the surface depending on two auxiliary variables and . Thus a parametric surface is (in today's terms) a vector-valued function
depending on an ordered pair of real variables , and defined in an open set in the -plane. One of the chief aims of Gauss's investigations was to deduce those features of the surface which could be described by a function which would remain unchanged if the surface underwent a transformation in space (such as bending the surface without stretching it), or a change in the particular parametric form of the same geometrical surface.
One natural such invariant quantity is the length of a curve drawn along the surface. Another is the angle between a pair of curves drawn along the surface and meeting at a common point. A third such quantity is the area of a piece of the
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https://en.wikipedia.org/wiki/Function%20composition
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In mathematics, function composition is an operation that takes two functions and , and produces a function such that . In this operation, the function is applied to the result of applying the function to . That is, the functions and are composed to yield a function that maps in domain to in codomain .
Intuitively, if is a function of , and is a function of , then is a function of . The resulting composite function is denoted , defined by for all in .
The notation is read as " of ", " after ", " circle ", " round ", " about ", " composed with ", " following ", " then ", or " on ", or "the composition of and ". Intuitively, composing functions is a chaining process in which the output of function feeds the input of function .
The composition of functions is a special case of the composition of relations, sometimes also denoted by . As a result, all properties of composition of relations are true of composition of functions, such as the property of associativity.
Composition of functions is different from multiplication of functions (if defined at all), and has some quite different properties; in particular, composition of functions is not commutative.
Examples
Composition of functions on a finite set: If , and , then , as shown in the figure.
Composition of functions on an infinite set: If (where is the set of all real numbers) is given by and is given by , then:
If an airplane's altitude at time is , and the air pressure at altitude is , then is the pressure around the plane at time .
Properties
The composition of functions is always associative—a property inherited from the composition of relations. That is, if , , and are composable, then . Since the parentheses do not change the result, they are generally omitted.
In a strict sense, the composition is only meaningful if the codomain of equals the domain of ; in a wider sense, it is sufficient that the former be an improper subset of the latter.
Moreover, it is often convenient to tacitly restrict the domain of , such that produces only values in the domain of . For example, the composition of the functions defined by and defined by can be defined on the interval .
The functions and are said to commute with each other if . Commutativity is a special property, attained only by particular functions, and often in special circumstances. For example, only when . The picture shows another example.
The composition of one-to-one (injective) functions is always one-to-one. Similarly, the composition of onto (surjective) functions is always onto. It follows that the composition of two bijections is also a bijection. The inverse function of a composition (assumed invertible) has the property that .
Derivatives of compositions involving differentiable functions can be found using the chain rule. Higher derivatives of such functions are given by Faà di Bruno's formula.
Composition monoids
Suppose one has two (or more) functions having the sam
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https://en.wikipedia.org/wiki/Levi-Civita%20symbol
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In mathematics, particularly in linear algebra, tensor analysis, and differential geometry, the Levi-Civita symbol or Levi-Civita epsilon represents a collection of numbers; defined from the sign of a permutation of the natural numbers , for some positive integer . It is named after the Italian mathematician and physicist Tullio Levi-Civita. Other names include the permutation symbol, antisymmetric symbol, or alternating symbol, which refer to its antisymmetric property and definition in terms of permutations.
The standard letters to denote the Levi-Civita symbol are the Greek lower case epsilon or , or less commonly the Latin lower case . Index notation allows one to display permutations in a way compatible with tensor analysis: where each index takes values . There are indexed values of , which can be arranged into an -dimensional array. The key defining property of the symbol is total antisymmetry in the indices. When any two indices are interchanged, equal or not, the symbol is negated:
If any two indices are equal, the symbol is zero. When all indices are unequal, we have: where (called the parity of the permutation) is the number of pairwise interchanges of indices necessary to unscramble into the order , and the factor is called the sign, or signature of the permutation. The value must be defined, else the particular values of the symbol for all permutations are indeterminate. Most authors choose , which means the Levi-Civita symbol equals the sign of a permutation when the indices are all unequal. This choice is used throughout this article.
The term "-dimensional Levi-Civita symbol" refers to the fact that the number of indices on the symbol matches the dimensionality of the vector space in question, which may be Euclidean or non-Euclidean, for example, or Minkowski space. The values of the Levi-Civita symbol are independent of any metric tensor and coordinate system. Also, the specific term "symbol" emphasizes that it is not a tensor because of how it transforms between coordinate systems; however it can be interpreted as a tensor density.
The Levi-Civita symbol allows the determinant of a square matrix, and the cross product of two vectors in three-dimensional Euclidean space, to be expressed in Einstein index notation.
Definition
The Levi-Civita symbol is most often used in three and four dimensions, and to some extent in two dimensions, so these are given here before defining the general case.
Two dimensions
In two dimensions, the Levi-Civita symbol is defined by:
The values can be arranged into a 2 × 2 antisymmetric matrix:
Use of the two-dimensional symbol is common in condensed matter, and in certain specialized high-energy topics like supersymmetry and twistor theory, where it appears in the context of 2-spinors.
Three dimensions
In three dimensions, the Levi-Civita symbol is defined by:
That is, is if is an even permutation of , if it is an odd permutation, and 0 if any index is repeated. In three dime
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https://en.wikipedia.org/wiki/Hermite%20polynomials
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In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence.
The polynomials arise in:
signal processing as Hermitian wavelets for wavelet transform analysis
probability, such as the Edgeworth series, as well as in connection with Brownian motion;
combinatorics, as an example of an Appell sequence, obeying the umbral calculus;
numerical analysis as Gaussian quadrature;
physics, where they give rise to the eigenstates of the quantum harmonic oscillator; and they also occur in some cases of the heat equation (when the term is present);
systems theory in connection with nonlinear operations on Gaussian noise.
random matrix theory in Gaussian ensembles.
Hermite polynomials were defined by Pierre-Simon Laplace in 1810, though in scarcely recognizable form, and studied in detail by Pafnuty Chebyshev in 1859. Chebyshev's work was overlooked, and they were named later after Charles Hermite, who wrote on the polynomials in 1864, describing them as new. They were consequently not new, although Hermite was the first to define the multidimensional polynomials in his later 1865 publications.
Definition
Like the other classical orthogonal polynomials, the Hermite polynomials can be defined from several different starting points. Noting from the outset that there are two different standardizations in common use, one convenient method is as follows:
The "probabilist's Hermite polynomials" are given by
while the "physicist's Hermite polynomials" are given by
These equations have the form of a Rodrigues' formula and can also be written as,
The two definitions are not exactly identical; each is a rescaling of the other:
These are Hermite polynomial sequences of different variances; see the material on variances below.
The notation and is that used in the standard references.
The polynomials are sometimes denoted by , especially in probability theory, because
is the probability density function for the normal distribution with expected value 0 and standard deviation 1.
The first eleven probabilist's Hermite polynomials are:
The first eleven physicist's Hermite polynomials are:
Properties
The th-order Hermite polynomial is a polynomial of degree . The probabilist's version has leading coefficient 1, while the physicist's version has leading coefficient .
Symmetry
From the Rodrigues formulae given above, we can see that and are even or odd functions depending on :
Orthogonality
and are th-degree polynomials for . These polynomials are orthogonal with respect to the weight function (measure)
or
i.e., we have
Furthermore,
or
where is the Kronecker delta.
The probabilist polynomials are thus orthogonal with respect to the standard normal probability density function.
Completeness
The Hermite polynomials (probabilist's or physicist's) form an orthogonal basis of the Hilbert space of functions satisfying
in which the inner product is given by the integral
including the Gaussian weight function de
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https://en.wikipedia.org/wiki/Polynomial%20sequence
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In mathematics, a polynomial sequence is a sequence of polynomials indexed by the nonnegative integers 0, 1, 2, 3, ..., in which each index is equal to the degree of the corresponding polynomial. Polynomial sequences are a topic of interest in enumerative combinatorics and algebraic combinatorics, as well as applied mathematics.
Examples
Some polynomial sequences arise in physics and approximation theory as the solutions of certain ordinary differential equations:
Laguerre polynomials
Chebyshev polynomials
Legendre polynomials
Jacobi polynomials
Others come from statistics:
Hermite polynomials
Many are studied in algebra and combinatorics:
Monomials
Rising factorials
Falling factorials
All-one polynomials
Abel polynomials
Bell polynomials
Bernoulli polynomials
Cyclotomic polynomials
Dickson polynomials
Fibonacci polynomials
Lagrange polynomials
Lucas polynomials
Spread polynomials
Touchard polynomials
Rook polynomials
Classes of polynomial sequences
Polynomial sequences of binomial type
Orthogonal polynomials
Secondary polynomials
Sheffer sequence
Sturm sequence
Generalized Appell polynomials
See also
Umbral calculus
References
Aigner, Martin. "A course in enumeration", GTM Springer, 2007, p21.
Roman, Steven "The Umbral Calculus", Dover Publications, 2005, .
Williamson, S. Gill "Combinatorics for Computer Science", Dover Publications, (2002) p177.
Polynomials
Sequences and series
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https://en.wikipedia.org/wiki/Discrete%20sine%20transform
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In mathematics, the discrete sine transform (DST) is a Fourier-related transform similar to the discrete Fourier transform (DFT), but using a purely real matrix. It is equivalent to the imaginary parts of a DFT of roughly twice the length, operating on real data with odd symmetry (since the Fourier transform of a real and odd function is imaginary and odd), where in some variants the input and/or output data are shifted by half a sample.
The DST is related to the discrete cosine transform (DCT), which is equivalent to a DFT of real and even functions. See the DCT article for a general discussion of how the boundary conditions relate the various DCT and DST types. Generally, the DST is derived from the DCT by replacing the Neumann condition at x=0 with a Dirichlet condition. Both the DCT and the DST were described by Nasir Ahmed, T. Natarajan, and K.R. Rao in 1974. The type-I DST (DST-I) was later described by Anil K. Jain in 1976, and the type-II DST (DST-II) was then described by H.B. Kekra and J.K. Solanka in 1978.
Applications
DSTs are widely employed in solving partial differential equations by spectral methods, where the different variants of the DST correspond to slightly different odd/even boundary conditions at the two ends of the array.
Informal overview
Like any Fourier-related transform, discrete sine transforms (DSTs) express a function or a signal in terms of a sum of sinusoids with different frequencies and amplitudes. Like the discrete Fourier transform (DFT), a DST operates on a function at a finite number of discrete data points. The obvious distinction between a DST and a DFT is that the former uses only sine functions, while the latter uses both cosines and sines (in the form of complex exponentials). However, this visible difference is merely a consequence of a deeper distinction: a DST implies different boundary conditions than the DFT or other related transforms.
The Fourier-related transforms that operate on a function over a finite domain, such as the DFT or DST or a Fourier series, can be thought of as implicitly defining an extension of that function outside the domain. That is, once you write a function as a sum of sinusoids, you can evaluate that sum at any , even for where the original was not specified. The DFT, like the Fourier series, implies a periodic extension of the original function. A DST, like a sine transform, implies an odd extension of the original function.
However, because DSTs operate on finite, discrete sequences, two issues arise that do not apply for the continuous sine transform. First, one has to specify whether the function is even or odd at both the left and right boundaries of the domain (i.e. the min-n and max-n boundaries in the definitions below, respectively). Second, one has to specify around what point the function is even or odd. In particular, consider a sequence (a,b,c) of three equally spaced data points, and say that we specify an odd left boundary. There are two sensible po
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https://en.wikipedia.org/wiki/Legendre%20function
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In physical science and mathematics, the Legendre functions , and associated Legendre functions , , and Legendre functions of the second kind, , are all solutions of Legendre's differential equation. The Legendre polynomials and the associated Legendre polynomials are also solutions of the differential equation in special cases, which, by virtue of being polynomials, have a large number of additional properties, mathematical structure, and applications. For these polynomial solutions, see the separate Wikipedia articles.
Legendre's differential equation
The general Legendre equation reads
where the numbers and may be complex, and are called the degree and order of the relevant function, respectively. The polynomial solutions when is an integer (denoted ), and are the Legendre polynomials ; and when
is an integer (denoted ), and is also an integer with are the associated Legendre polynomials. All other cases of and can be discussed as one, and the solutions are written , . If , the superscript is omitted, and one writes just , . However, the solution when is an integer is often discussed separately as Legendre's function of the second kind, and denoted .
This is a second order linear equation with three regular singular points (at , , and ). Like all such equations, it can be converted into a hypergeometric differential equation by a change of variable, and its solutions can be expressed using hypergeometric functions.
Solutions of the differential equation
Since the differential equation is linear, homogeneous (the right hand side =zero) and of second order, it has two linearly independent solutions, which can both be expressed in terms of the hypergeometric function, . With being the gamma function, the first solution is
and the second is,
These are generally known as Legendre functions of the first and second kind of noninteger degree, with the additional qualifier 'associated' if is non-zero. A useful relation between the and solutions is Whipple's formula.
Positive integer order
For positive integer the evaluation of above involves cancellation of singular terms. We can find the limit valid for as
with the (rising) Pochhammer symbol.
Legendre functions of the second kind ()
The nonpolynomial solution for the special case of integer degree , and , is often discussed separately.
It is given by
This solution is necessarily singular when .
The Legendre functions of the second kind can also be defined recursively via Bonnet's recursion formula
Associated Legendre functions of the second kind
The nonpolynomial solution for the special case of integer degree , and is given by
Integral representations
The Legendre functions can be written as contour integrals. For example,
where the contour winds around the points and in the positive direction and does not wind around .
For real , we have
Legendre function as characters
The real integral representation of are very useful in the study of harmonic analysis
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https://en.wikipedia.org/wiki/Nicholas%20Saunderson
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Nicholas Saunderson (20 January 1682 – 19 April 1739) was a blind English scientist and mathematician. According to one historian of statistics, he may have been the earliest discoverer of Bayes' theorem. He worked as Lucasian Professor of Mathematics at Cambridge University, a post also held by Isaac Newton, Charles Babbage and Stephen Hawking.
Biography
Saunderson was born at Thurlstone, Yorkshire, in January 1682. His parents were John and Ann Sanderson (or Saunderson), and his father made a living as an excise man. When he was about a year old, he lost his sight through smallpox; but this did not prevent him from learning arithmetic through assisting his father. As a child, he is also thought to have learnt to read by tracing the engravings on tombstones around St John the Baptist Church in Penistone with his fingers. His early education was at the free school, Penistone Grammar School where he learnt French, Latin and Greek. In 1700 a tutor taught him algebra and geometry, and in 1702 he attended Attercliffe Academy, near Sheffield, for logic and metaphysics. He was introduced to Cambridge via meetings with the local gentry at Underbank Hall, near Penistone.
In 1707, he arrived in Cambridge with his friend Joshua Dunn from Attercliffe Academy, a fellow-commoner at Christ's College. During this time, he resided in Christ's and could make use of the library but was not admitted to the university. He wanted to teach and with the permission of the Lucasian professor, William Whiston, Saunderson was allowed to teach, lecturing on mathematics, astronomy and optics. His teaching was highly appreciated.
Whiston was expelled from his chair on 30 October 1710; at the appeal of the heads of colleges, Queen Anne awarded Saunderson a Master of Arts degree on 19 November 1711 so that he would be eligible to succeed Whiston as Lucasian professor. He was chosen as the fourth Lucasian professor the next day, defeating the Trinity College candidate Christopher Hussey, backed by Richard Bentley, when the electors split 6 to 4 in his favour. On 6 November 1718 Saunderson was elected a fellow of the Royal Society. He was also a member of the Spitalfields Mathematical Society.
He was resident at Christ's College until 1723 when he married Abigail Dickons, daughter of William Dickons who was the rector of Boxworth, Cambridgeshire. They lived in Cambridge with their children John and Anne. He was created doctor of laws in 1728 by command of George II during a visit by the monarch to Cambridge. He died of scurvy, on 19 April 1739 and was buried in the chancel of the parish church at Boxworth near Cambridge.
Saunderson possessed the friendship of leading mathematicians of the time: Isaac Newton, Edmond Halley, Abraham De Moivre and Roger Cotes. His senses of hearing and touch were acute, and he was a good flautist. He could carry out mentally long and intricate mathematical calculations. He devised a calculating machine or abacus, by which he could perform ar
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https://en.wikipedia.org/wiki/Heptadecagon
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In geometry, a heptadecagon, septadecagon or 17-gon is a seventeen-sided polygon.
Regular heptadecagon
A regular heptadecagon is represented by the Schläfli symbol {17}.
Construction
As 17 is a Fermat prime, the regular heptadecagon is a constructible polygon (that is, one that can be constructed using a compass and unmarked straightedge): this was shown by Carl Friedrich Gauss in 1796 at the age of 19. This proof represented the first progress in regular polygon construction in over 2000 years. Gauss's proof relies firstly on the fact that constructibility is equivalent to expressibility of the trigonometric functions of the common angle in terms of arithmetic operations and square root extractions, and secondly on his proof that this can be done if the odd prime factors of , the number of sides of the regular polygon, are distinct Fermat primes, which are of the form for some nonnegative integer . Constructing a regular heptadecagon thus involves finding the cosine of in terms of square roots. Gauss's book Disquisitiones Arithmeticae gives this (in modern notation) as
Constructions for the regular triangle, pentagon, pentadecagon, and polygons with 2h times as many sides had been given by Euclid, but constructions based on the Fermat primes other than 3 and 5 were unknown to the ancients. (The only known Fermat primes are Fn for n = 0, 1, 2, 3, 4. They are 3, 5, 17, 257, and 65537.)
The explicit construction of a heptadecagon was given by Herbert William Richmond in 1893. The following method of construction uses Carlyle circles, as shown below. Based on the construction of the regular 17-gon, one can readily construct n-gons with n being the product of 17 with 3 or 5 (or both) and any power of 2: a regular 51-gon, 85-gon or 255-gon and any regular n-gon with 2h times as many sides.
Another construction of the regular heptadecagon using straightedge and compass is the following:
T. P. Stowell of Rochester, N. Y., responded to Query, by W.E. Heal, Wheeling, Indiana in The Analyst in the year 1874:
"To construct a regular polygon of seventeen sides in a circle.
Draw the radius CO at right-angles to the diameter AB: On OC and OB, take OQ equal to the half, and OD equal to the eighth part of the radius: Make DE and DF each equal to DQ and EG and FH respectively equal to EQ and FQ; take OK a mean proportional between OH and OQ, and through K, draw KM parallel to AB, meeting the semicircle described on OG in M; draw MN parallel to OC, cutting the given circle in N – the arc AN is the seventeenth part of the whole circumference."
The following simple design comes from Herbert William Richmond from the year 1893:
"LET OA, OB (fig. 6) be two perpendicular radii of a circle. Make OI one-fourth of OB, and the angle OIE one-fourth of OIA; also find in OA produced a point F such that EIF is 45°. Let the circle on AF as diameter cut OB in K, and let the circle whose centre is E and radius EK cut OA in N3 and N5; then if ordinates N3P3, N5P5
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https://en.wikipedia.org/wiki/Long%20line
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Long line or longline may refer to:
Long Line, an album by Peter Wolf
Long line (topology), or Alexandroff line, a topological space
Long line (telecommunications), a transmission line in a long-distance communications network
Longline fishing, a commercial fishing technique
AT&T Long Lines, a telecommunications network
AT&T Long Lines Building, now known as 33 Thomas Street, a building in New York City
Aerial crane or longline, a transport method
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https://en.wikipedia.org/wiki/Hyperspace%20%28disambiguation%29
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Hyperspace is a faster-than-light method of traveling used in science fiction.
Hyperspace or HyperSpace may also refer to:
Mathematics
Hypertopology, a topological space within which some of its elements form another topological space
Higher dimensions, including Kaluza–Klein's 4-dimensional space and Superstring theory's 9-dimensional space and Supergravity/M-theory's 10-dimensional space
n-dimensional space, the original meaning of the word hyperspace, common in late nineteenth century British books
Non-Euclidean space
Media
Books
Hyperspace (book), a 1994 book by Michio Kaku that attempts to explain the possibility of 10-dimensional space using string theory
Hyperspace (gamebook), a book in the Choose Your Own Adventure series
Film and television
Hyperspace (film), a 1984 3D science fiction comedy film
Hyperspace, the U.S. title of Space, a 2001 BBC documentary
Music
Hyperspace, the opening song on the Nada Surf album The Proximity Effect
Hyperspace, 2019 studio album by American producer and musician Beck
Hyperspace, a song by Buckner & Garcia from the album Pac-Man Fever
Other uses
The hypertextual or architectural aspect of cyberspace
Lost in hyperspace
HyperSpace (software), an operating system by Phoenix Technologies
See also
Hyper (disambiguation)
Hiperspace, IBM High Performance Space
Hyper Scape
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https://en.wikipedia.org/wiki/Probabilistic%20Turing%20machine
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In theoretical computer science, a probabilistic Turing machine is a non-deterministic Turing machine that chooses between the available transitions at each point according to some probability distribution. As a consequence, a probabilistic Turing machine can—unlike a deterministic Turing Machine—have stochastic results; that is, on a given input and instruction state machine, it may have different run times, or it may not halt at all; furthermore, it may accept an input in one execution and reject the same input in another execution.
In the case of equal probabilities for the transitions, probabilistic Turing machines can be defined as deterministic Turing machines having an additional "write" instruction where the value of the write is uniformly distributed in the Turing Machine's alphabet (generally, an equal likelihood of writing a "1" or a "0" on to the tape). Another common reformulation is simply a deterministic Turing machine with an added tape full of random bits called the "random tape".
A quantum computer is another model of computation that is inherently probabilistic.
Description
A probabilistic Turing machine is a type of nondeterministic Turing machine in which each nondeterministic step is a "coin-flip", that is, at each step there are two possible next moves and the Turing machine probabilistically selects which move to take.
Formal definition
A probabilistic Turing machine can be formally defined as the 7-tuple , where
is a finite set of states
is the input alphabet
is a tape alphabet, which includes the blank symbol #
is the initial state
is the set of accepting (final) states
is the first probabilistic transition function. is a movement one cell to the left on the Turing machine's tape and is a movement one cell to the right.
is the second probabilistic transition function.
At each step, the Turing machine probabilistically applies either the transition function or the transition function . This choice is made independently of all prior choices. In this way, the process of selecting a transition function at each step of the computation resembles a coin flip.
