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Torus-Torus-Real-world objects that approximate a torus of revolution include swim rings, inner tubes and ringette rings. A torus should not be confused with a solid torus, which is formed by rotating a disk, rather than a circle, around an axis. A solid torus is a torus plus the volume inside the torus. Real-world ob... | milkshake721/2.1M-wiki-STEM |
Torus-Torus-In topology, a ring torus is homeomorphic to the Cartesian product of two circles: S1×S1 , and the latter is taken to be the definition in that context. It is a compact 2-manifold of genus 1. The ring torus is one way to embed this space into Euclidean space, but another way to do this is the Cartesian pro... | milkshake721/2.1M-wiki-STEM |
Torus-Torus-In the field of topology, a torus is any topological space that is homeomorphic to a torus. The surface of a coffee cup and a doughnut are both topological tori with genus one.
An example of a torus can be constructed by taking a rectangular strip of flexible material such as rubber, and joining the top edg... | milkshake721/2.1M-wiki-STEM |
Torus-Geometry-A torus can be defined parametrically by: where θ, φ are angles which make a full circle, so their values start and end at the same point, R is the distance from the center of the tube to the center of the torus, r is the radius of the tube.Angle θ represents rotation around the tube, whereas φ represent... | milkshake721/2.1M-wiki-STEM |
Torus-Geometry-An implicit equation in Cartesian coordinates for a torus radially symmetric about the z-axis is or the solution of f(x, y, z) = 0, where Algebraically eliminating the square root gives a quartic equation, The three classes of standard tori correspond to the three possible aspect ratios between R and r: ... | milkshake721/2.1M-wiki-STEM |
Torus-Geometry-R = r corresponds to the horn torus, which in effect is a torus with no "hole". | milkshake721/2.1M-wiki-STEM |
Torus-Geometry-R < r describes the self-intersecting spindle torus; its inner shell is a lemon and its outer shell is an apple When R = 0, the torus degenerates to the sphere.When R ≥ r, the interior of this torus is diffeomorphic (and, hence, homeomorphic) to a product of a Euclidean open disk and a circle. The volume... | milkshake721/2.1M-wiki-STEM |
Torus-Geometry-Expressing the surface area and the volume by the distance p of an outermost point on the surface of the torus to the center, and the distance q of an innermost point to the center (so that R = p + q/2 and r = p − q/2), yields As a torus is the product of two circles, a modified version of the spherical ... | milkshake721/2.1M-wiki-STEM |
Torus-Geometry-In traditional spherical coordinates there are three measures, R, the distance from the center of the coordinate system, and θ and φ, angles measured from the center point. | milkshake721/2.1M-wiki-STEM |
Torus-Geometry-As a torus has, effectively, two center points, the centerpoints of the angles are moved; φ measures the same angle as it does in the spherical system, but is known as the "toroidal" direction. The center point of θ is moved to the center of r, and is known as the "poloidal" direction. These terms were f... | milkshake721/2.1M-wiki-STEM |
Torus-Topology-Topologically, a torus is a closed surface defined as the product of two circles: S1 × S1. This can be viewed as lying in C2 and is a subset of the 3-sphere S3 of radius √2. This topological torus is also often called the Clifford torus. In fact, S3 is filled out by a family of nested tori in this manner... | milkshake721/2.1M-wiki-STEM |
Torus-Topology-The surface described above, given the relative topology from R3 , is homeomorphic to a topological torus as long as it does not intersect its own axis. A particular homeomorphism is given by stereographically projecting the topological torus into R3 from the north pole of S3.
