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Ingeometry,close-packing of equalspheresis a dense arrangement of congruent spheres in an infinite, regular arrangement (orlattice).Carl Friedrich Gaussproved that the highest average density – that is, the greatest fraction of space occupied by spheres – that can be achieved by alatticepacking is
The samepacking densitycan also be achieved by alternate stackings of the same close-packed planes of spheres, including structures that are aperiodic in the stacking direction. TheKepler conjecturestates that this is the highest density that can be achieved by any arrangement of spheres, either regular or irregular. This conjecture was proven byThomas Hales.[1][2]Highest density is known only for 1, 2, 3, 8, and 24 dimensions.[3]
Manycrystalstructures are based on a close-packing of a single kind of atom, or a close-packing of large ions with smaller ions filling the spaces between them. The cubic and hexagonal arrangements are very close to one another in energy, and it may be difficult to predict which form will be preferred from first principles.
There are two simple regular lattices that achieve this highest average density. They are calledface-centered cubic(FCC) (also calledcubicclose packed) andhexagonalclose-packed(HCP), based on theirsymmetry. Both are based upon sheets of spheres arranged at the vertices of a triangular tiling; they differ in how the sheets are stacked upon one another. The FCC lattice is also known to mathematicians as that generated by the A3root system.[4]
The problem of close-packing of spheres was first mathematically analyzed byThomas Harriotaround 1587, after a question on piling cannonballs on ships was posed to him by SirWalter Raleighon their expedition to America.[5]Cannonballs were usually piled in a rectangular or triangular wooden frame, forming a three-sided or four-sided pyramid. Both arrangements produce a face-centered cubic lattice – with different orientation to the ground. Hexagonal close-packing would result in a six-sided pyramid with a hexagonal base.
Thecannonball problemasks which flat square arrangements of cannonballs can be stacked into a square pyramid.Édouard Lucasformulated the problem as theDiophantine equation∑n=1Nn2=M2{\displaystyle \sum _{n=1}^{N}n^{2}=M^{2}}or16N(N+1)(2N+1)=M2{\displaystyle {\frac {1}{6}}N(N+1)(2N+1)=M^{2}}and conjectured that the only solutions areN=1,M=1,{\displaystyle N=1,M=1,}andN=24,M=70{\displaystyle N=24,M=70}. HereN{\displaystyle N}is the number of layers in the pyramidal stacking arrangement andM{\displaystyle M}is the number of cannonballs along an edge in the flat square arrangement.
In both the FCC and HCP arrangements each sphere has twelve neighbors. For every sphere there is one gap surrounded by six spheres (octahedral) and two smaller gaps surrounded by four spheres (tetrahedral). The distances to the centers of these gaps from the centers of the surrounding spheres is√3⁄2for the tetrahedral, and√2for the octahedral, when the sphere radius is 1.
Relative to a reference layer with positioning A, two more positionings B and C are possible. Every sequence of A, B, and C without immediate repetition of the same one is possible and gives an equally dense packing for spheres of a given radius.
The most regular ones are
There is an uncountably infinite number of disordered arrangements of planes (e.g. ABCACBABABAC...) that are sometimes collectively referred to as "Barlow packings", after crystallographerWilliam Barlow.[6]
In close-packing, the center-to-center spacing of spheres in thexyplane is a simple honeycomb-like tessellation with a pitch (distance between sphere centers) of one sphere diameter. The distance between sphere centers, projected on thez(vertical) axis, is:
wheredis the diameter of a sphere; this follows from the tetrahedral arrangement of close-packed spheres.
Thecoordination numberof HCP and FCC is 12 and theiratomic packing factors(APFs) are equal to the number mentioned above, 0.74.
When forming any sphere-packing lattice, the first fact to notice is that whenever two spheres touch a straight line may be drawn from the center of one sphere to the center of the other intersecting the point of contact. The distance between the centers along the shortest path namely that straight line will therefore ber1+r2wherer1is the radius of the first sphere andr2is the radius of the second. In close packing all of the spheres share a common radius,r. Therefore, two centers would simply have a distance 2r.
To form an A-B-A-B-... hexagonal close packing of spheres, the coordinate points of the lattice will be the spheres' centers. Suppose, the goal is to fill a box with spheres according to HCP. The box would be placed on thex-y-zcoordinate space.
First form a row of spheres. The centers will all lie on a straight line. Theirx-coordinate will vary by 2rsince the distance between each center of the spheres are touching is 2r. They-coordinate and z-coordinate will be the same. For simplicity, say that the balls are the first row and that theiry- andz-coordinates are simplyr, so that their surfaces rest on the zero-planes. Coordinates of the centers of the first row will look like (2r,r,r), (4r,r,r), (6r,r,r), (8r,r,r), ... .
Now, form the next row of spheres. Again, the centers will all lie on a straight line withx-coordinate differences of 2r, but there will be a shift of distancerin thex-direction so that the center of every sphere in this row aligns with thex-coordinate of where two spheres touch in the first row. This allows the spheres of the new row to slide in closer to the first row until all spheres in the new row are touching two spheres of the first row. Since the new spherestouchtwo spheres, their centers form an equilateral triangle with those two neighbors' centers. The side lengths are all 2r, so the height ory-coordinate difference between the rows is√3r. Thus, this row will have coordinates like this:
The first sphere of this row only touches one sphere in the original row, but its location follows suit with the rest of the row.
The next row follows this pattern of shifting thex-coordinate byrand they-coordinate by√3. Add rows until reaching thexandymaximum borders of the box.
In an A-B-A-B-... stacking pattern, the odd numberedplanesof spheres will have exactly the same coordinates save for a pitch difference in thez-coordinates and the even numberedplanesof spheres will share the samex- andy-coordinates. Both types of planes are formed using the pattern mentioned above, but the starting place for thefirstrow's first sphere will be different.
Using the plane described precisely above as plane #1, the A plane, place a sphere on top of this plane so that it lies touching three spheres in the A-plane. The three spheres are all already touching each other, forming an equilateral triangle, and since they all touch the new sphere, the four centers form aregular tetrahedron.[7]All of the sides are equal to 2rbecause all of the sides are formed by two spheres touching. The height of which or thez-coordinate difference between the two "planes" is√6r2/3. This, combined with the offsets in thexandy-coordinates gives the centers of the first row in the B plane:
The second row's coordinates follow the pattern first described above and are:
The difference to the next plane, the A plane, is again√6r2/3in thez-direction and a shift in thexandyto match thosex- andy-coordinates of the first A plane.[8]
In general, the coordinates of sphere centers can be written as:
wherei,jandkare indices starting at 0 for thex-,y- andz-coordinates.
Crystallographic features of HCP systems, such as vectors and atomic plane families, can be described using a four-valueMiller indexnotation (hkil) in which the third indexidenotes a degenerate but convenient component which is equal to −h−k. Theh,iandkindex directions are separated by 120°, and are thus not orthogonal; thelcomponent is mutually perpendicular to theh,iandkindex directions.
The FCC and HCP packings are the densest known packings of equal spheres with the highest symmetry (smallest repeat units).
Densersphere packingsare known, but they involveunequal sphere packing.
A packing density of 1, filling space completely, requires non-spherical shapes, such ashoneycombs.
Replacing each contact point between two spheres with an edge connecting the centers of the touching spheres produces tetrahedrons and octahedrons of equal edge lengths.
The FCC arrangement produces thetetrahedral-octahedral honeycomb.
The HCP arrangement produces thegyrated tetrahedral-octahedral honeycomb.
If, instead, every sphere is augmented with the points in space that are closer to it than to any other sphere, the duals of these honeycombs are produced: therhombic dodecahedral honeycombfor FCC, and thetrapezo-rhombic dodecahedral honeycombfor HCP.
Spherical bubbles appear in soapy water in a FCC or HCP arrangement when the water in the gaps between the bubbles drains out. This pattern also approaches therhombic dodecahedral honeycombortrapezo-rhombic dodecahedral honeycomb. However, such FCC or HCP foams of very small liquid content are unstable, as they do not satisfyPlateau's laws. TheKelvin foamand theWeaire–Phelan foamare more stable, having smaller interfacial energy in the limit of a very small liquid content.[9]
There are two types ofinterstitial holesleft by hcp and fcc conformations; tetrahedral and octahedral void. Four spheres surround the tetrahedral hole with three spheres being in one layer and one sphere from the next layer. Six spheres surround an octahedral voids with three spheres coming from one layer and three spheres coming from the next layer. Structures of many simple chemical compounds, for instance, are often described in terms of small atoms occupying tetrahedral or octahedral holes in closed-packed systems that are formed from larger atoms.
Layered structures are formed by alternating empty and filled octahedral planes. Two octahedral layers usually allow for four structural arrangements that can either be filled by an hpc of fcc packing systems. In filling tetrahedral holes a complete filling leads to fcc field array. In unit cells, hole filling can sometimes lead to polyhedral arrays with a mix of hcp and fcc layering.[10] | https://en.wikipedia.org/wiki/Close-packing_of_equal_spheres |
Apollonian sphere packingis the three-dimensional equivalent of theApollonian gasket. The principle of construction is very similar: with any four spheres that arecotangentto each other, it is then possible to construct two more spheres that are cotangent to four of them, resulting in an infinitesphere packing.
Thefractal dimensionis approximately 2.473946 (±1 in the last digit).[1]
ApolFrac is software for generating and visualization of Apollonian sphere packing.[2]
Thisfractal–related article is astub. You can help Wikipedia byexpanding it.
Thishyperbolic geometry-related article is astub. You can help Wikipedia byexpanding it. | https://en.wikipedia.org/wiki/Apollonian_sphere_packing |
In mathematics, the theory offinite sphere packingconcerns the question of how a finite number of equally-sizedspherescan be most efficiently packed. The question of packing finitely many spheres has only been investigated in detail in recent decades, with much of the groundwork being laid byLászló Fejes Tóth.
The similar problem for infinitely many spheres has a longer history of investigation, from which theKepler conjectureis most well-known. Atoms incrystal structurescan be simplistically viewed asclosely-packed spheresand treated as infinite sphere packings thanks to their large number.
Sphere packing problems are distinguished between packings in given containers and free packings. This article primarily discusses free packings.
In general, apackingrefers to any arrangement of a set of spatially-connected, possibly differently-sized or differently-shaped objects in space such that none of them overlap. In the case of the finite sphere packing problem, these objects are restricted to equally-sized spheres. Such a packing of spheres determines a specific volume known as theconvex hullof the packing, defined as the smallestconvex setthat includes all the spheres.
There are many possible ways to arrange spheres, which can be classified into three basic groups: sausage, pizza, and cluster packing.[1]
An arrangement in which the midpoint of all the spheres lie on a single straight line is called asausage packing, as the convex hull has a sausage-like shape. An approximate example in real life is the packing of tennis balls in a tube, though the ends must be rounded for the tube to coincide with the actual convex hull.
If all the midpoints lie on a plane, the packing is apizza packing. Approximate real-life examples of this kind of packing include billiard balls being packed in a triangle as they are set up. This holds for packings in three-dimensional Euclidean space.
If the midpoints of the spheres are arranged throughout 3D space, the packing is termed acluster packing. Real-life approximations include fruit being packed in multiple layers in a box.
By the given definitions, any sausage packing is technically also a pizza packing, and any pizza packing is technically also a cluster packing. In the more general case ofd{\displaystyle d}dimensions, "sausages" refer to one-dimensional arrangements, "clusters" tod{\displaystyle d}-dimensional arrangements, and "pizzas" to those with an in-between number of dimensions.[1]
One or two spheres always make a sausage. With three, a pizza packing (that is not also a sausage) becomes possible, and with four or more, clusters (that are not also pizzas) become possible.
The empty space between spheres varies depending on the type of packing. The amount of empty space is measured in thepacking density, which is defined as the ratio of the volume of the spheres to the volume of the total convex hull. The higher the packing density, the less empty space there is in the packing and thus the smaller the volume of the hull (in comparison to other packings with the same number and size of spheres).
To pack the spheres efficiently, it might be asked which packing has the highest possible density. It is easy to see that such a packing should have the property that the spheres lie next to each other, that is, each sphere should touch another on the surface. A more exact phrasing is to form agraphwhich assigns avertexfor each sphere and connects vertices with edges whenever the corresponding spheres if their surfaces touch. Then the highest-density packing must satisfy the property that the corresponding graph isconnected.
With three or four spheres, the sausage packing is optimal. It is believed that this holds true for anyn{\displaystyle n}up to55{\displaystyle 55}along withn=57,58,63,64{\displaystyle n=57,58,63,64}.[1][2]Forn=56,59,60,61,62{\displaystyle n=56,59,60,61,62}andn≥65{\displaystyle n\geq 65}, a cluster packing exists that is more efficient than the sausage packing, as shown in 1992 by Jörg Wills and Pier Mario Gandini.[2][3]It remains unknown what these most efficient cluster packings look like. For example, in the casen=56{\displaystyle n=56}, it is known that the optimal packing is not atetrahedralpacking like the classical packing of cannon balls, but is likely some kind ofoctahedralshape.[1]
The sudden transition in optimal packing shape is jokingly known by some mathematicians as thesausage catastrophe[4][5]The designationcatastrophecomes from the fact that the optimal packing shape suddenly shifts from the orderly sausage packing to the relatively unordered cluster packing and vice versa as one goes from one number to another, without a satisfying explanation as to why this happens. Even so, the transition in three dimensions is relatively tame; in four dimensions, the sudden transition is conjectured to happen around 377,000 spheres.[1]
For dimensionsd≤10{\displaystyle d\leq 10}, the optimal packing is always either a sausage or a cluster, and never a pizza. It is an open problem whether this holds true for all dimensions. This result only concerns spheres and not other convex bodies; in fact Gritzmann and Arhelger observed that for any dimensiond≥3{\displaystyle d\geq 3}there exists a convex shape for which the closest packing is a pizza.[1]
In the following section it is shown that for 455 spheres the sausage packing is non-optimal, and that there instead exists a special cluster packing that occupies a smaller volume.
The volume of a convex hull of a sausage packing withn{\displaystyle n}spheres of radiusr{\displaystyle r}is calculable with elementary geometry. The middle part of the hull is acylinderwith lengthh=2r⋅(n−1){\displaystyle h=2r\cdot (n-1)}while the caps at the end are half-spheres with radiusr{\displaystyle r}. The total volumeVW{\displaystyle V_{W}}is therefore given by.
Similarly, it is possible to find the volume of the convex hull of a tetrahedral packing, in which the spheres are arranged so that they form atetrahedralshape, which only leads to completely filled tetrahedra for specific numbers of spheres. If there arex{\displaystyle x}spheres along one edge of the tetrahedron, the total number of spheresn{\displaystyle n}is given by
Now theinradiusr{\displaystyle r}of a tetrahedral with side lengtha{\displaystyle a}is
From this we have
The volumeVT{\displaystyle V_{T}}of the tetrahedron is then given by the formula
In the case of many spheres being arranged inside a tetrahedron, the length of an edgea{\displaystyle a}increases by twice the radius of a sphere for each new layer, meaning that forx{\displaystyle x}layers the side length becomes
Substituting this value into the volume formula for the tetrahedron, we know that the volumeV{\displaystyle V}of the convex hull must be smaller than the tetrahedron itself, so that
Taking the number of spheres in a tetrahedron ofn{\displaystyle n}layers and substituting into the earlier expression to get the volumeVW{\displaystyle V_{\text{W}}}of the convex hull of a sausage packing with the same number of spheres, we have
Forx=13{\displaystyle x=13}, which translates ton=455{\displaystyle n=455}spheres the coefficient in front ofr3{\displaystyle r^{3}}is about 2845 for the tetrahedral packing and 2856 for the sausage packing, which implies that for this number of spheres the tetrahedron is more closely packed.
It is also possible with some more effort to derive the exact formula for the volume of the tetrahedral convex hullV{\displaystyle V}, which would involve subtracting the excess volume at the corners and edges of the tetrahedron. This allows the sausage packing to be proved non-optimal for smaller values ofx{\displaystyle x}and thereforen{\displaystyle n}.
The termsausagecomes from the mathematicianLászló Fejes Tóth, who posited thesausage conjecturein 1975,[6]which concerns a generalized version of the problem to spheres, convex hulls, and volume in higher dimensions. A generalized sphere ind{\displaystyle d}dimensions is ad{\displaystyle d}-dimensional body in which every boundary point lies equally far away from the midpoint. Fejes Tóth's sausage conjecture then states that fromd=5{\displaystyle d=5}upwards it is always optimal to arrange the spheres along a straight line. That is, the sausage catastrophe no longer occurs once we go above 4 dimensions. The overall conjecture remains open. The best results so far are those of Ulrich Betke and Martin Henk,[7]who proved the conjecture for dimensions 42 and above.
While it may be proved that the sausage packing is not optimal for 56 spheres, and that there must be some other packing that is optimal, it is not known what the optimal packing looks like. It is difficult to find the optimal packing as there is no "simple" formula for the volume of an arbitrarily shaped cluster. Optimality (and non-optimality) is shown through appropriate estimates of the volume, using methods fromconvex geometry, such as theBrunn-Minkowski inequality, mixed Minkowski volumes andSteiner's formula. A crucial step towards a unified theory of both finite and infinite (lattice and non-lattice) sphere packings was the introduction of parametric densities by Jörg Wills in 1992. The parametric density takes into account the influence of the edges of the packing.[8]
The definition of density used earlier concerns the volume of the convex hull of the spheres (or convex bodies)K{\displaystyle K}:
whereCn{\displaystyle C_{n}}is the convex hull of then{\displaystyle n}midpointsci{\displaystyle c_{i}}of the spheresKi{\displaystyle K_{i}}(instead of the sphere, we can also take an arbitrary convex body forK{\displaystyle K}). For a linear arrangement (sausage), the convex hull is a line segment through all the midpoints of the spheres. The plus sign in the formula refers toMinkowski additionof sets, so thatV(Cn+K){\displaystyle V(C_{n}+K)}refers to the volume of the convex hull of the spheres.
This definition works in two dimensions, where Laszlo Fejes-Toth,Claude Rogersand others used it to formulate a unified theory of finite and infinite packings. In three dimensions, Wills gives a simple argument that such a unified theory is not possible based on this definition: The densest finite arrangement of coins in three dimensions is the sausage withδ=1{\displaystyle \delta =1}. However, the optimal infinite arrangement is a hexagonal arrangement withδ≈0.9{\displaystyle \delta \approx 0.9}, so the infinite value cannot be obtained as a limit of finite values. To solve this issue, Wills introduces a modification to the definition by adding a positive parameterρ{\displaystyle \rho }:
ρ{\displaystyle \rho }allows the influence of the edges to be considered (giving the convex hull a certain thickness). This is then combined with methods from the theory ofmixed volumesandgeometry of numbersbyHermann Minkowski.
For each dimensiond≥2{\displaystyle d\geq 2}there are parameter valuesρs(d){\displaystyle \rho _{s}(d)}andρc(d){\displaystyle \rho _{c}(d)}such that forρ≤ρs(d){\displaystyle \rho \leq \rho _{s}(d)}the sausage is the densenst packing (for all integersn{\displaystyle n}), while forρ≥ρc(d){\displaystyle \rho \geq \rho _{c}(d)}and suffiricently largen{\displaystyle n}the cluster is densest. These parameters are dimension-specific. In two dimensions,ρc(2)=ρs(2)=32{\displaystyle \rho _{c}(2)=\rho _{s}(2)={\frac {\sqrt {3}}{2}}}so that there is a transition from sausages to clusters (sausage catastrophe).
There holds an inequality:[8]
where the volume of the unit ballBd{\displaystyle B^{d}}ind{\displaystyle d}dimensions isV(Bd){\displaystyle V(B^{d})}. Ford=2{\displaystyle d=2}, we haveρs(d)=ρc(d){\displaystyle \rho _{s}(d)=\rho _{c}(d)}and it is predicted that this holds for all dimensions, in which case the value ofρc(d){\displaystyle \rho _{c}(d)}can be found from that ofδ(Bd){\displaystyle \delta (B^{d})}. | https://en.wikipedia.org/wiki/Finite_sphere_packing |
Inmathematics, theHermite constant, named afterCharles Hermite, determines how long a shortest element of alatticeinEuclidean spacecan be.
The constantγnfor integersn> 0 is defined as follows. For a latticeLin Euclidean spaceRnwith unit covolume, i.e. vol(Rn/L) = 1, letλ1(L) denote the least length of a nonzero element ofL. Then√γnis the maximum ofλ1(L) over all such latticesL.
Thesquare rootin the definition of the Hermite constant is a matter of historical convention.
Alternatively, the Hermite constantγncan be defined as the square of the maximalsystoleof a flatn-dimensionaltorusof unit volume.
The Hermite constant is known in dimensions 1–8 and 24.
Forn= 2, one hasγ2=2/√3. This value is attained by thehexagonal latticeof theEisenstein integers, scaled to have afundamental parallelogramwith unit area.[1]
The constants for the missingnvalues are conjectured.[2]
It is known that[3]
A stronger estimate due toHans Frederick Blichfeldt[4]is[5]
whereΓ(x){\displaystyle \Gamma (x)}is thegamma function. | https://en.wikipedia.org/wiki/Hermite_constant |
Ingeometry, theinscribed sphereorinsphereof aconvex polyhedronis aspherethat is contained within the polyhedron andtangentto each of the polyhedron's faces. It is the largest sphere that is contained wholly within the polyhedron, and isdualto thedual polyhedron'scircumsphere.
The radius of the sphere inscribed in a polyhedronPis called theinradiusofP.
Allregular polyhedrahave inscribed spheres, but most irregular polyhedra do not have all facets tangent to a common sphere, although it is still possible to define the largest contained sphere for such shapes. For such cases, the notion of aninspheredoes not seem to have been properly defined and various interpretations of aninsphereare to be found:
Often these spheres coincide, leading to confusion as to exactly what properties define the insphere for polyhedra where they do not coincide.
For example, the regularsmall stellated dodecahedronhas a sphere tangent to all faces, while a larger sphere can still be fitted inside the polyhedron. Which is the insphere? Important authorities such as Coxeter or Cundy & Rollett are clear enough that the face-tangent sphere is the insphere. Again, such authorities agree that theArchimedean polyhedra(having regular faces and equivalent vertices) have no inspheres while the Archimedean dual orCatalanpolyhedra do have inspheres. But many authors fail to respect such distinctions and assume other definitions for the 'inspheres' of their polyhedra. | https://en.wikipedia.org/wiki/Inscribed_sphere |
Ingeometry, thekissing numberof amathematical spaceis defined as the greatest number of non-overlapping unitspheresthat can be arranged in that space such that they each touch a common unit sphere. For a givensphere packing(arrangement of spheres) in a given space, a kissing number can also be defined for each individual sphere as the number of spheres it touches. For alatticepacking the kissing number is the same for every sphere, but for an arbitrary sphere packing the kissing number may vary from one sphere to another.
Other names for kissing number that have been used areNewton number(after the originator of the problem), andcontact number.
In general, thekissing number problemseeks the maximum possible kissing number forn-dimensional spheresin (n+ 1)-dimensionalEuclidean space. Ordinary spheres correspond to two-dimensional closed surfaces in three-dimensional space.
Finding the kissing number when centers of spheres are confined to a line (the one-dimensional case) or a plane (two-dimensional case) is trivial. Proving a solution to the three-dimensional case, despite being easy to conceptualise and model in the physical world, eluded mathematicians until the mid-20th century.[1][2]Solutions in higher dimensions are considerably more challenging, and only a handful of cases have been solved exactly. For others, investigations have determined upper and lower bounds, but not exact solutions.[3]
In one dimension,[4]the kissing number is 2:
In two dimensions, the kissing number is 6:
Proof: Consider a circle with centerCthat is touched by circles with centersC1,C2, .... Consider the raysCCi. These rays all emanate from the same centerC, so the sum of angles between adjacent rays is 360°.
Assume by contradiction that there are more than six touching circles. Then at least two adjacent rays, sayCC1andCC2, are separated by an angle of less than 60°. The segmentsC Cihave the same length – 2r– for alli. Therefore, the triangleCC1C2isisosceles, and its third side –C1C2– has a side length of less than 2r. Therefore, the circles 1 and 2 intersect – a contradiction.[5]
In three dimensions, the kissing number is 12, but the correct value was much more difficult to establish than in dimensions one and two. It is easy to arrange 12 spheres so that each touches a central sphere, with a lot of space left over, and it is not obvious that there is no way to pack in a 13th sphere. (In fact, there is so much extra space that any two of the 12 outer spheres can exchange places through a continuous movement without any of the outer spheres losing contact with the center one.) This was the subject of a famous disagreement between mathematiciansIsaac NewtonandDavid Gregory. Newton correctly thought that the limit was 12; Gregory thought that a 13th could fit. Some incomplete proofs that Newton was correct were offered in the nineteenth century, most notably one byReinhold Hoppe, but the first correct proof (according to Brass, Moser, and Pach) did not appear until 1953.[1][2][6]
The twelve neighbors of the central sphere correspond to the maximum bulkcoordination numberof an atom in acrystal latticein which all atoms have the same size (as in a chemical element). A coordination number of 12 is found in acubic close-packedor ahexagonal close-packedstructure.
In four dimensions, the kissing number is 24. This was proven in 2003 by Oleg Musin.[7][8]Previously, the answer was thought to be either 24 or 25: it is straightforward to produce a packing of 24 spheres around a central sphere (one can place the spheres at the vertices of a suitably scaled24-cellcentered at the origin), but, as in the three-dimensional case, there is a lot of space left over — even more, in fact, than forn= 3 — so the situation was even less clear.
The existence of the highly symmetricalE8latticeandLeech latticehas allowed known results forn= 8 (where the kissing number is 240), andn= 24 (where it is 196,560).[9][10]The kissing number inndimensionsis unknown for other dimensions.
If arrangements are restricted tolatticearrangements, in which the centres of the spheres all lie on points in alattice, then this restricted kissing number is known forn= 1 to 9 andn= 24 dimensions.[11]For 5, 6, and 7 dimensions the arrangement with the highest known kissing number found so far is the optimal lattice arrangement, but the existence of a non-lattice arrangement with a higher kissing number has not been excluded.
The following table lists some known bounds on the kissing number in various dimensions.[12]The dimensions in which the kissing number is known are listed in boldface.
The kissing number problem can be generalized to the problem of finding the maximum number of non-overlappingcongruentcopies of anyconvex bodythat touch a given copy of the body. There are different versions of the problem depending on whether the copies are only required to be congruent to the original body,translatesof the original body, or translated by a lattice. For theregular tetrahedron, for example, it is known that both the lattice kissing number and the translative kissing number are equal to 18, whereas the congruent kissing number is at least 56.[14]
There are severalapproximation algorithmsonintersection graphswhere the approximation ratio depends on the kissing number.[15]For example, there is
a polynomial-time 10-approximation algorithm to find a maximum non-intersecting subset of a set of rotated unit squares.
The kissing number problem can be stated as the existence of a solution to a set ofinequalities. Letxn{\displaystyle x_{n}}be a set ofND-dimensional position vectors of the centres of the spheres. The condition that this set of spheres can lie round the centre sphere without overlapping is:[16]∃x{∀n{xnTxn=1}∧∀m,n:m≠n{(xn−xm)T(xn−xm)≥1}}{\displaystyle \exists x\ \left\{\forall _{n}\{x_{n}^{\textsf {T}}x_{n}=1\}\land \forall _{m,n:m\neq n}\{(x_{n}-x_{m})^{\textsf {T}}(x_{n}-x_{m})\geq 1\}\right\}}
Thus the problem for each dimension can be expressed in theexistential theory of the reals. However, general methods of solving problems in this form take at leastexponential timewhich is why this problem has only been solved up to four dimensions. By adding additional variables,ynm{\displaystyle y_{nm}}this can be converted to a singlequartic equationinN(N− 1)/2 +DNvariables:[17]∃xy{∑n(xnTxn−1)2+∑m,n:m<n((xn−xm)T(xn−xm)−1−(ynm)2)2=0}{\displaystyle \exists xy\ \left\{\sum _{n}\left(x_{n}^{\textsf {T}}x_{n}-1\right)^{2}+\sum _{m,n:m<n}{\Big (}(x_{n}-x_{m})^{\textsf {T}}(x_{n}-x_{m})-1-(y_{nm})^{2}{\Big )}^{2}=0\right\}}
Therefore, to solve the case inD= 5 dimensions andN=40+ 1 vectors would be equivalent to determining the existence of real solutions to a quartic polynomial in 1025 variables. For theD= 24 dimensions andN=196560+ 1, the quartic would have 19,322,732,544 variables. An alternative statement in terms ofdistance geometryis given by the distances squaredRmn{\displaystyle R_{mn}}between themthandnthsphere:∃R{∀n{R0n=1}∧∀m,n:m<n{Rmn≥1}}{\displaystyle \exists R\ \{\forall _{n}\{R_{0n}=1\}\land \forall _{m,n:m<n}\{R_{mn}\geq 1\}\}}
This must be supplemented with the condition that theCayley–Menger determinantis zero for any set of points which forms a (D+ 1) simplex inDdimensions, since that volume must be zero. SettingRmn=1+ymn2{\displaystyle R_{mn}=1+{y_{mn}}^{2}}gives a set of simultaneous polynomial equations in justywhich must be solved for real values only. The two methods, being entirely equivalent, have various different uses. For example, in the second case one can randomly alter the values of theyby small amounts to try to minimise the polynomial in terms of they. | https://en.wikipedia.org/wiki/Kissing_number |
Inmathematicsandcomputer science, in the field ofcoding theory, theHamming boundis a limit on the parameters of an arbitraryblock code: it is also known as thesphere-packing boundor thevolume boundfrom an interpretation in terms ofpacking ballsin theHamming metricinto thespaceof all possible words. It gives an important limitation on theefficiencywith which anyerror-correcting codecan utilize the space in which itscode wordsare embedded. A code that attains the Hamming bound is said to be aperfect code.
An original message and an encoded version are both composed in an alphabet ofqletters. Eachcode wordcontainsnletters. The original message (of lengthm) is shorter thannletters. The message is converted into ann-letter codeword by an encoding algorithm, transmitted over a noisychannel, and finally decoded by the receiver. The decoding process interprets a garbled codeword, referred to as simply aword, as the valid codeword "nearest" then-letter received string.
Mathematically, there are exactlyqmpossible messages of lengthm, and each message can be regarded as avectorof lengthm. The encoding scheme converts anm-dimensional vector into ann-dimensional vector. Exactlyqmvalid codewords are possible, but any one ofqnwords can be received because the noisy channel might distort one or more of thenletters when a codeword is transmitted.
An alphabet setAq{\displaystyle {\mathcal {A}}_{q}}is a set of symbols withq{\displaystyle q}elements. The set of strings of lengthn{\displaystyle n}on the alphabet setAq{\displaystyle {\mathcal {A}}_{q}}are denotedAqn{\displaystyle {\mathcal {A}}_{q}^{n}}. (There areqn{\displaystyle q^{n}}distinct strings in this set of strings.) Aq{\displaystyle q}-ary block code of lengthn{\displaystyle n}is a subset of the strings ofAqn{\displaystyle {\mathcal {A}}_{q}^{n}}, where the alphabet setAq{\displaystyle {\mathcal {A}}_{q}}is any alphabet set havingq{\displaystyle q}elements. (The choice of alphabet setAq{\displaystyle {\mathcal {A}}_{q}}makes no difference to the result, provided the alphabet is of sizeq{\displaystyle q}.)
LetAq(n,d){\displaystyle \ A_{q}(n,d)}denote the maximum possible size of aq{\displaystyle q}-ary block codeC{\displaystyle \ C}of lengthn{\displaystyle n}and minimumHamming distanced{\displaystyle d}between elements of the block code (necessarily positive forqn>1{\displaystyle q^{n}>1}).
Then, the Hamming bound is:
where
It follows from the definition ofd{\displaystyle d}that if at most
errors are made during transmission of acodewordthenminimum distance decodingwill decode it correctly (i.e., it decodes the received word as the codeword that was sent). Thus the code is said to be capable of correctingt{\displaystyle t}errors.
For each codewordc∈C{\displaystyle c\in C}, consider aballof fixed radiust{\displaystyle t}aroundc{\displaystyle c}. Every pair of these balls (Hamming balls) are non-intersecting by thet{\displaystyle t}-error-correcting property. Letm{\displaystyle m}be the number of words in each ball (in other words, the volume of the ball). A word that is in such a ball can deviate in at mostt{\displaystyle t}components from those of the ball'scentre, which is a codeword. The number of such words is then obtained bychoosingup tot{\displaystyle t}of then{\displaystyle n}components of a codeword to deviate to one of(q−1){\displaystyle (q-1)}possible other values (recall, the code isq{\displaystyle q}-ary: it takes values inAqn{\displaystyle {\mathcal {A}}_{q}^{n}}). Thus,
Aq(n,d){\displaystyle A_{q}(n,d)}is the (maximum) total number of codewords inC{\displaystyle C}, and so, by the definition oft{\displaystyle t}, the greatest number of balls with no two balls having a word in common. Taking theunionof the words in these balls centered at codewords, results in a set of words, each counted precisely once, that is a subset ofAqn{\displaystyle {\mathcal {A}}_{q}^{n}}(where|Aqn|=qn{\displaystyle |{\mathcal {A}}_{q}^{n}|=q^{n}}words) and so:
Whence:
For anAq(n,d){\displaystyle A_{q}(n,d)}codeC(a subset ofAqn{\displaystyle {\mathcal {A}}_{q}^{n}}), thecovering radiusofCis the smallest value ofrsuch that every element ofAqn{\displaystyle {\mathcal {A}}_{q}^{n}}is contained in at least one ball of radiusrcentered at each codeword ofC. Thepacking radiusofCis the largest value ofssuch that the set of balls of radiusscentered at each codeword ofCare mutuallydisjoint.
From the proof of the Hamming bound, it can be seen that fort=⌊12(d−1)⌋{\displaystyle t\,=\,\left\lfloor {\frac {1}{2}}(d-1)\right\rfloor }, we have:
Therefore,s≤rand if equality holds thens=r=t. The case of equality means that the Hamming bound is attained.
Codes that attain the Hamming bound are calledperfect codes. Examples include codes that have only one codeword, and codes that are the whole ofAqn{\displaystyle \scriptstyle {\mathcal {A}}_{q}^{n}}. Another example is given by therepeat codes, where each symbol of the message is repeated an odd fixed number of times to obtain a codeword whereq= 2. All of these examples are often called thetrivialperfect codes.
In 1973, Tietäväinen proved[1]that any non-trivial perfect code over a prime-power alphabet has the parameters of aHamming codeor aGolay code.
A perfect code may be interpreted as one in which the balls of Hamming radiustcentered on codewords exactly fill out the space (tis the covering radius = packing radius). Aquasi-perfect codeis one in which the balls of Hamming radiustcentered on codewords are disjoint and the balls of radiust+1 cover the space, possibly with some overlaps.[2]Another way to say this is that a code isquasi-perfectif its covering radius is one greater than its packing radius.[3] | https://en.wikipedia.org/wiki/Sphere-packing_bound |
Random close packing(RCP) of spheres is an empirical parameter used to characterize the maximumvolume fractionofsolidobjects obtained when they are packed randomly. For example, when a solid container is filled withgrain, shaking the container will reduce the volume taken up by the objects, thus allowing more grain to be added to the container. In other words, shaking increases the density of packed objects. But shaking cannot increase the density indefinitely, a limit is reached, and if this is reached without obvious packing into an ordered structure, such as a regular crystal lattice, this is the empirical random close-packed density for this particular procedure of packing. The random close packing is the highest possible volume fraction out of all possible packing procedures.
Experiments and computer simulations have shown that the most compact way to pack hard perfect same-size spheres randomly gives a maximum volume fraction of about 64%, i.e., approximately 64% of the volume of a container is occupied by the spheres. The problem of predicting theoretically the random close pack of spheres is difficult mainly because of the absence of a unique definition of randomness or disorder.[1]The random close packing value is significantly below the maximum possibleclose-packing of same-size hard spheres into a regular crystalline arrangements, which is 74.04%.[2]Both theface-centred cubic (fcc)and hexagonal close packed (hcp) crystal lattices have maximum densities equal to this upper limit, which can occur through the process ofgranular crystallisation.
The random close packing fraction of discs in the plane has also been considered a theoretically unsolved problem because of similar difficulties. An analytical, though not in closed form, solution to this problem was found in 2021 by R. Blumenfeld.[3]The solution was found by limiting the probability of growth of ordered clusters to be exponentially small and relating it to the distribution of `cells', which are the smallest voids surrounded by connected discs. The derived maximum volume fraction is 85.3542%, if only hexagonal lattice clusters are disallowed, and 85.2514% if one disallows also deformed square lattice clusters.
An analytical and closed-form solution for both 2D and 3D, mechanically stable, random packings of spheres has been found byA. Zacconein 2022 using the assumption that the most random branch of jammed states (maximally random jammed packings, extending up to the fcc closest packing) undergo crowding in a way qualitatively similar to an equilibrium liquid.[4][5]The reasons for the effectiveness of this solution are the object of ongoing debate.[6]
Random close packing of spheres does not have yet a precise geometric definition. It is defined statistically, and results are empirical. A container is randomly filled with objects, and then the container is shaken or tapped until the objects do not compact any further, at this point the packing state is RCP. The definition of packing fraction can be given as: "the volume taken by number of particles in a given space of volume". In other words, packing fraction defines the packing density. It has been shown that the filling fraction increases with the number of taps until the saturation density is reached.[7][8]Also, the saturation density increases as the tappingamplitudedecreases. Thus, RCP is the packing fraction given by thelimitas the tapping amplitude goes to zero, and the limit as the number of taps goes toinfinity.
The particle volume fraction at RCP depends on the objects being packed. If the objects arepolydispersedthen the volume fraction depends non-trivially on the size-distribution and can be arbitrarily close to 1 in the limit of very largestandard deviationof the size-distribution. In practice, available analytical solutions and numerical simulations display a saturation plateau at volume fractions 0.96-0.99 for large values of the standard deviation.[9][10]Still for (relatively) monodisperse objects the value for RCP depends on the object shape; for spheres it is 0.64, forM&M'scandy it is 0.68.[11]
Products containing loosely packed items are often labeled with this message: 'Contents May Settle During Shipping'.
Usually during shipping, the container will be bumped numerous times, which will increase the packing density.
The message is added to assure the consumer that the container is full on a mass basis, even though the container appears slightly empty. Systems of packed particles are also used as a basic model ofporous media. | https://en.wikipedia.org/wiki/Random_close_pack |
Sphere packing in a cylinderis a three-dimensionalpacking problemwith the objective of packing a given number of identicalspheresinside acylinderof specified diameter and length. For cylinders with diameters on the same order of magnitude as the spheres, such packings result in what are calledcolumnar structures.
These problems are studied extensively in the context ofbiology,nanoscience,materials science, and so forth due to the analogous assembly of small particles (likecellsandatoms) into cylindricalcrystalline structures.
The book "Columnar Structures of Spheres: Fundamentals and Applications"[1]serves as a notable contributions to this field of study. Authored by Winkelmann and Chan, the book reviews theoretical foundations and practical applications of densely packed spheres within cylindrical confinements.
Columnar structures appear in various research fields on a broad range of length scales from metres down to the nanoscale. On the largest scale, such structures can be found inbotanywhere seeds of a plant assemble around the stem. On a smaller scale bubbles of equal size crystallise to columnarfoamstructures when confined in a glass tube. Innanosciencesuch structures can be found in man-made objects which are on length scales from a micron to the nanoscale.
Columnar structures were first studied in botany due to their diverse appearances in plants.[2]D'Arcy Thompsonanalysed such arrangement of plant parts around the stem in his book "On Growth and Form" (1917). But they are also of interest in other biological areas, including bacteria,[3]viruses,[4]microtubules,[5]and thenotochordof thezebra fish.[6]
One of the largest flowers where the berries arrange in a regular cylindrical form is thetitan arum. This flower can be up to 3m in height and is natively solely found in western Sumatra and western Java.
On smaller length scales, the berries of theArum maculatumform a columnar structure in autumn. Its berries are similar to that of the corpse flower, since the titan arum is its larger relative. However, the cuckoo-pint is much smaller in height (height ≈ 20 cm). The berry arrangement varies with the stem to berry size.
Another plant that can be found in many gardens of residential areas is theAustralian bottlebrush. It assembles its seed capsules around a branch of the plant. The structure depends on the seed capsule size to branch size.
A further occurrence of ordered columnar arrangement on the macroscale arefoamstructures confined inside a glass tube. They can be realised experimentally with equal-sized soap bubbles inside a glass tube, produced by blowing air of constant gas flow through a needle dipped in a surfactant solution.[7]By putting the resulting foam column under forced drainage (feeding it with surfactant solution from the top), the foam can be adjusted to either a dry (bubbles shaped aspolyhedrons) or wet (spherical bubbles) structure.[8]
Due to this simple experimental set-up, many columnar structures have been discovered and investigated in the context of foams with experiments as well as simulation. Many simulations have been carried out using theSurface Evolverto investigate dry structure or thehard sphere modelfor the wet limit where the bubbles are spherical.
In the zigzag structure the bubbles are stacked on top of each other in a continuous w-shape. For this particular structure a moving interface with increasing liquid fraction was reported by Hutzleret al.in 1997.[9]This included an unexpected 180° twist interface, whose explanation is still lacking.
The first experimental observation of aline-slip structurewas discovered by Winkelmannet al.in a system of bubbles.[10]
Further discovered structures include complex structures with internal spheres/foam cells. Some dry foam structures with interior cells were found to consist of a chain of pentagonaldodecahedraorKelvin cellsin the centre of the tube.[11]For many more arrangements of this type, it was observed that the outside bubble layer is ordered, with each internal layer resembling a different, simpler columnar structure by usingX-ray tomography.[7]
Columnar structures have also been studied intensively in the context ofnanotubes. Their physical or chemical properties can be altered by trapping identical particles inside them.[12][13][14]These are usually done by self-assembling fullerenes such asC60, C70, or C78 into carbon nanotubes,[12]but also boron nitride nanotubes[15]
Such structures also assemble when particles are coated on the surface of a spherocylinder as in the context of pharmaceutical research. Lazároet al.examined the morphologies of virus capsid proteins self-assembled around metal nanorods.[16]Drug particles were coated as densely as possible on a spherocylinder to provide the best medical treatment.
Wuet al.built rods of the size of several microns. These microrods are created by densely packing silica colloidal particles inside cylindrical pores. By solidifying the assembled structures the microrods were imaged and examined using scanning electron microscopy (SEM).[17]
Columnar arrangements are also investigated as a possible candidate ofoptical metamaterials(i.e. materials with a negative refractive index) which find applications in super lenses[18]or optical cloaking.[19]Tanjeemet al.are constructing such a resonator by self-assembling nanospheres on the surface of the cylinder.[20][21]The nanospheres are suspended in anSDSsolution together with a cylinder of diameterD{\textstyle D}, much larger than the diameter of the nanospheresd{\displaystyle d}(D/d≈3to5{\textstyle D/d\approx 3{\text{ to }}5}). The nanospheres then stick to the surface of the cylinders by adepletion force.
The most common way of classifyingorderedcolumnar structures uses thephyllotactic notation, adopted from botany. It is used to describe arrangements of leaves of a plant, pine cones, or pineapples, but also planar patterns of florets in a sunflower head. While the arrangement in the former are cylindrical, the spirals in the latter are arranged on a disk. For columnar structures phyllotaxis in the context of cylindrical structures is adopted.
The phyllotactic notation describes such structures by a triplet of positive integers(l=m+n,m,n){\displaystyle (l=m+n,m,n)}withl≥m≥n{\textstyle l\geq m\geq n}. Each numberl{\textstyle l},m{\displaystyle m}, andn{\textstyle n}describes a family of spirals in the 3-dimensional packing. They count the number of spirals in each direction until the spiral repeats. This notation, however, only applies to triangular lattices and is therefore restricted to the ordered structures without internal spheres.
Ordered columnar structures without internal spheres are categorised into two separate classes:uniformandline-slipstructures. For each structure that can be identified with the triplet(l,m,n){\textstyle (l,m,n)}, there exist a uniform structure and at least one line slip.
A uniform structure is identified by each sphere having the same number of contacting neighbours.[22][1]This gives each sphere an identical neighbourhood. In the example image on the side each sphere has six neighbouring contacts.
The number of contacts is best visualised in the rolled-out contact network. It is created by rolling out the contact network into a plane of heightz{\textstyle z}and azimuthal angleθ{\textstyle \theta }of each sphere. For a uniform structure such as the one in the example image, this leads to a regularhexagonal lattice. Each dot in this pattern represents a sphere of the packing and each line a contact between adjacent spheres.
For all uniform structures above a diameter ratio ofD/d>2.0{\displaystyle D/d>2.0}, the regular hexagonal lattice is its characterising feature since this lattice type has the maximum number of contacts.[22][1]For different uniform structures(l,m,n){\displaystyle (l,m,n)}the rolled-out contact pattern only varies by a rotation in thez-θ{\textstyle z{\text{-}}\theta }plane. Each uniform structure is thus distinguished by its periodicity vectorV{\textstyle V}, which is defined by the phyllotactic triplet(l,m,n){\displaystyle (l,m,n)}.
For each uniform structure, there also exists a related but different structure, called a line-slip arrangement.[22][1]
The differences between uniform and line-slip structures are marginal and difficult to spot from images of the sphere packings. However, by comparing their rolled-out contact networks, one can spot that certain lines (which represent contacts) are missing.
All spheres in a uniform structure have the same number of contacts, but the number of contacts for spheres in a line slip may differ from sphere to sphere. For the example line slip in the image on the right side, some spheres count five and others six contacts. Thus a line slip structure is characterised by these gaps or loss of contacts.
Such a structure is termed line slip because the losses of contacts occur along a line in the rolled-out contact network. It was first identified by Picketet al., but not termed line slip.[23]
The direction, in which the loss of contacts occur can be denoted in the phyllotactic notation(l,m,n){\textstyle (l,m,n)}, since each number represents one of the lattice vectors in the hexagonal lattice.[22][1]This is usually indicated by a bold number.
By shearing the row of spheres below the loss of contact against a row above the loss of contact, one can regenerate two uniform structures related to this line slip. Thus, each line slip is related to two adjacent uniform structures, one at a higher and one at a lower diameter ratioD/d{\textstyle D/d}.[22][1][24]
Winkelmannet al.were the first to experimentally realise such a structure using soap bubbles in a system of deformable spheres.[10]
Columnar structures arise naturally in the context of dense hard sphere packings inside a cylinder. Mughalet al.studied such packings usingsimulated annealingup to the diameter ratio ofD/d=2.873{\textstyle D/d=2.873}for cylinder diameterD{\textstyle D}to sphere diameterd{\textstyle d}.[24]This includes some structures with internal spheres that are not in contact with the cylinder wall.
They calculated the packing fraction for all these structures as a function of the diameter ratio. At the peaks of this curve lie the uniform structures. In-between these discrete diameter ratios are the line slips at a lower packing density. Their packing fraction is significantly smaller than that of an unconfined lattice packing such asfcc, bcc, or hcp due to the free volume left by the cylindrical confinement.
The rich variety of such ordered structures can also be obtained by sequential depositioning the spheres into the cylinder.[25]Chan reproduced all dense sphere packings up toD/d<2.7013{\textstyle D/d<2.7013}using an algorithm, in which the spheres are placed sequentially dropped inside the cylinder.
Mughalet al.also discovered that such structures can be related to disk packings on a surface of a cylinder.[24]The contact network of both packings are identical. For both packing types, it was found that different uniform structures are connected with each other by line slips.[24]
Fuet al.extended this work to higher diameter ratiosD/d<4.0{\textstyle D/d<4.0}usinglinear programmingand discovered 17 new dense structures with internal spheres that are not in contact with the cylinder wall.[26]
A similar variety of dense crystalline structures have also been discovered for columnar packings ofspheroidsthroughMonte Carlo simulations.[27]Such packings include achiral structures with specific spheroid orientations and chiral helical structures with rotating spheroid orientations.
A further dynamic method to assemble such structures was introduced by Leeet al.[28]Here, polymeric beads are placed together with a fluid of higher density inside a rotatinglathe.
When the lathe is static, the beads float on top of the liquid. With increasing rotational speed, thecentripetal forcethen pushes the fluidoutwardsand the beadstowardthe central axis. Hence, the beads are essentially confined by a potential given by therotational energyErot=12mR2ω2,{\displaystyle E_{\text{rot}}={\frac {1}{2}}mR^{2}\omega ^{2},}wherem{\textstyle m}is the mass of the beads,R{\textstyle R}the distance from the central axis, andω{\textstyle \omega }the rotational speed. Due to theR2{\textstyle R^{2}}proportionality, the confining potential resembles that of a cylindricalharmonic oscillator.
Depending on number of spheres and rotational speed, a variety of ordered structures that are comparable to the dense sphere packings were discovered.
A comprehensive theory to this experiment was developed by Winkelmannet al.[29]It is based on analytic energy calculations using a generic sphere model and predictsperitectoidstructure transitions. | https://en.wikipedia.org/wiki/Cylinder_sphere_packing |
Sphere packing in a sphereis a three-dimensionalpacking problemwith the objective of packing a given number of equal spheres inside aunit sphere. It is the three-dimensional equivalent of thecircle packing in a circleproblem in two dimensions. | https://en.wikipedia.org/wiki/Sphere_packing_in_a_sphere |
Inmathematics,computer science,telecommunication,information theory, andsearching theory,error-correcting codes with feedbackareerror correcting codesdesigned to work in the presence of feedback from the receiver to the sender.[1]
Alice (the sender) wishes to send a valuexto Bob (the receiver). The communication channel between Alice and Bob is imperfect, and can introduce errors.
An error-correcting code is a way ofencodingxas a message such that Bob will successfully understand the valuexas intended by Alice, even if the message Alice sends and the message Bob receives differ. In an error-correcting code with feedback, the channel istwo-way: Bob can send feedback to Alice about the message he received.
In an error-correcting code withoutnoisy feedback, the feedback received by the sender is always free of errors. In an error-correcting code with noisy feedback, errors can occur in the feedback, as well as in the message.
An error-correcting code withnoiseless feedbackis equivalent to anadaptivesearchstrategy with errors.[1]
In 1956,Claude Shannonintroduced thediscretememorylesschannel with noiseless feedback. In 1961,Alfréd Rényiintroduced theBar-Kochba game(also known asTwenty questions), with a given percentage of wrong answers, and calculated the minimum number of randomly chosen questions to determine the answer.
In his 1964 dissertation,Elwyn Berlekampconsidered error correcting codes with noiseless feedback.[2][3]In Berlekamp's scenario, the receiver chose a subset of possible messages and asked the sender whether the given message was in this subset, a 'yes' or 'no' answer. Based on this answer, the receiver then chose a new subset and repeated the process. The game is further complicated due to noise; some of the answers will be wrong. | https://en.wikipedia.org/wiki/Error-correcting_codes_with_feedback |
IEEE 802.11n-2009, or802.11n, is a wireless-networking standard that uses multiple antennas to increase data rates. TheWi-Fi Alliancehas also retroactively labelled the technology for the standard asWi-Fi 4.[4][5]It standardized support formultiple-input multiple-output(MIMO),frame aggregation, and security improvements, among other features, and can be used in the 2.4 GHz or 5 GHz frequency bands.
Being the firstWi-Fistandard to introduceMIMOsupport, devices and systems which supported the 802.11n standard (or draft versions thereof) were sometimes referred to as MIMO Wi-Fi products, especially prior to the introduction of the next generation standard.[6]The use of MIMO-OFDM(orthogonal frequency division multiplexing) to increase the data rate while maintaining the same spectrum as 802.11a was first demonstrated by Airgo Networks.[7]
The purpose of the standard is to improve network throughput over the two previous standards—802.11aand802.11g—with a significant increase in the maximumnet data ratefrom 54 Mbit/s to 72 Mbit/s with a single spatial stream in a 20 MHz channel, and 600 Mbit/s (slightly highergross bit rateincluding for example error-correction codes, and slightly lower maximumthroughput) with the use of four spatial streams at a channel width of 40 MHz.[8][9]
IEEE 802.11n-2009 is an amendment to theIEEE 802.11-2007wireless-networking standard.802.11is a set ofIEEEstandards that govern wireless networking transmission methods. They are commonly used today in their802.11a,802.11b,802.11g, 802.11n,802.11acand802.11axversions to provide wireless connectivity in homes and businesses. Development of 802.11n began in 2002, seven years before publication. The 802.11n protocol is now Clause 20 of the publishedIEEE 802.11-2012standard and subsequently renamed to clause 19 of the published IEEE 802.11-2020 standard.
IEEE 802.11n is an amendment to IEEE 802.11-2007 as amended byIEEE 802.11k-2008,IEEE 802.11r-2008,IEEE 802.11y-2008, andIEEE 802.11w-2009, and builds on previous 802.11 standards by adding amultiple-input multiple-output(MIMO) system and 40 MHz channels to thePHY (physical layer)andframe aggregationto theMAC layer. There were older proprietary implementations of MIMO and 40MHz channels such asXpress,Super GandNitrowhich were based upon 802.11g and 802.11a technology, but this was the first time it was standardized across all radio manufacturers.
MIMO is a technology that uses multiple antennas to coherently resolve more information than possible using a single antenna. One way it provides this is throughspatial division multiplexing(SDM), which spatially multiplexes multiple independent data streams, transferred simultaneously within one spectral channel of bandwidth. MIMO SDM can significantly increase data throughput as the number of resolved spatial data streams is increased. Each spatial stream requires a discrete antenna at both the transmitter and the receiver. In addition, MIMO technology requires a separate radio-frequency chain and analog-to-digital converter for each antenna, making it more expensive to implement than non-MIMO systems.
Channels operating with a width of 40 MHz are another feature incorporated into 802.11n; this doubles the channel width from 20 MHz in previous 802.11 PHYs to transmit data, and provides twice the PHY data rate available over a single 20 MHz channel. It can be enabled in the 5 GHz mode, or within the 2.4 GHz mode if there is knowledge that it will not interfere with any other 802.11 or non-802.11 (such as Bluetooth) system using the same frequencies.[10]The MIMO architecture, together with the wider channels, offers increased physical transfer rate over standard802.11a(5 GHz) and802.11g(2.4 GHz).[11]
The transmitter and receiver useprecodingand postcoding techniques, respectively, to achieve the capacity of a MIMO link. Precoding includesspatial beamformingand spatial coding, where spatial beamforming improves the received signal quality at the decoding stage. Spatial coding can increase data throughput viaspatial multiplexingand increase range by exploiting the spatial diversity, through techniques such asAlamouti coding.
The number of simultaneous data streams is limited by the minimum number of antennas in use on both sides of the link. However, the individual radios often further limit the number of spatial streams that may carry unique data. Thea×b:cnotation helps identify what a given radio is capable of. The first number (a) is the maximum number of transmit antennas or transmitting TF chains that can be used by the radio. The second number (b) is the maximum number of receive antennas or receiving RF chains that can be used by the radio. The third number (c) is the maximum number of data spatial streams the radio can use. For example, a radio that can transmit on two antennas and receive on three, but can only send or receive two data streams, would be2 × 3 : 2.
The 802.11n draft allows up to4 × 4 : 4.Common configurations of 11n devices are2 × 2 : 2,2 × 3 : 2, and3 × 2 : 2. All three configurations have the same maximum throughputs and features, and differ only in the amount of diversity the antenna systems provide. In addition, a fourth configuration,3 × 3 : 3is becoming common, which has a higher throughput, due to the additional data stream.[12]
Assuming equal operating parameters to an 802.11g network achieving 54 megabits per second (on a single 20 MHz channel with one antenna), an 802.11n network can achieve 72 megabits per second (on a single 20 MHz channel with one antenna and 400 nsguard interval); 802.11n's speed may go up to 150 megabits per second if there are not other Bluetooth, microwave or Wi-Fi emissions in the neighborhood by using two 20 MHz channels in 40 MHz mode. If more antennas are used, then 802.11n can go up to 288 megabits per second in 20 MHz mode with four antennas, or 600 megabits per second in 40 MHz mode with four antennas and 400 ns guard interval. Because the 2.4 GHz band is seriously congested in most urban areas, 802.11n networks usually have more success in increasing data rate by utilizing more antennas in 20 MHz mode rather than by operating in the 40 MHz mode, as the 40 MHz mode requires a relatively free radio spectrum which is only available in rural areas away from cities. Thus, network engineers installing an 802.11n network should strive to select routers and wireless clients with the most antennas possible (one, two, three or four as specified by the 802.11n standard) and try to make sure that the network's bandwidth will be satisfactory even on the 20 MHz mode.
Data rates up to 600 Mbit/s are achieved only with the maximum of four spatial streams using one 40 MHz-wide channel. Various modulation schemes and coding rates are defined by the standard, which also assigns an arbitrary number to each; this number is themodulation and coding schemeindex, orMCS index. The table below shows the relationships between the variables that allow for the maximum data rate. GI (Guard Interval): Timing between symbols.[13]
20 MHz channel uses anFFTof 64, of which: 56OFDMsubcarriers, 52 are for data and 4 arepilot toneswith a carrier separation of 0.3125 MHz (20 MHz/64) (3.2 μs). Each of these subcarriers can be aBPSK,QPSK, 16-QAMor 64-QAM. The total bandwidth is 20 MHz with an occupied bandwidth of 17.8 MHz. Total symbol duration is 3.6 or 4microseconds, whichincludesa guard interval of 0.4 (also known as short guard interval (SGI)) or 0.8 microseconds.
PHY level data rate does not match user level throughput because of 802.11 protocol overheads, like the contention process, interframe spacing, PHY level headers (Preamble + PLCP) and acknowledgment frames. The mainmedia access control(MAC) feature that provides a performance improvement is aggregation. Two types of aggregation are defined:
Frame aggregationis a process of packing multiple MSDUs or MPDUs together to reduce the overheads and average them over multiple frames, thereby increasing the user level data rate. A-MPDU aggregation requires the use ofblock acknowledgementor BlockAck, which was introduced in 802.11e and has been optimized in 802.11n.
When 802.11g was released to share the band with existing 802.11b devices, it provided ways of ensuringbackward compatibilitybetween legacy and successor devices. 802.11n extends the coexistence management to protect its transmissions from legacy devices, which include802.11g,802.11band802.11a. There are MAC and PHY level protection mechanisms as listed below:
To achieve maximum output, a pure 802.11n 5 GHz network is recommended. The 5 GHz band has substantial capacity due to many non-overlapping radio channels and less radio interference as compared to the 2.4 GHz band.[14]An 802.11n-only network may be impractical for many users because they need to support legacy equipment that still is 802.11b/g only. In a mixed-mode system, an optimal solution would be to use a dual-radio access point and place the 802.11b/g traffic on the 2.4 GHz radio and the 802.11n traffic on the 5 GHz radio.[15]This setup assumes that all the 802.11n clients are 5 GHz capable, which is not a requirement of the standard. 5 GHz is optional on Wi-Fi 4; quite some Wi-Fi 4 capable devices only support 2.4 GHz and there is no practical way to upgrade them to support 5 GHz. Some enterprise-grade APs useband steeringto send 802.11n clients to the 5 GHz band, leaving the 2.4 GHz band for legacy clients. Band steering works by responding only to 5 GHz association requests and not the 2.4 GHz requests from dual-band clients.[16]
The 2.4 GHzISM bandis fairly congested. With 802.11n, there is the option to double the bandwidth per channel to 40 MHz (fat channel) which results in slightly more than double the data rate. However, in North America, when in 2.4 GHz, enabling this option takes up to 82% of the unlicensed band. For example, channel 3 SCA (secondary channel above), also known as 3+7, reserves the first 9 out of the 11 channels available. In Europe and other places where channels 1–13 are available, allocating 1+5 uses slightly more than 50% of the channels, but the overlap with 9+13 is not usually significant as it lies at the edges of the bands, and so two 40 MHz bands typically work unless the transmitters are physically very closely spaced.[original research?]
The specification calls for requiring one primary 20 MHz channel as well as a secondary adjacent channel spaced ±20 MHz away. The primary channel is used for communications with clients incapable of 40 MHz mode. When in 40 MHz mode, the center frequency is actually themeanof the primary and secondary channels.
Local regulations may restrict certain channels from operation. For example, Channels 12 and 13 are normally unavailable for use as either a primary or secondary channel in North America. For further information, seeList of WLAN channels.
TheWi-Fi Alliancehas upgraded its suite of compatibility tests for some enhancements that were finalized after a 2.0. Furthermore, it has affirmed that all draft-n certified products remain compatible with the products conforming to the final standards.[17]
After the first draft of the IEEE 802.11n standard was published in 2006, many manufacturers began producing so-called "draft-n" products that claimed to comply with the standard draft, even before standard finalization which mean they might not be inter-operational with products produced according to IEEE 802.11 standard after the standard publication, nor even among themselves.[18]The Wi-Fi Alliance began certifying products based on IEEE 802.11n draft 2.0 mid-2007.[19][20]This certification program established a set of features and a level of interoperability across vendors supporting those features, thus providing one definition of "draft n" to ensure compatibility and interoperability. The baseline certification covers both 20 MHz and 40 MHz wide channels, and up to two spatial streams, for maximum throughputs of 144.4 Mbit/s for 20 MHz and 300 Mbit/s for 40 MHz (with shortguard interval). A number of vendors in both the consumer and enterprise spaces have built products that have achieved this certification.[21]
The following are milestones in the development of 802.11n:[22] | https://en.wikipedia.org/wiki/IEEE_802.11n-2009#Data_rates |
IEEE 802.11ac-2013or802.11acis awireless networkingstandard in theIEEE 802.11set of protocols (which is part of theWi-Finetworking family), providing high-throughputwireless local area networks(WLANs) on the5 GHz band.[d]The standard has been retroactively labelled asWi-Fi 5byWi-Fi Alliance.[4][5]
The specification has multi-station throughput of at least 1.1gigabit per second(1.1 Gbit/s) and single-link throughput of at least 500 megabits per second (0.5 Gbit/s).[6]This is accomplished by extending the air-interface concepts embraced by802.11n: wider RF bandwidth (up to 160 MHz), moreMIMOspatial streams(up to eight), downlinkmulti-user MIMO(up to four clients), and high-density modulation (up to256-QAM).[7][8]
The Wi-Fi Alliance separated the introduction of 802.11ac wireless products into two phases ("waves"), named "Wave 1" and "Wave 2".[9][10]From mid-2013, the alliance started certifyingWave 1802.11ac products shipped by manufacturers, based on the IEEE 802.11ac Draft 3.0 (the IEEE standard was not finalized until later that year).[11]Subsequently in 2016, Wi-Fi Alliance introduced theWave 2certification, which includes additional features likeMU-MIMO(downlink only), 160 MHz channel width support, support for more 5 GHz channels, and four spatial streams (with four antennas; compared to three in Wave 1 and 802.11n, and eight in IEEE's802.11axspecification).[12]It meant Wave 2 products would have higher bandwidth and capacity than Wave 1 products.[13]
New technologies introduced with 802.11ac include the following:[8][14]
The single-link and multi-station enhancements supported by 802.11ac enable several new WLAN usage scenarios, such as simultaneous streaming of HD video to multiple clients throughout the home, rapid synchronization and backup of large data files, wireless display, large campus/auditorium deployments, and manufacturing floor automation.[15]
To fully utilize their WLAN capacities, 802.11ac access points and routers have sufficient throughput to require the inclusion of aUSB 3.0interface to provide various services such as video streaming,FTPservers, and personalcloudservices.[16]With storage locally attached throughUSB 2.0, filling the bandwidth made available by 802.11ac was not easily accomplished.
All rates assume 256-QAM, rate 5/6:
Wave 2, referring to products introduced in 2016, offers a higher throughput than legacy Wave 1 products, those introduced starting in 2013. The maximumphysical layertheoretical rate for Wave 1 is 1.3 Gbit/s, while Wave 2 can reach 2.34 Gbit/s. Wave 2 can therefore achieve 1 Gbit/s even if the real world throughput turns out to be only 50% of the theoretical rate. Wave 2 also supports a higher number of connected devices.[13]
Several companies are currently offering 802.11ac chipsets with higher modulation rates: MCS-10 and MCS-11 (1024-QAM), supported by Quantenna and Broadcom. Although technically not part of 802.11ac, these new MCS indices became official in the 802.11ax standard, ratified in 2021.
160 MHz channels are unavailable in some countries due to regulatory issues that allocated some frequencies for other purposes.
802.11ac-class device wireless speeds are often advertised as AC followed by a number, that number being the highest link rates in Mbit/s of all the simultaneously-usable radios in the device added up. For example, an AC1900 access point might have 600 Mbit/s capability on its 2.4 GHz radio and 1300 Mbit/s capability on its 5 GHz radio. No single client device could connect and achieve 1900 Mbit/s of throughput, but separate devices each connecting to the 2.4 GHz and 5 GHz radios could achieve combined throughput approaching 1900 Mbit/s. Different possible stream configurations can add up to the same AC number. | https://en.wikipedia.org/wiki/IEEE_802.11ac#Data_rates_and_speed |
Wi-Fi 6, orIEEE 802.11ax, is anIEEEstandard from theWi-Fi Alliance, for wireless networks (WLANs). It operates in the 2.4 GHz and 5 GHz bands,[4]with an extended version,Wi-Fi 6E, that adds the 6 GHz band.[5]It is an upgrade from Wi-Fi 5 (802.11ac), with improvements for better performance in crowded places. Wi-Fi 6 covers frequencies inlicense-exempt bandsbetween 1 and 7.125 GHz, including the commonly used 2.4 GHz and 5 GHz, as well as the broader6 GHz band.[6]
This standard aims to boost data speed (throughput-per-area[d]) in crowded places like offices and malls. Though the nominal data rate is only 37%[7]better than 802.11ac, the total network speed increases by 300%,[8]making it more efficient and reducing latency by 75%.[9]The quadrupling of overall throughput is made possible by a higherspectral efficiency.
802.11ax Wi-Fi has a main feature calledOFDMA, similar to howcell technologyworks withWi-Fi.[7]This brings better spectrum use, improved power control to avoid interference, and enhancements like 1024‑QAM,MIMOandMU-MIMOfor faster speeds. There are also reliability improvements such as lower power consumption and security protocols likeTarget Wake TimeandWPA3.
The 802.11ax standard was approved on September 1, 2020, with Draft 8 getting 95% approval. Subsequently, on February 1, 2021, the standard received official endorsement from the IEEE Standards Board.[10]
Notes
In 802.11ac (802.11's previous amendment),multi-user MIMOwas introduced, which is aspatial multiplexingtechnique. MU-MIMO allows the access point to form beams towards eachclient, while transmitting information simultaneously. By doing so, the interference between clients is reduced, and the overall throughput is increased, since multiple clients can receive data simultaneously.
With 802.11ax, a similar multiplexing is introduced in thefrequency-division multiplexing:OFDMA. With OFDMA, multiple clients are assigned to differentResource Unitsin the available spectrum. By doing so, an 80 MHz channel can be split into multiple Resource Units, so that multiple clients receive different types of data over the same spectrum, simultaneously.
To supportOFDMA, 802.11ax needs four times as many subcarriers as 802.11ac. Specifically, for 20, 40, 80, and 160 MHz channels, the 802.11ac standard has, respectively, 64, 128, 256 and 512 subcarriers while the 802.11ax standard has 256, 512, 1024, and 2048 subcarriers. Since the available bandwidths have not changed and the number of subcarriers increases by a factor of four, thesubcarrier spacingis reduced by the same factor. This introduces OFDM symbols that are four times longer: in 802.11ac, an OFDM symbol takes 3.2 microseconds to transmit. In 802.11ax, it takes 12.8 microseconds (both withoutguard intervals).
The 802.11ax amendment brings several key improvements over802.11ac. 802.11ax addresses frequency bands between 1 GHz and 6 GHz.[11]Therefore, unlike 802.11ac, 802.11ax also operates in the unlicensed 2.4 GHz band. Wi-Fi 6E introduces operation at frequencies of or near 6 GHz, and superwide channels that are 160 MHz wide,[12]the frequency ranges these channels can occupy and the number of these channels depends on the country the Wi-Fi 6 network operates in.[13]To meet the goal of supporting dense 802.11 deployments, the following features have been approved. | https://en.wikipedia.org/wiki/IEEE_802.11ax#Rate_set |
Hierarchical modulation, also calledlayered modulation, is one of thesignal processingtechniques formultiplexingandmodulatingmultiple data streams into one single symbol stream, where base-layer symbols and enhancement-layer symbols are synchronously overlaid before transmission.
Hierarchical modulation is particularly used to mitigate thecliff effectindigital televisionbroadcast, particularlymobile TV, by providing a (lower quality) fallback signal in case of weak signals, allowinggraceful degradationinstead of complete signal loss. It has been widely proven and included in various standards, such asDVB-T,MediaFLO, UMB (Ultra Mobile Broadband, a new 3.5th generation mobile network standard developed by 3GPP2), and is under study forDVB-H.
Hierarchical modulation is also taken as one of the practical implementations ofsuperposition precoding, which can help achieve the maximum sum rate of broadcast channels. When hierarchical-modulated signals are transmitted, users with good reception and advanced receivers can demodulate multiple layers. For a user with a conventional receiver or poor reception, it may only demodulate the data stream embedded in the base layer. With hierarchical modulation, a network operator can target users of different types with different services orQoS.
However, traditional hierarchical modulation suffers from serious inter-layer interference (ILI) with impact on the achievable symbol rate.
For example, the figure depicts a layering scheme withQPSKbase layer, and a64QAMenhancement layer. The first layer is 2 bits (represented by the green circles). The signal detector only needs to establish which quadrant the signal is in, to recover the value (which is '10', the green circle in the lower right corner). In better signal conditions, the detector can establish the phase and amplitude more precisely, to recover four more bits of data ('1101'). Thus, the base layer carries '10', and the enhancement layer carries '1101'.
For a hierarchically-modulated symbol with QPSK base layer and 16QAM enhancement layer, the base-layer throughput loss is up to about 1.5 bits/symbol with the total receivesignal-to-noise ratio(SNR) at about 23dB, about the minimum needed for the comparable non-hierarchical modulation, 64QAM. But unlayered 16QAM with the same SNR would approach full throughput. This means, due to ILI, about 1.5/4 = 37.5% loss of the base-layer achievable throughput. Furthermore, due to ILI and the imperfect demodulation of base-layer symbols, the demodulation error rate of higher-layer symbols increases too. | https://en.wikipedia.org/wiki/Hierarchical_modulation |
This is a list of well-knowndata structures. For a wider list of terms, seelist of terms relating to algorithms and data structures. For a comparison ofrunning timesfor a subset of this list seecomparison of data structures.
Some properties of abstract data types:
"Ordered" means that the elements of the data type have some kind of explicit order to them, where an element can be considered "before" or "after" another element. This order is usually determined by the order in which the elements are added to the structure, but the elements can be rearranged in some contexts, such assortinga list. For a structure that isn't ordered, on the other hand, no assumptions can be made about the ordering of the elements (although a physical implementation of these data types will often apply some kind of arbitrary ordering). "Uniqueness" means that duplicate elements are not allowed. Depending on the implementation of the data type, attempting to add a duplicate element may either be ignored, overwrite the existing element, or raise an error. The detection for duplicates is based on some inbuilt (or alternatively, user-defined) rule for comparing elements.
A data structure is said to be linear if its elements form a sequence.
Trees are a subset ofdirected acyclic graphs.
In these data structures each tree node compares a bit slice of key values.
These are data structures used forspace partitioningorbinary space partitioning.
Manygraph-based data structures are used in computer science and related fields: | https://en.wikipedia.org/wiki/List_of_data_structures |
The followingoutlineis provided as an overview of, and topical guide to, machine learning:
Machine learning(ML) is a subfield ofartificial intelligencewithincomputer sciencethat evolved from the study ofpattern recognitionandcomputational learning theory.[1]In 1959,Arthur Samueldefined machine learning as a "field of study that gives computers the ability to learn without being explicitly programmed".[2]ML involves the study and construction ofalgorithmsthat canlearnfrom and make predictions ondata.[3]These algorithms operate by building amodelfrom atraining setof example observations to make data-driven predictions or decisions expressed as outputs, rather than following strictly static program instructions.
Dimensionality reduction
Ensemble learning
Meta-learning
Reinforcement learning
Supervised learning
Bayesian statistics
Decision tree algorithm
Linear classifier
Unsupervised learning
Artificial neural network
Association rule learning
Hierarchical clustering
Cluster analysis
Anomaly detection
Semi-supervised learning
Deep learning
History of machine learning
Machine learning projects: | https://en.wikipedia.org/wiki/List_of_machine_learning_algorithms |
Pathfindingorpathingis the search, by a computer application, for the shortest route between two points. It is a more practical variant onsolving mazes. This field of research is based heavily onDijkstra's algorithmfor finding the shortest path on aweighted graph.
Pathfinding is closely related to theshortest path problem, withingraph theory, which examines how to identify the path that best meets some criteria (shortest, cheapest, fastest, etc) between two points in a large network.
At its core, a pathfinding method searches agraphby starting at onevertexand exploring adjacentnodesuntil the destination node is reached, generally with the intent of finding the cheapest route. Although graph searching methods such as abreadth-first searchwould find a route if given enough time, other methods, which "explore" the graph, would tend to reach the destination sooner. An analogy would be a person walking across a room; rather than examining every possible route in advance, the person would generally walk in the direction of the destination and only deviate from the path to avoid an obstruction, and make deviations as minor as possible.
Two primary problems of pathfinding are (1) to find a path between two nodes in agraph; and (2) theshortest path problem—to find theoptimal shortest path. Basic algorithms such asbreadth-firstanddepth-firstsearch address the first problem byexhaustingall possibilities; starting from the given node, they iterate over all potential paths until they reach the destination node. These algorithms run inO(|V|+|E|){\displaystyle O(|V|+|E|)}, or linear time, where V is the number of vertices, and E is the number ofedgesbetween vertices.
The more complicated problem is finding the optimal path. The exhaustive approach in this case is known as theBellman–Ford algorithm, which yields a time complexity ofO(|V||E|){\displaystyle O(|V||E|)}, or quadratic time. However, it is not necessary to examine all possible paths to find the optimal one. Algorithms such asA*andDijkstra's algorithmstrategically eliminate paths, either throughheuristicsor throughdynamic programming. By eliminating impossible paths, these algorithms can achieve time complexities as low asO(|E|log(|V|)){\displaystyle O(|E|\log(|V|))}.[1]
The above algorithms are among the best general algorithms which operate on a graph without preprocessing. However, in practical travel-routing systems, even better time complexities can be attained by algorithms which can pre-process the graph to attain better performance.[2]One such algorithm iscontraction hierarchies.
A common example of a graph-based pathfinding algorithm isDijkstra's algorithm.[3]This algorithm begins with a start node and an "open set" of candidate nodes. At each step, the node in the open set with the lowest distance from the start is examined. The node is marked "closed", and all nodes adjacent to it are added to the open set if they have not already been examined. This process repeats until a path to the destination has been found. Since the lowest distance nodes are examined first, the first time the destination is found, the path to it will be the shortest path.[4]
Dijkstra's algorithm fails if there is a negativeedgeweight. In the hypothetical situation where Nodes A, B, and C form a connected undirected graph with edges AB = 3, AC = 4, and BC = −2, the optimal path from A to C costs 1, and the optimal path from A to B costs 2. Dijkstra's Algorithm starting from A will first examine B, as that is the closest. It will assign a cost of 3 to it, and mark it closed, meaning that its cost will never be reevaluated. Therefore, Dijkstra's cannot evaluate negative edge weights. However, since for many practical purposes there will never be a negative edgeweight, Dijkstra's algorithm is largely suitable for the purpose of pathfinding.
A*is a variant of Dijkstra's algorithm with a wide variety of use cases. A* assigns a weight to each open node equal to the weight of the edge to that node plus the approximate distance between that node and the finish. This approximate distance is found by theheuristic, and represents a minimum possible distance between that node and the end. This allows it to eliminate longer paths once an initial path is found. If there is a path of length x between the start and finish, and the minimum distance between a node and the finish is greater than x, that node need not be examined.[5]
A* uses this heuristic to improve on the behavior relative to Dijkstra's algorithm. When the heuristic evaluates to zero, A* is equivalent to Dijkstra's algorithm. As the heuristic estimate increases and gets closer to the true distance, A* continues to find optimal paths, but runs faster (by virtue of examining fewer nodes). When the value of the heuristic is exactly the true distance, A* examines the fewest nodes. (However, it is generally impractical to write a heuristic function that always computes the true distance, as the same comparison result can often be reached using simpler calculations – for example, usingChebyshev distanceoverEuclidean distanceintwo-dimensional space.) As the value of the heuristic increases, A* examines fewer nodes but no longer guarantees an optimal path. In many applications (such as video games) this is acceptable and even desirable, in order to keep the algorithm running quickly.
Pathfinding has a history of being included in video games with moving objects or NPCs.Chris Crawfordin 1982 described how he "expended a great deal of time" trying to solve a problem with pathfinding inTanktics, in which computer tanks became trapped on land within U-shaped lakes. "After much wasted effort I discovered a better solution: delete U-shaped lakes from the map", he said.[6]
The concept of hierarchical pathfinding predates its adoption by thevideo game industryand has its roots in classical artificial intelligence research. One of the earliest formal descriptions appears in Sacerdoti's work on ABSTRIPS (Abstraction-BasedSTRIPS) in 1974,[7]which explored hierarchical search strategies in logic-based planning. Later research, such as Hierarchical A* by Holte et al., further developed the theory of abstraction hierarchies in search problems.[8]
In the context of video games, the need for efficient planning on large maps with limitedCPU timeled to the practical implementation of hierarchical pathfinding algorithms. A notable advancement was the introduction ofHierarchical Path-Finding A*(HPA*) by Botea et al. in 2004.[9]HPA* partitions the map into clusters and precomputes optimal local paths between entrance points of adjacent clusters. At runtime, it plans an abstract path through the cluster graph, then refines that path within each cluster. This significantly reduces the search space and allows for near-optimal planning with much faster performance.
Partial-Refinement A* (PRA*), developed by Sturtevant and Buro, takes a similar approach but emphasizes interleaved planning and acting. Instead of refining the entire path immediately, PRA* refines only the first few steps and continues refining the rest as needed during execution. This is especially useful in dynamic environments.
Similar techniques includenavigation meshes(navmesh), used for geometric planning in games, and multimodaltransportation planning, such as in variations of thetravelling salesman problemthat involve multiple transport types.
A hierarchical planner performs pathfinding in two phases: first, between clusters at a high level; then, within individual clusters at a low level.[10]This structure enablesguided local searchwith fewernodes, resulting in high performance. The main drawback is the implementation complexity of maintaining abstraction layers and refinements.
Amapwith a size of 3000×2000 nodes contains 6 million tiles. Planning a path directly on this scale, even with an optimizedalgorithm, is computationally intensive due to the vast number of graph nodes and possible paths. A hierarchical approach divides the map into 300×200 node clusters, forming a 10×10 grid (100 clusters total). The high-level abstract graph now contains only 100 nodes. A path is planned between these clusters, which is computationally cheap. Once the abstract path is found, each cluster on the path is processed using a regularA* plannerto find the exact low-level route within. This two-stage process significantly improves efficiency while maintaining near-optimal path quality.
Multi-agent pathfinding is to find the paths for multiple agents from their current locations to their target locations without colliding with each other, while at the same time optimizing a cost function, such as the sum of the path lengths of all agents. It is a generalization of pathfinding. Many multi-agent pathfinding algorithms are generalized from A*, or based on reduction to other well studied problems such as integer linear programming.[11]However, such algorithms are typically incomplete; in other words, not proven to produce a solution within polynomial time. Some parallel approaches, such asCollaborative Diffusion, are based onembarrassingly parallelalgorithms spreading multi-agent pathfinding into computational grid structures, e.g., cells similar tocellular automata. A different category of algorithms sacrifice optimality for performance by either making use of known navigation patterns (such as traffic flow) or the topology of the problem space.[12] | https://en.wikipedia.org/wiki/List_of_pathfinding_algorithms |
This is alist ofalgorithmgeneral topics. | https://en.wikipedia.org/wiki/List_of_algorithm_general_topics |
The NISTDictionary of Algorithms and Data Structures[1]is a reference work maintained by the U.S.National Institute of Standards and Technology. It defines a large number ofterms relating to algorithms and data structures. For algorithms and data structures not necessarily mentioned here, seelist of algorithmsandlist of data structures.
This list of terms was originally derived from the index of that document, and is in thepublic domain, as it was compiled by a Federal Government employee as part of a Federal Government work. Some of the terms defined are: | https://en.wikipedia.org/wiki/List_of_terms_relating_to_algorithms_and_data_structures |
Collective intelligenceCollective actionSelf-organized criticalityHerd mentalityPhase transitionAgent-based modellingSynchronizationAnt colony optimizationParticle swarm optimizationSwarm behaviour
Social network analysisSmall-world networksCentralityMotifsGraph theoryScalingRobustnessSystems biologyDynamic networks
Evolutionary computationGenetic algorithmsGenetic programmingArtificial lifeMachine learningEvolutionary developmental biologyArtificial intelligenceEvolutionary robotics
Reaction–diffusion systemsPartial differential equationsDissipative structuresPercolationCellular automataSpatial ecologySelf-replication
Conversation theoryEntropyFeedbackGoal-orientedHomeostasisInformation theoryOperationalizationSecond-order cyberneticsSelf-referenceSystem dynamicsSystems scienceSystems thinkingSensemakingVariety
Ordinary differential equationsPhase spaceAttractorsPopulation dynamicsChaosMultistabilityBifurcation
Rational choice theoryBounded rationality
Aheuristic[1]orheuristic technique(problem solving,mental shortcut,rule of thumb)[2][3][4][5]is any approach toproblem solvingthat employs apragmaticmethod that is not fullyoptimized, perfected, orrationalized, but is nevertheless "good enough" as anapproximationorattribute substitution.[6][7]Where finding an optimal solution is impossible or impractical, heuristic methods can be used to speed up the process of finding a satisfactory solution.[8][9]Heuristics can be mental shortcuts that ease thecognitive loadofmaking a decision.[10][11][12]
Heuristic reasoning is often based oninduction, or onanalogy... Induction is the process of discovering general laws... Induction tries to find regularity and coherence... Its most conspicuous instruments aregeneralization,specialization, analogy.[...] Heuristic discusses human behavior in the face of problems [... that have been] preserved in thewisdom ofproverbs.[13]
Gigerenzer & Gaissmaier (2011) state thatsub-setsofstrategyincludeheuristics,regression analysis, andBayesian inference.[14]
A heuristic is a strategy that ignores part of the information, with the goal of making decisions more quickly, frugally, and/or accurately than more complex methods (Gigerenzer and Gaissmaier [2011], p. 454; see also Todd et al. [2012], p. 7).[15]
Heuristics are strategies based on rules to generateoptimal decisions, like theanchoring effectandutility maximization problem.[16]These strategies depend on using readily accessible, though loosely applicable, information to controlproblem solvingin human beings, machines and abstract issues.[17][18]When an individual applies a heuristic in practice, it generally performs as expected. However it can alternatively create systematic errors.[19]
The most fundamental heuristic is trial and error, which can be used in everything from matching nuts and bolts to finding the values of variables in algebra problems. In mathematics, some common heuristics involve the use of visual representations, additional assumptions, forward/backward reasoning and simplification.
Dual process theoryconcernsembodied heuristics.[20]
Lakatosian heuristics is based on the key term:Justification (epistemology).[21]
One-reason decisions arealgorithmsthat are made of three rules: search rules,confirmation rules(stopping), and decision rules[22][23][24]
A class whose function is to determine and filter out superfluous things.[32]
Tracking heuristicsis aclassof heuristics.[37]
Social heuristics– Decision-making processes in social environments[42]
George Polyastudied and published on heuristics in 1945.[67]Polya (1945) citesPappus of Alexandriaas having written atextthat Polya dubsHeuristic.[68]Pappus' heuristic problem-solving methods consist ofanalysisandsynthesis.[69]
The study of heuristics in humandecision-makingwas developed in the 1970s and the 1980s, by the psychologistsAmos TverskyandDaniel Kahneman,[81]although the concept had been originally introduced by theNobel laureateHerbert A. Simon. Simon's original primary object of research was problem solving that showed that we operate within what he callsbounded rationality. He coined the termsatisficing, which denotes a situation in which people seek solutions, or accept choices or judgements, that are "good enough" for their purposes although they could be optimised.[82]
Rudolf Groneranalysed the history of heuristics from its roots in ancient Greece up to contemporary work incognitive psychologyandartificial intelligence,[83]proposing a cognitive style "heuristic versus algorithmic thinking", which can be assessed by means of a validatedquestionnaire.[84]
Theadaptive toolboxcontains strategies for fabricating heuristic devices.[85]The core mental capacities arerecall (memory),frequency,object permanence, andimitation.[86]Gerd Gigerenzerand his research group argued that models of heuristics need to be formal to allow for predictions of behavior that can be tested.[87]They study the fast and frugal heuristics in the "adaptive toolbox" of individuals or institutions, and theecological rationalityof these heuristics; that is, the conditions under which a given heuristic is likely to be successful.[88]The descriptive study of the "adaptive toolbox" is done by observation and experiment, while the prescriptive study ofecological rationalityrequires mathematical analysis and computer simulation. Heuristics – such as therecognition heuristic, thetake-the-best heuristicandfast-and-frugal trees– have been shown to be effective in predictions, particularly in situations of uncertainty. It is often said that heuristics trade accuracy for effort but this is only the case in situations of risk. Risk refers to situations where all possible actions, their outcomes and probabilities are known. In the absence of this information, that is under uncertainty, heuristics can achieve higher accuracy with lower effort.[89]This finding, known as aless-is-more effect, would not have been found without formal models. The valuable insight of this program is that heuristics are effective not despite their simplicity – but because of it. Furthermore,Gigerenzerand Wolfgang Gaissmaier found that both individuals and organisations rely on heuristics in an adaptive way.[90]
Heuristics, through greater refinement and research, have begun to be applied to other theories, or be explained by them. For example, thecognitive-experiential self-theory(CEST) is also an adaptive view of heuristic processing. CEST breaks down two systems that process information. At some times, roughly speaking, individuals consider issues rationally, systematically, logically, deliberately, effortfully, and verbally. On other occasions, individuals consider issues intuitively, effortlessly, globally, and emotionally.[91]From this perspective, heuristics are part of a larger experiential processing system that is often adaptive, but vulnerable to error in situations that require logical analysis.[92]
In 2002,Daniel KahnemanandShane Frederickproposed that cognitive heuristics work by a process calledattribute substitution, which happens without conscious awareness.[93]According to this theory, when somebody makes a judgement (of a "target attribute") that is computationally complex, a more easily calculated "heuristic attribute" is substituted. In effect, a cognitively difficult problem is dealt with by answering a rather simpler problem, without being aware of this happening.[93]This theory explains cases where judgements fail to showregression toward the mean.[94]Heuristics can be considered to reduce the complexity of clinical judgments in health care.[95]
Inpsychology, heuristics are simple, efficient rules, either learned or inculcated by evolutionary processes. These psychological heuristics have been proposed to explain how people make decisions, come to judgements, and solve problems. These rules typically come into play when people face complex problems or incomplete information. Researchers employ various methods to test whether people use these rules. The rules have been shown to work well under most circumstances, but in certain cases can lead to systematic errors orcognitive biases.[96]
Aheuristic deviceis used when an entityXexists to enable understanding of, or knowledge concerning, some other entityY.
A good example is amodelthat, asit is never identical with what it models, is a heuristic device to enable understanding of what it models. Stories, metaphors, etc., can also be termed heuristic in this sense. A classic example is the notion ofutopiaas described inPlato's best-known work,The Republic. This means that the "ideal city" as depicted inThe Republicis not given as something to be pursued, or to present an orientation-point for development. Rather, it shows how things would have to be connected, and how one thing would lead to another (often with highly problematic results), if one opted for certain principles and carried them through rigorously.
Heuristicis also often used as anounto describe arule of thumb, procedure, or method.[97]Philosophers of science have emphasised the importance of heuristics in creative thought and the construction of scientific theories.[98]Seminal works includeKarl Popper'sThe Logic of Scientific Discoveryand others byImre Lakatos,[99]Lindley Darden, andWilliam C. Wimsatt.
Inlegal theory, especially in the theory oflaw and economics, heuristics are used in thelawwhencase-by-case analysiswould be impractical, insofar as "practicality" is defined by the interests of a governing body.[100]
The present securities regulation regime largely assumes that all investors act as perfectly rational persons. In truth, actual investors face cognitive limitations from biases, heuristics, and framing effects. For instance, in all states in the United States thelegal drinking agefor unsupervised persons is 21 years, because it is argued that people need to be mature enough to make decisions involving the risks ofalcoholconsumption. However, assuming people mature at different rates, the specific age of 21 would be too late for some and too early for others. In this case, the somewhat arbitrary delineation is used because it is impossible or impractical to tell whether an individual is sufficiently mature for society to trust them with that kind of responsibility. Some proposed changes, however, have included the completion of an alcohol education course rather than the attainment of 21 years of age as the criterion for legal alcohol possession. This would put youth alcohol policy more on a case-by-case basis and less on a heuristic one, since the completion of such a course would presumably be voluntary and not uniform across the population.
The same reasoning applies topatent law.Patentsare justified on the grounds that inventors must be protected so they have incentive to invent. It is therefore argued that it is in society's best interest that inventors receive a temporary government-grantedmonopolyon their idea, so that they can recoup investment costs and make economic profit for a limited period. In the United States, the length of this temporary monopoly is 20 years from the date the patent application was filed, though the monopoly does not actually begin until the application has matured into a patent. However, like the drinking age problem above, the specific length of time would need to be different for every product to be efficient. A 20-year term is used because it is difficult to tell what the number should be for any individual patent. More recently, some, includingUniversity of North Dakotalaw professor Eric E. Johnson, have argued that patents in different kinds of industries – such assoftware patents– should be protected for different lengths of time.[101]
Thebias–variance tradeoffgives insight into describing the less-is-more strategy.[102]Aheuristiccan be used inartificial intelligencesystems while searching asolution space. The heuristic is derived by using some function that is put into the system by the designer, or by adjusting the weight of branches based on how likely each branch is to lead to agoal node.
Heuristics refers to the cognitive shortcuts that individuals use to simplify decision-making processes in economic situations.Behavioral economicsis a field that integrates insights from psychology and economics to better understand how people make decisions.
Anchoring and adjustment is one of the most extensively researched heuristics in behavioural economics. Anchoring is the tendency of people to make future judgements or conclusions based too heavily on the original information supplied to them. This initial knowledge functions as an anchor, and it can influence future judgements even if the anchor is entirely unrelated to the decisions at hand. Adjustment, on the other hand, is the process through which individuals make gradual changes to their initial judgements or conclusions.
Anchoring and adjustmenthas been observed in a wide range of decision-making contexts, including financial decision-making, consumer behavior, and negotiation. Researchers have identified a number of strategies that can be used to mitigate the effects of anchoring and adjustment, including providing multiple anchors, encouraging individuals to generate alternative anchors, and providing cognitive prompts to encourage more deliberative decision-making.
Other heuristics studied in behavioral economics include therepresentativeness heuristic, which refers to the tendency of individuals to categorize objects or events based on how similar they are to typical examples,[103]and theavailability heuristic, which refers to the tendency of individuals to judge the likelihood of an event based on how easily it comes to mind.[104]
Stereotypingis a type of heuristic that people use to form opinions or make judgements about things they have never seen or experienced.[105]They work as a mental shortcut to assess everything from the social status of a person (based on their actions),[12]to classifying a plant as a tree based on it being tall, having a trunk, and that it has leaves (even though the person making the evaluation might never have seen that particular type of tree before).
Stereotypes, as first described by journalistWalter Lippmannin his bookPublic Opinion(1922), are the pictures we have in our heads that are built around experiences as well as what we are told about the world.[106][107] | https://en.wikipedia.org/wiki/Heuristic |
Indigital logic, adon't-care term[1][2](abbreviatedDC, historically also known asredundancies,[2]irrelevancies,[2]optional entries,[3][4]invalid combinations,[5][4][6]vacuous combinations,[7][4]forbidden combinations,[8][2]unused statesorlogical remainders[9]) for a function is an input-sequence (a series of bits) for which the function output does not matter. An input that is known never to occur is acan't-happen term.[10][11][12][13]Both these types of conditions are treated the same way in logic design and may be referred to collectively asdon't-care conditionsfor brevity.[14]The designer of a logic circuit to implement the function need not care about such inputs, but can choose the circuit's output arbitrarily, usually such that the simplest, smallest, fastest or cheapest circuit results (minimization) or the power-consumption is minimized.[15][16]
Don't-care terms are important to consider in minimizing logic circuit design, including graphical methods likeKarnaugh–Veitch mapsand algebraic methods such as theQuine–McCluskey algorithm. In 1958,Seymour Ginsburgproved that minimization of states of afinite-state machinewith don't-care conditions does not necessarily yield a minimization of logic elements. Direct minimization of logic elements in such circuits was computationally impractical (for large systems) with the computing power available to Ginsburg in 1958.[17]
Examples of don't-care terms are the binary values 1010 through 1111 (10 through 15 in decimal) for a function that takes abinary-coded decimal(BCD) value, because a BCD value never takes on such values (so calledpseudo-tetrades); in the pictures, the circuit computing the lower left bar of a7-segment displaycan be minimized toab+acby an appropriate choice of circuit outputs fordcba= 1010…1111.
