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Cylinder set measure
In mathematics, cylinder set measure (or promeasure, or premeasure, or quasi-measure, or CSM) is a kind of prototype for a measure on an infinite-dimensional vector space. An example is the Gaussian cylinder set measure on Hilbert space.
Cylinder set measures are in general not measures (and in particular need not be countably additive but only finitely additive), but can be used to define measures, such as classical Wiener measure on the set of continuous paths starting at the origin in Euclidean space.
Definition
Let $E$ be a separable real topological vector space. Let ${\mathcal {A}}(E)$ denote the collection of all surjective continuous linear maps $T:E\to F_{T}$ defined on $E$ whose image is some finite-dimensional real vector space $F_{T}$:
${\mathcal {A}}(E):=\left\{T\in \mathrm {Lin} (E;F_{T}):T{\mbox{ surjective and }}\dim _{\mathbb {R} }F_{T}<+\infty \right\}.$
A cylinder set measure on $E$ is a collection of probability measures
$\left\{\mu _{T}:T\in {\mathcal {A}}(E)\right\}.$
where $\mu _{T}$ is a probability measure on $F_{T}.$ These measures are required to satisfy the following consistency condition: if $\pi _{ST}:F_{S}\to F_{T}$ is a surjective projection, then the push forward of the measure is as follows:
$\mu _{T}=\left(\pi _{ST}\right)_{*}\left(\mu _{S}\right).$
Remarks
The consistency condition
$\mu _{T}=\left(\pi _{ST}\right)_{*}(\mu _{S})$
is modelled on the way that true measures push forward (see the section cylinder set measures versus true measures). However, it is important to understand that in the case of cylinder set measures, this is a requirement that is part of the definition, not a result.
A cylinder set measure can be intuitively understood as defining a finitely additive function on the cylinder sets of the topological vector space $E.$ The cylinder sets are the pre-images in $E$ of measurable sets in $F_{T}$: if ${\mathcal {B}}_{T}$ denotes the $\sigma $-algebra on $F_{T}$ on which $\mu _{T}$ is defined, then
$\mathrm {Cyl} (E):=\left\{T^{-1}(B):B\in {\mathcal {B}}_{T},T\in {\mathcal {A}}(E)\right\}.$
In practice, one often takes ${\mathcal {B}}_{T}$ to be the Borel $\sigma $-algebra on $F_{T}.$ In this case, one can show that when $E$ is a separable Banach space, the σ-algebra generated by the cylinder sets is precisely the Borel $\sigma $-algebra of $E$:
$\mathrm {Borel} (E)=\sigma \left(\mathrm {Cyl} (E)\right).$
Cylinder set measures versus measures
A cylinder set measure on $E$ is not actually a measure on $E$: it is a collection of measures defined on all finite-dimensional images of $E.$ If $E$ has a probability measure $\mu $ already defined on it, then $\mu $ gives rise to a cylinder set measure on $E$ using the push forward: set $\mu _{T}=T_{*}(\mu )$on $F_{T}.$
When there is a measure $\mu $ on $E$ such that $\mu _{T}=T_{*}(\mu )$ in this way, it is customary to abuse notation slightly and say that the cylinder set measure $\left\{\mu _{T}:T\in {\mathcal {A}}(E)\right\}$ "is" the measure $\mu .$
Cylinder set measures on Hilbert spaces
When the Banach space $E$ is actually a Hilbert space $H,$ there is a canonical Gaussian cylinder set measure $\gamma ^{H}$ arising from the inner product structure on $H.$ Specifically, if $\langle \cdot ,\cdot \rangle $ denotes the inner product on $H,$ let $\langle \cdot ,\cdot \rangle _{T}$ denote the quotient inner product on $F_{T}.$ The measure $\gamma _{T}^{H}$ on $F_{T}$ is then defined to be the canonical Gaussian measure on $F_{T}$:
$\gamma _{T}^{H}:=i_{*}\left(\gamma ^{\dim F_{T}}\right),$
where $i:\mathbb {R} ^{\dim(F_{T})}\to F_{T}$ is an isometry of Hilbert spaces taking the Euclidean inner product on $\mathbb {R} ^{\dim(F_{T})}$ to the inner product $\langle \cdot ,\cdot \rangle _{T}$ on $F_{T},$ and $\gamma ^{n}$ is the standard Gaussian measure on $\mathbb {R} ^{n}.$
The canonical Gaussian cylinder set measure on an infinite-dimensional separable Hilbert space $H$ does not correspond to a true measure on $H.$ The proof is quite simple: the ball of radius $r$ (and center 0) has measure at most equal to that of the ball of radius $r$ in an $n$-dimensional Hilbert space, and this tends to 0 as $n$ tends to infinity. So the ball of radius $r$ has measure 0; as the Hilbert space is a countable union of such balls it also has measure 0, which is a contradiction.
An alternative proof that the Gaussian cylinder set measure is not a measure uses the Cameron–Martin theorem and a result on the quasi-invariance of measures. If $\gamma ^{H}=\gamma $ really were a measure, then the identity function on $H$ would radonify that measure, thus making $\operatorname {id} :H\to H$ into an abstract Wiener space. By the Cameron–Martin theorem, $\gamma $ would then be quasi-invariant under translation by any element of $H,$ which implies that either $H$ is finite-dimensional or that $\gamma $ is the zero measure. In either case, we have a contradiction.
Sazonov's theorem gives conditions under which the push forward of a canonical Gaussian cylinder set measure can be turned into a true measure.
Nuclear spaces and cylinder set measures
A cylinder set measure on the dual of a nuclear Fréchet space automatically extends to a measure if its Fourier transform is continuous.
Example: Let $S$ be the space of Schwartz functions on a finite dimensional vector space; it is nuclear. It is contained in the Hilbert space $H$ of $L^{2}$ functions, which is in turn contained in the space of tempered distributions $S^{\prime },$ the dual of the nuclear Fréchet space $S$:
$S\subseteq H\subseteq S^{\prime }.$
The Gaussian cylinder set measure on $H$ gives a cylinder set measure on the space of tempered distributions, which extends to a measure on the space of tempered distributions, $S^{\prime }.$
The Hilbert space $H$ has measure 0 in $S^{\prime },$ by the first argument used above to show that the canonical Gaussian cylinder set measure on $H$ does not extend to a measure on $H.$
See also
• Abstract Wiener space – separable Banach space equipped with a Hilbert subspace such that the standard cylinder set measure on the Hilbert subspace induces a Gaussian measure on the whole Banach spacePages displaying wikidata descriptions as a fallback
• Cylindrical σ-algebra
• Radonifying function
• Structure theorem for Gaussian measures
References
• I.M. Gel'fand, N.Ya. Vilenkin, Generalized functions. Applications of harmonic analysis, Vol 4, Acad. Press (1968)
• R.A. Minlos (2001) [1994], "cylindrical measure", Encyclopedia of Mathematics, EMS Press
• R.A. Minlos (2001) [1994], "cylinder set", Encyclopedia of Mathematics, EMS Press
• L. Schwartz, Radon measures.
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Wikipedia
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Quasimorphism
In group theory, given a group $G$, a quasimorphism (or quasi-morphism) is a function $f:G\to \mathbb {R} $ which is additive up to bounded error, i.e. there exists a constant $D\geq 0$ such that $|f(gh)-f(g)-f(h)|\leq D$ for all $g,h\in G$. The least positive value of $D$ for which this inequality is satisfied is called the defect of $f$, written as $D(f)$. For a group $G$, quasimorphisms form a subspace of the function space $\mathbb {R} ^{G}$.
Examples
• Group homomorphisms and bounded functions from $G$ to $\mathbb {R} $ are quasimorphisms. The sum of a group homomorphism and a bounded function is also a quasimorphism, and functions of this form are sometimes referred to as "trivial" quasimorphisms.[1]
• Let $G=F_{S}$ be a free group over a set $S$. For a reduced word $w$ in $S$, we first define the big counting function $C_{w}:F_{S}\to \mathbb {Z} _{\geq 0}$, which returns for $g\in G$ the number of copies of $w$ in the reduced representative of $g$. Similarly, we define the little counting function $c_{w}:F_{S}\to \mathbb {Z} _{\geq 0}$, returning the maximum number of non-overlapping copies in the reduced representative of $g$. For example, $C_{aa}(aaaa)=3$ and $c_{aa}(aaaa)=2$. Then, a big counting quasimorphism (resp. little counting quasimorphism) is a function of the form $H_{w}(g)=C_{w}(g)-C_{w^{-1}}(g)$ (resp. $h_{w}(g)=c_{w}(g)-c_{w^{-1}}(g))$.
• The rotation number ${\text{rot}}:{\text{Homeo}}^{+}(S^{1})\to \mathbb {R} $ is a quasimorphism, where ${\text{Homeo}}^{+}(S^{1})$ denotes the orientation-preserving homeomorphisms of the circle.
Homogeneous
A quasimorphism is homogeneous if $f(g^{n})=nf(g)$ for all $g\in G,n\in \mathbb {Z} $. It turns out the study of quasimorphisms can be reduced to the study of homogeneous quasimorphisms, as every quasimorphism $f:G\to \mathbb {R} $ is a bounded distance away from a unique homogeneous quasimorphism ${\overline {f}}:G\to \mathbb {R} $, given by :
${\overline {f}}(g)=\lim _{n\to \infty }{\frac {f(g^{n})}{n}}$.
A homogeneous quasimorphism $f:G\to \mathbb {R} $ has the following properties:
• It is constant on conjugacy classes, i.e. $f(g^{-1}hg)=f(h)$ for all $g,h\in G$,
• If $G$ is abelian, then $f$ is a group homomorphism. The above remark implies that in this case all quasimorphisms are "trivial".
Integer-valued
One can also define quasimorphisms similarly in the case of a function $f:G\to \mathbb {Z} $. In this case, the above discussion about homogeneous quasimorphisms does not hold anymore, as the limit $\lim _{n\to \infty }f(g^{n})/n$ does not exist in $\mathbb {Z} $ in general.
For example, for $\alpha \in \mathbb {R} $, the map $\mathbb {Z} \to \mathbb {Z} :n\mapsto \lfloor \alpha n\rfloor $ is a quasimorphism. There is a construction of the real numbers as a quotient of quasimorphisms $\mathbb {Z} \to \mathbb {Z} $ by an appropriate equivalence relation, see Construction of the reals numbers from integers (Eudoxus reals).
Notes
1. Frigerio (2017), p. 12.
References
• Calegari, Danny (2009), scl, MSJ Memoirs, vol. 20, Mathematical Society of Japan, Tokyo, pp. 17–25, doi:10.1142/e018, ISBN 978-4-931469-53-2
• Frigerio, Roberto (2017), Bounded cohomology of discrete groups, Mathematical Surveys and Monographs, vol. 227, American Mathematical Society, Providence, RI, pp. 12–15, arXiv:1610.08339, doi:10.1090/surv/227, ISBN 978-1-4704-4146-3, S2CID 53640921
Further reading
• What is a Quasi-morphism? by D. Kotschick
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Wikipedia
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Nilpotent operator
In operator theory, a bounded operator T on a Hilbert space is said to be nilpotent if Tn = 0 for some n. It is said to be quasinilpotent or topologically nilpotent if its spectrum σ(T) = {0}.
Examples
In the finite-dimensional case, i.e. when T is a square matrix with complex entries, σ(T) = {0} if and only if T is similar to a matrix whose only nonzero entries are on the superdiagonal, by the Jordan canonical form. In turn this is equivalent to Tn = 0 for some n. Therefore, for matrices, quasinilpotency coincides with nilpotency.
This is not true when H is infinite-dimensional. Consider the Volterra operator, defined as follows: consider the unit square X = [0,1] × [0,1] ⊂ R2, with the Lebesgue measure m. On X, define the (kernel) function K by
$K(x,y)=\left\{{\begin{matrix}1,&{\mbox{if}}\;x\geq y\\0,&{\mbox{otherwise}}.\end{matrix}}\right.$
The Volterra operator is the corresponding integral operator T on the Hilbert space L2(0,1) given by
$Tf(x)=\int _{0}^{1}K(x,y)f(y)dy.$
The operator T is not nilpotent: take f to be the function that is 1 everywhere and direct calculation shows that Tn f ≠ 0 (in the sense of L2) for all n. However, T is quasinilpotent. First notice that K is in L2(X, m), therefore T is compact. By the spectral properties of compact operators, any nonzero λ in σ(T) is an eigenvalue. But it can be shown that T has no nonzero eigenvalues, therefore T is quasinilpotent.
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Wikipedia
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Quasinorm
In linear algebra, functional analysis and related areas of mathematics, a quasinorm is similar to a norm in that it satisfies the norm axioms, except that the triangle inequality is replaced by
$\|x+y\|\leq K(\|x\|+\|y\|)$
Not to be confused with seminorm or pseudonorm.
for some $K>0.$
Definition
A quasi-seminorm[1] on a vector space $X$ is a real-valued map $p$ on $X$ that satisfies the following conditions:
1. Non-negativity: $p\geq 0;$
2. Absolute homogeneity: $p(sx)=|s|p(x)$ for all $x\in X$ and all scalars $s;$
3. there exists a real $k\geq 1$ such that $p(x+y)\leq k[p(x)+p(y)]$ for all $x,y\in X.$
• If $k=1$ then this inequality reduces to the triangle inequality. It is in this sense that this condition generalizes the usual triangle inequality.
A quasinorm[1] is a quasi-seminorm that also satisfies:
1. Positive definite/Point-separating: if $x\in X$ satisfies $p(x)=0,$ then $x=0.$
A pair $(X,p)$ consisting of a vector space $X$ and an associated quasi-seminorm $p$ is called a quasi-seminormed vector space. If the quasi-seminorm is a quasinorm then it is also called a quasinormed vector space.
Multiplier
The infimum of all values of $k$ that satisfy condition (3) is called the multiplier of $p.$ The multiplier itself will also satisfy condition (3) and so it is the unique smallest real number that satisfies this condition. The term $k$-quasi-seminorm is sometimes used to describe a quasi-seminorm whose multiplier is equal to $k.$
A norm (respectively, a seminorm) is just a quasinorm (respectively, a quasi-seminorm) whose multiplier is $1.$ Thus every seminorm is a quasi-seminorm and every norm is a quasinorm (and a quasi-seminorm).
Topology
If $p$ is a quasinorm on $X$ then $p$ induces a vector topology on $X$ whose neighborhood basis at the origin is given by the sets:[2]
$\{x\in X:p(x)<1/n\}$
as $n$ ranges over the positive integers. A topological vector space with such a topology is called a quasinormed topological vector space or just a quasinormed space.
Every quasinormed topological vector space is pseudometrizable.
A complete quasinormed space is called a quasi-Banach space. Every Banach space is a quasi-Banach space, although not conversely.
Related definitions
See also: Banach algebra
A quasinormed space $(A,\|\,\cdot \,\|)$ is called a quasinormed algebra if the vector space $A$ is an algebra and there is a constant $K>0$ such that
$\|xy\|\leq K\|x\|\cdot \|y\|$
for all $x,y\in A.$
A complete quasinormed algebra is called a quasi-Banach algebra.
Characterizations
A topological vector space (TVS) is a quasinormed space if and only if it has a bounded neighborhood of the origin.[2]
Examples
Since every norm is a quasinorm, every normed space is also a quasinormed space.
$L^{p}$ spaces with $0<p<1$
The $L^{p}$ spaces for $0<p<1$ are quasinormed spaces (indeed, they are even F-spaces) but they are not, in general, normable (meaning that there might not exist any norm that defines their topology). For $0<p<1,$ the Lebesgue space $L^{p}([0,1])$ is a complete metrizable TVS (an F-space) that is not locally convex (in fact, its only convex open subsets are itself $L^{p}([0,1])$ and the empty set) and the only continuous linear functional on $L^{p}([0,1])$ is the constant $0$ function (Rudin 1991, §1.47). In particular, the Hahn-Banach theorem does not hold for $L^{p}([0,1])$ when $0<p<1.$
See also
• Metrizable topological vector space – A topological vector space whose topology can be defined by a metric
• Norm (mathematics) – Length in a vector space
• 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
• Topological vector space – Vector space with a notion of nearness
References
1. Kalton 1986, pp. 297–324.
2. Wilansky 2013, p. 55.
• Aull, Charles E.; Robert Lowen (2001). Handbook of the History of General Topology. Springer. ISBN 0-7923-6970-X.
• Conway, John B. (1990). A Course in Functional Analysis. Springer. ISBN 0-387-97245-5.
• Kalton, N. (1986). "Plurisubharmonic functions on quasi-Banach spaces" (PDF). Studia Mathematica. Institute of Mathematics, Polish Academy of Sciences. 84 (3): 297–324. doi:10.4064/sm-84-3-297-324. ISSN 0039-3223.
• Nikolʹskiĭ, Nikolaĭ Kapitonovich (1992). Functional Analysis I: Linear Functional Analysis. Encyclopaedia of Mathematical Sciences. Vol. 19. Springer. ISBN 3-540-50584-9.
• 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.
• Swartz, Charles (1992). An Introduction to Functional Analysis. CRC Press. ISBN 0-8247-8643-2.
• 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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Quasinormal operator
In operator theory, quasinormal operators is a class of bounded operators defined by weakening the requirements of a normal operator.
Every quasinormal operator is a subnormal operator. Every quasinormal operator on a finite-dimensional Hilbert space is normal.
Definition and some properties
Definition
Let A be a bounded operator on a Hilbert space H, then A is said to be quasinormal if A commutes with A*A, i.e.
$A(A^{*}A)=(A^{*}A)A.\,$
Properties
A normal operator is necessarily quasinormal.
Let A = UP be the polar decomposition of A. If A is quasinormal, then UP = PU. To see this, notice that the positive factor P in the polar decomposition is of the form (A*A)1⁄2, the unique positive square root of A*A. Quasinormality means A commutes with A*A. As a consequence of the continuous functional calculus for self-adjoint operators, A commutes with P = (A*A)1⁄2 also, i.e.
$UPP=PUP.\,$
So UP = PU on the range of P. On the other hand, if h ∈ H lies in kernel of P, clearly UP h = 0. But PU h = 0 as well. because U is a partial isometry whose initial space is closure of range P. Finally, the self-adjointness of P implies that H is the direct sum of its range and kernel. Thus the argument given proves UP = PU on all of H.
On the other hand, one can readily verify that if UP = PU, then A must be quasinormal. Thus the operator A is quasinormal if and only if UP = PU.
When H is finite dimensional, every quasinormal operator A is normal. This is because that in the finite dimensional case, the partial isometry U in the polar decomposition A = UP can be taken to be unitary. This then gives
$A^{*}A=(UP)^{*}UP=PU(PU)^{*}=AA^{*}.\,$
In general, a partial isometry may not be extendable to a unitary operator and therefore a quasinormal operator need not be normal. For example, consider the unilateral shift T. T is quasinormal because T*T is the identity operator. But T is clearly not normal.
Quasinormal invariant subspaces
It is not known that, in general, whether a bounded operator A on a Hilbert space H has a nontrivial invariant subspace. However, when A is normal, an affirmative answer is given by the spectral theorem. Every normal operator A is obtained by integrating the identity function with respect to a spectral measure E = {EB} on the spectrum of A, σ(A):
$A=\int _{\sigma (A)}\lambda \,dE(\lambda ).\,$
For any Borel set B ⊂ σ(A), the projection EB commutes with A and therefore the range of EB is an invariant subspace of A.
The above can be extended directly to quasinormal operators. To say A commutes with A*A is to say that A commutes with (A*A)1⁄2. But this implies that A commutes with any projection EB in the spectral measure of (A*A)1⁄2, which proves the invariant subspace claim. In fact, one can conclude something stronger. The range of EB is actually a reducing subspace of A, i.e. its orthogonal complement is also invariant under A.
References
• P. Halmos, A Hilbert Space Problem Book, Springer, New York 1982.
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Quasiperfect number
In mathematics, a quasiperfect number is a natural number n for which the sum of all its divisors (the divisor function σ(n)) is equal to 2n + 1. Equivalently, n is the sum of its non-trivial divisors (that is, its divisors excluding 1 and n). No quasiperfect numbers have been found so far.
The quasiperfect numbers are the abundant numbers of minimal abundance (which is 1).
Theorems
If a quasiperfect number exists, it must be an odd square number greater than 1035 and have at least seven distinct prime factors.[1]
Related
Numbers do exist where the sum of all the divisors σ(n) is equal to 2n + 2: 20, 104, 464, 650, 1952, 130304, 522752 ... (sequence A088831 in the OEIS). Many of these numbers are of the form 2n−1(2n − 3) where 2n − 3 is prime (instead of 2n − 1 with perfect numbers). In addition, numbers exist where the sum of all the divisors σ(n) is equal to 2n − 1, such as the powers of 2.
Betrothed numbers relate to quasiperfect numbers like amicable numbers relate to perfect numbers.
Notes
1. Hagis, Peter; Cohen, Graeme L. (1982). "Some results concerning quasiperfect numbers". J. Austral. Math. Soc. Ser. A. 33 (2): 275–286. doi:10.1017/S1446788700018401. MR 0668448.
References
• Brown, E.; Abbott, H.; Aull, C.; Suryanarayana, D. (1973). "Quasiperfect numbers" (PDF). Acta Arith. 22 (4): 439–447. doi:10.4064/aa-22-4-439-447. MR 0316368.
• Kishore, Masao (1978). "Odd integers N with five distinct prime factors for which 2−10−12 < σ(N)/N < 2+10−12" (PDF). Mathematics of Computation. 32 (141): 303–309. doi:10.2307/2006281. ISSN 0025-5718. JSTOR 2006281. MR 0485658. Zbl 0376.10005.
• Cohen, Graeme L. (1980). "On odd perfect numbers (ii), multiperfect numbers and quasiperfect numbers". J. Austral. Math. Soc. Ser. A. 29 (3): 369–384. doi:10.1017/S1446788700021376. ISSN 0263-6115. MR 0569525. S2CID 120459203. Zbl 0425.10005.
• James J. Tattersall (1999). Elementary number theory in nine chapters. Cambridge University Press. pp. 147. ISBN 0-521-58531-7. Zbl 0958.11001.
• Guy, Richard (2004). Unsolved Problems in Number Theory, third edition. Springer-Verlag. p. 74. ISBN 0-387-20860-7.
• Sándor, József; Mitrinović, Dragoslav S.; Crstici, Borislav, eds. (2006). Handbook of number theory I. Dordrecht: Springer-Verlag. pp. 109–110. ISBN 1-4020-4215-9. Zbl 1151.11300.
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Quasiperiodic function
In mathematics, a quasiperiodic function is a function that has a certain similarity to a periodic function.[1] A function $f$ is quasiperiodic with quasiperiod $\omega $ if $f(z+\omega )=g(z,f(z))$, where $g$ is a "simpler" function than $f$. What it means to be "simpler" is vague.
Not to be confused with Almost periodic function or Quasi-periodic oscillation.
A simple case (sometimes called arithmetic quasiperiodic) is if the function obeys the equation:
$f(z+\omega )=f(z)+C$
Another case (sometimes called geometric quasiperiodic) is if the function obeys the equation:
$f(z+\omega )=Cf(z)$
An example of this is the Jacobi theta function, where
$\vartheta (z+\tau ;\tau )=e^{-2\pi iz-\pi i\tau }\vartheta (z;\tau ),$ ;\tau )=e^{-2\pi iz-\pi i\tau }\vartheta (z;\tau ),}
shows that for fixed $\tau $ it has quasiperiod $\tau $; it also is periodic with period one. Another example is provided by the Weierstrass sigma function, which is quasiperiodic in two independent quasiperiods, the periods of the corresponding Weierstrass ℘ function.
Functions with an additive functional equation
$f(z+\omega )=f(z)+az+b\ $
are also called quasiperiodic. An example of this is the Weierstrass zeta function, where
$\zeta (z+\omega ,\Lambda )=\zeta (z,\Lambda )+\eta (\omega ,\Lambda )\ $
for a z-independent η when ω is a period of the corresponding Weierstrass ℘ function.
In the special case where $f(z+\omega )=f(z)\ $ we say f is periodic with period ω in the period lattice $\Lambda $.
Quasiperiodic signals
Quasiperiodic signals in the sense of audio processing are not quasiperiodic functions in the sense defined here; instead they have the nature of almost periodic functions and that article should be consulted. The more vague and general notion of quasiperiodicity has even less to do with quasiperiodic functions in the mathematical sense.
A useful example is the function:
$f(z)=\sin(Az)+\sin(Bz)$
If the ratio A/B is rational, this will have a true period, but if A/B is irrational there is no true period, but a succession of increasingly accurate "almost" periods.
See also
• Quasiperiodic motion
References
1. Mitropolsky, Yu A. (1993). Systems of Evolution Equations with Periodic and Quasiperiodic Coefficients. A. M. Samoilenko, D. I. Martinyuk. Dordrecht: Springer Netherlands. p. 108. ISBN 978-94-011-2728-8. OCLC 840309575.
External links
• Quasiperiodic function at PlanetMath
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Wikipedia
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Quasi-polynomial
In mathematics, a quasi-polynomial (pseudo-polynomial) is a generalization of polynomials. While the coefficients of a polynomial come from a ring, the coefficients of quasi-polynomials are instead periodic functions with integral period. Quasi-polynomials appear throughout much of combinatorics as the enumerators for various objects.
For quasi-polynomial time complexity of algorithms, see Quasi-polynomial time. For functions that are characteristic-functions of linear time-delay systems, see Delay_differential_equation § The_characteristic_equation.
A quasi-polynomial can be written as $q(k)=c_{d}(k)k^{d}+c_{d-1}(k)k^{d-1}+\cdots +c_{0}(k)$, where $c_{i}(k)$ is a periodic function with integral period. If $c_{d}(k)$ is not identically zero, then the degree of $q$ is $d$. Equivalently, a function $f\colon \mathbb {N} \to \mathbb {N} $ is a quasi-polynomial if there exist polynomials $p_{0},\dots ,p_{s-1}$ such that $f(n)=p_{i}(n)$ when $i\equiv n{\bmod {s}}$. The polynomials $p_{i}$ are called the constituents of $f$.
Examples
• Given a $d$-dimensional polytope $P$ with rational vertices $v_{1},\dots ,v_{n}$, define $tP$ to be the convex hull of $tv_{1},\dots ,tv_{n}$. The function $L(P,t)=\#(tP\cap \mathbb {Z} ^{d})$ is a quasi-polynomial in $t$ of degree $d$. In this case, $L(P,t)$ is a function $\mathbb {N} \to \mathbb {N} $. This is known as the Ehrhart quasi-polynomial, named after Eugène Ehrhart.
• Given two quasi-polynomials $F$ and $G$, the convolution of $F$ and $G$ is
$(F*G)(k)=\sum _{m=0}^{k}F(m)G(k-m)$
which is a quasi-polynomial with degree $\leq \deg F+\deg G+1.$
See also
• Ehrhart polynomial
References
• Stanley, Richard P. (1997). Enumerative Combinatorics, Volume 1. Cambridge University Press. ISBN 0-521-55309-1, 0-521-56069-1.
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Wikipedia
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Metzler matrix
In mathematics, a Metzler matrix is a matrix in which all the off-diagonal components are nonnegative (equal to or greater than zero):
$\forall _{i\neq j}\,x_{ij}\geq 0.$
It is named after the American economist Lloyd Metzler.
Metzler matrices appear in stability analysis of time delayed differential equations and positive linear dynamical systems. Their properties can be derived by applying the properties of nonnegative matrices to matrices of the form M + aI, where M is a Metzler matrix.
Definition and terminology
In mathematics, especially linear algebra, a matrix is called Metzler, quasipositive (or quasi-positive) or essentially nonnegative if all of its elements are non-negative except for those on the main diagonal, which are unconstrained. That is, a Metzler matrix is any matrix A which satisfies
$A=(a_{ij});\quad a_{ij}\geq 0,\quad i\neq j.$
Metzler matrices are also sometimes referred to as $Z^{(-)}$-matrices, as a Z-matrix is equivalent to a negated quasipositive matrix.
Properties
The exponential of a Metzler (or quasipositive) matrix is a nonnegative matrix because of the corresponding property for the exponential of a nonnegative matrix. This is natural, once one observes that the generator matrices of continuous-time finite-state Markov processes are always Metzler matrices, and that probability distributions are always non-negative.
A Metzler matrix has an eigenvector in the nonnegative orthant because of the corresponding property for nonnegative matrices.
Relevant theorems
• Perron–Frobenius theorem
See also
• Nonnegative matrices
• Delay differential equation
• M-matrix
• P-matrix
• Z-matrix
• Hurwitz matrix
• Stochastic matrix
• Positive systems
Bibliography
• Berman, Abraham; Plemmons, Robert J. (1994). Nonnegative Matrices in the Mathematical Sciences. SIAM. ISBN 0-89871-321-8.
• Farina, Lorenzo; Rinaldi, Sergio (2000). Positive Linear Systems: Theory and Applications. New York: Wiley Interscience.
• Berman, Abraham; Neumann, Michael; Stern, Ronald (1989). Nonnegative Matrices in Dynamical Systems. Pure and Applied Mathematics. New York: Wiley Interscience.
• Kaczorek, Tadeusz (2002). Positive 1D and 2D Systems. London: Springer.
• Luenberger, David (1979). Introduction to Dynamic Systems: Theory, Modes & Applications. John Wiley & Sons. pp. 204–206. ISBN 0-471-02594-1.
• Kemp, Murray C.; Kimura, Yoshio (1978). Introduction to Mathematical Economics. New York: Springer. pp. 102–114. ISBN 0-387-90304-6.
Matrix classes
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Quasirandom group
In mathematics, a quasirandom group is a group that does not contain a large product-free subset. Such groups are precisely those without a small non-trivial irreducible representation. The namesake of these groups stems from their connection to graph theory: bipartite Cayley graphs over any subset of a quasirandom group are always bipartite quasirandom graphs.
Motivation
The notion of quasirandom groups arises when considering subsets of groups for which no two elements in the subset have a product in the subset; such subsets are termed product-free. László Babai and Vera Sós asked about the existence of a constant $c$ for which every finite group $G$ with order $n$ has a product-free subset with size at least $cn$.[1] A well-known result of Paul Erdős about sum-free sets of integers can be used to prove that $ c={\frac {1}{3}}$ suffices for abelian groups, but it turns out that such a constant does not exist for non-abelian groups.[2]
Both non-trivial lower and upper bounds are now known for the size of the largest product-free subset of a group with order $n$. A lower bound of $ cn^{\frac {11}{14}}$ can be proved by taking a large subset of a union of sufficiently many cosets,[3] and an upper bound of $ cn^{\frac {8}{9}}$ is given by considering the projective special linear group $\operatorname {PSL} (2,p)$ for any prime $p$.[4] In the process of proving the upper bound, Timothy Gowers defined the notion of a quasirandom group to encapsulate the product-free condition and proved equivalences involving quasirandomness in graph theory.
Graph quasirandomness
Formally, it does not make sense to talk about whether or not a single group is quasirandom. The strict definition of quasirandomness will apply to sequences of groups, but first bipartite graph quasirandomness must be defined. The motivation for considering sequences of groups stems from its connections with graphons, which are defined as limits of graphs in a certain sense.
Fix a real number $p\in [0,1].$ A sequence of bipartite graphs $(G_{n})$ (here $n$ is allowed to skip integers as long as $n$ tends to infinity) with $G_{n}$ having $n$ vertices, vertex parts $A_{n}$ and $B_{n}$, and $(p+o(1))|A_{n}||B_{n}|$ edges is quasirandom if any of the following equivalent conditions hold:
• For every bipartite graph $H$ with vertex parts $A'$ and $B'$, the number of labeled copies of $H$ in $G_{n}$ with $A'$ embedded in $A$ and $B'$ embedded in $B$ is $ \left(p^{e(H)}+o(1)\right)|A|^{|A'|}|B|^{|B'|}.$ Here, the function $o(1)$ is allowed to depend on $H.$
• The number of closed, labeled walks of length 4 in $G_{n}$ starting in $A_{n}$ is $(p^{4}+o(1))|A_{n}|^{2}|B_{n}|^{2}.$
• The number of edges between $A'$ and $B'$ is $p|A'||B'|+n^{2}o(1)$ for any pair of subsets $A'\subseteq A_{n}$ and $B'\subseteq B_{n}.$
• $\sum \limits _{a_{1},a_{2}\in A_{n}}N(a_{1},a_{2})^{2}=(p^{4}+o(1))|A_{n}|^{2}|B_{n}|^{2}$, where $N(u,v)$ denotes the number of common neighbors of $u$ and $v.$
• $\sum \limits _{b_{1},b_{2}\in B_{n}}N(b_{1},b_{2})^{2}=(p^{4}+o(1))|A_{n}|^{2}|B_{n}|^{2}.$
• The largest eigenvalue of $G_{n}$'s adjacency matrix is $ (p+o(1)){\sqrt {|A||B|}}$ and all other eigenvalues have magnitude at most $ {\sqrt {|A||B|}}o(1).$
It is a result of Chung–Graham–Wilson that each of the above conditions is equivalent.[5] Such graphs are termed quasirandom because each condition asserts that the quantity being considered is approximately what one would expect if the bipartite graph was generated according to the Erdős–Rényi random graph model; that is, generated by including each possible edge between $A_{n}$ and $B_{n}$ independently with probability $p.$
Though quasirandomness can only be defined for sequences of graphs, a notion of $c$-quasirandomness can be defined for a specific graph by allowing an error tolerance in any of the above definitions of graph quasirandomness. To be specific, given any of the equivalent definitions of quasirandomness, the $o(1)$ term can be replaced by a small constant $c>0$, and any graph satisfying that particular modified condition can be termed $c$-quasirandom. It turns out that $c$-quasirandomness under any condition is equivalent to $c^{k}$-quasirandomness under any other condition for some absolute constant $k\geq 1.$
The next step for defining group quasirandomness is the Cayley graph. Bipartite Cayley graphs give a way from translating quasirandomness in the graph-theoretic setting to the group-theoretic setting.
Given a finite group $\Gamma $ and a subset $S\subseteq \Gamma $, the bipartite Cayley graph $\operatorname {BiCay} (\Gamma ,S)$ is the bipartite graph with vertex sets $A$ and $B$, each labeled by elements of $G$, whose edges $a\sim b$ are between vertices whose ratio $ab^{-1}$ is an element of $S.$
Definition
With the tools defined above, one can now define group quasirandomness. A sequence of groups $(\Gamma _{n})$ with $|\Gamma _{n}|=n$ (again, $n$ is allowed to skip integers) is quasirandom if for every real number $p\in [0,1]$ and choice of subsets $S_{n}\in \Gamma _{n}$ with $|S_{n}|=(p+o(1))|\Gamma _{n}|$, the sequence of graphs $\operatorname {BiCay} (\Gamma _{n},S_{n})$ is quasirandom.[4]
Though quasirandomness can only be defined for sequences of groups, the concept of $c$-quasirandomness for specific groups can be extended to groups using the definition of $c$-quasirandomness for specific graphs.
Properties
As proved by Gowers, group quasirandomness turns out to be equivalent to a number of different conditions.
To be precise, given a sequence of groups $(\Gamma _{n})$, the following are equivalent:
• $(\Gamma _{n})$ is quasirandom; that is, all sequences of Cayley graphs defined by $(\Gamma _{n})$ are quasirandom.
• The dimension of the smallest non-trivial representation of $\Gamma _{n}$ is unbounded.
• The size of the largest product-free subset of $\Gamma _{n}$ is $o(|\Gamma _{n}|).$
• The size of the smallest non-trivial quotient of $\Gamma _{n}$ is unbounded.[4]
Cayley graphs generated from pseudorandom groups have strong mixing properties; that is, $\operatorname {BiCay} (\Gamma _{n},S)$ is a bipartite $(n,d,\lambda )$-graph for some $\lambda $ tending to zero as $n$ tends to infinity. (Recall that an $(n,d,\lambda )$ graph is a graph with $n$ vertices, each with degree $d$, whose adjacency matrix has a second largest eigenvalue of at most $\lambda .$)
In fact, it can be shown that for any $c$-quasirandom group $\Gamma $, the number of solutions to $xy=z$ with $x\in X$, $y\in Y$, and $z\in Z$ is approximately equal to what one might expect if $S$ was chosen randomly; that is, approximately equal to ${\tfrac {|X||Y||Z|}{|\Gamma |}}.$ This result follows from a direct application of the expander mixing lemma.
Examples
There are several notable families of quasirandom groups. In each case, the quasirandomness properties are most easily verified by checking the dimension of its smallest non-trivial representation.
• The projective special linear groups $\operatorname {PSL} (2,p)$ for prime $p$ form a sequence of quasirandom groups, since a classic result of Frobenius states that its smallest non-trivial representation has dimension at least ${\tfrac {1}{2}}(p-1).$ In fact, these groups are the groups with the largest known minimal non-trivial representation, as a function of group order.
• The alternating groups $(A_{n})$ are quasirandom, since its smallest non-trivial representation has dimension $n-1.$
• Any sequence of non-cyclic simple groups with increasing order is quasirandom, since its smallest non-trivial representation has dimension at least ${\tfrac {1}{2}}{\sqrt {\log n}}$, where $n$ is the order of the group.[4]
References
1. Babai, László; Sós, Vera T. (1985), "Sidon sets in groups and induced subgraphs of Cayley graphs", European Journal of Combinatorics, 6: 101–114, doi:10.1016/S0195-6698(85)80001-9
2. Erdős, P. (1965), "Extremal problems in number theory", Proceedings of the Symp. Pure Math. VIII, American Mathematical Society, pp. 181–189
3. Kedlaya, Kiran S. (1997), "Large product-free subsets of finite groups", Journal of Combinatorial Theory, Series A, 77: 339–343, doi:10.1006/jcta.1997.2715
4. Gowers, W.T. (2008), "Quasirandom Groups", Combinatorics, Probability and Computing, 17 (3): 363–387, doi:10.1017/S0963548307008826
5. Chung, F. R. K.; Graham, R. L.; Wilson, R. M. (1989), "Quasi-Random Graphs", Combinatorica, 9 (4): 345–362, doi:10.1007/BF02125347, S2CID 17166765
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Wikipedia
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Reflexive space
In the area of mathematics known as functional analysis, a reflexive space is a locally convex topological vector space (TVS) for which the canonical evaluation map from $X$ into its bidual (which is the strong dual of the strong dual of $X$) is an isomorphism of TVSs. Since a normable TVS is reflexive if and only if it is semi-reflexive, every normed space (and so in particular, every Banach space) $X$ is reflexive if and only if the canonical evaluation map from $X$ into its bidual is surjective; in this case the normed space is necessarily also a Banach space. In 1951, R. C. James discovered a Banach space, now known as James' space, that is not reflexive but is nevertheless isometrically isomorphic to its bidual (any such isomorphism is thus necessarily not the canonical evaluation map).
Reflexive spaces play an important role in the general theory of locally convex TVSs and in the theory of Banach spaces in particular. Hilbert spaces are prominent examples of reflexive Banach spaces. Reflexive Banach spaces are often characterized by their geometric properties.
Definition
Definition of the bidual
Main article: Bidual
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 $X$ (that is, for any $x\in X,x\neq 0$ there exists some $x^{\prime }\in X^{\prime }$ such that $x^{\prime }(x)\neq 0$). Let $X_{b}^{\prime }$ and $X_{b}^{\prime }$ 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] If $X$ is a normed space, then $X^{\prime \prime }$ is the continuous dual space of the Banach space $X_{b}^{\prime }$ with its usual norm topology.
Definitions of the evaluation map and reflexive spaces
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 a linear map 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 linear 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. Call $X$ semi-reflexive 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] A normable space is reflexive if and only if it is semi-reflexive or equivalently, if and only if the evaluation map is surjective.
Reflexive Banach spaces
Suppose $X$ is a normed vector space over the number field $\mathbb {F} =\mathbb {R} $ or $\mathbb {F} =\mathbb {C} $ (the real numbers or the complex numbers), with a norm $\|\,\cdot \,\|.$ Consider its dual normed space $X^{\prime },$ that consists of all continuous linear functionals $f:X\to \mathbb {F} $ and is equipped with the dual norm $\|\,\cdot \,\|^{\prime }$ defined by
$\|f\|^{\prime }=\sup\{|f(x)|\,:\,x\in X,\ \|x\|=1\}.$
The dual $X^{\prime }$ is a normed space (a Banach space to be precise), and its dual normed space $X^{\prime \prime }=\left(X^{\prime }\right)^{\prime }$ is called bidual space for $X.$ The bidual consists of all continuous linear functionals $h:X^{\prime }\to \mathbb {F} $ and is equipped with the norm $\|\,\cdot \,\|^{\prime \prime }$ dual to $\|\,\cdot \,\|^{\prime }.$ Each vector $x\in X$ generates a scalar function $J(x):X^{\prime }\to \mathbb {F} $ by the formula:
$J(x)(f)=f(x)\qquad {\text{ for all }}f\in X^{\prime },$
and $J(x)$ is a continuous linear functional on $X^{\prime },$ that is, $J(x)\in X^{\prime \prime }.$ One obtains in this way a map
$J:X\to X^{\prime \prime }$
called evaluation map, that is linear. It follows from the Hahn–Banach theorem that $J$ is injective and preserves norms:
${\text{ for all }}x\in X\qquad \|J(x)\|^{\prime \prime }=\|x\|,$
that is, $J$ maps $X$ isometrically onto its image $J(X)$ in $X^{\prime \prime }.$ Furthermore, the image $J(X)$ is closed in $X^{\prime \prime },$ but it need not be equal to $X^{\prime \prime }.$
A normed space $X$ is called reflexive if it satisfies the following equivalent conditions:
1. the evaluation map $J:X\to X^{\prime \prime }$ is surjective,
2. the evaluation map $J:X\to X^{\prime \prime }$ is an isometric isomorphism of normed spaces,
3. the evaluation map $J:X\to X^{\prime \prime }$ is an isomorphism of normed spaces.
A reflexive space $X$ is a Banach space, since $X$ is then isometric to the Banach space $X^{\prime \prime }.$
Remark
A Banach space $X$ is reflexive if it is linearly isometric to its bidual under this canonical embedding $J.$ James' space is an example of a non-reflexive space which is linearly isometric to its bidual. Furthermore, the image of James' space under the canonical embedding $J$ has codimension one in its bidual. [2] A Banach space $X$ is called quasi-reflexive (of order $d$) if the quotient $X^{\prime \prime }/J(X)$ has finite dimension $d.$
Examples
1. Every finite-dimensional normed space is reflexive, simply because in this case, the space, its dual and bidual all have the same linear dimension, hence the linear injection $J$ from the definition is bijective, by the rank–nullity theorem.
2. The Banach space $c_{0}$ of scalar sequences tending to 0 at infinity, equipped with the supremum norm, is not reflexive. It follows from the general properties below that $\ell ^{1}$ and $\ell ^{\infty }$ are not reflexive, because $\ell ^{1}$ is isomorphic to the dual of $c_{0}$ and $\ell ^{\infty }$ is isomorphic to the dual of $\ell ^{1}.$
3. All Hilbert spaces are reflexive, as are the Lp spaces $L^{p}$ for $1<p<\infty .$ More generally: all uniformly convex Banach spaces are reflexive according to the Milman–Pettis theorem. The $L^{1}(\mu )$ and $L^{\infty }(\mu )$ spaces are not reflexive (unless they are finite dimensional, which happens for example when $\mu $ is a measure on a finite set). Likewise, the Banach space $C([0,1])$ of continuous functions on $[0,1]$ is not reflexive.
4. The spaces $S_{p}(H)$ of operators in the Schatten class on a Hilbert space $H$ are uniformly convex, hence reflexive, when $1<p<\infty .$ When the dimension of $H$ is infinite, then $S_{1}(H)$ (the trace class) is not reflexive, because it contains a subspace isomorphic to $\ell ^{1},$ and $S_{\infty }(H)=L(H)$ (the bounded linear operators on $H$) is not reflexive, because it contains a subspace isomorphic to $\ell ^{\infty }.$ In both cases, the subspace can be chosen to be the operators diagonal with respect to a given orthonormal basis of $H.$
Properties
Since every finite-dimensional normed space is a reflexive Banach space, only infinite-dimensional spaces can be non-reflexive.
If a Banach space $Y$ is isomorphic to a reflexive Banach space $X$ then $Y$ is reflexive.[3]
Every closed linear subspace of a reflexive space is reflexive. The continuous dual of a reflexive space is reflexive. Every quotient of a reflexive space by a closed subspace is reflexive.[4]
Let $X$ be a Banach space. The following are equivalent.
1. The space $X$ is reflexive.
2. The continuous dual of $X$ is reflexive.[5]
3. The closed unit ball of $X$ is compact in the weak topology. (This is known as Kakutani's Theorem.)[6]
4. Every bounded sequence in $X$ has a weakly convergent subsequence.[7]
5. The statement of Riesz's lemma holds when the real number[note 1] is exactly $1.$[8] Explicitly, for every closed proper vector subspace $Y$ of $X,$ there exists some vector $u\in X$ of unit norm $\|u\|=1$ such that $\|u-y\|\geq 1$ for all $y\in Y.$
• Using $d(u,Y):=\inf _{y\in Y}\|u-y\|$ to denote the distance between the vector $u$ and the set $Y,$ this can be restated in simpler language as: $X$ is reflexive if and only if for every closed proper vector subspace $Y,$ there is some vector $u$ on the unit sphere of $X$ that is always at least a distance of $1=d(u,Y)$ away from the subspace.
• For example, if the reflexive Banach space $X=\mathbb {R} ^{3}$ is endowed with the usual Euclidean norm and $Y=\mathbb {R} \times \mathbb {R} \times \{0\}$ is the $x-y$ plane then the points $u=(0,0,\pm 1)$ satisfy the conclusion $d(u,Y)=1.$ If $Y$ is instead the $z$-axis then every point belonging to the unit circle in the $x-y$ plane satisfies the conclusion.
6. Every continuous linear functional on $X$ attains its supremum on the closed unit ball in $X.$[9] (James' theorem)
Since norm-closed convex subsets in a Banach space are weakly closed,[10] it follows from the third property that closed bounded convex subsets of a reflexive space $X$ are weakly compact. Thus, for every decreasing sequence of non-empty closed bounded convex subsets of $X,$ the intersection is non-empty. As a consequence, every continuous convex function $f$ on a closed convex subset $C$ of $X,$ such that the set
$C_{t}=\{x\in C\,:\,f(x)\leq t\}$
is non-empty and bounded for some real number $t,$ attains its minimum value on $C.$
The promised geometric property of reflexive Banach spaces is the following: if $C$ is a closed non-empty convex subset of the reflexive space $X,$ then for every $x\in X$ there exists a $c\in C$ such that $\|x-c\|$ minimizes the distance between $x$ and points of $C.$ This follows from the preceding result for convex functions, applied to$f(y)+\|y-x\|.$ Note that while the minimal distance between $x$ and $C$ is uniquely defined by $x,$ the point $c$ is not. The closest point $c$ is unique when $X$ is uniformly convex.
A reflexive Banach space is separable if and only if its continuous dual is separable. This follows from the fact that for every normed space $Y,$ separability of the continuous dual $Y^{\prime }$ implies separability of $Y.$[11]
Super-reflexive space
Informally, a super-reflexive Banach space $X$ has the following property: given an arbitrary Banach space $Y,$ if all finite-dimensional subspaces of $Y$ have a very similar copy sitting somewhere in $X,$ then $Y$ must be reflexive. By this definition, the space $X$ itself must be reflexive. As an elementary example, every Banach space $Y$ whose two dimensional subspaces are isometric to subspaces of $X=\ell ^{2}$ satisfies the parallelogram law, hence[12] $Y$ is a Hilbert space, therefore $Y$ is reflexive. So $\ell ^{2}$ is super-reflexive.
The formal definition does not use isometries, but almost isometries. A Banach space $Y$ is finitely representable[13] in a Banach space $X$ if for every finite-dimensional subspace $Y_{0}$ of $Y$ and every $\epsilon >0,$ there is a subspace $X_{0}$ of $X$ such that the multiplicative Banach–Mazur distance between $X_{0}$ and $Y_{0}$ satisfies
$d\left(X_{0},Y_{0}\right)<1+\varepsilon .$
A Banach space finitely representable in $\ell ^{2}$ is a Hilbert space. Every Banach space is finitely representable in $c_{0}.$ The Lp space $L^{p}([0,1])$ is finitely representable in $\ell ^{p}.$
A Banach space $X$ is super-reflexive if all Banach spaces $Y$ finitely representable in $X$ are reflexive, or, in other words, if no non-reflexive space $Y$ is finitely representable in $X.$ The notion of ultraproduct of a family of Banach spaces[14] allows for a concise definition: the Banach space $X$ is super-reflexive when its ultrapowers are reflexive.
James proved that a space is super-reflexive if and only if its dual is super-reflexive.[13]
Finite trees in Banach spaces
One of James' characterizations of super-reflexivity uses the growth of separated trees.[15] The description of a vectorial binary tree begins with a rooted binary tree labeled by vectors: a tree of height $n$ in a Banach space $X$ is a family of $2^{n+1}-1$ vectors of $X,$ that can be organized in successive levels, starting with level 0 that consists of a single vector $x_{\varnothing },$ the root of the tree, followed, for $k=1,\ldots ,n,$ by a family of $s^{k}$2 vectors forming level $k:$
$\left\{x_{\varepsilon _{1},\ldots ,\varepsilon _{k}}\right\},\quad \varepsilon _{j}=\pm 1,\quad j=1,\ldots ,k,$
that are the children of vertices of level $k-1.$ In addition to the tree structure, it is required here that each vector that is an internal vertex of the tree be the midpoint between its two children:
$x_{\emptyset }={\frac {x_{1}+x_{-1}}{2}},\quad x_{\varepsilon _{1},\ldots ,\varepsilon _{k}}={\frac {x_{\varepsilon _{1},\ldots ,\varepsilon _{k},1}+x_{\varepsilon _{1},\ldots ,\varepsilon _{k},-1}}{2}},\quad 1\leq k<n.$
Given a positive real number $t,$ the tree is said to be $t$-separated if for every internal vertex, the two children are $t$-separated in the given space norm:
$\left\|x_{1}-x_{-1}\right\|\geq t,\quad \left\|x_{\varepsilon _{1},\ldots ,\varepsilon _{k},1}-x_{\varepsilon _{1},\ldots ,\varepsilon _{k},-1}\right\|\geq t,\quad 1\leq k<n.$
Theorem.[15] The Banach space $X$ is super-reflexive if and only if for every $t\in (0,2\pi ],$ there is a number $n(t)$ such that every $t$-separated tree contained in the unit ball of $X$ has height less than $n(t).$
Uniformly convex spaces are super-reflexive.[15] Let $X$ be uniformly convex, with modulus of convexity $\delta _{X}$ and let $t$ be a real number in $(0,2].$ By the properties of the modulus of convexity, a $t$-separated tree of height $n,$ contained in the unit ball, must have all points of level $n-1$ contained in the ball of radius $1-\delta _{X}(t)<1.$ By induction, it follows that all points of level $n-k$ are contained in the ball of radius
$\left(1-\delta _{X}(t)\right)^{j},\ j=1,\ldots ,n.$
If the height $n$ was so large that
$\left(1-\delta _{X}(t)\right)^{n-1}<t/2,$
then the two points $x_{1},x_{-1}$ of the first level could not be $t$-separated, contrary to the assumption. This gives the required bound $n(t),$ function of $\delta _{X}(t)$ only.
Using the tree-characterization, Enflo proved[16] that super-reflexive Banach spaces admit an equivalent uniformly convex norm. Trees in a Banach space are a special instance of vector-valued martingales. Adding techniques from scalar martingale theory, Pisier improved Enflo's result by showing[17] that a super-reflexive space $X$ admits an equivalent uniformly convex norm for which the modulus of convexity satisfies, for some constant $c>0$ and some real number $q\geq 2,$
$\delta _{X}(t)\geq c\,t^{q},\quad {\text{ whenever }}t\in [0,2].$
Reflexive locally convex spaces
The notion of reflexive Banach space can be generalized to topological vector spaces in the following way.
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:
$J(x)(f)=f(x),\qquad f\in X^{\prime }.$
This is a continuous linear functional on $X_{b}^{\prime },$ that is,, $J(x)\in \left(X_{b}^{\prime }\right)_{b}^{\prime }.$ This induces a map called the evaluation map:
$J:X\to \left(X_{b}^{\prime }\right)_{b}^{\prime }.$
This map is linear. 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),
• reflexive if the evaluation map $J:X\to \left(X_{b}^{\prime }\right)_{b}^{\prime }$ is surjective and continuous (in this case $J$ is an isomorphism of topological vector spaces[18]).
Theorem[19] — A locally convex Hausdorff space $X$ is semi-reflexive if and only if $X$ with the $\sigma (X,X^{*})$-topology has the Heine–Borel property (i.e. weakly closed and bounded subsets of $X$ are weakly compact).
Theorem[20][21] — A locally convex space $X$ is reflexive if and only if it is semi-reflexive and barreled.
Theorem[22] — The strong dual of a semireflexive space is barrelled.
Theorem[23] — 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 infrabarreled.
Semireflexive spaces
Main article: Semi-reflexive space
Characterizations
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;[24]
4. $X_{\tau }^{\prime }$ is barreled;[24]
5. $X$ with the weak topology $\sigma \left(X,X^{\prime }\right)$ is quasi-complete.[24]
Characterizations of reflexive spaces
If $X$ is a Hausdorff locally convex space then the following are equivalent:
1. $X$ is reflexive;
2. $X$ is semireflexive and infrabarreled;[23]
3. $X$ is semireflexive and barreled;
4. $X$ is barreled and 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]
5. $X$ is semireflexive and quasibarrelled.[25]
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 $\sigma \left(X,X^{\prime }\right).$[26]
3. $X$ is a Banach space and $X_{b}^{\prime }$ is reflexive.[27]
4. Every sequence $\left(C_{n}\right)_{n=1}^{\infty },$ with $C_{n+1}\subseteq C_{n}$ for all $n$ of nonempty closed bounded convex subsets of $X$ has nonempty intersection.[28]
Theorem[29] — A real Banach space is reflexive if and only if every pair of non-empty disjoint closed convex subsets, one of which is bounded, can be strictly separated by a hyperplane.
James' theorem — A Banach space $B$ is reflexive if and only if every continuous linear functional on $B$ attains its supremum on the closed unit ball in $B.$
Sufficient conditions
Normed spaces
A normed space that is semireflexive is a reflexive Banach space.[30] A closed vector subspace of a reflexive Banach space is reflexive.[23]
Let $X$ be a Banach space and $M$ a closed vector subspace of $X.$ If two of $X,M,$ and $X/M$ are reflexive then they all are.[23] This is why reflexivity is referred to as a three-space property.[23]
Topological vector spaces
If a barreled locally convex Hausdorff space is semireflexive then it is reflexive.[1]
The strong dual of a reflexive space is reflexive.[31]Every Montel space is reflexive.[26] And the strong dual of a Montel space is a Montel space (and thus is reflexive).[26]
Properties
A locally convex Hausdorff reflexive space is barrelled. If $X$ is a normed space then $I:X\to X^{\prime \prime }$ is an isometry onto a closed subspace of $X^{\prime \prime }.$[30] This isometry can be expressed by:
$\|x\|=\sup _{\stackrel {x^{\prime }\in X^{\prime },}{\|x^{\prime }\|\leq 1}}\left|\left\langle x^{\prime },x\right\rangle \right|.$
Suppose that $X$ is a normed space and $X^{\prime \prime }$ is its bidual equipped with the bidual norm. Then the unit ball of $X,$ $I(\{x\in X:\|x\|\leq 1\})$ is dense in the unit ball $\left\{x^{\prime \prime }\in X^{\prime \prime }:\left\|x^{\prime \prime }\right\|\leq 1\right\}$ of $X^{\prime \prime }$ for the weak topology $\sigma \left(X^{\prime \prime },X^{\prime }\right).$[30]
Examples
1. Every finite-dimensional Hausdorff topological vector space is reflexive, because $J$ is bijective by linear algebra, and because there is a unique Hausdorff vector space topology on a finite dimensional vector space.
2. A normed space $X$ is reflexive as a normed space if and only if it is reflexive as a locally convex space. This follows from the fact that for a normed space $X$ its dual normed space $X^{\prime }$ coincides as a topological vector space with the strong dual space $X_{b}^{\prime }.$ As a corollary, the evaluation map $J:X\to X^{\prime \prime }$ coincides with the evaluation map $J:X\to \left(X_{b}^{\prime }\right)_{b}^{\prime },$ and the following conditions become equivalent:
1. $X$ is a reflexive normed space (that is, $J:X\to X^{\prime \prime }$ is an isomorphism of normed spaces),
2. $X$ is a reflexive locally convex space (that is, $J:X\to \left(X_{b}^{\prime }\right)_{b}^{\prime }$ is an isomorphism of topological vector spaces[18]),
3. $X$ is a semi-reflexive locally convex space (that is, $J:X\to \left(X_{b}^{\prime }\right)_{b}^{\prime }$ is surjective).
3. A (somewhat artificial) example of a semi-reflexive space that is not reflexive is obtained as follows: let $Y$ be an infinite dimensional reflexive Banach space, and let $X$ be the topological vector space $\left(Y,\sigma \left(Y,Y^{\prime }\right)\right),$ that is, the vector space $Y$ equipped with the weak topology. Then the continuous dual of $X$ and $Y^{\prime }$ are the same set of functionals, and bounded subsets of $X$ (that is, weakly bounded subsets of $Y$) are norm-bounded, hence the Banach space $Y^{\prime }$ is the strong dual of $X.$ Since $Y$ is reflexive, the continuous dual of $X^{\prime }=Y^{\prime }$ is equal to the image $J(X)$ of $X$ under the canonical embedding $J,$ but the topology on $X$ (the weak topology of $Y$) is not the strong topology $\beta \left(X,X^{\prime }\right),$ that is equal to the norm topology of $Y.$
4. Montel spaces are reflexive locally convex topological vector spaces. In particular, the following functional spaces frequently used in functional analysis are reflexive locally convex spaces:[32]
• the space $C^{\infty }(M)$ of smooth functions on arbitrary (real) smooth manifold $M,$ and its strong dual space $\left(C^{\infty }\right)^{\prime }(M)$ of distributions with compact support on $M,$
• the space ${\mathcal {D}}(M)$ of smooth functions with compact support on arbitrary (real) smooth manifold $M,$ and its strong dual space ${\mathcal {D}}^{\prime }(M)$ of distributions on $M,$
• the space ${\mathcal {O}}(M)$ of holomorphic functions on arbitrary complex manifold $M,$ and its strong dual space ${\mathcal {O}}^{\prime }(M)$ of analytic functionals on $M,$
• the Schwartz space ${\mathcal {S}}\left(\mathbb {R} ^{n}\right)$ on $\mathbb {R} ^{n},$ and its strong dual space ${\mathcal {S}}^{\prime }\left(\mathbb {R} ^{n}\right)$ of tempered distributions on $\mathbb {R} ^{n}.$
Counter-examples
• There exists a non-reflexive locally convex TVS whose strong dual is reflexive.[33]
Other types of reflexivity
A stereotype space, or polar reflexive space, is defined as a topological vector space (TVS) satisfying a similar condition of reflexivity, but with the topology of uniform convergence on totally bounded subsets (instead of bounded subsets) in the definition of dual space $X^{\prime }.$ More precisely, a TVS $X$ is called polar reflexive[34] or stereotype if the evaluation map into the second dual space
$J:X\to X^{\star \star },\quad J(x)(f)=f(x),\quad x\in X,\quad f\in X^{\star }$
is an isomorphism of topological vector spaces.[18] Here the stereotype dual space $X^{\star }$ is defined as the space of continuous linear functionals $X^{\prime }$ endowed with the topology of uniform convergence on totally bounded sets in $X$ (and the stereotype second dual space $X^{\star \star }$ is the space dual to $X^{\star }$ in the same sense).
In contrast to the classical reflexive spaces the class Ste of stereotype spaces is very wide (it contains, in particular, all Fréchet spaces and thus, all Banach spaces), it forms a closed monoidal category, and it admits standard operations (defined inside of Ste) of constructing new spaces, like taking closed subspaces, quotient spaces, projective and injective limits, the space of operators, tensor products, etc. The category Ste have applications in duality theory for non-commutative groups.
Similarly, one can replace the class of bounded (and totally bounded) subsets in $X$ in the definition of dual space $X^{\prime },$ by other classes of subsets, for example, by the class of compact subsets in $X$ – the spaces defined by the corresponding reflexivity condition are called reflective,[35][36] and they form an even wider class than Ste, but it is not clear (2012), whether this class forms a category with properties similar to those of Ste.
See also
• Grothendieck space
• A generalization which has some of the properties of reflexive spaces and includes many spaces of practical importance is the concept of Grothendieck space.
• Reflexive operator algebra
References
Notes
1. The statement of Riesz's lemma involves only one real number, which is denoted by $\alpha $ in the article on Riesz's lemma. The lemma always holds for all real $\alpha <1.$ But for a Banach space, the lemma holds for all $\alpha \leq 1$ if and only if the space is reflexive.
Citations
1. Trèves 2006, pp. 372–374.
2. Robert C. James (1951). "A non-reflexive Banach space isometric with its second conjugate space". Proc. Natl. Acad. Sci. U.S.A. 37 (3): 174–177. Bibcode:1951PNAS...37..174J. doi:10.1073/pnas.37.3.174. PMC 1063327. PMID 16588998.
3. Proposition 1.11.8 in Megginson (1998, p. 99).
4. Megginson (1998, pp. 104–105).
5. Corollary 1.11.17, p. 104 in Megginson (1998).
6. Conway 1985, Theorem V.4.2, p. 135.
7. Since weak compactness and weak sequential compactness coincide by the Eberlein–Šmulian theorem.
8. Diestel 1984, p. 6.
9. Theorem 1.13.11 in Megginson (1998, p. 125).
10. Theorem 2.5.16 in Megginson (1998, p. 216).
11. Theorem 1.12.11 and Corollary 1.12.12 in Megginson (1998, pp. 112–113).
12. see this characterization of Hilbert space among Banach spaces
13. James, Robert C. (1972), "Super-reflexive Banach spaces", Can. J. Math. 24:896–904.
14. Dacunha-Castelle, Didier; Krivine, Jean-Louis (1972), "Applications des ultraproduits à l'étude des espaces et des algèbres de Banach" (in French), Studia Math. 41:315–334.
15. see James (1972).
16. Enflo, Per (1972). "Banach spaces which can be given an equivalent uniformly convex norm". Israel Journal of Mathematics. 13: 281–288. doi:10.1007/BF02762802.
17. Pisier, Gilles (1975). "Martingales with values in uniformly convex spaces". Israel Journal of Mathematics. 20: 326–350. doi:10.1007/BF02760337.
18. An isomorphism of topological vector spaces is a linear and a homeomorphic map $\varphi :X\to Y.$
19. Edwards 1965, 8.4.2.
20. Schaefer 1966, 5.6, 5.5.
21. Edwards 1965, 8.4.5.
22. Edwards 1965, 8.4.3.
23. Narici & Beckenstein 2011, pp. 488–491.
24. Schaefer & Wolff 1999, p. 144.
25. Khaleelulla 1982, pp. 32–63.
26. Trèves 2006, p. 376.
27. Trèves 2006, p. 377.
28. Bernardes 2012.
29. Narici & Beckenstein 2011, pp. 212.
30. Trèves 2006, p. 375.
31. Schaefer & Wolff 1999, p. 145.
32. Edwards 1965, 8.4.7.
33. Schaefer & Wolff 1999, pp. 190–202.
34. Köthe, Gottfried (1983). Topological Vector Spaces I. Springer Grundlehren der mathematischen Wissenschaften. Springer. ISBN 978-3-642-64988-2.
35. Garibay Bonales, F.; Trigos-Arrieta, F. J.; Vera Mendoza, R. (2002). "A characterization of Pontryagin-van Kampen duality for locally convex spaces". Topology and Its Applications. 121 (1–2): 75–89. doi:10.1016/s0166-8641(01)00111-0.
36. Akbarov, S. S.; Shavgulidze, E. T. (2003). "On two classes of spaces reflexive in the sense of Pontryagin". Mat. Sbornik. 194 (10): 3–26.
General references
• Bernardes, Nilson C. Jr. (2012), On nested sequences of convex sets in Banach spaces, vol. 389, Journal of Mathematical Analysis and Applications, pp. 558–561 .
• Conway, John B. (1985). A Course in Functional Analysis. Springer.
• Diestel, Joe (1984). Sequences and series in Banach spaces. New York: Springer-Verlag. ISBN 0-387-90859-5. OCLC 9556781.
• Edwards, R. E. (1965). Functional analysis. Theory and applications. New York: Holt, Rinehart and Winston. ISBN 0030505356.
• 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.
• 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.
• Schaefer, Helmut H. (1966). Topological vector spaces. New York: The Macmillan Company.
• 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.
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Wikipedia
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Regular complex polygon
In geometry, a regular complex polygon is a generalization of a regular polygon in real space to an analogous structure in a complex Hilbert space, where each real dimension is accompanied by an imaginary one. A regular polygon exists in 2 real dimensions, $\mathbb {R} ^{2}$, while a complex polygon exists in two complex dimensions, $\mathbb {C} ^{2}$, which can be given real representations in 4 dimensions, $\mathbb {R} ^{4}$, which then must be projected down to 2 or 3 real dimensions to be visualized. A complex polygon is generalized as a complex polytope in $\mathbb {C} ^{n}$.
Three views of regular complex polygon 4{4}2,
This complex polygon has 8 edges (complex lines), labeled as a..h, and 16 vertices. Four vertices lie in each edge and two edges intersect at each vertex. In the left image, the outlined squares are not elements of the polytope but are included merely to help identify vertices lying in the same complex line. The octagonal perimeter of the left image is not an element of the polytope, but it is a petrie polygon.[1] In the middle image, each edge is represented as a real line and the four vertices in each line can be more clearly seen.
A perspective sketch representing the 16 vertex points as large black dots and the 8 4-edges as bounded squares within each edge. The green path represents the octagonal perimeter of the left hand image.
A complex polygon may be understood as a collection of complex points, lines, planes, and so on, where every point is the junction of multiple lines, every line of multiple planes, and so on.
The regular complex polygons have been completely characterized, and can be described using a symbolic notation developed by Coxeter.
Regular complex polygons
While 1-polytopes can have unlimited p, finite regular complex polygons, excluding the double prism polygons p{4}2, are limited to 5-edge (pentagonal edges) elements, and infinite regular aperiogons also include 6-edge (hexagonal edges) elements.
Shephard's modified Schläfli notation
Shephard originally devised a modified form of Schläfli's notation for regular polytopes. For a polygon bounded by p1-edges, with a p2-set as vertex figure and overall symmetry group of order g, we denote the polygon as p1(g)p2.
The number of vertices V is then g/p2 and the number of edges E is g/p1.
The complex polygon illustrated above has eight square edges (p1=4) and sixteen vertices (p2=2). From this we can work out that g = 32, giving the modified Schläfli symbol 4(32)2.
Coxeter's revised modified Schläfli notation
A more modern notation p1{q}p2 is due to Coxeter,[2] and is based on group theory. As a symmetry group, its symbol is p1[q]p2.
The symmetry group p1[q]p2 is represented by 2 generators R1, R2, where: R1p1 = R2p2 = I. If q is even, (R2R1)q/2 = (R1R2)q/2. If q is odd, (R2R1)(q−1)/2R2 = (R1R2)(q−1)/2R1. When q is odd, p1=p2.
For 4[4]2 has R14 = R22 = I, (R2R1)2 = (R1R2)2.
For 3[5]3 has R13 = R23 = I, (R2R1)2R2 = (R1R2)2R1.
Coxeter–Dynkin diagrams
Coxeter also generalised the use of Coxeter–Dynkin diagrams to complex polytopes, for example the complex polygon p{q}r is represented by and the equivalent symmetry group, p[q]r, is a ringless diagram . The nodes p and r represent mirrors producing p and r images in the plane. Unlabeled nodes in a diagram have implicit 2 labels. For example, a real regular polygon is 2{q}2 or {q} or .
One limitation, nodes connected by odd branch orders must have identical node orders. If they do not, the group will create "starry" polygons, with overlapping element. So and are ordinary, while is starry.
12 Irreducible Shephard groups
12 irreducible Shephard groups with their subgroup index relations.[3]
Subgroups from <5,3,2>30, <4,3,2>12 and <3,3,2>6
Subgroups relate by removing one reflection:
p[2q]2 --> p[q]p, index 2 and p[4]q --> p[q]p, index q.
Coxeter enumerated this list of regular complex polygons in $\mathbb {C} ^{2}$. A regular complex polygon, p{q}r or , has p-edges, and r-gonal vertex figures. p{q}r is a finite polytope if (p + r)q > pr(q − 2).
Its symmetry is written as p[q]r, called a Shephard group, analogous to a Coxeter group, while also allowing unitary reflections.
For nonstarry groups, the order of the group p[q]r can be computed as $g=8/q\cdot (1/p+2/q+1/r-1)^{-2}$.[4]
The Coxeter number for p[q]r is $h=2/(1/p+2/q+1/r-1)$, so the group order can also be computed as $g=2h^{2}/q$. A regular complex polygon can be drawn in orthogonal projection with h-gonal symmetry.
The rank 2 solutions that generate complex polygons are:
Group G3 = G(q,1,1)G2 = G(p,1,2)G4G6G5G8G14G9G10G20G16G21G17G18
2[q]2, q = 3,4...p[4]2, p = 2,3...3[3]33[6]23[4]34[3]43[8]24[6]24[4]33[5]35[3]53[10]25[6]25[4]3
Order 2q2p22448729614419228836060072012001800
h q2p612243060
Excluded solutions with odd q and unequal p and r are: 6[3]2, 6[3]3, 9[3]3, 12[3]3, ..., 5[5]2, 6[5]2, 8[5]2, 9[5]2, 4[7]2, 9[5]2, 3[9]2, and 3[11]2.
Other whole q with unequal p and r, create starry groups with overlapping fundamental domains: , , , , , and .
The dual polygon of p{q}r is r{q}p. A polygon of the form p{q}p is self-dual. Groups of the form p[2q]2 have a half symmetry p[q]p, so a regular polygon is the same as quasiregular . As well, regular polygon with the same node orders, , have an alternated construction , allowing adjacent edges to be two different colors.[5]
The group order, g, is used to compute the total number of vertices and edges. It will have g/r vertices, and g/p edges. When p=r, the number of vertices and edges are equal. This condition is required when q is odd.
Matrix generators
The group p[q]r, , can be represented by two matrices:[6]
NameR1
R2
Order p r
Matrix
$\left[{\begin{smallmatrix}e^{2\pi i/p}&0\\(e^{2\pi i/p}-1)k&1\\\end{smallmatrix}}\right]$
$\left[{\begin{smallmatrix}1&(e^{2\pi i/r}-1)k\\0&e^{2\pi i/r}\end{smallmatrix}}\right]$
With
$k={\sqrt {\frac {\cos({\frac {\pi }{p}}-{\frac {\pi }{r}})+\cos({\frac {2\pi }{q}})}{2\sin {\frac {\pi }{p}}\sin {\frac {\pi }{r}}}}}$
Examples
NameR1
R2
Order p q
Matrix
$\left[{\begin{smallmatrix}e^{2\pi i/p}&0\\0&1\\\end{smallmatrix}}\right]$
$\left[{\begin{smallmatrix}1&0\\0&e^{2\pi i/q}\\\end{smallmatrix}}\right]$
NameR1
R2
Order p 2
Matrix
$\left[{\begin{smallmatrix}e^{2\pi i/p}&0\\0&1\\\end{smallmatrix}}\right]$
$\left[{\begin{smallmatrix}0&1\\1&0\\\end{smallmatrix}}\right]$
NameR1
R2
Order 3 3
Matrix
$\left[{\begin{smallmatrix}{\frac {-1+{\sqrt {3}}i}{2}}&0\\{\frac {-3+{\sqrt {3}}i}{2}}&1\\\end{smallmatrix}}\right]$
$\left[{\begin{smallmatrix}1&{\frac {-3+{\sqrt {3}}i}{2}}\\0&{\frac {-1+{\sqrt {3}}i}{2}}\\\end{smallmatrix}}\right]$
NameR1
R2
Order 4 4
Matrix
$\left[{\begin{smallmatrix}i&0\\0&1\\\end{smallmatrix}}\right]$
$\left[{\begin{smallmatrix}1&0\\0&i\\\end{smallmatrix}}\right]$
NameR1
R2
Order 4 2
Matrix
$\left[{\begin{smallmatrix}i&0\\0&1\\\end{smallmatrix}}\right]$
$\left[{\begin{smallmatrix}0&1\\1&0\\\end{smallmatrix}}\right]$
NameR1
R2
Order 3 2
Matrix
$\left[{\begin{smallmatrix}{\frac {-1+{\sqrt {3}}i}{2}}&0\\{\frac {-3+{\sqrt {3}}i}{2}}&1\\\end{smallmatrix}}\right]$
$\left[{\begin{smallmatrix}1&-2\\0&-1\\\end{smallmatrix}}\right]$
Enumeration of regular complex polygons
Coxeter enumerated the complex polygons in Table III of Regular Complex Polytopes.[7]
GroupOrderCoxeter
number
PolygonVerticesEdgesNotes
G(q,q,2)
2[q]2 = [q]
q = 2,3,4,...
2qq2{q}2qq{}Real regular polygons
Same as
Same as if q even
GroupOrderCoxeter
number
PolygonVerticesEdgesNotes
G(p,1,2)
p[4]2
p=2,3,4,...
2p22pp(2p2)2p{4}2
p22pp{}same as p{}×p{} or
$\mathbb {R} ^{4}$ representation as p-p duoprism
2(2p2)p2{4}p2pp2{}$\mathbb {R} ^{4}$ representation as p-p duopyramid
G(2,1,2)
2[4]2 = [4]
842{4}2 = {4}44{}same as {}×{} or
Real square
G(3,1,2)
3[4]2
1866(18)23{4}2963{}same as 3{}×3{} or
$\mathbb {R} ^{4}$ representation as 3-3 duoprism
2(18)32{4}369{}$\mathbb {R} ^{4}$ representation as 3-3 duopyramid
G(4,1,2)
4[4]2
3288(32)24{4}21684{}same as 4{}×4{} or
$\mathbb {R} ^{4}$ representation as 4-4 duoprism or {4,3,3}
2(32)42{4}4816{}$\mathbb {R} ^{4}$ representation as 4-4 duopyramid or {3,3,4}
G(5,1,2)
5[4]2
50255(50)25{4}225105{}same as 5{}×5{} or
$\mathbb {R} ^{4}$ representation as 5-5 duoprism
2(50)52{4}51025{}$\mathbb {R} ^{4}$ representation as 5-5 duopyramid
G(6,1,2)
6[4]2
72366(72)26{4}236126{}same as 6{}×6{} or
$\mathbb {R} ^{4}$ representation as 6-6 duoprism
2(72)62{4}61236{}$\mathbb {R} ^{4}$ representation as 6-6 duopyramid
G4=G(1,1,2)
3[3]3
<2,3,3>
2463(24)33{3}3883{}Möbius–Kantor configuration
self-dual, same as
$\mathbb {R} ^{4}$ representation as {3,3,4}
G6
3[6]2
48123(48)23{6}224163{}same as
3{3}2starry polygon
2(48)32{6}31624{}
2{3}3starry polygon
G5
3[4]3
72123(72)33{4}324243{}self-dual, same as
$\mathbb {R} ^{4}$ representation as {3,4,3}
G8
4[3]4
96124(96)44{3}424244{}self-dual, same as
$\mathbb {R} ^{4}$ representation as {3,4,3}
G14
3[8]2
144243(144)23{8}272483{}same as
3{8/3}2starry polygon, same as
2(144)32{8}34872{}
2{8/3}3starry polygon
G9
4[6]2
192244(192)24{6}296484{}same as
2(192)42{6}44896{}
4{3}29648{}starry polygon
2{3}44896{}starry polygon
G10
4[4]3
288244(288)34{4}396724{}
124{8/3}3starry polygon
243(288)43{4}472963{}
123{8/3}4starry polygon
G20
3[5]3
360303(360)33{5}31201203{}self-dual, same as
$\mathbb {R} ^{4}$ representation as {3,3,5}
3{5/2}3self-dual, starry polygon
G16
5[3]5
600305(600)55{3}51201205{}self-dual, same as
$\mathbb {R} ^{4}$ representation as {3,3,5}
105{5/2}5self-dual, starry polygon
G21
3[10]2
720603(720)23{10}23602403{}same as
3{5}2starry polygon
3{10/3}2starry polygon, same as
3{5/2}2starry polygon
2(720)32{10}3240360{}
2{5}3starry polygon
2{10/3}3starry polygon
2{5/2}3starry polygon
G17
5[6]2
1200605(1200)25{6}26002405{}same as
205{5}2starry polygon
205{10/3}2starry polygon
605{3}2starry polygon
602(1200)52{6}5240600{}
202{5}5starry polygon
202{10/3}5starry polygon
602{3}5starry polygon
G18
5[4]3
1800605(1800)35{4}36003605{}
155{10/3}3starry polygon
305{3}3starry polygon
305{5/2}3starry polygon
603(1800)53{4}53606003{}
153{10/3}5starry polygon
303{3}5starry polygon
303{5/2}5starry polygon
2D graphs
Polygons of the form p{2r}q can be visualized by q color sets of p-edge. Each p-edge is seen as a regular polygon, while there are no faces.
Complex polygons 2{r}q
Polygons of the form 2{4}q are called generalized orthoplexes. They share vertices with the 4D q-q duopyramids, vertices connected by 2-edges.
• 2{4}2, , with 4 vertices, and 4 edges
• 2{4}3, , with 6 vertices, and 9 edges[8]
• 2{4}4, , with 8 vertices, and 16 edges
• 2{4}5, , with 10 vertices, and 25 edges
• 2{4}6, , with 12 vertices, and 36 edges
• 2{4}7, , with 14 vertices, and 49 edges
• 2{4}8, , with 16 vertices, and 64 edges
• 2{4}9, , with 18 vertices, and 81 edges
• 2{4}10, , with 20 vertices, and 100 edges
Complex polygons p{4}2
Polygons of the form p{4}2 are called generalized hypercubes (squares for polygons). They share vertices with the 4D p-p duoprisms, vertices connected by p-edges. Vertices are drawn in green, and p-edges are drawn in alternate colors, red and blue. The perspective is distorted slightly for odd dimensions to move overlapping vertices from the center.
• 2{4}2, or , with 4 vertices, and 4 2-edges
• 3{4}2, or , with 9 vertices, and 6 (triangular) 3-edges[9]
• 4{4}2, or , with 16 vertices, and 8 (square) 4-edges
• 5{4}2, or , with 25 vertices, and 10 (pentagonal) 5-edges
• 6{4}2, or , with 36 vertices, and 12 (hexagonal) 6-edges
• 7{4}2, or , with 49 vertices, and 14 (heptagonal)7-edges
• 8{4}2, or , with 64 vertices, and 16 (octagonal) 8-edges
• 9{4}2, or , with 81 vertices, and 18 (enneagonal) 9-edges
• 10{4}2, or , with 100 vertices, and 20 (decagonal) 10-edges
Complex polygons p{r}2
• 3{6}2, or , with 24 vertices in black, and 16 3-edges colored in 2 sets of 3-edges in red and blue[10]
• 3{8}2, or , with 72 vertices in black, and 48 3-edges colored in 2 sets of 3-edges in red and blue[11]
Complex polygons, p{r}p
Polygons of the form p{r}p have equal number of vertices and edges. They are also self-dual.
• 3{3}3, or , with 8 vertices in black, and 8 3-edges colored in 2 sets of 3-edges in red and blue[12]
• 3{4}3, or , with 24 vertices and 24 3-edges shown in 3 sets of colors, one set filled[13]
• 4{3}4, or , with 24 vertices and 24 4-edges shown in 4 sets of colors[14]
• 3{5}3, or , with 120 vertices and 120 3-edges[15]
• 5{3}5, or , with 120 vertices and 120 5-edges[16]
3D perspective
3D perspective projections of complex polygons p{4}2 can show the point-edge structure of a complex polygon, while scale is not preserved.
The duals 2{4}p: are seen by adding vertices inside the edges, and adding edges in place of vertices.
• 2{4}3, with 6 vertices, 9 edges in 3 sets
• 3{4}2, with 9 vertices, 6 3-edges in 2 sets of colors as
• 4{4}2, with 16 vertices, 8 4-edges in 2 sets of colors and filled square 4-edges as
• 5{4}2, with 25 vertices, 10 5-edges in 2 sets of colors as
Quasiregular polygons
A quasiregular polygon is a truncation of a regular polygon. A quasiregular polygon contains alternate edges of the regular polygons and . The quasiregular polygon has p vertices on the p-edges of the regular form.
Example quasiregular polygons
p[q]r2[4]23[4]24[4]25[4]26[4]27[4]28[4]23[3]33[4]3
Regular
4 2-edges
9 3-edges
16 4-edges
25 5-edges
36 6-edges
49 7-edges
64 8-edges
Quasiregular
=
4+4 2-edges
6 2-edges
9 3-edges
8 2-edges
16 4-edges
10 2-edges
25 5-edges
12 2-edges
36 6-edges
14 2-edges
49 7-edges
16 2-edges
64 8-edges
=
=
Regular
4 2-edges
6 2-edges
8 2-edges
10 2-edges
12 2-edges
14 2-edges
16 2-edges
Notes
1. Coxeter, Regular Complex Polytopes, 11.3 Petrie Polygon, a simple h-gon formed by the orbit of the flag (O0,O0O1) for the product of the two generating reflections of any nonstarry regular complex polygon, p1{q}p2.
2. Coxeter, Regular Complex Polytopes, p. xiv
3. Coxeter, Complex Regular Polytopes, p. 177, Table III
4. Lehrer & Taylor 2009, p. 87
5. Coxeter, Regular Complex Polytopes, Table IV. The regular polygons. pp. 178–179
6. Complex Polytopes, 8.9 The Two-Dimensional Case, p. 88
7. Regular Complex Polytopes, Coxeter, pp. 177–179
8. Coxeter, Regular Complex Polytopes, p. 108
9. Coxeter, Regular Complex Polytopes, p. 108
10. Coxeter, Regular Complex Polytopes, p. 109
11. Coxeter, Regular Complex Polytopes, p. 111
12. Coxeter, Regular Complex Polytopes, p. 30 diagram and p. 47 indices for 8 3-edges
13. Coxeter, Regular Complex Polytopes, p. 110
14. Coxeter, Regular Complex Polytopes, p. 110
15. Coxeter, Regular Complex Polytopes, p. 48
16. Coxeter, Regular Complex Polytopes, p. 49
References
• Coxeter, H.S.M. and Moser, W. O. J.; Generators and Relations for Discrete Groups (1965), esp pp 67–80.
• Coxeter, H.S.M. (1991), Regular Complex Polytopes, Cambridge University Press, ISBN 0-521-39490-2
• Coxeter, H.S.M. and Shephard, G.C.; Portraits of a family of complex polytopes, Leonardo Vol 25, No 3/4, (1992), pp 239–244,
• Shephard, G.C.; Regular complex polytopes, Proc. London math. Soc. Series 3, Vol 2, (1952), pp 82–97.
• G. C. Shephard, J. A. Todd, Finite unitary reflection groups, Canadian Journal of Mathematics. 6(1954), 274–304
• Gustav I. Lehrer and Donald E. Taylor, Unitary Reflection Groups, Cambridge University Press 2009
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Wikipedia
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Quasiregular element
In mathematics, specifically ring theory, the notion of quasiregularity provides a computationally convenient way to work with the Jacobson radical of a ring.[1] In this article, we primarily concern ourselves with the notion of quasiregularity for unital rings. However, one section is devoted to the theory of quasiregularity in non-unital rings, which constitutes an important aspect of noncommutative ring theory.
This article addresses the notion of quasiregularity in the context of ring theory, a branch of modern algebra. For other notions of quasiregularity in mathematics, see the disambiguation page quasiregular.
Definition
Let R be a ring (with unity) and let r be an element of R. Then r is said to be quasiregular, if 1 − r is a unit in R; that is, invertible under multiplication.[1] The notions of right or left quasiregularity correspond to the situations where 1 − r has a right or left inverse, respectively.[1]
An element x of a non-unital ring is said to be right quasiregular if there is y such that $x+y-xy=0$.[2] The notion of a left quasiregular element is defined in an analogous manner. The element y is sometimes referred to as a right quasi-inverse of x.[3] If the ring is unital, this definition of quasiregularity coincides with that given above.[4] If one writes $x\cdot y=x+y-xy$, then this binary operation $\cdot $ is associative.[5] In fact, the map $(R,\cdot )\to (R,\times );x\mapsto 1-x$ (where × denotes the multiplication of the ring R) is a monoid isomorphism.[4] Therefore, if an element possesses both a left and right quasi-inverse, they are equal.[6]
Note that some authors use different definitions. They call an element x right quasiregular if there exists y such that $x+y+xy=0$,[7] which is equivalent to saying that 1 + x has a right inverse when the ring is unital. If we write $x\circ y=x+y+xy$, then $(-x)\circ (-y)=-(x\cdot y)$, so we can easily go from one set-up to the other by changing signs.[8] For example, x is right quasiregular in one set-up iff −x is right quasiregular in the other set-up.[8]
Examples
• If R is a ring, then the additive identity of R is always quasiregular.
• If $x^{2}$ is right (resp. left) quasiregular, then $x$ is right (resp. left) quasiregular.[9]
• If R is a rng, every nilpotent element of R is quasiregular.[10] This fact is supported by an elementary computation:
If $x^{n+1}=0$, then
$(1-x)(1+x+x^{2}+\dotsb +x^{n})=1$ (or $(1+x)(1-x+x^{2}-\dotsb +(-x)^{n})=1$ if we follow the second convention).
From this we see easily that the quasi-inverse of x is $-x-x^{2}-\dotsb -x^{n}$ (or $-x+x^{2}-\dotsb +(-x)^{n}$).
• In the second convention, a matrix is quasiregular in a matrix ring if it does not possess −1 as an eigenvalue. More generally, a bounded operator is quasiregular if −1 is not in its spectrum.
• In a unital Banach algebra, if $\|x\|<1$, then the geometric series $\sum _{0}^{\infty }x^{n}$ converges. Consequently, every such x is quasiregular.
• If R is a ring and S = R[[X1, ..., Xn]] denotes the ring of formal power series in n indeterminants over R, an element of S is quasiregular if and only its constant term is quasiregular as an element of R.
Properties
• Every element of the Jacobson radical of a (not necessarily commutative) ring is quasiregular.[11] In fact, the Jacobson radical of a ring can be characterized as the unique right ideal of the ring, maximal with respect to the property that every element is right quasiregular.[12][13] However, a right quasiregular element need not necessarily be a member of the Jacobson radical.[14] This justifies the remark in the beginning of the article – "bad elements" are quasiregular, although quasiregular elements are not necessarily "bad". Elements of the Jacobson radical of a ring are often deemed to be "bad".
• If an element of a ring is nilpotent and central, then it is a member of the ring's Jacobson radical.[15] This is because the principal right ideal generated by that element consists of quasiregular (in fact, nilpotent) elements only.
• If an element, r, of a ring is idempotent, it cannot be a member of the ring's Jacobson radical.[16] This is because idempotent elements cannot be quasiregular. This property, as well as the one above, justify the remark given at the top of the article that the notion of quasiregularity is computationally convenient when working with the Jacobson radical.[1]
Generalization to semirings
The notion of quasiregular element readily generalizes to semirings. If a is an element of a semiring S, then an affine map from S to itself is $\mu _{a}(r)=ra+1$. An element a of S is said to be right quasiregular if $\mu _{a}$ has a fixed point, which need not be unique. Each such fixed point is called a left quasi-inverse of a. If b is a left quasi-inverse of a and additionally b = ab + 1, then b it is called a quasi-inverse of a; any element of the semiring that has a quasi-inverse is said to be quasiregular. It is possible that some but not all elements of a semiring be quasiregular; for example, in the semiring of nonnegative reals with the usual addition and multiplication of reals, $\mu _{a}$ has the fixed point ${\frac {1}{1-a}}$ for all a < 1, but has no fixed point for a ≥ 1.[17] If every element of a semiring is quasiregular then the semiring is called a quasi-regular semiring, closed semiring,[18] or occasionally a Lehmann semiring[17] (the latter honoring the paper of Daniel J. Lehmann.[19])
Examples of quasi-regular semirings are provided by the Kleene algebras (prominently among them, the algebra of regular expressions), in which the quasi-inverse is lifted to the role of a unary operation (denoted by a*) defined as the least fixedpoint solution. Kleene algebras are additively idempotent but not all quasi-regular semirings are so. We can extend the example of nonegative reals to include infinity and it becomes a quasi-regular semiring with the quasi-inverse of any element a ≥ 1 being the infinity. This quasi-regular semiring is not additively idempotent however, so it is not a Kleene algebra.[18] It is however a complete semiring.[20] More generally, all complete semirings are quasiregular.[21] The term closed semiring is actually used by some authors to mean complete semiring rather than just quasiregular.[22][23]
Conway semirings are also quasiregular; the two Conway axioms are actually independent, i.e. there are semirings satisfying only the product-star [Conway] axiom, (ab)* = 1+a(ba)*b, but not the sum-star axiom, (a+b)* = (a*b)*a* and vice versa; it is the product-star [Conway] axiom that implies that a semiring is quasiregular. Additionally, a commutative semiring is quasiregular if and only if it satisfies the product-star Conway axiom.[17]
Quasiregular semirings appear in algebraic path problems, a generalization of the shortest path problem.[18]
See also
• inverse element
Notes
1. Isaacs, p. 180
2. Lam, Ex. 4.2, p. 50
3. Polcino & Sehgal (2002), p. 298.
4. Lam, Ex. 4.2(3), p. 50
5. Lam, Ex. 4.1, p. 50
6. Since 0 is the multiplicative identity, if $x\cdot y=0=y'\cdot x$, then $y=(y'\cdot x)\cdot y=y'\cdot (x\cdot y)=y'$. Quasiregularity does not require the ring to have a multiplicative identity.
7. Kaplansky, p. 85
8. Lam, p. 51
9. Kaplansky, p. 108
10. Lam, Ex. 4.2(2), p. 50
11. Isaacs, Theorem 13.4(a), p. 180
12. Isaacs, Theorem 13.4(b), p. 180
13. Isaacs, Corollary 13.7, p. 181
14. Isaacs, p. 181
15. Isaacs, Corollary 13.5, p. 181
16. Isaacs, Corollary 13.6, p. 181
17. Jonathan S. Golan (30 June 2003). Semirings and Affine Equations over Them. Springer Science & Business Media. pp. 157–159 and 164–165. ISBN 978-1-4020-1358-4.
18. Marc Pouly; Jürg Kohlas (2011). Generic Inference: A Unifying Theory for Automated Reasoning. John Wiley & Sons. pp. 232 and 248–249. ISBN 978-1-118-01086-0.
19. Lehmann, D. J. (1977). "Algebraic structures for transitive closure" (PDF). Theoretical Computer Science. 4: 59–76. doi:10.1016/0304-3975(77)90056-1.
20. 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
21. U. Zimmermann (1981). Linear and combinatorial optimization in ordered algebraic structures. Elsevier. p. 141. ISBN 978-0-08-086773-1.
22. Dexter Kozen (1992). The Design and Analysis of Algorithms. Springer Science & Business Media. p. 31. ISBN 978-0-387-97687-7.
23. J.A. Storer (2001). An Introduction to Data Structures and Algorithms. Springer Science & Business Media. p. 336. ISBN 978-0-8176-4253-2.
References
• I. Martin Isaacs (1993). Algebra, a graduate course (1st ed.). Brooks/Cole Publishing Company. ISBN 0-534-19002-2.
• Irving Kaplansky (1969). Fields and Rings. The University of Chicago Press.
• Lam, Tsit-Yuen (2003). Exercises in Classical Ring Theory. Problem Books in Mathematics (2nd ed.). Springer-Verlag. ISBN 978-0387005003.
• Milies, César Polcino; Sehgal, Sudarshan K. (2002). An introduction to group rings. Springer. ISBN 978-1-4020-0238-0.
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Wikipedia
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Quasiregular polyhedron
In geometry, a quasiregular polyhedron is a uniform polyhedron that has exactly two kinds of regular faces, which alternate around each vertex. They are vertex-transitive and edge-transitive, hence a step closer to regular polyhedra than the semiregular, which are merely vertex-transitive.
Quasiregular figures
Right triangle domains (p q 2), = r{p,q}
r{4,3} r{5,3} r{6,3} r{7,3}... r{∞,3}
(3.4)2
(3.5)2
(3.6)2
(3.7)2
(3.∞)2
Isosceles triangle domains (p p 3), = = h{6,p}
h{6,4} h{6,5} h{6,6} h{6,7}... h{6,∞}
= = = = =
(4.3)4
(5.3)5
(6.3)6
(7.3)7
(∞.3)∞
Isosceles triangle domains (p p 4), = = h{8,p}
h{8,3} h{8,5} h{8,6} h{8,7}... h{8,∞}
= = = = =
(4.3)3
(4.5)5
(4.6)6
(4.7)7
(4.∞)∞
Scalene triangle domain (5 4 3),
(3.5)4
(4.5)3
(3.4)5
A quasiregular polyhedron or tiling has exactly two kinds of regular face, which alternate around each vertex. Their vertex figures are isogonal polygons.
Regular and quasiregular figures
Right triangle domains (p p 2), = = r{p,p} = {p,4}1⁄2
{3,4}1⁄2
r{3,3}
{4,4}1⁄2
r{4,4}
{5,4}1⁄2
r{5,5}
{6,4}1⁄2
r{6,6}...
{∞,4}1⁄2
r{∞,∞}
= = = = =
(3.3)2
(4.4)2
(5.5)2
(6.6)2
(∞.∞)2
Isosceles triangle domains (p p 3), = = {p,6}1⁄2
{3,6}1⁄2 {4,6}1⁄2 {5,6}1⁄2 {6,6}1⁄2... {∞,6}1⁄2
= = = = =
(3.3)3
(4.4)3
(5.5)3
(6.6)3
(∞.∞)3
Isosceles triangle domains (p p 4), = = {p,8}1⁄2
{3,8}1⁄2 {4,8}1⁄2 {5,8}1⁄2 {6,8}1⁄2... {∞,8}1⁄2
= = = = =
(3.3)4
(4.4)4
(5.5)4
(6.6)4
(∞.∞)4
A regular polyhedron or tiling can be considered quasiregular if it has an even number of faces around each vertex (and thus can have alternately colored faces).
Their dual figures are face-transitive and edge-transitive; they have exactly two kinds of regular vertex figures, which alternate around each face. They are sometimes also considered quasiregular.
There are only two convex quasiregular polyhedra: the cuboctahedron and the icosidodecahedron. Their names, given by Kepler, come from recognizing that their faces are all the faces (turned differently) of the dual-pair cube and octahedron, in the first case, and of the dual-pair icosahedron and dodecahedron, in the second case.
These forms representing a pair of a regular figure and its dual can be given a vertical Schläfli symbol ${\begin{Bmatrix}p\\q\end{Bmatrix}}$ or r{p,q}, to represent that their faces are all the faces (turned differently) of both the regular {p,q} and the dual regular {q,p}. A quasiregular polyhedron with this symbol will have a vertex configuration p.q.p.q (or (p.q)2).
More generally, a quasiregular figure can have a vertex configuration (p.q)r, representing r (2 or more) sequences of the faces around the vertex.
Tilings of the plane can also be quasiregular, specifically the trihexagonal tiling, with vertex configuration (3.6)2. Other quasiregular tilings exist on the hyperbolic plane, like the triheptagonal tiling, (3.7)2. Or more generally: (p.q)2, with 1/p + 1/q < 1/2.
