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Wikipedia:Local boundedness#0	In mathematics, a function is locally bounded if it is bounded around every point. A family of functions is locally bounded if for any point in their domain all the functions are bounded around that point and by the same number. == Locally bounded function == A real-valued or complex-valued function f {\displaystyle f} defined on some topological space X {\displaystyle X} is called a locally bounded functional if for any x 0 ∈ X {\displaystyle x_{0}\in X} there exists a neighborhood A {\displaystyle A} of x 0 {\displaystyle x_{0}} such that f ( A ) {\displaystyle f(A)} is a bounded set. That is, for some number M > 0 {\displaystyle M>0} one has \| f ( x ) \| ≤ M for all x ∈ A . {\displaystyle \|f(x)\|\leq M\quad {\text{ for all }}x\in A.} In other words, for each x {\displaystyle x} one can find a constant, depending on x , {\displaystyle x,} which is larger than all the values of the function in the neighborhood of x . {\displaystyle x.} Compare this with a bounded function, for which the constant does not depend on x . {\displaystyle x.} Obviously, if a function is bounded then it is locally bounded. The converse is not true in general (see below). This definition can be extended to the case when f : X → Y {\displaystyle f:X\to Y} takes values in some metric space ( Y , d ) . {\displaystyle (Y,d).} Then the inequality above needs to be replaced with d ( f ( x ) , y ) ≤ M for all x ∈ A , {\displaystyle d(f(x),y)\leq M\quad {\text{ for all }}x\in A,} where y ∈ Y {\displaystyle y\in Y} is some point in the metric space. The choice of y {\displaystyle y} does not affect the definition; choosing a different y {\displaystyle y} will at most increase the constant r {\displaystyle r} for which this inequality is true. == Examples == The function f : R → R {\displaystyle f:\mathbb {R} \to \mathbb {R} } defined by f ( x ) = 1 x 2 + 1 {\displaystyle f(x)={\frac {1}{x^{2}+1}}} is bounded, because 0 ≤ f ( x ) ≤ 1 {\displaystyle 0\leq f(x)\leq 1} for all x . {\displaystyle x.} Therefore, it is also locally bounded. The function f : R → R {\displaystyle f:\mathbb {R} \to \mathbb {R} } defined by f ( x ) = 2 x + 3 {\displaystyle f(x)=2x+3} is not bounded, as it becomes arbitrarily large. However, it is locally bounded because for each a , {\displaystyle a,} \| f ( x ) \| ≤ M {\displaystyle \|f(x)\|\leq M} in the neighborhood ( a − 1 , a + 1 ) , {\displaystyle (a-1,a+1),} where M = 2 \| a \| + 5. {\displaystyle M=2\|a\|+5.} The function f : R → R {\displaystyle f:\mathbb {R} \to \mathbb {R} } defined by f ( x ) = { 1 x , if x ≠ 0 , 0 , if x = 0 {\displaystyle f(x)={\begin{cases}{\frac {1}{x}},&{\mbox{if }}x\neq 0,\\0,&{\mbox{if }}x=0\end{cases}}} is neither bounded nor locally bounded. In any neighborhood of 0 this function takes values of arbitrarily large magnitude. Any continuous function is locally bounded. Here is a proof for functions of a real variable. Let f : U → R {\displaystyle f:U\to \mathbb {R} } be continuous where U ⊆ R , {\displaystyle U\subseteq \mathbb {R} ,} and we will show that f {\displaystyle f} is locally bounded at a {\displaystyle a} for all a ∈ U {\displaystyle a\in U} Taking ε = 1 in the definition of continuity, there exists δ > 0 {\displaystyle \delta >0} such that \| f ( x ) − f ( a ) \| < 1 {\displaystyle \|f(x)-f(a)\|<1} for all x ∈ U {\displaystyle x\in U} with \| x − a \| < δ {\displaystyle \|x-a\|<\delta } . Now by the triangle inequality, \| f ( x ) \| = \| f ( x ) − f ( a ) + f ( a ) \| ≤ \| f ( x ) − f ( a ) \| + \| f ( a ) \| < 1 + \| f ( a ) \| , {\displaystyle \|f(x)\|=\|f(x)-f(a)+f(a)\|\leq \|f(x)-f(a)\|+\|f(a)\|<1+\|f(a)\|,} which means that f {\displaystyle f} is locally bounded at a {\displaystyle a} (taking M = 1 + \| f ( a ) \| {\displaystyle M=1+\|f(a)\|} and the neighborhood ( a − δ , a + δ ) {\displaystyle (a-\delta ,a+\delta )} ). This argument generalizes easily to when the domain of f {\displaystyle f} is any topological space. The converse of the above result is not true however; that is, a discontinuous function may be locally bounded. For example consider the function f : R → R {\displaystyle f:\mathbb {R} \to \mathbb {R} } given by f ( 0 ) = 1 {\displaystyle f(0)=1} and f ( x ) = 0 {\displaystyle f(x)=0} for all x ≠ 0. {\displaystyle x\neq 0.