Write-only registers, as frequently found in older hardware, are often a consequence of don't-care optimizations in the trade-off between functionality and the number of necessary logic gates.[18]
Don't-care states can also occur inencoding schemesandcommunication protocols.[nb 1]
"Don't care" may also refer to an unknown value in amulti-valued logicsystem, in which case it may also be called anX valueordon't know.[19]In theVeriloghardware description languagesuch values are denoted by the letter "X". In theVHDLhardware description language such values are denoted (in the standard logic package) by the letter "X" (forced unknown) or the letter "W" (weak unknown).[20]
An X value does not exist in hardware. In simulation, an X value can result from two or more sources driving a signal simultaneously, or the stable output of aflip-flopnot having been reached. In synthesized hardware, however, the actual value of such a signal will be either 0 or 1, but will not be determinable from the circuit's inputs.[20]
Further considerations are needed for logic circuits that involve somefeedback. That is, those circuits that depend on the previous output(s) of the circuit as well as its current external inputs. Such circuits can be represented by astate machine. It is sometimes possible that some states that are nominally can't-happen conditions can accidentally be generated during power-up of the circuit or else by random interference (likecosmic radiation,electrical noiseor heat). This is also calledforbidden input.[21]In some cases, there is no combination of inputs that can exit the state machine into a normal operational state. The machine remains stuck in the power-up state or can be moved only between other can't-happen states in a walled garden of states. This is also called ahardware lockuporsoft error. Such states, while nominally can't-happen, are not don't-care, and designers take steps either to ensure that they are really made can't-happen, or else if they do happen, that they create adon't-care alarmindicating an emergency state[21]forerror detection, or they are transitory and lead to a normal operational state.[22][23][24] | https://en.wikipedia.org/wiki/Don%27t_care_alarm |
Indigital logic, adon't-care term[1][2](abbreviatedDC, historically also known asredundancies,[2]irrelevancies,[2]optional entries,[3][4]invalid combinations,[5][4][6]vacuous combinations,[7][4]forbidden combinations,[8][2]unused statesorlogical remainders[9]) for a function is an input-sequence (a series of bits) for which the function output does not matter. An input that is known never to occur is acan't-happen term.[10][11][12][13]Both these types of conditions are treated the same way in logic design and may be referred to collectively asdon't-care conditionsfor brevity.[14]The designer of a logic circuit to implement the function need not care about such inputs, but can choose the circuit's output arbitrarily, usually such that the simplest, smallest, fastest or cheapest circuit results (minimization) or the power-consumption is minimized.[15][16]
Don't-care terms are important to consider in minimizing logic circuit design, including graphical methods likeKarnaugh–Veitch mapsand algebraic methods such as theQuine–McCluskey algorithm. In 1958,Seymour Ginsburgproved that minimization of states of afinite-state machinewith don't-care conditions does not necessarily yield a minimization of logic elements. Direct minimization of logic elements in such circuits was computationally impractical (for large systems) with the computing power available to Ginsburg in 1958.[17]
Examples of don't-care terms are the binary values 1010 through 1111 (10 through 15 in decimal) for a function that takes abinary-coded decimal(BCD) value, because a BCD value never takes on such values (so calledpseudo-tetrades); in the pictures, the circuit computing the lower left bar of a7-segment displaycan be minimized toab+acby an appropriate choice of circuit outputs fordcba= 1010…1111.
Write-only registers, as frequently found in older hardware, are often a consequence of don't-care optimizations in the trade-off between functionality and the number of necessary logic gates.[18]
Don't-care states can also occur inencoding schemesandcommunication protocols.[nb 1]
"Don't care" may also refer to an unknown value in amulti-valued logicsystem, in which case it may also be called anX valueordon't know.[19]In theVeriloghardware description languagesuch values are denoted by the letter "X". In theVHDLhardware description language such values are denoted (in the standard logic package) by the letter "X" (forced unknown) or the letter "W" (weak unknown).[20]
An X value does not exist in hardware. In simulation, an X value can result from two or more sources driving a signal simultaneously, or the stable output of aflip-flopnot having been reached. In synthesized hardware, however, the actual value of such a signal will be either 0 or 1, but will not be determinable from the circuit's inputs.[20]
Further considerations are needed for logic circuits that involve somefeedback. That is, those circuits that depend on the previous output(s) of the circuit as well as its current external inputs. Such circuits can be represented by astate machine. It is sometimes possible that some states that are nominally can't-happen conditions can accidentally be generated during power-up of the circuit or else by random interference (likecosmic radiation,electrical noiseor heat). This is also calledforbidden input.[21]In some cases, there is no combination of inputs that can exit the state machine into a normal operational state. The machine remains stuck in the power-up state or can be moved only between other can't-happen states in a walled garden of states. This is also called ahardware lockuporsoft error. Such states, while nominally can't-happen, are not don't-care, and designers take steps either to ensure that they are really made can't-happen, or else if they do happen, that they create adon't-care alarmindicating an emergency state[21]forerror detection, or they are transitory and lead to a normal operational state.[22][23][24] | https://en.wikipedia.org/wiki/Forbidden_input |
Random number generatorsare important in many kinds of technical applications, includingphysics,engineeringormathematicalcomputer studies (e.g.,Monte Carlosimulations),cryptographyandgambling(ongame servers).
This list includes many common types, regardless of quality or applicability to a given use case.
The following algorithms arepseudorandom number generators.
Simple to implement, fast, but not widely known. With appropriate initialisations, passes all current empirical test suites, and is formally proven to converge. Easy to extend for arbitrary period length and improved statistical performance over higher dimensions and with higher precision.
Cipheralgorithms andcryptographic hashescan be used as very high-quality pseudorandom number generators. However, generally they are considerably slower (typically by a factor 2–10) than fast, non-cryptographic random number generators.
These include:
A few cryptographically secure pseudorandom number generators do not rely on cipher algorithms but try to link mathematically the difficulty of distinguishing their output from a `true' random stream to a computationally difficult problem. These approaches are theoretically important but are too slow to be practical in most applications. They include:
These approaches combine a pseudo-random number generator (often in the form of a block or stream cipher) with an external source of randomness (e.g., mouse movements, delay between keyboard presses etc.). | https://en.wikipedia.org/wiki/List_of_random_number_generators |
Alinear congruential generator(LCG) is analgorithmthat yields a sequence of pseudo-randomized numbers calculated with a discontinuouspiecewise linear equation. The method represents one of the oldest and best-knownpseudorandom number generatoralgorithms. The theory behind them is relatively easy to understand, and they are easily implemented and fast, especially on computer hardware which can providemodular arithmeticby storage-bit truncation.
The generator is defined by therecurrence relation:
whereX{\displaystyle X}is thesequenceof pseudo-random values, and
areintegerconstants that specify the generator. Ifc= 0, the generator is often called amultiplicative congruential generator(MCG), orLehmer RNG. Ifc≠ 0, the method is called amixed congruential generator.[1]: 4-
Whenc≠ 0, a mathematician would call the recurrence anaffine transformation, not alinearone, but the misnomer is well-established in computer science.[2]: 1
The Lehmer generator was published in 1951[3]and the Linear congruential generator was published in 1958 by W. E. Thomson and A. Rotenberg.[4][5]
A benefit of LCGs is that an appropriate choice of parameters results in a period which is both known and long. Although not the only criterion, too short a period is a fatal flaw in a pseudorandom number generator.[6]
While LCGs are capable of producingpseudorandom numberswhich can pass formaltests for randomness, the quality of the output is extremely sensitive to the choice of the parametersmanda.[1][2][7][8][9][10]For example,a= 1 andc= 1 produces a simple modulo-mcounter, which has a long period, but is obviously non-random. Other values ofccoprimetomproduce aWeyl sequence, which is better distributed but still obviously non-random.
Historically, poor choices forahave led to ineffective implementations of LCGs. A particularly illustrative example of this isRANDU, which was widely used in the early 1970s and led to many results which are currently being questioned because of the use of this poor LCG.[11][8]: 1198–9
There are three common families of parameter choice:
This is the original Lehmer RNG construction. The period ism−1 if the multiplierais chosen to be aprimitive elementof the integers modulom. The initial state must be chosen between 1 andm−1.
One disadvantage of a prime modulus is that the modular reduction requires a double-width product and an explicit reduction step. Often a prime just less than a power of 2 is used (theMersenne primes231−1 and 261−1 are popular), so that the reduction modulom= 2e−dcan be computed as (axmod 2e) +d⌊ax/2e⌋. This must be followed by a conditional subtraction ofmif the result is too large, but the number of subtractions is limited toad/m, which can be easily limited to one ifdis small.
If a double-width product is unavailable, and the multiplier is chosen carefully,Schrage's method[12][13]may be used. To do this, factorm=qa+r, i.e.q=⌊m/a⌋andr=mmoda. Then computeaxmodm=a(xmodq) −r⌊x/q⌋. Sincexmodq<q≤m/a, the first term is strictly less thanam/a=m. Ifais chosen so thatr≤q(and thusr/q≤ 1), then the second term is also less thanm:r⌊x/q⌋≤rx/q=x(r/q) ≤x<m. Thus, both products can be computed with a single-width product, and the difference between them lies in the range [1−m,m−1], so can be reduced to [0,m−1] with a single conditional add.[14]
The most expensive operation in Schrage's method is the division (with remainder) ofxbyq; fastalgorithms for division by a constantare not available since they also rely on double-width products.
A second disadvantage of a prime modulus is that it is awkward to convert the value 1 ≤x<mto uniform random bits. If a prime just less than a power of 2 is used, sometimes the missing values are simply ignored.
Choosingmto be apower of two, most oftenm= 232orm= 264, produces a particularly efficient LCG, because this allows the modulus operation to be computed by simply truncating the binary representation. In fact, the most significant bits are usually not computed at all. There are, however, disadvantages.
This form has maximal periodm/4, achieved ifa≡ ±3 (mod 8) and the initial stateX0is odd. Even in this best case, the low three bits ofXalternate between two values and thus only contribute one bit to the state.Xis always odd (the lowest-order bit never changes), and only one of the next two bits ever changes. Ifa≡ +3,Xalternates ±1↔±3, while ifa≡ −3,Xalternates ±1↔∓3 (all modulo 8).
It can be shown that this form is equivalent to a generator with modulusm/4 andc≠ 0.[1]
A more serious issue with the use of a power-of-two modulus is that the low bits have a shorter period than the high bits. Its simplicity of implementation comes from the fact that bits are never affected by higher-order bits, so the lowbbits of such a generator form a modulo-2bLCG by themselves, repeating with a period of 2b−2. Only the most significant bit ofXachieves the full period.
Whenc≠ 0, correctly chosen parameters allow a period equal tom, for all seed values. This will occurif and only if:[1]: 17–19
These three requirements are referred to as the Hull–Dobell Theorem.[15][16]
This form may be used with anym, but only works well formwith many repeated prime factors, such as a power of 2; using a computer'sword sizeis the most common choice. Ifmwere asquare-free integer, this would only allowa≡ 1 (modm), which makes a very poor PRNG; a selection of possible full-period multipliers is only available whenmhas repeated prime factors.
Although the Hull–Dobell theorem provides maximum period, it is not sufficient to guarantee agoodgenerator.[8]: 1199For example, it is desirable fora− 1 to not be any more divisible by prime factors ofmthan necessary. Ifmis a power of 2, thena− 1 should be divisible by 4 but not divisible by 8, i.e.a≡ 5 (mod 8).[1]: §3.2.1.3
Indeed, most multipliers produce a sequence which fails one test for non-randomness or another, and finding a multiplier which is satisfactory to all applicable criteria[1]: §3.3.3is quite challenging.[8]Thespectral testis one of the most important tests.[17]
Note that a power-of-2 modulus shares the problem as described above forc= 0: the lowkbits form a generator with modulus 2kand thus repeat with a period of 2k; only the most significant bit achieves the full period. If a pseudorandom number less thanris desired,⌊rX/m⌋is a much higher-quality result thanXmodr. Unfortunately, most programming languages make the latter much easier to write (X % r), so it is very commonly used.
The generator isnotsensitive to the choice ofc, as long as it is relatively prime to the modulus (e.g. ifmis a power of 2, thencmust be odd), so the valuec=1 is commonly chosen.
The sequence produced by other choices ofccan be written as a simple function of the sequence whenc=1.[1]: 11Specifically, ifYis the prototypical sequence defined byY0= 0 andYn+1=aYn+ 1 mod m, then a general sequenceXn+1=aXn+cmodmcan be written as an affine function ofY:
More generally, any two sequencesXandZwith the same multiplier and modulus are related by
In the common case wheremis a power of 2 anda≡ 5 (mod 8) (a desirable property for other reasons), it is always possible to find an initial valueX0so that the denominatorX1−X0≡ ±1 (modm), producing an even simpler relationship. With this choice ofX0,Xn=X0±Ynwill remain true for alln.[2]: 10-11The sign is determined byc≡ ±1 (mod 4), and the constantX0is determined by 1 ∓c≡ (1 −a)X0(modm).
As a simple example, consider the generatorsXn+1= 157Xn+ 3 mod 256 andYn+1= 157Yn+ 1 mod 256; i.e.m= 256,a= 157, andc= 3. Because 3 ≡ −1 (mod 4), we are searching for a solution to 1 + 3 ≡ (1 − 157)X0(mod 256). This is satisfied byX0≡ 41 (mod 64), so if we start with that, thenXn≡X0−Yn(mod 256) for alln.
For example, usingX0= 233 = 3×64 + 41:
The following table lists the parameters of LCGs in common use, including built-inrand()functions inruntime librariesof variouscompilers. This table is to show popularity, not examples to emulate;many of these parameters are poor.Tables of good parameters are available.[10][2]
As shown above, LCGs do not always use all of the bits in the values they produce. In general, they return the most significant bits. For example, theJavaimplementation operates with 48-bit values at each iteration but returns only their 32 most significant bits. This is because the higher-order bits have longer periods than the lower-order bits (see below). LCGs that use this truncation technique produce statistically better values than those that do not. This is especially noticeable in scripts that use the mod operation to reduce range; modifying the random number mod 2 will lead to alternating 0 and 1 without truncation.
Contrarily, some libraries use an implicit power-of-two modulus but never output or otherwise use the most significant bit, in order to limit the output to positivetwo's complementintegers. The output isas ifthe modulus were one bit less than the internal word size, and such generators are described as such in the table above.
LCGs are fast and require minimal memory (one modulo-mnumber, often 32 or 64 bits) to retain state. This makes them valuable for simulating multiple independent streams. LCGs are not intended, and must not be used, for cryptographic applications; use acryptographically secure pseudorandom number generatorfor such applications.
Although LCGs have a few specific weaknesses, many of their flaws come from having too small a state. The fact that people have been lulled for so many years into using them with such small moduli can be seen as a testament to the strength of the technique. A LCG with large enough state can pass even stringent statistical tests; a modulo-264LCG which returns the high 32 bits passesTestU01's SmallCrush suite,[citation needed]and a 96-bit LCG passes the most stringent BigCrush suite.[38]
For a specific example, an ideal random number generator with 32 bits of output is expected (by theBirthday theorem) to begin duplicating earlier outputs after√m≈ 216results.AnyPRNG whose output is its full, untruncated state will not produce duplicates until its full period elapses, an easily detectable statistical flaw.[39]For related reasons, any PRNG should have a period longer than the square of the number of outputs required. Given modern computer speeds, this means a period of 264for all but the least demanding applications, and longer for demanding simulations.
One flaw specific to LCGs is that, if used to choose points in an n-dimensional space, the points will lie on, at most,n√n!⋅mhyperplanes(Marsaglia's theorem, developed byGeorge Marsaglia).[7]This is due to serial correlation between successive values of the sequenceXn. Carelessly chosen multipliers will usually have far fewer, widely spaced planes, which can lead to problems. Thespectral test, which is a simple test of an LCG's quality, measures this spacing and allows a good multiplier to be chosen.
The plane spacing depends both on the modulus and the multiplier. A large enough modulus can reduce this distance below the resolution of double precision numbers. The choice of the multiplier becomes less important when the modulus is large. It is still necessary to calculate the spectral index and make sure that the multiplier is not a bad one, but purely probabilistically it becomes extremely unlikely to encounter a bad multiplier when the modulus is larger than about 264.
Another flaw specific to LCGs is the short period of the low-order bits whenmis chosen to be a power of 2. This can be mitigated by using a modulus larger than the required output, and using the most significant bits of the state.
Nevertheless, for some applications LCGs may be a good option. For instance, in an embedded system, the amount of memory available is often severely limited. Similarly, in an environment such as avideo game consoletaking a small number of high-order bits of an LCG may well suffice. (The low-order bits of LCGs when m is a power of 2 should never be relied on for any degree of randomness whatsoever.) The low order bits go through very short cycles. In particular, any full-cycle LCG, when m is a power of 2, will produce alternately odd and even results.
LCGs should be evaluated very carefully for suitability in non-cryptographic applications where high-qualityrandomnessis critical. For Monte Carlo simulations, an LCG must use a modulus greater and preferably much greater than the cube of the number of random samples which are required. This means, for example, that a (good) 32-bit LCG can be used to obtain about a thousand random numbers; a 64-bit LCG is good for about 221random samples (a little over two million), etc. For this reason, in practice LCGs are not suitable for large-scale Monte Carlo simulations.
The following is an implementation of an LCG inPython, in the form of agenerator:
The following is an implementation of an LCG inHaskellutilizing alazy evaluationstrategy to generate an infinite stream of output values in a list:
Free Pascal uses aMersenne Twisteras its default pseudo random number generator whereas Delphi uses a LCG. Here is a Delphi compatible example inFree Pascalbased on the information in the table above. Given the same RandSeed value it generates the same sequence of random numbers as Delphi.
Like all pseudorandom number generators, a LCG needs to store state and alter it each time it generates a new number. Multiple threads may access this state simultaneously causing a race condition. Implementations should use different state each with unique initialization for different threads to avoid equal sequences of random numbers on simultaneously executing threads.
There are several generators which are linear congruential generators in a different form, and thus the techniques used to analyze LCGs can be applied to them.
One method of producing a longer period is to sum the outputs of several LCGs of different periods having a largeleast common multiple; theWichmann–Hillgenerator is an example of this form. (We would prefer them to be completelycoprime, but a prime modulus implies an even period, so there must be a common factor of 2, at least.) This can be shown to be equivalent to a single LCG with a modulus equal to the product of the component LCG moduli.
Marsaglia's add-with-carry andsubtract-with-borrowPRNGs with a word size ofb=2wand lagsrands(r>s) are equivalent to LCGs with a modulus ofbr±bs± 1.[40][41]
Multiply-with-carryPRNGs with a multiplier ofaare equivalent to LCGs with a large prime modulus ofabr−1 and a power-of-2 multiplierb.
Apermuted congruential generatorbegins with a power-of-2-modulus LCG and applies an output transformation to eliminate the short period problem in the low-order bits.
The other widely used primitive for obtaining long-period pseudorandom sequences is thelinear-feedback shift registerconstruction, which is based on arithmetic in GF(2)[x], thepolynomial ringoverGF(2). Rather than integer addition and multiplication, the basic operations areexclusive-orandcarry-less multiplication, which is usually implemented as a sequence oflogical shifts. These have the advantage that all of their bits are full-period; they do not suffer from the weakness in the low-order bits that plagues arithmetic modulo 2k.[42]
Examples of this family includexorshiftgenerators and theMersenne twister. The latter provides a very long period (219937−1) and variate uniformity, but it fails some statistical tests.[43]Lagged Fibonacci generatorsalso fall into this category; although they use arithmetic addition, their period is ensured by an LFSR among the least-significant bits.
It is easy to detect the structure of a linear-feedback shift register with appropriate tests[44]such as the linear complexity test implemented in theTestU01suite; a Booleancirculant matrixinitialized from consecutive bits of an LFSR will never haverankgreater than the degree of the polynomial. Adding a non-linear output mixing function (as in thexoshiro256**andpermuted congruential generatorconstructions) can greatly improve the performance on statistical tests.
Another structure for a PRNG is a very simple recurrence function combined with a powerful output mixing function. This includescounter modeblock ciphers and non-cryptographic generators such asSplitMix64.
A structure similar to LCGs, butnotequivalent, is the multiple-recursive generator:Xn= (a1Xn−1+a2Xn−2+ ··· +akXn−k) modmfork≥ 2. With a prime modulus, this can generate periods up tomk−1, so is a useful extension of the LCG structure to larger periods.
A powerful technique for generating high-quality pseudorandom numbers is to combine two or more PRNGs of different structure; the sum of an LFSR and an LCG (as in theKISSorxorwowconstructions) can do very well at some cost in speed. | https://en.wikipedia.org/wiki/Linear_congruential_generator |
An approach to nonlinear congruential methods ofgenerating uniform pseudorandom numbersin the interval [0,1) is theInversive congruential generatorwith prime modulus. A generalization for arbitrary composite modulim=p1,…pr{\displaystyle m=p_{1},\dots p_{r}}with arbitrary distinctprimesp1,…,pr≥5{\displaystyle p_{1},\dots ,p_{r}\geq 5}will be present here.
LetZm={0,1,...,m−1}{\displaystyle \mathbb {Z} _{m}=\{0,1,...,m-1\}}. Forintegersa,b∈Zm{\displaystyle a,b\in \mathbb {Z} _{m}}with gcd (a,m) = 1 a generalized inversive congruential sequence(yn)n⩾0{\displaystyle (y_{n})_{n\geqslant 0}}of elements ofZm{\displaystyle \mathbb {Z} _{m}}is defined by
whereφ(m)=(p1−1)…(pr−1){\displaystyle \varphi (m)=(p_{1}-1)\dots (p_{r}-1)}denotes the number of positive integers less thanmwhich arerelatively primetom.
Let take m = 15 =3×5a=2,b=3{\displaystyle 3\times 5\,a=2,b=3}andy0=1{\displaystyle y_{0}=1}. Henceφ(m)=2×4=8{\displaystyle \varphi (m)=2\times 4=8\,}and the sequence(yn)n⩾0=(1,5,13,2,4,7,1,…){\displaystyle (y_{n})_{n\geqslant 0}=(1,5,13,2,4,7,1,\dots )}is not maximum.
The result below shows that these sequences are closely related to the following inversive congruential sequence with prime moduli.
For1≤i≤r{\displaystyle 1\leq i\leq r}letZpi={0,1,…,pi−1},mi=m/pi{\displaystyle \mathbb {Z} _{p_{i}}=\{0,1,\dots ,p_{i}-1\},m_{i}=m/p_{i}}andai,bi∈Zpi{\displaystyle a_{i},b_{i}\in \mathbb {Z} _{p_{i}}}be integers with
Let(yn)n⩾0{\displaystyle (y_{n})_{n\geqslant 0}}be a sequence of elements ofZpi{\displaystyle \mathbb {Z} _{p_{i}}}, given by
Let(yn(i))n⩾0{\displaystyle (y_{n}^{(i)})_{n\geqslant 0}}for1≤i≤r{\displaystyle 1\leq i\leq r}be defined as above.
Then
This theorem shows that an implementation of Generalized Inversive Congruential Generator is possible, where exact integer computations have to be performed only inZp1,…,Zpr{\displaystyle \mathbb {Z} _{p_{1}},\dots ,\mathbb {Z} _{p_{r}}}but not inZm.{\displaystyle \mathbb {Z} _{m}.}
Proof:
First, observe thatmi≡0(modpj),fori≠j,{\displaystyle m_{i}\equiv 0{\pmod {p_{j}}},\;{\text{for}}\;i\neq j,}and henceyn≡m1yn(1)+m2yn(2)+⋯+mryn(r)(modm){\displaystyle y_{n}\equiv m_{1}y_{n}^{(1)}+m_{2}y_{n}^{(2)}+\dots +m_{r}y_{n}^{(r)}{\pmod {m}}}if and only ifyn≡mi(yn(i))(modpi){\displaystyle y_{n}\equiv m_{i}(y_{n}^{(i)}){\pmod {p_{i}}}}, for1≤i≤r{\displaystyle 1\leq i\leq r}which will be shown on induction onn⩾0{\displaystyle n\geqslant 0}.
Recall thaty0≡mi(y0(i))(modpi){\displaystyle y_{0}\equiv m_{i}(y_{0}^{(i)}){\pmod {p_{i}}}}is assumed for1≤i≤r{\displaystyle 1\leq i\leq r}. Now, suppose that1≤i≤r{\displaystyle 1\leq i\leq r}andyn≡mi(yn(i))(modpi){\displaystyle y_{n}\equiv m_{i}(y_{n}^{(i)}){\pmod {p_{i}}}}for some integern⩾0{\displaystyle n\geqslant 0}. Then straightforward calculations andFermat's Theoremyield
which implies the desired result.
Generalized Inversive Congruential Pseudorandom Numbers are well equidistributed in one dimension. A reliable theoretical approach for assessing their statistical independence properties is based on the discrepancy ofs-tuples of pseudorandom numbers.
We use the notationDms=Dm(x0,…,xm−1){\displaystyle D_{m}^{s}=D_{m}(x_{0},\dots ,x_{m}-1)}wherexn=(xn,xn+1,…,xn+s−1){\displaystyle x_{n}=(x_{n},x_{n}+1,\dots ,x_{n}+s-1)}∈[0,1)s{\displaystyle [0,1)^{s}}of Generalized Inversive Congruential Pseudorandom Numbers fors≥2{\displaystyle s\geq 2}.
Higher bound
Lower bound:
For a fixed numberrof prime factors ofm, Theorem 2 shows thatDm(s)=O(m−1/2(logm)s){\displaystyle D_{m}^{(s)}=O(m^{-1/2}(\log m)^{s})}for any Generalized Inversive Congruential Sequence. In this case Theorem 3 implies that there exist Generalized Inversive Congruential Generators having a discrepancyDm(s){\displaystyle D_{m}^{(s)}}which is at least of the order of magnitudem−1/2{\displaystyle m^{-1/2}}for all dimensions≥2{\displaystyle s\geq 2}. However, ifmis composed only of small primes, thenrcan be of an order of magnitude(logm)/loglogm{\displaystyle (\log m)/\log \log m}and hence∏i=1r(2s−2+s(pi)−1/2)=O(mϵ){\displaystyle \textstyle \prod _{i=1}^{r}(2s-2+s(p_{i})^{-1/2})=O{(m^{\epsilon })}}for everyϵ>0{\displaystyle \epsilon >0}.[1]Therefore, one obtains in the general caseDms=O(m−1/2+ϵ){\displaystyle D_{m}^{s}=O(m^{-1/2+\epsilon })}for everyϵ>0{\displaystyle \epsilon >0}.
Since∏i=1r((pi−3)/(pi−1))1/2⩾2−r/2{\displaystyle \textstyle \prod _{i=1}^{r}((p_{i}-3)/(p_{i}-1))^{1/2}\geqslant 2^{-r/2}}, similar arguments imply that in the general case the lower bound in Theorem 3 is at least of the order of magnitudem−1/2−ϵ{\displaystyle m^{-1/2-\epsilon }}for everyϵ>0{\displaystyle \epsilon >0}. It is this range of magnitudes where one also finds the discrepancy of m independent and uniformly distributed random points which almost always has the order of magnitudem−1/2(loglogm)1/2{\displaystyle m^{-1/2}(\log \log m)^{1/2}}according to the law of the iterated logarithm for discrepancies.[2]In this sense, Generalized Inversive Congruential Pseudo-random Numbers model true random numbers very closely. | https://en.wikipedia.org/wiki/Generalized_inversive_congruential_pseudorandom_numbers |
In 1997,Moni NaorandOmer Reingolddescribed efficient constructions for variouscryptographic primitivesin private key as well aspublic-key cryptography. Their result is the construction of an efficientpseudorandom function. Letpandlbeprime numberswithl|p−1. Select an elementg∈Fp∗{\displaystyle {\mathbb {F} _{p}}^{*}}ofmultiplicative orderl. Then for each(n+1)-dimensionalvectora= (a0,a1, ...,an)∈(Fl)n+1{\displaystyle (\mathbb {F} _{l})^{n+1}}they define the function
wherex=x1...xnis thebit representationof integerx, 0 ≤x≤ 2n−1, with some extra leading zeros if necessary.[1]
Letp= 7 andl= 3; sol|p−1. Selectg= 4 ∈F7∗{\displaystyle {\mathbb {F} _{7}}^{*}}of multiplicative order 3 (since 43= 64 ≡ 1 mod 7). Forn= 3,a= (1, 1, 2, 1) andx= 5 (the bit representation of 5 is 101), we can computefa(5){\displaystyle f_{a}(5)}as follows:
The evaluation of functionfa(x){\displaystyle f_{a}(x)}in theNaor–Reingoldconstruction can be done very efficiently. Computing the value of the functionfa(x){\displaystyle f_{a}(x)}at any given point is comparable with onemodular exponentiationand n-modular multiplications. This function can be computed in parallel by threshold circuits of bounded depth and polynomial size.
TheNaor–Reingoldfunction can be used as the basis of manycryptographicschemes includingsymmetric encryption,authenticationanddigital signatures.
Assume that an attacker sees several outputs of the function, e.g.fa(1)=ga1,fa(2)=ga2,fa(3)=ga1a2{\displaystyle f_{a}(1)=g^{a_{1}},f_{a}(2)=g^{a_{2}},f_{a}(3)=g^{a_{1}a_{2}}}, ...fa(k)=ga1x1a2x2...anxn{\displaystyle f_{a}(k)=g^{a_{1}^{x_{1}}a_{2}^{x_{2}}...a_{n}^{x_{n}}}}and wants to computefa(k+1){\displaystyle f_{a}(k+1)}. Assume for simplicity thatx1= 0, then the attacker needs to solve thecomputational Diffie–Hellman (CDH)betweenfa(1)=ga1{\displaystyle f_{a}(1)=g^{a_{1}}}andfa(k)=ga2x2...anxn{\displaystyle f_{a}(k)=g^{a_{2}^{x_{2}}...a_{n}^{x_{n}}}}to getfa(k+1)=ga1a2x2…anxn{\displaystyle f_{a}(k+1)=g^{a_{1}a_{2}^{x_{2}}\dots a_{n}^{x_{n}}}}. In general, moving fromktok+ 1 changes the bit pattern and unlessk+ 1 is a power of 2 one can split the exponent infa(k+1){\displaystyle f_{a}(k+1)}so that the computation corresponds to computing theDiffie–Hellmankey between two of the earlier results. This attacker wants to predict the nextsequenceelement. Such an attack would be very bad—but it's also possible to fight it off by working ingroupswith a hardDiffie–Hellman problem(DHP).
Example:An attacker sees several outputs of the function e.g.fa(5)=4112011=41=4{\displaystyle f_{a}(5)=4^{1^{1}2^{0}1^{1}}=4^{1}=4}, as in the previous example, andfa(1)=4102011=41=4{\displaystyle f_{a}(1)=4^{1^{0}2^{0}1^{1}}=4^{1}=4}. Then, the attacker wants to predict the next sequence element of this function,fa(6){\displaystyle f_{a}(6)}. However, the attacker cannot predict the outcome offa(6){\displaystyle f_{a}(6)}from knowingfa(1){\displaystyle f_{a}(1)}andfa(5){\displaystyle f_{a}(5)}.
There are other attacks that would be very bad for apseudorandom number generator: the user expects to get random numbers from the output, so of course the stream should not be predictable, but even more, it should be indistinguishable from a random string. LetAf{\displaystyle {\mathcal {A}}^{f}}denote the algorithmA{\displaystyle {\mathcal {A}}}with access to an oracle for evaluating the functionfa(x){\displaystyle f_{a}(x)}. Suppose thedecisional Diffie–Hellman assumptionholds forFp{\displaystyle \mathbb {F} _{p}},Naor and Reingoldshow that for everyprobabilistic polynomial timealgorithmA{\displaystyle {\mathcal {A}}}and sufficiently largen
The first probability is taken over the choice of the seed s = (p, g, a) and the second probability is taken over the random distribution induced on p, g byIG(n){\displaystyle {\mathcal {I}}{\mathcal {G}}(n)}, instance generator, and the random choice of the functionRa(x){\displaystyle R_{a}(x)}among the set of all{0,1}n→Fp{\displaystyle \{0,1\}^{n}\to \mathbb {F} _{p}}functions.[2]
One natural measure of how useful a sequence may be forcryptographicpurposes is the size of itslinear complexity. The linear complexity of ann-element sequence W(x),x= 0,1,2,...,n– 1, over a ringR{\displaystyle {\mathcal {R}}}is the lengthlof the shortest linearrecurrence relationW(x+l) = Al−1W(x+l−1) + ... + A0W(x),x= 0,1,2,...,n–l−1 with A0, ..., Al−1∈R{\displaystyle {\mathcal {R}}}, which is satisfied by this sequence.
For someγ{\displaystyle \gamma }> 0,n≥ (1+γ{\displaystyle \gamma })logl{\displaystyle \log l}, for anyδ>0{\displaystyle \delta >0}, sufficiently largel, the linear complexity of the sequencefa(x){\displaystyle f_{a}(x)},0 ≤ x ≤ 2n-1, denoted byLa{\displaystyle L_{a}}satisfies
for all except possibly at most3(l−1)n−δ{\displaystyle 3(l-1)^{n-\delta }}vectors a ∈(Fl)n{\displaystyle (\mathbb {F} _{l})^{n}}.[3]The bound of this work has disadvantages, namely it does not apply to the very interesting caselogp≈logn≈n.{\displaystyle \log p\approx \log n\approx {n.}}
The statistical distribution offa(x){\displaystyle f_{a}(x)}is exponentially close touniform distributionfor almost all vectorsa∈(Fl)n{\displaystyle (\mathbb {F} _{l})^{n}}.
LetDa{\displaystyle {\mathbf {D} }_{a}}be thediscrepancyof the set{fa(x)|0≤x≤2n−1}{\displaystyle \{f_{a}(x)|0\leq x\leq 2^{n-1}\}}. Thus, ifn=logp{\displaystyle n=\log p}is the bit length ofpthen for all vectors a ∈(Fl)n{\displaystyle (\mathbb {F} _{l})^{n}}the boundDa≤Δ(l,p){\displaystyle {\mathbf {D} }_{a}\leq \Delta (l,p)}holds, where
and
Although this property does not seem to have any immediate cryptographic implications, the inverse fact, namely non uniform distribution, if true would have disastrous consequences for applications of this function.[4]
Theelliptic curveversion of this function is of interest as well. In particular, it may help to improve the cryptographic security of the corresponding system. Letp> 3 be prime and let E be an elliptic curve overFp{\displaystyle \mathbb {F} _{p}}, then each vectoradefines afinite sequencein thesubgroup⟨G⟩{\displaystyle \langle G\rangle }as:
wherex=x1…xn{\displaystyle x=x_{1}\dots x_{n}}is the bit representation of integerx,0≤x≤2n−1{\displaystyle x,0\leq x\leq 2^{n-1}}.
TheNaor–Reingoldelliptic curve sequence is defined as
If thedecisional Diffie–Hellman assumptionholds, the indexkis not enough to computeuk{\displaystyle u_{k}}in polynomial time, even if an attacker performs polynomially many queries to a random oracle.https://en.wikipedia.org/wiki/Elliptic_curve | https://en.wikipedia.org/wiki/Naor-Reingold_Pseudorandom_Function |
TheReeds–Sloane algorithm, named afterJames ReedsandNeil Sloane, is an extension of theBerlekamp–Massey algorithm, an algorithm for finding the shortestlinear-feedback shift register(LFSR) for a given outputsequence, for use on sequences that take their values from theintegers modn.
This cryptography-related article is astub. You can help Wikipedia byexpanding it. | https://en.wikipedia.org/wiki/Reeds%E2%80%93Sloane_algorithm |
Inmathematicsandelectronics engineering, abinary Golay codeis a type of linearerror-correcting codeused indigital communications. The binary Golay code, along with theternary Golay code, has a particularly deep and interesting connection to the theory offinite sporadic groupsin mathematics.[1]These codes are named in honor ofMarcel J. E. Golaywhose 1949 paper[2]introducing them has been called, byE. R. Berlekamp, the "best single published page" incoding theory.[3]
There are two closely related binary Golay codes. Theextended binary Golay code,G24(sometimes just called the "Golay code" in finite group theory) encodes 12 bits of data in a 24-bit word in such a way that any 3-bit errors can be corrected or any 4-bit errors can be detected.
The other, theperfect binary Golay code,G23, has codewords of length 23 and is obtained from the extended binary Golay code by deleting one coordinate position (conversely, the extended binary Golay code is obtained from the perfect binary Golay code by adding aparity bit). In standard coding notation, the codes have parameters [24, 12, 8] and [23, 12, 7], corresponding to the length of the codewords, thedimensionof the code, and the minimumHamming distancebetween two codewords, respectively.
In mathematical terms, the extended binary Golay codeG24consists of a 12-dimensionallinear subspaceWof the spaceV=F242of 24-bit words such that any two distinct elements ofWdiffer in at least 8 coordinates.Wis called a linear code because it is a vector space. In all,Wcomprises4096 = 212elements.
The binary Golay code,G23is aperfect code. That is, the spheres of radius three around code words form a partition of the vector space.G23is a 12-dimensionalsubspaceof the spaceF232.
The automorphism group of the perfect binary Golay codeG23(meaning the subgroup of the groupS23of permutations of the coordinates ofF232which leaveG23invariant), is theMathieu groupM23{\displaystyle M_{23}}. Theautomorphism groupof the extended binary Golay code is theMathieu groupM24{\displaystyle M_{24}}, of order210× 33× 5 × 7 × 11 × 23.M24{\displaystyle M_{24}}is transitive on octads and on dodecads. The other Mathieu groups occur asstabilizersof one or several elements ofW.
There is a single word of weight 24, which is a 1-dimensional invariant subspace.M24{\displaystyle M_{24}}therefore has an 11-dimensional irreducible representation on the field with 2 elements. In addition, since the binary golay code is a 12-dimensional subspace of a 24-dimensional space,M24{\displaystyle M_{24}}also acts on the 12-dimensionalquotient space, called thebinary Golay cocode. A word in the cocode is in the samecosetas a word of length 0, 1, 2, 3, or 4. In the last case, 6 (disjoint) cocode words all lie in the same coset. There is an 11-dimensional invariant subspace, consisting of cocode words with odd weight, which givesM24{\displaystyle M_{24}}a second 11-dimensional representation on the field with 2 elements.
It is convenient to use the "Miracle Octad Generator" format, with coordinates in an array of 4 rows, 6 columns. Addition is taking the symmetric difference. All 6 columns have the same parity, which equals that of the top row.
A partition of the 6 columns into 3 pairs of adjacent ones constitutes atrio. This is a partition into 3 octad sets. A subgroup, theprojective special linear groupPSL(2,7) x S3of a trio subgroup of M24is useful for generating a basis. PSL(2,7) permutes the octads internally, in parallel. S3permutes the 3 octads bodily.
The basis begins with octad T:
and 5 similar octads. The sumNof all 6 of these code words consists of all 1's. Adding N to a code word produces its complement.
Griess (p. 59) uses the labeling:
PSL(2,7) is naturally the linear fractional group generated by (0123456) and (0∞)(16)(23)(45). The 7-cycle acts on T to give a subspace including also the basis elements
and
The resulting 7-dimensional subspace has a 3-dimensional quotient space upon ignoring the latter 2 octads.
There are 4 other code words of similar structure that complete the basis of 12 code words for this representation of W.
W has a subspace of dimension 4, symmetric under PSL(2,7) x S3, spanned by N and 3 dodecads formed of subsets {0,3,5,6}, {0,1,4,6}, and {0,1,2,5}.
Error correction was vital to data transmission in theVoyager1 and 2 spacecraft particularly because memory constraints dictated offloading data virtually instantly leaving no second chances. Hundreds of color pictures ofJupiterandSaturnin their 1979, 1980, and 1981 fly-bys would be transmitted within a constrained telecommunications bandwidth. Color image transmission required three times as much data as black and white images, so the 7-error correctingReed–Muller codethat had been used to transmit the black and white Mariner images was replaced with the much higher data rate Golay (24,12,8) code.[9]
TheMIL-STD-188American military standards forautomatic link establishmentinhigh frequencyradio systems specify the use of an extended (24,12) Golay code forforward error correction.[10][11]
Intwo-way radiocommunicationdigital-coded squelch (DCS, CDCSS)system uses 23-bit Golay (23,12) code word which has the ability to detect and correct errors of 3 or fewer bits. | https://en.wikipedia.org/wiki/Binary_Golay_code |
Eugene August Prange(July 30, 1917 – February 12, 2006)[1][2]was an Americancoding theorist, a researcher at theAir Force Cambridge Research Laboratory(AFCRL) inMassachusettswho "introduced many of the early fundamental ideas of algebraic coding theory"[3]and was the first to investigatecyclic codesin 1957.[4][5]WithAndrew Gleason, he is the namesake of theGleason–Prange theoremon the symmetries of the extendedquadratic residue code.[6]
Prange was born in Illinois to August Prange and Eugenia Livingston.[7]He graduated from theUniversity of Illinoisand spentWorld War IIserving his country in England as an intelligence officer. He then studied atHarvard Universitybefore joiningAFCRL.[2] | https://en.wikipedia.org/wiki/Eugene_Prange |
Reed–Muller codesareerror-correcting codesthat are used in wireless communications applications, particularly in deep-space communication.[1]Moreover, the proposed5G standard[2]relies on the closely relatedpolar codes[3]for error correction in the control channel. Due to their favorable theoretical and mathematical properties, Reed–Muller codes have also been extensively studied intheoretical computer science.
Reed–Muller codes generalize theReed–Solomon codesand theWalsh–Hadamard code. Reed–Muller codes arelinear block codesthat arelocally testable,locally decodable, andlist decodable. These properties make them particularly useful in the design ofprobabilistically checkable proofs.
Traditional Reed–Muller codes are binary codes, which means that messages and codewords are binary strings. Whenrandmare integers with 0 ≤r≤m, the Reed–Muller code with parametersrandmis denoted as RM(r,m). When asked to encode a message consisting ofkbits, wherek=∑i=0r(mi){\displaystyle \textstyle k=\sum _{i=0}^{r}{\binom {m}{i}}}holds, the RM(r,m) code produces a codeword consisting of 2mbits.
Reed–Muller codes are named afterDavid E. Muller, who discovered the codes in 1954,[4]andIrving S. Reed, who proposed the first efficient decoding algorithm.[5]
Reed–Muller codes can be described in several different (but ultimately equivalent) ways. The description that is based on low-degree polynomials is quite elegant and particularly suited for their application aslocally testable codesandlocally decodable codes.[6]
Ablock codecan have one or more encoding functionsC:{0,1}k→{0,1}n{\textstyle C:\{0,1\}^{k}\to \{0,1\}^{n}}that map messagesx∈{0,1}k{\textstyle x\in \{0,1\}^{k}}to codewordsC(x)∈{0,1}n{\textstyle C(x)\in \{0,1\}^{n}}. The Reed–Muller codeRM(r,m)hasmessage lengthk=∑i=0r(mi){\displaystyle \textstyle k=\sum _{i=0}^{r}{\binom {m}{i}}}andblock lengthn=2m{\displaystyle \textstyle n=2^{m}}. One way to define an encoding for this code is based on the evaluation ofmultilinear polynomialswithmvariables andtotal degreeat mostr. Every multilinear polynomial over thefinite fieldwith two elements can be written as follows:pc(Z1,…,Zm)=∑S⊆{1,…,m}|S|≤rcS⋅∏i∈SZi.{\displaystyle p_{c}(Z_{1},\dots ,Z_{m})=\sum _{\underset {|S|\leq r}{S\subseteq \{1,\dots ,m\}}}c_{S}\cdot \prod _{i\in S}Z_{i}\,.}TheZ1,…,Zm{\textstyle Z_{1},\dots ,Z_{m}}are the variables of the polynomial, and the valuescS∈{0,1}{\textstyle c_{S}\in \{0,1\}}are the coefficients of the polynomial. Note that there are exactlyk=∑i=0r(mi){\textstyle k=\sum _{i=0}^{r}{\binom {m}{i}}}coefficients. With this in mind, an input message consists ofk{\textstyle k}valuesx∈{0,1}k{\textstyle x\in \{0,1\}^{k}}which are used as these coefficients. In this way, each messagex{\textstyle x}gives rise to a unique polynomialpx{\textstyle p_{x}}inmvariables. To construct the codewordC(x){\textstyle C(x)}, the encoder evaluates the polynomialpx{\textstyle p_{x}}at all pointsZ=(Z1,…,Zm)∈{0,1}m{\textstyle Z=(Z_{1},\ldots ,Z_{m})\in \{0,1\}^{m}}, where the polynomial is taken with multiplication and addition mod 2(px(Z)mod2)∈{0,1}{\textstyle (p_{x}(Z){\bmod {2}})\in \{0,1\}}. That is, the encoding function is defined viaC(x)=(px(Z)mod2)Z∈{0,1}m.{\displaystyle C(x)=\left(p_{x}(Z){\bmod {2}}\right)_{Z\in \{0,1\}^{m}}\,.}
The fact that the codewordC(x){\displaystyle C(x)}suffices to uniquely reconstructx{\displaystyle x}follows fromLagrange interpolation, which states that the coefficients of a polynomial are uniquely determined when sufficiently many evaluation points are given. SinceC(0)=0{\displaystyle C(0)=0}andC(x+y)=C(x)+C(y)mod2{\displaystyle C(x+y)=C(x)+C(y){\bmod {2}}}holds for all messagesx,y∈{0,1}k{\displaystyle x,y\in \{0,1\}^{k}}, the functionC{\displaystyle C}is alinear map. Thus the Reed–Muller code is alinear code.
For the codeRM(2,4), the parameters are as follows:
r=2m=4k=(42)+(41)+(40)=6+4+1=11n=2m=16{\textstyle {\begin{aligned}r&=2\\m&=4\\k&=\textstyle {\binom {4}{2}}+{\binom {4}{1}}+{\binom {4}{0}}=6+4+1=11\\n&=2^{m}=16\\\end{aligned}}}
LetC:{0,1}11→{0,1}16{\textstyle C:\{0,1\}^{11}\to \{0,1\}^{16}}be the encoding function just defined. To encode the string x = 1 1010 010101 of length 11, the encoder first constructs the polynomialpx{\textstyle p_{x}}in 4 variables:px(Z1,Z2,Z3,Z4)=1+(1⋅Z1+0⋅Z2+1⋅Z3+0⋅Z4)+(0⋅Z1Z2+1⋅Z1Z3+0⋅Z1Z4+1⋅Z2Z3+0⋅Z2Z4+1⋅Z3Z4)=1+Z1+Z3+Z1Z3+Z2Z3+Z3Z4{\displaystyle {\begin{aligned}p_{x}(Z_{1},Z_{2},Z_{3},Z_{4})&=1+(1\cdot Z_{1}+0\cdot Z_{2}+1\cdot Z_{3}+0\cdot Z_{4})+(0\cdot Z_{1}Z_{2}+1\cdot Z_{1}Z_{3}+0\cdot Z_{1}Z_{4}+1\cdot Z_{2}Z_{3}+0\cdot Z_{2}Z_{4}+1\cdot Z_{3}Z_{4})\\&=1+Z_{1}+Z_{3}+Z_{1}Z_{3}+Z_{2}Z_{3}+Z_{3}Z_{4}\end{aligned}}}Then it evaluates this polynomial at all 16 evaluation points (0101 meansZ1=0,Z2=1,Z3=0,Z4=1){\displaystyle Z_{1}=0,Z_{2}=1,Z_{3}=0,Z_{4}=1)}:px(0000)=1,px(0001)=1,px(0010)=0,px(0011)=1,{\displaystyle p_{x}(0000)=1,\;p_{x}(0001)=1,\;p_{x}(0010)=0,\;p_{x}(0011)=1,\;}
px(0100)=1,px(0101)=1,px(0110)=1,px(0111)=0,{\displaystyle p_{x}(0100)=1,\;p_{x}(0101)=1,\;p_{x}(0110)=1,\;p_{x}(0111)=0,\;}
px(1000)=0,px(1001)=0,px(1010)=0,px(1011)=1,{\displaystyle p_{x}(1000)=0,\;p_{x}(1001)=0,\;p_{x}(1010)=0,\;p_{x}(1011)=1,\;}
px(1100)=0,px(1101)=0,px(1110)=1,px(1111)=0.{\displaystyle p_{x}(1100)=0,\;p_{x}(1101)=0,\;p_{x}(1110)=1,\;p_{x}(1111)=0\,.}As a result, C(1 1010 010101) = 1101 1110 0001 0010 holds.
As was already mentioned, Lagrange interpolation can be used to efficiently retrieve the message from a codeword. However, a decoder needs to work even if the codeword has been corrupted in a few positions, that is, when the received word is different from any codeword. In this case, a local decoding procedure can help.
The algorithm from Reed is based on the following property:
you start from the code word, that is a sequence of evaluation points from an unknown polynomialpx{\textstyle p_{x}}ofF2[X1,X2,...,Xm]{\textstyle {\mathbb {F} }_{2}[X_{1},X_{2},...,X_{m}]}of degree at mostr{\textstyle r}that you want to find. The sequence may contains any number of errors up to2m−r−1−1{\textstyle 2^{m-r-1}-1}included.
If you consider a monomialμ{\textstyle \mu }of the highest degreed{\textstyle d}inpx{\textstyle p_{x}}and sum all the evaluation points of the polynomial where all variables inμ{\textstyle \mu }have the values 0 or 1, and all the other variables have value 0, you get the value of the coefficient (0 or 1) ofμ{\textstyle \mu }inpx{\textstyle p_{x}}(There are2d{\textstyle 2^{d}}such points). This is due to the fact that all lower monomial divisors ofμ{\textstyle \mu }appears an even number of time in the sum, and onlyμ{\textstyle \mu }appears once.
To take into account the possibility of errors, you can also remark that you can fix the value of other variables to any value. So instead of doing the sum only once for other variables not inμ{\textstyle \mu }with 0 value, you do it2m−d{\textstyle 2^{m-d}}times for each fixed valuations of the other variables. If there is no error, all those sums should be equals to the value of the coefficient searched.
The algorithm consists here to take the majority of the answers as the value searched. If the minority is larger than the maximum number of errors possible, the decoding step fails knowing there are too many errors in the input code.
Once a coefficient is computed, if it's 1, update the code to remove the monomialμ{\textstyle \mu }from the input code and continue to next monomial, in reverse order of their degree.
Let's consider the previous example and start from the code. Withr=2,m=4{\textstyle r=2,m=4}we can fix at most 1 error in the code.
Consider the input code as 1101 1110 0001 0110 (this is the previous code with one error).
We know the degree of the polynomialpx{\textstyle p_{x}}is at mostr=2{\textstyle r=2}, we start by searching for monomial of degree 2.
The four sums don't agree (so we know there is an error), but the minority report is not larger than the maximum number of error allowed (1), so we take the majority and the coefficient ofμ{\textstyle \mu }is 1.
We removeμ{\textstyle \mu }from the code before continue : code : 1101 1110 0001 0110, valuation ofμ{\textstyle \mu }is 0001000100010001, the new code is 1100 1111 0000 0111
One error detected, coefficient is 0, no change to current code.
One error detected, coefficient is 0, no change to current code.
One error detected, coefficient is 1, valuation ofμ{\textstyle \mu }is 0000 0011 0000 0011, current code is now 1100 1100 0000 0100.
One error detected, coefficient is 1, valuation ofμ{\textstyle \mu }is 0000 0000 0011 0011, current code is now 1100 1100 0011 0111.
One error detected, coefficient is 0, no change to current code.
We know now all coefficient of degree 2 for the polynomial, we can start mononials of degree 1. Notice that for each next degree, there are twice as much sums, and each sums is half smaller.
One error detected, coefficient is 0, no change to current code.
One error detected, coefficient is 1, valuation ofμ{\textstyle \mu }is 0011 0011 0011 0011, current code is now 1111 1111 0000 0100.
Then we'll find 0 forμ=X2{\textstyle \mu =X_{2}}, 1 forμ=X1{\textstyle \mu =X_{1}}and the current code become 1111 1111 1111 1011.
For the degree 0, we have 16 sums of only 1 bit. The minority is still of size 1, and we foundpx=1+X1+X3+X1X3+X2X3+X3X4{\textstyle p_{x}=1+X_{1}+X_{3}+X_{1}X_{3}+X_{2}X_{3}+X_{3}X_{4}}and the corresponding initial word 1 1010 010101
Using low-degree polynomials over a finite fieldF{\displaystyle \mathbb {F} }of sizeq{\displaystyle q}, it is possible to extend the definition of Reed–Muller codes to alphabets of sizeq{\displaystyle q}. Letm{\displaystyle m}andd{\displaystyle d}be positive integers, wherem{\displaystyle m}should be thought of as larger thand{\displaystyle d}. To encode a messagex∈Fk{\textstyle x\in \mathbb {F} ^{k}}of widthk=(m+dm){\displaystyle k=\textstyle {\binom {m+d}{m}}}, the message is again interpreted as anm{\displaystyle m}-variate polynomialpx{\displaystyle p_{x}}of total degree at mostd{\displaystyle d}and with coefficient fromF{\displaystyle \mathbb {F} }. Such a polynomial indeed has(m+dm){\displaystyle \textstyle {\binom {m+d}{m}}}coefficients. The Reed–Muller encoding ofx{\displaystyle x}is the list of all evaluations ofpx(a){\displaystyle p_{x}(a)}over alla∈Fm{\displaystyle a\in \mathbb {F} ^{m}}. Thus the block length isn=qm{\displaystyle n=q^{m}}.
Agenerator matrixfor a Reed–Muller codeRM(r,m)of lengthN= 2mcan be constructed as follows. Let us write the set of allm-dimensional binary vectors as:
We define inN-dimensional spaceF2N{\displaystyle \mathbb {F} _{2}^{N}}theindicator vectors
on subsetsA⊂X{\displaystyle A\subset X}by:
together with, also inF2N{\displaystyle \mathbb {F} _{2}^{N}}, the binary operation
referred to as thewedge product(not to be confused with thewedge productdefined in exterior algebra). Here,w=(w1,w2,…,wN){\displaystyle w=(w_{1},w_{2},\ldots ,w_{N})}andz=(z1,z2,…,zN){\displaystyle z=(z_{1},z_{2},\ldots ,z_{N})}are points inF2N{\displaystyle \mathbb {F} _{2}^{N}}(N-dimensional binary vectors), and the operation⋅{\displaystyle \cdot }is the usual multiplication in the fieldF2{\displaystyle \mathbb {F} _{2}}.
F2m{\displaystyle \mathbb {F} _{2}^{m}}is anm-dimensional vector space over the fieldF2{\displaystyle \mathbb {F} _{2}}, so it is possible to write
(F2)m={(ym,…,y1)∣yi∈F2}.{\displaystyle (\mathbb {F} _{2})^{m}=\{(y_{m},\ldots ,y_{1})\mid y_{i}\in \mathbb {F} _{2}\}.}
We define inN-dimensional spaceF2N{\displaystyle \mathbb {F} _{2}^{N}}the following vectors with lengthN:v0=(1,1,…,1){\displaystyle N:v_{0}=(1,1,\ldots ,1)}and
where 1 ≤ i ≤mand theHiarehyperplanesin(F2)m{\displaystyle (\mathbb {F} _{2})^{m}}(with dimensionm− 1):
The Reed–MullerRM(r,m)code of orderrand lengthN= 2mis the code generated byv0and the wedge products of up torof thevi,1 ≤i≤m(where by convention a wedge product of fewer than one vector is the identity for the operation). In other words, we can build a generator matrix for theRM(r,m)code, using vectors and their wedge product permutations up torat a timev0,v1,…,vn,…,(vi1∧vi2),…(vi1∧vi2…∧vir){\displaystyle {v_{0},v_{1},\ldots ,v_{n},\ldots ,(v_{i_{1}}\wedge v_{i_{2}}),\ldots (v_{i_{1}}\wedge v_{i_{2}}\ldots \wedge v_{i_{r}})}}, as the rows of the generator matrix, where1 ≤ik≤m.
Letm= 3. ThenN= 8, and
and
The RM(1,3) code is generated by the set
or more explicitly by the rows of the matrix:
The RM(2,3) code is generated by the set:
or more explicitly by the rows of the matrix:
The following properties hold:
such vectors andF2N{\displaystyle \mathbb {F} _{2}^{N}}have dimensionNso it is sufficient to check that theNvectors span; equivalently it is sufficient to check thatRM(m,m)=F2N{\displaystyle \mathrm {RM} (m,m)=\mathbb {F} _{2}^{N}}.
Letxbe a binary vector of lengthm, an element ofX. Let (x)idenote theithelement ofx. Define
where 1 ≤i≤m.
ThenI{x}=y1∧⋯∧ym{\displaystyle \mathbb {I} _{\{x\}}=y_{1}\wedge \cdots \wedge y_{m}}
RM(r,m) codes can be decoded usingmajority logic decoding. The basic idea of majority logic decoding is
to build several checksums for each received code word element. Since each of the different checksums must all
have the same value (i.e. the value of the message word element weight), we can use a majority logic decoding to decipher
the value of the message word element. Once each order of the polynomial is decoded, the received word is modified
accordingly by removing the corresponding codewords weighted by the decoded message contributions, up to the present stage.
So for arth order RM code, we have to decode iteratively r+1, times before we arrive at the final
received code-word. Also, the values of the message bits are calculated through this scheme; finally we can calculate
the codeword by multiplying the message word (just decoded) with the generator matrix.
One clue if the decoding succeeded, is to have an all-zero modified received word, at the end of (r+ 1)-stage decoding
through the majority logic decoding. This technique was proposed by Irving S. Reed, and is more general when applied
to other finite geometry codes.
A Reed–Muller code RM(r,m) exists for any integersm≥0{\displaystyle m\geq 0}and0≤r≤m{\displaystyle 0\leq r\leq m}. RM(m,m) is defined as the universe (2m,2m,1{\displaystyle 2^{m},2^{m},1}) code. RM(−1,m) is defined as the trivial code (2m,0,∞{\displaystyle 2^{m},0,\infty }). The remaining RM codes may be constructed from these elementary codes using the length-doubling construction
From this construction, RM(r,m) is a binarylinear block code(n,k,d) with lengthn= 2m, dimensionk(r,m)=k(r,m−1)+k(r−1,m−1){\displaystyle k(r,m)=k(r,m-1)+k(r-1,m-1)}and minimum distanced=2m−r{\displaystyle d=2^{m-r}}forr≥0{\displaystyle r\geq 0}. Thedual codeto RM(r,m) is RM(m-r-1,m). This shows that repetition and SPC codes are duals, biorthogonal and extended Hamming codes are duals and that codes withk=n/2are self-dual.
AllRM(r,m)codes with0≤m≤5{\displaystyle 0\leq m\leq 5}and alphabet size 2 are displayed here, annotated with the standard [n,k,d]coding theory notationforblock codes. The codeRM(r,m)is a[2m,k,2m−r]2{\displaystyle \textstyle [2^{m},k,2^{m-r}]_{2}}-code, that is, it is alinear codeover abinary alphabet, hasblock length2m{\displaystyle \textstyle 2^{m}},message length(or dimension)k, andminimum distance2m−r{\displaystyle \textstyle 2^{m-r}}. | https://en.wikipedia.org/wiki/Reed%E2%80%93Muller_code |
Incoding theory, theternary Golay codesare two closely relatederror-correcting codes.
The code generally known simply as theternary Golay codeis an[11,6,5]3{\displaystyle [11,6,5]_{3}}-code, that is, it is alinear codeover aternaryalphabet; therelative distanceof the code is as large as it possibly can be for a ternary code, and hence, the ternary Golay code is aperfect code.
Theextended ternary Golay codeis a [12, 6, 6]linear codeobtained by adding a zero-sumcheck digitto the [11, 6, 5] code.
In finitegroup theory, the extended ternary Golay code is sometimes referred to as the ternary Golay code.[citation needed]
The ternary Golay code consists of 36= 729 codewords.
Itsparity check matrixis
Any two different codewords differ in at least 5 positions.
Every ternary word of length 11 has aHamming distanceof at most 2 from exactly one codeword.
The code can also be constructed as thequadratic residue codeof length 11 over thefinite fieldF3(i.e.,the Galois FieldGF(3)).
Used in afootball poolwith 11 games, the ternary Golay code corresponds to 729 bets and guarantees exactly one bet with at most 2 wrong outcomes.
The set of codewords withHamming weight5 is a 3-(11,5,4)design.
Thegenerator matrixgiven by Golay (1949, Table 1.) is
Theautomorphism groupof the (original) ternary Golay code is theMathieu groupM11, which is the smallest of the sporadic simple groups.
Thecomplete weight enumeratorof the extended ternary Golay code is
Theautomorphism groupof the extended ternary Golay code is 2.M12, whereM12is theMathieu groupM12.
The extended ternary Golay code can be constructed as the span of the rows of aHadamard matrixof order 12 over the fieldF3.
Consider all codewords of the extended code which have just six nonzero digits. The sets of positions at which these nonzero digits occur form theSteiner systemS(5, 6, 12).
Agenerator matrixfor the extended ternary Golay code is
The corresponding parity check matrix for this generator matrix is[−B|I6]T{\displaystyle [-B|I_{6}]^{T}}, whereT{\displaystyle T}denotes thetransposeof the matrix.
An alternative generator matrix for this code is
And its parity check matrix is[−Balt|I6]T{\displaystyle [-B_{\mathrm {alt} }|I_{6}]^{T}}.
The three elements of the underlying finite field are represented here by{0,1,−1}{\displaystyle \{0,1,-1\}}, rather than by{0,1,2}{\displaystyle \{0,1,2\}}.
It is also understood that2=1+1=−1{\displaystyle 2=1+1=-1}(i.e.,theadditive inverseof 1) and−2=(−1)+(−1)=1{\displaystyle -2=(-1)+(-1)=1}. Products of these finite field elements are identical to those of the integers. Row and column sums are evaluated modulo 3.
Linear combinations, orvector addition, of the rows of the matrix
produces all possiblewordscontained in the code. This is referred to as thespanof the rows. The inner product of any two rows of the generator matrix will always sum to zero. These rows, or vectors, are said to beorthogonal.
The matrix product of the generator and parity-check matrices,[I6|Balt][−Balt|I6]T{\displaystyle [I_{6}|B_{\mathrm {alt} }]\,[-B_{\mathrm {alt} }|I_{6}]^{T}}, is the6×6{\displaystyle 6\times 6}matrix of all zeroes, and by intent.
Indeed, this is an example of the very definition of any parity check matrix with respect to its generator matrix.
The ternary Golay code was published byGolay(1949). It was independently discovered two years earlier by theFinnishfootball poolenthusiast Juhani Virtakallio, who published it in 1947 in issues 27, 28 and 33 of the football magazineVeikkaaja. (Barg 1993, p.25)
The ternary Golay code has been shown to be useful for an approach to fault-tolerantquantum computingknown asmagic state distillation.[1] | https://en.wikipedia.org/wiki/Ternary_Golay_code |
Incoding theory, theSingleton bound, named after the American mathematician Richard Collom Singleton (1928–2007), is a relatively crude upper bound on the size of an arbitraryblock codeC{\displaystyle C}with block lengthn{\displaystyle n}, sizeM{\displaystyle M}and minimum distanced{\displaystyle d}. It is also known as theJoshibound[1]proved byJoshi (1958)and even earlier byKomamiya (1953).
The minimum distance of a setC{\displaystyle C}of codewords of lengthn{\displaystyle n}is defined asd=min{x,y∈C:x≠y}d(x,y){\displaystyle d=\min _{\{x,y\in C:x\neq y\}}d(x,y)}whered(x,y){\displaystyle d(x,y)}is theHamming distancebetweenx{\displaystyle x}andy{\displaystyle y}. The expressionAq(n,d){\displaystyle A_{q}(n,d)}represents the maximum number of possible codewords in aq{\displaystyle q}-ary block code of lengthn{\displaystyle n}and minimum distanced{\displaystyle d}.
Then the Singleton bound states thatAq(n,d)≤qn−d+1.{\displaystyle A_{q}(n,d)\leq q^{n-d+1}.}
First observe that the number ofq{\displaystyle q}-ary words of lengthn{\displaystyle n}isqn{\displaystyle q^{n}}, since each letter in such a word may take one ofq{\displaystyle q}different values, independently of the remaining letters.
Now letC{\displaystyle C}be an arbitraryq{\displaystyle q}-ary block code of minimum distanced{\displaystyle d}. Clearly, all codewordsc∈C{\displaystyle c\in C}are distinct. If wepuncturethe code by deleting the firstd−1{\displaystyle d-1}letters of each codeword, then all resulting codewords must still be pairwise different, since all of the original codewords inC{\displaystyle C}haveHamming distanceat leastd{\displaystyle d}from each other. Thus the size of the altered code is the same as the original code.
The newly obtained codewords each have lengthn−(d−1)=n−d+1,{\displaystyle n-(d-1)=n-d+1,}and thus, there can be at mostqn−d+1{\displaystyle q^{n-d+1}}of them. SinceC{\displaystyle C}was arbitrary, this bound must hold for the largest possible code with these parameters, thus:[2]|C|≤Aq(n,d)≤qn−d+1.{\displaystyle |C|\leq A_{q}(n,d)\leq q^{n-d+1}.}
IfC{\displaystyle C}is alinear codewith block lengthn{\displaystyle n}, dimensionk{\displaystyle k}and minimum distanced{\displaystyle d}over thefinite fieldwithq{\displaystyle q}elements, then the maximum number of codewords isqk{\displaystyle q^{k}}and the Singleton bound implies:qk≤qn−d+1,{\displaystyle q^{k}\leq q^{n-d+1},}so thatk≤n−d+1,{\displaystyle k\leq n-d+1,}which is usually written as[3]d≤n−k+1.{\displaystyle d\leq n-k+1.}
In the linear code case a different proof of the Singleton bound can be obtained by observing that rank of theparity check matrixisn−k{\displaystyle n-k}.[4]Another simple proof follows from observing that the rows of any generator matrix in standard form have weight at mostn−k+1{\displaystyle n-k+1}.
The usual citation given for this result isSingleton (1964), but it was proven earlier byJoshi (1958). Joshi notes that the result had been obtained earlier byKomamiya (1953)using a more complex proof.Welsh (1988, p. 72) also notes the same regardingKomamiya (1953).
Linear block codes that achieve equality in the Singleton bound are calledMDS (maximum distance separable) codes. Examples of such codes include codes that have onlyq{\displaystyle q}codewords (the all-x{\displaystyle x}word forx∈Fq{\displaystyle x\in \mathbb {F} _{q}}, having thus minimum distancen{\displaystyle n}), codes that use the whole of(Fq)n{\displaystyle (\mathbb {F} _{q})^{n}}(minimum distance 1), codes with a single parity symbol (minimum distance 2) and theirdual codes. These are often calledtrivialMDS codes.
In the case of binary alphabets, only trivial MDS codes exist.[5][6]
Examples of non-trivial MDS codes includeReed-Solomon codesand their extended versions.[7][8]
MDS codes are an important class of block codes since, for a fixedn{\displaystyle n}andk{\displaystyle k}, they have the greatest error correcting and detecting capabilities. There are several ways to characterize MDS codes:[9]
Theorem—LetC{\displaystyle C}be a linear [n,k,d{\displaystyle n,k,d}] code overFq{\displaystyle \mathbb {F} _{q}}. The following are equivalent:
The last of these characterizations permits, by using theMacWilliams identities, an explicit formula for the complete weight distribution of an MDS code.[10]
Theorem—LetC{\displaystyle C}be a linear [n,k,d{\displaystyle n,k,d}] MDS code overFq{\displaystyle \mathbb {F} _{q}}. IfAw{\displaystyle A_{w}}denotes the number of codewords inC{\displaystyle C}of weightw{\displaystyle w}, thenAw=(nw)∑j=0w−d(−1)j(wj)(qw−d+1−j−1)=(nw)(q−1)∑j=0w−d(−1)j(w−1j)qw−d−j.{\displaystyle A_{w}={\binom {n}{w}}\sum _{j=0}^{w-d}(-1)^{j}{\binom {w}{j}}(q^{w-d+1-j}-1)={\binom {n}{w}}(q-1)\sum _{j=0}^{w-d}(-1)^{j}{\binom {w-1}{j}}q^{w-d-j}.}
The linear independence of the columns of a generator matrix of an MDS code permits a construction of MDS codes from objects infiniteprojective geometry. LetPG(N,q){\displaystyle PG(N,q)}be the finiteprojective spaceof (geometric) dimensionN{\displaystyle N}over the finite fieldFq{\displaystyle \mathbb {F} _{q}}. LetK={P1,P2,…,Pm}{\displaystyle K=\{P_{1},P_{2},\dots ,P_{m}\}}be a set of points in this projective space represented withhomogeneous coordinates. Form the(N+1)×m{\displaystyle (N+1)\times m}matrixG{\displaystyle G}whose columns are the homogeneous coordinates of these points. Then,[11]
Theorem—K{\displaystyle K}is a (spatial)m{\displaystyle m}-arcif and only ifG{\displaystyle G}is the generator matrix of an[m,N+1,m−N]{\displaystyle [m,N+1,m-N]}MDS code overFq{\displaystyle \mathbb {F} _{q}}. | https://en.wikipedia.org/wiki/Singleton_bound |
Incoding theory, anerasure codeis aforward error correction(FEC) code under the assumption of bit erasures (rather than bit errors), which transforms a message ofksymbols into a longer message (code word) withnsymbols such that the original message can be recovered from a subset of thensymbols. The fractionr=k/nis called thecode rate. The fractionk’/k, wherek’denotes the number of symbols required for recovery, is calledreception efficiency. The recovery algorithm expects that it is known which of thensymbols are lost.
Erasure coding was invented byIrving ReedandGustave Solomonin 1960.[1]
There are many different erasure coding schemes. The most popular erasure codes areReed-Solomon coding,Low-density parity-check code(LDPC codes), andTurbo codes.[1]
As of 2023, modern data storage systems can be designed to tolerate the complete failure of a few disks without data loss, using one of 3 approaches:[2][3][4]
While technically RAID can be seen as a kind of erasure code,[5]"RAID" is generally applied to an array attached to a single host computer (which is a single point of failure), while "erasure coding" generally implies multiple hosts,[3]sometimes called aRedundant Array of Inexpensive Servers(RAIS).
The erasure code allows operations to continue when any one of those hosts stops.[4][6]
Compared to block-level RAID systems, object storage erasure coding has some significant differences that make it more resilient.[7][8][9][10][11]
Optimal erasure codes have the property that anykout of thencode word symbols are sufficient to recover the original message (i.e., they have optimal reception efficiency). Optimal erasure codes aremaximum distance separable codes(MDS codes).
Parity check is the special case wheren=k+ 1. From a set ofkvalues{vi}1≤i≤k{\displaystyle \{v_{i}\}_{1\leq i\leq k}}, a checksum is computed and appended to theksource values:
The set ofk+ 1 values{vi}1≤i≤k+1{\displaystyle \{v_{i}\}_{1\leq i\leq k+1}}is now consistent with regard to the checksum.
If one of these values,ve{\displaystyle v_{e}}, is erased, it can be easily recovered by summing the remaining variables:
RAID 5is a widely-used application of the parity check erasure code.[1]
In the simple case wherek= 2, redundancy symbols may be created by sampling different points along the line between the two original symbols. This is pictured with a simple example, called err-mail:
Alicewants to send her telephone number (555629) toBobusing err-mail. Err-mail works just like e-mail, except
Instead of asking Bob to acknowledge the messages she sends, Alice devises the following scheme.
Bob knows that the form off(k) isf(i)=a+(b−a)(i−1){\displaystyle f(i)=a+(b-a)(i-1)}, whereaandbare the two parts of the telephone number.
Now suppose Bob receives "D=777" and "E=851".
Bob can reconstruct Alice's phone number by computing the values ofaandbfrom the values (f(4) andf(5)) he has received.
Bob can perform this procedure using any two err-mails, so the erasure code in this example has a rate of 40%.
Note that Alice cannot encode her telephone number in just one err-mail, because it contains six characters, and that the maximum length of one err-mail message is five characters. If she sent her phone number in pieces, asking Bob to acknowledge receipt of each piece, at least four messages would have to be sent anyway (two from Alice, and two acknowledgments from Bob). So the erasure code in this example, which requires five messages, is quite economical.
This example is a little bit contrived. For truly generic erasure codes that work over any data set, we would need something other than thef(i) given.
The linear construction above can be generalized topolynomial interpolation. Additionally, points are now computed over afinite field.
First we choose a finite fieldFwith order of at leastn, but usually a power of 2. The sender numbers the data symbols from 0 tok− 1 and sends them. He then constructs a(Lagrange) polynomialp(x) of orderksuch thatp(i) is equal to data symboli. He then sendsp(k), ...,p(n− 1). The receiver can now also use polynomial interpolation to recover the lost packets, provided he receivesksymbols successfully. If the order ofFis less than 2b, where b is the number of bits in a symbol, then multiple polynomials can be used.
The sender can construct symbolskton− 1 'on the fly', i.e., distribute the workload evenly between transmission of the symbols. If the receiver wants to do his calculations 'on the fly', he can construct a new polynomialq, such thatq(i) =p(i) if symboli<kwas received successfully andq(i) = 0 when symboli<kwas not received. Now letr(i) =p(i) −q(i). Firstly we know thatr(i) = 0 if symboli<khas been received successfully. Secondly, if symboli≥khas been received successfully, thenr(i) =p(i) −q(i) can be calculated. So we have enough data points to constructrand evaluate it to find the lost packets. So both the sender and the receiver requireO(n(n−k)) operations and onlyO(n−k) space for operating 'on the fly'.
This process is implemented byReed–Solomon codes, with code words constructed over afinite fieldusing aVandermonde matrix.
Most practical erasure codes aresystematic codes-- each one of the originalksymbols can be found copied, unencoded, as one of thenmessage symbols.[12](Erasure codes that supportsecret sharingnever use a systematic code).
Near-optimal erasure codesrequire (1 + ε)ksymbols to recover the message (where ε>0). Reducing ε can be done at the cost of CPU time.Near-optimal erasure codestrade correction capabilities for computational complexity: practical algorithms can encode and decode with linear time complexity.
Fountain codes(also known asrateless erasure codes) are notable examples ofnear-optimal erasure codes. They can transform aksymbol message into a practically infinite encoded form, i.e., they can generate an arbitrary amount of redundancy symbols that can all be used for error correction. Receivers can start decoding after they have received slightly more thankencoded symbols.
Regenerating codesaddress the issue of rebuilding (also called repairing) lost encoded fragments from existing encoded fragments. This issue occurs in distributed
storage systems where communication to maintain encoded redundancy is a problem.[12]
Erasure coding is now standard practice for reliable data storage.[13][14][15]In particular, various implementations of Reed-Solomon erasure coding are used byApache Hadoop, theRAID-6built into Linux, Microsoft Azure, Facebook cold storage, and Backblaze Vaults.[15][12]
The classical way to recover from failures in storage systems was to use replication. However, replication incurs significant overhead in terms of wasted bytes. Therefore, increasingly large storage systems, such as those used in data centers use erasure-coded storage. The most common form of erasure coding used in storage systems isReed-Solomon (RS) code, an advanced mathematics formula used to enable regeneration of missing data from pieces of known data, called parity blocks. In a (k,m) RS code, a given set ofkdata blocks, called "chunks", are encoded into (k+m) chunks. The total set of chunks comprises astripe. The coding is done such that as long as at leastkout of (k+m) chunks are available, one can recover the entire data. This means a (k,m) RS-encoded storage can tolerate up tomfailures.
Example:In RS (10, 4) code, which is used in Facebook for theirHDFS,[16]10 MB of user data is divided into ten 1MB blocks. Then, four additional 1 MB parity blocks are created to provide redundancy. This can tolerate up to 4 concurrent failures. The storage overhead here is 14/10 = 1.4X.
In the case of a fully replicated system, the 10 MB of user data will have to be replicated 4 times to tolerate up to 4 concurrent failures. The storage overhead in that case will be 50/10 = 5 times.
This gives an idea of the lower storage overhead of erasure-coded storage compared to full replication and thus the attraction in today's storage systems.
Initially, erasure codes were used to reduce the cost of storing "cold" (rarely-accessed) data efficiently; but erasure codes can also be used to improve performance serving "hot" (more-frequently-accessed) data.[12]
RAID N+M divides data blocks across N+M drives, and can recover all the data when any M drives fail.[1]In particular, RAID 7.3 refers to triple-parity RAID, and can recover all the data when any 3 drives fail.[17]
Here are some examples of implementations of the various codes: | https://en.wikipedia.org/wiki/Erasure_code |
Quantum error correction(QEC) is a set of techniques used inquantum computingto protectquantum informationfrom errors due todecoherenceand otherquantum noise. Quantum error correction is theorised as essential to achievefault tolerant quantum computingthat can reduce the effects of noise on stored quantum information, faulty quantum gates, faulty quantum state preparation, and faulty measurements. Effective quantum error correction would allow quantum computers with low qubit fidelity to execute algorithms of higher complexity or greatercircuit depth.[1]
Classicalerror correctionoften employsredundancy. The simplest albeit inefficient approach is therepetition code. A repetition code stores the desired (logical) information as multiple copies, and—if these copies are later found to disagree due to errors introduced to the system—determines the most likely value for the original data by majority vote. For instance, suppose we copy a bit in the one (on) state three times. Suppose further that noise in the system introduces an error that corrupts the three-bit state so that one of the copied bits becomes zero (off) but the other two remain equal to one. Assuming that errors are independent and occur with some sufficiently low probabilityp, it is most likely that the error is a single-bit error and the intended message is three bits in the one state. It is possible that a double-bit error occurs and the transmitted message is equal to three zeros, but this outcome is less likely than the above outcome. In this example, the logical information is a single bit in the one state and the physical information are the three duplicate bits. Creating a physical state that represents the logical state is calledencodingand determining which logical state is encoded in the physical state is calleddecoding. Similar to classical error correction, QEC codes do not always correctly decode logical qubits, but instead reduce the effect of noise on the logical state.
Copying quantum information is not possible due to theno-cloning theorem. This theorem seems to present an obstacle to formulating a theory of quantum error correction. But it is possible tospreadthe (logical) information of one logicalqubitonto a highly entangled state of several (physical) qubits.Peter Shorfirst discovered this method of formulating aquantum error correcting codeby storing the information of one qubit onto a highly entangled state of nine qubits.[2]
In classical error correction,syndrome decodingis used to diagnose which error was the likely source of corruption on an encoded state. An error can then be reversed by applying a corrective operation based on the syndrome. Quantum error correction also employs syndrome measurements. It performs a multi-qubit measurement that does not disturb the quantum information in the encoded state but retrieves information about the error. Depending on the QEC code used, syndrome measurement can determine the occurrence, location and type of errors. In most QEC codes, the type of error is either a bit flip, or a sign (of thephase) flip, or both (corresponding to thePauli matricesX, Z, and Y). The measurement of the syndrome has theprojectiveeffect of aquantum measurement, so even if the error due to the noise was arbitrary, it can be expressed as a combination ofbasisoperations called the error basis (which is given by the Pauli matrices and theidentity). To correct the error, the Pauli operator corresponding to the type of error is used on the corrupted qubit to revert the effect of the error.
The syndrome measurement provides information about the error that has happened, but not about the information that is stored in the logical qubit—as otherwise the measurement would destroy anyquantum superpositionof this logical qubit with other qubits in thequantum computer, which would prevent it from being used to convey quantum information.
The repetition code works in aclassical channel, because classical bits are easy to measure and to repeat. This approach does not work for a quantum channel in which, due to theno-cloning theorem, it is not possible to repeat a single qubit three times. To overcome this, a different method has to be used, such as thethree-qubit bit-flip codefirst proposed by Asher Peres in 1985.[3]This technique usesentanglementand syndrome measurements and is comparable in performance with the repetition code.
Consider the situation in which we want to transmit the state of a single qubit|ψ⟩{\displaystyle \vert \psi \rangle }through a noisychannelE{\displaystyle {\mathcal {E}}}. Let us moreover assume that this channel either flips the state of the qubit, with probabilityp{\displaystyle p}, or leaves it unchanged. The action ofE{\displaystyle {\mathcal {E}}}on a general inputρ{\displaystyle \rho }can therefore be written asE(ρ)=(1−p)ρ+p⋅ρ{\displaystyle {\mathcal {E}}(\rho )=(1-p)\rho +p\cdot \rho }.
Let|ψ⟩=α0|0⟩+α1|1⟩{\displaystyle |\psi \rangle =\alpha _{0}|0\rangle +\alpha _{1}|1\rangle }be the quantum state to be transmitted. With no error-correcting protocol in place, the transmitted state will be correctly transmitted with probability1−p{\displaystyle 1-p}. We can however improve on this number byencodingthe state into a greater number of qubits, in such a way that errors in the corresponding logical qubits can be detected and corrected. In the case of the simple three-qubit repetition code, the encoding consists in the mappings|0⟩→|0L⟩≡|000⟩{\displaystyle \vert 0\rangle \rightarrow \vert 0_{\rm {L}}\rangle \equiv \vert 000\rangle }and|1⟩→|1L⟩≡|111⟩{\displaystyle \vert 1\rangle \rightarrow \vert 1_{\rm {L}}\rangle \equiv \vert 111\rangle }. The input state|ψ⟩{\displaystyle \vert \psi \rangle }is encoded into the state|ψ′⟩=α0|000⟩+α1|111⟩{\displaystyle \vert \psi '\rangle =\alpha _{0}\vert 000\rangle +\alpha _{1}\vert 111\rangle }. This mapping can be realized for example using two CNOT gates, entangling the system with twoancillary qubitsinitialized in the state|0⟩{\displaystyle \vert 0\rangle }.[4]The encoded state|ψ′⟩{\displaystyle \vert \psi '\rangle }is what is now passed through the noisy channel.
The channel acts on|ψ′⟩{\displaystyle \vert \psi '\rangle }by flipping some subset (possibly empty) of its qubits. No qubit is flipped with probability(1−p)3{\displaystyle (1-p)^{3}}, a single qubit is flipped with probability3p(1−p)2{\displaystyle 3p(1-p)^{2}}, two qubits are flipped with probability3p2(1−p){\displaystyle 3p^{2}(1-p)}, and all three qubits are flipped with probabilityp3{\displaystyle p^{3}}. Note that a further assumption about the channel is made here: we assume thatE{\displaystyle {\mathcal {E}}}acts equally and independently on each of the three qubits in which the state is now encoded. The problem is now how to detect and correct such errors, while not corrupting the transmitted state.
Let us assume for simplicity thatp{\displaystyle p}is small enough that the probability of more than a single qubit being flipped is negligible. One can then detect whether a qubit was flipped, without also querying for the values being transmitted, by asking whether one of the qubits differs from the others. This amounts to performing a measurement with four different outcomes, corresponding to the following four projective measurements:P0=|000⟩⟨000|+|111⟩⟨111|,P1=|100⟩⟨100|+|011⟩⟨011|,P2=|010⟩⟨010|+|101⟩⟨101|,P3=|001⟩⟨001|+|110⟩⟨110|.{\displaystyle {\begin{aligned}P_{0}&=|000\rangle \langle 000|+|111\rangle \langle 111|,\\P_{1}&=|100\rangle \langle 100|+|011\rangle \langle 011|,\\P_{2}&=|010\rangle \langle 010|+|101\rangle \langle 101|,\\P_{3}&=|001\rangle \langle 001|+|110\rangle \langle 110|.\end{aligned}}}This reveals which qubits are different from the others, without at the same time giving information about the state of the qubits themselves. If the outcome corresponding toP0{\displaystyle P_{0}}is obtained, no correction is applied, while if the outcome corresponding toPi{\displaystyle P_{i}}is observed, then the PauliXgate is applied to thei{\displaystyle i}-th qubit. Formally, this correcting procedure corresponds to the application of the following map to the output of the channel:Ecorr(ρ)=P0ρP0+∑i=13XiPiρPiXi.{\displaystyle {\mathcal {E}}_{\operatorname {corr} }(\rho )=P_{0}\rho P_{0}+\sum _{i=1}^{3}X_{i}P_{i}\rho \,P_{i}X_{i}.}
Note that, while this procedure perfectly corrects the output when zero or one flips are introduced by the channel, if more than one qubit is flipped then the output is not properly corrected. For example, if the first and second qubits are flipped, then the syndrome measurement gives the outcomeP3{\displaystyle P_{3}}, and the third qubit is flipped, instead of the first two. To assess the performance of this error-correcting scheme for a general input we can study thefidelityF(ψ′){\displaystyle F(\psi ')}between the input|ψ′⟩{\displaystyle \vert \psi '\rangle }and the outputρout≡Ecorr(E(|ψ′⟩⟨ψ′|)){\displaystyle \rho _{\operatorname {out} }\equiv {\mathcal {E}}_{\operatorname {corr} }({\mathcal {E}}(\vert \psi '\rangle \langle \psi '\vert ))}. Being the output stateρout{\displaystyle \rho _{\operatorname {out} }}correct when no more than one qubit is flipped, which happens with probability(1−p)3+3p(1−p)2{\displaystyle (1-p)^{3}+3p(1-p)^{2}}, we can write it as[(1−p)3+3p(1−p)2]|ψ′⟩⟨ψ′|+(...){\displaystyle [(1-p)^{3}+3p(1-p)^{2}]\,\vert \psi '\rangle \langle \psi '\vert +(...)}, where the dots denote components ofρout{\displaystyle \rho _{\operatorname {out} }}resulting from errors not properly corrected by the protocol. It follows thatF(ψ′)=⟨ψ′|ρout|ψ′⟩≥(1−p)3+3p(1−p)2=1−3p2+2p3.{\displaystyle F(\psi ')=\langle \psi '\vert \rho _{\operatorname {out} }\vert \psi '\rangle \geq (1-p)^{3}+3p(1-p)^{2}=1-3p^{2}+2p^{3}.}Thisfidelityis to be compared with the corresponding fidelity obtained when no error-correcting protocol is used, which was shown before to equal1−p{\displaystyle {1-p}}. A little algebra then shows that the fidelityaftererror correction is greater than the one without forp<1/2{\displaystyle p<1/2}. Note that this is consistent with the working assumption that was made while deriving the protocol (ofp{\displaystyle p}being small enough).
The bit flip is the only kind of error in classical computers. In quantum computers, however, another kind of error is possible: the sign flip. Through transmission in a channel, the relative sign between|0⟩{\displaystyle |0\rangle }and|1⟩{\displaystyle |1\rangle }can become inverted. For instance, a qubit in the state|−⟩=(|0⟩−|1⟩)/2{\displaystyle |-\rangle =(|0\rangle -|1\rangle )/{\sqrt {2}}}may have its sign flip to|+⟩=(|0⟩+|1⟩)/2.{\displaystyle |+\rangle =(|0\rangle +|1\rangle )/{\sqrt {2}}.}
The original state of the qubit|ψ⟩=α0|0⟩+α1|1⟩{\displaystyle |\psi \rangle =\alpha _{0}|0\rangle +\alpha _{1}|1\rangle }will be changed into the state|ψ′⟩=α0|+++⟩+α1|−−−⟩.{\displaystyle |\psi '\rangle =\alpha _{0}|{+}{+}{+}\rangle +\alpha _{1}|{-}{-}{-}\rangle .}
In the Hadamard basis, bit flips become sign flips and sign flips become bit flips. LetEphase{\displaystyle E_{\text{phase}}}be a quantum channel that can cause at most one phase flip. Then the bit-flip code from above can recover|ψ⟩{\displaystyle |\psi \rangle }by transforming into the Hadamard basis before and after transmission throughEphase{\displaystyle E_{\text{phase}}}.
The error channel may induce either a bit flip, a sign flip (i.e., a phase flip), or both. It is possible to correct for both types of errors on a logical qubit using a well-designed QEC code. One example of a code that does this is the Shor code, published in 1995.[2][5]: 10Since these two types of errors are the only types of errors that can result after a projective measurement, a Shor code corrects arbitrary single-qubit errors.
LetE{\displaystyle E}be aquantum channelthat can arbitrarily corrupt a single qubit. The 1st, 4th and 7th qubits are for the sign flip code, while the three groups of qubits (1,2,3), (4,5,6), and (7,8,9) are designed for the bit flip code. With the Shor code, a qubit state|ψ⟩=α0|0⟩+α1|1⟩{\displaystyle |\psi \rangle =\alpha _{0}|0\rangle +\alpha _{1}|1\rangle }will be transformed into the product of 9 qubits|ψ′⟩=α0|0S⟩+α1|1S⟩{\displaystyle |\psi '\rangle =\alpha _{0}|0_{S}\rangle +\alpha _{1}|1_{S}\rangle }, where|0S⟩=122(|000⟩+|111⟩)⊗(|000⟩+|111⟩)⊗(|000⟩+|111⟩){\displaystyle |0_{\rm {S}}\rangle ={\frac {1}{2{\sqrt {2}}}}(|000\rangle +|111\rangle )\otimes (|000\rangle +|111\rangle )\otimes (|000\rangle +|111\rangle )}|1S⟩=122(|000⟩−|111⟩)⊗(|000⟩−|111⟩)⊗(|000⟩−|111⟩){\displaystyle |1_{\rm {S}}\rangle ={\frac {1}{2{\sqrt {2}}}}(|000\rangle -|111\rangle )\otimes (|000\rangle -|111\rangle )\otimes (|000\rangle -|111\rangle )}
If a bit flip error happens to a qubit, the syndrome analysis will be performed on each block of qubits (1,2,3), (4,5,6), and (7,8,9) to detect and correct at most one bit flip error in each block.
If the three bit flip group (1,2,3), (4,5,6), and (7,8,9) are considered as three inputs, then the Shor code circuit can be reduced as a sign flip code. This means that the Shor code can also repair a sign flip error for a single qubit.
The Shor code also can correct for any arbitrary errors (both bit flip and sign flip) to a single qubit. If an error is modeled by a unitary transform U, which will act on a qubit|ψ⟩{\displaystyle |\psi \rangle }, thenU{\displaystyle U}can be described in the formU=c0I+c1X+c2Y+c3Z{\displaystyle U=c_{0}I+c_{1}X+c_{2}Y+c_{3}Z}wherec0{\displaystyle c_{0}},c1{\displaystyle c_{1}},c2{\displaystyle c_{2}}, andc3{\displaystyle c_{3}}are complex constants, I is the identity, and thePauli matricesare given byX=(0110);Y=(0−ii0);Z=(100−1).{\displaystyle {\begin{aligned}X&={\begin{pmatrix}0&1\\1&0\end{pmatrix}};\\Y&={\begin{pmatrix}0&-i\\i&0\end{pmatrix}};\\Z&={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}.\end{aligned}}}
IfUis equal toI, then no error occurs. IfU=X{\displaystyle U=X}, a bit flip error occurs. IfU=Z{\displaystyle U=Z}, a sign flip error occurs. IfU=iY{\displaystyle U=iY}then both a bit flip error and a sign flip error occur. In other words, the Shor code can correct any combination of bit or phase errors on a single qubit.
More generally, the error operatorUdoes not need to be unitary, but can be an Kraus operator from aquantum operationrepresenting a system interacting with its environment.
Several proposals have been made for storing error-correctable quantum information in bosonic modes.[clarification needed]Unlike a two-level system, aquantum harmonic oscillatorhas infinitely many energy levels in a single physical system. Codes for these systems include cat,[6][7][8]Gottesman-Kitaev-Preskill (GKP),[9]and binomial codes.[10][11]One insight offered by these codes is to take advantage of the redundancy within a single system, rather than to duplicate many two-level qubits.
Written in theFockbasis, the simplest binomial encoding is|0L⟩=|0⟩+|4⟩2,|1L⟩=|2⟩,{\displaystyle |0_{\rm {L}}\rangle ={\frac {|0\rangle +|4\rangle }{\sqrt {2}}},\quad |1_{\rm {L}}\rangle =|2\rangle ,}where the subscript L indicates a "logically encoded" state. Then if the dominant error mechanism of the system is the stochastic application of the bosoniclowering operatora^,{\displaystyle {\hat {a}},}the corresponding error states are|3⟩{\displaystyle |3\rangle }and|1⟩,{\displaystyle |1\rangle ,}respectively. Since the codewords involve only even photon number, and the error states involve only odd photon number, errors can be detected by measuring thephoton numberparity of the system.[10][12]Measuring the odd parity will allow correction by application of an appropriate unitary operation without knowledge of the specific logical state of the qubit. However, the particular binomial code above is not robust to two-photon loss.
Schrödinger cat states, superpositions of coherent states, can also be used as logical states for error correction codes. Cat code, realized by Ofek et al.[13]in 2016, defined two sets of logical states:{|0L+⟩,|1L+⟩}{\displaystyle \{|0_{L}^{+}\rangle ,|1_{L}^{+}\rangle \}}and{|0L−⟩,|1L−⟩}{\displaystyle \{|0_{L}^{-}\rangle ,|1_{L}^{-}\rangle \}}, where each of the states is a superposition ofcoherent stateas follows
|0L+⟩≡|α⟩+|−α⟩,|1L+⟩≡|iα⟩+|−iα⟩,|0L−⟩≡|α⟩−|−α⟩,|1L−⟩≡|iα⟩−|−iα⟩.{\displaystyle {\begin{aligned}|0_{L}^{+}\rangle &\equiv |\alpha \rangle +|-\alpha \rangle ,\\|1_{L}^{+}\rangle &\equiv |i\alpha \rangle +|-i\alpha \rangle ,\\|0_{L}^{-}\rangle &\equiv |\alpha \rangle -|-\alpha \rangle ,\\|1_{L}^{-}\rangle &\equiv |i\alpha \rangle -|-i\alpha \rangle .\end{aligned}}}
Those two sets of states differ from the photon number parity, as states denoted with+{\displaystyle ^{+}}only occupy even photon number states and states with−{\displaystyle ^{-}}indicate they have odd parity. Similar to the binomial code, if the dominant error mechanism of the system is the stochastic application of the bosoniclowering operatora^{\displaystyle {\hat {a}}}, the error takes the logical states from the even parity subspace to the odd one, and vice versa. Single-photon-loss errors can therefore be detected by measuring the photon number parity operatorexp(iπa^†a^){\displaystyle \exp(i\pi {\hat {a}}^{\dagger }{\hat {a}})}using a dispersively coupled ancillary qubit.[12]
Still, cat qubits are not protected against two-photon lossa^2{\displaystyle {\hat {a}}^{2}}, dephasing noisea^†a^{\displaystyle {\hat {a}}^{\dagger }{\hat {a}}}, photon-gain errora^†{\displaystyle {\hat {a}}^{\dagger }}, etc.[6][7][8]
In general, aquantum codefor aquantum channelE{\displaystyle {\mathcal {E}}}is a subspaceC⊆H{\displaystyle {\mathcal {C}}\subseteq {\mathcal {H}}}, whereH{\displaystyle {\mathcal {H}}}is the state Hilbert space, such that there exists another quantum channelR{\displaystyle {\mathcal {R}}}with(R∘E)(ρ)=ρ∀ρ=PCρPC,{\displaystyle ({\mathcal {R}}\circ {\mathcal {E}})(\rho )=\rho \quad \forall \rho =P_{\mathcal {C}}\rho P_{\mathcal {C}},}wherePC{\displaystyle P_{\mathcal {C}}}is theorthogonal projectionontoC{\displaystyle {\mathcal {C}}}. HereR{\displaystyle {\mathcal {R}}}is known as thecorrection operation.