The torus can also be des... | milkshake721/2.1M-wiki-STEM |
Torus-Topology-Intuitively speaking, this means that a closed path that circles the torus' "hole" (say, a circle that traces out a particular latitude) and then circles the torus' "body" (say, a circle that traces out a particular longitude) can be deformed to a path that circles the body and then the hole. So, strictl... | milkshake721/2.1M-wiki-STEM |
Torus-Topology-If a torus is punctured and turned inside out then another torus results, with lines of latitude and longitude interchanged. This is equivalent to building a torus from a cylinder, by joining the circular ends together, in two ways: around the outside like joining two ends of a garden hose, or through th... | milkshake721/2.1M-wiki-STEM |
Torus-Topology-The first homology group of the torus is isomorphic to the fundamental group (this follows from Hurewicz theorem since the fundamental group is abelian). | milkshake721/2.1M-wiki-STEM |
Torus-Two-sheeted cover-The 2-torus double-covers the 2-sphere, with four ramification points. Every conformal structure on the 2-torus can be represented as a two-sheeted cover of the 2-sphere. The points on the torus corresponding to the ramification points are the Weierstrass points. In fact, the conformal type of t... | milkshake721/2.1M-wiki-STEM |
Torus-n-dimensional torus-The torus has a generalization to higher dimensions, the n-dimensional torus, often called the n-torus or hypertorus for short. (This is the more typical meaning of the term "n-torus", the other referring to n holes or of genus n.) Recalling that the torus is the product space of two circles, ... | milkshake721/2.1M-wiki-STEM |
Torus-n-dimensional torus-The standard 1-torus is just the circle: T1=S1 . The torus discussed above is the standard 2-torus, T2 . And similar to the 2-torus, the n-torus, Tn can be described as a quotient of Rn under integral shifts in any coordinate. That is, the n-torus is Rn modulo the action of the integer la... | milkshake721/2.1M-wiki-STEM |
Torus-n-dimensional torus-An n-torus in this sense is an example of an n-dimensional compact manifold. It is also an example of a compact abelian Lie group. This follows from the fact that the unit circle is a compact abelian Lie group (when identified with the unit complex numbers with multiplication). Group multiplic... | milkshake721/2.1M-wiki-STEM |
Torus-n-dimensional torus-Toroidal groups play an important part in the theory of compact Lie groups. This is due in part to the fact that in any compact Lie group G one can always find a maximal torus; that is, a closed subgroup which is a torus of the largest possible dimension. Such maximal tori T have a controlling... | milkshake721/2.1M-wiki-STEM |
Torus-n-dimensional torus-Automorphisms of T are easily constructed from automorphisms of the lattice Zn , which are classified by invertible integral matrices of size n with an integral inverse; these are just the integral matrices with determinant ±1. Making them act on Rn in the usual way, one has the typical tora... | milkshake721/2.1M-wiki-STEM |
Torus-n-dimensional torus-The fundamental group of an n-torus is a free abelian group of rank n. The k-th homology group of an n-torus is a free abelian group of rank n choose k. It follows that the Euler characteristic of the n-torus is 0 for all n. The cohomology ring H•( Tn , Z) can be identified with the exterior a... | milkshake721/2.1M-wiki-STEM |
Torus-n-dimensional torus-Configuration space As the n-torus is the n-fold product of the circle, the n-torus is the configuration space of n ordered, not necessarily distinct points on the circle. Symbolically, Tn=(S1)n . The configuration space of unordered, not necessarily distinct points is accordingly the orbifol... | milkshake721/2.1M-wiki-STEM |
Torus-n-dimensional torus-For n = 2, the quotient is the Möbius strip, the edge corresponding to the orbifold points where the two coordinates coincide. For n = 3 this quotient may be described as a solid torus with cross-section an equilateral triangle, with a twist; equivalently, as a triangular prism whose top and b... | milkshake721/2.1M-wiki-STEM |
Torus-n-dimensional torus-These orbifolds have found significant applications to music theory in the work of Dmitri Tymoczko and collaborators (Felipe Posada, Michael Kolinas, et al.), being used to model musical triads. | milkshake721/2.1M-wiki-STEM |
Torus-Flat torus-A flat torus is a torus with the metric inherited from its representation as the quotient, R2 /L, where L is a discrete subgroup of R2 isomorphic to Z2 . This gives the quotient the structure of a Riemannian manifold. Perhaps the simplest example of this is when L = Z2 : R2/Z2 , which can also be ... | milkshake721/2.1M-wiki-STEM |
Torus-Flat torus-This metric of the square flat torus can also be realised by specific embeddings of the familiar 2-torus into Euclidean 4-space or higher dimensions. Its surface has zero Gaussian curvature everywhere. Its surface is flat in the same sense that the surface of a cylinder is flat. In 3 dimensions, one ca... | milkshake721/2.1M-wiki-STEM |