The probabilistic selection of the transition function at each step introduces error into the Turing machine; that is, strings which the Turing machine is meant to accept may on some occasions be rejected and strings which the Turing machine is meant to reject may on some occasions be accepted. To accommodate this, a language is said to be recognized with error probability by a probabilistic Turing machine if:
a string in implies that
a string not in implies that
Complexity classes
As a result of the error introduced by utilizing probabilistic coin tosses, the notion of acceptance of a string by a probabilistic Turing machine can be defined in different ways. One such notion that includes several important complexity classes is allowing for an error probability of 1/3. For instance, the complexity class BPP is defined as the class of languages
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https://en.wikipedia.org/wiki/Lagrangian
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Lagrangian may refer to:
Mathematics
Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier
Lagrangian relaxation, the method of approximating a difficult constrained problem with an easier problem having an enlarged feasible set
Lagrangian dual problem, the problem of maximizing the value of the Lagrangian function, in terms of the Lagrange-multiplier variable; See Dual problem
Lagrangian, a functional whose extrema are to be determined in the calculus of variations
Lagrangian submanifold, a class of submanifolds in symplectic geometry
Lagrangian system, a pair consisting of a smooth fiber bundle and a Lagrangian density
Physics
Lagrangian (physics), a function in Lagrangian mechanics
Lagrangian (field theory), a formalism in classical field theory
Lagrangian point, a position in an orbital configuration of two large bodies
Lagrangian coordinates, a way of describing the motions of particles of a solid or fluid in continuum mechanics
Lagrangian coherent structure, distinguished surfaces of trajectories in a dynamical system
See also
Joseph-Louis Lagrange (1736–1813), Italian mathematician and astronomer
Lagrange (disambiguation)
List of things named after Joseph-Louis Lagrange
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https://en.wikipedia.org/wiki/Mathematical%20Association%20of%20America
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The Mathematical Association of America (MAA) is a professional society that focuses on mathematics accessible at the undergraduate level. Members include university, college, and high school teachers; graduate and undergraduate students; pure and applied mathematicians; computer scientists; statisticians; and many others in academia, government, business, and industry.
The MAA was founded in 1915 and is headquartered at 1529 18th Street, Northwest in the Dupont Circle neighborhood of Washington, D.C. The organization publishes mathematics journals and books, including the American Mathematical Monthly (established in 1894 by Benjamin Finkel), the most widely read mathematics journal in the world according to records on JSTOR.
Meetings
The MAA sponsors the annual summer MathFest and cosponsors with the American Mathematical Society the Joint Mathematics Meeting, held in early January of each year. On occasion the Society for Industrial and Applied Mathematics joins in these meetings. Twenty-nine regional sections also hold regular meetings.
Publications
The association publishes multiple journals in partnership with Taylor & Francis:
The American Mathematical Monthly is expository, aimed at a broad audience from undergraduate students to research mathematicians.
Mathematics Magazine is expository, aimed at teachers of undergraduate mathematics, especially at the junior-senior level.
The College Mathematics Journal is expository, aimed at teachers of undergraduate mathematics, especially at the freshman-sophomore level.
Math Horizons is expository, aimed at undergraduate students.
MAA FOCUS is the association member newsletter. The Association publishes an online resource, Mathematical Sciences Digital Library (Math DL). The service launched in 2001 with the online-only Journal of Online Mathematics and its Applications (JOMA) and a set of classroom tools, Digital Classroom Resources. These were followed in 2004 by Convergence, an online-only history magazine, and in 2005 by MAA Reviews, an online book review service, and Classroom Capsules and Notes, a set of classroom notes.
Competitions
The MAA sponsors numerous competitions for students, including the William Lowell Putnam Competition for undergraduate students, the online competition series, and the American Mathematics Competitions (AMC) for middle- and high-school students. This series of competitions is as follows:
AMC 8: 25 multiple choice questions in 40 minutes
AMC 10/AMC 12: 25 multiple choice questions in 75 minutes
AIME: 15 short answer questions in a 3-hour period
USAMO/USAJMO: 6 questions, 2 days, 9 hours, proof-based olympiad
Through this program, outstanding students are identified and invited to participate in the Mathematical Olympiad Program. Ultimately, six high school students are chosen to represent the U.S. at the International Mathematics Olympiad.
Sections
The MAA is composed of the following twenty-nine regional sections:
Allegheny Mountain, EPADEL, Flor
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https://en.wikipedia.org/wiki/Antisymmetric
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Antisymmetric or skew-symmetric may refer to:
Antisymmetry in linguistics
Antisymmetry in physics
Antisymmetric relation in mathematics
Skew-symmetric graph
Self-complementary graph
In mathematics, especially linear algebra, and in theoretical physics, the adjective antisymmetric (or skew-symmetric) is used for matrices, tensors, and other objects that change sign if an appropriate operation (e.g. matrix transposition) is performed. See:
Skew-symmetric matrix (a matrix A for which )
Skew-symmetric bilinear form is a bilinear form B such that for all x and y.
Antisymmetric tensor in matrices and index subsets.
"antisymmetric function" – odd function
See also
Symmetry in mathematics
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https://en.wikipedia.org/wiki/Erlang%20distribution
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The Erlang distribution is a two-parameter family of continuous probability distributions with support . The two parameters are:
a positive integer the "shape", and
a positive real number the "rate". The "scale", the reciprocal of the rate, is sometimes used instead.
The Erlang distribution is the distribution of a sum of independent exponential variables with mean each. Equivalently, it is the distribution of the time until the kth event of a Poisson process with a rate of . The Erlang and Poisson distributions are complementary, in that while the Poisson distribution counts the number of events that occur in a fixed amount of time, the Erlang distribution counts the amount of time until the occurrence of a fixed number of events. When , the distribution simplifies to the exponential distribution. The Erlang distribution is a special case of the gamma distribution wherein the shape of the distribution is discretised.
The Erlang distribution was developed by A. K. Erlang to examine the number of telephone calls which might be made at the same time to the operators of the switching stations. This work on telephone traffic engineering has been expanded to consider waiting times in queueing systems in general. The distribution is also used in the field of stochastic processes.
Characterization
Probability density function
The probability density function of the Erlang distribution is
The parameter k is called the shape parameter, and the parameter is called the rate parameter.
An alternative, but equivalent, parametrization uses the scale parameter , which is the reciprocal of the rate parameter (i.e., ):
When the scale parameter equals 2, the distribution simplifies to the chi-squared distribution with 2k degrees of freedom. It can therefore be regarded as a generalized chi-squared distribution for even numbers of degrees of freedom.
Cumulative distribution function (CDF)
The cumulative distribution function of the Erlang distribution is
where is the lower incomplete gamma function and is the lower regularized gamma function.
The CDF may also be expressed as
Erlang-k
The Erlang-k distribution (where k is a positive integer) is defined by setting k in the PDF of the Erlang distribution. For instance, the Erlang-2 distribution is , which is the same as .
Median
An asymptotic expansion is known for the median of an Erlang distribution, for which coefficients can be computed and bounds are known. An approximation is i.e. below the mean
Generating Erlang-distributed random variates
Erlang-distributed random variates can be generated from uniformly distributed random numbers () using the following formula:
Applications
Waiting times
Events that occur independently with some average rate are modeled with a Poisson process. The waiting times between k occurrences of the event are Erlang distributed. (The related question of the number of events in a given amount of time is described by the Poisson distribution.)
T
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https://en.wikipedia.org/wiki/Bernoulli%20distribution
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In probability theory and statistics, the Bernoulli distribution, named after Swiss mathematician Jacob Bernoulli, is the discrete probability distribution of a random variable which takes the value 1 with probability and the value 0 with probability . Less formally, it can be thought of as a model for the set of possible outcomes of any single experiment that asks a yes–no question. Such questions lead to outcomes that are boolean-valued: a single bit whose value is success/yes/true/one with probability p and failure/no/false/zero with probability q. It can be used to represent a (possibly biased) coin toss where 1 and 0 would represent "heads" and "tails", respectively, and p would be the probability of the coin landing on heads (or vice versa where 1 would represent tails and p would be the probability of tails). In particular, unfair coins would have
The Bernoulli distribution is a special case of the binomial distribution where a single trial is conducted (so n would be 1 for such a binomial distribution). It is also a special case of the two-point distribution, for which the possible outcomes need not be 0 and 1.
Properties
If is a random variable with a Bernoulli distribution, then:
The probability mass function of this distribution, over possible outcomes k, is
This can also be expressed as
or as
The Bernoulli distribution is a special case of the binomial distribution with
The kurtosis goes to infinity for high and low values of but for the two-point distributions including the Bernoulli distribution have a lower excess kurtosis than any other probability distribution, namely −2.
The Bernoulli distributions for form an exponential family.
The maximum likelihood estimator of based on a random sample is the sample mean.
Mean
The expected value of a Bernoulli random variable is
This is due to the fact that for a Bernoulli distributed random variable with and we find
Variance
The variance of a Bernoulli distributed is
We first find
From this follows
With this result it is easy to prove that, for any Bernoulli distribution, its variance will have a value inside .
Skewness
The skewness is . When we take the standardized Bernoulli distributed random variable we find that this random variable attains with probability and attains with probability . Thus we get
Higher moments and cumulants
The raw moments are all equal due to the fact that and .
The central moment of order is given by
The first six central moments are
The higher central moments can be expressed more compactly in terms of and
The first six cumulants are
Related distributions
If are independent, identically distributed (i.i.d.) random variables, all Bernoulli trials with success probability p, then their sum is distributed according to a binomial distribution with parameters n and p:
(binomial distribution).
The Bernoulli distribution is simply , also written as
The categorical distribution is the generalization of the Bernoull
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https://en.wikipedia.org/wiki/Marko%20Petkov%C5%A1ek
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Marko Petkovšek (1955 – 24 March 2023) was a Slovenian mathematician working mainly in symbolic computation.
He was a professor of discrete and computational mathematics at the University of Ljubljana. He is best known for Petkovšek's algorithm, and for the book that he coauthored with Herbert Wilf and Doron Zeilberger, A = B.
Education and career
Petkovšek was born in 1955 in Ljubljana, Slovenia. He attended the University of Ljubljana for his bachelors and masters degrees, which he finished respectively in 1978 and 1986. He completed his PhD at Carnegie Mellon University under the supervision of Dana Scott, with a thesis titled Finding Closed-Form Solutions of Difference Equations by Symbolic Methods. After his PhD, he returned to Ljubljana and a job at the University of Ljubljana.
Petkovšek retired from the University of Ljubljana in 2021, and died in 2023.
Books
References
1955 births
2023 deaths
20th-century Slovenian mathematicians
21st-century Slovenian mathematicians
Carnegie Mellon University alumni
Academic staff of the University of Ljubljana
Scientists from Ljubljana
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https://en.wikipedia.org/wiki/Pierre%20%C3%89mile%20Levasseur
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Pierre Émile Levasseur, 3rd Baron Levasseur (8 December 1828 – 10 July 1911), was a French economist, historian, Professor of geography, history and statistics in the Collège de France, at the Conservatoire national des arts et métiers and at the École Libre des Sciences Politiques, known as one of the founders and promoters of the study of commercial geography.
Life and work
Levasseur was born in Paris, France, as son of the jewelry manufacturer Pierre Antoine Levasseur. He was educated at the École Normale Supérieure in Paris.
Levasseur began teaching in the lycée at Alençon in 1852, and in 1857 became professor of rhetoric at Besançon. He returned to Paris to become professor at the lycée Saint Louis. In 1868 he was chosen a member of the Academy of Moral and Political Sciences. In 1872 he was appointed professor of geography, history and statistics in the College de France, and subsequently became also professor at the Conservatoire des arts et métiers and at the École libre des sciences politiques, which later became known as the Institut d'Etudes Politiques de Paris.
He strongly believed in the value of using statistics, graphics, and maps to teach the social sciences at a deeper level, and is credited with inventing the cartogram as a teaching aid.
Levasseur was president of the Société d'économie politique.
Levasseur was one of the founders of the study of commercial geography, and became a member of the Council of Public Instruction and honorary president of the French geographical society.
In 1886, he was elected as a member to the American Philosophical Society. Levasseur was elected member of the Royal Swedish Academy of Sciences in 1894. He was elected a member of the American Antiquarian Society in 1905.
Selected publications
His numerous writings include:
La question de l'or (1858)
Histoire des classes ouvrières en France depuis la conquête de Jules-César jusqu'à la Révolution (1859)
Histoire des classes ouvrières en France depuis la Révolution jusqu'à nos jours (1867)
L'Étude et l'enseignement de la géographie (1871)
La Population française (1889–1892)
L'Agriculture aux États-Unis (1894)
L'Enseignement primaire dans les pays civilisés (1897)
L'Ouvrier américain (1898)
Questions ouvrières et industrielles sous la Troisième République (1907)
Histoire des classes ouvrières et de l'industrie en France de 1789 à 1870 (1903–1904)
Grand Atlas de géographie physique et politique (1890–1894).
References
Attribution
External links
Emile Levasseur at bnf.fr
1828 births
1911 deaths
French economists
École Normale Supérieure alumni
Members of the Royal Swedish Academy of Sciences
Members of the American Antiquarian Society
Members of the American Philosophical Society
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https://en.wikipedia.org/wiki/Binomial%20type
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In mathematics, a polynomial sequence, i.e., a sequence of polynomials indexed by non-negative integers in which the index of each polynomial equals its degree, is said to be of binomial type if it satisfies the sequence of identities
Many such sequences exist. The set of all such sequences forms a Lie group under the operation of umbral composition, explained below. Every sequence of binomial type may be expressed in terms of the Bell polynomials. Every sequence of binomial type is a Sheffer sequence (but most Sheffer sequences are not of binomial type). Polynomial sequences put on firm footing the vague 19th century notions of umbral calculus.
Examples
In consequence of this definition the binomial theorem can be stated by saying that the sequence is of binomial type.
The sequence of "lower factorials" is defined by(In the theory of special functions, this same notation denotes upper factorials, but this present usage is universal among combinatorialists.) The product is understood to be 1 if n = 0, since it is in that case an empty product. This polynomial sequence is of binomial type.
Similarly the "upper factorials"are a polynomial sequence of binomial type.
The Abel polynomialsare a polynomial sequence of binomial type.
The Touchard polynomialswhere is the number of partitions of a set of size into disjoint non-empty subsets, is a polynomial sequence of binomial type. Eric Temple Bell called these the "exponential polynomials" and that term is also sometimes seen in the literature. The coefficients are "Stirling numbers of the second kind". This sequence has a curious connection with the Poisson distribution: If is a random variable with a Poisson distribution with expected value then . In particular, when , we see that the th moment of the Poisson distribution with expected value is the number of partitions of a set of size , called the th Bell number. This fact about the th moment of that particular Poisson distribution is "Dobinski's formula".
Characterization by delta operators
It can be shown that a polynomial sequence { pn(x) : n = 0, 1, 2, … } is of binomial type if and only if all three of the following conditions hold:
The linear transformation on the space of polynomials in x that is characterized byis shift-equivariant, and
p0(x) = 1 for all x, and
pn(0) = 0 for n > 0.
(The statement that this operator is shift-equivariant is the same as saying that the polynomial sequence is a Sheffer sequence; the set of sequences of binomial type is properly included within the set of Sheffer sequences.)
Delta operators
That linear transformation is clearly a delta operator, i.e., a shift-equivariant linear transformation on the space of polynomials in x that reduces degrees of polynomials by 1. The most obvious examples of delta operators are difference operators and differentiation. It can be shown that every delta operator can be written as a power series of the form
where D is differentiation (note that th
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https://en.wikipedia.org/wiki/Quadratic%20residue
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In number theory, an integer q is called a quadratic residue modulo n if it is congruent to a perfect square modulo n; i.e., if there exists an integer x such that:
Otherwise, q is called a quadratic nonresidue modulo n.
Originally an abstract mathematical concept from the branch of number theory known as modular arithmetic, quadratic residues are now used in applications ranging from acoustical engineering to cryptography and the factoring of large numbers.
History, conventions, and elementary facts
Fermat, Euler, Lagrange, Legendre, and other number theorists of the 17th and 18th centuries established theorems and formed conjectures about quadratic residues, but the first systematic treatment is § IV of Gauss's Disquisitiones Arithmeticae (1801). Article 95 introduces the terminology "quadratic residue" and "quadratic nonresidue", and states that if the context makes it clear, the adjective "quadratic" may be dropped.
For a given n a list of the quadratic residues modulo n may be obtained by simply squaring the numbers 0, 1, ..., . Because a2 ≡ (n − a)2 (mod n), the list of squares modulo n is symmetric around n/2, and the list only needs to go that high. This can be seen in the table below.
Thus, the number of quadratic residues modulo n cannot exceed n/2 + 1 (n even) or (n + 1)/2 (n odd).
The product of two residues is always a residue.
Prime modulus
Modulo 2, every integer is a quadratic residue.
Modulo an odd prime number p there are (p + 1)/2 residues (including 0) and (p − 1)/2 nonresidues, by Euler's criterion. In this case, it is customary to consider 0 as a special case and work within the multiplicative group of nonzero elements of the field Z/pZ. (In other words, every congruence class except zero modulo p has a multiplicative inverse. This is not true for composite moduli.)
Following this convention, the multiplicative inverse of a residue is a residue, and the inverse of a nonresidue is a nonresidue.
Following this convention, modulo an odd prime number there are an equal number of residues and nonresidues.
Modulo a prime, the product of two nonresidues is a residue and the product of a nonresidue and a (nonzero) residue is a nonresidue.
The first supplement to the law of quadratic reciprocity is that if p ≡ 1 (mod 4) then −1 is a quadratic residue modulo p, and if p ≡ 3 (mod 4) then −1 is a nonresidue modulo p. This implies the following:
If p ≡ 1 (mod 4) the negative of a residue modulo p is a residue and the negative of a nonresidue is a nonresidue.
If p ≡ 3 (mod 4) the negative of a residue modulo p is a nonresidue and the negative of a nonresidue is a residue.
Prime power modulus
All odd squares are ≡ 1 (mod 8) and thus also ≡ 1 (mod 4). If a is an odd number and m = 8, 16, or some higher power of 2, then a is a residue modulo m if and only if a ≡ 1 (mod 8).
For example, mod (32) the odd squares are
12 ≡ 152 ≡ 1
32 ≡ 132 ≡ 9
52 ≡ 112 ≡ 25
72 ≡ 92 ≡ 49 ≡ 17
and the even ones are
02 ≡ 82 ≡ 162 ≡ 0
22 ≡ 62≡ 10
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https://en.wikipedia.org/wiki/Rolle%27s%20theorem
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In calculus, Rolle's theorem or Rolle's lemma essentially states that any real-valued differentiable function that attains equal values at two distinct points must have at least one stationary point somewhere between them—that is, a point where the first derivative (the slope of the tangent line to the graph of the function) is zero. The theorem is named after Michel Rolle.
Standard version of the theorem
If a real-valued function is continuous on a proper closed interval , differentiable on the open interval , and , then there exists at least one in the open interval such that
This version of Rolle's theorem is used to prove the mean value theorem, of which Rolle's theorem is indeed a special case. It is also the basis for the proof of Taylor's theorem.
History
Although the theorem is named after Michel Rolle, Rolle's 1691 proof covered only the case of polynomial functions. His proof did not use the methods of differential calculus, which at that point in his life he considered to be fallacious. The theorem was first proved by Cauchy in 1823 as a corollary of a proof of the mean value theorem. The name "Rolle's theorem" was first used by Moritz Wilhelm Drobisch of Germany in 1834 and by Giusto Bellavitis of Italy in 1846.
Examples
First example
For a radius , consider the function
Its graph is the upper semicircle centered at the origin. This function is continuous on the closed interval and differentiable in the open interval , but not differentiable at the endpoints and . Since , Rolle's theorem applies, and indeed, there is a point where the derivative of is zero. The theorem applies even when the function cannot be differentiated at the endpoints because it only requires the function to be differentiable in the open interval.
Second example
If differentiability fails at an interior point of the interval, the conclusion of Rolle's theorem may not hold. Consider the absolute value function
Then , but there is no between −1 and 1 for which the is zero. This is because that function, although continuous, is not differentiable at . The derivative of changes its sign at , but without attaining the value 0. The theorem cannot be applied to this function because it does not satisfy the condition that the function must be differentiable for every in the open interval. However, when the differentiability requirement is dropped from Rolle's theorem, will still have a critical number in the open interval , but it may not yield a horizontal tangent (as in the case of the absolute value represented in the graph).
Generalization
The second example illustrates the following generalization of Rolle's theorem:
Consider a real-valued, continuous function on a closed interval with . If for every in the open interval the right-hand limit
and the left-hand limit
exist in the extended real line , then there is some number in the open interval such that one of the two limits
is and the other one is (in the extended real lin
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https://en.wikipedia.org/wiki/Reflexive%20relation
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In mathematics, a binary relation R on a set X is reflexive if it relates every element of X to itself.
An example of a reflexive relation is the relation "is equal to" on the set of real numbers, since every real number is equal to itself. A reflexive relation is said to have the reflexive property or is said to possess reflexivity. Along with symmetry and transitivity, reflexivity is one of three properties defining equivalence relations.
Definitions
Let be a binary relation on a set which by definition is just a subset of
For any the notation means that while "not " means that
The relation is called if for every or equivalently, if where denotes the identity relation on
The of is the union which can equivalently be defined as the smallest (with respect to ) reflexive relation on that is a superset of A relation is reflexive if and only if it is equal to its reflexive closure.
The or of is the smallest (with respect to ) relation on that has the same reflexive closure as It is equal to The reflexive reduction of can, in a sense, be seen as a construction that is the "opposite" of the reflexive closure of
For example, the reflexive closure of the canonical strict inequality on the reals is the usual non-strict inequality whereas the reflexive reduction of is
Related definitions
There are several definitions related to the reflexive property.
The relation is called:
, or If it does not relate any element to itself; that is, if not for every A relation is irreflexive if and only if its complement in is reflexive. An asymmetric relation is necessarily irreflexive. A transitive and irreflexive relation is necessarily asymmetric.
If whenever are such that then necessarily
If whenever are such that then necessarily
If every element that is part of some relation is related to itself. Explicitly, this means that whenever are such that then necessarily Equivalently, a binary relation is quasi-reflexive if and only if it is both left quasi-reflexive and right quasi-reflexive. A relation is quasi-reflexive if and only if its symmetric closure is left (or right) quasi-reflexive.
Antisymmetric If whenever are such that then necessarily
If whenever are such that then necessarily A relation is coreflexive if and only if its symmetric closure is anti-symmetric.
A reflexive relation on a nonempty set can neither be irreflexive, nor asymmetric ( is called if implies not ), nor antitransitive ( is if implies not ).
Examples
Examples of reflexive relations include:
"is equal to" (equality)
"is a subset of" (set inclusion)
"divides" (divisibility)
"is greater than or equal to"
"is less than or equal to"
Examples of irreflexive relations include:
"is not equal to"
"is coprime to" on the integers larger than 1
"is a proper subset of"
"is greater than"
"is less than"
An example of an irreflexive relation, which means that it does not relate any element to itself,
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https://en.wikipedia.org/wiki/Transitive%20relation
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In mathematics, a relation on a set is transitive if, for all elements , , in , whenever relates to and to , then also relates to . Each partial order as well as each equivalence relation needs to be transitive.