Regular polyhedra and tilings with an even number of faces at each vertex can also be considered quasiregular by differentiating between faces of the same order, by representing them differently, like coloring them alternately (without defining any surface orientation). A regular figure with Schläfli symbol {p,q} can be considered quasiregular, with vertex configuration (p.p)q/2, if q is even.
Examples:
The regular octahedron, with Schläfli symbol {3,4} and 4 being even, can be considered quasiregular as a tetratetrahedron (2 sets of 4 triangles of the tetrahedron), with vertex configuration (3.3)4/2 = (3a.3b)2, alternating two colors of triangular faces.
The square tiling, with vertex configuration 44 and 4 being even, can be considered quasiregular, with vertex configuration (4.4)4/2 = (4a.4b)2, colored as a checkerboard.
The triangular tiling, with vertex configuration 36 and 6 being even, can be considered quasiregular, with vertex configuration (3.3)6/2 = (3a.3b)3, alternating two colors of triangular faces.
Wythoff construction
Regular (p | 2 q) and quasiregular polyhedra (2 | p q) are created from a Wythoff construction with the generator point at one of 3 corners of the fundamental domain. This defines a single edge within the fundamental domain.
Coxeter defines a quasiregular polyhedron as one having a Wythoff symbol in the form p | q r, and it is regular if q=2 or q=r.[1]
The Coxeter-Dynkin diagram is another symbolic representation that shows the quasiregular relation between the two dual-regular forms:
Schläfli symbol Coxeter diagram Wythoff symbol
${\begin{Bmatrix}p,q\end{Bmatrix}}${p,q}q | 2 p
${\begin{Bmatrix}q,p\end{Bmatrix}}${q,p}p | 2 q
${\begin{Bmatrix}p\\q\end{Bmatrix}}$r{p,q} or 2 | p q
The convex quasiregular polyhedra
Further information: Rectification (geometry)
There are two uniform convex quasiregular polyhedra:
1. The cuboctahedron ${\begin{Bmatrix}3\\4\end{Bmatrix}}$, vertex configuration (3.4)2, Coxeter-Dynkin diagram
2. The icosidodecahedron ${\begin{Bmatrix}3\\5\end{Bmatrix}}$, vertex configuration (3.5)2, Coxeter-Dynkin diagram
In addition, the octahedron, which is also regular, ${\begin{Bmatrix}3\\3\end{Bmatrix}}$, vertex configuration (3.3)2, can be considered quasiregular if alternate faces are given different colors. In this form it is sometimes known as the tetratetrahedron. The remaining convex regular polyhedra have an odd number of faces at each vertex so cannot be colored in a way that preserves edge transitivity. It has Coxeter-Dynkin diagram
Each of these forms the common core of a dual pair of regular polyhedra. The names of two of these give clues to the associated dual pair: respectively cube $\cap $ octahedron, and icosahedron $\cap $ dodecahedron. The octahedron is the common core of a dual pair of tetrahedra (a compound known as the stella octangula); when derived in this way, the octahedron is sometimes called the tetratetrahedron, as tetrahedron $\cap $ tetrahedron.
Regular Dual regular Quasiregular common core Vertex figure
Tetrahedron
{3,3}
3 | 2 3
Tetrahedron
{3,3}
3 | 2 3
Tetratetrahedron
r{3,3}
2 | 3 3
3.3.3.3
Cube
{4,3}
3 | 2 4
Octahedron
{3,4}
4 | 2 3
Cuboctahedron
r{3,4}
2 | 3 4
3.4.3.4
Dodecahedron
{5,3}
3 | 2 5
Icosahedron
{3,5}
5 | 2 3
Icosidodecahedron
r{3,5}
2 | 3 5
3.5.3.5
Each of these quasiregular polyhedra can be constructed by a rectification operation on either regular parent, truncating the vertices fully, until each original edge is reduced to its midpoint.
Quasiregular tilings
This sequence continues as the trihexagonal tiling, vertex figure (3.6)2 - a quasiregular tiling based on the triangular tiling and hexagonal tiling.
Regular Dual regular Quasiregular combination Vertex figure
Hexagonal tiling
{6,3}
6 | 2 3
Triangular tiling
{3,6}
3 | 2 6
Trihexagonal tiling
r{6,3}
2 | 3 6
(3.6)2
The checkerboard pattern is a quasiregular coloring of the square tiling, vertex figure (4.4)2:
Regular Dual regular Quasiregular combination Vertex figure
{4,4}
4 | 2 4
{4,4}
4 | 2 4
r{4,4}
2 | 4 4
(4.4)2
The triangular tiling can also be considered quasiregular, with three sets of alternating triangles at each vertex, (3.3)3:
h{6,3}
3 | 3 3
=
In the hyperbolic plane, this sequence continues further, for example the triheptagonal tiling, vertex figure (3.7)2 - a quasiregular tiling based on the order-7 triangular tiling and heptagonal tiling.
Regular Dual regular Quasiregular combination Vertex figure
Heptagonal tiling
{7,3}
7 | 2 3
Triangular tiling
{3,7}
3 | 2 7
Triheptagonal tiling
r{3,7}
2 | 3 7
(3.7)2
Nonconvex examples
Coxeter, H.S.M. et al. (1954) also classify certain star polyhedra, having the same characteristics, as being quasiregular.
Two are based on dual pairs of regular Kepler–Poinsot solids, in the same way as for the convex examples:
the great icosidodecahedron ${\begin{Bmatrix}3\\5/2\end{Bmatrix}}$, and the dodecadodecahedron ${\begin{Bmatrix}5\\5/2\end{Bmatrix}}$:
Regular Dual regular Quasiregular common core Vertex figure
Great stellated dodecahedron
{5/2,3}
3 | 2 5/2
Great icosahedron
{3,5/2}
5/2 | 2 3
Great icosidodecahedron
r{3,5/2}
2 | 3 5/2
3.5/2.3.5/2
Small stellated dodecahedron
{5/2,5}
5 | 2 5/2
Great dodecahedron
{5,5/2}
5/2 | 2 5
Dodecadodecahedron
r{5,5/2}
2 | 5 5/2
5.5/2.5.5/2
Nine more are the hemipolyhedra, which are faceted forms of the aforementioned quasiregular polyhedra derived from rectification of regular polyhedra. These include equatorial faces passing through the centre of the polyhedra:
Quasiregular (rectified)
Tetratetrahedron
Cuboctahedron
Icosidodecahedron
Great icosidodecahedron
Dodecadodecahedron
Quasiregular (hemipolyhedra)
Tetrahemihexahedron
3/2 3 | 2
Octahemioctahedron
3/2 3 | 3
Small icosihemidodecahedron
3/2 3 | 5
Great icosihemidodecahedron
3/2 3 | 5/3
Small dodecahemicosahedron
5/3 5/2 | 3
Vertex figure
3.4.3/2.4
3.6.3/2.6
3.10.3/2.10
3.10/3.3/2.10/3
5/2.6.5/3.6
Quasiregular (hemipolyhedra)
Cubohemioctahedron
4/3 4 | 3
Small dodecahemidodecahedron
5/4 5 | 5
Great dodecahemidodecahedron
5/3 5/2 | 5/3
Great dodecahemicosahedron
5/4 5 | 3
Vertex figure
4.6.4/3.6
5.10.5/4.10
5/2.10/3.5/3.10/3
5.6.5/4.6
Lastly there are three ditrigonal forms, all facetings of the regular dodecahedron, whose vertex figures contain three alternations of the two face types:
Image Faceted form
Wythoff symbol
Coxeter diagram
Vertex figure
Ditrigonal dodecadodecahedron
3 | 5/3 5
or
(5.5/3)3
Small ditrigonal icosidodecahedron
3 | 5/2 3
or
(3.5/2)3
Great ditrigonal icosidodecahedron
3/2 | 3 5
or
((3.5)3)/2
In the Euclidean plane, the sequence of hemipolyhedra continues with the following four star tilings, where apeirogons appear as the aforementioned equatorial polygons:
Original
rectified
tiling
Edge
diagram
SolidVertex
Config
WythoffSymmetry group
Square
tiling
4.∞.4/3.∞
4.∞.-4.∞
4/3 4 | ∞p4m
Triangular
tiling
(3.∞.3.∞.3.∞)/23/2 | 3 ∞p6m
Trihexagonal
tiling
6.∞.6/5.∞
6.∞.-6.∞
6/5 6 | ∞
∞.3.∞.3/2
∞.3.∞.-3
3/2 3 | ∞
Quasiregular duals
Some authorities argue that, since the duals of the quasiregular solids share the same symmetries, these duals should be called quasiregular too. But not everybody uses this terminology. These duals are transitive on their edges and faces (but not on their vertices); they are the edge-transitive Catalan solids. The convex ones are, in corresponding order as above:
1. The rhombic dodecahedron, with two types of alternating vertices, 8 with three rhombic faces, and 6 with four rhombic faces.
2. The rhombic triacontahedron, with two types of alternating vertices, 20 with three rhombic faces, and 12 with five rhombic faces.
In addition, by duality with the octahedron, the cube, which is usually regular, can be made quasiregular if alternate vertices are given different colors.
Their face configurations are of the form V3.n.3.n, and Coxeter-Dynkin diagram
Cube
V(3.3)2
Rhombic dodecahedron
V(3.4)2
Rhombic triacontahedron
V(3.5)2
Rhombille tiling
V(3.6)2
V(3.7)2
V(3.8)2
These three quasiregular duals are also characterised by having rhombic faces.
This rhombic-faced pattern continues as V(3.6)2, the rhombille tiling.
Quasiregular polytopes and honeycombs
In higher dimensions, Coxeter defined a quasiregular polytope or honeycomb to have regular facets and quasiregular vertex figures. It follows that all vertex figures are congruent and that there are two kinds of facets, which alternate.[2]
In Euclidean 4-space, the regular 16-cell can also be seen as quasiregular as an alternated tesseract, h{4,3,3}, Coxeter diagrams: = , composed of alternating tetrahedron and tetrahedron cells. Its vertex figure is the quasiregular tetratetrahedron (an octahedron with tetrahedral symmetry), .
The only quasiregular honeycomb in Euclidean 3-space is the alternated cubic honeycomb, h{4,3,4}, Coxeter diagrams: = , composed of alternating tetrahedral and octahedral cells. Its vertex figure is the quasiregular cuboctahedron, .[2]
In hyperbolic 3-space, one quasiregular honeycomb is the alternated order-5 cubic honeycomb, h{4,3,5}, Coxeter diagrams: = , composed of alternating tetrahedral and icosahedral cells. Its vertex figure is the quasiregular icosidodecahedron, . A related paracompact alternated order-6 cubic honeycomb, h{4,3,6} has alternating tetrahedral and hexagonal tiling cells with vertex figure is a quasiregular trihexagonal tiling, .
Quasiregular polychora and honeycombs: h{4,p,q}
Space Finite Affine Compact Paracompact
Schläfli
symbol
h{4,3,3} h{4,3,4} h{4,3,5} h{4,3,6} h{4,4,3} h{4,4,4}
$\left\{3,{3 \atop 3}\right\}$ $\left\{3,{4 \atop 3}\right\}$ $\left\{3,{5 \atop 3}\right\}$ $\left\{3,{6 \atop 3}\right\}$ $\left\{4,{4 \atop 3}\right\}$ $\left\{4,{4 \atop 4}\right\}$
Coxeter
diagram
↔ ↔ ↔ ↔ ↔ ↔
↔ ↔
Image
Vertex
figure
r{p,3}
Regular polychora or honeycombs of the form {p,3,4} or can have their symmetry cut in half as into quasiregular form , creating alternately colored {p,3} cells. These cases include the Euclidean cubic honeycomb {4,3,4} with cubic cells, and compact hyperbolic {5,3,4} with dodecahedral cells, and paracompact {6,3,4} with infinite hexagonal tiling cells. They have four cells around each edge, alternating in 2 colors. Their vertex figures are quasiregular tetratetrahedra, = .
Regular and Quasiregular honeycombs: {p,3,4} and {p,31,1}
Space Euclidean 4-space Euclidean 3-space Hyperbolic 3-space
Name {3,3,4}
{3,31,1} = $\left\{3,{3 \atop 3}\right\}$
{4,3,4}
{4,31,1} = $\left\{4,{3 \atop 3}\right\}$
{5,3,4}
{5,31,1} = $\left\{5,{3 \atop 3}\right\}$
{6,3,4}
{6,31,1} = $\left\{6,{3 \atop 3}\right\}$
Coxeter
diagram
= = = =
Image
Cells
{p,3}
Similarly regular hyperbolic honeycombs of the form {p,3,6} or can have their symmetry cut in half as into quasiregular form , creating alternately colored {p,3} cells. They have six cells around each edge, alternating in 2 colors. Their vertex figures are quasiregular triangular tilings, .
Hyperbolic uniform honeycombs: {p,3,6} and {p,3[3]}
Form Paracompact Noncompact
Name {3,3,6}
{3,3[3]}
{4,3,6}
{4,3[3]}
{5,3,6}
{5,3[3]}
{6,3,6}
{6,3[3]}
{7,3,6}
{7,3[3]}
{8,3,6}
{8,3[3]}
... {∞,3,6}
{∞,3[3]}
Image
Cells
{3,3}
{4,3}
{5,3}
{6,3}
{7,3}
{8,3}
{∞,3}
See also
• Chiral polytope
• Rectification (geometry)
Notes
1. 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. (Section 7, The regular and quasiregular polyhedra p | q r)
2. Coxeter, Regular Polytopes, 4.7 Other honeycombs. p.69, p.88
References
• Cromwell, P. Polyhedra, Cambridge University Press (1977).
• Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8, 2.3 Quasi-Regular Polyhedra. (p. 17), Quasi-regular honeycombs p.69
External links
• Weisstein, Eric W. "Quasiregular polyhedron". MathWorld.
• Weisstein, Eric W. "Uniform polyhedron". MathWorld. Quasi-regular polyhedra: (p.q)r
• George Hart, Quasiregular polyhedra
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Quasiregular representation
In mathematics, quasiregular representation is a concept of representation theory, for a locally compact group G and a homogeneous space G/H where H is a closed subgroup.
This article addresses the notion of quasiregularity in the context of representation theory and topological algebra. For other notions of quasiregularity in mathematics, see the disambiguation page quasiregular.
In line with the concepts of regular representation and induced representation, G acts on functions on G/H. If however Haar measures give rise only to a quasi-invariant measure on G/H, certain 'correction factors' have to be made to the action on functions, for
L2(G/H)
to afford a unitary representation of G on square-integrable functions. With appropriate scaling factors, therefore, introduced into the action of G, this is the quasiregular representation or modified induced representation.
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Quasisymmetric map
In mathematics, a quasisymmetric homeomorphism between metric spaces is a map that generalizes bi-Lipschitz maps. While bi-Lipschitz maps shrink or expand the diameter of a set by no more than a multiplicative factor, quasisymmetric maps satisfy the weaker geometric property that they preserve the relative sizes of sets: if two sets A and B have diameters t and are no more than distance t apart, then the ratio of their sizes changes by no more than a multiplicative constant. These maps are also related to quasiconformal maps, since in many circumstances they are in fact equivalent.[1]
For quasisymmetric functions in algebraic combinatorics, see quasisymmetric function.
Definition
Let (X, dX) and (Y, dY) be two metric spaces. A homeomorphism f:X → Y is said to be η-quasisymmetric if there is an increasing function η : [0, ∞) → [0, ∞) such that for any triple x, y, z of distinct points in X, we have
${\frac {d_{Y}(f(x),f(y))}{d_{Y}(f(x),f(z))}}\leq \eta \left({\frac {d_{X}(x,y)}{d_{X}(x,z)}}\right).$
Basic properties
Inverses are quasisymmetric
If f : X → Y is an invertible η-quasisymmetric map as above, then its inverse map is $\eta '$-quasisymmetric, where $ \eta '(t)=1/\eta ^{-1}(1/t).$
Quasisymmetric maps preserve relative sizes of sets
If $A$ and $B$ are subsets of $X$ and $A$ is a subset of $B$, then
${\frac {1}{2\eta ({\frac {\operatorname {diam} A}{\operatorname {diam} B}})}}\leq {\frac {\operatorname {diam} f(B)}{\operatorname {diam} f(A)}}\leq \eta \left({\frac {2\operatorname {diam} B}{\operatorname {diam} A}}\right).$
Examples
Weakly quasisymmetric maps
A map f:X→Y is said to be H-weakly-quasisymmetric for some $H>0$ if for all triples of distinct points $x,y,z$ in $X$, then
$|f(x)-f(y)|\leq H|f(x)-f(z)|\;\;\;{\text{ whenever }}\;\;\;|x-y|\leq |x-z|$
Not all weakly quasisymmetric maps are quasisymmetric. However, if $X$ is connected and $X$ and $Y$ are doubling, then all weakly quasisymmetric maps are quasisymmetric. The appeal of this result is that proving weak-quasisymmetry is much easier than proving quasisymmetry directly, and in many natural settings the two notions are equivalent.
δ-monotone maps
A monotone map f:H → H on a Hilbert space H is δ-monotone if for all x and y in H,
$\langle f(x)-f(y),x-y\rangle \geq \delta |f(x)-f(y)|\cdot |x-y|.$
To grasp what this condition means geometrically, suppose f(0) = 0 and consider the above estimate when y = 0. Then it implies that the angle between the vector x and its image f(x) stays between 0 and arccos δ < π/2.
These maps are quasisymmetric, although they are a much narrower subclass of quasisymmetric maps. For example, while a general quasisymmetric map in the complex plane could map the real line to a set of Hausdorff dimension strictly greater than one, a δ-monotone will always map the real line to a rotated graph of a Lipschitz function L:ℝ → ℝ.[2]
Doubling measures
The real line
Quasisymmetric homeomorphisms of the real line to itself can be characterized in terms of their derivatives.[3] An increasing homeomorphism f:ℝ → ℝ is quasisymmetric if and only if there is a constant C > 0 and a doubling measure μ on the real line such that
$f(x)=C+\int _{0}^{x}\,d\mu (t).$
Euclidean space
An analogous result holds in Euclidean space. Suppose C = 0 and we rewrite the above equation for f as
$f(x)={\frac {1}{2}}\int _{\mathbb {R} }\left({\frac {x-t}{|x-t|}}+{\frac {t}{|t|}}\right)d\mu (t).$
Writing it this way, we can attempt to define a map using this same integral, but instead integrate (what is now a vector valued integrand) over ℝn: if μ is a doubling measure on ℝn and
$\int _{|x|>1}{\frac {1}{|x|}}\,d\mu (x)<\infty $
then the map
$f(x)={\frac {1}{2}}\int _{\mathbb {R} ^{n}}\left({\frac {x-y}{|x-y|}}+{\frac {y}{|y|}}\right)\,d\mu (y)$
is quasisymmetric (in fact, it is δ-monotone for some δ depending on the measure μ).[4]
Quasisymmetry and quasiconformality in Euclidean space
Let $\Omega $ and $\Omega '$ be open subsets of ℝn. If f : Ω → Ω´ is η-quasisymmetric, then it is also K-quasiconformal, where $K>0$ is a constant depending on $\eta $.
Conversely, if f : Ω → Ω´ is K-quasiconformal and $B(x,2r)$ is contained in $\Omega $, then $f$ is η-quasisymmetric on $B(x,2r)$, where $\eta $ depends only on $K$.
Quasi-Möbius maps
A related but weaker condition is the notion of quasi-Möbius maps where instead of the ratio only the cross-ratio is considered:[5]
Definition
Let (X, dX) and (Y, dY) be two metric spaces and let η : [0, ∞) → [0, ∞) be an increasing function. An η-quasi-Möbius homeomorphism f:X → Y is a homeomorphism for which for every quadruple x, y, z, t of distinct points in X, we have
${\frac {d_{Y}(f(x),f(z))d_{Y}(f(y),f(t))}{d_{Y}(f(x),f(y))d_{Y}(f(z),f(t))}}\leq \eta \left({\frac {d_{X}(x,z)d_{X}(y,t)}{d_{X}(x,y)d_{X}(z,t)}}\right).$
See also
• Douady–Earle extension
References
1. Heinonen, Juha (2001). Lectures on Analysis on Metric Spaces. Universitext. New York: Springer-Verlag. pp. x+140. ISBN 978-0-387-95104-1.
2. Kovalev, Leonid V. (2007). "Quasiconformal geometry of monotone mappings". Journal of the London Mathematical Society. 75 (2): 391–408. CiteSeerX 10.1.1.194.2458. doi:10.1112/jlms/jdm008.
3. Beurling, A.; Ahlfors, L. (1956). "The boundary correspondence under quasiconformal mappings". Acta Math. 96: 125–142. doi:10.1007/bf02392360.
4. Kovalev, Leonid; Maldonado, Diego; Wu, Jang-Mei (2007). "Doubling measures, monotonicity, and quasiconformality". Math. Z. 257 (3): 525–545. arXiv:math/0611110. doi:10.1007/s00209-007-0132-5. S2CID 119716883.
5. Buyalo, Sergei; Schroeder, Viktor (2007). Elements of Asymptotic Geometry. EMS Monographs in Mathematics. American Mathematical Society. p. 209. ISBN 978-3-03719-036-4.
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Quasithin group
In mathematics, a quasithin group is a finite simple group that resembles a group of Lie type of rank at most 2 over a field of characteristic 2. More precisely it is a finite simple group of characteristic 2 type and width 2. Here characteristic 2 type means that its centralizers of involutions resemble those of groups of Lie type over fields of characteristic 2, and the width is roughly the maximal rank of an abelian group of odd order normalizing a non-trivial 2-subgroup of G. When G is a group of Lie type of characteristic 2 type, the width is usually the rank (the dimension of a maximal torus of the algebraic group).
Classification
The classification of quasithin groups is a crucial part of the classification of finite simple groups. The quasithin groups were classified in a 1221-page paper by Michael Aschbacher and Stephen D. Smith (2004, 2004b). An earlier announcement by Geoffrey Mason (1980) of the classification, on the basis of which the classification of finite simple groups was announced as finished in 1983, was premature as the unpublished manuscript (Mason 1981) of his work was incomplete and contained serious gaps.
According to Aschbacher & Smith (2004b, theorem 0.1.1), the finite simple quasithin groups of even characteristic are given by
• Groups of Lie type of characteristic 2 and rank 1 or 2, except that U5(q) only occurs for q = 4
• PSL4(2), PSL5(2), Sp6(2)
• The alternating groups on 5, 6, 8, 9, points
• PSL2(p) for p a Fermat or Mersenne prime, Lε
3
(3), Lε
4
(3), G2(3)
• The Mathieu groups M11, M12, M22, M23, M24, The Janko groups J2, J3, J4, the Higman-Sims group, the Held group, and the Rudvalis group.
If the condition "even characteristic" is relaxed to "even type" in the sense of the revision of the classification by Daniel Gorenstein, Richard Lyons, and Ronald Solomon, then the only extra group that appears is the Janko group J1.
References
• Aschbacher, Michael; Smith, Stephen D. (2004), The classification of quasithin groups. I Structure of Strongly Quasithin K-groups, Mathematical Surveys and Monographs, vol. 111, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-3410-7, MR 2097623
• Aschbacher, Michael; Smith, Stephen D. (2004b), The classification of quasithin groups. II Main theorems: the classification of simple QTKE-groups., Mathematical Surveys and Monographs, vol. 112, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-3411-4, MR 2097624
• Mason, Geoffrey (1980), "Quasithin groups", in Collins, Michael J. (ed.), Finite simple groups. II, London: Academic Press Inc. [Harcourt Brace Jovanovich Publishers], pp. 181–197, ISBN 978-0-12-181480-9, MR 0606048
• Mason, Geoffrey (1981), The classification of finite quasithin groups, U. California Santa Cruz, p. 800 (unpublished typescript)
• Solomon, Ronald (2006), "Review of The classification of quasithin groups. I, II by Aschbacher and Smith", Bulletin of the American Mathematical Society, 43: 115–121, doi:10.1090/s0273-0979-05-01071-2
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Quasitopological space
In mathematics, a quasi-topology on a set X is a function that associates to every compact Hausdorff space C a collection of mappings from C to X satisfying certain natural conditions. A set with a quasi-topology is called a quasitopological space.
They were introduced by Spanier, who showed that there is a natural quasi-topology on the space of continuous maps from one space to another.
References
• Spanier, E. (1963), "Quasi-topologies", Duke Mathematical Journal, 30 (1): 1–14, doi:10.1215/S0012-7094-63-03001-1, MR 0144300.
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Quasitoric manifold
In mathematics, a quasitoric manifold is a topological analogue of the nonsingular projective toric variety of algebraic geometry. A smooth $2n$-dimensional manifold is a quasitoric manifold if it admits a smooth, locally standard action of an $n$-dimensional torus, with orbit space an $n$-dimensional simple convex polytope.
Quasitoric manifolds were introduced in 1991 by M. Davis and T. Januszkiewicz,[1] who called them "toric manifolds". However, the term "quasitoric manifold" was eventually adopted to avoid confusion with the class of compact smooth toric varieties, which are known to algebraic geometers as toric manifolds.[2]
Quasitoric manifolds are studied in a variety of contexts in algebraic topology, such as complex cobordism theory, and the other oriented cohomology theories.[3]
Definitions
Denote the $i$-th subcircle of the $n$-torus $T^{n}$ by $T_{i}$ so that $T_{1}\times \ldots \times T_{n}=T^{n}$. Then coordinate-wise multiplication of $T^{n}$ on $\mathbb {C} ^{n}$ is called the standard representation.
Given open sets $X$ in $M^{2n}$ and $Y$ in $\mathbb {C} ^{n}$, that are closed under the action of $T^{n}$, a $T^{n}$-action on $M^{2n}$ is defined to be locally isomorphic to the standard representation if $h(tx)=\alpha (t)h(x)$, for all $t$ in $T^{n}$, $x$ in $X$, where $h$ is a homeomorphism $X\rightarrow Y$, and $\alpha $ is an automorphism of $T^{n}$.
Given a simple convex polytope $P^{n}$ with $m$ facets, a $T^{n}$-manifold $M^{2n}$ is a quasitoric manifold over $P^{n}$ if,
1. the $T^{n}$-action is locally isomorphic to the standard representation,
2. there is a projection $\pi :M^{2n}\rightarrow P^{n}$ that maps each $l$-dimensional orbit to a point in the interior of an $l$-dimensional face of $P^{n}$, for $l=0,$ $...,$ $n$.
The definition implies that the fixed points of $M^{2n}$ under the $T^{n}$-action are mapped to the vertices of $P^{n}$ by $\pi $, while points where the action is free project to the interior of the polytope.
The dicharacteristic function
A quasitoric manifold can be described in terms of a dicharacteristic function and an associated dicharacteristic matrix. In this setting it is useful to assume that the facets $F_{1},\dots ,F_{m}$ of $P^{n}$ are ordered so that the intersection $F_{1}\cap \dots \cap F_{n}$ is a vertex $v$ of $P^{n}$, called the initial vertex.
A dicharacteristic function is a homomorphism $\lambda :T^{m}\rightarrow T^{n}$, such that if $F_{i_{1}}\cap \dots \cap F_{i_{k}}$ is a codimension-$k$ face of $P^{n}$, then $\lambda $ is a monomorphism on restriction to the subtorus $T_{i_{1}}\times \dots \times T_{i_{k}}$ in $T^{m}$.
The restriction of λ to the subtorus $T_{1}\times \ldots \times T_{n}$ corresponding to the initial vertex $v$ is an isomorphism, and so $\lambda (T_{1}),\ldots ,\lambda (T_{n})$ can be taken to be a basis for the Lie algebra of $T^{n}$. The epimorphism of Lie algebras associated to λ may be described as a linear transformation $\mathbb {Z} ^{m}\rightarrow \mathbb {Z} ^{n}$, represented by the $n\times m$ dicharacteristic matrix $\Lambda $ given by
${\begin{bmatrix}1&0&\dots &0&\lambda _{1,n+1}&\dots &\lambda _{1,m}\\0&1&\dots &0&\lambda _{2,n+1}&\dots &\lambda _{2,m}\\\vdots &\vdots &\ddots &\vdots &\vdots &\ddots &\vdots \\0&0&\dots &1&\lambda _{n,n+1}&\dots &\lambda _{n,m}\end{bmatrix}}.$
The $i$th column of $\Lambda $ is a primitive vector $\lambda _{i}=(\lambda _{1,i},\dots ,\lambda _{n,i})$ in $\mathbb {Z} ^{n}$, called the facet vector. As each facet vector is primitive, whenever the facets $F_{i_{1}}\cap \dots \cap F_{i_{n}}$ meet in a vertex, the corresponding columns $\lambda _{i_{1}},\dots \lambda _{i_{n}}$ form a basis of $\mathbb {Z} ^{n}$, with determinant equal to $\pm 1$. The isotropy subgroup associated to each facet $F_{i}$ is described by
$\{(e^{2\pi i\theta \lambda _{1,i}},\ldots ,e^{2\pi i\theta \lambda _{n,i}})\in T^{n}\},$
for some $\theta $ in $\mathbb {R} $.
In their original treatment of quasitoric manifolds, Davis and Januskiewicz[1] introduced the notion of a characteristic function that mapped each facet of the polytope to a vector determining the isotropy subgroup of the facet, but this is only defined up to sign. In more recent studies of quasitoric manifolds, this ambiguity has been removed by the introduction of the dicharacteristic function and its insistence that each circle $\lambda (T_{i})$ be oriented, forcing a choice of sign for each vector $\lambda _{i}$. The notion of the dicharacteristic function was originally introduced V. Buchstaber and N. Ray[4] to enable the study of quasitoric manifolds in complex cobordism theory. This was further refined by introducing the ordering of the facets of the polytope to define the initial vertex, which eventually leads to the above neat representation of the dicharacteristic matrix $\Lambda $ as $(I_{n}\mid S)$, where $I_{n}$ is the identity matrix and $S$ is an $n\times (m-n)$ submatrix.[5]
Relation to the moment-angle complex
The kernel $K(\lambda )$ of the dicharacteristic function acts freely on the moment angle complex $Z_{P^{n}}$, and so defines a principal $K(\lambda )$-bundle $Z_{P^{n}}\rightarrow M^{2n}$ over the resulting quotient space $M^{2n}$. This quotient space can be viewed as
$T^{n}\times P^{n}/\sim ,$
where pairs $(t_{1},p_{1})$, $(t_{2},p_{2})$ of $T^{n}\times P^{n}$ are identified if and only if $p_{1}=p_{2}$ and $t_{1}^{-1}t_{2}$ is in the image of $\lambda $ on restriction to the subtorus $T_{i_{1}}\times \dots \times T_{i_{k}}$ that corresponds to the unique face $F_{i_{1}}\cap \dots \cap F_{i_{k}}$ of $P^{n}$ containing the point $p_{1}$, for some $1\leq k\leq n$.
It can be shown that any quasitoric manifold $M^{2n}$ over $P^{n}$ is equivariently diffeomorphic to a quasitoric manifold of the form of the quotient space above.[6]
Examples
• The $n$-dimensional complex projective space $\mathbb {C} P^{n}$ is a quasitoric manifold over the $n$-simplex $\Delta ^{n}$. If $\Delta ^{n}$ is embedded in $\mathbb {R} ^{n+1}$ so that the origin is the initial vertex, a dicharacteristic function can be chosen so that the associated dicharacteristic matrix is
${\begin{bmatrix}1&0&\dots &0&-1\\0&1&\dots &0&-1\\\vdots &\vdots &\ddots &\vdots &\vdots \\0&0&\dots &1&-1\end{bmatrix}}.$
The moment angle complex $Z_{\Delta ^{n}}$ is the $(2n+1)$-sphere $S^{2n+1}$, the kernel $K(\lambda )$ is the diagonal subgroup $\{(t,\dots ,t)\}<T^{n+1}$, so the quotient of $Z_{\Delta ^{n}}$ under the action of $K(\lambda )$ is $\mathbb {C} P^{n}$.[7]
• The Bott manifolds that form the stages in a Bott tower are quasitoric manifolds over $n$-cubes. The $n$-cube $I^{n}$ is embedded in $\mathbb {R} ^{2n}$ so that the origin is the initial vertex, and a dicharacteristic function is chosen so that the associated dicharacteristic matrix $(I_{n}\mid S)$ has $S$ given by
${\begin{bmatrix}1&0&\cdots &0&0&\cdots &0&0\\-a(1,2)&1&\cdots &0&0&\cdots &0&0\\\vdots &\vdots &&\vdots &\vdots &&\vdots &\vdots \\-a(1,i)&-a(2,i)&\cdots &-a(i-1,i)&1&\cdots &0&0\\\vdots &\vdots &&\vdots &\vdots &&\vdots &\vdots \\-a(1,n)&-a(2,n)&\cdots &-a(i-1,n)&-a(i,n)&\cdots &-a(n-1,n)&1\end{bmatrix}},$
for integers $a(i,j)$.
The moment angle complex $Z_{I^{n}}$ is a product of $n$ copies of 3-sphere embedded in $\mathbb {C} ^{2n}$, the kernel $K(\lambda )$ is given by
$\{(t_{1},t_{1}^{-a(1,2)}t_{2},\dots ,t_{1}^{-a(1,i)}\dots t_{i-1}^{-a(i-1,i)}t_{i},\dots ,t_{1}^{-a(1,n)}\dots t_{n-1}^{-a(n-1,n)}t_{n},t_{1}^{-1},\dots ,t_{n}^{-1}):t_{i}\in T,1\leq i\leq n\}<T^{2n}$,
so that the quotient of $Z_{I^{n}}$ under the action of $K(\lambda )$ is the $n$-th stage of a Bott tower.[8] The integer values $a(i,j)$ are the tensor powers of the line bundles whose product is used in the iterated sphere-bundle construction of the Bott tower.[9]
The cohomology ring of a quasitoric manifold
Canonical complex line bundles $\rho _{i}$ over $M^{2n}$ given by
$Z_{P^{n}}\times _{K(l)}\mathbb {C} _{i}\longrightarrow M^{2n}$,
can be associated with each facet $F_{i}$ of $P^{n}$, for $1\leq i\leq m$, where $K(\lambda )$ acts on $\mathbb {C} _{i}$, by the restriction of $K(\lambda )$ to the $i$-th subcircle of $T^{m}$ embedded in $\mathbb {C} $. These bundles are known as the facial bundles associated to the quasitoric manifold. By the definition of $M^{2n}$, the preimage of a facet $\pi ^{-1}(F_{i})$ is a $2(n-1)$-dimensional quasitoric facial submanifold $M_{i}$ over $F_{i}$, whose isotropy subgroup is the restriction of $\lambda $ on the subcircle $T_{i}$ of $T^{m}$. Restriction of $\rho _{i}$ to $M_{i}$ gives the normal 2-plane bundle of the embedding of $M_{i}$ in $M^{2n}$.
Let $x_{i}$ in $H^{2}(M^{2n};\mathbb {Z} )$ denote the first Chern class of $\rho _{i}$. The integral cohomology ring $H^{*}(M^{2n};\mathbb {Z} )$ is generated by $x_{i}$, for $1\leq i\leq m$, subject to two sets of relations. The first are the relations generated by the Stanley–Reisner ideal of $P^{n}$; linear relations determined by the dicharacterstic function comprise the second set:
$x_{i}=-\lambda _{i,n+1}x_{n+1}-\cdots -\lambda _{i,m}x_{m},{\mbox{ for }}1\leq i\leq n$.
Therefore only $x_{n+1},\dots ,x_{m}$ are required to generate $H^{*}(M^{2n};\mathbb {Z} )$ multiplicatively.[1]
Comparison with toric manifolds
• Any projective toric manifold is a quasitoric manifold, and in some cases non-projective toric manifolds are also quasitoric manifolds.
• Not all quasitoric manifolds are toric manifolds. For example, the connected sum $\mathbb {C} P^{2}\sharp \mathbb {C} P^{2}$ can be constructed as a quasitoric manifold, but it is not a toric manifold.[10]
Notes
1. M. Davis and T. Januskiewicz, 1991.
2. V. Buchstaber and T. Panov, 2002.
3. V. Buchstaber and N. Ray, 2008.
4. V. Buchstaber and N. Ray, 2001.
5. V. Buchstaber, T. Panov and N. Ray, 2007.
6. M. Davis and T. Januskiewicz, 1991, Proposition 1.8.
7. V. Buchstaber, T. Panov and N. Ray, 2007, Example 3.11.
8. V. Buchstaber, T. Panov and N. Ray, 2007, Example 3.12.
9. Y. Civan and N. Ray, 2005.
10. M. Masuda and D. Y. Suh 2007.
References
• Buchstaber, V.; Panov, T. (2002), Torus Actions and their Applications in Topology and Combinatorics, University Lecture Series, vol. 24, American Mathematical Society
• Buchstaber, V.; Panov, T.; Ray, N. (2007), "Spaces of polytopes and cobordism of quasitoric manifolds", Moscow Mathematical Journal, 7 (2): 219–242, arXiv:math/0609346, doi:10.17323/1609-4514-2007-7-2-219-242, S2CID 72554
• Buchstaber, V.; Ray, N. (2001), "Tangential structures on toric manifolds and connected sums of polytopes", International Mathematics Research Notices, 2001 (4): 193–219, doi:10.1155/S1073792801000125, S2CID 8030669
• Buchstaber, V.; Ray, N. (2008), "An Invitation to Toric Topology: Vertex Four of a Remarkable Tetrahedron", Proceedings of the International Conference in Toric Topology; Osaka City University 2006, Contemporary Mathematics, vol. 460, American Mathematical Society, pp. 1–27
• Civan, Y.; Ray, N. (2005), "Homotopy decompositions and K-theory of Bott towers", K-Theory, 34: 1–33, arXiv:math/0408261, doi:10.1007/s10977-005-1551-x, S2CID 15934494
• Davis, M.; Januskiewicz, T. (1991), "Convex polytopes, Coxeter orbifolds and torus actions", Duke Mathematical Journal, 62 (2): 417–451, doi:10.1215/s0012-7094-91-06217-4, S2CID 115132549
• Masuda, M.; Suh, D. Y. (2008), "Classification problems of toric manifolds via topology", Proceedings of the International Conference in Toric Topology; Osaka City University 2006, Contemporary Mathematics, vol. 460, American Mathematical Society, pp. 273–286
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Quasitrace
In mathematics, especially functional analysis, a quasitrace is a not necessarily additive tracial functional on a C*-algebra. An additive quasitrace is called a trace. It is a major open problem if every quasitrace is a trace.
Definition
A quasitrace on a C*-algebra A is a map $\tau \colon A_{+}\to [0,\infty ]$ such that:
• $\tau $ is homogeneous:
$\tau (\lambda a)=\lambda \tau (a)$ for every $a\in A_{+}$ and $\lambda \in [0,\infty )$.
• $\tau $ is tracial:
$\tau (xx^{*})=\tau (x^{*}x)$ for every $x\in A$.
• $\tau $ is additive on commuting elements:
$\tau (a+b)=\tau (a)+\tau (b)$ for every $a,b\in A_{+}$ that satisfy $ab=ba$.
• and such that for each $n\geq 1$ the induced map
$\tau _{n}\colon M_{n}(A)_{+}\to [0,\infty ],(a_{j,k})_{j,k=1,...,n}\mapsto \tau (a_{11})+...\tau (a_{nn})$
has the same properties.
A quasitrace $\tau $ is:
• bounded if
$\sup\{\tau (a):a\in A_{+},\|a\|\leq 1\}<\infty .$
• normalized if
$\sup\{\tau (a):a\in A_{+},\|a\|\leq 1\}=1.$
• lower semicontinuous if
$\{a\in A_{+}:\tau (a)\leq t\}$ is closed for each $t\in [0,\infty )$.
Variants
• A 1-quasitrace is a map $A_{+}\to [0,\infty ]$ that is just homogeneous, tracial and additive on commuting elements, but does not necessarily extend to such a map on matrix algebras over A. If a 1-quasitrace extends to the matrix algebra $M_{n}(A)$, then it is called a n-quasitrace. There are examples of 1-quasitraces that are not 2-quasitraces. One can show that every 2-quasitrace is automatically a n-quasitrace for every $n\geq 1$. Sometimes in the literature, a quasitrace means a 1-quasitrace and a 2-quasitrace means a quasitrace.
Properties
• A quasitrace that is additive on all elements is called a trace.
• Uffe Haagerup showed that every quasitrace on a unital, exact C*-algebra is additive and thus a trace. The article of Haagerup [1] was circulated as handwritten notes in 1991 and remained unpublished until 2014. Blanchard and Kirchberg removed the assumption of unitality in Haagerup's result.[2] As of today (August 2020) it remains an open problem if every quasitrace is additive.
• Joachim Cuntz showed that a simple, unital C*-algebra is stably finite if and only if it admits a dimension function. A simple, unital C*-algebra is stably finite if and only if it admits a normalized quasitrace. An important consequence is that every simple, unital, stably finite, exact C*-algebra admits a tracial state.
• Every quasitrace on a von Neumann algebra is a trace.
Notes
1. (Haagerup 2014)
2. Blanchard, Kirchberg, 2004, Remarks 2.29(i)
References
• Blanchard, Etienne; Kirchberg, Eberhard (February 2004). "Non-simple purely infinite C∗-algebras: the Hausdorff case" (PDF). Journal of Functional Analysis. 207 (2): 461–513. doi:10.1016/j.jfa.2003.06.008.
• Haagerup, Uffe (2014). "Quasitraces on Exact C*-algebras are Traces". C. R. Math. Rep. Acad. Sci. Canada. 36: 67–92.
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Quasitransitive relation
The mathematical notion of quasitransitivity is a weakened version of transitivity that is used in social choice theory and microeconomics. Informally, a relation is quasitransitive if it is symmetric for some values and transitive elsewhere. The concept was introduced by Sen (1969) to study the consequences of Arrow's theorem.
Formal definition
A binary relation T over a set X is quasitransitive if for all a, b, and c in X the following holds:
$(a\operatorname {T} b)\wedge \neg (b\operatorname {T} a)\wedge (b\operatorname {T} c)\wedge \neg (c\operatorname {T} b)\Rightarrow (a\operatorname {T} c)\wedge \neg (c\operatorname {T} a).$
If the relation is also antisymmetric, T is transitive.
Alternately, for a relation T, define the asymmetric or "strict" part P:
$(a\operatorname {P} b)\Leftrightarrow (a\operatorname {T} b)\wedge \neg (b\operatorname {T} a).$
Then T is quasitransitive if and only if P is transitive.
Examples
Preferences are assumed to be quasitransitive (rather than transitive) in some economic contexts. The classic example is a person indifferent between 7 and 8 grams of sugar and indifferent between 8 and 9 grams of sugar, but who prefers 9 grams of sugar to 7.[1] Similarly, the Sorites paradox can be resolved by weakening assumed transitivity of certain relations to quasitransitivity.
Properties
• A relation R is quasitransitive if, and only if, it is the disjoint union of a symmetric relation J and a transitive relation P.[2] J and P are not uniquely determined by a given R;[3] however, the P from the only-if part is minimal.[4]
• As a consequence, each symmetric relation is quasitransitive, and so is each transitive relation.[5] Moreover, an antisymmetric and quasitransitive relation is always transitive.[6]
• The relation from the above sugar example, {(7,7), (7,8), (7,9), (8,7), (8,8), (8,9), (9,8), (9,9)}, is quasitransitive, but not transitive.
• A quasitransitive relation needn't be acyclic: for every non-empty set A, the universal relation A×A is both cyclic and quasitransitive.
• A relation is quasitransitive if, and only if, its complement is.
• Similarly, a relation is quasitransitive if, and only if, its converse is.
See also
• Intransitivity
• Reflexive relation
References
1. Robert Duncan Luce (Apr 1956). "Semiorders and a Theory of Utility Discrimination" (PDF). Econometrica. 24 (2): 178–191. doi:10.2307/1905751. JSTOR 1905751. Here: p.179; Luce's original example consists in 400 comparisons (of coffee cups with different amounts of sugar) rather than just 2.
2. The naminig follows Bossert & Suzumura (2009), p.2-3. — For the only-if part, define xJy as xRy ∧ yRx, and define xPy as xRy ∧ ¬yRx. — For the if part, assume xRy ∧ ¬yRx ∧ yRz ∧ ¬zRy holds. Then xPy and yPz, since xJy or yJz would contradict ¬yRx or ¬zRy. Hence xPz by transitivity, ¬xJz by disjointness, ¬zJx by symmetry. Therefore, zRx would imply zPx, and, by transitivity, zPy, which contradicts ¬zRy. Altogether, this proves xRz ∧ ¬zRx.
3. For example, if R is an equivalence relation, J may be chosen as the empty relation, or as R itself, and P as its complement.
4. Given R, whenever xRy ∧ ¬yRx holds, the pair (x,y) can't belong to the symmetric part, but must belong to the transitive part.
5. Since the empty relation is trivially both transitive and symmetric.
6. The antisymmetry of R forces J to be coreflexive; hence the union of J and the transitive P is again transitive.
• Sen, A. (1969). "Quasi-transitivity, rational choice and collective decisions". Rev. Econ. Stud. 36 (3): 381–393. doi:10.2307/2296434. JSTOR 2296434. Zbl 0181.47302.
• Frederic Schick (Jun 1969). "Arrow's Proof and the Logic of Preference". Philosophy of Science. 36 (2): 127–144. doi:10.1086/288241. JSTOR 186166. S2CID 121427121.
• Amartya K. Sen (1970). Collective Choice and Social Welfare. Holden-Day, Inc.
• Amartya K. Sen (Jul 1971). "Choice Functions and Revealed Preference" (PDF). The Review of Economic Studies. 38 (3): 307–317. doi:10.2307/2296384. JSTOR 2296384.
• A. Mas-Colell and H. Sonnenschein (1972). "General Possibility Theorems for Group Decisions" (PDF). The Review of Economic Studies. 39 (2): 185–192. doi:10.2307/2296870. JSTOR 2296870. S2CID 7295776. Archived from the original (PDF) on 2018-04-12.
• D.H. Blair and R.A. Pollak (1982). "Acyclic Collective Choice Rules". Econometrica. 50 (4): 931–943. doi:10.2307/1912770. JSTOR 1912770.
• Bossert, Walter; Suzumura, Kotaro (Apr 2005). Rational Choice on Arbitrary Domains: A Comprehensive Treatment (PDF) (Technical Report). Université de Montréal, Hitotsubashi University Tokyo.
• Bossert, Walter; Suzumura, Kotaro (Mar 2009). Quasi-transitive and Suzumura consistent relations (PDF) (Technical Report). Université de Montréal, Waseda University Tokyo. doi:10.1007/s00355-011-0600-z. S2CID 38375142. Archived from the original (PDF) on 2018-04-12.
• Bossert, Walter; Suzumura, Kōtarō (2010). Consistency, choice and rationality. Harvard University Press. ISBN 978-0674052994.
• Alan D. Miller and Shiran Rachmilevitch (Feb 2014). Arrow's Theorem Without Transitivity (PDF) (Working paper). University of Haifa.
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Wikipedia
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Quasi-triangulation
A quasi-triangulation is a subdivision of a geometric object into simplices, where vertices are not points but arbitrary sloped line segments.[1] This division is not a triangulation in the geometric sense. It is a topological triangulation, however. A quasi-triangulation may have some of the characteristics of a Delaunay triangulation.
References
1. Luzin S.Y.; Lyachek Y.T.; Petrosyan G.S.; Polubasov O.B. (2010). Models and algorithms for automated design of electronic and computer equipment (in Russian). BHV-Petersburg. p. 224. ISBN 978-5-9775-0576-5.
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Wikipedia
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Quasivariety
In mathematics, a quasivariety is a class of algebraic structures generalizing the notion of variety by allowing equational conditions on the axioms defining the class.
Definition
A trivial algebra contains just one element. A quasivariety is a class K of algebras with a specified signature satisfying any of the following equivalent conditions.[1]
1. K is a pseudoelementary class closed under subalgebras and direct products.
2. K is the class of all models of a set of quasiidentities, that is, implications of the form $s_{1}\approx t_{1}\land \ldots \land s_{n}\approx t_{n}\rightarrow s\approx t$, where $s,s_{1},\ldots ,s_{n},t,t_{1},\ldots ,t_{n}$ are terms built up from variables using the operation symbols of the specified signature.
3. K contains a trivial algebra and is closed under isomorphisms, subalgebras, and reduced products.
4. K contains a trivial algebra and is closed under isomorphisms, subalgebras, direct products, and ultraproducts.
Examples
Every variety is a quasivariety by virtue of an equation being a quasiidentity for which n = 0.
The cancellative semigroups form a quasivariety.
Let K be a quasivariety. Then the class of orderable algebras from K forms a quasivariety, since the preservation-of-order axioms are Horn clauses.[2]
References
1. Stanley Burris; H.P. Sankappanavar (1981). A Course in Universal Algebra. Springer-Verlag. ISBN 0-387-90578-2.
2. Viktor A. Gorbunov (1998). Algebraic Theory of Quasivarieties. Siberian School of Algebra and Logic. Plenum Publishing. ISBN 0-306-11063-6.
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Quater-imaginary base
The quater-imaginary numeral system is a numeral system, first proposed by Donald Knuth in 1960. Unlike standard numeral systems, which use an integer (such as 10 in decimal, or 2 in binary) as their bases, it uses the imaginary number 2i (equivalent to ${\sqrt {-4}}$) as its base. It is able to (almost) uniquely represent every complex number using only the digits 0, 1, 2, and 3.[1] Numbers less than zero, which are ordinarily represented with a minus sign, are representable as digit strings in quater-imaginary; for example, the number −1 is represented as "103" in quater-imaginary notation.
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List of numeral systems
Decomposing the quater-imaginary
In a positional system with base $b$,
$\ldots d_{3}d_{2}d_{1}d_{0}.d_{-1}d_{-2}d_{-3}\ldots $
represents$\dots +d_{3}\cdot b^{3}+d_{2}\cdot b^{2}+d_{1}\cdot b+d_{0}+d_{-1}\cdot b^{-1}+d_{-2}\cdot b^{-2}+d_{-3}\cdot b^{-3}\dots $
In this numeral system, $b=2i$,
and because $(2i)^{2}=-4$, the entire series of powers can be separated into two different series, so that it simplifies to ${\begin{aligned}&{}[\dots +d_{4}\cdot (-4)^{2}+d_{2}\cdot (-4)^{1}+d_{0}+d_{-2}\cdot (-4)^{-1}+\dots ]\end{aligned}}$ for even-numbered digits (digits that simplify to the value of the digit times a power of -4), and ${\begin{aligned}2i\cdot [\dots +d_{3}\cdot (-4)^{1}+d_{1}+d_{-1}\cdot (-4)^{-1}+d_{-3}\cdot (-4)^{-2}+\dots ]\end{aligned}}$ for those digits that still have an imaginary factor. Adding these two series together then gives the total value of the number.
Because of the separation of these two series, the real and imaginary parts of complex numbers are readily expressed in base −4 as $\ldots d_{4}d_{2}d_{0}.d_{-2}\ldots $ and $2\cdot (\ldots d_{5}d_{3}d_{1}.d_{-1}d_{-3}\ldots )$ respectively.
Converting from quater-imaginary
Powers of 2i
k (2i)k
−5−i/32
−41/16
−3i/8
−2−1/4
−1−i/2
01
12i
2−4
3−8i
416
532i
6−64
7−128i
8256
To convert a digit string from the quater-imaginary system to the decimal system, the standard formula for positional number systems can be used. This says that a digit string $\ldots d_{3}d_{2}d_{1}d_{0}$ in base b can be converted to a decimal number using the formula
$\cdots +d_{3}\cdot b^{3}+d_{2}\cdot b^{2}+d_{1}\cdot b+d_{0}$
For the quater-imaginary system, $b=2i$.
Additionally, for a given string $d$ in the form $d_{w-1},d_{w-2},\dots d_{0}$, the formula below can be used for a given string length $w$ in base $b$
$Q2D_{w}{\vec {d}}\equiv \sum _{k=0}^{w-1}d_{k}\cdot b^{k}$
Example
To convert the string $1101_{2i}$ to a decimal number, fill in the formula above:
$1\cdot (2i)^{3}+1\cdot (2i)^{2}+0\cdot (2i)^{1}+1\cdot (2i)^{0}=-8i-4+0+1=-3-8i$
Another, longer example: $1030003_{2i}$ in base 10 is
$1\cdot (2i)^{6}+3\cdot (2i)^{4}+3\cdot (2i)^{0}=-64+3\cdot 16+3=-13$
Converting into quater-imaginary
It is also possible to convert a decimal number to a number in the quater-imaginary system. Every complex number (every number of the form a+bi) has a quater-imaginary representation. Most numbers have a unique quater-imaginary representation, but just as 1 has the two representations 1 = 0.9 in decimal notation, so, because of 0.00012i = 1/15, the number 1/5 has the two quater-imaginary representations 0.00032i = 3·1/15 = 1/5 = 1 + 3·–4/15 = 1.03002i.
To convert an arbitrary complex number to quater-imaginary, it is sufficient to split the number into its real and imaginary components, convert each of those separately, and then add the results by interleaving the digits. For example, since −1+4i is equal to −1 plus 4i, the quater-imaginary representation of −1+4i is the quater-imaginary representation of −1 (namely, 103) plus the quater-imaginary representation of 4i (namely, 20), which gives a final result of −1+4i = 1232i.
To find the quater-imaginary representation of the imaginary component, it suffices to multiply that component by 2i, which gives a real number; then find the quater-imaginary representation of that real number, and finally shift the representation by one place to the right (thus dividing by 2i). For example, the quater-imaginary representation of 6i is calculated by multiplying 6i × 2i = −12, which is expressed as 3002i, and then shifting by one place to the right, yielding: 6i = 302i.
Finding the quater-imaginary representation of an arbitrary real integer number can be done manually by solving a system of simultaneous equations, as shown below, but there are faster methods for both real and imaginary integers, as shown in the negative base article.
Example: Real number
As an example of an integer number we can try to find the quater-imaginary counterpart of the decimal number 7 (or 710 since the base of the decimal system is 10). Since it is hard to predict exactly how long the digit string will be for a given decimal number, it is safe to assume a fairly large string. In this case, a string of six digits can be chosen. When an initial guess at the size of the string eventually turns out to be insufficient, a larger string can be used.
To find the representation, first write out the general formula, and group terms:
${\begin{aligned}7_{10}&=d_{0}+d_{1}\cdot b+d_{2}\cdot b^{2}+d_{3}\cdot b^{3}+d_{4}\cdot b^{4}+d_{5}\cdot b^{5}\\&=d_{0}+2id_{1}-4d_{2}-8id_{3}+16d_{4}+32id_{5}\\&=d_{0}-4d_{2}+16d_{4}+i(2d_{1}-8d_{3}+32d_{5})\\\end{aligned}}$
Since 7 is a real number, it is allowed to conclude that d1, d3 and d5 should be zero. Now the value of the coefficients d0, d2 and d4, must be found. Because d0 − 4 d2 + 16 d4 = 7 and because—by the nature of the quater-imaginary system—the coefficients can only be 0, 1, 2 or 3 the value of the coefficients can be found. A possible configuration could be: d0 = 3, d2 = 3 and d4 = 1. This configuration gives the resulting digit string for 710.
$7_{10}=010303_{2i}=10303_{2i}.$
Example: Imaginary number
Finding a quater-imaginary representation of a purely imaginary integer number ∈ iZ is analogous to the method described above for a real number. For example, to find the representation of 6i, it is possible to use the general formula. Then all coefficients of the real part have to be zero and the complex part should make 6. However, for 6i it is easily seen by looking at the formula that if d1 = 3 and all other coefficients are zero, we get the desired string for 6i. That is:
${\begin{aligned}6i_{10}=30_{2i}\end{aligned}}$
Another conversion method
For real numbers the quater-imaginary representation is the same as negative quaternary (base −4). A complex number x+iy can be converted to quater-imaginary by converting x and y/2 separately to negative quaternary. If both x and y are finite binary fractions we can use the following algorithm using repeated Euclidean division:
For example: 35+23i=121003.22i
35 23i/2i=11.5 11=12−0.5
35÷(−4)=−8, remainder 3 12/(−4)=−3, remainder 0 (−0.5)×(−4)=2
−8÷(−4)= 2, remainder 0 −3/(−4)= 1, remainder 1
2÷(−4)= 0, remainder 2 1/(−4)= 0, remainder 1
20003 + 101000 + 0.2 = 121003.2
32i+16×2−8i−4×0+2i×0+1×3−2×i/2=35+23i
Radix point "."
A radix point in the decimal system is the usual . (dot) which marks the separation between the integer part and the fractional part of the number. In the quater-imaginary system a radix point can also be used. For a digit string $\dots d_{5}d_{4}d_{3}d_{2}d_{1}d_{0}.d_{-1}d_{-2}d_{-3}\dots $ the radix point marks the separation between non-negative and negative powers of b. Using the radix point the general formula becomes:
$d_{5}b^{5}+d_{4}b^{4}+d_{3}b^{3}+d_{2}b^{2}+d_{1}b+d_{0}+d_{-1}b^{-1}+d_{-2}b^{-2}+d_{-3}b^{-3}$
or
${\begin{aligned}32id_{5}+16d_{4}-8id_{3}-4d_{2}+2id_{1}+d_{0}+{\frac {1}{2i}}d_{-1}+{\frac {1}{-4}}d_{-2}+{\frac {1}{-8i}}d_{-3}\\=32id_{5}+16d_{4}-8id_{3}-4d_{2}+2id_{1}+d_{0}-{\frac {i}{2}}d_{-1}-{\frac {1}{4}}d_{-2}+{\frac {i}{8}}d_{-3}\end{aligned}}$
Example
If the quater-imaginary representation of the complex unit i has to be found, the formula without radix point will not suffice. Therefore, the above formula should be used. Hence:
${\begin{aligned}i&=32id_{5}+16d_{4}-8id_{3}-4d_{2}+2id_{1}+d_{0}+{\frac {1}{2i}}d_{-1}+{\frac {1}{-4}}d_{-2}+{\frac {1}{-8i}}d_{-3}\\&=i(32d_{5}-8d_{3}+2d_{1}-{\frac {1}{2}}d_{-1}+{\frac {1}{8}}d_{-3})+16d_{4}-4d_{2}+d_{0}-{\frac {1}{4}}d_{-2}\\\end{aligned}}$
for certain coefficients dk. Then because the real part has to be zero: d4 = d2 = d0 = d−2 = 0. For the imaginary part, if d5 = d3 = d−3 = 0 and when d1 = 1 and d−1 = 2 the digit string can be found. Using the above coefficients in the digit string the result is:
$i=10.2_{2i}.$
Addition and subtraction
It is possible to add and subtract numbers in the quater-imaginary system. In doing this, there are two basic rules that have to be kept in mind:
1. Whenever a number exceeds 3, subtract 4 and "carry" −1 two places to the left.
2. Whenever a number drops below 0, add 4 and "carry" +1 two places to the left.
Or for short: "If you add four, carry +1. If you subtract four, carry −1". This is the opposite of normal long addition, in which a "carry" in the current column requires adding 1 to the next column to the left, and a "borrow" requires subtracting. In quater-imaginary arithmetic, a "carry" subtracts from the next-but-one column, and a "borrow" adds.
Example: Addition
Below are two examples of adding in the quater-imaginary system:
1 − 2i 1031
1 − 2i 1031
------ + <=> ---- +
2 − 4i 1022
3 − 4i 1023
1 − 8i 1001
------ + <=> ----- +
4 −12i 12320
In the first example we start by adding the two 1s in the first column (the "ones' column"), giving 2. Then we add the two 3s in the second column (the "2is column"), giving 6; 6 is greater than 3, so we subtract 4 (giving 2 as the result in the second column) and carry −1 into the fourth column. Adding the 0s in the third column gives 0; and finally adding the two 1s and the carried −1 in the fourth column gives 1.
In the second example we first add 3+1, giving 4; 4 is greater than 3, so we subtract 4 (giving 0) and carry −1 into the third column (the "−4s column"). Then we add 2+0 in the second column, giving 2. In the third column, we have 0+0+(−1), because of the carry; −1 is less than 0, so we add 4 (giving 3 as the result in the third column) and "borrow" +1 into the fifth column. In the fourth column, 1+1 is 2; and the carry in the fifth column gives 1, for a result of $12320_{2i}$.
Example: Subtraction
Subtraction is analogous to addition in that it uses the same two rules described above. Below is an example:
− 2 − 8i 1102
1 − 6i 1011
------- <=> -----
− 3 − 2i 1131
In this example we have to subtract $1011_{2i}$ from $1102_{2i}$. The rightmost digit is 2−1 = 1. The second digit from the right would become −1, so add 4 to give 3 and then carry +1 two places to the left. The third digit from the right is 1−0 = 1. Then the leftmost digit is 1−1 plus 1 from the carry, giving 1. This gives a final answer of $1131_{2i}$.
Multiplication
For long multiplication in the quater-imaginary system, the two rules stated above are used as well. When multiplying numbers, multiply the first string by each digit in the second string consecutively and add the resulting strings. With every multiplication, a digit in the second string is multiplied with the first string. The multiplication starts with the rightmost digit in the second string and then moves leftward by one digit, multiplying each digit with the first string. Then the resulting partial products are added where each is shifted to the left by one digit. An example:
11201
20121 ×
---------------
11201 ←––– 1 × 11201
12002 ←––– 2 × 11201
11201 ←––– 1 × 11201
00000 ←––– 0 × 11201
12002 + ←––– 2 × 11201
---------------
120231321
This corresponds to a multiplication of $(9-8i)\cdot (29+4i)=293-196i$.
Tabulated conversions
Below is a table of some decimal and complex numbers and their quater-imaginary counterparts.
Base 10Base 2i
1 1
2 2
3 3
4 10300
5 10301
6 10302
7 10303
8 10200
9 10201
10 10202
11 10203
12 10100
13 10101
14 10102
15 10103
16 10000
Base 10Base 2i
−1 103
−2 102
−3 101
−4 100
−5 203
−6 202
−7 201
−8 200
−9 303
−10 302
−11 301
−12 300
−13 1030003
−14 1030002
−15 1030001
−16 1030000
Base 10Base 2i
1i10.2
2i10.0
3i20.2
4i20.0
5i30.2
6i30.0
7i103000.2
8i103000.0
9i103010.2
10i103010.0
11i103020.2
12i103020.0
13i103030.2
14i103030.0
15i102000.2
16i102000.0
Base 10Base 2i
−1i0.2
−2i1030.0
−3i1030.2
−4i1020.0
−5i1020.2
−6i1010.0
−7i1010.2
−8i1000.0
−9i1000.2
−10i2030.0
−11i2030.2
−12i2020.0
−13i2020.2
−14i2010.0
−15i2010.2
−16i2000.0
Examples
Below are some other examples of conversions from decimal numbers to quater-imaginary numbers.
$5=16+(3\cdot -4)+1=10301_{2i}$
$i=2i+2\left(-{\frac {1}{2}}i\right)=10.2_{2i}$
$7{\frac {3}{4}}-7{\frac {1}{2}}i=1(16)+1(-8i)+2(-4)+1(2i)+3\left(-{\frac {1}{2}}i\right)+1\left(-{\frac {1}{4}}\right)=11210.31_{2i}$
Z-order curve
The representation
$z=\sum _{k\geq n}z_{k}\cdot (2i)^{-k}$
of an arbitrary complex number $z\in \mathbb {C} $ with $z_{k}\in \{0,1,2,3\}$ gives rise to an injective mapping
$\textstyle {\begin{array}{llcl}\varphi \colon &\mathbb {C} &\to &\mathbb {R} \\&\sum _{k\geq n}z_{k}\cdot (2i)^{-k}&\mapsto &\sum _{k\geq n}z_{k}\cdot r^{-k}\\\end{array}}$
with some suitable $r\in \mathbb {Z} $. Here $r=4$ cannot be taken as base because of
$\textstyle \sum _{k>0}3\cdot (2i)^{-k}={\tfrac {-3-6i}{5}}\;\;\;\;\neq \;\;\;\;1=\sum _{k>0}3\cdot 4^{-k}.$
The image $\varphi (\mathbb {C} )\subset \mathbb {R} $ is a Cantor set which allows to linearly order $\mathbb {C} $ similar to a Z-order curve. Since the image is disconnected, $\varphi $ is not continuous.
See also
• Quaternary numeral system
• Complex-base system
• Negative base
References
1. Donald Knuth (April 1960). "An imaginary number system". Communications of the ACM. 3 (4): 245–247. doi:10.1145/367177.367233. S2CID 16513137.
Further reading
• Knuth, Donald Ervin. "Positional Number Systems". The Art of Computer Programming. Vol. 2 (3 ed.). Addison-Wesley. p. 205.
• Warren Jr., Henry S. (2013) [2002]. Hacker's Delight (2 ed.). Addison Wesley - Pearson Education, Inc. p. 309. ISBN 978-0-321-84268-8. 0-321-84268-5.
Donald Knuth
Publications
• The Art of Computer Programming
• "The Complexity of Songs"
• Computers and Typesetting
• Concrete Mathematics
• Surreal Numbers
• Things a Computer Scientist Rarely Talks About
• Selected papers series
Software
• TeX
• Metafont
• MIXAL (MIX
• MMIX)
Fonts
• AMS Euler
• Computer Modern
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Literate programming
• WEB
• CWEB
Algorithms
• Knuth's Algorithm X
• Knuth–Bendix completion algorithm
• Knuth–Morris–Pratt algorithm
• Knuth shuffle
• Robinson–Schensted–Knuth correspondence
• Trabb Pardo–Knuth algorithm
• Generalization of Dijkstra's algorithm
• Knuth's Simpath algorithm
Other
• Dancing Links
• Knuth reward check
• Knuth Prize
• Knuth's up-arrow notation
• Man or boy test
• Quater-imaginary base
• -yllion
• Potrzebie system of weights and measures
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Wikipedia
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Finitary relation
In mathematics, a finitary relation over sets X1, ..., Xn is a subset of the Cartesian product X1 × ⋯ × Xn; that is, it is a set of n-tuples (x1, ..., xn) consisting of elements xi in Xi.[1][2][3] Typically, the relation describes a possible connection between the elements of an n-tuple. For example, the relation "x is divisible by y and z" consists of the set of 3-tuples such that when substituted to x, y and z, respectively, make the sentence true.
The non-negative integer n giving the number of "places" in the relation is called the arity, adicity or degree of the relation. A relation with n "places" is variously called an n-ary relation, an n-adic relation or a relation of degree n. Relations with a finite number of places are called finitary relations (or simply relations if the context is clear). It is also possible to generalize the concept to infinitary relations with infinite sequences.[4]
An n-ary relation over sets X1, ..., Xn is an element of the power set of X1 × ⋯ × Xn.
0-ary relations count only two members: the one that always holds, and the one that never holds. This is because there is only one 0-tuple, the empty tuple (). They are sometimes useful for constructing the base case of an induction argument.
Unary relations can be viewed as a collection of members (such as the collection of Nobel laureates) having some property (such as that of having been awarded the Nobel prize).
Binary relations are the most commonly studied form of finitary relations. When X1 = X2 it is called a homogeneous relation, for example:
• Equality and inequality, denoted by signs such as = and < in statements such as "5 < 12", or
• Divisibility, denoted by the sign | in statements such as "13|143".
Otherwise it is a heterogeneous relation, for example:
• Set membership, denoted by the sign ∈ in statements such as "1 ∈ N".
Example
Consider the ternary relation R "x thinks that y likes z" over the set of people P = {Alice, Bob, Charles, Denise}, defined by:
R = {(Alice, Bob, Denise), (Charles, Alice, Bob), (Charles, Charles, Alice), (Denise, Denise, Denise)}.
R can be represented equivalently by the following table:
Relation R "x thinks that y likes z"
PPP
AliceBobDenise
CharlesAliceBob
CharlesCharlesAlice
DeniseDeniseDenise
Here, each row represents a triple of R, that is it makes a statement of the form "x thinks that y likes z". For instance, the first row states that "Alice thinks that Bob likes Denise". All rows are distinct. The ordering of rows is insignificant but the ordering of columns is significant.[1]
The above table is also a simple example of a relational database, a field with theory rooted in relational algebra and applications in data management.[5] Computer scientists, logicians, and mathematicians, however, tend to have different conceptions what a general relation is, and what it is consisted of. For example, databases are designed to deal with empirical data, which is by definition finite, whereas in mathematics, relations with infinite arity (i.e., infinitary relation) are also considered.
Definitions
When two objects, qualities, classes, or attributes, viewed together by the mind, are seen under some connexion, that connexion is called a relation.
— Augustus De Morgan[6]
The first definition of relations encountered in mathematics is:
Definition 1
An n-ary relation R over sets X1, ⋯, Xn is a subset of the Cartesian product X1 × ⋯ × Xn.[1]
The second definition of relations makes use of an idiom that is common in mathematics, stipulating that "such and such is an n-tuple" in order to ensure that such and such a mathematical object is determined by the specification of mathematical objects with n elements. In the case of a relation R over n sets, there are n + 1 things to specify, namely, the n sets plus a subset of their Cartesian product. In the idiom, this is expressed by saying that R is a (n + 1)-tuple.
Definition 2
An n-ary relation R over sets X1, ⋯, Xn is an (n + 1)-tuple (X1, ⋯, Xn, G) where G is a subset of the Cartesian product X1 × ⋯ × Xn called the graph of R.
As a rule, whatever definition best fits the application at hand will be chosen for that purpose, and if it ever becomes necessary to distinguish between the two definitions, then an entity satisfying the second definition may be called an embedded or included relation.
Both statements (x1, ⋯, xn) ∈ R (under the first definition) and (x1, ⋯, xn) ∈ G (under the second definition) read "x1, ⋯, xn are R-related" and are denoted using prefix notation by Rx1⋯xn and using postfix notation by x1⋯xnR. In the case where R is a binary relation, those statements are also denoted using infix notation by x1Rx2.
The following considerations apply under either definition:
• The set Xi is called the ith domain of R.[1] Under the first definition, the relation does not uniquely determine a given sequence of domains. In the case where R is a binary relation, X1 is also called simply the domain or set of departure of R, and X2 is also called the codomain or set of destination of R.
• When the elements of Xi are relations, Xi is called a nonsimple domain of R.[1]
• The set of ∀xi ∈ Xi for which there exists (x1, ⋯, xi − 1, xi + 1, ⋯, xn) ∈ X1 × ⋯ × Xi − 1 × Xi + 1 × ⋯ × Xn such that Rx1⋯xi − 1xixi + 1⋯xn is called the ith domain of definition or active domain of R.[1] In the case where R is a binary relation, its first domain of definition is also called simply the domain of definition or active domain of R, and its second domain of definition is also called the codomain of definition or active codomain of R.
• When the ith domain of definition of R is equal to Xi, R is said to be total on Xi. In the case where R is a binary relation, when R is total on X1, it is also said to be left-total or serial, and when R is total on X2, it is also said to be right-total or surjective.
• When ∀x ∀y ∈ Xi. ∀z ∈ Xj. xRijz ∧ yRijz ⇒ x = y, where i ∈ I, j ∈ J, Rij = πij R, and {I, J} is a partition of {1, ..., n}, R is said to be unique on {Xi}i ∈ I, and {Xi}i ∈ J is called a primary key[1] of R. In the case where R is a binary relation, when R is unique on {X1}, it is also said to be left-unique or injective, and when R is unique on {X2}, it is also said to be right-unique or functional.
• When all Xi are the same set X, it is simpler to refer to R as an n-ary relation over X, called a homogeneous relation. Otherwise R is called a heterogeneous relation.
• When any of Xi is empty, the defining Cartesian product is empty, and the only relation over such a sequence of domains is the empty relation R = ∅. Hence it is commonly stipulated that all of the domains be nonempty.
Let a Boolean domain B be a two-element set, say, B = {0, 1}, whose elements can be interpreted as logical values, typically 0 = false and 1 = true. The characteristic function of R, denoted by χR, is the Boolean-valued function χR: X1 × ⋯ × Xn → B, defined by χR((x1, ⋯, xn)) = 1 if Rx1⋯xn and χR((x1, ⋯, xn)) = 0 otherwise.
In applied mathematics, computer science and statistics, it is common to refer to a Boolean-valued function as an n-ary predicate. From the more abstract viewpoint of formal logic and model theory, the relation R constitutes a logical model or a relational structure, that serves as one of many possible interpretations of some n-ary predicate symbol.
Because relations arise in many scientific disciplines, as well as in many branches of mathematics and logic, there is considerable variation in terminology. Aside from the set-theoretic extension of a relational concept or term, the term "relation" can also be used to refer to the corresponding logical entity, either the logical comprehension, which is the totality of intensions or abstract properties shared by all elements in the relation, or else the symbols denoting these elements and intensions. Further, some writers of the latter persuasion introduce terms with more concrete connotations (such as "relational structure" for the set-theoretic extension of a given relational concept).
History
See also: Algebraic logic § History
The logician Augustus De Morgan, in work published around 1860, was the first to articulate the notion of relation in anything like its present sense. He also stated the first formal results in the theory of relations (on De Morgan and relations, see Merrill 1990).
Charles Peirce, Gottlob Frege, Georg Cantor, Richard Dedekind and others advanced the theory of relations. Many of their ideas, especially on relations called orders, were summarized in The Principles of Mathematics (1903) where Bertrand Russell made free use of these results.
In 1970, Edgar Codd proposed a relational model for databases, thus anticipating the development of data base management systems.[1]
See also
• Incidence structure
• Hypergraph
• Logic of relatives
• Logical matrix
• Partial order
• Predicate (mathematical logic)
• Projection (set theory)
• Reflexive relation
• Relation algebra
• Relational algebra
• Relational model
• Relations (philosophy)
References
1. Codd, Edgar Frank (June 1970). "A Relational Model of Data for Large Shared Data Banks" (PDF). Communications of the ACM. 13 (6): 377–387. doi:10.1145/362384.362685. S2CID 207549016. Retrieved 2020-04-29.
2. "Relation - Encyclopedia of Mathematics". www.encyclopediaofmath.org. Retrieved 2019-12-12.
3. "Definition of n-ary Relation". cs.odu.edu. Retrieved 2019-12-12.
4. Nivat, Maurice (1981). Astesiano, Egidio; Böhm, Corrado (eds.). "Infinitary relations". Caap '81. Lecture Notes in Computer Science. Springer Berlin Heidelberg. 112: 46–75. doi:10.1007/3-540-10828-9_54. ISBN 978-3-540-38716-9.
5. "Relations — CS441" (PDF). www.pitt.edu. Retrieved 2019-12-11.
6. De Morgan, A. (1858) "On the syllogism, part 3" in Heath, P., ed. (1966) On the syllogism and other logical writings. Routledge. P. 119,
Bibliography
• Codd, Edgar Frank (1990). The Relational Model for Database Management: Version 2 (PDF). Boston: Addison-Wesley. ISBN 978-0201141924.
• Bourbaki, N. (1994) Elements of the History of Mathematics, John Meldrum, trans. Springer-Verlag.
• Carnap, Rudolf (1958) Introduction to Symbolic Logic with Applications. Dover Publications.
• Halmos, P.R. (1960) Naive Set Theory. Princeton NJ: D. Van Nostrand Company.
• Lawvere, F.W., and R. Rosebrugh (2003) Sets for Mathematics, Cambridge Univ. Press.
• Lewis, C.I. (1918) A Survey of Symbolic Logic, Chapter 3: Applications of the Boole—Schröder Algebra, via Internet Archive
• Lucas, J. R. (1999) Conceptual Roots of Mathematics. Routledge.
• Maddux, R.D. (2006) Relation Algebras, vol. 150 in "Studies in Logic and the Foundations of Mathematics". Elsevier Science.
• Merrill, Dan D. (1990) Augustus De Morgan and the logic of relations. Kluwer.
• Peirce, C.S. (1870), "Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic", Memoirs of the American Academy of Arts and Sciences 9, 317–78, 1870. Reprinted, Collected Papers CP 3.45–149, Chronological Edition CE 2, 359–429.
• Peirce, C.S. (1984) Writings of Charles S. Peirce: A Chronological Edition, Volume 2, 1867-1871. Peirce Edition Project, eds. Indiana University Press.
• Russell, Bertrand (1903/1938) The Principles of Mathematics, 2nd ed. Cambridge Univ. Press.
• Suppes, Patrick (1960/1972) Axiomatic Set Theory. Dover Publications.
• Tarski, A. (1956/1983) Logic, Semantics, Metamathematics, Papers from 1923 to 1938, J.H. Woodger, trans. 1st edition, Oxford University Press. 2nd edition, J. Corcoran, ed. Indianapolis IN: Hackett Publishing.
• Ulam, S.M. and Bednarek, A.R. (1990), "On the Theory of Relational Structures and Schemata for Parallel Computation", pp. 477–508 in A.R. Bednarek and Françoise Ulam (eds.), Analogies Between Analogies: The Mathematical Reports of S.M. Ulam and His Los Alamos Collaborators, University of California Press, Berkeley, CA.
• Ulam, S.M. (1990) Analogies Between Analogies: The Mathematical Reports of S.M. Ulam and His Los Alamos Collaborators in A.R. Bednarek and Françoise Ulam, eds., University of California Press.
• Roland Fraïssé (2000) [1986] Theory of Relations, North Holland
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Quaternion Society
The Quaternion Society was a scientific society, self-described as an "International Association for Promoting the Study of Quaternions and Allied Systems of Mathematics". At its peak it consisted of about 60 mathematicians spread throughout the academic world that were experimenting with quaternions and other hypercomplex number systems. The group's guiding light was Alexander Macfarlane who served as its secretary initially, and became president in 1909. The association published a Bibliography in 1904 and a Bulletin (annual report) from 1900 to 1913.
The Bulletin became a review journal for topics in vector analysis and abstract algebra such as the theory of equipollence. The mathematical work reviewed pertained largely to matrices and linear algebra as the methods were in rapid development at the time.
Genesis
In 1895, Professor P. Molenbroek of The Hague, Holland, and Shinkichi Kimura studying at Yale put out a call for scholars to form the society in widely circulated journals: Nature,[1] Science,[2] and the Bulletin of the American Mathematical Society.[3] Giuseppe Peano also announced the society formation in his Rivista di Matematica.
The call to form an Association was encouraged by Macfarlane in 1896:
The logical harmony and unification of the whole of mathematical analysis ought to be kept in view. The algebra of space ought to include the algebra of the plane as a special case, just as the algebra of the plane includes the algebra of the line…When vector analysis is developed and presented...we may expect to see many zealous cultivators, many fruitful applications, and, finally, universal diffusion ... May the movement initiated by Messrs. Molenbroek and Kimura hasten the realization of this happy result.[4]
In 1897 the British Association met in Toronto where vector products were discussed:
Professor Henrici proposed a new notation to denote the different products of vectors, which consists in using square brackets for vector products and round brackets for scalar products. He likewise advocated adoption of Heaviside’s term "ort" for vector, the tensor of which is the number 1. Prof. A. Macfarlane read a communication on the solution of the cubic equation in which he explained how the two binomials in Cardano’s formula may be treated as complex quantities, either circular or hyperbolic, all the roots of the cubic can then be deduced by a general method.[5]
A system of national secretaries was announced in the AMS Bulletin in 1899: Alexander McAulay for Australasia, Victor Schlegel for Germany, Joly for Great Britain and Ireland, Giuseppe Peano for Italy, Kimura for Japan, Aleksandr Kotelnikov for Russia, F. Kraft for Switzerland, and Arthur Stafford Hathaway for the USA. For France the national secretary was Paul Genty, an engineer with the division of Ponts et Chaussees, and a quaternion collaborator with Charles-Ange Laisant, author of Methode des Quaterniones (1881).
Victor Schlegel reported[6] on the new institution in the Monatshefte für Mathematik.
Officers
When the society was organized in 1899, Peter Guthrie Tait was chosen as president, but he declined for reasons of poor health.
The first president was Robert Stawell Ball, and Alexander Macfarlane served as secretary and treasurer. In 1905 Charles Jasper Joly took over as president and L. van Elfrinkhof as treasurer, while Macfarlane continued as secretary. In 1909 Macfarlane became president, James Byrnie Shaw became secretary, and van Elfrinkhof continued as treasurer. The next year Macfarlane and Shaw continued in their posts, while Macfarlane also absorbed the office of treasurer. When Macfarlane died in 1913 after nearly completing the issue of the Bulletin, Shaw completed it and wound up the association.
The rules state that the president had the power of veto.
Bulletin
The Bulletin of the Association Promoting the Study of Quaternions and Allied Systems of Mathematics was issued nine times under the editorship of Alexander Macfarlane. Every issue listed the officers of the Association, governing council, rules, members, and a financial statement from the treasurer. Today HathiTrust provides access to these publications that are mainly of historical interest: [7][8]
• March 1900 Published in Toronto by Roswell-Hutchinson Press.
• March 1901 Published in Dublin at the University Press. President Charles J. Joly address.
• March 1903, Dublin. Macfarlane announces Bibliography.
• April 1905, Dublin. President C.J. Joly address.
• March 1908 Published in Lancaster, Pennsylvania, by New Era Printing. J.B. Shaw reports on bibliographic supplement.
• June 1909, Lancaster. President Macfarlane address on notation.
• October 1910, Lancaster. J.B. Shaw challenged by "inclusion or exclusion of certain papers which are only remotely connected with the theory of operations in the abstract."
• June 1912, Lancaster. Obituary: Ferdinand Ferber. "Comparative Notation for Vector Expressions" by J.B. Shaw. President Macfarlane address citing Duncan Sommerville's comments.
• June 1913, Lancaster. Secretary Shaw reports the death of A. Macfarlane and G. Combebiac.
Bibliography
Published in 1904 at Dublin, cradle of quaternions, the 86 page Bibliography of Quaternions and Allied Systems of Mathematics[9] cited some one thousand references. The publication set a professional standard; for instance the Manual of Quaternions (1905) of Joly has no bibliography beyond citation of Macfarlane. Furthermore, in 1967 when Michael J. Crowe published A History of Vector Analysis, he wrote in the preface (page ix) :
Concerning bibliography. No formal bibliographical section has been included with this book. ... the need for a bibliography is greatly diminished by the existence of a book that lists nearly all relevant primary documents published to about 1912, this is Alexander Macfarlane’s Bibliography ...
Every year more papers and books appeared that were of interest to Association members so it was necessary to update the Bibliography with supplements in the Bulletin. The categories used to group the items in the supplements give a sense of the changing focus of the Association:
• 1905 Supplement
• 1908 Supplement: Matrices, Linear substitutions, Quadratic forms, Bilinear forms, Complex numbers, Equipollences, Vector analysis, Commutative algebras, Quaternions, Biquaternions, Linear associative algebras, General algebra and operations, Additional.
• 1909 Supplement
• 1910 Supplement: Matrices, Linear groups, Complex numbers & equipollences, Vector analysis, Ausdehnungslehre, Quaternions, Linear associative algebras.
• 1912 Supplement: Equipollences, Commutative systems, Space-analysis, Dyadic systems, Vector analysis, Quaternions.
• 1913 Supplement: Commutative systems, Space analysis, Dyadic systems, Vector analysis, Other, Quaternions, Hypercomplex numbers, General algebra.
Aftermath
In 1913 Macfarlane died, and as related by Dirk Struik, the Society "became a victim of the first World War".[10]
James Byrnie Shaw, the surviving officer, wrote 50 book notices for American mathematical publications.[11] The final article review in the Bulletin was The Wilson and Lewis Algebra of Four-Dimensional Space written by J. B. Shaw. He summarizes,
This algebra is applied to the representation of the Minkowski time-space world. It enables all analytical work to be with reals, although the geometry becomes non-Euclidean.
The article reviewed was "The space-time manifold of relativity, the non-Euclidean geometry of mechanics, and electromagnetics".[12] However, when the textbook The Theory of Relativity by Ludwik Silberstein in 1914 was made available as an English understanding of Minkowski space, the algebra of biquaternions was applied, but without references to the British background or Macfarlane or other quaternionists of the Society. The language of quaternions had become international, providing content to set theory and expanded mathematical notation, and expressing mathematical physics.
See also
• Formulario mathematico
• Hypercomplex number
• Hyperbolic quaternion
• Non-Euclidean geometry
Notes and references
• The Quaternion Association at MacTutor History of Mathematics archive
1. S. Kimura & P. Molenbroek (1895) Friends and Fellow Workers in Quaternions Nature 52:545–6 (#1353)
2. S. Kimura & P. Molenbroek (1895) To those Interested in Quaternions and Allied Systems of Mathematics Science 2nd Ser, 2:524–25
3. "Notes" Bulletin of the American Mathematical Society 2:53, 182; 5:317
4. MacFarlane, Alexander (1896). "Quaternions". Science. 3 (55): 99–100. Bibcode:1896Sci.....3...99M. doi:10.1126/science.3.55.99. JSTOR 1624707. PMID 17802063. S2CID 243118533.
5. "Physics at the British Association" Nature 56:461,2 (# 1454)
6. Victor Schlegel (1899) "Internationaler Verein zur Beförderung des Studiums der Quaternionen und verwandter Systeme der Mathematik", Monatshefte für Mathematik 10(1):376
7. P.R. Girard (1984) "The Quaternion Group and Modern Physics", European Journal of Physics 5:25–32
8. M. J. Crowe (1967) A History of Vector Analysis
9. Alexander Macfarlane (1904) Bibliography of Quaternions and Allied Systems of Mathematics, weblink from Cornell University Historical Math Monographs.
• Review:Bibliography of Quaternions in Nature 69:604
10. Dirk Struik (1967) A Concise History of Mathematics, 3rd edition, page 172, Dover Books
11. See author=Shaw, James Byrnie at Mathematical Reviews
12. E. B. Wilson & G. N. Lewis (1912) Proceedings of the American Academy of Arts and Sciences 48: 389–507
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Wikipedia
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Quaternion algebra
In mathematics, a quaternion algebra over a field F is a central simple algebra A over F[1][2] that has dimension 4 over F. Every quaternion algebra becomes a matrix algebra by extending scalars (equivalently, tensoring with a field extension), i.e. for a suitable field extension K of F, $A\otimes _{F}K$ is isomorphic to the 2 × 2 matrix algebra over K.
The notion of a quaternion algebra can be seen as a generalization of Hamilton's quaternions to an arbitrary base field. The Hamilton quaternions are a quaternion algebra (in the above sense) over $F=\mathbb {R} $, and indeed the only one over $\mathbb {R} $ apart from the 2 × 2 real matrix algebra, up to isomorphism. When $F=\mathbb {C} $, then the biquaternions form the quaternion algebra over F.
Structure
Quaternion algebra here means something more general than the algebra of Hamilton's quaternions. When the coefficient field F does not have characteristic 2, every quaternion algebra over F can be described as a 4-dimensional F-vector space with basis $\{1,i,j,k\}$, with the following multiplication rules:
$i^{2}=a$
$j^{2}=b$
$ij=k$
$ji=-k$
where a and b are any given nonzero elements of F. From these rules we get:
$k^{2}=ijij=-iijj=-ab$
The classical instances where $F=\mathbb {R} $ are Hamilton's quaternions (a = b = −1) and split-quaternions (a = −1, b = +1). In split-quaternions, $k^{2}=+1$ and $jk=-i$, differing from Hamilton's equations.
The algebra defined in this way is denoted (a,b)F or simply (a,b).[3] When F has characteristic 2, a different explicit description in terms of a basis of 4 elements is also possible, but in any event the definition of a quaternion algebra over F as a 4-dimensional central simple algebra over F applies uniformly in all characteristics.
A quaternion algebra (a,b)F is either a division algebra or isomorphic to the matrix algebra of 2 × 2 matrices over F; the latter case is termed split.[4] The norm form
$N(t+xi+yj+zk)=t^{2}-ax^{2}-by^{2}+abz^{2}\ $
defines a structure of division algebra if and only if the norm is an anisotropic quadratic form, that is, zero only on the zero element. The conic C(a,b) defined by
$ax^{2}+by^{2}=z^{2}\ $
has a point (x,y,z) with coordinates in F in the split case.[5]
Application
Quaternion algebras are applied in number theory, particularly to quadratic forms. They are concrete structures that generate the elements of order two in the Brauer group of F. For some fields, including algebraic number fields, every element of order 2 in its Brauer group is represented by a quaternion algebra. A theorem of Alexander Merkurjev implies that each element of order 2 in the Brauer group of any field is represented by a tensor product of quaternion algebras.[6] In particular, over p-adic fields the construction of quaternion algebras can be viewed as the quadratic Hilbert symbol of local class field theory.
Classification
It is a theorem of Frobenius that there are only two real quaternion algebras: 2 × 2 matrices over the reals and Hamilton's real quaternions.
In a similar way, over any local field F there are exactly two quaternion algebras: the 2 × 2 matrices over F and a division algebra. But the quaternion division algebra over a local field is usually not Hamilton's quaternions over the field. For example, over the p-adic numbers Hamilton's quaternions are a division algebra only when p is 2. For odd prime p, the p-adic Hamilton quaternions are isomorphic to the 2 × 2 matrices over the p-adics. To see the p-adic Hamilton quaternions are not a division algebra for odd prime p, observe that the congruence x2 + y2 = −1 mod p is solvable and therefore by Hensel's lemma — here is where p being odd is needed — the equation
x2 + y2 = −1
is solvable in the p-adic numbers. Therefore the quaternion
xi + yj + k
has norm 0 and hence doesn't have a multiplicative inverse.
One way to classify the F-algebra isomorphism classes of all quaternion algebras for a given field F is to use the one-to-one correspondence between isomorphism classes of quaternion algebras over F and isomorphism classes of their norm forms.
To every quaternion algebra A, one can associate a quadratic form N (called the norm form) on A such that
$N(xy)=N(x)N(y)$
for all x and y in A. It turns out that the possible norm forms for quaternion F-algebras are exactly the Pfister 2-forms.
Quaternion algebras over the rational numbers
Quaternion algebras over the rational numbers have an arithmetic theory similar to, but more complicated than, that of quadratic extensions of $\mathbb {Q} $.
Let $B$ be a quaternion algebra over $\mathbb {Q} $ and let $\nu $ be a place of $\mathbb {Q} $, with completion $\mathbb {Q} _{\nu }$ (so it is either the p-adic numbers $\mathbb {Q} _{p}$ for some prime p or the real numbers $\mathbb {R} $). Define $B_{\nu }:=\mathbb {Q} _{\nu }\otimes _{\mathbb {Q} }B$, which is a quaternion algebra over $\mathbb {Q} _{\nu }$. So there are two choices for $B_{\nu }$: the 2 × 2 matrices over $\mathbb {Q} _{\nu }$ or a division algebra.
We say that $B$ is split (or unramified) at $\nu $ if $B_{\nu }$ is isomorphic to the 2 × 2 matrices over $\mathbb {Q} _{\nu }$. We say that B is non-split (or ramified) at $\nu $ if $B_{\nu }$ is the quaternion division algebra over $\mathbb {Q} _{\nu }$. For example, the rational Hamilton quaternions is non-split at 2 and at $\infty $ and split at all odd primes. The rational 2 × 2 matrices are split at all places.
A quaternion algebra over the rationals which splits at $\infty $ is analogous to a real quadratic field and one which is non-split at $\infty $ is analogous to an imaginary quadratic field. The analogy comes from a quadratic field having real embeddings when the minimal polynomial for a generator splits over the reals and having non-real embeddings otherwise. One illustration of the strength of this analogy concerns unit groups in an order of a rational quaternion algebra: it is infinite if the quaternion algebra splits at $\infty $ and it is finite otherwise, just as the unit group of an order in a quadratic ring is infinite in the real quadratic case and finite otherwise.
The number of places where a quaternion algebra over the rationals ramifies is always even, and this is equivalent to the quadratic reciprocity law over the rationals. Moreover, the places where B ramifies determines B up to isomorphism as an algebra. (In other words, non-isomorphic quaternion algebras over the rationals do not share the same set of ramified places.) The product of the primes at which B ramifies is called the discriminant of B.
See also
• Composition algebra
• Cyclic algebra
• Octonion algebra
• Hurwitz quaternion order
• Hurwitz quaternion
Notes
1. See Pierce. Associative algebras. Springer. Lemma at page 14.
2. See Milies & Sehgal, An introduction to group rings, exercise 17, chapter 2.
3. Gille & Szamuely (2006) p.2
4. Gille & Szamuely (2006) p.3
5. Gille & Szamuely (2006) p.7
6. Lam (2005) p.139
References
• Gille, Philippe; Szamuely, Tamás (2006). Central simple algebras and Galois cohomology. Cambridge Studies in Advanced Mathematics. Vol. 101. Cambridge: Cambridge University Press. doi:10.1017/CBO9780511607219. ISBN 0-521-86103-9. Zbl 1137.12001.
• Lam, Tsit-Yuen (2005). Introduction to Quadratic Forms over Fields. Graduate Studies in Mathematics. Vol. 67. American Mathematical Society. ISBN 0-8218-1095-2. MR 2104929. Zbl 1068.11023.
Further reading
The Wikibook Associative Composition Algebra has a page on the topic of: Quaternion algebras over R and C
• 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. MR 1632779. Zbl 0955.16001.
• Maclachlan, Colin; Ried, Alan W. (2003). The Arithmetic of Hyperbolic 3-Manifolds. New York: Springer-Verlag. doi:10.1007/978-1-4757-6720-9. ISBN 0-387-98386-4. MR 1937957. See chapter 2 (Quaternion Algebras I) and chapter 7 (Quaternion Algebras II).
• Chisholm, Hugh, ed. (1911). "Algebra" . Encyclopædia Britannica (11th ed.). Cambridge University Press. (See section on quaternions.)
• Quaternion algebra at Encyclopedia of Mathematics.
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Wikipedia
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Quaternionic analysis
In mathematics, quaternionic analysis is the study of functions with quaternions as the domain and/or range. Such functions can be called functions of a quaternion variable just as functions of a real variable or a complex variable are called.
As with complex and real analysis, it is possible to study the concepts of analyticity, holomorphy, harmonicity and conformality in the context of quaternions. Unlike the complex numbers and like the reals, the four notions do not coincide.
Properties
The projections of a quaternion onto its scalar part or onto its vector part, as well as the modulus and versor functions, are examples that are basic to understanding quaternion structure.
An important example of a function of a quaternion variable is
$f_{1}(q)=uqu^{-1}$
which rotates the vector part of q by twice the angle represented by u.
The quaternion multiplicative inverse $f_{2}(q)=q^{-1}$ is another fundamental function, but as with other number systems, $f_{2}(0)$ and related problems are generally excluded due to the nature of dividing by zero.
Affine transformations of quaternions have the form
$f_{3}(q)=aq+b,\quad a,b,q\in \mathbb {H} .$
Linear fractional transformations of quaternions can be represented by elements of the matrix ring $M_{2}(\mathbb {H} )$ operating on the projective line over $\mathbb {H} $. For instance, the mappings $q\mapsto uqv,$ where $u$ and $v$ are fixed versors serve to produce the motions of elliptic space.
Quaternion variable theory differs in some respects from complex variable theory. For example: The complex conjugate mapping of the complex plane is a central tool but requires the introduction of a non-arithmetic, non-analytic operation. Indeed, conjugation changes the orientation of plane figures, something that arithmetic functions do not change.
In contrast to the complex conjugate, the quaternion conjugation can be expressed arithmetically, as $f_{4}(q)=-{\tfrac {1}{2}}(q+iqi+jqj+kqk)$
This equation can be proven, starting with the basis {1, i, j, k}:
$f_{4}(1)=-{\tfrac {1}{2}}(1-1-1-1)=1,\quad f_{4}(i)=-{\tfrac {1}{2}}(i-i+i+i)=-i,\quad f_{4}(j)=-j,\quad f_{4}(k)=-k$.