} Then f {\displaystyle f} is discontinuous at 0 but f {\displaystyle f} is locally bounded; it is locally constant apart from at zero, where we can take M = 1 {\displaystyle M=1} and the neighborhood ( − 1 , 1 ) , {\displaystyle (-1,1),} for example. == Locally bounded family == A set (also called a family) U of real-valued or complex-valued functions defined on some topological space X {\displaystyle X} is called locally bounded if for any x 0 ∈ X {\displaystyle x_{0}\in X} there exists a neighborhood A {\displaystyle A} of x 0 {\displaystyle x_{0}} and a positive number M > 0 {\displaystyle M>0} such that \| f ( x ) \| ≤ M {\displaystyle \|f(x)\|\leq M} for all x ∈ A {\displaystyle x\in A} and f ∈ U . {\displaystyle f\in U.} In other words, all the functions in the family must be locally bounded, and around each point they need to be bounded by the same constant. This definition can also be extended to the case when the functions in the family U take values in some metric space, by again replacing the absolute value with the distance function. == Examples == The family of functions f n : R → R {\displaystyle f_{n}:\mathbb {R} \to \mathbb {R} } f n ( x ) = x n {\displaystyle f_{n}(x)={\frac {x}{n}}} where n = 1 , 2 , … {\displaystyle n=1,2,\ldots } is locally bounded. Indeed, if x 0 {\displaystyle x_{0}} is a real number, one can choose the neighborhood A {\displaystyle A} to be the interval ( x 0 − a , x 0 + 1 ) . {\displaystyle \left(x_{0}-a,x_{0}+1\right).} Then for all x {\displaystyle x} in this interval and for all n ≥ 1 {\displaystyle n\geq 1} one has \| f n ( x ) \| ≤ M {\displaystyle \|f_{n}(x)\|\leq M} with M = 1 + \| x 0 \| . {\displaystyle M=1+\|x_{0}\|.} Moreover, the family is uniformly bounded, because neither the neighborhood A {\displaystyle A} nor the constant M {\displaystyle M} depend on the index n . {\displaystyle n.} The family of functions f n : R → R {\displaystyle f_{n}:\mathbb {R} \to \mathbb {R} } f n ( x ) = 1 x 2 + n 2 {\displaystyle f_{n}(x)={\frac {1}{x^{2}+n^{2}}}} is locally bounded, if n {\displaystyle n} is greater than zero. For any x 0 {\displaystyle x_{0}} one can choose the neighborhood A {\displaystyle A} to be R {\displaystyle \mathbb {R} } itself. Then we have \| f n ( x ) \| ≤ M {\displaystyle \|f_{n}(x)\|\leq M} with M = 1. {\displaystyle M=1.} Note that the value of M {\displaystyle M} does not depend on the choice of x0 or its neighborhood A . {\displaystyle A.} This family is then not only locally bounded, it is also uniformly bounded. The family of functions f n : R → R {\displaystyle f_{n}:\mathbb {R} \to \mathbb {R} } f n ( x ) = x + n {\displaystyle f_{n}(x)=x+n} is not locally bounded. Indeed, for any x {\displaystyle x} the values f n ( x ) {\displaystyle f_{n}(x)} cannot be bounded as n {\displaystyle n} tends toward infinity. == Topological vector spaces == Local boundedness may also refer to a property of topological vector spaces, or of functions from a topological space into a topological vector space (TVS). === Locally bounded topological vector spaces === A subset B ⊆ X {\displaystyle B\subseteq X} of a topological vector space (TVS) X {\displaystyle X} is called bounded if for each neighborhood U {\displaystyle U} of the origin in X {\displaystyle X} there exists a real number s > 0 {\displaystyle s>0} such that B ⊆ t U for all t > s . {\displaystyle B\subseteq tU\quad {\text{ for all }}t>s.