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Segre's theorem In projective geometry, Segre's theorem, named after the Italian mathematician Beniamino Segre, is the statement: • Any oval in a finite pappian projective plane of odd order is a nondegenerate projective conic section. This statement was assumed 1949 by the two Finnish mathematicians G. Järnefelt and P. Kustaanheimo and its proof was published in 1955 by B. Segre. A finite pappian projective plane can be imagined as the projective closure of the real plane (by a line at infinity), where the real numbers are replaced by a finite field K. Odd order means that \|K\| = n is odd. An oval is a curve similar to a circle (see definition below): any line meets it in at most 2 points and through any point of it there is exactly one tangent. The standard examples are the nondegenerate projective conic sections. In pappian projective planes of even order greater than four there are ovals which are not conics. In an infinite plane there exist ovals, which are not conics. In the real plane one just glues a half of a circle and a suitable ellipse smoothly. The proof of Segre's theorem, shown below, uses the 3-point version of Pascal's theorem and a property of a finite field of odd order, namely, that the product of all the nonzero elements equals -1. Definition of an oval Main article: Oval (projective plane) • In a projective plane a set ${\mathfrak {o}}$ of points is called oval, if: (1) Any line $g$ meets ${\mathfrak {o}}$ in at most two points. If $\|g\cap {\mathfrak {o}}\|=0$ the line $g$ is an exterior (or passing) line; in case $\|g\cap {\mathfrak {o}}\|=1$ a tangent line and if $\|g\cap {\mathfrak {o}}\|=2$ the line is a secant line. (2) For any point $P\in {\mathfrak {o}}$ there exists exactly one tangent $t$ at P, i.e., $t\cap {\mathfrak {o}}=\{P\}$. For finite planes (i.e. the set of points is finite) we have a more convenient characterization: • For a finite projective plane of order n (i.e. any line contains n + 1 points) a set ${\mathfrak {o}}$ of points is an oval if and only if $\|{\mathfrak {o}}\|=n+1$ and no three points are collinear (on a common line). Pascal's 3-point version Theorem Let be ${\mathfrak {o}}$ an oval in a pappian projective plane of characteristic $\neq 2$. ${\mathfrak {o}}$ is a nondegenerate conic if and only if statement (P3) holds: (P3): Let be $P_{1},P_{2},P_{3}$ any triangle on ${\mathfrak {o}}$ and ${\overline {P_{i}P_{i}}}$ the tangent at point $P_{i}$ to ${\mathfrak {o}}$, then the points $P_{4}:={\overline {P_{1}P_{1}}}\cap {\overline {P_{2}P_{3}}},\ P_{5}:={\overline {P_{2}P_{2}}}\cap {\overline {P_{1}P_{3}}},\ P_{6}:={\overline {P_{3}P_{3}}}\cap {\overline {P_{1}P_{2}}}$ are collinear.[1] Proof Let the projective plane be coordinatized inhomogeneously over a field $K$ such that $P_{3}=(0),\;g_{\infty }$ is the tangent at $P_{3},\ (0,0)\in {\mathfrak {o}}$, the x-axis is the tangent at the point $(0,0)$ and ${\mathfrak {o}}$ contains the point $(1,1)$. Furthermore, we set $P_{1}=(x_{1},y_{1}),\;P_{2}=(x_{2},y_{2})\ .$ (s. image) The oval ${\mathfrak {o}}$ can be described by a function $f:K\mapsto K$ such that: ${\mathfrak {o}}=\{(x,y)\in K^{2}\;\|\;y=f(x)\}\ \cup \{(\infty )\}\;.$ The tangent at point $(x_{0},f(x_{0}))$ will be described using a function $f'$ such that its equation is $y=f'(x_{0})(x-x_{0})+f(x_{0})$ Hence (s. image) $P_{5}=(x_{1},f'(x_{2})(x_{1}-x_{2})+f(x_{2}))$ and $P_{4}=(x_{2},f'(x_{1})(x_{2}-x_{1})+f(x_{1}))\;.$ I: if ${\mathfrak {o}}$ is a non degenerate conic we have $f(x)=x^{2}$ and $f'(x)=2x$ and one calculates easily that $P_{4},P_{5},P_{6}$ are collinear. II: If ${\mathfrak {o}}$ is an oval with property (P3), the slope of the line ${\overline {P_{4}P_{5}}}$ is equal to the slope of the line ${\overline {P_{1}P_{2}}}$, that means: $f'(x_{2})+f'(x_{1})-{\frac {f(x_{2})-f(x_{1})}{x_{2}-x_{1}}}={\frac {f(x_{2})-f(x_{1})}{x_{2}-x_{1}}}$ and hence (i): $(f'(x_{2})+f'(x_{1}))(x_{2}-x_{1})=2(f(x_{2})-f(x_{1}))$ for all $x_{1},x_{2}\in K$. With $f(0)=f'(0)=0$ one gets (ii): $f'(x_{2})x_{2}=2f(x_{2})$ and from $f(1)=1$ we get (iii): $f'(1)=2\;.$ (i) and (ii) yield (iv): $f'(x_{2})x_{1}=f'(x_{1})x_{2}$ and with (iii) at least we get (v): $f'(x_{2})=2x_{2}$ for all $x_{2}\in K$. A consequence of (ii) and (v) is $f(x_{2})=x_{2}^{2},\;x_{2}\in K$. Hence ${\mathfrak {o}}$ is a nondegenerate conic. Remark: Property (P3) is fulfilled for any oval in a pappian projective plane of characteristic 2 with a nucleus (all tangents meet at the nucleus). Hence in this case (P3) is also true for non-conic ovals.[2] Segre's theorem and its proof Theorem Any oval ${\mathfrak {o}}$ in a finite pappian projective plane of odd order is a nondegenerate conic section. Proof [3] For the proof we show that the oval has property (P3) of the 3-point version of Pascal's theorem. Let be $P_{1},P_{2},P_{3}$ any triangle on ${\mathfrak {o}}$ and $P_{4},P_{5},P_{6}$ defined as described in (P3). The pappian plane will be coordinatized inhomogeneously over a finite field $K$, such that$P_{3}=(\infty ),\;P_{2}=(0),\;P_{1}=(1,1)$ and $(0,0)$ is the common point of the tangents at $P_{2}$ and $P_{3}$. The oval ${\mathfrak {o}}$ can be described using a bijective function $f:K^{}:=K\cup \setminus \{0\}\mapsto K^{}$: ${\mathfrak {o}}=\{(x,y)\in K^{2}\;\|\;y=f(x),\;x\neq 0\}\;\cup \;\{(0),(\infty )\}\;.$ For a point $P=(x,y),\;x\in K\setminus \{0,1\}$, the expression $m(x)={\tfrac {f(x)-1}{x-1}}$ is the slope of the secant ${\overline {PP_{1}}}\;.$ Because both the functions $x\mapsto f(x)-1$ and $x\mapsto x-1$ are bijections from $K\setminus \{0,1\}$ to $K\setminus \{0,-1\}$, and $x\mapsto m(x)$ a bijection from $K\setminus \{0,1\}$ onto $K\setminus \{0,m_{1}\}$, where $m_{1}$ is the slope of the tangent at $P_{1}$, for $K^{}:=K\setminus \{0,1\}\;:$ we get $\prod _{x\in K^{}}(f(x)-1)=\prod _{x\in K^{}}(x-1)=1\quad {\text{und}}\quad m_{1}\cdot \prod _{x\in K^{}}{\frac {f(x)-1}{x-1}}=-1\;.$ (Remark: For $K^{}:=K\setminus \{0\}$ we have: $\displaystyle \prod _{k\in K^{}}k=-1\;.$) Hence $-1=m_{1}\cdot \prod _{x\in K^{}}{\frac {f(x)-1}{x-1}}=m_{1}\cdot {\frac \prod _{x\in K^{}}(f(x)-1)}\prod _{x\in K^{**}}(x-1)}}=m_{1}\;.$ Because the slopes of line ${\overline {P_{5}P_{6}}}$ and tangent ${\overline {P_{1}P_{1}}}$ both are $-1$, it follows that ${\overline {P_{1}P_{1}}}\cap {\overline {P_{2}P_{3}}}=P_{4}\in {\overline {P_{5}P_{6}}}$. This is true for any triangle $P_{1},P_{2},P_{3}\in {\mathfrak {o}}$. So: (P3) of the 3-point Pascal theorem holds and the oval is a non degenerate conic. References 1. E. Hartmann: Planar Circle Geometries, an Introduction to Moebius-, Laguerre- and Minkowski Planes. Skript, TH Darmstadt (PDF; 891 kB), p. 34. 2. E. Hartmann: Planar Circle Geometries, an Introduction to Moebius-, Laguerre- and Minkowski Planes. Skript, TH Darmstadt (PDF; 891 kB), p. 35. 3. E. Hartmann: Planar Circle Geometries, an Introduction to Moebius-, Laguerre- and Minkowski Planes. Skript, TH Darmstadt (PDF; 891 kB), p. 41. Sources • B. Segre: Ovals in a finite projective plane, Canadian Journal of Mathematics 7 (1955), pp. 414–416. • G. Järnefelt & P. Kustaanheimo: An observation on finite Geometries, Den 11 te Skandinaviske Matematikerkongress, Trondheim (1949), pp. 166–182. • Albrecht Beutelspacher, Ute Rosenbaum: Projektive Geometrie. 2. Auflage. Vieweg, Wiesbaden 2004, ISBN 3-528-17241-X, p. 162. • P. Dembowski: Finite Geometries. Springer-Verlag, 1968, ISBN 3-540-61786-8, p. 149 External links • Simeon Ball and Zsuzsa Weiner: An Introduction to Finite Geometry	Wikipedia
Segre class In mathematics, the Segre class is a characteristic class used in the study of cones, a generalization of vector bundles. For vector bundles the total Segre class is inverse to the total Chern class, and thus provides equivalent information; the advantage of the Segre class is that it generalizes to more general cones, while the Chern class does not. The Segre class was introduced in the non-singular case by Segre (1953).[1]. In the modern treatment of intersection theory in algebraic geometry, as developed e.g. in the definitive book of Fulton (1998), Segre classes play a fundamental role.[2] Definition Suppose $C$ is a cone over $X$, $q$ is the projection from the projective completion $\mathbb {P} (C\oplus 1)$ of $C$ to $X$, and ${\mathcal {O}}(1)$ is the anti-tautological line bundle on $\mathbb {P} (C\oplus 1)$. Viewing the Chern class $c_{1}({\mathcal {O}}(1))$ as a group endomorphism of the Chow group of $\mathbb {P} (C\oplus 1)$, the total Segre class of $C$ is given by: $s(C)=q_{}\left(\sum _{i\geq 0}c_{1}({\mathcal {O}}(1))^{i}[\mathbb {P} (C\oplus 1)]\right).$ The $i$th Segre class $s_{i}(C)$ is simply the $i$th graded piece of $s(C)$. If $C$ is of pure dimension $r$ over $X$ then this is given by: $s_{i}(C)=q_{}\left(c_{1}({\mathcal {O}}(1))^{r+i}[\mathbb {P} (C\oplus 1)]\right).$ The reason for using $\mathbb {P} (C\oplus 1)$ rather than $\mathbb {P} (C)$ is that this makes the total Segre class stable under addition of the trivial bundle ${\mathcal {O}}$. If Z is a closed subscheme of an algebraic scheme X, then $s(Z,X)$ denote the Segre class of the normal cone to $Z\hookrightarrow X$. Relation to Chern classes for vector bundles For a holomorphic vector bundle $E$ over a complex manifold $M$ a total Segre class $s(E)$ is the inverse to the total Chern class $c(E)$, see e.g. Fulton (1998).[3] Explicitly, for a total Chern class $c(E)=1+c_{1}(E)+c_{2}(E)+\cdots \,$ one gets the total Segre class $s(E)=1+s_{1}(E)+s_{2}(E)+\cdots \,$ where $c_{1}(E)=-s_{1}(E),\quad c_{2}(E)=s_{1}(E)^{2}-s_{2}(E),\quad \dots ,\quad c_{n}(E)=-s_{1}(E)c_{n-1}(E)-s_{2}(E)c_{n-2}(E)-\cdots -s_{n}(E)$ Let $x_{1},\dots ,x_{k}$ be Chern roots, i.e. formal eigenvalues of ${\frac {i\Omega }{2\pi }}$ where $\Omega $ is a curvature of a connection on $E$. While the Chern class c(E) is written as $c(E)=\prod _{i=1}^{k}(1+x_{i})=c_{0}+c_{1}+\cdots +c_{k}\,$ where $c_{i}$ is an elementary symmetric polynomial of degree $i$ in variables $x_{1},\dots ,x_{k}$ the Segre for the dual bundle $E^{\vee }$ which has Chern roots $-x_{1},\dots ,-x_{k}$ is written as $s(E^{\vee })=\prod _{i=1}^{k}{\frac {1}{1-x_{i}}}=s_{0}+s_{1}+\cdots $ Expanding the above expression in powers of $x_{1},\dots x_{k}$ one can see that $s_{i}(E^{\vee })$ is represented by a complete homogeneous symmetric polynomial of $x_{1},\dots x_{k}$ Properties Here are some basic properties. • For any cone C (e.g., a vector bundle), $s(C\oplus 1)=s(C)$.[4] • For a cone C and a vector bundle E, $c(E)s(C\oplus E)=s(C).$[5] • If E is a vector bundle, then[6] $s_{i}(E)=0$ for $i<0$. $s_{0}(E)$ is the identity operator. $s_{i}(E)\circ s_{j}(F)=s_{j}(F)\circ s_{i}(E)$ for another vector bundle F. • If L is a line bundle, then $s_{1}(L)=-c_{1}(L)$, minus the first Chern class of L.[6] • If E is a vector bundle of rank $e+1$, then, for a line bundle L, $s_{p}(E\otimes L)=\sum _{i=0}^{p}(-1)^{p-i}{\binom {e+p}{e+i}}s_{i}(E)c_{1}(L)^{p-i}.$[7] A key property of a Segre class is birational invariance: this is contained in the following. Let $p:X\to Y$ be a proper morphism between algebraic schemes such that $Y$ is irreducible and each irreducible component of $X$ maps onto $Y$. Then, for each closed subscheme $W\subset Y$, $V=p^{-1}(W)$ and $p_{V}:V\to W$ the restriction of $p$, ${p_{V}}_{}(s(V,X))=\operatorname {deg} (p)\,s(W,Y).$[8] Similarly, if $f:X\to Y$ is a flat morphism of constant relative dimension between pure-dimensional algebraic schemes, then, for each closed subscheme $W\subset Y$, $V=f^{-1}(W)$ and $f_{V}:V\to W$ the restriction of $f$, ${f_{V}}^{}(s(W,Y))=s(V,X).$[9] A basic example of birational invariance is provided by a blow-up. Let $\pi :{\widetilde {X}}\to X$ :{\widetilde {X}}\to X} be a blow-up along some closed subscheme Z. Since the exceptional divisor $E:=\pi ^{-1}(Z)\hookrightarrow {\widetilde {X}}$ is an effective Cartier divisor and the normal cone (or normal bundle) to it is ${\mathcal {O}}_{E}(E):={\mathcal {O}}_{X}(E)\|_{E}$, ${\begin{aligned}s(E,{\widetilde {X}})&=c({\mathcal {O}}_{E}(E))^{-1}[E]\\&=[E]-E\cdot [E]+E\cdot (E\cdot [E])+\cdots ,\end{aligned}}$ where we used the notation $D\cdot \alpha =c_{1}({\mathcal {O}}(D))\alpha $.[10] Thus, $s(Z,X)=g_{}\left(\sum _{k=1}^{\infty }(-1)^{k-1}E^{k}\right)$ where $g:E=\pi ^{-1}(Z)\to Z$ is given by $\pi $. Examples Example 1 Let Z be a smooth curve that is a complete intersection of effective Cartier divisors $D_{1},\dots ,D_{n}$ on a variety X. Assume the dimension of X is n + 1. Then the Segre class of the normal cone $C_{Z/X}$ to $Z\hookrightarrow X$ is:[11] $s(C_{Z/X})=[Z]-\sum _{i=1}^{n}D_{i}\cdot [Z].$ Indeed, for example, if Z is regularly embedded into X, then, since $C_{Z/X}=N_{Z/X}$ is the normal bundle and $N_{Z/X}=\bigoplus _{i=1}^{n}N_{D_{i}/X}\|_{Z}$ (see Normal cone#Properties), we have: $s(C_{Z/X})=c(N_{Z/X})^{-1}[Z]=\prod _{i=1}^{d}(1-c_{1}({\mathcal {O}}_{X}(D_{i})))[Z]=[Z]-\sum _{i=1}^{n}D_{i}\cdot [Z].$ Example 2 The following is Example 3.2.22. of Fulton (1998).[12] It recovers some classical results from Schubert's book on enumerative geometry. Viewing the dual projective space ${\breve {\mathbb {P} ^{3}}}$ as the Grassmann bundle $p:{\breve {\mathbb {P} ^{3}}}\to $ parametrizing the 2-planes in $\mathbb {P} ^{3}$, consider the tautological exact sequence $0\to S\to p^{}\mathbb {C} ^{3}\to Q\to 0$ where $S,Q$ are the tautological sub and quotient bundles. With $E=\operatorname {Sym} ^{2}(S^{}\otimes Q^{})$, the projective bundle $q:X=\mathbb {P} (E)\to {\breve {\mathbb {P} ^{3}}}$ is the variety of conics in $\mathbb {P} ^{3}$. With $\beta =c_{1}(Q^{})$, we have $c(S^{}\otimes Q^{})=2\beta +2\beta ^{2}$ and so, using Chern class#Computation formulae, $c(E)=1+8\beta +30\beta ^{2}+60\beta ^{3}$ and thus $s(E)=1+8h+34h^{2}+92h^{3}$ where $h=-\beta =c_{1}(Q).$ The coefficients in $s(E)$ have the enumerative geometric meanings; for example, 92 is the number of conics meeting 8 general lines. See also: Residual intersection#Example: conics tangent to given five conics. Example 3 Let X be a surface and $A,B,D$ effective Cartier divisors on it. Let $Z\subset X$ be the scheme-theoretic intersection of $A+D$ and $B+D$ (viewing those divisors as closed subschemes). For simplicity, suppose $A,B$ meet only at a single point P with the same multiplicity m and that P is a smooth point of X. Then[13] $s(Z,X)=[D]+(m^{2}[P]-D\cdot [D]).$ To see this, consider the blow-up $\pi :{\widetilde {X}}\to X$ :{\widetilde {X}}\to X} of X along P and let $g:{\widetilde {Z}}=\pi ^{-1}Z\to Z$, the strict transform of Z. By the formula at #Properties, $s(Z,X)=g_{}([{\widetilde {Z}}])-g_{}({\widetilde {Z}}\cdot [{\widetilde {Z}}]).$ Since ${\widetilde {Z}}=\pi ^{*}D+mE$ where $E=\pi ^{-1}P$, the formula above results. Multiplicity along a subvariety Let $(A,{\mathfrak {m}})$ be the local ring of a variety X at a closed subvariety V codimension n (for example, V can be a closed point). Then $\operatorname {length} _{A}(A/{\mathfrak {m}}^{t})$ is a polynomial of degree n in t for large t; i.e., it can be written as ${e(A)^{n} \over n!}t^{n}+$ the lower-degree terms and the integer $e(A)$ is called the multiplicity of A. The Segre class $s(V,X)$ of $V\subset X$ encodes this multiplicity: the coefficient of $[V]$ in $s(V,X)$ is $e(A)$.[14] References 1. Segre 1953 2. Fulton 1998 3. Fulton 1998, p.50. 4. Fulton 1998, Example 4.1.1. 5. Fulton 1998, Example 4.1.5. 6. Fulton 1998, Proposition 3.1. 7. Fulton 1998, Example 3.1.1. 8. Fulton 1998, Proposition 4.2. (a) 9. Fulton 1998, Proposition 4.2. (b) 10. Fulton 1998, § 2.5. 11. Fulton 1998, Example 9.1.1. 12. Fulton 1998 13. Fulton 1998, Example 4.2.2. 14. Fulton 1998, Example 4.3.1. Bibliography • Fulton, William (1998), Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., vol. 2 (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-62046-4, MR 1644323 • Segre, Beniamino (1953), "Nuovi metodi e resultati nella geometria sulle varietà algebriche", Ann. Mat. Pura Appl. (in Italian), 35 (4): 1–127, MR 0061420	Wikipedia
Segre classification The Segre classification is an algebraic classification of rank two symmetric tensors. The resulting types are then known as Segre types. It is most commonly applied to the energy–momentum tensor (or the Ricci tensor) and primarily finds application in the classification of exact solutions in general relativity. See also • Corrado Segre • Jordan normal form • Petrov classification References • Stephani, Hans; Kramer, Dietrich; MacCallum, Malcolm; Hoenselaers, Cornelius & Herlt, Eduard (2003). Exact Solutions of Einstein's Field Equations. Cambridge: Cambridge University Press. ISBN 0-521-46136-7. See section 5.1 for the Segre classification. • Segre, C. (1884). "Sulla teoria e sulla classificazione delle omografie in uno spazio lineare ad uno numero qualunque di dimensioni". Memorie della R. Accademia dei Lincei. 3a: 127.	Wikipedia
Segre cubic In algebraic geometry, the Segre cubic is a cubic threefold embedded in 4 (or sometimes 5) dimensional projective space, studied by Corrado Segre (1887). Definition The Segre cubic is the set of points (x0:x1:x2:x3:x4:x5) of P5 satisfying the equations $\displaystyle x_{0}+x_{1}+x_{2}+x_{3}+x_{4}+x_{5}=0$ $\displaystyle x_{0}^{3}+x_{1}^{3}+x_{2}^{3}+x_{3}^{3}+x_{4}^{3}+x_{5}^{3}=0.$ Properties The intersection of the Segre cubic with any hyperplane xi = 0 is the Clebsch cubic surface. Its intersection with any hyperplane xi = xj is Cayley's nodal cubic surface. Its dual is the Igusa quartic 3-fold in P4. Its Hessian is the Barth–Nieto quintic. A cubic hypersurface in P4 has at most 10 nodes, and up to isomorphism the Segre cubic is the unique one with 10 nodes. Its nodes are the points conjugate to (1:1:1:−1:−1:−1) under permutations of coordinates. The Segre cubic is rational and furthermore birationally equivalent to a compactification of the Siegel modular variety A2(2).[1] References 1. Hulek, Klaus; Sankaran, G. K. (2002). "The Geometry of Siegel Modular Varieties". Advanced Studies in Pure Mathematics. 35: 89–156. • Hunt, Bruce (1996), The geometry of some special arithmetic quotients, Lecture Notes in Mathematics, vol. 1637, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0094399, ISBN 978-3-540-61795-2, MR 1438547 • Hunt, Bruce (2000), "Nice modular varieties", Experimental Mathematics, 9 (4): 613–622, doi:10.1080/10586458.2000.10504664, ISSN 1058-6458, MR 1806296 • Segre, Corrado (1887), "Sulla varietà cubica con dieci punti doppii dello spazio a quattro dimensioni.", Atti della Reale Accademia delle scienze di Torino (in Italian), XXII: 791–801, JFM 19.0673.01	Wikipedia
Segregated Runge–Kutta methods The Segregated Runge–Kutta (SRK) method[1] is a family of IMplicit–EXplicit (IMEX) Runge–Kutta methods[2][3] that were developed to approximate the solution of differential algebraic equations (DAE) of index 2. The SRK method were motivated as a numerical method for the time integration of the incompressible Navier–Stokes equations with two salient properties. First, velocity and pressure computations are segregated. Second, the method keeps the same order of accuracy for both velocities and pressures. However, the SRK method can also be applied to any other DAE of index 2. The Segregated Runge–Kutta method Consider an index 2 DAE defined as follows: ${\begin{aligned}{\dot {y}}(t)&=f(y(t),z(t)),\\0&=g(y(t)).\end{aligned}}$ where $y(t)\in \mathbb {R} ^{n}$, $z(t)\in \mathbb {R} ^{m}$, $f:\mathbb {R} ^{n+m}\to \mathbb {R} ^{n}$ and $g:\mathbb {R} ^{n}\to \mathbb {R} ^{m}.$ In the previous equations $y$ is known as the differential variable, while $z$ is known as the algebraic variable. The time derivative of the differential variable, ${\dot {y}}$, depends on itself, $y$, on the algebraic variable, $z$, and on the time, $t$. The second equation can be seen as a constraint on differential variable, $y$. Let us take the time derivative of the second equation. Assuming that the function $g$ is linear and does not depend on time, and that the function $f$ is linear with respect to $z$, we have that $0={\dot {g}}(y)=g({\dot {y}})=g(f(y,z))=g(f(y)+f(z))=g(f(y))+g(f(z))).$ A Runge–Kutta time integration scheme is defined as a multistage integration in which each stage is computed as a combination of the unknowns evaluated in other stages. Depending on the definition of the parameters, this combination can lead to an implicit scheme or an explicit scheme. Implicit and explicit schemes can be combined, leading to IMEX schemes. Suppose that the function $f$ can be split into two operators ${\mathcal {F}}$ and ${\mathcal {G}}$ such that ${\dot {y}}(t)={\mathcal {F}}(y(t))+{\mathcal {G}}(y(t),z(t)),$ where ${\mathcal {F}}(y(t))$ and ${\mathcal {G}}(y(t),z(t))$ are the terms to be treated implicitly and explicitly, respectively. The SRK method is based on the use of IMEX Runge–Kutta schemes and can be defined by the following scheme: Given a time step size $h>0$, at a time $t_{n+1}=t_{n}+h$, for each Runge-Kutta stage $i$, with $0\leq i\leq s$, solve: 1) $y_{i}=y_{n}+h\sum _{j=1}^{i}a_{ij}{\mathcal {F}}(y_{j})+h\sum _{j=1}^{i-1}{\hat {a}}_{ij}{\mathcal {G}}(y_{j},z_{j}),$ 2) $g(f(z_{i}))=-g(f(y_{i}))$. Update the variables at $t_{n+1}$ solving: 3) $y_{n+1}=y_{n}+h\sum _{i=1}^{s}b_{i}{\mathcal {F}}(y_{i})+h\sum _{i=1}^{s}{\hat {b}}_{i}{\mathcal {G}}(y_{i},z_{i}),$ 4) $g(f(z_{n+1}))=-g(f(y_{n+1}))$. References 1. Colomés, Oriol; Badia, Santiago (3 February 2016). "Segregated Runge–Kutta methods for the incompressible Navier–Stokes equations". International Journal for Numerical Methods in Engineering. 105 (5): 372–400. Bibcode:2016IJNME.105..372C. doi:10.1002/nme.4987. hdl:2117/86545. S2CID 34117796. 2. Ascher, Uri M.; Ruuth, Steven J.; Spiteri, Raymond J. (November 1997). "Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations". Applied Numerical Mathematics. 25 (2–3): 151–176. CiteSeerX 10.1.1.48.1525. doi:10.1016/S0168-9274(97)00056-1. 3. Boscarino, Sebastiano (July 2009). "On an accurate third order implicit-explicit Runge–Kutta method for stiff problems". Applied Numerical Mathematics. 59 (7): 1515–1528. doi:10.1016/j.apnum.2008.10.003.	Wikipedia
Floer homology In mathematics, Floer homology is a tool for studying symplectic geometry and low-dimensional topology. Floer homology is a novel invariant that arises as an infinite-dimensional analogue of finite-dimensional Morse homology. Andreas Floer introduced the first version of Floer homology, now called Lagrangian Floer homology, in his proof of the Arnold conjecture in symplectic geometry. Floer also developed a closely related theory for Lagrangian submanifolds of a symplectic manifold. A third construction, also due to Floer, associates homology groups to closed three-dimensional manifolds using the Yang–Mills functional. These constructions and their descendants play a fundamental role in current investigations into the topology of symplectic and contact manifolds as well as (smooth) three- and four-dimensional manifolds. Floer homology is typically defined by associating to the object of interest an infinite-dimensional manifold and a real valued function on it. In the symplectic version, this is the free loop space of a symplectic manifold with the symplectic action functional. For the (instanton) version for three-manifolds, it is the space of SU(2)-connections on a three-dimensional manifold with the Chern–Simons functional. Loosely speaking, Floer homology is the Morse homology of the function on the infinite-dimensional manifold. A Floer chain complex is formed from the abelian group spanned by the critical points of the function (or possibly certain collections of critical points). The differential of the chain complex is defined by counting the function's gradient flow lines connecting certain pairs of critical points (or collections thereof). Floer homology is the homology of this chain complex. The gradient flow line equation, in a situation where Floer's ideas can be successfully applied, is typically a geometrically meaningful and analytically tractable equation. For symplectic Floer homology, the gradient flow equation for a path in the loopspace is (a perturbed version of) the Cauchy–Riemann equation for a map of a cylinder (the total space of the path of loops) to the symplectic manifold of interest; solutions are known as pseudoholomorphic curves. The Gromov compactness theorem is then used to show that the differential is well-defined and squares to zero, so that the Floer homology is defined. For instanton Floer homology, the gradient flow equations is exactly the Yang–Mills equation on the three-manifold crossed with the real line. Symplectic Floer homology Symplectic Floer Homology (SFH) is a homology theory associated to a symplectic manifold and a nondegenerate symplectomorphism of it. If the symplectomorphism is Hamiltonian, the homology arises from studying the symplectic action functional on the (universal cover of the) free loop space of a symplectic manifold. SFH is invariant under Hamiltonian isotopy of the symplectomorphism. Here, nondegeneracy means that 1 is not an eigenvalue of the derivative of the symplectomorphism at any of its fixed points. This condition implies that the fixed points are isolated. SFH is the homology of the chain complex generated by the fixed points of such a symplectomorphism, where the differential counts certain pseudoholomorphic curves in the product of the real line and the mapping torus of the symplectomorphism. This itself is a symplectic manifold of dimension two greater than the original manifold. For an appropriate choice of almost complex structure, punctured holomorphic curves (of finite energy) in it have cylindrical ends asymptotic to the loops in the mapping torus corresponding to fixed points of the symplectomorphism. A relative index may be defined between pairs of fixed points, and the differential counts the number of holomorphic cylinders with relative index 1. The symplectic Floer homology of a Hamiltonian symplectomorphism of a compact manifold is isomorphic to the singular homology of the underlying manifold. Thus, the sum of the Betti numbers of that manifold yields the lower bound predicted by one version of the Arnold conjecture for the number of fixed points for a nondegenerate symplectomorphism. The SFH of a Hamiltonian symplectomorphism also has a pair of pants product that is a deformed cup product equivalent to quantum cohomology. A version of the product also exists for non-exact symplectomorphisms. For the cotangent bundle of a manifold M, the Floer homology depends on the choice of Hamiltonian due to its noncompactness. For Hamiltonians that are quadratic at infinity, the Floer homology is the singular homology of the free loop space of M (proofs of various versions of this statement are due to Viterbo, Salamon–Weber, Abbondandolo–Schwarz, and Cohen). There are more complicated operations on the Floer homology of a cotangent bundle that correspond to the string topology operations on the homology of the loop space of the underlying manifold. The symplectic version of Floer homology figures in a crucial way in the formulation of the homological mirror symmetry conjecture. PSS isomorphism In 1996 S. Piunikhin, D. Salamon and M. Schwarz summarized the results about the relation between Floer homology and quantum cohomology and formulated as the following.Piunikhin, Salamon & Schwarz (1996) • The Floer cohomology groups of the loop space of a semi-positive symplectic manifold (M,ω) are naturally isomorphic to the ordinary cohomology of M, tensored by a suitable Novikov ring associated the group of covering transformations. • This isomorphism intertwines the quantum cup product structure on the cohomology of M with the pair-of-pants product on Floer homology. The above condition of semi-positive and the compactness of symplectic manifold M is required for us to obtain Novikov ring and for the definition of both Floer homology and quantum cohomology. The semi-positive condition means that one of the following holds (note that the three cases are not disjoint): • $\langle [\omega ],A\rangle =\lambda \langle c_{1},A\rangle $ for every A in π2(M) where λ≥0 (M is monotone). • $\langle c_{1},A\rangle =0$ for every A in π2(M). • The minimal Chern Number N ≥ 0 defined by $\langle c_{1},\pi _{2}(M)\rangle =N\mathbb {Z} $ is greater than or equal to n − 2. The quantum cohomology group of symplectic manifold M can be defined as the tensor products of the ordinary cohomology with Novikov ring Λ, i.e. $QH_{}(M)=H_{}(M)\otimes \Lambda .$ This construction of Floer homology explains the independence on the choice of the almost complex structure on M and the isomorphism to Floer homology provided from the ideas of Morse theory and pseudoholomorphic curves, where we must recognize the Poincaré duality between homology and cohomology as the background. Floer homology of three-manifolds There are several equivalent Floer homologies associated to closed three-manifolds. Each yields three types of homology groups, which fit into an exact triangle. A knot in a three-manifold induces a filtration on the chain complex of each theory, whose chain homotopy type is a knot invariant. (Their homologies satisfy similar formal properties to the combinatorially-defined Khovanov homology.) These homologies are closely related to the Donaldson and Seiberg invariants of 4-manifolds, as well as to Taubes's Gromov invariant of symplectic 4-manifolds; the differentials of the corresponding three-manifold homologies to these theories are studied by considering solutions to the relevant differential equations (Yang–Mills, Seiberg–Witten, and Cauchy–Riemann, respectively) on the 3-manifold cross R. The 3-manifold Floer homologies should also be the targets of relative invariants for four-manifolds with boundary, related by gluing constructions to the invariants of a closed 4-manifold obtained by gluing together bounded 3-manifolds along their boundaries. (This is closely related to the notion of a topological quantum field theory.) For Heegaard Floer homology, the 3-manifold homology was defined first, and an invariant for closed 4-manifolds was later defined in terms of it. There are also extensions of the 3-manifold homologies to 3-manifolds with boundary: sutured Floer homology (Juhász 2008) and bordered Floer homology (Lipshitz, Ozsváth & Thurston 2008). These are related to the invariants for closed 3-manifolds by gluing formulas for the Floer homology of a 3-manifold described as the union along the boundary of two 3-manifolds with boundary. The three-manifold Floer homologies also come equipped with a distinguished element of the homology if the three-manifold is equipped with a contact structure. Kronheimer and Mrowka first introduced the contact element in the Seiberg–Witten case. Ozsvath and Szabo constructed it for Heegaard Floer homology using Giroux's relation between contact manifolds and open book decompositions, and it comes for free, as the homology class of the empty set, in embedded contact homology. (Which, unlike the other three, requires a contact homology for its definition. For embedded contact homology see Hutchings (2009). These theories all come equipped with a priori relative gradings; these have been lifted to absolute gradings (by homotopy classes of oriented 2-plane fields) by Kronheimer and Mrowka (for SWF), Gripp and Huang (for HF), and Hutchings (for ECH). Cristofaro-Gardiner has shown that Taubes' isomorphism between ECH and Seiberg–Witten Floer cohomology preserves these absolute gradings. Instanton Floer homology This is a three-manifold invariant connected to Donaldson theory introduced by Floer himself. It is obtained using the Chern–Simons functional on the space of connections on a principal SU(2)-bundle over the three-manifold (more precisely, homology 3-spheres). Its critical points are flat connections and its flow lines are instantons, i.e. anti-self-dual connections on the three-manifold crossed with the real line. Instanton Floer homology may be viewed as a generalization of the Casson invariant because the Euler characteristic of the Floer homology agrees with the Casson invariant. Soon after Floer's introduction of Floer homology, Donaldson realized that cobordisms induce maps. This was the first instance of the structure that came to be known as a topological quantum field theory. Seiberg–Witten Floer homology Seiberg–Witten Floer homology or monopole Floer homology is a homology theory of smooth 3-manifolds (equipped with a spinc structure). It may be viewed as the Morse homology of the Chern–Simons–Dirac functional on U(1) connections on the three-manifold. The associated gradient flow equation corresponds to the Seiberg–Witten equations on the 3-manifold crossed with the real line. Equivalently, the generators of the chain complex are translation-invariant solutions to Seiberg–Witten equations (known as monopoles) on the product of a 3-manifold and the real line, and the differential counts solutions to the Seiberg–Witten equations on the product of a three-manifold and the real line, which are asymptotic to invariant solutions at infinity and negative infinity. One version of Seiberg–Witten–Floer homology was constructed rigorously in the monograph Monopoles and Three-manifolds by Peter Kronheimer and Tomasz Mrowka, where it is known as monopole Floer homology. Taubes has shown that it is isomorphic to embedded contact homology. Alternate constructions of SWF for rational homology 3-spheres have been given by Manolescu (2003) and Frøyshov (2010); they are known to agree. Heegaard Floer homology Heegaard Floer homology // (listen) is an invariant due to Peter Ozsváth and Zoltán Szabó of a closed 3-manifold equipped with a spinc structure. It is computed using a Heegaard diagram of the space via a construction analogous to Lagrangian Floer homology. Kutluhan, Lee & Taubes (2020) announced a proof that Heegaard Floer homology is isomorphic to Seiberg–Witten Floer homology, and Colin, Ghiggini & Honda (2011) announced a proof that the plus-version of Heegaard Floer homology (with reverse orientation) is isomorphic to embedded contact homology. A knot in a three-manifold induces a filtration on the Heegaard Floer homology groups, and the filtered homotopy type is a powerful knot invariant, called knot Floer homology. It categorifies the Alexander polynomial. Knot Floer homology was defined by Ozsváth & Szabó (2004) and independently by Rasmussen (2003). It is known to detect knot genus. Using grid diagrams for the Heegaard splittings, knot Floer homology was given a combinatorial construction by Manolescu, Ozsváth & Sarkar (2009). The Heegaard Floer homology of the double cover of S^3 branched over a knot is related by a spectral sequence to Khovanov homology (Ozsváth & Szabó 2005). The "hat" version of Heegaard Floer homology was described combinatorially by Sarkar & Wang (2010). The "plus" and "minus" versions of Heegaard Floer homology, and the related Ozsváth–Szabó four-manifold invariants, can be described combinatorially as well (Manolescu, Ozsváth & Thurston 2009). Embedded contact homology Embedded contact homology, due to Michael Hutchings, is an invariant of 3-manifolds (with a distinguished second homology class, corresponding to the choice of a spinc structure in Seiberg–Witten Floer homology) isomorphic (by work of Clifford Taubes) to Seiberg–Witten Floer cohomology and consequently (by work announced by Kutluhan, Lee & Taubes 2020 and Colin, Ghiggini & Honda 2011) to the plus-version of Heegaard Floer homology (with reverse orientation). It may be seen as an extension of Taubes's Gromov invariant, known to be equivalent to the Seiberg–Witten invariant, from closed symplectic 4-manifolds to certain non-compact symplectic 4-manifolds (namely, a contact three-manifold cross R). Its construction is analogous to symplectic field theory, in that it is generated by certain collections of closed Reeb orbits and its differential counts certain holomorphic curves with ends at certain collections of Reeb orbits. It differs from SFT in technical conditions on the collections of Reeb orbits that generate it—and in not counting all holomorphic curves with Fredholm index 1 with given ends, but only those that also satisfy a topological condition given by the ECH index, which in particular implies that the curves considered are (mainly) embedded. The Weinstein conjecture that a contact 3-manifold has a closed Reeb orbit for any contact form holds on any manifold whose ECH is nontrivial, and was proved by Taubes using techniques closely related to ECH; extensions of this work yielded the isomorphism between ECH and SWF. Many constructions in ECH (including its well-definedness) rely upon this isomorphism (Taubes 2007). The contact element of ECH has a particularly nice form: it is the cycle associated to the empty collection of Reeb orbits. An analog of embedded contact homology may be defined for mapping tori of symplectomorphisms of a surface (possibly with boundary) and is known as periodic Floer homology, generalizing the symplectic Floer homology of surface symplectomorphisms. More generally, it may be defined with respect to any stable Hamiltonian structure on the 3-manifold; like contact structures, stable Hamiltonian structures define a nonvanishing vector field (the Reeb vector field), and Hutchings and Taubes have proven an analogue of the Weinstein conjecture for them, namely that they always have closed orbits (unless they are mapping tori of a 2-torus). Lagrangian intersection Floer homology The Lagrangian Floer homology of two transversely intersecting Lagrangian submanifolds of a symplectic manifold is the homology of a chain complex generated by the intersection points of the two submanifolds and whose differential counts pseudoholomorphic Whitney discs. Given three Lagrangian submanifolds L0, L1, and L2 of a symplectic manifold, there is a product structure on the Lagrangian Floer homology: $HF(L_{0},L_{1})\otimes HF(L_{1},L_{2})\rightarrow HF(L_{0},L_{2}),$ which is defined by counting holomorphic triangles (that is, holomorphic maps of a triangle whose vertices and edges map to the appropriate intersection points and Lagrangian submanifolds). Papers on this subject are due to Fukaya, Oh, Ono, and Ohta; the recent work on "cluster homology" of Lalonde and Cornea offer a different approach to it. The Floer homology of a pair of Lagrangian submanifolds may not always exist; when it does, it provides an obstruction to isotoping one Lagrangian away from the other using a Hamiltonian isotopy. Several kinds of Floer homology are special cases of Lagrangian Floer homology. The symplectic Floer homology of a symplectomorphism of M can be thought of as a case of Lagrangian Floer homology in which the ambient manifold is M crossed with M and the Lagrangian submanifolds are the diagonal and the graph of the symplectomorphism. The construction of Heegaard Floer homology is based on a variant of Lagrangian Floer homology for totally real submanifolds defined using a Heegaard splitting of a three-manifold. Seidel–Smith and Manolescu constructed a link invariant as a certain case of Lagrangian Floer homology, which conjecturally agrees with Khovanov homology, a combinatorially-defined link invariant. Atiyah–Floer conjecture The Atiyah–Floer conjecture connects the instanton Floer homology with the Lagrangian intersection Floer homology.[1] Consider a 3-manifold Y with a Heegaard splitting along a surface $\Sigma $. Then the space of flat connections on $\Sigma $ modulo gauge equivalence is a symplectic manifold $M(\Sigma )$ of dimension 6g − 6, where g is the genus of the surface $\Sigma $. In the Heegaard splitting, $\Sigma $ bounds two different 3-manifolds; the space of flat connections modulo gauge equivalence on each 3-manifold with boundary embeds into $M(\Sigma )$ as a Lagrangian submanifold. One can consider the Lagrangian intersection Floer homology. Alternately, we can consider the Instanton Floer homology of the 3-manifold Y. The Atiyah–Floer conjecture asserts that these two invariants are isomorphic. Salamon–Wehrheim and Daemi–Fukaya are working on their programs to prove this conjecture. Relations to mirror symmetry The homological mirror symmetry conjecture of Maxim Kontsevich predicts an equality between the Lagrangian Floer homology of Lagrangians in a Calabi–Yau manifold $X$ and the Ext groups of coherent sheaves on the mirror Calabi–Yau manifold. In this situation, one should not focus on the Floer homology groups but on the Floer chain groups. Similar to the pair-of-pants product, one can construct multi-compositions using pseudo-holomorphic n-gons. These compositions satisfy the $A_{\infty }$-relations making the category of all (unobstructed) Lagrangian submanifolds in a symplectic manifold into an $A_{\infty }$-category, called the Fukaya category. To be more precise, one must add additional data to the Lagrangian – a grading and a spin structure. A Lagrangian with a choice of these structures is often called a brane in homage to the underlying physics. The Homological Mirror Symmetry conjecture states there is a type of derived Morita equivalence between the Fukaya category of the Calabi–Yau $X$ and a dg category underlying the bounded derived category of coherent sheaves of the mirror, and vice versa. Symplectic field theory (SFT) This is an invariant of contact manifolds and symplectic cobordisms between them, originally due to Yakov Eliashberg, Alexander Givental and Helmut Hofer. The symplectic field theory as well as its subcomplexes, rational symplectic field theory and contact homology, are defined as homologies of differential algebras, which are generated by closed orbits of the Reeb vector field of a chosen contact form. The differential counts certain holomorphic curves in the cylinder over the contact manifold, where the trivial examples are the branched coverings of (trivial) cylinders over closed Reeb orbits. It further includes a linear homology theory, called cylindrical or linearized contact homology (sometimes, by abuse of notation, just contact homology), whose chain groups are vector spaces generated by closed orbits and whose differentials count only holomorphic cylinders. However, cylindrical contact homology is not always defined due to the presence of holomorphic discs and a lack of regularity and transversality results. In situations where cylindrical contact homology makes sense, it may be seen as the (slightly modified) Morse homology of the action functional on the free loop space, which sends a loop to the integral of the contact form alpha over the loop. Reeb orbits are the critical points of this functional. SFT also associates a relative invariant of a Legendrian submanifold of a contact manifold known as relative contact homology. Its generators are Reeb chords, which are trajectories of the Reeb vector field beginning and ending on a Lagrangian, and its differential counts certain holomorphic strips in the symplectization of the contact manifold whose ends are asymptotic to given Reeb chords. In SFT the contact manifolds can be replaced by mapping tori of symplectic manifolds with symplectomorphisms. While the cylindrical contact homology is well-defined and given by the symplectic Floer homologies of powers of the symplectomorphism, (rational) symplectic field theory and contact homology can be considered as generalized symplectic Floer homologies. In the important case when the symplectomorphism is the time-one map of a time-dependent Hamiltonian, it was however shown that these higher invariants do not contain any further information. Floer homotopy One conceivable way to construct a Floer homology theory of some object would be to construct a related spectrum whose ordinary homology is the desired Floer homology. Applying other homology theories to such a spectrum could yield other interesting invariants. This strategy was proposed by Ralph Cohen, John Jones, and Graeme Segal, and carried out in certain cases for Seiberg–Witten–Floer homology by Manolescu (2003) and for the symplectic Floer homology of cotangent bundles by Cohen. This approach was the basis of Manolescu's 2013 construction of Pin (2)-equivariant Seiberg–Witten Floer homology, with which he disproved the Triangulation Conjecture for manifolds of dimension 5 and higher. Analytic foundations Many of these Floer homologies have not been completely and rigorously constructed, and many conjectural equivalences have not been proved. Technical difficulties come up in the analysis involved, especially in constructing compactified moduli spaces of pseudoholomorphic curves. Hofer, in collaboration with Kris Wysocki and Eduard Zehnder, has developed new analytic foundations via their theory of polyfolds and a "general Fredholm theory". While the polyfold project is not yet fully completed, in some important cases transversality was shown using simpler methods. Computation Floer homologies are generally difficult to compute explicitly. For instance, the symplectic Floer homology for all surface symplectomorphisms was completed only in 2007. The Heegaard Floer homology has been a success story in this regard: researchers have exploited its algebraic structure to compute it for various classes of 3-manifolds and have found combinatorial algorithms for computation of much of the theory. It is also connected to existing invariants and structures and many insights into 3-manifold topology have resulted. References Footnotes 1. Atiyah 1988 Books and surveys • Atiyah, Michael (1988). "New invariants of 3- and 4-dimensional manifolds". The Mathematical Heritage of Hermann Weyl. Proceedings of Symposia in Pure Mathematics. Vol. 48. pp. 285–299. doi:10.1090/pspum/048/974342. ISBN 9780821814826. • Augustin Banyaga; David Hurtubise (2004). Lectures on Morse Homology. Kluwer Academic Publishers. ISBN 978-1-4020-2695-9. • Simon Donaldson; M. Furuta; D. Kotschick (2002). Floer homology groups in Yang–Mills theory. Cambridge Tracts in Mathematics. Vol. 147. Cambridge University Press. ISBN 978-0-521-80803-3. • Ellwood, David A.; Ozsváth, Peter S.; Stipsicz, András I.; Szabó, Zoltán, eds. (2006). Floer Homology, Gauge Theory, And Low-dimensional Topology. Clay Mathematics Proceedings. Vol. 5. Clay Mathematics Institute. ISBN 978-0-8218-3845-7. • Kronheimer, Peter; Mrowka, Tomasz (2007). Monopoles and Three-Manifolds. Cambridge University Press. ISBN 978-0-521-88022-0. • McDuff, Dusa; Salamon, Dietmar (1998). Introduction to Symplectic Topology. Oxford University Press. ISBN 978-0-19-850451-1. • McDuff, Dusa (2005). "Floer theory and low dimensional topology". Bulletin of the American Mathematical Society. 43: 25–42. doi:10.1090/S0273-0979-05-01080-3. MR 2188174. • Schwarz, Matthias (2012) [1993]. Morse Homology. Progress in Mathematics. Vol. 111. Birkhäuser. ISBN 978-3-0348-8577-5. • Seidel, Paul (2008). Fukaya Categories and Picard Lefschetz Theory. European Mathematical Society. ISBN 978-3037190630. Research articles • Colin, Vincent; Ghiggini, Paolo; Honda, Ko (2011). "Equivalence of Heegaard Floer homology and embedded contact homology via open book decompositions". PNAS. 108 (20): 8100–8105. Bibcode:2011PNAS..108.8100C. doi:10.1073/pnas.1018734108. PMC 3100941. PMID 21525415. • Floer, Andreas (1988). "The unregularized gradient flow of the symplectic action". Comm. Pure Appl. Math. 41 (6): 775–813. doi:10.1002/cpa.3160410603. • ——— (1988). "An instanton-invariant for 3-manifolds". Comm. Math. Phys. 118 (2): 215–240. Bibcode:1988CMaPh.118..215F. doi:10.1007/BF01218578. S2CID 122096068. Project Euclid • ——— (1988). "Morse theory for Lagrangian intersections". J. Differential Geom. 28 (3): 513–547. doi:10.4310/jdg/1214442477. MR 0965228. • ——— (1989). "Cuplength estimates on Lagrangian intersections". Comm. Pure Appl. Math. 42 (4): 335–356. doi:10.1002/cpa.3160420402. • ——— (1989). "Symplectic fixed points and holomorphic spheres". Comm. Math. Phys. 120 (4): 575–611. Bibcode:1988CMaPh.120..575F. doi:10.1007/BF01260388. S2CID 123345003. • ——— (1989). "Witten's complex and infinite dimensional Morse Theory" (PDF). J. Diff. Geom. 30 (1): 202–221. doi:10.4310/jdg/1214443291. • Frøyshov, Kim A. (2010). "Monopole Floer homology for rational homology 3-spheres". Duke Math. J. 155 (3): 519–576. arXiv:0809.4842. doi:10.1215/00127094-2010-060. S2CID 8073050. • Gromov, Mikhail (1985). "Pseudo holomorphic curves in symplectic manifolds". Inventiones Mathematicae. 82 (2): 307–347. Bibcode:1985InMat..82..307G. doi:10.1007/BF01388806. S2CID 4983969. • Hofer, Helmut; Wysocki, Kris; Zehnder, Eduard (2007). "A General Fredholm Theory I: A Splicing-Based Differential Geometry". Journal of the European Mathematical Society. 9 (4): 841–876. arXiv:math.FA/0612604. Bibcode:2006math.....12604H. doi:10.4171/JEMS/99. S2CID 14716262. • Juhász, András (2008). "Floer homology and surface decompositions". Geometry & Topology. 12 (1): 299–350. arXiv:math/0609779. doi:10.2140/gt.2008.12.299. S2CID 56418423. • Kutluhan, Cagatay; Lee, Yi-Jen; Taubes, Clifford Henry (2020). "HF=HM I: Heegaard Floer homology and Seiberg–Witten Floer homology". Geometry & Topology. 24 (6): 2829–2854. arXiv:1007.1979. doi:10.2140/gt.2020.24.2829. S2CID 118772589. • Lipshitz, Robert; Ozsváth, Peter; Thurston, Dylan (2008). "Bordered Heegaard Floer homology: Invariance and pairing". Memoirs of the American Mathematical Society. 254 (1216). arXiv:0810.0687. doi:10.1090/memo/1216. S2CID 115166724. • Manolescu, Ciprian (2003). "Seiberg–Witten–Floer stable homotopy type of three-manifolds with b1 = 0". Geom. Topol. 7 (2): 889–932. arXiv:math/0104024. doi:10.2140/gt.2003.7.889. S2CID 9130339. • Manolescu, Ciprian; Ozsváth, Peter S.; Sarkar, Sucharit (2009). "A combinatorial description of knot Floer homology". Ann. of Math. 169 (2): 633–660. arXiv:math/0607691. Bibcode:2006math......7691M. doi:10.4007/annals.2009.169.633. S2CID 15427272. • Manolescu, Ciprian; Ozsváth, Peter; Thurston, Dylan (2009). "Grid diagrams and Heegaard Floer invariants". arXiv:0910.0078 [math.GT]. • Ozsváth, Peter; Szabo, Zoltán (2004). "Holomorphic disks and topological invariants for closed three-manifolds". Ann. of Math. 159 (3): 1027–1158. arXiv:math/0101206. Bibcode:2001math......1206O. doi:10.4007/annals.2004.159.1027. S2CID 119143219. • ———; Szabo (2004). "Holomorphic disks and three-manifold invariants: properties and applications". Ann. of Math. 159 (3): 1159–1245. arXiv:math/0105202. Bibcode:2001math......5202O. doi:10.4007/annals.2004.159.1159. S2CID 8154024. • Ozsváth, Peter; Szabó, Zoltán (2004). "Holomorphic disks and knot invariants". Advances in Mathematics. 186 (1): 58–116. arXiv:math.GT/0209056. doi:10.1016/j.aim.2003.05.001. • Ozsváth, Peter; Szabó, Zoltán (2005). "On the Heegaard Floer homology of branched double-covers". Advances in Mathematics. 194 (1): 1–33. arXiv:math.GT/0209056. Bibcode:2003math......9170O. doi:10.1016/j.aim.2004.05.008. S2CID 17245314. • Rasmussen, Jacob (2003). "Floer homology and knot complements". arXiv:math/0306378. • Salamon, Dietmar; Wehrheim, Katrin (2008). "Instanton Floer homology with Lagrangian boundary conditions". Geometry & Topology. 12 (2): 747–918. arXiv:math/0607318. doi:10.2140/gt.2008.12.747. S2CID 119680541. • Sarkar, Sucharit; Wang, Jiajun (2010). "An algorithm for computing some Heegaard Floer homologies". Ann. of Math. 171 (2): 1213–1236. arXiv:math/0607777. doi:10.4007/annals.2010.171.1213. S2CID 55279928. • Hutchings (2009). The embedded contact homology index revisited. pp. 263–297. arXiv:0805.1240. Bibcode:2008arXiv0805.1240H. doi:10.1090/crmp/049/10. ISBN 9780821843567. S2CID 7751880. {{cite book}}: \|journal= ignored (help) • Taubes, Clifford (2007). "The Seiberg–Witten equations and the Weistein conjecture". Geom. Topol. 11 (4): 2117–2202. arXiv:math/0611007. doi:10.2140/gt.2007.11.2117. S2CID 119680690. • Piunikhin, Sergey; Salamon, Dietmar; Schwarz, Matthias (1996). "Symplectic Floer–Donaldson theory and quantum cohomology". Contact and Symplectic Geometry. Cambridge University Press. pp. 171–200. ISBN 978-0-521-57086-2. External links • "Atiyah-Floer conjecture", Encyclopedia of Mathematics, EMS Press, 2001 [1994] • "Heegaard Floer Knot Homology", The Knot Atlas. Authority control: National • Germany	Wikipedia
Seiberg–Witten theory In theoretical physics, Seiberg–Witten theory is an ${\mathcal {N}}=2$ supersymmetric gauge theory with an exact low-energy effective action (for massless degrees of freedom), of which the kinetic part coincides with the Kähler potential of the moduli space of vacua. Before taking the low-energy effective action, the theory is known as ${\mathcal {N}}=2$ supersymmetric Yang–Mills theory, as the field content is a single ${\mathcal {N}}=2$ vector supermultiplet, analogous to the field content of Yang–Mills theory being a single vector gauge field (in particle theory language) or connection (in geometric language). For applications to 4-manifolds, see Seiberg–Witten invariants. The theory was studied in detail by Nathan Seiberg and Edward Witten (Seiberg & Witten 1994). Seiberg–Witten curves In general, effective Lagrangians of supersymmetric gauge theories are largely determined by their holomorphic (really, meromorphic) properties and their behavior near the singularities. In gauge theory with ${\mathcal {N}}=2$ extended supersymmetry, the moduli space of vacua is a special Kähler manifold and its Kähler potential is constrained by above conditions. In the original approach,[1][2] by Seiberg and Witten, holomorphy and electric-magnetic duality constraints are strong enough to almost uniquely constrain the prepotential ${\mathcal {F}}$ (a holomorphic function which defines the theory), and therefore the metric of the moduli space of vacua, for theories with SU(2) gauge group. More generally, consider the example with gauge group SU(n). The classical potential is $V(x)={\frac {1}{g^{2}}}\operatorname {Tr} [\phi ,{\bar {\phi }}]^{2}\,$ (1) where $\phi $ is a scalar field appearing in an expansion of superfields in the theory. The potential must vanish on the moduli space of vacua by definition, but the $\phi $ need not. The vacuum expectation value of $\phi $ can be gauge rotated into the Cartan subalgebra, making it a traceless diagonal complex matrix $a$. Because the fields $\phi $ no longer have vanishing vacuum expectation value, other fields become massive due to the Higgs mechanism (spontaneous symmetry breaking). They are integrated out in order to find the effective ${\mathcal {N}}=2$ U(1) gauge theory. Its two-derivative, four-fermions low-energy action is given by a Lagrangian which can be expressed in terms of a single holomorphic function ${\mathcal {F}}$ on ${\mathcal {N}}=1$ superspace as follows: ${\mathcal {L}}_{\text{eff}}^{\mathrm {U} (1)}={\frac {1}{4\pi }}\operatorname {Im} \left[\int d^{4}\theta {\frac {d{\mathcal {F}}}{dA}}{\bar {A}}+\int d^{2}\theta {\frac {1}{2}}{\frac {d^{2}{\mathcal {F}}}{dA^{2}}}W_{\alpha }W^{\alpha }\right]\,$ (3) where ${\mathcal {F}}={\frac {i}{2\pi }}{\mathcal {A}}^{2}\operatorname {\ln } {\frac {{\mathcal {A}}^{2}}{\Lambda ^{2}}}+\sum _{k=1}^{\infty }{\mathcal {F}}_{k}{\frac {\Lambda ^{4k}}{{\mathcal {A}}^{4k}}}{\mathcal {A}}^{2}\,$ (4) and $A$ is a chiral superfield on ${\mathcal {N}}=1$ superspace which fits inside the ${\mathcal {N}}=2$ chiral multiplet ${\mathcal {A}}$. The first term is a perturbative loop calculation and the second is the instanton part where $k$ labels fixed instanton numbers. In theories whose gauge groups are products of unitary groups, ${\mathcal {F}}$ can be computed exactly using localization[3] and the limit shape techniques.[4] The Kähler potential is the kinetic part of the low energy action, and explicitly is written in terms of ${\mathcal {F}}$ as $K(A,{\bar {A}})=\mathrm {Im} \left({\frac {\partial {\mathcal {F}}}{\partial A}}{\bar {A}}\right).$ (5) From ${\mathcal {F}}$ we can get the mass of the BPS particles. $M\approx \|na+ma_{D}\|\,$ (6) $a_{D}={\frac {d{\mathcal {F}}}{da}}\,$ (7) One way to interpret this is that these variables $a$ and its dual can be expressed as periods of a meromorphic differential on a Riemann surface called the Seiberg–Witten curve. N = 2 supersymmetric Yang–Mills theory Before the low energy, or infrared, limit is taken, the action can be given in terms of a Lagrangian over ${\mathcal {N}}=2$ superspace with field content $\Psi $, which is a single ${\mathcal {N}}=2$ vector/chiral superfield in the adjoint representation of the gauge group, and a holomorphic function ${\mathcal {F}}$ of $\Psi $ called the prepotential. Then the Lagrangian is given by ${\mathcal {L}}_{SYM2}=\mathrm {Im} \mathrm {Tr} \left({\frac {1}{4\pi }}\int d^{2}\theta d^{2}\vartheta {\mathcal {F}}(\Psi )\right)$ where $\theta ,\vartheta $ are coordinates for the spinor directions of superspace.[5] Once the low energy limit is taken, the ${\mathcal {N}}=2$ superfield $\Psi $ is typically labelled by ${\mathcal {A}}$ instead. The so called minimal theory is given by a specific choice of ${\mathcal {F}}$, ${\mathcal {F}}(\psi )={\frac {1}{2}}\tau \Psi ^{2},$ where $\tau $ is the complex coupling constant. The minimal theory can be written on Minkowski spacetime as ${\mathcal {L}}={\frac {1}{g^{2}}}\mathrm {Tr} \left(-{\frac {1}{4}}F_{\mu \nu }F^{\mu \nu }+g^{2}{\frac {\theta }{32\pi ^{2}}}F_{\mu \nu }*F^{\mu \nu }+(D_{\mu }\phi )^{\dagger }(D^{\mu }\phi )-{\frac {1}{2}}[\phi ,\phi ^{\dagger }]^{2}-i\lambda \sigma ^{\mu }D_{\mu }{\bar {\lambda }}-i{\bar {\psi }}{\bar {\sigma }}^{\mu }D_{\mu }\psi -i{\sqrt {2}}[\lambda ,\psi ]\phi ^{\dagger }-i{\sqrt {2}}[{\bar {\lambda }},{\bar {\psi }}]\phi \right)$ with $A_{\mu },\lambda ,\psi ,\phi $ making up the ${\mathcal {N}}=2$ chiral multiplet. Geometry of the moduli space For this section fix the gauge group as $\mathrm {SU(2)} $. A low-energy vacuum solution is an ${\mathcal {N}}=2$ vector superfield ${\mathcal {A}}$ solving the equations of motion of the low-energy Lagrangian, for which the scalar part $\phi $ has vanishing potential, which as mentioned earlier holds if $[\phi ,\phi ^{\dagger }]=0$ (which exactly means $\phi $ is a normal operator, and therefore diagonalizable). The scalar $\phi $ transforms in the adjoint, that is, it can be identified as an element of ${\mathfrak {su}}(2)_{\mathbb {C} }\cong {\mathfrak {sl}}(2,\mathbb {C} )$, the complexification of ${\mathfrak {su}}(2)$. Thus $\phi $ is traceless and diagonalizable so can be gauge rotated to (is in the conjugacy class of) a matrix of the form ${\frac {1}{2}}a\sigma _{3}$ (where $\sigma _{3}$ is the third Pauli matrix) for $a\in \mathbb {C} $. However, $a$ and $-a$ give conjugate matrices (corresponding to the fact the Weyl group of $\mathrm {SU} (2)$ is $\mathbb {Z} _{2}$) so both label the same vacuum. Thus the gauge invariant quantity labelling inequivalent vacua is $u=a^{2}/2=\mathrm {Tr} \phi ^{2}$. The (classical) moduli space of vacua is a one-dimensional complex manifold (Riemann surface) parametrized by $u$, although the Kähler metric is given in terms of $a$ as $ds^{2}=\mathrm {Im} {\frac {\partial ^{2}{\mathcal {F}}}{\partial a^{2}}}dad{\bar {a}}=\mathrm {Im} da_{D}d{\bar {a}}=-{\frac {i}{2}}(da_{D}d{\bar {a}}-dad{\bar {a}}_{D})=:\mathrm {Im} \tau (a)dad{\bar {a}},$ where $a_{D}={\frac {\partial {\mathcal {F}}}{\partial a}}$. This is not invariant under an arbitrary change of coordinates, but due to symmetry in $a$ and $a_{D}$, switching to local coordinate $a_{D}$ gives a metric similar to the final form but with a different harmonic function replacing $\mathrm {Im} \tau (a)$. The switching of the two coordinates can be interpreted as an instance of electric-magnetic duality (Seiberg & Witten 1994). Under a minimal assumption of assuming there are only three singularities in the moduli space at $u=-1,+1$ and $\infty $, with prescribed monodromy data at each point derived from quantum field theoretic arguments, the moduli space ${\mathcal {M}}$ was found to be $H/\Gamma (2)$, where $H$ is the hyperbolic half-plane and $\Gamma (2)<\mathrm {SL} (2,\mathbb {Z} )$ is the second principal congruence subgroup, the subgroup of matrices congruent to 1 mod 2, generated by $M_{\infty }={\begin{pmatrix}-1&2\\0&-1\end{pmatrix}},M_{1}={\begin{pmatrix}1&0\\-2&1\end{pmatrix}},M_{-1}={\begin{pmatrix}-1&2\\-2&3\end{pmatrix}}.$ This space is a six-fold cover of the fundamental domain of the modular group and admits an explicit description as parametrizing a space of elliptic curves $E_{u}$ given by the vanishing of $y^{2}=(x-1)(x+1)(x-u),$ which are the Seiberg–Witten curves. The curve becomes singular precisely when $u=-1,+1$ or $\infty $. Monopole condensation and confinement The theory exhibits physical phenomena involving and linking magnetic monopoles, confinement, an attained mass gap and strong-weak duality, described in section 5.6 of Seiberg and Witten (1994). The study of these physical phenomena also motivated the theory of Seiberg–Witten invariants. The low-energy action is described by the ${\mathcal {N}}=2$ chiral multiplet ${\mathcal {A}}$ with gauge group $\mathrm {U} (1)$, the residual unbroken gauge from the original $\mathrm {SU} (2)$ symmetry. This description is weakly coupled for large $u$, but strongly coupled for small $u$. However, at the strongly coupled point the theory admits a dual description which is weakly coupled. The dual theory has different field content, with two ${\mathcal {N}}=1$ chiral superfields $M,{\tilde {M}}$, and gauge field the dual photon ${\mathcal {A}}_{D}$, with a potential that gives equations of motion which are Witten's monopole equations, also known as the Seiberg–Witten equations at the critical points $u=\pm u_{0}$ where the monopoles become massless. In the context of Seiberg–Witten invariants, one can view Donaldson invariants as coming from a twist of the original theory at $u=\infty $ giving a topological field theory. On the other hand, Seiberg–Witten invariants come from twisting the dual theory at $u=\pm u_{0}$. In theory, such invariants should receive contributions from all finite $u$ but in fact can be localized to the two critical points, and topological invariants can be read off from solution spaces to the monopole equations.[6] Relation to integrable systems The special Kähler geometry on the moduli space of vacua in Seiberg–Witten theory can be identified with the geometry of the base of complex completely integrable system. The total phase of this complex completely integrable system can be identified with the moduli space of vacua of the 4d theory compactified on a circle. The relation between Seiberg–Witten theory and integrable systems has been reviewed by Eric D'Hoker and D. H. Phong.[7] See Hitchin system. Seiberg–Witten prepotential via instanton counting Using supersymmetric localisation techniques, one can explicitly determine the instanton partition function of ${\mathcal {N}}=2$ super Yang–Mills theory. The Seiberg–Witten prepotential can then be extracted using the localization approach[8] of Nikita Nekrasov. It arises in the flat space limit $\varepsilon _{1}$, $\varepsilon _{2}\to 0$, of the partition function of the theory subject to the so-called $\Omega $-background. The latter is a specific background of four dimensional ${\mathcal {N}}=2$ supergravity. It can be engineered, formally by lifting the super Yang–Mills theory to six dimensions, then compactifying on 2-torus, while twisting the four dimensional spacetime around the two non-contractible cycles. In addition, one twists fermions so as to produce covariantly constant spinors generating unbroken supersymmetries. The two parameters $\varepsilon _{1}$, $\varepsilon _{2}$ of the $\Omega $-background correspond to the angles of the spacetime rotation. In Ω-background, all the non-zero modes can be integrated out, so the path integral with the boundary condition $\phi \to a$ at $x\to \infty $ can be expressed as a sum over instanton number of the products and ratios of fermionic and bosonic determinants, producing the so-called Nekrasov partition function. In the limit where $\varepsilon _{1}$, $\varepsilon _{2}$ approach 0, this sum is dominated by a unique saddle point. On the other hand, when $\varepsilon _{1}$, $\varepsilon _{2}$ approach 0, $Z(a;\varepsilon _{1},\varepsilon _{2},\Lambda )=\exp \left(-{\frac {1}{\varepsilon _{1}\varepsilon _{2}}}\left({\mathcal {F}}(a;\Lambda )+{\mathcal {O}}(\varepsilon _{1},\varepsilon _{2})\right)\right)\,$ (8) holds. See also • Ginzburg–Landau theory • Donaldson theory References 1. Seiberg, Nathan; Witten, Edward (1994). "Electric - magnetic duality, monopole condensation, and confinement in N=2 supersymmetric Yang-Mills theory". Nucl. Phys. B. 426 (1): 19–52. arXiv:hep-th/9407087. Bibcode:1994NuPhB.426...19S. doi:10.1016/0550-3213(94)90124-4. S2CID 14361074. 2. Seiberg, Nathan; Witten, Edward (1994). "Monopoles, duality and chiral symmetry breaking in N=2 supersymmetric QCD". Nucl. Phys. B. 431 (3): 484–550. arXiv:hep-th/9408099. Bibcode:1994NuPhB.431..484S. doi:10.1016/0550-3213(94)90214-3. S2CID 17584951. 3. Nekrasov, Nikita (2004). "Seiberg-Witten Prepotential from Instanton Counting". Advances in Theoretical and Mathematical Physics. 7 (5): 831–864. arXiv:hep-th/0206161. doi:10.4310/ATMP.2003.v7.n5.a4. S2CID 2285041. 4. Nekrasov, Nikita; Okounkov, Andrei (2003). "Seiberg-Witten theory and random partitions". Prog. Math. Progress in Mathematics. 244: 525–596. arXiv:hep-th/0306238. Bibcode:2003hep.th....6238N. doi:10.1007/0-8176-4467-9_15. ISBN 978-0-8176-4076-7. S2CID 14329429. 5. Seiberg, Nathan (May 1988). "Supersymmetry and non-perturbative beta functions". Physics Letters B. 206 (1): 75–80. doi:10.1016/0370-2693(88)91265-8. 6. Witten, Edward (1994). "Monopoles and four-manifolds". Mathematical Research Letters. 1 (6): 769–796. arXiv:hep-th/9411102. doi:10.4310/MRL.1994.v1.n6.a13. 7. D'Hoker, Eric; Phong, D. H. (1999-12-29). "Lectures on Supersymmetric Yang-Mills Theory and Integrable Systems". Theoretical Physics at the End of the Twentieth Century. pp. 1–125. arXiv:hep-th/9912271. Bibcode:1999hep.th...12271D. doi:10.1007/978-1-4757-3671-7_1. ISBN 978-1-4419-2948-8. S2CID 117202391. 8. Nekrasov, Nikita (2004). "Seiberg-Witten Prepotential from Instanton Counting". Advances in Theoretical and Mathematical Physics. 7 (5): 831–864. arXiv:hep-th/0206161. doi:10.4310/ATMP.2003.v7.n5.a4. S2CID 2285041. • Jost, Jürgen (2002). Riemannian Geometry and Geometric Analysis. Springer-Verlag. ISBN 3-540-42627-2. (See Section 7.2) • Hunter-Jones, Nicholas R. (September 2012). Seiberg–Witten Theory and Duality in N = 2 Supersymmetric Gauge Theories (Masters). Imperial College London. 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Seiberg–Witten invariants In mathematics, and especially gauge theory, Seiberg–Witten invariants are invariants of compact smooth oriented 4-manifolds introduced by Edward Witten (1994), using the Seiberg–Witten theory studied by Nathan Seiberg and Witten (1994a, 1994b) during their investigations of Seiberg–Witten gauge theory. Seiberg–Witten invariants are similar to Donaldson invariants and can be used to prove similar (but sometimes slightly stronger) results about smooth 4-manifolds. They are technically much easier to work with than Donaldson invariants; for example, the moduli spaces of solutions of the Seiberg–Witten equations tends to be compact, so one avoids the hard problems involved in compactifying the moduli spaces in Donaldson theory. For detailed descriptions of Seiberg–Witten invariants see (Donaldson 1996), (Moore 2001), (Morgan 1996), (Nicolaescu 2000), (Scorpan 2005, Chapter 10). For the relation to symplectic manifolds and Gromov–Witten invariants see (Taubes 2000). For the early history see (Jackson 1995). Spinc-structures The Spinc group (in dimension 4) is $(U(1)\times \mathrm {Spin} (4))/(\mathbb {Z} /2\mathbb {Z} ).$ where the $\mathbb {Z} /2\mathbb {Z} $ acts as a sign on both factors. The group has a natural homomorphism to SO(4) = Spin(4)/±1. Given a compact oriented 4 manifold, choose a smooth Riemannian metric $g$ with Levi Civita connection $\nabla ^{g}$. This reduces the structure group from the connected component GL(4)+ to SO(4) and is harmless from a homotopical point of view. A Spinc-structure or complex spin structure on M is a reduction of the structure group to Spinc, i.e. a lift of the SO(4) structure on the tangent bundle to the group Spinc. By a theorem of Hirzebruch and Hopf, every smooth oriented compact 4-manifold $M$ admits a Spinc structure.[1] The existence of a Spinc structure is equivalent to the existence of a lift of the second Stiefel–Whitney class $w_{2}(M)\in H^{2}(M,\mathbb {Z} /2\mathbb {Z} )$ to a class $K\in H^{2}(X,\mathbb {Z} ).$ Conversely such a lift determines the Spinc structure up to 2 torsion in $H^{2}(X,\mathbb {Z} ).$ A spin structure proper requires the more restrictive $w_{2}(M)=0.$ A Spinc structure determines (and is determined by) a spinor bundle $W=W^{+}\oplus W^{-}$ coming from the 2 complex dimensional positive and negative spinor representation of Spin(4) on which U(1) acts by multiplication. We have $K=c_{1}(W^{+})=c_{1}(W^{-})$. The spinor bundle $W$ comes with a graded Clifford algebra bundle representation i.e. a map $\gamma :\mathrm {Cliff} (M,g)\to {\mathcal {E}}{\mathit {nd}}(W)$ :\mathrm {Cliff} (M,g)\to {\mathcal {E}}{\mathit {nd}}(W)} such that for each 1 form $a$ we have $\gamma (a):W^{\pm }\to W^{\mp }$ and $\gamma (a)^{2}=-g(a,a)$. There is a unique hermitian metric $h$ on $W$ s.t. $\gamma (a)$ is skew Hermitian for real 1 forms $a$. It gives an induced action of the forms $\wedge ^{}M$ by anti-symmetrising. In particular this gives an isomorphism of $\wedge ^{+}M\cong {\mathcal {E}}{\mathit {nd}}_{0}^{sh}(W^{+})$ of the selfdual two forms with the traceless skew Hermitian endomorphisms of $W^{+}$ which are then identified. Seiberg–Witten equations Let $L=\det(W^{+})\equiv \det(W^{-})$ be the determinant line bundle with $c_{1}(L)=K$. For every connection $\nabla _{A}=\nabla _{0}+A$ with $A\in iA_{\mathbb {R} }^{1}(M)$ on $L$, there is a unique spinor connection $\nabla ^{A}$ on $W$ i.e. a connection such that $\nabla _{X}^{A}(\gamma (a)):=[\nabla _{X}^{A},\gamma (a)]=\gamma (\nabla _{X}^{g}a)$ for every 1-form $a$ and vector field $X$. The Clifford connection then defines a Dirac operator $D^{A}=\gamma \otimes 1\circ \nabla ^{A}=\gamma (dx^{\mu })\nabla _{\mu }^{A}$ on $W$. The group of maps ${\mathcal {G}}=\{u:M\to U(1)\}$ acts as a gauge group on the set of all connections on $L$. The action of ${\mathcal {G}}$ can be "gauge fixed" e.g. by the condition $d^{}A=0$, leaving an effective parametrisation of the space of all such connections of $H^{1}(M,\mathbb {R} )^{\mathrm {harm} }/H^{1}(M,\mathbb {Z} )\oplus d^{}A_{\mathbb {R} }^{+}(M)$ with a residual $U(1)$ gauge group action. Write $\phi $ for a spinor field of positive chirality, i.e. a section of $W^{+}$. The Seiberg–Witten equations for $(\phi ,\nabla ^{A})$ are now $D^{A}\phi =0$ $F_{A}^{+}=\sigma (\phi )+i\omega $ Here $F^{A}\in iA_{\mathbb {R} }^{2}(M)$ is the closed curvature 2-form of $\nabla ^{A}$, $F_{A}^{+}$ is its self-dual part, and σ is the squaring map $\phi \mapsto \left(\phi h(\phi ,-)-{\tfrac {1}{2}}h(\phi ,\phi )1_{W^{+}}\right)$ from $W^{+}$ to the a traceless Hermitian endomorphism of $W^{+}$ identified with an imaginary self-dual 2-form, and $\omega $ is a real selfdual two form, often taken to be zero or harmonic. The gauge group ${\mathcal {G}}$ acts on the space of solutions. After adding the gauge fixing condition $d^{}A=0$ the residual U(1) acts freely, except for "reducible solutions" with $\phi =0$. For technical reasons, the equations are in fact defined in suitable Sobolev spaces of sufficiently high regularity. An application of the Weitzenböck formula ${\nabla ^{A}}^{}\nabla ^{A}\phi =(D^{A})^{2}\phi -({\tfrac {1}{2}}\gamma (F_{A}^{+})+s)\phi $ and the identity $\Delta _{g}\|\phi \|_{h}^{2}=2h({\nabla ^{A}}^{}\nabla ^{A}\phi ,\phi )-2\|\nabla ^{A}\phi \|_{g\otimes h}$ to solutions of the equations gives an equality $\Delta \|\phi \|^{2}+\|\nabla ^{A}\phi \|^{2}+{\tfrac {1}{4}}\|\phi \|^{4}=(-s)\|\phi \|^{2}-{\tfrac {1}{2}}h(\phi ,\gamma (\omega )\phi )$. If $\|\phi \|^{2}$ is maximal $\Delta \|\phi \|^{2}\geq 0$, so this shows that for any solution, the sup norm $\\|\phi \\|_{\infty }$ is a priori bounded with the bound depending only on the scalar curvature $s$ of $(M,g)$ and the self dual form $\omega $. After adding the gauge fixing condition, elliptic regularity of the Dirac equation shows that solutions are in fact a priori bounded in Sobolev norms of arbitrary regularity, which shows all solutions are smooth, and that the space of all solutions up to gauge equivalence is compact. The solutions $(\phi ,\nabla ^{A})$ of the Seiberg–Witten equations are called monopoles, as these equations are the field equations of massless magnetic monopoles on the manifold $M$. The moduli space of solutions The space of solutions is acted on by the gauge group, and the quotient by this action is called the moduli space of monopoles. The moduli space is usually a manifold. For generic metrics, after gauge fixing, the equations cut out the solution space transversely and so define a smooth manifold. The residual U(1) "gauge fixed" gauge group U(1) acts freely except at reducible monopoles i.e. solutions with $\phi =0$. By the Atiyah–Singer index theorem the moduli space is finite dimensional and has "virtual dimension" $(K^{2}-2\chi _{\mathrm {top} }(M)-3\operatorname {sign} (M))/4$ which for generic metrics is the actual dimension away from the reducibles. It means that the moduli space is generically empty if the virtual dimension is negative. For a self dual 2 form $\omega $, the reducible solutions have $\phi =0$, and so are determined by connections $\nabla _{A}=\nabla _{0}+A$ on $L$ such that $F_{0}+dA=i(\alpha +\omega )$ for some anti selfdual 2-form $\alpha $. By the Hodge decomposition, since $F_{0}$ is closed, the only obstruction to solving this equation for $A$ given $\alpha $ and $\omega $, is the harmonic part of $\alpha $ and $\omega $, and the harmonic part, or equivalently, the (de Rham) cohomology class of the curvature form i.e. $[F_{0}]=F_{0}^{\mathrm {harm} }=i(\omega ^{\mathrm {harm} }+\alpha ^{\mathrm {harm} })\in H^{2}(M,\mathbb {R} )$. Thus, since the $[{\tfrac {1}{2\pi i}}F_{0}]=K$ the necessary and sufficient condition for a reducible solution is $\omega ^{\mathrm {harm} }\in 2\pi K+{\mathcal {H}}^{-}\in H^{2}(X,\mathbb {R} )$ where ${\mathcal {H}}^{-}$ is the space of harmonic anti-selfdual 2-forms. A two form $\omega $ is $K$-admissible if this condition is not met and solutions are necessarily irreducible. In particular, for $b^{+}\geq 1$, the moduli space is a (possibly empty) compact manifold for generic metrics and admissible $\omega $. Note that, if $b_{+}\geq 2$ the space of $K$-admissible two forms is connected, whereas if $b_{+}=1$ it has two connected components (chambers). The moduli space can be given a natural orientation from an orientation on the space of positive harmonic 2 forms, and the first cohomology. The a priori bound on the solutions, also gives a priori bounds on $F^{\mathrm {harm} }$. There are therefore (for fixed $\omega $) only finitely many $K\in H^{2}(M,\mathbb {Z} )$, and hence only finitely many Spinc structures, with a non empty moduli space. Seiberg–Witten invariants The Seiberg–Witten invariant of a four-manifold M with b2+(M) ≥ 2 is a map from the spinc structures on M to Z. The value of the invariant on a spinc structure is easiest to define when the moduli space is zero-dimensional (for a generic metric). In this case the value is the number of elements of the moduli space counted with signs. The Seiberg–Witten invariant can also be defined when b2+(M) = 1, but then it depends on the choice of a chamber. A manifold M is said to be of simple type if the Seiberg–Witten invariant vanishes whenever the expected dimension of the moduli space is nonzero. The simple type conjecture states that if M is simply connected and b2+(M) ≥ 2 then the manifold is of simple type. This is true for symplectic manifolds. If the manifold M has a metric of positive scalar curvature and b2+(M) ≥ 2 then all Seiberg–Witten invariants of M vanish. If the manifold M is the connected sum of two manifolds both of which have b2+ ≥ 1 then all Seiberg–Witten invariants of M vanish. If the manifold M is simply connected and symplectic and b2+(M) ≥ 2 then it has a spinc structure s on which the Seiberg–Witten invariant is 1. In particular it cannot be split as a connected sum of manifolds with b2+ ≥ 1. References 1. Hirzebruch, F.; Hopf, H. (1958). "Felder von Flächenelementen in 4-dimensionalen Mannigfaltigkeiten". Math. Ann. 136 (2): 156–172. doi:10.1007/BF01362296. hdl:21.11116/0000-0004-3A18-1. S2CID 120557396. • Donaldson, Simon K. (1996), "The Seiberg-Witten equations and 4-manifold topology.", Bulletin of the American Mathematical Society, (N.S.), 33 (1): 45–70, doi:10.1090/S0273-0979-96-00625-8, MR 1339810 • Jackson, Allyn (1995), A revolution in mathematics, archived from the original on April 26, 2010 • Morgan, John W. (1996), The Seiberg–Witten equations and applications to the topology of smooth four-manifolds, Mathematical Notes, vol. 44, Princeton, NJ: Princeton University Press, pp. viii+128, ISBN 978-0-691-02597-1, MR 1367507 • Moore, John Douglas (2001), Lectures on Seiberg-Witten invariants, Lecture Notes in Mathematics, vol. 1629 (2nd ed.), Berlin: Springer-Verlag, pp. viii+121, CiteSeerX 10.1.1.252.2658, doi:10.1007/BFb0092948, ISBN 978-3-540-41221-2, MR 1830497 • Nash, Ch. (2001) [1994], "Seiberg-Witten equations", Encyclopedia of Mathematics, EMS Press • Nicolaescu, Liviu I. (2000), Notes on Seiberg-Witten theory (PDF), Graduate Studies in Mathematics, vol. 28, Providence, RI: American Mathematical Society, pp. xviii+484, doi:10.1090/gsm/028, ISBN 978-0-8218-2145-9, MR 1787219 • Scorpan, Alexandru (2005), The wild world of 4-manifolds, American Mathematical Society, ISBN 978-0-8218-3749-8, MR 2136212. • Seiberg, Nathan; Witten, Edward (1994a), "Electric-magnetic duality, monopole condensation, and confinement in N=2 supersymmetric Yang-Mills theory", Nuclear Physics B, 426 (1): 19–52, arXiv:hep-th/9407087, Bibcode:1994NuPhB.426...19S, doi:10.1016/0550-3213(94)90124-4, MR 1293681, S2CID 14361074; "Erratum", Nuclear Physics B, 430 (2): 485–486, 1994, Bibcode:1994NuPhB.430..485., doi:10.1016/0550-3213(94)00449-8, MR 1303306 • Seiberg, N.; Witten, E. (1994b), "Monopoles, duality and chiral symmetry breaking in N=2 supersymmetric QCD", Nuclear Physics B, 431 (3): 484–550, arXiv:hep-th/9408099, Bibcode:1994NuPhB.431..484S, doi:10.1016/0550-3213(94)90214-3, MR 1306869, S2CID 17584951 • Taubes, Clifford Henry (2000), Wentworth, Richard (ed.), Seiberg Witten and Gromov invariants for symplectic 4-manifolds, First International Press Lecture Series, vol. 2, Somerville, MA: International Press, pp. vi+401, ISBN 978-1-57146-061-5, MR 1798809 • Witten, Edward (1994), "Monopoles and four-manifolds.", Mathematical Research Letters, 1 (6): 769–796, arXiv:hep-th/9411102, Bibcode:1994MRLet...1..769W, doi:10.4310/MRL.1994.v1.n6.a13, MR 1306021, S2CID 10611124, archived from the original on 2013-06-29	Wikipedia
Paul Seidel Paul Seidel (born 30 December 1970) is a Swiss-Italian mathematician. He is a faculty member at the Massachusetts Institute of Technology. Paul Seidel Born (1970-12-30) 30 December 1970 Florence, Italy Alma materUniversity of Oxford University of Heidelberg Awards • Simons Investigator (2012) • Fellow of the American Mathematical Society (2012) • Veblen Prize in Geometry (2010) • EMS Prize (2000) Scientific career FieldsMathematics InstitutionsUniversity of Chicago Massachusetts Institute of Technology Doctoral advisorSimon Donaldson Doctoral studentsAilsa Keating Career Seidel attended Heidelberg University, where he received his Diplom under supervision of Albrecht Dold in 1994. He then pursued his Ph.D. studies at the University of Oxford under supervision of Simon Donaldson (Thesis: Floer Homology and the Symplectic Isotopy Problem) in 1998. He was a chargé de recherche at the CNRS from 1999 to 2002, a professor at Imperial College London from 2002 to 2003, a professor at the University of Chicago from 2003 to 2007, and then a professor at the Massachusetts Institute of Technology from 2007 onwards.[1] Awards In 2000, Seidel was awarded the EMS Prize.[2] In 2010, he was awarded the Oswald Veblen Prize in Geometry "for his fundamental contributions to symplectic geometry and, in particular, for his development of advanced algebraic methods for computation of symplectic invariants."[3] In 2012, he became a fellow of the American Mathematical Society[4] and a Simons Investigator.[5] Personal life Seidel is married to Ju-Lee Kim, who is also a professor of mathematics at MIT.[6] Publications • Fukaya Categories and Picard-Lefschetz Theory, European Mathematical Society, 2008[7] References 1. "Curriculum Vitae" (PDF). Paul Seidel. Retrieved January 23, 2023. 2. "History of Prizes of the European Mathematical Society". Retrieved 2 October 2020. 3. https://www.ams.org/notices/201004/rtx100400521p.pdf 4. "List of Fellows of the American Mathematical Society". Retrieved 2013-07-15. 5. "Simons Investigators Awardees". Simons Foundation. 6. "Ju-Lee Kim". MIT Women in Mathematics. Massachusetts Institute of Technology. Archived from the original on 2015-11-21. Retrieved 2015-11-20.. 7. Smith, Ivan (2010). "Review: Fukaya categories and Picard-Lefschetz theory, by Paul Seidel". Bulletin of the American Mathematical Society. (N.S.). 47 (4): 735–742. doi:10.1090/s0273-0979-10-01289-9. External links • Website at MIT • Paul Seidel at the Mathematics Genealogy Project • Laudatio from the Veblen Prize, Notices AMS, April 2010 • Honors for the EMS Prize 2000 Recipients of the Oswald Veblen Prize in Geometry • 1964 Christos Papakyriakopoulos • 1964 Raoul Bott • 1966 Stephen Smale • 1966 Morton Brown and Barry Mazur • 1971 Robion Kirby • 1971 Dennis Sullivan • 1976 William Thurston • 1976 James Harris Simons • 1981 Mikhail Gromov • 1981 Shing-Tung Yau • 1986 Michael Freedman • 1991 Andrew Casson and Clifford Taubes • 1996 Richard S. Hamilton and Gang Tian • 2001 Jeff Cheeger, Yakov Eliashberg and Michael J. Hopkins • 2004 David Gabai • 2007 Peter Kronheimer and Tomasz Mrowka; Peter Ozsváth and Zoltán Szabó • 2010 Tobias Colding and William Minicozzi; Paul Seidel • 2013 Ian Agol and Daniel Wise • 2016 Fernando Codá Marques and André Neves • 2019 Xiuxiong Chen, Simon Donaldson and Song Sun Authority control International • ISNI • VIAF National • France • BnF data • Germany • Israel • United States • Netherlands Academics • MathSciNet • Mathematics Genealogy Project • ORCID • zbMATH Other • IdRef	Wikipedia
Seidel adjacency matrix In mathematics, in graph theory, the Seidel adjacency matrix of a simple undirected graph G is a symmetric matrix with a row and column for each vertex, having 0 on the diagonal, −1 for positions whose rows and columns correspond to adjacent vertices, and +1 for positions corresponding to non-adjacent vertices. It is also called the Seidel matrix or—its original name—the (−1,1,0)-adjacency matrix. It can be interpreted as the result of subtracting the adjacency matrix of G from the adjacency matrix of the complement of G. The multiset of eigenvalues of this matrix is called the Seidel spectrum. The Seidel matrix was introduced by J. H. van Lint and Johan Jacob Seidel in 1966 and extensively exploited by Seidel and coauthors. The Seidel matrix of G is also the adjacency matrix of a signed complete graph KG in which the edges of G are negative and the edges not in G are positive. It is also the adjacency matrix of the two-graph associated with G and KG. The eigenvalue properties of the Seidel matrix are valuable in the study of strongly regular graphs. References • van Lint, J. H., and Seidel, J. J. (1966), Equilateral point sets in elliptic geometry. Indagationes Mathematicae, vol. 28 (= Proc. Kon. Ned. Aka. Wet. Ser. A, vol. 69), pp. 335–348. • Seidel, J. J. (1976), A survey of two-graphs. In: Colloquio Internazionale sulle Teorie Combinatorie (Proceedings, Rome, 1973), vol. I, pp. 481–511. Atti dei Convegni Lincei, No. 17. Accademia Nazionale dei Lincei, Rome. • Seidel, J. J. (1991), ed. D.G. Corneil and R. Mathon, Geometry and Combinatorics: Selected Works of J. J. Seidel. Boston: Academic Press. Many of the articles involve the Seidel matrix. • Seidel, J. J. (1968), Strongly Regular Graphs with (−1,1,0) Adjacency Matrix Having Eigenvalue 3. Linear Algebra and its Applications 1, 281–298. Matrix classes Explicitly constrained entries • Alternant • Anti-diagonal • Anti-Hermitian • Anti-symmetric • Arrowhead • Band • Bidiagonal • Bisymmetric • Block-diagonal • Block • Block tridiagonal • Boolean • Cauchy • Centrosymmetric • Conference • Complex Hadamard • Copositive • Diagonally dominant • Diagonal • Discrete Fourier Transform • Elementary • Equivalent • Frobenius • Generalized permutation • Hadamard • Hankel • Hermitian • Hessenberg • Hollow • Integer • Logical • Matrix unit • Metzler • Moore • Nonnegative • Pentadiagonal • Permutation • Persymmetric • Polynomial • Quaternionic • Signature • Skew-Hermitian • Skew-symmetric • Skyline • Sparse • Sylvester • Symmetric • Toeplitz • Triangular • Tridiagonal • Vandermonde • Walsh • Z Constant • Exchange • Hilbert • Identity • Lehmer • Of ones • Pascal • Pauli • Redheffer • Shift • Zero Conditions on eigenvalues or eigenvectors • Companion • Convergent • Defective • Definite • Diagonalizable • Hurwitz • Positive-definite • Stieltjes Satisfying conditions on products or inverses • Congruent • Idempotent or Projection • Invertible • Involutory • Nilpotent • Normal • Orthogonal • Unimodular • Unipotent • Unitary • Totally unimodular • Weighing With specific applications • Adjugate • Alternating sign • Augmented • Bézout • Carleman • Cartan • Circulant • Cofactor • Commutation • Confusion • Coxeter • Distance • Duplication and elimination • Euclidean distance • Fundamental (linear differential equation) • Generator • Gram • Hessian • Householder • Jacobian • Moment • Payoff • Pick • Random • Rotation • Seifert • Shear • Similarity • Symplectic • Totally positive • Transformation Used in statistics • Centering • Correlation • Covariance • Design • Doubly stochastic • Fisher information • Hat • Precision • Stochastic • Transition Used in graph theory • Adjacency • Biadjacency • Degree • Edmonds • Incidence • Laplacian • Seidel adjacency • Tutte Used in science and engineering • Cabibbo–Kobayashi–Maskawa • Density • Fundamental (computer vision) • Fuzzy associative • Gamma • Gell-Mann • Hamiltonian • Irregular • Overlap • S • State transition • Substitution • Z (chemistry) Related terms • Jordan normal form • Linear independence • Matrix exponential • Matrix representation of conic sections • Perfect matrix • Pseudoinverse • Row echelon form • Wronskian • Mathematics portal • List of matrices • Category:Matrices	Wikipedia
Seifert–Van Kampen theorem In mathematics, the Seifert–Van Kampen theorem of algebraic topology (named after Herbert Seifert and Egbert van Kampen), sometimes just called Van Kampen's theorem, expresses the structure of the fundamental group of a topological space $X$ in terms of the fundamental groups of two open, path-connected subspaces that cover $X$. It can therefore be used for computations of the fundamental group of spaces that are constructed out of simpler ones. Van Kampen's theorem for fundamental groups Let X be a topological space which is the union of two open and path connected subspaces U1, U2. Suppose U1 ∩ U2 is path connected and nonempty, and let x0 be a point in U1 ∩ U2 that will be used as the base of all fundamental groups. The inclusion maps of U1 and U2 into X induce group homomorphisms $j_{1}:\pi _{1}(U_{1},x_{0})\to \pi _{1}(X,x_{0})$ and $j_{2}:\pi _{1}(U_{2},x_{0})\to \pi _{1}(X,x_{0})$. Then X is path connected and $j_{1}$ and $j_{2}$ form a commutative pushout diagram: The natural morphism k is an isomorphism. That is, the fundamental group of X is the free product of the fundamental groups of U1 and U2 with amalgamation of $\pi _{1}(U_{1}\cap U_{2},x_{0})$.[1] Usually the morphisms induced by inclusion in this theorem are not themselves injective, and the more precise version of the statement is in terms of pushouts of groups. Van Kampen's theorem for fundamental groupoids Unfortunately, the theorem as given above does not compute the fundamental group of the circle – which is the most important basic example in algebraic topology – because the circle cannot be realised as the union of two open sets with connected intersection. This problem can be resolved by working with the fundamental groupoid $\pi _{1}(X,A)$ on a set A of base points, chosen according to the geometry of the situation. Thus for the circle, one uses two base points.[2] This groupoid consists of homotopy classes relative to the end points of paths in X joining points of A ∩ X. In particular, if X is a contractible space, and A consists of two distinct points of X, then $\pi _{1}(X,A)$ is easily seen to be isomorphic to the groupoid often written ${\mathcal {I}}$ with two vertices and exactly one morphism between any two vertices. This groupoid plays a role in the theory of groupoids analogous to that of the group of integers in the theory of groups.[3] The groupoid ${\mathcal {I}}$ also allows for groupoids a notion of homotopy: it is a unit interval object in the category of groupoids. The category of groupoids admits all colimits, and in particular all pushouts. Theorem. Let the topological space X be covered by the interiors of two subspaces X1, X2 and let A be a set which meets each path component of X1, X2 and X0 = X1 ∩ X2. Then A meets each path component of X and the diagram P of morphisms induced by inclusion is a pushout diagram in the category of groupoids.[4] This theorem gives the transition from topology to algebra, in determining completely the fundamental groupoid $\pi _{1}(X,A)$; one then has to use algebra and combinatorics to determine a fundamental group at some basepoint. One interpretation of the theorem is that it computes homotopy 1-types. To see its utility, one can easily find cases where X is connected but is the union of the interiors of two subspaces, each with say 402 path components and whose intersection has say 1004 path components. The interpretation of this theorem as a calculational tool for "fundamental groups" needs some development of 'combinatorial groupoid theory'.[5][6] This theorem implies the calculation of the fundamental group of the circle as the group of integers, since the group of integers is obtained from the groupoid ${\mathcal {I}}$ by identifying, in the category of groupoids, its two vertices. There is a version of the last theorem when X is covered by the union of the interiors of a family $\{U_{\lambda }:\lambda \in \Lambda \}$ of subsets.[7][8] The conclusion is that if A meets each path component of all 1,2,3-fold intersections of the sets $U_{\lambda }$, then A meets all path components of X and the diagram $\bigsqcup _{(\lambda ,\mu )\in \Lambda ^{2}}\pi _{1}(U_{\lambda }\cap U_{\mu },A)\rightrightarrows \bigsqcup _{\lambda \in \Lambda }\pi _{1}(U_{\lambda },A)\rightarrow \pi _{1}(X,A)$ of morphisms induced by inclusions is a coequaliser in the category of groupoids. [...] people still obstinately persist, when calculating with fundamental groups, in fixing a single base point, instead of cleverly choosing a whole packet of points which is invariant under the symmetries of the situation, which thus get lost on the way. In certain situations (such as descent theorems for fundamental groups à la Van Kampen) it is much more elegant, even indispensable for understanding something, to work with fundamental groupoids with respect to a suitable packet of base points [...] — Alexander Grothendieck, Esquisse d'un Programme (Section 2, English translation) Equivalent formulations In the language of combinatorial group theory, if $X$ is a topological space; $U$ and $V$ are open, path connected subspaces of $X$; $U\cap V$ is nonempty and path-connected; and $w\in U\cap V$; then $\pi _{1}(X,w)$ is the free product with amalgamation of $\pi _{1}(U,w)$ and $\pi _{1}(V,w)$, with respect to the (not necessarily injective) homomorphisms $I:\pi _{1}(U\cap V,w)\to \pi _{1}(U,w)$ and $J:\pi _{1}(U\cap V,w)\to \pi _{1}(V,w)$. Given group presentations: ${\begin{aligned}\pi _{1}(U,w)&=\langle u_{1},\dots ,u_{k}\mid \alpha _{1},\dots ,\alpha _{l}\rangle \\\pi _{1}(V,w)&=\langle v_{1},\dots ,v_{m}\mid \beta _{1},\dots ,\beta _{n}\rangle \\\pi _{1}(U\cap V,w)&=\langle w_{1},\dots ,w_{p}\mid \gamma _{1},\dots ,\gamma _{q}\rangle \end{aligned}}$ the amalgamation can be presented[9] as $\pi _{1}(X,w)=\left\langle u_{1},\dots ,u_{k},v_{1},\dots ,v_{m}\left\|\alpha _{1},\dots ,\alpha _{l},\beta _{1},\dots ,\beta _{n},I(w_{1})J(w_{1})^{-1},\dots ,I(w_{p})J(w_{p})^{-1}\right.\right\rangle .$ In category theory, $\pi _{1}(X,w)$ is the pushout, in the category of groups, of the diagram: $\pi _{1}(U,w)\gets \pi _{1}(U\cap V,w)\to \pi _{1}(V,w).$ Examples 2-sphere One can use Van Kampen's theorem to calculate fundamental groups for topological spaces that can be decomposed into simpler spaces. For example, consider the sphere $S^{2}$. Pick open sets $A=S^{2}\setminus \{n\}$ and $B=S^{2}\setminus \{s\}$ where n and s denote the north and south poles respectively. Then we have the property that A, B and A ∩ B are open path connected sets. Thus we can see that there is a commutative diagram including A ∩ B into A and B and then another inclusion from A and B into $S^{2}$ and that there is a corresponding diagram of homomorphisms between the fundamental groups of each subspace. Applying Van Kampen's theorem gives the result $\pi _{1}(S^{2})=\pi _{1}(A)\cdot \pi _{1}(B)/\ker(\Phi ).$ However, A and B are both homeomorphic to R2 which is simply connected, so both A and B have trivial fundamental groups. It is clear from this that the fundamental group of $S^{2}$ is trivial. Wedge sum of spaces Given two pointed spaces $(X,x)$ and $(Y,y)$ we can form their wedge sum, $(X\vee Y,p)$, by taking the quotient of $X\coprod Y$ by identifying their two basepoints. If $x$ admits a contractible open neighborhood $U\subset X$ and $y$ admits a contractible open neighborhood $V\subset Y$ (which is the case if, for instance, $X$ and $Y$ are CW complexes), then we can apply the Van Kampen theorem to $X\vee Y$ by taking $X\vee V$ and $U\vee Y$ as the two open sets and we conclude that the fundamental group of the wedge is the free product of the fundamental groups of the two spaces we started with: $\pi _{1}(X\vee Y,p)\cong \pi _{1}(X,x)*\pi _{1}(Y,y)$. Orientable genus-g surfaces A more complicated example is the calculation of the fundamental group of a genus-n orientable surface S, otherwise known as the genus-n surface group. One can construct S using its standard fundamental polygon. For the first open set A, pick a disk within the center of the polygon. Pick B to be the complement in S of the center point of A. Then the intersection of A and B is an annulus, which is known to be homotopy equivalent to (and so has the same fundamental group as) a circle. Then $\pi _{1}(A\cap B)=\pi _{1}(S^{1})$, which is the integers, and $\pi _{1}(A)=\pi _{1}(D^{2})={1}$. Thus the inclusion of $\pi _{1}(A\cap B)$ into $\pi _{1}(A)$ sends any generator to the trivial element. However, the inclusion of $\pi _{1}(A\cap B)$ into $\pi _{1}(B)$ is not trivial. In order to understand this, first one must calculate $\pi _{1}(B)$. This is easily done as one can deformation retract B (which is S with one point deleted) onto the edges labeled by $A_{1}B_{1}A_{1}^{-1}B_{1}^{-1}A_{2}B_{2}A_{2}^{-1}B_{2}^{-1}\cdots A_{n}B_{n}A_{n}^{-1}B_{n}^{-1}.$ This space is known to be the wedge sum of 2n circles (also called a bouquet of circles), which further is known to have fundamental group isomorphic to the free group with 2n generators, which in this case can be represented by the edges themselves: $\{A_{1},B_{1},\dots ,A_{n},B_{n}\}$. We now have enough information to apply Van Kampen's theorem. The generators are the loops $\{A_{1},B_{1},\dots ,A_{n},B_{n}\}$ (A is simply connected, so it contributes no generators) and there is exactly one relation: $A_{1}B_{1}A_{1}^{-1}B_{1}^{-1}A_{2}B_{2}A_{2}^{-1}B_{2}^{-1}\cdots A_{n}B_{n}A_{n}^{-1}B_{n}^{-1}=1.$ Using generators and relations, this group is denoted $\left\langle A_{1},B_{1},\dots ,A_{n},B_{n}\left\|A_{1}B_{1}A_{1}^{-1}B_{1}^{-1}\cdots A_{n}B_{n}A_{n}^{-1}B_{n}^{-1}\right.\right\rangle .$ Simple-connectedness If X is space that can be written as the union of two open simply connected sets U and V with U ∩ V non-empty and path-connected, then X is simply connected.[10] Generalizations As explained above, this theorem was extended by Ronald Brown to the non-connected case by using the fundamental groupoid $\pi _{1}(X,A)$ on a set A of base points. The theorem for arbitrary covers, with the restriction that A meets all threefold intersections of the sets of the cover, is given in the paper by Brown and Abdul Razak Salleh.[11] The theorem and proof for the fundamental group, but using some groupoid methods, are also given in J. Peter May's book.[12] The version that allows more than two overlapping sets but with A a singleton is also given in Allen Hatcher's book below, theorem 1.20. Applications of the fundamental groupoid on a set of base points to the Jordan curve theorem, covering spaces, and orbit spaces are given in Ronald Brown's book.[13] In the case of orbit spaces, it is convenient to take A to include all the fixed points of the action. An example here is the conjugation action on the circle. References to higher-dimensional versions of the theorem which yield some information on homotopy types are given in an article on higher-dimensional group theories and groupoids.[14] Thus a 2-dimensional Van Kampen theorem which computes nonabelian second relative homotopy groups was given by Ronald Brown and Philip J. Higgins.[15] A full account and extensions to all dimensions are given by Brown, Higgins, and Rafael Sivera,[16] while an extension to n-cubes of spaces is given by Ronald Brown and Jean-Louis Loday.[17] Fundamental groups also appear in algebraic geometry and are the main topic of Alexander Grothendieck's first Séminaire de géométrie algébrique (SGA1). A version of Van Kampen's theorem appears there, and is proved along quite different lines than in algebraic topology, namely by descent theory. A similar proof works in algebraic topology.[18] See also • Higher-dimensional algebra • Higher category theory • Pseudocircle • Ronald Brown (mathematician) Notes 1. Lee, John M. (2011). Introduction to topological manifolds (2nd ed.). New York: Springer. ISBN 978-1-4419-7939-1. OCLC 697506452. pg. 252, Theorem 10.1. 2. R. Brown, Groupoids and Van Kampen's theorem, Proc. London Math. Soc. (3) 17 (1967) 385–401. 3. Ronald Brown. "Groupoids in Mathematics". http://groupoids.org.uk/gpdsweb.html 4. R. Brown. Topology and Groupoids., Booksurge PLC (2006). http://groupoids.org.uk/topgpds.html 5. P.J. Higgins, Categories and Groupoids, Van Nostrand, 1971, Reprints of Theory and Applications of Categories, No. 7 (2005),pp 1–195. 6. R. Brown, Topology and Groupoids., Booksurge PLC (2006). 7. Ronald Brown, Philip J. Higgins and Rafael Sivera. Nonabelian Algebraic Topology: filtered spaces, crossed complexes, cubical homotopy groupoids, European Mathematical Society Tracts vol 15, August, 2011. 8. "Higher-dimensional, generalized Van Kampen theorems (HD-GVKT)". 9. Lee 2011, p. 253, Theorem 10.3. 10. Greenberg & Harper 1981 11. Brown, Ronald; Salleh, Abdul Razak (1984). "A Van Kampen theorem for unions of nonconnected spaces". Archiv der Mathematik. Basel. 42 (1): 85–88. doi:10.1007/BF01198133. 12. May, J. Peter (1999). A Concise Introduction to Algebraic Topology. chapter 2. 13. Brown, Ronald, "Topology and Groupoids", Booksurge, (2006) 14. Ronald Brown. "Higher-dimensional group theory" . 2007. http://www.bangor.ac.uk/~mas010/hdaweb2.htm 15. Brown, Ronald; Higgins, Philip J. (1978). "On the connection between the second relative homotopy groups of some related spaces". Proceedings of the London Mathematical Society. 3. 36: 193–212. doi:10.1112/plms/s3-36.2.193. 16. Brown, Ronald, Higgins, Philip J., and Sivera, Rafael, "Nonabelian algebraic topology: filtered spaces, crossed complexes, cubical homotopy groupoids", EMS Tracts in Mathematics vol 15, 20011. http://groupoids.org.uk/nonab-a-t.html 17. Brown, Ronald; Loday, Jean-Louis (1987). "Van Kampen theorems for diagrams of spaces". Topology. 26: 311–334. doi:10.1016/0040-9383(87)90004-8. 18. Douady, Adrien and Douady, Régine, "Algèbre et théories galoisiennes", Cassini (2005) References • Allen Hatcher, Algebraic topology. (2002) Cambridge University Press, Cambridge, xii+544 pp. ISBN 0-521-79160-X and ISBN 0-521-79540-0 • Peter May, A Concise Course in Algebraic Topology. (1999) University of Chicago Press, ISBN 0-226-51183-9 (Section 2.7 provides a category-theoretic presentation of the theorem as a colimit in the category of groupoids). • Ronald Brown, Groupoids and Van Kampen's theorem, Proc. London Math. Soc. (3) 17 (1967) 385–401. • Mathoverflow discussion on many base points • Ronald Brown, Topology and groupoids (2006) Booksurge LLC ISBN 1-4196-2722-8 • R. Brown and A. Razak, A Van Kampen theorem for unions of non-connected spaces, Archiv. Math. 42 (1984) 85–88. (This paper gives probably the optimal version of the theorem, namely the groupoid version of the theorem for an arbitrary open cover and a set of base points which meets every path component of every 1-.2-3-fold intersections of the sets of the cover.) • P.J. Higgins, Categories and groupoids (1971) Van Nostrand Reinhold • Ronald Brown, Higher-dimensional group theory (2007) (Gives a broad view of higher-dimensional Van Kampen theorems involving multiple groupoids). • Greenberg, Marvin J.; Harper, John R. (1981), Algebraic topology. A first course, Mathematics Lecture Note Series, vol. 58, Benjamin/Cummings, ISBN 0805335579 • Seifert, H., Konstruction drei dimensionaler geschlossener Raume. Berichte Sachs. Akad. Leipzig, Math.-Phys. Kl. (83) (1931) 26–66. • E. R. van Kampen. On the connection between the fundamental groups of some related spaces. American Journal of Mathematics, vol. 55 (1933), pp. 261–267. • Brown, R., Higgins, P. J, On the connection between the second relative homotopy groups of some related spaces, Proc. London Math. Soc. (3) 36 (1978) 193–212. • Brown, R., Higgins, P. J. and Sivera, R.. 2011, EMS Tracts in Mathematics Vol.15 (2011) Nonabelian Algebraic Topology: filtered spaces, crossed complexes, cubical homotopy groupoids; (The first of three Parts discusses the applications of the 1- and 2-dimensional versions of the Seifert–van Kampen Theorem. The latter allows calculations of nonabelian second relative homotopy groups, and in fact of homotopy 2-types. The second part applies a Higher Homotopy van Kampen Theorem for crossed complexes, proved in Part III.) • "Van Kampen's theorem result". PlanetMath. • R. Brown, H. Kamps, T. Porter : A homotopy double groupoid of a Hausdorff space II: a Van Kampen theorem', Theory and Applications of Categories, 14 (2005) 200–220. • Dylan G.L. Allegretti, Simplicial Sets and Van Kampen's Theorem (Discusses generalized versions of Van Kampen's theorem applied to topological spaces and simplicial sets). • R. Brown and J.-L. Loday, "Van Kampen theorems for diagrams of spaces", Topology 26 (1987) 311–334. This article incorporates material from Van Kampen's theorem on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License. External links • Media related to Seifert–Van Kampen theorem at Wikimedia Commons Topology Fields • General (point-set) • Algebraic • Combinatorial • Continuum • Differential • Geometric • low-dimensional • Homology • cohomology • Set-theoretic • Digital Key concepts • Open set / Closed set • Interior • Continuity • Space • compact • Connected • Hausdorff • metric • uniform • Homotopy • homotopy group • fundamental group • Simplicial complex • CW complex • Polyhedral complex • Manifold • Bundle (mathematics) • Second-countable space • Cobordism Metrics and properties • Euler characteristic • Betti number • Winding number • Chern number • Orientability Key results • Banach fixed-point theorem • De Rham cohomology • Invariance of domain • Poincaré conjecture • Tychonoff's theorem • Urysohn's lemma • Category • Mathematics portal • Wikibook • Wikiversity • Topics • general • algebraic • geometric • Publications	Wikipedia
Seifert conjecture In mathematics, the Seifert conjecture states that every nonsingular, continuous vector field on the 3-sphere has a closed orbit. It is named after Herbert Seifert. In a 1950 paper, Seifert asked if such a vector field exists, but did not phrase non-existence as a conjecture. He also established the conjecture for perturbations of the Hopf fibration. The conjecture was disproven in 1974 by Paul Schweitzer, who exhibited a $C^{1}$ counterexample. Schweitzer's construction was then modified by Jenny Harrison in 1988 to make a $C^{2+\delta }$ counterexample for some $\delta >0$. The existence of smoother counterexamples remained an open question until 1993 when Krystyna Kuperberg constructed a very different $C^{\infty }$ counterexample. Later this construction was shown to have real analytic and piecewise linear versions. References • V. Ginzburg and B. Gürel, A $C^{2}$-smooth counterexample to the Hamiltonian Seifert conjecture in $R^{4}$, Ann. of Math. (2) 158 (2003), no. 3, 953–976 • Harrison, Jenny (1988). "$C^{2}$ counterexamples to the Seifert conjecture". Topology. 27 (3): 249–278. doi:10.1016/0040-9383(88)90009-2. MR 0963630. • Kuperberg, Greg (1996). "A volume-preserving counterexample to the Seifert conjecture". Commentarii Mathematici Helvetici. 71 (1): 70–97. arXiv:alg-geom/9405012. doi:10.1007/BF02566410. MR 1371679. • Kuperberg, Greg; Kuperberg, Krystyna (1996). "Generalized counterexamples to the Seifert conjecture". Annals of Mathematics. (2). 143 (3): 547–576. arXiv:math/9802040. doi:10.2307/2118536. MR 1394969. • Kuperberg, Krystyna (1994). "A smooth counterexample to the Seifert conjecture". Annals of Mathematics. (2). 140 (3): 723–732. doi:10.2307/2118623. MR 1307902. • P. A. Schweitzer, Counterexamples to the Seifert conjecture and opening closed leaves of foliations, Annals of Mathematics (2) 100 (1974), 386–400. • H. Seifert, Closed integral curves in 3-space and isotopic two-dimensional deformations, Proc. Amer. Math. Soc. 1, (1950). 287–302. Further reading • K. Kuperberg, Aperiodic dynamical systems. Notices Amer. Math. Soc. 46 (1999), no. 9, 1035–1040. Disproved conjectures • Borsuk's • Chinese hypothesis • Connes • Euler's sum of powers • Ganea • Hedetniemi's • Hauptvermutung • Hirsch • Kalman's • Keller's • Mertens • Ono's inequality • Pólya • Ragsdale • Schoen–Yau • Seifert • Tait's • Von Neumann • Weyl–Berry • Williamson	Wikipedia