Anon-degenerate codeis one for which different elements of the set of correctable errors produce linearly independent results when applied to elements of the code. If distinct of the set of correctable errors produce orthogonal results, the code is consideredpure.[14]
Over time, researchers have come up with several codes:
That these codes allow indeed for quantum computations of arbitrary length is the content of thequantum threshold theorem, found byMichael Ben-OrandDorit Aharonov, which asserts that you can correct for all errors if you concatenate quantum codes such as the CSS codes—i.e. re-encode each logical qubit by the same code again, and so on, on logarithmically many levels—providedthat the error rate of individualquantum gatesis below a certain threshold; as otherwise, the attempts to measure the syndrome and correct the errors would introduce more new errors than they correct for.
As of late 2004, estimates for this threshold indicate that it could be as high as 1–3%,[20]provided that there are sufficiently manyqubitsavailable.
There have been several experimental realizations of CSS-based codes. The first demonstration was withnuclear magnetic resonance qubits.[21]Subsequently, demonstrations have been made with linear optics,[22]trapped ions,[23][24]and superconducting (transmon) qubits.[25]
In 2016 for the first time the lifetime of a quantum bit was prolonged by employing a QEC code.[13]The error-correction demonstration was performed onSchrödinger-cat statesencoded in a superconducting resonator, and employed aquantum controllercapable of performing real-time feedback operations including read-out of the quantum information, its analysis, and the correction of its detected errors. The work demonstrated how the quantum-error-corrected system reaches the break-even point at which the lifetime of a logical qubit exceeds the lifetime of the underlying constituents of the system (the physical qubits).
Other error correcting codes have also been implemented, such as one aimed at correcting for photon loss, the dominant error source in photonic qubit schemes.[26][27]
In 2021, anentangling gatebetween two logical qubits encoded intopological quantum error-correction codeshas first been realized using 10 ions in atrapped-ion quantum computer.[28][29]2021 also saw the first experimental demonstration of fault-tolerant Bacon-Shor code in a single logical qubit of a trapped-ion system, i.e. a demonstration for which the addition of error correction is able to suppress more errors than is introduced by the overhead required to implement the error correction as well as fault tolerant Steane code.[30][31][32]In a different direction, using an encoding corresponding to the Jordan-Wigner mapped Majorana zero modes of a Kitaev chain, researchers were able to perform quantum teleportation of a logical qubit, where an improvement in fidelity from 71% to 85% was observed.[33]
In 2022, researchers at theUniversity of Innsbruckhave demonstrated a fault-tolerant universal set of gates on two logical qubits in a trapped-ion quantum computer. They have performed a logical two-qubit controlled-NOT gate between two instances of the seven-qubit colour code, and fault-tolerantly prepared a logicalmagic state.[34]
In February 2023, researchers at Google claimed to have decreased quantum errors by increasing the qubit number in experiments, they used a fault tolerantsurface codemeasuring an error rate of 3.028% and 2.914% for a distance-3 qubit array and a distance-5 qubit array respectively.[35][36][37]
In April 2024, researchers atMicrosoftclaimed to have successfully tested a quantum error correction code that allowed them to achieve an error rate with logical qubits that is 800 times better than the underlying physical error rate.[38]
This qubit virtualization system was used to create 4 logical qubits with 30 of the 32 qubits on Quantinuum's trapped-ion hardware. The system uses an active syndrome extraction technique to diagnose errors and correct them while calculations are underway without destroying the logical qubits.[39]
In January 2025, researchers atUNSW Sydneymanaged to develop an error correction method usingantimony-based materials, includingantimonides, leveraging high-dimensional quantum states (qudits) with up to eight states. By encoding quantum information in the nuclear spin of aphosphorusatom embedded insiliconand employing advanced pulse control techniques, they demonstrated enhanced error resilience.[40]
In 2022, research at University of Engineering and Technology Lahore demonstrated error cancellation by inserting single-qubit Z-axis rotation gates into strategically chosen locations of the superconductor quantum circuits.[41]The scheme has been shown to effectively correct errors that would otherwise rapidly add up under constructive interference of coherent noise. This is a circuit-level calibration scheme that traces deviations (e.g. sharp dips or notches) in the decoherence curve to detect and localize the coherent error, but does not require encoding or parity measurements.[42]However, further investigation is needed to establish the effectiveness of this method for the incoherent noise.[41] | https://en.wikipedia.org/wiki/Quantum_error_correction |
Ininformation theory, asoft-decision decoderis a kind ofdecoding method– a class ofalgorithmused to decode data that has been encoded with anerror correcting code. Whereas ahard-decision decoderoperates on data that take on a fixed set of possible values (typically 0 or 1 in a binary code), the inputs to a soft-decision decoder may take on a whole range of values in-between. This extra information indicates the reliability of each input data point, and is used to form better estimates of the original data. Therefore, a soft-decision decoder will typically perform better in the presence of corrupted data than its hard-decision counterpart.[1]
Soft-decision decoders are often used inViterbi decodersandturbo codedecoders.
This article related totelecommunicationsis astub. You can help Wikipedia byexpanding it. | https://en.wikipedia.org/wiki/Soft-decision_decoder |
Gödel's incompleteness theoremsare twotheoremsofmathematical logicthat are concerned with the limits of provability in formal axiomatic theories. These results, published byKurt Gödelin 1931, are important both in mathematical logic and in thephilosophy of mathematics. The theorems are widely, but not universally, interpreted as showing thatHilbert's programto find a complete and consistent set ofaxiomsfor allmathematicsis impossible.[1][additional citation(s) needed]
The first incompleteness theorem states that noconsistent systemofaxiomswhose theorems can be listed by aneffective procedure(i.e. analgorithm) is capable ofprovingall truths about the arithmetic ofnatural numbers. For any such consistent formal system, there will always be statements about natural numbers that are true, but that are unprovable within the system.
The second incompleteness theorem, an extension of the first, shows that the system cannot demonstrate its own consistency.
Employing adiagonal argument, Gödel's incompleteness theorems were among the first of several closely related theorems on the limitations of formal systems. They were followed byTarski's undefinability theoremon the formal undefinability of truth,Church's proof that Hilbert'sEntscheidungsproblemis unsolvable, andTuring's theorem that there is no algorithm to solve thehalting problem.
The incompleteness theorems apply toformal systemsthat are of sufficient complexity to express the basic arithmetic of the natural numbers and which are consistent and effectively axiomatized. Particularly in the context offirst-order logic, formal systems are also calledformal theories. In general, a formal system is a deductive apparatus that consists of a particular set of axioms along with rules of symbolic manipulation (or rules of inference) that allow for the derivation of new theorems from the axioms. One example of such a system is first-orderPeano arithmetic, a system in which all variables are intended to denote natural numbers. In other systems, such asset theory, only some sentences of the formal system express statements about the natural numbers. The incompleteness theorems are about formal provabilitywithinthese systems, rather than about "provability" in an informal sense.
There are several properties that a formal system may have, including completeness, consistency, and the existence of an effective axiomatization. The incompleteness theorems show that systems which contain a sufficient amount of arithmetic cannot possess all three of these properties.
A formal system is said to beeffectively axiomatized(also calledeffectively generated) if its set of theorems isrecursively enumerable. This means that there is a computer program that, in principle, could enumerate all the theorems of the system without listing any statements that are not theorems. Examples of effectively generated theories include Peano arithmetic andZermelo–Fraenkel set theory(ZFC).[2]
The theory known astrue arithmeticconsists of all true statements about the standard integers in the language of Peano arithmetic. This theory is consistent and complete, and contains a sufficient amount of arithmetic. However, it does not have a recursively enumerable set of axioms, and thus does not satisfy the hypotheses of the incompleteness theorems.
A set of axioms is (syntactically, ornegation-)completeif, for any statement in the axioms' language, that statement or its negation is provable from the axioms.[3]This is the notion relevant for Gödel's first Incompleteness theorem. It is not to be confused withsemanticcompleteness, which means that the set of axioms proves all the semantic tautologies of the given language. In hiscompleteness theorem(not to be confused with the incompleteness theorems described here), Gödel proved that first-order logic issemanticallycomplete. But it is not syntactically complete, since there are sentences expressible in the language of first-order logic that can be neither proved nor disproved from the axioms of logic alone.
In a system of mathematics, thinkers such as Hilbert believed that it was just a matter of time to find such an axiomatization that would allow one to either prove or disprove (by proving its negation) every mathematical formula.
A formal system might be syntactically incomplete by design, as logics generally are. Or it may be incomplete simply because not all the necessary axioms have been discovered or included. For example,Euclidean geometrywithout theparallel postulateis incomplete, because some statements in the language (such as the parallel postulate itself) can not be proved from the remaining axioms. Similarly, the theory ofdense linear ordersis not complete, but becomes complete with an extra axiom stating that there are no endpoints in the order. Thecontinuum hypothesisis a statement in the language ofZFCthat is not provable within ZFC, so ZFC is not complete. In this case, there is no obvious candidate for a new axiom that resolves the issue.
The theory of first-orderPeano arithmeticseems consistent. Assuming this is indeed the case, note that it has an infinite but recursively enumerable set of axioms, and can encode enough arithmetic for the hypotheses of the incompleteness theorem. Thus by the first incompleteness theorem, Peano Arithmetic is not complete. The theorem gives an explicit example of a statement of arithmetic that is neither provable nor disprovable in Peano's arithmetic. Moreover, this statement is true in the usualmodel. In addition, no effectively axiomatized, consistent extension of Peano arithmetic can be complete.
A set of axioms is (simply)consistentif there is no statement such that both the statement and its negation are provable from the axioms, andinconsistentotherwise. That is to say, a consistent axiomatic system is one that is free from contradiction.
Peano arithmetic is provably consistent from ZFC, but not from within itself. Similarly, ZFC is not provably consistent from within itself, but ZFC + "there exists aninaccessible cardinal" proves ZFC is consistent because ifκis the least such cardinal, thenVκsitting inside thevon Neumann universeis amodelof ZFC, and a theory is consistent if and only if it has a model.
If one takes all statements in the language ofPeano arithmeticas axioms, then this theory is complete, has a recursively enumerable set of axioms, and can describe addition and multiplication. However, it is not consistent.
Additional examples of inconsistent theories arise from theparadoxesthat result when theaxiom schema of unrestricted comprehensionis assumed in set theory.
The incompleteness theorems apply only to formal systems which are able to prove a sufficient collection of facts about the natural numbers. One sufficient collection is the set of theorems ofRobinson arithmeticQ. Some systems, such as Peano arithmetic, can directly express statements about natural numbers. Others, such as ZFC set theory, are able to interpret statements about natural numbers into their language. Either of these options is appropriate for the incompleteness theorems.
The theory ofalgebraically closed fieldsof a givencharacteristicis complete, consistent, and has an infinite but recursively enumerable set of axioms. However it is not possible to encode the integers into this theory, and the theory cannot describe arithmetic of integers. A similar example is the theory ofreal closed fields, which is essentially equivalent toTarski's axiomsforEuclidean geometry. So Euclidean geometry itself (in Tarski's formulation) is an example of a complete, consistent, effectively axiomatized theory.
The system ofPresburger arithmeticconsists of a set of axioms for the natural numbers with just the addition operation (multiplication is omitted). Presburger arithmetic is complete, consistent, and recursively enumerable and can encode addition but not multiplication of natural numbers, showing that for Gödel's theorems one needs the theory to encode not just addition but also multiplication.
Dan Willard(2001) has studied some weak families of arithmetic systems which allow enough arithmetic as relations to formalise Gödel numbering, but which are not strong enough to have multiplication as a function, and so fail to prove the second incompleteness theorem; that is to say, these systems are consistent and capable of proving their own consistency (seeself-verifying theories).
In choosing a set of axioms, one goal is to be able to prove as many correct results as possible, without proving any incorrect results. For example, we could imagine a set of true axioms which allow us to prove every true arithmetical claim about the natural numbers (Smith 2007, p. 2). In the standard system of first-order logic, an inconsistent set of axioms will prove every statement in its language (this is sometimes called theprinciple of explosion), and is thus automatically complete. A set of axioms that is both complete and consistent, however, proves amaximal setof non-contradictorytheorems.[citation needed]
The pattern illustrated in the previous sections with Peano arithmetic, ZFC, and ZFC + "there exists an inaccessible cardinal" cannot generally be broken. Here ZFC + "there exists an inaccessible cardinal" cannot from itself, be proved consistent. It is also not complete, as illustrated by the continuum hypothesis, which is unresolvable[4]in ZFC + "there exists an inaccessible cardinal".
The first incompleteness theorem shows that, in formal systems that can express basic arithmetic, a complete and consistent finite list of axioms can never be created: each time an additional, consistent statement is added as an axiom, there are other true statements that still cannot be proved, even with the new axiom. If an axiom is ever added that makes the system complete, it does so at the cost of making the system inconsistent. It is not even possible for an infinite list of axioms to be complete, consistent, and effectively axiomatized.
Gödel's first incompleteness theoremfirst appeared as "Theorem VI" in Gödel's 1931 paper "On Formally Undecidable Propositions of Principia Mathematica and Related SystemsI". The hypotheses of the theorem were improved shortly thereafter by J. Barkley Rosser (1936) usingRosser's trick. The resulting theorem (incorporating Rosser's improvement) may be paraphrased in English as follows, where "formal system" includes the assumption that the system is effectively generated.
First Incompleteness Theorem: "Any consistent formal systemFwithin which a certain amount of elementary arithmetic can be carried out is incomplete; i.e. there are statements of the language ofFwhich can neither be proved nor disproved inF." (Raatikainen 2020)
The unprovable statementGFreferred to by the theorem is often referred to as "the Gödel sentence" for the systemF. The proof constructs a particular Gödel sentence for the systemF, but there are infinitely many statements in the language of the system that share the same properties, such as the conjunction of the Gödel sentence and anylogically validsentence.
Each effectively generated system has its own Gödel sentence. It is possible to define a larger systemF'that contains the whole ofFplusGFas an additional axiom. This will not result in a complete system, because Gödel's theorem will also apply toF', and thusF'also cannot be complete. In this case,GFis indeed a theorem inF', because it is an axiom. BecauseGFstates only that it is not provable inF, no contradiction is presented by its provability withinF'. However, because the incompleteness theorem applies toF', there will be a new Gödel statementGF'forF', showing thatF'is also incomplete.GF'will differ fromGFin thatGF'will refer toF', rather thanF.
The Gödel sentence is designed to refer, indirectly, to itself. The sentence states that, when a particular sequence of steps is used to construct another sentence, that constructed sentence will not be provable inF. However, the sequence of steps is such that the constructed sentence turns out to beGFitself. In this way, the Gödel sentenceGFindirectly states its own unprovability withinF.[5]
To prove the first incompleteness theorem, Gödel demonstrated that the notion of provability within a system could be expressed purely in terms of arithmetical functions that operate onGödel numbersof sentences of the system. Therefore, the system, which can prove certain facts about numbers, can also indirectly prove facts about its own statements, provided that it is effectively generated. Questions about the provability of statements within the system are represented as questions about the arithmetical properties of numbers themselves, which would be decidable by the system if it were complete.
Thus, although the Gödel sentence refers indirectly to sentences of the systemF, when read as an arithmetical statement the Gödel sentence directly refers only to natural numbers. It asserts that no natural number has a particular property, where that property is given by aprimitive recursiverelation (Smith 2007, p. 141). As such, the Gödel sentence can be written in the language of arithmetic with a simple syntactic form. In particular, it can be expressed as a formula in the language of arithmetic consisting of a number of leading universal quantifiers followed by a quantifier-free body (these formulas are at levelΠ10{\displaystyle \Pi _{1}^{0}}of thearithmetical hierarchy). Via theMRDP theorem, the Gödel sentence can be re-written as a statement that a particular polynomial in many variables with integer coefficients never takes the value zero when integers are substituted for its variables (Franzén 2005, p. 71).
The first incompleteness theorem shows that the Gödel sentenceGFof an appropriate formal theoryFis unprovable inF. Because, when interpreted as a statement about arithmetic, this unprovability is exactly what the sentence (indirectly) asserts, the Gödel sentence is, in fact, true (Smoryński 1977, p. 825; also seeFranzén 2005, pp. 28–33). For this reason, the sentenceGFis often said to be "true but unprovable." (Raatikainen 2020). However, since the Gödel sentence cannot itself formally specify its intended interpretation, the truth of the sentenceGFmay only be arrived at via a meta-analysis from outside the system. In general, this meta-analysis can be carried out within the weak formal system known asprimitive recursive arithmetic, which proves the implicationCon(F)→GF, whereCon(F)is a canonical sentence asserting the consistency ofF(Smoryński 1977, p. 840,Kikuchi & Tanaka 1994, p. 403).
Although the Gödel sentence of a consistent theory is true as a statement about theintended interpretationof arithmetic, the Gödel sentence will be false in somenonstandard models of arithmetic, as a consequence of Gödel'scompleteness theorem(Franzén 2005, p. 135). That theorem shows that, when a sentence is independent of a theory, the theory will have models in which the sentence is true and models in which the sentence is false. As described earlier, the Gödel sentence of a systemFis an arithmetical statement which claims that no number exists with a particular property. The incompleteness theorem shows that this claim will be independent of the systemF, and the truth of the Gödel sentence follows from the fact that no standard natural number has the property in question. Any model in which the Gödel sentence is false must contain some element which satisfies the property within that model. Such a model must be "nonstandard" – it must contain elements that do not correspond to any standard natural number (Raatikainen 2020,Franzén 2005, p. 135).
Gödel specifically citesRichard's paradoxand theliar paradoxas semantical analogues to his syntactical incompleteness result in the introductory section of "On Formally Undecidable Propositions in Principia Mathematica and Related Systems I". Theliar paradoxis the sentence "This sentence is false." An analysis of the liar sentence shows that it cannot be true (for then, as it asserts, it is false), nor can it be false (for then, it is true). A Gödel sentenceGfor a systemFmakes a similar assertion to the liar sentence, but with truth replaced by provability:Gsays "Gis not provable in the systemF." The analysis of the truth and provability ofGis a formalized version of the analysis of the truth of the liar sentence.
It is not possible to replace "not provable" with "false" in a Gödel sentence because the predicate "Qis theGödel numberof a false formula" cannot be represented as a formula of arithmetic. This result, known asTarski's undefinability theorem, was discovered independently both by Gödel, when he was working on the proof of the incompleteness theorem, and by the theorem's namesake,Alfred Tarski.
Compared to the theorems stated in Gödel's 1931 paper, many contemporary statements of the incompleteness theorems are more general in two ways. These generalized statements are phrased to apply to a broader class of systems, and they are phrased to incorporate weaker consistency assumptions.
Gödel demonstrated the incompleteness of the system ofPrincipia Mathematica, a particular system of arithmetic, but a parallel demonstration could be given for any effective system of a certain expressiveness. Gödel commented on this fact in the introduction to his paper, but restricted the proof to one system for concreteness. In modern statements of the theorem, it is common to state the effectiveness and expressiveness conditions as hypotheses for the incompleteness theorem, so that it is not limited to any particular formal system. The terminology used to state these conditions was not yet developed in 1931 when Gödel published his results.
Gödel's original statement and proof of the incompleteness theorem requires the assumption that the system is not just consistent butω-consistent. A system isω-consistentif it is not ω-inconsistent, and is ω-inconsistent if there is a predicatePsuch that for every specific natural numbermthe system proves~P(m), and yet the system also proves that there exists a natural numbernsuch thatP(n). That is, the system says that a number with propertyPexists while denying that it has any specific value. The ω-consistency of a system implies its consistency, but consistency does not imply ω-consistency.J. Barkley Rosser(1936) strengthened the incompleteness theorem by finding a variation of the proof (Rosser's trick) that only requires the system to be consistent, rather than ω-consistent. This is mostly of technical interest, because all true formal theories of arithmetic (theories whose axioms are all true statements about natural numbers) are ω-consistent, and thus Gödel's theorem as originally stated applies to them. The stronger version of the incompleteness theorem that only assumes consistency, rather than ω-consistency, is now commonly known as Gödel's incompleteness theorem and as the Gödel–Rosser theorem.
For each formal systemFcontaining basic arithmetic, it is possible to canonically define a formula Cons(F) expressing the consistency ofF. This formula expresses the property that "there does not exist a natural number coding a formal derivation within the systemFwhose conclusion is a syntactic contradiction." The syntactic contradiction is often taken to be "0=1", in which case Cons(F) states "there is no natural number that codes a derivation of '0=1' from the axioms ofF."
Gödel's second incompleteness theoremshows that, under general assumptions, this canonical consistency statement Cons(F) will not be provable inF. The theorem first appeared as "Theorem XI" in Gödel's 1931 paper "On Formally Undecidable Propositions in Principia Mathematica and Related Systems I". In the following statement, the term "formalized system" also includes an assumption thatFis effectively axiomatized. This theorem states that for any consistent systemFwithin which a certain amount of elementary arithmetic can be carried out, the consistency ofFcannot be proved inFitself.[6]This theorem is stronger than the first incompleteness theorem because the statement constructed in the first incompleteness theorem does not directly express the consistency of the system. The proof of the second incompleteness theorem is obtained by formalizing the proof of the first incompleteness theorem within the systemFitself.
There is a technical subtlety in the second incompleteness theorem regarding the method of expressing the consistency ofFas a formula in the language ofF. There are many ways to express the consistency of a system, and not all of them lead to the same result. The formula Cons(F) from the second incompleteness theorem is a particular expression of consistency.
Other formalizations of the claim thatFis consistent may be inequivalent inF, and some may even be provable. For example, first-order Peano arithmetic (PA) can prove that "the largest consistentsubsetof PA" is consistent. But, because PA is consistent, the largest consistent subset of PA is just PA, so in this sense PA "proves that it is consistent". What PA does not prove is that the largest consistent subset of PA is, in fact, the whole of PA. (The term "largest consistent subset of PA" is meant here to be the largest consistent initial segment of the axioms of PA under some particular effective enumeration.)
The standard proof of the second incompleteness theorem assumes that the provability predicateProvA(P)satisfies theHilbert–Bernays provability conditions. Letting#(P)represent the Gödel number of a formulaP, the provability conditions say:
There are systems, such as Robinson arithmetic, which are strong enough to meet the assumptions of the first incompleteness theorem, but which do not prove the Hilbert–Bernays conditions. Peano arithmetic, however, is strong enough to verify these conditions, as are all theories stronger than Peano arithmetic.
Gödel's second incompleteness theorem also implies that a systemF1satisfying the technical conditions outlined above cannot prove the consistency of any systemF2that proves the consistency ofF1. This is because such a systemF1can prove that ifF2proves the consistency ofF1, thenF1is in fact consistent. For the claim thatF1is consistent has form "for all numbersn,nhas the decidable property of not being a code for a proof of contradiction inF1". IfF1were in fact inconsistent, thenF2would prove for somenthatnis the code of a contradiction inF1. But ifF2also proved thatF1is consistent (that is, that there is no suchn), then it would itself be inconsistent. This reasoning can be formalized inF1to show that ifF2is consistent, thenF1is consistent. Since, by second incompleteness theorem,F1does not prove its consistency, it cannot prove the consistency ofF2either.
This corollary of the second incompleteness theorem shows that there is no hope of proving, for example, the consistency of Peano arithmetic using any finitistic means that can be formalized in a system the consistency of which is provable in Peano arithmetic (PA). For example, the system ofprimitive recursive arithmetic(PRA), which is widely accepted as an accurate formalization of finitistic mathematics, is provably consistent in PA. Thus PRA cannot prove the consistency of PA. This fact is generally seen to imply thatHilbert's program, which aimed to justify the use of "ideal" (infinitistic) mathematical principles in the proofs of "real" (finitistic) mathematical statements by giving a finitistic proof that the ideal principles are consistent, cannot be carried out.[7]
The corollary also indicates the epistemological relevance of the second incompleteness theorem. It would provide no interesting information if a systemFproved its consistency. This is because inconsistent theories prove everything, including their consistency. Thus a consistency proof ofFinFwould give us no clue as to whetherFis consistent; no doubts about the consistency ofFwould be resolved by such a consistency proof. The interest in consistency proofs lies in the possibility of proving the consistency of a systemFin some systemF'that is in some sense less doubtful thanFitself, for example, weaker thanF. For many naturally occurring theoriesFandF', such asF= Zermelo–Fraenkel set theory andF'= primitive recursive arithmetic, the consistency ofF'is provable inF, and thusF'cannot prove the consistency ofFby the above corollary of the second incompleteness theorem.
The second incompleteness theorem does not rule out altogether the possibility of proving the consistency of a different system with different axioms. For example,Gerhard Gentzenproved the consistency of Peano arithmetic in a different system that includes an axiom asserting that theordinalcalledε0iswellfounded; seeGentzen's consistency proof. Gentzen's theorem spurred the development ofordinal analysisin proof theory.
There are two distinct senses of the word "undecidable" in mathematics and computer science. The first of these is theproof-theoreticsense used in relation to Gödel's theorems, that of a statement being neither provable nor refutable in a specifieddeductive system. The second sense, which will not be discussed here, is used in relation tocomputability theoryand applies not to statements but todecision problems, which are countably infinite sets of questions each requiring a yes or no answer. Such a problem is said to be undecidable if there is nocomputable functionthat correctly answers every question in the problem set (seeundecidable problem).
Because of the two meanings of the word undecidable, the termindependentis sometimes used instead of undecidable for the "neither provable nor refutable" sense.
Undecidability of a statement in a particular deductive system does not, in and of itself, address the question of whether thetruth valueof the statement is well-defined, or whether it can be determined by other means. Undecidability only implies that the particular deductive system being considered does not prove the truth or falsity of the statement. Whether there exist so-called "absolutely undecidable" statements, whose truth value can never be known or is ill-specified, is a controversial point in thephilosophy of mathematics.
The combined work of Gödel andPaul Cohenhas given two concrete examples of undecidable statements (in the first sense of the term): Thecontinuum hypothesiscan neither be proved nor refuted inZFC(the standard axiomatization ofset theory), and theaxiom of choicecan neither be proved nor refuted in ZF (which is all the ZFC axiomsexceptthe axiom of choice). These results do not require the incompleteness theorem. Gödel proved in 1940 that neither of these statements could be disproved in ZF or ZFC set theory. In the 1960s, Cohen proved that neither is provable from ZF, and the continuum hypothesis cannot be proved from ZFC.
Shelah (1974)showed that theWhitehead problemingroup theoryis undecidable, in the first sense of the term, in standard set theory.[8]
Gregory Chaitinproduced undecidable statements inalgorithmic information theoryand proved another incompleteness theorem in that setting.Chaitin's incompleteness theoremstates that for any system that can represent enough arithmetic, there is an upper boundcsuch that no specific number can be proved in that system to haveKolmogorov complexitygreater thanc. While Gödel's theorem is related to theliar paradox, Chaitin's result is related toBerry's paradox.
These are natural mathematical equivalents of the Gödel "true but undecidable" sentence. They can be proved in a larger system which is generally accepted as a valid form of reasoning, but are undecidable in a more limited system such as Peano Arithmetic.
In 1977,ParisandHarringtonproved that theParis–Harrington principle, a version of the infiniteRamsey theorem, is undecidable in (first-order)Peano arithmetic, but can be proved in the stronger system ofsecond-order arithmetic. Kirby and Paris later showed thatGoodstein's theorem, a statement about sequences of natural numbers somewhat simpler than the Paris–Harrington principle, is also undecidable in Peano arithmetic.
Kruskal's tree theorem, which has applications in computer science, is also undecidable from Peano arithmetic but provable in set theory. In fact Kruskal's tree theorem (or its finite form) is undecidable in a much stronger system ATR0codifying the principles acceptable based on a philosophy of mathematics calledpredicativism.[9]The related but more generalgraph minor theorem(2003) has consequences forcomputational complexity theory.
The incompleteness theorem is closely related to several results aboutundecidable setsinrecursion theory.
Kleene (1943)presented a proof of Gödel's incompleteness theorem using basic results of computability theory. One such result shows that thehalting problemis undecidable: no computer program can correctly determine, given any programPas input, whetherPeventually halts when run with a particular given input. Kleene showed that the existence of a complete effective system of arithmetic with certain consistency properties would force the halting problem to be decidable, a contradiction.[10]This method of proof has also been presented byShoenfield (1967);Charlesworth (1981); andHopcroft & Ullman (1979).[11]
Franzén (2005)explains howMatiyasevich's solutiontoHilbert's 10th problemcan be used to obtain a proof to Gödel's first incompleteness theorem.[12]Matiyasevichproved that there is no algorithm that, given a multivariate polynomialp(x1,x2,...,xk)with integer coefficients, determines whether there is an integer solution to the equationp= 0. Because polynomials with integer coefficients, and integers themselves, are directly expressible in the language of arithmetic, if a multivariate integer polynomial equationp= 0 does have a solution in the integers then any sufficiently strong system of arithmeticTwill prove this. Moreover, suppose the systemTis ω-consistent. In that case, it will never prove that a particular polynomial equation has a solution when there is no solution in the integers. Thus, ifTwere complete and ω-consistent, it would be possible to determine algorithmically whether a polynomial equation has a solution by merely enumerating proofs ofTuntil either "phas a solution" or "phas no solution" is found, in contradiction to Matiyasevich's theorem. Hence it follows thatTcannot be ω-consistent and complete. Moreover, for each consistent effectively generated systemT, it is possible to effectively generate a multivariate polynomialpover the integers such that the equationp= 0 has no solutions over the integers, but the lack of solutions cannot be proved inT.[13]
Smoryński (1977)shows how the existence ofrecursively inseparable setscan be used to prove the first incompleteness theorem. This proof is often extended to show that systems such as Peano arithmetic areessentially undecidable.[14]
Chaitin's incompleteness theoremgives a different method of producing independent sentences, based onKolmogorov complexity. Like the proof presented by Kleene that was mentioned above, Chaitin's theorem only applies to theories with the additional property that all their axioms are true in the standard model of the natural numbers. Gödel's incompleteness theorem is distinguished by its applicability to consistent theories that nonetheless include false statements in the standard model; these theories are known asω-inconsistent.
Theproof by contradictionhas three essential parts. To begin, choose a formal system that meets the proposed criteria:
The main problem in fleshing out the proof described above is that it seems at first that to construct a statementpthat is equivalent to "pcannot be proved",pwould somehow have to contain a reference top, which could easily give rise to an infinite regress. Gödel's technique is to show that statements can be matched with numbers (often called the arithmetization ofsyntax) in such a way that"proving a statement"can be replaced with"testing whether a number has a given property". This allows a self-referential formula to be constructed in a way that avoids any infinite regress of definitions. The same technique was later used byAlan Turingin his work on theEntscheidungsproblem.
In simple terms, a method can be devised so that every formula or statement that can be formulated in the system gets a unique number, called itsGödel number, in such a way that it is possible to mechanically convert back and forth between formulas and Gödel numbers. The numbers involved might be very long indeed (in terms of number of digits), but this is not a barrier; all that matters is that such numbers can be constructed. A simple example is how English can be stored as asequence of numbers for each letterand then combined into a single larger number:
In principle, proving a statement true or false can be shown to be equivalent to proving that the number matching the statement does or does not have a given property. Because the formal system is strong enough to support reasoning aboutnumbers in general, it can support reasoning aboutnumbers that represent formulae and statementsas well. Crucially, because the system can support reasoning aboutproperties of numbers, the results are equivalent to reasoning aboutprovability of their equivalent statements.
Having shown that in principle the system can indirectly make statements about provability, by analyzing properties of those numbers representing statements it is now possible to show how to create a statement that actually does this.
A formulaF(x)that contains exactly one free variablexis called astatement formorclass-sign. As soon asxis replaced by a specific number, the statement form turns into abona fidestatement, and it is then either provable in the system, or not. For certain formulas one can show that for every natural numbern,F(n){\displaystyle F(n)}is true if and only if it can be proved (the precise requirement in the original proof is weaker, but for the proof sketch this will suffice). In particular, this is true for every specific arithmetic operation between a finite number of natural numbers, such as "2×3 = 6".
Statement forms themselves are not statements and therefore cannot be proved or disproved. But every statement formF(x)can be assigned a Gödel number denoted byG(F). The choice of the free variable used in the formF(x) is not relevant to the assignment of the Gödel numberG(F).
The notion of provability itself can also be encoded by Gödel numbers, in the following way: since a proof is a list of statements which obey certain rules, the Gödel number of a proof can be defined. Now, for every statementp, one may ask whether a numberxis the Gödel number of its proof. The relation between the Gödel number ofpandx, the potential Gödel number of its proof, is an arithmetical relation between two numbers. Therefore, there is a statement formBew(y)that uses this arithmetical relation to state that a Gödel number of a proof ofyexists:
The nameBewis short forbeweisbar, the German word for "provable"; this name was originally used by Gödel to denote the provability formula just described. Note that "Bew(y)" is merely an abbreviation that represents a particular, very long, formula in the original language ofT; the string "Bew" itself is not claimed to be part of this language.
An important feature of the formulaBew(y)is that if a statementpis provable in the system thenBew(G(p))is also provable. This is because any proof ofpwould have a corresponding Gödel number, the existence of which causesBew(G(p))to be satisfied.
The next step in the proof is to obtain a statement which, indirectly, asserts its own unprovability. Although Gödel constructed this statement directly, the existence of at least one such statement follows from thediagonal lemma, which says that for any sufficiently strong formal system and any statement formFthere is a statementpsuch that the system proves
By lettingFbe the negation ofBew(x), we obtain the theorem
and thepdefined by this roughly states that its own Gödel number is the Gödel number of an unprovable formula.
The statementpis not literally equal to~Bew(G(p)); rather,pstates that if a certain calculation is performed, the resulting Gödel number will be that of an unprovable statement. But when this calculation is performed, the resulting Gödel number turns out to be the Gödel number ofpitself. This is similar to the following sentence in English:
This sentence does not directly refer to itself, but when the stated transformation is made the original sentence is obtained as a result, and thus this sentence indirectly asserts its own unprovability. The proof of the diagonal lemma employs a similar method.
Now, assume that the axiomatic system isω-consistent, and letpbe the statement obtained in the previous section.
Ifpwere provable, thenBew(G(p))would be provable, as argued above. Butpasserts the negation ofBew(G(p)). Thus the system would be inconsistent, proving both a statement and its negation. This contradiction shows thatpcannot be provable.
If the negation ofpwere provable, thenBew(G(p))would be provable (becausepwas constructed to be equivalent to the negation ofBew(G(p))). However, for each specific numberx,xcannot be the Gödel number of the proof ofp, becausepis not provable (from the previous paragraph). Thus on one hand the system proves there is a number with a certain property (that it is the Gödel number of the proof ofp), but on the other hand, for every specific numberx, we can prove that it does not have this property. This is impossible in an ω-consistent system. Thus the negation ofpis not provable.
Thus the statementpis undecidable in our axiomatic system: it can neither be proved nor disproved within the system.
In fact, to show thatpis not provable only requires the assumption that the system is consistent. The stronger assumption of ω-consistency is required to show that the negation ofpis not provable. Thus, ifpis constructed for a particular system:
If one tries to "add the missing axioms" to avoid the incompleteness of the system, then one has to add eitherpor "notp" as axioms. But then the definition of "being a Gödel number of a proof" of a statement changes. which means that the formulaBew(x)is now different. Thus when we apply the diagonal lemma to this new Bew, we obtain a new statementp, different from the previous one, which will be undecidable in the new system if it is ω-consistent.
Boolos (1989)sketches an alternative proof of the first incompleteness theorem that usesBerry's paradoxrather than theliar paradoxto construct a true but unprovable formula. A similar proof method was independently discovered bySaul Kripke.[15]Boolos's proof proceeds by constructing, for anycomputably enumerablesetSof true sentences of arithmetic, another sentence which is true but not contained inS. This gives the first incompleteness theorem as a corollary. According to Boolos, this proof is interesting because it provides a "different sort of reason" for the incompleteness of effective, consistent theories of arithmetic.[16]
The incompleteness theorems are among a relatively small number of nontrivial theorems that have been transformed into formalized theorems that can be completely verified byproof assistantsoftware. Gödel's original proofs of the incompleteness theorems, like most mathematical proofs, were written innatural languageintended for human readers.
Computer-verified proofs of versions of the first incompleteness theorem were announced byNatarajan Shankarin 1986 usingNqthm(Shankar 1994), by Russell O'Connor in 2003 usingRocq(previously known asCoq) (O'Connor 2005) and by John Harrison in 2009 usingHOL Light(Harrison 2009). A computer-verified proof of both incompleteness theorems was announced byLawrence Paulsonin 2013 usingIsabelle(Paulson 2014).
The main difficulty in proving the second incompleteness theorem is to show that various facts about provability used in the proof of the first incompleteness theorem can be formalized within a systemSusing a formal predicatePfor provability. Once this is done, the second incompleteness theorem follows by formalizing the entire proof of the first incompleteness theorem within the systemSitself.
Letpstand for the undecidable sentence constructed above, and assume for purposes of obtaining a contradiction that the consistency of the systemScan be proved from within the systemSitself. This is equivalent to proving the statement "SystemSis consistent".
Now consider the statementc, wherec= "If the systemSis consistent, thenpis not provable". The proof of sentenceccan be formalized within the systemS, and therefore the statementc, "pis not provable", (or identically, "notP(p)") can be proved in the systemS.
Observe then, that if we can prove that the systemSis consistent (ie. the statement in the hypothesis ofc), then we have proved thatpis not provable. But this is a contradiction since by the 1st Incompleteness Theorem, this sentence (ie. what is implied in the sentencec, ""p" is not provable") is what we construct to be unprovable. Notice that this is why we require formalizing the first Incompleteness Theorem inS: to prove the 2nd Incompleteness Theorem, we obtain a contradiction with the 1st Incompleteness Theorem which can do only by showing that the theorem holds inS. So we cannot prove that the systemSis consistent. And the 2nd Incompleteness Theorem statement follows.
The incompleteness results affect thephilosophy of mathematics, particularly versions offormalism, which use a single system of formal logic to define their principles.
The incompleteness theorem is sometimes thought to have severe consequences for the program oflogicismproposed byGottlob FregeandBertrand Russell, which aimed to define the natural numbers in terms of logic.[17]Bob HaleandCrispin Wrightargue that it is not a problem for logicism because the incompleteness theorems apply equally to first-order logic as they do to arithmetic. They argue that only those who believe that the natural numbers are to be defined in terms of first order logic have this problem.
Many logicians believe that Gödel's incompleteness theorems struck a fatal blow toDavid Hilbert'ssecond problem, which asked for a finitary consistency proof for mathematics. The second incompleteness theorem, in particular, is often viewed as making the problem impossible. Not all mathematicians agree with this analysis, however, and the status of Hilbert's second problem is not yet decided (see "Modern viewpoints on the status of the problem").
Authors including the philosopherJ. R. Lucasand physicistRoger Penrosehave debated what, if anything, Gödel's incompleteness theorems imply about human intelligence. Much of the debate centers on whether the human mind is equivalent to aTuring machine, or by theChurch–Turing thesis, any finite machine at all. If it is, and if the machine is consistent, then Gödel's incompleteness theorems would apply to it.
Putnam (1960)suggested that while Gödel's theorems cannot be applied to humans, since they make mistakes and are therefore inconsistent, it may be applied to the human faculty of science or mathematics in general. Assuming that it is consistent, either its consistency cannot be proved or it cannot be represented by a Turing machine.[18]
Wigderson (2010)has proposed that the concept of mathematical "knowability" should be based oncomputational complexityrather than logical decidability. He writes that "whenknowabilityis interpreted by modern standards, namely via computational complexity, the Gödel phenomena are very much with us."[19]
Douglas Hofstadter, in his booksGödel, Escher, BachandI Am a Strange Loop, cites Gödel's theorems as an example of what he calls astrange loop, a hierarchical, self-referential structure existing within an axiomatic formal system. He argues that this is the same kind of structure that gives rise to consciousness, the sense of "I", in the human mind. While the self-reference in Gödel's theorem comes from the Gödel sentence asserting its unprovability within the formal system of Principia Mathematica, the self-reference in the human mind comes from how the brain abstracts and categorises stimuli into "symbols", or groups of neurons which respond to concepts, in what is effectively also a formal system, eventually giving rise to symbols modeling the concept of the very entity doing the perception. Hofstadter argues that a strange loop in a sufficiently complex formal system can give rise to a "downward" or "upside-down" causality, a situation in which the normal hierarchy of cause-and-effect is flipped upside-down. In the case of Gödel's theorem, this manifests, in short, as the following:
Merely from knowing the formula's meaning, one can infer its truth or falsity without any effort to derive it in the old-fashioned way, which requires one to trudge methodically "upwards" from the axioms. This is not just peculiar; it is astonishing. Normally, one cannot merely look at what a mathematical conjecture says and simply appeal to the content of that statement on its own to deduce whether the statement is true or false.[20]
In the case of the mind, a far more complex formal system, this "downward causality" manifests, in Hofstadter's view, as the ineffable human instinct that the causality of our minds lies on the high level of desires, concepts, personalities, thoughts, and ideas, rather than on the low level of interactions between neurons or even fundamental particles, even though according to physics the latter seems to possess the causal power.
There is thus a curious upside-downness to our normal human way of perceiving the world: we are built to perceive “big stuff” rather than “small stuff”, even though the domain of the tiny seems to be where the actual motors driving reality reside.[20]
Although Gödel's theorems are usually studied in the context of classical logic, they also have a role in the study ofparaconsistent logicand of inherently contradictory statements (dialetheia). Priest (1984,2006) argues that replacing the notion of formal proof in Gödel's theorem with the usual notion of informal proof can be used to show that naive mathematics is inconsistent, and uses this as evidence fordialetheism.[21]The cause of this inconsistency is the inclusion of a truth predicate for a system within the language of the system.[22]Shapiro (2002)gives a more mixed appraisal of the applications of Gödel's theorems to dialetheism.[23]
Appeals and analogies are sometimes made to the incompleteness of theorems in support of arguments that go beyond mathematics and logic. Several authors have commented negatively on such extensions and interpretations, includingFranzén (2005),Raatikainen (2005),Sokal & Bricmont (1999); andStangroom & Benson (2006).[24]Sokal & Bricmont (1999)andStangroom & Benson (2006), for example, quote fromRebecca Goldstein's comments on the disparity between Gödel's avowedPlatonismand theanti-realistuses to which his ideas are sometimes put.Sokal & Bricmont (1999)criticizeRégis Debray's invocation of the theorem in the context of sociology; Debray has defended this use as metaphorical (ibid.).[25]
After Gödel published his proof of thecompleteness theoremas his doctoral thesis in 1929, he turned to a second problem for hishabilitation. His original goal was to obtain a positive solution toHilbert's second problem.[26]At the time, theories of natural numbers and real numbers similar tosecond-order arithmeticwere known as "analysis", while theories of natural numbers alone were known as "arithmetic".
Gödel was not the only person working on the consistency problem.Ackermannhad published a flawed consistency proof for analysis in 1925, in which he attempted to use the method ofε-substitutionoriginally developed by Hilbert. Later that year,von Neumannwas able to correct the proof for a system of arithmetic without any axioms of induction. By 1928, Ackermann had communicated a modified proof to Bernays; this modified proof led Hilbert to announce his belief in 1929 that the consistency of arithmetic had been demonstrated and that a consistent proof of analysis would likely soon follow. After the publication of the incompleteness theorems showed that Ackermann's modified proof must be erroneous, von Neumann produced a concrete example showing that its main technique was unsound.[27]
In the course of his research, Gödel discovered that although a sentence, asserting its falsehood leads to paradox, a sentence that asserts its non-provability does not. In particular, Gödel was aware of the result now calledTarski's indefinability theorem, although he never published it. Gödel announced his first incompleteness theorem to Carnap, Feigel, and Waismann on August 26, 1930; all four would attend theSecond Conference on the Epistemology of the Exact Sciences, a key conference inKönigsbergthe following week.
The 1930Königsberg conferencewas a joint meeting of three academic societies, with many of the key logicians of the time in attendance. Carnap, Heyting, and von Neumann delivered one-hour addresses on the mathematical philosophies of logicism, intuitionism, and formalism, respectively.[28]The conference also included Hilbert's retirement address, as he was leaving his position at the University of Göttingen. Hilbert used the speech to argue his belief that all mathematical problems can be solved. He ended his address by saying,
For the mathematician there is noIgnorabimus, and, in my opinion, not at all for natural science either. ... The true reason why [no one] has succeeded in finding an unsolvable problem is, in my opinion, that there is no unsolvable problem. In contrast to the foolishIgnorabimus, our credo avers: We must know. We shall know!
This speech quickly became known as a summary of Hilbert's beliefs on mathematics (its final six words, "Wir müssen wissen. Wir werden wissen!", were used as Hilbert's epitaph in 1943). Although Gödel was likely in attendance for Hilbert's address, the two never met face to face.[29]
Gödel announced his first incompleteness theorem at a roundtable discussion session on the third day of the conference. The announcement drew little attention apart from that of von Neumann, who pulled Gödel aside for a conversation. Later that year, working independently with knowledge of the first incompleteness theorem, von Neumann obtained a proof of the second incompleteness theorem, which he announced to Gödel in a letter dated November 20, 1930.[30]Gödel had independently obtained the second incompleteness theorem and included it in his submitted manuscript, which was received byMonatshefte für Mathematikon November 17, 1930.
Gödel's paper was published in theMonatsheftein 1931 under the title "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I" ("On Formally Undecidable Propositions in Principia Mathematica and Related Systems I"). As the title implies, Gödel originally planned to publish a second part of the paper in the next volume of theMonatshefte; the prompt acceptance of the first paper was one reason he changed his plans.[31]
Gödel gave a series of lectures on his theorems at Princeton in 1933–1934 to an audience that included Church, Kleene, and Rosser. By this time, Gödel had grasped that the key property his theorems required is that the system must be effective (at the time, the term "general recursive" was used). Rosser proved in 1936 that the hypothesis of ω-consistency, which was an integral part of Gödel's original proof, could be replaced by simple consistency if the Gödel sentence was changed appropriately. These developments left the incompleteness theorems in essentially their modern form.
Gentzen published hisconsistency prooffor first-order arithmetic in 1936. Hilbert accepted this proof as "finitary" although (as Gödel's theorem had already shown) it cannot be formalized within the system of arithmetic that is being proved consistent.
The impact of the incompleteness theorems on Hilbert's program was quickly realized. Bernays included a full proof of the incompleteness theorems in the second volume ofGrundlagen der Mathematik(1939), along with additional results of Ackermann on the ε-substitution method and Gentzen's consistency proof of arithmetic. This was the first full published proof of the second incompleteness theorem.
Finsler (1926)used a version ofRichard's paradoxto construct an expression that was false but unprovable in a particular, informal framework he had developed.[32]Gödel was unaware of this paper when he proved the incompleteness theorems (Collected Works Vol. IV., p. 9). Finsler wrote to Gödel in 1931 to inform him about this paper, which Finsler felt had priority for an incompleteness theorem. Finsler's methods did not rely on formalized provability and had only a superficial resemblance to Gödel's work.[33]Gödel read the paper but found it deeply flawed, and his response to Finsler laid out concerns about the lack of formalization.[34]Finsler continued to argue for his philosophy of mathematics, which eschewed formalization, for the remainder of his career.
In September 1931,Ernst Zermelowrote to Gödel to announce what he described as an "essential gap" in Gödel's argument.[35]In October, Gödel replied with a 10-page letter, where he pointed out that Zermelo mistakenly assumed that the notion of truth in a system is definable in that system; it is not true in general byTarski's undefinability theorem.[36]However, Zermelo did not relent and published his criticisms in print with "a rather scathing paragraph on his young competitor".[37]Gödel decided that pursuing the matter further was pointless, and Carnap agreed.[38]Much of Zermelo's subsequent work was related to logic stronger than first-order logic, with which he hoped to show both the consistency and categoricity of mathematical theories.
Ludwig Wittgensteinwrote several passages about the incompleteness theorems that were published posthumously in his 1953Remarks on the Foundations of Mathematics, particularly, one section sometimes called the "notorious paragraph" where he seems to confuse the notions of "true" and "provable" in Russell's system. Gödel was a member of theVienna Circleduring the period in which Wittgenstein's earlyideal language philosophyandTractatus Logico-Philosophicusdominated the circle's thinking. There has been some controversy about whether Wittgenstein misunderstood the incompleteness theorem or just expressed himself unclearly. Writings in Gödel'sNachlassexpress the belief that Wittgenstein misread his ideas.
Multiple commentators have read Wittgenstein as misunderstandingGödel, althoughFloyd & Putnam (2000)as well asPriest (2004)have provided textual readings arguing that most commentary misunderstands Wittgenstein.[39]On their release, Bernays, Dummett, and Kreisel wrote separate reviews on Wittgenstein's remarks, all of which were extremely negative.[40]The unanimity of this criticism caused Wittgenstein's remarks on the incompleteness theorems to have little impact on the logic community. In 1972, Gödel stated: "Has Wittgenstein lost his mind? Does he mean it seriously? He intentionally utters trivially nonsensical statements", and wrote toKarl Mengerthat Wittgenstein's comments demonstrate a misunderstanding of the incompleteness theorems writing:
It is clear from the passages you cite that Wittgenstein didnotunderstand [the first incompleteness theorem] (or pretended not to understand it). He interpreted it as a kind of logical paradox, while in fact is just the opposite, namely a mathematical theorem within an absolutely uncontroversial part of mathematics (finitary number theory or combinatorics).[41]
Since the publication of Wittgenstein'sNachlassin 2000, a series of papers in philosophy have sought to evaluate whether the original criticism of Wittgenstein's remarks was justified.Floyd & Putnam (2000)argue that Wittgenstein had a more complete understanding of the incompleteness theorem than was previously assumed. They are particularly concerned with the interpretation of a Gödel sentence for an ω-inconsistent system as saying "I am not provable", since the system has no models in which the provability predicate corresponds to actual provability.Rodych (2003)argues that their interpretation of Wittgenstein is not historically justified.Berto (2009)explores the relationship between Wittgenstein's writing and theories of paraconsistent logic.[42]
None of the following agree in all translated words and in typography. The typography is a serious matter, because Gödel expressly wished to emphasize "those metamathematical notions that had been defined in their usual sense before . . ." (van Heijenoort 1967, p. 595). Three translations exist. Of the first John Dawson states that: "The Meltzer translation was seriously deficient and received a devastating review in theJournal of Symbolic Logic; "Gödel also complained about Braithwaite's commentary (Dawson 1997, p. 216). "Fortunately, the Meltzer translation was soon supplanted by a better one prepared by Elliott Mendelson for Martin Davis's anthologyThe Undecidable. . . he found the translation "not quite so good" as he had expected . . . [but because of time constraints he] agreed to its publication" (ibid). (In a footnote Dawson states that "he would regret his compliance, for the published volume was marred throughout by sloppy typography and numerous misprints" (ibid)). Dawson states that "The translation that Gödel favored was that by Jean van Heijenoort" (ibid). For the serious student another version exists as a set of lecture notes recorded by Stephen Kleene and J. B. Rosser "during lectures given by Gödel at to the Institute for Advanced Study during the spring of 1934" (cf commentary byDavis 1965, p. 39 and beginning on p. 41); this version is titled "On Undecidable Propositions of Formal Mathematical Systems". In their order of publication:
` | https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems |
Inalgorithmic information theory(a subfield ofcomputer scienceandmathematics), theKolmogorov complexityof an object, such as a piece of text, is the length of a shortestcomputer program(in a predeterminedprogramming language) that produces the object as output. It is a measure of thecomputationalresources needed to specify the object, and is also known asalgorithmic complexity,Solomonoff–Kolmogorov–Chaitin complexity,program-size complexity,descriptive complexity, oralgorithmic entropy. It is named afterAndrey Kolmogorov, who first published on the subject in 1963[1][2]and is a generalization of classical information theory.
The notion of Kolmogorov complexity can be used to state andprove impossibilityresults akin toCantor's diagonal argument,Gödel's incompleteness theorem, andTuring's halting problem.
In particular, no programPcomputing alower boundfor each text's Kolmogorov complexity can return a value essentially larger thanP's own length (see section§ Chaitin's incompleteness theorem); hence no single program can compute the exact Kolmogorov complexity for infinitely many texts. Kolmogorov complexity is the length of the ultimately compressed version of a file (i.e., anything which can be put in a computer). Formally, it is the length of a shortest program from which the file can be reconstructed. While Kolmogorov complexity is uncomputable, various approaches have been proposed and reviewed.[3]
Consider the following twostringsof 32 lowercase letters and digits:
The first string has a short English-language description, namely "write ab 16 times", which consists of17characters. The second one has no obvious simple description (using the same character set) other than writing down the string itself, i.e., "write 4c1j5b2p0cv4w1x8rx2y39umgw5q85s7" which has38characters. Hence the operation of writing the first string can be said to have "less complexity" than writing the second.
More formally, thecomplexityof a string is the length of the shortest possible description of the string in some fixeduniversaldescription language (the sensitivity of complexity relative to the choice of description language is discussed below). It can be shown that the Kolmogorov complexity of any string cannot be more than a few bytes larger than the length of the string itself. Strings like theababexample above, whose Kolmogorov complexity is small relative to the string's size, are not considered to be complex.
The Kolmogorov complexity can be defined for any mathematical object, but for simplicity the scope of this article is restricted to strings. We must first specify a description language for strings. Such a description language can be based on any computer programming language, such asLisp,Pascal, orJava. IfPis a program which outputs a stringx, thenPis a description ofx. The length of the description is just the length ofPas a character string, multiplied by the number of bits in a character (e.g., 7 forASCII).
We could, alternatively, choose an encoding forTuring machines, where anencodingis a function which associates to each Turing MachineMa bitstring <M>. IfMis a Turing Machine which, on inputw, outputs stringx, then the concatenated string <M>wis a description ofx. For theoretical analysis, this approach is more suited for constructing detailed formal proofs and is generally preferred in the research literature. In this article, an informal approach is discussed.
Any stringshas at least one description. For example, the second string above is output by thepseudo-code:
whereas the first string is output by the (much shorter) pseudo-code:
If a descriptiond(s) of a stringsis of minimal length (i.e., using the fewest bits), it is called aminimal descriptionofs, and the length ofd(s) (i.e. the number of bits in the minimal description) is theKolmogorov complexityofs, writtenK(s). Symbolically,
The length of the shortest description will depend on the choice of description language; but the effect of changing languages is bounded (a result called theinvariance theorem).
There are two definitions of Kolmogorov complexity:plainandprefix-free. The plain complexity is the minimal description length of any program, and denotedC(x){\displaystyle C(x)}while the prefix-free complexity is the minimal description length of any program encoded in aprefix-free code, and denotedK(x){\displaystyle K(x)}. The plain complexity is more intuitive, but the prefix-free complexity is easier to study.
By default, all equations hold only up to an additive constant. For example,f(x)=g(x){\displaystyle f(x)=g(x)}really means thatf(x)=g(x)+O(1){\displaystyle f(x)=g(x)+O(1)}, that is,∃c,∀x,|f(x)−g(x)|≤c{\displaystyle \exists c,\forall x,|f(x)-g(x)|\leq c}.
LetU:2∗→2∗{\displaystyle U:2^{*}\to 2^{*}}be a computable function mapping finite binary strings to binary strings. It is a universal function if, and only if, for any computablef:2∗→2∗{\displaystyle f:2^{*}\to 2^{*}}, we can encode the function in a "program"sf{\displaystyle s_{f}}, such that∀x∈2∗,U(sfx)=f(x){\displaystyle \forall x\in 2^{*},U(s_{f}x)=f(x)}. We can think ofU{\displaystyle U}as a program interpreter, which takes in an initial segment describing the program, followed by data that the program should process.
One problem with plain complexity is thatC(xy)≮C(x)+C(y){\displaystyle C(xy)\not <C(x)+C(y)}, because intuitively speaking, there is no general way to tell where to divide an output string just by looking at the concatenated string. We can divide it by specifying the length ofx{\displaystyle x}ory{\displaystyle y}, but that would takeO(min(lnx,lny)){\displaystyle O(\min(\ln x,\ln y))}extra symbols. Indeed, for anyc>0{\displaystyle c>0}there existsx,y{\displaystyle x,y}such thatC(xy)≥C(x)+C(y)+c{\displaystyle C(xy)\geq C(x)+C(y)+c}.[4]
Typically, inequalities with plain complexity have a term likeO(min(lnx,lny)){\displaystyle O(\min(\ln x,\ln y))}on one side, whereas the same inequalities with prefix-free complexity have onlyO(1){\displaystyle O(1)}.
The main problem with plain complexity is that there is something extra sneaked into a program. A program not only represents for something with its code, but also represents its own length. In particular, a programx{\displaystyle x}may represent a binary number up tolog2|x|{\displaystyle \log _{2}|x|}, simply by its own length. Stated in another way, it is as if we are using a termination symbol to denote where a word ends, and so we are not using 2 symbols, but 3. To fix this defect, we introduce the prefix-free Kolmogorov complexity.[5]
A prefix-free code is a subset of2∗{\displaystyle 2^{*}}such that given any two different wordsx,y{\displaystyle x,y}in the set, neither is a prefix of the other. The benefit of a prefix-free code is that we can build a machine that reads words from the code forward in one direction, and as soon as it reads the last symbol of the word, itknowsthat the word is finished, and does not need to backtrack or a termination symbol.
Define aprefix-free Turing machineto be a Turing machine that comes with a prefix-free code, such that the Turing machine can read any string from the code in one direction, and stop reading as soon as it reads the last symbol. Afterwards, it may compute on a work tape and write to a write tape, but it cannot move its read-head anymore.
This gives us the following formal way to describeK.[6]
Note that some universal Turing machines may not be programmable with prefix codes. We must pick only a prefix-free universal Turing machine.
The prefix-free complexity of a stringx{\displaystyle x}is the shortest prefix code that makes the machine outputx{\displaystyle x}:K(x):=min{|c|:c∈S,U(c)=x}{\displaystyle K(x):=\min\{|c|:c\in S,U(c)=x\}}
There are some description languages which are optimal, in the following sense: given any description of an object in a description language, said description may be used in the optimal description language with a constant overhead. The constant depends only on the languages involved, not on the description of the object, nor the object being described.
Here is an example of an optimal description language. A description will have two parts:
In more technical terms, the first part of a description is a computer program (specifically: a compiler for the object's language, written in the description language), with the second part being the input to that computer program which produces the object as output.
The invariance theorem follows:Given any description languageL, the optimal description language is at least as efficient asL, with some constant overhead.
Proof:Any descriptionDinLcan be converted into a description in the optimal language by first describingLas a computer programP(part 1), and then using the original descriptionDas input to that program (part 2). The
total length of this new descriptionD′is (approximately):
The length ofPis a constant that doesn't depend onD. So, there is at most a constant overhead, regardless of the object described. Therefore, the optimal language is universalup tothis additive constant.
Theorem: IfK1andK2are the complexity functions relative toTuring completedescription languagesL1andL2, then there is a constantc– which depends only on the languagesL1andL2chosen – such that
Proof: By symmetry, it suffices to prove that there is some constantcsuch that for all stringss
Now, suppose there is a program in the languageL1which acts as aninterpreterforL2:
wherepis a program inL2. The interpreter is characterized by the following property:
Thus, ifPis a program inL2which is a minimal description ofs, thenInterpretLanguage(P) returns the strings. The length of this description ofsis the sum of
This proves the desired upper bound.
Algorithmic information theoryis the area of computer science that studies Kolmogorov complexity and other complexity measures on strings (or otherdata structures).
The concept and theory of Kolmogorov Complexity is based on a crucial theorem first discovered byRay Solomonoff, who published it in 1960, describing it in "A Preliminary Report on a General Theory of Inductive Inference"[7]as part of his invention ofalgorithmic probability. He gave a more complete description in his 1964 publications, "A Formal Theory of Inductive Inference," Part 1 and Part 2 inInformation and Control.[8][9]
Andrey Kolmogorov laterindependently publishedthis theorem inProblems Inform. Transmission[10]in 1965.Gregory Chaitinalso presents this theorem inJ. ACM– Chaitin's paper was submitted October 1966 and revised in December 1968, and cites both Solomonoff's and Kolmogorov's papers.[11]
The theorem says that, among algorithms that decode strings from their descriptions (codes), there exists an optimal one. This algorithm, for all strings, allows codes as short as allowed by any other algorithm up to an additive constant that depends on the algorithms, but not on the strings themselves. Solomonoff used this algorithm and the code lengths it allows to define a "universal probability" of a string on which inductive inference of the subsequent digits of the string can be based. Kolmogorov used this theorem to define several functions of strings, including complexity, randomness, and information.
When Kolmogorov became aware of Solomonoff's work, he acknowledged Solomonoff's priority.[12]For several years, Solomonoff's work was better known in the Soviet Union than in the Western World. The general consensus in the scientific community, however, was to associate this type of complexity with Kolmogorov, who was concerned with randomness of a sequence, while Algorithmic Probability became associated with Solomonoff, who focused on prediction using his invention of the universal prior probability distribution. The broader area encompassing descriptional complexity and probability is often called Kolmogorov complexity. The computer scientistMing Liconsiders this an example of theMatthew effect: "...to everyone who has, more will be given..."[13]
There are several other variants of Kolmogorov complexity or algorithmic information. The most widely used one is based onself-delimiting programs, and is mainly due toLeonid Levin(1974).
An axiomatic approach to Kolmogorov complexity based onBlum axioms(Blum 1967) was introduced by Mark Burgin in the paper presented for publication by Andrey Kolmogorov.[14]
In the late 1990s and early 2000s, methods developed to approximate Kolmogorov complexity relied on popular compression algorithms like LZW,[3]which made difficult or impossible to provide any estimation to short strings until a method based onAlgorithmic probabilitywas introduced,[15][16]offering the only alternative to compression-based methods.[17]
We writeK(x,y){\displaystyle K(x,y)}to beK((x,y)){\displaystyle K((x,y))}, where(x,y){\displaystyle (x,y)}means some fixed way to code for a tuple of strings x and y.
We omit additive factors ofO(1){\displaystyle O(1)}. This section is based on.[6]
Theorem.K(x)≤C(x)+2log2C(x){\displaystyle K(x)\leq C(x)+2\log _{2}C(x)}
Proof.Take any program for the universal Turing machine used to define plain complexity, and convert it to a prefix-free program by first coding the length of the program in binary, then convert the length to prefix-free coding. For example, suppose the program has length 9, then we can convert it as follows:9↦1001↦11−00−00−11−01{\displaystyle 9\mapsto 1001\mapsto 11-00-00-11-\color {red}{01}}where we double each digit, then add a termination code. The prefix-free universal Turing machine can then read in any program for the other machine as follows:[code for simulating the other machine][coded length of the program][the program]{\displaystyle [{\text{code for simulating the other machine}}][{\text{coded length of the program}}][{\text{the program}}]}The first part programs the machine to simulate the other machine, and is a constant overheadO(1){\displaystyle O(1)}. The second part has length≤2log2C(x)+3{\displaystyle \leq 2\log _{2}C(x)+3}. The third part has lengthC(x){\displaystyle C(x)}.
Theorem: There existsc{\displaystyle c}such that∀x,C(x)≤|x|+c{\displaystyle \forall x,C(x)\leq |x|+c}. More succinctly,C(x)≤|x|{\displaystyle C(x)\leq |x|}. Similarly,K(x)≤|x|+2log2|x|{\displaystyle K(x)\leq |x|+2\log _{2}|x|}, andK(x||x|)≤|x|{\displaystyle K(x||x|)\leq |x|}.[clarification needed]
Proof.For the plain complexity, just write a program that simply copies the input to the output. For the prefix-free complexity, we need to first describe the length of the string, before writing out the string itself.
Theorem. (extra information bounds, subadditivity)
Note that there is no way to compareK(xy){\displaystyle K(xy)}andK(x|y){\displaystyle K(x|y)}orK(x){\displaystyle K(x)}orK(y|x){\displaystyle K(y|x)}orK(y){\displaystyle K(y)}. There are strings such that the whole stringxy{\displaystyle xy}is easy to describe, but its substrings are very hard to describe.
Theorem. (symmetry of information)K(x,y)=K(x|y,K(y))+K(y)=K(y,x){\displaystyle K(x,y)=K(x|y,K(y))+K(y)=K(y,x)}.
Proof.One side is simple. For the other side withK(x,y)≥K(x|y,K(y))+K(y){\displaystyle K(x,y)\geq K(x|y,K(y))+K(y)}, we need to use a counting argument (page 38[18]).
Theorem. (information non-increase)For any computable functionf{\displaystyle f}, we haveK(f(x))≤K(x)+K(f){\displaystyle K(f(x))\leq K(x)+K(f)}.
Proof.Program the Turing machine to read two subsequent programs, one describing the function and one describing the string. Then run both programs on the work tape to producef(x){\displaystyle f(x)}, and write it out.
At first glance it might seem trivial to write a program which can computeK(s) for anys, such as the following:
This program iterates through all possible programs (by iterating through all possible strings and only considering those which are valid programs), starting with the shortest. Each program is executed to find the result produced by that program, comparing it to the inputs. If the result matches then the length of the program is returned.
However this will not work because some of the programsptested will not terminate, e.g. if they contain infinite loops. There is no way to avoid all of these programs by testing them in some way before executing them due to the non-computability of thehalting problem.
What is more, no program at all can compute the functionK, be it ever so sophisticated. This is proven in the following.
Theorem: There exist strings of arbitrarily large Kolmogorov complexity. Formally: for each natural numbern, there is a stringswithK(s) ≥n.[note 1]
Proof:Otherwise all of the infinitely many possible finite strings could be generated by the finitely many[note 2]programs with a complexity belownbits.
Theorem:Kis not acomputable function. In other words, there is no program which takes any stringsas input and produces the integerK(s) as output.
The followingproofby contradictionuses a simplePascal-like language to denote programs; for sake of proof simplicity assume its description (i.e. aninterpreter) to have a length of1400000bits.
Assume for contradiction there is a program
which takes as input a stringsand returnsK(s). All programs are of finite length so, for sake of proof simplicity, assume it to be7000000000bits.
Now, consider the following program of length1288bits:
UsingKolmogorovComplexityas a subroutine, the program tries every string, starting with the shortest, until it returns a string with Kolmogorov complexity at least8000000000bits,[note 3]i.e. a string that cannot be produced by any program shorter than8000000000bits. However, the overall length of the above program that producedsis only7001401288bits,[note 4]which is a contradiction. (If the code ofKolmogorovComplexityis shorter, the contradiction remains. If it is longer, the constant used inGenerateComplexStringcan always be changed appropriately.)[note 5]
The above proof uses a contradiction similar to that of theBerry paradox: "1The2smallest3positive4integer5that6cannot7be8defined9in10fewer11than12twenty13English14words". It is also possible to show the non-computability ofKby reduction from the non-computability of the halting problemH, sinceKandHareTuring-equivalent.[19]
There is a corollary, humorously called the "full employment theorem" in the programming language community, stating that there is no perfect size-optimizing compiler.
The chain rule[20]for Kolmogorov complexity states that there exists a constantcsuch that for allXandY:
It states that the shortest program that reproducesXandYisno morethan a logarithmic term larger than a program to reproduceXand a program to reproduceYgivenX. Using this statement, one can definean analogue of mutual information for Kolmogorov complexity.
It is straightforward to compute upper bounds forK(s) – simplycompressthe stringswith some method, implement the corresponding decompressor in the chosen language, concatenate the decompressor to the compressed string, and measure the length of the resulting string – concretely, the size of aself-extracting archivein the given language.
A stringsis compressible by a numbercif it has a description whose length does not exceed |s| −cbits. This is equivalent to saying thatK(s) ≤ |s| −c. Otherwise,sis incompressible byc. A string incompressible by 1 is said to be simplyincompressible– by thepigeonhole principle, which applies because every compressed string maps to only one uncompressed string,incompressible stringsmust exist, since there are 2nbit strings of lengthn, but only 2n− 1 shorter strings, that is, strings of length less thann, (i.e. with length 0, 1, ...,n− 1).[note 6]
For the same reason, most strings are complex in the sense that they cannot be significantly compressed – theirK(s) is not much smaller than |s|, the length ofsin bits. To make this precise, fix a value ofn. There are 2nbitstrings of lengthn. Theuniformprobabilitydistribution on the space of these bitstrings assigns exactly equal weight 2−nto each string of lengthn.
Theorem: With the uniform probability distribution on the space of bitstrings of lengthn, the probability that a string is incompressible bycis at least1 − 2−c+1+ 2−n.
To prove the theorem, note that the number of descriptions of length not exceedingn−cis given by the geometric series:
There remain at least
bitstrings of lengthnthat are incompressible byc. To determine the probability, divide by 2n.
By the above theorem (§ Compression), most strings are complex in the sense that they cannot be described in any significantly "compressed" way. However, it turns out that the fact that a specific string is complex cannot be formally proven, if the complexity of the string is above a certain threshold. The precise formalization is as follows. First, fix a particularaxiomatic systemSfor thenatural numbers. The axiomatic system has to be powerful enough so that, to certain assertionsAabout complexity of strings, one can associate a formulaFAinS. This association must have the following property:
IfFAis provable from the axioms ofS, then the corresponding assertionAmust be true. This "formalization" can be achieved based on aGödel numbering.
Theorem: There exists a constantL(which only depends onSand on the choice of description language) such that there does not exist a stringsfor which the statement
can be proven withinS.[21][22]
Proof Idea: The proof of this result is modeled on a self-referential construction used inBerry's paradox. We firstly obtain a program which enumerates the proofs withinSand we specify a procedurePwhich takes as an input an integerLand prints the stringsxwhich are within proofs withinSof the statementK(x) ≥L. By then settingLto greater than the length of this procedureP, we have that the required length of a program to printxas stated inK(x) ≥Las being at leastLis then less than the amountLsince the stringxwas printed by the procedureP. This is a contradiction. So it is not possible for the proof systemSto proveK(x) ≥LforLarbitrarily large, in particular, forLlarger than the length of the procedureP, (which is finite).
Proof:
We can find an effective enumeration of all the formal proofs inSby some procedure
which takes as inputnand outputs some proof. This function enumerates all proofs. Some of these are proofs for formulas we do not care about here, since every possible proof in the language ofSis produced for somen. Some of these are complexity formulas of the formK(s) ≥nwheresandnare constants in the language ofS. There is a procedure
which determines whether thenth proof actually proves a complexity formulaK(s) ≥L. The stringss, and the integerLin turn, are computable by procedure:
Consider the following procedure:
Given ann, this procedure tries every proof until it finds a string and a proof in the formal systemSof the formulaK(s) ≥Lfor someL≥n; if no such proof exists, it loops forever.
Finally, consider the program consisting of all these procedure definitions, and a main call:
where the constantn0will be determined later on. The overall program length can be expressed asU+log2(n0), whereUis some constant and log2(n0) represents the length of the integer valuen0, under the reasonable assumption that it is encoded in binary digits. We will choosen0to be greater than the program length, that is, such thatn0>U+log2(n0). This is clearly true forn0sufficiently large, because the left hand side grows linearly inn0whilst the right hand side grows logarithmically inn0up to the fixed constantU.
Then no proof of the form "K(s)≥L" withL≥n0can be obtained inS, as can be seen by anindirect argument:
IfComplexityLowerBoundNthProof(i)could return a value ≥n0, then the loop insideGenerateProvablyComplexStringwould eventually terminate, and that procedure would return a stringssuch that
This is a contradiction,Q.E.D.
As a consequence, the above program, with the chosen value ofn0, must loop forever.
Similar ideas are used to prove the properties ofChaitin's constant.
The minimum message length principle of statistical and inductive inference and machine learning was developed byC.S. Wallaceand D.M. Boulton in 1968. MML isBayesian(i.e. it incorporates prior beliefs) and information-theoretic. It has the desirable properties of statistical invariance (i.e. the inference transforms with a re-parametrisation, such as from polar coordinates to Cartesian coordinates), statistical consistency (i.e. even for very hard problems, MML will converge to any underlying model) and efficiency (i.e. the MML model will converge to any true underlying model about as quickly as is possible). C.S. Wallace and D.L. Dowe (1999) showed a formal connection between MML and algorithmic information theory (or Kolmogorov complexity).[23]
Kolmogorov randomnessdefines a string (usually ofbits) as beingrandomif and only if everycomputer programthat can produce that string is at least as long as the string itself. To make this precise, a universal computer (oruniversal Turing machine) must be specified, so that "program" means a program for this universal machine. A random string in this sense is "incompressible" in that it is impossible to "compress" the string into a program that is shorter than the string itself. For every universal computer, there is at least one algorithmically random string of each length.[24]Whether a particular string is random, however, depends on the specific universal computer that is chosen. This is because a universal computer can have a particular string hard-coded in itself, and a program running on this universal computer can then simply refer to this hard-coded string using a short sequence of bits (i.e. much shorter than the string itself).
This definition can be extended to define a notion of randomness forinfinitesequences from a finite alphabet. Thesealgorithmically random sequencescan be defined in three equivalent ways. One way uses an effective analogue ofmeasure theory; another uses effectivemartingales. The third way defines an infinite sequence to be random if the prefix-free Kolmogorov complexity of its initial segments grows quickly enough — there must be a constantcsuch that the complexity of an initial segment of lengthnis always at leastn−c. This definition, unlike the definition of randomness for a finite string, is not affected by which universal machine is used to define prefix-free Kolmogorov complexity.[25]
For dynamical systems, entropy rate and algorithmic complexity of the trajectories are related by a theorem of Brudno, that the equalityK(x;T)=h(T){\displaystyle K(x;T)=h(T)}holds for almost allx{\displaystyle x}.[26]
It can be shown[27]that for the output ofMarkov information sources, Kolmogorov complexity is related to theentropyof the information source. More precisely, the Kolmogorov complexity of the output of a Markov information source, normalized by the length of the output, converges almost surely (as the length of the output goes to infinity) to theentropyof the source.
Theorem.(Theorem 14.2.5[28]) The conditional Kolmogorov complexity of a binary stringx1:n{\displaystyle x_{1:n}}satisfies1nK(x1:n|n)≤Hb(1n∑ixi)+logn2n+O(1/n){\displaystyle {\frac {1}{n}}K(x_{1:n}|n)\leq H_{b}\left({\frac {1}{n}}\sum _{i}x_{i}\right)+{\frac {\log n}{2n}}+O(1/n)}whereHb{\displaystyle H_{b}}is thebinary entropy function(not to be confused with the entropy rate).
The Kolmogorov complexity function is equivalent to deciding the halting problem.
If we have a halting oracle, then the Kolmogorov complexity of a string can be computed by simply trying every halting program, in lexicographic order, until one of them outputs the string.
The other direction is much more involved.[29][30]It shows that given a Kolmogorov complexity function, we can construct a functionp{\displaystyle p}, such thatp(n)≥BB(n){\displaystyle p(n)\geq BB(n)}for all largen{\displaystyle n}, whereBB{\displaystyle BB}is theBusy Beavershift function (also denoted asS(n){\displaystyle S(n)}). By modifying the function at lower values ofn{\displaystyle n}we get an upper bound onBB{\displaystyle BB}, which solves the halting problem.
Consider this programpK{\textstyle p_{K}}, which takes input asn{\textstyle n}, and usesK{\textstyle K}.
We prove by contradiction thatpK(n)≥BB(n){\textstyle p_{K}(n)\geq BB(n)}for all largen{\textstyle n}.
Letpn{\textstyle p_{n}}be a Busy Beaver of lengthn{\displaystyle n}. Consider this (prefix-free) program, which takes no input:
Let the string output by the program bex{\textstyle x}.
The program has length≤n+2log2n+O(1){\textstyle \leq n+2\log _{2}n+O(1)}, wheren{\displaystyle n}comes from the length of the Busy Beaverpn{\textstyle p_{n}},2log2n{\displaystyle 2\log _{2}n}comes from using the (prefix-free)Elias delta codefor the numbern{\displaystyle n}, andO(1){\displaystyle O(1)}comes from the rest of the program. Therefore,K(x)≤n+2log2n+O(1)≤2n{\displaystyle K(x)\leq n+2\log _{2}n+O(1)\leq 2n}for all bign{\textstyle n}. Further, since there are only so many possible programs with length≤2n{\textstyle \leq 2n}, we havel(x)≤2n+1{\textstyle l(x)\leq 2n+1}bypigeonhole principle.
By assumption,pK(n)<BB(n){\textstyle p_{K}(n)<BB(n)}, so every string of length≤2n+1{\textstyle \leq 2n+1}has a minimal program with runtime<BB(n){\textstyle <BB(n)}. Thus, the stringx{\textstyle x}has a minimal program with runtime<BB(n){\textstyle <BB(n)}. Further, that program has lengthK(x)≤2n{\textstyle K(x)\leq 2n}. This contradicts howx{\textstyle x}was constructed.
Fix a universal Turing machineU{\displaystyle U}, the same one used to define the (prefix-free) Kolmogorov complexity. Define the (prefix-free) universal probability of a stringx{\displaystyle x}to beP(x)=∑U(p)=x2−l(p){\displaystyle P(x)=\sum _{U(p)=x}2^{-l(p)}}In other words, it is the probability that, given a uniformly random binary stream as input, the universal Turing machine would halt after reading a certain prefix of the stream, and outputx{\displaystyle x}.
Note.U(p)=x{\displaystyle U(p)=x}does not mean that the input stream isp000⋯{\displaystyle p000\cdots }, but that the universal Turing machine would halt at some point after reading the initial segmentp{\displaystyle p}, without reading any further input, and that, when it halts, its has writtenx{\displaystyle x}to the output tape.
Theorem.(Theorem 14.11.1[28])log1P(x)=K(x)+O(1){\displaystyle \log {\frac {1}{P(x)}}=K(x)+O(1)}
The conditional Kolmogorov complexity of two stringsK(x|y){\displaystyle K(x|y)}is, roughly speaking, defined as the Kolmogorov complexity ofxgivenyas an auxiliary input to the procedure.[31][32]
There is also a length-conditional complexityK(x|L(x)){\displaystyle K(x|L(x))}, which is the complexity ofxgiven the length ofxas known/input.[33][34]
Time-bounded Kolmogorov complexity is a modified version of Kolmogorov complexity where the space of programs to be searched for a solution is confined to only programs that can run within some pre-defined number of steps.[35]It is hypothesised that the possibility of the existence of an efficient algorithm for determining approximate time-bounded Kolmogorov complexity is related to the question of whether trueone-way functionsexist.[36][37] | https://en.wikipedia.org/wiki/Kolmogorov_complexity |
Ininformation theory,data compression,source coding,[1]orbit-rate reductionis the process of encodinginformationusing fewerbitsthan the original representation.[2]Any particular compression is eitherlossyorlossless. Lossless compression reduces bits by identifying and eliminatingstatistical redundancy. No information is lost in lossless compression. Lossy compression reduces bits by removing unnecessary or less important information.[3]Typically, a device that performs data compression is referred to as an encoder, and one that performs the reversal of the process (decompression) as a decoder.
The process of reducing the size of adata fileis often referred to as data compression. In the context ofdata transmission, it is called source coding: encoding is done at the source of the data before it is stored or transmitted.[4]Source coding should not be confused withchannel coding, for error detection and correction orline coding, the means for mapping data onto a signal.
Data Compression algorithms present aspace-time complexity trade-offbetween the bytes needed to store or transmit information, and theComputational resourcesneeded to perform the encoding and decoding. The design of data compression schemes involves balancing the degree of compression, the amount of distortion introduced (when usinglossy data compression), and the computational resources or time required to compress and decompress the data.[5]
Lossless data compressionalgorithmsusually exploitstatistical redundancyto represent data without losing anyinformation, so that the process is reversible. Lossless compression is possible because most real-world data exhibits statistical redundancy. For example, an image may have areas of color that do not change over several pixels; instead of coding "red pixel, red pixel, ..." the data may be encoded as "279 red pixels". This is a basic example ofrun-length encoding; there are many schemes to reduce file size by eliminating redundancy.
TheLempel–Ziv(LZ) compression methods are among the most popular algorithms for lossless storage.[6]DEFLATEis a variation on LZ optimized for decompression speed and compression ratio,[7]but compression can be slow. In the mid-1980s, following work byTerry Welch, theLempel–Ziv–Welch(LZW) algorithm rapidly became the method of choice for most general-purpose compression systems. LZW is used inGIFimages, programs such asPKZIP, and hardware devices such as modems.[8]LZ methods use a table-based compression model where table entries are substituted for repeated strings of data. For most LZ methods, this table is generated dynamically from earlier data in the input. The table itself is oftenHuffman encoded.Grammar-based codeslike this can compress highly repetitive input extremely effectively, for instance, a biologicaldata collectionof the same or closely related species, a huge versioned document collection, internet archival, etc. The basic task of grammar-based codes is constructing a context-free grammar deriving a single string. Other practical grammar compression algorithms includeSequiturandRe-Pair.
The strongest modern lossless compressors useprobabilisticmodels, such asprediction by partial matching. TheBurrows–Wheeler transformcan also be viewed as an indirect form of statistical modelling.[citation needed]In a further refinement of the direct use ofprobabilistic modelling, statistical estimates can be coupled to an algorithm calledarithmetic coding. Arithmetic coding is a more modern coding technique that uses the mathematical calculations of afinite-state machineto produce a string of encoded bits from a series of input data symbols. It can achieve superior compression compared to other techniques such as the better-known Huffman algorithm. It uses an internal memory state to avoid the need to perform a one-to-one mapping of individual input symbols to distinct representations that use an integer number of bits, and it clears out the internal memory only after encoding the entire string of data symbols. Arithmetic coding applies especially well to adaptive data compression tasks where the statistics vary and are context-dependent, as it can be easily coupled with an adaptive model of theprobability distributionof the input data. An early example of the use of arithmetic coding was in an optional (but not widely used) feature of theJPEGimage coding standard.[9]It has since been applied in various other designs includingH.263,H.264/MPEG-4 AVCandHEVCfor video coding.[10]
Archive software typically has the ability to adjust the "dictionary size", where a larger size demands morerandom-access memoryduring compression and decompression, but compresses stronger, especially on repeating patterns in files' content.[11][12]
In the late 1980s, digital images became more common, and standards for losslessimage compressionemerged. In the early 1990s, lossy compression methods began to be widely used.[13]In these schemes, some loss of information is accepted as dropping nonessential detail can save storage space. There is a correspondingtrade-offbetween preserving information and reducing size. Lossy data compression schemes are designed by research on how people perceive the data in question. For example, the human eye is more sensitive to subtle variations inluminancethan it is to the variations in color. JPEG image compression works in part by rounding off nonessential bits of information.[14]A number of popular compression formats exploit these perceptual differences, includingpsychoacousticsfor sound, andpsychovisualsfor images and video.
Most forms of lossy compression are based ontransform coding, especially thediscrete cosine transform(DCT). It was first proposed in 1972 byNasir Ahmed, who then developed a working algorithm with T. Natarajan andK. R. Raoin 1973, before introducing it in January 1974.[15][16]DCT is the most widely used lossy compression method, and is used in multimedia formats for images (such as JPEG andHEIF),[17]video(such asMPEG,AVCand HEVC) and audio (such asMP3,AACandVorbis).
Lossy image compression is used indigital cameras, to increase storage capacities. Similarly,DVDs,Blu-rayandstreaming videouse lossyvideo coding formats. Lossy compression is extensively used in video.
In lossy audio compression, methods of psychoacoustics are used to remove non-audible (or less audible) components of theaudio signal. Compression of human speech is often performed with even more specialized techniques;speech codingis distinguished as a separate discipline from general-purpose audio compression. Speech coding is used ininternet telephony, for example, audio compression is used for CD ripping and is decoded by the audio players.[citation needed]
Lossy compression can causegeneration loss.
The theoretical basis for compression is provided byinformation theoryand, more specifically,Shannon's source coding theorem; domain-specific theories includealgorithmic information theoryfor lossless compression andrate–distortion theoryfor lossy compression. These areas of study were essentially created byClaude Shannon, who published fundamental papers on the topic in the late 1940s and early 1950s. Other topics associated with compression includecoding theoryandstatistical inference.[18]
There is a close connection betweenmachine learningand compression. A system that predicts theposterior probabilitiesof a sequence given its entire history can be used for optimal data compression (by usingarithmetic codingon the output distribution). Conversely, an optimal compressor can be used for prediction (by finding the symbol that compresses best, given the previous history). This equivalence has been used as a justification for using data compression as a benchmark for "general intelligence".[19][20][21]
An alternative view can show compression algorithms implicitly map strings into implicitfeature space vectors, and compression-based similarity measures compute similarity within these feature spaces. For each compressor C(.) we define an associated vector space ℵ, such that C(.) maps an input string x, corresponding to the vector norm ||~x||. An exhaustive examination of the feature spaces underlying all compression algorithms is precluded by space; instead, feature vectors chooses to examine three representative lossless compression methods, LZW, LZ77, and PPM.[22]
According toAIXItheory, a connection more directly explained inHutter Prize, the best possible compression of x is the smallest possible software that generates x. For example, in that model, a zip file's compressed size includes both the zip file and the unzipping software, since you can not unzip it without both, but there may be an even smaller combined form.
Examples of AI-powered audio/video compression software includeNVIDIA Maxine, AIVC.[23]Examples of software that can perform AI-powered image compression includeOpenCV,TensorFlow,MATLAB's Image Processing Toolbox (IPT) and High-Fidelity Generative Image Compression.[24]
Inunsupervised machine learning,k-means clusteringcan be utilized to compress data by grouping similar data points into clusters. This technique simplifies handling extensive datasets that lack predefined labels and finds widespread use in fields such asimage compression.[25]
Data compression aims to reduce the size of data files, enhancing storage efficiency and speeding up data transmission. K-means clustering, an unsupervised machine learning algorithm, is employed to partition a dataset into a specified number of clusters, k, each represented by thecentroidof its points. This process condenses extensive datasets into a more compact set of representative points. Particularly beneficial inimageandsignal processing, k-means clustering aids in data reduction by replacing groups of data points with their centroids, thereby preserving the core information of the original data while significantly decreasing the required storage space.[26]
Large language models(LLMs) are also efficient lossless data compressors on some data sets, as demonstrated byDeepMind's research with the Chinchilla 70B model. Developed by DeepMind, Chinchilla 70B effectively compressed data, outperforming conventional methods such asPortable Network Graphics(PNG) for images andFree Lossless Audio Codec(FLAC) for audio. It achieved compression of image and audio data to 43.4% and 16.4% of their original sizes, respectively. There is, however, some reason to be concerned that the data set used for testing overlaps the LLM training data set, making it possible that the Chinchilla 70B model is only an efficient compression tool on data it has already been trained on.[27][28]
Data compression can be viewed as a special case ofdata differencing.[29][30]Data differencing consists of producing adifferencegiven asourceand atarget,with patching reproducing thetargetgiven asourceand adifference.Since there is no separate source and target in data compression, one can consider data compression as data differencing with empty source data, the compressed file corresponding to a difference from nothing. This is the same as considering absoluteentropy(corresponding to data compression) as a special case ofrelative entropy(corresponding to data differencing) with no initial data.
The termdifferential compressionis used to emphasize the data differencing connection.
Entropy codingoriginated in the 1940s with the introduction ofShannon–Fano coding,[31]the basis forHuffman codingwhich was developed in 1950.[32]Transform codingdates back to the late 1960s, with the introduction offast Fourier transform(FFT) coding in 1968 and theHadamard transformin 1969.[33]
An important image compression technique is thediscrete cosine transform(DCT), a technique developed in the early 1970s.[15]DCT is the basis for JPEG, alossy compressionformat which was introduced by theJoint Photographic Experts Group(JPEG) in 1992.[34]JPEG greatly reduces the amount of data required to represent an image at the cost of a relatively small reduction in image quality and has become the most widely usedimage file format.[35][36]Its highly efficient DCT-based compression algorithm was largely responsible for the wide proliferation ofdigital imagesanddigital photos.[37]
Lempel–Ziv–Welch(LZW) is alossless compressionalgorithm developed in 1984. It is used in theGIFformat, introduced in 1987.[38]DEFLATE, a lossless compression algorithm specified in 1996, is used in thePortable Network Graphics(PNG) format.[39]
Wavelet compression, the use ofwaveletsin image compression, began after the development of DCT coding.[40]TheJPEG 2000standard was introduced in 2000.[41]In contrast to the DCT algorithm used by the original JPEG format, JPEG 2000 instead usesdiscrete wavelet transform(DWT) algorithms.[42][43][44]JPEG 2000 technology, which includes theMotion JPEG 2000extension, was selected as thevideo coding standardfordigital cinemain 2004.[45]
Audio data compression, not to be confused withdynamic range compression, has the potential to reduce the transmissionbandwidthand storage requirements of audio data.Audio compression formats compression algorithmsare implemented insoftwareas audiocodecs. In both lossy and lossless compression,information redundancyis reduced, using methods such ascoding,quantization, DCT andlinear predictionto reduce the amount of information used to represent the uncompressed data.
Lossy audio compression algorithms provide higher compression and are used in numerous audio applications includingVorbisandMP3. These algorithms almost all rely onpsychoacousticsto eliminate or reduce fidelity of less audible sounds, thereby reducing the space required to store or transmit them.[2][46]
The acceptable trade-off between loss of audio quality and transmission or storage size depends upon the application. For example, one 640 MBcompact disc(CD) holds approximately one hour of uncompressedhigh fidelitymusic, less than 2 hours of music compressed losslessly, or 7 hours of music compressed in theMP3format at a mediumbit rate. A digital sound recorder can typically store around 200 hours of clearly intelligible speech in 640 MB.[47]
Lossless audio compression produces a representation of digital data that can be decoded to an exact digital duplicate of the original. Compression ratios are around 50–60% of the original size,[48]which is similar to those for generic lossless data compression. Lossless codecs usecurve fittingor linear prediction as a basis for estimating the signal. Parameters describing the estimation and the difference between the estimation and the actual signal are coded separately.[49]
A number of lossless audio compression formats exist. Seelist of lossless codecsfor a listing. Some formats are associated with a distinct system, such asDirect Stream Transfer, used inSuper Audio CDandMeridian Lossless Packing, used inDVD-Audio,Dolby TrueHD,Blu-rayandHD DVD.
Someaudio file formatsfeature a combination of a lossy format and a lossless correction; this allows stripping the correction to easily obtain a lossy file. Such formats includeMPEG-4 SLS(Scalable to Lossless),WavPack, andOptimFROG DualStream.
When audio files are to be processed, either by further compression or forediting, it is desirable to work from an unchanged original (uncompressed or losslessly compressed). Processing of a lossily compressed file for some purpose usually produces a final result inferior to the creation of the same compressed file from an uncompressed original. In addition to sound editing or mixing, lossless audio compression is often used for archival storage, or as master copies.
Lossy audio compression is used in a wide range of applications. In addition to standalone audio-only applications of file playback in MP3 players or computers, digitally compressed audio streams are used in most video DVDs, digital television, streaming media on theInternet, satellite and cable radio, and increasingly in terrestrial radio broadcasts. Lossy compression typically achieves far greater compression than lossless compression, by discarding less-critical data based onpsychoacousticoptimizations.[50]
Psychoacoustics recognizes that not all data in an audio stream can be perceived by the humanauditory system. Most lossy compression reduces redundancy by first identifying perceptually irrelevant sounds, that is, sounds that are very hard to hear. Typical examples include high frequencies or sounds that occur at the same time as louder sounds. Those irrelevant sounds are coded with decreased accuracy or not at all.
Due to the nature of lossy algorithms,audio qualitysuffers adigital generation losswhen a file is decompressed and recompressed. This makes lossy compression unsuitable for storing the intermediate results in professional audio engineering applications, such as sound editing and multitrack recording. However, lossy formats such asMP3are very popular with end-users as the file size is reduced to 5-20% of the original size and a megabyte can store about a minute's worth of music at adequate quality.
Several proprietary lossy compression algorithms have been developed that provide higher quality audio performance by using a combination of lossless and lossy algorithms with adaptive bit rates and lower compression ratios. Examples includeaptX,LDAC,LHDC,MQAandSCL6.
To determine what information in an audio signal is perceptually irrelevant, most lossy compression algorithms use transforms such as themodified discrete cosine transform(MDCT) to converttime domainsampled waveforms into a transform domain, typically thefrequency domain. Once transformed, component frequencies can be prioritized according to how audible they are. Audibility of spectral components is assessed using theabsolute threshold of hearingand the principles ofsimultaneous masking—the phenomenon wherein a signal is masked by another signal separated by frequency—and, in some cases,temporal masking—where a signal is masked by another signal separated by time.Equal-loudness contoursmay also be used to weigh the perceptual importance of components. Models of the human ear-brain combination incorporating such effects are often calledpsychoacoustic models.[51]
Other types of lossy compressors, such as thelinear predictive coding(LPC) used with speech, are source-based coders. LPC uses a model of the human vocal tract to analyze speech sounds and infer the parameters used by the model to produce them moment to moment. These changing parameters are transmitted or stored and used to drive another model in the decoder which reproduces the sound.
Lossy formats are often used for the distribution of streaming audio or interactive communication (such as in cell phone networks). In such applications, the data must be decompressed as the data flows, rather than after the entire data stream has been transmitted. Not all audio codecs can be used for streaming applications.[50]
Latencyis introduced by the methods used to encode and decode the data. Some codecs will analyze a longer segment, called aframe, of the data to optimize efficiency, and then code it in a manner that requires a larger segment of data at one time to decode. The inherent latency of the coding algorithm can be critical; for example, when there is a two-way transmission of data, such as with a telephone conversation, significant delays may seriously degrade the perceived quality.
