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https://en.wikipedia.org/wiki/Kernel%20%28linear%20algebra%29
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In mathematics, the kernel of a linear map, also known as the null space or nullspace, is the linear subspace of the domain of the map which is mapped to the zero vector. That is, given a linear map between two vector spaces and , the kernel of is the vector space of all elements of such that , where denotes the zero vector in , or more symbolically:
Properties
The kernel of is a linear subspace of the domain .
In the linear map two elements of have the same image in if and only if their difference lies in the kernel of , that is,
From this, it follows that the image of is isomorphic to the quotient of by the kernel:
In the case where is finite-dimensional, this implies the rank–nullity theorem:
where the term refers the dimension of the image of , while refers to the dimension of the kernel of ,
That is,
so that the rank–nullity theorem can be restated as
When is an inner product space, the quotient can be identified with the orthogonal complement in of This is the generalization to linear operators of the row space, or coimage, of a matrix.
Application to modules
The notion of kernel also makes sense for homomorphisms of modules, which are generalizations of vector spaces where the scalars are elements of a ring, rather than a field. The domain of the mapping is a module, with the kernel constituting a submodule. Here, the concepts of rank and nullity do not necessarily apply.
In functional analysis
If V and W are topological vector spaces such that W is finite-dimensional, then a linear operator L: V → W is continuous if and only if the kernel of L is a closed subspace of V.
Representation as matrix multiplication
Consider a linear map represented as a m × n matrix A with coefficients in a field K (typically or ), that is operating on column vectors x with n components over K.
The kernel of this linear map is the set of solutions to the equation , where 0 is understood as the zero vector. The dimension of the kernel of A is called the nullity of A. In set-builder notation,
The matrix equation is equivalent to a homogeneous system of linear equations:
Thus the kernel of A is the same as the solution set to the above homogeneous equations.
Subspace properties
The kernel of a matrix A over a field K is a linear subspace of Kn. That is, the kernel of A, the set Null(A), has the following three properties:
Null(A) always contains the zero vector, since .
If and , then . This follows from the distributivity of matrix multiplication over addition.
If and c is a scalar , then , since .
The row space of a matrix
The product Ax can be written in terms of the dot product of vectors as follows:
Here, a1, ... , am denote the rows of the matrix A. It follows that x is in the kernel of A, if and only if x is orthogonal (or perpendicular) to each of the row vectors of A (since orthogonality is defined as having a dot product of 0).
The row space, or coimage, of a matrix A is the span of the row vectors of A. B
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https://en.wikipedia.org/wiki/Forward%20algorithm
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The forward algorithm, in the context of a hidden Markov model (HMM), is used to calculate a 'belief state': the probability of a state at a certain time, given the history of evidence. The process is also known as filtering. The forward algorithm is closely related to, but distinct from, the Viterbi algorithm.
The forward and backward algorithms should be placed within the context of probability as they appear to simply be names given to a set of standard mathematical procedures within a few fields. For example, neither "forward algorithm" nor "Viterbi" appear in the Cambridge encyclopedia of mathematics. The main observation to take away from these algorithms is how to organize Bayesian updates and inference to be efficient in the context of directed graphs of variables (see sum-product networks).
For an HMM such as this one:
this probability is written as . Here is the hidden state which is abbreviated as and are the observations to .
The backward algorithm complements the forward algorithm by taking into account the future history if one wanted to improve the estimate for past times. This is referred to as smoothing and the forward/backward algorithm computes for . Thus, the full forward/backward algorithm takes into account all evidence. Note that a belief state can be calculated at each time step, but doing this does not, in a strict sense, produce the most likely state sequence, but rather the most likely state at each time step, given the previous history. In order to achieve the most likely sequence, the Viterbi algorithm is required. It computes the most likely state sequence given the history of observations, that is, the state sequence that maximizes .
History
The forward algorithm is one of the algorithms used to solve the decoding problem. Since the development of speech recognition and pattern recognition and related fields like computational biology which use HMMs, the forward algorithm has gained popularity.
Algorithm
The goal of the forward algorithm is to compute the joint probability , where for notational convenience we have abbreviated as and as . Computing directly would require marginalizing over all possible state sequences , the number of which grows exponentially with . Instead, the forward algorithm takes advantage of the conditional independence rules of the hidden Markov model (HMM) to perform the calculation recursively.
To demonstrate the recursion, let
.
Using the chain rule to expand , we can then write
.
Because is conditionally independent of everything but , and is conditionally independent of everything but , this simplifies to
.
Thus, since and are given by the model's emission distributions and transition probabilities, one can quickly calculate from and avoid incurring exponential computation time.
The initial condition is set as some prior probability over as
such that
Once the joint probability has been computed using the forward algorithm, we can easily obtain the
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https://en.wikipedia.org/wiki/Marcel%20Riesz
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Marcel Riesz ( ; 16 November 1886 – 4 September 1969) was a Hungarian mathematician, known for work on summation methods, potential theory, and other parts of analysis, as well as number theory, partial differential equations, and Clifford algebras. He spent most of his career in Lund (Sweden).
Marcel is the younger brother of Frigyes Riesz, who was also an important mathematician and at times they worked together (see F. and M. Riesz theorem).
Biography
Marcel Riesz was born in Győr, Austria-Hungary. He was the younger brother of the mathematician Frigyes Riesz. In 1904, he won the Loránd Eötvös competition. Upon entering the Budapest University, he also studied in Göttingen, and the academic year 1910-11 he spent in Paris. Earlier, in 1908, he attended the
1908 International Congress of Mathematicians in Rome. There he met Gösta Mittag-Leffler, in three years, Mittag-Leffler would offer Riesz to come to Sweden.
Riesz obtained his PhD at Eötvös Loránd University under the supervision of Lipót Fejér. In 1911, he moved to Sweden, where from 1911 to 1925 he taught at Stockholm University.
From 1926 to 1952, he was a professor at Lund University. According to Lars Gårding, Riesz arrived in Lund as a renowned star of mathematics, and for a time his appointment may have seemed like an exile. Indeed, there was no established school of mathematics in Lund at the time. However, Riesz managed to turn the tide and make the academic atmosphere more active.
Retired from the Lund University, he spent 10 years at universities in the United States. As a visiting research professor, he worked in Maryland, Chicago, etc.
After ten years of intense work with little rest, he suffered a breakdown. Riesz returned to Lund in 1962. After a long illness, he died there in 1969.
Riesz was elected a member of the Royal Swedish Academy of Sciences in 1936.
Mathematical work
Classical analysis
The work of Riesz as a student of Fejér in Budapest was devoted to trigonometric series:
One of his results states that if
and if the Fejer means of the series tend to zero, then all the coefficients an and bn are zero.
His results on summability of trigonometric series include a generalisation of Fejér's theorem to Cesàro means of arbitrary order. He also studied the summability of power and Dirichlet series, and coauthored a book on the latter with G.H. Hardy.
In 1916, he introduced the Riesz interpolation formula for trigonometric polynomials, which allowed him to give a new proof of Bernstein's inequality.
He also introduced the Riesz function Riesz(x), and showed that the Riemann hypothesis is equivalent to the bound as for any
Together with his brother Frigyes Riesz, he proved the F. and M. Riesz theorem, which implies, in particular, that if μ is a complex measure on the unit circle such that
then the variation |μ| of μ and the Lebesgue measure on the circle are mutually absolutely continuous.
Functional-analytic methods
Part of the analytic work of Rie
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https://en.wikipedia.org/wiki/Eilenberg%E2%80%93MacLane%20space
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In mathematics, specifically algebraic topology, an Eilenberg–MacLane space is a topological space with a single nontrivial homotopy group.
Let G be a group and n a positive integer. A connected topological space X is called an Eilenberg–MacLane space of type , if it has n-th homotopy group isomorphic to G and all other homotopy groups trivial. Assuming that G is abelian in the case that , Eilenberg–MacLane spaces of type always exist, and are all weak homotopy equivalent. Thus, one may consider as referring to a weak homotopy equivalence class of spaces. It is common to refer to any representative as "a " or as "a model of ". Moreover, it is common to assume that this space is a CW-complex (which is always possible via CW approximation).
The name is derived from Samuel Eilenberg and Saunders Mac Lane, who introduced such spaces in the late 1940s.
As such, an Eilenberg–MacLane space is a special kind of topological space that in homotopy theory can be regarded as a building block for CW-complexes via fibrations in a Postnikov system. These spaces are important in many contexts in algebraic topology, including computations of homotopy groups of spheres, definition of cohomology operations, and for having a strong connection to singular cohomology.
A generalised Eilenberg–Maclane space is a space which has the homotopy type of a product of Eilenberg–Maclane spaces
.
Examples
The unit circle is a .
The infinite-dimensional complex projective space is a model of .
The infinite-dimensional real projective space is a .
The wedge sum of k unit circles is a , where is the free group on k generators.
The complement to any connected knot or graph in a 3-dimensional sphere is of type ; this is called the "asphericity of knots", and is a 1957 theorem of Christos Papakyriakopoulos.
Any compact, connected, non-positively curved manifold M is a , where is the fundamental group of M. This is a consequence of the Cartan–Hadamard theorem.
An infinite lens space given by the quotient of by the free action for is a . This can be shown using covering space theory and the fact that the infinite dimensional sphere is contractible. Note this includes as a .
The configuration space of points in the plane is a , where is the pure braid group on strands.
Correspondingly, the th unordered configuration space of is a , where denotes the -strand braid group.
The infinite symmetric product of a n-sphere is a . More generally is a for all Moore spaces .
Some further elementary examples can be constructed from these by using the fact that the product is . For instance the -dimensional Torus is a .
Remark on constructing Eilenberg–MacLane spaces
For and an arbitrary group the construction of is identical to that of the classifying space of the group . Note that if G has a torsion element, then every CW-complex of type K(G,1) has to be infinite-dimensional.
There are multiple techniques for constructing higher Eilenberg-Maclane s
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https://en.wikipedia.org/wiki/Sigma-additive%20set%20function
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In mathematics, an additive set function is a function mapping sets to numbers, with the property that its value on a union of two disjoint sets equals the sum of its values on these sets, namely, If this additivity property holds for any two sets, then it also holds for any finite number of sets, namely, the function value on the union of k disjoint sets (where k is a finite number) equals the sum of its values on the sets. Therefore, an additive set function is also called a finitely additive set function (the terms are equivalent). However, a finitely additive set function might not have the additivity property for a union of an infinite number of sets. A σ-additive set function is a function that has the additivity property even for countably infinite many sets, that is,
Additivity and sigma-additivity are particularly important properties of measures. They are abstractions of how intuitive properties of size (length, area, volume) of a set sum when considering multiple objects. Additivity is a weaker condition than σ-additivity; that is, σ-additivity implies additivity.
The term modular set function is equivalent to additive set function; see modularity below.
Additive (or finitely additive) set functions
Let be a set function defined on an algebra of sets with values in (see the extended real number line). The function is called or , if whenever and are disjoint sets in then
A consequence of this is that an additive function cannot take both and as values, for the expression is undefined.
One can prove by mathematical induction that an additive function satisfies
for any disjoint sets in
σ-additive set functions
Suppose that is a σ-algebra. If for every sequence of pairwise disjoint sets in
holds then is said to be or .
Every -additive function is additive but not vice versa, as shown below.
τ-additive set functions
Suppose that in addition to a sigma algebra we have a topology If for every directed family of measurable open sets
we say that is -additive. In particular, if is inner regular (with respect to compact sets) then it is τ-additive.
Properties
Useful properties of an additive set function include the following.
Value of empty set
Either or assigns to all sets in its domain, or assigns to all sets in its domain. Proof: additivity implies that for every set If then this equality can be satisfied only by plus or minus infinity.
Monotonicity
If is non-negative and then That is, is a . Similarly, If is non-positive and then
Modularity
A set function on a family of sets is called a and a if whenever and are elements of then
The above property is called and the argument below proves that modularity is equivalent to additivity.
Given and Proof: write and and where all sets in the union are disjoint. Additivity implies that both sides of the equality equal
However, the related properties of submodularity and subadditivity are not equivalent to each other.
N
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https://en.wikipedia.org/wiki/Brown%27s%20representability%20theorem
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In mathematics, Brown's representability theorem in homotopy theory gives necessary and sufficient conditions for a contravariant functor F on the homotopy category Hotc of pointed connected CW complexes, to the category of sets Set, to be a representable functor.
More specifically, we are given
F: Hotcop → Set,
and there are certain obviously necessary conditions for F to be of type Hom(—, C), with C a pointed connected CW-complex that can be deduced from category theory alone. The statement of the substantive part of the theorem is that these necessary conditions are then sufficient. For technical reasons, the theorem is often stated for functors to the category of pointed sets; in other words the sets are also given a base point.
Brown representability theorem for CW complexes
The representability theorem for CW complexes, due to Edgar H. Brown, is the following. Suppose that:
The functor F maps coproducts (i.e. wedge sums) in Hotc to products in Set:
The functor F maps homotopy pushouts in Hotc to weak pullbacks. This is often stated as a Mayer–Vietoris axiom: for any CW complex W covered by two subcomplexes U and V, and any elements u ∈ F(U), v ∈ F(V) such that u and v restrict to the same element of F(U ∩ V), there is an element w ∈ F(W) restricting to u and v, respectively.
Then F is representable by some CW complex C, that is to say there is an isomorphism
F(Z) ≅ HomHotc(Z, C)
for any CW complex Z, which is natural in Z in that for any morphism from Z to another CW complex Y the induced maps F(Y) → F(Z) and HomHot(Y, C) → HomHot(Z, C) are compatible with these isomorphisms.
The converse statement also holds: any functor represented by a CW complex satisfies the above two properties. This direction is an immediate consequence of basic category theory, so the deeper and more interesting part of the equivalence is the other implication.
The representing object C above can be shown to depend functorially on F: any natural transformation from F to another functor satisfying the conditions of the theorem necessarily induces a map of the representing objects. This is a consequence of Yoneda's lemma.
Taking F(X) to be the singular cohomology group Hi(X,A) with coefficients in a given abelian group A, for fixed i > 0; then the representing space for F is the Eilenberg–MacLane space K(A, i). This gives a means of showing the existence of Eilenberg-MacLane spaces.
Variants
Since the homotopy category of CW-complexes is equivalent to the localization of the category of all topological spaces at the weak homotopy equivalences, the theorem can equivalently be stated for functors on a category defined in this way.
However, the theorem is false without the restriction to connected pointed spaces, and an analogous statement for unpointed spaces is also false.
A similar statement does, however, hold for spectra instead of CW complexes. Brown also proved a general categorical version of the representability theorem, which includes both
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https://en.wikipedia.org/wiki/Cotton%20tensor
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In differential geometry, the Cotton tensor on a (pseudo)-Riemannian manifold of dimension n is a third-order tensor concomitant of the metric. The vanishing of the Cotton tensor for is necessary and sufficient condition for the manifold to be conformally flat. By contrast, in dimensions ,
the vanishing of the Cotton tensor is necessary but not sufficient for the metric to be conformally flat; instead, the corresponding necessary and sufficient condition in these higher dimensions is the vanishing of the Weyl tensor, while the Cotton tensor just becomes a constant times
the divergence of the Weyl tensor. For the Cotton tensor is identically zero. The concept is named after Émile Cotton.
The proof of the classical result that for the vanishing of the Cotton tensor is equivalent to the metric being conformally flat is given by Eisenhart using a standard integrability argument. This tensor density is uniquely characterized by its conformal properties coupled with the demand that it be differentiable for arbitrary metrics, as shown by .
Recently, the study of three-dimensional spaces is becoming of great interest, because the Cotton tensor restricts the relation between the Ricci tensor and the energy–momentum tensor of matter in the Einstein equations and plays an important role in the Hamiltonian formalism of general relativity.
Definition
In coordinates, and denoting the Ricci tensor by Rij and the scalar curvature by R, the components of the Cotton tensor are
The Cotton tensor can be regarded as a vector valued 2-form, and for n = 3 one can use the Hodge star operator to convert this into a second order trace free tensor density
sometimes called the Cotton–York tensor.
Properties
Conformal rescaling
Under conformal rescaling of the metric for some scalar function . We see that the Christoffel symbols transform as
where is the tensor
The Riemann curvature tensor transforms as
In -dimensional manifolds, we obtain the Ricci tensor by contracting the transformed Riemann tensor to see it transform as
Similarly the Ricci scalar transforms as
Combining all these facts together permits us to conclude the Cotton-York tensor transforms as
or using coordinate independent language as
where the gradient is contracted with the Weyl tensor W.
Symmetries
The Cotton tensor has the following symmetries:
and therefore
In addition the Bianchi formula for the Weyl tensor can be rewritten as
where is the positive divergence in the first component of W.
References
A. Garcia, F.W. Hehl, C. Heinicke, A. Macias (2004) "The Cotton tensor in Riemannian spacetimes", Classical and Quantum Gravity 21: 1099–1118, Eprint arXiv:gr-qc/0309008
Riemannian geometry
Tensors in general relativity
Tensors
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https://en.wikipedia.org/wiki/Lebesgue%20space
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Lebesgue space may refer to:
Lp space, a special Banach space of functions (or rather, equivalence classes of functions)
Standard probability space, a non-pathological probability space
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https://en.wikipedia.org/wiki/Dedekind-infinite%20set
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In mathematics, a set A is Dedekind-infinite (named after the German mathematician Richard Dedekind) if some proper subset B of A is equinumerous to A. Explicitly, this means that there exists a bijective function from A onto some proper subset B of A. A set is Dedekind-finite if it is not Dedekind-infinite (i.e., no such bijection exists). Proposed by Dedekind in 1888, Dedekind-infiniteness was the first definition of "infinite" that did not rely on the definition of the natural numbers.
A simple example is , the set of natural numbers. From Galileo's paradox, there exists a bijection that maps every natural number n to its square n2. Since the set of squares is a proper subset of , is Dedekind-infinite.
Until the foundational crisis of mathematics showed the need for a more careful treatment of set theory, most mathematicians assumed that a set is infinite if and only if it is Dedekind-infinite. In the early twentieth century, Zermelo–Fraenkel set theory, today the most commonly used form of axiomatic set theory, was proposed as an axiomatic system to formulate a theory of sets free of paradoxes such as Russell's paradox. Using the axioms of Zermelo–Fraenkel set theory with the originally highly controversial axiom of choice included (ZFC) one can show that a set is Dedekind-finite if and only if it is finite in the usual sense. However, there exists a model of Zermelo–Fraenkel set theory without the axiom of choice (ZF) in which there exists an infinite, Dedekind-finite set, showing that the axioms of ZF are not strong enough to prove that every set that is Dedekind-finite is finite. There are definitions of finiteness and infiniteness of sets besides the one given by Dedekind that do not depend on the axiom of choice.
A vaguely related notion is that of a Dedekind-finite ring.
Comparison with the usual definition of infinite set
This definition of "infinite set" should be compared with the usual definition: a set A is infinite when it cannot be put in bijection with a finite ordinal, namely a set of the form for some natural number n – an infinite set is one that is literally "not finite", in the sense of bijection.
During the latter half of the 19th century, most mathematicians simply assumed that a set is infinite if and only if it is Dedekind-infinite. However, this equivalence cannot be proved with the axioms of Zermelo–Fraenkel set theory without the axiom of choice (AC) (usually denoted "ZF"). The full strength of AC is not needed to prove the equivalence; in fact, the equivalence of the two definitions is strictly weaker than the axiom of countable choice (CC). (See the references below.)
Dedekind-infinite sets in ZF
A set A is Dedekind-infinite if it satisfies any, and then all, of the following equivalent (over ZF) conditions:
it has a countably infinite subset;
there exists an injective map from a countably infinite set to A;
there is a function that is injective but not surjective;
there is an injective function , where
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https://en.wikipedia.org/wiki/Hyperbolic
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Hyperbolic is an adjective describing something that resembles or pertains to a hyperbola (a curve), to hyperbole (an overstatement or exaggeration), or to hyperbolic geometry.
The following phenomena are described as hyperbolic because they manifest hyperbolas, not because something about them is exaggerated.
Hyperbolic angle, an unbounded variable referring to a hyperbola instead of a circle
Hyperbolic coordinates, location by geometric mean and hyperbolic angle in quadrant I
Hyperbolic distribution, a probability distribution characterized by the logarithm of the probability density function being a hyperbola
Hyperbolic equilibrium point, a fixed point that does not have any center manifolds
Hyperbolic function, an analog of an ordinary trigonometric or circular function
Hyperbolic geometric graph, a random network generated by connecting nearby points sprinkled in a hyperbolic space
Hyperbolic geometry, a non-Euclidean geometry
Hyperbolic group, a finitely generated group equipped with a word metric satisfying certain properties characteristic of hyperbolic geometry
Hyperbolic growth, growth of a quantity toward a finite-time singularity
Hyperbolic logarithm, original designation of natural logarithm (1647–1748) before Euler's formulation with e
Hyperbolic manifold, a complete Riemannian n-manifold of constant sectional curvature −1
Hyperbolic motion, an isometry in a hyperbolic space
Hyperbolic navigation, a class of radio navigation systems based on the difference in timing between the reception of two signals, without reference to a common clock
Hyperbolic number, a synonym for split-complex number
Hyperbolic orthogonality, an orthogonality found in pseudo-Euclidean space
Hyperbolic paraboloid, a doubly ruled surface shaped like a saddle
Hyperbolic partial differential equation, a partial differential equation (PDE) of order n that has a well-posed initial value problem for the first n−1 derivatives
Hyperbolic plane can refer to:
The 2-dimensional plane in hyperbolic geometry (a non-Euclidean geometry)
The hyperbolic plane as isotropic quadratic form
Hyperbolic quaternions, a non-associative algebra, precursor to Minkowski space
Hyperbolic rotation, a synonym for squeeze mapping
Hyperbolic sector, a planar region demarcated by radial lines and a hyperbola
Hyperbolic soccerball, a tessellation of the hyperbolic plane
Hyperbolic space, hyperbolic spatial geometry in which every point is a saddle point
Hyperbolic trajectory, a Kepler orbit with eccentricity greater than 1
Hyperbolic versor, a versor parameterized by a hyperbolic angle
Hyperbolic unit, a non-real quantity with square equal to +1
See also
Exaggeration
Hyperboloid
Hyperboloid structure
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https://en.wikipedia.org/wiki/Hans%20Hahn%20%28mathematician%29
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Hans Hahn (; 27 September 1879 – 24 July 1934) was an Austrian mathematician and philosopher who made contributions to functional analysis, topology, set theory, the calculus of variations, real analysis, and order theory. In philosophy he was among the main logical positivists of the Vienna Circle.
Biography
Born in Vienna as the son of a higher government official of the K.K. Telegraphen-Korrespondenz Bureau (since 1946 named "Austria Presse Agentur"), in 1898 Hahn became a student at the Universität Wien starting with a study of law. In 1899 he switched over to mathematics and spent some time at the universities of Strasbourg, Munich and Göttingen. In 1902 he took his Ph.D. in Vienna, on the subject "Zur Theorie der zweiten Variation einfacher Integrale". He was a student of Gustav von Escherich.
He was appointed to the teaching staff (Habilitation) in Vienna in 1905. After 1905/1906 as a stand-in for Otto Stolz at Innsbruck and some further years as a Privatdozent in Vienna, he was nominated in 1909 Professor extraordinarius in Czernowitz, at that time a town within the empire of Austria. After joining the Austrian army in 1915, he was badly wounded in 1916 and became again Professor extraordinarius, now in Bonn. In 1917 he was nominated a regular Professor there and in 1921 he returned to Vienna with this title, where he stayed until his rather early death in 1934 at the age of 54, following cancer surgery.