Definition
A homogeneous relation on the set is a transitive relation if,
for all , if and , then .
Or in terms of first-order logic:
,
where is the infix notation for .
Examples
As a non-mathematical example, the relation "is an ancestor of" is transitive. For example, if Amy is an ancestor of Becky, and Becky is an ancestor of Carrie, then Amy, too, is an ancestor of Carrie.
On the other hand, "is the birth parent of" is not a transitive relation, because if Alice is the birth parent of Brenda, and Brenda is the birth parent of Claire, then this does not imply that Alice is the birth parent of Claire. What is more, it is antitransitive: Alice can never be the birth parent of Claire.
Non-transitive, non-antitransitive relations include sports fixtures (playoff schedules), 'knows' and 'talks to'.
"Is greater than", "is at least as great as", and "is equal to" (equality) are transitive relations on various sets, for instance, the set of real numbers or the set of natural numbers:
whenever x > y and y > z, then also x > z
whenever x ≥ y and y ≥ z, then also x ≥ z
whenever x = y and y = z, then also x = z.
More examples of transitive relations:
"is a subset of" (set inclusion, a relation on sets)
"divides" (divisibility, a relation on natural numbers)
"implies" (implication, symbolized by "⇒", a relation on propositions)
Examples of non-transitive relations:
"is the successor of" (a relation on natural numbers)
"is a member of the set" (symbolized as "∈")
"is perpendicular to" (a relation on lines in Euclidean geometry)
The empty relation on any set is transitive because there are no elements such that and , and hence the transitivity condition is vacuously true. A relation containing only one ordered pair is also transitive: if the ordered pair is of the form for some the only such elements are , and indeed in this case , while if the ordered pair is not of the form then there are no such elements and hence is vacuously transitive.
Properties
Closure properties
The converse (inverse) of a transitive relation is always transitive. For instance, knowing that "is a subset of" is transitive and "is a superset of" is its converse, one can conclude that the latter is transitive as well.
The intersection of two transitive relations is always transitive. For instance, knowing that "was born before" and "has the same first name as" are transitive, one can conclude that "was born before and also has the same first name as" is also transitive.
The union of two transitive relations need not be transitive. For instance, "was born before or has the same first name as" is not a transitive relation, since e.g. Herbert Hoover is related to Franklin D. Roosevelt, who is in turn related to Franklin Pierce, whi
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https://en.wikipedia.org/wiki/Baire%20space
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In mathematics, a topological space is said to be a Baire space if countable unions of closed sets with empty interior also have empty interior.
According to the Baire category theorem, compact Hausdorff spaces and complete metric spaces are examples of Baire spaces.
The Baire category theorem combined with the properties of Baire spaces has numerous applications in topology, geometry, analysis, in particular functional analysis. For more motivation and applications, see the article Baire category theorem. The current article focuses more on characterizations and basic properties of Baire spaces per se.
Bourbaki introduced the term "Baire space" in honor of René Baire, who investigated the Baire category theorem in the context of Euclidean space in his 1899 thesis.
Definition
The definition that follows is based on the notions of meagre (or first category) set (namely, a set that is a countable union of sets whose closure has empty interior) and nonmeagre (or second category) set (namely, a set that is not meagre). See the corresponding article for details.
A topological space is called a Baire space if it satisfies any of the following equivalent conditions:
Every countable intersection of dense open sets is dense.
Every countable union of closed sets with empty interior has empty interior.
Every meagre set has empty interior.
Every nonempty open set is nonmeagre.
Every comeagre set is dense.
Whenever a countable union of closed sets has an interior point, at least one of the closed sets has an interior point.
The equivalence between these definitions is based on the associated properties of complementary subsets of (that is, of a set and of its complement ) as given in the table below.
Baire category theorem
The Baire category theorem gives sufficient conditions for a topological space to be a Baire space.
(BCT1) Every complete pseudometric space is a Baire space. In particular, every completely metrizable topological space is a Baire space.
(BCT2) Every locally compact regular space is a Baire space. In particular, every locally compact Hausdorff space is a Baire space.
BCT1 shows that the following are Baire spaces:
The space of real numbers.
The space of irrational numbers, which is homeomorphic to the Baire space of set theory.
Every Polish space.
BCT2 shows that the following are Baire spaces:
Every compact Hausdorff space; for example, the Cantor set (or Cantor space).
Every manifold, even if it is not paracompact (hence not metrizable), like the long line.
One should note however that there are plenty of spaces that are Baire spaces without satisfying the conditions of the Baire category theorem, as shown in the Examples section below.
Properties
Every nonempty Baire space is nonmeagre. In terms of countable intersections of dense open sets, being a Baire space is equivalent to such intersections being dense, while being a nonmeagre space is equivalent to the weaker condition that such intersect
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https://en.wikipedia.org/wiki/Bell%20number
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In combinatorial mathematics, the Bell numbers count the possible partitions of a set. These numbers have been studied by mathematicians since the 19th century, and their roots go back to medieval Japan. In an example of Stigler's law of eponymy, they are named after Eric Temple Bell, who wrote about them in the 1930s.
The Bell numbers are denoted , where is an integer greater than or equal to zero. Starting with , the first few Bell numbers are
1, 1, 2, 5, 15, 52, 203, 877, 4140, ... .
The Bell number counts the number of different ways to partition a set that has exactly elements, or equivalently, the number of equivalence relations on it. also counts the number of different rhyme schemes for -line poems.
As well as appearing in counting problems, these numbers have a different interpretation, as moments of probability distributions. In particular, is the -th moment of a Poisson distribution with mean 1.
Counting
Set partitions
In general, is the number of partitions of a set of size . A partition of a set is defined as a family of nonempty, pairwise disjoint subsets of whose union is . For example, because the 3-element set can be partitioned in 5 distinct ways:
As suggested by the set notation above, the ordering of subsets within the family is not considered; ordered partitions are counted by a different sequence of numbers, the ordered Bell numbers. is 1 because there is exactly one partition of the empty set. This partition is itself the empty set; it can be interpreted as a family of subsets of the empty set, consisting of zero subsets. It is vacuously true that all of the subsets in this family are non-empty subsets of the empty set and that they are pairwise disjoint subsets of the empty set, because there are no subsets to have these unlikely properties.
The partitions of a set correspond one-to-one with its equivalence relations. These are binary relations that are reflexive, symmetric, and transitive. The equivalence relation corresponding to a partition defines two elements as being equivalent when they belong to the same partition subset as each other. Conversely, every equivalence relation corresponds to a partition into equivalence classes. Therefore, the Bell numbers also count the equivalence relations.
Factorizations
If a number is a squarefree positive integer, meaning that it is the product of some number of distinct prime numbers), then gives the number of different multiplicative partitions of . These are factorizations of into numbers greater than one, treating two factorizations as the same if they have the same factors in a different order. For instance, 30 is the product of the three primes 2, 3, and 5, and has = 5 factorizations:
Rhyme schemes
The Bell numbers also count the rhyme schemes of an n-line poem or stanza. A rhyme scheme describes which lines rhyme with each other, and so may be interpreted as a partition of the set of lines into rhyming subsets. Rhyme schemes are usually written as
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https://en.wikipedia.org/wiki/Squaring%20the%20circle
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Squaring the circle is a problem in geometry first proposed in Greek mathematics. It is the challenge of constructing a square with the area of a given circle by using only a finite number of steps with a compass and straightedge. The difficulty of the problem raised the question of whether specified axioms of Euclidean geometry concerning the existence of lines and circles implied the existence of such a square.
In 1882, the task was proven to be impossible, as a consequence of the Lindemann–Weierstrass theorem, which proves that pi () is a transcendental number.
That is, is not the root of any polynomial with rational coefficients. It had been known for decades that the construction would be impossible if were transcendental, but that fact was not proven until 1882. Approximate constructions with any given non-perfect accuracy exist, and many such constructions have been found.
Despite the proof that it is impossible, attempts to square the circle have been common in pseudomathematics (i.e. the work of mathematical cranks). The expression "squaring the circle" is sometimes used as a metaphor for trying to do the impossible.
The term quadrature of the circle is sometimes used as a synonym for squaring the circle. It may also refer to approximate or numerical methods for finding the area of a circle. In general, quadrature or squaring may also be applied to other plane figures.
History
Methods to calculate the approximate area of a given circle, which can be thought of as a precursor problem to squaring the circle, were known already in many ancient cultures. These methods can be summarized by stating the approximation to that they produce. In around 2000 BCE, the Babylonian mathematicians used the approximation and at approximately the same time the ancient Egyptian mathematicians used Over 1000 years later, the Old Testament Books of Kings used the simpler approximation Ancient Indian mathematics, as recorded in the Shatapatha Brahmana and Shulba Sutras, used several different approximations Archimedes proved a formula for the area of a circle, according to which . In Chinese mathematics, in the third century CE, Liu Hui found even more accurate approximations using a method similar to that of Archimedes, and in the fifth century Zu Chongzhi found , an approximation known as Milü.
The problem of constructing a square whose area is exactly that of a circle, rather than an approximation to it, comes from Greek mathematics. Greek mathematicians found compass and straightedge constructions to convert any polygon into a square of equivalent area. They used this construction to compare areas of polygons geometrically, rather than by the numerical computation of area that would be more typical in modern mathematics. As Proclus wrote many centuries later, this motivated the search for methods that would allow comparisons with non-polygonal shapes:
The first known Greek to study the problem was Anaxagoras, who worked on it while in prison.
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https://en.wikipedia.org/wiki/Gradient%20descent
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In mathematics, gradient descent (also often called steepest descent) is a first-order iterative optimization algorithm for finding a local minimum of a differentiable function. The idea is to take repeated steps in the opposite direction of the gradient (or approximate gradient) of the function at the current point, because this is the direction of steepest descent. Conversely, stepping in the direction of the gradient will lead to a local maximum of that function; the procedure is then known as gradient ascent.
It is particularly useful in machine learning for minimizing the cost or loss function. Gradient descent should not be confused with local search algorithms, although both are iterative methods for optimization.
Gradient descent is generally attributed to Augustin-Louis Cauchy, who first suggested it in 1847. Jacques Hadamard independently proposed a similar method in 1907. Its convergence properties for non-linear optimization problems were first studied by Haskell Curry in 1944, with the method becoming increasingly well-studied and used in the following decades.
A simple extension of gradient descent, stochastic gradient descent, serves as the most basic algorithm used for training most deep networks today.
Description
Gradient descent is based on the observation that if the multi-variable function is defined and differentiable in a neighborhood of a point , then decreases fastest if one goes from in the direction of the negative gradient of at . It follows that, if
for a small enough step size or learning rate , then . In other words, the term is subtracted from because we want to move against the gradient, toward the local minimum. With this observation in mind, one starts with a guess for a local minimum of , and considers the sequence such that
We have a monotonic sequence
so, hopefully, the sequence converges to the desired local minimum. Note that the value of the step size is allowed to change at every iteration.
It is possible to guarantee the convergence to a local minimum under certain assumptions on the function (for example, convex and Lipschitz) and particular choices of . Those include the sequence
as in the Barzilai-Borwein method, or a sequence satisfying the Wolfe conditions (which can be found by using line search). When the function is convex, all local minima are also global minima, so in this case gradient descent can converge to the global solution.
This process is illustrated in the adjacent picture. Here, is assumed to be defined on the plane, and that its graph has a bowl shape. The blue curves are the contour lines, that is, the regions on which the value of is constant. A red arrow originating at a point shows the direction of the negative gradient at that point. Note that the (negative) gradient at a point is orthogonal to the contour line going through that point. We see that gradient descent leads us to the bottom of the bowl, that is, to the point where the value of the func
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https://en.wikipedia.org/wiki/Orthogonalization
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In linear algebra, orthogonalization is the process of finding a set of orthogonal vectors that span a particular subspace. Formally, starting with a linearly independent set of vectors {v1, ... , vk} in an inner product space (most commonly the Euclidean space Rn), orthogonalization results in a set of orthogonal vectors {u1, ... , uk} that generate the same subspace as the vectors v1, ... , vk. Every vector in the new set is orthogonal to every other vector in the new set; and the new set and the old set have the same linear span.
In addition, if we want the resulting vectors to all be unit vectors, then we normalize each vector and the procedure is called orthonormalization.
Orthogonalization is also possible with respect to any symmetric bilinear form (not necessarily an inner product, not necessarily over real numbers), but standard algorithms may encounter division by zero in this more general setting.
Orthogonalization algorithms
Methods for performing orthogonalization include:
Gram–Schmidt process, which uses projection
Householder transformation, which uses reflection
Givens rotation
Symmetric orthogonalization, which uses the Singular value decomposition
When performing orthogonalization on a computer, the Householder transformation is usually preferred over the Gram–Schmidt process since it is more numerically stable, i.e. rounding errors tend to have less serious effects.
On the other hand, the Gram–Schmidt process produces the jth orthogonalized vector after the jth iteration, while orthogonalization using Householder reflections produces all the vectors only at the end. This makes only the Gram–Schmidt process applicable for iterative methods like the Arnoldi iteration.
The Givens rotation is more easily parallelized than Householder transformations.
Symmetric orthogonalization was formulated by Per-Olov Löwdin.
Local orthogonalization
To compensate for the loss of useful signal in traditional noise attenuation approaches because of incorrect parameter selection or inadequacy of denoising assumptions, a weighting operator can be applied on the initially denoised section for the retrieval of useful signal from the initial noise section. The new denoising process is referred to as the local orthogonalization of signal and noise.
It has a wide range of applications in many signals processing and seismic exploration fields.
See also
Orthogonality
Biorthogonal system
Orthogonal basis
References
Linear algebra
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https://en.wikipedia.org/wiki/Principle%20of%20maximum%20entropy
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The principle of maximum entropy states that the probability distribution which best represents the current state of knowledge about a system is the one with largest entropy, in the context of precisely stated prior data (such as a proposition that expresses testable information).
Another way of stating this: Take precisely stated prior data or testable information about a probability distribution function. Consider the set of all trial probability distributions that would encode the prior data. According to this principle, the distribution with maximal information entropy is the best choice.
History
The principle was first expounded by E. T. Jaynes in two papers in 1957 where he emphasized a natural correspondence between statistical mechanics and information theory. In particular, Jaynes offered a new and very general rationale why the Gibbsian method of statistical mechanics works. He argued that the entropy of statistical mechanics and the information entropy of information theory are basically the same thing. Consequently, statistical mechanics should be seen just as a particular application of a general tool of logical inference and information theory.
Overview
In most practical cases, the stated prior data or testable information is given by a set of conserved quantities (average values of some moment functions), associated with the probability distribution in question. This is the way the maximum entropy principle is most often used in statistical thermodynamics. Another possibility is to prescribe some symmetries of the probability distribution. The equivalence between conserved quantities and corresponding symmetry groups implies a similar equivalence for these two ways of specifying the testable information in the maximum entropy method.
The maximum entropy principle is also needed to guarantee the uniqueness and consistency of probability assignments obtained by different methods, statistical mechanics and logical inference in particular.
The maximum entropy principle makes explicit our freedom in using different forms of prior data. As a special case, a uniform prior probability density (Laplace's principle of indifference, sometimes called the principle of insufficient reason), may be adopted. Thus, the maximum entropy principle is not merely an alternative way to view the usual methods of inference of classical statistics, but represents a significant conceptual generalization of those methods.
However these statements do not imply that thermodynamical systems need not be shown to be ergodic to justify treatment as a statistical ensemble.
In ordinary language, the principle of maximum entropy can be said to express a claim of epistemic modesty, or of maximum ignorance. The selected distribution is the one that makes the least claim to being informed beyond the stated prior data, that is to say the one that admits the most ignorance beyond the stated prior data.
Testable information
The principle of maximum entropy is usefu
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https://en.wikipedia.org/wiki/Mean%20squared%20error
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In statistics, the mean squared error (MSE) or mean squared deviation (MSD) of an estimator (of a procedure for estimating an unobserved quantity) measures the average of the squares of the errors—that is, the average squared difference between the estimated values and the actual value. MSE is a risk function, corresponding to the expected value of the squared error loss. The fact that MSE is almost always strictly positive (and not zero) is because of randomness or because the estimator does not account for information that could produce a more accurate estimate. In machine learning, specifically empirical risk minimization, MSE may refer to the empirical risk (the average loss on an observed data set), as an estimate of the true MSE (the true risk: the average loss on the actual population distribution).
The MSE is a measure of the quality of an estimator. As it is derived from the square of Euclidean distance, it is always a positive value that decreases as the error approaches zero.
The MSE is the second moment (about the origin) of the error, and thus incorporates both the variance of the estimator (how widely spread the estimates are from one data sample to another) and its bias (how far off the average estimated value is from the true value). For an unbiased estimator, the MSE is the variance of the estimator. Like the variance, MSE has the same units of measurement as the square of the quantity being estimated. In an analogy to standard deviation, taking the square root of MSE yields the root-mean-square error or root-mean-square deviation (RMSE or RMSD), which has the same units as the quantity being estimated; for an unbiased estimator, the RMSE is the square root of the variance, known as the standard error.
Definition and basic properties
The MSE either assesses the quality of a predictor (i.e., a function mapping arbitrary inputs to a sample of values of some random variable), or of an estimator (i.e., a mathematical function mapping a sample of data to an estimate of a parameter of the population from which the data is sampled). In the context of prediction, understanding the prediction interval can also be useful as it provides a range within which a future observation will fall, with a certain probability..The definition of an MSE differs according to whether one is describing a predictor or an estimator.
Predictor
If a vector of predictions is generated from a sample of data points on all variables, and is the vector of observed values of the variable being predicted, with being the predicted values (e.g. as from a least-squares fit), then the within-sample MSE of the predictor is computed as
In other words, the MSE is the mean of the squares of the errors . This is an easily computable quantity for a particular sample (and hence is sample-dependent).
In matrix notation,
where is and is a column vector.
The MSE can also be computed on q data points that were not used in estimating the model, either because th
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https://en.wikipedia.org/wiki/Trapezoid
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In geometry, a trapezoid () in North American English, or trapezium () in British English, is a quadrilateral that has at least one pair of parallel sides.
The parallel sides are called the bases of the trapezoid. The other two sides are called the legs (or the lateral sides) if they are not parallel; otherwise, the trapezoid is a parallelogram, and there are two pairs of bases. A scalene trapezoid is a trapezoid with no sides of equal measure, in contrast with the special cases below.
A trapezoid is usually considered to be a convex quadrilateral in Euclidean geometry, but there are also crossed cases. If ABCD is a convex trapezoid, then ABDC is a crossed trapezoid. The metric formulas in this article apply in convex trapezoids.
Etymology and trapezium versus trapezoid
The ancient Greek mathematician Euclid defined five types of quadrilateral, of which four had two sets of parallel sides (known in English as square, rectangle, rhombus and rhomboid) and the last did not have two sets of parallel sides – a τραπέζια (trapezia literally 'table', itself from τετράς (tetrás) 'four' + πέζα (péza) 'foot; end, border, edge').
Two types of trapezia were introduced by Proclus (AD 412 to 485) in his commentary on the first book of Euclid's Elements:
one pair of parallel sides – a trapezium (τραπέζιον), divided into isosceles (equal legs) and scalene (unequal) trapezia
no parallel sides – trapezoid (τραπεζοειδή, trapezoeidé, literally 'trapezium-like' (εἶδος means 'resembles'), in the same way as cuboid means 'cube-like' and rhomboid means 'rhombus-like')
All European languages follow Proclus's structure as did English until the late 18th century, until an influential mathematical dictionary published by Charles Hutton in 1795 supported without explanation a transposition of the terms. This was reversed in British English in about 1875, but it has been retained in American English to the present.
The following table compares usages, with the most specific definitions at the top to the most general at the bottom.
Inclusive versus exclusive definition
There is some disagreement whether parallelograms, which have two pairs of parallel sides, should be regarded as trapezoids.
Some define a trapezoid as a quadrilateral having only one pair of parallel sides (the exclusive definition), thereby excluding parallelograms. Some sources use the term proper trapezoid to describe trapezoids under the exclusive definition, analogous to uses of the word proper in some other mathematical objects.
Others define a trapezoid as a quadrilateral with at least one pair of parallel sides (the inclusive definition), making the parallelogram a special type of trapezoid. The latter definition is consistent with its uses in higher mathematics such as calculus. This article uses the inclusive definition and considers parallelograms as special cases of a trapezoid. This is also advocated in the taxonomy of quadrilaterals.
Under the inclusive definition, all parallelograms
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https://en.wikipedia.org/wiki/MATH-MATIC
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MATH-MATIC is the marketing name for the AT-3 (Algebraic Translator 3) compiler, an early programming language for the UNIVAC I and UNIVAC II.
MATH-MATIC was written beginning around 1955 by a team led by Charles Katz under the direction of Grace Hopper. A preliminary manual was produced in 1957 and a final manual the following year.
Syntactically, MATH-MATIC was similar to Univac's contemporaneous business-oriented language, FLOW-MATIC, differing in providing algebraic-style expressions and floating-point arithmetic, and arrays rather than record structures.
Notable features
Expressions in MATH-MATIC could contain numeric exponents, including decimals and fractions, by way of a custom typewriter.
MATH-MATIC programs could include inline assembler sections of ARITH-MATIC code and UNIVAC machine code.
The UNIVAC I had only 1000 words of memory, and the successor UNIVAC II as little as 2000. MATH-MATIC allowed for larger programs, automatically generating code to read overlay segments from UNISERVO tape as required. The compiler attempted to avoid splitting loops across segments.
Influence
In proposing the collaboration with the ACM that led to ALGOL 58, the Gesellschaft für Angewandte Mathematik und Mechanik wrote that it considered MATH-MATIC the closest available language to its own proposal.
In contrast to Backus' FORTRAN, MATH-MATIC did not emphasise execution speed of compiled programs. The UNIVAC machines did not have floating-point hardware, and MATH-MATIC was translated via A-3 (ARITH-MATIC) pseudo-assembler code rather than directly to UNIVAC machine code, limiting its usefulness.
MATH-MATIC Sample program
A sample MATH-MATIC program:
Notes
References
Numerical programming languages
Programming languages
Programming languages created in 1957
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https://en.wikipedia.org/wiki/Standardized%20moment
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In probability theory and statistics, a standardized moment of a probability distribution is a moment (often a higher degree central moment) that is normalized, typically by a power of the standard deviation, rendering the moment scale invariant. The shape of different probability distributions can be compared using standardized moments.
Standard normalization
Let X be a random variable with a probability distribution P and mean value (i.e. the first raw moment or moment about zero), the operator E denoting the expected value of X. Then the standardized moment of degree k is that is, the ratio of the kth moment about the mean
to the kth power of the standard deviation,
The power of k is because moments scale as meaning that they are homogeneous functions of degree k, thus the standardized moment is scale invariant. This can also be understood as being because moments have dimension; in the above ratio defining standardized moments, the dimensions cancel, so they are dimensionless numbers.