Seifert–Weber space In mathematics, Seifert–Weber space (introduced by Herbert Seifert and Constantin Weber) is a closed hyperbolic 3-manifold. It is also known as Seifert–Weber dodecahedral space and hyperbolic dodecahedral space. It is one of the first discovered examples of closed hyperbolic 3-manifolds. It is constructed by gluing each face of a dodecahedron to its opposite in a way that produces a closed 3-manifold. There are three ways to do this gluing consistently. Opposite faces are misaligned by 1/10 of a turn, so to match them they must be rotated by 1/10, 3/10 or 5/10 turn; a rotation of 3/10 gives the Seifert–Weber space. Rotation of 1/10 gives the Poincaré homology sphere, and rotation by 5/10 gives 3-dimensional real projective space. With the 3/10-turn gluing pattern, the edges of the original dodecahedron are glued to each other in groups of five. Thus, in the Seifert–Weber space, each edge is surrounded by five pentagonal faces, and the dihedral angle between these pentagons is 72°. This does not match the 117° dihedral angle of a regular dodecahedron in Euclidean space, but in hyperbolic space there exist regular dodecahedra with any dihedral angle between 60° and 117°, and the hyperbolic dodecahedron with dihedral angle 72° may be used to give the Seifert–Weber space a geometric structure as a hyperbolic manifold. It is a (finite volume) quotient space of the (non-finite volume) order-5 dodecahedral honeycomb, a regular tessellation of hyperbolic 3-space by dodecahedra with this dihedral angle. The Seifert–Weber space is a rational homology sphere, and its first homology group is isomorphic to $\mathbb {Z} _{5}^{3}$. William Thurston conjectured that the Seifert–Weber space is not a Haken manifold, that is, it does not contain any incompressible surfaces; Burton, Rubinstein & Tillmann (2012) proved the conjecture with the aid of their computer software Regina. References • Barbieri, Elena; Cavicchioli, Alberto; Spaggiari, Fulvia (2009). "Some series of honey-comb spaces". The Rocky Mountain Journal of Mathematics. 39 (2): 381–398. • Weber, Constantin; Seifert, Herbert (1933). "Die beiden Dodekaederräume". Mathematische Zeitschrift. 37 (1): 237–253. doi:10.1007/BF01474572. MR 1545392. • Thurston, William (1997), Levy, Silvio (ed.), Three-dimensional geometry and topology. Vol. 1, Princeton Mathematical Series, vol. 35, Princeton, NJ: Princeton University Press, ISBN 0-691-08304-5 • Burton, Benjamin A.; Rubinstein, J. Hyam; Tillmann, Stephan (2012). "The Weber–Seifert dodecahedral space is non-Haken". Transactions of the American Mathematical Society. 364: 911–932. arXiv:0909.4625. doi:10.1090/S0002-9947-2011-05419-X. • Weeks, Jeffrey. The shape of space (2nd ed.). Marcel Dekker. pp. 219. ISBN 978-0824707095. External links • Regina – Support Data: Weber-Seifert dodecahedral space • The Weber–Seifert dodecahedral space: Answering a computational challenge Manifolds (Glossary) Basic concepts • Topological manifold • Atlas • Differentiable/Smooth manifold • Differential structure • Smooth atlas • Submanifold • Riemannian manifold • Smooth map • Submersion • Pushforward • Tangent space • Differential form • Vector field Main results (list) • Atiyah–Singer index • Darboux's • De Rham's • Frobenius • Generalized Stokes • Hopf–Rinow • Noether's • Sard's • Whitney embedding Maps • Curve • Diffeomorphism • Local • Geodesic • Exponential map • in Lie theory • Foliation • Immersion • Integral curve • Lie derivative • Section • Submersion Types of manifolds • Closed • (Almost) Complex • (Almost) Contact • Fibered • Finsler • Flat • G-structure • Hadamard • Hermitian • Hyperbolic • Kähler • Kenmotsu • Lie group • Lie algebra • Manifold with boundary • Oriented • Parallelizable • Poisson • Prime • Quaternionic • Hypercomplex • (Pseudo−, Sub−) Riemannian • Rizza • (Almost) Symplectic • Tame Tensors Vectors • Distribution • Lie bracket • Pushforward • Tangent space • bundle • Torsion • Vector field • Vector flow Covectors • Closed/Exact • Covariant derivative • Cotangent space • bundle • De Rham cohomology • Differential form • Vector-valued • Exterior derivative • Interior product • Pullback • Ricci curvature • flow • Riemann curvature tensor • Tensor field • density • Volume form • Wedge product Bundles • Adjoint • Affine • Associated • Cotangent • Dual • Fiber • (Co) Fibration • Jet • Lie algebra • (Stable) Normal • Principal • Spinor • Subbundle • Tangent • Tensor • Vector Connections • Affine • Cartan • Ehresmann • Form • Generalized • Koszul • Levi-Civita • Principal • Vector • Parallel transport Related • Classification of manifolds • Gauge theory • History • Morse theory • Moving frame • Singularity theory Generalizations • Banach manifold • Diffeology • Diffiety • Fréchet manifold • K-theory • Orbifold • Secondary calculus • over commutative algebras • Sheaf • Stratifold • Supermanifold • Stratified space	Wikipedia
Seiffert's spiral Seiffert's spherical spiral is a curve on a sphere made by moving on the sphere with constant speed and angular velocity with respect to a fixed diameter. If the selected diameter is the line from the north pole to the south pole, then the requirement of constant angular velocity means that the longitude of the moving point changes at a constant rate.[1] The cylindrical coordinates of the varying point on this curve are given by the Jacobian elliptic functions. Formulation Symbols $r$ cylindrical radius $\theta $ angle of curve from beginning of spiral to a particular point on the spiral $\operatorname {sn} (s,k)$ $\operatorname {cn} (s,k)$ basic Jacobi Elliptic Function[2] $\vartheta _{i}(s)$ Jacobi Theta Functions (where $i$ the kind of Theta Functions show)[3] $k$ elliptic modulus (any positive real constant)[4] Representation via equations The Seiffert's spherical spiral can be expressed in cylindrical coordinates as $r=\operatorname {sn} (s,k),\,\theta =k\cdot s{\text{ and }}z=\operatorname {cn} (s,k)$ or expressed as Jacobi theta functions $r={\frac {\vartheta _{3}(0)\cdot \vartheta _{1}(s\cdot \vartheta _{3}^{-2}(0))}{\vartheta _{2}(0)\cdot \vartheta _{4}(s\cdot \vartheta _{3}^{-2}(0))}},\,\theta ={\frac {\vartheta _{2}^{2}(q)}{\vartheta _{3}^{2}(q)}}\cdot s{\text{ and }}z={\frac {\vartheta _{4}(0)\cdot \vartheta _{3}(s\cdot \vartheta _{3}^{-2}(0))}{\vartheta _{3}(0)\cdot \vartheta _{4}(s\cdot \vartheta _{3}^{-2}(0))}}$.[5] See also • Rhumb line References 1. Bowman, F (1961). Introduction to Elliptic Functions with Applications. New York: Dover. 2. Weisstein, Eric W. "Jacobi Elliptic Functions". mathworld.wolfram.com. Retrieved 2023-01-31. 3. Weisstein, Eric W. "Jacobi Theta Functions". mathworld.wolfram.com. Retrieved 2023-01-31. 4. W., Weisstein, Eric. "Elliptic Modulus -- from Wolfram MathWorld". mathworld.wolfram.com. Retrieved 2023-01-31.{{cite web}}: CS1 maint: multiple names: authors list (link) 5. Weisstein, Eric W. "Seiffert's Spherical Spiral". mathworld.wolfram.com. Retrieved 2023-01-31. • Seiffert, A. (1896), Ueber eine neue geometrische Einführung in die Theorie der elliptischen Functionen, vol. 127, Wissenschaftliche Beilage zum Jahresbericht der Städtischen Realschule zu Charlottenburg, Ostern, JFM 27.0337.02 • Erdös, Paul (2000), "Spiraling the Earth with C. G. J. Jacobi", American Journal of Physics, 88 (10): 888–895, doi:10.1119/1.1285882 External links • Weisstein, Eric W. "Seiffert's Spherical Spiral". MathWorld.	Wikipedia
Selberg's 1/4 conjecture In mathematics, Selberg's conjecture, also known as Selberg's eigenvalue conjecture, conjectured by Selberg (1965, p. 13), states that the eigenvalues of the Laplace operator on Maass wave forms of congruence subgroups are at least 1/4. Selberg showed that the eigenvalues are at least 3/16. Subsequent works improved the bound, and the best bound currently known is 975/4096≈0.238..., due to Kim & Sarnak (2003). For the conjecture about the Riemann zeta function, see Selberg's zeta function conjecture. The generalized Ramanujan conjecture for the general linear group implies Selberg's conjecture. More precisely, Selberg's conjecture is essentially the generalized Ramanujan conjecture for the group GL2 over the rationals at the infinite place, and says that the component at infinity of the corresponding representation is a principal series representation of GL2(R) (rather than a complementary series representation). The generalized Ramanujan conjecture in turn follows from the Langlands functoriality conjecture, and this has led to some progress on Selberg's conjecture. References • Gelbart, S. (2001) [1994], "Selberg conjecture", Encyclopedia of Mathematics, EMS Press • Kim, Henry H.; Sarnak, Peter (2003), "Functoriality for the exterior square of GL4 and the symmetric fourth of GL2. Appendix 2.", Journal of the American Mathematical Society, 16 (1): 139–183, doi:10.1090/S0894-0347-02-00410-1, ISSN 0894-0347, MR 1937203 • Selberg, Atle (1965), "On the estimation of Fourier coefficients of modular forms", in Whiteman, Albert Leon (ed.), Theory of Numbers, Proceedings of Symposia in Pure Mathematics, vol. VIII, Providence, R.I.: American Mathematical Society, pp. 1–15, ISBN 978-0-8218-1408-6, MR 0182610 • Luo, W.; Rudnick, Z.; Sarnak, P. (1995-03-01). "On Selberg's eigenvalue conjecture". Geometric & Functional Analysis. 5 (2): 387–401. doi:10.1007/BF01895672. ISSN 1420-8970. External links • "Selberg conjecture - Encyclopedia of Mathematics". encyclopediaofmath.org. Retrieved 2022-06-08.	Wikipedia
Selberg's zeta function conjecture In mathematics, the Selberg conjecture, named after Atle Selberg, is a theorem about the density of zeros of the Riemann zeta function ζ(1/2 + it). It is known that the function has infinitely many zeroes on this line in the complex plane: the point at issue is how densely they are clustered. Results on this can be formulated in terms of N(T), the function counting zeroes on the line for which the value of t satisfies 0 ≤ t ≤ T. Background In 1942 Atle Selberg investigated the problem of the Hardy–Littlewood conjecture 2; and he proved that for any $\varepsilon >0$ there exist $T_{0}=T_{0}(\varepsilon )>0$ and $c=c(\varepsilon )>0,$ such that for $T\geq T_{0}$ and $H=T^{0.5+\varepsilon }$ the inequality $N(T+H)-N(T)\geq cH\log T$ holds true. In his turn, Selberg stated a conjecture relating to shorter intervals,[1] namely that it is possible to decrease the value of the exponent a = 0.5 in $H=T^{0.5+\varepsilon }.$ Proof of the conjecture In 1984 Anatolii Karatsuba proved[2][3][4] that for a fixed $\varepsilon $ satisfying the condition $0<\varepsilon <0.001,$ a sufficiently large T and $H=T^{a+\varepsilon },$ $a={\tfrac {27}{82}}={\tfrac {1}{3}}-{\tfrac {1}{246}},$ the interval in the ordinate t (T, T + H) contains at least cH ln T real zeros of the Riemann zeta function $\zeta {\Bigl (}{\tfrac {1}{2}}+it{\Bigr )};$ and thereby confirmed the Selberg conjecture. The estimates of Selberg and Karatsuba cannot be improved in respect of the order of growth as T → +∞. Further work In 1992 Karatsuba proved[5] that an analog of the Selberg conjecture holds for "almost all" intervals (T, T + H], H = Tε, where ε is an arbitrarily small fixed positive number. The Karatsuba method permits one to investigate zeroes of the Riemann zeta-function on "supershort" intervals of the critical line, that is, on the intervals (T, T + H], the length H of which grows slower than any, even arbitrarily small degree T. In particular, he proved that for any given numbers ε, ε1 satisfying the conditions 0 < ε, ε1< 1 almost all intervals (T, T + H] for H ≥ exp[(ln T)ε] contain at least H (ln T)1 −ε1 zeros of the function ζ(1/2 + it). This estimate is quite close to the conditional result that follows from the Riemann hypothesis. References 1. Selberg, A. (1942). "On the zeros of Riemann's zeta-function". Shr. Norske Vid. Akad. Oslo (10): 1–59. 2. Karatsuba, A. A. (1984). "On the zeros of the function ζ(s) on short intervals of the critical line". Izv. Akad. Nauk SSSR, Ser. Mat. (48:3): 569–584. 3. Karatsuba, A. A. (1984). "The distribution of zeros of the function ζ(1/2 + it)". Izv. Akad. Nauk SSSR, Ser. Mat. (48:6): 1214–1224. 4. Karatsuba, A. A. (1985). "On the zeros of the Riemann zeta-function on the critical line". Proc. Steklov Inst. Math. (167): 167–178. 5. Karatsuba, A. A. (1992). "On the number of zeros of the Riemann zeta-function lying in almost all short intervals of the critical line". Izv. Ross. Akad. Nauk, Ser. Mat. (56:2): 372–397.	Wikipedia
Arthur–Selberg trace formula In mathematics, the Arthur–Selberg trace formula is a generalization of the Selberg trace formula from the group SL2 to arbitrary reductive groups over global fields, developed by James Arthur in a long series of papers from 1974 to 2003. It describes the character of the representation of G(A) on the discrete part L2 0 (G(F)\G(A)) of L2(G(F)\G(A)) in terms of geometric data, where G is a reductive algebraic group defined over a global field F and A is the ring of adeles of F. There are several different versions of the trace formula. The first version was the unrefined trace formula, whose terms depend on truncation operators and have the disadvantage that they are not invariant. Arthur later found the invariant trace formula and the stable trace formula which are more suitable for applications. The simple trace formula (Flicker & Kazhdan 1988) is less general but easier to prove. The local trace formula is an analogue over local fields. Jacquet's relative trace formula is a generalization where one integrates the kernel function over non-diagonal subgroups. Notation • F is a global field, such as the field of rational numbers. • A is the ring of adeles of F. • G is a reductive algebraic group defined over F. The compact case In the case when G(F)\G(A) is compact the representation splits as a direct sum of irreducible representations, and the trace formula is similar to the Frobenius formula for the character of the representation induced from the trivial representation of a subgroup of finite index. In the compact case, which is essentially due to Selberg, the groups G(F) and G(A) can be replaced by any discrete subgroup Γ of a locally compact group G with Γ\G compact. The group G acts on the space of functions on Γ\G by the right regular representation R, and this extends to an action of the group ring of G, considered as the ring of functions f on G. The character of this representation is given by a generalization of the Frobenius formula as follows. The action of a function f on a function φ on Γ\G is given by $\displaystyle R(f)(\phi )(x)=\int _{G}f(y)\phi (xy)\,dy=\int _{\Gamma \backslash G}\sum _{\gamma \in \Gamma }f(x^{-1}\gamma y)\phi (y)\,dy.$ In other words, R(f) is an integral operator on L2(Γ\G) (the space of functions on Γ\G) with kernel $\displaystyle K_{f}(x,y)=\sum _{\gamma \in \Gamma }f(x^{-1}\gamma y).$ Therefore, the trace of R(f) is given by $\displaystyle \operatorname {Tr} (R(f))=\int _{\Gamma \backslash G}K_{f}(x,x)\,dx.$ The kernel K can be written as $K_{f}(x,y)=\sum _{o\in O}K_{o}(x,y)$ where O is the set of conjugacy classes in Γ, and $K_{o}(x,y)=\sum _{\gamma \in o}f(x^{-1}\gamma y)=\sum _{\delta \in \Gamma _{\gamma }\backslash \Gamma }f(x^{-1}\delta ^{-1}\gamma \delta y)$ where γ is an element of the conjugacy class o, and Γγ is its centralizer in Γ. On the other hand, the trace is also given by $\displaystyle \operatorname {Tr} (R(f))=\sum _{\pi }m(\pi )\operatorname {Tr} (R(f)\|\pi )$ where m(π) is the multiplicity of the irreducible unitary representation π of G in L2(Γ\G). Examples • If Γ and G are both finite, the trace formula is equivalent to the Frobenius formula for the character of an induced representation. • If G is the group R of real numbers and Γ the subgroup Z of integers, then the trace formula becomes the Poisson summation formula. Difficulties in the non-compact case In most cases of the Arthur–Selberg trace formula, the quotient G(F)\G(A) is not compact, which causes the following (closely related) problems: • The representation on L2(G(F)\G(A)) contains not only discrete components, but also continuous components. • The kernel is no longer integrable over the diagonal, and the operators R(f) are no longer of trace class. Arthur dealt with these problems by truncating the kernel at cusps in such a way that the truncated kernel is integrable over the diagonal. This truncation process causes many problems; for example, the truncated terms are no longer invariant under conjugation. By manipulating the terms further, Arthur was able to produce an invariant trace formula whose terms are invariant. The original Selberg trace formula studied a discrete subgroup Γ of a real Lie group G(R) (usually SL2(R)). In higher rank it is more convenient to replace the Lie group with an adelic group G(A). One reason for this that the discrete group can be taken as the group of points G(F) for F a (global) field, which is easier to work with than discrete subgroups of Lie groups. It also makes Hecke operators easier to work with. The trace formula in the non-compact case One version of the trace formula (Arthur 1983) asserts the equality of two distributions on G(A): $\sum _{o\in O}J_{o}^{T}=\sum _{\chi \in X}J_{\chi }^{T}.$ The left hand side is the geometric side of the trace formula, and is a sum over equivalence classes in the group of rational points G(F) of G, while the right hand side is the spectral side of the trace formula and is a sum over certain representations of subgroups of G(A). Distributions The invariant trace formula The version of the trace formula above is not particularly easy to use in practice, one of the problems being that the terms in it are not invariant under conjugation. Arthur (1981) found a modification in which the terms are invariant. The invariant trace formula states $\sum _{M}{\frac {\|W_{0}^{M}\|}{\|W_{0}^{G}\|}}\sum _{\gamma \in (M(Q))}a^{M}(\gamma )I_{M}(\gamma ,f)=\sum _{M}{\frac {\|W_{0}^{M}\|}{\|W_{0}^{G}\|}}\int _{\Pi (M)}a^{M}(\pi )I_{M}(\pi ,f)\,d\pi $ where • f is a test function on G(A) • M ranges over a finite set of rational Levi subgroups of G • (M(Q)) is the set of conjugacy classes of M(Q) • Π(M) is the set of irreducible unitary representations of M(A) • aM(γ) is related to the volume of M(Q,γ)\M(A,γ) • aM(π) is related to the multiplicity of the irreducible representation π in L2(M(Q)\M(A)) • $\displaystyle I_{M}(\gamma ,f)$ is related to $\displaystyle \int _{M(A,\gamma )\backslash M(A)}f(x^{-1}\gamma x)\,dx$ • $\displaystyle I_{M}(\pi ,f)$ is related to trace $\displaystyle \int _{M(A)}f(x)\pi (x)\,dx$ • W0(M) is the Weyl group of M. Stable trace formula Langlands (1983) suggested the possibility a stable refinement of the trace formula that can be used to compare the trace formula for two different groups. Such a stable trace formula was found and proved by Arthur (2002). Two elements of a group G(F) are called stably conjugate if they are conjugate over the algebraic closure of the field F. The point is that when one compares elements in two different groups, related for example by inner twisting, one does not usually get a good correspondence between conjugacy classes, but only between stable conjugacy classes. So to compare the geometric terms in the trace formulas for two different groups, one would like the terms to be not just invariant under conjugacy, but also to be well behaved on stable conjugacy classes; these are called stable distributions. The stable trace formula writes the terms in the trace formula of a group G in terms of stable distributions. However these stable distributions are not distributions on the group G, but are distributions on a family of quasisplit groups called the endoscopic groups of G. Unstable orbital integrals on the group G correspond to stable orbital integrals on its endoscopic groups H. Simple trace formula There are several simple forms of the trace formula, which restrict the compactly supported test functions f in some way (Flicker & Kazhdan 1988). The advantage of this is that the trace formula and its proof become much easier, and the disadvantage is that the resulting formula is less powerful. For example, if the functions f are cuspidal, which means that $\int _{n\in N(A)}f(xny)\,dn=0$ for any unipotent radical N of a proper parabolic subgroup (defined over F) and any x, y in G(A), then the operator R(f) has image in the space of cusp forms so is compact. Applications Jacquet & Langlands (1970) used the Selberg trace formula to prove the Jacquet–Langlands correspondence between automorphic forms on GL2 and its twisted forms. The Arthur–Selberg trace formula can be used to study similar correspondences on higher rank groups. It can also be used to prove several other special cases of Langlands functoriality, such as base change, for some groups. Kottwitz (1988) used the Arthur–Selberg trace formula to prove the Weil conjecture on Tamagawa numbers. Lafforgue (2002) described how the trace formula is used in his proof of the Langlands conjecture for general linear groups over function fields. See also • Maass wave form • Harmonic Maass form • Arthur's conjectures References • Arthur, James (1981), "The trace formula in invariant form", Annals of Mathematics, Second Series, 114 (1): 1–74, doi:10.2307/1971376, JSTOR 1971376, MR 0625344 • Arthur, James (1983), "The trace formula for reductive groups" (PDF), Conference on automorphic theory (Dijon, 1981), Publ. Math. Univ. Paris VII, vol. 15, Paris: Univ. Paris VII, pp. 1–41, CiteSeerX 10.1.1.207.4897, doi:10.1007/978-1-4684-6730-7_1, ISBN 978-0-8176-3135-2, MR 0723181 • Arthur, James (2002), "A stable trace formula. I. General expansions" (PDF), Journal of the Institute of Mathematics of Jussieu, 1 (2): 175–277, doi:10.1017/S1474-748002000051, MR 1954821, archived from the original (PDF) on 2008-05-09 • Arthur, James (2005), "An introduction to the trace formula" (PDF), Harmonic analysis, the trace formula, and Shimura varieties, Clay Math. Proc., vol. 4, Providence, R.I.: American Mathematical Society, pp. 1–263, MR 2192011, archived from the original (PDF) on 2008-05-09 • Flicker, Yuval Z.; Kazhdan, David A. (1988), "A simple trace formula", Journal d'Analyse Mathématique, 50: 189–200, doi:10.1007/BF02796122 • Gelbart, Stephen (1996), Lectures on the Arthur-Selberg trace formula, University Lecture Series, vol. 9, Providence, R.I.: American Mathematical Society, arXiv:math.RT/9505206, doi:10.1090/ulect/009, ISBN 978-0-8218-0571-8, MR 1410260, S2CID 118372096 • Jacquet, H.; Langlands, Robert P. (1970), Automorphic forms on GL(2), Lecture Notes in Mathematics, vol. 114, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0058988, ISBN 978-3-540-04903-6, MR 0401654, S2CID 122773458 • Konno, Takuya (2000), "A survey on the Arthur-Selberg trace formula" (PDF), Surikaisekikenkyusho Kõkyuroku (1173): 243–288, MR 1840082 • Kottwitz, Robert E. (1988), "Tamagawa numbers", Ann. of Math., 2, 127 (3): 629–646, doi:10.2307/2007007, JSTOR 2007007, MR 0942522 • Labesse, Jean-Pierre (1986), "La formule des traces d'Arthur-Selberg", Astérisque (133): 73–88, MR 0837215 • Langlands, Robert P. (2001), "The trace formula and its applications: an introduction to the work of James Arthur", Canadian Mathematical Bulletin, 44 (2): 160–209, doi:10.4153/CMB-2001-020-8, ISSN 0008-4395, MR 1827854 • Lafforgue, Laurent (2002), "Chtoucas de Drinfeld, formule des traces d'Arthur-Selberg et correspondance de Langlands", Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002), Beijing: Higher Ed. Press, pp. 383–400, MR 1989194 • Langlands, Robert P. (1983), Les débuts d'une formule des traces stable, Publications Mathématiques de l'Université Paris VII [Mathematical Publications of the University of Paris VII], vol. 13, Paris: Université de Paris VII U.E.R. de Mathématiques, MR 0697567 • Shokranian, Salahoddin (1992), The Selberg-Arthur trace formula, Lecture Notes in Mathematics, vol. 1503, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0092305, ISBN 978-3-540-55021-1, MR 1176101 External links • Works of James Arthur at the Clay institute • Archive of Collected Works of James Arthur at the University of Toronto Department of Mathematics	Wikipedia
Selberg class In mathematics, the Selberg class is an axiomatic definition of a class of L-functions. The members of the class are Dirichlet series which obey four axioms that seem to capture the essential properties satisfied by most functions that are commonly called L-functions or zeta functions. Although the exact nature of the class is conjectural, the hope is that the definition of the class will lead to a classification of its contents and an elucidation of its properties, including insight into their relationship to automorphic forms and the Riemann hypothesis. The class was defined by Atle Selberg in (Selberg 1992), who preferred not to use the word "axiom" that later authors have employed.[1] Definition The formal definition of the class S is the set of all Dirichlet series $F(s)=\sum _{n=1}^{\infty }{\frac {a_{n}}{n^{s}}}$ absolutely convergent for Re(s) > 1 that satisfy four axioms (or assumptions as Selberg calls them): 1. Analyticity: $F(s)$ has a meromorphic continuation to the entire complex plane, with the only possible pole (if any) when s equals 1. 2. Ramanujan conjecture: a1 = 1 and $a_{n}\ll _{\varepsilon }n^{\varepsilon }$ for any ε > 0; 3. Functional equation: there is a gamma factor of the form $\gamma (s)=Q^{s}\prod _{i=1}^{k}\Gamma (\omega _{i}s+\mu _{i})$ where Q is real and positive, Γ the gamma function, the ωi real and positive, and the μi complex with non-negative real part, as well as a so-called root number $\alpha \in \mathbb {C} ,\;\|\alpha \|=1$, such that the function $\Phi (s)=\gamma (s)F(s)\,$ satisfies $\Phi (s)=\alpha \,{\overline {\Phi (1-{\overline {s}})}};$ 4. Euler product: For Re(s) > 1, F(s) can be written as a product over primes: $F(s)=\prod _{p}F_{p}(s)$ with $F_{p}(s)=\exp \left(\sum _{n=1}^{\infty }{\frac {b_{p^{n}}}{p^{ns}}}\right)$ and, for some θ < 1/2, $b_{p^{n}}=O(p^{n\theta }).\,$ Comments on definition The condition that the real part of μi be non-negative is because there are known L-functions that do not satisfy the Riemann hypothesis when μi is negative. Specifically, there are Maass forms associated with exceptional eigenvalues, for which the Ramanujan–Peterssen conjecture holds, and have a functional equation, but do not satisfy the Riemann hypothesis. The condition that θ < 1/2 is important, as the θ = 1 case includes $(1-2^{-s})(1-2^{1-s})$ whose zeros are not on the critical line. Without the condition $a_{n}\ll _{\varepsilon }n^{\varepsilon }$ there would be $L(s+1/3,\chi _{4})L(s-1/3,\chi _{4})$ which violates the Riemann hypothesis. It is a consequence of 4. that the an are multiplicative and that $F_{p}(s)=\sum _{n=0}^{\infty }{\frac {a_{p^{n}}}{p^{ns}}}{\text{ for Re}}(s)>0.$ Examples The prototypical example of an element in S is the Riemann zeta function.[2] Another example, is the L-function of the modular discriminant Δ $L(s,\Delta )=\sum _{n=1}^{\infty }{\frac {a_{n}}{n^{s}}}$ where $a_{n}=\tau (n)/n^{11/2}$ and τ(n) is the Ramanujan tau function.[3] All known examples are automorphic L-functions, and the reciprocals of Fp(s) are polynomials in p−s of bounded degree.[4] The best results on the structure of the Selberg class are due to Kaczorowski and Perelli, who show that the Dirichlet L-functions (including the Riemann zeta-function) are the only examples with degree less than 2.[5] Basic properties As with the Riemann zeta function, an element F of S has trivial zeroes that arise from the poles of the gamma factor γ(s). The other zeroes are referred to as the non-trivial zeroes of F. These will all be located in some strip 1 − A ≤ Re(s) ≤ A. Denoting the number of non-trivial zeroes of F with 0 ≤ Im(s) ≤ T by NF(T),[6] Selberg showed that $N_{F}(T)=d_{F}{\frac {T\log(T+C)}{2\pi }}+O(\log T).$ Here, dF is called the degree (or dimension) of F. It is given by[7] $d_{F}=2\sum _{i=1}^{k}\omega _{i}.$ It can be shown that F = 1 is the only function in S whose degree is less than 1. If F and G are in the Selberg class, then so is their product and $d_{FG}=d_{F}+d_{G}.$ A function F ≠ 1 in S is called primitive if whenever it is written as F = F1F2, with Fi in S, then F = F1 or F = F2. If dF = 1, then F is primitive. Every function F ≠ 1 of S can be written as a product of primitive functions. Selberg's conjectures, described below, imply that the factorization into primitive functions is unique. Examples of primitive functions include the Riemann zeta function and Dirichlet L-functions of primitive Dirichlet characters. Assuming conjectures 1 and 2 below, L-functions of irreducible cuspidal automorphic representations that satisfy the Ramanujan conjecture are primitive.[8] Selberg's conjectures In (Selberg 1992), Selberg made conjectures concerning the functions in S: • Conjecture 1: For all F in S, there is an integer nF such that $\sum _{p\leq x}{\frac {\|a_{p}\|^{2}}{p}}=n_{F}\log \log x+O(1)$ and nF = 1 whenever F is primitive. • Conjecture 2: For distinct primitive F, F′ ∈ S, $\sum _{p\leq x}{\frac {a_{p}{\overline {a_{p}^{\prime }}}}{p}}=O(1).$ • Conjecture 3: If F is in S with primitive factorization $F=\prod _{i=1}^{m}F_{i},$ χ is a primitive Dirichlet character, and the function $F^{\chi }(s)=\sum _{n=1}^{\infty }{\frac {\chi (n)a_{n}}{n^{s}}}$ is also in S, then the functions Fiχ are primitive elements of S (and consequently, they form the primitive factorization of Fχ). • Riemann hypothesis for S: For all F in S, the non-trivial zeroes of F all lie on the line Re(s) = 1/2. Consequences of the conjectures Conjectures 1 and 2 imply that if F has a pole of order m at s = 1, then F(s)/ζ(s)m is entire. In particular, they imply Dedekind's conjecture.[9] M. Ram Murty showed in (Murty 1994) that conjectures 1 and 2 imply the Artin conjecture. In fact, Murty showed that Artin L-functions corresponding to irreducible representations of the Galois group of a solvable extension of the rationals are automorphic as predicted by the Langlands conjectures.[10] The functions in S also satisfy an analogue of the prime number theorem: F(s) has no zeroes on the line Re(s) = 1. As mentioned above, conjectures 1 and 2 imply the unique factorization of functions in S into primitive functions. Another consequence is that the primitivity of F is equivalent to nF = 1.[11] See also • List of zeta functions Notes 1. The title of Selberg's paper is somewhat a spoof on Paul Erdős, who had many papers named (approximately) "(Some) Old and new problems and results about...". Indeed, the 1989 Amalfi conference was quite surprising in that both Selberg and Erdős were present, with the story being that Selberg did not know that Erdős was to attend. 2. Murty 2008 3. Murty 2008 4. Murty 1994 5. Jerzy Kaczorowski & Alberto Perelli (2011). "On the structure of the Selberg class, VII" (PDF). Annals of Mathematics. 173: 1397–1441. doi:10.4007/annals.2011.173.3.4. 6. The zeroes on the boundary are counted with half-multiplicity. 7. While the ωi are not uniquely defined by F, Selberg's result shows that their sum is well-defined. 8. Murty 1994, Lemma 4.2 9. A celebrated conjecture of Dedekind asserts that for any finite algebraic extension $F$ of $\mathbb {Q} $, the zeta function $\zeta _{F}(s)$ is divisible by the Riemann zeta function $\zeta (s)$. That is, the quotient $\zeta _{F}(s)/\zeta (s)$ is entire. More generally, Dedekind conjectures that if $K$ is a finite extension of $F$, then $\zeta _{K}(s)/\zeta _{F}(s)$ should be entire. This conjecture is still open. 10. Murty 1994, Theorem 4.3 11. Conrey & Ghosh 1993, § 4 References • Selberg, Atle (1992), "Old and new conjectures and results about a class of Dirichlet series", Proceedings of the Amalfi Conference on Analytic Number Theory (Maiori, 1989), Salerno: Univ. Salerno, pp. 367–385, MR 1220477, Zbl 0787.11037 Reprinted in Collected Papers, vol 2, Springer-Verlag, Berlin (1991) • Conrey, J. Brian; Ghosh, Amit (1993), "On the Selberg class of Dirichlet series: small degrees", Duke Mathematical Journal, 72 (3): 673–693, arXiv:math.NT/9204217, doi:10.1215/s0012-7094-93-07225-0, MR 1253620, Zbl 0796.11037 • Murty, M. Ram (1994), "Selberg's conjectures and Artin L-functions", Bulletin of the American Mathematical Society, New Series, 31 (1): 1–14, arXiv:math/9407219, doi:10.1090/s0273-0979-1994-00479-3, MR 1242382, S2CID 265909, Zbl 0805.11062 • Murty, M. Ram (2008), Problems in analytic number theory, Graduate Texts in Mathematics, Readings in Mathematics, vol. 206 (Second ed.), Springer-Verlag, Chapter 8, doi:10.1007/978-0-387-72350-1, ISBN 978-0-387-72349-5, MR 2376618, Zbl 1190.11001 L-functions in number theory Analytic examples • Riemann zeta function • Dirichlet L-functions • L-functions of Hecke characters • Automorphic L-functions • Selberg class Algebraic examples • Dedekind zeta functions • Artin L-functions • Hasse–Weil L-functions • Motivic L-functions Theorems • Analytic class number formula • Riemann–von Mangoldt formula • Weil conjectures Analytic conjectures • Riemann hypothesis • Generalized Riemann hypothesis • Lindelöf hypothesis • Ramanujan–Petersson conjecture • Artin conjecture Algebraic conjectures • Birch and Swinnerton-Dyer conjecture • Deligne's conjecture • Beilinson conjectures • Bloch–Kato conjecture • Langlands conjecture p-adic L-functions • Main conjecture of Iwasawa theory • Selmer group • Euler system	Wikipedia
Selberg sieve In number theory, the Selberg sieve is a technique for estimating the size of "sifted sets" of positive integers which satisfy a set of conditions which are expressed by congruences. It was developed by Atle Selberg in the 1940s. Description In terms of sieve theory the Selberg sieve is of combinatorial type: that is, derives from a careful use of the inclusion–exclusion principle. Selberg replaced the values of the Möbius function which arise in this by a system of weights which are then optimised to fit the given problem. The result gives an upper bound for the size of the sifted set. Let $A$ be a set of positive integers $\leq x$ and let $P$ be a set of primes. Let $A_{d}$ denote the set of elements of $A$ divisible by $d$ when $d$ is a product of distinct primes from $P$. Further let $A_{1}$ denote $A$ itself. Let $z$ be a positive real number and $P(z)$ denote the product of the primes in $P$ which are $\leq z$. The object of the sieve is to estimate $S(A,P,z)=\left\vert A\setminus \bigcup _{p\mid P(z)}A_{p}\right\vert .$ We assume that \|Ad\| may be estimated by $\left\vert A_{d}\right\vert ={\frac {1}{f(d)}}X+R_{d}.$ where f is a multiplicative function and X = \|A\|. Let the function g be obtained from f by Möbius inversion, that is $g(n)=\sum _{d\mid n}\mu (d)f(n/d)$ $f(n)=\sum _{d\mid n}g(d)$ where μ is the Möbius function. Put $V(z)=\sum _{\begin{smallmatrix}d<z\\d\mid P(z)\end{smallmatrix}}{\frac {1}{g(d)}}.$ Then $S(A,P,z)\leq {\frac {X}{V(z)}}+O\left({\sum _{\begin{smallmatrix}d_{1},d_{2}<z\\d_{1},d_{2}\mid P(z)\end{smallmatrix}}\left\vert R_{[d_{1},d_{2}]}\right\vert }\right)$ where $[d_{1},d_{2}]$ denotes the least common multiple of $d_{1}$ and $d_{2}$. It is often useful to estimate $V(z)$ by the bound $V(z)\geq \sum _{d\leq z}{\frac {1}{f(d)}}.\,$ Applications • The Brun–Titchmarsh theorem on the number of primes in arithmetic progression; • The number of n ≤ x such that n is coprime to φ(n) is asymptotic to e−γ x / log log log (x) . References • Cojocaru, Alina Carmen; Murty, M. Ram (2005). An introduction to sieve methods and their applications. London Mathematical Society Student Texts. Vol. 66. Cambridge University Press. pp. 113–134. ISBN 0-521-61275-6. Zbl 1121.11063. • Diamond, Harold G.; Halberstam, Heini (2008). A Higher-Dimensional Sieve Method: with Procedures for Computing Sieve Functions. Cambridge Tracts in Mathematics. Vol. 177. With William F. Galway. Cambridge: Cambridge University Press. ISBN 978-0-521-89487-6. Zbl 1207.11099. • Greaves, George (2001). Sieves in number theory. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. Vol. 43. Berlin: Springer-Verlag. ISBN 3-540-41647-1. Zbl 1003.11044. • Halberstam, Heini; Richert, H.E. (1974). Sieve Methods. London Mathematical Society Monographs. Vol. 4. Academic Press. ISBN 0-12-318250-6. Zbl 0298.10026. • Hooley, Christopher (1976). Applications of sieve methods to the theory of numbers. Cambridge Tracts in Mathematics. Vol. 70. Cambridge University Press. pp. 7–12. ISBN 0-521-20915-3. Zbl 0327.10044. • Selberg, Atle (1947). "On an elementary method in the theory of primes". Norske Vid. Selsk. Forh. Trondheim. 19: 64–67. ISSN 0368-6302. Zbl 0041.01903.	Wikipedia
Selberg trace formula In mathematics, the Selberg trace formula, introduced by Selberg (1956), is an expression for the character of the unitary representation of a Lie group G on the space L2(Γ\G) of square-integrable functions, where Γ is a cofinite discrete group. The character is given by the trace of certain functions on G. The simplest case is when Γ is cocompact, when the representation breaks up into discrete summands. Here the trace formula is an extension of the Frobenius formula for the character of an induced representation of finite groups. When Γ is the cocompact subgroup Z of the real numbers G = R, the Selberg trace formula is essentially the Poisson summation formula. The case when Γ\G is not compact is harder, because there is a continuous spectrum, described using Eisenstein series. Selberg worked out the non-compact case when G is the group SL(2, R); the extension to higher rank groups is the Arthur–Selberg trace formula. When Γ is the fundamental group of a Riemann surface, the Selberg trace formula describes the spectrum of differential operators such as the Laplacian in terms of geometric data involving the lengths of geodesics on the Riemann surface. In this case the Selberg trace formula is formally similar to the explicit formulas relating the zeros of the Riemann zeta function to prime numbers, with the zeta zeros corresponding to eigenvalues of the Laplacian, and the primes corresponding to geodesics. Motivated by the analogy, Selberg introduced the Selberg zeta function of a Riemann surface, whose analytic properties are encoded by the Selberg trace formula. Early history Cases of particular interest include those for which the space is a compact Riemann surface S. The initial publication in 1956 of Atle Selberg dealt with this case, its Laplacian differential operator and its powers. The traces of powers of a Laplacian can be used to define the Selberg zeta function. The interest of this case was the analogy between the formula obtained, and the explicit formulae of prime number theory. Here the closed geodesics on S play the role of prime numbers. At the same time, interest in the traces of Hecke operators was linked to the Eichler–Selberg trace formula, of Selberg and Martin Eichler, for a Hecke operator acting on a vector space of cusp forms of a given weight, for a given congruence subgroup of the modular group. Here the trace of the identity operator is the dimension of the vector space, i.e. the dimension of the space of modular forms of a given type: a quantity traditionally calculated by means of the Riemann–Roch theorem. Applications The trace formula has applications to arithmetic geometry and number theory. For instance, using the trace theorem, Eichler and Shimura calculated the Hasse–Weil L-functions associated to modular curves; Goro Shimura's methods by-passed the analysis involved in the trace formula. The development of parabolic cohomology (from Eichler cohomology) provided a purely algebraic setting based on group cohomology, taking account of the cusps characteristic of non-compact Riemann surfaces and modular curves. The trace formula also has purely differential-geometric applications. For instance, by a result of Buser, the length spectrum of a Riemann surface is an isospectral invariant, essentially by the trace formula. Later work The general theory of Eisenstein series was largely motivated by the requirement to separate out the continuous spectrum, which is characteristic of the non-compact case.[1] The trace formula is often given for algebraic groups over the adeles rather than for Lie groups, because this makes the corresponding discrete subgroup Γ into an algebraic group over a field which is technically easier to work with. The case of SL2(C) is discussed in Gel'fand, Graev & Pyatetskii-Shapiro (1990) and Elstrodt, Grunewald & Mennicke (1998). Gel'fand et al also treat SL2(F) where F is a locally compact topological field with ultrametric norm, so a finite extension of the p-adic numbers Qp or of the formal Laurent series Fq((T)); they also handle the adelic case in characteristic 0, combining all completions R and Qp of the rational numbers Q. Contemporary successors of the theory are the Arthur–Selberg trace formula applying to the case of general semisimple G, and the many studies of the trace formula in the Langlands philosophy (dealing with technical issues such as endoscopy). The Selberg trace formula can be derived from the Arthur–Selberg trace formula with some effort. Selberg trace formula for compact hyperbolic surfaces A compact hyperbolic surface X can be written as the space of orbits $\Gamma \backslash \mathbf {H} ,$ where Γ is a subgroup of PSL(2, R), and H is the upper half plane, and Γ acts on H by linear fractional transformations. The Selberg trace formula for this case is easier than the general case because the surface is compact so there is no continuous spectrum, and the group Γ has no parabolic or elliptic elements (other than the identity). Then the spectrum for the Laplace–Beltrami operator on X is discrete and real, since the Laplace operator is self adjoint with compact resolvent; that is $0=\mu _{0}<\mu _{1}\leq \mu _{2}\leq \cdots $ where the eigenvalues μn correspond to Γ-invariant eigenfunctions u in C∞(H) of the Laplacian; in other words ${\begin{cases}u(\gamma z)=u(z),\qquad \forall \gamma \in \Gamma \\y^{2}\left(u_{xx}+u_{yy}\right)+\mu _{n}u=0.\end{cases}}$ Using the variable substitution $\mu =s(1-s),\qquad s={\tfrac {1}{2}}+ir$ the eigenvalues are labeled $r_{n},n\geq 0.$ Then the Selberg trace formula is given by $\sum _{n=0}^{\infty }h(r_{n})={\frac {\mu (X)}{4\pi }}\int _{-\infty }^{\infty }r\,h(r)\tanh(\pi r)\,dr+\sum _{\{T\}}{\frac {\log N(T_{0})}{N(T)^{\frac {1}{2}}-N(T)^{-{\frac {1}{2}}}}}g(\log N(T)).$ The right hand side is a sum over conjugacy classes of the group Γ, with the first term corresponding to the identity element and the remaining terms forming a sum over the other conjugacy classes {T } (which are all hyperbolic in this case). The function h has to satisfy the following: • be analytic on \|Im(r)\| ≤ 1/2 + δ; • h(−r) = h(r); • there exist positive constants δ and M such that: $\vert h(r)\vert \leq M\left(1+\left\|\operatorname {Re} (r)\right\|\right)^{-2-\delta }.$ The function g is the Fourier transform of h, that is, $h(r)=\int _{-\infty }^{\infty }g(u)e^{iru}\,du.$ Notes 1. Lax & Phillips 1980 References • Borel, Armand (1997). Automorphic forms on SL2(R). Cambridge Tracts in Mathematics. Vol. 130. Cambridge University Press. ISBN 0-521-58049-8. MR 1482800. • Chavel, Isaac; Randol, Burton (1984). "XI. The Selberg Trace Formula". Eigenvalues in Riemannian geometry. Pure and Applied Mathematics. Vol. 115. Academic Press. ISBN 0-12-170640-0. MR 0768584. • Elstrodt, Jürgen (1981). "Die Selbergsche Spurformel für kompakte Riemannsche Flächen" (PDF). Jahresber. Deutsch. Math.-Verein. (in German). Bielefeld: Deutsche Mathematiker-Vereinigung. 83: 45–77. MR 0612411. • Elstrodt, J.; Grunewald, F.; Mennicke, J. (1998). Groups acting on hyperbolic space: Harmonic analysis and number theory. Springer Monographs in Mathematics. Springer-Verlag. ISBN 3-540-62745-6. MR 1483315. • Fischer, Jürgen (1987), An approach to the Selberg trace formula via the Selberg zeta-function, Lecture Notes in Mathematics, vol. 1253, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0077696, ISBN 978-3-540-15208-8, MR 0892317 • Gel'fand, I. M.; Graev, M. I.; Pyatetskii-Shapiro, I. I. (1990), Representation theory and automorphic functions, Generalized Functions, vol. 6, Boston, MA: Academic Press, ISBN 978-0-12-279506-0, MR 1071179 • Godement, Roger (1966). "The decomposition of L2(G/Γ) for Γ=SL(2,Z)". Algebraic Groups and Discontinuous Subgroups. Proc. Sympos. Pure Math. Providence: American Mathematical Society. pp. 211–224. MR 0210827. • Hejhal, Dennis A. (1976), "The Selberg trace formula and the Riemann zeta function", Duke Mathematical Journal, 43 (3): 441–482, doi:10.1215/S0012-7094-76-04338-6, ISSN 0012-7094, MR 0414490 • Hejhal, Dennis A. (1976), The Selberg trace formula for PSL(2,R). Vol. I, Lecture Notes in Mathematics, Vol. 548, vol. 548, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0079608, ISBN 978-3-540-07988-0, MR 0439755 • Hejhal, Dennis A. (1983), The Selberg trace formula for PSL(2,R). Vol. 2, Lecture Notes in Mathematics, vol. 1001, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0061302, ISBN 978-3-540-12323-1, MR 0711197 • Iwaniec, Henryk (2002). Spectral methods of automorphic forms. Graduate Studies in Mathematics. Vol. 53 (Second ed.). American Mathematical Society. ISBN 0-8218-3160-7. MR 1942691. • Lax, Peter D.; Phillips, Ralph S. (1980). "Scattering theory for automorphic functions" (PDF). Bull. Amer. Math. Soc. 2: 261–295. MR 0555264. • McKean, H. P. (1972), "Selberg's trace formula as applied to a compact Riemann surface", Communications on Pure and Applied Mathematics, 25 (3): 225–246, doi:10.1002/cpa.3160250302, ISSN 0010-3640, MR 0473166 • Selberg, Atle (1956), "Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series", J. Indian Math. Soc., New Series, 20: 47–87, MR 0088511 • Sunada, Toshikazu (1991), Trace formulae in spectral geometry, Proc. ICM-90 Kyoto, Springer-Verlag, pp. 577–585 External links • Selberg trace formula resource page Authority control International • FAST National • France • BnF data • Germany • Israel • United States Other • IdRef	Wikipedia
Selberg zeta function The Selberg zeta-function was introduced by Atle Selberg (1956). It is analogous to the famous Riemann zeta function $\zeta (s)=\prod _{p\in \mathbb {P} }{\frac {1}{1-p^{-s}}}$ where $\mathbb {P} $ is the set of prime numbers. The Selberg zeta-function uses the lengths of simple closed geodesics instead of the primes numbers. If $\Gamma $ is a subgroup of SL(2,R), the associated Selberg zeta function is defined as follows, $\zeta _{\Gamma }(s)=\prod _{p}(1-N(p)^{-s})^{-1},$ or $Z_{\Gamma }(s)=\prod _{p}\prod _{n=0}^{\infty }(1-N(p)^{-s-n}),$ where p runs over conjugacy classes of prime geodesics (equivalently, conjugacy classes of primitive hyperbolic elements of $\Gamma $), and N(p) denotes the length of p (equivalently, the square of the bigger eigenvalue of p). For any hyperbolic surface of finite area there is an associated Selberg zeta-function; this function is a meromorphic function defined in the complex plane. The zeta function is defined in terms of the closed geodesics of the surface. The zeros and poles of the Selberg zeta-function, Z(s), can be described in terms of spectral data of the surface. The zeros are at the following points: 1. For every cusp form with eigenvalue $s_{0}(1-s_{0})$ there exists a zero at the point $s_{0}$. The order of the zero equals the dimension of the corresponding eigenspace. (A cusp form is an eigenfunction to the Laplace–Beltrami operator which has Fourier expansion with zero constant term.) 2. The zeta-function also has a zero at every pole of the determinant of the scattering matrix, $\phi (s)$. The order of the zero equals the order of the corresponding pole of the scattering matrix. The zeta-function also has poles at $1/2-\mathbb {N} $, and can have zeros or poles at the points $-\mathbb {N} $. The Ihara zeta function is considered a p-adic (and a graph-theoretic) analogue of the Selberg zeta function. Selberg zeta-function for the modular group For the case where the surface is $\Gamma \backslash \mathbb {H} ^{2}$, where $\Gamma $ is the modular group, the Selberg zeta-function is of special interest. For this special case the Selberg zeta-function is intimately connected to the Riemann zeta-function. In this case the determinant of the scattering matrix is given by: $\varphi (s)=\pi ^{1/2}{\frac {\Gamma (s-1/2)\zeta (2s-1)}{\Gamma (s)\zeta (2s)}}.$ In particular, we see that if the Riemann zeta-function has a zero at $s_{0}$, then the determinant of the scattering matrix has a pole at $s_{0}/2$, and hence the Selberg zeta-function has a zero at $s_{0}/2$. See also • Selberg trace fromula References • Fischer, Jürgen (1987), An approach to the Selberg trace formula via the Selberg zeta-function, Lecture Notes in Mathematics, vol. 1253, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0077696, ISBN 978-3-540-15208-8, MR 0892317 • Hejhal, Dennis A. (1976), The Selberg trace formula for PSL(2,R). Vol. I, Lecture Notes in Mathematics, Vol. 548, vol. 548, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0079608, MR 0439755 • Hejhal, Dennis A. (1983), The Selberg trace formula for PSL(2,R). Vol. 2, Lecture Notes in Mathematics, vol. 1001, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0061302, ISBN 978-3-540-12323-1, MR 0711197 • Iwaniec, H. Spectral methods of automorphic forms, American Mathematical Society, second edition, 2002. • Selberg, Atle (1956), "Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series", J. Indian Math. Soc., New Series, 20: 47–87, MR 0088511 • Venkov, A. B. Spectral theory of automorphic functions. Proc. Steklov. Inst. Math, 1982. • Sunada, T., L-functions in geometry and some applications, Proc. Taniguchi Symp. 1985, "Curvature and Topology of Riemannian Manifolds", Springer Lect. Note in Math. 1201(1986), 266-284.	Wikipedia
Selection sort In computer science, selection sort is an in-place comparison sorting algorithm. It has an O(n2) time complexity, which makes it inefficient on large lists, and generally performs worse than the similar insertion sort. Selection sort is noted for its simplicity and has performance advantages over more complicated algorithms in certain situations, particularly where auxiliary memory is limited. Selection sort ClassSorting algorithm Data structureArray Worst-case performance$O(n^{2})$ comparisons, $O(n)$ swaps Best-case performance$O(n^{2})$ comparisons, $O(1)$ swap Average performance$O(n^{2})$ comparisons, $O(n)$ swaps Worst-case space complexity$O(1)$ auxiliary The algorithm divides the input list into two parts: a sorted sublist of items which is built up from left to right at the front (left) of the list and a sublist of the remaining unsorted items that occupy the rest of the list. Initially, the sorted sublist is empty and the unsorted sublist is the entire input list. The algorithm proceeds by finding the smallest (or largest, depending on sorting order) element in the unsorted sublist, exchanging (swapping) it with the leftmost unsorted element (putting it in sorted order), and moving the sublist boundaries one element to the right. The time efficiency of selection sort is quadratic, so there are a number of sorting techniques which have better time complexity than selection sort. Example Here is an example of this sort algorithm sorting five elements: Sorted sublist Unsorted sublist Least element in unsorted list () (11, 25, 12, 22, 64) 11 (11) (25, 12, 22, 64) 12 (11, 12) (25, 22, 64) 22 (11, 12, 22) (25, 64) 25 (11, 12, 22, 25) (64) 64 (11, 12, 22, 25, 64) () (Nothing appears changed on these last two lines because the last two numbers were already in order.) Selection sort can also be used on list structures that make add and remove efficient, such as a linked list. In this case it is more common to remove the minimum element from the remainder of the list, and then insert it at the end of the values sorted so far. For example: arr[] = 64 25 12 22 11 // Find the minimum element in arr[0...4] // and place it at beginning 11 25 12 22 64 // Find the minimum element in arr[1...4] // and place it at beginning of arr[1...4] 11 12 25 22 64 // Find the minimum element in arr[2...4] // and place it at beginning of arr[2...4] 11 12 22 25 64 // Find the minimum element in arr[3...4] // and place it at beginning of arr[3...4] 11 12 22 25 64 Implementations Below is an implementation in C. /* a[0] to a[aLength-1] is the array to sort / int i,j; int aLength; // initialise to a's length / advance the position through the entire array / / (could do i < aLength-1 because single element is also min element) / for (i = 0; i < aLength-1; i++) { / find the min element in the unsorted a[i .. aLength-1] / / assume the min is the first element / int jMin = i; / test against elements after i to find the smallest / for (j = i+1; j < aLength; j++) { / if this element is less, then it is the new minimum / if (a[j] < a[jMin]) { / found new minimum; remember its index */ jMin = j; } } if (jMin != i) { swap(a[i], a[jMin]); } } Complexity Selection sort is not difficult to analyze compared to other sorting algorithms, since none of the loops depend on the data in the array. Selecting the minimum requires scanning $n$ elements (taking $n-1$ comparisons) and then swapping it into the first position. Finding the next lowest element requires scanning the remaining $n-2$ elements and so on. Therefore, the total number of comparisons is $(n-1)+(n-2)+...+1=\sum _{i=1}^{n-1}i$ By arithmetic progression, $\sum _{i=1}^{n-1}i={\frac {(n-1)+1}{2}}(n-1)={\frac {1}{2}}n(n-1)={\frac {1}{2}}(n^{2}-n)$ which is of complexity $O(n^{2})$ in terms of number of comparisons. Each of these scans requires one swap for $n-1$ elements (the final element is already in place). Comparison to other sorting algorithms Among quadratic sorting algorithms (sorting algorithms with a simple average-case of Θ(n2)), selection sort almost always outperforms bubble sort and gnome sort. Insertion sort is very similar in that after the kth iteration, the first $k$ elements in the array are in sorted order. Insertion sort's advantage is that it only scans as many elements as it needs in order to place the $k+1$st element, while selection sort must scan all remaining elements to find the $k+1$st element. Simple calculation shows that insertion sort will therefore usually perform about half as many comparisons as selection sort, although it can perform just as many or far fewer depending on the order the array was in prior to sorting. It can be seen as an advantage for some real-time applications that selection sort will perform identically regardless of the order of the array, while insertion sort's running time can vary considerably. However, this is more often an advantage for insertion sort in that it runs much more efficiently if the array is already sorted or "close to sorted." While selection sort is preferable to insertion sort in terms of number of writes ($n-1$ swaps versus up to $n(n-1)/2$ swaps, with each swap being two writes), this is roughly twice the theoretical minimum achieved by cycle sort, which performs at most n writes. This can be important if writes are significantly more expensive than reads, such as with EEPROM or Flash memory, where every write lessens the lifespan of the memory. Selection sort can be implemented without unpredictable branches for the benefit of CPU branch predictors, by finding the location of the minimum with branch-free code and then performing the swap unconditionally. Finally, selection sort is greatly outperformed on larger arrays by $\Theta (n\log n)$ divide-and-conquer algorithms such as mergesort. However, insertion sort or selection sort are both typically faster for small arrays (i.e. fewer than 10–20 elements). A useful optimization in practice for the recursive algorithms is to switch to insertion sort or selection sort for "small enough" sublists. Variants Heapsort greatly improves the basic algorithm by using an implicit heap data structure to speed up finding and removing the lowest datum. If implemented correctly, the heap will allow finding the next lowest element in $\Theta (\log n)$ time instead of $\Theta (n)$ for the inner loop in normal selection sort, reducing the total running time to $\Theta (n\log n)$. A bidirectional variant of selection sort (called double selection sort or sometimes cocktail sort due to its similarity to cocktail shaker sort) finds both the minimum and maximum values in the list in every pass. This requires three comparisons per two items (a pair of elements is compared, then the greater is compared to the maximum and the lesser is compared to the minimum) rather than regular selection sort's one comparison per item, but requires only half as many passes, a net 25% savings. Selection sort can be implemented as a stable sort if, rather than swapping in step 2, the minimum value is inserted into the first position and the intervening values shifted up. However, this modification either requires a data structure that supports efficient insertions or deletions, such as a linked list, or it leads to performing $\Theta (n^{2})$ writes. In the bingo sort variant, items are sorted by repeatedly looking through the remaining items to find the greatest value and moving all items with that value to their final location.[1] Like counting sort, this is an efficient variant if there are many duplicate values: selection sort does one pass through the remaining items for each item moved, while Bingo sort does one pass for each value. After an initial pass to find the greatest value, subsequent passes move every item with that value to its final location while finding the next value as in the following pseudocode (arrays are zero-based and the for-loop includes both the top and bottom limits, as in Pascal): bingo(array A) { This procedure sorts in ascending order by repeatedly moving maximal items to the end. } begin last := length(A) - 1; { The first iteration is written to look very similar to the subsequent ones, but without swaps. } nextMax := A[last]; for i := last - 1 downto 0 do if A[i] > nextMax then nextMax := A[i]; while (last > 0) and (A[last] = nextMax) do last := last - 1; while last > 0 do begin prevMax := nextMax; nextMax := A[last]; for i := last - 1 downto 0 do if A[i] > nextMax then if A[i] <> prevMax then nextMax := A[i]; else begin swap(A[i], A[last]); last := last - 1; end while (last > 0) and (A[last] = nextMax) do last := last - 1; end; end; Thus, if on average there are more than two items with the same value, bingo sort can be expected to be faster because it executes the inner loop fewer times than selection sort. See also • Selection algorithm References 1. This article incorporates public domain material from Paul E. Black. "Bingo sort". Dictionary of Algorithms and Data Structures. NIST. • Donald Knuth. The Art of Computer Programming, Volume 3: Sorting and Searching, Third Edition. Addison–Wesley, 1997. ISBN 0-201-89685-0. Pages 138–141 of Section 5.2.3: Sorting by Selection. • Anany Levitin. Introduction to the Design & Analysis of Algorithms, 2nd Edition. ISBN 0-321-35828-7. Section 3.1: Selection Sort, pp 98–100. • Robert Sedgewick. Algorithms in C++, Parts 1–4: Fundamentals, Data Structure, Sorting, Searching: Fundamentals, Data Structures, Sorting, Searching Pts. 1–4, Second Edition. Addison–Wesley Longman, 1998. ISBN 0-201-35088-2. Pages 273–274 External links The Wikibook Algorithm implementation has a page on the topic of: Selection sort • Animated Sorting Algorithms: Selection Sort at the Wayback Machine (archived 7 March 2015) – graphical demonstration Sorting algorithms Theory • Computational complexity theory • Big O notation • Total order • Lists • Inplacement • Stability • Comparison sort • Adaptive sort • Sorting network • Integer sorting • X + Y sorting • Transdichotomous model • Quantum sort Exchange sorts • Bubble sort • Cocktail shaker sort • Odd–even sort • Comb sort • Gnome sort • Proportion extend sort • Quicksort Selection sorts • Selection sort • Heapsort • Smoothsort • Cartesian tree sort • Tournament sort • Cycle sort • Weak-heap sort Insertion sorts • Insertion sort • Shellsort • Splaysort • Tree sort • Library sort • Patience sorting Merge sorts • Merge sort • Cascade merge sort • Oscillating merge sort • Polyphase merge sort Distribution sorts • American flag sort • Bead sort • Bucket sort • Burstsort • Counting sort • Interpolation sort • Pigeonhole sort • Proxmap sort • Radix sort • Flashsort Concurrent sorts • Bitonic sorter • Batcher odd–even mergesort • Pairwise sorting network • Samplesort Hybrid sorts • Block merge sort • Kirkpatrick–Reisch sort • Timsort • Introsort • Spreadsort • Merge-insertion sort Other • Topological sorting • Pre-topological order • Pancake sorting • Spaghetti sort Impractical sorts • Stooge sort • Slowsort • Bogosort	Wikipedia
Choice function A choice function (selector, selection) is a mathematical function f that is defined on some collection X of nonempty sets and assigns some element of each set S in that collection to S by f(S); f(S) maps S to some element of S. In other words, f is a choice function for X if and only if it belongs to the direct product of X. For the combinatorial choice function C(n, k), see Combination and Binomial coefficient. An example Let X = { {1,4,7}, {9}, {2,7} }. Then the function that assigns 7 to the set {1,4,7}, 9 to {9}, and 2 to {2,7} is a choice function on X. History and importance Ernst Zermelo (1904) introduced choice functions as well as the axiom of choice (AC) and proved the well-ordering theorem,[1] which states that every set can be well-ordered. AC states that every set of nonempty sets has a choice function. A weaker form of AC, the axiom of countable choice (ACω) states that every countable set of nonempty sets has a choice function. However, in the absence of either AC or ACω, some sets can still be shown to have a choice function. • If $X$ is a finite set of nonempty sets, then one can construct a choice function for $X$ by picking one element from each member of $X.$ This requires only finitely many choices, so neither AC or ACω is needed. • If every member of $X$ is a nonempty set, and the union $\bigcup X$ is well-ordered, then one may choose the least element of each member of $X$. In this case, it was possible to simultaneously well-order every member of $X$ by making just one choice of a well-order of the union, so neither AC nor ACω was needed. (This example shows that the well-ordering theorem implies AC. The converse is also true, but less trivial.) Choice function of a multivalued map Given two sets X and Y, let F be a multivalued map from X to Y (equivalently, $F:X\rightarrow {\mathcal {P}}(Y)$ is a function from X to the power set of Y). A function $f:X\rightarrow Y$ is said to be a selection of F, if: $\forall x\in X\,(f(x)\in F(x))\,.$ The existence of more regular choice functions, namely continuous or measurable selections is important in the theory of differential inclusions, optimal control, and mathematical economics.[2] See Selection theorem. Bourbaki tau function Nicolas Bourbaki used epsilon calculus for their foundations that had a $\tau $ symbol that could be interpreted as choosing an object (if one existed) that satisfies a given proposition. So if $P(x)$ is a predicate, then $\tau _{x}(P)$ is one particular object that satisfies $P$ (if one exists, otherwise it returns an arbitrary object). Hence we may obtain quantifiers from the choice function, for example $P(\tau _{x}(P))$ was equivalent to $(\exists x)(P(x))$.[3] However, Bourbaki's choice operator is stronger than usual: it's a global choice operator. That is, it implies the axiom of global choice.[4] Hilbert realized this when introducing epsilon calculus.[5] See also • Axiom of countable choice • Axiom of dependent choice • Hausdorff paradox • Hemicontinuity Notes 1. Zermelo, Ernst (1904). "Beweis, dass jede Menge wohlgeordnet werden kann". Mathematische Annalen. 59 (4): 514–16. doi:10.1007/BF01445300. 2. Border, Kim C. (1989). Fixed Point Theorems with Applications to Economics and Game Theory. Cambridge University Press. ISBN 0-521-26564-9. 3. Bourbaki, Nicolas. Elements of Mathematics: Theory of Sets. ISBN 0-201-00634-0. 4. John Harrison, "The Bourbaki View" eprint. 5. "Here, moreover, we come upon a very remarkable circumstance, namely, that all of these transfinite axioms are derivable from a single axiom, one that also contains the core of one of the most attacked axioms in the literature of mathematics, namely, the axiom of choice: $A(a)\to A(\varepsilon (A))$, where $\varepsilon $ is the transfinite logical choice function." Hilbert (1925), “On the Infinite”, excerpted in Jean van Heijenoort, From Frege to Gödel, p. 382. From nCatLab. References This article incorporates material from Choice function on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.	Wikipedia
Selection principle In mathematics, a selection principle is a rule asserting the possibility of obtaining mathematically significant objects by selecting elements from given sequences of sets. The theory of selection principles studies these principles and their relations to other mathematical properties. Selection principles mainly describe covering properties, measure- and category-theoretic properties, and local properties in topological spaces, especially function spaces. Often, the characterization of a mathematical property using a selection principle is a nontrivial task leading to new insights on the characterized property. The main selection principles In 1924, Karl Menger [1] introduced the following basis property for metric spaces: Every basis of the topology contains a sequence of sets with vanishing diameters that covers the space. Soon thereafter, Witold Hurewicz[2] observed that Menger's basis property is equivalent to the following selective property: for every sequence of open covers of the space, one can select finitely many open sets from each cover in the sequence, such that the family of all selected sets covers the space. Topological spaces having this covering property are called Menger spaces. Hurewicz's reformulation of Menger's property was the first important topological property described by a selection principle. Let $\mathbf {A} $ and $\mathbf {B} $ be classes of mathematical objects. In 1996, Marion Scheepers[3] introduced the following selection hypotheses, capturing a large number of classic mathematical properties: • ${\text{S}}_{1}(\mathbf {A} ,\mathbf {B} )$: For every sequence ${\mathcal {U}}_{1},{\mathcal {U}}_{2},\ldots $ of elements from the class $\mathbf {A} $, there are elements $U_{1}\in {\mathcal {U}}_{1},U_{2}\in {\mathcal {U}}_{2},\dots $ such that $\{U_{n}:n\in \mathbb {N} \}\in \mathbf {B} $. • ${\text{S}}_{\text{fin}}(\mathbf {A} ,\mathbf {B} )$: For every sequence ${\mathcal {U}}_{1},{\mathcal {U}}_{2},\ldots $ of elements from the class $\mathbf {A} $, there are finite subsets ${\mathcal {F}}_{1}\subseteq {\mathcal {U}}_{1},{\mathcal {F}}_{2}\subseteq {\mathcal {U}}_{2},\dots $ such that $\bigcup _{n=1}^{\infty }{\mathcal {F}}_{n}\in \mathbf {B} $. In the case where the classes $\mathbf {A} $ and $\mathbf {B} $ consist of covers of some ambient space, Scheepers also introduced the following selection principle. • ${\text{U}}_{\text{fin}}(\mathbf {A} ,\mathbf {B} )$: For every sequence ${\mathcal {U}}_{1},{\mathcal {U}}_{2},\ldots $ of elements from the class $\mathbf {A} $, none containing a finite subcover, there are finite subsets ${\mathcal {F}}_{1}\subseteq {\mathcal {U}}_{1},{\mathcal {F}}_{2}\subseteq {\mathcal {U}}_{2},\dots $ such that $\{\bigcup {\mathcal {F}}_{1},\bigcup {\mathcal {F}}_{2},\dotsc \}\in \mathbf {B} $. Later, Boaz Tsaban identified the prevalence of the following related principle: • ${\binom {\mathbf {A} }{\mathbf {B} }}$: Every member of the class $\mathbf {A} $ includes a member of the class $\mathbf {B} $. The notions thus defined are selection principles. An instantiation of a selection principle, by considering specific classes $\mathbf {A} $ and $\mathbf {B} $, gives a selection (or: selective) property. However, these terminologies are used interchangeably in the literature. Variations For a set $A\subset X$ and a family ${\mathcal {F}}$ of subsets of $X$, the star of $A$ in ${\mathcal {F}}$ is the set ${\text{St}}(A,{\mathcal {F}})=\bigcup \{F\in {\mathcal {F}}:A\cap F\neq \emptyset \}$. In 1999, Ljubisa D.R. Kocinac introduced the following star selection principles:[4] • ${\text{S}}_{1}^{}(\mathbf {A} ,\mathbf {B} )$: For every sequence ${\mathcal {U}}_{1},{\mathcal {U}}_{2},\ldots $ of elements from the class $\mathbf {A} $, there are elements $U_{1}\in {\mathcal {U}}_{1},U_{2}\in {\mathcal {U}}_{2},\dots $ such that $\{{\text{St}}(U_{n},{\mathcal {U}}_{n}):n\in \mathbb {N} \}\in \mathbf {B} $. • ${\text{S}}_{\text{fin}}^{}(\mathbf {A} ,\mathbf {B} )$: For every sequence ${\mathcal {U}}_{1},{\mathcal {U}}_{2},\ldots $ of elements from the class $\mathbf {A} $, there are finite subsets ${\mathcal {F}}_{1}\subseteq {\mathcal {U}}_{1},{\mathcal {F}}_{2}\subseteq {\mathcal {U}}_{2},\dots $ such that $\{{\text{St}}(\bigcup {\mathcal {F}}_{n},{\mathcal {U}}_{n}):n\in \mathbb {N} \}\in \mathbf {B} $. The star selection principles are special cases of the general selection principles. This can be seen by modifying the definition of the family $\mathbf {B} $ accordingly. Covering properties Covering properties form the kernel of the theory of selection principles. Selection properties that are not covering properties are often studied by using implications to and from selective covering properties of related spaces. Let $X$ be a topological space. An open cover of $X$ is a family of open sets whose union is the entire space $X.$ For technical reasons, we also request that the entire space $X$ is not a member of the cover. The class of open covers of the space $X$ is denoted by $\mathbf {O} $. (Formally, $\mathbf {O} (X)$, but usually the space $X$ is fixed in the background.) The above-mentioned property of Menger is, thus, ${\text{S}}_{\text{fin}}(\mathbf {O} ,\mathbf {O} )$. In 1942, Fritz Rothberger considered Borel's strong measure zero sets, and introduced a topological variation later called Rothberger space (also known as C$''$ space). In the notation of selections, Rothberger's property is the property ${\text{S}}_{1}(\mathbf {O} ,\mathbf {O} )$. An open cover ${\mathcal {U}}$ of $X$ is point-cofinite if it has infinitely many elements, and every point $x\in X$ belongs to all but finitely many sets $U\in {\mathcal {U}}$. (This type of cover was considered by Gerlits and Nagy, in the third item of a certain list in their paper. The list was enumerated by Greek letters, and thus these covers are often called $\gamma $-covers.) The class of point-cofinite open covers of $X$ is denoted by $\mathbf {\Gamma } $. A topological space is a Hurewicz space if it satisfies ${\text{U}}_{\text{fin}}(\mathbf {O} ,\mathbf {\Gamma } )$. An open cover ${\mathcal {U}}$ of $X$ is an $\omega $-cover if every finite subset of $X$ is contained in some member of ${\mathcal {U}}$. The class of $\omega $-covers of $X$ is denoted by $\mathbf {\Omega } $. A topological space is a γ-space if it satisfies ${\binom {\mathbf {\Omega } }{\mathbf {\Gamma } }}$. By using star selection hypotheses one obtains properties such as star-Menger (${\text{S}}_{\text{fin}}^{}(\mathbf {O} ,\mathbf {O} )$), star-Rothberger (${\text{S}}_{1}^{}(\mathbf {O} ,\mathbf {O} )$) and star-Hurewicz (${\text{S}}_{\text{fin}}^{}(\mathbf {O} ,\mathbf {\Gamma } )$). The Scheepers Diagram There are 36 selection properties of the form $\Pi (\mathbf {A} ,\mathbf {B} )$, for $\Pi \in \{{\text{S}}_{1},{\text{S}}_{\text{fin}},{\text{U}}_{\text{fin}},{\bigl (}~~{\bigr )}\}$ and $\mathbf {A} ,\mathbf {B} \in \{\mathbf {O} ,\mathbf {\Gamma } ,\mathbf {\Omega } \}$. Some of them are trivial (hold for all spaces, or fail for all spaces). Restricting attention to Lindelöf spaces, the diagram below, known as the Scheepers Diagram,[3][5] presents nontrivial selection properties of the above form, and every nontrivial selection property is equivalent to one in the diagram. Arrows denote implications. Local properties Selection principles also capture important local properties. Let $Y$ be a topological space, and $y\in Y$. The class of sets $A$ in the space $Y$ that have the point $y$ in their closure is denoted by $\mathbf {\Omega _{y}} $. The class $\mathbf {\Omega _{y}^{\text{ctbl}}} $ consists of the countable elements of the class $\mathbf {\Omega _{y}} $. The class of sequences in $Y$ that converge to $y$ is denoted by $\mathbf {\Gamma _{y}} $. • A space $Y$ is Fréchet–Urysohn if and only if it satisfies ${\binom {\mathbf {\Omega _{y}} }{\mathbf {\Gamma _{y}} }}$ for all points $y\in Y$. • A space $Y$ is strongly Fréchet–Urysohn if and only if it satisfies ${\text{S}}_{1}(\mathbf {\Omega _{y}} ,\mathbf {\Gamma _{y}} )$ for all points $y\in Y$. • A space $Y$ has countable tightness if and only if it satisfies ${\binom {\mathbf {\Omega _{y}} }{\mathbf {\Omega _{y}^{\text{ctbl}}} }}$ for all points $y\in Y$. • A space $Y$ has countable fan tightness if and only if it satisfies ${\text{S}}_{\text{fin}}(\mathbf {\Omega _{y}} ,\mathbf {\Omega _{y}} )$ for all points $y\in Y$. • A space $Y$ has countable strong fan tightness if and only if it satisfies ${\text{S}}_{1}(\mathbf {\Omega _{y}} ,\mathbf {\Omega _{y}} )$ for all points $y\in Y$. Topological games There are close connections between selection principles and topological games. The Menger game Let $X$ be a topological space. The Menger game ${\text{G}}_{\text{fin}}(\mathbf {O} ,\mathbf {O} )$ played on $X$ is a game for two players, Alice and Bob. It has an inning per each natural number $n$. At the $n^{th}$ inning, Alice chooses an open cover ${\mathcal {U}}_{n}$ of $X$, and Bob chooses a finite subset ${\mathcal {F}}_{n}$ of ${\mathcal {U}}$. If the family $\bigcup _{n=1}^{\infty }{\mathcal {F}}_{n}$ is a cover of the space $X$, then Bob wins the game. Otherwise, Alice wins. A strategy for a player is a function determining the move of the player, given the earlier moves of both players. A strategy for a player is a winning strategy if each play where this player sticks to this strategy is won by this player. • A topological space is ${\text{S}}_{\text{fin}}(\mathbf {O} ,\mathbf {O} )$ if and only if Alice has no winning strategy in the game ${\text{G}}_{\text{fin}}(\mathbf {O} ,\mathbf {O} )$ played on this space.[2][3] • Let $X$ be a metric space. Bob has a winning strategy in the game ${\text{G}}_{\text{fin}}(\mathbf {O} ,\mathbf {O} )$ played on the space $X$ if and only if the space $X$ is $\sigma $-compact.[6][7] Note that among Lindelöf spaces, metrizable is equivalent to regular and second-countable, and so the previous result may alternatively be obtained by considering limited information strategies.[8] A Markov strategy is one that only uses the most recent move of the opponent and the current round number. • Let $X$ be a regular space. Bob has a winning Markov strategy in the game ${\text{G}}_{\text{fin}}(\mathbf {O} ,\mathbf {O} )$ played on the space $X$ if and only if the space $X$ is $\sigma $-compact. • Let $X$ be a second-countable space. Bob has a winning Markov strategy in the game ${\text{G}}_{\text{fin}}(\mathbf {O} ,\mathbf {O} )$ played on the space $X$ if and only if he has a winning perfect-information strategy. In a similar way, we define games for other selection principles from the given Scheepers Diagram. In all these cases a topological space has a property from the Scheepers Diagram if and only if Alice has no winning strategy in the corresponding game.[9] But this does not hold in general: Let $\mathbf {K} $ be the family of k-covers of a space. That is, such that every compact set in the space is covered by some member of the cover. Francis Jordan demonstrated a space where the selection principle ${\text{S}}_{1}(\mathbf {K} ,\mathbf {O} )$ holds, but Alice has a winning strategy for the game ${\text{G}}_{1}(\mathbf {K} ,\mathbf {O} )$ [10] Examples and properties • Every ${\text{S}}_{\text{fin}}(\mathbf {O} ,\mathbf {O} )$ space is a Lindelöf space. • Every σ-compact space (a countable union of compact spaces) is ${\text{U}}_{\text{fin}}(\mathbf {O} ,\mathbf {\Gamma } )$. • ${\binom {\mathbf {\Omega } }{\mathbf {\Gamma } }}\Rightarrow {\text{U}}_{\text{fin}}(\mathbf {O} ,\mathbf {\Gamma } )\Rightarrow {\text{S}}_{\text{fin}}(\mathbf {O} ,\mathbf {O} )$. • ${\binom {\mathbf {\Omega } }{\mathbf {\Gamma } }}\Rightarrow {\text{S}}_{1}(\mathbf {O} ,\mathbf {O} )\Rightarrow {\text{S}}_{\text{fin}}(\mathbf {O} ,\mathbf {O} )$. • Assuming the Continuum Hypothesis, there are sets of real numbers witnessing that the above implications cannot be reversed.[5] • Every Luzin set is ${\text{S}}_{\text{fin}}(\mathbf {O} ,\mathbf {O} )$ but no ${\text{U}}_{\text{fin}}(\mathbf {O} ,\mathbf {\Gamma } )$.[11][12] • Every Sierpiński set is Hurewicz.[13] Subsets of the real line $\mathbb {R} $ (with the induced subspace topology) holding selection principle properties, most notably Menger and Hurewicz spaces, can be characterized by their continuous images in the Baire space $\mathbb {N} ^{\mathbb {N} }$. For functions $f,g\in \mathbb {N} ^{\mathbb {N} }$, write $f\leq ^{}g$ if $f(n)\leq g(n)$ for all but finitely many natural numbers $n$. Let $A$ be a subset of $\mathbb {N} ^{\mathbb {N} }$. The set $A$ is bounded if there is a function $g\in \mathbb {N} ^{\mathbb {N} }$ such that $f\leq ^{}g$ for all functions $f\in A$. The set $A$ is dominating if for each function $f\in \mathbb {N} ^{\mathbb {N} }$ there is a function $g\in A$ such that $f\leq ^{}g$. • A subset of the real line is ${\text{S}}_{\text{fin}}(\mathbf {O} ,\mathbf {O} )$ if and only if every continuous image of that space into the Baire space is not dominating.[14] • A subset of the real line is ${\text{U}}_{\text{fin}}(\mathbf {O} ,\mathbf {\Gamma } )$ if and only if every continuous image of that space into the Baire space is bounded.[14] Connections with other fields General topology • Every ${\text{S}}_{\text{fin}}(\mathbf {O} ,\mathbf {O} )$ space is a D-space.[15] Let P be a property of spaces. A space $X$ is productively P if, for each space $Y$ with property P, the product space $X\times Y$ has property P. • Every separable productively paracompact space is ${\text{U}}_{\text{fin}}(\mathbf {O} ,\mathbf {\Gamma } )$. • Assuming the Continuum Hypothesis, every productively Lindelöf space is productively ${\text{U}}_{\text{fin}}(\mathbf {O} ,\mathbf {\Gamma } )$[16] • Let $A$ be a ${\binom {\mathbf {\Omega } }{\mathbf {\Gamma } }}$ subset of the real line, and $M$ be a meager subset of the real line. Then the set $A+M=\{a+x:a\in A,x\in M\}$ is meager.[17] Measure theory • Every ${\text{S}}_{1}(\mathbf {O} ,\mathbf {O} )$ subset of the real line is a strong measure zero set.[11] Function spaces Let $X$ be a Tychonoff space, and $C(X)$ be the space of continuous functions $f\colon X\to \mathbb {R} $ with pointwise convergence topology. • $X$ satisfies ${\binom {\mathbf {\Omega } }{\mathbf {\Gamma } }}$ if and only if $C(X)$ is Fréchet–Urysohn if and only if $C(X)$ is strong Fréchet–Urysohn.[18] • $X$ satisfies ${\text{S}}_{1}(\mathbf {\Omega } ,\mathbf {\Omega } )$ if and only if $C(X)$ has countable strong fan tightness.[19] • $X$ satisfies ${\text{S}}_{\text{fin}}(\mathbf {\Omega } ,\mathbf {\Omega } )$ if and only if $C(X)$ has countable fan tightness.[20][5] See also • Compact space • Sigma-compact • Menger space • Hurewicz space • Rothberger space References 1. Menger, Karl (1924). "Einige Überdeckungssätze der Punktmengenlehre". Selecta Mathematica. pp. 421–444. doi:10.1007/978-3-7091-6110-4_14. ISBN 978-3-7091-7282-7. {{cite book}}: \|journal= ignored (help) 2. Hurewicz, Witold (1926). "Über eine verallgemeinerung des Borelschen Theorems". Mathematische Zeitschrift. 24 (1): 401–421. doi:10.1007/bf01216792. S2CID 119867793. 3. Scheepers, Marion (1996). "Combinatorics of open covers I: Ramsey theory". Topology and Its Applications. 69: 31–62. doi:10.1016/0166-8641(95)00067-4. 4. Kocinac, Ljubisa D. R. (2015). "Star selection principles: a survey". Khayyam Journal of Mathematics. 1: 82–106. 5. Just, Winfried; Miller, Arnold; Scheepers, Marion; Szeptycki, Paul (1996). "Combinatorics of open covers II". Topology and Its Applications. 73 (3): 241–266. arXiv:math/9509211. doi:10.1016/S0166-8641(96)00075-2. S2CID 14946860. 6. Scheepers, Marion (1995-01-01). "A direct proof of a theorem of Telgársky". Proceedings of the American Mathematical Society. 123 (11): 3483–3485. doi:10.1090/S0002-9939-1995-1273523-1. ISSN 0002-9939. 7. Telgársky, Rastislav (1984-06-01). "On games of Topsoe". Mathematica Scandinavica. 54: 170–176. doi:10.7146/math.scand.a-12050. ISSN 1903-1807. 8. Steven, Clontz (2017-07-31). "Applications of limited information strategies in Menger's game". Commentationes Mathematicae Universitatis Carolinae. Charles University in Prague, Karolinum Press. 58 (2): 225–239. doi:10.14712/1213-7243.2015.201. ISSN 0010-2628. 9. Pawlikowski, Janusz (1994). "Undetermined sets of point-open games". Fundamenta Mathematicae. 144 (3): 279–285. ISSN 0016-2736. 10. Jordan, Francis (2020). "On the instability of a topological game related to consonance". Topology and Its Applications. Elsevier BV. 271: 106990. doi:10.1016/j.topol.2019.106990. ISSN 0166-8641. S2CID 213386675. 11. Rothberger, Fritz (1938). "Eine Verschärfung der Eigenschaft C". Fundamenta Mathematicae. 30: 50–55. doi:10.4064/fm-30-1-50-55. 12. Hurewicz, Witold (1927). "Über Folgen stetiger Funktionen". Fundamenta Mathematicae. 9: 193–210. doi:10.4064/fm-9-1-193-210. 13. Fremlin, David; Miller, Arnold (1988). "On some properties of Hurewicz, Menger and Rothberger" (PDF). Fundamenta Mathematicae. 129: 17–33. doi:10.4064/fm-129-1-17-33. 14. Recław, Ireneusz (1994). "Every Lusin set is undetermined in the point-open game". Fundamenta Mathematicae. 144: 43–54. doi:10.4064/fm-144-1-43-54. 15. Aurichi, Leandro (2010). "D-Spaces, Topological Games, and Selection Principles" (PDF). Topology Proceedings. 36: 107–122. 16. Szewczak, Piotr; Tsaban, Boaz (2016). "Product of Menger spaces, II: general spaces". arXiv:1607.01687 [math.GN]. 17. Galvin, Fred; Miller, Arnold (1984). "$\gamma $-sets and other singular sets of real numbers". Topology and Its Applications. 17 (2): 145–155. doi:10.1016/0166-8641(84)90038-5. 18. Gerlits, J.; Nagy, Zs. (1982). "Some properties of $C(X)$, I". Topology and Its Applications. 14 (2): 151–161. doi:10.1016/0166-8641(82)90065-7. 19. Sakai, Masami (1988). "Property $C''$ and function spaces". Proceedings of the American Mathematical Society. 104 (9): 917–919. doi:10.1090/S0002-9939-97-03897-5. 20. Arhangel'skii, Alexander (1986). "Hurewicz spaces, analytic sets and fan-tightness of spaces of functions". Soviet Math. Dokl. 2: 396–399. 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Selection theorem In functional analysis, a branch of mathematics, a selection theorem is a theorem that guarantees the existence of a single-valued selection function from a given set-valued map. There are various selection theorems, and they are important in the theories of differential inclusions, optimal control, and mathematical economics.[1] Preliminaries Given two sets X and Y, let F be a set-valued function from X and Y. Equivalently, $F:X\rightarrow {\mathcal {P}}(Y)$ is a function from X to the power set of Y. A function $f:X\rightarrow Y$ is said to be a selection of F if $\forall x\in X:\,\,\,f(x)\in F(x)\,.$ In other words, given an input x for which the original function F returns multiple values, the new function f returns a single value. This is a special case of a choice function. The axiom of choice implies that a selection function always exists; however, it is often important that the selection have some "nice" properties, such as continuity or measurability. This is where the selection theorems come into action: they guarantee that, if F satisfies certain properties, then it has a selection f that is continuous or has other desirable properties. Selection theorems for set-valued functions The approximate selection theorem[2] says that the following conditions are sufficient for the existence of a continuous selection: • $X$: compact metric space • $Y$: nonempty compact, convex subset of a normed linear space • $F:X\to 2^{Y}$ a set-valued function, all values nonempty, compact, convex. • $F$ has closed graph. • For every $\varepsilon >0$ there exists a continuous function $f:X\rightarrow Y$ with $\operatorname {graph} (f)\subset [\operatorname {graph} (F)]_{\varepsilon }$, where $[S]_{\epsilon }$ is the $\epsilon $-dilation of $S$, that is, the union of radius-$\epsilon $ open balls centered on points in $S$. The Michael selection theorem[3] says that the following conditions are sufficient for the existence of a continuous selection: • X is a paracompact space; • Y is a Banach space; • F is lower hemicontinuous; • for all x in X, the set F(x) is nonempty, convex and closed. The Deutsch–Kenderov theorem[4] generalizes Michael's theorem as follows: • X is a paracompact space; • Y is a normed vector space; • F is almost lower hemicontinuous, that is, at each $x\in X$, for each neighborhood $V$ of $0$ there exists a neighborhood $U$ of $x$ such that $ \bigcap _{u\in U}\{F(u)+V\}\neq \emptyset $; • for all x in X, the set F(x) is nonempty and convex. These conditions guarantee that $F$ has a continuous approximate selection, that is, for each neighborhood $V$ of $0$ in $Y$ there is a continuous function $f\colon X\mapsto Y$ such that for each $x\in X$, $f(x)\in F(X)+V$.[4] In a later note, Xu proved that the Deutsch–Kenderov theorem is also valid if $Y$ is a locally convex topological vector space.[5] The Yannelis-Prabhakar selection theorem[6] says that the following conditions are sufficient for the existence of a continuous selection: • X is a paracompact Hausdorff space; • Y is a linear topological space; • for all x in X, the set F(x) is nonempty and convex; • for all y in Y, the inverse set F−1(y) is an open set in X. The Kuratowski and Ryll-Nardzewski measurable selection theorem says that if X is a Polish space and ${\mathcal {B}}$ its Borel σ-algebra, $\mathrm {Cl} (X)$ is the set of nonempty closed subsets of X, $(\Omega ,{\mathcal {F}})$ is a measurable space, and $F:\Omega \to \mathrm {Cl} (X)$ is an ${\mathcal {F}}$-weakly measurable map (that is, for every open subset $U\subseteq X$ we have $\{\omega \in \Omega :F(\omega )\cap U\neq \emptyset \}\in {\mathcal {F}}$), then $F$ has a selection that is $({\mathcal {F}},{\mathcal {B}})$-measurable.[7] Other selection theorems for set-valued functions include: • Bressan–Colombo directionally continuous selection theorem • Castaing representation theorem • Fryszkowski decomposable map selection • Helly's selection theorem • Zero-dimensional Michael selection theorem • Robert Aumann measurable selection theorem Selection theorems for set-valued sequences • Blaschke selection theorem References 1. Border, Kim C. (1989). Fixed Point Theorems with Applications to Economics and Game Theory. Cambridge University Press. ISBN 0-521-26564-9. 2. Shapiro, Joel H. (2016). A Fixed-Point Farrago. Springer International Publishing. pp. 68–70. ISBN 978-3-319-27978-7. OCLC 984777840. 3. Michael, Ernest (1956). "Continuous selections. I". Annals of Mathematics. Second Series. 63 (2): 361–382. doi:10.2307/1969615. hdl:10338.dmlcz/119700. JSTOR 1969615. MR 0077107. 4. Deutsch, Frank; Kenderov, Petar (January 1983). "Continuous Selections and Approximate Selection for Set-Valued Mappings and Applications to Metric Projections". SIAM Journal on Mathematical Analysis. 14 (1): 185–194. doi:10.1137/0514015. 5. Xu, Yuguang (December 2001). "A Note on a Continuous Approximate Selection Theorem". Journal of Approximation Theory. 113 (2): 324–325. doi:10.1006/jath.2001.3622. 6. Yannelis, Nicholas C.; Prabhakar, N. D. (1983-12-01). "Existence of maximal elements and equilibria in linear topological spaces". Journal of Mathematical Economics. 12 (3): 233–245. doi:10.1016/0304-4068(83)90041-1. ISSN 0304-4068. 7. V. I. Bogachev, "Measure Theory" Volume II, page 36. 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Nef polygon In mathematics Nef polygons and Nef polyhedra are the sets of polygons and polyhedra which can be obtained from a finite set of halfplanes (halfspaces) by Boolean operations of set intersection and set complement. The objects are named after the Swiss mathematician Walter Nef (1919–2013[1]), who introduced them in his 1978 book on polyhedra.[2][3] Since other Boolean operations, such as union or difference, may be expressed via intersection and complement operations, the sets of Nef polygons (polyhedra) are closed with respect to these operations as well.[4] In addition, the class of Nef polyhedra is closed with respect to the topological operations of taking closure, interior, exterior, and boundary. Boolean operations, such as difference or intersection, may produce non-regular sets. However the class of Nef polyhedra is also closed with respect to the operation of regularization.[5] Convex polytopes are a special subclass of Nef polyhedra, being the set of polyhedra which are the intersections of a finite set of half-planes.[6] Terminology In the language of Nef polyhedra you can refer to various objects as 'faces' with different dimensions. What would normally be called a 'corner' or 'vertex' of a shape is called a 'face' with dimension of 0. An 'edge' or 'segment' is a face with dimension 1. A flat shape in 3D space, like a triangle, is called a face with dimension 2 – or a 'facet'. A shape in 3D space, like a cube, is called a face with dimension 3 – or a 'volume'.[7] Implementations The Computational Geometry Algorithms Library, or CGAL, represents Nef Polyhedra by using two main data structures. The first is a 'Sphere map' and the second is a 'Selective Nef Complex' (or SNC). The 'sphere map' stores information about the polyhedron by creating an imaginary sphere around each vertex, and painting it with various points and lines representing how the polyhedron divides space. The SNC basically stores and organizes the sphere maps. Each face contains a 'label' or 'mark' telling whether it is part of the object or not.[7] See also • CGAL References 1. http://math.ch/archive/documents/WalterNef.pdf 2. Nef, W. (1978). Beiträge zur Theorie der Polyeder. Bern: Herbert Lang. 3. Bieri, Hanspeter (1995). "Nef Polyhedra: A Brief Introduction". Geometric Modelling. Computing Supplement. Vol. 10. pp. 43–60. doi:10.1007/978-3-7091-7584-2_3. ISBN 978-3-211-82666-9. 4. "2D Boolean Operations on Nef Polygons". the CGAL package overview. 5. Tammik, Jeremy (2007). "AutoCAD Nef Polyhedron Implementation". CiteSeerX 10.1.1.89.6020. 6. Hachenberger, Peter; Kettner, Lutz (June 2005). "Boolean Operations on 3D Selective Nef Complexes: Optimized Implementation and Experiments" (PDF). Proc. of 2005 ACM Symposium on Solid and Physical Modeling. SPM. Boston, MA. doi:10.1145/1060244.1060263. 7. Hachenberger, Peter; Kettner, Lutz; Mehlhorn, Kurt (2006). "Boolean Operations on 3D Selective Nef Complexes: Data Structure, Algorithms, Optimized Implementation and Experiments". CiteSeerX 10.1.1.73.157.	Wikipedia
Selenne Bañuelos Selenne Bañuelos (born January 29, 1985)[1] is an American mathematician and associate professor of mathematics at California State University Channel Islands. Her research is in the areas of differential and difference equations and dynamical systems, with a focus on their applications to mathematical biology. Selenne Bañuelos Born (1985-01-29) January 29, 1985 Los Angeles, California Alma materUniversity of California, Santa Barbara University of Southern California Awards Mathematical Association of America Henry L. Alder Award (2020) Scientific career FieldsMathematician InstitutionsCalifornia State University Channel Islands ThesisStructured two-stage population model with migration between multiple locations in a periodic environment (2013) Doctoral advisorRobert John Sacker Early life, education, and career Bañuelos was born Selenne Hayde Torres-Garcia to Alex Garcia and Georgina Torres, Mexican immigrants who raised her in the community of Boyle Heights, east of downtown Los Angeles.[2][3] Bañuelos earned her B.S. degree in mathematics from the University of California, Santa Barbara in 2007.[3] She was awarded a Ph.D. in applied mathematics from the University of Southern California (USC) in 2013. Her dissertation Structured two-stage population model with migration between multiple locations in a periodic environment was supervised by Robert John Sacker.[4][2] During her doctoral studies at USC, she was presented with the Department of Mathematics Denis Ray Estes Graduate Teaching Prize. She was a co-founder of the USC chapter of SACNAS.[3] Bañuelos was part of the SACNAS Chapter Leadership Institute Alumni for her work at USC.[5] Bañuelos joined the faculty at California State University Channel Islands in 2014 as an assistant professor of mathematics. She is currently an associate professor of mathematics. She is co-advisor to the SACNAS chapter at Channel Islands, a mentor for Math Alliance[6], and a mentor and advisor for the CSU Alliance PUMP (Preparing Undergraduates through Mentoring towards PhDs) Program.[7] In 2014, Bañuelos was a Linton Poodry SACNAS Leadership Institute fellow [8] and in 2015, she was a Project NExT, New Experiences in Teaching, fellow.[3] In 2018, Bañuelos was featured on the Lathisms calendar.[3] In 2020, she received the Mathematical Association of America Henry L. Alder Award for Distinguished Teaching.[9][10] References 1. "California Birth Index, 1905-1995". californiabirthindex.org. Retrieved 19 March 2021. 2. "Structured two-stage population model with migration between multiple locations in a periodic environment". University of Southern California Digital Library. Retrieved 19 March 2021. 3. "Selenne Bañuelos". www.lathisms.org. Retrieved 19 March 2021. 4. Selenne Bañuelos at the Mathematics Genealogy Project 5. "Chaper Leadership Institute Alumni". www.sacnas.org. Retrieved 19 March 2021. 6. "Selenne Bañuelos". MathAlliance.org. Retrieved 19 March 2021. 7. "Selenne Bañuelos - Faculty Biographies- CSU Channel Islands". 2021-07-25. Archived from the original on 25 July 2021. Retrieved 2022-07-22. 8. "Linton-Poodry SACNAS Leadership Institute (LPSLI) Alumni". www.sacnas.org. Retrieved 19 March 2021. 9. "Henry L. Adler Award". Mathematical Association of America. Retrieved 19 March 2021. 10. "CSUCI mathematics faculty wins national award for extraordinary teaching practices". California State University Channel Islands. Retrieved 19 March 2021. External links • Mathematics Faculty at CSU Channel Islands • Selenne Bañuelos Author Profile at MathSciNet • Accomplishments of Selenne Bañuelos at CSU Channel Islands Authority control: Academics • MathSciNet • Mathematics Genealogy Project	Wikipedia
Self-adjoint operator In mathematics, a self-adjoint operator on an infinite-dimensional complex vector space V with inner product $\langle \cdot ,\cdot \rangle $ (equivalently, a Hermitian operator in the finite-dimensional case) is a linear map A (from V to itself) that is its own adjoint. If V is finite-dimensional with a given orthonormal basis, this is equivalent to the condition that the matrix of A is a Hermitian matrix, i.e., equal to its conjugate transpose A∗. By the finite-dimensional spectral theorem, V has an orthonormal basis such that the matrix of A relative to this basis is a diagonal matrix with entries in the real numbers. This article deals with applying generalizations of this concept to operators on Hilbert spaces of arbitrary dimension. Self-adjoint operators are used in functional analysis and quantum mechanics. In quantum mechanics their importance lies in the Dirac–von Neumann formulation of quantum mechanics, in which physical observables such as position, momentum, angular momentum and spin are represented by self-adjoint operators on a Hilbert space. Of particular significance is the Hamiltonian operator ${\hat {H}}$ defined by ${\hat {H}}\psi =-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\psi +V\psi ,$ which as an observable corresponds to the total energy of a particle of mass m in a real potential field V. Differential operators are an important class of unbounded operators. The structure of self-adjoint operators on infinite-dimensional Hilbert spaces essentially resembles the finite-dimensional case. That is to say, operators are self-adjoint if and only if they are unitarily equivalent to real-valued multiplication operators. With suitable modifications, this result can be extended to possibly unbounded operators on infinite-dimensional spaces. Since an everywhere-defined self-adjoint operator is necessarily bounded, one needs be more attentive to the domain issue in the unbounded case. This is explained below in more detail. Definitions Let $A$ be an unbounded (i.e. not necessarily bounded) operator with a dense domain $\operatorname {Dom} A\subseteq H.$ This condition holds automatically when $H$ is finite-dimensional since $\operatorname {Dom} A=H$ for every linear operator on a finite-dimensional space. Let the inner product $\langle \cdot ,\cdot \rangle $ be conjugate-linear on the second argument. This applies to complex Hilbert spaces only. By definition, the adjoint operator $A^{}$ acts on the subspace $\operatorname {Dom} A^{}\subseteq H$ consisting of the elements $y$ for which there is a $z\in H$ such that $\langle Ax,y\rangle =\langle x,z\rangle ,$ for every $x\in \operatorname {Dom} A.$ Setting $A^{}y=z$ defines the linear operator $A^{}.$ The graph of an (arbitrary) operator $A$ is the set $G(A)=\{(x,Ax)\mid x\in \operatorname {Dom} A\}.$ An operator $B$ is said to extend $A$ if $G(A)\subseteq G(B).$ This is written as $A\subseteq B.$ The densely defined operator $A$ is called symmetric if $\langle Ax,y\rangle =\langle x,Ay\rangle ,$ for all $x,y\in \operatorname {Dom} A.$ As shown below, $A$ is symmetric if and only if $G(A)\subseteq G(A^{}).$ The unbounded densely defined operator $A$ is called self-adjoint if $G(A)=G(A^{}).$ Explicitly, $\operatorname {Dom} A=\operatorname {Dom} A^{}$ and $A=A^{}.$ Every self-adjoint operator is symmetric. Conversely, a symmetric operator $A$ for which $\operatorname {Dom} A=\operatorname {Dom} A^{}$ is self-adjoint. In physics, the term Hermitian refers to symmetric as well as self-adjoint operators alike. The subtle difference between the two is generally overlooked. A subset $\rho (A)\subseteq \mathbb {C} $ is called the resolvent set (or regular set) if for every $\lambda \in \rho (A),$ the (not-necessarily-bounded) operator $A-\lambda I$ has a bounded everywhere-defined inverse. The complement $\sigma (A)=\mathbb {C} \setminus \rho (A)$ is called spectrum. In finite dimensions, $\sigma (A)$ consists exclusively of eigenvalues. Bounded self-adjoint operators A bounded operator A is self-adjoint if $\langle Ax,y\rangle =\langle x,Ay\rangle $ for all $x$ and $y$ in H. If A is symmetric and $\mathrm {Dom} (A)=H$, then, by Hellinger–Toeplitz theorem, A is necessarily bounded.[1] Every bounded linear operator T : H → H on a Hilbert space H can be written in the form $T=A+iB$ where A : H → H and B : H → H are bounded self-adjoint operators.[2] Properties of bounded self-adjoint operators Let H be a Hilbert space and let $A:H\to H$ be a bounded self-adjoint linear operator defined on $\operatorname {D} \left(A\right)=H$. • $\left\langle h,Ah\right\rangle $ is real for all $h\in H$.[3] • $\left\\|A\right\\|=\sup \left\{\|\langle h,Ah\rangle \|:\\|h\\|=1\right\}$[3] if $\operatorname {dim} H\neq 0.$ • If the image of A, denoted by $\operatorname {Im} A$, is dense in H then $A:H\to \operatorname {Im} A$ is invertible. • The eigenvalues of A are real and eigenvectors belonging to different eigenvalues are orthogonal.[3] • If $\lambda $ is an eigenvalue of A then $\|\lambda \|\leq \\|A\\|$; in particular, $\|\lambda \|\leq \sup \left\{\|\langle h,Ah\rangle \|:\\|h\\|\leq 1\right\}$.[3] • In general, there may not exist any eigenvalue $\lambda $ such that $\|\lambda \|=\sup \left\{\|\langle h,Ah\rangle \|:\\|h\\|\leq 1\right\}$, but if in addition A is compact then there necessarily exists an eigenvalue $\lambda $, equal to either $\\|A\\|$ or $-\\|A\\|$,[4] such that $\|\lambda \|=\sup \left\{\|\langle h,Ah\rangle \|:\\|h\\|\leq 1\right\}$,[3] • If a sequence of bounded self-adjoint linear operators is convergent then the limit is self-adjoint.[2] • There exists a number $\lambda $, equal to either $\\|A\\|$ or $-\\|A\\|$, and a sequence $\left(x_{i}\right)_{i=1}^{\infty }\subseteq H$ such that $\lim _{i\to \infty }Ax_{i}-\lambda x_{i}=0$ and $\\|x_{i}\\|=1$ for all i.[4] Symmetric operators See also: Extensions of symmetric operators NOTE: symmetric operators are defined above. A is symmetric ⇔ A⊆A An unbounded, densely defined operator $A$ is symmetric if and only if $A\subseteq A^{}.$ Indeed, the if-part follows directly from the definition of the adjoint operator. For the only-if-part, assuming that $A$ is symmetric, the inclusion $\operatorname {Dom} (A)\subseteq \operatorname {Dom} (A^{})$ follows from the Cauchy–Bunyakovsky–Schwarz inequality: for every $x,y\in \operatorname {Dom} (A),$ $\|\langle Ax,y\rangle \|=\|\langle x,Ay\rangle \|\leq \\|x\\|\cdot \\|Ay\\|.$ The equality $A=A^{}\|_{\operatorname {Dom} (A)}$ holds due to the equality $\langle x,A^{}y\rangle =\langle Ax,y\rangle =\langle x,Ay\rangle ,$ for every $x,y\in \operatorname {Dom} A\subseteq \operatorname {Dom} A^{},$ the density of $\operatorname {Dom} A,$ and non-degeneracy of the inner product. The Hellinger–Toeplitz theorem says that an everywhere-defined symmetric operator is bounded and self-adjoint. A is symmetric ⇔ ∀x ⟨Ax, x⟩ ∈ R The only-if part follows directly from the definition (see above). To prove the if-part, assume without loss of generality that the inner product $\langle \cdot ,\cdot \rangle $ is anti-linear on the first argument and linear on the second. (In the reverse scenario, we work with $\langle x,y\rangle _{\text{op}}{\stackrel {\text{def}}{=}}\ \langle y,x\rangle $ instead). The symmetry of $A$ follows from the polarization identity ${\begin{aligned}4\langle Ax,y\rangle ={}&\langle A(x+y),x+y\rangle -\langle A(x-y),x-y\rangle \\[1mm]&{}-i\langle A(x+iy),x+iy\rangle +i\langle A(x-iy),x-iy\rangle \end{aligned}}$ which holds for every $x,y\in \operatorname {Dom} A.$ \|\|(A−λ)x\|\| ≥ d(λ)⋅\|\|x\|\| This property is used in the proof that the spectrum of a self-adjoint operator is real. Define $S=\{x\in \operatorname {Dom} A\mid \Vert x\Vert =1\},$ $\textstyle m=\inf _{x\in S}\langle Ax,x\rangle ,$ and $\textstyle M=\sup _{x\in S}\langle Ax,x\rangle .$ The values $m,M\in \mathbb {R} \cup \{\pm \infty \}$ are properly defined since $S\neq \emptyset ,$ and $\langle Ax,x\rangle \in \mathbb {R} ,$ due to symmetry. Then, for every $\lambda \in \mathbb {C} $ and every $x\in \operatorname {Dom} A,$ $\Vert A-\lambda x\Vert \geq d(\lambda )\cdot \Vert x\Vert ,$ where $\textstyle d(\lambda )=\inf _{r\in [m,M]}\|r-\lambda \|.$ Indeed, let $x\in \operatorname {Dom} A\setminus \{0\}.$ By Cauchy-Schwarz inequality, $\Vert A-\lambda x\Vert \geq {\frac {\|\langle A-\lambda x,x\rangle \|}{\Vert x\Vert }}=\left\|\left\langle A{\frac {x}{\Vert x\Vert }},{\frac {x}{\Vert x\Vert }}\right\rangle -\lambda \right\|\cdot \Vert x\Vert \geq d(\lambda )\cdot \Vert x\Vert .$ If $\lambda \notin [m,M],$ then $d(\lambda )>0,$ and $A-\lambda I$ is called bounded below. A simple example As noted above, the spectral theorem applies only to self-adjoint operators, and not in general to symmetric operators. Nevertheless, we can at this point give a simple example of a symmetric operator that has an orthonormal basis of eigenvectors. (This operator is actually "essentially self-adjoint.") The operator A below can be seen to have a compact inverse, meaning that the corresponding differential equation Af = g is solved by some integral (and therefore compact) operator G. The compact symmetric operator G then has a countable family of eigenvectors which are complete in L2. The same can then be said for A. Consider the complex Hilbert space L2[0,1] and the differential operator $A=-{\frac {d^{2}}{dx^{2}}}$ with $\mathrm {Dom} (A)$ consisting of all complex-valued infinitely differentiable functions f on [0, 1] satisfying the boundary conditions $f(0)=f(1)=0.$ Then integration by parts of the inner product shows that A is symmetric. The reader is invited to perform integration by parts twice and verify that the given boundary conditions for $\operatorname {Dom} (A)$ ensure that the boundary terms in the integration by parts vanish. The eigenfunctions of A are the sinusoids $f_{n}(x)=\sin(n\pi x)\qquad n=1,2,\ldots $ with the real eigenvalues n2π2; the well-known orthogonality of the sine functions follows as a consequence of the property of being symmetric. We consider generalizations of this operator below. Spectrum of self-adjoint operators Let $A$ be an unbounded symmetric operator. $A$ is self-adjoint if and only if $\sigma (A)\subseteq \mathbb {R} .$ Proof: self-adjoint operator has real spectrum Let $A$ be self-adjoint. Self-adjoint operators are symmetric. The initial steps of this proof are carried out based on the symmetry alone. Self-adjointness of $A$ is not used directly until step 1b(i). Let $\lambda \in \mathbb {C} .$ Denote $R_{\lambda }=A-\lambda I.$ Using the notations from the section on symmetric operators (see above), it suffices to prove that $\sigma (A)\subseteq [m,M].$ 1. Let $\lambda \in \mathbb {C} \setminus [m,M].$ The goal is to prove the existence and boundedness of the inverted resolvent operator $R_{\lambda }^{-1},$ and show that $\operatorname {Dom} R_{\lambda }^{-1}=H.$ We begin by showing that $\ker R_{\lambda }=\{0\}$ and $\operatorname {Im} R_{\lambda }=H.$ 1. As shown above, $R_{\lambda }$ is bounded below, i.e. $\Vert R_{\lambda }x\Vert \geq d(\lambda )\cdot \Vert x\Vert ,$ with $d(\lambda )>0.$ The triviality of $\ker R_{\lambda }$ follows. 2. It remains to show that $\operatorname {Im} R_{\lambda }=H.$ Indeed, 1. $\operatorname {Im} R_{\lambda }$ is closed. To prove this, pick a sequence $y_{n}=R_{\lambda }x_{n}\in \operatorname {Im} R_{\lambda }$ converging to some $y\in H.$ Since $\\|x_{n}-x_{m}\\|\leq {\frac {1}{d(\lambda )}}\\|y_{n}-y_{m}\\|,$ $x_{n}$ is fundamental. Hence, it converges to some $x\in H.$ Furthermore, $y_{n}+\lambda x_{n}=Ax_{n}$ and $y_{n}+\lambda x_{n}\to y+\lambda x.$ One should emphasize that the arguments made thus far hold for any symmetric but not necessarily self-adjoint operator. It now follows from self-adjointness that $A$ is closed, so $x\in \operatorname {Dom} A=\operatorname {Dom} R_{\lambda },$ $Ax=y+\lambda x\in \operatorname {Im} A,$ and consequently $y=R_{\lambda }x\in \operatorname {Im} R_{\lambda }.$ Finally, 2. $\operatorname {Im} R_{\lambda }$ is dense in $H.$ Indeed, the article about Adjoint operator points out that $\operatorname {Im} R_{\lambda }^{\perp }=\ker R_{\lambda }^{}.$ From self-adjointness of $A$ (i.e. $A^{}=A)$, $R_{\lambda }^{}=R_{\bar {\lambda }}.$ Since $\lambda \in \mathbb {C} \setminus [m,M],$ the inclusion ${\bar {\lambda }}\in \mathbb {C} \setminus [m,M]$ implies that $d({\bar {\lambda }})>0,$ and consequently, $\ker R_{\bar {\lambda }}=\{0\}.$ 2. The operator $R_{\lambda }\colon \operatorname {Dom} A\to H$ has now been proven to be bijective, so the set-theoretic inverse $R_{\lambda }^{-1}$ exists and is everywhere defined. The graph of $R_{\lambda }^{-1}$ is the set $\{(R_{\lambda }x,x)\mid x\in \operatorname {Dom} A\}.$ Since $R_{\lambda }$ is closed (because $A$ is), so is $R_{\lambda }^{-1}.$ By closed graph theorem, $R_{\lambda }^{-1}$ is bounded, so $\lambda \notin \sigma (A).$ Proof: Symmetric operator with real spectrum is self-adjoint 1. By assumption, $A$ is symmetric; therefore $A\subseteq A^{}.$ For every $\lambda \in \mathbb {C} ,$ $A-\lambda I\subseteq A^{}-\lambda I.$ Let $\sigma (A)\subseteq [m,M].$ (These constants are defined in the section on symmetic operators above). If $\lambda \notin [m,M],$ then ${\bar {\lambda }}\notin [m,M].$ Since $\lambda $ and ${\bar {\lambda }}$ are not in the spectrum, the operators $A-\lambda I,A-{\bar {\lambda }}I:\operatorname {Dom} A\to H$ are bijective. Moreover, 2. $A-\lambda I=A^{}-\lambda I.$ Indeed, $H=\operatorname {Im} (A-\lambda I)\subseteq \operatorname {Im} (A^{}-\lambda I).$ If one had $\operatorname {Dom} (A-\lambda I)\subsetneq \operatorname {Dom} (A^{}-\lambda I),$ then $A^{}-\lambda I$ would not be injective, i.e. one would have $\ker(A^{}-\lambda I)\neq \{0\}.$ As discussed in the article about Adjoint operator, $\operatorname {Im} (A-{\bar {\lambda }}I)^{\perp }=\ker(A^{}-\lambda I),$ and, hence, $\operatorname {Im} (A-{\bar {\lambda }}I)\neq H.$ This contradicts the bijectiveness. 3. The equality $A-\lambda I=A^{}-\lambda I$ shows that $A=A^{},$ i.e. $A$ is self-adjoint. Indeed, it suffices to prove that $A^{}\subseteq A.$ For every $x\in \operatorname {Dom} A^{}$ and $y=A^{}x,$ $A^{}x=y\Leftrightarrow (A^{}-\lambda I)x=y-\lambda x\Leftrightarrow (A-\lambda I)x=y-\lambda x\Leftrightarrow Ax=y.$ Essential self-adjointness A symmetric operator A is always closable; that is, the closure of the graph of A is the graph of an operator. A symmetric operator A is said to be essentially self-adjoint if the closure of A is self-adjoint. Equivalently, A is essentially self-adjoint if it has a unique self-adjoint extension. In practical terms, having an essentially self-adjoint operator is almost as good as having a self-adjoint operator, since we merely need to take the closure to obtain self-adjoint operator. Example: f(x) → x·f(x) Consider the complex Hilbert space L2(R), and the operator which multiplies a given function by x: $Af(x)=xf(x)$ The domain of A is the space of all L2 functions $f(x)$ for which $xf(x)$ is also square-integrable. Then A is self-adjoint.[5] On the other hand, A does not have any eigenfunctions. (More precisely, A does not have any normalizable eigenvectors, that is, eigenvectors that are actually in the Hilbert space on which A is defined.) As we will see later, self-adjoint operators have very important spectral properties; they are in fact multiplication operators on general measure spaces. Symmetric vs self-adjoint operators As has been discussed above, although the distinction between a symmetric operator and a self-adjoint (or essentially self-adjoint) operator is a subtle one, it is important since self-adjointness is the hypothesis in the spectral theorem. Here we discuss some concrete examples of the distinction; see the section below on extensions of symmetric operators for the general theory. A note regarding domains See also: Extensions of symmetric operators and Unbounded operator Every self-adjoint operator is symmetric. Conversely, every symmetric operator for which $\operatorname {Dom} (A^{})\subseteq \operatorname {Dom} (A)$ is self-adjoint. Symmetric operators for which $\operatorname {Dom} (A^{})$ is strictly greater than $\operatorname {Dom} (A)$ cannot be self-adjoint. Boundary conditions In the case where the Hilbert space is a space of functions on a bounded domain, these distinctions have to do with a familiar issue in quantum physics: One cannot define an operator—such as the momentum or Hamiltonian operator—on a bounded domain without specifying boundary conditions. In mathematical terms, choosing the boundary conditions amounts to choosing an appropriate domain for the operator. Consider, for example, the Hilbert space $L^{2}([0,1])$ (the space of square-integrable functions on the interval [0,1]). Let us define a "momentum" operator A on this space by the usual formula, setting Planck's constant equal to 1: $Af=-i{\frac {df}{dx}}.