He had married Eleonore ("Lilly") Minor in 1909 and they had a daughter, Nora (born 1910).
He was also interested in philosophy, and was part of a discussion group concerning Mach's positivism with Otto Neurath (who had married Hahn’s sister Olga Hahn-Neurath in 1912), and Phillip Frank prior to the First World War. In 1922, he helped arrange Moritz Schlick's entry into the group, which led to the founding of the Vienna Circle, the group that was at the center of logical positivist thought in the 1920s. His most famous student was Kurt Gödel, whose Ph.D. thesis was completed in 1929. After Anschluss the fact that Hans Hahn had been of partial Jewish origin caused Gödel's difficulties with getting a position at the University of Vienna. Within the Vienna Circle, Hahn was also known (and controversial) for using his mathematical and philosophical work to study psychic phenomena; according to Karl Menger he sometimes openly advocated further research into extrasensory perception while lecturing. Politically Hahn was a socialist and was chairman of the Association of Socialist University Teachers.
Hahn's contributions to mathematics include the Hahn–Banach theorem and (independently of Banach and Steinhaus) the uniform boundedness principle. Other theorems include:
the Hahn decomposition theorem;
the Hahn embedding theorem;
the Hahn–Kolmogorov theorem;
the Hahn–Mazurkiewicz theorem;
the Vitali–Hahn–Saks theorem.
Hahn authored the book : according to Arthur Rosenthal, "... (it) formed a great advance in the Theory of Real functio
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https://en.wikipedia.org/wiki/Vitali%E2%80%93Hahn%E2%80%93Saks%20theorem
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In mathematics, the Vitali–Hahn–Saks theorem, introduced by , , and , proves that under some conditions a sequence of measures converging point-wise does so uniformly and the limit is also a measure.
Statement of the theorem
If is a measure space with and a sequence of complex measures. Assuming that each is absolutely continuous with respect to and that a for all the finite limits exist Then the absolute continuity of the with respect to is uniform in that is, implies that uniformly in Also is countably additive on
Preliminaries
Given a measure space a distance can be constructed on the set of measurable sets with This is done by defining
where is the symmetric difference of the sets
This gives rise to a metric space by identifying two sets when Thus a point with representative is the set of all such that
Proposition: with the metric defined above is a complete metric space.
Proof: Let
Then
This means that the metric space can be identified with a subset of the Banach space .
Let , with
Then we can choose a sub-sequence such that exists almost everywhere and . It follows that for some (furthermore if and only if for large enough, then we have that the limit inferior of the sequence) and hence Therefore, is complete.
Proof of Vitali-Hahn-Saks theorem
Each defines a function on by taking . This function is well defined, this is it is independent on the representative of the class due to the absolute continuity of with respect to . Moreover is continuous.
For every the set
is closed in , and by the hypothesis we have that
By Baire category theorem at least one must contain a non-empty open set of . This means that there is and a such that
implies
On the other hand, any with can be represented as with and . This can be done, for example by taking and . Thus, if and then
Therefore, by the absolute continuity of with respect to , and since is arbitrary, we get that implies uniformly in In particular, implies
By the additivity of the limit it follows that is finitely-additive. Then, since it follows that is actually countably additive.
References
Theorems in measure theory
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https://en.wikipedia.org/wiki/Hahn%20embedding%20theorem
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In mathematics, especially in the area of abstract algebra dealing with ordered structures on abelian groups, the Hahn embedding theorem gives a simple description of all linearly ordered abelian groups. It is named after Hans Hahn.
Overview
The theorem states that every linearly ordered abelian group G can be embedded as an ordered subgroup of the additive group ℝΩ endowed with a lexicographical order, where ℝ is the additive group of real numbers (with its standard order), Ω is the set of Archimedean equivalence classes of G, and ℝΩ is the set of all functions from Ω to ℝ which vanish outside a well-ordered set.
Let 0 denote the identity element of G. For any nonzero element g of G, exactly one of the elements g or −g is greater than 0; denote this element by |g|. Two nonzero elements g and h of G are Archimedean equivalent if there exist natural numbers N and M such that N|g| > |h| and M|h| > |g|. Intuitively, this means that neither g nor h is "infinitesimal" with respect to the other. The group G is Archimedean if all nonzero elements are Archimedean-equivalent. In this case, Ω is a singleton, so ℝΩ is just the group of real numbers. Then Hahn's Embedding Theorem reduces to Hölder's theorem (which states that a linearly ordered abelian group is Archimedean if and only if it is a subgroup of the ordered additive group of the real numbers).
gives a clear statement and proof of the theorem. The papers of and together provide another proof. See also .
See also
Archimedean group
References
Ordered groups
Theorems in group theory
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https://en.wikipedia.org/wiki/AMPL
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AMPL (A Mathematical Programming Language) is an algebraic modeling language to describe and solve high-complexity problems for large-scale mathematical computing (i.e., large-scale optimization and scheduling-type problems).
It was developed by Robert Fourer, David Gay, and Brian Kernighan at Bell Laboratories.
AMPL supports dozens of solvers, both open source and commercial software, including CBC, CPLEX, FortMP, MOSEK, MINOS, IPOPT, SNOPT, KNITRO, and LGO. Problems are passed to solvers as nl files.
AMPL is used by more than 100 corporate clients, and by government agencies and academic institutions.
One advantage of AMPL is the similarity of its syntax to the mathematical notation of optimization problems. This allows for a very concise and readable definition of problems in the domain of optimization. Many modern solvers available on the NEOS Server (formerly hosted at the Argonne National Laboratory, currently hosted at the University of Wisconsin, Madison) accept AMPL input. According to the NEOS statistics AMPL is the most popular format for representing mathematical programming problems.
Features
AMPL features a mix of declarative and imperative programming styles. Formulating optimization models occurs via declarative language elements such as sets, scalar and multidimensional parameters, decision variables, objectives and constraints, which allow for concise description of most problems in the domain of mathematical optimization.
Procedures and control flow statements are available in AMPL for
the exchange of data with external data sources such as spreadsheets, databases, XML and text files
data pre- and post-processing tasks around optimization models
the construction of hybrid algorithms for problem types for which no direct efficient solvers are available.
To support re-use and simplify construction of large-scale optimization problems, AMPL allows separation of model and data.
AMPL supports a wide range of problem types, among them:
Linear programming
Quadratic programming
Nonlinear programming
Mixed-integer programming
Mixed-integer quadratic programming with or without convex quadratic constraints
Mixed-integer nonlinear programming
Second-order cone programming
Global optimization
Semidefinite programming problems with bilinear matrix inequalities
Complementarity theory problems (MPECs) in discrete or continuous variables
Constraint programming
AMPL invokes a solver in a separate process which has these advantages:
User can interrupt the solution process at any time
Solver errors do not affect the interpreter
32-bit version of AMPL can be used with a 64-bit solver and vice versa
Interaction with the solver is done through a well-defined nl interface.
Availability
AMPL is available for many popular 32 & 64-bit operating systems including Linux, macOS, Solaris, AIX, and Windows.
The translator is proprietary software maintained by AMPL Optimization LLC. However, several online services exist, providing fre
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https://en.wikipedia.org/wiki/List%20of%20homological%20algebra%20topics
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This is a list of homological algebra topics, by Wikipedia page.
Basic techniques
Cokernel
Exact sequence
Chain complex
Differential module
Five lemma
Short five lemma
Snake lemma
Nine lemma
Extension (algebra)
Central extension
Splitting lemma
Projective module
Injective module
Projective resolution
Injective resolution
Koszul complex
Exact functor
Derived functor
Ext functor
Tor functor
Filtration (abstract algebra)
Spectral sequence
Abelian category
Triangulated category
Derived category
Applications
Group cohomology
Galois cohomology
Lie algebra cohomology
Sheaf cohomology
Whitehead problem
Homological conjectures in commutative algebra
Homological algebra
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https://en.wikipedia.org/wiki/Beit%20Hanoun
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Beit Hanoun or Beit Hanun () is a city on the northeast edge of the Gaza Strip. According to the Palestinian Central Bureau of Statistics, the town had a population of 52,237 in 2017. It is administered by the Hamas administration. It is located by the Hanoun stream, just away from the Israeli town of Sderot.
History
The Ayyubids defeated the Crusaders at a battle in Umm al-Nasser hill, just west of Beit Hanoun in 1239, and built the Umm al-Naser Mosque ("Mother of Victories Mosque") there in commemoration of the victory. A Mamluk post office was located in Beit Hanoun as well.
Ottoman era
Incorporated into the Ottoman Empire in 1517 with all of Israel, Beit Hanoun appeared in the 1596 tax registers as being in the Nahiya of Gaza, part of Gaza Sanjak. It had a population of 36 Muslim households and paid a fixed tax rate of 33,3% on wheat, barley, summer crops, fruit trees, occasional revenues, goats and/ or beehives; a total of 9,300 akçe.
During the 17th and 18th centuries, the area of Beit Hanoun experienced a significant process of settlement decline due to nomadic pressures on local communities. The residents of abandoned villages moved to surviving settlements, but the land continued to be cultivated by neighboring villages. Beit Hanoun survived, and Pierre Jacotin named the village Deir Naroun on his map depicting Napoleon's Syrian campaign of 1799.
In 1838 Edward Robinson passed by, and described how "all were busy with the wheat harvest; the reapers were in the fields; donkeys and camels were moving homewards with their high loads of sheaves; while on the threshing-floors near the village I counted not less than thirty gangs of cattle.." He further noted it as a Muslim village, located in the Gaza district.
In May 1863, the French explorer Victor Guérin visited the village. Among the gardens he observed indications of ancient constructions in the shape of cut stones, fragments of columns, and bases. Socin found from an official Ottoman village list from about 1870 that Beit Hanoun had 94 houses and a population of 294, though the population count included men, only. Hartmann found that Bet Hanun had 95 houses.
In 1883 the PEF's Survey of Western Palestine described it as a small adobe village, "surrounded by gardens, with a well to the west. The ground is flat, and to the east is a pond beside the road."
British Mandate era
In the 1922 census of Palestine conducted by the British Mandate authorities, Beit Hanoun had a population of 885 inhabitants, all Muslim, decreasing in the 1931 census to 849, still all Muslims, in 194 houses.
In the 1945 statistics Beit Hanun had a population of 1,680 Muslims and 50 Jews, with 20,025 dunams of land, according to an official land and population survey. Of this, 2,768 dunams were for citrus and bananas, 697 were plantations and irrigable land, 13,186 used for cereals, while 59 dunams were built-up land.
Egyptian occupation
In the 1948 Arab–Israeli War, the vicinity of Bei
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https://en.wikipedia.org/wiki/Robert%20Lee%20Moore
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Robert Lee Moore (November 14, 1882 – October 4, 1974) was an American mathematician who taught for many years at the University of Texas. He is known for his work in general topology, for the Moore method of teaching university mathematics, and for his racist treatment of African-American mathematics students.
Life
Although Moore's father was reared in New England and was of New England ancestry, he fought in the American Civil War on the side of the Confederacy. After the war, he ran a hardware store in Dallas, then little more than a railway stop, and raised six children, of whom Robert, named after the commander of the Confederate Army of Northern Virginia, was the fifth.
Moore entered the University of Texas at the unusually youthful age of 15, in 1898, already knowing calculus thanks to self-study. He completed the B.Sc. in three years instead of the usual four; his teachers included G. B. Halsted and L. E. Dickson. After a year as a teaching fellow at Texas, he taught high school for a year in Marshall, Texas.
An assignment of Halsted's led Moore to prove that one of Hilbert's axioms for geometry was redundant. When E. H. Moore (no relation), who headed the Department of Mathematics at the University of Chicago, and whose research interests were on the foundations of geometry, heard of Robert's feat, he arranged for a scholarship that would allow Robert to study for a doctorate at Chicago. Oswald Veblen supervised Moore's 1905 thesis, titled Sets of Metrical Hypotheses for Geometry.
Moore then taught one year at the University of Tennessee, two years at Princeton University, and three years at Northwestern University. In 1910, he married Margaret MacLelland Key of Brenham, Texas; they had no children. In 1911, he took up a position at the University of Pennsylvania.
In 1920, Moore returned to the University of Texas at Austin as an associate professor and was promoted to full professor three years later. In 1951, he went on half pay but continued to teach five classes a year, including a section of freshman calculus, until the University authorities forced his definitive retirement in 1969, his 87th year.
A strong supporter of the American Mathematical Society, he presided over it, 1936–38. He edited its Colloquium Publications, 1929–33, and was the editor-in-chief, 1930–33. In 1931, he was elected to the National Academy of Sciences.
Topologist
According to the bibliography in Wilder (1976), Moore published 67 papers and one monograph, his 1932 Foundations of Point Set Theory. He is primarily remembered for his work on the foundations of topology, a topic he first touched on in his Ph.D. thesis. By the time Moore returned to the University of Texas, he had published 17 papers on point-set topology—a term he coined—including his 1915 paper "On a set of postulates which suffice to define a number-plane", giving an axiom system for plane topology.
The Moore plane, Moore's road space, Moore space, and the normal Moore space conjectu
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https://en.wikipedia.org/wiki/Herbrand%E2%80%93Ribet%20theorem
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In mathematics, the Herbrand–Ribet theorem is a result on the class group of certain number fields. It is a strengthening of Ernst Kummer's theorem to the effect that the prime p divides the class number of the cyclotomic field of p-th roots of unity if and only if p divides the numerator of the n-th Bernoulli number Bn
for some n, 0 < n < p − 1. The Herbrand–Ribet theorem specifies what, in particular, it means when p divides such an Bn.
Statement
The Galois group Δ of the cyclotomic field of pth roots of unity for an odd prime p, Q(ζ) with ζp = 1, consists of the p − 1 group elements σa, where . As a consequence of Fermat's little theorem, in the ring of p-adic integers we have p − 1 roots of unity, each of which is congruent mod p to some number in the range 1 to p − 1; we can therefore define a Dirichlet character ω (the Teichmüller character) with values in by requiring that for n relatively prime to p, ω(n) be congruent to n modulo p. The p part of the class group is a -module (since it is p-primary), hence a module over the group ring . We now define idempotent elements of the group ring for each n from 1 to p − 1, as
It is easy to see that and where is the Kronecker delta. This allows us to break up the p part of the ideal class group G of Q(ζ) by means of the idempotents; if G is the p-primary part of the ideal class group, then, letting Gn = εn(G), we have .
The Herbrand–Ribet theorem states that for odd n, Gn is nontrivial if and only if p divides the Bernoulli number Bp−n.
The theorem makes no assertion about even values of n, but there is no known p for which Gn is nontrivial for any even n: triviality for all p would be a consequence of Vandiver's conjecture.
Proofs
The part saying p divides Bp−n if Gn is not trivial is due to Jacques Herbrand. The converse, that if p divides Bp−n then Gn is not trivial is due to Kenneth Ribet, and is considerably more difficult. By class field theory, this can only be true if there is an unramified extension of the field of pth roots of unity by a cyclic extension of degree p which behaves in the specified way under the action of Σ; Ribet proves this by actually constructing such an extension using methods in the theory of modular forms. A more elementary proof of Ribet's converse to Herbrand's theorem, a consequence of the theory of Euler systems, can be found in Washington's book.
Generalizations
Ribet's methods were developed further by Barry Mazur and Andrew Wiles in order to prove the main conjecture of Iwasawa theory, a corollary of which is a strengthening of the Herbrand–Ribet theorem: the power of p dividing Bp−n is exactly the power of p dividing the order of Gn.
See also
Iwasawa theory
Notes
Cyclotomic fields
Theorems in algebraic number theory
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https://en.wikipedia.org/wiki/Beit%20Lahia
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Beit Lahia or Beit Lahiya () is a city in the Gaza Strip north of Jabalia, near Beit Hanoun and the 1949 Armistice Line with Israel. According to the Palestinian Central Bureau of Statistics, the city had a population of 89,838 in 2017. The political party Hamas is still administering the city, together with the entire Gaza Strip, after winning the 2005 municipal elections.
Geography
The word "Lahia" is Syriac and means "desert" or "fatigue". It is surrounded by sand dunes, some of which rise to above sea level. The area is renowned for its many large sycamore fig trees. The city is known for its fresh, sweet water, berries and citrus trees. According to Edward Henry Palmer, "Lahia" was from "Lahi", a personal name.
History
Beit Lahia has an ancient hill and nearby lay abandoned village ruins. It has been suggested that it was Bethelia, home town of Sozomen, where there was a temple. Ceramics from the Byzantine period have been found.
A mihrab, or mosque alcove indicating the direction of salaah (prayer), is all that remains of an ancient mosque to the west of Beit Lahia dating to the end of the Fatimid period and beginning of the Ayyubid Dynasty of Saladin, and two other mosques dating to the Ottoman period.
Yaqut al-Hamawi (d. 1229) described "Bait Lihya" as being located "near Ghazzah", and he further noted that "it is a village with many fruit-trees".
Mamluk period
A marble slab, deposited in the maqam of Salim Abu Musallam in Beit Lahia is inscribed in late Mamluk naskhi letters. It is an epitaph over four sons of the Governor of Gaza, Aqbay al-Ashrafi, who all died in the month of Rajab 897 (=29 April-9 May 1492 CE). It is assumed that the children died of the plague, described by Mujir al-Din, which ravaged Palestine in 1491–2.
Ottoman Empire
In 1517, the village was incorporated into the Ottoman Empire with the rest of Palestine, and in 1596, Beit Lahia appeared in Ottoman tax registers as being in nahiya (subdistrict) of Gaza under the Gaza Sanjak. It had a population of 70 Muslim households and paid a fix tax rate of 25% on various agricultural products, including wheat, barley, summer crops, vineyards, fruit trees, goats and/or beehives.
During the 17th and 18th centuries, the area of Beit Lahia experienced a significant process of settlement decline due to nomadic pressures on local communities. The residents of abandoned villages moved to surviving settlements, but the land continued to be cultivated by neighboring villages.
In 1838, Edward Robinson noted Beit Lehia as a Muslim village, located in the Gaza district.
In May 1863 Victor Guérin visited the village. He described it as "peopled by 250 inhabitants, it occupies an oblong valley, well cultivated, and surrounded by high sand-dunes, which cause a great heat. It is a little oasis, incessantly menaced by moving sand-hills, which surround it on every side, and would engulf it were it not for the continued struggle of man to arrest their progress". An Ottoman village l
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https://en.wikipedia.org/wiki/Bertrand%27s%20paradox
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There are three different paradoxes called Bertrand's paradox or the Bertrand paradox:
Bertrand paradox (probability)
Bertrand paradox (economics)
Bertrand's box paradox
Not to be confused with the famous paradox discovered by Bertrand Russell.
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https://en.wikipedia.org/wiki/Dirac%20operator
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In mathematics and quantum mechanics, a Dirac operator is a differential operator that is a formal square root, or half-iterate, of a second-order operator such as a Laplacian. The original case which concerned Paul Dirac was to factorise formally an operator for Minkowski space, to get a form of quantum theory compatible with special relativity; to get the relevant Laplacian as a product of first-order operators he introduced spinors. It was first published in 1928.
Formal definition
In general, let D be a first-order differential operator acting on a vector bundle V over a Riemannian manifold M. If
where ∆ is the Laplacian of V, then D is called a Dirac operator.
In high-energy physics, this requirement is often relaxed: only the second-order part of D2 must equal the Laplacian.
Examples
Example 1
D = −i ∂x is a Dirac operator on the tangent bundle over a line.
Example 2
Consider a simple bundle of notable importance in physics: the configuration space of a particle with spin confined to a plane, which is also the base manifold. It is represented by a wavefunction
where x and y are the usual coordinate functions on R2. χ specifies the probability amplitude for the particle to be in the spin-up state, and similarly for η. The so-called spin-Dirac operator can then be written
where σi are the Pauli matrices. Note that the anticommutation relations for the Pauli matrices make the proof of the above defining property trivial. Those relations define the notion of a Clifford algebra.
Solutions to the Dirac equation for spinor fields are often called harmonic spinors.
Example 3
Feynman's Dirac operator describes the propagation of a free fermion in three dimensions and is elegantly written
using the Feynman slash notation. In introductory textbooks to quantum field theory, this will appear in the form
where are the off-diagonal Dirac matrices , with and the remaining constants are the speed of light, being Planck's constant, and the mass of a fermion (for example, an electron). It acts on a four-component wave function , the Sobolev space of smooth, square-integrable functions. It can be extended to a self-adjoint operator on that domain. The square, in this case, is not the Laplacian, but instead (after setting )
Example 4
Another Dirac operator arises in Clifford analysis. In euclidean n-space this is
where {ej: j = 1, ..., n} is an orthonormal basis for euclidean n-space, and Rn is considered to be embedded in a Clifford algebra.
This is a special case of the Atiyah–Singer–Dirac operator acting on sections of a spinor bundle.
Example 5
For a spin manifold, M, the Atiyah–Singer–Dirac operator is locally defined as follows: For and e1(x), ..., ej(x) a local orthonormal basis for the tangent space of M at x, the Atiyah–Singer–Dirac operator is
where is the spin connection, a lifting of the Levi-Civita connection on M to the spinor bundle over M. The square in this case is not the Laplacian, but instead where is
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https://en.wikipedia.org/wiki/Adaptive%20Multi-Rate%20Wideband
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Adaptive Multi-Rate Wideband (AMR-WB) is a patented wideband speech audio coding standard developed based on Adaptive Multi-Rate encoding, using a similar methodology to algebraic code-excited linear prediction (ACELP). AMR-WB provides improved speech quality due to a wider speech bandwidth of 50–7000 Hz compared to narrowband speech coders which in general are optimized for POTS wireline quality of 300–3400 Hz. AMR-WB was developed by Nokia and VoiceAge and it was first specified by 3GPP.
AMR-WB is codified as G.722.2, an ITU-T standard speech codec, formally known as Wideband coding of speech at around 16 kbit/s using Adaptive Multi-Rate Wideband (AMR-WB). G.722.2 AMR-WB is the same codec as the 3GPP AMR-WB. The corresponding 3GPP specifications are TS 26.190 for the speech codec and TS 26.194 for the Voice Activity Detector.
The AMR-WB format has the following parameters:
Frequency bands processed: 50–6400 Hz (all modes) plus 6400–7000 Hz (23.85 kbit/s mode only)
Delay frame size: 20 ms
Look ahead: 5 ms
AMR-WB codec employs a bandsplitting filter; the one-way delay of this filter is 0.9375 ms
Complexity: 38 WMOPS, RAM 5.3 kilowords
Voice activity detection, discontinuous transmission, comfort noise generator
Fixed point: bit-exact C code
Floating point: under work
A common file extension for the AMR-WB file format is .awb. There also exists another storage format for AMR-WB that is suitable for applications with more advanced demands on the storage format, like random access or synchronization with video. This format is the 3GPP-specified 3GP container format, based on the ISO base media file format. 3GP also allows use of AMR-WB bit streams for stereo sound.
AMR modes
AMR-WB operates, like AMR, with nine different bit rates. The lowest bit rate providing excellent speech quality in a clean environment is 12.65 kbit/s. Higher bit rates are useful in background noise conditions and for music. Also, lower bit rates of 6.60 and 8.85 kbit/s provide reasonable quality, especially when compared to narrow-band codecs.
The frequencies from 6.4 kHz to 7 kHz are only transmitted in the highest bitrate mode (23.85 kbit/s), while in the rest of the modes the decoder generates sounds by using the lower frequency data (75–6400 Hz) along with random noise (in order to simulate the high frequency band).
All modes are sampled at 16 kHz (using 14-bit resolution) and processed at 12.8 kHz.