In contrast to the speed of compression, which is proportional to the number of operations required by the algorithm, here latency refers to the number of samples that must be analyzed before a block of audio is processed. In the minimum case, latency is zero samples (e.g., if the coder/decoder simply reduces the number of bits used to quantize the signal). Time domain algorithms such as LPC also often have low latencies, hence their popularity in speech coding for telephony. In algorithms such as MP3, however, a large number of samples have to be analyzed to implement a psychoacoustic model in the frequency domain, and latency is on the order of 23 ms.
Speech encodingis an important category of audio data compression. The perceptual models used to estimate what aspects of speech a human ear can hear are generally somewhat different from those used for music. The range of frequencies needed to convey the sounds of a human voice is normally far narrower than that needed for music, and the sound is normally less complex. As a result, speech can be encoded at high quality using a relatively low bit rate.
This is accomplished, in general, by some combination of two approaches:
The earliest algorithms used in speech encoding (and audio data compression in general) were theA-law algorithmand theμ-law algorithm.
Early audio research was conducted atBell Labs. There, in 1950,C. Chapin Cutlerfiled the patent ondifferential pulse-code modulation(DPCM).[52]In 1973,Adaptive DPCM(ADPCM) was introduced by P. Cummiskey,Nikil S. JayantandJames L. Flanagan.[53][54]
Perceptual codingwas first used forspeech codingcompression, withlinear predictive coding(LPC).[55]Initial concepts for LPC date back to the work ofFumitada Itakura(Nagoya University) and Shuzo Saito (Nippon Telegraph and Telephone) in 1966.[56]During the 1970s,Bishnu S. AtalandManfred R. SchroederatBell Labsdeveloped a form of LPC calledadaptive predictive coding(APC), a perceptual coding algorithm that exploited the masking properties of the human ear, followed in the early 1980s with thecode-excited linear prediction(CELP) algorithm which achieved a significantcompression ratiofor its time.[55]Perceptual coding is used by modern audio compression formats such asMP3[55]andAAC.
Discrete cosine transform(DCT), developed byNasir Ahmed, T. Natarajan andK. R. Raoin 1974,[16]provided the basis for themodified discrete cosine transform(MDCT) used by modern audio compression formats such as MP3,[57]Dolby Digital,[58][59]and AAC.[60]MDCT was proposed by J. P. Princen, A. W. Johnson and A. B. Bradley in 1987,[61]following earlier work by Princen and Bradley in 1986.[62]
The world's first commercialbroadcast automationaudio compression system was developed by Oscar Bonello, an engineering professor at theUniversity of Buenos Aires.[63]In 1983, using the psychoacoustic principle of the masking of critical bands first published in 1967,[64]he started developing a practical application based on the recently developedIBM PCcomputer, and the broadcast automation system was launched in 1987 under the nameAudicom.[65]35 years later, almost all the radio stations in the world were using this technology manufactured by a number of companies because the inventor refused to patent his work, preferring to publish it and leave it in the public domain.[66]
A literature compendium for a large variety of audio coding systems was published in the IEEE'sJournal on Selected Areas in Communications(JSAC), in February 1988. While there were some papers from before that time, this collection documented an entire variety of finished, working audio coders, nearly all of them using perceptual techniques and some kind of frequency analysis and back-end noiseless coding.[67]
Uncompressed videorequires a very highdata rate. Althoughlossless video compressioncodecs perform at a compression factor of 5 to 12, a typicalH.264lossy compression video has a compression factor between 20 and 200.[68]
The two key video compression techniques used invideo coding standardsare the DCT andmotion compensation(MC). Most video coding standards, such as theH.26xandMPEGformats, typically use motion-compensated DCT video coding (block motion compensation).[69][70]
Most video codecs are used alongside audio compression techniques to store the separate but complementary data streams as one combined package using so-calledcontainer formats.[71]
Video data may be represented as a series of still image frames. Such data usually contains abundant amounts of spatial and temporalredundancy. Video compression algorithms attempt to reduce redundancy and store information more compactly.
Mostvideo compression formatsandcodecsexploit both spatial and temporal redundancy (e.g. through difference coding withmotion compensation). Similarities can be encoded by only storing differences between e.g. temporally adjacent frames (inter-frame coding) or spatially adjacent pixels (intra-frame coding).Inter-framecompression (a temporaldelta encoding) (re)uses data from one or more earlier or later frames in a sequence to describe the current frame.Intra-frame coding, on the other hand, uses only data from within the current frame, effectively being still-image compression.[51]
Theintra-frame video coding formatsused in camcorders and video editing employ simpler compression that uses only intra-frame prediction. This simplifies video editing software, as it prevents a situation in which a compressed frame refers to data that the editor has deleted.
Usually, video compression additionally employslossy compressiontechniques likequantizationthat reduce aspects of the source data that are (more or less) irrelevant to the human visual perception by exploiting perceptual features of human vision. For example, small differences in color are more difficult to perceive than are changes in brightness. Compression algorithms can average a color across these similar areas in a manner similar to those used in JPEG image compression.[9]As in all lossy compression, there is atrade-offbetweenvideo qualityandbit rate, cost of processing the compression and decompression, and system requirements. Highly compressed video may present visible or distractingartifacts.
Other methods other than the prevalent DCT-based transform formats, such asfractal compression,matching pursuitand the use of adiscrete wavelet transform(DWT), have been the subject of some research, but are typically not used in practical products.Wavelet compressionis used in still-image coders and video coders without motion compensation. Interest in fractal compression seems to be waning, due to recent theoretical analysis showing a comparative lack of effectiveness of such methods.[51]
In inter-frame coding, individual frames of a video sequence are compared from one frame to the next, and thevideo compression codecrecords thedifferencesto the reference frame. If the frame contains areas where nothing has moved, the system can simply issue a short command that copies that part of the previous frame into the next one. If sections of the frame move in a simple manner, the compressor can emit a (slightly longer) command that tells the decompressor to shift, rotate, lighten, or darken the copy. This longer command still remains much shorter than data generated by intra-frame compression. Usually, the encoder will also transmit a residue signal which describes the remaining more subtle differences to the reference imagery. Using entropy coding, these residue signals have a more compact representation than the full signal. In areas of video with more motion, the compression must encode more data to keep up with the larger number of pixels that are changing. Commonly during explosions, flames, flocks of animals, and in some panning shots, the high-frequency detail leads to quality decreases or to increases in thevariable bitrate.
Many commonly used video compression methods (e.g., those in standards approved by theITU-TorISO) share the same basic architecture that dates back toH.261which was standardized in 1988 by the ITU-T. They mostly rely on the DCT, applied to rectangular blocks of neighboring pixels, and temporal prediction usingmotion vectors, as well as nowadays also an in-loop filtering step.
In the prediction stage, variousdeduplicationand difference-coding techniques are applied that help decorrelate data and describe new data based on already transmitted data.
Then rectangular blocks of remainingpixeldata are transformed to the frequency domain. In the main lossy processing stage, frequency domain data gets quantized in order to reduce information that is irrelevant to human visual perception.
In the last stage statistical redundancy gets largely eliminated by anentropy coderwhich often applies some form of arithmetic coding.
In an additional in-loop filtering stage various filters can be applied to the reconstructed image signal. By computing these filters also inside the encoding loop they can help compression because they can be applied to reference material before it gets used in the prediction process and they can be guided using the original signal. The most popular example aredeblocking filtersthat blur out blocking artifacts from quantization discontinuities at transform block boundaries.
In 1967, A.H. Robinson and C. Cherry proposed arun-length encodingbandwidth compression scheme for the transmission of analog television signals.[72]The DCT, which is fundamental to modern video compression,[73]was introduced byNasir Ahmed, T. Natarajan andK. R. Raoin 1974.[16][74]
H.261, which debuted in 1988, commercially introduced the prevalent basic architecture of video compression technology.[75]It was the firstvideo coding formatbased on DCT compression.[73]H.261 was developed by a number of companies, includingHitachi,PictureTel,NTT,BTandToshiba.[76]
The most popularvideo coding standardsused for codecs have been theMPEGstandards.MPEG-1was developed by theMotion Picture Experts Group(MPEG) in 1991, and it was designed to compressVHS-quality video. It was succeeded in 1994 byMPEG-2/H.262,[75]which was developed by a number of companies, primarilySony,ThomsonandMitsubishi Electric.[77]MPEG-2 became the standard video format forDVDandSD digital television.[75]In 1999, it was followed byMPEG-4/H.263.[75]It was also developed by a number of companies, primarily Mitsubishi Electric,HitachiandPanasonic.[78]
H.264/MPEG-4 AVCwas developed in 2003 by a number of organizations, primarily Panasonic,Godo Kaisha IP BridgeandLG Electronics.[79]AVC commercially introduced the moderncontext-adaptive binary arithmetic coding(CABAC) andcontext-adaptive variable-length coding(CAVLC) algorithms. AVC is the main video encoding standard forBlu-ray Discs, and is widely used by video sharing websites and streaming internet services such asYouTube,Netflix,Vimeo, andiTunes Store, web software such asAdobe Flash PlayerandMicrosoft Silverlight, and variousHDTVbroadcasts over terrestrial and satellite television.[citation needed]
Genetics compression algorithmsare the latest generation of lossless algorithms that compress data (typically sequences of nucleotides) using both conventional compression algorithms and genetic algorithms adapted to the specific datatype. In 2012, a team of scientists from Johns Hopkins University published a genetic compression algorithm that does not use a reference genome for compression. HAPZIPPER was tailored forHapMapdata and achieves over 20-fold compression (95% reduction in file size), providing 2- to 4-fold better compression and is less computationally intensive than the leading general-purpose compression utilities. For this, Chanda, Elhaik, and Bader introduced MAF-based encoding (MAFE), which reduces the heterogeneity of the dataset by sorting SNPs by their minor allele frequency, thus homogenizing the dataset.[80]Other algorithms developed in 2009 and 2013 (DNAZip and GenomeZip) have compression ratios of up to 1200-fold—allowing 6 billion basepair diploid human genomes to be stored in 2.5 megabytes (relative to a reference genome or averaged over many genomes).[81][82]For a benchmark in genetics/genomics data compressors, see[83]
It is estimated that the total amount of data that is stored on the world's storage devices could be further compressed with existing compression algorithms by a remaining average factor of 4.5:1.[84]It is estimated that the combined technological capacity of the world to store information provides 1,300exabytesof hardware digits in 2007, but when the corresponding content is optimally compressed, this only represents 295 exabytes ofShannon information.[85] | https://en.wikipedia.org/wiki/Source_coding |
Elias δ codeorElias delta codeis auniversal codeencoding the positive integers developed byPeter Elias.[1]: 200
To code a numberX≥ 1:
An equivalent way to express the same process:
To represent a numberx{\displaystyle x}, Elias delta (δ) uses⌊log2(x)⌋+2⌊log2(⌊log2(x)⌋+1)⌋+1{\displaystyle \lfloor \log _{2}(x)\rfloor +2\lfloor \log _{2}(\lfloor \log _{2}(x)\rfloor +1)\rfloor +1}bits.[1]: 200This is useful for very large integers, where the overall encoded representation's bits end up being fewer [than what one might obtain usingElias gamma coding] due to thelog2(⌊log2(x)⌋+1){\displaystyle \log _{2}(\lfloor \log _{2}(x)\rfloor +1)}portion of the previous expression.
The code begins, usingγ′{\displaystyle \gamma '}instead ofγ{\displaystyle \gamma }:
To decode an Elias delta-coded integer:
Example:
This code can be generalized to zero or negative integers in the same ways described inElias gamma coding.
Elias delta coding does not code zero or negative integers.
One way to code all non negative integers is to add 1 before coding and then subtract 1 after decoding.
One way to code all integers is to set up abijection, mapping integers all integers (0, 1, −1, 2, −2, 3, −3, ...) to strictly positive integers (1, 2, 3, 4, 5, 6, 7, ...) before coding.
This bijection can be performed using the"ZigZag" encoding from Protocol Buffers(not to be confused withZigzag code, nor theJPEG Zig-zag entropy coding). | https://en.wikipedia.org/wiki/Elias_delta_coding |
Equifinalityis the principle that inopen systemsa given end state can be reached by many potential means. The term and concept is due to the GermanHans Driesch, the developmental biologist, later applied by the AustrianLudwig von Bertalanffy, the founder of generalsystems theory, and byWilliam T. Powers, the founder ofperceptual control theory.Drieschandvon Bertalanffyprefer this term, in contrast to "goal", in describingcomplex systems' similar orconvergentbehavior. Powers simply emphasised the flexibility of response, since it emphasizes that the same end state may be achieved via many different paths ortrajectories.
Inclosed systems, a direct cause-and-effect relationship exists between the initial condition and the final state of the system: When a computer's 'on' switch is pushed, the system powers up. Open systems (such as biological and social systems), however, operate quite differently. The idea of equifinality suggests that similar results may be achieved with different initial conditions and in many different ways.[1]This phenomenon has also been referred to asisotelesis[2](fromGreekἴσοςisos"equal" and τέλεσιςtelesis: "the intelligent direction of effort toward the achievement of an end") when in games involvingsuperrationality.
Inbusiness, equifinality implies that firms may establish similar competitive advantages based on substantially different competencies.
Inpsychology, equifinality refers to how different early experiences in life (e.g., parentaldivorce,physical abuse, parental substance abuse) can lead to similar outcomes (e.g.,childhood depression). In other words, there are many different early experiences that can lead to the samepsychological disorder.
Inarchaeology, equifinality refers to how different historical processes may lead to a similar outcome or social formation. For example, the development of agriculture or the bow and arrow occurred independently in many different areas of the world, yet for different reasons and through different historical trajectories. This highlights that generalizations based on cross-cultural comparisons cannot be made uncritically.
InEarthandEnvironmentalSciences, two general types of equifinality are distinguished:process equifinality(concerned with real-world open systems) andmodel equifinality(concerned with conceptual open systems).[3]For example, process equifinality ingeomorphologyindicates that similar landforms might arise as a result of quite different sets of processes. Model equifinality refers to a condition where distinct configurations of model components (e.g. distinct model parameter values) can lead to similar or equally acceptable simulations (or representations of the real-world process of interest). This similarity or equal acceptability is conditional on the objective functions and criteria of acceptability defined by the modeler. While model equifinality has various facets, model parameter and structural equifinality are mostly known and focused in modeling studies.[3]Equifinality (particularly parameter equifinality) and Monte Carlo experiments are the foundation of theGLUE methodthat was the first generalised method for uncertainty assessment inhydrological modeling.[4]GLUE is now widely used within and beyondenvironmentalmodeling. | https://en.wikipedia.org/wiki/Equifinality |
Ergodic theoryis a branch ofmathematicsthat studiesstatisticalproperties of deterministicdynamical systems; it is the study ofergodicity. In this context, "statistical properties" refers to properties which are expressed through the behavior oftime averagesof various functions along trajectories of dynamical systems. The notion of deterministic dynamical systems assumes that the equations determining the dynamics do not contain anyrandomperturbations,noise, etc. Thus, the statistics with which we are concerned are properties of the dynamics.
Ergodic theory, likeprobability theory, is based on general notions ofmeasuretheory. Its initial development was motivated by problems ofstatistical physics.
A central concern of ergodic theory is the behavior of a dynamical system when it is allowed to run for a long time. The first result in this direction is thePoincaré recurrence theorem, which claims thatalmost allpoints in any subset of thephase spaceeventually revisit the set. Systems for which the Poincaré recurrence theorem holds areconservative systems; thus all ergodic systems are conservative.
More precise information is provided by variousergodic theoremswhich assert that, under certain conditions, the time average of a function along the trajectories existsalmost everywhereand is related to the space average. Two of the most important theorems are those ofBirkhoff(1931) andvon Neumannwhich assert the existence of a time average along each trajectory. For the special class ofergodic systems, this time average is the same for almost all initial points: statistically speaking, the system that evolves for a long time "forgets" its initial state. Stronger properties, such asmixingandequidistribution, have also been extensively studied.
The problem of metric classification of systems is another important part of the abstract ergodic theory. An outstanding role in ergodic theory and its applications tostochastic processesis played by the various notions ofentropyfor dynamical systems.
The concepts ofergodicityand theergodic hypothesisare central to applications of ergodic theory. The underlying idea is that for certain systems the time average of their properties is equal to the average over the entire space. Applications of ergodic theory to other parts of mathematics usually involve establishing ergodicity properties for systems of special kind. Ingeometry, methods of ergodic theory have been used to study thegeodesic flowonRiemannian manifolds, starting with the results ofEberhard HopfforRiemann surfacesof negative curvature.Markov chainsform a common context for applications inprobability theory. Ergodic theory has fruitful connections withharmonic analysis,Lie theory(representation theory,latticesinalgebraic groups), andnumber theory(the theory ofdiophantine approximations,L-functions).
Ergodic theory is often concerned withergodic transformations. The intuition behind such transformations, which act on a given set, is that they do a thorough job "stirring" the elements of that set. E.g. if the set is a quantity of hot oatmeal in a bowl, and if a spoonful of syrup is dropped into the bowl, then iterations of the inverse of an ergodic transformation of the oatmeal will not allow the syrup to remain in a local subregion of the oatmeal, but will distribute the syrup evenly throughout. At the same time, these iterations will not compress or dilate any portion of the oatmeal: they preserve the measure that is density.
The formal definition is as follows:
LetT:X→Xbe ameasure-preserving transformationon ameasure space(X,Σ,μ), withμ(X) = 1. ThenTisergodicif for everyEinΣwithμ(T−1(E) ΔE) = 0(that is,Eisinvariant), eitherμ(E) = 0orμ(E) = 1.
The operator Δ here is the symmetric difference of sets, equivalent to theexclusive-oroperation with respect to set membership. The condition that the symmetric difference be measure zero is called beingessentially invariant.
LetT:X→Xbe ameasure-preserving transformationon ameasure space(X, Σ,μ) and suppose ƒ is aμ-integrable function, i.e. ƒ ∈L1(μ). Then we define the followingaverages:
Time average:This is defined as the average (if it exists) over iterations ofTstarting from some initial pointx:
Space average:Ifμ(X) is finite and nonzero, we can consider thespaceorphaseaverage of ƒ:
In general the time average and space average may be different. But if the transformation is ergodic, and the measure is invariant, then the time average is equal to the space averagealmost everywhere. This is the celebrated ergodic theorem, in an abstract form due toGeorge David Birkhoff. (Actually, Birkhoff's paper considers not the abstract general case but only the case of dynamical systems arising from differential equations on a smooth manifold.) Theequidistribution theoremis a special case of the ergodic theorem, dealing specifically with the distribution of probabilities on the unit interval.
More precisely, thepointwiseorstrong ergodic theoremstates that the limit in the definition of the time average of ƒ exists for almost everyxand that the (almost everywhere defined) limit functionf^{\displaystyle {\hat {f}}}is integrable:
Furthermore,f^{\displaystyle {\hat {f}}}isT-invariant, that is to say
holds almost everywhere, and ifμ(X) is finite, then the normalization is the same:
In particular, ifTis ergodic, thenf^{\displaystyle {\hat {f}}}must be a constant (almost everywhere), and so one has that
almost everywhere. Joining the first to the last claim and assuming thatμ(X) is finite and nonzero, one has that
foralmost allx, i.e., for allxexcept for a set ofmeasurezero.
For an ergodic transformation, the time average equals the space average almost surely.
As an example, assume that the measure space (X, Σ,μ) models the particles of a gas as above, and let ƒ(x) denote thevelocityof the particle at positionx. Then the pointwise ergodic theorems says that the average velocity of all particles at some given time is equal to the average velocity of one particle over time.
A generalization of Birkhoff's theorem isKingman's subadditive ergodic theorem.
Birkhoff–Khinchin theorem. Let ƒ be measurable,E(|ƒ|) < ∞, andTbe a measure-preserving map. Thenwith probability 1:
whereE(f|C){\displaystyle E(f|{\mathcal {C}})}is theconditional expectationgiven the σ-algebraC{\displaystyle {\mathcal {C}}}of invariant sets ofT.
Corollary(Pointwise Ergodic Theorem): In particular, ifTis also ergodic, thenC{\displaystyle {\mathcal {C}}}is the trivial σ-algebra, and thus with probability 1:
Von Neumann's mean ergodic theorem, holds in Hilbert spaces.[1]
LetUbe aunitary operatoron aHilbert spaceH; more generally, an isometric linear operator (that is, a not necessarily surjective linear operator satisfying ‖Ux‖ = ‖x‖ for allxinH, or equivalently, satisfyingU*U= I, but not necessarilyUU* = I). LetPbe theorthogonal projectiononto {ψ∈H|Uψ= ψ} = ker(I−U).
Then, for anyxinH, we have:
where the limit is with respect to the norm onH. In other words, the sequence of averages
converges toPin thestrong operator topology.
Indeed, it is not difficult to see that in this case anyx∈H{\displaystyle x\in H}admits an orthogonal decomposition into parts fromker(I−U){\displaystyle \ker(I-U)}andran(I−U)¯{\displaystyle {\overline {\operatorname {ran} (I-U)}}}respectively. The former part is invariant in all the partial sums asN{\displaystyle N}grows, while for the latter part, from thetelescoping seriesone would have:
This theorem specializes to the case in which the Hilbert spaceHconsists ofL2functions on a measure space andUis an operator of the form
whereTis a measure-preserving endomorphism ofX, thought of in applications as representing a time-step of a discrete dynamical system.[2]The ergodic theorem then asserts that the average behavior of a function ƒ over sufficiently large time-scales is approximated by the orthogonal component of ƒ which is time-invariant.
In another form of the mean ergodic theorem, letUtbe a strongly continuousone-parameter groupof unitary operators onH. Then the operator
converges in the strong operator topology asT→ ∞. In fact, this result also extends to the case of strongly continuousone-parameter semigroupof contractive operators on a reflexive space.
Remark: Some intuition for the mean ergodic theorem can be developed by considering the case where complex numbers of unit length are regarded as unitary transformations on the complex plane (by left multiplication). If we pick a single complex number of unit length (which we think of asU), it is intuitive that its powers will fill up the circle. Since the circle is symmetric around 0, it makes sense that the averages of the powers ofUwill converge to 0. Also, 0 is the only fixed point ofU, and so the projection onto the space of fixed points must be the zero operator (which agrees with the limit just described).
Let (X, Σ,μ) be as above a probability space with a measure preserving transformationT, and let 1 ≤p≤ ∞. The conditional expectation with respect to the sub-σ-algebra ΣTof theT-invariant sets is a linear projectorETof norm 1 of the Banach spaceLp(X, Σ,μ) onto its closed subspaceLp(X, ΣT,μ). The latter may also be characterized as the space of allT-invariantLp-functions onX. The ergodic means, as linear operators onLp(X, Σ,μ) also have unit operator norm; and, as a simple consequence of the Birkhoff–Khinchin theorem, converge to the projectorETin thestrong operator topologyofLpif 1 ≤p≤ ∞, and in theweak operator topologyifp= ∞. More is true if 1 <p≤ ∞ then the Wiener–Yoshida–Kakutani ergodic dominated convergence theorem states that the ergodic means of ƒ ∈Lpare dominated inLp; however, if ƒ ∈L1, the ergodic means may fail to be equidominated inLp. Finally, if ƒ is assumed to be in the Zygmund class, that is |ƒ| log+(|ƒ|) is integrable, then the ergodic means are even dominated inL1.
Let (X, Σ,μ) be a measure space such thatμ(X) is finite and nonzero. The time spent in a measurable setAis called thesojourn time. An immediate consequence of the ergodic theorem is that, in an ergodic system, the relative measure ofAis equal to themean sojourn time:
for allxexcept for a set ofmeasurezero, where χAis theindicator functionofA.
Theoccurrence timesof a measurable setAis defined as the setk1,k2,k3, ..., of timesksuch thatTk(x) is inA, sorted in increasing order. The differences between consecutive occurrence timesRi=ki−ki−1are called therecurrence timesofA. Another consequence of the ergodic theorem is that the average recurrence time ofAis inversely proportional to the measure ofA, assuming[clarification needed]that the initial pointxis inA, so thatk0= 0.
(Seealmost surely.) That is, the smallerAis, the longer it takes to return to it.
The ergodicity of thegeodesic flowoncompactRiemann surfacesof variable negativecurvatureand on compactmanifolds of constant negative curvatureof any dimension was proved byEberhard Hopfin 1939, although special cases had been studied earlier: see for example,Hadamard's billiards(1898) andArtin billiard(1924). The relation between geodesic flows on Riemann surfaces and one-parameter subgroups onSL(2,R)was described in 1952 byS. V. FominandI. M. Gelfand. The article onAnosov flowsprovides an example of ergodic flows on SL(2,R) and on Riemann surfaces of negative curvature. Much of the development described there generalizes to hyperbolic manifolds, since they can be viewed as quotients of thehyperbolic spaceby theactionof alatticein the semisimple Lie groupSO(n,1). Ergodicity of the geodesic flow onRiemannian symmetric spaceswas demonstrated byF. I. Mautnerin 1957. In 1967D. V. AnosovandYa. G. Sinaiproved ergodicity of the geodesic flow on compact manifolds of variable negativesectional curvature. A simple criterion for the ergodicity of a homogeneous flow on ahomogeneous spaceof asemisimple Lie groupwas given byCalvin C. Moorein 1966. Many of the theorems and results from this area of study are typical ofrigidity theory.
In the 1930sG. A. Hedlundproved that the horocycle flow on a compact hyperbolic surface is minimal and ergodic. Unique ergodicity of the flow was established byHillel Furstenbergin 1972.Ratner's theoremsprovide a major generalization of ergodicity for unipotent flows on the homogeneous spaces of the form Γ \G, whereGis aLie groupand Γ is a lattice inG.
In the last 20 years, there have been many works trying to find a measure-classification theorem similar toRatner's theorems but for diagonalizable actions, motivated by conjectures of Furstenberg andMargulis. An important partial result (solving those conjectures with an extra assumption of positive entropy) was proved byElon Lindenstrauss, and he was awarded theFields medalin 2010 for this result. | https://en.wikipedia.org/wiki/Ergodic_theory |
Incomputer programming,machine codeiscomputer codeconsisting ofmachine languageinstructions, which are used to control a computer'scentral processing unit(CPU). For conventionalbinary computers, machine code is the binary[nb 1]representation of a computer program that is actually read and interpreted by the computer. A program in machine code consists of a sequence of machine instructions (possibly interspersed with data).[1]
Each machine code instruction causes the CPU to perform a specific task. Examples of such tasks include:
In general, each architecture family (e.g.,x86,ARM) has its owninstruction set architecture(ISA), and hence its own specific machine code language. There are exceptions, such as theVAXarchitecture, which includes optional support of thePDP-11instruction set; theIA-64architecture, which includes optional support of theIA-32instruction set; and thePowerPC 615microprocessor, which can natively process bothPowerPCand x86 instruction sets.
Machine code is a strictly numerical language, and it is the lowest-level interface to the CPU intended for a programmer.Assembly languageprovides a direct map between the numerical machine code and a human-readable mnemonic. In assembly, numericalopcodesand operands are replaced with mnemonics and labels. For example, thex86architecture has available the 0x90 opcode; it is represented asNOPin the assemblysource code. While it is possible to write programs directly in machine code, managing individual bits and calculating numericaladdressesis tedious and error-prone. Therefore, programs are rarely written directly in machine code. However, an existing machine code program may be edited if the assembly source code is not available.
The majority of programs today are written in ahigh-level language. A high-level program may be translated into machine code by acompiler.
Every processor or processor family has its owninstruction set. Machine instructions are patterns ofbits[nb 2]that specify some particular action.[2]An instruction set is described by itsinstruction format. Some ways in which instruction formats may differ:[2]
A processor's instruction set needs to execute the circuits of a computer'sdigital logic level. At the digital level, the program needs to control the computer's registers, bus, memory, ALU, and other hardware components.[3]To control a computer'sarchitecturalfeatures, machine instructions are created. Examples of features that are controlled using machine instructions:
The criteria for instruction formats include:
Determining the size of the address field is a choice between space and speed.[7]On some computers, the number of bits in the address field may be too small to access all of the physical memory. Also,virtual address spaceneeds to be considered. Another constraint may be a limitation on the size of registers used to construct the address. Whereas a shorter address field allows the instructions to execute more quickly, other physical properties need to be considered when designing the instruction format.
Instructions can be separated into two types: general-purpose and special-purpose. Special-purpose instructions exploit architectural features that are unique to a computer. General-purpose instructions control architectural features common to all computers.[8]
General-purpose instructions control:
A much more human-friendly rendition of machine language, namedassembly language, usesmnemonic codesto refer to machine code instructions, rather than using the instructions' numeric values directly, and usessymbolic namesto refer to storage locations and sometimesregisters.[9]For example, on theZilog Z80processor, the machine code00000101, which causes the CPU to decrement theBgeneral-purpose register, would be represented in assembly language asDEC B.[10]
TheIBM 704, 709, 704x and 709xstore one instruction in each instruction word; IBM numbers the bit from the left as S, 1, ..., 35. Most instructions have one of two formats:
For all but theIBM 7094and 7094 II, there are three index registers designated A, B and C; indexing with multiple 1 bits in the tag subtracts thelogical orof the selected index registers and loading with multiple 1 bits in the tag loads all of the selected index registers. The 7094 and 7094 II have seven index registers, but when they are powered on they are inmultiple tag mode, in which they use only the three of the index registers in a fashion compatible with earlier machines, and require a Leave Multiple Tag Mode (LMTM) instruction in order to access the other four index registers.
The effective address is normally Y-C(T), where C(T) is either 0 for a tag of 0, the logical or of the selected index registers in multiple tag mode or the selected index register if not in multiple tag mode. However, the effective address for index register control instructions is just Y.
A flag with both bits 1 selects indirect addressing; the indirect address word has both a tag and a Y field.
In addition totransfer(branch) instructions, these machines have skip instruction that conditionally skip one or two words, e.g., Compare Accumulator with Storage (CAS) does a three way compare and conditionally skips to NSI, NSI+1 or NSI+2, depending on the result.
TheMIPS architectureprovides a specific example for a machine code whose instructions are always 32 bits long.[11]: 299The general type of instruction is given by theop(operation) field, the highest 6 bits. J-type (jump) and I-type (immediate) instructions are fully specified byop. R-type (register) instructions include an additional fieldfunctto determine the exact operation. The fields used in these types are:
rs,rt, andrdindicate register operands;shamtgives a shift amount; and theaddressorimmediatefields contain an operand directly.[11]: 299–301
For example, adding the registers 1 and 2 and placing the result in register 6 is encoded:[11]: 554
Load a value into register 8, taken from the memory cell 68 cells after the location listed in register 3:[11]: 552
Jumping to the address 1024:[11]: 552
On processor architectures withvariable-length instruction sets[12](such asIntel'sx86processor family) it is, within the limits of the control-flowresynchronizingphenomenon known as theKruskal count,[13][12][14][15][16]sometimes possible through opcode-level programming to deliberately arrange the resulting code so that two code paths share a common fragment of opcode sequences.[nb 3]These are calledoverlapping instructions,overlapping opcodes,overlapping code,overlapped code,instruction scission, orjump into the middle of an instruction.[17][18][19]
In the 1970s and 1980s, overlapping instructions were sometimes used to preserve memory space. One example were in the implementation of error tables inMicrosoft'sAltair BASIC, whereinterleaved instructionsmutually shared their instruction bytes.[20][12][17]The technique is rarely used today, but might still be necessary to resort to in areas where extreme optimization for size is necessary on byte-level such as in the implementation ofboot loaderswhich have to fit intoboot sectors.[nb 4]
It is also sometimes used as acode obfuscationtechnique as a measure againstdisassemblyand tampering.[12][15]
The principle is also used in shared code sequences offat binarieswhich must run on multiple instruction-set-incompatible processor platforms.[nb 3]
This property is also used to findunintended instructionscalledgadgetsin existing code repositories and is used inreturn-oriented programmingas alternative tocode injectionfor exploits such asreturn-to-libc attacks.[21][12]
In some computers, the machine code of thearchitectureis implemented by an even more fundamental underlying layer calledmicrocode, providing a common machine language interface across a line or family of different models of computer with widely different underlyingdataflows. This is done to facilitateportingof machine language programs between different models. An example of this use is the IBMSystem/360family of computers and their successors.
Machine code is generally different frombytecode(also known as p-code), which is either executed by an interpreter or itself compiled into machine code for faster (direct) execution. An exception is when a processor is designed to use a particular bytecode directly as its machine code, such as is the case withJava processors.
Machine code and assembly code are sometimes callednativecodewhen referring to platform-dependent parts of language features or libraries.[22]
From the point of view of the CPU, machine code is stored in RAM, but is typically also kept in a set of caches for performance reasons. There may be different caches for instructions and data, depending on the architecture.
The CPU knows what machine code to execute, based on its internal program counter. The program counter points to a memory address and is changed based on special instructions which may cause programmatic branches. The program counter is typically set to a hard coded value when the CPU is first powered on, and will hence execute whatever machine code happens to be at this address.
Similarly, the program counter can be set to execute whatever machine code is at some arbitrary address, even if this is not valid machine code. This will typically trigger an architecture specific protection fault.
The CPU is oftentimes told, by page permissions in a paging based system, if the current page actually holds machine code by an execute bit — pages have multiple such permission bits (readable, writable, etc.) for various housekeeping functionality. E.g. onUnix-likesystems memory pages can be toggled to be executable with themprotect()system call, and on Windows,VirtualProtect()can be used to achieve a similar result. If an attempt is made to execute machine code on a non-executable page, an architecture specific fault will typically occur. Treatingdata as machine code, or finding new ways to use existing machine code, by various techniques, is the basis of some security vulnerabilities.
Similarly, in a segment based system, segment descriptors can indicate whether a segment can contain executable code and in whatringsthat code can run.
From the point of view of aprocess, thecode spaceis the part of itsaddress spacewhere the code in execution is stored. Inmultitaskingsystems this comprises the program'scode segmentand usuallyshared libraries. Inmulti-threadingenvironment, different threads of one process share code space along with data space, which reduces the overhead ofcontext switchingconsiderably as compared to process switching.
Machine code can be seen as a set of electrical pulses that make the instructions readable to the computer; it is not readable by humans,[23]withDouglas Hofstadtercomparing it to examining the atoms of aDNAmolecule.[24]However, various tools and methods exist to decode machine code to human-readablesource code. One such method isdisassembly, which easily decodes it back to its corresponding assembly languagesource codebecause assembly language forms a one-to-one mapping to machine code.[25]
Machine code may also be decoded tohigh-level languageunder two conditions. The first condition is to accept anobfuscatedreading of the source code. An obfuscated version of source code is displayed if the machine code is sent to adecompilerof the source language. The second condition requires the machine code to have information about the source code encoded within. The information includes asymbol tablethat containsdebug symbols. The symbol table may be stored within the executable, or it may exist in separate files. Adebuggercan then read the symbol table to help the programmer interactivelydebugthe machine code inexecution. | https://en.wikipedia.org/wiki/Overlapping_instructions |
Incoding theory, especially intelecommunications, aself-synchronizing codeis auniquely decodable codein which thesymbolstream formed by a portion of onecode word, or by the overlapped portion of any two adjacent code words, is not a valid code word.[1]Put another way, a set of strings (called "code words") over an alphabet is called a self-synchronizing code if for each string obtained by concatenating two code words, the substring starting at the second symbol and ending at the second-last symbol does not contain any code word as substring. Every self-synchronizing code is aprefix code, but not all prefix codes are self-synchronizing.
Other terms for self-synchronizing code aresynchronized code[2]or, ambiguously,comma-free code.[3]A self-synchronizing code permits the properframingof transmitted code words provided that no uncorrected errors occur in thesymbol stream; externalsynchronizationis not required. Self-synchronizing codes also allow recovery from uncorrected errors in the stream; with most prefix codes, an uncorrected error in a singlebitmay propagate errors further in the stream and make the subsequent datacorrupted.
Importance of self-synchronizing codes is not limited todata transmission. Self-synchronization also facilitates some cases ofdata recovery, for example of adigitally encoded text.
Counterexamples: | https://en.wikipedia.org/wiki/Self-synchronizing_code |
Aninstruction set architecture(ISA) is an abstract model of acomputer, also referred to ascomputer architecture. A realization of an ISA is called animplementation. An ISA permits multiple implementations that may vary inperformance, physical size, and monetary cost (among other things); because the ISA serves as theinterfacebetweensoftwareandhardware. Software that has been written for an ISA can run on different implementations of the same ISA. This has enabledbinary compatibilitybetween different generations of computers to be easily achieved, and the development of computer families. Both of these developments have helped to lower the cost of computers and to increase their applicability. For these reasons, the ISA is one of the most important abstractions incomputingtoday.
An ISA defines everything amachine languageprogrammerneeds to know in order to program a computer. What an ISA defines differs between ISAs; in general, ISAs define the supporteddata types, what state there is (such as themain memoryandregisters) and their semantics (such as thememory consistencyandaddressing modes), theinstruction set(the set ofmachine instructionsthat comprises a computer's machine language), and theinput/outputmodel.
In the early decades of computing, there were computers that usedbinary,decimal[1]and eventernary.[2][3]Contemporary computers are almost exclusively binary.
Characters are encoded as strings of bits or digits, using a wide variety of character sets; even within a single manufacturer there were character set differences.
Integers are encoded with a variety ofrepresentations, includingSign-magnitude,Ones' complement,Two's complement,Offset binary,Nines' complementandTen's complement.
Similarly, floating point numbers are encoded with a variety of representations for the sign,exponentandmantissa. In contemporary machinesIBM hexadecimal floating-pointandIEEE 754floating point have largely supplanted older formats.
Addresses are typically unsigned integers generated from a combination of fields in an instruction, data from registers and data from storage; the details vary depending on the architecture.
Computer architecturesare often described asn-bitarchitectures. In the first3⁄4of the 20th century,nis often12,18,24, 30,36,48or60. In the last1⁄3of the 20th century,nis often 8, 16, or 32, and in the 21st century,nis often 16, 32 or 64, but other sizes have been used (including 6,39,128). This is actually a simplification as computer architecture often has a few more or less "natural" data sizes in theinstruction set, but the hardware implementation of these may be very different. Many instruction set architectures have instructions that, on some implementations of that instruction set architecture, operate on half and/or twice the size of the processor's major internal datapaths. Examples of this are theZ80,MC68000, and theIBM System/360. On these types of implementations, a twice as wide operation typically also takes around twice as many clock cycles (which is not the case on high performance implementations). On the 68000, for instance, this means 8 instead of 4 clock ticks, and this particular chip may be described as a32-bitarchitecture with a16-bitimplementation. The IBM System/360 instruction set architecture is 32-bit, but several models of the System/360 series, such as theIBM System/360 Model 30, have smaller internal data paths, while others, such as the360/195, have larger internal data paths. The external databus width is not used to determine the width of the architecture; theNS32008, NS32016 and NS32032were basically the same 32-bit chip with different external data buses; the NS32764 had a64-bitbus, and used 32-bit register. Early 32-bit microprocessors often had a 24-bit address, as did the System/360 processors.
In the first3⁄4of the 20th century, word oriented decimal computers typically had 10 digit[4][5][6]words with a separate sign,[a]using all ten digits in integers and using two digits for exponents[7][5]in floating point numbers.
An architecture may use "big" or "little" endianness, or both, or be configurable to use either. Little-endian processors orderbytesin memory with the least significant byte of a multi-byte value in the lowest-numbered memory location. Big-endian architectures instead arrange bytes with the most significant byte at the lowest-numbered address. The x86 architecture as well as several8-bitarchitectures are little-endian. MostRISCarchitectures (SPARC, Power, PowerPC, MIPS) were originally big-endian (ARM was little-endian), but many (including ARM) are now configurable as either.
Endiannessonlyapplies to processors that allow individual addressing of units of data (such asbytes) that aresmallerthan some of the data formats.
In some architectures, an instruction has a single opcode. In others, some instructions have an opcode and one or more modifiers. E.g., on theIBM System/370, byte 0 is the opcode but when byte 0 is a B216then byte 1 selects a specific instruction, e.g., B20516is store clock (STCK).
Architectures typically allow instructions to include some combination of operandaddressing modes:
The number of operands is one of the factors that may give an indication about the performance of the instruction set.
A three-operand architecture (2-in, 1-out) will allow
to be computed in one instruction
A two-operand architecture (1-in, 1-in-and-out) will allow
to be computed in one instruction
but requires that
be done in two instructions
As can be seen in the table below some instructions sets keep to a very simple fixed encoding length, and other have variable-length. Usually it isRISCarchitectures that have fixed encoding length andCISCarchitectures that have variable length, but not always.
The table below compares basic information about instruction set architectures.
Notes:
8 data registers
8 pointer registers
4 index registers
4 buffer registers
1 multiplier quotient register
Test and branch | https://en.wikipedia.org/wiki/Comparison_of_instruction_set_architectures |
Acompressed instruction set, or simplycompressed instructions, are a variation on amicroprocessor'sinstruction set architecture(ISA) that allows instructions to be represented in a more compact format. In most real-world examples, compressed instructions are 16 bits long in a processor that would otherwise use 32-bit instructions. The 16-bit ISA is a subset of the full 32-bit ISA, not a separate instruction set. The smaller format requires some tradeoffs: generally, there are fewer instructions available, and fewerprocessor registerscan be used.
The concept was originally introduced byHitachias a way to improve thecode densityof theirSuperHRISCprocessor design as it moved from 16-bit to 32-bit instructions in the SH-5 version. The new design had two instruction sets, one giving access to the entire ISA of the new design, and a smaller 16-bit set known as SHcompact that allowed programs to run in smaller amounts ofmain memory. As the memory of even the smallest systems is noworders of magnitudelarger than the systems that spawned the concept, size is no longer the main concern. Today the advantage is that it reduces the number of accesses to main memory and thereby reduces energy use inmobile devices.
Hitachi's patents were licensed byArm Ltd.for their processors, where it was known as "Thumb". Similar systems are found in MIPS16e andPowerPCVLE. The original patents have expired and the concept can be found in a number of modern designs, includingRISC-V, which was designed from the outset to use it. The introduction of64-bit computinghas led to the term no longer being as widely used; these processors generally use 32-bit instructions and are technically a form of compressed ISA, but as they are mostly modified versions of an older ISA from a 32-bit version of the same processor family; there is no real compression.
Microprocessorsencode their instructions as a series ofbits, normally divided into a number of 8-bitbytes. For instance, in theMOS 6502, theADCinstruction performs binary addition between an operand value and the value already stored in theaccumulator. There are a variety of places the processor might find the operand; it might be located inmain memory, or in the specialzero page, or be an explicit constant like "10". Each of these variations used a different 8-bit instruction, oropcode; if one wanted to add the constant 10 to the accumulator the instruction would be encoded in memory as$69 $0A, with $0A beinghexadecimalfor the decimal value 10. If it was instead adding the value stored in main memory at location $4400, it would be$6D $00 $44, with alittle-endianaddress.[1]
Note that the second instruction requires three bytes because the memory address is 16 bits long. Depending on the instruction, it might use one, two, or three bytes.[1]This is now known as avariable length instruction set, although that term was not common at the time as most processors, includingmainframesandminicomputers, normally used some variation of this concept. Even in the late 1970s, as microprocessors began to move from 8-bit formats to 16, this concept remained common; theIntel 8086continued to use 8-bit opcodes which could be followed by zero to five additional bytes depending on theaddressing mode.[2]
It was during the move to 32-bit systems, and especially as theRISCconcept began to take over processor design, that variable length instructions began to go away. In theMIPS architecture, for instance, all instructions are a single 32-bit value, with a 6-bit opcode in themost significant bitsand the remaining 26 bits used in various ways representing its limited set of addressing modes. Most RISC designs are similar. Moving to a fixed-length instruction format was one of the key design concepts behind the performance of early RISC designs; in earlier systems the instruction might take one to six memory cycles to read, requiring wiring between various parts of the logic to ensure the processor didn't attempt to perform the instruction before the data was ready. In RISC designs, operations normally take one cycle, greatly simplifying the decoding. The savings in these interlocking circuits is instead applied to additional logic or addingprocessor registers, which have a direct impact on performance.[3]
The downside to the RISC approach is that many instructions simply do not require four bytes. For instance, theLogical Shift Leftinstruction shifts the bits in a register to the left. In the 6502, which has only a single arithmetic register A, this instruction can be represented entirely by its 8-bit opcode$06.[1]On processors with more registers, all that is needed is the opcode and register number, another 4 or 5 bits. On MIPS, for instance, the instruction needs only a 6-bit opcode and a 5-bit register number. But as is the case for most RISC designs, the instruction still takes up a full 32 bits. As these sorts of instructions are relatively common, RISC programs generally take up more memory than the same program on a variable length processor.[4]
One notable, and particularly early, exception amongst RISC designs is theIBM 801architecture which maintains five instruction formats: two utilising a 16-bit instruction length, and three utilising a 32-bit instruction length.[5]: 10For instructions requiring less space, such as shift instructions employing only register operands, the shorter 16-bit instruction formats are used.[5]: 51–58
In the 1980s, when the RISC concept was first emerging, increased program size was a common point of complaint. As the instructions took up more room, the system would have to spend more time reading instructions from memory. It was suggested these extra accesses might actually slow the program down. Extensivebenchmarkingeventually demonstrated RISC was faster in almost all cases, and this argument faded. However, there are cases where memory use remains a concern regardless of performance, and that is in small systems and embedded applications. Even in the early 2000s, the price ofDRAMwas enough that cost-sensitive devices had limited memory. It was for this market thatHitachideveloped theSuperHdesign.[6]
In the earlier SuperH designs, SH-1 through SH-4, instructions always take up 16 bits. The resulting instruction set has real-world limitations; for instance, it can only perform two-operand math of the formA = A + B, whereas most processors of the era used the three-operand format,A = B + C. By removing one operand, four bits are removed from the instruction (there are 16 registers, needing 4 bits), although this is at the cost of making math code somewhat more complex to write. For the markets targeted by the SuperH, this was an easy tradeoff to make. A significant advantage of the 16-bit format is that theinstruction cachenow holds twice as many instructions for any given amount ofSRAM. This allows the system to perform at higher speeds, although some of that might be mitigated by the use of additional instructions needed to perform operations that might be performed by a single 3-operand instruction.[7]
For the SH-5, Hitachi moved to a 32-bit instruction format. In order to providebackward compatibilitywith their earlier designs, they included a second instruction set, SHcompact. SHcompact mapped the original 16-bit instructions one-way onto the internal 32-bit instruction; it did not perform multiple instructions as would be the case in earliermicrocodedprocessors, it was simply a smaller format for the same instruction. This allowed the original small-format programs to be easily ported to the new SH-5, while adding little to the complexity of theinstruction decoder.[8]
ARM licensed a number of Hitachi's patents on aspects of the instruction design and used them to implement their Thumb instructions. ARM processors with a "T" in the name included this instruction set in addition to their original 32-bit versions, and could be switched from 32- to 16-bit mode on the fly using theBXcommand. When in Thumb mode, only the top eight registers of the ARM's normal sixteen registers are visible, but these are the same registers as in 32-bit mode and thus data can be passed between Thumb and normal code using those registers. Every Thumb instruction was a counterpart of a 32-bit version, so Thumb was a strict subset of the original ISA.[9]One key difference between ARM's model and SuperH is that Thumb retains some three-operand instructions in the 16-bit format, which it accomplished by reducing the visible register file to eight, so only 3 bits are required to select a register.[10]
TheMIPS architecturealso added a similar compressed set in their MIPS16e, which is very similar to Thumb. It too allows only eight registers to be used, although these are not simply the first eight; the MIPS design uses register 0 as thezero register, so registers 0 and 1 in 16-bit mode are instead mapped onto MIPS32 registers 16 and 17. Most other details of the system are similar to Thumb.[11]Likewise, the latest version of thePower ISA, formerlyPowerPC, include the "VLE" instructions which are essentially identical. These were added at the behest ofFreescale Semiconductor, whose interest in Power is mostly aimed at the embedded market.[12]
Starting around 2015, many processors have moved to a 64-bit format. These generally retained a 32-bit instruction format, while expanding the internal registers to a 64-bit format. By the original definition, these are compressed instructions, as they are smaller than the basic data word size. However, this term is not used in this context; references to compressed instructions invariably refer to 16-bit versions.[13] | https://en.wikipedia.org/wiki/Compressed_instruction_set |
Incomputer scienceandcomputer engineering,computer architectureis a description of the structure of acomputersystem made from component parts.[1]It can sometimes be a high-level description that ignores details of the implementation.[2]At a more detailed level, the description may include theinstruction set architecturedesign,microarchitecturedesign,logic design, andimplementation.[3]
The first documented computer architecture was in the correspondence betweenCharles BabbageandAda Lovelace, describing theanalytical engine. While building the computerZ1in 1936,Konrad Zusedescribed in two patent applications for his future projects that machine instructions could be stored in the same storage used for data, i.e., thestored-programconcept.[4][5]Two other early and important examples are:
The term "architecture" in computer literature can be traced to the work of Lyle R. Johnson andFrederick P. Brooks, Jr., members of the Machine Organization department in IBM's main research center in 1959. Johnson had the opportunity to write a proprietary research communication about theStretch, an IBM-developedsupercomputerforLos Alamos National Laboratory(at the time known as Los Alamos Scientific Laboratory). To describe the level of detail for discussing the luxuriously embellished computer, he noted that his description of formats, instruction types, hardware parameters, and speed enhancements were at the level of "system architecture", a term that seemed more useful than "machine organization".[8]
Subsequently, Brooks, a Stretch designer, opened Chapter 2 of a book calledPlanning a Computer System: Project Stretchby stating, "Computer architecture, like other architecture, is the art of determining the needs of the user of a structure and then designing to meet those needs as effectively as possible within economic and technological constraints."[9]
Brooks went on to help develop theIBM System/360line of computers, in which "architecture" became a noun defining "what the user needs to know".[10]The System/360 line was succeeded by several compatible lines of computers, including the currentIBM Zline. Later, computer users came to use the term in many less explicit ways.[11]
The earliest computer architectures were designed on paper and then directly built into the final hardware form.[12]Later, computer architecture prototypes were physically built in the form of atransistor–transistor logic(TTL) computer—such as the prototypes of the6800and thePA-RISC—tested, and tweaked, before committing to the final hardware form.
As of the 1990s, new computer architectures are typically "built", tested, and tweaked—inside some other computer architecture in acomputer architecture simulator; or inside a FPGA as asoft microprocessor; or both—before committing to the final hardware form.[13]
The discipline of computer architecture has three main subcategories:[14]
There are other technologies in computer architecture. The following technologies are used in bigger companies like Intel, and were estimated in 2002[14]to count for 1% of all of computer architecture:
Computer architecture is concerned with balancing the performance, efficiency, cost, and reliability of a computer system. The case of instruction set architecture can be used to illustrate the balance of these competing factors. More complexinstruction setsenable programmers to write more space efficient programs, since a single instruction can encode some higher-level abstraction (such as thex86 Loop instruction).[16]However, longer and more complex instructions take longer for theprocessorto decode and can be more costly to implement effectively. The increased complexity from a large instruction set also creates more room for unreliability when instructions interact in unexpected ways.
The implementation involvesintegrated circuit design, packaging,power, andcooling. Optimization of the design requires familiarity with topics fromcompilersandoperating systemstologic designand packaging.[17]
Aninstruction set architecture(ISA) is the interface between the computer's software and hardware and also can be viewed as the programmer's view of the machine. Computers do not understandhigh-level programming languagessuch asJava,C++, or most programming languages used. A processor only understands instructions encoded in some numerical fashion, usually asbinary numbers. Software tools, such ascompilers, translate those high level languages into instructions that the processor can understand.
Besides instructions, the ISA defines items in the computer that are available to a program—e.g.,data types,registers,addressing modes, andmemory. Instructions locate these available items with register indexes (or names) and memory addressing modes.
The ISA of a computer is usually described in a small instruction manual, which describes how the instructions are encoded. Also, it may define short (vaguely) mnemonic names for the instructions. The names can be recognized by a software development tool called anassembler. An assembler is a computer program that translates a human-readable form of the ISA into a computer-readable form.Disassemblersare also widely available, usually indebuggersand software programs to isolate and correct malfunctions in binary computer programs.
ISAs vary in quality and completeness. A good ISA compromises betweenprogrammerconvenience (how easy the code is to understand), size of the code (how much code is required to do a specific action), cost of thecomputerto interpret the instructions (more complexity means more hardware needed to decode and execute the instructions), and speed of the computer (with more complex decoding hardware comes longer decode time).Memory organizationdefines how instructions interact with the memory, and how memory interacts with itself.
During designemulation, emulators can run programs written in a proposed instruction set. Modern emulators can measure size, cost, and speed to determine whether a particular ISA is meeting its goals.
Computer organization helps optimize performance-based products. For example, software engineers need to know theprocessing powerofprocessors. They may need to optimize software in order to gain the most performance for the lowest price. This can require quite a detailed analysis of the computer's organization. For example, in anSD card, the designers might need to arrange the card so that the most data can be processed in the fastest possible way.
Computer organization also helps plan the selection of a processor for a particular project.Multimediaprojects may need very rapid data access, whilevirtual machinesmay need fast interrupts. Sometimes certain tasks need additional components as well. For example, a computer capable of running a virtual machine needsvirtual memoryhardware so that the memory of different virtual computers can be kept separated. Computer organization and features also affect power consumption and processor cost.
Once aninstruction setandmicroarchitecturehave been designed, a practical machine must be developed. This design process is called theimplementation. Implementation is usually not considered architectural design, but rather hardwaredesign engineering. Implementation can be further broken down into several steps:
ForCPUs, the entire implementation process is organized differently and is often referred to asCPU design.
The exact form of a computer system depends on the constraints and goals. Computer architectures usually trade off standards,powerversusperformance, cost, memory capacity,latency(latency is the amount of time that it takes for information from one node to travel to the source) and throughput. Sometimes other considerations, such as features, size, weight, reliability, and expandability are also factors.
The most common scheme does an in-depth power analysis and figures out how to keep power consumption low while maintaining adequate performance.
Modern computer performance is often described ininstructions per cycle(IPC), which measures the efficiency of the architecture at any clock frequency; a faster IPC rate means the computer is faster. Older computers had IPC counts as low as 0.1 while modern processors easily reach nearly 1.Superscalarprocessors may reach three to five IPC by executing several instructions per clock cycle.[citation needed]
Counting machine-language instructions would be misleading because they can do varying amounts of work in different ISAs. The "instruction" in the standard measurements is not a count of the ISA's machine-language instructions, but a unit of measurement, usually based on the speed of theVAXcomputer architecture.
Many people used to measure a computer's speed by theclock rate(usually inMHzor GHz). This refers to the cycles per second of the main clock of theCPU. However, this metric is somewhat misleading, as a machine with a higher clock rate may not necessarily have greater performance. As a result, manufacturers have moved away from clock speed as a measure of performance.
Other factors influence speed, such as the mix offunctional units,busspeeds, available memory, and the type and order of instructions in the programs.
There are two main types of speed:latencyandthroughput. Latency is the time between the start of a process and its completion. Throughput is the amount of work done per unit time.Interrupt latencyis the guaranteed maximum response time of the system to an electronic event (like when the disk drive finishes moving some data).
Performance is affected by a very wide range of design choices — for example,pipelininga processor usually makes latency worse, but makes throughput better. Computers that control machinery usually need low interrupt latencies. These computers operate in areal-timeenvironment and fail if an operation is not completed in a specified amount of time. For example, computer-controlled anti-lock brakes must begin braking within a predictable and limited time period after the brake pedal is sensed or else failure of the brake will occur.
Benchmarkingtakes all these factors into account by measuring the time a computer takes to run through a series of test programs. Although benchmarking shows strengths, it should not be how you choose a computer. Often the measured machines split on different measures. For example, one system might handle scientific applications quickly, while another might rendervideo gamesmore smoothly. Furthermore, designers may target and add special features to their products, through hardware or software, that permit a specific benchmark to execute quickly but do not offer similar advantages to general tasks.
Power efficiency is another important measurement in modern computers. Higher power efficiency can often be traded for lower speed or higher cost. The typical measurement when referring to power consumption in computer architecture is MIPS/W (millions of instructions per second per watt).
Modern circuits have less power required pertransistoras the number of transistors per chip grows.[18]This is because each transistor that is put in a new chip requires its own power supply and requires new pathways to be built to power it.[clarification needed]However, the number of transistors per chip is starting to increase at a slower rate. Therefore, power efficiency is starting to become as important, if not more important than fitting more and more transistors into a single chip. Recent processor designs have shown this emphasis as they put more focus on power efficiency rather than cramming as many transistors into a single chip as possible.[19]In the world ofembedded computers, power efficiency has long been an important goal next to throughput and latency.
Increases in clock frequency have grown more slowly over the past few years, compared to power reduction improvements. This has been driven by the end ofMoore's Lawand demand for longerbattery lifeand reductions in size formobile technology. This change in focus from higher clock rates to power consumption and miniaturization can be shown by the significant reductions in power consumption, as much as 50%, that were reported byIntelin their release of theHaswell microarchitecture; where they dropped their power consumption benchmark from 30–40wattsdown to 10–20 watts.[20]Comparing this to the processing speed increase of 3 GHz to 4 GHz (2002 to 2006), it can be seen that the focus in research and development is shifting away from clock frequency and moving towards consuming less power and taking up less space.[21] | https://en.wikipedia.org/wiki/Computer_architecture |
[1]Incomputing, anemulatorishardwareorsoftwarethat enables onecomputer system(called thehost) to behave like another computer system (called theguest). An emulator typically enables the host system to run software or use peripheral devices designed for the guest system.
Emulation refers to the ability of acomputer programin an electronic device to emulate (or imitate) another program or device.
Manyprinters, for example, are designed to emulateHPLaserJetprinters because a significant amount of software is written specifically for HP models. If a non-HP printer emulates an HP printer, any software designed for an actual HP printer will also function on the non-HP device, producing equivalent print results. Since at least the 1990s, manyvideo gameenthusiasts and hobbyists have used emulators to play classicarcade gamesfrom the 1980s using the games' original 1980s machine code and data, which is interpreted by a current-era system, and to emulate oldvideo game consoles(seevideo game console emulator).
A hardware emulator is an emulator which takes the form of a hardware device. Examples include the DOS-compatible card installed in some 1990s-eraMacintoshcomputers, such as theCentris 610orPerforma 630, that allowed them to runpersonal computer(PC) software programs andfield-programmable gate array-basedhardware emulators. TheChurch–Turing thesisimplies that theoretically, any operating environment can be emulated within any other environment, assuming memory limitations are ignored. However, in practice, it can be quite difficult, particularly when the exact behavior of the system to be emulated is not documented and has to be deduced throughreverse engineering. It also says nothing about timing constraints; if the emulator does not perform as quickly as it did using the original hardware, the software inside the emulation may run much more slowly (possibly triggering timer interrupts that alter behavior).
"Can aCommodore 64emulateMS-DOS?"
Yes, it's possible for a [Commodore] 64 to emulate an IBM PC [which uses MS-DOS], in the same sense that it's possible to bail outLake Michiganwith ateaspoon.
Most emulators just emulate a hardware architecture—if operating system firmware or software is required for the desired software, it must be provided as well (and may itself be emulated). Both the OS and the software will then beinterpretedby the emulator, rather than being run by native hardware. Apart from this interpreter for the emulated binarymachine's language, some other hardware (such as input or output devices) must be provided in virtual form as well; for example, if writing to a specific memory location should influence what is displayed on the screen, then this would need to be emulated. While emulation could, if taken to the extreme, go down to the atomic level, basing its output on a simulation of the actual circuitry from a virtual power source, this would be a highly unusual solution. Emulators typically stop at a simulation of the documented hardware specifications and digital logic. Sufficient emulation of some hardware platforms requires extreme accuracy, down to the level of individual clock cycles, undocumented features, unpredictable analog elements, and implementation bugs. This is particularly the case with classic home computers such as theCommodore 64, whose software often depends on highly sophisticated low-level programming tricks invented by game programmers and the "demoscene".
In contrast, some other platforms have had very little use of direct hardware addressing, such as an emulator for the PlayStation 4.[citation needed]In these cases, a simplecompatibility layermay suffice. This translates system calls for the foreign system into system calls for the host system e.g., the Linux compatibility layer used on *BSD to run closed source Linux native software onFreeBSDandNetBSD.[3]For example, while theNintendo 64graphic processor was fully programmable, most games used one of a few pre-made programs, which were mostly self-contained and communicated with the game viaFIFO; therefore, many emulators do not emulate the graphic processor at all, but simply interpret the commands received from the CPU as the original program would. Developers of software forembedded systemsorvideo game consolesoften design their software on especially accurate emulators calledsimulatorsbefore trying it on the real hardware. This is so that software can be produced and tested before the final hardware exists in large quantities, so that it can be tested without taking the time to copy the program to be debugged at a low level and without introducing the side effects of adebugger. In many cases, the simulator is actually produced by the company providing the hardware, which theoretically increases its accuracy. Math co-processor emulators allow programs compiled with math instructions to run on machines that do not have the co-processor installed, but the extra work done by the CPU may slow the system down. If a math coprocessor is not installed or present on the CPU, when the CPU executes any co-processor instruction it will make a determined interrupt (coprocessor not available), calling the math emulator routines. When the instruction is successfully emulated, the program continues executing.
Logic simulation is the use of a computer program to simulate the operation of a digital circuit such as a processor.[1]This is done after a digital circuit has been designed in logic equations, but before the circuit is fabricated in hardware.
Functional simulation is the use of a computer program to simulate the execution of a second computer program written in symbolicassembly languageorcompilerlanguage, rather than in binarymachine code. By using a functional simulator, programmers can execute and trace selected sections of source code to search for programming errors (bugs), without generating binary code. This is distinct from simulating execution of binary code, which is software emulation. The first functional simulator was written byAutoneticsabout 1960[citation needed]for testing assembly language programs for later execution in military computerD-17B. This made it possible for flight programs to be written, executed, and tested before D-17B computer hardware had been built. Autonetics also programmed a functional simulator for testing flight programs for later execution in the military computerD-37C.
Video game console emulators are programs that allow a personal computer or video game console to emulate another video game console. They are most often used to play older 1980s to 2000s-era video games on modern personal computers and more contemporary video game consoles. They are also used to translate games into other languages, to modify existing games, and in the development process of "home brew"DIYdemos and in the creation of new games for older systems. TheInternethas helped in the spread of console emulators, as most - if not all - would be unavailable for sale in retail outlets. Examples of console emulators that have been released in the last few decades are:RPCS3,Dolphin,Cemu,PCSX2,PPSSPP,ZSNES,Citra,ePSXe,Project64,Visual Boy Advance,Nestopia, andYuzu.
Due to their popularity, emulators have been impersonated by malware. Most of these emulators are for video game consoles like the Xbox 360, Xbox One, Nintendo 3DS, etc. Generally such emulators make currently impossible claims such as being able to runXbox OneandXbox 360games in a single program.[4]
As computers andglobal computer networkscontinued to advance and emulator developers grew more skilled in their work, the length of time between the commercial release of a console and its successful emulation began to shrink.Fifth generationconsoles such asNintendo 64,PlayStationandsixth generationhandhelds, such as theGame Boy Advance, saw significant progress toward emulation during their production. This led to an effort by console manufacturers to stop unofficial emulation, but consistent failures such asSega v. Accolade977 F.2d 1510 (9th Cir. 1992),Sony Computer Entertainment, Inc. v. Connectix Corporation203 F.3d 596 (2000), andSony Computer Entertainment America v. Bleem214 F.3d 1022 (2000),[5]have had the opposite effect. According to all legal precedents, emulation is legal within the United States. However, unauthorized distribution of copyrighted code remains illegal, according to both country-specificcopyrightand international copyright law under theBerne Convention.[6][better source needed]Under United States law, obtaining adumpedcopy of the original machine'sBIOSis legal under the rulingLewis Galoob Toys, Inc. v. Nintendo of America, Inc., 964 F.2d 965 (9th Cir. 1992) asfair useas long as the user obtained a legally purchased copy of the machine. To mitigate this however, several emulators for platforms such asGame Boy Advanceare capable of running without a BIOS file, using high-level emulation to simulate BIOS subroutines at a slight cost in emulation accuracy.[7][8][9]
Terminal emulators are software programs that provide modern computers and devices interactive access to applications running onmainframe computeroperating systems or other host systems such asHP-UXorOpenVMS. Terminals such as theIBM 3270orVT100and many others are no longer produced as physical devices. Instead, software running on modern operating systems simulates a "dumb" terminal and is able to render the graphical and text elements of the host application, send keystrokes and process commands using the appropriate terminal protocol. Some terminal emulation applications includeAttachmate Reflection,IBM Personal Communications, andMicro FocusRumba.
Other types of emulators include:
Typically, an emulator is divided intomodulesthat correspond roughly to the emulated computer's subsystems.
Most often, an emulator will be composed of the following modules:
Buses are often not emulated, either for reasons of performance or simplicity, and virtual peripherals communicate directly with the CPU or the memory subsystem.
It is possible for the memory subsystem emulation to be reduced to simply an array of elements each sized like an emulated word; however, this model fails very quickly as soon as any location in the computer's logical memory does not matchphysical memory. This clearly is the case whenever the emulated hardware allows for advanced memory management (in which case, theMMUlogic can be embedded in the memory emulator, made a module of its own, or sometimes integrated into the CPU simulator). Even if the emulated computer does not feature an MMU, though, there are usually other factors that break the equivalence between logical and physical memory: many (if not most) architectures offermemory-mapped I/O; even those that do not often have a block of logical memory mapped toROM, which means that the memory-array module must be discarded if the read-only nature of ROM is to be emulated. Features such asbank switchingorsegmentationmay also complicate memory emulation. As a result, most emulators implement at least two procedures for writing to and reading from logical memory, and it is these procedures' duty to map every access to the correct location of the correct object.
On abase-limit addressingsystem where memory from address0to addressROMSIZE-1is read-only memory, while the rest is RAM, something along the line of the following procedures would be typical:
The CPU simulator is often the most complicated part of an emulator. Many emulators are written using "pre-packaged" CPU simulators, in order to concentrate on good and efficient emulation of a specific machine. The simplest form of a CPU simulator is aninterpreter, which is a computer program that follows the execution flow of the emulated program code and, for every machine code instruction encountered, executes operations on the host processor that are semantically equivalent to the original instructions. This is made possible by assigning avariableto eachregisterandflagof the simulated CPU. The logic of the simulated CPU can then more or less be directly translated into software algorithms, creating a software re-implementation that basically mirrors the original hardware implementation.
The following example illustrates how CPU simulation can be accomplished by an interpreter. In this case, interrupts are checked-for before every instruction executed, though this behavior is rare in real emulators for performance reasons (it is generally faster to use a subroutine to do the work of an interrupt).
Interpreters are very popular as computer simulators, as they are much simpler to implement than more time-efficient alternative solutions, and their speed is more than adequate for emulating computers of more than roughly a decade ago on modern machines. However, the speed penalty inherent in interpretation can be a problem when emulating computers whose processor speed is on the sameorder of magnitudeas the host machine[dubious–discuss]. Until not many years ago, emulation in such situations was considered completely impractical by many[dubious–discuss].
What allowed breaking through this restriction were the advances indynamic recompilationtechniques[dubious–discuss]. Simplea prioritranslation of emulated program code into code runnable on the host architecture is usually impossible because of several reasons:
Various forms of dynamic recompilation, including the popularJust In Time compiler (JIT)technique, try to circumvent these problems by waiting until the processor control flow jumps into a location containing untranslated code, and only then ("just in time") translates a block of the code into host code that can be executed.
The translated code is kept in acodecache[dubious–discuss], and the original code is not lost or affected; this way, even data segments can be (meaninglessly) translated by the recompiler, resulting in no more than a waste of translation time. Speed may not be desirable as some older games were not designed with the speed of faster computers in mind. A game designed for a 30 MHz PC with a level timer of 300 game seconds might only give the player 30 seconds on a 300 MHz PC. Other programs, such as some DOS programs, may not even run on faster computers. Particularly when emulating computers which were "closed-box", in which changes to the core of the system were not typical, software may use techniques that depend on specific characteristics of the computer it ran on (e.g. its CPU's speed) and thus precise control of the speed of emulation is important for such applications to be properly emulated.
Most emulators do not, as mentioned earlier, emulate the mainsystem bus; each I/O device is thus often treated as a special case, and no consistent interface for virtual peripherals is provided. This can result in a performance advantage, since each I/O module can be tailored to the characteristics of the emulated device; designs based on a standard, unified I/OAPIcan, however, rival such simpler models, if well thought-out, and they have the additional advantage of "automatically" providing a plug-in service through which third-party virtual devices can be used within the emulator. A unified I/O API may not necessarily mirror the structure of the real hardware bus: bus design is limited by several electric constraints and a need for hardwareconcurrencymanagement that can mostly be ignored in a software implementation.
Even in emulators that treat each device as a special case, there is usually a common basic infrastructure for:
Emulation is one strategy in pursuit ofdigital preservationand combatingobsolescence. Emulation focuses on recreating an original computer environment, which can be time-consuming and difficult to achieve, but valuable because of its ability to maintain a closer connection to the authenticity of the digital object, operating system, or even gaming platform.[10]Emulation addresses the originalhardwareandsoftwareenvironment of the digital object, and recreates it on a current machine.[11]The emulator allows the user to have access to any kind ofapplicationoroperating systemon a currentplatform, while thesoftwareruns as it did in its original environment.[12]Jeffery Rothenberg, an early proponent of emulation as adigital preservationstrategy states, "the ideal approach would provide a singleextensible, long-term solution that can be designed once and for all and applied uniformly, automatically, and in organized synchrony (for example, at every refresh cycle) to all types of documents and media".[13]He further states that this should not only apply to out of date systems, but also be upwardly mobile to future unknown systems.[14]Practically speaking, when a certain application is released in a new version, rather than addresscompatibilityissues andmigrationfor every digital object created in the previous version of thatapplication, one could create an emulator for theapplication, allowing access to all of said digital objects.
Because of its primary use of digital formats,new media artrelies heavily on emulation as a preservation strategy. Artists such asCory Arcangelspecialize in resurrecting obsolete technologies in their artwork and recognize the importance of adecentralizedand deinstitutionalized process for the preservation of digital culture. In many cases, the goal of emulation in new media art is to preserve a digital medium so that it can be saved indefinitely and reproduced without error, so that there is no reliance on hardware that ages and becomes obsolete. The paradox is that the emulation and the emulator have to be made to work on future computers.[15]
Emulation techniques are commonly used during the design and development of new systems. It eases the development process by providing the ability to detect, recreate and repair flaws in the design even before the system is actually built.[16]It is particularly useful in the design ofmulti-coresystems, where concurrency errors can be very difficult to detect and correct without the controlled environment provided by virtual hardware.[17]This also allows the software development to take place before the hardware is ready,[18]thus helping to validate design decisions and give a little more control.
The word "emulator" was coined in 1963 at IBM[19]during development of the NPL (IBM System/360) product line, using a "newcombinationofsoftware,microcode, andhardware".[20]They discovered that simulation using additional instructions implemented inmicrocodeand hardware, instead of software simulation using only standard instructions, to execute programs written for earlier IBM computers dramatically increased simulation speed. Earlier, IBM providedsimulatorsfor, e.g., the650on the705.[21]In addition to simulators, IBM had compatibility features on the709and7090,[22]for which it provided the IBM 709 computer with a program to run legacy programs written for theIBM 704on the709and later on the IBM 7090. This program used the instructions added by the compatibility feature[23]to trap instructions requiring special handling; all other 704 instructions ran the same on a 7090. The compatibility feature on the1410[24]only required setting a console toggle switch, not a support program.
In 1963, when microcode was first used to speed up this simulation process, IBM engineers coined the term "emulator" to describe the concept. In the 2000s, it has become common to use the word "emulate" in the context of software. However, before 1980, "emulation" referred only to emulation with a hardware or microcode assist, while "simulation" referred to pure software emulation.[25]For example, a computer specially built for running programs designed for another architecture is an emulator. In contrast, a simulator could be a program which runs on a PC, so that old Atari games can be simulated on it. Purists continue to insist on this distinction, but currently the term "emulation" often means the complete imitation of a machine executing binary code while "simulation" often refers tocomputer simulation, where a computer program is used to simulate an abstract model. Computer simulation is used in virtually every scientific and engineering domain and Computer Science is no exception, with several projects simulating abstract models of computer systems, such asnetwork simulation, which both practically and semantically differs from network emulation.[26]
Hardware virtualization is thevirtualizationofcomputersas complete hardware platforms, certain logical abstractions of their components, or only the functionality required to run variousoperating systems. Virtualization hides the physical characteristics of a computing platform from the users, presenting instead an abstract computing platform.[27][28]At its origins, the software that controlled virtualization was called a "control program", but the terms "hypervisor" or "virtual machine monitor" became preferred over time.[29]Each hypervisor can manage or run multiplevirtual machines. | https://en.wikipedia.org/wiki/Emulator |
Incomputercentral processing units,micro-operations(also known asmicro-opsorμops, historically also asmicro-actions[2]) are detailed low-level instructions used in some designs to implement complex machine instructions (sometimes termedmacro-instructionsin this context).[3]: 8–9
Usually, micro-operations perform basic operations on data stored in one or moreregisters, including transferring data between registers or between registers and externalbusesof thecentral processing unit(CPU), and performing arithmetic or logical operations on registers. In a typicalfetch-decode-execute cycle, each step of a macro-instruction is decomposed during its execution so the CPU determines and steps through a series of micro-operations. The execution of micro-operations is performed under control of the CPU'scontrol unit, which decides on their execution while performing various optimizations such as reordering, fusion and caching.[1]
Various forms of μops have long been the basis for traditionalmicrocoderoutines used to simplify the implementation of a particularCPU designor perhaps just the sequencing of certain multi-step operations or addressing modes. More recently, μops have also been employed in a different way in order to let modernCISCprocessors more easily handle asynchronous parallel and speculative execution: As with traditional microcode, one or more table lookups (or equivalent) is done to locate the appropriate μop-sequence based on the encoding and semantics of the machine instruction (the decoding or translation step), however, instead of having rigid μop-sequences controlling the CPU directly from a microcode-ROM, μops are here dynamically buffered for rescheduling before being executed.[4]: 6–7, 9–11
This buffering means that the fetch and decode stages can be more detached from the execution units than is feasible in a more traditional microcoded (or hard-wired) design. As this allows a degree of freedom regarding execution order, it makes some extraction ofinstruction-level parallelismout of a normal single-threaded program possible (provided that dependencies are checked, etc.). It opens up for more analysis and therefore also for reordering of code sequences in order to dynamically optimize mapping and scheduling of μops onto machine resources (such asALUs, load/store units, etc.). As this happens on the μop-level, sub-operations of different machine (macro) instructions may often intermix in a particular μop-sequence, forming partially reordered machine instructions as a direct consequence of the out-of-order dispatching of microinstructions from several macro instructions. However, this is not the same as themicro-op fusion, which aims at the fact that a more complex microinstruction may replace a few simpler microinstructions in certain cases, typically in order to minimize state changes and usage of the queue andre-order bufferspace, therefore reducing power consumption. Micro-op fusion is used in some modern CPU designs.[3]: 89–91, 105–106[4]: 6–7, 9–15
Execution optimization has gone even further; processors not only translate many machine instructions into a series of μops, but also do the opposite when appropriate; they combine certain machine instruction sequences (such as a compare followed by a conditional jump) into a more complex μop which fits the execution model better and thus can be executed faster or with less machine resources involved. This is also known asmacro-op fusion.[3]: 106–107[4]: 12–13
Another way to try to improve performance is to cache the decoded micro-operations in amicro-operation cache, so that if the same macroinstruction is executed again, the processor can directly access the decoded micro-operations from the cache, instead of decoding them again. Theexecution trace cachefound inIntelNetBurstmicroarchitecture (Pentium 4) is a widespread example of this technique.[5]The size of this cache may be stated in terms of how many thousands (or strictly multiple of 1024) of micro-operations it can store:Kμops.[6] | https://en.wikipedia.org/wiki/Micro-operation |
No instruction set computing(NISC) is a computing architecture and compiler technology for designing highly efficient custom processors and hardware accelerators by allowing a compiler to have low-level control of hardware resources.
NISC is a statically scheduled horizontal nanocoded architecture (SSHNA). The term "statically scheduled" means that theoperation schedulingandHazardhandling are done by acompiler. The term "horizontal nanocoded" means that NISC does not have any predefinedinstruction setormicrocode. The compiler generates nanocodes which directly controlfunctional units,registersandmultiplexersof a givendatapath. Giving low-level control to the compiler enables better utilization of datapath resources, which ultimately result in better performance. The benefits of NISC technology are:
The instruction set and controller ofprocessorsare the most tedious and time-consuming parts to design. By eliminating these two, design of custom processing elements become significantly easier.
Furthermore, the datapath of NISC processors can even be generated automatically for a given application. Therefore, designer's productivity is improved significantly.
Since NISC datapaths are very efficient and can be generated automatically, NISC technology is comparable tohigh level synthesis(HLS) orC to HDLsynthesis approaches. In fact, one of the benefits of this architecture style is its capability to bridge these two technologies (custom processor design and HLS).
Incomputer science,zero instruction set computer(ZISC) refers to acomputer architecturebased solely onpattern matchingand absence of(micro-)instructionsin the classical[clarification needed]sense. These chips are known for being thought of as comparable to theneural networks, being marketed for the number of "synapses" and "neurons".[1]TheacronymZISC alludes toreduced instruction set computer(RISC).[citation needed]
ZISC is a hardware implementation ofKohonen networks(artificial neural networks) allowing massively parallel processing of very simple data (0 or 1). This hardware implementation was invented by Guy Paillet[2]and Pascal Tannhof (IBM),[3][2]developed in cooperation with the IBM chip factory ofEssonnes, in France, and was commercialized by IBM.
The ZISC architecture alleviates thememory bottleneck[clarification needed]by blending pattern memory with pattern learning and recognition logic.[how?]Their massivelyparallel computingsolves the"winner takes all problem in action selection"[clarification neededfromWinner-takes-allproblem inNeural Networks]by allotting each "neuron" its own memory and allowing simultaneous problem-solving the results of which are settled up disputing with each other.[4]
According toTechCrunch, software emulations of these types of chips are currently used for image recognition by many large tech companies, such asFacebookandGoogle. When applied to other miscellaneous pattern detection tasks, such as with text, results are said to be produced in microseconds even with chips released in 2007.[1]
Junko Yoshida, of theEE Times, compared the NeuroMem chip with "The Machine", a machine capable of being able to predict crimes from scanning people's faces from the television seriesPerson of Interest, describing it as "the heart ofbig data" and "foreshadow[ing] a real-life escalation in the era of massive data collection".[5]
In the past, microprocessor design technology evolved fromcomplex instruction set computer(CISC) toreduced instruction set computer(RISC). In the early days of the computer industry, compiler technology did not exist and programming was done inassembly language. To make programming easier, computer architects created complex instructions which were direct representations of high level functions of high level programming languages. Another force that encouraged instruction complexity was the lack of large memory blocks.
As compiler and memory technologies advanced, RISC architectures were introduced. RISC architectures need more instruction memory and require a compiler to translate high-level languages to RISC assembly code. Further advancement of compiler and memory technologies leads to emergingvery long instruction word(VLIW) processors, where the compiler controls the schedule of instructions and handles data hazards.
NISC is a successor of VLIW processors. In NISC, the compiler has both horizontal and vertical control of the operations in the datapath. Therefore, the hardware is much simpler. However the control memory size is larger than the previous generations. To address this issue, low-overhead compression techniques can be used. | https://en.wikipedia.org/wiki/No_instruction_set_computing |
Asimulationis an imitative representation of a process or system that could exist in the real world.[1][2][3]In this broad sense, simulation can often be used interchangeably withmodel.[2]Sometimes a clear distinction between the two terms is made, in which simulations require the use of models; the model represents the key characteristics or behaviors of the selected system or process, whereas the simulation represents the evolution of the model over time.[3]Another way to distinguish between the terms is to define simulation asexperimentationwith the help of a model.[4]This definition includes time-independent simulations. Often,computers are used to execute the simulation.
Simulation is used in many contexts, such as simulation of technology forperformance tuningor optimizing,safety engineering, testing, training, education, and video games. Simulation is also used withscientific modellingof natural systems or human systems to gain insight into their functioning,[5]as in economics. Simulation can be used to show the eventual real effects of alternative conditions and courses of action. Simulation is also used when the real system cannot be engaged, because it may not be accessible, or it may be dangerous or unacceptable to engage, or it is being designed but not yet built, or it may simply not exist.[6]
Key issues inmodeling and simulationinclude the acquisition of valid sources of information about the relevant selection of key characteristics and behaviors used to build the model, the use of simplifying approximations and assumptions within the model, and fidelity and validity of the simulation outcomes. Procedures and protocols formodel verification and validationare an ongoing field of academic study, refinement, research and development in simulations technology or practice, particularly in the work of computer simulation.
Historically, simulations used in different fields developed largely independently, but 20th-century studies ofsystems theoryandcyberneticscombined with spreading use of computers across all those fields have led to some unification and a more systematic view of the concept.
Physical simulationrefers to simulation in which physical objects are substituted for the real thing. These physical objects are often chosen because they are smaller or cheaper than the actual object or system. (See also:physical modelandscale model.)
Alternatively,physical simulationmay refer to computer simulations considering selected laws of physics, as inmultiphysics simulation.[7](See also:Physics engine.)
Interactive simulationis a special kind of physical simulation, often referred to as ahuman-in-the-loopsimulation, in which physical simulations include human operators, such as in aflight simulator,sailing simulator, ordriving simulator.
Continuous simulationis a simulation based oncontinuous-time rather than discrete-timesteps, using numerical integration ofdifferential equations.[8]
Discrete-event simulationstudies systems whose states change their values only at discrete times.[9]For example, a simulation of an epidemic could change the number of infected people at time instants when susceptible individuals get infected or when infected individuals recover.
Stochastic simulationis a simulation where some variable or process is subject to random variations and is projected usingMonte Carlotechniques using pseudo-random numbers. Thus replicated runs with the same boundary conditions will each produce different results within a specific confidence band.[8]
Deterministic simulationis a simulation which is not stochastic: thus the variables are regulated by deterministic algorithms. So replicated runs from the same boundary conditions always produce identical results.
Hybrid simulation(or combined simulation) corresponds to a mix between continuous and discrete event simulation and results in integrating numerically the differential equations between two sequential events to reduce the number of discontinuities.[10]
Astand-alone simulationis a simulation running on a single workstation by itself.
Adistributed simulationis one which uses more than one computer simultaneously, to guarantee access from/to different resources (e.g. multi-users operating different systems, or distributed data sets); a classical example isDistributed Interactive Simulation(DIS).[11]
Parallel simulationspeeds up a simulation's execution by concurrently distributing its workload over multiple processors, as inhigh-performance computing.[12]
Interoperable simulationis where multiple models, simulators (often defined as federates) interoperate locally, distributed over a network; a classical example isHigh-Level Architecture.[13][14]
Modeling and simulation as a serviceis where simulation is accessed as a service over the web.[15]
Modeling, interoperable simulation and serious gamesis whereserious gameapproaches (e.g. game engines and engagement methods) are integrated with interoperable simulation.[16]
Simulation fidelityis used to describe the accuracy of a simulation and how closely it imitates the real-life counterpart. Fidelity is broadly classified as one of three categories: low, medium, and high. Specific descriptions of fidelity levels are subject to interpretation, but the following generalizations can be made:
Asynthetic environmentis a computer simulation that can be included in human-in-the-loop simulations.[19]
Simulation in failure analysisrefers to simulation in which we create environment/conditions to identify the cause of equipment failure. This can be the best and fastest method to identify the failure cause.
A computer simulation (or "sim") is an attempt to model a real-life or hypothetical situation on a computer so that it can be studied to see how the system works. By changing variables in the simulation,predictionsmay be made about the behaviour of the system. It is a tool to virtually investigate the behaviour of the system under study.[3]
Computer simulation has become a useful part ofmodelingmany natural systems inphysics,chemistryandbiology,[20]and human systems in economics andsocial science(e.g.,computational sociology) as well as in engineering to gain insight into the operation of those systems. A good example of the usefulness of using computers to simulate can be found in the field ofnetwork traffic simulation. In such simulations, themodelbehaviour will change each simulation according to the set of initial parameters assumed for the environment.
Traditionally, the formal modeling of systems has been via amathematical model, which attempts to find analytical solutions enabling the prediction of the behaviour of the system from a set of parameters and initial conditions. Computer simulation is often used as an adjunct to, or substitution for, modeling systems for which simpleclosed form analytic solutionsare not possible. There are many different types of computer simulation, the common feature they all share is the attempt to generate a sample of representativescenariosfor a model in which a complete enumeration of all possible states would be prohibitive or impossible.
Several software packages exist for running computer-based simulation modeling (e.g.Monte Carlosimulation,stochasticmodeling, multimethod modeling) that makes all the modeling almost effortless.
Modern usage of the term "computer simulation" may encompass virtually any computer-based representation.
Incomputer science, simulation has some specialized meanings:Alan Turingused the termsimulationto refer to what happens when auniversal machineexecutes a state transition table (in modern terminology, a computer runs a program) that describes the state transitions, inputs and outputs of a subject discrete-state machine.[21]The computer simulates the subject machine. Accordingly, intheoretical computer sciencethe termsimulationis a relation betweenstate transition systems, useful in the study ofoperational semantics.
Less theoretically, an interesting application of computer simulation is to simulate computers using computers. Incomputer architecture, a type of simulator, typically called anemulator, is often used to execute a program that has to run on some inconvenient type of computer (for example, a newly designed computer that has not yet been built or an obsolete computer that is no longer available), or in a tightly controlled testing environment (seeComputer architecture simulatorandPlatform virtualization). For example, simulators have been used to debug amicroprogramor sometimes commercial application programs, before the program is downloaded to the target machine. Since the operation of the computer is simulated, all of the information about the computer's operation is directly available to the programmer, and the speed and execution of the simulation can be varied at will.
Simulators may also be used to interpretfault trees, or testVLSIlogic designs before they are constructed.Symbolic simulationuses variables to stand for unknown values.
In the field ofoptimization, simulations of physical processes are often used in conjunction withevolutionary computationto optimize control strategies.
Simulation is extensively used for educational purposes. It is used for cases where it is prohibitively expensive or simply too dangerous to allow trainees to use the real equipment in the real world. In such situations they will spend time learning valuable lessons in a "safe" virtual environment yet living a lifelike experience (or at least it is the goal). Often the convenience is to permit mistakes during training for a safety-critical system.
Simulations in education are somewhat like training simulations. They focus on specific tasks. The term 'microworld' is used to refer to educational simulations which model some abstract concept rather than simulating a realistic object or environment, or in some cases model a real-world environment in a simplistic way so as to help a learner develop an understanding of the key concepts. Normally, a user can create some sort of construction within the microworld that will behave in a way consistent with the concepts being modeled.Seymour Papertwas one of the first to advocate the value of microworlds, and theLogoprogramming environment developed by Papert is one of the most well-known microworlds.
Project management simulationis increasingly used to train students and professionals in the art and science of project management. Using simulation forproject managementtraining improves learning retention and enhances the learning process.[22][23]
Social simulationsmay be used in social science classrooms to illustrate social and political processes in anthropology, economics, history, political science, or sociology courses, typically at the high school or university level. These may, for example, take the form of civics simulations, in which participants assume roles in a simulated society, or international relations simulations in which participants engage in negotiations, alliance formation, trade, diplomacy, and the use of force. Such simulations might be based on fictitious political systems, or be based on current or historical events. An example of the latter would beBarnard College'sReacting to the Pastseries of historical educational games.[24]TheNational Science Foundationhas also supported the creation ofreacting gamesthat address science and math education.[25]In social media simulations, participants train communication with critics and other stakeholders in a private environment.
In recent years, there has been increasing use of social simulations for staff training in aid and development agencies. The Carana simulation, for example, was first developed by theUnited Nations Development Programme, and is now used in a very revised form by theWorld Bankfor training staff to deal with fragile and conflict-affected countries.[26]
Military uses for simulation often involve aircraft or armoured fighting vehicles, but can also target small arms and other weapon systems training. Specifically, virtual firearms ranges have become the norm in most military training processes and there is a significant amount of data to suggest this is a useful tool for armed professionals.[27]
Avirtual simulationis a category of simulation that uses simulation equipment to create asimulated worldfor the user. Virtual simulations allow users to interact with avirtual world. Virtual worlds operate on platforms of integrated software and hardware components. In this manner, the system can accept input from the user (e.g., body tracking, voice/sound recognition, physical controllers) and produce output to the user (e.g., visual display, aural display, haptic display) .[28]Virtual simulations use the aforementioned modes of interaction to produce a sense ofimmersionfor the user.
There is a wide variety of input hardware available to accept user input for virtual simulations. The following list briefly describes several of them:
Research in future input systems holds a great deal of promise for virtual simulations. Systems such asbrain–computer interfaces(BCIs) offer the ability to further increase the level of immersion for virtual simulation users. Lee, Keinrath, Scherer, Bischof, Pfurtscheller[29]proved that naïve subjects could be trained to use a BCI to navigate a virtual apartment with relative ease. Using the BCI, the authors found that subjects were able to freely navigate the virtual environment with relatively minimal effort. It is possible that these types of systems will become standard input modalities in future virtual simulation systems.
There is a wide variety of output hardware available to deliver a stimulus to users in virtual simulations. The following list briefly describes several of them:
Clinical healthcare simulatorsare increasingly being developed and deployed to teach therapeutic and diagnostic procedures as well as medical concepts and decision making to personnel in the health professions. Simulators have been developed for training procedures ranging from the basics such asblood draw, tolaparoscopicsurgery[31]and trauma care. They are also important to help on prototyping new devices[32]for biomedical engineering problems. Currently, simulators are applied to research and develop tools for new therapies,[33]treatments[34]and early diagnosis[35]in medicine.
Many medical simulators involve a computer connected to a plastic simulation of the relevant anatomy.[36]Sophisticated simulators of this type employ a life-size mannequin that responds to injected drugs and can be programmed to create simulations of life-threatening emergencies.
In other simulations, visual components of the procedure are reproduced bycomputer graphicstechniques, while touch-based components are reproduced byhapticfeedback devices combined with physical simulation routines computed in response to the user's actions. Medical simulations of this sort will often use 3DCTorMRIscans of patient data to enhance realism. Some medical simulations are developed to be widely distributed (such as web-enabled simulations[37]and procedural simulations[38]that can be viewed via standard web browsers) and can be interacted with using standard computer interfaces, such as thekeyboardandmouse.
An important medical application of a simulator—although, perhaps, denoting a slightly different meaning ofsimulator—is the use of aplacebodrug, a formulation that simulates the active drug in trials of drug efficacy.
Patient safety is a concern in the medical industry. Patients have been known to suffer injuries and even death due to management error, and lack of using best standards of care and training. According to Building a National Agenda for Simulation-Based Medical Education (Eder-Van Hook, Jackie, 2004), "a health care provider's ability to react prudently in an unexpected situation is one of the most critical factors in creating a positive outcome in medical emergency, regardless of whether it occurs on the battlefield, freeway, or hospital emergency room." Eder-Van Hook (2004) also noted that medical errors kill up to 98,000 with an estimated cost between $37 and $50 million and $17 to $29 billion for preventable adverse events dollars per year.
Simulation is being used to study patient safety, as well as train medical professionals.[39]Studying patient safety and safety interventions in healthcare is challenging, because there is a lack of experimental control (i.e., patient complexity, system/process variances) to see if an intervention made a meaningful difference (Groves & Manges, 2017).[40]An example of innovative simulation to study patient safety is from nursing research. Groves et al. (2016) used a high-fidelity simulation to examine nursing safety-oriented behaviors during times such aschange-of-shift report.[39]
However, the value of simulation interventions to translating to clinical practice are is still debatable.[41]As Nishisaki states, "there is good evidence that simulation training improves provider and teamself-efficacyand competence on manikins. There is also good evidence that procedural simulation improves actual operational performance in clinical settings."[41]However, there is a need to have improved evidence to show thatcrew resource managementtraining through simulation.[41]One of the largest challenges is showing that team simulation improves team operational performance at the bedside.[42]Although evidence that simulation-based training actually improves patient outcome has been slow to accrue, today the ability of simulation to provide hands-on experience that translates to the operating room is no longer in doubt.[43][44][45]
One of the largest factors that might impact the ability to have training impact the work of practitioners at the bedside is the ability to empower frontline staff (Stewart, Manges, Ward, 2015).[42][46]Another example of an attempt to improve patient safety through the use of simulations training is patient care to deliver just-in-time service or/and just-in-place. This training consists of 20 minutes of simulated training just before workers report to shift. One study found that just in time training improved the transition to the bedside. The conclusion as reported in Nishisaki (2008) work, was that the simulation training improved resident participation in real cases; but did not sacrifice the quality of service. It could be therefore hypothesized that by increasing the number of highly trained residents through the use of simulation training, that the simulation training does, in fact, increase patient safety.
The first medical simulators were simple models of human patients.[47]
Since antiquity, these representations in clay and stone were used to demonstrate clinical features of disease states and their effects on humans. Models have been found in many cultures and continents. These models have been used in some cultures (e.g., Chinese culture) as a "diagnostic" instrument, allowing women to consult male physicians while maintaining social laws of modesty. Models are used today to help students learn theanatomyof themusculoskeletalsystem and organ systems.[47]
In 2002, theSociety for Simulation in Healthcare(SSH) was formed to become a leader in international interprofessional advances the application of medical simulation in healthcare[48]
The need for a "uniform mechanism to educate, evaluate, and certify simulation instructors for the health care profession" was recognized by McGaghie et al. in their critical review of simulation-based medical education research.[49]In 2012 the SSH piloted two new certifications to provide recognition to educators in an effort to meet this need.[50]
Active models that attempt to reproduce living anatomy or physiology are recent developments. The famous"Harvey" mannequinwas developed at theUniversity of Miamiand is able to recreate many of the physical findings of thecardiologyexamination, includingpalpation,auscultation, andelectrocardiography.[51]
More recently, interactive models have been developed that respond to actions taken by a student or physician.[51]Until recently, these simulations were two dimensional computer programs that acted more like a textbook than a patient. Computer simulations have the advantage of allowing a student to make judgments, and also to make errors. The process of iterative learning through assessment, evaluation, decision making, and error correction creates a much stronger learning environment than passive instruction.