The first four standardized moments can be written as:
For skewness and kurtosis, alternative definitions exist, which are based on the third and fourth cumulant respectively.
Other normalizations
Another scale invariant, dimensionless measure for characteristics of a distribution is the coefficient of variation, . However, this is not a standardized moment, firstly because it is a reciprocal, and secondly because is the first moment about zero (the mean), not the first moment about the mean (which is zero).
See Normalization (statistics) for further normalizing ratios.
See also
Coefficient of variation
Moment (mathematics)
Central moment
References
Statistical deviation and dispersion
Statistical ratios
Moment (mathematics)
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https://en.wikipedia.org/wiki/Tensor%20contraction
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In multilinear algebra, a tensor contraction is an operation on a tensor that arises from the natural pairing of a finite-dimensional vector space and its dual. In components, it is expressed as a sum of products of scalar components of the tensor(s) caused by applying the summation convention to a pair of dummy indices that are bound to each other in an expression. The contraction of a single mixed tensor occurs when a pair of literal indices (one a subscript, the other a superscript) of the tensor are set equal to each other and summed over. In Einstein notation this summation is built into the notation. The result is another tensor with order reduced by 2.
Tensor contraction can be seen as a generalization of the trace.
Abstract formulation
Let V be a vector space over a field k. The core of the contraction operation, and the simplest case, is the natural pairing of V with its dual vector space V∗. The pairing is the linear transformation from the tensor product of these two spaces to the field k:
corresponding to the bilinear form
where f is in V∗ and v is in V. The map C defines the contraction operation on a tensor of type , which is an element of . Note that the result is a scalar (an element of k). Using the natural isomorphism between and the space of linear transformations from V to V, one obtains a basis-free definition of the trace.
In general, a tensor of type (with and ) is an element of the vector space
(where there are m factors V and n factors V∗). Applying the natural pairing to the kth V factor and the lth V∗ factor, and using the identity on all other factors, defines the (k, l) contraction operation, which is a linear map which yields a tensor of type . By analogy with the case, the general contraction operation is sometimes called the trace.
Contraction in index notation
In tensor index notation, the basic contraction of a vector and a dual vector is denoted by
which is shorthand for the explicit coordinate summation
(where are the components of in a particular basis and are the components of in the corresponding dual basis).
Since a general mixed dyadic tensor is a linear combination of decomposable tensors of the form , the explicit formula for the dyadic case follows: let
be a mixed dyadic tensor. Then its contraction is
.
A general contraction is denoted by labeling one covariant index and one contravariant index with the same letter, summation over that index being implied by the summation convention. The resulting contracted tensor inherits the remaining indices of the original tensor. For example, contracting a tensor T of type (2,2) on the second and third indices to create a new tensor U of type (1,1) is written as
By contrast, let
be an unmixed dyadic tensor. This tensor does not contract; if its base vectors are dotted, the result is the contravariant metric tensor,
,
whose rank is 2.
Metric contraction
As in the previous example, contraction on a pair of i
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https://en.wikipedia.org/wiki/Adjugate%20matrix
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In linear algebra, the adjugate or classical adjoint of a square matrix is the transpose of its cofactor matrix and is denoted by . It is also occasionally known as adjunct matrix, or "adjoint", though the latter term today normally refers to a different concept, the adjoint operator which for a matrix is the conjugate transpose.
The product of a matrix with its adjugate gives a diagonal matrix (entries not on the main diagonal are zero) whose diagonal entries are the determinant of the original matrix:
where is the identity matrix of the same size as . Consequently, the multiplicative inverse of an invertible matrix can be found by dividing its adjugate by its determinant.
Definition
The adjugate of is the transpose of the cofactor matrix of ,
In more detail, suppose is a unital commutative ring and is an matrix with entries from . The -minor of , denoted , is the determinant of the matrix that results from deleting row and column of . The cofactor matrix of is the matrix whose entry is the cofactor of , which is the -minor times a sign factor:
The adjugate of is the transpose of , that is, the matrix whose entry is the cofactor of ,
Important consequence
The adjugate is defined so that the product of with its adjugate yields a diagonal matrix whose diagonal entries are the determinant . That is,
where is the identity matrix. This is a consequence of the Laplace expansion of the determinant.
The above formula implies one of the fundamental results in matrix algebra, that is invertible if and only if is an invertible element of . When this holds, the equation above yields
Examples
1 × 1 generic matrix
Since the determinant of a 0 × 0 matrix is 1, the adjugate of any 1 × 1 matrix (complex scalar) is . Observe that
2 × 2 generic matrix
The adjugate of the 2 × 2 matrix
is
By direct computation,
In this case, it is also true that ((A)) = (A) and hence that ((A)) = A.
3 × 3 generic matrix
Consider a 3 × 3 matrix
Its cofactor matrix is
where
Its adjugate is the transpose of its cofactor matrix,
3 × 3 numeric matrix
As a specific example, we have
It is easy to check the adjugate is the inverse times the determinant, .
The in the second row, third column of the adjugate was computed as follows. The (2,3) entry of the adjugate is the (3,2) cofactor of A. This cofactor is computed using the submatrix obtained by deleting the third row and second column of the original matrix A,
The (3,2) cofactor is a sign times the determinant of this submatrix:
and this is the (2,3) entry of the adjugate.
Properties
For any matrix , elementary computations show that adjugates have the following properties:
, where is the identity matrix.
, where is the zero matrix, except that if then .
for any scalar .
.
.
If is invertible, then . It follows that:
is invertible with inverse .
.
is entrywise polynomial in . In particular, over the real or complex numbers, the adjugate is a smooth function
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https://en.wikipedia.org/wiki/Covariance%20and%20contravariance%20of%20vectors
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In physics, especially in multilinear algebra and tensor analysis, covariance and contravariance describe how the quantitative description of certain geometric or physical entities changes with a change of basis. In modern mathematical notation, the role is sometimes swapped.
A simple illustrative case is that of a vector. For a vector, once a set of basis vectors has been defined, then the components of that vector will always vary opposite to that of the basis vectors. A vector is therefore a contravariant tensor. Take a standard position vector for example. By changing the scale of the reference axes from meters to centimeters (that is, dividing the scale of the reference axes by 100, so that the basis vectors now are meters long), the components of the measured position vector are multiplied by 100. A vector's components change scale inversely to changes in scale to the reference axes, and consequently a vector is called a contravariant tensor.
In contrast, a covector, also called a dual vector, has components that vary with the basis vectors in the corresponding vector space. It is an example of a covariant tensor. A covector is an object that represents a linear map from vectors to scalars. It is actually not a vector, but an object that lives in a dual vector space. Some good examples of covectors are dot product operators involving vectors. For example if is a vector, then a corresponding object in the dual space would be the linear operator . Sometimes, the components of the covector are referred to as the covariant components of , although this is potentially misleading, (due to a vector having components that always vary in the contravariant sense). Despite potential confusion, this is what will be meant when the "covariant components of a vector" are referred to herein.
The gradient is often cited as an example of a covector, but this is incorrect. If the components of the gradient of a function , , are expressed in terms of a given basis, , then these components will in fact still vary oppositely to that of the basis vectors, as can be seen by observing (using the Einstein summation convention):
where is the kronecker delta symbol, and represents the components of some transformation matrix corresponding to the transformation . As can be seen, whatever transformation acts on the basis vectors, the inverse transformation must act on the components.
A third concept related to covariance and contravariance is invariance. A scalar (also called type-0 or rank-0 tensor) is an object that does not vary with the change in basis. An example of a physical observable that is a scalar is the mass of a particle. The single, scalar value of mass is independent to changes in basis vectors and consequently is called invariant. The magnitude of a vector (such as distance) is another example of an invariant, because it remains fixed even if geometrical vector components vary. (For example, for a position vector of leng
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https://en.wikipedia.org/wiki/Spherical%20harmonics
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In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields.
Since the spherical harmonics form a complete set of orthogonal functions and thus an orthonormal basis, each function defined on the surface of a sphere can be written as a sum of these spherical harmonics. This is similar to periodic functions defined on a circle that can be expressed as a sum of circular functions (sines and cosines) via Fourier series. Like the sines and cosines in Fourier series, the spherical harmonics may be organized by (spatial) angular frequency, as seen in the rows of functions in the illustration on the right. Further, spherical harmonics are basis functions for irreducible representations of SO(3), the group of rotations in three dimensions, and thus play a central role in the group theoretic discussion of SO(3).
Spherical harmonics originate from solving Laplace's equation in the spherical domains. Functions that are solutions to Laplace's equation are called harmonics. Despite their name, spherical harmonics take their simplest form in Cartesian coordinates, where they can be defined as homogeneous polynomials of degree in that obey Laplace's equation. The connection with spherical coordinates arises immediately if one uses the homogeneity to extract a factor of radial dependence from the above-mentioned polynomial of degree ; the remaining factor can be regarded as a function of the spherical angular coordinates and only, or equivalently of the orientational unit vector specified by these angles. In this setting, they may be viewed as the angular portion of a set of solutions to Laplace's equation in three dimensions, and this viewpoint is often taken as an alternative definition. Notice, however, that spherical harmonics are not functions on the sphere which are harmonic with respect to the Laplace-Beltrami operator for the standard round metric on the sphere: the only harmonic functions in this sense on the sphere are the constants, since harmonic functions satisfy the Maximum principle. Spherical harmonics, as functions on the sphere, are eigenfunctions of the Laplace-Beltrami operator (see the section Higher dimensions below).
A specific set of spherical harmonics, denoted or , are known as Laplace's spherical harmonics, as they were first introduced by Pierre Simon de Laplace in 1782. These functions form an orthogonal system, and are thus basic to the expansion of a general function on the sphere as alluded to above.
Spherical harmonics are important in many theoretical and practical applications, including the representation of multipole electrostatic and electromagnetic fields, electron configurations, gravitational fields, geoids, the magnetic fields of planetary bodies and stars, and the cosmic microwave background radiation. In 3D computer graphics, spherical harmonics play a role in a w
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https://en.wikipedia.org/wiki/List%20of%20order%20structures%20in%20mathematics
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In mathematics, and more specifically in order theory, several different types of ordered set have been studied.
They include:
Cyclic orders, orderings in which triples of elements are either clockwise or counterclockwise
Lattices, partial orders in which each pair of elements has a greatest lower bound and a least upper bound. Many different types of lattice have been studied; see map of lattices for a list.
Partially ordered sets (or posets), orderings in which some pairs are comparable and others might not be
Preorders, a generalization of partial orders allowing ties (represented as equivalences and distinct from incomparabilities)
Semiorders, partial orders determined by comparison of numerical values, in which values that are too close to each other are incomparable; a subfamily of partial orders with certain restrictions
Total orders, orderings that specify, for every two distinct elements, which one is less than the other
Weak orders, generalizations of total orders allowing ties (represented either as equivalences or, in strict weak orders, as transitive incomparabilities)
Well-orders, total orders in which every non-empty subset has a least element
Well-quasi-orderings, a class of preorders generalizing the well-orders
See also
Glossary of order theory
List of order theory topics
Mathematics-related lists
Order theory
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https://en.wikipedia.org/wiki/Predicate
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Predicate or predication may refer to:
Predicate (grammar), in linguistics
Predication (philosophy)
several closely related uses in mathematics and formal logic:
Predicate (mathematical logic)
Propositional function
Finitary relation, or n-ary predicate
Boolean-valued function
Syntactic predicate, in formal grammars and parsers
Functional predicate
Predication (computer architecture)
in United States law, the basis or foundation of something
Predicate crime
Predicate rules, in the U.S. Title 21 CFR Part 11
Predicate, a term used in some European context for either nobles' honorifics or for nobiliary particles
See also
Predicate logic
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https://en.wikipedia.org/wiki/Algebraic%20notation
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Algebraic notation may refer to:
In mathematics and computers, infix notation, the practice of representing a binary operator and operands with the operator between the two operands (as in "2 + 2")
Algebraic notation (chess), the standard system for recording movement of pieces in a chess game
In linguistics, recursive categorical syntax, also known as "algebraic syntax", a theory of how natural languages are structured
Mathematical notation for algebra
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https://en.wikipedia.org/wiki/Euler%20line
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In geometry, the Euler line, named after Leonhard Euler (), is a line determined from any triangle that is not equilateral. It is a central line of the triangle, and it passes through several important points determined from the triangle, including the orthocenter, the circumcenter, the centroid, the Exeter point and the center of the nine-point circle of the triangle.
The concept of a triangle's Euler line extends to the Euler line of other shapes, such as the quadrilateral and the tetrahedron.
Triangle centers on the Euler line
Individual centers
Euler showed in 1765 that in any triangle, the orthocenter, circumcenter and centroid are collinear. This property is also true for another triangle center, the nine-point center, although it had not been defined in Euler's time. In equilateral triangles, these four points coincide, but in any other triangle they are all distinct from each other, and the Euler line is determined by any two of them.
Other notable points that lie on the Euler line include the de Longchamps point, the Schiffler point, the Exeter point, and the Gossard perspector. However, the incenter generally does not lie on the Euler line; it is on the Euler line only for isosceles triangles, for which the Euler line coincides with the symmetry axis of the triangle and contains all triangle centers.
The tangential triangle of a reference triangle is tangent to the latter's circumcircle at the reference triangle's vertices. The circumcenter of the tangential triangle lies on the Euler line of the reference triangle. The center of similitude of the orthic and tangential triangles is also on the Euler line.
Proofs
A vector proof
Let be a triangle. A proof of the fact that the circumcenter , the centroid and the orthocenter are collinear relies on free vectors. We start by stating the prerequisites. First, satisfies the relation
This follows from the fact that the absolute barycentric coordinates of are . Further, the problem of Sylvester reads as
Now, using the vector addition, we deduce that
By adding these three relations, term by term, we obtain that
In conclusion, , and so the three points , and (in this order) are collinear.
In Dörrie's book, the Euler line and the problem of Sylvester are put together into a single proof. However, most of the proofs of the problem of Sylvester rely on the fundamental properties of free vectors, independently of the Euler line.
Properties
Distances between centers
On the Euler line the centroid G is between the circumcenter O and the orthocenter H and is twice as far from the orthocenter as it is from the circumcenter:
The segment GH is a diameter of the orthocentroidal circle.
The center N of the nine-point circle lies along the Euler line midway between the orthocenter and the circumcenter:
Thus the Euler line could be repositioned on a number line with the circumcenter O at the location 0, the centroid G at 2t, the nine-point center at 3t, and the orthocenter H at 6t for
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https://en.wikipedia.org/wiki/Liouville%E2%80%93Neumann%20series
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In mathematics, the Liouville–Neumann series is an infinite series that corresponds to the resolvent formalism technique of solving the Fredholm integral equations in Fredholm theory.
Definition
The Liouville–Neumann (iterative) series is defined as
which, provided that is small enough so that the series converges, is the unique continuous solution of the Fredholm integral equation of the second kind,
If the nth iterated kernel is defined as n−1 nested integrals of n operators ,
then
with
so K0 may be taken to be .
The resolvent (or solving kernel for the integral operator) is then given by a schematic analog "geometric series",
where K0 has been taken to be .
The solution of the integral equation thus becomes simply
Similar methods may be used to solve the Volterra equations.
See also
Neumann series
References
Mathews, Jon; Walker, Robert L. (1970), Mathematical methods of physics (2nd ed.), New York: W. A. Benjamin,
Fredholm theory
Mathematical series
Mathematical physics
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https://en.wikipedia.org/wiki/It%C3%B4%27s%20lemma
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In mathematics, Itô's lemma or Itô's formula (also called the Itô-Doeblin formula, especially in the French literature) is an identity used in Itô calculus to find the differential of a time-dependent function of a stochastic process. It serves as the stochastic calculus counterpart of the chain rule. It can be heuristically derived by forming the Taylor series expansion of the function up to its second derivatives and retaining terms up to first order in the time increment and second order in the Wiener process increment. The lemma is widely employed in mathematical finance, and its best known application is in the derivation of the Black–Scholes equation for option values.
Kiyoshi Itô published a proof of the formula in 1951.
Motivation
Suppose we are given the stochastic differential equation
where is a Wiener process and the functions are deterministic (not stochastic) functions of time. In general, it's not possible to write a solution directly in terms of However, we can formally write an integral solution
This expression lets us easily read off the mean and variance of (which has no higher moments). First, notice that every individually has mean 0, so the expectation value of is simply the integral of the drift function:
Similarly, because the terms have variance 1 and no correlation with one another, the variance of is simply the integral of the variance of each infinitesimal step in the random walk:
However, sometimes we are faced with a stochastic differential equation for a more complex process in which the process appears on both sides of the differential equation. That is, say
for some functions and In this case, we cannot immediately write a formal solution as we did for the simpler case above. Instead, we hope to write the process as a function of a simpler process taking the form above. That is, we want to identify three functions and such that and In practice, Ito's lemma is used in order to find this transformation. Finally, once we have transformed the problem into the simpler type of problem, we can determine the mean and higher moments of the process.
Informal derivation
A formal proof of the lemma relies on taking the limit of a sequence of random variables. This approach is not presented here since it involves a number of technical details. Instead, we give a sketch of how one can derive Itô's lemma by expanding a Taylor series and applying the rules of stochastic calculus.
Suppose is an Itô drift-diffusion process that satisfies the stochastic differential equation
where is a Wiener process.
If is a twice-differentiable scalar function, its expansion in a Taylor series is
Substituting for and therefore for gives
In the limit , the terms and tend to zero faster than , which is . Setting the and terms to zero, substituting for (due to the quadratic variation of a Wiener process), and collecting the and terms, we obtain
as required.
Geometric intuition
Suppose we know that
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https://en.wikipedia.org/wiki/Directed%20acyclic%20graph
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In mathematics, particularly graph theory, and computer science, a directed acyclic graph (DAG) is a directed graph with no directed cycles. That is, it consists of vertices and edges (also called arcs), with each edge directed from one vertex to another, such that following those directions will never form a closed loop. A directed graph is a DAG if and only if it can be topologically ordered, by arranging the vertices as a linear ordering that is consistent with all edge directions. DAGs have numerous scientific and computational applications, ranging from biology (evolution, family trees, epidemiology) to information science (citation networks) to computation (scheduling).
Directed acyclic graphs are sometimes instead called acyclic directed graphs or acyclic digraphs.
Definitions
A graph is formed by vertices and by edges connecting pairs of vertices, where the vertices can be any kind of object that is connected in pairs by edges. In the case of a directed graph, each edge has an orientation, from one vertex to another vertex. A path in a directed graph is a sequence of edges having the property that the ending vertex of each edge in the sequence is the same as the starting vertex of the next edge in the sequence; a path forms a cycle if the starting vertex of its first edge equals the ending vertex of its last edge. A directed acyclic graph is a directed graph that has no cycles.
A vertex of a directed graph is said to be reachable from another vertex when there exists a path that starts at and ends at . As a special case, every vertex is considered to be reachable from itself (by a path with zero edges). If a vertex can reach itself via a nontrivial path (a path with one or more edges), then that path is a cycle, so another way to define directed acyclic graphs is that they are the graphs in which no vertex can reach itself via a nontrivial path.
Mathematical properties
Reachability relation, transitive closure, and transitive reduction
The reachability relation of a DAG can be formalized as a partial order on the vertices of the DAG. In this partial order, two vertices and are ordered as exactly when there exists a directed path from to in the DAG; that is, when can reach (or is reachable from ). However, different DAGs may give rise to the same reachability relation and the same partial order. For example, a DAG with two edges and has the same reachability relation as the DAG with three edges , , and . Both of these DAGs produce the same partial order, in which the vertices are ordered as .
The transitive closure of a DAG is the graph with the most edges that has the same reachability relation as the DAG. It has an edge for every pair of vertices (, ) in the reachability relation of the DAG, and may therefore be thought of as a direct translation of the reachability relation into graph-theoretic terms. The same method of translating partial orders into DAGs works more generally: for every finite partially ordere
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https://en.wikipedia.org/wiki/The%20Nine%20Chapters%20on%20the%20Mathematical%20Art
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The Nine Chapters on the Mathematical Art is a Chinese mathematics book, composed by several generations of scholars from the 10th–2nd century BCE, its latest stage being from the 2nd century CE. This book is one of the earliest surviving mathematical texts from China, the first being the Suan shu shu (202 BCE – 186 BCE) and Zhoubi Suanjing (compiled throughout the Han until the late 2nd century CE). It lays out an approach to mathematics that centres on finding the most general methods of solving problems, which may be contrasted with the approach common to ancient Greek mathematicians, who tended to deduce propositions from an initial set of axioms.
Entries in the book usually take the form of a statement of a problem, followed by the statement of the solution and an explanation of the procedure that led to the solution. These were commented on by Liu Hui in the 3rd century.
History
Original book
The full title of The Nine Chapters on the Mathematical Art appears on two bronze standard measures which are dated to 179 CE, but there is speculation that the same book existed beforehand under different titles.
Most scholars believe that Chinese mathematics and the mathematics of the ancient Mediterranean world had developed more or less independently up to the time when The Nine Chapters reached its final form. The method of chapter 7 was not found in Europe until the 13th century, and the method of chapter 8 uses Gaussian elimination before Carl Friedrich Gauss (1777–1855). There is also the mathematical proof given in the treatise for the Pythagorean theorem. The influence of The Nine Chapters greatly assisted the development of ancient mathematics in the regions of Korea and Japan. Its influence on mathematical thought in China persisted until the Qing dynasty era.
Liu Hui wrote a very detailed commentary on this book in 263. He analyses the procedures of The Nine Chapters step by step, in a manner which is clearly designed to give the reader confidence that they are reliable, although he is not concerned to provide formal proofs in the Euclidean manner. Liu's commentary is of great mathematical interest in its own right. Liu credits the earlier mathematicians Zhang Cang (fl. 165 BCE – d. 142 BCE) and Geng Shouchang (fl. 75 BCE – 49 BCE) (see armillary sphere) with the initial arrangement and commentary on the book, yet Han dynasty records do not indicate the names of any authors of commentary, as they are not mentioned until the 3rd century.
The Nine Chapters is an anonymous work, and its origins are not clear. Until recent years, there was no substantial evidence of related mathematical writing that might have preceded it, with the exception of mathematical work by those such as Jing Fang (78–37 BCE), Liu Xin (d. 23), and Zhang Heng (78–139) and the geometry clauses of the Mozi of the 4th century BCE. This is no longer the case. The Suàn shù shū (算數書) or Writings on Reckonings is an ancient Chinese text on mathematics approximately seve
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https://en.wikipedia.org/wiki/Normal%20number
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In mathematics, a real number is said to be simply normal in an integer base b if its infinite sequence of digits is distributed uniformly in the sense that each of the b digit values has the same natural density 1/b. A number is said to be normal in base b if, for every positive integer n, all possible strings n digits long have density b−n.