$ We must now specify a domain for A, which amounts to choosing boundary conditions. If we choose $\operatorname {Dom} (A)=\left\{{\text{smooth functions}}\right\},$ then A is not symmetric (because the boundary terms in the integration by parts do not vanish). If we choose $\operatorname {Dom} (A)=\left\{{\text{smooth functions}}\,f\mid f(0)=f(1)=0\right\},$ then using integration by parts, one can easily verify that A is symmetric. This operator is not essentially self-adjoint,[6] however, basically because we have specified too many boundary conditions on the domain of A, which makes the domain of the adjoint too big. (This example is discussed also in the "Examples" section below.) Specifically, with the above choice of domain for A, the domain of the closure $A^{\mathrm {cl} }$ of A is $\operatorname {Dom} \left(A^{\mathrm {cl} }\right)=\left\{{\text{functions }}f{\text{ with two derivatives in }}L^{2}\mid f(0)=f(1)=0\right\},$ whereas the domain of the adjoint $A^{}$ of A is $\operatorname {Dom} \left(A^{}\right)=\left\{{\text{functions }}f{\text{ with two derivatives in }}L^{2}\right\}.$ That is to say, the domain of the closure has the same boundary conditions as the domain of A itself, just a less stringent smoothness assumption. Meanwhile, since there are "too many" boundary conditions on A, there are "too few" (actually, none at all in this case) for $A^{}$. If we compute $\langle g,Af\rangle $ for $f\in \operatorname {Dom} (A)$ using integration by parts, then since $f$ vanishes at both ends of the interval, no boundary conditions on $g$ are needed to cancel out the boundary terms in the integration by parts. Thus, any sufficiently smooth function $g$ is in the domain of $A^{}$, with $A^{}g=-i\,dg/dx$.[7] Since the domain of the closure and the domain of the adjoint do not agree, A is not essentially self-adjoint. After all, a general result says that the domain of the adjoint of $A^{\mathrm {cl} }$ is the same as the domain of the adjoint of A. Thus, in this case, the domain of the adjoint of $A^{\mathrm {cl} }$ is bigger than the domain of $A^{\mathrm {cl} }$ itself, showing that $A^{\mathrm {cl} }$ is not self-adjoint, which by definition means that A is not essentially self-adjoint. The problem with the preceding example is that we imposed too many boundary conditions on the domain of A. A better choice of domain would be to use periodic boundary conditions: $\operatorname {Dom} (A)=\{{\text{smooth functions}}\,f\mid f(0)=f(1)\}.$ With this domain, A is essentially self-adjoint.[8] In this case, we can understand the implications of the domain issues for the spectral theorem. If we use the first choice of domain (with no boundary conditions), all functions $f_{\beta }(x)=e^{\beta x}$ for $\beta \in \mathbb {C} $ are eigenvectors, with eigenvalues $-i\beta $, and so the spectrum is the whole complex plane. If we use the second choice of domain (with Dirichlet boundary conditions), A has no eigenvectors at all. If we use the third choice of domain (with periodic boundary conditions), we can find an orthonormal basis of eigenvectors for A, the functions $f_{n}(x):=e^{2\pi inx}$. Thus, in this case finding a domain such that A is self-adjoint is a compromise: the domain has to be small enough so that A is symmetric, but large enough so that $D(A^{})=D(A)$. Schrödinger operators with singular potentials A more subtle example of the distinction between symmetric and (essentially) self-adjoint operators comes from Schrödinger operators in quantum mechanics. If the potential energy is singular—particularly if the potential is unbounded below—the associated Schrödinger operator may fail to be essentially self-adjoint. In one dimension, for example, the operator ${\hat {H}}:={\frac {P^{2}}{2m}}-X^{4}$ is not essentially self-adjoint on the space of smooth, rapidly decaying functions.[9] In this case, the failure of essential self-adjointness reflects a pathology in the underlying classical system: A classical particle with a $-x^{4}$ potential escapes to infinity in finite time. This operator does not have a unique self-adjoint, but it does admit self-adjoint extensions obtained by specifying "boundary conditions at infinity". (Since ${\hat {H}}$ is a real operator, it commutes with complex conjugation. Thus, the deficiency indices are automatically equal, which is the condition for having a self-adjoint extension. See the discussion of extensions of symmetric operators below.) In this case, if we initially define ${\hat {H}}$ on the space of smooth, rapidly decaying functions, the adjoint will be "the same" operator (i.e., given by the same formula) but on the largest possible domain, namely $\operatorname {Dom} \left({\hat {H}}^{}\right)=\left\{{\text{twice differentiable functions }}f\in L^{2}(\mathbb {R} )\left\|\left(-{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}f}{dx^{2}}}-x^{4}f(x)\right)\in L^{2}(\mathbb {R} )\right.\right\}.$ It is then possible to show that ${\hat {H}}^{}$ is not a symmetric operator, which certainly implies that ${\hat {H}}$ is not essentially self-adjoint. Indeed, ${\hat {H}}^{}$ has eigenvectors with pure imaginary eigenvalues,[10][11] which is impossible for a symmetric operator. This strange occurrence is possible because of a cancellation between the two terms in ${\hat {H}}^{}$: There are functions $f$ in the domain of ${\hat {H}}^{}$ for which neither $d^{2}f/dx^{2}$ nor $x^{4}f(x)$ is separately in $L^{2}(\mathbb {R} )$, but the combination of them occurring in ${\hat {H}}^{}$ is in $L^{2}(\mathbb {R} )$. This allows for ${\hat {H}}^{}$ to be nonsymmetric, even though both $d^{2}/dx^{2}$ and $X^{4}$ are symmetric operators. This sort of cancellation does not occur if we replace the repelling potential $-x^{4}$ with the confining potential $x^{4}$. Conditions for Schrödinger operators to be self-adjoint or essentially self-adjoint can be found in various textbooks, such as those by Berezin and Shubin, Hall, and Reed and Simon listed in the references. Spectral theorem Main article: Spectral theorem In the physics literature, the spectral theorem is often stated by saying that a self-adjoint operator has an orthonormal basis of eigenvectors. Physicists are well aware, however, of the phenomenon of "continuous spectrum"; thus, when they speak of an "orthonormal basis" they mean either an orthonormal basis in the classic sense or some continuous analog thereof. In the case of the momentum operator $ P=-i{\frac {d}{dx}}$, for example, physicists would say that the eigenvectors are the functions $f_{p}(x):=e^{ipx}$, which are clearly not in the Hilbert space $L^{2}(\mathbb {R} )$. (Physicists would say that the eigenvectors are "non-normalizable.") Physicists would then go on to say that these "eigenvectors" are orthonormal in a continuous sense, where the usual Kronecker delta $\delta _{i,j}$ is replaced by a Dirac delta function $\delta \left(p-p'\right)$. Although these statements may seem disconcerting to mathematicians, they can be made rigorous by use of the Fourier transform, which allows a general $L^{2}$ function to be expressed as a "superposition" (i.e., integral) of the functions $e^{ipx}$, even though these functions are not in $L^{2}$. The Fourier transform "diagonalizes" the momentum operator; that is, it converts it into the operator of multiplication by $p$, where $p$ is the variable of the Fourier transform. The spectral theorem in general can be expressed similarly as the possibility of "diagonalizing" an operator by showing it is unitarily equivalent to a multiplication operator. Other versions of the spectral theorem are similarly intended to capture the idea that a self-adjoint operator can have "eigenvectors" that are not actually in the Hilbert space in question. Statement of the spectral theorem Partially defined operators A, B on Hilbert spaces H, K are unitarily equivalent if and only if there is a unitary transformation U : H → K such that • U maps dom A bijectively onto dom B, • $BU\xi =UA\xi ,\qquad \forall \xi \in \operatorname {dom} A.$ A multiplication operator is defined as follows: Let (X, Σ, μ) be a countably additive measure space and f a real-valued measurable function on X. An operator $T_{f}$ of the form $[T_{f}\psi ](x)=f(x)\psi (x)$ whose domain is the space of ψ for which the right-hand side above is in L2 is called a multiplication operator. One version of the spectral theorem can be stated as follows. Theorem — Any multiplication operator is a (densely defined) self-adjoint operator. Any self-adjoint operator is unitarily equivalent to a multiplication operator.[12] Other versions of the spectral theorem can be found in the spectral theorem article linked to above. The spectral theorem for unbounded self-adjoint operators can be proved by reduction to the spectral theorem for unitary (hence bounded) operators.[13] This reduction uses the Cayley transform for self-adjoint operators which is defined in the next section. We might note that if T is multiplication by f, then the spectrum of T is just the essential range of f. Functional calculus One important application of the spectral theorem is to define a "functional calculus." That is to say, if $h$ is a function on the real line and $T$ is a self-adjoint operator, we wish to define the operator $h(T)$. If $T$ has a true orthonormal basis of eigenvectors $e_{j}$ with eigenvalues $\lambda _{j}$, then $h(T)$ is the operator with eigenvectors $e_{j}$ and eigenvalues $h\left(\lambda _{j}\right)$. The goal of functional calculus is to extend this idea to the case where $T$ has continuous spectrum. Of particular importance in quantum physics is the case in which $T$ is the Hamiltonian operator ${\hat {H}}$ and $h(x):=e^{-itx/\hbar }$ is an exponential. In this case, the functional calculus should allow us to define the operator $U(t):=h\left({\hat {H}}\right)=e^{\frac {-it{\hat {H}}}{\hbar }},$ which is the operator defining the time-evolution in quantum mechanics. Given the representation of T as the operator of multiplication by $f$—as guaranteed by the spectral theorem—it is easy to characterize the functional calculus: If h is a bounded real-valued Borel function on R, then h(T) is the operator of multiplication by the composition $h\circ f$. Resolution of the identity It has been customary to introduce the following notation $\operatorname {E} _{T}(\lambda )=\mathbf {1} _{(-\infty ,\lambda ]}(T)$ where $\mathbf {1} _{(-\infty ,\lambda ]}$ is the characteristic function (indicator function)of the interval $(-\infty ,\lambda ]$. The family of projection operators ET(λ) is called resolution of the identity for T. Moreover, the following Stieltjes integral representation for T can be proved: $T=\int _{-\infty }^{+\infty }\lambda d\operatorname {E} _{T}(\lambda ).$ The definition of the operator integral above can be reduced to that of a scalar valued Stieltjes integral using the weak operator topology. In more modern treatments however, this representation is usually avoided, since most technical problems can be dealt with by the functional calculus. Formulation in the physics literature In physics, particularly in quantum mechanics, the spectral theorem is expressed in a way which combines the spectral theorem as stated above and the Borel functional calculus using Dirac notation as follows: If H is self-adjoint and f is a Borel function, $f(H)=\int dE\left\|\Psi _{E}\rangle f(E)\langle \Psi _{E}\right\|$ with $H\left\|\Psi _{E}\right\rangle =E\left\|\Psi _{E}\right\rangle $ where the integral runs over the whole spectrum of H. The notation suggests that H is diagonalized by the eigenvectors ΨE. Such a notation is purely formal. One can see the similarity between Dirac's notation and the previous section. The resolution of the identity (sometimes called projection valued measures) formally resembles the rank-1 projections $\left\|\Psi _{E}\right\rangle \left\langle \Psi _{E}\right\|$. In the Dirac notation, (projective) measurements are described via eigenvalues and eigenstates, both purely formal objects. As one would expect, this does not survive passage to the resolution of the identity. In the latter formulation, measurements are described using the spectral measure of $\|\Psi \rangle $, if the system is prepared in $\|\Psi \rangle $ prior to the measurement. Alternatively, if one would like to preserve the notion of eigenstates and make it rigorous, rather than merely formal, one can replace the state space by a suitable rigged Hilbert space. If f = 1, the theorem is referred to as resolution of unity: $I=\int dE\left\|\Psi _{E}\right\rangle \left\langle \Psi _{E}\right\|$ In the case $H_{\text{eff}}=H-i\Gamma $ is the sum of an Hermitian H and a skew-Hermitian (see skew-Hermitian matrix) operator $-i\Gamma $, one defines the biorthogonal basis set $H_{\text{eff}}^{}\left\|\Psi _{E}^{}\right\rangle =E^{}\left\|\Psi _{E}^{}\right\rangle $ and write the spectral theorem as: $f\left(H_{\text{eff}}\right)=\int dE\left\|\Psi _{E}\right\rangle f(E)\left\langle \Psi _{E}^{}\right\|$ (See Feshbach–Fano partitioning method for the context where such operators appear in scattering theory). Extensions of symmetric operators Further information: Extensions of symmetric operators and Unbounded operator The following question arises in several contexts: if an operator A on the Hilbert space H is symmetric, when does it have self-adjoint extensions? An operator that has a unique self-adjoint extension is said to be essentially self-adjoint; equivalently, an operator is essentially self-adjoint if its closure (the operator whose graph is the closure of the graph of A) is self-adjoint. In general, a symmetric operator could have many self-adjoint extensions or none at all. Thus, we would like a classification of its self-adjoint extensions. The first basic criterion for essential self-adjointness is the following:[14] Theorem — If A is a symmetric operator on H, then A is essentially self-adjoint if and only if the range of the operators $A-i$ and $A+i$ are dense in H. Equivalently, A is essentially self-adjoint if and only if the operators $A^{}-i$ and $A^{}+i$ have trivial kernels.[15] That is to say, A fails to be self-adjoint if and only if $A^{}$ has an eigenvector with eigenvalue $i$ or $-i$. Another way of looking at the issue is provided by the Cayley transform of a self-adjoint operator and the deficiency indices. (It is often of technical convenience to deal with closed operators. In the symmetric case, the closedness requirement poses no obstacles, since it is known that all symmetric operators are closable.) Theorem — Suppose A is a symmetric operator. Then there is a unique partially defined linear operator $\operatorname {W} (A):\operatorname {ran} (A+i)\to \operatorname {ran} (A-i)$ such that $\operatorname {W} (A)(Ax+ix)=Ax-ix,\qquad x\in \operatorname {dom} (A).$ Here, ran and dom denote the image (in other words, range) and the domain, respectively. W(A) is isometric on its domain. Moreover, the range of 1 − W(A) is dense in H. Conversely, given any partially defined operator U which is isometric on its domain (which is not necessarily closed) and such that 1 − U is dense, there is a (unique) operator S(U) $\operatorname {S} (U):\operatorname {ran} (1-U)\to \operatorname {ran} (1+U)$ such that $\operatorname {S} (U)(x-Ux)=i(x+Ux)\qquad x\in \operatorname {dom} (U).$ The operator S(U) is densely defined and symmetric. The mappings W and S are inverses of each other. The mapping W is called the Cayley transform. It associates a partially defined isometry to any symmetric densely defined operator. Note that the mappings W and S are monotone: This means that if B is a symmetric operator that extends the densely defined symmetric operator A, then W(B) extends W(A), and similarly for S. Theorem — A necessary and sufficient condition for A to be self-adjoint is that its Cayley transform W(A) be unitary. This immediately gives us a necessary and sufficient condition for A to have a self-adjoint extension, as follows: Theorem — A necessary and sufficient condition for A to have a self-adjoint extension is that W(A) have a unitary extension. A partially defined isometric operator V on a Hilbert space H has a unique isometric extension to the norm closure of dom(V). A partially defined isometric operator with closed domain is called a partial isometry. Given a partial isometry V, the deficiency indices of V are defined as the dimension of the orthogonal complements of the domain and range: ${\begin{aligned}n_{+}(V)&=\dim \operatorname {dom} (V)^{\perp }\\n_{-}(V)&=\dim \operatorname {ran} (V)^{\perp }\end{aligned}}$ Theorem — A partial isometry V has a unitary extension if and only if the deficiency indices are identical. Moreover, V has a unique unitary extension if and only if the deficiency indices are both zero. We see that there is a bijection between symmetric extensions of an operator and isometric extensions of its Cayley transform. The symmetric extension is self-adjoint if and only if the corresponding isometric extension is unitary. A symmetric operator has a unique self-adjoint extension if and only if both its deficiency indices are zero. Such an operator is said to be essentially self-adjoint. Symmetric operators which are not essentially self-adjoint may still have a canonical self-adjoint extension. Such is the case for non-negative symmetric operators (or more generally, operators which are bounded below). These operators always have a canonically defined Friedrichs extension and for these operators we can define a canonical functional calculus. Many operators that occur in analysis are bounded below (such as the negative of the Laplacian operator), so the issue of essential adjointness for these operators is less critical. Self-adjoint extensions in quantum mechanics In quantum mechanics, observables correspond to self-adjoint operators. By Stone's theorem on one-parameter unitary groups, self-adjoint operators are precisely the infinitesimal generators of unitary groups of time evolution operators. However, many physical problems are formulated as a time-evolution equation involving differential operators for which the Hamiltonian is only symmetric. In such cases, either the Hamiltonian is essentially self-adjoint, in which case the physical problem has unique solutions or one attempts to find self-adjoint extensions of the Hamiltonian corresponding to different types of boundary conditions or conditions at infinity. Example. The one-dimensional Schrödinger operator with the potential $V(x)=-(1+\|x\|)^{\alpha }$, defined initially on smooth compactly supported functions, is essentially self-adjoint (that is, has a self-adjoint closure) for 0 < α ≤ 2 but not for α > 2. See Berezin and Schubin, pages 55 and 86, or Section 9.10 in Hall. The failure of essential self-adjointness for $\alpha >2$ has a counterpart in the classical dynamics of a particle with potential $V(x)$: The classical particle escapes to infinity in finite time.[16] Example. There is no self-adjoint momentum operator p for a particle moving on a half-line. Nevertheless, the Hamiltonian $p^{2}$ of a "free" particle on a half-line has several self-adjoint extensions corresponding to different types of boundary conditions. Physically, these boundary conditions are related to reflections of the particle at the origin (see Reed and Simon, vol.2). Von Neumann's formulas Suppose A is symmetric densely defined. Then any symmetric extension of A is a restriction of A. Indeed, A ⊆ B and B symmetric yields B ⊆ A by applying the definition of dom(A). Theorem — Suppose A is a densely defined symmetric operator. Let $N_{\pm }=\operatorname {ran} (A\pm i)^{\perp },$ Then $N_{\pm }=\operatorname {ker} (A^{}\mp i),$ and $\operatorname {dom} \left(A^{}\right)=\operatorname {dom} \left({\overline {A}}\right)\oplus N_{+}\oplus N_{-},$ where the decomposition is orthogonal relative to the graph inner product of dom(A): $\langle \xi \mid \eta \rangle _{\text{graph}}=\langle \xi \mid \eta \rangle +\left\langle A^{}\xi \mid A^{}\eta \right\rangle .$ These are referred to as von Neumann's formulas in the Akhiezer and Glazman reference. Examples A symmetric operator that is not essentially self-adjoint We first consider the Hilbert space $L^{2}[0,1]$ and the differential operator $D:\phi \mapsto {\frac {1}{i}}\phi '$ defined on the space of continuously differentiable complex-valued functions on [0,1], satisfying the boundary conditions $\phi (0)=\phi (1)=0.$ Then D is a symmetric operator as can be shown by integration by parts. The spaces N+, N− (defined below) are given respectively by the distributional solutions to the equation ${\begin{aligned}-iu'&=iu\\-iu'&=-iu\end{aligned}}$ which are in L2[0, 1]. One can show that each one of these solution spaces is 1-dimensional, generated by the functions x → e−x and x → ex respectively. This shows that D is not essentially self-adjoint,[17] but does have self-adjoint extensions. These self-adjoint extensions are parametrized by the space of unitary mappings N+ → N−, which in this case happens to be the unit circle T. In this case, the failure of essential self-adjointenss is due to an "incorrect" choice of boundary conditions in the definition of the domain of $D$. Since $D$ is a first-order operator, only one boundary condition is needed to ensure that $D$ is symmetric. If we replaced the boundary conditions given above by the single boundary condition $\phi (0)=\phi (1)$, then D would still be symmetric and would now, in fact, be essentially self-adjoint. This change of boundary conditions gives one particular essentially self-adjoint extension of D. Other essentially self-adjoint extensions come from imposing boundary conditions of the form $\phi (1)=e^{i\theta }\phi (0)$. This simple example illustrates a general fact about self-adjoint extensions of symmetric differential operators P on an open set M. They are determined by the unitary maps between the eigenvalue spaces $N_{\pm }=\left\{u\in L^{2}(M):P_{\operatorname {dist} }u=\pm iu\right\}$ where Pdist is the distributional extension of P. Constant-coefficient operators We next give the example of differential operators with constant coefficients. Let $P\left({\vec {x}}\right)=\sum _{\alpha }c_{\alpha }x^{\alpha }$ be a polynomial on Rn with real coefficients, where α ranges over a (finite) set of multi-indices. Thus $\alpha =(\alpha _{1},\alpha _{2},\ldots ,\alpha _{n})$ and $x^{\alpha }=x_{1}^{\alpha _{1}}x_{2}^{\alpha _{2}}\cdots x_{n}^{\alpha _{n}}.$ We also use the notation $D^{\alpha }={\frac {1}{i^{\|\alpha \|}}}\partial _{x_{1}}^{\alpha _{1}}\partial _{x_{2}}^{\alpha _{2}}\cdots \partial _{x_{n}}^{\alpha _{n}}.$ Then the operator P(D) defined on the space of infinitely differentiable functions of compact support on Rn by $P(\operatorname {D} )\phi =\sum _{\alpha }c_{\alpha }\operatorname {D} ^{\alpha }\phi $ is essentially self-adjoint on L2(Rn). Theorem — Let P a polynomial function on Rn with real coefficients, F the Fourier transform considered as a unitary map L2(Rn) → L2(Rn). Then FP(D)F is essentially self-adjoint and its unique self-adjoint extension is the operator of multiplication by the function P. More generally, consider linear differential operators acting on infinitely differentiable complex-valued functions of compact support. If M is an open subset of Rn $P\phi (x)=\sum _{\alpha }a_{\alpha }(x)\left[D^{\alpha }\phi \right](x)$ where aα are (not necessarily constant) infinitely differentiable functions. P is a linear operator $C_{0}^{\infty }(M)\to C_{0}^{\infty }(M).$ Corresponding to P there is another differential operator, the formal adjoint of P $P^{\mathrm {form} }\phi =\sum _{\alpha }D^{\alpha }\left({\overline {a_{\alpha }}}\phi \right)$ Theorem — The adjoint P* of P is a restriction of the distributional extension of the formal adjoint to an appropriate subspace of $L^{2}$. Specifically: $\operatorname {dom} P^{}=\left\{u\in L^{2}(M):P^{\mathrm {form} }u\in L^{2}(M)\right\}.$ Spectral multiplicity theory The multiplication representation of a self-adjoint operator, though extremely useful, is not a canonical representation. This suggests that it is not easy to extract from this representation a criterion to determine when self-adjoint operators A and B are unitarily equivalent. The finest grained representation which we now discuss involves spectral multiplicity. This circle of results is called the Hahn–Hellinger theory of spectral multiplicity. Uniform multiplicity We first define uniform multiplicity: Definition. A self-adjoint operator A has uniform multiplicity n where n is such that 1 ≤ n ≤ ω if and only if A is unitarily equivalent to the operator Mf of multiplication by the function f(λ) = λ on $L_{\mu }^{2}\left(\mathbf {R} ,\mathbf {H} _{n}\right)=\left\{\psi :\mathbf {R} \to \mathbf {H} _{n}:\psi {\mbox{ measurable and }}\int _{\mathbf {R} }\\|\psi (t)\\|^{2}d\mu (t)<\infty \right\}$ :\mathbf {R} \to \mathbf {H} _{n}:\psi {\mbox{ measurable and }}\int _{\mathbf {R} }\\|\psi (t)\\|^{2}d\mu (t)<\infty \right\}} where Hn is a Hilbert space of dimension n. The domain of Mf consists of vector-valued functions ψ on R such that $\int _{\mathbf {R} }\|\lambda \|^{2}\ \\|\psi (\lambda )\\|^{2}\,d\mu (\lambda )<\infty .$ Non-negative countably additive measures μ, ν are mutually singular if and only if they are supported on disjoint Borel sets. Theorem — Let A be a self-adjoint operator on a separable Hilbert space H. Then there is an ω sequence of countably additive finite measures on R (some of which may be identically 0) $\left\{\mu _{\ell }\right\}_{1\leq \ell \leq \omega }$ such that the measures are pairwise singular and A is unitarily equivalent to the operator of multiplication by the function f(λ) = λ on $\bigoplus _{1\leq \ell \leq \omega }L_{\mu _{\ell }}^{2}\left(\mathbf {R} ,\mathbf {H} _{\ell }\right).$ This representation is unique in the following sense: For any two such representations of the same A, the corresponding measures are equivalent in the sense that they have the same sets of measure 0. Direct integrals The spectral multiplicity theorem can be reformulated using the language of direct integrals of Hilbert spaces: Theorem — [18] Any self-adjoint operator on a separable Hilbert space is unitarily equivalent to multiplication by the function λ ↦ λ on $\int _{\mathbf {R} }^{\oplus }H_{\lambda }\,d\mu (\lambda ).$ Unlike the multiplication-operator version of the spectral theorem, the direct-integral version is unique in the sense that the measure equivalence class of μ (or equivalently its sets of measure 0) is uniquely determined and the measurable function $\lambda \mapsto \mathrm {dim} (H_{\lambda })$ is determined almost everywhere with respect to μ.[19] The function $\lambda \mapsto \operatorname {dim} \left(H_{\lambda }\right)$ is the spectral multiplicity function of the operator. We may now state the classification result for self-adjoint operators: Two self-adjoint operators are unitarily equivalent if and only if (1) their spectra agree as sets, (2) the measures appearing in their direct-integral representations have the same sets of measure zero, and (3) their spectral multiplicity functions agree almost everywhere with respect to the measure in the direct integral.[20] Example: structure of the Laplacian The Laplacian on Rn is the operator $\Delta =\sum _{i=1}^{n}\partial _{x_{i}}^{2}.$ As remarked above, the Laplacian is diagonalized by the Fourier transform. Actually it is more natural to consider the negative of the Laplacian −Δ since as an operator it is non-negative; (see elliptic operator). Theorem — If n = 1, then −Δ has uniform multiplicity ${\text{mult}}=2$, otherwise −Δ has uniform multiplicity ${\text{mult}}=\omega $. Moreover, the measure μmult may be taken to be Lebesgue measure on [0, ∞). Pure point spectrum A self-adjoint operator A on H has pure point spectrum if and only if H has an orthonormal basis {ei}i ∈ I consisting of eigenvectors for A. Example. The Hamiltonian for the harmonic oscillator has a quadratic potential V, that is $-\Delta +\|x\|^{2}.$ This Hamiltonian has pure point spectrum; this is typical for bound state Hamiltonians in quantum mechanics. As was pointed out in a previous example, a sufficient condition that an unbounded symmetric operator has eigenvectors which form a Hilbert space basis is that it has a compact inverse. See also • Compact operator on Hilbert space • Theoretical and experimental justification for the Schrödinger equation • Unbounded operator • Hermitian adjoint • Positive operator • Non-Hermitian quantum mechanics Citations 1. Hall 2013 Corollary 9.9 2. Griffel 2002, p. 238. 3. Griffel 2002, pp. 224–230. 4. Griffel 2002, pp. 240–245. 5. Hall 2013 Proposition 9.30 6. Hall 2013 Proposition 9.27 7. Hall 2013 Proposition 9.28 8. Hall 2013 Example 9.25 9. Hall 2013 Theorem 9.41 10. Berezin & Shubin 1991 p. 85 11. Hall 2013 Section 9.10 12. Hall 2013 Theorems 7.20 and 10.10 13. Hall 2013 Section 10.4 14. Hall 2013 Theorem 9.21 15. Hall 2013 Corollary 9.22 16. Hall 2013 Chapter 2, Exercise 4 17. Hall 2013 Section 9.6 18. Hall 2013 Theorems 7.19 and 10.9 19. Hall 2013 Proposition 7.22 20. Hall 2013 Proposition 7.24 References • Akhiezer, N. I.; Glazman, I. M. (1981), Theory of Linear Operators in Hilbert Space, Two volumes, Pitman, ISBN 9780486318653 • Berezin, F. A.; Shubin, M. A. (1991), The Schrödinger Equation, Kluwer • Carey, R. W.; Pincus, J. D. (May 1974). "An Invariant for Certain Operator Algebras". Proceedings of the National Academy of Sciences. 71 (5): 1952–1956. Bibcode:1974PNAS...71.1952C. doi:10.1073/pnas.71.5.1952. PMC 388361. PMID 16592156. • Carey, R. W.; Pincus, J. D. (1973). "The structure of intertwining isometries". Indiana University Mathematics Journal. 7 (22): 679–703. doi:10.1512/iumj.1973.22.22056. • Griffel, D. H. (2002). Applied functional analysis. Mineola, N.Y: Dover. ISBN 0-486-42258-5. OCLC 49250076. • Hall, B. C. (2013), Quantum Theory for Mathematicians, Graduate Texts in Mathematics, vol. 267, Springer, ISBN 978-1461471158 • Kato, T. (1966), Perturbation Theory for Linear Operators, New York: Springer • Moretti, V. (2018), Spectral Theory and Quantum Mechanics:Mathematical Foundations of Quantum Theories, Symmetries and Introduction to the Algebraic Formulation, Springer-Verlag, ISBN 978-3-319-70706-8 • Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. • Reed, M.; Simon, B. (1972), Methods of Mathematical Physics, Vol 2, Academic Press • Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. • Teschl, G. (2009), Mathematical Methods in Quantum Mechanics; With Applications to Schrödinger Operators, Providence: American Mathematical Society • Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. • Yosida, K. (1965), Functional Analysis, Academic Press Functional analysis (topics – glossary) Spaces • Banach • Besov • Fréchet • Hilbert • Hölder • Nuclear • Orlicz • Schwartz • Sobolev • Topological vector Properties • Barrelled • Complete • Dual (Algebraic/Topological) • Locally convex • Reflexive • Reparable Theorems • Hahn–Banach • Riesz representation • Closed graph • Uniform boundedness principle • Kakutani fixed-point • Krein–Milman • Min–max • Gelfand–Naimark • Banach–Alaoglu Operators • Adjoint • Bounded • Compact • Hilbert–Schmidt • Normal • Nuclear • Trace class • Transpose • Unbounded • Unitary Algebras • Banach algebra • C-algebra • Spectrum of a C-algebra • Operator algebra • Group algebra of a locally compact group • Von Neumann algebra Open problems • Invariant subspace problem • Mahler's conjecture Applications • Hardy space • Spectral theory of ordinary differential equations • Heat kernel • Index theorem • Calculus of variations • Functional calculus • Integral operator • Jones polynomial • Topological quantum field theory • Noncommutative geometry • Riemann hypothesis • Distribution (or Generalized functions) Advanced topics • Approximation property • Balanced set • Choquet theory • Weak topology • Banach–Mazur distance • Tomita–Takesaki theory • Mathematics portal • Category • Commons Hilbert spaces Basic concepts • Adjoint • Inner product and L-semi-inner product • Hilbert space and Prehilbert space • Orthogonal complement • Orthonormal basis Main results • Bessel's inequality • Cauchy–Schwarz inequality • Riesz representation Other results • Hilbert projection theorem • Parseval's identity • Polarization identity (Parallelogram law) Maps • Compact operator on Hilbert space • Densely defined • Hermitian form • Hilbert–Schmidt • Normal • Self-adjoint • Sesquilinear form • Trace class • Unitary Examples • Cn(K) with K compact & n<∞ • Segal–Bargmann F Authority control: National • Israel • United States	Wikipedia
Self-avoiding walk In mathematics, a self-avoiding walk (SAW) is a sequence of moves on a lattice (a lattice path) that does not visit the same point more than once. This is a special case of the graph theoretical notion of a path. A self-avoiding polygon (SAP) is a closed self-avoiding walk on a lattice. Very little is known rigorously about the self-avoiding walk from a mathematical perspective, although physicists have provided numerous conjectures that are believed to be true and are strongly supported by numerical simulations. In computational physics, a self-avoiding walk is a chain-like path in R2 or R3 with a certain number of nodes, typically a fixed step length and has the property that it doesn't cross itself or another walk. A system of SAWs satisfies the so-called excluded volume condition. In higher dimensions, the SAW is believed to behave much like the ordinary random walk. SAWs and SAPs play a central role in the modeling of the topological and knot-theoretic behavior of thread- and loop-like molecules such as proteins. Indeed, SAWs may have first been introduced by the chemist Paul Flory[1] in order to model the real-life behavior of chain-like entities such as solvents and polymers, whose physical volume prohibits multiple occupation of the same spatial point. SAWs are fractals. For example, in d = 2 the fractal dimension is 4/3, for d = 3 it is close to 5/3 while for d ≥ 4 the fractal dimension is 2. The dimension is called the upper critical dimension above which excluded volume is negligible. A SAW that does not satisfy the excluded volume condition was recently studied to model explicit surface geometry resulting from expansion of a SAW.[2] The properties of SAWs cannot be calculated analytically, so numerical simulations are employed. The pivot algorithm is a common method for Markov chain Monte Carlo simulations for the uniform measure on n-step self-avoiding walks. The pivot algorithm works by taking a self-avoiding walk and randomly choosing a point on this walk, and then applying symmetrical transformations (rotations and reflections) on the walk after the nth step to create a new walk. Calculating the number of self-avoiding walks in any given lattice is a common computational problem. There is currently no known formula, although there are rigorous methods of approximation.[3][4] Universality One of the phenomena associated with self-avoiding walks and statistical physics models in general is the notion of universality, that is, independence of macroscopic observables from microscopic details, such as the choice of the lattice. One important quantity that appears in conjectures for universal laws is the connective constant, defined as follows. Let cn denote the number of n-step self-avoiding walks. Since every (n + m)-step self avoiding walk can be decomposed into an n-step self-avoiding walk and an m-step self-avoiding walk, it follows that cn+m ≤ cncm. Therefore, the sequence {log cn} is subadditive and we can apply Fekete's lemma to show that the following limit exists: $\mu =\lim _{n\to \infty }c_{n}^{\frac {1}{n}}.$ μ is called the connective constant, since cn depends on the particular lattice chosen for the walk so does μ. The exact value of μ is only known for the hexagonal lattice, where it is equal to:[5] ${\sqrt {2+{\sqrt {2}}}}.$ For other lattices, μ has only been approximated numerically, and is believed not to even be an algebraic number. It is conjectured that[6] $c_{n}\approx \mu ^{n}n^{\frac {11}{32}}$ as n → ∞, where μ depends on the lattice, but the power law correction $n^{\frac {11}{32}}$ does not; in other words, this law is believed to be universal. On networks Self-avoiding walks have also been studied in the context of network theory.[7] In this context, it is customary to treat the SAW as a dynamical process, such that in every time-step a walker randomly hops between neighboring nodes of the network. The walk ends when the walker reaches a dead-end state, such that it can no longer progress to newly un-visited nodes. It was recently found that on Erdős–Rényi networks, the distribution of path lengths of such dynamically grown SAWs can be calculated analytically, and follows the Gompertz distribution.[8] Limits Consider the uniform measure on n-step self-avoiding walks in the full plane. It is currently unknown whether the limit of the uniform measure as n → ∞ induces a measure on infinite full-plane walks. However, Harry Kesten has shown that such a measure exists for self-avoiding walks in the half-plane. One important question involving self-avoiding walks is the existence and conformal invariance of the scaling limit, that is, the limit as the length of the walk goes to infinity and the mesh of the lattice goes to zero. The scaling limit of the self-avoiding walk is conjectured to be described by Schramm–Loewner evolution with parameter κ = 8/3. See also • Critical phenomena – Physics associated with critical points • Hamiltonian path – Path in a graph that visits each vertex exactly once • Knight's tour – Mathematical problem set on a chessboard • Random walk – Mathematical formalization of a path that consists of a succession of random steps • Snake – Video game genre • Universality – Properties of systems that are independent of the dynamical detailsPages displaying wikidata descriptions as a fallback References 1. P. Flory (1953). Principles of Polymer Chemistry. Cornell University Press. p. 672. ISBN 9780801401343. 2. A. Bucksch; G. Turk; J.S. Weitz (2014). "The Fiber Walk: A Model of Tip-Driven Growth with Lateral Expansion". PLOS ONE. 9 (1): e85585. arXiv:1304.3521. Bibcode:2014PLoSO...985585B. doi:10.1371/journal.pone.0085585. PMC 3899046. PMID 24465607. 3. Hayes B (Jul–Aug 1998). "How to Avoid Yourself" (PDF). American Scientist. 86 (4): 314. doi:10.1511/1998.31.3301. 4. Liśkiewicz M; Ogihara M; Toda S (July 2003). "The complexity of counting self-avoiding walks in subgraphs of two-dimensional grids and hypercubes". Theoretical Computer Science. 304 (1–3): 129–56. doi:10.1016/S0304-3975(03)00080-X. 5. Duminil-Copin, Hugo; Smirnov, Stanislav (1 May 2012). "The connective constant of the honeycomb lattice equals sqrt(2+sqrt 2)". Annals of Mathematics. 175 (3): 1653–1665. arXiv:1007.0575. doi:10.4007/annals.2012.175.3.14. S2CID 59164280. 6. Lawler, Gregory F.; Schramm, Oded; Werner, Wendelin (2004). "On the scaling limit of planar self-avoiding walk". Proceedings of Symposia in Pure Mathematics. American Mathematical Society. 72 (2): 339–364. arXiv:math/0204277. doi:10.1090/pspum/072.2/2112127. ISBN 0-8218-3638-2. S2CID 16710180. 7. Carlos P. Herrero (2005). "Self-avoiding walks on scale-free networks". Phys. Rev. E. 71 (3): 1728. arXiv:cond-mat/0412658. Bibcode:2005PhRvE..71a6103H. doi:10.1103/PhysRevE.71.016103. PMID 15697654. S2CID 2707668. 8. Tishby, I.; Biham, O.; Katzav, E. (2016). "The distribution of path lengths of self avoiding walks on Erdős–Rényi networks". Journal of Physics A: Mathematical and Theoretical. 49 (28): 285002. arXiv:1603.06613. Bibcode:2016JPhA...49B5002T. doi:10.1088/1751-8113/49/28/285002. S2CID 119182848. Further reading 1. Madras, N.; Slade, G. (1996). The Self-Avoiding Walk. Birkhäuser. ISBN 978-0-8176-3891-7. 2. Lawler, G. F. (1991). Intersections of Random Walks. Birkhäuser. ISBN 978-0-8176-3892-4. 3. Madras, N.; Sokal, A. D. (1988). "The pivot algorithm – A highly efficient Monte-Carlo method for the self-avoiding walk". Journal of Statistical Physics. 50 (1–2): 109–186. Bibcode:1988JSP....50..109M. doi:10.1007/bf01022990. S2CID 123272694. 4. Fisher, M. E. (1966). "Shape of a self-avoiding walk or polymer chain". Journal of Chemical Physics. 44 (2): 616–622. Bibcode:1966JChPh..44..616F. doi:10.1063/1.1726734. External links • OEIS sequence A007764 (Number of nonintersecting (or self-avoiding) rook paths joining opposite corners of an n X n grid)—the number of self-avoiding paths joining opposite corners of an N × N grid, for N from 0 to 12. Also includes an extended list up to N = 21. • Weisstein, Eric W. "Self-Avoiding Walk". MathWorld. • Java applet of a 2D self-avoiding walk • Generic python implementation to simulate SAWs and expanding FiberWalks on a square lattices in n-dimensions. • Norris software to generate SAWs on the Diamond cubic. 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Self-complementary graph In the mathematical field of graph theory, a self-complementary graph is a graph which is isomorphic to its complement. The simplest non-trivial self-complementary graphs are the 4-vertex path graph and the 5-vertex cycle graph. There is no known characterization of self-complementary graphs. Examples Every Paley graph is self-complementary.[1] For example, the 3 × 3 rook's graph (the Paley graph of order nine) is self-complementary, by a symmetry that keeps the center vertex in place but exchanges the roles of the four side midpoints and four corners of the grid.[2] All strongly regular self-complementary graphs with fewer than 37 vertices are Paley graphs; however, there are strongly regular graphs on 37, 41, and 49 vertices that are not Paley graphs.[3] The Rado graph is an infinite self-complementary graph.[4] Properties An n-vertex self-complementary graph has exactly half number of edges of the complete graph, i.e., n(n – 1)/4 edges, and (if there is more than one vertex) it must have diameter either 2 or 3.[1] Since n(n – 1) must be divisible by 4, n must be congruent to 0 or 1 mod 4; for instance, a 6-vertex graph cannot be self-complementary. Computational complexity The problems of checking whether two self-complementary graphs are isomorphic and of checking whether a given graph is self-complementary are polynomial-time equivalent to the general graph isomorphism problem.[5] References 1. Sachs, Horst (1962), "Über selbstkomplementäre Graphen", Publicationes Mathematicae Debrecen, 9: 270–288, MR 0151953. 2. Shpectorov, S. (1998), "Complementary l1-graphs", Discrete Mathematics, 192 (1–3): 323–331, doi:10.1016/S0012-365X(98)0007X-1, MR 1656740. 3. Rosenberg, I. G. (1982), "Regular and strongly regular selfcomplementary graphs", Theory and practice of combinatorics, North-Holland Math. Stud., vol. 60, Amsterdam: North-Holland, pp. 223–238, MR 0806985. 4. Cameron, Peter J. (1997), "The random graph", The mathematics of Paul Erdős, II, Algorithms Combin., vol. 14, Berlin: Springer, pp. 333–351, arXiv:1301.7544, Bibcode:2013arXiv1301.7544C, MR 1425227. See in particular Proposition 5. 5. Colbourn, Marlene J.; Colbourn, Charles J. (1978), "Graph isomorphism and self-complementary graphs", SIGACT News, 10 (1): 25–29, doi:10.1145/1008605.1008608. External links • Weisstein, Eric W., "Self-Complementary Graph", MathWorld	Wikipedia
Self-concordant function In optimization, a self-concordant function is a function $f:\mathbb {R} \rightarrow \mathbb {R} $ for which $\|f'''(x)\|\leq 2f''(x)^{3/2}$ or, equivalently, a function $f:\mathbb {R} \rightarrow \mathbb {R} $ that, wherever $f''(x)>0$, satisfies $\left\|{\frac {d}{dx}}{\frac {1}{\sqrt {f''(x)}}}\right\|\leq 1$ and which satisfies $f'''(x)=0$ elsewhere. More generally, a multivariate function $f(x):\mathbb {R} ^{n}\rightarrow \mathbb {R} $ is self-concordant if $\left.{\frac {d}{d\alpha }}\nabla ^{2}f(x+\alpha y)\right\|_{\alpha =0}\preceq 2{\sqrt {y^{T}\nabla ^{2}f(x)\,y}}\,\nabla ^{2}f(x)$ or, equivalently, if its restriction to any arbitrary line is self-concordant.[1] History As mentioned in the "Bibliography Comments"[2] of their 1994 book,[3] self-concordant functions were introduced in 1988 by Yurii Nesterov[4][5] and further developed with Arkadi Nemirovski.[6] As explained in[7] their basic observation was that the Newton method is affine invariant, in the sense that if for a function $f(x)$ we have Newton steps $x_{k+1}=x_{k}-[f''(x_{k})]^{-1}f'(x_{k})$ then for a function $\phi (y)=f(Ay)$ where $A$ is a non-degenerate linear transformation, starting from $y_{0}=A^{-1}x_{0}$ we have the Newton steps $y_{k}=A^{-1}x_{k}$ which can be shown recursively $y_{k+1}=y_{k}-[\phi ''(y_{k})]^{-1}\phi '(y_{k})=y_{k}-[A^{T}f''(Ay_{k})A]^{-1}A^{T}f'(Ay_{k})=A^{-1}x_{k}-A^{-1}[f''(x_{k})]^{-1}f'(x_{k})=A^{-1}x_{k+1}$. However, the standard analysis of the Newton method supposes that the Hessian of $f$ is Lipschitz continuous, that is $\\|f''(x)-f''(y)\\|\leq M\\|x-y\\|$ for some constant $M$. If we suppose that $f$ is 3 times continuously differentiable, then this is equivalent to $\|\langle f'''(x)[u]v,v\rangle \|\leq M\\|u\\|\\|v\\|^{2}$for all $u,v\in \mathbb {R} ^{n}$ where $f'''(x)[u]=\lim _{\alpha \to 0}\alpha ^{-1}[f''(x+\alpha u)-f''(x)]$ . Then the left hand side of the above inequality is invariant under the affine transformation $f(x)\to \phi (y)=f(Ay),u\to A^{-1}u,v\to A^{-1}v$, however the right hand side is not. The authors note that the right hand side can be made also invariant if we replace the Euclidean metric by the scalar product defined by the Hessian of $f$ defined as $\\|w\\|_{f''(x)}=\langle f''(x)w,w\rangle ^{1/2}$ for $w\in \mathbb {R} ^{n}$. They then arrive at the definition of a self concordant function as $\|\langle f'''(x)[u]u,u\rangle \|\leq M\langle f''(x)u,u\rangle ^{3/2}$. Properties Linear combination If $f_{1}$ and $f_{2}$ are self-concordant with constants $M_{1}$ and $M_{2}$ and $\alpha ,\beta >0$, then $\alpha f_{1}+\beta f_{2}$ is self-concordant with constant $\max(\alpha ^{-1/2}M_{1},\beta ^{-1/2}M_{2})$. Affine transformation If $f$ is self-concordant with constant $M$ and $Ax+b$ is an affine transformation of $\mathbb {R} ^{n}$, then $\phi (x)=f(Ax+b)$ is also self-concordant with parameter $M$. Convex conjugate If $f$ is self-concordant, then its convex conjugate $f^{}$ is also self-concordant.[8][9] Non-singular Hessian If $f$ is self-concordant and the domain of $f$ contains no straight line (infinite in both directions), then $f''$ is non-singular. Conversely, if for some $x$ in the domain of $f$ and $u\in \mathbb {R} ^{n},u\neq 0$ we have $\langle f''(x)u,u\rangle =0$, then $\langle f''(x+\alpha u)u,u\rangle =0$ for all $\alpha $ for which $x+\alpha u$ is in the domain of $f$ and then $f(x+\alpha u)$ is linear and cannot have a maximum so all of $x+\alpha u,\alpha \in \mathbb {R} $ is in the domain of $f$. We note also that $f$ cannot have a minimum inside its domain. Applications Among other things, self-concordant functions are useful in the analysis of Newton's method. Self-concordant barrier functions are used to develop the barrier functions used in interior point methods for convex and nonlinear optimization. The usual analysis of the Newton method would not work for barrier functions as their second derivative cannot be Lipschitz continuous, otherwise they would be bounded on any compact subset of $\mathbb {R} ^{n}$. Self-concordant barrier functions • are a class of functions that can be used as barriers in constrained optimization methods • can be minimized using the Newton algorithm with provable convergence properties analogous to the usual case (but these results are somewhat more difficult to derive) • to have both of the above, the usual constant bound on the third derivative of the function (required to get the usual convergence results for the Newton method) is replaced by a bound relative to the Hessian Minimizing a self-concordant function A self-concordant function may be minimized with a modified Newton method where we have a bound on the number of steps required for convergence. We suppose here that $f$ is a standard self-concordant function, that is it is self-concordant with parameter $M=2$. We define the Newton decrement $\lambda _{f}(x)$ of $f$ at $x$ as the size of the Newton step $[f''(x)]^{-1}f'(x)$ in the local norm defined by the Hessian of $f$ at $x$ $\lambda _{f}(x)=\langle f''(x)[f''(x)]^{-1}f'(x),[f''(x)]^{-1}f'(x)\rangle ^{1/2}=\langle [f''(x)]^{-1}f'(x),f'(x)\rangle ^{1/2}$ Then for $x$ in the domain of $f$, if $\lambda _{f}(x)<1$ then it is possible to prove that the Newton iterate $x_{+}=x-[f''(x)]^{-1}f'(x)$ will be also in the domain of $f$. This is because, based on the self-concordance of $f$, it is possible to give some finite bounds on the value of $f(x_{+})$. We further have $\lambda _{f}(x_{+})\leq {\Bigg (}{\frac {\lambda _{f}(x)}{1-\lambda _{f}(x)}}{\Bigg )}^{2}$ Then if we have $\lambda _{f}(x)<{\bar {\lambda }}={\frac {3-{\sqrt {5}}}{2}}$ then it is also guaranteed that $\lambda _{f}(x_{+})<\lambda _{f}(x)$, so that we can continue to use the Newton method until convergence. Note that for $\lambda _{f}(x_{+})<\beta $ for some $\beta \in (0,{\bar {\lambda }})$ we have quadratic convergence of $\lambda _{f}$ to 0 as $\lambda _{f}(x_{+})\leq (1-\beta )^{-2}\lambda _{f}(x)^{2}$. This then gives quadratic convergence of $f(x_{k})$ to $f(x^{})$ and of $x$ to $x^{}$, where $x^{}=\arg \min f(x)$, by the following theorem. If $\lambda _{f}(x)<1$ then $\omega (\lambda _{f}(x))\leq f(x)-f(x^{})\leq \omega _{}(\lambda _{f}(x))$ $\omega '(\lambda _{f}(x))\leq \\|x-x^{}\\|_{x}\leq \omega _{}'(\lambda _{f}(x))$ with the following definitions $\omega (t)=t-\log(1+t)$ $\omega _{*}(t)=-t-\log(1-t)$ $\\|u\\|_{x}=\langle f''(x)u,u\rangle ^{1/2}$ If we start the Newton method from some $x_{0}$ with $\lambda _{f}(x_{0})\geq {\bar {\lambda }}$ then we have to start by using a damped Newton method defined by $x_{k+1}=x_{k}-{\frac {1}{1+\lambda _{f}(x_{k})}}[f''(x_{k})]^{-1}f'(x_{k})$ For this it can be shown that $f(x_{k+1})\leq f(x_{k})-\omega (\lambda _{f}(x_{k}))$ with $\omega $ as defined previously. Note that $\omega (t)$ is an increasing function for $t>0$ so that $\omega (t)\geq \omega ({\bar {\lambda }})$ for any $t\geq {\bar {\lambda }}$, so the value of $f$ is guaranteed to decrease by a certain amount in each iteration, which also proves that $x_{k+1}$ is in the domain of $f$. Examples Self-concordant functions • Linear and convex quadratic functions are self-concordant with $M=0$ since their third derivative is zero. • Any function $f(x)=-\log(-g(x))-\log x$ where $g(x)$ is defined and convex for all $x>0$ and verifies $\|g'''(x)\|\leq 3g''(x)/x$, is self concordant on its domain which is $\{x\mid x>0,g(x)<0\}$. Some examples are • $g(x)=-x^{p}$ for $0<p\leq 1$ • $g(x)=-\log x$ • $g(x)=x^{p}$ for $-1\leq p\leq 0$ • $g(x)=(ax+b)^{2}/x$ • for any function $g$ satisfying the conditions, the function $g(x)+ax^{2}+bx+c$ with $a\geq 0$ also satisfies the conditions Functions that are not self-concordant • $f(x)=e^{x}$ • $f(x)={\frac {1}{x^{p}}},x>0,p>0$ • $f(x)=\|x^{p}\|,p>2$ Self-concordant barriers • positive half-line $\mathbb {R} _{+}$: $f(x)=-\log x$ with domain $x>0$ is a self-concordant barrier with $M=1$. • positive orthant $\mathbb {R} _{+}^{n}$: $f(x)=-\sum _{i=1}^{n}\log x_{i}$ with $M=n$ • the logarithmic barrier $f(x)=-\log \phi (x)$ for the quadratic region defined by $\phi (x)>0$ where $\phi (x)=\alpha +\langle a,x\rangle -{\frac {1}{2}}\langle Ax,x\rangle $ where $A=A^{T}\geq 0$ is a positive semi-definite symmetric matrix self-concordant for $M=2$ • second order cone $\{(x,y)\in \mathbb {R} ^{n-1}\times \mathbb {R} \mid \\|x\\|\leq y\}$: $f(x,y)=-\log(y^{2}-x^{T}x)$ • semi-definite cone $A=A^{T}\geq 0$: $f(A)=-\log \det A$ • exponential cone $\{(x,y,z)\in \mathbb {R} ^{3}\mid ye^{x/y}\leq z,y>0\}$: $f(x,y,z)=-\log(y\log(z/y)-x)-\log z-\log y$ • power cone $\{(x_{1},x_{2},y)\in \mathbb {R} _{+}^{2}\times \mathbb {R} \mid \|y\|\leq x_{1}^{\alpha }x_{2}^{1-\alpha }\}$: $f(x_{1},x_{2},y)=-\log(x_{1}^{2\alpha }x_{2}^{2(1-\alpha )}-y^{2})-\log x_{1}-\log x_{2}$ References 1. Boyd, Stephen P.; Vandenberghe, Lieven (2004). Convex Optimization (PDF). Cambridge University Press. ISBN 978-0-521-83378-3. Retrieved October 15, 2011. 2. Nesterov, Yurii; Nemirovskii, Arkadii (January 1994). Interior-Point Polynomial Algorithms in Convex Programming (Bibliography Comments). Society for Industrial and Applied Mathematics. doi:10.1137/1.9781611970791.bm. ISBN 978-0-89871-319-0. 3. Nesterov, I︠U︡. E. (1994). Interior-point polynomial algorithms in convex programming. Nemirovskiĭ, Arkadiĭ Semenovich. Philadelphia: Society for Industrial and Applied Mathematics. ISBN 0-89871-319-6. OCLC 29310677. 4. Yu. E. NESTEROV, Polynomial time methods in linear and quadratic programming, Izvestija AN SSSR, Tekhnitcheskaya kibernetika, No. 3, 1988, pp. 324-326. (In Russian.) 5. Yu. E. NESTEROV, Polynomial time iterative methods in linear and quadratic programming, Voprosy kibernetiki, Moscow,1988, pp. 102-125. (In Russian.) 6. Y.E. Nesterov and A.S. Nemirovski, Self–concordant functions and polynomial–time methods in convex programming, Technical report, Central Economic and Mathematical Institute, USSR Academy of Science, Moscow, USSR, 1989. 7. Nesterov, I︠U︡. E. Introductory lectures on convex optimization : a basic course. Boston. ISBN 978-1-4419-8853-9. OCLC 883391994. 8. Nesterov, Yurii; Nemirovskii, Arkadii (1994). "nterior-Point Polynomial Algorithms in Convex Programming". Studies in Applied and Numerical Mathematics. doi:10.1137/1.9781611970791. ISBN 978-0-89871-319-0. 9. Sun, Tianxiao; Tran-Dinh, Quoc (2018). "Generalized Self-Concordant Functions: A Recipe for Newton-Type Methods". Mathematical Programming: Proposition 6. arXiv:1703.04599.	Wikipedia
Consistency In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction.[1] The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent if it has a model, i.e., there exists an interpretation under which all formulas in the theory are true. This is the sense used in traditional Aristotelian logic, although in contemporary mathematical logic the term satisfiable is used instead. The syntactic definition states a theory $T$ is consistent if there is no formula $\varphi $ such that both $\varphi $ and its negation $\lnot \varphi $ are elements of the set of consequences of $T$. Let $A$ be a set of closed sentences (informally "axioms") and $\langle A\rangle $ the set of closed sentences provable from $A$ under some (specified, possibly implicitly) formal deductive system. The set of axioms $A$ is consistent when $\varphi ,\lnot \varphi \in \langle A\rangle $ for no formula $\varphi $.[2] If there exists a deductive system for which these semantic and syntactic definitions are equivalent for any theory formulated in a particular deductive logic, the logic is called complete. The completeness of the sentential calculus was proved by Paul Bernays in 1918[3] and Emil Post in 1921,[4] while the completeness of predicate calculus was proved by Kurt Gödel in 1930,[5] and consistency proofs for arithmetics restricted with respect to the induction axiom schema were proved by Ackermann (1924), von Neumann (1927) and Herbrand (1931).[6] Stronger logics, such as second-order logic, are not complete. A consistency proof is a mathematical proof that a particular theory is consistent.[7] The early development of mathematical proof theory was driven by the desire to provide finitary consistency proofs for all of mathematics as part of Hilbert's program. Hilbert's program was strongly impacted by the incompleteness theorems, which showed that sufficiently strong proof theories cannot prove their consistency (provided that they are consistent). Although consistency can be proved using model theory, it is often done in a purely syntactical way, without any need to reference some model of the logic. The cut-elimination (or equivalently the normalization of the underlying calculus if there is one) implies the consistency of the calculus: since there is no cut-free proof of falsity, there is no contradiction in general. Consistency and completeness in arithmetic and set theory In theories of arithmetic, such as Peano arithmetic, there is an intricate relationship between the consistency of the theory and its completeness. A theory is complete if, for every formula φ in its language, at least one of φ or ¬φ is a logical consequence of the theory. Presburger arithmetic is an axiom system for the natural numbers under addition. It is both consistent and complete. Gödel's incompleteness theorems show that any sufficiently strong recursively enumerable theory of arithmetic cannot be both complete and consistent. Gödel's theorem applies to the theories of Peano arithmetic (PA) and primitive recursive arithmetic (PRA), but not to Presburger arithmetic. Moreover, Gödel's second incompleteness theorem shows that the consistency of sufficiently strong recursively enumerable theories of arithmetic can be tested in a particular way. Such a theory is consistent if and only if it does not prove a particular sentence, called the Gödel sentence of the theory, which is a formalized statement of the claim that the theory is indeed consistent. Thus the consistency of a sufficiently strong, recursively enumerable, consistent theory of arithmetic can never be proven in that system itself. The same result is true for recursively enumerable theories that can describe a strong enough fragment of arithmetic—including set theories such as Zermelo–Fraenkel set theory (ZF). These set theories cannot prove their own Gödel sentence—provided that they are consistent, which is generally believed. Because consistency of ZF is not provable in ZF, the weaker notion relative consistency is interesting in set theory (and in other sufficiently expressive axiomatic systems). If T is a theory and A is an additional axiom, T + A is said to be consistent relative to T (or simply that A is consistent with T) if it can be proved that if T is consistent then T + A is consistent. If both A and ¬A are consistent with T, then A is said to be independent of T. First-order logic Notation In the following context of mathematical logic, the turnstile symbol $\vdash $ means "provable from". That is, $a\vdash b$ reads: b is provable from a (in some specified formal system). Definition • A set of formulas $\Phi $ in first-order logic is consistent (written $\operatorname {Con} \Phi $) if there is no formula $\varphi $ such that $\Phi \vdash \varphi $ and $\Phi \vdash \lnot \varphi $. Otherwise $\Phi $ is inconsistent (written $\operatorname {Inc} \Phi $). • $\Phi $ is said to be simply consistent if for no formula $\varphi $ of $\Phi $, both $\varphi $ and the negation of $\varphi $ are theorems of $\Phi $. • $\Phi $ is said to be absolutely consistent or Post consistent if at least one formula in the language of $\Phi $ is not a theorem of $\Phi $. • $\Phi $ is said to be maximally consistent if $\Phi $ is consistent and for every formula $\varphi $, $\operatorname {Con} (\Phi \cup \{\varphi \})$ implies $\varphi \in \Phi $. • $\Phi $ is said to contain witnesses if for every formula of the form $\exists x\,\varphi $ there exists a term $t$ such that $(\exists x\,\varphi \to \varphi {t \over x})\in \Phi $, where $\varphi {t \over x}$ denotes the substitution of each $x$ in $\varphi $ by a $t$; see also First-order logic. Basic results 1. The following are equivalent: 1. $\operatorname {Inc} \Phi $ 2. For all $\varphi ,\;\Phi \vdash \varphi .$ 2. Every satisfiable set of formulas is consistent, where a set of formulas $\Phi $ is satisfiable if and only if there exists a model ${\mathfrak {I}}$ such that ${\mathfrak {I}}\vDash \Phi $. 3. For all $\Phi $ and $\varphi $: 1. if not $\Phi \vdash \varphi $, then $\operatorname {Con} \left(\Phi \cup \{\lnot \varphi \}\right)$; 2. if $\operatorname {Con} \Phi $ and $\Phi \vdash \varphi $, then $\operatorname {Con} \left(\Phi \cup \{\varphi \}\right)$; 3. if $\operatorname {Con} \Phi $, then $\operatorname {Con} \left(\Phi \cup \{\varphi \}\right)$ or $\operatorname {Con} \left(\Phi \cup \{\lnot \varphi \}\right)$. 4. Let $\Phi $ be a maximally consistent set of formulas and suppose it contains witnesses. For all $\varphi $ and $\psi $: 1. if $\Phi \vdash \varphi $, then $\varphi \in \Phi $, 2. either $\varphi \in \Phi $ or $\lnot \varphi \in \Phi $, 3. $(\varphi \lor \psi )\in \Phi $ if and only if $\varphi \in \Phi $ or $\psi \in \Phi $, 4. if $(\varphi \to \psi )\in \Phi $ and $\varphi \in \Phi $, then $\psi \in \Phi $, 5. $\exists x\,\varphi \in \Phi $ if and only if there is a term $t$ such that $\varphi {t \over x}\in \Phi $. Henkin's theorem Let $S$ be a set of symbols. Let $\Phi $ be a maximally consistent set of $S$-formulas containing witnesses. Define an equivalence relation $\sim $ on the set of $S$-terms by $t_{0}\sim t_{1}$ if $\;t_{0}\equiv t_{1}\in \Phi $, where $\equiv $ denotes equality. Let ${\overline {t}}$ denote the equivalence class of terms containing $t$; and let $T_{\Phi }:=\{\;{\overline {t}}\mid t\in T^{S}\}$ where $T^{S}$ is the set of terms based on the set of symbols $S$. Define the $S$-structure ${\mathfrak {T}}_{\Phi }$ over $T_{\Phi }$, also called the term-structure corresponding to $\Phi $, by: 1. for each $n$-ary relation symbol $R\in S$, define $R^{{\mathfrak {T}}_{\Phi }}{\overline {t_{0}}}\ldots {\overline {t_{n-1}}}$ if $\;Rt_{0}\ldots t_{n-1}\in \Phi ;$ ;} [8] 2. for each $n$-ary function symbol $f\in S$, define $f^{{\mathfrak {T}}_{\Phi }}({\overline {t_{0}}}\ldots {\overline {t_{n-1}}}):={\overline {ft_{0}\ldots t_{n-1}}};$ 3. for each constant symbol $c\in S$, define $c^{{\mathfrak {T}}_{\Phi }}:={\overline {c}}.$ Define a variable assignment $\beta _{\Phi }$ by $\beta _{\Phi }(x):={\bar {x}}$ for each variable $x$. Let ${\mathfrak {I}}_{\Phi }:=({\mathfrak {T}}_{\Phi },\beta _{\Phi })$ be the term interpretation associated with $\Phi $. Then for each $S$-formula $\varphi $: ${\mathfrak {I}}_{\Phi }\vDash \varphi $ if and only if $\;\varphi \in \Phi .$ Sketch of proof There are several things to verify. First, that $\sim $ is in fact an equivalence relation. Then, it needs to be verified that (1), (2), and (3) are well defined. This falls out of the fact that $\sim $ is an equivalence relation and also requires a proof that (1) and (2) are independent of the choice of $t_{0},\ldots ,t_{n-1}$ class representatives. Finally, ${\mathfrak {I}}_{\Phi }\vDash \varphi $ can be verified by induction on formulas. Model theory In ZFC set theory with classical first-order logic,[9] an inconsistent theory $T$ is one such that there exists a closed sentence $\varphi $ such that $T$ contains both $\varphi $ and its negation $\varphi '$. A consistent theory is one such that the following logically equivalent conditions hold 1. $\{\varphi ,\varphi '\}\not \subseteq T$[10] 2. $\varphi '\not \in T\lor \varphi \not \in T$ See also Wikiquote has quotations related to Consistency. • Cognitive dissonance • Equiconsistency • Hilbert's problems • Hilbert's second problem • Jan Łukasiewicz • Paraconsistent logic • ω-consistency • Gentzen's consistency proof • Proof by contradiction Footnotes 1. Tarski 1946 states it this way: "A deductive theory is called consistent or non-contradictory if no two asserted statements of this theory contradict each other, or in other words, if of any two contradictory sentences … at least one cannot be proved," (p. 135) where Tarski defines contradictory as follows: "With the help of the word not one forms the negation of any sentence; two sentences, of which the first is a negation of the second, are called contradictory sentences" (p. 20). This definition requires a notion of "proof". Gödel 1931 defines the notion this way: "The class of provable formulas is defined to be the smallest class of formulas that contains the axioms and is closed under the relation "immediate consequence", i.e., formula c of a and b is defined as an immediate consequence in terms of modus ponens or substitution; cf Gödel 1931, van Heijenoort 1967, p. 601. Tarski defines "proof" informally as "statements follow one another in a definite order according to certain principles … and accompanied by considerations intended to establish their validity [true conclusion for all true premises – Reichenbach 1947, p. 68]" cf Tarski 1946, p. 3. Kleene 1952 defines the notion with respect to either an induction or as to paraphrase) a finite sequence of formulas such that each formula in the sequence is either an axiom or an "immediate consequence" of the preceding formulas; "A proof is said to be a proof of its last formula, and this formula is said to be (formally) provable or be a (formal) theorem" cf Kleene 1952, p. 83. 2. Hodges, Wilfrid (1997). A Shorter Model Theory. New York: Cambridge University Press. p. 37. Let $L$ be a signature, $T$ a theory in $L_{\infty \omega }$ and $\varphi $ a sentence in $L_{\infty \omega }$. We say that $\varphi $ is a consequence of $T$, or that $T$ entails $\varphi $, in symbols $T\vdash \varphi $, if every model of $T$ is a model of $\varphi $. (In particular if $T$ has no models then $T$ entails $\varphi $.) Warning: we don't require that if $T\vdash \varphi $ then there is a proof of $\varphi $ from $T$. In any case, with infinitary languages, it's not always clear what would constitute proof. Some writers use $T\vdash \varphi $ to mean that $\varphi $ is deducible from $T$ in some particular formal proof calculus, and they write $T\models \varphi $ for our notion of entailment (a notation which clashes with our $A\models \varphi $). For first-order logic, the two kinds of entailment coincide by the completeness theorem for the proof calculus in question. We say that $\varphi $ is valid, or is a logical theorem, in symbols $\vdash \varphi $, if $\varphi $ is true in every $L$-structure. We say that $\varphi $ is consistent if $\varphi $ is true in some $L$-structure. Likewise, we say that a theory $T$ is consistent if it has a model. We say that two theories S and T in L infinity omega are equivalent if they have the same models, i.e. if Mod(S) = Mod(T). (Please note the definition of Mod(T) on p. 30 ...) 3. van Heijenoort 1967, p. 265 states that Bernays determined the independence of the axioms of Principia Mathematica, a result not published until 1926, but he says nothing about Bernays proving their consistency. 4. Post proves both consistency and completeness of the propositional calculus of PM, cf van Heijenoort's commentary and Post's 1931 Introduction to a general theory of elementary propositions in van Heijenoort 1967, pp. 264ff. Also Tarski 1946, pp. 134ff. 5. cf van Heijenoort's commentary and Gödel's 1930 The completeness of the axioms of the functional calculus of logic in van Heijenoort 1967, pp. 582ff. 6. cf van Heijenoort's commentary and Herbrand's 1930 On the consistency of arithmetic in van Heijenoort 1967, pp. 618ff. 7. Informally, Zermelo–Fraenkel set theory is ordinarily assumed; some dialects of informal mathematics customarily assume the axiom of choice in addition. 8. This definition is independent of the choice of $t_{i}$ due to the substitutivity properties of $\equiv $ and the maximal consistency of $\Phi $. 9. the common case in many applications to other areas of mathematics as well as the ordinary mode of reasoning of informal mathematics in calculus and applications to physics, chemistry, engineering 10. according to De Morgan's laws References • Gödel, Kurt (1 December 1931). "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I". Monatshefte für Mathematik und Physik. 38 (1): 173–198. doi:10.1007/BF01700692. • Kleene, Stephen (1952). Introduction to Metamathematics. New York: North-Holland. ISBN 0-7204-2103-9. 10th impression 1991. • Reichenbach, Hans (1947). Elements of Symbolic Logic. New York: Dover. ISBN 0-486-24004-5. • Tarski, Alfred (1946). Introduction to Logic and to the Methodology of Deductive Sciences (Second ed.). New York: Dover. ISBN 0-486-28462-X. • van Heijenoort, Jean (1967). From Frege to Gödel: A Source Book in Mathematical Logic. Cambridge, MA: Harvard University Press. ISBN 0-674-32449-8. (pbk.) • "Consistency". The Cambridge Dictionary of Philosophy. • Ebbinghaus, H. D.; Flum, J.; Thomas, W. Mathematical Logic. • Jevons, W. S. (1870). Elementary Lessons in Logic. External links Look up consistency in Wiktionary, the free dictionary. • Mortensen, Chris (2017). "Inconsistent Mathematics". Stanford Encyclopedia of Philosophy. ‌Logical truth ⊤ Functional: • truth value • truth function • ⊨ tautology Formal: • theory • formal proof • theorem Negation • ⊥ false • contradiction • inconsistency Mathematical logic General • Axiom • list • Cardinality • First-order logic • Formal proof • Formal semantics • Foundations of mathematics • Information theory • Lemma • Logical consequence • Model • Theorem • Theory • Type theory Theorems (list) & Paradoxes • Gödel's completeness and incompleteness theorems • Tarski's undefinability • Banach–Tarski paradox • Cantor's theorem, paradox and diagonal argument • Compactness • Halting problem • Lindström's • Löwenheim–Skolem • Russell's paradox Logics Traditional • Classical logic • Logical truth • Tautology • Proposition • Inference • Logical equivalence • Consistency • Equiconsistency • Argument • Soundness • Validity • Syllogism • Square of opposition • Venn diagram Propositional • Boolean algebra • Boolean functions • Logical connectives • Propositional calculus • Propositional formula • Truth tables • Many-valued logic • 3 • Finite • ∞ Predicate • First-order • list • Second-order • Monadic • Higher-order • Free • Quantifiers • Predicate • Monadic predicate calculus Set theory • Set • Hereditary • Class • (Ur-)Element • Ordinal number • Extensionality • Forcing • Relation • Equivalence • Partition • Set operations: • Intersection • Union • Complement • Cartesian product • Power set • Identities Types of Sets • Countable • Uncountable • Empty • Inhabited • Singleton • Finite • Infinite • Transitive • Ultrafilter • Recursive • Fuzzy • Universal • Universe • Constructible • Grothendieck • Von Neumann Maps & Cardinality • Function/Map • Domain • Codomain • Image • In/Sur/Bi-jection • Schröder–Bernstein theorem • Isomorphism • Gödel numbering • Enumeration • Large cardinal • Inaccessible • Aleph number • Operation • Binary Set theories • Zermelo–Fraenkel • Axiom of choice • Continuum hypothesis • General • Kripke–Platek • Morse–Kelley • Naive • New Foundations • Tarski–Grothendieck • Von Neumann–Bernays–Gödel • Ackermann • Constructive Formal systems (list), Language & Syntax • Alphabet • Arity • Automata • Axiom schema • Expression • Ground • Extension • by definition • Conservative • Relation • Formation rule • Grammar • Formula • Atomic • Closed • Ground • Open • Free/bound variable • Language • Metalanguage • Logical connective • ¬ • ∨ • ∧ • → • ↔ • = • Predicate • Functional • Variable • Propositional variable • Proof • Quantifier • ∃ • ! • ∀ • rank • Sentence • Atomic • Spectrum • Signature • String • Substitution • Symbol • Function • Logical/Constant • Non-logical • Variable • Term • Theory • list Example axiomatic systems (list) • of arithmetic: • Peano • second-order • elementary function • primitive recursive • Robinson • Skolem • of the real numbers • Tarski's axiomatization • of Boolean algebras • canonical • minimal axioms • of geometry: • Euclidean: • Elements • Hilbert's • Tarski's • non-Euclidean • Principia Mathematica Proof theory • Formal proof • Natural deduction • Logical consequence • Rule of inference • Sequent calculus • Theorem • Systems • Axiomatic • Deductive • Hilbert • list • Complete theory • Independence (from ZFC) • Proof of impossibility • Ordinal analysis • Reverse mathematics • Self-verifying theories Model theory • Interpretation • Function • of models • Model • Equivalence • Finite • Saturated • Spectrum • Submodel • Non-standard model • of arithmetic • Diagram • Elementary • Categorical theory • Model complete theory • Satisfiability • Semantics of logic • Strength • Theories of truth • Semantic • Tarski's • Kripke's • T-schema • Transfer principle • Truth predicate • Truth value • Type • Ultraproduct • Validity Computability theory • Church encoding • Church–Turing thesis • Computably enumerable • Computable function • Computable set • Decision problem • Decidable • Undecidable • P • NP • P versus NP problem • Kolmogorov complexity • Lambda calculus • Primitive recursive function • Recursion • Recursive set • Turing machine • Type theory Related • Abstract logic • Category theory • Concrete/Abstract Category • Category of sets • History of logic • History of mathematical logic • timeline • Logicism • Mathematical object • Philosophy of mathematics • Supertask Mathematics portal Metalogic and metamathematics • Cantor's theorem • Entscheidungsproblem • Church–Turing thesis • Consistency • Effective method • Foundations of mathematics • of geometry • Gödel's completeness theorem • Gödel's incompleteness theorems • Soundness • Completeness • Decidability • Interpretation • Löwenheim–Skolem theorem • Metatheorem • Satisfiability • Independence • Type–token distinction • Use–mention distinction Authority control: National • Germany	Wikipedia
List of self-intersecting polygons Self-intersecting polygons, crossed polygons, or self-crossing polygons are polygons some of whose edges cross each other. They contrast with simple polygons, whose edges never cross. Some types of self-intersecting polygons are: • the crossed quadrilateral, with four edges • the antiparallelogram, a crossed quadrilateral with alternate edges of equal length • the crossed rectangle, an antiparallelogram whose edges are two opposite sides and the two diagonals of a rectangle, hence having two edges parallel • Star polygons • pentagram, with five edges • heptagram, with seven edges • octagram, with eight edges • enneagram or nonagram, with nine edges • decagram, with ten edges • hendecagram, with eleven edges • dodecagram, with twelve edges • icositetragram, with twenty four edges • 257-gram, with two hundred and fifty seven edges See also • List of regular polytopes and compounds § Stars • Complex polygon	Wikipedia
Self-dissimilarity Self-dissimilarity is a measure of complexity defined in a series of papers by David Wolpert and William G. Macready.[1][2] The degrees of self-dissimilarity between the patterns of a system observed at various scales (e.g. the average matter density of a physical body for volumes at different orders of magnitude) constitute a complexity "signature" of that system. See also • Diversity index • Index of dissimilarity • Jensen–Shannon divergence • Self-similarity • Similarity measure • Variance References 1. Wolpert, David H.; Macready, William (2004). "Self-dissimilarity as a high dimensional complexity measure". In Y. Bar-Yam (ed.). International Conference on Complex Systems (PDF). Perseus books, in press. 2. Wolpert, D.H. & Macready, W.G. (2000). "Self-Dissimilarity: An Empirically Observable Measure of Complexity". In Y. Bar-Yam (ed.). Unifying Themes in Complex Systems (PDF). Perseus books.	Wikipedia
Dual code In coding theory, the dual code of a linear code $C\subset \mathbb {F} _{q}^{n}$ is the linear code defined by $C^{\perp }=\{x\in \mathbb {F} _{q}^{n}\mid \langle x,c\rangle =0\;\forall c\in C\}$ where $\langle x,c\rangle =\sum _{i=1}^{n}x_{i}c_{i}$ is a scalar product. In linear algebra terms, the dual code is the annihilator of C with respect to the bilinear form $\langle \cdot \rangle $. The dimension of C and its dual always add up to the length n: $\dim C+\dim C^{\perp }=n.$ A generator matrix for the dual code is the parity-check matrix for the original code and vice versa. The dual of the dual code is always the original code. Self-dual codes A self-dual code is one which is its own dual. This implies that n is even and dim C = n/2. If a self-dual code is such that each codeword's weight is a multiple of some constant $c>1$, then it is of one of the following four types:[1] • Type I codes are binary self-dual codes which are not doubly even. Type I codes are always even (every codeword has even Hamming weight). • Type II codes are binary self-dual codes which are doubly even. • Type III codes are ternary self-dual codes. Every codeword in a Type III code has Hamming weight divisible by 3. • Type IV codes are self-dual codes over F4. These are again even. Codes of types I, II, III, or IV exist only if the length n is a multiple of 2, 8, 4, or 2 respectively. If a self-dual code has a generator matrix of the form $G=[I_{k}\|A]$, then the dual code $C^{\perp }$ has generator matrix $[-{\bar {A}}^{T}\|I_{k}]$, where $I_{k}$ is the $(n/2)\times (n/2)$ identity matrix and ${\bar {a}}=a^{q}\in \mathbb {F} _{q}$. References 1. Conway, J.H.; Sloane,N.J.A. (1988). Sphere packings, lattices and groups. Grundlehren der mathematischen Wissenschaften. Vol. 290. Springer-Verlag. p. 77. ISBN 0-387-96617-X. • Hill, Raymond (1986). A first course in coding theory. Oxford Applied Mathematics and Computing Science Series. Oxford University Press. p. 67. ISBN 0-19-853803-0. • Pless, Vera (1982). Introduction to the theory of error-correcting codes. Wiley-Interscience Series in Discrete Mathematics. John Wiley & Sons. p. 8. ISBN 0-471-08684-3. • J.H. van Lint (1992). Introduction to Coding Theory. GTM. Vol. 86 (2nd ed.). Springer-Verlag. p. 34. ISBN 3-540-54894-7. External links • MATH32031: Coding Theory - Dual Code - pdf with some examples and explanations	Wikipedia
Configuration (geometry) In mathematics, specifically projective geometry, a configuration in the plane consists of a finite set of points, and a finite arrangement of lines, such that each point is incident to the same number of lines and each line is incident to the same number of points.[1] This article is about points and lines. For incidences of polytopes, see Configuration (polytope). Although certain specific configurations had been studied earlier (for instance by Thomas Kirkman in 1849), the formal study of configurations was first introduced by Theodor Reye in 1876, in the second edition of his book Geometrie der Lage, in the context of a discussion of Desargues' theorem. Ernst Steinitz wrote his dissertation on the subject in 1894, and they were popularized by Hilbert and Cohn-Vossen's 1932 book Anschauliche Geometrie, reprinted in English as Hilbert & Cohn-Vossen (1952). Configurations may be studied either as concrete sets of points and lines in a specific geometry, such as the Euclidean or projective planes (these are said to be realizable in that geometry), or as a type of abstract incidence geometry. In the latter case they are closely related to regular hypergraphs and biregular bipartite graphs, but with some additional restrictions: every two points of the incidence structure can be associated with at most one line, and every two lines can be associated with at most one point. That is, the girth of the corresponding bipartite graph (the Levi graph of the configuration) must be at least six. Notation A configuration in the plane is denoted by (pγ ℓπ), where p is the number of points, ℓ the number of lines, γ the number of lines per point, and π the number of points per line. These numbers necessarily satisfy the equation $p\gamma =\ell \pi \,$ as this product is the number of point-line incidences (flags). Configurations having the same symbol, say (pγ ℓπ), need not be isomorphic as incidence structures. For instance, there exist three different (93 93) configurations: the Pappus configuration and two less notable configurations. In some configurations, p = ℓ and consequently, γ = π. These are called symmetric or balanced configurations[2] and the notation is often condensed to avoid repetition. For example, (93 93) abbreviates to (93). Examples Notable projective configurations include the following: • (11), the simplest possible configuration, consisting of a point incident to a line. Often excluded as being trivial. • (32), the triangle. Each of its three sides meets two of its three vertices, and vice versa. More generally any polygon of n sides forms a configuration of type (n2) • (43 62) and (62 43), the complete quadrangle and complete quadrilateral respectively. • (73), the Fano plane. This configuration exists as an abstract incidence geometry, but cannot be constructed in the Euclidean plane. • (83), the Möbius–Kantor configuration. This configuration describes two quadrilaterals that are simultaneously inscribed and circumscribed in each other. It cannot be constructed in Euclidean plane geometry but the equations defining it have nontrivial solutions in complex numbers. • (93), the Pappus configuration. • (94 123), the Hesse configuration of nine inflection points of a cubic curve in the complex projective plane and the twelve lines determined by pairs of these points. This configuration shares with the Fano plane the property that it contains every line through its points; configurations with this property are known as Sylvester–Gallai configurations due to the Sylvester–Gallai theorem that shows that they cannot be given real-number coordinates.[3] • (103), the Desargues configuration. • (124 163), the Reye configuration. • (125 302), the Schläfli double six, formed by 12 of the 27 lines on a cubic surface • (153), the Cremona–Richmond configuration, formed by the 15 lines complementary to a double six and their 15 tangent planes • (166), the Kummer configuration. • (214), the Grünbaum–Rigby configuration. • (273), the Gray configuration • (354), Danzer's configuration.[4] • (6015), the Klein configuration. Duality of configurations The projective dual of a configuration (pγ ℓπ) is a (ℓπ pγ) configuration in which the roles of "point" and "line" are exchanged. Types of configurations therefore come in dual pairs, except when taking the dual results in an isomorphic configuration. These exceptions are called self-dual configurations and in such cases p = ℓ.[5] The number of (n3) configurations The number of nonisomorphic configurations of type (n3), starting at n = 7, is given by the sequence 1, 1, 3, 10, 31, 229, 2036, 21399, 245342, ... (sequence A001403 in the OEIS) These numbers count configurations as abstract incidence structures, regardless of realizability.[6] As Gropp (1997) discusses, nine of the ten (103) configurations, and all of the (113) and (123) configurations, are realizable in the Euclidean plane, but for each n ≥ 16 there is at least one nonrealizable (n3) configuration. Gropp also points out a long-lasting error in this sequence: an 1895 paper attempted to list all (123) configurations, and found 228 of them, but the 229th configuration, the Gropp configuration, was not discovered until 1988. Constructions of symmetric configurations There are several techniques for constructing configurations, generally starting from known configurations. Some of the simplest of these techniques construct symmetric (pγ) configurations. Any finite projective plane of order n is an ((n2 + n + 1)n + 1) configuration. Let Π be a projective plane of order n. Remove from Π a point P and all the lines of Π which pass through P (but not the points which lie on those lines except for P) and remove a line ℓ not passing through P and all the points that are on line ℓ. The result is a configuration of type ((n2 – 1)n). If, in this construction, the line ℓ is chosen to be a line which does pass through P, then the construction results in a configuration of type ((n2)n). Since projective planes are known to exist for all orders n which are powers of primes, these constructions provide infinite families of symmetric configurations. Not all configurations are realizable, for instance, a (437) configuration does not exist.[7] However, Gropp (1990) has provided a construction which shows that for k ≥ 3, a (pk) configuration exists for all p ≥ 2 ℓk + 1, where ℓk is the length of an optimal Golomb ruler of order k. Unconventional configurations Higher dimensions The concept of a configuration may be generalized to higher dimensions,[8] for instance to points and lines or planes in space. In such cases, the restrictions that no two points belong to more than one line may be relaxed, because it is possible for two points to belong to more than one plane. Notable three-dimensional configurations are the Möbius configuration, consisting of two mutually inscribed tetrahedra, Reye's configuration, consisting of twelve points and twelve planes, with six points per plane and six planes per point, the Gray configuration consisting of a 3×3×3 grid of 27 points and the 27 orthogonal lines through them, and the Schläfli double six, a configuration with 30 points, 12 lines, two lines per point, and five points per line. Topological configurations Configuration in the projective plane that is realized by points and pseudolines is called topological configuration.[2] For instance, it is known that there exists no point-line (194) configurations, however, there exists a topological configuration with these parameters. Configurations of points and circles Another generalization of the concept of a configuration concerns configurations of points and circles, a notable example being the (83 64) Miquel configuration.[2] See also • Perles configuration, a set of 9 points and 9 lines which do not all have equal numbers of incidences to each other Notes 1. In the literature, the terms projective configuration (Hilbert & Cohn-Vossen 1952) and tactical configuration of type (1,1) (Dembowski 1968) are also used to describe configurations as defined here. 2. Grünbaum 2009. 3. Kelly 1986. 4. Grünbaum 2008, Boben, Gévay & Pisanski 2015 5. Coxeter 1999, pp. 106–149 6. Betten, Brinkmann & Pisanski 2000. 7. This configuration would be a projective plane of order 6 which does not exist by the Bruck–Ryser theorem. 8. Gévay 2014. References • Berman, Leah W., "Movable (n4) configurations", The Electronic Journal of Combinatorics, 13 (1): R104. • Betten, A; Brinkmann, G.; Pisanski, T. (2000), "Counting symmetric configurations", Discrete Applied Mathematics, 99 (1–3): 331–338, doi:10.1016/S0166-218X(99)00143-2. • Boben, Marko; Gévay, Gábor; Pisanski, T. (2015), "Danzer's configuration revisited", Advances in Geometry, 15 (4): 393–408. • Coxeter, H.S.M. (1999), "Self-dual configurations and regular graphs", The Beauty of Geometry, Dover, ISBN 0-486-40919-8 • Dembowski, Peter (1968), Finite geometries, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 44, Berlin, New York: Springer-Verlag, ISBN 3-540-61786-8, MR 0233275 • Gévay, Gábor (2014), "Constructions for large point-line (nk) configurations", Ars Mathematica Contemporanea, 7: 175-199. • Gropp, Harald (1990), "On the existence and non-existence of configurations nk", Journal of Combinatorics and Information System Science, 15: 34–48 • Gropp, Harald (1997), "Configurations and their realization", Discrete Mathematics, 174 (1–3): 137–151, doi:10.1016/S0012-365X(96)00327-5. • Grünbaum, Branko (2006), "Configurations of points and lines", in Davis, Chandler; Ellers, Erich W. (eds.), The Coxeter Legacy: Reflections and Projections, American Mathematical Society, pp. 179–225. • Grünbaum, Branko (2008), "Musing on an example of Danzer's", European Journal of Combinatorics, 29: 1910-1918. • Grünbaum, Branko (2009), Configurations of Points and Lines, Graduate Studies in Mathematics, vol. 103, American Mathematical Society, ISBN 978-0-8218-4308-6. • Hilbert, David; Cohn-Vossen, Stephan (1952), Geometry and the Imagination (2nd ed.), Chelsea, pp. 94–170, ISBN 0-8284-1087-9. • Kelly, L. M. (1986), "A resolution of the Sylvester–Gallai problem of J. P. Serre", Discrete and Computational Geometry, 1 (1): 101–104, doi:10.1007/BF02187687. • Pisanski, Tomaž; Servatius, Brigitte (2013), Configurations from a Graphical Viewpoint, Springer, ISBN 9780817683641. External links • Weisstein, Eric W., "Configuration", MathWorld	Wikipedia
Uniform tiling In geometry, a uniform tiling is a tessellation of the plane by regular polygon faces with the restriction of being vertex-transitive. Uniform tilings can exist in both the Euclidean plane and hyperbolic plane. Uniform tilings are related to the finite uniform polyhedra which can be considered uniform tilings of the sphere. Most uniform tilings can be made from a Wythoff construction starting with a symmetry group and a singular generator point inside of the fundamental domain. A planar symmetry group has a polygonal fundamental domain and can be represented by the group name represented by the order of the mirrors in sequential vertices. A fundamental domain triangle is (p q r), and a right triangle (p q 2), where p, q, r are whole numbers greater than 1. The triangle may exist as a spherical triangle, a Euclidean plane triangle, or a hyperbolic plane triangle, depending on the values of p, q and r. There are a number of symbolic schemes for naming these figures, from a modified Schläfli symbol for right triangle domains: (p q 2) → {p, q}. The Coxeter-Dynkin diagram is a triangular graph with p, q, r labeled on the edges. If r = 2, the graph is linear since order-2 domain nodes generate no reflections. The Wythoff symbol takes the 3 integers and separates them by a vertical bar (\|). If the generator point is off the mirror opposite a domain node, it is given before the bar. Finally tilings can be described by their vertex configuration, the sequence of polygons around each vertex. All uniform tilings can be constructed from various operations applied to regular tilings. These operations as named by Norman Johnson are called truncation (cutting vertices), rectification (cutting vertices until edges disappear), and cantellation (cutting edges). Omnitruncation is an operation that combines truncation and cantellation. Snubbing is an operation of alternate truncation of the omnitruncated form. (See Uniform polyhedron#Wythoff construction operators for more details.) Coxeter groups Coxeter groups for the plane define the Wythoff construction and can be represented by Coxeter-Dynkin diagrams: For groups with whole number orders, including: Euclidean plane Orbifold symmetry Coxeter group Coxeter diagram notes Compact 333 (3 3 3) ${\tilde {A}}_{2}$ [3[3]] 3 reflective forms, 1 snub 442 (4 4 2) ${\tilde {B}}_{2}$ [4,4] 5 reflective forms, 1 snub 632 (6 3 2) ${\tilde {G}}_{2}$ [6,3] 7 reflective forms, 1 snub 2222 (∞ 2 ∞ 2) ${\tilde {I}}_{1}$ × ${\tilde {I}}_{1}$ [∞,2,∞] 3 reflective forms, 1 snub Noncompact (frieze) ∞∞ (∞) ${\tilde {I}}_{1}$ [∞] 22∞ (2 2 ∞) ${\tilde {I}}_{1}$ × ${\tilde {A}}_{2}$ [∞,2] 2 reflective forms, 1 snub Hyperbolic plane Orbifold symmetry Coxeter group Coxeter diagram notes Compact pq2 (p q 2) [p,q] 2(p+q) < pq pqr (p q r) [(p,q,r)] pq+pr+qr < pqr Paracompact ∞p2 (p ∞ 2) [p,∞] p>=3 ∞pq (p q ∞) [(p,q,∞)] p,q>=3, p+q>6 ∞∞p (p ∞ ∞) [(p,∞,∞)] p>=3 ∞∞∞ (∞ ∞ ∞) [(∞,∞,∞)] Uniform tilings of the Euclidean plane Further information: List of k-uniform tilings There are symmetry groups on the Euclidean plane constructed from fundamental triangles: (4 4 2), (6 3 2), and (3 3 3). Each is represented by a set of lines of reflection that divide the plane into fundamental triangles. These symmetry groups create 3 regular tilings, and 7 semiregular ones. A number of the semiregular tilings are repeated from different symmetry constructors. A prismatic symmetry group represented by (2 2 2 2) represents by two sets of parallel mirrors, which in general can have a rectangular fundamental domain. It generates no new tilings. A further prismatic symmetry group represented by (∞ 2 2) which has an infinite fundamental domain. It constructs two uniform tilings, the apeirogonal prism and apeirogonal antiprism. The stacking of the finite faces of these two prismatic tilings constructs one non-Wythoffian uniform tiling of the plane. It is called the elongated triangular tiling, composed of alternating layers of squares and triangles. Right angle fundamental triangles: (p q 2) (p q 2) Fund. triangles Parent Truncated Rectified Bitruncated Birectified (dual) Cantellated Omnitruncated (Cantitruncated) Snub Wythoff symbol q \| p 2 2 q \| p 2 \| p q 2 p \| q p \| q 2 p q \| 2 p q 2 \| \| p q 2 Schläfli symbol {p,q} t{p,q} r{p,q} 2t{p,q}=t{q,p} 2r{p,q}={q,p} rr{p,q} tr{p,q} sr{p,q} Coxeter diagram Vertex config. pq q.2p.2p (p.q)2 p. 2q.2q qp p. 4.q.4 4.2p.2q 3.3.p. 3.q Square tiling (4 4 2) 0 {4,4} 4.8.8 4.4.4.4 4.8.8 {4,4} 4.4.4.4 4.8.8 3.3.4.3.4 Hexagonal tiling (6 3 2) 0 {6,3} 3.12.12 3.6.3.6 6.6.6 {3,6} 3.4.6.4 4.6.12 3.3.3.3.6 General fundamental triangles: (p q r) Wythoff symbol (p q r) Fund. triangles q \| p r r q \| p r \| p q r p \| q p \| q r p q \| r p q r \| \| p q r Coxeter diagram Vertex config. (p.q)r r.2p.q.2p (p.r)q q.2r.p. 2r (q.r)p q.2r.p. 2r r.2q.p. 2q 3.r.3.q.3.p Triangular (3 3 3) 0 (3.3)3 3.6.3.6 (3.3)3 3.6.3.6 (3.3)3 3.6.3.6 6.6.6 3.3.3.3.3.3 Non-simplical fundamental domains The only possible fundamental domain in Euclidean 2-space that is not a simplex is the rectangle (∞ 2 ∞ 2), with Coxeter diagram: . All forms generated from it become a square tiling. Uniform tilings of the hyperbolic plane Further information: Uniform tilings in hyperbolic plane There are infinitely many uniform tilings of convex regular polygons on the hyperbolic plane, each based on a different reflective symmetry group (p q r). A sampling is shown here with a Poincaré disk projection. The Coxeter-Dynkin diagram is given in a linear form, although it is actually a triangle, with the trailing segment r connecting to the first node. Further symmetry groups exist in the hyperbolic plane with quadrilateral fundamental domains starting with (2 2 2 3), etc., that can generate new forms. As well there's fundamental domains that place vertices at infinity, such as (∞ 2 3), etc. Right angle fundamental triangles: (p q 2) (p q 2) Fund. triangles Parent Truncated Rectified Bitruncated Birectified (dual) Cantellated Omnitruncated (Cantitruncated) Snub Wythoff symbol q \| p 2 2 q \| p 2 \| p q 2 p \| q p \| q 2 p q \| 2 p q 2 \| \| p q 2 Schläfli symbol t{p,q} t{p,q} r{p,q} 2t{p,q}=t{q,p} 2r{p,q}={q,p} rr{p,q} tr{p,q} sr{p,q} Coxeter diagram Vertex figure pq (q.2p.2p) (p.q.p.q) (p. 2q.2q) qp (p. 4.q.4) (4.2p.2q) (3.3.p. 3.q) (5 4 2) V4.8.10 {5,4} 4.10.10 4.5.4.5 5.8.8 {4,5} 4.4.5.4 4.8.10 3.3.4.3.5 (5 5 2) V4.10.10 {5,5} 5.10.10 5.5.5.5 5.10.10 {5,5} 5.4.5.4 4.10.10 3.3.5.3.5 (7 3 2) V4.6.14 {7,3} 3.14.14 3.7.3.7 7.6.6 {3,7} 3.4.7.4 4.6.14 3.3.3.3.7 (8 3 2) V4.6.16 {8,3} 3.16.16 3.8.3.8 8.6.6 {3,8} 3.4.8.4 4.6.16 3.3.3.3.8 General fundamental triangles (p q r) Wythoff symbol (p q r) Fund. triangles q \| p r r q \| p r \| p q r p \| q p \| q r p q \| r p q r \| \| p q r Coxeter diagram Vertex figure (p.r)q (r.2p.q.2p) (p.q)r (q.2r.p. 2r) (q.r)p (r.2q.p. 2q) (2p.2q.2r) (3.r.3.q.3.p) (4 3 3) V6.6.8 (3.4)3 3.8.3.8 (3.4)3 3.6.4.6 (3.3)4 3.6.4.6 6.6.8 3.3.3.3.3.4 (4 4 3) V6.8.8 (3.4)4 3.8.4.8 (4.4)3 3.6.4.6 (3.4)4 4.6.4.6 6.8.8 3.3.3.4.3.4 (4 4 4) V8.8.8 (4.4)4 4.8.4.8 (4.4)4 4.8.4.8 (4.4)4 4.8.4.8 8.8.8 3.4.3.4.3.4 Expanded lists of uniform tilings There are a number ways the list of uniform tilings can be expanded: 1. Vertex figures can have retrograde faces and turn around the vertex more than once. 2. Star polygon tiles can be included. 3. Apeirogons, {∞}, can be used as tiling faces. 4. Zigzags (apeirogons alternating between two angles) can also be used. 5. The restriction that tiles meet edge-to-edge can be relaxed, allowing additional tilings such as the Pythagorean tiling. Symmetry group triangles with retrogrades include: (4/3 4/3 2) (6 3/2 2) (6/5 3 2) (6 6/5 3) (6 6 3/2) Symmetry group triangles with infinity include: (4 4/3 ∞) (3/2 3 ∞) (6 6/5 ∞) (3 3/2 ∞) Branko Grünbaum and G. C. Shephard, in the 1987 book Tilings and patterns, in section 12.3 enumerates a list of 25 uniform tilings, including the 11 convex forms, and adds 14 more they call hollow tilings which included the first two expansions above, star polygon faces and vertex figures.[1] H.S.M. Coxeter, M. S. Longuet-Higgins, and J. C. P. Miller, in the 1954 paper 'Uniform polyhedra', in Table 8: Uniform Tessellations, use the first three expansions and enumerates a total of 38 uniform tilings. If a tiling made of 2 apeirogons is also counted, the total can be considered 39 uniform tilings. In 1981, Grünbaum, Miller, and Shephard in their paper Uniform Tilings with Hollow Tiles listed 25 tilings using the first two expansions and 28 more when the third is added (making 53 using Coxeter et al.'s definition). When the fourth is added, they list an additional 23 uniform tilings and 10 families (8 depending on continuous parameters and 2 on discrete parameters).[2] Besides the 11 convex solutions, the 28 uniform star tilings listed by Coxeter et al., grouped by shared edge graphs, are shown below, followed by 15 more listed by Grünbaum et al. that meet Coxeter et al.'s definition but were missed by them. This set is not proved complete. By "2.25" is meant tiling 25 in Grünbaum et al.'s table 2 from 1981. The following three tilings are exceptional in that there is only finitely many of one face type: two apeirogons in each. Sometimes the order-2 apeirogonal tiling is not included, as its two faces meet at more than one edge. Frieze group symmetry McNeill[3]DiagramVertex Config WythoffSymmetryNotes I1∞.∞p1m1(Two half-plane tiles, order-2 apeirogonal tiling) I24.4.∞∞ 2 \| 2p1m1Apeirogonal prism I33.3.3.∞\| 2 2 ∞p11gApeirogonal antiprism For clarity, apeirogons are not coloured from here onward. A set of polygons round one vertex is highlighted. McNeill only lists tilings given by Coxeter et al. (1954). The eleven convex uniform tilings have been repeated for reference. Wallpaper group symmetry McNeill[3]Grünbaum et al 1981[2]Edge diagram HighlightedVertex Config WythoffSymmetry Convex1.9 4.4.4.44 \| 2 4p4m I42.14 4.∞.4/3.∞ 4.∞.-4.∞ 4/3 4 \| ∞p4m Convex1.24 6.6.63 \| 2 6p6m Convex1.25 3.3.3.3.3.36 \| 2 3p6m I52.26 (3.∞.3.∞.3.∞)/23/2 \| 3 ∞p3m1 Convex1.23 3.6.3.62 \| 3 6p6m I62.25 6.∞.6/5.∞ 6.∞.-6.∞ 6/5 6 \| ∞p6m I72.24 ∞.3.∞.3/2 3.∞.-3.∞ 3/2 3 \| ∞p6m Convex1.14 3.4.6.43 6 \| 2p6m 11.15 3/2.12.6.12 -3.12.6.12 3/2 6 \| 6p6m 1.16 4.12.4/3.12/11 4.12.-4.-12 2 6 (3/2 6/2) \|p6m Convex1.5 4.8.82 4 \| 4p4m 22.7 4.8/3.∞.8/34 ∞ \| 4/3p4m 1.7 8/3.8.8/5.8/7 8.8/3.-8.-8/3 4/3 4 (4/2 ∞/2) \|p4m 2.6 8.4/3.8.∞ -4.8.∞.8 4/3 ∞ \| 4p4m Convex1.20 3.12.122 3 \| 6p6m 32.17 6.12/5.∞.12/56 ∞ \| 6/5p6m 1.21 12/5.12.12/7.12/11 12.12/5.-12.-12/5 6/5 6 (6/2 ∞/2) \|p6m 2.16 12.6/5.12.∞ -6.12.∞.12 6/5 ∞ \| 6p6m 41.18 12/5.3.12/5.6/5 3.12/5.-6.12/5 3 6 \| 6/5p6m 1.19 12/5.4.12/7.4/3 4.12/5.-4.-12/5 2 6/5 (3/2 6/2) \|p6m 1.17 4.3/2.4.6/5 3.-4.6.-4 3/2 6 \| 2p6m 52.5 8.8/3.∞4/3 4 ∞ \|p4m 62.15 12.12/5.∞6/5 6 ∞ \|p6m 71.6 8.4/3.8/5 4.-8.8/3 2 4/3 4 \|p4m Convex1.11 4.6.122 3 6 \|p6m 81.13 6.4/3.12/7 4.-6.12/5 2 3 6/5 \|p6m 91.12 12.6/5.12/7 6.-12.12/5 3 6/5 6 \|p6m 101.8 4.8/5.8/5 -4.8/3.8/3 2 4 \| 4/3p4m 111.22 12/5.12/5.3/2 -3.12/5.12/5 2 3 \| 6/5p6m Convex1.1 3.3.3.4.4non-Wythoffiancmm 121.2 4.4.3/2.3/2.3/2 3.3.3.-4.-4 non-Wythoffiancmm Convex1.3 3.3.4.3.4\| 2 4 4p4g 131.4 4.3/2.4.3/2.3/2 3.3.-4.3.-4 \| 2 4/3 4/3p4g 142.4 3.4.3.4/3.3.∞ 3.4.3.-4.3.∞ \| 4/3 4 ∞p4 Convex1.10 3.3.3.3.6\| 2 3 6p6 2.1 3/2.∞.3/2.∞.3/2.4/3.4/3 3.4.4.3.∞.3.∞ non-Wythoffiancmm 2.2 3/2.∞.3/2.∞.3/2.4.4 3.-4.-4.3.∞.3.∞ non-Wythoffiancmm 2.3 3/2.∞.3/2.4.4.3/2.4/3.4/3 3.4.4.3.-4.-4.3.∞ non-Wythoffianp3 2.8 4.∞.4/3.8/3.8 4.8.8/3.-4.∞ non-Wythoffianp4m 2.9 4.∞.4.8.8/3 -4.8.8/3.4.∞ non-Wythoffianp4m 2.10 4.∞.4/3.8.4/3.8 4.8.-4.8.-4.∞ non-Wythoffianp4m 2.11 4.∞.4/3.8.4/3.8 4.8.-4.8.-4.∞ non-Wythoffianp4g 2.12 4.∞.4/3.8/3.4.8/3 4.8/3.4.8/3.-4.∞ non-Wythoffianp4m 2.13 4.∞.4/3.8/3.4.8/3 4.8/3.4.8/3.-4.∞ non-Wythoffianp4g 2.18 3/2.∞.3/2.4/3.4/3.3/2.4/3.4/3 3.4.4.3.4.4.3.∞ non-Wythoffianp6m 2.19 3/2.∞.3/2.4.4.3/2.4.4 3.-4.-4.3.-4.-4.3.∞ non-Wythoffianp6m 2.20 3/2.∞.3/2.∞.3/2.12/11.6.12/11 3.12.-6.12.3.∞.3.∞ non-Wythoffianp6m 2.21 3/2.∞.3/2.∞.3/2.12.6/5.12 3.-12.6.-12.3.∞.3.∞ non-Wythoffianp6m 2.22 3/2.∞.3/2.∞.3/2.12/7.6/5.12/7 3.12/5.6.12/5.3.∞.3.∞ non-Wythoffianp6m 2.23 3/2.∞.3/2.∞.3/2.12/5.6.12/5 3.-12/5.-6.-12/5.3.∞.3.∞ non-Wythoffianp6m There are two uniform tilings for the vertex figure 4.8.-4.8.-4.∞ (Grünbaum et al. 2.10 and 2.11) and also two uniform tilings for the vertex figure 4.8/3.4.8/3.-4.∞ (Grünbaum et al. 2.12 and 2.13), with different symmetries. There is also a third tiling for each vertex figure that is only pseudo-uniform (vertices come in two symmetry orbits). They use different sets of square faces. Hence, for star Euclidean tilings, the vertex figure does not necessarily determine the tiling.[2] In the pictures below, the included squares with horizontal and vertical edges are marked with a central dot. A single square has edges highlighted.[2] • 2.10 and 2.12 (p4m) • 2.11 and 2.13 (p4g) • Pseudo-uniform The tilings with zigzags are listed below. The notation {∞α} denotes a zigzag with angle 0 < α < π. The apeirogon can be considered as the special case α = π. The symmetries are given for the generic case: there are sometimes special values of α that increase the symmetry. Tilings 3.1 and 3.12 can even become regular; 3.32 already is (it has no free parameters). Sometimes there are special values of α that cause the tiling to degenerate.[2] Tilings with zigzags Grünbaum et al 1981[2]DiagramVertex Config Symmetry 3.1 ∞α.∞β.∞γ α+β+γ=2π p2 3.2 ∞α.∞β.-∞α+β 0<α+β≤π p2 3.3 3.3.∞π-α.-3.∞α+2π/3 0≤α≤π/6 pgg 3.4 3.3.-∞π-α.-3.∞−α+2π/3 0≤α<π/3 pgg 3.5 4.4.∞φ.4.4.-∞φ φ=2 arctan(n/k), nk even, (n,k)=1 drawn for φ=2 arctan 2 pmg 3.6 4.4.∞φ.-4.-4.∞φ φ=2 arctan(n/k), nk even, (n,k)=1 drawn for φ=2 arctan 1/2 pmg 3.7 3.4.4.3.-∞2π/3.-3.-∞2π/3cmm 3.8 3.-4.-4.3.-∞2π/3.-3.-∞2π/3cmm 3.9 4.4.∞π/3.∞.-∞π/3p2 3.10 4.4.∞2π/3.∞.-∞2π/3p2 3.11 ∞.∞α.∞.∞−α 0<α<π cmm 3.12 ∞α.∞π-α.∞α.∞π-α 0<α≤π/2 cmm 3.13 3.∞α.-3.-∞α π/3<α<π p31m 3.14 4.4.∞2π/3.4.4.-∞2π/3p31m 3.15 4.4.∞π/3.-4.-4.-∞π/3p31m 3.16 4.∞α.-4.-∞α 0<α<π, α≠π/2 p4g 3.17 4.-8.∞π/2.∞.-∞π/2.-8cmm 3.18 4.-8.∞π/2.∞.-∞π/2.-8p4 3.19 4.8/3.∞π/2.∞.-∞π/2.8/3cmm 3.20 4.8/3.∞π/2.∞.-∞π/2.8/3p4 3.21 6.-12.∞π/3.∞.-∞π/3.-12p6 3.22 6.-12.∞2π/3.∞.-∞2π/3.-12p6 3.23 6.12/5.∞π/3.∞.-∞π/3.12/5p6 3.24 6.12/5.∞2π/3.∞.-∞2π/3.12/5p6 3.25 3.3.3.∞2π/3.-3.∞2π/3p31m 3.26 3.∞.3.-∞2π/3.-3.-∞2π/3cm 3.27 3.∞.-∞2π/3.∞.-∞2π/3.∞p31m 3.28 3.∞2π/3.∞2π/3.-3.-∞2π/3.-∞2π/3p31m 3.29 ∞.∞π/3.∞π/3.∞.-∞π/3.-∞π/3cmm 3.30 ∞.∞π/3.-∞2π/3.∞.∞2π/3.-∞π/3p2 3.31 ∞.∞2π/3.∞2π/3.∞.-∞2π/3.-∞2π/3cmm 3.32 ∞π/3.∞π/3.∞π/3.∞π/3.∞π/3.∞π/3p6m 3.33 ∞π/3.-∞2π/3.-∞2π/3.∞π/3.-∞2π/3.-∞2π/3cmm The tiling pairs 3.17 and 3.18, as well as 3.19 and 3.20, have identical vertex configurations but different symmetries.[2] Tilings 3.7 through 3.10 have the same edge arrangement as 2.1 and 2.2; 3.17 through 3.20 have the same edge arrangement as 2.10 through 2.13; 3.21 through 3.24 have the same edge arrangement as 2.18 through 2.23; and 3.25 through 3.33 have the same edge arrangement as 1.25 (the regular triangular tiling).[2] Self-dual tilings Tilings can also be self-dual. The square tiling, with Schläfli symbol {4,4}, is self-dual; shown here are two square tilings (red and black), dual to each other. Uniform tilings using star polygons Seeing a star polygon as a nonconvex polygon with twice as many sides allows star polygons, and counting these as regular polygons allows them to be used in a uniform tiling. These polygons are labeled as {Nα} for a isotoxal nonconvex 2N-gon with external dihedral angle α. Its external vertices are labeled as N* α , and internal N** α . This expansion to the definition requires corners with only 2 polygons to not be considered vertices. The tiling is defined by its vertex configuration as a cyclic sequence of convex and nonconvex polygons around every vertex. There are 4 such uniform tilings with adjustable angles α, and 18 uniform tilings that only work with specific angles; yielding a total of 22 uniform tilings that use star polygons.[4] All of these tilings are topologically related to the ordinary uniform tilings with convex regular polygons, with 2-valence vertices ignored, and square faces as digons, reduced to a single edge. 4 uniform tilings with star polygons, angle α 3.6* α .6** α Topological 3.12.12 4.4* α .4** α Topological 4.8.8 6.3* α .3** α Topological 6.6.6 3.3* α .3.3** α Topological 3.6.3.6 18 uniform tilings with star polygons 4.6.4* π/6 .6 Topological 4.4.4.4 (8.4* π/4 )2 Topological 4.4.4.4 12.12.4* π/3 Topological 4.8.8 3.3.8* π/12 .4** π/3 .8* π/12 Topological 4.8.8 3.3.8* π/12 .3.4.3.8* π/12 Topological 4.8.8 3.4.8.3.8* π/12 Topological 4.8.8 5.5.4* 4π/10 .5.4* π/10 Topological 3.3.4.3.4 4.6* π/6 .6** π/2 .6* π/6 Topological 6.6.6 (4.6* π/6 )3 Topological 6.6.6 9.9.6* 4π/9 Topological 6.6.6 (6.6* π/3 )2 Topological 3.6.3.6 (12.3* π/6 )2 Topological 3.6.3.6 3.4.6.3.12* π/6 Topological 4.6.12 3.3.3.12* π/6 .3.3.12* π/6 Topological 3.12.12 18.18.3* 2π/9 Topological 3.12.12 3.6.6* π/3 .6 Topological 3.4.6.4 8.3* π/12 .8.6* 5π/12 Topological 3.4.6.4 9.3.9.3* π/9 Topological 3.6.3.6 Uniform tilings using alternating polygons Star polygons of the form {pα} can also represent convex 2p-gons alternating two angles, the simplest being a rhombus {2α}. Allowing these as regular polygons, creates more uniform tilings, with some example below. Examples 3.2.6.2* Topological 3.4.6.4 4.4.4.4 Topological 4.4.4.4 (2* π/6 .2** π/3 )2 Topological 4.4.4.4 2* π/6 .2* π/6 .2 π/3 .2 π/3 Topological 4.4.4.4 4.2* π/6 .4.2** π/3 Topological 4.4.4.4 See also Wikimedia Commons has media related to Uniform tilings. • Wythoff symbol • List of uniform tilings • Uniform tilings in hyperbolic plane • Uniform polytope References 1. Tiles and Patterns, Table 12.3.1 p.640 2. Grünbaum, Branko; Miller, J. C. P.; Shephard, G. C. (1981). "Uniform Tilings with Hollow Tiles". In Davis, Chandler; Grünbaum, Branko; Sherk, F. A. (eds.). The Geometric Vein: The Coxeter Festschrift. Springer. pp. 17–64. ISBN 978-1-4612-5650-2. 3. Jim McNeill 4. Tilings and Patterns Branko Gruenbaum, G.C. Shephard, 1987. 2.5 Tilings using star polygons, pp.82-85. • Norman Johnson Uniform Polytopes, Manuscript (1991) • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966 • Grünbaum, Branko; Shephard, G. C. (1987). Tilings and Patterns. W. H. Freeman and Company. ISBN 0-7167-1193-1. (Star tilings section 12.3) • H. S. M. Coxeter, M. S. Longuet-Higgins, J. C. P. Miller, Uniform polyhedra, Phil. Trans. 1954, 246 A, 401–50 JSTOR 91532 (Table 8) External links • Weisstein, Eric W. "Uniform tessellation". MathWorld. • Uniform Tessellations on the Euclid plane • Tessellations of the Plane • David Bailey's World of Tessellations • k-uniform tilings • n-uniform tilings • Klitzing, Richard. "4D Euclidean tilings". Fundamental convex regular and uniform honeycombs in dimensions 2–9 Space Family ${\tilde {A}}_{n-1}$ ${\tilde {C}}_{n-1}$ ${\tilde {B}}_{n-1}$ ${\tilde {D}}_{n-1}$ ${\tilde {G}}_{2}$ / ${\tilde {F}}_{4}$ / ${\tilde {E}}_{n-1}$ E2 Uniform tiling {3[3]} δ3 hδ3 qδ3 Hexagonal E3 Uniform convex honeycomb {3[4]} δ4 hδ4 qδ4 E4 Uniform 4-honeycomb {3[5]} δ5 hδ5 qδ5 24-cell honeycomb E5 Uniform 5-honeycomb {3[6]} δ6 hδ6 qδ6 E6 Uniform 6-honeycomb {3[7]} δ7 hδ7 qδ7 222 E7 Uniform 7-honeycomb {3[8]} δ8 hδ8 qδ8 133 • 331 E8 Uniform 8-honeycomb {3[9]} δ9 hδ9 qδ9 152 • 251 • 521 E9 Uniform 9-honeycomb {3[10]} δ10 hδ10 qδ10 E10 Uniform 10-honeycomb {3[11]} δ11 hδ11 qδ11 En-1 Uniform (n-1)-honeycomb {3[n]} δn hδn qδn 1k2 • 2k1 • k21 Tessellation Periodic • Pythagorean • Rhombille • Schwarz triangle • Rectangle • Domino • Uniform tiling and honeycomb • Coloring • Convex • Kisrhombille • Wallpaper group • Wythoff Aperiodic • Ammann–Beenker • Aperiodic set of prototiles • List • Einstein problem • Socolar–Taylor • Gilbert • Penrose • Pentagonal • Pinwheel • Quaquaversal • Rep-tile and Self-tiling • Sphinx • Socolar • Truchet Other • Anisohedral and Isohedral • Architectonic and catoptric • Circle Limit III • Computer graphics • Honeycomb • Isotoxal • List • Packing • Problems • Domino • Wang • Heesch's • Squaring • Dividing a square into similar rectangles • Prototile • Conway criterion • Girih • Regular Division of the Plane • Regular grid • Substitution • Voronoi • Voderberg By vertex type Spherical • 2n • 33.n • V33.n • 42.n • V42.n Regular • 2∞ • 36 • 44 • 63 Semi- regular • 32.4.3.4 • V32.4.3.4 • 33.42 • 33.∞ • 34.6 • V34.6 • 3.4.6.4 • (3.6)2 • 3.122 • 42.∞ • 4.6.12 • 4.82 Hyper- bolic • 32.4.3.5 • 32.4.3.6 • 32.4.3.7 • 32.4.3.8 • 32.4.3.∞ • 32.5.3.5 • 32.5.3.6 • 32.6.3.6 • 32.6.3.8 • 32.7.3.7 • 32.8.3.8 • 33.4.3.4 • 32.∞.3.∞ • 34.7 • 34.8 • 34.∞ • 35.4 • 37 • 38 • 3∞ • (3.4)3 • (3.4)4 • 3.4.62.4 • 3.4.7.4 • 3.4.8.4 • 3.4.∞.4 • 3.6.4.6 • (3.7)2 • (3.8)2 • 3.142 • 3.162 • (3.∞)2 • 3.∞2 • 42.5.4 • 42.6.4 • 42.7.4 • 42.8.4 • 42.∞.4 • 45 • 46 • 47 • 48 • 4∞ • (4.5)2 • (4.6)2 • 4.6.12 • 4.6.14 • V4.6.14 • 4.6.16 • V4.6.16 • 4.6.∞ • (4.7)2 • (4.8)2 • 4.8.10 • V4.8.10 • 4.8.12 • 4.8.14 • 4.8.16 • 4.8.∞ • 4.102 • 4.10.12 • 4.122 • 4.12.16 • 4.142 • 4.162 • 4.∞2 • (4.∞)2 • 54 • 55 • 56 • 5∞ • 5.4.6.4 • (5.6)2 • 5.82 • 5.102 • 5.122 • (5.∞)2 • 64 • 65 • 66 • 68 • 6.4.8.4 • (6.8)2 • 6.82 • 6.102 • 6.122 • 6.162 • 73 • 74 • 77 • 7.62 • 7.82 • 7.142 • 83 • 84 • 86 • 88 • 8.62 • 8.122 • 8.162 • ∞3 • ∞4 • ∞5 • ∞∞ • ∞.62 • ∞.82	Wikipedia