The bit rates are the following:
Mandatory multi-rate configuration
6.60 kbit/s (used for circuit switched GSM and UMTS connections; should only be used temporarily during bad radio connections and is not considered wideband speech)
8.85 kbit/s (used for circuit switched GSM and UMTS connections; should only be used temporarily during bad radio connections and is not considered wideband speech; provides quality equal to G.722 at 48 kbit/s for clean speech)
12.65 kbit/s (main anchor bitrate; used for circuit switched GSM and UMTS connections; offers superior
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https://en.wikipedia.org/wiki/Algebraic%20code-excited%20linear%20prediction
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Algebraic code-excited linear prediction (ACELP) is a speech coding algorithm in which a limited set of pulses is distributed as excitation to a linear prediction filter. It is a linear predictive coding (LPC) algorithm that is based on the code-excited linear prediction (CELP) method and has an algebraic structure. ACELP was developed in 1989 by the researchers at the Université de Sherbrooke in Canada.
The ACELP method is widely employed in current speech coding standards such as AMR, EFR, AMR-WB (G.722.2), VMR-WB, EVRC, EVRC-B, SMV, TETRA, PCS 1900, MPEG-4 CELP and ITU-T G-series standards G.729, G.729.1 (first coding stage) and G.723.1. The ACELP algorithm is also used in the proprietary ACELP.net codec. Audible Inc. use a modified version for their speaking books. It is also used in conference-calling software, speech compression tools and has become one of the 3GPP formats.
The ACELP patent expired in 2018 and is now royalty-free.
Features
The main advantage of ACELP is that the algebraic codebook it uses can be made very large (> 50 bits) without running into storage (RAM/ROM) or complexity (CPU time) problems.
Technology
The ACELP algorithm is based on that used in code-excited linear prediction (CELP), but ACELP codebooks have a specific algebraic structure imposed upon them.
A 16-bit algebraic codebook shall be used in the innovative codebook search, the aim of which is to find the best innovation and gain parameters. The innovation vector contains, at most, four non-zero pulses.
In ACELP, a block of N speech samples is synthesized by filtering an appropriate innovation sequence from a codebook, scaled by a gain factor g c, through two time-varying filters.
The long-term (pitch) synthesis filter is given by:
The short-term synthesis filter is given by:
References
Speech codecs
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https://en.wikipedia.org/wiki/Chisini%20mean
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In mathematics, a function f of n variables x1, ..., xn leads to a Chisini mean M if, for every vector ⟨x1, ..., xn⟩, there exists a unique M such that
f(M,M, ..., M) = f(x1,x2, ..., xn).
The arithmetic, harmonic, geometric, generalised, Heronian and quadratic means are all Chisini means, as are their weighted variants.
While Oscar Chisini was arguably the first to deal with "substitution means" in some depth in 1929, the idea of defining a mean as above is quite old, appearing (for example) in early works of Augustus De Morgan.
See also
Fréchet mean
Generalized mean
Jensen's inequality
Quasi-arithmetic mean
Stolarsky mean
References
Mathematical analysis
Means
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https://en.wikipedia.org/wiki/Complex%20polygon
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The term complex polygon can mean two different things:
In geometry, a polygon in the unitary plane, which has two complex dimensions.
In computer graphics, a polygon whose boundary is not simple.
Geometry
In geometry, a complex polygon is a polygon in the complex Hilbert plane, which has two complex dimensions.
A complex number may be represented in the form , where and are real numbers, and is the square root of . Multiples of such as are called imaginary numbers. A complex number lies in a complex plane having one real and one imaginary dimension, which may be represented as an Argand diagram. So a single complex dimension comprises two spatial dimensions, but of different kinds - one real and the other imaginary.
The unitary plane comprises two such complex planes, which are orthogonal to each other. Thus it has two real dimensions and two imaginary dimensions.
A complex polygon is a (complex) two-dimensional (i.e. four spatial dimensions) analogue of a real polygon. As such it is an example of the more general complex polytope in any number of complex dimensions.
In a real plane, a visible figure can be constructed as the real conjugate of some complex polygon.
Computer graphics
In computer graphics, a complex polygon is a polygon which has a boundary comprising discrete circuits, such as a polygon with a hole in it.
Self-intersecting polygons are also sometimes included among the complex polygons. Vertices are only counted at the ends of edges, not where edges intersect in space.
A formula relating an integral over a bounded region to a closed line integral may still apply when the "inside-out" parts of the region are counted negatively.
Moving around the polygon, the total amount one "turns" at the vertices can be any integer times 360°, e.g. 720° for a pentagram and 0° for an angular "eight".
See also
Regular polygon
Convex hull
Nonzero-rule
List of self-intersecting polygons
References
Citations
Bibliography
Coxeter, H. S. M., Regular Complex Polytopes, Cambridge University Press, 1974.
External links
Introduction to Polygons
Types of polygons
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https://en.wikipedia.org/wiki/Double%20exponential%20distribution
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In statistics, the double exponential distribution may refer to
Laplace distribution, or bilateral exponential distribution, consisting of two exponential distributions glued together on each side of a threshold
Gumbel distribution, the cumulative distribution function of which is an iterated exponential function (the exponential of an exponential function).
Continuous distributions
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https://en.wikipedia.org/wiki/Kronecker%E2%80%93Weber%20theorem
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In algebraic number theory, it can be shown that every cyclotomic field is an abelian extension of the rational number field Q, having Galois group of the form . The Kronecker–Weber theorem provides a partial converse: every finite abelian extension of Q is contained within some cyclotomic field. In other words, every algebraic integer whose Galois group is abelian can be expressed as a sum of roots of unity with rational coefficients. For example,
and
The theorem is named after Leopold Kronecker and Heinrich Martin Weber.
Field-theoretic formulation
The Kronecker–Weber theorem can be stated in terms of fields and field extensions.
Precisely, the Kronecker–Weber theorem states: every finite abelian extension of the rational numbers Q is a subfield of a cyclotomic field.
That is, whenever an algebraic number field has a Galois group over Q that is an abelian group, the field is a subfield of a field obtained by adjoining a root of unity to the rational numbers.
For a given abelian extension K of Q there is a minimal cyclotomic field that contains it. The theorem allows one to define the conductor of K as the smallest integer n such that K lies inside the field generated by the n-th roots of unity. For example the quadratic fields have as conductor the absolute value of their discriminant, a fact generalised in class field theory.
History
The theorem was first stated by though his argument was not complete for extensions of degree a power of 2.
published a proof, but this had some gaps and errors that were pointed out and corrected by . The first complete proof was given by .
Generalizations
proved the local Kronecker–Weber theorem which states that any abelian extension of a local field can be constructed using cyclotomic extensions and Lubin–Tate extensions. , and gave other proofs.
Hilbert's twelfth problem asks for generalizations of the Kronecker–Weber theorem to base fields other than the rational numbers, and asks for the analogues of the roots of unity for those fields. A different approach to abelian extensions is given by class field theory.
References
External links
Class field theory
Cyclotomic fields
Theorems in algebraic number theory
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https://en.wikipedia.org/wiki/Topology%20optimization
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Topology optimization (TO) is a mathematical method that optimizes material layout within a given design space, for a given set of loads, boundary conditions and constraints with the goal of maximizing the performance of the system. Topology optimization is different from shape optimization and sizing optimization in the sense that the design can attain any shape within the design space, instead of dealing with predefined configurations.
The conventional topology optimization formulation uses a finite element method (FEM) to evaluate the design performance. The design is optimized using either gradient-based mathematical programming techniques such as the optimality criteria algorithm and the method of moving asymptotes or non gradient-based algorithms such as genetic algorithms.
Topology optimization has a wide range of applications in aerospace, mechanical, bio-chemical and civil engineering. Currently, engineers mostly use topology optimization at the concept level of a design process. Due to the free forms that naturally occur, the result is often difficult to manufacture. For that reason the result emerging from topology optimization is often fine-tuned for manufacturability. Adding constraints to the formulation in order to increase the manufacturability is an active field of research. In some cases results from topology optimization can be directly manufactured using additive manufacturing; topology optimization is thus a key part of design for additive manufacturing.
Problem statement
A topology optimization problem can be written in the general form of an optimization problem as:
The problem statement includes the following:
An objective function . This function represents the quantity that is being minimized for best performance. The most common objective function is compliance, where minimizing compliance leads to maximizing the stiffness of a structure.
The material distribution as a problem variable. This is described by the density of the material at each location . Material is either present, indicated by a 1, or absent, indicated by a 0. is a state field that satisfies a linear or nonlinear state equation depending on .
The design space . This indicates the allowable volume within which the design can exist. Assembly and packaging requirements, human and tool accessibility are some of the factors that need to be considered in identifying this space . With the definition of the design space, regions or components in the model that cannot be modified during the course of the optimization are considered as non-design regions.
constraints a characteristic that the solution must satisfy. Examples are the maximum amount of material to be distributed (volume constraint) or maximum stress values.
Evaluating often includes solving a differential equation. This is most commonly done using the finite element method since these equations do not have a known analytical solution.
Implementation methodologies
There are vari
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https://en.wikipedia.org/wiki/Quadrilateral%20%28disambiguation%29
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A quadrilateral, in geometry, is a polygon with 4 sides.
Quadrilateral may also refer to:
Complete quadrilateral, in projective geometry, a configuration with 4 lines and 6 points
Chicago-Lambeth Quadrilateral, a four-point statement of fundamental doctrine, in the Anglican Communion
Wesleyan Quadrilateral, the four sources of doctrine in the Methodist Church
Golden Quadrilateral, a network of highways in India
Quadrilateral Security Dialogue, a strategic alliance of the United States, Japan, Australia and India within Asia.
Quadrilateral Treaty, a pact between the Argentine provinces of Buenos Aires, Santa Fe, Entre Ríos and Corrientes, signed on 25 January 1822.
Quadrilatero, in the Revolutions of 1848, in the Italian states - an area within the group of fortresses at Mantua, Verona, Peschiera and Legnago
Quadrilateral (horse), thoroughbred racehorse
"Quadrilateral", an alternative name for Southern Dobruja mostly used by Romanians ()
See also
Quadriliteral
Quadrangle (geography)
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https://en.wikipedia.org/wiki/Martin%20David%20Kruskal
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Martin David Kruskal (; September 28, 1925 – December 26, 2006) was an American mathematician and physicist. He made fundamental contributions in many areas of mathematics and science, ranging from plasma physics to general relativity and from nonlinear analysis to asymptotic analysis. His most celebrated contribution was in the theory of solitons.
He was a student at the University of Chicago and at New York University, where he completed his Ph.D. under Richard Courant in 1952. He spent much of his career at Princeton University, as a research scientist at the Plasma Physics Laboratory starting in 1951, and then as a professor of astronomy (1961), founder and chair of the Program in Applied and Computational Mathematics (1968), and professor of mathematics (1979). He retired from Princeton University in 1989 and joined the mathematics department of Rutgers University, holding the David Hilbert Chair of Mathematics.
Apart from serious mathematical work, Kruskal was known for mathematical diversions. For example, he invented the Kruskal Count, a magical effect that has been known to perplex professional magicians because it was based not on sleight of hand but on a mathematical phenomenon.
Personal life
Martin David Kruskal was born to a Jewish family in New York City and grew up in New Rochelle. He was generally known as Martin to the world and David to his family. His father, Joseph B. Kruskal Sr., was a successful fur wholesaler. His mother, Lillian Rose Vorhaus Kruskal Oppenheimer, became a noted promoter of the art of origami during the early era of television and founded the Origami Center of America in New York City, which later became OrigamiUSA. He was one of five children. His two brothers, both eminent mathematicians, were Joseph Kruskal (1928–2010; discoverer of multidimensional scaling, the Kruskal tree theorem, and Kruskal's algorithm) and William Kruskal (1919–2005; discoverer of the Kruskal–Wallis test).
Martin Kruskal's wife, Laura Kruskal, was a lecturer and writer about origami and originator of many new models. They were married for 56 years. Martin Kruskal also invented several origami models including an envelope for sending secret messages. The envelope could be easily unfolded, but it could not then be easily refolded to conceal the deed. Their three children are Karen (an attorney), Kerry (an author of children's books), and Clyde, a computer scientist.
Research
Martin Kruskal's scientific interests covered a wide range of topics in pure mathematics and applications of mathematics to the sciences. He had lifelong interests in many topics in partial differential equations and nonlinear analysis and developed fundamental ideas about asymptotic expansions, adiabatic invariants, and numerous related topics.
His Ph.D. dissertation, written under the direction of Richard Courant and Bernard Friedman at New York University, was on the topic "The Bridge Theorem For Minimal Surfaces". He received his Ph.D. in 1952.
In the
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https://en.wikipedia.org/wiki/Von%20Staudt%E2%80%93Clausen%20theorem
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In number theory, the von Staudt–Clausen theorem is a result determining the fractional part of Bernoulli numbers, found independently by
and .
Specifically, if n is a positive integer and we add 1/p to the Bernoulli number B2n for every prime p such that p − 1 divides 2n, we obtain an integer, i.e.,
This fact immediately allows us to characterize the denominators of the non-zero Bernoulli numbers B2n as the product of all primes p such that p − 1 divides 2n; consequently the denominators are square-free and divisible by 6.
These denominators are
6, 30, 42, 30, 66, 2730, 6, 510, 798, 330, 138, 2730, 6, 870, 14322, 510, 6, 1919190, 6, 13530, ... .
The sequence of integers is
1, 1, 1, 1, 1, 1, 2, -6, 56, -528, 6193, -86579, 1425518, -27298230, ... .
Proof
A proof of the Von Staudt–Clausen theorem follows from an explicit formula for Bernoulli numbers which is:
and as a corollary:
where are the Stirling numbers of the second kind.
Furthermore the following lemmas are needed:
Let p be a prime number then,
1. If p-1 divides 2n then,
2. If p-1 does not divide 2n then,
Proof of (1) and (2): One has from Fermat's little theorem,
for .
If p-1 divides 2n then one has,
for .
Thereafter one has,
from which (1) follows immediately.
If p-1 does not divide 2n then after Fermat's theorem one has,
If one lets (Greatest integer function) then after iteration one has,
for and .
Thereafter one has,
Lemma (2) now follows from the above and the fact that S(n,j)=0 for j>n.
(3). It is easy to deduce that for a>2 and b>2, ab divides (ab-1)!.
(4). Stirling numbers of second kind are integers.
Proof of the theorem: Now we are ready to prove Von-Staudt Clausen theorem,
If j+1 is composite and j>3 then from (3), j+1 divides j!.
For j=3,
If j+1 is prime then we use (1) and (2) and if j+1 is composite then we use (3) and (4) to deduce:
where is an integer, which is the Von-Staudt Clausen theorem.
See also
Kummer's congruence
References
External links
Theorems in number theory
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https://en.wikipedia.org/wiki/Agoh%E2%80%93Giuga%20conjecture
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In number theory the Agoh–Giuga conjecture on the Bernoulli numbers Bk postulates that p is a prime number if and only if
It is named after Takashi Agoh and Giuseppe Giuga.
Equivalent formulation
The conjecture as stated above is due to Takashi Agoh (1990); an equivalent formulation is due to Giuseppe Giuga, from 1950, to the effect that p is prime if and only if
which may also be written as
It is trivial to show that p being prime is sufficient for the second equivalence to hold, since if p is prime, Fermat's little theorem states that
for , and the equivalence follows, since
Status
The statement is still a conjecture since it has not yet been proven that if a number n is not prime (that is, n is composite), then the formula does not hold. It has been shown that a composite number n satisfies the formula if and only if it is both a Carmichael number and a Giuga number, and that if such a number exists, it has at least 13,800 digits (Borwein, Borwein, Borwein, Girgensohn 1996). Laerte Sorini, finally, in a work of 2001 showed that a possible counterexample should be a number n greater than 1036067 which represents the limit suggested by Bedocchi for the demonstration technique specified by Giuga to his own conjecture.
Relation to Wilson's theorem
The Agoh–Giuga conjecture bears a similarity to Wilson's theorem, which has been proven to be true. Wilson's theorem states that a number p is prime if and only if
which may also be written as
For an odd prime p we have
and for p=2 we have
So, the truth of the Agoh–Giuga conjecture combined with Wilson's theorem would give: a number p is prime if and only if
and
References
Conjectures about prime numbers
Unsolved problems in number theory
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https://en.wikipedia.org/wiki/Bateman%E2%80%93Horn%20conjecture
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In number theory, the Bateman–Horn conjecture is a statement concerning the frequency of prime numbers among the values of a system of polynomials, named after mathematicians Paul T. Bateman and Roger A. Horn who proposed it in 1962. It provides a vast generalization of such conjectures as the Hardy and Littlewood conjecture on the density of twin primes or their conjecture on primes of the form n2 + 1; it is also a strengthening of Schinzel's hypothesis H.
Definition
The Bateman–Horn conjecture provides a conjectured density for the positive integers at which a given set of polynomials all have prime values. For a set of m distinct irreducible polynomials ƒ1, ..., ƒm with integer coefficients, an obvious necessary condition for the polynomials to simultaneously generate prime values infinitely often is that they satisfy Bunyakovsky's property, that there does not exist a prime number p that divides their product f(n) for every positive integer n. For, if there were such a prime p, having all values of the polynomials simultaneously prime for a given n would imply that at least one of them must be equal to p, which can only happen for finitely many values of n or there would be a polynomial with infinitely many roots, whereas the conjecture is how to give conditions where the values are simultaneously prime for infinitely many n.
An integer n is prime-generating for the given system of polynomials if every polynomial ƒi(n) produces a prime number when given n as its argument. If P(x) is the number of prime-generating integers among the positive integers less than x, then the Bateman–Horn conjecture states that
where D is the product of the degrees of the polynomials and where C is the product over primes p
with the number of solutions to
Bunyakovsky's property implies for all primes p,
so each factor in the infinite product C is positive.
Intuitively one then naturally expects that the constant C is itself positive, and with some work this can be proved.
(Work is needed since some infinite products of positive numbers equal zero.)
Negative numbers
As stated above, the conjecture is not true: the single polynomial ƒ1(x) = −x produces only negative numbers when given a positive argument, so the fraction of prime numbers among its values is always zero. There are two equally valid ways of refining the conjecture to avoid this difficulty:
One may require all the polynomials to have positive leading coefficients, so that only a constant number of their values can be negative.
Alternatively, one may allow negative leading coefficients but count a negative number as being prime when its absolute value is prime.
It is reasonable to allow negative numbers to count as primes as a step towards formulating more general conjectures that apply to other systems of numbers than the integers, but at the same time it is easy
to just negate the polynomials if necessary to reduce to the case where the leading coefficients are positive.
Examples
If the sy
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https://en.wikipedia.org/wiki/Isogeny
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In mathematics, particularly in algebraic geometry, an isogeny is a morphism of algebraic groups (also known as group varieties) that is surjective and has a finite kernel.
If the groups are abelian varieties, then any morphism of the underlying algebraic varieties which is surjective with finite fibres is automatically an isogeny, provided that . Such an isogeny then provides a group homomorphism between the groups of -valued points of and , for any field over which is defined.
The terms "isogeny" and "isogenous" come from the Greek word ισογενη-ς, meaning "equal in kind or nature". The term "isogeny" was introduced by Weil; before this, the term "isomorphism" was somewhat confusingly used for what is now called an isogeny.
Case of abelian varieties
For abelian varieties, such as elliptic curves, this notion can also be formulated as follows:
Let E1 and E2 be abelian varieties of the same dimension over a field k. An isogeny between E1 and E2 is a dense morphism of varieties that preserves basepoints (i.e. f maps the identity point on E1 to that on E2).
This is equivalent to the above notion, as every dense morphism between two abelian varieties of the same dimension is automatically surjective with finite fibres, and if it preserves identities then it is a homomorphism of groups.
Two abelian varieties E1 and E2 are called isogenous if there is an isogeny . This can be shown to be an equivalence relation; in the case of elliptic curves, symmetry is due to the existence of the dual isogeny. As above, every isogeny induces homomorphisms of the groups of the k-valued points of the abelian varieties.
See also
Abelian varieties up to isogeny
Selmer group
References
Morphisms of schemes
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https://en.wikipedia.org/wiki/Critical%20value
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Critical value may refer to:
In differential topology, a critical value of a differentiable function between differentiable manifolds is the image (value of) ƒ(x) in N of a critical point x in M.
In statistical hypothesis testing, the critical values of a statistical test are the boundaries of the acceptance region of the test. The acceptance region is the set of values of the test statistic for which the null hypothesis is not rejected. Depending on the shape of the acceptance region, there can be one or more than one critical value.
In complex dynamics, a critical value is the image of a critical point.
In medicine, a critical value or panic value is a value of a laboratory test that indicates a serious risk to the patient. Laboratory staff may be required to directly notify a physician or clinical staff of these values.
References
Multivariable calculus
Differential topology
Critical phenomena
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https://en.wikipedia.org/wiki/Taut%20submanifold
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In mathematics, a (compact) taut submanifold N of a space form M is a compact submanifold with the property that for every the distance function
is a perfect Morse function.
If N is not compact, one needs to consider the restriction of the to any of their sublevel sets.
References
Differential geometry
Morse theory
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https://en.wikipedia.org/wiki/Space%20form
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In mathematics, a space form is a complete Riemannian manifold M of constant sectional curvature K. The three most fundamental examples are Euclidean n-space, the n-dimensional sphere, and hyperbolic space, although a space form need not be simply connected.
Reduction to generalized crystallography
The Killing–Hopf theorem of Riemannian geometry states that the universal cover of an n-dimensional space form with curvature is isometric to , hyperbolic space, with curvature is isometric to , Euclidean n-space, and with curvature is isometric to , the n-dimensional sphere of points distance 1 from the origin in .
By rescaling the Riemannian metric on , we may create a space of constant curvature for any . Similarly, by rescaling the Riemannian metric on , we may create a space of constant curvature for any . Thus the universal cover of a space form with constant curvature is isometric to .
This reduces the problem of studying space forms to studying discrete groups of isometries of which act properly discontinuously. Note that the fundamental group of , , will be isomorphic to . Groups acting in this manner on are called crystallographic groups. Groups acting in this manner on and are called Fuchsian groups and Kleinian groups, respectively.
See also
Borel conjecture
References
Riemannian geometry
Conjectures
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https://en.wikipedia.org/wiki/Bias%20%28disambiguation%29
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Bias is an inclination toward something, or a predisposition, partiality, prejudice, preference, or predilection.
Bias may also refer to:
Scientific method and statistics
The bias introduced into an experiment through a confounder
Algorithmic bias, machine learning algorithms that exhibit politically unacceptable behavior
Cultural bias, interpreting and judging phenomena in terms particular to one's own culture
Funding bias, bias relative to the commercial interests of a study's financial sponsor
Infrastructure bias, the influence of existing social or scientific infrastructure on scientific observations
Publication bias, bias toward publication of certain experimental results
Bias (statistics), the systematic distortion of a statistic
Biased sample, a sample falsely taken to be typical of a population
Estimator bias, a bias from an estimator whose expectation differs from the true value of the parameter
Personal equation, a concept in 19th- and early 20th-century science that each observer had an inherent bias when it came to measurements and observations
Reporting bias, a bias resulting from what is and is not reported in research, either by participants in the research or by the researcher.
Cognitive science
Cognitive bias, any of a wide range of effects identified in cognitive science.
Confirmation bias, tendency of people to favor information that confirm their beliefs of hypothesis
See List of cognitive biases for a comprehensive list
Mathematics and engineering
Exponent bias, the constant offset of an exponent's value
Inductive bias, the set of assumptions that a machine learner uses to predict outputs of given inputs that it has not encountered.