Consequently, since $f_{4}$ is linear,
$f_{4}(q)=f_{4}(w+xi+yj+zk)=wf_{4}(1)+xf_{4}(i)+yf_{4}(j)+zf_{4}(k)=w-xi-yj-zk=q^{*}.$
The success of complex analysis in providing a rich family of holomorphic functions for scientific work has engaged some workers in efforts to extend the planar theory, based on complex numbers, to a 4-space study with functions of a quaternion variable.[1] These efforts were summarized in Deavours (1973).[lower-alpha 1]
Though $\mathbb {H} $ appears as a union of complex planes, the following proposition shows that extending complex functions requires special care:
Let $f_{5}(z)=u(x,y)+iv(x,y)$ be a function of a complex variable, $z=x+iy$. Suppose also that $u$ is an even function of $y$ and that $v$ is an odd function of $y$. Then $f_{5}(q)=u(x,y)+rv(x,y)$ is an extension of $f_{5}$ to a quaternion variable $q=x+yr$ where $r^{2}=-1$ and $r\in \mathbb {H} $. Then, let $r^{*}$ represent the conjugate of $r$, so that $q=x-yr^{*}$. The extension to $\mathbb {H} $ will be complete when it is shown that $f_{5}(q)=f_{5}(x-yr^{*})$. Indeed, by hypothesis
$u(x,y)=u(x,-y),\quad v(x,y)=-v(x,-y)\quad $ one obtains
$f_{5}(x-yr^{*})=u(x,-y)+r^{*}v(x,-y)=u(x,y)+rv(x,y)=f_{5}(q).$
Homographies
In the following, colons and square brackets are used to denote homogeneous vectors.
The rotation about axis r is a classical application of quaternions to space mapping.[2] In terms of a homography, the rotation is expressed
$[q:1]{\begin{pmatrix}u&0\\0&u\end{pmatrix}}=[qu:u]\thicksim [u^{-1}qu:1],$
where $u=\exp(\theta r)=\cos \theta +r\sin \theta $ is a versor. If p * = −p, then the translation $q\mapsto q+p$ is expressed by
$[q:1]{\begin{pmatrix}1&0\\p&1\end{pmatrix}}=[q+p:1].$
Rotation and translation xr along the axis of rotation is given by
$[q:1]{\begin{pmatrix}u&0\\uxr&u\end{pmatrix}}=[qu+uxr:u]\thicksim [u^{-1}qu+xr:1].$
Such a mapping is called a screw displacement. In classical kinematics, Chasles' theorem states that any rigid body motion can be displayed as a screw displacement. Just as the representation of a Euclidean plane isometry as a rotation is a matter of complex number arithmetic, so Chasles' theorem, and the screw axis required, is a matter of quaternion arithmetic with homographies: Let s be a right versor, or square root of minus one, perpendicular to r, with t = rs.
Consider the axis passing through s and parallel to r. Rotation about it is expressed[3] by the homography composition
${\begin{pmatrix}1&0\\-s&1\end{pmatrix}}{\begin{pmatrix}u&0\\0&u\end{pmatrix}}{\begin{pmatrix}1&0\\s&1\end{pmatrix}}={\begin{pmatrix}u&0\\z&u\end{pmatrix}},$
where $z=us-su=\sin \theta (rs-sr)=2t\sin \theta .$
Now in the (s,t)-plane the parameter θ traces out a circle $u^{-1}z=u^{-1}(2t\sin \theta )=2\sin \theta (t\cos \theta -s\sin \theta )$ in the half-plane $\lbrace wt+xs:x>0\rbrace .$
Any p in this half-plane lies on a ray from the origin through the circle $\lbrace u^{-1}z:0<\theta <\pi \rbrace $ and can be written $p=au^{-1}z,\ \ a>0.$
Then up = az, with ${\begin{pmatrix}u&0\\az&u\end{pmatrix}}$ as the homography expressing conjugation of a rotation by a translation p.
The derivative for quaternions
Since the time of Hamilton, it has been realized that requiring the independence of the derivative from the path that a differential follows toward zero is too restrictive: it excludes even $f(q)=q^{2}$ from differentiation. Therefore, a direction-dependent derivative is necessary for functions of a quaternion variable.[4][5] Considering the increment of polynomial function of quaternionic argument shows that the increment is a linear map of increment of the argument. From this, a definition can be made:
A continuous map $f:\mathbb {H} \rightarrow \mathbb {H} $ is called differentiable on the set $U\subset \mathbb {H} $, if, at every point $x\in U$, the increment of the map $f$ can be represented as
$f(x+h)-f(x)={\frac {df(x)}{dx}}\circ h+o(h)$
where
${\frac {df(x)}{dx}}:\mathbb {H} \rightarrow \mathbb {H} $
is linear map of quaternion algebra $\mathbb {H} $ and $o:\mathbb {H} \rightarrow \mathbb {H} $ is a continuous map such that
$\lim _{a\rightarrow 0}{\frac {|o(a)|}{|a|}}=0$
The linear map ${\frac {df(x)}{dx}}$ is called the derivative of the map $f$.
On the quaternions, the derivative may be expressed as
${\frac {df(x)}{dx}}=\sum _{s}{\frac {d_{s0}f(x)}{dx}}\otimes {\frac {d_{s1}f(x)}{dx}}$
Therefore, the differential of the map $f$ may be expressed as follows with brackets on either side.
${\frac {df(x)}{dx}}\circ dx=\left(\sum _{s}{\frac {d_{s0}f(x)}{dx}}\otimes {\frac {d_{s1}f(x)}{dx}}\right)\circ dx=\sum _{s}{\frac {d_{s0}f(x)}{dx}}dx{\frac {d_{s1}f(x)}{dx}}$
The number of terms in the sum will depend on the function f. The expressions ${\frac {d_{sp}df(x)}{dx}},p=0,1$ are called components of derivative.
The derivative of a quaternionic function holds the following equalities
${\frac {df(x)}{dx}}\circ h=\lim _{t\to 0}(t^{-1}(f(x+th)-f(x)))$
${\frac {d(f(x)+g(x))}{dx}}={\frac {df(x)}{dx}}+{\frac {dg(x)}{dx}}$
${\frac {df(x)g(x)}{dx}}={\frac {df(x)}{dx}}\ g(x)+f(x)\ {\frac {dg(x)}{dx}}$
${\frac {df(x)g(x)}{dx}}\circ h=\left({\frac {df(x)}{dx}}\circ h\right)\ g(x)+f(x)\left({\frac {dg(x)}{dx}}\circ h\right)$
${\frac {daf(x)b}{dx}}=a\ {\frac {df(x)}{dx}}\ b$
${\frac {daf(x)b}{dx}}\circ h=a\left({\frac {df(x)}{dx}}\circ h\right)b$
For the function f(x) = axb, the derivative is
${\frac {daxb}{dx}}=a\otimes b$ $dy={\frac {daxb}{dx}}\circ dx=a\,dx\,b$
and so the components are:
${\frac {d_{10}axb}{dx}}=a$ ${\frac {d_{11}axb}{dx}}=b$
Similarly, for the function f(x) = x2, the derivative is
${\frac {dx^{2}}{dx}}=x\otimes 1+1\otimes x$ $dy={\frac {dx^{2}}{dx}}\circ dx=x\,dx+dx\,x$
and the components are:
${\frac {d_{10}x^{2}}{dx}}=x$ ${\frac {d_{11}x^{2}}{dx}}=1$
${\frac {d_{20}x^{2}}{dx}}=1$ ${\frac {d_{21}x^{2}}{dx}}=x$
Finally, for the function f(x) = x−1, the derivative is
${\frac {dx^{-1}}{dx}}=-x^{-1}\otimes x^{-1}$ $dy={\frac {dx^{-1}}{dx}}\circ dx=-x^{-1}dx\,x^{-1}$
and the components are:
${\frac {d_{10}x^{-1}}{dx}}=-x^{-1}$ ${\frac {d_{11}x^{-1}}{dx}}=x^{-1}$
See also
• Cayley transform
• Quaternionic manifold
Notes
1. Deavours (1973) recalls a 1935 issue of Commentarii Mathematici Helvetici where an alternative theory of "regular functions" was initiated by Fueter (1936) through the idea of Morera's theorem: quaternion function $F$ is "left regular at $q$" when the integral of $F$ vanishes over any sufficiently small hypersurface containing $q$. Then the analogue of Liouville's theorem holds: The only regular quaternion function with bounded norm in $\mathbb {E} ^{4}$ is a constant. One approach to construct regular functions is to use power series with real coefficients. Deavours also gives analogues for the Poisson integral, the Cauchy integral formula, and the presentation of Maxwell’s equations of electromagnetism with quaternion functions.
Citations
1. (Fueter 1936)
2. (Cayley 1848, especially page 198)
3. (Hamilton 1853, §287 pp. 273,4)
4. (Hamilton 1866, Chapter II, On differentials and developments of functions of quaternions, pp. 391–495)
5. (Laisant 1881, Chapitre 5: Différentiation des Quaternions, pp. 104–117)
References
• Arnold, Vladimir (1995), translated by Porteous, Ian R., "The geometry of spherical curves and the algebra of quaternions", Russian Mathematical Surveys, 50 (1): 1–68, doi:10.1070/RM1995v050n01ABEH001662, S2CID 250897899, Zbl 0848.58005
• Cayley, Arthur (1848), "On the application of quaternions to the theory of rotation", London and Edinburgh Philosophical Magazine, Series 3, 33 (221): 196–200, doi:10.1080/14786444808645844
• Deavours, C.A. (1973), "The quaternion calculus", American Mathematical Monthly, Washington, DC: Mathematical Association of America, 80 (9): 995–1008, doi:10.2307/2318774, ISSN 0002-9890, JSTOR 2318774, Zbl 0282.30040
• Du Val, Patrick (1964), Homographies, Quaternions and Rotations, Oxford Mathematical Monographs, Oxford: Clarendon Press, MR 0169108, Zbl 0128.15403
• Fueter, Rudolf (1936), "Über die analytische Darstellung der regulären Funktionen einer Quaternionenvariablen", Commentarii Mathematici Helvetici (in German), 8: 371–378, doi:10.1007/BF01199562, S2CID 121227604, Zbl 0014.16702
• Gentili, Graziano; Stoppato, Caterina; Struppa, Daniele C. (2013), Regular Functions of a Quaternionic Variable, Berlin: Springer, doi:10.1007/978-3-642-33871-7, ISBN 978-3-642-33870-0, S2CID 118710284, Zbl 1269.30001
• Gormley, P.G. (1947), "Stereographic projection and the linear fractional group of transformations of quaternions", Proceedings of the Royal Irish Academy, Section A, 51: 67–85, JSTOR 20488472
• Gürlebeck, Klaus; Sprößig, Wolfgang (1990), Quaternionic analysis and elliptic boundary value problems, Basel: Birkhäuser, ISBN 978-3-7643-2382-0, Zbl 0850.35001
• John C.Holladay (1957), "The Stone–Weierstrass theorem for quaternions" (PDF), Proc. Amer. Math. Soc., 8: 656, doi:10.1090/S0002-9939-1957-0087047-7.
• Hamilton, William Rowan (1853), Lectures on Quaternions, Dublin: Hodges and Smith, OL 23416635M
• Hamilton, William Rowan (1866), Hamilton, William Edwin (ed.), Elements of Quaternions, London: Longmans, Green, & Company, Zbl 1204.01046
• Joly, Charles Jasper (1903), "Quaternions and projective geometry", Philosophical Transactions of the Royal Society of London, 201 (331–345): 223–327, Bibcode:1903RSPTA.201..223J, doi:10.1098/rsta.1903.0018, JFM 34.0092.01, JSTOR 90902
• Laisant, Charles-Ange (1881), Introduction à la Méthode des Quaternions (in French), Paris: Gauthier-Villars, JFM 13.0524.02
• Porter, R. Michael (1998), "Möbius invariant quaternion geometry" (PDF), Conformal Geometry and Dynamics, 2 (6): 89–196, doi:10.1090/S1088-4173-98-00032-0, Zbl 0910.53005
• Sudbery, A. (1979), "Quaternionic analysis", Mathematical Proceedings of the Cambridge Philosophical Society, 85 (2): 199–225, Bibcode:1979MPCPS..85..199S, doi:10.1017/S0305004100055638, hdl:10338.dmlcz/101933, S2CID 7606387, Zbl 0399.30038
Analysis in topological vector spaces
Basic concepts
• Abstract Wiener space
• Classical Wiener space
• Bochner space
• Convex series
• Cylinder set measure
• Infinite-dimensional vector function
• Matrix calculus
• Vector calculus
Derivatives
• Differentiable vector–valued functions from Euclidean space
• Differentiation in Fréchet spaces
• Fréchet derivative
• Total
• Functional derivative
• Gateaux derivative
• Directional
• Generalizations of the derivative
• Hadamard derivative
• Holomorphic
• Quasi-derivative
Measurability
• Besov measure
• Cylinder set measure
• Canonical Gaussian
• Classical Wiener measure
• Measure like set functions
• infinite-dimensional Gaussian measure
• Projection-valued
• Vector
• Bochner / Weakly / Strongly measurable function
• Radonifying function
Integrals
• Bochner
• Direct integral
• Dunford
• Gelfand–Pettis/Weak
• Regulated
• Paley–Wiener
Results
• Cameron–Martin theorem
• Inverse function theorem
• Nash–Moser theorem
• Feldman–Hájek theorem
• No infinite-dimensional Lebesgue measure
• Sazonov's theorem
• Structure theorem for Gaussian measures
Related
• Crinkled arc
• Covariance operator
Functional calculus
• Borel functional calculus
• Continuous functional calculus
• Holomorphic functional calculus
Applications
• Banach manifold (bundle)
• Convenient vector space
• Choquet theory
• Fréchet manifold
• Hilbert manifold
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Quaternion-Kähler manifold
In differential geometry, a quaternion-Kähler manifold (or quaternionic Kähler manifold) is a Riemannian 4n-manifold whose Riemannian holonomy group is a subgroup of Sp(n)·Sp(1) for some $n\geq 2$. Here Sp(n) is the sub-group of $SO(4n)$ consisting of those orthogonal transformations that arise by left-multiplication by some quaternionic $n\times n$ matrix, while the group $Sp(1)=S^{3}$ of unit-length quaternions instead acts on quaternionic $n$-space ${\mathbb {H} }^{n}={\mathbb {R} }^{4n}$ by right scalar multiplication. The Lie group $Sp(n)\cdot Sp(1)\subset SO(4n)$ generated by combining these actions is then abstractly isomorphic to $[Sp(n)\times Sp(1)]/{\mathbb {Z} }_{2}$.
Although the above loose version of the definition includes hyperkähler manifolds, the standard convention of excluding these will be followed by also requiring that the scalar curvature be non-zero— as is automatically true if the holonomy group equals the entire group Sp(n)·Sp(1).
Early history
Marcel Berger's 1955 paper[1] on the classification of Riemannian holonomy groups first raised the issue of the existence of non-symmetric manifolds with holonomy Sp(n)·Sp(1). Although no examples of such manifolds were constructed until the 1980s, certain interesting results were proved in the mid-1960s in pioneering work by Edmond Bonan[2] and Kraines[3] who have independently proven that any such manifold admits a parallel 4-form $\Omega $.The long awaited analog of strong Lefschetz theorem was published [4] in 1982 : $\Omega ^{n-k}\wedge \bigwedge ^{2k}T^{*}M=\bigwedge ^{4n-2k}T^{*}M.$
In the context of Berger's classification of Riemannian holonomies, quaternion-Kähler manifolds constitute the only class of irreducible, non-symmetric manifolds of special holonomy that are automatically Einstein, but not automatically Ricci-flat. If the Einstein constant of a simply connected manifold with holonomy in $Sp(n)Sp(1)$ is zero, where $n\geq 2$, then the holonomy is actually contained in $Sp(n)$, and the manifold is hyperkähler. This case is excluded from the definition by declaring quaternion-Kähler to mean not only that the holonomy group is contained in $Sp(n)Sp(1)$, but also that the manifold has non-zero (constant) scalar curvature.
With this convention, quaternion-Kähler manifolds can thus be naturally divided into those for which the Ricci curvature is positive, and those for which it is instead negative.
Examples
There are no known examples of compact quaternion-Kähler manifolds that are not locally symmetric. (Again, hyperkähler manifolds are excluded from the discussion by fiat.) On the other hand, there are many symmetric quaternion-Kähler manifolds; these were first classified by Joseph A. Wolf,[5] and so are known as Wolf spaces. For any simple Lie group G, there is a unique Wolf space G/K obtained as a quotient of G by a subgroup $K=K_{0}\cdot \operatorname {SU} (2)$, where $SU(2)$ is the subgroup associated with the highest root of G, and K0 is its centralizer in G. The Wolf spaces with positive Ricci curvature are compact and simply connected. For example, if $G=Sp(n+1)$, the corresponding Wolf space is the quaternionic projective space $\mathbb {HP} _{n}$ of (right) quaternionic lines through the origin in $\mathbb {H} ^{n+1}$.
A conjecture often attributed to LeBrun and Salamon (see below) asserts that all complete quaternion-Kähler manifolds of positive scalar curvature are symmetric. By contrast, however, constructions of Galicki-Lawson [6] and of LeBrun[7] show that complete, non-locally-symmetric quaternion-Kähler manifolds of negative scalar curvature exist in great profusion. The Galicki-Lawson construction just cited also gives rise to vast numbers of compact non-locally-symmetric orbifold examples with positive Einstein constant, and many of these in turn give rise[8] to compact, non-singular 3-Sasakian Einstein manifolds of dimension $4n+3$.
Twistor spaces
Questions about quaternion-Kähler manifolds can be translated into the language of complex geometry using the methods of twistor theory; this fact is encapsulated in a theorem discovered independently by Salamon and Bérard-Bergery, and inspired by earlier work of Penrose. Let $M$ be a quaternion-Kähler manifold, and $H$ be the sub-bundle of $End(TM)$ arising from the holonomy action of ${\mathfrak {sp}}(1)\subset {\mathfrak {sp}}(n)\oplus {\mathfrak {sp}}(1)$. Then $H$ contains an $S^{2}$-bundle $Z\to M$ consisting of all $j\in H$ that satisfy $j^{2}=-1$. The points of $Z$ thus represent complex structures on tangent spaces of $M$. Using this, the total space $Z$ can then be equipped with a tautological almost complex structure. Salamon[9] (and, independently, Bérard-Bergery[10]) proved that this almost complex structure is integrable, thereby making $Z$ into a complex manifold.
When the Ricci curvature of M is positive, Z is a Fano manifold, and so, in particular, is a smooth projective algebraic complex variety. Moreover, it admits a Kähler–Einstein metric, and, more importantly, comes equipped with a holomorphic contact structure, corresponding to the horizontal spaces of the Riemannian connection on H. These facts were used by LeBrun and Salamon[11] to prove that, up to isometry and rescaling, there are only finitely many positive-scalar-curvature compact quaternion-Kähler manifolds in any given dimension. This same paper also shows that any such manifold is actually a symmetric space unless its second homology is a finite group with non-trivial 2-torsion. Related techniques had also been used previously by Poon and Salamon[12] to show that there are no non-symmetric examples at all in dimension 8.
In the converse direction, a result of LeBrun[13] shows that any Fano manifold that admits both a Kähler–Einstein metric and a holomorphic contact structure is actually the twistor space of a quaternion-Kähler manifold of positive scalar curvature, which is moreover unique up to isometries and rescalings.
References
1. Berger, Marcel (1955). "Sur les groups d'holonomie des variétés à connexion affine et des variétés riemanniennes" (PDF). Bull. Soc. Math. France. 83: 279–330. doi:10.24033/bsmf.1464.
2. Bonan, Edmond (1965). "Structure presque quaternale sur une variété differentiable". Comptes Rendus de l'Académie des Sciences. 261: 5445–8.
3. Kraines, Vivian Yoh (1966). "Topology of quaternionic manifolds" (PDF). Transactions of the American Mathematical Society. 122 (2): 357–367. doi:10.1090/S0002-9947-1966-0192513-X. JSTOR 1994553.
4. Bonan, Edmond (1982). "Sur l'algèbre extérieure d'une variété presque hermitienne quaternionique". Comptes Rendus de l'Académie des Sciences. 295: 115–118.
5. Wolf, Joseph A. (1965). "Complex homogeneous contact manifolds and quaternionic symmetric spaces". J. Math. Mech. 14 (6): 1033–47. JSTOR 24901319.
6. Galicki, K.; Lawson, H.B., Jr. (1988). "Quaternionic reduction and quaternionic orbifolds" (PDF). Math. Ann. 282: 1–21. doi:10.1007/BF01457009. S2CID 120748113.{{cite journal}}: CS1 maint: multiple names: authors list (link)
7. LeBrun, Claude (1991). "On complete quaternionic-Kähler manifolds" (PDF). Duke Math. J. 63 (3): 723–743. doi:10.1215/S0012-7094-91-06331-3.
8. Boyer, Charles; Galicki, Krzysztof (2008). Sasakian Geometry. Oxford Mathematical Monographs. Oxford University Press. ISBN 978-0-19-856495-9.
9. Salamon, Simon (1982). "Quaternionic Kähler manifolds". Invent. Math. 67: 143–171. Bibcode:1982InMat..67..143S. doi:10.1007/BF01393378. S2CID 118575943.
10. Besse 1987
11. LeBrun, Claude; Salamon, Simon (1994). "Strong rigidity of positive quaternion-Kähler manifolds". Invent. Math. 118: 109–132. Bibcode:1994InMat.118..109L. doi:10.1007/BF01231528. S2CID 121184428.
12. Poon, Y.S.; Salamon, S.M. (1991). "Quaternionic Kähler 8-manifolds with positive scalar curvature". J. Differential Geom. 33 (2): 363–378. doi:10.4310/jdg/1214446322.
13. LeBrun, Claude (1995). "Fano manifolds, contact structures, and quaternionic geometry". Internat. J. Math. 6 (3): 419–437. arXiv:dg-ga/9409001. CiteSeerX 10.1.1.251.3603. doi:10.1142/S0129167X95000146. S2CID 18361986.
• Besse, Arthur L. (2007) [1987]. Einstein Manifolds. Springer. ISBN 978-3-540-74120-6.
• Salamon, Simon (1982). "Quaternionic Kähler manifolds". Invent. Math. 67: 143–171. Bibcode:1982InMat..67..143S. doi:10.1007/bf01393378. S2CID 118575943.
• Joyce, Dominic D. (2000). Compact Manifolds with Special Holonomy. Oxford Mathematical Monographs. Oxford University Press. ISBN 978-0-19-850601-0.
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Quaternionic discrete series representation
In mathematics, a quaternionic discrete series representation is a discrete series representation of a semisimple Lie group G associated with a quaternionic structure on the symmetric space of G. They were introduced by Gross and Wallach (1994, 1996).
Quaternionic discrete series representations exist when the maximal compact subgroup of the group G has a normal subgroup isomorphic to SU(2). Every complex simple Lie group has a real form with quaternionic discrete series representations. In particular the classical groups SU(2,n), SO(4,n), and Sp(1,n) have quaternionic discrete series representations.
Quaternionic representations are analogous to holomorphic discrete series representations, which exist when the symmetric space of the group has a complex structure. The groups SU(2,n) have both holomorphic and quaternionic discrete series representations.
See also
• Quaternionic symmetric space
References
• Gross, Benedict H.; Wallach, Nolan R (1994), "A distinguished family of unitary representations for the exceptional groups of real rank =4", in Brylinski, Jean-Luc; Brylinski, Ranee; Guillemin, Victor; Kac, Victor (eds.), Lie theory and geometry, Progr. Math., vol. 123, Boston, MA: Birkhäuser Boston, pp. 289–304, ISBN 978-0-8176-3761-3, MR 1327538
• Gross, Benedict H.; Wallach, Nolan R (1996), "On quaternionic discrete series representations, and their continuations", Journal für die reine und angewandte Mathematik, 1996 (481): 73–123, doi:10.1515/crll.1996.481.73, ISSN 0075-4102, MR 1421947, S2CID 116031362
External links
• Garrett, Paul (2004), Some facts about discrete series (holomorphic, quaternionic) (PDF)
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Quaternionic polytope
In geometry, a quaternionic polytope is a generalization of a polytope in real space to an analogous structure in a quaternionic module, where each real dimension is accompanied by three imaginary ones. Similarly to complex polytopes, points are not ordered and there is no sense of "between", and thus a quaternionic polytope may be understood as an arrangement of connected points, lines, planes and so on, where every point is the junction of multiple lines, every line of multiple planes, and so on. Likewise, each line must contain multiple points, each plane multiple lines, and so on. Since the quaternions are non-commutative, a convention must be made for the multiplication of vectors by scalars, which is usually in favour of left-multiplication.[1]
As is the case for the complex polytopes, the only quaternionic polytopes to have been systematically studied are the regular ones. Like the real and complex regular polytopes, their symmetry groups may be described as reflection groups. For example, the regular quaternionic lines are in a one-to-one correspondence with the finite subgroups of U1(H): the binary cyclic groups, binary dihedral groups, binary tetrahedral group, binary octahedral group, and binary icosahedral group.[2]
References
1. Davis, C.; Grünbaum, B.; Sherk, F.A (2012-12-06). The Geometric Vein: The Coxeter Festschrift - Google Books. ISBN 9781461256489. Retrieved 2016-04-15.
2. Hans Cuypers (September 1995). "Regular quaternionic polytopes". Linear Algebra and Its Applications. 226–228: 311–329. doi:10.1016/0024-3795(95)00149-L.
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Quaternionic projective space
In mathematics, quaternionic projective space is an extension of the ideas of real projective space and complex projective space, to the case where coordinates lie in the ring of quaternions $\mathbb {H} .$ Quaternionic projective space of dimension n is usually denoted by
$\mathbb {HP} ^{n}$
and is a closed manifold of (real) dimension 4n. It is a homogeneous space for a Lie group action, in more than one way. The quaternionic projective line $\mathbb {HP} ^{1}$ is homeomorphic to the 4-sphere.
In coordinates
Its direct construction is as a special case of the projective space over a division algebra. The homogeneous coordinates of a point can be written
$[q_{0},q_{1},\ldots ,q_{n}]$
where the $q_{i}$ are quaternions, not all zero. Two sets of coordinates represent the same point if they are 'proportional' by a left multiplication by a non-zero quaternion c; that is, we identify all the
$[cq_{0},cq_{1}\ldots ,cq_{n}]$.
In the language of group actions, $\mathbb {HP} ^{n}$ is the orbit space of $\mathbb {H} ^{n+1}\setminus \{(0,\ldots ,0)\}$ by the action of $\mathbb {H} ^{\times }$, the multiplicative group of non-zero quaternions. By first projecting onto the unit sphere inside $\mathbb {H} ^{n+1}$ one may also regard $\mathbb {HP} ^{n}$ as the orbit space of $S^{4n+3}$ by the action of ${\text{Sp}}(1)$, the group of unit quaternions.[1] The sphere $S^{4n+3}$ then becomes a principal Sp(1)-bundle over $\mathbb {HP} ^{n}$:
$\mathrm {Sp} (1)\to S^{4n+3}\to \mathbb {HP} ^{n}.$
This bundle is sometimes called a (generalized) Hopf fibration.
There is also a construction of $\mathbb {HP} ^{n}$ by means of two-dimensional complex subspaces of $\mathbb {H} ^{2n}$, meaning that $\mathbb {HP} ^{n}$ lies inside a complex Grassmannian.
Topology
Homotopy theory
The space $\mathbb {HP} ^{\infty }$, defined as the union of all finite $\mathbb {HP} ^{n}$'s under inclusion, is the classifying space BS3. The homotopy groups of $\mathbb {HP} ^{\infty }$ are given by $\pi _{i}(\mathbb {HP} ^{\infty })=\pi _{i}(BS^{3})\cong \pi _{i-1}(S^{3}).$ These groups are known to be very complex and in particular they are non-zero for infinitely many values of $i$. However, we do have that
$\pi _{i}(\mathbb {HP} ^{\infty })\otimes \mathbb {Q} \cong {\begin{cases}\mathbb {Q} &i=4\\0&i\neq 4\end{cases}}$
It follows that rationally, i.e. after localisation of a space, $\mathbb {HP} ^{\infty }$ is an Eilenberg–Maclane space $K(\mathbb {Q} ,4)$. That is $\mathbb {HP} _{\mathbb {Q} }^{\infty }\simeq K(\mathbb {Z} ,4)_{\mathbb {Q} }.$ (cf. the example K(Z,2)). See rational homotopy theory.
In general, $\mathbb {HP} ^{n}$ has a cell structure with one cell in each dimension which is a multiple of 4, up to $4n$. Accordingly, its cohomology ring is $\mathbb {Z} [v]/v^{n+1}$, where $v$ is a 4-dimensional generator. This is analogous to complex projective space. It also follows from rational homotopy theory that $\mathbb {HP} ^{n}$ has infinite homotopy groups only in dimensions 4 and $4n+3$.
Differential geometry
$\mathbb {HP} ^{n}$ carries a natural Riemannian metric analogous to the Fubini-Study metric on $\mathbb {CP} ^{n}$, with respect to which it is a compact quaternion-Kähler symmetric space with positive curvature.
Quaternionic projective space can be represented as the coset space
$\mathbb {HP} ^{n}=\operatorname {Sp} (n+1)/\operatorname {Sp} (n)\times \operatorname {Sp} (1)$
where $\operatorname {Sp} (n)$ is the compact symplectic group.
Characteristic classes
Since $\mathbb {HP} ^{1}=S^{4}$, its tangent bundle is stably trivial. The tangent bundles of the rest have nontrivial Stiefel–Whitney and Pontryagin classes. The total classes are given by the following formulas:
$w(\mathbb {HP} ^{n})=(1+u)^{n+1}$
$p(\mathbb {HP} ^{n})=(1+v)^{2n+2}(1+4v)^{-1}$
where $v$ is the generator of $H^{4}(\mathbb {HP} ^{n};\mathbb {Z} )$ and $u$ is its reduction mod 2.[2]
Special cases
Quaternionic projective line
The one-dimensional projective space over $\mathbb {H} $ is called the "projective line" in generalization of the complex projective line. For example, it was used (implicitly) in 1947 by P. G. Gormley to extend the Möbius group to the quaternion context with linear fractional transformations. For the linear fractional transformations of an associative ring with 1, see projective line over a ring and the homography group GL(2,A).
From the topological point of view the quaternionic projective line is the 4-sphere, and in fact these are diffeomorphic manifolds. The fibration mentioned previously is from the 7-sphere, and is an example of a Hopf fibration.
Explicit expressions for coordinates for the 4-sphere can be found in the article on the Fubini–Study metric.
Quaternionic projective plane
The 8-dimensional $\mathbb {HP} ^{2}$ has a circle action, by the group of complex scalars of absolute value 1 acting on the other side (so on the right, as the convention for the action of c above is on the left). Therefore, the quotient manifold
$\mathbb {HP} ^{2}/\mathrm {U} (1)$
may be taken, writing U(1) for the circle group. It has been shown that this quotient is the 7-sphere, a result of Vladimir Arnold from 1996, later rediscovered by Edward Witten and Michael Atiyah.
References
1. Naber, Gregory L. (2011) [1997]. "Physical and Geometrical Motivation". Topology, Geometry and Gauge fields. Texts in Applied Mathematics. Vol. 25. Springer. p. 50. doi:10.1007/978-1-4419-7254-5_0. ISBN 978-1-4419-7254-5.
2. Szczarba, R.H. (1964). "On tangent bundles of fibre spaces and quotient spaces" (PDF). American Journal of Mathematics. 86 (4): 685–697. doi:10.2307/2373152. JSTOR 2373152.
Further reading
• Arnol'd, V.I. (1999). "Relatives of the Quotient of the Complex Projective Plane by the Complex Conjugation". Tr. Mat. Inst. Steklova. 224: 56–6. CiteSeerX 10.1.1.50.6421. Treats the analogue of the result mentioned for quaternionic projective space and the 13-sphere.
• Gormley, P.G. (1947), "Stereographic projection and the linear fractional group of transformations of quaternions", Proceedings of the Royal Irish Academy, Section A, 51: 67–85, JSTOR 20488472
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Quaternionic structure
In mathematics, a quaternionic structure or Q-structure is an axiomatic system that abstracts the concept of a quaternion algebra over a field.
A quaternionic structure is a triple (G, Q, q) where G is an elementary abelian group of exponent 2 with a distinguished element −1, Q is a pointed set with distinguished element 1, and q is a symmetric surjection G×G → Q satisfying axioms
${\begin{aligned}{\text{1.}}\quad &q(a,(-1)a)=1,\\{\text{2.}}\quad &q(a,b)=q(a,c)\Leftrightarrow q(a,bc)=1,\\{\text{3.}}\quad &q(a,b)=q(c,d)\Rightarrow \exists x\mid q(a,b)=q(a,x),q(c,d)=q(c,x)\end{aligned}}.$
Every field F gives rise to a Q-structure by taking G to be F∗/F∗2, Q the set of Brauer classes of quaternion algebras in the Brauer group of F with the split quaternion algebra as distinguished element and q(a,b) the quaternion algebra (a,b)F.
References
• Lam, Tsit-Yuen (2005). Introduction to Quadratic Forms over Fields. Graduate Studies in Mathematics. Vol. 67. American Mathematical Society. ISBN 0-8218-1095-2. MR 2104929. Zbl 1068.11023.
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Quaternion-Kähler symmetric space
In differential geometry, a quaternion-Kähler symmetric space or Wolf space is a quaternion-Kähler manifold which, as a Riemannian manifold, is a Riemannian symmetric space. Any quaternion-Kähler symmetric space with positive Ricci curvature is compact and simply connected, and is a Riemannian product of quaternion-Kähler symmetric spaces associated to compact simple Lie groups.
For any compact simple Lie group G, there is a unique G/H obtained as a quotient of G by a subgroup
$H=K\cdot \mathrm {Sp} (1).\,$
Here, Sp(1) is the compact form of the SL(2)-triple associated with the highest root of G, and K its centralizer in G. These are classified as follows.
G H quaternionic dimension geometric interpretation
$\mathrm {SU} (p+2)\,$ $\mathrm {S} (\mathrm {U} (p)\times \mathrm {U} (2))$ p Grassmannian of complex 2-dimensional subspaces of $\mathbb {C} ^{p+2}$
$\mathrm {SO} (p+4)\,$ $\mathrm {SO} (p)\cdot \mathrm {SO} (4)$ p Grassmannian of oriented real 4-dimensional subspaces of $\mathbb {R} ^{p+4}$
$\mathrm {Sp} (p+1)\,$ $\mathrm {Sp} (p)\cdot \mathrm {Sp} (1)$ p Grassmannian of quaternionic 1-dimensional subspaces of $\mathbb {H} ^{p+1}$
$E_{6}\,$ $\mathrm {SU} (6)\cdot \mathrm {SU} (2)$ 10 Space of symmetric subspaces of $(\mathbb {C} \otimes \mathbb {O} )P^{2}$ isometric to $(\mathbb {C} \otimes \mathbb {H} )P^{2}$
$E_{7}\,$ $\mathrm {Spin} (12)\cdot \mathrm {Sp} (1)$ 16 Rosenfeld projective plane $(\mathbb {H} \otimes \mathbb {O} )P^{2}$ over $\mathbb {H} \otimes \mathbb {O} $
$E_{8}\,$ $E_{7}\cdot \mathrm {Sp} (1)$ 28 Space of symmetric subspaces of $(\mathbb {O} \otimes \mathbb {O} )P^{2}$ isomorphic to $(\mathbb {H} \otimes \mathbb {O} )P^{2}$
$F_{4}\,$ $\mathrm {Sp} (3)\cdot \mathrm {Sp} (1)$ 7 Space of the symmetric subspaces of $\mathbb {OP} ^{2}$ which are isomorphic to $\mathbb {HP} ^{2}$
$G_{2}\,$ $\mathrm {SO} (4)\,$ 2 Space of the subalgebras of the octonion algebra $\mathbb {O} $ which are isomorphic to the quaternion algebra $\mathbb {H} $
The twistor spaces of quaternion-Kähler symmetric spaces are the homogeneous holomorphic contact manifolds, classified by Boothby: they are the adjoint varieties of the complex semisimple Lie groups.
These spaces can be obtained by taking a projectivization of a minimal nilpotent orbit of the respective complex Lie group. The holomorphic contact structure is apparent, because the nilpotent orbits of semisimple Lie groups are equipped with the Kirillov-Kostant holomorphic symplectic form. This argument also explains how one can associate a unique Wolf space to each of the simple complex Lie groups.
See also
• Quaternionic discrete series representation
References
• Besse, Arthur L. (2008), Einstein Manifolds, Classics in Mathematics, Berlin: Springer-Verlag, ISBN 978-3-540-74120-6, MR 2371700. Reprint of the 1987 edition.
• Salamon, Simon (1982), "Quaternionic Kähler manifolds", Inventiones Mathematicae, 67 (1): 143–171, Bibcode:1982InMat..67..143S, doi:10.1007/BF01393378, MR 0664330, S2CID 118575943.
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Quaternionic vector space
In mathematics, a left (or right) quaternionic vector space is a left (or right) H-module where H is the (non-commutative) division ring of quaternions.
The space Hn of n-tuples of quaternions is both a left and right H-module using the componentwise left and right multiplication:
$q(q_{1},q_{2},\ldots q_{n})=(qq_{1},qq_{2},\ldots qq_{n})$
$(q_{1},q_{2},\ldots q_{n})q=(q_{1}q,q_{2}q,\ldots q_{n}q)$
for quaternions q and q1, q2, ... qn.
Since H is a division algebra, every finitely generated (left or right) H-module has a basis, and hence is isomorphic to Hn for some n.
See also
• Vector space
• General linear group
• Special linear group
• SL(n,H)
• Symplectic group
References
• Harvey, F. Reese (1990). Spinors and Calibrations. San Diego: Academic Press. ISBN 0-12-329650-1.
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Freudenthal magic square
In mathematics, the Freudenthal magic square (or Freudenthal–Tits magic square) is a construction relating several Lie algebras (and their associated Lie groups). It is named after Hans Freudenthal and Jacques Tits, who developed the idea independently. It associates a Lie algebra to a pair of division algebras A, B. The resulting Lie algebras have Dynkin diagrams according to the table at right. The "magic" of the Freudenthal magic square is that the constructed Lie algebra is symmetric in A and B, despite the original construction not being symmetric, though Vinberg's symmetric method gives a symmetric construction.
Not to be confused with magic square.
A \ B $\mathbb {R} $ $\mathbb {C} $ $\mathbb {H} $ $\mathbb {O} $
$\mathbb {R} $ A1
A2
C3
F4
$\mathbb {C} $ A2
A2 × A2
A5
E6
$\mathbb {H} $ C3
A5
D6
E7
$\mathbb {O} $ F4
E6
E7
E8
The Freudenthal magic square includes all of the exceptional Lie groups apart from G2, and it provides one possible approach to justify the assertion that "the exceptional Lie groups all exist because of the octonions": G2 itself is the automorphism group of the octonions (also, it is in many ways like a classical Lie group because it is the stabilizer of a generic 3-form on a 7-dimensional vector space – see prehomogeneous vector space).
Constructions
See history for context and motivation. These were originally constructed circa 1958 by Freudenthal and Tits, with more elegant formulations following in later years.[1]
Tits' approach
Tits' approach, discovered circa 1958 and published in (Tits 1966), is as follows.
Associated with any normed real division algebra A (i.e., R, C, H or O) there is a Jordan algebra, J3(A), of 3 × 3 A-Hermitian matrices. For any pair (A, B) of such division algebras, one can define a Lie algebra
$L=\left({\mathfrak {der}}(A)\oplus {\mathfrak {der}}(J_{3}(B))\right)\oplus \left(A_{0}\otimes J_{3}(B)_{0}\right)$
where ${\mathfrak {der}}$ denotes the Lie algebra of derivations of an algebra, and the subscript 0 denotes the trace-free part. The Lie algebra L has ${\mathfrak {der}}(A)\oplus {\mathfrak {der}}(J_{3}(B))$ as a subalgebra, and this acts naturally on $A_{0}\otimes J_{3}(B)_{0}$. The Lie bracket on $A_{0}\otimes J_{3}(B)_{0}$ (which is not a subalgebra) is not obvious, but Tits showed how it could be defined, and that it produced the following table of compact Lie algebras.
BRCHO
Ader(A/B)00${\mathfrak {sp}}_{1}$${\mathfrak {g}}_{2}$
R0 ${\mathfrak {so}}_{3}$${\mathfrak {su}}_{3}$${\mathfrak {sp}}_{3}$${\mathfrak {f}}_{4}$
C0 ${\mathfrak {su}}_{3}$${\mathfrak {su}}_{3}\oplus {\mathfrak {su}}_{3}$${\mathfrak {su}}_{6}$${\mathfrak {e}}_{6}$
H${\mathfrak {sp}}_{1}$ ${\mathfrak {sp}}_{3}$${\mathfrak {su}}_{6}$${\mathfrak {so}}_{12}$${\mathfrak {e}}_{7}$
O${\mathfrak {g}}_{2}$ ${\mathfrak {f}}_{4}$${\mathfrak {e}}_{6}$${\mathfrak {e}}_{7}$${\mathfrak {e}}_{8}$
By construction, the row of the table with A=R gives ${\mathfrak {der}}(J_{3}(B))$, and similarly vice versa.
Vinberg's symmetric method
The "magic" of the Freudenthal magic square is that the constructed Lie algebra is symmetric in A and B. This is not obvious from Tits' construction. Ernest Vinberg gave a construction which is manifestly symmetric, in (Vinberg 1966). Instead of using a Jordan algebra, he uses an algebra of skew-hermitian trace-free matrices with entries in A ⊗ B, denoted ${\mathfrak {sa}}_{3}(A\otimes B)$. Vinberg defines a Lie algebra structure on
${\mathfrak {der}}(A)\oplus {\mathfrak {der}}(B)\oplus {\mathfrak {sa}}_{3}(A\otimes B).$
When A and B have no derivations (i.e., R or C), this is just the Lie (commutator) bracket on ${\mathfrak {sa}}_{3}(A\otimes B)$. In the presence of derivations, these form a subalgebra acting naturally on ${\mathfrak {sa}}_{3}(A\otimes B)$ as in Tits' construction, and the tracefree commutator bracket on ${\mathfrak {sa}}_{3}(A\otimes B)$ is modified by an expression with values in ${\mathfrak {der}}(A)\oplus {\mathfrak {der}}(B)$.
Triality
A more recent construction, due to Pierre Ramond (Ramond 1976) and Bruce Allison (Allison 1978) and developed by Chris Barton and Anthony Sudbery, uses triality in the form developed by John Frank Adams; this was presented in (Barton & Sudbery 2000), and in streamlined form in (Barton & Sudbery 2003). Whereas Vinberg's construction is based on the automorphism groups of a division algebra A (or rather their Lie algebras of derivations), Barton and Sudbery use the group of automorphisms of the corresponding triality. The triality is the trilinear map
$A_{1}\times A_{2}\times A_{3}\to \mathbf {R} $
obtained by taking three copies of the division algebra A, and using the inner product on A to dualize the multiplication. The automorphism group is the subgroup of SO(A1) × SO(A2) × SO(A3) preserving this trilinear map. It is denoted Tri(A). The following table compares its Lie algebra to the Lie algebra of derivations.
A: R C H O
${\mathfrak {der}}(A)$ 0 0 ${\mathfrak {sp}}_{1}$ ${\mathfrak {g}}_{2}$
${\mathfrak {tri}}(A)$ 0 ${\mathfrak {u}}_{1}\oplus {\mathfrak {u}}_{1}$ ${\mathfrak {sp}}_{1}\oplus {\mathfrak {sp}}_{1}\oplus {\mathfrak {sp}}_{1}$ ${\mathfrak {so}}_{8}$
Barton and Sudbery then identify the magic square Lie algebra corresponding to (A,B) with a Lie algebra structure on the vector space
${\mathfrak {tri}}(A)\oplus {\mathfrak {tri}}(B)\oplus (A_{1}\otimes B_{1})\oplus (A_{2}\otimes B_{2})\oplus (A_{3}\otimes B_{3}).$
The Lie bracket is compatible with a Z2 × Z2 grading, with tri(A) and tri(B) in degree (0,0), and the three copies of A ⊗ B in degrees (0,1), (1,0) and (1,1). The bracket preserves tri(A) and tri(B) and these act naturally on the three copies of A ⊗ B, as in the other constructions, but the brackets between these three copies are more constrained.
For instance when A and B are the octonions, the triality is that of Spin(8), the double cover of SO(8), and the Barton-Sudbery description yields
${\mathfrak {e}}_{8}\cong {\mathfrak {so}}_{8}\oplus {\widehat {\mathfrak {so}}}_{8}\oplus (V\otimes {\widehat {V}})\oplus (S_{+}\otimes {\widehat {S}}_{+})\oplus (S_{-}\otimes {\widehat {S}}_{-})$
where V, S+ and S− are the three 8-dimensional representations of ${\mathfrak {so}}_{8}$ (the fundamental representation and the two spin representations), and the hatted objects are an isomorphic copy.
With respect to one of the Z2 gradings, the first three summands combine to give ${\mathfrak {so}}_{16}$ and the last two together form one of its spin representations Δ+128 (the superscript denotes the dimension). This is a well known symmetric decomposition of E8.
The Barton–Sudbery construction extends this to the other Lie algebras in the magic square. In particular, for the exceptional Lie algebras in the last row (or column), the symmetric decompositions are:
${\mathfrak {f}}_{4}\cong {\mathfrak {so}}_{9}\oplus \Delta ^{16}$
${\mathfrak {e}}_{6}\cong ({\mathfrak {so}}_{10}\oplus {\mathfrak {u}}_{1})\oplus \Delta ^{32}$
${\mathfrak {e}}_{7}\cong ({\mathfrak {so}}_{12}\oplus {\mathfrak {sp}}_{1})\oplus \Delta _{+}^{64}$
${\mathfrak {e}}_{8}\cong {\mathfrak {so}}_{16}\oplus \Delta _{+}^{128}.$
Generalizations
Split composition algebras
In addition to the normed division algebras, there are other composition algebras over R, namely the split-complex numbers, the split-quaternions and the split-octonions. If one uses these instead of the complex numbers, quaternions, and octonions, one obtains the following variant of the magic square (where the split versions of the division algebras are denoted by a prime).
A\B R C' H' O'
R ${\mathfrak {so}}_{3}$ ${\mathfrak {sl}}_{3}(\mathbf {R} )$ ${\mathfrak {sp}}_{6}(\mathbf {R} )$ ${\mathfrak {f}}_{4(4)}$
C' ${\mathfrak {sl}}_{3}(\mathbf {R} )$ ${\mathfrak {sl}}_{3}(\mathbf {R} )\oplus {\mathfrak {sl}}_{3}(\mathbf {R} )$ ${\mathfrak {sl}}_{6}(\mathbf {R} )$ ${\mathfrak {e}}_{6(6)}$
H' ${\mathfrak {sp}}_{6}(\mathbf {R} )$ ${\mathfrak {sl}}_{6}(\mathbf {R} )$ ${\mathfrak {so}}_{6,6}$ ${\mathfrak {e}}_{7(7)}$
O' ${\mathfrak {f}}_{4(4)}$ ${\mathfrak {e}}_{6(6)}$ ${\mathfrak {e}}_{7(7)}$ ${\mathfrak {e}}_{8(8)}$
Here all the Lie algebras are the split real form except for so3, but a sign change in the definition of the Lie bracket can be used to produce the split form so2,1. In particular, for the exceptional Lie algebras, the maximal compact subalgebras are as follows:
Split form ${\mathfrak {f}}_{4(4)}$ ${\mathfrak {e}}_{6(6)}$ ${\mathfrak {e}}_{7(7)}$ ${\mathfrak {e}}_{8(8)}$
Maximal compact ${\mathfrak {sp}}_{3}\oplus {\mathfrak {sp}}_{1}$ ${\mathfrak {sp}}_{4}$ ${\mathfrak {su}}_{8}$ ${\mathfrak {so}}_{16}$
A non-symmetric version of the magic square can also be obtained by combining the split algebras with the usual division algebras. According to Barton and Sudbery, the resulting table of Lie algebras is as follows.
A\B R C H O
R ${\mathfrak {so}}_{3}$ ${\mathfrak {su}}_{3}$ ${\mathfrak {sp}}_{3}$ ${\mathfrak {f}}_{4}$
C' ${\mathfrak {sl}}_{3}(\mathbf {R} )$ ${\mathfrak {sl}}_{3}(\mathbf {C} )$ ${\mathfrak {sl}}_{3}(\mathbf {H} )$ ${\mathfrak {e}}_{6(-26)}$
H' ${\mathfrak {sp}}_{6}(\mathbf {R} )$ ${\mathfrak {su}}_{3,3}$ ${\mathfrak {so}}_{6}^{*}(\mathbf {H} )$ ${\mathfrak {e}}_{7(-25)}$
O' ${\mathfrak {f}}_{4(4)}$ ${\mathfrak {e}}_{6(2)}$ ${\mathfrak {e}}_{7(-5)}$ ${\mathfrak {e}}_{8(-24)}$
The real exceptional Lie algebras appearing here can again be described by their maximal compact subalgebras.
Lie algebra ${\mathfrak {e}}_{6(2)}$ ${\mathfrak {e}}_{6(-26)}$ ${\mathfrak {e}}_{7(-5)}$ ${\mathfrak {e}}_{7(-25)}$ ${\mathfrak {e}}_{8(-24)}$
Maximal compact ${\mathfrak {su}}_{6}\oplus {\mathfrak {sp}}_{1}$ ${\mathfrak {f}}_{4}$ ${\mathfrak {su}}_{12}\oplus {\mathfrak {sp}}_{1}$ ${\mathfrak {e}}_{6}\oplus {\mathfrak {u}}_{1}$ ${\mathfrak {e}}_{7}\oplus {\mathfrak {sp}}_{1}$
Arbitrary fields
The split forms of the composition algebras and Lie algebras can be defined over any field K. This yields the following magic square.
${\mathfrak {so}}_{3}(\mathbf {K} )$ ${\mathfrak {sl}}_{3}(\mathbf {K} )$ ${\mathfrak {sp}}_{6}(\mathbf {K} )$ ${\mathfrak {f}}_{4}(\mathbf {K} )$
${\mathfrak {sl}}_{3}(\mathbf {K} )$ ${\mathfrak {sl}}_{3}(\mathbf {K} )\oplus {\mathfrak {sl}}_{3}(\mathbf {K} )$ ${\mathfrak {sl}}_{6}(\mathbf {K} )$ ${\mathfrak {e}}_{6}(\mathbf {K} )$
${\mathfrak {sp}}_{6}(\mathbf {K} )$ ${\mathfrak {sl}}_{6}(\mathbf {K} )$ ${\mathfrak {so}}_{12}(\mathbf {K} )$ ${\mathfrak {e}}_{7}(\mathbf {K} )$
${\mathfrak {f}}_{4}(\mathbf {K} )$ ${\mathfrak {e}}_{6}(\mathbf {K} )$ ${\mathfrak {e}}_{7}(\mathbf {K} )$ ${\mathfrak {e}}_{8}(\mathbf {K} )$
There is some ambiguity here if K is not algebraically closed. In the case K = C, this is the complexification of the Freudenthal magic squares for R discussed so far.
More general Jordan algebras
The squares discussed so far are related to the Jordan algebras J3(A), where A is a division algebra. There are also Jordan algebras Jn(A), for any positive integer n, as long as A is associative. These yield split forms (over any field K) and compact forms (over R) of generalized magic squares.
${\mathfrak {so}}_{n}(\mathbf {K} )$ ${\mathfrak {sl}}_{n}(\mathbf {K} ){\text{ or }}{\mathfrak {su}}_{n}$ ${\mathfrak {sp}}_{2n}(\mathbf {K} ){\text{ or }}{\mathfrak {sp}}_{n}$
${\mathfrak {sl}}_{n}(\mathbf {K} ){\text{ or }}{\mathfrak {su}}_{n}$ ${\mathfrak {sl}}_{n}(\mathbf {K} )\oplus {\mathfrak {sl}}_{n}(\mathbf {K} ){\text{ or }}{\mathfrak {su}}_{n}\oplus {\mathfrak {su}}_{n}$ ${\mathfrak {sl}}_{2n}(\mathbf {K} ){\text{ or }}{\mathfrak {su}}_{2n}$
${\mathfrak {sp}}_{2n}(\mathbf {K} ){\text{ or }}{\mathfrak {sp}}_{n}$ ${\mathfrak {sl}}_{2n}(\mathbf {K} ){\text{ or }}{\mathfrak {su}}_{2n}$ ${\mathfrak {so}}_{4n}(\mathbf {K} )$
For n = 2, J2(O) is also a Jordan algebra. In the compact case (over R) this yields a magic square of orthogonal Lie algebras.
A\B R C H O
R ${\mathfrak {so}}_{2}$ ${\mathfrak {so}}_{3}$ ${\mathfrak {so}}_{5}$ ${\mathfrak {so}}_{9}$
C ${\mathfrak {so}}_{3}$ ${\mathfrak {so}}_{4}$ ${\mathfrak {so}}_{6}$ ${\mathfrak {so}}_{10}$
H ${\mathfrak {so}}_{5}$ ${\mathfrak {so}}_{6}$ ${\mathfrak {so}}_{8}$ ${\mathfrak {so}}_{12}$
O ${\mathfrak {so}}_{9}$ ${\mathfrak {so}}_{10}$ ${\mathfrak {so}}_{12}$ ${\mathfrak {so}}_{16}$
The last row and column here are the orthogonal algebra part of the isotropy algebra in the symmetric decomposition of the exceptional Lie algebras mentioned previously.
These constructions are closely related to hermitian symmetric spaces – cf. prehomogeneous vector spaces.
Symmetric spaces
Riemannian symmetric spaces, both compact and non-compact, can be classified uniformly using a magic square construction, in (Huang & Leung 2010). The irreducible compact symmetric spaces are, up to finite covers, either a compact simple Lie group, a Grassmannian, a Lagrangian Grassmannian, or a double Lagrangian Grassmannian of subspaces of $(\mathbf {A} \otimes \mathbf {B} )^{n},$ for normed division algebras A and B. A similar construction produces the irreducible non-compact symmetric spaces.
History
Rosenfeld projective planes
Following Ruth Moufang's discovery in 1933 of the Cayley projective plane or "octonionic projective plane" P2(O), whose symmetry group is the exceptional Lie group F4, and with the knowledge that G2 is the automorphism group of the octonions, it was proposed by Rozenfeld (1956) that the remaining exceptional Lie groups E6, E7, and E8 are isomorphism groups of projective planes over certain algebras over the octonions:[1]
• the bioctonions, C ⊗ O,
• the quateroctonions, H ⊗ O,
• the octooctonions, O ⊗ O.
This proposal is appealing, as there are certain exceptional compact Riemannian symmetric spaces with the desired symmetry groups and whose dimension agree with that of the putative projective planes (dim(P2(K ⊗ K′)) = 2 dim(K)dim(K′)), and this would give a uniform construction of the exceptional Lie groups as symmetries of naturally occurring objects (i.e., without an a priori knowledge of the exceptional Lie groups). The Riemannian symmetric spaces were classified by Cartan in 1926 (Cartan's labels are used in sequel); see classification for details, and the relevant spaces are:
• the octonionic projective plane – FII, dimension 16 = 2 × 8, F4 symmetry, Cayley projective plane P2(O),
• the bioctonionic projective plane – EIII, dimension 32 = 2 × 2 × 8, E6 symmetry, complexified Cayley projective plane, P2(C ⊗ O),
• the "quateroctonionic projective plane"[2] – EVI, dimension 64 = 2 × 4 × 8, E7 symmetry, P2(H ⊗ O),
• the "octooctonionic projective plane"[3] – EVIII, dimension 128 = 2 × 8 × 8, E8 symmetry, P2(O ⊗ O).
The difficulty with this proposal is that while the octonions are a division algebra, and thus a projective plane is defined over them, the bioctonions, quateroctonions and octooctonions are not division algebras, and thus the usual definition of a projective plane does not work. This can be resolved for the bioctonions, with the resulting projective plane being the complexified Cayley plane, but the constructions do not work for the quateroctonions and octooctonions, and the spaces in question do not obey the usual axioms of projective planes,[1] hence the quotes on "(putative) projective plane". However, the tangent space at each point of these spaces can be identified with the plane (H ⊗ O)2, or (O ⊗ O)2 further justifying the intuition that these are a form of generalized projective plane.[2][3] Accordingly, the resulting spaces are sometimes called Rosenfeld projective planes and notated as if they were projective planes. More broadly, these compact forms are the Rosenfeld elliptic projective planes, while the dual non-compact forms are the Rosenfeld hyperbolic projective planes. A more modern presentation of Rosenfeld's ideas is in (Rosenfeld 1997), while a brief note on these "planes" is in (Besse 1987, pp. 313–316).[4]
The spaces can be constructed using Tits' theory of buildings, which allows one to construct a geometry with any given algebraic group as symmetries, but this requires starting with the Lie groups and constructing a geometry from them, rather than constructing a geometry independently of a knowledge of the Lie groups.[1]
Magic square
While at the level of manifolds and Lie groups, the construction of the projective plane P2(K ⊗ K′) of two normed division algebras does not work, the corresponding construction at the level of Lie algebras does work. That is, if one decomposes the Lie algebra of infinitesimal isometries of the projective plane P2(K) and applies the same analysis to P2(K ⊗ K′), one can use this decomposition, which holds when P2(K ⊗ K′) can actually be defined as a projective plane, as a definition of a "magic square Lie algebra" M(K,K′). This definition is purely algebraic, and holds even without assuming the existence of the corresponding geometric space. This was done independently circa 1958 in (Tits 1966) and by Freudenthal in a series of 11 papers, starting with (Freudenthal 1954a) and ending with (Freudenthal 1963), though the simplified construction outlined here is due to (Vinberg 1966).[1]
See also
• E6 (mathematics)
• E7 (mathematics)
• E8 (mathematics)
• F4 (mathematics)
• G2 (mathematics)
• Euclidean Hurwitz algebra
• Euclidean Jordan algebra
• Jordan triple system
Notes
1. (Baez 2002, 4.3 The Magic Square)
2. (Baez 2002, 4.5 E7)
3. (Baez 2002, 4.6 E8)
4. "This Week's Finds in Mathematical Physics – Week 106", John Baez July 23, 1997
References
• Adams, John Frank (1996). Mahmud, Zafer; Mimura, Mamora (eds.). Lectures on Exceptional Lie Groups. Chicago Lectures in Mathematics. University of Chicago Press. ISBN 978-0-226-00527-0.
• Allison, B.N. (1978). "Structurable Algebras". Math. Ann. 237 (2): 133–156. doi:10.1007/bf01351677. S2CID 120322064.
• Baez, John C. (2002). "The Octonions". Bulletin of the American Mathematical Society. 39 (2): 145–205. arXiv:math/0105155. doi:10.1090/S0273-0979-01-00934-X. ISSN 0273-0979. MR 1886087. S2CID 586512. – 4.3: The Magic Square
• Baez, John C. (2005). "Errata for The Octonions" (PDF). Bulletin of the American Mathematical Society. 42 (2): 213–214. doi:10.1090/S0273-0979-05-01052-9.
• Barton, C. H.; Sudbery, A. (2000). "Magic Squares of Lie Algebras". arXiv:math/0001083.
• Barton, C. H.; Sudbery, A. (2003). "Magic squares and matrix models of Lie algebras". Advances in Mathematics. 180 (2): 596–647. arXiv:math.RA/0203010. doi:10.1016/S0001-8708(03)00015-X. S2CID 119621987.
• Besse, Arthur L. (1987). Einstein Manifolds. Berlin: Springer. ISBN 978-3-540-15279-8.
• Freudenthal, Hans (1954a). "Beziehungen der E7 und E8 zur Oktavenebene. I". Indagationes Mathematicae (in German). 16: 218–230. doi:10.1016/S1385-7258(54)50032-6. MR 0063358.
• Freudenthal, Hans (1954b). "Beziehungen der E7 und E8 zur Oktavenebene. II". Indagationes Mathematicae (in German). 16: 363–368. doi:10.1016/S1385-7258(54)50045-4. MR 0068549.
• Freudenthal, Hans (1955a). "Beziehungen der E7 und E8 zur Oktavenebene. III". Indagationes Mathematicae (in German). 17: 151–157. doi:10.1016/S1385-7258(55)50020-5. MR 0068550.
• Freudenthal, Hans (1955b). "Beziehungen der E7 und E8 zur Oktavenebene. IV". Indagationes Mathematicae (in German). 17: 277–285. doi:10.1016/S1385-7258(55)50039-4. MR 0068551.
• Freudenthal, Hans (1959). "Beziehungen der E7 und E8 zur Oktavenebene. V–IX". Indagationes Mathematicae (in German). 21: 165–201, 447–474. doi:10.1016/S1385-7258(59)50019-0.
• Freudenthal, Hans (1963). "Beziehungen der E7 und E8 zur Oktavenebene. X, XI". Indagationes Mathematicae (in German). 25: 457–471, 472–487. doi:10.1016/S1385-7258(63)50046-8. MR 0163203.
• Freudenthal, Hans (1951), Oktaven, Ausnahmegruppen und Oktavengeometrie, Mathematisch Instituut der Rijksuniversiteit te Utrecht
• Freudenthal, Hans (1985), "Oktaven, Ausnahmegruppen und Oktavengeometrie", Geometriae Dedicata, 19: 7–63, doi:10.1007/bf00233101, S2CID 121496094 (reprint of 1951 article)
• Huang, Yongdong; Leung, Naichung Conan (30 July 2010). "A uniform description of compact symmetric spaces as Grassmannians using the magic square" (PDF). Mathematische Annalen. 350 (May 2011): 79–106. doi:10.1007/s00208-010-0549-8. S2CID 121427210.
• Landsberg, J. M.; Manivel, L. (2001). "The Projective Geometry of Freudenthal's Magic Square". Journal of Algebra. 239 (2): 477–512. arXiv:math.AG/9908039. doi:10.1006/jabr.2000.8697. S2CID 16320642.
• Postnikov, M. (1986), Lie groups and Lie algebras. Lectures in geometry. Semester V, Mir
• Ramond, Pierre (Dec 1976). Introduction to Exceptional Lie Groups and Algebras (Report). Pasadena: California Institute of Technology. CALT-68-577.
• Rozenfeld, Boris A. (1956). "[Geometrical interpretation of compact simple Lie groups of class E]". Dokl. Akad. Nauk SSSR (in Russian). 106: 600–603.
• Rosenfeld, Boris A. (1997). Geometry of Lie groups. Mathematics and its Applications. Vol. 393. Dordrecht: Kluwer Academic Publishers Group. pp. xviii+393. ISBN 978-0-7923-4390-5.
• Tits, Jacques (1966). "Algèbres alternatives, algèbres de Jordan et algèbres de Lie exceptionnelles" [Alternative algebras, Jordan algebras and exceptional Lie algebras]. Indagationes Mathematicae (in French). 28: 223–237. doi:10.1016/S1385-7258(66)50028-2. MR 0219578.
• Vinberg, E.B. (1966). "[Construction of the exceptional simple Lie algebras]". Trudy Sem. Vekt. Tenz. Anal. (in Russian). 13: 7–9.
• Vinberg, E.B. (2005). "Construction of the exceptional simple Lie algebras". Amer. Math. Soc. Transl. 213: 241–242.
• Yokota, Ichiro (1985). "Non-symmetry of the Freudenthal's magic square". J. Fac. Sci. Shinshu Univ. 20: 13.
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Wikipedia
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Queen's graph
In mathematics, a queen's graph is an undirected graph that represents all legal moves of the queen—a chess piece—on a chessboard. In the graph, each vertex represents a square on a chessboard, and each edge is a legal move the queen can make, that is, a horizontal, vertical or diagonal move by any number of squares. If the chessboard has dimensions $m\times n$, then the induced graph is called the $m\times n$ queen's graph.
Queen's graph
abcdefgh
8
8
77
66
55
44
33
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abcdefgh
In an $8\times 8$ queen's graph, each square of the chessboard above is a vertex. There is an edge between any two vertices that a queen could move between; as an example, the vertices adjacent to d4 are marked with a white dot (i.e. there is an edge from d4 to each marked vertex).
Vertices$mn$
Chromatic numbern if $m=n\equiv 1,5{\pmod {6}}$
PropertiesBiconnected, Hamiltonian
Table of graphs and parameters
Independent sets of the graphs correspond to placements of multiple queens where no two queens are attacking each other. They are studied in the eight queens puzzle, where eight non-attacking queens are placed on a standard $8\times 8$ chessboard. Dominating sets represent arrangements of queens where every square is attacked or occupied by a queen; five queens, but no fewer, can dominate the $8\times 8$ chessboard.
Colourings of the graphs represent ways to colour each square so that a queen cannot move between any two squares of the same colour; at least n colours are needed for an $n\times n$ chessboard, but 9 colours are needed for the $8\times 8$ board.
Properties
There is a Hamiltonian cycle for each queen's graph, and the graphs are biconnected (they remain connected if any single vertex is removed). The special cases of the $1\times n$ and $2\times 2$ queen's graphs are complete.[1]
Independence
abcdefgh
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abcdefgh
An independent set of size 8 for an $8\times 8$ chessboard (such sets are necessarily also dominating).[2]
Main article: Eight queens puzzle
An independent set of the graph corresponds to a placement of several queens on a chessboard such that no two queens are attacking each other. In an $n\times n$ chessboard, the largest independent set contains at most n vertices, as no two queens can be in the same row or column.[2] This upper bound can be achieved for all n except n=2 and n=3.[3] In the case of n=8, this is the traditional eight queens puzzle.[2]
Domination
A dominating set of the queen's graph corresponds to a placement of queens such that every square on the chessboard is either attacked or occupied by a queen. On an $8\times 8$ chessboard, five queens can dominate, and this is the minimum number possible[4]: 113–114 (four queens leave at least two squares unattacked). There are 4,860 such placements of five queens, including ones where the queens control also all occupied squares, i.e. they attack respectively protect each other. In this subgroup, there are also positions where the queens occupy squares on the main diagonal only[4]: 113–114 (e.g. from a1 to h8), or all on a subdiagonal (e.g. from a2 to g8).[5][6]
abcdefgh
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abcdefgh
A dominating (and independent) set of size 5.
abcdefgh
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abcdefgh
A dominating set on the main diagonal.
abcdefgh
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abcdefgh
A dominating set on a sub diagonal.
Modifying the graph by replacing the non-looping rectangular $8\times 8$ chessboard with a torus or cylinder reduces the minimum dominating set size to four.[4]: 139
Dotted squares are adjacent to the centre square. The 8 non-adjacent squares are adjacent in the corresponding knight's graph.[4]: 117
The $3\times 3$ queen's graph is dominated by the single vertex at the centre of the board. The centre vertex of the $5\times 5$ queen's graph is adjacent to all but 8 vertices: those vertices that are adjacent to the centre vertex of the $5\times 5$ knight's graph.[4]: 117
Domination numbers
Define the domination number d(n) of an $n\times n$ queen's graph to be the size of the smallest dominating set, and the diagonal domination number dd(n) to be the size of the smallest dominating set that is a subset of the long diagonal. Note that $d(n)\leq dd(n)$ for all n. The bound is attained for $d(8)=dd(8)=5$, but not for $d(11)=5,dd(11)=7$.[4]: 119
The domination number is linear in n, with bounds given by:[4]: 119, 121
${\frac {n-1}{2}}\leq d(n)\leq n-\left\lfloor {\frac {n}{3}}\right\rfloor .$
Initial values of d(n), for $n=1,2,3,\dots $, are 1, 1, 1, 2, 3, 3, 4, 5, 5, 5, 5 (sequence A075458 in the OEIS).
Let Kn be the maximum size of a subset of $\{1,2,3,\dots ,n\}$ such that every number has the same parity and no three numbers form an arithmetic progression (the set is "midpoint-free"). The diagonal domination number of an $n\times n$ queen's graph is $n-K_{n}$.[4]: 116
Define the independent domination number ID(n) to be the size of the smallest independent, dominant set in an $n\times n$ queen's graph. It is known that $ID(n)<0.705n+0.895$.[7]
Colouring
A colouring of the queen's graph is an assignment of colours to each vertex such that no two adjacent vertices are given the same colour. For instance, if a8 is coloured red then no other square on the a-file, eighth rank or long diagonal can be coloured red, as a queen can move from a8 to any of these squares. The chromatic number of the graph is the smallest number of colours that can be used to colour it.
In the case of an $n\times n$ queen's graph, at least n colours are required, as each square in a rank or file needs a different colour (i.e. the rows and columns are cliques).[1] The chromatic number is exactly n if $n\equiv 1,5{\pmod {6}}$ (i.e. n is one more or one less than a multiple of 6).[9]
The chromatic number of an $8\times 8$ queen's graph is 9.[10]
Irredundance
A set of vertices is irredundant if removing any vertex from the set changes the neighbourhood of the set i.e. for each vertex, there is an adjacent vertex that is not adjacent to any other vertex in the set. This corresponds to a set of queens which each uniquely control at least one square. The maximum size IR(n) of an irredundant set on the $n\times n$ queen's graph is difficult to characterise; known values include $IR(5)=5,IR(6)=7,IR(7)=9,IR(8)=11.$[4]: 206–207
Pursuit–evasion game
Consider the pursuit–evasion game on an $8\times 8$ queen's graph played according to the following rules: a white queen starts in one corner and a black queen in the opposite corner. Players alternate moves, which consist of moving the queen to an adjacent vertex that can be reached without passing over (horizontally, vertically or diagonally) or landing on a vertex that is adjacent to the opposite queen. This game can be won by white with a pairing strategy.[11]
See also
• King's graph
• Knight's graph
• Rook's graph
• Bishop's graph
• Chess portal
References
1. Weisstein, Eric W. "Queen Graph". MathWorld.
2. Averbach, Bonnie; Chein, Orin (2000). Problem Solving Through Recreational Mathematics. Dover Publications. pp. 211–212. ISBN 9780486131740.
3. Bernhardsson, Bo (1991). "Explicit Solutions to the N-Queens Problem for All N". ACM Sigart. 2 (2): 7. doi:10.1145/122319.122322. S2CID 10644706.
4. Watkins, John J. (2012). Across the Board: The Mathematics of Chessboard Problems. Princeton University Press.
5. Dominating queens - in researchgate.net
6. 5 Queens on a Chessboard
7. Cockayne, E. J. (1990). "Chessboard domination problems". Discrete Mathematics. 86 (1–3): 13–20. doi:10.1016/0012-365X(90)90344-H.
8. Iyer, M. R.; Menon, V. V. (1966). "On Coloring the $n\times n$ Chessboard". The American Mathematical Monthly. 72 (7): 723.
9. Chvátal, Václav. "Colouring the queen graphs". Retrieved 14 February 2022.
10. Bell, Jordan; Stevens, Brett (2009). "A survey of known results and research areas for n-queens". Discrete Mathematics. 309 (1): 1–31. doi:10.1016/j.disc.2007.12.043.
11. Averbach & Chein 2000, pp. 257–258, 443.
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Nonconvex great rhombicuboctahedron
In geometry, the nonconvex great rhombicuboctahedron is a nonconvex uniform polyhedron, indexed as U17. It has 26 faces (8 triangles and 18 squares), 48 edges, and 24 vertices.[1] It is represented by the Schläfli symbol rr{4,3⁄2} and Coxeter-Dynkin diagram of . Its vertex figure is a crossed quadrilateral.
Nonconvex great rhombicuboctahedron
TypeUniform star polyhedron
ElementsF = 26, E = 48
V = 24 (χ = 2)
Faces by sides8{3}+(6+12){4}
Coxeter diagram
Wythoff symbol3/2 4 | 2
3 4/3 | 2
Symmetry groupOh, [4,3], *432
Index referencesU17, C59, W85
Dual polyhedronGreat deltoidal icositetrahedron
Vertex figure
4.4.4.3/2
Bowers acronymQuerco
This model shares the name with the convex great rhombicuboctahedron, also called the truncated cuboctahedron.
An alternative name for this figure is quasirhombicuboctahedron. From that derives its Bowers acronym: querco.
Orthographic projections
Cartesian coordinates
Cartesian coordinates for the vertices of a nonconvex great rhombicuboctahedron centered at the origin with edge length 1 are all the permutations of
(±ξ, ±1, ±1),
where ξ = √2 − 1.
Related polyhedra
It shares the vertex arrangement with the convex truncated cube. It additionally shares its edge arrangement with the great cubicuboctahedron (having the triangular faces and 6 square faces in common), and with the great rhombihexahedron (having 12 square faces in common). It has the same vertex figure as the pseudo great rhombicuboctahedron, which is not a uniform polyhedron.
Truncated cube
Great rhombicuboctahedron
Great cubicuboctahedron
Great rhombihexahedron
Pseudo great rhombicuboctahedron
Great deltoidal icositetrahedron
Great deltoidal icositetrahedron
TypeStar polyhedron
Face
ElementsF = 24, E = 48
V = 26 (χ = 2)
Symmetry groupOh, [4,3], *432
Index referencesDU17
dual polyhedronNonconvex great rhombicuboctahedron
The great deltoidal icositetrahedron is the dual of the nonconvex great rhombicuboctahedron.
References
1. Maeder, Roman. "17: great rhombicuboctahedron". MathConsult.{{cite web}}: CS1 maint: url-status (link)
• Wenninger, Magnus (1983), Dual Models, Cambridge University Press, doi:10.1017/CBO9780511569371, ISBN 978-0-521-54325-5, MR 0730208
External links
Weisstein, Eric W. "Great Deltoidal Icositetrahedron". MathWorld.
• Weisstein, Eric W. "Uniform great rhombicuboctahedron". MathWorld.
• Great Rhombicuboctahedron Paper model
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Mixed volume
In mathematics, more specifically, in convex geometry, the mixed volume is a way to associate a non-negative number to a tuple of convex bodies in $\mathbb {R} ^{n}$. This number depends on the size and shape of the bodies, and their relative orientation to each other.
Definition
Let $K_{1},K_{2},\dots ,K_{r}$ be convex bodies in $\mathbb {R} ^{n}$ and consider the function
$f(\lambda _{1},\ldots ,\lambda _{r})=\mathrm {Vol} _{n}(\lambda _{1}K_{1}+\cdots +\lambda _{r}K_{r}),\qquad \lambda _{i}\geq 0,$
where ${\text{Vol}}_{n}$ stands for the $n$-dimensional volume, and its argument is the Minkowski sum of the scaled convex bodies $K_{i}$. One can show that $f$ is a homogeneous polynomial of degree $n$, so can be written as
$f(\lambda _{1},\ldots ,\lambda _{r})=\sum _{j_{1},\ldots ,j_{n}=1}^{r}V(K_{j_{1}},\ldots ,K_{j_{n}})\lambda _{j_{1}}\cdots \lambda _{j_{n}},$
where the functions $V$ are symmetric. For a particular index function $j\in \{1,\ldots ,r\}^{n}$, the coefficient $V(K_{j_{1}},\dots ,K_{j_{n}})$ is called the mixed volume of $K_{j_{1}},\dots ,K_{j_{n}}$.
Properties
• The mixed volume is uniquely determined by the following three properties:
1. $V(K,\dots ,K)={\text{Vol}}_{n}(K)$;
2. $V$ is symmetric in its arguments;
3. $V$ is multilinear: $V(\lambda K+\lambda 'K',K_{2},\dots ,K_{n})=\lambda V(K,K_{2},\dots ,K_{n})+\lambda 'V(K',K_{2},\dots ,K_{n})$ for $\lambda ,\lambda '\geq 0$.
• The mixed volume is non-negative and monotonically increasing in each variable: $V(K_{1},K_{2},\ldots ,K_{n})\leq V(K_{1}',K_{2},\ldots ,K_{n})$ for $K_{1}\subseteq K_{1}'$.
• The Alexandrov–Fenchel inequality, discovered by Aleksandr Danilovich Aleksandrov and Werner Fenchel:
$V(K_{1},K_{2},K_{3},\ldots ,K_{n})\geq {\sqrt {V(K_{1},K_{1},K_{3},\ldots ,K_{n})V(K_{2},K_{2},K_{3},\ldots ,K_{n})}}.$
Numerous geometric inequalities, such as the Brunn–Minkowski inequality for convex bodies and Minkowski's first inequality, are special cases of the Alexandrov–Fenchel inequality.
Quermassintegrals
Let $K\subset \mathbb {R} ^{n}$ be a convex body and let $B=B_{n}\subset \mathbb {R} ^{n}$ be the Euclidean ball of unit radius. The mixed volume
$W_{j}(K)=V({\overset {n-j{\text{ times}}}{\overbrace {K,K,\ldots ,K} }},{\overset {j{\text{ times}}}{\overbrace {B,B,\ldots ,B} }})$
is called the j-th quermassintegral of $K$.[1]
The definition of mixed volume yields the Steiner formula (named after Jakob Steiner):
$\mathrm {Vol} _{n}(K+tB)=\sum _{j=0}^{n}{\binom {n}{j}}W_{j}(K)t^{j}.$
Intrinsic volumes
The j-th intrinsic volume of $K$ is a different normalization of the quermassintegral, defined by
$V_{j}(K)={\binom {n}{j}}{\frac {W_{n-j}(K)}{\kappa _{n-j}}},$ or in other words $\mathrm {Vol} _{n}(K+tB)=\sum _{j=0}^{n}V_{j}(K)\,\mathrm {Vol} _{n-j}(tB_{n-j}).$
where $\kappa _{n-j}={\text{Vol}}_{n-j}(B_{n-j})$ is the volume of the $(n-j)$-dimensional unit ball.
Hadwiger's characterization theorem
Main article: Hadwiger's theorem
Hadwiger's theorem asserts that every valuation on convex bodies in $\mathbb {R} ^{n}$ that is continuous and invariant under rigid motions of $\mathbb {R} ^{n}$ is a linear combination of the quermassintegrals (or, equivalently, of the intrinsic volumes).[2]
Notes
1. McMullen, Peter (1991). "Inequalities between intrinsic volumes". Monatshefte für Mathematik. 111 (1): 47–53. doi:10.1007/bf01299276. MR 1089383.
2. Klain, Daniel A. (1995). "A short proof of Hadwiger's characterization theorem". Mathematika. 42 (2): 329–339. doi:10.1112/s0025579300014625. MR 1376731.
External links
Burago, Yu.D. (2001) [1994], "Mixed volume theory", Encyclopedia of Mathematics, EMS Press
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Decision tree model
In computational complexity the decision tree model is the model of computation in which an algorithm is considered to be basically a decision tree, i.e., a sequence of queries or tests that are done adaptively, so the outcome of previous tests can influence the tests performed next.
Typically, these tests have a small number of outcomes (such as a yes–no question) and can be performed quickly (say, with unit computational cost), so the worst-case time complexity of an algorithm in the decision tree model corresponds to the depth of the corresponding decision tree. This notion of computational complexity of a problem or an algorithm in the decision tree model is called its decision tree complexity or query complexity.
Decision trees models are instrumental in establishing lower bounds for complexity theory for certain classes of computational problems and algorithms. Several variants of decision tree models have been introduced, depending on the computational model and type of query algorithms are allowed to perform.
For example, a decision tree argument is used to show that a comparison sort of $n$ items must take $n\log(n)$ comparisons. For comparison sorts, a query is a comparison of two items $a,\,b$, with two outcomes (assuming no items are equal): either $a<b$ or $a>b$. Comparison sorts can be expressed as a decision tree in this model, since such sorting algorithms only perform these types of queries.
Comparison trees and lower bounds for sorting
Decision trees are often employed to understand algorithms for sorting and other similar problems; this was first done by Ford and Johnson.[1]
For example, many sorting algorithms are comparison sorts, which means that they only gain information about an input sequence $x_{1},x_{2},\ldots ,x_{n}$ via local comparisons: testing whether $x_{i}<x_{j}$, $x_{i}=x_{j}$, or $x_{i}>x_{j}$. Assuming that the items to be sorted are all distinct and comparable, this can be rephrased as a yes-or-no question: is $x_{i}>x_{j}$?
These algorithms can be modeled as binary decision trees, where the queries are comparisons: an internal node corresponds to a query, and the node's children correspond to the next query when the answer to the question is yes or no. For leaf nodes, the output corresponds to a permutation $\pi $ that describes how the input sequence was scrambled from the fully ordered list of items. (The inverse of this permutation, $\pi ^{-1}$, re-orders the input sequence.)
One can show that comparison sorts must use $\Omega (n\log(n))$ comparisons through a simple argument: for an algorithm to be correct, it must be able to output every possible permutation of $n$ elements; otherwise, the algorithm would fail for that particular permutation as input. So, its corresponding decision tree must have at least as many leaves as permutations: $n!$ leaves. Any binary tree with at least $n!$ leaves has depth at least $\log _{2}(n!)=\Omega (n\log _{2}(n))$, so this is a lower bound on the run time of a comparison sorting algorithm. In this case, the existence of numerous comparison-sorting algorithms having this time complexity, such as mergesort and heapsort, demonstrates that the bound is tight.[2]: 91
This argument does not use anything about the type of query, so it in fact proves a lower bound for any sorting algorithm that can be modeled as a binary decision tree. In essence, this is a rephrasing of the information-theoretic argument that a correct sorting algorithm must learn at least $\log _{2}(n!)$ bits of information about the input sequence. As a result, this also works for randomized decision trees as well.
Other decision tree lower bounds do use that the query is a comparison. For example, consider the task of only using comparisons to find the smallest number among $n$ numbers. Before the smallest number can be determined, every number except the smallest must "lose" (compare greater) in at least one comparison. So, it takes at least $n-1$ comparisons to find the minimum. (The information-theoretic argument here only gives a lower bound of $\log(n)$.) A similar argument works for general lower bounds for computing order statistics.[2]: 214
Linear and algebraic decision trees
Linear decision trees generalize the above comparison decision trees to computing functions that take real vectors $x\in \mathbb {R} ^{n}$ as input. The tests in linear decision trees are linear functions: for a particular choice of real numbers $a_{0},\dots ,a_{n}$, output the sign of $a_{0}+\textstyle \sum _{i=1}^{n}a_{i}x_{i}$. (Algorithms in this model can only depend on the sign of the output.) Comparison trees are linear decision trees, because the comparison between $x_{i}$ and $x_{j}$ corresponds to the linear function $x_{i}-x_{j}$. From its definition, linear decision trees can only specify functions $f$ whose fibers can be constructed by taking unions and intersections of half-spaces.
Algebraic decision trees are a generalization of linear decision trees that allow the test functions to be polynomials of degree $d$. Geometrically, the space is divided into semi-algebraic sets (a generalization of hyperplane).
These decision tree models, defined by Rabin[3] and Reingold,[4] are often used for proving lower bounds in computational geometry.[5] For example, Ben-Or showed that element uniqueness (the task of computing $f:\mathbb {R} ^{n}\to \{0,1\}$, where $f(x)$ is 0 if and only if there exist distinct coordinates $i,j$ such that $x_{i}=x_{j}$) requires an algebraic decision tree of depth $\Omega (n\log(n))$.[6] This was first showed for linear decision models by Dobkin and Lipton.[7] They also show a $n^{2}$ lower bound for linear decision trees on the knapsack problem, generalized to algebraic decision trees by Steele and Yao.[8]
Boolean decision tree complexities
For Boolean decision trees, the task is to compute the value of an n-bit Boolean function $f:\{0,1\}^{n}\to \{0,1\}$ for an input $x\in \{0,1\}^{n}$. The queries correspond to reading a bit of the input, $x_{i}$, and the output is $f(x)$. Each query may be dependent on previous queries. There are many types of computational models using decision trees that could be considered, admitting multiple complexity notions, called complexity measures.
Deterministic decision tree
If the output of a decision tree is $f(x)$, for all $x\in \{0,1\}^{n}$, the decision tree is said to "compute" $f$. The depth of a tree is the maximum number of queries that can happen before a leaf is reached and a result obtained. $D(f)$, the deterministic decision tree complexity of $f$ is the smallest depth among all deterministic decision trees that compute $f$.
Randomized decision tree
One way to define a randomized decision tree is to add additional nodes to the tree, each controlled by a probability $p_{i}$. Another equivalent definition is to define it as a distribution over deterministic decision trees. Based on this second definition, the complexity of the randomized tree is defined as the largest depth among all the trees in the support of the underlying distribution. $R_{2}(f)$ is defined as the complexity of the lowest-depth randomized decision tree whose result is $f(x)$ with probability at least $2/3$ for all $x\in \{0,1\}^{n}$ (i.e., with bounded 2-sided error).
$R_{2}(f)$ is known as the Monte Carlo randomized decision-tree complexity, because the result is allowed to be incorrect with bounded two-sided error. The Las Vegas decision-tree complexity $R_{0}(f)$ measures the expected depth of a decision tree that must be correct (i.e., has zero-error). There is also a one-sided bounded-error version which is denoted by $R_{1}(f)$.
Nondeterministic decision tree
The nondeterministic decision tree complexity of a function is known more commonly as the certificate complexity of that function. It measures the number of input bits that a nondeterministic algorithm would need to look at in order to evaluate the function with certainty.
Formally, the certificate complexity of $f$ at $x$ is the size of the smallest subset of indices $S\subset [n]$ such that, for all $y\in \{0,1\}^{n}$, if $y_{i}=x_{i}$ for all $i\in S$, then $f(y)=f(x)$. The certificate complexity of $f$ is the maximum certificate complexity over all $x$. The analogous notion where one only requires the verifier to be correct with 2/3 probability is denoted $RC(f)$.
Quantum decision tree
The quantum decision tree complexity $Q_{2}(f)$ is the depth of the lowest-depth quantum decision tree that gives the result $f(x)$ with probability at least $2/3$ for all $x\in \{0,1\}^{n}$. Another quantity, $Q_{E}(f)$, is defined as the depth of the lowest-depth quantum decision tree that gives the result $f(x)$ with probability 1 in all cases (i.e. computes $f$ exactly). $Q_{2}(f)$ and $Q_{E}(f)$ are more commonly known as quantum query complexities, because the direct definition of a quantum decision tree is more complicated than in the classical case. Similar to the randomized case, we define $Q_{0}(f)$ and $Q_{1}(f)$.
These notions are typically bounded by the notions of degree and approximate degree. The degree of $f$, denoted $\deg(f)$, is the smallest degree of any polynomial $p$ satisfying $f(x)=p(x)$ for all $x\in \{0,1\}^{n}$. The approximate degree of $f$, denoted ${\widetilde {\deg }}(f)$, is the smallest degree of any polynomial $p$ satisfying $p(x)\in [0,1/3]$ whenever $f(x)=0$ and $p(x)\in [2/3,1]$ whenever $f(x)=1$.
Beals et al. established that $Q_{0}(f)\geq \deg(f)/2$ and $Q_{2}(f)\geq {\widetilde {\deg }}(f)/2$.[9]
Relationships between boolean function complexity measures
It follows immediately from the definitions that for all $n$-bit Boolean functions $f$,$Q_{2}(f)\leq R_{2}(f)\leq R_{1}(f)\leq R_{0}(f)\leq D(f)\leq n$, and $Q_{2}(f)\leq Q_{0}(f)\leq D(f)\leq n$. Finding the best upper bounds in the converse direction is a major goal in the field of query complexity.
All of these types of query complexity are polynomially related. Blum and Impagliazzo,[10] Hartmanis and Hemachandra,[11] and Tardos[12] independently discovered that $D(f)\leq R_{0}(f)^{2}$. Noam Nisan found that the Monte Carlo randomized decision tree complexity is also polynomially related to deterministic decision tree complexity: $D(f)=O(R_{2}(f)^{3})$.[13] (Nisan also showed that $D(f)=O(R_{1}(f)^{2})$.) A tighter relationship is known between the Monte Carlo and Las Vegas models: $R_{0}(f)=O(R_{2}(f)^{2}\log R_{2}(f))$.[14] This relationship is optimal up to polylogarithmic factors.[15] As for quantum decision tree complexities, $D(f)=O(Q_{2}(f)^{4})$, and this bound is tight.[16][15] Midrijanis showed that $D(f)=O(Q_{0}(f)^{3})$,[17][18] improving a quartic bound due to Beals et al.[9]
It is important to note that these polynomial relationships are valid only for total Boolean functions. For partial Boolean functions, that have a domain a subset of $\{0,1\}^{n}$, an exponential separation between $Q_{0}(f)$ and $D(f)$ is possible; the first example of such a problem was discovered by Deutsch and Jozsa.
Sensitivity conjecture
For a Boolean function $f:\{0,1\}^{n}\to \{0,1\}$, the sensitivity of $f$ is defined to be the maximum sensitivity of $f$ over all $x$, where the sensitivity of $f$ at $x$ is the number of single-bit changes in $x$ that change the value of $f(x)$. Sensitivity is related to the notion of total influence from the analysis of Boolean functions, which is equal to average sensitivity over all $x$.
The sensitivity conjecture is the conjecture that sensitivity is polynomially related to query complexity; that is, there exists exponent $c,c'$ such that, for all $f$, $D(f)=O(s(f)^{c})$ and $s(f)=O(D(f)^{c'})$. One can show through a simple argument that $s(f)\leq D(f)$, so the conjecture is specifically concerned about finding a lower bound for sensitivity. Since all of the previously-discussed complexity measures are polynomially related, the precise type of complexity measure is not relevant. However, this is typically phrased as the question of relating sensitivity with block sensitivity.
The block sensitivity of $f$, denoted $bs(f)$, is defined to be the maximum block sensitivity of $f$ over all $x$. The block sensitivity of $f$ at $x$ is the maximum number $t$ of disjoint subsets $S_{1},\ldots ,S_{t}\subset [n]$ such that, for any of the subsets $S_{i}$, flipping the bits of $x$ corresponding to $S_{i}$ changes the value of $f(x)$.[13]
Since block sensitivity takes a maximum over more choices of subsets, $s(f)\leq bs(f)$. Further, block sensitivity is polynomially related to the previously discussed complexity measures; for example, Nisan's paper introducing block-sensitivity showed that $bs(f)\leq D(f)=O(bs(f)^{4})$.[13] So, one could rephrase the sensitivity conjecture as showing that, for some $c$, $bs(f)=O(s(f)^{c})$. In 1992, Nisan and Szegedy conjectured that $c=2$ suffices.[19] This would be tight, as Rubinstein in 1995 showed a quadratic separation between sensitivity and block sensitivity.[20]
In July 2019, 27 years after the conjecture was initially posed, Hao Huang from Emory University proved the sensitivity conjecture, showing that $bs(f)=O(s(f)^{4})$.[21] This proof is notably succinct, proving this statement in two pages when prior progress towards the sensitivity conjecture had been limited.[22][23]
Summary of known results
Best-known separations for complexity measures as of October 2020[16]
$D$ $R_{0}$ $R_{2}$ $C$ $RC$ $bs$ $s$ $Q_{0}$ $\deg $ $Q$ ${\widetilde {\deg }}$
$D$ 22, 322, 32, 33, 62, 32, 344
$R_{0}$ 1222, 32, 33, 62, 32, 33, 44
$R$ 1122, 32, 33, 61.5, 32, 33, 44
$C$ 111, 2222.22, 51.15, 31.63, 32, 42, 4
$RC$ 11111.5, 22, 41.15, 21.63, 222
$bs$ 111112, 41.15, 21.63, 222
$s$ 1111111.15, 21.63, 222
$Q_{0}$ 11.33, 21.33, 322, 32, 33, 62, 32, 44
$\deg $ 11.33, 21.33, 22222122
$Q$ 11122, 32, 33, 612, 34
${\widetilde {\deg }}$ 1112222111
This table summarizes results on separations between Boolean function complexity measures. The complexity measures are, in order, deterministic, zero-error randomized, two-sided-error randomized, certificate, randomized certificate, block sensitivity, sensitivity, exact quantum, degree, quantum, and approximate degree complexities.
The number in the $A$-th row and $B$-th column denotes bounds on the exponent $c$, which is the infimum of all $k$ satisfying $A(f)=O(B(f)^{k})$ for all boolean functions $f$. For example, the entry in the D-th row and s-th column is "3, 6", so $D(f)=O(\operatorname {s} (f)^{6+o(1)})$ for all $f$, and there exists a function $g$ such that $D(g)=\Omega (\operatorname {s} (g)^{3-o(1)})$.
See also
• Comparison sort
• Decision tree
• Aanderaa–Karp–Rosenberg conjecture
• Minimum spanning tree#Decision trees
References
1. Ford, Lester R. Jr.; Johnson, Selmer M. (1959-05-01). "A Tournament Problem". The American Mathematical Monthly. 66 (5): 387–389. doi:10.1080/00029890.1959.11989306. ISSN 0002-9890.
2. Introduction to algorithms. Cormen, Thomas H. (Third ed.). Cambridge, Mass.: MIT Press. 2009. ISBN 978-0-262-27083-0. OCLC 676697295.{{cite book}}: CS1 maint: others (link)
3. Rabin, Michael O. (1972-12-01). "Proving simultaneous positivity of linear forms". Journal of Computer and System Sciences. 6 (6): 639–650. doi:10.1016/S0022-0000(72)80034-5. ISSN 0022-0000.
4. Reingold, Edward M. (1972-10-01). "On the Optimality of Some Set Algorithms". Journal of the ACM. 19 (4): 649–659. doi:10.1145/321724.321730. ISSN 0004-5411. S2CID 18605212.
5. Preparata, Franco P. (1985). Computational geometry : an introduction. Shamos, Michael Ian. New York: Springer-Verlag. ISBN 0-387-96131-3. OCLC 11970840.
6. Ben-Or, Michael (1983-12-01). "Lower bounds for algebraic computation trees". Proceedings of the fifteenth annual ACM symposium on Theory of computing - STOC '83. STOC '83. New York, NY, USA: Association for Computing Machinery. pp. 80–86. doi:10.1145/800061.808735. ISBN 978-0-89791-099-6. S2CID 1499957.
7. Dobkin, David; Lipton, Richard J. (1976-06-01). "Multidimensional Searching Problems". SIAM Journal on Computing. 5 (2): 181–186. doi:10.1137/0205015. ISSN 0097-5397.
8. Michael Steele, J; Yao, Andrew C (1982-03-01). "Lower bounds for algebraic decision trees". Journal of Algorithms. 3 (1): 1–8. doi:10.1016/0196-6774(82)90002-5. ISSN 0196-6774.
9. Beals, R.; Buhrman, H.; Cleve, R.; Mosca, M.; de Wolf, R. (2001). "Quantum lower bounds by polynomials". Journal of the ACM. 48 (4): 778–797. arXiv:quant-ph/9802049. doi:10.1145/502090.502097. S2CID 1078168.
10. Blum, M.; Impagliazzo, R. (1987). "Generic oracles and oracle classes". Proceedings of 18th IEEE FOCS. pp. 118–126.
11. Hartmanis, J.; Hemachandra, L. (1987), "One-way functions, robustness, and non-isomorphism of NP-complete sets", Technical Report DCS TR86-796, Cornell University
12. Tardos, G. (1989). "Query complexity, or why is it difficult to separate NPA ∩ coNPA from PA by random oracles A?". Combinatorica. 9 (4): 385–392. doi:10.1007/BF02125350. S2CID 45372592.
13. Nisan, N. (1989). "CREW PRAMs and decision trees". Proceedings of 21st ACM STOC. pp. 327–335.
14. Kulkarni, R. and Tal, A. On Fractional Block Sensitivity. Electronic Colloquium on Computational Complexity (ECCC). Vol. 20. 2013.
15. Ambainis, Andris; Balodis, Kaspars; Belovs, Aleksandrs; Lee, Troy; Santha, Miklos; Smotrovs, Juris (2017-09-04). "Separations in Query Complexity Based on Pointer Functions". Journal of the ACM. 64 (5): 32:1–32:24. arXiv:1506.04719. doi:10.1145/3106234. ISSN 0004-5411. S2CID 10214557.
16. Aaronson, Scott; Ben-David, Shalev; Kothari, Robin; Rao, Shravas; Tal, Avishay (2020-10-23). "Degree vs. Approximate Degree and Quantum Implications of Huang's Sensitivity Theorem". arXiv:2010.12629 [quant-ph].