Simulators have been proposed as an ideal tool for assessment of students for clinical skills.[52]For patients, "cybertherapy" can be used for sessions simulating traumatic experiences, from fear of heights to social anxiety.[53]
Programmed patients and simulated clinical situations, including mock disaster drills, have been used extensively for education and evaluation. These "lifelike" simulations are expensive, and lack reproducibility. A fully functional "3Di" simulator would be the most specific tool available for teaching and measurement of clinical skills.Gaming platformshave been applied to create these virtual medical environments to create an interactive method for learning and application of information in a clinical context.[54][55]
Immersive disease state simulations allow a doctor or HCP to experience what a disease actually feels like. Using sensors and transducers symptomatic effects can be delivered to a participant allowing them to experience the patients disease state.
Such a simulator meets the goals of an objective and standardized examination for clinical competence.[56]This system is superior to examinations that use "standard patients" because it permits the quantitative measurement of competence, as well as reproducing the same objective findings.[57]
Simulation in entertainmentencompasses many large and popular industries such as film, television, video games (includingserious games) and rides in theme parks. Although modern simulation is thought to have its roots in training and the military, in the 20th century it also became a conduit for enterprises which were more hedonistic in nature.
The first simulation game may have been created as early as 1947 by Thomas T. Goldsmith Jr. and Estle Ray Mann. This was a straightforward game that simulated a missile being fired at a target. The curve of the missile and its speed could be adjusted using several knobs. In 1958, a computer game calledTennis for Twowas created by Willy Higginbotham which simulated a tennis game between two players who could both play at the same time using hand controls and was displayed on an oscilloscope.[58]This was one of the first electronic video games to use a graphical display.
Computer-generated imagerywas used in the film to simulate objects as early as 1972 inA Computer Animated Hand, parts of which were shown on the big screen in the 1976 filmFutureworld. This was followed by the "targeting computer" that young Skywalker turns off in the 1977 filmStar Wars.
The filmTron(1982) was the first film to use computer-generated imagery for more than a couple of minutes.[59]
Advances in technology in the 1980s caused 3D simulation to become more widely used and it began to appear in movies and in computer-based games such as Atari'sBattlezone(1980) andAcornsoft'sElite(1984), one of the firstwire-frame 3D graphics gamesforhome computers.
Advances in technology in the 1980s made the computer more affordable and more capable than they were in previous decades,[60]which facilitated the rise of computer such as the Xbox gaming. The firstvideo game consolesreleased in the 1970s and early 1980s fell prey to theindustry crashin 1983, but in 1985,Nintendoreleased the Nintendo Entertainment System (NES) which became one of the best selling consoles in video game history.[61]In the 1990s, computer games became widely popular with the release of such game asThe SimsandCommand & Conquerand the still increasing power of desktop computers. Today, computer simulation games such asWorld of Warcraftare played by millions of people around the world.
In 1993, the filmJurassic Parkbecame the first popular film to use computer-generated graphics extensively, integrating the simulated dinosaurs almost seamlessly into live action scenes.
This event transformed the film industry; in 1995, the filmToy Storywas the first film to use only computer-generated images and by the new millennium computer generated graphics were the leading choice for special effects in films.[62]
The advent ofvirtual cinematographyin the early 2000s has led to an explosion of movies that would have been impossible to shoot without it. Classic examples are thedigital look-alikesof Neo, Smith and other characters in theMatrixsequels and the extensive use of physically impossible camera runs inThe Lord of the Ringstrilogy.
The terminal in thePan Am (TV series)no longer existed during the filming of this 2011–2012 aired series, which was no problem as they created it in virtual cinematography usingautomatedviewpointfinding and matching in conjunction with compositing real and simulated footage, which has been the bread and butter of the movie artist in and aroundfilm studiossince the early 2000s.
Computer-generated imageryis "the application of the field of 3D computer graphics to special effects". This technology is used for visual effects because they are high in quality, controllable, and can create effects that would not be feasible using any other technology either because of cost, resources or safety.[63]Computer-generated graphics can be seen in many live-action movies today, especially those of the action genre. Further, computer-generated imagery has almost completely supplanted hand-drawn animation in children's movies which are increasingly computer-generated only. Examples of movies that use computer-generated imagery includeFinding Nemo,300andIron Man.
Simulation games, as opposed to other genres of video and computer games, represent or simulate an environment accurately. Moreover, they represent the interactions between the playable characters and the environment realistically. These kinds of games are usually more complex in terms of gameplay.[64]Simulation games have become incredibly popular among people of all ages.[65]Popular simulation games includeSimCityandTiger Woods PGA Tour. There are alsoflight simulatoranddriving simulatorgames.
Simulators have been used for entertainment since theLink Trainerin the 1930s.[66]The first modern simulator ride to open at a theme park was Disney'sStar Toursin 1987 soon followed by Universal'sThe Funtastic World of Hanna-Barberain 1990 which was the first ride to be done entirely with computer graphics.[67]
Simulator rides are the progeny of military training simulators and commercial simulators, but they are different in a fundamental way. While military training simulators react realistically to the input of the trainee in real time, ride simulators only feel like they move realistically and move according to prerecorded motion scripts.[67]One of the first simulator rides, Star Tours, which cost $32 million, used a hydraulic motion based cabin. The movement was programmed by a joystick. Today's simulator rides, such asThe Amazing Adventures of Spider-Maninclude elements to increase the amount of immersion experienced by the riders such as: 3D imagery, physical effects (spraying water or producing scents), and movement through an environment.[68]
Manufacturing simulationrepresents one of the most important applications of simulation. This technique represents a valuable tool used by engineers when evaluating the effect of capital investment in equipment and physical facilities like factory plants, warehouses, and distribution centers. Simulation can be used to predict the performance of an existing or planned system and to compare alternative solutions for a particular design problem.[69]
Another important goal ofsimulation in manufacturing systemsis to quantify system performance. Common measures of system performance include the following:[70]
An automobile simulator provides an opportunity to reproduce the characteristics of real vehicles in a virtual environment. It replicates the external factors and conditions with which a vehicle interacts enabling a driver to feel as if they are sitting in the cab of their own vehicle. Scenarios and events are replicated with sufficient reality to ensure that drivers become fully immersed in the experience rather than simply viewing it as an educational experience.
The simulator provides a constructive experience for the novice driver and enables more complex exercises to be undertaken by the more mature driver. For novice drivers, truck simulators provide an opportunity to begin their career by applying best practice. For mature drivers, simulation provides the ability to enhance good driving or to detect poor practice and to suggest the necessary steps for remedial action. For companies, it provides an opportunity to educate staff in the driving skills that achieve reduced maintenance costs, improved productivity and, most importantly, to ensure the safety of their actions in all possible situations.
Abiomechanics simulatoris a simulation platform for creating dynamic mechanical models built from combinations of rigid and deformable bodies, joints, constraints, and various force actuators. It is specialized for creating biomechanical models of human anatomical structures, with the intention to study their function and eventually assist in the design and planning of medical treatment.
A biomechanics simulator is used to analyze walking dynamics, study sports performance, simulate surgical procedures, analyze joint loads, design medical devices, and animate human and animal movement.
A neuromechanical simulator that combines biomechanical and biologically realistic neural network simulation. It allows the user to test hypotheses on the neural basis of behavior in a physically accurate 3-D virtual environment.
A city simulator can be acity-building gamebut can also be a tool used by urban planners to understand how cities are likely to evolve in response to various policy decisions.AnyLogicis an example of modern, large-scale urban simulators designed for use by urban planners. City simulators are generallyagent-based simulations with explicit representations forland useand transportation.UrbanSimandLEAMare examples of large-scale urban simulation models that are used by metropolitan planning agencies and military bases for land use andtransportation planning.
Several Christmas-themed simulations exist, many of which are centred aroundSanta Claus. An example of these simulations are websites which claim to allow the user to track Santa Claus. Due to the fact that Santa is alegendarycharacter and not a real, living person, it is impossible to provide actual information on his location, and services such asNORAD Tracks Santaand theGoogle Santa Tracker(the former of which claims to useradarand other technologies to track Santa)[71]display fake, predetermined location information to users. Another example of these simulations are websites that claim to allow the user to email or send messages to Santa Claus. Websites such asemailSanta.comor Santa's former page on the now-defunctWindows Live SpacesbyMicrosoftuse automatedprogramsor scripts to generate personalized replies claimed to be from Santa himself based on user input.[72][73][74][75]
The classroom of the future will probably contain several kinds of simulators, in addition to textual and visual learning tools. This will allow students to enter the clinical years better prepared, and with a higher skill level. The advanced student or postgraduate will have a more concise and comprehensive method of retraining—or of incorporating new clinical procedures into their skill set—and regulatory bodies and medical institutions will find it easier to assess the proficiency andcompetencyof individuals.
The classroom of the future will also form the basis of a clinical skills unit for continuing education of medical personnel; and in the same way that the use of periodic flight training assists airline pilots, this technology will assist practitioners throughout their career.[citation needed]
The simulator will be more than a "living" textbook, it will become an integral a part of the practice of medicine.[citation needed]The simulator environment will also provide a standard platform for curriculum development in institutions of medical education.
Modern satellite communications systems (SATCOM) are often large and complex with many interacting parts and elements. In addition, the need for broadband connectivity on a moving vehicle has increased dramatically in the past few years for both commercial and military applications. To accurately predict and deliver high quality of service, SATCOM system designers have to factor in terrain as well as atmospheric and meteorological conditions in their planning. To deal with such complexity, system designers and operators increasingly turn towards computer models of their systems to simulate real-world operating conditions and gain insights into usability and requirements prior to final product sign-off. Modeling improves the understanding of the system by enabling the SATCOM system designer or planner to simulate real-world performance by injecting the models with multiple hypothetical atmospheric and environmental conditions. Simulation is often used in the training of civilian and military personnel. This usually occurs when it is prohibitively expensive or simply too dangerous to allow trainees to use the real equipment in the real world. In such situations, they will spend time learning valuable lessons in a "safe" virtual environment yet living a lifelike experience (or at least it is the goal). Often the convenience is to permit mistakes during training for a safety-critical system.
Simulation solutions are being increasingly integrated withcomputer-aidedsolutions and processes (computer-aided designor CAD,computer-aided manufacturingor CAM,computer-aided engineeringor CAE, etc.). The use of simulation throughout theproduct lifecycle, especially at the earlier concept and design stages, has the potential of providing substantial benefits. These benefits range from direct cost issues such as reduced prototyping and shorter time-to-market to better performing products and higher margins. However, for some companies, simulation has not provided the expected benefits.
The successful use of simulation, early in the lifecycle, has been largely driven by increased integration of simulation tools with the entire set of CAD, CAM and product-lifecycle management solutions. Simulation solutions can now function across the extended enterprise in amulti-CAD environment, and include solutions for managing simulation data and processes and ensuring that simulation results are made part of the product lifecycle history.
Simulation training has become a method for preparing people for disasters. Simulations can replicate emergency situations and track how learners respond thanks to a lifelike experience. Disaster preparedness simulations can involve training on how to handleterrorismattacks, natural disasters,pandemicoutbreaks, or other life-threatening emergencies.
One organization that has used simulation training for disaster preparedness is CADE (Center for Advancement of Distance Education). CADE[76]has used a video game to prepare emergency workers for multiple types of attacks. As reported by News-Medical.Net, "The video game is the first in a series of simulations to address bioterrorism, pandemic flu, smallpox, and other disasters that emergency personnel must prepare for.[77]" Developed by a team from theUniversity of Illinois at Chicago(UIC), the game allows learners to practice their emergency skills in a safe, controlled environment.
The Emergency Simulation Program (ESP) at the British Columbia Institute of Technology (BCIT), Vancouver, British Columbia, Canada is another example of an organization that uses simulation to train for emergency situations. ESP uses simulation to train on the following situations: forest fire fighting, oil or chemical spill response, earthquake response, law enforcement, municipal firefighting, hazardous material handling, military training, and response to terrorist attack[78]One feature of the simulation system is the implementation of "Dynamic Run-Time Clock," which allows simulations to run a 'simulated' time frame, "'speeding up' or 'slowing down' time as desired"[78]Additionally, the system allows session recordings, picture-icon based navigation, file storage of individual simulations, multimedia components, and launch external applications.
At the University of Québec in Chicoutimi, a research team at the outdoor research and expertise laboratory (Laboratoire d'Expertise et de Recherche en Plein Air – LERPA) specializes in using wilderness backcountry accident simulations to verify emergency response coordination.
Instructionally, the benefits of emergency training through simulations are that learner performance can be tracked through the system. This allows the developer to make adjustments as necessary or alert the educator on topics that may require additional attention. Other advantages are that the learner can be guided or trained on how to respond appropriately before continuing to the next emergency segment—this is an aspect that may not be available in the live environment. Some emergency training simulators also allow for immediate feedback, while other simulations may provide a summary and instruct the learner to engage in the learning topic again.
In a live-emergency situation, emergency responders do not have time to waste. Simulation-training in this environment provides an opportunity for learners to gather as much information as they can and practice their knowledge in a safe environment. They can make mistakes without risk of endangering lives and be given the opportunity to correct their errors to prepare for the real-life emergency.
Simulations in economicsand especially inmacroeconomics, judge the desirability of the effects of proposed policy actions, such asfiscal policychanges ormonetary policychanges. A mathematical model of the economy, having been fitted to historical economic data, is used as a proxy for the actual economy; proposed values ofgovernment spending, taxation,open market operations, etc. are used as inputs to the simulation of the model, and various variables of interest such as theinflation rate, theunemployment rate, thebalance of tradedeficit, the governmentbudget deficit, etc. are the outputs of the simulation. The simulated values of these variables of interest are compared for different proposed policy inputs to determine which set of outcomes is most desirable.[79]
Simulation is an important feature in engineering systems or any system that involves many processes. For example, inelectrical engineering, delay lines may be used to simulatepropagation delayandphase shiftcaused by an actualtransmission line. Similarly,dummy loadsmay be used to simulateimpedancewithout simulating propagation and is used in situations where propagation is unwanted. A simulator may imitate only a few of the operations and functions of the unit it simulates.Contrast with:emulate.[80]
Most engineering simulations entail mathematical modeling and computer-assisted investigation. There are many cases, however, where mathematical modeling is not reliable. Simulation offluid dynamicsproblems often require both mathematical and physical simulations. In these cases the physical models requiredynamic similitude. Physical and chemical simulations have also direct realistic uses, rather than research uses; inchemical engineering, for example,process simulationsare used to give the process parameters immediately used for operating chemical plants, such as oil refineries. Simulators are also used for plant operator training. It is called Operator Training Simulator (OTS) and has been widely adopted by many industries from chemical to oil&gas and to the power industry. This created a safe and realistic virtual environment to train board operators and engineers.Mimicis capable of providing high fidelity dynamic models of nearly all chemical plants for operator training and control system testing.
Ergonomic simulationinvolves the analysis of virtual products or manual tasks within a virtual environment. In the engineering process, the aim of ergonomics is to develop and to improve the design of products and work environments.[81]Ergonomic simulation utilizes an anthropometric virtual representation of the human, commonly referenced as a mannequin or Digital Human Models (DHMs), to mimic the postures, mechanical loads, and performance of a human operator in a simulated environment such as an airplane, automobile, or manufacturing facility. DHMs are recognized as evolving and valuable tool for performing proactive ergonomics analysis and design.[82]The simulations employ 3D-graphics and physics-based models to animate the virtual humans. Ergonomics software uses inverse kinematics (IK) capability for posing the DHMs.[81]
Software tools typically calculate biomechanical properties including individual muscle forces, joint forces and moments. Most of these tools employ standard ergonomic evaluation methods such as the NIOSH lifting equation and Rapid Upper Limb Assessment (RULA). Some simulations also analyze physiological measures including metabolism, energy expenditure, and fatigue limits Cycle time studies, design and process validation, user comfort, reachability, and line of sight are other human-factors that may be examined in ergonomic simulation packages.[83]
Modeling and simulation of a task can be performed by manually manipulating the virtual human in the simulated environment. Some ergonomicssimulation softwarepermits interactive,real-time simulationand evaluation through actual human input via motion capture technologies. However, motion capture for ergonomics requires expensive equipment and the creation of props to represent the environment or product.
Some applications of ergonomic simulation in include analysis of solid waste collection, disaster management tasks, interactive gaming,[84]automotive assembly line,[85]virtual prototyping of rehabilitation aids,[86]and aerospace product design.[87]Ford engineers use ergonomics simulation software to perform virtual product design reviews. Using engineering data, the simulations assist evaluation of assembly ergonomics. The company uses Siemen's Jack and Jill ergonomics simulation software in improving worker safety and efficiency, without the need to build expensive prototypes.[88]
In finance, computer simulations are often used for scenario planning.Risk-adjustednet present value, for example, is computed from well-defined but not always known (or fixed) inputs. By imitating the performance of the project under evaluation, simulation can provide a distribution of NPV over a range ofdiscount ratesand other variables. Simulations are also often used to test a financial theory or the ability of a financial model.[89]
Simulations are frequently used in financial training to engage participants in experiencing various historical as well as fictional situations. There are stock market simulations, portfolio simulations, risk management simulations or models and forex simulations. Such simulations are typically based onstochastic asset models. Using these simulations in a training program allows for the application of theory into a something akin to real life. As with other industries, the use of simulations can be technology or case-study driven.
Flight simulation is mainly used to train pilots outside of the aircraft.[90]In comparison to training in flight, simulation-based training allows for practicing maneuvers or situations that may be impractical (or even dangerous) to perform in the aircraft while keeping the pilot and instructor in a relatively low-risk environment on the ground. For example, electrical system failures, instrument failures, hydraulic system failures, and even flight control failures can be simulated without risk to the crew or equipment.[91]
Instructors can also provide students with a higher concentration of training tasks in a given period of time than is usually possible in the aircraft. For example, conducting multipleinstrument approachesin the actual aircraft may require significant time spent repositioning the aircraft, while in a simulation, as soon as one approach has been completed, the instructor can immediately reposition the simulated aircraft to a location from which the next approach can be begun.
Flight simulation also provides an economic advantage over training in an actual aircraft. Once fuel, maintenance, and insurance costs are taken into account, the operating costs of an FSTD are usually substantially lower than the operating costs of the simulated aircraft. For some large transport category airplanes, the operating costs may be several times lower for the FSTD than the actual aircraft. Another advantage is reduced environmental impact, as simulators don't contribute directly to carbon or noise emissions.[92]
There also exist "engineering flight simulators" which are a key element of theaircraft design process.[93]Many benefits that come from a lower number of test flights like cost and safety improvements are described above, but there are some unique advantages. Having a simulator available allows for faster design iteration cycle or using more test equipment than could be fit into a real aircraft.[94]
Bearing resemblance toflight simulators, amarine simulatoris meant for training of ship personnel. The most common marine simulators include:[95]
Simulators like these are mostly used within maritime colleges, training institutions, and navies. They often consist of a replication of a ships' bridge, with the operating console(s), and a number of screens on which the virtual surroundings are projected.
Military simulations, also known informally as war games, are models in which theories of warfare can be tested and refined without the need for actual hostilities. They exist in many different forms, with varying degrees of realism. In recent times, their scope has widened to include not only military but also political and social factors (for example, theNationlabseries of strategic exercises in Latin America).[97]While many governments make use of simulation, both individually and collaboratively, little is known about the model's specifics outside professional circles.
Network and distributed systems have been extensively simulated in other to understand the impact of new protocols and algorithms before their deployment in the actual systems. The simulation can focus on different levels (physical layer,network layer,application layer), and evaluate different metrics (network bandwidth, resource consumption, service time, dropped packets, system availability). Examples of simulation scenarios of network and distributed systems are:
Simulation techniques have also been applied to payment and securities settlement systems. Among the main users are central banks who are generally responsible for the oversight of market infrastructure and entitled to contribute to the smooth functioning of the payment systems.
Central banks have been using payment system simulations to evaluate things such as the adequacy or sufficiency of liquidity available ( in the form of account balances and intraday credit limits) to participants (mainly banks) to allow efficient settlement of payments.[102][103]The need for liquidity is also dependent on the availability and the type of netting procedures in the systems, thus some of the studies have a focus on system comparisons.[104]
Another application is to evaluate risks related to events such as communication network breakdowns or the inability of participants to send payments (e.g. in case of possible bank failure).[105]This kind of analysis falls under the concepts ofstress testingorscenario analysis.
A common way to conduct these simulations is to replicate the settlement logics of the real payment or securities settlement systems under analysis and then use real observed payment data. In case of system comparison or system development, naturally, also the other settlement logics need to be implemented.
To perform stress testing and scenario analysis, the observed data needs to be altered, e.g. some payments delayed or removed. To analyze the levels of liquidity, initial liquidity levels are varied. System comparisons (benchmarking) or evaluations of new netting algorithms or rules are performed by running simulations with a fixed set of data and varying only the system setups.
An inference is usually done by comparing the benchmark simulation results to the results of altered simulation setups by comparing indicators such as unsettled transactions or settlement delays.
Project management simulation is simulation used for project management training and analysis. It is often used as a training simulation for project managers. In other cases, it is used for what-if analysis and for supporting decision-making in real projects. Frequently the simulation is conducted using software tools.
A robotics simulator is used to create embedded applications for a specific (or not) robot without being dependent on the 'real' robot. In some cases, these applications can be transferred to the real robot (or rebuilt) without modifications. Robotics simulators allow reproducing situations that cannot be 'created' in the real world because of cost, time, or the 'uniqueness' of a resource. A simulator also allows fast robot prototyping. Many robot simulators featurephysics enginesto simulate a robot's dynamics.
Simulation of production systemsis used mainly to examine the effect of improvements or investments in aproduction system. Most often this is done using a static spreadsheet with process times and transportation times. For more sophisticated simulationsDiscrete Event Simulation(DES) is used with the advantages to simulate dynamics in the production system. A production system is very much dynamic depending on variations in manufacturing processes, assembly times, machine set-ups, breaks, breakdowns and small stoppages.[106]There is muchsoftwarecommonly used for discrete event simulation. They differ in usability and markets but do often share the same foundation.
Simulations are useful in modeling the flow of transactions through business processes, such as in the field ofsales process engineering, to study and improve the flow of customer orders through various stages of completion (say, from an initial proposal for providing goods/services through order acceptance and installation). Such simulations can help predict the impact of how improvements in methods might impact variability, cost, labor time, and the number of transactions at various stages in the process. A full-featured computerized process simulator can be used to depict such models, as can simpler educational demonstrations using spreadsheet software, pennies being transferred between cups based on the roll of a die, or dipping into a tub of colored beads with a scoop.[107]
In sports,computer simulationsare often done to predict the outcome of events and the performance of individual sportspeople. They attempt to recreate the event through models built from statistics. The increase in technology has allowed anyone with knowledge of programming the ability to run simulations of their models. The simulations are built from a series of mathematicalalgorithms, or models, and can vary with accuracy. Accuscore, which is licensed by companies such asESPN, is a well-known simulation program for all majorsports. It offers a detailed analysis of games through simulated betting lines, projected point totals and overall probabilities.
With the increased interest infantasy sportssimulation models that predict individual player performance have gained popularity. Companies like What If Sports and StatFox specialize in not only using their simulations for predicting game results but how well individual players will do as well. Many people use models to determine whom to start in their fantasy leagues.
Another way simulations are helping the sports field is in the use ofbiomechanics. Models are derived and simulations are run from data received from sensors attached to athletes and video equipment.Sports biomechanicsaided by simulation models answer questions regarding training techniques such as the effect of fatigue on throwing performance (height of throw) and biomechanical factors of the upper limbs (reactive strength index; hand contact time).[108]
Computer simulations allow their users to take models which before were too complex to run, and give them answers. Simulations have proven to be some of the best insights into both play performance and team predictability.
Simulation was used atKennedy Space Center(KSC) to train and certifySpace Shuttleengineers during simulated launch countdown operations. The Space Shuttle engineering community would participate in a launch countdown integrated simulation before each Shuttle flight. This simulation is a virtual simulation where real people interact with simulated Space Shuttle vehicle and Ground Support Equipment (GSE) hardware. The Shuttle Final Countdown Phase Simulation, also known as S0044, involved countdown processes that would integrate many of the Space Shuttle vehicle and GSE systems. Some of the Shuttle systems integrated in the simulation are the main propulsion system,RS-25,solid rocket boosters, ground liquid hydrogen and liquid oxygen,external tank, flight controls, navigation, and avionics.[109]The high-level objectives of the Shuttle Final Countdown Phase Simulation are:
The Shuttle Final Countdown Phase Simulation took place at theKennedy Space CenterLaunch Control Centerfiring rooms. The firing room used during the simulation is the same control room where real launch countdown operations are executed. As a result, equipment used for real launch countdown operations is engaged. Command and control computers, application software, engineering plotting and trending tools, launch countdown procedure documents, launch commit criteria documents, hardware requirement documents, and any other items used by the engineering launch countdown teams during real launch countdown operations are used during the simulation.
The Space Shuttle vehicle hardware and related GSE hardware is simulated bymathematical models(written in Shuttle Ground Operations Simulator (SGOS) modeling language[111]) that behave and react like real hardware. During the Shuttle Final Countdown Phase Simulation, engineers command and control hardware via real application software executing in the control consoles – just as if they were commanding real vehicle hardware. However, these real software applications do not interface with real Shuttle hardware during simulations. Instead, the applications interface with mathematical model representations of the vehicle and GSE hardware. Consequently, the simulations bypass sensitive and even dangerous mechanisms while providing engineering measurements detailing how the hardware would have reacted. Since these math models interact with the command and control application software, models and simulations are also used to debug and verify the functionality of application software.[112]
The only true way to testGNSSreceivers (commonly known as Sat-Nav's in the commercial world) is by using an RF Constellation Simulator. A receiver that may, for example, be used on an aircraft, can be tested under dynamic conditions without the need to take it on a real flight. The test conditions can be repeated exactly, and there is full control over all the test parameters. this is not possible in the 'real-world' using the actual signals. For testing receivers that will use the newGalileo (satellite navigation)there is no alternative, as the real signals do not yet exist.
Predicting weather conditions by extrapolating/interpolating previous data is one of the real use of simulation. Most of the weather forecasts use this information published by Weather bureaus. This kind of simulations helps in predicting and forewarning about extreme weather conditions like the path of an active hurricane/cyclone.Numerical weather predictionfor forecasting involves complicated numeric computer models to predict weather accurately by taking many parameters into account.
Strategy games—both traditional and modern—may be viewed as simulations of abstracted decision-making for the purpose of training military and political leaders (seeHistory of Gofor an example of such a tradition, orKriegsspielfor a more recent example).
Many other video games are simulators of some kind. Such games can simulate various aspects of reality, frombusiness, togovernment, toconstruction, topiloting vehicles(see above).
Historically, the word had negative connotations:
...therefore a general custom of simulation (which is this last degree) is a vice, using either of a natural falseness or fearfulness...
...for Distinction Sake, a Deceiving by Words, is commonly called a Lye, and a Deceiving by Actions, Gestures, or Behavior, is called Simulation...
However, the connection between simulation anddissemblinglater faded out and is now only of linguistic interest.[113] | https://en.wikipedia.org/wiki/Simulation |
Incomputer science,register transfer language(RTL) is a kind ofintermediate representation(IR) that is very close toassembly language, such as that which is used in acompiler. It is used to describe data flow at theregister-transfer levelof anarchitecture.[1]Academic papers and textbooks often use a form of RTL as an architecture-neutral assembly language.RTLis used as the name of a specific intermediate representation in several compilers, including theGNU Compiler Collection(GCC), Zephyr, and the European compiler projects CerCo andCompCert.
The idea behind RTL was first described inThe Design and Application of a Retargetable Peephole Optimizer.[2]
In GCC, RTL is generated from theGIMPLErepresentation, transformed by various passes in the GCCmiddle-end, and then converted to assembly language.
GCC's RTL is usually written in a form that looks like aLispS-expression:
Thisside-effect expressionsays "sum the contents of register 138 with the contents of register 139 and store the result in register 140". The SI specifies the access mode for each register. In the example, it is "SImode", i.e. "access the register as 32-bit integer".
The sequence of RTL generated has some dependency on the characteristics of the processor for which GCC is generating code. However, the meaning of the RTL is more or less independent of the target: it would usually be possible to read and understand a piece of RTL without knowing what processor it was generated for. Similarly, the meaning of the RTL doesn't usually depend on the program's original high-level language.
A register transfer language is a system for expressing in symbolic form the microoperation sequences among the registers of a digital module. It is a convenient tool for describing the internal organization of digital computers in a concise and precise manner. It can also be used to facilitate the design process of digital systems.[3] | https://en.wikipedia.org/wiki/Register_transfer_language |
Inmathematics, theKrylov–Bogolyubov theorem(also known as theexistence of invariant measures theorem) may refer to either of the two related fundamentaltheoremswithin the theory ofdynamical systems. The theorems guarantee the existence ofinvariant measuresfor certain "nice" maps defined on "nice" spaces and were named afterRussian-Ukrainianmathematiciansandtheoretical physicistsNikolay KrylovandNikolay Bogolyubovwho proved the theorems.[1]
Theorem (Krylov–Bogolyubov). LetXbe acompact,metrizabletopological spaceandF:X→Xacontinuous map. ThenFadmits an invariantBorelprobability measure.
That is, if Borel(X) denotes theBorelσ-algebragenerated by the collectionTofopen subsetsofX, then there exists a probability measureμ: Borel(X) → [0, 1] such that for any subsetA∈ Borel(X),
In terms of thepush forward, this states that
LetXbe aPolish spaceand letPt,t≥0,{\displaystyle P_{t},t\geq 0,}be the transition probabilities for a time-homogeneousMarkovsemigrouponX, i.e.
Theorem (Krylov–Bogolyubov). If there exists a pointx∈X{\displaystyle x\in X}for which the family of probability measures {Pt(x, ·) |t> 0 } isuniformly tightand the semigroup (Pt) satisfies theFeller property, then there exists at least one invariant measure for (Pt), i.e. a probability measureμonXsuch that
This article incorporates material fromKrylov-Bogolubov theoremonPlanetMath, which is licensed under theCreative Commons Attribution/Share-Alike License. | https://en.wikipedia.org/wiki/Krylov%E2%80%93Bogolyubov_theorem |
Inmathematicsandphysics, thePoincaré recurrence theoremstates that certaindynamical systemswill, after a sufficiently long but finite time, return to a state arbitrarily close to (for continuous state systems), or exactly the same as (for discrete state systems), their initial state.
ThePoincaré recurrence timeis the length of time elapsed until the recurrence. This time may vary greatly depending on the exact initial state and required degree of closeness. The result applies to isolated mechanical systems subject to some constraints, e.g., all particles must be bound to a finite volume. The theorem is commonly discussed in the context ofergodic theory,dynamical systemsandstatistical mechanics. Systems to which the Poincaré recurrence theorem applies are calledconservative systems.
The theorem is named afterHenri Poincaré, who discussed it in 1890.[1][2]A proof was presented byConstantin Carathéodoryusingmeasure theoryin 1919.[3][4]
Anydynamical systemdefined by anordinary differential equationdetermines aflow mapftmappingphase spaceon itself. The system is said to bevolume-preservingif the volume of a set in phase space is invariant under the flow. For instance, allHamiltonian systemsare volume-preserving because ofLiouville's theorem. The theorem is then: If aflowpreserves volume and has only bounded orbits, then, for eachopen set, any orbit that intersects this open set intersects it infinitely often.[5]
The proof, speaking qualitatively, hinges on two premises:[6]
Imagine any finite starting volumeD1{\displaystyle D_{1}}of thephase spaceand to follow its path under the dynamics of the system. The volume evolves through a "phase tube" in the phase space, keeping its size constant. Assuming a finite phase space, after some number of stepsk1{\displaystyle k_{1}}the phase tube must intersect itself. This means that at least a finite fractionR1{\displaystyle R_{1}}of the starting volume is recurring.
Now, consider the size of the non-returning portionD2{\displaystyle D_{2}}of the starting phase volume – that portion that never returns to the starting volume. Using the principle just discussed in the last paragraph, we know that if the non-returning portion is finite, then a finite partR2{\displaystyle R_{2}}of it must return afterk2{\displaystyle k_{2}}steps. But that would be a contradiction, since in a numberk3={\displaystyle k_{3}=}lcm(k1,k2){\displaystyle (k_{1},k_{2})}of step, bothR1{\displaystyle R_{1}}andR2{\displaystyle R_{2}}would be returning, against the hypothesis that onlyR1{\displaystyle R_{1}}was. Thus, the non-returning portion of the starting volume cannot be theempty set, i.e. allD1{\displaystyle D_{1}}is recurring after some number of steps.
The theorem does not comment on certain aspects of recurrence which this proof cannot guarantee:
Let
be a finitemeasure spaceand let
be ameasure-preserving transformation. Below are two alternative statements of the theorem.
For anyE∈Σ{\displaystyle E\in \Sigma }, the set of those pointsx{\displaystyle x}ofE{\displaystyle E}for which there existsN∈N{\displaystyle N\in \mathbb {N} }such thatfn(x)∉E{\displaystyle f^{n}(x)\notin E}for alln>N{\displaystyle n>N}has zero measure.
In other words, almost every point ofE{\displaystyle E}returns toE{\displaystyle E}. In fact, almost every point returns infinitely often;i.e.
The following is a topological version of this theorem:
IfX{\displaystyle X}is asecond-countableHausdorff spaceandΣ{\displaystyle \Sigma }contains theBorel sigma-algebra, then the set ofrecurrent pointsoff{\displaystyle f}has full measure. That is, almost every point is recurrent.
More generally, the theorem applies toconservative systems, and not just to measure-preserving dynamical systems. Roughly speaking, one can say that conservative systems are precisely those to which the recurrence theorem applies.
For time-independent quantum mechanical systems with discrete energy eigenstates, a similar theorem holds. For everyε>0{\displaystyle \varepsilon >0}andT0>0{\displaystyle T_{0}>0}there exists a timeTlarger thanT0{\displaystyle T_{0}}, such that||ψ(T)⟩−|ψ(0)⟩|<ε{\displaystyle ||\psi (T)\rangle -|\psi (0)\rangle |<\varepsilon }, where|ψ(t)⟩{\displaystyle |\psi (t)\rangle }denotes the state vector of the system at timet.[7][8][9]
The essential elements of the proof are as follows. The system evolves in time according to:
where theEn{\displaystyle E_{n}}are the energy eigenvalues (we usenatural units, soℏ=1{\displaystyle \hbar =1}), and the|ϕn⟩{\displaystyle |\phi _{n}\rangle }are the energyeigenstates. The squared norm of the difference of the state vector at timeT{\displaystyle T}and time zero, can be written as:
We can truncate the summation at somen=Nindependent ofT, because
∑n=N+1∞|cn|2[1−cos(EnT)]≤2∑n=N+1∞|cn|2{\displaystyle \sum _{n=N+1}^{\infty }|c_{n}|^{2}[1-\cos(E_{n}T)]\leq 2\sum _{n=N+1}^{\infty }|c_{n}|^{2}}
which can be made arbitrarily small by increasingN, as the summation∑n=0∞|cn|2{\displaystyle \sum _{n=0}^{\infty }|c_{n}|^{2}}, being the squared norm of the initial state, converges to 1.
The finite sum
can be made arbitrarily small for specific choices of the timeT, according to the following construction. Choose an arbitraryδ>0{\displaystyle \delta >0}, and then chooseTsuch that there are integerskn{\displaystyle k_{n}}that satisfies
for all numbers0≤n≤N{\displaystyle 0\leq n\leq N}. For this specific choice ofT,
As such, we have:
The state vector|ψ(T)⟩{\displaystyle |\psi (T)\rangle }thus returns arbitrarily close to the initial state|ψ(0)⟩{\displaystyle |\psi (0)\rangle }.
This article incorporates material from Poincaré recurrence theorem onPlanetMath, which is licensed under theCreative Commons Attribution/Share-Alike License. | https://en.wikipedia.org/wiki/Poincar%C3%A9_recurrence_theorem |
Operating signalsare a type ofbrevity codeused in operational communication among radio and telegraph operators. For example: | https://en.wikipedia.org/wiki/Operating_signals |
SINPO, anacronymforSignal, Interference, Noise, Propagation, and Overall, is a Signal Reporting Code used to describe the quality of broadcast andradiotelegraphtransmissions. SINPFEMO, an acronym forSignal, Interference, Noise, Propagation, frequency of Fading, dEpth, Modulation, and Overallis used to describe the quality ofradiotelephonytransmissions. SINPFEMO code consists of the SINPO code plus the addition of three letters to describe additional features of radiotelephony transmissions. These codes are defined byRecommendation ITU-R Sm.1135, SINPO and SINPFEMO codes.[1]
SINPO code is most frequently used inreception reportswritten byshortwave listeners. Each letter of the code stands for a specific factor of the signal, and each item is graded on a 1 to 5 scale (where 1 stands for nearly undetectable/severe/unusable and 5 for excellent/nil/extremely strong).
The code originated with theCCIR(a predecessor to the ITU-R) in 1951, and was widely used by BBC shortwave listeners to submit signal reports, with many going so far as to mail audio recordings to the BBC's offices.[2]
SINPO and SINPFEMO are the official signal reporting codes for international civil aviation[3]and ITU-R.[1]
The use of the SINPO code can be subjective and may vary from person to person. Not allshortwavelisteners are conversant with the SINPO code and prefer using plain language instead.
Each category is rated from 1 to 5 with 1 beingunusableorsevereand 5 beingperfectornil. A higher number always indicates a better result. Wheresignalandoverall meritare graded in terms of strength (where 1 is the lowest and 5 is the highest),interference,noiseandpropagationare graded in the opposite way – theirdegrading effectis measured. For example, a 5 forsignal strengthrepresents a very strong signal, but a 5 fornoiseindicates very little noise.
Many raters misunderstand the code and will rate everything either 55555 or 11111 when in reality both extremes are unusual in the extreme. 55555 essentially means perfect reception akin to a local station while that is occasionally possible, when talking about long-distance short-wave reception, it is rarely the case.
Another common mistake in rating is presenting an O rating higher than any previously rated element. By definition, a reading cannot present perfect reception if there is any noise or interference or fading present. In other words, it cannot be considered perfect local quality reception if any of those things are present.
An extension of SINPO code, for use in radiotelephony (voice over radio) communications, SINPFEMO is an acronym forSignal, Interference, Noise, Propagation, frequency of Fading, dEpth, Modulation, and Overall.
In responding to a shortwave reception, the SINPO indicates to the transmitting station the overall quality of the reception.
The SINPO code in normal use consists of the 5 rating numbers listed without the letters, as in the examples below:
Generally, a SINPO with a code number starting with a 2 or lower would not be worth reporting, unless there is no noise, interference or loss of propagation, since it would be likely the signal would be unintelligible.
Although the original SINPO code established technical specifications for each number (i.e., a number 3 in the P column meant a fixed number of fades per minute), these are rarely adhered to by reporters. The 'S' meter displays the relative strength of the received RF signal in decibels; however, this should not be used as the sole indication of signal strength, as no two S meters are calibrated exactly alike, and many lower-priced receivers omit the S meter altogether. References to a "SINFO" code may also be found in some literature. In this case, the 'F' stands for Fading, instead of 'P' for Propagation, but the two codes are interchangeable. It was presumed that the average listener would be more familiar with the meaning of "fading" than "propagation". A simple way to ensure the rating applied is useful is to rate the "O" column first based on the intelligibility of the station. If you can understand everything easily, the station will rate a 4 or higher. If you have to work hard, but can understand everything '3' is the appropriate rating. If you cannot understand everything although you put great effort into it, a '2' is appropriate, and if you cannot understand the programming at all '1' is the appropriate rating.
Some listeners may not know how to distinguish between the 'I' which indicates interference from adjacent stations, and the 'N' which describes natural atmospheric or man-made noise; also for some listeners, the rating for 'Propagation' may not be completely understood. As a result of this confusion, many stations suggest the SIO code – a simpler code which makes the limitations noted above not relevant. Despite this, some books and periodicals maintain the SINPO code is the best for DX reporters.[4]
SINPO is said to have evolved from the BBC's RAISO format (full version: RAFISBEMVO, which measured:[5] | https://en.wikipedia.org/wiki/SINPO_code |
Telegram style,telegraph style,telegraphic style, ortelegraphese[1]is a clipped way of writing which abbreviates words and packs information into the smallest possible number of words or characters. It originated in thetelegraphage when telecommunication consisted only of short messages transmitted by hand over the telegraph wire. The telegraph companies charged for their service by the number of words in a message, with a maximum of 15 characters per word for a plain-languagetelegram, and 10 per word for one written in code. The style developed to minimize costs but still convey the message clearly and unambiguously.
The related termcablesedescribes the style of press messages sent uncoded but in a highly condensed style oversubmarine communications cables. In the U.S. Foreign Service, cablese referred to condensed telegraphic messaging that made heavy use of abbreviations and avoided use of definite or indefinite articles, punctuation, and other words unnecessary for comprehension of the message.
Before the telegraph age military dispatches from overseas were made by letters transported by rapid sailing ships. Clarity and concision were often considered important in such correspondence.
An apocryphal story about the briefest correspondence in history has a writer (variously identified asVictor HugoorOscar Wilde) inquiring about the sales of his new book by sending the message "?" to his publisher, and receiving "!" in reply.[2]
Through the history of telegraphy, very many dictionaries of telegraphese,codesorcipherswere developed, each serving to minimise the number of characters or words which needed to be transmitted in order to impart a message; the drivers for this economy were, for telegraph operators, the resource cost and limitedbandwidthof the system; and for the consumer, the cost of sending messages.
Examples of telegraphic code-words and their equivalent expressions, taken fromThe Adams Cable Codex(1894)[3]are:
Note that in the Adams code, the code-words are all actual English words; some telegraph companies charged more for coded messages, or had shorter word-size limits (10-character maximum vs. 15 characters). Compare these to the following examples from theA.B.C. Universal Commercial Electric Telegraphic Code(1901)[4]all of which are English-like, but invented words:
In some ways, telegram style was the precursor to the abbreviated language used intext messagingor short message standard (SMS) services such asTwitter, referred to asSMS language
For telegrams, space was at a premium—economically speaking—and abbreviations were used as necessity. This motivation was revived for compressing information into the 160-character limit of a costly SMS before the advent of multi-message capabilities. Length constraints, and the initial handicap of having to enter each individual letter using multiple keypresses on a numeric pad, drove re-adoption of telegraphic style. Continued space limits and high per-message cost meant the practice persisted for some time after the introduction of built-inpredictive textassistance. Some[who?]who favor predictive entry claim that telegraphing persists, despite it then needing more effort to write (and read); however, many others[who?]assert that predictive text generation is usually wrong, and hence find it more tedious and vexing to erase-and-correct predicted text than to turn off auto-text generation and directly enter their messages "telegraph style".
In Japanese, telegrams are printed using thekatakanascript, one of the few instances in which this script is used for entire sentences. This is a rare context in which someone might see the particle katakana ヲ instead of the equivalent hiragana を; these are virtually never used in words, so they are not in the parts of speech that get substituted into katakana.[citation needed]
The average length of a telegram in the 1900s in the US was 11.93 words; more than half of the messages were 10 words or fewer.[5]
According to another study, the mean length of the telegrams sent in the UK before 1950 was 14.6 words or 78.8 characters.[6]
For German telegrams, the mean length is 11.5 words or 72.4 characters.[6]At the end of the 19th century the average length of a German telegram was calculated as 14.2 words.[6] | https://en.wikipedia.org/wiki/Telegraphese |
Hypertext Transfer Protocol(HTTP) response status codes are issued by a server in response to aclient's requestmade to the server. It includes codes from IETFRequest for Comments(RFCs), other specifications, and some additional codes used in some common applications of the HTTP. The first digit of the status code specifies one of five standard classes of responses. The optional message phrases shown are typical, but any human-readable alternative may be provided, or none at all.
Unless otherwise stated, the status code is part of the HTTP standard.[1]
TheInternet Assigned Numbers Authority(IANA) maintains the official registry of HTTP status codes.[2]
All HTTP response status codes are separated into five classes or categories. The first digit of the status code defines the class of response, while the last two digits do not have any classifying or categorization role. There are five classes defined by the standard:
An informational response indicates that the request was received and understood. It is issued on a provisional basis while request processing continues. It alerts the client to wait for a final response. The message consists only of the status line and optional header fields, and is terminated by an empty line. As the HTTP/1.0 standard did not define any 1xx status codes, serversmust not[note 1]send a 1xx response to an HTTP/1.0 compliant client except under experimental conditions.
This class of status codes indicates the action requested by the client was received, understood, and accepted.[2]
This class of status code indicates the client must take additional action to complete the request. Many of these status codes are used inURL redirection.[2]
A user agent may carry out the additional action with no user interaction only if the method used in the second request is GET or HEAD. A user agent may automatically redirect a request. A user agent should detect and intervene to prevent cyclical redirects.[1]: §15.4
This class of status code is intended for situations in which the error seems to have been caused by the client. Except when responding to a HEAD request, the servershouldinclude an entity containing an explanation of the error situation, and whether it is a temporary or permanent condition. These status codes are applicable to anyrequest method. User agentsshoulddisplay any included entity to the user.
Theserverfailed to fulfill a request.
Response status codes beginning with the digit "5" indicate cases in which the server is aware that it has encountered an error or is otherwise incapable of performing the request. Except when responding to a HEAD request, the servershouldinclude an entity containing an explanation of the error situation, and indicate whether it is a temporary or permanent condition. Likewise, user agentsshoulddisplay any included entity to the user. These response codes are applicable to anyrequest method.
The following codes are not specified by any standard.
Microsoft'sInternet Information Services(IIS) web server expands the 4xx error space to signal errors with the client's request.
IIS sometimes uses additional decimal sub-codes for more specific information,[47]however these sub-codes only appear in the response payload and in documentation, not in the place of an actual HTTP status code.
Thenginxweb server software expands the 4xx error space to signal issues with the client's request.[48][49]
Cloudflare's reverse proxy service expands the 5xx series of errors space to signal issues with the origin server.[51]
Amazon Web Services'Elastic Load Balancingadds a few custom return codes to signal issues either with the client request or with the origin server.[55]
The followingcachingrelated warning codes were specified underRFC7234. Unlike the other status codes above, these were not sent as the response status in the HTTP protocol, but as part of the "Warning" HTTP header.[56][57]
Since this "Warning" header is often neither sent by servers nor acknowledged by clients, this header and its codes were obsoleted by the HTTP Working Group in 2022 withRFC9111.[58] | https://en.wikipedia.org/wiki/List_of_HTTP_status_codes |
Policeunits in theUnited Statestend to use atactical designator(ortactical callsign) consisting of a letter of the policeradio alphabetfollowed by one or two numbers. For example, "Mary One" might identify the head of a city'shomicidedivision. Police and fire department radio systems are assigned officialcallsigns, however. Examples are KQY672 and KYX556. The official headquarters callsigns are usually announced at least hourly, and more frequently byMorse code.
TheUnited States Armyuses tactical designators that change daily. They normally consist of letter-number-letter prefixes identifying a unit, followed by a number-number suffix identifying the role of the person using the callsign. | https://en.wikipedia.org/wiki/Tactical_designator |
The Medical Priority Dispatch System(MPDS), sometimes referred to as theAdvanced Medical Priority Dispatch System(AMPDS) is a unified system used to dispatch appropriate aid to medical emergencies including systematized caller interrogation and pre-arrival instructions. Priority Dispatch Corporation is licensed to design and publish MPDS and its various products, with research supported by the International Academy of Emergency Medical Dispatch (IAEMD). Priority Dispatch Corporation, in conjunction with the International Academies of Emergency Dispatch, have also produced similar systems for Police (Police Priority Dispatch System, PPDS) and Fire (Fire Priority Dispatch System, FPDS)
MPDS was developed by Jeff Clawson from 1976 to 1979 when he worked as anemergency medical technicianand dispatcher prior to medical school. He designed a set of standardized protocols to triage patients via the telephone and thus improve the emergency response system. Protocols were first alphabetized by chief complaint that included key questions to ask the caller, pre-arrival instructions, and dispatch priorities. After many revisions, these simple cards have evolved into MPDS.
MPDS today still starts with the dispatcher asking the caller key questions. These questions allow the dispatchers to categorize the call by chief complaint and set a determinant level ranging fromA(minor) toE(immediately life-threatening) relating to the severity of the patient's condition. The system also uses the determinantOwhich may be a referral to another service or other situation that may not actually require an ambulance response. Another sub-category code is used to further categorize the patient.
The system is often used in the form of a software system called ProQA, which is also produced by Priority Dispatch Corp.
Each dispatch determinant is made up of three pieces of information, which builds the determinant in a number-letter-number format. The first component, a number from 1 to 36, indicates a complaint or specific protocol from the MPDS: the selection of this card is based on the initial questions asked by the emergency dispatcher. The second component, a letter A through E (including the Greek character Ω), is the response determinant indicating the potential severity of injury or illness based on information provided by the caller and the recommended type of response. The third component, a number, is the sub-determinant and provides more specific information about the patient's specific condition. For instance, a suspected cardiac or respiratory arrest where the patient is not breathing is given the MPDS code 9-E-1, whereas a superficial animal bite has the code 3-A-3. The MPDS codes allow emergency medical service providers to determine the appropriate response mode (e.g. "routine" or "lights and sirens") and resources to be assigned to the event. Some protocols also utilise a single-letter suffix which may be added to the end of the code to provide additional information, e.g. the code 6-D-1 is a patient with breathing difficulties who is not alert, 6-D-1A is a patient with breathing difficulties who is not alert and also has asthma, and 6-D-1E is a patient with breathing difficulties who is not alert and hasemphysema/COAD/COPD.
[1]
This Protocol was created to handle the influx of emergency calls during the H1N1 pandemic: it directed that Standard EMS Resources be delayed until patients could be assessed by a Flu Response Unit (FRU), a single provider that could attend a patient and determine what additional resources were required for patient care to reduce the risk of pandemic exposure to EMS Personnel. In March 2020 the protocol was revised to assist with mitigating theCOVID-19 pandemic.[2]
[3]
As well as triaging emergency calls, MPDS also provides instructions for the dispatcher to give to the caller whilst assistance is en route. These post-dispatch and pre-arrival instructions are intended both to keep the caller and the patient safe, but also, where necessary, to turn the caller into the "first first responder" by giving them potentially life-saving instructions. They include:
Whilst MPDS uses the determinants to provide a recommendation as to the type of response that may be appropriate, some countries use a different response approach. For example, in the United Kingdom, most, but not all front-line emergency ambulances have advanced life support trained crews, meaning that the ALS/BLS distinction becomes impossible to implement. Instead, each individual response code is assigned to one of several categories, as determined by the Government, with associated response targets for each.
[4]
* This may include an emergency ambulance, a rapid response car, ambulance officers, or specialist crews e.g.HART. Other basic life support responses may also be sent, e.g.Community First Responder.
** If an emergency ambulance is unlikely to reach the patient within the average response time, a rapid response car and/or Community First Responder may also be dispatched.
The exact nature of the response sent may vary slightly betweenAmbulance Trusts. Following a Category 2, 3, or 5 telephone triage, the patient may receive an ambulance response (which could be Category 1-4 depending on the outcome of the triage), may be referred to another service or provider, or treatment may be completed over the phone.
In an independent report into the emergency response to theManchester Arena bombing, an Advanced Paramedic for theNorth West Ambulance Servicestated it was "very much understood" that MPDS "vastly underemphasises the priority of traumatic calls."[5] | https://en.wikipedia.org/wiki/Advanced_Medical_Priority_Dispatch_System |
Emergency service response codesare predefined systems used byemergency servicesto describe the priority and response assigned tocalls for service. Response codes vary from country to country, jurisdiction to jurisdiction, and even agency to agency, with different methods used to categorize responses to reported events.
In theUnited States, response codes are used to describe a mode of response for an emergency unit responding to a call. They generally vary but often have three basic tiers:
Some agencies may use the terms "upgrade" and "downgrade" to denote an increase or decrease in priority. For example, if a police unit is conducting a Code 1 response to an argument, and the dispatcher reports that the argument has escalated to a fight, the unit may report an "upgrade" to a Code 3 response. The term downgrade may be used in the opposite situation.
A similar variation, generally used by units instead of dispatchers, is to "increase code" and "reduce code". For example, if multiple units are responding Code 3 to a call, but the units already at the scene have mostly resolved the situation, the scene units may request that the responding units "reduce code". In this example, to "reduce code" would mean to continue responding, but at Code 2 or Code 1, rather than discontinue altogether.
Multiple analyses link “Code 3” operations with crashes involving responders. Accurate use of protocols establishing the priority of various cases is critical. The standard for emergency dispatcher training is becoming very high.
Someemergency medical services - (EMS) dispatch agenciesuse "Priority" dispatching to establish the urgency of a given request for service, or ”call”. They ask the caller a series of questions to establish how urgently help is required. They ask: is the patient alert? Talking? Breathing? The answers help establish who needs to respond and the priority of the response.
TheNational Incident Management System(NIMS) states "it is required that plain language be used for multi-agency, multi-jurisdiction and multi-discipline events, such as major disasters and exercises", and federal grants became contingent on this beginning fiscal year 2006.[1]NIMS also strongly encourages the use of plain language for internal use within a single agency.
Plain language avoids confusion resulting from varying meanings assigned to different codes.
Historically “10-codes” and “signal codes” were used when radios were less reliable and frequent repetition was required. These codes were rarely uniform even between local agencies. Most used “10-4” to mean “acknowledged”, for example, but some agencies used it as “message ends”. A “Signal 30” could be a fatal car crash or any death, depending on local usage.
Plain language helps ensure critical clarity in emergencies. It is the clear standard.
The use of lights and sirens is up to the individual police officer driving to the call. The nature of the call is an aggravating factor when deciding when to use them. Calls are graded by either the control room direct (in the case of emergency calls) or by some sort of first contact centre (nonemergency calls). Grading is affected by such factors as the use or threat of violence at the incident being reported. Even though the grading is done by the control room, officers can request an incident be upgraded if they feel in their judgement they are needed immediately. They can also request to downgrade an incident if they feel they cannot justify using emergency equipment like blue lights and sirens.
There is no nationally agreed call grading system with a number of different systems being used across the UK and attendance times given the grade varies between forces, depending on how rural the county is. For example, Suffolk Constabulary break down Grade A emergencies into further sub-categories of Grade A Urban and Rural, with Urban attendance times attracting a 15-minute arrival time and Grade A Rural attendance would attract a 20-minute arrival time. Some of these are listed below but is not exhaustive.[2]
Another variant in use within the UK.
A numerical grading system is used in some forces.[3][failed verification]
Ambulance responses in the UK are as follows. Some ambulance services allow driver discretion for Category 3/4 calls; this may be dependent on the type of call or how long it has been waiting for a response for. 999 calls to the ambulance service are triaged using either theNHS Pathwayssystem or theMedical Priority Dispatch System.
Will be attended by single responders and ambulance crews
Response time measured with arrival of transporting vehicle
[5]
The use of flashing lights and sirens is colloquially known asblues and twos, which refers to the blue lights and the two-tone siren once commonplace (although most sirens now use a range of tones). In the UK, only blue lights are used to denote emergency vehicles (although other colours may be used as sidelights, stop indicators, etc.). A call requiring the use of lights and sirens is often colloquially known as ablue light run.
Code 1: A time critical case with a lights and sirens ambulance response. An example is a cardiac arrest or serious traffic accident.
Code 2:An acute but non-time critical response. The ambulance does not use lights and sirens to respond. An example of this response code is a broken leg.
Code 3:A non-urgent routine case. These include cases such as a person with ongoing back pain but no recent injury.
Source
Additional codes are used for internal purposes.
Country Fire AuthorityThere are two types of response for the Country Fire Authority which cover the outer Melbourne Area. These are similar to those used byAmbulance Victoria, minus the use of Code 2.
Code 1:A time critical event with response requiring lights and siren. This usually is a known and going fire or a rescue incident.
Code 2:Unused within theCountry Fire Authority
Code 3:Non-urgent event, such as a previously extinguished fire or community service cases (such as animal rescue or changing of smoke alarm batteries for the elderly).
Marine Rescue NSW
Code 1 Urgent Response - Use warning devices
Code 2 Semi Urgent Response - Use of Warning devices at skippers discretion
Code 3 Non Urgent Response - Warning Devices not needed
Code 4 Training - No Warning devices to be used unless specifically needed for training
TheNew South Wales Rural Fire Serviceand theNew South Wales State Emergency Serviceuse two levels of response, depending on what the call-out is and what has been directed of the crew attending the incident by orders of the duty officer:
TheNew South Wales Police Forceuses two distinct classifications for responding to incidents. In order to respond 'code red' a driver must be suitably trained and have qualified in appropriate police driver training courses.[7]
New South Wales Ambulanceuse 2 priorities similar to both SES and RFS.
SA Ambulance Serviceuse a Priority system.
Note: Priority 0 has been reserved for future use. Priority 9 is used for administration taskings and non-patient related vehicle movements.
TheSouth Australian Metropolitan Fire Service,Country Fire Serviceand South AustralianState Emergency Serviceuse a Priority System which has been recently updated.
All calls are routed through the Metropolitan Fire Service (Call Sign "Adelaide Fire") includingState Emergency Service132 500 calls.
During significant weather events the State Communication Centre (SCC) unit of the SES take over call taking responsibly. This operations centre is staffed by volunteers routing calls for assistance to the closest unit who will dispatch the events to individual teams.
Queensland Policeuses the priority system:
For Queensland Police code 1 and code 2 are exactly the same response time. Rarely will a job be given a priority code 1, instead officers will (in most cases) be told to respond code 2.
St John Ambulance Northern Territoryuses terms to determine the response:[8]
St John Ambulance Western Australiauses the following codes to determine a response:[9]
TheWestern Australia Police Forceuses the following Priority codes from 1 to 6 to determine the urgency of Police response:
TheDepartment of Fire and Emergency Serviceshave two response codes:[10]
BC EHS Clinical Response Modelimplemented as of May 30, 2018 by BC Emergency Health Services, updating how they assign paramedics, ambulance and other resources to 9-1-1 calls.[11]
In Sweden, emergency services use specific response codes as in many other countries to categorize the urgency of incidents and determine the appropriate response. These codes vary among the police, ambulance services, and fire services.
TheSwedish Police Authoritycategorizes emergency responses into three urgency levels, known asangelägenhetsgrader(urgency levels). Each level determines the extent to which officers may disregard traffic regulations while responding to incidents. However, all exemptions require caution and must only be exercised if necessary.[12]
However, even in an emergency (trängande fall), police must obey the instructions oftraffic officersortraffic guard. The use ofblue lights and sirensto request the right of way is only permitted duringträngande fall
The Swedish ambulance services classify emergencies into four priority levels:[13]
The Swedish fire services operate with two primary response priorities similarly to the ambulance service,:[14] | https://en.wikipedia.org/wiki/Emergency_service_response_codes |
CB slangis the distinctiveanti-language,argot, orcantwhich developed among users ofCitizens Band radio(CB), especiallytruck driversin the United States during the 1970s and early 1980s,[1]when it was an important part of theculture of the trucking industry.
The slang itself is not only cyclical, but also geographical. Through time, certain terms are added or dropped as attitudes towards it changed. For example, in the early days of the CB radio, the term "Good buddy" was widely used.[2]
Nicknames orcall signsgiven or adopted by CB radio users are known as "handles".[2][3]Many truck drivers will call each other "Hand,"[4]or by the name of the company for which they drive.[citation needed]
CB and its distinctive language started in theUnited Statesbut was then exported to other countries includingMexico,Germany, andCanada. | https://en.wikipedia.org/wiki/List_of_CB_slang |
Alist of international common and basictechnical standards, which have been established worldwide and are related by their frequent and widespread use. These standards are conventionally accepted asbest practiceand used globally byindustryandorganizations.
In circumstances and situations there are certain methods and systems that are used as benchmarks, guidelines or protocols forcommunication,measurement,orientation,reference for information,science,symbolsandtime. These standards are employed to universally convey meaning, classification and to relate details of information.
The standards listed may be formal or informal and some might not be recognised by all governments or organizations.
Inradio communications
Inelectronics
Ingeographical location
Inelectronics | https://en.wikipedia.org/wiki/List_of_international_common_standards |
The Medical Priority Dispatch System(MPDS), sometimes referred to as theAdvanced Medical Priority Dispatch System(AMPDS) is a unified system used to dispatch appropriate aid to medical emergencies including systematized caller interrogation and pre-arrival instructions. Priority Dispatch Corporation is licensed to design and publish MPDS and its various products, with research supported by the International Academy of Emergency Medical Dispatch (IAEMD). Priority Dispatch Corporation, in conjunction with the International Academies of Emergency Dispatch, have also produced similar systems for Police (Police Priority Dispatch System, PPDS) and Fire (Fire Priority Dispatch System, FPDS)
MPDS was developed by Jeff Clawson from 1976 to 1979 when he worked as anemergency medical technicianand dispatcher prior to medical school. He designed a set of standardized protocols to triage patients via the telephone and thus improve the emergency response system. Protocols were first alphabetized by chief complaint that included key questions to ask the caller, pre-arrival instructions, and dispatch priorities. After many revisions, these simple cards have evolved into MPDS.
MPDS today still starts with the dispatcher asking the caller key questions. These questions allow the dispatchers to categorize the call by chief complaint and set a determinant level ranging fromA(minor) toE(immediately life-threatening) relating to the severity of the patient's condition. The system also uses the determinantOwhich may be a referral to another service or other situation that may not actually require an ambulance response. Another sub-category code is used to further categorize the patient.
The system is often used in the form of a software system called ProQA, which is also produced by Priority Dispatch Corp.
Each dispatch determinant is made up of three pieces of information, which builds the determinant in a number-letter-number format. The first component, a number from 1 to 36, indicates a complaint or specific protocol from the MPDS: the selection of this card is based on the initial questions asked by the emergency dispatcher. The second component, a letter A through E (including the Greek character Ω), is the response determinant indicating the potential severity of injury or illness based on information provided by the caller and the recommended type of response. The third component, a number, is the sub-determinant and provides more specific information about the patient's specific condition. For instance, a suspected cardiac or respiratory arrest where the patient is not breathing is given the MPDS code 9-E-1, whereas a superficial animal bite has the code 3-A-3. The MPDS codes allow emergency medical service providers to determine the appropriate response mode (e.g. "routine" or "lights and sirens") and resources to be assigned to the event. Some protocols also utilise a single-letter suffix which may be added to the end of the code to provide additional information, e.g. the code 6-D-1 is a patient with breathing difficulties who is not alert, 6-D-1A is a patient with breathing difficulties who is not alert and also has asthma, and 6-D-1E is a patient with breathing difficulties who is not alert and hasemphysema/COAD/COPD.
[1]
This Protocol was created to handle the influx of emergency calls during the H1N1 pandemic: it directed that Standard EMS Resources be delayed until patients could be assessed by a Flu Response Unit (FRU), a single provider that could attend a patient and determine what additional resources were required for patient care to reduce the risk of pandemic exposure to EMS Personnel. In March 2020 the protocol was revised to assist with mitigating theCOVID-19 pandemic.[2]
[3]
As well as triaging emergency calls, MPDS also provides instructions for the dispatcher to give to the caller whilst assistance is en route. These post-dispatch and pre-arrival instructions are intended both to keep the caller and the patient safe, but also, where necessary, to turn the caller into the "first first responder" by giving them potentially life-saving instructions. They include:
Whilst MPDS uses the determinants to provide a recommendation as to the type of response that may be appropriate, some countries use a different response approach. For example, in the United Kingdom, most, but not all front-line emergency ambulances have advanced life support trained crews, meaning that the ALS/BLS distinction becomes impossible to implement. Instead, each individual response code is assigned to one of several categories, as determined by the Government, with associated response targets for each.
[4]
* This may include an emergency ambulance, a rapid response car, ambulance officers, or specialist crews e.g.HART. Other basic life support responses may also be sent, e.g.Community First Responder.
** If an emergency ambulance is unlikely to reach the patient within the average response time, a rapid response car and/or Community First Responder may also be dispatched.
The exact nature of the response sent may vary slightly betweenAmbulance Trusts. Following a Category 2, 3, or 5 telephone triage, the patient may receive an ambulance response (which could be Category 1-4 depending on the outcome of the triage), may be referred to another service or provider, or treatment may be completed over the phone.
In an independent report into the emergency response to theManchester Arena bombing, an Advanced Paramedic for theNorth West Ambulance Servicestated it was "very much understood" that MPDS "vastly underemphasises the priority of traumatic calls."[5] | https://en.wikipedia.org/wiki/Medical_Priority_Dispatch_System |
The Password Gameis a 2023puzzlebrowser gamedeveloped by Neal Agarwal, where the player creates apasswordthat follows increasingly unusual and complicated rules. Based on Agarwal's experience withpassword policies,[1]the game was developed in two months, releasing on June 27, 2023. The gamewent viraland was recognized in the media for the gameplay's absurdity and commentary on the user experience of generating a password. It has been played over 10 million times.
The Password Gameis aweb-basedpuzzle video game.[2]The player is tasked with typing apasswordin an input box.[3]The game has a total of 35 rules that the password must follow and which appear in a specific order.[4]As the player changes the password to comply with the first rule, a second one appears, and so on.[2][5]For each additional rule, the player must follow all the previous ones to progress, which can cause conflict.[5][6]When all 35 rules are fulfilled, the player is able to confirm it as the final password and then must retype the password to complete the game.[4]
Although the initial requirements include setting a minimum of characters or including numbers, uppercase letters, or special characters,[1][7]the rules gradually become more unusual and complex.[3][6]These can involve managing having Roman numerals in the string to multiply,[6][8]adding the name of a country that players have to guess from randomGoogle Street Viewimagery (as a reference toGeoGuessr),[6][9][10]inserting the day'sWordleanswer,[8]typing the best move in a generatedchessposition usingalgebraic notation,[6][11]inserting the URL of aYouTubevideo of a randomly generated length,[4][6][11]and adjusting boldface, italics, font types, and text sizes.[4]
Other game rules involveemojisin the password. One demands inclusion of the emoji representing themoon phaseat that point in time.[12]Because of two other rules, the player is required to insert an egg emoji named Paul, and once it hatches, it is replaced by a chicken emoji. The player then must keep it fed using caterpillar emojis that must be replenished over time.[13][14]If it starves, the player overfeeds it, or the Paul emoji is deleted in any way, the game ends. Red text subsequently appears over a black background, referencing the death screen characteristic of theDark Soulsaction role-playing game series.[11][13]At some point during the game, a flame emoji will appear, spreading through the password by replacing characters, including the egg, with flames that must be removed.[15]
The Password Gamewas developed by Neal Agarwal, who posts his games on his website, neal.fun.[2][16]Agarwal had conceptualized the idea of the game as a parody ofpassword policiesas they got "weirder".[3]According to Agarwal, "the final straw" that made him start to work on the game may have been when he was trying to create an account on a service and was told that his password was too long, mocking the notion of a password being "too secure".[1]Development started in late April 2023 and took two months.[3]Agarwal mentioned that implementingregular expressions("find" operations instrings) was hard, especially due to features of the game's text editor that show up as the player progresses, like making text bold or italic.[1]Some of the game's password requirements were suggested to him onTwitter. Before release, Agarwal was unsure whether winning the game was possible; he attempted it unsuccessfully multiple times.[3]The game was released on his website on June 27, 2023.[3][1]
The Password Gamewent viralonline soon after release.[20]After its first day, thetweetannouncing the game was retweeted over 11,000 times, and according to the developer, the game's website received over one million visits. The tweet received multiple comments discussing numbers that people reached in the game.[3]As reported byEngadget, Twitter mentions of Agarwal were "full of people cursing him for creating" the game and people exclaiming having beaten it, to the surprise of the developer.[8]As of October 2023, the game was visited over 10 million times.[21]
Many critics have contrasted the standardness and simplicity of the game's initial password rules to the absurdity of the following ones.[22]The sixteenth rule of the game, which is about finding the best chess move in a specific position, was considered the most challenging byPCGamesN[11]and made other reviewers give up the game.[3][18][23]WhileTechRadarandThe Indian ExpressdeemedThe Password Gameto be a good way to kill time,[12][18]PC Gamercalled it "the evilest will-breaking browser game to exist".[9]The game was regarded byPCGamesNas possibly "one of the most inventive experiences of the year".[11]Polygondescribed it as a "comedy set in a user interface" that incorporates many secrets behind its apparent simplicity.[3]Rock Paper Shotgundiscussed the gameplay loop of the game, finding they frequently experienced amusement, followed by effort to fulfill the rule, and feeling satisfied.[2]PCWorldfelt it emphasized the usefulness ofpassword managers,[7]whileTechRadarfound it outdated due to tools likepassword generators.[5] | https://en.wikipedia.org/wiki/The_Password_Game |
Code wordmay refer to: | https://en.wikipedia.org/wiki/Code_word_(disambiguation) |
Infunctional analysisand related areas ofmathematicsanabsorbing setin avector spaceis asetS{\displaystyle S}which can be "inflated" or "scaled up" to eventually always include any given point of the vector space.
Alternative terms areradialorabsorbent set.
Everyneighborhood of the originin everytopological vector spaceis an absorbing subset.
Notation for scalars
Suppose thatX{\displaystyle X}is a vector space over thefieldK{\displaystyle \mathbb {K} }ofreal numbersR{\displaystyle \mathbb {R} }orcomplex numbersC,{\displaystyle \mathbb {C} ,}and for any−∞≤r≤∞,{\displaystyle -\infty \leq r\leq \infty ,}letBr={a∈K:|a|<r}andB≤r={a∈K:|a|≤r}{\displaystyle B_{r}=\{a\in \mathbb {K} :|a|<r\}\quad {\text{ and }}\quad B_{\leq r}=\{a\in \mathbb {K} :|a|\leq r\}}denote theopen ball(respectively, theclosed ball) of radiusr{\displaystyle r}inK{\displaystyle \mathbb {K} }centered at0.{\displaystyle 0.}Define the product of a setK⊆K{\displaystyle K\subseteq \mathbb {K} }of scalars with a setA{\displaystyle A}of vectors asKA={ka:k∈K,a∈A},{\displaystyle KA=\{ka:k\in K,a\in A\},}and define the product ofK⊆K{\displaystyle K\subseteq \mathbb {K} }with a single vectorx{\displaystyle x}asKx={kx:k∈K}.{\displaystyle Kx=\{kx:k\in K\}.}
Balanced core and balanced hull
A subsetS{\displaystyle S}ofX{\displaystyle X}is said to bebalancedifas∈S{\displaystyle as\in S}for alls∈S{\displaystyle s\in S}and all scalarsa{\displaystyle a}satisfying|a|≤1;{\displaystyle |a|\leq 1;}this condition may be written more succinctly asB≤1S⊆S,{\displaystyle B_{\leq 1}S\subseteq S,}and it holds if and only ifB≤1S=S.{\displaystyle B_{\leq 1}S=S.}
Given a setT,{\displaystyle T,}the smallestbalanced setcontainingT,{\displaystyle T,}denoted bybalT,{\displaystyle \operatorname {bal} T,}is called thebalanced hullofT{\displaystyle T}while the largest balanced set contained withinT,{\displaystyle T,}denoted bybalcoreT,{\displaystyle \operatorname {balcore} T,}is called thebalanced coreofT.{\displaystyle T.}These sets are given by the formulasbalT=⋃|c|≤1cT=B≤1T{\displaystyle \operatorname {bal} T~=~{\textstyle \bigcup \limits _{|c|\leq 1}}c\,T=B_{\leq 1}T}andbalcoreT={⋂|c|≥1cTif0∈T∅if0∉T,{\displaystyle \operatorname {balcore} T~=~{\begin{cases}{\textstyle \bigcap \limits _{|c|\geq 1}}c\,T&{\text{ if }}0\in T\\\varnothing &{\text{ if }}0\not \in T,\\\end{cases}}}(these formulas show that the balanced hull and the balanced core always exist and are unique).
A setT{\displaystyle T}is balanced if and only if it is equal to its balanced hull (T=balT{\displaystyle T=\operatorname {bal} T}) or to its balanced core (T=balcoreT{\displaystyle T=\operatorname {balcore} T}), in which case all three of these sets are equal:T=balT=balcoreT.{\displaystyle T=\operatorname {bal} T=\operatorname {balcore} T.}
Ifc{\displaystyle c}is any scalar thenbal(cT)=cbalT=|c|balT{\displaystyle \operatorname {bal} (c\,T)=c\,\operatorname {bal} T=|c|\,\operatorname {bal} T}while ifc≠0{\displaystyle c\neq 0}is non-zero or if0∈T{\displaystyle 0\in T}then alsobalcore(cT)=cbalcoreT=|c|balcoreT.{\displaystyle \operatorname {balcore} (c\,T)=c\,\operatorname {balcore} T=|c|\,\operatorname {balcore} T.}
IfS{\displaystyle S}andA{\displaystyle A}are subsets ofX,{\displaystyle X,}thenA{\displaystyle A}is said toabsorbS{\displaystyle S}if it satisfies any of the following equivalent conditions:
IfA{\displaystyle A}is abalanced setthen this list can be extended to include:
If0∈A{\displaystyle 0\in A}(a necessary condition forA{\displaystyle A}to be anabsorbing set, or to be a neighborhood of the origin in a topology) then this list can be extended to include:
If0∉S{\displaystyle 0\not \in S}or0∈A{\displaystyle 0\in A}then this list can be extended to include:
A set absorbing a point
A set is said toabsorb a pointx{\displaystyle x}if it absorbs thesingleton set{x}.{\displaystyle \{x\}.}A setA{\displaystyle A}absorbs the origin if and only if it contains the origin; that is, if and only if0∈A.{\displaystyle 0\in A.}As detailed below, a set is said to beabsorbing inX{\displaystyle X}if it absorbs every point ofX.{\displaystyle X.}
This notion of one set absorbing another is also used in other definitions:
A subset of atopological vector spaceX{\displaystyle X}is calledboundedif it is absorbed by every neighborhood of the origin.
A set is calledbornivorousif it absorbs every bounded subset.
First examples
Every set absorbs the empty set but the empty set does not absorb any non-empty set. The singleton set{0}{\displaystyle \{\mathbf {0} \}}containing the origin is the one and only singleton subset that absorbs itself.
Suppose thatX{\displaystyle X}is equal to eitherR2{\displaystyle \mathbb {R} ^{2}}orC.{\displaystyle \mathbb {C} .}IfA:=S1∪{0}{\displaystyle A:=S^{1}\cup \{\mathbf {0} \}}is theunit circle(centered at the origin0{\displaystyle \mathbf {0} }) together with the origin, then{0}{\displaystyle \{\mathbf {0} \}}is the one and only non-empty set thatA{\displaystyle A}absorbs. Moreover, there doesnotexistanynon-empty subset ofX{\displaystyle X}that is absorbed by the unit circleS1.{\displaystyle S^{1}.}In contrast, everyneighborhoodof the origin absorbs everybounded subsetofX{\displaystyle X}(and so in particular, absorbs every singleton subset/point).
A subsetA{\displaystyle A}of a vector spaceX{\displaystyle X}over a fieldK{\displaystyle \mathbb {K} }is called anabsorbing(orabsorbent)subsetofX{\displaystyle X}and is said to beabsorbing inX{\displaystyle X}if it satisfies any of the following equivalent conditions (here ordered so that each condition is an easy consequence of the previous one, starting with the definition):
IfK=R{\displaystyle \mathbb {K} =\mathbb {R} }then to this list can be appended:
IfA{\displaystyle A}isbalancedthen to this list can be appended:
IfA{\displaystyle A}isconvexorbalancedthen to this list can be appended:
If0∈A{\displaystyle 0\in A}(which is necessary forA{\displaystyle A}to be absorbing) then it suffices to check any of the above conditions for all non-zerox∈X,{\displaystyle x\in X,}rather than allx∈X.{\displaystyle x\in X.}
LetF:X→Y{\displaystyle F:X\to Y}be a linear map between vector spaces and letB⊆X{\displaystyle B\subseteq X}andC⊆Y{\displaystyle C\subseteq Y}be balanced sets. ThenC{\displaystyle C}absorbsF(B){\displaystyle F(B)}if and only ifF−1(C){\displaystyle F^{-1}(C)}absorbsB.{\displaystyle B.}[2]
If a setA{\displaystyle A}absorbs another setB{\displaystyle B}then any superset ofA{\displaystyle A}also absorbsB.{\displaystyle B.}A setA{\displaystyle A}absorbs the origin if and only if the origin is an element ofA.{\displaystyle A.}
A setA{\displaystyle A}absorbs a finite unionB1∪⋯∪Bn{\displaystyle B_{1}\cup \cdots \cup B_{n}}of sets if and only it absorbs each set individuality (that is, if and only ifA{\displaystyle A}absorbsBi{\displaystyle B_{i}}for everyi=1,…,n{\displaystyle i=1,\ldots ,n}). In particular, a setA{\displaystyle A}is an absorbing subset ofX{\displaystyle X}if and only if it absorbs every finite subset ofX.{\displaystyle X.}
Theunit ballof anynormed vector space(orseminormed vector space) is absorbing.
More generally, ifX{\displaystyle X}is atopological vector space(TVS) then any neighborhood of the origin inX{\displaystyle X}is absorbing inX.{\displaystyle X.}This fact is one of the primary motivations for defining the property "absorbing inX.{\displaystyle X.}"
Every superset of an absorbing set is absorbing. Consequently, the union of any family of (one or more) absorbing sets is absorbing. The intersection of finitely many absorbing subsets is once again an absorbing subset. However, the open balls(−rn,−rn){\displaystyle (-r_{n},-r_{n})}of radiusrn=1,1/2,1/3,…{\displaystyle r_{n}=1,1/2,1/3,\ldots }are all absorbing inX:=R{\displaystyle X:=\mathbb {R} }although their intersection⋂n∈N(−1/n,1/n)={0}{\displaystyle \bigcap _{n\in \mathbb {N} }(-1/n,1/n)=\{0\}}is not absorbing.
IfD≠∅{\displaystyle D\neq \varnothing }is adisk(a convex and balanced subset) thenspanD=⋃n=1∞nD;{\displaystyle \operatorname {span} D={\textstyle \bigcup \limits _{n=1}^{\infty }}nD;}and so in particular, a diskD≠∅{\displaystyle D\neq \varnothing }is always an absorbing subset ofspanD.{\displaystyle \operatorname {span} D.}[3]Thus ifD{\displaystyle D}is a disk inX,{\displaystyle X,}thenD{\displaystyle D}is absorbing inX{\displaystyle X}if and only ifspanD=X.{\displaystyle \operatorname {span} D=X.}This conclusion is not guaranteed if the setD≠∅{\displaystyle D\neq \varnothing }is balanced but not convex; for example, the unionD{\displaystyle D}of thex{\displaystyle x}andy{\displaystyle y}axes inX=R2{\displaystyle X=\mathbb {R} ^{2}}is a non-convex balanced set that is not absorbing inspanD=R2.{\displaystyle \operatorname {span} D=\mathbb {R} ^{2}.}
The image of an absorbing set under a surjective linear operator is again absorbing. The inverse image of an absorbing subset (of the codomain) under a linear operator is again absorbing (in the domain).
IfA{\displaystyle A}absorbing then the same is true of thesymmetric set⋂|u|=1uA⊆A.{\displaystyle {\textstyle \bigcap \limits _{|u|=1}}uA\subseteq A.}
Auxiliary normed spaces
IfW{\displaystyle W}isconvexand absorbing inX{\displaystyle X}then thesymmetric setD:=⋂|u|=1uW{\displaystyle D:={\textstyle \bigcap \limits _{|u|=1}}uW}will be convex andbalanced(also known as anabsolutely convex setor adisk) in addition to being absorbing inX.{\displaystyle X.}This guarantees that theMinkowski functionalpD:X→R{\displaystyle p_{D}:X\to \mathbb {R} }ofD{\displaystyle D}will be aseminormonX,{\displaystyle X,}thereby making(X,pD){\displaystyle \left(X,p_{D}\right)}into aseminormed spacethat carries its canonicalpseduometrizabletopology. The set of scalar multiplesrD{\displaystyle rD}asr{\displaystyle r}ranges over{12,13,14,…}{\displaystyle \left\{{\tfrac {1}{2}},{\tfrac {1}{3}},{\tfrac {1}{4}},\ldots \right\}}(or over any other set of non-zero scalars having0{\displaystyle 0}as a limit point) forms aneighborhood basisof absorbingdisksat the origin for thislocally convextopology. IfX{\displaystyle X}is atopological vector spaceand if this convex absorbing subsetW{\displaystyle W}is also abounded subsetofX,{\displaystyle X,}then all this will also be true of the absorbing diskD:=⋂|u|=1uW;{\displaystyle D:={\textstyle \bigcap \limits _{|u|=1}}uW;}if in additionD{\displaystyle D}does not contain any non-trivial vector subspace thenpD{\displaystyle p_{D}}will be anormand(X,pD){\displaystyle \left(X,p_{D}\right)}will form what is known as anauxiliary normed space.[4]If this normed space is aBanach spacethenD{\displaystyle D}is called aBanach disk.
Every absorbing set contains the origin.
IfD{\displaystyle D}is an absorbingdiskin a vector spaceX{\displaystyle X}then there exists an absorbing diskE{\displaystyle E}inX{\displaystyle X}such thatE+E⊆D.{\displaystyle E+E\subseteq D.}[5]
IfA{\displaystyle A}is an absorbing subset ofX{\displaystyle X}thenX=⋃n=1∞nA{\displaystyle X={\textstyle \bigcup \limits _{n=1}^{\infty }}nA}and more generally,X=⋃n=1∞snA{\displaystyle X={\textstyle \bigcup \limits _{n=1}^{\infty }}s_{n}A}for any sequence of scalarss1,s2,…{\displaystyle s_{1},s_{2},\ldots }such that|sn|→∞.{\displaystyle \left|s_{n}\right|\to \infty .}Consequently, if atopological vector spaceX{\displaystyle X}is anon-meager subsetof itself (or equivalently for TVSs, if it is aBaire space) and ifA{\displaystyle A}is a closed absorbing subset ofX{\displaystyle X}thenA{\displaystyle A}necessarily contains a non-empty open subset ofX{\displaystyle X}(in other words,A{\displaystyle A}'stopological interiorwill not be empty), which guarantees thatA−A{\displaystyle A-A}is aneighborhood of the origininX.{\displaystyle X.}
Every absorbing set is atotal set, meaning that every absorbing subspace isdense.
Proofs | https://en.wikipedia.org/wiki/Absorbing_set |
Annihilator(s)may refer to: | https://en.wikipedia.org/wiki/Annihilator_(disambiguation) |
Inmathematics, theannihilatorof asubsetSof amoduleover aringis theidealformed by the elements of the ring that give always zero when multiplied by each element ofS.
Over anintegral domain, a module that has a nonzero annihilator is atorsion module, and afinitely generatedtorsion module has a nonzero annihilator.
The above definition applies also in the case ofnoncommutative rings, where theleft annihilatorof a left module is a left ideal, and theright-annihilator, of a right module is a right ideal.
LetRbe aring, and letMbe a leftR-module. Choose anon-emptysubsetSofM. TheannihilatorofS, denoted AnnR(S), is the set of all elementsrinRsuch that, for allsinS,rs= 0.[1]In set notation,
It is the set of all elements ofRthat "annihilate"S(the elements for whichSis a torsion set). Subsets of right modules may be used as well, after the modification of "sr= 0" in the definition.
The annihilator of a single elementxis usually written AnnR(x) instead of AnnR({x}). If the ringRcan be understood from the context, the subscriptRcan be omitted.
SinceRis a module over itself,Smay be taken to be a subset ofRitself, and sinceRis both a right and a leftR-module, the notation must be modified slightly to indicate the left or right side. Usuallyℓ.AnnR(S){\displaystyle \ell .\!\mathrm {Ann} _{R}(S)\,}andr.AnnR(S){\displaystyle r.\!\mathrm {Ann} _{R}(S)\,}or some similar subscript scheme are used to distinguish the left and right annihilators, if necessary.
IfMis anR-module andAnnR(M) = 0, thenMis called afaithful module.
IfSis a subset of a leftR-moduleM, then Ann(S) is a leftidealofR.[2]
IfSis asubmoduleofM, then AnnR(S) is even a two-sided ideal: (ac)s=a(cs) = 0, sincecsis another element ofS.[3]
IfSis a subset ofMandNis the submodule ofMgenerated byS, then in general AnnR(N) is a subset of AnnR(S), but they are not necessarily equal. IfRiscommutative, then the equality holds.
Mmay be also viewed as anR/AnnR(M)-module using the actionr¯m:=rm{\displaystyle {\overline {r}}m:=rm\,}. Incidentally, it is not always possible to make anR-module into anR/I-module this way, but if the idealIis a subset of the annihilator ofM, then this action is well-defined. Considered as anR/AnnR(M)-module,Mis automatically a faithful module.
Throughout this section, letR{\displaystyle R}be a commutative ring andM{\displaystyle M}afinitely generatedR{\displaystyle R}-module.
Thesupport of a moduleis defined as
Then, when the module is finitely generated, there is the relation
whereV(⋅){\displaystyle V(\cdot )}is the set ofprime idealscontaining the subset.[4]
Given ashort exact sequenceof modules,
the support property
together with the relation with the annihilator implies
More specifically, the relations
If the sequence splits then the inequality on the left is always an equality. This holds for arbitrarydirect sumsof modules, as
Given an idealI⊆R{\displaystyle I\subseteq R}and letM{\displaystyle M}be a finitely generated module, then there is the relation
on the support. Using the relation to support, this gives the relation with the annihilator[6]
OverZ{\displaystyle \mathbb {Z} }any finitely generated module is completely classified as the direct sum of itsfreepart with its torsion part from the fundamental theorem of abelian groups. Then the annihilator of a finitely generated module is non-trivial only if it is entirely torsion. This is because
since the only element killing each of theZ{\displaystyle \mathbb {Z} }is0{\displaystyle 0}. For example, the annihilator ofZ/2⊕Z/3{\displaystyle \mathbb {Z} /2\oplus \mathbb {Z} /3}is
the ideal generated by(6){\displaystyle (6)}. In fact the annihilator of a torsion module
isisomorphicto the ideal generated by theirleast common multiple,(lcm(a1,…,an)){\displaystyle (\operatorname {lcm} (a_{1},\ldots ,a_{n}))}. This shows the annihilators can be easily be classified over the integers.
There is a similar computation that can be done for anyfinitely presented moduleover a commutative ringR{\displaystyle R}. The definition of finite presentedness ofM{\displaystyle M}implies there exists an exact sequence, called a presentation, given by
whereϕ{\displaystyle \phi }is inMatk,l(R){\displaystyle {\text{Mat}}_{k,l}(R)}. Writingϕ{\displaystyle \phi }explicitly as amatrixgives it as
henceM{\displaystyle M}has the direct sum decomposition
If each of these ideals is written as
then the idealI{\displaystyle I}given by
presents the annihilator.
Over the commutative ringk[x,y]{\displaystyle k[x,y]}for afieldk{\displaystyle k}, the annihilator of the module
is given by the ideal
Thelatticeof ideals of the formℓ.AnnR(S){\displaystyle \ell .\!\mathrm {Ann} _{R}(S)}whereSis a subset ofRis acomplete latticewhenpartially orderedbyinclusion. There is interest in studying rings for which this lattice (or its right counterpart) satisfies theascending chain conditionordescending chain condition.
Denote the lattice of left annihilator ideals ofRasLA{\displaystyle {\mathcal {LA}}\,}and the lattice of right annihilator ideals ofRasRA{\displaystyle {\mathcal {RA}}\,}. It is known thatLA{\displaystyle {\mathcal {LA}}\,}satisfies the ascending chain conditionif and only ifRA{\displaystyle {\mathcal {RA}}\,}satisfies the descending chain condition, and symmetricallyRA{\displaystyle {\mathcal {RA}}\,}satisfies the ascending chain condition if and only ifLA{\displaystyle {\mathcal {LA}}\,}satisfies the descending chain condition. If either lattice has either of these chain conditions, thenRhas no infinite pairwise orthogonal sets ofidempotents.[7][8]
IfRis a ring for whichLA{\displaystyle {\mathcal {LA}}\,}satisfies the A.C.C. andRRhas finiteuniform dimension, thenRis called a leftGoldie ring.[8]
WhenRis commutative andMis anR-module, we may describe AnnR(M) as thekernelof the action mapR→ EndR(M)determined by theadjunct mapof theidentityM→Malong theHom-tensor adjunction.
More generally, given abilinear mapof modulesF:M×N→P{\displaystyle F\colon M\times N\to P}, the annihilator of a subsetS⊆M{\displaystyle S\subseteq M}is the set of all elements inN{\displaystyle N}that annihilateS{\displaystyle S}:
Conversely, givenT⊆N{\displaystyle T\subseteq N}, one can define an annihilator as a subset ofM{\displaystyle M}.
The annihilator gives aGalois connectionbetween subsets ofM{\displaystyle M}andN{\displaystyle N}, and the associatedclosure operatoris stronger than the span.
In particular:
An important special case is in the presence of anondegenerate formon avector space, particularly aninner product: then the annihilator associated to the mapV×V→K{\displaystyle V\times V\to K}is called theorthogonal complement.
Given a moduleMover aNoetheriancommutative ringR, a prime ideal ofRthat is an annihilator of a nonzero element ofMis called anassociated primeofM. | https://en.wikipedia.org/wiki/Annihilator_(ring_theory) |
Inring theory, a branch ofmathematics, anidempotent elementor simplyidempotentof aringis an elementasuch thata2=a.[1][a]That is, the element isidempotentunder the ring's multiplication.Inductivelythen, one can also conclude thata=a2=a3=a4= ... =anfor any positiveintegern. For example, an idempotent element of amatrix ringis precisely anidempotent matrix.
For general rings, elements idempotent under multiplication are involved in decompositions ofmodules, and connected tohomologicalproperties of the ring. InBoolean algebra, the main objects of study are rings in which all elements are idempotent under both addition and multiplication.
One may consider thering of integers modulon, wherenissquare-free. By theChinese remainder theorem, this ring factors into theproduct of ringsof integers modulop, wherepisprime. Now each of these factors is afield, so it is clear that the factors' only idempotents will be0and1. That is, each factor has two idempotents. So if there aremfactors, there will be2midempotents.
We can check this for the integersmod 6,R=Z/ 6Z. Since6has two prime factors (2and3) it should have22idempotents.
From these computations,0,1,3, and4are idempotents of this ring, while2and5are not. This also demonstrates the decomposition properties described below: because3 + 4 ≡ 1 (mod 6), there is a ring decomposition3Z/ 6Z⊕ 4Z/ 6Z. In3Z/ 6Zthe multiplicative identity is3 + 6Zand in4Z/ 6Zthe multiplicative identity is4 + 6Z.
Given a ringRand an elementf∈Rsuch thatf2≠ 0, thequotient ring
has the idempotentf. For example, this could be applied tox∈Z[x], or anypolynomialf∈k[x1, ...,xn].
There is acircleof idempotents in the ring ofsplit-quaternions. Split quaternions have the structure of areal algebra, so elements can be writtenw+xi +yj +zk over abasis{1, i, j, k}, with j2= k2= +1. For any θ,
The elementsis called ahyperbolic unitand so far, the i-coordinate has been taken as zero. When this coordinate is non-zero, then there is ahyperboloid of one sheetofhyperbolic units in split-quaternions. The same equality shows the idempotent property of1+s2{\displaystyle {\frac {1+s}{2}}}wheresis on the hyperboloid.
A partial list of important types of idempotents includes:
Any non-trivial idempotentais azero divisor(becauseab= 0with neitheranorbbeing zero, whereb= 1 −a). This shows thatintegral domainsanddivision ringsdo not have such idempotents.Local ringsalso do not have such idempotents, but for a different reason. The only idempotent contained in theJacobson radicalof a ring is0.
The idempotents ofRhave an important connection to decomposition ofR-modules. IfMis anR-module andE= EndR(M)is itsring of endomorphisms, thenA⊕B=Mif and only if there is a unique idempotenteinEsuch thatA=eMandB= (1 −e)M. Clearly then,Mis directly indecomposable if and only if0and1are the only idempotents inE.[2]
In the case whenM=R(assumed unital), the endomorphism ringEndR(R) =R, where eachendomorphismarises as left multiplication by a fixed ring element. With this modification of notation,A⊕B=Ras right modules if and only if there exists a unique idempotentesuch thateR=Aand(1 −e)R=B. Thus every direct summand ofRis generated by an idempotent.
Ifais a central idempotent, then the corner ringaRa=Rais a ring with multiplicative identitya. Just as idempotents determine the direct decompositions ofRas a module, the central idempotents ofRdetermine the decompositions ofRas adirect sumof rings. IfRis the direct sum of the ringsR1, ...,Rn, then the identity elements of the ringsRiare central idempotents inR, pairwise orthogonal, and their sum is1. Conversely, given central idempotentsa1, ...,aninRthat are pairwise orthogonal and have sum1, thenRis the direct sum of the ringsRa1, ...,Ran. So in particular, every central idempotentainRgives rise to a decomposition ofRas a direct sum of the corner ringsaRaand(1 −a)R(1 −a). As a result, a ringRis directly indecomposable as a ring if and only if the identity1is centrally primitive.
Working inductively, one can attempt to decompose1into a sum of centrally primitive elements. If1is centrally primitive, we are done. If not, it is a sum of central orthogonal idempotents, which in turn are primitive or sums of more central idempotents, and so on. The problem that may occur is that this may continue without end, producing an infinite family of central orthogonal idempotents. The condition "Rdoes not contain infinite sets of central orthogonal idempotents" is a type of finiteness condition on the ring. It can be achieved in many ways, such as requiring the ring to be rightNoetherian. If a decompositionR=c1R⊕c2R⊕ ... ⊕cnRexists with eachcia centrally primitive idempotent, thenRis a direct sum of the corner ringsciRci, each of which is ring irreducible.[3]
Forassociative algebrasorJordan algebrasover a field, thePeirce decompositionis a decomposition of an algebra as a sum of eigenspaces of commuting idempotent elements.