Duality (mathematics) In mathematics, a duality translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often (but not always) by means of an involution operation: if the dual of A is B, then the dual of B is A. Such involutions sometimes have fixed points, so that the dual of A is A itself. For example, Desargues' theorem is self-dual in this sense under the standard duality in projective geometry. For the property of optimization problems, see Duality (optimization). In mathematical contexts, duality has numerous meanings.[1] It has been described as "a very pervasive and important concept in (modern) mathematics"[2] and "an important general theme that has manifestations in almost every area of mathematics".[3] Many mathematical dualities between objects of two types correspond to pairings, bilinear functions from an object of one type and another object of the second type to some family of scalars. For instance, linear algebra duality corresponds in this way to bilinear maps from pairs of vector spaces to scalars, the duality between distributions and the associated test functions corresponds to the pairing in which one integrates a distribution against a test function, and Poincaré duality corresponds similarly to intersection number, viewed as a pairing between submanifolds of a given manifold.[4] From a category theory viewpoint, duality can also be seen as a functor, at least in the realm of vector spaces. This functor assigns to each space its dual space, and the pullback construction assigns to each arrow f: V → W its dual f∗: W∗ → V∗. Introductory examples In the words of Michael Atiyah, Duality in mathematics is not a theorem, but a "principle".[5] The following list of examples shows the common features of many dualities, but also indicates that the precise meaning of duality may vary from case to case. Complement of a subset A simple, maybe the most simple, duality arises from considering subsets of a fixed set S. To any subset A ⊆ S, the complement Ac[6] consists of all those elements in S that are not contained in A. It is again a subset of S. Taking the complement has the following properties: • Applying it twice gives back the original set, i.e., (Ac)c = A. This is referred to by saying that the operation of taking the complement is an involution. • An inclusion of sets A ⊆ B is turned into an inclusion in the opposite direction Bc ⊆ Ac. • Given two subsets A and B of S, A is contained in Bc if and only if B is contained in Ac. This duality appears in topology as a duality between open and closed subsets of some fixed topological space X: a subset U of X is closed if and only if its complement in X is open. Because of this, many theorems about closed sets are dual to theorems about open sets. For example, any union of open sets is open, so dually, any intersection of closed sets is closed. The interior of a set is the largest open set contained in it, and the closure of the set is the smallest closed set that contains it. Because of the duality, the complement of the interior of any set U is equal to the closure of the complement of U. Dual cone A duality in geometry is provided by the dual cone construction. Given a set $C$ of points in the plane $\mathbb {R} ^{2}$ (or more generally points in $\mathbb {R} ^{n}$), the dual cone is defined as the set $C^{}\subseteq \mathbb {R} ^{2}$ consisting of those points $(x_{1},x_{2})$ satisfying $x_{1}c_{1}+x_{2}c_{2}\geq 0$ for all points $(c_{1},c_{2})$ in $C$, as illustrated in the diagram. Unlike for the complement of sets mentioned above, it is not in general true that applying the dual cone construction twice gives back the original set $C$. Instead, $C^{}$ is the smallest cone[7] containing $C$ which may be bigger than $C$. Therefore this duality is weaker than the one above, in that • Applying the operation twice gives back a possibly bigger set: for all $C$, $C$ is contained in $C^{}$. (For some $C$, namely the cones, the two are actually equal.) The other two properties carry over without change: • It is still true that an inclusion $C\subseteq D$ is turned into an inclusion in the opposite direction ($D^{}\subseteq C^{}$). • Given two subsets $C$ and $D$ of the plane, $C$ is contained in $D^{}$ if and only if $D$ is contained in $C^{}$. Dual vector space A very important example of a duality arises in linear algebra by associating to any vector space V its dual vector space V. Its elements are the linear functionals $\varphi :V\to K$, where K is the field over which V is defined. The three properties of the dual cone carry over to this type of duality by replacing subsets of $\mathbb {R} ^{2}$ by vector space and inclusions of such subsets by linear maps. That is: • Applying the operation of taking the dual vector space twice gives another vector space V. There is always a map V → V. For some V, namely precisely the finite-dimensional vector spaces, this map is an isomorphism. • A linear map V → W gives rise to a map in the opposite direction (W* → V). • Given two vector spaces V and W, the maps from V to W correspond to the maps from W to V. A particular feature of this duality is that V and V are isomorphic for certain objects, namely finite-dimensional vector spaces. However, this is in a sense a lucky coincidence, for giving such an isomorphism requires a certain choice, for example the choice of a basis of V. This is also true in the case if V is a Hilbert space, via the Riesz representation theorem. Galois theory In all the dualities discussed before, the dual of an object is of the same kind as the object itself. For example, the dual of a vector space is again a vector space. Many duality statements are not of this kind. Instead, such dualities reveal a close relation between objects of seemingly different nature. One example of such a more general duality is from Galois theory. For a fixed Galois extension K / F, one may associate the Galois group Gal(K/E) to any intermediate field E (i.e., F ⊆ E ⊆ K). This group is a subgroup of the Galois group G = Gal(K/F). Conversely, to any such subgroup H ⊆ G there is the fixed field KH consisting of elements fixed by the elements in H. Compared to the above, this duality has the following features: • An extension F ⊆ F′ of intermediate fields gives rise to an inclusion of Galois groups in the opposite direction: Gal(K/F′) ⊆ Gal(K/F). • Associating Gal(K/E) to E and KH to H are inverse to each other. This is the content of the fundamental theorem of Galois theory. Order-reversing dualities Main article: Duality (order theory) Given a poset P = (X, ≤) (short for partially ordered set; i.e., a set that has a notion of ordering but in which two elements cannot necessarily be placed in order relative to each other), the dual poset Pd = (X, ≥) comprises the same ground set but the converse relation. Familiar examples of dual partial orders include • the subset and superset relations ⊂ and ⊃ on any collection of sets, such as the subsets of a fixed set S. This gives rise to the first example of a duality mentioned above. • the divides and multiple-of relations on the integers. • the descendant-of and ancestor-of relations on the set of humans. A duality transform is an involutive antiautomorphism f of a partially ordered set S, that is, an order-reversing involution f : S → S.[8][9] In several important cases these simple properties determine the transform uniquely up to some simple symmetries. For example, if f1, f2 are two duality transforms then their composition is an order automorphism of S; thus, any two duality transforms differ only by an order automorphism. For example, all order automorphisms of a power set S = 2R are induced by permutations of R. A concept defined for a partial order P will correspond to a dual concept on the dual poset Pd. For instance, a minimal element of P will be a maximal element of Pd: minimality and maximality are dual concepts in order theory. Other pairs of dual concepts are upper and lower bounds, lower sets and upper sets, and ideals and filters. In topology, open sets and closed sets are dual concepts: the complement of an open set is closed, and vice versa. In matroid theory, the family of sets complementary to the independent sets of a given matroid themselves form another matroid, called the dual matroid. Dimension-reversing dualities There are many distinct but interrelated dualities in which geometric or topological objects correspond to other objects of the same type, but with a reversal of the dimensions of the features of the objects. A classical example of this is the duality of the Platonic solids, in which the cube and the octahedron form a dual pair, the dodecahedron and the icosahedron form a dual pair, and the tetrahedron is self-dual. The dual polyhedron of any of these polyhedra may be formed as the convex hull of the center points of each face of the primal polyhedron, so the vertices of the dual correspond one-for-one with the faces of the primal. Similarly, each edge of the dual corresponds to an edge of the primal, and each face of the dual corresponds to a vertex of the primal. These correspondences are incidence-preserving: if two parts of the primal polyhedron touch each other, so do the corresponding two parts of the dual polyhedron. More generally, using the concept of polar reciprocation, any convex polyhedron, or more generally any convex polytope, corresponds to a dual polyhedron or dual polytope, with an i-dimensional feature of an n-dimensional polytope corresponding to an (n − i − 1)-dimensional feature of the dual polytope. The incidence-preserving nature of the duality is reflected in the fact that the face lattices of the primal and dual polyhedra or polytopes are themselves order-theoretic duals. Duality of polytopes and order-theoretic duality are both involutions: the dual polytope of the dual polytope of any polytope is the original polytope, and reversing all order-relations twice returns to the original order. Choosing a different center of polarity leads to geometrically different dual polytopes, but all have the same combinatorial structure. From any three-dimensional polyhedron, one can form a planar graph, the graph of its vertices and edges. The dual polyhedron has a dual graph, a graph with one vertex for each face of the polyhedron and with one edge for every two adjacent faces. The same concept of planar graph duality may be generalized to graphs that are drawn in the plane but that do not come from a three-dimensional polyhedron, or more generally to graph embeddings on surfaces of higher genus: one may draw a dual graph by placing one vertex within each region bounded by a cycle of edges in the embedding, and drawing an edge connecting any two regions that share a boundary edge. An important example of this type comes from computational geometry: the duality for any finite set S of points in the plane between the Delaunay triangulation of S and the Voronoi diagram of S. As with dual polyhedra and dual polytopes, the duality of graphs on surfaces is a dimension-reversing involution: each vertex in the primal embedded graph corresponds to a region of the dual embedding, each edge in the primal is crossed by an edge in the dual, and each region of the primal corresponds to a vertex of the dual. The dual graph depends on how the primal graph is embedded: different planar embeddings of a single graph may lead to different dual graphs. Matroid duality is an algebraic extension of planar graph duality, in the sense that the dual matroid of the graphic matroid of a planar graph is isomorphic to the graphic matroid of the dual graph. A kind of geometric duality also occurs in optimization theory, but not one that reverses dimensions. A linear program may be specified by a system of real variables (the coordinates for a point in Euclidean space $\mathbb {R} ^{n}$), a system of linear constraints (specifying that the point lie in a halfspace; the intersection of these halfspaces is a convex polytope, the feasible region of the program), and a linear function (what to optimize). Every linear program has a dual problem with the same optimal solution, but the variables in the dual problem correspond to constraints in the primal problem and vice versa. Duality in logic and set theory In logic, functions or relations A and B are considered dual if A(¬x) = ¬B(x), where ¬ is logical negation. The basic duality of this type is the duality of the ∃ and ∀ quantifiers in classical logic. These are dual because ∃x.¬P(x) and ¬∀x.P(x) are equivalent for all predicates P in classical logic: if there exists an x for which P fails to hold, then it is false that P holds for all x (but the converse does not hold constructively). From this fundamental logical duality follow several others: • A formula is said to be satisfiable in a certain model if there are assignments to its free variables that render it true; it is valid if every assignment to its free variables makes it true. Satisfiability and validity are dual because the invalid formulas are precisely those whose negations are satisfiable, and the unsatisfiable formulas are those whose negations are valid. This can be viewed as a special case of the previous item, with the quantifiers ranging over interpretations. • In classical logic, the ∧ and ∨ operators are dual in this sense, because (¬x ∧ ¬y) and ¬(x ∨ y) are equivalent. This means that for every theorem of classical logic there is an equivalent dual theorem. De Morgan's laws are examples. More generally, ∧ (¬ xi) = ¬∨ xi. The left side is true if and only if ∀i.¬xi, and the right side if and only if ¬∃i.xi. • In modal logic, □p means that the proposition p is "necessarily" true, and ◊p that p is "possibly" true. Most interpretations of modal logic assign dual meanings to these two operators. For example in Kripke semantics, "p is possibly true" means "there exists some world W such that p is true in W", while "p is necessarily true" means "for all worlds W, p is true in W". The duality of □ and ◊ then follows from the analogous duality of ∀ and ∃. Other dual modal operators behave similarly. For example, temporal logic has operators denoting "will be true at some time in the future" and "will be true at all times in the future" which are similarly dual. Other analogous dualities follow from these: • Set-theoretic union and intersection are dual under the set complement operator ⋅C. That is, AC ∩ BC = (A ∪ B)C, and more generally, ∩ AC α = (∪ Aα)C . This follows from the duality of ∀ and ∃: an element x is a member of ∩ AC α if and only if ∀α.¬x ∈ Aα, and is a member of (∪ Aα)C if and only if ¬∃α. x ∈ Aα. Dual objects A group of dualities can be described by endowing, for any mathematical object X, the set of morphisms Hom (X, D) into some fixed object D, with a structure similar to that of X. This is sometimes called internal Hom. In general, this yields a true duality only for specific choices of D, in which case X* = Hom (X, D) is referred to as the dual of X. There is always a map from X to the bidual, that is to say, the dual of the dual, $X\to X^{*}:=(X^{})^{}=\operatorname {Hom} (\operatorname {Hom} (X,D),D).$ It assigns to some x ∈ X the map that associates to any map f : X → D (i.e., an element in Hom(X, D)) the value f(x). Depending on the concrete duality considered and also depending on the object X, this map may or may not be an isomorphism. Dual vector spaces revisited The construction of the dual vector space $V^{}=\operatorname {Hom} (V,K)$ mentioned in the introduction is an example of such a duality. Indeed, the set of morphisms, i.e., linear maps, forms a vector space in its own right. The map V → V** mentioned above is always injective. It is surjective, and therefore an isomorphism, if and only if the dimension of V is finite. This fact characterizes finite-dimensional vector spaces without referring to a basis. Isomorphisms of V and V∗ and inner product spaces A vector space V is isomorphic to V∗ precisely if V is finite-dimensional. In this case, such an isomorphism is equivalent to a non-degenerate bilinear form $\varphi :V\times V\to K$ In this case V is called an inner product space. For example, if K is the field of real or complex numbers, any positive definite bilinear form gives rise to such an isomorphism. In Riemannian geometry, V is taken to be the tangent space of a manifold and such positive bilinear forms are called Riemannian metrics. Their purpose is to measure angles and distances. Thus, duality is a foundational basis of this branch of geometry. Another application of inner product spaces is the Hodge star which provides a correspondence between the elements of the exterior algebra. For an n-dimensional vector space, the Hodge star operator maps k-forms to (n − k)-forms. This can be used to formulate Maxwell's equations. In this guise, the duality inherent in the inner product space exchanges the role of magnetic and electric fields. Duality in projective geometry In some projective planes, it is possible to find geometric transformations that map each point of the projective plane to a line, and each line of the projective plane to a point, in an incidence-preserving way.[10] For such planes there arises a general principle of duality in projective planes: given any theorem in such a plane projective geometry, exchanging the terms "point" and "line" everywhere results in a new, equally valid theorem.[11] A simple example is that the statement "two points determine a unique line, the line passing through these points" has the dual statement that "two lines determine a unique point, the intersection point of these two lines". For further examples, see Dual theorems. A conceptual explanation of this phenomenon in some planes (notably field planes) is offered by the dual vector space. In fact, the points in the projective plane $\mathbb {RP} ^{2}$ correspond to one-dimensional subvector spaces $V\subset \mathbb {R} ^{3}$[12] while the lines in the projective plane correspond to subvector spaces $W$ of dimension 2. The duality in such projective geometries stems from assigning to a one-dimensional $V$ the subspace of $(\mathbb {R} ^{3})^{}$ consisting of those linear maps $f:\mathbb {R} ^{3}\to \mathbb {R} $ which satisfy $f(V)=0$. As a consequence of the dimension formula of linear algebra, this space is two-dimensional, i.e., it corresponds to a line in the projective plane associated to $(\mathbb {R} ^{3})^{}$. The (positive definite) bilinear form $\langle \cdot ,\cdot \rangle :\mathbb {R} ^{3}\times \mathbb {R} ^{3}\to \mathbb {R} ,\langle x,y\rangle =\sum _{i=1}^{3}x_{i}y_{i}$ :\mathbb {R} ^{3}\times \mathbb {R} ^{3}\to \mathbb {R} ,\langle x,y\rangle =\sum _{i=1}^{3}x_{i}y_{i}} yields an identification of this projective plane with the $\mathbb {RP} ^{2}$. Concretely, the duality assigns to $V\subset \mathbb {R} ^{3}$ its orthogonal $\left\{w\in \mathbb {R} ^{3},\langle v,w\rangle =0{\text{ for all }}v\in V\right\}$. The explicit formulas in duality in projective geometry arise by means of this identification. Topological vector spaces and Hilbert spaces In the realm of topological vector spaces, a similar construction exists, replacing the dual by the topological dual vector space. There are several notions of topological dual space, and each of them gives rise to a certain concept of duality. A topological vector space $X$ that is canonically isomorphic to its bidual $X''$ is called a reflexive space: $X\cong X''.$ Examples: • As in the finite-dimensional case, on each Hilbert space H its inner product ⟨⋅, ⋅⟩ defines a map $H\to H^{},v\mapsto (w\mapsto \langle w,v\rangle ),$ which is a bijection due to the Riesz representation theorem. As a corollary, every Hilbert space is a reflexive Banach space. • The dual normed space of an Lp-space is Lq where 1/p + 1/q = 1 provided that 1 ≤ p < ∞, but the dual of L∞ is bigger than L1. Hence L1 is not reflexive. • Distributions are linear functionals on appropriate spaces of functions. They are an important technical means in the theory of partial differential equations (PDE): instead of solving a PDE directly, it may be easier to first solve the PDE in the "weak sense", i.e., find a distribution that satisfies the PDE and, second, to show that the solution must, in fact, be a function.[13] All the standard spaces of distributions — ${\mathcal {D}}'(U)$, ${\mathcal {S}}'(\mathbb {R} ^{n})$, ${\mathcal {C}}^{\infty }(U)'$ — are reflexive locally convex spaces.[14] Further dual objects The dual lattice of a lattice L is given by $\operatorname {Hom} (L,\mathbf {Z} ),$ which is used in the construction of toric varieties.[15] The Pontryagin dual of locally compact topological groups G is given by $\operatorname {Hom} (G,S^{1}),$ continuous group homomorphisms with values in the circle (with multiplication of complex numbers as group operation). Dual categories Main article: Dual (category theory) Opposite category and adjoint functors In another group of dualities, the objects of one theory are translated into objects of another theory and the maps between objects in the first theory are translated into morphisms in the second theory, but with direction reversed. Using the parlance of category theory, this amounts to a contravariant functor between two categories C and D: F: C → D which for any two objects X and Y of C gives a map HomC(X, Y) → HomD(F(Y), F(X)) That functor may or may not be an equivalence of categories. There are various situations, where such a functor is an equivalence between the opposite category Cop of C, and D. Using a duality of this type, every statement in the first theory can be translated into a "dual" statement in the second theory, where the direction of all arrows has to be reversed.[16] Therefore, any duality between categories C and D is formally the same as an equivalence between C and Dop (Cop and D). However, in many circumstances the opposite categories have no inherent meaning, which makes duality an additional, separate concept.[17] A category that is equivalent to its dual is called self-dual. An example of self-dual category is the category of Hilbert spaces.[18] Many category-theoretic notions come in pairs in the sense that they correspond to each other while considering the opposite category. For example, Cartesian products Y1 × Y2 and disjoint unions Y1 ⊔ Y2 of sets are dual to each other in the sense that Hom (X, Y1 × Y2) = Hom (X, Y1) × Hom (X, Y2) and Hom (Y1 ⊔ Y2, X) = Hom (Y1, X) × Hom (Y2, X) for any set X. This is a particular case of a more general duality phenomenon, under which limits in a category C correspond to colimits in the opposite category Cop; further concrete examples of this are epimorphisms vs. monomorphism, in particular factor modules (or groups etc.) vs. submodules, direct products vs. direct sums (also called coproducts to emphasize the duality aspect). Therefore, in some cases, proofs of certain statements can be halved, using such a duality phenomenon. Further notions displaying related by such a categorical duality are projective and injective modules in homological algebra,[19] fibrations and cofibrations in topology and more generally model categories.[20] Two functors F: C → D and G: D → C are adjoint if for all objects c in C and d in D HomD(F(c), d) ≅ HomC(c, G(d)), in a natural way. Actually, the correspondence of limits and colimits is an example of adjoints, since there is an adjunction colim: CI ↔ C: Δ between the colimit functor that assigns to any diagram in C indexed by some category I its colimit and the diagonal functor that maps any object c of C to the constant diagram which has c at all places. Dually, Δ: C ↔ CI: lim. Spaces and functions Gelfand duality is a duality between commutative C-algebras A and compact Hausdorff spaces X is the same: it assigns to X the space of continuous functions (which vanish at infinity) from X to C, the complex numbers. Conversely, the space X can be reconstructed from A as the spectrum of A. Both Gelfand and Pontryagin duality can be deduced in a largely formal, category-theoretic way.[21] In a similar vein there is a duality in algebraic geometry between commutative rings and affine schemes: to every commutative ring A there is an affine spectrum, Spec A. Conversely, given an affine scheme S, one gets back a ring by taking global sections of the structure sheaf OS. In addition, ring homomorphisms are in one-to-one correspondence with morphisms of affine schemes, thereby there is an equivalence (Commutative rings)op ≅ (affine schemes)[22] Affine schemes are the local building blocks of schemes. The previous result therefore tells that the local theory of schemes is the same as commutative algebra, the study of commutative rings. Noncommutative geometry draws inspiration from Gelfand duality and studies noncommutative C-algebras as if they were functions on some imagined space. Tannaka–Krein duality is a non-commutative analogue of Pontryagin duality.[23] Galois connections In a number of situations, the two categories which are dual to each other are actually arising from partially ordered sets, i.e., there is some notion of an object "being smaller" than another one. A duality that respects the orderings in question is known as a Galois connection. An example is the standard duality in Galois theory mentioned in the introduction: a bigger field extension corresponds—under the mapping that assigns to any extension L ⊃ K (inside some fixed bigger field Ω) the Galois group Gal (Ω / L) —to a smaller group.[24] The collection of all open subsets of a topological space X forms a complete Heyting algebra. There is a duality, known as Stone duality, connecting sober spaces and spatial locales. • Birkhoff's representation theorem relating distributive lattices and partial orders Pontryagin duality Pontryagin duality gives a duality on the category of locally compact abelian groups: given any such group G, the character group χ(G) = Hom (G, S1) given by continuous group homomorphisms from G to the circle group S1 can be endowed with the compact-open topology. Pontryagin duality states that the character group is again locally compact abelian and that G ≅ χ(χ(G)).[25] Moreover, discrete groups correspond to compact abelian groups; finite groups correspond to finite groups. On the one hand, Pontryagin is a special case of Gelfand duality. On the other hand, it is the conceptual reason of Fourier analysis, see below. Analytic dualities In analysis, problems are frequently solved by passing to the dual description of functions and operators. Fourier transform switches between functions on a vector space and its dual: ${\widehat {f}}(\xi ):=\int _{-\infty }^{\infty }f(x)\ e^{-2\pi ix\xi }\,dx,$ and conversely $f(x)=\int _{-\infty }^{\infty }{\widehat {f}}(\xi )\ e^{2\pi ix\xi }\,d\xi .$ If f is an L2-function on R or RN, say, then so is ${\widehat {f}}$ and $f(-x)={\widehat {\widehat {f}}}(x)$. Moreover, the transform interchanges operations of multiplication and convolution on the corresponding function spaces. A conceptual explanation of the Fourier transform is obtained by the aforementioned Pontryagin duality, applied to the locally compact groups R (or RN etc.): any character of R is given by ξ ↦ e−2πixξ. The dualizing character of Fourier transform has many other manifestations, for example, in alternative descriptions of quantum mechanical systems in terms of coordinate and momentum representations. • Laplace transform is similar to Fourier transform and interchanges operators of multiplication by polynomials with constant coefficient linear differential operators. • Legendre transformation is an important analytic duality which switches between velocities in Lagrangian mechanics and momenta in Hamiltonian mechanics. Homology and cohomology Theorems showing that certain objects of interest are the dual spaces (in the sense of linear algebra) of other objects of interest are often called dualities. Many of these dualities are given by a bilinear pairing of two K-vector spaces A ⊗ B → K. For perfect pairings, there is, therefore, an isomorphism of A to the dual of B. Poincaré duality Poincaré duality of a smooth compact complex manifold X is given by a pairing of singular cohomology with C-coefficients (equivalently, sheaf cohomology of the constant sheaf C) Hi(X) ⊗ H2n−i(X) → C, where n is the (complex) dimension of X.[26] Poincaré duality can also be expressed as a relation of singular homology and de Rham cohomology, by asserting that the map $(\gamma ,\omega )\mapsto \int _{\gamma }\omega $ (integrating a differential k-form over an 2n−k-(real) -dimensional cycle) is a perfect pairing. Poincaré duality also reverses dimensions; it corresponds to the fact that, if a topological manifold is represented as a cell complex, then the dual of the complex (a higher-dimensional generalization of the planar graph dual) represents the same manifold. In Poincaré duality, this homeomorphism is reflected in an isomorphism of the kth homology group and the (n − k)th cohomology group. Duality in algebraic and arithmetic geometry The same duality pattern holds for a smooth projective variety over a separably closed field, using l-adic cohomology with Qℓ-coefficients instead.[27] This is further generalized to possibly singular varieties, using intersection cohomology instead, a duality called Verdier duality.[28] Serre duality or coherent duality are similar to the statements above, but applies to cohomology of coherent sheaves instead.[29] With increasing level of generality, it turns out, an increasing amount of technical background is helpful or necessary to understand these theorems: the modern formulation of these dualities can be done using derived categories and certain direct and inverse image functors of sheaves (with respect to the classical analytical topology on manifolds for Poincaré duality, l-adic sheaves and the étale topology in the second case, and with respect to coherent sheaves for coherent duality). Yet another group of similar duality statements is encountered in arithmetics: étale cohomology of finite, local and global fields (also known as Galois cohomology, since étale cohomology over a field is equivalent to group cohomology of the (absolute) Galois group of the field) admit similar pairings. The absolute Galois group G(Fq) of a finite field, for example, is isomorphic to ${\widehat {\mathbf {Z} }}$, the profinite completion of Z, the integers. Therefore, the perfect pairing (for any G-module M) Hn(G, M) × H1−n (G, Hom (M, Q/Z)) → Q/Z[30] is a direct consequence of Pontryagin duality of finite groups. For local and global fields, similar statements exist (local duality and global or Poitou–Tate duality).[31] See also • Adjoint functor • Autonomous category • Convex body and polar body. • Dual abelian variety • Dual basis • Dual (category theory) • Dual code • Duality (electrical engineering) • Duality (optimization) • Dualizing module • Dualizing sheaf • Dual lattice • Dual norm • Dual numbers, a certain associative algebra; the term "dual" here is synonymous with double, and is unrelated to the notions given above. • Dual system • Koszul duality • Langlands dual • Linear programming#Duality • List of dualities • Matlis duality • Petrie duality • Pontryagin duality • S-duality • T-duality, Mirror symmetry Notes 1. Atiyah 2007, p. 1 2. Kostrikin 2001, This quote is the first sentence of the final section named comments in this single-paged-document 3. Gowers 2008, p. 187, col. 1 4. Gowers 2008, p. 189, col. 2 5. Atiyah 2007, p. 1 6. The complement is also denoted as S \ A. 7. More precisely, $C^{*}$ is the smallest closed convex cone containing $C$. 8. Artstein-Avidan & Milman 2007 9. Artstein-Avidan & Milman 2008 10. Veblen & Young 1965. 11. (Veblen & Young 1965, Ch. I, Theorem 11) 12. More generally, one can consider the projective planes over any field, such as the complex numbers or finite fields or even division rings. 13. See elliptic regularity. 14. Edwards (1965, 8.4.7). 15. Fulton 1993 16. Mac Lane 1998, Ch. II.1. 17. (Lam 1999, §19C) 18. Jiří Adámek; J. Rosicky (1994). Locally Presentable and Accessible Categories. Cambridge University Press. p. 62. ISBN 978-0-521-42261-1. 19. Weibel (1994) 20. Dwyer and Spaliński (1995) 21. Negrepontis 1971. 22. Hartshorne 1966, Ch. II.2, esp. Prop. II.2.3 23. Joyal and Street (1991) 24. See (Lang 2002, Theorem VI.1.1) for finite Galois extensions. 25. (Loomis 1953, p. 151, section 37D) 26. Griffiths & Harris 1994, p. 56 27. Milne 1980, Ch. VI.11 28. Iversen 1986, Ch. VII.3, VII.5 29. Hartshorne 1966, Ch. III.7 30. Milne (2006, Example I.1.10) 31. Mazur (1973); Milne (2006) References Duality in general • Atiyah, Michael (2007). "Duality in Mathematics and Physics lecture notes from the Institut de Matematica de la Universitat de Barcelona (IMUB)" (PDF). • Kostrikin, A. I. (2001) [1994], "Duality", Encyclopedia of Mathematics, EMS Press. • Gowers, Timothy (2008), "III.19 Duality", The Princeton Companion to Mathematics, Princeton University Press, pp. 187–190. • Cartier, Pierre (2001), "A mad day's work: from Grothendieck to Connes and Kontsevich. The evolution of concepts of space and symmetry", Bulletin of the American Mathematical Society, New Series, 38 (4): 389–408, doi:10.1090/S0273-0979-01-00913-2, ISSN 0002-9904, MR 1848254 (a non-technical overview about several aspects of geometry, including dualities) Duality in algebraic topology • James C. Becker and Daniel Henry Gottlieb, A History of Duality in Algebraic Topology Specific dualities • Artstein-Avidan, Shiri; Milman, Vitali (2008), "The concept of duality for measure projections of convex bodies", Journal of Functional Analysis, 254 (10): 2648–66, doi:10.1016/j.jfa.2007.11.008. Also author's site. • Artstein-Avidan, Shiri; Milman, Vitali (2007), "A characterization of the concept of duality", Electronic Research Announcements in Mathematical Sciences, 14: 42–59, archived from the original on 2011-07-24, retrieved 2009-05-30. Also author's site. • Dwyer, William G.; Spaliński, Jan (1995), "Homotopy theories and model categories", Handbook of algebraic topology, Amsterdam: North-Holland, pp. 73–126, MR 1361887 • Fulton, William (1993), Introduction to toric varieties, Princeton University Press, ISBN 978-0-691-00049-7 • Griffiths, Phillip; Harris, Joseph (1994), Principles of algebraic geometry, Wiley Classics Library, New York: John Wiley & Sons, ISBN 978-0-471-05059-9, MR 1288523 • Hartshorne, Robin (1966), Residues and Duality, Lecture Notes in Mathematics, vol. 20, Springer-Verlag, pp. 20–48, ISBN 978-3-540-34794-1 • Hartshorne, Robin (1977), Algebraic Geometry, Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157, OCLC 13348052 • Iversen, Birger (1986), Cohomology of sheaves, Universitext, Springer-Verlag, ISBN 978-3-540-16389-3, MR 0842190 • Joyal, André; Street, Ross (1991), "An introduction to Tannaka duality and quantum groups" (PDF), Category theory, Lecture Notes in Mathematics, vol. 1488, Springer-Verlag, pp. 413–492, doi:10.1007/BFb0084235, ISBN 978-3-540-46435-8, MR 1173027 • Lam, Tsit-Yuen (1999), Lectures on modules and rings, Graduate Texts in Mathematics, vol. 189, Springer-Verlag, ISBN 978-0-387-98428-5, MR 1653294 • Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, vol. 211, Springer-Verlag, ISBN 978-0-387-95385-4, MR 1878556 • Loomis, Lynn H. (1953), An introduction to abstract harmonic analysis, D. Van Nostrand, pp. x+190, hdl:2027/uc1.b4250788 • Mac Lane, Saunders (1998), Categories for the Working Mathematician (2nd ed.), Springer-Verlag, ISBN 978-0-387-98403-2 • Mazur, Barry (1973), "Notes on étale cohomology of number fields", Annales Scientifiques de l'École Normale Supérieure, Série 4, 6 (4): 521–552, doi:10.24033/asens.1257, ISSN 0012-9593, MR 0344254 • Milne, James S. (1980), Étale cohomology, Princeton University Press, ISBN 978-0-691-08238-7 • Milne, James S. (2006), Arithmetic duality theorems (2nd ed.), Charleston, South Carolina: BookSurge, LLC, ISBN 978-1-4196-4274-6, MR 2261462 • Negrepontis, Joan W. (1971), "Duality in analysis from the point of view of triples", Journal of Algebra, 19 (2): 228–253, doi:10.1016/0021-8693(71)90105-0, ISSN 0021-8693, MR 0280571 • Veblen, Oswald; Young, John Wesley (1965), Projective geometry. Vols. 1, 2, Blaisdell Publishing Co. Ginn and Co., MR 0179666 • Weibel, Charles A. (1994), An introduction to homological algebra, Cambridge University Press, ISBN 978-0-521-55987-4, MR 1269324 • Edwards, R. E. (1965). Functional analysis. Theory and applications. New York: Holt, Rinehart and Winston. ISBN 0030505356.	Wikipedia
Homeomorphism group In mathematics, particularly topology, the homeomorphism group of a topological space is the group consisting of all homeomorphisms from the space to itself with function composition as the group operation. Homeomorphism groups are very important in the theory of topological spaces and in general are examples of automorphism groups. Homeomorphism groups are topological invariants in the sense that the homeomorphism groups of homeomorphic topological spaces are isomorphic as groups. Properties and examples There is a natural group action of the homeomorphism group of a space on that space. Let $X$ be a topological space and denote the homeomorphism group of $X$ by $G$. The action is defined as follows: ${\begin{aligned}G\times X&\longrightarrow X\\(\varphi ,x)&\longmapsto \varphi (x)\end{aligned}}$ This is a group action since for all $\varphi ,\psi \in G$, $\varphi \cdot (\psi \cdot x)=\varphi (\psi (x))=(\varphi \circ \psi )(x)$ where $\cdot $ denotes the group action, and the identity element of $G$ (which is the identity function on $X$) sends points to themselves. If this action is transitive, then the space is said to be homogeneous. Topology As with other sets of maps between topological spaces, the homeomorphism group can be given a topology, such as the compact-open topology. In the case of regular, locally compact spaces the group multiplication is then continuous. If the space is compact and Hausdorff, the inversion is continuous as well and $\operatorname {Homeo} (X)$ becomes a topological group. If $X$ is Hausdorff, locally compact and locally connected this holds as well.[1] However there are locally compact separable metric spaces for which the inversion map is not continuous and $\operatorname {Homeo} (X)$ therefore not a topological group.[1] In the category of topological spaces with homeomorphisms, group objects are exactly homeomorphism groups. Mapping class group Main article: Mapping class group In geometric topology especially, one considers the quotient group obtained by quotienting out by isotopy, called the mapping class group: ${\rm {MCG}}(X)={\rm {Homeo}}(X)/{\rm {Homeo}}_{0}(X)$ The MCG can also be interpreted as the 0th homotopy group, ${\rm {MCG}}(X)=\pi _{0}({\rm {Homeo}}(X))$. This yields the short exact sequence: $1\rightarrow {\rm {Homeo}}_{0}(X)\rightarrow {\rm {Homeo}}(X)\rightarrow {\rm {MCG}}(X)\rightarrow 1.$ In some applications, particularly surfaces, the homeomorphism group is studied via this short exact sequence, and by first studying the mapping class group and group of isotopically trivial homeomorphisms, and then (at times) the extension. See also • Mapping class group References 1. Dijkstra, Jan J. (2005), "On homeomorphism groups and the compact-open topology" (PDF), American Mathematical Monthly, 112 (10): 910–912, doi:10.2307/30037630, MR 2186833 • "homeomorphism group", Encyclopedia of Mathematics, EMS Press, 2001 [1994]	Wikipedia
Involution (mathematics) In mathematics, an involution, involutory function, or self-inverse function[1] is a function f that is its own inverse, f(f(x)) = x For the archaic use of this term, see exponentiation. for all x in the domain of f.[2] Equivalently, applying f twice produces the original value. General properties Any involution is a bijection. The identity map is a trivial example of an involution. Examples of nontrivial involutions include negation ($x\mapsto -x$), reciprocation ($x\mapsto 1/x$), and complex conjugation ($z\mapsto {\bar {z}}$) in arithmetic; reflection, half-turn rotation, and circle inversion in geometry; complementation in set theory; and reciprocal ciphers such as the ROT13 transformation and the Beaufort polyalphabetic cipher. The composition g ∘ f of two involutions f and g is an involution if and only if they commute: g ∘ f = f ∘ g.[3] Involutions on finite sets The number of involutions, including the identity involution, on a set with n = 0, 1, 2, ... elements is given by a recurrence relation found by Heinrich August Rothe in 1800: $a_{0}=a_{1}=1$ and $a_{n}=a_{n-1}+(n-1)a_{n-2}$ for $n>1.$ The first few terms of this sequence are 1, 1, 2, 4, 10, 26, 76, 232 (sequence A000085 in the OEIS); these numbers are called the telephone numbers, and they also count the number of Young tableaux with a given number of cells.[4] The number $a_{n}$ can also be expressed by non-recursive formulas, such as the sum $a_{n}=\sum _{m=0}^{\lfloor {\frac {n}{2}}\rfloor }{\frac {n!}{2^{m}m!(n-2m)!}}.$ The number of fixed points of an involution on a finite set and its number of elements have the same parity. Thus the number of fixed points of all the involutions on a given finite set have the same parity. In particular, every involution on an odd number of elements has at least one fixed point. This can be used to prove Fermat's two squares theorem.[5] Involution throughout the fields of mathematics Pre-calculus Some basic examples of involutions include the functions ${\begin{alignedat}{4}f_{1}(x)&=-x,\\f_{2}(x)&={\frac {1}{x}},\\f_{3}(x)&={\frac {x}{x-1}},\\\end{alignedat}}$ the composition $f_{4}(x):=(f_{1}\circ f_{2})(x)=(f_{2}\circ f_{1})(x)=-{\frac {1}{x}},$ and more generally the function $g(x)={\frac {b-x}{1+cx}}$ is an involution for constants $b$ and $c$ that satisfy $bc\neq -1.$ These are not the only pre-calculus involutions. Another one within the positive reals is $f(x)=\ln \left({\frac {e^{x}+1}{e^{x}-1}}\right).$ The graph of an involution (on the real numbers) is symmetric across the line $y=x$. This is due to the fact that the inverse of any general function will be its reflection over the line $y=x$. This can be seen by "swapping" $x$ with $y$. If, in particular, the function is an involution, then its graph is its own reflection. Other elementary involutions are useful in solving functional equations. Euclidean geometry A simple example of an involution of the three-dimensional Euclidean space is reflection through a plane. Performing a reflection twice brings a point back to its original coordinates. Another involution is reflection through the origin; not a reflection in the above sense, and so, a distinct example. These transformations are examples of affine involutions. Projective geometry An involution is a projectivity of period 2, that is, a projectivity that interchanges pairs of points.[6]: 24 • Any projectivity that interchanges two points is an involution. • The three pairs of opposite sides of a complete quadrangle meet any line (not through a vertex) in three pairs of an involution. This theorem has been called Desargues's Involution Theorem.[7] Its origins can be seen in Lemma IV of the lemmas to the Porisms of Euclid in Volume VII of the Collection of Pappus of Alexandria.[8] • If an involution has one fixed point, it has another, and consists of the correspondence between harmonic conjugates with respect to these two points. In this instance the involution is termed "hyperbolic", while if there are no fixed points it is "elliptic". In the context of projectivities, fixed points are called double points.[6]: 53 Another type of involution occurring in projective geometry is a polarity which is a correlation of period 2. [9] Linear algebra Further information: Involutory matrix In linear algebra, an involution is a linear operator T on a vector space, such that $T^{2}=I$. Except for in characteristic 2, such operators are diagonalizable for a given basis with just 1s and −1s on the diagonal of the corresponding matrix. If the operator is orthogonal (an orthogonal involution), it is orthonormally diagonalizable. For example, suppose that a basis for a vector space V is chosen, and that e1 and e2 are basis elements. There exists a linear transformation f which sends e1 to e2, and sends e2 to e1, and which is the identity on all other basis vectors. It can be checked that f(f(x)) = x for all x in V. That is, f is an involution of V. For a specific basis, any linear operator can be represented by a matrix T. Every matrix has a transpose, obtained by swapping rows for columns. This transposition is an involution on the set of matrices. Since elementwise complex conjugation is an independent involution, the conjugate transpose or Hermitian adjoint is also an involution. The definition of involution extends readily to modules. Given a module M over a ring R, an R endomorphism f of M is called an involution if f 2 is the identity homomorphism on M. Involutions are related to idempotents; if 2 is invertible then they correspond in a one-to-one manner. In functional analysis, Banach -algebras and C-algebras are special types of Banach algebras with involutions. Quaternion algebra, groups, semigroups In a quaternion algebra, an (anti-)involution is defined by the following axioms: if we consider a transformation $x\mapsto f(x)$ then it is an involution if • $f(f(x))=x$ (it is its own inverse) • $f(x_{1}+x_{2})=f(x_{1})+f(x_{2})$ and $f(\lambda x)=\lambda f(x)$ (it is linear) • $f(x_{1}x_{2})=f(x_{1})f(x_{2})$ An anti-involution does not obey the last axiom but instead • $f(x_{1}x_{2})=f(x_{2})f(x_{1})$ This former law is sometimes called antidistributive. It also appears in groups as ${\left(xy\right)}^{-1}={\left(y\right)}^{-1}{\left(x\right)}^{-1}$. Taken as an axiom, it leads to the notion of semigroup with involution, of which there are natural examples that are not groups, for example square matrix multiplication (i.e. the full linear monoid) with transpose as the involution. Ring theory Further information: *-algebra In ring theory, the word involution is customarily taken to mean an antihomomorphism that is its own inverse function. Examples of involutions in common rings: • complex conjugation on the complex plane • multiplication by j in the split-complex numbers • taking the transpose in a matrix ring. Group theory In group theory, an element of a group is an involution if it has order 2; i.e. an involution is an element $a$ such that $a\neq e$ and a2 = e, where e is the identity element.[10] Originally, this definition agreed with the first definition above, since members of groups were always bijections from a set into itself; i.e., group was taken to mean permutation group. By the end of the 19th century, group was defined more broadly, and accordingly so was involution. A permutation is an involution precisely if and only if it can be written as a finite product of disjoint transpositions. The involutions of a group have a large impact on the group's structure. The study of involutions was instrumental in the classification of finite simple groups. An element $x$ of a group $G$ is called strongly real if there is an involution $t$ with $x^{t}=x^{-1}$ (where $x^{t}=x^{-1}=t^{-1}\cdot x\cdot t$). Coxeter groups are groups generated by involutions with the relations determined only by relations given for pairs of the generating involutions. Coxeter groups can be used, among other things, to describe the possible regular polyhedra and their generalizations to higher dimensions. Mathematical logic The operation of complement in Boolean algebras is an involution. Accordingly, negation in classical logic satisfies the law of double negation: ¬¬A is equivalent to A. Generally in non-classical logics, negation that satisfies the law of double negation is called involutive. In algebraic semantics, such a negation is realized as an involution on the algebra of truth values. Examples of logics which have involutive negation are Kleene and Bochvar three-valued logics, Łukasiewicz many-valued logic, fuzzy logic IMTL, etc. Involutive negation is sometimes added as an additional connective to logics with non-involutive negation; this is usual, for example, in t-norm fuzzy logics. The involutiveness of negation is an important characterization property for logics and the corresponding varieties of algebras. For instance, involutive negation characterizes Boolean algebras among Heyting algebras. Correspondingly, classical Boolean logic arises by adding the law of double negation to intuitionistic logic. The same relationship holds also between MV-algebras and BL-algebras (and so correspondingly between Łukasiewicz logic and fuzzy logic BL), IMTL and MTL, and other pairs of important varieties of algebras (resp. corresponding logics). In the study of binary relations, every relation has a converse relation. Since the converse of the converse is the original relation, the conversion operation is an involution on the category of relations. Binary relations are ordered through inclusion. While this ordering is reversed with the complementation involution, it is preserved under conversion. Computer science The XOR bitwise operation with a given value for one parameter is an involution. XOR masks were once used to draw graphics on images in such a way that drawing them twice on the background reverts the background to its original state. The NOT bitwise operation is also an involution, and is a special case of the XOR operation where one parameter has all bits set to 1. Another example is a bit mask and shift function operating on color values stored as integers, say in the form RGB, that swaps R and B, resulting in the form BGR. f(f(RGB))=RGB, f(f(BGR))=BGR. The RC4 cryptographic cipher is an involution, as encryption and decryption operations use the same function. Practically all mechanical cipher machines implement a reciprocal cipher, an involution on each typed-in letter. Instead of designing two kinds of machines, one for encrypting and one for decrypting, all the machines can be identical and can be set up (keyed) the same way.[11] See also • Automorphism • Idempotence • ROT13 References 1. Robert Alexander Adams, Calculus: Single Variable, 2006, ISBN 0321307143, p. 165 2. Russell, Bertrand (1903), Principles of mathematics (2nd ed.), W. W. Norton & Company, Inc, p. 426, ISBN 9781440054167 3. Kubrusly, Carlos S. (2011), The Elements of Operator Theory, Springer Science & Business Media, Problem 1.11(a), p. 27, ISBN 9780817649982. 4. Knuth, Donald E. (1973), The Art of Computer Programming, Volume 3: Sorting and Searching, Reading, Mass.: Addison-Wesley, pp. 48, 65, MR 0445948. 5. Zagier, D. (1990), "A one-sentence proof that every prime p≡ 1 (mod 4) is a sum of two squares", American Mathematical Monthly, 97 (2): 144, doi:10.2307/2323918, JSTOR 2323918, MR 1041893. 6. A.G. Pickford (1909) Elementary Projective Geometry, Cambridge University Press via Internet Archive 7. J. V. Field and J. J. Gray (1987) The Geometrical Work of Girard Desargues, (New York: Springer), p. 54 8. Ivor Thomas (editor) (1980) Selections Illustrating the History of Greek Mathematics, Volume II, number 362 in the Loeb Classical Library (Cambridge and London: Harvard and Heinemann), pp. 610–3 9. H. S. M. Coxeter (1969) Introduction to Geometry, pp 244–8, John Wiley & Sons 10. John S. Rose. "A Course on Group Theory". p. 10, section 1.13. 11. Greg Goebel. "The Mechanization of Ciphers". 2018. Further reading • Ell, Todd A.; Sangwine, Stephen J. (2007). "Quaternion involutions and anti-involutions". Computers & Mathematics with Applications. 53 (1): 137–143. arXiv:math/0506034. doi:10.1016/j.camwa.2006.10.029. S2CID 45639619. • Knus, Max-Albert; Merkurjev, Alexander; Rost, Markus; Tignol, Jean-Pierre (1998), The book of involutions, Colloquium Publications, vol. 44, With a preface by J. Tits, Providence, RI: American Mathematical Society, ISBN 0-8218-0904-0, Zbl 0955.16001 • "Involution", Encyclopedia of Mathematics, EMS Press, 2001 [1994] External links • Media related to Involution at Wikimedia Commons	Wikipedia
Self-linking number In knot theory, the self-linking number is an invariant of framed knots. It is related to the linking number of curves. A framing of a knot is a choice of a non-zero non-tangent vector at each point of the knot. More precisely, a framing is a choice of a non-zero section in the normal bundle of the knot, i.e. a (non-zero) normal vector field. Given a framed knot C, the self-linking number is defined to be the linking number of C with a new curve obtained by pushing points of C along the framing vectors. Given a Seifert surface for a knot, the associated Seifert framing is obtained by taking a tangent vector to the surface pointing inwards and perpendicular to the knot. The self-linking number obtained from a Seifert framing is always zero.[1] The blackboard framing of a knot is the framing where each of the vectors points in the vertical (z) direction. The self-linking number obtained from the blackboard framing is called the Kauffman self-linking number of the knot. This is not a knot invariant because it is only well-defined up to regular isotopy. References 1. Sumners, De Witt L.; Cruz-White, Irma I.; Ricca, Renzo L. (2021). "Zero helicity of Seifert framed defects". J. Phys. A. 54 (29): 295203. Bibcode:2021JPhA...54C5203S. doi:10.1088/1751-8121/abf45c. S2CID 233533506. • Chernov, Vladimir (2005), "Framed knots in 3-manifolds and affine self-linking numbers", Journal of Knot Theory and its Ramifications, 14 (6): 791–818, arXiv:math/0105139, doi:10.1142/S0218216505004056, MR 2172898. • Moskovich, Daniel (2004), "Framing and the self-linking integral", Far East Journal of Mathematical Sciences, 14 (2): 165–183, arXiv:math/0211223, Bibcode:2002math.....11223M, MR 2105976 Knot theory (knots and links) Hyperbolic • Figure-eight (41) • Three-twist (52) • Stevedore (61) • 62 • 63 • Endless (74) • Carrick mat (818) • Perko pair (10161) • (−2,3,7) pretzel (12n242) • Whitehead (52 1 ) • Borromean rings (63 2 ) • L10a140 • Conway knot (11n34) Satellite • Composite knots • Granny • Square • Knot sum Torus • Unknot (01) • Trefoil (31) • Cinquefoil (51) • Septafoil (71) • Unlink (02 1 ) • Hopf (22 1 ) • Solomon's (42 1 ) Invariants • Alternating • Arf invariant • Bridge no. • 2-bridge • Brunnian • Chirality • Invertible • Crosscap no. • Crossing no. • Finite type invariant • Hyperbolic volume • Khovanov homology • Genus • Knot group • Link group • Linking no. • Polynomial • Alexander • Bracket • HOMFLY • Jones • Kauffman • Pretzel • Prime • list • Stick no. • Tricolorability • Unknotting no. and problem Notation and operations • Alexander–Briggs notation • Conway notation • Dowker–Thistlethwaite notation • Flype • Mutation • Reidemeister move • Skein relation • Tabulation Other • Alexander's theorem • Berge • Braid theory • Conway sphere • Complement • Double torus • Fibered • Knot • List of knots and links • Ribbon • Slice • Sum • Tait conjectures • Twist • Wild • Writhe • Surgery theory • Category • Commons	Wikipedia
Loop (graph theory) In graph theory, a loop (also called a self-loop or a buckle) is an edge that connects a vertex to itself. A simple graph contains no loops. Depending on the context, a graph or a multigraph may be defined so as to either allow or disallow the presence of loops (often in concert with allowing or disallowing multiple edges between the same vertices): • Where graphs are defined so as to allow loops and multiple edges, a graph without loops or multiple edges is often distinguished from other graphs by calling it a simple graph. • Where graphs are defined so as to disallow loops and multiple edges, a graph that does have loops or multiple edges is often distinguished from the graphs that satisfy these constraints by calling it a multigraph or pseudograph. In a graph with one vertex, all edges must be loops. Such a graph is called a bouquet. Degree For an undirected graph, the degree of a vertex is equal to the number of adjacent vertices. A special case is a loop, which adds two to the degree. This can be understood by letting each connection of the loop edge count as its own adjacent vertex. In other words, a vertex with a loop "sees" itself as an adjacent vertex from both ends of the edge thus adding two, not one, to the degree. For a directed graph, a loop adds one to the in degree and one to the out degree. See also In graph theory • Cycle (graph theory) • Graph theory • Glossary of graph theory In topology • Möbius ladder • Möbius strip • Strange loop • Klein bottle References • Balakrishnan, V. K.; Graph Theory, McGraw-Hill; 1 edition (February 1, 1997). ISBN 0-07-005489-4. • Bollobás, Béla; Modern Graph Theory, Springer; 1st edition (August 12, 2002). ISBN 0-387-98488-7. • Diestel, Reinhard; Graph Theory, Springer; 2nd edition (February 18, 2000). ISBN 0-387-98976-5. • Gross, Jonathon L, and Yellen, Jay; Graph Theory and Its Applications, CRC Press (December 30, 1998). ISBN 0-8493-3982-0. • Gross, Jonathon L, and Yellen, Jay; (eds); Handbook of Graph Theory. CRC (December 29, 2003). ISBN 1-58488-090-2. • Zwillinger, Daniel; CRC Standard Mathematical Tables and Formulae, Chapman & Hall/CRC; 31st edition (November 27, 2002). ISBN 1-58488-291-3. External links • This article incorporates public domain material from Paul E. Black. "Self loop". Dictionary of Algorithms and Data Structures. NIST.	Wikipedia
Self-powered dynamic systems A self-powered dynamic system[1][2][3][4][5][6][7] is defined as a dynamic system powered by its own excessive kinetic energy, renewable energy or a combination of both. The particular area of work is the concept of fully or partially self-powered dynamic systems requiring zero or reduced external energy inputs. The exploited technologies are particularly associated with self-powered sensors, regenerative actuators, human powered devices, and dynamic systems powered by renewable resources (e.g. solar-powered airships[8][9]) as self-sustained systems. Various strategies can be employed to improve the design of a self-powered system and among them adopting a bio-inspired design is investigated to demonstrate the advantage of biomimetics in improving power density. The concept of "self-powered dynamic systems" in the figure is described as follows. I. Input power (e.g. fuel energy powering a vehicle engine or propulsion system), or input excitation (e.g. vibration excitation to a structure) to the system. The source of this input energy can be of renewable energy source (e.g. solar power to a dynamic system). II. The kinetic energy in the direction of motion of a dynamic system is only recovered if the system is stationary (e.g. a bridge structure), or the recoverable energy is negligible in comparison with the power required for motion (e.g. a low powered sensor). III. The movement of the dynamic system perpendicular to the desired direction of the motion is usually the wasted kinetic energy in the system (e.g. the vertical motion of an automobile suspension is wasted to heat energy in the shock absorbers, or vibration of an aircraft wing is converted into heat energy through structural damping). IV. The vertical movement of the dynamic system is a source of recoverable kinetic energy. V. The recoverable kinetic energy can be converted to electrical energy through an energy conversion mechanism such as an electromagnetic scheme (e.g. replacing the viscous damper of a car shock absorber with regenerative actuator), piezoelectric (e.g. embedding piezoelectric material in aircraft wings), or electrostatic (e.g. vibration of a micro cantilever in a MEMS sensor). VI. The recovered electrical power can be stored or used as a power source. VII. The recovered electrical energy can power subsystems of the dynamic system such as sensors and actuators. VIII. The recovered electrical power can be realized as an input to the dynamic system itself. Such self-powered schemes are particularly beneficial in development of self-powered sensors[10] and self-powered actuators[11] by employing energy harvesting techniques,[12][13][14] where kinetic energy is converted to electrical energy through piezoelectric, electromagnetic or electrostatic electromechanical mechanisms.[15] Developing a self-powered sensor eliminates the use of an external source of power such as a battery and therefore can be considered as a self-sustained system. A self-sustained system does not required maintenance (e.g. replacing the battery of the sensor at the end of the battery life). This is particularly beneficial in remote sensing and applications in hostile or inaccessible environments. References 1. Farbod Khoshnoud, David J. Dell, Y. K. Chen, R. K. Calay, Clarence W. de Silva, Houman Owhadi, Self-Powered Dynamic Systems, European Conference for Aeronautics and Space Sciences, Munich, Germany, 1–5 July 2013. 2. Farbod Khoshnoud, Ibrahim I. Esat, Clarence W. de Silva, Michael M. McKerns and Houman Owhadi, Self-Powered Dynamic Systems in the Framework of Optimal Uncertainty Quantification, ASME Journal of Dynamic Systems, Measurement, and Control, Volume 139, Issue 9, 2017, doi: 10.1115/1.4036367. 3. Farbod Khoshnoud, Ibrahim I. Esat, Clarence W. De Silva, Jason Rhodes, Alina Kiessling, Marco B. Quadrelli, Self-powered Solar Aerial Vehicles: towards infinite endurance UAVs, Unmanned Systems Journal, Vol. 8, No. 2, 2020, pp. 1–23. DOI: 10.1142/S2301385020500077. 4. Farbod Khoshnoud, Y. Zhang, R. Shimura, A. Shahba, G. Jin, G. Pissanidis, Y.K. Chen, Clarence W. De Silva, Energy regeneration from suspension dynamic modes and self-powered actuation, IEEE/ASME transaction on Mechatronics, Volume: 20, Issue: 5, pp. 2513 - 2524, 2015. 5. Clarence W. de Silva, Farbod Khoshnoud, Maoqing Li, Saman K. Halgamuge, Mechatronics: Fundamentals and Applications, (Chapter 12: Self-Powered and Bio-Inspired Dynamic Systems), CRC Press, 2015, ISBN 9781482239317. 6. Farbod Khoshnoud, Michael McKerns, Clarence W. De Silva, Ibrahim Esat, Richard H.C. Bonser, Houman Owhadi, Self-powered and Bio-inspired Dynamic Systems: Research and Education, ASME International Mechanical Engineering Congress and Exposition, Phoenix, Arizona, USA, 2016. 7. Vladimir V. Vantsevich, Michael V. Blundell, Advanced Autonomous Vehicle Design for Severe Environments, (Chapter: Farbod Khoshnoud, Clarence W. de Silva, Mechatronics of vehicle control and self-powered systems), IOS Press, sponsored by NATO Advanced Science Institute, ISBN online 978-1-61499-576-0, 2015. 8. Brunel Solar Powered Robotic Airship - Octoship. 9. Brunel Solar Powered Autonomous Airship. 10. Farbod Khoshnoud, Houman Owhadi, Clarence W. de Silva, Weidong Zhu and Carlos E. Ventura, Energy harvesting from ambient vibration with a nanotube based oscillator for remote vibration monitoring, Proc. of the Canadian Congress of Applied Mechanics, Vancouver, BC, pp. 805 – 808, June 2011. 11. Farbod Khoshnoud, Dinesh B. Sundar, Nuri M. Badi, Yong K. Chen, Rajnish K. Calay and Clarence W. de Silva, Energy harvesting from suspension systems using regenerative force actuators, International Journal of Vehicle Noise and Vibration, Vol. 9, Nos. 3/4, pp. 294 - 311, 2013. 12. Williams, C. B., and R. B. Yates. 1996. Analysis of a micro-electric generator for Microsystems, Sensors and Actuators A. 52, pp. 8–11. 13. James, E. P., M. J. Tudor, S. P. Beeby, N. R. Harris, P. Glynne-Jones, J. N. Ross, N. M. White. 2004. An investigation of self-powered systems for condition monitoring, applications. Sensors and Actuators A, 110, 171–176. 14. Roundy, S., P. K. Wright, and J. Rabaey. 2003. A study of low level vibrations as a power source for wireless sensor nodes. Computer Communications, 26, pp. 1131–1144. 15. Clarence W. De Silva, Mechatronics—A Foundation Course, CRC Press/Taylor&Francis. Boca Raton, FL, 2010.	Wikipedia
Regular extension In field theory, a branch of algebra, a field extension $L/k$ is said to be regular if k is algebraically closed in L (i.e., $k={\hat {k}}$ where ${\hat {k}}$ is the set of elements in L algebraic over k) and L is separable over k, or equivalently, $L\otimes _{k}{\overline {k}}$ is an integral domain when ${\overline {k}}$ is the algebraic closure of $k$ (that is, to say, $L,{\overline {k}}$ are linearly disjoint over k).[1][2] Properties • Regularity is transitive: if F/E and E/K are regular then so is F/K.[3] • If F/K is regular then so is E/K for any E between F and K.[3] • The extension L/k is regular if and only if every subfield of L finitely generated over k is regular over k.[2] • Any extension of an algebraically closed field is regular.[3][4] • An extension is regular if and only if it is separable and primary.[5] • A purely transcendental extension of a field is regular. Self-regular extension There is also a similar notion: a field extension $L/k$ is said to be self-regular if $L\otimes _{k}L$ is an integral domain. A self-regular extension is relatively algebraically closed in k.[6] However, a self-regular extension is not necessarily regular. References 1. Fried & Jarden (2008) p.38 2. Cohn (2003) p.425 3. Fried & Jarden (2008) p.39 4. Cohn (2003) p.426 5. Fried & Jarden (2008) p.44 6. Cohn (2003) p.427 • Fried, Michael D.; Jarden, Moshe (2008). Field arithmetic. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. Vol. 11 (3rd revised ed.). Springer-Verlag. pp. 38–41. ISBN 978-3-540-77269-9. Zbl 1145.12001. • M. Nagata (1985). Commutative field theory: new edition, Shokado. (Japanese) • Cohn, P. M. (2003). Basic Algebra. Groups, Rings, and Fields. Springer-Verlag. ISBN 1-85233-587-4. Zbl 1003.00001. • A. Weil, Foundations of algebraic geometry.	Wikipedia
Gömböc The gömböc (Hungarian: [ˈɡømbøt͡s] GUHM-buhts) is the first known physical example of a class of convex three-dimensional homogeneous bodies, called mono-monostatic, which, when resting on a flat surface have just one stable and one unstable point of equilibrium. The existence of this class was conjectured by the Russian mathematician Vladimir Arnold in 1995 and proven in 2006 by the Hungarian scientists Gábor Domokos and Péter Várkonyi by constructing at first a mathematical example and subsequently a physical example. Mono-monostatic shapes exist in countless varieties, most of which are close to a sphere, with a stringent shape tolerance (about one part in a thousand). The gömböc is the first mono-monostatic shape which has been constructed physically. It has a sharpened top, as shown in the photo. Its shape helped to explain the body structure of some tortoises in relation to their ability to return to an equilibrium position after being placed upside down.[1][2][3] Copies of the gömböc have been donated to institutions and museums, and the largest one was presented at the World Expo 2010 in Shanghai, China.[4] Name If analyzed quantitatively in terms of flatness and thickness, the discovered mono-monostatic bodies are the most sphere-like, apart from the sphere itself. Because of this, the first physical example was named gömböc, a diminutive form of gömb ("sphere" in Hungarian). History In geometry, a body with a single stable resting position is called monostatic, and the term mono-monostatic has been coined to describe a body which additionally has only one unstable point of balance. (The previously known monostatic polyhedron does not qualify, as it has several unstable equilibria.) A sphere weighted so that its center of mass is shifted from the geometrical center is mono-monostatic. However, it is inhomogeneous; that is, its material density varies across its body. Another example of an inhomogeneous mono-monostatic body is the Comeback Kid, Weeble or roly-poly toy (see left figure). At equilibrium, the center of mass and the contact point are on the line perpendicular to the ground. When the toy is pushed, its center of mass rises and shifts away from that line. This produces a righting moment, which returns the toy to the equilibrium position. The above examples of mono-monostatic objects are necessarily inhomogeneous. The question of whether it is possible to construct a three-dimensional body which is mono-monostatic but also homogeneous and convex was raised by Russian mathematician Vladimir Arnold in 1995. Being convex is essential as it is trivial to construct a mono-monostatic non-convex body: an example would be a ball with a cavity inside it. It was already well known, from a geometrical and topological generalization of the classical four-vertex theorem, that a plane curve has at least four extrema of curvature, specifically, at least two local maxima and at least two local minima (see right figure), meaning that a (convex) mono-monostatic object does not exist in two dimensions. Whereas a common anticipation was that a three-dimensional body should also have at least four extrema, Arnold conjectured that this number could be smaller.[5] Mathematical solution The problem was solved in 2006 by Gábor Domokos and Péter Várkonyi. Domokos met Arnold in 1995 at a major mathematics conference in Hamburg, where Arnold presented a plenary talk illustrating that most geometrical problems have four solutions or extremal points. In a personal discussion, however, Arnold questioned whether four is a requirement for mono-monostatic bodies and encouraged Domokos to seek examples with fewer equilibria.[6] The rigorous proof of the solution can be found in references of their work.[5] The summary of the results is that the three-dimensional homogeneous convex (mono-monostatic) body, which has one stable and one unstable equilibrium point, does exist and is not unique. Such bodies are hard to visualize, describe or identify. Their form is dissimilar to any typical representative of any other equilibrium geometrical class. They should have minimal "flatness" and, to avoid having two unstable equilibria, must also have minimal "thinness". They are the only non-degenerate objects having simultaneously minimal flatness and thinness. The shape of those bodies is susceptible to small variation, outside which it is no longer mono-monostatic. For example, the first solution of Domokos and Várkonyi closely resembled a sphere, with a shape deviation of only 10−5. It was dismissed as it was tough to test experimentally.[7] The gömböc, as the first physical example, is less sensitive; yet it has a shape tolerance of 10−3, that is 0.1 mm for a 10 cm size. Domokos developed a classification system for shapes based on their points of equilibrium by analyzing pebbles and noting their equilibrium points.[8] In one experiment, Domokos and his wife tested 2000 pebbles collected on the beaches of the Greek island of Rhodes and found not a single mono-monostatic body among them, illustrating the difficulty of finding or constructing such a body.[5][7] The solution of Domokos and Várkonyi has curved edges and resembles a sphere with a squashed top. In the top figure, it rests in its stable equilibrium. Its unstable equilibrium position is obtained by rotating the figure 180° about a horizontal axis. Theoretically, it will rest there, but the smallest perturbation will bring it back to the stable point. All mono-monostatic shapes (including the gömböc shape) have sphere-like properties. In particular, its flatness and thinness are minimal, and this is the only type of nondegenerate object with this property.[5] Domokos and Várkonyi are interested in finding a polyhedral solution with a surface consisting of a minimal number of flat planes. There is a prize[3] to anyone who finds the respective minimal numbers of F, E, and V faces, edges and vertices for such a polyhedron, which amounts to $10,000 divided by the number C = F + E + V − 2, which is called the mechanical complexity of mono-monostatic polyhedra. It has been proved that one can approximate a curvilinear mono-monostatic shape with a finite number of discrete surfaces;[9] however, they estimate that it would take thousands of planes to achieve that. By offering this prize, they hope to stimulate finding a radically different solution from their own.[3] Relation to animals The balancing properties of the gömböc are associated with the "righting response" ⁠— the ability to turn back when placed upside down⁠ ⁠— of shelled animals such as tortoises and beetles. These animals may become flipped over in a fight or predator attack, and so the righting response is crucial for survival. In order to right themselves, relatively flat animals (such as beetles) heavily rely on momentum and thrust developed by moving their limbs and wings. However, the limbs of many dome-shaped tortoises are too short to be of use in for righting. Domokos and Várkonyi spent a year measuring tortoises in the Budapest Zoo, Hungarian Museum of Natural History and various pet shops in Budapest, digitizing and analyzing their shells, and attempting to "explain" their body shapes and functions from their geometry work published by the biology journal Proceedings of the Royal Society.[10] It was then immediately popularized in several science news reports, including the science journals Nature[2] and Science.[3] The reported model can be summarized as flat shells in tortoises are advantageous for swimming and digging. However, the sharp shell edges hinder the rolling. Those tortoises usually have long legs and necks and actively use them to push the ground to return to the normal position if placed upside down. On the contrary, "rounder" tortoises easily roll on their own; those have shorter limbs and use them little when recovering from lost balance. (Some limb movement would always be needed because of imperfect shell shape, ground conditions, etc.) Round shells also resist better the crushing jaws of a predator and are better for thermal regulation.[1][2][3] Art In the fall of 2020, the Korzo Theatre in The Hague and the Theatre Municipal in Biarritz presented the solo dance production "Gömböc" [11] by French choreographer Antonin Comestaz [12] A 2021 solo exhibition of conceptual artist Ryan Gander evolved around the theme of self-righting and featured seven large gömböc shapes gradually covered by black volcanic sand.[13] Media For their discovery, Domokos and Várkonyi were decorated with the Knight's Cross of the Republic of Hungary.[14] The New York Times Magazine selected the gömböc as one of the 70 most interesting ideas of the year 2007.[15] The Stamp News website[16] shows Hungary's new stamps issued on 30 April 2010, which illustrate a gömböc in different positions. The stamp booklets are arranged so that the gömböc appears to come to life when the booklet is flipped. The stamps were issued in association with the gömböc on display at the World Expo 2010 (1 May to 31 October). This was also covered by the Linn's Stamp News magazine.[17] See also • Instability • Monostatic polytope • Self-righting watercraft References 1. Summers, Adam (March 2009). "The Living Gömböc. Some tortoise shells evolved the ideal shape for staying upright". Natural History. 118 (2): 22–23. 2. Ball, Philip (16 October 2007). "How tortoises turn right-side up". Nature News. doi:10.1038/news.2007.170. S2CID 178518465. 3. Rehmeyer, Julie (5 April 2007). "Can't Knock It Down". Science News. 4. Hungary Pavilion features Gomboc, expo.shanghaidaily.com (12 July 2010) 5. Varkonyi, P.L.; Domokos, G. (2006). "Mono-monostatic bodies: the answer to Arnold's question" (PDF). The Mathematical Intelligencer. 28 (4): 34–38. doi:10.1007/bf02984701. S2CID 15720880. 6. Domokos, Gábor (2008). "My Lunch with Arnold" (PDF). The Mathematical Intelligencer. 28 (4): 31–33. doi:10.1007/BF02984700. S2CID 120684940. 7. Freiberger, Marianne (May 2009). "The Story of the Gömböc". Plus magazine. 8. Varkonyi, P.L.; Domokos, G. (2006). "Static Equilibria of Rigid Bodies: Dice, Pebbles, and the Poincare-Hopf Theorem". Journal of Nonlinear Science. 16 (3): 255. Bibcode:2006JNS....16..255V. doi:10.1007/s00332-005-0691-8. S2CID 17412564. 9. Lángi, Zsolt (2022). "A solution to some problems of Conway and Guy on monostable polyhedra" (PDF). Bulletin of the London Mathematical Society. 54 (2): 501–516. doi:10.1112/blms.12579. S2CID 220968924. 10. Domokos, G.; Varkonyi, P.L. (2008). "Geometry and self-righting of turtles" (free download pdf). Proc. R. Soc. B. 275 (1630): 11–17. doi:10.1098/rspb.2007.1188. PMC 2562404. PMID 17939984. 11. "" Gömböc " d'Antonin Comestaz". dansercanalhistorique. 22 September 2020. 12. "Categorie:Choreografie Antonin Comestaz". TheaterEncyclopedie. 30 January 2018. 13. "Exhibition \| Ryan Gander, 'The Self Righting of All Things' at Lisson Gallery, Lisson Street, London, United Kingdom". ocula.com. 14 November 2021. 14. A gömböc for the Whipple. News, University of Cambridge (27 April 2009) 15. Per-Lee, Myra (9 December 2007) Whose Bright Idea Was That? The New York Times Magazine Ideas of 2007. Inventorspot.com. 16. Better City – Better Life: Shanghai World Expo 2010 Archived 16 August 2017 at the Wayback Machine. Stampnews.com (22 November 2010). Retrieved on 20 October 2016. 17. McCarty, Denise (28 June 2010) "World of New Issues: Expo stamps picture Hungary's gömböc, Iceland's ice cube". Linn's Stamp News p. 14 External links • Non-technical description of development, with short video • Expo 2010 presentation of a gömböc shape, with photos	Wikipedia
Self-similar process Self-similar processes are types of stochastic processes that exhibit the phenomenon of self-similarity. A self-similar phenomenon behaves the same when viewed at different degrees of magnification, or different scales on a dimension (space or time). Self-similar processes can sometimes be described using heavy-tailed distributions, also known as long-tailed distributions. Examples of such processes include traffic processes, such as packet inter-arrival times and burst lengths. Self-similar processes can exhibit long-range dependency. Overview The design of robust and reliable networks and network services has become an increasingly challenging task in today's Internet world. To achieve this goal, understanding the characteristics of Internet traffic plays a more and more critical role. Empirical studies of measured traffic traces have led to the wide recognition of self-similarity in network traffic.[1] Self-similar Ethernet traffic exhibits dependencies over a long range of time scales. This is to be contrasted with telephone traffic which is Poisson in its arrival and departure process.[2] In traditional Poisson traffic, the short-term fluctuations would average out, and a graph covering a large amount of time would approach a constant value. Heavy-tailed distributions have been observed in many natural phenomena including both physical and sociological phenomena. Mandelbrot established the use of heavy-tailed distributions to model real-world fractal phenomena, e.g. Stock markets, earthquakes, climate, and the weather. Ethernet, WWW, SS7, TCP, FTP, TELNET and VBR video (digitised video of the type that is transmitted over ATM networks) traffic is self-similar. Self-similarity in packetised data networks can be caused by the distribution of file sizes, human interactions and/ or Ethernet dynamics. Self-similar and long-range dependent characteristics in computer networks present a fundamentally different set of problems to people doing analysis and/or design of networks, and many of the previous assumptions upon which systems have been built are no longer valid in the presence of self-similarity.[3] The Poisson distribution Before the heavy-tailed distribution is introduced mathematically, the Poisson process with a memoryless waiting-time distribution, used to model (among many things) traditional telephony networks, is briefly reviewed below. Assuming pure-chance arrivals and pure-chance terminations leads to the following: • The number of call arrivals in a given time has a Poisson distribution, i.e.: $P(a)=\left({\frac {\mu ^{a}}{a!}}\right)e^{-\mu },$ where a is the number of call arrivals in time T, and $\mu $ is the mean number of call arrivals in time T. For this reason, pure-chance traffic is also known as Poisson traffic. • The number of call departures in a given time, also has a Poisson distribution, i.e.: $P(d)=\left({\frac {\lambda ^{d}}{d!}}\right)e^{-\lambda },$ where d is the number of call departures in time T and $\lambda $ is the mean number of call departures in time T. • The intervals, T, between call arrivals and departures are intervals between independent, identically distributed random events. It can be shown that these intervals have a negative exponential distribution, i.e.: $P[T\geq \ t]=e^{-t/h},\,$ where h is the mean holding time (MHT). The heavy-tail distribution Main article: Heavy-tailed distribution A distribution is said to have a heavy tail if $\lim _{x\to \infty }e^{\lambda x}\Pr[X>x]=\infty \quad {\mbox{for all }}\lambda >0.\,$ One simple example of a heavy-tailed distribution is the Pareto distribution. Modelling self-similar traffic Since (unlike traditional telephony traffic) packetised traffic exhibits self-similar or fractal characteristics, conventional traffic models do not apply to networks which carry self-similar traffic. With the convergence of voice and data, the future multi-service network will be based on packetised traffic, and models which accurately reflect the nature of self-similar traffic will be required to develop, design and dimension future multi-service networks. Previous analytic work done in Internet studies adopted assumptions such as exponentially-distributed packet inter-arrivals, and conclusions reached under such assumptions may be misleading or incorrect in the presence of heavy-tailed distributions.[2] Deriving mathematical models which accurately represent long-range dependent traffic is a fertile area of research. Self-similar stochastic processes modeled by Tweedie distributions Leland et al have provided a mathematical formalism to describe self-similar stochastic processes.[4] For the sequence of numbers $Y=(Y_{i}:i=0,1,2,...,N)$ with mean ${\hat {\mu }}={\text{E}}(Y_{i})$, deviations $y_{i}=Y_{i}-{\hat {\mu }}$, variance ${\hat {\sigma }}^{2}={\text{E}}(y_{i}^{2})$, and autocorrelation function $r(k)={\text{E}}(y_{i},y_{i+k})/{\text{E}}(y_{i}^{2})$ with lag k, if the autocorrelation of this sequence has the long range behavior $r(k)\sim k^{-d}L(k)$ as k→∞ and where L(k) is a slowly varying function at large values of k, this sequence is called a self-similar process. The method of expanding bins can be used to analyze self-similar processes. Consider a set of equal-sized non-overlapping bins that divides the original sequence of N elements into groups of m equal-sized segments (N/m is integer) so that new reproductive sequences, based on the mean values, can be defined: $Y_{i}^{(m)}=(Y_{im-m+1}+...+Y_{im})/m$. The variance determined from this sequence will scale as the bin size changes such that ${\text{var}}[Y^{(m)}]={\hat {\sigma }}^{2}m^{-d}$ if and only if the autocorrelation has the limiting form[5] $\lim _{k\to \infty }r(k)/k^{-d}=(2-d)(1-d)/2$. One can also construct a set of corresponding additive sequences $Z_{i}^{(m)}=mY_{i}^{(m)}$, based on the expanding bins, $Z_{i}^{(m)}=(Y_{im-m+1}+...+Y_{im})$. Provided the autocorrelation function exhibits the same behavior, the additive sequences will obey the relationship ${\text{var}}[Z_{i}^{(m)}]=m^{2}{\text{var}}[Y^{(m)}]=({\hat {\sigma }}^{2}/{\hat {\mu }}^{2-d}){\text{E}}[Z_{i}^{(m)}]^{2-d}$ Since ${\hat {\mu }}$ and ${\hat {\sigma }}^{2}$ are constants this relationship constitutes a variance-to-mean power law (Taylor's law), with p=2-d.[6] Tweedie distributions are a special case of exponential dispersion models, a class of models used to describe error distributions for the generalized linear model.[7] These Tweedie distributions are characterized by an inherent scale invariance and thus for any random variable Y that obeys a Tweedie distribution, the variance var(Y) relates to the mean E(Y) by the power law, ${\text{var}}\,(Y)=a[{\text{E}}\,(Y)]^{p},$ where a and p are positive constants. The exponent p for the variance to mean power law associated with certain self-similar stochastic processes ranges between 1 and 2 and thus may be modeled in part by a Tweedie compound Poisson–gamma distribution.[6] The additive form of the Tweedie compound Poisson-gamma model has the cumulant generating function (CGF), $K_{p}^{}(s;\theta ,\lambda )=\lambda \kappa _{p}(\theta )[(1+s/\theta )^{\alpha }-1]$, where $\kappa _{p}(\theta )={\dfrac {\alpha -1}{\alpha }}\left({\dfrac {\theta }{\alpha -1}}\right)^{\alpha }$, is the cumulant function, α is the Tweedie exponent $\alpha ={\dfrac {p-2}{p-1}}$, s is the generating function variable, θ is the canonical parameter and λ is the index parameter. The first and second derivatives of the CGF, with s=0, yields the mean and variance, respectively. One can thus confirm that for the additive models the variance relates to the mean by the power law, $\mathrm {var} (Z)\propto \mathrm {E} (Z)^{p}$. Whereas this Tweedie compound Poisson-gamma CGF will represent the probability density function for certain self-similar stochastic processes, it does not return information regarding the long range correlations inherent to the sequence Y. Nonetheless, the Tweedie distributions provide a means understand the possible origins of self-similar stochastic processes for reason of their role as foci for a central limit-like convergence effect known as the Tweedie convergence theorem. In nontechnical terms this theorem tells us that any exponential dispersion model that asymptotically manifests a variance-to-mean power law is required to have a variance function that comes within the domain of attraction of a Tweedie model. The Tweedie convergence theorem can be used to explain the origin of the variance to mean power law, 1/f noise and multifractality, features associated with self-similar processes.[6] Network performance Network performance degrades gradually with increasing self-similarity. The more self-similar the traffic, the longer the queue size. The queue length distribution of self-similar traffic decays more slowly than with Poisson sources. However, long-range dependence implies nothing about its short-term correlations which affect performance in small buffers. Additionally, aggregating streams of self-similar traffic typically intensifies the self-similarity ("burstiness") rather than smoothing it, compounding the problem. Self-similar traffic exhibits the persistence of clustering which has a negative impact on network performance. • With Poisson traffic (found in conventional telephony networks), clustering occurs in the short term but smooths out over the long term. • With self-similar traffic, the bursty behaviour may itself be bursty, which exacerbates the clustering phenomena, and degrades network performance. Many aspects of network quality of service depend on coping with traffic peaks that might cause network failures, such as • Cell/packet loss and queue overflow • Violation of delay bounds e.g. in video • Worst cases in statistical multiplexing Poisson processes are well-behaved because they are stateless, and peak loading is not sustained, so queues do not fill. With long-range order, peaks last longer and have greater impact: the equilibrium shifts for a while.[8] See also • Long-tail traffic References 1. Park, Kihong; Willinger, Walter (2000), Self-Similar Network Traffic and Performance Evaluation, New York, NY, USA: John Wiley & Sons, Inc., ISBN 0471319740. 2. "Appendix: Heavy-tailed Distributions". Cs.bu.edu. 2001-04-12. Retrieved 2012-06-25. 3. "The Self-Similarity and Long Range Dependence in Networks Web site". Cs.bu.edu. Retrieved 2012-06-25. 4. Leland, W E; Leland, W. E.; M. S. Taqqu; W. Willinger; D. V. Wilson (1994). "On the self-similar nature of ethernet traffic". IEEE/ACM Trans. Netw. 2: 1–15. doi:10.1109/90.282603. S2CID 6011907. 5. Tsybakov B & Georganas ND (1997) On self-similar traffic in ATM queues: definitions, overflow probability bound, and cell delay distribution. IEEE/ACM Trans. Netw. 5, 397–409 6. Kendal, Wayne S.; Jørgensen, Bent (2011-12-27). "Tweedie convergence: A mathematical basis for Taylor's power law, 1/f noise, and multifractality". Physical Review E. American Physical Society (APS). 84 (6): 066120. Bibcode:2011PhRvE..84f6120K. doi:10.1103/physreve.84.066120. ISSN 1539-3755. PMID 22304168. 7. Jørgensen, Bent (1997). The theory of dispersion models. Chapman & Hall. ISBN 978-0412997112. 8. "Everything you always wanted to know about Self-Similar Network Traffic and Long-Range Dependency, but were ashamed to ask". Cs.kent.ac.uk. Retrieved 2012-06-25. External links • A site offering numerous links to articles written on the effect of self-similar traffic on network performance. 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Self-similarity matrix In data analysis, the self-similarity matrix is a graphical representation of similar sequences in a data series. Similarity can be explained by different measures, like spatial distance (distance matrix), correlation, or comparison of local histograms or spectral properties (e.g. IXEGRAM[1]). This technique is also applied for the search of a given pattern in a long data series as in gene matching. A similarity plot can be the starting point for dot plots or recurrence plots. Definition To construct a self-similarity matrix, one first transforms a data series into an ordered sequence of feature vectors $V=(v_{1},v_{2},\ldots ,v_{n})$, where each vector $v_{i}$ describes the relevant features of a data series in a given local interval. Then the self-similarity matrix is formed by computing the similarity of pairs of feature vectors $S(j,k)=s(v_{j},v_{k})\quad j,k\in (1,\ldots ,n)$ where $s(v_{j},v_{k})$ is a function measuring the similarity of the two vectors, for instance, the inner product $s(v_{j},v_{k})=v_{j}\cdot v_{k}$. Then similar segments of feature vectors will show up as path of high similarity along diagonals of the matrix.[2] Similarity plots are used for action recognition that is invariant to point of view [3] and for audio segmentation using spectral clustering of the self-similarity matrix.[4] Example See also • Recurrence plot • Distance matrix • Similarity matrix • Substitution matrix • Dot plot (bioinformatics) References 1. M. A. Casey; A. Westner (July 2000). "Separation of mixed audio sources by independent subspace analysis" (PDF). Proc. Int. Comput. Music Conf. Retrieved 2013-11-19. 2. Müller, Meinard; Michael Clausen (2007). "Transposition-invariant self-similarity matrices" (PDF). Proceedings of the 8th International Conference on Music Information Retrieval (ISMIR 2007): 47–50. Retrieved 2013-11-19. 3. I.N. Junejo; E. Dexter; I. Laptev; Patrick Pérez (2008). "Cross-View Action Recognition from Temporal Self-similarities". Computer Vision – ECCV 2008. Lecture Notes in Computer Science. Vol. 5303. pp. 293–306. CiteSeerX 10.1.1.405.1518. doi:10.1007/978-3-540-88688-4_22. ISBN 978-3-540-88685-3. 4. Dubnov, Shlomo; Ted Apel (2004). "Audio segmentation by singular value clustering". Proceedings of Computer Music Conference (ICMC 2004). CiteSeerX 10.1.1.324.4298. 5. Cross-View Action Recognition from Temporal Self-Similarities (2008), I. Junejo, E. Dexter, I. Laptev, and Patrick Pérez) Further reading • N. Marwan; M. C. Romano; M. Thiel; J. Kurths (2007). "Recurrence Plots for the Analysis of Complex Systems". Physics Reports. 438 (5–6): 237. Bibcode:2007PhR...438..237M. doi:10.1016/j.physrep.2006.11.001. • J. Foote (1999). "Visualizing music and audio using self-similarity". Proceedings of the seventh ACM international conference on Multimedia (Part 1). pp. 77–80. CiteSeerX 10.1.1.223.194. doi:10.1145/319463.319472. ISBN 978-1581131512. S2CID 3329298.{{cite book}}: CS1 maint: date and year (link) • M. A. Casey (2002). B.S. Manjunath; P. Salembier; T. Sikora (eds.). Sound Classification and Similarity Tools. pp. 309–323. ISBN 978-0471486787. {{cite book}}: \|journal= ignored (help) External links • http://www.recurrence-plot.tk/related_methods.php	Wikipedia
Self-verifying theories Self-verifying theories are consistent first-order systems of arithmetic, much weaker than Peano arithmetic, that are capable of proving their own consistency. Dan Willard was the first to investigate their properties, and he has described a family of such systems. According to Gödel's incompleteness theorem, these systems cannot contain the theory of Peano arithmetic nor its weak fragment Robinson arithmetic; nonetheless, they can contain strong theorems. In outline, the key to Willard's construction of his system is to formalise enough of the Gödel machinery to talk about provability internally without being able to formalise diagonalisation. Diagonalisation depends upon being able to prove that multiplication is a total function (and in the earlier versions of the result, addition also). Addition and multiplication are not function symbols of Willard's language; instead, subtraction and division are, with the addition and multiplication predicates being defined in terms of these. Here, one cannot prove the $\Pi _{2}^{0}$ sentence expressing totality of multiplication: $(\forall x,y)\ (\exists z)\ {\rm {multiply}}(x,y,z).$ where ${\rm {multiply}}$ is the three-place predicate which stands for $z/y=x.$ When the operations are expressed in this way, provability of a given sentence can be encoded as an arithmetic sentence describing termination of an analytic tableau. Provability of consistency can then simply be added as an axiom. The resulting system can be proven consistent by means of a relative consistency argument with respect to ordinary arithmetic. One can further add any true $\Pi _{1}^{0}$ sentence of arithmetic to the theory while still retaining consistency of the theory. References • Solovay, Robert M. (9 October 1989). "Injecting Inconsistencies into Models of PA". Annals of Pure and Applied Logic. 44 (1–2): 101–132. doi:10.1016/0168-0072(89)90048-1. • Willard, Dan E. (Jun 2001). "Self-Verifying Axiom Systems, the Incompleteness Theorem and Related Reflection Principles". The Journal of Symbolic Logic. 66 (2): 536–596. doi:10.2307/2695030. JSTOR 2695030. S2CID 2822314. • Willard, Dan E. (Mar 2002). "How to Extend the Semantic Tableaux and Cut-Free Versions of the Second Incompleteness Theorem almost to Robinson's Arithmetic Q". The Journal of Symbolic Logic. 67 (1): 465–496. doi:10.2178/jsl/1190150055. JSTOR 2695021. S2CID 8311827. External links • Dan Willard's home page. 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Sturm–Liouville theory In mathematics and its applications, a Sturm–Liouville problem is a second-order linear ordinary differential equation of the form: ${\frac {\mathrm {d} }{\mathrm {d} x}}\!\!\left[\,p(x){\frac {\mathrm {d} y}{\mathrm {d} x}}\right]+q(x)y=-\lambda \,w(x)y,$ for given functions $p(x)$, $q(x)$ and $w(x)$, together with some boundary conditions at extreme values of $x$. The goals of a given Sturm–Liouville problem are: • To find the λ for which there exists a non-trivial solution to the problem. Such values λ are called the eigenvalues of the problem. • For each eigenvalue λ, to find the corresponding solution $y=y(x)$ of the problem. Such functions $y$ are called the eigenfunctions associated to each λ. Sturm–Liouville theory is the general study of Sturm–Liouville problems. In particular, for a "regular" Sturm–Liouville problem, it can be shown that there are an infinite number of eigenvalues each with a unique eigenfunction, and that these eigenfunctions form an orthonormal basis of a certain Hilbert space of functions. This theory is important in applied mathematics, where Sturm–Liouville problems occur very frequently, particularly when dealing with separable linear partial differential equations. For example, in quantum mechanics, the one-dimensional time-independent Schrödinger equation is a Sturm–Liouville problem. Sturm–Liouville theory is named after Jacques Charles François Sturm (1803–1855) and Joseph Liouville (1809–1882) who developed the theory. Main results The main results in Sturm–Liouville theory apply to a Sturm–Liouville problem ${\frac {\mathrm {d} }{\mathrm {d} x}}\!\!\left[\,p(x){\frac {\mathrm {d} y}{\mathrm {d} x}}\right]+q(x)y=-\lambda \,w(x)y,$ (1) on a finite interval $[a,b]$ that is "regular". The problem is said to be regular if: • the coefficient functions $p,q,w$ and the derivative $p'$ are all continuous on $[a,b]$; • $p(x)>0$ and $w(x)>0$ for all $x\in [a,b]$; • the problem has separated boundary conditions of the form: $\alpha _{1}y(a)+\alpha _{2}y'(a)=0\qquad \alpha _{1},\alpha _{2}{\text{ not both }}0$ (2) $\beta _{1}y(b)\,+\,\beta _{2}y'(b)=0\qquad \beta _{1},\beta _{2}{\text{ not both }}0.$ (3) The function $w=w(x)$, sometimes denoted $r=r(x)$, is called the weight or density function. The goals of a Sturm–Liouville problem are: • to find the eigenvalues: those λ for which there exists a non-trivial solution; • for each eigenvalue λ, to find the corresponding eigenfunction $y=y(x)$. For a regular Sturm–Liouville problem, a function $y=y(x)$ is called a solution if it is continuously differentiable and satisfies the equation (1) at every $x\in (a,b)$. In the case of more general $p,q,w$, the solutions must be understood in a weak sense. The terms eigenvalue and eigenvector are used because the solutions correspond to the eigenvalues and eigenfunctions of a Hermitian differential operator in an appropriate Hilbert space of functions with inner product defined using the weight function. Sturm–Liouville theory studies the existence and asymptotic behavior of the eigenvalues, the corresponding qualitative theory of the eigenfunctions and their completeness in the function space. The main result of Sturm–Liouville theory states that, for any regular Sturm–Liouville problem: • The eigenvalues $\lambda _{1},\lambda _{2},\dots $ are real and can be numbered so that $\lambda _{1}<\lambda _{2}<\cdots <\lambda _{n}<\cdots \to \infty ;$ ;} • Corresponding to each eigenvalue $\lambda _{n}$ is a unique (up to constant multiple) eigenfunction $y_{n}=y_{n}(x)$ with exactly $n-1$ zeros in $[a,b]$, called the nth fundamental solution. • The normalized eigenfunctions $y_{n}$ form an orthonormal basis under the w-weighted inner product in the Hilbert space $L^{2}([a,b],w(x)\,\mathrm {d} x)$; that is, $\langle y_{n},y_{m}\rangle =\int _{a}^{b}y_{n}(x)y_{m}(x)w(x)\,\mathrm {d} x=\delta _{nm},$ where $\delta _{nm}$ is the Kronecker delta. Reduction to Sturm–Liouville form The differential equation (1) is said to be in Sturm–Liouville form or self-adjoint form. All second-order linear homogenous ordinary differential equations can be recast in the form on the left-hand side of (1) by multiplying both sides of the equation by an appropriate integrating factor (although the same is not true of second-order partial differential equations, or if y is a vector). Some examples are below. Bessel equation $x^{2}y''+xy'+\left(x^{2}-\nu ^{2}\right)y=0$ which can be written in Sturm–Liouville form (first by dividing through by x, then by collapsing the first two terms on the left into one term) as $\left(xy'\right)'+\left(x-{\frac {\nu ^{2}}{x}}\right)y=0.$ Legendre equation $\left(1-x^{2}\right)y''-2xy'+\nu (\nu +1)y=0$ which can easily be put into Sturm–Liouville form, since d/dx(1 − x2) = −2x, so the Legendre equation is equivalent to $\left(\left(1-x^{2}\right)y'\right)'+\nu (\nu +1)y=0$ Example using an integrating factor $x^{3}y''-xy'+2y=0$ Divide throughout by x3: $y''-{\frac {1}{x^{2}}}y'+{\frac {2}{x^{3}}}y=0$ Multiplying throughout by an integrating factor of $\mu (x)=\exp \left(\int -{\frac {dx}{x^{2}}}\right)=e^{{1}/{x}},$ gives $e^{{1}/{x}}y''-{\frac {e^{{1}/{x}}}{x^{2}}}y'+{\frac {2e^{{1}/{x}}}{x^{3}}}y=0$ which can be easily put into Sturm–Liouville form since ${\frac {d}{dx}}e^{{1}/{x}}=-{\frac {e^{{1}/{x}}}{x^{2}}}$ so the differential equation is equivalent to $\left(e^{{1}/{x}}y'\right)'+{\frac {2e^{{1}/{x}}}{x^{3}}}y=0.$ Integrating factor for general second-order homogenous equation $P(x)y''+Q(x)y'+R(x)y=0$ Multiplying through by the integrating factor $\mu (x)={\frac {1}{P(x)}}\exp \left(\int {\frac {Q(x)}{P(x)}}\,dx\right),$ and then collecting gives the Sturm–Liouville form: ${\frac {d}{dx}}\left(\mu (x)P(x)y'\right)+\mu (x)R(x)y=0,$ or, explicitly: ${\frac {d}{dx}}\left(\exp \left(\int {\frac {Q(x)}{P(x)}}\,dx\right)y'\right)+{\frac {R(x)}{P(x)}}\exp \left(\int {\frac {Q(x)}{P(x)}}\,dx\right)y=0.$ Sturm–Liouville equations as self-adjoint differential operators The mapping defined by: $Lu=-{\frac {1}{w(x)}}\left({\frac {d}{dx}}\left[p(x)\,{\frac {du}{dx}}\right]+q(x)u\right)$ can be viewed as a linear operator L mapping a function u to another function Lu, and it can be studied in the context of functional analysis. In fact, equation (1) can be written as $Lu=\lambda u.$ This is precisely the eigenvalue problem; that is, one seeks eigenvalues λ1, λ2, λ3,... and the corresponding eigenvectors u1, u2, u3,... of the L operator. The proper setting for this problem is the Hilbert space $L^{2}([a,b],w(x)\,dx)$ with scalar product $\langle f,g\rangle =\int _{a}^{b}{\overline {f(x)}}g(x)w(x)\,dx.$ In this space L is defined on sufficiently smooth functions which satisfy the above regular boundary conditions. Moreover, L is a self-adjoint operator: $\langle Lf,g\rangle =\langle f,Lg\rangle .$ This can be seen formally by using integration by parts twice, where the boundary terms vanish by virtue of the boundary conditions. It then follows that the eigenvalues of a Sturm–Liouville operator are real and that eigenfunctions of L corresponding to different eigenvalues are orthogonal. However, this operator is unbounded and hence existence of an orthonormal basis of eigenfunctions is not evident. To overcome this problem, one looks at the resolvent $\left(L-z\right)^{-1},\qquad z\in \mathbb {R} ,$ where z is not an eigenvalue. Then, computing the resolvent amounts to solving a nonhomogeneous equation, which can be done using the variation of parameters formula. This shows that the resolvent is an integral operator with a continuous symmetric kernel (the Green's function of the problem). As a consequence of the Arzelà–Ascoli theorem, this integral operator is compact and existence of a sequence of eigenvalues αn which converge to 0 and eigenfunctions which form an orthonormal basis follows from the spectral theorem for compact operators. Finally, note that $\left(L-z\right)^{-1}u=\alpha u,\qquad Lu=\left(z+\alpha ^{-1}\right)u,$ are equivalent, so we may take $\lambda =z+\alpha ^{-1}$ with the same eigenfunctions. If the interval is unbounded, or if the coefficients have singularities at the boundary points, one calls L singular. In this case, the spectrum no longer consists of eigenvalues alone and can contain a continuous component. There is still an associated eigenfunction expansion (similar to Fourier series versus Fourier transform). This is important in quantum mechanics, since the one-dimensional time-independent Schrödinger equation is a special case of a Sturm–Liouville equation. Application to inhomogeneous second-order boundary value problems Consider a general inhomogeneous second-order linear differential equation $P(x)y''+Q(x)y'+R(x)y=f$ for given functions $P(x),Q(x),R(x),f(x)$. As before, this can be reduced to the Sturm–Liouville form $Ly=f$: writing a general Sturm–Liouville operator as: $Lu={\frac {p}{w(x)}}u''+{\frac {p'}{w(x)}}u'+{\frac {q}{w(x)}}u,$ one solves the system: $p=Pw,\quad p'=Qw,\quad q=Rw.$ It suffices to solve the first two equations, which amounts to solving (Pw)′ = Qw, or $w'={\frac {Q-P'}{P}}w:=\alpha w.$ A solution is: $w=\exp \left(\int \alpha \,dx\right),\quad p=P\exp \left(\int \alpha \,dx\right),\quad q=R\exp \left(\int \alpha \,dx\right).$ Given this transformation, one is left to solve: $Ly=f.$ In general, if initial conditions at some point are specified, for example y(a) = 0 and y′(a) = 0, a second order differential equation can be solved using ordinary methods and the Picard–Lindelöf theorem ensures that the differential equation has a unique solution in a neighbourhood of the point where the initial conditions have been specified. But if in place of specifying initial values at a single point, it is desired to specify values at two different points (so-called boundary values), e.g. y(a) = 0 and y(b) = 1, the problem turns out to be much more difficult. Notice that by adding a suitable known differentiable function to y, whose values at a and b satisfy the desired boundary conditions, and injecting inside the proposed differential equation, it can be assumed without loss of generality that the boundary conditions are of the form y(a) = 0 and y(b) = 0. Here, the Sturm–Liouville theory comes in play: indeed, a large class of functions f can be expanded in terms of a series of orthonormal eigenfunctions ui of the associated Liouville operator with corresponding eigenvalues λi: $f(x)=\sum _{i}\alpha _{i}u_{i}(x),\quad \alpha _{i}\in {\mathbb {R} }.$ Then a solution to the proposed equation is evidently: $y=\sum _{i}{\frac {\alpha _{i}}{\lambda _{i}}}u_{i}.$ This solution will be valid only over the open interval a < x < b, and may fail at the boundaries. Example: Fourier series Consider the Sturm–Liouville problem: $Lu=-{\frac {d^{2}u}{dx^{2}}}=\lambda u$ (4) for the unknowns are λ and u(x). For boundary conditions, we take for example: $u(0)=u(\pi )=0.$ Observe that if k is any integer, then the function $u_{k}(x)=\sin kx$ is a solution with eigenvalue λ = k2. We know that the solutions of a Sturm–Liouville problem form an orthogonal basis, and we know from Fourier series that this set of sinusoidal functions is an orthogonal basis. Since orthogonal bases are always maximal (by definition) we conclude that the Sturm–Liouville problem in this case has no other eigenvectors. Given the preceding, let us now solve the inhomogeneous problem $Ly=x,\qquad x\in (0,\pi )$ with the same boundary conditions $y(0)=y(\pi )=0$. In this case, we must expand f(x) = x as a Fourier series. The reader may check, either by integrating ∫ eikxx dx or by consulting a table of Fourier transforms, that we thus obtain $Ly=\sum _{k=1}^{\infty }-2{\frac {\left(-1\right)^{k}}{k}}\sin kx.$ This particular Fourier series is troublesome because of its poor convergence properties. It is not clear a priori whether the series converges pointwise. Because of Fourier analysis, since the Fourier coefficients are "square-summable", the Fourier series converges in L2 which is all we need for this particular theory to function. We mention for the interested reader that in this case we may rely on a result which says that Fourier series converge at every point of differentiability, and at jump points (the function x, considered as a periodic function, has a jump at π) converges to the average of the left and right limits (see convergence of Fourier series). Therefore, by using formula (4), we obtain the solution: $y=\sum _{k=1}^{\infty }2{\frac {(-1)^{k}}{k^{3}}}\sin kx={\tfrac {1}{6}}(x^{3}-\pi ^{2}x).$ In this case, we could have found the answer using antidifferentiation, but this is no longer useful in most cases when the differential equation is in many variables. Application to partial differential equations Normal modes Certain partial differential equations can be solved with the help of Sturm–Liouville theory. Suppose we are interested in the vibrational modes of a thin membrane, held in a rectangular frame, 0 ≤ x ≤ L1, 0 ≤ y ≤ L2. The equation of motion for the vertical membrane's displacement, W(x,y,t) is given by the wave equation: ${\frac {\partial ^{2}W}{\partial x^{2}}}+{\frac {\partial ^{2}W}{\partial y^{2}}}={\frac {1}{c^{2}}}{\frac {\partial ^{2}W}{\partial t^{2}}}.$ The method of separation of variables suggests looking first for solutions of the simple form W = X(x) × Y(y) × T(t). For such a function W the partial differential equation becomes X″/X + Y″/Y = 1/c2 T″/T. Since the three terms of this equation are functions of x, y, t separately, they must be constants. For example, the first term gives X″ = λX for a constant λ. The boundary conditions ("held in a rectangular frame") are W = 0 when x = 0, L1 or y = 0, L2 and define the simplest possible Sturm–Liouville eigenvalue problems as in the example, yielding the "normal mode solutions" for W with harmonic time dependence, $W_{mn}(x,y,t)=A_{mn}\sin \left({\frac {m\pi x}{L_{1}}}\right)\sin \left({\frac {n\pi y}{L_{2}}}\right)\cos \left(\omega _{mn}t\right)$ where m and n are non-zero integers, Amn are arbitrary constants, and $\omega _{mn}^{2}=c^{2}\left({\frac {m^{2}\pi ^{2}}{L_{1}^{2}}}+{\frac {n^{2}\pi ^{2}}{L_{2}^{2}}}\right).$ The functions Wmn form a basis for the Hilbert space of (generalized) solutions of the wave equation; that is, an arbitrary solution W can be decomposed into a sum of these modes, which vibrate at their individual frequencies ωmn. This representation may require a convergent infinite sum. Second-order linear equation Consider a linear second-order differential equation in one spatial dimension and first-order in time of the form: $f(x){\frac {\partial ^{2}u}{\partial x^{2}}}+g(x){\frac {\partial u}{\partial x}}+h(x)u={\frac {\partial u}{\partial t}}+k(t)u,$ $u(a,t)=u(b,t)=0,\qquad u(x,0)=s(x).$ Separating variables, we assume that $u(x,t)=X(x)T(t).$ Then our above partial differential equation may be written as: ${\frac {{\hat {L}}X(x)}{X(x)}}={\frac {{\hat {M}}T(t)}{T(t)}}$ where ${\hat {L}}=f(x){\frac {d^{2}}{dx^{2}}}+g(x){\frac {d}{dx}}+h(x),\qquad {\hat {M}}={\frac {d}{dt}}+k(t).$ Since, by definition, L̂ and X(x) are independent of time t and M̂ and T(t) are independent of position x, then both sides of the above equation must be equal to a constant: ${\hat {L}}X(x)=\lambda X(x),\qquad X(a)=X(b)=0,\qquad {\hat {M}}T(t)=\lambda T(t).$ The first of these equations must be solved as a Sturm–Liouville problem in terms of the eigenfunctions Xn(x) and eigenvalues λn. The second of these equations can be analytically solved once the eigenvalues are known. ${\frac {d}{dt}}T_{n}(t)={\bigl (}\lambda _{n}-k(t){\bigr )}T_{n}(t)$ $T_{n}(t)=a_{n}\exp \left(\lambda _{n}t-\int _{0}^{t}k(\tau )\,d\tau \right)$ $u(x,t)=\sum _{n}a_{n}X_{n}(x)\exp \left(\lambda _{n}t-\int _{0}^{t}k(\tau )\,d\tau \right)$ $a_{n}={\frac {{\bigl \langle }X_{n}(x),s(x){\bigr \rangle }}{{\bigl \langle }X_{n}(x),X_{n}(x){\bigr \rangle }}}$ where ${\bigl \langle }y(x),z(x){\bigr \rangle }=\int _{a}^{b}y(x)z(x)w(x)\,dx,$ $w(x)={\frac {\exp \left(\int {\frac {g(x)}{f(x)}}\,dx\right)}{f(x)}}.$ Representation of solutions and numerical calculation The Sturm–Liouville differential equation (1) with boundary conditions may be solved analytically, which can be exact or provide an approximation, by the Rayleigh–Ritz method, or by the matrix-variational method of Gerck et al.[1][2][3] Numerically, a variety of methods are also available. In difficult cases, one may need to carry out the intermediate calculations to several hundred decimal places of accuracy in order to obtain the eigenvalues correctly to a few decimal places. • Shooting methods[4][5] • Finite difference method • Spectral parameter power series method[6] Shooting methods Shooting methods proceed by guessing a value of λ, solving an initial value problem defined by the boundary conditions at one endpoint, say, a, of the interval [a,b], comparing the value this solution takes at the other endpoint b with the other desired boundary condition, and finally increasing or decreasing λ as necessary to correct the original value. This strategy is not applicable for locating complex eigenvalues. Spectral parameter power series method The spectral parameter power series (SPPS) method makes use of a generalization of the following fact about homogeneous second-order linear ordinary differential equations: if y is a solution of equation (1) that does not vanish at any point of [a,b], then the function $y(x)\int _{a}^{x}{\frac {dt}{p(t)y(t)^{2}}}$ is a solution of the same equation and is linearly independent from y. Further, all solutions are linear combinations of these two solutions. In the SPPS algorithm, one must begin with an arbitrary value λ∗ 0 (often λ∗ 0 = 0 ; it does not need to be an eigenvalue) and any solution y0 of (1) with λ = λ∗ 0 which does not vanish on [a,b]. (Discussion below of ways to find appropriate y0 and λ∗ 0 .) Two sequences of functions X(n)(t), X̃(n)(t) on [a,b], referred to as iterated integrals, are defined recursively as follows. First when n = 0, they are taken to be identically equal to 1 on [a,b]. To obtain the next functions they are multiplied alternately by 1/py2 0 and wy2 0 and integrated, specifically, for n > 0: $X^{(n)}(t)={\begin{cases}\displaystyle -\int _{a}^{x}X^{(n-1)}(t)p(t)^{-1}y_{0}(t)^{-2}\,dt&n{\text{ odd}},\\[6pt]\displaystyle \quad \int _{a}^{x}X^{(n-1)}(t)y_{0}(t)^{2}w(t)\,dt&n{\text{ even}}\end{cases}}$ (5) ${\tilde {X}}^{(n)}(t)={\begin{cases}\displaystyle \quad \int _{a}^{x}{\tilde {X}}^{(n-1)}(t)y_{0}(t)^{2}w(t)\,dt&n{\text{ odd}},\\[6pt]\displaystyle -\int _{a}^{x}{\tilde {X}}^{(n-1)}(t)p(t)^{-1}y_{0}(t)^{-2}\,dt&n{\text{ even.}}\end{cases}}$ (6) The resulting iterated integrals are now applied as coefficients in the following two power series in λ: $u_{0}=y_{0}\sum _{k=0}^{\infty }\left(\lambda -\lambda _{0}^{}\right)^{k}{\tilde {X}}^{(2k)},$ $u_{1}=y_{0}\sum _{k=0}^{\infty }\left(\lambda -\lambda _{0}^{}\right)^{k}X^{(2k+1)}.$ Then for any λ (real or complex), u0 and u1 are linearly independent solutions of the corresponding equation (1). (The functions p(x) and q(x) take part in this construction through their influence on the choice of y0.) Next one chooses coefficients c0 and c1 so that the combination y = c0u0 + c1u1 satisfies the first boundary condition (2). This is simple to do since X(n)(a) = 0 and X̃(n)(a) = 0, for n > 0. The values of X(n)(b) and X̃(n)(b) provide the values of u0(b) and u1(b) and the derivatives u′0(b) and u′0(b), so the second boundary condition (3) becomes an equation in a power series in λ. For numerical work one may truncate this series to a finite number of terms, producing a calculable polynomial in λ whose roots are approximations of the sought-after eigenvalues. When λ = λ0, this reduces to the original construction described above for a solution linearly independent to a given one. The representations (5) and (6) also have theoretical applications in Sturm–Liouville theory.[6] Construction of a nonvanishing solution The SPPS method can, itself, be used to find a starting solution y0. Consider the equation (py′)′ = μqy; i.e., q, w, and λ are replaced in (1) by 0, −q, and μ respectively. Then the constant function 1 is a nonvanishing solution corresponding to the eigenvalue μ0 = 0. While there is no guarantee that u0 or u1 will not vanish, the complex function y0 = u0 + iu1 will never vanish because two linearly-independent solutions of a regular Sturm–Liouville equation cannot vanish simultaneously as a consequence of the Sturm separation theorem. This trick gives a solution y0 of (1) for the value λ0 = 0. In practice if (1) has real coefficients, the solutions based on y0 will have very small imaginary parts which must be discarded. See also • Normal mode • Oscillation theory • Self-adjoint • Variation of parameters • Spectral theory of ordinary differential equations • Atkinson–Mingarelli theorem References 1. Ed Gerck, A. B. d'Oliveira, H. F. de Carvalho. "Heavy baryons as bound states of three quarks." Lettere al Nuovo Cimento 38(1):27–32, Sep 1983. 2. Augusto B. d'Oliveira, Ed Gerck, Jason A. C. Gallas. "Solution of the Schrödinger equation for bound states in closed form." Physical Review A, 26:1(1), June 1982. 3. Robert F. O'Connell, Jason A. C. Gallas, Ed Gerck. "Scaling Laws for Rydberg Atoms in Magnetic Fields." Physical Review Letters 50(5):324–327, January 1983. 4. Pryce, J. D. (1993). Numerical Solution of Sturm–Liouville Problems. Oxford: Clarendon Press. ISBN 0-19-853415-9. 5. Ledoux, V.; Van Daele, M.; Berghe, G. Vanden (2009). "Efficient computation of high index Sturm–Liouville eigenvalues for problems in physics". Comput. Phys. Commun. 180 (2): 532–554. arXiv:0804.2605. Bibcode:2009CoPhC.180..241L. doi:10.1016/j.cpc.2008.10.001. S2CID 13955991. 6. Kravchenko, V. V.; Porter, R. M. (2010). "Spectral parameter power series for Sturm–Liouville problems". Mathematical Methods in the Applied Sciences. 33 (4): 459–468. arXiv:0811.4488. Bibcode:2010MMAS...33..459K. doi:10.1002/mma.1205. S2CID 17029224. Further reading • "Sturm–Liouville theory", Encyclopedia of Mathematics, EMS Press, 2001 [1994] • Hartman, Philip (2002). Ordinary Differential Equations (2 ed.). Philadelphia: SIAM. ISBN 978-0-89871-510-1. • Polyanin, A. D. & Zaitsev, V. F. (2003). Handbook of Exact Solutions for Ordinary Differential Equations (2 ed.). Boca Raton: Chapman & Hall/CRC Press. ISBN 1-58488-297-2. • Teschl, Gerald (2012). Ordinary Differential Equations and Dynamical Systems. Providence: American Mathematical Society. ISBN 978-0-8218-8328-0. (Chapter 5) • Teschl, Gerald (2009). Mathematical Methods in Quantum Mechanics; With Applications to Schrödinger Operators. Providence: American Mathematical Society. ISBN 978-0-8218-4660-5. (see Chapter 9 for singular Sturm–Liouville operators and connections with quantum mechanics) • Zettl, Anton (2005). Sturm–Liouville Theory. Providence: American Mathematical Society. ISBN 0-8218-3905-5. • Birkhoff, Garrett (1973). A source book in classical analysis. Cambridge, Massachusetts: Harvard University Press. ISBN 0-674-82245-5. (See Chapter 8, part B, for excerpts from the works of Sturm and Liouville and commentary on them.) • Kravchenko, Vladislav (2020). Direct and Inverse Sturm-Liouville Problems: A Method of Solution. Cham: Birkhäuser. ISBN 978-3-030-47848-3. Functional analysis (topics – glossary) Spaces • Banach • Besov • Fréchet • Hilbert • Hölder • Nuclear • Orlicz • Schwartz • Sobolev • Topological vector Properties • Barrelled • Complete • Dual (Algebraic/Topological) • Locally convex • Reflexive • Reparable Theorems • Hahn–Banach • Riesz representation • Closed graph • Uniform boundedness principle • Kakutani fixed-point • Krein–Milman • Min–max • Gelfand–Naimark • Banach–Alaoglu Operators • Adjoint • Bounded • Compact • Hilbert–Schmidt • Normal • Nuclear • Trace class • Transpose • Unbounded • Unitary Algebras • Banach algebra • C-algebra • Spectrum of a C-algebra • Operator algebra • Group algebra of a locally compact group • Von Neumann algebra Open problems • Invariant subspace problem • Mahler's conjecture Applications • Hardy space • Spectral theory of ordinary differential equations • Heat kernel • Index theorem • Calculus of variations • Functional calculus • Integral operator • Jones polynomial • Topological quantum field theory • Noncommutative geometry • Riemann hypothesis • Distribution (or Generalized functions) Advanced topics • Approximation property • Balanced set • Choquet theory • Weak topology • Banach–Mazur distance • Tomita–Takesaki theory • Mathematics portal • Category • Commons	Wikipedia