Seat bias, any bias in a method of apportionment that favors either large or small parties over the other
Electricity
Biasing, a voltage or current added to an electronic device to move its operating point to a desired part of its transfer function
Grid bias of a vacuum tube, used to control the electron flow from the heated cathode to the positively charged anode
Tape bias (also AC bias), a high-frequency signal (generally from 40 to 150 kHz) added to the audio signal recorded on an analog tape recorder
Places
Bias, Landes, on the coast in southwestern France
Bias, Lot-et-Garonne, in southwestern France
Bias, West Virginia, a community in the United States
Bias Bay, now called Daya Bay, in Guangdong Province, China
Bias River, a river in north-western India
People
Bias (mythology), multiple figures in Greek mythology
Bias Brahmin, a Brahmin community found in India
Bias of Priene, one of the Seven Sages of Greece
Bias, a Spartan commander caught in an ambush by the Athenian general Iphicrates
Fanny Bias (1789–1825), French dancer, one of the first who raised on pointes
Len Bias (1963–1986), American basketball player
Oliver Bias (born 2001), footballer
Tiffany Bias (born 1992), Thai basketball player
Organisations
BIAS (Berkley Integrated Audio S
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https://en.wikipedia.org/wiki/Projectionless%20C%2A-algebra
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In mathematics, a projectionless C*-algebra is a C*-algebra with no nontrivial projections. For a unital C*-algebra, the projections 0 and 1 are trivial. While for a non-unital C*-algebra, only 0 is considered trivial. The problem of whether simple infinite-dimensional C*-algebras with this property exist was posed in 1958 by Irving Kaplansky, and the first example of one was published in 1981 by Bruce Blackadar. For commutative C*-algebras, being projectionless is equivalent to its spectrum being connected. Due to this, being projectionless can be considered as a noncommutative analogue of a connected space.
Examples
C, the algebra of complex numbers.
The reduced group C*-algebra of the free group on finitely many generators.
The Jiang-Su algebra is simple, projectionless, and KK-equivalent to C.
Dimension drop algebras
Let be the class consisting of the C*-algebras for each , and let be the class of all C*-algebras of the form
,
where are integers, and where belong to .
Every C*-algebra A in is projectionless, moreover, its only projection is 0.
References
C*-algebras
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https://en.wikipedia.org/wiki/Frobenius%20normal%20form
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In linear algebra, the Frobenius normal form or rational canonical form of a square matrix A with entries in a field F is a canonical form for matrices obtained by conjugation by invertible matrices over F. The form reflects a minimal decomposition of the vector space into subspaces that are cyclic for A (i.e., spanned by some vector and its repeated images under A). Since only one normal form can be reached from a given matrix (whence the "canonical"), a matrix B is similar to A if and only if it has the same rational canonical form as A. Since this form can be found without any operations that might change when extending the field F (whence the "rational"), notably without factoring polynomials, this shows that whether two matrices are similar does not change upon field extensions. The form is named after German mathematician Ferdinand Georg Frobenius.
Some authors use the term rational canonical form for a somewhat different form that is more properly called the primary rational canonical form. Instead of decomposing into a minimum number of cyclic subspaces, the primary form decomposes into a maximum number of cyclic subspaces. It is also defined over F, but has somewhat different properties: finding the form requires factorization of polynomials, and as a consequence the primary rational canonical form may change when the same matrix is considered over an extension field of F. This article mainly deals with the form that does not require factorization, and explicitly mentions "primary" when the form using factorization is meant.
Motivation
When trying to find out whether two square matrices A and B are similar, one approach is to try, for each of them, to decompose the vector space as far as possible into a direct sum of stable subspaces, and compare the respective actions on these subspaces. For instance if both are diagonalizable, then one can take the decomposition into eigenspaces (for which the action is as simple as it can get, namely by a scalar), and then similarity can be decided by comparing eigenvalues and their multiplicities. While in practice this is often a quite insightful approach, there are various drawbacks this has as a general method. First, it requires finding all eigenvalues, say as roots of the characteristic polynomial, but it may not be possible to give an explicit expression for them. Second, a complete set of eigenvalues might exist only in an extension of the field one is working over, and then one does not get a proof of similarity over the original field. Finally A and B might not be diagonalizable even over this larger field, in which case one must instead use a decomposition into generalized eigenspaces, and possibly into Jordan blocks.
But obtaining such a fine decomposition is not necessary to just decide whether two matrices are similar. The rational canonical form is based on instead using a direct sum decomposition into stable subspaces that are as large as possible, while still allowing a very si
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https://en.wikipedia.org/wiki/Frobenius%20endomorphism
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In commutative algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with prime characteristic , an important class which includes finite fields. The endomorphism maps every element to its -th power. In certain contexts it is an automorphism, but this is not true in general.
Definition
Let be a commutative ring with prime characteristic (an integral domain of positive characteristic always has prime characteristic, for example). The Frobenius endomorphism F is defined by
for all r in R. It respects the multiplication of R:
and is 1 as well. Moreover, it also respects the addition of . The expression can be expanded using the binomial theorem. Because is prime, it divides but not any for ; it therefore will divide the numerator, but not the denominator, of the explicit formula of the binomial coefficients
if . Therefore, the coefficients of all the terms except and are divisible by , and hence they vanish. Thus
This shows that F is a ring homomorphism.
If is a homomorphism of rings of characteristic , then
If and are the Frobenius endomorphisms of and , then this can be rewritten as:
This means that the Frobenius endomorphism is a natural transformation from the identity functor on the category of characteristic rings to itself.
If the ring is a ring with no nilpotent elements, then the Frobenius endomorphism is injective: means , which by definition means that is nilpotent of order at most . In fact, this is necessary and sufficient, because if is any nilpotent, then one of its powers will be nilpotent of order at most . In particular, if is a field then the Frobenius endomorphism is injective.
The Frobenius morphism is not necessarily surjective, even when is a field. For example, let be the finite field of elements together with a single transcendental element; equivalently, is the field of rational functions with coefficients in . Then the image of does not contain . If it did, then there would be a rational function whose -th power would equal . But the degree of this -th power is , which is a multiple of . In particular, it can't be 1, which is the degree of . This is a contradiction; so is not in the image of .
A field is called perfect if either it is of characteristic zero or it is of positive characteristic and its Frobenius endomorphism is an automorphism. For example, all finite fields are perfect.
Fixed points of the Frobenius endomorphism
Consider the finite field . By Fermat's little theorem, every element of satisfies . Equivalently, it is a root of the polynomial . The elements of therefore determine roots of this equation, and because this equation has degree it has no more than roots over any extension. In particular, if is an algebraic extension of (such as the algebraic closure or another finite field), then is the fixed field of the Frobenius automorphism of .
Let be a ring of char
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https://en.wikipedia.org/wiki/Radical%20of%20an%20algebraic%20group
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The radical of an algebraic group is the identity component of its maximal normal solvable subgroup.
For example, the radical of the general linear group (for a field K) is the subgroup consisting of scalar matrices, i.e. matrices with and for .
An algebraic group is called semisimple if its radical is trivial, i.e., consists of the identity element only. The group is semi-simple, for example.
The subgroup of unipotent elements in the radical is called the unipotent radical, it serves to define reductive groups.
See also
Reductive group
Unipotent group
References
"Radical of a group", Encyclopaedia of Mathematics
Algebraic groups
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https://en.wikipedia.org/wiki/Reductive%20group
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In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group G over a perfect field is reductive if it has a representation that has a finite kernel and is a direct sum of irreducible representations. Reductive groups include some of the most important groups in mathematics, such as the general linear group GL(n) of invertible matrices, the special orthogonal group SO(n), and the symplectic group Sp(2n). Simple algebraic groups and (more generally) semisimple algebraic groups are reductive.
Claude Chevalley showed that the classification of reductive groups is the same over any algebraically closed field. In particular, the simple algebraic groups are classified by Dynkin diagrams, as in the theory of compact Lie groups or complex semisimple Lie algebras. Reductive groups over an arbitrary field are harder to classify, but for many fields such as the real numbers R or a number field, the classification is well understood. The classification of finite simple groups says that most finite simple groups arise as the group G(k) of k-rational points of a simple algebraic group G over a finite field k, or as minor variants of that construction.
Reductive groups have a rich representation theory in various contexts. First, one can study the representations of a reductive group G over a field k as an algebraic group, which are actions of G on k-vector spaces. But also, one can study the complex representations of the group G(k) when k is a finite field, or the infinite-dimensional unitary representations of a real reductive group, or the automorphic representations of an adelic algebraic group. The structure theory of reductive groups is used in all these areas.
Definitions
A linear algebraic group over a field k is defined as a smooth closed subgroup scheme of GL(n) over k, for some positive integer n. Equivalently, a linear algebraic group over k is a smooth affine group scheme over k.
With the unipotent radical
A connected linear algebraic group over an algebraically closed field is called semisimple if every smooth connected solvable normal subgroup of is trivial. More generally, a connected linear algebraic group over an algebraically closed field is called reductive if the largest smooth connected unipotent normal subgroup of is trivial. This normal subgroup is called the unipotent radical and is denoted . (Some authors do not require reductive groups to be connected.) A group over an arbitrary field k is called semisimple or reductive if the base change is semisimple or reductive, where is an algebraic closure of k. (This is equivalent to the definition of reductive groups in the introduction when k is perfect.) Any torus over k, such as the multiplicative group Gm, is reductive.
With representation theory
Over fields of characteristic zero another equivalent definition of a reductive group is a connected group admitting a faithful semisimple representa
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https://en.wikipedia.org/wiki/Mahler%20measure
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In mathematics, the Mahler measure of a polynomial with complex coefficients is defined as
where factorizes over the complex numbers as
The Mahler measure can be viewed as a kind of height function. Using Jensen's formula, it can be proved that this measure is also equal to the geometric mean of for on the unit circle (i.e., ):
By extension, the Mahler measure of an algebraic number is defined as the Mahler measure of the minimal polynomial of over . In particular, if is a Pisot number or a Salem number, then its Mahler measure is simply .
The Mahler measure is named after the German-born Australian mathematician Kurt Mahler.
Properties
The Mahler measure is multiplicative:
where is the norm of .
Kronecker's Theorem: If is an irreducible monic integer polynomial with , then either or is a cyclotomic polynomial.
(Lehmer's conjecture) There is a constant such that if is an irreducible integer polynomial, then either or .
The Mahler measure of a monic integer polynomial is a Perron number.
Higher-dimensional Mahler measure
The Mahler measure of a multi-variable polynomial is defined similarly by the formula
It inherits the above three properties of the Mahler measure for a one-variable polynomial.
The multi-variable Mahler measure has been shown, in some cases, to be related to special values
of zeta-functions and -functions. For example, in 1981, Smyth proved the formulas
where is the Dirichlet L-function, and
where is the Riemann zeta function. Here is called the logarithmic Mahler measure.
Some results by Lawton and Boyd
From the definition, the Mahler measure is viewed as the integrated values of polynomials over the torus (also see Lehmer's conjecture). If vanishes on the torus , then the convergence of the integral defining is not obvious, but it is known that does converge and is equal to a limit of one-variable Mahler measures, which had been conjectured by Boyd.
This is formulated as follows: Let denote the integers and define . If is a polynomial in variables and define the polynomial of one variable by
and define by
where .
Boyd's proposal
Boyd provided more general statements than the above theorem. He pointed out that the classical Kronecker's theorem, which characterizes monic polynomials with integer coefficients all of whose roots are inside the unit disk, can be regarded as characterizing those polynomials of one variable whose measure is exactly 1, and that this result extends to polynomials in several variables.
Define an extended cyclotomic polynomial to be a polynomial of the form
where is the m-th cyclotomic polynomial, the are integers, and the are chosen minimally so that is a polynomial in the . Let be the set of polynomials that are products of monomials and extended cyclotomic polynomials.
This led Boyd to consider the set of values
and the union . He made the far-reaching conjecture that the set of is a closed subset of . An immediate consequence of t
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https://en.wikipedia.org/wiki/Identity%20component
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In mathematics, specifically group theory, the identity component of a group G refers to several closely related notions of the largest connected subgroup of G containing the identity element.
In point set topology, the identity component of a topological group G is the connected component G0 of G that contains the identity element of the group. The identity path component of a topological group G is the path component of G that contains the identity element of the group.
In algebraic geometry, the identity component of an algebraic group G over a field k is the identity component of the underlying topological space. The identity component of a group scheme G over a base scheme S is, roughly speaking, the group scheme G0 whose fiber over the point s of S is the connected component (Gs)0 of the fiber Gs, an algebraic group.
Properties
The identity component G0 of a topological or algebraic group G is a closed normal subgroup of G. It is closed since components are always closed. It is a subgroup since multiplication and inversion in a topological or algebraic group are continuous maps by definition. Moreover, for any continuous automorphism a of G we have
a(G0) = G0.
Thus, G0 is a characteristic subgroup of G, so it is normal.
The identity component G0 of a topological group G need not be open in G. In fact, we may have G0 = {e}, in which case G is totally disconnected. However, the identity component of a locally path-connected space (for instance a Lie group) is always open, since it contains a path-connected neighbourhood of {e}; and therefore is a clopen set.
The identity path component of a topological group may in general be smaller than the identity component (since path connectedness is a stronger condition than connectedness), but these agree if G is locally path-connected.
Component group
The quotient group G/G0 is called the group of components or component group of G. Its elements are just the connected components of G. The component group G/G0 is a discrete group if and only if G0 is open. If G is an algebraic group of finite type, such as an affine algebraic group, then G/G0 is actually a finite group.
One may similarly define the path component group as the group of path components (quotient of G by the identity path component), and in general the component group is a quotient of the path component group, but if G is locally path connected these groups agree. The path component group can also be characterized as the zeroth homotopy group,
Examples
The group of non-zero real numbers with multiplication (R*,•) has two components and the group of components is ({1,−1},•).
Consider the group of units U in the ring of split-complex numbers. In the ordinary topology of the plane {z = x + j y : x, y ∈ R}, U is divided into four components by the lines y = x and y = − x where z has no inverse. Then U0 = { z : |y| < x } . In this case the group of components of U is isomorphic to the Klein four-group.
The identity component
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https://en.wikipedia.org/wiki/Euler%E2%80%93Tricomi%20equation
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In mathematics, the Euler–Tricomi equation is a linear partial differential equation useful in the study of transonic flow. It is named after mathematicians Leonhard Euler and Francesco Giacomo Tricomi.
It is elliptic in the half plane x > 0, parabolic at x = 0 and hyperbolic in the half plane x < 0.
Its characteristics are
which have the integral
where C is a constant of integration. The characteristics thus comprise two families of semicubical parabolas, with cusps on the line x = 0, the curves lying on the right hand side of the y-axis.
Particular solutions
A general expression for particular solutions to the Euler–Tricomi equations is:
where
These can be linearly combined to form further solutions such as:
for k = 0:
for k = 1:
etc.
The Euler–Tricomi equation is a limiting form of Chaplygin's equation.
See also
Burgers equation
Chaplygin's equation
Bibliography
A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC Press, 2002.
External links
Tricomi and Generalized Tricomi Equations at EqWorld: The World of Mathematical Equations.
Partial differential equations
Equations of fluid dynamics
Leonhard Euler
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https://en.wikipedia.org/wiki/Modular%20lattice
|
In the branch of mathematics called order theory, a modular lattice is a lattice that satisfies the following self-dual condition,
Modular law implies
where are arbitrary elements in the lattice, ≤ is the partial order, and ∨ and ∧ (called join and meet respectively) are the operations of the lattice. This phrasing emphasizes an interpretation in terms of projection onto the sublattice , a fact known as the diamond isomorphism theorem. An alternative but equivalent condition stated as an equation (see below) emphasizes that modular lattices form a variety in the sense of universal algebra.
Modular lattices arise naturally in algebra and in many other areas of mathematics. In these scenarios, modularity is an abstraction of the 2nd Isomorphism Theorem. For example, the subspaces of a vector space (and more generally the submodules of a module over a ring) form a modular lattice.
In a not necessarily modular lattice, there may still be elements for which the modular law holds in connection with arbitrary elements and (for ). Such an element is called a modular element. Even more generally, the modular law may hold for any and a fixed pair . Such a pair is called a modular pair, and there are various generalizations of modularity related to this notion and to semimodularity.
Modular lattices are sometimes called Dedekind lattices after Richard Dedekind, who discovered the modular identity in several motivating examples.
Introduction
The modular law can be seen as a restricted associative law that connects the two lattice operations similarly to the way in which the associative law λ(μx) = (λμ)x for vector spaces connects multiplication in the field and scalar multiplication.
The restriction is clearly necessary, since it follows from . In other words, no lattice with more than one element satisfies the unrestricted consequent of the modular law.
It is easy to see that implies in every lattice. Therefore, the modular law can also be stated as
Modular law (variant) implies .
The modular law can be expressed as an equation that is required to hold unconditionally. Since implies and since , replace with in the defining equation of the modular law to obtain:
Modular identity .
This shows that, using terminology from universal algebra, the modular lattices form a subvariety of the variety of lattices. Therefore, all homomorphic images, sublattices and direct products of modular lattices are again modular.
Examples
The lattice of submodules of a module over a ring is modular. As a special case, the lattice of subgroups of an abelian group is modular.
The lattice of normal subgroups of a group is modular. But in general the lattice of all subgroups of a group is not modular. For an example, the lattice of subgroups of the dihedral group of order 8 is not modular.
The smallest non-modular lattice is the "pentagon" lattice N5 consisting of five elements 0, 1, x, a, b such that 0 < x < b < 1, 0 < a < 1, and a is not com
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https://en.wikipedia.org/wiki/Geometry%20pipelines
|
Geometric manipulation of modelling primitives, such as that performed by a geometry pipeline, is the first stage in computer graphics systems which perform image generation based on geometric models. While geometry pipelines were originally implemented in software, they have become highly amenable to hardware implementation, particularly since the advent of very-large-scale integration (VLSI) in the early 1980s. A device called the Geometry Engine developed by Jim Clark and Marc Hannah at Stanford University in about 1981 was the watershed for what has since become an increasingly commoditized function in contemporary image-synthetic raster display systems.
Geometric transformations are applied to the vertices of polygons, or other geometric objects used as modelling primitives, as part of the first stage in a classical geometry-based graphic image rendering pipeline. Geometric computations may also be applied to transform polygon or repair surface normals, and then to perform the lighting and shading computations used in their subsequent rendering.
History
Hardware implementations of the geometry pipeline were introduced in the early Evans & Sutherland Picture System, but perhaps received broader recognition when later applied in the broad range of graphics systems products introduced by Silicon Graphics (SGI). Initially the SGI geometry hardware performed simple model space to screen space viewing transformations with all the lighting and shading handled by a separate hardware implementation stage. In later, much higher performance applications, such as the RealityEngine, they began to be applied to perform part of the rendering support as well.
More recently, perhaps dating from the late 1990s, the hardware support required to perform the manipulation and rendering of quite complex scenes has become accessible to the consumer market.
Companies such as Nvidia and AMD Graphics (formerly ATI) are two current leading representatives of hardware vendors in this space. The GeForce line of graphics cards from Nvidia was the first to support full OpenGL and Direct3D hardware geometry processing in the consumer PC market, while some earlier products such as Rendition Verite incorporated hardware geometry processing through proprietary programming interfaces. On the whole, earlier graphics accelerators by 3Dfx, Matrox and others relied on the CPU for geometry processing.
This subject matter is part of the technical foundation for modern computer graphics, and is a comprehensive topic taught at both the undergraduate and graduate levels as part of a computer science education.
See also
Vertex pipeline
Graphics pipeline (include Pixel pipeline)
Rasterisation
Open Graphics Project
References
3D computer graphics
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https://en.wikipedia.org/wiki/Biorthogonal%20system
|
In mathematics, a biorthogonal system is a pair of indexed families of vectors
such that
where and form a pair of topological vector spaces that are in duality, is a bilinear mapping and is the Kronecker delta.
An example is the pair of sets of respectively left and right eigenvectors of a matrix, indexed by eigenvalue, if the eigenvalues are distinct.
A biorthogonal system in which and is an orthonormal system.
Projection
Related to a biorthogonal system is the projection
where its image is the linear span of and the kernel is
Construction
Given a possibly non-orthogonal set of vectors and the projection related is
where is the matrix with entries
and then is a biorthogonal system.
See also
References
Jean Dieudonné, On biorthogonal systems Michigan Math. J. 2 (1953), no. 1, 7–20
Topological vector spaces
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https://en.wikipedia.org/wiki/DFT%20matrix
|
In applied mathematics, a DFT matrix is an expression of a discrete Fourier transform (DFT) as a transformation matrix, which can be applied to a signal through matrix multiplication.
Definition
An N-point DFT is expressed as the multiplication , where is the original input signal, is the N-by-N square DFT matrix, and is the DFT of the signal.
The transformation matrix can be defined as , or equivalently:
,
where is a primitive Nth root of unity in which . We can avoid writing large exponents for using the fact that for any exponent we have the identity This is the Vandermonde matrix for the roots of unity, up to the normalization factor. Note that the normalization factor in front of the sum ( ) and the sign of the exponent in ω are merely conventions, and differ in some treatments. All of the following discussion applies regardless of the convention, with at most minor adjustments. The only important thing is that the forward and inverse transforms have opposite-sign exponents, and that the product of their normalization factors be 1/N. However, the choice here makes the resulting DFT matrix unitary, which is convenient in many circumstances.
Fast Fourier transform algorithms utilize the symmetries of the matrix to reduce the time of multiplying a vector by this matrix, from the usual . Similar techniques can be applied for multiplications by matrices such as Hadamard matrix and the Walsh matrix.
Examples
Two-point
The two-point DFT is a simple case, in which the first entry is the DC (sum) and the second entry is the AC (difference).
The first row performs the sum, and the second row performs the difference.
The factor of is to make the transform unitary (see below).
Four-point
The four-point clockwise DFT matrix is as follows:
where .
Eight-point
The first non-trivial integer power of two case is for eight points:
where
(Note that .)
Evaluating for the value of , gives:
The following image depicts the DFT as a matrix multiplication, with elements of the matrix depicted by samples of complex exponentials:
The real part (cosine wave) is denoted by a solid line, and the imaginary part (sine wave) by a dashed line.
The top row is all ones (scaled by for unitarity), so it "measures" the DC component in the input signal. The next row is eight samples of negative one cycle of a complex exponential, i.e., a signal with a fractional frequency of −1/8, so it "measures" how much "strength" there is at fractional frequency +1/8 in the signal. Recall that a matched filter compares the signal with a time reversed version of whatever we're looking for, so when we're looking for fracfreq. 1/8 we compare with fracfreq. −1/8 so that is why this row is a negative frequency. The next row is negative two cycles of a complex exponential, sampled in eight places, so it has a fractional frequency of −1/4, and thus "measures" the extent to which the signal has a fractional frequency of +1/4.
The following summarizes how the 8-point
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https://en.wikipedia.org/wiki/Non-well-founded%20set%20theory
|
Non-well-founded set theories are variants of axiomatic set theory that allow sets to be elements of themselves and otherwise violate the rule of well-foundedness. In non-well-founded set theories, the foundation axiom of ZFC is replaced by axioms implying its negation.
The study of non-well-founded sets was initiated by Dmitry Mirimanoff in a series of papers between 1917 and 1920, in which he formulated the distinction between well-founded and non-well-founded sets; he did not regard well-foundedness as an axiom. Although a number of axiomatic systems of non-well-founded sets were proposed afterwards, they did not find much in the way of applications until Peter Aczel’s hyperset theory in 1988.