Ifais an idempotent of the endomorphism ringEndR(M), then the endomorphismf= 1 − 2ais anR-moduleinvolutionofM. That is,fis anR-module homomorphismsuch thatf2is the identity endomorphism ofM.
An idempotent elementaofRand its associated involutionfgives rise to two involutions of the moduleR, depending on viewingRas a left or right module. Ifrrepresents an arbitrary element ofR,fcan be viewed as a rightR-module homomorphismr↦frso thatffr=r, orfcan also be viewed as a leftR-module homomorphismr↦rf, whererff=r.
This process can be reversed if2is aninvertible elementofR:[b]ifbis an involution, then2−1(1 −b)and2−1(1 +b)are orthogonal idempotents, corresponding toaand1 −a. Thus for a ring in which2is invertible, the idempotent elementscorrespondto involutions in a one-to-one manner.
Lifting idempotents also has major consequences for thecategory ofR-modules. All idempotents lift moduloIif and only if everyRdirect summand ofR/Ihas aprojective coveras anR-module.[4]Idempotents always lift modulonil idealsand rings for whichRisI-adically complete.
Lifting is most important whenI= J(R), theJacobson radicalofR. Yet another characterization ofsemiperfect ringsis that they aresemilocal ringswhose idempotents lift moduloJ(R).[5]
One may define apartial orderon the idempotents of a ring as follows: ifaandbare idempotents, we writea≤bif and only ifab=ba=a. With respect to this order,0is the smallest and1the largest idempotent. For orthogonal idempotentsaandb,a+bis also idempotent, and we havea≤a+bandb≤a+b. Theatomsof this partial order are precisely the primitive idempotents.[6]
When the above partial order is restricted to the central idempotents ofR, alatticestructure, or even aBoolean algebrastructure, can be given. For two central idempotentseandf, thecomplementis given by
themeetis given by
and thejoinis given by
The ordering now becomes simplye≤fif and only ifeR⊆fR, and the join and meet satisfy(e∨f)R=eR+fRand(e∧f)R=eR∩fR= (eR)(fR). It is shown inGoodearl 1991, p. 99 that ifRisvon Neumann regularand rightself-injective, then the lattice is acomplete lattice. | https://en.wikipedia.org/wiki/Idempotent_(ring_theory) |
Inmathematics, anull semigroup(also called azero semigroup) is asemigroupwith anabsorbing element, calledzero, in which the product of any two elements is zero.[1]If every element of a semigroup is aleft zerothen the semigroup is called aleft zero semigroup; aright zero semigroupis defined analogously.[2]
According toA. H. CliffordandG. B. Preston, "In spite of their triviality, these semigroups arise naturally in a number of investigations."[1]
LetSbe a semigroup with zero element 0. ThenSis called anull semigroupifxy= 0 for allxandyinS.
LetS= {0,a,b,c} be (the underlying set of) a null semigroup. Then theCayley tableforSis as given below:
A semigroup in which every element is aleft zeroelement is called aleft zero semigroup. Thus a semigroupSis a left zero semigroup ifxy=xfor allxandyinS.
LetS= {a,b,c} be a left zero semigroup. Then the Cayley table forSis as given below:
A semigroup in which every element is aright zeroelement is called aright zero semigroup. Thus a semigroupSis a right zero semigroup ifxy=yfor allxandyinS.
LetS= {a,b,c} be a right zero semigroup. Then the Cayley table forSis as given below:
A non-trivial null (left/right zero) semigroup does not contain anidentity element. It follows that the only null (left/right zero)monoidis the trivial monoid.
The class of null semigroups is:
It follows that the class of null (left/right zero) semigroups is avariety of universal algebra, and thus avariety of finite semigroups. The variety of finite null semigroups is defined by the identityab=cd. | https://en.wikipedia.org/wiki/Null_semigroup |
Inmathematics, theabsolute valueormodulusof areal numberx{\displaystyle x},denoted|x|{\displaystyle |x|},is thenon-negativevalueofx{\displaystyle x}without regard to itssign. Namely,|x|=x{\displaystyle |x|=x}ifx{\displaystyle x}is apositive number, and|x|=−x{\displaystyle |x|=-x}ifx{\displaystyle x}isnegative(in which case negatingx{\displaystyle x}makes−x{\displaystyle -x}positive), and|0|=0{\displaystyle |0|=0}.For example, the absolute value of 3is 3,and the absolute value of −3 isalso 3.The absolute value of a number may be thought of as itsdistancefrom zero.
Generalisations of the absolute value for real numbers occur in a wide variety of mathematical settings. For example, an absolute value is also defined for thecomplex numbers, thequaternions,ordered rings,fieldsandvector spaces. The absolute value is closely related to the notions ofmagnitude,distance, andnormin various mathematical and physical contexts.
In 1806,Jean-Robert Argandintroduced the termmodule, meaningunit of measurein French, specifically for thecomplexabsolute value,[1][2]and it was borrowed into English in 1866 as the Latin equivalentmodulus.[1]The termabsolute valuehas been used in this sense from at least 1806 in French[3]and 1857 in English.[4]The notation|x|, with avertical baron each side, was introduced byKarl Weierstrassin 1841.[5]Other names forabsolute valueincludenumerical value[1]andmagnitude.[1]The absolute value ofx{\displaystyle x}has also been denotedabsx{\displaystyle \operatorname {abs} x}in some mathematical publications,[6]and inspreadsheets, programming languages, and computational software packages, the absolute value ofx{\textstyle x}is generally represented byabs(x), or a similar expression,[7]as it has been since the earliest days ofhigh-level programming languages.[8]
The vertical bar notation also appears in a number of other mathematical contexts: for example, when applied to a set, it denotes itscardinality; when applied to amatrix, it denotes itsdeterminant.[9]Vertical bars denote the absolute value only for algebraic objects for which the notion of an absolute value is defined, notably anelementof anormed division algebra, for example a real number, a complex number, or a quaternion. A closely related but distinct notation is the use of vertical bars for either theEuclidean norm[10]orsup norm[11]of a vectorinRn{\displaystyle \mathbb {R} ^{n}},although double vertical bars with subscripts(‖⋅‖2{\displaystyle \|\cdot \|_{2}}and‖⋅‖∞{\displaystyle \|\cdot \|_{\infty }},respectively) are a more common and less ambiguous notation.
For anyreal numberx{\displaystyle x},theabsolute valueormodulusofx{\displaystyle x}is denotedby|x|{\displaystyle |x|}, with avertical baron each side of the quantity, and is defined as[12]|x|={x,ifx≥0−x,ifx<0.{\displaystyle |x|={\begin{cases}x,&{\text{if }}x\geq 0\\-x,&{\text{if }}x<0.\end{cases}}}
The absolute valueofx{\displaystyle x}is thus always either apositive numberorzero, but nevernegative. Whenx{\displaystyle x}itself is negative(x<0{\displaystyle x<0}),then its absolute value is necessarily positive(|x|=−x>0{\displaystyle |x|=-x>0}).
From ananalytic geometrypoint of view, the absolute value of a real number is that number'sdistancefrom zero along thereal number line, and more generally the absolute value of the difference of two real numbers (theirabsolute difference) is the distance between them.[13]The notion of an abstractdistance functionin mathematics can be seen to be a generalisation of the absolute value of the difference (see"Distance"below).
Since thesquare root symbolrepresents the uniquepositivesquare root, when applied to a positive number, it follows that|x|=x2.{\displaystyle |x|={\sqrt {x^{2}}}.}This is equivalent to the definition above, and may be used as an alternative definition of the absolute value of real numbers.[14]
The absolute value has the following four fundamental properties (a{\textstyle a},b{\textstyle b}are real numbers), that are used for generalization of this notion to other domains:
Non-negativity, positive definiteness, and multiplicativity are readily apparent from the definition. To see that subadditivity holds, first note that|a+b|=s(a+b){\displaystyle |a+b|=s(a+b)}wheres=±1{\displaystyle s=\pm 1},with its sign chosen to make the result positive. Now, since−1⋅x≤|x|{\displaystyle -1\cdot x\leq |x|}and+1⋅x≤|x|{\displaystyle +1\cdot x\leq |x|},it follows that, whichever of±1{\displaystyle \pm 1}is the valueofs{\displaystyle s},one hass⋅x≤|x|{\displaystyle s\cdot x\leq |x|}for allrealx{\displaystyle x}.Consequently,|a+b|=s⋅(a+b)=s⋅a+s⋅b≤|a|+|b|{\displaystyle |a+b|=s\cdot (a+b)=s\cdot a+s\cdot b\leq |a|+|b|}, as desired.
Some additional useful properties are given below. These are either immediate consequences of the definition or implied by the four fundamental properties above.
Two other useful properties concerning inequalities are:
These relations may be used to solve inequalities involving absolute values. For example:
The absolute value, as "distance from zero", is used to define the absolute difference between arbitrary real numbers, the standardmetricon the real numbers.
Since thecomplex numbersare notordered, the definition given at the top for the real absolute value cannot be directly applied to complex numbers. However, the geometric interpretation of the absolute value of a real number as its distance from 0 can be generalised. The absolute value of a complex number is defined by the Euclidean distance of its corresponding point in thecomplex planefrom theorigin. This can be computed using thePythagorean theorem: for any complex numberz=x+iy,{\displaystyle z=x+iy,}wherex{\displaystyle x}andy{\displaystyle y}are real numbers, theabsolute valueormodulusofz{\displaystyle z}isdenoted|z|{\displaystyle |z|}and is defined by[15]|z|=Re(z)2+Im(z)2=x2+y2,{\displaystyle |z|={\sqrt {\operatorname {Re} (z)^{2}+\operatorname {Im} (z)^{2}}}={\sqrt {x^{2}+y^{2}}},}thePythagorean additionofx{\displaystyle x}andy{\displaystyle y}, whereRe(z)=x{\displaystyle \operatorname {Re} (z)=x}andIm(z)=y{\displaystyle \operatorname {Im} (z)=y}denote the real and imaginary partsofz{\displaystyle z},respectively. When theimaginary party{\displaystyle y}is zero, this coincides with the definition of the absolute value of thereal numberx{\displaystyle x}.
When a complex numberz{\displaystyle z}is expressed in itspolar formasz=reiθ,{\displaystyle z=re^{i\theta },}its absolute valueis|z|=r.{\displaystyle |z|=r.}
Since the product of any complex numberz{\displaystyle z}and itscomplex conjugatez¯=x−iy{\displaystyle {\bar {z}}=x-iy},with the same absolute value, is always the non-negative real number(x2+y2){\displaystyle \left(x^{2}+y^{2}\right)},the absolute value of a complex numberz{\displaystyle z}is the square rootofz⋅z¯,{\displaystyle z\cdot {\overline {z}},}which is therefore called theabsolute squareorsquared modulusofz{\displaystyle z}:|z|=z⋅z¯.{\displaystyle |z|={\sqrt {z\cdot {\overline {z}}}}.}This generalizes the alternative definition for reals:|x|=x⋅x{\textstyle |x|={\sqrt {x\cdot x}}}.
The complex absolute value shares the four fundamental properties given above for the real absolute value. The identity|z|2=|z2|{\displaystyle |z|^{2}=|z^{2}|}is a special case of multiplicativity that is often useful by itself.
The real absolute value function iscontinuouseverywhere. It isdifferentiableeverywhere except forx= 0. It ismonotonically decreasingon theinterval(−∞, 0]and monotonically increasing on the interval[0, +∞). Since a real number and itsoppositehave the same absolute value, it is aneven function, and is hence notinvertible. The real absolute value function is apiecewise linear,convex function.
For both real and complex numbers the absolute value function isidempotent(meaning that the absolute value of any absolute value is itself).
The absolute value function of a real number returns its value irrespective of its sign, whereas thesign (or signum) functionreturns a number's sign irrespective of its value. The following equations show the relationship between these two functions:
or
and forx≠ 0,
Lets,t∈R{\displaystyle s,t\in \mathbb {R} }, then the following relationship to theminimumandmaximumfunctions hold:
and
The formulas can be derived by considering each cases>t{\displaystyle s>t}andt>s{\displaystyle t>s}separately.
From the last formula one can derive also|t|=max(t,−t){\displaystyle |t|=\max(t,-t)}.
The real absolute value function has aderivativefor everyx≠ 0, but is notdifferentiableatx= 0. Its derivative forx≠ 0is given by thestep function:[16][17]
The real absolute value function is an example of a continuous function that achieves aglobal minimumwhere the derivative does not exist.
Thesubdifferentialof|x|atx= 0is the interval[−1, 1].[18]
Thecomplexabsolute value function is continuous everywhere butcomplex differentiablenowherebecause it violates theCauchy–Riemann equations.[16]
The second derivative of|x|with respect toxis zero everywhere except zero, where it does not exist. As ageneralised function, the second derivative may be taken as two times theDirac delta function.
Theantiderivative(indefiniteintegral) of the real absolute value function is
whereCis an arbitraryconstant of integration. This is not acomplex antiderivativebecause complex antiderivatives can only exist for complex-differentiable (holomorphic) functions, which the complex absolute value function is not.
The following two formulae are special cases of thechain rule:
ddxf(|x|)=x|x|(f′(|x|)){\displaystyle {d \over dx}f(|x|)={x \over |x|}(f'(|x|))}
if the absolute value is inside a function, and
ddx|f(x)|=f(x)|f(x)|f′(x){\displaystyle {d \over dx}|f(x)|={f(x) \over |f(x)|}f'(x)}
if another function is inside the absolute value. In the first case, the derivative is always discontinuous atx=0{\textstyle x=0}in the first case and wheref(x)=0{\textstyle f(x)=0}in the second case.
The absolute value is closely related to the idea ofdistance. As noted above, the absolute value of a real or complex number is the distance from that number to the origin, along the real number line, for real numbers, or in the complex plane, for complex numbers, and more generally, the absolute value of the difference of two real or complex numbers is the distance between them.
The standardEuclidean distancebetween two points
and
inEuclideann-spaceis defined as:
This can be seen as a generalisation, since fora1{\displaystyle a_{1}}andb1{\displaystyle b_{1}}real, i.e. in a 1-space, according to the alternative definition of the absolute value,
and fora=a1+ia2{\displaystyle a=a_{1}+ia_{2}}andb=b1+ib2{\displaystyle b=b_{1}+ib_{2}}complex numbers, i.e. in a 2-space,
The above shows that the "absolute value"-distance, for real and complex numbers, agrees with the standard Euclidean distance, which they inherit as a result of considering them as one and two-dimensional Euclidean spaces, respectively.
The properties of the absolute value of the difference of two real or complex numbers: non-negativity, identity of indiscernibles, symmetry and the triangle inequality given above, can be seen to motivate the more general notion of adistance functionas follows:
A real valued functiondon a setX×Xis called ametric(or adistance function) onX, if it satisfies the following four axioms:[19]
The definition of absolute value given for real numbers above can be extended to anyordered ring. That is, ifais an element of an ordered ringR, then theabsolute valueofa, denoted by|a|, is defined to be:[20]
where−ais theadditive inverseofa, 0 is theadditive identity, and < and ≥ have the usual meaning with respect to the ordering in the ring.
The four fundamental properties of the absolute value for real numbers can be used to generalise the notion of absolute value to an arbitrary field, as follows.
A real-valued functionvon afieldFis called anabsolute value(also amodulus,magnitude,value, orvaluation)[21]if it satisfies the following four axioms:
Where0denotes theadditive identityofF. It follows from positive-definiteness and multiplicativity thatv(1) = 1, where1denotes themultiplicative identityofF. The real and complex absolute values defined above are examples of absolute values for an arbitrary field.
Ifvis an absolute value onF, then the functiondonF×F, defined byd(a,b) =v(a−b), is a metric and the following are equivalent:
An absolute value which satisfies any (hence all) of the above conditions is said to benon-Archimedean, otherwise it is said to beArchimedean.[22]
Again the fundamental properties of the absolute value for real numbers can be used, with a slight modification, to generalise the notion to an arbitrary vector space.
A real-valued function on avector spaceVover a fieldF, represented as‖ · ‖, is called anabsolute value, but more usually anorm, if it satisfies the following axioms:
For allainF, andv,uinV,
The norm of a vector is also called itslengthormagnitude.
In the case ofEuclidean spaceRn{\displaystyle \mathbb {R} ^{n}}, the function defined by
is a norm called the Euclidean norm. When the real numbersR{\displaystyle \mathbb {R} }are considered as the one-dimensional vector spaceR1{\displaystyle \mathbb {R} ^{1}}, the absolute value is anorm, and is thep-norm (seeLpspace) for anyp. In fact the absolute value is the "only" norm onR1{\displaystyle \mathbb {R} ^{1}}, in the sense that, for every norm‖ · ‖onR1{\displaystyle \mathbb {R} ^{1}},‖x‖ = ‖1‖ ⋅ |x|.
The complex absolute value is a special case of the norm in aninner product space, which is identical to the Euclidean norm when the complex plane is identified as theEuclidean planeR2{\displaystyle \mathbb {R} ^{2}}.
Every composition algebraAhas aninvolutionx→x* called itsconjugation. The product inAof an elementxand its conjugatex* is writtenN(x) =x x* and called thenorm of x.
The real numbersR{\displaystyle \mathbb {R} }, complex numbersC{\displaystyle \mathbb {C} }, and quaternionsH{\displaystyle \mathbb {H} }are all composition algebras with norms given bydefinite quadratic forms. The absolute value in thesedivision algebrasis given by the square root of the composition algebra norm.
In general the norm of a composition algebra may be aquadratic formthat is not definite and hasnull vectors. However, as in the case of division algebras, when an elementxhas a non-zero norm, thenxhas amultiplicative inversegiven byx*/N(x). | https://en.wikipedia.org/wiki/Absolute_value |
Inmathematics, theinverse functionof afunctionf(also called theinverseoff) is afunctionthat undoes the operation off. The inverse offexistsif and only iffisbijective, and if it exists, is denoted byf−1.{\displaystyle f^{-1}.}
For a functionf:X→Y{\displaystyle f\colon X\to Y}, its inversef−1:Y→X{\displaystyle f^{-1}\colon Y\to X}admits an explicit description: it sends each elementy∈Y{\displaystyle y\in Y}to the unique elementx∈X{\displaystyle x\in X}such thatf(x) =y.
As an example, consider thereal-valuedfunction of a real variable given byf(x) = 5x− 7. One can think offas the function which multiplies its input by 5 then subtracts 7 from the result. To undo this, one adds 7 to the input, then divides the result by 5. Therefore, the inverse offis the functionf−1:R→R{\displaystyle f^{-1}\colon \mathbb {R} \to \mathbb {R} }defined byf−1(y)=y+75.{\displaystyle f^{-1}(y)={\frac {y+7}{5}}.}
Letfbe a function whosedomainis thesetX, and whosecodomainis the setY. Thenfisinvertibleif there exists a functiongfromYtoXsuch thatg(f(x))=x{\displaystyle g(f(x))=x}for allx∈X{\displaystyle x\in X}andf(g(y))=y{\displaystyle f(g(y))=y}for ally∈Y{\displaystyle y\in Y}.[1]
Iffis invertible, then there is exactly one functiongsatisfying this property. The functiongis called the inverse off, and is usually denoted asf−1, a notation introduced byJohn Frederick William Herschelin 1813.[2][3][4][5][6][nb 1]
The functionfis invertible if and only if it is bijective. This is because the conditiong(f(x))=x{\displaystyle g(f(x))=x}for allx∈X{\displaystyle x\in X}implies thatfisinjective, and the conditionf(g(y))=y{\displaystyle f(g(y))=y}for ally∈Y{\displaystyle y\in Y}implies thatfissurjective.
The inverse functionf−1tofcan be explicitly described as the function
Recall that iffis an invertible function with domainXand codomainY, then
Using thecomposition of functions, this statement can be rewritten to the following equations between functions:
whereidXis theidentity functionon the setX; that is, the function that leaves its argument unchanged. Incategory theory, this statement is used as the definition of an inversemorphism.
Considering function composition helps to understand the notationf−1. Repeatedly composing a functionf:X→Xwith itself is callediteration. Iffis appliedntimes, starting with the valuex, then this is written asfn(x); sof2(x) =f(f(x)), etc. Sincef−1(f(x)) =x, composingf−1andfnyieldsfn−1, "undoing" the effect of one application off.
While the notationf−1(x)might be misunderstood,[1](f(x))−1certainly denotes themultiplicative inverseoff(x)and has nothing to do with the inverse function off.[6]The notationf⟨−1⟩{\displaystyle f^{\langle -1\rangle }}might be used for the inverse function to avoid ambiguity with themultiplicative inverse.[7]
In keeping with the general notation, some English authors use expressions likesin−1(x)to denote the inverse of the sine function applied tox(actually apartial inverse; see below).[8][6]Other authors feel that this may be confused with the notation for the multiplicative inverse ofsin (x), which can be denoted as(sin (x))−1.[6]To avoid any confusion, aninverse trigonometric functionis often indicated by the prefix "arc" (for Latinarcus).[9][10]For instance, the inverse of the sine function is typically called thearcsinefunction, written asarcsin(x).[9][10]Similarly, the inverse of ahyperbolic functionis indicated by the prefix "ar" (for Latinārea).[10]For instance, the inverse of thehyperbolic sinefunction is typically written asarsinh(x).[10]The expressions likesin−1(x)can still be useful to distinguish themultivaluedinverse from the partial inverse:sin−1(x)={(−1)narcsin(x)+πn:n∈Z}{\displaystyle \sin ^{-1}(x)=\{(-1)^{n}\arcsin(x)+\pi n:n\in \mathbb {Z} \}}. Other inverse special functions are sometimes prefixed with the prefix "inv", if the ambiguity of thef−1notation should be avoided.[11][10]
The functionf:R→ [0,∞)given byf(x) =x2is not injective because(−x)2=x2{\displaystyle (-x)^{2}=x^{2}}for allx∈R{\displaystyle x\in \mathbb {R} }. Therefore,fis not invertible.
If the domain of the function is restricted to the nonnegative reals, that is, we take the functionf:[0,∞)→[0,∞);x↦x2{\displaystyle f\colon [0,\infty )\to [0,\infty );\ x\mapsto x^{2}}with the sameruleas before, then the function is bijective and so, invertible.[12]The inverse function here is called the(positive) square root functionand is denoted byx↦x{\displaystyle x\mapsto {\sqrt {x}}}.
The following table shows several standard functions and their inverses:
Many functions given by algebraic formulas possess a formula for their inverse. This is because the inversef−1{\displaystyle f^{-1}}of an invertible functionf:R→R{\displaystyle f\colon \mathbb {R} \to \mathbb {R} }has an explicit description as
This allows one to easily determine inverses of many functions that are given by algebraic formulas. For example, iffis the function
then to determinef−1(y){\displaystyle f^{-1}(y)}for a real numbery, one must find the unique real numberxsuch that(2x+ 8)3=y. This equation can be solved:
Thus the inverse functionf−1is given by the formula
Sometimes, the inverse of a function cannot be expressed by aclosed-form formula. For example, iffis the function
thenfis a bijection, and therefore possesses an inverse functionf−1. Theformula for this inversehas an expression as an infinite sum:
Since a function is a special type ofbinary relation, many of the properties of an inverse function correspond to properties ofconverse relations.
If an inverse function exists for a given functionf, then it is unique.[13]This follows since the inverse function must be the converse relation, which is completely determined byf.
There is a symmetry between a function and its inverse. Specifically, iffis an invertible function with domainXand codomainY, then its inversef−1has domainYand imageX, and the inverse off−1is the original functionf. In symbols, for functionsf:X→Yandf−1:Y→X,[13]
This statement is a consequence of the implication that forfto be invertible it must be bijective. Theinvolutorynature of the inverse can be concisely expressed by[14]
The inverse of a composition of functions is given by[15]
Notice that the order ofgandfhave been reversed; to undoffollowed byg, we must first undog, and then undof.
For example, letf(x) = 3xand letg(x) =x+ 5. Then the compositiong∘fis the function that first multiplies by three and then adds five,
To reverse this process, we must first subtract five, and then divide by three,
This is the composition(f−1∘g−1)(x).
IfXis a set, then theidentity functiononXis its own inverse:
More generally, a functionf:X→Xis equal to its own inverse, if and only if the compositionf∘fis equal toidX. Such a function is called aninvolution.
Iffis invertible, then the graph of the function
is the same as the graph of the equation
This is identical to the equationy=f(x)that defines the graph off, except that the roles ofxandyhave been reversed. Thus the graph off−1can be obtained from the graph offby switching the positions of thexandyaxes. This is equivalent toreflectingthe graph across the liney=x.[16][1]
By theinverse function theorem, acontinuous functionof a single variablef:A→R{\displaystyle f\colon A\to \mathbb {R} }(whereA⊆R{\displaystyle A\subseteq \mathbb {R} }) is invertible on its range (image) if and only if it is either strictlyincreasing or decreasing(with no localmaxima or minima). For example, the function
is invertible, since thederivativef′(x) = 3x2+ 1is always positive.
If the functionfisdifferentiableon an intervalIandf′(x) ≠ 0for eachx∈I, then the inversef−1is differentiable onf(I).[17]Ify=f(x), the derivative of the inverse is given by the inverse function theorem,
UsingLeibniz's notationthe formula above can be written as
This result follows from thechain rule(see the article oninverse functions and differentiation).
The inverse function theorem can be generalized to functions of several variables. Specifically, a continuously differentiablemultivariable functionf:Rn→Rnis invertible in a neighborhood of a pointpas long as theJacobian matrixoffatpisinvertible. In this case, the Jacobian off−1atf(p)is thematrix inverseof the Jacobian offatp.
Even if a functionfis not one-to-one, it may be possible to define apartial inverseoffbyrestrictingthe domain. For example, the function
is not one-to-one, sincex2= (−x)2. However, the function becomes one-to-one if we restrict to the domainx≥ 0, in which case
(If we instead restrict to the domainx≤ 0, then the inverse is the negative of the square root ofy.)
Alternatively, there is no need to restrict the domain if we are content with the inverse being amultivalued function:
Sometimes, this multivalued inverse is called thefull inverseoff, and the portions (such as√xand −√x) are calledbranches. The most important branch of a multivalued function (e.g. the positive square root) is called theprincipal branch, and its value atyis called theprincipal valueoff−1(y).
For a continuous function on the real line, one branch is required between each pair oflocal extrema. For example, the inverse of acubic functionwith a local maximum and a local minimum has three branches (see the adjacent picture).
The above considerations are particularly important for defining the inverses oftrigonometric functions. For example, thesine functionis not one-to-one, since
for every realx(and more generallysin(x+ 2πn) = sin(x)for everyintegern). However, the sine is one-to-one on the interval[−π/2,π/2], and the corresponding partial inverse is called thearcsine. This is considered the principal branch of the inverse sine, so the principal value of the inverse sine is always between −π/2andπ/2. The following table describes the principal branch of each inverse trigonometric function:[19]
Function compositionon the left and on the right need not coincide. In general, the conditions
imply different properties off. For example, letf:R→[0, ∞)denote the squaring map, such thatf(x) =x2for allxinR, and letg:[0, ∞)→Rdenote the square root map, such thatg(x) =√xfor allx≥ 0. Thenf(g(x)) =xfor allxin[0, ∞); that is,gis a right inverse tof. However,gis not a left inverse tof, since, e.g.,g(f(−1)) = 1 ≠ −1.
Iff:X→Y, aleft inverseforf(orretractionoff) is a functiong:Y→Xsuch that composingfwithgfrom the left gives the identity function[20]g∘f=idX.{\displaystyle g\circ f=\operatorname {id} _{X}{\text{.}}}That is, the functiongsatisfies the rule
The functiongmust equal the inverse offon the image off, but may take any values for elements ofYnot in the image.
A functionfwith nonempty domain is injective if and only if it has a left inverse.[21]An elementary proof runs as follows:
If nonemptyf:X→Yis injective, construct a left inverseg:Y→Xas follows: for ally∈Y, ifyis in the image off, then there existsx∈Xsuch thatf(x) =y. Letg(y) =x; this definition is unique becausefis injective. Otherwise, letg(y)be an arbitrary element ofX.
For allx∈X,f(x)is in the image off. By construction,g(f(x)) =x, the condition for a left inverse.
In classical mathematics, every injective functionfwith a nonempty domain necessarily has a left inverse; however, this may fail inconstructive mathematics. For instance, a left inverse of theinclusion{0,1} →Rof the two-element set in the reals violatesindecomposabilityby giving aretractionof the real line to the set{0,1}.[22]
Aright inverseforf(orsectionoff) is a functionh:Y→Xsuch that
That is, the functionhsatisfies the rule
Thus,h(y)may be any of the elements ofXthat map toyunderf.
A functionfhas a right inverse if and only if it issurjective(though constructing such an inverse in general requires theaxiom of choice).
An inverse that is both a left and right inverse (atwo-sided inverse), if it exists, must be unique. In fact, if a function has a left inverse and a right inverse, they are both the same two-sided inverse, so it can be calledthe inverse.
A function has a two-sided inverse if and only if it is bijective.
Iff:X→Yis any function (not necessarily invertible), thepreimage(orinverse image) of an elementy∈Yis defined to be the set of all elements ofXthat map toy:
The preimage ofycan be thought of as theimageofyunder the (multivalued) full inverse of the functionf.
The notion can be generalized to subsets of the range. Specifically, ifSis anysubsetofY, the preimage ofS, denoted byf−1(S){\displaystyle f^{-1}(S)}, is the set of all elements ofXthat map toS:
For example, take the functionf:R→R;x↦x2. This function is not invertible as it is not bijective, but preimages may be defined for subsets of the codomain, e.g.
The original notion and its generalization are related by the identityf−1(y)=f−1({y}),{\displaystyle f^{-1}(y)=f^{-1}(\{y\}),}The preimage of a single elementy∈Y– asingleton set{y}– is sometimes called thefiberofy. WhenYis the set of real numbers, it is common to refer tof−1({y})as alevel set. | https://en.wikipedia.org/wiki/Inverse_function |
Inmathematics, areflection(also spelledreflexion)[1]is amappingfrom aEuclidean spaceto itself that is anisometrywith ahyperplaneas the set offixed points; this set is called theaxis(in dimension 2) orplane(in dimension 3) of reflection. The image of a figure by a reflection is itsmirror imagein the axis or plane of reflection. For example the mirror image of the small Latin letterpfor a reflection with respect to avertical axis(avertical reflection) would look likeq. Its image by reflection in ahorizontal axis(ahorizontal reflection) would look likeb. A reflection is aninvolution: when applied twice in succession, every point returns to its original location, and every geometrical object is restored to its original state.
The termreflectionis sometimes used for a larger class of mappings from a Euclidean space to itself, namely the non-identity isometries that are involutions. The set of fixed points (the "mirror") of such an isometry is anaffine subspace, but is possibly smaller than a hyperplane. For instance areflection through a pointis an involutive isometry with just one fixed point; the image of the letterpunder it
would look like ad. This operation is also known as acentral inversion(Coxeter 1969, §7.2), and exhibits Euclidean space as asymmetric space. In aEuclidean vector space, the reflection in the point situated at the origin is the same as vector negation. Other examples include reflections in a line in three-dimensional space. Typically, however, unqualified use of the term "reflection" means reflection in ahyperplane.
Some mathematicians use "flip" as a synonym for "reflection".[2][3][4]
In a plane (or, respectively, 3-dimensional) geometry, to find the reflection of a point drop aperpendicularfrom the point to the line (plane) used for reflection, and extend it the same distance on the other side. To find the reflection of a figure, reflect each point in the figure.
To reflect pointPthrough the lineABusingcompass and straightedge, proceed as follows (see figure):
PointQis then the reflection of pointPthrough lineAB.
Thematrixfor a reflection isorthogonalwithdeterminant−1 andeigenvalues−1, 1, 1, ..., 1. The product of two such matrices is a special orthogonal matrix that represents a rotation. Everyrotationis the result of reflecting in an even number of reflections in hyperplanes through the origin, and everyimproper rotationis the result of reflecting in an odd number. Thus reflections generate theorthogonal group, and this result is known as theCartan–Dieudonné theorem.
Similarly theEuclidean group, which consists of all isometries of Euclidean space, is generated by reflections in affine hyperplanes. In general, agroupgenerated by reflections in affine hyperplanes is known as areflection group. Thefinite groupsgenerated in this way are examples ofCoxeter groups.
Reflection across an arbitrary line through the origin intwo dimensionscan be described by the following formula
wherev{\displaystyle v}denotes the vector being reflected,l{\displaystyle l}denotes any vector in the line across which the reflection is performed, andv⋅l{\displaystyle v\cdot l}denotes thedot productofv{\displaystyle v}withl{\displaystyle l}. Note the formula above can also be written as
saying that a reflection ofv{\displaystyle v}acrossl{\displaystyle l}is equal to 2 times theprojectionofv{\displaystyle v}onl{\displaystyle l}, minus the vectorv{\displaystyle v}. Reflections in a line have the eigenvalues of 1, and −1.
Given a vectorv{\displaystyle v}inEuclidean spaceRn{\displaystyle \mathbb {R} ^{n}}, the formula for the reflection in thehyperplanethrough the origin,orthogonaltoa{\displaystyle a}, is given by
wherev⋅a{\displaystyle v\cdot a}denotes thedot productofv{\displaystyle v}witha{\displaystyle a}. Note that the second term in the above equation is just twice thevector projectionofv{\displaystyle v}ontoa{\displaystyle a}. One can easily check that
Using thegeometric product, the formula is
Since these reflections are isometries of Euclidean space fixing the origin they may be represented byorthogonal matrices. The orthogonal matrix corresponding to the above reflection is thematrix
whereI{\displaystyle I}denotes then×n{\displaystyle n\times n}identity matrixandaT{\displaystyle a^{T}}is thetransposeof a. Its entries are
whereδijis theKronecker delta.
The formula for the reflection in the affine hyperplanev⋅a=c{\displaystyle v\cdot a=c}not through the origin is | https://en.wikipedia.org/wiki/Reflection_(mathematics) |
Inmathematics,reflection symmetry,line symmetry,mirror symmetry, ormirror-image symmetryissymmetrywith respect to areflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry.
In2-dimensional space, there is a line/axis of symmetry, in3-dimensional space, there is aplaneof symmetry. An object or figure which is indistinguishable from its transformed image is calledmirror symmetric.
In formal terms, amathematical objectis symmetric with respect to a givenoperationsuch as reflection,rotation, ortranslation, if, when applied to the object, this operation preserves some property of the object.[1]The set of operations that preserve a given property of the object form agroup. Two objects are symmetric to each other with respect to a given group of operations if one is obtained from the other by some of the operations (and vice versa).
The symmetric function of a two-dimensional figure is a line such that, for eachperpendicularconstructed, if the perpendicular intersects the figure at a distance 'd' from the axis along the perpendicular, then there exists another intersection of the shape and the perpendicular at the same distance 'd' from the axis, in the opposite direction along the perpendicular.
Another way to think about the symmetric function is that if the shape were to be folded in half over the axis, the two halves would be identical: the two halves are each other'smirror images.[1]Thus, a square has four axes of symmetry because there are four different ways to fold it and have the edges all match. A circle has infinitely many axes of symmetry, while aconeandspherehave infinitely many planes of symmetry.
Triangleswith reflection symmetry areisosceles.Quadrilateralswith reflection symmetry arekites, (concave) deltoids,rhombi,[2]andisosceles trapezoids. All even-sided polygons have two simple reflective forms, one with lines of reflections through vertices, and one through edges. For an arbitrary shape, theaxialityof the shape measures how close it is to being bilaterally symmetric. It equals 1 for shapes with reflection symmetry, and between two-thirds and 1 for anyconvex shape.
In 3D, the cube in which the plane can configure in all of the three axes that can reflect the cube has 9 planes of reflective symmetry.[3]
For more general types ofreflectionthere are correspondingly more general types of reflection symmetry. For example:
Animals that are bilaterally symmetrichave reflection symmetry around thesagittal plane, which divides the body vertically into left and right halves, with one of each sense organ and limb pair on either side. Most animals are bilaterally symmetric, likely because this supportsforward movementandstreamlining.[4][5][6]
Mirror symmetry is often used inarchitecture, as in the facade ofSanta Maria Novella,Florence.[7]It is also found in the design of ancient structures such asStonehenge.[8]Symmetry was a core element in some styles of architecture, such asPalladianism.[9] | https://en.wikipedia.org/wiki/Reflection_symmetry |
Inmathematics, ablock matrix pseudoinverseis a formula for thepseudoinverseof apartitioned matrix. This is useful for decomposing or approximating many algorithms updating parameters insignal processing, which are based on theleast squaresmethod.
Consider a column-wise partitioned matrix:
If the above matrix is full column rank, theMoore–Penrose inversematrices of it and its transpose are
This computation of the pseudoinverse requires (n+p)-square matrix inversion and does not take advantage of the block form.
To reduce computational costs ton- andp-square matrix inversions and to introduce parallelism, treating the blocks separately, one derives[1]
whereorthogonal projectionmatrices are defined by
The above formulas are not necessarily valid if[AB]{\displaystyle {\begin{bmatrix}\mathbf {A} &\mathbf {B} \end{bmatrix}}}does not have full rank – for example, ifA≠0{\displaystyle \mathbf {A} \neq 0}, then
Given the same matrices as above, we consider the following least squares problems, which
appear as multiple objective optimizations or constrained problems in signal processing.
Eventually, we can implement a parallel algorithm for least squares based on the following results.
Suppose a solutionx=[x1x2]{\displaystyle \mathbf {x} ={\begin{bmatrix}\mathbf {x} _{1}\\\mathbf {x} _{2}\\\end{bmatrix}}}solves an over-determined system:
Using the block matrix pseudoinverse, we have
Therefore, we have a decomposed solution:
Suppose a solutionx{\displaystyle \mathbf {x} }solves an under-determined system:
The minimum-norm solution is given by
Using the block matrix pseudoinverse, we have
Instead of([AB]T[AB])−1{\displaystyle \mathbf {\left({\begin{bmatrix}\mathbf {A} &\mathbf {B} \end{bmatrix}}^{\textsf {T}}{\begin{bmatrix}\mathbf {A} &\mathbf {B} \end{bmatrix}}\right)} ^{-1}}, we need to calculate directly or indirectly[citation needed][original research?]
In a dense and small system, we can usesingular value decomposition,QR decomposition, orCholesky decompositionto replace the matrix inversions with numerical routines. In a large system, we may employiterative methodssuch as Krylov subspace methods.
Consideringparallel algorithms, we can compute(ATA)−1{\displaystyle \left(\mathbf {A} ^{\textsf {T}}\mathbf {A} \right)^{-1}}and(BTB)−1{\displaystyle \left(\mathbf {B} ^{\textsf {T}}\mathbf {B} \right)^{-1}}in parallel. Then, we finish to compute(ATPB⊥A)−1{\displaystyle \left(\mathbf {A} ^{\textsf {T}}\mathbf {P} _{B}^{\perp }\mathbf {A} \right)^{-1}}and(BTPA⊥B)−1{\displaystyle \left(\mathbf {B} ^{\textsf {T}}\mathbf {P} _{A}^{\perp }\mathbf {B} \right)^{-1}}also in parallel. | https://en.wikipedia.org/wiki/Block_matrix_pseudoinverse |
Inmathematics, aregular semigroupis asemigroupSin which every element isregular, i.e., for each elementainSthere exists an elementxinSsuch thataxa=a.[1]Regular semigroups are one of the most-studied classes of semigroups, and their structure is particularly amenable to study viaGreen's relations.[2]
Regular semigroups were introduced byJ. A. Greenin his influential 1951 paper "On the structure of semigroups"; this was also the paper in whichGreen's relationswere introduced. The concept ofregularityin a semigroup was adapted from an analogous condition forrings, already considered byJohn von Neumann.[3]It was Green's study of regular semigroups which led him to define his celebratedrelations. According to a footnote in Green 1951, the suggestion that the notion of regularity be applied tosemigroupswas first made byDavid Rees.
The terminversive semigroup(French: demi-groupe inversif) was historically used as synonym in the papers ofGabriel Thierrin(a student ofPaul Dubreil) in the 1950s,[4][5]and it is still used occasionally.[6]
There are two equivalent ways in which to define a regular semigroupS:
To see the equivalence of these definitions, first suppose thatSis defined by (2). Thenbserves as the requiredxin (1). Conversely, ifSis defined by (1), thenxaxis an inverse fora, sincea(xax)a=axa(xa) =axa=aand (xax)a(xax) =x(axa)(xax) =xa(xax) =x(axa)x=xax.[8]
The set of inverses (in the above sense) of an elementain an arbitrarysemigroupSis denoted byV(a).[9]Thus, another way of expressing definition (2) above is to say that in a regular semigroup,V(a) is nonempty, for everyainS. The product of any elementawith anybinV(a) is alwaysidempotent:abab=ab, sinceaba=a.[10]
A regular semigroup in which idempotents commute (with idempotents) is aninverse semigroup, or equivalently, every element has auniqueinverse. To see this, letSbe a regular semigroup in which idempotents commute. Then every element ofShas at least one inverse. Suppose thatainShas two inversesbandc, i.e.,
Then
So, by commuting the pairs of idempotentsab&acandba&ca, the inverse ofais shown to be unique. Conversely, it can be shown that anyinverse semigroupis a regular semigroup in which idempotents commute.[12]
The existence of a unique pseudoinverse implies the existence of a unique inverse, but the opposite is not true. For example, in thesymmetric inverse semigroup, the empty transformation Ø does not have a unique pseudoinverse, because Ø = ØfØ for any transformationf. The inverse of Ø is unique however, because only onefsatisfies the additional constraint thatf=fØf, namelyf= Ø. This remark holds more generally in any semigroup with zero. Furthermore, if every element has a unique pseudoinverse, then the semigroup is agroup, and the unique pseudoinverse of an element coincides with the group inverse.
Recall that theprincipal idealsof a semigroupSare defined in terms ofS1, thesemigroup with identity adjoined; this is to ensure that an elementabelongs to the principal right, left and two-sidedidealswhich it generates. In a regular semigroupS, however, an elementa=axaautomatically belongs to these ideals, without recourse to adjoining an identity.Green's relationscan therefore be redefined for regular semigroups as follows:
In a regular semigroupS, everyL{\displaystyle {\mathcal {L}}}- andR{\displaystyle {\mathcal {R}}}-class contains at least oneidempotent. Ifais any element ofSanda′is any inverse fora, thenaisL{\displaystyle {\mathcal {L}}}-related toa′aandR{\displaystyle {\mathcal {R}}}-related toaa′.[14]
Theorem.LetSbe a regular semigroup; letaandbbe elements ofS, and letV(x)denote the set of inverses ofxinS. Then
IfSis aninverse semigroup, then the idempotent in eachL{\displaystyle {\mathcal {L}}}- andR{\displaystyle {\mathcal {R}}}-class is unique.[12]
Some special classes of regular semigroups are:[16]
Theclassof generalised inverse semigroups is theintersectionof the class of locally inverse semigroups and the class of orthodox semigroups.[17]
All inverse semigroups are orthodox and locally inverse. The converse statements do not hold. | https://en.wikipedia.org/wiki/Regular_semigroup |
Inaccounting,financeandeconomics, anaccounting identityis an equality that must be true regardless of the value of its variables, or a statement that by definition (or construction) must be true.[1][2]Where an accounting identity applies, any deviation from numerical equality signifies an error in formulation, calculation or measurement.[3]
The termaccounting identitymay be used to distinguish between propositions that are theories (which may or may not be true, or relationships that may or may not always hold) and statements that are by definition true. Despite the fact that the statements are by definition true, the underlying figures as measured or estimated may notadd updue to measurement error, particularly for certain identities in macroeconomics.[4]
The most basic identity in accounting is that thebalance sheetmust balance, that is, thatassetsmust equal the sum ofliabilities(debts) andequity(the value of the firm to the owner). In its most common formulation it is known as theaccounting equation:
where debt includes non-financial liabilities. Balance sheets are commonly presented as two parallel columns, each summing to the same total, with the assets on the left, and liabilities and owners' equity on the right. The parallel columns of Assets and Equities are, in effect, two views of the same set of business facts.
The balance of the balance sheet reflects the conventions ofdouble-entry bookkeeping, by which business transactions are recorded. In double-entry bookkeeping, every transaction is recorded by paired entries, and typically a transaction will result in two or more pairs of entries. The sale of product, for example, would record both a receipt of cash (or the creation of a trade receivable in the case of an extension of credit to the buyer) and a reduction in the inventory of goods for sale; the receipt of cash or a trade receivable is an addition to revenue, and the reduction in goods inventory is an addition to expense. In this case, an "expense" is the "expending" of an asset. Thus, there are two pairs of entries: an addition to revenue balanced by an addition to cash; a subtraction from inventory balanced by an addition to expense. The cash and inventory accounts are asset accounts; the revenue and expense accounts will close at the end of the accounting period to affect equity.
Double-entry bookkeeping conventions are employed as well for theNational Accounts. Economic concepts such as national product, aggregate income, investment and savings, as well as the balance of payments and balance of trade, involve accounting identities. The application of double-entry bookkeeping conventions in measuring aggregate economic activity derives from the recognition that: every purchase is also a sale, every payment made translates income received, and every act of lending also an act of borrowing.
Here the termidentityis amathematical identityor alogical tautology, since it defines an equivalence which does not depend on the particular values of the variables.
Accounting has a number of identities in common usage, and since many identities can be decomposed into others, no comprehensive listing is possible.
Accounting identities also apply between accounting periods, such as changes in cash balances. For example:
Any asset recorded in a firm's balance sheet will have acarrying value. By definition, the carrying value must equal the historic cost (or acquisition cost) of the asset, plus (or minus) any subsequent adjustments in the value of the asset, such asdepreciation.
In economics, there are numerous accounting identities.
One of the most commonly known is thebalance of paymentsidentity,[6]where:
A common problem with the balance of payments identity is that, due to measurement error, the balance of payments may not total correctly. For example, in the context of the identity that the sum of all countries' current accounts must be zero,The Economistmagazine has noted that "In theory, individual countries’ current-account deficits and surpluses should cancel each other out. But because of statistical errors and omissions they never do."[7]
The basic equation forgross domestic productis also an identity, and is sometimes referred to as theNational Income Identity:[8]
This identity holds because investment refers to the sum of intended and unintended investment, the latter being unintended accumulations of inventories; unintended inventory accumulation necessarily equals output produced (GDP) minus intended uses of that output—consumption, intended investment in machinery, inventories, etc., government spending, and net exports.
A key identity that is used in explaining the multiple expansion of themoney supplyis:
Here the liabilities include deposits of customers, against which reserves often must be held. | https://en.wikipedia.org/wiki/Accounting_identity |
This article listsmathematical identities, that is,identically true relationsholding inmathematics. | https://en.wikipedia.org/wiki/List_of_mathematical_identities |
Inmathematics, alawis aformulathat is always true within a given context.[1]Laws describe arelationship, between two or moreexpressionsorterms(which may containvariables), usually usingequalityorinequality,[2]or between formulas themselves, for instance, inmathematical logic. For example, the formulaa2≥0{\displaystyle a^{2}\geq 0}is true for allreal numbersa, and is therefore a law. Laws over an equality are calledidentities.[3]For example,(a+b)2=a2+2ab+b2{\displaystyle (a+b)^{2}=a^{2}+2ab+b^{2}}andcos2θ+sin2θ=1{\displaystyle \cos ^{2}\theta +\sin ^{2}\theta =1}are identities.[4]Mathematical laws are distinguished fromscientific lawswhich are based onobservations, and try to describe orpredicta range ofnatural phenomena.[5]The more significant laws are often calledtheorems.
Triangle inequality: Ifa,b, andcare the lengths of the sides of atrianglethen the triangle inequality states that
with equality only in the degenerate case of a triangle with zeroarea. InEuclidean geometryand some other geometries, the triangle inequality is a theorem about vectors and vector lengths (norms):
where the length of the third side has been replaced by the length of the vector sumu+v. Whenuandvare real numbers, they can be viewed as vectors inR1{\displaystyle \mathbb {R} ^{1}}, and the triangle inequality expresses a relationship betweenabsolute values.
Pythagorean theorem: It states that the area of thesquarewhose side is thehypotenuse(the side opposite theright angle) is equal to the sum of the areas of the squares on the other two sides. Thetheoremcan be written as anequationrelating the lengths of the sidesa,band the hypotenusec, sometimes called the Pythagorean equation:[6]
Geometrically,trigonometric identitiesare identities involving certain functions of one or moreangles.[7]They are distinct fromtriangle identities, which are identities involving both angles and side lengths of a triangle. Only the former are covered in this article.
These identities are useful whenever expressions involving trigonometric functions need to be simplified. Another important application is theintegrationof non-trigonometric functions: a common technique which involves first using thesubstitution rule with a trigonometric function, and then simplifying the resulting integral with a trigonometric identity.
One of the most prominent examples of trigonometric identities involves the equationsin2θ+cos2θ=1,{\displaystyle \sin ^{2}\theta +\cos ^{2}\theta =1,}which is true for allrealvalues ofθ{\displaystyle \theta }. On the other hand, the equation
is only true for certain values ofθ{\displaystyle \theta }, not all. For example, this equation is true whenθ=0,{\displaystyle \theta =0,}but false whenθ=2{\displaystyle \theta =2}.
Another group of trigonometric identities concerns the so-called addition/subtraction formulas (e.g. the double-angle identitysin(2θ)=2sinθcosθ{\displaystyle \sin(2\theta )=2\sin \theta \cos \theta }, the addition formula fortan(x+y){\displaystyle \tan(x+y)}), which can be used to break down expressions of larger angles into those with smaller constituents.
Cauchy–Schwarz inequality: An upper bound on theinner productbetween twovectorsin an inner product space in terms of the product of the vectornorms. It is considered one of the most important and widely usedinequalitiesin mathematics.[8]
The Cauchy–Schwarz inequality states that for all vectorsu{\displaystyle \mathbf {u} }andv{\displaystyle \mathbf {v} }of aninner product space
where⟨⋅,⋅⟩{\displaystyle \langle \cdot ,\cdot \rangle }is theinner product. Examples of inner products include the real and complexdot product; see theexamples in inner product. Every inner product gives rise to a Euclideanl2{\displaystyle l_{2}}norm, called thecanonicalorinducednorm, where the norm of a vectoru{\displaystyle \mathbf {u} }is denoted and defined by
where⟨u,u⟩{\displaystyle \langle \mathbf {u} ,\mathbf {u} \rangle }is always a non-negative real number (even if the inner product is complex-valued). By taking the square root of both sides of the above inequality, the Cauchy–Schwarz inequality can be written in its more familiar form in terms of the norm:[9][10]
Moreover, the two sides are equal if and only ifu{\displaystyle \mathbf {u} }andv{\displaystyle \mathbf {v} }arelinearly dependent.[11][12][13]
Pigeonhole principle: Ifnitems are put intomcontainers, withn>m, then at least one container must contain more than one item.[14]For example, of three gloves (none of which is ambidextrous/reversible), at least two must be right-handed or at least two must be left-handed, because there are three objects but only two categories of handedness to put them into.
De Morgan's laws: Inpropositional logicandBoolean algebra,De Morgan's laws,[15][16][17]also known asDe Morgan's theorem,[18]are a pair of transformation rules that are bothvalidrules of inference. They are named afterAugustus De Morgan, a 19th-century British mathematician. The rules allow the expression ofconjunctionsanddisjunctionspurely in terms of each other vianegation. The rules can be expressed in English as:
The threeLaws of thoughtare:
Benford's lawis an observation that in many real-life sets of numericaldata, theleading digitis likely to be small.[21]In sets that obey the law, the number 1 appears as the leading significant digit about 30% of the time, while 9 appears as the leading significant digit less than 5% of the time. Uniformly distributed digits would each occur about 11.1% of the time.[22]
Strong law of small numbers, in a humorous way, states any given small number appears in far more contexts than may seem reasonable, leading to many apparently surprising coincidences in mathematics, simply because small numbers appear so often and yet are so few. | https://en.wikipedia.org/wiki/Law_(mathematics) |
Inlinear algebra, theidentity matrixof sizen{\displaystyle n}is then×n{\displaystyle n\times n}square matrixwithoneson themain diagonalandzeroselsewhere. It has unique properties, for example when the identity matrix represents ageometric transformation, the object remains unchanged by the transformation. In other contexts, it is analogous to multiplying by the number 1.
The identity matrix is often denoted byIn{\displaystyle I_{n}}, or simply byI{\displaystyle I}if the size is immaterial or can be trivially determined by the context.[1]
I1=[1],I2=[1001],I3=[100010001],…,In=[100⋯0010⋯0001⋯0⋮⋮⋮⋱⋮000⋯1].{\displaystyle I_{1}={\begin{bmatrix}1\end{bmatrix}},\ I_{2}={\begin{bmatrix}1&0\\0&1\end{bmatrix}},\ I_{3}={\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}},\ \dots ,\ I_{n}={\begin{bmatrix}1&0&0&\cdots &0\\0&1&0&\cdots &0\\0&0&1&\cdots &0\\\vdots &\vdots &\vdots &\ddots &\vdots \\0&0&0&\cdots &1\end{bmatrix}}.}
The termunit matrixhas also been widely used,[2][3][4][5]but the termidentity matrixis now standard.[6]The termunit matrixis ambiguous, because it is also used for amatrix of onesand for anyunitof thering of alln×n{\displaystyle n\times n}matrices.[7]
In some fields, such asgroup theoryorquantum mechanics, the identity matrix is sometimes denoted by a boldface one,1{\displaystyle \mathbf {1} }, or called "id" (short for identity). Less frequently, some mathematics books useU{\displaystyle U}orE{\displaystyle E}to represent the identity matrix, standing for "unit matrix"[2]and the German wordEinheitsmatrixrespectively.[8]
In terms of a notation that is sometimes used to concisely describediagonal matrices, the identity matrix can be written asIn=diag(1,1,…,1).{\displaystyle I_{n}=\operatorname {diag} (1,1,\dots ,1).}The identity matrix can also be written using theKronecker deltanotation:[8](In)ij=δij.{\displaystyle (I_{n})_{ij}=\delta _{ij}.}
WhenA{\displaystyle A}is anm×n{\displaystyle m\times n}matrix, it is a property ofmatrix multiplicationthatImA=AIn=A.{\displaystyle I_{m}A=AI_{n}=A.}In particular, the identity matrix serves as themultiplicative identityof thematrix ringof alln×n{\displaystyle n\times n}matrices, and as theidentity elementof thegeneral linear groupGL(n){\displaystyle GL(n)}, which consists of allinvertiblen×n{\displaystyle n\times n}matrices under the matrix multiplication operation. In particular, the identity matrix is invertible. It is aninvolutory matrix, equal to its own inverse. In this group, two square matrices have the identity matrix as their product exactly when they are the inverses of each other.
Whenn×n{\displaystyle n\times n}matrices are used to representlinear transformationsfrom ann{\displaystyle n}-dimensional vector space to itself, the identity matrixIn{\displaystyle I_{n}}represents theidentity function, for whateverbasiswas used in this representation.
Thei{\displaystyle i}th column of an identity matrix is theunit vectorei{\displaystyle e_{i}}, a vector whosei{\displaystyle i}th entry is 1 and 0 elsewhere. Thedeterminantof the identity matrix is 1, and itstraceisn{\displaystyle n}.
The identity matrix is the onlyidempotent matrixwith non-zero determinant. That is, it is the only matrix such that:
Theprincipal square rootof an identity matrix is itself, and this is its onlypositive-definitesquare root. However, every identity matrix with at least two rows and columns has an infinitude of symmetric square roots.[9]
Therankof an identity matrixIn{\displaystyle I_{n}}equals the sizen{\displaystyle n}, i.e.:rank(In)=n.{\displaystyle \operatorname {rank} (I_{n})=n.} | https://en.wikipedia.org/wiki/Identity_matrix |
Inmathematics, ifA{\displaystyle A}is asubsetofB,{\displaystyle B,}then theinclusion mapis thefunctionι{\displaystyle \iota }that sends each elementx{\displaystyle x}ofA{\displaystyle A}tox,{\displaystyle x,}treated as an element ofB:{\displaystyle B:}ι:A→B,ι(x)=x.{\displaystyle \iota :A\rightarrow B,\qquad \iota (x)=x.}
An inclusion map may also be referred to as aninclusion function, aninsertion,[1]or acanonical injection.
A "hooked arrow" (U+21AA↪RIGHTWARDS ARROW WITH HOOK)[2]is sometimes used in place of the function arrow above to denote an inclusion map; thus:ι:A↪B.{\displaystyle \iota :A\hookrightarrow B.}
(However, some authors use this hooked arrow for anyembedding.)
This and other analogousinjectivefunctions[3]fromsubstructuresare sometimes callednatural injections.
Given anymorphismf{\displaystyle f}betweenobjectsX{\displaystyle X}andY{\displaystyle Y}, if there is an inclusion mapι:A→X{\displaystyle \iota :A\to X}into thedomainX{\displaystyle X}, then one can form therestrictionf∘ι{\displaystyle f\circ \iota }off.{\displaystyle f.}In many instances, one can also construct a canonical inclusion into thecodomainR→Y{\displaystyle R\to Y}known as therangeoff.{\displaystyle f.}
Inclusion maps tend to behomomorphismsofalgebraic structures; thus, such inclusion maps areembeddings. More precisely, given a substructure closed under some operations, the inclusion map will be an embedding for tautological reasons. For example, for some binary operation⋆,{\displaystyle \star ,}to require thatι(x⋆y)=ι(x)⋆ι(y){\displaystyle \iota (x\star y)=\iota (x)\star \iota (y)}is simply to say that⋆{\displaystyle \star }is consistently computed in the sub-structure and the large structure. The case of aunary operationis similar; but one should also look atnullaryoperations, which pick out aconstantelement. Here the point is thatclosuremeans such constants must already be given in the substructure.
Inclusion maps are seen inalgebraic topologywhere ifA{\displaystyle A}is astrong deformation retractofX,{\displaystyle X,}the inclusion map yields anisomorphismbetween allhomotopy groups(that is, it is ahomotopy equivalence).
Inclusion maps ingeometrycome in different kinds: for exampleembeddingsofsubmanifolds.Contravariantobjects (which is to say, objects that havepullbacks; these are calledcovariantin an older and unrelated terminology) such asdifferential formsrestrictto submanifolds, giving a mapping in theother direction. Another example, more sophisticated, is that ofaffine schemes, for which the inclusionsSpec(R/I)→Spec(R){\displaystyle \operatorname {Spec} \left(R/I\right)\to \operatorname {Spec} (R)}andSpec(R/I2)→Spec(R){\displaystyle \operatorname {Spec} \left(R/I^{2}\right)\to \operatorname {Spec} (R)}may be differentmorphisms, whereR{\displaystyle R}is acommutative ringandI{\displaystyle I}is anidealofR.{\displaystyle R.} | https://en.wikipedia.org/wiki/Inclusion_map |
Inmathematics, anindicator functionor acharacteristic functionof asubsetof asetis afunctionthat maps elements of the subset to one, and all other elements to zero. That is, ifAis a subset of some setX, then the indicator function ofAis the function1A{\displaystyle \mathbf {1} _{A}}defined by1A(x)=1{\displaystyle \mathbf {1} _{A}\!(x)=1}ifx∈A,{\displaystyle x\in A,}and1A(x)=0{\displaystyle \mathbf {1} _{A}\!(x)=0}otherwise. Other common notations are𝟙AandχA.{\displaystyle \chi _{A}.}[a]
The indicator function ofAis theIverson bracketof the property of belonging toA; that is,
1A(x)=[x∈A].{\displaystyle \mathbf {1} _{A}(x)=\left[\ x\in A\ \right].}
For example, theDirichlet functionis the indicator function of therational numbersas a subset of thereal numbers.
Given an arbitrary setX, the indicator function of a subsetAofXis the function1A:X↦{0,1}{\displaystyle \mathbf {1} _{A}\colon X\mapsto \{0,1\}}defined by1A(x)={1ifx∈A0ifx∉A.{\displaystyle \operatorname {\mathbf {1} } _{A}\!(x)={\begin{cases}1&{\text{if }}x\in A\\0&{\text{if }}x\notin A\,.\end{cases}}}
TheIverson bracketprovides the equivalent notation[x∈A]{\displaystyle \left[\ x\in A\ \right]}or⟦x∈A⟧,that can be used instead of1A(x).{\displaystyle \mathbf {1} _{A}\!(x).}
The function1A{\displaystyle \mathbf {1} _{A}}is sometimes denoted𝟙A,IA,χA[a]or even justA.[b]
The notationχA{\displaystyle \chi _{A}}is also used to denote thecharacteristic functioninconvex analysis, which is defined as if using thereciprocalof the standard definition of the indicator function.
A related concept instatisticsis that of adummy variable. (This must not be confused with "dummy variables" as that term is usually used in mathematics, also called abound variable.)
The term "characteristic function" has an unrelated meaning inclassic probability theory. For this reason,traditional probabilistsuse the termindicator functionfor the function defined here almost exclusively, while mathematicians in other fields are more likely to use the termcharacteristic functionto describe the function that indicates membership in a set.
Infuzzy logicandmodern many-valued logic, predicates are thecharacteristic functionsof aprobability distribution. That is, the strict true/false valuation of the predicate is replaced by a quantity interpreted as the degree of truth.
Theindicatororcharacteristicfunctionof a subsetAof some setXmapselements ofXto thecodomain{0,1}.{\displaystyle \{0,\,1\}.}
This mapping issurjectiveonly whenAis a non-emptyproper subsetofX. IfA=X,{\displaystyle A=X,}then1A≡1.{\displaystyle \mathbf {1} _{A}\equiv 1.}By a similar argument, ifA=∅{\displaystyle A=\emptyset }then1A≡0.{\displaystyle \mathbf {1} _{A}\equiv 0.}
IfA{\displaystyle A}andB{\displaystyle B}are two subsets ofX,{\displaystyle X,}then1A∩B(x)=min{1A(x),1B(x)}=1A(x)⋅1B(x),1A∪B(x)=max{1A(x),1B(x)}=1A(x)+1B(x)−1A(x)⋅1B(x),{\displaystyle {\begin{aligned}\mathbf {1} _{A\cap B}(x)~&=~\min {\bigl \{}\mathbf {1} _{A}(x),\ \mathbf {1} _{B}(x){\bigr \}}~~=~\mathbf {1} _{A}(x)\cdot \mathbf {1} _{B}(x),\\\mathbf {1} _{A\cup B}(x)~&=~\max {\bigl \{}\mathbf {1} _{A}(x),\ \mathbf {1} _{B}(x){\bigr \}}~=~\mathbf {1} _{A}(x)+\mathbf {1} _{B}(x)-\mathbf {1} _{A}(x)\cdot \mathbf {1} _{B}(x)\,,\end{aligned}}}
and the indicator function of thecomplementofA{\displaystyle A}i.e.A∁{\displaystyle A^{\complement }}is:1A∁=1−1A.{\displaystyle \mathbf {1} _{A^{\complement }}=1-\mathbf {1} _{A}.}
More generally, supposeA1,…,An{\displaystyle A_{1},\dotsc ,A_{n}}is a collection of subsets ofX. For anyx∈X:{\displaystyle x\in X:}
∏k∈I(1−1Ak(x)){\displaystyle \prod _{k\in I}\left(\ 1-\mathbf {1} _{A_{k}}\!\left(x\right)\ \right)}
is a product of0s and1s. This product has the value1at precisely thosex∈X{\displaystyle x\in X}that belong to none of the setsAk{\displaystyle A_{k}}and is 0 otherwise. That is
∏k∈I(1−1Ak)=1X−⋃kAk=1−1⋃kAk.{\displaystyle \prod _{k\in I}(1-\mathbf {1} _{A_{k}})=\mathbf {1} _{X-\bigcup _{k}A_{k}}=1-\mathbf {1} _{\bigcup _{k}A_{k}}.}
Expanding the product on the left hand side,
1⋃kAk=1−∑F⊆{1,2,…,n}(−1)|F|1⋂FAk=∑∅≠F⊆{1,2,…,n}(−1)|F|+11⋂FAk{\displaystyle \mathbf {1} _{\bigcup _{k}A_{k}}=1-\sum _{F\subseteq \{1,2,\dotsc ,n\}}(-1)^{|F|}\mathbf {1} _{\bigcap _{F}A_{k}}=\sum _{\emptyset \neq F\subseteq \{1,2,\dotsc ,n\}}(-1)^{|F|+1}\mathbf {1} _{\bigcap _{F}A_{k}}}
where|F|{\displaystyle |F|}is thecardinalityofF. This is one form of the principle ofinclusion-exclusion.
As suggested by the previous example, the indicator function is a useful notational device incombinatorics. The notation is used in other places as well, for instance inprobability theory: ifXis aprobability spacewith probability measureP{\displaystyle \mathbb {P} }andAis ameasurable set, then1A{\displaystyle \mathbf {1} _{A}}becomes arandom variablewhoseexpected valueis equal to the probability ofA:
EX{1A(x)}=∫X1A(x)dP(x)=∫AdP(x)=P(A).{\displaystyle \operatorname {\mathbb {E} } _{X}\left\{\ \mathbf {1} _{A}(x)\ \right\}\ =\ \int _{X}\mathbf {1} _{A}(x)\ \operatorname {d\ \mathbb {P} } (x)=\int _{A}\operatorname {d\ \mathbb {P} } (x)=\operatorname {\mathbb {P} } (A).}
This identity is used in a simple proof ofMarkov's inequality.
In many cases, such asorder theory, the inverse of the indicator function may be defined. This is commonly called thegeneralized Möbius function, as a generalization of the inverse of the indicator function in elementarynumber theory, theMöbius function. (See paragraph below about the use of the inverse in classical recursion theory.)
Given aprobability space(Ω,F,P){\displaystyle \textstyle (\Omega ,{\mathcal {F}},\operatorname {P} )}withA∈F,{\displaystyle A\in {\mathcal {F}},}the indicator random variable1A:Ω→R{\displaystyle \mathbf {1} _{A}\colon \Omega \rightarrow \mathbb {R} }is defined by1A(ω)=1{\displaystyle \mathbf {1} _{A}(\omega )=1}ifω∈A,{\displaystyle \omega \in A,}otherwise1A(ω)=0.{\displaystyle \mathbf {1} _{A}(\omega )=0.}
Kurt Gödeldescribed therepresenting functionin his 1934 paper "On undecidable propositions of formal mathematical systems" (the symbol "¬" indicates logical inversion, i.e. "NOT"):[1]: 42
There shall correspond to each class or relationRa representing functionϕ(x1,…xn)=0{\displaystyle \phi (x_{1},\ldots x_{n})=0}ifR(x1,…xn){\displaystyle R(x_{1},\ldots x_{n})}andϕ(x1,…xn)=1{\displaystyle \phi (x_{1},\ldots x_{n})=1}if¬R(x1,…xn).{\displaystyle \neg R(x_{1},\ldots x_{n}).}
Kleeneoffers up the same definition in the context of theprimitive recursive functionsas a functionφof a predicatePtakes on values0if the predicate is true and1if the predicate is false.[2]
For example, because the product of characteristic functionsϕ1∗ϕ2∗⋯∗ϕn=0{\displaystyle \phi _{1}*\phi _{2}*\cdots *\phi _{n}=0}whenever any one of the functions equals0, it plays the role of logical OR: IFϕ1=0{\displaystyle \phi _{1}=0\ }ORϕ2=0{\displaystyle \ \phi _{2}=0}OR ... ORϕn=0{\displaystyle \phi _{n}=0}THEN their product is0. What appears to the modern reader as the representing function's logical inversion, i.e. the representing function is0when the functionRis "true" or satisfied", plays a useful role in Kleene's definition of the logical functions OR, AND, and IMPLY,[2]: 228the bounded-[2]: 228and unbounded-[2]: 279 ffmu operatorsand the CASE function.[2]: 229
In classical mathematics, characteristic functions of sets only take values1(members) or0(non-members). Infuzzy set theory, characteristic functions are generalized to take value in the real unit interval[0, 1], or more generally, in somealgebraorstructure(usually required to be at least aposetorlattice). Such generalized characteristic functions are more usually calledmembership functions, and the corresponding "sets" are calledfuzzysets. Fuzzy sets model the gradual change in the membershipdegreeseen in many real-worldpredicateslike "tall", "warm", etc.
In general, the indicator function of a set is not smooth; it is continuous if and only if itssupportis aconnected component. In thealgebraic geometryoffinite fields, however, everyaffine varietyadmits a (Zariski) continuous indicator function.[3]Given afinite setof functionsfα∈Fq[x1,…,xn]{\displaystyle f_{\alpha }\in \mathbb {F} _{q}\left[\ x_{1},\ldots ,x_{n}\right]}letV={x∈Fqn:fα(x)=0}{\displaystyle V={\bigl \{}\ x\in \mathbb {F} _{q}^{n}:f_{\alpha }(x)=0\ {\bigr \}}}be their vanishing locus. Then, the functionP(x)=∏(1−fα(x)q−1){\textstyle \mathbb {P} (x)=\prod \left(\ 1-f_{\alpha }(x)^{q-1}\right)}acts as an indicator function forV.{\displaystyle V.}Ifx∈V{\displaystyle x\in V}thenP(x)=1,{\displaystyle \mathbb {P} (x)=1,}otherwise, for somefα,{\displaystyle f_{\alpha },}we havefα(x)≠0{\displaystyle f_{\alpha }(x)\neq 0}which implies thatfα(x)q−1=1,{\displaystyle f_{\alpha }(x)^{q-1}=1,}henceP(x)=0.{\displaystyle \mathbb {P} (x)=0.}
Although indicator functions are not smooth, they admitweak derivatives. For example, considerHeaviside step functionH(x)≡I(x>0){\displaystyle H(x)\equiv \operatorname {\mathbb {I} } \!{\bigl (}x>0{\bigr )}}Thedistributional derivativeof the Heaviside step function is equal to theDirac delta function, i.e.dH(x)dx=δ(x){\displaystyle {\frac {\mathrm {d} H(x)}{\mathrm {d} x}}=\delta (x)}and similarly the distributional derivative ofG(x):=I(x<0){\displaystyle G(x):=\operatorname {\mathbb {I} } \!{\bigl (}x<0{\bigr )}}isdG(x)dx=−δ(x).{\displaystyle {\frac {\mathrm {d} G(x)}{\mathrm {d} x}}=-\delta (x).}
Thus the derivative of the Heaviside step function can be seen as theinward normal derivativeat theboundaryof the domain given by the positive half-line. In higher dimensions, the derivative naturally generalises to the inward normal derivative, while the Heaviside step function naturally generalises to the indicator function of some domainD. The surface ofDwill be denoted byS. Proceeding, it can be derived that the inwardnormal derivativeof the indicator gives rise to asurface delta function, which can be indicated byδS(x){\displaystyle \delta _{S}(\mathbf {x} )}:δS(x)=−nx⋅∇xI(x∈D){\displaystyle \delta _{S}(\mathbf {x} )=-\mathbf {n} _{x}\cdot \nabla _{x}\operatorname {\mathbb {I} } \!{\bigl (}\ \mathbf {x} \in D\ {\bigr )}\ }wherenis the outwardnormalof the surfaceS. This 'surface delta function' has the following property:[4]−∫Rnf(x)nx⋅∇xI(x∈D)dnx=∮Sf(β)dn−1β.{\displaystyle -\int _{\mathbb {R} ^{n}}f(\mathbf {x} )\,\mathbf {n} _{x}\cdot \nabla _{x}\operatorname {\mathbb {I} } \!{\bigl (}\ \mathbf {x} \in D\ {\bigr )}\;\operatorname {d} ^{n}\mathbf {x} =\oint _{S}\,f(\mathbf {\beta } )\;\operatorname {d} ^{n-1}\mathbf {\beta } .}
By setting the functionfequal to one, it follows that theinward normal derivative of the indicatorintegrates to the numerical value of thesurface areaS. | https://en.wikipedia.org/wiki/Indicator_function |
In mathematics, theexterior algebraorGrassmann algebraof avector spaceV{\displaystyle V}is anassociative algebrathat containsV,{\displaystyle V,}which has a product, calledexterior productorwedge productand denoted with∧{\displaystyle \wedge }, such thatv∧v=0{\displaystyle v\wedge v=0}for every vectorv{\displaystyle v}inV.{\displaystyle V.}The exterior algebra is named afterHermann Grassmann,[3]and the names of the product come from the "wedge" symbol∧{\displaystyle \wedge }and the fact that the product of two elements ofV{\displaystyle V}is "outside"V.{\displaystyle V.}
The wedge product ofk{\displaystyle k}vectorsv1∧v2∧⋯∧vk{\displaystyle v_{1}\wedge v_{2}\wedge \dots \wedge v_{k}}is called abladeof degreek{\displaystyle k}ork{\displaystyle k}-blade. The wedge product was introduced originally as an algebraic construction used ingeometryto studyareas,volumes, and their higher-dimensional analogues: themagnitudeof a2-bladev∧w{\displaystyle v\wedge w}is the area of theparallelogramdefined byv{\displaystyle v}andw,{\displaystyle w,}and, more generally, the magnitude of ak{\displaystyle k}-blade is the (hyper)volume of theparallelotopedefined by the constituent vectors. Thealternating propertythatv∧v=0{\displaystyle v\wedge v=0}implies a skew-symmetric property thatv∧w=−w∧v,{\displaystyle v\wedge w=-w\wedge v,}and more generally any blade flips sign whenever two of its constituent vectors are exchanged, corresponding to a parallelotope of opposite orientation.
The full exterior algebra contains objects that are not themselves blades, butlinear combinationsof blades; a sum of blades of homogeneous degreek{\displaystyle k}is called ak-vector, while a more general sum of blades of arbitrary degree is called amultivector.[4]Thelinear spanof thek{\displaystyle k}-blades is called thek{\displaystyle k}-th exterior powerofV.{\displaystyle V.}The exterior algebra is thedirect sumof thek{\displaystyle k}-th exterior powers ofV,{\displaystyle V,}and this makes the exterior algebra agraded algebra.
The exterior algebra isuniversalin the sense that every equation that relates elements ofV{\displaystyle V}in the exterior algebra is also valid in every associative algebra that containsV{\displaystyle V}and in which the square of every element ofV{\displaystyle V}is zero.
The definition of the exterior algebra can be extended for spaces built from vector spaces, such asvector fieldsandfunctionswhosedomainis a vector space. Moreover, the field ofscalarsmay be any field. More generally, the exterior algebra can be defined formodulesover acommutative ring. In particular, the algebra ofdifferential formsink{\displaystyle k}variables is an exterior algebra over the ring of thesmooth functionsink{\displaystyle k}variables.
The two-dimensionalEuclidean vector spaceR2{\displaystyle \mathbf {R} ^{2}}is arealvector space equipped with abasisconsisting of a pair of orthogonalunit vectorse1=[10],e2=[01].{\displaystyle \mathbf {e} _{1}={\begin{bmatrix}1\\0\end{bmatrix}},\quad \mathbf {e} _{2}={\begin{bmatrix}0\\1\end{bmatrix}}.}
Suppose thatv=[ab]=ae1+be2,w=[cd]=ce1+de2{\displaystyle \mathbf {v} ={\begin{bmatrix}a\\b\end{bmatrix}}=a\mathbf {e} _{1}+b\mathbf {e} _{2},\quad \mathbf {w} ={\begin{bmatrix}c\\d\end{bmatrix}}=c\mathbf {e} _{1}+d\mathbf {e} _{2}}are a pair of given vectors inR2{\displaystyle \mathbf {R} ^{2}}, written in components. There is a unique parallelogram havingv{\displaystyle \mathbf {v} }andw{\displaystyle \mathbf {w} }as two of its sides. Theareaof this parallelogram is given by the standarddeterminantformula:Area=|det[vw]|=|det[acbd]|=|ad−bc|.{\displaystyle {\text{Area}}=\left|\det {\begin{bmatrix}\mathbf {v} &\mathbf {w} \end{bmatrix}}\right|=\left|\det {\begin{bmatrix}a&c\\b&d\end{bmatrix}}\right|=\left|ad-bc\right|.}
Consider now the exterior product ofv{\displaystyle \mathbf {v} }andw{\displaystyle \mathbf {w} }:v∧w=(ae1+be2)∧(ce1+de2)=ace1∧e1+ade1∧e2+bce2∧e1+bde2∧e2=(ad−bc)e1∧e2,{\displaystyle {\begin{aligned}\mathbf {v} \wedge \mathbf {w} &=(a\mathbf {e} _{1}+b\mathbf {e} _{2})\wedge (c\mathbf {e} _{1}+d\mathbf {e} _{2})\\&=ac\mathbf {e} _{1}\wedge \mathbf {e} _{1}+ad\mathbf {e} _{1}\wedge \mathbf {e} _{2}+bc\mathbf {e} _{2}\wedge \mathbf {e} _{1}+bd\mathbf {e} _{2}\wedge \mathbf {e} _{2}\\&=\left(ad-bc\right)\mathbf {e} _{1}\wedge \mathbf {e} _{2},\end{aligned}}}where the first step uses the distributive law for theexterior product, and the last uses the fact that the exterior product is analternating map, and in particulare2∧e1=−(e1∧e2).{\displaystyle \mathbf {e} _{2}\wedge \mathbf {e} _{1}=-(\mathbf {e} _{1}\wedge \mathbf {e} _{2}).}(The fact that the exterior product is an alternating map also forcese1∧e1=e2∧e2=0.{\displaystyle \mathbf {e} _{1}\wedge \mathbf {e} _{1}=\mathbf {e} _{2}\wedge \mathbf {e} _{2}=0.}) Note that the coefficient in this last expression is precisely the determinant of the matrix[vw]. The fact that this may be positive or negative has the intuitive meaning thatvandwmay be oriented in a counterclockwise or clockwise sense as the vertices of the parallelogram they define. Such an area is called thesigned areaof the parallelogram: theabsolute valueof the signed area is the ordinary area, and the sign determines its orientation.
The fact that this coefficient is the signed area is not an accident. In fact, it is relatively easy to see that the exterior product should be related to the signed area if one tries to axiomatize this area as an algebraic construct. In detail, ifA(v,w)denotes the signed area of the parallelogram of which the pair of vectorsvandwform two adjacent sides, then A must satisfy the following properties:
With the exception of the last property, the exterior product of two vectors satisfies the same properties as the area. In a certain sense, the exterior product generalizes the final property by allowing the area of a parallelogram to be compared to that of any chosen parallelogram in a parallel plane (here, the one with sidese1ande2). In other words, the exterior product provides abasis-independentformulation of area.[5]
For vectors inR3, the exterior algebra is closely related to thecross productandtriple product. Using the standard basis{e1,e2,e3}, the exterior product of a pair of vectors
and
is
where {e1∧e2,e3∧e1,e2∧e3} is the basis for the three-dimensional space ⋀2(R3). The coefficients above are the same as those in the usual definition of thecross productof vectors in three dimensions, the only difference being that the exterior product is not an ordinary vector, but instead is abivector.
Bringing in a third vector
the exterior product of three vectors is
wheree1∧e2∧e3is the basis vector for the one-dimensional space ⋀3(R3). The scalar coefficient is thetriple productof the three vectors.
The cross product and triple product in three dimensions each admit both geometric and algebraic interpretations. The cross productu×vcan be interpreted as a vector which is perpendicular to bothuandvand whose magnitude is equal to the area of the parallelogram determined by the two vectors. It can also be interpreted as the vector consisting of theminorsof the matrix with columnsuandv. The triple product ofu,v, andwis geometrically a (signed) volume. Algebraically, it is the determinant of the matrix with columnsu,v, andw. The exterior product in three dimensions allows for similar interpretations. In fact, in the presence of a positively orientedorthonormal basis, the exterior product generalizes these notions to higher dimensions.
The exterior algebra⋀(V){\displaystyle \bigwedge (V)}of a vector spaceV{\displaystyle V}over afieldK{\displaystyle K}is defined as thequotient algebraof thetensor algebraT(V), where
by the two-sidedidealI{\displaystyle I}generated by all elements of the formx⊗x{\displaystyle x\otimes x}such thatx∈V{\displaystyle x\in V}.[6]Symbolically,
The exterior product∧{\displaystyle \wedge }of two elements of⋀(V){\displaystyle \bigwedge (V)}is defined by
The exterior product is by constructionalternatingon elements ofV{\displaystyle V}, which means thatx∧x=0{\displaystyle x\wedge x=0}for allx∈V,{\displaystyle x\in V,}by the above construction. It follows that the product is alsoanticommutativeon elements ofV{\displaystyle V}, for supposing thatx,y∈V{\displaystyle x,y\in V},
hence
More generally, ifσ{\displaystyle \sigma }is apermutationof the integers[1,…,k]{\displaystyle [1,\dots ,k]}, andx1{\displaystyle x_{1}},x2{\displaystyle x_{2}}, ...,xk{\displaystyle x_{k}}are elements ofV{\displaystyle V}, it follows that
wheresgn(σ){\displaystyle \operatorname {sgn}(\sigma )}is thesignature of the permutationσ{\displaystyle \sigma }.[7]
In particular, ifxi=xj{\displaystyle x_{i}=x_{j}}for somei≠j{\displaystyle i\neq j}, then the following generalization of the alternating property also holds:
Together with the distributive property of the exterior product, one further generalization is that a necessary and sufficient condition for{x1,x2,…,xk}{\displaystyle \{x_{1},x_{2},\dots ,x_{k}\}}to be a linearly dependent set of vectors is that
Thekthexterior powerofV{\displaystyle V}, denoted⋀k(V){\displaystyle {\textstyle \bigwedge }^{\!k}(V)}, is thevector subspaceof⋀(V){\displaystyle {\textstyle \bigwedge }(V)}spannedby elements of the form
Ifα∈⋀k(V){\displaystyle \alpha \in {\textstyle \bigwedge }^{\!k}(V)}, thenα{\displaystyle \alpha }is said to be ak-vector. If, furthermore,α{\displaystyle \alpha }can be expressed as an exterior product ofk{\displaystyle k}elements ofV{\displaystyle V}, thenα{\displaystyle \alpha }is said to bedecomposable(orsimple, by some authors; or ablade, by others). Although decomposablek{\displaystyle k}-vectors span⋀k(V){\displaystyle {\textstyle \bigwedge }^{\!k}(V)}, not every element of⋀k(V){\displaystyle {\textstyle \bigwedge }^{\!k}(V)}is decomposable. For example, givenR4{\displaystyle \mathbf {R} ^{4}}with a basis{e1,e2,e3,e4}{\displaystyle \{e_{1},e_{2},e_{3},e_{4}\}}, the following 2-vector is not decomposable:
If thedimensionofV{\displaystyle V}isn{\displaystyle n}and{e1,…,en}{\displaystyle \{e_{1},\dots ,e_{n}\}}is abasisforV{\displaystyle V}, then the set
is a basis for⋀k(V){\displaystyle {\textstyle \bigwedge }^{\!k}(V)}. The reason is the following: given any exterior product of the form
every vectorvj{\displaystyle v_{j}}can be written as alinear combinationof the basis vectorsei{\displaystyle e_{i}}; using the bilinearity of the exterior product, this can be expanded to a linear combination of exterior products of those basis vectors. Any exterior product in which the same basis vector appears more than once is zero; any exterior product in which the basis vectors do not appear in the proper order can be reordered, changing the sign whenever two basis vectors change places. In general, the resulting coefficients of the basisk-vectors can be computed as theminorsof thematrixthat describes the vectorsvj{\displaystyle v_{j}}in terms of the basisei{\displaystyle e_{i}}.
By counting the basis elements, the dimension of⋀k(V){\displaystyle {\textstyle \bigwedge }^{\!k}(V)}is equal to abinomial coefficient:
wheren{\displaystyle n}is the dimension of thevectors, andk{\displaystyle k}is the number of vectors in the product. The binomial coefficient produces the correct result, even for exceptional cases; in particular,⋀k(V)={0}{\displaystyle {\textstyle \bigwedge }^{\!k}(V)=\{0\}}fork>n{\displaystyle k>n}.
Any element of the exterior algebra can be written as a sum ofk-vectors. Hence, as a vector space the exterior algebra is adirect sum
(where, by convention,⋀0(V)=K{\displaystyle {\textstyle \bigwedge }^{\!0}(V)=K}, thefieldunderlyingV{\displaystyle V}, and⋀1(V)=V{\displaystyle {\textstyle \bigwedge }^{\!1}(V)=V}), and therefore its dimension is equal to the sum of the binomial coefficients, which is2n{\displaystyle 2^{n}}.
Ifα∈⋀k(V){\displaystyle \alpha \in {\textstyle \bigwedge }^{\!k}(V)}, then it is possible to expressα{\displaystyle \alpha }as a linear combination of decomposablek-vectors:
where eachα(i){\displaystyle \alpha ^{(i)}}is decomposable, say
Therankof thek-vectorα{\displaystyle \alpha }is the minimal number of decomposablek-vectors in such an expansion ofα{\displaystyle \alpha }. This is similar to the notion oftensor rank.
Rank is particularly important in the study of 2-vectors (Sternberg 1964, §III.6) (Bryant et al. 1991). The rank of a 2-vectorα{\displaystyle \alpha }can be identified with half therank of the matrixof coefficients ofα{\displaystyle \alpha }in a basis. Thus ifei{\displaystyle e_{i}}is a basis forV{\displaystyle V}, thenα{\displaystyle \alpha }can be expressed uniquely as
whereaij=−aji{\displaystyle a_{ij}=-a_{ji}}(the matrix of coefficients isskew-symmetric). The rank of the matrixaij{\displaystyle a_{ij}}is therefore even, and is twice the rank of the formα{\displaystyle \alpha }.
In characteristic 0, the 2-vectorα{\displaystyle \alpha }has rankp{\displaystyle p}if and only if
The exterior product of ak-vector with ap-vector is a(k+p){\displaystyle (k+p)}-vector, once again invoking bilinearity. As a consequence, the direct sum decomposition of the preceding section
gives the exterior algebra the additional structure of agraded algebra, that is
Moreover, ifKis the base field, we have
The exterior product is graded anticommutative, meaning that ifα∈⋀k(V){\displaystyle \alpha \in {\textstyle \bigwedge }^{\!k}(V)}andβ∈⋀p(V){\displaystyle \beta \in {\textstyle \bigwedge }^{\!p}(V)}, then
In addition to studying the graded structure on the exterior algebra,Bourbaki (1989)studies additional graded structures on exterior algebras, such as those on the exterior algebra of agraded module(a module that already carries its own gradation).
LetVbe a vector space over the fieldK. Informally, multiplication in⋀(V){\displaystyle {\textstyle \bigwedge }(V)}is performed by manipulating symbols and imposing adistributive law, anassociative law, and using the identityv∧v=0{\displaystyle v\wedge v=0}forv∈V. Formally,⋀(V){\displaystyle {\textstyle \bigwedge }(V)}is the "most general" algebra in which these rules hold for the multiplication, in the sense that any unital associativeK-algebra containingVwith alternating multiplication onVmust contain a homomorphic image of⋀(V){\displaystyle {\textstyle \bigwedge }(V)}. In other words, the exterior algebra has the followinguniversal property:[8]
Given any unital associativeK-algebraAand anyK-linear mapj:V→A{\displaystyle j:V\to A}such thatj(v)j(v)=0{\displaystyle j(v)j(v)=0}for everyvinV, then there existsprecisely oneunitalalgebra homomorphismf:⋀(V)→A{\displaystyle f:{\textstyle \bigwedge }(V)\to A}such thatj(v) =f(i(v))for allvinV(hereiis the natural inclusion ofVin⋀(V){\displaystyle {\textstyle \bigwedge }(V)}, see above).
To construct the most general algebra that containsVand whose multiplication is alternating onV, it is natural to start with the most general associative algebra that containsV, thetensor algebraT(V), and then enforce the alternating property by taking a suitablequotient. We thus take the two-sidedidealIinT(V)generated by all elements of the formv⊗vforvinV, and define⋀(V){\displaystyle {\textstyle \bigwedge }(V)}as the quotient
(and use∧as the symbol for multiplication in⋀(V){\displaystyle {\textstyle \bigwedge }(V)}). It is then straightforward to show that⋀(V){\displaystyle {\textstyle \bigwedge }(V)}containsVand satisfies the above universal property.
As a consequence of this construction, the operation of assigning to a vector spaceVits exterior algebra⋀(V){\displaystyle {\textstyle \bigwedge }(V)}is afunctorfrom thecategoryof vector spaces to the category of algebras.
Rather than defining⋀(V){\displaystyle {\textstyle \bigwedge }(V)}first and then identifying the exterior powers⋀k(V){\displaystyle {\textstyle \bigwedge }^{\!k}(V)}as certain subspaces, one may alternatively define the spaces⋀k(V){\displaystyle {\textstyle \bigwedge }^{\!k}(V)}first and then combine them to form the algebra⋀(V){\displaystyle {\textstyle \bigwedge }(V)}. This approach is often used in differential geometry and is described in the next section.
Given acommutative ringR{\displaystyle R}and anR{\displaystyle R}-moduleM{\displaystyle M}, we can define the exterior algebra⋀(M){\displaystyle {\textstyle \bigwedge }(M)}just as above, as a suitable quotient of the tensor algebraT(M){\displaystyle \mathrm {T} (M)}. It will satisfy the analogous universal property. Many of the properties of⋀(M){\displaystyle {\textstyle \bigwedge }(M)}also require thatM{\displaystyle M}be aprojective module. Where finite dimensionality is used, the properties further require thatM{\displaystyle M}befinitely generatedand projective. Generalizations to the most common situations can be found inBourbaki (1989).
Exterior algebras ofvector bundlesare frequently considered in geometry and topology. There are no essential differences between the algebraic properties of the exterior algebra of finite-dimensional vector bundles and those of the exterior algebra of finitely generated projective modules, by theSerre–Swan theorem. More general exterior algebras can be defined forsheavesof modules.
For a field of characteristic not 2,[9]the exterior algebra of a vector spaceV{\displaystyle V}overK{\displaystyle K}can be canonically identified with the vector subspace ofT(V){\displaystyle \mathrm {T} (V)}that consists ofantisymmetric tensors. For characteristic 0 (or higher thandimV{\displaystyle \dim V}), the vector space ofk{\displaystyle k}-linear antisymmetric tensors is transversal to the idealI{\displaystyle I}, hence, a good choice to represent the quotient. But for nonzero characteristic, the vector space ofK{\displaystyle K}-linear antisymmetric tensors could be not transversal to the ideal (actually, fork≥charK{\displaystyle k\geq \operatorname {char} K}, the vector space ofK{\displaystyle K}-linear antisymmetric tensors is contained inI{\displaystyle I}); nevertheless, transversal or not, a product can be defined on this space such that the resulting algebra is isomorphic to the exterior algebra: in the first case the natural choice for the product is just the quotient product (using the available projection), in the second case, this product must be slightly modified as given below (along Arnold setting), but such that the algebra stays isomorphic with the exterior algebra, i.e. the quotient ofT(V){\displaystyle \mathrm {T} (V)}by the idealI{\displaystyle I}generated by elements of the formx⊗x{\displaystyle x\otimes x}. Of course, for characteristic0{\displaystyle 0}(or higher than the dimension of the vector space), one or the other definition of the product could be used, as the two algebras are isomorphic (see V. I. Arnold or Kobayashi-Nomizu).
LetTr(V){\displaystyle \mathrm {T} ^{r}(V)}be the space of homogeneous tensors of degreer{\displaystyle r}. This is spanned by decomposable tensors
Theantisymmetrization(or sometimes theskew-symmetrization) of a decomposable tensor is defined by
and, whenr!≠0{\displaystyle r!\neq 0}(for nonzero characteristic fieldr!{\displaystyle r!}might be 0):
where the sum is taken over thesymmetric groupof permutations on the symbols{1,…,r}{\displaystyle \{1,\dots ,r\}}. This extends by linearity and homogeneity to an operation, also denoted byA{\displaystyle {\mathcal {A}}}andAlt{\displaystyle {\rm {Alt}}}, on the full tensor algebraT(V){\displaystyle \mathrm {T} (V)}.
Note that
Such that, when defined,Alt(r){\displaystyle \operatorname {Alt} ^{(r)}}is the projection for the exterior (quotient) algebra onto the r-homogeneous alternating tensor subspace.
On the other hand, the imageA(T(V)){\displaystyle {\mathcal {A}}(\mathrm {T} (V))}is always thealternating tensor graded subspace(not yet an algebra, as product is not yet defined), denotedA(V){\displaystyle A(V)}. This is a vector subspace ofT(V){\displaystyle \mathrm {T} (V)}, and it inherits the structure of a graded vector space from that onT(V){\displaystyle \mathrm {T} (V)}. Moreover, the kernel ofA(r){\displaystyle {\mathcal {A}}^{(r)}}is preciselyI(r){\displaystyle I^{(r)}}, the homogeneous subset of the idealI{\displaystyle I}, or the kernel ofA{\displaystyle {\mathcal {A}}}isI{\displaystyle I}. WhenAlt{\displaystyle \operatorname {Alt} }is defined,A(V){\displaystyle A(V)}carries an associative graded product⊗^{\displaystyle {\widehat {\otimes }}}defined by (the same as the wedge product)
AssumingK{\displaystyle K}has characteristic 0, asA(V){\displaystyle A(V)}is a supplement ofI{\displaystyle I}inT(V){\displaystyle \mathrm {T} (V)}, with the above given product, there is a canonical isomorphism
When the characteristic of the field is nonzero,A{\displaystyle {\mathcal {A}}}will do whatAlt{\displaystyle {\rm {Alt}}}did before, but the product cannot be defined as above. In such a case, isomorphismA(V)≅⋀(V){\displaystyle A(V)\cong {\textstyle \bigwedge }(V)}still holds, in spite ofA(V){\displaystyle A(V)}not being a supplement of the idealI{\displaystyle I}, but then, the product should be modified as given below (∧˙{\displaystyle {\dot {\wedge }}}product, Arnold setting).