Self-oscillation Self-oscillation is the generation and maintenance of a periodic motion by a source of power that lacks any corresponding periodicity. The oscillator itself controls the phase with which the external power acts on it. Self-oscillators are therefore distinct from forced and parametric resonators, in which the power that sustains the motion must be modulated externally. In linear systems, self-oscillation appears as an instability associated with a negative damping term, which causes small perturbations to grow exponentially in amplitude. This negative damping is due to a positive feedback between the oscillation and the modulation of the external source of power. The amplitude and waveform of steady self-oscillations are determined by the nonlinear characteristics of the system. Self-oscillations are important in physics, engineering, biology, and economics. History of the subject The study of self-oscillators dates back to Robert Willis, George Biddell Airy, James Clerk Maxwell, and Lord Rayleigh in the 19th century. The term itself (also translated as "auto-oscillation") was coined by the Soviet physicist Aleksandr Andronov, who studied them in the context of the mathematical theory of the structural stability of dynamical systems. Other important work on the subject, both theoretical and experimental, was due to André Blondel, Balthasar van der Pol, Alfred-Marie Liénard, and Philippe Le Corbeiller in the 20th century.[1] The same phenomenon is sometimes labelled as "maintained", "sustained", "self-exciting", "self-induced", "spontaneous", or "autonomous" oscillation. Unwanted self-oscillations are known in the mechanical engineering literature as hunting, and in electronics as parasitic oscillations.[1] Important early studied examples of self-oscillation include the centrifugal governor[2] and railroad wheels. Mathematical basis Main article: Oscillation (differential equation) Self-oscillation is manifested as a linear instability of a dynamical system's static equilibrium. Two mathematical tests that can be used to diagnose such an instability are the Routh–Hurwitz and Nyquist criteria. The amplitude of the oscillation of an unstable system grows exponentially with time (i.e., small oscillations are negatively damped), until nonlinearities become important and limit the amplitude. This can produce a steady and sustained oscillation. In some cases, self-oscillation can be seen as resulting from a time lag in a closed loop system, which makes the change in variable xt dependent on the variable xt-1 evaluated at an earlier time.[1] Examples in engineering Railway and automotive wheels Hunting oscillation in railway wheels and shimmy in automotive tires can cause an uncomfortable wobbling effect, which in extreme cases can derail trains and cause cars to lose grip. Central heating thermostats Early central heating thermostats were guilty of self-exciting oscillation because they responded too quickly. The problem was overcome by hysteresis, i.e., making them switch state only when the temperature varied from the target by a specified minimum amount. Automatic transmissions Self-exciting oscillation occurred in early automatic transmission designs when the vehicle was traveling at a speed which was between the ideal speeds of 2 gears. In these situations the transmission system would switch almost continuously between the 2 gears, which was both annoying and hard on the transmission. Such behavior is now inhibited by introducing hysteresis into the system. Steering of vehicles when course corrections are delayed There are many examples of self-exciting oscillation caused by delayed course corrections, ranging from light aircraft in a strong wind to erratic steering of road vehicles by a driver who is inexperienced or drunk. SEIG (self-excited induction generator) If an induction motor is connected to a capacitor and the shaft turns above synchronous speed, it operates as a self-excited induction generator. Self-exciting transmitters Many early radio systems tuned their transmitter circuit, so the system automatically created radio waves of the desired frequency. This design has given way to designs that use a separate oscillator to provide a signal that is then amplified to the desired power. Examples in other fields Population cycles in biology For example, a reduction in population of an herbivore species because of predation, this makes the populations of predators of that species decline, the reduced level of predation allows the herbivore population to increase, this allows the predator population to increase, etc. Closed loops of time-lagged differential equations are a sufficient explanation for such cycles - in this case the delays are caused mainly by the breeding cycles of the species involved. See also • Hopf bifurcation • Limit cycle • Van der Pol oscillator • Hidden oscillation References 1. Jenkins, Alejandro (2013). "Self-oscillation". Physics Reports. 525 (2): 167–222. arXiv:1109.6640. Bibcode:2013PhR...525..167J. doi:10.1016/j.physrep.2012.10.007. S2CID 227438422. 2. Maxwell, J. Clerk (1867). "On Governors". Proceedings of the Royal Society of London. 16: 270–283. JSTOR 112510. .	Wikipedia
Function problem In computational complexity theory, a function problem is a computational problem where a single output (of a total function) is expected for every input, but the output is more complex than that of a decision problem. For function problems, the output is not simply 'yes' or 'no'. Formal definition A functional problem $P$ is defined by a relation $R$ over strings of an arbitrary alphabet $\Sigma $: $R\subseteq \Sigma ^{}\times \Sigma ^{}.$ An algorithm solves $P$ if for every input $x$ such that there exists a $y$ satisfying $(x,y)\in R$, the algorithm produces one such $y$, and if there are no such $y$, it rejects. A promise function problem is allowed to do anything (thus may not terminate) if no such $y$ exists. Examples A well-known function problem is given by the Functional Boolean Satisfiability Problem, FSAT for short. The problem, which is closely related to the SAT decision problem, can be formulated as follows: Given a boolean formula $\varphi $ with variables $x_{1},\ldots ,x_{n}$, find an assignment $x_{i}\rightarrow \{{\text{TRUE}},{\text{FALSE}}\}$ such that $\varphi $ evaluates to ${\text{TRUE}}$ or decide that no such assignment exists. In this case the relation $R$ is given by tuples of suitably encoded boolean formulas and satisfying assignments. While a SAT algorithm, fed with a formula $\varphi $, only needs to return "unsatisfiable" or "satisfiable", an FSAT algorithm needs to return some satisfying assignment in the latter case. Other notable examples include the travelling salesman problem, which asks for the route taken by the salesman, and the integer factorization problem, which asks for the list of factors. Relationship to other complexity classes Consider an arbitrary decision problem $L$ in the class NP. By the definition of NP, each problem instance $x$ that is answered 'yes' has a polynomial-size certificate $y$ which serves as a proof for the 'yes' answer. Thus, the set of these tuples $(x,y)$ forms a relation, representing the function problem "given $x$ in $L$, find a certificate $y$ for $x$". This function problem is called the function variant of $L$; it belongs to the class FNP. FNP can be thought of as the function class analogue of NP, in that solutions of FNP problems can be efficiently (i.e., in polynomial time in terms of the length of the input) verified, but not necessarily efficiently found. In contrast, the class FP, which can be thought of as the function class analogue of P, consists of function problems whose solutions can be found in polynomial time. Self-reducibility Observe that the problem FSAT introduced above can be solved using only polynomially many calls to a subroutine which decides the SAT problem: An algorithm can first ask whether the formula $\varphi $ is satisfiable. After that the algorithm can fix variable $x_{1}$ to TRUE and ask again. If the resulting formula is still satisfiable the algorithm keeps $x_{1}$ fixed to TRUE and continues to fix $x_{2}$, otherwise it decides that $x_{1}$ has to be FALSE and continues. Thus, FSAT is solvable in polynomial time using an oracle deciding SAT. In general, a problem in NP is called self-reducible if its function variant can be solved in polynomial time using an oracle deciding the original problem. Every NP-complete problem is self-reducible. It is conjectured that the integer factorization problem is not self-reducible, because deciding whether an integer is prime is in P (easy),[1] while the integer factorization problem is believed to be hard for a classical computer. There are several (slightly different) notions of self-reducibility.[2][3][4] Reductions and complete problems Function problems can be reduced much like decision problems: Given function problems $\Pi _{R}$ and $\Pi _{S}$ we say that $\Pi _{R}$ reduces to $\Pi _{S}$ if there exists polynomially-time computable functions $f$ and $g$ such that for all instances $x$ of $R$ and possible solutions $y$ of $S$, it holds that • If $x$ has an $R$-solution, then $f(x)$ has an $S$-solution. • $(f(x),y)\in S\implies (x,g(x,y))\in R.$ It is therefore possible to define FNP-complete problems analogous to the NP-complete problem: A problem $\Pi _{R}$ is FNP-complete if every problem in FNP can be reduced to $\Pi _{R}$. The complexity class of FNP-complete problems is denoted by FNP-C or FNPC. Hence the problem FSAT is also an FNP-complete problem, and it holds that $\mathbf {P} =\mathbf {NP} $ if and only if $\mathbf {FP} =\mathbf {FNP} $. Total function problems The relation $R(x,y)$ used to define function problems has the drawback of being incomplete: Not every input $x$ has a counterpart $y$ such that $(x,y)\in R$. Therefore the question of computability of proofs is not separated from the question of their existence. To overcome this problem it is convenient to consider the restriction of function problems to total relations yielding the class TFNP as a subclass of FNP. This class contains problems such as the computation of pure Nash equilibria in certain strategic games where a solution is guaranteed to exist. In addition, if TFNP contains any FNP-complete problem it follows that $\mathbf {NP} ={\textbf {co-NP}}$. See also • Decision problem • Search problem • Counting problem (complexity) • Optimization problem References • Raymond Greenlaw, H. James Hoover, Fundamentals of the theory of computation: principles and practice, Morgan Kaufmann, 1998, ISBN 1-55860-474-X, p. 45-51 • Elaine Rich, Automata, computability and complexity: theory and applications, Prentice Hall, 2008, ISBN 0-13-228806-0, section 28.10 "The problem classes FP and FNP", pp. 689–694 1. Agrawal, Manindra; Kayal, Neeraj; Saxena, Nitin (2004). "PRIMES is in P" (PDF). Annals of Mathematics. 160 (2): 781–793. doi:10.4007/annals.2004.160.781. JSTOR 3597229. 2. Ko, K. (1983). "On self-reducibility and weak P-selectivity". Journal of Computer and System Sciences. 26 (2): 209–221. 3. Schnorr, C. (1976). "Optimal algorithms for self-reducible problems". In S. Michaelson and R. Milner, editors, Proceedings of the 3rd International Colloquium on Automata, Languages, and Programming: 322–337. 4. Selman, A. (1988). "Natural self-reducible sets". SIAM Journal on Computing. 17 (5): 989–996.	Wikipedia
Self-adjoint In mathematics, and more specifically in abstract algebra, an element x of a -algebra is self-adjoint if $x^{}=x$. A self-adjoint element is also Hermitian, though the reverse doesn't necessarily hold. A collection C of elements of a star-algebra is self-adjoint if it is closed under the involution operation. For example, if $x^{}=y$ then since $y^{}=x^{*}=x$ in a star-algebra, the set {x,y} is a self-adjoint set even though x and y need not be self-adjoint elements. In functional analysis, a linear operator $A:H\to H$ on a Hilbert space is called self-adjoint if it is equal to its own adjoint A∗. See self-adjoint operator for a detailed discussion. If the Hilbert space is finite-dimensional and an orthonormal basis has been chosen, then the operator A is self-adjoint if and only if the matrix describing A with respect to this basis is Hermitian, i.e. if it is equal to its own conjugate transpose. Hermitian matrices are also called self-adjoint. In a dagger category, a morphism $f$ is called self-adjoint if $f=f^{\dagger }$; this is possible only for an endomorphism $f\colon a\to a$. See also • Hermitian matrix • Normal element • Symmetric matrix • Self-adjoint operator • Unitary element References • Reed, M.; Simon, B. (1972). Methods of Mathematical Physics. Vol 2. Academic Press. • Teschl, G. (2009). Mathematical Methods in Quantum Mechanics; With Applications to Schrödinger Operators. Providence: American Mathematical Society. Functional analysis (topics – glossary) Spaces • Banach • Besov • Fréchet • Hilbert • Hölder • Nuclear • Orlicz • Schwartz • Sobolev • Topological vector Properties • Barrelled • Complete • Dual (Algebraic/Topological) • Locally convex • Reflexive • Reparable Theorems • Hahn–Banach • Riesz representation • Closed graph • Uniform boundedness principle • Kakutani fixed-point • Krein–Milman • Min–max • Gelfand–Naimark • Banach–Alaoglu Operators • Adjoint • Bounded • Compact • Hilbert–Schmidt • Normal • Nuclear • Trace class • Transpose • Unbounded • Unitary Algebras • Banach algebra • C-algebra • Spectrum of a C-algebra • Operator algebra • Group algebra of a locally compact group • Von Neumann algebra Open problems • Invariant subspace problem • Mahler's conjecture Applications • Hardy space • Spectral theory of ordinary differential equations • Heat kernel • Index theorem • Calculus of variations • Functional calculus • Integral operator • Jones polynomial • Topological quantum field theory • Noncommutative geometry • Riemann hypothesis • Distribution (or Generalized functions) Advanced topics • Approximation property • Balanced set • Choquet theory • Weak topology • Banach–Mazur distance • Tomita–Takesaki theory • Mathematics portal • Category • Commons Spectral theory and -algebras Basic concepts • Involution/-algebra • Banach algebra • B-algebra • C-algebra • Noncommutative topology • Projection-valued measure • Spectrum • Spectrum of a C-algebra • Spectral radius • Operator space Main results • Gelfand–Mazur theorem • Gelfand–Naimark theorem • Gelfand representation • Polar decomposition • Singular value decomposition • Spectral theorem • Spectral theory of normal C-algebras Special Elements/Operators • Isospectral • Normal operator • Hermitian/Self-adjoint operator • Unitary operator • Unit Spectrum • Krein–Rutman theorem • Normal eigenvalue • Spectrum of a C-algebra • Spectral radius • Spectral asymmetry • Spectral gap Decomposition • Decomposition of a spectrum • Continuous • Point • Residual • Approximate point • Compression • Direct integral • Discrete • Spectral abscissa Spectral Theorem • Borel functional calculus • Min-max theorem • Positive operator-valued measure • Projection-valued measure • Riesz projector • Rigged Hilbert space • Spectral theorem • Spectral theory of compact operators • Spectral theory of normal C-algebras Special algebras • Amenable Banach algebra • With an Approximate identity • Banach function algebra • Disk algebra • Nuclear C-algebra • Uniform algebra • Von Neumann algebra • Tomita–Takesaki theory Finite-Dimensional • Alon–Boppana bound • Bauer–Fike theorem • Numerical range • Schur–Horn theorem Generalizations • Dirac spectrum • Essential spectrum • Pseudospectrum • Structure space (Shilov boundary) Miscellaneous • Abstract index group • Banach algebra cohomology • Cohen–Hewitt factorization theorem • Extensions of symmetric operators • Fredholm theory • Limiting absorption principle • Schröder–Bernstein theorems for operator algebras • Sherman–Takeda theorem • Unbounded operator Examples • Wiener algebra Applications • Almost Mathieu operator • Corona theorem • Hearing the shape of a drum (Dirichlet eigenvalue) • Heat kernel • Kuznetsov trace formula • Lax pair • Proto-value function • Ramanujan graph • Rayleigh–Faber–Krahn inequality • Spectral geometry • Spectral method • Spectral theory of ordinary differential equations • Sturm–Liouville theory • Superstrong approximation • Transfer operator • Transform theory • Weyl law • Wiener–Khinchin theorem	Wikipedia
Selfridge–Conway procedure The Selfridge–Conway procedure is a discrete procedure that produces an envy-free cake-cutting for three partners.[1]: 13–14 It is named after John Selfridge and John Horton Conway. Selfridge discovered it in 1960, and told it to Richard Guy, who told many people, but Selfridge did not publish it. John Conway later discovered it independently, and also never published it.[2] This procedure was the first envy-free discrete procedure devised for three partners, and it paved the way for more advanced procedures for n partners (see envy-free cake-cutting). A procedure is envy-free if each recipient believes that (according to their own measure) no other recipient has received a larger share. The maximal number of cuts in the procedure is five. The pieces are not always contiguous. The Procedure Suppose we have three players P1, P2 and P3. Where the procedure gives a criterion for a decision it means that criterion gives an optimum choice for the player. 1. P1 divides the cake into three pieces they consider of equal size. 2. Let's call A the largest piece according to P2. 3. P2 cuts off a bit of A to make it the same size as the second largest. Now A is divided into: the trimmed piece A1 and the trimmings A2. Leave the trimmings A2 to the side for now. • If P2 thinks that the two largest parts are equal (such that no trimming is needed), then each player chooses a part in this order: P3, P2 and finally P1. 4. P3 chooses a piece among A1 and the two other pieces. 5. P2 chooses a piece with the limitation that if P3 didn't choose A1, P2 must choose it. 6. P1 chooses the last piece leaving just the trimmings A2 to be divided. It remains to divide the trimmings A2. The trimmed piece A1 has been chosen by either P2 or P3; let's call the player who chose it PA and the other player PB. 1. PB cuts A2 into three equal pieces. 2. PA chooses a piece of A2 - we name it A21. 3. P1 chooses a piece of A2 - we name it A22. 4. PB chooses the last remaining piece of A2 - we name it A23. Analysis Let's see why the procedure is envy-free. It must be shown that each player believes that no other player received a larger share. Without loss of generality, we can write (see illustration above): • PA received: A1 + A21. • PB received: B + A23. • P1 received: C + A22. In the following analysis "largest" means "largest according to that player": • PA received A1 + A21. For them, A1 ≥ B and A1 ≥ C. And they consider their choice A21 to be the largest piece of A2. So no other player received a larger share: A1 + A21 ≥ B + A23, C + A22. • PB received B + A23. For them, B ≥ A1 and B ≥ C since they chose B. Also, they are the one that cut A2 in 3 pieces, so for them all those pieces are equal. • P1 received C + A22. For them, C ≥ A1 and C = B. • P1 believes that PB didn't receive a larger share. In other words: C + A22 ≥ B + A23. Remember that P1 chose their piece of A2 before PB, thus A22 ≥ A23 in their view. • P1 believes that PA didn't receive a larger share. In other words: C + A22 ≥ A1 + A21. Remember that for P1, C is equal to A since they cut the cake in the first round. Also, A = A1 + A2 = A1 + (A21 + A22 + A23); therefore C ≥ A1 + A21. (Even if PA took the whole A2 and P1 did not receive A22, P1 would not envy PA.) Generalizations Note that if all we want is an envy-free division for a part of the cake (i.e. we allow free disposal), then we only need to use the first part of the Selfridge–Conway procedure, i.e.: • P1 divides the cake into three equal pieces; • P2 trims at most one piece such that the two largest pieces are equal; • P3 takes a piece, then P2, then P1. This guarantees that there is no envy. This procedure can be generalized to 4 partners in the following way:[3] • P1 divides the cake into 5 equal pieces; • P2 trims at most 2 pieces, such that the 3 largest pieces are equal; • P3 trims at most 1 piece, such that the 2 largest pieces are equal; • P4 takes a piece, then P3, then P2, then P1. This guarantees that there is no envy. By induction, the procedure can be generalized to n partners, the first one dividing the cake to $2^{n-2}+1$ equal pieces and the other partners follow by trimming. The resulting division is envy-free. We can apply the same procedure again on the remainders. By doing so an infinite number of times, we get an envy-free division of the entire cake.[4] A refinement of this infinite procedure yields a finite envy-free division procedure: the Brams–Taylor procedure. References 1. Robertson, Jack; Webb, William (1998). Cake-Cutting Algorithms: Be Fair If You Can. Natick, Massachusetts: A. K. Peters. ISBN 978-1-56881-076-8. LCCN 97041258. OL 2730675W. 2. Brams, Steven J.; Taylor, Alan D. (1996). Fair Division [From cake-cutting to dispute resolution]. pp. 116–120. ISBN 0521556449. 3. Brams, Steven J.; Taylor, Alan D. (1996). Fair Division [From cake-cutting to dispute resolution]. pp. 131–137. ISBN 0521556449. 4. Brams, Steven J.; Taylor, Alan D. (1996). Fair Division [From cake-cutting to dispute resolution]. p. 137. ISBN 0521556449.	Wikipedia
George Seligman George Benham Seligman (born April 30, 1927)[1] is an American mathematician who works on Lie algebras, especially semi-simple Lie algebras. George B. Seligman Born(1927-04-30)April 30, 1927 Attica, New York, U.S. NationalityAmerican Alma materUniversity of Rochester Yale University Scientific career FieldsMathematics InstitutionsPrinceton University Yale University ThesisLie algebras of prime characteristic (1954) Doctoral advisorNathan Jacobson Doctoral studentsJames E. Humphreys Daniel K. Nakano Biography Seligman received his bachelor's degree in 1950 from the University of Rochester and his PhD in 1954 from Yale University under Nathan Jacobson with thesis Lie algebras of prime characteristic.[2] After he received his PhD he was a Henry Burchard Fine Instructor at Princeton University from 1954–1956. In 1956 he became an instructor and from 1965 a full professor at Yale, where he was chair of the mathematics department from 1974 to 1977. For the academic year 1958/59 he was a Fulbright Lecturer at the University of Münster. His doctoral students include James E. Humphreys and Daniel K. Nakano. Since 1959 he has been married to Irene Schwieder and the couple has two daughters. Selected works Books • On Lie algebras of prime characteristic, American Mathematical Society, 1956 • Liesche Algebren, Schriftenreihe des Mathematischen Instituts der Universität Münster, 1959 • Modular Lie Algebras, Springer Verlag 1967[3] • Rational methods in Lie algebras, Marcel Dekker 1976[4] • Rational constructions of modules for simple Lie algebras, American Mathematical Society 1981 • Construction of Lie Algebras and their Modules, Springer Verlag 1988 Articles • Seligman, G. B. (1954). "On a class of semisimple restricted Lie algebras". Proceedings of the National Academy of Sciences of the United States of America. 40 (8): 726–728. Bibcode:1954PNAS...40..726S. doi:10.1073/pnas.40.8.726. PMC 534151. PMID 16589548. • Seligman, George B. (1957). "Characteristic ideals and the structure of Lie algebras". Proceedings of the American Mathematical Society. 8: 159–164. doi:10.1090/s0002-9939-1957-0082974-9. MR 0082974. • Seligman, George B. (1959). "On automorphisms of Lie algebras of classical type". Transactions of the American Mathematical Society. 92 (3): 430–448. doi:10.1090/s0002-9947-1959-0106965-0. MR 0106965. • Seligman, George B. (1960). "On automorphisms of Lie algebras of classical type. II". Trans. Amer. Math. Soc. 94 (3): 452–482. doi:10.1090/s0002-9947-1960-0113969-9. MR 0113969. • Seligman, George B. (1960). "On automorphisms of Lie algebras of classical type. III". Trans. Amer. Math. Soc. 97 (2): 286–312. doi:10.1090/s0002-9947-1960-0123644-2. MR 0123644. • Seligman, George B. (1967). "Some results on Lie p-algebras". Bulletin of the American Mathematical Society. 73 (4): 528–530. doi:10.1090/s0002-9904-1967-11731-2. MR 0219585. • "Algebraic Lie groups". Bull. Amer. Math. Soc. 74: 1051–1065. 1968. doi:10.1090/s0002-9904-1968-12046-4. MR 0232810. • Seligman, George B. (2003). "On idempotents in reduced enveloping algebras". Trans. Amer. Math. Soc. 355 (8): 3291–3300. doi:10.1090/s0002-9947-03-03314-2. MR 1974688. References 1. biographical information American Men and Women of Science, Thomson Gale 2004 2. George Seligman at the Mathematics Genealogy Project 3. Schafer, R. D. (1971). "Review: Modular Lie algebras by George B. Seligman" (PDF). Bull. Amer. Math. Soc. 77 (5): 689–694. doi:10.1090/s0002-9904-1971-12772-6. 4. Humphreys, James E. (1977). "Review: Rational methods in Lie algebras by George B. Seligman" (PDF). Bulletin of the American Mathematical Society. 83 (5): 993–997. doi:10.1090/S0002-9904-1977-14348-6. Authority control International • ISNI • VIAF National • France • BnF data • Germany • Israel • Belgium • United States • Czech Republic • Netherlands Academics • MathSciNet • Mathematics Genealogy Project • zbMATH Other • IdRef	Wikipedia
Selman Akbulut Selman Akbulut (born 1949) is a Turkish mathematician, specializing in research in topology, and geometry. He was a professor at Michigan State University until February 2020. Selman Akbulut Selman Akbulut at Oberwolfach in 2012. Born1949 Balıkesir, Turkey NationalityTurkish EducationUniversity of California OccupationMathematician Known forAkbulut cork Career In 1975 he earned his Ph.D. from the University of California, Berkeley as a student of Robion Kirby. In topology, he has worked on handlebody theory, low-dimensional manifolds,[1] symplectic topology, G2 manifolds. In the topology of real-algebraic sets, he and Henry C. King proved that every compact piecewise-linear manifold is a real-algebraic set; they discovered new topological invariants of real-algebraic sets.[2] He was a visiting scholar several times at the Institute for Advanced Study (in 1975-76, 1980–81, 2002, and 2005).[3] On February 14, 2020, Akbulut was removed from his tenured position at MSU by the Board of Trustees, after complaints regarding his teaching attendance and communications with colleagues.[4][5][6] Contributions He has developed 4-dimensional handlebody techniques, settling conjectures and solving problems about 4-manifolds, such as a conjecture of Christopher Zeeman,[7] the Harer–Kas–Kirby conjecture, a problem of Martin Scharlemann,[8] and problems of Sylvain Cappell and Julius Shaneson.[9][10][11] He constructed an exotic compact 4-manifold (with boundary) from which he discovered "Akbulut corks".[12][13][14][15] His most recent results concern the 4-dimensional smooth Poincaré conjecture.[16] He has supervised 14 Ph.D students as of 2019. He has more than 100 papers and three books published, and several books edited. Notes 1. Akbulut, Selman (2016). 4-manifolds. Oxford University Press. ISBN 9780198784869. Retrieved 13 August 2019. 2. S. Akbulut and H.C. King, Topology of real algebraic sets, MSRI Publications, 25. Springer-Verlag, New York (1992) ISBN 0-387-97744-9 3. Institute for Advanced Study: A Community of Scholars Archived 2013-01-06 at the Wayback Machine 4. Graham, Karly; Monroe, Maddie. "Board of Trustees fires tenured professor for cause". The State News. 5. Stanley, Samuel L. "Dismissal of Tenured Faculty for Cause" (PDF). MSU Board of Trustees. Retrieved 14 February 2020. 6. Frost, Mikenzie (14 February 2020). "MSU Trustees dismiss tenured professor, address Title IX investigation delays". WWMT. Retrieved 14 February 2020. 7. S. Akbulut, A solution to a conjecture of Zeeman, Topology, vol.30, no.3, (1991), 513-515. 8. S. Akbulut, Scharlemann's manifold is standard, Ann of Math., 149 (1999) 497-510. 9. S. Akbulut, Cappell-Shaneson homotopy spheres are standard Ann. of Math., 171 (2010) 2171-2175. 10. S. Akbulut, Cappell-Shaneson's 4-dimensional s-cobordism, Geometry-Topology, vol.6, (2002), 425-494. 11. M. Freedman, R. Gompf, S. Morrison, K. Walker, Man and machine thinking about the smooth 4-dimensional Poincaré conjecture. Quantum Topol. 1 (2010), no. 2, 171–208 12. S. Akbulut, A Fake compact contractible 4-manifold, Journal of Differential Geometry 33, (1991), 335-356 13. S. Akbulut, An exotic 4-manifold, Journ. of Diff. Geom. 33, (1991), 357-361 14. B. Ozbagci and A.I. Stipsicz. Surgery on contact 3-manifolds and Stein surfaces (p. 14), Springer ISBN 3-540-22944-2 15. A. Scorpan, The wild world of 4-manifolds (p.90), AMS Pub. ISBN 0-8218-3749-4 16. Morrison, Scott. "Poincaré conjecture". Secret Blogging Seminar. Retrieved 13 August 2019. External links • Selman Akbulut at the Mathematics Genealogy Project • Akbulut's homepage • Akbulut's papers at ArXiv • Akbulut-King invariants • Real algebraic geometry • Akbulut cork Authority control International • ISNI • VIAF National • France • BnF data • Germany • Israel • United States • Australia • Netherlands Academics • Google Scholar • MathSciNet • Mathematics Genealogy Project • zbMATH Other • IdRef	Wikipedia
Selmer group In arithmetic geometry, the Selmer group, named in honor of the work of Ernst Sejersted Selmer (1951) by John William Scott Cassels (1962), is a group constructed from an isogeny of abelian varieties. The Selmer group of an isogeny The Selmer group of an abelian variety A with respect to an isogeny f : A → B of abelian varieties can be defined in terms of Galois cohomology as $\operatorname {Sel} ^{(f)}(A/K)=\bigcap _{v}\ker(H^{1}(G_{K},\ker(f))\rightarrow H^{1}(G_{K_{v}},A_{v}[f])/\operatorname {im} (\kappa _{v}))$ where Av[f] denotes the f-torsion of Av and $\kappa _{v}$ is the local Kummer map $B_{v}(K_{v})/f(A_{v}(K_{v}))\rightarrow H^{1}(G_{K_{v}},A_{v}[f])$. Note that $H^{1}(G_{K_{v}},A_{v}[f])/\operatorname {im} (\kappa _{v})$ is isomorphic to $H^{1}(G_{K_{v}},A_{v})[f]$. Geometrically, the principal homogeneous spaces coming from elements of the Selmer group have Kv-rational points for all places v of K. The Selmer group is finite. This implies that the part of the Tate–Shafarevich group killed by f is finite due to the following exact sequence 0 → B(K)/f(A(K)) → Sel(f)(A/K) → Ш(A/K)[f] → 0. The Selmer group in the middle of this exact sequence is finite and effectively computable. This implies the weak Mordell–Weil theorem that its subgroup B(K)/f(A(K)) is finite. There is a notorious problem about whether this subgroup can be effectively computed: there is a procedure for computing it that will terminate with the correct answer if there is some prime p such that the p-component of the Tate–Shafarevich group is finite. It is conjectured that the Tate–Shafarevich group is in fact finite, in which case any prime p would work. However, if (as seems unlikely) the Tate–Shafarevich group has an infinite p-component for every prime p, then the procedure may never terminate. Ralph Greenberg (1994) has generalized the notion of Selmer group to more general p-adic Galois representations and to p-adic variations of motives in the context of Iwasawa theory. The Selmer group of a finite Galois module More generally one can define the Selmer group of a finite Galois module M (such as the kernel of an isogeny) as the elements of H1(GK,M) that have images inside certain given subgroups of H1(GKv,M). References • Cassels, John William Scott (1962), "Arithmetic on curves of genus 1. III. The Tate–Šafarevič and Selmer groups", Proceedings of the London Mathematical Society, Third Series, 12: 259–296, doi:10.1112/plms/s3-12.1.259, ISSN 0024-6115, MR 0163913 • Cassels, John William Scott (1991), Lectures on elliptic curves, London Mathematical Society Student Texts, vol. 24, Cambridge University Press, doi:10.1017/CBO9781139172530, ISBN 978-0-521-41517-0, MR 1144763 • Greenberg, Ralph (1994), "Iwasawa Theory and p-adic Deformation of Motives", in Serre, Jean-Pierre; Jannsen, Uwe; Kleiman, Steven L. (eds.), Motives, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-1637-0, MR 1265554 • Selmer, Ernst S. (1951), "The Diophantine equation ax3 + by3 + cz3 = 0", Acta Mathematica, 85: 203–362, doi:10.1007/BF02395746, ISSN 0001-5962, MR 0041871 See also • Wiles's proof of Fermat's Last Theorem L-functions in number theory Analytic examples • Riemann zeta function • Dirichlet L-functions • L-functions of Hecke characters • Automorphic L-functions • Selberg class Algebraic examples • Dedekind zeta functions • Artin L-functions • Hasse–Weil L-functions • Motivic L-functions Theorems • Analytic class number formula • Riemann–von Mangoldt formula • Weil conjectures Analytic conjectures • Riemann hypothesis • Generalized Riemann hypothesis • Lindelöf hypothesis • Ramanujan–Petersson conjecture • Artin conjecture Algebraic conjectures • Birch and Swinnerton-Dyer conjecture • Deligne's conjecture • Beilinson conjectures • Bloch–Kato conjecture • Langlands conjecture p-adic L-functions • Main conjecture of Iwasawa theory • Selmer group • Euler system	Wikipedia
Sema Salur Sema Salur is a Turkish-American mathematician, currently serving as a Professor of Mathematics at the University of Rochester.[1] She was awarded the Ruth I. Michler Memorial Prize for 2014–2015,[2] a prize intended to give a recently promoted associate professor a year-long fellowship at Cornell University;[3] and has been the recipient of a National Science Foundation Research Award beginning in 2017.[4] She specialises in the "geometry and topology of the moduli spaces of calibrated submanifolds inside Calabi–Yau, G2 and Spin(7) manifolds",[2][5] which are important to certain aspects of string theory and M-theory in physics, theories that attempt to unite gravity, electromagnetism, and the strong and weak nuclear forces into one coherent Theory of Everything.[5] Education • 1993: B.S. in Mathematics, Boğaziçi University, Turkey.[2] • 2000: PhD in Mathematics, Michigan State University[2] References 1. "Sema Salur". University of Rochester. 2. "Ruth I. Michler Prize 2014-2015". Association for Women in Mathematics. 3. "The Ruth I Michler Memorial Prize of the AWM". St Andrews University. 4. "Award Abstract #1711178: Manifolds with Special Holonomy and Applications". National Science Foundation. 5. "Professor Sema Salur receives NSF Research Award". University of Rochester. Authority control International • ISNI Academics • MathSciNet • Mathematics Genealogy Project	Wikipedia
Semën Samsonovich Kutateladze Semën Samsonovich Kutateladze (born October 2, 1945 in Leningrad, now St. Petersburg) is a mathematician. He is known for contributions to functional analysis and its applications to vector lattices and optimization.[1][2] In particular, he has made contributions to the calculus of subdifferentials for vector-lattice valued functions, to whose study he introduced methods of Boolean-valued models and infinitesimals. Semën Kutateladze Semën Kutateladze Born (1945-10-02) October 2, 1945 Leningrad, USSR Alma materNovosibirsk State University Scientific career FieldsMathematics InstitutionsSobolev Institute of Mathematics, Novosibirsk State University Doctoral advisorG.Sh. Rubinstein He is professor of mathematics at Novosibirsk State University,[3] where he has continued and enriched the scientific tradition of Leonid Kantorovich.[4][5][6][7] His father was the heat physicist Samson Kutateladze. Selected books and articles • Bair, Jacques; Błaszczyk, Piotr; Ely, Robert; Henry, Valérie; Kanovei, Vladimir; Katz, Karin; Katz, Mikhail; Kutateladze, Semen; McGaffey, Thomas; Schaps, David; Sherry, David; Shnider, Steve (2013), "Is mathematical history written by the victors?" (PDF), Notices of the American Mathematical Society, 60 (7): 886–904, arXiv:1306.5973, doi:10.1090/noti1001. • Gordon, E. I.; Kusraev, A. G.; Kutateladze, S. S. Infinitesimal Analysis. Updated and revised translation of the 2001 Russian original. Translated by Kutateladze. Mathematics and its Applications, 544. Kluwer Academic Publishers, Dordrecht, 2002.[8] • Kutateladze, S. S. Fundamentals of Functional Analysis. Translated from the second (1995) edition. Kluwer Texts in the Mathematical Sciences, 12. Kluwer Academic Publishers Group, Dordrecht, 1996. • Kusraev, A. G.; Kutateladze, S. S. Subdifferentials: Theory and Applications. Translated from the Russian. Mathematics and its Applications, 323. Kluwer Academic Publishers Group, Dordrecht, 1995. • Kutateladze, S. S.; Rubinov, A. M. Minkowski Duality, and Its Applications. Russian Math. Surveys, 1972, Vol. 27, No. 3, 137–191. • Kutateladze S.S. Choquet boundaries in K-spaces. Russian Math. Surveys, 1975, Vol. 30, No. 4, 115–155. • Kutateladze S.S. Convex operators. Russian Math. Surveys, 1979, Vol. 34, No. 1, 181–214. • Kutateladze S.S. On Grothendieck subspaces. Siberian Math. J., 2005, Vol. 46. No. 3, 489–493. • Kutateladze S.S. What is Boolean valued analysis? Siberian Advances in Mathematics, 2007, Vol. 17, No. 2, 91–111. • Kutateladze S.S. The tragedy of mathematics in Russia.Siberian Electronic Mathematical Reports, 2012, Vol. 9, A85–A100. • Kutateladze S.S. Harpedonaptae and abstract convexity. Journal of Applied and Industrial Mathematics, 2008, Vol. 2, No. 2, 215–221. • Kutateladze S.S. The Farkas lemma revisited. Siberian Math. J., 2010, Vol. 51, No. 1, 78–87. • Kutateladze S.S. Leibnizian, Robinsonian, and Boolean valued monads, Journal of Applied and Industrial Mathematics, 2011, Vol. 5, No. 3, 365–373. • Kutateladze S.S. Nonstandard analysis: its creator and place. Journal of Applied and Industrial Mathematics, 2013, Vol. 7, No. 3, 287–297. See also • John L. Bell • Paul J. Cohen • Forcing • Jerome Keisler • Model theory • Influence of non-standard analysis • Nikolai Luzin • Leonid Kantorovich • Sergei Sobolev • Aleksandr Danilovich Aleksandrov • Dana Scott References 1. Aleksandrov, A. D.; Ladyzhenskaya, O. A.; Reshetnyak, Yu. G. (1997). "Semën Samsonovich Kutateladze (on his 50th birthday)". Russian Math. Surveys. 52 (2): 447–450. Bibcode:1997RuMaS..52..447A. doi:10.1070/RM1997v052n02ABEH001807. MR 1480167. S2CID 250918620. 2. Gutman, A. E.; Kusraev, A. G.; Reshetnyak, Yu. G. (2005). "On the nth birthday of Semën Samsonovich Kutateladze for n=60". Sib. Èlektron. Mat. Izv. 2: A.12–A.33. MR 2178006. 3. About Kutateladze in Russian 4. Leifman, Lev J., ed. (1990). Functional analysis, optimization, and mathematical economics: A collection of papers dedicated to the memory of Leonid Vitalʹevich Kantorovich. New York: The Clarendon Press, Oxford University Press. pp. xvi+341. ISBN 978-0-19-505729-4. MR 1082562.{{cite book}}: CS1 maint: multiple names: authors list (link) 5. Kutateladze, S.S., "The World Line of Kantorovich", Notices of the ISMS, International Society for Mathematical Sciences, Osaka, Japan, January 2007. 6. Kutateladze, S.S., "Kantorovich's Phenomenon", Siberian Mathematical Journal '' (Сибирский мат. журн.), 2007, V. 48, No. 1, 3–4, November 29, 2006. 7. Kutateladze, S.S., "Mathematics and Economics of Kantorovich" 8. E.I. Gordon; A.G. Kusraev; Semën Samsonovich Kutateladze (14 March 2013). Infinitesimal Analysis. Springer Science & Business Media. ISBN 978-94-017-0063-4. External links • Webpage at the Sobolev Institute of Mathematics • An Heir, Pavel Golovkin's film • Semёn Kutateladze about science, innovation and education (in Russian) • Semёn Kutateladze about science, innovation and education (in English) • Profile at zbMATH • Semën Kutateladze on Twitter Authority control International • ISNI • VIAF National • France • BnF data • Germany • Czech Republic • Netherlands • Poland Academics • CiNii • DBLP • Google Scholar • MathSciNet • Mathematics Genealogy Project • ORCID • ResearcherID • Scopus • zbMATH Other • IdRef	Wikipedia
Semi-Lagrangian scheme The Semi-Lagrangian scheme (SLS) is a numerical method that is widely used in numerical weather prediction models for the integration of the equations governing atmospheric motion. A Lagrangian description of a system (such as the atmosphere) focuses on following individual air parcels along their trajectories as opposed to the Eulerian description, which considers the rate of change of system variables fixed at a particular point in space. A semi-Lagrangian scheme uses Eulerian framework but the discrete equations come from the Lagrangian perspective. Some background The Lagrangian rate of change of a quantity $F$ is given by ${\frac {DF}{Dt}}={\frac {\partial F}{\partial t}}+(\mathbf {v} \cdot {\vec {\nabla }})F,$ where $F$ can be a scalar or vector field and $\mathbf {v} $ is the velocity field. The first term on the right-hand side of the above equation is the local or Eulerian rate of change of $F$ and the second term is often called the advection term. Note that the Lagrangian rate of change is also known as the material derivative. It can be shown that the equations governing atmospheric motion can be written in the Lagrangian form ${\frac {D\mathbf {V} }{Dt}}=\mathbf {S} (\mathbf {V} ),$ where the components of the vector $\mathbf {V} $ are the (dependent) variables describing a parcel of air (such as velocity, pressure, temperature etc.) and the function $\mathbf {S} (\mathbf {V} )$ represents source and/or sink terms. In a Lagrangian scheme, individual air parcels are traced but there are clearly certain drawbacks: the number of parcels can be very large indeed and it may often happen for a large number of parcels to cluster together, leaving relatively large regions of space completely empty. Such voids can cause computational problems, e.g. when calculating spatial derivatives of various quantities. There are ways round this, such as the technique known as Smoothed Particle Hydrodynamics, where a dependent variable is expressed in non-local form, i.e. as an integral of itself times a kernel function. Semi-Lagrangian schemes avoid the problem of having regions of space essentially free of parcels. The Semi-Lagrangian scheme Semi-Lagrangian schemes use a regular (Eulerian) grid, just like finite difference methods. The idea is this: at every time step the point where a parcel originated from is calculated. An interpolation scheme is then utilized to estimate the value of the dependent variable at the grid points surrounding the point where the particle originated from. The references listed contain more details on how the Semi-Lagrangian scheme is applied. See also • Lagrangian and Eulerian specification of the flow field • Contour advection • Trajectory (fluid mechanics) External links • ctraj: C++ trajectory library, including semi-Lagrangian tracer codes. References • E. Kalnay, Atmospheric Modeling, Data Assimilation and Predictability (Chapter 3, Section 3.3.3), Cambridge University Press, Cambridge, 2003. • A. Persson, User Guide to ECMWF forecast products (Section 2.1.3), http://www.ecmwf.int/sites/default/files/User_Guide_V1.2_20151123.pdf • D.A. Randall, Atmospheric Modeling (AT604, Chapter 5, Section 5.11), http://kiwi.atmos.colostate.edu/group/dave/at604.html	Wikipedia
Semi-Thue system In theoretical computer science and mathematical logic a string rewriting system (SRS), historically called a semi-Thue system, is a rewriting system over strings from a (usually finite) alphabet. Given a binary relation $R$ between fixed strings over the alphabet, called rewrite rules, denoted by $s\rightarrow t$, an SRS extends the rewriting relation to all strings in which the left- and right-hand side of the rules appear as substrings, that is $usv\rightarrow utv$, where $s$, $t$, $u$, and $v$ are strings. The notion of a semi-Thue system essentially coincides with the presentation of a monoid. Thus they constitute a natural framework for solving the word problem for monoids and groups. An SRS can be defined directly as an abstract rewriting system. It can also be seen as a restricted kind of a term rewriting system. As a formalism, string rewriting systems are Turing complete.[1] The semi-Thue name comes from the Norwegian mathematician Axel Thue, who introduced systematic treatment of string rewriting systems in a 1914 paper.[2] Thue introduced this notion hoping to solve the word problem for finitely presented semigroups. Only in 1947 was the problem shown to be undecidable— this result was obtained independently by Emil Post and A. A. Markov Jr.[3][4] Definition A string rewriting system or semi-Thue system is a tuple $(\Sigma ,R)$ where • Σ is an alphabet, usually assumed finite.[5] The elements of the set $\Sigma ^{}$ ( is the Kleene star here) are finite (possibly empty) strings on Σ, sometimes called words in formal languages; we will simply call them strings here. • R is a binary relation on strings from Σ, i.e., $R\subseteq \Sigma ^{}\times \Sigma ^{}.$ Each element $(u,v)\in R$ is called a (rewriting) rule and is usually written $u\rightarrow v$. If the relation R is symmetric, then the system is called a Thue system. The rewriting rules in R can be naturally extended to other strings in $\Sigma ^{}$ by allowing substrings to be rewritten according to R. More formally, the one-step rewriting relation relation ${\xrightarrow[{R}]{}}$ induced by R on $\Sigma ^{}$ for any strings $s,t\in \Sigma ^{}$: $s{\xrightarrow[{R}]{}}t$ if and only if there exist $x,y,u,v\in \Sigma ^{}$ such that $s=xuy$, $t=xvy$, and $u\rightarrow v$. Since ${\xrightarrow[{R}]{}}$ is a relation on $\Sigma ^{}$, the pair $(\Sigma ^{},{\xrightarrow[{R}]{}})$ fits the definition of an abstract rewriting system. Obviously R is a subset of ${\xrightarrow[{R}]{}}$. Some authors use a different notation for the arrow in ${\xrightarrow[{R}]{}}$ (e.g. ${\xrightarrow[{R}]{}}$) in order to distinguish it from R itself ($\rightarrow $) because they later want to be able to drop the subscript and still avoid confusion between R and the one-step rewrite induced by R. Clearly in a semi-Thue system we can form a (finite or infinite) sequence of strings produced by starting with an initial string $s_{0}\in \Sigma ^{}$ and repeatedly rewriting it by making one substring-replacement at a time: $s_{0}\ {\xrightarrow[{R}]{}}\ s_{1}\ {\xrightarrow[{R}]{}}\ s_{2}\ {\xrightarrow[{R}]{}}\ \ldots $ A zero-or-more-steps rewriting like this is captured by the reflexive transitive closure of ${\xrightarrow[{R}]{}}$, denoted by ${\xrightarrow[{R}]{}}$ (see abstract rewriting system#Basic notions). This is called the rewriting relation or reduction relation on $\Sigma ^{}$ induced by R. Thue congruence In general, the set $\Sigma ^{}$ of strings on an alphabet forms a free monoid together with the binary operation of string concatenation (denoted as $\cdot $ and written multiplicatively by dropping the symbol). In a SRS, the reduction relation ${\xrightarrow[{R}]{}}$ is compatible with the monoid operation, meaning that $x{\xrightarrow[{R}]{}}y$ implies $uxv{\xrightarrow[{R}]{}}uyv$ for all strings $x,y,u,v\in \Sigma ^{}$. Since ${\xrightarrow[{R}]{}}$ is by definition a preorder, $\left(\Sigma ^{},\cdot ,{\xrightarrow[{R}]{}}\right)$ forms a monoidal preorder. Similarly, the reflexive transitive symmetric closure of ${\xrightarrow[{R}]{}}$, denoted ${\overset {}{\underset {R}{\leftrightarrow }}}$ (see abstract rewriting system#Basic notions), is a congruence, meaning it is an equivalence relation (by definition) and it is also compatible with string concatenation. The relation ${\overset {}{\underset {R}{\leftrightarrow }}}$ is called the Thue congruence generated by R. In a Thue system, i.e. if R is symmetric, the rewrite relation ${\xrightarrow[{R}]{}}$ coincides with the Thue congruence ${\overset {}{\underset {R}{\leftrightarrow }}}$. Factor monoid and monoid presentations Since ${\overset {}{\underset {R}{\leftrightarrow }}}$ is a congruence, we can define the factor monoid ${\mathcal {M}}_{R}=\Sigma ^{}/{\overset {}{\underset {R}{\leftrightarrow }}}$ of the free monoid $\Sigma ^{}$ by the Thue congruence in the usual manner. If a monoid ${\mathcal {M}}$ is isomorphic with ${\mathcal {M}}_{R}$, then the semi-Thue system $(\Sigma ,R)$ is called a monoid presentation of ${\mathcal {M}}$. We immediately get some very useful connections with other areas of algebra. For example, the alphabet {a, b} with the rules { ab → ε, ba → ε }, where ε is the empty string, is a presentation of the free group on one generator. If instead the rules are just { ab → ε }, then we obtain a presentation of the bicyclic monoid. The importance of semi-Thue systems as presentation of monoids is made stronger by the following: Theorem: Every monoid has a presentation of the form $(\Sigma ,R)$, thus it may be always be presented by a semi-Thue system, possibly over an infinite alphabet.[6] In this context, the set $\Sigma $ is called the set of generators of ${\mathcal {M}}$, and $R$ is called the set of defining relations ${\mathcal {M}}$. We can immediately classify monoids based on their presentation. ${\mathcal {M}}$ is called • finitely generated if $\Sigma $ is finite. • finitely presented if both $\Sigma $ and $R$ are finite. Undecidability of the word problem Post proved the word problem (for semigroups) to be undecidable in general, essentially by reducing the halting problem[7] for Turing machines to an instance of the word problem. Concretely, Post devised an encoding as a finite string of the state of a Turing machine plus tape, such that the actions of this machine can be carried out by a string rewrite system acting on this string encoding. The alphabet of the encoding has one set of letters $S_{0},S_{1},\dotsc ,S_{m}$ for symbols on the tape (where $S_{0}$ means blank), another set of letters $q_{1},\dotsc ,q_{r}$ for states of the Turing machine, and finally three letters $q_{r+1},q_{r+2},h$ that have special roles in the encoding. $q_{r+1}$ and $q_{r+2}$ are intuitively extra internal states of the Turing machine which it transitions to when halting, whereas $h$ marks the end of the non-blank part of the tape; a machine reaching an $h$ should behave the same as if there was a blank there, and the $h$ was in the next cell. The strings that are valid encodings of Turing machine states start with an $h$, followed by zero or more symbol letters, followed by exactly one internal state letter $q_{i}$ (which encodes the state of the machine), followed by one or more symbol letters, followed by an ending $h$. The symbol letters are straight off the contents of the tape, and the internal state letter marks the position of the head; the symbol after the internal state letter is that in the cell currently under the head of the Turing machine. A transition where the machine upon being in state $q_{i}$ and seeing the symbol $S_{k}$ writes back symbol $S_{l}$, moves right, and transitions to state $q_{j}$ is implemented by the rewrite $q_{i}S_{k}\to S_{l}q_{j}$ whereas that transition instead moving to the left is implemented by the rewrite $S_{p}q_{i}S_{k}\to q_{j}S_{p}S_{l}$ with one instance for each symbol $S_{p}$ in that cell to the left. In the case that we reach the end of the visited portion of the tape, we use instead $hq_{i}S_{k}\to hq_{j}S_{0}S_{l}$, lengthening the string by one letter. Because all rewrites involve one internal state letter $q_{i}$, the valid encodings only contain one such letter, and each rewrite produces exactly one such letter, the rewrite process exactly follows the run of the Turing machine encoded. This proves that string rewrite systems are Turing complete. The reason for having two halted symbols $q_{r+1}$ and $q_{r+2}$ is that we want all halting Turing machines to terminate at the same total state, not just a particular internal state. This requires clearing the tape after halting, so $q_{r+1}$ eats the symbol on it left until reaching the $h$, where it transitions into $q_{r+2}$ which instead eats the symbol on its right. (In this phase the string rewrite system no longer simulates a Turing machine, since that cannot remove cells from the tape.) After all symbols are gone, we have reached the terminal string $hq_{r+2}h$. A decision procedure for the word problem would then also yield a procedure for deciding whether the given Turing machine terminates when started in a particular total state $t$, by testing whether $t$ and $hq_{r+2}h$ belong to the same congruence class with respect to this string rewrite system. Technically, we have the following: Lemma. Let $M$ be a deterministic Turing machine and $R$ be the string rewrite system implementing $M$, as described above. Then $M$ will halt when started from the total state encoded as $t$ if and only if $t\mathrel {\overset {}{\underset {R}{\leftrightarrow }}} hq_{r+2}h$ (that is to say, if and only if $t$ and $hq_{r+2}h$ are Thue congruent for $R$). That $t\mathrel {\overset {}{\underset {R}{\rightarrow }}} hq_{r+2}h$ if $M$ halts when started from $t$ is immediate from the construction of $R$ (simply running $M$ until it halts constructs a proof of $t\mathrel {\overset {}{\underset {R}{\rightarrow }}} hq_{r+2}h$), but ${\overset {}{\underset {R}{\leftrightarrow }}}$ also allows the Turing machine $M$ to take steps backwards. Here it becomes relevant that $M$ is deterministic, because then the forward steps are all unique; in a ${\overset {}{\underset {R}{\leftrightarrow }}}$ walk from $t$ to $hq_{r+2}h$ the last backward step must be followed by its counterpart as a forward step, so these two cancel, and by induction all backward steps can be eliminated from such a walk. Hence if $M$ does not halt when started from $t$, i.e., if we do not have $t\mathrel {\overset {}{\underset {R}{\rightarrow }}} hq_{r+2}h$, then we also do not have $t\mathrel {\overset {}{\underset {R}{\leftrightarrow }}} hq_{r+2}h$. Therefore, deciding ${\overset {}{\underset {R}{\leftrightarrow }}}$ tells us the answer to the halting problem for $M$. An apparent limitation of this argument is that in order to produce a semigroup $\Sigma ^{}{\big /}{\overset {}{\underset {R}{\leftrightarrow }}}$ with undecidable word problem, one must first have a concrete example of a Turing machine $M$ for which the halting problem is undecidable, but the various Turing machines figuring in the proof of the undecidability of the general halting problem all have as component a hypothetical Turing machine solving the halting problem, so none of those machines can actually exist; all that proves is that there is some Turing machine for which the decision problem is undecidable. However, that there is some Turing machine with undecidable halting problem means that the halting problem for a universal Turing machine is undecidable (since that can simulate any Turing machine), and concrete examples of universal Turing machines have been constructed. Connections with other notions A semi-Thue system is also a term-rewriting system—one that has monadic words (functions) ending in the same variable as the left- and right-hand side terms,[8] e.g. a term rule $f_{2}(f_{1}(x))\rightarrow g(x)$ is equivalent to the string rule $f_{1}f_{2}\rightarrow g$. A semi-Thue system is also a special type of Post canonical system, but every Post canonical system can also be reduced to an SRS. Both formalisms are Turing complete, and thus equivalent to Noam Chomsky's unrestricted grammars, which are sometimes called semi-Thue grammars.[9] A formal grammar only differs from a semi-Thue system by the separation of the alphabet into terminals and non-terminals, and the fixation of a starting symbol amongst non-terminals. A minority of authors actually define a semi-Thue system as a triple $(\Sigma ,A,R)$, where $A\subseteq \Sigma ^{}$ is called the set of axioms. Under this "generative" definition of semi-Thue system, an unrestricted grammar is just a semi-Thue system with a single axiom in which one partitions the alphabet into terminals and non-terminals, and makes the axiom a nonterminal.[10] The simple artifice of partitioning the alphabet into terminals and non-terminals is a powerful one; it allows the definition of the Chomsky hierarchy based on what combination of terminals and non-terminals the rules contain. This was a crucial development in the theory of formal languages. In quantum computing, the notion of a quantum Thue system can be developed.[11] Since quantum computation is intrinsically reversible, the rewriting rules over the alphabet $\Sigma $ are required to be bidirectional (i.e. the underlying system is a Thue system, not a semi-Thue system). On a subset of alphabet characters $Q\subseteq \Sigma $ one can attach a Hilbert space $\mathbb {C} ^{d}$, and a rewriting rule taking a substring to another one can carry out a unitary operation on the tensor product of the Hilbert space attached to the strings; this implies that they preserve the number of characters from the set $Q$. Similar to the classical case one can show that a quantum Thue system is a universal computational model for quantum computation, in the sense that the executed quantum operations correspond to uniform circuit classes (such as those in BQP when e.g. guaranteeing termination of the string rewriting rules within polynomially many steps in the input size), or equivalently a Quantum Turing machine. History and importance Semi-Thue systems were developed as part of a program to add additional constructs to logic, so as to create systems such as propositional logic, that would allow general mathematical theorems to be expressed in a formal language, and then proven and verified in an automatic, mechanical fashion. The hope was that the act of theorem proving could then be reduced to a set of defined manipulations on a set of strings. It was subsequently realized that semi-Thue systems are isomorphic to unrestricted grammars, which in turn are known to be isomorphic to Turing machines. This method of research succeeded and now computers can be used to verify the proofs of mathematic and logical theorems. At the suggestion of Alonzo Church, Emil Post in a paper published in 1947, first proved "a certain Problem of Thue" to be unsolvable, what Martin Davis states as "...the first unsolvability proof for a problem from classical mathematics -- in this case the word problem for semigroups."[12] Davis also asserts that the proof was offered independently by A. A. Markov.[13] See also • L-system • Markov algorithm — a variant of string rewriting systems • MU puzzle Notes 1. See section "Undecidability of the word problem" in this article. 2. Book and Otto, p. 36 3. Abramsky et al. p. 416 4. Salomaa et al., p.444 5. In Book and Otto a semi-Thue system is defined over a finite alphabet through most of the book, except chapter 7 when monoid presentation are introduced, when this assumption is quietly dropped. 6. Book and Otto, Theorem 7.1.7, p. 149 7. Post, following Turing, technically makes use of the undecidability of the printing problem (whether a Turing machine ever prints a particular symbol), but the two problems reduce to each other. Indeed, Post includes an extra step in his construction that effectively converts printing the watched symbol into halting. 8. Nachum Dershowitz and Jean-Pierre Jouannaud. Rewrite Systems (1990) p. 6 9. D.I.A. Cohen, Introduction to Computer Theory, 2nd ed., Wiley-India, 2007, ISBN 81-265-1334-9, p.572 10. Dan A. Simovici, Richard L. Tenney, Theory of formal languages with applications, World Scientific, 1999 ISBN 981-02-3729-4, chapter 4 11. J. Bausch, T. Cubitt, M. Ozols, The Complexity of Translationally-Invariant Spin Chains with Low Local Dimension, Ann. Henri Poincare 18(11), 2017 doi:10.1007/s00023-017-0609-7 pp. 3449-3513 12. Martin Davis (editor) (1965), The Undecidable: Basic Papers on Undecidable Propositions, Unsolvable Problems and Computable Functions, after page 292, Raven Press, New York 13. A. A. Markov (1947) Doklady Akademii Nauk SSSR (N.S.) 55: 583–586 References Monographs • Ronald V. Book and Friedrich Otto, String-rewriting Systems, Springer, 1993, ISBN 0-387-97965-4. • Matthias Jantzen, Confluent string rewriting, Birkhäuser, 1988, ISBN 0-387-13715-7. Textbooks • Martin Davis, Ron Sigal, Elaine J. Weyuker, Computability, complexity, and languages: fundamentals of theoretical computer science, 2nd ed., Academic Press, 1994, ISBN 0-12-206382-1, chapter 7 • Elaine Rich, Automata, computability and complexity: theory and applications, Prentice Hall, 2007, ISBN 0-13-228806-0, chapter 23.5. Surveys • Samson Abramsky, Dov M. Gabbay, Thomas S. E. Maibaum (ed.), Handbook of Logic in Computer Science: Semantic modelling, Oxford University Press, 1995, ISBN 0-19-853780-8. • Grzegorz Rozenberg, Arto Salomaa (ed.), Handbook of Formal Languages: Word, language, grammar, Springer, 1997, ISBN 3-540-60420-0. Landmark papers • Post, Emil (1947). "Recursive Unsolvability of a Problem of Thue". The Journal of Symbolic Logic. 12 (1): 1–11. doi:10.2307/2267170. JSTOR 2267170. S2CID 30320278.{{cite journal}}: CS1 maint: url-status (link) Automata theory: formal languages and formal grammars Chomsky hierarchyGrammarsLanguagesAbstract machines • Type-0 • — • Type-1 • — • — • — • — • — • Type-2 • — • — • Type-3 • — • — • Unrestricted • (no common name) • Context-sensitive • Positive range concatenation • Indexed • — • Linear context-free rewriting systems • Tree-adjoining • Context-free • Deterministic context-free • Visibly pushdown • Regular • — • Non-recursive • Recursively enumerable • Decidable • Context-sensitive • Positive range concatenation* • Indexed* • — • Linear context-free rewriting language • Tree-adjoining • Context-free • Deterministic context-free • Visibly pushdown • Regular • Star-free • Finite • Turing machine • Decider • Linear-bounded • PTIME Turing Machine • Nested stack • Thread automaton • restricted Tree stack automaton • Embedded pushdown • Nondeterministic pushdown • Deterministic pushdown • Visibly pushdown • Finite • Counter-free (with aperiodic finite monoid) • Acyclic finite Each category of languages, except those marked by a *, is a proper subset of the category directly above it. Any language in each category is generated by a grammar and by an automaton in the category in the same line.	Wikipedia
Semi-abelian category In mathematics, specifically in category theory, a semi-abelian category is a pre-abelian category in which the induced morphism ${\overline {f}}:\operatorname {coim} f\rightarrow \operatorname {im} f$ is a bimorphism, i.e., a monomorphism and an epimorphism, for every morphism $f$. Properties The two properties used in the definition can be characterized by several equivalent conditions.[1] Every semi-abelian category has a maximal exact structure. If a semi-abelian category is not quasi-abelian, then the class of all kernel-cokernel pairs does not form an exact structure. Examples Every quasi-abelian category is semi-abelian. In particular, every abelian category is semi-abelian. Non quasi-abelian examples are the following. • The category of (possibly non Hausdorff) bornological spaces is semi-abelian.[2][3][4] • Let $Q$ be the quiver ${\begin{array}{ccc}1&{\xrightarrow {}}&2&{\xleftarrow {}}&3\\\downarrow {}&&\downarrow {}&&\downarrow {}\\4&{\xrightarrow {}}&5&{\xleftarrow {}}&6\\\end{array}}$ and $k$ be a field. The category of finitely generated projective modules over the algebra $kQ$ is semi-abelian.[5] History The concept of a semi-abelian category was developed in the 1960s. Raikov conjectured that the notion of a quasi-abelian category is equivalent to that of a semi-abelian category. Around 2005 it turned out that the conjecture is false.[6] Left and right semi-abelian categories By dividing the two conditions on the induced map in the definition, one can define left semi-abelian categories by requiring that ${\overline {f}}$ is a monomorphism for each morphism $f$. Accordingly, right semi-abelian categories are pre-abelian categories such that ${\overline {f}}$ is an epimorphism for each morphism $f$.[7] If a category is left semi-abelian and right quasi-abelian, then it is already quasi-abelian. The same holds, if the category is right semi-abelian and left quasi-abelian.[8] Citations 1. Kopylov et al., 2012. 2. Bonet et al., 2004/2005. 3. Sieg et al., 2011, Example 4.1. 4. Rump, 2011, p. 44. 5. Rump, 2008, p. 993. 6. Rump, 2011, p. 44f. 7. Rump, 2001. 8. Rump, 2001. References • José Bonet, J., Susanne Dierolf, The pullback for bornological and ultrabornological spaces. Note Mat. 25(1), 63–67 (2005/2006). • Yaroslav Kopylov and Sven-Ake Wegner, On the notion of a semi-abelian category in the sense of Palamodov, Appl. Categ. Structures 20 (5) (2012) 531–541. • Wolfgang Rump, A counterexample to Raikov's conjecture, Bull. London Math. Soc. 40, 985–994 (2008). • Wolfgang Rump, Almost abelian categories, Cahiers Topologie Géom. Différentielle Catég. 42(3), 163–225 (2001). • Wolfgang Rump, Analysis of a problem of Raikov with applications to barreled and bornological spaces, J. Pure and Appl. Algebra 215, 44–52 (2011). • Dennis Sieg and Sven-Ake Wegner, Maximal exact structures on additive categories, Math. Nachr. 284 (2011), 2093–2100.	Wikipedia
Abelian variety In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions. Abelian varieties are at the same time among the most studied objects in algebraic geometry and indispensable tools for much research on other topics in algebraic geometry and number theory. Algebraic structure → Group theory Group theory Basic notions • Subgroup • Normal subgroup • Quotient group • (Semi-)direct product Group homomorphisms • kernel • image • direct sum • wreath product • simple • finite • infinite • continuous • multiplicative • additive • cyclic • abelian • dihedral • nilpotent • solvable • action • Glossary of group theory • List of group theory topics Finite groups • Cyclic group Zn • Symmetric group Sn • Alternating group An • Dihedral group Dn • Quaternion group Q • Cauchy's theorem • Lagrange's theorem • Sylow theorems • Hall's theorem • p-group • Elementary abelian group • Frobenius group • Schur multiplier Classification of finite simple groups • cyclic • alternating • Lie type • sporadic • Discrete groups • Lattices • Integers ($\mathbb {Z} $) • Free group Modular groups • PSL(2, $\mathbb {Z} $) • SL(2, $\mathbb {Z} $) • Arithmetic group • Lattice • Hyperbolic group Topological and Lie groups • Solenoid • Circle • General linear GL(n) • Special linear SL(n) • Orthogonal O(n) • Euclidean E(n) • Special orthogonal SO(n) • Unitary U(n) • Special unitary SU(n) • Symplectic Sp(n) • G2 • F4 • E6 • E7 • E8 • Lorentz • Poincaré • Conformal • Diffeomorphism • Loop Infinite dimensional Lie group • O(∞) • SU(∞) • Sp(∞) Algebraic groups • Linear algebraic group • Reductive group • Abelian variety • Elliptic curve An abelian variety can be defined by equations having coefficients in any field; the variety is then said to be defined over that field. Historically the first abelian varieties to be studied were those defined over the field of complex numbers. Such abelian varieties turn out to be exactly those complex tori that can be embedded into a complex projective space. Abelian varieties defined over algebraic number fields are a special case, which is important also from the viewpoint of number theory. Localization techniques lead naturally from abelian varieties defined over number fields to ones defined over finite fields and various local fields. Since a number field is the fraction field of a Dedekind domain, for any nonzero prime of your Dedekind domain, there is a map from the Dedekind domain to the quotient of the Dedekind domain by the prime, which is a finite field for all finite primes. This induces a map from the fraction field to any such finite field. Given a curve with equation defined over the number field, we can apply this map to the coefficients to get a curve defined over some finite field, where the choices of finite field correspond to the finite primes of the number field. Abelian varieties appear naturally as Jacobian varieties (the connected components of zero in Picard varieties) and Albanese varieties of other algebraic varieties. The group law of an abelian variety is necessarily commutative and the variety is non-singular. An elliptic curve is an abelian variety of dimension 1. Abelian varieties have Kodaira dimension 0. History and motivation Further information: History of manifolds and varieties In the early nineteenth century, the theory of elliptic functions succeeded in giving a basis for the theory of elliptic integrals, and this left open an obvious avenue of research. The standard forms for elliptic integrals involved the square roots of cubic and quartic polynomials. When those were replaced by polynomials of higher degree, say quintics, what would happen? In the work of Niels Abel and Carl Jacobi, the answer was formulated: this would involve functions of two complex variables, having four independent periods (i.e. period vectors). This gave the first glimpse of an abelian variety of dimension 2 (an abelian surface): what would now be called the Jacobian of a hyperelliptic curve of genus 2. After Abel and Jacobi, some of the most important contributors to the theory of abelian functions were Riemann, Weierstrass, Frobenius, Poincaré and Picard. The subject was very popular at the time, already having a large literature. By the end of the 19th century, mathematicians had begun to use geometric methods in the study of abelian functions. Eventually, in the 1920s, Lefschetz laid the basis for the study of abelian functions in terms of complex tori. He also appears to be the first to use the name "abelian variety". It was André Weil in the 1940s who gave the subject its modern foundations in the language of algebraic geometry. Today, abelian varieties form an important tool in number theory, in dynamical systems (more specifically in the study of Hamiltonian systems), and in algebraic geometry (especially Picard varieties and Albanese varieties). Analytic theory Definition A complex torus of dimension g is a torus of real dimension 2g that carries the structure of a complex manifold. It can always be obtained as the quotient of a g-dimensional complex vector space by a lattice of rank 2g. A complex abelian variety of dimension g is a complex torus of dimension g that is also a projective algebraic variety over the field of complex numbers. By invoking the Kodaira embedding theorem and Chow's theorem one may equivalently define a complex abelian variety of dimension g to be a complex torus of dimension g that admits a positive line bundle. Since they are complex tori, abelian varieties carry the structure of a group. A morphism of abelian varieties is a morphism of the underlying algebraic varieties that preserves the identity element for the group structure. An isogeny is a finite-to-one morphism. When a complex torus carries the structure of an algebraic variety, this structure is necessarily unique. In the case g = 1, the notion of abelian variety is the same as that of elliptic curve, and every complex torus gives rise to such a curve; for g > 1 it has been known since Riemann that the algebraic variety condition imposes extra constraints on a complex torus. Riemann conditions The following criterion by Riemann decides whether or not a given complex torus is an abelian variety, i.e. whether or not it can be embedded into a projective space. Let X be a g-dimensional torus given as X = V/L where V is a complex vector space of dimension g and L is a lattice in V. Then X is an abelian variety if and only if there exists a positive definite hermitian form on V whose imaginary part takes integral values on L×L. Such a form on X is usually called a (non-degenerate) Riemann form. Choosing a basis for V and L, one can make this condition more explicit. There are several equivalent formulations of this; all of them are known as the Riemann conditions. The Jacobian of an algebraic curve Every algebraic curve C of genus g ≥ 1 is associated with an abelian variety J of dimension g, by means of an analytic map of C into J. As a torus, J carries a commutative group structure, and the image of C generates J as a group. More accurately, J is covered by Cg:[1] any point in J comes from a g-tuple of points in C. The study of differential forms on C, which give rise to the abelian integrals with which the theory started, can be derived from the simpler, translation-invariant theory of differentials on J. The abelian variety J is called the Jacobian variety of C, for any non-singular curve C over the complex numbers. From the point of view of birational geometry, its function field is the fixed field of the symmetric group on g letters acting on the function field of Cg. Abelian functions An abelian function is a meromorphic function on an abelian variety, which may be regarded therefore as a periodic function of n complex variables, having 2n independent periods; equivalently, it is a function in the function field of an abelian variety. For example, in the nineteenth century there was much interest in hyperelliptic integrals that may be expressed in terms of elliptic integrals. This comes down to asking that J is a product of elliptic curves, up to an isogeny. See also: abelian integral Important theorems One important structure theorem of abelian varieties is Matsusaka's theorem. It states that over an algebraically closed field every abelian variety $A$ is the quotient of the Jacobian of some curve; that is, there is some surjection of abelian varieties $J\to A$ where $J$ is a Jacobian. This theorem remains true if the ground field is infinite.[2] Algebraic definition Two equivalent definitions of abelian variety over a general field k are commonly in use: • a connected and complete algebraic group over k • a connected and projective algebraic group over k. When the base is the field of complex numbers, these notions coincide with the previous definition. Over all bases, elliptic curves are abelian varieties of dimension 1. In the early 1940s, Weil used the first definition (over an arbitrary base field) but could not at first prove that it implied the second. Only in 1948 did he prove that complete algebraic groups can be embedded into projective space. Meanwhile, in order to make the proof of the Riemann hypothesis for curves over finite fields that he had announced in 1940 work, he had to introduce the notion of an abstract variety and to rewrite the foundations of algebraic geometry to work with varieties without projective embeddings (see also the history section in the Algebraic Geometry article). Structure of the group of points By the definitions, an abelian variety is a group variety. Its group of points can be proven to be commutative. For C, and hence by the Lefschetz principle for every algebraically closed field of characteristic zero, the torsion group of an abelian variety of dimension g is isomorphic to (Q/Z)2g. Hence, its n-torsion part is isomorphic to (Z/nZ)2g, i.e. the product of 2g copies of the cyclic group of order n. When the base field is an algebraically closed field of characteristic p, the n-torsion is still isomorphic to (Z/nZ)2g when n and p are coprime. When n and p are not coprime, the same result can be recovered provided one interprets it as saying that the n-torsion defines a finite flat group scheme of rank 2g. If instead of looking at the full scheme structure on the n-torsion, one considers only the geometric points, one obtains a new invariant for varieties in characteristic p (the so-called p-rank when n = p). The group of k-rational points for a global field k is finitely generated by the Mordell-Weil theorem. Hence, by the structure theorem for finitely generated abelian groups, it is isomorphic to a product of a free abelian group Zr and a finite commutative group for some non-negative integer r called the rank of the abelian variety. Similar results hold for some other classes of fields k. Products The product of an abelian variety A of dimension m, and an abelian variety B of dimension n, over the same field, is an abelian variety of dimension m + n. An abelian variety is simple if it is not isogenous to a product of abelian varieties of lower dimension. Any abelian variety is isogenous to a product of simple abelian varieties. Polarisation and dual abelian variety Dual abelian variety Main article: Dual abelian variety To an abelian variety A over a field k, one associates a dual abelian variety Av (over the same field), which is the solution to the following moduli problem. A family of degree 0 line bundles parametrised by a k-variety T is defined to be a line bundle L on A×T such that 1. for all t in T, the restriction of L to A×{t} is a degree 0 line bundle, 2. the restriction of L to {0}×T is a trivial line bundle (here 0 is the identity of A). Then there is a variety Av and a family of degree 0 line bundles P, the Poincaré bundle, parametrised by Av such that a family L on T is associated a unique morphism f: T → Av so that L is isomorphic to the pullback of P along the morphism 1A×f: A×T → A×Av. Applying this to the case when T is a point, we see that the points of Av correspond to line bundles of degree 0 on A, so there is a natural group operation on Av given by tensor product of line bundles, which makes it into an abelian variety. This association is a duality in the sense that there is a natural isomorphism between the double dual Avv and A (defined via the Poincaré bundle) and that it is contravariant functorial, i.e. it associates to all morphisms f: A → B dual morphisms fv: Bv → Av in a compatible way. The n-torsion of an abelian variety and the n-torsion of its dual are dual to each other when n is coprime to the characteristic of the base. In general - for all n - the n-torsion group schemes of dual abelian varieties are Cartier duals of each other. This generalises the Weil pairing for elliptic curves. Polarisations A polarisation of an abelian variety is an isogeny from an abelian variety to its dual that is symmetric with respect to double-duality for abelian varieties and for which the pullback of the Poincaré bundle along the associated graph morphism is ample (so it is analogous to a positive-definite quadratic form). Polarised abelian varieties have finite automorphism groups. A principal polarisation is a polarisation that is an isomorphism. Jacobians of curves are naturally equipped with a principal polarisation as soon as one picks an arbitrary rational base point on the curve, and the curve can be reconstructed from its polarised Jacobian when the genus is > 1. Not all principally polarised abelian varieties are Jacobians of curves; see the Schottky problem. A polarisation induces a Rosati involution on the endomorphism ring $\mathrm {End} (A)\otimes \mathbb {Q} $ of A. Polarisations over the complex numbers Over the complex numbers, a polarised abelian variety can also be defined as an abelian variety A together with a choice of a Riemann form H. Two Riemann forms H1 and H2 are called equivalent if there are positive integers n and m such that nH1=mH2. A choice of an equivalence class of Riemann forms on A is called a polarisation of A. A morphism of polarised abelian varieties is a morphism A → B of abelian varieties such that the pullback of the Riemann form on B to A is equivalent to the given form on A. Abelian scheme One can also define abelian varieties scheme-theoretically and relative to a base. This allows for a uniform treatment of phenomena such as reduction mod p of abelian varieties (see Arithmetic of abelian varieties), and parameter-families of abelian varieties. An abelian scheme over a base scheme S of relative dimension g is a proper, smooth group scheme over S whose geometric fibers are connected and of dimension g. The fibers of an abelian scheme are abelian varieties, so one could think of an abelian scheme over S as being a family of abelian varieties parametrised by S. For an abelian scheme A / S, the group of n-torsion points forms a finite flat group scheme. The union of the pn-torsion points, for all n, forms a p-divisible group. Deformations of abelian schemes are, according to the Serre–Tate theorem, governed by the deformation properties of the associated p-divisible groups. Example Let $A,B\in \mathbb {Z} $ be such that $x^{3}+Ax+B$ has no repeated complex roots. Then the discriminant $\Delta =-16(4A^{3}+27B^{2})$ is nonzero. Let $R=\mathbb {Z} [1/\Delta ]$, so $\operatorname {Spec} R$ is an open subscheme of $\operatorname {Spec} \mathbb {Z} $. Then $\operatorname {Proj} R[x,y,z]/(y^{2}z-x^{3}-Axz^{2}-Bz^{3})$ is an abelian scheme over $\operatorname {Spec} R$. It can be extended to a Néron model over $\operatorname {Spec} \mathbb {Z} $, which is a smooth group scheme over $\operatorname {Spec} \mathbb {Z} $, but the Néron model is not proper and hence is not an abelian scheme over $\operatorname {Spec} \mathbb {Z} $. Non-existence V. A. Abrashkin[3] and Jean-Marc Fontaine[4] independently proved that there are no nonzero abelian varieties over Q with good reduction at all primes. Equivalently, there are no nonzero abelian schemes over Spec Z. The proof involves showing that the coordinates of pn-torsion points generate number fields with very little ramification and hence of small discriminant, while, on the other hand, there are lower bounds on discriminants of number fields.[5] Semiabelian variety A semiabelian variety is a commutative group variety which is an extension of an abelian variety by a torus. See also • Complex torus • Motives • Timeline of abelian varieties • Moduli of abelian varieties • Equations defining abelian varieties • Horrocks–Mumford bundle References 1. Bruin, N. "N-Covers of Hyperelliptic Curves" (PDF). Math Department Oxford University. Retrieved 14 January 2015. J is covered by Cg: 2. Milne, J.S., Jacobian varieties, in Arithmetic Geometry, eds Cornell and Silverman, Springer-Verlag, 1986 3. "V. A. Abrashkin, "Group schemes of period $p$ over the ring of Witt vectors", Dokl. Akad. Nauk SSSR, 283:6 (1985), 1289–1294". www.mathnet.ru. Retrieved 2020-08-23. 4. Fontaine, Jean-Marc. Il n'y a pas de variété abélienne sur Z. OCLC 946402079. 5. "There is no Abelian scheme over Z" (PDF). Archived (PDF) from the original on 23 Aug 2020. Sources • Birkenhake, Christina; Lange, H. (1992), Complex Abelian Varieties, Berlin, New York: Springer-Verlag, ISBN 978-0-387-54747-3. A comprehensive treatment of the complex theory, with an overview of the history of the subject. • Dolgachev, I.V. (2001) [1994], "Abelian scheme", Encyclopedia of Mathematics, EMS Press • Faltings, Gerd; Chai, Ching-Li (1990), Degeneration of Abelian Varieties, Springer Verlag, ISBN 3-540-52015-5 • Milne, James, Abelian Varieties, retrieved 6 October 2016. Online course notes. • Mumford, David (2008) [1970], Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, vol. 5, Providence, R.I.: American Mathematical Society, ISBN 978-81-85931-86-9, MR 0282985, OCLC 138290 • Venkov, B.B.; Parshin, A.N. (2001) [1994], "Abelian_variety", Encyclopedia of Mathematics, EMS Press • Bruin, N; Flynn, E.V., N-COVERS OF HYPERELLIPTIC CURVES (PDF), Oxford: Mathematical Institute, University of Oxford. Description of the Jacobian of the Covering Curves Authority control: National • France • BnF data • Germany • Israel • United States • Japan	Wikipedia
Semialgebraic set In mathematics, a semialgebraic set is a finite union of sets defined by polynomial equalities and polynomial inequalities. A semialgebraic function is a function with a semialgebraic graph. Such sets and functions are mainly studied in real algebraic geometry which is the appropriate framework for algebraic geometry over the real numbers. Definition Let $\mathbb {F} $ be a real closed field. (For example $\mathbb {F} $ could be the field of real numbers $\mathbb {R} $.) A subset $S$ of $\mathbb {F} ^{n}$ is a semialgebraic set if it is a finite union of sets defined by polynomial equalities of the form $\{(x_{1},...,x_{n})\in \mathbb {F} ^{n}\mid P(x_{1},...,x_{n})=0\}$ and of sets defined by polynomial inequalities of the form $\{(x_{1},...,x_{n})\in \mathbb {F} ^{n}\mid Q(x_{1},...,x_{n})>0\}.$ Properties Similarly to algebraic subvarieties, finite unions and intersections of semialgebraic sets are still semialgebraic sets. Furthermore, unlike subvarieties, the complement of a semialgebraic set is again semialgebraic. Finally, and most importantly, the Tarski–Seidenberg theorem says that they are also closed under the projection operation: in other words a semialgebraic set projected onto a linear subspace yields another semialgebraic set (as is the case for quantifier elimination). These properties together mean that semialgebraic sets form an o-minimal structure on R. A semialgebraic set (or function) is said to be defined over a subring A of R if there is some description as in the definition, where the polynomials can be chosen to have coefficients in A. On a dense open subset of the semialgebraic set S, it is (locally) a submanifold. One can define the dimension of S to be the largest dimension at points at which it is a submanifold. It is not hard to see that a semialgebraic set lies inside an algebraic subvariety of the same dimension. See also • Łojasiewicz inequality • Existential theory of the reals • Subanalytic set • piecewise algebraic space References • Bochnak, J.; Coste, M.; Roy, M.-F. (1998), Real algebraic geometry, Berlin: Springer-Verlag, ISBN 9783662037188. • Bierstone, Edward; Milman, Pierre D. (1988), "Semianalytic and subanalytic sets", Inst. Hautes Études Sci. Publ. Math., 67: 5–42, doi:10.1007/BF02699126, MR 0972342, S2CID 56006439. • van den Dries, L. (1998), Tame topology and o-minimal structures, Cambridge University Press, ISBN 9780521598385. External links • PlanetMath page	Wikipedia
RE (complexity) In computability theory and computational complexity theory, RE (recursively enumerable) is the class of decision problems for which a 'yes' answer can be verified by a Turing machine in a finite amount of time.[1] Informally, it means that if the answer to a problem instance is 'yes', then there is some procedure that takes finite time to determine this, and this procedure never falsely reports 'yes' when the true answer is 'no'. However, when the true answer is 'no', the procedure is not required to halt; it may go into an "infinite loop" for some 'no' cases. Such a procedure is sometimes called a semi-algorithm, to distinguish it from an algorithm, defined as a complete solution to a decision problem.[2] Similarly, co-RE is the set of all languages that are complements of a language in RE. In a sense, co-RE contains languages of which membership can be disproved in a finite amount of time, but proving membership might take forever. Equivalent definition Equivalently, RE is the class of decision problems for which a Turing machine can list all the 'yes' instances, one by one (this is what 'enumerable' means). Each member of RE is a recursively enumerable set and therefore a Diophantine set. To show this is equivalent, note that if there is a machine $E$ that enumerates all accepted inputs, another machine that takes in a string can run $E$ and accept if the string is enumerated. Conversely, if a machine $M$ accepts when an input is in a language, another machine can enumerate all strings in the language by interleaving simulations of $M$ on every input and outputting strings that are accepted (there is an order of execution that will eventually get to every execution step because there are countably many ordered pairs of inputs and steps). Relations to other classes The set of recursive languages (R) is a subset of both RE and co-RE.[3] In fact, it is the intersection of those two classes, because we can decide any problem for which there exists a recogniser and also a co-recogniser by simply interleaving them until one obtains a result. Therefore: ${\mbox{R}}={\mbox{RE}}\cap {\mbox{co-RE}}$. Conversely, the set of languages that are neither RE nor co-RE is known as NRNC. These are the set of languages for which neither membership nor non-membership can be proven in a finite amount of time, and contain all other languages that are not in either RE or co-RE. That is: ${\mbox{NRNC}}={\mbox{ALL}}-({\mbox{RE}}\cup {\mbox{co-RE}})$. Not only are these problems undecidable, but neither they nor their complement are recursively enumerable. In January of 2020, a preprint announced a proof that RE was equivalent to the class MIP* (the class where a classical verifier interacts with multiple all-powerful quantum provers who share entanglement);[4] a revised, but not yet fully reviewed, proof was published in Communications of the ACM in November 2021. The proof implies that the Connes embedding problem and Tsirelson's problem are false.[5] RE-complete RE-complete is the set of decision problems that are complete for RE. In a sense, these are the "hardest" recursively enumerable problems. Generally, no constraint is placed on the reductions used except that they must be many-one reductions. Examples of RE-complete problems: 1. Halting problem: Whether a program given a finite input finishes running or will run forever. 2. By Rice's Theorem, deciding membership in any nontrivial subset of the set of recursive functions is RE-hard. It will be complete whenever the set is recursively enumerable. 3. John Myhill (1955)[6] proved that all creative sets are RE-complete. 4. The uniform word problem for groups or semigroups. (Indeed, the word problem for some individual groups is RE-complete.) 5. Deciding membership in a general unrestricted formal grammar. (Again, certain individual grammars have RE-complete membership problems.) 6. The validity problem for first-order logic. 7. Post correspondence problem: Given a list of pairs of strings, determine if there is a selection from these pairs (allowing repeats) such that the concatenation of the first items (of the pairs) is equal to the concatenation of the second items. 8. Determining if a Diophantine equation has any integer solutions. co-RE-complete co-RE-complete is the set of decision problems that are complete for co-RE. In a sense, these are the complements of the hardest recursively enumerable problems. Examples of co-RE-complete problems: 1. The domino problem for Wang tiles. 2. The satisfiability problem for first-order logic. See also • Knuth–Bendix completion algorithm • List of undecidable problems • Polymorphic recursion • Risch algorithm • Semidecidability References 1. Complexity Zoo: Class RE 2. Korfhage, Robert R. (1966). Logic and Algorithms, With Applications to the Computer and Information Sciences. Wiley. p. 89. A method of solution will be called a semi-algorithm for [a problem] P on [a device] M if the solution to P (if one exists) appears after the performance of finitely many steps. A semi-algorithm will be called an algorithm if, in addition, whenever the problem has no solution the method enables the device to determine this after a finite number of steps and halts. 3. Complexity Zoo: Class co-RE 4. Ji, Zhengfeng; Natarajan, Anand; Vidick, Thomas; Wright, John; Yuen, Henry (2020). "MIP=RE". arXiv:2001.04383 [quant-ph]. 5. Ji, Zhengfeng; Natarajan, Anand; Vidick, Thomas; Wright, John; Yuen, Henry (November 2021). "MIP = RE". Communications of the ACM. 64 (11): 131–138. doi:10.1145/3485628. S2CID 210165045. 6. Myhill, John (1955), "Creative sets", Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 1 (2): 97–108, doi:10.1002/malq.19550010205, MR 0071379. Important complexity classes Considered feasible • DLOGTIME • AC0 • ACC0 • TC0 • L • SL • RL • NL • NL-complete • NC • SC • CC • P • P-complete • ZPP • RP • BPP • BQP • APX • FP Suspected infeasible • UP • NP • NP-complete • NP-hard • co-NP • co-NP-complete • AM • QMA • PH • ⊕P • PP • #P • #P-complete • IP • PSPACE • PSPACE-complete Considered infeasible • EXPTIME • NEXPTIME • EXPSPACE • 2-EXPTIME • ELEMENTARY • PR • R • RE • ALL Class hierarchies • Polynomial hierarchy • Exponential hierarchy • Grzegorczyk hierarchy • Arithmetical hierarchy • Boolean hierarchy Families of classes • DTIME • NTIME • DSPACE • NSPACE • Probabilistically checkable proof • Interactive proof system List of complexity classes	Wikipedia
Nef line bundle In algebraic geometry, a line bundle on a projective variety is nef if it has nonnegative degree on every curve in the variety. The classes of nef line bundles are described by a convex cone, and the possible contractions of the variety correspond to certain faces of the nef cone. In view of the correspondence between line bundles and divisors (built from codimension-1 subvarieties), there is an equivalent notion of a nef divisor. Definition More generally, a line bundle L on a proper scheme X over a field k is said to be nef if it has nonnegative degree on every (closed irreducible) curve in X.[1] (The degree of a line bundle L on a proper curve C over k is the degree of the divisor (s) of any nonzero rational section s of L.) A line bundle may also be called an invertible sheaf. The term "nef" was introduced by Miles Reid as a replacement for the older terms "arithmetically effective" (Zariski 1962, definition 7.6) and "numerically effective", as well as for the phrase "numerically eventually free".[2] The older terms were misleading, in view of the examples below. Every line bundle L on a proper curve C over k which has a global section that is not identically zero has nonnegative degree. As a result, a basepoint-free line bundle on a proper scheme X over k has nonnegative degree on every curve in X; that is, it is nef.[3] More generally, a line bundle L is called semi-ample if some positive tensor power $L^{\otimes a}$ is basepoint-free. It follows that a semi-ample line bundle is nef. Semi-ample line bundles can be considered the main geometric source of nef line bundles, although the two concepts are not equivalent; see the examples below. A Cartier divisor D on a proper scheme X over a field is said to be nef if the associated line bundle O(D) is nef on X. Equivalently, D is nef if the intersection number $D\cdot C$ is nonnegative for every curve C in X. To go back from line bundles to divisors, the first Chern class is the isomorphism from the Picard group of line bundles on a variety X to the group of Cartier divisors modulo linear equivalence. Explicitly, the first Chern class $c_{1}(L)$ is the divisor (s) of any nonzero rational section s of L.[4] The nef cone To work with inequalities, it is convenient to consider R-divisors, meaning finite linear combinations of Cartier divisors with real coefficients. The R-divisors modulo numerical equivalence form a real vector space $N^{1}(X)$ of finite dimension, the Néron–Severi group tensored with the real numbers.[5] (Explicitly: two R-divisors are said to be numerically equivalent if they have the same intersection number with all curves in X.) An R-divisor is called nef if it has nonnegative degree on every curve. The nef R-divisors form a closed convex cone in $N^{1}(X)$, the nef cone Nef(X). The cone of curves is defined to be the convex cone of linear combinations of curves with nonnegative real coefficients in the real vector space $N_{1}(X)$ of 1-cycles modulo numerical equivalence. The vector spaces $N^{1}(X)$ and $N_{1}(X)$ are dual to each other by the intersection pairing, and the nef cone is (by definition) the dual cone of the cone of curves.[6] A significant problem in algebraic geometry is to analyze which line bundles are ample, since that amounts to describing the different ways a variety can be embedded into projective space. One answer is Kleiman's criterion (1966): for a projective scheme X over a field, a line bundle (or R-divisor) is ample if and only if its class in $N^{1}(X)$ lies in the interior of the nef cone.[7] (An R-divisor is called ample if it can be written as a positive linear combination of ample Cartier divisors.) It follows from Kleiman's criterion that, for X projective, every nef R-divisor on X is a limit of ample R-divisors in $N^{1}(X)$. Indeed, for D nef and A ample, D + cA is ample for all real numbers c > 0. Metric definition of nef line bundles Let X be a compact complex manifold with a fixed Hermitian metric, viewed as a positive (1,1)-form $\omega $. Following Jean-Pierre Demailly, Thomas Peternell and Michael Schneider, a holomorphic line bundle L on X is said to be nef if for every $\epsilon >0$ there is a smooth Hermitian metric $h_{\epsilon }$ on L whose curvature satisfies $\Theta _{h_{\epsilon }}(L)\geq -\epsilon \omega $. When X is projective over C, this is equivalent to the previous definition (that L has nonnegative degree on all curves in X).[8] Even for X projective over C, a nef line bundle L need not have a Hermitian metric h with curvature $\Theta _{h}(L)\geq 0$, which explains the more complicated definition just given.[9] Examples • If X is a smooth projective surface and C is an (irreducible) curve in X with self-intersection number $C^{2}\geq 0$, then C is nef on X, because any two distinct curves on a surface have nonnegative intersection number. If $C^{2}<0$, then C is effective but not nef on X. For example, if X is the blow-up of a smooth projective surface Y at a point, then the exceptional curve E of the blow-up $\pi \colon X\to Y$ has $E^{2}=-1$. • Every effective divisor on a flag manifold or abelian variety is nef, using that these varieties have a transitive action of a connected algebraic group.[10] • Every line bundle L of degree 0 on a smooth complex projective curve X is nef, but L is semi-ample if and only if L is torsion in the Picard group of X. For X of genus g at least 1, most line bundles of degree 0 are not torsion, using that the Jacobian of X is an abelian variety of dimension g. • Every semi-ample line bundle is nef, but not every nef line bundle is even numerically equivalent to a semi-ample line bundle. For example, David Mumford constructed a line bundle L on a suitable ruled surface X such that L has positive degree on all curves, but the intersection number $c_{1}(L)^{2}$ is zero.[11] It follows that L is nef, but no positive multiple of $c_{1}(L)$ is numerically equivalent to an effective divisor. In particular, the space of global sections $H^{0}(X,L^{\otimes a})$ is zero for all positive integers a. Contractions and the nef cone A contraction of a normal projective variety X over a field k is a surjective morphism $f\colon X\to Y$ with Y a normal projective variety over k such that $f_{}O_{X}=O_{Y}$. (The latter condition implies that f has connected fibers, and it is equivalent to f having connected fibers if k has characteristic zero.[12]) A contraction is called a fibration if dim(Y) < dim(X). A contraction with dim(Y) = dim(X) is automatically a birational morphism.[13] (For example, X could be the blow-up of a smooth projective surface Y at a point.) A face F of a convex cone N means a convex subcone such that any two points of N whose sum is in F must themselves be in F. A contraction of X determines a face F of the nef cone of X, namely the intersection of Nef(X) with the pullback $f^{}(N^{1}(Y))\subset N^{1}(X)$. Conversely, given the variety X, the face F of the nef cone determines the contraction $f\colon X\to Y$ up to isomorphism. Indeed, there is a semi-ample line bundle L on X whose class in $N^{1}(X)$ is in the interior of F (for example, take L to be the pullback to X of any ample line bundle on Y). Any such line bundle determines Y by the Proj construction:[14] $Y={\text{Proj }}\bigoplus _{a\geq 0}H^{0}(X,L^{\otimes a}).$ To describe Y in geometric terms: a curve C in X maps to a point in Y if and only if L has degree zero on C. As a result, there is a one-to-one correspondence between the contractions of X and some of the faces of the nef cone of X.[15] (This correspondence can also be formulated dually, in terms of faces of the cone of curves.) Knowing which nef line bundles are semi-ample would determine which faces correspond to contractions. The cone theorem describes a significant class of faces that do correspond to contractions, and the abundance conjecture would give more. Example: Let X be the blow-up of the complex projective plane $\mathbb {P} ^{2}$ at a point p. Let H be the pullback to X of a line on $\mathbb {P} ^{2}$, and let E be the exceptional curve of the blow-up $\pi \colon X\to \mathbb {P} ^{2}$. Then X has Picard number 2, meaning that the real vector space $N^{1}(X)$ has dimension 2. By the geometry of convex cones of dimension 2, the nef cone must be spanned by two rays; explicitly, these are the rays spanned by H and H − E.[16] In this example, both rays correspond to contractions of X: H gives the birational morphism $X\to \mathbb {P} ^{2}$, and H − E gives a fibration $X\to \mathbb {P} ^{1}$ with fibers isomorphic to $\mathbb {P} ^{1}$ (corresponding to the lines in $\mathbb {P} ^{2}$ through the point p). Since the nef cone of X has no other nontrivial faces, these are the only nontrivial contractions of X; that would be harder to see without the relation to convex cones. Notes 1. Lazarsfeld (2004), Definition 1.4.1. 2. Reid (1983), section 0.12f. 3. Lazarsfeld (2004), Example 1.4.5. 4. Lazarsfeld (2004), Example 1.1.5. 5. Lazarsfeld (2004), Example 1.3.10. 6. Lazarsfeld (2004), Definition 1.4.25. 7. Lazarsfeld (2004), Theorem 1.4.23. 8. Demailly et al. (1994), section 1. 9. Demailly et al. (1994), Example 1.7. 10. Lazarsfeld (2004), Example 1.4.7. 11. Lazarsfeld (2004), Example 1.5.2. 12. Lazarsfeld (2004), Definition 2.1.11. 13. Lazarsfeld (2004), Example 2.1.12. 14. Lazarsfeld (2004), Theorem 2.1.27. 15. Kollár & Mori (1998), Remark 1.26. 16. Kollár & Mori (1998), Lemma 1.22 and Example 1.23(1). References • Demailly, Jean-Pierre; Peternell, Thomas; Schneider, Michael (1994), "Compact complex manifolds with numerically effective tangent bundles" (PDF), Journal of Algebraic Geometry, 3: 295–345, MR 1257325 • Kollár, János; Mori, Shigefumi (1998), Birational geometry of algebraic varieties, Cambridge University Press, doi:10.1017/CBO9780511662560, ISBN 978-0-521-63277-5, MR 1658959 • Lazarsfeld, Robert (2004), Positivity in algebraic geometry, vol. 1, Berlin: Springer-Verlag, doi:10.1007/978-3-642-18808-4, ISBN 3-540-22533-1, MR 2095471 • Reid, Miles (1983), "Minimal models of canonical 3-folds", Algebraic varieties and analytic varieties (Tokyo, 1981), Advanced Studies in Pure Mathematics, vol. 1, North-Holland, pp. 131–180, doi:10.2969/aspm/00110131, ISBN 0-444-86612-4, MR 0715649 • Zariski, Oscar (1962), "The theorem of Riemann-Roch for high multiples of an effective divisor on an algebraic surface", Annals of Mathematics, 2, 76: 560–615, doi:10.2307/1970376, MR 0141668	Wikipedia