17. Midrijanis, Gatis (2004), "Exact quantum query complexity for total Boolean functions", arXiv:quant-ph/0403168
18. Midrijanis, Gatis (2005), "On Randomized and Quantum Query Complexities", arXiv:quant-ph/0501142
19. Nisan, Noam; Szegedy, Mario (1992-07-01). "On the degree of Boolean functions as real polynomials". Proceedings of the twenty-fourth annual ACM symposium on Theory of computing - STOC '92. STOC '92. Victoria, British Columbia, Canada: Association for Computing Machinery. pp. 462–467. doi:10.1145/129712.129757. ISBN 978-0-89791-511-3. S2CID 6919144.
20. Rubinstein, David (1995-06-01). "Sensitivity vs. block sensitivity of Boolean functions". Combinatorica. 15 (2): 297–299. doi:10.1007/BF01200762. ISSN 1439-6912. S2CID 41010711.
21. Huang, Hao (2019). "Induced subgraphs of hypercubes and a proof of the Sensitivity Conjecture". Annals of Mathematics. 190 (3): 949–955. arXiv:1907.00847. doi:10.4007/annals.2019.190.3.6. ISSN 0003-486X. JSTOR 10.4007/annals.2019.190.3.6. S2CID 195767594.
22. Klarreich, Erica. "Decades-Old Computer Science Conjecture Solved in Two Pages". Quanta Magazine. Retrieved 2019-07-26.
23. Hatami, Pooya; Kulkarni, Raghav; Pankratov, Denis (2011-06-22). "Variations on the Sensitivity Conjecture". Theory of Computing. 1: 1–27. doi:10.4086/toc.gs.2011.004. ISSN 1557-2862. S2CID 6918061.
Surveys
• Buhrman, Harry; de Wolf, Ronald (2002), "Complexity Measures and Decision Tree Complexity: A Survey" (PDF), Theoretical Computer Science, 288 (1): 21–43, doi:10.1016/S0304-3975(01)00144-X
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Brachistochrone curve
In physics and mathematics, a brachistochrone curve (from Ancient Greek βράχιστος χρόνος (brákhistos khrónos) 'shortest time'),[1] or curve of fastest descent, is the one lying on the plane between a point A and a lower point B, where B is not directly below A, on which a bead slides frictionlessly under the influence of a uniform gravitational field to a given end point in the shortest time. The problem was posed by Johann Bernoulli in 1696.
The brachistochrone curve is the same shape as the tautochrone curve; both are cycloids. However, the portion of the cycloid used for each of the two varies. More specifically, the brachistochrone can use up to a complete rotation of the cycloid (at the limit when A and B are at the same level), but always starts at a cusp. In contrast, the tautochrone problem can use only up to the first half rotation, and always ends at the horizontal.[2] The problem can be solved using tools from the calculus of variations[3] and optimal control.[4]
The curve is independent of both the mass of the test body and the local strength of gravity. Only a parameter is chosen so that the curve fits the starting point A and the ending point B.[5] If the body is given an initial velocity at A, or if friction is taken into account, then the curve that minimizes time differs from the tautochrone curve.
History
Johann Bernoulli posed the problem of the brachistochrone to the readers of Acta Eruditorum in June, 1696.[6][7] He said:
I, Johann Bernoulli, address the most brilliant mathematicians in the world. Nothing is more attractive to intelligent people than an honest, challenging problem, whose possible solution will bestow fame and remain as a lasting monument. Following the example set by Pascal, Fermat, etc., I hope to gain the gratitude of the whole scientific community by placing before the finest mathematicians of our time a problem which will test their methods and the strength of their intellect. If someone communicates to me the solution of the proposed problem, I shall publicly declare him worthy of praise
Bernoulli wrote the problem statement as:
Given two points A and B in a vertical plane, what is the curve traced out by a point acted on only by gravity, which starts at A and reaches B in the shortest time.
Johann and his brother Jakob Bernoulli derived the same solution, but Johann's derivation was incorrect, and he tried to pass off Jakob's solution as his own.[8] Johann published the solution in the journal in May of the following year, and noted that the solution is the same curve as Huygens's tautochrone curve. After deriving the differential equation for the curve by the method given below, he went on to show that it does yield a cycloid.[9][10] However, his proof is marred by his use of a single constant instead of the three constants, vm, 2g and D, below.
Bernoulli allowed six months for the solutions but none were received during this period. At the request of Leibniz, the time was publicly extended for a year and a half.[11] At 4 p.m. on 29 January 1697 when he arrived home from the Royal Mint, Isaac Newton found the challenge in a letter from Johann Bernoulli.[12] Newton stayed up all night to solve it and mailed the solution anonymously by the next post. Upon reading the solution, Bernoulli immediately recognized its author, exclaiming that he "recognizes a lion from his claw mark". This story gives some idea of Newton's power, since Johann Bernoulli took two weeks to solve it.[5][13] Newton also wrote, "I do not love to be dunned [pestered] and teased by foreigners about mathematical things...", and Newton had already solved Newton's minimal resistance problem, which is considered the first of the kind in calculus of variations.
In the end, five mathematicians responded with solutions: Newton, Jakob Bernoulli, Gottfried Leibniz, Ehrenfried Walther von Tschirnhaus and Guillaume de l'Hôpital. Four of the solutions (excluding l'Hôpital's) were published in the same edition of the journal as Johann Bernoulli's. In his paper, Jakob Bernoulli gave a proof of the condition for least time similar to that below before showing that its solution is a cycloid.[9] According to Newtonian scholar Tom Whiteside, in an attempt to outdo his brother, Jakob Bernoulli created a harder version of the brachistochrone problem. In solving it, he developed new methods that were refined by Leonhard Euler into what the latter called (in 1766) the calculus of variations. Joseph-Louis Lagrange did further work that resulted in modern infinitesimal calculus.
Earlier, in 1638, Galileo had tried to solve a similar problem for the path of the fastest descent from a point to a wall in his Two New Sciences. He draws the conclusion that the arc of a circle is faster than any number of its chords,[14]
From the preceding it is possible to infer that the quickest path of all [lationem omnium velocissimam], from one point to another, is not the shortest path, namely, a straight line, but the arc of a circle.
...
Consequently the nearer the inscribed polygon approaches a circle the shorter the time required for descent from A to C. What has been proven for the quadrant holds true also for smaller arcs; the reasoning is the same.
Just after Theorem 6 of Two New Sciences, Galileo warns of possible fallacies and the need for a "higher science". In this dialogue Galileo reviews his own work. Galileo studied the cycloid and gave it its name, but the connection between it and his problem had to wait for advances in mathematics.
Galileo’s conjecture is that “The shortest time of all [for a movable body] will be that of its fall along the arc ADB [of a quarter circle] and similar properties are to be understood as holding for all lesser arcs taken upward from the lowest limit B.”
In Fig.1, from the “Dialogue Concerning the Two Chief World Systems”, Galileo claims that the body sliding along the circular arc of a quarter circle, from A to B will reach B in less time than if it took any other path from A to B. Similarly, in Fig. 2, from any point D on the arc AB, he claims that the time along the lesser arc DB will be less than for any other path from D to B. In fact, the quickest path from A to B or from D to B, the brachistochrone, is a cycloidal arc, which is shown in Fig. 3 for the path from A to B, and Fig.4 for the path from D to B, superposed on the respective circular arc. [15]
Johann Bernoulli's solution
Introduction
In a letter to L’Hôpital, (21/12/1696), Bernoulli stated that when considering the problem of the curve of quickest descent, after only 2 days he noticed a curious affinity or connection with another no less remarkable problem leading to an ‘indirect method’ of solution. Then shortly afterwards he discovered a ‘direct method’. [16]
Direct method
In a letter to Henri Basnage, held at the University of Basel Public Library, dated 30 March 1697, Johann Bernoulli stated that he had found two methods (always referred to as "direct" and "indirect") to show that the Brachistochrone was the "common cycloid", also called the "roulette". Following advice from Leibniz, he included only the indirect method in the Acta Eruditorum Lipsidae of May 1697. He wrote that this was partly because he believed it was sufficient to convince anyone who doubted the conclusion, partly because it also resolved two famous problems in optics that "the late Mr. Huygens" had raised in his treatise on light. In the same letter he criticised Newton for concealing his method.
In addition to his indirect method he also published the five other replies to the problem that he received.
Johann Bernoulli's direct method is historically important as a proof that the brachistochrone is the cycloid. The method is to determine the curvature of the curve at each point. All the other proofs, including Newton's (which was not revealed at the time) are based on finding the gradient at each point.
In 1718, Bernoulli explained how he solved the brachistochrone problem by his direct method.[17][18]
He explained that he had not published it in 1697, for reasons that no longer applied in 1718. This paper was largely ignored until 1904 when the depth of the method was first appreciated by Constantin Carathéodory, who stated that it shows that the cycloid is the only possible curve of quickest descent. According to him, the other solutions simply implied that the time of descent is stationary for the cycloid, but not necessarily the minimum possible.
Analytic solution
A body is regarded as sliding along any small circular arc Ce between the radii KC and Ke, with centre K fixed. The first stage of the proof involves finding the particular circular arc, Mm, which the body traverses in the minimum time.
The line KNC intersects AL at N, and line Kne intersects it at n, and they make a small angle CKe at K. Let NK = a, and define a variable point, C on KN extended. Of all the possible circular arcs Ce, it is required to find the arc Mm, which requires the minimum time to slide between the 2 radii, KM and Km. To find Mm Bernoulli argues as follows.
Let MN = x. He defines m so that MD = mx, and n so that Mm = nx + na and notes that x is the only variable and that m is finite and n is infinitely small. The small time to travel along arc Mm is ${\frac {Mm}{MD^{\frac {1}{2}}}}={\frac {n(x+a)}{(mx)^{\frac {1}{2}}}}$, which has to be a minimum (‘un plus petit’). He does not explain that because Mm is so small the speed along it can be assumed to be the speed at M, which is as the square root of MD, the vertical distance of M below the horizontal line AL.
It follows that, when differentiated this must give
${\frac {(x-a)dx}{2x^{\frac {3}{2}}}}=0$ so that x = a.
This condition defines the curve that the body slides along in the shortest time possible. For each point, M on the curve, the radius of curvature, MK is cut in 2 equal parts by its axis AL. This property, which Bernoulli says had been known for a long time, is unique to the cycloid.
Finally, he considers the more general case where the speed is an arbitrary function X(x), so the time to be minimised is ${\frac {(x+a)}{X}}$. The minimum condition then becomes $X={\frac {(x+a)dX}{dx}}$ which he writes as :$X=(x+a)\Delta x$ and which gives MN (=x) as a function of NK (= a). From this the equation of the curve could be obtained from the integral calculus, though he does not demonstrate this.
Synthetic solution
He then proceeds with what he called his Synthetic Solution, which was a classical, geometrical proof, that there is only a single curve that a body can slide down in the minimum time, and that curve is the cycloid.
The reason for the synthetic demonstration, in the manner of the ancients, is to convince Mr de la Hire. He has little time for our new analysis, describing it as false (He claims he has found 3 ways to prove that the curve is a cubic parabola) – Letter from Johan Bernoulli to Pierre Varignon dated 27 Jul 1697. [19]
Assume AMmB is the part of the cycloid joining A to B, which the body slides down in the minimum time. Let ICcJ be part of a different curve joining A to B, which can be closer to AL than AMmB. If the arc Mm subtends the angle MKm at its centre of curvature, K, let the arc on IJ that subtends the same angle be Cc. The circular arc through C with centre K is Ce. Point D on AL is vertically above M. Join K to D and point H is where CG intersects KD, extended if necessary.
Let $\tau $ and t be the times the body takes to fall along Mm and Ce respectively.
$\tau \propto {\frac {Mm}{MD^{\frac {1}{2}}}}$, $t\propto {\frac {Ce}{CG^{\frac {1}{2}}}}$,
Extend CG to point F where, $CF={\frac {CH^{2}}{MD}}$ and since ${\frac {Mm}{Ce}}={\frac {MD}{CH}}$, it follows that
${\frac {\tau }{t}}={\frac {Mm}{Ce}}.\left({\frac {CG}{MD}}\right)^{\frac {1}{2}}=\left({\frac {CG}{CF}}\right)^{\frac {1}{2}}$
Since MN = NK, for the cycloid:
$GH={\frac {MD.HD}{DK}}={\frac {MD.CM}{MK}}$, $CH={\frac {MD.CK}{MK}}={\frac {MD.(MK+CM)}{MK}}$, and $CG=CH+GH={\frac {MD.(MK+2CM)}{MK}}$
If Ce is closer to K than Mm then
$CH={\frac {MD.(MK-CM)}{MK}}$ and $CG=CH-GH={\frac {MD.(MK-2CM)}{MK}}$
In either case,
$CF={\frac {CH^{2}}{MD}}>CG$, and it follows that $\tau <t$
If the arc, Cc subtended by the angle infinitesimal angle MKm on IJ is not circular, it must be greater than Ce, since Cec becomes a right-triangle in the limit as angle MKm approaches zero.
Note, Bernoulli proves that CF > CG by a similar but different argument.
From this he concludes that a body traverses the cycloid AMB in less time than any other curve ACB.
Indirect method
According to Fermat’s principle, the actual path between two points taken by a beam of light is one that takes the least time. In 1697 Johann Bernoulli used this principle to derive the brachistochrone curve by considering the trajectory of a beam of light in a medium where the speed of light increases following a constant vertical acceleration (that of gravity g).[20]
By the conservation of energy, the instantaneous speed of a body v after falling a height y in a uniform gravitational field is given by:
$v={\sqrt {2gy}}$,
The speed of motion of the body along an arbitrary curve does not depend on the horizontal displacement.
Bernoulli noted that the law of refraction gives a constant of the motion for a beam of light in a medium of variable density:
${\frac {\sin {\theta }}{v}}={\frac {1}{v}}{\frac {dx}{ds}}={\frac {1}{v_{m}}}$,
where vm is the constant and $\theta $ represents the angle of the trajectory with respect to the vertical.
The equations above lead to two conclusions:
1. At the onset, the angle must be zero when the particle speed is zero. Hence, the brachistochrone curve is tangent to the vertical at the origin.
2. The speed reaches a maximum value when the trajectory becomes horizontal and the angle θ = 90°.
Assuming for simplicity that the particle (or the beam) with coordinates (x,y) departs from the point (0,0) and reaches maximum speed after falling a vertical distance D:
$v_{m}={\sqrt {2gD}}$.
Rearranging terms in the law of refraction and squaring gives:
$v_{m}^{2}dx^{2}=v^{2}ds^{2}=v^{2}(dx^{2}+dy^{2})$
which can be solved for dx in terms of dy:
$dx={\frac {v\,dy}{\sqrt {v_{m}^{2}-v^{2}}}}$.
Substituting from the expressions for v and vm above gives:
$dx={\sqrt {\frac {y}{D-y}}}\,dy\,,$
which is the differential equation of an inverted cycloid generated by a circle of diameter D=2r, whose parametric equation is:
${\begin{aligned}x&=r(\varphi -\sin \varphi )\\y&=r(1-\cos \varphi ).\end{aligned}}$
where φ is a real parameter, corresponding to the angle through which the rolling circle has rotated. For given φ, the circle's centre lies at (x, y) = (rφ, r).
In the brachistochrone problem, the motion of the body is given by the time evolution of the parameter:
$\varphi (t)=\omega t\,,\omega ={\sqrt {\frac {g}{r}}}$
where t is the time since the release of the body from the point (0,0).
Jakob Bernoulli's solution
Johann's brother Jakob showed how 2nd differentials can be used to obtain the condition for least time. A modernized version of the proof is as follows. If we make a negligible deviation from the path of least time, then, for the differential triangle formed by the displacement along the path and the horizontal and vertical displacements,
$ds^{2}=dx^{2}+dy^{2}$.
On differentiation with dy fixed we get,
$2ds\ d^{2}s=2dx\ d^{2}x$.
And finally rearranging terms gives,
${\frac {dx}{ds}}d^{2}x=d^{2}s=v\ d^{2}t$
where the last part is the displacement for given change in time for 2nd differentials. Now consider the changes along the two neighboring paths in the figure below for which the horizontal separation between paths along the central line is d2x (the same for both the upper and lower differential triangles). Along the old and new paths, the parts that differ are,
$d^{2}t_{1}={\frac {1}{v_{1}}}{\frac {dx_{1}}{ds_{1}}}d^{2}x$
$d^{2}t_{2}={\frac {1}{v_{2}}}{\frac {dx_{2}}{ds_{2}}}d^{2}x$
For the path of least times these times are equal so for their difference we get,
$d^{2}t_{2}-d^{2}t_{1}=0={\bigg (}{\frac {1}{v_{2}}}{\frac {dx_{2}}{ds_{2}}}-{\frac {1}{v_{1}}}{\frac {dx_{1}}{ds_{1}}}{\bigg )}d^{2}x$
And the condition for least time is,
${\frac {1}{v_{2}}}{\frac {dx_{2}}{ds_{2}}}={\frac {1}{v_{1}}}{\frac {dx_{1}}{ds_{1}}}$
which agrees with Johann's assumption based on the law of refraction.
Newton's solution
Introduction
In June 1696, Johann Bernoulli had used the pages of the Acta Eruditorum Lipsidae to pose a challenge to the international mathematical community: to find the form of the curve joining two fixed points so that a mass will slide down along it, under the influence of gravity alone, in the minimum amount of time. The solution was originally to be submitted within six months. At the suggestion of Leibniz, Bernoulli extended the challenge until Easter 1697, by means of a printed text called "Programma", published in Groningen, in the Netherlands.
The Programma is dated 1 January 1697, in the Gregorian Calendar. This was 22 December 1696 in the Julian Calendar, in use in Britain. According to Newton's niece, Catherine Conduitt, Newton learned of the challenge at 4 pm on 29 January and had solved it by 4 am the following morning. His solution, communicated to the Royal Society, is dated 30 January. This solution, later published anonymously in the Philosophical Transactions, is correct but does not indicate the method by which Newton arrived at his conclusion. Bernoulli, writing to Henri Basnage in March 1697, indicated that even though its author, "by an excess of modesty", had not revealed his name, yet even from the scant details supplied it could be recognised as Newton's work, "as the lion by its claw" (in Latin, ex ungue Leonem).
D. T. Whiteside characteristically explains the origin of the Latin expression, originally from Greek, in considerable detail. The letter in French has ‘ex ungue Leonem’ preceded by the French word ‘comme’. The much quoted version ‘tanquam ex ungue Leonem’ is due to David Brewster’s book on the life and works of Newton in 1855. Bernoulli's intention was simply that he could tell the anonymous solution was Newton’s, just as it was possible to tell that an animal was a lion given its claw. It was not meant to suggest that Bernoulli considered Newton to be the lion among mathematicians, as it has since come to be interpreted. [21]
John Wallis, who was 80 years old at the time, had learned of the problem in September 1696 from Johann Bernoulli's youngest brother Hieronymus, and had spent three months attempting a solution before passing it in December to David Gregory, who also failed to solve it. After Newton had submitted his solution, Gregory asked him for the details and made notes from their conversation. These can be found in the University of Edinburgh Library, manuscript A $78^{1}$, dated 7 March 1697. Either Gregory did not understand Newton's argument, or Newton's explanation was very brief. However, it is possible, with a high degree of confidence, to construct Newton's proof from Gregory's notes, by analogy with his method to determine the solid of minimum resistance (Principia, Book 2, Proposition 34, Scholium 2). A detailed description of his solution of this latter problem is included in the draft of a letter in 1694, also to David Gregory.[22] In addition to the minimum time curve problem, there was a second problem that Newton also solved at the same time. Both solutions appeared anonymously in Philosophical Transactions of the Royal Society, for January 1697.
The Brachistochrone problem
Fig. 1, shows Gregory’s diagram (except the additional line IF is absent from it, and Z, the start point has been added). The curve ZVA is a cycloid and CHV is its generating circle. Since it appears that the body is moving upward from e to E, it must be assumed that a small body is released from Z and slides along the curve to A, without friction, under the action of gravity.
Consider a small arc eE, which the body is ascending. Assume that it traverses the straight line eL to point L, horizontally displaced from E by a small distance, o, instead of the arc eE. Note, that eL is not the tangent at e, and that o is negative when L is between B and E. Draw the line through E parallel to CH, cutting eL at n. From a property of the cycloid, En is the normal to the tangent at E, and similarly the tangent at E is parallel to VH.
Since the displacement EL is small, it differs little in direction from the tangent at E so that the angle EnL is close to a right-angle. In the limit as the arc eE approaches zero, eL becomes parallel to VH, provided o is small compared to eE making the triangles EnL and CHV similar.
Also en approaches the length of chord eE, and the increase in length, $eL-eE=nL={\frac {o.CH}{CV}}$, ignoring terms in $o^{2}$ and higher, which represent the error due to the approximation that eL and VH are parallel.
The speed along eE or eL can be taken as that at E, proportional to ${\sqrt {CB}}$, which is as CH, since $CH={\sqrt {CB.CV}}$
This appears to be all that Gregory’s note contains.
Let t be the additional time to reach L,
$t\propto {\frac {nL}{\sqrt {CB}}}={\frac {o.CH}{CV.{\sqrt {CB}}}}={\frac {o}{\sqrt {CV}}}$
Therefore, the increase in time to traverse a small arc displaced at one endpoint depends only on the displacement at the endpoint and is independent of the position of the arc. However, by Newton’s method, this is just the condition required for the curve to be traversed in the minimum time possible. Therefore, he concludes that the minimum curve must be the cycloid.
He argues as follows.
Assuming now that Fig. 1 is the minimum curve not yet determined, with vertical axis CV, and the circle CHV removed, and Fig. 2 shows part of the curve between the infinitesimal arc eE and a further infinitesimal arc Ff a finite distance along the curve. The extra time, t, to traverse eL (rather than eE) is nL divided by the speed at E (proportional to ${\sqrt {CB}}$), ignoring terms in $o^{2}$ and higher:
$t\propto {\frac {o.DE}{eE.{\sqrt {CB}}}}$,
At L the particle continues along a path LM, parallel to the original EF, to some arbitrary point M. As it has the same speed at L as at E, the time to traverse LM is the same as it would have been along the original curve EF. At M it returns to the original path at point f. By the same reasoning, the reduction in time, T, to reach f from M rather than from F is
$T\propto {\frac {o.FG}{Ff.{\sqrt {CI}}}}$
The difference (t – T) is the extra time it takes along the path eLMf compared to the original eEFf :
$(t-T)\propto \left({\frac {DE}{eE{\sqrt {CB}}}}-{\frac {FG}{Ff{\sqrt {CI}}}}\right).o$ plus terms in $o^{2}$ and higher (1)
Because eEFf is the minimum curve, (t – T) is must be greater than zero, whether o is positive or negative. It follows that the coefficient of o in (1) must be zero:
${\frac {DE}{eE{\sqrt {CB}}}}={\frac {FG}{Ff{\sqrt {CI}}}}$ (2) in the limit as eE and fF approach zero. Note since eEFf is the minimum curve it has to be assumed that the coefficient of $o^{2}$ is greater than zero.
Clearly there has to be 2 equal and opposite displacements, or the body would not return to the endpoint, A, of the curve.
If e is fixed, and if f is considered a variable point higher up the curve, then for all such points, f, ${\frac {FG}{Ff{\sqrt {CI}}}}$ is constant (equal to ${\frac {DE}{eE{\sqrt {CB}}}}$). By keeping f fixed and making e variable it is clear that ${\frac {DE}{eE{\sqrt {CB}}}}$ is also constant.
But, since points, e and f are arbitrary, equation (2) can be true only if ${\frac {DE}{eE{\sqrt {CB}}}}={\text{constant}}$, everywhere, and this condition characterises the curve that is sought. This is the same technique he uses to find the form of the Solid of Least Resistance.
For the cycloid, ${\frac {DE}{eE}}={\frac {BH}{VH}}={\frac {CH}{CV}}$ , so that ${\frac {DE}{eE{\sqrt {CB}}}}={\frac {CH}{CV.{\sqrt {CB}}}}$, which was shown above to be constant, and the Brachistochrone is the cycloid.
Newton gives no indication of how he discovered that the cycloid satisfied this last relation. It may have been by trial and error, or he may have recognised immediately that it implied the curve was the cycloid.
See also
• Aristotle's wheel paradox
• Beltrami identity
• Calculus of variations
• Catenary
• Cycloid
• Newton's minimal resistance problem
• Tautochrone curve
• Trochoid
• Uniformly accelerated motion
References
1. Chisholm, Hugh, ed. (1911). "Brachistochrone" . Encyclopædia Britannica (11th ed.). Cambridge University Press.
2. Stewart, James. "Section 10.1 - Curves Defined by Parametric Equations." Calculus: Early Transcendentals. 7th ed. Belmont, CA: Thomson Brooks/Cole, 2012. 640. Print.
3. Weisstein, Eric W. "Brachistochrone Problem". MathWorld.
4. Ross, I. M. The Brachistochrone Paradigm, in Primer on Pontryagin's Principle in Optimal Control, Collegiate Publishers, 2009. ISBN 978-0-9843571-0-9.
5. Hand, Louis N., and Janet D. Finch. "Chapter 2: Variational Calculus and Its Application to Mechanics." Analytical Mechanics. Cambridge: Cambridge UP, 1998. 45, 70. Print.
6. Johann Bernoulli (June 1696) "Problema novum ad cujus solutionem Mathematici invitantur." (A new problem to whose solution mathematicians are invited.), Acta Eruditorum, 18 : 269. From p. 269: "Datis in plano verticali duobus punctis A & B (vid Fig. 5) assignare Mobili M, viam AMB, per quam gravitate sua descendens & moveri incipiens a puncto A, brevissimo tempore perveniat ad alterum punctum B." (Given in a vertical plane two points A and B (see Figure 5), assign to the moving [body] M, the path AMB, by means of which — descending by its own weight and beginning to be moved [by gravity] from point A — it would arrive at the other point B in the shortest time.)
7. Solutions to Johann Bernoulli's problem of 1696:
• Isaac Newton (January 1697) "De ratione temporis quo grave labitur per rectam data duo puncta conjungentem, ad tempus brevissimum quo, vi gravitatis, transit ab horum uno ad alterum per arcum cycloidis" (On a proof [that] the time in which a weight slides by a line joining two given points [is] the shortest in terms of time when it passes, via gravitational force, from one of these [points] to the other through a cycloidal arc), Philosophical Transactions of the Royal Society of London, 19 : 424-425.
• G.G.L. (Gottfried Wilhelm Leibniz) (May 1697) "Communicatio suae pariter, duarumque alienarum ad edendum sibi primum a Dn. Jo. Bernoullio, deinde a Dn. Marchione Hospitalio communicatarum solutionum problematis curva celerrimi descensus a Dn. Jo. Bernoullio Geometris publice propositi, una cum solutione sua problematis alterius ab eodem postea propositi." (His communication together with [those] of two others in a report to him first from Johann Bernoulli, [and] then from the Marquis de l'Hôpital, of reported solutions of the problem of the curve of quickest descent, [which was] publicly proposed by Johann Bernoulli, geometer — one with a solution of his other problem proposed afterward by the same [person].), Acta Eruditorum, 19 : 201–205.
• Johann Bernoulli (May 1697) "Curvatura radii in diaphanis non uniformibus, Solutioque Problematis a se in Actis 1696, p. 269, propositi, de invenienda Linea Brachystochrona, id est, in qua grave a dato puncto ad datum punctum brevissimo tempore decurrit, & de curva Synchrona seu radiorum unda construenda." (The curvature of [light] rays in non-uniform media, and a solution of the problem [which was] proposed by me in the Acta Eruditorum of 1696, p. 269, from which is to be found the brachistochrone line [i.e., curve], that is, in which a weight descends from a given point to a given point in the shortest time, and on constructing the tautochrone or the wave of [light] rays.), Acta Eruditorum, 19 : 206–211.
• Jacob Bernoulli (May 1697) "Solutio problematum fraternorum, … " (A solution of [my] brother's problems, … ), Acta Eruditorum, 19 : 211–214.
• Marquis de l'Hôpital (May 1697) "Domini Marchionis Hospitalii solutio problematis de linea celerrimi descensus" (Lord Marquis de l'Hôpital's solution of the problem of the line of fastest descent), Acta Eruditorum, 19 : 217-220.
• reprinted: Isaac Newton (May 1697) "Excerpta ex Transactionibus Philos. Anglic. M. Jan. 1697." (Excerpt from the English Philosophical Transactions of the month of January in 1697), Acta Eruditorum, 19 : 223–224.
8. Livio, Mario (2003) [2002]. The Golden Ratio: The Story of Phi, the World's Most Astonishing Number (First trade paperback ed.). New York City: Broadway Books. p. 116. ISBN 0-7679-0816-3.
9. Struik, J. D. (1969), A Source Book in Mathematics, 1200-1800, Harvard University Press, ISBN 0-691-02397-2
10. Herman Erlichson (1999), "Johann Bernoulli's brachistochrone solution using Fermat's principle of least time", Eur. J. Phys., 20 (5): 299–304, Bibcode:1999EJPh...20..299E, doi:10.1088/0143-0807/20/5/301, S2CID 250741844
11. Sagan, Carl (2011). Cosmos. Random House Publishing Group. p. 94. ISBN 9780307800985. Retrieved 2 June 2016.
12. Katz, Victor J. (1998). A History of Mathematics: An Introduction (2nd ed.). Addison Wesley Longman. p. 547. ISBN 978-0-321-01618-8.
13. D.T. Whiteside, Newton the Mathematician, in Bechler, Contemporary Newtonian Research, p. 122.
14. Galileo Galilei (1638), "Third Day, Theorem 22, Prop. 36", Discourses regarding two new sciences, p. 239 This conclusion had appeared six years earlier in Galileo's Dialogue Concerning the Two Chief World Systems (Day 4).
15. Galilei, Galileo (1967). "Dialogue Concerning the Two Chief World Systems – Ptolemaic and Copernican translated by Stillman Drake, foreword by Albert Einstein " (Hardback ed.). University of California Press Berkeley and Los Angeles. p. 451. ISBN 0520004493.
16. Costabel, Pierre; Peiffer, Jeanne (1988). Der Briefwechsel von Johann I Bernoulli", Vol. II: "Der Briefwechsel mit Pierre Varignon, Erster Teil: 1692-1702" (Hardback ed.). Springer Basel Ag. p. 329. ISBN 978-3-0348-5068-1.
17. Bernoulli, Johann. Mémoires de l'Académie des Sciences (French Academy of Sciences) Vol. 3, 1718, pp. 135–138
18. The Early Period of the Calculus of Variations, by P. Freguglia and M. Giaquinta, pp. 53–57, ISBN 978-3-319-38945-5.
19. Costabel, Pierre; Peiffer, Jeanne (1988). "Der Briefwechsel von Johann I Bernoulli", Vol. II: "Der Briefwechsel mit Pierre Varignon, Erster Teil: 1692-1702" (Hardback ed.). Springer Basel Aktiengesellschaft. pp. 117–118. ISBN 978-3-0348-5068-1.
20. Babb, Jeff; Currie, James (July 2008), "The Brachistochrone Problem: Mathematics for a Broad Audience via a Large Context Problem" (PDF), The Montana Mathematics Enthusiast, 5 (2&3): 169–184, doi:10.54870/1551-3440.1099, S2CID 8923709, archived from the original (PDF) on 2011-07-27
21. Whiteside, Derek Thomas (2008). The Mathematical Papers of Isaac Newton Vol. 8 (Paperback ed.). Cambridge University Press. pp. 9–10, notes (21) and (22). ISBN 978-0-521-20103-2.
22. Dubois, Jacques (1991). "Chute d'une bille le long d'une gouttière cycloïdale; Tautochrone et brachistochrone; Propriétés et historique" (PDF). Bulletin de l'Union des Physiciens. 85 (737): 1251–1289.
External links
Wikimedia Commons has media related to Brachistochrone.
• "Brachistochrone", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Weisstein, Eric W. "Brachistochrone Problem". MathWorld.
• Brachistochrone ( at MathCurve, with excellent animated examples)
• The Brachistochrone, Whistler Alley Mathematics.
• Table IV from Bernoulli's article in Acta Eruditorum 1697
• Brachistochrones by Michael Trott and Brachistochrone Problem by Okay Arik, Wolfram Demonstrations Project.
• The Brachistochrone problem at MacTutor
• Geodesics Revisited — Introduction to geodesics including two ways of derivation of the equation of geodesic with brachistochrone as a special case of a geodesic.
• Optimal control solution to the Brachistochrone problem in Python.
• The straight line, the catenary, the brachistochrone, the circle, and Fermat Unified approach to some geodesics.
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Quillen's lemma
In algebra, Quillen's lemma states that an endomorphism of a simple module over the enveloping algebra of a finite-dimensional Lie algebra over a field k is algebraic over k. In contrast to a version of Schur's lemma due to Dixmier, it does not require k to be uncountable. Quillen's original short proof uses generic flatness.
References
• Quillen, D. (1969). "On the endomorphism ring of a simple module over an enveloping algebra". Proceedings of the American Mathematical Society. 21: 171–172. doi:10.1090/S0002-9939-1969-0238892-4.
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Plus construction
In mathematics, the plus construction is a method for simplifying the fundamental group of a space without changing its homology and cohomology groups.
Explicitly, if $X$ is a based connected CW complex and $P$ is a perfect normal subgroup of $\pi _{1}(X)$ then a map $f\colon X\to Y$ is called a +-construction relative to $P$ if $f$ induces an isomorphism on homology, and $P$ is the kernel of $\pi _{1}(X)\to \pi _{1}(Y)$.[1]
The plus construction was introduced by Michel Kervaire (1969), and was used by Daniel Quillen to define algebraic K-theory. Given a perfect normal subgroup of the fundamental group of a connected CW complex $X$, attach two-cells along loops in $X$ whose images in the fundamental group generate the subgroup. This operation generally changes the homology of the space, but these changes can be reversed by the addition of three-cells.
The most common application of the plus construction is in algebraic K-theory. If $R$ is a unital ring, we denote by $\operatorname {GL} _{n}(R)$ the group of invertible $n$-by-$n$ matrices with elements in $R$. $\operatorname {GL} _{n}(R)$ embeds in $\operatorname {GL} _{n+1}(R)$ by attaching a $1$ along the diagonal and $0$s elsewhere. The direct limit of these groups via these maps is denoted $\operatorname {GL} (R)$ and its classifying space is denoted $B\operatorname {GL} (R)$. The plus construction may then be applied to the perfect normal subgroup $E(R)$ of $\operatorname {GL} (R)=\pi _{1}(B\operatorname {GL} (R))$, generated by matrices which only differ from the identity matrix in one off-diagonal entry. For $n>0$, the $n$-th homotopy group of the resulting space, $B\operatorname {GL} (R)^{+}$, is isomorphic to the $n$-th $K$-group of $R$, that is,
$\pi _{n}\left(B\operatorname {GL} (R)^{+}\right)\cong K_{n}(R).$
See also
• Semi-s-cobordism
References
1. Charles Weibel, An introduction to algebraic K-theory IV, Definition 1.4.1
• Adams, J. Frank (1978), Infinite loop spaces, Princeton, N.J.: Princeton University Press, pp. 82–95, ISBN 0-691-08206-5
• Kervaire, Michel A. (1969), "Smooth homology spheres and their fundamental groups", Transactions of the American Mathematical Society, 144: 67–72, doi:10.2307/1995269, ISSN 0002-9947, MR 0253347
• Quillen, Daniel (1971), "The Spectrum of an Equivariant Cohomology Ring: I", Annals of Mathematics, Second Series, 94 (3): 549–572, doi:10.2307/1970770.
• Quillen, Daniel (1971), "The Spectrum of an Equivariant Cohomology Ring: II", Annals of Mathematics, Second Series, 94 (3): 573–602, doi:10.2307/1970771.
• Quillen, Daniel (1972), "On the cohomology and K-theory of the general linear groups over a finite field", Annals of Mathematics, Second Series, 96 (3): 552–586, doi:10.2307/1970825.
External links
• "Plus-construction", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
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Quillen–Suslin theorem
The Quillen–Suslin theorem, also known as Serre's problem or Serre's conjecture, is a theorem in commutative algebra concerning the relationship between free modules and projective modules over polynomial rings. In the geometric setting it is a statement about the triviality of vector bundles on affine space.
Quillen–Suslin theorem
FieldCommutative algebra
Conjectured byJean-Pierre Serre
Conjectured in1955
First proof byDaniel Quillen
Andrei Suslin
First proof in1976
The theorem states that every finitely generated projective module over a polynomial ring is free.
History
Background
Geometrically, finitely generated projective modules over the ring $R[x_{1},\dots ,x_{n}]$ correspond to vector bundles over affine space $\mathbb {A} _{R}^{n}$, where free modules correspond to trivial vector bundles. This correspondence (from modules to (algebraic) vector bundles) is given by the 'globalisation' or 'twiddlification' functor, sending $M\to {\widetilde {M}}$ (cite Hartshorne II.5, page 110). Affine space is topologically contractible, so it admits no non-trivial topological vector bundles. A simple argument using the exponential exact sequence and the d-bar Poincaré lemma shows that it also admits no non-trivial holomorphic vector bundles.
Jean-Pierre Serre, in his 1955 paper Faisceaux algébriques cohérents, remarked that the corresponding question was not known for algebraic vector bundles: "It is not known whether there exist projective A-modules of finite type which are not free."[1] Here $A$ is a polynomial ring over a field, that is, $A$ = $k[x_{1},\dots ,x_{n}]$.
To Serre's dismay, this problem quickly became known as Serre's conjecture. (Serre wrote, "I objected as often as I could [to the name]."[2]) The statement does not immediately follow from the proofs given in the topological or holomorphic case. These cases only guarantee that there is a continuous or holomorphic trivialization, not an algebraic trivialization.
Serre made some progress towards a solution in 1957 when he proved that every finitely generated projective module over a polynomial ring over a field was stably free, meaning that after forming its direct sum with a finitely generated free module, it became free. The problem remained open until 1976, when Daniel Quillen and Andrei Suslin independently proved the result. Quillen was awarded the Fields Medal in 1978 in part for his proof of the Serre conjecture. Leonid Vaseršteĭn later gave a simpler and much shorter proof of the theorem which can be found in Serge Lang's Algebra.
Generalization
A generalization relating projective modules over regular Noetherian rings A and their polynomial rings is known as the Bass–Quillen conjecture.
Note that although $GL_{n}$-bundles on affine space are all trivial, this is not true for G-bundles where G is a general reductive algebraic group.
Notes
1. "On ignore s'il existe des A-modules projectifs de type fini qui ne soient pas libres." Serre, FAC, p. 243.
2. Lam, p. 1
References
• Serre, Jean-Pierre (March 1955), "Faisceaux algébriques cohérents", Annals of Mathematics, Second Series, 61 (2): 197–278, doi:10.2307/1969915, JSTOR 1969915, MR 0068874
• Serre, Jean-Pierre (1958), "Modules projectifs et espaces fibrés à fibre vectorielle", Séminaire P. Dubreil, M.-L. Dubreil-Jacotin et C. Pisot, 1957/58, Fasc. 2, Exposé 23 (in French), MR 0177011
• Quillen, Daniel (1976), "Projective modules over polynomial rings", Inventiones Mathematicae, 36 (1): 167–171, doi:10.1007/BF01390008, MR 0427303
• Suslin, Andrei A. (1976), Проективные модули над кольцами многочленов свободны [Projective modules over polynomial rings are free], Doklady Akademii Nauk SSSR (in Russian), 229 (5): 1063–1066, MR 0469905. Translated in "Projective modules over polynomial rings are free", Soviet Mathematics, 17 (4): 1160–1164, 1976.
• Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, vol. 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR 1878556
An account of this topic is provided by:
• Lam, T. Y. (2006), Serre's problem on projective modules, Springer Monographs in Mathematics, Berlin; New York: Springer Science+Business Media, pp. 300pp., ISBN 978-3-540-23317-6, MR 2235330
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Quillen determinant line bundle
In mathematics, the Quillen determinant line bundle is a line bundle over the space of Cauchy–Riemann operators of a vector bundle over a Riemann surface, introduced by Quillen (1985). Quillen proved the existence of the Quillen metric on the determinant line bundle, a Hermitian metric defined using the analytic torsion of a family of differential operators.
See also
• Quillen metric
References
• Quillen, Daniel (1985), "Determinants of Cauchy-Riemann operators over a Riemann surface", Functional Analysis and Its Applications, Springer New York, 19: 31–34, doi:10.1007/BF01086022, ISSN 0016-2663, MR 0783704
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Quillen adjunction
In homotopy theory, a branch of mathematics, a Quillen adjunction between two closed model categories C and D is a special kind of adjunction between categories that induces an adjunction between the homotopy categories Ho(C) and Ho(D) via the total derived functor construction. Quillen adjunctions are named in honor of the mathematician Daniel Quillen.
Formal definition
Given two closed model categories C and D, a Quillen adjunction is a pair
(F, G): C $\leftrightarrows $ D
of adjoint functors with F left adjoint to G such that F preserves cofibrations and trivial cofibrations or, equivalently by the closed model axioms, such that G preserves fibrations and trivial fibrations. In such an adjunction F is called the left Quillen functor and G is called the right Quillen functor.
Properties
It is a consequence of the axioms that a left (right) Quillen functor preserves weak equivalences between cofibrant (fibrant) objects. The total derived functor theorem of Quillen says that the total left derived functor
LF: Ho(C) → Ho(D)
is a left adjoint to the total right derived functor
RG: Ho(D) → Ho(C).
This adjunction (LF, RG) is called the derived adjunction.
If (F, G) is a Quillen adjunction as above such that
F(c) → d
with c cofibrant and d fibrant is a weak equivalence in D if and only if
c → G(d)
is a weak equivalence in C then it is called a Quillen equivalence of the closed model categories C and D. In this case the derived adjunction is an adjoint equivalence of categories so that
LF(c) → d
is an isomorphism in Ho(D) if and only if
c → RG(d)
is an isomorphism in Ho(C).
References
• Goerss, Paul G. [in German]; Jardine, John F. (1999). Simplicial Homotopy Theory. Progress in Mathematics. Vol. 174. Basel, Boston, Berlin: Birkhäuser. ISBN 978-3-7643-6064-1.
• Philip S. Hirschhorn, Model Categories and Their Localizations, American Mathematical Soc., Aug 24, 2009 - Mathematics - 457 pages
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Quillen metric
In mathematics, and especially differential geometry, the Quillen metric is a metric on the determinant line bundle of a family of operators. It was introduced by Daniel Quillen[1] for certain elliptic operators over a Riemann surface, and generalized to higher-dimensional manifolds by Jean-Michel Bismut and Dan Freed.[2]
The Quillen metric was used by Quillen to give a differential-geometric interpretation of the ample line bundle over the moduli space of vector bundles on a compact Riemann surface, known as the Quillen determinant line bundle. It can be seen as defining the Chern–Weil representative of the first Chern class of this ample line bundle. The Quillen metric construction and its generalizations were used by Bismut and Freed to compute the holonomy of certain determinant line bundles of Dirac operators, and this holonomy is associated to certain anomaly cancellations in Chern–Simons theory predicted by Edward Witten.[3][4]
The Quillen metric was also used by Simon Donaldson in 1987 in a new inductive proof of the Hitchin–Kobayashi correspondence for projective algebraic manifolds, published one year after the resolution of the correspondence by Shing-Tung Yau and Karen Uhlenbeck for arbitrary compact Kähler manifolds.[5]
Determinant line bundle of a family of operators
Suppose $D_{t}$ are a family of Fredholm operators $D_{t}:V\to W$ between Hilbert spaces, varying continuously with respect to $t\in X$ for some topological space $X$. Since each of these operators is Fredholm, the kernel and cokernel are finite-dimensional. Thus there are assignments
$t\mapsto \ker D_{t},\quad t\mapsto {\text{coker}}D_{t}$
which define families of vector spaces over $X$. Despite the assumption that the operators $D_{t}$ vary continuously in $t$, these assignments of vector spaces do not form vector bundles over the topological space $X$, because the dimension of the kernel and cokernel may jump discontinuously for a family of differential operators. However, the index of a differential operator, the dimension of the kernel subtracted by the dimension of the cokernel, is an invariant up to continuous deformations. That is, the assignment
$t\mapsto {\text{ind}}(D_{t}):=\dim \ker D_{t}-\dim {\text{coker}}D_{t}$
is a constant function on $X$. Since it is not possible to take a difference of vector bundles, it is not possible to combine the families of kernels and cokernels of $D_{t}$ into a vector bundle. However, in the K-theory of $X$, formal differences of vector bundles may be taken, and associated to the family $D_{t}$ is an element
${\text{ind}}(D_{t})=[t\mapsto \ker D_{t}-{\text{coker}}D_{t}]\in K(X).$
This virtual index bundle contains information about the analytical properties of the family $D_{t}$, and its virtual rank, the difference of dimensions, may be computed using the Atiyah–Singer index theorem, provided the operators $D_{t}$ are elliptic differential operators.
Whilst the virtual index bundle is not a genuine vector bundle over the parameter space $X$, it is possible to pass to a genuine line bundle constructed out of ${\text{ind}}(D_{t})$. For any $t$, the determinant line of $D_{t}:V\to W$ is defined as the one-dimensional vector space
$\det D_{t}:=\left(\Lambda ^{\dim {\text{coker}}D_{t}}{\text{coker}}D_{t}\right)^{*}\otimes \Lambda ^{\dim \ker D_{t}}\ker D_{t}.$
One defines the determinant line bundle of the family $D_{t}$ as the fibrewise determinant of the virtual index bundle,
${\mathcal {L}}=\det {\text{ind}}(D_{t})$
which over each $t\in X$ has fibre given by the determinant line $\det D_{t}$.[6] This genuine line bundle over the topological space $X$ has the same first Chern class as the virtual index bundle, and this may be computed from the index theorem.
Quillen metric
The Quillen metric was introduced by Quillen, and is a Hermitian metric on the determinant line bundle of a certain family of differential operators parametrised by the space of unitary connections on a complex vector bundle over a compact Riemann surface. In this section the construction is sketched.
Given a Fredholm operator $D:V\to W$ between complex Hilbert spaces, one naturally obtains Hermitian inner products on the finite-dimensional vector spaces $\ker D$ and ${\text{coker}}D$ by restriction. These combine to give a Hermitian inner product, $h$ say, on the determinant line $\det D$, a one-dimensional complex vector space. However, when one has a family $D_{t}$ of such operators parametrised by a smooth manifold $X$, the assignment $t\mapsto h_{t}$ of Hermitian inner products on each fibre of the determinant line bundle ${\mathcal {L}}$ does not define a smooth Hermitian metric. Indeed, in this setting care needs to be taken that the line bundle ${\mathcal {L}}$ is in fact a smooth line bundle, and Quillen showed that one can construct a smooth trivialisation of ${\mathcal {L}}$.[1]
The natural Hermitian metrics $h_{t}$ may develop singular behaviour whenever the eigenvalues $\lambda $ of the Laplacian operators $D_{t}^{*}D_{t}$ cross or become equal, combining smaller eigenspaces into larger eigenspaces. In order to cancel out this singular behaviour, one must regularise the Hermitian metric $h$ by multiplying by an infinite determinant
$\Pi \lambda =\exp(-\zeta '(0))$
where $\zeta (s)$ is the zeta function operator of the Laplacian $D_{t}^{*}D_{t}$, defined by as the meromorphic continuation to $s=0$ of
$\zeta (s)=\sum _{\lambda }\lambda ^{-s}$
which is defined for ${\text{Re}}(s)>1$. This zeta function and infinite determinant is intimately related to the analytic torsion of the Laplacian $D_{t}^{*}D_{t}$. In the general setting studied by Bismut and Freed, some care needs to be taken in the definition of this infinite determinant, which is defined in terms of a supertrace.
Quillen considered the affine space ${\mathcal {A}}$ of unitary connections on a smooth complex vector bundle $E\to \Sigma $ over a compact Riemann surface, and the family of differential operators ${\bar {\partial }}_{A}:L_{1}^{2}(E)\to L^{2}(\Omega ^{0,1}(E))$, the Dolbeault operators of the Chern connections $A\in {\mathcal {A}}$, acting between Sobolev spaces of sections of $E$, which are Hilbert spaces. Each operator ${\bar {\partial }}_{A}$ is elliptic, and so by elliptic regularity its kernel consists of smooth sections of $E$. Indeed $\ker {\bar {\partial }}_{A}$ consists of the holomorphic sections of $E$ with respect to the holomorphic structure induced by the Dolbeault operator ${\bar {\partial }}_{A}$. Quillen's construction produces a metric on the determinant line bundle of this family, ${\mathcal {L}}\to {\mathcal {A}}$, and Quillen showed that the curvature form of the Chern connection associated to the Quillen metric is given by the Atiyah–Bott symplectic form on the space of unitary connections, previously discovered by Michael Atiyah and Raoul Bott in their study of the Yang–Mills equations over Riemann surfaces.[7]
Curvature
Associated to the Quillen metric and its generalised construction by Bismut and Freed is a unitary connection, and to this unitary connection is associated its curvature form. The associated cohomology class of this curvature form is predicted by the families version of the Atiyah–Singer index theorem, and the agreement of this prediction with the curvature form was proven by Bismut and Freed.[3] In the setting of Riemann surfaces studied by Quillen, this curvature is shown to be given by
$\Omega _{A}(a,b)=\int _{\Sigma }{\text{trace}}(a\wedge b)$
where $A\in {\mathcal {A}}$ is a unitary connection and $a,b\in \Omega ^{1}({\text{End}}(E))$ are tangent vectors to ${\mathcal {A}}$ at $A$. This symplectic form is the Atiyah–Bott symplectic form first discovered by Atiyah and Bott. Using this symplectic form, Atiyah and Bott demonstrated that the Narasimhan–Seshadri theorem could be interpreted as an infinite-dimensional version of the Kempf–Ness theorem from geometric invariant theory, and in this setting the Quillen metric plays the role of the Kähler metric which allows the symplectic reduction of ${\mathcal {A}}$ to be taken.
In Donaldson's new proof of the Hitchin–Kobayashi correspondence for projective algebraic manifolds, he explained how to construct a determinant line bundle over the space of unitary connections on a vector bundle over an arbitrary algebraic manifold which has the higher-dimensional Atiyah–Bott symplectic form as its curvature:[5]
$\Omega _{A}(a,b)=\int _{M}{\text{trace}}(a\wedge b)\wedge \omega ^{n-1}$
where $(M,\omega )$ is a projective algebraic manifold. This construction was used by Donaldson in an inductive proof of the correspondence.
Generalisations and alternate notions
The Quillen metric is primarily considered in the study of holomorphic vector bundles over Riemann surfaces or higher dimensional complex manifolds, and in Bismut and Freeds generalisation to the study of families of elliptic operators. In the study of moduli spaces of algebraic varieties and complex manifolds, it is possible to construct determinant line bundles on the space of almost-complex structures on a fixed smooth manifold $(M,\omega )$ which induce a Kähler structure with form $\omega $.[8][9] Just as the Quillen metric for vector bundles was related to the stability of vector bundles in the work of Atiyah and Bott and Donaldson, one may relate the Quillen metric for the determinant bundle for manifolds to the stability theory of manifolds. Indeed, the K-energy functional defined by Toshiki Mabuchi, which has critical points given by constant scalar curvature Kähler metrics, can be interpreted as the log-norm functional for a Quillen metric on the space of Kähler metrics.
References
1. Quillen, D. (1985), "Determinants of Cauchy-Riemann operators over a Riemann surface", Functional Analysis and Its Applications, 19 (1): 31–34, doi:10.1007/BF01086022, MR 0783704, S2CID 122340883
2. Bismut, Jean-Michel; Freed, Daniel S. (1986), "The analysis of elliptic families. I. Metrics and connections on determinant bundles.", Comm. Math. Phys., 106 (1): 159–176, doi:10.1007/BF01210930, MR 0853982, S2CID 55389271
3. Bismut, J.M. and Freed, D.S., 1986. The analysis of elliptic families. II. Dirac operators, eta invariants, and the holonomy theorem. Communications in mathematical physics, 107(1), pp.103-163.
4. Witten, E., 1985. Global gravitational anomalies. Communications in Mathematical Physics, 100(2), pp.197-229.
5. Donaldson, S.K., 1987. Infinite determinants, stable bundles and curvature. Duke Mathematical Journal, 54(1), pp.231-247.
6. Freed, D.S., 1987. On determinant line bundles. Mathematical aspects of string theory, 1, pp.189-238.
7. Atiyah, M.F. and Bott, R., 1983. The yang-mills equations over riemann surfaces. Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, 308(1505), pp.523-615.
8. Thomas, R.P., 2005. Notes on GIT and symplectic reduction for bundles and varieties. Surveys in Differential Geometry, 10(1), pp.221-273.
9. Werner Müller, Katrin Wendland. Extremal Kaehler metrics and Ray-Singer analytic torsion. Geometric Aspects of Partial Differential Equations, Contemp. Math. 242 (1999), pp. 135-160. math.DG/9904048
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Quillen spectral sequence
In the area of mathematics known as K-theory, the Quillen spectral sequence, also called the Brown–Gersten–Quillen or BGQ spectral sequence (named after Kenneth Brown, Stephen Gersten, and Daniel Quillen), is a spectral sequence converging to the sheaf cohomology of a type of topological space that occurs in algebraic geometry.[1][2] It is used in calculating the homotopy properties of a simplicial group.
References
1. Srinivas, Vasudevan (2013). Algebraic K-Theory. Springer Science & Business Media. ISBN 9781489967350.
2. Friedlander, Eric; Grayson, Daniel R. (2005). Handbook of K-Theory. Springer Science & Business Media. ISBN 9783540230199.
• Quillen, Daniel (1973). "Higher algebraic K-theory: I". Algebraic K-Theory I. Proceedings of the Conference Held at the Seattle Research Center of Battelle Memorial Institute, August 28 - September 8, 1972. Springer-Verlag. pp. 85–147.
• Brown, Kenneth S.; Gersten, Stephen M. (1973). "Algebraic K-theory as generalized sheaf cohomology". Algebraic K-theory, I: Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972). Lecture Notes in Math. Vol. 341. Berlin: Springer. pp. 266–292. MR 0347943.
External links
• A spectral sequence of Quillen at the Stacks Project
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Quillen's theorems A and B
In topology, a branch of mathematics, Quillen's Theorem A gives a sufficient condition for the classifying spaces of two categories to be homotopy equivalent. Quillen's Theorem B gives a sufficient condition for a square consisting of classifying spaces of categories to be homotopy Cartesian. The two theorems play central roles in Quillen's Q-construction in algebraic K-theory and are named after Daniel Quillen.
The precise statements of the theorems are as follows.[1]
Quillen's Theorem A — If $f:C\to D$ is a functor such that the classifying space $B(d\downarrow f)$ of the comma category $d\downarrow f$ is contractible for any object d in D, then f induces a homotopy equivalence $BC\to BD$.
Quillen's Theorem B — If $f:C\to D$ is a functor that induces a homotopy equivalence $B(d'\downarrow f)\to B(d\downarrow f)$ for any morphism $d\to d'$ in D, then there is an induced long exact sequence:
$\cdots \to \pi _{i+1}BD\to \pi _{i}B(d\downarrow f)\to \pi _{i}BC\to \pi _{i}BD\to \cdots .$
In general, the homotopy fiber of $Bf:BC\to BD$ is not naturally the classifying space of a category: there is no natural category $Ff$ such that $FBf=BFf$. Theorem B constructs $Ff$ in a case when $f$ is especially nice.
References
1. Weibel 2013, Ch. IV. Theorem 3.7 and Theorem 3.8
• Ara, Dimitri; Maltsiniotis, Georges (April 2018). "Un théorème A de Quillen pour les ∞-catégories strictes I : La preuve simpliciale". Advances in Mathematics. 328: 446–500. arXiv:1703.04689. doi:10.1016/j.aim.2018.01.018.
• Quillen, Daniel (1973), "Higher algebraic K-theory. I", Algebraic K-theory, I: Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), Lecture Notes in Math, vol. 341, Berlin, New York: Springer-Verlag, pp. 85–147, doi:10.1007/BFb0067053, ISBN 978-3-540-06434-3, MR 0338129
• Srinivas, V. (2008), Algebraic K-theory, Modern Birkhäuser Classics (Paperback reprint of the 1996 2nd ed.), Boston, MA: Birkhäuser, ISBN 978-0-8176-4736-0, Zbl 1125.19300
• Weibel, Charles (2013). The K-book: an introduction to algebraic K-theory. Graduate Studies in Math. Vol. 145. AMS. ISBN 978-0-8218-9132-2.
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Quincunx
A quincunx (/ˈkwɪn.kʌŋks/) is a geometric pattern consisting of five points arranged in a cross, with four of them forming a square or rectangle and a fifth at its center.[1] The same pattern has other names, including "in saltire" or "in cross" in heraldry (depending on the orientation of the outer square), the five-point stencil in numerical analysis, and the five dots tattoo. It forms the arrangement of five units in the pattern corresponding to the five-spot on six-sided dice, playing cards, and dominoes. It is represented in Unicode as U+2059 ⁙ FIVE DOT PUNCTUATION or (for the die pattern) U+2684 ⚄ DIE FACE-5.
Historical origins of the name
The quincunx was originally a coin issued by the Roman Republic c. 211–200 BC, whose value was five twelfths (quinque and uncia) of an as, the Roman standard bronze coin. On the Roman quincunx coins, the value was sometimes indicated by a pattern of five dots or pellets. However, these dots were not always arranged in a quincunx pattern.
The Oxford English Dictionary (OED) dates the first appearances of the Latin word in English as 1545 and 1574 ("in the sense 'five-twelfths of a pound or as'"; i.e. 100 old pence). The first citation for a geometric meaning, as "a pattern used for planting trees", dates from 1606. The OED also cites a 1647 reference to the German astronomer Kepler for an astronomical/astrological meaning, an angle of 5/12 of a whole circle. When used to describe a tree-planting pattern, the same word can also refer to groups of more than five trees, arranged in a square grid but aligned diagonally to the dimensions of the surrounding plot of land; however, this article considers only five-point patterns and not their extension to larger square grids.
Examples
Quincunx patterns occur in many contexts:
• In heraldry, groups of five elements (charges) are often arranged in a quincunx pattern. This arrangement is called, in heraldic terminology, in saltire for its usual orientation with the sides of the square vertical or horizontal, or in cross when the square is diagonally oriented.. The flag of the Solomon Islands features this pattern, with its five stars representing the five main island groups in the Solomon Islands. Another instance of this pattern occurred in the flag of the 19th-century Republic of Yucatán, where it signified the five departments into which the republic was divided. The coat of arms of Portugal includes both orientations of the same patterns, nested within each other.[2]
• Quincunxes are used in modern computer graphics as a pattern for multisample anti-aliasing. Quincunx antialiasing samples scenes at the corners and centers of each pixel. These five sample points, in the shape of a quincunx, are combined to produce each displayed pixel. However, samples at the corner points are shared with adjacent pixels, so the number of samples needed is only twice the number of displayed pixels.[3]
• In numerical analysis, the quincunx pattern describes the two-dimensional five-point stencil, a sampling pattern used to derive finite difference approximations to derivatives. The five points of the five-point stencil are arranged directly above, below, and to the two sides of the center point, rather than (as in quincunx sampling) diagonally with respect to it.[4]
• In Khmer architecture, the towers of a temple, such as Angkor Wat, are sometimes arranged in a quincunx to represent the five peaks of Mount Meru.[5]
• A quincunx is one of the quintessential designs of Cosmatesque inlay stonework.[6]
• A quincuncial map is a conformal map projection that maps the poles of the sphere to the centre and four corners of a square, thus forming a quincunx.
• The points on each face of a unit cell of a face-centred cubic lattice form a quincunx.
• The quincunx as a tattoo is known as the five dots tattoo. It has been variously interpreted as a fertility symbol,[7] a reminder of sayings on how to treat women or police,[8] a recognition symbol among the Romani people,[8] a group of close friends,[9] standing alone in the world,[10] or time spent in prison (with the outer four dots representing the prison walls and the inner dot representing the prisoner).[11] Thomas Edison, whose many inventions included an electric pen which later became the basis of a tattooing machine created by Samuel O'Reilly, had this pattern tattooed on his forearm.[12]
• The first two stages of the Saturn V super heavy-lift rocket had engines in a quincunx arrangement.[13]
• A baseball diamond forms a quincunx with the four bases and the pitcher's mound.[14]
Literary and symbolic references
Various literary works use or refer to the quincunx pattern:
• Quincunx (1564) was the name of the political treatise by a Polish-Ruthenian writer Stanisław Orzechowski: here, the five points symbolized five pillars of Polish state, with the Church at the very top.[15]
• The Garden of Cyrus, or The Quincuncial Lozenge, or Network Plantations of the Ancients, naturally, artificially, mystically considered, is an essay by Sir Thomas Browne, published in 1658. Browne elaborates upon evidence of the quincunx pattern in art, nature and mystically as evidence of "the wisdom of God". Although writing about the quincunx in its geometric meaning, he may have been influenced by English astrology, as the astrological meaning of "quincunx" (unrelated to the pattern) was introduced by the astronomer Kepler in 1604. The Victorian critic Edmund Gosse complained that "gathering his forces it is Quincunx, Quincunx, all the way until the very sky itself is darkened with revolving Chess-boards", while conceding that "this radically bad book contains some of the most lovely paragraphs which passed from an English pen during the seventeenth Century".[16]
• James Joyce uses the term in "Grace", a short story in Dubliners of 1914, to describe the seating arrangement of five men in a church service. Lobner[17] argues that in this context the pattern serves as a symbol both of the wounds of Christ and of the Greek cross.
• Lawrence Durrell's novel sequence The Avignon Quintet is arranged in the form of a quincunx, according to the author; the final novel in the sequence is called Quinx, the plot of which includes the discovery of a quincunx of stones.[18]
• The Quincunx is the title of a lengthy and elaborate novel by Charles Palliser set in 19th-century England, published in 1989; the pattern appears in the text as a heraldic device, and is also reflected in the structure of the book.[19]
• In the first chapter of The Rings of Saturn, W. G. Sebald's narrator cites Browne's writing on the quincunx. The quincunx in turn becomes a model for the way in which the rest of the novel unfolds.[20]
• Séamus Heaney describes Ireland's historical provinces as together forming a quincunx, as the Irish word for province cúige (literally: "fifth part") also explicates. The five provinces of Ireland were Ulster (north), Leinster (east), Connacht (west), Munster (south) and Meath (center, and now a county within Leinster). More specifically, in his essay Frontiers of Writing, Heaney creates an image of five towers forming a quincunx pattern within Ireland, one tower for each of the five provinces, each having literary significance.[21][22]
• Early African American scientist Benjamin Banneker describes a dream in which he is asked to measure the shape of the soul after death. The answer is "quincunx". Research locates his ancestry in Senegal, where the quincunx is a common religious symbol.[23]
References
Wikimedia Commons has media related to Quincunxes.
Look up quincunx in Wiktionary, the free dictionary.
1. Webster's Collegiate Dictionary, 10th ed., as quoted by Pajares-Ayuela (2001).
2. Fox-Davies, Arthur Charles (1915), The book of public arms : a complete encyclopædia of all royal, territorial, municipal, corporate, official, and impersonal arms, T.C. & E.C. Jack, p. 624
3. Chambers, Mike (February 27, 2001), "NVIDIA GeForce3 Preview", NV News, archived from the original on November 13, 2009.
4. Knabner, Peter; Angermann, Lutz (2003), "1.2 The Finite Difference Method", Numerical Methods for Elliptic and Parabolic Partial Differential Equations, Texts in Applied Mathematics, vol. 44, Springer-Verlag, pp. 21–29, ISBN 978-0-387-95449-3.
5. "Angkor Wat". earthobservatory.nasa.gov. 2004-12-25. Retrieved 2022-03-04.
6. Pajares-Ayuela, Paloma (2001), "The Signification — The Cosmatesque Quincunx: A Double-Cross Motif", Cosmatesque ornament: flat polychrome geometric patterns in architecture, W. W. Norton & Company, pp. 196–246, ISBN 978-0-393-73037-1.
7. Gilbert, Steve (2000), Tattoo history: a source book : an anthology of historical records of tattooing throughout the world, Juno Books, p. 153, ISBN 978-1-890451-06-6.
8. Turner, Robert (2005), Kishkindha, Osiris Press Ltd, p. 53, ISBN 978-1-905315-05-5.
9. Daye, Douglas D. (1997), A law enforcement sourcebook of Asian crime and cultures: tactics and mindsets, CRC Press, p. 113, ISBN 978-0-8493-8116-4.
10. Vigil, James Diego (2002), A rainbow of gangs: street cultures in the mega-city, University of Texas Press, p. 115, ISBN 978-0-292-78749-0.
11. Baldayev, Danzig (2006), Russian criminal tattoo encyclopedia, Volume 3, FUEL Publishing, p. 214.
12. Sherwood, Dane; Wood, Sandy; Kovalchik, Kara (2006), The Pocket Idiot's Guide to Not So Useless Facts, Penguin, p. 48, ISBN 978-1-59257-567-1.
13. Allday, Jonathan (2000), Apollo in Perspective: Spaceflight Then and Now, CRC Press, p. 77, ISBN 9780750306454, The engines were arranged across the base of the stage in the same pattern as the dots on a number 5 domino.
14. di Milo, Brisbane (2002), "Telepathic letter to Alfred Jarry", in Clements, Cal (ed.), Pataphysica, Writers Club Press, pp. 60–68, ISBN 9780595236046. See in particular p. 62.
15. Jerzy Ziomek: Renesans. Wyd. XI – 5 dodruk. Warszawa: Wydawnictwo Naukowe PWN, 2012, s. 201-204, seria: Wielka Historia Literatury Polskiej. ISBN 978-83-01-13843-1.
16. "That Vulcan gave Arrows unto Apollo and Diana" Aquarium of Vulcanblog; Sir Thomas Browne: A Study in Religious Philosophy, Dunn, William P., pp 126-129, 1950
17. Lobner, Corinna del Greco (1989), "Equivocation As Stylistic Device: Joyce's 'Grace' and Dante", Lectura Dantis, 4; for additional work on this instance of the quincunx pattern, see Duffy, Charles F. (1972), "The Seating Arrangement in 'Grace'", James Joyce Quarterly, 9: 487–489.
18. Gifford, James (1999), "Reading Orientalism and the Crisis of Epistemology in the Novels of Lawrence Durrell", CLCWeb: Comparative Literature and Culture, 1 (2), doi:10.7771/1481-4374.1036, the most dominant formal element expressing this state of multiplicity in The Avignon Quintet is its quincunx structure.
19. Onega, Susana (2000), "Mirror games and hidden narratives in The Quincunx", in Todd, Richard; Flora, Luisa (eds.), Theme Parks, Rainforests and Sprouting Wastelands: European Essays on Theory and Performance in Contemporary British Fiction, Costerus New Series, vol. 123, Rodopi, pp. 151–163, ISBN 9789042005020.
20. Horstkotte, Silke (2005), "The double dynamics of focalization in W. G. Sebald's The Rings of Saturn", in Meister, Jan Christoph (ed.), Narratology Beyond Literary Criticism: Mediality, Disciplinarity, Narratologia : contributions to narrative theory, vol. 6, Walter de Gruyter, pp. 25–44, ISBN 9783110183528.
21. Heaney, Séamus (1995), "Frontiers of Writing", The Redress of Poetry: Oxford Lectures, Faber and Faber, pp. 186–202.
22. Corcoran, Neil (1999), Poets of Modern Ireland, SIU Press, p. 62, ISBN 9780809322909.
23. Eglash, Ron (April 1997), "The African heritage of Benjamin Banneker", Social Studies of Science, 27 (2): 307–315, doi:10.1177/030631297027002004, JSTOR 285472, S2CID 143652183
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Wikipedia
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Quincunx matrix
In mathematics, the matrix
${\begin{pmatrix}1&-1\\1&1\end{pmatrix}}$
is sometimes called the quincunx matrix. It is a 2×2 Hadamard matrix, and its rows form the basis of a diagonal square lattice consisting of the integer points whose coordinates both have the same parity; this lattice is a two-dimensional analogue of the three-dimensional body-centered cubic lattice.[1]
See also
• Quincunx
Notes
1. Van De Ville, D.; Blu, T.; Unser, M. (2005), "On the multidimensional extension of the quincunx subsampling matrix" (PDF), IEEE Signal Processing Letters, 12 (2): 112–115, CiteSeerX 10.1.1.649.9329, doi:10.1109/LSP.2004.839697.
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Wikipedia
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Quine–Putnam indispensability argument
The Quine–Putnam indispensability argument[lower-alpha 1] is an argument in the philosophy of mathematics for the existence of abstract mathematical objects such as numbers and sets, a position known as mathematical platonism. It was named after the philosophers Willard Quine and Hilary Putnam, and is one of the most important arguments in the philosophy of mathematics.
Willard Quine
Hilary Putnam
Although elements of the indispensability argument may have originated with thinkers such as Gottlob Frege and Kurt Gödel, Quine's development of the argument was unique for introducing to it a number of his philosophical positions such as naturalism, confirmational holism, and the criterion of ontological commitment. Putnam gave Quine's argument its first detailed formulation in his 1971 book Philosophy of Logic. He later came to disagree with various aspects of Quine's thinking, however, and formulated his own indispensability argument based on the no miracles argument in the philosophy of science. A standard form of the argument in contemporary philosophy is credited to Mark Colyvan; whilst being influenced by both Quine and Putnam, it differs in important ways from their formulations. It is presented in the Stanford Encyclopedia of Philosophy:[2]
• We ought to have ontological commitment to all and only the entities that are indispensable to our best scientific theories.
• Mathematical entities are indispensable to our best scientific theories.
• Therefore, we ought to have ontological commitment to mathematical entities.
Nominalists, philosophers who reject the existence of abstract objects, have argued against both premises of this argument. An influential argument by Hartry Field claims that mathematical entities are dispensable to science. This argument has been supported by attempts to demonstrate that scientific and mathematical theories can be reformulated to remove all references to mathematical entities. Other philosophers, including Penelope Maddy, Mary Leng, Elliott Sober, and Joseph Melia, have argued that we do not need to believe in all of the entities that are indispensable to science. The arguments of these writers inspired a new explanatory version of the argument, which Alan Baker and Mark Colyvan support, that argues mathematics is indispensable to specific scientific explanations as well as whole theories.
Background
In his 1973 paper "Mathematical Truth", Paul Benacerraf raised a problem for the philosophy of mathematics.[lower-alpha 2] According to Benacerraf, mathematical sentences such as "two is a prime number" seem to imply the existence of mathematical objects.[5] He supported this claim with the idea that mathematics should not have its own special semantics, or in other words, the meaning of mathematical sentences should follow the same rules as non-mathematical sentences. For example, according to this reasoning, if the sentence "Mars is a planet" implies the existence of the planet Mars, then the sentence "two is a prime number" should also imply the existence of the number two.[6] But according to Benacerraf, if mathematical objects existed, they would be unknowable to us.[5] This is because mathematical objects, if they exist, are abstract objects; objects that cannot cause things to happen and that have no spatio-temporal location.[7] Benacerraf argued, on the basis of the causal theory of knowledge, that we would not be able to know about such objects because they cannot come into causal contact with us.[lower-alpha 3][8] This is called Benacerraf's epistemological problem because it concerns the epistemology of mathematics, that is, how we come to know what we do about mathematics.[9]
The philosophy of mathematics is split into two main strands; platonism and nominalism. Platonism holds that there exist abstract mathematical objects such as numbers and sets whilst nominalism denies their existence.[10] Each of these views faces issues due to the problem raised by Benacerraf. Because nominalism rejects the existence of mathematical objects, it faces no epistemological problem but it does face problems concerning the idea that mathematics should not have its own special semantics. Platonism does not face problems concerning the semantic half of the dilemma but it has difficulty explaining how we can have any knowledge about mathematical objects.[11]
The indispensability argument aims to overcome the epistemological problem posed against platonism by providing a justification for belief in abstract mathematical objects.[5] It is part of a broad class of indispensability arguments most commonly applied in the philosophy of mathematics, but which also includes arguments in the philosophy of language and ethics.[12] In the most general sense, indispensability arguments aim to support their conclusion based on the claim that the truth of the conclusion is indispensable or necessary for a certain purpose.[13] When applied in the field of ontology—the study of what exists—they exemplify a Quinean strategy for establishing the existence of controversial entities that cannot be directly investigated. According to this strategy, the indispensability of these entities for formulating a theory of other less-controversial entities counts as evidence for their existence.[14] In the case of philosophy of mathematics, the indispensability of mathematical entities for formulating scientific theories is taken as evidence for the existence of those mathematical entities.[15]
Overview of the argument
Mark Colyvan presents the argument in the Stanford Encyclopedia of Philosophy in the following form:[2]
• We ought to have ontological commitment to all and only the entities that are indispensable to our best scientific theories.
• Mathematical entities are indispensable to our best scientific theories.
• Therefore, we ought to have ontological commitment to mathematical entities.
Here, an ontological commitment to an entity is a commitment to believing that that entity exists.[16] The first premise is based on two fundamental assumptions; naturalism and confirmational holism. According to naturalism, we should look to our best scientific theories to determine what we have best reason to believe exists.[17] Quine summarized naturalism as "the recognition that it is within science itself, and not in some prior philosophy, that reality is to be identified and described".[18] Confirmational holism is the view that scientific theories cannot be confirmed in isolation and must be confirmed as wholes. Therefore, according to confirmational holism, if we should believe in science, then we should believe in all of science, including any of the mathematics that is assumed by our best scientific theories.[17] The argument is mainly aimed at nominalists that are scientific realists as it attempts to justify belief in mathematical entities in a manner similar to the justification for belief in theoretical entities such as electrons or quarks; Quine held that such nominalists have a "double standard" with regards to ontology.[2]
The indispensability argument differs from other arguments for platonism because it only argues for belief in the parts of mathematics that are indispensable to science. It does not necessarily justify belief in the most abstract parts of set theory, which Quine called "mathematical recreation … without ontological rights".[19] Some philosophers infer from the argument that mathematical knowledge is a posteriori because it implies mathematical truths can only be established via the empirical confirmation of scientific theories to which they are indispensable. This also indicates mathematical truths are contingent since empirically known truths are generally contingent. Such a position is controversial because it contradicts the traditional view of mathematical knowledge as a priori knowledge of necessary truths.[20]
Whilst Quine's original argument is an argument for platonism, indispensability arguments can also be constructed to argue for the weaker claim of sentence realism—the claim that mathematical theory is objectively true. This is a weaker claim because it does not necessarily imply there are abstract mathematical objects.[21]
Major concepts
Indispensability
The second premise of the indispensability argument states mathematical objects are indispensable to our best scientific theories. In this context, indispensability is not the same as ineliminability because any entity can be eliminated from a theoretical system given appropriate adjustments to the other parts of the system.[22] Therefore, dispensability requires an entity is eliminable without sacrificing the attractiveness of the theory. The attractiveness of the theory can be evaluated in terms of theoretical virtues such as explanatory power, empirical adequacy and simplicity.[23] Furthermore, if an entity is dispensable to a theory, an equivalent theory can be formulated without it.[24] This is the case, for example, if each sentence in one theory is a paraphrase of a sentence in another or if the two theories predict the same empirical observations.[25]
According to the Stanford Encyclopedia of Philosophy, one of the most influential argument against the indispensability argument comes from Hartry Field.[26] It rejects the claim that mathematical objects are indispensable to science;[27] Field has supported this argument by reformulating or "nominalizing" scientific theories so they do not refer to mathematical objects.[28] As part of this project, Field has offered a reformulation of Newtonian physics in terms of the relationships between space-time points. Instead of referring to numerical distances, Field's reformulation uses relationships such as "between" and "congruent" to recover the theory without implying the existence of numbers.[29] John Burgess and Mark Balaguer have taken steps to extend this nominalizing project to areas of modern physics, including quantum mechanics.[30] Philosophers such as David Malament and Otávio Bueno dispute whether such reformulations are successful or even possible, particularly in the case of quantum mechanics.[31]
Field's alternative to platonism is mathematical fictionalism, according to which mathematical theories are false because they make claims about abstract mathematical objects even though abstract objects do not exist.[32] As part of his argument against the indispensability argument, Field has tried to explain how it is possible for false mathematical statements to be used by science without making scientific predictions false.[33] His argument is based on the idea that mathematics is conservative. A mathematical theory is conservative if, when combined with a scientific theory, it does not imply anything about the physical world that the scientific theory alone would not have already.[34] This explains how it is possible for mathematics to be used by scientific theories without making the predictions of science false. In addition, Field has attempted to specify how exactly mathematics is useful in application.[26] Field thinks mathematics is useful for science because mathematical language provides a useful shorthand for talking about complex physical systems.[30]
Another approach to denying that mathematical entities are indispensable to science is to reformulate mathematical theories themselves so they do not imply the existence of mathematical objects. Charles Chihara, Geoffrey Hellman, and Putnam have offered modal reformulations of mathematics that replace all references to mathematical objects with claims about possibilities.[30]
Naturalism
The naturalism underlying the indispensability argument is a form of methodological naturalism, as opposed to metaphysical naturalism, that asserts the primacy of the scientific method for determining the truth.[35] In other words, according to Quine's naturalism, our best scientific theories are the best guide to what exists.[17] This form of naturalism rejects the idea that philosophy precedes and ultimately justifies belief in science, instead holding that science and philosophy are continuous with one another as part of a single, unified investigation into the world.[36] As such, this form of naturalism precludes the idea of a prior philosophy that can overturn the ontological commitments of science.[37] This is in contrast to alternative forms of naturalism, such as a form supported by David Armstrong that holds a principle called the Eleatic principle. According to this principle there are only causal entities and no non-causal entities.[38] Quine's naturalism claims such a principle cannot be used to overturn our best scientific theories' ontological commitment to mathematical entities because philosophical principles cannot overrule science.[39]
Quine held his naturalism as a fundamental assumption but later philosophers have provided arguments to support it. The most common arguments in support of Quinean naturalism are track-record arguments. These are arguments that appeal to science's successful track record compared to philosophy and other disciplines.[40] David Lewis famously made such an argument in a passage from his 1991 book Parts of Classes, deriding the track record of philosophy compared to mathematics and arguing that the idea of philosophy overriding science is absurd.[41] Critics of the track record argument have argued that it goes too far, discrediting philosophical arguments and methods entirely, and contest the idea that philosophy can be uniformly judged to have had a bad track record.[42]
Quine's naturalism has also been criticized by Penelope Maddy for contradicting mathematical practice.[43] According to the indispensability argument, mathematics is subordinated to the natural sciences in the sense that its legitimacy depends on them.[44] But Maddy argues mathematicians do not seem to believe their practice is restricted in any way by the activity of the natural sciences. For example, mathematicians' arguments over the axioms of Zermelo–Fraenkel set theory do not appeal to their applications to the natural sciences. Similarly, Charles Parsons has argued that mathematical truths seem immediately obvious in a way that suggests they do not depend on the results of our best theories.[45]
Confirmational holism
Confirmational holism is the view that scientific theories and hypotheses cannot be confirmed in isolation and must be confirmed together as part of a larger cluster of theories.[46] An example of this idea provided by Michael Resnik is of the hypothesis that an observer will see oil and water separate out if they are added together because they do not mix. This hypothesis cannot be confirmed in isolation because it relies on assumptions such as the absence of any chemical that will interfere with their separation and that the eyes of the observer are functioning well enough to observe the separation.[47] Because mathematical theories are likewise assumed by scientific theories, confirmational holism implies the empirical confirmations of scientific theories also support these mathematical theories.[48]
According to a counterargument by Maddy, the theses of naturalism and confirmational holism that make up the first premise of the indispensability argument are in tension with one another. Maddy said naturalism tells us that we should respect the methods used by scientists as the best method for uncovering the truth, but scientists do not seem to act as though we should believe in all of the entities that are indispensable to science.[49] To illustrate this point, Maddy uses the example of atomic theory; she said that despite the atom being indispensable to scientists' best theories by 1860, their reality was not universally accepted until 1913 when they were put to a direct experimental test.[50] Maddy also appeals to the fact that scientists use mathematical idealizations, such as assuming bodies of water to be infinitely deep without regard for the trueness of such applications of mathematics. According to Maddy, this indicates that scientists do not view the indispensable use of mathematics for science as justification for the belief in mathematics or mathematical entities. Overall, Maddy said we should side with naturalism and reject confirmational holism, meaning we do not need to believe in all of the entities that are indispensable to science.[26]
Another counterargument due to Elliott Sober claims that mathematical theories are not tested in the same way as scientific theories. Whilst scientific theories compete with alternatives to find which theory has the most empirical support, there are no alternatives for mathematical theory to compete with because all scientific theories share the same mathematical core. As a result, according to Sober, mathematical theories do not share the empirical support of our best scientific theories so we should reject confirmational holism.[51]
Since these counterarguments have been raised, a number of philosophers—including Resnik, Alan Baker, Patrick Dieveney, David Liggins, Jacob Busch, and Andrea Sereni—have argued that confirmational holism can be eliminated from the argument.[52] For example, Resnik has offered a pragmatic indispensability argument that "claims that the justification for doing science ... also justifies our accepting as true such mathematics as science uses".[53]
Ontological commitment
Another key part of the argument is the concept of ontological commitment. To say that we should have an ontological commitment to an entity means we should believe that entity exists. Quine believed that we should have ontological commitment to all the entities to which our best scientific theories are themselves committed.[54] According to Quine's "criterion of ontological commitment", the commitments of a theory can be found by translating or "regimenting" the theory from ordinary language into first-order logic. This criterion says that the ontological commitments of the theory are all of the objects over which the regimented theory quantifies; the existential quantifier for Quine was the natural equivalent of the ordinary language term "there is", which he believed obviously carries ontological commitment.[55] Quine thought it is important to translate our best scientific theories into first-order logic because ordinary language is ambiguous, whereas logic can make the commitments of a theory more precise. Translating theories to first-order logic also has advantages over translating them to higher-order logics such as second-order logic. Whilst second-order logic has the same expressive power as first-order logic, it lacks some of the technical strengths of first-order logic such as completeness and compactness. Second-order logic also allows quantification over properties like "redness", but whether we have ontological commitment to properties is controversial.[16] According to Quine, such quantification is simply ungrammatical.[56]
Jody Azzouni has objected to Quine's criterion of ontological commitment, saying that the existential quantifier in first-order logic need not be interpreted as always carrying ontological commitment.[57] According to Azzouni, the ordinary language equivalent of existential quantification "there is" is often used in sentences without implying ontological commitment. In particular, Azzouni points to the use of "there is" when referring to fictional objects in sentences such as "there are fictional detectives who are admired by some real detectives".[58] According to Azzouni, for us to have ontological commitment to an entity, we must have the right level of epistemic access to it. This means, for example, that it must overcome some epistemic burdens for us to be able to postulate it. But according to Azzouni, mathematical entities are "mere posits" that can be postulated by anyone at any time by "simply writing down a set of axioms", so we do not need to treat them as real.[59]
More modern presentations of the argument do not necessarily accept Quine's criterion of ontological commitment and may allow for ontological commitments to be directly determined from ordinary language.[60][lower-alpha 4]
Mathematical explanation
In his counterargument, Joseph Melia argues that the role of mathematics in science is not genuinely explanatory and is solely used to "make more things sayable about concrete objects".[62] He appeals to a practice he calls weaseling, which occurs when a person makes a statement and then later withdraws something implied by that statement. An example of weaseling is the statement: "Everyone who came to the seminar had a handout. But the person who came in late didn't get one."[63] Whilst this statement can be interpreted as being self-contradictory, it is more charitable to interpret it as coherently making the claim: "Except for the person who came in late, everyone who came to the seminar had a handout."[63] Melia said a similar situation occurs in scientists' use of statements that imply the existence of mathematical objects. According to Melia, whilst scientists use statements that imply the existence of mathematics in their theories, "almost all scientists ... deny that there are such things as mathematical objects".[63] As in the seminar-handout example, Melia said it is most charitable to interpret scientists not as contradicting themselves, but rather as weaseling away their commitment to mathematical objects. According to Melia, because this weaseling is not a genuinely explanatory use of mathematical language, it is acceptable to not believe in the mathematical objects that scientists weasel away.[62]
Inspired by Maddy's and Sober's arguments against confirmational holism,[64] as well as Melia's argument that we can suspend belief in mathematics if it does not play a genuinely explanatory role in science,[65] Colyvan and Baker have defended an explanatory version of the argument.[66][lower-alpha 5] This version of the argument attempts to remove the reliance on confirmational holism by replacing it with an inference to the best explanation. It states we are justified in believing in mathematical objects because they appear in our best scientific explanations, not because they inherit the empirical support of our best theories.[69] It is presented by the Internet Encyclopedia of Philosophy in the following form:[66]
• There are genuinely mathematical explanations of empirical phenomena.
• We ought to be committed to the theoretical posits in such explanations.
• Therefore, we ought to be committed to the entities postulated by the mathematics in question.
An example of mathematics' explanatory indispensability presented by Baker is the periodic cicada, a type of insect that has life cycles of 13 or 17 years. It is hypothesized that this is an evolutionary advantage because 13 and 17 are prime numbers. Because prime numbers have no non-trivial factors, this means it is less likely predators can synchronize with the cicadas' life cycles. Baker said that this is an explanation in which mathematics, specifically number theory, plays a key role in explaining an empirical phenomenon.[70] Other important examples are explanations of the hexagonal structure of bee honeycombs, the existence of antipodes on the Earth's surface that have identical temperature and pressure, the connection between Minkowski space and Lorentz contraction, and the impossibility of crossing all seven bridges of Königsberg only once in a walk across the city.[71] The main response to this form of the argument, which philosophers such as Melia, Chris Daly, Simon Langford, and Juha Saatsi adopted, is to deny there are genuinely mathematical explanations of empirical phenomena, instead framing the role of mathematics as representational or indexical.[72]
Historical development
Precursors and influences on Quine
The argument is historically associated with Willard Quine and Hilary Putnam but it can be traced to earlier thinkers such as Gottlob Frege and Kurt Gödel. In his arguments against mathematical formalism—a view that argues that mathematics is akin to a game like chess with rules about how mathematical symbols such as "2" can be manipulated—Frege said in 1903 that "it is applicability alone which elevates arithmetic from a game to the rank of a science".[73] Gödel, concerned about the axioms of set theory, said in a 1947 paper that if a new axiom were to have enough verifiable consequences, it "would have to be accepted at least in the same sense as any well‐established physical theory".[73] Frege's and Gödel's arguments differ from the later Quinean indispensability argument because they lack features such as naturalism and subordination of practice, leading some philosophers, including Pieranna Garavaso, to say that they are not genuine examples of the indispensability argument.[74]
Whilst developing his philosophical view of confirmational holism, Quine was influenced by Pierre Duhem.[75] At the beginning of the twentieth century, Duhem defended the law of inertia from critics who said that it is devoid of empirical content and unfalsifiable.[47] These critics based this claim on the fact that the law does not make any observable predictions without positing some observational frame of reference and that falsifying instances can always be avoided by changing the choice of reference frame. Duhem responded by saying that the law produces predictions when paired with auxiliary hypotheses fixing the frame of reference and is therefore no different from any other physical theory.[76] Duhem said that although individual hypotheses may make no observable predictions alone, they can be confirmed as parts of systems of hypotheses. Quine extended this idea to mathematical hypotheses, claiming that although mathematical hypotheses hold no empirical content on their own, they can share in the empirical confirmations of the systems of hypotheses in which they are contained.[77] This thesis later came to be known as the Duhem–Quine thesis.[78]
Quine described his naturalism as the "abandonment of the goal of a first philosophy. It sees natural science as an inquiry into reality, fallible and corrigible but not answerable to any supra-scientific tribunal, and not in need of any justification beyond observation and the hypothetico-deductive method."[79] The term "first philosophy" is used in reference to Descartes' Meditations on First Philosophy, in which Descartes used his method of doubt in an attempt to secure the foundations of science. Quine said that Descartes' attempts to provide the foundations for science had failed and that the project of finding a foundational justification for science should be rejected because he believed philosophy could never provide a method of justification more convincing than the scientific method.[80] Quine was also influenced by the logical positivists, such as his teacher Rudolf Carnap; his naturalism was formulated in response to many of their ideas.[81] For the logical positivists, all justified beliefs were reducible to sense data, including our knowledge of ordinary objects such as trees.[82] Quine criticized sense data as self-defeating, saying that we must believe in ordinary objects to organize our experiences of the world. He also said that because science is our best theory of how sense-experience gives us beliefs about ordinary objects, we should believe in it as well.[83] Whilst the logical positivists said that individual claims must be supported by sense data, Quine's confirmational holism means scientific theory is inherently tied up with mathematical theory and so evidence for scientific theories can justify belief in mathematical objects despite them not being directly perceived.[82]
Quine and Putnam
Whilst he eventually became a platonist due to his formulation of the indispensability argument,[84] Quine was sympathetic to nominalism from the early stages of his career.[85] In a 1946 lecture, he said: "I will put my cards on the table now and avow my prejudices: I should like to be able to accept nominalism."[86] In 1947, he released a paper with Nelson Goodman titled "Steps toward a Constructive Nominalism" as part of a joint project to "set up a nominalistic language in which all of natural science can be expressed".[87] In a letter to Joseph Henry Woodger the following year, however, Quine said that he was becoming more convinced "the assumption of abstract entities and the assumptions of the external world are assumptions of the same sort".[88] He subsequently released the 1948 paper "On What There Is", in which he said that "[t]he analogy between the myth of mathematics and the myth of physics is ... strikingly close", marking a shift towards his eventual acceptance of a "reluctant platonism".[89]
Throughout the 1950s, Quine regularly mentioned platonism, nominalism, and constructivism as plausible views, and he had not yet reached a definitive conclusion about which is correct.[90] It is unclear exactly when Quine accepted platonism; in 1953, he distanced himself from the claims of nominalism in his 1947 paper with Goodman, but by 1956, Goodman was still describing Quine's "defection" from nominalism as "still somewhat tentative".[91] According to Lieven Decock, Quine had accepted the need for abstract mathematical entities by the publication of his 1960 book Word and Object, in which he wrote "a thoroughgoing nominalist doctrine is too much to live up to".[92] However, whilst he released suggestions of the indispensability argument in a number of papers, he never gave it a detailed formulation.[93]
Putnam gave the argument its first explicit presentation in his 1971 book Philosophy of Logic in which he attributed it to Quine.[94] He stated the argument as "quantification over mathematical entities is indispensable for science, both formal and physical; therefore we should accept such quantification; but this commits us to accepting the existence of the mathematical entities in question."[95] He also wrote Quine had "for years stressed both the indispensability of quantification over mathematical entities and the intellectual dishonesty of denying the existence of what one daily presupposes".[95] Putnam's endorsement of Quine's version of the argument is disputed. The Internet Encyclopedia of Philosophy states: "In his early work, Hilary Putnam accepted Quine's version of the indispensability argument."[96] Liggins also states that the argument has been attributed to Putnam by many philosophers of mathematics. Liggins and Bueno, however, said Putnam never endorsed the argument and only presented it as an argument from Quine.[97] Putnam has said he differed with Quine in his attitude to the argument from at least 1975.[98] Features of the argument that Putnam came to disagree with include its reliance on a single, regimented, best theory.[96]
In 1975, Putnam formulated his own indispensability argument based on the no miracles argument in the philosophy of science, which argues the success of science can only be explained by scientific realism without being rendered miraculous. He wrote that year: "I believe that the positive argument for realism [in science] has an analogue in the case of mathematical realism. Here too, I believe, realism is the only philosophy that doesn't make the success of the science a miracle."[99] The Internet Encyclopedia of Philosophy terms this version of the argument "Putnam's success argument" and presents it in the following form:[96]
• Mathematics succeeds as the language of science.
• There must be a reason for the success of mathematics as the language of science.
• No positions other than realism in mathematics provide a reason.