Finally, we always getA(V){\displaystyle A(V)}isomorphic with⋀(V){\displaystyle {\textstyle \bigwedge }(V)}, but the product could (or should) be chosen in two ways (or only one). Actually, the product could be chosen in many ways, rescaling it on homogeneous spaces asc(r+p)/c(r)c(p){\displaystyle c(r+p)/c(r)c(p)}for an arbitrary sequencec(r){\displaystyle c(r)}in the field, as long as the division makes sense (this is such that the redefined product is also associative, i.e. defines an algebra onA(V){\displaystyle A(V)}). Also note, the interior product definition should be changed accordingly, in order to keep its skew derivation property.
Suppose thatVhas finite dimensionn, and that a basise1, ...,enofVis given. Then any alternating tensort∈ Ar(V) ⊂Tr(V)can be written inindex notationwith theEinstein summation conventionas
whereti1⋅⋅⋅iriscompletely antisymmetricin its indices.
The exterior product of two alternating tensorstandsof ranksrandpis given by
The components of this tensor are precisely the skew part of the components of the tensor products⊗t, denoted by square brackets on the indices:
Theinterior productmay also be described in index notation as follows. Lett=ti0i1⋯ir−1{\displaystyle t=t^{i_{0}i_{1}\cdots i_{r-1}}}be an antisymmetric tensor of rankr{\displaystyle r}. Then, forα∈V∗,ιαt{\displaystyle \iota _{\alpha }t}is an alternating tensor of rankr−1{\displaystyle r-1}, given by
wherenis the dimension ofV.
Given two vector spacesVandXand a natural numberk, analternating operatorfromVktoXis amultilinear map
such that wheneverv1, ...,vkarelinearly dependentvectors inV, then
The map
which associates tok{\displaystyle k}vectors fromV{\displaystyle V}their exterior product, i.e. their correspondingk{\displaystyle k}-vector, is also alternating. In fact, this map is the "most general" alternating operator defined onVk;{\displaystyle V^{k};}given any other alternating operatorf:Vk→X,{\displaystyle f:V^{k}\rightarrow X,}there exists a uniquelinear mapϕ:⋀k(V)→X{\displaystyle \phi :{\textstyle \bigwedge }^{\!k}(V)\rightarrow X}withf=ϕ∘w.{\displaystyle f=\phi \circ w.}Thisuniversal propertycharacterizes the space of alternating operators onVk{\displaystyle V^{k}}and can serve as its definition.
The above discussion specializes to the case whenX=K{\displaystyle X=K}, the base field. In this case an alternating multilinear function
is called analternating multilinear form. The set of allalternatingmultilinear formsis a vector space, as the sum of two such maps, or the product of such a map with a scalar, is again alternating. By the universal property of the exterior power, the space of alternating forms of degreek{\displaystyle k}onV{\displaystyle V}isnaturallyisomorphic with thedual vector space(⋀k(V))∗{\displaystyle {\bigl (}{\textstyle \bigwedge }^{\!k}(V){\bigr )}^{*}}. IfV{\displaystyle V}is finite-dimensional, then the latter isnaturally isomorphic[clarification needed]to⋀k(V∗){\displaystyle {\textstyle \bigwedge }^{\!k}\left(V^{*}\right)}. In particular, ifV{\displaystyle V}isn{\displaystyle n}-dimensional, the dimension of the space of alternating maps fromVk{\displaystyle V^{k}}toK{\displaystyle K}is thebinomial coefficient(nk){\displaystyle \textstyle {\binom {n}{k}}}.
Under such identification, the exterior product takes a concrete form: it produces a new anti-symmetric map from two given ones. Supposeω:Vk→Kandη:Vm→Kare two anti-symmetric maps. As in the case oftensor productsof multilinear maps, the number of variables of their exterior product is the sum of the numbers of their variables. Depending on the choice of identification of elements of exterior power with multilinear forms, the exterior product is defined as
or as
where, if the characteristic of the base fieldK{\displaystyle K}is 0, the alternation Alt of a multilinear map is defined to be the average of the sign-adjusted values over all thepermutationsof its variables:
When thefieldK{\displaystyle K}hasfinite characteristic, an equivalent version of the second expression without any factorials or any constants is well-defined:
where hereShk,m⊂Sk+mis the subset of(k,m)shuffles:permutationsσof the set{1, 2, ...,k+m}such thatσ(1) <σ(2) < ⋯ <σ(k), andσ(k+ 1) <σ(k+ 2) < ... <σ(k+m). As this might look very specific and fine tuned, an equivalent raw version is to sum in the above formula over permutations in left cosets ofSk+m/ (Sk×Sm).
Suppose thatV{\displaystyle V}is finite-dimensional. IfV∗{\displaystyle V^{*}}denotes thedual spaceto the vector spaceV{\displaystyle V}, then for eachα∈V∗{\displaystyle \alpha \in V^{*}}, it is possible to define anantiderivationon the algebra⋀(V){\displaystyle {\textstyle \bigwedge }(V)},
This derivation is called theinterior productwithα{\displaystyle \alpha }, or sometimes theinsertion operator, orcontractionbyα{\displaystyle \alpha }.
Suppose thatw∈⋀k(V){\displaystyle w\in {\textstyle \bigwedge }^{\!k}(V)}. Thenw{\displaystyle w}is a multilinear mapping ofV∗{\displaystyle V^{*}}toK{\displaystyle K}, so it is defined by its values on thek-foldCartesian productV∗×V∗×⋯×V∗{\displaystyle V^{*}\times V^{*}\times \dots \times V^{*}}. Ifu1,u2, ...,uk−1arek−1{\displaystyle k-1}elements ofV∗{\displaystyle V^{*}}, then define
Additionally, letιαf=0{\displaystyle \iota _{\alpha }f=0}wheneverf{\displaystyle f}is a pure scalar (i.e., belonging to⋀0(V){\displaystyle {\textstyle \bigwedge }^{\!0}(V)}).
The interior product satisfies the following properties:
These three properties are sufficient to characterize the interior product as well as define it in the general infinite-dimensional case.
Further properties of the interior product include:
Suppose thatV{\displaystyle V}has finite dimensionn{\displaystyle n}. Then the interior product induces a canonical isomorphism of vector spaces
by the recursive definition
In the geometrical setting, a non-zero element of the top exterior power⋀n(V){\displaystyle {\textstyle \bigwedge }^{\!n}(V)}(which is a one-dimensional vector space) is sometimes called avolume form(ororientation form, although this term may sometimes lead to ambiguity). The name orientation form comes from the fact that a choice of preferred top element determines an orientation of the whole exterior algebra, since it is tantamount to fixing an ordered basis of the vector space. Relative to the preferred volume formσ{\displaystyle \sigma }, the isomorphism is given explicitly by
If, in addition to a volume form, the vector spaceVis equipped with aninner productidentifyingV{\displaystyle V}withV∗{\displaystyle V^{*}}, then the resulting isomorphism is called theHodge star operator, which maps an element to itsHodge dual:
The composition of⋆{\displaystyle \star }with itself maps⋀k(V)→⋀k(V){\displaystyle {\textstyle \bigwedge }^{\!k}(V)\to {\textstyle \bigwedge }^{\!k}(V)}and is always a scalar multiple of the identity map. In most applications, the volume form is compatible with the inner product in the sense that it is an exterior product of anorthonormal basisofV{\displaystyle V}. In this case,
where id is the identity mapping, and the inner product hasmetric signature(p,q)—ppluses andqminuses.
ForV{\displaystyle V}a finite-dimensional space, aninner product(or apseudo-Euclideaninner product) onV{\displaystyle V}defines an isomorphism ofV{\displaystyle V}withV∗{\displaystyle V^{*}}, and so also an isomorphism of⋀k(V){\displaystyle {\textstyle \bigwedge }^{\!k}(V)}with(⋀kV)∗{\displaystyle {\bigl (}{\textstyle \bigwedge }^{\!k}V{\bigr )}^{*}}. The pairing between these two spaces also takes the form of an inner product. On decomposablek{\displaystyle k}-vectors,
the determinant of the matrix of inner products. In the special casevi=wi, the inner product is the square norm of thek-vector, given by the determinant of theGramian matrix(⟨vi,vj⟩). This is then extended bilinearly (or sesquilinearly in the complex case) to a non-degenerate inner product on⋀k(V).{\displaystyle {\textstyle \bigwedge }^{\!k}(V).}Ifei,i= 1, 2, ...,n, form anorthonormal basisofV{\displaystyle V}, then the vectors of the form
constitute an orthonormal basis for⋀k(V){\displaystyle {\textstyle \bigwedge }^{\!k}(V)}, a statement equivalent to theCauchy–Binet formula.
With respect to the inner product, exterior multiplication and the interior product are mutually adjoint. Specifically, forv∈⋀k−1(V){\displaystyle \mathbf {v} \in {\textstyle \bigwedge }^{\!k-1}(V)},w∈⋀k(V){\displaystyle \mathbf {w} \in {\textstyle \bigwedge }^{\!k}(V)}, andx∈V{\displaystyle x\in V},
wherex♭∈V∗is themusical isomorphism, the linear functional defined by
for ally∈V{\displaystyle y\in V}. This property completely characterizes the inner product on the exterior algebra.
Indeed, more generally forv∈⋀k−l(V){\displaystyle \mathbf {v} \in {\textstyle \bigwedge }^{\!k-l}(V)},w∈⋀k(V){\displaystyle \mathbf {w} \in {\textstyle \bigwedge }^{\!k}(V)}, andx∈⋀l(V){\displaystyle \mathbf {x} \in {\textstyle \bigwedge }^{\!l}(V)}, iteration of the above adjoint properties gives
where nowx♭∈⋀l(V∗)≃(⋀l(V))∗{\displaystyle \mathbf {x} ^{\flat }\in {\textstyle \bigwedge }^{\!l}\left(V^{*}\right)\simeq {\bigl (}{\textstyle \bigwedge }^{\!l}(V){\bigr )}^{*}}is the duall{\displaystyle l}-vector defined by
for ally∈⋀l(V){\displaystyle \mathbf {y} \in {\textstyle \bigwedge }^{\!l}(V)}.
There is a correspondence between the graded dual of the graded algebra⋀(V){\displaystyle {\textstyle \bigwedge }(V)}and alternating multilinear forms onV{\displaystyle V}. The exterior algebra (as well as thesymmetric algebra) inherits a bialgebra structure, and, indeed, aHopf algebrastructure, from thetensor algebra. See the article ontensor algebrasfor a detailed treatment of the topic.
The exterior product of multilinear forms defined above is dual to acoproductdefined on⋀(V){\displaystyle {\textstyle \bigwedge }(V)}, giving the structure of acoalgebra. Thecoproductis a linear functionΔ:⋀(V)→⋀(V)⊗⋀(V){\displaystyle \Delta :{\textstyle \bigwedge }(V)\to {\textstyle \bigwedge }(V)\otimes {\textstyle \bigwedge }(V)}, which is given by
on elementsv∈V{\displaystyle v\in V}. The symbol1{\displaystyle 1}stands for the unit element of the fieldK{\displaystyle K}. Recall thatK≃⋀0(V)⊆⋀(V){\displaystyle K\simeq {\textstyle \bigwedge }^{\!0}(V)\subseteq {\textstyle \bigwedge }(V)}, so that the above really does lie in⋀(V)⊗⋀(V){\displaystyle {\textstyle \bigwedge }(V)\otimes {\textstyle \bigwedge }(V)}. This definition of the coproduct is lifted to the full space⋀(V){\displaystyle {\textstyle \bigwedge }(V)}by (linear) homomorphism.
The correct form of this homomorphism is not what one might naively write, but has to be the one carefully defined in thecoalgebraarticle. In this case, one obtains
Expanding this out in detail, one obtains the following expression on decomposable elements:
where the second summation is taken over all(p,k−p)-shuffles. By convention, one takes that Sh(k,0) and Sh(0,k) equals {id: {1, ...,k} → {1, ...,k}}. It is also convenient to take the pure wedge productsvσ(1)∧⋯∧vσ(p){\displaystyle v_{\sigma (1)}\wedge \dots \wedge v_{\sigma (p)}}andvσ(p+1)∧⋯∧vσ(k){\displaystyle v_{\sigma (p+1)}\wedge \dots \wedge v_{\sigma (k)}}to equal 1 forp= 0 andp=k, respectively (the empty product in⋀(V){\displaystyle {\textstyle \bigwedge }(V)}). The shuffle follows directly from the first axiom of a co-algebra: the relative order of the elementsxk{\displaystyle x_{k}}ispreservedin the riffle shuffle: the riffle shuffle merely splits the ordered sequence into two ordered sequences, one on the left, and one on the right.
Observe that the coproduct preserves the grading of the algebra. Extending to the full space⋀(V),{\textstyle {\textstyle \bigwedge }(V),}one has
The tensor symbol ⊗ used in this section should be understood with some caution: it isnotthe same tensor symbol as the one being used in the definition of the alternating product. Intuitively, it is perhaps easiest to think it as just another, but different, tensor product: it is still (bi-)linear, as tensor products should be, but it is the product that is appropriate for the definition of a bialgebra, that is, for creating the object⋀(V)⊗⋀(V){\displaystyle {\textstyle \bigwedge }(V)\otimes {\textstyle \bigwedge }(V)}. Any lingering doubt can be shaken by pondering the equalities(1 ⊗v) ∧ (1 ⊗w) = 1 ⊗ (v∧w)and(v⊗ 1) ∧ (1 ⊗w) =v⊗w, which follow from the definition of the coalgebra, as opposed to naive manipulations involving the tensor and wedge symbols. This distinction is developed in greater detail in the article ontensor algebras. Here, there is much less of a problem, in that the alternating product∧{\displaystyle \wedge }clearly corresponds to multiplication in the exterior algebra, leaving the symbol⊗{\displaystyle \otimes }free for use in the definition of the bialgebra. In practice, this presents no particular problem, as long as one avoids the fatal trap of replacing alternating sums of⊗{\displaystyle \otimes }by the wedge symbol, with one exception. One can construct an alternating product from⊗{\displaystyle \otimes }, with the understanding that it works in a different space. Immediately below, an example is given: the alternating product for thedual spacecan be given in terms of the coproduct. The construction of the bialgebra here parallels the construction in thetensor algebraarticle almost exactly, except for the need to correctly track the alternating signs for the exterior algebra.
In terms of the coproduct, the exterior product on the dual space is just the graded dual of the coproduct:
where the tensor product on the right-hand side is of multilinear linear maps (extended by zero on elements of incompatible homogeneous degree: more precisely,α∧β=ε∘ (α⊗β) ∘ Δ, whereε{\displaystyle \varepsilon }is the counit, as defined presently).
Thecounitis the homomorphismε:⋀(V)→K{\displaystyle \varepsilon :{\textstyle \bigwedge }(V)\to K}that returns the 0-graded component of its argument. The coproduct and counit, along with the exterior product, define the structure of abialgebraon the exterior algebra.
With anantipodedefined on homogeneous elements byS(x)=(−1)(degx+12)x{\displaystyle S(x)=(-1)^{\binom {{\text{deg}}\,x\,+1}{2}}x}, the exterior algebra is furthermore aHopf algebra.[12]
Suppose thatV{\displaystyle V}andW{\displaystyle W}are a pair of vector spaces andf:V→W{\displaystyle f:V\to W}is alinear map. Then, by the universal property, there exists a unique homomorphism of graded algebras
such that
In particular,⋀(f){\displaystyle {\textstyle \bigwedge }(f)}preserves homogeneous degree. Thek-graded components of⋀(f){\textstyle \bigwedge \left(f\right)}are given on decomposable elements by
Let
The components of the transformation⋀k(f){\displaystyle {\textstyle \bigwedge }^{\!k}(f)}relative to a basis ofV{\displaystyle V}andW{\displaystyle W}is the matrix ofk×k{\displaystyle k\times k}minors off{\displaystyle f}. In particular, ifV=W{\displaystyle V=W}andV{\displaystyle V}is of finite dimensionn{\displaystyle n}, then⋀n(f){\displaystyle {\textstyle \bigwedge }^{\!n}(f)}is a mapping of a one-dimensional vector space⋀n(V){\displaystyle {\textstyle \bigwedge }^{\!n}(V)}to itself, and is therefore given by a scalar: thedeterminantoff{\displaystyle f}.
If0→U→V→W→0{\displaystyle 0\to U\to V\to W\to 0}is ashort exact sequenceof vector spaces, then
is an exact sequence of graded vector spaces,[13]as is
In particular, the exterior algebra of a direct sum is isomorphic to the tensor product of the exterior algebras:
This is a graded isomorphism; i.e.,
In greater generality, for a short exact sequence of vector spaces0→U→fV→gW→0,{\textstyle 0\to U\mathrel {\overset {f}{\to }} V\mathrel {\overset {g}{\to }} W\to 0,}there is a naturalfiltration
whereFp{\displaystyle F^{p}}forp≥1{\displaystyle p\geq 1}is spanned by elements of the formu1∧…∧uk+1−p∧v1∧…vp−1{\displaystyle u_{1}\wedge \ldots \wedge u_{k+1-p}\wedge v_{1}\wedge \ldots v_{p-1}}forui∈U{\displaystyle u_{i}\in U}andvi∈V.{\displaystyle v_{i}\in V.}The corresponding quotients admit a natural isomorphism
In particular, ifUis 1-dimensional then
is exact, and ifWis 1-dimensional then
is exact.[15]
The natural setting for (oriented)k{\displaystyle k}-dimensional volume and exterior algebra isaffine space. This is also the intimate connection between exterior algebra anddifferential forms, as to integrate we need a 'differential' object to measure infinitesimal volume. IfA{\displaystyle \mathbb {A} }is an affine space over the vector spaceV{\displaystyle V}, and a (simplex) collection of orderedk+1{\displaystyle k+1}pointsA0,A1,...,Ak{\displaystyle A_{0},A_{1},...,A_{k}}, we can define its orientedk{\displaystyle k}-dimensional volume as the exterior product of vectorsA0A1∧A0A2∧⋯∧A0Ak={\displaystyle A_{0}A_{1}\wedge A_{0}A_{2}\wedge \cdots \wedge A_{0}A_{k}={}}(−1)jAjA0∧AjA1∧AjA2∧⋯∧AjAk{\displaystyle (-1)^{j}A_{j}A_{0}\wedge A_{j}A_{1}\wedge A_{j}A_{2}\wedge \cdots \wedge A_{j}A_{k}}(using concatenationPQ{\displaystyle PQ}to mean thedisplacement vectorfrom pointP{\displaystyle P}toQ{\displaystyle Q}); if the order of the points is changed, the oriented volume changes by a sign, according to the parity of the permutation. Inn{\displaystyle n}-dimensional space, the volume of anyn{\displaystyle n}-dimensional simplex is a scalar multiple of any other.
The sum of the(k−1){\displaystyle (k-1)}-dimensional oriented areas of the boundary simplexes of ak{\displaystyle k}-dimensional simplex is zero, as for the sum of vectors around a triangle or the oriented triangles bounding the tetrahedron in the previous section.
The vector space structure on⋀(V){\displaystyle {\textstyle \bigwedge }(V)}generalises addition of vectors inV{\displaystyle V}: we have(u1+u2)∧v=u1∧v+u2∧v{\displaystyle (u_{1}+u_{2})\wedge v=u_{1}\wedge v+u_{2}\wedge v}and similarly ak-bladev1∧⋯∧vk{\displaystyle v_{1}\wedge \dots \wedge v_{k}}is linear in each factor.
In applications tolinear algebra, the exterior product provides an abstract algebraic manner for describing thedeterminantand theminorsof amatrix. For instance, it is well known that the determinant of a square matrix is equal to the volume of the parallelotope whose sides are the columns of the matrix (with a sign to track orientation). This suggests that the determinant can bedefinedin terms of the exterior product of the column vectors. Likewise, thek×kminors of a matrix can be defined by looking at the exterior products of column vectors chosenkat a time. These ideas can be extended not just to matrices but tolinear transformationsas well: the determinant of a linear transformation is the factor by which it scales the oriented volume of any given reference parallelotope. So the determinant of a linear transformation can be defined in terms of what the transformation does to the top exterior power. The action of a transformation on the lesser exterior powers gives abasis-independent way to talk about the minors of the transformation.
In physics, many quantities are naturally represented by alternating operators. For example, if the motion of a charged particle is described by velocity and acceleration vectors in four-dimensional spacetime, then normalization of the velocity vector requires that the electromagnetic force must be an alternating operator on the velocity. Its six degrees of freedom are identified with the electric and magnetic fields.
InEinstein's theories of relativity, theelectromagnetic fieldis generally given as adifferential 2-formF=dA{\displaystyle F=dA}in4-spaceor as the equivalentalternating tensor fieldFij=A[i,j]=A[i;j],{\displaystyle F_{ij}=A_{[i,j]}=A_{[i;j]},}theelectromagnetic tensor. ThendF=ddA=0{\displaystyle dF=ddA=0}or the equivalent Bianchi identityF[ij,k]=F[ij;k]=0.{\displaystyle F_{[ij,k]}=F_{[ij;k]}=0.}None of this requires a metric.
Adding theLorentz metricand anorientationprovides theHodge star operator⋆{\displaystyle \star }and thus makes it possible to defineJ=⋆d⋆F{\displaystyle J={\star }d{\star }F}or the equivalent tensordivergenceJi=F,jij=F;jij{\displaystyle J^{i}=F_{,j}^{ij}=F_{;j}^{ij}}whereFij=gikgjlFkl.{\displaystyle F^{ij}=g^{ik}g^{jl}F_{kl}.}
The decomposablek-vectors have geometric interpretations: the bivectoru∧v{\displaystyle u\wedge v}represents the plane spanned by the vectors, "weighted" with a number, given by the area of the orientedparallelogramwith sidesu{\displaystyle u}andv{\displaystyle v}. Analogously, the 3-vectoru∧v∧w{\displaystyle u\wedge v\wedge w}represents the spanned 3-space weighted by the volume of the orientedparallelepipedwith edgesu{\displaystyle u},v{\displaystyle v}, andw{\displaystyle w}.
Decomposablek-vectors in⋀k(V){\displaystyle {\textstyle \bigwedge }^{\!k}(V)}correspond to weightedk-dimensionallinear subspacesofV{\displaystyle V}. In particular, theGrassmannianofk-dimensional subspaces ofV{\displaystyle V}, denotedGrk(V){\displaystyle \operatorname {Gr} _{k}(V)}, can be naturally identified with analgebraic subvarietyof theprojective spaceP(⋀k(V)){\textstyle \mathbf {P} {\bigl (}{\textstyle \bigwedge }^{\!k}(V){\bigr )}}. This is called thePlücker embedding, and the image of the embedding can be characterized by thePlücker relations.
The exterior algebra has notable applications indifferential geometry, where it is used to definedifferential forms.[16]Differential forms are mathematical objects that evaluate the length of vectors, areas of parallelograms, and volumes ofhigher-dimensional bodies, so they can beintegratedover curves, surfaces and higher dimensionalmanifoldsin a way that generalizes theline integralsandsurface integralsfrom calculus. Adifferential format a point of adifferentiable manifoldis an alternating multilinear form on thetangent spaceat the point. Equivalently, a differential form of degreekis alinear functionalon thekth exterior power of the tangent space. As a consequence, the exterior product of multilinear forms defines a natural exterior product for differential forms. Differential forms play a major role in diverse areas of differential geometry.
Analternate approachdefines differential forms in terms ofgerms of functions.
In particular, theexterior derivativegives the exterior algebra of differential forms on a manifold the structure of adifferential graded algebra. The exterior derivative commutes withpullbackalong smooth mappings between manifolds, and it is therefore anaturaldifferential operator. The exterior algebra of differential forms, equipped with the exterior derivative, is acochain complexwhose cohomology is called thede Rham cohomologyof the underlying manifold and plays a vital role in thealgebraic topologyof differentiable manifolds.
Inrepresentation theory, the exterior algebra is one of the two fundamentalSchur functorson the category of vector spaces, the other being thesymmetric algebra. Together, these constructions are used to generate theirreducible representationsof thegeneral linear group(seeFundamental representation).
The exterior algebra over the complex numbers is the archetypal example of asuperalgebra, which plays a fundamental role in physical theories pertaining tofermionsandsupersymmetry. A single element of the exterior algebra is called asupernumber[17]orGrassmann number. The exterior algebra itself is then just a one-dimensionalsuperspace: it is just the set of all of the points in the exterior algebra. The topology on this space is essentially theweak topology, theopen setsbeing thecylinder sets. Ann-dimensional superspace is just then{\displaystyle n}-fold product of exterior algebras.
LetL{\displaystyle L}be a Lie algebra over a fieldK{\displaystyle K}, then it is possible to define the structure of achain complexon the exterior algebra ofL{\displaystyle L}. This is aK{\displaystyle K}-linear mapping
defined on decomposable elements by
TheJacobi identityholds if and only if1{\displaystyle {1}}, and so this is a necessary and sufficient condition for an anticommutative nonassociative algebraL{\displaystyle L}to be a Lie algebra. Moreover, in that case⋀(L){\textstyle {\textstyle \bigwedge }(L)}is achain complexwith boundary operator∂{\displaystyle \partial }. Thehomologyassociated to this complex is theLie algebra homology.
The exterior algebra is the main ingredient in the construction of theKoszul complex, a fundamental object inhomological algebra.
The exterior algebra was first introduced byHermann Grassmannin 1844 under the blanket term ofAusdehnungslehre, orTheory of Extension.[18]This referred more generally to an algebraic (or axiomatic) theory of extended quantities and was one of the early precursors to the modern notion of avector space.Saint-Venantalso published similar ideas of exterior calculus for which he claimed priority over Grassmann.[19]
The algebra itself was built from a set of rules, or axioms, capturing the formal aspects of Cayley and Sylvester's theory of multivectors. It was thus acalculus, much like thepropositional calculus, except focused exclusively on the task of formal reasoning in geometrical terms.[20]In particular, this new development allowed for anaxiomaticcharacterization of dimension, a property that had previously only been examined from the coordinate point of view.
The import of this new theory of vectors andmultivectorswas lost to mid-19th-century mathematicians,[21]until being thoroughly vetted byGiuseppe Peanoin 1888. Peano's work also remained somewhat obscure until the turn of the century, when the subject was unified by members of the French geometry school (notablyHenri Poincaré,Élie Cartan, andGaston Darboux) who applied Grassmann's ideas to the calculus ofdifferential forms.
A short while later,Alfred North Whitehead, borrowing from the ideas of Peano and Grassmann, introduced hisuniversal algebra. This then paved the way for the 20th-century developments ofabstract algebraby placing the axiomatic notion of an algebraic system on a firm logical footing. | https://en.wikipedia.org/wiki/Exterior_algebra |
Inalgebra, agraded-commutative ring(also called askew-commutative ring) is agraded ringthat is commutative in the graded sense; that is,homogeneous elementsx,ysatisfy
where |x| and |y| denote the degrees ofxandy.
Acommutative (non-graded) ring, with trivial grading, is a basic example. For a nontrivial example, anexterior algebrais generally not acommutative ringbut is agraded-commutative ring.
Acup productoncohomologysatisfies the skew-commutative relation; hence, acohomology ringis graded-commutative. In fact, many examples of graded-commutative rings come fromalgebraic topologyandhomological algebra.
Thisabstract algebra-related article is astub. You can help Wikipedia byexpanding it. | https://en.wikipedia.org/wiki/Graded-commutative_ring |
Inmathematics, anoperationis afunctionfrom asetto itself. For example, an operation onreal numberswill take in real numbers and return a real number. An operation can take zero or more input values (also called "operands" or "arguments") to a well-defined output value. The number of operands is thearityof the operation.
The most commonly studied operations arebinary operations(i.e., operations of arity 2), such asadditionandmultiplication, andunary operations(i.e., operations of arity 1), such asadditive inverseandmultiplicative inverse. An operation ofarityzero, ornullary operation, is aconstant.[1][2]Themixed productis an example of an operation of arity 3, also calledternary operation.
Generally, the arity is taken to be finite. However,infinitary operationsare sometimes considered,[1]in which case the "usual" operations of finite arity are calledfinitary operations.
Apartial operationis defined similarly to an operation, but with apartial functionin place of a function.
There are two common types of operations:unaryandbinary. Unary operations involve only one value, such asnegationandtrigonometric functions.[3]Binary operations, on the other hand, take two values, and includeaddition,subtraction,multiplication,division, andexponentiation.[4]
Operations can involve mathematical objects other than numbers. Thelogical valuestrueandfalsecan be combined usinglogic operations, such asand,or,andnot.Vectorscan be added and subtracted.[5]Rotationscan be combined using thefunction compositionoperation, performing the first rotation and then the second. Operations onsetsinclude the binary operationsunionandintersectionand the unary operation ofcomplementation.[6][7][8]Operations onfunctionsincludecompositionandconvolution.[9][10]
Operations may not be defined for every possible value of itsdomain. For example, in the real numbers one cannot divide by zero[11]or take square roots of negative numbers. The values for which an operation is defined form a set called itsdomain of definitionoractive domain. The set which contains the values produced is called thecodomain, but the set of actual values attained by the operation is its codomain of definition, active codomain,imageorrange.[12]For example, in the real numbers, the squaring operation only produces non-negative numbers; the codomain is the set of real numbers, but the range is the non-negative numbers.
Operations can involve dissimilar objects: a vector can be multiplied by ascalarto form another vector (an operation known asscalar multiplication),[13]and theinner productoperation on two vectors produces a quantity that is scalar.[14][15]An operation may or may not have certain properties, for example it may beassociative,commutative,anticommutative,idempotent, and so on.
The values combined are calledoperands,arguments, orinputs, and the value produced is called thevalue,result, oroutput. Operations can have fewer or more than two inputs (including the case of zero input and infinitely many inputs[1]).
Anoperatoris similar to an operation in that it refers to the symbol or the process used to denote the operation. Hence, their point of view is different. For instance, one often speaks of "the operation of addition" or "the addition operation," when focusing on the operands and result, but one switch to "addition operator" (rarely "operator of addition"), when focusing on the process, or from the more symbolic viewpoint, the function+:X×X→X(where X is a set such as the set of real numbers).
Ann-ary operationωon asetXis afunctionω:Xn→X. The setXnis called thedomainof the operation, the output set is called thecodomainof the operation, and the fixed non-negative integern(the number of operands) is called thearityof the operation. Thus aunary operationhas arity one, and abinary operationhas arity two. An operation of arity zero, called anullaryoperation, is simply an element of the codomainY. Ann-ary operation can also be viewed as an(n+ 1)-aryrelationthat istotalon itsninput domains anduniqueon its output domain.
Ann-ary partial operationωfromXntoXis apartial functionω:Xn→X. Ann-ary partial operation can also be viewed as an(n+ 1)-ary relation that is unique on its output domain.
The above describes what is usually called afinitary operation, referring to the finite number of operands (the valuen). There are obvious extensions where the arity is taken to be an infiniteordinalorcardinal,[1]or even an arbitrary set indexing the operands.
Often, the use of the termoperationimplies that the domain of the function includes a power of the codomain (i.e. theCartesian productof one or more copies of the codomain),[16]although this is by no means universal, as in the case ofdot product, where vectors are multiplied and result in a scalar. Ann-ary operationω:Xn→Xis called aninternal operation. Ann-ary operationω:Xi×S×Xn−i− 1→Xwhere0 ≤i<nis called anexternal operationby thescalar setoroperator setS. In particular for a binary operation,ω:S×X→Xis called aleft-external operationbyS, andω:X×S→Xis called aright-external operationbyS. An example of an internal operation isvector addition, where two vectors are added and result in a vector. An example of an external operation isscalar multiplication, where a vector is multiplied by a scalar and result in a vector.
Ann-ary multifunctionormultioperationωis a mapping from a Cartesian power of a set into the set of subsets of that set, formallyω:Xn→P(X){\displaystyle \omega :X^{n}\rightarrow {\mathcal {P}}(X)}.[17] | https://en.wikipedia.org/wiki/Operation_(mathematics) |
Symmetryoccurs not only ingeometry, but also in other branches of mathematics. Symmetry is a type ofinvariance: the property that a mathematical object remains unchanged under a set ofoperationsortransformations.[1]
Given a structured objectXof any sort, asymmetryis amappingof the object onto itself which preserves the structure. This can occur in many ways; for example, ifXis a set with no additional structure, a symmetry is abijectivemap from the set to itself, giving rise topermutation groups. If the objectXis a set of points in the plane with itsmetricstructure or any othermetric space, a symmetry is abijectionof the set to itself which preserves the distance between each pair of points (i.e., anisometry).
In general, every kind of structure in mathematics will have its own kind of symmetry, many of which are listed in the given points mentioned above.
The types of symmetry considered in basic geometry includereflectional symmetry,rotational symmetry,translational symmetryandglide reflection symmetry, which are described more fully in the main articleSymmetry (geometry).
Letf(x) be areal-valued function of a real variable, thenfisevenif the following equation holds for allxand-xin the domain off:
Geometrically speaking, the graph face of an even function issymmetricwith respect to they-axis, meaning that itsgraphremains unchanged afterreflectionabout they-axis. Examples of even functions include|x|,x2,x4,cos(x), andcosh(x).
Again, letfbe areal-valued function of a real variable, thenfisoddif the following equation holds for allxand-xin the domain off:
That is,
Geometrically, the graph of an odd function has rotational symmetry with respect to theorigin, meaning that itsgraphremains unchanged afterrotationof 180degreesabout the origin. Examples of odd functions arex,x3,sin(x),sinh(x), anderf(x).
Theintegralof an odd function from −Ato +Ais zero, provided thatAis finite and that the function is integrable (e.g., has no vertical asymptotes between −AandA).[3]
The integral of an even function from −Ato +Ais twice the integral from 0 to +A, provided thatAis finite and the function is integrable (e.g., has no vertical asymptotes between −AandA).[3]This also holds true whenAis infinite, but only if the integral converges.
Inlinear algebra, asymmetric matrixis asquare matrixthat is equal to itstranspose(i.e., it is invariant under matrix transposition). Formally, matrixAis symmetric if
By the definition of matrix equality, which requires that the entries in all corresponding positions be equal, equal matrices must have the same dimensions (as matrices of different sizes or shapes cannot be equal). Consequently, only square matrices can be symmetric.
The entries of a symmetric matrix are symmetric with respect to themain diagonal. So if the entries are written asA= (aij), thenaij= aji, for all indicesiandj.
For example, the following 3×3 matrix is symmetric:
Every squarediagonal matrixis symmetric, since all off-diagonal entries are zero. Similarly, each diagonal element of askew-symmetric matrixmust be zero, since each is its own negative.
In linear algebra, arealsymmetric matrix represents aself-adjoint operatorover arealinner product space. The corresponding object for acomplexinner product space is aHermitian matrixwith complex-valued entries, which is equal to itsconjugate transpose. Therefore, in linear algebra over the complex numbers, it is often assumed that a symmetric matrix refers to one which has real-valued entries. Symmetric matrices appear naturally in a variety of applications, and typical numerical linear algebra software makes special accommodations for them.
Thesymmetric groupSn(on afinite setofnsymbols) is thegroupwhose elements are all thepermutationsof thensymbols, and whosegroup operationis thecompositionof such permutations, which are treated asbijective functionsfrom the set of symbols to itself.[4]Since there aren! (nfactorial) possible permutations of a set ofnsymbols, it follows that theorder(i.e., the number of elements) of the symmetric groupSnisn!.
Asymmetric polynomialis apolynomialP(X1,X2, ...,Xn) innvariables, such that if any of the variables are interchanged, one obtains the same polynomial. Formally,Pis asymmetric polynomialif for anypermutationσ of the subscripts 1, 2, ...,n, one hasP(Xσ(1),Xσ(2), ...,Xσ(n)) =P(X1,X2, ...,Xn).
Symmetric polynomials arise naturally in the study of the relation between the roots of a polynomial in one variable and its coefficients, since the coefficients can be given by polynomial expressions in the roots, and all roots play a similar role in this setting. From this point of view, theelementary symmetric polynomialsare the most fundamental symmetric polynomials. Atheoremstates that any symmetric polynomial can be expressed in terms of elementary symmetric polynomials, which implies that everysymmetricpolynomial expressionin the roots of amonic polynomialcan alternatively be given as a polynomial expression in the coefficients of the polynomial.
In two variablesX1andX2, one has symmetric polynomials such as:
and in three variablesX1,X2andX3, one has as a symmetric polynomial:
Inmathematics, asymmetric tensoristensorthat is invariant under apermutationof its vector arguments:
for every permutation σ of the symbols {1,2,...,r}.
Alternatively, anrthorder symmetric tensor represented in coordinates as a quantity withrindices satisfies
The space of symmetric tensors of rankron a finite-dimensionalvector spaceisnaturally isomorphicto the dual of the space ofhomogeneous polynomialsof degreeronV. Overfieldsofcharacteristic zero, thegraded vector spaceof all symmetric tensors can be naturally identified with thesymmetric algebraonV. A related concept is that of theantisymmetric tensororalternating form. Symmetric tensors occur widely inengineering,physicsandmathematics.
Given a polynomial, it may be that some of the roots are connected by variousalgebraic equations. For example, it may be that for two of the roots, sayAandB, thatA2+ 5B3= 7. The central idea of Galois theory is to consider thosepermutations(or rearrangements) of the roots having the property thatanyalgebraic equation satisfied by the roots isstill satisfiedafter the roots have been permuted. An important proviso is that we restrict ourselves to algebraic equations whose coefficients arerational numbers. Thus, Galois theory studies the symmetries inherent in algebraic equations.
Inabstract algebra, anautomorphismis anisomorphismfrom amathematical objectto itself. It is, in some sense, asymmetryof the object, and a way ofmappingthe object to itself while preserving all of its structure. The set of all automorphisms of an object forms agroup, called theautomorphism group. It is, loosely speaking, thesymmetry groupof the object.
In quantum mechanics, bosons have representatives that are symmetric under permutation operators, and fermions have antisymmetric representatives.
This implies the Pauli exclusion principle for fermions. In fact, the Pauli exclusion principle with a single-valued many-particle wavefunction is equivalent to requiring the wavefunction to be antisymmetric. An antisymmetric two-particle state is represented as asum of statesin which one particle is in state|x⟩{\displaystyle \scriptstyle |x\rangle }and the other in state|y⟩{\displaystyle \scriptstyle |y\rangle }:
and antisymmetry under exchange means thatA(x,y) = −A(y,x). This implies thatA(x,x) = 0, which is Pauli exclusion. It is true in any basis, since unitary changes of basis keep antisymmetric matrices antisymmetric, although strictly speaking, the quantityA(x,y)is not a matrix but an antisymmetric rank-twotensor.
Conversely, if the diagonal quantitiesA(x,x)are zeroin every basis, then the wavefunction component:
is necessarily antisymmetric. To prove it, consider the matrix element:
This is zero, because the two particles have zero probability to both be in the superposition state|x⟩+|y⟩{\displaystyle \scriptstyle |x\rangle +|y\rangle }. But this is equal to
The first and last terms on the right hand side are diagonal elements and are zero, and the whole sum is equal to zero. So the wavefunction matrix elements obey:
or
We call a relation symmetric if every time the relation stands from A to B, it stands too from B to A.
Note that symmetry is not the exact opposite ofantisymmetry.
Anisometryis adistance-preserving map betweenmetric spaces. Given a metric space, or a set and scheme for assigning distances between elements of the set, an isometry is a transformation which maps elements to another metric space such that the distance between the elements in the new metric space is equal to the distance between the elements in the original metric space. In a two-dimensional or three-dimensional space, two geometric figures arecongruentif they are related by an isometry: related by either arigid motion, or acompositionof a rigid motion and areflection. Up to a relation by a rigid motion, they are equal if related by adirect isometry.
Isometries have been used to unify the working definition of symmetry in geometry and for functions, probability distributions, matrices, strings, graphs, etc.[7]
A symmetry of adifferential equationis a transformation that leaves the differential equation invariant. Knowledge of such symmetries may help solve the differential equation.
ALine symmetryof asystem of differential equationsis a continuous symmetry of the system of differential equations. Knowledge of a Line symmetry can be used to simplify an ordinary differential equation throughreduction of order.[8]
Forordinary differential equations, knowledge of an appropriate set of Lie symmetries allows one to explicitly calculate a set of first integrals, yielding a complete solution without integration.
Symmetries may be found by solving a related set of ordinary differential equations.[8]Solving these equations is often much simpler than solving the original differential equations.
In the case of a finite number of possible outcomes, symmetry with respect to permutations (relabelings) implies adiscrete uniform distribution.
In the case of a real interval of possible outcomes, symmetry with respect to interchanging sub-intervals of equal length corresponds to acontinuous uniform distribution.
In other cases, such as "taking a random integer" or "taking a random real number", there are no probability distributions at all symmetric with respect to relabellings or to exchange of equally long subintervals. Other reasonable symmetries do not single out one particular distribution, or in other words, there is not a unique probability distribution providing maximum symmetry.
There is one type ofisometry in one dimensionthat may leave the probability distribution unchanged, that is reflection in a point, for example zero.
A possible symmetry for randomness with positive outcomes is that the former applies for the logarithm, i.e., the outcome and its reciprocal have the same distribution. However this symmetry does not single out any particular distribution uniquely.
For a "random point" in a plane or in space, one can choose an origin, and consider a probability distribution with circular or spherical symmetry, respectively. | https://en.wikipedia.org/wiki/Symmetry_in_mathematics |
Inphysics,canonical quantizationis a procedure forquantizingaclassical theory, while attempting to preserve the formal structure, such assymmetries, of the classical theory to the greatest extent possible.
Historically, this was not quiteWerner Heisenberg's route to obtainingquantum mechanics, butPaul Diracintroduced it in his 1926 doctoral thesis, the "method of classical analogy" for quantization,[1]and detailed it in his classic textPrinciples of Quantum Mechanics.[2]The wordcanonicalarises from theHamiltonianapproach to classical mechanics, in which a system's dynamics is generated via canonicalPoisson brackets, a structure which isonly partially preservedin canonical quantization.
This method was further used by Paul Dirac in the context ofquantum field theory, in his construction ofquantum electrodynamics. In the field theory context, it is also called thesecond quantizationof fields, in contrast to the semi-classicalfirst quantizationof single particles.
When it was first developed,quantum physicsdealt only with thequantizationof themotionof particles, leaving theelectromagnetic fieldclassical, hence the namequantum mechanics.[3]
Later the electromagnetic field was also quantized, and even the particles themselves became represented through quantized fields, resulting in the development ofquantum electrodynamics(QED) andquantum field theoryin general.[4]Thus, by convention, the original form of particle quantum mechanics is denotedfirst quantization, while quantum field theory is formulated in the language ofsecond quantization.
The following exposition is based onDirac'streatise on quantum mechanics.[2]In theclassical mechanicsof a particle, there are dynamic variables which are called coordinates (x) and momenta (p). These specify thestateof a classical system. Thecanonical structure(also known as thesymplecticstructure) ofclassical mechanicsconsists ofPoisson bracketsenclosing these variables, such as{x,p} = 1. All transformations of variables which preserve these brackets are allowed ascanonical transformationsin classical mechanics. Motion itself is such a canonical transformation.
By contrast, inquantum mechanics, all significant features of a particle are contained in astate|ψ⟩{\displaystyle |\psi \rangle }, called aquantum state. Observables are represented byoperatorsacting on aHilbert spaceof suchquantum states.
The eigenvalue of an operator acting on one of its eigenstates represents the value of a measurement on the particle thus represented. For example, theenergyis read off by theHamiltonianoperatorH^{\displaystyle {\hat {H}}}acting on a state|ψn⟩{\displaystyle |\psi _{n}\rangle }, yieldingH^|ψn⟩=En|ψn⟩,{\displaystyle {\hat {H}}|\psi _{n}\rangle =E_{n}|\psi _{n}\rangle ,}whereEnis the characteristic energy associated to this|ψn⟩{\displaystyle |\psi _{n}\rangle }eigenstate.
Any state could be represented as alinear combinationof eigenstates of energy; for example,|ψ⟩=∑n=0∞an|ψn⟩,{\displaystyle |\psi \rangle =\sum _{n=0}^{\infty }a_{n}|\psi _{n}\rangle ,}whereanare constant coefficients.
As in classical mechanics, all dynamical operators can be represented by functions of the position and momentum ones,X^{\displaystyle {\hat {X}}}andP^{\displaystyle {\hat {P}}}, respectively. The connection between this representation and the more usualwavefunctionrepresentation is given by the eigenstate of the position operatorX^{\displaystyle {\hat {X}}}representing a particle at positionx{\displaystyle x}, which is denoted by an element|x⟩{\displaystyle |x\rangle }in the Hilbert space, and which satisfiesX^|x⟩=x|x⟩{\displaystyle {\hat {X}}|x\rangle =x|x\rangle }. Then,ψ(x)=⟨x|ψ⟩{\displaystyle \psi (x)=\langle x|\psi \rangle }.
Likewise, the eigenstates|p⟩{\displaystyle |p\rangle }of the momentum operatorP^{\displaystyle {\hat {P}}}specify themomentum representation:ψ(p)=⟨p|ψ⟩{\displaystyle \psi (p)=\langle p|\psi \rangle }.
The central relation between these operators is a quantum analog of the abovePoisson bracketof classical mechanics, thecanonical commutation relation,[X^,P^]=X^P^−P^X^=iℏ.{\displaystyle [{\hat {X}},{\hat {P}}]={\hat {X}}{\hat {P}}-{\hat {P}}{\hat {X}}=i\hbar .}
This relation encodes (and formally leads to) theuncertainty principle, in the formΔxΔp≥ħ/2. This algebraic structure may be thus considered as the quantum analog of thecanonical structureof classical mechanics.
When turning to N-particle systems, i.e., systems containing Nidentical particles(particles characterized by the samequantum numberssuch asmass,chargeandspin), it is necessary to extend the single-particle state functionψ(r){\displaystyle \psi (\mathbf {r} )}to the N-particle state functionψ(r1,r2,…,rN){\displaystyle \psi (\mathbf {r} _{1},\mathbf {r} _{2},\dots ,\mathbf {r} _{N})}. A fundamental difference between classical and quantum mechanics concerns the concept ofindistinguishabilityof identical particles. Only two species of particles are thus possible in quantum physics, the so-calledbosonsandfermionswhich obey the following rules for each kind of particle:
where we have interchanged two coordinates(rj,rk){\displaystyle (\mathbf {r} _{j},\mathbf {r} _{k})}of the state function. The usual wave function is obtained using theSlater determinantand theidentical particlestheory. Using this basis, it is possible to solve various many-particle problems.
Dirac's book[2]details his popular rule of supplantingPoisson bracketsbycommutators:
{A,B}⟼1iℏ[A^,B^].{\displaystyle \{A,B\}\longmapsto {\tfrac {1}{i\hbar }}[{\hat {A}},{\hat {B}}]~.}
One might interpret this proposal as saying that we should seek a "quantization map"Q{\displaystyle Q}mapping a functionf{\displaystyle f}on the classical phase space to an operatorQf{\displaystyle Q_{f}}on the quantum Hilbert space such thatQ{f,g}=1iℏ[Qf,Qg]{\displaystyle Q_{\{f,g\}}={\frac {1}{i\hbar }}[Q_{f},Q_{g}]}It is now known that there is no reasonable such quantization map satisfying the above identity exactly for all functionsf{\displaystyle f}andg{\displaystyle g}.[citation needed]
One concrete version of the above impossibility claim is Groenewold's theorem (after Dutch theoretical physicistHilbrand J. Groenewold), which we describe for a system with one degree of freedom for simplicity. Let us accept the following "ground rules" for the mapQ{\displaystyle Q}. First,Q{\displaystyle Q}should send the constant function 1 to the identity operator. Second,Q{\displaystyle Q}should takex{\displaystyle x}andp{\displaystyle p}to the usual position and momentum operatorsX{\displaystyle X}andP{\displaystyle P}. Third,Q{\displaystyle Q}should take a polynomial inx{\displaystyle x}andp{\displaystyle p}to a "polynomial" inX{\displaystyle X}andP{\displaystyle P}, that is, a finite linear combinations of products ofX{\displaystyle X}andP{\displaystyle P}, which may be taken in any desired order. In its simplest form, Groenewold's theorem says that there is no map satisfying the above ground rules and also the bracket conditionQ{f,g}=1iℏ[Qf,Qg]{\displaystyle Q_{\{f,g\}}={\frac {1}{i\hbar }}[Q_{f},Q_{g}]}for all polynomialsf{\displaystyle f}andg{\displaystyle g}.
Actually, the nonexistence of such a map occurs already by the time we reach polynomials of degree four. Note that the Poisson bracket of two polynomials of degree four has degree six, so it does not exactly make sense to require a map on polynomials of degree four to respect the bracket condition. Wecan, however, require that the bracket condition holds whenf{\displaystyle f}andg{\displaystyle g}have degree three. Groenewold's theorem[5]can be stated as follows:
Theorem—There is no quantization mapQ{\displaystyle Q}(following the above ground rules) on polynomials of degree less than or equal to four that satisfiesQ{f,g}=1iℏ[Qf,Qg]{\displaystyle Q_{\{f,g\}}={\frac {1}{i\hbar }}[Q_{f},Q_{g}]}wheneverf{\displaystyle f}andg{\displaystyle g}have degree less than or equal to three. (Note that in this case,{f,g}{\displaystyle \{f,g\}}has degree less than or equal to four.)
The proof can be outlined as follows.[6][7]Suppose we first try to find a quantization map on polynomials of degree less than or equal to three satisfying the bracket condition wheneverf{\displaystyle f}has degree less than or equal to two andg{\displaystyle g}has degree less than or equal to two. Then there is precisely one such map, and it is theWeyl quantization. The impossibility result now is obtained by writing the same polynomial of degree four as a Poisson bracket of polynomials of degree threein two different ways. Specifically, we havex2p2=19{x3,p3}=13{x2p,xp2}{\displaystyle x^{2}p^{2}={\frac {1}{9}}\{x^{3},p^{3}\}={\frac {1}{3}}\{x^{2}p,xp^{2}\}}On the other hand, we have already seen that if there is going to be a quantization map on polynomials of degree three, it must be the Weyl quantization; that is, we have already determined the only possible quantization of all the cubic polynomials above.
The argument is finished by computing by brute force that19[Q(x3),Q(p3)]{\displaystyle {\frac {1}{9}}[Q(x^{3}),Q(p^{3})]}does not coincide with13[Q(x2p),Q(xp2)].{\displaystyle {\frac {1}{3}}[Q(x^{2}p),Q(xp^{2})].}Thus, we have two incompatible requirements for the value ofQ(x2p2){\displaystyle Q(x^{2}p^{2})}.
IfQrepresents the quantization map that acts on functionsfin classical phase space, then the following properties are usually considered desirable:[8]
However, not only are these four properties mutually inconsistent,any threeof them are also inconsistent![9]As it turns out, the only pairs of these properties that lead to self-consistent, nontrivial solutions are 2 & 3, and possibly 1 & 3 or 1 & 4. Accepting properties 1 & 2, along with a weaker condition that 3 be true only asymptotically in the limitħ→0(seeMoyal bracket), leads todeformation quantization, and some extraneous information must be provided, as in the standard theories utilized in most of physics. Accepting properties 1 & 2 & 3 but restricting the space of quantizable observables to exclude terms such as the cubic ones in the above example amounts togeometric quantization.
Quantum mechanicswas successful at describing non-relativistic systems with fixed numbers of particles, but a new framework was needed to describe systems in which particles can be created or destroyed, for example, the electromagnetic field, considered as a collection of photons. It was eventually realized thatspecial relativitywas inconsistent with single-particle quantum mechanics, so that all particles are now described relativistically byquantum fields.
When the canonical quantization procedure is applied to a field, such as the electromagnetic field, the classicalfieldvariables becomequantum operators. Thus, the normal modes comprising the amplitude of the field are simple oscillators, each of which isquantizedin standard first quantization, above, without ambiguity. The resulting quanta are identified with individual particles or excitations. For example, the quanta of the electromagnetic field are identified with photons. Unlike first quantization, conventional second quantization is completely unambiguous, in effect afunctor, since the constituent set of its oscillators are quantized unambiguously.
Historically, quantizing the classical theory of a single particle gave rise to a wavefunction. The classical equations of motion of a field are typically identical in form to the (quantum) equations for the wave-function ofone of its quanta. For example, theKlein–Gordon equationis the classical equation of motion for a free scalar field, but also the quantum equation for a scalar particle wave-function. This meant that quantizing a fieldappearedto be similar to quantizing a theory that was already quantized, leading to the fanciful termsecond quantizationin the early literature, which is still used to describe field quantization, even though the modern interpretation detailed is different.
One drawback to canonical quantization for a relativistic field is that by relying on the Hamiltonian to determine time dependence,relativistic invarianceis no longer manifest. Thus it is necessary to check thatrelativistic invarianceis not lost. Alternatively, theFeynman integral approachis available for quantizing relativistic fields, and is manifestly invariant. For non-relativistic field theories, such as those used incondensed matter physics, Lorentz invariance is not an issue.
Quantum mechanically, the variables of a field (such as the field's amplitude at a given point) are represented by operators on aHilbert space. In general, all observables are constructed as operators on the Hilbert space, and the time-evolution of the operators is governed by theHamiltonian, which must be a positive operator. A state|0⟩{\displaystyle |0\rangle }annihilated by the Hamiltonian must be identified as thevacuum state, which is the basis for building all other states. In a non-interacting (free) field theory, the vacuum is normally identified as a state containing zero particles. In a theory with interacting particles, identifying the vacuum is more subtle, due tovacuum polarization, which implies that the physical vacuum in quantum field theory is never really empty. For further elaboration, see the articles onthe quantum mechanical vacuumandthe vacuum of quantum chromodynamics. The details of the canonical quantization depend on the field being quantized, and whether it is free or interacting.
Ascalar field theoryprovides a good example of the canonical quantization procedure.[10]Classically, a scalar field is a collection of an infinity ofoscillatornormal modes. It suffices to consider a 1+1-dimensional space-timeR×S1,{\displaystyle \mathbb {R} \times S_{1},}in which the spatial direction iscompactifiedto a circle of circumference 2π, rendering the momenta discrete.
The classicalLagrangiandensity describes aninfinity of coupled harmonic oscillators, labelled byxwhich is now alabel(and not the displacement dynamical variable to be quantized), denoted by the classical fieldφ,L(ϕ)=12(∂tϕ)2−12(∂xϕ)2−12m2ϕ2−V(ϕ),{\displaystyle {\mathcal {L}}(\phi )={\tfrac {1}{2}}(\partial _{t}\phi )^{2}-{\tfrac {1}{2}}(\partial _{x}\phi )^{2}-{\tfrac {1}{2}}m^{2}\phi ^{2}-V(\phi ),}whereV(φ)is a potential term, often taken to be a polynomial or monomial of degree 3 or higher. The action functional isS(ϕ)=∫L(ϕ)dxdt=∫L(ϕ,∂tϕ)dt.{\displaystyle S(\phi )=\int {\mathcal {L}}(\phi )dxdt=\int L(\phi ,\partial _{t}\phi )dt\,.}The canonical momentum obtained via theLegendre transformationusing the actionLisπ=∂tϕ{\displaystyle \pi =\partial _{t}\phi }, and the classicalHamiltonianis found to beH(ϕ,π)=∫dx[12π2+12(∂xϕ)2+12m2ϕ2+V(ϕ)].{\displaystyle H(\phi ,\pi )=\int dx\left[{\tfrac {1}{2}}\pi ^{2}+{\tfrac {1}{2}}(\partial _{x}\phi )^{2}+{\tfrac {1}{2}}m^{2}\phi ^{2}+V(\phi )\right].}
Canonical quantization treats the variablesφandπas operators withcanonical commutation relationsat timet= 0, given by[ϕ(x),ϕ(y)]=0,[π(x),π(y)]=0,[ϕ(x),π(y)]=iℏδ(x−y).{\displaystyle [\phi (x),\phi (y)]=0,\ \ [\pi (x),\pi (y)]=0,\ \ [\phi (x),\pi (y)]=i\hbar \delta (x-y).}Operators constructed fromφandπcan then formally be defined at other times via the time-evolution generated by the Hamiltonian,O(t)=eitHOe−itH.{\displaystyle {\mathcal {O}}(t)=e^{itH}{\mathcal {O}}e^{-itH}.}
However, sinceφandπno longer commute, this expression is ambiguous at the quantum level. The problem is to construct a representation of the relevant operatorsO{\displaystyle {\mathcal {O}}}on aHilbert spaceH{\displaystyle {\mathcal {H}}}and to construct a positive operatorHas aquantum operatoron this Hilbert space in such a way that it gives this evolution for the operatorsO{\displaystyle {\mathcal {O}}}as given by the preceding equation, and to show thatH{\displaystyle {\mathcal {H}}}contains a vacuum state|0⟩{\displaystyle |0\rangle }on whichHhas zero eigenvalue. In practice, this construction is a difficult problem for interacting field theories, and has been solved completely only in a few simple cases via the methods ofconstructive quantum field theory. Many of these issues can be sidestepped using the Feynman integral as described for a particularV(φ)in the article onscalar field theory.
In the case of a free field, withV(φ) = 0, the quantization procedure is relatively straightforward. It is convenient toFourier transformthe fields, so thatϕk=∫ϕ(x)e−ikxdx,πk=∫π(x)e−ikxdx.{\displaystyle \phi _{k}=\int \phi (x)e^{-ikx}dx,\ \ \pi _{k}=\int \pi (x)e^{-ikx}dx.}The reality of the fields implies thatϕ−k=ϕk†,π−k=πk†.{\displaystyle \phi _{-k}=\phi _{k}^{\dagger },~~~\pi _{-k}=\pi _{k}^{\dagger }.}The classical Hamiltonian may be expanded in Fourier modes asH=12∑k=−∞∞[πkπk†+ωk2ϕkϕk†],{\displaystyle H={\frac {1}{2}}\sum _{k=-\infty }^{\infty }\left[\pi _{k}\pi _{k}^{\dagger }+\omega _{k}^{2}\phi _{k}\phi _{k}^{\dagger }\right],}whereωk=k2+m2{\displaystyle \omega _{k}={\sqrt {k^{2}+m^{2}}}}.
This Hamiltonian is thus recognizable as an infinite sum of classicalnormal modeoscillator excitationsφk, each one of which is quantized in thestandardmanner, so the free quantum Hamiltonian looks identical. It is theφks that have become operators obeying the standard commutation relations,[φk,πk†] = [φk†,πk] =iħ, with all others vanishing. The collective Hilbert space of all these oscillators is thus constructed using creation and annihilation operators constructed from these modes,ak=12ℏωk(ωkϕk+iπk),ak†=12ℏωk(ωkϕk†−iπk†),{\displaystyle a_{k}={\frac {1}{\sqrt {2\hbar \omega _{k}}}}\left(\omega _{k}\phi _{k}+i\pi _{k}\right),\ \ a_{k}^{\dagger }={\frac {1}{\sqrt {2\hbar \omega _{k}}}}\left(\omega _{k}\phi _{k}^{\dagger }-i\pi _{k}^{\dagger }\right),}for which[ak,ak†] = 1for allk, with all other commutators vanishing.
The vacuum|0⟩{\displaystyle |0\rangle }is taken to be annihilated by all of theak, andH{\displaystyle {\mathcal {H}}}is the Hilbert space constructed by applying any combination of the infinite collection of creation operatorsak†to|0⟩{\displaystyle |0\rangle }. This Hilbert space is calledFock space. For eachk, this construction is identical to aquantum harmonic oscillator. The quantum field is an infinite array of quantum oscillators. The quantum Hamiltonian then amounts toH=∑k=−∞∞ℏωkak†ak=∑k=−∞∞ℏωkNk,{\displaystyle H=\sum _{k=-\infty }^{\infty }\hbar \omega _{k}a_{k}^{\dagger }a_{k}=\sum _{k=-\infty }^{\infty }\hbar \omega _{k}N_{k},}whereNkmay be interpreted as thenumber operatorgiving thenumber of particlesin a state with momentumk.
This Hamiltonian differs from the previous expression by the subtraction of the zero-point energyħωk/2of each harmonic oscillator. This satisfies the condition thatHmust annihilate the vacuum, without affecting the time-evolution of operators via the above exponentiation operation. This subtraction of the zero-point energy may be considered to be a resolution of the quantum operator ordering ambiguity, since it is equivalent to requiring thatall creation operators appear to the left of annihilation operatorsin the expansion of the Hamiltonian. This procedure is known asWick orderingornormal ordering.
All other fields can be quantized by a generalization of this procedure. Vector or tensor fields simply have more components, and independent creation and destruction operators must be introduced for each independent component. If a field has anyinternal symmetry, then creation and destruction operators must be introduced for each component of the field related to this symmetry as well. If there is agauge symmetry, then the number of independent components of the field must be carefully analyzed to avoid over-counting equivalent configurations, andgauge-fixingmay be applied if needed.
It turns out that commutation relations are useful only for quantizingbosons, for which the occupancy number of any state is unlimited. To quantizefermions, which satisfy thePauli exclusion principle, anti-commutators are needed. These are defined by{A,B} =AB+BA.
When quantizing fermions, the fields are expanded in creation and annihilation operators,θk†,θk, which satisfy{θk,θl†}=δkl,{θk,θl}=0,{θk†,θl†}=0.{\displaystyle \{\theta _{k},\theta _{l}^{\dagger }\}=\delta _{kl},\ \ \{\theta _{k},\theta _{l}\}=0,\ \ \{\theta _{k}^{\dagger },\theta _{l}^{\dagger }\}=0.}
The states are constructed on a vacuum|0⟩{\displaystyle |0\rangle }annihilated by theθk, and theFock spaceis built by applying all products of creation operatorsθk†to|0⟩. Pauli's exclusion principle is satisfied, because(θk†)2|0⟩=0{\displaystyle (\theta _{k}^{\dagger })^{2}|0\rangle =0}, by virtue of the anti-commutation relations.
The construction of the scalar field states above assumed that the potential was minimized atφ= 0, so that the vacuum minimizing the Hamiltonian satisfies⟨φ⟩ = 0, indicating that thevacuum expectation value(VEV) of the field is zero. In cases involvingspontaneous symmetry breaking, it is possible to have a non-zero VEV, because the potential is minimized for a valueφ=v. This occurs for example, ifV(φ) =gφ4− 2m2φ2withg> 0andm2> 0, for which the minimum energy is found atv= ±m/√g. The value ofvin one of these vacua may be considered ascondensateof the fieldφ. Canonical quantization then can be carried out for theshifted fieldφ(x,t) −v, and particle states with respect to the shifted vacuum are defined by quantizing the shifted field. This construction is utilized in theHiggs mechanismin thestandard modelofparticle physics.
The classical theory is described using aspacelikefoliationofspacetimewith the state at each slice being described by an element of asymplectic manifoldwith the time evolution given by thesymplectomorphismgenerated by aHamiltonianfunction over the symplectic manifold. Thequantum algebraof "operators" is anħ-deformation of the algebra of smooth functionsover the symplectic space such that theleading termin the Taylor expansion overħof thecommutator[A,B]expressed in thephase space formulationisiħ{A,B}. (Here, the curly braces denote thePoisson bracket. The subleading terms are all encoded in theMoyal bracket, the suitable quantum deformation of the Poisson bracket.) In general, for the quantities (observables) involved,
and providing the arguments of such brackets,ħ-deformations are highly nonunique—quantization is an "art", and is specified by the physical context.
(Twodifferentquantum systems may represent two different, inequivalent, deformations of the sameclassical limit,ħ→ 0.)
Now, one looks forunitary representationsof this quantum algebra. With respect to such a unitary representation, a symplectomorphism in the classical theory would now deform to a (metaplectic)unitary transformation. In particular, the time evolution symplectomorphism generated by the classical Hamiltonian deforms to a unitary transformation generated by the corresponding quantum Hamiltonian.
A further generalization is to consider aPoisson manifoldinstead of a symplectic space for the classical theory and perform anħ-deformation of the correspondingPoisson algebraor evenPoisson supermanifolds.
In contrast to the theory of deformation quantization described above, geometric quantization seeks to construct an actual Hilbert space and operators on it. Starting with a symplectic manifoldM{\displaystyle M}, one first constructs a prequantum Hilbert space consisting of the space of square-integrable sections of an appropriate line bundle overM{\displaystyle M}. On this space, one can mapallclassical observables to operators on the prequantum Hilbert space, with the commutator corresponding exactly to the Poisson bracket. The prequantum Hilbert space, however, is clearly too big to describe the quantization ofM{\displaystyle M}.
One then proceeds by choosing a polarization, that is (roughly), a choice ofn{\displaystyle n}variables on the2n{\displaystyle 2n}-dimensional phase space. ThequantumHilbert space is then the space of sections that depend only on then{\displaystyle n}chosen variables, in the sense that they are covariantly constant in the othern{\displaystyle n}directions. If the chosen variables are real, we get something like the traditional Schrödinger Hilbert space. If the chosen variables are complex, we get something like theSegal–Bargmann space. | https://en.wikipedia.org/wiki/Canonical_quantization |
InmathematicsandphysicsCCR algebras(aftercanonical commutation relations) andCAR algebras(after canonical anticommutation relations) arise from thequantum mechanicalstudy ofbosonsandfermions, respectively. They play a prominent role inquantum statistical mechanics[1]andquantum field theory.
LetV{\displaystyle V}be arealvector spaceequipped with anonsingularrealantisymmetricbilinear form(⋅,⋅){\displaystyle (\cdot ,\cdot )}(i.e. asymplectic vector space). Theunital*-algebragenerated by elements ofV{\displaystyle V}subject to the relations
for anyf,g{\displaystyle f,~g}inV{\displaystyle V}is called thecanonical commutation relations (CCR) algebra. The uniqueness of the representations of this algebra whenV{\displaystyle V}isfinite dimensionalis discussed in theStone–von Neumann theorem.
IfV{\displaystyle V}is equipped with anonsingularrealsymmetric bilinear form(⋅,⋅){\displaystyle (\cdot ,\cdot )}instead, the unital *-algebra generated by the elements ofV{\displaystyle V}subject to the relations
for anyf,g{\displaystyle f,~g}inV{\displaystyle V}is called thecanonical anticommutation relations (CAR) algebra.
There is a distinct, but closely related meaning of CCR algebra, called the CCR C*-algebra. LetH{\displaystyle H}be a real symplectic vector space with nonsingular symplectic form(⋅,⋅){\displaystyle (\cdot ,\cdot )}. In the theory ofoperator algebras, the CCR algebra overH{\displaystyle H}is the unitalC*-algebragenerated by elements{W(f):f∈H}{\displaystyle \{W(f):~f\in H\}}subject to
These are called the Weyl form of the canonical commutation relations and, in particular, they imply that eachW(f){\displaystyle W(f)}isunitaryandW(0)=1{\displaystyle W(0)=1}. It is well known that the CCR algebra is a simple (unless the sympletic form is degenerate) non-separable algebra and is unique up to isomorphism.[2]
WhenH{\displaystyle H}is a complexHilbert spaceand(⋅,⋅){\displaystyle (\cdot ,\cdot )}is given by the imaginary part of the inner-product, the CCR algebra isfaithfully representedon thesymmetric Fock spaceoverH{\displaystyle H}by setting
for anyf,g∈H{\displaystyle f,g\in H}. The field operatorsB(f){\displaystyle B(f)}are defined for eachf∈H{\displaystyle f\in H}as thegeneratorof the one-parameter unitary group(W(tf))t∈R{\displaystyle (W(tf))_{t\in \mathbb {R} }}on the symmetric Fock space. These areself-adjointunbounded operators, however they formally satisfy
As the assignmentf↦B(f){\displaystyle f\mapsto B(f)}is real-linear, so the operatorsB(f){\displaystyle B(f)}define a CCR algebra over(H,2Im⟨⋅,⋅⟩){\displaystyle (H,2\operatorname {Im} \langle \cdot ,\cdot \rangle )}in the sense ofSection 1.
LetH{\displaystyle H}be a Hilbert space. In the theory of operator algebras the CAR algebra is the uniqueC*-completionof the complex unital *-algebra generated by elements{b(f),b∗(f):f∈H}{\displaystyle \{b(f),b^{*}(f):~f\in H\}}subject to the relations
for anyf,g∈H{\displaystyle f,g\in H},λ∈C{\displaystyle \lambda \in \mathbb {C} }.
WhenH{\displaystyle H}is separable the CAR algebra is anAF algebraand in the special caseH{\displaystyle H}is infinite dimensional it is often written asM2∞(C){\displaystyle {M_{2^{\infty }}(\mathbb {C} )}}.[3]
LetFa(H){\displaystyle F_{a}(H)}be theantisymmetric Fock spaceoverH{\displaystyle H}and letPa{\displaystyle P_{a}}be the orthogonal projection onto antisymmetric vectors:
The CAR algebra is faithfully represented onFa(H){\displaystyle F_{a}(H)}by setting
for allf,g1,…,gn∈H{\displaystyle f,g_{1},\ldots ,g_{n}\in H}andn∈N{\displaystyle n\in \mathbb {N} }. The fact that these form a C*-algebra is due to the fact that creation and annihilation operators on antisymmetric Fock space are bona-fidebounded operators. Moreover, the field operatorsB(f):=b∗(f)+b(f){\displaystyle B(f):=b^{*}(f)+b(f)}satisfy
giving the relationship withSection 1.
LetV{\displaystyle V}be a realZ2{\displaystyle \mathbb {Z} _{2}}-graded vector spaceequipped with a nonsingular antisymmetric bilinear superform(⋅,⋅){\displaystyle (\cdot ,\cdot )}(i.e.(g,f)=−(−1)|f||g|(f,g){\displaystyle (g,f)=-(-1)^{|f||g|}(f,g)}) such that(f,g){\displaystyle (f,g)}is real if eitherf{\displaystyle f}org{\displaystyle g}is an even element andimaginaryif both of them are odd. The unital *-algebra generated by the elements ofV{\displaystyle V}subject to the relations
for any two pure elementsf,g{\displaystyle f,~g}inV{\displaystyle V}is the obvioussuperalgebrageneralization which unifies CCRs with CARs: if all pure elements are even, one obtains a CCR, while if all pure elements are odd, one obtains a CAR.
In mathematics, the abstract structure of the CCR and CAR algebras, over any field, not just the complex numbers, is studied by the name ofWeylandClifford algebras, where many significant results have accrued. One of these is that thegradedgeneralizations ofWeylandCliffordalgebras allow the basis-free formulation of the canonical commutation and anticommutation relations in terms of a symplectic and a symmetric non-degenerate bilinear form. In addition, the binary elements in this graded Weyl algebra give a basis-free version of the commutation relations of thesymplecticandindefinite orthogonal Lie algebras.[4] | https://en.wikipedia.org/wiki/CCR_and_CAR_algebras |
Conformastatic spacetimesrefer to a special class of static solutions toEinstein's equationingeneral relativity.
Theline elementfor the conformastatic class of solutions in Weyl's canonical coordinates reads[1][2][3][4][5][6](1)ds2=−e2Ψ(ρ,ϕ,z)dt2+e−2Ψ(ρ,ϕ,z)(dρ2+dz2+ρ2dϕ2),{\displaystyle (1)\qquad ds^{2}=-e^{2\Psi (\rho ,\phi ,z)}dt^{2}+e^{-2\Psi (\rho ,\phi ,z)}{\Big (}d\rho ^{2}+dz^{2}+\rho ^{2}d\phi ^{2}{\Big )}\;,}as a solution to the field equation(2)Rab−12Rgab=8πTab.{\displaystyle (2)\qquad R_{ab}-{\frac {1}{2}}Rg_{ab}=8\pi T_{ab}\;.}Eq(1) has only one metric functionΨ(ρ,ϕ,z){\displaystyle \Psi (\rho ,\phi ,z)}to be identified, and for each concreteΨ(ρ,ϕ,z){\displaystyle \Psi (\rho ,\phi ,z)}, Eq(1) would yields aspecificconformastatic spacetime.
In consistency with the conformastatic geometry Eq(1), the electrostatic field would arise from an electrostatic potentialAa{\displaystyle A_{a}}without spatial symmetry:[3][4][5](3)Aa=Φ(ρ,z,ϕ)[dt]a,{\displaystyle (3)\qquad A_{a}=\Phi (\rho ,z,\phi )[dt]_{a}\;,}which would yield the electromagnetic field tensorFab{\displaystyle F_{ab}}by(4)Fab=Ab;a−Aa;b,{\displaystyle (4)\qquad F_{ab}=A_{b\,;a}-A_{a\,;b}\;,}as well as the correspondingstress–energy tensorby(5)Tab(EM)=14π(FacFbc−14gabFcdFcd).{\displaystyle (5)\qquad T_{ab}^{(EM)}={\frac {1}{4\pi }}{\Big (}F_{ac}F_{b}^{\;\;c}-{\frac {1}{4}}g_{ab}F_{cd}F^{cd}{\Big )}\;.}
Plug Eq(1) and Eqs(3)(4)(5) into "trace-free" (R=0)Einstein's field equation, and one could obtain the reduced field equations for the metric functionΨ(ρ,ϕ,z){\displaystyle \Psi (\rho ,\phi ,z)}:[3][5]
(6)∇2Ψ=e−2Ψ∇Φ∇Φ{\displaystyle (6)\qquad \nabla ^{2}\Psi \,=\,e^{-2\Psi }\,\nabla \Phi \,\nabla \Phi }(7)ΨiΨj=e−2ΨΦiΦj{\displaystyle (7)\qquad \Psi _{i}\Psi _{j}=e^{-2\Psi }\Phi _{i}\Phi _{j}}
where∇2=∂ρρ+1ρ∂ρ+1ρ2∂ϕϕ+∂zz{\displaystyle \nabla ^{2}=\partial _{\rho \rho }+{\frac {1}{\rho }}\,\partial _{\rho }+{\frac {1}{\rho ^{2}}}\partial _{\phi \phi }+\partial _{zz}}and∇=∂ρe^ρ+1ρ∂ϕe^ϕ+∂ze^z{\displaystyle \nabla =\partial _{\rho }\,{\hat {e}}_{\rho }+{\frac {1}{\rho }}\partial _{\phi }\,{\hat {e}}_{\phi }+\partial _{z}\,{\hat {e}}_{z}}are respectively the genericLaplaceandgradientoperators. in Eq(7),i,j{\displaystyle i\,,j}run freely over the coordinates[ρ,z,ϕ]{\displaystyle [\rho ,z,\phi ]}.
The extremal Reissner–Nordström spacetime is a typical conformastatic solution. In this case, the metric function is identified as[4][5]
(8)ΨERN=lnLL+M,L=ρ2+z2,{\displaystyle (8)\qquad \Psi _{ERN}\,=\,\ln {\frac {L}{L+M}}\;,\quad L={\sqrt {\rho ^{2}+z^{2}}}\;,}
which put Eq(1) into the concrete form
(9)ds2=−L2(L+M)2dt2+(L+M)2L2(dρ2+dz2+ρ2dφ2).{\displaystyle (9)\qquad ds^{2}=-{\frac {L^{2}}{(L+M)^{2}}}dt^{2}+{\frac {(L+M)^{2}}{L^{2}}}\,{\big (}d\rho ^{2}+dz^{2}+\rho ^{2}d\varphi ^{2}{\big )}\;.}
Applying the transformations
(10)L=r−M,z=(r−M)cosθ,ρ=(r−M)sinθ,{\displaystyle (10)\;\;\quad L=r-M\;,\quad z=(r-M)\cos \theta \;,\quad \rho =(r-M)\sin \theta \;,}
one obtains the usual form of the line element of extremal Reissner–Nordström solution,
(11)ds2=−(1−Mr)2dt2+(1−Mr)−2dr2+r2(dθ2+sin2θdϕ2).{\displaystyle (11)\;\;\quad ds^{2}=-{\Big (}1-{\frac {M}{r}}{\Big )}^{2}dt^{2}+{\Big (}1-{\frac {M}{r}}{\Big )}^{-2}dr^{2}+r^{2}{\Big (}d\theta ^{2}+\sin ^{2}\theta \,d\phi ^{2}{\Big )}\;.}
Some conformastatic solutions have been adopted to describe charged dust disks.[3]
Many solutions, such as the extremal Reissner–Nordström solution discussed above, can be treated as either a conformastatic metric orWeyl metric, so it would be helpful to make a comparison between them. The Weyl spacetimes refer to the static, axisymmetric class of solutions to Einstein's equation, whose line element takes the following form (still in Weyl's canonical coordinates):(12)ds2=−e2ψ(ρ,z)dt2+e2γ(ρ,z)−2ψ(ρ,z)(dρ2+dz2)+e−2ψ(ρ,z)ρ2dϕ2.{\displaystyle (12)\;\;\quad ds^{2}=-e^{2\psi (\rho ,z)}dt^{2}+e^{2\gamma (\rho ,z)-2\psi (\rho ,z)}(d\rho ^{2}+dz^{2})+e^{-2\psi (\rho ,z)}\rho ^{2}d\phi ^{2}\,.}Hence, a Weyl solution become conformastatic if the metric functionγ(ρ,z){\displaystyle \gamma (\rho ,z)}vanishes, and the other metric functionψ(ρ,z){\displaystyle \psi (\rho ,z)}drops the axial symmetry:(13)γ(ρ,z)≡0,ψ(ρ,z)↦Ψ(ρ,ϕ,z).{\displaystyle (13)\;\;\quad \gamma (\rho ,z)\equiv 0\;,\quad \psi (\rho ,z)\mapsto \Psi (\rho ,\phi ,z)\,.}TheWeyl electrovac field equationswould reduce to the following ones withγ(ρ,z){\displaystyle \gamma (\rho ,z)}:
(14.a)∇2ψ=(∇ψ)2{\displaystyle (14.a)\quad \nabla ^{2}\psi =\,(\nabla \psi )^{2}}(14.b)∇2ψ=e−2ψ(∇Φ)2{\displaystyle (14.b)\quad \nabla ^{2}\psi =\,e^{-2\psi }(\nabla \Phi )^{2}}(14.c)ψ,ρ2−ψ,z2=e−2ψ(Φ,ρ2−Φ,z2){\displaystyle (14.c)\quad \psi _{,\,\rho }^{2}-\psi _{,\,z}^{2}=e^{-2\psi }{\big (}\Phi _{,\,\rho }^{2}-\Phi _{,\,z}^{2}{\big )}}(14.d)2ψ,ρψ,z=2e−2ψΦ,ρΦ,z{\displaystyle (14.d)\quad 2\psi _{,\,\rho }\psi _{,\,z}=2e^{-2\psi }\Phi _{,\,\rho }\Phi _{,\,z}}(14.e)∇2Φ=2∇ψ∇Φ,{\displaystyle (14.e)\quad \nabla ^{2}\Phi =\,2\nabla \psi \nabla \Phi \,,}
where∇2=∂ρρ+1ρ∂ρ+∂zz{\displaystyle \nabla ^{2}=\partial _{\rho \rho }+{\frac {1}{\rho }}\,\partial _{\rho }+\partial _{zz}}and∇=∂ρe^ρ+∂ze^z{\displaystyle \nabla =\partial _{\rho }\,{\hat {e}}_{\rho }+\partial _{z}\,{\hat {e}}_{z}}are respectively the reducedcylindrically symmetricLaplace and gradient operators.
It is also noticeable that, Eqs(14) for Weyl areconsistent but not identicalwith the conformastatic Eqs(6)(7) above. | https://en.wikipedia.org/wiki/Conformastatic_spacetimes |
Indifferential geometry, theLie derivative(/liː/LEE), named afterSophus LiebyWładysław Ślebodziński,[1][2]evaluates the change of atensor field(including scalar functions,vector fieldsandone-forms), along theflowdefined by another vector field. This change is coordinate invariant and therefore the Lie derivative is defined on anydifferentiable manifold.
Functions, tensor fields and forms can be differentiated with respect to a vector field. IfTis a tensor field andXis a vector field, then the Lie derivative ofTwith respect toXis denotedLXT{\displaystyle {\mathcal {L}}_{X}T}. Thedifferential operatorT↦LXT{\displaystyle T\mapsto {\mathcal {L}}_{X}T}is aderivationof the algebra oftensor fieldsof the underlying manifold.
The Lie derivative commutes withcontractionand theexterior derivativeondifferential forms.
Although there are many concepts of taking a derivative in differential geometry, they all agree when the expression being differentiated is a function orscalar field. Thus in this case the word "Lie" is dropped, and one simply speaks of the derivative of a function.
The Lie derivative of a vector fieldYwith respect to another vector fieldXis known as the "Lie bracket" ofXandY, and is often denoted [X,Y] instead ofLXY{\displaystyle {\mathcal {L}}_{X}Y}. The space of vector fields forms aLie algebrawith respect to this Lie bracket. The Lie derivative constitutes an infinite-dimensionalLie algebra representationof this Lie algebra, due to the identity
valid for any vector fieldsXandYand any tensor fieldT.
Considering vector fields asinfinitesimal generatorsofflows(i.e. one-dimensionalgroupsofdiffeomorphisms) onM, the Lie derivative is thedifferentialof the representation of thediffeomorphism groupon tensor fields, analogous to Lie algebra representations asinfinitesimal representationsassociated togroup representationinLie grouptheory.
Generalisations exist forspinorfields,fibre bundleswith aconnectionandvector-valued differential forms.
A 'naïve' attempt to define the derivative of atensor fieldwith respect to avector fieldwould be to take thecomponentsof the tensor field and take thedirectional derivativeof each component with respect to the vector field. However, this definition is undesirable because it is not invariant underchanges of coordinate system, e.g. the naive derivative expressed inpolarorspherical coordinatesdiffers from the naive derivative of the components inCartesian coordinates. On an abstractmanifoldsuch a definition is meaningless and ill defined.
Indifferential geometry, there are three main coordinate independent notions of differentiation of tensor fields:
The main difference between the Lie derivative and a derivative with respect to a connection is that the latter derivative of a tensor field with respect to atangent vectoris well-defined even if it is not specified how to extend that tangent vector to a vector field. However, a connection requires the choice of an additional geometric structure (e.g. aRiemannian metricin the case ofLevi-Civita connection, or just an abstractconnection) on the manifold. In contrast, when taking a Lie derivative, no additional structure on the manifold is needed, but it is impossible to talk about the Lie derivative of a tensor field with respect to a single tangent vector, since the value of the Lie derivative of a tensor field with respect to a vector fieldXat a pointpdepends on the value ofXin a neighborhood ofp, not just atpitself. Finally, the exterior derivative of differential forms does not require any additional choices, but is only a well defined derivative of differential forms (including functions), thus excluding vectors and other tensors that are not purely differential forms.
The idea of Lie derivatives is to use a vector field to define a notion of transport (Lie transport). A smooth vector field defines a smooth flow on the manifold, which allows vectors to be transported between two points on the same line of flow (This contrasts with connections, which allows transport between arbitrary points). Intuitively, a vectorY(p){\displaystyle Y(p)}based at pointp{\displaystyle p}is transported by flowing its base point top′{\displaystyle p'}, while flowing its tip pointp+Y(p)δ{\displaystyle p+Y(p)\delta }top′+δp′{\displaystyle p'+\delta p'}.
The Lie derivative may be defined in several equivalent ways. To keep things simple, we begin by defining the Lie derivative acting on scalar functions and vector fields, before moving on to the definition for general tensors.
Defining the derivative of a functionf:M→R{\displaystyle f\colon M\to {\mathbb {R} }}on a manifold takes care because thedifference quotient(f(x+h)−f(x))/h{\displaystyle \textstyle (f(x+h)-f(x))/h}cannot be determined while the displacementx+h{\displaystyle x+h}is undefined.
The Lie derivative of a functionf:M→R{\displaystyle f\colon M\to {\mathbb {R} }}with respect to avector fieldX{\displaystyle X}at a pointp∈M{\displaystyle p\in M}is the function
whereΦXt(p){\displaystyle \Phi _{X}^{t}(p)}is the point to which theflowdefined by the vector fieldX{\displaystyle X}maps the pointp{\displaystyle p}at time instantt.{\displaystyle t.}In the vicinity oft=0,{\displaystyle t=0,}ΦXt(p){\displaystyle \Phi _{X}^{t}(p)}is the unique solution of the system
of first-order autonomous (i.e. time-independent) differential equations, withΦX0(p)=p.{\displaystyle \Phi _{X}^{0}(p)=p.}
SettingLXf=∇Xf{\displaystyle {\mathcal {L}}_{X}f=\nabla _{X}f}identifies the Lie derivative of a function with thedirectional derivative, which is also denoted byX(f):=LXf=∇Xf{\displaystyle X(f):={\mathcal {L}}_{X}f=\nabla _{X}f}.
IfXandYare both vector fields, then the Lie derivative ofYwith respect toXis also known as theLie bracketofXandY, and is sometimes denoted[X,Y]{\displaystyle [X,Y]}. There are several approaches to defining the Lie bracket, all of which are equivalent. We list two definitions here, corresponding to the two definitions of a vector field given above:
The Lie derivative is the speed with which the tensor field changes under the space deformation caused by the flow.
Formally, given a differentiable (time-independent) vector fieldX{\displaystyle X}on a smooth manifoldM,{\displaystyle M,}letΦXt:M→M{\displaystyle \Phi _{X}^{t}:M\to M}be the corresponding local flow. SinceΦXt{\displaystyle \Phi _{X}^{t}}is a local diffeomorphism for eacht{\displaystyle t}, it gives rise to apullback of tensor fields. For covariant tensors, this is just the multi-linear extension of thepullback map
(ΦXt)p∗:TΦXt(p)∗M→Tp∗M,((ΦXt)p∗α)(Y)=α(TpΦXt(Y)),α∈TΦXt(p)∗M,Y∈TpM{\displaystyle \left(\Phi _{X}^{t}\right)_{p}^{*}:T_{\Phi _{X}^{t}(p)}^{*}M\to T_{p}^{*}M,\qquad \left(\left(\Phi _{X}^{t}\right)_{p}^{*}\alpha \right)(Y)=\alpha {\bigl (}T_{p}\Phi _{X}^{t}(Y){\bigr )},\quad \alpha \in T_{\Phi _{X}^{t}(p)}^{*}M,Y\in T_{p}M}For contravariant tensors, one extends the inverse
of thedifferentialTpΦXt{\displaystyle T_{p}\Phi _{X}^{t}}. For everyt,{\displaystyle t,}there is, consequently, a tensor field(ΦXt)∗T{\displaystyle (\Phi _{X}^{t})^{*}T}of the same type asT{\displaystyle T}'s.
IfT{\displaystyle T}is an(r,0){\displaystyle (r,0)}- or(0,s){\displaystyle (0,s)}-type tensor field, then the Lie derivativeLXT{\displaystyle {\cal {L}}_{X}T}ofT{\displaystyle T}along a vector fieldX{\displaystyle X}is defined at pointp∈M{\displaystyle p\in M}to be
The resulting tensor fieldLXT{\displaystyle {\cal {L}}_{X}T}is of the same type asT{\displaystyle T}'s.
More generally, for every smooth 1-parameter familyΦt{\displaystyle \Phi _{t}}of diffeomorphisms that integrate a vector fieldX{\displaystyle X}in the sense thatddt|t=0Φt=X∘Φ0{\displaystyle {d \over dt}{\biggr |}_{t=0}\Phi _{t}=X\circ \Phi _{0}}, one hasLXT=(Φ0−1)∗ddt|t=0Φt∗T=−ddt|t=0(Φt−1)∗Φ0∗T.{\displaystyle {\mathcal {L}}_{X}T={\bigl (}\Phi _{0}^{-1}{\bigr )}^{*}{d \over dt}{\biggr |}_{t=0}\Phi _{t}^{*}T=-{d \over dt}{\biggr |}_{t=0}{\bigl (}\Phi _{t}^{-1}{\bigr )}^{*}\Phi _{0}^{*}T\,.}
We now give an algebraic definition. The algebraic definition for the Lie derivative of a tensor field follows from the following four axioms:
Using the first and third axioms, applying the Lie derivativeLX{\displaystyle {\mathcal {L}}_{X}}toY(f){\displaystyle Y(f)}shows that
which is one of the standard definitions for theLie bracket.
The Lie derivative acting on a differential form is theanticommutatorof theinterior productwith the exterior derivative. So if α is a differential form,
This follows easily by checking that the expression commutes with exterior derivative, is a derivation (being an anticommutator of graded derivations) and does the right thing on functions. This isCartan's magic formula. Seeinterior productfor details.
Explicitly, letTbe a tensor field of type(p,q). ConsiderTto be a differentiablemultilinear mapofsmoothsectionsα1,α2, ...,αpof the cotangent bundleT∗Mand of sectionsX1,X2, ...,Xqof thetangent bundleTM, writtenT(α1,α2, ...,X1,X2, ...) intoR. Define the Lie derivative ofTalongYby the formula
The analytic and algebraic definitions can be proven to be equivalent using the properties of the pushforward and theLeibniz rulefor differentiation. The Lie derivative commutes with the contraction.
A particularly important class of tensor fields is the class ofdifferential forms. The restriction of the Lie derivative to the space of differential forms is closely related to theexterior derivative. Both the Lie derivative and the exterior derivative attempt to capture the idea of a derivative in different ways. These differences can be bridged by introducing the idea of aninterior product, after which the relationships falls out as an identity known asCartan's formula. Cartan's formula can also be used as a definition of the Lie derivative on the space of differential forms.
LetMbe a manifold andXa vector field onM. Letω∈Λk(M){\displaystyle \omega \in \Lambda ^{k}(M)}be ak-form, i.e., for eachp∈M{\displaystyle p\in M},ω(p){\displaystyle \omega (p)}is analternatingmultilinear mapfrom(TpM)k{\displaystyle (T_{p}M)^{k}}to the real numbers. Theinterior productofXandωis the(k− 1)-formiXω{\displaystyle i_{X}\omega }defined as
The differential formiXω{\displaystyle i_{X}\omega }is also called thecontractionofωwithX, and
is a∧{\displaystyle \wedge }-antiderivationwhere∧{\displaystyle \wedge }is thewedge product on differential forms. That is,iX{\displaystyle i_{X}}isR-linear, and
forω∈Λk(M){\displaystyle \omega \in \Lambda ^{k}(M)}and η another differential form. Also, for a functionf∈Λ0(M){\displaystyle f\in \Lambda ^{0}(M)}, that is, a real- or complex-valued function onM, one has
wherefX{\displaystyle fX}denotes the product offandX.
The relationship betweenexterior derivativesand Lie derivatives can then be summarized as follows. First, since the Lie derivative of a functionfwith respect to a vector fieldXis the same as the directional derivativeX(f), it is also the same as thecontractionof the exterior derivative offwithX:
For a general differential form, the Lie derivative is likewise a contraction, taking into account the variation inX:
This identity is known variously asCartan formula,Cartan homotopy formulaorCartan's magic formula. Seeinterior productfor details. The Cartan formula can be used as a definition of the Lie derivative of a differential form. Cartan's formula shows in particular that
The Lie derivative also satisfies the relation
In localcoordinatenotation, for a type(r,s)tensor fieldT{\displaystyle T}, the Lie derivative alongX{\displaystyle X}is
here, the notation∂a=∂∂xa{\displaystyle \partial _{a}={\frac {\partial }{\partial x^{a}}}}means taking the partial derivative with respect to the coordinatexa{\displaystyle x^{a}}. Alternatively, if we are using atorsion-freeconnection(e.g., theLevi Civita connection), then the partial derivative∂a{\displaystyle \partial _{a}}can be replaced with thecovariant derivativewhich means replacing∂aXb{\displaystyle \partial _{a}X^{b}}with (by abuse of notation)∇aXb=X;ab:=(∇X)ab=∂aXb+ΓacbXc{\displaystyle \nabla _{a}X^{b}=X_{;a}^{b}:=(\nabla X)_{a}^{\ b}=\partial _{a}X^{b}+\Gamma _{ac}^{b}X^{c}}where theΓbca=Γcba{\displaystyle \Gamma _{bc}^{a}=\Gamma _{cb}^{a}}are theChristoffel coefficients.
The Lie derivative of a tensor is another tensor of the same type, i.e., even though the individual terms in the expression depend on the choice of coordinate system, the expression as a whole results in a tensor
which is independent of any coordinate system and of the same type asT{\displaystyle T}.
The definition can be extended further totensor densities. IfTis a tensor density of some real number valued weightw(e.g. the volume density of weight 1), then its Lie derivative is a tensor density of the same type and weight.
Notice the new term at the end of the expression.
For alinear connectionΓ=(Γbca){\displaystyle \Gamma =(\Gamma _{bc}^{a})}, the Lie derivative alongX{\displaystyle X}is[3]
For clarity we now show the following examples in localcoordinatenotation.
For ascalar fieldϕ(xc)∈F(M){\displaystyle \phi (x^{c})\in {\mathcal {F}}(M)}we have:
Hence for the scalar fieldϕ(x,y)=x2−sin(y){\displaystyle \phi (x,y)=x^{2}-\sin(y)}and the vector fieldXa∂a=sin(x)∂y−y2∂x{\displaystyle X^{a}\partial _{a}=\sin(x)\partial _{y}-y^{2}\partial _{x}}the corresponding Lie derivative becomesLXϕ=(sin(x)∂y−y2∂x)(x2−sin(y))=sin(x)∂y(x2−sin(y))−y2∂x(x2−sin(y))=−sin(x)cos(y)−2xy2{\displaystyle {\begin{alignedat}{3}{\mathcal {L}}_{X}\phi &=(\sin(x)\partial _{y}-y^{2}\partial _{x})(x^{2}-\sin(y))\\&=\sin(x)\partial _{y}(x^{2}-\sin(y))-y^{2}\partial _{x}(x^{2}-\sin(y))\\&=-\sin(x)\cos(y)-2xy^{2}\\\end{alignedat}}}
For an example of higher rank differential form, consider the 2-formω=(x2+y2)dx∧dz{\displaystyle \omega =(x^{2}+y^{2})dx\wedge dz}and the vector fieldX{\displaystyle X}from the previous example. Then,LXω=d(isin(x)∂y−y2∂x((x2+y2)dx∧dz))+isin(x)∂y−y2∂x(d((x2+y2)dx∧dz))=d(−y2(x2+y2)dz)+isin(x)∂y−y2∂x(2ydy∧dx∧dz)=(−2xy2dx+(−2yx2−4y3)dy)∧dz+(2ysin(x)dx∧dz+2y3dy∧dz)=(−2xy2+2ysin(x))dx∧dz+(−2yx2−2y3)dy∧dz{\displaystyle {\begin{aligned}{\mathcal {L}}_{X}\omega &=d(i_{\sin(x)\partial _{y}-y^{2}\partial _{x}}((x^{2}+y^{2})dx\wedge dz))+i_{\sin(x)\partial _{y}-y^{2}\partial _{x}}(d((x^{2}+y^{2})dx\wedge dz))\\&=d(-y^{2}(x^{2}+y^{2})dz)+i_{\sin(x)\partial _{y}-y^{2}\partial _{x}}(2ydy\wedge dx\wedge dz)\\&=\left(-2xy^{2}dx+(-2yx^{2}-4y^{3})dy\right)\wedge dz+(2y\sin(x)dx\wedge dz+2y^{3}dy\wedge dz)\\&=\left(-2xy^{2}+2y\sin(x)\right)dx\wedge dz+(-2yx^{2}-2y^{3})dy\wedge dz\end{aligned}}}
Some more abstract examples.