Intuitively, a number being simply normal means that no digit occurs more frequently than any other. If a number is normal, no finite combination of digits of a given length occurs more frequently than any other combination of the same length. A normal number can be thought of as an infinite sequence of coin flips (binary) or rolls of a die (base 6). Even though there will be sequences such as 10, 100, or more consecutive tails (binary) or fives (base 6) or even 10, 100, or more repetitions of a sequence such as tail-head (two consecutive coin flips) or 6-1 (two consecutive rolls of a die), there will also be equally many of any other sequence of equal length. No digit or sequence is "favored".
A number is said to be normal (sometimes called absolutely normal) if it is normal in all integer bases greater than or equal to 2.
While a general proof can be given that almost all real numbers are normal (meaning that the set of non-normal numbers has Lebesgue measure zero), this proof is not constructive, and only a few specific numbers have been shown to be normal. For example, any Chaitin's constant is normal (and uncomputable). It is widely believed that the (computable) numbers , , and e are normal, but a proof remains elusive.
Definitions
Let be a finite alphabet of -digits, the set of all infinite sequences that may be drawn from that alphabet, and the set of finite sequences, or strings. Let be such a sequence. For each in let denote the number of times the digit appears in the first digits of the sequence . We say that is simply normal if the limit
for each . Now let be any finite string in and let be the number of times the string appears as a substring in the first digits of the sequence . (For instance, if , then .) is normal if, for all finite strings ,
where denotes the length of the string . In other words, is normal if all strings of equal length occur with equal asymptotic frequency. For example, in a normal binary sequence (a sequence over the alphabet ), and each occur with frequency ; , , , and each occur with frequency ; , , , , , , , and each occur with frequency ; etc. Roughly speaking, the probability of finding the string in any given position in is precisely that expected if the sequence had been produced at random.
Suppose now that is an integer greater than 1 and is a real number. Consider the infinite digit sequence expansion of in the base positional number system (we ignore the decimal point). We say that is simply normal in base if the sequence is simply normal and that is normal in base if the sequence is normal. The number is called a n
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https://en.wikipedia.org/wiki/Translation%20%28geometry%29
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In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure, shape or space by the same distance in a given direction. A translation can also be interpreted as the addition of a constant vector to every point, or as shifting the origin of the coordinate system. In a Euclidean space, any translation is an isometry.
As a function
If is a fixed vector, known as the translation vector, and is the initial position of some object, then the translation function will work as .
If is a translation, then the image of a subset under the function is the translate of by . The translate of by is often written .
Horizontal and vertical translations
In geometry, a vertical translation (also known as vertical shift) is a translation of a geometric object in a direction parallel to the vertical axis of the Cartesian coordinate system.
Often, vertical translations are considered for the graph of a function. If f is any function of x, then the graph of the function f(x) + c (whose values are given by adding a constant c to the values of f) may be obtained by a vertical translation of the graph of f(x) by distance c. For this reason the function f(x) + c is sometimes called a vertical translate of f(x). For instance, the antiderivatives of a function all differ from each other by a constant of integration and are therefore vertical translates of each other.
In function graphing, a horizontal translation is a transformation which results in a graph that is equivalent to shifting the base graph left or right in the direction of the x-axis. A graph is translated k units horizontally by moving each point on the graph k units horizontally.
For the base function f(x) and a constant k, the function given by g(x) = f(x − k), can be sketched f(x) shifted k units horizontally.
If function transformation was talked about in terms of geometric transformations it may be clearer why functions translate horizontally the way they do. When addressing translations on the Cartesian plane it is natural to introduce translations in this type of notation:
or
where and are horizontal and vertical changes respectively.
Example
Taking the parabola y = x2 , a horizontal translation 5 units to the right would be represented by T(x, y) = (x + 5, y). Now we must connect this transformation notation to an algebraic notation. Consider the point (a, b) on the original parabola that moves to point (c, d) on the translated parabola. According to our translation, c = a + 5 and d = b. The point on the original parabola was b = a2. Our new point can be described by relating d and c in the same equation. b = d and a = c − 5.
So d = b = a2 = (c − 5)2. Since this is true for all the points on our new parabola, the new equation is y = (x − 5)2.
Application in classical physics
In classical physics, translational motion is movement that changes the position of an object, as opposed to rotation. For example, according to Whittaker:
A trans
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https://en.wikipedia.org/wiki/List%20of%20mathematical%20probabilists
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See probabilism for the followers of such a theory in theology or philosophy.
This list contains only probabilists in the sense of mathematicians specializing in probability theory.
This list is incomplete; please add to it.
David Aldous (born 1952)
Siva Athreya
Thomas Bayes (1702–1761) - British mathematician and Presbyterian minister, known for Bayes' theorem
Gerard Ben-Arous (born 1957) - Courant Institute of Mathematical Sciences
Itai Benjamini
Jakob Bernoulli (1654–1705) - Switzerland, known for Bernoulli trials
Joseph Louis François Bertrand (1822–1900)
Abram Samoilovitch Besicovitch (1891–1970)
Patrick Billingsley (1925–2011)
Erwin Bolthausen (born 1945)
Carlo Emilio Bonferroni (1892–1960)
Émile Borel (1871–1956)
Sourav Chatterjee
Kai Lai Chung (1917–2009)
Erhan Cinlar (born 1941)
Harald Cramér (1893–1985)
Amir Dembo (born 1958)
Persi Diaconis (born 1945)
Hugo Duminil-Copin (born 1985)
Joseph Leo Doob (1910–2004)
Lester Dubins (1920–2010)
Eugene Dynkin (1924–2014)
Robert J. Elliott (born 1940)
Paul Erdős (1913–1996)
Alison Etheridge (born 1964)
Steve Evans (born 1960)
William Feller (1906–1970)
Bruno de Finetti (1906–1985) - Italian probabilist and statistician
Geoffrey Grimmett (born 1950)
Alice Guionnet (born 1969)
Ian Hacking (born 1936)
Paul Halmos (1916–2006)
Joseph Halpern (born 1953)
David Heath (c.1943–2011)
Wassily Hoeffding (1914–1991)
Kiyoshi Itô (1915–2008)
Jean Jacod (1944–)
Edwin Thompson Jaynes (1922–1998)
Mark Kac (1914–1984)
Olav Kallenberg (born 1939)
Rudolf E. Kálmán (1930–2016)
Samuel Karlin (1924–2007)
David George Kendall (1918–2007)
Richard Kenyon (born 1964) - Yale University
Harry Kesten (1931–2019)
John Maynard Keynes (1883–1946) - best known for his pioneering work in economics
Aleksandr Khinchin (1894–1959)
Andrey Kolmogorov (1903–1987)
Pierre-Simon Laplace (1749–1827)
Gregory Lawler (born 1955)
Lucien Le Cam (1924–2000)
Jean-François Le Gall (born 1959)
Paul Lévy (1886–1971)
Jarl Waldemar Lindeberg (1876–1932)
Andrey Markov (1856–1922)
Stefan Mazurkiewicz (1888–1945)
Henry McKean (born 1930)
Paul-André Meyer (1934–2003)
Richard von Mises (1883–1953)
Abraham de Moivre (1667–1754)
Octav Onicescu (1892–1983)
K. R. Parthasarathy (born 1936)
Blaise Pascal (1623–1662)
Charles E. M. Pearce (1940–2012)
Judea Pearl (born 1936)
Yuval Peres (born 1963)
Edwin A. Perkins (born 1953)
Siméon Denis Poisson (1781–1840)
Yuri Vasilevich Prokhorov (1929–2013)
Frank P. Ramsey (1903–1930)
Alfréd Rényi (1921–1970)
Oded Schramm (1961–2008)
Romano Scozzafava (born 1935)
Scott Sheffield (born 1973)
Albert Shiryaev (born 1934)
Yakov Sinai (born 1935)
Ray Solomonoff (1926–2009)
Frank Spitzer (1926–1992)
Ruslan L. Stratonovich (1930–1997)
Daniel W. Stroock (born 1940)
Tibor Szele (1918–1955)
Alain-Sol Sznitman (born 1955)
Michel Talagrand (born 1952)
Heinrich Emil Timerding (1873–1945)
Andrei Toom (born 1942)
S. R. Srinivasa Varadhan (born 1940) - 2007 Abel Prize laureate
Bálint Virág (born 1973)
Wendelin Werner (born 1968)
Norbert Wien
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https://en.wikipedia.org/wiki/Jean-Pierre%20Serre
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Jean-Pierre Serre (; born 15 September 1926) is a French mathematician who has made contributions to algebraic topology, algebraic geometry and algebraic number theory. He was awarded the Fields Medal in 1954, the Wolf Prize in 2000 and the inaugural Abel Prize in 2003.
Biography
Personal life
Born in Bages, Pyrénées-Orientales, to pharmacist parents, Serre was educated at the Lycée de Nîmes and then from 1945 to 1948 at the École Normale Supérieure in Paris. He was awarded his doctorate from the Sorbonne in 1951. From 1948 to 1954 he held positions at the Centre National de la Recherche Scientifique in Paris. In 1956 he was elected professor at the Collège de France, a position he held until his retirement in 1994. His wife, Professor Josiane Heulot-Serre, was a chemist; she also was the director of the Ecole Normale Supérieure de Jeunes Filles. Their daughter is the former French diplomat, historian and writer Claudine Monteil. The French mathematician Denis Serre is his nephew. He practices skiing, table tennis, and rock climbing (in Fontainebleau).
Career
From a very young age he was an outstanding figure in the school of Henri Cartan, working on algebraic topology, several complex variables and then commutative algebra and algebraic geometry, where he introduced sheaf theory and homological algebra techniques. Serre's thesis concerned the Leray–Serre spectral sequence associated to a fibration. Together with Cartan, Serre established the technique of using Eilenberg–MacLane spaces for computing homotopy groups of spheres, which at that time was one of the major problems in topology.
In his speech at the Fields Medal award ceremony in 1954, Hermann Weyl gave high praise to Serre, and also made the point that the award was for the first time awarded to a non-analyst. Serre subsequently changed his research focus.
Algebraic geometry
In the 1950s and 1960s, a fruitful collaboration between Serre and the two-years-younger Alexander Grothendieck led to important foundational work, much of it motivated by the Weil conjectures. Two major foundational papers by Serre were Faisceaux Algébriques Cohérents (FAC, 1955), on coherent cohomology, and Géométrie Algébrique et Géométrie Analytique (GAGA, 1956).
Even at an early stage in his work Serre had perceived a need to construct more general and refined cohomology theories to tackle the Weil conjectures. The problem was that the cohomology of a coherent sheaf over a finite field could not capture as much topology as singular cohomology with integer coefficients. Amongst Serre's early candidate theories of 1954–55 was one based on Witt vector coefficients.
Around 1958 Serre suggested that isotrivial principal bundles on algebraic varieties – those that become trivial after pullback by a finite étale map – are important. This acted as one important source of inspiration for Grothendieck to develop the étale topology and the corresponding theory of étale cohomology. These tools, developed in full b
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https://en.wikipedia.org/wiki/George%20Peacock
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George Peacock FRS (9 April 1791 – 8 November 1858) was an English mathematician and Anglican cleric. He founded what has been called the British algebra of logic.
Early life
Peacock was born on 9 April 1791 at Thornton Hall, Denton, near Darlington, County Durham. His father, Thomas Peacock, was a priest of the Church of England, incumbent and for 50 years curate of the parish of Denton, where he also kept a school. In early life Peacock did not show any precocity of genius, and was more remarkable for daring feats of climbing than for any special attachment to study. Initially, he received his elementary education from his father and then at Sedbergh School, and at 17 years of age, he was sent to Richmond School under James Tate, a graduate of Cambridge University. At this school he distinguished himself greatly both in classics and in the rather elementary mathematics then required for entrance at Cambridge. In 1809 he became a student of Trinity College, Cambridge.
In 1812 Peacock took the rank of Second Wrangler, and the second Smith's prize, the senior wrangler being John Herschel. Two years later he became a candidate for a fellowship in his college and won it immediately, partly by means of his extensive and accurate knowledge of the classics. A fellowship then meant about 200 pounds a year, tenable for seven years provided the Fellow did not marry meanwhile, and capable of being extended after the seven years provided the Fellow took clerical orders, which Peacock did in 1819.
Mathematical career
The year after taking a Fellowship, Peacock was appointed a tutor and lecturer of his college, which position he continued to hold for many years. Peacock, in common with many other students of his own standing, was profoundly impressed with the need of reforming Cambridge's position ignoring the differential notation for calculus, and while still an undergraduate formed a league with Babbage and Herschel to adopt measures to bring it about. In 1815 they formed what they called the Analytical Society, the object of which was stated to be to advocate the d 'ism of the Continent versus the dot-age of the university.
The first movement on the part of the Analytical Society was to translate from the French the smaller work of Lacroix on the differential and integral calculus; it was published in 1816. At that time the French language had the best manuals, as well as the greatest works on mathematics. Peacock followed up the translation with a volume containing a copious Collection of Examples of the Application of the Differential and Integral Calculus, which was published in 1820. The sale of both books was rapid, and contributed materially to further the object of the Society. In that time, high wranglers of one year became the examiners of the mathematical tripos three or four years afterwards. Peacock was appointed an examiner in 1817, and he did not fail to make use of the position as a powerful lever to advance the cause of reform. In his
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https://en.wikipedia.org/wiki/Dyirbal%20language
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Dyirbal (also Djirubal) is an Australian Aboriginal language spoken in northeast Queensland by the Dyirbal people. In 2016, the Australian Bureau of Statistics reported that there were 8 speakers of the language. It is a member of the small Dyirbalic branch of the Pama–Nyungan family. It possesses many outstanding features that have made it well known among linguists.
In the years since the Dyirbal grammar by Robert Dixon was published in 1972, Dyirbal has steadily moved closer to extinction as younger community members have failed to learn it.
Dialects
There are many different groups speaking dialects of Dyirbal language. Researcher Robert Dixon estimates that Dyirbal had, at its peak, 10 dialects.
Dialects include:
Dyirbal (or Jirrbal) spoken by the Dyirbalŋan
Mamu, spoken by the Waɽibara, Dulgubara, Bagiɽgabara, Dyiɽibara, and Mandubara (There are also different types of Mamu spoken by individual groups, such as Warribara Mamu, and Dulgubara Mamu)
Giramay (Or Girramay), spoken by the Giramaygan
Gulŋay (or Gulngay), spoken by the Malanbara
Dyiru (or Djirru), spoken by the Dyirubagala
Ngadyan (or Ngadjan), spoken by the Ngadyiandyi
Walmalbarra
The speakers of these dialects largely regard their dialects as different languages. They were classified as dialects by researcher Robert Dixon, who classified them as such based on linguistic criteria and their similarities, some dialects sharing as much as 90% of their vocabularies. Since the dialects were viewed by speakers as different languages, the language had no formal name, so Dixon assigned the language the name Dyirbal, naming it after Jirrbal, which was the dialect with the largest number of speakers at the time he was studying it.
Neighbouring languages
Languages neighbouring the many Dyirbal dialects include:
Ngaygungu
Mbabaram
Muluriji
Yidiny
Warungu
Warrgamay
Nyawaygi
Phonology
Consonants
Dyirbal has only four places of articulation for the stop and nasals, whereas most other Australian Aboriginal languages have five or six. This is because Dyirbal lacks the dental/alveolar/retroflex split typically found in these languages. Like the majority of Australian languages, it does not make a distinction between voiced consonants (such as b, d, g, etc.) and voiceless consonants (the corresponding p, t, and k, etc. respectively). Like Pinyin, standard Dyirbal orthography uses voiced consonants, which seem to be preferred by speakers of most Australian languages since the sounds (which can often be semi-voiced) are closer to English semi-voiced b, d, g than aspirated p, t, k.
Vowels
The Dyirbal vowel system is typical of Australia, with three vowels: , and , though is realised as in certain environments and can be realised as , also depending on the environment in which the phoneme appears. Thus the actual inventory of sounds is greater than the inventory of phonemes would suggest. Stress always falls on the first syllable of a word and usually on subsequent odd-numbe
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https://en.wikipedia.org/wiki/Spectrum%20of%20a%20matrix
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In mathematics, the spectrum of a matrix is the set of its eigenvalues. More generally, if is a linear operator on any finite-dimensional vector space, its spectrum is the set of scalars such that is not invertible. The determinant of the matrix equals the product of its eigenvalues. Similarly, the trace of the matrix equals the sum of its eigenvalues.
From this point of view, we can define the pseudo-determinant for a singular matrix to be the product of its nonzero eigenvalues (the density of multivariate normal distribution will need this quantity).
In many applications, such as PageRank, one is interested in the dominant eigenvalue, i.e. that which is largest in absolute value. In other applications, the smallest eigenvalue is important, but in general, the whole spectrum provides valuable information about a matrix.
Definition
Let V be a finite-dimensional vector space over some field K and suppose T : V → V is a linear map. The spectrum of T, denoted σT, is the multiset of roots of the characteristic polynomial of T. Thus the elements of the spectrum are precisely the eigenvalues of T, and the multiplicity of an eigenvalue λ in the spectrum equals the dimension of the generalized eigenspace of T for λ (also called the algebraic multiplicity of λ).
Now, fix a basis B of V over K and suppose M ∈ MatK (V) is a matrix. Define the linear map T : V → V pointwise by Tx = Mx, where on the right-hand side x is interpreted as a column vector and M acts on x by matrix multiplication. We now say that x ∈ V is an eigenvector of M if x is an eigenvector of T. Similarly, λ ∈ K is an eigenvalue of M if it is an eigenvalue of T, and with the same multiplicity, and the spectrum of M, written σM, is the multiset of all such eigenvalues.
Related notions
The eigendecomposition (or spectral decomposition) of a diagonalizable matrix is a decomposition of a diagonalizable matrix into a specific canonical form whereby the matrix is represented in terms of its eigenvalues and eigenvectors.
The spectral radius of a square matrix is the largest absolute value of its eigenvalues. In spectral theory, the spectral radius of a bounded linear operator is the supremum of the absolute values of the elements in the spectrum of that operator.
Notes
References
Matrix theory
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https://en.wikipedia.org/wiki/Spectrum%20%28functional%20analysis%29
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In mathematics, particularly in functional analysis, the spectrum of a bounded linear operator (or, more generally, an unbounded linear operator) is a generalisation of the set of eigenvalues of a matrix. Specifically, a complex number is said to be in the spectrum of a bounded linear operator if
either has no set-theoretic inverse;
or the set-theoretic inverse is either unbounded or defined on a non-dense subset.
Here, is the identity operator.
By the closed graph theorem, is in the spectrum if and only if the bounded operator is non-bijective on .
The study of spectra and related properties is known as spectral theory, which has numerous applications, most notably the mathematical formulation of quantum mechanics.
The spectrum of an operator on a finite-dimensional vector space is precisely the set of eigenvalues. However an operator on an infinite-dimensional space may have additional elements in its spectrum, and may have no eigenvalues. For example, consider the right shift operator R on the Hilbert space ℓ2,
This has no eigenvalues, since if Rx=λx then by expanding this expression we see that x1=0, x2=0, etc. On the other hand, 0 is in the spectrum because although the operator R − 0 (i.e. R itself) is invertible, the inverse is defined on a set which is not dense in ℓ2. In fact every bounded linear operator on a complex Banach space must have a non-empty spectrum.
The notion of spectrum extends to unbounded (i.e. not necessarily bounded) operators. A complex number λ is said to be in the spectrum of an unbounded operator defined on domain if there is no bounded inverse defined on the whole of If T is closed (which includes the case when T is bounded), boundedness of follows automatically from its existence.
The space of bounded linear operators B(X) on a Banach space X is an example of a unital Banach algebra. Since the definition of the spectrum does not mention any properties of B(X) except those that any such algebra has, the notion of a spectrum may be generalised to this context by using the same definition verbatim.
Spectrum of a bounded operator
Definition
Let be a bounded linear operator acting on a Banach space over the complex scalar field , and be the identity operator on . The spectrum of is the set of all for which the operator does not have an inverse that is a bounded linear operator.
Since is a linear operator, the inverse is linear if it exists; and, by the bounded inverse theorem, it is bounded. Therefore, the spectrum consists precisely of those scalars for which is not bijective.
The spectrum of a given operator is often denoted , and its complement, the resolvent set, is denoted . ( is sometimes used to denote the spectral radius of )
Relation to eigenvalues
If is an eigenvalue of , then the operator is not one-to-one, and therefore its inverse is not defined. However, the converse statement is not true: the operator may not have an inverse, even if is not an eigenvalu
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https://en.wikipedia.org/wiki/Analytical%20Society
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The Analytical Society was a group of individuals in early-19th-century Britain whose aim was to promote the use of Leibnizian notation for differentiation in calculus as opposed to the Newton notation for differentiation. The latter system came into being in the 18th century as a convention of Sir Isaac Newton, and was in use throughout Great Britain. According to a mathematical historian:
In 1800, English mathematics was trapped in the doldrums of fluxional notation and of an intuitive geometric-physical approach to mathematics designed to prepare the student for reading Newton's Principia...The study of any mathematics not pertinent to the traditional questions of Tripos was not only ignored, but actually discouraged. Cambridge was isolated, and its students remained ignorant of continental developments.
The Society was first envisioned by Charles Babbage as a parody on the debate of whether Bible texts should be annotated, with Babbage having the notion that his textbook by Sylvestre Lacroix was without need for interpretation once translated.
Its membership originally consisted of a group of Cambridge students led by Babbage and including Edward Bromhead.
The Cambridge mathematician Robert Woodhouse had brought the Leibniz notation to England with his book Principles of Analytical Calculation in 1803. While Newton's notation was unsuitable for a function of several variables, Woodhouse showed, for instance, how to find the total differential of where φ is a function of p and q:
The slow uptake of the continental methods in calculus led to the formation of the Analytical Society by Charles Babbage, John Herschel and George Peacock.
Though the Society was disbanded by 1814 when most of the original members had graduated, its influence continued to be felt. The evidence of Analytical Society work appeared in 1816 when Peacock and Herschel completed the translation of Sylvestre Lacroix's textbook An Elementary Treatise on Differential and Integral Calculus that had been started by Babbage. In 1817 Peacock introduced Leibnizian symbols in that year's examinations in the local senate-house.
Both the exam and the textbook met with little criticism until 1819, when both were criticised by D.M. Peacock, vicar of Sedbergh, 1796 to 1840. He wrote:
The University should be more on its guard ... against the introduction of merely algebraic or analytical speculations into its public examinations.