Semidirect product In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. There are two closely related concepts of semidirect product: • an inner semidirect product is a particular way in which a group can be made up of two subgroups, one of which is a normal subgroup. • an outer semidirect product is a way to construct a new group from two given groups by using the Cartesian product as a set and a particular multiplication operation. Algebraic structure → Group theory Group theory Basic notions • Subgroup • Normal subgroup • Quotient group • (Semi-)direct product Group homomorphisms • kernel • image • direct sum • wreath product • simple • finite • infinite • continuous • multiplicative • additive • cyclic • abelian • dihedral • nilpotent • solvable • action • Glossary of group theory • List of group theory topics Finite groups • Cyclic group Zn • Symmetric group Sn • Alternating group An • Dihedral group Dn • Quaternion group Q • Cauchy's theorem • Lagrange's theorem • Sylow theorems • Hall's theorem • p-group • Elementary abelian group • Frobenius group • Schur multiplier Classification of finite simple groups • cyclic • alternating • Lie type • sporadic • Discrete groups • Lattices • Integers ($\mathbb {Z} $) • Free group Modular groups • PSL(2, $\mathbb {Z} $) • SL(2, $\mathbb {Z} $) • Arithmetic group • Lattice • Hyperbolic group Topological and Lie groups • Solenoid • Circle • General linear GL(n) • Special linear SL(n) • Orthogonal O(n) • Euclidean E(n) • Special orthogonal SO(n) • Unitary U(n) • Special unitary SU(n) • Symplectic Sp(n) • G2 • F4 • E6 • E7 • E8 • Lorentz • Poincaré • Conformal • Diffeomorphism • Loop Infinite dimensional Lie group • O(∞) • SU(∞) • Sp(∞) Algebraic groups • Linear algebraic group • Reductive group • Abelian variety • Elliptic curve As with direct products, there is a natural equivalence between inner and outer semidirect products, and both are commonly referred to simply as semidirect products. For finite groups, the Schur–Zassenhaus theorem provides a sufficient condition for the existence of a decomposition as a semidirect product (also known as splitting extension). Inner semidirect product definitions Given a group G with identity element e, a subgroup H, and a normal subgroup N ◁ G, the following statements are equivalent: • G is the product of subgroups, G = NH, and these subgroups have trivial intersection: N ∩ H = {e}. • For every g ∈ G, there are unique n ∈ N and h ∈ H such that g = nh. • For every g ∈ G, there are unique n ∈ N and h ∈ H such that g = hn. • The composition π ∘ i of the natural embedding i: H → G with the natural projection π: G → G/N is an isomorphism between H and the quotient group G/N. • There exists a homomorphism G → H that is the identity on H and whose kernel is N. In other words, there is a split exact sequence $1\to N\to G\to H\to 1$ of groups (which is also known as group extension of $H$ by $N$). If any of these statements holds (and hence all of them hold, by their equivalence), we say G is the semidirect product of N and H, written $G=N\rtimes H$ or $G=H\ltimes N,$ or that G splits over N; one also says that G is a semidirect product of H acting on N, or even a semidirect product of H and N. To avoid ambiguity, it is advisable to specify which is the normal subgroup. If $G=H\ltimes N$, then there is a group homomorphism $\varphi \colon H\rightarrow \mathrm {Aut} (N)$ given by $\varphi _{h}(n)=hnh^{-1}$, and for $g=hn,g'=h'n'$, we have $gg'=hnh'n'=hh'h'^{-1}nh'n'=hh'\varphi _{{h'}^{-1}}(n)n'=h^{}n^{}$. Inner and outer semidirect products Let us first consider the inner semidirect product. In this case, for a group $G$, consider its normal subgroup N and the subgroup H (not necessarily normal). Assume that the conditions on the list above hold. Let $\operatorname {Aut} (N)$ denote the group of all automorphisms of N, which is a group under composition. Construct a group homomorphism $\varphi \colon H\to \operatorname {Aut} (N)$ defined by conjugation, $\varphi _{h}(n)=hnh^{-1}$, for all h in H and n in N. In this way we can construct a group $G'=(N,H)$ with group operation defined as $(n_{1},h_{1})\cdot (n_{2},h_{2})=(n_{1}\varphi _{h_{1}}(n_{2}),\,h_{1}h_{2})$ for n1, n2 in N and h1, h2 in H. The subgroups N and H determine G up to isomorphism, as we will show later. In this way, we can construct the group G from its subgroups. This kind of construction is called an inner semidirect product (also known as internal semidirect product[1]). Let us now consider the outer semidirect product. Given any two groups N and H and a group homomorphism φ: H → Aut(N), we can construct a new group N ⋊φ H, called the outer semidirect product of N and H with respect to φ, defined as follows:[2] • The underlying set is the Cartesian product N × H. • The group operation $\bullet $ is determined by the homomorphism φ: ${\begin{aligned}\bullet :(N\rtimes _{\varphi }H)\times (N\rtimes _{\varphi }H)&\to N\rtimes _{\varphi }H\\(n_{1},h_{1})\bullet (n_{2},h_{2})&=(n_{1}\varphi _{h_{1}}(n_{2}),\,h_{1}h_{2})\end{aligned}}$ :(N\rtimes _{\varphi }H)\times (N\rtimes _{\varphi }H)&\to N\rtimes _{\varphi }H\\(n_{1},h_{1})\bullet (n_{2},h_{2})&=(n_{1}\varphi _{h_{1}}(n_{2}),\,h_{1}h_{2})\end{aligned}}} for n1, n2 in N and h1, h2 in H. This defines a group in which the identity element is (eN, eH) and the inverse of the element (n, h) is (φh−1(n−1), h−1). Pairs (n, eH) form a normal subgroup isomorphic to N, while pairs (eN, h) form a subgroup isomorphic to H. The full group is a semidirect product of those two subgroups in the sense given earlier. Conversely, suppose that we are given a group G with a normal subgroup N and a subgroup H, such that every element g of G may be written uniquely in the form g = nh where n lies in N and h lies in H. Let φ: H → Aut(N) be the homomorphism (written φ(h) = φh) given by $\varphi _{h}(n)=hnh^{-1}$ for all n ∈ N, h ∈ H. Then G is isomorphic to the semidirect product N ⋊φ H. The isomorphism λ: G → N ⋊φ H is well defined by λ(a) = λ(nh) = (n, h) due to the uniqueness of the decomposition a = nh. In G, we have $(n_{1}h_{1})(n_{2}h_{2})=n_{1}h_{1}n_{2}(h_{1}^{-1}h_{1})h_{2}=(n_{1}\varphi _{h_{1}}(n_{2}))(h_{1}h_{2})$ Thus, for a = n1h1 and b = n2h2 we obtain ${\begin{aligned}\lambda (ab)&=\lambda (n_{1}h_{1}n_{2}h_{2})=\lambda (n_{1}\varphi _{h_{1}}(n_{2})h_{1}h_{2})=(n_{1}\varphi _{h_{1}}(n_{2}),h_{1}h_{2})=(n_{1},h_{1})\bullet (n_{2},h_{2})\\[5pt]&=\lambda (n_{1}h_{1})\bullet \lambda (n_{2}h_{2})=\lambda (a)\bullet \lambda (b),\end{aligned}}$ which proves that λ is a homomorphism. Since λ is obviously an epimorphism and monomorphism, then it is indeed an isomorphism. This also explains the definition of the multiplication rule in N ⋊φ H. The direct product is a special case of the semidirect product. To see this, let φ be the trivial homomorphism (i.e., sending every element of H to the identity automorphism of N) then N ⋊φ H is the direct product N × H. A version of the splitting lemma for groups states that a group G is isomorphic to a semidirect product of the two groups N and H if and only if there exists a short exact sequence $1\longrightarrow N\,{\overset {\beta }{\longrightarrow }}\,G\,{\overset {\alpha }{\longrightarrow }}\,H\longrightarrow 1$ and a group homomorphism γ: H → G such that α ∘ γ = idH, the identity map on H. In this case, φ: H → Aut(N) is given by φ(h) = φh, where $\varphi _{h}(n)=\beta ^{-1}(\gamma (h)\beta (n)\gamma (h^{-1})).$ Examples Dihedral group The dihedral group D2n with 2n elements is isomorphic to a semidirect product of the cyclic groups Cn and C2.[3] Here, the non-identity element of C2 acts on Cn by inverting elements; this is an automorphism since Cn is abelian. The presentation for this group is: $\langle a,\;b\mid a^{2}=e,\;b^{n}=e,\;aba^{-1}=b^{-1}\rangle .$ Cyclic groups More generally, a semidirect product of any two cyclic groups Cm with generator a and Cn with generator b is given by one extra relation, aba−1 = bk, with k and n coprime, and $k^{m}\equiv 1{\pmod {n}}$;[3] that is, the presentation:[3] $\langle a,\;b\mid a^{m}=e,\;b^{n}=e,\;aba^{-1}=b^{k}\rangle .$ If r and m are coprime, ar is a generator of Cm and arba−r = bkr, hence the presentation: $\langle a,\;b\mid a^{m}=e,\;b^{n}=e,\;aba^{-1}=b^{k^{r}}\rangle $ gives a group isomorphic to the previous one. Holomorph of a group One canonical example of a group expressed as a semi-direct product is the holomorph of a group. This is defined as $\operatorname {Hol} (G)=G\rtimes \operatorname {Aut} (G)$ where ${\text{Aut}}(G)$ is the automorphism group of a group $G$ and the structure map $\varphi $ comes from the right action of ${\text{Aut}}(G)$ on $G$. In terms of multiplying elements, this gives the group structure $(g,\alpha )(h,\beta )=(g(\varphi (\alpha )\cdot h),\alpha \beta ).$ Fundamental group of the Klein bottle The fundamental group of the Klein bottle can be presented in the form $\langle a,\;b\mid aba^{-1}=b^{-1}\rangle .$ and is therefore a semidirect product of the group of integers, $\mathbb {Z} $, with $\mathbb {Z} $. The corresponding homomorphism φ: $\mathbb {Z} $ → Aut($\mathbb {Z} $) is given by φ(h)(n) = (−1)hn. Upper triangular matrices The group $\mathbb {T} _{n}$ of upper triangular matrices with non-zero determinant, that is with non-zero entries on the diagonal, has a decomposition into the semidirect product $\mathbb {T} _{n}\cong \mathbb {U} _{n}\rtimes \mathbb {D} _{n}$[4] where $\mathbb {U} _{n}$ is the subgroup of matrices with only $1$'s on the diagonal, which is called the upper unitriangular matrix group, and $\mathbb {D} _{n}$ is the subgroup of diagonal matrices. The group action of $\mathbb {D} _{n}$ on $\mathbb {U} _{n}$ is induced by matrix multiplication. If we set $A={\begin{bmatrix}x_{1}&0&\cdots &0\\0&x_{2}&\cdots &0\\\vdots &\vdots &&\vdots \\0&0&\cdots &x_{n}\end{bmatrix}}$ and $B={\begin{bmatrix}1&a_{12}&a_{13}&\cdots &a_{1n}\\0&1&a_{23}&\cdots &a_{2n}\\\vdots &\vdots &\vdots &&\vdots \\0&0&0&\cdots &1\end{bmatrix}}$ then their matrix product is $AB={\begin{bmatrix}x_{1}&x_{1}a_{12}&x_{1}a_{13}&\cdots &x_{1}a_{1n}\\0&x_{2}&x_{2}a_{23}&\cdots &x_{2}a_{2n}\\\vdots &\vdots &\vdots &&\vdots \\0&0&0&\cdots &x_{n}\end{bmatrix}}.$ This gives the induced group action $m:\mathbb {D} _{n}\times \mathbb {U} _{n}\to \mathbb {U} _{n}$ $m(A,B)={\begin{bmatrix}1&x_{1}a_{12}&x_{1}a_{13}&\cdots &x_{1}a_{1n}\\0&1&x_{2}a_{23}&\cdots &x_{2}a_{2n}\\\vdots &\vdots &\vdots &&\vdots \\0&0&0&\cdots &1\end{bmatrix}}.$ A matrix in $\mathbb {T} _{n}$ can be represented by matrices in $\mathbb {U} _{n}$ and $\mathbb {D} _{n}$. Hence $\mathbb {T} _{n}\cong \mathbb {U} _{n}\rtimes \mathbb {D} _{n}$. Group of isometries on the plane The Euclidean group of all rigid motions (isometries) of the plane (maps f: $\mathbb {R} $2 → $\mathbb {R} $2 such that the Euclidean distance between x and y equals the distance between f(x) and f(y) for all x and y in $\mathbb {R} ^{2}$) is isomorphic to a semidirect product of the abelian group $\mathbb {R} ^{2}$ (which describes translations) and the group O(2) of orthogonal 2 × 2 matrices (which describes rotations and reflections that keep the origin fixed). Applying a translation and then a rotation or reflection has the same effect as applying the rotation or reflection first and then a translation by the rotated or reflected translation vector (i.e. applying the conjugate of the original translation). This shows that the group of translations is a normal subgroup of the Euclidean group, that the Euclidean group is a semidirect product of the translation group and O(2), and that the corresponding homomorphism φ: O(2) → Aut($\mathbb {R} $2) is given by matrix multiplication: φ(h)(n) = hn. Orthogonal group O(n) The orthogonal group O(n) of all orthogonal real n × n matrices (intuitively the set of all rotations and reflections of n-dimensional space that keep the origin fixed) is isomorphic to a semidirect product of the group SO(n) (consisting of all orthogonal matrices with determinant 1, intuitively the rotations of n-dimensional space) and C2. If we represent C2 as the multiplicative group of matrices {I, R}, where R is a reflection of n-dimensional space that keeps the origin fixed (i.e., an orthogonal matrix with determinant –1 representing an involution), then φ: C2 → Aut(SO(n)) is given by φ(H)(N) = HNH−1 for all H in C2 and N in SO(n). In the non-trivial case (H is not the identity) this means that φ(H) is conjugation of operations by the reflection (in 3-dimensional space a rotation axis and the direction of rotation are replaced by their "mirror image"). Semi-linear transformations The group of semilinear transformations on a vector space V over a field $\mathbb {K} $, often denoted ΓL(V), is isomorphic to a semidirect product of the linear group GL(V) (a normal subgroup of ΓL(V)), and the automorphism group of $\mathbb {K} $. Crystallographic groups In crystallography, the space group of a crystal splits as the semidirect product of the point group and the translation group if and only if the space group is symmorphic. Non-symmorphic space groups have point groups that are not even contained as subset of the space group, which is responsible for much of the complication in their analysis.[5] Non-examples Of course, no simple group can be expressed as a semi-direct product (because they do not have nontrivial normal subgroups), but there are a few common counterexamples of groups containing a non-trivial normal subgroup that nonetheless cannot be expressed as a semi-direct product. Note that although not every group $G$ can be expressed as a split extension of $H$ by $A$, it turns out that such a group can be embedded into the wreath product $A\wr H$ by the universal embedding theorem. Z4 The cyclic group $\mathbb {Z} _{4}$ is not a simple group since it has a subgroup of order 2, namely $\{0,2\}\cong \mathbb {Z} _{2}$ is a subgroup and their quotient is $\mathbb {Z} _{2}$, so there's an extension $0\to \mathbb {Z} _{2}\to \mathbb {Z} _{4}\to \mathbb {Z} _{2}\to 0$ If the extension was split, then the group $G$ in $0\to \mathbb {Z} _{2}\to G\to \mathbb {Z} _{2}\to 0$ would be isomorphic to $\mathbb {Z} _{2}\times \mathbb {Z} _{2}$. Q8 The group of the eight quaternions $\{\pm 1,\pm i,\pm j,\pm k\}$ where $ijk=-1$ and $i^{2}=j^{2}=k^{2}=-1$, is another example of a group[6] which has non-trivial normal subgroups yet is still not split. For example, the subgroup generated by $i$ is isomorphic to $\mathbb {Z} _{4}$ and is normal. It also has a subgroup of order $2$ generated by $-1$. This would mean $Q_{8}$ would have to be a split extension in the following hypothetical exact sequence of groups: $0\to \mathbb {Z} _{4}\to Q_{8}\to \mathbb {Z} _{2}\to 0$, but such an exact sequence does not exist. This can be shown by computing the first group cohomology group of $\mathbb {Z} _{2}$ with coefficients in $\mathbb {Z} _{4}$, so $H^{1}(\mathbb {Z} _{2},\mathbb {Z} _{4})\cong \mathbb {Z} /2$ and noting the two groups in these extensions are $\mathbb {Z} _{2}\times \mathbb {Z} _{4}$ and the dihedral group $D_{8}$. But, as neither of these groups is isomorphic with $Q_{8}$, the quaternion group is not split. This non-existence of isomorphisms can be checked by noting the trivial extension is abelian while $Q_{8}$ is non-abelian, and noting the only normal subgroups are $\mathbb {Z} _{2}$ and $\mathbb {Z} _{4}$, but $Q_{8}$ has three subgroups isomorphic to $\mathbb {Z} _{4}$. Properties If G is the semidirect product of the normal subgroup N and the subgroup H, and both N and H are finite, then the order of G equals the product of the orders of N and H. This follows from the fact that G is of the same order as the outer semidirect product of N and H, whose underlying set is the Cartesian product N × H. Relation to direct products Suppose G is a semidirect product of the normal subgroup N and the subgroup H. If H is also normal in G, or equivalently, if there exists a homomorphism G → N that is the identity on N with kernel H, then G is the direct product of N and H. The direct product of two groups N and H can be thought of as the semidirect product of N and H with respect to φ(h) = idN for all h in H. Note that in a direct product, the order of the factors is not important, since N × H is isomorphic to H × N. This is not the case for semidirect products, as the two factors play different roles. Furthermore, the result of a (proper) semidirect product by means of a non-trivial homomorphism is never an abelian group, even if the factor groups are abelian. Non-uniqueness of semidirect products (and further examples) As opposed to the case with the direct product, a semidirect product of two groups is not, in general, unique; if G and G′ are two groups that both contain isomorphic copies of N as a normal subgroup and H as a subgroup, and both are a semidirect product of N and H, then it does not follow that G and G′ are isomorphic because the semidirect product also depends on the choice of an action of H on N. For example, there are four non-isomorphic groups of order 16 that are semidirect products of C8 and C2; in this case, C8 is necessarily a normal subgroup because it has index 2. One of these four semidirect products is the direct product, while the other three are non-abelian groups: • the dihedral group of order 16 • the quasidihedral group of order 16 • the Iwasawa group of order 16 If a given group is a semidirect product, then there is no guarantee that this decomposition is unique. For example, there is a group of order 24 (the only one containing six elements of order 4 and six elements of order 6) that can be expressed as semidirect product in the following ways: (D8 ⋉ C3) ≅ (C2 ⋉ Q12) ≅ (C2 ⋉ D12) ≅ (D6 ⋉ V).[7] Existence Main article: Schur–Zassenhaus theorem In general, there is no known characterization (i.e., a necessary and sufficient condition) for the existence of semidirect products in groups. However, some sufficient conditions are known, which guarantee existence in certain cases. For finite groups, the Schur–Zassenhaus theorem guarantees existence of a semidirect product when the order of the normal subgroup is coprime to the order of the quotient group. For example, the Schur–Zassenhaus theorem implies the existence of a semi-direct product among groups of order 6; there are two such products, one of which is a direct product, and the other a dihedral group. In contrast, the Schur–Zassenhaus theorem does not say anything about groups of order 4 or groups of order 8 for instance. Generalizations Within group theory, the construction of semidirect products can be pushed much further. The Zappa–Szep product of groups is a generalization that, in its internal version, does not assume that either subgroup is normal. There is also a construction in ring theory, the crossed product of rings. This is constructed in the natural way from the group ring for a semidirect product of groups. The ring-theoretic approach can be further generalized to the semidirect sum of Lie algebras. For geometry, there is also a crossed product for group actions on a topological space; unfortunately, it is in general non-commutative even if the group is abelian. In this context, the semidirect product is the space of orbits of the group action. The latter approach has been championed by Alain Connes as a substitute for approaches by conventional topological techniques; c.f. noncommutative geometry. The semidirect product is a special case of the Grothendieck construction in category theory. Specifically, an action of $H$ on $N$ (respecting the group, or even just monoid structure) is the same thing as a functor $F:BH\to Cat$ from the groupoid $BH$ associated to H (having a single object *, whose endomorphisms are H) to the category of categories such that the unique object in $BH$ is mapped to $BN$. The Grothendieck construction of this functor is equivalent to $B(H\rtimes N)$, the (groupoid associated to) semidirect product.[8] Groupoids Another generalization is for groupoids. This occurs in topology because if a group G acts on a space X it also acts on the fundamental groupoid π1(X) of the space. The semidirect product π1(X) ⋊ G is then relevant to finding the fundamental groupoid of the orbit space X/G. For full details see Chapter 11 of the book referenced below, and also some details in semidirect product[9] in ncatlab. Abelian categories Non-trivial semidirect products do not arise in abelian categories, such as the category of modules. In this case, the splitting lemma shows that every semidirect product is a direct product. Thus the existence of semidirect products reflects a failure of the category to be abelian. Notation Usually the semidirect product of a group H acting on a group N (in most cases by conjugation as subgroups of a common group) is denoted by N ⋊ H or H ⋉ N. However, some sources[10] may use this symbol with the opposite meaning. In case the action φ: H → Aut(N) should be made explicit, one also writes N ⋊φ H. One way of thinking about the N ⋊ H symbol is as a combination of the symbol for normal subgroup (◁) and the symbol for the product (×). Barry Simon, in his book on group representation theory,[11] employs the unusual notation $N\mathbin {\circledS _{\varphi }} H$ for the semidirect product. Unicode lists four variants:[12] ValueMathMLUnicode description ⋉U+22C9ltimesLEFT NORMAL FACTOR SEMIDIRECT PRODUCT ⋊U+22CArtimesRIGHT NORMAL FACTOR SEMIDIRECT PRODUCT ⋋U+22CBlthreeLEFT SEMIDIRECT PRODUCT ⋌U+22CCrthreeRIGHT SEMIDIRECT PRODUCT Here the Unicode description of the rtimes symbol says "right normal factor", in contrast to its usual meaning in mathematical practice. In LaTeX, the commands \rtimes and \ltimes produce the corresponding characters. With the AMS symbols package loaded, \leftthreetimes produces ⋋ and \rightthreetimes produces ⋌. See also • Affine Lie algebra • Grothendieck construction, a categorical construction that generalizes the semidirect product • Holomorph • Lie algebra semidirect sum • Subdirect product • Wreath product • Zappa–Szép product • Crossed product Notes 1. DS Dummit and RM Foote (1991), Abstract algebra, Englewood Cliffs, NJ: Prentice Hall, 142. 2. Robinson, Derek John Scott (2003). An Introduction to Abstract Algebra. Walter de Gruyter. pp. 75–76. ISBN 9783110175448. 3. Mac Lane, Saunders; Birkhoff, Garrett (1999). Algebra (3rd ed.). American Mathematical Society. pp. 414–415. ISBN 0-8218-1646-2. 4. Milne. Algebraic Groups (PDF). pp. 45, semi-direct products. Archived (PDF) from the original on 2016-03-07. 5. Thompson, Nick. "Irreducible Brillouin Zones and Band Structures". bandgap.io. Retrieved 13 December 2017. 6. "abstract algebra - Can every non-simple group $G$ be written as a semidirect product?". Mathematics Stack Exchange. Retrieved 2020-10-29. 7. H.E. Rose (2009). A Course on Finite Groups. Springer Science & Business Media. p. 183. ISBN 978-1-84882-889-6. Note that Rose uses the opposite notation convention than the one adopted on this page (p. 152). 8. Barr & Wells (2012, §12.2) 9. "Ncatlab.org". 10. e.g., E. B. Vinberg (2003). A Course in Algebra. Providence, RI: American Mathematical Society. p. 389. ISBN 0-8218-3413-4. 11. B. Simon (1996). Representations of Finite and Compact Groups. Providence, RI: American Mathematical Society. p. 6. ISBN 0-8218-0453-7. 12. See unicode.org References • Barr, Michael; Wells, Charles (2012), Category theory for computing science, Reprints in Theory and Applications of Categories, vol. 2012, p. 558, Zbl 1253.18001 • Brown, R. (2006), Topology and groupoids, Booksurge, ISBN 1-4196-2722-8	Wikipedia
Semi-infinite programming In optimization theory, semi-infinite programming (SIP) is an optimization problem with a finite number of variables and an infinite number of constraints, or an infinite number of variables and a finite number of constraints. In the former case the constraints are typically parameterized.[1] Mathematical formulation of the problem The problem can be stated simply as: $\min _{x\in X}\;\;f(x)$ ${\text{subject to: }}$ $g(x,y)\leq 0,\;\;\forall y\in Y$ where $f:R^{n}\to R$ $g:R^{n}\times R^{m}\to R$ $X\subseteq R^{n}$ $Y\subseteq R^{m}.$ SIP can be seen as a special case of bilevel programs in which the lower-level variables do not participate in the objective function. Methods for solving the problem In the meantime, see external links below for a complete tutorial. Examples In the meantime, see external links below for a complete tutorial. See also • Optimization • Generalized semi-infinite programming (GSIP) References • Bonnans, J. Frédéric; Shapiro, Alexander (2000). "5.4 and 7.4.4 Semi-infinite programming". Perturbation analysis of optimization problems. Springer Series in Operations Research. New York: Springer-Verlag. pp. 496–526 and 581. ISBN 978-0-387-98705-7. MR 1756264. • M. A. Goberna and M. A. López, Linear Semi-Infinite Optimization, Wiley, 1998. • Hettich, R.; Kortanek, K. O. (1993). "Semi-infinite programming: Theory, methods, and applications". SIAM Review. 35 (3): 380–429. doi:10.1137/1035089. JSTOR 2132425. MR 1234637. • Edward J. Anderson and Peter Nash, Linear Programming in Infinite-Dimensional Spaces, Wiley, 1987. • Bonnans, J. Frédéric; Shapiro, Alexander (2000). "5.4 and 7.4.4 Semi-infinite programming". Perturbation analysis of optimization problems. Springer Series in Operations Research. New York: Springer-Verlag. pp. 496–526 and 581. ISBN 978-0-387-98705-7. MR 1756264. • M. A. Goberna and M. A. López, Linear Semi-Infinite Optimization, Wiley, 1998. • Hettich, R.; Kortanek, K. O. (1993). "Semi-infinite programming: Theory, methods, and applications". SIAM Review. 35 (3): 380–429. doi:10.1137/1035089. JSTOR 2132425. MR 1234637. • David Luenberger (1997). Optimization by Vector Space Methods. John Wiley & Sons. ISBN 0-471-18117-X. • Rembert Reemtsen and Jan-J. Rückmann (Editors), Semi-Infinite Programming (Nonconvex Optimization and Its Applications). Springer, 1998, ISBN 0-7923-5054-5, 1998 External links • Description of semi-infinite programming from INFORMS (Institute for Operations Research and Management Science). • A complete, free, open source Semi Infinite Programming Tutorial is available here from Elsevier as a pdf download from their Journal of Computational and Applied Mathematics, Volume 217, Issue 2, 1 August 2008, Pages 394–419	Wikipedia
Semi-invariant of a quiver In mathematics, given a quiver Q with set of vertices Q0 and set of arrows Q1, a representation of Q assigns a vector space Vi to each vertex and a linear map V(α): V(s(α)) → V(t(α)) to each arrow α, where s(α), t(α) are, respectively, the starting and the ending vertices of α. Given an element d ∈ $\mathbb {N} $Q0, the set of representations of Q with dim Vi = d(i) for each i has a vector space structure. It is naturally endowed with an action of the algebraic group Πi∈Q0 GL(d(i)) by simultaneous base change. Such action induces one on the ring of functions. The ones which are invariants up to a character of the group are called semi-invariants. They form a ring whose structure reflects representation-theoretical properties of the quiver. Definitions Let Q = (Q0,Q1,s,t) be a quiver. Consider a dimension vector d, that is an element in $\mathbb {N} $Q0. The set of d-dimensional representations is given by $\operatorname {Rep} (Q,\mathbf {d} ):=\{V\in \operatorname {Rep} (Q):V_{i}=\mathbf {d} (i)\}$ Once fixed bases for each vector space Vi this can be identified with the vector space $\bigoplus _{\alpha \in Q_{1}}\operatorname {Hom} _{k}(k^{\mathbf {d} (s(\alpha ))},k^{\mathbf {d} (t(\alpha ))})$ Such affine variety is endowed with an action of the algebraic group GL(d) := Πi∈ Q0 GL(d(i)) by simultaneous base change on each vertex: ${\begin{array}{ccc}GL(\mathbf {d} )\times \operatorname {Rep} (Q,\mathbf {d} )&\longrightarrow &\operatorname {Rep} (Q,\mathbf {d} )\\{\Big (}(g_{i}),(V_{i},V(\alpha )){\Big )}&\longmapsto &(V_{i},g_{t(\alpha )}\cdot V(\alpha )\cdot g_{s(\alpha )}^{-1})\end{array}}$ By definition two modules M,N ∈ Rep(Q,d) are isomorphic if and only if their GL(d)-orbits coincide. We have an induced action on the coordinate ring k[Rep(Q,d)] by defining: ${\begin{array}{ccc}GL(\mathbf {d} )\times k[\operatorname {Rep} (Q,\mathbf {d} )]&\longrightarrow &k[\operatorname {Rep} (Q,\mathbf {d} )]\\(g,f)&\longmapsto &g\cdot f(-):=f(g^{-1}.-)\end{array}}$ Polynomial invariants An element f ∈ k[Rep(Q,d)] is called an invariant (with respect to GL(d)) if g⋅f = f for any g ∈ GL(d). The set of invariants $I(Q,\mathbf {d} ):=k[\operatorname {Rep} (Q,\mathbf {d} )]^{GL(\mathbf {d} )}$ is in general a subalgebra of k[Rep(Q,d)]. Example Consider the 1-loop quiver Q: For d = (n) the representation space is End(kn) and the action of GL(n) is given by usual conjugation. The invariant ring is $I(Q,\mathbf {d} )=k[c_{1},\ldots ,c_{n}]$ where the cis are defined, for any A ∈ End(kn), as the coefficients of the characteristic polynomial $\det(A-t\mathbb {I} )=t^{n}-c_{1}(A)t^{n-1}+\cdots +(-1)^{n}c_{n}(A)$ Semi-invariants In case Q has neither loops nor cycles the variety k[Rep(Q,d)] has a unique closed orbit corresponding to the unique d-dimensional semi-simple representation, therefore any invariant function is constant. Elements which are invariants with respect to the subgroup SL(d) := Π{i ∈ Q0} SL(d(i)) form a ring, SI(Q,d), with a richer structure called ring of semi-invariants. It decomposes as $SI(Q,\mathbf {d} )=\bigoplus _{\sigma \in \mathbb {Z} ^{Q_{0}}}SI(Q,\mathbf {d} )_{\sigma }$ where $SI(Q,\mathbf {d} )_{\sigma }:=\{f\in k[\operatorname {Rep} (Q,\mathbf {d} )]:g\cdot f=\prod _{i\in Q_{0}}\det(g_{i})^{\sigma _{i}}f,\forall g\in GL(\mathbf {d} )\}.$ A function belonging to SI(Q,d)σ is called semi-invariant of weight σ. Example Consider the quiver Q: $1{\xrightarrow {\ \ \alpha \ }}2$ Fix d = (n,n). In this case k[Rep(Q,(n,n))] is congruent to the set of square matrices of size n: M(n). The function defined, for any B ∈ M(n), as detu(B(α)) is a semi-invariant of weight (u,−u) in fact $(g_{1},g_{2})\cdot {\det }^{u}(B)={\det }^{u}(g_{2}^{-1}Bg_{1})={\det }^{u}(g_{1}){\det }^{-u}(g_{2}){\det }^{u}(B)$ The ring of semi-invariants equals the polynomial ring generated by det, i.e. ${\mathsf {SI}}(Q,\mathbf {d} )=k[\det ]$ Characterization of representation type through semi-invariant theory For quivers of finite representation-type, that is to say Dynkin quivers, the vector space k[Rep(Q,d)] admits an open dense orbit. In other words, it is a prehomogenous vector space. Sato and Kimura described the ring of semi-invariants in such case. Sato–Kimura theorem Let Q be a Dynkin quiver, d a dimension vector. Let Σ be the set of weights σ such that there exists fσ ∈ SI(Q,d)σ non-zero and irreducible. Then the following properties hold true. i) For every weight σ we have dimk SI(Q,d)σ ≤ 1. ii) All weights in Σ are linearly independent over $\mathbb {Q} $. iii) SI(Q,d) is the polynomial ring generated by the fσ's, σ ∈ Σ. Furthermore, we have an interpretation for the generators of this polynomial algebra. Let O be the open orbit, then k[Rep(Q,d)] \ O = Z1 ∪ ... ∪ Zt where each Zi is closed and irreducible. We can assume that the Zis are arranged in increasing order with respect to the codimension so that the first l have codimension one and Zi is the zero-set of the irreducible polynomial f1, then SI(Q,d) = k[f1, ..., fl]. Example In the example above the action of GL(n,n) has an open orbit on M(n) consisting of invertible matrices. Then we immediately recover SI(Q,(n,n)) = k[det]. Skowronski–Weyman provided a geometric characterization of the class of tame quivers (i.e. Dynkin and Euclidean quivers) in terms of semi-invariants. Skowronski–Weyman theorem Let Q be a finite connected quiver. The following are equivalent: i) Q is either a Dynkin quiver or a Euclidean quiver. ii) For each dimension vector d, the algebra SI(Q,d) is complete intersection. iii) For each dimension vector d, the algebra SI(Q,d) is either a polynomial algebra or a hypersurface. Example Consider the Euclidean quiver Q: Pick the dimension vector d = (1,1,1,1,2). An element V ∈ k[Rep(Q,d)] can be identified with a 4-ple (A1, A2, A3, A4) of matrices in M(1,2). Call Di,j the function defined on each V as det(Ai,Aj). Such functions generate the ring of semi-invariants: $SI(Q,\mathbf {d} )={\frac {k[D_{1,2},D_{3,4},D_{1,4},D_{2,3},D_{1,3},D_{2,4}]}{D_{1,2}D_{3,4}+D_{1,4}D_{2,3}-D_{1,3}D_{2,4}}}$ See also • Wild problem References • Derksen, H.; Weyman, J. (2000), "Semi-invariants of quivers and saturation for Littlewood–Richardson coefficients.", J. Amer. Math. Soc., 3 (13): 467–479, doi:10.1090/S0894-0347-00-00331-3, MR 1758750 • Sato, M.; Kimura, T. (1977), "A classification of irreducible prehomogeneous vector spaces and their relative invariants.", Nagoya Math. J., 65: 1–155, doi:10.1017/S0027763000017633, MR 0430336 • Skowronski, A.; Weyman, J. (2000), "The algebras of semi-invariants of quivers.", Transform. Groups, 5 (4): 361–402, doi:10.1007/bf01234798, MR 1800533, S2CID 120708005	Wikipedia
Semi-locally simply connected In mathematics, specifically algebraic topology, semi-locally simply connected is a certain local connectedness condition that arises in the theory of covering spaces. Roughly speaking, a topological space X is semi-locally simply connected if there is a lower bound on the sizes of the “holes” in X. This condition is necessary for most of the theory of covering spaces, including the existence of a universal cover and the Galois correspondence between covering spaces and subgroups of the fundamental group. Most “nice” spaces such as manifolds and CW complexes are semi-locally simply connected, and topological spaces that do not satisfy this condition are considered somewhat pathological. The standard example of a non-semi-locally simply connected space is the Hawaiian earring. Definition A space X is called semi-locally simply connected if every point in X has a neighborhood U with the property that every loop in U can be contracted to a single point within X (i.e. every loop in U is nullhomotopic in X). The neighborhood U need not be simply connected: though every loop in U must be contractible within X, the contraction is not required to take place inside of U. For this reason, a space can be semi-locally simply connected without being locally simply connected. Equivalent to this definition, a space X is semi-locally simply connected if every point in X has a neighborhood U for which the homomorphism from the fundamental group of U to the fundamental group of X, induced by the inclusion map of U into X, is trivial. Most of the main theorems about covering spaces, including the existence of a universal cover and the Galois correspondence, require a space to be path-connected, locally path-connected, and semi-locally simply connected, a condition known as unloopable (délaçable in French).[1] In particular, this condition is necessary for a space to have a simply connected covering space. Examples A simple example of a space that is not semi-locally simply connected is the Hawaiian earring: the union of the circles in the Euclidean plane with centers (1/n, 0) and radii 1/n, for n a natural number. Give this space the subspace topology. Then all neighborhoods of the origin contain circles that are not nullhomotopic. The Hawaiian earring can also be used to construct a semi-locally simply connected space that is not locally simply connected. In particular, the cone on the Hawaiian earring is contractible and therefore semi-locally simply connected, but it is clearly not locally simply connected. Topology of fundamental group In terms of the natural topology on the fundamental group, a locally path-connected space is semi-locally simply connected if and only if its quasitopological fundamental group is discrete. References 1. Bourbaki 2016, p. 340. • Bourbaki, Nicolas (2016). Topologie algébrique: Chapitres 1 à 4. Springer. Ch. IV pp. 339 -480. ISBN 978-3662493601. • J.S. Calcut, J.D. McCarthy Discreteness and homogeneity of the topological fundamental group Topology Proceedings, Vol. 34,(2009), pp. 339–349 • Hatcher, Allen (2002). Algebraic Topology. Cambridge University Press. ISBN 0-521-79540-0.	Wikipedia
Semi-orthogonal matrix In linear algebra, a semi-orthogonal matrix is a non-square matrix with real entries where: if the number of columns exceeds the number of rows, then the rows are orthonormal vectors; but if the number of rows exceeds the number of columns, then the columns are orthonormal vectors. Equivalently, a non-square matrix A is semi-orthogonal if either $A^{\operatorname {T} }A=I{\text{ or }}AA^{\operatorname {T} }=I.\,$[1][2][3] In the following, consider the case where A is an m × n matrix for m > n. Then $A^{\operatorname {T} }A=I_{n},{\text{ and}}$ $AA^{\operatorname {T} }={\text{the matrix of the orthogonal projection onto the column space of }}A.$ The fact that $ A^{\operatorname {T} }A=I_{n}$ implies the isometry property $\\|Ax\\|_{2}=\\|x\\|_{2}\,$ for all x in Rn. For example, ${\begin{bmatrix}1\\0\end{bmatrix}}$ is a semi-orthogonal matrix. A semi-orthogonal matrix A is semi-unitary (either A†A = I or AA† = I) and either left-invertible or right-invertible (left-invertible if it has more rows than columns, otherwise right invertible). As a linear transformation applied from the left, a semi-orthogonal matrix with more rows than columns preserves the dot product of vectors, and therefore acts as an isometry of Euclidean space, such as a rotation or reflection. References 1. Abadir, K.M., Magnus, J.R. (2005). Matrix Algebra. Cambridge University Press. 2. Zhang, Xian-Da. (2017). Matrix analysis and applications. Cambridge University Press. 3. Povey, Daniel, et al. (2018). "Semi-Orthogonal Low-Rank Matrix Factorization for Deep Neural Networks." Interspeech.	Wikipedia
Semi-reflexive space In the area of mathematics known as functional analysis, a semi-reflexive space is a locally convex topological vector space (TVS) X such that the canonical evaluation map from X into its bidual (which is the strong dual of the strong dual of X) is bijective. If this map is also an isomorphism of TVSs then it is called reflexive. Semi-reflexive spaces play an important role in the general theory of locally convex TVSs. Since a normable TVS is semi-reflexive if and only if it is reflexive, the concept of semi-reflexivity is primarily used with TVSs that are not normable. Definition and notation Brief definition Suppose that X is a topological vector space (TVS) over the field $\mathbb {F} $ (which is either the real or complex numbers) whose continuous dual space, $X^{\prime }$, separates points on X (i.e. for any $x\in X$ there exists some $x^{\prime }\in X^{\prime }$ such that $x^{\prime }(x)\neq 0$). Let $X_{b}^{\prime }$ and $X_{\beta }^{\prime }$ both denote the strong dual of X, which is the vector space $X^{\prime }$ of continuous linear functionals on X endowed with the topology of uniform convergence on bounded subsets of X; this topology is also called the strong dual topology and it is the "default" topology placed on a continuous dual space (unless another topology is specified). If X is a normed space, then the strong dual of X is the continuous dual space $X^{\prime }$ with its usual norm topology. The bidual of X, denoted by $X^{\prime \prime }$, is the strong dual of $X_{b}^{\prime }$; that is, it is the space $\left(X_{b}^{\prime }\right)_{b}^{\prime }$.[1] For any $x\in X,$ let $J_{x}:X^{\prime }\to \mathbb {F} $ be defined by $J_{x}\left(x^{\prime }\right)=x^{\prime }(x)$, where $J_{x}$ is called the evaluation map at x; since $J_{x}:X_{b}^{\prime }\to \mathbb {F} $ is necessarily continuous, it follows that $J_{x}\in \left(X_{b}^{\prime }\right)^{\prime }$. Since $X^{\prime }$ separates points on X, the map $J:X\to \left(X_{b}^{\prime }\right)^{\prime }$ defined by $J(x):=J_{x}$ is injective where this map is called the evaluation map or the canonical map. This map was introduced by Hans Hahn in 1927.[2] We call X semireflexive if $J:X\to \left(X_{b}^{\prime }\right)^{\prime }$ is bijective (or equivalently, surjective) and we call X reflexive if in addition $J:X\to X^{\prime \prime }=\left(X_{b}^{\prime }\right)_{b}^{\prime }$ is an isomorphism of TVSs.[1] If X is a normed space then J is a TVS-embedding as well as an isometry onto its range; furthermore, by Goldstine's theorem (proved in 1938), the range of J is a dense subset of the bidual $\left(X^{\prime \prime },\sigma \left(X^{\prime \prime },X^{\prime }\right)\right)$.[2] A normable space is reflexive if and only if it is semi-reflexive. A Banach space is reflexive if and only if its closed unit ball is $\sigma \left(X^{\prime },X\right)$-compact.[2] Detailed definition Let X be a topological vector space over a number field $\mathbb {F} $ (of real numbers $\mathbb {R} $ or complex numbers $\mathbb {C} $). Consider its strong dual space $X_{b}^{\prime }$, which consists of all continuous linear functionals $f:X\to \mathbb {F} $ and is equipped with the strong topology $b\left(X^{\prime },X\right)$, that is, the topology of uniform convergence on bounded subsets in X. The space $X_{b}^{\prime }$ is a topological vector space (to be more precise, a locally convex space), so one can consider its strong dual space $\left(X_{b}^{\prime }\right)_{b}^{\prime }$, which is called the strong bidual space for X. It consists of all continuous linear functionals $h:X_{b}^{\prime }\to {\mathbb {F} }$ and is equipped with the strong topology $b\left(\left(X_{b}^{\prime }\right)^{\prime },X_{b}^{\prime }\right)$. Each vector $x\in X$ generates a map $J(x):X_{b}^{\prime }\to \mathbb {F} $ by the following formula: $J(x)(f)=f(x),\qquad f\in X'.$ This is a continuous linear functional on $X_{b}^{\prime }$, that is, $J(x)\in \left(X_{b}^{\prime }\right)_{b}^{\prime }$. One obtains a map called the evaluation map or the canonical injection: $J:X\to \left(X_{b}^{\prime }\right)_{b}^{\prime }.$ which is a linear map. If X is locally convex, from the Hahn–Banach theorem it follows that J is injective and open (that is, for each neighbourhood of zero $U$ in X there is a neighbourhood of zero V in $\left(X_{b}^{\prime }\right)_{b}^{\prime }$ such that $J(U)\supseteq V\cap J(X)$). But it can be non-surjective and/or discontinuous. A locally convex space $X$ is called semi-reflexive if the evaluation map $J:X\to \left(X_{b}^{\prime }\right)_{b}^{\prime }$ is surjective (hence bijective); it is called reflexive if the evaluation map $J:X\to \left(X_{b}^{\prime }\right)_{b}^{\prime }$ is surjective and continuous, in which case J will be an isomorphism of TVSs). Characterizations of semi-reflexive spaces If X is a Hausdorff locally convex space then the following are equivalent: 1. X is semireflexive; 2. the weak topology on X had the Heine-Borel property (that is, for the weak topology $\sigma \left(X,X^{\prime }\right)$, every closed and bounded subset of $X_{\sigma }$ is weakly compact).[1] 3. If linear form on $X^{\prime }$ that continuous when $X^{\prime }$ has the strong dual topology, then it is continuous when $X^{\prime }$ has the weak topology;[3] 4. $X_{\tau }^{\prime }$ is barrelled, where the $\tau $ indicates the Mackey topology on $X^{\prime }$;[3] 5. X weak the weak topology $\sigma \left(X,X^{\prime }\right)$ is quasi-complete.[3] Theorem[4] — A locally convex Hausdorff space $X$ is semi-reflexive if and only if $X$ with the $\sigma \left(X,X^{\prime }\right)$-topology has the Heine–Borel property (i.e. weakly closed and bounded subsets of $X$ are weakly compact). Sufficient conditions Every semi-Montel space is semi-reflexive and every Montel space is reflexive. Properties If $X$ is a Hausdorff locally convex space then the canonical injection from $X$ into its bidual is a topological embedding if and only if $X$ is infrabarrelled.[5] The strong dual of a semireflexive space is barrelled. Every semi-reflexive space is quasi-complete.[3] Every semi-reflexive normed space is a reflexive Banach space.[6] The strong dual of a semireflexive space is barrelled.[7] Reflexive spaces Main article: Reflexive space If X is a Hausdorff locally convex space then the following are equivalent: 1. X is reflexive; 2. X is semireflexive and barrelled; 3. X is barrelled and the weak topology on X had the Heine-Borel property (which means that for the weak topology $\sigma \left(X,X^{\prime }\right)$, every closed and bounded subset of $X_{\sigma }$ is weakly compact).[1] 4. X is semireflexive and quasibarrelled.[8] If X is a normed space then the following are equivalent: 1. X is reflexive; 2. the closed unit ball is compact when X has the weak topology $\sigma \left(X,X^{\prime }\right)$.[9] 3. X is a Banach space and $X_{b}^{\prime }$ is reflexive.[10] Examples Every non-reflexive infinite-dimensional Banach space is a distinguished space that is not semi-reflexive.[11] If $X$ is a dense proper vector subspace of a reflexive Banach space then $X$ is a normed space that not semi-reflexive but its strong dual space is a reflexive Banach space.[11] There exists a semi-reflexive countably barrelled space that is not barrelled.[11] See also • Grothendieck space - A generalization which has some of the properties of reflexive spaces and includes many spaces of practical importance. • Reflexive operator algebra • Reflexive space Citations 1. Trèves 2006, pp. 372–374. 2. Narici & Beckenstein 2011, pp. 225–273. 3. Schaefer & Wolff 1999, p. 144. 4. Edwards 1965, 8.4.2. 5. Narici & Beckenstein 2011, pp. 488–491. 6. Schaefer & Wolff 1999, p. 145. 7. Edwards 1965, 8.4.3. 8. Khaleelulla 1982, pp. 32–63. 9. Trèves 2006, p. 376. 10. Trèves 2006, p. 377. 11. Khaleelulla 1982, pp. 28–63. Bibliography • Edwards, Robert E. (1995). Functional Analysis: Theory and Applications. New York: Dover Publications. ISBN 978-0-486-68143-6. OCLC 30593138. • Edwards, R. E. (1965). Functional analysis. Theory and applications. New York: Holt, Rinehart and Winston. ISBN 0030505356. • John B. Conway, A Course in Functional Analysis, Springer, 1985. • James, Robert C. (1972), Some self-dual properties of normed linear spaces. Symposium on Infinite-Dimensional Topology (Louisiana State Univ., Baton Rouge, La., 1967), Ann. of Math. Studies, vol. 69, Princeton, NJ: Princeton Univ. Press, pp. 159–175. • Khaleelulla, S. M. (1982). Counterexamples in Topological Vector Spaces. Lecture Notes in Mathematics. Vol. 936. Berlin, Heidelberg, New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370. • Kolmogorov, A. N.; Fomin, S. V. (1957). Elements of the Theory of Functions and Functional Analysis, Volume 1: Metric and Normed Spaces. Rochester: Graylock Press. • Megginson, Robert E. (1998), An introduction to Banach space theory, Graduate Texts in Mathematics, vol. 183, New York: Springer-Verlag, pp. xx+596, ISBN 0-387-98431-3. • Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. • Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. • Schechter, Eric (1996). Handbook of Analysis and Its Foundations. San Diego, CA: Academic Press. ISBN 978-0-12-622760-4. OCLC 175294365. • Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. • Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114. Functional analysis (topics – glossary) Spaces • Banach • Besov • Fréchet • Hilbert • Hölder • Nuclear • Orlicz • Schwartz • Sobolev • Topological vector Properties • Barrelled • Complete • Dual (Algebraic/Topological) • Locally convex • Reflexive • Reparable Theorems • Hahn–Banach • Riesz representation • Closed graph • Uniform boundedness principle • Kakutani fixed-point • Krein–Milman • Min–max • Gelfand–Naimark • Banach–Alaoglu Operators • Adjoint • Bounded • Compact • Hilbert–Schmidt • Normal • Nuclear • Trace class • Transpose • Unbounded • Unitary Algebras • Banach algebra • C-algebra • Spectrum of a C-algebra • Operator algebra • Group algebra of a locally compact group • Von Neumann algebra Open problems • Invariant subspace problem • Mahler's conjecture Applications • Hardy space • Spectral theory of ordinary differential equations • Heat kernel • Index theorem • Calculus of variations • Functional calculus • Integral operator • Jones polynomial • Topological quantum field theory • Noncommutative geometry • Riemann hypothesis • Distribution (or Generalized functions) Advanced topics • Approximation property • Balanced set • Choquet theory • Weak topology • Banach–Mazur distance • Tomita–Takesaki theory • Mathematics portal • Category • Commons Topological vector spaces (TVSs) Basic concepts • Banach space • Completeness • Continuous linear operator • Linear functional • Fréchet space • Linear map • Locally convex space • Metrizability • Operator topologies • Topological vector space • Vector space Main results • Anderson–Kadec • Banach–Alaoglu • Closed graph theorem • F. Riesz's • Hahn–Banach (hyperplane separation • Vector-valued Hahn–Banach) • Open mapping (Banach–Schauder) • Bounded inverse • Uniform boundedness (Banach–Steinhaus) Maps • Bilinear operator • form • Linear map • Almost open • Bounded • Continuous • Closed • Compact • Densely defined • Discontinuous • Topological homomorphism • Functional • Linear • Bilinear • Sesquilinear • Norm • Seminorm • Sublinear function • Transpose Types of sets • Absolutely convex/disk • Absorbing/Radial • Affine • Balanced/Circled • Banach disks • Bounding points • Bounded • Complemented subspace • Convex • Convex cone (subset) • Linear cone (subset) • Extreme point • Pre-compact/Totally bounded • Prevalent/Shy • Radial • Radially convex/Star-shaped • Symmetric Set operations • Affine hull • (Relative) Algebraic interior (core) • Convex hull • Linear span • Minkowski addition • Polar • (Quasi) Relative interior Types of TVSs • Asplund • B-complete/Ptak • Banach • (Countably) Barrelled • BK-space • (Ultra-) Bornological • Brauner • Complete • Convenient • (DF)-space • Distinguished • F-space • FK-AK space • FK-space • Fréchet • tame Fréchet • Grothendieck • Hilbert • Infrabarreled • Interpolation space • K-space • LB-space • LF-space • Locally convex space • Mackey • (Pseudo)Metrizable • Montel • Quasibarrelled • Quasi-complete • Quasinormed • (Polynomially • Semi-) Reflexive • Riesz • Schwartz • Semi-complete • Smith • Stereotype • (B • Strictly • Uniformly) convex • (Quasi-) Ultrabarrelled • Uniformly smooth • Webbed • With the approximation property • Mathematics portal • Category • Commons	Wikipedia
Semiregular polyhedron In geometry, the term semiregular polyhedron (or semiregular polytope) is used variously by different authors. Semiregular polyhedra: Archimedean solids, prisms, and antiprisms Definitions In its original definition, it is a polyhedron with regular polygonal faces, and a symmetry group which is transitive on its vertices; today, this is more commonly referred to as a uniform polyhedron (this follows from Thorold Gosset's 1900 definition of the more general semiregular polytope).[1][2] These polyhedra include: • The thirteen Archimedean solids. • The elongated square gyrobicupola, also called a pseudo-rhombicuboctahedron, a Johnson solid, has identical vertex figures 3.4.4.4, but is not vertex-transitive including a twist has been argued for inclusion as a 14th Archimedean solid by Branko Grünbaum. • An infinite series of convex prisms. • An infinite series of convex antiprisms (their semiregular nature was first observed by Kepler). These semiregular solids can be fully specified by a vertex configuration: a listing of the faces by number of sides, in order as they occur around a vertex. For example: 3.5.3.5 represents the icosidodecahedron, which alternates two triangles and two pentagons around each vertex. In contrast: 3.3.3.5 is a pentagonal antiprism. These polyhedra are sometimes described as vertex-transitive. Since Gosset, other authors have used the term semiregular in different ways in relation to higher dimensional polytopes. E. L. Elte[3] provided a definition which Coxeter found too artificial. Coxeter himself dubbed Gosset's figures uniform, with only a quite restricted subset classified as semiregular.[4] Yet others have taken the opposite path, categorising more polyhedra as semiregular. These include: • Three sets of star polyhedra which meet Gosset's definition, analogous to the three convex sets listed above. • The duals of the above semiregular solids, arguing that since the dual polyhedra share the same symmetries as the originals, they too should be regarded as semiregular. These duals include the Catalan solids, the convex dipyramids, and the convex antidipyramids or trapezohedra, and their nonconvex analogues. A further source of confusion lies in the way that the Archimedean solids are defined, again with different interpretations appearing. Gosset's definition of semiregular includes figures of higher symmetry: the regular and quasiregular polyhedra. Some later authors prefer to say that these are not semiregular, because they are more regular than that - the uniform polyhedra are then said to include the regular, quasiregular, and semiregular ones. This naming system works well, and reconciles many (but by no means all) of the confusions. In practice even the most eminent authorities can get themselves confused, defining a given set of polyhedra as semiregular and/or Archimedean, and then assuming (or even stating) a different set in subsequent discussions. Assuming that one's stated definition applies only to convex polyhedra is probably the most common failing. Coxeter, Cromwell,[5] and Cundy & Rollett[6] are all guilty of such slips. General remarks Rhombic semiregular polyhedra (Kepler) Trigonal trapezohedron (V(3.3)2) Rhombic dodecahedron V(3.4)2 Rhombic triacontahedron V(3.5)2 Johannes Kepler coined the category semiregular in his book Harmonices Mundi (1619), including the 13 Archimedean solids, two infinite families (prisms and antiprisms on regular bases), and two edge-transitive Catalan solids, the rhombic dodecahedron and rhombic triacontahedron. He also considered a rhombus as a semiregular polygon (being equilateral and alternating two angles) as well as star polygons, now called isotoxal figures which he used in planar tilings. The trigonal trapezohedron, a topological cube with congruent rhombic faces, would also qualify as semiregular, though Kepler did not mention it specifically. In many works semiregular polyhedron is used as a synonym for Archimedean solid.[7] For example, Cundy & Rollett (1961). We can distinguish between the facially-regular and vertex-transitive figures based on Gosset, and their vertically-regular (or versi-regular) and facially-transitive duals. Coxeter et al. (1954) use the term semiregular polyhedra to classify uniform polyhedra with Wythoff symbol of the form p q \| r, a definition encompassing only six of the Archimedean solids, as well as the regular prisms (but not the regular antiprisms) and numerous nonconvex solids. Later, Coxeter (1973) would quote Gosset's definition without comment, thus accepting it by implication. Eric Weisstein, Robert Williams and others use the term to mean the convex uniform polyhedra excluding the five regular polyhedra – including the Archimedean solids, the uniform prisms, and the uniform antiprisms (overlapping with the cube as a prism and regular octahedron as an antiprism).[8][9] Peter Cromwell (1997) writes in a footnote to Page 149 that, "in current terminology, 'semiregular polyhedra' refers to the Archimedean and Catalan (Archimedean dual) solids". On Page 80 he describes the thirteen Archimedeans as semiregular, while on Pages 367 ff. he discusses the Catalans and their relationship to the 'semiregular' Archimedeans. By implication this treats the Catalans as not semiregular, thus effectively contradicting (or at least confusing) the definition he provided in the earlier footnote. He ignores nonconvex polyhedra. See also • Semiregular polytope • Regular polyhedron References 1. Thorold Gosset On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900 2. Coxeter, H.S.M. Regular polytopes, 3rd Edn, Dover (1973) 3. Elte, E. L. (1912), The Semiregular Polytopes of the Hyperspaces, Groningen: University of Groningen 4. Coxeter, H.S.M., Longuet-Higgins, M.S. and Miller, J.C.P. Uniform Polyhedra, Philosophical Transactions of the Royal Society of London 246 A (1954), pp. 401-450. (JSTOR archive, subscription required). 5. Cromwell, P. Polyhedra, Cambridge University Press (1977) 6. Cundy H.M and Rollett, A.P. Mathematical models, 2nd Edn. Oxford University Press (1961) 7. "Archimedes". (2006). In Encyclopædia Britannica. Retrieved 19 Dec 2006, from Encyclopædia Britannica Online (subscription required). 8. Weisstein, Eric W. "Semiregular polyhedron". MathWorld. The definition here does not exclude the case of all faces being congruent, but the Platonic solids are not included in the article's enumeration. 9. Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X. (Chapter 3: Polyhedra) External links • Weisstein, Eric W. "Semiregular polyhedron". MathWorld. • George Hart: Archimedean Semi-regular Polyhedra • David Darling: semi-regular polyhedron • polyhedra.mathmos.net: Semi-Regular Polyhedron • Encyclopaedia of Mathematics: Semi-regular polyhedra, uniform polyhedra, Archimedean solids	Wikipedia
Semi-s-cobordism In mathematics, a cobordism (W, M, M−) of an (n + 1)-dimensional manifold (with boundary) W between its boundary components, two n-manifolds M and M−, is called a semi-s-cobordism if (and only if) the inclusion $M\hookrightarrow W$ is a simple homotopy equivalence (as in an s-cobordism), with no further requirement on the inclusion $M^{-}\hookrightarrow W$ (not even being a homotopy equivalence). Other notations The original creator of this topic, Jean-Claude Hausmann, used the notation M− for the right-hand boundary of the cobordism. Properties A consequence of (W, M, M−) being a semi-s-cobordism is that the kernel of the derived homomorphism on fundamental groups $K=\ker(\pi _{1}(M^{-})\twoheadrightarrow \pi _{1}(W))$ is perfect. A corollary of this is that $\pi _{1}(M^{-})$ solves the group extension problem $1\rightarrow K\rightarrow \pi _{1}(M^{-})\rightarrow \pi _{1}(M)\rightarrow 1$. The solutions to the group extension problem for prescribed quotient group $\pi _{1}(M)$ and kernel group K are classified up to congruence by group cohomology (see Mac Lane's Homology pp. 124-129), so there are restrictions on which n-manifolds can be the right-hand boundary of a semi-s-cobordism with prescribed left-hand boundary M and superperfect kernel group K. Relationship with Plus cobordisms Note that if (W, M, M−) is a semi-s-cobordism, then (W, M−, M) is a plus cobordism. (This justifies the use of M− for the right-hand boundary of a semi-s-cobordism, a play on the traditional use of M+ for the right-hand boundary of a plus cobordism.) Thus, a semi-s-cobordism may be thought of as an inverse to Quillen's Plus construction in the manifold category. Note that (M−)+ must be diffeomorphic (respectively, piecewise-linearly (PL) homeomorphic) to M but there may be a variety of choices for (M+)− for a given closed smooth (respectively, PL) manifold M. References • MacLane (1963), Homology, pp. 124–129, ISBN 0-387-58662-8 • Hausmann, Jean-Claude (1976), "Homological Surgery", Annals of Mathematics, Second Series, 104 (3): 573–584, doi:10.2307/1970967, JSTOR 1970967. • Hausmann, Jean-Claude; Vogel, Pierre (1978), "The Plus Construction and Lifting Maps from Manifolds", Proceedings of Symposia in Pure Mathematics, 32: 67–76. • Hausmann, Jean-Claude (1978), "Manifolds with a Given Homology and Fundamental Group", Commentarii Mathematici Helvetici, 53 (1): 113–134, doi:10.1007/BF02566068.	Wikipedia
Delta set In mathematics, a Δ-set S, often called a Δ-complex or a semi-simplicial set, is a combinatorial object that is useful in the construction and triangulation of topological spaces, and also in the computation of related algebraic invariants of such spaces. A Δ-set is somewhat more general than a simplicial complex, yet not quite as general as a simplicial set. As an example, suppose we want to triangulate the 1-dimensional circle $S^{1}$. To do so with a simplicial complex, we need at least three vertices, and edges connecting them. But delta-sets allow for a simpler triangulation: thinking of $S^{1}$ as the interval [0,1] with the two endpoints identified, we can define a triangulation with a single vertex 0, and a single edge looping between 0 and 0. Definition and related data Formally, a Δ-set is a sequence of sets $\{S_{n}\}_{n=0}^{\infty }$ together with maps $d_{i}\colon S_{n+1}\rightarrow S_{n}$ with $i=0,1,\ldots ,n+1$ for $n\geq 1$ that satisfy $d_{i}\circ d_{j}=d_{j-1}\circ d_{i}$ whenever $i<j$. This definition generalizes the notion of a simplicial complex, where the $S_{n}$ are the sets of n-simplices, and the $d_{i}$ are the face maps. It is not as general as a simplicial set, since it lacks "degeneracies." Given Δ-sets S and T, a map of Δ-sets is a collection of set-maps $\{f_{n}\colon S_{n}\rightarrow T_{n}\}_{n=0}^{\infty }$ such that $f_{n}\circ d_{i}=d_{i}\circ f_{n+1}$ whenever both sides of the equation are defined. With this notion, we can define the category of Δ-sets, whose objects are Δ-sets and whose morphisms are maps of Δ-sets. Each Δ-set has a corresponding geometric realization, defined as $\|S\|=\left(\coprod _{n=0}^{\infty }S_{n}\times \Delta ^{n}\right)/_{\sim }$ where we declare that $(\sigma ,d^{i}t)\sim (d_{i}\sigma ,t)\quad {\text{ for all }}\sigma \in S_{n},t\in \Delta ^{n-1}.$ Here, $\Delta ^{n}$ denotes the standard n-simplex, and $d^{i}\colon \Delta ^{n-1}\rightarrow \Delta ^{n}$ is the inclusion of the i-th face. The geometric realization is a topological space with the quotient topology. The geometric realization of a Δ-set S has a natural filtration $\|S\|_{0}\subset \|S\|_{1}\subset \cdots \subset \|S\|,$ where $\|S\|_{N}=\left(\coprod _{n=0}^{N}S_{n}\times \Delta ^{n}\right)/_{\sim }$ is a "restricted" geometric realization. Related functors The geometric realization of a Δ-set described above defines a covariant functor from the category of Δ-sets to the category of topological spaces. Geometric realization takes a Δ-set to a topological space, and carries maps of Δ-sets to induced continuous maps between geometric realizations. If S is a Δ-set, there is an associated free abelian chain complex, denoted $(\mathbb {Z} S,\partial )$, whose n-th group is the free abelian group $(\mathbb {Z} S)_{n}=\mathbb {Z} \langle S_{n}\rangle ,$ generated by the set $S_{n}$, and whose n-th differential is defined by $\partial _{n}=d_{0}-d_{1}+d_{2}-\cdots +(-1)^{n}d_{n}.$ This defines a covariant functor from the category of Δ-sets to the category of chain complexes of abelian groups. A Δ-set is carried to the chain complex just described, and a map of Δ-sets is carried to a map of chain complexes, which is defined by extending the map of Δ-sets in the standard way using the universal property of free abelian groups. Given any topological space X, one can construct a Δ-set $\mathrm {sing} (X)$ as follows. A singular n-simplex in X is a continuous map $\sigma \colon \Delta ^{n}\rightarrow X.$ Define $\mathrm {sing} _{n}^{}(X)$ to be the collection of all singular n-simplicies in X, and define $d_{i}\colon \mathrm {sing} _{i+1}(X)\rightarrow \mathrm {sing} _{i}(X)$ by $d_{i}(\sigma )=\sigma \circ d^{i},$ where again $d^{i}$ is the $i$-th face map. One can check that this is in fact a Δ-set. This defines a covariant functor from the category of topological spaces to the category of Δ-sets. A topological space is carried to the Δ-set just described, and a continuous map of spaces is carried to a map of Δ-sets, which is given by composing the map with the singular n-simplices. Examples This example illustrates the constructions described above. We can create a Δ-set S whose geometric realization is the unit circle $S^{1}$, and use it to compute the homology of this space. Thinking of $S^{1}$ as an interval with the endpoints identified, define $S_{0}=\{v\},\quad S_{1}=\{e\},$ with $S_{n}=\varnothing $ for all $n\geq 2$. The only possible maps $d_{0},d_{1}\colon S_{1}\rightarrow S_{0},$ are $d_{0}(e)=d_{1}(e)=v.\quad $ It is simple to check that this is a Δ-set, and that $\|S\|\cong S^{1}$. Now, the associated chain complex $(\mathbb {Z} S,\partial )$ is $0\longrightarrow \mathbb {Z} \langle e\rangle {\stackrel {\partial _{1}}{\longrightarrow }}\mathbb {Z} \langle v\rangle \longrightarrow 0,$ where $\partial _{1}(e)=d_{0}(e)-d_{1}(e)=v-v=0.$ In fact, $\partial _{n}=0$ for all n. The homology of this chain complex is also simple to compute: $H_{0}(\mathbb {Z} S)={\frac {\ker \partial _{0}}{\mathrm {im} \partial _{1}}}=\mathbb {Z} \langle v\rangle \cong \mathbb {Z} ,$ $H_{1}(\mathbb {Z} S)={\frac {\ker \partial _{1}}{\mathrm {im} \partial _{2}}}=\mathbb {Z} \langle e\rangle \cong \mathbb {Z} .$ All other homology groups are clearly trivial. The following example is from section 2.1 of Hatcher's Algebraic Topology.[1] Consider the Δ-set structure given to the torus in the figure, which has one vertex, three edges, and two 2-simplices. The boundary map $\partial _{1}$ is 0 because there is only one vertex, so $H_{0}(T^{2})={\text{ker }}\partial _{0}/{\text{ im }}\partial _{1}=\mathbb {Z} $. Let $\{e_{0}^{1},e_{1}^{1},e_{0}^{1}+e_{1}^{1}-e_{2}^{1}\}$ be a basis for $\Delta _{1}(T^{2})$. Then $\partial _{2}(e_{0}^{2})=e_{0}^{1}+e_{1}^{1}-e_{2}^{1}=\partial _{2}(e_{1}^{2})$, so ${\text{im }}\partial _{2}=\langle e_{0}^{1}+e_{1}^{1}-e_{2}^{1}\rangle $, and hence $H_{1}(T^{2})={\text{ker }}\partial _{1}/{\text{ im }}\partial _{2}=\mathbb {Z} ^{3}/\mathbb {Z} =\mathbb {Z} ^{2}.$ Since there are no 3-simplices, $H_{2}(T^{2})={\text{ker }}\partial _{2}$. We have that $\partial _{2}(pe_{0}^{2}+qe_{1}^{2})=(p+q)(e_{0}^{1}+e_{1}^{1}-e_{2}^{1})$ which is 0 if and only if $p=-q$. Hence ${\text{ker }}\partial _{2}$ is infinite cyclic generated by $e_{0}^{2}-e_{1}^{2}$. So $H_{2}(T^{2})=\mathbb {Z} $. Clearly $H_{n}(T^{2})=0$ for $n\geq 3.$ Thus, $H_{n}(T^{2})={\begin{cases}\mathbb {Z} &n=0,2\\\mathbb {Z} ^{2}&n=1\\0&n\geq 3.\end{cases}}$ It is worth highlighting that the minimum number of simplices needed to endow $T^{2}$ with the structure of a simplicial complex is 7 vertices, 21 edges, and 14 2-simplices, for a total of 42 simplices. This would make the above calculations, which only used 6 simplices, much harder for someone to do by hand. This is a non-example. Consider a line segment. This is a 1-dimensional Δ-set and a 1-dimensional simplicial set. However, if we view the line segment as a 2-dimensional simplicial set, in which the 2-simplex is viewed as degenerate, then the line segment is not a Δ-set, as we do not allow for such degeneracies. Abstract nonsense We now inspect the relation between Δ-sets and simplicial sets. Consider the simplex category $\Delta $, whose objects are the finite totally ordered sets $[n]:=\{0,1,\cdots ,n\}$ and whose morphisms are monotone maps. A simplicial set is defined to be a presheaf on $\Delta $, i.e. a (contravariant) functor $S:\Delta ^{\text{op}}\to {\text{Set}}$. On the other hand, consider the subcategory ${\hat {\Delta }}$ of $\Delta $ whose morphisms are only the strict monotone maps. Note that the morphisms in ${\hat {\Delta }}$ are precisely the injections in $\Delta $, and one can prove that these are generated by the monotone maps of the form $\delta ^{i}:[n]\to [n+1]$ which "skip" the element $i\in [n+1]$. From this we see that a presheaf $S:{\hat {\Delta }}^{\text{op}}\to {\text{Set}}$ on ${\hat {\Delta }}$ is determined by a sequence of sets $\{S_{n}\}_{n=0}^{\infty }$ (where we denote $S([n])$ by $S_{n}$ for simplicity) together with maps $d_{i}:S_{n+1}\to S_{n}$ for $i=0,1,\ldots ,n+1$ (where we denote $S(\delta ^{i})$ by $d_{i}$ for simplicity as well). In fact, after checking that $\delta ^{j}\circ \delta ^{i}=\delta ^{i}\circ \delta ^{j-1}$ in ${\hat {\Delta }}$, one concludes that $d_{i}\circ d_{j}=d_{j-1}\circ d_{i}$ whenever $i<j$. Thus, a presheaf on ${\hat {\Delta }}$ determines the data of a Δ-set and, conversely, all Δ-sets arise in this way.[2] Moreover, Δ-maps $f:S\to T$ between Δ-sets correspond to natural transformations when we view $S$ and $T$ as (contravariant) functors. In this sense, Δ-sets are presheaves on ${\hat {\Delta }}$ while simplicial sets are presheaves on $\Delta $. From this perspective, it is now easy to see that every simplicial set is a Δ-set. Indeed, notice there is an inclusion ${\hat {\Delta }}\hookrightarrow \Delta $; so that every simplicial set $S:\Delta ^{\text{op}}\to {\text{Set}}$ naturally gives rise to a Δ-set, namely the composite $ {\hat {\Delta }}^{\text{op}}\hookrightarrow \Delta ^{\text{op}}\xrightarrow {S} {\text{Set}}$. Pros and cons One advantage of using Δ-sets in this way is that the resulting chain complex is generally much simpler than the singular chain complex. For reasonably simple spaces, all of the groups will be finitely generated, whereas the singular chain groups are, in general, not even countably generated. One drawback of this method is that one must prove that the geometric realization of the Δ-set is actually homeomorphic to the topological space in question. This can become a computational challenge as the Δ-set increases in complexity. See also • Simplicial complexes • Simplicial sets • Singular homology References 1. Hatcher, Allen (2002). Algebraic topology. Cambridge: Cambridge University Press. ISBN 0-521-79160-X. OCLC 45420394. 2. Friedman, Greg (2012). "Survey article: An elementary illustrated introduction to simplicial sets". The Rocky Mountain Journal of Mathematics. 42 (2): 353–423. arXiv:0809.4221. doi:10.1216/RMJ-2012-42-2-353. MR 2915498. • Ranicki, Andrew A. (1993). Algebraic L-theory and Topological Manifolds (PDF). Cambridge Tracts in Mathematics. Vol. 102. Cambridge Univ. Press. ISBN 978-0-521-42024-2. • Ranicki, Andrew; Weiss, Michael (2012). "On The Algebraic L-theory of Δ-sets". Pure and Applied Mathematics Quarterly. 8 (2): 423–450. arXiv:math.AT/0701833. doi:10.4310/pamq.2012.v8.n2.a3. MR 2900173. • Rourke, Colin P.; Sanderson, Brian J. (1971). "Δ-Sets I: Homotopy Theory". Quarterly Journal of Mathematics. 22 (3): 321–338. Bibcode:1971QJMat..22..321R. doi:10.1093/qmath/22.3.321.	Wikipedia
Stable vector bundle In mathematics, a stable vector bundle is a (holomorphic or algebraic) vector bundle that is stable in the sense of geometric invariant theory. Any holomorphic vector bundle may be built from stable ones using Harder–Narasimhan filtration. Stable bundles were defined by David Mumford in Mumford (1963) and later built upon by David Gieseker, Fedor Bogomolov, Thomas Bridgeland and many others. Motivation One of the motivations for analyzing stable vector bundles is their nice behavior in families. In fact, Moduli spaces of stable vector bundles can be constructed using the Quot scheme in many cases, whereas the stack of vector bundles $\mathbf {B} GL_{n}$ is an Artin stack whose underlying set is a single point. Here's an example of a family of vector bundles which degenerate poorly. If we tensor the Euler sequence of $\mathbb {P} ^{1}$ by ${\mathcal {O}}(1)$ there is an exact sequence $0\to {\mathcal {O}}(-1)\to {\mathcal {O}}\oplus {\mathcal {O}}\to {\mathcal {O}}(1)\to 0$[1] which represents a non-zero element $v\in {\text{Ext}}^{1}({\mathcal {O}}(1),{\mathcal {O}}(-1))\cong k$[2] since the trivial exact sequence representing the $0$ vector is $0\to {\mathcal {O}}(-1)\to {\mathcal {O}}(-1)\oplus {\mathcal {O}}(1)\to {\mathcal {O}}(1)\to 0$ If we consider the family of vector bundles $E_{t}$ in the extension from $t\cdot v$ for $t\in \mathbb {A} ^{1}$, there are short exact sequences $0\to {\mathcal {O}}(-1)\to E_{t}\to {\mathcal {O}}(1)\to 0$ which have Chern classes $c_{1}=0,c_{2}=0$ generically, but have $c_{1}=0,c_{2}=-1$ at the origin. This kind of jumping of numerical invariants does not happen in moduli spaces of stable vector bundles.[3] Stable vector bundles over curves A slope of a holomorphic vector bundle W over a nonsingular algebraic curve (or over a Riemann surface) is a rational number μ(W) = deg(W)/rank(W). A bundle W is stable if and only if $\mu (V)<\mu (W)$ for all proper non-zero subbundles V of W and is semistable if $\mu (V)\leq \mu (W)$ for all proper non-zero subbundles V of W. Informally this says that a bundle is stable if it is "more ample" than any proper subbundle, and is unstable if it contains a "more ample" subbundle. If W and V are semistable vector bundles and μ(W) >μ(V), then there are no nonzero maps W → V. Mumford proved that the moduli space of stable bundles of given rank and degree over a nonsingular curve is a quasiprojective algebraic variety. The cohomology of the moduli space of stable vector bundles over a curve was described by Harder & Narasimhan (1975) using algebraic geometry over finite fields and Atiyah & Bott (1983) using Narasimhan-Seshadri approach. Stable vector bundles in higher dimensions If X is a smooth projective variety of dimension m and H is a hyperplane section, then a vector bundle (or a torsion-free sheaf) W is called stable (or sometimes Gieseker stable) if ${\frac {\chi (V(nH))}{{\hbox{rank}}(V)}}<{\frac {\chi (W(nH))}{{\hbox{rank}}(W)}}{\text{ for }}n{\text{ large}}$ for all proper non-zero subbundles (or subsheaves) V of W, where χ denotes the Euler characteristic of an algebraic vector bundle and the vector bundle V(nH) means the n-th twist of V by H. W is called semistable if the above holds with < replaced by ≤. Slope stability For bundles on curves the stability defined by slopes and by growth of Hilbert polynomial coincide. In higher dimensions, these two notions are different and have different advantages. Gieseker stability has an interpretation in terms of geometric invariant theory, while μ-stability has better properties for tensor products, pullbacks, etc. Let X be a smooth projective variety of dimension n, H its hyperplane section. A slope of a vector bundle (or, more generally, a torsion-free coherent sheaf) E with respect to H is a rational number defined as $\mu (E):={\frac {c_{1}(E)\cdot H^{n-1}}{\operatorname {rk} (E)}}$ where c1 is the first Chern class. The dependence on H is often omitted from the notation. A torsion-free coherent sheaf E is μ-semistable if for any nonzero subsheaf F ⊆ E the slopes satisfy the inequality μ(F) ≤ μ(E). It's μ-stable if, in addition, for any nonzero subsheaf F ⊆ E of smaller rank the strict inequality μ(F) < μ(E) holds. This notion of stability may be called slope stability, μ-stability, occasionally Mumford stability or Takemoto stability. For a vector bundle E the following chain of implications holds: E is μ-stable ⇒ E is stable ⇒ E is semistable ⇒ E is μ-semistable. Harder-Narasimhan filtration Main article: Harder–Narasimhan stratification Let E be a vector bundle over a smooth projective curve X. Then there exists a unique filtration by subbundles $0=E_{0}\subset E_{1}\subset \ldots \subset E_{r+1}=E$ such that the associated graded components Fi := Ei+1/Ei are semistable vector bundles and the slopes decrease, μ(Fi) > μ(Fi+1). This filtration was introduced in Harder & Narasimhan (1975) and is called the Harder-Narasimhan filtration. Two vector bundles with isomorphic associated gradeds are called S-equivalent. On higher-dimensional varieties the filtration also always exist and is unique, but the associated graded components may no longer be bundles. For Gieseker stability the inequalities between slopes should be replaced with inequalities between Hilbert polynomials. Kobayashi–Hitchin correspondence Main article: Kobayashi–Hitchin correspondence Narasimhan–Seshadri theorem says that stable bundles on a projective nonsingular curve are the same as those that have projectively flat unitary irreducible connections. For bundles of degree 0 projectively flat connections are flat and thus stable bundles of degree 0 correspond to irreducible unitary representations of the fundamental group. Kobayashi and Hitchin conjectured an analogue of this in higher dimensions. It was proved for projective nonsingular surfaces by Donaldson (1985), who showed that in this case a vector bundle is stable if and only if it has an irreducible Hermitian–Einstein connection. Generalizations It's possible to generalize (μ-)stability to non-smooth projective schemes and more general coherent sheaves using the Hilbert polynomial. Let X be a projective scheme, d a natural number, E a coherent sheaf on X with dim Supp(E) = d. Write the Hilbert polynomial of E as PE(m) = Σd i=0 αi(E)/(i!) mi. Define the reduced Hilbert polynomial pE := PE/αd(E). A coherent sheaf E is semistable if the following two conditions hold:[4] • E is pure of dimension d, i.e. all associated primes of E have dimension d; • for any proper nonzero subsheaf F ⊆ E the reduced Hilbert polynomials satisfy pF(m) ≤ pE(m) for large m. A sheaf is called stable if the strict inequality pF(m) < pE(m) holds for large m. Let Cohd(X) be the full subcategory of coherent sheaves on X with support of dimension ≤ d. The slope of an object F in Cohd may be defined using the coefficients of the Hilbert polynomial as ${\hat {\mu }}_{d}(F)=\alpha _{d-1}(F)/\alpha _{d}(F)$ if αd(F) ≠ 0 and 0 otherwise. The dependence of ${\hat {\mu }}_{d}$ on d is usually omitted from the notation. A coherent sheaf E with $\operatorname {dim} \,\operatorname {Supp} (E)=d$ is called μ-semistable if the following two conditions hold:[5] • the torsion of E is in dimension ≤ d-2; • for any nonzero subobject F ⊆ E in the quotient category Cohd(X)/Cohd-1(X) we have ${\hat {\mu }}(F)\leq {\hat {\mu }}(E)$. E is μ-stable if the strict inequality holds for all proper nonzero subobjects of E. Note that Cohd is a Serre subcategory for any d, so the quotient category exists. A subobject in the quotient category in general doesn't come from a subsheaf, but for torsion-free sheaves the original definition and the general one for d = n are equivalent. There are also other directions for generalizations, for example Bridgeland's stability conditions. One may define stable principal bundles in analogy with stable vector bundles. See also • Kobayashi–Hitchin correspondence • Corlette–Simpson correspondence • Quot scheme References 1. Note $\Omega _{\mathbb {P} ^{1}}^{1}\cong {\mathcal {O}}(-2)$ from the Adjunction formula on the canonical sheaf. 2. Since there are isomorphisms${\begin{aligned}{\text{Ext}}^{1}({\mathcal {O}}(1),{\mathcal {O}}(-1))&\cong {\text{Ext}}^{1}({\mathcal {O}},{\mathcal {O}}(-2))\\&\cong H^{1}(\mathbb {P} ^{1},\omega _{\mathbb {P} ^{1}})\end{aligned}}$ 3. Faltings, Gerd. "Vector bundles on curves" (PDF). Archived (PDF) from the original on 4 March 2020. 4. Huybrechts, Daniel; Lehn, Manfred (1997). The Geometry of Moduli Spaces of Sheaves (PDF)., Definition 1.2.4 5. Huybrechts, Daniel; Lehn, Manfred (1997). The Geometry of Moduli Spaces of Sheaves (PDF)., Definition 1.6.9 • Atiyah, Michael Francis; Bott, Raoul (1983), "The Yang-Mills equations over Riemann surfaces", Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences, 308 (1505): 523–615, doi:10.1098/rsta.1983.0017, ISSN 0080-4614, JSTOR 37156, MR 0702806 • Donaldson, S. K. (1985), "Anti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles", Proceedings of the London Mathematical Society, Third Series, 50 (1): 1–26, doi:10.1112/plms/s3-50.1.1, ISSN 0024-6115, MR 0765366 • Friedman, Robert (1998), Algebraic surfaces and holomorphic vector bundles, Universitext, Berlin, New York: Springer-Verlag, ISBN 978-0-387-98361-5, MR 1600388 • Harder, G.; Narasimhan, M. S. (1975), "On the cohomology groups of moduli spaces of vector bundles on curves", Mathematische Annalen, 212 (3): 215–248, doi:10.1007/BF01357141, ISSN 0025-5831, MR 0364254 • Huybrechts, Daniel; Lehn, Manfred (2010), The Geometry of Moduli Spaces of Sheaves, Cambridge Mathematical Library (2nd ed.), Cambridge University Press, ISBN 978-0521134200 • Mumford, David (1963), "Projective invariants of projective structures and applications", Proc. Internat. Congr. Mathematicians (Stockholm, 1962), Djursholm: Inst. Mittag-Leffler, pp. 526–530, MR 0175899 • Mumford, David; Fogarty, J.; Kirwan, F. (1994), Geometric invariant theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], vol. 34 (3rd ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-56963-3, MR 1304906 especially appendix 5C. • Narasimhan, M. S.; Seshadri, C. S. (1965), "Stable and unitary vector bundles on a compact Riemann surface", Annals of Mathematics, Second Series, The Annals of Mathematics, Vol. 82, No. 3, 82 (3): 540–567, doi:10.2307/1970710, ISSN 0003-486X, JSTOR 1970710, MR 0184252 Topics in algebraic curves Rational curves • Five points determine a conic • Projective line • Rational normal curve • Riemann sphere • Twisted cubic Elliptic curves Analytic theory • Elliptic function • Elliptic integral • Fundamental pair of periods • Modular form Arithmetic theory • Counting points on elliptic curves • Division polynomials • Hasse's theorem on elliptic curves • Mazur's torsion theorem • Modular elliptic curve • Modularity theorem • Mordell–Weil theorem • Nagell–Lutz theorem • Supersingular elliptic curve • Schoof's algorithm • Schoof–Elkies–Atkin algorithm Applications • Elliptic curve cryptography • Elliptic curve primality Higher genus • De Franchis theorem • Faltings's theorem • Hurwitz's automorphisms theorem • Hurwitz surface • Hyperelliptic curve Plane curves • AF+BG theorem • Bézout's theorem • Bitangent • Cayley–Bacharach theorem • Conic section • Cramer's paradox • Cubic plane curve • Fermat curve • Genus–degree formula • Hilbert's sixteenth problem • Nagata's conjecture on curves • Plücker formula • Quartic plane curve • Real plane curve Riemann surfaces • Belyi's theorem • Bring's curve • Bolza surface • Compact Riemann surface • Dessin d'enfant • Differential of the first kind • Klein quartic • Riemann's existence theorem • Riemann–Roch theorem • Teichmüller space • Torelli theorem Constructions • Dual curve • Polar curve • Smooth completion Structure of curves Divisors on curves • Abel–Jacobi map • Brill–Noether theory • Clifford's theorem on special divisors • Gonality of an algebraic curve • Jacobian variety • Riemann–Roch theorem • Weierstrass point • Weil reciprocity law Moduli • ELSV formula • Gromov–Witten invariant • Hodge bundle • Moduli of algebraic curves • Stable curve Morphisms • Hasse–Witt matrix • Riemann–Hurwitz formula • Prym variety • Weber's theorem (Algebraic curves) Singularities • Acnode • Crunode • Cusp • Delta invariant • Tacnode Vector bundles • Birkhoff–Grothendieck theorem • Stable vector bundle • Vector bundles on algebraic curves	Wikipedia
Semi-major and semi-minor axes In geometry, the major axis of an ellipse is its longest diameter: a line segment that runs through the center and both foci, with ends at the two most widely separated points of the perimeter. The semi-major axis (major semiaxis) is the longest semidiameter or one half of the major axis, and thus runs from the centre, through a focus, and to the perimeter. The semi-minor axis (minor semiaxis) of an ellipse or hyperbola is a line segment that is at right angles with the semi-major axis and has one end at the center of the conic section. For the special case of a circle, the lengths of the semi-axes are both equal to the radius of the circle. Part of a series on Astrodynamics Orbital mechanics Orbital elements • Apsis • Argument of periapsis • Eccentricity • Inclination • Mean anomaly • Orbital nodes • Semi-major axis • True anomaly Types of two-body orbits by eccentricity • Circular orbit • Elliptic orbit Transfer orbit • (Hohmann transfer orbit • Bi-elliptic transfer orbit) • Parabolic orbit • Hyperbolic orbit • Radial orbit • Decaying orbit Equations • Dynamical friction • Escape velocity • Kepler's equation • Kepler's laws of planetary motion • Orbital period • Orbital velocity • Surface gravity • Specific orbital energy • Vis-viva equation Celestial mechanics Gravitational influences • Barycenter • Hill sphere • Perturbations • Sphere of influence N-body orbits Lagrangian points • (Halo orbits) • Lissajous orbits • Lyapunov orbits Engineering and efficiency Preflight engineering • Mass ratio • Payload fraction • Propellant mass fraction • Tsiolkovsky rocket equation Efficiency measures • Gravity assist • Oberth effect The length of the semi-major axis a of an ellipse is related to the semi-minor axis's length b through the eccentricity e and the semi-latus rectum $\ell $, as follows: ${\begin{aligned}b&=a{\sqrt {1-e^{2}}},\\\ell &=a(1-e^{2}),\\a\ell &=b^{2}.\end{aligned}}$ The semi-major axis of a hyperbola is, depending on the convention, plus or minus one half of the distance between the two branches. Thus it is the distance from the center to either vertex of the hyperbola. A parabola can be obtained as the limit of a sequence of ellipses where one focus is kept fixed as the other is allowed to move arbitrarily far away in one direction, keeping $\ell $ fixed. Thus a and b tend to infinity, a faster than b. The major and minor axes are the axes of symmetry for the curve: in an ellipse, the minor axis is the shorter one; in a hyperbola, it is the one that does not intersect the hyperbola. Ellipse The equation of an ellipse is ${\frac {(x-h)^{2}}{a^{2}}}+{\frac {(y-k)^{2}}{b^{2}}}=1,$ where (h, k) is the center of the ellipse in Cartesian coordinates, in which an arbitrary point is given by (x, y). The semi-major axis is the mean value of the maximum and minimum distances $r_{\text{max}}$ and $r_{\text{min}}$ of the ellipse from a focus — that is, of the distances from a focus to the endpoints of the major axis $a={\frac {r_{\text{max}}+r_{\text{min}}}{2}}.$ In astronomy these extreme points are called apsides.[1] The semi-minor axis of an ellipse is the geometric mean of these distances: $b={\sqrt {r_{\text{max}}r_{\text{min}}}}.$ The eccentricity of an ellipse is defined as $e={\sqrt {1-{\frac {b^{2}}{a^{2}}}}},$ so $r_{\text{min}}=a(1-e),\quad r_{\text{max}}=a(1+e).$ Now consider the equation in polar coordinates, with one focus at the origin and the other on the $\theta =\pi $ direction: $r(1+e\cos \theta )=\ell .$ The mean value of $r=\ell /(1-e)$ and $r=\ell /(1+e)$, for $\theta =\pi $ and $\theta =0$ is $a={\frac {\ell }{1-e^{2}}}.$ In an ellipse, the semi-major axis is the geometric mean of the distance from the center to either focus and the distance from the center to either directrix. The semi-minor axis of an ellipse runs from the center of the ellipse (a point halfway between and on the line running between the foci) to the edge of the ellipse. The semi-minor axis is half of the minor axis. The minor axis is the longest line segment perpendicular to the major axis that connects two points on the ellipse's edge. The semi-minor axis b is related to the semi-major axis a through the eccentricity e and the semi-latus rectum $\ell $, as follows: ${\begin{aligned}b&=a{\sqrt {1-e^{2}}},\\a\ell &=b^{2}.\end{aligned}}$ A parabola can be obtained as the limit of a sequence of ellipses where one focus is kept fixed as the other is allowed to move arbitrarily far away in one direction, keeping $\ell $ fixed. Thus a and b tend to infinity, a faster than b. The length of the semi-minor axis could also be found using the following formula:[2] $2b={\sqrt {(p+q)^{2}-f^{2}}},$ where f is the distance between the foci, p and q are the distances from each focus to any point in the ellipse. Hyperbola The semi-major axis of a hyperbola is, depending on the convention, plus or minus one half of the distance between the two branches; if this is a in the x-direction the equation is: ${\frac {\left(x-h\right)^{2}}{a^{2}}}-{\frac {\left(y-k\right)^{2}}{b^{2}}}=1.$ In terms of the semi-latus rectum and the eccentricity we have $a={\ell \over e^{2}-1}.$ The transverse axis of a hyperbola coincides with the major axis.[3] In a hyperbola, a conjugate axis or minor axis of length $2b$, corresponding to the minor axis of an ellipse, can be drawn perpendicular to the transverse axis or major axis, the latter connecting the two vertices (turning points) of the hyperbola, with the two axes intersecting at the center of the hyperbola. The endpoints $(0,\pm b)$ of the minor axis lie at the height of the asymptotes over/under the hyperbola's vertices. Either half of the minor axis is called the semi-minor axis, of length b. Denoting the semi-major axis length (distance from the center to a vertex) as a, the semi-minor and semi-major axes' lengths appear in the equation of the hyperbola relative to these axes as follows: ${\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}=1.$ The semi-minor axis is also the distance from one of focuses of the hyperbola to an asymptote. Often called the impact parameter, this is important in physics and astronomy, and measure the distance a particle will miss the focus by if its journey is unperturbed by the body at the focus. The semi-minor axis and the semi-major axis are related through the eccentricity, as follows: $b=a{\sqrt {e^{2}-1}}.$[4] Note that in a hyperbola b can be larger than a.[5] Astronomy Orbital period In astrodynamics the orbital period T of a small body orbiting a central body in a circular or elliptical orbit is:[1] $T=2\pi {\sqrt {\frac {a^{3}}{\mu }}},$ where: a is the length of the orbit's semi-major axis, $\mu $ is the standard gravitational parameter of the central body. Note that for all ellipses with a given semi-major axis, the orbital period is the same, disregarding their eccentricity. The specific angular momentum h of a small body orbiting a central body in a circular or elliptical orbit is[1] $h={\sqrt {a\mu (1-e^{2})}},$ where: a and $\mu $ are as defined above, e is the eccentricity of the orbit. In astronomy, the semi-major axis is one of the most important orbital elements of an orbit, along with its orbital period. For Solar System objects, the semi-major axis is related to the period of the orbit by Kepler's third law (originally empirically derived):[1] $T^{2}\propto a^{3},$ where T is the period, and a is the semi-major axis. This form turns out to be a simplification of the general form for the two-body problem, as determined by Newton:[1] $T^{2}={\frac {4\pi ^{2}}{G(M+m)}}a^{3},$ where G is the gravitational constant, M is the mass of the central body, and m is the mass of the orbiting body. Typically, the central body's mass is so much greater than the orbiting body's, that m may be ignored. Making that assumption and using typical astronomy units results in the simpler form Kepler discovered. The orbiting body's path around the barycenter and its path relative to its primary are both ellipses.[1] The semi-major axis is sometimes used in astronomy as the primary-to-secondary distance when the mass ratio of the primary to the secondary is significantly large ($M\gg m$); thus, the orbital parameters of the planets are given in heliocentric terms. The difference between the primocentric and "absolute" orbits may best be illustrated by looking at the Earth–Moon system. The mass ratio in this case is 81.30059. The Earth–Moon characteristic distance, the semi-major axis of the geocentric lunar orbit, is 384,400 km. (Given the lunar orbit's eccentricity e = 0.0549, its semi-minor axis is 383,800 km. Thus the Moon's orbit is almost circular.) The barycentric lunar orbit, on the other hand, has a semi-major axis of 379,730 km, the Earth's counter-orbit taking up the difference, 4,670 km. The Moon's average barycentric orbital speed is 1.010 km/s, whilst the Earth's is 0.012 km/s. The total of these speeds gives a geocentric lunar average orbital speed of 1.022 km/s; the same value may be obtained by considering just the geocentric semi-major axis value. Average distance It is often said that the semi-major axis is the "average" distance between the primary focus of the ellipse and the orbiting body. This is not quite accurate, because it depends on what the average is taken over. • averaging the distance over the eccentric anomaly indeed results in the semi-major axis. • averaging over the true anomaly (the true orbital angle, measured at the focus) results in the semi-minor axis $b=a{\sqrt {1-e^{2}}}$. • averaging over the mean anomaly (the fraction of the orbital period that has elapsed since pericentre, expressed as an angle) gives the time-average $a\left(1+{\frac {e^{2}}{2}}\right)\,$. The time-averaged value of the reciprocal of the radius, $r^{-1}$, is $a^{-1}$. Energy; calculation of semi-major axis from state vectors In astrodynamics, the semi-major axis a can be calculated from orbital state vectors: $a=-{\frac {\mu }{2\varepsilon }}$ for an elliptical orbit and, depending on the convention, the same or $a={\frac {\mu }{2\varepsilon }}$ for a hyperbolic trajectory, and $\varepsilon ={\frac {v^{2}}{2}}-{\frac {\mu }{\|\mathbf {r} \|}}$ (specific orbital energy) and $\mu =GM,$ (standard gravitational parameter), where: v is orbital velocity from velocity vector of an orbiting object, r is a cartesian position vector of an orbiting object in coordinates of a reference frame with respect to which the elements of the orbit are to be calculated (e.g. geocentric equatorial for an orbit around Earth, or heliocentric ecliptic for an orbit around the Sun), G is the gravitational constant, M is the mass of the gravitating body, and $\varepsilon $ is the specific energy of the orbiting body. Note that for a given amount of total mass, the specific energy and the semi-major axis are always the same, regardless of eccentricity or the ratio of the masses. Conversely, for a given total mass and semi-major axis, the total specific orbital energy is always the same. This statement will always be true under any given conditions. Semi-major and semi-minor axes of the planets' orbits Planet orbits are always cited as prime examples of ellipses (Kepler's first law). However, the minimal difference between the semi-major and semi-minor axes shows that they are virtually circular in appearance. That difference (or ratio) is based on the eccentricity and is computed as ${\frac {a}{b}}={\frac {1}{\sqrt {1-e^{2}}}}$, which for typical planet eccentricities yields very small results. The reason for the assumption of prominent elliptical orbits lies probably in the much larger difference between aphelion and perihelion. That difference (or ratio) is also based on the eccentricity and is computed as ${\frac {r_{\text{a}}}{r_{\text{p}}}}={\frac {1+e}{1-e}}$. Due to the large difference between aphelion and perihelion, Kepler's second law is easily visualized. Eccentricity Semi-major axis a (AU) Semi-minor axis b (AU) Difference (%) Perihelion (AU) Aphelion (AU) Difference (%) Mercury 0.206 0.38700 0.37870 2.2 0.307 0.467 52 Venus 0.007 0.72300 0.72298 0.002 0.718 0.728 1.4 Earth 0.017 1.00000 0.99986 0.014 0.983 1.017 3.5 Mars 0.093 1.52400 1.51740 0.44 1.382 1.666 21 Jupiter 0.049 5.20440 5.19820 0.12 4.950 5.459 10 Saturn 0.057 9.58260 9.56730 0.16 9.041 10.124 12 Uranus 0.046 19.21840 19.19770 0.11 18.330 20.110 9.7 Neptune 0.010 30.11000 30.10870 0.004 29.820 30.400 1.9 1 AU (astronomical unit) equals 149.6 million km. References 1. Lissauer, Jack J.; de Pater, Imke (2019). Fundamental Planetary Sciences: physics, chemistry, and habitability. New York: Cambridge University Press. pp. 24–31. ISBN 9781108411981. 2. "Major / Minor axis of an ellipse", Math Open Reference, 12 May 2013. 3. "7.1 Alternative Characterization". www.geom.uiuc.edu. Archived from the original on 2018-10-24. Retrieved 2007-09-06. 4. "The Geometry of Orbits: Ellipses, Parabolas, and Hyperbolas". www.bogan.ca. 5. "7.1 Alternative Characterization". Archived from the original on 2018-10-24. Retrieved 2007-09-06. External links • Semi-major and semi-minor axes of an ellipse With interactive animation Gravitational orbits Types General • Box • Capture • Circular • Elliptical / Highly elliptical • Escape • Horseshoe • Hyperbolic trajectory • Inclined / Non-inclined • Kepler • Lagrange point • Osculating • Parabolic trajectory • Parking • Prograde / Retrograde • Synchronous • semi • sub • Transfer orbit Geocentric • Geosynchronous • Geostationary • Geostationary transfer • Graveyard • High Earth • Low Earth • Medium Earth • Molniya • Near-equatorial • Orbit of the Moon • Polar • Sun-synchronous • Tundra About other points • Mars • Areocentric • Areosynchronous • Areostationary • Lagrange points • Distant retrograde • Halo • Lissajous • Lunar • Sun • Heliocentric • Earth's orbit • Mars cycler • Heliosynchronous • Other • Lunar cycler Parameters • Shape • Size • e Eccentricity • a Semi-major axis • b Semi-minor axis • Q, q Apsides Orientation • i Inclination • Ω Longitude of the ascending node • ω Argument of periapsis • ϖ Longitude of the periapsis Position • M Mean anomaly • ν, θ, f True anomaly • E Eccentric anomaly • L Mean longitude • l True longitude Variation • T Orbital period • n Mean motion • v Orbital speed • t0 Epoch Maneuvers • Bi-elliptic transfer • Collision avoidance (spacecraft) • Delta-v • Delta-v budget • Gravity assist • Gravity turn • Hohmann transfer • Inclination change • Low-energy transfer • Oberth effect • Phasing • Rocket equation • Rendezvous • Transposition, docking, and extraction Orbital mechanics • Astronomical coordinate systems • Characteristic energy • Escape velocity • Ephemeris • Equatorial coordinate system • Ground track • Hill sphere • Interplanetary Transport Network • Kepler's laws of planetary motion • Lagrangian point • n-body problem • Orbit equation • Orbital state vectors • Perturbation • Retrograde and prograde motion • Specific orbital energy • Specific angular momentum • Two-line elements • List of orbits	Wikipedia