Torus-Flat torus-A simple 4-dimensional Euclidean embedding of a rectangular flat torus (more general than the square one) is as follows: cos sin cos sin v) where R and P are positive constants determining the aspect ratio. It is diffeomorphic to a regular torus but not isometric. It can not be analytically embedded (... | milkshake721/2.1M-wiki-STEM |
Torus-Flat torus-If R and P in the above flat torus parametrization form a unit vector (R, P) = (cos(η), sin(η)) then u, v, and 0 < η < π/2 parameterize the unit 3-sphere as Hopf coordinates. In particular, for certain very specific choices of a square flat torus in the 3-sphere S3, where η = π/4 above, the torus will ... | milkshake721/2.1M-wiki-STEM |
Torus-Flat torus-Other tori in S3 having this partitioning property include the square tori of the form Q⋅T, where Q is a rotation of 4-dimensional space R4 , or in other words Q is a member of the Lie group SO(4). | milkshake721/2.1M-wiki-STEM |
Torus-Flat torus-It is known that there exists no C2 (twice continuously differentiable) embedding of a flat torus into 3-space. (The idea of the proof is to take a large sphere containing such a flat torus in its interior, and shrink the radius of the sphere until it just touches the torus for the first time. Such a p... | milkshake721/2.1M-wiki-STEM |
Torus-Flat torus-In April 2012, an explicit C1 (continuously differentiable) embedding of a flat torus into 3-dimensional Euclidean space R3 was found. It is a flat torus in the sense that as metric spaces, it is isometric to a flat square torus. It is similar in structure to a fractal as it is constructed by repeated... | milkshake721/2.1M-wiki-STEM |
Torus-Genus g surface-In the theory of surfaces there is another object, the "genus" g surface. Instead of the product of n circles, a genus g surface is the connected sum of g two-tori. To form a connected sum of two surfaces, remove from each the interior of a disk and "glue" the surfaces together along the boundary ... | milkshake721/2.1M-wiki-STEM |
Torus-Genus g surface-As examples, a genus zero surface (without boundary) is the two-sphere while a genus one surface (without boundary) is the ordinary torus. The surfaces of higher genus are sometimes called n-holed tori (or, rarely, n-fold tori). The terms double torus and triple torus are also occasionally used.
T... | milkshake721/2.1M-wiki-STEM |
Torus-Toroidal polyhedra-Polyhedra with the topological type of a torus are called toroidal polyhedra, and have Euler characteristic V − E + F = 0. For any number of holes, the formula generalizes to V − E + F = 2 − 2N, where N is the number of holes.
The term "toroidal polyhedron" is also used for higher-genus polyhed... | milkshake721/2.1M-wiki-STEM |
Torus-Automorphisms-The homeomorphism group (or the subgroup of diffeomorphisms) of the torus is studied in geometric topology. Its mapping class group (the connected components of the homeomorphism group) is surjective onto the group GL (n,Z) of invertible integer matrices, which can be realized as linear maps on th... | milkshake721/2.1M-wiki-STEM |
Torus-Automorphisms-At the level of homotopy and homology, the mapping class group can be identified as the action on the first homology (or equivalently, first cohomology, or on the fundamental group, as these are all naturally isomorphic; also the first cohomology group generates the cohomology algebra: MCG Ho Aut Au... | milkshake721/2.1M-wiki-STEM |
Torus-Coloring a torus-The torus's chromatic number is seven, meaning every graph that can be embedded on the torus has a chromatic number of at most seven. (Since the complete graph K7 can be embedded on the torus, and χ(K7)=7 , the upper bound is tight.) Equivalently, in a torus divided into regions, it is always p... | milkshake721/2.1M-wiki-STEM |
Torus-de Bruijn torus-In combinatorial mathematics, a de Bruijn torus is an array of symbols from an alphabet (often just 0 and 1) that contains every m-by-n matrix exactly once. It is a torus because the edges are considered wraparound for the purpose of finding matrices. Its name comes from the De Bruijn sequence, wh... | milkshake721/2.1M-wiki-STEM |
Torus-Cutting a torus-A solid torus of revolution can be cut by n (> 0) planes into maximally (n+2n−1)+(nn−1)=16(n3+3n2+8n) parts.The first 11 numbers of parts, for 0 ≤ n ≤ 10 (including the case of n = 0, not covered by the above formulas), are as follows: 1, 2, 6, 13, 24, 40, 62, 91, 128, 174, 230, ... (sequence A003... | milkshake721/2.1M-wiki-STEM |
Torus-Notes-Nociones de Geometría Analítica y Álgebra Lineal, ISBN 978-970-10-6596-9, Author: Kozak Ana Maria, Pompeya Pastorelli Sonia, Verdanega Pedro Emilio, Editorial: McGraw-Hill, Edition 2007, 744 pages, language: Spanish Allen Hatcher. Algebraic Topology. Cambridge University Press, 2002. ISBN 0-521-79540-0.