Nevertheless, the reforms were encouraged by younger members of Cambridge University. George Peacock successfully encouraged a colleague, Richard Gwatkin of St John's College at Cambridge University, to adopt the new notation in his exams.
Use of Leibnizian notation began to spread after this. In 1820, the notation was used by William Whewell, a previously neutral but influential Cambridge University faculty member, in his examinations. In 1821, Peacock again used Leibnizian notation in his examinations, and the notation became well established.
The So
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https://en.wikipedia.org/wiki/Constraint
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Constraint may refer to:
Constraint (computer-aided design), a demarcation of geometrical characteristics between two or more entities or solid modeling bodies
Constraint (mathematics), a condition of an optimization problem that the solution must satisfy
Constraint (classical mechanics), a relation between coordinates and momenta
Constraint (information theory), the degree of statistical dependence between or among variables
Constraints (journal), a scientific journal
Constraint (database), a concept in relational database
See also
Biological constraints, factors which make populations resistant to evolutionary change
Carrier's constraint
Constrained optimization, in finance, linear programming, economics and cost modeling
Constrained writing, in literature
Constraint algorithm, such as SHAKE, or LINCS
Constraint satisfaction, in computer science
Finite domain constraint
First class constraint in Hamiltonian mechanics
Integrity constraints
Loading gauge, a constraint in engineering
Optimality theory, in linguistics, a constraint-based theory which is primarily influential in phonology
Primary constraint in Hamiltonian mechanics
Restraint (disambiguation)
Second class constraint in Hamiltonian mechanics
Secondary constraint in Hamiltonian mechanics
Structure gauge, a constraint in engineering
Theory of constraints, in business management
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https://en.wikipedia.org/wiki/Isaac%20Todhunter
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Isaac Todhunter FRS (23 November 1820 – 1 March 1884), was an English mathematician who is best known today for the books he wrote on mathematics and its history.
Life and work
The son of George Todhunter, a Nonconformist minister, and Mary née Hume, he was born at Rye, Sussex. He was educated at Hastings, where his mother had opened a school after the death of his father in 1826. He was at first at a school run by Robert Carr, moving then to one opened by John Baptist Austin.
Todhunter became an assistant master at a school at Peckham, attending at the same time evening classes at the University College, London where he was influenced by Augustus De Morgan. In 1842 he obtained a mathematical scholarship and graduated as B.A. at London University, where he was awarded the gold medal on the M.A. examination. About this time he became mathematical master at a school at Wimbledon.
In 1844 Todhunter entered St John's College, Cambridge, where he was senior wrangler in 1848, and gained the first Smith's Prize and the Burney Prize; and in 1849 he was elected to a fellowship, and began his life of college lecturer and private tutor. In 1862 he was made a fellow of the Royal Society, and in 1865 a member of the Mathematical Society of London. In 1871 he gained the Adams Prize and was elected to the council of the Royal Society. He was elected honorary fellow of St John's in 1874, having resigned his fellowship on his marriage in 1864. In 1880 his eyesight began to fail, and shortly afterwards he was attacked with paralysis.
He is buried in the Mill Road cemetery, Cambridge.
Personal life
Todhunter married 13 August 1864 Louisa Anna Maria, eldest daughter of Captain (afterwards Admiral) George Davies, R.N. (at that time head of the county constabulary force). He died on 1 March 1884, at his residence, 6 Brookside, Cambridge. A mural tablet and medallion portrait were placed in the ante-chapel of his college by his widow, who, with four sons and one daughter, survived him.
He was a sound Latin and Greek scholar, familiar with French, German, Spanish, Italian, and also Russian, Hebrew, and Sanskrit. He was well versed in the history of philosophy, and on three occasions acted as examiner for the moral sciences tripos.
Selected writings
Treatise on the Differential Calculus and the Elements of the Integral Calculus (1852, 6th ed., 1873)
Treatise on Analytical Statics (1853, 4th ed., 1874)
Treatise on the Integral Calculus (1857, 4th ed., 1874)
Treatise on Algebra (1858, 6th ed., 1871)
Treatise on differential Calculus
Treatise on Plane Coordinate Geometry (1858, 3rd ed., 1861)
Plane Trigonometry (1859, 4th ed., 1869)
Spherical Trigonometry (1859)
History of the Calculus of Variations (1861)
Theory of Equations (1861, 2nd ed. 1875)
Examples of Analytical Geometry of Three Dimensions (1858, 3rd ed., 1873)
Mechanics for Beginners (1867)
A History of the Mathematical Theory of Probability from the Time of Pascal to that of Laplace (1865)
Researches in
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https://en.wikipedia.org/wiki/William%20Feller
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William "Vilim" Feller (July 7, 1906 – January 14, 1970), born Vilibald Srećko Feller, was a Croatian–American mathematician specializing in probability theory.
Early life and education
Feller was born in Zagreb to Ida Oemichen-Perc, a Croatian–Austrian Catholic, and Eugen Viktor Feller, son of a Polish–Jewish father (David Feller) and an Austrian mother (Elsa Holzer).
Eugen Feller was a famous chemist and created Elsa fluid named after his mother. According to Gian-Carlo Rota, Eugen Feller's surname was a "Slavic tongue twister", which William changed at the age of twenty. This claim appears to be false. His forename, Vilibald, was chosen by his Catholic mother for the saint day of his birthday.
Work
Feller held a docent position at the University of Kiel beginning in 1928. Because he refused to sign a Nazi oath, he fled the Nazis and went to Copenhagen, Denmark in 1933. He also lectured in Sweden (Stockholm and Lund). As a refugee in Sweden, Feller reported being troubled by increasing fascism at the universities. He reported that the mathematician Torsten Carleman would offer his opinion that Jews and foreigners should be executed.
Finally, in 1939 he arrived in the U.S., where he became a citizen in 1944 and was on the faculty at Brown and Cornell. In 1950 he became a professor at Princeton University.
The works of Feller are contained in 104 papers and two books on a variety of topics such as mathematical analysis, theory of measurement, functional analysis, geometry, and differential equations in addition to his work in mathematical statistics and probability.
Feller was one of the greatest probabilists of the twentieth century. He is remembered for his championing of probability theory as a branch of mathematical analysis in Sweden and the United States. In the middle of the 20th century, probability theory was popular in France and Russia, while mathematical statistics was more popular in the United Kingdom and the United States, according to the Swedish statistician, Harald Cramér. His two-volume textbook on probability theory and its applications was called "the most successful treatise on probability ever written" by Gian-Carlo Rota. By stimulating his colleagues and students in Sweden and then in the United States, Feller helped establish research groups studying the analytic theory of probability. In his research, Feller contributed to the study of the relationship between Markov chains and differential equations, where his theory of generators of one-parameter semigroups of stochastic processes gave rise to the theory of "Feller operators".
Results
Numerous topics relating to probability are named after him, including Feller processes, Feller's explosion test, Feller–Brown movement, and the Lindeberg–Feller theorem. Feller made fundamental contributions to renewal theory, Tauberian theorems, random walks, diffusion processes, and the law of the iterated logarithm. Feller was among those early editors who launched the jour
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https://en.wikipedia.org/wiki/Weibull%20distribution
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In probability theory and statistics, the Weibull distribution is a continuous probability distribution. It models a broad range of random variables, largely in the nature of a time to failure or time between events. Examples are maximum one-day rainfalls and the time a user spends on a web page.
The distribution is named after Swedish mathematician Waloddi Weibull, who described it in detail in 1939, although it was first identified by Maurice René Fréchet and first applied by to describe a particle size distribution.
Definition
Standard parameterization
The probability density function of a Weibull random variable is
where k > 0 is the shape parameter and λ > 0 is the scale parameter of the distribution. Its complementary cumulative distribution function is a stretched exponential function. The Weibull distribution is related to a number of other probability distributions; in particular, it interpolates between the exponential distribution (k = 1) and the Rayleigh distribution (k = 2 and ).
If the quantity X is a "time-to-failure", the Weibull distribution gives a distribution for which the failure rate is proportional to a power of time. The shape parameter, k, is that power plus one, and so this parameter can be interpreted directly as follows:
A value of indicates that the failure rate decreases over time (like in case of the Lindy effect, which however corresponds to Pareto distributions rather than Weibull distributions). This happens if there is significant "infant mortality", or defective items failing early and the failure rate decreasing over time as the defective items are weeded out of the population. In the context of the diffusion of innovations, this means negative word of mouth: the hazard function is a monotonically decreasing function of the proportion of adopters;
A value of indicates that the failure rate is constant over time. This might suggest random external events are causing mortality, or failure. The Weibull distribution reduces to an exponential distribution;
A value of indicates that the failure rate increases with time. This happens if there is an "aging" process, or parts that are more likely to fail as time goes on. In the context of the diffusion of innovations, this means positive word of mouth: the hazard function is a monotonically increasing function of the proportion of adopters. The function is first convex, then concave with an inflexion point at .
In the field of materials science, the shape parameter k of a distribution of strengths is known as the Weibull modulus. In the context of diffusion of innovations, the Weibull distribution is a "pure" imitation/rejection model.
Alternative parameterizations
First alternative
Applications in medical statistics and econometrics often adopt a different parameterization. The shape parameter k is the same as above, while the scale parameter is . In this case, for x ≥ 0, the probability density function is
the cumulative distribution function
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https://en.wikipedia.org/wiki/Beta%20distribution
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In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval [0, 1] or (0, 1) in terms of two positive parameters, denoted by alpha (α) and beta (β), that appear as exponents of the variable and its complement to 1, respectively, and control the shape of the distribution.
The beta distribution has been applied to model the behavior of random variables limited to intervals of finite length in a wide variety of disciplines. The beta distribution is a suitable model for the random behavior of percentages and proportions.
In Bayesian inference, the beta distribution is the conjugate prior probability distribution for the Bernoulli, binomial, negative binomial and geometric distributions.
The formulation of the beta distribution discussed here is also known as the beta distribution of the first kind, whereas beta distribution of the second kind is an alternative name for the beta prime distribution. The generalization to multiple variables is called a Dirichlet distribution.
Definitions
Probability density function
The probability density function (PDF) of the beta distribution, for or , and shape parameters α, β > 0, is a power function of the variable x and of its reflection as follows:
where Γ(z) is the gamma function. The beta function, , is a normalization constant to ensure that the total probability is 1. In the above equations x is a realization—an observed value that actually occurred—of a random variable X.
Several authors, including N. L. Johnson and S. Kotz, use the symbols p and q (instead of α and β) for the shape parameters of the beta distribution, reminiscent of the symbols traditionally used for the parameters of the Bernoulli distribution, because the beta distribution approaches the Bernoulli distribution in the limit when both shape parameters α and β approach the value of zero.
In the following, a random variable X beta-distributed with parameters α and β will be denoted by:
Other notations for beta-distributed random variables used in the statistical literature are and .
Cumulative distribution function
The cumulative distribution function is
where is the incomplete beta function and is the regularized incomplete beta function.
Alternative parameterizations
Two parameters
Mean and sample size
The beta distribution may also be reparameterized in terms of its mean μ and the sum of the two shape parameters ( p. 83). Denoting by αPosterior and βPosterior the shape parameters of the posterior beta distribution resulting from applying Bayes theorem to a binomial likelihood function and a prior probability, the interpretation of the addition of both shape parameters to be sample size = ν = α·Posterior + β·Posterior is only correct for the Haldane prior probability Beta(0,0). Specifically, for the Bayes (uniform) prior Beta(1,1) the correct interpretation would be sample size = α·Posterior + β Posterior − 2, or ν = (sample size) + 2.
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https://en.wikipedia.org/wiki/Gamma%20distribution
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In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-squared distribution are special cases of the gamma distribution. There are two equivalent parameterizations in common use:
With a shape parameter and a scale parameter .
With a shape parameter and an inverse scale parameter , called a rate parameter.
In each of these forms, both parameters are positive real numbers.
The gamma distribution is the maximum entropy probability distribution (both with respect to a uniform base measure and a base measure) for a random variable for which E[X] = kθ = α/β is fixed and greater than zero, and E[ln(X)] = ψ(k) + ln(θ) = ψ(α) − ln(β) is fixed (ψ is the digamma function).
Definitions
The parameterization with k and θ appears to be more common in econometrics and other applied fields, where the gamma distribution is frequently used to model waiting times. For instance, in life testing, the waiting time until death is a random variable that is frequently modeled with a gamma distribution. See Hogg and Craig for an explicit motivation.
The parameterization with and is more common in Bayesian statistics, where the gamma distribution is used as a conjugate prior distribution for various types of inverse scale (rate) parameters, such as the λ of an exponential distribution or a Poisson distribution – or for that matter, the β of the gamma distribution itself. The closely related inverse-gamma distribution is used as a conjugate prior for scale parameters, such as the variance of a normal distribution.
If k is a positive integer, then the distribution represents an Erlang distribution; i.e., the sum of k independent exponentially distributed random variables, each of which has a mean of θ.
Characterization using shape α and rate β
The gamma distribution can be parameterized in terms of a shape parameter α = k and an inverse scale parameter β = 1/θ, called a rate parameter. A random variable X that is gamma-distributed with shape α and rate β is denoted
The corresponding probability density function in the shape-rate parameterization is
where is the gamma function.
For all positive integers, .
The cumulative distribution function is the regularized gamma function:
where is the lower incomplete gamma function.
If α is a positive integer (i.e., the distribution is an Erlang distribution), the cumulative distribution function has the following series expansion:
Characterization using shape k and scale θ
A random variable X that is gamma-distributed with shape k and scale θ is denoted by
The probability density function using the shape-scale parametrization is
Here Γ(k) is the gamma function evaluated at k.
The cumulative distribution function is the regularized gamma function:
where is the lower incomplete gamma function.
It can also be expressed as follows, if k is a positive integer (i.e., the distribut
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https://en.wikipedia.org/wiki/Triangulation
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In trigonometry and geometry, triangulation is the process of determining the location of a point by forming triangles to the point from known points.
Applications
In surveying
Specifically in surveying, triangulation involves only angle measurements at known points, rather than measuring distances to the point directly as in trilateration; the use of both angles and distance measurements is referred to as triangulateration.
In computer vision
Computer stereo vision and optical 3D measuring systems use this principle to determine the spatial dimensions and the geometry of an item. Basically, the configuration consists of two sensors observing the item. One of the sensors is typically a digital camera device, and the other one can also be a camera or a light projector. The projection centers of the sensors and the considered point on the object's surface define a (spatial) triangle. Within this triangle, the distance between the sensors is the base b and must be known. By determining the angles between the projection rays of the sensors and the basis, the intersection point, and thus the 3D coordinate, is calculated from the triangular relations.
History
Triangulation today is used for many purposes, including surveying, navigation, metrology, astrometry, binocular vision, model rocketry and, in the military, the gun direction, the trajectory and distribution of fire power of weapons.
The use of triangles to estimate distances dates to antiquity. In the 6th century BC, about 250 years prior to the establishment of the Ptolemaic dynasty, the Greek philosopher Thales is recorded as using similar triangles to estimate the height of the pyramids of ancient Egypt. He measured the length of the pyramids' shadows and that of his own at the same moment, and compared the ratios to his height (intercept theorem). Thales also estimated the distances to ships at sea as seen from a clifftop by measuring the horizontal distance traversed by the line-of-sight for a known fall, and scaling up to the height of the whole cliff. Such techniques would have been familiar to the ancient Egyptians. Problem 57 of the Rhind papyrus, a thousand years earlier, defines the seqt or seked as the ratio of the run to the rise of a slope, i.e. the reciprocal of gradients as measured today. The slopes and angles were measured using a sighting rod that the Greeks called a dioptra, the forerunner of the Arabic alidade. A detailed contemporary collection of constructions for the determination of lengths from a distance using this instrument is known, the Dioptra of Hero of Alexandria (–70 AD), which survived in Arabic translation; but the knowledge became lost in Europe until in 1615 Snellius, after the work of Eratosthenes, reworked the technique for an attempt to measure the circumference of the earth. In China, Pei Xiu (224–271) identified "measuring right angles and acute angles" as the fifth of his six principles for accurate map-making, necessary to accurately establi
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https://en.wikipedia.org/wiki/Put%E2%80%93call%20parity
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In financial mathematics, the put–call parity defines a relationship between the price of a European call option and European put option, both with the identical strike price and expiry, namely that a portfolio of a long call option and a short put option is equivalent to (and hence has the same value as) a single forward contract at this strike price and expiry. This is because if the price at expiry is above the strike price, the call will be exercised, while if it is below, the put will be exercised, and thus in either case one unit of the asset will be purchased for the strike price, exactly as in a forward contract.
The validity of this relationship requires that certain assumptions be satisfied; these are specified and the relationship is derived below. In practice transaction costs and financing costs (leverage) mean this relationship will not exactly hold, but in liquid markets the relationship is close to exact.
Assumptions
Put–call parity is a static replication, and thus requires minimal assumptions, namely the existence of a forward contract. In the absence of traded forward contracts, the forward contract can be replaced (indeed, itself replicated) by the ability to buy the underlying asset and finance this by borrowing for fixed term (e.g., borrowing bonds), or conversely to borrow and sell (short) the underlying asset and loan the received money for term, in both cases yielding a self-financing portfolio.
These assumptions do not require any transactions between the initial date and expiry, and are thus significantly weaker than those of the Black–Scholes model, which requires dynamic replication and continual transaction in the underlying.
Replication assumes one can enter into derivative transactions, which requires leverage (and capital costs to back this), and buying and selling entails transaction costs, notably the bid–ask spread. The relationship thus only holds exactly in an ideal frictionless market with unlimited liquidity. However, real world markets may be sufficiently liquid that the relationship is close to exact, most significantly FX markets in major currencies or major stock indices, in the absence of market turbulence.
Statement
Put–call parity can be stated in a number of equivalent ways, most tersely as:
where is the (current) value of a call, is the (current) value of a put, is the discount factor, is the forward price of the underlying asset, and is the strike price. The left side corresponds to a portfolio of a long call and a short put; the right side corresponds to a forward contract. The assets and on the left side are given in present values, while the assets and are given in future values (forward price of asset, and strike price paid at expiry), which the discount factor converts to present values.
Now the spot price can be obtained by discounting the forward price by the factor . Using spot price instead of forward price gives us:
Rearranging the terms gives a first interpretation
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https://en.wikipedia.org/wiki/Th%C4%81bit%20ibn%20Qurra
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Thābit ibn Qurra (full name: , , ); 826 or 836 – February 19, 901, was a polymath known for his work in mathematics, medicine, astronomy, and translation. He lived in Baghdad in the second half of the ninth century during the time of the Abbasid Caliphate.
Thābit ibn Qurra made important discoveries in algebra, geometry, and astronomy. In astronomy, Thābit is considered one of the first reformers of the Ptolemaic system, and in mechanics he was a founder of statics. Thābit also wrote extensively on medicine and produced philosophical treatises.
Biography
Thābit was born in Harran in Upper Mesopotamia, which at the time was part of the Diyar Mudar subdivision of the al-Jazira region of the Abbasid Caliphate. Thābit belonged to the Sabians of Harran, a Hellenized Semitic polytheistic astral religion that still existed in ninth-century Harran.
As a youth, Thābit worked as money changer in a marketplace in Harran until meeting Muḥammad ibn Mūsā, the oldest of three mathematicians and astronomers known as the Banū Mūsā. Thābit displayed such exceptional linguistic skills that ibn Mūsā chose him to come to Baghdad to be trained in mathematics, astronomy, and philosophy under the tutelage of the Banū Mūsā. Here, Thābit was introduced to not only a community of scholars but also to those who had significant power and influence in Baghdad.
Thābit and his pupils lived in the midst of the most intellectually vibrant, and probably the largest, city of the time, Baghdad. Thābit came to Baghdad in the first place to work for the Banū Mūsā becoming a part of their circle and helping them translate Greek mathematical texts. What is unknown is how Banū Mūsā and Thābit occupied himself with mathematics, astronomy, astrology, magic, mechanics, medicine, and philosophy. Later in his life, Thābit's patron was the Abbasid Caliph al-Mu'tadid (reigned 892–902), whom he became a court astronomer for. Thābit became the Caliph's personal friend and courtier. Thābit died in Baghdad in 901. His son, Sinan ibn Thabit and grandson, Ibrahim ibn Sinan would also make contributions to the medicine and science. By the end of his life, Thābit had managed to write 150 works on mathematics, astronomy, and medicine. With all the work done by Thābit, most of his work has not lasted time. There are less than a dozen works by him that have survived.
Translation
Thābit's native language was Syriac, which was the Middle Aramaic variety from Edessa, and he was fluent in both Medieval Greek and Arabic. He was the author to multiple treaties. Due to him being trilingual, Thābit was able to have a major role during the Graeco-Arabic translation movement. He would also make a school of translation in Baghdad.
Thābit translated from Greek into Arabic works by Apollonius of Perga, Archimedes, Euclid and Ptolemy. He revised the translation of Euclid's Elements of Hunayn ibn Ishaq. He also rewrote Ishaq ibn Hunayn's translation of Ptolemy's Almagest and translated Ptolemy's Geography.Thābit
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https://en.wikipedia.org/wiki/6
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6 (six) is the natural number following 5 and preceding 7. It is a composite number and the smallest perfect number.
In mathematics
Six is the smallest positive integer which is neither a square number nor a prime number. It is the second smallest composite number after four, equal to the sum and the product of its three proper divisors (, and ). As such, 6 is the only number that is both the sum and product of three consecutive positive numbers. It is the smallest perfect number, which are numbers that are equal to their aliquot sum, or sum of their proper divisors. It is also the largest of the four all-Harshad numbers (1, 2, 4, and 6).
6 is a pronic number and the only semiprime to be. It is the first discrete biprime (2 × 3) which makes it the first member of the (2 × q) discrete biprime family, where q is a higher prime. All primes above 3 are of the form 6n ± 1 for n ≥ 1.
As a perfect number:
6 is related to the Mersenne prime 3, since . (The next perfect number is 28.)
6 is the only even perfect number that is not the sum of successive odd cubes.
6 is the root of the 6-aliquot tree, and is itself the aliquot sum of only one other number; the square number, .
Six is the first unitary perfect number, since it is the sum of its positive proper unitary divisors, without including itself. Only five such numbers are known to exist; sixty (10 × 6) and ninety (15 × 6) are the next two.
All integers that are multiples of 6 are pseudoperfect (all multiples of a pseudoperfect number are pseudoperfect). Six is also the smallest Granville number, or -perfect number.
Unrelated to 6's being a perfect number, a Golomb ruler of length 6 is a "perfect ruler". Six is a congruent number.
6 is the second primary pseudoperfect number, and harmonic divisor number. It is also the second superior highly composite number, and the last to also be a primorial.
There are 6 non-equivalent ways in which 100 can be expressed as the sum of two prime numbers: (3 + 97), (11 + 89), (17 + 83), (29 + 71), (41 + 59) and (47 + 53).