Hence for acovector field, i.e., adifferential form,A=Aa(xb)dxa{\displaystyle A=A_{a}(x^{b})dx^{a}}we have:
The coefficient of the last expression is the local coordinate expression of the Lie derivative.
For a covariant rank 2 tensor fieldT=Tab(xc)dxa⊗dxb{\displaystyle T=T_{ab}(x^{c})dx^{a}\otimes dx^{b}}we have:(LXT)=(LXT)abdxa⊗dxb=X(Tab)dxa⊗dxb+TcbLX(dxc)⊗dxb+Tacdxa⊗LX(dxc)=(Xc∂cTab+Tcb∂aXc+Tac∂bXc)dxa⊗dxb{\displaystyle {\begin{aligned}({\mathcal {L}}_{X}T)&=({\mathcal {L}}_{X}T)_{ab}dx^{a}\otimes dx^{b}\\&=X(T_{ab})dx^{a}\otimes dx^{b}+T_{cb}{\mathcal {L}}_{X}(dx^{c})\otimes dx^{b}+T_{ac}dx^{a}\otimes {\mathcal {L}}_{X}(dx^{c})\\&=(X^{c}\partial _{c}T_{ab}+T_{cb}\partial _{a}X^{c}+T_{ac}\partial _{b}X^{c})dx^{a}\otimes dx^{b}\\\end{aligned}}}
IfT=g{\displaystyle T=g}is the symmetric metric tensor, it is parallel with respect to theLevi-Civita connection(akacovariant derivative), and it becomes fruitful to use the connection. This has the effect of replacing all derivatives with covariant derivatives, giving
The Lie derivative has a number of properties. LetF(M){\displaystyle {\mathcal {F}}(M)}be thealgebraof functions defined on themanifoldM. Then
is aderivationon the algebraF(M){\displaystyle {\mathcal {F}}(M)}. That is,LX{\displaystyle {\mathcal {L}}_{X}}isR-linear and
Similarly, it is a derivation onF(M)×X(M){\displaystyle {\mathcal {F}}(M)\times {\mathcal {X}}(M)}whereX(M){\displaystyle {\mathcal {X}}(M)}is the set of vector fields onM:[4]
which may also be written in the equivalent notation
where thetensor productsymbol⊗{\displaystyle \otimes }is used to emphasize the fact that the product of a function times a vector field is being taken over the entire manifold.
Additional properties are consistent with that of theLie bracket. Thus, for example, considered as a derivation on a vector field,
one finds the above to be just theJacobi identity. Thus, one has the important result that the space of vector fields overM, equipped with the Lie bracket, forms aLie algebra.
The Lie derivative also has important properties when acting on differential forms. Letαandβbe two differential forms onM, and letXandYbe two vector fields. Then
Various generalizations of the Lie derivative play an important role in differential geometry.
A definition for Lie derivatives ofspinorsalong generic spacetime vector fields, not necessarilyKillingones, on a general (pseudo)Riemannian manifoldwas already proposed in 1971 byYvette Kosmann.[5]Later, it was provided a geometric framework which justifies herad hocprescription within the general framework of Lie derivatives onfiber bundles[6]in the explicit context of gauge natural bundles which turn out to be the most appropriate arena for (gauge-covariant) field theories.[7]
In a givenspin manifold, that is in a Riemannian manifold(M,g){\displaystyle (M,g)}admitting aspin structure, the Lie derivative of aspinorfieldψ{\displaystyle \psi }can be defined by first defining it with respect to infinitesimal isometries (Killing vector fields) via theAndré Lichnerowicz's local expression given in 1963:[8]
where∇aXb=∇[aXb]{\displaystyle \nabla _{a}X_{b}=\nabla _{[a}X_{b]}}, asX=Xa∂a{\displaystyle X=X^{a}\partial _{a}}is assumed to be aKilling vector field, andγa{\displaystyle \gamma ^{a}}areDirac matrices.
It is then possible to extend Lichnerowicz's definition to all vector fields (generic infinitesimal transformations) by retaining Lichnerowicz's local expression for agenericvector fieldX{\displaystyle X}, but explicitly taking the antisymmetric part of∇aXb{\displaystyle \nabla _{a}X_{b}}only.[5]More explicitly, Kosmann's local expression given in 1972 is:[5]
where[γa,γb]=γaγb−γbγa{\displaystyle [\gamma ^{a},\gamma ^{b}]=\gamma ^{a}\gamma ^{b}-\gamma ^{b}\gamma ^{a}}is the commutator,d{\displaystyle d}isexterior derivative,X♭=g(X,−){\displaystyle X^{\flat }=g(X,-)}is the dual 1 form corresponding toX{\displaystyle X}under the metric (i.e. with lowered indices) and⋅{\displaystyle \cdot }is Clifford multiplication.
It is worth noting that the spinor Lie derivative is independent of the metric, and hence also of theconnection. This is not obvious from the right-hand side of Kosmann's local expression, as the right-hand side seems to depend on the metric through the spin connection (covariant derivative), the dualisation of vector fields (lowering of the indices) and the Clifford multiplication on thespinor bundle. Such is not the case: the quantities on the right-hand side of Kosmann's local expression combine so as to make all metric and connection dependent terms cancel.
To gain a better understanding of the long-debated concept of Lie derivative of spinor fields one may refer to the original article,[9][10]where the definition of a Lie derivative of spinor fields is placed in the more general framework of the theory of Lie derivatives of sections of fiber bundles and the direct approach by Y. Kosmann to the spinor case is generalized to gauge natural bundles in the form of a new geometric concept called theKosmann lift.
As for the tensor counterpart, also for spinors the vanishing of the Lie derivative along a Killing vector implements on the spinor the symmetries encoded by that Killing vector. However, differently from tensors, from spinors it is possible to build bi-linear quantities (such as the velocity vectorψ¯γaψ{\displaystyle {\overline {\psi }}\gamma ^{a}\psi }or the spin axial-vectorψ¯γaγ5ψ{\displaystyle {\overline {\psi }}\gamma ^{a}\gamma ^{5}\psi }) which are tensors. A natural question that now arises is whether the vanishing of the Lie derivative along a Killing vector of a spinor is equivalent to the vanishing of the Lie derivative along the same Killing vector of all the spinor bi-linear quantities. While a spinor that is Lie-invariant implies that all its bi-linear quantities are also Lie invariant, the converse is in general not true.[11]
If we have a principal bundle over the manifold M with G as the structure group, and we pick X to be a covariant vector field as section of the tangent space of the principal bundle (i.e. it has horizontal and vertical components), then the covariant Lie derivative is just the Lie derivative with respect to X over the principal bundle.
Now, if we're given a vector fieldYoverM(but not the principal bundle) but we also have aconnectionover the principal bundle, we can define a vector field X over the principal bundle such that its horizontal component matchesYand its vertical component agrees with the connection. This is the covariant Lie derivative.
Seeconnection formfor more details.
Another generalization, due toAlbert Nijenhuis, allows one to define the Lie derivative of a differential form along any section of the bundle Ωk(M, TM) of differential forms with values in the tangent bundle. IfK∈ Ωk(M, TM) and α is a differentialp-form, then it is possible to define the interior productiKα ofKand α. The Nijenhuis–Lie derivative is then the anticommutator of the interior product and the exterior derivative:
In 1931,Władysław Ślebodzińskiintroduced a new differential operator, later called byDavid van Dantzigthat of Lie derivation, which can be applied to scalars, vectors, tensors and affine connections and which proved to be a powerful instrument in the study of groups of automorphisms.
The Lie derivatives of general geometric objects (i.e., sections ofnatural fiber bundles) were studied byA. Nijenhuis, Y. Tashiro andK. Yano.
For a quite long time, physicists had been using Lie derivatives, without reference to the work of mathematicians. In 1940,Léon Rosenfeld[12]—and before him (in 1921)Wolfgang Pauli[13]—introduced what he called a ‘local variation’δ∗A{\displaystyle \delta ^{\ast }A}of a geometric objectA{\displaystyle A\,}induced by an infinitesimal transformation of coordinates generated by a vector fieldX{\displaystyle X\,}. One can easily prove that hisδ∗A{\displaystyle \delta ^{\ast }A}is−LX(A){\displaystyle -{\mathcal {L}}_{X}(A)\,}. | https://en.wikipedia.org/wiki/Lie_derivative |
Inphysics, theMoyal bracketis the suitably normalized antisymmetrization of the phase-spacestar product.
The Moyal bracket was developed in about 1940 byJosé Enrique Moyal, but Moyal only succeeded in publishing his work in 1949 after a lengthy dispute withPaul Dirac.[1][2]In the meantime this idea was independently introduced in 1946 byHip Groenewold.[3]
The Moyal bracket is a way of describing thecommutatorof observables in thephase space formulationofquantum mechanicswhen these observables are described as functions onphase space. It relies on schemes for identifying functions on phase space with quantum observables, the most famous of these schemes being theWigner–Weyl transform. It underliesMoyal’s dynamical equation, an equivalent formulation ofHeisenberg’s quantum equation of motion, thereby providing the quantum generalization ofHamilton’s equations.
Mathematically, it is adeformationof the phase-spacePoisson bracket(essentially anextensionof it), the deformation parameter being the reducedPlanck constantħ. Thus, itsgroup contractionħ→0yields thePoisson bracketLie algebra.
Up to formal equivalence, the Moyal Bracket is theunique one-parameter Lie-algebraic deformationof the Poisson bracket. Its algebraic isomorphism to the algebra of commutators bypasses the negative result of the Groenewold–van Hove theorem, which precludes such an isomorphism for the Poisson bracket, a question implicitly raised by Dirac in his 1926 doctoral thesis,[4]the "method of classical analogy" for quantization.[5]
For instance, in a two-dimensional flatphase space, and for theWeyl-map correspondence, the Moyal bracket reads,
where★is the star-product operator in phase space (cf.Moyal product), whilefandgare differentiable phase-space functions, and{f,g}is their Poisson bracket.[6]
More specifically, inoperational calculuslanguage, this equals
{{f,g}}=2ℏf(x,p)sin(ℏ2(∂←x∂→p−∂←p∂→x))g(x,p).{\displaystyle \{\{f,g\}\}\ ={\frac {2}{\hbar }}~f(x,p)\ \sin \left({{\tfrac {\hbar }{2}}({\overleftarrow {\partial }}_{x}{\overrightarrow {\partial }}_{p}-{\overleftarrow {\partial }}_{p}{\overrightarrow {\partial }}_{x})}\right)\ g(x,p).}
The left & right arrows over the partial derivatives denote the left & right partial derivatives. Sometimes the Moyal bracket is referred to as theSine bracket.
A popular (Fourier) integral representation for it, introduced by George Baker[7]is
Each correspondence map from phase space to Hilbert space induces a characteristic "Moyal" bracket (such as the one illustrated here for the Weyl map). All such Moyal brackets areformally equivalentamong themselves, in accordance with a systematic theory.[8]
The Moyal bracket specifies the eponymous infinite-dimensionalLie algebra—it is antisymmetric in its argumentsfandg, and satisfies theJacobi identity.
The corresponding abstractLie algebrais realized byTf≡ f★, so that
On a 2-torus phase space,T2, with periodic
coordinatesxandp, each in[0,2π], and integer mode indicesmi, for basis functionsexp(i(m1x+m2p)), this Lie algebra reads,[9]
which reduces toSU(N) for integerN≡ 4π/ħ.SU(N) then emerges as a deformation ofSU(∞), with deformation parameter 1/N.
Generalization of the Moyal bracket for quantum systems withsecond-class constraintsinvolves an operation on equivalence classes of functions in phase space,[10]which can be considered as aquantum deformationof theDirac bracket.
Next to the sine bracket discussed, Groenewold further introduced[3]the cosine bracket, elaborated by Baker,[7][11]
Here, again,★is the star-product operator in phase space,fandgare differentiable phase-space functions, andfgis the ordinary product.
The sine and cosine brackets are, respectively, the results of antisymmetrizing and symmetrizing the star product. Thus, as the sine bracket is theWigner mapof the commutator, the cosine bracket is the Wigner image of theanticommutatorin standard quantum mechanics. Similarly, as the Moyal bracket equals the Poisson bracket up to higher orders ofħ, the cosine bracket equals the ordinary product up to higher orders ofħ. In theclassical limit, the Moyal bracket helps reduction to theLiouville equation (formulated in terms of the Poisson bracket), as the cosine bracket leads to the classicalHamilton–Jacobi equation.[12]
The sine and cosine bracket also stand in relation to equations ofa purely algebraic descriptionof quantum mechanics.[12][13] | https://en.wikipedia.org/wiki/Moyal_bracket |
Inmathematicsand intheoretical physics, theStone–von Neumann theoremrefers to any one of a number of different formulations of theuniquenessof thecanonical commutation relationsbetweenpositionandmomentumoperators. It is named afterMarshall StoneandJohn von Neumann.[1][2][3][4]
Inquantum mechanics, physicalobservablesare represented mathematically bylinear operatorsonHilbert spaces.
For a single particle moving on thereal lineR{\displaystyle \mathbb {R} }, there are two important observables: position andmomentum. In the Schrödinger representation quantum description of such a particle, theposition operatorxandmomentum operatorp{\displaystyle p}are respectively given by[xψ](x0)=x0ψ(x0)[pψ](x0)=−iℏ∂ψ∂x(x0){\displaystyle {\begin{aligned}[][x\psi ](x_{0})&=x_{0}\psi (x_{0})\\[][p\psi ](x_{0})&=-i\hbar {\frac {\partial \psi }{\partial x}}(x_{0})\end{aligned}}}on the domainV{\displaystyle V}of infinitely differentiable functions of compact support onR{\displaystyle \mathbb {R} }. Assumeℏ{\displaystyle \hbar }to be a fixednon-zeroreal number—in quantum theoryℏ{\displaystyle \hbar }is thereduced Planck constant, which carries units of action (energytimestime).
The operatorsx{\displaystyle x},p{\displaystyle p}satisfy thecanonical commutation relationLie algebra,[x,p]=xp−px=iℏ.{\displaystyle [x,p]=xp-px=i\hbar .}
Already in his classic book,[5]Hermann Weylobserved that this commutation law wasimpossible to satisfyfor linear operatorsp,xacting onfinite-dimensionalspaces unlessħvanishes. This is apparent from taking thetraceover both sides of the latter equation and using the relationTrace(AB) = Trace(BA); the left-hand side is zero, the right-hand side is non-zero. Further analysis shows that any two self-adjoint operators satisfying the above commutation relation cannot be bothbounded(in fact, a theorem ofWielandtshows the relation cannot be satisfied by elements ofanynormed algebra[note 1]). For notational convenience, the nonvanishing square root ofℏmay be absorbed into the normalization ofpandx, so that, effectively, it is replaced by 1. We assume this normalization in what follows.
The idea of the Stone–von Neumann theorem is that any two irreducible representations of the canonical commutation relations are unitarily equivalent. Since, however, the operators involved are necessarily unbounded (as noted above), there are tricky domain issues that allow for counter-examples.[6]: Example 14.5To obtain a rigorous result, one must require that the operators satisfy the exponentiated form of the canonical commutation relations, known as the Weyl relations. The exponentiated operators are bounded and unitary. Although, as noted below, these relations are formally equivalent to the standard canonical commutation relations, this equivalence is not rigorous, because (again) of the unbounded nature of the operators. (There is also a discrete analog of the Weyl relations, which can hold in a finite-dimensional space,[6]: Chapter 14, Exercise 5namelySylvester'sclock and shift matricesin the finite Heisenberg group, discussed below.)
One would like to classify representations of the canonical commutation relation by two self-adjoint operators acting on separable Hilbert spaces,up to unitary equivalence. ByStone's theorem, there is a one-to-one correspondence between self-adjoint operators and (strongly continuous) one-parameter unitary groups.
LetQandPbe two self-adjoint operators satisfying the canonical commutation relation,[Q,P] =i, andsandttwo real parameters. IntroduceeitQandeisP, the corresponding unitary groups given byfunctional calculus. (For the explicit operatorsxandpdefined above, these are multiplication byeitxand pullback by translationx→x+s.) A formal computation[6]: Section 14.2(using a special case of theBaker–Campbell–Hausdorff formula) readily yieldseitQeisP=e−isteisPeitQ.{\displaystyle e^{itQ}e^{isP}=e^{-ist}e^{isP}e^{itQ}.}
Conversely, given two one-parameter unitary groupsU(t)andV(s)satisfying the braiding relation
U(t)V(s)=e−istV(s)U(t)∀s,t,{\displaystyle U(t)V(s)=e^{-ist}V(s)U(t)\qquad \forall s,t,}(E1)
formally differentiating at 0 shows that the two infinitesimal generators satisfy the above canonical commutation relation. This braiding formulation of the canonical commutation relations (CCR) for one-parameter unitary groups is called theWeyl form of the CCR.
It is important to note that the preceding derivation is purely formal. Since the operators involved are unbounded, technical issues prevent application of the Baker–Campbell–Hausdorff formula without additional domain assumptions. Indeed, there exist operators satisfying the canonical commutation relation but not the Weyl relations (E1).[6]: Example 14.5Nevertheless, in "good" cases, we expect that operators satisfying the canonical commutation relation will also satisfy the Weyl relations.
The problem thus becomes classifying two jointlyirreducibleone-parameter unitary groupsU(t)andV(s)which satisfy the Weyl relation on separable Hilbert spaces. The answer is the content of theStone–von Neumann theorem:all such pairs of one-parameter unitary groups are unitarily equivalent.[6]: Theorem 14.8In other words, for any two suchU(t)andV(s)acting jointly irreducibly on a Hilbert spaceH, there is a unitary operatorW:L2(R) →Hso thatW∗U(t)W=eitxandW∗V(s)W=eisp,{\displaystyle W^{*}U(t)W=e^{itx}\quad {\text{and}}\quad W^{*}V(s)W=e^{isp},}wherepandxare the explicit position and momentum operators from earlier. WhenWisUin this equation, so, then, in thex-representation, it is evident thatPis unitarily equivalent toe−itQPeitQ=P+t, and the spectrum ofPmust range along the entire real line. The analog argument holds forQ.
There is also a straightforward extension of the Stone–von Neumann theorem tondegrees of freedom.[6]: Theorem 14.8Historically, this result was significant, because it was a key step in proving thatHeisenberg'smatrix mechanics, which presents quantum mechanical observables and dynamics in terms of infinite matrices, is unitarily equivalent toSchrödinger's wave mechanical formulation (seeSchrödinger picture),[U(t)ψ](x)=eitxψ(x),[V(s)ψ](x)=ψ(x+s).{\displaystyle [U(t)\psi ](x)=e^{itx}\psi (x),\qquad [V(s)\psi ](x)=\psi (x+s).}
In terms of representation theory, the Stone–von Neumann theorem classifies certain unitary representations of theHeisenberg group. This is discussed in more detail inthe Heisenberg group section, below.
Informally stated, with certain technical assumptions, every representation of the Heisenberg groupH2n+ 1is equivalent to the position operators and momentum operators onRn. Alternatively, that they are all equivalent to theWeyl algebra(orCCR algebra) on a symplectic space of dimension2n.
More formally, there is aunique(up to scale) non-trivial central strongly continuous unitary representation.
This was later generalized byMackey theory– and was the motivation for the introduction of the Heisenberg group in quantum physics.
In detail:
In all cases, if one has a representationH2n+ 1→A, whereAis an algebra[clarification needed]and thecentermaps to zero, then one simply has a representation of the corresponding abelian group or algebra, which isFourier theory.[clarification needed]
If the center does not map to zero, one has a more interesting theory, particularly if one restricts oneself tocentralrepresentations.
Concretely, by a central representation one means a representation such that the center of the Heisenberg group maps into thecenter of the algebra: for example, if one is studying matrix representations or representations by operators on a Hilbert space, then the center of the matrix algebra or the operator algebra is thescalar matrices. Thus the representation of the center of the Heisenberg group is determined by a scale value, called thequantizationvalue (in physics terms, the Planck constant), and if this goes to zero, one gets a representation of the abelian group (in physics terms, this is the classical limit).
More formally, thegroup algebraof the Heisenberg group over its field ofscalarsK, writtenK[H], has centerK[R], so rather than simply thinking of the group algebra as an algebra over the fieldK, one may think of it as an algebra over the commutative algebraK[R]. As the center of a matrix algebra or operator algebra is the scalar matrices, aK[R]-structure on the matrix algebra is a choice of scalar matrix – a choice of scale. Given such a choice of scale, a central representation of the Heisenberg group is a map ofK[R]-algebrasK[H] →A, which is the formal way of saying that it sends the center to a chosen scale.
Then the Stone–von Neumann theorem is that, given the standard quantum mechanical scale (effectively, the value of ħ), every strongly continuous unitary representation is unitarily equivalent to the standard representation with position and momentum.
LetGbe alocally compact abelian groupandG^be thePontryagin dualofG. TheFourier–Plancherel transformdefined byf↦f^(γ)=∫Gγ(t)¯f(t)dμ(t){\displaystyle f\mapsto {\hat {f}}(\gamma )=\int _{G}{\overline {\gamma (t)}}f(t)d\mu (t)}extends to a C*-isomorphism from thegroup C*-algebraC*(G)ofGandC0(G^), i.e. thespectrumofC*(G)is preciselyG^. WhenGis the real lineR, this is Stone's theorem characterizing one-parameter unitary groups. The theorem of Stone–von Neumann can also be restated using similar language.
The groupGacts on theC*-algebraC0(G)by right translationρ: forsinGandfinC0(G),(s⋅f)(t)=f(t+s).{\displaystyle (s\cdot f)(t)=f(t+s).}
Under the isomorphism given above, this action becomes the natural action ofGonC*(G^):(s⋅f)^(γ)=γ(s)f^(γ).{\displaystyle {\widehat {(s\cdot f)}}(\gamma )=\gamma (s){\hat {f}}(\gamma ).}
So a covariant representation corresponding to theC*-crossed productC∗(G^)⋊ρ^G{\displaystyle C^{*}\left({\hat {G}}\right)\rtimes _{\hat {\rho }}G}is a unitary representationU(s)ofGandV(γ)ofG^such thatU(s)V(γ)U∗(s)=γ(s)V(γ).{\displaystyle U(s)V(\gamma )U^{*}(s)=\gamma (s)V(\gamma ).}
It is a general fact that covariant representations are in one-to-one correspondence with *-representation of the corresponding crossed product. On the other hand, allirreducible representationsofC0(G)⋊ρG{\displaystyle C_{0}(G)\rtimes _{\rho }G}are unitarily equivalent to theK(L2(G)){\displaystyle {\mathcal {K}}\left(L^{2}(G)\right)}, thecompact operatorsonL2(G)). Therefore, all pairs{U(s),V(γ)}are unitarily equivalent. Specializing to the case whereG=Ryields the Stone–von Neumann theorem.
The above canonical commutation relations forP,Qare identical to the commutation relations that specify theLie algebraof the generalHeisenberg groupH2n+1forna positive integer. This is theLie groupof(n+ 2) × (n+ 2)square matrices of the formM(a,b,c)=[1ac01nb001].{\displaystyle \mathrm {M} (a,b,c)={\begin{bmatrix}1&a&c\\0&1_{n}&b\\0&0&1\end{bmatrix}}.}
In fact, using the Heisenberg group, one can reformulate the Stone von Neumann theorem in the language of representation theory.
Note that the center ofH2n+1consists of matricesM(0, 0,c). However, this center isnottheidentity operatorin Heisenberg's original CCRs. The Heisenberg group Lie algebra generators, e.g. forn= 1, areP=[010000000],Q=[000001000],z=[001000000],{\displaystyle {\begin{aligned}P&={\begin{bmatrix}0&1&0\\0&0&0\\0&0&0\end{bmatrix}},&Q&={\begin{bmatrix}0&0&0\\0&0&1\\0&0&0\end{bmatrix}},&z&={\begin{bmatrix}0&0&1\\0&0&0\\0&0&0\end{bmatrix}},\end{aligned}}}and the central generatorz= logM(0, 0, 1) = exp(z) − 1is not the identity.
Theorem—For each non-zero real numberhthere is anirreducible representationUhacting on the Hilbert spaceL2(Rn)by[Uh(M(a,b,c))]ψ(x)=ei(b⋅x+hc)ψ(x+ha).{\displaystyle \left[U_{h}(\mathrm {M} (a,b,c))\right]\psi (x)=e^{i(b\cdot x+hc)}\psi (x+ha).}
All these representations areunitarily inequivalent; and any irreducible representation which is not trivial on the center ofHnis unitarily equivalent to exactly one of these.
Note thatUhis a unitary operator because it is the composition of two operators which are easily seen to be unitary: the translation to theleftbyhaand multiplication by a function ofabsolute value1. To showUhis multiplicative is a straightforward calculation. The hard part of the theorem is showing the uniqueness; this claim, nevertheless, follows easily from the Stone–von Neumann theorem as stated above. We will sketch below a proof of the corresponding Stone–von Neumann theorem for certainfiniteHeisenberg groups.
In particular, irreducible representationsπ,π′of the Heisenberg groupHnwhich are non-trivial on the center ofHnare unitarily equivalent if and only ifπ(z) =π′(z)for anyzin the center ofHn.
One representation of the Heisenberg group which is important innumber theoryand the theory ofmodular formsis thetheta representation, so named because theJacobi theta functionis invariant under the action of the discrete subgroup of the Heisenberg group.
For any non-zeroh, the mappingαh:M(a,b,c)→M(−h−1b,ha,c−a⋅b){\displaystyle \alpha _{h}:\mathrm {M} (a,b,c)\to \mathrm {M} \left(-h^{-1}b,ha,c-a\cdot b\right)}is anautomorphismofHnwhich is the identity on the center ofHn. In particular, the representationsUhandUhαare unitarily equivalent. This means that there is a unitary operatorWonL2(Rn)such that, for anyginHn,WUh(g)W∗=Uhα(g).{\displaystyle WU_{h}(g)W^{*}=U_{h}\alpha (g).}
Moreover, by irreducibility of the representationsUh, it follows thatup to a scalar, such an operatorWis unique (cf.Schur's lemma). SinceWis unitary, this scalar multiple is uniquely determined and hence such an operatorWis unique.
Theorem—The operatorWis theFourier transformonL2(Rn).
This means that, ignoring the factor of(2π)n/2in the definition of the Fourier transform,∫Rne−ix⋅pei(b⋅x+hc)ψ(x+ha)dx=ei(ha⋅p+h(c−b⋅a))∫Rne−iy⋅(p−b)ψ(y)dy.{\displaystyle \int _{\mathbf {R} ^{n}}e^{-ix\cdot p}e^{i(b\cdot x+hc)}\psi (x+ha)\ dx=e^{i(ha\cdot p+h(c-b\cdot a))}\int _{\mathbf {R} ^{n}}e^{-iy\cdot (p-b)}\psi (y)\ dy.}
This theorem has the immediate implication that the Fourier transform isunitary, also known as thePlancherel theorem. Moreover,(αh)2M(a,b,c)=M(−a,−b,c).{\displaystyle (\alpha _{h})^{2}\mathrm {M} (a,b,c)=\mathrm {M} (-a,-b,c).}
Theorem—The operatorW1such thatW1UhW1∗=Uhα2(g){\displaystyle W_{1}U_{h}W_{1}^{*}=U_{h}\alpha ^{2}(g)}is the reflection operator[W1ψ](x)=ψ(−x).{\displaystyle [W_{1}\psi ](x)=\psi (-x).}
From this fact theFourier inversion formulaeasily follows.
TheSegal–Bargmann spaceis the space of holomorphic functions onCnthat are square-integrable with respect to a Gaussian measure. Fock observed in 1920s that the operatorsaj=∂∂zj,aj∗=zj,{\displaystyle a_{j}={\frac {\partial }{\partial z_{j}}},\qquad a_{j}^{*}=z_{j},}acting on holomorphic functions, satisfy the same commutation relations as the usual annihilation and creation operators, namely,[aj,ak∗]=δj,k.{\displaystyle \left[a_{j},a_{k}^{*}\right]=\delta _{j,k}.}
In 1961, Bargmann showed thata∗jis actually the adjoint ofajwith respect to the inner product coming from the Gaussian measure. By taking appropriate linear combinations ofajanda∗j, one can then obtain "position" and "momentum" operators satisfying the canonical commutation relations. It is not hard to show that the exponentials of these operators satisfy the Weyl relations and that the exponentiated operators act irreducibly.[6]: Section 14.4The Stone–von Neumann theorem therefore applies and implies the existence of a unitary map fromL2(Rn)to the Segal–Bargmann space that intertwines the usual annihilation and creation operators with the operatorsajanda∗j. This unitary map is theSegal–Bargmann transform.
The Heisenberg groupHn(K)is defined for any commutative ringK. In this section let us specialize to the fieldK=Z/pZforpa prime. This field has the property that there is an embeddingωofKas anadditive groupinto the circle groupT. Note thatHn(K)is finite withcardinality|K|2n+ 1. For finite Heisenberg groupHn(K)one can give a simple proof of the Stone–von Neumann theorem using simple properties ofcharacter functionsof representations. These properties follow from theorthogonality relationsfor characters of representations of finite groups.
For any non-zerohinKdefine the representationUhon the finite-dimensionalinner product spaceℓ2(Kn)by[UhM(a,b,c)ψ](x)=ω(b⋅x+hc)ψ(x+ha).{\displaystyle \left[U_{h}\mathrm {M} (a,b,c)\psi \right](x)=\omega (b\cdot x+hc)\psi (x+ha).}
Theorem—For a fixed non-zeroh, the character functionχofUhis given by:χ(M(a,b,c))={|K|nω(hc)ifa=b=00otherwise{\displaystyle \chi (\mathrm {M} (a,b,c))={\begin{cases}|K|^{n}\,\omega (hc)&{\text{if }}a=b=0\\0&{\text{otherwise}}\end{cases}}}
It follows that1|Hn(K)|∑g∈Hn(K)|χ(g)|2=1|K|2n+1|K|2n|K|=1.{\displaystyle {\frac {1}{\left|H_{n}(\mathbf {K} )\right|}}\sum _{g\in H_{n}(K)}|\chi (g)|^{2}={\frac {1}{|K|^{2n+1}}}|K|^{2n}|K|=1.}
By the orthogonality relations for characters of representations of finite groups this fact implies the corresponding Stone–von Neumann theorem for Heisenberg groupsHn(Z/pZ), particularly:
Actually, all irreducible representations ofHn(K)on which the center acts nontrivially arise in this way.[6]: Chapter 14, Exercise 5
The Stone–von Neumann theorem admits numerous generalizations. Much of the early work ofGeorge Mackeywas directed at obtaining a formulation[7]of the theory ofinduced representationsdeveloped originally byFrobeniusfor finite groups to the context of unitary representations of locally compact topological groups. | https://en.wikipedia.org/wiki/Stone%E2%80%93von_Neumann_theorem |
Inmathematics,multipliers and centralizersare algebraic objects in the study ofBanach spaces. They are used, for example, in generalizations of theBanach–Stone theorem.
Let (X, ‖·‖) be a Banach space over a fieldK(either therealorcomplex numbers), and let Ext(X) be the set ofextreme pointsof theclosed unit ballof thecontinuous dual spaceX∗.
Acontinuous linear operatorT:X→Xis said to be amultiplierif every pointpin Ext(X) is aneigenvectorfor theadjoint operatorT∗:X∗→X∗. That is, there exists a functionaT: Ext(X) →Ksuch that
makingaT(p){\displaystyle a_{T}(p)}the eigenvalue corresponding top. Given two multipliersSandTonX,Sis said to be anadjointforTif
i.e.aSagrees withaTin the real case, and with thecomplex conjugateofaTin the complex case.
Thecentralizer(orcommutant) ofX, denotedZ(X), is the set of all multipliers onXfor which an adjoint exists. | https://en.wikipedia.org/wiki/Multipliers_and_centralizers_(Banach_spaces) |
Inmathematics, agroup actionof a groupG{\displaystyle G}on asetS{\displaystyle S}is agroup homomorphismfromG{\displaystyle G}to some group (underfunction composition) of functions fromS{\displaystyle S}to itself. It is said thatG{\displaystyle G}actsonS{\displaystyle S}.
Many sets oftransformationsform agroupunderfunction composition; for example, therotationsaround a point in the plane. It is often useful to consider the group as anabstract group, and to say that one has a group action of the abstract group that consists of performing the transformations of the group of transformations. The reason for distinguishing the group from the transformations is that, generally, a group of transformations of astructureacts also on various related structures; for example, the above rotation group also acts on triangles by transforming triangles into triangles.
If a group acts on a structure, it will usually also act on objects built from that structure. For example, the group ofEuclidean isometriesacts onEuclidean spaceand also on the figures drawn in it; in particular, it acts on the set of alltriangles. Similarly, the group ofsymmetriesof apolyhedronacts on thevertices, theedges, and thefacesof the polyhedron.
A group action on avector spaceis called arepresentationof the group. In the case of a finite-dimensional vector space, it allows one to identify many groups withsubgroupsof thegeneral linear groupGL(n,K){\displaystyle \operatorname {GL} (n,K)}, the group of theinvertible matricesofdimensionn{\displaystyle n}over afieldK{\displaystyle K}.
Thesymmetric groupSn{\displaystyle S_{n}}acts on anysetwithn{\displaystyle n}elements by permuting the elements of the set. Although the group of allpermutationsof a set depends formally on the set, the concept of group action allows one to consider a single group for studying the permutations of all sets with the samecardinality.
IfG{\displaystyle G}is agroupwithidentity elemente{\displaystyle e}, andX{\displaystyle X}is a set, then a (left)group actionα{\displaystyle \alpha }ofG{\displaystyle G}onXis afunction
that satisfies the following twoaxioms:[1]
for allgandhinGand allxinX{\displaystyle X}.
The groupG{\displaystyle G}is then said to act onX{\displaystyle X}(from the left). A setX{\displaystyle X}together with an action ofG{\displaystyle G}is called a (left)G{\displaystyle G}-set.
It can be notationally convenient tocurrythe actionα{\displaystyle \alpha }, so that, instead, one has a collection oftransformationsαg:X→X, with one transformationαgfor each group elementg∈G. The identity and compatibility relations then read
and
The second axiom states that the function composition is compatible with the group multiplication; they form acommutative diagram. This axiom can be shortened even further, and written asαg∘αh=αgh{\displaystyle \alpha _{g}\circ \alpha _{h}=\alpha _{gh}}.
With the above understanding, it is very common to avoid writingα{\displaystyle \alpha }entirely, and to replace it with either a dot, or with nothing at all. Thus,α(g,x)can be shortened tog⋅xorgx, especially when the action is clear from context. The axioms are then
From these two axioms, it follows that for any fixedginG{\displaystyle G}, the function fromXto itself which mapsxtog⋅xis abijection, with inverse bijection the corresponding map forg−1. Therefore, one may equivalently define a group action ofGonXas a group homomorphism fromGinto the symmetric groupSym(X)of all bijections fromXto itself.[2]
Likewise, aright group actionofG{\displaystyle G}onX{\displaystyle X}is a function
that satisfies the analogous axioms:[3]
(withα(x,g)often shortened toxgorx⋅gwhen the action being considered is clear from context)
for allgandhinGand allxinX.
The difference between left and right actions is in the order in which a productghacts onx. For a left action,hacts first, followed bygsecond. For a right action,gacts first, followed byhsecond. Because of the formula(gh)−1=h−1g−1, a left action can be constructed from a right action by composing with the inverse operation of the group. Also, a right action of a groupGonXcan be considered as a left action of itsopposite groupGoponX.
Thus, for establishing general properties of group actions, it suffices to consider only left actions. However, there are cases where this is not possible. For example, the multiplication of a groupinducesboth a left action and a right action on the group itself—multiplication on the left and on the right, respectively.
LetGbe a group acting on a setX. The action is calledfaithfuloreffectiveifg⋅x=xfor allx∈Ximplies thatg=eG. Equivalently, thehomomorphismfromGto the group of bijections ofXcorresponding to the action isinjective.
The action is calledfree(orsemiregularorfixed-point free) if the statement thatg⋅x=xfor somex∈Xalready implies thatg=eG. In other words, no non-trivial element ofGfixes a point ofX. This is a much stronger property than faithfulness.
For example, the action of any group on itself by left multiplication is free. This observation impliesCayley's theoremthat any group can beembeddedin a symmetric group (which is infinite when the group is). A finite group may act faithfully on a set of size much smaller than its cardinality (however such an action cannot be free). For instance the abelian 2-group(Z/ 2Z)n(of cardinality2n) acts faithfully on a set of size2n. This is not always the case, for example thecyclic groupZ/ 2nZcannot act faithfully on a set of size less than2n.
In general the smallest set on which a faithful action can be defined can vary greatly for groups of the same size. For example, three groups of size 120 are the symmetric groupS5, the icosahedral groupA5×Z/ 2Zand the cyclic groupZ/ 120Z. The smallest sets on which faithful actions can be defined for these groups are of size 5, 7, and 16 respectively.
The action ofGonXis calledtransitiveif for any two pointsx,y∈Xthere exists ag∈Gso thatg⋅x=y.
The action issimply transitive(orsharply transitive, orregular) if it is both transitive and free. This means that givenx,y∈Xthere is exactly oneg∈Gsuch thatg⋅x=y. IfXis acted upon simply transitively by a groupGthen it is called aprincipal homogeneous spaceforGor aG-torsor.
For an integern≥ 1, the action isn-transitiveifXhas at leastnelements, and for any pair ofn-tuples(x1, ...,xn), (y1, ...,yn) ∈Xnwith pairwise distinct entries (that isxi≠xj,yi≠yjwheni≠j) there exists ag∈Gsuch thatg⋅xi=yifori= 1, ...,n. In other words, the action on the subset ofXnof tuples without repeated entries is transitive. Forn= 2, 3this is often called double, respectively triple, transitivity. The class of2-transitive groups(that is, subgroups of a finite symmetric group whose action is 2-transitive) and more generallymultiply transitive groupsis well-studied in finite group theory.
An action issharplyn-transitivewhen the action on tuples without repeated entries inXnis sharply transitive.
The action of the symmetric group ofXis transitive, in factn-transitive for anynup to the cardinality ofX. IfXhas cardinalityn, the action of thealternating groupis(n− 2)-transitive but not(n− 1)-transitive.
The action of thegeneral linear groupof a vector spaceVon the setV∖ {0}of non-zero vectors is transitive, but not 2-transitive (similarly for the action of thespecial linear groupif the dimension ofvis at least 2). The action of theorthogonal groupof a Euclidean space is not transitive on nonzero vectors but it is on theunit sphere.
The action ofGonXis calledprimitiveif there is nopartitionofXpreserved by all elements ofGapart from the trivial partitions (the partition in a single piece and itsdual, the partition intosingletons).
Assume thatXis atopological spaceand the action ofGis byhomeomorphisms.
The action iswanderingif everyx∈Xhas aneighbourhoodUsuch that there are only finitely manyg∈Gwithg⋅U∩U≠ ∅.[4]
More generally, a pointx∈Xis called a point of discontinuity for the action ofGif there is an open subsetU∋xsuch that there are only finitely manyg∈Gwithg⋅U∩U≠ ∅. Thedomain of discontinuityof the action is the set of all points of discontinuity. Equivalently it is the largestG-stable open subsetΩ ⊂Xsuch that the action ofGonΩis wandering.[5]In a dynamical context this is also called awandering set.
The action isproperly discontinuousif for everycompactsubsetK⊂Xthere are only finitely manyg∈Gsuch thatg⋅K∩K≠ ∅. This is strictly stronger than wandering; for instance the action ofZonR2∖ {(0, 0)}given byn⋅(x,y) = (2nx, 2−ny)is wandering and free but not properly discontinuous.[6]
The action bydeck transformationsof thefundamental groupof a locallysimply connected spaceon auniversal coveris wandering and free. Such actions can be characterized by the following property: everyx∈Xhas a neighbourhoodUsuch thatg⋅U∩U= ∅for everyg∈G∖ {eG}.[7]Actions with this property are sometimes calledfreely discontinuous, and the largest subset on which the action is freely discontinuous is then called thefree regular set.[8]
An action of a groupGon alocally compact spaceXis calledcocompactif there exists a compact subsetA⊂Xsuch thatX=G⋅A. For a properly discontinuous action, cocompactness is equivalent to compactness of thequotient spaceX/G.
Now assumeGis atopological groupandXa topological space on which it acts by homeomorphisms. The action is said to becontinuousif the mapG×X→Xis continuous for theproduct topology.
The action is said to beproperif the mapG×X→X×Xdefined by(g,x) ↦ (x,g⋅x)isproper.[9]This means that given compact setsK,K′the set ofg∈Gsuch thatg⋅K∩K′ ≠ ∅is compact. In particular, this is equivalent to proper discontinuity ifGis adiscrete group.
It is said to belocally freeif there exists a neighbourhoodUofeGsuch thatg⋅x≠xfor allx∈Xandg∈U∖ {eG}.
The action is said to bestrongly continuousif the orbital mapg↦g⋅xis continuous for everyx∈X. Contrary to what the name suggests, this is a weaker property than continuity of the action.[citation needed]
IfGis aLie groupandXadifferentiable manifold, then the subspace ofsmooth pointsfor the action is the set of pointsx∈Xsuch that the mapg↦g⋅xissmooth. There is a well-developed theory ofLie group actions, i.e. action which are smooth on the whole space.
Ifgacts bylinear transformationson amoduleover acommutative ring, the action is said to beirreducibleif there are no proper nonzerog-invariant submodules. It is said to besemisimpleif it decomposes as adirect sumof irreducible actions.
Consider a groupGacting on a setX. Theorbitof an elementxinXis the set of elements inXto whichxcan be moved by the elements ofG. The orbit ofxis denoted byG⋅x:G⋅x={g⋅x:g∈G}.{\displaystyle G{\cdot }x=\{g{\cdot }x:g\in G\}.}
The defining properties of a group guarantee that the set of orbits of (pointsxin)Xunder the action ofGform apartitionofX. The associatedequivalence relationis defined by sayingx~yif and only ifthere exists aginGwithg⋅x=y. The orbits are then theequivalence classesunder this relation; two elementsxandyare equivalent if and only if their orbits are the same, that is,G⋅x=G⋅y.
The group action istransitiveif and only if it has exactly one orbit, that is, if there existsxinXwithG⋅x=X. This is the case if and only ifG⋅x=XforallxinX(given thatXis non-empty).
The set of all orbits ofXunder the action ofGis written asX/G(or, less frequently, asG\X), and is called thequotientof the action. In geometric situations it may be called theorbit space, while in algebraic situations it may be called the space ofcoinvariants, and writtenXG, by contrast with the invariants (fixed points), denotedXG: the coinvariants are aquotientwhile the invariants are asubset. The coinvariant terminology and notation are used particularly ingroup cohomologyandgroup homology, which use the same superscript/subscript convention.
IfYis asubsetofX, thenG⋅Ydenotes the set{g⋅y:g∈Gandy∈Y}. The subsetYis said to beinvariant underGifG⋅Y=Y(which is equivalentG⋅Y⊆Y). In that case,Galso operates onYbyrestrictingthe action toY. The subsetYis calledfixed underGifg⋅y=yfor allginGand allyinY. Every subset that is fixed underGis also invariant underG, but not conversely.
Every orbit is an invariant subset ofXon whichGactstransitively. Conversely, any invariant subset ofXis a union of orbits. The action ofGonXistransitiveif and only if all elements are equivalent, meaning that there is only one orbit.
AG-invariantelement ofXisx∈Xsuch thatg⋅x=xfor allg∈G. The set of all suchxis denotedXGand called theG-invariantsofX. WhenXis aG-module,XGis the zerothcohomologygroup ofGwith coefficients inX, and the higher cohomology groups are thederived functorsof thefunctorofG-invariants.
GivenginGandxinXwithg⋅x=x, it is said that "xis a fixed point ofg" or that "gfixesx". For everyxinX, thestabilizer subgroupofGwith respect tox(also called theisotropy grouporlittle group[10]) is the set of all elements inGthat fixx:Gx={g∈G:g⋅x=x}.{\displaystyle G_{x}=\{g\in G:g{\cdot }x=x\}.}This is asubgroupofG, though typically not a normal one. The action ofGonXisfreeif and only if all stabilizers are trivial. The kernelNof the homomorphism with the symmetric group,G→ Sym(X), is given by theintersectionof the stabilizersGxfor allxinX. IfNis trivial, the action is said to be faithful (or effective).
Letxandybe two elements inX, and letgbe a group element such thaty=g⋅x. Then the two stabilizer groupsGxandGyare related byGy=gGxg−1. Proof: by definition,h∈Gyif and only ifh⋅(g⋅x) =g⋅x. Applyingg−1to both sides of this equality yields(g−1hg)⋅x=x; that is,g−1hg∈Gx. An opposite inclusion follows similarly by takingh∈Gxandx=g−1⋅y.
The above says that the stabilizers of elements in the same orbit areconjugateto each other. Thus, to each orbit, we can associate aconjugacy classof a subgroup ofG(that is, the set of all conjugates of the subgroup). Let(H)denote the conjugacy class ofH. Then the orbitOhas type(H)if the stabilizerGxof some/anyxinObelongs to(H). A maximal orbit type is often called aprincipal orbit type.
Orbits and stabilizers are closely related. For a fixedxinX, consider the mapf:G→Xgiven byg↦g⋅x. By definition the imagef(G)of this map is the orbitG⋅x. The condition for two elements to have the same image isf(g)=f(h)⟺g⋅x=h⋅x⟺g−1h⋅x=x⟺g−1h∈Gx⟺h∈gGx.{\displaystyle f(g)=f(h)\iff g{\cdot }x=h{\cdot }x\iff g^{-1}h{\cdot }x=x\iff g^{-1}h\in G_{x}\iff h\in gG_{x}.}In other words,f(g) =f(h)if and only ifgandhlie in the samecosetfor the stabilizer subgroupGx. Thus, thefiberf−1({y})offover anyyinG⋅xis contained in such a coset, and every such coset also occurs as a fiber. Thereforefinduces abijectionbetween the setG/Gxof cosets for the stabilizer subgroup and the orbitG⋅x, which sendsgGx↦g⋅x.[11]This result is known as theorbit-stabilizer theorem.
IfGis finite then the orbit-stabilizer theorem, together withLagrange's theorem, gives|G⋅x|=[G:Gx]=|G|/|Gx|,{\displaystyle |G\cdot x|=[G\,:\,G_{x}]=|G|/|G_{x}|,}in other words the length of the orbit ofxtimes the order of its stabilizer is theorder of the group. In particular that implies that the orbit length is a divisor of the group order.
This result is especially useful since it can be employed for counting arguments (typically in situations whereXis finite as well).
A result closely related to the orbit-stabilizer theorem isBurnside's lemma:|X/G|=1|G|∑g∈G|Xg|,{\displaystyle |X/G|={\frac {1}{|G|}}\sum _{g\in G}|X^{g}|,}whereXgis the set of points fixed byg. This result is mainly of use whenGandXare finite, when it can be interpreted as follows: the number of orbits is equal to the average number of points fixed per group element.
Fixing a groupG, the set of formal differences of finiteG-sets forms a ring called theBurnside ringofG, where addition corresponds todisjoint union, and multiplication toCartesian product.
The notion of group action can be encoded by theactiongroupoidG′ =G⋉Xassociated to the group action. The stabilizers of the action are the vertex groups of the groupoid and the orbits of the action are its components.
IfXandYare twoG-sets, amorphismfromXtoYis a functionf:X→Ysuch thatf(g⋅x) =g⋅f(x)for allginGand allxinX. Morphisms ofG-sets are also calledequivariant mapsorG-maps.
The composition of two morphisms is again a morphism. If a morphismfis bijective, then its inverse is also a morphism. In this casefis called anisomorphism, and the twoG-setsXandYare calledisomorphic; for all practical purposes, isomorphicG-sets are indistinguishable.
Some example isomorphisms:
With this notion of morphism, the collection of allG-sets forms acategory; this category is aGrothendieck topos(in fact, assuming a classicalmetalogic, thistoposwill even be Boolean).
We can also consider actions ofmonoidson sets, by using the same two axioms as above. This does not define bijective maps and equivalence relations however. Seesemigroup action.
Instead of actions on sets, we can define actions of groups and monoids on objects of an arbitrary category: start with an objectXof some category, and then define an action onXas a monoid homomorphism into the monoid ofendomorphismsofX. IfXhas an underlying set, then all definitions and facts stated above can be carried over. For example, if we take the category of vector spaces, we obtaingroup representationsin this fashion.
We can view a groupGas a category with a single object in which every morphism isinvertible.[15]A (left) group action is then nothing but a (covariant)functorfromGto thecategory of sets, and a group representation is a functor fromGto thecategory of vector spaces.[16]A morphism betweenG-sets is then anatural transformationbetween the group action functors.[17]In analogy, an action of agroupoidis a functor from the groupoid to the category of sets or to some other category.
In addition tocontinuous actionsof topological groups on topological spaces, one also often considerssmooth actionsof Lie groups onsmooth manifolds, regular actions ofalgebraic groupsonalgebraic varieties, andactionsofgroup schemesonschemes. All of these are examples ofgroup objectsacting on objects of their respective category. | https://en.wikipedia.org/wiki/Stabilizer_subgroup |
Mathematical diagrams, such aschartsandgraphs, are mainly designed to convey mathematical relationships—for example, comparisons over time.[1]
Acomplex numbercan be visually represented as a pair of numbers forming a vector on a diagram called anArgand diagramThecomplex planeis sometimes called theArgand planebecause it is used inArgand diagrams. These are named afterJean-Robert Argand(1768–1822), although they were first described by Norwegian-Danish land surveyor and mathematicianCaspar Wessel(1745–1818).[2]Argand diagrams are frequently used to plot the positions of thepolesandzeroesof afunctionin the complex plane.
The concept of the complex plane allows ageometricinterpretation of complex numbers. Underaddition, they add likevectors. Themultiplicationof two complex numbers can be expressed most easily inpolar coordinates— the magnitude ormodulusof the product is the product of the twoabsolute values, or moduli, and the angle orargumentof the product is the sum of the two angles, or arguments. In particular, multiplication by a complex number of modulus 1 acts as a rotation.
In the context offast Fourier transformalgorithms, abutterflyis a portion of the computation that combines the results of smallerdiscrete Fourier transforms(DFTs) into a larger DFT, or vice versa (breaking a larger DFT up into subtransforms). The name "butterfly" comes from the shape of the data-flow diagram in the radix-2 case, as described below. The same structure can also be found in theViterbi algorithm, used for finding the most likely sequence of hidden states.
Thebutterfly diagramshow a data-flow diagram connecting the inputsx(left) to the outputsythat depend on them (right) for a "butterfly" step of a radix-2Cooley–Tukey FFT algorithm. This diagram resembles abutterflyas in theMorpho butterflyshown for comparison, hence the name.
In mathematics, and especially incategory theory, a commutative diagram is a diagram ofobjects, also known as vertices, andmorphisms, also known as arrows or edges, such that when selecting two objects any directed path through the diagram leads to the same result by composition.
Commutative diagrams play the role in category theory that equations play in algebra.
AHasse diagramis a simple picture of a finitepartially ordered set, forming adrawingof the partial order'stransitive reduction. Concretely, one represents each element of the set as a vertex on the page and draws a line segment or curve that goes upward fromxtoyprecisely whenx<yand there is nozsuch thatx<z<y. In this case, we say ycoversx, or y is an immediate successor of x. In a Hasse diagram, it is required that the curves be drawn so that each meets exactly two vertices: its two endpoints. Any such diagram (given that the vertices are labeled) uniquely determines a partial order, and any partial order has a unique transitive reduction, but there are many possible placements of elements in the plane, resulting in different Hasse diagrams for a given order that may have widely varying appearances.
InKnot theorya useful way to visualise and manipulate knots is to project the knot onto a plane—;think of the knot casting a shadow on the wall. A small perturbation in the choice of projection will ensure that it isone-to-oneexcept at the double points, calledcrossings, where the "shadow" of the knot crosses itself once transversely[3]
At each crossing we must indicate which section is "over" and which is "under", so as to be able to recreate the original knot. This is often done by creating a break in the strand going underneath. If by following the diagram the knot alternately crosses itself "over" and "under", then the diagram represents a particularly well-studied class of knot,alternating knots.
AVenn diagramis a representation of mathematical sets: a mathematical diagram representing sets as circles, with their relationships to each other expressed through their overlapping positions, so that all possible relationships between the sets are shown.[4]
The Venn diagram is constructed with a collection of simple closed curves drawn in the plane. The principle of these diagrams is that classes be represented by regions in such relation to one another that all the possible logical relations of these classes can be indicated in the same diagram. That is, the diagram initially leaves room for any possible relation of the classes, and the actual or given relation, can then be specified by indicating that some particular region is null or is not null.[5]
AVoronoi diagramis a special kind of decomposition of ametric spacedetermined by distances to a specified discrete set of objects in the space, e.g., by adiscrete setof points. This diagram is named afterGeorgy Voronoi, also called a Voronoitessellation, a Voronoi decomposition, or a Dirichlet tessellation afterPeter Gustav Lejeune Dirichlet.
In the simplest case, we are given a set of points S in the plane, which are the Voronoi sites. Each site s has a Voronoi cell V(s) consisting of all points closer to s than to any other site. The segments of the Voronoi diagram are all the points in the plane that are equidistant to two sites. The Voronoi nodes are the points equidistant to three (or more) sites
Awallpaper grouporplane symmetry grouporplane crystallographic groupis a mathematical classification of a two-dimensional repetitive pattern, based on the symmetries in the pattern. Such patterns occur frequently in architecture and decorative art. There are 17 possible distinctgroups.
Wallpaper groups are two-dimensionalsymmetry groups, intermediate in complexity between the simplerfrieze groupsand the three-dimensionalcrystallographic groups, also calledspace groups. Wallpaper groups categorize patterns by their symmetries. Subtle differences may place similar patterns in different groups, while patterns which are very different in style, color, scale or orientation may belong to the same group.
AYoung diagramorYoung tableau, also calledFerrers diagram, is a finite collection of boxes, or cells, arranged in left-justified rows, with the row sizes weakly decreasing (each row has the same or shorter length than its predecessor).
Listing the number of boxes in each row gives apartitionλ{\displaystyle \lambda }of a positive integern, the total number of boxes of the diagram. The Young diagram is said to be of shapeλ{\displaystyle \lambda }, and it carries the same information as that partition. Listing the number of boxes in each column gives another partition, theconjugateortransposepartition ofλ{\displaystyle \lambda }; one obtains a Young diagram of that shape by reflecting the original diagram along its main diagonal.
Young tableaux were introduced byAlfred Young, amathematicianatCambridge University, in 1900. They were then applied to the study of symmetric group byGeorg Frobeniusin 1903. Their theory was further developed by many mathematicians. | https://en.wikipedia.org/wiki/Mathematical_diagram |
Inmathematics,anticommutativityis a specific property of some non-commutativemathematicaloperations. Swapping the position oftwo argumentsof an antisymmetric operation yields a result which is theinverseof the result with unswapped arguments. The notioninverserefers to agroup structureon the operation'scodomain, possibly with another operation.Subtractionis an anticommutative operation because commuting the operands ofa−bgivesb−a= −(a−b);for example,2 − 10 = −(10 − 2) = −8.Another prominent example of an anticommutative operation is theLie bracket.
Inmathematical physics, wheresymmetryis of central importance, or even just inmultilinear algebrathese operations are mostly (multilinear with respect to somevector structuresand then) calledantisymmetric operations, and when they are not already ofaritygreater than two, extended in anassociativesetting to cover more than twoarguments.
IfA,B{\displaystyle A,B}are twoabelian groups, abilinear mapf:A2→B{\displaystyle f\colon A^{2}\to B}isanticommutativeif for allx,y∈A{\displaystyle x,y\in A}we have
More generally, amultilinear mapg:An→B{\displaystyle g:A^{n}\to B}is anticommutative if for allx1,…xn∈A{\displaystyle x_{1},\dots x_{n}\in A}we have
wheresgn(σ){\displaystyle {\text{sgn}}(\sigma )}is thesignof thepermutationσ{\displaystyle \sigma }.
If the abelian groupB{\displaystyle B}has no 2-torsion, implying that ifx=−x{\displaystyle x=-x}thenx=0{\displaystyle x=0}, then any anticommutative bilinear mapf:A2→B{\displaystyle f\colon A^{2}\to B}satisfies
More generally, bytransposingtwo elements, any anticommutative multilinear mapg:An→B{\displaystyle g\colon A^{n}\to B}satisfies
if any of thexi{\displaystyle x_{i}}are equal; such a map is said to bealternating. Conversely, using multilinearity, any alternating map is anticommutative. In the binary case this works as follows: iff:A2→B{\displaystyle f\colon A^{2}\to B}is alternating then by bilinearity we have
and the proof in the multilinear case is the same but in only two of the inputs.
Examples of anticommutative binary operations include: | https://en.wikipedia.org/wiki/Anticommutativity |
Inabstract algebra, the termassociatoris used in different ways as a measure of thenon-associativityof analgebraic structure. Associators are commonly studied astriple systems.
For anon-associative ringoralgebraR, theassociatoris themultilinear map[⋅,⋅,⋅]:R×R×R→R{\displaystyle [\cdot ,\cdot ,\cdot ]:R\times R\times R\to R}given by
Just as thecommutator
measures the degree ofnon-commutativity, the associator measures the degree of non-associativity ofR.
For anassociative ringoralgebrathe associator is identically zero.
The associator in any ring obeys the identity
The associator isalternatingprecisely whenRis analternative ring.
The associator is symmetric in its two rightmost arguments whenRis apre-Lie algebra.
Thenucleusis thesetof elements that associate with all others: that is, theninRsuch that
The nucleus is an associative subring ofR.
AquasigroupQis a set with abinary operation⋅:Q×Q→Q{\displaystyle \cdot :Q\times Q\to Q}such that for eacha,binQ,
the equationsa⋅x=b{\displaystyle a\cdot x=b}andy⋅a=b{\displaystyle y\cdot a=b}have unique solutionsx,yinQ. In a quasigroupQ, the associator is the map(⋅,⋅,⋅):Q×Q×Q→Q{\displaystyle (\cdot ,\cdot ,\cdot ):Q\times Q\times Q\to Q}defined by the equation
for alla,b,cinQ. As with its ring theory analog, the quasigroup associator is a measure of nonassociativity ofQ.
Inhigher-dimensional algebra, where there may be non-identitymorphismsbetween algebraic expressions, anassociatoris anisomorphism
Incategory theory, the associator expresses the associative properties of the internal productfunctorinmonoidal categories.
Thisalgebra-related article is astub. You can help Wikipedia byexpanding it. | https://en.wikipedia.org/wiki/Associator |
Inmathematics, theBaker–Campbell–Hausdorff formulagives the value ofZ{\displaystyle Z}that solves the equationeXeY=eZ{\displaystyle e^{X}e^{Y}=e^{Z}}for possiblynoncommutativeXandYin theLie algebraof aLie group. There are various ways of writing the formula, but all ultimately yield an expression forZ{\displaystyle Z}in Lie algebraic terms, that is, as a formal series (not necessarily convergent) inX{\displaystyle X}andY{\displaystyle Y}and iterated commutators thereof. The first few terms of this series are:Z=X+Y+12[X,Y]+112[X,[X,Y]]+112[Y,[Y,X]]+⋯,{\displaystyle Z=X+Y+{\frac {1}{2}}[X,Y]+{\frac {1}{12}}[X,[X,Y]]+{\frac {1}{12}}[Y,[Y,X]]+\cdots \,,}where "⋯{\displaystyle \cdots }" indicates terms involving highercommutatorsofX{\displaystyle X}andY{\displaystyle Y}. IfX{\displaystyle X}andY{\displaystyle Y}are sufficiently small elements of the Lie algebrag{\displaystyle {\mathfrak {g}}}of a Lie groupG{\displaystyle G}, the series is convergent. Meanwhile, every elementg{\displaystyle g}sufficiently close to the identity inG{\displaystyle G}can be expressed asg=eX{\displaystyle g=e^{X}}for a smallX{\displaystyle X}ing{\displaystyle {\mathfrak {g}}}. Thus, we can say thatnear the identitythe group multiplication inG{\displaystyle G}—written aseXeY=eZ{\displaystyle e^{X}e^{Y}=e^{Z}}—can be expressed in purely Lie algebraic terms. The Baker–Campbell–Hausdorff formula can be used to give comparatively simple proofs of deep results in theLie group–Lie algebra correspondence.
IfX{\displaystyle X}andY{\displaystyle Y}are sufficiently smalln×n{\displaystyle n\times n}matrices, thenZ{\displaystyle Z}can be computed as the logarithm ofeXeY{\displaystyle e^{X}e^{Y}}, where the exponentials and the logarithm can be computed aspower series. The point of the Baker–Campbell–Hausdorff formula is then the highly nonobvious claim thatZ:=log(eXeY){\displaystyle Z:=\log \left(e^{X}e^{Y}\right)}can be expressed as a series in repeated commutators ofX{\displaystyle X}andY{\displaystyle Y}.
Modern expositions of the formula can be found in, among other places, the books of Rossmann[1]and Hall.[2]
The formula is named afterHenry Frederick Baker,John Edward Campbell, andFelix Hausdorffwho stated its qualitative form, i.e. that only commutators and commutators of commutators, ad infinitum, are needed to express the solution. An earlier statement of the form was adumbrated byFriedrich Schurin 1890[3]where a convergent power series is given, with terms recursively defined.[4]This qualitative form is what is used in the most important applications, such as the relatively accessible proofs of theLie correspondenceand inquantum field theory. Following Schur, it was noted in print by Campbell[5](1897); elaborated byHenri Poincaré[6](1899) and Baker (1902);[7]and systematized geometrically, and linked to theJacobi identityby Hausdorff (1906).[8]The first actual explicit formula, with all numerical coefficients, is due toEugene Dynkin(1947).[9]The history of the formula is described in detail in the article of Achilles and Bonfiglioli[10]and in the book of Bonfiglioli and Fulci.[11]
For many purposes, it is only necessary to know that an expansion forZ{\displaystyle Z}in terms of iterated commutators ofX{\displaystyle X}andY{\displaystyle Y}exists; the exact coefficients are often irrelevant. (See, for example, the discussion of the relationship betweenLie group and Lie algebra homomorphismsin Section 5.2 of Hall's book,[2]where the precise coefficients play no role in the argument.) A remarkably direct existence proof was given byMartin Eichler,[12]see also the "Existence results" section below.
In other cases, one may need detailed information aboutZ{\displaystyle Z}and it is therefore desirable to computeZ{\displaystyle Z}as explicitly as possible. Numerous formulas exist; we will describe two of the main ones (Dynkin's formula and the integral formula of Poincaré) in this section.
LetGbe a Lie group with Lie algebrag{\displaystyle {\mathfrak {g}}}. Letexp:g→G{\displaystyle \exp :{\mathfrak {g}}\to G}be theexponential map.
The following general combinatorial formula was introduced byEugene Dynkin(1947),[13][14]log(expXexpY)=∑n=1∞(−1)n−1n∑r1+s1>0⋮rn+sn>0[Xr1Ys1Xr2Ys2⋯XrnYsn](∑j=1n(rj+sj))⋅∏i=1nri!si!,{\displaystyle \log(\exp X\exp Y)=\sum _{n=1}^{\infty }{\frac {(-1)^{n-1}}{n}}\sum _{\begin{smallmatrix}r_{1}+s_{1}>0\\\vdots \\r_{n}+s_{n}>0\end{smallmatrix}}{\frac {[X^{r_{1}}Y^{s_{1}}X^{r_{2}}Y^{s_{2}}\dotsm X^{r_{n}}Y^{s_{n}}]}{\left(\sum _{j=1}^{n}(r_{j}+s_{j})\right)\cdot \prod _{i=1}^{n}r_{i}!s_{i}!}},}where the sum is performed over all nonnegative values ofsi{\displaystyle s_{i}}andri{\displaystyle r_{i}}, and the following notation has been used:[Xr1Ys1⋯XrnYsn]=[X,[X,⋯[X⏟r1,[Y,[Y,⋯[Y⏟s1,⋯[X,[X,⋯[X⏟rn,[Y,[Y,⋯Y⏟sn]]⋯]]{\displaystyle [X^{r_{1}}Y^{s_{1}}\dotsm X^{r_{n}}Y^{s_{n}}]=[\underbrace {X,[X,\dotsm [X} _{r_{1}},[\underbrace {Y,[Y,\dotsm [Y} _{s_{1}},\,\dotsm \,[\underbrace {X,[X,\dotsm [X} _{r_{n}},[\underbrace {Y,[Y,\dotsm Y} _{s_{n}}]]\dotsm ]]}with the understanding that[X] :=X.
The series is not convergent in general; it is convergent (and the stated formula is valid) for all sufficiently smallX{\displaystyle X}andY{\displaystyle Y}.
Since[A,A] = 0, the term is zero ifsn>1{\displaystyle s_{n}>1}or ifsn=0{\displaystyle s_{n}=0}andrn>1{\displaystyle r_{n}>1}.[15]
The first few terms are well-known, with all higher-order terms involving[X,Y]andcommutatornestings thereof (thus in theLie algebra):
Z(X,Y)=log(expXexpY)=X+Y+12[X,Y]+112([X,[X,Y]]+[Y,[Y,X]])−124[Y,[X,[X,Y]]]−1720([Y,[Y,[Y,[Y,X]]]]+[X,[X,[X,[X,Y]]]])+1360([X,[Y,[Y,[Y,X]]]]+[Y,[X,[X,[X,Y]]]])+1120([Y,[X,[Y,[X,Y]]]]+[X,[Y,[X,[Y,X]]]])+1240([X,[Y,[X,[Y,[X,Y]]]]])+1720([X,[Y,[X,[X,[X,Y]]]]]−[X,[X,[Y,[Y,[X,Y]]]]])+11440([X,[Y,[Y,[Y,[X,Y]]]]]−[X,[X,[Y,[X,[X,Y]]]]])+⋯{\displaystyle {\begin{aligned}Z(X,Y)&=\log(\exp X\exp Y)\\&{}=X+Y+{\frac {1}{2}}[X,Y]+{\frac {1}{12}}\left([X,[X,Y]]+[Y,[Y,X]]\right)\\&{}\quad -{\frac {1}{24}}[Y,[X,[X,Y]]]\\&{}\quad -{\frac {1}{720}}\left([Y,[Y,[Y,[Y,X]]]]+[X,[X,[X,[X,Y]]]]\right)\\&{}\quad +{\frac {1}{360}}\left([X,[Y,[Y,[Y,X]]]]+[Y,[X,[X,[X,Y]]]]\right)\\&{}\quad +{\frac {1}{120}}\left([Y,[X,[Y,[X,Y]]]]+[X,[Y,[X,[Y,X]]]]\right)\\&{}\quad +{\frac {1}{240}}\left([X,[Y,[X,[Y,[X,Y]]]]]\right)\\&{}\quad +{\frac {1}{720}}\left([X,[Y,[X,[X,[X,Y]]]]]-[X,[X,[Y,[Y,[X,Y]]]]]\right)\\&{}\quad +{\frac {1}{1440}}\left([X,[Y,[Y,[Y,[X,Y]]]]]-[X,[X,[Y,[X,[X,Y]]]]]\right)+\cdots \end{aligned}}}
The above lists all summands of order 6 or lower (i.e. those containing 6 or fewerX's andY's). TheX↔Y(anti-)/symmetry in alternating orders of the expansion, follows fromZ(Y,X) = −Z(−X, −Y). A complete elementary proof of this formula can be found in the article on thederivative of the exponential map.
There are numerous other expressions forZ{\displaystyle Z}, many of which are used in the physics literature.[16][17]A popular integral formula is[18][19]log(eXeY)=X+(∫01ψ(eadXetadY)dt)Y,{\displaystyle \log \left(e^{X}e^{Y}\right)=X+\left(\int _{0}^{1}\psi \left(e^{\operatorname {ad} _{X}}~e^{t\operatorname {ad} _{Y}}\right)dt\right)Y,}involving thegenerating function for the Bernoulli numbers,ψ(x)=defxlogxx−1=1−∑n=1∞(1−x)nn(n+1),{\displaystyle \psi (x)~{\stackrel {\text{def}}{=}}~{\frac {x\log x}{x-1}}=1-\sum _{n=1}^{\infty }{(1-x)^{n} \over n(n+1)}~,}utilized by Poincaré and Hausdorff.[nb 1]
For a matrix Lie groupG⊂GL(n,R){\displaystyle G\subset {\mbox{GL}}(n,\mathbb {R} )}the Lie algebra is thetangent spaceof the identityI, and the commutator is simply[X,Y] =XY−YX; the exponential map is thestandard exponential map of matrices,expX=eX=∑n=0∞Xnn!.{\displaystyle \exp X=e^{X}=\sum _{n=0}^{\infty }{\frac {X^{n}}{n!}}.}
When one solves forZineZ=eXeY,{\displaystyle e^{Z}=e^{X}e^{Y},}using the series expansions forexpandlogone obtains a simpler formula:Z=∑n>0(−1)n−1n∑1≤i≤nri+si>0Xr1Ys1⋯XrnYsnr1!s1!⋯rn!sn!,‖X‖+‖Y‖<log2,‖Z‖<log2.{\displaystyle Z=\sum _{n>0}{\frac {(-1)^{n-1}}{n}}\sum _{\stackrel {r_{i}+s_{i}>0}{1\leq i\leq n}}{\frac {X^{r_{1}}Y^{s_{1}}\cdots X^{r_{n}}Y^{s_{n}}}{r_{1}!s_{1}!\cdots r_{n}!s_{n}!}},\quad \|X\|+\|Y\|<\log 2,\|Z\|<\log 2.}[nb 2]The first, second, third, and fourth order terms are:
The formulas for the variouszj{\displaystyle z_{j}}'s isnotthe Baker–Campbell–Hausdorff formula. Rather, the Baker–Campbell–Hausdorff formula is one of various expressions forzj{\displaystyle z_{j}}'sin terms of repeated commutators ofX{\displaystyle X}andY{\displaystyle Y}. The point is that it is far from obvious that it is possible to express eachzj{\displaystyle z_{j}}in terms of commutators. (The reader is invited, for example, to verify by direct computation thatz3{\displaystyle z_{3}}is expressible as a linear combination of the two nontrivial third-order commutators ofX{\displaystyle X}andY{\displaystyle Y}, namely[X,[X,Y]]{\displaystyle [X,[X,Y]]}and[Y,[X,Y]]{\displaystyle [Y,[X,Y]]}.) The general result that eachzj{\displaystyle z_{j}}is expressible as a combination of commutators was shown in an elegant, recursive way by Eichler.[12]
A consequence of the Baker–Campbell–Hausdorff formula is the following result about thetrace:trlog(eXeY)=trX+trY.{\displaystyle \operatorname {tr} \log \left(e^{X}e^{Y}\right)=\operatorname {tr} X+\operatorname {tr} Y.}That is to say, since eachzj{\displaystyle z_{j}}withj≥2{\displaystyle j\geq 2}is expressible as a linear combination of commutators, the trace of each such terms is zero.
SupposeX{\displaystyle X}andY{\displaystyle Y}are the following matrices in the Lie algebrasl(2;C){\displaystyle {\mathfrak {sl}}(2;\mathbb {C} )}(the space of2×2{\displaystyle 2\times 2}matrices with trace zero):X=(0iπiπ0);Y=(0100).{\displaystyle X={\begin{pmatrix}0&i\pi \\i\pi &0\end{pmatrix}};\quad Y={\begin{pmatrix}0&1\\0&0\end{pmatrix}}.}TheneXeY=(−100−1)(1101)=(−1−10−1).{\displaystyle e^{X}e^{Y}={\begin{pmatrix}-1&0\\0&-1\end{pmatrix}}{\begin{pmatrix}1&1\\0&1\end{pmatrix}}={\begin{pmatrix}-1&-1\\0&-1\end{pmatrix}}.}It is then not hard to show[20]that there does not exist a matrixZ{\displaystyle Z}insl(2;C){\displaystyle \operatorname {sl} (2;\mathbb {C} )}witheXeY=eZ{\displaystyle e^{X}e^{Y}=e^{Z}}. (Similar examples may be found in the article of Wei.[21])
This simple example illustrates that the various versions of the Baker–Campbell–Hausdorff formula, which give expressions forZin terms of iterated Lie-brackets ofXandY, describeformalpower series whose convergence is not guaranteed. Thus, if one wantsZto be an actual element of the Lie algebra containingXandY(as opposed to a formal power series), one has to assume thatXandYare small. Thus, the conclusion that the product operation on a Lie group is determined by the Lie algebra is only a local statement. Indeed, the result cannot be global, because globally one can have nonisomorphic Lie groups with isomorphic Lie algebras.
Concretely, if working with a matrix Lie algebra and‖⋅‖{\displaystyle \|\cdot \|}is a givensubmultiplicative matrix norm, convergence is guaranteed[14][22]if‖X‖+‖Y‖<ln22.{\displaystyle \|X\|+\|Y\|<{\frac {\ln 2}{2}}.}
IfX{\displaystyle X}andY{\displaystyle Y}commute, that is[X,Y]=0{\displaystyle [X,Y]=0}, the Baker–Campbell–Hausdorff formula reduces toeXeY=eX+Y{\displaystyle e^{X}e^{Y}=e^{X+Y}}.
Another case assumes that[X,Y]{\displaystyle [X,Y]}commutes with bothX{\displaystyle X}andY{\displaystyle Y}, as for thenilpotentHeisenberg group. Then the formula reduces to itsfirst three terms.
Theorem([23])—IfX{\displaystyle X}andY{\displaystyle Y}commute with their commutator,[X,[X,Y]]=[Y,[X,Y]]=0{\displaystyle [X,[X,Y]]=[Y,[X,Y]]=0}, theneXeY=eX+Y+12[X,Y]{\displaystyle e^{X}e^{Y}=e^{X+Y+{\frac {1}{2}}[X,Y]}}.
This is the degenerate case used routinely inquantum mechanics, as illustrated below and is sometimes known as thedisentangling theorem.[24]In this case, there are no smallness restrictions onX{\displaystyle X}andY{\displaystyle Y}. This result is behind the "exponentiated commutation relations" that enter into theStone–von Neumann theorem. A simple proof of this identity is given below.
Another useful form of the general formula emphasizes expansion in terms ofYand uses theadjointmapping notationadX(Y)=[X,Y]{\displaystyle \operatorname {ad} _{X}(Y)=[X,Y]}:log(expXexpY)=X+adX1−e−adXY+O(Y2)=X+adX/2(1+cothadX/2)Y+O(Y2),{\displaystyle \log(\exp X\exp Y)=X+{\frac {\operatorname {ad} _{X}}{1-e^{-\operatorname {ad} _{X}}}}~Y+O\left(Y^{2}\right)=X+\operatorname {ad} _{X/2}(1+\coth \operatorname {ad} _{X/2})~Y+O\left(Y^{2}\right),}which is evident from the integral formula above. (The coefficients of the nested commutators with a singleY{\displaystyle Y}are normalized Bernoulli numbers.)
Now assume that the commutator is a multiple ofY{\displaystyle Y}, so that[X,Y]=sY{\displaystyle [X,Y]=sY}. Then all iterated commutators will be multiples ofY{\displaystyle Y}, and no quadratic or higher terms inY{\displaystyle Y}appear. Thus, theO(Y2){\displaystyle O\left(Y^{2}\right)}term above vanishes and we obtain:
Theorem([25])—If[X,Y]=sY{\displaystyle [X,Y]=sY}, wheres{\displaystyle s}is a complex number withs≠2πin{\displaystyle s\neq 2\pi in}for all integersn{\displaystyle n}, then we haveeXeY=exp(X+s1−e−sY).{\displaystyle e^{X}e^{Y}=\exp \left(X+{\frac {s}{1-e^{-s}}}Y\right).}
Again, in this case there are no smallness restriction onX{\displaystyle X}andY{\displaystyle Y}. The restriction ons{\displaystyle s}guarantees that the expression on the right side makes sense. (Whens=0{\displaystyle s=0}we may interpretlims→0s/(1−e−s)=1{\textstyle \lim _{s\to 0}s/(1-e^{-s})=1}.) We also obtain a simple "braiding identity":eXeY=eexp(s)YeX,{\displaystyle e^{X}e^{Y}=e^{\exp(s)Y}e^{X},}which may be written as an adjoint dilation:eXeYe−X=eexp(s)Y.{\displaystyle e^{X}e^{Y}e^{-X}=e^{\exp(s)\,Y}.}
IfX{\displaystyle X}andY{\displaystyle Y}are matrices, one can computeZ:=log(eXeY){\displaystyle Z:=\log \left(e^{X}e^{Y}\right)}using the power series for the exponential and logarithm, with convergence of the series ifX{\displaystyle X}andY{\displaystyle Y}are sufficiently small. It is natural to collect together all terms where the total degree inX{\displaystyle X}andY{\displaystyle Y}equals a fixed numberk{\displaystyle k}, giving an expressionzk{\displaystyle z_{k}}. (See the section "Matrix Lie group illustration" above for formulas for the first severalzk{\displaystyle z_{k}}'s.) A remarkably direct and concise, recursive proof that eachzk{\displaystyle z_{k}}is expressible in terms of repeated commutators ofX{\displaystyle X}andY{\displaystyle Y}was given byMartin Eichler.[12]
Alternatively, we can give an existence argument as follows. The Baker–Campbell–Hausdorff formula implies that ifXandYare in someLie algebrag,{\displaystyle {\mathfrak {g}},}defined over any field ofcharacteristic 0likeR{\displaystyle \mathbb {R} }orC{\displaystyle \mathbb {C} }, thenZ=log(exp(X)exp(Y)),{\displaystyle Z=\log(\exp(X)\exp(Y)),}can formally be written as an infinite sum of elements ofg{\displaystyle {\mathfrak {g}}}. [This infinite series may or may not converge, so it need not define an actual elementZing{\displaystyle {\mathfrak {g}}}.] For many applications, the mere assurance of the existence of this formal expression is sufficient, and an explicit expression for this infinite sum is not needed. This is for instance the case in theLorentzian[26]construction of a Lie group representation from a Lie algebra representation. Existence can be seen as follows.
We consider the ringS=R[[X,Y]]{\displaystyle S=\mathbb {R} [[X,Y]]}of allnon-commuting formal power serieswith real coefficients in the non-commuting variablesXandY. There is aring homomorphismfromSto thetensor productofSwithSoverR,Δ:S→S⊗S,{\displaystyle \Delta \colon S\to S\otimes S,}called thecoproduct, such thatΔ(X)=X⊗1+1⊗X{\displaystyle \Delta (X)=X\otimes 1+1\otimes X}andΔ(Y)=Y⊗1+1⊗Y.{\displaystyle \Delta (Y)=Y\otimes 1+1\otimes Y.}(The definition of Δ is extended to the other elements ofSby requiringR-linearity, multiplicativity and infinite additivity.)
One can then verify the following properties:
The existence of the Campbell–Baker–Hausdorff formula can now be seen as follows:[13]The elementsXandYare primitive, soexp(X){\displaystyle \exp(X)}andexp(Y){\displaystyle \exp(Y)}are grouplike; so their productexp(X)exp(Y){\displaystyle \exp(X)\exp(Y)}is also grouplike; so its logarithmlog(exp(X)exp(Y)){\displaystyle \log(\exp(X)\exp(Y))}is primitive; and hence can be written as an infinite sum of elements of the Lie algebra generated byXandY.
Theuniversal enveloping algebraof thefree Lie algebragenerated byXandYis isomorphic to the algebra of allnon-commuting polynomialsinXandY. In common with all universal enveloping algebras, it has a natural structure of aHopf algebra, with a coproductΔ. The ringSused above is just a completion of this Hopf algebra.
A related combinatoric expansion that is useful in dual[16]applications iset(X+Y)=etXetYe−t22[X,Y]et36(2[Y,[X,Y]]+[X,[X,Y]])e−t424([[[X,Y],X],X]+3[[[X,Y],X],Y]+3[[[X,Y],Y],Y])⋯{\displaystyle e^{t(X+Y)}=e^{tX}~e^{tY}~e^{-{\frac {t^{2}}{2}}[X,Y]}~e^{{\frac {t^{3}}{6}}(2[Y,[X,Y]]+[X,[X,Y]])}~e^{{\frac {-t^{4}}{24}}([[[X,Y],X],X]+3[[[X,Y],X],Y]+3[[[X,Y],Y],Y])}\cdots }where the exponents of higher order intare likewise nested commutators, i.e., homogeneous Lie polynomials.[27]These exponents,Cninexp(−tX) exp(t(X+Y)) = Πnexp(tnCn), follow recursively by application of the above BCH expansion.
As a corollary of this, theSuzuki–Trotter decompositionfollows.
The following identity (Campbell 1897) leads to a special case of the Baker–Campbell–Hausdorff formula.
LetGbe a matrix Lie group andgits corresponding Lie algebra. LetadXbe the linear operator ongdefined byadXY= [X,Y] =XY−YXfor some fixedX∈g. (Theadjoint endomorphismencountered above.) Denote withAdAfor fixedA∈Gthe linear transformation ofggiven byAdAY=AYA−1.
A standard combinatorial lemma which is utilized[18]in producing the above explicit expansions is given by[28]
Lemma(Campbell 1897)—AdeX=eadX,{\displaystyle \operatorname {Ad} _{e^{X}}=e^{\operatorname {ad} _{X}},}so, explicitly,AdeXY=eXYe−X=eadXY=Y+[X,Y]+12![X,[X,Y]]+13![X,[X,[X,Y]]]+⋯.{\displaystyle \operatorname {Ad} _{e^{X}}Y=e^{X}Ye^{-X}=e^{\operatorname {ad} _{X}}Y=Y+\left[X,Y\right]+{\frac {1}{2!}}[X,[X,Y]]+{\frac {1}{3!}}[X,[X,[X,Y]]]+\cdots .}
This is a particularly useful formula which is commonly used to conductunitary transforms in quantum mechanics. By defining the iterated commutator,[X,Y]n≡[X,⋯[X,[X⏟ntimes,Y]]⋯],[X,Y]0≡Y,{\displaystyle [X,Y]_{n}\equiv \underbrace {[X,\dotsb [X,[X} _{n{\text{ times }}},Y]]\dotsb ],\quad [X,Y]_{0}\equiv Y,}we can write this formula more compactly as,eXYe−X=∑n=0∞[X,Y]nn!.{\displaystyle e^{X}Ye^{-X}=\sum _{n=0}^{\infty }{\frac {[X,Y]_{n}}{n!}}.}
Evaluate the derivative with respect tosoff(s)Y≡esXYe−sX, solution of the resulting differential equation and evaluation ats= 1,ddsf(s)Y=dds(esXYe−sX)=XesXYe−sX−esXYe−sXX=adX(esXYe−sX){\displaystyle {\frac {d}{ds}}f(s)Y={\frac {d}{ds}}\left(e^{sX}Ye^{-sX}\right)=Xe^{sX}Ye^{-sX}-e^{sX}Ye^{-sX}X=\operatorname {ad} _{X}(e^{sX}Ye^{-sX})}or[29]f′(s)=adXf(s),f(0)=1⟹f(s)=esadX.{\displaystyle f'(s)=\operatorname {ad} _{X}f(s),\qquad f(0)=1\qquad \Longrightarrow \qquad f(s)=e^{s\operatorname {ad} _{X}}.}
For[X,Y]central, i.e., commuting with bothXandY,esXYe−sX=Y+s[X,Y].{\displaystyle e^{sX}Ye^{-sX}=Y+s[X,Y]~.}Consequently, forg(s) ≡esXesY, it follows thatdgds=(X+esXYe−sX)g(s)=(X+Y+s[X,Y])g(s),{\displaystyle {\frac {dg}{ds}}={\Bigl (}X+e^{sX}Ye^{-sX}{\Bigr )}g(s)=(X+Y+s[X,Y])~g(s)~,}whose solution isg(s)=es(X+Y)+s22[X,Y].{\displaystyle g(s)=e^{s(X+Y)+{\frac {s^{2}}{2}}[X,Y]}~.}Takings=1{\displaystyle s=1}gives one of the special cases of the Baker–Campbell–Hausdorff formula described above:eXeY=eX+Y+12[X,Y].{\displaystyle e^{X}e^{Y}=e^{X+Y+{\frac {1}{2}}[X,Y]}~.}
More generally, for non-central[X,Y], we haveeXeYe−X=eeXYe−X=eeadXY,{\displaystyle e^{X}e^{Y}e^{-X}=e^{e^{X}Ye^{-X}}=e^{e^{{\text{ad}}_{X}}Y},}which can be written as the following braiding identity:eXeY=e(Y+[X,Y]+12![X,[X,Y]]+13![X,[X,[X,Y]]]+⋯)eX.{\displaystyle e^{X}e^{Y}=e^{(Y+\left[X,Y\right]+{\frac {1}{2!}}[X,[X,Y]]+{\frac {1}{3!}}[X,[X,[X,Y]]]+\cdots )}~e^{X}.}
A particularly useful variant of the above is the infinitesimal form. This is commonly written ase−XdeX=dX−12![X,dX]+13![X,[X,dX]]−14![X,[X,[X,dX]]]+⋯{\displaystyle e^{-X}de^{X}=dX-{\frac {1}{2!}}\left[X,dX\right]+{\frac {1}{3!}}[X,[X,dX]]-{\frac {1}{4!}}[X,[X,[X,dX]]]+\cdots }This variation is commonly used to write coordinates andvielbeinsas pullbacks of the metric on a Lie group.
For example, writingX=Xiei{\displaystyle X=X^{i}e_{i}}for some functionsXi{\displaystyle X^{i}}and a basisei{\displaystyle e_{i}}for the Lie algebra, one readily computes thate−XdeX=dXiei−12!XidXjfijkek+13!XiXjdXkfjklfilmem−⋯,{\displaystyle e^{-X}de^{X}=dX^{i}e_{i}-{\frac {1}{2!}}X^{i}dX^{j}{f_{ij}}^{k}e_{k}+{\frac {1}{3!}}X^{i}X^{j}dX^{k}{f_{jk}}^{l}{f_{il}}^{m}e_{m}-\cdots ,}for[ei,ej]=fijkek{\displaystyle [e_{i},e_{j}]={f_{ij}}^{k}e_{k}}thestructure constantsof the Lie algebra.
The series can be written more compactly (cf. main article) ase−XdeX=eiWijdXj,{\displaystyle e^{-X}de^{X}=e_{i}{W^{i}}_{j}dX^{j},}with the infinite seriesW=∑n=0∞(−1)nMn(n+1)!=(I−e−M)M−1.{\displaystyle W=\sum _{n=0}^{\infty }{\frac {(-1)^{n}M^{n}}{(n+1)!}}=(I-e^{-M})M^{-1}.}Here,Mis a matrix whose matrix elements areMjk=Xifijk{\displaystyle {M_{j}}^{k}=X^{i}{f_{ij}}^{k}}.
The usefulness of this expression comes from the fact that the matrixMis a vielbein. Thus, given some mapN→G{\displaystyle N\to G}from some manifoldNto some manifoldG, themetric tensoron the manifoldNcan be written as the pullback of the metric tensorBmn{\displaystyle B_{mn}}on the Lie groupG,gij=WimWjnBmn.{\displaystyle g_{ij}={W_{i}}^{m}{W_{j}}^{n}B_{mn}.}The metric tensorBmn{\displaystyle B_{mn}}on the Lie group is the Cartan metric, theKilling form. ForNa (pseudo-)Riemannian manifold, the metric is a (pseudo-)Riemannian metric.
A special case of the Baker–Campbell–Hausdorff formula is useful inquantum mechanicsand especiallyquantum optics, whereXandYareHilbert spaceoperators, generating theHeisenberg Lie algebra. Specifically, the position and momentum operators in quantum mechanics, usually denotedX{\displaystyle X}andP{\displaystyle P}, satisfy thecanonical commutation relation:[X,P]=iℏI{\displaystyle [X,P]=i\hbar I}whereI{\displaystyle I}is the identity operator. It follows thatX{\displaystyle X}andP{\displaystyle P}commute with their commutator. Thus, if weformallyapplied a special case of the Baker–Campbell–Hausdorff formula (even thoughX{\displaystyle X}andP{\displaystyle P}are unbounded operators and not matrices), we would conclude thateiaXeibP=ei(aX+bP−abℏ2).{\displaystyle e^{iaX}e^{ibP}=e^{i\left(aX+bP-{\frac {ab\hbar }{2}}\right)}.}This "exponentiated commutation relation" does indeed hold, and forms the basis of theStone–von Neumann theorem.
Further,ei(aX+bP)=eiaX/2eibPeiaX/2.{\displaystyle e^{i(aX+bP)}=e^{iaX/2}e^{ibP}e^{iaX/2}.}
A related application is theannihilation and creation operators,âandâ†. Their commutator[â†,â] = −Iiscentral, that is, it commutes with bothâandâ†. As indicated above, the expansion then collapses to the semi-trivial degenerate form:eva^†−v∗a^=eva^†e−v∗a^e−|v|2/2,{\displaystyle e^{v{\hat {a}}^{\dagger }-v^{*}{\hat {a}}}=e^{v{\hat {a}}^{\dagger }}e^{-v^{*}{\hat {a}}}e^{-|v|^{2}/2},}wherevis just a complex number.
This example illustrates the resolution of thedisplacement operator,exp(vâ†−v*â), into exponentials of annihilation and creation operators and scalars.[30]
This degenerate Baker–Campbell–Hausdorff formula then displays the product of two displacement operators as another displacement operator (up to a phase factor), with the resultant displacement equal to the sum of the two displacements,eva^†−v∗a^eua^†−u∗a^=e(v+u)a^†−(v∗+u∗)a^e(vu∗−uv∗)/2,{\displaystyle e^{v{\hat {a}}^{\dagger }-v^{*}{\hat {a}}}e^{u{\hat {a}}^{\dagger }-u^{*}{\hat {a}}}=e^{(v+u){\hat {a}}^{\dagger }-(v^{*}+u^{*}){\hat {a}}}e^{(vu^{*}-uv^{*})/2},}since theHeisenberg groupthey provide a representation of isnilpotent. The degenerate Baker–Campbell–Hausdorff formula is frequently used inquantum field theoryas well.[31] | https://en.wikipedia.org/wiki/Baker%E2%80%93Campbell%E2%80%93Hausdorff_formula |
Inmathematics, thePincherle derivative[1]T′{\displaystyle T'}of alinear operatorT:K[x]→K[x]{\displaystyle T:\mathbb {K} [x]\to \mathbb {K} [x]}on thevector spaceofpolynomialsin the variablexover afieldK{\displaystyle \mathbb {K} }is thecommutatorofT{\displaystyle T}with the multiplication byxin thealgebra of endomorphismsEnd(K[x]){\displaystyle \operatorname {End} (\mathbb {K} [x])}. That is,T′{\displaystyle T'}is another linear operatorT′:K[x]→K[x]{\displaystyle T':\mathbb {K} [x]\to \mathbb {K} [x]}
(for the origin of thead{\displaystyle \operatorname {ad} }notation, see the article on theadjoint representation) so that
This concept is named after the Italian mathematicianSalvatore Pincherle(1853–1936).
The Pincherle derivative, like anycommutator, is aderivation, meaning it satisfies the sum and products rules: given twolinear operatorsS{\displaystyle S}andT{\displaystyle T}belonging toEnd(K[x]),{\displaystyle \operatorname {End} \left(\mathbb {K} [x]\right),}
One also has[T,S]′=[T′,S]+[T,S′]{\displaystyle [T,S]^{\prime }=[T^{\prime },S]+[T,S^{\prime }]}where[T,S]=TS−ST{\displaystyle [T,S]=TS-ST}is the usualLie bracket, which follows from theJacobi identity.
The usualderivative,D=d/dx, is an operator on polynomials. By straightforward computation, its Pincherle derivative is
This formula generalizes to
byinduction. This proves that the Pincherle derivative of adifferential operator
is also a differential operator, so that the Pincherle derivative is a derivation ofDiff(K[x]){\displaystyle \operatorname {Diff} (\mathbb {K} [x])}.
WhenK{\displaystyle \mathbb {K} }hascharacteristiczero, the shift operator
can be written as
by theTaylor formula. Its Pincherle derivative is then
In other words, the shift operators areeigenvectorsof the Pincherle derivative, whose spectrum is the whole space of scalarsK{\displaystyle \mathbb {K} }.
IfTisshift-equivariant, that is, ifTcommutes withShor[T,Sh]=0{\displaystyle [T,S_{h}]=0}, then we also have[T′,Sh]=0{\displaystyle [T',S_{h}]=0}, so thatT′{\displaystyle T'}is also shift-equivariant and for the same shifth{\displaystyle h}.
The "discrete-time delta operator"
is the operator
whose Pincherle derivative is the shift operatorδ′=Sh{\displaystyle \delta '=S_{h}}. | https://en.wikipedia.org/wiki/Pincherle_derivative |
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