V. ... | milkshake721/2.1M-wiki-STEM |
Flagellation-Flagellation-Flagellation (Latin flagellum, 'whip'), flogging or whipping is the act of beating the human body with special implements such as whips, rods, switches, the cat o' nine tails, the sjambok, the knout, etc. Typically, flogging has been imposed on an unwilling subject as a punishment; however, it... | milkshake721/2.1M-wiki-STEM |
Flagellation-Flagellation-The strokes are typically aimed at the unclothed back of a person, though they can be administered to other areas of the body. For a moderated subform of flagellation, described as bastinado, the soles of a person's bare feet are used as a target for beating (see foot whipping). | milkshake721/2.1M-wiki-STEM |
Flagellation-Flagellation-In some circumstances the word flogging is used loosely to include any sort of corporal punishment, including birching and caning. However, in British legal terminology, a distinction was drawn (and still is, in one or two colonial territories) between flogging (with a cat o' nine tails) and w... | milkshake721/2.1M-wiki-STEM |
Flagellation-Current use as punishment-Officially abolished in most countries, flogging or whipping, including foot whipping in some countries, is still a common punishment in some parts of the world, particularly in countries using Islamic law and in some territories which were former British colonies. Medically super... | milkshake721/2.1M-wiki-STEM |
Flagellation-Historical use as punishment-Judaism According to the Torah (Deuteronomy 25:1–3) and Rabbinic law lashes may be given for offenses that do not merit capital punishment, and may not exceed 40. However, in the absence of a Sanhedrin, corporal punishment is not practiced in Jewish law. Halakha specifies the l... | milkshake721/2.1M-wiki-STEM |
Flagellation-Historical use as punishment-Antiquity In the Roman Empire, flagellation was often used as a prelude to crucifixion, and in this context is sometimes referred to as scourging. Most famously according to the gospel accounts, this occurred prior to the crucifixion of Jesus Christ. Due to the context of the f... | milkshake721/2.1M-wiki-STEM |
Flagellation-Historical use as punishment-Whips with small pieces of metal or bone at the tips were commonly used. Such a device could easily cause disfigurement and serious trauma, such as ripping pieces of flesh from the body or loss of an eye. In addition to causing severe pain, the victim would approach a state of ... | milkshake721/2.1M-wiki-STEM |
Flagellation-Historical use as punishment-The Romans reserved this treatment for non-citizens, as stated in the lex Porcia and lex Sempronia, dating from 195 and 123 BC. The poet Horace refers to the horribile flagellum (horrible whip) in his Satires. Typically, the one to be punished was stripped naked and bound to a ... | milkshake721/2.1M-wiki-STEM |
Flagellation-Historical use as punishment-From Middle Ages to modern times The Whipping Act was passed in England in 1530. Under this legislation, vagrants were to be taken to a nearby populated area "and there tied to the end of a cart naked and beaten with whips throughout such market town till the body shall be bloo... | milkshake721/2.1M-wiki-STEM |
Flagellation-Historical use as punishment-In the Russian Empire, knouts were used to flog criminals and political offenders. Sentences of a hundred lashes would usually result in death. Whipping was used as a punishment for Russian serfs.In April 2020, Saudi Arabia said it would replace flogging with prison sentences o... | milkshake721/2.1M-wiki-STEM |