There is not a prime such that the multiplicative order of 2 modulo is 6, that is,
By Zsigmondy's theorem, if is a natural number that is not 1 or 6, then there is a prime such that . See for such .
The ring of integer of the sixth cyclotomic field , which is called Eisenstein integer, has 6 units: ±1, ±ω, ±ω2, where .
The six exponentials theorem guarantees (given the right conditions on the exponents) the transcendence of at least one of a set of exponentials.
There are six basic trigonometric functions: sin, cos, sec, csc, tan, and cot.
The smallest non-abelian group is the symmetric group which has 3! = 6 elements.
Six is a triangular number and so is its square (). It is the first octahedral number, preceding 19.
A six-sided polygon is a hexagon, one of the three regular polygons capable of tiling the plane. Figurate numbers representing hexagons (including six) are called hexagonal numbers. Because 6 is the product of
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https://en.wikipedia.org/wiki/Lucky%20number
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In number theory, a lucky number is a natural number in a set which is generated by a certain "sieve". This sieve is similar to the Sieve of Eratosthenes that generates the primes, but it eliminates numbers based on their position in the remaining set, instead of their value (or position in the initial set of natural numbers).
The term was introduced in 1956 in a paper by Gardiner, Lazarus, Metropolis and Ulam. In the same work they also suggested calling another sieve, "the sieve of Josephus Flavius" because of its similarity with the counting-out game in the Josephus problem.
Lucky numbers share some properties with primes, such as asymptotic behaviour according to the prime number theorem; also, a version of Goldbach's conjecture has been extended to them. There are infinitely many lucky numbers. Twin lucky numbers and twin primes also appear to occur with similar frequency. However, if Ln denotes the n-th lucky number, and pn the n-th prime, then Ln > pn for all sufficiently large n.
Because of their apparent similarities with the prime numbers, some mathematicians have suggested that some of their common properties may also be found in other sets of numbers generated by sieves of a certain unknown form, but there is little theoretical basis for this conjecture.
The sieving process
Continue removing the nth remaining numbers, where n is the next number in the list after the last surviving number. Next in this example is 9.
One way that the application of the procedure differs from that of the Sieve of Eratosthenes is that for n being the number being multiplied on a specific pass, the first number eliminated on the pass is the n-th remaining number that has not yet been eliminated, as opposed to the number 2n. That is to say, the list of numbers this sieve counts through is different on each pass (for example 1, 3, 7, 9, 13, 15, 19... on the third pass), whereas in the Sieve of Eratosthenes, the sieve always counts through the entire original list (1, 2, 3...).
When this procedure has been carried out completely, the remaining integers are the lucky numbers (those that happen to be prime are in bold):
1, 3, 7, 9, 13, 15, 21, 25, 31, 33, 37, 43, 49, 51, 63, 67, 69, 73, 75, 79, 87, 93, 99, 105, 111, 115, 127, 129, 133, 135, 141, 151, 159, 163, 169, 171, 189, 193, 195, 201, 205, 211, 219, 223, 231, 235, 237, 241, 259, 261, 267, 273, 283, 285, 289, 297, 303, 307, 319, 321, 327, ... .
The lucky number which removes n from the list of lucky numbers is: (0 if n is a lucky number)
0, 2, 0, 2, 3, 2, 0, 2, 0, 2, 3, 2, 0, 2, 0, 2, 3, 2, 7, 2, 0, 2, 3, 2, 0, 2, 9, 2, 3, 2, 0, 2, 0, 2, 3, 2, 0, 2, 7, 2, 3, 2, 0, 2, 13, 2, 3, 2, 0, 2, 0, 2, 3, 2, 15, 2, 9, 2, 3, 2, 7, 2, 0, 2, 3, 2, 0, 2, 0, 2, 3, 2, 0, 2, 0, 2, 3, 2, 0, 2, 7, 2, 3, 2, 21, 2, ...
Lucky primes
A "lucky prime" is a lucky number that is prime. They are:
3, 7, 13, 31, 37, 43, 67, 73, 79, 127, 151, 163, 193, 211, 223, 241, 283, 307, 331, 349, 367, 409, 421, 433, 463, 487, 541, 577
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https://en.wikipedia.org/wiki/15%20%28number%29
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15 (fifteen) is the natural number following 14 and preceding 16.
Mathematics
15 is:
The eighth composite number and the sixth semiprime and the first odd and fourth discrete semiprime; its proper divisors are , , and , so the first of the form (3.q), where q is a higher prime.
a deficient number, a lucky number, a bell number (i.e., the number of partitions for a set of size 4), a pentatope number, and a repdigit in binary (1111) and quaternary (33). In hexadecimal, and higher bases, it is represented as F.
with an aliquot sum of 9; within an aliquot sequence of three composite numbers (15,9,4,3,1,0) to the Prime in the 3-aliquot tree.
the second member of the first cluster of two discrete semiprimes (14, 15); the next such cluster is (21, 22).
a triangular number, a hexagonal number, and a centered tetrahedral number.
the number of partitions of 7.
the smallest number that can be factorized using Shor's quantum algorithm.
the magic constant of the unique order-3 normal magic square.
the number of supersingular primes.
the smallest positive number that can be expressed as the difference of two positive squares in more than one way: or (see image).
Furthermore,
15's prime factors, (3 and 5), form the first twin-prime pair.
The first 15 superabundant numbers are the same as the first 15 colossally abundant numbers.
In decimal, 15 contains the digits 1 and 5 and is the result of adding together the integers from 1 to 5 (1 + 2 + 3 + 4 + 5 = 15). The only other number with this property (in decimal) is 27.
There are 15 truncatable primes that are both right-truncatable and left-truncatable:
2, 3, 5, 7, 23, 37, 53, 73, 313, 317, 373, 797, 3137, 3797, 739397
There are 15 perfect matchings of the complete graph K6 and 15 rooted binary trees with four labeled leaves, both of these being among the types of objects counted by double factorials.
With only two exceptions, all prime quadruplets enclose a multiple of 15, with 15 itself being enclosed by the quadruplet (11, 13, 17, 19).
If a positive definite quadratic form with integer matrix represents all positive integers up to 15, then it represents all positive integers via the 15 and 290 theorems.
15 is the product of distinct Fermat primes, 3 and 5; hence, a regular pentadecagon is constructible with a compass and unmarked straightedge, and is expressible in terms of square roots.
There are 15 monohedral convex pentagonal tilings, with eight being edge-to-edge.
There are 15 regular and semiregular tilings when infinite (improper) apeirogonal forms are counted: three are regular (with one self-dual), eight are semiregular (with one chiral), and four are apeirogonal (from a total of 8, in-which 4 are duplicates).
Full icosahedral symmetry contains 15 mirror planes (2-fold axes). Specifically, the symmetry order for both the regular icosahedron and regular dodecahedron (which is made of regular pentagons) is 120: equal to sum of the first 15 integers, and the factorial of 5,
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https://en.wikipedia.org/wiki/20%20%28number%29
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20 (twenty; Roman numeral XX) is the natural number following 19 and preceding 21. A group of twenty units may also be referred to as a score.
In mathematics
Twenty is a pronic number, as it is the product of consecutive integers, namely 4 and 5. It is the third composite number to be the product of a squared prime and a prime, and also the second member of the 22 × q family in this form.
20 has an aliquot sum of 22; a semiprime, within an aliquot sequence of four composite numbers (20, 22, 14, 10, 8) that belong to the prime 7-aliquot tree.
20 is the smallest primitive abundant number.
20 is the third tetrahedral number.
20 is the basis for vigesimal number systems.
20 is the number of parallelogram polyominoes with 5 cells.
20 is the number of moves (quarter or half turns) required to optimally solve a Rubik's Cube in the worst case.
20 is the length of a side of the fifth smallest right triangle that forms a primitive Pythagorean triple, (20,21,29). This is the second Pythagorean triple that can be formed using Pell numbers where and are one unit apart.
20 is the smallest non-trivial decimal neon number equal to the sum of its digits when raised to the thirteenth power (2013 = 8192 × 1013).
There are twenty edge-to-edge 2-uniform tilings by convex regular polygons, which are uniform tessellations of the plane containing 2 orbits of vertices.
The largest number of faces a Platonic solid can have is twenty faces, which make up a regular icosahedron. A dodecahedron, on the other hand, has twenty vertices, likewise the most a regular polyhedron can have. There are a total of 20 regular and semiregular polyhedra, aside from the infinite family of semiregular prisms and antiprisms that exists in the third dimension: the 5 Platonic solids, and 15 Archimedean solids (including chiral forms of the snub cube and snub dodecahedron). There are also four uniform compound polyhedra that contain twenty polyhedra (UC13, UC14, UC19, UC33), which is the most any such solids can have; while another twenty uniform compounds contain five polyhedra. The compound of twenty octahedra can be obtained by orienting two pairs of compounds of ten octahedra, which can also coincide to yield a regular compound of five octahedra.
In total, there are 20 semiregular polytopes that only exist up through the 8th dimension, which include 13 Archimedean solids and 7 Gosset polytopes (without counting enantiomorphs, or semiregular prisms and antiprisms).
Bring's curve is a Riemann surface of genus four, whose fundamental polygon is a regular hyperbolic 20-sided icosagon, with an area equal to by the Gauss-Bonnet theorem.
The Happy Family of sporadic groups is made up of twenty finite simple groups that are all subquotients of the friendly giant, the largest sporadic group. The largest supersingular prime factor that divides the order of the friendly giant is 71, which is the 20th indexed prime number. Both 71 and 20 represent self-convolved Fibonacci numbers
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https://en.wikipedia.org/wiki/17%20%28number%29
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17 (seventeen) is the natural number following 16 and preceding 18. It is a prime number.
Seventeen is the sum of the first four prime numbers.
In mathematics
Seventeen is the seventh prime number, which makes it the fourth super-prime, as seven is itself prime. It forms a twin prime with 19, a cousin prime with 13, and a sexy prime with both 11 and 23. Seventeen is the only prime number which is the sum of four consecutive primes (2, 3, 5, and 7), as any other four consecutive primes that are added always generate an even number divisible by two. It is one of six lucky numbers of Euler which produce primes of the form , and the sixth Mersenne prime exponent, which yields 131,071. It is also the minimum possible number of givens for a sudoku puzzle with a unique solution. 17 can be written in the form and ; and as such, it is a Leyland prime and Leyland prime of the second kind:
17 is the third Fermat prime, as it is of the form with . On the other hand, the seventeenth Jacobsthal–Lucas number — that is part of a sequence which includes four Fermat primes (except for 3) — is the fifth and largest known Fermat prime: 65,537. It is one more than the smallest number with exactly seventeen divisors, 65,536 = 216. Since seventeen is a Fermat prime, regular heptadecagons can be constructed with a compass and unmarked ruler. This was proven by Carl Friedrich Gauss and ultimately led him to choose mathematics over philology for his studies.
Either 16 or 18 unit squares can be formed into rectangles with perimeter equal to the area; and there are no other natural numbers with this property. The Platonists regarded this as a sign of their peculiar propriety; and Plutarch notes it when writing that the Pythagoreans "utterly abominate" 17, which "bars them off from each other and disjoins them".
17 is the minimum number of vertices on a graph such that, if the edges are colored with three different colors, there is bound to be a monochromatic triangle; see Ramsey's theorem.
There are also:
17 crystallographic space groups in two dimensions. These are sometimes called wallpaper groups, as they represent the seventeen possible symmetry types that can be used for wallpaper.
17 combinations of regular polygons that completely fill a plane vertex. Eleven of these belong to regular and semiregular tilings, while 6 of these (3.7.42, 3.8.24, 3.9.18, 3.10.15, 4.5.20, and 5.5.10) exclusively surround a point in the plane and fill it only when irregular polygons are included.
17 orthogonal curvilinear coordinate systems (to within a conformal symmetry) in which the three-variable Laplace equation can be solved using the separation of variables technique.
17 distinct fully supported stellations generated by an icosahedron. The seventeenth prime number is 59, which is equal to the total number of stellations of the icosahedron by Miller's rules. Without counting the icosahedron as a zeroth stellation, this total becomes 58, a count equal to the sum of
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https://en.wikipedia.org/wiki/19%20%28number%29
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19 (nineteen) is the natural number following 18 and preceding 20. It is a prime number.
Mathematics
is the eighth prime number, and forms a sexy prime with 13, a twin prime with 17, and a cousin prime with 23. It is the third full reptend prime in decimal, the fifth central trinomial coefficient, and the seventh Mersenne prime exponent. 19 is the second Keith number, and more specifically the first Keith prime. It is also the second octahedral number, after 6.
R19 is the second base-10 repunit prime, short for the number 1111111111111111111.
19 is the maximum number of fourth powers needed to sum up to any natural number, and in the context of Waring's problem, 19 is the fourth value of g(k).
The sum of the squares of the first 19 primes is divisible by 19.
19 is the sixth Heegner number. 67 and 163, respectively the 19th and 38th prime numbers, are the two largest Heegner numbers, of nine total.
19 is the third centered triangular number as well as the third centered hexagonal number.
The 19th triangular number is 190, equivalently the sum of the first 19 non-zero integers, that is also the sixth centered nonagonal number.
19 is the first number in an infinite sequence of numbers in decimal whose digits start with 1 and have trailing 9's, that form triangular numbers containing trailing zeroes in proportion to 9s present in the original number; i.e. 19900 is the 199th triangular number, and 1999000 is the 1999th.
Like 19, 199 and 1999 are also both prime, as are 199999 and 19999999. In fact, a number of the form 19n, where n is the number of nines that terminate in the number, is prime for:
n = {1, 2, 3, 5, 7, 26, 27, 53, 147, 236, 248, 386, 401}.
19, alongside 109, 1009, and 10009, are all prime (with 109 also full reptend), and form part of a sequence of numbers where inserting a digit inside the previous term produces the next smallest prime possible, up to scale, with the composite number 9 as root. 100019 is the next such smallest prime number, by the insertion of a 1.
Numbers of the form 10n9 equivalent to 10x + 9 with x = n + 1, where n is the number of zeros in the term, are prime for n = {0, 1, 2, 3, 8, 17, 21, 44, 48, 55, 68, 145, 201, 271, 2731, 4563}, and probably prime for n = {31811, 43187, 48109, 92691}.
The Collatz sequence for nine requires nineteen steps to return back to one, more than any other number below it. On the other hand, nineteen requires twenty steps, like eighteen. Less than ten thousand, only thirty-one other numbers require nineteen steps to return back to one:
{56, 58, 60, 61, 352, 360, 362, 368, 369, 372, 373, 401, 402, 403, 2176, ..., and 2421}.
19 is the first prime number that is not a permutable prime in decimal, as its reverse (91) is composite; where 91 is also the fourth centered nonagonal number.
19 × 91 = 1729, the first Hardy-Ramanujan number or taxicab number, also a Harshad number in base-ten, as it's divisible by the sum of its digits, 19.
The number of nodes i
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https://en.wikipedia.org/wiki/18%20%28number%29
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18 (eighteen) is the natural number following 17 and preceding 19.
In mathematics
Eighteen is the tenth composite number, its divisors being 1, 2, 3, 6 and 9. Three of these divisors (3, 6 and 9) add up to 18, hence 18 is a semiperfect number. Eighteen is the first inverted square-prime of the form p·q2.
In base ten, it is a Harshad number.
It is an abundant number, as the sum of its proper divisors is greater than itself (1+2+3+6+9 = 21). It is known to be a solitary number, despite not being coprime to this sum.
It is the number of one-sided pentominoes.
It is the only number where the sum of its written digits in base 10 (1+8 = 9) is equal to half of itself (18/2 = 9).
It is a Fine number.
In science
Chemistry
Eighteen is the atomic number of argon.
Group 18 of the periodic table is called the noble gases.
The 18-electron rule is a rule of thumb in transition metal chemistry for characterising and predicting the stability of metal complexes.
In religion and literature
The Hebrew word for "life" is (chai), which has a numerical value of 18. Consequently, the custom has arisen in Jewish circles to give donations and monetary gifts in multiples of 18 as an expression of blessing for long life.
In Judaism, in the Talmud; Pirkei Avot (5:25), Rabbi Yehudah ben Teime gives the age of 18 as the appropriate age to get married ("Ben shmonah esra lechupah", at eighteen years old to the Chupah (marriage canopy)). (See Coming of age, Age of majority).
Shemoneh Esrei (sh'MOH-nuh ES-ray) is a prayer that is the center of any Jewish religious service. Its name means "eighteen". The prayer is also known as the Amidah.
In Ancient Roman custom the number 18 can symbolise a blood relative.
Joseph Heller's novel Catch-22 was originally named Catch-18 because of the Hebrew meaning of the number, but was amended to the published title to avoid confusion with another war novel, Mila 18.
There are 18 chapters in the Bhagavad Gita, which is contained in the Mahabharata, which has 18 books. The Kurukshetra War which the epic depicts, is between 18 armies (11 on the Kuru side, 7 on the Pandava). The war itself lasts for 18 days. In the other Hindu epic, the Ramayana, the war between Rama and the demons also lasted 18 days.
In Babism the first 18 disciples of the Báb were known as the Letters of the Living.
As lucky or unlucky number
In Chinese tradition, 18 is pronounced and is considered a lucky number due to similarity with 'definitely get rich', 'to get rich for sure'.
According to applications of numerology in Judaism, the letters of the word chai ("living") add up to 18. Thus, 18 is considered a lucky number and many gifts for B'nai Mitzvot and weddings are in $18 increments.
Age 18
In most countries, 18 is the age of majority, in which a minor becomes a legal adult. It is also the voting age, marriageable age, drinking age and smoking age in most countries, though sometimes these ages are different than the age of majority. Many websites
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https://en.wikipedia.org/wiki/Menger%20sponge
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In mathematics, the Menger sponge (also known as the Menger cube, Menger universal curve, Sierpinski cube, or Sierpinski sponge) is a fractal curve. It is a three-dimensional generalization of the one-dimensional Cantor set and two-dimensional Sierpinski carpet. It was first described by Karl Menger in 1926, in his studies of the concept of topological dimension.
Construction
The construction of a Menger sponge can be described as follows:
Begin with a cube.
Divide every face of the cube into nine squares, like a Rubik's Cube. This sub-divides the cube into 27 smaller cubes.
Remove the smaller cube in the middle of each face, and remove the smaller cube in the center of the more giant cube, leaving 20 smaller cubes. This is a level-1 Menger sponge (resembling a void cube).
Repeat steps two and three for each of the remaining smaller cubes, and continue to iterate ad infinitum.
The second iteration gives a level-2 sponge, the third iteration gives a level-3 sponge, and so on. The Menger sponge itself is the limit of this process after an infinite number of iterations.
Properties
The th stage of the Menger sponge, , is made up of smaller cubes, each with a side length of (1/3)n. The total volume of is thus . The total surface area of is given by the expression . Therefore, the construction's volume approaches zero while its surface area increases without bound. Yet any chosen surface in the construction will be thoroughly punctured as the construction continues so that the limit is neither a solid nor a surface; it has a topological dimension of 1 and is accordingly identified as a curve.
Each face of the construction becomes a Sierpinski carpet, and the intersection of the sponge with any diagonal of the cube or any midline of the faces is a Cantor set. The cross-section of the sponge through its centroid and perpendicular to a space diagonal is a regular hexagon punctured with hexagrams arranged in six-fold symmetry. The number of these hexagrams, in descending size, is given by , with .
The sponge's Hausdorff dimension is ≅ 2.727. The Lebesgue covering dimension of the Menger sponge is one, the same as any curve. Menger showed, in the 1926 construction, that the sponge is a universal curve, in that every curve is homeomorphic to a subset of the Menger sponge, where a curve means any compact metric space of Lebesgue covering dimension one; this includes trees and graphs with an arbitrary countable number of edges, vertices and closed loops, connected in arbitrary ways. Similarly, the Sierpinski carpet is a universal curve for all curves that can be drawn on the two-dimensional plane. The Menger sponge constructed in three dimensions extends this idea to graphs that are not planar and might be embedded in any number of dimensions.
The Menger sponge is a closed set; since it is also bounded, the Heine–Borel theorem implies that it is compact. It has Lebesgue measure 0. Because it contains continuous paths, it is an uncountable set.
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https://en.wikipedia.org/wiki/Quotient%20of%20a%20formal%20language
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In mathematics and computer science, the right quotient (or simply quotient) of a language with respect to language is the language consisting of strings w such that wx is in for some string x in Formally:
In other words, we take all the strings in that have a suffix in , and remove this suffix.
Similarly, the left quotient of with respect to is the language consisting of strings w such that xw is in for some string x in . Formally:
In other words, we take all the strings in that have a prefix in , and remove this prefix.
Note that the operands of are in reverse order: the first operand is and is second.
Example
Consider
and
Now, if we insert a divider into an element of , the part on the right is in only if the divider is placed adjacent to a b (in which case i ≤ n and j = n) or adjacent to a c (in which case i = 0 and j ≤ n). The part on the left, therefore, will be either or ; and can be written as
Properties
Some common closure properties of the quotient operation include:
The quotient of a regular language with any other language is regular.
The quotient of a context free language with a regular language is context free.
The quotient of two context free languages can be any recursively enumerable language.
The quotient of two recursively enumerable languages is recursively enumerable.
These closure properties hold for both left and right quotients.
See also
Brzozowski derivative
References
Formal languages
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https://en.wikipedia.org/wiki/Quotient%20rule
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In calculus, the quotient rule is a method of finding the derivative of a function that is the ratio of two differentiable functions. Let , where both and are differentiable and The quotient rule states that the derivative of is
It is provable in many ways by using other derivative rules.
Examples
Example 1: Basic example
Given , let , then using the quotient rule:
Example 2: Derivative of tangent function
The quotient rule can be used to find the derivative of as follows:
Reciprocal rule
The reciprocal rule is a special case of the quotient rule in which the numerator . Applying the quotient rule gives
Note that utilizing the chain rule yields the same result.
Proofs
Proof from derivative definition and limit properties
Let Applying the definition of the derivative and properties of limits gives the following proof, with the term added and subtracted to allow splitting and factoring in subsequent steps without affecting the value:The limit evaluation is justified by the differentiability of , implying continuity, which can be expressed as .
Proof using implicit differentiation
Let so that
The product rule then gives
Solving for and substituting back for gives:
Proof using the reciprocal rule or chain rule
Let
Then the product rule gives
To evaluate the derivative in the second term, apply the reciprocal rule, or the power rule along with the chain rule:
Substituting the result into the expression gives
Proof by logarithmic differentiation
Let Taking the absolute value and natural logarithm of both sides of the equation gives
Applying properties of the absolute value and logarithms,
Taking the logarithmic derivative of both sides,
Solving for and substituting back for gives:
Note: Taking the absolute value of the functions is necessary for the logarithmic differentiation of functions that may have negative values, as logarithms are only real-valued for positive arguments. This works because , which justifies taking the absolute value of the functions for logarithmic differentiation.