The theory of non-well-founded sets has been applied in the logical modelling of non-terminating computational processes in computer science (process algebra and final semantics), linguistics and natural language semantics (situation theory), philosophy (work on the Liar Paradox), and in a different setting, non-standard analysis.
Details
In 1917, Dmitry Mirimanoff introduced the concept of well-foundedness of a set:
A set, x0, is well-founded if it has no infinite descending membership sequence
In ZFC, there is no infinite descending ∈-sequence by the axiom of regularity. In fact, the axiom of regularity is often called the foundation axiom since it can be proved within ZFC− (that is, ZFC without the axiom of regularity) that well-foundedness implies regularity. In variants of ZFC without the axiom of regularity, the possibility of non-well-founded sets with set-like ∈-chains arises. For example, a set A such that A ∈ A is non-well-founded.
Although Mirimanoff also introduced a notion of isomorphism between possibly non-well-founded sets, he considered neither an axiom of foundation nor of anti-foundation. In 1926, Paul Finsler introduced the first axiom that allowed non-well-founded sets. After Zermelo adopted Foundation into his own system in 1930 (from previous work of von Neumann 1925–1929) interest in non-well-founded sets waned for decades. An early non-well-founded set theory was Willard Van Orman Quine’s New Foundations, although it is not merely ZF with a replacement for Foundation.
Several proofs of the independence of Foundation from the rest of ZF were published in 1950s particularly by Paul Bernays (1954), following an announcement of the result in an earlier paper of his from 1941, and by Ernst Specker who gave a different proof in his Habilitationsschrift of 1951, proof which was published in 1957. Then in 1957 Rieger's theorem was published, which gave a general method for such proof to be carried out, rekindling some interest in non-well-founded axiomatic systems. The next axiom proposal came in a 1960 congress talk of Dana Scott (never published as a paper), proposing an alternative axiom now called SAFA. Another axiom proposed in the late 1960s was Maurice Boffa's axiom of superuniversality, described by Aczel as the highpoint
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https://en.wikipedia.org/wiki/Igor%20Shafarevich
|
Igor Rostislavovich Shafarevich (; 3 June 1923 – 19 February 2017) was a Soviet and Russian mathematician who contributed to algebraic number theory and algebraic geometry. Outside mathematics, he wrote books and articles that criticised socialism and other books which were described as anti-semitic.
Mathematics
From his early years, Shafarevich made fundamental contributions to several parts of mathematics
including algebraic number theory, algebraic geometry and arithmetic algebraic geometry. In particular, in algebraic number theory, the Shafarevich–Weil theorem extends the commutative reciprocity map to the case of Galois groups, which are central extensions of abelian groups by finite groups.
Shafarevich was the first mathematician to give a completely self-contained formula for the Hilbert pairing, thus initiating an important branch of the study of explicit formulas in number theory. Another famous (and slightly incomplete) result is Shafarevich's theorem on solvable Galois groups, giving the realization of every finite solvable group as a Galois group over the rationals.
Another development is the Golod–Shafarevich theorem on towers of unramified extensions of number fields.
Shafarevich and his school greatly contributed to the study of algebraic geometry of surfaces. He started a famous Moscow seminar on classification of algebraic surfaces that updated the treatment of birational geometry around 1960 and was largely responsible for the early introduction of the scheme theory approach to algebraic geometry in the Soviet school. His investigation in arithmetic of elliptic curves led him, independently of John Tate, to the introduction of the group related to elliptic curves over number fields, the Tate–Shafarevich group (usually called 'Sha', and denoted as 'Ш', the first Cyrillic letter of his surname).
He contributed the Grothendieck–Ogg–Shafarevich formula and to the Néron–Ogg–Shafarevich criterion.
With former student Ilya Piatetski-Shapiro, he proved a version of the Torelli theorem for K3 surfaces.
He formulated the Shafarevich conjecture, which stated the finiteness of the set of Abelian varieties over a number field having fixed dimension and prescribed set of primes of bad reduction. The conjecture was proved by Gerd Faltings as a partial step in his proof of the Mordell conjecture.
Shafarevich's students included Yuri Manin, Alexey Parshin, Igor Dolgachev, Evgeny Golod, Alexei Kostrikin, Suren Arakelov, G. V. Belyi, Victor Abrashkin, Andrey Todorov, Andrey N. Tyurin, and Victor Kolyvagin.
He was a member of the Serbian Academy of Sciences and Arts in the department of Mathematics, Physics and Earth Sciences.
In 1960, he was elected a Member of the German Academy of Sciences Leopoldina. In 1981, he was elected as a foreign member of the Royal Society.
In 2017, Shafarevich was awarded the Leonhard Euler Gold Medal by the Russian Academy of Sciences.
Soviet politics
Shafarevich came into conflict with the Soviet auth
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https://en.wikipedia.org/wiki/Negative%20frequency
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In mathematics, signed frequency (negative and positive frequency) expands upon the concept of frequency, from just an absolute value representing how often some repeating event occurs, to also have a positive or negative sign representing one of two opposing orientations for occurrences of those events. The following examples help illustrate the concept:
For a rotating object, the absolute value of its frequency of rotation indicates how many rotations the object completes per unit of time, while the sign could indicate whether it is rotating clockwise or counterclockwise.
Mathematically speaking, the vector has a positive frequency of +1 radian per unit of time and rotates counterclockwise around the unit circle, while the vector has a negative frequency of -1 radian per unit of time, which rotates clockwise instead.
For a harmonic oscillator such as a pendulum, the absolute value of its frequency indicates how many times it swings back and forth per unit of time, while the sign could indicate in which of the two opposite directions it started moving.
For a periodic function represented in a Cartesian coordinate system, the absolute value of its frequency indicates how often in its domain it repeats its values, while changing the sign of its frequency could represent a reflection around its y-axis.
Sinusoids
Let be a nonnegative angular frequency with units of radians per unit of time and let be a phase in radians. A function has slope When used as the argument of a sinusoid, can represent a negative frequency.
Because cosine is an even function, the negative frequency sinusoid is indistinguishable from the positive frequency sinusoid
Similarly, because sine is an odd function, the negative frequency sinusoid is indistinguishable from the positive frequency sinusoid or
Thus any sinusoid can be represented in terms of positive frequencies only.
The sign of the underlying phase slope is ambiguous. Because leads by radians (or cycle) for positive frequencies and lags by the same amount for negative frequencies, the ambiguity about the phase slope is resolved simply by observing a cosine and sine operator simultaneously and seeing which one leads the other.
The sign of is also preserved in the complex-valued function:
since and can be separately observed and compared. A common interpretation is that is a simpler function than either of its components, because it simplifies multiplicative trigonometric calculations, which leads to its formal description as the analytic representation of .
The sum of an analytic representation with its complex conjugate extracts the actual real-valued function they represent. For instance:
which gives rise to the somewhat misleading interpretation that comprises both a positive and a negative frequency. But the "sum" involves a cancellation of all imaginary components . That cancellation merely results in an ambiguity about the sign of the frequency. Using either sign provides an equi
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https://en.wikipedia.org/wiki/Scorer%27s%20function
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In mathematics, the Scorer's functions are special functions studied by and denoted Gi(x) and Hi(x).
Hi(x) and -Gi(x) solve the equation
and are given by
The Scorer's functions can also be defined in terms of Airy functions:
References
Special functions
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https://en.wikipedia.org/wiki/Approximation%20theory
|
In mathematics, approximation theory is concerned with how functions can best be approximated with simpler functions, and with quantitatively characterizing the errors introduced thereby. What is meant by best and simpler will depend on the application.
A closely related topic is the approximation of functions by generalized Fourier series, that is, approximations based upon summation of a series of terms based upon orthogonal polynomials.
One problem of particular interest is that of approximating a function in a computer mathematical library, using operations that can be performed on the computer or calculator (e.g. addition and multiplication), such that the result is as close to the actual function as possible. This is typically done with polynomial or rational (ratio of polynomials) approximations.
The objective is to make the approximation as close as possible to the actual function, typically with an accuracy close to that of the underlying computer's floating point arithmetic. This is accomplished by using a polynomial of high degree, and/or narrowing the domain over which the polynomial has to approximate the function.
Narrowing the domain can often be done through the use of various addition or scaling formulas for the function being approximated. Modern mathematical libraries often reduce the domain into many tiny segments and use a low-degree polynomial for each segment.
Optimal polynomials
Once the domain (typically an interval) and degree of the polynomial are chosen, the polynomial itself is chosen in such a way as to minimize the worst-case error. That is, the goal is to minimize the maximum value of , where P(x) is the approximating polynomial, f(x) is the actual function, and x varies over the chosen interval. For well-behaved functions, there exists an Nth-degree polynomial that will lead to an error curve that oscillates back and forth between and a total of N+2 times, giving a worst-case error of . It is seen that there exists an Nth-degree polynomial that can interpolate N+1 points in a curve. That such a polynomial is always optimal is asserted by the equioscillation theorem. It is possible to make contrived functions f(x) for which no such polynomial exists, but these occur rarely in practice.
For example, the graphs shown to the right show the error in approximating log(x) and exp(x) for N = 4. The red curves, for the optimal polynomial, are level, that is, they oscillate between and exactly. In each case, the number of extrema is N+2, that is, 6. Two of the extrema are at the end points of the interval, at the left and right edges of the graphs.
To prove this is true in general, suppose P is a polynomial of degree N having the property described, that is, it gives rise to an error function that has N + 2 extrema, of alternating signs and equal magnitudes. The red graph to the right shows what this error function might look like for N = 4. Suppose Q(x) (whose error function is shown in blue to the right) is ano
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https://en.wikipedia.org/wiki/Stanley%27s%20reciprocity%20theorem
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In combinatorial mathematics, Stanley's reciprocity theorem, named after MIT mathematician Richard P. Stanley, states that a certain functional equation is satisfied by the generating function of any rational cone (defined below) and the generating function of the cone's interior.
Definitions
A rational cone is the set of all d-tuples
(a1, ..., ad)
of nonnegative integers satisfying a system of inequalities
where M is a matrix of integers. A d-tuple satisfying the corresponding strict inequalities, i.e., with ">" rather than "≥", is in the interior of the cone.
The generating function of such a cone is
The generating function Fint(x1, ..., xd) of the interior of the cone is defined in the same way, but one sums over d-tuples in the interior rather than in the whole cone.
It can be shown that these are rational functions.
Formulation
Stanley's reciprocity theorem states that for a rational cone as above, we have
Matthias Beck and Mike Develin have shown how to prove this by using the calculus of residues. Develin has said that this amounts to proving the result "without doing any work".
Stanley's reciprocity theorem generalizes Ehrhart-Macdonald reciprocity for Ehrhart polynomials of rational convex polytopes.
See also
Ehrhart polynomial
References
Algebraic combinatorics
Theorems in combinatorics
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https://en.wikipedia.org/wiki/Hough%20function
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In applied mathematics, the Hough functions are the eigenfunctions of Laplace's tidal equations which govern fluid motion on a rotating sphere. As such, they are relevant in geophysics and meteorology where they form part of the solutions for atmospheric and ocean waves. These functions are named in honour of Sydney Samuel Hough.
Each Hough mode is a function of latitude and may be expressed as an infinite sum of associated Legendre polynomials; the functions are orthogonal over the sphere in the continuous case. Thus they can also be thought of as a generalized Fourier series in which the basis functions are the normal modes of an atmosphere at rest.
See also
Secondary circulation
Legendre polynomials
Primitive equations
References
Further reading
Atmospheric dynamics
Physical oceanography
Fluid mechanics
Special functions
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https://en.wikipedia.org/wiki/Parabolic%20fractal%20distribution
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In probability and statistics, the parabolic fractal distribution is a type of discrete probability distribution in which the logarithm of the frequency or size of entities in a population is a quadratic polynomial of the logarithm of the rank (with the largest example having rank 1). This can markedly improve the fit over a simple power-law relationship (see references below).
In the Laherrère/Deheuvels paper below, examples include galaxy sizes (ordered by luminosity), towns (in the USA, France, and world), spoken languages (by number of speakers) in the world, and oil fields in the world (by size). They also mention utility for this distribution in fitting seismic events (no example). The authors assert the advantage of this distribution is that it can be fitted using the largest known examples of the population being modeled, which are often readily available and complete, then the fitted parameters found can be used to compute the size of the entire population. So, for example, the populations of the hundred largest cities on the planet can be sorted and fitted, and the parameters found used to extrapolate to the smallest villages, to estimate the population of the planet. Another example is estimating total world oil reserves using the largest fields.
In a number of applications, there is a so-called King effect where the top-ranked item(s) have a significantly greater frequency or size than the model predicts on the basis of the other items. The Laherrère/Deheuvels paper shows the example of Paris, when sorting the sizes of towns in France. When the paper was written Paris was the largest city with about ten million inhabitants, but the next largest town had only about 1.5 million. Towns in France excluding Paris closely follow a parabolic distribution, well enough that the 56 largest gave a very good estimate of the population of the country. But that distribution would predict the largest city to have about two million inhabitants, not 10 million. The King Effect is named after the notion that a King must defeat all rivals for the throne and takes their wealth, estates and power, thereby creating a buffer between himself and the next-richest of his subjects. That specific effect (intentionally created) may apply to corporate sizes, where the largest businesses use their wealth to buy up smaller rivals. Absent intent, the King Effect may occur as a result of some persistent growth advantage due to scale, or to some unique advantage. Larger cities are more efficient connectors of people, talent and other resources. Unique advantages might include being a port city, or a Capital city where law is made, or a center of activity where physical proximity increases opportunity and creates a feedback loop. An example is the motion picture industry; where actors, writers and other workers move to where the most studios are, and new studios are founded in the same place because that is where the most talent resides.
To test for the King Effect,
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https://en.wikipedia.org/wiki/Morley%27s%20trisector%20theorem
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In plane geometry, Morley's trisector theorem states that in any triangle, the three points of intersection of the adjacent angle trisectors form an equilateral triangle, called the first Morley triangle or simply the Morley triangle. The theorem was discovered in 1899 by Anglo-American mathematician Frank Morley. It has various generalizations; in particular, if all the trisectors are intersected, one obtains four other equilateral triangles.
Proofs
There are many proofs of Morley's theorem, some of which are very technical.
Several early proofs were based on delicate trigonometric calculations. Recent proofs include an algebraic proof by extending the theorem to general fields other than characteristic three, and John Conway's elementary geometry proof. The latter starts with an equilateral triangle and shows that a triangle may be built around it which will be similar to any selected triangle. Morley's theorem does not hold in spherical and hyperbolic geometry.
One proof uses the trigonometric identity
which, by using of the sum of two angles identity, can be shown to be equal to
The last equation can be verified by applying the sum of two angles identity to the left side twice and eliminating the cosine.
Points are constructed on as shown. We have , the sum of any triangle's angles, so Therefore, the angles of triangle are and
From the figure
and
Also from the figure
and
The law of sines applied to triangles and yields
and
Express the height of triangle in two ways
and
where equation (1) was used to replace and in these two equations. Substituting equations (2) and (5) in the equation and equations (3) and (6) in the equation gives
and
Since the numerators are equal
or
Since angle and angle are equal and the sides forming these angles are in the same ratio, triangles and are similar.
Similar angles and equal , and similar angles and equal Similar arguments yield the base angles of triangles and
In particular angle is found to be and from the figure we see that
Substituting yields
where equation (4) was used for angle and therefore
Similarly the other angles of triangle are found to be
Side and area
The first Morley triangle has side lengths
where R is the circumradius of the original triangle and A, B, and C are the angles of the original triangle. Since the area of an equilateral triangle is the area of Morley's triangle can be expressed as
Morley's triangles
Morley's theorem entails 18 equilateral triangles. The triangle described in the trisector theorem above, called the first Morley triangle, has vertices given in trilinear coordinates relative to a triangle ABC as follows:
Another of Morley's equilateral triangles that is also a central triangle is called the second Morley triangle and is given by these vertices:
The third of Morley's 18 equilateral triangles that is also a central triangle is called the third Morley triangle and is given by these vertices:
The first, se
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https://en.wikipedia.org/wiki/Morley%27s%20theorem
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Morley's theorem may refer to:
Morley's trisector theorem, a theorem in geometry, discovered by Frank Morley
Morley's categoricity theorem, a theorem in model theory, discovered by Michael D. Morley
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https://en.wikipedia.org/wiki/De%20Morgan%20Medal
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The De Morgan Medal is a prize for outstanding contribution to mathematics, awarded by the London Mathematical Society. The Society's most prestigious award, it is given in memory of Augustus De Morgan, who was the first President of the society.
The medal is awarded every third year (in years divisible by 3) to a mathematician who is normally resident in the United Kingdom on 1 January of the relevant year. The only grounds for the award of the medal are the candidate's contributions to mathematics.
In 1968, Mary Cartwright became the first woman to receive the award.
De Morgan Medal winners
Recipients of the De Morgan Medal include the following:
See also
Whitehead Prize
Fröhlich Prize
Senior Whitehead Prize
Berwick Prize
Naylor Prize and Lectureship
Pólya Prize (LMS)
List of mathematics awards
References
British science and technology awards
Awards established in 1884
Triennial events
Awards of the London Mathematical Society
1884 establishments in the United Kingdom
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https://en.wikipedia.org/wiki/Logistic
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Logistic may refer to:
Mathematics
Logistic function, a sigmoid function used in many fields
Logistic map, a recurrence relation that sometimes exhibits chaos
Logistic regression, a statistical model using the logistic function
Logit, the inverse of the logistic function
Logistic distribution, the derivative of the logistic function, a continuous probability distribution, used in probability theory and statistics
Mathematical logic, subfield of mathematics exploring the applications of formal logic to mathematics
Other uses
Logistics, the management of resources and their distributions
Logistic engineering, the scientific study of logistics
Military logistics, the study of logistics at the service of military units and operations
See also
Logic (disambiguation)
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https://en.wikipedia.org/wiki/Adams%20Prize
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The Adams Prize is one of the most prestigious prizes awarded by the University of Cambridge. It is awarded each year by the Faculty of Mathematics at the University of Cambridge and St John's College to a UK-based mathematician for distinguished research in the Mathematical Sciences.
The prize is named after the mathematician John Couch Adams. It was endowed by members of St John's College and was approved by the senate of the university in 1848 to commemorate Adams' controversial role in the discovery of the planet Neptune. Originally open only to Cambridge graduates, the current stipulation is that the mathematician must reside in the UK and must be under forty years of age. Each year applications are invited from mathematicians who have worked in a specific area of mathematics. the Adams Prize is worth approximately £14,000. The prize is awarded in three parts. The first third is paid directly to the candidate; another third is paid to the candidate's institution to fund research expenses; and the final third is paid on publication of a survey paper in the winner's field in a major mathematics journal.
The prize has been awarded to many well known mathematicians, including
James Clerk Maxwell and Sir William Hodge. The first time it was awarded to a female mathematician was in 2002 when it was awarded to Susan Howson, then a lecturer at the University of Nottingham for her work on number theory and elliptic curves.
Subject area
2014–15: "Algebraic Geometry"
2015–16: "Applied Analysis".
2016–17: "Statistical Analysis of Big Data".
2017–18: “The Mathematics of Astronomy and Cosmology”
2018–19: “The Mathematics of Networks”
List of prize winners
The complete list of prize winners can be found on the Adams Prize webpage on the University of Cambridge's website. The following partial list is compiled from internet sources:
See also
List of mathematics awards
References
Mathematical awards and prizes of the University of Cambridge
Awards established in 1848
British science and technology awards
Early career awards
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https://en.wikipedia.org/wiki/List%20of%20mathematical%20societies
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This article provides a list of mathematical societies.
International
African Mathematical Union
Association for Women in Mathematics
Circolo Matematico di Palermo
European Mathematical Society
European Women in Mathematics
Foundations of Computational Mathematics
International Association for Cryptologic Research
International Association of Mathematical Physics
International Linear Algebra Society
International Mathematical Union
International Statistical Institute
International Society for Analysis, its Applications and Computation
International Society for Mathematical Sciences
Kurt Gödel Society
Mathematical Council of the Americas (MCofA)
Mathematical Society of South Eastern Europe (MASSEE)
Mathematical Optimization Society
Maths Society
Ramanujan Mathematical Society
Quaternion Society
Society for Industrial and Applied Mathematics
Southeast Asian Mathematical Society (SEAMS)
Spectra (mathematical association)
Unión Matemática de América Latina y el Caribe (UMALCA)
Young Mathematicians Network
Honor societies
Kappa Mu Epsilon
Mu Alpha Theta
Pi Mu Epsilon
National and subnational
Arranged as follows: Society name in English (Society name in home-language; Abbreviation if used), Country and/or subregion/city if not specified in name.
This list is sorted by continent.
Africa
Algeria Mathematical Society
Gabon Mathematical Society
South African Mathematical Society
Asia
Bangladesh Mathematical Society
Calcutta Mathematical Society (CalMathSoc), Kolkata, India
Chinese Mathematical Society
Indian Mathematical Society
Iranian Mathematical Society
Israel Mathematical Union
Jadavpur University Mathematical Society (JMS), Jadavpur, India
Kerala Mathematical Association, Kerala State, India
Korean Mathematical Society, South Korea
Mathematical Society of Japan
Mathematical Society of the Philippines
Pakistan Mathematical Society
Turkish Mathematical Society
Europe
Albanian Mathematical Association
Armenian Mathematical Union
Austrian Mathematical Society (Österreichische Mathematische Gesellschaft; ÖMG)
Trinity Mathematical Society, Cambridge, UK
Cyprus Mathematical Society
Danish Mathematical Society
French Mathematical Society (Société Mathématique de France; SMF)
Society of Applied Mathematics and Mechanics (Gesellschaft für Angewandte Mathematik und Mechanik; GAMM), Germany
German Mathematical Society (Deutsche Mathematiker-Vereinigung; DMV)
Hellenic Mathematical Society (Ελληνική Μαθηματική Εταιρεία; EME)
Icelandic Mathematical Society
Institute of Mathematics and its Applications, UK
Irish Mathematical Society
Italian Mathematical Union
János Bolyai Mathematical Society, Hungary
Kharkov Mathematical Society, Kharkiv, Ukraine
Latvian Mathematical Society
London Mathematical Society, London, UK
Luxembourg Mathematical Society
Malta Mathematical Society
Mathematical Association, UK
Moscow Mathematical Society, Moscow, Russia
Norwegian Mathematical Society
Norwegian Statistical Associat
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https://en.wikipedia.org/wiki/Hilbert%27s%20Theorem%2090
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In abstract algebra, Hilbert's Theorem 90 (or Satz 90) is an important result on cyclic extensions of fields (or to one of its generalizations) that leads to Kummer theory. In its most basic form, it states that if L/K is an extension of fields with cyclic Galois group G = Gal(L/K) generated by an element and if is an element of L of relative norm 1, that isthen there exists in L such thatThe theorem takes its name from the fact that it is the 90th theorem in David Hilbert's Zahlbericht , although it is originally due to .
Often a more general theorem due to is given the name, stating that if L/K is a finite Galois extension of fields with arbitrary Galois group G = Gal(L/K), then the first cohomology group of G, with coefficients in the multiplicative group of L, is trivial:
Examples
Let be the quadratic extension . The Galois group is cyclic of order 2, its generator acting via conjugation:
An element in has norm . An element of norm one thus corresponds to a rational solution of the equation or in other words, a point with rational coordinates on the unit circle. Hilbert's Theorem 90 then states that every such element a of norm one can be written as
where is as in the conclusion of the theorem, and c and d are both integers. This may be viewed as a rational parametrization of the rational points on the unit circle. Rational points on the unit circle correspond to Pythagorean triples, i.e. triples of integers satisfying .