• Therefore, realism in mathematics must be correct.[lower-alpha 6]
According to the Internet Encyclopedia of Philosophy, the first and second premises of the argument have been seen as uncontroversial, so discussion of this argument has been focused on the third premise. Other positions that have attempted to provide a reason for the success of mathematics include Field's reformulations of science, which explain the usefulness of mathematics as a conservative shorthand.[96] Putnam has criticized Field's reformulations for only applying to classical physics and for being unlikely to be able to be extended to future fundamental physics.[102]
Continued development of the argument
Chihara, in his 1973 nominalist book Ontology and the Vicious Circle Principle, was one of the earliest philosophers to attempt to reformulate mathematics in response to Quine's arguments.[103] Field's Science Without Numbers followed in 1980 and dominated discussion about the indispensability argument throughout the 1980s and 1990s.[104] With the introduction of arguments against the first premise of the argument, initially by Maddy in the 1990s and continued by Melia and others in the 2000s, Field's approach has come to be known as "Hard Road Nominalism" due to the difficulty of creating technical reconstructions of science that it requires. Approaches attacking the first premise, in contrast, have come to be known as "Easy Road Nominalism".[105]
Colyvan's formulation in his 1998 paper "In Defence of Indispensability" and his 2001 book The Indispensability of Mathematics is often seen as the standard or "canonical" formulation of the argument within more-recent philosophical work.[106] Colyvan's version of the argument has been influential in debates in contemporary philosophy of mathematics.[107] It differs in key ways from the arguments presented by Quine and Putnam. Quine's version of the argument relies on translating scientific theories from ordinary language into first-order logic to determine its ontological commitments whereas the modern version allows ontological commitments to be directly determined from ordinary language. Putnam's arguments were for the objectivity of mathematics but not necessarily for mathematical objects.[108] Putnam has explicitly distanced himself from this version of the argument, saying, "from my point of view, Colyvan's description of my argument(s) is far from right", and has contrasted his indispensability argument with "the fictitious 'Quine–Putnam indispensability argument'".[109] Colyvan has said "the attribution to Quine and Putnam [is] an acknowledgement of intellectual debts rather than an indication that the argument, as presented, would be endorsed in every detail by either Quine or Putnam".[110]
Influence
According to James Franklin, the indispensability argument is widely considered to be the best argument for platonism in the philosophy of mathematics.[111] The Stanford Encyclopedia of Philosophy identifies it as one of the major arguments in the debate between mathematical realism and mathematical anti-realism; according to the Stanford Encyclopedia of Philosophy, some within the field see it as the only good argument for platonism.[112]
Quine's and Putnam's arguments have also been influential outside philosophy of mathematics, inspiring indispensability arguments in other areas of philosophy. For example, David Lewis, who was a student of Quine, used an indispensability argument to argue for modal realism in his 1986 book On the Plurality of Worlds. According to his argument, quantification over possible worlds is indispensable to our best philosophical theories, so we should believe in their concrete existence.[113] Other indispensability arguments in metaphysics are defended by philosophers such as David Armstrong, Graeme Forbes, and Alvin Plantinga, who have argued for the existence of states of affairs due to the indispensable theoretical role they play in our best philosophical theories of truthmakers, modality, and possible worlds.[114] In the field of ethics, David Enoch has expanded the criterion of ontological commitment used in the Quine–Putnam indispensability argument to argue for moral realism. According to Enoch's "deliberative indispensability argument", indispensability to deliberation is just as ontologically committing as indispensability to science, and moral facts are indispensable to deliberation. Therefore, according to Enoch, we should believe in moral facts.[115]
Notes
1. Also referred to as the Putnam–Quine indispensability argument, holism–naturalism indispensability argument[1] or simply the indispensability argument
2. The concerns Benacerraf raised date back at least to Plato and Socrates, and were given detailed attention in the late nineteenth century prior to Quine and Putnam's arguments, which were raised in the 1960s and 1970s.[3] In contemporary philosophy, however, Benacerraf's presentation of these problems is considered to be the classic one.[4]
3. Subsequent philosophers have generalized this problem beyond the causal theory of knowledge; for Hartry Field, the general problem is to provide a mechanism explaining how mathematical beliefs can accurately reflect the properties of abstract mathematical objects.[8]
4. Non-Quinean forms of the argument can also be constructed using alternative criteria of ontological commitment. For example, Sam Baron defends a version of the argument that depends on a criterion of ontological commitment based on truthmaker theory.[61]
5. Baker identifies Field as originating this form of the argument in 1989, while other philosophers argue he was the first to raise the connection between indispensability and explanation but did not fully formulate an explanatory version of the indispensability argument.[67] Other thinkers who anticipated certain details of the explanatory form of the argument include Mark Steiner in 1978 and J. J. C. Smart in 1990.[68]
6. According to the Internet Encyclopedia of Philosophy, this version of the argument can be used to argue for platonism or sentence realism.[96] However, Putnam himself used it to argue for sentence realism.[100] Putnam's view is a reformulation of mathematics in terms of modal logic that maintains mathematical objectivity without being committed to mathematical objects.[101]
References
Citations
1. Decock 2002, p. 236.
2. Colyvan 2019, §1.
3. Molinini, Pataut & Sereni 2016, p. 318.
4. Balaguer 2018, §1.5.
5. Marcus, Introduction.
6. Colyvan 2012, pp. 9–10.
7. Paseau & Baker 2023, p. 2; Colyvan 2012, p. 1.
8. Colyvan 2012, pp. 10–12.
9. Horsten 2019, §3.4; Colyvan 2019, §6.
10. Colyvan 2012, pp. 8–9.
11. Shapiro 2000, pp. 31–32; Colyvan 2012, pp. 9–10.
12. Panza & Sereni 2015, pp. 470–471; Sinclair & Leibowitz 2016, pp. 10–18.
13. Colyvan 2019, Introduction.
14. Panza & Sereni 2016, p. 470.
15. Colyvan 2019.
16. Marcus, §2.
17. Colyvan 2019, §3.
18. Maddy 2005, p. 437. Primary source: Quine 1981a, p. 21.
19. Colyvan 2019, §2; Marcus, §7; Bostock 2009, pp. 276–277. Primary source: Quine 1998, p. 400.
20. Marcus, §7; Colyvan 2001, Ch. 6.
21. Panza & Sereni 2013, p. 201.
22. Colyvan 2019, §2. See also footnote 3 there.
23. Colyvan 2019, §2.
24. Busch & Sereni 2012, p. 347.
25. Panza & Sereni 2013, pp. 205–207.
26. Colyvan 2019, §4.
27. Colyvan 2019, §4; Colyvan 2001, p. 69; Linnebo 2017, pp. 105–106.
28. Linnebo 2017, pp. 105–106.
29. Colyvan 2001, p. 72.
30. Marcus, §7.
31. Balaguer 2018, §2.1; Bueno 2020, §3.3.2.
32. Balaguer 2018, Introduction.
33. Colyvan 2019, §4; Colyvan 2001, pp. 70–71; Linnebo 2017, pp. 105–106.
34. Colyvan 2001, p. 71; Paseau & Baker 2023, p. 14.
35. Paseau & Baker 2023, p. 4.
36. Colyvan 2001, pp. 23–24.
37. Colyvan 2001, p. 25.
38. Colyvan 2001, pp. 32–33.
39. Colyvan 2001, pp. 32–33; Bangu 2012, pp. 16–17.
40. Paseau & Baker 2023, p. 6.
41. Paseau & Baker 2023, p. 6; Weatherson 2021, §7.1.
42. Paseau & Baker 2023, p. 7.
43. Colyvan 2001, p. 93.
44. Marcus, §6; Colyvan 2001, p. 93.
45. Horsten 2019, §3.2; Colyvan 2019, §4; Bostock 2009, p. 278.
46. Resnik 2005, p. 414; Paseau & Baker 2023, p. 9.
47. Resnik 2005, p. 414.
48. Horsten 2019, §3.2.
49. Colyvan 2019, §4; Paseau & Baker 2023, p. 23.
50. Colyvan 2001, p. 92; Paseau & Baker 2023, pp. 22–23.
51. Colyvan 2019, §4; Bostock 2009, p. 278; Resnik 2005, p. 419.
52. Marcus 2014.
53. Colyvan 2001, p. 14–15. Primary source: Resnik 1995, pp. 171.
54. Leng 2010, pp. 39–40.
55. Marcus, §2; Bangu 2012, pp. 26–28.
56. Burgess 2013, p. 287.
57. Bangu 2012, p. 28; Bueno 2020, §5.
58. Antunes 2018, p. 16. Primary source: Azzouni 2004, pp. 68–69.
59. Bueno 2020, §5; Colyvan 2012, p. 64; Shapiro 2000, p. 251. Primary source: Azzouni 2004, p. 127
60. Liggins 2008, §5.
61. Asay 2020, p. 226. Primary source: Baron 2013.
62. Liggins 2012, pp. 998–999; Knowles & Liggins 2015, pp. 3398–3399; Daly & Langford 2009, pp. 641–644. Primary source: Melia 1998, pp. 70–71.
63. Liggins 2012, pp. 998–999; Knowles & Liggins 2015, pp. 3398–3399. Primary source: Melia 2000, p. 489.
64. Colyvan 2019, §5.
65. Mancosu 2018, §3.2; Bangu 2013, pp. 256–258.
66. Marcus, §5.
67. Molinini, Pataut & Sereni 2016, p. 320; Bangu 2013, pp. 255–256; Marcus 2015, Ch. 7, §3.
68. Colyvan 2019, Bibliography.
69. Marcus 2014, pp. 3583–3584; Leng 2005; Paseau & Baker 2023, p. 37.
70. Colyvan 2019, §5; Paseau & Baker 2023, pp. 35–36.
71. Molinini, Pataut & Sereni 2016, p. 321; Bangu 2012, pp. 152–153; Ginammi 2016, p. 64.
72. Molinini 2016, p. 405.
73. Colyvan 2001, pp. 8–9. Primary sources: Frege 2017, §91; Gödel 1947, §3.
74. Marcus, §6; Sereni 2015.
75. Maddy 2007, p. 91.
76. Resnik 2005, p. 415.
77. Resnik 2005, pp. 414–415.
78. Paseau & Baker 2023, p. 10.
79. Marcus, §2a; Shapiro 2000, p. 212. Primary source: Quine 1981b, p. 67.
80. Maddy 2005, p. 438.
81. Shapiro 2000, p. 212; Marcus, §2a.
82. Marcus, §2a.
83. Maddy 2007, p. 442; Marcus, §2a.
84. Putnam 2012, p. 223; Paseau & Baker 2023, p. 2.
85. Mancosu 2010; Decock 2002, p. 235.
86. Mancosu 2010, p. 398. Primary sources: Quine 2008, p. 6.
87. Mancosu 2010, p. 398; Verhaegh 2018, p. 112; Paseau & Baker 2023, pp. 2–3. Primary sources: Goodman & Quine 1947; Quine 1939, p. 708.
88. Mancosu 2010, p. 402.
89. Verhaegh 2018, p. 113; Mancosu 2010, p. 403. Primary source: Quine 1948, p. 37.
90. Decock 2002, p. 235.
91. Burgess 2013, p. 290. Primary source: Goodman 1956.
92. Decock 2002, p. 235. Primary source: Quine 1960, p. 269..
93. Marcus, §2; Paseau & Baker 2023, p. 1.
94. Bueno 2018, pp. 202–203; Shapiro 2000, p. 216; Sereni 2015, footnote 2.
95. Bueno 2018, p. 205; Liggins 2008, §4; Decock 2002, p. 231. Primary source: Putnam 1971, p. 347.
96. Marcus, §3.
97. Liggins 2008, pp. 115, 123; Bueno 2018, pp. 202–203.
98. Putnam 2012, p. 183.
99. Marcus, §3. Primary source: Putnam 1979, p. 73..
100. Colyvan 2001, pp. 2–3.
101. Bueno 2013, p. 227; Bueno 2018, pp. 201–202; Colyvan 2001, pp. 2–3; Putnam 2012, pp. 182–183.
102. Putnam 2012, pp. 190–192.
103. Burgess & Rosen 1997, p. 196.
104. Knowles & Liggins 2015, p. 3398.
105. Paseau & Baker 2023, pp. 30–31.
106. Molinini, Pataut & Sereni 2016, p. 320; Bueno 2018, p. 203.
107. Sereni 2015, §2.1; Marcus 2014, p. 3576.
108. Colyvan 2019, Introduction; Liggins 2008, §5.
109. Putnam 2012, pp. 182, 186.
110. Colyvan 2019, footnote 1.
111. Franklin 2009, p. 134.
112. Colyvan 2019, §6.
113. Weatherson 2021, §6.1; Nolan 2005, pp. 204–205.
114. Melia 2017, p. 96.
115. Sinclair & Leibowitz 2016, pp. 15–16; McPherson & Plunkett 2015, pp. 104–105.
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• Weatherson, Brian (2021). "David Lewis". In Zalta, Edward N. (ed.). The Stanford Encyclopedia of Philosophy (Winter 2021 ed.). Metaphysics Research Lab, Stanford University. ISSN 1095-5054.
Primary sources
This section provides a list of the primary sources that are referred to or quoted in the article but not used to source content.
• Azzouni, Jody (2004). Deflating Existential Consequence: A Case for Nominalism. Oxford University Press. ISBN 978-1-4294-3096-8.
• Baron, Sam (2013). "A Truthmaker Indispensability Argument". Synthese. 190 (12): 2413–2427. doi:10.1007/s11229-011-9989-2. ISSN 0039-7857. S2CID 255061951.
• Benacerraf, Paul (1973). "Mathematical Truth". The Journal of Philosophy. 70 (19): 661–679. doi:10.2307/2025075. JSTOR 2025075.
• Baker, Alan (2005). "Are there Genuine Mathematical Explanations of Physical Phenomena?". Mind. 114 (454): 223–238. doi:10.1093/mind/fzi223. ISSN 0026-4423.
• Chihara, Charles (1978). Ontology and the Vicious Circle Principle. Cornell University Press. ISBN 0-8014-0727-3.
• Colyvan, Mark (1998). "In Defence of Indispensability". Philosophia Mathematica. 6 (1): 39–62. doi:10.1093/philmat/6.1.39. ISSN 0031-8019.
• Descartes, René (1993) [1641]. Meditations on First Philosophy. Translated by Cress, Donald A. Hackett Publishing. ISBN 978-1-60384-481-9.
• Enoch, David (2011). Taking Morality Seriously. Oxford University Press. ISBN 978-0-19-957996-9.
• Field, Hartry (1980). Science Without Numbers. Oxford University Press. ISBN 978-0-19-877791-5.
• Field, Hartry (1989). "Indispensability Arguments and Inference to the Best Explanation". Realism, Mathematics, and Modality. Wiley-Blackwell. pp. 14–20. ISBN 0-631-16303-4.
• Frege, Gottlob (2017) [1903]. Grundgesetze der Arithmetik. Vol. II. ISBN 978-3-7446-0231-0. OCLC 1189408701.
• Gödel, Kurt (1947). "What is Cantor's Continuum Problem?". The American Mathematical Monthly. 54 (9): 515–525. doi:10.2307/2304666. ISSN 0002-9890. JSTOR 2304666.
• Goodman, Nelson; Quine, W. V. (1947). "Steps Toward a Constructive Nominalism". Journal of Symbolic Logic. 12 (4): 105–122. doi:10.2307/2266485. ISSN 0022-4812. JSTOR 2266485. S2CID 46182517.
• Goodman, Nelson (1956). "A World of Individuals". In Bochenski, Innocenty; Church, Alonzo; Goodman, Nelson (eds.). The Problem of Universals. University of Notre Dame Press. pp. 15–31. OCLC 81970496.
• Lewis, David (1986). On the Plurality of Worlds. Wiley-Blackwell. ISBN 0-631-13993-1.
• Lewis, David (1991). Parts of Classes. Wiley-Blackwell. ISBN 978-0-631-17656-5.
• Maddy, Penelope (1992). "Indispensability and Practice". The Journal of Philosophy. 89 (6): 275–289. doi:10.2307/2026712. JSTOR 2026712.
• Melia, Joseph (1998). "Field's Programme: Some Interference". Analysis. 58 (2): 63–71. doi:10.1093/analys/58.2.63. ISSN 0003-2638.
• Melia, Joseph (2000). "Weaseling Away the Indispensability Argument". Mind. 109 (435): 455–480. doi:10.1093/mind/109.435.455. ISSN 0026-4423.
• Putnam, Hilary (1971). Philosophy of Logic. Harper & Row. ISBN 978-0-061-36042-8.
• Putnam, Hilary (1979) [1975]. "What Is Mathematical Truth?". Mathematics, Matter and Method: Philosophical Papers. Vol. I (2nd ed.). Cambridge University Press. pp. 60–78. ISBN 978-0-521-29550-5.
• Quine, W. V. (1939). "Designation and Existence". The Journal of Philosophy. 36 (26): 701–709. doi:10.2307/2017667. ISSN 0022-362X. JSTOR 2017667.
• Quine, W. V. (1948). "On What There Is". The Review of Metaphysics. 2 (5): 21–38. ISSN 0034-6632. JSTOR 20123117.
• Quine, W. V. (1951). "Two Dogmas of Empiricism". Philosophical Review. 60 (1): 20–43. doi:10.2307/2181906. ISSN 0031-8108. JSTOR 2181906.
• Quine, W. V. (1958). "Speaking of Objects". Proceedings and Addresses of the American Philosophical Association. 31: 5–22. doi:10.2307/3129242. ISSN 0065-972X. JSTOR 3129242.
• Quine, W. V. (1960). Word and Object. MIT Press. OCLC 1159745436.
• Quine, W. V. (1966) [1955]. "Posits and Reality". The Ways of Paradox and Other Essays. Harvard University Press. pp. 233–241. ISBN 978-0-674-94837-2.
• Quine, W. V. (1969). "Existence and Quantification". Ontological Relativity and Other Essays. Columbia University Press. pp. 91–113. OCLC 1104870602.
• Quine, W. V. (1981a). "Things and Their Place in Theories". Theories and Things. Harvard University Press. pp. 1–23. ISBN 0-674-87925-2. OCLC 7278383.
• Quine, W. V. (1981b). "Five Milestones of Empiricism". Theories and Things. Harvard University Press. pp. 67–72. ISBN 0-674-87925-2. OCLC 7278383.
• Quine, W. V. (1983) [1963]. "Carnap and Logical Truth". In Benacerraf, Paul; Putnam, Hilary (eds.). Philosophy of Mathematics: Selected Readings (2nd ed.). Cambridge University Press. pp. 355–376. ISBN 978-0-521-29648-9.
• Quine, W. V. (1998). "Reply to Charles Parsons". In Hahn, Lewis Edwin; Schilpp, Paul Arthur (eds.). The Philosophy of W.V. Quine (2nd expanded ed.). Open Court Publishing Company. pp. 396–403. ISBN 0-8126-9371-X. OCLC 37935049.
• Quine, W. V. (2008) [1946]. "Nominalism". In Zimmerman, Dean (ed.). Oxford Studies in Metaphysics. Vol. 4. Oxford University Press. pp. 6–21. ISBN 978-0-199-54298-7.
• Resnik, Michael (1995). "Scientific vs. Mathematical Realism: The Indispensability Argument". Philosophia Mathematica. 3 (2): 166–174. doi:10.1093/philmat/3.2.166. ISSN 0031-8019.
• Smart, J. J. C. (1990). "Explanation–Opening Address". Royal Institute of Philosophy Supplement. 27: 1–19. doi:10.1017/S1358246100005014. ISSN 1358-2461. S2CID 143223945.
• Sober, Elliott (1993). "Mathematics and Indispensability". Philosophical Review. 102 (1): 35–58. doi:10.2307/2185652. JSTOR 2185652.
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Wikipedia
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Quinn McNemar
Quinn Michael McNemar (February 20, 1900 – July 3, 1986)[1] was an American psychologist and statistician. He is known for his work on IQ tests, for his book Psychological Statistics (1949) and for McNemar's test, the statistical test he introduced in 1947.[2][3]
Quinn McNemar
BornFebruary 20, 1900
Greenland, West Virginia
DiedJuly 3, 1986(1986-07-03) (aged 86)
Palo Alto, California
CitizenshipUS
Alma materJuniata College
Stanford University
Known forMcNemar's test
Revising the Stanford-Binet IQ test
Scientific career
Fieldspsychology, statistics
InstitutionsStanford University
University of Texas
Doctoral advisorLewis Terman
Life
McNemar was born in Greenland, West Virginia in 1900. He obtained his bachelor's degree in mathematics in 1925 from Juniata College, studied for his doctorate in psychology under Lewis Terman at Stanford University, and joined the faculty at Stanford in 1931. In 1942 he published The Revision of the Stanford–Binet Scale, the IQ test released in 1916 by Terman. By the time he retired from Stanford in 1965 he held professorships in psychology, statistics and education. He taught for another five years at the University of Texas before retiring to Palo Alto, where he died in 1986.[3]
He was president of the Psychometric Society in 1951 and of the American Psychological Association in 1964.[4][5][6]
References
1. Hastorf, A. H.; Hilgard, E. R.; Sears, R. R. (1988). "Quinn McNemar (1900–1986)". American Psychologist. 43 (3): 196–197. doi:10.1037/h0091955.
2. McNemar, Quinn (1947-06-18). "Note on the sampling error of the difference between correlated proportions or percentages". Psychometrika. 12 (2): 153–157. doi:10.1007/BF02295996. PMID 20254758. S2CID 46226024.
3. "Quinn McNemar, reviser of IQ test" (PDF). Sandstone and Tile. Stanford Historical Society. 10 (3–4). Spring–Summer 1986.
4. "Quinn McNemar". A Dictionary of Statistics. Oxford University Press. 2008.
5. "Psychologists Honor Scientists at Sub Base". The Courier-Journal. September 3, 1962. p. 31. Retrieved March 29, 2020 – via Newspapers.com.
6. "Former APA Presidents". American Psychological Association. Retrieved March 29, 2020.
Presidents of the American Psychological Association
1892–1900
• G. Stanley Hall (1892)
• George Trumbull Ladd (1893)
• William James (1894)
• James McKeen Cattell (1895)
• George Stuart Fullerton (1896)
• James Mark Baldwin (1897)
• Hugo Münsterberg (1898)
• John Dewey (1899)
• Joseph Jastrow (1900)
1901–1925
• Josiah Royce (1901)
• Edmund Sanford (1902)
• William Lowe Bryan (1903)
• William James (1904)
• Mary Whiton Calkins (1905)
• James Rowland Angell (1906)
• Henry Rutgers Marshall (1907)
• George M. Stratton (1908)
• Charles Hubbard Judd (1909)
• Walter Bowers Pillsbury (1910)
• Carl Seashore (1911)
• Edward Thorndike (1912)
• Howard C. Warren (1913)
• Robert S. Woodworth (1914)
• John B. Watson (1915)
• Raymond Dodge (1916)
• Robert Yerkes (1917)
• John Wallace Baird (1918)
• Walter Dill Scott (1919)
• Shepherd Ivory Franz (1920)
• Margaret Floy Washburn (1921)
• Knight Dunlap (1922)
• Lewis Terman (1923)
• G. Stanley Hall (1924)
• I. Madison Bentley (1925)
1926–1950
• Harvey A. Carr (1926)
• Harry Levi Hollingworth (1927)
• Edwin Boring (1928)
• Karl Lashley (1929)
• Herbert Langfeld (1930)
• Walter Samuel Hunter (1931)
• Walter Richard Miles (1932)
• Louis Leon Thurstone (1933)
• Joseph Peterson (1934)
• Albert Poffenberger (1935)
• Clark L. Hull (1936)
• Edward C. Tolman (1937)
• John Dashiell (1938)
• Gordon Allport (1939)
• Leonard Carmichael (1940)
• Herbert Woodrow (1941)
• Calvin Perry Stone (1942)
• John Edward Anderson (1943)
• Gardner Murphy (1944)
• Edwin Ray Guthrie (1945)
• Henry Garrett (1946)
• Carl Rogers (1947)
• Donald Marquis (1948)
• Ernest Hilgard (1949)
• J. P. Guilford (1950)
1951–1975
• Robert Richardson Sears (1951)
• J. McVicker Hunt (1952)
• Laurance F. Shaffer (1953)
• Orval Hobart Mowrer (1954)
• E. Lowell Kelly (1955)
• Theodore Newcomb (1956)
• Lee Cronbach (1957)
• Harry Harlow (1958)
• Wolfgang Köhler (1959)
• Donald O. Hebb (1960)
• Neal E. Miller (1961)
• Paul E. Meehl (1962)
• Charles E. Osgood (1963)
• Quinn McNemar (1964)
• Jerome Bruner (1965)
• Nicholas Hobbs (1966)
• Gardner Lindzey (1967)
• Abraham Maslow (1968)
• George Armitage Miller (1969)
• George Albee (1970)
• Kenneth B. Clark (1971)
• Anne Anastasi (1972)
• Leona E. Tyler (1973)
• Albert Bandura (1974)
• Donald T. Campbell (1975)
1976–2000
• Wilbert J. McKeachie (1976)
• Theodore H. Blau (1977)
• M. Brewster Smith (1978)
• Nicholas Cummings (1979)
• Florence Denmark (1980)
• John J. Conger (1981)
• William Bevan (1982)
• Max Siegel (1983)
• Janet Taylor Spence (1984)
• Robert Perloff (1985)
• Logan Wright (1986)
• Bonnie Strickland (1987)
• Raymond D. Fowler (1988)
• Joseph Matarazzo (1989)
• Stanley Graham (1990)
• Charles Spielberger (1991)
• Jack Wiggins Jr. (1992)
• Frank H. Farley (1993)
• Ronald E. Fox (1994)
• Robert J. Resnick (1995)
• Dorothy Cantor (1996)
• Norman Abeles (1997)
• Martin Seligman (1998)
• Richard Suinn (1999)
• Patrick H. DeLeon (2000)
2001–Present
• Norine G. Johnson (2001)
• Philip Zimbardo (2002)
• Robert Sternberg (2003)
• Diane F. Halpern (2004)
• Ronald F. Levant (2005)
• Gerald Koocher (2006)
• Sharon Brehm (2007)
• Alan E. Kazdin (2008)
• James H. Bray (2009)
• Carol D. Goodheart (2010)
• Melba J. T. Vasquez (2011)
• Suzanne Bennett Johnson (2012)
• Donald N. Bersoff (2013)
• Nadine Kaslow (2014)
• Barry S. Anton (2015)
• Susan H. McDaniel (2016)
• Antonio Puente (2017)
• Jessica Henderson Daniel (2018)
• Rosie Phillips Davis (2019)
• Sandra Shullman (2020)
• Jennifer F. Kelly (2021)
• Frank C. Worrell (2022)
• Thema Bryant (2023)
Authority control
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Wikipedia
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Quintic threefold
In mathematics, a quintic threefold is a 3-dimensional hypersurface of degree 5 in 4-dimensional projective space $\mathbb {P} ^{4}$. Non-singular quintic threefolds are Calabi–Yau manifolds.
The Hodge diamond of a non-singular quintic 3-fold is
1
00
010
11011011
010
00
1
Mathematician Robbert Dijkgraaf said "One number which every algebraic geometer knows is the number 2,875 because obviously, that is the number of lines on a quintic."[1]
Definition
A quintic threefold is a special class of Calabi–Yau manifolds defined by a degree $5$ projective variety in $\mathbb {P} ^{4}$. Many examples are constructed as hypersurfaces in $\mathbb {P} ^{4}$, or complete intersections lying in $\mathbb {P} ^{4}$, or as a smooth variety resolving the singularities of another variety. As a set, a Calabi-Yau manifold is
$X=\{x=[x_{0}:x_{1}:x_{2}:x_{3}:x_{4}]\in \mathbb {CP} ^{4}:p(x)=0\}$
where $p(x)$ is a degree $5$ homogeneous polynomial. One of the most studied examples is from the polynomial
$p(x)=x_{0}^{5}+x_{1}^{5}+x_{2}^{5}+x_{3}^{5}+x_{4}^{5}$
called a Fermat polynomial. Proving that such a polynomial defines a Calabi-Yau requires some more tools, like the Adjunction formula and conditions for smoothness.
Hypersurfaces in P4
Recall that a homogeneous polynomial $f\in \Gamma (\mathbb {P} ^{4},{\mathcal {O}}(d))$ (where ${\mathcal {O}}(d)$ is the Serre-twist of the hyperplane line bundle) defines a projective variety, or projective scheme, $X$, from the algebra
${\frac {k[x_{0},\ldots ,x_{4}]}{(f)}}$
where $k$ is a field, such as $\mathbb {C} $. Then, using the adjunction formula to compute its canonical bundle, we have
${\begin{aligned}\Omega _{X}^{3}&=\omega _{X}\\&=\omega _{\mathbb {P} ^{4}}\otimes {\mathcal {O}}(d)\\&\cong {\mathcal {O}}(-(4+1))\otimes {\mathcal {O}}(d)\\&\cong {\mathcal {O}}(d-5)\end{aligned}}$
hence in order for the variety to be Calabi-Yau, meaning it has a trivial canonical bundle, its degree must be $5$. It is then a Calabi-Yau manifold if in addition this variety is smooth. This can be checked by looking at the zeros of the polynomials
$\partial _{0}f,\ldots ,\partial _{4}f$
and making sure the set
$\{x=[x_{0}:\cdots :x_{4}]|f(x)=\partial _{0}f(x)=\cdots =\partial _{4}f(x)=0\}$
is empty.
Examples
Fermat Quintic
One of the easiest examples to check of a Calabi-Yau manifold is given by the Fermat quintic threefold, which is defined by the vanishing locus of the polynomial
$f=x_{0}^{5}+x_{1}^{5}+x_{2}^{5}+x_{3}^{5}+x_{4}^{5}$
Computing the partial derivatives of $f$ gives the four polynomials
${\begin{aligned}\partial _{0}f=5x_{0}^{4}\\\partial _{1}f=5x_{1}^{4}\\\partial _{2}f=5x_{2}^{4}\\\partial _{3}f=5x_{3}^{4}\\\partial _{4}f=5x_{4}^{4}\\\end{aligned}}$
Since the only points where they vanish is given by the coordinate axes in $\mathbb {P} ^{4}$, the vanishing locus is empty since $[0:0:0:0:0]$ is not a point in $\mathbb {P} ^{4}$.
As a Hodge Conjecture testbed
Another application of the quintic threefold is in the study of the infinitesimal generalized Hodge conjecture where this difficult problem can be solved in this case.[2] In fact, all of the lines on this hypersurface can be found explicitly.
Dwork family of quintic three-folds
Another popular class of examples of quintic three-folds, studied in many contexts, is the Dwork family. One popular study of such a family is from Candelas, De La Ossa, Green, and Parkes,[3] when they discovered mirror symmetry. This is given by the family[4] pages 123-125
$f_{\psi }=x_{0}^{5}+x_{1}^{5}+x_{2}^{5}+x_{3}^{5}+x_{4}^{5}-5\psi x_{0}x_{1}x_{2}x_{3}x_{4}$
where $\psi $ is a single parameter not equal to a 5-th root of unity. This can be found by computing the partial derivates of $f_{\psi }$ and evaluating their zeros. The partial derivates are given by
${\begin{aligned}\partial _{0}f_{\psi }=5x_{0}^{4}-5\psi x_{1}x_{2}x_{3}x_{4}\\\partial _{1}f_{\psi }=5x_{1}^{4}-5\psi x_{0}x_{2}x_{3}x_{4}\\\partial _{2}f_{\psi }=5x_{2}^{4}-5\psi x_{0}x_{1}x_{3}x_{4}\\\partial _{3}f_{\psi }=5x_{3}^{4}-5\psi x_{0}x_{1}x_{2}x_{4}\\\partial _{4}f_{\psi }=5x_{4}^{4}-5\psi x_{0}x_{1}x_{2}x_{3}\\\end{aligned}}$
At a point where the partial derivatives are all zero, this gives the relation $x_{i}^{5}=\psi x_{0}x_{1}x_{2}x_{3}x_{4}$. For example, in $\partial _{0}f_{\psi }$ we get
${\begin{aligned}5x_{0}^{4}&=5\psi x_{1}x_{2}x_{3}x_{4}\\x_{0}^{4}&=\psi x_{1}x_{2}x_{3}x_{4}\\x_{0}^{5}&=\psi x_{0}x_{1}x_{2}x_{3}x_{4}\end{aligned}}$
by dividing out the $5$ and multiplying each side by $x_{0}$. From multiplying these families of equations $x_{i}^{5}=\psi x_{0}x_{1}x_{2}x_{3}x_{4}$ together we have the relation
$\prod x_{i}^{5}=\psi ^{5}\prod x_{i}^{5}$
showing a solution is either given by an $x_{i}=0$ or $\psi ^{5}=1$. But in the first case, these give a smooth sublocus since the varying term in $f_{\psi }$ vanishes, so a singular point must lie in $\psi ^{5}=1$. Given such a $\psi $, the singular points are then of the form
$[\mu _{5}^{a_{0}}:\cdots :\mu _{5}^{a_{4}}]$ :\mu _{5}^{a_{4}}]}
such that
$\mu _{5}^{\sum a_{i}}=\psi ^{-1}$
where $\mu _{5}=e^{2\pi i/5}$. For example, the point
$[\mu _{5}^{4}:\mu _{5}^{-1}:\mu _{5}^{-1}:\mu _{5}^{-1}:\mu _{5}^{-1}]$
is a solution of both $f_{1}$ and its partial derivatives since $(\mu _{5}^{i})^{5}=(\mu _{5}^{5})^{i}=1^{i}=1$, and $\psi =1$.
Other examples
• Barth–Nieto quintic
• Consani–Scholten quintic
Curves on a quintic threefold
Computing the number of rational curves of degree $1$ can be computed explicitly using Schubert calculus. Let $T^{*}$ be the rank $2$ vector bundle on the Grassmannian $G(2,5)$ of $2$-planes in some rank $5$ vector space. Projectivizing $G(2,5)$ to $\mathbb {G} (1,4)$ gives the projective grassmannian of degree 1 lines in $\mathbb {P} ^{4}$ and $T^{*}$ descends to a vector bundle on this projective Grassmannian. Its total chern class is
$c(T^{*})=1+\sigma _{1}+\sigma _{1,1}$
in the Chow ring $A^{\bullet }(\mathbb {G} (1,4))$. Now, a section $l\in \Gamma (\mathbb {G} (1,4),T^{*})$ of the bundle corresponds to a linear homogeneous polynomial, ${\tilde {l}}\in \Gamma (\mathbb {P} ^{4},{\mathcal {O}}(1))$, so a section of ${\text{Sym}}^{5}(T^{*})$ corresponds to a quintic polynomial, a section of $\Gamma (\mathbb {P} ^{4},{\mathcal {O}}(5))$. Then, in order to calculate the number of lines on a generic quintic threefold, it suffices to compute the integral[5]
$\int _{\mathbb {G} (1,4)}c({\text{Sym}}^{5}(T^{*}))=2875$
This can be done by using the splitting principle. Since
${\begin{aligned}c(T^{*})&=(1+\alpha )(1+\beta )\\&=1+(\alpha +\beta )+\alpha \beta \end{aligned}}$
and for a dimension $2$ vector space, $V=V_{1}\oplus V_{2}$,
${\text{Sym}}^{5}(V)=\bigoplus _{i=0}^{5}(V_{1}^{\otimes 5-i}\otimes V_{2}^{\otimes i})$
so the total chern class of ${\text{Sym}}^{5}(T^{*})$ is given by the product
$c({\text{Sym}}^{5}(T^{*}))=\prod _{i=0}^{5}(1+(5-i)\alpha +i\beta )$
Then, the Euler class, or the top class is
$5\alpha (4\alpha +\beta )(3\alpha +2\beta )(2\alpha +3\beta )(\alpha +4\beta )5\beta $
expanding this out in terms of the original chern classes gives
${\begin{aligned}c_{6}({\text{Sym}}^{5}(T^{*}))&=25\sigma _{1,1}(4\sigma _{1}^{2}+9\sigma _{1,1})(6\sigma _{1}^{2}+\sigma _{1,1})\\&=(100\sigma _{2,2}+225\sigma _{2,2})(6\sigma _{1}^{2}+\sigma _{1,1})\\&=325\sigma _{2,2}(6\sigma _{1}^{2}+\sigma _{1,1})\end{aligned}}$
using the relations $\sigma _{1,1}\cdot \sigma _{1}^{2}=\sigma _{2,2}$, $\sigma _{1,1}^{2}=\sigma _{2,2}$.
Rational curves
Herbert Clemens (1984) conjectured that the number of rational curves of a given degree on a generic quintic threefold is finite. (Some smooth but non-generic quintic threefolds have infinite families of lines on them.) This was verified for degrees up to 7 by Sheldon Katz (1986) who also calculated the number 609250 of degree 2 rational curves. Philip Candelas, Xenia C. de la Ossa, and Paul S. Green et al. (1991) conjectured a general formula for the virtual number of rational curves of any degree, which was proved by Givental (1996) (the fact that the virtual number equals the actual number relies on confirmation of Clemens' conjecture, currently known for degree at most 11 Cotterill (2012)). The number of rational curves of various degrees on a generic quintic threefold is given by
2875, 609250, 317206375, 242467530000, ...(sequence A076912 in the OEIS).
Since the generic quintic threefold is a Calabi–Yau threefold and the moduli space of rational curves of a given degree is a discrete, finite set (hence compact), these have well-defined Donaldson–Thomas invariants (the "virtual number of points"); at least for degree 1 and 2, these agree with the actual number of points.
See also
• Mirror symmetry (string theory)
• Gromov–Witten invariant
• Jacobian ideal - gives an explicit basis for the Hodge-decomposition
• Deformation theory
• Hodge structure
• Schubert calculus - techniques for determining the number of lines on a quintic threefold
References
1. Robbert Dijkgraaf (29 March 2015). "The Unreasonable Effectiveness of Quantum Physics in Modern Mathematics". youtube.com. Trev M. Archived from the original on 2021-12-21. Retrieved 10 September 2015. see 29 minutes 57 seconds
2. Albano, Alberto; Katz, Sheldon (1991). "Lines on the Fermat quintic threefold and the infinitesimal generalized Hodge conjecture". Transactions of the American Mathematical Society. 324 (1): 353–368. doi:10.1090/S0002-9947-1991-1024767-6. ISSN 0002-9947.
3. Candelas, Philip; De La Ossa, Xenia C.; Green, Paul S.; Parkes, Linda (1991-07-29). "A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory". Nuclear Physics B. 359 (1): 21–74. Bibcode:1991NuPhB.359...21C. doi:10.1016/0550-3213(91)90292-6. ISSN 0550-3213.
4. Gross, Mark; Huybrechts, Daniel; Joyce, Dominic (2003). Ellingsrud, Geir; Olson, Loren; Ranestad, Kristian; Stromme, Stein A. (eds.). Calabi-Yau Manifolds and Related Geometries: Lectures at a Summer School in Nordfjordeid, Norway, June 2001. Universitext. Berlin Heidelberg: Springer-Verlag. pp. 123–125. ISBN 978-3-540-44059-8.
5. Katz, Sheldon. Enumerative Geometry and String Theory. p. 108.
• Arapura, Donu, "Computing Some Hodge Numbers" (PDF)
• Candelas, Philip; de la Ossa, Xenia C.; Green, Paul S.; Parkes, Linda (1991), "A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory", Nuclear Physics B, 359 (1): 21–74, Bibcode:1991NuPhB.359...21C, doi:10.1016/0550-3213(91)90292-6, MR 1115626
• Clemens, Herbert (1984), "Some results about Abel-Jacobi mappings", Topics in transcendental algebraic geometry (Princeton, N.J., 1981/1982), Ann. of Math. Stud., vol. 106, Princeton University Press, pp. 289–304, MR 0756858
• Cotterill, Ethan (2012), "Rational curves of degree 11 on a general quintic 3-fold", The Quarterly Journal of Mathematics, 63 (3): 539–568, doi:10.1093/qmath/har001, MR 2967162
• Cox, David A.; Katz, Sheldon (1999), Mirror symmetry and algebraic geometry, Mathematical Surveys and Monographs, vol. 68, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-1059-0, MR 1677117
• Givental, Alexander B. (1996), "Equivariant Gromov-Witten invariants", International Mathematics Research Notices, 1996 (13): 613–663, doi:10.1155/S1073792896000414, MR 1408320
• Katz, Sheldon (1986), "On the finiteness of rational curves on quintic threefolds", Compositio Mathematica, 60 (2): 151–162, MR 0868135
• Pandharipande, Rahul (1998), "Rational curves on hypersurfaces (after A. Givental)", Astérisque, 1997/98 (252): 307–340, arXiv:math/9806133, Bibcode:1998math......6133P, MR 1685628
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Quintuple product identity
In mathematics the Watson quintuple product identity is an infinite product identity introduced by Watson (1929) and rediscovered by Bailey (1951) and Gordon (1961). It is analogous to the Jacobi triple product identity, and is the Macdonald identity for a certain non-reduced affine root system. It is related to Euler's pentagonal number theorem.
Statement
$\prod _{n\geq 1}(1-s^{n})(1-s^{n}t)(1-s^{n-1}t^{-1})(1-s^{2n-1}t^{2})(1-s^{2n-1}t^{-2})=\sum _{n\in \mathbf {Z} }s^{(3n^{2}+n)/2}(t^{3n}-t^{-3n-1})$
References
• Bailey, W. N. (1951), "On the simplification of some identities of the Rogers-Ramanujan type", Proceedings of the London Mathematical Society, Third Series, 1: 217–221, doi:10.1112/plms/s3-1.1.217, ISSN 0024-6115, MR 0043839
• Carlitz, L.; Subbarao, M. V. (1972), "A simple proof of the quintuple product identity", Proceedings of the American Mathematical Society, 32: 42–44, doi:10.2307/2038301, ISSN 0002-9939, JSTOR 2038301, MR 0289316
• Gordon, Basil (1961), "Some identities in combinatorial analysis", The Quarterly Journal of Mathematics, Second Series, 12: 285–290, doi:10.1093/qmath/12.1.285, ISSN 0033-5606, MR 0136551
• Watson, G. N. (1929), "Theorems stated by Ramanujan. VII: Theorems on continued fractions.", Journal of the London Mathematical Society, 4 (1): 39–48, doi:10.1112/jlms/s1-4.1.39, ISSN 0024-6107, JFM 55.0273.01
• Foata, D., & Han, G. N. (2001). The triple, quintuple and septuple product identities revisited. In The Andrews Festschrift (pp. 323–334). Springer, Berlin, Heidelberg.
• Cooper, S. (2006). The quintuple product identity. International Journal of Number Theory, 2(01), 115-161.
Further reading
• Subbarao, M. V., & Vidyasagar, M. (1970). On Watson’s quintuple product identity. Proceedings of the American Mathematical Society, 26(1), 23-27.
• Hirschhorn, M. D. (1988). A generalisation of the quintuple product identity. Journal of the Australian Mathematical Society, 44(1), 42-45.
• Alladi, K. (1996). The quintuple product identity and shifted partition functions. Journal of Computational and Applied Mathematics, 68(1-2), 3-13.
• Farkas, H., & Kra, I. (1999). On the quintuple product identity. Proceedings of the American Mathematical Society, 127(3), 771-778.
• Chen, W. Y., Chu, W., & Gu, N. S. (2005). Finite form of the quintuple product identity. arXiv preprint math/0504277.
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Quasi-isomorphism
In homological algebra, a branch of mathematics, a quasi-isomorphism or quism is a morphism A → B of chain complexes (respectively, cochain complexes) such that the induced morphisms
$H_{n}(A_{\bullet })\to H_{n}(B_{\bullet })\ ({\text{respectively, }}H^{n}(A^{\bullet })\to H^{n}(B^{\bullet }))$
of homology groups (respectively, of cohomology groups) are isomorphisms for all n.
In the theory of model categories, quasi-isomorphisms are sometimes used as the class of weak equivalences when the objects of the category are chain or cochain complexes. This results in a homology-local theory, in the sense of Bousfield localization in homotopy theory.
See also
• Derived category
References
• Gelfand, Sergei I., Manin, Yuri I. Methods of Homological Algebra, 2nd ed. Springer, 2000.
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Great stellated truncated dodecahedron
In geometry, the great stellated truncated dodecahedron (or quasitruncated great stellated dodecahedron or great stellatruncated dodecahedron) is a nonconvex uniform polyhedron, indexed as U66. It has 32 faces (20 triangles and 12 decagrams), 90 edges, and 60 vertices.[1] It is given a Schläfli symbol t0,1{5/3,3}.
Great stellated truncated dodecahedron
TypeUniform star polyhedron
ElementsF = 32, E = 90
V = 60 (χ = 2)
Faces by sides20{3}+12{10/3}
Coxeter diagram
Wythoff symbol2 3 | 5/3
Symmetry groupIh, [5,3], *532
Index referencesU66, C83, W104
Dual polyhedronGreat triakis icosahedron
Vertex figure
3.10/3.10/3
Bowers acronymQuit Gissid
Related polyhedra
It shares its vertex arrangement with three other uniform polyhedra: the small icosicosidodecahedron, the small ditrigonal dodecicosidodecahedron, and the small dodecicosahedron:
Great stellated truncated dodecahedron
Small icosicosidodecahedron
Small ditrigonal dodecicosidodecahedron
Small dodecicosahedron
Cartesian coordinates
Cartesian coordinates for the vertices of a great stellated truncated dodecahedron are all the even permutations of
(0, ±τ, ±(2−1/τ))
(±τ, ±1/τ, ±2/τ)
(±1/τ2, ±1/τ, ±2)
where τ = (1+√5)/2 is the golden ratio (sometimes written φ).
See also
• List of uniform polyhedra
References
1. Maeder, Roman. "66: great stellated truncated dodecahedron". MathConsult.{{cite web}}: CS1 maint: url-status (link)
External links
• Weisstein, Eric W. "Great stellated truncated dodecahedron". MathWorld.
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Small stellated truncated dodecahedron
In geometry, the small stellated truncated dodecahedron (or quasitruncated small stellated dodecahedron or small stellatruncated dodecahedron) is a nonconvex uniform polyhedron, indexed as U58. It has 24 faces (12 pentagons and 12 decagrams), 90 edges, and 60 vertices.[1] It is given a Schläfli symbol t{5⁄3,5}, and Coxeter diagram .
Small stellated truncated dodecahedron
TypeUniform star polyhedron
ElementsF = 24, E = 90
V = 60 (χ = −6)
Faces by sides12{5}+12{10/3}
Coxeter diagram
Wythoff symbol2 5 | 5/3
2 5/4 | 5/3
Symmetry groupIh, [5,3], *532
Index referencesU58, C74, W97
Dual polyhedronGreat pentakis dodecahedron
Vertex figure
5.10/3.10/3
Bowers acronymQuit Sissid
Related polyhedra
It shares its vertex arrangement with three other uniform polyhedra: the convex rhombicosidodecahedron, the small dodecicosidodecahedron and the small rhombidodecahedron.
It also has the same vertex arrangement as the uniform compounds of 6 or 12 pentagrammic prisms.
Rhombicosidodecahedron
Small dodecicosidodecahedron
Small rhombidodecahedron
Small stellated truncated dodecahedron
Compound of six pentagrammic prisms
Compound of twelve pentagrammic prisms
See also
• List of uniform polyhedra
References
1. Maeder, Roman. "58: small stellated truncated dodecahedron". MathConsult.{{cite web}}: CS1 maint: url-status (link)
External links
• Weisstein, Eric W. "Small stellated truncated dodecahedron". MathWorld.
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Great truncated cuboctahedron
In geometry, the great truncated cuboctahedron (or quasitruncated cuboctahedron or stellatruncated cuboctahedron) is a nonconvex uniform polyhedron, indexed as U20. It has 26 faces (12 squares, 8 hexagons and 6 octagrams), 72 edges, and 48 vertices.[1] It is represented by the Schläfli symbol tr{4/3,3}, and Coxeter-Dynkin diagram . It is sometimes called the quasitruncated cuboctahedron because it is related to the truncated cuboctahedron, , except that the octagonal faces are replaced by {8/3} octagrams.
Great truncated cuboctahedron
TypeUniform star polyhedron
ElementsF = 26, E = 72
V = 48 (χ = 2)
Faces by sides12{4}+8{6}+6{8/3}
Coxeter diagram
Wythoff symbol2 3 4/3 |
Symmetry groupOh, [4,3], *432
Index referencesU20, C67, W93
Dual polyhedronGreat disdyakis dodecahedron
Vertex figure
4.6/5.8/3
Bowers acronymQuitco
Convex hull
Its convex hull is a nonuniform truncated cuboctahedron. The truncated cuboctahedron and the great truncated cuboctahedron form isomorphic graphs despite their different geometric structure.
Convex hull
Great truncated cuboctahedron
Orthographic projections
Cartesian coordinates
Cartesian coordinates for the vertices of a great truncated cuboctahedron centered at the origin are all permutations of
(±1, ±(1−√2), ±(1−2√2)).
See also
• List of uniform polyhedra
References
1. Maeder, Roman. "20: great truncated cuboctahedron". MathConsult. Archived from the original on 2020-02-17.
External links
• Weisstein, Eric W. "Great truncated cuboctahedron". MathWorld.
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Truncated dodecadodecahedron
In geometry, the truncated dodecadodecahedron (or stellatruncated dodecadodecahedron) is a nonconvex uniform polyhedron, indexed as U59. It is given a Schläfli symbol t0,1,2{5⁄3,5}. It has 54 faces (30 squares, 12 decagons, and 12 decagrams), 180 edges, and 120 vertices.[1] The central region of the polyhedron is connected to the exterior via 20 small triangular holes.
Truncated dodecadodecahedron
TypeUniform star polyhedron
ElementsF = 54, E = 180
V = 120 (χ = −6)
Faces by sides30{4}+12{10}+12{10/3}
Coxeter diagram
Wythoff symbol2 5 5/3 |
Symmetry groupIh, [5,3], *532
Index referencesU59, C75, W98
Dual polyhedronMedial disdyakis triacontahedron
Vertex figure
4.10/9.10/3
Bowers acronymQuitdid
The name truncated dodecadodecahedron is somewhat misleading: truncation of the dodecadodecahedron would produce rectangular faces rather than squares, and the pentagram faces of the dodecadodecahedron would turn into truncated pentagrams rather than decagrams. However, it is the quasitruncation of the dodecadodecahedron, as defined by Coxeter, Longuet-Higgins & Miller (1954).[2] For this reason, it is also known as the quasitruncated dodecadodecahedron.[3] Coxeter et al. credit its discovery to a paper published in 1881 by Austrian mathematician Johann Pitsch.[4]
Cartesian coordinates
Cartesian coordinates for the vertices of a truncated dodecadodecahedron are all the triples of numbers obtained by circular shifts and sign changes from the following points (where $\tau ={\frac {1+{\sqrt {5}}}{2}}$ is the golden ratio):
$(1,1,3);\quad ({\frac {1}{\tau }},{\frac {1}{\tau ^{2}}},2\tau );\quad (\tau ,{\frac {2}{\tau }},\tau ^{2});\quad (\tau ^{2},{\frac {1}{\tau ^{2}}},2);\quad ({\sqrt {5}},1,{\sqrt {5}}).$
Each of these five points has eight possible sign patterns and three possible circular shifts, giving a total of 120 different points.
As a Cayley graph
The truncated dodecadodecahedron forms a Cayley graph for the symmetric group on five elements, as generated by two group members: one that swaps the first two elements of a five-tuple, and one that performs a circular shift operation on the last four elements. That is, the 120 vertices of the polyhedron may be placed in one-to-one correspondence with the 5! permutations on five elements, in such a way that the three neighbors of each vertex are the three permutations formed from it by swapping the first two elements or circularly shifting (in either direction) the last four elements.[5]
Related polyhedra
Medial disdyakis triacontahedron
Medial disdyakis triacontahedron
TypeStar polyhedron
Face
ElementsF = 120, E = 180
V = 54 (χ = −6)
Symmetry groupIh, [5,3], *532
Index referencesDU59
dual polyhedronTruncated dodecadodecahedron
The medial disdyakis triacontahedron is a nonconvex isohedral polyhedron. It is the dual of the uniform truncated dodecadodecahedron.
See also
• List of uniform polyhedra
References
1. Maeder, Roman. "59: truncated dodecadodecahedron". MathConsult.{{cite web}}: CS1 maint: url-status (link)
2. Coxeter, H. S. M.; Longuet-Higgins, M. S.; Miller, J. C. P. (1954), "Uniform polyhedra", Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences, 246: 401–450, Bibcode:1954RSPTA.246..401C, doi:10.1098/rsta.1954.0003, JSTOR 91532, MR 0062446. See especially the description as a quasitruncation on p. 411 and the photograph of a model of its skeleton in Fig. 114, Plate IV.
3. Wenninger writes "quasitruncated dodecahedron", but this appears to be a mistake. Wenninger, Magnus J. (1971), "98 Quasitruncated dodecahedron", Polyhedron Models, Cambridge University Press, pp. 152–153.
4. Pitsch, Johann (1881), "Über halbreguläre Sternpolyeder", Zeitschrift für das Realschulwesen, 6: 9–24, 72–89, 216. According to Coxeter, Longuet-Higgins & Miller (1954), the truncated dodecadodecahedron appears as no. XII on p.86.
5. Eppstein, David (2008), "The topology of bendless three-dimensional orthogonal graph drawing", in Tollis, Ioannis G.; Patrignani, Marizio (eds.), Proc. 16th Int. Symp. Graph Drawing, Lecture Notes in Computer Science, vol. 5417, Heraklion, Crete: Springer-Verlag, pp. 78–89, arXiv:0709.4087, doi:10.1007/978-3-642-00219-9_9.
• Wenninger, Magnus (1983), Dual Models, Cambridge University Press, doi:10.1017/CBO9780511569371, ISBN 978-0-521-54325-5, MR 0730208
External links
• Weisstein, Eric W. "Truncated dodecadodecahedron". MathWorld.
• Weisstein, Eric W. "Medial disdyakis triacontahedron". MathWorld.
Star-polyhedra navigator
Kepler-Poinsot
polyhedra
(nonconvex
regular polyhedra)
• small stellated dodecahedron
• great dodecahedron
• great stellated dodecahedron
• great icosahedron
Uniform truncations
of Kepler-Poinsot
polyhedra
• dodecadodecahedron
• truncated great dodecahedron
• rhombidodecadodecahedron
• truncated dodecadodecahedron
• snub dodecadodecahedron
• great icosidodecahedron
• truncated great icosahedron
• nonconvex great rhombicosidodecahedron
• great truncated icosidodecahedron
Nonconvex uniform
hemipolyhedra
• tetrahemihexahedron
• cubohemioctahedron
• octahemioctahedron
• small dodecahemidodecahedron
• small icosihemidodecahedron
• great dodecahemidodecahedron
• great icosihemidodecahedron
• great dodecahemicosahedron
• small dodecahemicosahedron
Duals of nonconvex
uniform polyhedra
• medial rhombic triacontahedron
• small stellapentakis dodecahedron
• medial deltoidal hexecontahedron
• small rhombidodecacron
• medial pentagonal hexecontahedron
• medial disdyakis triacontahedron
• great rhombic triacontahedron
• great stellapentakis dodecahedron
• great deltoidal hexecontahedron
• great disdyakis triacontahedron
• great pentagonal hexecontahedron
Duals of nonconvex
uniform polyhedra with
infinite stellations
• tetrahemihexacron
• hexahemioctacron
• octahemioctacron
• small dodecahemidodecacron
• small icosihemidodecacron
• great dodecahemidodecacron
• great icosihemidodecacron
• great dodecahemicosacron
• small dodecahemicosacron
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Stellated truncated hexahedron
In geometry, the stellated truncated hexahedron (or quasitruncated hexahedron, and stellatruncated cube[1]) is a uniform star polyhedron, indexed as U19. It has 14 faces (8 triangles and 6 octagrams), 36 edges, and 24 vertices.[2] It is represented by Schläfli symbol t'{4,3} or t{4/3,3}, and Coxeter-Dynkin diagram, . It is sometimes called quasitruncated hexahedron because it is related to the truncated cube, , except that the square faces become inverted into {8/3} octagrams.
Stellated truncated hexahedron
TypeUniform star polyhedron
ElementsF = 14, E = 36
V = 24 (χ = 2)
Faces by sides8{3}+6{8/3}
Coxeter diagram{{{stH-Coxeter}}}
Wythoff symbol2 3 | 4/3
2 3/2 | 4/3
Symmetry groupOh, [4,3], *432
Index referencesU19, C66, W92
Dual polyhedronGreat triakis octahedron
Vertex figure
3.8/3.8/3
Bowers acronymQuith
Even though the stellated truncated hexahedron is a stellation of the truncated hexahedron, its core is a regular octahedron.
Orthographic projections
Related polyhedra
It shares the vertex arrangement with three other uniform polyhedra: the convex rhombicuboctahedron, the small rhombihexahedron, and the small cubicuboctahedron.
Rhombicuboctahedron
Small cubicuboctahedron
Small rhombihexahedron
Stellated truncated hexahedron
See also
• List of uniform polyhedra
References
1. Weisstein, Eric W. "Uniform Polyhedron". MathWorld.{{cite web}}: CS1 maint: url-status (link)
2. Maeder, Roman. "19: stellated truncated hexahedron". MathConsult.{{cite web}}: CS1 maint: url-status (link)
External links
• Weisstein, Eric W. "Stellated truncated hexahedron". MathWorld.
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Q.E.D.
Q.E.D. or QED is an initialism of the Latin phrase quod erat demonstrandum, meaning "which was to be demonstrated". Literally it states "what was to be shown".[1] Traditionally, the abbreviation is placed at the end of mathematical proofs and philosophical arguments in print publications, to indicate that the proof or the argument is complete.
Etymology and early use
The phrase quod erat demonstrandum is a translation into Latin from the Greek ὅπερ ἔδει δεῖξαι (hoper edei deixai; abbreviated as ΟΕΔ). Translating from the Latin phrase into English yields "what was to be demonstrated". However, translating the Greek phrase ὅπερ ἔδει δεῖξαι can produce a slightly different meaning. In particular, since the verb "δείκνυμι" also means to show or to prove,[2] a different translation from the Greek phrase would read "The very thing it was required to have shown."[3]
The Greek phrase was used by many early Greek mathematicians, including Euclid[4] and Archimedes.
The Latin phrase is attested in a 1501 Euclid translation of Giorgio Valla.[5] Its abbreviation q.e.d. is used once in 1598 by Johannes Praetorius,[6] more in 1643 by Anton Deusing,[7] extensively in 1655 by Isaac Barrow in the form Q.E.D.,[8] and subsequently by many post-Renaissance mathematicians and philosophers.[9]
Modern philosophy
During the European Renaissance, scholars often wrote in Latin, and phrases such as Q.E.D. were often used to conclude proofs.
Perhaps the most famous use of Q.E.D. in a philosophical argument is found in the Ethics of Baruch Spinoza, published posthumously in 1677.[11] Written in Latin, it is considered by many to be Spinoza's magnum opus. The style and system of the book are, as Spinoza says, "demonstrated in geometrical order", with axioms and definitions followed by propositions. For Spinoza, this is a considerable improvement over René Descartes's writing style in the Meditations, which follows the form of a diary.[12]
Difference from Q.E.F.
There is another Latin phrase with a slightly different meaning, usually shortened similarly, but being less common in use. Quod erat faciendum, originating from the Greek geometers' closing ὅπερ ἔδει ποιῆσαι (hoper edei poiēsai), meaning "which had to be done".[13] Because of the difference in meaning, the two phrases should not be confused.
Euclid used the Greek original of Quod Erat Faciendum (Q.E.F.) to close propositions that were not proofs of theorems, but constructions of geometric objects.[14] For example, Euclid's first proposition showing how to construct an equilateral triangle, given one side, is concluded this way.[15]
English equivalent
There is no common formal English equivalent, although the end of a proof may be announced with a simple statement such as “thus it is proved,” "this completes the proof", "as required", "as desired", "as expected", "hence proved", "ergo", "so correct", or other similar locutions.
Typographical forms used symbolically
Due to the paramount importance of proofs in mathematics, mathematicians since the time of Euclid have developed conventions to demarcate the beginning and end of proofs. In printed English language texts, the formal statements of theorems, lemmas, and propositions are set in italics by tradition. The beginning of a proof usually follows immediately thereafter, and is indicated by the word "proof" in boldface or italics. On the other hand, several symbolic conventions exist to indicate the end of a proof.
While some authors still use the classical abbreviation, Q.E.D., it is relatively uncommon in modern mathematical texts. Paul Halmos claims to have pioneered the use of a solid black square (or rectangle) at the end of a proof as a Q.E.D. symbol,[16] a practice which has become standard, although not universal. Halmos noted that he adopted this use of a symbol from magazine typography customs in which simple geometric shapes had been used to indicate the end of an article, so-called end marks.[17][18] This symbol was later called the tombstone, the Halmos symbol, or even a halmos by mathematicians. Often the Halmos symbol is drawn on chalkboard to signal the end of a proof during a lecture, although this practice is not so common as its use in printed text.
The tombstone symbol appears in TeX as the character $\blacksquare $ (filled square, \blacksquare) and sometimes, as a $\square $ (hollow square, \square or \Box).[19] In the AMS Theorem Environment for LaTeX, the hollow square is the default end-of-proof symbol. Unicode explicitly provides the "end of proof" character, U+220E (∎). Some authors use other Unicode symbols to note the end of a proof, including, ▮ (U+25AE, a black vertical rectangle), and ‣ (U+2023, a triangular bullet). Other authors have adopted two forward slashes (//, $//$) or four forward slashes (////, $////$).[20] In other cases, authors have elected to segregate proofs typographically—by displaying them as indented blocks.[21]
Modern humorous use
In Joseph Heller's 1961 book Catch-22, the Chaplain, having been told to examine a forged letter allegedly signed by him (which he knew he didn't sign), verified that his name was in fact there. His investigator replied, "Then you wrote it. Q.E.D." The chaplain said he did not write it and that it was not his handwriting, to which the investigator replied, "Then you signed your name in somebody else's handwriting again."[22]
In the 1978 science-fiction radio comedy, and later in the television, novel, and film adaptations of The Hitchhiker's Guide to the Galaxy, "Q.E.D." is referred to in the Guide's entry for the babel fish, when it is claimed that the babel fish – which serves the "mind-bogglingly" useful purpose of being able to translate any spoken language when inserted into a person's ear – is used as evidence for existence and non-existence of God. The exchange from the novel is as follows: "'I refuse to prove I exist,' says God, 'for proof denies faith, and without faith I am nothing.' 'But,' says Man, 'The babel fish is a dead giveaway, isn't it? It could not have evolved by chance. It proves you exist, and so therefore, by your own arguments, you don't. QED.' 'Oh dear,' says God, 'I hadn't thought of that,' and promptly vanishes in a puff of logic."[23]
In Neal Stephenson's 1999 novel Cryptonomicon, Q.E.D. is used as a punchline to several humorous anecdotes, in which characters go to great lengths to prove something non-mathematical.[24]
Singer-songwriter Thomas Dolby's 1988 song "Airhead" includes the lyric, "Quod erat demonstrandum, baby," referring to the self-evident vacuousness of the eponymous subject; and in response, a female voice delightedly squeals, "Oooh... you speak French!" [25]
See also
• List of Latin abbreviations
• A priori and a posteriori
• Bob's your uncle
• Ipso facto
• Q.E.A.
• List of Latin phrases (E) § ergo
References
1. "Definition of QUOD ERAT DEMONSTRANDUM". www.merriam-webster.com. Retrieved 2017-09-03.
2. Entry δείκνυμι at LSJ.
3. Euclid's Elements translated from Greek by Thomas L. Heath. 2003 Green Lion Press pg. xxiv
4. Elements 2.5 by Euclid (ed. J. L. Heiberg), retrieved 16 July 2005
5. Valla, Giorgio. "Georgii Vallae Placentini viri clariss. De expetendis, et fugiendis rebus opus. 1".
6. Praetorius, Johannes. "Ioannis Praetorii Ioachimici Problema, quod iubet ex Quatuor rectis lineis datis quadrilaterum fieri, quod sit in Circulo".
7. Deusing, Anton. "Antonii Deusingii Med. ac Philos. De Vero Systemate Mundi Dissertatio Mathematica : Quâ Copernici Systema Mundi reformatur: Sublatis interim infinitis penè orbibus, quibus in Systemate Ptolemaico humana mens distrahitur".
8. Barrow, Isaac. "Elementa geometrie : libri XV".
9. "Earliest Known Uses of some of the Words of Mathematics (Q)". jeff560.tripod.com. Retrieved 2019-11-04.
10. Philippe van Lansberge (1604). Triangulorum Geometriæ. Apud Zachariam Roman. pp. 1–5. quod-erat-demonstrandum 0-1700.
11. "Baruch Spinoza (1632–1677) – Modern Philosophy". opentextbc.ca. Retrieved 2019-11-04.
12. The Chief Works of Benedict De Spinoza, translated by R. H. M. Elwes, 1951. ISBN 0-486-20250-X.
13. Gauss, Carl Friedrich; Waterhouse, William C. (7 February 2018). Disquisitiones Arithmeticae. ISBN 9781493975600.
14. Weisstein, Eric W. "Q.E.F." mathworld.wolfram.com. Retrieved 2019-11-04.
15. "Euclid's Elements, Book I, Proposition 1". mathcs.clarku.edu. Retrieved 2019-11-04.
16. This (generally accepted) claim was made in Halmos's autobiography, I Want to Be a Mathematician. The first usage of the solid black rectangle as an end-of-proof symbol appears to be in Halmos's Measure Theory (1950). The intended meaning of the symbol is explicitly given in the preface.
17. Halmos, Paul R. (1985). I Want to Be a Mathematician: An Automathography. p. 403. ISBN 9781461210849.
18. Felici, James (2003). "The complete manual of typography : a guide to setting perfect type". Berkeley, CA : Peachpit Press.
19. See, for example, list of mathematical symbols for more.
20. Rudin, Walter (1987). Real and Complex Analysis. McGraw-Hill. ISBN 0-07-100276-6.
21. Rudin, Walter (1976). Principles of Mathematical Analysis. New York: McGraw-Hill. ISBN 0-07-054235-X.
22. Heller, Joseph (1971). Catch-22. ISBN 978-0-573-60685-4. Retrieved 15 July 2011.
23. Adams, Douglas (2005). The Hitchhiker's Guide to the Galaxy. The Hitchhiker's Guide to the Galaxy (Film tie-in ed.). Basingstoke and Oxford: Pan Macmillan. pp. 62–64. ISBN 0-330-43798-4.
24. Stephenson, Neal (1999). Cryptonomicon. New York, NY: Avon Books. ISBN 978-0-06-051280-4.
25. "Airhead – Thomas Dolby". play.google.com. Retrieved 2016-09-15.
External links
Look up quod erat demonstrandum or QED in Wiktionary, the free dictionary.
• Earliest Known Uses of Some of the Words of Mathematics (Q)
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Quota rule
In mathematics and political science, the quota rule describes a desired property of a proportional apportionment or election method. It states that the number of seats that should be allocated to a given party should be between the upper or lower roundings (called upper and lower quotas) of its fractional proportional share (called natural quota).[1] As an example, if a party deserves 10.56 seats out of 15, the quota rule states that when the seats are allotted, the party may get 10 or 11 seats, but not lower or higher. Many common election methods, such as all highest averages methods, violate the quota rule.
Mathematics
If $P$ is the population of the party, $T$ is the total population, and $S$ is the number of available seats, then the natural quota for that party (the number of seats the party would ideally get) is
${\frac {P}{T}}\cdot S$
The lower quota is then the natural quota rounded down to the nearest integer while the upper quota is the natural quota rounded up. The quota rule states that the only two allocations that a party can receive should be either the lower or upper quota.[1] If at any time an allocation gives a party a greater or lesser number of seats than the upper or lower quota, that allocation (and by extension, the method used to allocate it) is said to be in violation of the quota rule. Another way to state this is to say that a given method only satisfies the quota rule if each party's allocation differs from its natural quota by less than one, where each party's allocation is an integer value.[2]
Example
If there are 5 available seats in the council of a club with 300 members, and party A has 106 members, then the natural quota for party A is ${\frac {106}{300}}\cdot 5\approx 1.8$. The lower quota for party A is 1, because 1.8 rounded down equal 1. The upper quota, 1.8 rounded up, is 2. Therefore, the quota rule states that the only two allocations allowed for party A are 1 or 2 seats on the council. If there is a second party, B, that has 137 members, then the quota rule states that party B gets ${\frac {137}{300}}\cdot 5\approx 2.3$, rounded up and down equals either 2 or 3 seats. Finally, a party C with the remaining 57 members of the club has a natural quota of ${\frac {57}{300}}\cdot 5\approx 0.95$, which means its allocated seats should be either 0 or 1. In all cases, the method for actually allocating the seats determines whether an allocation violates the quota rule, which in this case would mean giving party A any seats other than 1 or 2, giving party B any other than 2 or 3, or giving party C any other than 0 or 1 seat.
Relation to apportionment paradoxes
The Balinski–Young theorem proved in 1980 that if an apportionment method satisfies the quota rule, it must fail to satisfy some apportionment paradox.[3] For instance, although Largest remainder method satisfies the quota rule, it violates the Alabama paradox and the population paradox.[4] The theorem itself is broken up into several different proofs that cover a wide number of circumstances.[5]
Specifically, there are two main statements that apply to the quota rule:
• Any method that follows the quota rule must fail the population paradox.[5]
• Any method that is free of both the Alabama paradox and the population paradox must necessarily fail the quota rule for some circumstances.[5]
Use in apportionment methods
Different methods for allocating seats may or may not satisfy the quota rule. While many methods do violate the quota rule, it is sometimes preferable to violate the rule very rarely than to violate some other apportionment paradox; some sophisticated methods violate the rule so rarely that it has not ever happened in a real apportionment, while some methods that never violate the quota rule violate other paradoxes in much more serious fashions.
The Largest remainder method does satisfy the quota rule. The method works by proportioning seats equally until a fractional value is reached; the surplus seats are then given to the party with the largest fractional parts until there are no more surplus seats. Because it is impossible to give more than one surplus seat to a party, every party will always get either its lower or upper quota.[6]
The D'Hondt method, also known as the Jefferson method[7] sometimes violates the quota rule by allocating more seats than the upper quota allowed.[8] Since Jefferson was the first method used for Congressional apportionment in the United States, this violation led to a growing problem where larger states receive more representatives than smaller states, which was not corrected until the Webster/Sainte-Laguë method was implemented in 1842; even though Webster/Sainte-Laguë does violate the quota rule, it happens extremely rarely.[9]
References
1. Michael J. Caulfield. "Apportioning Representatives in the United States Congress - The Quota Rule". MAA Publications. Retrieved October 22, 2018
2. Alan Stein. Apportionment Methods Retrieved December 9, 2018
3. Beth-Allyn Osikiewicz, Ph.D. Impossibilities of Apportionment Retrieved October 23, 2018.
4. Warren D. Smith. (2007).Apportionment and rounding schemes Retrieved October 23, 2018
5. M.L. Balinski and H.P. Young. (1980). "The Theory of Apportionment". Retrieved October 23 2018
6. Hilary Freeman. "Apportionment". Retrieved October 22 2018
7. "Apportionment 2" Retrieved October 22, 2018.
8. Jefferson’s Method Retrieved October 22, 2018.
9. Ghidewon Abay Asmerom. Apportionment. Lecture 4. Retrieved October 23, 2018.
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Wikipedia
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Quota sampling
Quota sampling is a method for selecting survey participants that is a non-probabilistic version of stratified sampling.
Process
In quota sampling, a population is first segmented into mutually exclusive sub-groups, just as in stratified sampling. Then judgment is used to select the subjects or units from each segment based on a specified proportion. For example, an interviewer may be told to sample 200 females and 300 males between the age of 45 and 60. This means that individuals can put a demand on who they want to sample (targeting).
This second step makes the technique non-probability sampling. In quota sampling, there is non-random sample selection and this can be unreliable. For example, interviewers might be tempted to interview those people in the street who look most helpful, or may choose to use accidental sampling to question those closest to them, to save time. The problem is these samples may be biased in a way that is difficult to quantify or adjust for. For example, if interviewers decide to question the first person they see, they may oversample tall respondents (who are more easily visible from a distance), which could lead to an overestimate of average income. This non-random element is a source of uncertainty about the nature of the actual sample.[1]
Uses
Quota sampling is useful when time is limited, a sampling frame is not available, the research budget is very tight or detailed accuracy is not important. Subsets are chosen and then either convenience or judgment sampling is used to choose people from each subset. The researcher decides how many of each category are selected.
Connection to stratified sampling
Quota sampling is the non-probability version of stratified sampling. In stratified sampling, subsets of the population are created so that each subset has a common characteristic, such as gender. Random sampling chooses a number of subjects from each subset with, unlike a quota sample, each potential subject having a known probability of being selected.[2]
See also
• Coefficient of variation
• Standard deviation
References
• Dodge, Y. (2003) The Oxford Dictionary of Statistical Terms, OUP. ISBN 0-19-920613-9
1. Marketing Research and Information Systems. (Marketing and Agribusiness Texts – 4)
2. Kenneth F Warren (2018-02-15). In Defense Of Public Opinion Polling. Routledge, 2018. ISBN 9780429979538.
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Wikipedia
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Quotient (disambiguation)
Quotient is the result of division in mathematics.
Look up quotient in Wiktionary, the free dictionary.
Quotient may also refer to:
Mathematics
• Quotient set by an equivalence relation
• Quotient group
• Quotient ring
• Quotient module
• Quotient space (linear algebra)
• Quotient space (topology), by an equivalence relation in the case of a topological space
• Quotient (universal algebra)
• Quotient object in a category
• Quotient category
• Quotient of a formal language
• Quotient type
Other uses
• Intelligence quotient, a psychological measurement of human intelligence
• Quotient Technology, the parent company of Coupons.com
• Quotients (EP), music by the band Hyland
• Runs Per Wicket Ratio, a statistic used to rank teams in league tables in cricket, also known as Quotient
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Wikipedia
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Quotient group
A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored" out). For example, the cyclic group of addition modulo n can be obtained from the group of integers under addition by identifying elements that differ by a multiple of $n$ and defining a group structure that operates on each such class (known as a congruence class) as a single entity. It is part of the mathematical field known as group 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
For a congruence relation on a group, the equivalence class of the identity element is always a normal subgroup of the original group, and the other equivalence classes are precisely the cosets of that normal subgroup. The resulting quotient is written $G\,/\,N$, where $G$ is the original group and $N$ is the normal subgroup. (This is pronounced $G{\bmod {N}}$, where ${\mbox{mod}}$ is short for modulo.)
Much of the importance of quotient groups is derived from their relation to homomorphisms. The first isomorphism theorem states that the image of any group G under a homomorphism is always isomorphic to a quotient of $G$. Specifically, the image of $G$ under a homomorphism $\varphi :G\rightarrow H$ is isomorphic to $G\,/\,\ker(\varphi )$ where $\ker(\varphi )$ denotes the kernel of $\varphi $.
The dual notion of a quotient group is a subgroup, these being the two primary ways of forming a smaller group from a larger one. Any normal subgroup has a corresponding quotient group, formed from the larger group by eliminating the distinction between elements of the subgroup. In category theory, quotient groups are examples of quotient objects, which are dual to subobjects.
For other examples of quotient objects, see quotient ring, quotient space (linear algebra), quotient space (topology), and quotient set.
Definition and illustration
Given a group $G$ and a subgroup $H$, and a fixed element $a\in G$, one can consider the corresponding left coset: $aH:=\left\{ah:h\in H\right\}$. Cosets are a natural class of subsets of a group; for example consider the abelian group G of integers, with operation defined by the usual addition, and the subgroup $H$ of even integers. Then there are exactly two cosets: $0+H$, which are the even integers, and $1+H$, which are the odd integers (here we are using additive notation for the binary operation instead of multiplicative notation).
For a general subgroup $H$, it is desirable to define a compatible group operation on the set of all possible cosets, $\left\{aH:a\in G\right\}$. This is possible exactly when $H$ is a normal subgroup, see below. A subgroup $N$ of a group $G$ is normal if and only if the coset equality $aN=Na$ holds for all $a\in G$. A normal subgroup of $G$ is denoted $N$.
Definition
Let $N$ be a normal subgroup of a group $G$ . Define the set $G\,/\,N$ to be the set of all left cosets of $N$ in $G$ . That is, $G\,/\,N=\left\{aN:a\in G\right\}$. Since the identity element $e\in N$, $a\in aN$. Define a binary operation on the set of cosets, $G\,/\,N$, as follows. For each $aN$ and $bN$ in $G\,/\,N$, the product of $aN$ and $bN$, $(aN)(bN)$, is $(ab)N$. This works only because $(ab)N$ does not depend on the choice of the representatives, 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/":): a and $b$, of each left coset, $aN$ and $bN$. To prove this, suppose $xN=aN$ and $yN=bN$ for some $x,y,a,b\in G$. Then
$ (ab)N=a(bN)=a(yN)=a(Ny)=(aN)y=(xN)y=x(Ny)=x(yN)=(xy)N$.
This depends on the fact that N is a normal subgroup. It still remains to be shown that this condition is not only sufficient but necessary to define the operation on G/N.
To show that it is necessary, consider that for a subgroup $N$ of $G$, we have been given that the operation is well defined. That is, for all $xN=aN$ and $yN=bN$, for $x,y,a,b\in G,\;(ab)N=(xy)N$.
Let $n\in N$ and $g\in G$. Since $eN=nN$, we have $gN=(eg)N=(eN)(gN)=(nN)(gN)=(ng)N$.
Now, $gN=(ng)N\Leftrightarrow N=(g^{-1}ng)N\Leftrightarrow g^{-1}ng\in N,\;\forall \,n\in N$ and $g\in G$.
Hence $N$ is a normal subgroup of $G$ .
It can also be checked that this operation on $G\,/\,N$ is always associative, $G\,/\,N$ has identity element $N$, and the inverse of element $aN$ can always be represented by $a^{-1}N$. Therefore, the set $G\,/\,N$ together with the operation defined by $(aN)(bN)=(ab)N$ forms a group, the quotient group of $G$ by $N$.
Due to the normality of $N$, the left cosets and right cosets of $N$ in $G$ are the same, and so, $G\,/\,N$ could have been defined to be the set of right cosets of $N$ in $G$ .
Example: Addition modulo 6
For example, consider the group with addition modulo 6: $G=\left\{0,1,2,3,4,5\right\}$. Consider the subgroup $N=\left\{0,3\right\}$, which is normal because $G$ is abelian. Then the set of (left) cosets is of size three:
$G\,/\,N=\left\{a+N:a\in G\right\}=\left\{\left\{0,3\right\},\left\{1,4\right\},\left\{2,5\right\}\right\}=\left\{0+N,1+N,2+N\right\}$.
The binary operation defined above makes this set into a group, known as the quotient group, which in this case is isomorphic to the cyclic group of order 3.
Motivation for the name "quotient"
The reason $G\,/\,N$ is called a quotient group comes from division of integers. When dividing 12 by 3 one obtains the answer 4 because one can regroup 12 objects into 4 subcollections of 3 objects. The quotient group is the same idea, although we end up with a group for a final answer instead of a number because groups have more structure than an arbitrary collection of objects.
To elaborate, when looking at $G\,/\,N$ with $N$ a normal subgroup of $G$, the group structure is used to form a natural "regrouping". These are the cosets of $N$ in $G$. Because we started with a group and normal subgroup, the final quotient contains more information than just the number of cosets (which is what regular division yields), but instead has a group structure itself.
Examples
Even and odd integers
Consider the group of integers $\mathbb {Z} $ (under addition) and the subgroup $2\mathbb {Z} $ consisting of all even integers. This is a normal subgroup, because $\mathbb {Z} $ is abelian. There are only two cosets: the set of even integers and the set of odd integers, and therefore the quotient group $\mathbb {Z} \,/\,2\mathbb {Z} $ is the cyclic group with two elements. This quotient group is isomorphic with the set $\left\{0,1\right\}$ with addition modulo 2; informally, it is sometimes said that $\mathbb {Z} \,/\,2\mathbb {Z} $ equals the set $\left\{0,1\right\}$ with addition modulo 2.
Example further explained...
Let $\gamma (m)$ be the remainders of $m\in \mathbb {Z} $ when dividing by $2$. Then, $\gamma (m)=0$ when $m$ is even and $\gamma (m)=1$ when $m$ is odd.
By definition of $\gamma $, the kernel of $\gamma $, $\ker(\gamma )$ $=\{m\in \mathbb {Z} :\gamma (m)=0\}$ :\gamma (m)=0\}} , is the set of all even integers.
Let $H=$ $\ker(\gamma )$. Then, $H$ is a subgroup, because the identity in $\mathbb {Z} $, which is $0$, is in $H$, the sum of two even integers is even and hence if $m$ and $n$ are in $H$, $m+n$ is in $H$ (closure) and if $m$ is even, $-m$ is also even and so $H$ contains its inverses.
Define $\mu :\mathbb {Z} /H\to \mathbb {Z} _{2}$ :\mathbb {Z} /H\to \mathbb {Z} _{2}} as $\mu (aH)=\gamma (a)$ for $a\in \mathbb {Z} $ and $\mathbb {Z} /H$ is the quotient group of left cosets; $\mathbb {Z} /H=\{H,1+H\}$.
Note that we have defined $\mu $, $\mu (aH)$ is $1$ if $a$ is odd and $0$ if $a$ is even.
Thus, $\mu $ is an isomorphism from $\mathbb {Z} /H$ to $\mathbb {Z} _{2}$.
Remainders of integer division
A slight generalization of the last example. Once again consider the group of integers $\mathbb {Z} $ under addition. Let n be any positive integer. We will consider the subgroup $n\mathbb {Z} $ of $\mathbb {Z} $ consisting of all multiples of $n$. Once again $n\mathbb {Z} $ is normal in $\mathbb {Z} $ because $\mathbb {Z} $ is abelian. The cosets are the collection $\left\{n\mathbb {Z} ,1+n\mathbb {Z} ,\;\ldots ,(n-2)+n\mathbb {Z} ,(n-1)+n\mathbb {Z} \right\}$. An integer $k$ belongs to the coset $r+n\mathbb {Z} $, where $r$ is the remainder when dividing $k$ by $n$. The quotient $\mathbb {Z} \,/\,n\mathbb {Z} $ can be thought of as the group of "remainders" modulo $n$. This is a cyclic group of order $n$.
Complex integer roots of 1
The twelfth roots of unity, which are points on the complex unit circle, form a multiplicative abelian group $G$, shown on the picture on the right as colored balls with the number at each point giving its complex argument. Consider its subgroup $N$ made of the fourth roots of unity, shown as red balls. This normal subgroup splits the group into three cosets, shown in red, green and blue. One can check that the cosets form a group of three elements (the product of a red element with a blue element is blue, the inverse of a blue element is green, etc.). Thus, the quotient group $G\,/\,N$ is the group of three colors, which turns out to be the cyclic group with three elements.
The real numbers modulo the integers
Consider the group of real numbers $\mathbb {R} $ under addition, and the subgroup $\mathbb {Z} $ of integers. Each coset of $\mathbb {Z} $ in $\mathbb {R} $ is a set of the form $a+\mathbb {Z} $, where $a$ is a real number. Since $a_{1}+\mathbb {Z} $ and $a_{2}+\mathbb {Z} $ are identical sets when the non-integer parts of $a_{1}$ and $a_{2}$ are equal, one may impose the restriction $0\leq a<1$ without change of meaning. Adding such cosets is done by adding the corresponding real numbers, and subtracting 1 if the result is greater than or equal to 1. The quotient group $\mathbb {R} \,/\,\mathbb {Z} $ is isomorphic to the circle group, the group of complex numbers of absolute value 1 under multiplication, or correspondingly, the group of rotations in 2D about the origin, that is, the special orthogonal group ${\mbox{SO}}(2)$. An isomorphism is given by $f(a+\mathbb {Z} )=\exp(2\pi ia)$ (see Euler's identity).
Matrices of real numbers
If $G$ is the group of invertible $3\times 3$ real matrices, and $N$ is the subgroup of $3\times 3$ real matrices with determinant 1, then $N$ is normal in $G$ (since it is the kernel of the determinant homomorphism). The cosets of $N$ are the sets of matrices with a given determinant, and hence $G\,/\,N$ is isomorphic to the multiplicative group of non-zero real numbers. The group $N$ is known as the special linear group ${\mbox{SL}}(3)$.
Integer modular arithmetic
Consider the abelian group $\mathbb {Z} _{4}=\mathbb {Z} \,/\,4\mathbb {Z} $ (that is, the set $\left\{0,1,2,3\right\}$ with addition modulo 4), and its subgroup $\left\{0,2\right\}$. The quotient group $\mathbb {Z} _{4}\,/\,\left\{0,2\right\}$ is $\left\{\left\{0,2\right\},\left\{1,3\right\}\right\}$. This is a group with identity element $\left\{0,2\right\}$, and group operations such as $\left\{0,2\right\}+\left\{1,3\right\}=\left\{1,3\right\}$. Both the subgroup $\left\{0,2\right\}$ and the quotient group $\left\{\left\{0,2\right\},\left\{1,3\right\}\right\}$ are isomorphic with $\mathbb {Z} _{2}$.
Integer multiplication
Consider the multiplicative group $G=(\mathbb {Z} _{n^{2}})^{\times }$. The set $N$ of $n$th residues is a multiplicative subgroup isomorphic to $(\mathbb {Z} _{n})^{\times }$. Then $N$ is normal in $G$ and the factor group $G\,/\,N$ has the cosets $N,(1+n)N,(1+n)2N,\;\ldots ,(1+n)n-1N$. The Paillier cryptosystem is based on the conjecture that it is difficult to determine the coset of a random element of $G$ without knowing the factorization of $n$.
Properties
The quotient group $G\,/\,G$ is isomorphic to the trivial group (the group with one element), and $G\,/\,\left\{e\right\}$ is isomorphic to $G$.
The order of $G\,/\,N$, by definition the number of elements, is equal to $\vert G:N\vert $, the index of $N$ in $G$. If $G$ is finite, the index is also equal to the order of $G$ divided by the order of $N$. The set $G\,/\,N$ may be finite, although both $G$ and $N$ are infinite (for example, $\mathbb {Z} \,/\,2\mathbb {Z} $).
There is a "natural" surjective group homomorphism $\pi :G\rightarrow G\,/\,N$, sending each element $g$ of $G$ to the coset of $N$ to which $g$ belongs, that is: $\pi (g)=gN$. The mapping $\pi $ is sometimes called the canonical projection of $G$ onto $G\,/\,N$. Its kernel is $N$.
There is a bijective correspondence between the subgroups of $G$ that contain $N$ and the subgroups of $G\,/\,N$; if $H$ is a subgroup of $G$ containing $N$, then the corresponding subgroup of $G\,/\,N$ is $\pi (H)$. This correspondence holds for normal subgroups of $G$ and $G\,/\,N$ as well, and is formalized in the lattice theorem.
Several important properties of quotient groups are recorded in the fundamental theorem on homomorphisms and the isomorphism theorems.
If $G$ is abelian, nilpotent, solvable, cyclic or finitely generated, then so is $G\,/\,N$.
If $H$ is a subgroup in a finite group $G$, and the order of $H$ is one half of the order of $G$, then $H$ is guaranteed to be a normal subgroup, so $G\,/\,H$ exists and is isomorphic to $C_{2}$. This result can also be stated as "any subgroup of index 2 is normal", and in this form it applies also to infinite groups. Furthermore, if $p$ is the smallest prime number dividing the order of a finite group, $G$, then if $G\,/\,H$ has order $p$, $H$ must be a normal subgroup of $G$.[1]
Given $G$ and a normal subgroup $N$, then $G$ is a group extension of $G\,/\,N$ by $N$. One could ask whether this extension is trivial or split; in other words, one could ask whether $G$ is a direct product or semidirect product of $N$ and $G\,/\,N$. This is a special case of the extension problem. An example where the extension is not split is as follows: Let $G=\mathbb {Z} _{4}=\left\{0,1,2,3\right\}$, and $N=\left\{0,2\right\}$, which is isomorphic to $\mathbb {Z} _{2}$. Then $G\,/\,N$ is also isomorphic to $\mathbb {Z} _{2}$. But $\mathbb {Z} _{2}$ has only the trivial automorphism, so the only semi-direct product of $N$ and $G\,/\,N$ is the direct product. Since $\mathbb {Z} _{4}$ is different from $\mathbb {Z} _{2}\times \mathbb {Z} _{2}$, we conclude that $G$ is not a semi-direct product of $N$ and $G\,/\,N$.
Quotients of Lie groups
If $G$ is a Lie group and $N$ is a normal and closed (in the topological rather than the algebraic sense of the word) Lie subgroup of $G$, the quotient $G$ / $N$ is also a Lie group. In this case, the original group $G$ has the structure of a fiber bundle (specifically, a principal $N$-bundle), with base space $G$ / $N$ and fiber $N$. The dimension of $G$ / $N$ equals $\dim G-\dim N$.[2]
Note that the condition that $N$ is closed is necessary. Indeed, if $N$ is not closed then the quotient space is not a T1-space (since there is a coset in the quotient which cannot be separated from the identity by an open set), and thus not a Hausdorff space.
For a non-normal Lie subgroup $N$, the space $G\,/\,N$ of left cosets is not a group, but simply a differentiable manifold on which $G$ acts. The result is known as a homogeneous space.
See also
• Group extension
• Quotient category
• Short exact sequence
Notes
1. Dummit & Foote (2003, p. 120)
2. John M. Lee, Introduction to Smooth Manifolds, Second Edition, theorem 21.17
References
• Dummit, David S.; Foote, Richard M. (2003), Abstract Algebra (3rd ed.), New York: Wiley, ISBN 978-0-471-43334-7
• Herstein, I. N. (1975), Topics in Algebra (2nd ed.), New York: Wiley, ISBN 0-471-02371-X
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Wikipedia
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Quotient space (topology)
In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient topology, that is, with the finest topology that makes continuous the canonical projection map (the function that maps points to their equivalence classes). In other words, a subset of a quotient space is open if and only if its preimage under the canonical projection map is open in the original topological space.
For quotient spaces in linear algebra, see quotient space (linear algebra).
Intuitively speaking, the points of each equivalence class are identified or "glued together" for forming a new topological space. For example, identifying the points of a sphere that belong to the same diameter produces the projective plane as a quotient space.
Definition
Let $X$ be a topological space, and let $\sim $ be an equivalence relation on $X.$ The quotient set $Y=X/{\sim }$ is the set of equivalence classes of elements of $X.$ The equivalence class of $x\in X$ is denoted $[x].$
The construction of $Y$ defines a canonical surjection $ q:X\ni x\mapsto [x]\in Y.$ As discussed below, $q$ is a quotient mapping, commonly called the canonical quotient map, or canonical projection map, associated to $X/{\sim }.$
The quotient space under $\sim $ is the set $Y$ equipped with the quotient topology, whose open sets are those subsets $ U\subseteq Y$ whose preimage $q^{-1}(U)$ is open. In other words, $U$ is open in the quotient topology on $X/{\sim }$ if and only if $ \{x\in X:[x]\in U\}$ is open in $X.$ Similarly, a subset $S\subseteq Y$ is closed if and only if $\{x\in X:[x]\in S\}$ is closed in $X.$
The quotient topology is the final topology on the quotient set, with respect to the map $x\mapsto [x].$
Quotient map
A map $f:X\to Y$ is a quotient map (sometimes called an identification map[1]) if it is surjective and $Y$ is equipped with the final topology induced by $f.$ The latter condition admits two more-elementary phrasings: a subset $V\subseteq Y$ is open (closed) if and only if $f^{-1}(V)$ is open (resp. closed). Every quotient map is continuous but not every continuous map is a quotient map.
Saturated sets
A subset $S$ of $X$ is called saturated (with respect to $f$) if it is of the form $S=f^{-1}(T)$ for some set $T,$ which is true if and only if $f^{-1}(f(S))=S.$ The assignment $T\mapsto f^{-1}(T)$ establishes a one-to-one correspondence (whose inverse is $S\mapsto f(S)$) between subsets $T$ of $Y=f(X)$ and saturated subsets of $X.$ With this terminology, a surjection $f:X\to Y$ is a quotient map if and only if for every saturated subset $S$ of $X,$ $S$ is open in $X$ if and only if $f(S)$ is open in $Y.$ In particular, open subsets of $X$ that are not saturated have no impact on whether the function $f$ is a quotient map (or, indeed, continuous: a function $f:X\to Y$ is continuous if and only if, for every saturated $ S\subseteq X$ such that $f(S)$ is open in $ f(X)$, the set $S$ is open in $ X$).
Indeed, if $\tau $ is a topology on $X$ and $f:X\to Y$ is any map then set $\tau _{f}$ of all $U\in \tau $ that are saturated subsets of $X$ forms a topology on $X.$ If $Y$ is also a topological space then $f:(X,\tau )\to Y$ is a quotient map (respectively, continuous) if and only if the same is true of $f:\left(X,\tau _{f}\right)\to Y.$
Quotient space of fibers characterization
Given an equivalence relation $\,\sim \,$ on $X,$ denote the equivalence class of a point $x\in X$ by $[x]:=\{z\in X:z\sim x\}$ and let $X/{\sim }:=\{[x]:x\in X\}$ denote the set of equivalence classes. The map $q:X\to X/{\sim }$ that sends points to their equivalence classes (that is, it is defined by $q(x):=[x]$ for every $x\in X$) is called the canonical map. It is a surjective map and for all $a,b\in X,$ $a\,\sim \,b$ if and only if $q(a)=q(b);$ consequently, $q(x)=q^{-1}(q(x))$ for all $x\in X.$ In particular, this shows that the set of equivalence class $X/{\sim }$ is exactly the set of fibers of the canonical map $q.$ If $X$ is a topological space then giving $X/{\sim }$ the quotient topology induced by $q$ will make it into a quotient space and make $q:X\to X/{\sim }$ into a quotient map. Up to a homeomorphism, this construction is representative of all quotient spaces; the precise meaning of this is now explained.
Let $f:X\to Y$ be a surjection between topological spaces (not yet assumed to be continuous or a quotient map) and declare for all $a,b\in X$ that $a\,\sim \,b$ if and only if $f(a)=f(b).$ Then $\,\sim \,$ is an equivalence relation on $X$ such that for every $x\in X,$ $[x]=f^{-1}(f(x)),$ which implies that $f([x])$ (defined by $f([x])=\{\,f(z)\,:z\in [x]\}$) is a singleton set; denote the unique element in $f([x])$ by ${\hat {f}}([x])$ (so by definition, $f([x])=\{\,{\hat {f}}([x])\,\}$). The assignment $[x]\mapsto {\hat {f}}([x])$ defines a bijection ${\hat {f}}:X/{\sim }\;\;\to \;Y$ between the fibers of $f$ and points in $Y.$
Define the map $q:X\to X/{\sim }$ as above (by $q(x):=[x]$) and give $X/\sim $ the quotient topology induced by $q$ (which makes $q$ a quotient map). These maps are related by:
$f={\hat {f}}\circ q\quad {\text{ and }}\quad q={\hat {f}}^{-1}\circ f.$
From this and the fact that $q:X\to X/\sim $ is a quotient map, it follows that $f:X\to Y$ is continuous if and only if this is true of ${\hat {f}}:X/\sim \;\;\to \;Y.$ Furthermore, $f:X\to Y$ is a quotient map if and only if ${\hat {f}}:X/\sim \;\;\to \;Y$ is a homeomorphism (or equivalently, if and only if both ${\hat {f}}$ and its inverse are continuous).
Related definitions
A hereditarily quotient map is a surjective map $f:X\to Y$ with the property that for every subset $T\subseteq Y,$ the restriction $f{\big \vert }_{f^{-1}(T)}~:~f^{-1}(T)\to T$ is also a quotient map. There exist quotient maps that are not hereditarily quotient.
Examples
• Gluing. Topologists talk of gluing points together. If $X$ is a topological space, gluing the points $x$ and $y$ in $X$ means considering the quotient space obtained from the equivalence relation $a\sim b$ if and only if $a=b$ or $a=x,b=y$ (or $a=y,b=x$).
• Consider the unit square $I^{2}=[0,1]\times [0,1]$ and the equivalence relation ~ generated by the requirement that all boundary points be equivalent, thus identifying all boundary points to a single equivalence class. Then $I^{2}/\sim $ is homeomorphic to the sphere $S^{2}.$
• Adjunction space. More generally, suppose $X$ is a space and $A$ is a subspace of $X.$ One can identify all points in $A$ to a single equivalence class and leave points outside of $A$ equivalent only to themselves. The resulting quotient space is denoted $X/A.$ The 2-sphere is then homeomorphic to a closed disc with its boundary identified to a single point: $D^{2}/\partial {D^{2}}.$
• Consider the set $\mathbb {R} $ of real numbers with the ordinary topology, and write $x\sim y$ if and only if $x-y$ is an integer. Then the quotient space $X/\sim $ is homeomorphic to the unit circle $S^{1}$ via the homeomorphism which sends the equivalence class of $x$ to $\exp(2\pi ix).$
• A generalization of the previous example is the following: Suppose a topological group $G$ acts continuously on a space $X.$ One can form an equivalence relation on $X$ by saying points are equivalent if and only if they lie in the same orbit. The quotient space under this relation is called the orbit space, denoted $X/G.$ In the previous example $G=\mathbb {Z} $ acts on $\mathbb {R} $ by translation. The orbit space $\mathbb {R} /\mathbb {Z} $ is homeomorphic to $S^{1}.$
• Note: The notation $\mathbb {R} /\mathbb {Z} $ is somewhat ambiguous. If $\mathbb {Z} $ is understood to be a group acting on $\mathbb {R} $ via addition, then the quotient is the circle. However, if $\mathbb {Z} $ is thought of as a topological subspace of $\mathbb {R} $ (that is identified as a single point) then the quotient $\{\mathbb {Z} \}\cup \{\,\{r\}:r\in \mathbb {R} \setminus \mathbb {Z} \}$ (which is identifiable with the set $\{\mathbb {Z} \}\cup (\mathbb {R} \setminus \mathbb {Z} )$) is a countably infinite bouquet of circles joined at a single point $\mathbb {Z} .$
• This next example shows that it is in general not true that if $q:X\to Y$ is a quotient map then every convergent sequence (respectively, every convergent net) in $Y$ has a lift (by $q$) to a convergent sequence (or convergent net) in $X.$ Let $X=[0,1]$ and $\,\sim ~=~\{\,\{0,1\}\,\}~\cup ~\left\{\{x\}:x\in (0,1)\,\right\}.$ Let $Y:=X/\,\sim \,$ and let $q:X\to X/\sim \,$ be the quotient map $q(x):=[x],$ so that $q(0)=q(1)=\{0,1\}$ and $q(x)=\{x\}$ for every $x\in (0,1).$ The map $h:X/\,\sim \,\to S^{1}\subseteq \mathbb {C} $ defined by $h([x]):=e^{2\pi ix}$ is well-defined (because $e^{2\pi i(0)}=1=e^{2\pi i(1)}$) and a homeomorphism. Let $I=\mathbb {N} $ and let $a_{\bullet }:=\left(a_{i}\right)_{i\in I}{\text{ and }}b_{\bullet }:=\left(b_{i}\right)_{i\in I}$ be any sequences (or more generally, any nets) valued in $(0,1)$ such that $a_{\bullet }\to 0{\text{ and }}b_{\bullet }\to 1$ in $X=[0,1].$ Then the sequence
$y_{1}:=q\left(a_{1}\right),y_{2}:=q\left(b_{1}\right),y_{3}:=q\left(a_{2}\right),y_{4}:=q\left(b_{2}\right),\ldots $
converges to $[0]=[1]$ in $X/\sim \,$ but there does not exist any convergent lift of this sequence by the quotient map $q$ (that is, there is no sequence $s_{\bullet }=\left(s_{i}\right)_{i\in I}$ in $X$ that both converges to some $x\in X$ and satisfies $y_{i}=q\left(s_{i}\right)$ for every $i\in I$). This counterexample can be generalized to nets by letting $(A,\leq )$ be any directed set, and making $I:=A\times \{1,2\}$ into a net by declaring that for any $(a,m),(b,n)\in I,$ $(m,a)\;\leq \;(n,b)$ holds if and only if both (1) $a\leq b,$ and (2) if $a=b{\text{ then }}m\leq n;$ then the $A$-indexed net defined by letting $y_{(a,m)}$ equal $a_{i}{\text{ if }}m=1$ and equal to $b_{i}{\text{ if }}m=2$ has no lift (by $q$) to a convergent $A$-indexed net in $X=[0,1].$
Properties
Quotient maps $q:X\to Y$ are characterized among surjective maps by the following property: if $Z$ is any topological space and $f:Y\to Z$ is any function, then $f$ is continuous if and only if $f\circ q$ is continuous.