} A locally bounded TVS is a TVS that possesses a bounded neighborhood of the origin. By Kolmogorov's normability criterion, this is true of a locally convex space if and only if the topology of the TVS is induced by some seminorm. In particular, every locally bounded TVS is pseudometrizable. === Locally bounded functions === Let f : X → Y {\displaystyle f:X\to Y} a function between topological vector spaces is said to be a locally bounded function if every point of X {\displaystyle X} has a neighborhood whose image under f {\displaystyle f} is bounded. The following theorem relates local boundedness of functions with the local boundedness of topological vector spaces: Theorem. A topological vector space X {\displaystyle X} is locally bounded if and only if the identity map id X : X → X {\displaystyle \operatorname {id} _{X}:X\to X} is locally bounded. == See also == Bornological space – Space where bounded operators are continuous Bounded operator – Linear transformation between topological vector spaces Bounded set (topological vector space) – Generalization of boundedness == External links == PlanetMath entry for Locally Bounded nLab entry for Locally Bounded Category
Wikipedia:Local diffeomorphism#0	In mathematics, more specifically differential topology, a local diffeomorphism is intuitively a map between smooth manifolds that preserves the local differentiable structure. The formal definition of a local diffeomorphism is given below. == Formal definition == Let X {\displaystyle X} and Y {\displaystyle Y} be differentiable manifolds. A function f : X → Y {\displaystyle f:X\to Y} is a local diffeomorphism if, for each point x ∈ X {\displaystyle x\in X} , there exists an open set U {\displaystyle U} containing x {\displaystyle x} such that the image f ( U ) {\displaystyle f(U)} is open in Y {\displaystyle Y} and f \| U : U → f ( U ) {\displaystyle f\vert _{U}:U\to f(U)} is a diffeomorphism. A local diffeomorphism is a special case of an immersion f : X → Y {\displaystyle f:X\to Y} . In this case, for each x ∈ X {\displaystyle x\in X} , there exists an open set U {\displaystyle U} containing x {\displaystyle x} such that the image f ( U ) {\displaystyle f(U)} is an embedded submanifold, and f \| U : U → f ( U ) {\displaystyle f\|_{U}:U\to f(U)} is a diffeomorphism. Here X {\displaystyle X} and f ( U ) {\displaystyle f(U)} have the same dimension, which may be less than the dimension of Y {\displaystyle Y} . === Characterizations === A map is a local diffeomorphism if and only if it is a smooth immersion (smooth local embedding) and an open map. The inverse function theorem implies that a smooth map f : X → Y {\displaystyle f:X\to Y} is a local diffeomorphism if and only if the derivative D f x : T x X → T f ( x ) Y {\displaystyle Df_{x}:T_{x}X\to T_{f(x)}Y} is a linear isomorphism for all points x ∈ X {\displaystyle x\in X} . This implies that X {\displaystyle X} and Y {\displaystyle Y} have the same dimension. It follows that a map f : X → Y {\displaystyle f:X\to Y} between two manifolds of equal dimension ( dim ⁡ X = dim ⁡ Y {\displaystyle \operatorname {dim} X=\operatorname {dim} Y} ) is a local diffeomorphism if and only if it is a smooth immersion (smooth local embedding), or equivalently, if and only if it is a smooth submersion. This is because, for any x ∈ X {\displaystyle x\in X} , both T x X {\displaystyle T_{x}X} and T f ( x ) Y {\displaystyle T_{f(x)}Y} have the same dimension, thus D f x {\displaystyle Df_{x}} is a linear isomorphism if and only if it is injective, or equivalently, if and only if it is surjective. Here is an alternative argument for the case of an immersion: every smooth immersion is a locally