Flagellation-Historical use as punishment-As critics of flogging aboard the ships and vessels of the United States Navy became more vocal, the Department of the Navy began in 1846 to require annual reports of discipline including flogging, and limited the maximum number of lashes to 12. These annual reports were requir... | milkshake721/2.1M-wiki-STEM |
Flagellation-Historical use as punishment-United Kingdom Flagellation was so common in England as punishment that caning (and spanking and whipping) are called "the English vice".Flogging was a common disciplinary measure in the Royal Navy that became associated with a seaman's manly disregard for pain. Aboard ships, k... | milkshake721/2.1M-wiki-STEM |
Flagellation-Historical use as punishment-The 3rd battalion's Royal Anglian Regiment nickname of "The Steelbacks" is taken from one of its former regiments, the 48th (Northamptonshire) Regiment of Foot who earned the nickname for their stoicism when being flogged with the cat o' nine tails ("Not a whimper under the las... | milkshake721/2.1M-wiki-STEM |
Flagellation-Historical use as punishment-Shortly after the establishment of Northern Ireland the Special Powers Act of 1922 was enacted by the Parliament of Northern Ireland. This Act enabled the government to 'take all such steps and issue all such orders as may be necessary for preserving the peace and maintaining o... | milkshake721/2.1M-wiki-STEM |
Flagellation-Historical use as punishment-France Meanwhile, during the French Revolutionary Wars the French Army stopped floggings altogether. The King's German Legion (KGL), which were German units in British pay, did not flog. In one case, a British soldier on detached duty with the KGL was sentenced to be flogged, b... | milkshake721/2.1M-wiki-STEM |
Flagellation-Historical use as punishment-Australian penal colonies Once common in the British Army and British Royal Navy as a means of discipline, flagellation also featured prominently in the British penal colonies in early colonial Australia. Given that convicts in Australia were already "imprisoned", punishments f... | milkshake721/2.1M-wiki-STEM |
Flagellation-Historical use as punishment-Flagellation took place either with a single whip or, more notoriously, with the cat o' nine tails. Typically, the offender's upper half was bared and he was suspended by the wrists beneath a tripod of wooden beams (known as 'the triangle'). In many cases, the offender's feet b... | milkshake721/2.1M-wiki-STEM |
Flagellation-Historical use as punishment-With the prisoner thus stripped and bound, either one or two floggers administered the prescribed number of strokes, or "lashes," to the victim's back. During the flogging, a doctor or other medical worker was consulted at regular intervals as to the condition of the prisoner. ... | milkshake721/2.1M-wiki-STEM |
Flagellation-Historical use as punishment-Female convicts were also subject to flogging as punishment, both on the convict ships and in the penal colonies. Although they were generally given fewer lashes than males (usually limited to 40 in each flogging), there was no other difference between the manner in which males... | milkshake721/2.1M-wiki-STEM |
Flagellation-Historical use as punishment-Floggings of both male and female convicts were public, administered before the whole colony's company, assembled especially for the purpose. In addition to the infliction of pain, one of the principal purposes of the flogging was to humiliate the offender in front of his mates... | milkshake721/2.1M-wiki-STEM |
Flagellation-Historical use as punishment-At the conclusion of the whipping, the prisoner's lacerated back was normally rinsed with brine, which served as a crude and painful disinfectant.