The quotient space $X/{\sim }$ together with the quotient map $q:X\to X/{\sim }$ is characterized by the following universal property: if $g:X\to Z$ is a continuous map such that $a\sim b$ implies $g(a)=g(b)$ for all $a,b\in X,$ then there exists a unique continuous map $f:X/{\sim }\to Z$ such that $g=f\circ q.$ In other words, the following diagram commutes:
One says that $g$ descends to the quotient for expressing this, that is that it factorizes through the quotient space. The continuous maps defined on $X/{\sim }$ are, therefore, precisely those maps which arise from continuous maps defined on $X$ that respect the equivalence relation (in the sense that they send equivalent elements to the same image). This criterion is copiously used when studying quotient spaces.
Given a continuous surjection $q:X\to Y$ it is useful to have criteria by which one can determine if $q$ is a quotient map. Two sufficient criteria are that $q$ be open or closed. Note that these conditions are only sufficient, not necessary. It is easy to construct examples of quotient maps that are neither open nor closed. For topological groups, the quotient map is open.
Compatibility with other topological notions
Separation
• In general, quotient spaces are ill-behaved with respect to separation axioms. The separation properties of $X$ need not be inherited by $X/\sim ,$ and $X/\sim $ may have separation properties not shared by $X.$
• $X/\sim $ is a T1 space if and only if every equivalence class of $\,\sim \,$ is closed in $X.$
• If the quotient map is open, then $X/\sim $ is a Hausdorff space if and only if ~ is a closed subset of the product space $X\times X.$
Connectedness
• If a space is connected or path connected, then so are all its quotient spaces.
• A quotient space of a simply connected or contractible space need not share those properties.
Compactness
• If a space is compact, then so are all its quotient spaces.
• A quotient space of a locally compact space need not be locally compact.
Dimension
• The topological dimension of a quotient space can be more (as well as less) than the dimension of the original space; space-filling curves provide such examples.
See also
Topology
• Covering space – Type of continuous map in topology
• Disjoint union (topology) – space formed by equipping the disjoint union of the underlying sets with a natural topology called the disjoint union topologyPages displaying wikidata descriptions as a fallback
• Final topology – Finest topology making some functions continuous
• Mapping cone (topology) – topological constructionPages displaying wikidata descriptions as a fallback
• Product space – Topology on Cartesian products of topological spacesPages displaying short descriptions of redirect targets
• Subspace (topology) – Inherited topologyPages displaying short descriptions of redirect targets
• Topological space – Mathematical space with a notion of closeness
Algebra
• Quotient category
• Quotient group – Group obtained by aggregating similar elements of a larger group
• Quotient space (linear algebra) – Vector space consisting of affine subsets
• Mapping cone (homological algebra) – Tool in homological algebra
Notes
1. Brown 2006, p. 103.
References
• Bourbaki, Nicolas (1989) [1966]. General Topology: Chapters 1–4 [Topologie Générale]. Éléments de mathématique. Berlin New York: Springer Science & Business Media. ISBN 978-3-540-64241-1. OCLC 18588129.
• Bourbaki, Nicolas (1989) [1967]. General Topology 2: Chapters 5–10 [Topologie Générale]. Éléments de mathématique. Vol. 4. Berlin New York: Springer Science & Business Media. ISBN 978-3-540-64563-4. OCLC 246032063.
• Brown, Ronald (2006), Topology and Groupoids, Booksurge, ISBN 1-4196-2722-8
• Dixmier, Jacques (1984). General Topology. Undergraduate Texts in Mathematics. Translated by Berberian, S. K. New York: Springer-Verlag. ISBN 978-0-387-90972-1. OCLC 10277303.
• Dugundji, James (1966). Topology. Boston: Allyn and Bacon. ISBN 978-0-697-06889-7. OCLC 395340485.
• Kelley, John L. (1975). General Topology. Graduate Texts in Mathematics. Vol. 27. New York: Springer Science & Business Media. ISBN 978-0-387-90125-1. OCLC 338047.
• Munkres, James R. (2000). Topology (Second ed.). Upper Saddle River, NJ: Prentice Hall, Inc. ISBN 978-0-13-181629-9. OCLC 42683260.
• Willard, Stephen (2004) [1970]. General Topology. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-43479-7. OCLC 115240.
• Willard, Stephen (1970). General Topology. Reading, MA: Addison-Wesley. ISBN 0-486-43479-6.
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Wikipedia
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Lie algebra
In mathematics, a Lie algebra (pronounced /liː/ LEE) is a vector space ${\mathfrak {g}}$ together with an operation called the Lie bracket, an alternating bilinear map ${\mathfrak {g}}\times {\mathfrak {g}}\rightarrow {\mathfrak {g}}$, that satisfies the Jacobi identity. Otherwise said, a Lie algebra is an algebra over a field where the multiplication operation is now called Lie bracket and has two additional properties: it is alternating and satisfies the Jacobi identity. The Lie bracket of two vectors $x$ and $y$ is denoted $[x,y]$. The Lie bracket does not need to be associative, meaning that the Lie algebra can be non associative. Given an associative algebra (like for example the space of square matrices), a Lie bracket can be and is often defined through the commutator, namely defining $[x,y]=xy-yx$ correctly defines a Lie bracket in addition to the already existing multiplication operation.
"Lie bracket" redirects here. For the operation on vector fields, see Lie bracket of vector fields.
Lie groups and Lie algebras
Classical groups
• General linear GL(n)
• Special linear SL(n)
• Orthogonal O(n)
• Special orthogonal SO(n)
• Unitary U(n)
• Special unitary SU(n)
• Symplectic Sp(n)
Simple Lie groups
Classical
• An
• Bn
• Cn
• Dn
Exceptional
• G2
• F4
• E6
• E7
• E8
Other Lie groups
• Circle
• Lorentz
• Poincaré
• Conformal group
• Diffeomorphism
• Loop
• Euclidean
Lie algebras
• Lie group–Lie algebra correspondence
• Exponential map
• Adjoint representation
• Killing form
• Index
• Simple Lie algebra
• Loop algebra
• Affine Lie algebra
Semisimple Lie algebra
• Dynkin diagrams
• Cartan subalgebra
• Root system
• Weyl group
• Real form
• Complexification
• Split Lie algebra
• Compact Lie algebra
Representation theory
• Lie group representation
• Lie algebra representation
• Representation theory of semisimple Lie algebras
• Representations of classical Lie groups
• Theorem of the highest weight
• Borel–Weil–Bott theorem
Lie groups in physics
• Particle physics and representation theory
• Lorentz group representations
• Poincaré group representations
• Galilean group representations
Scientists
• Sophus Lie
• Henri Poincaré
• Wilhelm Killing
• Élie Cartan
• Hermann Weyl
• Claude Chevalley
• Harish-Chandra
• Armand Borel
• Glossary
• Table of Lie groups
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
Lie algebras are closely related to Lie groups, which are groups that are also smooth manifolds: any Lie group gives rise to a Lie algebra, which is its tangent space at the identity. Conversely, to any finite-dimensional Lie algebra over real or complex numbers, there is a corresponding connected Lie group unique up to finite coverings (Lie's third theorem). This correspondence allows one to study the structure and classification of Lie groups in terms of Lie algebras.
In physics, Lie groups appear as symmetry groups of physical systems, and their Lie algebras (tangent vectors near the identity) may be thought of as infinitesimal symmetry motions. Thus Lie algebras and their representations are used extensively in physics, notably in quantum mechanics and particle physics.
An elementary example (that is not derived from an associative algebra) is the space of three dimensional vectors ${\mathfrak {g}}=\mathbb {R} ^{3}$ with the Lie bracket operation defined by the cross product $[x,y]=x\times y.$ This is skew-symmetric since $x\times y=-y\times x$, and instead of associativity it satisfies the Jacobi identity:
$x\times (y\times z)\ =\ (x\times y)\times z\ +\ y\times (x\times z).$
This is the Lie algebra of the Lie group of rotations of space, and each vector $v\in \mathbb {R} ^{3}$ may be pictured as an infinitesimal rotation around the axis $v$, with velocity equal to the magnitude of $v$. The Lie bracket is a measure of the non-commutativity between two rotations: since a rotation commutes with itself, we have the alternating property $[x,x]=x\times x=0$.
History
Lie algebras were introduced to study the concept of infinitesimal transformations by Marius Sophus Lie in the 1870s,[1] and independently discovered by Wilhelm Killing[2] in the 1880s. The name Lie algebra was given by Hermann Weyl in the 1930s; in older texts, the term infinitesimal group is used.
Definitions
Definition of a Lie algebra
A Lie algebra is a vector space $\,{\mathfrak {g}}$ over some field $F$ together with a binary operation $[\,\cdot \,,\cdot \,]:{\mathfrak {g}}\times {\mathfrak {g}}\to {\mathfrak {g}}$ called the Lie bracket satisfying the following axioms:[lower-alpha 1]
• Bilinearity,
$[ax+by,z]=a[x,z]+b[y,z],$
$[z,ax+by]=a[z,x]+b[z,y]$
for all scalars $a$, $b$ in $F$ and all elements $x$, $y$, $z$ in ${\mathfrak {g}}$.
• Alternativity,
$[x,x]=0\ $
for all $x$ in ${\mathfrak {g}}$.
• The Jacobi identity,
$[x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0\ $
for all $x$, $y$, $z$ in ${\mathfrak {g}}$.
Using bilinearity to expand the Lie bracket $[x+y,x+y]$ and using alternativity shows that $[x,y]+[y,x]=0\ $ for all elements $x$, $y$ in ${\mathfrak {g}}$, showing that bilinearity and alternativity together imply
• Anticommutativity,
$[x,y]=-[y,x],\ $
for all elements $x$, $y$ in ${\mathfrak {g}}$. If the field's characteristic is not 2 then anticommutativity implies alternativity, since it implies $[x,x]=-[x,x].$[3]
It is customary to denote a Lie algebra by a lower-case fraktur letter such as ${\mathfrak {g,h,b,n}}$. If a Lie algebra is associated with a Lie group, then the algebra is denoted by the fraktur version of the group: for example the Lie algebra of SU(n) is ${\mathfrak {su}}(n)$.
Generators and dimension
Elements of a Lie algebra ${\mathfrak {g}}$ are said to generate it if the smallest subalgebra containing these elements is ${\mathfrak {g}}$ itself. The dimension of a Lie algebra is its dimension as a vector space over $F$. The cardinality of a minimal generating set of a Lie algebra is always less than or equal to its dimension.
See the classification of low-dimensional real Lie algebras for other small examples.
Subalgebras, ideals and homomorphisms
The Lie bracket is not required to be associative, meaning that $[[x,y],z]$ need not equal $[x,[y,z]]$. However, it is flexible. Nonetheless, much of the terminology of associative rings and algebras is commonly applied to Lie algebras. A Lie subalgebra is a subspace ${\mathfrak {h}}\subseteq {\mathfrak {g}}$ which is closed under the Lie bracket. An ideal ${\mathfrak {i}}\subseteq {\mathfrak {g}}$ is a subalgebra satisfying the stronger condition:[4]
$[{\mathfrak {g}},{\mathfrak {i}}]\subseteq {\mathfrak {i}}.$
A Lie algebra homomorphism is a linear map compatible with the respective Lie brackets:
$\phi :{\mathfrak {g}}\to {\mathfrak {g'}},\quad \phi ([x,y])=[\phi (x),\phi (y)]\ {\text{for all}}\ x,y\in {\mathfrak {g}}.$ :{\mathfrak {g}}\to {\mathfrak {g'}},\quad \phi ([x,y])=[\phi (x),\phi (y)]\ {\text{for all}}\ x,y\in {\mathfrak {g}}.}
As for associative rings, ideals are precisely the kernels of homomorphisms; given a Lie algebra ${\mathfrak {g}}$ and an ideal ${\mathfrak {i}}$ in it, one constructs the factor algebra or quotient algebra ${\mathfrak {g}}/{\mathfrak {i}}$, and the first isomorphism theorem holds for Lie algebras.
Since the Lie bracket is a kind of infinitesimal commutator of the corresponding Lie group, we say that two elements $x,y\in {\mathfrak {g}}$ commute if their bracket vanishes: $[x,y]=0$.
The centralizer subalgebra of a subset $S\subset {\mathfrak {g}}$ is the set of elements commuting with $S$: that is, ${\mathfrak {z}}_{\mathfrak {g}}(S)=\{x\in {\mathfrak {g}}\ \mid \ [x,s]=0\ {\text{ for all }}s\in S\}$. The centralizer of ${\mathfrak {g}}$ itself is the center ${\mathfrak {z}}({\mathfrak {g}})$. Similarly, for a subspace S, the normalizer subalgebra of $S$ is ${\mathfrak {n}}_{\mathfrak {g}}(S)=\{x\in {\mathfrak {g}}\ \mid \ [x,s]\in S\ {\text{ for all}}\ s\in S\}$.[5] Equivalently, if $S$ is a Lie subalgebra, ${\mathfrak {n}}_{\mathfrak {g}}(S)$ is the largest subalgebra such that $S$ is an ideal of ${\mathfrak {n}}_{\mathfrak {g}}(S)$.
Examples
For ${\mathfrak {d}}(2)\subset {\mathfrak {gl}}(2)$, the commutator of two elements $g\in {\mathfrak {gl}}(2)$ and $d\in {\mathfrak {d}}(2)$:
${\begin{aligned}\left[{\begin{bmatrix}a&b\\c&d\end{bmatrix}},{\begin{bmatrix}x&0\\0&y\end{bmatrix}}\right]&={\begin{bmatrix}ax&by\\cx&dy\\\end{bmatrix}}-{\begin{bmatrix}ax&bx\\cy&dy\\\end{bmatrix}}\\&={\begin{bmatrix}0&b(y-x)\\c(x-y)&0\end{bmatrix}}\end{aligned}}$
shows ${\mathfrak {d}}(2)$ is a subalgebra, but not an ideal. In fact, every one-dimensional linear subspace of a Lie algebra has an induced abelian Lie algebra structure, which is generally not an ideal. For any simple Lie algebra, all abelian Lie algebras can never be ideals.
Direct sum and semidirect product
For two Lie algebras ${\mathfrak {g^{}}}$ and ${\mathfrak {g'}}$, their direct sum Lie algebra is the vector space ${\mathfrak {g}}\oplus {\mathfrak {g'}}$consisting of all pairs ${\mathfrak {}}(x,x'),\,x\in {\mathfrak {g}},\ x'\in {\mathfrak {g'}}$, with the operation
$[(x,x'),(y,y')]=([x,y],[x',y']),$
so that the copies of ${\mathfrak {g}},{\mathfrak {g}}'$ commute with each other: $[(x,0),(0,x')]=0.$
Let ${\mathfrak {g}}$ be a Lie algebra and ${\mathfrak {i}}$ an ideal of ${\mathfrak {g}}$. If the canonical map ${\mathfrak {g}}\to {\mathfrak {g}}/{\mathfrak {i}}$ splits (i.e., admits a section), then ${\mathfrak {g}}$ is said to be a semidirect product of ${\mathfrak {i}}$ and ${\mathfrak {g}}/{\mathfrak {i}}$, ${\mathfrak {g}}={\mathfrak {g}}/{\mathfrak {i}}\ltimes {\mathfrak {i}}$. See also semidirect sum of Lie algebras.
Levi's theorem says that a finite-dimensional Lie algebra is a semidirect product of its radical and the complementary subalgebra (Levi subalgebra).
Derivations
A derivation on the Lie algebra ${\mathfrak {g}}$ (or on any non-associative algebra) is a linear map $\delta \colon {\mathfrak {g}}\rightarrow {\mathfrak {g}}$ that obeys the Leibniz law, that is,
$\delta ([x,y])=[\delta (x),y]+[x,\delta (y)]$
for all $x,y\in {\mathfrak {g}}$. The inner derivation associated to any $x\in {\mathfrak {g}}$ is the adjoint mapping $\mathrm {ad} _{x}$ defined by $\mathrm {ad} _{x}(y):=[x,y]$. (This is a derivation as a consequence of the Jacobi identity.) The outer derivations are derivations which do not come from the adjoint representation of the Lie algebra. If ${\mathfrak {g}}$ is semisimple, every derivation is inner.
The derivations form a vector space $\mathrm {Der} ({\mathfrak {g}})$, which is a Lie subalgebra of ${\mathfrak {gl}}({\mathfrak {g}})$; the bracket is commutator. The inner derivations form a Lie subalgebra of $\mathrm {Der} ({\mathfrak {g}})$.
Examples
For example, given a Lie algebra ideal ${\mathfrak {i}}\subset {\mathfrak {g}}$ the adjoint representation ${\mathfrak {ad}}_{\mathfrak {g}}$ of ${\mathfrak {g}}$ acts as outer derivations on ${\mathfrak {i}}$ since $[x,i]\subset {\mathfrak {i}}$ for any $x\in {\mathfrak {g}}$ and $i\in {\mathfrak {i}}$. For the Lie algebra ${\mathfrak {b}}_{n}$ of upper triangular matrices in ${\mathfrak {gl}}(n)$, it has an ideal ${\mathfrak {n}}_{n}$ of strictly upper triangular matrices (where the only non-zero elements are above the diagonal of the matrix). For instance, the commutator of elements in ${\mathfrak {b}}_{3}$ and ${\mathfrak {n}}_{3}$ gives
${\begin{aligned}\left[{\begin{bmatrix}a&b&c\\0&d&e\\0&0&f\end{bmatrix}},{\begin{bmatrix}0&x&y\\0&0&z\\0&0&0\end{bmatrix}}\right]&={\begin{bmatrix}0&ax&ay+bz\\0&0&dz\\0&0&0\end{bmatrix}}-{\begin{bmatrix}0&dx&ex+yf\\0&0&fz\\0&0&0\end{bmatrix}}\\&={\begin{bmatrix}0&(a-d)x&(a-f)y-ex+bz\\0&0&(d-f)z\\0&0&0\end{bmatrix}}\end{aligned}}$
shows there exist outer derivations from ${\mathfrak {b}}_{3}$ in ${\text{Der}}({\mathfrak {n}}_{3})$.
Split Lie algebra
Let V be a finite-dimensional vector space over a field F, ${\mathfrak {gl}}(V)$ the Lie algebra of linear transformations and ${\mathfrak {g}}\subseteq {\mathfrak {gl}}(V)$ a Lie subalgebra. Then ${\mathfrak {g}}$ is said to be split if the roots of the characteristic polynomials of all linear transformations in ${\mathfrak {g}}$ are in the base field F.[6] More generally, a finite-dimensional Lie algebra ${\mathfrak {g}}$ is said to be split if it has a Cartan subalgebra whose image under the adjoint representation $\operatorname {ad} :{\mathfrak {g}}\to {\mathfrak {gl}}({\mathfrak {g}})$ :{\mathfrak {g}}\to {\mathfrak {gl}}({\mathfrak {g}})} is a split Lie algebra. A split real form of a complex semisimple Lie algebra (cf. #Real form and complexification) is an example of a split real Lie algebra. See also split Lie algebra for further information.
Vector space basis
For practical calculations, it is often convenient to choose an explicit vector space basis for the algebra. A common construction for this basis is sketched in the article structure constants.
Definition using category-theoretic notation
Although the definitions above are sufficient for a conventional understanding of Lie algebras, once this is understood, additional insight can be gained by using notation common to category theory, that is, by defining a Lie algebra in terms of linear maps—that is, morphisms of the category of vector spaces—without considering individual elements. (In this section, the field over which the algebra is defined is supposed to be of characteristic different from two.)
For the category-theoretic definition of Lie algebras, two braiding isomorphisms are needed. If A is a vector space, the interchange isomorphism $\tau :A\otimes A\to A\otimes A$ is defined by
$\tau (x\otimes y)=y\otimes x.$
The cyclic-permutation braiding $\sigma :A\otimes A\otimes A\to A\otimes A\otimes A$ is defined as
$\sigma =(\mathrm {id} \otimes \tau )\circ (\tau \otimes \mathrm {id} ),$
where $\mathrm {id} $ is the identity morphism. Equivalently, $\sigma $ is defined by
$\sigma (x\otimes y\otimes z)=y\otimes z\otimes x.$
With this notation, a Lie algebra can be defined as an object $A$ in the category of vector spaces together with a morphism
$[\cdot ,\cdot ]:A\otimes A\rightarrow A$
that satisfies the two morphism equalities
$[\cdot ,\cdot ]\circ (\mathrm {id} +\tau )=0,$
and
$[\cdot ,\cdot ]\circ ([\cdot ,\cdot ]\otimes \mathrm {id} )\circ (\mathrm {id} +\sigma +\sigma ^{2})=0.$
Examples
Vector spaces
Any vector space $V$ endowed with the identically zero Lie bracket becomes a Lie algebra. Such Lie algebras are called abelian, cf. below. Any one-dimensional Lie algebra over a field of characteristic different from 2 is abelian, by the alternating property of the Lie bracket.
Associative algebra with commutator bracket
• On an associative algebra $A$ over a field $F$ with multiplication $(x,y)\mapsto xy$, a Lie bracket may be defined by the commutator $[x,y]=xy-yx$. With this bracket, $A$ is a Lie algebra.[7] The associative algebra A is called an enveloping algebra of the Lie algebra $(A,[\,\cdot \,,\cdot \,])$. Every Lie algebra can be embedded into one that arises from an associative algebra in this fashion; see universal enveloping algebra.
• The associative algebra of the endomorphisms of an F-vector space $V$ with the above Lie bracket is denoted ${\mathfrak {gl}}(V)$.
• For a finite dimensional vector space $V=F^{n}$, the previous example is exactly the Lie algebra of n × n matrices, denoted ${\mathfrak {gl}}(n,F)$ or ${\mathfrak {gl}}_{n}(F)$,[8] and with bracket $[X,Y]=XY-YX$ where adjacency indicates matrix multiplication. This is the Lie algebra of the general linear group, consisting of invertible matrices.
Special matrices
Two important subalgebras of ${\mathfrak {gl}}_{n}(F)$ are:
• The matrices of trace zero form the special linear Lie algebra ${\mathfrak {sl}}_{n}(F)$, the Lie algebra of the special linear group $\mathrm {SL} _{n}(F)$.[9]
• The skew-hermitian matrices form the unitary Lie algebra ${\mathfrak {u}}(n)$, the Lie algebra of the unitary group U(n).
Matrix Lie algebras
A complex matrix group is a Lie group consisting of matrices, $G\subset M_{n}(\mathbb {C} )$, where the multiplication of G is matrix multiplication. The corresponding Lie algebra ${\mathfrak {g}}$ is the space of matrices which are tangent vectors to G inside the linear space $M_{n}(\mathbb {C} )$: this consists of derivatives of smooth curves in G at the identity:
${\mathfrak {g}}=\{X=c'(0)\in M_{n}(\mathbb {C} )\ \mid \ {\text{ smooth }}c:\mathbb {R} \to G,\ c(0)=I\}.$
The Lie bracket of ${\mathfrak {g}}$ is given by the commutator of matrices, $[X,Y]=XY-YX$. Given the Lie algebra, one can recover the Lie group as the image of the matrix exponential mapping $\exp :M_{n}(\mathbb {C} )\to M_{n}(\mathbb {C} )$ defined by $\exp(X)=I+X+{\tfrac {1}{2!}}X^{2}+\cdots $, which converges for every matrix $X$: that is, $G=\exp({\mathfrak {g}})$.
The following are examples of Lie algebras of matrix Lie groups:[10]
• The special linear group ${\rm {SL}}_{n}(\mathbb {C} )$, consisting of all n × n matrices with determinant 1. Its Lie algebra ${\mathfrak {sl}}_{n}(\mathbb {C} )$consists of all n × n matrices with complex entries and trace 0. Similarly, one can define the corresponding real Lie group ${\rm {SL}}_{n}(\mathbb {R} )$ and its Lie algebra ${\mathfrak {sl}}_{n}(\mathbb {R} )$.
• The unitary group $U(n)$ consists of n × n unitary matrices (satisfying $U^{*}=U^{-1}$). Its Lie algebra ${\mathfrak {u}}(n)$ consists of skew-self-adjoint matrices ($X^{*}=-X$).
• The special orthogonal group $\mathrm {SO} (n)$, consisting of real determinant-one orthogonal matrices ($A^{\mathrm {T} }=A^{-1}$). Its Lie algebra ${\mathfrak {so}}(n)$ consists of real skew-symmetric matrices ($X^{\rm {T}}=-X$). The full orthogonal group $\mathrm {O} (n)$, without the determinant-one condition, consists of $\mathrm {SO} (n)$ and a separate connected component, so it has the same Lie algebra as $\mathrm {SO} (n)$. See also infinitesimal rotations with skew-symmetric matrices. Similarly, one can define a complex version of this group and algebra, simply by allowing complex matrix entries.
Two dimensions
• On any field $F$ there is, up to isomorphism, a single two-dimensional nonabelian Lie algebra. With generators x, y, its bracket is defined as $\left[x,y\right]=y$. It generates the affine group in one dimension.
This can be realized by the matrices:
$x=\left({\begin{array}{cc}1&0\\0&0\end{array}}\right),\qquad y=\left({\begin{array}{cc}0&1\\0&0\end{array}}\right).$
Since
$\left({\begin{array}{cc}1&c\\0&0\end{array}}\right)^{n+1}=\left({\begin{array}{cc}1&c\\0&0\end{array}}\right)$
for any natural number $n$ and any $c$, one sees that the resulting Lie group elements are upper triangular 2×2 matrices with unit lower diagonal:
$\exp(a\cdot {}x+b\cdot {}y)=\left({\begin{array}{cc}e^{a}&{\tfrac {b}{a}}(e^{a}-1)\\0&1\end{array}}\right)=1+{\tfrac {e^{a}-1}{a}}\left(a\cdot {}x+b\cdot {}y\right).$
Three dimensions
• The Heisenberg algebra ${\rm {H}}_{3}(\mathbb {R} )$ is a three-dimensional Lie algebra generated by elements x, y, and z with Lie brackets
$[x,y]=z,\quad [x,z]=0,\quad [y,z]=0$.
It is usually realized as the space of 3×3 strictly upper-triangular matrices, with the commutator Lie bracket and the basis
$x=\left({\begin{array}{ccc}0&1&0\\0&0&0\\0&0&0\end{array}}\right),\quad y=\left({\begin{array}{ccc}0&0&0\\0&0&1\\0&0&0\end{array}}\right),\quad z=\left({\begin{array}{ccc}0&0&1\\0&0&0\\0&0&0\end{array}}\right)~.\quad $
Any element of the Heisenberg group has a representation as a product of group generators, i.e., matrix exponentials of these Lie algebra generators,
$\left({\begin{array}{ccc}1&a&c\\0&1&b\\0&0&1\end{array}}\right)=e^{by}e^{cz}e^{ax}~.$
• The Lie algebra ${\mathfrak {so}}(3)$ of the group SO(3) is spanned by the three matrices[11]
$F_{1}=\left({\begin{array}{ccc}0&0&0\\0&0&-1\\0&1&0\end{array}}\right),\quad F_{2}=\left({\begin{array}{ccc}0&0&1\\0&0&0\\-1&0&0\end{array}}\right),\quad F_{3}=\left({\begin{array}{ccc}0&-1&0\\1&0&0\\0&0&0\end{array}}\right)~.\quad $
The commutation relations among these generators are
$[F_{1},F_{2}]=F_{3},$
$[F_{2},F_{3}]=F_{1},$
$[F_{3},F_{1}]=F_{2}.$
The three-dimensional Euclidean space $\mathbb {R} ^{3}$ with the Lie bracket given by the cross product of vectors has the same commutation relations as above: thus, it is isomorphic to ${\mathfrak {so}}(3)$. This Lie algebra is unitarily equivalent to the usual Spin (physics) angular-momentum component operators for spin-1 particles in quantum mechanics.
Infinite dimensions
• An important class of infinite-dimensional real Lie algebras arises in differential topology. The space of smooth vector fields on a differentiable manifold M forms a Lie algebra, where the Lie bracket is defined to be the commutator of vector fields. One way of expressing the Lie bracket is through the formalism of Lie derivatives, which identifies a vector field X with a first order partial differential operator LX acting on smooth functions by letting LX(f) be the directional derivative of the function f in the direction of X. The Lie bracket [X,Y] of two vector fields is the vector field defined through its action on functions by the formula:
$L_{[X,Y]}f=L_{X}(L_{Y}f)-L_{Y}(L_{X}f).\,$
• Kac–Moody algebras are a large class of infinite-dimensional Lie algebras whose structure is very similar to the finite-dimensional cases above.
• The Moyal algebra is an infinite-dimensional Lie algebra that contains all classical Lie algebras as subalgebras.
• The Virasoro algebra is of paramount importance in string theory.
Representations
Main article: Lie algebra representation
Definitions
Given a vector space V, let ${\mathfrak {gl}}(V)$ denote the Lie algebra consisting of all linear endomorphisms of V, with bracket given by $[X,Y]=XY-YX$. A representation of a Lie algebra ${\mathfrak {g}}$ on V is a Lie algebra homomorphism
$\pi :{\mathfrak {g}}\to {\mathfrak {gl}}(V).$ :{\mathfrak {g}}\to {\mathfrak {gl}}(V).}
A representation is said to be faithful if its kernel is zero. Ado's theorem[12] states that every finite-dimensional Lie algebra has a faithful representation on a finite-dimensional vector space.
Adjoint representation
For any Lie algebra ${\mathfrak {g}}$, we can define a representation
$\operatorname {ad} \colon {\mathfrak {g}}\to {\mathfrak {gl}}({\mathfrak {g}})$
given by $\operatorname {ad} (x)(y)=[x,y]$; it is a representation on the vector space ${\mathfrak {g}}$ called the adjoint representation.
Goals of representation theory
One important aspect of the study of Lie algebras (especially semisimple Lie algebras) is the study of their representations. (Indeed, most of the books listed in the references section devote a substantial fraction of their pages to representation theory.) Although Ado's theorem is an important result, the primary goal of representation theory is not to find a faithful representation of a given Lie algebra ${\mathfrak {g}}$. Indeed, in the semisimple case, the adjoint representation is already faithful. Rather the goal is to understand all possible representation of ${\mathfrak {g}}$, up to the natural notion of equivalence. In the semisimple case over a field of characteristic zero, Weyl's theorem[13] says that every finite-dimensional representation is a direct sum of irreducible representations (those with no nontrivial invariant subspaces). The irreducible representations, in turn, are classified by a theorem of the highest weight.
Representation theory in physics
The representation theory of Lie algebras plays an important role in various parts of theoretical physics. There, one considers operators on the space of states that satisfy certain natural commutation relations. These commutation relations typically come from a symmetry of the problem—specifically, they are the relations of the Lie algebra of the relevant symmetry group. An example would be the angular momentum operators, whose commutation relations are those of the Lie algebra ${\mathfrak {so}}(3)$ of the rotation group SO(3). Typically, the space of states is very far from being irreducible under the pertinent operators, but one can attempt to decompose it into irreducible pieces. In doing so, one needs to know the irreducible representations of the given Lie algebra. In the study of the quantum hydrogen atom, for example, quantum mechanics textbooks give (without calling it that) a classification of the irreducible representations of the Lie algebra ${\mathfrak {so}}(3)$.
Structure theory and classification
Lie algebras can be classified to some extent. In particular, this has an application to the classification of Lie groups.
Abelian, nilpotent, and solvable
Analogously to abelian, nilpotent, and solvable groups, defined in terms of the derived subgroups, one can define abelian, nilpotent, and solvable Lie algebras.
A Lie algebra ${\mathfrak {g}}$ is abelian if the Lie bracket vanishes, i.e. [x,y] = 0, for all x and y in ${\mathfrak {g}}$. Abelian Lie algebras correspond to commutative (or abelian) connected Lie groups such as vector spaces $\mathbb {K} ^{n}$ or tori $\mathbb {T} ^{n}$, and are all of the form ${\mathfrak {k}}^{n},$ meaning an n-dimensional vector space with the trivial Lie bracket.
A more general class of Lie algebras is defined by the vanishing of all commutators of given length. A Lie algebra ${\mathfrak {g}}$ is nilpotent if the lower central series
${\mathfrak {g}}>[{\mathfrak {g}},{\mathfrak {g}}]>[[{\mathfrak {g}},{\mathfrak {g}}],{\mathfrak {g}}]>[[[{\mathfrak {g}},{\mathfrak {g}}],{\mathfrak {g}}],{\mathfrak {g}}]>\cdots $
becomes zero eventually. By Engel's theorem, a Lie algebra is nilpotent if and only if for every u in ${\mathfrak {g}}$ the adjoint endomorphism
$\operatorname {ad} (u):{\mathfrak {g}}\to {\mathfrak {g}},\quad \operatorname {ad} (u)v=[u,v]$
is nilpotent.
More generally still, a Lie algebra ${\mathfrak {g}}$ is said to be solvable if the derived series:
${\mathfrak {g}}>[{\mathfrak {g}},{\mathfrak {g}}]>[[{\mathfrak {g}},{\mathfrak {g}}],[{\mathfrak {g}},{\mathfrak {g}}]]>[[[{\mathfrak {g}},{\mathfrak {g}}],[{\mathfrak {g}},{\mathfrak {g}}]],[[{\mathfrak {g}},{\mathfrak {g}}],[{\mathfrak {g}},{\mathfrak {g}}]]]>\cdots $
becomes zero eventually.
Every finite-dimensional Lie algebra has a unique maximal solvable ideal, called its radical. Under the Lie correspondence, nilpotent (respectively, solvable) connected Lie groups correspond to nilpotent (respectively, solvable) Lie algebras.
Simple and semisimple
Main article: Semisimple Lie algebra
A Lie algebra is "simple" if it has no non-trivial ideals and is not abelian. (This implies that a one-dimensional—necessarily abelian—Lie algebra is by definition not simple, even though it has no nontrivial ideals.) A Lie algebra ${\mathfrak {g}}$ is called semisimple if it is isomorphic to a direct sum of simple algebras. There are several equivalent characterizations of semisimple algebras, such as having no nonzero solvable ideals.
The concept of semisimplicity for Lie algebras is closely related with the complete reducibility (semisimplicity) of their representations. When the ground field F has characteristic zero, any finite-dimensional representation of a semisimple Lie algebra is semisimple (i.e., direct sum of irreducible representations). In general, a Lie algebra is called reductive if the adjoint representation is semisimple. Thus, a semisimple Lie algebra is reductive.
Cartan's criterion
Cartan's criterion gives conditions for a Lie algebra to be nilpotent, solvable, or semisimple. It is based on the notion of the Killing form, a symmetric bilinear form on ${\mathfrak {g}}$ defined by the formula
$K(u,v)=\operatorname {tr} (\operatorname {ad} (u)\operatorname {ad} (v)),$
where tr denotes the trace of a linear operator. A Lie algebra ${\mathfrak {g}}$ is semisimple if and only if the Killing form is nondegenerate. A Lie algebra ${\mathfrak {g}}$ is solvable if and only if $K({\mathfrak {g}},[{\mathfrak {g}},{\mathfrak {g}}])=0.$
Classification
The Levi decomposition expresses an arbitrary Lie algebra as a semidirect sum of its solvable radical and a semisimple Lie algebra, almost in a canonical way. (Such a decomposition exists for a finite-dimensional Lie algebra over a field of characteristic zero.[14]) Furthermore, semisimple Lie algebras over an algebraically closed field have been completely classified through their root systems.
Relation to Lie groups
Main article: Lie group–Lie algebra correspondence
Although Lie algebras are often studied in their own right, historically they arose as a means to study Lie groups.
We now briefly outline the relationship between Lie groups and Lie algebras. Any Lie group gives rise to a canonically determined Lie algebra (concretely, the tangent space at the identity). Conversely, for any finite-dimensional Lie algebra ${\mathfrak {g}}$, there exists a corresponding connected Lie group $G$ with Lie algebra ${\mathfrak {g}}$. This is Lie's third theorem; see the Baker–Campbell–Hausdorff formula. This Lie group is not determined uniquely; however, any two Lie groups with the same Lie algebra are locally isomorphic, and in particular, have the same universal cover. For instance, the special orthogonal group SO(3) and the special unitary group SU(2) give rise to the same Lie algebra, which is isomorphic to $\mathbb {R} ^{3}$ with the cross-product, but SU(2) is a simply-connected twofold cover of SO(3).
If we consider simply connected Lie groups, however, we have a one-to-one correspondence: For each (finite-dimensional real) Lie algebra ${\mathfrak {g}}$, there is a unique simply connected Lie group $G$ with Lie algebra ${\mathfrak {g}}$.
The correspondence between Lie algebras and Lie groups is used in several ways, including in the classification of Lie groups and the related matter of the representation theory of Lie groups. Every representation of a Lie algebra lifts uniquely to a representation of the corresponding connected, simply connected Lie group, and conversely every representation of any Lie group induces a representation of the group's Lie algebra; the representations are in one-to-one correspondence. Therefore, knowing the representations of a Lie algebra settles the question of representations of the group.
As for classification, it can be shown that any connected Lie group with a given Lie algebra is isomorphic to the universal cover mod a discrete central subgroup. So classifying Lie groups becomes simply a matter of counting the discrete subgroups of the center, once the classification of Lie algebras is known (solved by Cartan et al. in the semisimple case).
If the Lie algebra is infinite-dimensional, the issue is more subtle. In many instances, the exponential map is not even locally a homeomorphism (for example, in Diff(S1), one may find diffeomorphisms arbitrarily close to the identity that are not in the image of exp). Furthermore, some infinite-dimensional Lie algebras are not the Lie algebra of any group.
Real form and complexification
Given a complex Lie algebra ${\mathfrak {g}}$, a real Lie algebra ${\mathfrak {g}}_{0}$ is said to be a real form of ${\mathfrak {g}}$ if the complexification ${\mathfrak {g}}_{0}\otimes _{\mathbb {R} }\mathbb {C} \simeq {\mathfrak {g}}$ is isomorphic to ${\mathfrak {g}}$.[15] A real form need not be unique; for example, ${\mathfrak {sl}}_{2}\mathbb {C} $ has two real forms ${\mathfrak {sl}}_{2}\mathbb {R} $ and ${\mathfrak {su}}_{2}$.[15]
Given a semisimple finite-dimensional complex Lie algebra ${\mathfrak {g}}$, a split form of it is a real form that splits; i.e., it has a Cartan subalgebra which acts via an adjoint representation with real eigenvalues. A split form exists and is unique (up to isomorphisms).[15] A compact form is a real form that is the Lie algebra of a compact Lie group. A compact form exists and is also unique.[15]
Lie algebra with additional structures
A Lie algebra can be equipped with some additional structures that are assumed to be compatible with the bracket. For example, a graded Lie algebra is a Lie algebra with a graded vector space structure. If it also comes with differential (so that the underlying graded vector space is a chain complex), then it is called a differential graded Lie algebra.
A simplicial Lie algebra is a simplicial object in the category of Lie algebras; in other words, it is obtained by replacing the underlying set with a simplicial set (so it might be better thought of as a family of Lie algebras).
Lie ring
A Lie ring arises as a generalisation of Lie algebras, or through the study of the lower central series of groups. A Lie ring is defined as a nonassociative ring with multiplication that is anticommutative and satisfies the Jacobi identity. More specifically we can define a Lie ring $L$ to be an abelian group with an operation $[\cdot ,\cdot ]$ that has the following properties:
• Bilinearity:
$[x+y,z]=[x,z]+[y,z],\quad [z,x+y]=[z,x]+[z,y]$
for all x, y, z ∈ L.
• The Jacobi identity:
$[x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0\quad $
for all x, y, z in L.
• For all x in L:
$[x,x]=0\quad $
Lie rings need not be Lie groups under addition. Any Lie algebra is an example of a Lie ring. Any associative ring can be made into a Lie ring by defining a bracket operator $[x,y]=xy-yx$. Conversely to any Lie algebra there is a corresponding ring, called the universal enveloping algebra.
Lie rings are used in the study of finite p-groups through the Lazard correspondence. The lower central factors of a p-group are finite abelian p-groups, so modules over Z/pZ. The direct sum of the lower central factors is given the structure of a Lie ring by defining the bracket to be the commutator of two coset representatives. The Lie ring structure is enriched with another module homomorphism, the pth power map, making the associated Lie ring a so-called restricted Lie ring.
Lie rings are also useful in the definition of a p-adic analytic groups and their endomorphisms by studying Lie algebras over rings of integers such as the p-adic integers. The definition of finite groups of Lie type due to Chevalley involves restricting from a Lie algebra over the complex numbers to a Lie algebra over the integers, and then reducing modulo p to get a Lie algebra over a finite field.
Examples
• Any Lie algebra over a general ring instead of a field is an example of a Lie ring. Lie rings are not Lie groups under addition, despite the name.
• Any associative ring can be made into a Lie ring by defining a bracket operator
$[x,y]=xy-yx.$
• For an example of a Lie ring arising from the study of groups, let $G$ be a group with $[x,y]=x^{-1}y^{-1}xy$ the commutator operation, and let $G=G_{0}\supseteq G_{1}\supseteq G_{2}\supseteq \cdots \supseteq G_{n}\supseteq \cdots $ be a central series in $G$ — that is the commutator subgroup $[G_{i},G_{j}]$ is contained in $G_{i+j}$ for any $i,j$. Then
$L=\bigoplus G_{i}/G_{i+1}$
is a Lie ring with addition supplied by the group operation (which is abelian in each homogeneous part), and the bracket operation given by
$[xG_{i},yG_{j}]=[x,y]G_{i+j}\ $
extended linearly. The centrality of the series ensures that the commutator $[x,y]$ gives the bracket operation the appropriate Lie theoretic properties.
See also
• Adjoint representation of a Lie algebra
• Affine Lie algebra
• Anyonic Lie algebra
• Automorphism of a Lie algebra
• Chiral Lie algebra
• Free Lie algebra
• Index of a Lie algebra
• Lie algebra cohomology
• Lie algebra extension
• Lie algebra representation
• Lie bialgebra
• Lie coalgebra
• Lie operad
• Particle physics and representation theory
• Lie superalgebra
• Poisson algebra
• Pre-Lie algebra
• Quantum groups
• Moyal algebra
• Quasi-Frobenius Lie algebra
• Quasi-Lie algebra
• Restricted Lie algebra
• Serre relations
• Symmetric Lie algebra
• Gelfand–Fuks cohomology
Remarks
1. Bourbaki (1989, Section 2.) allows more generally for a module over a commutative ring; in this article, this is called a Lie ring.
References
1. O'Connor & Robertson 2000
2. O'Connor & Robertson 2005
3. Humphreys 1978, p. 1
4. Due to the anticommutativity of the commutator, the notions of a left and right ideal in a Lie algebra coincide.
5. Jacobson 1962, p. 28
6. Jacobson 1962, p. 42
7. Bourbaki 1989, §1.2. Example 1.
8. Bourbaki 1989, §1.2. Example 2.
9. Humphreys 1978, p. 2
10. Hall 2015, §3.4
11. Hall 2015, Example 3.27
12. Jacobson 1962, Ch. VI
13. Hall 2015, Theorem 10.9
14. Jacobson 1962, Ch. III, § 9.
15. Fulton & Harris 1991, §26.1.
Sources
• Beltiţă, Daniel (2006). Smooth Homogeneous Structures in Operator Theory. CRC Monographs and Surveys in Pure and Applied Mathematics. Vol. 137. CRC Press. ISBN 978-1-4200-3480-6. MR 2188389.
• Boza, Luis; Fedriani, Eugenio M.; Núñez, Juan (2001-06-01). "A new method for classifying complex filiform Lie algebras". Applied Mathematics and Computation. 121 (2–3): 169–175. doi:10.1016/s0096-3003(99)00270-2. ISSN 0096-3003.
• Bourbaki, Nicolas (1989). Lie Groups and Lie Algebras: Chapters 1-3. Springer. ISBN 978-3-540-64242-8.
• Erdmann, Karin; Wildon, Mark (2006). Introduction to Lie Algebras. Springer. ISBN 1-84628-040-0.
• Fulton, William; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. Vol. 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. MR 1153249. OCLC 246650103.
• Hall, Brian C. (2015). Lie groups, Lie algebras, and Representations: An Elementary Introduction. Graduate Texts in Mathematics. Vol. 222 (2nd ed.). Springer. doi:10.1007/978-3-319-13467-3. ISBN 978-3319134666. ISSN 0072-5285.
• Hofmann, Karl H.; Morris, Sidney A (2007). The Lie Theory of Connected Pro-Lie Groups. European Mathematical Society. ISBN 978-3-03719-032-6.
• Humphreys, James E. (1978). Introduction to Lie Algebras and Representation Theory. Graduate Texts in Mathematics. Vol. 9 (2nd ed.). Springer-Verlag. ISBN 978-0-387-90053-7.
• Jacobson, Nathan (1979) [1962]. Lie algebras. Dover. ISBN 978-0-486-63832-4.
• Kac, Victor G.; et al. Course notes for MIT 18.745: Introduction to Lie Algebras. Archived from the original on 2010-04-20.{{cite book}}: CS1 maint: bot: original URL status unknown (link)
• Mubarakzyanov, G.M. (1963). "On solvable Lie algebras". Izv. Vys. Ucheb. Zaved. Matematika (in Russian). 1 (32): 114–123. MR 0153714. Zbl 0166.04104.
• O'Connor, J.J; Robertson, E.F. (2000). "Biography of Sophus Lie". MacTutor History of Mathematics Archive.
• O'Connor, J.J; Robertson, E.F. (2005). "Biography of Wilhelm Killing". MacTutor History of Mathematics Archive.
• Popovych, R.O.; Boyko, V.M.; Nesterenko, M.O.; Lutfullin, M.W.; et al. (2003). "Realizations of real low-dimensional Lie algebras". J. Phys. A: Math. Gen. 36 (26): 7337–60. arXiv:math-ph/0301029. Bibcode:2003JPhA...36.7337P. doi:10.1088/0305-4470/36/26/309. S2CID 9800361.
• Serre, Jean-Pierre (2006). Lie Algebras and Lie Groups (2nd ed.). Springer. ISBN 978-3-540-55008-2.
• Steeb, Willi-Hans (2007). Continuous Symmetries, Lie Algebras, Differential Equations and Computer Algebra (2nd ed.). World Scientific. doi:10.1142/6515. ISBN 978-981-270-809-0. MR 2382250.
• Varadarajan, Veeravalli S. (2004). Lie Groups, Lie Algebras, and Their Representations (1st ed.). Springer. ISBN 978-0-387-90969-1.
External links
• "Lie algebra", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• McKenzie, Douglas (2015). "An Elementary Introduction to Lie Algebras for Physicists".
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Quotient rule
In calculus, the quotient rule is a method of finding the derivative of a function that is the ratio of two differentiable functions.[1][2][3] Let $h(x)={\frac {f(x)}{g(x)}}$, where both f and g are differentiable and $g(x)\neq 0.$ The quotient rule states that the derivative of h(x) is
$h'(x)={\frac {f'(x)g(x)-f(x)g'(x)}{g(x)^{2}}}.$
Part of a series of articles about
Calculus
• Fundamental theorem
• Limits
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It is provable in many ways by using other derivative rules.
Examples
Example 1: Basic example
Given $h(x)={\frac {e^{x}}{x^{2}}}$, let $f(x)=e^{x},g(x)=x^{2}$, then using the quotient rule:
${\begin{aligned}{\frac {d}{dx}}\left({\frac {e^{x}}{x^{2}}}\right)&={\frac {\left({\frac {d}{dx}}e^{x}\right)(x^{2})-(e^{x})\left({\frac {d}{dx}}x^{2}\right)}{(x^{2})^{2}}}\\&={\frac {(e^{x})(x^{2})-(e^{x})(2x)}{x^{4}}}\\&={\frac {x^{2}e^{x}-2xe^{x}}{x^{4}}}\\&={\frac {xe^{x}-2e^{x}}{x^{3}}}\\&={\frac {e^{x}(x-2)}{x^{3}}}.\end{aligned}}$
Example 2: Derivative of tangent function
The quotient rule can be used to find the derivative of $\tan x={\frac {\sin x}{\cos x}}$ as follows:
${\begin{aligned}{\frac {d}{dx}}\tan x&={\frac {d}{dx}}\left({\frac {\sin x}{\cos x}}\right)\\&={\frac {\left({\frac {d}{dx}}\sin x\right)(\cos x)-(\sin x)\left({\frac {d}{dx}}\cos x\right)}{\cos ^{2}x}}\\&={\frac {(\cos x)(\cos x)-(\sin x)(-\sin x)}{\cos ^{2}x}}\\&={\frac {\cos ^{2}x+\sin ^{2}x}{\cos ^{2}x}}\\&={\frac {1}{\cos ^{2}x}}=\sec ^{2}x.\end{aligned}}$
Reciprocal rule
The reciprocal rule is a special case of the quotient rule in which the numerator $f(x)=1$. Applying the quotient rule gives
$h'(x)={\frac {d}{dx}}\left[{\frac {1}{g(x)}}\right]={\frac {0\cdot g(x)-1\cdot g'(x)}{g(x)^{2}}}={\frac {-g'(x)}{g(x)^{2}}}.$
Note that utilizing the chain rule yields the same result.
Proofs
Proof from derivative definition and limit properties
Let $h(x)={\frac {f(x)}{g(x)}}.$ Applying the definition of the derivative and properties of limits gives the following proof, with the term $f(x)g(x)$ added and subtracted to allow splitting and factoring in subsequent steps without affecting the value:
${\begin{aligned}h'(x)&=\lim _{k\to 0}{\frac {h(x+k)-h(x)}{k}}\\&=\lim _{k\to 0}{\frac {{\frac {f(x+k)}{g(x+k)}}-{\frac {f(x)}{g(x)}}}{k}}\\&=\lim _{k\to 0}{\frac {f(x+k)g(x)-f(x)g(x+k)}{k\cdot g(x)g(x+k)}}\\&=\lim _{k\to 0}{\frac {f(x+k)g(x)-f(x)g(x+k)}{k}}\cdot \lim _{k\to 0}{\frac {1}{g(x)g(x+k)}}\\&=\lim _{k\to 0}\left[{\frac {f(x+k)g(x)-f(x)g(x)+f(x)g(x)-f(x)g(x+k)}{k}}\right]\cdot {\frac {1}{g(x)^{2}}}\\&=\left[\lim _{k\to 0}{\frac {f(x+k)g(x)-f(x)g(x)}{k}}-\lim _{k\to 0}{\frac {f(x)g(x+k)-f(x)g(x)}{k}}\right]\cdot {\frac {1}{g(x)^{2}}}\\&=\left[\lim _{k\to 0}{\frac {f(x+k)-f(x)}{k}}\cdot g(x)-f(x)\cdot \lim _{k\to 0}{\frac {g(x+k)-g(x)}{k}}\right]\cdot {\frac {1}{g(x)^{2}}}\\&={\frac {f'(x)g(x)-f(x)g'(x)}{g(x)^{2}}}.\end{aligned}}$
The limit evaluation $\lim _{k\to 0}{\frac {1}{g(x+k)g(x)}}={\frac {1}{g(x)^{2}}}$ is justified by the differentiability of $g(x)$, implying continuity, which can be expressed as $\lim _{k\to 0}g(x+k)=g(x)$.
Proof using implicit differentiation
Let $h(x)={\frac {f(x)}{g(x)}},$ so that $f(x)=g(x)h(x).$
The product rule then gives $f'(x)=g'(x)h(x)+g(x)h'(x).$
Solving for $h'(x)$ and substituting back for $h(x)$ gives:
${\begin{aligned}h'(x)&={\frac {f'(x)-g'(x)h(x)}{g(x)}}\\&={\frac {f'(x)-g'(x)\cdot {\frac {f(x)}{g(x)}}}{g(x)}}\\&={\frac {f'(x)g(x)-f(x)g'(x)}{g(x)^{2}}}.\end{aligned}}$
Proof using the reciprocal rule or chain rule
Let $h(x)={\frac {f(x)}{g(x)}}=f(x)\cdot {\frac {1}{g(x)}}.$
Then the product rule gives $h'(x)=f'(x)\cdot {\frac {1}{g(x)}}+f(x)\cdot {\frac {d}{dx}}\left[{\frac {1}{g(x)}}\right].$
To evaluate the derivative in the second term, apply the reciprocal rule, or the power rule along with the chain rule:
${\frac {d}{dx}}\left[{\frac {1}{g(x)}}\right]=-{\frac {1}{g(x)^{2}}}\cdot g'(x)={\frac {-g'(x)}{g(x)^{2}}}.$
Substituting the result into the expression gives
${\begin{aligned}h'(x)&=f'(x)\cdot {\frac {1}{g(x)}}+f(x)\cdot \left[{\frac {-g'(x)}{g(x)^{2}}}\right]\\&={\frac {f'(x)}{g(x)}}-{\frac {f(x)g'(x)}{g(x)^{2}}}\\&={\frac {g(x)}{g(x)}}\cdot {\frac {f'(x)}{g(x)}}-{\frac {f(x)g'(x)}{g(x)^{2}}}\\&={\frac {f'(x)g(x)-f(x)g'(x)}{g(x)^{2}}}.\end{aligned}}$
Proof by logarithmic differentiation
Let $h(x)={\frac {f(x)}{g(x)}}.$ Taking the absolute value and natural logarithm of both sides of the equation gives
$\ln |h(x)|=\ln \left|{\frac {f(x)}{g(x)}}\right|$
Applying properties of the absolute value and logarithms,
$\ln |h(x)|=\ln |f(x)|-\ln |g(x)|$
Taking the logarithmic derivative of both sides,
${\frac {h'(x)}{h(x)}}={\frac {f'(x)}{f(x)}}-{\frac {g'(x)}{g(x)}}$
Solving for $h'(x)$ and substituting back ${\tfrac {f(x)}{g(x)}}$ for $h(x)$ gives:
${\begin{aligned}h'(x)&=h(x)\left[{\frac {f'(x)}{f(x)}}-{\frac {g'(x)}{g(x)}}\right]\\&={\frac {f(x)}{g(x)}}\left[{\frac {f'(x)}{f(x)}}-{\frac {g'(x)}{g(x)}}\right]\\&={\frac {f'(x)}{g(x)}}-{\frac {f(x)g'(x)}{g(x)^{2}}}\\&={\frac {f'(x)g(x)-f(x)g'(x)}{g(x)^{2}}}.\end{aligned}}$
Note: Taking the absolute value of the functions is necessary for the logarithmic differentiation of functions that may have negative values, as logarithms are only real-valued for positive arguments. This works because ${\tfrac {d}{dx}}(\ln |u|)={\tfrac {u'}{u}}$, which justifies taking the absolute value of the functions for logarithmic differentiation.
Higher order derivatives
Implicit differentiation can be used to compute the nth derivative of a quotient (partially in terms of its first n − 1 derivatives). For example, differentiating $f=gh$ twice (resulting in $f''=g''h+2g'h'+gh''$) and then solving for $h''$ yields
$h''=\left({\frac {f}{g}}\right)''={\frac {f''-g''h-2g'h'}{g}}.$
See also
• Chain rule – For derivatives of composed functions
• Differentiation of integrals – Problem in mathematics
• Differentiation rules – Rules for computing derivatives of functions
• General Leibniz rule – Generalization of the product rule in calculus
• Inverse functions and differentiation – Calculus identityPages displaying short descriptions of redirect targets
• Linearity of differentiation – Calculus property
• Product rule – Formula for the derivative of a product
• Reciprocal rule – differentiation rulePages displaying wikidata descriptions as a fallback
• Table of derivatives – Rules for computing derivatives of functionsPages displaying short descriptions of redirect targets
• Vector calculus identities – Mathematical identities
References
1. Stewart, James (2008). Calculus: Early Transcendentals (6th ed.). Brooks/Cole. ISBN 978-0-495-01166-8.
2. Larson, Ron; Edwards, Bruce H. (2009). Calculus (9th ed.). Brooks/Cole. ISBN 978-0-547-16702-2.
3. Thomas, George B.; Weir, Maurice D.; Hass, Joel (2010). Thomas' Calculus: Early Transcendentals (12th ed.). Addison-Wesley. ISBN 978-0-321-58876-0.
Calculus
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Wikipedia
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Quotient by an equivalence relation
In mathematics, given a category C, a quotient of an object X by an equivalence relation $f:R\to X\times X$ is a coequalizer for the pair of maps
$R\ {\overset {f}{\to }}\ X\times X\ {\overset {\operatorname {pr} _{i}}{\to }}\ X,\ \ i=1,2,$
This article is about a generalization to category theory, used in scheme theory. For the common meaning, see Equivalence class.
where R is an object in C and "f is an equivalence relation" means that, for any object T in C, the image (which is a set) of $f:R(T)=\operatorname {Mor} (T,R)\to X(T)\times X(T)$ is an equivalence relation; that is, a reflexive, symmetric and transitive relation.
The basic case in practice is when C is the category of all schemes over some scheme S. But the notion is flexible and one can also take C to be the category of sheaves.
Examples
• Let X be a set and consider some equivalence relation on it. Let Q be the set of all equivalence classes in X. Then the map $q:X\to Q$ that sends an element x to the equivalence class to which x belongs is a quotient.
• In the above example, Q is a subset of the power set H of X. In algebraic geometry, one might replace H by a Hilbert scheme or disjoint union of Hilbert schemes. In fact, Grothendieck constructed a relative Picard scheme of a flat projective scheme X[1] as a quotient Q (of the scheme Z parametrizing relative effective divisors on X) that is a closed scheme of a Hilbert scheme H. The quotient map $q:Z\to Q$ can then be thought of as a relative version of the Abel map.
See also
• Categorical quotient, a special case
Notes
1. One also needs to assume the geometric fibers are integral schemes; Mumford's example shows the "integral" cannot be omitted.
References
• Nitsure, N. Construction of Hilbert and Quot schemes. Fundamental algebraic geometry: Grothendieck’s FGA explained, Mathematical Surveys and Monographs 123, American Mathematical Society 2005, 105–137.
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Quotient category
In mathematics, a quotient category is a category obtained from another category by identifying sets of morphisms. Formally, it is a quotient object in the category of (locally small) categories, analogous to a quotient group or quotient space, but in the categorical setting.
For the quotient of an abelian category by a Serre subcategory, see Quotient of an abelian category.
Definition
Let C be a category. A congruence relation R on C is given by: for each pair of objects X, Y in C, an equivalence relation RX,Y on Hom(X,Y), such that the equivalence relations respect composition of morphisms. That is, if
$f_{1},f_{2}:X\to Y\,$
are related in Hom(X, Y) and
$g_{1},g_{2}:Y\to Z\,$
are related in Hom(Y, Z), then g1f1 and g2f2 are related in Hom(X, Z).
Given a congruence relation R on C we can define the quotient category C/R as the category whose objects are those of C and whose morphisms are equivalence classes of morphisms in C. That is,
$\mathrm {Hom} _{{\mathcal {C}}/R}(X,Y)=\mathrm {Hom} _{\mathcal {C}}(X,Y)/R_{X,Y}.$
Composition of morphisms in C/R is well-defined since R is a congruence relation.
Properties
There is a natural quotient functor from C to C/R which sends each morphism to its equivalence class. This functor is bijective on objects and surjective on Hom-sets (i.e. it is a full functor).
Every functor F : C → D determines a congruence on C by saying f ~ g iff F(f) = F(g). The functor F then factors through the quotient functor C → C/~ in a unique manner. This may be regarded as the "first isomorphism theorem" for categories.
Examples
• Monoids and groups may be regarded as categories with one object. In this case the quotient category coincides with the notion of a quotient monoid or a quotient group.
• The homotopy category of topological spaces hTop is a quotient category of Top, the category of topological spaces. The equivalence classes of morphisms are homotopy classes of continuous maps.
• Let k be a field and consider the abelian category Mod(k) of all vector spaces over k with k-linear maps as morphisms. To "kill" all finite-dimensional spaces, we can call two linear maps f,g : X → Y congruent iff their difference has finite-dimensional image. In the resulting quotient category, all finite-dimensional vector spaces are isomorphic to 0. [This is actually an example of a quotient of additive categories, see below.]
Related concepts
Quotients of additive categories modulo ideals
If C is an additive category and we require the congruence relation ~ on C to be additive (i.e. if f1, f2, g1 and g2 are morphisms from X to Y with f1 ~ f2 and g1 ~g2, then f1 + g1 ~ f2 + g2), then the quotient category C/~ will also be additive, and the quotient functor C → C/~ will be an additive functor.
The concept of an additive congruence relation is equivalent to the concept of a two-sided ideal of morphisms: for any two objects X and Y we are given an additive subgroup I(X,Y) of HomC(X, Y) such that for all f ∈ I(X,Y), g ∈ HomC(Y, Z) and h∈ HomC(W, X), we have gf ∈ I(X,Z) and fh ∈ I(W,Y). Two morphisms in HomC(X, Y) are congruent iff their difference is in I(X,Y).
Every unital ring may be viewed as an additive category with a single object, and the quotient of additive categories defined above coincides in this case with the notion of a quotient ring modulo a two-sided ideal.
Localization of a category
The localization of a category introduces new morphisms to turn several of the original category's morphisms into isomorphisms. This tends to increase the number of morphisms between objects, rather than decrease it as in the case of quotient categories. But in both constructions it often happens that two objects become isomorphic that weren't isomorphic in the original category.
Serre quotients of abelian categories
The Serre quotient of an abelian category by a Serre subcategory is a new abelian category which is similar to a quotient category but also in many cases has the character of a localization of the category.
References
• Mac Lane, Saunders (1998). Categories for the Working Mathematician. Graduate Texts in Mathematics. Vol. 5 (Second ed.). Springer-Verlag.
Category theory
Key concepts
Key concepts
• Category
• Adjoint functors
• CCC
• Commutative diagram
• Concrete category
• End
• Exponential
• Functor
• Kan extension
• Morphism
• Natural transformation
• Universal property
Universal constructions
Limits
• Terminal objects
• Products
• Equalizers
• Kernels
• Pullbacks
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Colimits
• Initial objects
• Coproducts
• Coequalizers
• Cokernels and quotients
• Pushout
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Algebraic categories
• Sets
• Relations
• Magmas
• Groups
• Abelian groups
• Rings (Fields)
• Modules (Vector spaces)
Constructions on categories
• Free category
• Functor category
• Kleisli category
• Opposite category
• Quotient category
• Product category
• Comma category
• Subcategory
Higher category theory
Key concepts
• Categorification
• Enriched category
• Higher-dimensional algebra
• Homotopy hypothesis
• Model category
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n-categories
Weak n-categories
• Bicategory (pseudofunctor)
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• ∞-groupoid
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Strict n-categories
• 2-category (2-functor)
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Categorified concepts
• 2-group
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• En-ring
• (Traced)(Symmetric) monoidal category
• n-group
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• Category
• Outline
• Glossary
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Lie group action
In differential geometry, a Lie group action is a group action adapted to the smooth setting: G is a Lie group, M is a smooth manifold, and the action map is differentiable.
Definition and first properties
Let $\sigma :G\times M\to M,(g,x)\mapsto g\cdot x$ be a (left) group action of a Lie group G on a smooth manifold M; it is called a Lie group action (or smooth action) if the map $\sigma $ is differentiable. Equivalently, a Lie group action of G on M consists of a Lie group homomorphism $G\to \mathrm {Diff} (M)$. A smooth manifold endowed with a Lie group action is also called a G-manifold.
The fact that the action map $\sigma $ is smooth has a couple of immediate consequences:
• the stabilizers $G_{x}\subseteq G$ of the group action are closed, thus are Lie subgroups of G
• the orbits $G\cdot x\subseteq M$ of the group action are immersed submanifolds.
Forgetting the smooth structure, a Lie group action is a particular case of a continuous group action.
Examples
For every Lie group G, the following are Lie group actions:
• the trivial action of G on any manifold
• the action of G on itself by left multiplication, right multiplication or conjugation
• the action of any Lie subgroup $H\subseteq G$ on G by left multiplication, right multiplication or conjugation
• the adjoint action of G on its Lie algebra ${\mathfrak {g}}$.
Other examples of Lie group actions include:
• the action of $\mathbb {R} $ on M given by the flow of any complete vector field
• the actions of the general linear group $GL(n,\mathbb {R} )$ and of its Lie subgroups $G\subseteq GL(n,\mathbb {R} )$ on $\mathbb {R} ^{n}$ by matrix multiplication
• more generally, any Lie group representation on a vector space
• any Hamiltonian group action on a symplectic manifold
• the transitive action underlying any homogeneous space
• more generally, the group action underlying any principal bundle
Infinitesimal Lie algebra action
Following the spirit of the Lie group-Lie algebra correspondence, Lie group actions can also be studied from the infinitesimal point of view. Indeed, any Lie group action $\sigma :G\times M\to M$ induces an infinitesimal Lie algebra action on M, i.e. a Lie algebra homomorphism ${\mathfrak {g}}\to {\mathfrak {X}}(M)$. Intuitively, this is obtained by differentiating at the identity the Lie group homomorphism $G\to \mathrm {Diff} (M)$, and interpreting the set of vector fields ${\mathfrak {X}}(M)$ as the Lie algebra of the (infinite-dimensional) Lie group $\mathrm {Diff} (M)$.
More precisely, fixing any $x\in M$, the orbit map $\sigma _{x}:G\to M,g\mapsto g\cdot x$ is differentiable and one can compute its differential at the identity $e\in G$. If $X\in {\mathfrak {g}}$, then its image under $d_{e}\sigma _{x}:{\mathfrak {g}}\to T_{x}M$ is a tangent vector at x, and varying x one obtains a vector field on M. The minus of this vector field, denoted by $X^{\#}$, is also called the fundamental vector field associated with X (the minus sign ensures that ${\mathfrak {g}}\to {\mathfrak {X}}(M),X\mapsto X^{\#}$ is a Lie algebra homomorphism).
Conversely, by Lie–Palais theorem, any abstract infinitesimal action of a (finite-dimensional) Lie algebra on a compact manifold can be integrated to a Lie group action.[1]
Moreover, an infinitesimal Lie algebra action ${\mathfrak {g}}\to {\mathfrak {X}}(M)$ is injective if and only if the corresponding global Lie group action is free. This follows from the fact that the kernel of $d_{e}\sigma _{x}:{\mathfrak {g}}\to T_{x}M$ is the Lie algebra ${\mathfrak {g}}_{x}\subseteq {\mathfrak {g}}$ of the stabilizer $G_{x}\subseteq G$. On the other hand, ${\mathfrak {g}}\to {\mathfrak {X}}(M)$ in general not surjective. For instance, let $\pi :P\to M$ be a principal G-bundle: the image of the infinitesimal action is actually equal to the vertical subbundle $T^{\pi }P\subset TP$.
Proper actions
An important (and common) class of Lie group actions is that of proper ones. Indeed, such a topological condition implies that
• the stabilizers $G_{x}\subseteq G$ are compact
• the orbits $G\cdot x\subseteq M$ are embedded submanifolds
• the orbit space $M/G$ is Hausdorff
In general, if a Lie group G is compact, any smooth G-action is automatically proper. An example of proper action by a not necessarily compact Lie group is given by the action a Lie subgroup $H\subseteq G$ on G.
Structure of the orbit space
Given a Lie group action of G on M, the orbit space $M/G$ does not admit in general a manifold structure. However, if the action is free and proper, then $M/G$ has a unique smooth structure such that the projection $M\to M/G$ is a submersion (in fact, $M\to M/G$ is a principal G-bundle).[2]
The fact that $M/G$ is Hausdorff depends only on the properness of the action (as discussed above); the rest of the claim requires freeness and is a consequence of the slice theorem. If the "free action" condition (i.e. "having zero stabilizers") is relaxed to "having finite stabilizers", $M/G$ becomes instead an orbifold (or quotient stack).
An application of this principle is the Borel construction from algebraic topology. Assuming that G is compact, let $EG$ denote the universal bundle, which we can assume to be a manifold since G is compact, and let G act on $EG\times M$ diagonally. The action is free since it is so on the first factor and is proper since G is compact; thus, one can form the quotient manifold $M_{G}=(EG\times M)/G$ and define the equivariant cohomology of M as
$H_{G}^{*}(M)=H_{\text{dr}}^{*}(M_{G})$,
where the right-hand side denotes the de Rham cohomology of the manifold $M_{G}$.
See also
• Hamiltonian group action
• Equivariant differential form
• isotropy representation
Notes
1. Palais, Richard S. (1957). "A global formulation of the Lie theory of transformation groups". Memoirs of the American Mathematical Society (22): 0. doi:10.1090/memo/0022. ISSN 0065-9266.
2. Lee, John M. (2012). Introduction to smooth manifolds (2nd ed.). New York: Springer. ISBN 978-1-4419-9982-5. OCLC 808682771.
References
• Michele Audin, Torus actions on symplectic manifolds, Birkhauser, 2004
• John Lee, Introduction to smooth manifolds, chapter 9, ISBN 978-1-4419-9981-8
• Frank Warner, Foundations of differentiable manifolds and Lie groups, chapter 3, ISBN 978-0-387-90894-6
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Quotient module
In algebra, given a module and a submodule, one can construct their quotient module.[1][2] This construction, described below, is very similar to that of a quotient vector space.[3] It differs from analogous quotient constructions of rings and groups by the fact that in these cases, the subspace that is used for defining the quotient is not of the same nature as the ambient space (that is, a quotient ring is the quotient of a ring by an ideal, not a subring, and a quotient group is the quotient of a group by a normal subgroup, not by a general subgroup).
Given a module A over a ring R, and a submodule B of A, the quotient space A/B is defined by the equivalence relation
$a\sim b$ if and only if $b-a\in B,$
for any a, b in A.[4] The elements of A/B are the equivalence classes $[a]=a+B=\{a+b:b\in B\}.$ The function $\pi :A\to A/B$ sending a in A to its equivalence class a + B is called the quotient map or the projection map, and is a module homomorphism.
The addition operation on A/B is defined for two equivalence classes as the equivalence class of the sum of two representatives from these classes; and scalar multiplication of elements of A/B by elements of R is defined similarly. Note that it has to be shown that these operations are well-defined. Then A/B becomes itself an R-module, called the quotient module. In symbols, for all a, b in A and r in R:
${\begin{aligned}&(a+B)+(b+B):=(a+b)+B,\\&r\cdot (a+B):=(r\cdot a)+B.\end{aligned}}$
Examples
Consider the polynomial ring, $\mathbb {R} [X]$ with real coefficients, and the $\mathbb {R} [X]$-module $A=\mathbb {R} [X],$ . Consider the submodule
$B=(X^{2}+1)\mathbb {R} [X]$
of A, that is, the submodule of all polynomials divisible by X 2 + 1. It follows that the equivalence relation determined by this module will be
P(X) ~ Q(X) if and only if P(X) and Q(X) give the same remainder when divided by X 2 + 1.
Therefore, in the quotient module A/B, X 2 + 1 is the same as 0; so one can view A/B as obtained from $\mathbb {R} [X]$ by setting X 2 + 1 = 0. This quotient module is isomorphic to the complex numbers, viewed as a module over the real numbers $\mathbb {R} .$
See also
• Quotient group
• Quotient ring
• Quotient (universal algebra)
References
1. Dummit, David S.; Foote, Richard M. (2004). Abstract Algebra (3rd ed.). John Wiley & Sons. ISBN 0-471-43334-9.
2. Lang, Serge (2002). Algebra. Graduate Texts in Mathematics. Springer. ISBN 0-387-95385-X.
3. Roman, Steven (2008). Advanced linear algebra (3rd ed.). New York: Springer Science + Business Media. p. 117. ISBN 978-0-387-72828-5.
4. Roman 2008, p. 118 Theorem 4.7
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Quotient of a formal language
In mathematics and computer science, the right quotient (or simply quotient) of a language $L_{1}$ with respect to language $L_{2}$ is the language consisting of strings w such that wx is in $L_{1}$ for some string x in $L_{2}$.[1] Formally:
$L_{1}/L_{2}=\{w\in \Sigma ^{*}\mid \exists x\in L_{2}\colon \ wx\in L_{1}\}$
In other words, we take all the strings in $L_{1}$ that have a suffix in $L_{2}$, and remove this suffix.
Similarly, the left quotient of $L_{1}$ with respect to $L_{2}$ is the language consisting of strings w such that xw is in $L_{1}$ for some string x in $L_{2}$. Formally:
$L_{2}\backslash L_{1}=\{w\in \Sigma ^{*}\mid \exists x\in L_{2}\colon \ xw\in L_{1}\}$
In other words, we take all the strings in $L_{1}$ that have a prefix in $L_{2}$, and remove this prefix.
Note that the operands of $\backslash $ are in reverse order: the first operand is $L_{2}$ and $L_{1}$ is second.
Example
Consider
$L_{1}=\{a^{n}b^{n}c^{n}\mid n\geq 0\}$
and
$L_{2}=\{b^{i}c^{j}\mid i,j\geq 0\}.$
Now, if we insert a divider into an element of $L_{1}$, the part on the right is in $L_{2}$ only if the divider is placed adjacent to a b (in which case i ≤ n and j = n) or adjacent to a c (in which case i = 0 and j ≤ n). The part on the left, therefore, will be either $a^{n}b^{n-i}$ or $a^{n}b^{n}c^{n-j}$; and $L_{1}/L_{2}$ can be written as
$\{\ a^{p}b^{q}c^{r}\ \mid \ p=q\geq r\ \ {\text{or}}\ \ (p\geq q{\text{ and }}r=0)\ \}.$
Properties
Some common closure properties of the quotient operation include:
• The quotient of a regular language with any other language is regular.
• The quotient of a context free language with a regular language is context free.
• The quotient of two context free languages can be any recursively enumerable language.
• The quotient of two recursively enumerable languages is recursively enumerable.
These closure properties hold for both left and right quotients.
See also
• Brzozowski derivative
References
1. Linz, Peter (2011). An Introduction to Formal Languages and Automata. Jones & Bartlett Publishers. pp. 104–108. ISBN 9781449615529. Retrieved 7 July 2014.
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Wikipedia
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Equivalence class
In mathematics, when the elements of some set $S$ have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set $S$ into equivalence classes. These equivalence classes are constructed so that elements $a$ and $b$ belong to the same equivalence class if, and only if, they are equivalent.
"Quotient map" redirects here. For Quotient map in topology, see Quotient map (topology).
Formally, given a set $S$ and an equivalence relation $\,\sim \,$ on $S,$ the equivalence class of an element $a$ in $S,$ denoted by $[a],$[1] is the set[2]
$\{x\in S:x\sim a\}$
of elements which are equivalent to $a.$ It may be proven, from the defining properties of equivalence relations, that the equivalence classes form a partition of $S.$ This partition—the set of equivalence classes—is sometimes called the quotient set or the quotient space of $S$ by $\,\sim \,,$ and is denoted by $S/{\sim }$ .
When the set $S$ has some structure (such as a group operation or a topology) and the equivalence relation $\,\sim \,$ is compatible with this structure, the quotient set often inherits a similar structure from its parent set. Examples include quotient spaces in linear algebra, quotient spaces in topology, quotient groups, homogeneous spaces, quotient rings, quotient monoids, and quotient categories.
Examples
• Let $X$ be the set of all rectangles in a plane, and $\,\sim \,$ the equivalence relation "has the same area as", then for each positive real number $A,$ there will be an equivalence class of all the rectangles that have area $A.$[3]
• Consider the modulo 2 equivalence relation on the set of integers, $\mathbb {Z} ,$ such that $x\sim y$ if and only if their difference $x-y$ is an even number. This relation gives rise to exactly two equivalence classes: one class consists of all even numbers, and the other class consists of all odd numbers. Using square brackets around one member of the class to denote an equivalence class under this relation, $[7],[9],$ and $[1]$ all represent the same element of $\mathbb {Z} /{\sim }.$[4]
• Let $X$ be the set of ordered pairs of integers $(a,b)$ with non-zero $b,$ and define an equivalence relation $\,\sim \,$ on $X$ such that $(a,b)\sim (c,d)$ if and only if $ad=bc,$ then the equivalence class of the pair $(a,b)$ can be identified with the rational number $a/b,$ and this equivalence relation and its equivalence classes can be used to give a formal definition of the set of rational numbers.[5] The same construction can be generalized to the field of fractions of any integral domain.
• If $X$ consists of all the lines in, say, the Euclidean plane, and $L\sim M$ means that $L$ and $M$ are parallel lines, then the set of lines that are parallel to each other form an equivalence class, as long as a line is considered parallel to itself. In this situation, each equivalence class determines a point at infinity.
Definition and notation
An equivalence relation on a set $X$ is a binary relation $\,\sim \,$ on $X$ satisfying the three properties:[6][7]
• $a\sim a$ for all $a\in X$ (reflexivity),
• $a\sim b$ implies $b\sim a$ for all $a,b\in X$ (symmetry),
• if $a\sim b$ and $b\sim c$ then $a\sim c$ for all $a,b,c\in X$ (transitivity).
The equivalence class of an element $a$ is often denoted $[a]$ or $[a]_{\sim },$ and is defined as the set $\{x\in X:a\sim x\}$ of elements that are related to $a$ by $\,\sim .$[2] The word "class" in the term "equivalence class" may generally be considered as a synonym of "set", although some equivalence classes are not sets but proper classes. For example, "being isomorphic" is an equivalence relation on groups, and the equivalence classes, called isomorphism classes, are not sets.
The set of all equivalence classes in $X$ with respect to an equivalence relation $R$ is denoted as $X/R,$ and is called $X$ modulo $R$ (or the quotient set of $X$ by $R$).[8] The surjective map $x\mapsto [x]$ from $X$ onto $X/R,$ which maps each element to its equivalence class, is called the canonical surjection, or the canonical projection.
Every element of an equivalence class characterizes the class, and may be used to represent it. When such an element is chosen, it is called a representative of the class. The choice of a representative in each class defines an injection from $X/R$ to X. Since its composition with the canonical surjection is the identity of $X/R,$ such an injection is called a section, when using the terminology of category theory.
Sometimes, there is a section that is more "natural" than the other ones. In this case, the representatives are called canonical representatives. For example, in modular arithmetic, for every integer m greater than 1, the congruence modulo m is an equivalence relation on the integers, for which two integers a and b are equivalent—in this case, one says congruent —if m divides $a-b;$ this is denoted $ a\equiv b{\pmod {m}}.$ Each class contains a unique non-negative integer smaller than $m,$ and these integers are the canonical representatives.
The use of representatives for representing classes allows avoiding to consider explicitly classes as sets. In this case, the canonical surjection that maps an element to its class is replaced by the function that maps an element to the representative of its class. In the preceding example, this function is denoted $a{\bmod {m}},$ and produces the remainder of the Euclidean division of a by m.
Properties
Every element $x$ of $X$ is a member of the equivalence class $[x].$ Every two equivalence classes $[x]$ and $[y]$ are either equal or disjoint. Therefore, the set of all equivalence classes of $X$ forms a partition of $X$: every element of $X$ belongs to one and only one equivalence class.[9] Conversely, every partition of $X$ comes from an equivalence relation in this way, according to which $x\sim y$ if and only if $x$ and $y$ belong to the same set of the partition.[10]
It follows from the properties of an equivalence relation that
$x\sim y$
if and only if $[x]=[y].$
In other words, if $\,\sim \,$ is an equivalence relation on a set $X,$ and $x$ and $y$ are two elements of $X,$ then these statements are equivalent:
• $x\sim y$
• $[x]=[y]$
• $[x]\cap [y]\neq \emptyset .$
Graphical representation
An undirected graph may be associated to any symmetric relation on a set $X,$ where the vertices are the elements of $X,$ and two vertices $s$ and $t$ are joined if and only if $s\sim t.$ Among these graphs are the graphs of equivalence relations. These graphs, called cluster graphs, are characterized as the graphs such that the connected components are cliques.[4]
Invariants
If $\,\sim \,$ is an equivalence relation on $X,$ and $P(x)$ is a property of elements of $X$ such that whenever $x\sim y,$ $P(x)$ is true if $P(y)$ is true, then the property $P$ is said to be an invariant of $\,\sim \,,$ or well-defined under the relation $\,\sim .$
A frequent particular case occurs when $f$ is a function from $X$ to another set $Y$; if $f\left(x_{1}\right)=f\left(x_{2}\right)$ whenever $x_{1}\sim x_{2},$ then $f$ is said to be class invariant under $\,\sim \,,$ or simply invariant under $\,\sim .$ This occurs, for example, in the character theory of finite groups. Some authors use "compatible with $\,\sim \,$" or just "respects $\,\sim \,$" instead of "invariant under $\,\sim \,$".
Any function $f:X\to Y$ is class invariant under $\,\sim \,,$ according to which $x_{1}\sim x_{2}$ if and only if $f\left(x_{1}\right)=f\left(x_{2}\right).$ The equivalence class of $x$ is the set of all elements in $X$ which get mapped to $f(x),$ that is, the class $[x]$ is the inverse image of $f(x).$ This equivalence relation is known as the kernel of $f.$
More generally, a function may map equivalent arguments (under an equivalence relation $\sim _{X}$ on $X$) to equivalent values (under an equivalence relation $\sim _{Y}$ on $Y$). Such a function is a morphism of sets equipped with an equivalence relation.
Quotient space in topology
In topology, a quotient space is a topological space formed on the set of equivalence classes of an equivalence relation on a topological space, using the original space's topology to create the topology on the set of equivalence classes.
In abstract algebra, congruence relations on the underlying set of an algebra allow the algebra to induce an algebra on the equivalence classes of the relation, called a quotient algebra. In linear algebra, a quotient space is a vector space formed by taking a quotient group, where the quotient homomorphism is a linear map. By extension, in abstract algebra, the term quotient space may be used for quotient modules, quotient rings, quotient groups, or any quotient algebra. However, the use of the term for the more general cases can as often be by analogy with the orbits of a group action.
The orbits of a group action on a set may be called the quotient space of the action on the set, particularly when the orbits of the group action are the right cosets of a subgroup of a group, which arise from the action of the subgroup on the group by left translations, or respectively the left cosets as orbits under right translation.
A normal subgroup of a topological group, acting on the group by translation action, is a quotient space in the senses of topology, abstract algebra, and group actions simultaneously.
Although the term can be used for any equivalence relation's set of equivalence classes, possibly with further structure, the intent of using the term is generally to compare that type of equivalence relation on a set $X,$ either to an equivalence relation that induces some structure on the set of equivalence classes from a structure of the same kind on $X,$ or to the orbits of a group action. Both the sense of a structure preserved by an equivalence relation, and the study of invariants under group actions, lead to the definition of invariants of equivalence relations given above.
See also
• Equivalence partitioning, a method for devising test sets in software testing based on dividing the possible program inputs into equivalence classes according to the behavior of the program on those inputs
• Homogeneous space, the quotient space of Lie groups
• Partial equivalence relation – Mathematical concept for comparing objects
• Quotient by an equivalence relation
• Setoid – Mathematical construction of a set with an equivalence relation
• Transversal (combinatorics) – Set that intersects every one of a family of sets
Notes
1. "7.3: Equivalence Classes". Mathematics LibreTexts. 2017-09-20. Retrieved 2020-08-30.
2. Weisstein, Eric W. "Equivalence Class". mathworld.wolfram.com. Retrieved 2020-08-30.
3. Avelsgaard 1989, p. 127
4. Devlin 2004, p. 123
5. Maddox 2002, pp. 77–78
6. Devlin 2004, p. 122
7. Weisstein, Eric W. "Equivalence Relation". mathworld.wolfram.com. Retrieved 2020-08-30.
8. Wolf 1998, p. 178
9. Maddox 2002, p. 74, Thm. 2.5.15
10. Avelsgaard 1989, p. 132, Thm. 3.16
References
• Avelsgaard, Carol (1989), Foundations for Advanced Mathematics, Scott Foresman, ISBN 0-673-38152-8
• Devlin, Keith (2004), Sets, Functions, and Logic: An Introduction to Abstract Mathematics (3rd ed.), Chapman & Hall/ CRC Press, ISBN 978-1-58488-449-1
• Maddox, Randall B. (2002), Mathematical Thinking and Writing, Harcourt/ Academic Press, ISBN 0-12-464976-9
• Wolf, Robert S. (1998), Proof, Logic and Conjecture: A Mathematician's Toolbox, Freeman, ISBN 978-0-7167-3050-7
Further reading
• Sundstrom (2003), Mathematical Reasoning: Writing and Proof, Prentice-Hall
• Smith; Eggen; St.Andre (2006), A Transition to Advanced Mathematics (6th ed.), Thomson (Brooks/Cole)
• Schumacher, Carol (1996), Chapter Zero: Fundamental Notions of Abstract Mathematics, Addison-Wesley, ISBN 0-201-82653-4
• O'Leary (2003), The Structure of Proof: With Logic and Set Theory, Prentice-Hall
• Lay (2001), Analysis with an introduction to proof, Prentice Hall
• Morash, Ronald P. (1987), Bridge to Abstract Mathematics, Random House, ISBN 0-394-35429-X
• Gilbert; Vanstone (2005), An Introduction to Mathematical Thinking, Pearson Prentice-Hall
• Fletcher; Patty, Foundations of Higher Mathematics, PWS-Kent
• Iglewicz; Stoyle, An Introduction to Mathematical Reasoning, MacMillan
• D'Angelo; West (2000), Mathematical Thinking: Problem Solving and Proofs, Prentice Hall
• Cupillari, The Nuts and Bolts of Proofs, Wadsworth
• Bond, Introduction to Abstract Mathematics, Brooks/Cole
• Barnier; Feldman (2000), Introduction to Advanced Mathematics, Prentice Hall
• Ash, A Primer of Abstract Mathematics, MAA
External links
• Media related to Equivalence classes at Wikimedia Commons
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Quotient space (linear algebra)
In linear algebra, the quotient of a vector space $V$ by a subspace $N$ is a vector space obtained by "collapsing" $N$ to zero. The space obtained is called a quotient space and is denoted $V/N$ (read "$V$ mod $N$" or "$V$ by $N$").
This article is about quotients of vector spaces. For quotients of topological spaces, see Quotient space (topology).
Definition
Formally, the construction is as follows.[1] Let $V$ be a vector space over a field $\mathbb {K} $, and let $N$ be a subspace of $V$. We define an equivalence relation $\sim $ on $V$ by stating that $x\sim y$ if $x-y\in N$. That is, $x$ is related to $y$ if one can be obtained from the other by adding an element of $N$. From this definition, one can deduce that any element of $N$ is related to the zero vector; more precisely, all the vectors in $N$ get mapped into the equivalence class of the zero vector.
The equivalence class – or, in this case, the coset – of $x$ is often denoted
$[x]=x+N$
since it is given by
$[x]=\{x+n:n\in N\}$
The quotient space $V/N$ is then defined as $V/_{\sim }$, the set of all equivalence classes induced by $\sim $ on $V$. Scalar multiplication and addition are defined on the equivalence classes by[2][3]
• $\alpha [x]=[\alpha x]$ for all $\alpha \in \mathbb {K} $, and
• $[x]+[y]=[x+y]$.
It is not hard to check that these operations are well-defined (i.e. do not depend on the choice of representatives). These operations turn the quotient space $V/N$ into a vector space over $\mathbb {K} $ with $N$ being the zero class, $[0]$.
The mapping that associates to $v\in V$ the equivalence class $[v]$ is known as the quotient map.
Alternatively phrased, the quotient space $V/N$ is the set of all affine subsets of $V$ which are parallel to $N$.[4]
Examples
Lines in Cartesian Plane
Let X = R2 be the standard Cartesian plane, and let Y be a line through the origin in X. Then the quotient space X/Y can be identified with the space of all lines in X which are parallel to Y. That is to say that, the elements of the set X/Y are lines in X parallel to Y. Note that the points along any one such line will satisfy the equivalence relation because their difference vectors belong to Y. This gives a way to visualize quotient spaces geometrically. (By re-parameterising these lines, the quotient space can more conventionally be represented as the space of all points along a line through the origin that is not parallel to Y. Similarly, the quotient space for R3 by a line through the origin can again be represented as the set of all co-parallel lines, or alternatively be represented as the vector space consisting of a plane which only intersects the line at the origin.)
Subspaces of Cartesian Space
Another example is the quotient of Rn by the subspace spanned by the first m standard basis vectors. The space Rn consists of all n-tuples of real numbers (x1, ..., xn). The subspace, identified with Rm, consists of all n-tuples such that the last n − m entries are zero: (x1, ..., xm, 0, 0, ..., 0). Two vectors of Rn are in the same equivalence class modulo the subspace if and only if they are identical in the last n − m coordinates. The quotient space Rn/Rm is isomorphic to Rn−m in an obvious manner.
Polynomial Vector Space
Let ${\mathcal {P}}_{3}(\mathbb {R} )$ be the vector space of all cubic polynomials over the real numbers. Then ${\mathcal {P}}_{3}(\mathbb {R} )/\langle x^{2}\rangle $ is a quotient space, where each element is the set corresponding to polynomials that differ by a quadratic term only. For example, one element of the quotient space is $\{x^{3}+ax^{2}-2x+3:a\in \mathbb {R} \}$, while another element of the quotient space is $\{ax^{2}+2.7x:a\in \mathbb {R} \}$.
General Subspaces
More generally, if V is an (internal) direct sum of subspaces U and W,
$V=U\oplus W$
then the quotient space V/U is naturally isomorphic to W.[5]
Lebesgue Integrals
An important example of a functional quotient space is an Lp space.
Properties
There is a natural epimorphism from V to the quotient space V/U given by sending x to its equivalence class [x]. The kernel (or nullspace) of this epimorphism is the subspace U. This relationship is neatly summarized by the short exact sequence
$0\to U\to V\to V/U\to 0.\,$
If U is a subspace of V, the dimension of V/U is called the codimension of U in V. Since a basis of V may be constructed from a basis A of U and a basis B of V/U by adding a representative of each element of B to A, the dimension of V is the sum of the dimensions of U and V/U. If V is finite-dimensional, it follows that the codimension of U in V is the difference between the dimensions of V and U:[6][7]
$\mathrm {codim} (U)=\dim(V/U)=\dim(V)-\dim(U).$
Let T : V → W be a linear operator. The kernel of T, denoted ker(T), is the set of all x in V such that Tx = 0. The kernel is a subspace of V. The first isomorphism theorem for vector spaces says that the quotient space V/ker(T) is isomorphic to the image of V in W. An immediate corollary, for finite-dimensional spaces, is the rank–nullity theorem: the dimension of V is equal to the dimension of the kernel (the nullity of T) plus the dimension of the image (the rank of T).
The cokernel of a linear operator T : V → W is defined to be the quotient space W/im(T).
Quotient of a Banach space by a subspace
If X is a Banach space and M is a closed subspace of X, then the quotient X/M is again a Banach space. The quotient space is already endowed with a vector space structure by the construction of the previous section. We define a norm on X/M by
$\|[x]\|_{X/M}=\inf _{m\in M}\|x-m\|_{X}=\inf _{m\in M}\|x+m\|_{X}=\inf _{y\in [x]}\|y\|_{X}.$
When X is complete, then the quotient space X/M is complete with respect to the norm, and therefore a Banach space.
Examples
Let C[0,1] denote the Banach space of continuous real-valued functions on the interval [0,1] with the sup norm. Denote the subspace of all functions f ∈ C[0,1] with f(0) = 0 by M. Then the equivalence class of some function g is determined by its value at 0, and the quotient space C[0,1]/M is isomorphic to R.
If X is a Hilbert space, then the quotient space X/M is isomorphic to the orthogonal complement of M.
Generalization to locally convex spaces
The quotient of a locally convex space by a closed subspace is again locally convex.[8] Indeed, suppose that X is locally convex so that the topology on X is generated by a family of seminorms {pα | α ∈ A} where A is an index set. Let M be a closed subspace, and define seminorms qα on X/M by
$q_{\alpha }([x])=\inf _{v\in [x]}p_{\alpha }(v).$
Then X/M is a locally convex space, and the topology on it is the quotient topology.
If, furthermore, X is metrizable, then so is X/M. If X is a Fréchet space, then so is X/M.[9]
See also
• Quotient group
• Quotient module
• Quotient set
• Quotient space (topology)
References
1. Halmos (1974) pp. 33-34 §§ 21-22
2. Katznelson & Katznelson (2008) p. 9 § 1.2.4
3. Roman (2005) p. 75-76, ch. 3
4. Axler (2015) p. 95, § 3.83
5. Halmos (1974) p. 34, § 22, Theorem 1
6. Axler (2015) p. 97, § 3.89
7. Halmos (1974) p. 34, § 22, Theorem 2
8. Dieudonné (1976) p. 65, § 12.14.8
9. Dieudonné (1976) p. 54, § 12.11.3
Sources
• Axler, Sheldon (2015). Linear Algebra Done Right. Undergraduate Texts in Mathematics (3rd ed.). Springer. ISBN 978-3-319-11079-0.
• Dieudonné, Jean (1976), Treatise on Analysis, vol. 2, Academic Press, ISBN 978-0122155024
• Halmos, Paul Richard (1974) [1958]. Finite-Dimensional Vector Spaces. Undergraduate Texts in Mathematics (2nd ed.). Springer. ISBN 0-387-90093-4.
• Katznelson, Yitzhak; Katznelson, Yonatan R. (2008). A (Terse) Introduction to Linear Algebra. American Mathematical Society. ISBN 978-0-8218-4419-9.
• Roman, Steven (2005). Advanced Linear Algebra. Graduate Texts in Mathematics (2nd ed.). Springer. ISBN 0-387-24766-1.
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Quotient space of an algebraic stack
In algebraic geometry, the quotient space of an algebraic stack F, denoted by |F|, is a topological space which as a set is the set of all integral substacks of F and which then is given a "Zariski topology": an open subset has a form $|U|\subset |F|$ for some open substack U of F.[1]
The construction $X\mapsto |X|$ is functorial; i.e., each morphism $f:X\to Y$ of algebraic stacks determines a continuous map $f:|X|\to |Y|$.
An algebraic stack X is punctual if $|X|$ is a point.
When X is a moduli stack, the quotient space $|X|$ is called the moduli space of X. If $f:X\to Y$ is a morphism of algebraic stacks that induces a homeomorphism $f:|X|{\overset {\sim }{\to }}|Y|$, then Y is called a coarse moduli stack of X. ("The" coarse moduli requires a universality.)
References
1. In other words, there is a natural bijection between the set of all open immersions to F and the set of all open subsets of $|F|$.
• H. Gillet, Intersection theory on algebraic stacks and Q-varieties, J. Pure Appl. Algebra 34 (1984), 193–240, Proceedings of the Luminy conference on algebraic K-theory (Luminy, 1983).
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Geometric quotient
In algebraic geometry, a geometric quotient of an algebraic variety X with the action of an algebraic group G is a morphism of varieties $\pi :X\to Y$ such that[1]
(i) For each y in Y, the fiber $\pi ^{-1}(y)$ is an orbit of G.
(ii) The topology of Y is the quotient topology: a subset $U\subset Y$ is open if and only if $\pi ^{-1}(U)$ is open.
(iii) For any open subset $U\subset Y$, $\pi ^{\#}:k[U]\to k[\pi ^{-1}(U)]^{G}$ is an isomorphism. (Here, k is the base field.)
The notion appears in geometric invariant theory. (i), (ii) say that Y is an orbit space of X in topology. (iii) may also be phrased as an isomorphism of sheaves ${\mathcal {O}}_{Y}\simeq \pi _{*}({\mathcal {O}}_{X}^{G})$. In particular, if X is irreducible, then so is Y and $k(Y)=k(X)^{G}$: rational functions on Y may be viewed as invariant rational functions on X (i.e., rational-invariants of X).