injective function, while invariance of domain guarantees that any continuous injective function between manifolds of equal dimensions is necessarily an open map. == Discussion == All manifolds of the same dimension are "locally diffeomorphic," in the following sense: if X {\displaystyle X} and Y {\displaystyle Y} have the same dimension, and x ∈ X {\displaystyle x\in X} and y ∈ Y {\displaystyle y\in Y} , then there exist open neighbourhoods U {\displaystyle U} of x {\displaystyle x} and V {\displaystyle V} of y {\displaystyle y} and a diffeomorphism f : U → V {\displaystyle f:U\to V} . However, this map f {\displaystyle f} need not extend to a smooth map defined on all of X {\displaystyle X} , let alone extend to a local diffeomorphism. Thus the existence of a local diffeomorphism f : X → Y {\displaystyle f:X\to Y} is a stronger condition than "to be locally diffeomophic." Indeed, although locally-defined diffeomorphisms preserve differentiable structure locally, one must be able to "patch up" these (local) diffeomorphisms to ensure that the domain is the entire smooth manifold. For example, one can impose two different differentiable structures on R 4 {\displaystyle \mathbb {R} ^{4}} that each make R 4 {\displaystyle \mathbb {R} ^{4}} into a differentiable manifold, but both structures are not locally diffeomorphic (see Exotic R 4 {\displaystyle \mathbb {R} ^{4}} ). As another example, there can be no local diffeomorphism from the 2-sphere to Euclidean 2-space, although they do indeed have the same local differentiable structure. This is because all local diffeomorphisms are continuous, the continuous image of a compact space is compact, and the 2-sphere is compact whereas Euclidean 2-space is not. === Properties === If a local diffeomorphism between two manifolds exists then their dimensions must be equal. Every local diffeomorphism is also a local homeomorphism and therefore a locally injective open map. A local diffeomorphism has constant rank of n . {\displaystyle n.} == Examples == A diffeomorphism is a bijective local diffeomorphism. A smooth covering map is a local diffeomorphism such that every point in the target has a neighborhood that is evenly covered by the map. == Local flow diffeomorphisms == == See also == Diffeomorphism – Isomorphism of differentiable manifolds Homeomorphism – Mapping which preserves all topological properties of a given space Invariance of domain – Theorem in topology about homeomorphic subsets of Euclidean space Large diffeomorphism – Class of diffeomorphism Local homeomorphism – Mathematical function revertible near each point Spacetime symmetries – features of space-time representing symmetriesPages displaying wikidata descriptions as a fallback == Notes == == References == Michor, Peter W. (2008), Topics in differential geometry, Graduate Studies in Mathematics, vol. 93, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-2003-2, MR 2428390. Lee, John M. (2013), Introduction to smooth manifolds, Graduate Texts in Mathematics, vol. 218 (Second ed.), New York, NY.: Springer, ISBN 978-1-4419-9981-8, MR 2954043 Axler, Sheldon (2024), Linear algebra done right, Undergraduate Texts in Mathematics (Fourth ed.), Springer, Cham, doi:10.1007/978-3-031-41026-0, ISBN 978-3-031-41026-0, MR 4696768