Flogging still continued for years after independence. The last person flogged in Australia was William John O'Meally in 1958 in Me... | milkshake721/2.1M-wiki-STEM |
Flagellation-Historical use as punishment-Contemporary Syria In Syria where torture of political dissidents, POWs and civilians is extremely common, flagellation has become one of the most common forms of torture. Flagellation is used by both the Free Syrian Army and the Syrian Arab Army, but is not practiced by the Sy... | milkshake721/2.1M-wiki-STEM |
Flagellation-As a religious practice-Antiquity During the Ancient Roman festival of Lupercalia, young men ran through the streets with thongs cut from the hide of goats which had just been sacrificed, whipping people with the thongs as they ran. According to Plutarch, women would put themselves in their way to receive ... | milkshake721/2.1M-wiki-STEM |
Flagellation-As a religious practice-Christianity The Flagellation, in a Christian context, refers to an episode in the Passion of Christ prior to Jesus' crucifixion. The practice of mortification of the flesh for religious purposes has been utilised by members of various Christian denominations since the time of the G... | milkshake721/2.1M-wiki-STEM |
Flagellation-As a religious practice-Some members of strict monastic orders, and some members of the Catholic lay organization Opus Dei, practice mild self-flagellation using the discipline. Pope John Paul II took the discipline regularly. Self-flagellation remains common in Colombia, the Philippines, Mexico, Spain and... | milkshake721/2.1M-wiki-STEM |
Flagellation-As a religious practice-Shi'a Islam As suffering and cutting the body with knives or chains (matam) have been prohibited by Shi'a marjas like Ali Khamenei, Supreme Leader of Iran, some Shi'a observe mourning with blood donation which is called "Qame Zani" and flailing. Yet some Shi'ite men and boys continu... | milkshake721/2.1M-wiki-STEM |
Flagellation-As a sexual practice-Flagellation is also used as a sexual practice in the context of BDSM. The intensity of the beating is usually far less than used for punishment. | milkshake721/2.1M-wiki-STEM |
Flagellation-As a sexual practice-There are anecdotal reports of people willingly being bound or whipped, as a prelude to or substitute for sex, during the 14th century. Flagellation practiced within an erotic setting has been recorded from at least the 1590s evidenced by a John Davies epigram, and references to "flogg... | milkshake721/2.1M-wiki-STEM |
Abandoned pets-Abandoned pets-Abandoned pets are companion animals that are either inadvertently or deliberately abandoned by their owners, by either dumping the animals on the streets, leaving them alone in a vacant property, or relinquishing them at an animal shelter. Animal welfare laws in many states of the United... | milkshake721/2.1M-wiki-STEM |
Liouville's formula-Liouville's formula-In mathematics, Liouville's formula, also known as the Abel-Jacobi-Liouville Identity, is an equation that expresses the determinant of a square-matrix solution of a first-order system of homogeneous linear differential equations in terms of the sum of the diagonal coefficients o... | milkshake721/2.1M-wiki-STEM |
Liouville's formula-Liouville's formula-Liouville's formula is a generalization of Abel's identity and can be used to prove it. Since Liouville's formula relates the different linearly independent solutions of the system of differential equations, it can help to find one solution from the other(s), see the example appl... | milkshake721/2.1M-wiki-STEM |
Liouville's formula-Statement of Liouville's formula-Consider the n-dimensional first-order homogeneous linear differential equation y′=A(t)y on an interval I of the real line, where A(t) for t ∈ I denotes a square matrix of dimension n with real or complex entries. Let Φ denote a matrix-valued solution on I, meaning t... | milkshake721/2.1M-wiki-STEM |
Liouville's formula-Statement of Liouville's formula-Let trA(s)=∑i=1nai,i(s),s∈I, denote the trace of A(s) = (ai, j (s))i, j ∈ {1,...,n}, the sum of its diagonal entries. If the trace of A is a continuous function, then the determinant of Φ satisfies det det exp (∫t0ttrA(s)ds) for all t and t0 in I. | milkshake721/2.1M-wiki-STEM |
Liouville's formula-Example application-This example illustrates how Liouville's formula can help to find the general solution of a first-order system of homogeneous linear differential equations. Consider y′=(1−1/x1+x−1)⏟=A(x)y on the open interval I = (0, ∞). Assume that the easy solution y(x)=(1x),x∈I, is already fo... | milkshake721/2.1M-wiki-STEM |
Liouville's formula-Example application-Therefore, by integration, we see that ln x+c2,x∈I, involving the natural logarithm and the constant of integration c2. Solving equation (1) for y2(x) and substituting for y1(x) gives ln x+c2x−c1,x∈I, which is the general solution for y. With the special choice c1 = 0 and c2 = ... | milkshake721/2.1M-wiki-STEM |
Liouville's formula-Proof of Liouville's formula-We omit the argument x for brevity. By the Leibniz formula for determinants, the derivative of the determinant of Φ = (Φi, j )i, j ∈ {0,...,n} can be calculated by differentiating one row at a time and taking the sum, i.e.