Cohomology
The theorem can be stated in terms of group cohomology: if L× is the multiplicative group of any (not necessarily finite) Galois extension L of a field K with corresponding Galois group G, then
Specifically, group cohomology is the cohomology of the complex whose i-cochains are arbitrary functions from i-tuples of group elements to the multiplicative coefficient group, , with differentials defined in dimensions by:
where denotes the image of the -module element under the action of the group element .
Note that in the first of these we have identified a 0-cochain , with its unique image value .
The triviality of the first cohomology group is then equivalent to the 1-cocycles being equal to the 1-coboundaries , viz.:
For cyclic , a 1-cocycle is determined by , with and:On the other hand, a 1-coboundary is determined by . Equating these gives the original version of the Theorem.
A further generalization is to cohomology with non-abelian coefficients: that if H is either the general or special linear group over L, including , then Another generalization is to a scheme X:
where is the group of isomorphism classes of locally free sheaves of -modules of rank 1 for the Zariski topology, and is the sheaf defined by the affine line without the origin considered as a group under multiplication.
There is yet another generalization to Milnor K-theory which plays a role in Voevodsky's proof of the Milnor conjecture.
Proof
Let be cyclic of degree and generate . Pick any of norm
By clea
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https://en.wikipedia.org/wiki/Salem%20Prize
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The Salem Prize, in memory of Raphael Salem, is awarded each year to young researchers for outstanding contributions to the field of analysis. It is awarded by the School of Mathematics at the Institute for Advanced Study in Princeton and was founded by the widow of Raphael Salem in his memory. The prize is considered highly prestigious and many Fields Medalists previously received it. The prize was 5000 French Francs in 1990.
Past winners
(Note: a F symbol denotes mathematicians who later earned a Fields Medal).
See also
List of mathematics awards
References
Mathematics awards
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https://en.wikipedia.org/wiki/Eduard%20Vogel
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Eduard Vogel (7 March 1829February 1856) was a German explorer in Central Africa.
Early career
Vogel was born in Krefeld. He studied mathematics, botany and astronomy at Leipzig and Berlin, studying with Encke at the latter institution. In 1851, he was engaged as assistant astronomer to director John Russel Hind at George Bishop's private observatory in London. That year August Heinrich Petermann introduced Vogel to the Royal Geographical Society.
Africa commission
In 1853 Petermann arranged for Vogel to be chosen by the British government to join the Richardson, Overweg and Barth expedition with supplies. That expedition had been sent to Africa in 1849 to find a trade route that bypassed the Arabs. Vogel was to be a replacement for Richardson who had died two years earlier and was tasked to make geographical and meteorological observations and to collect botanical specimens. In 1853, the expedition was in the western Sudan.
Vogel sailed from England on 20 February 1853. The day Vogel left London, news had arrived that Overweg had also died, leaving Barth on his own.
Meeting Barth
On 25 July, Vogel left Tripoli with a caravan to catch up with Barth. Vogel arrived at the end of the Trans-Saharan trade route, Kuka, the capital of Bornu on 13 January 1854. Vogel's specimens, and the fact that both expedition engineers were soldiers, made the king there suspicious of his intentions, and Vogel's movements were severely restricted.
Instead of waiting for Barth to return, on 19 July, Vogel joined a steamboat expedition heading up the Niger and Benue Rivers to the Mandara Mountains where he was imprisoned by the king of Mora who had received a message about the suspicious stranger from Bornu. Vogel eventually escaped to Marghi in Nigeria where he waited for news of Barth.
Upon hearing of a change of king in Bornu, Vogel returned to wait for Barth, whom he met December 1854.
By some accounts Vogel was disliked by the other members of the expedition due to his poor attitude, difficult personality and unwillingness to learn Arabic, the lingua franca of north Africa. The arrival of Barth helped defuse some of the conflict, although one of the two engineers refused to travel any further while Vogel was part of the expedition. Barth himself contemplated getting rid of Vogel and stealing his equipment.
Further exploration
Vogel left Barth, and taking one engineer and four servants headed for Bauchi where he ingratiated himself with the Emir by killing a man the Emir disliked. He then became the first European to cross the Muri mountains angering the Tangale people in the process as he desecrated their shrines by sleeping in them during the journey. He penetrated south to the upper course of the Benue, returning to Kuka 1 December 1855. From this date, the notes of his expedition cease.
Death
Vogel left Kuka for the Nile Valley, leaving his engineer, MacGuire, with his notes and specimen collections. Vogel got as far as Wadai (also spelled Ouaddai)
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https://en.wikipedia.org/wiki/Isometry%20group
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In mathematics, the isometry group of a metric space is the set of all bijective isometries (that is, bijective, distance-preserving maps) from the metric space onto itself, with the function composition as group operation. Its identity element is the identity function. The elements of the isometry group are sometimes called motions of the space.
Every isometry group of a metric space is a subgroup of isometries. It represents in most cases a possible set of symmetries of objects/figures in the space, or functions defined on the space. See symmetry group.
A discrete isometry group is an isometry group such that for every point of the space the set of images of the point under the isometries is a discrete set.
In pseudo-Euclidean space the metric is replaced with an isotropic quadratic form; transformations preserving this form are sometimes called "isometries", and the collection of them is then said to form an isometry group of the pseudo-Euclidean space.
Examples
The isometry group of the subspace of a metric space consisting of the points of a scalene triangle is the trivial group. A similar space for an isosceles triangle is the cyclic group of order two, C2. A similar space for an equilateral triangle is D3, the dihedral group of order 6.
The isometry group of a two-dimensional sphere is the orthogonal group O(3).
The isometry group of the n-dimensional Euclidean space is the Euclidean group E(n).
The isometry group of the Poincaré disc model of the hyperbolic plane is the projective special unitary group PSU(1,1).
The isometry group of the Poincaré half-plane model of the hyperbolic plane is PSL(2,R).
The isometry group of Minkowski space is the Poincaré group.
Riemannian symmetric spaces are important cases where the isometry group is a Lie group.
See also
Point group
Point groups in two dimensions
Point groups in three dimensions
Fixed points of isometry groups in Euclidean space
References
Metric geometry
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https://en.wikipedia.org/wiki/Difference%20of%20two%20squares
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In mathematics, the difference of two squares is a squared (multiplied by itself) number subtracted from another squared number. Every difference of squares may be factored according to the identity
in elementary algebra.
Proof
The proof of the factorization identity is straightforward. Starting from the left-hand side, apply the distributive law to get
By the commutative law, the middle two terms cancel:
leaving
The resulting identity is one of the most commonly used in mathematics. Among many uses, it gives a simple proof of the AM–GM inequality in two variables.
The proof holds in any commutative ring.
Conversely, if this identity holds in a ring R for all pairs of elements a and b, then R is commutative. To see this, apply the distributive law to the right-hand side of the equation and get
.
For this to be equal to , we must have
for all pairs a, b, so R is commutative.
Geometrical demonstrations
The difference of two squares can also be illustrated geometrically as the difference of two square areas in a plane. In the diagram, the shaded part represents the difference between the areas of the two squares, i.e. . The area of the shaded part can be found by adding the areas of the two rectangles; , which can be factorized to . Therefore, .
Another geometric proof proceeds as follows: We start with the figure shown in the first diagram below, a large square with a smaller square removed from it. The side of the entire square is a, and the side of the small removed square is b. The area of the shaded region is . A cut is made, splitting the region into two rectangular pieces, as shown in the second diagram. The larger piece, at the top, has width a and height a-b. The smaller piece, at the bottom, has width a-b and height b. Now the smaller piece can be detached, rotated, and placed to the right of the larger piece. In this new arrangement, shown in the last diagram below, the two pieces together form a rectangle, whose width is and whose height is . This rectangle's area is . Since this rectangle came from rearranging the original figure, it must have the same area as the original figure. Therefore, .
Uses
Factorization of polynomials and simplification of expressions
The formula for the difference of two squares can be used for factoring polynomials that contain the square of a first quantity minus the square of a second quantity. For example, the polynomial can be factored as follows:
As a second example, the first two terms of can be factored as , so we have:
Moreover, this formula can also be used for simplifying expressions:
Complex number case: sum of two squares
The difference of two squares is used to find the linear factors of the sum of two squares, using complex number coefficients.
For example, the complex roots of can be found using difference of two squares:
(since )
Therefore, the linear factors are and .
Since the two factors found by this method are complex conjugates, we can use this in reverse
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https://en.wikipedia.org/wiki/Fractional%20Fourier%20transform
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In mathematics, in the area of harmonic analysis, the fractional Fourier transform (FRFT) is a family of linear transformations generalizing the Fourier transform. It can be thought of as the Fourier transform to the n-th power, where n need not be an integer — thus, it can transform a function to any intermediate domain between time and frequency. Its applications range from filter design and signal analysis to phase retrieval and pattern recognition.
The FRFT can be used to define fractional convolution, correlation, and other operations, and can also be further generalized into the linear canonical transformation (LCT). An early definition of the FRFT was introduced by Condon, by solving for the Green's function for phase-space rotations, and also by Namias, generalizing work of Wiener on Hermite polynomials.
However, it was not widely recognized in signal processing until it was independently reintroduced around 1993 by several groups. Since then, there has been a surge of interest in extending Shannon's sampling theorem for signals which are band-limited in the Fractional Fourier domain.
A completely different meaning for "fractional Fourier transform" was introduced by Bailey and Swartztrauber as essentially another name for a z-transform, and in particular for the case that corresponds to a discrete Fourier transform shifted by a fractional amount in frequency space (multiplying the input by a linear chirp) and evaluating at a fractional set of frequency points (e.g. considering only a small portion of the spectrum). (Such transforms can be evaluated efficiently by Bluestein's FFT algorithm.) This terminology has fallen out of use in most of the technical literature, however, in preference to the FRFT. The remainder of this article describes the FRFT.
Introduction
The continuous Fourier transform of a function is a unitary operator of space that maps the function to its frequential version (all expressions are taken in the sense, rather than pointwise):
and is determined by via the inverse transform
Let us study its n-th iterated defined by
and when n is a non-negative integer, and . Their sequence is finite since is a 4-periodic automorphism: for every function , .
More precisely, let us introduce the parity operator that inverts , . Then the following properties hold:
The FRFT provides a family of linear transforms that further extends this definition to handle non-integer powers of the FT.
Definition
Note: some authors write the transform in terms of the "order " instead of the "angle ", in which case the is usually times . Although these two forms are equivalent, one must be careful about which definition the author uses.
For any real , the -angle fractional Fourier transform of a function ƒ is denoted by and defined by
Formally, this formula is only valid when the input function is in a sufficiently nice space (such as L1 or Schwartz space), and is defined via a density argument, in a way similar to tha
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https://en.wikipedia.org/wiki/Homotopy%20groups%20of%20spheres
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In the mathematical field of algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other. They are examples of topological invariants, which reflect, in algebraic terms, the structure of spheres viewed as topological spaces, forgetting about their precise geometry. Unlike homology groups, which are also topological invariants, the homotopy groups are surprisingly complex and difficult to compute.
The -dimensional unit sphere — called the -sphere for brevity, and denoted as — generalizes the familiar circle () and the ordinary sphere (). The -sphere may be defined geometrically as the set of points in a Euclidean space of dimension located at a unit distance from the origin. The -th homotopy group summarizes the different ways in which the -dimensional sphere can be mapped continuously into the sphere . This summary does not distinguish between two mappings if one can be continuously deformed to the other; thus, only equivalence classes of mappings are summarized. An "addition" operation defined on these equivalence classes makes the set of equivalence classes into an abelian group.
The problem of determining falls into three regimes, depending on whether is less than, equal to, or greater than :
For , any mapping from to is homotopic (i.e., continuously deformable) to a constant mapping, i.e., a mapping that maps all of to a single point of . Therefore the homotopy group is the trivial group.
When , every map from to itself has a degree that measures how many times the sphere is wrapped around itself. This degree identifies the homotopy group with the group of integers under addition. For example, every point on a circle can be mapped continuously onto a point of another circle; as the first point is moved around the first circle, the second point may cycle several times around the second circle, depending on the particular mapping.
The most interesting and surprising results occur when . The first such surprise was the discovery of a mapping called the Hopf fibration, which wraps the 3-sphere around the usual sphere in a non-trivial fashion, and so is not equivalent to a one-point mapping.
The question of computing the homotopy group for positive turned out to be a central question in algebraic topology that has contributed to development of many of its fundamental techniques and has served as a stimulating focus of research. One of the main discoveries is that the homotopy groups are independent of for . These are called the stable homotopy groups of spheres and have been computed for values of up to 64. The stable homotopy groups form the coefficient ring of an extraordinary cohomology theory, called stable cohomotopy theory. The unstable homotopy groups (for ) are more erratic; nevertheless, they have been tabulated for . Most modern computations use spectral sequences, a technique first applied to homotopy groups of spheres by Jean-Pierre Serre. Several
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https://en.wikipedia.org/wiki/Efim%20Zelmanov
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Efim Isaakovich Zelmanov (; born 7 September 1955 in Khabarovsk) is a Russian-American mathematician, known for his work on combinatorial problems in nonassociative algebra and group theory, including his solution of the restricted Burnside problem. He was awarded a Fields Medal at the International Congress of Mathematicians in Zürich in 1994.
Zelmanov was born into a Jewish family in Khabarovsk, Soviet Union (now in Russia). He entered Novosibirsk State University in 1972, when he was 17 years old. He obtained a doctoral degree at Novosibirsk State University in 1980, and a higher degree at Leningrad State University in 1985. He had a position in Novosibirsk until 1987, when he left the Soviet Union.In 1990 he moved to the United States, becoming a professor at the University of Wisconsin–Madison. He was at the University of Chicago in 1994/5, then at Yale University. In 2011, he became a professor at the University of California, San Diego and a Distinguished Professor at the Korea Institute for Advanced Study. In 2022, he moved to the People's Republic of China and joined the Southern University of Science and Technology in Shenzhen, China. He served as a chair professor and the scientific director of the SUSTech International Center for Mathematics.
Zelmanov was elected a member of the U.S. National Academy of Sciences in 2001, becoming, at the age of 47, the youngest member of the mathematics section of the academy.
He is also an elected member of the American Academy of Arts and Sciences (1996) and a foreign member of the Korean Academy of Science and Engineering and of the Spanish Royal Academy of Sciences. In 2012 he became a fellow of the American Mathematical Society.
Zelmanov gave invited talks at the International Congress of Mathematicians in Warsaw (1983), Kyoto (1990) and Zurich (1994). He delivered the 2004 Turán Memorial Lectures. He was awarded Honorary Doctor degrees from the University of Hagen, Germany (1997),, the University of Alberta, Canada (2011), Taras Shevchenko National University of Kyiv, Ukraine (2012), the Universidad Internacional Menéndez Pelayo in Santander, Spain (2015) and the University of Lincoln, UK (2016).
Zelmanov's early work was on Jordan algebras in the case of infinite dimensions. He was able to show that Glennie's identity in a certain sense generates all identities that hold. He then showed that the Engel identity for Lie algebras implies nilpotence, in the case of infinite dimensions.
Notable publications
Zelʹmanov, E.I. Solution of the restricted Burnside problem for groups of odd exponent. Izv. Akad. Nauk SSSR Ser. Mat. 54 (1990), no. 1, 42–59, 221. English translation in Math. USSR-Izv. 36 (1991), no. 1, 41–60. doi:10.1070/IM1991v036n01ABEH001946
Zelʹmanov, E.I. Solution of the restricted Burnside problem for 2-groups. Mat. Sb. 182 (1991), no. 4, 568–592. English translation in Math. USSR-Sb. 72 (1992), no. 2, 543–565. doi:10.1070/SM1992v072n02ABEH001272
References
External links
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https://en.wikipedia.org/wiki/Hill%20cipher
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In classical cryptography, the Hill cipher is a polygraphic substitution cipher based on linear algebra. Invented by Lester S. Hill in 1929, it was the first polygraphic cipher in which it was practical (though barely) to operate on more than three symbols at once.
The following discussion assumes an elementary knowledge of matrices.
Encryption
Each letter is represented by a number modulo 26. Though this is not an essential feature of the cipher, this simple scheme is often used:
To encrypt a message, each block of n letters (considered as an n-component vector) is multiplied by an invertible n × n matrix, against modulus 26. To decrypt the message, each block is multiplied by the inverse of the matrix used for encryption.
The matrix used for encryption is the cipher key, and it should be chosen randomly from the set of invertible n × n matrices (modulo 26). The cipher can, of course, be adapted to an alphabet with any number of letters; all arithmetic just needs to be done modulo the number of letters instead of modulo 26.
Consider the message 'ACT', and the key below (or GYBNQKURP in letters):
Since 'A' is 0, 'C' is 2 and 'T' is 19, the message is the vector:
Thus the enciphered vector is given by:
which corresponds to a ciphertext of 'POH'. Now, suppose that our message is instead 'CAT', or:
This time, the enciphered vector is given by:
which corresponds to a ciphertext of 'FIN'. Every letter has changed. The Hill cipher has achieved Shannon's diffusion, and an n-dimensional Hill cipher can diffuse fully across n symbols at once.
Decryption
In order to decrypt, we turn the ciphertext back into a vector, then simply multiply by the inverse matrix of the key matrix (IFKVIVVMI in letters). We find that, modulo 26, the inverse of the matrix used in the previous example is:
Taking the previous example ciphertext of 'POH', we get:
which gets us back to 'ACT', as expected.
Two complications exist in picking the encrypting matrix:
Not all matrices have an inverse. The matrix will have an inverse if and only if its determinant is not zero.
The determinant of the encrypting matrix must not have any common factors with the modular base.
Thus, if we work modulo 26 as above, the determinant must be nonzero, and must not be divisible by 2 or 13. If the determinant is 0, or has common factors with the modular base, then the matrix cannot be used in the Hill cipher, and another matrix must be chosen (otherwise it will not be possible to decrypt). Fortunately, matrices which satisfy the conditions to be used in the Hill cipher are fairly common.
For our example key matrix:
So, modulo 26, the determinant is 25. Since and , 25 has no common factors with 26, and this matrix can be used for the Hill cipher.
The risk of the determinant having common factors with the modulus can be eliminated by making the modulus prime. Consequently, a useful variant of the Hill cipher adds 3 extra symbols (such as a space, a period and a question mark) to inc
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https://en.wikipedia.org/wiki/Classification%20of%20Clifford%20algebras
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In abstract algebra, in particular in the theory of nondegenerate quadratic forms on vector spaces, the structures of finite-dimensional real and complex Clifford algebras for a nondegenerate quadratic form have been completely classified. In each case, the Clifford algebra is algebra isomorphic to a full matrix ring over R, C, or H (the quaternions), or to a direct sum of two copies of such an algebra, though not in a canonical way. Below it is shown that distinct Clifford algebras may be algebra-isomorphic, as is the case of Cl1,1(R) and Cl2,0(R), which are both isomorphic as rings to the ring of two-by-two matrices over the real numbers.
The significance of this result is that the additional structure on a Clifford algebra relative to the "underlying" associative algebra — namely, the structure given by the grade involution automorphism and reversal anti-automorphism (and their composition, the Clifford conjugation) — is in general an essential part of its definition, not a procedural artifact of its construction as the quotient of a tensor algebra by an ideal. The category of Clifford algebras is not just a selection from the category of matrix rings, picking out those in which the ring product can be constructed as the Clifford product for some vector space and quadratic form. With few exceptions, "forgetting" the additional structure (in the category theory sense of a forgetful functor) is not reversible.
Continuing the example above: Cl1,1(R) and Cl2,0(R) share the same associative algebra structure, isomorphic to (and commonly denoted as) the matrix algebra M2(R). But they are distinguished by different choices of grade involution — of which two-dimensional subring, closed under the ring product, to designate as the even subring — and therefore of which of the various anti-automorphisms of M2(R) can accurately represent the reversal anti-automorphism of the Clifford algebra. These distinguished (anti-)automorphisms are structures on the tensor algebra which are preserved by the "quotient by ideal" construction of the Clifford algebra. The matrix algebra representation admits them, but only the Clifford algebra distinguishes them from other elements of the matrix algebra's (anti-)automorphism group.
Of the four real degrees of freedom in M2(R)), only one, generated by
,
is reversed by the obvious anti-automorphism of M2(R), the matrix transpose. (The designation refers to the Pauli matrices, which are here just a conventional way of naming the degrees of freedom in a two by two matrix.) In the case of Cl2,0(R), where both degrees of freedom in the odd part have positive norm, the obvious assignment of the even subalgebra to the matrices spanned by 1 and (and the odd part to the other two non-identity symmetric degrees of freedom) is consistent with selecting the matrix transpose as the reversal anti-automorphism. In this representation, the grade involution can be expressed as the inner automorphism .
But in the case of Cl1,1(R), t
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https://en.wikipedia.org/wiki/Kirby%20calculus
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In mathematics, the Kirby calculus in geometric topology, named after Robion Kirby, is a method for modifying framed links in the 3-sphere using a finite set of moves, the Kirby moves. Using four-dimensional Cerf theory, he proved that if M and N are 3-manifolds, resulting from Dehn surgery on framed links L and J respectively, then they are homeomorphic if and only if L and J are related by a sequence of Kirby moves. According to the Lickorish–Wallace theorem any closed orientable 3-manifold is obtained by such surgery on some link in the 3-sphere.
Some ambiguity exists in the literature on the precise use of the term "Kirby moves". Different presentations of "Kirby calculus" have a different set of moves and these are sometimes called Kirby moves. Kirby's original formulation involved two kinds of move, the "blow-up" and the "handle slide"; Roger Fenn and Colin Rourke exhibited an equivalent construction in terms of a single move, the Fenn–Rourke move, that appears in many expositions and extensions of the Kirby calculus. Dale Rolfsen's book, Knots and Links, from which many topologists have learned the Kirby calculus, describes a set of two moves: 1) delete or add a component with surgery coefficient infinity 2) twist along an unknotted component and modify surgery coefficients appropriately (this is called the Rolfsen twist). This allows an extension of the Kirby calculus to rational surgeries.
There are also various tricks to modify surgery diagrams. One such useful move is the slam-dunk.
An extended set of diagrams and moves are used for describing 4-manifolds. A framed link in the 3-sphere encodes instructions for attaching 2-handles to the 4-ball. (The 3-dimensional boundary of this manifold is the 3-manifold interpretation of the link diagram mentioned above.) 1-handles are denoted by either
a pair of 3-balls (the attaching region of the 1-handle) or, more commonly,
unknotted circles with dots.
The dot indicates that a neighborhood of a standard 2-disk with boundary the dotted circle is to be excised from the interior of the 4-ball. Excising this 2-handle is equivalent to adding a 1-handle; 3-handles and 4-handles are usually not indicated in the diagram.
Handle decomposition
A closed, smooth 4-manifold is usually described by a handle decomposition.
A 0-handle is just a ball, and the attaching map is disjoint union.
A 1-handle is attached along two disjoint 3-balls.
A 2-handle is attached along a solid torus; since this solid torus is embedded in a 3-manifold, there is a relation between handle decompositions on 4-manifolds, and knot theory in 3-manifolds.
A pair of handles with index differing by 1, whose cores link each other in a sufficiently simple way can be cancelled without changing the underlying manifold. Similarly, such a cancelling pair can be created.