For example, if H is a closed subgroup of G, then $G/H$ is a geometric quotient. A GIT quotient may or may not be a geometric quotient: but both are categorical quotients, which is unique; in other words, one cannot have both types of quotients (without them being the same).
Relation to other quotients
A geometric quotient is a categorical quotient. This is proved in Mumford's geometric invariant theory.
A geometric quotient is precisely a good quotient whose fibers are orbits of the group.
Examples
• The canonical map $\mathbb {A} ^{n+1}\setminus 0\to \mathbb {P} ^{n}$ is a geometric quotient.
• If L is a linearized line bundle on an algebraic G-variety X, then, writing $X_{(0)}^{s}$ for the set of stable points with respect to L, the quotient
$X_{(0)}^{s}\to X_{(0)}^{s}/G$
is a geometric quotient.
References
1. Brion, M. "Introduction to actions of algebraic groups" (PDF). Definition 1.18.
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Wikipedia
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Quotientable automorphism
In mathematics, in the realm of group theory, a quotientable automorphism of a group is an automorphism that takes every normal subgroup to within itself. As a result, it gives a corresponding automorphism for every quotient group.
All family automorphisms are quotientable, and particularly, all class automorphisms and power automorphisms are. As well, all inner automorphisms are quotientable, and more generally, any automorphism defined by an algebraic formula is quotientable.
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Quotient
In arithmetic, a quotient (from Latin: quotiens 'how many times', pronounced /ˈkwoʊʃənt/) is a quantity produced by the division of two numbers.[1] The quotient has widespread use throughout mathematics. It has two definitions: either the integer part of a division (in the case of Euclidean division),[2] or as a fraction or a ratio (in the case of a general division). For example, when dividing 20 (the dividend) by 3 (the divisor), the quotient is 6 (with a remainder of 2) in the first sense, and $6{\tfrac {2}{3}}=6.66...$ (a repeating decimal) in the second sense. Ratios can be defined as dimensionless quotients;[3] non-dimensionless quotients are also known as rates.[4]
For other uses, see Quotient (disambiguation).
Arithmetic operations
Addition (+)
$\scriptstyle \left.{\begin{matrix}\scriptstyle {\text{term}}\,+\,{\text{term}}\\\scriptstyle {\text{summand}}\,+\,{\text{summand}}\\\scriptstyle {\text{addend}}\,+\,{\text{addend}}\\\scriptstyle {\text{augend}}\,+\,{\text{addend}}\end{matrix}}\right\}\,=\,$ $\scriptstyle {\text{sum}}$
Subtraction (−)
$\scriptstyle \left.{\begin{matrix}\scriptstyle {\text{term}}\,-\,{\text{term}}\\\scriptstyle {\text{minuend}}\,-\,{\text{subtrahend}}\end{matrix}}\right\}\,=\,$ $\scriptstyle {\text{difference}}$
Multiplication (×)
$\scriptstyle \left.{\begin{matrix}\scriptstyle {\text{factor}}\,\times \,{\text{factor}}\\\scriptstyle {\text{multiplier}}\,\times \,{\text{multiplicand}}\end{matrix}}\right\}\,=\,$ $\scriptstyle {\text{product}}$
Division (÷)
$\scriptstyle \left.{\begin{matrix}\scriptstyle {\frac {\scriptstyle {\text{dividend}}}{\scriptstyle {\text{divisor}}}}\\[1ex]\scriptstyle {\frac {\scriptstyle {\text{numerator}}}{\scriptstyle {\text{denominator}}}}\end{matrix}}\right\}\,=\,$ $\scriptstyle \left\{{\begin{matrix}\scriptstyle {\text{fraction}}\\\scriptstyle {\text{quotient}}\\\scriptstyle {\text{ratio}}\end{matrix}}\right.$
Exponentiation (^)
$\scriptstyle {\text{base}}^{\text{exponent}}\,=\,$ $\scriptstyle {\text{power}}$
nth root (√)
$\scriptstyle {\sqrt[{\text{degree}}]{\scriptstyle {\text{radicand}}}}\,=\,$ $\scriptstyle {\text{root}}$
Logarithm (log)
$\scriptstyle \log _{\text{base}}({\text{anti-logarithm}})\,=\,$ $\scriptstyle {\text{logarithm}}$
Notation
Main article: Division (mathematics) § Notation
The quotient is most frequently encountered as two numbers, or two variables, divided by a horizontal line. The words "dividend" and "divisor" refer to each individual part, while the word "quotient" refers to the whole.
${\dfrac {1}{2}}\quad {\begin{aligned}&\leftarrow {\text{dividend or numerator}}\\&\leftarrow {\text{divisor or denominator}}\end{aligned}}{\Biggr \}}\leftarrow {\text{quotient}}$
Integer part definition
The quotient is also less commonly defined as the greatest whole number of times a divisor may be subtracted from a dividend—before making the remainder negative. For example, the divisor 3 may be subtracted up to 6 times from the dividend 20, before the remainder becomes negative:
20 − 3 − 3 − 3 − 3 − 3 − 3 ≥ 0,
while
20 − 3 − 3 − 3 − 3 − 3 − 3 − 3 < 0.
In this sense, a quotient is the integer part of the ratio of two numbers.[5]
Quotient of two integers
Main article: Rational number
A rational number can be defined as the quotient of two integers (as long as the denominator is non-zero).
A more detailed definition goes as follows:[6]
A real number r is rational, if and only if it can be expressed as a quotient of two integers with a nonzero denominator. A real number that is not rational is irrational.
Or more formally:
Given a real number r, r is rational if and only if there exists integers a and b such that $r={\tfrac {a}{b}}$ and $b\neq 0$.
The existence of irrational numbers—numbers that are not a quotient of two integers—was first discovered in geometry, in such things as the ratio of the diagonal to the side in a square.[7]
More general quotients
Outside of arithmetic, many branches of mathematics have borrowed the word "quotient" to describe structures built by breaking larger structures into pieces. Given a set with an equivalence relation defined on it, a "quotient set" may be created which contains those equivalence classes as elements. A quotient group may be formed by breaking a group into a number of similar cosets, while a quotient space may be formed in a similar process by breaking a vector space into a number of similar linear subspaces.
See also
• Product (mathematics)
• Quotient category
• Quotient graph
• Integer division
• Quotient module
• Quotient object
• Quotient of a formal language, also left and right quotient
• Quotient ring
• Quotient set
• Quotient space (topology)
• Quotient type
• Quotition and partition
References
1. "Quotient". Dictionary.com.
2. Weisstein, Eric W. "Integer Division". mathworld.wolfram.com. Retrieved 2020-08-27.
3. "ISO 80000-1:2022(en) Quantities and units — Part 1: General". iso.org. Retrieved 2023-07-23.
4. "The quotient of two numbers (or quantities); the relative sizes of two numbers (or quantities)", "The Mathematics Dictionary"
5. Weisstein, Eric W. "Quotient". MathWorld.
6. Epp, Susanna S. (2011-01-01). Discrete mathematics with applications. Brooks/Cole. p. 163. ISBN 9780495391326. OCLC 970542319.
7. "Irrationality of the square root of 2". www.math.utah.edu. Retrieved 2020-08-27.
Fractions and ratios
Division and ratio
• Dividend ÷ Divisor = Quotient
Fraction
• Numerator/Denominator = Quotient
• Algebraic
• Aspect
• Binary
• Continued
• Decimal
• Dyadic
• Egyptian
• Golden
• Silver
• Integer
• Irreducible
• Reduction
• Just intonation
• LCD
• Musical interval
• Paper size
• Percentage
• Unit
Authority control: National
• Germany
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Qvist's theorem
In projective geometry, Qvist's theorem, named after the Finnish mathematician Bertil Qvist, is a statement on ovals in finite projective planes. Standard examples of ovals are non-degenerate (projective) conic sections. The theorem gives an answer to the question How many tangents to an oval can pass through a point in a finite projective plane? The answer depends essentially upon the order (number of points on a line −1) of the plane.
Definition of an oval
Main article: Oval (projective plane)
• In a projective plane a set Ω of points is called an oval, if:
1. Any line l meets Ω in at most two points, and
2. For any point P ∈ Ω there exists exactly one tangent line t through P, i.e., t ∩ Ω = {P}.
When |l ∩ Ω | = 0 the line l is an exterior line (or passant),[1] if |l ∩ Ω| = 1 a tangent line and if |l ∩ Ω| = 2 the line is a secant line.
For finite planes (i.e. the set of points is finite) we have a more convenient characterization:[2]
• For a finite projective plane of order n (i.e. any line contains n + 1 points) a set Ω of points is an oval if and only if |Ω| = n + 1 and no three points are collinear (on a common line).
Statement and proof of Qvist's theorem
Qvist's theorem[3][4]
Let Ω be an oval in a finite projective plane of order n.
(a) If n is odd,
every point P ∉ Ω is incident with 0 or 2 tangents.
(b) If n is even,
there exists a point N, the nucleus or knot, such that, the set of tangents to oval Ω is the pencil of all lines through N.
Proof
(a) Let tR be the tangent to Ω at point R and let P1, ... , Pn be the remaining points of this line. For each i, the lines through Pi partition Ω into sets of cardinality 2 or 1 or 0. Since the number |Ω| = n + 1 is even, for any point Pi, there must exist at least one more tangent through that point. The total number of tangents is n + 1, hence, there are exactly two tangents through each Pi, tR and one other. Thus, for any point P not in oval Ω, if P is on any tangent to Ω it is on exactly two tangents.
(b) Let s be a secant, s ∩ Ω = {P0, P1} and s= {P0, P1,...,Pn}. Because |Ω| = n + 1 is odd, through any Pi, i = 2,...,n, there passes at least one tangent ti. The total number of tangents is n + 1. Hence, through any point Pi for i = 2,...,n there is exactly one tangent. If N is the point of intersection of two tangents, no secant can pass through N. Because n + 1, the number of tangents, is also the number of lines through any point, any line through N is a tangent.
Example in a pappian plane of even order
Using inhomogeneous coordinates over a field K, |K| = n even, the set
Ω1 = {(x, y) | y = x2} ∪ {(∞)},
the projective closure of the parabola y = x2, is an oval with the point N = (0) as nucleus (see image), i.e., any line y = c, with c ∈ K, is a tangent.
Definition and property of hyperovals
• Any oval Ω in a finite projective plane of even order n has a nucleus N.
The point set Ω := Ω ∪ {N} is called a hyperoval or (n + 2)-arc. (A finite oval is an (n + 1)-arc.)
One easily checks the following essential property of a hyperoval:
• For a hyperoval Ω and a point R ∈ Ω the pointset Ω \ {R} is an oval.
This property provides a simple means of constructing additional ovals from a given oval.
Example
For a projective plane over a finite field K, |K| = n even and n > 4, the set
Ω1 = {(x, y) | y = x2} ∪ {(∞)} is an oval (conic section) (see image),
Ω1 = {(x, y) | y = x2} ∪ {(0), (∞)} is a hyperoval and
Ω2 = {(x, y) | y = x2} ∪ {(0)} is another oval that is not a conic section. (Recall that a conic section is determined uniquely by 5 points.)
Notes
1. In the English literature this term is usually rendered in French (or German) rather than translating it as a passing line.
2. Dembowski 1968, p. 147
3. Bertil Qvist: Some remarks concerning curves of the second degree in a finite plane, Helsinki (1952), Ann. Acad. Sci Fenn Nr. 134, 1–27
4. Dembowski 1968, pp. 147–8
References
• Beutelspacher, Albrecht; Rosenbaum, Ute (1998), Projective Geometry / from foundations to applications, Cambridge University Press, ISBN 978-0-521-48364-3
• 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
External links
• E. Hartmann: Planar Circle Geometries, an Introduction to Moebius-, Laguerre- and Minkowski Planes. Skript, TH Darmstadt (PDF; 891 kB), p. 40.
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Qāḍī Zāda al-Rūmī
Qāḍī Zāda al-Rūmī (1364 in Bursa, Ottoman Empire – 1436 in Samarqand, Timurid Empire), whose actual name was Salah al-Din Musa Pasha (qāḍī zāda means "son of the judge", al-rūmī "the Roman" indicating he came from Asia Minor, which was once Roman), was a Turkish astronomer and mathematician who worked at the observatory in Samarkand. He computed sin 1° to an accuracy of 10−12.
Together with Ulugh Beg, al-Kāshī and a few other astronomers, Qāḍī Zāda produced the Zij-i-Sultani, the first comprehensive stellar catalogue since the Maragheh observatory's Zij-i Ilkhani two centuries earlier. The Zij-i Sultani contained the positions of 992 stars.
His works
• Sharh al-Mulakhkhas (Commentary on Jaghmini's compendium on the science of Astronomy)
• Sharh Ashkal al-Ta'sis (Commentary on Samarkandi's Arithmetics)
Further reading
• Adivar, A. (1943). Osmanli Türklarinde Ilim. Istanbul.{{cite book}}: CS1 maint: location missing publisher (link)
• Ataev, U. (1972). "The commentary of Kazi-zade ar-Rumi on the astronomical treatise of Nasir ad-Din at-Tusi". Questions on the History of Mathematics and Astronomy I. Trudy Samarkand. Gos. Univ. (N.S.) Vyp. (in Russian). 229: 124–127.
• Dilgan, H. (1970–1980). "Qāḍī Zāda al-Rūmī". Dictionary of Scientific Biography. New York: Charles Scribner's Sons. ISBN 978-0-684-10114-9.
• O'Connor, John J.; Robertson, Edmund F., "Qadi Zada al-Rumi", MacTutor History of Mathematics Archive, University of St Andrews
• Pasaev, Z. A. (1972). "The commentary of Kazi-zade ar-Rumi on the astronomical treatise of al-Djagmini (a brief survey)". Questions on Differential Equations, Number Theory, Algebra and Geometry. Trudy Samarkand. Gos. Univ. (N.S.) Vyp. (in Russian). 202: 202–204.
• Rozenfel'd, B. A.; A. P. Yushkevich (1960). "The treatise of Qadi Zada al-Rumi on the determination of the sine of one degree". Istor.-Mat. Issled. (in Russian). 13: 33–538.
• Rozenfel'd, B. A.; A. P. Yushkevich (1960). "Notes on a treatise of Qadi Zada al-Rumi". Istor.-Mat. Issled. (in Russian). 13: 552–556.
External links
• Ragep, F. Jamil (2007). "Qāḍīzāde al‐Rūmī: Ṣalāḥ al‐Dīn Mūsā ibn Muḥammad ibn Maḥmūd al‐Rūmī". In Thomas Hockey; et al. (eds.). The Biographical Encyclopedia of Astronomers. New York: Springer. p. 942. ISBN 978-0-387-31022-0. (PDF version)
Astronomy in the medieval Islamic world
Astronomers
• by century
8th
• Ahmad Nahavandi
• Al-Fadl ibn Naubakht
• Muḥammad ibn Ibrāhīm al-Fazārī
• Ibrāhīm al-Fazārī
• Mashallah ibn Athari
• Yaʿqūb ibn Ṭāriq
9th
• Abu Ali al-Khayyat
• Abu Ma'shar al-Balkhi
• Abu Said Gorgani
• Al-Farghani
• Al-Kindi
• Al-Mahani
• Abu Hanifa Dinawari
• Al-Ḥajjāj ibn Yūsuf
• Al-Marwazi
• Ali ibn Isa al-Asturlabi
• Banū Mūsā brothers
• Iranshahri
• Khalid ibn Abd al‐Malik al‐Marwarrudhi
• Al-Khwarizmi
• Sahl ibn Bishr
• Thābit ibn Qurra
• Yahya ibn Abi Mansur
10th
• al-Sufi
• Ibn
• Al-Adami
• al-Khojandi
• al-Khazin
• al-Qūhī
• Abu al-Wafa
• Ahmad ibn Yusuf
• al-Battani
• Al-Qabisi
• Ibn al-A'lam
• Al-Nayrizi
• Al-Saghani
• Aṣ-Ṣaidanānī
• Ibn Yunus
• Ibrahim ibn Sinan
• Ma Yize
• al-Sijzi
• Al-ʻIjliyyah
• Nastulus
• Abolfadl Harawi
• Haseb-i Tabari
• al-Majriti
• Abu al-Hasan al-Ahwazi
11th
• Abu Nasr Mansur
• al-Biruni
• Ali ibn Ridwan
• Al-Zarqālī
• Ibn al-Samh
• Alhazen
• Avicenna
• Ibn al-Saffar
• Kushyar Gilani
• Said al-Andalusi
• Ibrahim ibn Said al-Sahli
• Ibn Mu'adh al-Jayyani
• Al-Isfizari
• Ali ibn Khalaf
12th
• Al-Bitruji
• Avempace
• Ibn Tufail
• Al-Kharaqī
• Al-Khazini
• Al-Samawal al-Maghribi
• Abu al-Salt
• Averroes
• Ibn al-Kammad
• Jabir ibn Aflah
• Omar Khayyam
• Sharaf al-Din al-Tusi
13th
• Ibn al-Banna' al-Marrakushi
• Ibn al‐Ha'im al‐Ishbili
• Jamal ad-Din
• Alam al-Din al-Hanafi
• Najm al‐Din al‐Misri
• Muhyi al-Din al-Maghribi
• Nasir al-Din al-Tusi
• Qutb al-Din al-Shirazi
• Shams al-Din al-Samarqandi
• Zakariya al-Qazwini
• al-Urdi
• al-Abhari
• Muhammad ibn Abi Bakr al‐Farisi
• Abu Ali al-Hasan al-Marrakushi
• Ibn Ishaq al-Tunisi
• Ibn al‐Raqqam
• Al-Ashraf Umar II
• Fakhr al-Din al-Akhlati
14th
• Ibn al-Shatir
• Al-Khalili
• Ibn Shuayb
• al-Battiwi
• Abū al‐ʿUqūl
• Al-Wabkanawi
• Nizam al-Din al-Nisapuri
• al-Jadiri
• Sadr al-Shari'a al-Asghar
• Fathullah Shirazi
15th
• Ali Kuşçu
• Abd al‐Wajid
• Jamshīd al-Kāshī
• Kadızade Rumi
• Ulugh Beg
• Sibt al-Maridini
• Ibn al-Majdi
• al-Wafa'i
• al-Kubunani
• 'Abd al-'Aziz al-Wafa'i
16th
• Al-Birjandi
• al-Khafri
• Baha' al-din al-'Amili
• Piri Reis
• Takiyüddin
17th
• Yang Guangxian
• Ehmedê Xanî
• Al Achsasi al Mouakket
• Muhammad al-Rudani
Topics
Works
• Arabic star names
• Islamic calendar
• Aja'ib al-Makhluqat
• Encyclopedia of the Brethren of Purity
• Tabula Rogeriana
• The Book of Healing
• The Remaining Signs of Past Centuries
Zij
• Alfonsine tables
• Huihui Lifa
• Book of Fixed Stars
• Toledan Tables
• Zij-i Ilkhani
• Zij-i Sultani
• Sullam al-sama'
Instruments
• Alidade
• Analog computer
• Aperture
• Armillary sphere
• Astrolabe
• Astronomical clock
• Celestial globe
• Compass
• Compass rose
• Dioptra
• Equatorial ring
• Equatorium
• Globe
• Graph paper
• Magnifying glass
• Mural instrument
• Navigational astrolabe
• Nebula
• Octant
• Planisphere
• Quadrant
• Sextant
• Shadow square
• Sundial
• Schema for horizontal sundials
• Triquetrum
Concepts
• Almucantar
• Apogee
• Astrology
• Astrophysics
• Axial tilt
• Azimuth
• Celestial mechanics
• Celestial spheres
• Circular orbit
• Deferent and epicycle
• Earth's rotation
• Eccentricity
• Ecliptic
• Elliptic orbit
• Equant
• Galaxy
• Geocentrism
• Gravitational energy
• Gravity
• Heliocentrism
• Inertia
• Islamic cosmology
• Moonlight
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• Muwaqqit
• Obliquity
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• Sublunary sphere
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• Supernova
• Temporal finitism
• Trepidation
• Triangulation
• Tusi couple
• Universe
Institutions
• Al-Azhar University
• House of Knowledge
• House of Wisdom
• University of al-Qarawiyyin
• Observatories
• Constantinople (Taqi al-Din)
• Maragheh
• Samarkand (Ulugh Beg)
Influences
• Babylonian astronomy
• Egyptian astronomy
• Hellenistic astronomy
• Indian astronomy
Influenced
• Byzantine science
• Chinese astronomy
• Medieval European science
• Indian astronomy
Mathematics in the medieval Islamic world
Mathematicians
9th century
• 'Abd al-Hamīd ibn Turk
• Sanad ibn Ali
• al-Jawharī
• Al-Ḥajjāj ibn Yūsuf
• Al-Kindi
• Qusta ibn Luqa
• Al-Mahani
• al-Dinawari
• Banū Mūsā
• Hunayn ibn Ishaq
• Al-Khwarizmi
• Yusuf al-Khuri
• Ishaq ibn Hunayn
• Na'im ibn Musa
• Thābit ibn Qurra
• al-Marwazi
• Abu Said Gorgani
10th century
• Abu al-Wafa
• al-Khazin
• Al-Qabisi
• Abu Kamil
• Ahmad ibn Yusuf
• Aṣ-Ṣaidanānī
• Sinān ibn al-Fatḥ
• al-Khojandi
• Al-Nayrizi
• Al-Saghani
• Brethren of Purity
• Ibn Sahl
• Ibn Yunus
• al-Uqlidisi
• Al-Battani
• Sinan ibn Thabit
• Ibrahim ibn Sinan
• Al-Isfahani
• Nazif ibn Yumn
• al-Qūhī
• Abu al-Jud
• Al-Sijzi
• Al-Karaji
• al-Majriti
• al-Jabali
11th century
• Abu Nasr Mansur
• Alhazen
• Kushyar Gilani
• Al-Biruni
• Ibn al-Samh
• Abu Mansur al-Baghdadi
• Avicenna
• al-Jayyānī
• al-Nasawī
• al-Zarqālī
• ibn Hud
• Al-Isfizari
• Omar Khayyam
• Muhammad al-Baghdadi
12th century
• Jabir ibn Aflah
• Al-Kharaqī
• Al-Khazini
• Al-Samawal al-Maghribi
• al-Hassar
• Sharaf al-Din al-Tusi
• Ibn al-Yasamin
13th century
• Ibn al‐Ha'im al‐Ishbili
• Ahmad al-Buni
• Ibn Munim
• Alam al-Din al-Hanafi
• Ibn Adlan
• al-Urdi
• Nasir al-Din al-Tusi
• al-Abhari
• Muhyi al-Din al-Maghribi
• al-Hasan al-Marrakushi
• Qutb al-Din al-Shirazi
• Shams al-Din al-Samarqandi
• Ibn al-Banna'
• Kamāl al-Dīn al-Fārisī
14th century
• Nizam al-Din al-Nisapuri
• Ibn al-Shatir
• Ibn al-Durayhim
• Al-Khalili
• al-Umawi
15th century
• Ibn al-Majdi
• al-Rūmī
• al-Kāshī
• Ulugh Beg
• Ali Qushji
• al-Wafa'i
• al-Qalaṣādī
• Sibt al-Maridini
• Ibn Ghazi al-Miknasi
16th century
• Al-Birjandi
• Muhammad Baqir Yazdi
• Taqi ad-Din
• Ibn Hamza al-Maghribi
• Ahmad Ibn al-Qadi
Mathematical
works
• The Compendious Book on Calculation by Completion and Balancing
• De Gradibus
• Principles of Hindu Reckoning
• Book of Optics
• The Book of Healing
• Almanac
• Book on the Measurement of Plane and Spherical Figures
• Encyclopedia of the Brethren of Purity
• Toledan Tables
• Tabula Rogeriana
• Zij
Concepts
• Alhazen's problem
• Islamic geometric patterns
Centers
• Al-Azhar University
• Al-Mustansiriya University
• House of Knowledge
• House of Wisdom
• Constantinople observatory of Taqi ad-Din
• Madrasa
• Maragheh observatory
• University of al-Qarawiyyin
Influences
• Babylonian mathematics
• Greek mathematics
• Indian mathematics
Influenced
• Byzantine mathematics
• European mathematics
• Indian mathematics
Related
• Hindu–Arabic numeral system
• Arabic numerals (Eastern Arabic numerals, Western Arabic numerals)
• Trigonometric functions
• History of trigonometry
• History of algebra
Authority control
International
• ISNI
• VIAF
National
• Germany
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Academics
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Other
• İslâm Ansiklopedisi
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Associative algebra
In mathematics, an associative algebra A is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field K. The addition and multiplication operations together give A the structure of a ring; the addition and scalar multiplication operations together give A the structure of a vector space over K. In this article we will also use the term K-algebra to mean an associative algebra over the field K. A standard first example of a K-algebra is a ring of square matrices over a field K, with the usual matrix multiplication.
This article is about a particular kind of algebra over a commutative ring. For other uses of the term "algebra", see Algebra (disambiguation).
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
A commutative algebra is an associative algebra that has a commutative multiplication, or, equivalently, an associative algebra that is also a commutative ring.
In this article associative algebras are assumed to have a multiplicative identity, denoted 1; they are sometimes called unital associative algebras for clarification. In some areas of mathematics this assumption is not made, and we will call such structures non-unital associative algebras. We will also assume that all rings are unital, and all ring homomorphisms are unital.
Many authors consider the more general concept of an associative algebra over a commutative ring R, instead of a field: An R-algebra is an R-module with an associative R-bilinear binary operation, which also contains a multiplicative identity. For examples of this concept, if S is any ring with center C, then S is an associative C-algebra.
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
Definition
Let R be a commutative ring (so R could be a field). An associative R-algebra (or more simply, an R-algebra) is a ring that is also an R-module in such a way that the two additions (the ring addition and the module addition) are the same operation, and scalar multiplication satisfies
$r\cdot (xy)=(r\cdot x)y=x(r\cdot y)$
for all r in R and x, y in the algebra. (This definition implies that the algebra is unital, since rings are supposed to have a multiplicative identity.)
Equivalently, an associative algebra A is a ring together with a ring homomorphism from R to the center of A. If f is such a homomorphism, the scalar multiplication is $(r,x)\mapsto f(r)x$ (here the multiplication is the ring multiplication); if the scalar multiplication is given, the ring homomorphism is given by $r\mapsto r\cdot 1_{A}$ (See also § From ring homomorphisms below).
Every ring is an associative $\mathbb {Z} $-algebra, where $\mathbb {Z} $ denotes the ring of the integers.
A commutative algebra is an associative algebra that is also a commutative ring.
As a monoid object in the category of modules
The definition is equivalent to saying that a unital associative R-algebra is a monoid object in R-Mod (the monoidal category of R-modules). By definition, a ring is a monoid object in the category of abelian groups; thus, the notion of an associative algebra is obtained by replacing the category of abelian groups with the category of modules.
Pushing this idea further, some authors have introduced a "generalized ring" as a monoid object in some other category that behaves like the category of modules. Indeed, this reinterpretation allows one to avoid making an explicit reference to elements of an algebra A. For example, the associativity can be expressed as follows. By the universal property of a tensor product of modules, the multiplication (the R-bilinear map) corresponds to a unique R-linear map
$m:A\otimes _{R}A\to A$.
The associativity then refers to the identity:
$m\circ ({\operatorname {id} }\otimes m)=m\circ (m\otimes \operatorname {id} ).$
From ring homomorphisms
An associative algebra amounts to a ring homomorphism whose image lies in the center. Indeed, starting with a ring A and a ring homomorphism $\eta \colon R\to A$ whose image lies in the center of A, we can make A an R-algebra by defining
$r\cdot x=\eta (r)x$
for all r ∈ R and x ∈ A. If A is an R-algebra, taking x = 1, the same formula in turn defines a ring homomorphism $\eta \colon R\to A$ whose image lies in the center.
If a ring is commutative then it equals its center, so that a commutative R-algebra can be defined simply as a commutative ring A together with a commutative ring homomorphism $\eta \colon R\to A$.
The ring homomorphism η appearing in the above is often called a structure map. In the commutative case, one can consider the category whose objects are ring homomorphisms R → A; i.e., commutative R-algebras and whose morphisms are ring homomorphisms A → A' that are under R; i.e., R → A → A' is R → A' (i.e., the coslice category of the category of commutative rings under R.) The prime spectrum functor Spec then determines an anti-equivalence of this category to the category of affine schemes over Spec R.
How to weaken the commutativity assumption is a subject matter of noncommutative algebraic geometry and, more recently, of derived algebraic geometry. See also: generic matrix ring.
Algebra homomorphisms
Main article: algebra homomorphism
A homomorphism between two R-algebras is an R-linear ring homomorphism. Explicitly, $\varphi :A_{1}\to A_{2}$ is an associative algebra homomorphism if
${\begin{aligned}\varphi (r\cdot x)&=r\cdot \varphi (x)\\\varphi (x+y)&=\varphi (x)+\varphi (y)\\\varphi (xy)&=\varphi (x)\varphi (y)\\\varphi (1)&=1\end{aligned}}$
The class of all R-algebras together with algebra homomorphisms between them form a category, sometimes denoted R-Alg.
The subcategory of commutative R-algebras can be characterized as the coslice category R/CRing where CRing is the category of commutative rings.
Examples
The most basic example is a ring itself; it is an algebra over its center or any subring lying in the center. In particular, any commutative ring is an algebra over any of its subrings. Other examples abound both from algebra and other fields of mathematics.
Algebra
• Any ring A can be considered as a Z-algebra. The unique ring homomorphism from Z to A is determined by the fact that it must send 1 to the identity in A. Therefore, rings and Z-algebras are equivalent concepts, in the same way that abelian groups and Z-modules are equivalent.
• Any ring of characteristic n is a (Z/nZ)-algebra in the same way.
• Given an R-module M, the endomorphism ring of M, denoted EndR(M) is an R-algebra by defining (r·φ)(x) = r·φ(x).
• Any ring of matrices with coefficients in a commutative ring R forms an R-algebra under matrix addition and multiplication. This coincides with the previous example when M is a finitely-generated, free R-module.
• In particular, the square n-by-n matrices with entries from the field K form an associative algebra over K.
• The complex numbers form a 2-dimensional commutative algebra over the real numbers.
• The quaternions form a 4-dimensional associative algebra over the reals (but not an algebra over the complex numbers, since the complex numbers are not in the center of the quaternions).
• Every polynomial ring R[x1, ..., xn] is a commutative R-algebra. In fact, this is the free commutative R-algebra on the set {x1, ..., xn}.
• The free R-algebra on a set E is an algebra of "polynomials" with coefficients in R and noncommuting indeterminates taken from the set E.
• The tensor algebra of an R-module is naturally an associative R-algebra. The same is true for quotients such as the exterior and symmetric algebras. Categorically speaking, the functor that maps an R-module to its tensor algebra is left adjoint to the functor that sends an R-algebra to its underlying R-module (forgetting the multiplicative structure).
• The following ring is used in the theory of λ-rings. Given a commutative ring A, let $G(A)=1+tA[\![t]\!],$ the set of formal power series with constant term 1. It is an abelian group with the group operation that is the multiplication of power series. It is then a ring with the multiplication, denoted by $\circ $, such that $(1+at)\circ (1+bt)=1+abt,$ determined by this condition and the ring axioms. The additive identity is 1 and the multiplicative identity is $1+t$. Then $A$ has a canonical structure of a $G(A)$-algebra given by the ring homomorphism
${\begin{cases}G(A)\to A\\1+\sum _{i>0}a_{i}t^{i}\mapsto a_{1}\end{cases}}$
On the other hand, if A is a λ-ring, then there is a ring homomorphism
${\begin{cases}A\to G(A)\\a\mapsto 1+\sum _{i>0}\lambda ^{i}(a)t^{i}\end{cases}}$
giving $G(A)$ a structure of an A-algebra.
• Given a module M over a commutative ring R, the direct sum of modules $R\oplus M$ has a structure of an R-algebra by thinking M consists of infinitesimal elements; i.e., the multiplication is given as $(a+x)(b+y)=ab+ay+bx.$ The notion is sometimes called the algebra of dual numbers.
• A quasi-free algebra, introduced by Cuntz and Quillen, is a sort of generalization of a free algebra and a semisimple algebra over an algebraically closed field.
Representation theory
• The universal enveloping algebra of a Lie algebra is an associative algebra that can be used to study the given Lie algebra.
• If G is a group and R is a commutative ring, the set of all functions from G to R with finite support form an R-algebra with the convolution as multiplication. It is called the group algebra of G. The construction is the starting point for the application to the study of (discrete) groups.
• If G is an algebraic group (e.g., semisimple complex Lie group), then the coordinate ring of G is the Hopf algebra A corresponding to G. Many structures of G translate to those of A.
• A quiver algebra (or a path algebra) of a directed graph is the free associative algebra over a field generated by the paths in the graph.
Analysis
• Given any Banach space X, the continuous linear operators A : X → X form an associative algebra (using composition of operators as multiplication); this is a Banach algebra.
• Given any topological space X, the continuous real- or complex-valued functions on X form a real or complex associative algebra; here the functions are added and multiplied pointwise.
• The set of semimartingales defined on the filtered probability space (Ω, F, (Ft)t ≥ 0, P) forms a ring under stochastic integration.
• The Weyl algebra
• An Azumaya algebra
Geometry and combinatorics
• The Clifford algebras, which are useful in geometry and physics.
• Incidence algebras of locally finite partially ordered sets are associative algebras considered in combinatorics.
• The partition algebra and its subalgebras, including the Brauer algebra and the Temperley-Lieb algebra.
• A differential graded algebra is an associative algebra together with a grading and a differential. For example, the de Rham algebra $\Omega (M)=\bigoplus _{p=0}^{n}\Omega ^{p}(M)$, where $\Omega ^{p}(M)$ consists of differential p-forms on a manifold M, is a differential graded algebra.
Mathematical physics
• A Poisson algebra is a commutative associative algebra over a field together with a structure of a Lie algebra so that the Lie bracket $\{,\}$ satisfies the Leibniz rule; i.e., $\{fg,h\}=f\{g,h\}+g\{f,h\}$.
• Given a Poisson algebra ${\mathfrak {a}}$, consider the vector space ${\mathfrak {a}}[\![u]\!]$ of formal power series over ${\mathfrak {a}}$. If ${\mathfrak {a}}[\![u]\!]$ has a structure of an associative algebra with multiplication $*$ such that, for $f,g\in {\mathfrak {a}}$,
$f*g=fg-{\frac {1}{2}}\{f,g\}u+\cdots $,
then ${\mathfrak {a}}[\![u]\!]$ is called a deformation quantization of ${\mathfrak {a}}$.
• A quantized enveloping algebra. The dual of such an algebra turns out to be an associative algebra (see § Dual of an associative algebra) and is, philosophically speaking, the (quantized) coordinate ring of a quantum group.
• Gerstenhaber algebra
Constructions
Subalgebras
A subalgebra of an R-algebra A is a subset of A which is both a subring and a submodule of A. That is, it must be closed under addition, ring multiplication, scalar multiplication, and it must contain the identity element of A.
Quotient algebras
Let A be an R-algebra. Any ring-theoretic ideal I in A is automatically an R-module since r · x = (r1A)x. This gives the quotient ring A / I the structure of an R-module and, in fact, an R-algebra. It follows that any ring homomorphic image of A is also an R-algebra.
Direct products
The direct product of a family of R-algebras is the ring-theoretic direct product. This becomes an R-algebra with the obvious scalar multiplication.
Free products
One can form a free product of R-algebras in a manner similar to the free product of groups. The free product is the coproduct in the category of R-algebras.
Tensor products
The tensor product of two R-algebras is also an R-algebra in a natural way. See tensor product of algebras for more details. Given a commutative ring R and any ring A the tensor product R ⊗Z A can be given the structure of an R-algebra by defining r · (s ⊗ a) = (rs ⊗ a). The functor which sends A to R ⊗Z A is left adjoint to the functor which sends an R-algebra to its underlying ring (forgetting the module structure). See also: Change of rings.
Free algebra
A free algebra is an algebra generated by symbols. If one imposes commutativity; i.e., take the quotient by commutators, then one gets a polynomial algebra.
Dual of an associative algebra
Let A be an associative algebra over a commutative ring R. Since A is in particular a module, we can take the dual module A* of A. A priori, the dual A* need not have a structure of an associative algebra. However, A may come with an extra structure (namely, that of a Hopf algebra) so that the dual is also an associative algebra.
For example, take A to be the ring of continuous functions on a compact group G. Then, not only A is an associative algebra, but it also comes with the co-multiplication $\Delta (f)(g,h)=f(gh)$ and co-unit $\epsilon (f)=f(1)$.[1] The "co-" refers to the fact that they satisfy the dual of the usual multiplication and unit in the algebra axiom. Hence, the dual $A^{*}$ is an associative algebra. The co-multiplication and co-unit are also important in order to form a tensor product of representations of associative algebras (see § Representations below).
Enveloping algebra
Given an associative algebra A over a commutative ring R, the enveloping algebra $A^{e}$ of A is the algebra $A\otimes _{R}A^{op}$ or $A^{op}\otimes _{R}A$, depending on authors.[2]
Note that a bimodule over A is exactly a left module over $A^{e}$.
Separable algebra
Main article: Separable algebra
Let A be an algebra over a commutative ring R. Then the algebra A is a right[3] module over $A^{e}:=A^{op}\otimes _{R}A$ with the action $x\cdot (a\otimes b)=axb$. Then, by definition, A is said to separable if the multiplication map $A\otimes _{R}A\to A,\,x\otimes y\mapsto xy$ splits as an $A^{e}$-linear map,[4] where $A\otimes A$ is an $A^{e}$-module by $(x\otimes y)\cdot (a\otimes b)=ax\otimes yb$. Equivalently,[5] $A$ is separable if it is a projective module over $A^{e}$; thus, the $A^{e}$-projective dimension of A, sometimes called the bidimension of A, measures the failure of separability.
Finite-dimensional algebra
See also: Central simple algebra
Let A be a finite-dimensional algebra over a field k. Then A is an Artinian ring.
Commutative case
As A is Artinian, if it is commutative, then it is a finite product of Artinian local rings whose residue fields are algebras over the base field k. Now, a reduced Artinian local ring is a field and thus the following are equivalent[6]
1. $A$ is separable.
2. $A\otimes {\overline {k}}$ is reduced, where ${\overline {k}}$ is some algebraic closure of k.
3. $A\otimes {\overline {k}}={\overline {k}}^{n}$ for some n.
4. $\dim _{k}A$ is the number of $k$-algebra homomorphisms $A\to {\overline {k}}$.
Let $\Gamma =\operatorname {Gal} (k_{s}/k)=\varprojlim \operatorname {Gal} (k'/k)$, the profinite group of finite Galois extensions of k. Then $A\mapsto X_{A}=\{k-{\text{algebra homomorphisms }}A\to k_{s}\}$ is an anti-equivalence of the category of finite-dimensional separable k-algebras to the category of finite sets with continuous $\Gamma $-actions.[7]
Noncommutative case
Since a simple Artinian ring is a (full) matrix ring over a division ring, if A is a simple algebra, then A is a (full) matrix algebra over a division algebra D over k; i.e., $A=M_{n}(D)$. More generally, if A is a semisimple algebra, then it is a finite product of matrix algebras (over various division k-algebras), the fact known as the Artin–Wedderburn theorem.
The fact that A is Artinian simplifies the notion of a Jacobson radical; for an Artinian ring, the Jacobson radical of A is the intersection of all (two-sided) maximal ideals (in contrast, in general, a Jacobson radical is the intersection of all left maximal ideals or the intersection of all right maximal ideals.)
The Wedderburn principal theorem states:[8] for a finite-dimensional algebra A with a nilpotent ideal I, if the projective dimension of $A/I$ as a module over the enveloping algebra $(A/I)^{e}$ is at most one, then the natural surjection $p:A\to A/I$ splits; i.e., $A$ contains a subalgebra $B$ such that $p|_{B}:B{\overset {\sim }{\to }}A/I$ is an isomorphism. Taking I to be the Jacobson radical, the theorem says in particular that the Jacobson radical is complemented by a semisimple algebra. The theorem is an analog of Levi's theorem for Lie algebras.
Lattices and orders
Main articles: Lattice (order) and Order (ring theory)
Let R be a Noetherian integral domain with field of fractions K (for example, they can be $\mathbb {Z} ,\mathbb {Q} $). A lattice L in a finite-dimensional K-vector space V is a finitely generated R-submodule of V that spans V; in other words, $L\otimes _{R}K=V$.
Let $A_{K}$ be a finite-dimensional K-algebra. An order in $A_{K}$ is an R-subalgebra that is a lattice. In general, there are a lot fewer orders than lattices; e.g., ${1 \over 2}\mathbb {Z} $ is a lattice in $\mathbb {Q} $ but not an order (since it is not an algebra).[9]
A maximal order is an order that is maximal among all the orders.
Related concepts
Coalgebras
Main article: Coalgebra
An associative algebra over K is given by a K-vector space A endowed with a bilinear map A × A → A having two inputs (multiplicator and multiplicand) and one output (product), as well as a morphism K → A identifying the scalar multiples of the multiplicative identity. If the bilinear map A × A → A is reinterpreted as a linear map (i. e., morphism in the category of K-vector spaces) A ⊗ A → A (by the universal property of the tensor product), then we can view an associative algebra over K as a K-vector space A endowed with two morphisms (one of the form A ⊗ A → A and one of the form K → A) satisfying certain conditions that boil down to the algebra axioms. These two morphisms can be dualized using categorial duality by reversing all arrows in the commutative diagrams that describe the algebra axioms; this defines the structure of a coalgebra.
There is also an abstract notion of F-coalgebra, where F is a functor. This is vaguely related to the notion of coalgebra discussed above.
Representations
Main article: Algebra representation
A representation of an algebra A is an algebra homomorphism ρ : A → End(V) from A to the endomorphism algebra of some vector space (or module) V. The property of ρ being an algebra homomorphism means that ρ preserves the multiplicative operation (that is, ρ(xy) = ρ(x)ρ(y) for all x and y in A), and that ρ sends the unit of A to the unit of End(V) (that is, to the identity endomorphism of V).
If A and B are two algebras, and ρ : A → End(V) and τ : B → End(W) are two representations, then there is a (canonical) representation A $\otimes $ B → End(V $\otimes $ W) of the tensor product algebra A $\otimes $ B on the vector space V $\otimes $ W. However, there is no natural way of defining a tensor product of two representations of a single associative algebra in such a way that the result is still a representation of that same algebra (not of its tensor product with itself), without somehow imposing additional conditions. Here, by tensor product of representations, the usual meaning is intended: the result should be a linear representation of the same algebra on the product vector space. Imposing such additional structure typically leads to the idea of a Hopf algebra or a Lie algebra, as demonstrated below.
Motivation for a Hopf algebra
Consider, for example, two representations $\sigma :A\rightarrow \mathrm {End} (V)$ and $\tau :A\rightarrow \mathrm {End} (W)$. One might try to form a tensor product representation $\rho :x\mapsto \sigma (x)\otimes \tau (x)$ according to how it acts on the product vector space, so that
$\rho (x)(v\otimes w)=(\sigma (x)(v))\otimes (\tau (x)(w)).$
However, such a map would not be linear, since one would have
$\rho (kx)=\sigma (kx)\otimes \tau (kx)=k\sigma (x)\otimes k\tau (x)=k^{2}(\sigma (x)\otimes \tau (x))=k^{2}\rho (x)$
for k ∈ K. One can rescue this attempt and restore linearity by imposing additional structure, by defining an algebra homomorphism Δ: A → A ⊗ A, and defining the tensor product representation as
$\rho =(\sigma \otimes \tau )\circ \Delta .$
Such a homomorphism Δ is called a comultiplication if it satisfies certain axioms. The resulting structure is called a bialgebra. To be consistent with the definitions of the associative algebra, the coalgebra must be co-associative, and, if the algebra is unital, then the co-algebra must be co-unital as well. A Hopf algebra is a bialgebra with an additional piece of structure (the so-called antipode), which allows not only to define the tensor product of two representations, but also the Hom module of two representations (again, similarly to how it is done in the representation theory of groups).
Motivation for a Lie algebra
See also: Lie algebra representation
One can try to be more clever in defining a tensor product. Consider, for example,
$x\mapsto \rho (x)=\sigma (x)\otimes {\mbox{Id}}_{W}+{\mbox{Id}}_{V}\otimes \tau (x)$
so that the action on the tensor product space is given by
$\rho (x)(v\otimes w)=(\sigma (x)v)\otimes w+v\otimes (\tau (x)w)$.
This map is clearly linear in x, and so it does not have the problem of the earlier definition. However, it fails to preserve multiplication:
$\rho (xy)=\sigma (x)\sigma (y)\otimes {\mbox{Id}}_{W}+{\mbox{Id}}_{V}\otimes \tau (x)\tau (y)$.
But, in general, this does not equal
$\rho (x)\rho (y)=\sigma (x)\sigma (y)\otimes {\mbox{Id}}_{W}+\sigma (x)\otimes \tau (y)+\sigma (y)\otimes \tau (x)+{\mbox{Id}}_{V}\otimes \tau (x)\tau (y)$.
This shows that this definition of a tensor product is too naive; the obvious fix is to define it such that it is antisymmetric, so that the middle two terms cancel. This leads to the concept of a Lie algebra.
Non-unital algebras
Some authors use the term "associative algebra" to refer to structures which do not necessarily have a multiplicative identity, and hence consider homomorphisms which are not necessarily unital.
One example of a non-unital associative algebra is given by the set of all functions f: R → R whose limit as x nears infinity is zero.
Another example is the vector space of continuous periodic functions, together with the convolution product.
See also
• Abstract algebra
• Algebraic structure
• Algebra over a field
• Sheaf of algebras, a sort of an algebra over a ringed space
• Deligne's conjecture on Hochschild cohomology
Notes
1. Example 1 in Tjin, T. (October 10, 1992). "An introduction to quantized Lie groups and algebras". International Journal of Modern Physics A. 07 (25): 6175–6213. arXiv:hep-th/9111043. Bibcode:1992IJMPA...7.6175T. doi:10.1142/S0217751X92002805. ISSN 0217-751X. S2CID 119087306.
2. Vale 2009, Definition 3.1.
3. Editorial note: as it turns, $A^{e}$ is a full matrix ring in interesting cases and it is more conventional to let matrices act from the right.
4. Cohn 2003, § 4.7.
5. To see the equivalence, note a section of $A\otimes _{R}A\to A$ can be used to construct a section of a surjection.
6. Waterhouse 1979, § 6.2.
7. Waterhouse 1979, § 6.3.
8. Cohn 2003, Theorem 4.7.5.
9. Artin 1999, Ch. IV, § 1.
References
• Artin, Michael (1999). "Noncommutative Rings" (PDF). Archived (PDF) from the original on October 9, 2022.
• Bourbaki, N. (1989). Algebra I. Springer. ISBN 3-540-64243-9.
• Cohn, P.M. (2003). Further Algebra and Applications (2nd ed.). Springer. ISBN 1852336676. Zbl 1006.00001.
• Nathan Jacobson, Structure of Rings
• James Byrnie Shaw (1907) A Synopsis of Linear Associative Algebra, link from Cornell University Historical Math Monographs.
• Ross Street (1998) Quantum Groups: an entrée to modern algebra, an overview of index-free notation.
• Vale, R. (2009). "notes on quasi-free algebras" (PDF).
• Waterhouse, William (1979), Introduction to affine group schemes, Graduate Texts in Mathematics, vol. 66, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-4612-6217-6, ISBN 978-0-387-90421-4, MR 0547117
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R-algebroid
In mathematics, R-algebroids are constructed starting from groupoids. These are more abstract concepts than the Lie algebroids that play a similar role in the theory of Lie groupoids to that of Lie algebras in the theory of Lie groups. (Thus, a Lie algebroid can be thought of as 'a Lie algebra with many objects ').
Definition
An R-algebroid, $R{\mathsf {G}}$, is constructed from a groupoid ${\mathsf {G}}$ as follows. The object set of $R{\mathsf {G}}$ is the same as that of ${\mathsf {G}}$ and $R{\mathsf {G}}(b,c)$ is the free R-module on the set ${\mathsf {G}}(b,c)$, with composition given by the usual bilinear rule, extending the composition of ${\mathsf {G}}$.[1]
R-category
A groupoid ${\mathsf {G}}$ can be regarded as a category with invertible morphisms. Then an R-category is defined as an extension of the R-algebroid concept by replacing the groupoid ${\mathsf {G}}$ in this construction with a general category C that does not have all morphisms invertible.
R-algebroids via convolution products
One can also define the R-algebroid, ${\bar {R}}{\mathsf {G}}:=R{\mathsf {G}}(b,c)$, to be the set of functions ${\mathsf {G}}(b,c){\longrightarrow }R$ with finite support, and with the convolution product defined as follows: $\displaystyle (f*g)(z)=\sum \{(fx)(gy)\mid z=x\circ y\}$ .[2]
Only this second construction is natural for the topological case, when one needs to replace 'function' by 'continuous function with compact support', and in this case $R\cong \mathbb {C} $.
Examples
• Every Lie algebra is a Lie algebroid over the one point manifold.
• The Lie algebroid associated to a Lie groupoid.
See also
• Algebraic category
• Algebroid (disambiguation)
• Bialgebra
• Bicategory
• Convolution product
• Crossed module
• Double groupoid
• Higher-dimensional algebra
• Hopf algebra
• Module (mathematics)
• Ring (mathematics)
References
1. Mosa 1986
2. Brown & Mosa 1986
This article incorporates material from Algebroid Structures and Algebroid Extended Symmetries on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
Sources
• Brown, R.; Mosa, G. H. (1986). "Double algebroids and crossed modules of algebroids". Maths Preprint. University of Wales-Bangor.
• Mosa, G.H. (1986). Higher dimensional algebroids and Crossed complexes (PhD). University of Wales. uk.bl.ethos.815719.
• Mackenzie, Kirill C.H. (1987). Lie Groupoids and Lie Algebroids in Differential Geometry. London Mathematical Society Lecture Note Series. Vol. 124. Cambridge University Press. ISBN 978-0-521-34882-9.
• Mackenzie, Kirill C.H. (2005). General Theory of Lie Groupoids and Lie Algebroids. London Mathematical Society Lecture Note Series. Vol. 213. Cambridge University Press. ISBN 978-0-521-49928-6.
• Marle, Charles-Michel (2002). "Differential calculus on a Lie algebroid and Poisson manifolds". arXiv:0804.2451 [math.DG].
• Weinstein, Alan (1996). "Groupoids: unifying internal and external symmetry". AMS Notices. 43: 744–752. arXiv:math/9602220. Bibcode:1996math......2220W. CiteSeerX 10.1.1.29.5422.
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Wikipedia
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R-matrix
The term R-matrix has several meanings, depending on the field of study.
The term R-matrix is used in connection with the Yang–Baxter equation. This is an equation which was first introduced in the field of statistical mechanics, taking its name from independent work of C. N. Yang and R. J. Baxter. The classical R-matrix arises in the definition of the classical Yang–Baxter equation.[1]
In quasitriangular Hopf algebra, the R-matrix is a solution of the Yang–Baxter equation.
The numerical modeling of diffraction gratings in optical science can be performed using the R-matrix propagation algorithm.[2]
R-matrix method in quantum mechanics
There is a method in computational quantum mechanics for studying scattering known as the R-matrix. This method was originally formulated for studying resonances in nuclear scattering by Wigner and Eisenbud.[3] Using that work as a basis, an R-matrix method was developed for electron, positron and photon scattering by atoms.[4] This approach was later adapted for electron, positron and photon scattering by molecules.[5][6][7]
R-matrix method is used in UKRmol[8] and UKRmol+[9] code suits. The user-friendly software Quantemol Electron Collisions (Quantemol-EC) and its predecessor Quantemol-N are based on UKRmol/UKRmol+ and employ MOLPRO package for electron configuration calculations.
See also
• UK Molecular R-matrix Codes
References
1. Kupershmidt, Boris A. (1999). "What a Classical r-Matrix Really Is". Journal of Nonlinear Mathematical Physics. Informa UK Limited. 6 (4): 448–488. arXiv:math/9910188. Bibcode:1999JNMP....6..448K. doi:10.2991/jnmp.1999.6.4.5. ISSN 1402-9251.
2. Li, Lifeng (1994-11-01). "Bremmer series, R-matrix propagation algorithm, and numerical modeling of diffraction gratings". Journal of the Optical Society of America A. The Optical Society. 11 (11): 2829–2836. Bibcode:1994JOSAA..11.2829L. doi:10.1364/josaa.11.002829. ISSN 1084-7529.
3. Wigner, E. P.; Eisenbud, L. (1947-07-01). "Higher Angular Momenta and Long Range Interaction in Resonance Reactions". Physical Review. American Physical Society (APS). 72 (1): 29–41. Bibcode:1947PhRv...72...29W. doi:10.1103/physrev.72.29. ISSN 0031-899X.
4. Burke, P G; Hibbert, A; Robb, W D (1971). "Electron scattering by complex atoms". Journal of Physics B: Atomic and Molecular Physics. IOP Publishing. 4 (2): 153–161. Bibcode:1971JPhB....4..153B. doi:10.1088/0022-3700/4/2/002. ISSN 0022-3700.
5. Schneider, Barry (1975). "R-matrix theory for electron-atom and electron-molecule collisions using analytic basis set expansions". Chemical Physics Letters. Elsevier BV. 31 (2): 237–241. Bibcode:1975CPL....31..237S. doi:10.1016/0009-2614(75)85010-x. ISSN 0009-2614.
6. Schneider, Barry I. (1975-06-01). "R-matrix theory for electron-molecule collisions using analytic basis set expansions. II. Electron-H2 scattering in the static-exchange model". Physical Review A. American Physical Society (APS). 11 (6): 1957–1962. Bibcode:1975PhRvA..11.1957S. doi:10.1103/physreva.11.1957. ISSN 0556-2791.
7. C J Gillan, J Tennyson, and P G Burke, in Computational Methods for Electron-Molecule Collisions, eds. W M Huo and F A Gianturco, (Plenum, New York, 1995), p. 239
8. Carr, J.M.; Galiatsatos, P.G.; Gorfinkiel, J.D.; Harvey, A.G.; Lysaght, M.A.; Madden, D.; Mašín, Z.; Plummer, M.; Tennyson, J. (2012). "The UKRmol program suite". Eur. Phys. J. D (66): 58. doi:10.1140/epjd/e2011-20653-6.
9. Mašín, Zdeněk; Benda, Jakub; Gorfinkiel, Jimena D.; Harvey, Alex G.; Tennyson, Jonathan (2019-12-07). "UKRmol+: A suite for modelling electronic processes in molecules interacting with electrons, positrons and photons using the R-matrix method". Computer Physics Communications. 249: 107092. arXiv:1908.03018. doi:10.1016/j.cpc.2019.107092.
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Generalized flag variety
In mathematics, a generalized flag variety (or simply flag variety) is a homogeneous space whose points are flags in a finite-dimensional vector space V over a field F. When F is the real or complex numbers, a generalized flag variety is a smooth or complex manifold, called a real or complex flag manifold. Flag varieties are naturally projective varieties.
Flag varieties can be defined in various degrees of generality. A prototype is the variety of complete flags in a vector space V over a field F, which is a flag variety for the special linear group over F. Other flag varieties arise by considering partial flags, or by restriction from the special linear group to subgroups such as the symplectic group. For partial flags, one needs to specify the sequence of dimensions of the flags under consideration. For subgroups of the linear group, additional conditions must be imposed on the flags.
In the most general sense, a generalized flag variety is defined to mean a projective homogeneous variety, that is, a smooth projective variety X over a field F with a transitive action of a reductive group G (and smooth stabilizer subgroup; that is no restriction for F of characteristic zero). If X has an F-rational point, then it is isomorphic to G/P for some parabolic subgroup P of G. A projective homogeneous variety may also be realised as the orbit of a highest weight vector in a projectivized representation of G. The complex projective homogeneous varieties are the compact flat model spaces for Cartan geometries of parabolic type. They are homogeneous Riemannian manifolds under any maximal compact subgroup of G, and they are precisely the coadjoint orbits of compact Lie groups.
Flag manifolds can be symmetric spaces. Over the complex numbers, the corresponding flag manifolds are the Hermitian symmetric spaces. Over the real numbers, an R-space is a synonym for a real flag manifold and the corresponding symmetric spaces are called symmetric R-spaces.
Flags in a vector space
Main article: flag (linear algebra)
A flag in a finite dimensional vector space V over a field F is an increasing sequence of subspaces, where "increasing" means each is a proper subspace of the next (see filtration):
$\{0\}=V_{0}\subset V_{1}\subset V_{2}\subset \cdots \subset V_{k}=V.$
If we write the dim Vi = di then we have
$0=d_{0}<d_{1}<d_{2}<\cdots <d_{k}=n,$
where n is the dimension of V. Hence, we must have k ≤ n. A flag is called a complete flag if di = i for all i, otherwise it is called a partial flag. The signature of the flag is the sequence (d1, ..., dk).
A partial flag can be obtained from a complete flag by deleting some of the subspaces. Conversely, any partial flag can be completed (in many different ways) by inserting suitable subspaces.
Prototype: the complete flag variety
According to basic results of linear algebra, any two complete flags in an n-dimensional vector space V over a field F are no different from each other from a geometric point of view. That is to say, the general linear group acts transitively on the set of all complete flags.
Fix an ordered basis for V, identifying it with Fn, whose general linear group is the group GL(n,F) of n × n invertible matrices. The standard flag associated with this basis is the one where the ith subspace is spanned by the first i vectors of the basis. Relative to this basis, the stabilizer of the standard flag is the group of nonsingular lower triangular matrices, which we denote by Bn. The complete flag variety can therefore be written as a homogeneous space GL(n,F) / Bn, which shows in particular that it has dimension n(n−1)/2 over F.
Note that the multiples of the identity act trivially on all flags, and so one can restrict attention to the special linear group SL(n,F) of matrices with determinant one, which is a semisimple algebraic group; the set of lower triangular matrices of determinant one is a Borel subgroup.
If the field F is the real or complex numbers we can introduce an inner product on V such that the chosen basis is orthonormal. Any complete flag then splits into a direct sum of one-dimensional subspaces by taking orthogonal complements. It follows that the complete flag manifold over the complex numbers is the homogeneous space
$U(n)/T^{n}$
where U(n) is the unitary group and Tn is the n-torus of diagonal unitary matrices. There is a similar description over the real numbers with U(n) replaced by the orthogonal group O(n), and Tn by the diagonal orthogonal matrices (which have diagonal entries ±1).
Partial flag varieties
The partial flag variety
$F(d_{1},d_{2},\ldots d_{k},\mathbb {F} )$
is the space of all flags of signature (d1, d2, ... dk) in a vector space V of dimension n = dk over F. The complete flag variety is the special case that di = i for all i. When k=2, this is a Grassmannian of d1-dimensional subspaces of V.
This is a homogeneous space for the general linear group G of V over F. To be explicit, take V = Fn so that G = GL(n,F). The stabilizer of a flag of nested subspaces Vi of dimension di can be taken to be the group of nonsingular block lower triangular matrices, where the dimensions of the blocks are ni := di − di−1 (with d0 = 0).
Restricting to matrices of determinant one, this is a parabolic subgroup P of SL(n,F), and thus the partial flag variety is isomorphic to the homogeneous space SL(n,F)/P.
If F is the real or complex numbers, then an inner product can be used to split any flag into a direct sum, and so the partial flag variety is also isomorphic to the homogeneous space
$U(n)/U(n_{1})\times \cdots \times U(n_{k})$
in the complex case, or
$O(n)/O(n_{1})\times \cdots \times O(n_{k})$
in the real case.
Generalization to semisimple groups
The upper triangular matrices of determinant one are a Borel subgroup of SL(n,F), and hence the stabilizers of partial flags are parabolic subgroups. Furthermore, a partial flag is determined by the parabolic subgroup which stabilizes it.
Hence, more generally, if G is a semisimple algebraic or Lie group, then the (generalized) flag variety for G is G/P where P is a parabolic subgroup of G. The correspondence between parabolic subgroups and generalized flag varieties allows each to be understood in terms of the other.
The extension of the terminology "flag variety" is reasonable, because points of G/P can still be described using flags. When G is a classical group, such as a symplectic group or orthogonal group, this is particularly transparent. If (V, ω) is a symplectic vector space then a partial flag in V is isotropic if the symplectic form vanishes on proper subspaces of V in the flag. The stabilizer of an isotropic flag is a parabolic subgroup of the symplectic group Sp(V,ω). For orthogonal groups there is a similar picture, with a couple of complications. First, if F is not algebraically closed, then isotropic subspaces may not exist: for a general theory, one needs to use the split orthogonal groups. Second, for vector spaces of even dimension 2m, isotropic subspaces of dimension m come in two flavours ("self-dual" and "anti-self-dual") and one needs to distinguish these to obtain a homogeneous space.
Cohomology
If G is a compact, connected Lie group, it contains a maximal torus T and the space G/T of left cosets with the quotient topology is a compact real manifold. If H is any other closed, connected subgroup of G containing T, then G/H is another compact real manifold. (Both are actually complex homogeneous spaces in a canonical way through complexification.)
The presence of a complex structure and cellular (co)homology make it easy to see that the cohomology ring of G/H is concentrated in even degrees, but in fact, something much stronger can be said. Because G → G/H is a principal H-bundle, there exists a classifying map G/H → BH with target the classifying space BH. If we replace G/H with the homotopy quotient GH in the sequence G → G/H → BH, we obtain a principal G-bundle called the Borel fibration of the right multiplication action of H on G, and we can use the cohomological Serre spectral sequence of this bundle to understand the fiber-restriction homomorphism H*(G/H) → H*(G) and the characteristic map H*(BH) → H*(G/H), so called because its image, the characteristic subring of H*(G/H), carries the characteristic classes of the original bundle H → G → G/H.
Let us now restrict our coefficient ring to be a field k of characteristic zero, so that, by Hopf's theorem, H*(G) is an exterior algebra on generators of odd degree (the subspace of primitive elements). It follows that the edge homomorphisms
$E_{r+1}^{0,r}\to E_{r+1}^{r+1,0}$
of the spectral sequence must eventually take the space of primitive elements in the left column H*(G) of the page E2 bijectively into the bottom row H*(BH): we know G and H have the same rank, so if the collection of edge homomorphisms were not full rank on the primitive subspace, then the image of the bottom row H*(BH) in the final page H*(G/H) of the sequence would be infinite-dimensional as a k-vector space, which is impossible, for instance by cellular cohomology again, because a compact homogeneous space admits a finite CW structure.
Thus the ring map H*(G/H) → H*(G) is trivial in this case, and the characteristic map is surjective, so that H*(G/H) is a quotient of H*(BH). The kernel of the map is the ideal generated by the images of primitive elements under the edge homomorphisms, which is also the ideal generated by positive-degree elements in the image of the canonical map H*(BG) → H*(BH) induced by the inclusion of H in G.
The map H*(BG) → H*(BT) is injective, and likewise for H, with image the subring H*(BT)W(G) of elements invariant under the action of the Weyl group, so one finally obtains the concise description
$H^{*}(G/H)\cong H^{*}(BT)^{W(H)}/{\big (}{\widetilde {H}}^{*}(BT)^{W(G)}{\big )},$
where ${\widetilde {H}}^{*}$ denotes positive-degree elements and the parentheses the generation of an ideal. For example, for the complete complex flag manifold U(n)/Tn, one has
$H^{*}{\big (}U(n)/T^{n}{\big )}\cong \mathbb {Q} [t_{1},\ldots ,t_{n}]/(\sigma _{1},\ldots ,\sigma _{n}),$
where the tj are of degree 2 and the σj are the first n elementary symmetric polynomials in the variables tj. For a more concrete example, take n = 2, so that U(2)/[U(1) × U(1)] is the complex Grassmannian Gr(1,$\mathbb {C} $2) ≈ $\mathbb {C} $P1 ≈ S2. Then we expect the cohomology ring to be an exterior algebra on a generator of degree two (the fundamental class), and indeed,
$H^{*}{\big (}U(2)/T^{2}{\big )}\cong \mathbb {Q} [t_{1},t_{2}]/(t_{1}+t_{2},t_{1}t_{2})\cong \mathbb {Q} [t_{1}]/(t_{1}^{2}),$
as hoped.
Highest weight orbits and projective homogeneous varieties
If G is a semisimple algebraic group (or Lie group) and V is a (finite dimensional) highest weight representation of G, then the highest weight space is a point in the projective space P(V) and its orbit under the action of G is a projective algebraic variety. This variety is a (generalized) flag variety, and furthermore, every (generalized) flag variety for G arises in this way.
Armand Borel showed that this characterizes the flag varieties of a general semisimple algebraic group G: they are precisely the complete homogeneous spaces of G, or equivalently (in this context), the projective homogeneous G-varieties.
Symmetric spaces
Main article: Symmetric space
Let G be a semisimple Lie group with maximal compact subgroup K. Then K acts transitively on any conjugacy class of parabolic subgroups, and hence the generalized flag variety G/P is a compact homogeneous Riemannian manifold K/(K∩P) with isometry group K. Furthermore, if G is a complex Lie group, G/P is a homogeneous Kähler manifold.
Turning this around, the Riemannian homogeneous spaces
M = K/(K∩P)
admit a strictly larger Lie group of transformations, namely G. Specializing to the case that M is a symmetric space, this observation yields all symmetric spaces admitting such a larger symmetry group, and these spaces have been classified by Kobayashi and Nagano.
If G is a complex Lie group, the symmetric spaces M arising in this way are the compact Hermitian symmetric spaces: K is the isometry group, and G is the biholomorphism group of M.
Over the real numbers, a real flag manifold is also called an R-space, and the R-spaces which are Riemannian symmetric spaces under K are known as symmetric R-spaces. The symmetric R-spaces which are not Hermitian symmetric are obtained by taking G to be a real form of the biholomorphism group Gc of a Hermitian symmetric space Gc/Pc such that P := Pc∩G is a parabolic subgroup of G. Examples include projective spaces (with G the group of projective transformations) and spheres (with G the group of conformal transformations).
See also
• Parabolic Lie algebra
• Bruhat decomposition
References
• Robert J. Baston and Michael G. Eastwood, The Penrose Transform: its Interaction with Representation Theory, Oxford University Press, 1989.
• Jürgen Berndt, Lie group actions on manifolds, Lecture notes, Tokyo, 2002.
• Jürgen Berndt, Sergio Console and Carlos Olmos, Submanifolds and Holonomy, Chapman & Hall/CRC Press, 2003.
• Michel Brion, Lectures on the geometry of flag varieties, Lecture notes, Varsovie, 2003.
• James E. Humphreys, Linear Algebraic Groups, Graduate Texts in Mathematics, 21, Springer-Verlag, 1972.
• S. Kobayashi and T. Nagano, On filtered Lie algebras and geometric structures I, II, J. Math. Mech. 13 (1964), 875–907, 14 (1965) 513–521.
Authority control: National
• Germany
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Wikipedia
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Anand Kumar
Anand Kumar (born 1 January 1973) is an Indian Mathematics educator, best known for his Super 30 programme, which he started in Patna, Bihar in 2002, known for coaching underprivileged students for JEE- Main & JEE-Advanced, the entrance examination for the Indian Institutes of Technology (IITs). Kumar was named in Time magazine's list of Best of Asia 2010.[3] In 2023, he was awarded the Padma Shri, country's fourth highest civilian award by the Government of India for his contributions in the field of literature and education.[4]
Anand Kumar
Born (1973-01-01) 1 January 1973
Patna, Bihar, India
Alma mater
• Bihar National College
• Patna University
Occupations
• Educationalist
• mathematician
Years active2002–present
Known forSuper 30 program
SpouseRitu Rashmi
ChildrenJagat Kumar (son)
Parents
• Jayanti Devi (mother)
• Rajendra Prasad (father)
RelativesPranav Kumar (brother)
Awards
• Padma Shri (2023)
• S. Ramanujan award (2010)[1]
• Maulana Abul Kalam Azad Shiksha Puraskar (2010)[2]
WebsiteSuper 30
By 2018, 422 out of 510 students had made it to the IITs and Discovery Channel showcased his work in a documentary.[5][6][7][8][9] Kumar has spoken at MIT and Harvard about his programs for students from the underprivileged sections of Indian society.[10] Kumar and his school have been the subject of several smear campaigns, some of which have been carried in Indian media sources.[11] His life and work had been portrayed in the 2019 film, Super 30, where Kumar is played by Hrithik Roshan.[12]
Early life
Anand Kumar was born in Bihar, India. His father was a clerk[13] in the postal department of India. His father could not afford private schooling for his children, and Anand attended a Hindi medium government school, where he developed his deep interest in mathematics.[14][15] In his childhood, he studied at Patna High School, in Patna, Bihar. During his graduation, Kumar submitted papers on number theory, which were published in the Mathematical Spectrum.[16] Kumar secured admission to the University of Cambridge, but could not attend because of his father's death and his financial condition.[14][17][18]
Teaching career
In 1992, Kumar began teaching mathematics.[14][15] He rented a classroom for Rs. 300 per month, and began his own institute, the Ramanujan School of Mathematics (RSM).[14][19] Within the span of year, his class grew from two students to thirty-six, and after three years almost 500 students had enrolled.[14] Then in early 2000, when a poor student came to him seeking coaching for IIT-JEE, who could not afford the annual admission fee due to poverty, Kumar was motivated to start the Super 30 programme in 2002, for which he is now well-known.[14][18]
Since 2002, every May, the Ramanujan School of Mathematics holds a competitive test to select 30 students for the Super 30 program. Many students appear at the test, and eventually, he takes thirty intelligent students from economically backward sections, tutors them, and provides study materials and lodging for a year.[14] He prepares them for the Joint Entrance Examination for the Indian Institutes of Technology (IIT). His mother, Jayanti Devi, cooks for the students, and his brother Pranav Kumar takes care of the management.[14][20]
During 2003 to 2017, 391 students out of 450 passed the IITs.[14][21] In 2010, all the students of Super 30 cleared IIT JEE entrance making it a three in a row for the institution.[22] Kumar has no financial support for Super 30 from any government as well as private agencies, and manages on the tuition fee he earns from the Ramanujan Institute.[18] After the success of Super 30 and its growing popularity, he received offers from the private sector – both national and international companies – as well as the government for financial help, but he has refused it; Kumar wanted to sustain Super 30 through his own efforts.[18][20]
From 2006 to 2010, 30 out of 30 students cleared the IIT-JEE. In subsequent years, the pass rates for the 30 students at the IIT-JEE examinations were: 2011 – 24, 2012–27, 2013–28, 2014–27, 2015–25, 2016–28, 2017–30, and 2018–26.[20][23][24]
Personal life
In 2019, Kumar revealed that he has been suffering from Acoustic neuroma, a rare kind of brain tumor, and lost 80-90% hearing ability of his right ear due to it. He is under treatment of renowned neurosurgeon B. K. Misra at the Hinduja Hospital in Mumbai.[25]
Recognition
In March 2009, Discovery Channel broadcast a one-hour-long programme on Super 30,[6][17][26] and half a page was devoted to Kumar in The New York Times.[17] Actress and former Miss Japan Norika Fujiwara visited Patna to make a documentary on Kumar's initiatives.[19] Kumar has been featured in programmes by the BBC.[17] He has spoken about his experiences at various global-level institutes including Indian Institute of Management Ahmedabad, a number of IITs, University of British Columbia, Tokyo University and Stanford University.[17][27][28] He was also inducted in the Limca Book of Records (2009) for his contribution in helping poor students pass the IIT-JEE by providing them free coaching.[29] Time magazine included Super 30 in the list of Best of Asia 2010. Kumar was awarded the S. Ramanujan Award for 2010 by the Institute for Research and Documentation in Social Sciences (IRDS) in July 2010.[1]
Super 30 received praise from United States President Barack Obama's special envoy, Rashad Hussain, who termed it the "best" institute in the country.[30] Newsweek Magazine has taken note of the initiative of mathematician Anand Kumar's Super 30 and included his school in the list of four most innovative schools in the world.[31] Kumar was given the top award of Bihar government, "Maulana Abdul Kalam Azad Shiksha Puraskar", in November 2010.[2] He was awarded the Prof. Yashwantrao Kelkar Yuva Puraskar award in 2010 by Akhil Bharatiya Vidyarthi Parishad (ABVP) in Bangalore.
In April 2011, Kumar was selected by Europe's magazine Focus as "one of the global personalities who have the ability to shape exceptionally talented people."[32] Kumar also helped Amitabh Bachchan in preparing for his role in the film Aarakshan.[33] Many people from Bollywood including renowned directors and producers are interested in making a movie on the life of Anand Kumar.[34] He was named by UK based magazine Monocle among the list of 20 pioneering teachers of the world.[35] He was also honoured by government of British Columbia, Canada.[36] Kumar was given the Baroda Sun Life Achievement Award by Bank of Baroda in Mumbai.[37] Kumar was conferred with the Ramanujan Mathematics Award at the Eighth National Mathematics Convention at function in Rajkot.[38] He was conferred with an Honorary Doctorate of Science (DSc) by Karpagam University, Coimbatore.[39] He was also awarded Maharishi Ved Vyas by Madhya Pradesh Government for extraordinary contribution in education[40] Anand Kumar was honored by ministry of education of Saxony of Germany.[41]
Kumar presented his biography to then-President of India, Pranab Mukherjee, which was written by Canada-based psychiatrist Biju Mathew.[42] Kumar was awarded "Rashtriya Bal Kalyan Award" by president of India Ram Nath Kovind.[43] A Canadian MP Marc Dalton has praised Anand Kumar's "inspiring work" with underprivileged children as a successful model for education in Canadian parliament.[44] Anand Kumar was conferred an honorary PhD by the National Institute of Technology.[45]
Awards
On 8 November 2018, Anand Kumar was honoured with the Global Education Award[46] 2018 by Malabar Gold & Diamonds in Dubai. His efforts in the field of education are considered "pioneering". Anand Kumar has been felicitated in the US with "Education Excellence Award 2019 " by the Foundation For Excellence in Education (FFE) at a function in San Jose, California.[47] Anand Kumar received "Mahaveer Award" in Chennai.[48] Anand Kumar recently awarded with Bharat Ganit Ratna Award 2022- a special award declared by DASA India - National VO operated from Agartala Tripura on 10 March 2022, which has been handed over by Anjan Banik, National Chairman of DASA India.[49]
In 2023, he was awarded the Padma Shri by the Government of India for his contributions in the field of literature and education.[50]
In 2018, he was awarded Mahaveer Awards for excellence in human endeavour in the field of Education, presented by Bhagwan Mahaveer Foundation.
In popular culture
Bollywood director Vikas Bahl has directed a film titled Super 30, with Hrithik Roshan as Anand Kumar, based on his life and works. The movie was successful at the box office and became a great inspiration for many students.[51][12]
Smear campaigns
On 23 July 2018, an article in Dainik Jagran cited former Super 30 students who said that only three students from the program had passed the IIT JEE exam that year, contrary to Kumar's claim that 26 had passed.[52] The report also claimed that students who sought to enroll in Super 30 were pushed to enroll in another coaching center called Ramanujan Classes, a for-profit institution, on the pretext that Kumar would coach them if they performed well. Furthermore, the article alleged that by asking IIT aspirants to enroll in Ramanujan Classes, Kumar made over Rs 1 crore annually.[52]
Deputy Chief Minister of Bihar Tejashwi Yadav spoke in favour of Anand Kumar and said that "propaganda is being run in media influenced by feudal mindset to discredit and defame Anand Kumar." Union Cabinet Minister and former actor, Shatrughan Sinha, has also spoken in Kumar's favour on Twitter.[53][54] In August 2018, The Hindu reported that Kumar and his school are frequently the target of smear campaigns, and identified the potential sources of the fabricated stories that appeared in the Dainik Jagran newspaper in July.[11][55] Dainik Jagran made amends and presented its ‘Editor’s Choice Award’ to Super 30 founder Anand Kumar in 2021 recognising his work in the field of education.[56]
References
1. "IRDS Awards 2010".
2. "Bihar honours Super 30 founder with top award". The Times of India. Archived from the original on 18 May 2013. Retrieved 20 February 2012.
3. Thottam, Jyoti (13 May 2010). "The Best of Asia 2010 - TIME". Time.
4. "Padma Awards 2023 announced". Press Information Buereau. Ministry of Home Affairs, Govt of India. Retrieved 26 January 2023.
5. "JEE Advanced result 2017: It is 30/30 for Anand Kumar's Super 30". Hindustan Times. 12 June 2017. Archived from the original on 13 June 2017. Retrieved 13 June 2017.
6. Chaudhary, Pranava K (14 March 2009). "Discovery to showcase Super-30 today". The Times of India. Archived from the original on 30 June 2013. Retrieved 30 June 2013.
7. "Living the vilayat dream at home – Teacher who failed to study abroad coaches students to do so". The Telegraph. 22 February 2005. Archived from the original on 23 September 2009. Retrieved 15 May 2010.
8. Tewary, Amarnath (21 September 2006). "Helping poor Indians crack toughest test". BBC News. Archived from the original on 29 May 2010. Retrieved 15 May 2010.
9. "Super 30 does it again: 28 of its students crack JEE Advanced". Hindustan Times. 12 June 2016. Archived from the original on 22 June 2016. Retrieved 27 June 2016.
10. "Super 30 founder Anand Kumar invited to speak at MIT and Harvard". The Economic Times India. 29 September 2014. Archived from the original on 1 October 2014. Retrieved 29 September 2014.
11. Amarnath Tewary (22 August 2018). "Bihar Super-30 founder faces smear campaign". The Hindu. Archived from the original on 13 July 2019. Retrieved 5 June 2019. When asked who is behind all this, he quipped, "everyone knows in Patna who is he…why should I take his name?" Is he former DGP Abhyanand who was earlier associated with Super-30 for five years?, Mr. Kumar did not respond.
12. "Super 30 trailer out. Hrithik Roshan stuns as math wizard Anand Kumar in new film". India Today. 4 June 2019. Archived from the original on 4 June 2019. Retrieved 4 June 2019.
13. "He trains India's poorest students for the IIT: Rediff.com Get Ahead". Archived from the original on 17 December 2009. Retrieved 15 December 2009.
14. "He trains India's poorest students for the IIT". Careers360. 15 December 2009. Archived from the original on 17 December 2009. Retrieved 15 December 2009.
15. "Super 30 Founder Anand Kumar, a Mathematician on a Mission". Success Stories. 28 May 2012. Archived from the original on 7 November 2012. Retrieved 25 September 2012.
16. "Letter to the Editor: Happy numbers (page 122 and 123)".
17. Sengupta, Uttam (14 June 2009). "Genius at work". The Sunday Tribune. Archived from the original on 23 September 2009. Retrieved 11 July 2009.
18. "Mr. Cent Per Cent". The Hindu. Chennai, India. 14 November 2009. Archived from the original on 30 April 2012. Retrieved 26 June 2013.
19. Kumar, Abhay (6 June 2009). "I am planning expansion of Super 30". Deccan Herald. Archived from the original on 24 September 2009. Retrieved 18 December 2009.
20. "26 students from Anand Kumar's Super 30 academy crack IIT-JEE". The Economic Times. 10 June 2018. Archived from the original on 23 March 2019. Retrieved 5 June 2019.
21. Patna, PTI (19 June 2014). "JEE advance result: Anand Kumar's Super 30 wins laurels again, 27 out of 30 qualify for IITs'". Financial Express. Archived from the original on 22 June 2014. Retrieved 19 June 2014.
22. "Super 30's super record in IIT-JEE". The Hindu. Chennai, India. 26 May 2010. Archived from the original on 31 July 2013. Retrieved 25 September 2010.
23. "JEE Advanced 2017 Results". Patna, India: NDTV. 17 June 2017. Archived from the original on 11 June 2017. Retrieved 12 June 2017.
24. "26 students from Anand Kumar's Super 30 academy crack IIT-JEE". TOI. Patna, India. 10 June 2018. Archived from the original on 11 June 2018. Retrieved 11 June 2018.
25. "Anand Kumar reveals he has brain tumour, says 'wanted Super 30 to be made while I am alive'". 11 July 2019.
26. "100/100 for Super 30 in IIT-JEE". The Hindu. Chennai, India. 26 May 2009. Archived from the original on 10 June 2009. Retrieved 11 July 2009.
27. "Use skills to ensure country's growth: Super30 founder to IIT students". Economics Times. 26 August 2014. Archived from the original on 1 November 2014. Retrieved 15 October 2014.
28. "Anand focus on teachers for excellence". The Telegraph. 8 October 2014. Archived from the original on 19 October 2014. Retrieved 14 October 2014.
29. "Kumar of Super-30 finds place in Limca Book". The Times of India. 15 September 2009. Archived from the original on 5 July 2013. Retrieved 18 December 2009.
30. "Obama's special envoy hails Super 30". The Hindu. Chennai, India. 8 August 2010. Archived from the original on 21 October 2012. Retrieved 25 September 2010.
31. "Super-30-incredible-says-Newsweek". The Times of India. 19 September 2010. Archived from the original on 5 July 2013. Retrieved 25 September 2010.
32. "Super 30 founder is Europe journal's global personality". Hindu Business Line. 20 April 2011. Archived from the original on 7 October 2012. Retrieved 22 April 2011.
33. "Big B gets teaching tips from Super 30's Anand". The Times of India. 31 July 2011. Archived from the original on 4 January 2012. Retrieved 31 July 2011.
34. "Bollywood movie on Bihar's Super-30 on cards". The Times of India. 23 November 2011. Archived from the original on 1 May 2013. Retrieved 23 November 2011..
35. "Math wizard Anand Kumar in list of world's 20 top teachers". Indian Express. 4 December 2011. Archived from the original on 1 November 2012. Retrieved 1 April 2012.
36. "Bihar's Super 30 coaching idea wins high praise in Canada". The Times of India. 21 February 2012. Archived from the original on 8 July 2012.
37. "Founder of 'Super-30' programme Anand Kumar honoured". The Economic Times. 21 July 2012. Archived from the original on 17 May 2013. Retrieved 20 September 2012.
38. "Anand Kumar, Founder of Super30 Classes, Gets Ramanujan Mathematics Award". Indiatimes. 29 January 2014. Archived from the original on 30 January 2014. Retrieved 30 January 2014.
39. "Honorary doctorate conferred on 'Super 30' founder". The Times of India. 14 December 2014. Archived from the original on 15 December 2014. Retrieved 15 December 2014.
40. "ANAND KUMAR FOUNDER OF 'SUPER 30′ AWARDED MAHARISHI VED VYAS NATIONAL AWARD". Dreamiit. 17 August 2015. Archived from the original on 22 December 2015. Retrieved 10 September 2015.
41. "Super 30 founding mathematician honoured in Germany". Economics Times. 1 December 2015. Archived from the original on 22 December 2015. Retrieved 18 December 2015.
42. "Mukherjee lauds Anand Kumar's work". Business Standard. 10 June 2016. Archived from the original on 13 August 2016. Retrieved 27 June 2016.
43. "Rashtriya Bal Kalyan Award presented to Super-30 founder". Hindu. 15 November 2017.
44. "Super 30 founder Anand Kumar's 'inspiring work' lauded in Canadian Parliament". The Hindustan Times. 23 February 2021.
45. "Super 30 founder Anand Kumar awarded honorary doctorate by NIT Delhi". The print. 7 August 2022.
46. "Home – Prime Time Research Media Global Education Awards, Market Research Company". globaleducationawards.com. Archived from the original on 7 March 2019. Retrieved 7 March 2019.
47. "Rashtriya Bal Kalyan Award presented to Super-30 founder". The Economic Times. 19 September 2019.
48. "Super 30 Founder Anand Kumar To Receive Mahaveer Award". Outlook. 27 January 2021.
49. "Anand Kumar awarded as Ganit Ratna". Northeast Color. 12 March 2022.
50. "Padma Awards 2023 announced". Press Information Buereau. Ministry of Home Affairs, Govt of India. Retrieved 26 January 2023.
51. "Confirmed! Hrithik Roshan is playing mathematician Anand Kumar in Vikas Bahl's Super 30". The Indian Express. 25 September 2017. Archived from the original on 24 July 2018. Retrieved 25 September 2017.
52. "Patna's Super 30 mentor Anand Kumar accused of deceit to gain popularity". India Today. 23 July 2018. Archived from the original on 25 July 2018. Retrieved 23 July 2018.
53. Subhash K Jha (3 August 2018). "Anand Kumar fighting a smear campaign". Deccan Chronicle. Archived from the original on 5 June 2019. Retrieved 5 June 2019. But ever since the announcement of the project, it seems people are out to discredit Anand's works.
54. "Super 30 mentor Anand Kumar gets support from Tejashwi Yadav, Shatrughan Sinha amid charges of fabrication". The Financial Express. 31 July 2018. Archived from the original on 27 March 2019. Retrieved 7 October 2018.
55. "Banker jailed for 86 days for 'no crime' in Bihar". The Times of India. 10 October 2018. Archived from the original on 1 March 2019. Retrieved 5 June 2019.
56. "Jagran Josh Education Awards 2021". Dainik Jagran. 25 March 2021.
External links
• Super30 website
Recipients of Padma Shri in Literature & Education
1950s
• K. Shankar Pillai (1954)
• Krishna Kanta Handique (1955)
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Robert Creighton Buck
Robert Creighton Buck (30 August 1920 Cincinnati – 1 February 1998 Wisconsin), usually cited as R. Creighton Buck, was an American mathematician who, with Ralph Boas, introduced Boas–Buck polynomials.[1] He taught at University of Wisconsin–Madison for 40 years. In addition, he was a writer.[2]
Biography
Buck was born in Cincinnati.[3] He studied at the University of Cincinnati and then earned his PhD in 1947[3] at Harvard University under David Widder and Ralph Boas with dissertation Uniqueness, Interpolation and Characterization Theorems for Functions of Exponential Type. For three years he was an assistant professor at Brown University, before he became in 1950 an associate professor at the University of Wisconsin, Madison, where he was promoted to professor in 1954. In 1973, he became the acting director of the University of Wisconsin Army Mathematics Research Center when J. Barkley Rosser retired.[4] At Madison he became in 1980 "Hilldale Professor" and from 1964 to 1966 he was chair of the mathematics department. In 1990 he retired as professor emeritus but remained mathematically active.[3]
Buck worked on approximation theory, complex analysis, topological algebra, and operations research. He worked for six years for the Institute for Defense Analyses in operations research. Buck wrote, in collaboration with Ellen F. Buck,[5] a textbook Advanced Calculus, commonly used in U.S. colleges and universities. He also worked on the history of mathematics. For his essay Sherlock Holmes in Babylon[6] he won the Lester Randolph Ford Award. His doctoral students include Lee Rubel and Thomas W. Hawkins, a well-known historian of mathematics.
Buck was vice-president of the American Mathematical Society and the Mathematical Association of America (MAA), whose "Committee on the Undergraduate Program in Mathematics“ (CUPM) he founded and from 1959 to 1963 chaired. In 1962 he was an invited speaker (Global solutions of differential equations) at the International Congress of Mathematicians in Stockholm.
Buck was an accomplished amateur pianist and at age 18 won a prize for composition for piano. He wrote several science fiction stories.[3]
Publications
• Advanced Calculus, McGraw Hill, New York 1956, 3rd edn. Waveland Press, 2003
• with Ralph Boas: Polynomial expansions of analytic functions, Springer 1958,[7] 2nd edn, Academic Press, Springer 1964
• with Ellen F. Buck: Introduction to differential equations, Boston, Houghton Mifflin 1978
• with Alfred Willcox: Calculus of several variables, Houghton Mifflin 1971
• “Sherlock Holmes in Babylon”, AMM 1980
References
1. Boas; Buck (1958). Polynomial expansions of analytic functions. Springer.
2. Robert Creighton Buck, U. W. Madison biography
3. "Buck, R. Creighton". The Capital Times. February 2, 1998. p. 17. Retrieved November 9, 2019 – via Newspapers.com.
4. "25 Years ago Today". The Capital Times. January 15, 1998. p. 40. Retrieved November 10, 2019 – via Newspapers.com.
5. Obituary for: Ellen F. Buck | Crandall Funeral Home
6. American Mathematical Monthly, Vol.87, 1980, pp. 335–345. Reprinted in Marlow Anderson, Victor Katz, Robin Wilson (eds.) Sherlock Holmes in Babylon and other tales of mathematical history, MAA 2004
7. Rainville, Earl (1959). "Review: Polynomial expansions of analytic functions. By Ralph P. Boas, Jr. and R. Creighton Buck" (PDF). Bull. Amer. Math. Soc. 65 (3): 150–151. doi:10.1090/s0002-9904-1959-10304-9.
External links
• R. Creighton (Robert) Buck at the Mathematics Genealogy Project
• Literature by and about Robert Creighton Buck in the German National Library catalogue
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James's theorem
In mathematics, particularly functional analysis, James' theorem, named for Robert C. James, states that a Banach space $X$ is reflexive if and only if every continuous linear functional's norm on $X$ attains its supremum on the closed unit ball in $X.$
A stronger version of the theorem states that a weakly closed subset $C$ of a Banach space $X$ is weakly compact if and only if the dual norm each continuous linear functional on $X$ attains a maximum on $C.$
The hypothesis of completeness in the theorem cannot be dropped.[1]
Statements
The space $X$ considered can be a real or complex Banach space. Its continuous dual space is denoted by $X^{\prime }.$ The topological dual of ℝ-Banach space deduced from $X$ by any restriction scalar will be denoted $X_{\mathbb {R} }^{\prime }.$ (It is of interest only if $X$ is a complex space because if $X$ is a $\mathbb {R} $-space then $X_{\mathbb {R} }^{\prime }=X^{\prime }.$)
James compactness criterion — Let $X$ be a Banach space and $A$ a weakly closed nonempty subset of $X.$ The following conditions are equivalent:
• $A$ is weakly compact.
• For every $f\in X^{\prime },$ there exists an element $a_{0}\in A$ such that $\left|f\left(a_{0}\right)\right|=\sup _{a\in A}|f(a)|.$
• For any $f\in X_{\mathbb {R} }^{\prime },$ there exists an element $a_{0}\in A$ such that $f\left(a_{0}\right)=\sup _{a\in A}|f(a)|.$
• For any $f\in X_{\mathbb {R} }^{\prime },$ there exists an element $a_{0}\in A$ such that $f\left(a_{0}\right)=\sup _{a\in A}f(a).$
A Banach space being reflexive if and only if its closed unit ball is weakly compact one deduces from this, since the norm of a continuous linear form is the upper bound of its modulus on this ball:
James' theorem — A Banach space $X$ is reflexive if and only if for all $f\in X^{\prime },$ there exists an element $a\in X$ of norm $\|a\|\leq 1$ such that $f(a)=\|f\|.$
History
Historically, these sentences were proved in reverse order. In 1957, James had proved the reflexivity criterion for separable Banach spaces[2] and 1964 for general Banach spaces.[3] Since the reflexivity is equivalent to the weak compactness of the unit sphere, Victor L. Klee reformulated this as a compactness criterion for the unit sphere in 1962 and assumes that this criterion characterizes any weakly compact quantities.[4] This was then actually proved by James in 1964.[5]
See also
• Banach–Alaoglu theorem – Theorem in functional analysis
• Bishop–Phelps theorem
• Dual norm – Measurement on a normed vector space
• Eberlein–Šmulian theorem – Relates three different kinds of weak compactness in a Banach space
• Goldstine theorem
• Mazur's lemma – On strongly convergent combinations of a weakly convergent sequence in a Banach space
• Operator norm – Measure of the "size" of linear operators
Notes
1. James (1971)
2. James (1957)
3. James (1964)
4. Klee (1962)
5. James (1964)
References
• James, Robert C. (1957), "Reflexivity and the supremum of linear functionals", Annals of Mathematics, 66 (1): 159–169, doi:10.2307/1970122, JSTOR 1970122, MR 0090019
• Klee, Victor (1962), "A conjecture on weak compactness", Transactions of the American Mathematical Society, 104 (3): 398–402, doi:10.1090/S0002-9947-1962-0139918-7, MR 0139918.
• James, Robert C. (1964), "Weakly compact sets", Transactions of the American Mathematical Society, 113 (1): 129–140, doi:10.2307/1994094, JSTOR 1994094, MR 0165344.
• James, Robert C. (1971), "A counterexample for a sup theorem in normed space", Israel Journal of Mathematics, 9 (4): 511–512, doi:10.1007/BF02771466, MR 0279565.
• James, Robert C. (1972), "Reflexivity and the sup of linear functionals", Israel Journal of Mathematics, 13 (3–4): 289–300, doi:10.1007/BF02762803, MR 0338742.
• Megginson, Robert E. (1998), An introduction to Banach space theory, Graduate Texts in Mathematics, vol. 183, Springer-Verlag, ISBN 0-387-98431-3
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Richard Borcherds
Richard Ewen Borcherds (/ˈbɔːrtʃərdz/; born 29 November 1959)[1] is a British[4] mathematician currently working in quantum field theory. He is known for his work in lattices, group theory, and infinite-dimensional algebras,[5][6] for which he was awarded the Fields Medal in 1998.
Richard Borcherds
Borcherds in 1993
Born
Richard Ewen Borcherds
(1959-11-29) 29 November 1959[1]
Cape Town, South Africa
NationalityBritish[2]
Alma materTrinity College, Cambridge
Known forBorcherds algebra
Awards
• Whitehead Prize (1992)
• EMS Prize (1992)
• FRS (1994)
• Fields Medal (1998)
Scientific career
FieldsMathematics
Institutions
• University of California, Berkeley
• University of Cambridge
ThesisThe leech lattice and other lattices (1984)
Doctoral advisorJohn Horton Conway[3]
Doctoral studentsDaniel Allcock[3]
Websitemath.berkeley.edu/~reb
Early life
Borcherds was born in Cape Town, South Africa, but the family moved to Birmingham in the United Kingdom when he was six months old.[7]
Education
Borcherds was educated at King Edward's School, Birmingham, and Trinity College, Cambridge,[8] where he studied under John Horton Conway.[9]
Career
After receiving his doctorate in 1985, Borcherds has held various alternating positions at Cambridge and the University of California, Berkeley, serving as Morrey Assistant Professor of Mathematics at Berkeley from 1987 to 1988. He was a Royal Society University Research Fellow.[10][8] From 1996 he held a Royal Society Research Professorship at Cambridge before returning to Berkeley in 1999 as Professor of Mathematics.[8]
An interview with Simon Singh for The Guardian, in which Borcherds suggested he might have some traits associated with Asperger syndrome,[7] subsequently led to a chapter about him in a book on autism by Simon Baron-Cohen.[11][12] Baron-Cohen concluded that while Borcherds had many autistic traits, he did not merit a formal diagnosis of Asperger syndrome.[11]
Awards and honours
In 1992 Borcherds was one of the first recipients of the EMS prizes awarded at the first European Congress of Mathematics in Paris, and in 1994 he was an invited speaker at the International Congress of Mathematicians in Zurich.[9] In 1994, he was elected to be a Fellow of the Royal Society.[13] In 1998 at the 23rd International Congress of Mathematicians in Berlin, Germany he received the Fields Medal together with Maxim Kontsevich, William Timothy Gowers and Curtis T. McMullen.[9] The award cited him "for his contributions to algebra, the theory of automorphic forms, and mathematical physics, including the introduction of vertex algebras and Borcherds' Lie algebras, the proof of the Conway-Norton moonshine conjecture[14] and the discovery of a new class of automorphic infinite products." In 2012 he became a fellow of the American Mathematical Society,[15] and in 2014 he was elected to the National Academy of Sciences.[16]
References
1. "BORCHERDS, Prof. Richard Ewen". Who's Who 2014, A & C Black, an imprint of Bloomsbury Publishing plc, 2014; online edn, Oxford University Press.(subscription required)
2. Goddard, Peter (1998). "The work of Richard Ewen Borcherds". Proceedings of the International Congress of Mathematicians, Vol. I (Berlin, 1998). pp. 99–108. arXiv:math/9808136. Bibcode:1998math......8136G. ISSN 1431-0635. {{cite book}}: |journal= ignored (help).
3. Richard Borcherds at the Mathematics Genealogy Project
4. "Richard Borcherds".