Norm (mathematics) In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is zero only at the origin. In particular, the Euclidean distance in a Euclidean space is defined by a norm on the associated Euclidean vector space, called the Euclidean norm, the 2-norm, or, sometimes, the magnitude of the vector. This norm can be defined as the square root of the inner product of a vector with itself. This article is about norms of normed vector spaces. For field theory, see Field norm. For ideals, see Ideal norm. For group theory, see Norm (group). For norms in descriptive set theory, see prewellordering. A seminorm satisfies the first two properties of a norm, but may be zero for vectors other than the origin.[1] A vector space with a specified norm is called a normed vector space. In a similar manner, a vector space with a seminorm is called a seminormed vector space. The term pseudonorm has been used for several related meanings. It may be a synonym of "seminorm".[1] A pseudonorm may satisfy the same axioms as a norm, with the equality replaced by an inequality "$\,\leq \,$" in the homogeneity axiom.[2] It can also refer to a norm that can take infinite values,[3] or to certain functions parametrised by a directed set.[4] Definition Given a vector space $X$ over a subfield $F$ of the complex numbers $\mathbb {C} ,$ a norm on $X$ is a real-valued function $p:X\to \mathbb {R} $ with the following properties, where $\|s\|$ denotes the usual absolute value of a scalar $s$:[5] 1. Subadditivity/Triangle inequality: $p(x+y)\leq p(x)+p(y)$ for all $x,y\in X.$ 2. Absolute homogeneity: $p(sx)=\|s\|p(x)$ for all $x\in X$ and all scalars $s.$ 3. Positive definiteness/positiveness[6]/Point-separating: for all $x\in X,$ if $p(x)=0$ then $x=0.$ • Because property (2.) implies $p(0)=0,$ some authors replace property (3.) with the equivalent condition: for every $x\in X,$ $p(x)=0$ if and only if $x=0.$ A seminorm on $X$ is a function $p:X\to \mathbb {R} $ that has properties (1.) and (2.)[7] so that in particular, every norm is also a seminorm (and thus also a sublinear functional). However, there exist seminorms that are not norms. Properties (1.) and (2.) imply that if $p$ is a norm (or more generally, a seminorm) then $p(0)=0$ and that $p$ also has the following property: 1. Non-negativity:[6] $p(x)\geq 0$ for all $x\in X.$ Some authors include non-negativity as part of the definition of "norm", although this is not necessary. Although this article defined "positive" to be a synonym of "positive definite", some authors instead define "positive" to be a synonym of "non-negative";[8] these definitions are not equivalent. Equivalent norms Suppose that $p$ and $q$ are two norms (or seminorms) on a vector space $X.$ Then $p$ and $q$ are called equivalent, if there exist two positive real constants $c$ and $C$ with $c>0$ such that for every vector $x\in X,$ $cq(x)\leq p(x)\leq Cq(x).$ The relation "$p$ is equivalent to $q$" is reflexive, symmetric ($cq\leq p\leq Cq$ implies ${\tfrac {1}{C}}p\leq q\leq {\tfrac {1}{c}}p$), and transitive and thus defines an equivalence relation on the set of all norms on $X.$ The norms $p$ and $q$ are equivalent if and only if they induce the same topology on $X.$[9] Any two norms on a finite-dimensional space are equivalent but this does not extend to infinite-dimensional spaces.[9] Notation If a norm $p:X\to \mathbb {R} $ is given on a vector space $X,$ then the norm of a vector $z\in X$ is usually denoted by enclosing it within double vertical lines: $\\|z\\|=p(z).$ Such notation is also sometimes used if $p$ is only a seminorm. For the length of a vector in Euclidean space (which is an example of a norm, as explained below), the notation $\|x\|$ with single vertical lines is also widespread. Examples Every (real or complex) vector space admits a norm: If $x_{\bullet }=\left(x_{i}\right)_{i\in I}$ is a Hamel basis for a vector space $X$ then the real-valued map that sends $x=\sum _{i\in I}s_{i}x_{i}\in X$ (where all but finitely many of the scalars $s_{i}$ are $0$) to $\sum _{i\in I}\left\|s_{i}\right\|$ is a norm on $X.$[10] There are also a large number of norms that exhibit additional properties that make them useful for specific problems. Absolute-value norm The absolute value $\\|x\\|=\|x\|$ is a norm on the one-dimensional vector spaces formed by the real or complex numbers. Any norm $p$ on a one-dimensional vector space $X$ is equivalent (up to scaling) to the absolute value norm, meaning that there is a norm-preserving isomorphism of vector spaces $f:\mathbb {F} \to X,$ where $\mathbb {F} $ is either $\mathbb {R} $ or $\mathbb {C} ,$ and norm-preserving means that $\|x\|=p(f(x)).$ This isomorphism is given by sending $1\in \mathbb {F} $ to a vector of norm $1,$ which exists since such a vector is obtained by multiplying any non-zero vector by the inverse of its norm. Euclidean norm Further information: Euclidean norm and Euclidean distance On the $n$-dimensional Euclidean space $\mathbb {R} ^{n},$ the intuitive notion of length of the vector ${\boldsymbol {x}}=\left(x_{1},x_{2},\ldots ,x_{n}\right)$ is captured by the formula[11] $\\|{\boldsymbol {x}}\\|_{2}:={\sqrt {x_{1}^{2}+\cdots +x_{n}^{2}}}.$ This is the Euclidean norm, which gives the ordinary distance from the origin to the point X—a consequence of the Pythagorean theorem. This operation may also be referred to as "SRSS", which is an acronym for the square root of the sum of squares.[12] The Euclidean norm is by far the most commonly used norm on $\mathbb {R} ^{n},$[11] but there are other norms on this vector space as will be shown below. However, all these norms are equivalent in the sense that they all define the same topology. The inner product of two vectors of a Euclidean vector space is the dot product of their coordinate vectors over an orthonormal basis. Hence, the Euclidean norm can be written in a coordinate-free way as $\\|{\boldsymbol {x}}\\|:={\sqrt {{\boldsymbol {x}}\cdot {\boldsymbol {x}}}}.$ The Euclidean norm is also called the $L^{2}$ norm,[13] $\ell ^{2}$ norm, 2-norm, or square norm; see $L^{p}$ space. It defines a distance function called the Euclidean length, $L^{2}$ distance, or $\ell ^{2}$ distance. The set of vectors in $\mathbb {R} ^{n+1}$ whose Euclidean norm is a given positive constant forms an $n$-sphere. Euclidean norm of complex numbers See also: Dot product § Complex vectors The Euclidean norm of a complex number is the absolute value (also called the modulus) of it, if the complex plane is identified with the Euclidean plane $\mathbb {R} ^{2}.$ This identification of the complex number $x+iy$ as a vector in the Euclidean plane, makes the quantity $ {\sqrt {x^{2}+y^{2}}}$ (as first suggested by Euler) the Euclidean norm associated with the complex number. Quaternions and octonions See also: Quaternion and Octonion There are exactly four Euclidean Hurwitz algebras over the real numbers. These are the real numbers $\mathbb {R} ,$ the complex numbers $\mathbb {C} ,$ the quaternions $\mathbb {H} ,$ and lastly the octonions $\mathbb {O} ,$ where the dimensions of these spaces over the real numbers are $1,2,4,{\text{ and }}8,$ respectively. The canonical norms on $\mathbb {R} $ and $\mathbb {C} $ are their absolute value functions, as discussed previously. The canonical norm on $\mathbb {H} $ of quaternions is defined by $\lVert q\rVert ={\sqrt {\,qq^{}~}}={\sqrt {\,q^{}q~}}={\sqrt {\,a^{2}+b^{2}+c^{2}+d^{2}~}}$ for every quaternion $q=a+b\,\mathbf {i} +c\,\mathbf {j} +d\,\mathbf {k} $ in $\mathbb {H} .$ This is the same as the Euclidean norm on $\mathbb {H} $ considered as the vector space $\mathbb {R} ^{4}.$ Similarly, the canonical norm on the octonions is just the Euclidean norm on $\mathbb {R} ^{8}.$ Finite-dimensional complex normed spaces On an $n$-dimensional complex space $\mathbb {C} ^{n},$ the most common norm is $\\|{\boldsymbol {z}}\\|:={\sqrt {\left\|z_{1}\right\|^{2}+\cdots +\left\|z_{n}\right\|^{2}}}={\sqrt {z_{1}{\bar {z}}_{1}+\cdots +z_{n}{\bar {z}}_{n}}}.$ In this case, the norm can be expressed as the square root of the inner product of the vector and itself: $\\|{\boldsymbol {x}}\\|:={\sqrt {{\boldsymbol {x}}^{H}~{\boldsymbol {x}}}},$ where ${\boldsymbol {x}}$ is represented as a column vector ${\begin{bmatrix}x_{1}\;x_{2}\;\dots \;x_{n}\end{bmatrix}}^{\rm {T}}$ and ${\boldsymbol {x}}^{H}$ denotes its conjugate transpose. This formula is valid for any inner product space, including Euclidean and complex spaces. For complex spaces, the inner product is equivalent to the complex dot product. Hence the formula in this case can also be written using the following notation: $\\|{\boldsymbol {x}}\\|:={\sqrt {{\boldsymbol {x}}\cdot {\boldsymbol {x}}}}.$ Taxicab norm or Manhattan norm Main article: Taxicab geometry $\\|{\boldsymbol {x}}\\|_{1}:=\sum _{i=1}^{n}\left\|x_{i}\right\|.$ The name relates to the distance a taxi has to drive in a rectangular street grid (like that of the New York borough of Manhattan) to get from the origin to the point $x.$ The set of vectors whose 1-norm is a given constant forms the surface of a cross polytope of dimension equivalent to that of the norm minus 1. The Taxicab norm is also called the $\ell ^{1}$ norm. The distance derived from this norm is called the Manhattan distance or $\ell _{1}$ distance. The 1-norm is simply the sum of the absolute values of the columns. In contrast, $\sum _{i=1}^{n}x_{i}$ is not a norm because it may yield negative results. p-norm Main article: Lp space Let $p\geq 1$ be a real number. The $p$-norm (also called $\ell _{p}$-norm) of vector $\mathbf {x} =(x_{1},\ldots ,x_{n})$ is[11] $\\|\mathbf {x} \\|_{p}:=\left(\sum _{i=1}^{n}\left\|x_{i}\right\|^{p}\right)^{1/p}.$ For $p=1,$ we get the taxicab norm, for $p=2$ we get the Euclidean norm, and as $p$ approaches $\infty $ the $p$-norm approaches the infinity norm or maximum norm: $\\|\mathbf {x} \\|_{\infty }:=\max _{i}\left\|x_{i}\right\|.$ The $p$-norm is related to the generalized mean or power mean. For $p=2,$ the $\\|\,\cdot \,\\|_{2}$-norm is even induced by a canonical inner product $\langle \,\cdot ,\,\cdot \rangle ,$ meaning that $ \\|\mathbf {x} \\|_{2}={\sqrt {\langle \mathbf {x} ,\mathbf {x} \rangle }}$ for all vectors $\mathbf {x} .$ This inner product can be expressed in terms of the norm by using the polarization identity. On $\ell ^{2},$ this inner product is the Euclidean inner product defined by $\langle \left(x_{n}\right)_{n},\left(y_{n}\right)_{n}\rangle _{\ell ^{2}}~=~\sum _{n}{\overline {x_{n}}}y_{n}$ while for the space $L^{2}(X,\mu )$ associated with a measure space $(X,\Sigma ,\mu ),$ which consists of all square-integrable functions, this inner product is $\langle f,g\rangle _{L^{2}}=\int _{X}{\overline {f(x)}}g(x)\,\mathrm {d} x.$ This definition is still of some interest for $0<p<1,$ but the resulting function does not define a norm,[14] because it violates the triangle inequality. What is true for this case of $0<p<1,$ even in the measurable analog, is that the corresponding $L^{p}$ class is a vector space, and it is also true that the function $\int _{X}\|f(x)-g(x)\|^{p}~\mathrm {d} \mu $ (without $p$th root) defines a distance that makes $L^{p}(X)$ into a complete metric topological vector space. These spaces are of great interest in functional analysis, probability theory and harmonic analysis. However, aside from trivial cases, this topological vector space is not locally convex, and has no continuous non-zero linear forms. Thus the topological dual space contains only the zero functional. The partial derivative of the $p$-norm is given by ${\frac {\partial }{\partial x_{k}}}\\|\mathbf {x} \\|_{p}={\frac {x_{k}\left\|x_{k}\right\|^{p-2}}{\\|\mathbf {x} \\|_{p}^{p-1}}}.$ The derivative with respect to $x,$ therefore, is ${\frac {\partial \\|\mathbf {x} \\|_{p}}{\partial \mathbf {x} }}={\frac {\mathbf {x} \circ \|\mathbf {x} \|^{p-2}}{\\|\mathbf {x} \\|_{p}^{p-1}}}.$ where $\circ $ denotes Hadamard product and $\|\cdot \|$ is used for absolute value of each component of the vector. For the special case of $p=2,$ this becomes ${\frac {\partial }{\partial x_{k}}}\\|\mathbf {x} \\|_{2}={\frac {x_{k}}{\\|\mathbf {x} \\|_{2}}},$ or ${\frac {\partial }{\partial \mathbf {x} }}\\|\mathbf {x} \\|_{2}={\frac {\mathbf {x} }{\\|\mathbf {x} \\|_{2}}}.$ Maximum norm (special case of: infinity norm, uniform norm, or supremum norm) Main article: Maximum norm If $\mathbf {x} $ is some vector such that $\mathbf {x} =(x_{1},x_{2},\ldots ,x_{n}),$ then: $\\|\mathbf {x} \\|_{\infty }:=\max \left(\left\|x_{1}\right\|,\ldots ,\left\|x_{n}\right\|\right).$ The set of vectors whose infinity norm is a given constant, $c,$ forms the surface of a hypercube with edge length $2c.$ Zero norm In probability and functional analysis, the zero norm induces a complete metric topology for the space of measurable functions and for the F-space of sequences with F–norm $ (x_{n})\mapsto \sum _{n}{2^{-n}x_{n}/(1+x_{n})}.$[15] Here we mean by F-norm some real-valued function $\lVert \cdot \rVert $ on an F-space with distance $d,$ such that $\lVert x\rVert =d(x,0).$ The F-norm described above is not a norm in the usual sense because it lacks the required homogeneity property. Hamming distance of a vector from zero See also: Hamming distance and discrete metric In metric geometry, the discrete metric takes the value one for distinct points and zero otherwise. When applied coordinate-wise to the elements of a vector space, the discrete distance defines the Hamming distance, which is important in coding and information theory. In the field of real or complex numbers, the distance of the discrete metric from zero is not homogeneous in the non-zero point; indeed, the distance from zero remains one as its non-zero argument approaches zero. However, the discrete distance of a number from zero does satisfy the other properties of a norm, namely the triangle inequality and positive definiteness. When applied component-wise to vectors, the discrete distance from zero behaves like a non-homogeneous "norm", which counts the number of non-zero components in its vector argument; again, this non-homogeneous "norm" is discontinuous. In signal processing and statistics, David Donoho referred to the zero "norm" with quotation marks. Following Donoho's notation, the zero "norm" of $x$ is simply the number of non-zero coordinates of $x,$ or the Hamming distance of the vector from zero. When this "norm" is localized to a bounded set, it is the limit of $p$-norms as $p$ approaches 0. Of course, the zero "norm" is not truly a norm, because it is not positive homogeneous. Indeed, it is not even an F-norm in the sense described above, since it is discontinuous, jointly and severally, with respect to the scalar argument in scalar–vector multiplication and with respect to its vector argument. Abusing terminology, some engineers omit Donoho's quotation marks and inappropriately call the number-of-non-zeros function the $L^{0}$ norm, echoing the notation for the Lebesgue space of measurable functions. Infinite dimensions The generalization of the above norms to an infinite number of components leads to $\ell ^{p}$ and $L^{p}$ spaces, with norms $\\|x\\|_{p}={\bigg (}\sum _{i\in \mathbb {N} }\left\|x_{i}\right\|^{p}{\bigg )}^{1/p}{\text{ and }}\ \\|f\\|_{p,X}={\bigg (}\int _{X}\|f(x)\|^{p}~\mathrm {d} x{\bigg )}^{1/p}$ for complex-valued sequences and functions on $X\subseteq \mathbb {R} ^{n}$ respectively, which can be further generalized (see Haar measure). Any inner product induces in a natural way the norm $ \\|x\\|:={\sqrt {\langle x,x\rangle }}.$ Other examples of infinite-dimensional normed vector spaces can be found in the Banach space article. Composite norms Other norms on $\mathbb {R} ^{n}$ can be constructed by combining the above; for example $\\|x\\|:=2\left\|x_{1}\right\|+{\sqrt {3\left\|x_{2}\right\|^{2}+\max(\left\|x_{3}\right\|,2\left\|x_{4}\right\|)^{2}}}$ is a norm on $\mathbb {R} ^{4}.$ For any norm and any injective linear transformation $A$ we can define a new norm of $x,$ equal to $\\|Ax\\|.$ In 2D, with $A$ a rotation by 45° and a suitable scaling, this changes the taxicab norm into the maximum norm. Each $A$ applied to the taxicab norm, up to inversion and interchanging of axes, gives a different unit ball: a parallelogram of a particular shape, size, and orientation. In 3D, this is similar but different for the 1-norm (octahedrons) and the maximum norm (prisms with parallelogram base). There are examples of norms that are not defined by "entrywise" formulas. For instance, the Minkowski functional of a centrally-symmetric convex body in $\mathbb {R} ^{n}$ (centered at zero) defines a norm on $\mathbb {R} ^{n}$ (see § Classification of seminorms: absolutely convex absorbing sets below). All the above formulas also yield norms on $\mathbb {C} ^{n}$ without modification. There are also norms on spaces of matrices (with real or complex entries), the so-called matrix norms. In abstract algebra Main article: Field norm Let $E$ be a finite extension of a field $k$ of inseparable degree $p^{\mu },$ and let $k$ have algebraic closure $K.$ If the distinct embeddings of $E$ are $\left\{\sigma _{j}\right\}_{j},$ then the Galois-theoretic norm of an element $\alpha \in E$ is the value $ \left(\prod _{j}{\sigma _{k}(\alpha )}\right)^{p^{\mu }}.$ As that function is homogeneous of degree $[E:k]$, the Galois-theoretic norm is not a norm in the sense of this article. However, the $[E:k]$-th root of the norm (assuming that concept makes sense) is a norm.[16] Composition algebras The concept of norm $N(z)$ in composition algebras does not share the usual properties of a norm as it may be negative or zero for $z\neq 0.$ A composition algebra $(A,{}^{},N)$ consists of an algebra over a field $A,$ an involution ${}^{},$ and a quadratic form $N(z)=zz^{}$ called the "norm". The characteristic feature of composition algebras is the homomorphism property of $N$: for the product $wz$ of two elements $w$ and $z$ of the composition algebra, its norm satisfies $N(wz)=N(w)N(z).$ For $\mathbb {R} ,$ $\mathbb {C} ,$ $\mathbb {H} ,$ and O the composition algebra norm is the square of the norm discussed above. In those cases the norm is a definite quadratic form. In other composition algebras the norm is an isotropic quadratic form. Properties For any norm $p:X\to \mathbb {R} $ on a vector space $X,$ the reverse triangle inequality holds: $p(x\pm y)\geq \|p(x)-p(y)\|{\text{ for all }}x,y\in X.$ If $u:X\to Y$ is a continuous linear map between normed spaces, then the norm of $u$ and the norm of the transpose of $u$ are equal.[17] For the $L^{p}$ norms, we have Hölder's inequality[18] $\|\langle x,y\rangle \|\leq \\|x\\|_{p}\\|y\\|_{q}\qquad {\frac {1}{p}}+{\frac {1}{q}}=1.$ A special case of this is the Cauchy–Schwarz inequality:[18] $\left\|\langle x,y\rangle \right\|\leq \\|x\\|_{2}\\|y\\|_{2}.$ Every norm is a seminorm and thus satisfies all properties of the latter. In turn, every seminorm is a sublinear function and thus satisfies all properties of the latter. In particular, every norm is a convex function. Equivalence The concept of unit circle (the set of all vectors of norm 1) is different in different norms: for the 1-norm, the unit circle is a square, for the 2-norm (Euclidean norm), it is the well-known unit circle, while for the infinity norm, it is a different square. For any $p$-norm, it is a superellipse with congruent axes (see the accompanying illustration). Due to the definition of the norm, the unit circle must be convex and centrally symmetric (therefore, for example, the unit ball may be a rectangle but cannot be a triangle, and $p\geq 1$ for a $p$-norm). In terms of the vector space, the seminorm defines a topology on the space, and this is a Hausdorff topology precisely when the seminorm can distinguish between distinct vectors, which is again equivalent to the seminorm being a norm. The topology thus defined (by either a norm or a seminorm) can be understood either in terms of sequences or open sets. A sequence of vectors $\{v_{n}\}$ is said to converge in norm to $v,$ if $\left\\|v_{n}-v\right\\|\to 0$ as $n\to \infty .$ Equivalently, the topology consists of all sets that can be represented as a union of open balls. If $(X,\\|\cdot \\|)$ is a normed space then[19] $\\|x-y\\|=\\|x-z\\|+\\|z-y\\|{\text{ for all }}x,y\in X{\text{ and }}z\in [x,y].$ Two norms $\\|\cdot \\|_{\alpha }$ and $\\|\cdot \\|_{\beta }$ on a vector space $X$ are called equivalent if they induce the same topology,[9] which happens if and only if there exist positive real numbers $C$ and $D$ such that for all $x\in X$ $C\\|x\\|_{\alpha }\leq \\|x\\|_{\beta }\leq D\\|x\\|_{\alpha }.$ For instance, if $p>r\geq 1$ on $\mathbb {C} ^{n},$ then[20] $\\|x\\|_{p}\leq \\|x\\|_{r}\leq n^{(1/r-1/p)}\\|x\\|_{p}.$ In particular, $\\|x\\|_{2}\leq \\|x\\|_{1}\leq {\sqrt {n}}\\|x\\|_{2}$ $\\|x\\|_{\infty }\leq \\|x\\|_{2}\leq {\sqrt {n}}\\|x\\|_{\infty }$ $\\|x\\|_{\infty }\leq \\|x\\|_{1}\leq n\\|x\\|_{\infty },$ That is, $\\|x\\|_{\infty }\leq \\|x\\|_{2}\leq \\|x\\|_{1}\leq {\sqrt {n}}\\|x\\|_{2}\leq n\\|x\\|_{\infty }.$ If the vector space is a finite-dimensional real or complex one, all norms are equivalent. On the other hand, in the case of infinite-dimensional vector spaces, not all norms are equivalent. Equivalent norms define the same notions of continuity and convergence and for many purposes do not need to be distinguished. To be more precise the uniform structure defined by equivalent norms on the vector space is uniformly isomorphic. Classification of seminorms: absolutely convex absorbing sets Main article: Seminorm All seminorms on a vector space $X$ can be classified in terms of absolutely convex absorbing subsets $A$ of $X.$ To each such subset corresponds a seminorm $p_{A}$ called the gauge of $A,$ defined as $p_{A}(x):=\inf\{r\in \mathbb {R} :r>0,x\in rA\}$ where $\inf _{}$ is the infimum, with the property that $\left\{x\in X:p_{A}(x)<1\right\}~\subseteq ~A~\subseteq ~\left\{x\in X:p_{A}(x)\leq 1\right\}.$ Conversely: Any locally convex topological vector space has a local basis consisting of absolutely convex sets. A common method to construct such a basis is to use a family $(p)$ of seminorms $p$ that separates points: the collection of all finite intersections of sets $\{p<1/n\}$ turns the space into a locally convex topological vector space so that every p is continuous. Such a method is used to design weak and weak topologies. norm case: Suppose now that $(p)$ contains a single $p:$ since $(p)$ is separating, $p$ is a norm, and $A=\{p<1\}$ is its open unit ball. Then $A$ is an absolutely convex bounded neighbourhood of 0, and $p=p_{A}$ is continuous. The converse is due to Andrey Kolmogorov: any locally convex and locally bounded topological vector space is normable. Precisely: If $X$ is an absolutely convex bounded neighbourhood of 0, the gauge $g_{X}$ (so that $X=\{g_{X}<1\}$ is a norm. See also • Asymmetric norm – Generalization of the concept of a norm • F-seminorm – A topological vector space whose topology can be defined by a metricPages displaying short descriptions of redirect targets • Gowers norm • Kadec norm – All infinite-dimensional, separable Banach spaces are homeomorphicPages displaying short descriptions of redirect targets • Least-squares spectral analysis – Periodicity computation method • Mahalanobis distance – Statistical distance measure • Magnitude (mathematics) – Property determining comparison and ordering • Matrix norm – Norm on a vector space of matrices • Minkowski distance – Mathematical metric in normed vector space • Minkowski functional – Function made from a set • Operator norm – Measure of the "size" of linear operators • Paranorm – A topological vector space whose topology can be defined by a metricPages displaying short descriptions of redirect targets • Relation of norms and metrics – Mathematical space with a notion of distancePages displaying short descriptions of redirect targets • Seminorm – nonnegative-real-valued function on a real or complex vector space that satisfies the triangle inequality and is absolutely homogenousPages displaying wikidata descriptions as a fallback • Sublinear function References 1. Knapp, A.W. (2005). Basic Real Analysis. Birkhäuser. p. . ISBN 978-0-817-63250-2. 2. "Pseudo-norm - Encyclopedia of Mathematics". encyclopediaofmath.org. Retrieved 2022-05-12. 3. "Pseudonorm". www.spektrum.de (in German). Retrieved 2022-05-12. 4. Hyers, D. H. (1939-09-01). "Pseudo-normed linear spaces and Abelian groups". Duke Mathematical Journal. 5 (3). doi:10.1215/s0012-7094-39-00551-x. ISSN 0012-7094. 5. Pugh, C.C. (2015). Real Mathematical Analysis. Springer. p. page 28. ISBN 978-3-319-17770-0. Prugovečki, E. (1981). Quantum Mechanics in Hilbert Space. p. page 20. 6. Kubrusly 2011, p. 200. 7. Rudin, W. (1991). Functional Analysis. p. 25. 8. Narici & Beckenstein 2011, pp. 120–121. 9. Conrad, Keith. "Equivalence of norms" (PDF). kconrad.math.uconn.edu. Retrieved September 7, 2020. 10. Wilansky 2013, pp. 20–21. 11. Weisstein, Eric W. "Vector Norm". mathworld.wolfram.com. Retrieved 2020-08-24. 12. Chopra, Anil (2012). Dynamics of Structures, 4th Ed. Prentice-Hall. ISBN 978-0-13-285803-8. 13. Weisstein, Eric W. "Norm". mathworld.wolfram.com. Retrieved 2020-08-24. 14. Except in $\mathbb {R} ^{1},$ where it coincides with the Euclidean norm, and $\mathbb {R} ^{0},$ where it is trivial. 15. Rolewicz, Stefan (1987), Functional analysis and control theory: Linear systems, Mathematics and its Applications (East European Series), vol. 29 (Translated from the Polish by Ewa Bednarczuk ed.), Dordrecht; Warsaw: D. Reidel Publishing Co.; PWN—Polish Scientific Publishers, pp. xvi, 524, doi:10.1007/978-94-015-7758-8, ISBN 90-277-2186-6, MR 0920371, OCLC 13064804 16. Lang, Serge (2002) [1993]. Algebra (Revised 3rd ed.). New York: Springer Verlag. p. 284. ISBN 0-387-95385-X. 17. Trèves 2006, pp. 242–243. 18. Golub, Gene; Van Loan, Charles F. (1996). Matrix Computations (Third ed.). Baltimore: The Johns Hopkins University Press. p. 53. ISBN 0-8018-5413-X. 19. Narici & Beckenstein 2011, pp. 107–113. 20. "Relation between p-norms". Mathematics Stack Exchange. Bibliography • Bourbaki, Nicolas (1987) [1981]. Topological Vector Spaces: Chapters 1–5. Éléments de mathématique. Translated by Eggleston, H.G.; Madan, S. Berlin New York: Springer-Verlag. ISBN 3-540-13627-4. OCLC 17499190. • Khaleelulla, S. M. (1982). Counterexamples in Topological Vector Spaces. Lecture Notes in Mathematics. Vol. 936. Berlin, Heidelberg, New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370. • Kubrusly, Carlos S. (2011). The Elements of Operator Theory (Second ed.). Boston: Birkhäuser Basel. ISBN 978-0-8176-4998-2. OCLC 710154895. • Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. • Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. • Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. • Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114. 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Normed vector space In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers on which a norm is defined.[1] A norm is a generalization of the intuitive notion of "length" in the physical world. If $V$ is a vector space over $K$, where $K$ is a field equal to $\mathbb {R} $ or to $\mathbb {C} $, then a norm on $V$ is a map $V\to \mathbb {R} $, typically denoted by $\lVert \cdot \rVert $, satisfying the following four axioms: 1. Non-negativity: for every $x\in V$,$\;\lVert x\rVert \geq 0$. 2. Positive definiteness: for every $x\in V$, $\;\lVert x\rVert =0$ if and only if $x$ is the zero vector. 3. Absolute homogeneity: for every $\lambda \in K$ and $x\in V$, $\lVert \lambda x\rVert =\|\lambda \|\,\lVert x\rVert $ 4. Triangle inequality: for every $x\in V$ and $y\in V$, $\\|x+y\\|\leq \\|x\\|+\\|y\\|.$ If $V$ is a real or complex vector space as above, and $\lVert \cdot \rVert $ is a norm on $V$, then the ordered pair $(V,\lVert \cdot \rVert )$ is called a normed vector space. If it is clear from context which norm is intended, then it is common to denote the normed vector space simply by $V$. A norm induces a distance, called its (norm) induced metric, by the formula $d(x,y)=\\|y-x\\|.$ which makes any normed vector space into a metric space and a topological vector space. If this metric space is complete then the normed space is a Banach space. Every normed vector space can be "uniquely extended" to a Banach space, which makes normed spaces intimately related to Banach spaces. Every Banach space is a normed space but converse is not true. For example, the set of the finite sequences of real numbers can be normed with the Euclidean norm, but it is not complete for this norm. An inner product space is a normed vector space whose norm is the square root of the inner product of a vector and itself. The Euclidean norm of a Euclidean vector space is a special case that allows defining Euclidean distance by the formula $d(A,B)=\\|{\overrightarrow {AB}}\\|.$ The study of normed spaces and Banach spaces is a fundamental part of functional analysis, a major subfield of mathematics. Definition See also: Seminormed space A normed vector space is a vector space equipped with a norm. A seminormed vector space is a vector space equipped with a seminorm. A useful variation of the triangle inequality is $\\|x-y\\|\geq \|\\|x\\|-\\|y\\|\|$ for any vectors $x$ and $y.$ This also shows that a vector norm is a (uniformly) continuous function. Property 3 depends on a choice of norm $\|\alpha \|$ on the field of scalars. When the scalar field is $\mathbb {R} $ (or more generally a subset of $\mathbb {C} $), this is usually taken to be the ordinary absolute value, but other choices are possible. For example, for a vector space over $\mathbb {Q} $ one could take $\|\alpha \|$ to be the $p$-adic absolute value. Topological structure If $(V,\\|\,\cdot \,\\|)$ is a normed vector space, the norm $\\|\,\cdot \,\\|$ induces a metric (a notion of distance) and therefore a topology on $V.$ This metric is defined in the natural way: the distance between two vectors $\mathbf {u} $ and $\mathbf {v} $ is given by $\\|\mathbf {u} -\mathbf {v} \\|.$ This topology is precisely the weakest topology which makes $\\|\,\cdot \,\\|$ continuous and which is compatible with the linear structure of $V$ in the following sense: 1. The vector addition $\,+\,:V\times V\to V$ is jointly continuous with respect to this topology. This follows directly from the triangle inequality. 2. The scalar multiplication $\,\cdot \,:\mathbb {K} \times V\to V,$ where $\mathbb {K} $ is the underlying scalar field of $V,$ is jointly continuous. This follows from the triangle inequality and homogeneity of the norm. Similarly, for any seminormed vector space we can define the distance between two vectors $\mathbf {u} $ and $\mathbf {v} $ as $\\|\mathbf {u} -\mathbf {v} \\|.$ This turns the seminormed space into a pseudometric space (notice this is weaker than a metric) and allows the definition of notions such as continuity and convergence. To put it more abstractly every seminormed vector space is a topological vector space and thus carries a topological structure which is induced by the semi-norm. Of special interest are complete normed spaces, which are known as Banach spaces. Every normed vector space $V$ sits as a dense subspace inside some Banach space; this Banach space is essentially uniquely defined by $V$ and is called the completion of $V.$ Two norms on the same vector space are called equivalent if they define the same topology. On a finite-dimensional vector space, all norms are equivalent but this is not true for infinite dimensional vector spaces. All norms on a finite-dimensional vector space are equivalent from a topological viewpoint as they induce the same topology (although the resulting metric spaces need not be the same).[2] And since any Euclidean space is complete, we can thus conclude that all finite-dimensional normed vector spaces are Banach spaces. A normed vector space $V$ is locally compact if and only if the unit ball $B=\{x:\\|x\\|\leq 1\}$ is compact, which is the case if and only if $V$ is finite-dimensional; this is a consequence of Riesz's lemma. (In fact, a more general result is true: a topological vector space is locally compact if and only if it is finite-dimensional. The point here is that we don't assume the topology comes from a norm.) The topology of a seminormed vector space has many nice properties. Given a neighbourhood system ${\mathcal {N}}(0)$ around 0 we can construct all other neighbourhood systems as ${\mathcal {N}}(x)=x+{\mathcal {N}}(0):=\{x+N:N\in {\mathcal {N}}(0)\}$ with $x+N:=\{x+n:n\in N\}.$ Moreover, there exists a neighbourhood basis for the origin consisting of absorbing and convex sets. As this property is very useful in functional analysis, generalizations of normed vector spaces with this property are studied under the name locally convex spaces. A norm (or seminorm) $\\|\cdot \\|$ on a topological vector space $(X,\tau )$ is continuous if and only if the topology $\tau _{\\|\cdot \\|}$ that $\\|\cdot \\|$ induces on $X$ is coarser than $\tau $ (meaning, $\tau _{\\|\cdot \\|}\subseteq \tau $), which happens if and only if there exists some open ball $B$ in $(X,\\|\cdot \\|)$ (such as maybe $\{x\in X:\\|x\\|<1\}$ for example) that is open in $(X,\tau )$ (said different, such that $B\in \tau $). Normable spaces See also: Metrizable topological vector space § Normability A topological vector space $(X,\tau )$ is called normable if there exists a norm $\\|\cdot \\|$ on $X$ such that the canonical metric $(x,y)\mapsto \\|y-x\\|$ induces the topology $\tau $ on $X.$ The following theorem is due to Kolmogorov:[3] Kolmogorov's normability criterion: A Hausdorff topological vector space is normable if and only if there exists a convex, von Neumann bounded neighborhood of $0\in X.$ A product of a family of normable spaces is normable if and only if only finitely many of the spaces are non-trivial (that is, $\neq \{0\}$).[3] Furthermore, the quotient of a normable space $X$ by a closed vector subspace $C$ is normable, and if in addition $X$'s topology is given by a norm $\\|\,\cdot ,\\|$ then the map $X/C\to \mathbb {R} $ given by $ x+C\mapsto \inf _{c\in C}\\|x+c\\|$ is a well defined norm on $X/C$ that induces the quotient topology on $X/C.$[4] If $X$ is a Hausdorff locally convex topological vector space then the following are equivalent: 1. $X$ is normable. 2. $X$ has a bounded neighborhood of the origin. 3. the strong dual space $X_{b}^{\prime }$ of $X$ is normable.[5] 4. the strong dual space $X_{b}^{\prime }$ of $X$ is metrizable.[5] Furthermore, $X$ is finite dimensional if and only if $X_{\sigma }^{\prime }$ is normable (here $X_{\sigma }^{\prime }$ denotes $X^{\prime }$ endowed with the weak-* topology). The topology $\tau $ of the Fréchet space $C^{\infty }(K),$ as defined in the article on spaces of test functions and distributions, is defined by a countable family of norms but it is not a normable space because there does not exist any norm $\\|\cdot \\|$ on $C^{\infty }(K)$ such that the topology that this norm induces is equal to $\tau .$ Even if a metrizable topological vector space has a topology that is defined by a family of norms, then it may nevertheless still fail to be normable space (meaning that its topology can not be defined by any single norm). An example of such a space is the Fréchet space $C^{\infty }(K),$ whose definition can be found in the article on spaces of test functions and distributions, because its topology $\tau $ is defined by a countable family of norms but it is not a normable space because there does not exist any norm $\\|\cdot \\|$ on $C^{\infty }(K)$ such that the topology this norm induces is equal to $\tau .$ In fact, the topology of a locally convex space $X$ can be a defined by a family of norms on $X$ if and only if there exists at least one continuous norm on $X.$[6] Linear maps and dual spaces The most important maps between two normed vector spaces are the continuous linear maps. Together with these maps, normed vector spaces form a category. The norm is a continuous function on its vector space. All linear maps between finite dimensional vector spaces are also continuous. An isometry between two normed vector spaces is a linear map $f$ which preserves the norm (meaning $\\|f(\mathbf {v} )\\|=\\|\mathbf {v} \\|$ for all vectors $\mathbf {v} $). Isometries are always continuous and injective. A surjective isometry between the normed vector spaces $V$ and $W$ is called an isometric isomorphism, and $V$ and $W$ are called isometrically isomorphic. Isometrically isomorphic normed vector spaces are identical for all practical purposes. When speaking of normed vector spaces, we augment the notion of dual space to take the norm into account. The dual $V^{\prime }$ of a normed vector space $V$ is the space of all continuous linear maps from $V$ to the base field (the complexes or the reals) — such linear maps are called "functionals". The norm of a functional $\varphi $ is defined as the supremum of $\|\varphi (\mathbf {v} )\|$ where $\mathbf {v} $ ranges over all unit vectors (that is, vectors of norm $1$) in $V.$ This turns $V^{\prime }$ into a normed vector space. An important theorem about continuous linear functionals on normed vector spaces is the Hahn–Banach theorem. Normed spaces as quotient spaces of seminormed spaces The definition of many normed spaces (in particular, Banach spaces) involves a seminorm defined on a vector space and then the normed space is defined as the quotient space by the subspace of elements of seminorm zero. For instance, with the $L^{p}$ spaces, the function defined by $\\|f\\|_{p}=\left(\int \|f(x)\|^{p}\;dx\right)^{1/p}$ is a seminorm on the vector space of all functions on which the Lebesgue integral on the right hand side is defined and finite. However, the seminorm is equal to zero for any function supported on a set of Lebesgue measure zero. These functions form a subspace which we "quotient out", making them equivalent to the zero function. Finite product spaces Given $n$ seminormed spaces $\left(X_{i},q_{i}\right)$ with seminorms $q_{i}:X_{i}\to \mathbb {R} ,$ denote the product space by $X:=\prod _{i=1}^{n}X_{i}$ where vector addition defined as $\left(x_{1},\ldots ,x_{n}\right)+\left(y_{1},\ldots ,y_{n}\right):=\left(x_{1}+y_{1},\ldots ,x_{n}+y_{n}\right)$ and scalar multiplication defined as $\alpha \left(x_{1},\ldots ,x_{n}\right):=\left(\alpha x_{1},\ldots ,\alpha x_{n}\right).$ Define a new function $q:X\to \mathbb {R} $ by $q\left(x_{1},\ldots ,x_{n}\right):=\sum _{i=1}^{n}q_{i}\left(x_{i}\right),$ which is a seminorm on $X.$ The function $q$ is a norm if and only if all $q_{i}$ are norms. More generally, for each real $p\geq 1$ the map $q:X\to \mathbb {R} $ defined by $q\left(x_{1},\ldots ,x_{n}\right):=\left(\sum _{i=1}^{n}q_{i}\left(x_{i}\right)^{p}\right)^{\frac {1}{p}}$ is a semi norm. For each $p$ this defines the same topological space. A straightforward argument involving elementary linear algebra shows that the only finite-dimensional seminormed spaces are those arising as the product space of a normed space and a space with trivial seminorm. Consequently, many of the more interesting examples and applications of seminormed spaces occur for infinite-dimensional vector spaces. See also • Banach space, normed vector spaces which are complete with respect to the metric induced by the norm • Banach–Mazur compactum – Set of n-dimensional subspaces of a normed space made into a compact metric space. • Finsler manifold, where the length of each tangent vector is determined by a norm • Inner product space, normed vector spaces where the norm is given by an inner product • Kolmogorov's normability criterion – Characterization of normable spaces • Locally convex topological vector space – a vector space with a topology defined by convex open sets • Space (mathematics) – mathematical set with some added structure • Topological vector space – Vector space with a notion of nearness References 1. Callier, Frank M. (1991). Linear System Theory. New York: Springer-Verlag. ISBN 0-387-97573-X. 2. Kedlaya, Kiran S. (2010), p-adic differential equations, Cambridge Studies in Advanced Mathematics, vol. 125, Cambridge University Press, CiteSeerX 10.1.1.165.270, ISBN 978-0-521-76879-5, Theorem 1.3.6 3. Schaefer 1999, p. 41. 4. Schaefer 1999, p. 42. 5. Trèves 2006, pp. 136–149, 195–201, 240–252, 335–390, 420–433. 6. Jarchow 1981, p. 130. sfn error: no target: CITEREFJarchow1981 (help) Bibliography • Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277. • Banach, Stefan (1932). Théorie des Opérations Linéaires [Theory of Linear Operations] (PDF). Monografie Matematyczne (in French). Vol. 1. Warszawa: Subwencji Funduszu Kultury Narodowej. Zbl 0005.20901. Archived from the original (PDF) on 2014-01-11. Retrieved 2020-07-11. • Rolewicz, Stefan (1987), Functional analysis and control theory: Linear systems, Mathematics and its Applications (East European Series), vol. 29 (Translated from the Polish by Ewa Bednarczuk ed.), Dordrecht; Warsaw: D. Reidel Publishing Co.; PWN—Polish Scientific Publishers, pp. xvi+524, doi:10.1007/978-94-015-7758-8, ISBN 90-277-2186-6, MR 0920371, OCLC 13064804 • Schaefer, H. H. (1999). Topological Vector Spaces. New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. • Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. External links • Media related to Normed spaces at Wikimedia Commons Banach space topics Types of Banach spaces • Asplund • Banach • list • Banach lattice • Grothendieck • Hilbert • Inner product space • Polarization identity • (Polynomially) Reflexive • Riesz • L-semi-inner product • (B • Strictly • Uniformly) convex • Uniformly smooth • (Injective • Projective) Tensor product (of Hilbert spaces) Banach spaces are: • Barrelled • Complete • F-space • Fréchet • tame • Locally convex • Seminorms/Minkowski functionals • Mackey • Metrizable • Normed • norm • Quasinormed • Stereotype Function space Topologies • Banach–Mazur compactum • Dual • Dual space • Dual norm • Operator • Ultraweak • Weak • polar • operator • Strong • polar • operator • Ultrastrong • Uniform convergence Linear operators • Adjoint • Bilinear • form • operator • sesquilinear • (Un)Bounded • Closed • Compact • on Hilbert spaces • (Dis)Continuous • Densely defined • Fredholm • kernel • operator • Hilbert–Schmidt • Functionals • positive • Pseudo-monotone • Normal • Nuclear • Self-adjoint • Strictly singular • Trace class • Transpose • Unitary Operator theory • Banach algebras • C-algebras • Operator space • Spectrum • C-algebra • radius • Spectral theory • of ODEs • Spectral theorem • Polar decomposition • Singular value decomposition Theorems • Anderson–Kadec • Banach–Alaoglu • Banach–Mazur • Banach–Saks • Banach–Schauder (open mapping) • Banach–Steinhaus (Uniform boundedness) • Bessel's inequality • Cauchy–Schwarz inequality • Closed graph • Closed range • Eberlein–Šmulian • Freudenthal spectral • Gelfand–Mazur • Gelfand–Naimark • Goldstine • Hahn–Banach • hyperplane separation • Kakutani fixed-point • Krein–Milman • Lomonosov's invariant subspace • Mackey–Arens • Mazur's lemma • M. 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Open coloring axiom The open coloring axiom (abbreviated OCA) is an axiom about coloring edges of a graph whose vertices are a subset of the real numbers: two different versions were introduced by Abraham, Rubin & Shelah (1985) and by Todorčević (1989). Statement Suppose that X is a subset of the reals, and each pair of elements of X is colored either black or white, with the set of white pairs being open in the complete graph on X. The open coloring axiom states that either: 1. X has an uncountable subset such that any pair from this subset is white; or 2. X can be partitioned into a countable number of subsets such that any pair from the same subset is black. A weaker version, OCAP, replaces the uncountability condition in the first case with being a compact perfect set in X. Both OCA and OCAP can be stated equivalently for arbitrary separable spaces. Relation to other axioms OCAP can be proved in ZFC for analytic subsets of a Polish space, and from the axiom of determinacy. The full OCA is consistent with (but independent of) ZFC, and follows from the proper forcing axiom. OCA implies that the smallest unbounded set of Baire space has cardinality $\aleph _{2}$. Moreover, assuming OCA, Baire space contains few "gaps" between sets of sequences — more specifically, that the only possible gaps are Hausdorff gaps and analogous (κ,ω)-gaps where κ is an initial ordinal not less than ω2. References • Abraham, Uri; Rubin, Matatyahu; Shelah, Saharon (1985), "On the consistency of some partition theorems for continuous colorings, and the structure of ℵ1-dense real order types", Ann. Pure Appl. Logic, 29 (2): 123–206, doi:10.1016/0168-0072(84)90024-1, Zbl 0585.03019 • Carotenuto, Gemma (2013), An introduction to OCA (PDF), notes on lectures by Matteo Viale • Kunen, Kenneth (2011), Set theory, Studies in Logic, vol. 34, London: College Publications, ISBN 978-1-84890-050-9, Zbl 1262.03001 • Moore, Justin Tatch (2011), "Logic and foundations the proper forcing axiom", in Bhatia, Rajendra (ed.), Proceedings of the international congress of mathematicians (ICM 2010), Hyderabad, India, August 19–27, 2010. Vol. II: Invited lectures (PDF), Hackensack, NJ: World Scientific, pp. 3–29, ISBN 978-981-4324-30-4, Zbl 1258.03075 • Todorčević, Stevo (1989), Partition problems in topology, Contemporary Mathematics, vol. 84, Providence, RI: American Mathematical Society, ISBN 0-8218-5091-1, MR 0980949, Zbl 0659.54001	Wikipedia
Semiring In abstract algebra, a semiring is an algebraic structure. It is a generalization of a ring, dropping the requirement that each element must have an additive inverse. At the same time, it is a generalization of bounded distributive lattices. Algebraic structure → Ring theory Ring theory Basic concepts Rings • Subrings • Ideal • Quotient ring • Fractional ideal • Total ring of fractions • Product of rings • Free product of associative algebras • Tensor product of algebras Ring homomorphisms • Kernel • Inner automorphism • Frobenius endomorphism Algebraic structures • Module • Associative algebra • Graded ring • Involutive ring • Category of rings • Initial ring $\mathbb {Z} $ • Terminal ring $0=\mathbb {Z} _{1}$ Related structures • Field • Finite field • Non-associative ring • Lie ring • Jordan ring • Semiring • Semifield Commutative algebra Commutative rings • Integral domain • Integrally closed domain • GCD domain • Unique factorization domain • Principal ideal domain • Euclidean domain • Field • Finite field • Composition ring • Polynomial ring • Formal power series ring Algebraic number theory • Algebraic number field • Ring of integers • Algebraic independence • Transcendental number theory • Transcendence degree p-adic number theory and decimals • Direct limit/Inverse limit • Zero ring $\mathbb {Z} _{1}$ • Integers modulo pn $\mathbb {Z} /p^{n}\mathbb {Z} $ • Prüfer p-ring $\mathbb {Z} (p^{\infty })$ • Base-p circle ring $\mathbb {T} $ • Base-p integers $\mathbb {Z} $ • p-adic rationals $\mathbb {Z} [1/p]$ • Base-p real numbers $\mathbb {R} $ • p-adic integers $\mathbb {Z} _{p}$ • p-adic numbers $\mathbb {Q} _{p}$ • p-adic solenoid $\mathbb {T} _{p}$ Algebraic geometry • Affine variety Noncommutative algebra Noncommutative rings • Division ring • Semiprimitive ring • Simple ring • Commutator Noncommutative algebraic geometry Free algebra Clifford algebra • Geometric algebra Operator algebra The smallest semiring that is not a ring is the two-element Boolean algebra, e.g. with logical disjunction $\lor $ as addition. A motivating example that is neither a ring nor a lattice is the set of natural numbers $\mathbb {N} $ under ordinary addition and multiplication, when including the number zero. Semirings are abundant, because a suitable multiplication operation arises as the function composition of endomorphism over any commutative monoid. The theory of (associative) algebras over commutative rings can be generalized to one over commutative semirings. Algebraic structures Group-like • Group • Semigroup / Monoid • Rack and quandle • Quasigroup and loop • Abelian group • Magma • Lie group Group theory Ring-like • Ring • Rng • Semiring • Near-ring • Commutative ring • Domain • Integral domain • Field • Division ring • Lie ring Ring theory Lattice-like • Lattice • Semilattice • Complemented lattice • Total order • Heyting algebra • Boolean algebra • Map of lattices • Lattice theory Module-like • Module • Group with operators • Vector space • Linear algebra Algebra-like • Algebra • Associative • Non-associative • Composition algebra • Lie algebra • Graded • Bialgebra • Hopf algebra Terminology Some authors call semiring the structure without the requirement for there to be a $0$ or $1$. This makes the analogy between ring and semiring on the one hand and group and semigroup on the other hand work more smoothly. These authors often use rig for the concept defined here.[1][note 1] This originated as a joke, suggesting that rigs are rings without negative elements. (And this is similar to using rng to mean a ring without a multiplicative identity.) The term dioid (for "double monoid") has been used to mean semirings or other structures. It was used by Kuntzman in 1972 to denote a semiring.[2] (It is alternatively sometimes used for naturally ordered semirings,[3] but the term was also used for idempotent subgroups by Baccelli et al. in 1992.[4]) Definition A semiring is a set $R$ equipped with two binary operations $\,+\,$ and $\,\cdot ,\,$ called addition and multiplication, such that:[5][6][7] • $(R,+)$ is a monoid with identity element called $0$: • $(a+b)+c=a+(b+c)$ • $0+a=a=a+0$ • $(R,\,\cdot \,)$ is a monoid with identity element called $1$: • $(a\cdot b)\cdot c=a\cdot (b\cdot c)$ • $1\cdot a=a=a\cdot 1$ • Addition is commutative: • $a+b=b+a$ • Multiplication by the additive identity $0$ annihilates $R$: • $a\cdot 0=0=0\cdot a$ • Multiplication left- and right-distributes over addition: • $a\cdot (b+c)=(a\cdot b)+(a\cdot c)$ • $(b+c)\cdot a=(b\cdot a)+(c\cdot a)$ Explicitly stated, $(R,+)$ is a commutative monoid. Notation The symbol $\cdot $ is usually omitted from the notation; that is, $a\cdot b$ is just written $ab.$ Similarly, an order of operations is conventional, in which $\cdot $ is applied before $+$. That is, $a+b\cdot c$ denotes $a+(b\cdot c)$. For the purpose of disambiguation, one may write $0_{R}$ or $1_{R}$ to emphasize which structure the units at hand belong to. If $x\in R$ is an element of a semiring and $n\in {\mathbb {N} }$, then $n$-times repeated multiplication of $x$ with itself is denoted $x^{n}$, and one similarly writes $x\,n:=x+x+\cdots +x$ for the $n$-times repeated addition. Construction of new semirings The zero ring with underlying set $\{0\}$ is also a semiring, called the trivial semiring. This triviality can be characterized via $0=1$ and so $0\neq 1$ is often silently assumed as if it were an additional axiom. Now given any semiring, there are several ways to define new ones. As noted, the natural numbers ${\mathbb {N} }$ with its arithmetic structure form a semiring. The set $\{x\in R\mid x=0_{R}\lor \exists p.x=p+1_{R}\}$ equipped with the operations inherited from a semiring $R$, is always a sub-semiring of $R$. If $M$ is a commutative monoid, function composition provides the multiplication to form a semiring: The set $\operatorname {End} (M)$ of endomorphisms $M\to M$ forms a semiring, where addition is defined from pointwise addition in $M$. The zero morphism and the identity are the respective neutral elements. If $M=R^{n}$ with $R$ a semiring, we obtain a semiring that can be associated with the square $n\times n$ matrices ${\mathcal {M}}_{n}(R)$ with coefficients in $R$, the matrix semiring using ordinary addition and multiplication rules of matrices. Yet more abstractly, given $n\in {\mathbb {N} }$ and $R$ a semiring, ${\mathcal {M}}_{n}(R)$ is always a semiring also. It is generally non-commutative even if $R$ was commutative. Dorroh extensions: If $R$ is a semiring, then $R\times {\mathbb {N} }$ with pointwise addition and multiplication given by $\langle x,n\rangle \bullet \langle y,m\rangle :=\langle x\cdot y+(x\,m+y\,n),n\cdot m\rangle $ :=\langle x\cdot y+(x\,m+y\,n),n\cdot m\rangle } defines another semiring with mulitplicative unit $1_{R\times {\mathbb {N} }}:=\langle 0_{R},1_{\mathbb {N} }\rangle $. Very similarly, if $N$ is any sub-semiring of $R$, one may also define a semiring on $R\times N$, just by replacing the repeated addition in the formula by multiplication. Indeed, these constructions even work under looser conditions, as the structure $R$ is not actually required to have a multiplicative unit. Zerosumfree semirings are in a sense furthest away from being rings. Given a semiring, one may adjoin a new zero $0'$ to the underlying set and thus obtain such a zerosumfree semiring that also lacks zero divisors. In particular, now $0\cdot 0'=0'$ and the old semiring is actually not a sub-semiring. One may then go on and adjoin new elements "on top" one at a time, while always respecting the zero. These two strategies also work under looser conditions. Sometimes the notations $-\infty $ resp. $+\infty $ are used when performing these constructions. Adjoining a new zero to the trivial semiring, in this way, results in another semiring which may be expressed in terms of the logical connectives of disjunction and conjunction: $\langle \{0,1\},+,\cdot ,\langle 0,1\rangle \rangle =\langle \{\bot ,\top \},\lor ,\land ,\langle \bot ,\top \rangle \rangle $. Consequently, this is the smallest semiring that is not a ring. Explicitly, it violates the ring axioms as $\top \lor P=\top $ for all $P$, i.e. $1$ has no additive inverse. In the self-dual definition, the fault is with $\bot \land P=\bot $. (This is not to be conflated with the ring $\mathbb {Z} _{2}$, whos addition functions as xor $\veebar $.) In the von Neumann model of the naturals, $0_{\omega }:=\{\}$, $1_{\omega }:=\{0_{\omega }\}$ and $2_{\omega }:=\{0_{\omega },1_{\omega }\}={\mathcal {P}}1_{\omega }$. The two-element semiring may be presented in terms of the set theoretic union and intersection as $\langle {\mathcal {P}}1_{\omega },\cup ,\cap ,\langle \{\},1_{\omega }\rangle \rangle $. Now this structure in fact still constitutes a semiring when $1_{\omega }$ is replaced by any set whatsoever. The ideals on a semiring $R$, with their standard operations on subset, form a lattice-ordered, simple and zerosumfree semiring. The ideals of ${\mathcal {M}}_{n}(R)$ are in bijection with the ideals of $R$. The collection of left ideals of $R$ (and likewise the right ideals) also have much of that algebraic structure, except that then $R$ does not function as a two-sided multiplicative identity. If $R$ is a semiring and $A$ is an inhabited set, $A^{}$ denotes the free monoid and the formal polynomials $R[A^{}]$ over its words form another semiring. For small sets, the generating elements are conventionally used to denote the polynomial semiring. For example, in case of a singleton $A=\{X\}$ such that $A^{}=\{\varepsilon ,X,X^{2},X^{3},\dots \}$, one writes $R[X]$. Zerosumfree sub-semirings of $R$ can be used to determine sub-semirings of $R[A^{}]$. Given a set $A$, not necessarily just a singleton, adjoining a default element to the set underlying a semiring $R$ one may define the semiring of partial functions from $A$ to $R$. Given a derivation ${\mathrm {d} }$ on a semirings $R$, another semiring with multiplication "$\bullet $" fulfilling $x\bullet y=y\bullet x+{\mathrm {d} }(y)$ may be established. The above is by no means an exhaustive list of systematic constructions. Derivations Derivations on a semiring $R$ are the maps ${\mathrm {d} }\colon R\to R$ with ${\mathrm {d} }(x+y)={\mathrm {d} }(x)+{\mathrm {d} }(y)$ and ${\mathrm {d} }(x\cdot y)={\mathrm {d} }(x)\cdot y+x\cdot {\mathrm {d} }(y)$. For example, if $E$ is the $2\times 2$ unit matrix and $U={\bigl (}{\begin{smallmatrix}0&1\\0&0\end{smallmatrix}}{\bigr )}$, then the subset of ${\mathcal {M}}_{2}(R)$ given by the matrices $a\,E+b\,U$ with $a,b\in R$ is a semiring with derivation $a\,E+b\,U\mapsto b\,U$. Properties A basic property of semirings is that $1$ is not a left or right zero divisor, and that $1$ but also $0$ squares to itself, i.e. these have $u^{2}=u$. Some notable properties are inherited from the monoid structures: The monoid axioms demand unit existence, and so the set underlying a semiring cannot be empty. Also, the 2-ary predicate $x\leq _{\mathrm {pre} }y$ defined as $\exists d.x+d=y$, here defined for the addition operation, always constitutes the right canonical preorder relation. Reflexivity $y\leq _{\mathrm {pre} }y$ is witnessed by the identity. Further, $0\leq _{\mathrm {pre} }y$ is always valid, and so zero is the least element with respect to this preorder. Considering it for the commutative addition in particular, the distinction of "right" may be disregarded. In the non-negative integers $\mathbb {N} $, for example, this relation is anti-symmetric and strongly connected, and thus in fact a (non-strict) total order. Below, more conditional properties are discussed. Semifields Any field is also a semifield, which in turn is a semiring in which also multiplicative inverses exist. Rings Any field is also a ring, which in turn is a semiring in which also additive inverses exist. Note that a semiring omits such a requirement, i.e., it requires only a commutative monoid, not a commutative group. The extra requirement for a ring itself already implies the existence of a multiplicative zero. This contrast is also why for the theory of semirings, the multiplicative zero must be specified explicitly. Here $-1$, the additive inverse of $1$, squares to $1$. As additive differences $d=y-x$ always exist in a ring, $x\leq _{\mathrm {pre} }y$ is a trivial relation. Commutative semirings A semiring is called a commutative semiring if also the multiplication is commutative.[8] Its axioms can be stated concisely: It consists of two commutative monoids $\langle +,0\rangle $ and $\langle \cdot ,1\rangle $ on one set such that $a\cdot 0=0$ and $a\cdot (b+c)=a\cdot b+a\cdot c$. The center of a semiring is a sub-semiring and being commutative is equivalent to being its own center. The commutative semiring of natural numbers is the initial object among its kind, meaning there is a unique structure preserving map of ${\mathbb {N} }$ into any commutative semiring. The bounded distributive lattices are partially ordered, commutative semirings fulfilling certain algebraic equations relating to distributivity and idempotence. Thus so are their duals. Ordered semirings Notions or order can be defined using strict, non-strict or second-order formulations. Additional properties such as commutativity simplify the axioms. Given a strict total order (also sometimes called linear order, or pseudo-order in a constructive formulation), then by definition, the positive and negative elements fulfill $0<x$ resp. $x<0$. By irreflexivity of a strict order, if $s$ is a left zero divisor, then $s\cdot x<s\cdot y$ is false. The non-negative elements are characterized by $\neg (x<0)$, which is then written $0\leq x$. Generally, the strict total order can be negated to define an associated partial order. The asymmetry of the former manifests as $x<y\to x\leq y$. In fact in classical mathematics the latter is a (non-strict) total order and such that $0\leq x$ implies $x=0\lor 0<x$. Likewise, given any (non-strict) total order, its negation is irreflexive and transitive, and those two properties found together are sometimes called strict quasi-order. Classically this defines a strict total order - indeed strict total order and total order can there be defined in terms of one another. Recall that "$\leq _{\mathrm {pre} }$" defined above is trivial in any ring. The existence of rings that admit a non-trivial non-strict order shows that these need not necessarily coincide with "$\leq _{\mathrm {pre} }$". Additively idempotent semirings A semiring in which every element is an additive idempotent, that is, $x+x=x$ for all elements $x$, is called an (additively) idempotent semiring.[9] Establishing $1+1=1$ suffices. Be aware that sometimes this is just called idempotent semiring, regardless of rules for multiplication. In such a semiring, $x\leq _{\mathrm {pre} }y$ is equivalent to $x+y=y$ and always constitutes a partial order, here now denoted $x\leq y$. In particular, here $x\leq 0\leftrightarrow x=0$. So additively idempotent semirings are zerosumfree and, indeed, the only additively idempotent semiring that has all additive inverses is the trivial ring and so this property is specific to semiring theory. Addition and multiplication respect the ordering in the sense that $x\leq y$ implies $x+t\leq y+t$, and furthermore implies $s\cdot x\leq s\cdot y$ as well as $x\cdot s\leq y\cdot s$, for all $x,y,t$ and $s$. If $R$ is addtively idempotent, then so are the polynomials in $R[X^{}]$. A semiring such that there is a lattice structure on its underlying set is lattice-ordered if the sum coincides with the meet, $x+y=x\lor y$, and the product lies beneath the join $x\cdot y\leq x\land y$. The lattice-ordered semiring of ideals on a semiring is not necessarily distributive with respect to the lattice structure. More strictly than just additive idempotence, a semiring is called simple iff $x+1=1$ for all $x$. Then also $1+1=1$ and $x\leq 1$ for all $x$. Here $1$ then functions akin to an additively infinite element. If $R$ is an additively idempotent semiring, then $\{x\in R\mid x+1=1\}$ with the inherited operations is its simple sub-semiring. An example of an additively idempotent semiring that is not simple is the tropical semiring on ${\mathbb {R} }\cup \{-\infty \}$ with the 2-ary maximum function, with respect to the standard order, as addition. Its simple sub-semiring is trivial. A c-semiring is an idempotent semiring and with addition defined over arbitrary sets. An additively idempotent semiring with idempotent multiplication, $x^{2}=x$, is called additively and multiplicatively idempotent semiring, but sometimes also just idempotent semiring. The commutative, simple semirings with that property are exactly the bounded distributive lattices with unique minimal and maximal element (which then are the units). Heyting algebras are such semirings and the Boolean algebras are a special case. Further, given two bounded distributive lattices, there are constructions resulting in commutative additively-idempotent semirings, which are more complicated than just the direct sum of structures. Number lines In a model of the ring ${\mathbb {R} }$, one can define a non-trivial positivity predicate $0<x$ and a predicate $x<y$ as $0<(y-x)$ that constitutes a strict total order, which fulfills properties such as $\neg (x<0\lor 0<x)\to x=0$, or classically the law of trichotomy. With its standard addition and multiplication, this structure forms the strictly ordered field that is Dedekind-complete. By definition, all first-order properties proven in the theory of the reals are also provable in the decidable theory of the real closed field. For example, here $x<y$ is mutually exclusive with $\exists d.y+d^{2}=x$. But beyond just ordered fields, the four properties listed below are also still valid in many sub-semirings of ${\mathbb {R} }$, including the rationals, the integers, as well as the non-negative parts of each of these structures. In particular, the non-negative reals, the non-negative rationals and the non-negative integers are such a semirings. The first two properties are analogous to the property valid in the idempotent semirings: Translation and scaling respect these ordered rings, in the sense that addition and multiplication in this ring validate • $(x<y)\,\to \,x+t<y+t$ • $(x<y\land 0<s)\,\to \,s\cdot x<s\cdot y$ In particular, $(0<y\land 0<s)\to 0<s\cdot y$ and so squaring of elements preserves positivity. Take note of two more properties that are always valid in a ring. Firstly, trivially $P\,\to \,x\leq _{\mathrm {pre} }y$ for any $P$. In particular, the positive additive difference existence can be expressed as • $(x<y)\,\to \,x\leq _{\mathrm {pre} }y$ Secondly, in the presence of a trichotomous order, the non-zero elements of the additive group are partitioned into positive and negative elements, with the inversion operation moving between them. With $(-1)^{2}=1$, all squares are proven non-negative. Consequently, non-trivial rings have a positive multiplicative unit, • $0<1$ Having discussed a strict order, it follows that $0\neq 1$ and $1\neq 1+1$, etc. Discretely ordered semirings There are a few conflicting notions of discreteness in order theory. Given some strict order on a semiring, one such notion is given by $1$ being positive and covering $0$, i.e. there being no element $x$ between the units, $\neg (0<x\land x<1)$. Now in the present context, an order shall be called discrete if this is fulfilled and, furthermore, all elements of the semiring are non-negative, so that the semiring starts out with the units. Denote by ${\mathsf {PA}}^{-}$ the theory of a commutative, discretly ordered semiring also validating the above four properties relating a strict order with the algebraic structure. All of its models have the model $\mathbb {N} $ as its initial segment and Gödel incompleteness and Tarski undefinability already apply to ${\mathsf {PA}}^{-}$. The non-negative elements of a commutative, discretely ordered ring always validate the axioms of ${\mathsf {PA}}^{-}$. So a slightly more exotic model of the theory is given by the positive elements in the polynomial ring ${\mathbb {Z} }[X]$, with positivity predicate for $p= \sum }_{k=0}^{n}a_{k}X^{k}$ defined in terms of the last non-zero coefficient, $0<p:=(0<a_{n})$, and $p<q:=(0<q-p)$ as above. While ${\mathsf {PA}}^{-}$ proves all $\Sigma _{1}$-sentences that are true about $\mathbb {N} $, beyond this complexity one can find simple such statements that are independent of ${\mathsf {PA}}^{-}$. For example, while $\Pi _{1}$-sentences true about $\mathbb {N} $ are still true for the other model just defined, inspection of the polynomial $X$ demonstrates ${\mathsf {PA}}^{-}$-independence of the $\Pi _{2}$-claim that all numbers are of the form $2q$ or $2q+1$ ("odd or even"). Showing that also ${\mathbb {Z} }[X,Y]/(X^{2}-2Y^{2})$ can be discretely ordered demonstrates that the $\Pi _{1}$-claim $x^{2}\neq 2y^{2}$ for non-zero $x$ ("no rational squared equals $2$") is independent. Likewise, analysis for ${\mathbb {Z} }[X,Y,Z]/(XZ-Y^{2})$ demonstrates independence of some statements about factorization true in $\mathbb {N} $. There are ${\mathsf {PA}}$ characterizations of primality that ${\mathsf {PA}}^{-}$ does not validate for $2$. In the other direction, from any model of ${\mathsf {PA}}^{-}$ one may construct an ordered ring, which then has elements that are negative with respect to the order, that is still discrete the sense that $1$ covers $0$. This is by done adding all additive inverses, i.e. by working towards $\forall x.x\leq _{\mathrm {pre} }0$. The initial ring $\mathbb {Z} $ may of course be defined through pairs of elements from $\mathbb {N} $. Beyond the size of the two-element algebra, no simple semiring starts out with the units. Being discretly ordered also stands in contrast to, e.g., the standard ordering on the semiring of non-negative rationals ${\mathbb {Q} }_{\geq 0}$, which is dense between the units. For another example, ${\mathbb {Z} }[X]/(2X^{2}-1)$ can be ordered, but not discretely so. Natural numbers ${\mathsf {PA}}^{-}$ plus mathematical induction gives a theory equivalent to first-order Peano arithmetic ${\mathsf {PA}}$. The theory is also famously not categorical, but $\mathbb {N} $ is of course the intended model. ${\mathsf {PA}}$ proves that there are no zero divisors and it is zerosumfree and so no model of it is a ring. The standard axiomatization of ${\mathsf {PA}}$ is more concise and the theory uses "$\leq _{\mathrm {pre} }$". However, just removing the potent induction principle from that axiomatization does not leave a workable algebraic theory. Indeed, even Robinson arithmetic ${\mathsf {Q}}$, which removes induction but adds back the predecessor existence postulate, does not prove the monoid axiom $\forall y.(0+y=y)$. Complete semirings A complete semiring is a semiring for which the additive monoid is a complete monoid, meaning that it has an infinitary sum operation $\Sigma _{I}$ for any index set $I$ and that the following (infinitary) distributive laws must hold:[10][11][12] $ \sum }_{i\in I}{\left(a\cdot a_{i}\right)}=a\cdot \left( \sum }_{i\in I}{a_{i}}\right),\qquad \sum }_{i\in I}{\left(a_{i}\cdot a\right)}=\left( \sum }_{i\in I}{a_{i}}\right)\cdot a.$ Examples of a complete semiring are the power set of a monoid under union and the matrix semiring over a complete semiring.[13] For commutative, additively idempotent and simple semirings, this property is related to residuated lattices. Continuous semirings A continuous semiring is similarly defined as one for which the addition monoid is a continuous monoid. That is, partially ordered with the least upper bound property, and for which addition and multiplication respect order and suprema. The semiring $\mathbb {N} \cup \{\infty \}$ with usual addition, multiplication and order extended is a continuous semiring.[14] Any continuous semiring is complete:[10] this may be taken as part of the definition.[13] Star semirings A star semiring (sometimes spelled starsemiring) is a semiring with an additional unary operator ∗,[9][11][15][16] satisfying $a^{}=1+aa^{}=1+a^{}a.$ A Kleene algebra is a star semiring with idempotent addition and some additional axioms. They are important in the theory of formal languages and regular expressions.[11] Complete star semirings In a complete star semiring, the star operator behaves more like the usual Kleene star: for a complete semiring we use the infinitary sum operator to give the usual definition of the Kleene star:[11] $a^{}= \sum }_{j\geq 0}{a^{j}},$ where $a^{j}={\begin{cases}1,&j=0,\\a\cdot a^{j-1}=a^{j-1}\cdot a,&j>0.\end{cases}}$ Note that star semirings are not related to -algebra, where the star operation should instead be thought of as complex conjugation. Conway semiring A Conway semiring is a star semiring satisfying the sum-star and product-star equations:[9][17] ${\begin{aligned}(a+b)^{}&=\left(a^{}b\right)^{}a^{},\\(ab)^{}&=1+a(ba)^{}b.\end{aligned}}$ Every complete star semiring is also a Conway semiring,[18] but the converse does not hold. An example of Conway semiring that is not complete is the set of extended non-negative rational numbers $\mathbb {Q} _{\geq 0}\cup \{\infty \}$ with the usual addition and multiplication (this is a modification of the example with extended non-negative reals given in this section by eliminating irrational numbers).[11] An iteration semiring is a Conway semiring satisfying the Conway group axioms,[9] associated by John Conway to groups in star-semirings.[19] Examples • By definition, any ring and any semifield is also a semiring. • The non-negative elements of a commutative, discretely ordered ring form a commutative, discretely (in the sense defined above) ordered semiring. This includes the non-negative integers $\mathbb {N} $. • Also the non-negative rational numbers as well as the non-negative real numbers form commutative, ordered semirings.[20][21][22] The latter is called probability semiring.[6] Neither are rings or distributive lattices. These examples also have multiplicative inverses. • New semirings can conditionally be constructed from existing ones, as described. The extended natural numbers $\mathbb {N} \cup \{\infty \}$ with addition and multiplication extended so that $0\cdot \infty =0$.[21] • The set of polynomials with natural number coefficients, denoted $\mathbb {N} [x],$ forms a commutative semiring. In fact, this is the free commutative semiring on a single generator $\{x\}.$ Also polynomials with coefficients in other semirings may be defined, as discussed. • The non-negative terminating fractions ${\tfrac {\mathbb {N} }{b^{\mathbb {N} }}}:=\left\{mb^{-n}\mid m,n\in \mathbb {N} \right\}$, in a positional number system to a given base $b\in \mathbb {N} $, form a sub-semiring of the rationals. One has ${\tfrac {\mathbb {N} }{b^{\mathbb {N} }}}\subseteq {\tfrac {\mathbb {N} }{c^{\mathbb {N} }}}$‍ if $b$ divides $c$. For $\|b\|>1.$, the set ${\tfrac {\mathbb {Z} _{0}}{b^{\mathbb {Z} _{0}}}}:={\tfrac {\mathbb {N} }{b^{\mathbb {N} }}}\cup \left(-{\tfrac {\mathbb {N} _{0}}{b^{\mathbb {N} }}}\right)$ is the ring of all terminating fractions to base $b,$ and is dense in $\mathbb {Q} $. • The log semiring on $\mathbb {R} \cup \{\pm \infty \}$ with addition given by $x\oplus y=-\log \left(e^{-x}+e^{-y}\right)$ with multiplication $\,+,\,$ zero element $+\infty ,$ and unit element $0.$[6] Similarly, in the presence of an appropriate order with bottom element, • Tropical semirings are variously defined. The max-plus semiring $\mathbb {R} \cup \{-\infty \}$ is a commutative semiring with $\max(a,b)$ serving as semiring addition (identity $-\infty $) and ordinary addition (identity 0) serving as semiring multiplication. In an alternative formulation, the tropical semiring is $\mathbb {R} \cup \{\infty \},$ and min replaces max as the addition operation.[23] A related version has $\mathbb {R} \cup \{\pm \infty \}$ as the underlying set.[6][10] They are an active area of research, linking algebraic varieties with piecewise linear structures.[24] • The Łukasiewicz semiring: the closed interval $[0,1]$ with addition given by taking the maximum of the arguments ($a+b=\max(a,b)$) and multiplication $ab$ given by $\max(0,a+b-1)$ appears in multi-valued logic.[11] • The Viterbi semiring is also defined over the base set $[0,1]$ and has the maximum as its addition, but its multiplication is the usual multiplication of real numbers. It appears in probabilistic parsing.[11] Note that $\max(x,x)=x$. More regarding additively idempotent semirings, • The set of all ideals of a given semiring form a semiring under addition and multiplication of ideals. • Any bounded, distributive lattice is a commutative, semiring under join and meet. A Boolean algebra is a special case of these. A Boolean ring is also a semiring (indeed, a ring) but it is not idempotent under addition. A Boolean semiring is a semiring isomorphic to a sub-semiring of a Boolean algebra.[20] • The commutative semiring formed by the two-element Boolean algebra and defined by $1+1=1$. It is also called the Boolean semiring.[6][21][22][9] Now given two sets $X$ and $Y,$ binary relations between $X$ and $Y$ correspond to matrices indexed by $X$ and $Y$ with entries in the Boolean semiring, matrix addition corresponds to union of relations, and matrix multiplication corresponds to composition of relations.[25] • Any unital quantale is a semiring under join and multiplication. • A normal skew lattice in a ring $R$ is a semiring for the operations multiplication and nabla, where the latter operation is defined by $a\nabla b=a+b+ba-aba-bab$ More using monoids, • The construction of semirings $\operatorname {End} (M)$ from a commutative monoid $M$ has been described. As noted, give a semiring $R$, the $n\times n$ matrices form another semiring. For example, the matrices with non-negative entries, ${\mathcal {M}}_{n}(\mathbb {N} ),$ form a matrix semiring.[20] • Given an alphabet (finite set) Σ, the set of formal languages over $\Sigma $ (subsets of $\Sigma ^{}$) is a semiring with product induced by string concatenation $L_{1}\cdot L_{2}=\left\{w_{1}w_{2}\mid w_{1}\in L_{1},w_{2}\in L_{2}\right\}$ and addition as the union of languages (that is, ordinary union as sets). The zero of this semiring is the empty set (empty language) and the semiring's unit is the language containing only the empty string.[11] • Generalizing the previous example (by viewing $\Sigma ^{}$ as the free monoid over $\Sigma $), take $M$ to be any monoid; the power set $\wp (M)$ of all subsets of $M$ forms a semiring under set-theoretic union as addition and set-wise multiplication: $U\cdot V=\{u\cdot v\mid u\in U,\ v\in V\}.$[22] • Similarly, if $(M,e,\cdot )$ is a monoid, then the set of finite multisets in $M$ forms a semiring. That is, an element is a function $f\mid M\to \mathbb {N} $; given an element of $M,$ the function tells you how many times that element occurs in the multiset it represents. The additive unit is the constant zero function. The multiplicative unit is the function mapping $e$ to $1,$ and all other elements of $M$ to $0.$ The sum is given by $(f+g)(x)=f(x)+g(x)$ and the product is given by $(fg)(x)=\sum \{f(y)g(z)\mid y\cdot z=x\}.$ Regarding sets and similar abstractions, • Given a set $U,$ the set of binary relations over $U$ is a semiring with addition the union (of relations as sets) and multiplication the composition of relations. The semiring's zero is the empty relation and its unit is the identity relation.[11] These relations correspond to the matrix semiring (indeed, matrix semialgebra) of square matrices indexed by $U$ with entries in the Boolean semiring, and then addition and multiplication are the usual matrix operations, while zero and the unit are the usual zero matrix and identity matrix. • The set of cardinal numbers smaller than any given infinite cardinal form a semiring under cardinal addition and multiplication. The class of all cardinals of an inner model form a (class) semiring under (inner model) cardinal addition and multiplication. • The family of (isomorphism equivalence classes of) combinatorial classes (sets of countably many objects with non-negative integer sizes such that there are finitely many objects of each size) with the empty class as the zero object, the class consisting only of the empty set as the unit, disjoint union of classes as addition, and Cartesian product of classes as multiplication.[26] • Isomorphism classes of objects in any distributive category, under coproduct and product operations, form a semiring known as a Burnside rig.[27] A Burnside rig is a ring if and only if the category is trivial. Semiring of sets See also: Ring of sets § semiring A semiring (of sets)[28] is a (non-empty) collection ${\mathcal {S}}$ of subsets of $X$ such that 1. $\varnothing \in {\mathcal {S}}.$ • If (3) holds, then $\varnothing \in {\mathcal {S}}$ if and only if ${\mathcal {S}}\neq \varnothing .$ 2. If $E,F\in {\mathcal {S}}$ then $E\cap F\in {\mathcal {S}}.$ 3. If $E,F\in {\mathcal {S}}$ then there exists a finite number of mutually disjoint sets $C_{1},\ldots ,C_{n}\in {\mathcal {S}}$ such that $E\setminus F=\bigcup _{i=1}^{n}C_{i}.$ Conditions (2) and (3) together with $S\neq \varnothing $ imply that $\varnothing \in S.$ Such semirings are used in measure theory. An example of a semiring of sets is the collection of half-open, half-closed real intervals $[a,b)\subset \mathbb {R} .$ A semialgebra[29] or elementary family [30] is a collection ${\mathcal {S}}$ of subsets of $X$ satisfying the semiring properties except with (3) replaced with: • If $E\in {\mathcal {S}}$ then there exists a finite number of mutually disjoint sets $C_{1},\ldots ,C_{n}\in {\mathcal {S}}$ such that $X\setminus E=\bigcup _{i=1}^{n}C_{i}.$ This condition is stronger than (3), which can be seen as follows. If ${\mathcal {S}}$ is a semialgebra and $E,F\in {\mathcal {S}}$, then we can write $F^{c}=F_{1}\cup ...\cup F_{n}$ for disjoint $F_{i}\in S$. Then: $E\setminus F=E\cap F^{c}=E\cap (F_{1}\cup ...\cup F_{n})=(E\cap F_{1})\cup ...\cup (E\cap F_{n})$ and every $E\cap F_{i}\in S$ since it is closed under intersection, and disjoint since they are contained in the disjoint $F_{i}$'s. Moreover the condition is strictly stronger: any $S$ that is both a ring and a semialgebra is an algebra, hence any ring that is not an algebra is also not a semialgebra (e.g. the collection of finite sets on an infinite set $X$). Star semirings Several structures mentioned above can be equipped with a star operation. • The aforementioned semiring of binary relations over some base set $U$ in which $R^{}=\bigcup _{n\geq 0}R^{n}$ for all $R\subseteq U\times U.$ This star operation is actually the reflexive and transitive closure of $R$ (that is, the smallest reflexive and transitive binary relation over $U$ containing $R.$).[11] • The semiring of formal languages is also a complete star semiring, with the star operation coinciding with the Kleene star (for sets/languages).[11] • The set of non-negative extended reals $[0,\infty ]$ together with the usual addition and multiplication of reals is a complete star semiring with the star operation given by $a^{}={\tfrac {1}{1-a}}$ for $0\leq a<1$ (that is, the geometric series) and $a^{}=\infty $ for $a\geq 1.$[11] • The Boolean semiring with $0^{}=1^{}=1.$[lower-alpha 1][11] • The semiring on $\mathbb {N} \cup \{\infty \},$ with extended addition and multiplication, and $0^{}=1,a^{*}=\infty $ for $a\geq 1.$[lower-alpha 1][11] Applications The $(\max ,+)$ and $(\min ,+)$ tropical semirings on the reals are often used in performance evaluation on discrete event systems. The real numbers then are the "costs" or "arrival time"; the "max" operation corresponds to having to wait for all prerequisites of an events (thus taking the maximal time) while the "min" operation corresponds to being able to choose the best, less costly choice; and + corresponds to accumulation along the same path. The Floyd–Warshall algorithm for shortest paths can thus be reformulated as a computation over a $(\min ,+)$ algebra. Similarly, the Viterbi algorithm for finding the most probable state sequence corresponding to an observation sequence in a hidden Markov model can also be formulated as a computation over a $(\max ,\times )$ algebra on probabilities. These dynamic programming algorithms rely on the distributive property of their associated semirings to compute quantities over a large (possibly exponential) number of terms more efficiently than enumerating each of them.[31][32] Generalizations A generalization of semirings does not require the existence of a multiplicative identity, so that multiplication is a semigroup rather than a monoid. Such structures are called hemirings[33] or pre-semirings.[34] A further generalization are left-pre-semirings,[35] which additionally do not require right-distributivity (or right-pre-semirings, which do not require left-distributivity). Yet a further generalization are near-semirings: in addition to not requiring a neutral element for product, or right-distributivity (or left-distributivity), they do not require addition to be commutative. Just as cardinal numbers form a (class) semiring, so do ordinal numbers form a near-semiring, when the standard ordinal addition and multiplication are taken into account. However, the class of ordinals can be turned into a semiring by considering the so-called natural (or Hessenberg) operations instead. In category theory, a 2-rig is a category with functorial operations analogous to those of a rig. That the cardinal numbers form a rig can be categorified to say that the category of sets (or more generally, any topos) is a 2-rig. See also • Ring of sets – Family closed under unions and relative complements • Valuation algebra – Algebra describing information processingPages displaying short descriptions of redirect targets Notes 1. For an example see the definition of rig on Proofwiki.org 1. This is a complete star semiring and thus also a Conway semiring.[11] Citations 1. Głazek (2002) p.7 2. Kuntzmann, J. (1972). Théorie des réseaux (graphes) (in French). Paris: Dunod. Zbl 0239.05101. 3. Semirings for breakfast, slide 17 4. Baccelli, François Louis; Olsder, Geert Jan; Quadrat, Jean-Pierre; Cohen, Guy (1992). Synchronization and linearity. An algebra for discrete event systems. Wiley Series on Probability and Mathematical Statistics. Chichester: Wiley. Zbl 0824.93003. 5. Berstel & Perrin (1985), p. 26 6. Lothaire (2005) p.211 7. Sakarovitch (2009) pp.27–28 8. Lothaire (2005) p.212 9. Ésik, Zoltán (2008). "Iteration semirings". In Ito, Masami (ed.). Developments in language theory. 12th international conference, DLT 2008, Kyoto, Japan, September 16–19, 2008. Proceedings. Lecture Notes in Computer Science. Vol. 5257. Berlin: Springer-Verlag. pp. 1–20. doi:10.1007/978-3-540-85780-8_1. ISBN 978-3-540-85779-2. Zbl 1161.68598. 10. Kuich, Werner (2011). "Algebraic systems and pushdown automata". In Kuich, Werner (ed.). Algebraic foundations in computer science. Essays dedicated to Symeon Bozapalidis on the occasion of his retirement. Lecture Notes in Computer Science. Vol. 7020. Berlin: Springer-Verlag. pp. 228–256. ISBN 978-3-642-24896-2. Zbl 1251.68135. 11. Droste, M., & Kuich, W. (2009). Semirings and Formal Power Series. Handbook of Weighted Automata, 3–28. doi:10.1007/978-3-642-01492-5_1, pp. 7-10 12. Kuich, Werner (1990). "ω-continuous semirings, algebraic systems and pushdown automata". In Paterson, Michael S. (ed.). Automata, Languages and Programming: 17th International Colloquium, Warwick University, England, July 16-20, 1990, Proceedings. Lecture Notes in Computer Science. Vol. 443. Springer-Verlag. pp. 103–110. ISBN 3-540-52826-1. 13. Sakaraovich (2009) p.471 14. Ésik, Zoltán; Leiß, Hans (2002). "Greibach normal form in algebraically complete semirings". In Bradfield, Julian (ed.). Computer science logic. 16th international workshop, CSL 2002, 11th annual conference of the EACSL, Edinburgh, Scotland, September 22-25, 2002. Proceedings. Lecture Notes in Computer Science. Vol. 2471. Berlin: Springer-Verlag. pp. 135–150. Zbl 1020.68056. 15. Lehmann, Daniel J. "Algebraic structures for transitive closure." Theoretical Computer Science 4, no. 1 (1977): 59-76. 16. Berstel & Reutenauer (2011) p.27 17. Ésik, Zoltán; Kuich, Werner (2004). "Equational axioms for a theory of automata". In Martín-Vide, Carlos (ed.). Formal languages and applications. Studies in Fuzziness and Soft Computing. Vol. 148. Berlin: Springer-Verlag. pp. 183–196. ISBN 3-540-20907-7. Zbl 1088.68117. 18. Droste, M., & Kuich, W. (2009). Semirings and Formal Power Series. Handbook of Weighted Automata, 3–28. doi:10.1007/978-3-642-01492-5_1, Theorem 3.4 p. 15 19. Conway, J.H. (1971). Regular algebra and finite machines. London: Chapman and Hall. ISBN 0-412-10620-5. Zbl 0231.94041. 20. Guterman, Alexander E. (2008). "Rank and determinant functions for matrices over semirings". In Young, Nicholas; Choi, Yemon (eds.). Surveys in Contemporary Mathematics. London Mathematical Society Lecture Note Series. Vol. 347. Cambridge University Press. pp. 1–33. ISBN 978-0-521-70564-6. ISSN 0076-0552. Zbl 1181.16042. 21. Sakarovitch (2009) p.28 22. Berstel & Reutenauer (2011) p. 4 23. Speyer, David; Sturmfels, Bernd (2009) [2004]. "Tropical Mathematics". Math. Mag. 82 (3): 163–173. arXiv:math/0408099. doi:10.4169/193009809x468760. S2CID 119142649. Zbl 1227.14051. 24. Speyer, David; Sturmfels, Bernd (2009). "Tropical Mathematics". Mathematics Magazine. 82 (3): 163–173. doi:10.1080/0025570X.2009.11953615. ISSN 0025-570X. S2CID 15278805. 25. John C. Baez (6 Nov 2001). "quantum mechanics over a commutative rig". Newsgroup: sci.physics.research. Usenet: [email protected]. Retrieved November 25, 2018. 26. Bard, Gregory V. (2009), Algebraic Cryptanalysis, Springer, Section 4.2.1, "Combinatorial Classes", ff., pp. 30–34, ISBN 9780387887579. 27. Schanuel S.H. (1991) Negative sets have Euler characteristic and dimension. In: Carboni A., Pedicchio M.C., Rosolini G. (eds) Category Theory. Lecture Notes in Mathematics, vol 1488. Springer, Berlin, Heidelberg 28. Noel Vaillant, Caratheodory's Extension, on probability.net. 29. Durrett 2019, pp. 3–4. sfn error: no target: CITEREFDurrett2019 (help) 30. Folland 1999, p. 23. sfn error: no target: CITEREFFolland1999 (help) 31. Pair, Claude (1967), "Sur des algorithmes pour des problèmes de cheminement dans les graphes finis (On algorithms for path problems in finite graphs)", in Rosentiehl (ed.), Théorie des graphes (journées internationales d'études) -- Theory of Graphs (international symposium), Rome (Italy), July 1966: Dunod (Paris) et Gordon and Breach (New York), p. 271{{citation}}: CS1 maint: location (link) 32. Derniame, Jean Claude; Pair, Claude (1971), Problèmes de cheminement dans les graphes (Path Problems in Graphs), Dunod (Paris) 33. Jonathan S. Golan, Semirings and their applications, Chapter 1, p1 34. Michel Gondran, Michel Minoux, Graphs, Dioids, and Semirings: New Models and Algorithms, Chapter 1, Section 4.2, p22 35. Michel Gondran, Michel Minoux, Graphs, Dioids, and Semirings: New Models and Algorithms, Chapter 1, Section 4.1, p20 Bibliography • Derniame, Jean Claude; Pair, Claude (1971), Problèmes de cheminement dans les graphes (Path Problems in Graphs), Dunod (Paris) • François Baccelli, Guy Cohen, Geert Jan Olsder, Jean-Pierre Quadrat, Synchronization and Linearity (online version), Wiley, 1992, ISBN 0-471-93609-X • Golan, Jonathan S., Semirings and their applications. Updated and expanded version of The theory of semirings, with applications to mathematics and theoretical computer science (Longman Sci. Tech., Harlow, 1992, MR1163371. Kluwer Academic Publishers, Dordrecht, 1999. xii+381 pp. ISBN 0-7923-5786-8 MR1746739 • Berstel, Jean; Perrin, Dominique (1985). Theory of codes. Pure and applied mathematics. Vol. 117. Academic Press. ISBN 978-0-12-093420-1. Zbl 0587.68066. • Durrett, Richard (2019). Probability: Theory and Examples (PDF). Cambridge Series in Statistical and Probabilistic Mathematics. Vol. 49 (5th ed.). Cambridge New York, NY: Cambridge University Press. ISBN 978-1-108-47368-2. OCLC 1100115281. Retrieved November 5, 2020. • Lothaire, M. (2005). Applied combinatorics on words. Encyclopedia of Mathematics and Its Applications. Vol. 105. A collective work by Jean Berstel, Dominique Perrin, Maxime Crochemore, Eric Laporte, Mehryar Mohri, Nadia Pisanti, Marie-France Sagot, Gesine Reinert, Sophie Schbath, Michael Waterman, Philippe Jacquet, Wojciech Szpankowski, Dominique Poulalhon, Gilles Schaeffer, Roman Kolpakov, Gregory Koucherov, Jean-Paul Allouche and Valérie Berthé. Cambridge: Cambridge University Press. ISBN 0-521-84802-4. Zbl 1133.68067. • Głazek, Kazimierz (2002). A guide to the literature on semirings and their applications in mathematics and information sciences. With complete bibliography. Dordrecht: Kluwer Academic. ISBN 1-4020-0717-5. Zbl 1072.16040. • Sakarovitch, Jacques (2009). Elements of automata theory. Translated from the French by Reuben Thomas. Cambridge: Cambridge University Press. ISBN 978-0-521-84425-3. Zbl 1188.68177. • Berstel, Jean; Reutenauer, Christophe (2011). Noncommutative rational series with applications. Encyclopedia of Mathematics and Its Applications. Vol. 137. Cambridge: Cambridge University Press. ISBN 978-0-521-19022-0. Zbl 1250.68007. Further reading • Golan, Jonathan S. (2003). Semirings and Affine Equations over Them. Springer Science & Business Media. ISBN 978-1-4020-1358-4. Zbl 1042.16038. • Gondran, Michel; Minoux, Michel (2008). Graphs, Dioids and Semirings: New Models and Algorithms. Operations Research/Computer Science Interfaces Series. Vol. 41. Dordrecht: Springer Science & Business Media. ISBN 978-0-387-75450-5. Zbl 1201.16038. • Grillet, Mireille P. (1970). "Green's relations in a semiring". Port. Math. 29: 181–195. Zbl 0227.16029. • Gunawardena, Jeremy (1998). "An introduction to idempotency". In Gunawardena, Jeremy (ed.). Idempotency. Based on a workshop, Bristol, UK, October 3–7, 1994 (PDF). Cambridge: Cambridge University Press. pp. 1–49. Zbl 0898.16032. • Jipsen, P. (2004). "From semirings to residuated Kleene lattices". Studia Logica. 76 (2): 291–303. doi:10.1023/B:STUD.0000032089.54776.63. S2CID 9946523. Zbl 1045.03049. • Steven Dolan (2013) Fun with Semirings, doi:10.1145/2500365.2500613 Authority control International • FAST National • France • BnF data • Germany • Israel • United States	Wikipedia
Semialgebraic space In mathematics, especially in real algebraic geometry, a semialgebraic space is a space which is locally isomorphic to a semialgebraic set. Definition Let U be an open subset of Rn for some n. A semialgebraic function on U is defined to be a continuous real-valued function on U whose restriction to any semialgebraic set contained in U has a graph which is a semialgebraic subset of the product space Rn×R. This endows Rn with a sheaf ${\mathcal {O}}_{\mathbf {R} ^{n}}$ of semialgebraic functions. (For example, any polynomial mapping between semialgebraic sets is a semialgebraic function, as is the maximum of two semialgebraic functions.) A semialgebraic space is a locally ringed space $(X,{\mathcal {O}}_{X})$ which is locally isomorphic to Rn with its sheaf of semialgebraic functions. See also • Semialgebraic set • Real algebraic geometry • Real closed ring	Wikipedia
Semicircle law The semicircle law may refer to: • The Wigner semicircle distribution, which describes the eigenvalues of a random matrix, or • The Semicircle law (quantum Hall effect), which describes a relationship between components of the macroscopic conductivity tensor.	Wikipedia
Semiperimeter In geometry, the semiperimeter of a polygon is half its perimeter. Although it has such a simple derivation from the perimeter, the semiperimeter appears frequently enough in formulas for triangles and other figures that it is given a separate name. When the semiperimeter occurs as part of a formula, it is typically denoted by the letter s. Motivation: triangles The semiperimeter is used most often for triangles; the formula for the semiperimeter of a triangle with side lengths a, b, c $s={\frac {a+b+c}{2}}.$ Properties In any triangle, any vertex and the point where the opposite excircle touches the triangle partition the triangle's perimeter into two equal lengths, thus creating two paths each of which has a length equal to the semiperimeter. If A, B, B', C' are as shown in the figure, then the segments connecting a vertex with the opposite excircle tangency (AA', BB', CC', shown in red in the diagram) are known as splitters, and ${\begin{aligned}s&=\|AB\|+\|A'B\|=\|AB\|+\|AB'\|=\|AC\|+\|A'C\|\\&=\|AC\|+\|AC'\|=\|BC\|+\|B'C\|=\|BC\|+\|BC'\|.\end{aligned}}$ The three splitters concur at the Nagel point of the triangle. A cleaver of a triangle is a line segment that bisects the perimeter of the triangle and has one endpoint at the midpoint of one of the three sides. So any cleaver, like any splitter, divides the triangle into two paths each of whose length equals the semiperimeter. The three cleavers concur at the center of the Spieker circle, which is the incircle of the medial triangle; the Spieker center is the center of mass of all the points on the triangle's edges. A line through the triangle's incenter bisects the perimeter if and only if it also bisects the area. A triangle's semiperimeter equals the perimeter of its medial triangle. By the triangle inequality, the longest side length of a triangle is less than the semiperimeter. Formulas invoking the semiperimeter For triangles The area A of any triangle is the product of its inradius (the radius of its inscribed circle) and its semiperimeter: $A=rs.$ The area of a triangle can also be calculated from its semiperimeter and side lengths a, b, c using Heron's formula: $A={\sqrt {s\left(s-a\right)\left(s-b\right)\left(s-c\right)}}.$ The circumradius R of a triangle can also be calculated from the semiperimeter and side lengths: $R={\frac {abc}{4{\sqrt {s(s-a)(s-b)(s-c)}}}}.$ This formula can be derived from the law of sines. The inradius is $r={\sqrt {\frac {(s-a)(s-b)(s-c)}{s}}}.$ The law of cotangents gives the cotangents of the half-angles at the vertices of a triangle in terms of the semiperimeter, the sides, and the inradius. The length of the internal bisector of the angle opposite the side of length a is[1] $t_{a}={\frac {2{\sqrt {bcs(s-a)}}}{b+c}}.$ In a right triangle, the radius of the excircle on the hypotenuse equals the semiperimeter. The semiperimeter is the sum of the inradius and twice the circumradius. The area of the right triangle is $(s-a)(s-b)$ where a, b are the legs. For quadrilaterals The formula for the semiperimeter of a quadrilateral with side lengths a, b, c, d is $s={\frac {a+b+c+d}{2}}.$ One of the triangle area formulas involving the semiperimeter also applies to tangential quadrilaterals, which have an incircle and in which (according to Pitot's theorem) pairs of opposite sides have lengths summing to the semiperimeter—namely, the area is the product of the inradius and the semiperimeter: $K=rs.$ The simplest form of Brahmagupta's formula for the area of a cyclic quadrilateral has a form similar to that of Heron's formula for the triangle area: $K={\sqrt {\left(s-a\right)\left(s-b\right)\left(s-c\right)\left(s-d\right)}}.$ Bretschneider's formula generalizes this to all convex quadrilaterals: $K={\sqrt {(s-a)(s-b)(s-c)(s-d)-abcd\cdot \cos ^{2}\left({\frac {\alpha +\gamma }{2}}\right)}},$ in which α and γ are two opposite angles. The four sides of a bicentric quadrilateral are the four solutions of a quartic equation parametrized by the semiperimeter, the inradius, and the circumradius. Regular polygons The area of a convex regular polygon is the product of its semiperimeter and its apothem. Circles The semiperimeter of a circle, also called the semicircumference, is directly proportional to its radius r: $s=\pi \cdot r.\!$ The constant of proportionality is the number pi, π. See also • Semidiameter References 1. Johnson, Roger A. (2007). Advanced Euclidean Geometry. Mineola, New York: Dover. p. 70. ISBN 9780486462370. External links • Weisstein, Eric W. "Semiperimeter". MathWorld.	Wikipedia
Semiconductor luminescence equations The semiconductor luminescence equations (SLEs)[1][2] describe luminescence of semiconductors resulting from spontaneous recombination of electronic excitations, producing a flux of spontaneously emitted light. This description established the first step toward semiconductor quantum optics because the SLEs simultaneously includes the quantized light–matter interaction and the Coulomb-interaction coupling among electronic excitations within a semiconductor. The SLEs are one of the most accurate methods to describe light emission in semiconductors and they are suited for a systematic modeling of semiconductor emission ranging from excitonic luminescence to lasers. Due to randomness of the vacuum-field fluctuations, semiconductor luminescence is incoherent whereas the extensions of the SLEs include[2] the possibility to study resonance fluorescence resulting from optical pumping with coherent laser light. At this level, one is often interested to control and access higher-order photon-correlation effects, distinct many-body states, as well as light–semiconductor entanglement. Such investigations are the basis of realizing and developing the field of quantum-optical spectroscopy which is a branch of quantum optics. Starting point The derivation of the SLEs starts from a system Hamiltonian that fully includes many-body interactions, quantized light field, and quantized light–matter interaction. Like almost always in many-body physics, it is most convenient to apply the second-quantization formalism. For example, a light field corresponding to frequency $\omega $ is then described through Boson creation and annihilation operators ${\hat {B}}_{\omega }^{\dagger }$ and ${\hat {B}}_{\omega }$, respectively, where the "hat" over $B$ signifies the operator nature of the quantity. The operator-combination ${\hat {B}}_{\omega }^{\dagger }\,{\hat {B}}_{\omega }$ determines the photon-number operator. When the photon coherences, here the expectation value $\langle {\hat {B}}_{\omega }\rangle $, vanish and the system becomes quasistationary, semiconductors emit incoherent light spontaneously, commonly referred to as luminescence (L). (This is the underlying principle behind light-emitting diodes.) The corresponding luminescence flux is proportional to the temporal change in photon number,[2] $\mathrm {L} (\omega )={\frac {\partial }{\partial t}}\langle {\hat {B}}_{\omega }^{\dagger }{\hat {B}}_{\omega }\rangle =2\,\mathrm {Re} \left[\sum _{\mathbf {k} }{\mathcal {F}}_{\omega }^{\star }\,\Pi _{\mathbf {k} ,\omega }\right]\,.$ As a result, the luminescence becomes directly generated by a photon-assisted electron–hole recombination, $\Pi _{\mathbf {k} ,\omega }\equiv \Delta \langle {\hat {B}}_{\omega }^{\dagger }{\hat {P}}_{\mathbf {k} }\rangle $ that describes a correlated emission of a photon $({\hat {B}}_{\omega }^{\dagger })$ when an electron with wave vector $\mathbf {k} $ recombines with a hole, i.e., an electronic vacancy. Here, ${\hat {P}}_{\mathbf {k} }$ determines the corresponding electron–hole recombination operator defining also the microscopic polarization within semiconductor. Therefore, $\Pi _{\mathbf {k} ,\omega }$ can also be viewed as photon-assisted polarization. Many electron–hole pairs contribute to the photon emission at frequency $\omega $; the explicit $\Delta $ notation within $\Pi _{\mathbf {k} ,\omega }$ denotes that the correlated part of the expectation value $\langle {\hat {B}}_{\omega }^{\dagger }P_{\mathbf {k} }\rangle $ is constructed using the cluster-expansion approach. The quantity ${\mathcal {F}}_{\omega }$ contains the dipole-matrix element for interband transition, light-mode's mode function, and vacuum-field amplitude. Principal structure of SLEs In general, the SLEs includes all single- and two-particle correlations needed to compute the luminescence spectrum self-consistently. More specifically, a systematic derivation produces a set of equations involving photon-number-like correlations Semiconductor luminescence equations (photon-number-like correlations) $\mathrm {i} \hbar {\frac {\partial }{\partial t}}\Delta \langle {\hat {B}}_{\omega }^{\dagger }{\hat {B}}_{\omega '}\rangle =(\hbar \omega '-\hbar \omega )\,\Delta \langle {\hat {B}}_{\omega }^{\dagger }{\hat {B}}_{\omega '}\rangle +\mathrm {i} \sum \limits _{\mathbf {k} }\left[{\mathcal {F}}_{\omega '}^{\star }\Pi _{\mathbf {k} ,\omega }+{\mathcal {F}}_{\omega }\Pi _{\mathbf {k} ,\omega '}^{\star }\right]$ whose diagonal form reduces to the luminescence formula above. The dynamics of photon-assisted correlations follows from Semiconductor luminescence equations (photon-assisted correlations) $\mathrm {i} \hbar {\frac {\partial }{\partial t}}\Pi _{\mathbf {k} ,\omega }=\left({\tilde {\epsilon }}_{\mathbf {k} }-\hbar \omega \right)\Pi _{\mathbf {k} ,\omega }+\Omega _{\mathbf {k} ,\omega }^{\mathrm {spont} }-\left(1-f_{\mathbf {k} }^{e}-f_{\mathbf {k} }^{h}\right)\left[\Omega _{\omega }^{\mathrm {stim} }+\sum \limits _{\mathbf {k'} }V_{\mathbf {k} -\mathbf {k} '}\,\Pi _{\mathbf {k'} ,\omega }\right]+T[\Pi ]$ where the first contribution, ${\tilde {\epsilon }}_{\mathbf {k} }$, contains the Coulomb-renormalized single-particle energy that is determined by the bandstructure of the solid. The Coulomb renormalization are identical to those that appear in the semiconductor Bloch equations (SBEs), showing that all photon-assisted polarizations are coupled with each other via the unscreened Coulomb-interaction $V_{\mathbf {k} }$. The three-particle correlations that appear are indicated symbolically via the $T[\Pi ]$ contributions – they introduce excitation-induced dephasing, screening of Coulomb interaction, and additional highly correlated contributions such as phonon-sideband emission. The explicit form of a spontaneous-emission source $\Omega _{\mathbf {k} ,\omega }^{\mathrm {spont} }$ and a stimulated contribution $\Omega _{\omega }^{\mathrm {stim} }$ are discussed below. The excitation level of a semiconductor is characterized by electron and hole occupations, $f_{\mathbf {k} }^{e}$ and $f_{\mathbf {k} }^{h}$, respectively. They modify the $\Pi _{\mathbf {k} ,\omega }$ via the Coulomb renormalizations and the Pauli-blocking factor, $\left(1-f_{\mathbf {k} }^{e}-f_{\mathbf {k} }^{h}\right)$. These occupations are changed by spontaneous recombination of electrons and holes, yielding $\left.{\frac {\partial }{\partial t}}f_{\mathbf {k} }^{e}\right\|_{\mathrm {L} }=\left.{\frac {\partial }{\partial t}}f_{\mathbf {k} }^{h}\right\|_{\mathrm {L} }=-2\,\mathrm {Re} \left[\sum _{\omega }{\mathcal {F}}_{\omega }^{\star }\,\Pi _{\mathbf {k} ,\omega }\right]\,.$ In its full form, the occupation dynamics also contains Coulomb-correlation terms.[2] It is straight forward to verify that the photon-assisted recombination[3][4][5] destroys as many electron–hole pairs as it creates photons because due to the general conservation law ${\frac {\partial }{\partial t}}\sum _{\omega }\langle {\hat {B}}_{\omega }^{\dagger }{\hat {B}}_{\omega }\rangle =-{\frac {\partial }{\partial t}}\sum _{\mathbf {k} }f_{\mathbf {k} }^{e}$. Besides the terms already described above, the photon-assisted polarization dynamics contains a spontaneous-emission source $\Omega _{\mathbf {k} ,\omega }^{\mathrm {spont} }=\mathrm {i} {\mathcal {F}}_{\omega }{\Bigl (}f_{\mathbf {k} }^{e}f_{\mathbf {k} }^{h}+\sum _{\mathbf {k'} }c_{\mathrm {X} }^{\mathbf {k} ,\mathbf {k'} }{\Bigr )}\,.$ Intuitively, $f_{\mathbf {k} }^{e}\,f_{\mathbf {k} }^{h}$ describes the probability to find electron and hole with same $\mathbf {k} $ when electrons and holes are uncorrelated, i.e., plasma. Such form is to be expected for a probability of two uncorrelated events to occur simultaneously at a desired $\mathbf {k} $ value. The possibility to have truly correlated electron–hole pairs is defined by a two-particle correlation $c_{\mathrm {X} }^{\mathbf {k} ,\mathbf {k'} }$; the corresponding probability is directly proportional to the correlation. In practice, $c_{\mathrm {X} }^{\mathbf {k} ,\mathbf {k'} }$ becomes large when electron–hole pairs are bound as excitons via their mutual Coulomb attraction. Nevertheless, both the presence of electron–hole plasma and excitons can equivalently induce the spontaneous-emission source. As the semiconductor emits light spontaneously, the luminescence is further altered by a stimulated contribution $\Delta \Omega _{\omega }^{\mathrm {stim} }=\mathrm {i} \sum _{\omega }{\mathcal {F}}_{\omega '}\,\Delta \langle {\hat {B}}_{\omega }^{\dagger }{\hat {B}}_{\omega '}\rangle $ that is particularly important when describing spontaneous emission in semiconductor microcavities and lasers because then spontaneously emitted light can return to the emitter (i.e., the semiconductor), either stimulating or inhibiting further spontaneous-emission processes. This term is also responsible for the Purcell effect. To complete the SLEs, one must additionally solve the quantum dynamics of exciton correlations ${\begin{aligned}\mathrm {i} \hbar {\frac {\partial }{\partial t}}c_{\mathrm {X} }^{\mathbf {k} ,\mathbf {k'} }=&\left({\tilde {\epsilon }}_{\mathbf {k} }-{\tilde {\epsilon }}_{\mathbf {k'} }\right)\,c_{\mathrm {X} }^{\mathbf {k} ,\mathbf {k'} }+S_{\mathrm {X} }^{\mathbf {k} ,\mathbf {k'} }\\&+{\Bigl (}1-f_{\mathbf {k'} }^{e}-f_{\mathbf {k'} }^{h}{\Bigr )}\sum _{\mathbf {l} }V_{\mathbf {l} -\mathbf {k} '}\,c_{\mathrm {X} }^{\mathbf {k} ,\mathbf {l} }-{\Bigl (}1-f_{\mathbf {k} }^{e}-f_{\mathbf {k} }^{h}{\Bigr )}\sum _{\mathbf {l} }V_{\mathbf {l} -\mathbf {k} '}\,c_{\mathrm {X} }^{\mathbf {l} ,\mathbf {k'} }\\&+D_{\mathrm {X,\,rest} }^{\mathbf {k} ,\mathbf {k'} }+T_{\mathrm {X} }^{\mathbf {k} ,\mathbf {k'} }\,.\end{aligned}}$ The first line contains the Coulomb-renormalized kinetic energy of electron–hole pairs and the second line defines a source that results from a Boltzmann-type in- and out-scattering of two electrons and two holes due to the Coulomb interaction. The second line contains the main Coulomb sums that correlate electron–hole pairs into excitons whenever the excitation conditions are suitable. The remaining two- and three-particle correlations are presented symbolically by $D_{\mathrm {X,\,rest} }^{\mathbf {k} ,\mathbf {k'} }$ and $T_{\mathrm {X} }^{\mathbf {k} ,\mathbf {k'} }$, respectively.[2][6] Interpretation and consequences Microscopically, the luminescence processes are initiated whenever the semiconductor is excited because at least the electron and hole distributions, that enter the spontaneous-emission source, are nonvanishing. As a result, $\Omega _{\mathbf {k} ,\omega }^{\mathrm {spont} }$ is finite and it drives the photon-assisted processes for all those $\mathbf {k} $ values that correspond to the excited states. This means that $\Pi _{\omega ,\mathbf {k} }$ is simultaneously generated for many $\mathbf {k} $ values. Since the Coulomb interaction couples $\Pi _{\omega ,\mathbf {k} }$ with all $\mathbf {k} $ values, the characteristic transition energy follows from the exciton energy, not the bare kinetic energy of an electron–hole pair. More mathematically, the homogeneous part of the $\Pi _{\omega ,\mathbf {k} }$ dynamics has eigenenergies that are defined by the generalized Wannier equation not the free-carrier energies. For low electron–hole densities, the Wannier equation produces a set of bound eigenstates which define the exciton resonances. Therefore, $\Pi _{\omega ,\mathbf {k} }$ shows a discrete set of exciton resonances regardless which many-body state initiated the emission through the spontaneous-emission source. These resonances are directly transferred to excitonic peaks in the luminescence itself. This yields an unexpected consequence; the excitonic resonance can equally well originate from an electron–hole plasma or the presence of excitons.[7] At first, this consequence of SLEs seems counterintuitive because in few-particle picture an unbound electron–hole pair cannot recombine and release energy corresponding to the exciton resonance because that energy is well below the energy an unbound electron–hole pair possesses. However, the excitonic plasma luminescence is a genuine many-body effect where plasma emits collectively to the exciton resonance. Namely, when a high number of electronic states participate in the emission of a single photon, one can always distribute the energy of initial many-body state between the one photon at exciton energy and remaining many-body state (with one electron–hole pair removed) without violating the energy conservation. The Coulomb interaction mediates such energy rearrangements very efficiently. A thorough analysis of energy and many-body state rearrangement is given in Ref.[2] In general, excitonic plasma luminescence explains many nonequilibrium emission properties observed in present-day semiconductor luminescence experiments. In fact, the dominance of excitonic plasma luminescence has been measured in both quantum-well[8] and quantum-dot systems.[9] Only when excitons are present abundantly, the role of excitonic plasma luminescence can be ignored. Connections and generalizations Structurally, the SLEs resemble the semiconductor Bloch equations (SBEs) if the $\Pi _{\omega ,\mathbf {k} }$ are compared with the microscopic polarization within the SBEs. As the main difference, $\Pi _{\omega ,\mathbf {k} }$ also has a photon index $\omega $, its dynamics is driven spontaneously, and it is directly coupled to three-particle correlations. Technically, the SLEs are more difficult to solve numerically than the SBEs due to the additional $\omega $ degree of freedom. However, the SLEs often are the only (at low carrier densities) or more convenient (lasing regime) to compute luminescence accurately. Furthermore, the SLEs not only yield a full predictability without the need for phenomenological approximations but they also can be used as a systematic starting point for more general investigations such as laser design[10][11] and disorder studies.[12] The presented SLEs discussion does not specify the dimensionality or the band structure of the system studied. As one analyses a specified system, one often has to explicitly include the electronic bands involved, the dimensionality of wave vectors, photon, and exciton center-of-mass momentum. Many explicit examples are given in Refs.[6][13] for quantum-well and quantum-wire systems, and in Refs.[4][14][15] for quantum-dot systems. Semiconductors also can show several resonances well below the fundamental exciton resonance when phonon-assisted electron–hole recombination takes place. These processes are describable by three-particle correlations (or higher) where photon, electron–hole pair, and a lattice vibration, i.e., a phonon, become correlated. The dynamics of phonon-assisted correlations are similar to the phonon-free SLEs. Like for the excitonic luminescence, also excitonic phonon sidebands can equally well be initiated by either electron–hole plasma or excitons.[16] The SLEs can also be used as a systematic starting point for semiconductor quantum optics.[2][17][18] As a first step, one also includes two-photon absorption correlations, $\Delta \langle {\hat {B}}_{\omega }{\hat {B}}_{\omega '}\rangle $, and then continues toward higher-order photon-correlation effects. This approach can be applied to analyze the resonance fluorescence effects and to realize and understand the quantum-optical spectroscopy. See also • Coherent effects in semiconductor optics • Cluster-expansion approach • Photoluminescence • Quantum-optical spectroscopy • Elliott formula • Semiconductor laser theory References 1. Kira, M.; Jahnke, F.; Koch, S.; Berger, J.; Wick, D.; Nelson, T.; Khitrova, G.; Gibbs, H. (1997). "Quantum Theory of Nonlinear Semiconductor Microcavity Luminescence Explaining "Boser" Experiments". Physical Review Letters 79 (25): 5170–5173. doi:10.1103/PhysRevLett.79.5170 2. Kira, M.; Koch, S. W. (2011). Semiconductor Quantum Optics. Cambridge University Press. ISBN 978-0521875097. 3. Li, Jianzhong (2007). "Laser cooling of semiconductor quantum wells: Theoretical framework and strategy for deep optical refrigeration by luminescence upconversion". Physical Review B 75 (15). doi:10.1103/PhysRevB.75.155315 4. Berstermann, T.; Auer, T.; Kurtze, H.; Schwab, M.; Yakovlev, D.; Bayer, M.; Wiersig, J.; Gies, C.; Jahnke, F.; Reuter, D.; Wieck, A. (2007). "Systematic study of carrier correlations in the electron–hole recombination dynamics of quantum dots". Physical Review B 76 (16). doi:10.1103/PhysRevB.76.165318 5. Shuvayev, V.; Kuskovsky, I.; Deych, L.; Gu, Y.; Gong, Y.; Neumark, G.; Tamargo, M.; Lisyansky, A. (2009). "Dynamics of the radiative recombination in cylindrical nanostructures with type-II band alignment". Physical Review B 79 (11). doi:10.1103/PhysRevB.79.115307 6. Kira, M.; Koch, S.W. (2006). "Many-body correlations and excitonic effects in semiconductor spectroscopy". Progress in Quantum Electronics 30 (5): 155–296. doi:10.1016/j.pquantelec.2006.12.002 7. Kira, M.; Jahnke, F.; Koch, S. (1998). "Microscopic Theory of Excitonic Signatures in Semiconductor Photoluminescence". Physical Review Letters 81 (15): 3263–3266. doi:10.1103/PhysRevLett.81.3263 8. Chatterjee, S.; Ell, C.; Mosor, S.; Khitrova, G.; Gibbs, H.; Hoyer, W.; Kira, M.; Koch, S.; Prineas, J.; Stolz, H. (2004). "Excitonic Photoluminescence in Semiconductor Quantum Wells: Plasma versus Excitons". Physical Review Letters 92 (6). doi:10.1103/PhysRevLett.92.067402 9. Schwab, M.; Kurtze, H.; Auer, T.; Berstermann, T.; Bayer, M.; Wiersig, J.; Baer, N.; Gies, C.; Jahnke, F.; Reithmaier, J.; Forchel, A.; Benyoucef, M.; Michler, P. (2006). "Radiative emission dynamics of quantum dots in a single cavity micropillar". Physical Review B 74 (4). doi:10.1103/PhysRevB.74.045323 10. Hader, J.; Moloney, J. V.; Koch, S. W. (2006). "Influence of internal fields on gain and spontaneous emission in InGaN quantum wells". Applied Physics Letters 89 (17): 171120. doi:10.1063/1.2372443 11. Hader, J.; Hardesty, G.; Wang, T.; Yarborough, M. J.; Kaneda, Y.; Moloney, J. V.; Kunert, B.; Stolz, W. et al. (2010). "Predictive Microscopic Modeling of VECSELs". IEEE J. Quantum Electron. 46: 810. doi:10.1109/JQE.2009.2035714 12. Rubel, O.; Baranovskii, S. D.; Hantke, K.; Heber, J. D.; Koch, J.; Thomas, P. V.; Marshall, J. M.; Stolz, W. et al. (2005). "On the theoretical description of luminescence in disordered quantum structures". J. Optoelectron. Adv. M. 7 (1): 115. 13. Imhof, S.; Bückers, C.; Thränhardt, A.; Hader, J.; Moloney, J. V.; Koch, S. W. (2008). "Microscopic theory of the optical properties of Ga(AsBi)/GaAs quantum wells". Semicond. Sci. Technol. 23 (12): 125009. 14. Feldtmann, T.; Schneebeli, L.; Kira, M.; Koch, S. (2006). "Quantum theory of light emission from a semiconductor quantum dot". Physical Review B 73 (15). doi:10.1103/PhysRevB.73.155319 15. Baer, N.; Gies, C.; Wiersig, J.; Jahnke, F. (2006). "Luminescence of a semiconductor quantum dot system". The European Physical Journal B 50 (3): 411–418. doi:10.1140/epjb/e2006-00164-3 16. Böttge, C. N.; Kira, M.; Koch, S. W. (2012). "Enhancement of the phonon-sideband luminescence in semiconductor microcavities". Physical Review B 85 (9). doi:10.1103/PhysRevB.85.094301 17. Gies, Christopher; Wiersig, Jan; Jahnke, Frank (2008). "Output Characteristics of Pulsed and Continuous-Wave-Excited Quantum-Dot Microcavity Lasers". Physical Review Letters 101 (6). doi:10.1103/PhysRevLett.101.067401 18. Aßmann, M.; Veit, F.; Bayer, M.; Gies, C.; Jahnke, F.; Reitzenstein, S.; Höfling, S.; Worschech, L. et al. (2010). "Ultrafast tracking of second-order photon correlations in the emission of quantum-dot microresonator lasers". Physical Review B 81 (16). doi:10.1103/PhysRevB.81.165314 Further reading • Jahnke, F. (2012). Quantum Optics with Semiconductor Nanostructures. Woodhead Publishing Ltd. ISBN 978-0857092328. • Kira, M.; Koch, S. W. (2011). Semiconductor Quantum Optics. Cambridge University Press. ISBN 978-0521875097. • Haug, H.; Koch, S. W. (2009). Quantum Theory of the Optical and Electronic Properties of Semiconductors (5th ed.). World Scientific. p. 216. ISBN 978-9812838841. • Piprek, J. (2007). Nitride Semiconductor Devices: Principles and Simulation. Wiley-VCH Verlag GmbH \& Co. KGaA. ISBN 978-3527406678. • Klingshirn, C. F. (2006). Semiconductor Optics. Springer. ISBN 978-3540383451. • Kalt, H.; Hetterich, M. (2004). Optics of Semiconductors and Their Nanostructures. Springer. ISBN 978-3540383451.	Wikipedia