Two different smooth handlebody decompositions of a smooth 4-manifold are related by a finite sequence of isotopies of the attaching maps, and the creation
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https://en.wikipedia.org/wiki/Reciprocal%20polynomial
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In algebra, given a polynomial
with coefficients from an arbitrary field, its reciprocal polynomial or reflected polynomial, denoted by or , is the polynomial
That is, the coefficients of are the coefficients of in reverse order. Reciprocal polynomials arise naturally in linear algebra as the characteristic polynomial of the inverse of a matrix.
In the special case where the field is the complex numbers, when
the conjugate reciprocal polynomial, denoted , is defined by,
where denotes the complex conjugate of , and is also called the reciprocal polynomial when no confusion can arise.
A polynomial is called self-reciprocal or palindromic if .
The coefficients of a self-reciprocal polynomial satisfy for all .
Properties
Reciprocal polynomials have several connections with their original polynomials, including:
.
is a root of a polynomial if and only if is a root of .
If then is irreducible if and only if is irreducible.
is primitive if and only if is primitive.
Other properties of reciprocal polynomials may be obtained, for instance:
A self-reciprocal polynomial of odd degree is divisible by x+1, hence is not irreducible if its degree is > 1.
Palindromic and antipalindromic polynomials
A self-reciprocal polynomial is also called palindromic because its coefficients, when the polynomial is written in the order of ascending or descending powers, form a palindrome. That is, if
is a polynomial of degree , then is palindromic if for .
Similarly, a polynomial of degree is called antipalindromic if for . That is, a polynomial is antipalindromic if .
Examples
From the properties of the binomial coefficients, it follows that the polynomials are palindromic for all positive integers , while the polynomials are palindromic when is even and antipalindromic when is odd.
Other examples of palindromic polynomials include cyclotomic polynomials and Eulerian polynomials.
Properties
If is a root of a polynomial that is either palindromic or antipalindromic, then is also a root and has the same multiplicity.
The converse is true: If a polynomial is such that is a root then if is also a root of the same multiplicity, then the polynomial is either palindromic or antipalindromic.
For any polynomial , the polynomial is palindromic and the polynomial is antipalindromic.
It follows that any polynomial can be written as the sum of a palindromic and an antipalindromic polynomial, since .
The product of two palindromic or antipalindromic polynomials is palindromic.
The product of a palindromic polynomial and an antipalindromic polynomial is antipalindromic.
A palindromic polynomial of odd degree is a multiple of (it has –1 as a root) and its quotient by is also palindromic.
An antipalindromic polynomial over a field with odd characteristic is a multiple of (it has 1 as a root) and its quotient by is palindromic.
An antipalindromic polynomial of even degree is a multiple of (it has −1 and 1 as roots) and its
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https://en.wikipedia.org/wiki/Laplace%20distribution
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In probability theory and statistics, the Laplace distribution is a continuous probability distribution named after Pierre-Simon Laplace. It is also sometimes called the double exponential distribution, because it can be thought of as two exponential distributions (with an additional location parameter) spliced together along the abscissa, although the term is also sometimes used to refer to the Gumbel distribution. The difference between two independent identically distributed exponential random variables is governed by a Laplace distribution, as is a Brownian motion evaluated at an exponentially distributed random time. Increments of Laplace motion or a variance gamma process evaluated over the time scale also have a Laplace distribution.
Definitions
Probability density function
A random variable has a distribution if its probability density function is
Here, is a location parameter and , which is sometimes referred to as the "diversity", is a scale parameter. If and , the positive half-line is exactly an exponential distribution scaled by 1/2.
The probability density function of the Laplace distribution is also reminiscent of the normal distribution; however, whereas the normal distribution is expressed in terms of the squared difference from the mean , the Laplace density is expressed in terms of the absolute difference from the mean. Consequently, the Laplace distribution has fatter tails than the normal distribution.
Cumulative distribution function
The Laplace distribution is easy to integrate (if one distinguishes two symmetric cases) due to the use of the absolute value function. Its cumulative distribution function is as follows:
The inverse cumulative distribution function is given by
Properties
Moments
Related distributions
If then .
If then .
If then (exponential distribution).
If then ．
If then .
If then (exponential power distribution).
If (normal distribution) then and .
If then (chi-squared distribution).
If then . (F-distribution)
If (uniform distribution) then .
If and (Bernoulli distribution) independent of , then .
If and independent of , then ．
If has a Rademacher distribution and then .
If and independent of , then .
If (geometric stable distribution) then .
The Laplace distribution is a limiting case of the hyperbolic distribution.
If with (Rayleigh distribution) then . Note that if , then with , which in turn equals the exponential distribution .
Given an integer , if (gamma distribution, using characterization), then (infinite divisibility)
If X has a Laplace distribution, then Y = eX has a log-Laplace distribution; conversely, if X has a log-Laplace distribution, then its logarithm has a Laplace distribution.
Probability of a Laplace being greater than another
Let be independent laplace random variables: and , and we want to compute .
The probability of can be reduced (using the properties below) to , where . This probability is equal to
When , both expressions a
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https://en.wikipedia.org/wiki/AWStats
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AWStats (Advanced Web Statistics) is an open source Web analytics reporting tool, suitable for analyzing data from Internet services such as web, streaming media, mail, and FTP servers. AWStats parses and analyzes server log files, producing HTML reports. Data is visually presented within reports by tables and bar graphs. Static reports can be created through a command line interface, and on-demand reporting is supported through a Web browser CGI program.
AWStats supports most major web server log file formats including Apache (NCSA combined/XLF/ELF log format or Common Log Format (CLF)), WebStar, IIS (W3C log format), and many other common web server log formats.
Development was moved from SourceForge to GitHub in 2014.
Cross-platform availability
Written in Perl, AWStats can be deployed on almost any operating system. It is a server-based website log analysis tool, with packages available for most Linux distributions. AWStats can be installed on a workstation, such as Microsoft Windows, for local use in situations where log files can be downloaded from a remote server.
Licensing
AWStats is licensed under the GNU General Public License (GPL).
Support
Proper web log analysis tool configuration and report interpretation requires a bit of technical and business knowledge. AWStats support resources include documentation and user community forums
Security considerations
The on-demand CGI program has been the object of security exploits, as is the case of many CGI programs. Organizations wishing to provide public access to their Web analytics reports should consider generating static HTML reports. The on-demand facility can still be used by restricting its use to internal users. Precautions should be taken against referrer spam. Referrer spam filtering functionality was added in version 6.5.
See also
List of web analytics software
References
External links
Official Website
Free software programmed in Perl
Free web analytics software
Perl software
Web analytics
Web log analysis software
Web technology
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https://en.wikipedia.org/wiki/Smith%20normal%20form
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In mathematics, the Smith normal form (sometimes abbreviated SNF) is a normal form that can be defined for any matrix (not necessarily square) with entries in a principal ideal domain (PID). The Smith normal form of a matrix is diagonal, and can be obtained from the original matrix by multiplying on the left and right by invertible square matrices. In particular, the integers are a PID, so one can always calculate the Smith normal form of an integer matrix. The Smith normal form is very useful for working with finitely generated modules over a PID, and in particular for deducing the structure of a quotient of a free module. It is named after the Irish mathematician Henry John Stephen Smith.
Definition
Let A be a nonzero m×n matrix over a principal ideal domain R. There exist invertible and -matrices S, T (with coefficients in R) such that the product S A T is
and the diagonal elements satisfy for all . This is the Smith normal form of the matrix A. The elements are unique up to multiplication by a unit and are called the elementary divisors, invariants, or invariant factors. They can be computed (up to multiplication by a unit) as
where (called i-th determinant divisor) equals the greatest common divisor of the determinants of all minors of the matrix A and .
Algorithm
The first goal is to find invertible square matrices and such that the product is diagonal. This is the hardest part of the algorithm. Once diagonality is achieved, it becomes relatively easy to put the matrix into Smith normal form. Phrased more abstractly, the goal is to show that, thinking of as a map from (the free -module of rank ) to (the free -module of rank ), there are isomorphisms and such that has the simple form of a diagonal matrix. The matrices and can be found by starting out with identity matrices of the appropriate size, and modifying each time a row operation is performed on in the algorithm by the corresponding column operation (for example, if row is added to row of , then column should be subtracted from column of to retain the product invariant), and similarly modifying for each column operation performed. Since row operations are left-multiplications and column operations are right-multiplications, this preserves the invariant where denote current values and denotes the original matrix; eventually the matrices in this invariant become diagonal. Only invertible row and column operations are performed, which ensures that and remain invertible matrices.
For , write for the number of prime factors of (these exist and are unique since any PID is also a unique factorization domain). In particular, is also a Bézout domain, so it is a gcd domain and the gcd of any two elements satisfies a Bézout's identity.
To put a matrix into Smith normal form, one can repeatedly apply the following, where loops from 1 to .
Step I: Choosing a pivot
Choose to be the smallest column index of with a non-zero entry, starting the search
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https://en.wikipedia.org/wiki/Annihilator
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Annihilator(s) may refer to:
Mathematics
Annihilator (ring theory)
Annihilator (linear algebra), the annihilator of a subset of a vector subspace
Annihilator method, a type of differential operator, used in a particular method for solving differential equations
Annihilator matrix, in regression analysis
Music
Annihilator (band), a Canadian heavy metal band
Annihilator (album), a 2010 album by the aforementioned band
Other media
Annihilator (Justice League), an automaton in the fictional series Justice League Unlimited
Annihilators (Marvel Comics), a team of superheroes
Annihilator, a 2015 science fiction comic by Grant Morrison and Frazer Irving
Annihilator (film), a 1986 television film starring Mark Lindsay Chapman
The Annihilators (film), a 1985 action film by Charles E. Sellier Jr.
The Annihilators (novel), a 1983 novel by Donald Hamilton
See also
Annihilation (disambiguation)
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https://en.wikipedia.org/wiki/Linear%20approximation
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In mathematics, a linear approximation is an approximation of a general function using a linear function (more precisely, an affine function). They are widely used in the method of finite differences to produce first order methods for solving or approximating solutions to equations.
Definition
Given a twice continuously differentiable function of one real variable, Taylor's theorem for the case states that
where is the remainder term. The linear approximation is obtained by dropping the remainder:
This is a good approximation when is close enough to since a curve, when closely observed, will begin to resemble a straight line. Therefore, the expression on the right-hand side is just the equation for the tangent line to the graph of at . For this reason, this process is also called the tangent line approximation. Linear approximations in this case are further improved when the second derivative of a, , is sufficiently small (close to zero) (i.e., at or near an inflection point).
If is concave down in the interval between and , the approximation will be an overestimate (since the derivative is decreasing in that interval). If is concave up, the approximation will be an underestimate.
Linear approximations for vector functions of a vector variable are obtained in the same way, with the derivative at a point replaced by the Jacobian matrix. For example, given a differentiable function with real values, one can approximate for close to by the formula
The right-hand side is the equation of the plane tangent to the graph of at
In the more general case of Banach spaces, one has
where is the Fréchet derivative of at .
Applications
Optics
Gaussian optics is a technique in geometrical optics that describes the behaviour of light rays in optical systems by using the paraxial approximation, in which only rays which make small angles with the optical axis of the system are considered. In this approximation, trigonometric functions can be expressed as linear functions of the angles. Gaussian optics applies to systems in which all the optical surfaces are either flat or are portions of a sphere. In this case, simple explicit formulae can be given for parameters of an imaging system such as focal distance, magnification and brightness, in terms of the geometrical shapes and material properties of the constituent elements.
Period of oscillation
The period of swing of a simple gravity pendulum depends on its length, the local strength of gravity, and to a small extent on the maximum angle that the pendulum swings away from vertical, , called the amplitude. It is independent of the mass of the bob. The true period T of a simple pendulum, the time taken for a complete cycle of an ideal simple gravity pendulum, can be written in several different forms (see pendulum), one example being the infinite series:
where L is the length of the pendulum and g is the local acceleration of gravity.
However, if one takes the linear approximation (i.
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https://en.wikipedia.org/wiki/Alexander%20polynomial
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In mathematics, the Alexander polynomial is a knot invariant which assigns a polynomial with integer coefficients to each knot type. James Waddell Alexander II discovered this, the first knot polynomial, in 1923. In 1969, John Conway showed a version of this polynomial, now called the Alexander–Conway polynomial, could be computed using a skein relation, although its significance was not realized until the discovery of the Jones polynomial in 1984. Soon after Conway's reworking of the Alexander polynomial, it was realized that a similar skein relation was exhibited in Alexander's paper on his polynomial.
Definition
Let K be a knot in the 3-sphere. Let X be the infinite cyclic cover of the knot complement of K. This covering can be obtained by cutting the knot complement along a Seifert surface of K and gluing together infinitely many copies of the resulting manifold with boundary in a cyclic manner. There is a covering transformation t acting on X. Consider the first homology (with integer coefficients) of X, denoted . The transformation t acts on the homology and so we can consider a module over the ring of Laurent polynomials . This is called the Alexander invariant or Alexander module.
The module is finitely presentable; a presentation matrix for this module is called the Alexander matrix. If the number of generators, , is less than or equal to the number of relations, , then we consider the ideal generated by all minors of the matrix; this is the zeroth Fitting ideal or Alexander ideal and does not depend on choice of presentation matrix. If , set the ideal equal to 0. If the Alexander ideal is principal, take a generator; this is called an Alexander polynomial of the knot. Since this is only unique up to multiplication by the Laurent monomial , one often fixes a particular unique form. Alexander's choice of normalization is to make the polynomial have a positive constant term.
Alexander proved that the Alexander ideal is nonzero and always principal. Thus an Alexander polynomial always exists, and is clearly a knot invariant, denoted . It turns out that the Alexander polynomial of a knot is the same polynomial for the mirror image knot. In other words, it cannot distinguish between a knot and its mirror image.
Computing the polynomial
The following procedure for computing the Alexander polynomial was given by J. W. Alexander in his paper.
Take an oriented diagram of the knot with crossings; there are regions of the knot diagram. To work out the Alexander polynomial, first one must create an incidence matrix of size . The rows correspond to the crossings, and the columns to the regions. The values for the matrix entries are either .
Consider the entry corresponding to a particular region and crossing. If the region is not adjacent to the crossing, the entry is 0. If the region is adjacent to the crossing, the entry depends on its location. The following table gives the entry, determined by the location of the
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https://en.wikipedia.org/wiki/Felix%20Iversen
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Felix Christian Herbert Iversen (22 October 1887 – 31 July 1973) was a Finnish mathematician and a pacifist. He was a student of Ernst Lindelöf, and later an associate professor of mathematics at the University of Helsinki. Although he stopped performing serious research in mathematics around 1922, he continued working as a professor until his retirement in 1954 and published a textbook on mathematics in 1950. The Soviet Union awarded Felix Iversen the Stalin Peace Prize in 1954.
References
1887 births
1973 deaths
Finnish mathematicians
Academic staff of the University of Helsinki
Stalin Peace Prize recipients
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https://en.wikipedia.org/wiki/Hasse%20norm%20theorem
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In number theory, the Hasse norm theorem states that if L/K is a cyclic extension of number fields, then if a nonzero element of K is a local norm everywhere, then it is a global norm.
Here to be a global norm means to be an element k of K such that there is an element l of L with ; in other words k is a relative norm of some element of the extension field L. To be a local norm means that for some prime p of K and some prime P of L lying over K, then k is a norm from LP; here the "prime" p can be an archimedean valuation, and the theorem is a statement about completions in all valuations, archimedean and non-archimedean.
The theorem is no longer true in general if the extension is abelian but not cyclic. Hasse gave the counterexample that 3 is a local norm everywhere for the extension but is not a global norm. Serre and Tate showed that another counterexample is given by the field where every rational square is a local norm everywhere but is not a global norm.
This is an example of a theorem stating a local-global principle.
The full theorem is due to . The special case when the degree n of the extension is 2 was proved by , and the special case when n is prime was proved by Furtwangler in 1902.
The Hasse norm theorem can be deduced from the theorem that an element of the Galois cohomology group H2(L/K) is trivial if it is trivial locally everywhere, which is in turn equivalent to the deep theorem that the first cohomology of the idele class group vanishes. This is true for all finite Galois extensions of number fields, not just cyclic ones. For cyclic extensions the group H2(L/K) is isomorphic to the Tate cohomology group H0(L/K) which describes which elements are norms, so for cyclic extensions it becomes Hasse's theorem that an element is a norm if it is a local norm everywhere.
See also
Grunwald–Wang theorem, about when an element that is a power everywhere locally is a power.
References
H. Hasse, "A history of class field theory", in J.W.S. Cassels and A. Frohlich (edd), Algebraic number theory, Academic Press, 1973. Chap.XI.
G. Janusz, Algebraic number fields, Academic Press, 1973. Theorem V.4.5, p. 156
Class field theory
Theorems in algebraic number theory
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https://en.wikipedia.org/wiki/Tollerton%2C%20Nottinghamshire
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Tollerton is an English village and civil parish in the Rushcliffe district of Nottinghamshire, just south-east of Nottingham. Statistics from the 2021 census show the population of the village has increased to 2,486.
Governance
Tollerton has a parish council and is represented on Rushcliffe Borough Council. The Member of Parliament (MP) for Rushcliffe since December 2019 is the Conservative Ruth Edwards.
Tollerton Hall
St Hugh's College was founded in 1948 at Tollerton Hall by the Roman Catholic Diocese of Nottingham as its junior seminary, accepting boys from aged 11 upwards, but by 1969 it had opened its doors to secular students. It closed in 1986, leaving the hall as a corporate HQ until June 2017, when it was bought by a businessman, Ian Kershaw, for use as a private home again.
Event and amenities
There is an annual village fayre held in June. This gathers residents around craft stalls, entertainments, refreshments and small exhibitions. The money generated helps local charities.
The pub named the Air Hostess recalls Nottingham Airport at Tollerton. It is unusual in having a piste for playing pétanque. It used to have a large carved sign depicting an air hostess. However, in 2011, the then owners of the premises, Everards Brewery, decided to refurbish them and in doing so replaced the original sign with a conventional hanging sign depicting an air hostess. This changed to a hanging sign depicting Tollerton Hall, an aircraft and fields. A scheme was put in operation to save the pub from closure by introducing local ownership. This led to the establishment of a community trust, in which shares were issued, and the pub freehold was purchased. It reopened with a new tenant landlord in July 2020.
Tollerton has two churches. St Peter's Anglican Church dates from the end of the 12th century. Developments in 1909 resulted in the church as it is today. There is a Methodist church located in a modern building at the village's southern border.
The few shops include a post office, a petrol station which includes a mini Waitrose, and a restaurant serving oriental-style food, The Charde. The village is host to a hairdressers, a pet shop and a pet salon. The Parish Rooms at the end of the parade of shops in Burnside Grove serve as a centre for local community activities and meetings. Regular pop up shops visit the village.
Outdoor amenities include a park with an multi-use games area for five-a-side football, basketball and tennis and a full-sized grass main football pitch.
The park was refurbished with new children's play equipment in April 2008 and again more recently through the active Tollerton project. This added the first interactive play equipment in Rushcliffe, the largest single piece of adventure play equipment produced for a UK park by manufacturer Produlic and a community gym.
Education
Tollerton Primary School takes children of 5–11 years of age. It is a member of Equals Trust Multi Academy Trust, based in nearby Keyworth.
Tollert
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https://en.wikipedia.org/wiki/Hasse%E2%80%93Minkowski%20theorem
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The Hasse–Minkowski theorem is a fundamental result in number theory which states that two quadratic forms over a number field are equivalent if and only if they are equivalent locally at all places, i.e. equivalent over every completion of the field (which may be real, complex, or p-adic). A related result is that a quadratic space over a number field is isotropic if and only if it is isotropic locally everywhere, or equivalently, that a quadratic form over a number field nontrivially represents zero if and only if this holds for all completions of the field. The theorem was proved in the case of the field of rational numbers by Hermann Minkowski and generalized to number fields by Helmut Hasse. The same statement holds even more generally for all global fields.
Importance
The importance of the Hasse–Minkowski theorem lies in the novel paradigm it presented for answering arithmetical questions: in order to determine whether an equation of a certain type has a solution in rational numbers, it is sufficient to test whether it has solutions over complete fields of real and p-adic numbers, where analytic considerations, such as Newton's method and its p-adic analogue, Hensel's lemma, apply. This is encapsulated in the idea of a local-global principle, which is one of the most fundamental techniques in arithmetic geometry.
Application to the classification of quadratic forms
The Hasse–Minkowski theorem reduces the problem of classifying quadratic forms over a number field K up to equivalence to the set of analogous but much simpler questions over local fields. Basic invariants of a nonsingular quadratic form are its dimension, which is a positive integer, and its discriminant modulo the squares in K, which is an element of the multiplicative group K*/K*2. In addition, for every place v of K, there is an invariant coming from the completion Kv. Depending on the choice of v, this completion may be the real numbers R, the complex numbers C, or a p-adic number field, each of which has different kinds of invariants:
Case of R. By Sylvester's law of inertia, the signature (or, alternatively, the negative index of inertia) is a complete invariant.
Case of C. All nonsingular quadratic forms of the same dimension are equivalent.
Case of Qp and its algebraic extensions. Forms of the same dimension are classified up to equivalence by their Hasse invariant.
These invariants must satisfy some compatibility conditions: a parity relation (the sign of the discriminant must match the negative index of inertia) and a product formula (a local–global relation). Conversely, for every set of invariants satisfying these relations, there is a quadratic form over K with these invariants.
References
Quadratic forms
Theorems in number theory
Hermann Minkowski
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https://en.wikipedia.org/wiki/Hasse%27s%20theorem
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In mathematics, there are several theorems of Helmut Hasse that are sometimes called Hasse's theorem:
Hasse norm theorem
Hasse's theorem on elliptic curves
Hasse–Arf theorem
Hasse–Minkowski theorem
See also
Hasse principle, the principle that an integer equation can be solved by piecing together modular solutions
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https://en.wikipedia.org/wiki/Calculus%20%28disambiguation%29
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Calculus (from Latin calculus meaning ‘pebble’, plural calculī) in its most general sense is any method or system of calculation.
Calculus may refer to:
Biology
Calculus (spider), a genus of the family Oonopidae
Caseolus calculus, a genus and species of small land snails
Mathematics
Infinitesimal calculus (or simply Calculus), which investigate motion and rates of change
Differential calculus
Integral calculus
Non-standard calculus, an approach to infinitesimal calculus using Robinson's infinitesimals
Calculus of sums and differences (difference operator), also called the finite-difference calculus, a discrete analogue of "calculus"
Functional calculus, a way to apply various types of functions to operators
Schubert calculus, a branch of algebraic geometry
Tensor calculus (also called tensor analysis), a generalization of vector calculus that encompasses tensor fields
Vector calculus (also called vector analysis), comprising specialized notations for multivariable analysis of vectors in an inner-product space
Matrix calculus, a specialized notation for multivariable calculus over spaces of matrices
Numerical calculus (also called numerical analysis), the study of numerical approximations
Umbral calculus, the combinatorics of certain operations on polynomials
The calculus of variations, a field of study that deals with extremizing functionals
Itô calculus An extension of calculus to stochastic processes.