5. James Lepowsky, "The Work of Richard Borcherds", Notices of the American Mathematical Society, Volume 46, Number 1 (January 1999).
6. Borcherds, Richard E. (1998). "What is moonshine?". Doc. Math. (Bielefeld) Extra Vol. ICM Berlin, 1998, vol. I. pp. 607–616.
7. Simon Singh, "Interview with Richard Borcherds", The Guardian (28 August 1998)
8. "UC Berkeley professor wins highest honor in mathematics, the prestigious Fields Medal". University of California, Berkeley. 19 August 1998. Retrieved 22 July 2009.
9. Jackson, Allyn (November 1998). "Borcherds, Gowers, Kontsevich, and McMullen Receive Fields Medals" (PDF). Notices of the American Mathematical Society. American Mathematical Society. 45 (10).
10. Cook, Alan (2000). "URFs become FRS: Frances Ashcroft, Athene Donald, and John Pethica". Notes and Records of the Royal Society. London: Royal Society. 54 (3): 409–411. doi:10.1098/rsnr.2000.0181. S2CID 58095147.
11. Baron-Cohen, Simon (2004). "A Professor of Mathematics". The Essential Difference: Male and Female Brains and the Truth about Autism. Basic Books. ISBN 0-465-00556-X. (see external links) records conversations with Richard Borcherds and his family.
12. High flying obsessives, The Guardian, December 2000
13. "EC/1994/05: Borcherds, Richard Ewen". London: The Royal Society. Archived from the original on 8 July 2019.
14. Borcherds, Richard E. (1992). "Monstrous moonshine and monstrous Lie superalgebras". Inventiones Mathematicae. Springer Science and Business Media LLC. 109 (1): 405–444. Bibcode:1992InMat.109..405B. doi:10.1007/bf01232032. ISSN 0020-9910. S2CID 16145482.
15. List of Fellows of the American Mathematical Society. Retrieved 10 November 2012.
16. National Academy of Sciences Members and Foreign Associates Elected Archived 18 August 2015 at the Wayback Machine, National Academy of Sciences, 29 April 2014.
Further reading
• Conway and Sloane, Sphere Packings, Lattices, and Groups, Third Edition, Springer, 1998 ISBN 0-387-98585-9.
• Frenkel, Lepowsky and Meurman, Vertex Operator Algebras and the Monster, Academic Press, 1988 ISBN 0-12-267065-5.
• Kac, Victor, Vertex Algebras for Beginners, Second Edition, AMS 1997 ISBN 0-8218-0643-2.
• O'Connor, John J.; Robertson, Edmund F., "Richard Borcherds", MacTutor History of Mathematics Archive, University of St Andrews
External links
• Richard Borcherds's publications indexed by the Scopus bibliographic database. (subscription required)
• Richard Borcherds's results at International Mathematical Olympiad
Fields Medalists
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Robert James Blattner
Robert James Blattner (6 August 1931 – 13 June 2015) was a mathematics professor at UCLA[1][2] working on harmonic analysis, representation theory, and geometric quantization, who introduced Blattner's conjecture. Born in Milwaukee,[3] Blattner received his bachelor's degree from Harvard University in 1953[1] and his Ph.D. from the University of Chicago in 1957.[4] He joined the UCLA mathematics department in 1957 and remained on the staff until his retirement as professor emeritus in 1992.[1]
He was most widely known for a conjecture that he made, contained in the so-called Blattner formula, which suggested that a certain deep property of the discrete series of representations of a semi simple real Lie group was true. He made this conjecture in the mid 1960s. The discrete series, constructed by Harish-Chandra, which is basic to most central questions in harmonic analysis and arithmetic, was still very new and very difficult to penetrate. The conjecture was later proved and the solution was published in 1975 by Wilfried Schmid and Henryk Hecht by analytic methods, and later, in 1979 by Thomas Enright who used algebraic methods; both proofs were quite deep, giving an indication of the insight that led Blattner to this conjecture.[1]
Blattner was a visiting scholar at the Institute for Advanced Study in 1964–65.[5] In 2012 he became a fellow of the American Mathematical Society.[6]
References
1. Varadarajan, V. S. (20 October 2015). "In Memoriam: Robert J. Blattner". UCLA Mathematics (math.ucla.edu).
2. UCLA page about Robert Blattner
3. "A Community of Scholars: Faculty and Members, 1930-1980". 1980.
4. Robert James Blattner at the Mathematics Genealogy Project
5. Institute for Advanced Study: A Community of Scholars
6. List of Fellows of the American Mathematical Society, retrieved 2012-11-10.
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Robert Daverman
Robert Jay Daverman (born 28 September 1941) is an American topologist.
Daverman was born in Grand Rapids, Michigan, on 28 September 1941. He earned a bachelor's degree in 1963 from Calvin College and pursued doctoral study under R. H. Bing at the University of Wisconsin–Madison. After completing his thesis Locally Fenced 2-spheres in S3 in 1967, Daverman began teaching at the University of Tennessee–Knoxville.[1] While on the Knoxville faculty, Daverman served on the American Mathematical Society's Committee on Science Policy.[2] By the time he was selected as one of the inaugural fellows of the AMS in 2012, Daverman had gained emeritus status.[3]
Selected publications
• Daverman, Robert J. (1986). Decompositions of Manifolds. Academic Press. ISBN 9780122042201.[4]
• Daverman, Robert J. (2009). Embeddings in Manifolds. American Mathematical Society. ISBN 9780821836972.[5]
References
1. Guilbault, Craig (May 2002). "Robert J. Daverman: a short mathematical tribute" (PDF). {{cite journal}}: Cite journal requires |journal= (help)
2. "UA Dean Robert Olin Named Chair of American Mathematical Society Committee". University of Alabama. 30 November 2004. Retrieved 3 April 2022.
3. "Four Faculty Members Named American Mathematics Society Fellows". University of Tennessee–Knoxville. 5 November 2012. Retrieved 3 April 2022.
4. Cannon, James W. (1988). "Decompositions of Manifolds. By Robert J. Daverman". The American Mathematical Monthly. 95 (5): 471–475. doi:10.1080/00029890.1988.11972035.
5. Cannon, James W. (2011). "Embeddings in manifolds, by Robert J. Daverman and Gerard A. Venema, Graduate Studies in Mathematics, Vol. 106, American Mathematical Society, Providence, RI, 2009, xviii+468 pp., ISBN 978-0-8218-3697-2, hardcover, US $75.00" (PDF). Bulletin of the American Mathematical Society. 48 (3): 485–490. doi:10.1090/S0273-0979-2011-01320-9.
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R. K. Rubugunday
Raghunath Krishna Rubugunday (1918–2000) was an Indian mathematician specializing in number theory notable for his contribution to Waring's problem.[1]
Raghunath Krishna Rubugunday
Born1918
Madras
Died2000
NationalityIndian
Known forWaring's problem
Scientific career
FieldsMathematics
Rubugunday was born in Madras in 1918. The famous mathematician K. Ananda Rau was an uncle on his father's side. He completed his B.A. Hons from Presidency College, Madras and Tripos from Cambridge in 1938. He returned to India and among other positions he was the Head of the Department of Mathematics at Saugar university.[1]
References
1. Rajendra Bhatia (2010). Proceedings of the International Congress of Mathematicians, v.I. World Scientific. pp. 199–. ISBN 978-981-4324-35-9. Retrieved 21 July 2013.
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Robert Lee Moore
Robert Lee Moore (November 14, 1882 – October 4, 1974) was an American mathematician who taught for many years at the University of Texas. He is known for his work in general topology, for the Moore method of teaching university mathematics, and for his racist treatment of African-American mathematics students.
Robert Lee Moore
R. L. Moore in 1904
Born(1882-11-14)November 14, 1882
Dallas, Texas, US
DiedOctober 4, 1974(1974-10-04) (aged 91)
Austin, Texas, US
Alma materUniversity of Chicago (Ph.D., 1905)
Scientific career
FieldsMathematics
InstitutionsUniversity of Texas at Austin
ThesisSets of Metrical Hypotheses for Geometry (1905)
Doctoral advisorOswald Veblen
E. H. Moore
Doctoral students
• Richard Anderson
• R. H. Bing
• Mary-Elizabeth Hamstrom
• F. Burton Jones
• John Kline
• Edwin E Moise
• Anna Mullikin
• Mary Ellen Rudin
• Gordon Whyburn
• Raymond Wilder
Life
Although Moore's father was reared in New England and was of New England ancestry, he fought in the American Civil War on the side of the Confederacy. After the war, he ran a hardware store in Dallas, then little more than a railway stop, and raised six children, of whom Robert, named after the commander of the Confederate Army of Northern Virginia, was the fifth.
Moore entered the University of Texas at the unusually youthful age of 15, in 1898, already knowing calculus thanks to self-study. He completed the B.Sc. in three years instead of the usual four; his teachers included G. B. Halsted and L. E. Dickson. After a year as a teaching fellow at Texas, he taught high school for a year in Marshall, Texas.
An assignment of Halsted's led Moore to prove that one of Hilbert's axioms for geometry was redundant. When E. H. Moore (no relation), who headed the Department of Mathematics at the University of Chicago, and whose research interests were on the foundations of geometry, heard of Robert's feat, he arranged for a scholarship that would allow Robert to study for a doctorate at Chicago. Oswald Veblen supervised Moore's 1905 thesis, titled Sets of Metrical Hypotheses for Geometry.
Moore then taught one year at the University of Tennessee, two years at Princeton University, and three years at Northwestern University. In 1910, he married Margaret MacLelland Key of Brenham, Texas; they had no children. In 1911, he took up a position at the University of Pennsylvania.
In 1920, Moore returned to the University of Texas at Austin as an associate professor and was promoted to full professor three years later. In 1951, he went on half pay but continued to teach five classes a year, including a section of freshman calculus, until the University authorities forced his definitive retirement in 1969, his 87th year.
A strong supporter of the American Mathematical Society, he presided over it, 1936–38. He edited its Colloquium Publications, 1929–33, and was the editor-in-chief, 1930–33. In 1931, he was elected to the National Academy of Sciences.
Topologist
According to the bibliography in Wilder (1976), Moore published 67 papers and one monograph, his 1932 Foundations of Point Set Theory. He is primarily remembered for his work on the foundations of topology, a topic he first touched on in his Ph.D. thesis. By the time Moore returned to the University of Texas, he had published 17 papers on point-set topology—a term he coined—including his 1915 paper "On a set of postulates which suffice to define a number-plane", giving an axiom system for plane topology. The Moore plane, Moore's road space, Moore space, Moore's quotient theorem and the normal Moore space conjecture are named in his honor.
Unusual teacher
Robert Lee Moore is known to have supervised 50 doctoral dissertations, almost all at Texas, including those of R. H. Bing, F. Burton Jones, John R. Kline, Edwin Evariste Moise, Mary Ellen Rudin, Gordon Whyburn, Richard Davis Anderson, and Raymond Louis Wilder. Moore has been described as having been one of the most charismatic and inspiring university teachers of mathematics ever active in the United States. Accounts have been given of his ability to teach students who had never previously distinguished themselves in mathematics how to do proofs. He went out of his way to teach elementary and service courses every year, and actually forbade his pre-doctoral level students from consulting the mathematical literature.[1][2]
It was while attending lectures at the University of Chicago that Moore first hit on his original teaching methods. Finding these lectures rather boring, even mind dulling, he would liven up a lecture by running a race in his mind with the lecturer, by trying to discover the proof of an announced theorem before the lecturer had finished his presentation. Moore often won this silent race, and when he did not, he felt that he was better off from having made the attempt. It was at the University of Pennsylvania, while teaching a course on the foundations of geometry, that Moore first tried out the teaching methods that came to be known as the Moore method. The success of this method led others to adopt it and similar methods.
Prejudice
Moore's record as a teacher of mathematics has been tarnished by his racism towards black students.[3][4] Most of Moore's career was spent in a racially segregated part of the United States. African-American students were prohibited from even enrolling at the University of Texas until the late 1950s,[5] and Moore himself was strongly in favor of segregation.[6][7][8][9] After the University of Texas began admitting African-American students, he refused to allow them into his classes, even for mathematics graduate students such as Vivienne Malone-Mayes.[3][4][7][8] He told another African-American mathematics student, Walker E. Hunt, "you are welcome to take my course but you start with a C and can only go down from there".[9] On one occasion he walked out of a talk by a student, his academic grandchild, after discovering that the speaker was African-American.[4][7][8]
Moore was also known for repeatedly claiming that female students were inferior to male students, and, though "less pronounced than his racism", for his antisemitism.[4] However, while Moore's racism is confirmed by several first-hand accounts of his refusal to teach African-American students, the often-repeated description of him as a misogynist and antisemite is based largely on his oral remarks. Some of the sources reporting these remarks, such as Mary Ellen Rudin, also point out that in fact he encouraged females who showed mathematical talent and that he had Jewish students, such as Edwin E. Moise (who was asked about Moore's anti-Semitic reputation in an interview)[10] and Martin Ettlinger, and close colleagues, such as Hyman J. Ettlinger. His encouragement of Rudin and other white female students is documented[11] and between 1949 and 1970 (the earliest period when national data are known) 4 of Moore's 31 doctoral students (13%) were female, while nationally 175 were female out of 2646 doctoral graduates in mathematics and statistics (7%).[12]
Honors
• The Robert Lee Moore Hall, a classroom building at the University of Texas, was named after Moore from its construction in 1972–2020. In 2020, following complaints and petitions in 2019 and then the 2020 George Floyd protests and Black Lives Matter movement, it was renamed to the "Physics, Math and Astronomy Building".[13]
Quotations
• "That student is taught the best who is told the least." Moore, quote in Parker (2005: vii).[2]
• "I hear, I forget. I see, I remember. I do, I understand." (Chinese proverb that was a favorite of Moore's. Quoted in Halmos, P.R. (1985) I want to be a mathematician: an automathography. Springer-Verlag: 258)
Notes
1. Devlin, Keith (2015). "Devlin's Angle: The Greatest Math Teacher Ever?". Devlin's Angle. Retrieved August 29, 2018.
2. Parker, John, 2005. R. L. Moore: Mathematician and Teacher. Mathematical Association of America. ISBN 0-88385-550-X.
3. Schaffer, Karl (2017). "The Daughters of Hypatia: Dancing the Stories of Women in Mathematics". In Beery, Janet L.; Greenwald, Sarah J.; Jensen-Vallin, Jacqueline A.; Mast, Maura B. (eds.). Women in Mathematics: Celebrating the Centennial of the Mathematical Association of America. Association for Women in Mathematics Series. Springer. pp. 375–395. doi:10.1007/978-3-319-66694-5_21.
4. Ross, Peter (September 2007). "Review of R. L. Moore: mathematician & teacher". The Mathematical Intelligencer. 29 (4): 75–79. doi:10.1007/bf02986178. S2CID 123211850.
5. Parker (2005), p. 288.
6. Lewis, Albert C. (2004). "The beginnings of the R. L. Moore school of topology" (PDF). Historia Mathematica. 31 (3): 279–295. doi:10.1016/s0315-0860(03)00050-8. Retrieved December 24, 2011.
7. Hersh, Reuben; John-Steiner, Vera (2010). Loving and Hating Mathematics: Challenging the Myths of Mathematical Life. Princeton University Press. p. 405. ISBN 9781400836116..
8. McCann, Mac (May 29, 2015). "Written in Stone: History of racism lives on in UT monuments". The Austin Chronicle.
9. Corry, Leo (2007). "A clash of mathematical titans in Austin: Harry S. Vandiver and Robert Lee Moore (1924–1974)". The Mathematical Intelligencer. 29 (4): 62–74. doi:10.1007/BF02986177. MR 2361623. S2CID 119957075.
10. "An Interview of Edwin Moise". Topology Atlas. 2000. Archived from the original on January 7, 2020. Retrieved August 29, 2018.
11. Parker, John (2005). "His Female Students". R. L. Moore: Mathematician and Teacher. Mathematical Association of America. pp. 241–256. ISBN 0-88385-550-X.
12. "Degrees in mathematics and statistics conferred by postsecondary institutions, by level of degree and sex of student: Selected years, 1949-50 through 2014-15". Digest of Education Statistics. 2017. Retrieved August 29, 2018.
13. Girgis, Lauren (September 11, 2019). "People for PMA hold grassroots discussion on renaming Robert Lee Moore Hall". The Daily Texan.
References
• Jones, F. Burton, 1997, "The Beginning of Topology in the United States and the Moore School" in C. E. Aull and R. Louwen, eds., Handbook of the History of General Topology, Vol. 1. Kluwer: 97–103.
• Lewis, Albert C., 1990, "R. L. Moore" in Dictionary of Scientific Biography, vol. 18. Charles Scribner's Sons: 651–53.
• Moore, R. L., 1970 (1932). Foundations of Point Set Theory. Vol. 13 of the AMS Colloquium Publications. American Mathematical Society.
• Wilder, R. L., 1976, "Robert Lee Moore 1882–1974," Bulletin of the AMS 82: 417–27. Includes a complete bibliography of Moores writings.
Further reading
• Traylor, D. Reginald (1972). Creative Teaching: The Heritage of R.L. Moore. with William Bane and Madeline Jones. Houston, Texas: University of Houston. OCLC 735304. Retrieved September 7, 2008. Biography, with extensive list of academic descendants of R. L. Moore and their publications.
External links
• Robert Lee Moore at the Mathematics Genealogy Project
• O'Connor, John J.; Robertson, Edmund F. "Robert Lee Moore". MacTutor History of Mathematics Archive. University of St Andrews.
• The Legacy of Robert Lee Moore Project.
• Links to biographical material and the Moore method.
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1901–1924
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R. Leonard Brooks
Rowland Leonard Brooks (February 6, 1916 – June 18, 1993)[1] was an English mathematician, known for proving Brooks's theorem on the relation between the chromatic number and the degree of graphs. He was born in Lincolnshire, England, studied at Trinity College, Cambridge University, and also worked with fellow Trinity students W. T. Tutte, Cedric Smith, and Arthur Harold Stone on the problem of "Squaring the square" (partitioning rectangles and squares into unequal squares), both under their own names and under the pseudonym Blanche Descartes.[2]
R. Leonard Brooks
After leaving Cambridge, he worked as a full-time tax inspector.[1]
References
1. Brooks, Smith, Stone, Tutte, squaring.net, retrieved 2010-07-30.
2. Soifer, Alexander (2008), The Mathematical Coloring Book, Springer-Verlag, pp. 82–83, ISBN 978-0-387-74640-1.
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Ralph P. Boas Jr.
Ralph Philip Boas Jr. (August 8, 1912 – July 25, 1992) was a mathematician, teacher, and journal editor. He wrote over 200 papers, mainly in the fields of real and complex analysis.[3]
Ralph P. Boas Jr.
Born(1912-08-08)August 8, 1912
Walla Walla, Washington
DiedJuly 25, 1992(1992-07-25) (aged 79)
Seattle, Washington
NationalityAmerican
Alma materHarvard University
AwardsLester R. Ford Award (1970, 1978)[1][2]
Scientific career
FieldsMathematics
InstitutionsNorthwestern University
Duke University
Doctoral advisorDavid Widder
Doctoral studentsCreighton Buck
Philip J. Davis
Christopher Imoru
Dale Mugler
Biography
He was born in Walla Walla, Washington, the son of an English professor at Whitman College, but moved frequently as a child; his younger sister, Marie Boas Hall, later to become a historian of science, was born in Springfield, Massachusetts, where his father had become a high school teacher.[4] He was home-schooled until the age of eight, began his formal schooling in the sixth grade, and graduated from high school while still only 15.[4] After a gap year auditing classes at Mount Holyoke College (where his father had become a professor) he entered Harvard, intending to major in chemistry and go into medicine, but ended up studying mathematics instead.[4] His first mathematics publication was written as an undergraduate, after he discovered an incorrect proof in another paper.[4] He got his A.B. degree in 1933, received a Sheldon Fellowship for a year of travel, and returned to Harvard for his doctoral studies in 1934.[4] He earned his doctorate there in 1937, under the supervision of David Widder.[3][4][5]
After postdoctoral studies at Princeton University with Salomon Bochner, and then the University of Cambridge in England, he began a two-year instructorship at Duke University, where he met his future wife, Mary Layne, also a mathematics instructor at Duke. They were married in 1941, and when the United States entered World War II later that year, Boas moved to the Navy Pre-flight School in Chapel Hill, North Carolina. In 1942, he interviewed for a position in the Manhattan Project, at the Los Alamos National Laboratory, but ended up returning to Harvard to teach in a Navy instruction program there, while his wife taught at Tufts University.[4]
Beginning when he was an instructor at Duke University, Boas had become a prolific reviewer for Mathematical Reviews, and at the end of the war he took a position as its full-time editor.[4] In the academic year 1950–1951 he was a Guggenheim Fellow.[6] In 1950 he became Professor of Mathematics at Northwestern University, without ever previously having been an assistant or associate professor; his wife became a professor of physics at nearby DePaul University, due to the anti-nepotism rules then in place at Northwestern.[3][4] He stayed at Northwestern until his retirement in 1980, and was chair there from 1957 to 1972.[3][4] He was president of the Mathematical Association of America from 1973 to 1974, and as president launched the Dolciani Mathematical Expositions series of books.[7] He was also editor of the American Mathematical Monthly from 1976 to 1981.[3] He continued mathematical work after retiring, for instance as co-editor (with George Leitmann) of the Journal of Mathematical Analysis and Applications from 1985 to 1991.[3]
Along with his mathematical education, Boas was educated in many languages: Latin in junior high school, French and German in high school, Greek at Mount Holyoke, Sanskrit as a Harvard undergraduate, and later self-taught Russian while at Duke University.[4]
Boas' son Harold P. Boas is also a noted mathematician.
The hunting of big game
Boas, Frank Smithies, and colleagues were behind the 1938 paper A Contribution to the Mathematical Theory of Big Game Hunting published in the American Mathematical Monthly under the pseudonym H. Pétard (referring to Hamlet's "hoist by his own petard"). The paper offers short spoofs of theorems and proofs from mathematics and physics, in the form of applications to the hunting of lions in the Sahara desert. One "proof" parodies the Bolzano–Weierstrass theorem:
The Bolzano-Weierstrass Method. Bisect the desert by a line running N-S. The lion is either in the E portion or in the W portion; let us suppose him to be in the W portion. Bisect this portion by a line running E-W. The lion is either in the N portion or in the S portion; let us suppose him to be in the N portion. We continue this process indefinitely, constructing a sufficiently strong fence about the chosen portion at each step. The diameter of the chosen portions approaches zero, so that the lion is ultimately surrounded by a fence of arbitrarily small perimeter.
The paper became a classic of mathematical humor and spawned various follow-ons over the years with theories or methods from other scientific areas adapted to hunting lions.
The paper and later work is published in Lion Hunting and Other Mathematical Pursuits : A Collection of Mathematics, Verse, and Stories by the Late Ralph P. Boas Jr., edited by Gerald L. Alexanderson and Dale H. Mugler, ISBN 0-88385-323-X. Various online collections of the lion hunting methods exist too.
Pondiczery
E. S. Pondiczery was another pseudonym invented by Boas and Smithies as the fictional person behind the "H. Pétard" pseudonym,[4] and later used again by Boas, this time for a serious paper on topology, Power problems in abstract spaces, Duke Mathematical Journal, 11 (1944), 835–837. This paper and the name became part of the Hewitt-Marczewski-Pondiczery theorem.
The name, revealed in Lion Hunting and Other Mathematical Pursuits cited above, came from Pondicherry (a place in India disputed by the Dutch, English and French) and a slavic twist. The initials "E.S." were a plan to write a spoof on extra-sensory perception (ESP).
Other
His best-known books are the lion-hunting book previously mentioned and the monograph A Primer of Real Functions.[8] The current edition of the primer has been revised and edited by his son, mathematician Harold P. Boas.
The best-known of his 13 doctoral students is Philip J. Davis, who is also his only advisee who did not graduate from Northwestern. Boas advised Davis, who was at Harvard University, while Boas was visiting at Brown University.
References
1. Boas, Ralph P. (1969). "Inequalities for the derivatives of polynomials". Mathematics Magazine. 42 (4): 165–174. doi:10.2307/2688534. JSTOR 2688534.
2. Boas, Ralph P. (1977). "Partial sums of infinite series, and how they grow". Amer. Math. Monthly. 84 (4): 237–258. doi:10.2307/2318865. JSTOR 2318865.
3. Gasper, George (1993), "In memoriam Ralph P. Boas Jr", Journal of Mathematical Analysis and Applications, 173: 1–2, doi:10.1006/jmaa.1993.1048.
4. Albers, Donald J.; Alexanderson, Gerald L.; Reid, Constance, eds. (1990), "Ralph P. Boas Jr.", More Mathematical People, Harcourt Brace Jovanovich, pp. 22–41.
5. Ralph P. Boas Jr. at the Mathematics Genealogy Project
6. Ralph P. Boas Jr. – John Simon Guggenheim Memorial Foundation
7. MAA presidents: Ralph Philip Boas Jr., Mathematical Association of America, retrieved 2013-04-02.
8. Gál, I. S. (1962). "Review: A Primer of Real Functions by Ralph B. Boas Jr., Carus Monograph No. 13. Wiley, New York, 1960". Bulletin of the American Mathematical Society. 68 (1): 10–12. doi:10.1090/S0002-9904-1962-10672-7.
• A Contribution to the Mathematical Theory of Big Game Hunting, American Mathematical Monthly, August–September 1938, page 446, archived May 13, 2021, at the Wayback Machine.
• Pondiczery was Ralph Boas — A Historical Vignette, Melvin Henriksen.
Further reading
• Some Modern Mathematical Methods in the Theory of Lion Hunting, O. Morphy, American Mathematical Monthly, volume 75 (1968), pages 185–187.
• Linguistic Contributions To The Formal Theory Of Big-Game Hunting, R. Mathiesen, Lingua Pranca, 1978.
External links
• Ralph Philip Boas Jr MacTutor History of Mathematics Archive, University of St Andrews
• Quotations related to Ralph P. Boas Jr. at Wikiquote
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Richard Hamming
Richard Wesley Hamming (February 11, 1915 – January 7, 1998) was an American mathematician whose work had many implications for computer engineering and telecommunications. His contributions include the Hamming code (which makes use of a Hamming matrix), the Hamming window, Hamming numbers, sphere-packing (or Hamming bound), Hamming graph concepts, and the Hamming distance.
Richard Hamming
Born
Richard Wesley Hamming
(1915-02-11)February 11, 1915
Chicago, Illinois, U.S.
DiedJanuary 7, 1998(1998-01-07) (aged 82)
Monterey, California, U.S.
Alma materUniversity of Chicago (B.S. 1937)
University of Nebraska (M.A. 1939)
University of Illinois at Urbana–Champaign (Ph.D. 1942)
Known for
• Hamming code
• Hamming window
• Hamming numbers
• Hamming distance
• Hamming weight
• Association for Computing Machinery
AwardsTuring Award (1968)
IEEE Emanuel R. Piore Award (1979)
Harold Pender Award (1981)
IEEE Hamming Medal (1988)
Scientific career
FieldsMathematics
Institutions
• University of Louisville
• Manhattan Project (Los Alamos Laboratory, (1945-1946))
• Bell Telephone Laboratories (1946–1976)
• Naval Postgraduate School (1976–1998)
ThesisSome Problems in the Boundary Value Theory of Linear Differential Equations (1942)
Doctoral advisorWaldemar Trjitzinsky
InfluencedDavid J. Farber
Born in Chicago, Hamming attended University of Chicago, University of Nebraska and the University of Illinois at Urbana–Champaign, where he wrote his doctoral thesis in mathematics under the supervision of Waldemar Trjitzinsky (1901–1973). In April 1945, he joined the Manhattan Project at the Los Alamos Laboratory, where he programmed the IBM calculating machines that computed the solution to equations provided by the project's physicists. He left to join the Bell Telephone Laboratories in 1946. Over the next fifteen years, he was involved in nearly all of the laboratories' most prominent achievements. For his work, he received the Turing Award in 1968, being its third recipient.[1]
After retiring from the Bell Labs in 1976, Hamming took a position at the Naval Postgraduate School in Monterey, California, where he worked as an adjunct professor and senior lecturer in computer science, and devoted himself to teaching and writing books. He delivered his last lecture in December 1997, just a few weeks before he died from a heart attack on January 7, 1998.
Early life
Richard Wesley Hamming was born in Chicago, Illinois, on February 11, 1915,[2] the son of Richard J. Hamming, a credit manager, and Mabel G. Redfield.[3] He grew up in Chicago, where he attended Crane Technical High School and Crane Junior College.[3]
Hamming initially wanted to study engineering, but money was scarce during the Great Depression, and the only scholarship offer he received came from the University of Chicago, which had no engineering school. Instead, he became a science student, majoring in mathematics,[4] and received his Bachelor of Science degree in 1937.[2] He later considered this a fortunate turn of events. "As an engineer," he said, "I would have been the guy going down manholes instead of having the excitement of frontier research work."[2]
He went on to earn a Master of Arts degree from the University of Nebraska in 1939, and then entered the University of Illinois at Urbana–Champaign, where he wrote his doctoral thesis on Some Problems in the Boundary Value Theory of Linear Differential Equations under the supervision of Waldemar Trjitzinsky.[4] His thesis was an extension of Trjitzinsky's work in that area. He looked at Green's function and further developed Jacob Tamarkin's methods for obtaining characteristic solutions.[5] While he was a graduate student, he discovered and read George Boole's The Laws of Thought.[6]
The University of Illinois at Urbana–Champaign awarded Hamming his Doctor of Philosophy in 1942, and he became an instructor in mathematics there. He married Wanda Little, a fellow student, on September 5, 1942,[4] immediately after she was awarded her own Master of Arts in English literature. They would remain married until his death, and had no children.[3] In 1944, he became an assistant professor at the J.B. Speed Scientific School at the University of Louisville in Louisville, Kentucky.[4]
Manhattan Project
With World War II still ongoing, Hamming left Louisville in April 1945 to work on the Manhattan Project at the Los Alamos Laboratory, in Hans Bethe's division, programming the IBM calculating machines that computed the solution to equations provided by the project's physicists. His wife Wanda soon followed, taking a job at Los Alamos as a human computer, working for Bethe and Edward Teller.[4] Hamming later recalled that:
Shortly before the first field test (you realize that no small scale experiment can be done—either you have a critical mass or you do not), a man asked me to check some arithmetic he had done, and I agreed, thinking to fob it off on some subordinate. When I asked what it was, he said, "It is the probability that the test bomb will ignite the whole atmosphere." I decided I would check it myself! The next day when he came for the answers I remarked to him, "The arithmetic was apparently correct but I do not know about the formulas for the capture cross sections for oxygen and nitrogen—after all, there could be no experiments at the needed energy levels." He replied, like a physicist talking to a mathematician, that he wanted me to check the arithmetic not the physics, and left. I said to myself, "What have you done, Hamming, you are involved in risking all of life that is known in the Universe, and you do not know much of an essential part?" I was pacing up and down the corridor when a friend asked me what was bothering me. I told him. His reply was, "Never mind, Hamming, no one will ever blame you."[6]
Hamming remained at Los Alamos until 1946, when he accepted a post at the Bell Telephone Laboratories (BTL). For the trip to New Jersey, he bought Klaus Fuchs's old car. When he later sold it just weeks before Fuchs was unmasked as a spy, the FBI regarded the timing as suspicious enough to interrogate Hamming.[3] Although Hamming described his role at Los Alamos as being that of a "computer janitor",[7] he saw computer simulations of experiments that would have been impossible to perform in a laboratory. "And when I had time to think about it," he later recalled, "I realized that it meant that science was going to be changed".[2]
Bell Laboratories
At the Bell Labs Hamming shared an office for a time with Claude Shannon. The Mathematical Research Department also included John Tukey and Los Alamos veterans Donald Ling and Brockway McMillan. Shannon, Ling, McMillan and Hamming came to call themselves the Young Turks.[4] "We were first-class troublemakers," Hamming later recalled. "We did unconventional things in unconventional ways and still got valuable results. Thus management had to tolerate us and let us alone a lot of the time."[2]
Although Hamming had been hired to work on elasticity theory, he still spent much of his time with the calculating machines.[7] Before he went home on one Friday in 1947, he set the machines to perform a long and complex series of calculations over the weekend, only to find when he arrived on Monday morning that an error had occurred early in the process and the calculation had errored off.[8] Digital machines manipulated information as sequences of zeroes and ones, units of information that Tukey would christen "bits".[9] If a single bit in a sequence was wrong, then the whole sequence would be. To detect this, a parity bit was used to verify the correctness of each sequence. "If the computer can tell when an error has occurred," Hamming reasoned, "surely there is a way of telling where the error is so that the computer can correct the error itself."[8]
Hamming set himself the task of solving this problem,[3] which he realised would have an enormous range of applications. Each bit can only be a zero or a one, so if you know which bit is wrong, then it can be corrected. In a landmark paper published in 1950, he introduced a concept of the number of positions in which two code words differ, and therefore how many changes are required to transform one code word into another, which is today known as the Hamming distance.[10] Hamming thereby created a family of mathematical error-correcting codes, which are called Hamming codes. This not only solved an important problem in telecommunications and computer science, it opened up a whole new field of study.[10][11]
The Hamming bound, also known as the sphere-packing or volume bound is a limit on the parameters of an arbitrary block code. It is from an interpretation in terms of sphere packing in the Hamming distance into the space of all possible words. It gives an important limitation on the efficiency with which any error-correcting code can utilize the space in which its code words are embedded. A code which attains the Hamming bound is said to be a perfect code. Hamming codes are perfect codes.[12][13]
Returning to differential equations, Hamming studied means of numerically integrating them. A popular approach at the time was Milne's Method, attributed to Arthur Milne.[14] This had the drawback of being unstable, so that under certain conditions the result could be swamped by roundoff noise. Hamming developed an improved version, the Hamming predictor-corrector. This was in use for many years, but has since been superseded by the Adams method.[15] He did extensive research into digital filters, devising a new filter, the Hamming window, and eventually writing an entire book on the subject, Digital Filters (1977).[16]
During the 1950s, he programmed one of the earliest computers, the IBM 650, and with Ruth A. Weiss developed the L2 programming language, one of the earliest computer languages, in 1956. It was widely used within the Bell Labs, and also by external users, who knew it as Bell 2. It was superseded by Fortran when the Bell Labs' IBM 650 were replaced by the IBM 704 in 1957.[17]
In A Discipline of Programming (1976), Edsger Dijkstra attributed to Hamming the problem of efficiently finding regular numbers.[18] The problem became known as "Hamming's problem", and the regular numbers are often referred to as Hamming numbers in Computer Science, although he did not discover them.[19]
Throughout his time at Bell Labs, Hamming avoided management responsibilities. He was promoted to management positions several times, but always managed to make these only temporary. "I knew in a sense that by avoiding management," he later recalled, "I was not doing my duty by the organization. That is one of my biggest failures."[2]
Later life
Hamming served as president of the Association for Computing Machinery from 1958 to 1960.[7] In 1960, he predicted that one day half of the Bell Labs budget would be spent on computing. None of his colleagues thought that it would ever be so high, but his forecast actually proved to be too low.[20] His philosophy on scientific computing appeared as the motto of his Numerical Methods for Scientists and Engineers (1962):
The purpose of computing is insight, not numbers.[21]
In later life, Hamming became interested in teaching. Between 1960 and 1976, when he left Bell Labs, he held visiting or adjunct professorships at Stanford University, Stevens Institute of Technology, the City College of New York, the University of California at Irvine and Princeton University.[22] As a Young Turk, Hamming had resented older scientists who had used up space and resources that would have been put to much better use by the young Turks. Looking at a commemorative poster of the Bell Labs' valued achievements, he noted that he had worked on or been associated with nearly all of those listed in the first half of his career at Bell Labs, but none in the second. He therefore resolved to retire in 1976, after thirty years.[2]
In 1976 he moved to the Naval Postgraduate School in Monterey, California, where he worked as an adjunct professor and senior lecturer in computer science.[3] He gave up research, and concentrated on teaching and writing books.[4] He noted that:
The way mathematics is currently taught it is exceedingly dull. In the calculus book we are currently using on my campus, I found no single problem whose answer I felt the student would care about! The problems in the text have the dignity of solving a crossword puzzle – hard to be sure, but the result is of no significance in life.[4]
Hamming attempted to rectify the situation with a new text, Methods of Mathematics Applied to Calculus, Probability, and Statistics (1985).[4] In 1993, he remarked that "when I left BTL, I knew that that was the end of my scientific career. When I retire from here, in another sense, it's really the end."[2] And so it proved. He became Professor Emeritus in June 1997,[23] and delivered his last lecture in December 1997, just a few weeks before his death from a heart attack on January 7, 1998.[7] He was survived by his wife Wanda.[23]
Hamming's final recorded lecture series[24] is maintained by Naval Postgraduate School along with ongoing work[25] that preserves his insights and extends his legacy.
Appearances
• Hamming takes part in the 1962 TV series The Computer and the Mind of Man[26]
Awards and professional recognition
• Turing Award, Association for Computing Machinery, 1968.[27]
• IEEE Emanuel R. Piore Award – [28]
1979 "For introduction of error correcting codes, pioneering work in operating systems and programming languages, and the advancement of numerical computation."
• Member of the National Academy of Engineering, 1980.[29]
• Harold Pender Award, University of Pennsylvania, 1981.[30]
• IEEE Richard W. Hamming Medal, 1988.[31]
• Fellow of the Association for Computing Machinery, 1994.[32]
• Basic Research Award, Eduard Rhein Foundation, 1996.[33]
The IEEE Richard W. Hamming Medal, named after him, is an award given annually by the Institute of Electrical and Electronics Engineers (IEEE), for "exceptional contributions to information sciences, systems and technology", and he was the first recipient of this medal.[34] The reverse side of the medal depicts a Hamming parity check matrix for a Hamming error-correcting code.[7]
Bibliography
• Hamming, Richard W. (1962). Numerical Methods for Scientists and Engineers. New York: McGraw-Hill.; second edition 1973
• — (1968). Calculus and the Computer Revolution. Boston: Houghton-Mifflin.
• — (1971). Introduction To Applied Numerical Analysis. New York: McGraw-Hill. ISBN 9780070258891.; Hemisphere Pub. Corp reprint 1989; Dover reprint 2012
• — (1972). Computers and Society. New York: McGraw-Hill.
• — (1977). Digital Filters. Englewood Cliffs, New Jersey: Prentice Hall. ISBN 978-0-13-212571-0.; second edition 1983; third edition 1989.
• — (1980). The Unreasonable Effectiveness of Mathematics. Washington, D.C.: The American Mathematical Monthly.
• — (1980). Coding and Information Theory. Englewood Cliffs, New Jersey: Prentice Hall. ISBN 978-0-13-139139-0.; second edition 1986.
• — (1985). Methods of Mathematics Applied to Calculus, Probability, and Statistics. Englewood Cliffs, New Jersey: Prentice Hall. ISBN 978-0-13-578899-8.
• — (1991). The Art of Probability for Scientists and Engineers. Redwood City, California: Addison-Wesley. ISBN 978-0-201-51058-4.
• — (1997). The Art of Doing Science and Engineering: Learning to Learn. Australia: Gordon and Breach. ISBN 978-90-5699-500-3.
Lectures
• 1991 - You and Your Research. Lecture sponsored by the Dept. of Electrical and Computer engineering, University of California, San Diego. Electrical and Computer Engineering Distinguished Lecture Series. Digital Object Made Available by Special Collections & Archives, UC San Diego.
Notes
1. "A.M. Turing Award, Richard W. Hamming". Association for Computing Machinery. Retrieved August 1, 2022.
2. "Computer Pioneers – Richard Wesley Hamming". IEEE Computer Society. Archived from the original on September 3, 2014. Retrieved August 30, 2014.
3. Carnes 2005, pp. 220–221.
4. "Richard W. Hamming – A.M. Turing Award Winner". Association for Computing Machinery. Retrieved August 30, 2014.
5. "Hamming biography". University of St Andrews. Retrieved August 30, 2014.
6. Hamming 1998, p. 643.
7. Morgan 1998, p. 972.
8. "Richard W. Hamming Additional Materials". Association for Computing Machinery. Retrieved August 30, 2014.
9. Shannon 1948, p. 379.
10. Morgan 1998, pp. 973–975.
11. Hamming 1950, pp. 147–160.
12. Ling & Xing 2004, pp. 82–88.
13. Pless 1982, pp. 21–24.
14. Weisstein, Eric W. "Milne's Method". MathWorld. Retrieved September 2, 2014.
15. Morgan 1998, p. 975.
16. Morgan 1998, p. 976–977.
17. Holbrook, Bernard D.; Brown, W. Stanley. "Computing Science Technical Report No. 99 – A History of Computing Research at Bell Laboratories (1937–1975)". Bell Labs. Archived from the original on September 2, 2014. Retrieved September 2, 2014.
18. Dijkstra 1976, pp. 129–134.
19. "Hamming Problem". Cunningham & Cunningham, Inc. Retrieved September 2, 2014.
20. Morgan 1998, p. 977.
21. Hamming 1962, pp. vii, 276, 395.
22. Carnes 2005, p. 220–221; Tveito, Bruaset & Lysne 2009, p. 59.
23. Fisher, Lawrence (January 11, 1998). "Richard Hamming, 82, Dies; Pioneer in Digital Technology". The New York Times. Retrieved August 30, 2014.
24. "Learning to Learn: The Art of Doing Science and Engineering lecture videos". Naval Postgraduate School, YouTube. Retrieved July 31, 2022.
25. "Hamming Resources at NPS". Naval Postgraduate School. Retrieved July 31, 2022.
26. "readers' and editor's forum: New Computer TV Series" (PDF). Computers and Automation. XII (1): 46–47. January 1963.
27. "A. M. Turing Award". Association for Computing Machinery. Archived from the original on December 12, 2009. Retrieved February 5, 2011.
28. "IEEE Emanuel R. Piore Award Recipients" (PDF). IEEE. Archived from the original (PDF) on November 24, 2010. Retrieved March 20, 2021.
29. "NAE Members Directory – Dr. Richard W. Hamming". National Academy of Engineering. Retrieved February 5, 2011.
30. "The Harold Pender Award". School of Engineering and Applied Science, University of Pennsylvania. Archived from the original on February 22, 2012. Retrieved February 5, 2011.
31. "IEEE Richard W. Hamming Medal Recipients" (PDF). IEEE. Retrieved February 5, 2011.
32. "ACM Fellows – H". Association for Computing Machinery. Archived from the original on January 24, 2011. Retrieved February 5, 2011.
33. "Award Winners (chronological)". Eduard Rhein Foundation. Archived from the original on July 18, 2011. Retrieved February 5, 2011.
34. "IEEE Richard W. Hamming Medal". IEEE. Retrieved February 5, 2011.
References
• Carnes, Mark C. (2005). American National Biography. Supplement 2. New York: Oxford University Press. ISBN 978-0-19-522202-9.
• Dijkstra, Edsger W. (1976). A Discipline of Programming. Englewood Cliffs, New Jersey: Prentice-Hall. ISBN 978-0-13-215871-8. Retrieved September 2, 2014.
• Hamming, Richard W. (1950). "Error detecting and error correcting codes" (PDF). Bell System Technical Journal. 29 (2): 147–160. doi:10.1002/j.1538-7305.1950.tb00463.x. MR 0035935. S2CID 61141773. Archived from the original (PDF) on May 25, 2006.
• Hamming, Richard (1962). Numerical Methods for Scientists and Engineers. New York: McGraw-Hill. ISBN 978-0-486-65241-2.
• Hamming, Richard (1980). "The Unreasonable Effectiveness of Mathematics". American Mathematical Monthly. 87 (2): 81–90. doi:10.2307/2321982. JSTOR 2321982. Archived from the original on February 3, 2007. Retrieved September 12, 2006.
• Hamming, Richard (August–September 1998). "Mathematics on a Distant Planet" (PDF). American Mathematical Monthly. 105 (7): 640–650. doi:10.2307/2589247. JSTOR 2589247.
• Ling, San; Xing, Chaoping (2004). Coding Theory: a First Course. Cambridge: Cambridge University Press. ISBN 978-0-521-82191-9.
• Morgan, Samuel P. (September 1998). "Richard Wesley Hamming (1915–1998)" (PDF). Notices of the AMS. 45 (8): 972–977. ISSN 0002-9920. Retrieved August 30, 2014.
• Pless, Vera (1982). Introduction to the Theory of Error-Correcting Codes. New York: Wiley. ISBN 978-0-471-08684-0.
• Shannon, Claude (July 1948). "A Mathematical Theory of Communication" (PDF). The Bell System Technical Journal. 27 (3): 379–423, 623–656. doi:10.1002/j.1538-7305.1948.tb01338.x. hdl:11858/00-001M-0000-002C-4314-2. Archived from the original (PDF) on March 28, 2015. Retrieved September 2, 2014.
• Tveito, Aslak; Bruaset, Are Magnus; Lysne, Olav (2009). Simula Research Laboratory: By Thinking Constantly about it. New York: Springer Science & Business Media. p. 59. ISBN 978-3-642-01156-6.
External links
Wikiquote has quotations related to Richard Hamming.
• O'Connor, John J.; Robertson, Edmund F., "Richard Hamming", MacTutor History of Mathematics Archive, University of St Andrews
• Richard Hamming at the Mathematics Genealogy Project
IEEE Richard W. Hamming Medal
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Wikipedia
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RAC drawing
In graph drawing, a RAC drawing of a graph is a drawing in which the vertices are represented as points, the edges are represented as straight line segments or polylines, at most two edges cross at any point, and when two edges cross they do so at right angles to each other. In the name of this drawing style, "RAC" stands for "right angle crossing".
The right-angle crossing style and the name "RAC drawing" for this style were both formulated by Didimo, Eades & Liotta (2009),[1] motivated by previous user studies showing that crossings with large angles are much less harmful to the readability of drawings than shallow crossings.[2] Even for planar graphs, allowing some right-angle crossings in a drawing of the graph can significantly improve measures of the drawing quality such as its area or angular resolution.[3]
Examples
The complete graph K5 has a RAC drawing with straight edges, but K6 does not. Every 6-vertex RAC drawing has at most 14 edges, but K6 has 15 edges, too many to have a RAC drawing.[1]
A complete bipartite graph Ka,b has a RAC drawing with straight edges if and only if either min(a,b) ≤ 2 or a + b ≤ 7. If min(a,b) ≤ 2, then the graph is a planar graph, and (by Fáry's theorem) every planar graph has a straight-line drawing with no crossings. Such a drawing is automatically a RAC drawing. The only two cases remaining are the graphs K3,3 and K3,4. A drawing of K3,4 is shown; K3,3 can be formed from it by deleting one vertex. Neither of the next two larger graphs, K4,4 and K3,5, has a RAC drawing.[4]
Edges and bends
If an n-vertex graph (n ≥ 4) has a RAC drawing with straight edges, it can have at most 4n − 10 edges. This is tight: there exist RAC-drawable graphs with exactly 4n − 10 edges.[1] For drawings with polyline edges, the bound on the number of edges in the graph depends on the number of bends that are allowed per edge. The graphs that have RAC drawings with one or two bends per edge have O(n) edges; more specifically, with one bend there are at most 5.5n edges[5] and with two bends there are at most 74.2n edges.[6] Every graph has a RAC drawing with three bends per edge.[1]
Relation to 1-planarity
A graph is 1-planar if it has a drawing with at most one crossing per edge. Intuitively, this restriction makes it easier to cause this crossing to be at right angles, and the 4n − 10 bound on the number of edges of straight-line RAC drawings is close to the bounds of 4n − 8 on the number of edges in a 1-planar graph, and of 4n − 9 on the number of edges in a straight-line 1-planar graph. Every RAC drawing with 4n − 10 edges is 1-planar.[7][8] Additionally, every outer-1-planar graph (that is, a graph drawn with one crossing per edge with all vertices on the outer face of the drawing) has a RAC drawing.[9] However, there exist 1-planar graphs with 4n − 10 edges that do not have RAC drawings.[7]
Computational complexity
It is NP-hard to determine whether a given graph has a RAC drawing with straight edges,[10] even if the input graph is 1-planar and the output RAC drawing must be 1-planar as well.[11] More specifically, RAC drawing is complete for the existential theory of the reals.[12] The RAC drawing problem remains NP-hard for upward drawing of directed acyclic graphs.[13] However, in the special case of outer-1-planar graphs, a RAC drawing can be constructed in linear time.[14]
References
1. Didimo, Walter; Eades, Peter; Liotta, Giuseppe (2009), "Drawing graphs with right angle crossings", Algorithms and Data Structures: 11th International Symposium, WADS 2009, Banff, Canada, August 21–23, 2009. Proceedings, Lecture Notes in Computer Science, vol. 5664, pp. 206–217, doi:10.1007/978-3-642-03367-4_19.
2. Huang, Weidong; Hong, Seok-Hee; Eades, Peter (2008), "Effects of crossing angles", IEEE Pacific Visualization Symposium (PacificVIS '08), pp. 41–46, doi:10.1109/PACIFICVIS.2008.4475457.
3. van Kreveld, Marc (2011), "The quality ratio of RAC drawings and planar drawings of planar graphs", Graph Drawing: 18th International Symposium, GD 2010, Konstanz, Germany, September 21–24, 2010, Revised Selected Papers, Lecture Notes in Computer Science, vol. 6502, pp. 371–376, doi:10.1007/978-3-642-18469-7_34.
4. Didimo, Walter; Eades, Peter; Liotta, Giuseppe (2010), "A characterization of complete bipartite RAC graphs", Information Processing Letters, 110 (16): 687–691, doi:10.1016/j.ipl.2010.05.023, MR 2676805.
5. Angelini, Patrizio; Bekos, Michael; Förster, Henry; Kaufmann, Michael (2018), On RAC Drawings of Graphs with one Bend per Edge, arXiv:1808.10470
6. Arikushi, Karin; Fulek, Radoslav; Keszegh, Balázs; Morić, Filip; Tóth, Csaba D. (2012), "Graphs that admit right angle crossing drawings", Computational Geometry Theory & Applications, 45 (4): 169–177, arXiv:1001.3117, doi:10.1016/j.comgeo.2011.11.008, MR 2876688.
7. Eades, Peter; Liotta, Giuseppe (2013), "Right angle crossing graphs and 1-planarity", Discrete Applied Mathematics, 161 (7–8): 961–969, doi:10.1016/j.dam.2012.11.019, MR 3030582.
8. Ackerman, Eyal (2014), "A note on 1-planar graphs", Discrete Applied Mathematics, 175: 104–108, doi:10.1016/j.dam.2014.05.025, MR 3223912.
9. Dehkordi, Hooman Reisi; Eades, Peter (2012), "Every outer-1-plane graph has a right angle crossing drawing", International Journal of Computational Geometry & Applications, 22 (6): 543–557, doi:10.1142/S021819591250015X, MR 3042921.
10. Argyriou, Evmorfia N.; Bekos, Michael A.; Symvonis, Antonios (2011), "The straight-line RAC drawing problem is NP-hard", SOFSEM 2011: 37th Conference on Current Trends in Theory and Practice of Computer Science, Nový Smokovec, Slovakia, January 22-28, 2011, Proceedings, Lecture Notes in Computer Science, vol. 6543, pp. 74–85, arXiv:1009.5227, Bibcode:2011LNCS.6543...74A, doi:10.1007/978-3-642-18381-2_6
11. Bekos, Michael A.; Didimo, Walter; Liotta, Giuseppe; Mehrabi, Saeed; Montecchiani, Fabrizio (2017), "On RAC drawings of 1-planar graphs", Theoretical Computer Science, 689: 48–57, arXiv:1608.08418, doi:10.1016/j.tcs.2017.05.039
12. Schaefer, Marcus (2021), "RAC-drawability is $\exists \mathbb {R} $-complete", Proceedings of the 29th International Symposium on Graph Drawing and Network Visualization (GD 2021), arXiv:2107.11663
13. Angelini, Patrizio; Cittadini, Luca; Di Battista, Giuseppe; Didimo, Walter; Frati, Fabrizio; Kaufmann, Michael; Symvonis, Antonios (2010), "On the perspectives opened by right angle crossing drawings", Graph Drawing: 17th International Symposium, GD 2009, Chicago, IL, USA, September 22–25, 2009, Revised Papers, Lecture Notes in Computer Science, vol. 5849, pp. 21–32, doi:10.1007/978-3-642-11805-0_5.
14. Auer, Christopher; Bachmaier, Christian; Brandenburg, Franz J.; Hanauer, Kathrin; Gleißner, Andreas; Neuwirth, Daniel; Reislhuber, Josef (2013), "Recognizing Outer 1-Planar Graphs in Linear Time", Graph Drawing LNCS, 8284: 107–118, doi:10.1007/978-3-319-03841-4
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Wikipedia
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Rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction ${\tfrac {p}{q}}$ of two integers, a numerator p and a non-zero denominator q.[1] For example, ${\tfrac {-3}{7}}$ is a rational number, as is every integer (e.g., 5 = 5/1). The set of all rational numbers, also referred to as "the rationals",[2] the field of rationals[3] or the field of rational numbers is usually denoted by boldface Q,[4] or blackboard bold $\mathbb {Q} .$[5]
A rational number is a real number. The real numbers that are rational are those whose decimal expansion either terminates after a finite number of digits (example: 3/4 = 0.75), or eventually begins to repeat the same finite sequence of digits over and over (example: 9/44 = 0.20454545...).[6] This statement is true not only in base 10, but also in every other integer base, such as the binary and hexadecimal ones (see Repeating decimal § Extension to other bases).
A real number that is not rational is called irrational.[7] Irrational numbers include the square root of 2 (${\sqrt {2}}$), π, e, and the golden ratio (φ). Since the set of rational numbers is countable, and the set of real numbers is uncountable, almost all real numbers are irrational.[1]
Rational numbers can be formally defined as equivalence classes of pairs of integers (p, q) with q ≠ 0, using the equivalence relation defined as follows:
$(p_{1},q_{1})\sim (p_{2},q_{2})\iff p_{1}q_{2}=p_{2}q_{1}.$
The fraction ${\tfrac {p}{q}}$ then denotes the equivalence class of (p, q).[8]
Rational numbers together with addition and multiplication form a field which contains the integers, and is contained in any field containing the integers. In other words, the field of rational numbers is a prime field, and a field has characteristic zero if and only if it contains the rational numbers as a subfield. Finite extensions of $\mathbb {Q} $ are called algebraic number fields, and the algebraic closure of $\mathbb {Q} $ is the field of algebraic numbers.[9]
In mathematical analysis, the rational numbers form a dense subset of the real numbers. The real numbers can be constructed from the rational numbers by completion, using Cauchy sequences, Dedekind cuts, or infinite decimals (see Construction of the real numbers).
Terminology
The term rational in reference to the set $\mathbb {Q} $ refers to the fact that a rational number represents a ratio of two integers. In mathematics, "rational" is often used as a noun abbreviating "rational number". The adjective rational sometimes means that the coefficients are rational numbers. For example, a rational point is a point with rational coordinates (i.e., a point whose coordinates are rational numbers); a rational matrix is a matrix of rational numbers; a rational polynomial may be a polynomial with rational coefficients, although the term "polynomial over the rationals" is generally preferred, to avoid confusion between "rational expression" and "rational function" (a polynomial is a rational expression and defines a rational function, even if its coefficients are not rational numbers). However, a rational curve is not a curve defined over the rationals, but a curve which can be parameterized by rational functions.
Etymology
Although nowadays rational numbers are defined in terms of ratios, the term rational is not a derivation of ratio. On the contrary, it is ratio that is derived from rational: the first use of ratio with its modern meaning was attested in English about 1660,[10] while the use of rational for qualifying numbers appeared almost a century earlier, in 1570.[11] This meaning of rational came from the mathematical meaning of irrational, which was first used in 1551, and it was used in "translations of Euclid (following his peculiar use of ἄλογος)".[12][13]
This unusual history originated in the fact that ancient Greeks "avoided heresy by forbidding themselves from thinking of those [irrational] lengths as numbers".[14] So such lengths were irrational, in the sense of illogical, that is "not to be spoken about" (ἄλογος in Greek).[15]
This etymology is similar to that of imaginary numbers and real numbers.
Arithmetic
See also: Fraction (mathematics) § Arithmetic with fractions
Irreducible fraction
Every rational number may be expressed in a unique way as an irreducible fraction ${\tfrac {a}{b}},$ where a and b are coprime integers and b > 0. This is often called the canonical form of the rational number.
Starting from a rational number ${\tfrac {a}{b}},$ its canonical form may be obtained by dividing a and b by their greatest common divisor, and, if b < 0, changing the sign of the resulting numerator and denominator.
Embedding of integers
Any integer n can be expressed as the rational number ${\tfrac {n}{1}},$ which is its canonical form as a rational number.
Equality
${\frac {a}{b}}={\frac {c}{d}}$ if and only if $ad=bc$
If both fractions are in canonical form, then:
${\frac {a}{b}}={\frac {c}{d}}$ if and only if $a=c$ and $b=d$[8]
Ordering
If both denominators are positive (particularly if both fractions are in canonical form):
${\frac {a}{b}}<{\frac {c}{d}}$ if and only if $ad<bc.$
On the other hand, if either denominator is negative, then each fraction with a negative denominator must first be converted into an equivalent form with a positive denominator—by changing the signs of both its numerator and denominator.[8]
Addition
Two fractions are added as follows:
${\frac {a}{b}}+{\frac {c}{d}}={\frac {ad+bc}{bd}}.$
If both fractions are in canonical form, the result is in canonical form if and only if b, d are coprime integers.[8][16]
Subtraction
${\frac {a}{b}}-{\frac {c}{d}}={\frac {ad-bc}{bd}}.$
If both fractions are in canonical form, the result is in canonical form if and only if b, d are coprime integers.[16]
Multiplication
The rule for multiplication is:
${\frac {a}{b}}\cdot {\frac {c}{d}}={\frac {ac}{bd}}.$
where the result may be a reducible fraction—even if both original fractions are in canonical form.[8][16]
Inverse
Every rational number ${\tfrac {a}{b}}$ has an additive inverse, often called its opposite,
$-\left({\frac {a}{b}}\right)={\frac {-a}{b}}.$
If ${\tfrac {a}{b}}$ is in canonical form, the same is true for its opposite.
A nonzero rational number ${\tfrac {a}{b}}$ has a multiplicative inverse, also called its reciprocal,
$\left({\frac {a}{b}}\right)^{-1}={\frac {b}{a}}.$
If ${\tfrac {a}{b}}$ is in canonical form, then the canonical form of its reciprocal is either ${\tfrac {b}{a}}$ or ${\tfrac {-b}{-a}},$ depending on the sign of a.
Division
If b, c, d are nonzero, the division rule is
${\frac {\,{\dfrac {a}{b}}\,}{\dfrac {c}{d}}}={\frac {ad}{bc}}.$
Thus, dividing ${\tfrac {a}{b}}$ by ${\tfrac {c}{d}}$ is equivalent to multiplying ${\tfrac {a}{b}}$ by the reciprocal of ${\tfrac {c}{d}}:$[16]
${\frac {ad}{bc}}={\frac {a}{b}}\cdot {\frac {d}{c}}.$
Exponentiation to integer power
If n is a non-negative integer, then
$\left({\frac {a}{b}}\right)^{n}={\frac {a^{n}}{b^{n}}}.$
The result is in canonical form if the same is true for ${\tfrac {a}{b}}.$ In particular,
$\left({\frac {a}{b}}\right)^{0}=1.$
If a ≠ 0, then
$\left({\frac {a}{b}}\right)^{-n}={\frac {b^{n}}{a^{n}}}.$
If ${\tfrac {a}{b}}$ is in canonical form, the canonical form of the result is ${\tfrac {b^{n}}{a^{n}}}$ if a > 0 or n is even. Otherwise, the canonical form of the result is ${\tfrac {-b^{n}}{-a^{n}}}.$
Continued fraction representation
Main article: Continued fraction
A finite continued fraction is an expression such as
$a_{0}+{\cfrac {1}{a_{1}+{\cfrac {1}{a_{2}+{\cfrac {1}{\ddots +{\cfrac {1}{a_{n}}}}}}}}},$
where an are integers. Every rational number ${\tfrac {a}{b}}$ can be represented as a finite continued fraction, whose coefficients an can be determined by applying the Euclidean algorithm to (a, b).
Other representations
• common fraction: ${\tfrac {8}{3}}$
• mixed numeral: $2{\tfrac {2}{3}}$
• repeating decimal using a vinculum: $2.{\overline {6}}$
• repeating decimal using parentheses: $2.(6)$
• continued fraction using traditional typography: $2+{\tfrac {1}{1+{\tfrac {1}{2}}}}$
• continued fraction in abbreviated notation: $[2;1,2]$
• Egyptian fraction: $2+{\tfrac {1}{2}}+{\tfrac {1}{6}}$
• prime power decomposition: $2^{3}\times 3^{-1}$
• quote notation: $3'6$
are different ways to represent the same rational value.
Formal construction
The rational numbers may be built as equivalence classes of ordered pairs of integers.[8][16]
More precisely, let $(\mathbb {Z} \times (\mathbb {Z} \setminus \{0\}))$ be the set of the pairs (m, n) of integers such n ≠ 0. An equivalence relation is defined on this set by
$(m_{1},n_{1})\sim (m_{2},n_{2})\iff m_{1}n_{2}=m_{2}n_{1}.$[8][16]
Addition and multiplication can be defined by the following rules:
$(m_{1},n_{1})+(m_{2},n_{2})\equiv (m_{1}n_{2}+n_{1}m_{2},n_{1}n_{2}),$
$(m_{1},n_{1})\times (m_{2},n_{2})\equiv (m_{1}m_{2},n_{1}n_{2}).$[8]
This equivalence relation is a congruence relation, which means that it is compatible with the addition and multiplication defined above; the set of rational numbers $\mathbb {Q} $ is the defined as the quotient set by this equivalence relation, $(\mathbb {Z} \times (\mathbb {Z} \backslash \{0\}))/\sim ,$ equipped with the addition and the multiplication induced by the above operations. (This construction can be carried out with any integral domain and produces its field of fractions.)[8]
The equivalence class of a pair (m, n) is denoted ${\tfrac {m}{n}}.$ Two pairs (m1, n1) and (m2, n2) belong to the same equivalence class (that is are equivalent) if and only if
$m_{1}n_{2}=m_{2}n_{1}.$
This means that
${\frac {m_{1}}{n_{1}}}={\frac {m_{2}}{n_{2}}}$
if and only if[8][16]
$m_{1}n_{2}=m_{2}n_{1}.$
Every equivalence class ${\tfrac {m}{n}}$ may be represented by infinitely many pairs, since
$\cdots ={\frac {-2m}{-2n}}={\frac {-m}{-n}}={\frac {m}{n}}={\frac {2m}{2n}}=\cdots .$
Each equivalence class contains a unique canonical representative element. The canonical representative is the unique pair (m, n) in the equivalence class such that m and n are coprime, and n > 0. It is called the representation in lowest terms of the rational number.
The integers may be considered to be rational numbers identifying the integer n with the rational number ${\tfrac {n}{1}}.$
A total order may be defined on the rational numbers, that extends the natural order of the integers. One has
${\frac {m_{1}}{n_{1}}}\leq {\frac {m_{2}}{n_{2}}}$
If
${\begin{aligned}&(n_{1}n_{2}>0\quad {\text{and}}\quad m_{1}n_{2}\leq n_{1}m_{2})\\&\qquad {\text{or}}\\&(n_{1}n_{2}<0\quad {\text{and}}\quad m_{1}n_{2}\geq n_{1}m_{2}).\end{aligned}}$
Properties
The set $\mathbb {Q} $ of all rational numbers, together with the addition and multiplication operations shown above, forms a field.[8]
$\mathbb {Q} $ has no field automorphism other than the identity. (A field automorphism must fix 0 and 1; as it must fix the sum and the difference of two fixed elements, it must fix every integer; as it must fix the quotient of two fixed elements, it must fix every rational number, and is thus the identity.)
$\mathbb {Q} $ is a prime field, which is a field that has no subfield other than itself.[17] The rationals are the smallest field with characteristic zero. Every field of characteristic zero contains a unique subfield isomorphic to $\mathbb {Q} .$
With the order defined above, $\mathbb {Q} $ is an ordered field[16] that has no subfield other than itself, and is the smallest ordered field, in the sense that every ordered field contains a unique subfield isomorphic to $\mathbb {Q} .$
$\mathbb {Q} $ is the field of fractions of the integers $\mathbb {Z} .$[18] The algebraic closure of $\mathbb {Q} ,$ i.e. the field of roots of rational polynomials, is the field of algebraic numbers.
The rationals are a densely ordered set: between any two rationals, there sits another one, and, therefore, infinitely many other ones.[8] For example, for any two fractions such that
${\frac {a}{b}}<{\frac {c}{d}}$
(where $b,d$ are positive), we have
${\frac {a}{b}}<{\frac {a+c}{b+d}}<{\frac {c}{d}}.$
Any totally ordered set which is countable, dense (in the above sense), and has no least or greatest element is order isomorphic to the rational numbers.[19]
Countability
The set of all rational numbers is countable, as is illustrated in the figure to the right. As a rational number can be expressed as a ratio of two integers, it is possible to assign two integers to any point on a square lattice as in a Cartesian coordinate system, such that any grid point corresponds to a rational number. This method, however, exhibits a form of redundancy, as several different grid points will correspond to the same rational number; these are highlighted in red on the provided graphic. An obvious example can be seen in the line going diagonally towards the bottom right; such ratios will always equal 1, as any non-zero number divided by itself will always equal one.
It is possible to generate all of the rational numbers without such redundancies: examples include the Calkin–Wilf tree and Stern–Brocot tree.
As the set of all rational numbers is countable, and the set of all real numbers (as well as the set of irrational numbers) is uncountable, the set of rational numbers is a null set, that is, almost all real numbers are irrational, in the sense of Lebesgue measure.
Real numbers and topological properties
The rationals are a dense subset of the real numbers; every real number has rational numbers arbitrarily close to it.[8] A related property is that rational numbers are the only numbers with finite expansions as regular continued fractions.[20]
In the usual topology of the real numbers, the rationals are neither an open set nor a closed set.[21]
By virtue of their order, the rationals carry an order topology. The rational numbers, as a subspace of the real numbers, also carry a subspace topology. The rational numbers form a metric space by using the absolute difference metric $d(x,y)=|x-y|,$ and this yields a third topology on $\mathbb {Q} .$ All three topologies coincide and turn the rationals into a topological field. The rational numbers are an important example of a space which is not locally compact. The rationals are characterized topologically as the unique countable metrizable space without isolated points. The space is also totally disconnected. The rational numbers do not form a complete metric space, and the real numbers are the completion of $\mathbb {Q} $ under the metric $d(x,y)=|x-y|$ above.[16]
p-adic numbers
Main article: p-adic number
In addition to the absolute value metric mentioned above, there are other metrics which turn $\mathbb {Q} $ into a topological field:
Let p be a prime number and for any non-zero integer a, let $|a|_{p}=p_{-n},$ where pn is the highest power of p dividing a.
In addition set $|0|_{p}=0.$ For any rational number ${\frac {a}{b}},$ we set
$\left|{\frac {a}{b}}\right|_{p}={\frac {|a|_{p}}{|b|_{p}}}.$
Then
$d_{p}(x,y)=|x-y|_{p}$
defines a metric on $\mathbb {Q} .$[22]
The metric space $(\mathbb {Q} ,d_{p})$ is not complete, and its completion is the p-adic number field $\mathbb {Q} _{p}.$ Ostrowski's theorem states that any non-trivial absolute value on the rational numbers $\mathbb {Q} $ is equivalent to either the usual real absolute value or a p-adic absolute value.
See also
• Dyadic rational
• Floating point
• Ford circles
• Gaussian rational
• Naive height—height of a rational number in lowest term
• Niven's theorem
• Rational data type
• Divine Proportions: Rational Trigonometry to Universal Geometry
Number systems
Complex $:\;\mathbb {C} $ :\;\mathbb {C} }
Real $:\;\mathbb {R} $ :\;\mathbb {R} }
Rational $:\;\mathbb {Q} $ :\;\mathbb {Q} }
Integer $:\;\mathbb {Z} $ :\;\mathbb {Z} }
Natural $:\;\mathbb {N} $ :\;\mathbb {N} }
Zero: 0
One: 1
Prime numbers
Composite numbers
Negative integers
Fraction
Finite decimal
Dyadic (finite binary)
Repeating decimal
Irrational
Algebraic irrational
Transcendental
Imaginary
References
1. Rosen, Kenneth (2007). Discrete Mathematics and its Applications (6th ed.). New York, NY: McGraw-Hill. pp. 105, 158–160. ISBN 978-0-07-288008-3.
2. Lass, Harry (2009). Elements of Pure and Applied Mathematics (illustrated ed.). Courier Corporation. p. 382. ISBN 978-0-486-47186-0. Extract of page 382
3. Robinson, Julia (1996). The Collected Works of Julia Robinson. American Mathematical Soc. p. 104. ISBN 978-0-8218-0575-6. Extract of page 104
4. It was thus denoted in 1895 by Giuseppe Peano after quoziente, Italian for "quotient",
5. It first appeared in Bourbaki's Algèbre.
6. "Rational number". Encyclopedia Britannica. Retrieved 2020-08-11.
7. Weisstein, Eric W. "Rational Number". mathworld.wolfram.com. Retrieved 2020-08-11.
8. Biggs, Norman L. (2002). Discrete Mathematics. India: Oxford University Press. pp. 75–78. ISBN 978-0-19-871369-2.
9. Gilbert, Jimmie; Linda, Gilbert (2005). Elements of Modern Algebra (6th ed.). Belmont, CA: Thomson Brooks/Cole. pp. 243–244. ISBN 0-534-40264-X.
10. Oxford English Dictionary (2nd ed.). Oxford University Press. 1989. Entry ratio, n., sense 2.a.
11. Oxford English Dictionary (2nd ed.). Oxford University Press. 1989. Entry rational, a. (adv.) and n.1, sense 5.a.
12. Oxford English Dictionary (2nd ed.). Oxford University Press. 1989. Entry irrational, a. and n., sense 3.
13. Shor, Peter (2017-05-09). "Does rational come from ratio or ratio come from rational". Stack Exchange. Retrieved 2021-03-19.
14. Coolman, Robert (2016-01-29). "How a Mathematical Superstition Stultified Algebra for Over a Thousand Years". Retrieved 2021-03-20.
15. Kramer, Edna (1983). The Nature and Growth of Modern Mathematics. Princeton University Press. p. 28.
16. "Fraction - Encyclopedia of Mathematics". encyclopediaofmath.org. Retrieved 2021-08-17.
17. Sūgakkai, Nihon (1993). Encyclopedic Dictionary of Mathematics, Volume 1. London, England: MIT Press. p. 578. ISBN 0-2625-9020-4.
18. Bourbaki, N. (2003). Algebra II: Chapters 4 - 7. Springer Science & Business Media. p. A.VII.5.
19. Giese, Martin; Schönegge, Arno (December 1995). Any two countable densely ordered sets without endpoints are isomorphic - a formal proof with KIV (PDF) (Technical report). Retrieved 17 August 2021.
20. Anthony Vazzana; David Garth (2015). Introduction to Number Theory (2nd, revised ed.). CRC Press. p. 1. ISBN 978-1-4987-1752-6. Extract of page 1
21. Richard A. Holmgren (2012). A First Course in Discrete Dynamical Systems (2nd, illustrated ed.). Springer Science & Business Media. p. 26. ISBN 978-1-4419-8732-7. Extract of page 26
22. Weisstein, Eric W. "p-adic Number". mathworld.wolfram.com. Retrieved 2021-08-17.
External links
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• "Rational number", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• "Rational Number" From MathWorld – A Wolfram Web Resource
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Radial basis function network
In the field of mathematical modeling, a radial basis function network is an artificial neural network that uses radial basis functions as activation functions. The output of the network is a linear combination of radial basis functions of the inputs and neuron parameters. Radial basis function networks have many uses, including function approximation, time series prediction, classification, and system control. They were first formulated in a 1988 paper by Broomhead and Lowe, both researchers at the Royal Signals and Radar Establishment.[1][2][3]
Network architecture
Radial basis function (RBF) networks typically have three layers: an input layer, a hidden layer with a non-linear RBF activation function and a linear output layer. The input can be modeled as a vector of real numbers $\mathbf {x} \in \mathbb {R} ^{n}$. The output of the network is then a scalar function of the input vector, $\varphi :\mathbb {R} ^{n}\to \mathbb {R} $ :\mathbb {R} ^{n}\to \mathbb {R} } , and is given by
$\varphi (\mathbf {x} )=\sum _{i=1}^{N}a_{i}\rho (||\mathbf {x} -\mathbf {c} _{i}||)$
where $N$ is the number of neurons in the hidden layer, $\mathbf {c} _{i}$ is the center vector for neuron $i$, and $a_{i}$ is the weight of neuron $i$ in the linear output neuron. Functions that depend only on the distance from a center vector are radially symmetric about that vector, hence the name radial basis function. In the basic form, all inputs are connected to each hidden neuron. The norm is typically taken to be the Euclidean distance (although the Mahalanobis distance appears to perform better with pattern recognition[4][5]) and the radial basis function is commonly taken to be Gaussian
$\rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}=\exp \left[-\beta _{i}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert ^{2}\right]$.
The Gaussian basis functions are local to the center vector in the sense that
$\lim _{||x||\to \infty }\rho (\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert )=0$
i.e. changing parameters of one neuron has only a small effect for input values that are far away from the center of that neuron.
Given certain mild conditions on the shape of the activation function, RBF networks are universal approximators on a compact subset of $\mathbb {R} ^{n}$.[6] This means that an RBF network with enough hidden neurons can approximate any continuous function on a closed, bounded set with arbitrary precision.
The parameters $a_{i}$, $\mathbf {c} _{i}$, and $\beta _{i}$ are determined in a manner that optimizes the fit between $\varphi $ and the data.
Normalized
Two normalized radial basis functions in one input dimension (sigmoids). The basis function centers are located at $c_{1}=0.75$ and $c_{2}=3.25$.
Three normalized radial basis functions in one input dimension. The additional basis function has center at $c_{3}=2.75$
Four normalized radial basis functions in one input dimension. The fourth basis function has center at $c_{4}=0$. Note that the first basis function (dark blue) has become localized.
Normalized architecture
In addition to the above unnormalized architecture, RBF networks can be normalized. In this case the mapping is
$\varphi (\mathbf {x} )\ {\stackrel {\mathrm {def} }{=}}\ {\frac {\sum _{i=1}^{N}a_{i}\rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}}{\sum _{i=1}^{N}\rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}}}=\sum _{i=1}^{N}a_{i}u{\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}$
where
$u{\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}\ {\stackrel {\mathrm {def} }{=}}\ {\frac {\rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}}{\sum _{j=1}^{N}\rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{j}\right\Vert {\big )}}}$
is known as a normalized radial basis function.
Theoretical motivation for normalization
There is theoretical justification for this architecture in the case of stochastic data flow. Assume a stochastic kernel approximation for the joint probability density
$P\left(\mathbf {x} \land y\right)={1 \over N}\sum _{i=1}^{N}\,\rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}\,\sigma {\big (}\left\vert y-e_{i}\right\vert {\big )}$
where the weights $\mathbf {c} _{i}$ and $e_{i}$ are exemplars from the data and we require the kernels to be normalized
$\int \rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}\,d^{n}\mathbf {x} =1$
and
$\int \sigma {\big (}\left\vert y-e_{i}\right\vert {\big )}\,dy=1$.
The probability densities in the input and output spaces are
$P\left(\mathbf {x} \right)=\int P\left(\mathbf {x} \land y\right)\,dy={1 \over N}\sum _{i=1}^{N}\,\rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}$
and
The expectation of y given an input $\mathbf {x} $ is
$\varphi \left(\mathbf {x} \right)\ {\stackrel {\mathrm {def} }{=}}\ E\left(y\mid \mathbf {x} \right)=\int y\,P\left(y\mid \mathbf {x} \right)dy$
where
$P\left(y\mid \mathbf {x} \right)$
is the conditional probability of y given $\mathbf {x} $. The conditional probability is related to the joint probability through Bayes theorem
$P\left(y\mid \mathbf {x} \right)={\frac {P\left(\mathbf {x} \land y\right)}{P\left(\mathbf {x} \right)}}$
which yields
$\varphi \left(\mathbf {x} \right)=\int y\,{\frac {P\left(\mathbf {x} \land y\right)}{P\left(\mathbf {x} \right)}}\,dy$.
This becomes
$\varphi \left(\mathbf {x} \right)={\frac {\sum _{i=1}^{N}e_{i}\rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}}{\sum _{i=1}^{N}\rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}}}=\sum _{i=1}^{N}e_{i}u{\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}$
when the integrations are performed.
Local linear models
It is sometimes convenient to expand the architecture to include local linear models. In that case the architectures become, to first order,
$\varphi \left(\mathbf {x} \right)=\sum _{i=1}^{N}\left(a_{i}+\mathbf {b} _{i}\cdot \left(\mathbf {x} -\mathbf {c} _{i}\right)\right)\rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}$
and
$\varphi \left(\mathbf {x} \right)=\sum _{i=1}^{N}\left(a_{i}+\mathbf {b} _{i}\cdot \left(\mathbf {x} -\mathbf {c} _{i}\right)\right)u{\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}$
in the unnormalized and normalized cases, respectively. Here $\mathbf {b} _{i}$ are weights to be determined. Higher order linear terms are also possible.
This result can be written
$\varphi \left(\mathbf {x} \right)=\sum _{i=1}^{2N}\sum _{j=1}^{n}e_{ij}v_{ij}{\big (}\mathbf {x} -\mathbf {c} _{i}{\big )}$
where
$e_{ij}={\begin{cases}a_{i},&{\mbox{if }}i\in [1,N]\\b_{ij},&{\mbox{if }}i\in [N+1,2N]\end{cases}}$
and
$v_{ij}{\big (}\mathbf {x} -\mathbf {c} _{i}{\big )}\ {\stackrel {\mathrm {def} }{=}}\ {\begin{cases}\delta _{ij}\rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )},&{\mbox{if }}i\in [1,N]\\\left(x_{ij}-c_{ij}\right)\rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )},&{\mbox{if }}i\in [N+1,2N]\end{cases}}$
in the unnormalized case and
$v_{ij}{\big (}\mathbf {x} -\mathbf {c} _{i}{\big )}\ {\stackrel {\mathrm {def} }{=}}\ {\begin{cases}\delta _{ij}u{\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )},&{\mbox{if }}i\in [1,N]\\\left(x_{ij}-c_{ij}\right)u{\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )},&{\mbox{if }}i\in [N+1,2N]\end{cases}}$
in the normalized case.
Here $\delta _{ij}$ is a Kronecker delta function defined as
$\delta _{ij}={\begin{cases}1,&{\mbox{if }}i=j\\0,&{\mbox{if }}i\neq j\end{cases}}$.
Training
RBF networks are typically trained from pairs of input and target values $\mathbf {x} (t),y(t)$, $t=1,\dots ,T$ by a two-step algorithm.
In the first step, the center vectors $\mathbf {c} _{i}$ of the RBF functions in the hidden layer are chosen. This step can be performed in several ways; centers can be randomly sampled from some set of examples, or they can be determined using k-means clustering. Note that this step is unsupervised.
The second step simply fits a linear model with coefficients $w_{i}$ to the hidden layer's outputs with respect to some objective function. A common objective function, at least for regression/function estimation, is the least squares function:
$K(\mathbf {w} )\ {\stackrel {\mathrm {def} }{=}}\ \sum _{t=1}^{T}K_{t}(\mathbf {w} )$
where
$K_{t}(\mathbf {w} )\ {\stackrel {\mathrm {def} }{=}}\ {\big [}y(t)-\varphi {\big (}\mathbf {x} (t),\mathbf {w} {\big )}{\big ]}^{2}$.
We have explicitly included the dependence on the weights. Minimization of the least squares objective function by optimal choice of weights optimizes accuracy of fit.
There are occasions in which multiple objectives, such as smoothness as well as accuracy, must be optimized. In that case it is useful to optimize a regularized objective function such as
$H(\mathbf {w} )\ {\stackrel {\mathrm {def} }{=}}\ K(\mathbf {w} )+\lambda S(\mathbf {w} )\ {\stackrel {\mathrm {def} }{=}}\ \sum _{t=1}^{T}H_{t}(\mathbf {w} )$
where
$S(\mathbf {w} )\ {\stackrel {\mathrm {def} }{=}}\ \sum _{t=1}^{T}S_{t}(\mathbf {w} )$
and
$H_{t}(\mathbf {w} )\ {\stackrel {\mathrm {def} }{=}}\ K_{t}(\mathbf {w} )+\lambda S_{t}(\mathbf {w} )$
where optimization of S maximizes smoothness and $\lambda $ is known as a regularization parameter.
A third optional backpropagation step can be performed to fine-tune all of the RBF net's parameters.[3]
Interpolation
RBF networks can be used to interpolate a function $y:\mathbb {R} ^{n}\to \mathbb {R} $ when the values of that function are known on finite number of points: $y(\mathbf {x} _{i})=b_{i},i=1,\ldots ,N$. Taking the known points $\mathbf {x} _{i}$ to be the centers of the radial basis functions and evaluating the values of the basis functions at the same points $g_{ij}=\rho (||\mathbf {x} _{j}-\mathbf {x} _{i}||)$ the weights can be solved from the equation
$\left[{\begin{matrix}g_{11}&g_{12}&\cdots &g_{1N}\\g_{21}&g_{22}&\cdots &g_{2N}\\\vdots &&\ddots &\vdots \\g_{N1}&g_{N2}&\cdots &g_{NN}\end{matrix}}\right]\left[{\begin{matrix}w_{1}\\w_{2}\\\vdots \\w_{N}\end{matrix}}\right]=\left[{\begin{matrix}b_{1}\\b_{2}\\\vdots \\b_{N}\end{matrix}}\right]$
It can be shown that the interpolation matrix in the above equation is non-singular, if the points $\mathbf {x} _{i}$ are distinct, and thus the weights $w$ can be solved by simple linear algebra:
$\mathbf {w} =\mathbf {G} ^{-1}\mathbf {b} $
where $G=(g_{ij})$.
Function approximation
If the purpose is not to perform strict interpolation but instead more general function approximation or classification the optimization is somewhat more complex because there is no obvious choice for the centers. The training is typically done in two phases first fixing the width and centers and then the weights. This can be justified by considering the different nature of the non-linear hidden neurons versus the linear output neuron.
Training the basis function centers
Basis function centers can be randomly sampled among the input instances or obtained by Orthogonal Least Square Learning Algorithm or found by clustering the samples and choosing the cluster means as the centers.
The RBF widths are usually all fixed to same value which is proportional to the maximum distance between the chosen centers.
Pseudoinverse solution for the linear weights
After the centers $c_{i}$ have been fixed, the weights that minimize the error at the output can be computed with a linear pseudoinverse solution:
$\mathbf {w} =\mathbf {G} ^{+}\mathbf {b} $,
where the entries of G are the values of the radial basis functions evaluated at the points $x_{i}$: $g_{ji}=\rho (||x_{j}-c_{i}||)$.
The existence of this linear solution means that unlike multi-layer perceptron (MLP) networks, RBF networks have an explicit minimizer (when the centers are fixed).
Gradient descent training of the linear weights
Another possible training algorithm is gradient descent. In gradient descent training, the weights are adjusted at each time step by moving them in a direction opposite from the gradient of the objective function (thus allowing the minimum of the objective function to be found),
$\mathbf {w} (t+1)=\mathbf {w} (t)-\nu {\frac {d}{d\mathbf {w} }}H_{t}(\mathbf {w} )$
where $\nu $ is a "learning parameter."
For the case of training the linear weights, $a_{i}$, the algorithm becomes
$a_{i}(t+1)=a_{i}(t)+\nu {\big [}y(t)-\varphi {\big (}\mathbf {x} (t),\mathbf {w} {\big )}{\big ]}\rho {\big (}\left\Vert \mathbf {x} (t)-\mathbf {c} _{i}\right\Vert {\big )}$
in the unnormalized case and
$a_{i}(t+1)=a_{i}(t)+\nu {\big [}y(t)-\varphi {\big (}\mathbf {x} (t),\mathbf {w} {\big )}{\big ]}u{\big (}\left\Vert \mathbf {x} (t)-\mathbf {c} _{i}\right\Vert {\big )}$
in the normalized case.
For local-linear-architectures gradient-descent training is
$e_{ij}(t+1)=e_{ij}(t)+\nu {\big [}y(t)-\varphi {\big (}\mathbf {x} (t),\mathbf {w} {\big )}{\big ]}v_{ij}{\big (}\mathbf {x} (t)-\mathbf {c} _{i}{\big )}$
Projection operator training of the linear weights
For the case of training the linear weights, $a_{i}$ and $e_{ij}$, the algorithm becomes
$a_{i}(t+1)=a_{i}(t)+\nu {\big [}y(t)-\varphi {\big (}\mathbf {x} (t),\mathbf {w} {\big )}{\big ]}{\frac {\rho {\big (}\left\Vert \mathbf {x} (t)-\mathbf {c} _{i}\right\Vert {\big )}}{\sum _{i=1}^{N}\rho ^{2}{\big (}\left\Vert \mathbf {x} (t)-\mathbf {c} _{i}\right\Vert {\big )}}}$
in the unnormalized case and
$a_{i}(t+1)=a_{i}(t)+\nu {\big [}y(t)-\varphi {\big (}\mathbf {x} (t),\mathbf {w} {\big )}{\big ]}{\frac {u{\big (}\left\Vert \mathbf {x} (t)-\mathbf {c} _{i}\right\Vert {\big )}}{\sum _{i=1}^{N}u^{2}{\big (}\left\Vert \mathbf {x} (t)-\mathbf {c} _{i}\right\Vert {\big )}}}$
in the normalized case and
$e_{ij}(t+1)=e_{ij}(t)+\nu {\big [}y(t)-\varphi {\big (}\mathbf {x} (t),\mathbf {w} {\big )}{\big ]}{\frac {v_{ij}{\big (}\mathbf {x} (t)-\mathbf {c} _{i}{\big )}}{\sum _{i=1}^{N}\sum _{j=1}^{n}v_{ij}^{2}{\big (}\mathbf {x} (t)-\mathbf {c} _{i}{\big )}}}$
in the local-linear case.
For one basis function, projection operator training reduces to Newton's method.
Examples
Logistic map
The basic properties of radial basis functions can be illustrated with a simple mathematical map, the logistic map, which maps the unit interval onto itself. It can be used to generate a convenient prototype data stream. The logistic map can be used to explore function approximation, time series prediction, and control theory. The map originated from the field of population dynamics and became the prototype for chaotic time series. The map, in the fully chaotic regime, is given by
$x(t+1)\ {\stackrel {\mathrm {def} }{=}}\ f\left[x(t)\right]=4x(t)\left[1-x(t)\right]$
where t is a time index. The value of x at time t+1 is a parabolic function of x at time t. This equation represents the underlying geometry of the chaotic time series generated by the logistic map.
Generation of the time series from this equation is the forward problem. The examples here illustrate the inverse problem; identification of the underlying dynamics, or fundamental equation, of the logistic map from exemplars of the time series. The goal is to find an estimate
$x(t+1)=f\left[x(t)\right]\approx \varphi (t)=\varphi \left[x(t)\right]$
for f.
Unnormalized radial basis functions
The architecture is
$\varphi (\mathbf {x} )\ {\stackrel {\mathrm {def} }{=}}\ \sum _{i=1}^{N}a_{i}\rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}$
where
$\rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}=\exp \left[-\beta _{i}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert ^{2}\right]=\exp \left[-\beta _{i}\left(x(t)-c_{i}\right)^{2}\right]$.
Since the input is a scalar rather than a vector, the input dimension is one. We choose the number of basis functions as N=5 and the size of the training set to be 100 exemplars generated by the chaotic time series. The weight $\beta $ is taken to be a constant equal to 5. The weights $c_{i}$ are five exemplars from the time series. The weights $a_{i}$ are trained with projection operator training:
$a_{i}(t+1)=a_{i}(t)+\nu {\big [}x(t+1)-\varphi {\big (}\mathbf {x} (t),\mathbf {w} {\big )}{\big ]}{\frac {\rho {\big (}\left\Vert \mathbf {x} (t)-\mathbf {c} _{i}\right\Vert {\big )}}{\sum _{i=1}^{N}\rho ^{2}{\big (}\left\Vert \mathbf {x} (t)-\mathbf {c} _{i}\right\Vert {\big )}}}$
where the learning rate $\nu $ is taken to be 0.3. The training is performed with one pass through the 100 training points. The rms error is 0.15.
Normalized radial basis functions
The normalized RBF architecture is
$\varphi (\mathbf {x} )\ {\stackrel {\mathrm {def} }{=}}\ {\frac {\sum _{i=1}^{N}a_{i}\rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}}{\sum _{i=1}^{N}\rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}}}=\sum _{i=1}^{N}a_{i}u{\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}$
where
$u{\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}\ {\stackrel {\mathrm {def} }{=}}\ {\frac {\rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}}{\sum _{i=1}^{N}\rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}}}$.
Again:
$\rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}=\exp \left[-\beta \left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert ^{2}\right]=\exp \left[-\beta \left(x(t)-c_{i}\right)^{2}\right]$.
Again, we choose the number of basis functions as five and the size of the training set to be 100 exemplars generated by the chaotic time series. The weight $\beta $ is taken to be a constant equal to 6. The weights $c_{i}$ are five exemplars from the time series. The weights $a_{i}$ are trained with projection operator training:
$a_{i}(t+1)=a_{i}(t)+\nu {\big [}x(t+1)-\varphi {\big (}\mathbf {x} (t),\mathbf {w} {\big )}{\big ]}{\frac {u{\big (}\left\Vert \mathbf {x} (t)-\mathbf {c} _{i}\right\Vert {\big )}}{\sum _{i=1}^{N}u^{2}{\big (}\left\Vert \mathbf {x} (t)-\mathbf {c} _{i}\right\Vert {\big )}}}$
where the learning rate $\nu $ is again taken to be 0.3. The training is performed with one pass through the 100 training points. The rms error on a test set of 100 exemplars is 0.084, smaller than the unnormalized error. Normalization yields accuracy improvement. Typically accuracy with normalized basis functions increases even more over unnormalized functions as input dimensionality increases.
Time series prediction
Once the underlying geometry of the time series is estimated as in the previous examples, a prediction for the time series can be made by iteration:
$\varphi (0)=x(1)$
${x}(t)\approx \varphi (t-1)$
${x}(t+1)\approx \varphi (t)=\varphi [\varphi (t-1)]$.
A comparison of the actual and estimated time series is displayed in the figure. The estimated times series starts out at time zero with an exact knowledge of x(0). It then uses the estimate of the dynamics to update the time series estimate for several time steps.
Note that the estimate is accurate for only a few time steps. This is a general characteristic of chaotic time series. This is a property of the sensitive dependence on initial conditions common to chaotic time series. A small initial error is amplified with time. A measure of the divergence of time series with nearly identical initial conditions is known as the Lyapunov exponent.
Control of a chaotic time series
We assume the output of the logistic map can be manipulated through a control parameter $c[x(t),t]$ such that
${x}_{}^{}(t+1)=4x(t)[1-x(t)]+c[x(t),t]$.
The goal is to choose the control parameter in such a way as to drive the time series to a desired output $d(t)$. This can be done if we choose the control parameter to be
$c_{}^{}[x(t),t]\ {\stackrel {\mathrm {def} }{=}}\ -\varphi [x(t)]+d(t+1)$
where
$y[x(t)]\approx f[x(t)]=x(t+1)-c[x(t),t]$
is an approximation to the underlying natural dynamics of the system.
The learning algorithm is given by
$a_{i}(t+1)=a_{i}(t)+\nu \varepsilon {\frac {u{\big (}\left\Vert \mathbf {x} (t)-\mathbf {c} _{i}\right\Vert {\big )}}{\sum _{i=1}^{N}u^{2}{\big (}\left\Vert \mathbf {x} (t)-\mathbf {c} _{i}\right\Vert {\big )}}}$
where
$\varepsilon \ {\stackrel {\mathrm {def} }{=}}\ f[x(t)]-\varphi [x(t)]=x(t+1)-c[x(t),t]-\varphi [x(t)]=x(t+1)-d(t+1)$.
See also
• Radial basis function kernel
• instance-based learning
• In Situ Adaptive Tabulation
• Predictive analytics
• Chaos theory
• Hierarchical RBF
• Cerebellar model articulation controller
• Instantaneously trained neural networks
References
1. Broomhead, D. S.; Lowe, David (1988). Radial basis functions, multi-variable functional interpolation and adaptive networks (Technical report). RSRE. 4148. Archived from the original on April 9, 2013.
2. Broomhead, D. S.; Lowe, David (1988). "Multivariable functional interpolation and adaptive networks" (PDF). Complex Systems. 2: 321–355.
3. Schwenker, Friedhelm; Kestler, Hans A.; Palm, Günther (2001). "Three learning phases for radial-basis-function networks". Neural Networks. 14 (4–5): 439–458. CiteSeerX 10.1.1.109.312. doi:10.1016/s0893-6080(01)00027-2. PMID 11411631.
4. Beheim, Larbi; Zitouni, Adel; Belloir, Fabien (January 2004). "New RBF neural network classifier with optimized hidden neurons number".
5. Ibrikci, Turgay; Brandt, M.E.; Wang, Guanyu; Acikkar, Mustafa (23–26 October 2002). Mahalanobis distance with radial basis function network on protein secondary structures. Proceedings of the Second Joint 24th Annual Conference and the Annual Fall Meeting of the Biomedical Engineering Society. Engineering in Medicine and Biology Society, Proceedings of the Annual International Conference of the IEEE. Vol. 3. Houston, TX, USA (published 6 January 2003). pp. 2184–5. doi:10.1109/IEMBS.2002.1053230. ISBN 0-7803-7612-9. ISSN 1094-687X.
6. Park, J.; I. W. Sandberg (Summer 1991). "Universal Approximation Using Radial-Basis-Function Networks". Neural Computation. 3 (2): 246–257. doi:10.1162/neco.1991.3.2.246. PMID 31167308. S2CID 34868087.
Further reading
• J. Moody and C. J. Darken, "Fast learning in networks of locally tuned processing units," Neural Computation, 1, 281-294 (1989). Also see Radial basis function networks according to Moody and Darken
• T. Poggio and F. Girosi, "Networks for approximation and learning," Proc. IEEE 78(9), 1484-1487 (1990).
• Roger D. Jones, Y. C. Lee, C. W. Barnes, G. W. Flake, K. Lee, P. S. Lewis, and S. Qian, ?Function approximation and time series prediction with neural networks,? Proceedings of the International Joint Conference on Neural Networks, June 17–21, p. I-649 (1990).
• Martin D. Buhmann (2003). Radial Basis Functions: Theory and Implementations. Cambridge University. ISBN 0-521-63338-9.
• Yee, Paul V. & Haykin, Simon (2001). Regularized Radial Basis Function Networks: Theory and Applications. John Wiley. ISBN 0-471-35349-3.
• John R. Davies, Stephen V. Coggeshall, Roger D. Jones, and Daniel Schutzer, "Intelligent Security Systems," in Freedman, Roy S., Flein, Robert A., and Lederman, Jess, Editors (1995). Artificial Intelligence in the Capital Markets. Chicago: Irwin. ISBN 1-55738-811-3. {{cite book}}: |author= has generic name (help)CS1 maint: multiple names: authors list (link)
• Simon Haykin (1999). Neural Networks: A Comprehensive Foundation (2nd ed.). Upper Saddle River, NJ: Prentice Hall. ISBN 0-13-908385-5.
• S. Chen, C. F. N. Cowan, and P. M. Grant, "Orthogonal Least Squares Learning Algorithm for Radial Basis Function Networks", IEEE Transactions on Neural Networks, Vol 2, No 2 (Mar) 1991.
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