Higher order derivatives
Implicit differentiation can be used to compute the th derivative of a quotient (partially in terms of its first derivatives). For example, differentiating twice (resulting in ) and then solving for yields
See also
References
Articles containing proofs
Differentiation rules
Theorems in analysis
Theorems in calculus
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https://en.wikipedia.org/wiki/21%20%28number%29
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21 (twenty-one) is the natural number following 20 and preceding 22.
The current century is the 21st century AD, under the Gregorian calendar.
In mathematics
Twenty-one is the fifth distinct semiprime, and the second of the form where is a higher prime.
As a biprime with proper divisors 1, 3 and 7, twenty-one has a prime aliquot sum of 11 within an aliquot sequence containing only one composite number (21, 11, 1, 0); it is the second composite member of the 11-aliquot tree, following 18. 21 is the first member of the second cluster of two discrete semiprimes (21, 22), where the next such cluster is (38, 39). 21 is the smallest natural number that is not close to a power of two , where the range of nearness is . 21 is a Harshad number in base ten, and a repdigit in quaternary (1114).
While 21 is the sixth triangular number, it is also the sum of the divisors of the first five positive integers:
21 is the fifth Motzkin number, and the eighth Fibonacci number, equal to the sum of the preceding terms in the sequence, 8 and 13. It is the smallest non-trivial example in decimal of a Fibonacci number whose digits are Fibonacci numbers and whose digit sum is also a Fibonacci number. It is also an octagonal number, and a Padovan number (preceded by the terms (9, 12, 16), where it is the sum of the first two of these). 21 is a Blum integer, since it is a semiprime with both its prime factors being Gaussian primes.
While the twenty-first prime number 73 is the largest member of Bhargava's definite quadratic 17–integer matrix representative of all prime numbers,
the twenty-first composite number 33 is the largest member of a like definite quadratic 9–integer matrix
representative of all odd numbers.
21 is also a positive integer that has the following property (see brief proof below):
Note that a necessary condition for is that for any coprime to , and must satisfy the condition above, therefore at least one of and must only have factor 2 and 5.
Let denote the quantity of the numbers smaller than that only have factor 2 and 5 and that are coprime to , we instantly have .
We can easily see that for sufficiently large , , but , as goes to infinity, thus fails to hold for sufficiently large .
In fact, For every , we have
and
so fails to hold when (actually, when ).
Just check a few numbers to see that '= 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 15, 21.
Twenty-one is the smallest number of differently sized squares needed to square the square.
In science
The atomic number of scandium.
It is very often the day of the solstices in both June and December, though the precise date varies by year.
Age 21
In thirteen countries, 21 is the age of majority. See also: Coming of age.
In eight countries, 21 is the minimum age to purchase tobacco products.
In seventeen countries, 21 is the drinking age.
In nine countries, it is the voting age.
In the United States:
21 is the minimum age at which a person may gamble or enter casinos in most states
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https://en.wikipedia.org/wiki/Su%20Buqing
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Su Buqing, also spelled Su Buchin (; September 23, 1902 – March 17, 2003), was a Chinese mathematician, educator and poet. He was the founder of differential geometry in China, and served as president of Fudan University and honorary chairman of the Chinese Mathematical Society.
Early Life
Su was born in Pingyang County, Zhejiang Province in 1902, with ancestry from Quanzhou.
Su's academic journey began in 1911 when he transferred to the First Primary School in Pingyang County and was subsequently admitted to the Zhejiang Provincial Tenth Middle School in 1914. In 1919, Su was offered a scholarship by the principal of his middle school to study in Japan. Su successfully passed the entrance examination for Tokyo Higher Technical College in February 1920 and enrolled in the Department of Electrical Engineering where he graduated from in March 1924. After that, he was admitted to the Department of Mathematics of Tohoku Imperial University in Japan and graduated in 1927. Under the supervision of Tadahiko Kubota, Su received his Ph.D. from the university in 1931 for dissertation entitled The relation between affine and projective differential geometry.
During his time in Tohoku Imperial University, Su met and married Yonako Matsumoto in 1928.
Career
After returning to China, he first served as a professor and dean at Zhejiang University (he established the Chen-Su School with Chen Jiangong). When universities across China underwent reorganization in 1952, Su and his students from Zhejiang University's Department of Mathematics relocated to Fudan University where he initially served as the Provost and later became the honorary President of Fudan University in 1978.
He was honorary chairman of the Chinese Mathematical Society (CMS) and elected to Academia Sinica and the Chinese Academy of Sciences in 1948 and 1955 respectively.
In 1985, the Propaganda Department of the Zhejiang Provincial Party Committee appointed Su as the honorary President of Wenzhou University. Additionally, he provided guidance and supervision in the establishment of the Shanghai Society of Industrial and Applied Mathematics in 1989 and the China Society of Industrial and Applied Mathematics (CSIAM) in 1990, serving as a consultant for the latter.
Some other positions Su held included member of the sixth (1983-88) and later vice chairman of the seventh and eighth (1988-98) National Committee of the Chinese People's Political Consultative Conference (CPPCC), honorary chairman of the China Democratic League Central Committee (1997), and Conference and deputy to the National People's Congress (NPC).
Research
Praised commonly in mathematical field as the "first geometer in the orient", Su was engaged in research, teaching and education in differential geometry and computational geometry. In his early years, he made excellent contributions to affine differential geometry and projective differential geometry. He obtained extraordinary achievements in general space different
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https://en.wikipedia.org/wiki/Empty%20sum
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In mathematics, an empty sum, or nullary sum, is a summation where the number of terms is zero.
The natural way to extend non-empty sums is to let the empty sum be the additive identity.
Let , , , ... be a sequence of numbers, and let
be the sum of the first m terms of the sequence. This satisfies the recurrence
provided that we use the following natural convention: .
In other words, a "sum" with only one term evaluates to that one term, while a "sum" with no terms evaluates to 0.
Allowing a "sum" with only 1 or 0 terms reduces the number of cases to be considered in many mathematical formulas. Such "sums" are natural starting points in induction proofs, as well as in algorithms. For these reasons, the "empty sum is zero" extension is standard practice in mathematics and computer programming (assuming the domain has a zero element).
For the same reason, the empty product is taken to be the multiplicative identity.
For sums of other objects (such as vectors, matrices, polynomials), the value of an empty summation is taken to be its additive identity.
Examples
Empty linear combinations
In linear algebra, a basis of a vector space V is a linearly independent subset B such that every element of V is a linear combination of B.
The empty sum convention allows the zero-dimensional vector space V={0} to have a basis, namely the empty set.
See also
Empty product
Iterated binary operation
Empty function
References
Operations on numbers
0 (number)
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https://en.wikipedia.org/wiki/Reflexive%20space
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In the area of mathematics known as functional analysis, a reflexive space is a locally convex topological vector space (TVS) for which the canonical evaluation map from into its bidual (which is the strong dual of the strong dual of ) is an isomorphism of TVSs.
Since a normable TVS is reflexive if and only if it is semi-reflexive, every normed space (and so in particular, every Banach space) is reflexive if and only if the canonical evaluation map from into its bidual is surjective;
in this case the normed space is necessarily also a Banach space.
In 1951, R. C. James discovered a Banach space, now known as James' space, that is reflexive but is nevertheless isometrically isomorphic to its bidual (any such isomorphism is thus necessarily the canonical evaluation map).
Reflexive spaces play an important role in the general theory of locally convex TVSs and in the theory of Banach spaces in particular. Hilbert spaces are prominent examples of reflexive Banach spaces. Reflexive Banach spaces are often characterized by their geometric properties.
Definition
Definition of the bidual
Suppose that is a topological vector space (TVS) over the field (which is either the real or complex numbers) whose continuous dual space, separates points on (that is, for any there exists some such that ).
Let and both denote the strong dual of which is the vector space of continuous linear functionals on endowed with the topology of uniform convergence on bounded subsets of ;
this topology is also called the strong dual topology and it is the "default" topology placed on a continuous dual space (unless another topology is specified).
If is a normed space, then the strong dual of is the continuous dual space with its usual norm topology.
The bidual of denoted by is the strong dual of ; that is, it is the space
If is a normed space, then is the continuous dual space of the Banach space with its usual norm topology.
Definitions of the evaluation map and reflexive spaces
For any let be defined by where is a linear map called the evaluation map at ;
since is necessarily continuous, it follows that
Since separates points on the linear map defined by is injective where this map is called the evaluation map or the canonical map.
Call semi-reflexive if is bijective (or equivalently, surjective) and we call reflexive if in addition is an isomorphism of TVSs.
A normable space is reflexive if and only if it is semi-reflexive or equivalently, if and only if the evaluation map is surjective.
Reflexive Banach spaces
Suppose is a normed vector space over the number field or (the real numbers or the complex numbers), with a norm Consider its dual normed space that consists of all continuous linear functionals and is equipped with the dual norm defined by
The dual is a normed space (a Banach space to be precise), and its dual normed space is called bidual space for The bidual consists of all continuous linear fun
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https://en.wikipedia.org/wiki/Invariance%20of%20domain
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Invariance of domain is a theorem in topology about homeomorphic subsets of Euclidean space .
It states:
If is an open subset of and is an injective continuous map, then is open in and is a homeomorphism between and .
The theorem and its proof are due to L. E. J. Brouwer, published in 1912.
The proof uses tools of algebraic topology, notably the Brouwer fixed point theorem.
Notes
The conclusion of the theorem can equivalently be formulated as: " is an open map".
Normally, to check that is a homeomorphism, one would have to verify that both and its inverse function are continuous;
the theorem says that if the domain is an subset of and the image is also in then continuity of is automatic.
Furthermore, the theorem says that if two subsets and of are homeomorphic, and is open, then must be open as well.
(Note that is open as a subset of and not just in the subspace topology.
Openness of in the subspace topology is automatic.)
Both of these statements are not at all obvious and are not generally true if one leaves Euclidean space.
It is of crucial importance that both domain and image of are contained in Euclidean space .
Consider for instance the map defined by
This map is injective and continuous, the domain is an open subset of , but the image is not open in
A more extreme example is the map defined by because here is injective and continuous but does not even yield a homeomorphism onto its image.
The theorem is also not generally true in infinitely many dimensions. Consider for instance the Banach Lp space of all bounded real sequences.
Define as the shift
Then is injective and continuous, the domain is open in , but the image is not.
Consequences
An important consequence of the domain invariance theorem is that cannot be homeomorphic to if
Indeed, no non-empty open subset of can be homeomorphic to any open subset of in this case.
Generalizations
The domain invariance theorem may be generalized to manifolds: if and are topological -manifolds without boundary and is a continuous map which is locally one-to-one (meaning that every point in has a neighborhood such that restricted to this neighborhood is injective), then is an open map (meaning that is open in whenever is an open subset of ) and a local homeomorphism.
There are also generalizations to certain types of continuous maps from a Banach space to itself.
See also
for other conditions that ensure that a given continuous map is open.
Notes
References
(see p. 72–73 for Hirsch's proof utilizing non-existence of a differentiable retraction)
External links
Algebraic topology
Theory of continuous functions
Homeomorphisms
Theorems in topology
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https://en.wikipedia.org/wiki/Direct%20limit
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In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way. These objects may be groups, rings, vector spaces or in general objects from any category. The way they are put together is specified by a system of homomorphisms (group homomorphism, ring homomorphism, or in general morphisms in the category) between those smaller objects. The direct limit of the objects , where ranges over some directed set , is denoted by . (This is a slight abuse of notation as it suppresses the system of homomorphisms that is crucial for the structure of the limit.)
Direct limits are a special case of the concept of colimit in category theory. Direct limits are dual to inverse limits, which are a special case of limits in category theory.
Formal definition
We will first give the definition for algebraic structures like groups and modules, and then the general definition, which can be used in any category.
Direct limits of algebraic objects
In this section objects are understood to consist of underlying sets equipped with a given algebraic structure, such as groups, rings, modules (over a fixed ring), algebras (over a fixed field), etc. With this in mind, homomorphisms are understood in the corresponding setting (group homomorphisms, etc.).
Let be a directed set. Let be a family of objects indexed by and be a homomorphism for all with the following properties:
is the identity of , and
for all .
Then the pair is called a direct system over .
The direct limit of the direct system is denoted by and is defined as follows. Its underlying set is the disjoint union of the 's modulo a certain :
Here, if and , then if and only if there is some with and such that .
Intuitively, two elements in the disjoint union are equivalent if and only if they "eventually become equal" in the direct system. An equivalent formulation that highlights the duality to the inverse limit is that an element is equivalent to all its images under the maps of the direct system, i.e. whenever .
One obtains from this definition canonical functions sending each element to its equivalence class. The algebraic operations on are defined such that these maps become homomorphisms. Formally, the direct limit of the direct system consists of the object together with the canonical homomorphisms .
Direct limits in an arbitrary category
The direct limit can be defined in an arbitrary category by means of a universal property. Let be a direct system of objects and morphisms in (as defined above). A target is a pair where is an object in and are morphisms for each such that whenever . A direct limit of the direct system is a universally repelling target in the sense that is a target and for each target , there is a unique morphism such that for each i. The following diagram
will then commute for all i, j.
The direct limit is often denoted
with the direct system and
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https://en.wikipedia.org/wiki/Tangent%20bundle
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In differential geometry, the tangent bundle of a differentiable manifold is a manifold which assembles all the tangent vectors in . As a set, it is given by the disjoint union of the tangent spaces of . That is,
where denotes the tangent space to at the point . So, an element of can be thought of as a pair , where is a point in and is a tangent vector to at .
There is a natural projection
defined by . This projection maps each element of the tangent space to the single point .
The tangent bundle comes equipped with a natural topology (described in a section below). With this topology, the tangent bundle to a manifold is the prototypical example of a vector bundle (which is a fiber bundle whose fibers are vector spaces). A section of is a vector field on , and the dual bundle to is the cotangent bundle, which is the disjoint union of the cotangent spaces of . By definition, a manifold is parallelizable if and only if the tangent bundle is trivial. By definition, a manifold is framed if and only if the tangent bundle is stably trivial, meaning that for some trivial bundle the Whitney sum is trivial. For example, the n-dimensional sphere Sn is framed for all n, but parallelizable only for (by results of Bott-Milnor and Kervaire).
Role
One of the main roles of the tangent bundle is to provide a domain and range for the derivative of a smooth function. Namely, if is a smooth function, with and smooth manifolds, its derivative is a smooth function .
Topology and smooth structure
The tangent bundle comes equipped with a natural topology (not the disjoint union topology) and smooth structure so as to make it into a manifold in its own right. The dimension of is twice the dimension of .
Each tangent space of an n-dimensional manifold is an n-dimensional vector space. If is an open contractible subset of , then there is a diffeomorphism which restricts to a linear isomorphism from each tangent space to . As a manifold, however, is not always diffeomorphic to the product manifold . When it is of the form , then the tangent bundle is said to be trivial. Trivial tangent bundles usually occur for manifolds equipped with a 'compatible group structure'; for instance, in the case where the manifold is a Lie group. The tangent bundle of the unit circle is trivial because it is a Lie group (under multiplication and its natural differential structure). It is not true however that all spaces with trivial tangent bundles are Lie groups; manifolds which have a trivial tangent bundle are called parallelizable. Just as manifolds are locally modeled on Euclidean space, tangent bundles are locally modeled on , where is an open subset of Euclidean space.
If M is a smooth n-dimensional manifold, then it comes equipped with an atlas of charts , where is an open set in and
is a diffeomorphism. These local coordinates on give rise to an isomorphism for all . We may then define a map
by
We use these maps to define the topology
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https://en.wikipedia.org/wiki/Enumeration
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An enumeration is a complete, ordered listing of all the items in a collection. The term is commonly used in mathematics and computer science to refer to a listing of all of the elements of a set. The precise requirements for an enumeration (for example, whether the set must be finite, or whether the list is allowed to contain repetitions) depend on the discipline of study and the context of a given problem.
Some sets can be enumerated by means of a natural ordering (such as 1, 2, 3, 4, ... for the set of positive integers), but in other cases it may be necessary to impose a (perhaps arbitrary) ordering. In some contexts, such as enumerative combinatorics, the term enumeration is used more in the sense of counting – with emphasis on determination of the number of elements that a set contains, rather than the production of an explicit listing of those elements.
Combinatorics
In combinatorics, enumeration means counting, i.e., determining the exact number of elements of finite sets, usually grouped into infinite families, such as the family of sets each consisting of all permutations of some finite set. There are flourishing subareas in many branches of mathematics concerned with enumerating in this sense objects of special kinds. For instance, in partition enumeration and graph enumeration the objective is to count partitions or graphs that meet certain conditions.
Set theory
In set theory, the notion of enumeration has a broader sense, and does not require the set being enumerated to be finite.
Listing
When an enumeration is used in an ordered list context, we impose some sort of ordering structure requirement on the index set. While we can make the requirements on the ordering quite lax in order to allow for great generality, the most natural and common prerequisite is that the index set be well-ordered. According to this characterization, an ordered enumeration is defined to be a surjection (an onto relationship) with a well-ordered domain. This definition is natural in the sense that a given well-ordering on the index set provides a unique way to list the next element given a partial enumeration.
Countable vs. uncountable
Unless otherwise specified, an enumeration is done by means of natural numbers. That is, an enumeration of a set is a bijective function from the natural numbers or an initial segment of the natural numbers to .
A set is countable if it can be enumerated, that is, if there exists an enumeration of it. Otherwise, it is uncountable. For example, the set of the real numbers is uncountable.
A set is finite if it can be enumerated by means of a proper initial segment of the natural numbers, in which case, its cardinality is . The empty set is finite, as it can be enumerated by means of the empty initial segment of the natural numbers.
The term set is sometimes used for countable sets. However it is also often used for computably enumerable sets, which are the countable sets for which an enumeration function can
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https://en.wikipedia.org/wiki/Algebraic%20enumeration
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Algebraic enumeration is a subfield of enumeration that deals with finding exact formulas for the number of combinatorial objects of a given type, rather than estimating this number asymptotically. Methods of finding these formulas include generating functions and the solution of recurrence relations.
References
Enumerative combinatorics
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https://en.wikipedia.org/wiki/Homotopy
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In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from "same, similar" and "place") if one can be "continuously deformed" into the other, such a deformation being called a homotopy (, ; , ) between the two functions. A notable use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology.
In practice, there are technical difficulties in using homotopies with certain spaces. Algebraic topologists work with compactly generated spaces, CW complexes, or spectra.
Formal definition
Formally, a homotopy between two continuous functions f and g from a
topological space X to a topological space Y is defined to be a continuous function from the product of the space X with the unit interval [0, 1] to Y such that and for all .
If we think of the second parameter of H as time then H describes a continuous deformation of f into g: at time 0 we have the function f and at time 1 we have the function g. We can also think of the second parameter as a "slider control" that allows us to smoothly transition from f to g as the slider moves from 0 to 1, and vice versa.
An alternative notation is to say that a homotopy between two continuous functions is a family of continuous functions for such that and , and the map is continuous from to . The two versions coincide by setting . It is not sufficient to require each map to be continuous.
The animation that is looped above right provides an example of a homotopy between two embeddings, f and g, of the torus into . X is the torus, Y is , f is some continuous function from the torus to R3 that takes the torus to the embedded surface-of-a-doughnut shape with which the animation starts; g is some continuous function that takes the torus to the embedded surface-of-a-coffee-mug shape. The animation shows the image of ht(X) as a function of the parameter t, where t varies with time from 0 to 1 over each cycle of the animation loop. It pauses, then shows the image as t varies back from 1 to 0, pauses, and repeats this cycle.
Properties
Continuous functions f and g are said to be homotopic if and only if there is a homotopy H taking f to g as described above. Being homotopic is an equivalence relation on the set of all continuous functions from X to Y.
This homotopy relation is compatible with function composition in the following sense: if are homotopic, and are homotopic, then their compositions and are also homotopic.
Examples
If are given by and , then the map given by is a homotopy between them.
More generally, if is a convex subset of Euclidean space and are paths with the same endpoints, then there is a linear homotopy (or straight-line homotopy) given by
Let be the identity function on the unit n-disk; i.e. the set . Let be the constant function which sends every point to the origin. Then the following is a homotopy between them:
Homotopy equiv
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https://en.wikipedia.org/wiki/Whitehead%20problem
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In group theory, a branch of abstract algebra, the Whitehead problem is the following question:
Saharon Shelah proved that Whitehead's problem is independent of ZFC, the standard axioms of set theory.
Refinement
Assume that A is an abelian group such that every short exact sequence
must split if B is also abelian. The Whitehead problem then asks: must A be free? This splitting requirement is equivalent to the condition Ext1(A, Z) = 0. Abelian groups A satisfying this condition are sometimes called Whitehead groups, so Whitehead's problem asks: is every Whitehead group free? It should be mentioned that if this condition is strengthened by requiring that the exact sequence
must split for any abelian group C, then it is well known that this is equivalent to A being free.
Caution: The converse of Whitehead's problem, namely that every free abelian group is Whitehead, is a well known group-theoretical fact. Some authors call Whitehead group only a non-free group A satisfying Ext1(A, Z) = 0. Whitehead's problem then asks: do Whitehead groups exist?
Shelah's proof
Saharon Shelah showed that, given the canonical ZFC axiom system, the problem is independent of the usual axioms of set theory. More precisely, he showed that:
If every set is constructible, then every Whitehead group is free;
If Martin's axiom and the negation of the continuum hypothesis both hold, then there is a non-free Whitehead group.
Since the consistency of ZFC implies the consistency of both of the following:
The axiom of constructibility (which asserts that all sets are constructible);
Martin's axiom plus the negation of the continuum hypothesis,
Whitehead's problem cannot be resolved in ZFC.
Discussion
J. H. C. Whitehead, motivated by the second Cousin problem, first posed the problem in the 1950s. Stein answered the question in the affirmative for countable groups. Progress for larger groups was slow, and the problem was considered an important one in algebra for some years.
Shelah's result was completely unexpected. While the existence of undecidable statements had been known since Gödel's incompleteness theorem of 1931, previous examples of undecidable statements (such as the continuum hypothesis) had all been in pure set theory. The Whitehead problem was the first purely algebraic problem to be proved undecidable.
Shelah later showed that the Whitehead problem remains undecidable even if one assumes the continuum hypothesis. The Whitehead conjecture is true if all sets are constructible. That this and other statements about uncountable abelian groups are provably independent of ZFC shows that the theory of such groups is very sensitive to the assumed underlying set theory.
See also
Free abelian group
Whitehead torsion
List of statements undecidable in ZFC
Statements true in L
References
Further reading
An expository account of Shelah's proof.
Independence results
Group theory
Mathematical problems
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