Logic
Logical calculus, a formal system that defines a language and rules to derive an expression from premises
Propositional calculus, specifies the rules of inference governing the logic of propositions
Predicate calculus, specifies the rules of inference governing the logic of predicates
Proof calculus, a framework for expressing systems of logical inference
Sequent calculus, a proof calculus for first-order logic
Cirquent calculus, a proof calculus based on graph-style structures called cirquents
Situation calculus, a framework for describing relations within a dynamic system
Event calculus, a model for reasoning about events and their effects
Fluent calculus, a model for describing relations within a dynamic system
Calculus of relations, the manipulation of binary relations with the algebra of sets, composition of relations, and transpose relations
Epsilon calculus, a logical language which replaces quantifiers with the epsilon operator
Fitch-style calculus, a method for constructing formal proofs used in first-order logic
Modal μ-calculus, a common temporal logic used by formal verification methods such as model checking
Medicine
Calculus (dental), deposits of calcium phosphate salts on teeth, also known as tartar
Calculus (medicine), a stone formed in the body such as a gall stone or kidney stone
Physics
Bondi k-calculus, a method used in relativity theory
Jones calculus, used in optics to describe polarized light
Mueller calculus, used in optics to handle Stokes vectors, which describe the
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https://en.wikipedia.org/wiki/Math%20League
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Math League is a math competition for elementary, middle, and high school students in the United States, Canada, and other countries. The Math League was founded in 1977 by two high school mathematics teachers, Steven R. Conrad and Daniel Flegler. Math Leagues, Inc. publishes old contests through a series of books entitled Math League Press. The purpose of the Math League Contests is to provide students "an enriching opportunity to participate in an academically-oriented activity" and to let students "gain recognition for mathematical achievement".
Math League runs three contest formats:
Grades 4-5: 30 multiple-choice questions to solve in 30 minutes, covering arithmetic and basic principles
Grades 6-8: 35 multiple-choice questions to solve in 30 minutes, covering advanced arithmetic and basic topics in geometry and algebra
Grades 9-12: Series of 6 contests. Each contest contains 6 short-answer questions to solve in 30 minutes, covering geometry, algebra, trigonometry, and other advanced pre-calculus topics.
Only plain paper, pencil or pen, and a calculator without QWERTY keyboard are allowed.
Students who score above 12 points in grades 4 and 5, and above 15 points in grades 6-8 are awarded a 'Certificate of Merit." Which means they win
References
External links
Math League Homepage
Mathematics competitions
Recurring events established in 1977
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https://en.wikipedia.org/wiki/Geometry%20%28Robert%20Rich%20album%29
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Geometry (1991) is an album by the American ambient and electronic musician Robert Rich. Although completed in 1988, this album was not released until three years later.
This album is more active and structured than any of his previous works. The music was inspired in part by the complex patterns of Islamic designs like those found at the Alhambra, and features complex just intonation. The first three tracks consist of complex layered sequences of electronic notes in rich, organic-sounding chime tones. Rich revisited this style in Gaudí (1991) and Electric Ladder (2006). Tracks 4 through 7 are slow textures that more common to Robert Rich’s work. The album ends with another active piece similar to the first three tracks.
Tracks 1, 2, 4 and 8 were mixed by Robert Rich and future collaborator Steve Roach at Roach’s studio in Venice, California. This album was released in a two CD set with Numena in 1997.
Track listing
"Primes, Part 1" – 5:20
"Primes, Part 2" – 6:34
"Interlocking Circles" – 12:35
"Geometry of the Skies" 13:48
"Nesting Ground" – 6:13
"Geomancy" – 10:35
"Amrita (Water of Life)" – 6:39
"Logos" – 9:57
1988 albums
Robert Rich (musician) albums
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https://en.wikipedia.org/wiki/Loop-erased%20random%20walk
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In mathematics, loop-erased random walk is a model for a random simple path with important applications in combinatorics, physics and quantum field theory. It is intimately connected to the uniform spanning tree, a model for a random tree. See also random walk for more general treatment of this topic.
Definition
Assume G is some graph and is some path of length n on G. In other words, are vertices of G such that and are connected by an edge. Then the loop erasure of is a new simple path created by erasing all the loops of in chronological order. Formally, we define indices inductively using
where "max" here means up to the length of the path . The induction stops when for some we have . Assume this happens at J i.e. is the last . Then the loop erasure of , denoted by is a simple path of length J defined by
Now let G be some graph, let v be a vertex of G, and let R be a random walk on G starting from v. Let T be some stopping time for R. Then the loop-erased random walk until time T is LE(R([1,T])). In other words, take R from its beginning until T — that's a (random) path — erase all the loops in chronological order as above — you get a random simple path.
The stopping time T may be fixed, i.e. one may perform n steps and then loop-erase. However, it is usually more natural to take T to be the hitting time in some set. For example, let G be the graph Z2 and let R be a random walk starting from the point (0,0). Let T be the time when R first hits the circle of radius 100 (we mean here of course a discretized circle). LE(R) is called the loop-erased random walk starting at (0,0) and stopped at the circle.
Uniform spanning tree
For any graph G, a spanning tree of G is a subgraph of G containing all vertices and some of the edges, which is a tree, i.e. connected and with no cycles. A spanning tree chosen randomly from among all possible spanning trees with equal probability is called a uniform spanning tree. There are typically exponentially many spanning trees (too many to generate them all and then choose one randomly); instead, uniform spanning trees can be generated more efficiently by an algorithm called Wilson's algorithm which uses loop-erased random walks.
The algorithm proceeds according to the following steps. First, construct a single-vertex tree T by choosing (arbitrarily) one vertex. Then, while the tree T constructed so far does not yet include all of the vertices of the graph, let v be an arbitrary vertex that is not in T, perform a loop-erased random walk from v until reaching a vertex in T, and add the resulting path to T. Repeating this process until all vertices are included produces a uniformly distributed tree, regardless of the arbitrary choices of vertices at each step.
A connection in the other direction is also true. If v and w are two vertices in G then, in any spanning tree, they are connected by a unique path. Taking this path in the uniform spanning tree gives a random simple path. It turns out that the d
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https://en.wikipedia.org/wiki/Modular%20representation%20theory
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Modular representation theory is a branch of mathematics, and is the part of representation theory that studies linear representations of finite groups over a field K of positive characteristic p, necessarily a prime number. As well as having applications to group theory, modular representations arise naturally in other branches of mathematics, such as algebraic geometry, coding theory, combinatorics and number theory.
Within finite group theory, character-theoretic results proved by Richard Brauer using modular representation theory played an important role in early progress towards the classification of finite simple groups, especially for simple groups whose characterization was not amenable to purely group-theoretic methods because their Sylow 2-subgroups were too small in an appropriate sense. Also, a general result on embedding of elements of order 2 in finite groups called the Z* theorem, proved by George Glauberman using the theory developed by Brauer, was particularly useful in the classification program.
If the characteristic p of K does not divide the order |G|, then modular representations are completely reducible, as with ordinary (characteristic 0) representations, by virtue of Maschke's theorem. In the other case, when |G| ≡ 0 mod p, the process of averaging over the group needed to prove Maschke's theorem breaks down, and representations need not be completely reducible. Much of the discussion below implicitly assumes that the field K is sufficiently large (for example, K algebraically closed suffices), otherwise some statements need refinement.
History
The earliest work on representation theory over finite fields is by who showed that when p does not divide the order of the group, the representation theory is similar to that in characteristic 0. He also investigated modular invariants of some finite groups. The systematic study of modular representations, when the characteristic p divides the order of the group, was started by and was continued by him for the next few decades.
Example
Finding a representation of the cyclic group of two elements over F2 is equivalent to the problem of finding matrices whose square is the identity matrix. Over every field of characteristic other than 2, there is always a basis such that the matrix can be written as a diagonal matrix with only 1 or −1 occurring on the diagonal, such as
Over F2, there are many other possible matrices, such as
Over an algebraically closed field of positive characteristic, the representation theory of a finite cyclic group is fully explained by the theory of the Jordan normal form. Non-diagonal Jordan forms occur when the characteristic divides the order of the group.
Ring theory interpretation
Given a field K and a finite group G, the group algebra K[G] (which is the K-vector space with K-basis consisting of the elements of G, endowed with algebra multiplication by extending the multiplication of G by linearity) is an Artinian ring.
When the order of
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https://en.wikipedia.org/wiki/Multiplicative%20group
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In mathematics and group theory, the term multiplicative group refers to one of the following concepts:
the group under multiplication of the invertible elements of a field, ring, or other structure for which one of its operations is referred to as multiplication. In the case of a field F, the group is , where 0 refers to the zero element of F and the binary operation • is the field multiplication,
the algebraic torus GL(1)..
Examples
The multiplicative group of integers modulo n is the group under multiplication of the invertible elements of . When n is not prime, there are elements other than zero that are not invertible.
The multiplicative group of positive real numbers is an abelian group with 1 its identity element. The logarithm is a group isomorphism of this group to the additive group of real numbers, .
The multiplicative group of a field is the set of all nonzero elements: , under the multiplication operation. If is finite of order q (for example q = p a prime, and ), then the multiplicative group is cyclic: .
Group scheme of roots of unity
The group scheme of n-th roots of unity is by definition the kernel of the n-power map on the multiplicative group GL(1), considered as a group scheme. That is, for any integer n > 1 we can consider the morphism on the multiplicative group that takes n-th powers, and take an appropriate fiber product of schemes, with the morphism e that serves as the identity.
The resulting group scheme is written μn (or ). It gives rise to a reduced scheme, when we take it over a field K, if and only if the characteristic of K does not divide n. This makes it a source of some key examples of non-reduced schemes (schemes with nilpotent elements in their structure sheaves); for example μp over a finite field with p elements for any prime number p.
This phenomenon is not easily expressed in the classical language of algebraic geometry. For example, it turns out to be of major importance in expressing the duality theory of abelian varieties in characteristic p (theory of Pierre Cartier). The Galois cohomology of this group scheme is a way of expressing Kummer theory.
See also
Multiplicative group of integers modulo n
Additive group
Notes
References
Michiel Hazewinkel, Nadiya Gubareni, Nadezhda Mikhaĭlovna Gubareni, Vladimir V. Kirichenko. Algebras, rings and modules. Volume 1. 2004. Springer, 2004.
Algebraic structures
Group theory
Field (mathematics)
|
https://en.wikipedia.org/wiki/Lyons%20group
|
In the area of modern algebra known as group theory, the Lyons group Ly or Lyons-Sims group LyS is a sporadic simple group of order
283756711313767
= 51765179004000000
≈ 5.
History
Ly is one of the 26 sporadic groups and was discovered by Richard Lyons and Charles Sims in 1972-73. Lyons characterized 51765179004000000 as the unique possible order of any finite simple group where the centralizer of some involution is isomorphic to the nontrivial central extension of the alternating group A11 of degree 11 by the cyclic group C2. proved the existence of such a group and its uniqueness up to isomorphism with a combination of permutation group theory and machine calculations.
When the McLaughlin sporadic group was discovered, it was noticed that a centralizer of one of its involutions was the perfect double cover of the alternating group A8. This suggested considering the double covers of the other alternating groups An as possible centralizers of involutions in simple groups. The cases n ≤ 7 are ruled out by the Brauer–Suzuki theorem, the case n = 8 leads to the McLaughlin group, the case n = 9 was ruled out by Zvonimir Janko, Lyons himself ruled out the case n = 10 and found the Lyons group for n = 11, while the cases n ≥ 12 were ruled out by J.G. Thompson and Ronald Solomon.
The Schur multiplier and the outer automorphism group are both trivial.
Since 37 and 67 are not supersingular primes, the Lyons group cannot be a subquotient of the monster group. Thus it is one of the 6 sporadic groups called the pariahs.
Representations
showed that the Lyons group has a modular representation of dimension 111 over the field of five elements, which is the smallest dimension of any faithful linear representation and is one of the easiest ways of calculating with it. It has also been given by several complicated presentations in terms of generators and relations, for instance those given by or .
The smallest faithful permutation representation is a rank 5 permutation representation on 8835156 points with stabilizer G2(5). There is also a slightly larger rank 5 permutation representation on 9606125 points with stabilizer 3.McL:2.
Maximal subgroups
found the 9 conjugacy classes of maximal subgroups of Ly as follows:
G2(5)
3.McL:2
53.PSL3(5)
2.A11
51+4:4.S6
35:(2 × M11)
32+4:2.A5.D8
67:22
37:18
References
Richard Lyons (1972,5) "Evidence for a new finite simple group", Journal of Algebra 20:540–569 and 34:188–189.
External links
MathWorld: Lyons group
Atlas of Finite Group Representations: Lyons group
Sporadic groups
|
https://en.wikipedia.org/wiki/Held%20group
|
In the area of modern algebra known as group theory, the Held group He is a sporadic simple group of order
21033527317 = 4030387200
≈ 4.
History
He is one of the 26 sporadic groups and was found by during an investigation of simple groups containing an involution whose centralizer is isomorphic to that of an involution in the Mathieu group M24. A second such group is the linear group L5(2). The Held group is the third possibility, and its construction was completed by John McKay and Graham Higman.
The outer automorphism group has order 2 and the Schur multiplier is trivial.
Representations
The smallest faithful complex representation has dimension 51; there are two such representations that are duals of each other.
It centralizes an element of order 7 in the Monster group. As a result the prime 7 plays a special role in the theory of the group; for example, the smallest representation of the Held group over any field is the 50-dimensional representation over the field with 7 elements, and it acts naturally on a vertex operator algebra over the field with 7 elements.
The smallest permutation representation is a rank 5 action on 2058 points with point stabilizer Sp4(4):2.
The automorphism group He:2 of the Held group He is a subgroup of the Fischer group Fi24.
Generalized monstrous moonshine
Conway and Norton suggested in their 1979 paper that monstrous moonshine is not limited to the monster, but that similar phenomena may be found for other groups. Larissa Queen and others subsequently found that one can construct the expansions of many Hauptmoduln from simple combinations of dimensions of sporadic groups. For He, the relevant McKay-Thompson series is where one can set the constant term a(0) = 10 (),
and η(τ) is the Dedekind eta function.
Presentation
It can be defined in terms of the generators a and b and relations
Maximal subgroups
found the 11 conjugacy classes of maximal subgroups of He as follows:
S4(4):2
22.L3(4).S3
26:3.S6
26:3.S6
21+6.L3(2)
72:2.L2(7)
3.S7
71+2:(3 × S3)
S4 × L3(2)
7:3 × L3(2)
52:4A4
References
External links
MathWorld: Held group
Atlas of Finite Group Representations: Held group\
Sporadic groups
|
https://en.wikipedia.org/wiki/Galois%20cohomology
|
In mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups. A Galois group G associated to a field extension L/K acts in a natural way on some abelian groups, for example those constructed directly from L, but also through other Galois representations that may be derived by more abstract means. Galois cohomology accounts for the way in which taking Galois-invariant elements fails to be an exact functor.
History
The current theory of Galois cohomology came together around 1950, when it was realised that the Galois cohomology of ideal class groups in algebraic number theory was one way to formulate class field theory, at the time it was in the process of ridding itself of connections to L-functions. Galois cohomology makes no assumption that Galois groups are abelian groups, so this was a non-abelian theory. It was formulated abstractly as a theory of class formations. Two developments of the 1960s turned the position around. Firstly, Galois cohomology appeared as the foundational layer of étale cohomology theory (roughly speaking, the theory as it applies to zero-dimensional schemes). Secondly, non-abelian class field theory was launched as part of the Langlands philosophy.
The earliest results identifiable as Galois cohomology had been known long before, in algebraic number theory and the arithmetic of elliptic curves. The normal basis theorem implies that the first cohomology group of the additive group of L will vanish; this is a result on general field extensions, but was known in some form to Richard Dedekind. The corresponding result for the multiplicative group is known as Hilbert's Theorem 90, and was known before 1900. Kummer theory was another such early part of the theory, giving a description of the connecting homomorphism coming from the m-th power map.
In fact, for a while the multiplicative case of a 1-cocycle for groups that are not necessarily cyclic was formulated as the solubility of Noether's equations, named for Emmy Noether; they appear under this name in Emil Artin's treatment of Galois theory, and may have been folklore in the 1920s. The case of 2-cocycles for the multiplicative group is that of the Brauer group, and the implications seem to have been well known to algebraists of the 1930s.
In another direction, that of torsors, these were already implicit in the infinite descent arguments of Fermat for elliptic curves. Numerous direct calculations were done, and the proof of the Mordell–Weil theorem had to proceed by some surrogate of a finiteness proof for a particular H1 group. The 'twisted' nature of objects over fields that are not algebraically closed, which are not isomorphic but become so over the algebraic closure, was also known in many cases linked to other algebraic groups (such as quadratic forms, simple algebras, Severi–Brauer varieties), in the 1930s, before the general theory arrived.
The needs
|
https://en.wikipedia.org/wiki/Unordered%20pair
|
In mathematics, an unordered pair or pair set is a set of the form {a, b}, i.e. a set having two elements a and b with no particular relation between them, where {a, b} = {b, a}. In contrast, an ordered pair (a, b) has a as its first element and b as its second element, which means (a, b) ≠ (b, a).
While the two elements of an ordered pair (a, b) need not be distinct, modern authors only call {a, b} an unordered pair if a ≠ b.
But for a few authors a singleton is also considered an unordered pair, although today, most would say that {a, a} is a multiset. It is typical to use the term unordered pair even in the situation where the elements a and b could be equal, as long as this equality has not yet been established.
A set with precisely two elements is also called a 2-set or (rarely) a binary set.
An unordered pair is a finite set; its cardinality (number of elements) is 2 or (if the two elements are not distinct) 1.
In axiomatic set theory, the existence of unordered pairs is required by an axiom, the axiom of pairing.
More generally, an unordered n-tuple is a set of the form {a1, a2,... an}.
Notes
References
.
Basic concepts in set theory
|
https://en.wikipedia.org/wiki/Monte%20Carlo%20integration
|
In mathematics, Monte Carlo integration is a technique for numerical integration using random numbers. It is a particular Monte Carlo method that numerically computes a definite integral. While other algorithms usually evaluate the integrand at a regular grid, Monte Carlo randomly chooses points at which the integrand is evaluated. This method is particularly useful for higher-dimensional integrals.
There are different methods to perform a Monte Carlo integration, such as uniform sampling, stratified sampling, importance sampling, sequential Monte Carlo (also known as a particle filter), and mean-field particle methods.
Overview
In numerical integration, methods such as the trapezoidal rule use a deterministic approach. Monte Carlo integration, on the other hand, employs a non-deterministic approach: each realization provides a different outcome. In Monte Carlo, the final outcome is an approximation of the correct value with respective error bars, and the correct value is likely to be within those error bars.
The problem Monte Carlo integration addresses is the computation of a multidimensional definite integral
where Ω, a subset of Rm, has volume
The naive Monte Carlo approach is to sample points uniformly on Ω: given N uniform samples,
I can be approximated by
.
This is because the law of large numbers ensures that
.
Given the estimation of I from QN, the error bars of QN can be estimated by the sample variance using the unbiased estimate of the variance.
which leads to
.
As long as the sequence
is bounded, this variance decreases asymptotically to zero as 1/N. The estimation of the error of QN is thus
which decreases as . This is standard error of the mean multiplied with .
This result does not depend on the number of dimensions of the integral, which is the promised advantage of Monte Carlo integration against most deterministic methods that depend exponentially on the dimension.
It is important to notice that, unlike in deterministic methods, the estimate of the error is not a strict error bound; random sampling may not uncover all the important features of the integrand that can result in an underestimate of the error.
While the naive Monte Carlo works for simple examples, an improvement over deterministic algorithms can only be accomplished with algorithms that use problem-specific sampling distributions.
With an appropriate sample distribution it is possible to exploit the fact that almost all higher-dimensional integrands are very localized and only small subspace notably contributes to the integral.
A large part of the Monte Carlo literature is dedicated in developing strategies to improve the error estimates. In particular, stratified sampling—dividing the region in sub-domains—and importance sampling—sampling from non-uniform distributions—are two examples of such techniques.
Example
A paradigmatic example of a Monte Carlo integration is the estimation of π. Consider the function
and the set Ω = [−1,1] × [−1,1]
|
https://en.wikipedia.org/wiki/Paraconsistent%20mathematics
|
Paraconsistent mathematics, sometimes called inconsistent mathematics, represents an attempt to develop the classical infrastructure of mathematics (e.g. analysis) based on a foundation of paraconsistent logic instead of classical logic. A number of reformulations of analysis can be developed, for example functions which both do and do not have a given value simultaneously.
Chris Mortensen claims (see references):
One could hardly ignore the examples of analysis and its special case, the calculus. There prove to be many places where there are distinctive inconsistent insights; see Mortensen (1995) for example. (1) Robinson's non-standard analysis was based on infinitesimals, quantities smaller than any real number, as well as their reciprocals, the infinite numbers. This has an inconsistent version, which has some advantages for calculation in being able to discard higher-order infinitesimals. The theory of differentiation turned out to have these advantages, while the theory of integration did not. (2)
References
McKubre-Jordens, M. and Weber, Z. (2012). "Real analysis in paraconsistent logic". Journal of Philosophical Logic 41 (5):901–922. doi: 10.1017/S1755020309990281
Mortensen, C. (1995). Inconsistent Mathematics. Dordrecht: Kluwer.
Weber, Z. (2010). "Transfinite numbers in paraconsistent set theory". Review of Symbolic Logic 3 (1):71–92. doi:10.1017/S1755020309990281
External links
Entry in the Internet Encyclopedia of Philosophy
Entry in the Stanford Encyclopedia of Philosophy
Lectures by Manuel Bremer of the University of Düsseldorf
Philosophy of mathematics
Proof theory
Paraconsistent logic
|
https://en.wikipedia.org/wiki/GNU%20Scientific%20Library
|
The GNU Scientific Library (or GSL) is a software library for numerical computations in applied mathematics and science. The GSL is written in C; wrappers are available for other programming languages. The GSL is part of the GNU Project and is distributed under the GNU General Public License.
Project history
The GSL project was initiated in 1996 by physicists Mark Galassi and James Theiler of Los Alamos National Laboratory. They aimed at writing a modern replacement for widely used but somewhat outdated Fortran libraries such as Netlib. They carried out the overall design and wrote early modules; with that ready they recruited other scientists to contribute.
The "overall development of the library and the design and implementation of the major modules" was carried out by Brian Gough and Gerard Jungman. Other major contributors were Jim Davies, Reid Priedhorsky, M. Booth, and F. Rossi.
Version 1.0 was released in 2001. In the following years, the library expanded only slowly; as the documentation stated, the maintainers were more interested in stability than in additional functionality. Major version 1 ended with release 1.16 of July 2013; this was the only public activity in the three years 2012–2014.
Vigorous development resumed with publication of version 2.0 in October 2015. The latest version 2.7 was released in June 2021.
Example
The following example program calculates the value of the Bessel function of the first kind and order zero for 5:
#include <stdio.h>
#include <gsl/gsl_sf_bessel.h>
int main(void)
{
double x = 5.0;
double y = gsl_sf_bessel_J0(x);
printf("J0(%g) = %.18e\n", x, y);
return 0;
}
The example program has to be linked to the GSL library
upon compilation:
$ gcc $(gsl-config --cflags) example.c $(gsl-config --libs)
The output is shown below and should be correct to double-precision accuracy:
J0(5) = -1.775967713143382920e-01
Features
The software library provides facilities for:
Programming-language bindings
Since the GSL is written in C, it is straightforward to provide wrappers for other programming languages. Such wrappers currently exist for
AMPL
C++
Fortran
Haskell
Java
Julia
Common Lisp
OCaml
Octave
Perl Data Language
Python
R
Ruby
Rust
C++ support
The GSL can be used in C++ classes, but not using pointers to member functions, because the type of pointer to member function is different from pointer to function. Instead, pointers to static functions have to be used. Another common workaround is using a functor.
C++ wrappers for GSL are available. Not all of these are regularly maintained. They do offer access to matrix and vector classes without having to use GSL's interface to malloc and free functions. Some also offer support for also creating workspaces that behave like Smart pointer classes. Finally, there is (limited, as of April 2020) support for allowing the user to create classes to represent a parameterised function as a functor.
While not strictly wrappers, there are som
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