Since the matrix-valued solution Φ satisfies the... | milkshake721/2.1M-wiki-STEM |
Liouville's formula-Proof of Liouville's formula-When we subtract from the i-th row the linear combination ∑j=1j≠inai,j(Φj,1,…,Φj,n), of all the other rows, then the value of the determinant remains unchanged, hence det det det Φ for every i ∈ {1, . . . , n} by the linearity of the determinant with respect to every row... | milkshake721/2.1M-wiki-STEM |
Liouville's formula-Proof of Liouville's formula-Fix x0 ∈ I. Since the trace of A is assumed to be continuous function on I, it is bounded on every closed and bounded subinterval of I and therefore integrable, hence := det exp (−∫x0xtrA(ξ)dξ),x∈I, is a well defined function. Differentiating both sides, using the produ... | milkshake721/2.1M-wiki-STEM |
Choi's theorem on completely positive maps-Choi's theorem on completely positive maps-In mathematics, Choi's theorem on completely positive maps is a result that classifies completely positive maps between finite-dimensional (matrix) C*-algebras. An infinite-dimensional algebraic generalization of Choi's theorem is kno... | milkshake721/2.1M-wiki-STEM |
Choi's theorem on completely positive maps-Statement-Choi's theorem. Let Φ:Cn×n→Cm×m be a linear map. The following are equivalent: (i) Φ is n-positive (i.e. id n⊗Φ)(A)∈Cn×n⊗Cm×m is positive whenever A∈Cn×n⊗Cn×n is positive).
(ii) The matrix with operator entries id n⊗Φ)(∑ijEij⊗Eij)=∑ijEij⊗Φ(Eij)∈Cnm×nm is positive,... | milkshake721/2.1M-wiki-STEM |
Choi's theorem on completely positive maps-Proof-(i) implies (ii) We observe that if E=∑ijEij⊗Eij, then E=E* and E2=nE, so E=n−1EE* which is positive. Therefore CΦ =(In ⊗ Φ)(E) is positive by the n-positivity of Φ.
(iii) implies (i) This holds trivially.
(ii) implies (iii) This mainly involves chasing the different way... | milkshake721/2.1M-wiki-STEM |
Choi's theorem on completely positive maps-Consequences-Kraus operators In the context of quantum information theory, the operators {Vi} are called the Kraus operators (after Karl Kraus) of Φ. Notice, given a completely positive Φ, its Kraus operators need not be unique. For example, any "square root" factorization of ... | milkshake721/2.1M-wiki-STEM |
Choi's theorem on completely positive maps-Consequences-When the Kraus operators are obtained from the eigenvector decomposition of the Choi matrix, because the eigenvectors form an orthogonal set, the corresponding Kraus operators are also orthogonal in the Hilbert–Schmidt inner product. This is not true in general fo... | milkshake721/2.1M-wiki-STEM |
Choi's theorem on completely positive maps-Consequences-This can be viewed as a special case of the result relating two minimal Stinespring representations.
Alternatively, there is an isometry scalar matrix {uij}ij ∈ Cnm × nm such that Ai=∑j=1uijBj.