Convection–diffusion equation The convection–diffusion equation is a combination of the diffusion and convection (advection) equations, and describes physical phenomena where particles, energy, or other physical quantities are transferred inside a physical system due to two processes: diffusion and convection. Depending on context, the same equation can be called the advection–diffusion equation, drift–diffusion equation,[1] or (generic) scalar transport equation.[2] Equation General The general equation is[3][4] ${\frac {\partial c}{\partial t}}=\mathbf {\nabla } \cdot (D\mathbf {\nabla } c)-\mathbf {\nabla } \cdot (\mathbf {v} c)+R$ where • c is the variable of interest (species concentration for mass transfer, temperature for heat transfer), • D is the diffusivity (also called diffusion coefficient), such as mass diffusivity for particle motion or thermal diffusivity for heat transport, • v is the velocity field that the quantity is moving with. It is a function of time and space. For example, in advection, c might be the concentration of salt in a river, and then v would be the velocity of the water flow as a function of time and location. Another example, c might be the concentration of small bubbles in a calm lake, and then v would be the velocity of bubbles rising towards the surface by buoyancy (see below) depending on time and location of the bubble. For multiphase flows and flows in porous media, v is the (hypothetical) superficial velocity. • R describes sources or sinks of the quantity c. For example, for a chemical species, R > 0 means that a chemical reaction is creating more of the species, and R < 0 means that a chemical reaction is destroying the species. For heat transport, R > 0 might occur if thermal energy is being generated by friction. • ∇ represents gradient and ∇ ⋅ represents divergence. In this equation, ∇c represents concentration gradient. Understanding the terms involved The right-hand side of the equation is the sum of three contributions. • The first, ∇ ⋅ (D∇c), describes diffusion. Imagine that c is the concentration of a chemical. When concentration is low somewhere compared to the surrounding areas (e.g. a local minimum of concentration), the substance will diffuse in from the surroundings, so the concentration will increase. Conversely, if concentration is high compared to the surroundings (e.g. a local maximum of concentration), then the substance will diffuse out and the concentration will decrease. The net diffusion is proportional to the Laplacian (or second derivative) of concentration if the diffusivity D is a constant. • The second contribution, −∇ ⋅ (vc), describes convection (or advection). Imagine standing on the bank of a river, measuring the water's salinity (amount of salt) each second. Upstream, somebody dumps a bucket of salt into the river. A while later, you would see the salinity suddenly rise, then fall, as the zone of salty water passes by. Thus, the concentration at a given location can change because of the flow. • The final contribution, R, describes the creation or destruction of the quantity. For example, if c is the concentration of a molecule, then R describes how the molecule can be created or destroyed by chemical reactions. R may be a function of c and of other parameters. Often there are several quantities, each with its own convection–diffusion equation, where the destruction of one quantity entails the creation of another. For example, when methane burns, it involves not only the destruction of methane and oxygen but also the creation of carbon dioxide and water vapor. Therefore, while each of these chemicals has its own convection–diffusion equation, they are coupled together and must be solved as a system of simultaneous differential equations. Common simplifications In a common situation, the diffusion coefficient is constant, there are no sources or sinks, and the velocity field describes an incompressible flow (i.e., it has zero divergence). Then the formula simplifies to:[5][6][7] ${\frac {\partial c}{\partial t}}=D\nabla ^{2}c-\mathbf {v} \cdot \nabla c.$ In this form, the convection–diffusion equation combines both parabolic and hyperbolic partial differential equations. In non-interacting material, D=0 (for example, when temperature is close to absolute zero, dilute gas has almost zero mass diffusivity), hence the transport equation is simply: ${\frac {\partial c}{\partial t}}+\mathbf {v} \cdot \nabla c=0.$ Using Fourier transform in both temporal and spatial domain (that is, with integral kernel $e^{j\omega t+j\mathbf {k} \cdot \mathbf {x} }$), its characteristic equation can be obtained: $j\omega {\tilde {c}}+\mathbf {v} \cdot j\mathbf {k} {\tilde {c}}=0\rightarrow \omega =-\mathbf {k} \cdot \mathbf {v} ,$ which gives the general solution: $c=f(\mathbf {x} -\mathbf {v} t),$ where $f$ is any differentiable scalar function. This is the basis of temperature measurement for near Bose–Einstein condensate[8] via time of flight method.[9] Stationary version The stationary convection–diffusion equation describes the steady-state behavior of a convective-diffusive system. In a steady state, ∂c/∂t = 0, so the formula is: $0=\nabla \cdot (D\nabla c)-\nabla \cdot (\mathbf {v} c)+R.$ Derivation The convection–diffusion equation can be derived in a straightforward way[4] from the continuity equation, which states that the rate of change for a scalar quantity in a differential control volume is given by flow and diffusion into and out of that part of the system along with any generation or consumption inside the control volume: ${\frac {\partial c}{\partial t}}+\nabla \cdot \mathbf {j} =R,$ where j is the total flux and R is a net volumetric source for c. There are two sources of flux in this situation. First, diffusive flux arises due to diffusion. This is typically approximated by Fick's first law: $\mathbf {j} _{\text{diff}}=-D\nabla c$ i.e., the flux of the diffusing material (relative to the bulk motion) in any part of the system is proportional to the local concentration gradient. Second, when there is overall convection or flow, there is an associated flux called advective flux: $\mathbf {j} _{\text{adv}}=\mathbf {v} c$ The total flux (in a stationary coordinate system) is given by the sum of these two: $\mathbf {j} =\mathbf {j} _{\text{diff}}+\mathbf {j} _{\text{adv}}=-D\nabla c+\mathbf {v} c.$ Plugging into the continuity equation: ${\frac {\partial c}{\partial t}}+\nabla \cdot \left(-D\nabla c+\mathbf {v} c\right)=R.$ Complex mixing phenomena In general, D, v, and R may vary with space and time. In cases in which they depend on concentration as well, the equation becomes nonlinear, giving rise to many distinctive mixing phenomena such as Rayleigh–Bénard convection when v depends on temperature in the heat transfer formulation and reaction–diffusion pattern formation when R depends on concentration in the mass transfer formulation. Velocity in response to a force In some cases, the average velocity field v exists because of a force; for example, the equation might describe the flow of ions dissolved in a liquid, with an electric field pulling the ions in some direction (as in gel electrophoresis). In this situation, it is usually called the drift–diffusion equation or the Smoluchowski equation,[1] after Marian Smoluchowski who described it in 1915[10] (not to be confused with the Einstein–Smoluchowski relation or Smoluchowski coagulation equation). Typically, the average velocity is directly proportional to the applied force, giving the equation:[11][12] ${\frac {\partial c}{\partial t}}=\nabla \cdot (D\nabla c)-\nabla \cdot \left(\zeta ^{-1}\mathbf {F} c\right)+R$ where F is the force, and ζ characterizes the friction or viscous drag. (The inverse ζ−1 is called mobility.) Derivation of Einstein relation When the force is associated with a potential energy F = −∇U (see conservative force), a steady-state solution to the above equation (i.e. 0 = R = ∂c/∂t) is: $c\propto \exp \left(-D^{-1}\zeta ^{-1}U\right)$ (assuming D and ζ are constant). In other words, there are more particles where the energy is lower. This concentration profile is expected to agree with the Boltzmann distribution (more precisely, the Gibbs measure). From this assumption, the Einstein relation can be proven:[12] $D\zeta =k_{\mathrm {B} }T.$ Smoluchowski convection-diffusion equation The Smoluchowski convective-diffusion equation is a stochastic (Smoluchowski) diffusion equation with an additional convective flow-field,[13] ${\frac {\partial c}{\partial t}}=\nabla \cdot (D\nabla c)-\mathbf {\nabla } \cdot (\mathbf {v} c)-\nabla \cdot \left(\zeta ^{-1}\mathbf {F} c\right)$ In this case, the force F describes the conservative interparticle interaction force between two colloidal particles or the intermolecular interaction force between two molecules in the fluid, and it is unrelated to the externally imposed flow velocity v. The steady-state version of this equation is the basis to provide a description of the pair distribution function (which can be identified with c) of colloidal suspensions under shear flows.[13] An approximate solution to the steady-state version of this equation has been found using the method of matched asymptotic expansions.[14] This solution provides a theory for the transport-controlled reaction rate of two molecules in a shear flow, and also provides a way to extend the DLVO theory of colloidal stability to colloidal systems subject to shear flows (e.g. in microfluidics, chemical reactors, environmental flows). The full solution to the steady-state equation, obtained using the method of matched asymptotic expansions, has been developed by Alessio Zaccone and L. Banetta to compute the pair distribution function of Lennard-Jones interacting particles in shear flow[15] and subsequently extended to compute the pair distribution function of charge-stabilized (Yukawa or Debye–Hückel) colloidal particles in shear flows.[16] As a stochastic differential equation The convection–diffusion equation (with no sources or drains, R = 0) can be viewed as a stochastic differential equation, describing random motion with diffusivity D and bias v. For example, the equation can describe the Brownian motion of a single particle, where the variable c describes the probability distribution for the particle to be in a given position at a given time. The reason the equation can be used that way is because there is no mathematical difference between the probability distribution of a single particle, and the concentration profile of a collection of infinitely many particles (as long as the particles do not interact with each other). The Langevin equation describes advection, diffusion, and other phenomena in an explicitly stochastic way. One of the simplest forms of the Langevin equation is when its "noise term" is Gaussian; in this case, the Langevin equation is exactly equivalent to the convection–diffusion equation.[12] However, the Langevin equation is more general.[12] Numerical solution Main article: Numerical solution of the convection–diffusion equation The convection–diffusion equation can only rarely be solved with a pen and paper. More often, computers are used to numerically approximate the solution to the equation, typically using the finite element method. For more details and algorithms see: Numerical solution of the convection–diffusion equation. Similar equations in other contexts The convection–diffusion equation is a relatively simple equation describing flows, or alternatively, describing a stochastically-changing system. Therefore, the same or similar equation arises in many contexts unrelated to flows through space. • It is formally identical to the Fokker–Planck equation for the velocity of a particle. • It is closely related to the Black–Scholes equation and other equations in financial mathematics.[17] • It is closely related to the Navier–Stokes equations, because the flow of momentum in a fluid is mathematically similar to the flow of mass or energy. The correspondence is clearest in the case of an incompressible Newtonian fluid, in which case the Navier–Stokes equation is: ${\frac {\partial \mathbf {M} }{\partial t}}=\mu \nabla ^{2}\mathbf {M} -\mathbf {v} \cdot \nabla \mathbf {M} +(\mathbf {f} -\nabla P)$ where M is the momentum of the fluid (per unit volume) at each point (equal to the density ρ multiplied by the velocity v), μ is viscosity, P is fluid pressure, and f is any other body force such as gravity. In this equation, the term on the left-hand side describes the change in momentum at a given point; the first term on the right describes the diffusion of momentum by viscosity; the second term on the right describes the advective flow of momentum; and the last two terms on the right describes the external and internal forces which can act as sources or sinks of momentum. In semiconductor physics In semiconductor physics, this equation is called the drift–diffusion equation. The word "drift" is related to drift current and drift velocity. The equation is normally written:[18] ${\begin{aligned}{\frac {\mathbf {J} _{n}}{-q}}&=-D_{n}\nabla n-n\mu _{n}\mathbf {E} \\{\frac {\mathbf {J} _{p}}{q}}&=-D_{p}\nabla p+p\mu _{p}\mathbf {E} \\{\frac {\partial n}{\partial t}}&=-\nabla \cdot {\frac {\mathbf {J} _{n}}{-q}}+R\\{\frac {\partial p}{\partial t}}&=-\nabla \cdot {\frac {\mathbf {J} _{p}}{q}}+R\end{aligned}}$ where • n and p are the concentrations (densities) of electrons and holes, respectively, • q > 0 is the elementary charge, • Jn and Jp are the electric currents due to electrons and holes respectively, • Jn/−q and Jp/q are the corresponding "particle currents" of electrons and holes respectively, • R represents carrier generation and recombination (R > 0 for generation of electron-hole pairs, R < 0 for recombination.) • E is the electric field vector • $\mu _{n}$ and $\mu _{p}$ are electron and hole mobility. The diffusion coefficient and mobility are related by the Einstein relation as above: ${\begin{aligned}D_{n}&={\frac {\mu _{n}k_{\mathrm {B} }T}{q}},\\D_{p}&={\frac {\mu _{p}k_{\mathrm {B} }T}{q}},\end{aligned}}$ where kB is the Boltzmann constant and T is absolute temperature. The drift current and diffusion current refer separately to the two terms in the expressions for J, namely: ${\begin{aligned}{\frac {\mathbf {J} _{n,{\text{drift}}}}{-q}}&=-n\mu _{n}\mathbf {E} ,\\{\frac {\mathbf {J} _{p,{\text{drift}}}}{q}}&=p\mu _{p}\mathbf {E} ,\\{\frac {\mathbf {J} _{n,{\text{diff}}}}{-q}}&=-D_{n}\nabla n,\\{\frac {\mathbf {J} _{p,{\text{diff}}}}{q}}&=-D_{p}\nabla p.\end{aligned}}$ This equation can be solved together with Poisson's equation numerically.[19] An example of results of solving the drift diffusion equation is shown on the right. When light shines on the center of semiconductor, carriers are generated in the middle and diffuse towards two ends. The drift–diffusion equation is solved in this structure and electron density distribution is displayed in the figure. One can see the gradient of carrier from center towards two ends. See also • Advanced Simulation Library • Conservation equations • Incompressible Navier–Stokes equations • Nernst–Planck equation • Double diffusive convection • Natural convection • Buckley–Leverett equation References 1. Chandrasekhar (1943). "Stochastic Problems in Physics and Astronomy". Rev. Mod. Phys. 15 (1): 1. Bibcode:1943RvMP...15....1C. doi:10.1103/RevModPhys.15.1. See equation (312) 2. Baukal; Gershtein; Li, eds. (2001). Computational Fluid Dynamics in Industrial Combustion. CRC Press. p. 67. ISBN 0-8493-2000-3 – via Google Books. 3. Stocker, Thomas (2011). Introduction to Climate Modelling. Berlin: Springer. p. 57. ISBN 978-3-642-00772-9 – via Google Books. 4. Socolofsky, Scott A.; Jirka, Gerhard H. "Advective Diffusion Equation" (PDF). Lecture notes. Archived from the original (PDF) on June 25, 2010. Retrieved April 18, 2012. 5. Bejan A (2004). Convection Heat Transfer. 6. Bird, Stewart, Lightfoot (1960). Transport Phenomena.{{cite book}}: CS1 maint: multiple names: authors list (link) 7. Probstein R (1994). Physicochemical Hydrodynamics. 8. Ketterle, W.; Durfee, D. S.; Stamper-Kurn, D. M. (1999-04-01). "Making, probing and understanding Bose-Einstein condensates". arXiv:cond-mat/9904034. 9. Brzozowski, Tomasz M; Maczynska, Maria; Zawada, Michal; Zachorowski, Jerzy; Gawlik, Wojciech (2002-01-14). "Time-of-flight measurement of the temperature of cold atoms for short trap-probe beam distances". Journal of Optics B: Quantum and Semiclassical Optics. 4 (1): 62–66. Bibcode:2002JOptB...4...62B. doi:10.1088/1464-4266/4/1/310. ISSN 1464-4266. S2CID 67796405. 10. Smoluchowski, M. v. (1915). "Über Brownsche Molekularbewegung unter Einwirkung äußerer Kräfte und den Zusammenhang mit der verallgemeinerten Diffusionsgleichung" (PDF). Ann. Phys. 4. Folge. 353 (48): 1103–1112. Bibcode:1915AnP...353.1103S. doi:10.1002/andp.19163532408. 11. "Smoluchowski Diffusion Equation" (PDF). 12. Doi & Edwards (1988). The Theory of Polymer Dynamics. pp. 46–52. ISBN 978-0-19-852033-7 – via Google Books. 13. Dhont, J. K. G. (1996). An Introduction to the Dynamics of Colloids. Elsevier. p. 195. ISBN 0-444-82009-4 – via Google Books. 14. Zaccone, A.; Gentili, D.; Wu, H.; Morbidelli, M. (2009). "Theory of activated-rate processes under shear with application to shear-induced aggregation of colloids". Physical Review E. 80 (5): 051404. arXiv:0906.4879. Bibcode:2009PhRvE..80e1404Z. doi:10.1103/PhysRevE.80.051404. hdl:2434/653702. PMID 20364982. S2CID 22763509. 15. Banetta, L.; Zaccone, A. (2019). "Radial distribution function of Lennard-Jones fluids in shear flows from intermediate asymptotics". Physical Review E. 99 (5): 052606. arXiv:1901.05175. Bibcode:2019PhRvE..99e2606B. doi:10.1103/PhysRevE.99.052606. PMID 31212460. S2CID 119011235. 16. Banetta, L.; Zaccone, A. (2020). "Pair correlation function of charge-stabilized colloidal systems under sheared conditions". Colloid and Polymer Science. 298 (7): 761–771. arXiv:2006.00246. doi:10.1007/s00396-020-04609-4. 17. Arabas, S.; Farhat, A. (2020). "Derivative pricing as a transport problem: MPDATA solutions to Black-Scholes-type equations". J. Comput. Appl. Math. 373: 112275. arXiv:1607.01751. doi:10.1016/j.cam.2019.05.023. S2CID 128273138. 18. Hu, Yue (2015). "Simulation of a partially depleted absorber (PDA) photodetector". Optics Express. 23 (16): 20402–20417. Bibcode:2015OExpr..2320402H. doi:10.1364/OE.23.020402. hdl:11603/11470. PMID 26367895. 19. Hu, Yue (2014). "Modeling sources of nonlinearity in a simple pin photodetector". Journal of Lightwave Technology. 32 (20): 3710–3720. Bibcode:2014JLwT...32.3710H. CiteSeerX 10.1.1.670.2359. doi:10.1109/JLT.2014.2315740. S2CID 9882873. Further reading • Sewell, Granville (1988). The Numerical Solution of Ordinary and Partial Differential Equations. Academic Press. ISBN 0-12-637475-9.	Wikipedia
Connected relation In mathematics, a relation on a set is called connected or complete or total if it relates (or "compares") all distinct pairs of elements of the set in one direction or the other while it is called strongly connected if it relates all pairs of elements. As described in the terminology section below, the terminology for these properties is not uniform. This notion of "total" should not be confused with that of a total relation in the sense that for all $x\in X$ there is a $y\in X$ so that $x\mathrel {R} y$ (see serial relation). Transitive binary relations Symmetric Antisymmetric Connected Well-founded Has joins Has meets Reflexive Irreflexive Asymmetric Total, Semiconnex Anti- reflexive Equivalence relation Y ✗ ✗ ✗ ✗ ✗ Y ✗ ✗ Preorder (Quasiorder) ✗ ✗ ✗ ✗ ✗ ✗ Y ✗ ✗ Partial order ✗ Y ✗ ✗ ✗ ✗ Y ✗ ✗ Total preorder ✗ ✗ Y ✗ ✗ ✗ Y ✗ ✗ Total order ✗ Y Y ✗ ✗ ✗ Y ✗ ✗ Prewellordering ✗ ✗ Y Y ✗ ✗ Y ✗ ✗ Well-quasi-ordering ✗ ✗ ✗ Y ✗ ✗ Y ✗ ✗ Well-ordering ✗ Y Y Y ✗ ✗ Y ✗ ✗ Lattice ✗ Y ✗ ✗ Y Y Y ✗ ✗ Join-semilattice ✗ Y ✗ ✗ Y ✗ Y ✗ ✗ Meet-semilattice ✗ Y ✗ ✗ ✗ Y Y ✗ ✗ Strict partial order ✗ Y ✗ ✗ ✗ ✗ ✗ Y Y Strict weak order ✗ Y ✗ ✗ ✗ ✗ ✗ Y Y Strict total order ✗ Y Y ✗ ✗ ✗ ✗ Y Y Symmetric Antisymmetric Connected Well-founded Has joins Has meets Reflexive Irreflexive Asymmetric Definitions, for all $a,b$ and $S\neq \varnothing :$ :} ${\begin{aligned}&aRb\\\Rightarrow {}&bRa\end{aligned}}$ ${\begin{aligned}aRb{\text{ and }}&bRa\\\Rightarrow a={}&b\end{aligned}}$ ${\begin{aligned}a\neq {}&b\Rightarrow \\aRb{\text{ or }}&bRa\end{aligned}}$ ${\begin{aligned}\min S\\{\text{exists}}\end{aligned}}$ ${\begin{aligned}a\vee b\\{\text{exists}}\end{aligned}}$ ${\begin{aligned}a\wedge b\\{\text{exists}}\end{aligned}}$ $aRa$ ${\text{not }}aRa$ ${\begin{aligned}aRb\Rightarrow \\{\text{not }}bRa\end{aligned}}$ Y indicates that the column's property is always true the row's term (at the very left), while ✗ indicates that the property is not guaranteed in general (it might, or might not, hold). For example, that every equivalence relation is symmetric, but not necessarily antisymmetric, is indicated by Y in the "Symmetric" column and ✗ in the "Antisymmetric" column, respectively. All definitions tacitly require the homogeneous relation $R$ be transitive: for all $a,b,c,$ if $aRb$ and $bRc$ then $aRc.$ A term's definition may require additional properties that are not listed in this table. Connectedness features prominently in the definition of total orders: a total (or linear) order is a partial order in which any two elements are comparable; that is, the order relation is connected. Similarly, a strict partial order that is connected is a strict total order. A relation is a total order if and only if it is both a partial order and strongly connected. A relation is a strict total order if, and only if, it is a strict partial order and just connected. A strict total order can never be strongly connected (except on an empty domain). Formal definition A relation $R$ on a set $X$ is called connected when for all $x,y\in X,$ ${\text{ if }}x\neq y{\text{ then }}x\mathrel {R} y\quad {\text{or}}\quad y\mathrel {R} x,$ or, equivalently, when for all $x,y\in X,$ $x\mathrel {R} y\quad {\text{or}}\quad y\mathrel {R} x\quad {\text{or}}\quad x=y.$ A relation with the property that for all $x,y\in X,$ $x\mathrel {R} y\quad {\text{or}}\quad y\mathrel {R} x$ is called strongly connected.[1][2][3] Terminology The main use of the notion of connected relation is in the context of orders, where it is used to define total, or linear, orders. In this context, the property is often not specifically named. Rather, total orders are defined as partial orders in which any two elements are comparable.[4][5] Thus, total is used more generally for relations that are connected or strongly connected.[6] However, this notion of "total relation" must be distinguished from the property of being serial, which is also called total. Similarly, connected relations are sometimes called complete,[7] although this, too, can lead to confusion: The universal relation is also called complete,[8] and "complete" has several other meanings in order theory. Connected relations are also called connex[9][10] or said to satisfy trichotomy[11] (although the more common definition of trichotomy is stronger in that exactly one of the three options $x\mathrel {R} y,y\mathrel {R} x,x=y$ must hold). When the relations considered are not orders, being connected and being strongly connected are importantly different properties. Sources which define both then use pairs of terms such as weakly connected and connected,[12] complete and strongly complete,[13] total and complete,[6] semiconnex and connex,[14] or connex and strictly connex,[15] respectively, as alternative names for the notions of connected and strongly connected as defined above. Characterizations Let $R$ be a homogeneous relation. The following are equivalent:[14] • $R$ is strongly connected; • $U\subseteq R\cup R^{\top }$; • ${\overline {R}}\subseteq R^{\top }$; • ${\overline {R}}$ is asymmetric, where $U$ is the universal relation and $R^{\top }$ is the converse relation of $R.$ The following are equivalent:[14] • $R$ is connected; • ${\overline {I}}\subseteq R\cup R^{\top }$; • ${\overline {R}}\subseteq R^{\top }\cup I$; • ${\overline {R}}$ is antisymmetric, where ${\overline {I}}$ is the complementary relation of the identity relation $I$ and $R^{\top }$ is the converse relation of $R.$ Introducing progressions, Russell invoked the axiom of connection: Whenever a series is originally given by a transitive asymmetrical relation, we can express connection by the condition that any two terms of our series are to have the generating relation. — Bertrand Russell, The Principles of Mathematics, page 239 Properties • The edge relation[note 1] $E$ of a tournament graph $G$ is always a connected relation on the set of $G$'s vertices. • If a strongly connected relation is symmetric, it is the universal relation. • A relation is strongly connected if, and only if, it is connected and reflexive.[proof 1] • A connected relation on a set $X$ cannot be antitransitive, provided $X$ has at least 4 elements.[16] On a 3-element set $\{a,b,c\},$ for example, the relation $\{(a,b),(b,c),(c,a)\}$ has both properties. • If $R$ is a connected relation on $X,$ then all, or all but one, elements of $X$ are in the range of $R.$[proof 2] Similarly, all, or all but one, elements of $X$ are in the domain of $R.$ Notes 1. Defined formally by $vEw$ if a graph edge leads from vertex $v$ to vertex $w$ Proofs 1. For the only if direction, both properties follow trivially. — For the if direction: when $x\neq y,$ then $x\mathrel {R} y\lor y\mathrel {R} x$ follows from connectedness; when $x=y,$ $x\mathrel {R} y$ follows from reflexivity. 2. If $x,y\in X\setminus \operatorname {ran} (R),$ then $x\mathrel {R} y$ and $y\mathrel {R} x$ are impossible, so $x=y$ follows from connectedness. References 1. Clapham, Christopher; Nicholson, James (2014-09-18). "connected". The Concise Oxford Dictionary of Mathematics. Oxford University Press. ISBN 978-0-19-967959-1. Retrieved 2021-04-12. 2. Nievergelt, Yves (2015-10-13). Logic, Mathematics, and Computer Science: Modern Foundations with Practical Applications. Springer. p. 182. ISBN 978-1-4939-3223-8. 3. Causey, Robert L. (1994). Logic, Sets, and Recursion. Jones & Bartlett Learning. ISBN 0-86720-463-X., p. 135 4. Paul R. Halmos (1968). Naive Set Theory. Princeton: Nostrand. Here: Ch.14. Halmos gives the names of reflexivity, anti-symmetry, and transitivity, but not of connectedness. 5. Patrick Cousot (1990). "Methods and Logics for Proving Programs". In Jan van Leeuwen (ed.). Formal Models and Semantics. Handbook of Theoretical Computer Science. Vol. B. Elsevier. pp. 841–993. ISBN 0-444-88074-7. Here: Sect.6.3, p.878 6. Aliprantis, Charalambos D.; Border, Kim C. (2007-05-02). Infinite Dimensional Analysis: A Hitchhiker's Guide. Springer. ISBN 978-3-540-32696-0., p. 6 7. Makinson, David (2012-02-27). Sets, Logic and Maths for Computing. Springer. ISBN 978-1-4471-2500-6., p. 50 8. Whitehead, Alfred North; Russell, Bertrand (1910). Principia Mathematica. Cambridge: Cambridge University Press.{{cite book}}: CS1 maint: date and year (link) 9. Wall, Robert E. (1974). Introduction to Mathematical Linguistics. Prentice-Hall. page 114. 10. Carl Pollard. "Relations and Functions" (PDF). Ohio State University. Retrieved 2018-05-28. Page 7. 11. Kunen, Kenneth (2009). The Foundations of Mathematics. College Publications. ISBN 978-1-904987-14-7. p. 24 12. Fishburn, Peter C. (2015-03-08). The Theory of Social Choice. Princeton University Press. p. 72. ISBN 978-1-4008-6833-9. 13. Roberts, Fred S. (2009-03-12). Measurement Theory: Volume 7: With Applications to Decisionmaking, Utility, and the Social Sciences. Cambridge University Press. ISBN 978-0-521-10243-8. page 29 14. Schmidt, Gunther; Ströhlein, Thomas (1993). Relations and Graphs: Discrete Mathematics for Computer Scientists. Berlin: Springer. ISBN 978-3-642-77970-1. 15. Ganter, Bernhard; Wille, Rudolf (2012-12-06). Formal Concept Analysis: Mathematical Foundations. Springer Science & Business Media. ISBN 978-3-642-59830-2. p. 86 16. Jochen Burghardt (Jun 2018). Simple Laws about Nonprominent Properties of Binary Relations (Technical Report). arXiv:1806.05036. Bibcode:2018arXiv180605036B. Lemma 8.2, p.8. Order theory • Topics • Glossary • Category Key concepts • Binary relation • Boolean algebra • Cyclic order • Lattice • Partial order • Preorder • Total order • Weak ordering Results • Boolean prime ideal theorem • Cantor–Bernstein theorem • Cantor's isomorphism theorem • Dilworth's theorem • Dushnik–Miller theorem • Hausdorff maximal principle • Knaster–Tarski theorem • Kruskal's tree theorem • Laver's theorem • Mirsky's theorem • Szpilrajn extension theorem • Zorn's lemma Properties & Types (list) • Antisymmetric • Asymmetric • Boolean algebra • topics • Completeness • Connected • Covering • Dense • Directed • (Partial) Equivalence • Foundational • Heyting algebra • Homogeneous • Idempotent • Lattice • Bounded • Complemented • Complete • Distributive • Join and meet • Reflexive • Partial order • Chain-complete • Graded • Eulerian • Strict • Prefix order • Preorder • Total • Semilattice • Semiorder • Symmetric • Total • Tolerance • Transitive • Well-founded • Well-quasi-ordering (Better) • (Pre) Well-order Constructions • Composition • Converse/Transpose • Lexicographic order • Linear extension • Product order • Reflexive closure • Series-parallel partial order • Star product • Symmetric closure • Transitive closure Topology & Orders • Alexandrov topology & Specialization preorder • Ordered topological vector space • Normal cone • Order topology • Order topology • Topological vector lattice • Banach • Fréchet • Locally convex • Normed Related • Antichain • Cofinal • Cofinality • Comparability • Graph • Duality • Filter • Hasse diagram • Ideal • Net • Subnet • Order morphism • Embedding • Isomorphism • Order type • Ordered field • Ordered vector space • Partially ordered • Positive cone • Riesz space • Upper set • Young's lattice	Wikipedia
Semicubical parabola In mathematics, a cuspidal cubic or semicubical parabola is an algebraic plane curve that has an implicit equation of the form $y^{2}-a^{2}x^{3}=0$ (with a ≠ 0) in some Cartesian coordinate system. Solving for y leads to the explicit form $y=\pm ax^{\frac {3}{2}},$ which imply that every real point satisfies x ≥ 0. The exponent explains the term semicubical parabola. (A parabola can be described by the equation y = ax2.) Solving the implicit equation for x yields a second explicit form $x=\left({\frac {y}{a}}\right)^{\frac {2}{3}}.$ The parametric equation $\quad x=t^{2},\quad y=at^{3}$ can also be deduced from the implicit equation by putting $ t={\frac {y}{ax}}.$ [1] The semicubical parabolas have a cuspidal singularity; hence the name of cuspidal cubic. The arc length of the curve was calculated by the English mathematician William Neile and published in 1657 (see section History).[2] Properties of semicubical parabolas Similarity Any semicubical parabola $(t^{2},at^{3})$ is similar to the semicubical unit parabola $(u^{2},u^{3})$. Proof: The similarity $(x,y)\rightarrow (a^{2}x,a^{2}y)$ (uniform scaling) maps the semicubical parabola $(t^{2},at^{3})$ onto the curve $((at)^{2},(at)^{3})=(u^{2},u^{3})$ with $u=at$. Singularity The parametric representation $(t^{2},at^{3})$ is regular except at point $(0,0)$. At point $(0,0)$ the curve has a singularity (cusp). The proof follows from the tangent vector $(2t,3t^{2})$. Only for $t=0$ this vector has zero length. Tangents Differentiating the semicubical unit parabola $y=\pm x^{3/2}$ one gets at point $(x_{0},y_{0})$ of the upper branch the equation of the tangent: $y={\frac {\sqrt {x_{0}}}{2}}\left(3x-x_{0}\right).$ This tangent intersects the lower branch at exactly one further point with coordinates [3] $\left({\frac {x_{0}}{4}},-{\frac {y_{0}}{8}}\right).$ (Proving this statement one should use the fact, that the tangent meets the curve at $(x_{0},y_{0})$ twice.) Arclength Determining the arclength of a curve $(x(t),y(t))$ one has to solve the integral $ \int {\sqrt {x'(t)^{2}+y'(t)^{2}}}\;dt.$ For the semicubical parabola $(t^{2},at^{3}),\;0\leq t\leq b,$ one gets $\int _{0}^{b}{\sqrt {x'(t)^{2}+y'(t)^{2}}}\;dt=\int _{0}^{b}t{\sqrt {4+9a^{2}t^{2}}}\;dt=\cdots =\left[{\frac {1}{27a^{2}}}\left(4+9a^{2}t^{2}\right)^{\frac {3}{2}}\right]_{0}^{b}\;.$ (The integral can be solved by the substitution $u=4+9a^{2}t^{2}$.) Example: For a = 1 (semicubical unit parabola) and b = 2, which means the length of the arc between the origin and point (4,8), one gets the arc length 9.073. Evolute of the unit parabola The evolute of the parabola $(t^{2},t)$ is a semicubical parabola shifted by 1/2 along the x-axis: $ \left({\frac {1}{2}}+t^{2},{\frac {4}{{\sqrt {3}}^{3}}}t^{3}\right).$ Polar coordinates In order to get the representation of the semicubical parabola $(t^{2},at^{3})$ in polar coordinates, one determines the intersection point of the line $y=mx$ with the curve. For $m\neq 0$ there is one point different from the origin: $ \left({\frac {m^{2}}{a^{2}}},{\frac {m^{3}}{a^{2}}}\right).$ This point has distance $ {\frac {m^{2}}{a^{2}}}{\sqrt {1+m^{2}}}$ from the origin. With $m=\tan \varphi $ and $\sec ^{2}\varphi =1+\tan ^{2}\varphi $ ( see List of identities) one gets [4] $r=\left({\frac {\tan \varphi }{a}}\right)^{2}\sec \varphi \;,\quad -{\frac {\pi }{2}}<\varphi <{\frac {\pi }{2}}.$ Relation between a semicubical parabola and a cubic function Mapping the semicubical parabola $(t^{2},t^{3})$ by the projective map $ (x,y)\rightarrow \left({\frac {x}{y}},{\frac {1}{y}}\right)$ (involutoric perspectivity with axis $y=1$ and center $(0,-1)$) yields $ \left({\frac {1}{t}},{\frac {1}{t^{3}}}\right),$ hence the cubic function $y=x^{3}.$ The cusp (origin) of the semicubical parabola is exchanged with the point at infinity of the y-axis. This property can be derived, too, if one represents the semicubical parabola by homogeneous coordinates: In equation (A) the replacement $x={\tfrac {x_{1}}{x_{3}}},\;y={\tfrac {x_{2}}{x_{3}}}$ (the line at infinity has equation $x_{3}=0$.) and the multiplication by $x_{3}^{3}$ is performed. One gets the equation of the curve • in homogeneous coordinates: $x_{3}x_{2}^{2}-x_{1}^{3}=0.$ Choosing line $x_{\color {red}2}=0$ as line at infinity and introducing $x={\tfrac {x_{1}}{x_{2}}},\;y={\tfrac {x_{3}}{x_{2}}}$ yields the (affine) curve $y=x^{3}.$ Isochrone curve An additional defining property of the semicubical parabola is that it is an isochrone curve, meaning that a particle following its path while being pulled down by gravity travels equal vertical intervals in equal time periods. In this way it is related to the tautochrone curve, for which particles at different starting points always take equal time to reach the bottom, and the brachistochrone curve, the curve that minimizes the time it takes for a falling particle to travel from its start to its end. History The semicubical parabola was discovered in 1657 by William Neile who computed its arc length. Although the lengths of some other non-algebraic curves including the logarithmic spiral and cycloid had already been computed (that is, those curves had been rectified), the semicubical parabola was the first algebraic curve (excluding the line and circle) to be rectified.[1] References 1. Pickover, Clifford A. (2009), "The Length of Neile's Semicubical Parabola", The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics, Sterling Publishing Company, Inc., p. 148, ISBN 9781402757969. 2. August Pein: Die semicubische oder Neil'sche Parabel, ihre Sekanten und Tangenten ,p.2 3. August Pein: Die semicubische oder Neil'sche Parabel, ihre Sekanten und Tangenten ,p.26 4. August Pein: Die semicubische oder Neil'sche Parabel, ihre Sekanten und Tangenten ,p. 10 • August Pein: Die semicubische oder Neil'sche Parabel, ihre Sekanten und Tangenten , 1875, Dissertation • Clifford A. Pickover: The Length of Neile's Semicubical Parabola External links • O'Connor, John J.; Robertson, Edmund F., "Neile's Semi-cubical Parabola", MacTutor History of Mathematics Archive, University of St Andrews	Wikipedia
Decidability (logic) In logic, a true/false decision problem is decidable if there exists an effective method for deriving the correct answer. Zeroth-order logic (propositional logic) is decidable, whereas first-order and higher-order logic are not. Logical systems are decidable if membership in their set of logically valid formulas (or theorems) can be effectively determined. A theory (set of sentences closed under logical consequence) in a fixed logical system is decidable if there is an effective method for determining whether arbitrary formulas are included in the theory. Many important problems are undecidable, that is, it has been proven that no effective method for determining membership (returning a correct answer after finite, though possibly very long, time in all cases) can exist for them. Decidability of a logical system Each logical system comes with both a syntactic component, which among other things determines the notion of provability, and a semantic component, which determines the notion of logical validity. The logically valid formulas of a system are sometimes called the theorems of the system, especially in the context of first-order logic where Gödel's completeness theorem establishes the equivalence of semantic and syntactic consequence. In other settings, such as linear logic, the syntactic consequence (provability) relation may be used to define the theorems of a system. A logical system is decidable if there is an effective method for determining whether arbitrary formulas are theorems of the logical system. For example, propositional logic is decidable, because the truth-table method can be used to determine whether an arbitrary propositional formula is logically valid. First-order logic is not decidable in general; in particular, the set of logical validities in any signature that includes equality and at least one other predicate with two or more arguments is not decidable.[1] Logical systems extending first-order logic, such as second-order logic and type theory, are also undecidable. The validities of monadic predicate calculus with identity are decidable, however. This system is first-order logic restricted to those signatures that have no function symbols and whose relation symbols other than equality never take more than one argument. Some logical systems are not adequately represented by the set of theorems alone. (For example, Kleene's logic has no theorems at all.) In such cases, alternative definitions of decidability of a logical system are often used, which ask for an effective method for determining something more general than just validity of formulas; for instance, validity of sequents, or the consequence relation {(Г, A) \| Г ⊧ A} of the logic. Decidability of a theory A theory is a set of formulas, often assumed to be closed under logical consequence. Decidability for a theory concerns whether there is an effective procedure that decides whether the formula is a member of the theory or not, given an arbitrary formula in the signature of the theory. The problem of decidability arises naturally when a theory is defined as the set of logical consequences of a fixed set of axioms. There are several basic results about decidability of theories. Every (non-paraconsistent) inconsistent theory is decidable, as every formula in the signature of the theory will be a logical consequence of, and thus a member of, the theory. Every complete recursively enumerable first-order theory is decidable. An extension of a decidable theory may not be decidable. For example, there are undecidable theories in propositional logic, although the set of validities (the smallest theory) is decidable. A consistent theory that has the property that every consistent extension is undecidable is said to be essentially undecidable. In fact, every consistent extension will be essentially undecidable. The theory of fields is undecidable but not essentially undecidable. Robinson arithmetic is known to be essentially undecidable, and thus every consistent theory that includes or interprets Robinson arithmetic is also (essentially) undecidable. Examples of decidable first-order theories include the theory of real closed fields, and Presburger arithmetic, while the theory of groups and Robinson arithmetic are examples of undecidable theories. Some decidable theories Some decidable theories include (Monk 1976, p. 234):[2] • The set of first-order logical validities in the signature with only equality, established by Leopold Löwenheim in 1915. • The set of first-order logical validities in a signature with equality and one unary function, established by Ehrenfeucht in 1959. • The first-order theory of the natural numbers in the signature with equality and addition, also called Presburger arithmetic. The completeness was established by Mojżesz Presburger in 1929. • The first-order theory of the natural numbers in the signature with equality and multiplication, also called Skolem arithmetic. • The first-order theory of Boolean algebras, established by Alfred Tarski in 1940 (found in 1940 but announced in 1949). • The first-order theory of algebraically closed fields of a given characteristic, established by Tarski in 1949. • The first-order theory of real-closed ordered fields, established by Tarski in 1949 (see also Tarski's exponential function problem). • The first-order theory of Euclidean geometry, established by Tarski in 1949. • The first-order theory of Abelian groups, established by Szmielew in 1955. • The first-order theory of hyperbolic geometry, established by Schwabhäuser in 1959. • Specific decidable sublanguages of set theory investigated in the 1980s through today.(Cantone et al., 2001) • The monadic second-order theory of trees (see S2S). Methods used to establish decidability include quantifier elimination, model completeness, and the Łoś-Vaught test. Some undecidable theories Some undecidable theories include (Monk 1976, p. 279):[2] • The set of logical validities in any first-order signature with equality and either: a relation symbol of arity no less than 2, or two unary function symbols, or one function symbol of arity no less than 2, established by Trakhtenbrot in 1953. • The first-order theory of the natural numbers with addition, multiplication, and equality, established by Tarski and Andrzej Mostowski in 1949. • The first-order theory of the rational numbers with addition, multiplication, and equality, established by Julia Robinson in 1949. • The first-order theory of groups, established by Alfred Tarski in 1953.[3] Remarkably, not only the general theory of groups is undecidable, but also several more specific theories, for example (as established by Mal'cev 1961) the theory of finite groups. Mal'cev also established that the theory of semigroups and the theory of rings are undecidable. Robinson established in 1949 that the theory of fields is undecidable. • Robinson arithmetic (and therefore any consistent extension, such as Peano arithmetic) is essentially undecidable, as established by Raphael Robinson in 1950. • The first-order theory with equality and two function symbols[4] The interpretability method is often used to establish undecidability of theories. If an essentially undecidable theory T is interpretable in a consistent theory S, then S is also essentially undecidable. This is closely related to the concept of a many-one reduction in computability theory. Semidecidability A property of a theory or logical system weaker than decidability is semidecidability. A theory is semidecidable if there is an effective method which, given an arbitrary formula, will always tell correctly when the formula is in the theory, but may give either a negative answer or no answer at all when the formula is not in the theory. A logical system is semidecidable if there is an effective method for generating theorems (and only theorems) such that every theorem will eventually be generated. This is different from decidability because in a semidecidable system there may be no effective procedure for checking that a formula is not a theorem. Every decidable theory or logical system is semidecidable, but in general the converse is not true; a theory is decidable if and only if both it and its complement are semi-decidable. For example, the set of logical validities V of first-order logic is semi-decidable, but not decidable. In this case, it is because there is no effective method for determining for an arbitrary formula A whether A is not in V. Similarly, the set of logical consequences of any recursively enumerable set of first-order axioms is semidecidable. Many of the examples of undecidable first-order theories given above are of this form. Relationship with completeness Decidability should not be confused with completeness. For example, the theory of algebraically closed fields is decidable but incomplete, whereas the set of all true first-order statements about nonnegative integers in the language with + and × is complete but undecidable. Unfortunately, as a terminological ambiguity, the term "undecidable statement" is sometimes used as a synonym for independent statement. Relationship to computability As with the concept of a decidable set, the definition of a decidable theory or logical system can be given either in terms of effective methods or in terms of computable functions. These are generally considered equivalent per Church's thesis. Indeed, the proof that a logical system or theory is undecidable will use the formal definition of computability to show that an appropriate set is not a decidable set, and then invoke Church's thesis to show that the theory or logical system is not decidable by any effective method (Enderton 2001, pp. 206ff.). In context of games Some games have been classified as to their decidability: • Chess is decidable.[5][6] The same holds for all other finite two-player games with perfect information. • Mate in n in infinite chess (with limitations on rules and gamepieces) is decidable.[7][8] However, there are positions (with finitely many pieces) that are forced wins, but not mate in n for any finite n.[9] • Some team games with imperfect information on a finite board (but with unlimited time) are undecidable.[10] See also • Entscheidungsproblem • Existential quantification References Notes 1. Trakhtenbrot, 1953 . 2. Monk, Donald (1976). Mathematical Logic. Springer. ISBN 9780387901701. 3. Tarski, A.; Mostovski, A.; Robinson, R. (1953), Undecidable Theories, Studies in Logic and the Foundation of Mathematics, North-Holland, Amsterdam, ISBN 9780444533784 4. Gurevich, Yuri (1976). "The Decision Problem for Standard Classes". J. Symb. Log. 41 (2): 460–464. CiteSeerX 10.1.1.360.1517. doi:10.1017/S0022481200051513. S2CID 798307. Retrieved 5 August 2014. 5. Stack Exchange Computer Science. "Is chess game movement TM decidable?" 6. Undecidable Chess Problem? 7. Mathoverflow.net/Decidability-of-chess-on-an-infinite-board Decidability-of-chess-on-an-infinite-board 8. Brumleve, Dan; Hamkins, Joel David; Schlicht, Philipp (2012). "The Mate-in-n Problem of Infinite Chess Is Decidable". Conference on Computability in Europe. Lecture Notes in Computer Science. Vol. 7318. Springer. pp. 78–88. arXiv:1201.5597. doi:10.1007/978-3-642-30870-3_9. ISBN 978-3-642-30870-3. S2CID 8998263. 9. "Lo.logic – Checkmate in $\omega$ moves?". 10. Poonen, Bjorn (2014). "10. Undecidable Problems: A Sampler: §14.1 Abstract Games". In Kennedy, Juliette (ed.). Interpreting Gödel: Critical Essays. Cambridge University Press. pp. 211–241 See p. 239. arXiv:1204.0299. CiteSeerX 10.1.1.679.3322. ISBN 9781107002661.} Bibliography • Barwise, Jon (1982), "Introduction to first-order logic", in Barwise, Jon (ed.), Handbook of Mathematical Logic, Studies in Logic and the Foundations of Mathematics, Amsterdam: North-Holland, ISBN 978-0-444-86388-1 • Cantone, D.; Omodeo, E. G.; Policriti, A. (2013) [2001], Set Theory for Computing. From Decision Procedures to Logic Programming with Sets, Monographs in Computer Science, Springer, ISBN 9781475734522 • Chagrov, Alexander; Zakharyaschev, Michael (1997), Modal logic, Oxford Logic Guides, vol. 35, Oxford University Press, ISBN 978-0-19-853779-3, MR 1464942 • Davis, Martin (2013) [1958], Computability and Unsolvability, Dover, ISBN 9780486151069 • Enderton, Herbert (2001), A mathematical introduction to logic (2nd ed.), Academic Press, ISBN 978-0-12-238452-3 • Keisler, H. J. (1982), "Fundamentals of model theory", in Barwise, Jon (ed.), Handbook of Mathematical Logic, Studies in Logic and the Foundations of Mathematics, Amsterdam: North-Holland, ISBN 978-0-444-86388-1 • Monk, J. Donald (2012) [1976], Mathematical Logic, Springer-Verlag, ISBN 9781468494525 Mathematical logic General • Axiom • list • Cardinality • First-order logic • Formal proof • Formal semantics • Foundations of mathematics • Information theory • Lemma • Logical consequence • Model • Theorem • Theory • Type theory Theorems (list) & Paradoxes • Gödel's completeness and incompleteness theorems • Tarski's undefinability • Banach–Tarski paradox • Cantor's theorem, paradox and diagonal argument • Compactness • Halting problem • Lindström's • Löwenheim–Skolem • Russell's paradox Logics Traditional • Classical logic • Logical truth • Tautology • Proposition • Inference • Logical equivalence • Consistency • Equiconsistency • Argument • Soundness • Validity • Syllogism • Square of opposition • Venn diagram Propositional • Boolean algebra • Boolean functions • Logical connectives • Propositional calculus • Propositional formula • Truth tables • Many-valued logic • 3 • Finite • ∞ Predicate • First-order • list • Second-order • Monadic • Higher-order • Free • Quantifiers • Predicate • Monadic predicate calculus Set theory • Set • Hereditary • Class • (Ur-)Element • Ordinal number • Extensionality • Forcing • Relation • Equivalence • Partition • Set operations: • Intersection • Union • Complement • Cartesian product • Power set • Identities Types of Sets • Countable • Uncountable • Empty • Inhabited • Singleton • Finite • Infinite • Transitive • Ultrafilter • Recursive • Fuzzy • Universal • Universe • Constructible • Grothendieck • Von Neumann Maps & Cardinality • Function/Map • Domain • Codomain • Image • In/Sur/Bi-jection • Schröder–Bernstein theorem • Isomorphism • Gödel numbering • Enumeration • Large cardinal • Inaccessible • Aleph number • Operation • Binary Set theories • Zermelo–Fraenkel • Axiom of choice • Continuum hypothesis • General • Kripke–Platek • Morse–Kelley • Naive • New Foundations • Tarski–Grothendieck • Von Neumann–Bernays–Gödel • Ackermann • Constructive Formal systems (list), Language & Syntax • Alphabet • Arity • Automata • Axiom schema • Expression • Ground • Extension • by definition • Conservative • Relation • Formation rule • Grammar • Formula • Atomic • Closed • Ground • Open • Free/bound variable • Language • Metalanguage • Logical connective • ¬ • ∨ • ∧ • → • ↔ • = • Predicate • Functional • Variable • Propositional variable • Proof • Quantifier • ∃ • ! • ∀ • rank • Sentence • Atomic • Spectrum • Signature • String • Substitution • Symbol • Function • Logical/Constant • Non-logical • Variable • Term • Theory • list Example axiomatic systems (list) • of arithmetic: • Peano • second-order • elementary function • primitive recursive • Robinson • Skolem • of the real numbers • Tarski's axiomatization • of Boolean algebras • canonical • minimal axioms • of geometry: • Euclidean: • Elements • Hilbert's • Tarski's • non-Euclidean • Principia Mathematica Proof theory • Formal proof • Natural deduction • Logical consequence • Rule of inference • Sequent calculus • Theorem • Systems • Axiomatic • Deductive • Hilbert • list • Complete theory • Independence (from ZFC) • Proof of impossibility • Ordinal analysis • Reverse mathematics • Self-verifying theories Model theory • Interpretation • Function • of models • Model • Equivalence • Finite • Saturated • Spectrum • Submodel • Non-standard model • of arithmetic • Diagram • Elementary • Categorical theory • Model complete theory • Satisfiability • Semantics of logic • Strength • Theories of truth • Semantic • Tarski's • Kripke's • T-schema • Transfer principle • Truth predicate • Truth value • Type • Ultraproduct • Validity Computability theory • Church encoding • Church–Turing thesis • Computably enumerable • Computable function • Computable set • Decision problem • Decidable • Undecidable • P • NP • P versus NP problem • Kolmogorov complexity • Lambda calculus • Primitive recursive function • Recursion • Recursive set • Turing machine • Type theory Related • Abstract logic • Category theory • Concrete/Abstract Category • Category of sets • History of logic • History of mathematical logic • timeline • Logicism • Mathematical object • Philosophy of mathematics • Supertask Mathematics portal Metalogic and metamathematics • Cantor's theorem • Entscheidungsproblem • Church–Turing thesis • Consistency • Effective method • Foundations of mathematics • of geometry • Gödel's completeness theorem • Gödel's incompleteness theorems • Soundness • Completeness • Decidability • Interpretation • Löwenheim–Skolem theorem • Metatheorem • Satisfiability • Independence • Type–token distinction • Use–mention distinction Authority control International • FAST National • France • BnF data • Germany • Israel • United States Other • IdRef	Wikipedia
Semidiameter In geometry, the semidiameter or semi-diameter of a set of points may be one half of its diameter; or, sometimes, one half of its extent along a particular direction. Special cases The semi-diameter of a sphere, circle, or interval is the same thing as its radius — namely, any line segment from the center to its boundary. The semi-diameters of a non-circular ellipse are the halves of its extents along the two axes of symmetry. They are the parameters a, b of the implicit equation $\left({\frac {x}{a}}\right)^{2}+\left({\frac {y}{b}}\right)^{2}=1.\,\!$ Likewise, the semi-diameters of an ellipsoid are the parameters a, b, and c of its implicit equation $\left({\frac {x}{a}}\right)^{2}+\left({\frac {y}{b}}\right)^{2}+\left({\frac {z}{c}}\right)^{2}=1.\,\!$ The semi-diameters of a superellipse, superellipsoid, or superquadric can be identified in the same way. See also • Flattening • Semi-major and semi-minor axes • Semiperimeter	Wikipedia
Disk (mathematics) In geometry, a disk (also spelled disc)[1] is the region in a plane bounded by a circle. A disk is said to be closed if it contains the circle that constitutes its boundary, and open if it does not.[2] For a radius, $r$, an open disk is usually denoted as $D_{r}$ and a closed disk is ${\overline {D_{r}}}$. However in the field of topology the closed disk is usually denoted as $D^{2}$ while the open disk is $\operatorname {Int} D^{2}$. Formulas In Cartesian coordinates, the open disk of center $(a,b)$ and radius R is given by the formula:[1] $D=\{(x,y)\in {\mathbb {R} ^{2}}:(x-a)^{2}+(y-b)^{2}<R^{2}\}$ while the closed disk of the same center and radius is given by: ${\overline {D}}=\{(x,y)\in {\mathbb {R} ^{2}}:(x-a)^{2}+(y-b)^{2}\leq R^{2}\}.$ The area of a closed or open disk of radius R is πR2 (see area of a disk).[3] Properties The disk has circular symmetry.[4] The open disk and the closed disk are not topologically equivalent (that is, they are not homeomorphic), as they have different topological properties from each other. For instance, every closed disk is compact whereas every open disk is not compact.[5] However from the viewpoint of algebraic topology they share many properties: both of them are contractible[6] and so are homotopy equivalent to a single point. This implies that their fundamental groups are trivial, and all homology groups are trivial except the 0th one, which is isomorphic to Z. The Euler characteristic of a point (and therefore also that of a closed or open disk) is 1.[7] Every continuous map from the closed disk to itself has at least one fixed point (we don't require the map to be bijective or even surjective); this is the case n=2 of the Brouwer fixed point theorem.[8] The statement is false for the open disk:[9] Consider for example the function $f(x,y)=\left({\frac {x+{\sqrt {1-y^{2}}}}{2}},y\right)$ which maps every point of the open unit disk to another point on the open unit disk to the right of the given one. But for the closed unit disk it fixes every point on the half circle $x^{2}+y^{2}=1,x>0.$ As a statistical distribution A uniform distribution on a unit circular disk is occasionally encountered in statistics. It most commonly occurs in operations research in the mathematics of urban planning, where it may be used to model a population within a city. Other uses may take advantage of the fact that it is a distribution for which it is easy to compute the probability that a given set of linear inequalities will be satisfied. (Gaussian distributions in the plane require numerical quadrature.) "An ingenious argument via elementary functions" shows the mean Euclidean distance between two points in the disk to be 128/45π ≈ 0.90541,[10] while direct integration in polar coordinates shows the mean squared distance to be 1. If we are given an arbitrary location at a distance q from the center of the disk, it is also of interest to determine the average distance b(q) from points in the distribution to this location and the average square of such distances. The latter value can be computed directly as q2+1/2. Average distance to an arbitrary internal point To find b(q) we need to look separately at the cases in which the location is internal or external, i.e. in which q ≶ 1, and we find that in both cases the result can only be expressed in terms of complete elliptic integrals. If we consider an internal location, our aim (looking at the diagram) is to compute the expected value of r under a distribution whose density is 1/π for 0 ≤ r ≤ s(θ), integrating in polar coordinates centered on the fixed location for which the area of a cell is r dr dθ ; hence $b(q)={\frac {1}{\pi }}\int _{0}^{2\pi }{\textrm {d}}\theta \int _{0}^{s(\theta )}r^{2}{\textrm {d}}r={\frac {1}{3\pi }}\int _{0}^{2\pi }s(\theta )^{3}{\textrm {d}}\theta .$ Here s(θ) can be found in terms of q and θ using the Law of cosines. The steps needed to evaluate the integral, together with several references, will be found in the paper by Lew et al.;[10] the result is that $b(q)={\frac {4}{9\pi }}{\biggl \{}4(q^{2}-1)K(q^{2})+(q^{2}+7)E(q^{2}){\biggr \}}$ where K and E are complete elliptic integrals of the first and second kinds.[11] b(0) = 2/3; b(1) = 32/9π ≈ 1.13177. Average distance to an arbitrary external point Turning to an external location, we can set up the integral in a similar way, this time obtaining $b(q)={\frac {2}{3\pi }}\int _{0}^{{\textrm {sin}}^{-1}{\tfrac {1}{q}}}{\biggl \{}s_{+}(\theta )^{3}-s_{-}(\theta )^{3}{\biggr \}}{\textrm {d}}\theta $ where the law of cosines tells us that s+(θ) and s–(θ) are the roots for s of the equation $s^{2}-2qs\,{\textrm {cos}}\theta +q^{2}\!-\!1=0.$ Hence $b(q)={\frac {4}{3\pi }}\int _{0}^{{\textrm {sin}}^{-1}{\tfrac {1}{q}}}{\biggl \{}3q^{2}{\textrm {cos}}^{2}\theta {\sqrt {1-q^{2}{\textrm {sin}}^{2}\theta }}+{\Bigl (}1-q^{2}{\textrm {sin}}^{2}\theta {\Bigr )}^{\tfrac {3}{2}}{\biggl \}}{\textrm {d}}\theta .$ We may substitute u = q sinθ to get ${\begin{aligned}b(q)&={\frac {4}{3\pi }}\int _{0}^{1}{\biggl \{}3{\sqrt {q^{2}-u^{2}}}{\sqrt {1-u^{2}}}+{\frac {(1-u^{2})^{\tfrac {3}{2}}}{\sqrt {q^{2}-u^{2}}}}{\biggr \}}{\textrm {d}}u\\[0.6ex]&={\frac {4}{3\pi }}\int _{0}^{1}{\biggl \{}4{\sqrt {q^{2}-u^{2}}}{\sqrt {1-u^{2}}}-{\frac {q^{2}-1}{q}}{\frac {\sqrt {1-u^{2}}}{\sqrt {q^{2}-u^{2}}}}{\biggr \}}{\textrm {d}}u\\[0.6ex]&={\frac {4}{3\pi }}{\biggl \{}{\frac {4q}{3}}{\biggl (}(q^{2}+1)E({\tfrac {1}{q^{2}}})-(q^{2}-1)K({\tfrac {1}{q^{2}}}){\biggr )}-(q^{2}-1){\biggl (}qE({\tfrac {1}{q^{2}}})-{\frac {q^{2}-1}{q}}K({\tfrac {1}{q^{2}}}){\biggr )}{\biggr \}}\\[0.6ex]&={\frac {4}{9\pi }}{\biggl \{}q(q^{2}+7)E({\tfrac {1}{q^{2}}})-{\frac {q^{2}-1}{q}}(q^{2}+3)K({\tfrac {1}{q^{2}}}){\biggr \}}\end{aligned}}$ using standard integrals.[12] Hence again b(1) = 32/9π, while also[13] $\lim _{q\to \infty }b(q)=q+{\tfrac {1}{8q}}.$ See also • Unit disk, a disk with radius one • Annulus (mathematics), the region between two concentric circles • Ball (mathematics), the usual term for the 3-dimensional analogue of a disk • Disk algebra, a space of functions on a disk • Circular segment • Orthocentroidal disk, containing certain centers of a triangle References 1. Clapham, Christopher; Nicholson, James (2014), The Concise Oxford Dictionary of Mathematics, Oxford University Press, p. 138, ISBN 9780199679591. 2. Arnold, B. H. (2013), Intuitive Concepts in Elementary Topology, Dover Books on Mathematics, Courier Dover Publications, p. 58, ISBN 9780486275765. 3. Rotman, Joseph J. (2013), Journey into Mathematics: An Introduction to Proofs, Dover Books on Mathematics, Courier Dover Publications, p. 44, ISBN 9780486151687. 4. Altmann, Simon L. (1992). Icons and Symmetries. Oxford University Press. ISBN 9780198555995. disc circular symmetry. 5. Maudlin, Tim (2014), New Foundations for Physical Geometry: The Theory of Linear Structures, Oxford University Press, p. 339, ISBN 9780191004551. 6. Cohen, Daniel E. (1989), Combinatorial Group Theory: A Topological Approach, London Mathematical Society Student Texts, vol. 14, Cambridge University Press, p. 79, ISBN 9780521349369. 7. In higher dimensions, the Euler characteristic of a closed ball remains equal to +1, but the Euler characteristic of an open ball is +1 for even-dimensional balls and −1 for odd-dimensional balls. See Klain, Daniel A.; Rota, Gian-Carlo (1997), Introduction to Geometric Probability, Lezioni Lincee, Cambridge University Press, pp. 46–50. 8. Arnold (2013), p. 132. 9. Arnold (2013), Ex. 1, p. 135. 10. J. S. Lew et al., "On the Average Distances in a Circular Disc" (1977). 11. Abramowitz and Stegun, 17.3. 12. Gradshteyn and Ryzhik 3.155.7 and 3.169.9, taking due account of the difference in notation from Abramowitz and Stegun. (Compare A&S 17.3.11 with G&R 8.113.) This article follows A&S's notation. 13. Abramowitz and Stegun, 17.3.11 et seq. Compact topological surfaces and their immersions in 3D Without boundary Orientable • Sphere (genus 0) • Torus (genus 1) • Number 8 (genus 2) • Pretzel (genus 3) ... Non-orientable • Real projective plane • genus 1; Boy's surface • Roman surface • Klein bottle (genus 2) • Dyck's surface (genus 3) ... With boundary • Disk • Semisphere • Ribbon • Annulus • Cylinder • Möbius strip • Cross-cap • Sphere with three holes ... Related notions Properties • Connectedness • Compactness • Triangulatedness or smoothness • Orientability Characteristics • Number of boundary components • Genus • Euler characteristic Operations • Connected sum • Making a hole • Gluing a handle • Gluing a cross-cap • Immersion	Wikipedia
Semifield In mathematics, a semifield is an algebraic structure with two binary operations, addition and multiplication, which is similar to a field, but with some axioms relaxed. Algebraic structure → Ring theory Ring theory Basic concepts Rings • Subrings • Ideal • Quotient ring • Fractional ideal • Total ring of fractions • Product of rings • Free product of associative algebras • Tensor product of algebras Ring homomorphisms • Kernel • Inner automorphism • Frobenius endomorphism Algebraic structures • Module • Associative algebra • Graded ring • Involutive ring • Category of rings • Initial ring $\mathbb {Z} $ • Terminal ring $0=\mathbb {Z} _{1}$ Related structures • Field • Finite field • Non-associative ring • Lie ring • Jordan ring • Semiring • Semifield Commutative algebra Commutative rings • Integral domain • Integrally closed domain • GCD domain • Unique factorization domain • Principal ideal domain • Euclidean domain • Field • Finite field • Composition ring • Polynomial ring • Formal power series ring Algebraic number theory • Algebraic number field • Ring of integers • Algebraic independence • Transcendental number theory • Transcendence degree p-adic number theory and decimals • Direct limit/Inverse limit • Zero ring $\mathbb {Z} _{1}$ • Integers modulo pn $\mathbb {Z} /p^{n}\mathbb {Z} $ • Prüfer p-ring $\mathbb {Z} (p^{\infty })$ • Base-p circle ring $\mathbb {T} $ • Base-p integers $\mathbb {Z} $ • p-adic rationals $\mathbb {Z} [1/p]$ • Base-p real numbers $\mathbb {R} $ • p-adic integers $\mathbb {Z} _{p}$ • p-adic numbers $\mathbb {Q} _{p}$ • p-adic solenoid $\mathbb {T} _{p}$ Algebraic geometry • Affine variety Noncommutative algebra Noncommutative rings • Division ring • Semiprimitive ring • Simple ring • Commutator Noncommutative algebraic geometry Free algebra Clifford algebra • Geometric algebra Operator algebra Overview The term semifield has two conflicting meanings, both of which include fields as a special case. • In projective geometry and finite geometry (MSC 51A, 51E, 12K10), a semifield is a nonassociative division ring with multiplicative identity element.[1] More precisely, it is a nonassociative ring whose nonzero elements form a loop under multiplication. In other words, a semifield is a set S with two operations + (addition) and · (multiplication), such that • (S,+) is an abelian group, • multiplication is distributive on both the left and right, • there exists a multiplicative identity element, and • division is always possible: for every a and every nonzero b in S, there exist unique x and y in S for which b·x = a and y·b = a. Note in particular that the multiplication is not assumed to be commutative or associative. A semifield that is associative is a division ring, and one that is both associative and commutative is a field. A semifield by this definition is a special case of a quasifield. If S is finite, the last axiom in the definition above can be replaced with the assumption that there are no zero divisors, so that a·b = 0 implies that a = 0 or b = 0.[2] Note that due to the lack of associativity, the last axiom is not equivalent to the assumption that every nonzero element has a multiplicative inverse, as is usually found in definitions of fields and division rings. • In ring theory, combinatorics, functional analysis, and theoretical computer science (MSC 16Y60), a semifield is a semiring (S,+,·) in which all nonzero elements have a multiplicative inverse.[3][4] These objects are also called proper semifields. A variation of this definition arises if S contains an absorbing zero that is different from the multiplicative unit e, it is required that the non-zero elements be invertible, and a·0 = 0·a = 0. Since multiplication is associative, the (non-zero) elements of a semifield form a group. However, the pair (S,+) is only a semigroup, i.e. additive inverse need not exist, or, colloquially, 'there is no subtraction'. Sometimes, it is not assumed that the multiplication is associative. Primitivity of semifields A semifield D is called right (resp. left) primitive if it has an element w such that the set of nonzero elements of D* is equal to the set of all right (resp. left) principal powers of w. Examples We only give examples of semifields in the second sense, i.e. additive semigroups with distributive multiplication. Moreover, addition is commutative and multiplication is associative in our examples. • Positive rational numbers with the usual addition and multiplication form a commutative semifield. This can be extended by an absorbing 0. • Positive real numbers with the usual addition and multiplication form a commutative semifield. This can be extended by an absorbing 0, forming the probability semiring, which is isomorphic to the log semiring. • Rational functions of the form f /g, where f and g are polynomials in one variable with positive coefficients, form a commutative semifield. This can be extended to include 0. • The real numbers R can be viewed a semifield where the sum of two elements is defined to be their maximum and the product to be their ordinary sum; this semifield is more compactly denoted (R, max, +). Similarly (R, min, +) is a semifield. These are called the tropical semiring. This can be extended by −∞ (an absorbing 0); this is the limit (tropicalization) of the log semiring as the base goes to infinity. • Generalizing the previous example, if (A,·,≤) is a lattice-ordered group then (A,+,·) is an additively idempotent semifield with the semifield sum defined to be the supremum of two elements. Conversely, any additively idempotent semifield (A,+,·) defines a lattice-ordered group (A,·,≤), where a≤b if and only if a + b = b. • The boolean semifield B = {0, 1} with addition defined by logical or, and multiplication defined by logical and. See also • Planar ternary ring (first sense) References 1. Donald Knuth, Finite semifields and projective planes. J. Algebra, 2, 1965, 182--217 MR0175942. 2. Landquist, E.J., "On Nonassociative Division Rings and Projective Planes", Copyright 2000. 3. Golan, Jonathan S., Semirings and their applications. Updated and expanded version of The theory of semirings, with applications to mathematics and theoretical computer science (Longman Sci. Tech., Harlow, 1992, MR1163371. Kluwer Academic Publishers, Dordrecht, 1999. xii+381 pp. ISBN 0-7923-5786-8 MR1746739. 4. Hebisch, Udo; Weinert, Hanns Joachim, Semirings and semifields. Handbook of algebra, Vol. 1, 425--462, North-Holland, Amsterdam, 1996. MR1421808.	Wikipedia
Free ideal ring In mathematics, especially in the field of ring theory, a (right) free ideal ring, or fir, is a ring in which all right ideals are free modules with unique rank. A ring such that all right ideals with at most n generators are free and have unique rank is called an n-fir. A semifir is a ring in which all finitely generated right ideals are free modules of unique rank. (Thus, a ring is semifir if it is n-fir for all n ≥ 0.) The semifir property is left-right symmetric, but the fir property is not. Properties and examples It turns out that a left and right fir is a domain. Furthermore, a commutative fir is precisely a principal ideal domain, while a commutative semifir is precisely a Bézout domain. These last facts are not generally true for noncommutative rings, however (Cohn 1971). Every principal right ideal domain R is a right fir, since every nonzero principal right ideal of a domain is isomorphic to R. In the same way, a right Bézout domain is a semifir. Since all right ideals of a right fir are free, they are projective. So, any right fir is a right hereditary ring, and likewise a right semifir is a right semihereditary ring. Because projective modules over local rings are free, and because local rings have invariant basis number, it follows that a local, right hereditary ring is a right fir, and a local, right semihereditary ring is a right semifir. Unlike a principal right ideal domain, a right fir is not necessarily right Noetherian, however in the commutative case, R is a Dedekind domain since it is a hereditary domain, and so is necessarily Noetherian. Another important and motivating example of a free ideal ring are the free associative (unital) k-algebras for division rings k, also called non-commutative polynomial rings (Cohn 2000, §5.4). Semifirs have invariant basis number and every semifir is a Sylvester domain. References • Cohn, P. M. (1971), "Free ideal rings and free products of rings", Actes du Congrès International des Mathématiciens (Nice, 1970), vol. 1, Gauthier-Villars, pp. 273–278, MR 0506389, archived from the original on 2017-11-25, retrieved 2010-11-26 • Cohn, P. M. (2006), Free ideal rings and localization in general rings, New Mathematical Monographs, vol. 3, Cambridge University Press, ISBN 978-0-521-85337-8, MR 2246388 • Cohn, P. M. (1985), Free rings and their relations, London Mathematical Society Monographs, vol. 19 (2nd ed.), Boston, MA: Academic Press, ISBN 978-0-12-179152-0, MR 0800091 • Cohn, P. M. (2000), Introduction to ring theory, Springer Undergraduate Mathematics Series, Berlin, New York: Springer-Verlag, ISBN 978-1-85233-206-8, MR 1732101 • "Free ideal ring", Encyclopedia of Mathematics, EMS Press, 2001 [1994] Further reading • Cohn, P.M. (1995), Skew fields. Theory of general division rings, Encyclopedia of Mathematics and Its Applications, vol. 57, Cambridge: Cambridge University Press, ISBN 0-521-43217-0, Zbl 0840.16001	Wikipedia
Monoid ring In abstract algebra, a monoid ring is a ring constructed from a ring and a monoid, just as a group ring is constructed from a ring and a group. Definition Let R be a ring and let G be a monoid. The monoid ring or monoid algebra of G over R, denoted R[G] or RG, is the set of formal sums $\sum _{g\in G}r_{g}g$, where $r_{g}\in R$ for each $g\in G$ and rg = 0 for all but finitely many g, equipped with coefficient-wise addition, and the multiplication in which the elements of R commute with the elements of G. More formally, R[G] is the set of functions φ: G → R such that {g : φ(g) ≠ 0} is finite, equipped with addition of functions, and with multiplication defined by $(\phi \psi )(g)=\sum _{k\ell =g}\phi (k)\psi (\ell )$. If G is a group, then R[G] is also called the group ring of G over R. Universal property Given R and G, there is a ring homomorphism α: R → R[G] sending each r to r1 (where 1 is the identity element of G), and a monoid homomorphism β: G → R[G] (where the latter is viewed as a monoid under multiplication) sending each g to 1g (where 1 is the multiplicative identity of R). We have that α(r) commutes with β(g) for all r in R and g in G. The universal property of the monoid ring states that given a ring S, a ring homomorphism α': R → S, and a monoid homomorphism β': G → S to the multiplicative monoid of S, such that α'(r) commutes with β'(g) for all r in R and g in G, there is a unique ring homomorphism γ: R[G] → S such that composing α and β with γ produces α' and β '. Augmentation The augmentation is the ring homomorphism η: R[G] → R defined by $\eta \left(\sum _{g\in G}r_{g}g\right)=\sum _{g\in G}r_{g}.$ The kernel of η is called the augmentation ideal. It is a free R-module with basis consisting of 1 – g for all g in G not equal to 1. Examples Given a ring R and the (additive) monoid of natural numbers N (or {xn} viewed multiplicatively), we obtain the ring R[{xn}] =: R[x] of polynomials over R. The monoid Nn (with the addition) gives the polynomial ring with n variables: R[Nn] =: R[X1, ..., Xn]. Generalization If G is a semigroup, the same construction yields a semigroup ring R[G]. See also • Free algebra • Puiseux series References • Lang, Serge (2002). Algebra. Graduate Texts in Mathematics. Vol. 211 (Rev. 3rd ed.). New York: Springer-Verlag. ISBN 0-387-95385-X. Further reading • R.Gilmer. Commutative semigroup rings. University of Chicago Press, Chicago–London, 1984	Wikipedia
Semi-Hilbert space In mathematics, a semi-Hilbert space is a generalization of a Hilbert space in functional analysis, in which, roughly speaking, the inner product is required only to be positive semi-definite rather than positive definite, so that it gives rise to a seminorm rather than a vector space norm. The quotient of this space by the kernel of this seminorm is also required to be a Hilbert space in the usual sense. References • Optimal Interpolation in Semi-Hilbert Spaces	Wikipedia
Generalized arithmetic progression In mathematics, a generalized arithmetic progression (or multiple arithmetic progression) is a generalization of an arithmetic progression equipped with multiple common differences – whereas an arithmetic progression is generated by a single common difference, a generalized arithmetic progression can be generated by multiple common differences. For example, the sequence $17,20,22,23,25,26,27,28,29,\dots $ is not an arithmetic progression, but is instead generated by starting with 17 and adding either 3 or 5, thus allowing multiple common differences to generate it. A semilinear set generalizes this idea to multiple dimensions -- it is a set of vectors of integers, rather than a set of integers. Finite generalized arithmetic progression A finite generalized arithmetic progression, or sometimes just generalized arithmetic progression (GAP), of dimension d is defined to be a set of the form $\{x_{0}+l_{1}x_{1}+\cdots +l_{d}x_{d}:0\leq l_{1}<L_{1},\ldots ,0\leq l_{d}<L_{d}\}$ where $x_{0},x_{1},\dots ,x_{d},L_{1},\dots ,L_{d}\in \mathbb {Z} $. The product $L_{1}L_{2}\cdots L_{d}$ is called the size of the generalized arithmetic progression; the cardinality of the set can differ from the size if some elements of the set have multiple representations. If the cardinality equals the size, the progression is called proper. Generalized arithmetic progressions can be thought of as a projection of a higher dimensional grid into $\mathbb {Z} $. This projection is injective if and only if the generalized arithmetic progression is proper. Semilinear sets Formally, an arithmetic progression of $\mathbb {N} ^{d}$ is an infinite sequence of the form $\mathbf {v} ,\mathbf {v} +\mathbf {v} ',\mathbf {v} +2\mathbf {v} ',\mathbf {v} +3\mathbf {v} ',\ldots $, where $\mathbf {v} $ and $\mathbf {v} '$ are fixed vectors in $\mathbb {N} ^{d}$, called the initial vector and common difference respectively. A subset of $\mathbb {N} ^{d}$ is said to be linear if it is of the form $\left\{\mathbf {v} +\sum _{i=1}^{m}k_{i}\mathbf {v} _{i}\,\colon \,k_{1},\dots ,k_{m}\in \mathbb {N} \right\},$ where $m$ is some integer and $\mathbf {v} ,\mathbf {v} _{1},\dots ,\mathbf {v} _{m}$ are fixed vectors in $\mathbb {N} ^{d}$. A subset of $\mathbb {N} ^{d}$ is said to be semilinear if it is a finite union of linear sets. The semilinear sets are exactly the sets definable in Presburger arithmetic.[1] See also • Freiman's theorem References 1. Ginsburg, Seymour; Spanier, Edwin Henry (1966). "Semigroups, Presburger Formulas, and Languages". Pacific Journal of Mathematics. 16 (2): 285–296. doi:10.2140/pjm.1966.16.285. • Nathanson, Melvyn B. (1996). Additive Number Theory: Inverse Problems and Geometry of Sumsets. Graduate Texts in Mathematics. Vol. 165. Springer. ISBN 0-387-94655-1. Zbl 0859.11003.	Wikipedia
Semi-local ring In mathematics, a semi-local ring is a ring for which R/J(R) is a semisimple ring, where J(R) is the Jacobson radical of R. (Lam 2001, p. §20)(Mikhalev & Pilz 2002, p. C.7) For the older meaning of a Noetherian ring with a topology defined by an ideal in the Jacobson radical, see Zariski ring. The above definition is satisfied if R has a finite number of maximal right ideals (and finite number of maximal left ideals). When R is a commutative ring, the converse implication is also true, and so the definition of semi-local for commutative rings is often taken to be "having finitely many maximal ideals". Some literature refers to a commutative semi-local ring in general as a quasi-semi-local ring, using semi-local ring to refer to a Noetherian ring with finitely many maximal ideals. A semi-local ring is thus more general than a local ring, which has only one maximal (right/left/two-sided) ideal. Examples • Any right or left Artinian ring, any serial ring, and any semiperfect ring is semi-local. • The quotient $\mathbb {Z} /m\mathbb {Z} $ is a semi-local ring. In particular, if $m$ is a prime power, then $\mathbb {Z} /m\mathbb {Z} $ is a local ring. • A finite direct sum of fields $\bigoplus _{i=1}^{n}{F_{i}}$ is a semi-local ring. • In the case of commutative rings with unity, this example is prototypical in the following sense: the Chinese remainder theorem shows that for a semi-local commutative ring R with unit and maximal ideals m1, ..., mn $R/\bigcap _{i=1}^{n}m_{i}\cong \bigoplus _{i=1}^{n}R/m_{i}\,$. (The map is the natural projection). The right hand side is a direct sum of fields. Here we note that ∩i mi=J(R), and we see that R/J(R) is indeed a semisimple ring. • The classical ring of quotients for any commutative Noetherian ring is a semilocal ring. • The endomorphism ring of an Artinian module is a semilocal ring. • Semi-local rings occur for example in algebraic geometry when a (commutative) ring R is localized with respect to the multiplicatively closed subset S = ∩ (R \ pi), where the pi are finitely many prime ideals. Textbooks • Lam, T.Y. (2001), "7", A first course in noncommutative rings, Graduate Texts in Mathematics, vol. 131 (2 ed.), New York: Springer-Verlag, pp. xx+385, ISBN 0-387-95183-0, MR 1838439 • Mikhalev, Alexander V.; Pilz, Günter F., eds. (2002), The concise handbook of algebra, Dordrecht: Kluwer Academic Publishers, pp. xvi+618, ISBN 0-7923-7072-4, MR 1966155	Wikipedia
Semimodular lattice In the branch of mathematics known as order theory, a semimodular lattice, is a lattice that satisfies the following condition: Semimodular law a ∧ b <: a implies b <: a ∨ b. This article is about generalizations of modularity in terms of the atomic covering relation. For M-symmetry, the generalization of modularity in terms of modular pairs, see modular lattice. The notation a <: b means that b covers a, i.e. a < b and there is no element c such that a < c < b. An atomistic semimodular bounded lattice is called a matroid lattice because such lattices are equivalent to (simple) matroids. An atomistic semimodular bounded lattice of finite length is called a geometric lattice and corresponds to a matroid of finite rank.[1] Semimodular lattices are also known as upper semimodular lattices; the dual notion is that of a lower semimodular lattice. A finite lattice is modular if and only if it is both upper and lower semimodular. A finite lattice, or more generally a lattice satisfying the ascending chain condition or the descending chain condition, is semimodular if and only if it is M-symmetric. Some authors refer to M-symmetric lattices as semimodular lattices.[2] A semimodular lattice is one kind of algebraic lattice. Birkhoff's condition A lattice is sometimes called weakly semimodular if it satisfies the following condition due to Garrett Birkhoff: Birkhoff's condition If a ∧ b <: a and a ∧ b <: b, then a <: a ∨ b and b <: a ∨ b. Every semimodular lattice is weakly semimodular. The converse is true for lattices of finite length, and more generally for upper continuous (meets distribute over joins of chains) relatively atomic lattices. Mac Lane's condition The following two conditions are equivalent to each other for all lattices. They were found by Saunders Mac Lane, who was looking for a condition that is equivalent to semimodularity for finite lattices, but does not involve the covering relation. Mac Lane's condition 1 For any a, b, c such that b ∧ c < a < c < b ∨ a, there is an element d such that b ∧ c < d ≤ b and a = (a ∨ d) ∧ c. Mac Lane's condition 2 For any a, b, c such that b ∧ c < a < c < b ∨ c, there is an element d such that b ∧ c < d ≤ b and a = (a ∨ d) ∧ c. Every lattice satisfying Mac Lane's condition is semimodular. The converse is true for lattices of finite length, and more generally for relatively atomic lattices. Moreover, every upper continuous lattice satisfying Mac Lane's condition is M-symmetric. Notes 1. These definitions follow Stern (1999). Some authors use the term geometric lattice for the more general matroid lattices. Most authors only deal with the finite case, in which both definitions are equivalent to semimodular and atomistic. 2. For instance, Fofanova (2001). References • Fofanova, T. S. (2001) [1994], "Semi-modular lattice", Encyclopedia of Mathematics, EMS Press. (The article is about M-symmetric lattices.) • Stern, Manfred (1999), Semimodular Lattices, Cambridge University Press, ISBN 978-0-521-46105-4. External links • "Semimodular lattice". PlanetMath. • OEIS sequence A229202 (Number of unlabeled semimodular lattices with n elements) See also • Antimatroid	Wikipedia
Semimodule In mathematics, a semimodule over a semiring R is like a module over a ring except that it is only a commutative monoid rather than an abelian group. Definition Formally, a left R-semimodule consists of an additively-written commutative monoid M and a map from $R\times M$ to M satisfying the following axioms: 1. $r(m+n)=rm+rn$ 2. $(r+s)m=rm+sm$ 3. $(rs)m=r(sm)$ 4. $1m=m$ 5. $0_{R}m=r0_{M}=0_{M}$. A right R-semimodule can be defined similarly. For modules over a ring, the last axiom follows from the others. This is not the case with semimodules. Examples If R is a ring, then any R-module is an R-semimodule. Conversely, it follows from the second, fourth, and last axioms that (-1)m is an additive inverse of m for all $m\in M$, so any semimodule over a ring is in fact a module. Any semiring is a left and right semimodule over itself in the same way that a ring is a left and right module over itself. Every commutative monoid is uniquely an $\mathbb {N} $-semimodule in the same way that an abelian group is a $\mathbb {Z} $-module. References Golan, Jonathan S. (1999), "Semimodules over semirings", Semirings and their Applications, Dordrecht: Springer Netherlands, pp. 149–161, ISBN 978-90-481-5252-0, retrieved 2022-02-22	Wikipedia
Seminorm In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk and, conversely, the Minkowski functional of any such set is a seminorm. A topological vector space is locally convex if and only if its topology is induced by a family of seminorms. Definition Let $X$ be a vector space over either the real numbers $\mathbb {R} $ or the complex numbers $\mathbb {C} .$ A real-valued function $p:X\to \mathbb {R} $ is called a seminorm if it satisfies the following two conditions: 1. Subadditivity[1]/Triangle inequality: $p(x+y)\leq p(x)+p(y)$ for all $x,y\in X.$ 2. Absolute homogeneity:[1] $p(sx)=\|s\|p(x)$ for all $x\in X$ and all scalars $s.$ These two conditions imply that $p(0)=0$[proof 1] and that every seminorm $p$ also has the following property:[proof 2] 1. Nonnegativity:[1] $p(x)\geq 0$ for all $x\in X.$ Some authors include non-negativity as part of the definition of "seminorm" (and also sometimes of "norm"), although this is not necessary since it follows from the other two properties. By definition, a norm on $X$ is a seminorm that also separates points, meaning that it has the following additional property: 1. Positive definite/Positive[1]/Point-separating: whenever $x\in X$ satisfies $p(x)=0,$ then $x=0.$ A seminormed space is a pair $(X,p)$ consisting of a vector space $X$ and a seminorm $p$ on $X.$ If the seminorm $p$ is also a norm then the seminormed space $(X,p)$ is called a normed space. Since absolute homogeneity implies positive homogeneity, every seminorm is a type of function called a sublinear function. A map $p:X\to \mathbb {R} $ is called a sublinear function if it is subadditive and positive homogeneous. Unlike a seminorm, a sublinear function is not necessarily nonnegative. Sublinear functions are often encountered in the context of the Hahn–Banach theorem. A real-valued function $p:X\to \mathbb {R} $ is a seminorm if and only if it is a sublinear and balanced function. Examples • The trivial seminorm on $X,$ which refers to the constant $0$ map on $X,$ induces the indiscrete topology on $X.$ • Let $\mu $ be a measure on a space $\Omega $. For an arbitrary constant $c\geq 1$, let $X$ be the set of all functions $f:\Omega \rightarrow \mathbb {R} $ for which $\lVert f\rVert _{c}:=\left(\int _{\Omega }\|f\|^{c}\,d\mu \right)^{1/c}$ exists and is finite. It can be shown that $X$ is a vector space, and the functional $\lVert \cdot \rVert _{c}$ is a seminorm on $X$. However, it is not always a norm (e.g. if $\Omega =\mathbb {R} $ and $\mu $ is the Lebesgue measure) because $\lVert h\rVert _{c}=0$ does not always imply $h=0$. To make $\lVert \cdot \rVert _{c}$ a norm, quotient $X$ by the closed subspace of functions $h$ with $\lVert h\rVert _{c}=0$. The resulting space, $L^{c}(\mu )$, has a norm induced by $\lVert \cdot \rVert _{c}$. • If $f$ is any linear form on a vector space then its absolute value $\|f\|,$ defined by $x\mapsto \|f(x)\|,$ is a seminorm. • A sublinear function $f:X\to \mathbb {R} $ on a real vector space $X$ is a seminorm if and only if it is a symmetric function, meaning that $f(-x)=f(x)$ for all $x\in X.$ • Every real-valued sublinear function $f:X\to \mathbb {R} $ on a real vector space $X$ induces a seminorm $p:X\to \mathbb {R} $ defined by $p(x):=\max\{f(x),f(-x)\}.$[2] • Any finite sum of seminorms is a seminorm. The restriction of a seminorm (respectively, norm) to a vector subspace is once again a seminorm (respectively, norm). • If $p:X\to \mathbb {R} $ and $q:Y\to \mathbb {R} $ are seminorms (respectively, norms) on $X$ and $Y$ then the map $r:X\times Y\to \mathbb {R} $ defined by $r(x,y)=p(x)+q(y)$ is a seminorm (respectively, a norm) on $X\times Y.$ In particular, the maps on $X\times Y$ defined by $(x,y)\mapsto p(x)$ and $(x,y)\mapsto q(y)$ are both seminorms on $X\times Y.$ • If $p$ and $q$ are seminorms on $X$ then so are[3] $(p\vee q)(x)=\max\{p(x),q(x)\}$ and $(p\wedge q)(x):=\inf\{p(y)+q(z):x=y+z{\text{ with }}y,z\in X\}$ where $p\wedge q\leq p$ and $p\wedge q\leq q.$[4] • The space of seminorms on $X$ is generally not a distributive lattice with respect to the above operations. For example, over $\mathbb {R} ^{2}$, $p(x,y):=\max(\|x\|,\|y\|),q(x,y):=2\|x\|,r(x,y):=2\|y\|$ are such that $((p\vee q)\wedge (p\vee r))(x,y)=\inf\{\max(2\|x_{1}\|,\|y_{1}\|)+\max(\|x_{2}\|,2\|y_{2}\|):x=x_{1}+x_{2}{\text{ and }}y=y_{1}+y_{2}\}$ while $(p\vee q\wedge r)(x,y):=\max(\|x\|,\|y\|)$ • If $L:X\to Y$ is a linear map and $q:Y\to \mathbb {R} $ is a seminorm on $Y,$ then $q\circ L:X\to \mathbb {R} $ is a seminorm on $X.$ The seminorm $q\circ L$ will be a norm on $X$ if and only if $L$ is injective and the restriction $q{\big \vert }_{L(X)}$ is a norm on $L(X).$ Minkowski functionals and seminorms Main article: Minkowski functional Seminorms on a vector space $X$ are intimately tied, via Minkowski functionals, to subsets of $X$ that are convex, balanced, and absorbing. Given such a subset $D$ of $X,$ the Minkowski functional of $D$ is a seminorm. Conversely, given a seminorm $p$ on $X,$ the sets$\{x\in X:p(x)<1\}$ and $\{x\in X:p(x)\leq 1\}$ are convex, balanced, and absorbing and furthermore, the Minkowski functional of these two sets (as well as of any set lying "in between them") is $p.$[5] Algebraic properties Every seminorm is a sublinear function, and thus satisfies all properties of a sublinear function, including convexity, $p(0)=0,$ and for all vectors $x,y\in X$: the reverse triangle inequality: [2][6] $\|p(x)-p(y)\|\leq p(x-y)$ and also $ 0\leq \max\{p(x),p(-x)\}$ and $p(x)-p(y)\leq p(x-y).$[2][6] For any vector $x\in X$ and positive real $r>0:$[7] $x+\{y\in X:p(y)<r\}=\{y\in X:p(x-y)<r\}$ and furthermore, $\{x\in X:p(x)<r\}$ is an absorbing disk in $X.$[3] If $p$ is a sublinear function on a real vector space $X$ then there exists a linear functional $f$ on $X$ such that $f\leq p$[6] and furthermore, for any linear functional $g$ on $X,$ $g\leq p$ on $X$ if and only if $g^{-1}(1)\cap \{x\in X:p(x)<1=\varnothing \}.$[6] Other properties of seminorms Every seminorm is a balanced function. A seminorm $p$ is a norm on $X$ if and only if $\{x\in X:p(x)<1\}$ does not contain a non-trivial vector subspace. If $p:X\to [0,\infty )$ is a seminorm on $X$ then $\ker p:=p^{-1}(0)$ is a vector subspace of $X$ and for every $x\in X,$ $p$ is constant on the set $x+\ker p=\{x+k:p(k)=0\}$ and equal to $p(x).$[proof 3] Furthermore, for any real $r>0,$[3] $r\{x\in X:p(x)<1\}=\{x\in X:p(x)<r\}=\left\{x\in X:{\tfrac {1}{r}}p(x)<1\right\}.$ If $D$ is a set satisfying $\{x\in X:p(x)<1\}\subseteq D\subseteq \{x\in X:p(x)\leq 1\}$ then $D$ is absorbing in $X$ and $p=p_{D}$ where $p_{D}$ denotes the Minkowski functional associated with $D$ (that is, the gauge of $D$).[5] In particular, if $D$ is as above and $q$ is any seminorm on $X,$ then $q=p$ if and only if $\{x\in X:q(x)<1\}\subseteq D\subseteq \{x\in X:q(x)\leq \}.$[5] If $(X,\\|\,\cdot \,\\|)$ is a normed space and $x,y\in X$ then $\\|x-y\\|=\\|x-z\\|+\\|z-y\\|$ for all $z$ in the interval $[x,y].$[8] Every norm is a convex function and consequently, finding a global maximum of a norm-based objective function is sometimes tractable. Relationship to other norm-like concepts Let $p:X\to \mathbb {R} $ be a non-negative function. The following are equivalent: 1. $p$ is a seminorm. 2. $p$ is a convex $F$-seminorm. 3. $p$ is a convex balanced G-seminorm.[9] If any of the above conditions hold, then the following are equivalent: 1. $p$ is a norm; 2. $\{x\in X:p(x)<1\}$ does not contain a non-trivial vector subspace.[10] 3. There exists a norm on $X,$ with respect to which, $\{x\in X:p(x)<1\}$ is bounded. If $p$ is a sublinear function on a real vector space $X$ then the following are equivalent:[6] 1. $p$ is a linear functional; 2. $p(x)+p(-x)\leq 0{\text{ for every }}x\in X$; 3. $p(x)+p(-x)=0{\text{ for every }}x\in X$; Inequalities involving seminorms If $p,q:X\to [0,\infty )$ are seminorms on $X$ then: • $p\leq q$ if and only if $q(x)\leq 1$ implies $p(x)\leq 1.$[11] • If $a>0$ and $b>0$ are such that $p(x)<a$ implies $q(x)\leq b,$ then $aq(x)\leq bp(x)$ for all $x\in X.$ [12] • Suppose $a$ and $b$ are positive real numbers and $q,p_{1},\ldots ,p_{n}$ are seminorms on $X$ such that for every $x\in X,$ if $\max\{p_{1}(x),\ldots ,p_{n}(x)\}<a$ then $q(x)<b.$ Then $aq\leq b\left(p_{1}+\cdots +p_{n}\right).$[10] • If $X$ is a vector space over the reals and $f$ is a non-zero linear functional on $X,$ then $f\leq p$ if and only if $\varnothing =f^{-1}(1)\cap \{x\in X:p(x)<1\}.$[11] If $p$ is a seminorm on $X$ and $f$ is a linear functional on $X$ then: • $\|f\|\leq p$ on $X$ if and only if $\operatorname {Re} f\leq p$ on $X$ (see footnote for proof).[13][14] • $f\leq p$ on $X$ if and only if $f^{-1}(1)\cap \{x\in X:p(x)<1=\varnothing \}.$[6][11] • If $a>0$ and $b>0$ are such that $p(x)<a$ implies $f(x)\neq b,$ then $a\|f(x)\|\leq bp(x)$ for all $x\in X.$[12] Hahn–Banach theorem for seminorms Seminorms offer a particularly clean formulation of the Hahn–Banach theorem: If $M$ is a vector subspace of a seminormed space $(X,p)$ and if $f$ is a continuous linear functional on $M,$ then $f$ may be extended to a continuous linear functional $F$ on $X$ that has the same norm as $f.$[15] A similar extension property also holds for seminorms: Theorem[16][12] (Extending seminorms) — If $M$ is a vector subspace of $X,$ $p$ is a seminorm on $M,$ and $q$ is a seminorm on $X$ such that $p\leq q{\big \vert }_{M},$ then there exists a seminorm $P$ on $X$ such that $P{\big \vert }_{M}=p$ and $P\leq q.$ Proof: Let $S$ be the convex hull of $\{m\in M:p(m)\leq 1\}\cup \{x\in X:q(x)\leq 1\}.$ Then $S$ is an absorbing disk in $X$ and so the Minkowski functional $P$ of $S$ is a seminorm on $X.$ This seminorm satisfies $p=P$ on $M$ and $P\leq q$ on $X.$ $\blacksquare $ Topologies of seminormed spaces Pseudometrics and the induced topology A seminorm $p$ on $X$ induces a topology, called the seminorm-induced topology, via the canonical translation-invariant pseudometric $d_{p}:X\times X\to \mathbb {R} $; $d_{p}(x,y):=p(x-y)=p(y-x).$ This topology is Hausdorff if and only if $d_{p}$ is a metric, which occurs if and only if $p$ is a norm.[4] This topology makes $X$ into a locally convex pseudometrizable topological vector space that has a bounded neighborhood of the origin and a neighborhood basis at the origin consisting of the following open balls (or the closed balls) centered at the origin: $\{x\in X:p(x)<r\}\quad {\text{ or }}\quad \{x\in X:p(x)\leq r\}$ as $r>0$ ranges over the positive reals. Every seminormed space $(X,p)$ should be assumed to be endowed with this topology unless indicated otherwise. A topological vector space whose topology is induced by some seminorm is called seminormable. Equivalently, every vector space $X$ with seminorm $p$ induces a vector space quotient $X/W,$ where $W$ is the subspace of $X$ consisting of all vectors $x\in X$ with $p(x)=0.$ Then $X/W$ carries a norm defined by $p(x+W)=p(v).$ The resulting topology, pulled back to $X,$ is precisely the topology induced by $p.$ Any seminorm-induced topology makes $X$ locally convex, as follows. If $p$ is a seminorm on $X$ and $r\in \mathbb {R} ,$ call the set $\{x\in X:p(x)<r\}$ the open ball of radius $r$ about the origin; likewise the closed ball of radius $r$ is $\{x\in X:p(x)\leq r\}.$ The set of all open (resp. closed) $p$-balls at the origin forms a neighborhood basis of convex balanced sets that are open (resp. closed) in the $p$-topology on $X.$ Stronger, weaker, and equivalent seminorms The notions of stronger and weaker seminorms are akin to the notions of stronger and weaker norms. If $p$ and $q$ are seminorms on $X,$ then we say that $q$ is stronger than $p$ and that $p$ is weaker than $q$ if any of the following equivalent conditions holds: 1. The topology on $X$ induced by $q$ is finer than the topology induced by $p.$ 2. If $x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }$ is a sequence in $X,$ then $q\left(x_{\bullet }\right):=\left(q\left(x_{i}\right)\right)_{i=1}^{\infty }\to 0$ in $\mathbb {R} $ implies $p\left(x_{\bullet }\right)\to 0$ in $\mathbb {R} .$[4] 3. If $x_{\bullet }=\left(x_{i}\right)_{i\in I}$ is a net in $X,$ then $q\left(x_{\bullet }\right):=\left(q\left(x_{i}\right)\right)_{i\in I}\to 0$ in $\mathbb {R} $ implies $p\left(x_{\bullet }\right)\to 0$ in $\mathbb {R} .$ 4. $p$ is bounded on $\{x\in X:q(x)<1\}.$[4] 5. If $\inf {}\{q(x):p(x)=1,x\in X\}=0$ then $p(x)=0$ for all $x\in X.$[4] 6. There exists a real $K>0$ such that $p\leq Kq$ on $X.$[4] The seminorms $p$ and $q$ are called equivalent if they are both weaker (or both stronger) than each other. This happens if they satisfy any of the following conditions: 1. The topology on $X$ induced by $q$ is the same as the topology induced by $p.$ 2. $q$ is stronger than $p$ and $p$ is stronger than $q.$[4] 3. If $x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }$ is a sequence in $X$ then $q\left(x_{\bullet }\right):=\left(q\left(x_{i}\right)\right)_{i=1}^{\infty }\to 0$ if and only if $p\left(x_{\bullet }\right)\to 0.$ 4. There exist positive real numbers $r>0$ and $R>0$ such that $rq\leq p\leq Rq.$ Normability and seminormability See also: Normed space and Local boundedness § locally bounded topological vector space A topological vector space (TVS) is said to be a seminormable space (respectively, a normable space) if its topology is induced by a single seminorm (resp. a single norm). A TVS is normable if and only if it is seminormable and Hausdorff or equivalently, if and only if it is seminormable and T1 (because a TVS is Hausdorff if and only if it is a T1 space). A locally bounded topological vector space is a topological vector space that possesses a bounded neighborhood of the origin. Normability of topological vector spaces is characterized by Kolmogorov's normability criterion. A TVS is seminormable if and only if it has a convex bounded neighborhood of the origin.[17] Thus a locally convex TVS is seminormable if and only if it has a non-empty bounded open set.[18] A TVS is normable if and only if it is a T1 space and admits a bounded convex neighborhood of the origin. If $X$ is a Hausdorff locally convex TVS then the following are equivalent: 1. $X$ is normable. 2. $X$ is seminormable. 3. $X$ has a bounded neighborhood of the origin. 4. The strong dual $X_{b}^{\prime }$ of $X$ is normable.[19] 5. The strong dual $X_{b}^{\prime }$ of $X$ is metrizable.[19] Furthermore, $X$ is finite dimensional if and only if $X_{\sigma }^{\prime }$ is normable (here $X_{\sigma }^{\prime }$ denotes $X^{\prime }$ endowed with the weak-* topology). The product of infinitely many seminormable space is again seminormable if and only if all but finitely many of these spaces trivial (that is, 0-dimensional).[18] Topological properties • If $X$ is a TVS and $p$ is a continuous seminorm on $X,$ then the closure of $\{x\in X:p(x)<r\}$ in $X$ is equal to $\{x\in X:p(x)\leq r\}.$[3] • The closure of $\{0\}$ in a locally convex space $X$ whose topology is defined by a family of continuous seminorms ${\mathcal {P}}$ is equal to $\bigcap _{p\in {\mathcal {P}}}p^{-1}(0).$[11] • A subset $S$ in a seminormed space $(X,p)$ is bounded if and only if $p(S)$ is bounded.[20] • If $(X,p)$ is a seminormed space then the locally convex topology that $p$ induces on $X$ makes $X$ into a pseudometrizable TVS with a canonical pseudometric given by $d(x,y):=p(x-y)$ for all $x,y\in X.$[21] • The product of infinitely many seminormable spaces is again seminormable if and only if all but finitely many of these spaces are trivial (that is, 0-dimensional).[18] Continuity of seminorms If $p$ is a seminorm on a topological vector space $X,$ then the following are equivalent:[5] 1. $p$ is continuous. 2. $p$ is continuous at 0;[3] 3. $\{x\in X:p(x)<1\}$ is open in $X$;[3] 4. $\{x\in X:p(x)\leq 1\}$ is closed neighborhood of 0 in $X$;[3] 5. $p$ is uniformly continuous on $X$;[3] 6. There exists a continuous seminorm $q$ on $X$ such that $p\leq q.$[3] In particular, if $(X,p)$ is a seminormed space then a seminorm $q$ on $X$ is continuous if and only if $q$ is dominated by a positive scalar multiple of $p.$[3] If $X$ is a real TVS, $f$ is a linear functional on $X,$ and $p$ is a continuous seminorm (or more generally, a sublinear function) on $X,$ then $f\leq p$ on $X$ implies that $f$ is continuous.[6] Continuity of linear maps If $F:(X,p)\to (Y,q)$ is a map between seminormed spaces then let[15] $\\|F\\|_{p,q}:=\sup\{q(F(x)):p(x)\leq 1,x\in X\}.$ If $F:(X,p)\to (Y,q)$ is a linear map between seminormed spaces then the following are equivalent: 1. $F$ is continuous; 2. $\\|F\\|_{p,q}<\infty $;[15] 3. There exists a real $K\geq 0$ such that $p\leq Kq$;[15] • In this case, $\\|F\\|_{p,q}\leq K.$ If $F$ is continuous then $q(F(x))\leq \\|F\\|_{p,q}p(x)$ for all $x\in X.$[15] The space of all continuous linear maps $F:(X,p)\to (Y,q)$ between seminormed spaces is itself a seminormed space under the seminorm $\\|F\\|_{p,q}.$ This seminorm is a norm if $q$ is a norm.[15] Generalizations The concept of norm in composition algebras does not share the usual properties of a norm. A composition algebra $(A,,N)$ consists of an algebra over a field $A,$ an involution $\,,$ and a quadratic form $N,$ which is called the "norm". In several cases $N$ is an isotropic quadratic form so that $A$ has at least one null vector, contrary to the separation of points required for the usual norm discussed in this article. An ultraseminorm or a non-Archimedean seminorm is a seminorm $p:X\to \mathbb {R} $ that also satisfies $p(x+y)\leq \max\{p(x),p(y)\}{\text{ for all }}x,y\in X.$ Weakening subadditivity: Quasi-seminorms A map $p:X\to \mathbb {R} $ is called a quasi-seminorm if it is (absolutely) homogeneous and there exists some $b\leq 1$ such that $p(x+y)\leq bp(p(x)+p(y)){\text{ for all }}x,y\in X.$ The smallest value of $b$ for which this holds is called the multiplier of $p.$ A quasi-seminorm that separates points is called a quasi-norm on $X.$ Weakening homogeneity - $k$-seminorms A map $p:X\to \mathbb {R} $ is called a $k$-seminorm if it is subadditive and there exists a $k$ such that $0<k\leq 1$ and for all $x\in X$ and scalars $s,$ $p(sx)=\|s\|^{k}p(x)$ A $k$-seminorm that separates points is called a $k$-norm on $X.$ We have the following relationship between quasi-seminorms and $k$-seminorms: Suppose that $q$ is a quasi-seminorm on a vector space $X$ with multiplier $b.$ If $0<{\sqrt {k}}<\log _{2}b$ then there exists $k$-seminorm $p$ on $X$ equivalent to $q.$ See also • Asymmetric norm – Generalization of the concept of a norm • Banach space – Normed vector space that is complete • Contraction mapping – Function reducing distance between all points • Finest locally convex topology – A vector space with a topology defined by convex open setsPages displaying short descriptions of redirect targets • Hahn-Banach theorem – Theorem on extension of bounded linear functionalsPages displaying short descriptions of redirect targets • Gowers norm • Locally convex topological vector space – A vector space with a topology defined by convex open sets • Mahalanobis distance – Statistical distance measure • Matrix norm – Norm on a vector space of matrices • Minkowski functional – Function made from a set • Norm (mathematics) – Length in a vector space • Normed vector space – Vector space on which a distance is defined • Relation of norms and metrics – Mathematical space with a notion of distancePages displaying short descriptions of redirect targets • Sublinear function Notes Proofs 1. If $z\in X$ denotes the zero vector in $X$ while $0$ denote the zero scalar, then absolute homogeneity implies that $p(0)=p(0z)=\|0\|p(z)=0p(z)=0.$ $\blacksquare $ 2. Suppose $p:X\to \mathbb {R} $ is a seminorm and let $x\in X.$ Then absolute homogeneity implies $p(-x)=p((-1)x)=\|-1\|p(x)=p(x).$ The triangle inequality now implies $p(0)=p(x+(-x))\leq p(x)+p(-x)=p(x)+p(x)=2p(x).$ Because $x$ was an arbitrary vector in $X,$ it follows that $p(0)\leq 2p(0),$ which implies that $0\leq p(0)$ (by subtracting $p(0)$ from both sides). Thus $0\leq p(0)\leq 2p(x)$ which implies $0\leq p(x)$ (by multiplying thru by $1/2$). 3. Let $x\in X$ and $k\in p^{-1}(0).$ It remains to show that $p(x+k)=p(x).$ The triangle inequality implies $p(x+k)\leq p(x)+p(k)=p(x)+0=p(x).$ Since $p(-k)=0,$ $p(x)=p(x)-p(-k)\leq p(x-(-k))=p(x+k),$ as desired. $\blacksquare $ References 1. Kubrusly 2011, p. 200. 2. Narici & Beckenstein 2011, pp. 120–121. 3. Narici & Beckenstein 2011, pp. 116–128. 4. Wilansky 2013, pp. 15–21. 5. Schaefer & Wolff 1999, p. 40. 6. Narici & Beckenstein 2011, pp. 177–220. 7. Narici & Beckenstein 2011, pp. 116−128. 8. Narici & Beckenstein 2011, pp. 107–113. 9. Schechter 1996, p. 691. 10. Narici & Beckenstein 2011, p. 149. 11. Narici & Beckenstein 2011, pp. 149–153. 12. Wilansky 2013, pp. 18–21. 13. Obvious if $X$ is a real vector space. For the non-trivial direction, assume that $\operatorname {Re} f\leq p$ on $X$ and let $x\in X.$ Let $r\geq 0$ and $t$ be real numbers such that $f(x)=re^{it}.$ Then $\|f(x)\|=r=f\left(e^{-it}x\right)=\operatorname {Re} \left(f\left(e^{-it}x\right)\right)\leq p\left(e^{-it}x\right)=p(x).$ 14. Wilansky 2013, p. 20. 15. Wilansky 2013, pp. 21–26. 16. Narici & Beckenstein 2011, pp. 150. 17. Wilansky 2013, pp. 50–51. 18. Narici & Beckenstein 2011, pp. 156–175. 19. Trèves 2006, pp. 136–149, 195–201, 240–252, 335–390, 420–433. 20. Wilansky 2013, pp. 49–50. 21. Narici & Beckenstein 2011, pp. 115–154. • Adasch, Norbert; Ernst, Bruno; Keim, Dieter (1978). Topological Vector Spaces: The Theory Without Convexity Conditions. Lecture Notes in Mathematics. Vol. 639. Berlin New York: Springer-Verlag. ISBN 978-3-540-08662-8. OCLC 297140003. • Berberian, Sterling K. (1974). Lectures in Functional Analysis and Operator Theory. Graduate Texts in Mathematics. Vol. 15. New York: Springer. ISBN 978-0-387-90081-0. OCLC 878109401. • Bourbaki, Nicolas (1987) [1981]. Topological Vector Spaces: Chapters 1–5. Éléments de mathématique. Translated by Eggleston, H.G.; Madan, S. Berlin New York: Springer-Verlag. ISBN 3-540-13627-4. OCLC 17499190. • Conway, John (1990). A course in functional analysis. Graduate Texts in Mathematics. Vol. 96 (2nd ed.). New York: Springer-Verlag. ISBN 978-0-387-97245-9. OCLC 21195908. • Edwards, Robert E. (1995). Functional Analysis: Theory and Applications. New York: Dover Publications. ISBN 978-0-486-68143-6. OCLC 30593138. • Grothendieck, Alexander (1973). Topological Vector Spaces. Translated by Chaljub, Orlando. New York: Gordon and Breach Science Publishers. ISBN 978-0-677-30020-7. OCLC 886098. • Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342. • Khaleelulla, S. M. (1982). Counterexamples in Topological Vector Spaces. Lecture Notes in Mathematics. Vol. 936. Berlin, Heidelberg, New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370. • Köthe, Gottfried (1983) [1969]. Topological Vector Spaces I. Grundlehren der mathematischen Wissenschaften. Vol. 159. Translated by Garling, D.J.H. New York: Springer Science & Business Media. ISBN 978-3-642-64988-2. MR 0248498. OCLC 840293704. • Kubrusly, Carlos S. (2011). The Elements of Operator Theory (Second ed.). Boston: Birkhäuser Basel. ISBN 978-0-8176-4998-2. OCLC 710154895. • Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. • Prugovečki, Eduard (1981). Quantum mechanics in Hilbert space (2nd ed.). Academic Press. p. 20. ISBN 0-12-566060-X. • Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. • Schechter, Eric (1996). Handbook of Analysis and Its Foundations. San Diego, CA: Academic Press. ISBN 978-0-12-622760-4. OCLC 175294365. • Swartz, Charles (1992). An introduction to Functional Analysis. New York: M. Dekker. ISBN 978-0-8247-8643-4. OCLC 24909067. • Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. • Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114. 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Seminormal ring In algebra, a seminormal ring is a commutative reduced ring in which, whenever x, y satisfy $x^{3}=y^{2}$, there is s with $s^{2}=x$ and $s^{3}=y$. This definition was given by Swan (1980) as a simplification of the original definition of Traverso (1970). A basic example is an integrally closed domain, i.e., a normal ring. For an example which is not normal, one can consider the non-integral ring Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \mathbb{Z}[x, y]/xy} , or the ring of a nodal curve. In general, a reduced scheme $X$ can be said to be seminormal if every morphism $Y\to X$ which induces a homeomorphism of topological spaces, and an isomorphism on all residue fields, is an isomorphism of schemes. A semigroup is said to be seminormal if its semigroup algebra is seminormal. References • Swan, Richard G. (1980), "On seminormality", Journal of Algebra, 67 (1): 210–229, doi:10.1016/0021-8693(80)90318-X, ISSN 0021-8693, MR 0595029 • Traverso, Carlo (1970), "Seminormality and Picard group", Ann. Scuola Norm. Sup. Pisa (3), 24: 585–595, MR 0277542 • Vitulli, Marie A. (2011), "Weak normality and seminormality" (PDF), Commutative algebra---Noetherian and non-Noetherian perspectives, Berlin, New York: Springer-Verlag, pp. 441–480, arXiv:0906.3334, doi:10.1007/978-1-4419-6990-3_17, ISBN 978-1-4419-6989-7, MR 2762521 • Charles Weibel, The K-book: An introduction to algebraic K-theory	Wikipedia

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