This follows from the fact that for two square matrices M and N, M M* ... | milkshake721/2.1M-wiki-STEM |
Choi's theorem on completely positive maps-Consequences-Hermitian-preserving maps Choi's technique can be used to obtain a similar result for a more general class of maps. Φ is said to be Hermitian-preserving if A is Hermitian implies Φ(A) is also Hermitian. One can show Φ is Hermitian-preserving if and only if it is o... | milkshake721/2.1M-wiki-STEM |
Lambda diode-Lambda diode-A lambda diode is an electronic circuit that combines a complementary pair of junction gated field effect transistors into a two-terminal device that exhibits an area of differential negative resistance much like a tunnel diode. The term refers to the shape of the V–I curve of the device, whic... | milkshake721/2.1M-wiki-STEM |
Lambda diode-Lambda diode-Lambda diodes work at higher voltage than tunnel diodes. Whereas a typical tunnel diode may exhibit negative differential resistance approximately between 70 mV and 350 mV, this region occurs approximately between 1.5 V and 6 V in a lambda diode due to the higher pinch-off voltages of typical ... | milkshake721/2.1M-wiki-STEM |
Lambda diode-Lambda diode-Moreover, in a tunnel diode the current reaches a minimum of about 20% of the peak current before rising again towards higher voltages. The lambda diode current approaches zero as voltage increases, before rising quickly again at a voltage high enough to cause gate–source Zener breakdown in th... | milkshake721/2.1M-wiki-STEM |
Lambda diode-Lambda diode-A suggested modulatable variant but is a bit more difficult to build uses a PNP based optocoupler and can be tweaked by using its IR diode. This has the advantage that its properties can be fine tuned with a simple bias driver and used for high sensitivity radio applications, sometimes a modif... | milkshake721/2.1M-wiki-STEM |
Lambda diode-Applications-Like the tunnel diode, the negative resistance aspect of the lambda diode lends itself naturally to application in oscillator circuits and amplifiers. In addition, bistable circuits such as memory cells have been described. | milkshake721/2.1M-wiki-STEM |
Lambda diode-Literature-Graf, Rudolf F. (1999). Modern Dictionary of Electronics, 7th ed. Boston [etc.]: Newnes Press. p. 411. ISBN 0-7506-9866-7. | milkshake721/2.1M-wiki-STEM |
Casual wear-Casual wear-Casual wear (or casual attire or clothing) is a Western dress code that is relaxed, occasional, spontaneous and suited for everyday use. Casual wear became popular in the Western world following the counterculture of the 1960s. When emphasising casual wear's comfort, it may be referred to as lei... | milkshake721/2.1M-wiki-STEM |
Casual wear-Overview-Modern casual fashion can be traced to fashion sportswear from the 1920s, including tweed blazers, oxford shoes, and golf skirts. An increase in the popularity of bicycling brought about a need for culottes, a forerunner for casual shorts. As the century progressed, "casual" came to encompass more ... | milkshake721/2.1M-wiki-STEM |
Casual wear-Gender expression-Casual wear introduced a "unisexing" of fashion. By the 1960s, women adopted T-shirts, jeans, and collared shirts, and for the first time in nearly 200 years, it was fashionable for men to have long hair. Casual wear is typically the dress code in which forms of gender expression are exper... | milkshake721/2.1M-wiki-STEM |
Casual wear-Gender expression-Jeans, dress shirt (casually turn down collared), and a T-shirt or sleeveless shirt are typically considered casual wear for men in modern times. For men, the exposure of shoulders, thighs, and backs is still limited to casual wear. | milkshake721/2.1M-wiki-STEM |
Vulnerability assessment (computing)-Vulnerability assessment (computing)-Vulnerability assessment is a process of defining, identifying and classifying the security holes in information technology systems. An attacker can exploit a vulnerability to violate the security of a system. Some known vulnerabilities are Authe... | milkshake721/2.1M-wiki-STEM |
Vulnerability assessment (computing)-Purpose-Before deploying a system, it first must go through from a series of vulnerability assessments that will ensure that the build system is secure from all the known security risks. When a new vulnerability is discovered, the system administrator can again perform an assessment... | milkshake721/2.1M-wiki-STEM |
Vulnerability assessment (computing)-Purpose-The primary purpose of the assessment is to find the vulnerabilities in the system, but the assessment report conveys to stakeholders that the system is secured from these vulnerabilities. If an intruder gained access to a network consisting of vulnerable Web servers, it is ... | milkshake721/2.1M-wiki-STEM |
Vulnerability assessment (computing)-Assessment types-Depending on the system a vulnerability assessment can have many types and level. | milkshake721/2.1M-wiki-STEM |
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