Wikipedia:Local zeta function#0	In mathematics, the local zeta function Z(V, s) (sometimes called the congruent zeta function or the Hasse–Weil zeta function) is defined as Z ( V , s ) = exp ⁡ ( ∑ k = 1 ∞ N k k ( q − s ) k ) {\displaystyle Z(V,s)=\exp \left(\sum _{k=1}^{\infty }{\frac {N_{k}}{k}}(q^{-s})^{k}\right)} where V is a non-singular n-dimensional projective algebraic variety over the field Fq with q elements and Nk is the number of points of V defined over the finite field extension Fqk of Fq. Making the variable transformation t = q−s, gives Z ( V , t ) = exp ⁡ ( ∑ k = 1 ∞ N k t k k ) {\displaystyle {\mathit {Z}}(V,t)=\exp \left(\sum _{k=1}^{\infty }N_{k}{\frac {t^{k}}{k}}\right)} as the formal power series in the variable t {\displaystyle t} . Equivalently, the local zeta function is sometimes defined as follows: ( 1 ) Z ( V , 0 ) = 1 {\displaystyle (1)\ \ {\mathit {Z}}(V,0)=1\,} ( 2 ) d d t log ⁡ Z ( V , t ) = ∑ k = 1 ∞ N k t k − 1 . {\displaystyle (2)\ \ {\frac {d}{dt}}\log {\mathit {Z}}(V,t)=\sum _{k=1}^{\infty }N_{k}t^{k-1}\ .} In other words, the local zeta function Z(V, t) with coefficients in the finite field Fq is defined as a function whose logarithmic derivative generates the number Nk of solutions of the equation defining V in the degree k extension Fqk. == Formulation == Given a finite field F, there is, up to isomorphism, only one field Fk with [ F k : F ] = k {\displaystyle [F_{k}:F]=k\,} , for k = 1, 2, ... . When F is the unique field with q elements, Fk is the unique field with q k {\displaystyle q^{k}} elements. Given a set of polynomial equations — or an algebraic variety V — defined over F, we can count the number N k {\displaystyle N_{k}\,} of solutions in Fk and create the generating function G ( t ) = N 1 t + N 2 t 2 / 2 + N 3 t 3 / 3 + ⋯ {\displaystyle G(t)=N_{1}t+N_{2}t^{2}/2+N_{3}t^{3}/3+\cdots \,} . The correct definition for Z(t) is to set log Z equal to G, so Z = exp ⁡ ( G ( t ) ) {\displaystyle Z=\exp(G(t))\,} and Z(0) = 1, since G(0) = 0, and Z(t) is a priori a formal power series. The logarithmic derivative Z ′ ( t ) / Z ( t ) {\displaystyle Z'(t)/Z(t)\,} equals the generating function G ′ ( t ) = N 1 + N 2 t 1 + N 3 t 2 + ⋯ {\displaystyle G'(t)=N_{1}+N_{2}t^{1}+N_{3}t^{2}+\cdots \,} . == Examples == For example, assume all the Nk are 1; this happens for example if we start with an equation like X = 0, so that geometrically we are taking V to be a point. Then G ( t ) = − log ⁡ ( 1 − t ) {\displaystyle G(t)=-\log(1-t)} is the expansion of a logarithm (for \|t\| < 1). In this case we have Z ( t ) = 1 ( 1 − t ) . {\displaystyle Z(t)={\frac {1}{(1-t)}}\ .} To take something more interesting, let V be the projective line over F. If F has q elements, then this has q + 1 points, including the one point at infinity. Therefore, we have N k = q k + 1 {\displaystyle N_{k}=q^{k}+1} and G ( t ) = − log ⁡ ( 1 − t ) − log ⁡ ( 1 − q t ) {\displaystyle G(t)=-\log(1-t)-\log(1-qt)} for \|t\| small enough, and therefore Z ( t ) = 1 ( 1 − t ) ( 1 − q t ) . {\displaystyle Z(t)={\frac {1}{(1-t)(1-qt)}}\ .} The first study of these functions was in the 1923 dissertation of Emil Artin. He obtained results for the case of a hyperelliptic curve, and conjectured the further main points of the theory as applied to curves. The theory was then developed by F. K. Schmidt and Helmut Hasse. The earliest known nontrivial cases of local zeta functions were implicit in Carl Friedrich Gauss's Disquisitiones Arithmeticae, article 358. There, certain particular examples of elliptic curves over finite fields having complex multiplication have their points counted by means of cyclotomy. For the definition and some examples, see also. == Motivations == The relationship between the definitions of G and Z can be explained in a number of ways. (See for example the infinite product formula for Z below.) In practice it makes Z a rational function of t, something that is interesting even in the case of V an elliptic curve over a finite field. The local Z zeta functions are multiplied to get global ζ {\displaystyle \zeta } zeta functions, ζ = ∏ Z {\displaystyle \zeta =\prod Z} These generally involve different finite fields (for example the whole family of fields Z/pZ as p runs over all prime numbers). In these fields, the variable t is substituted by p−s, where s is the complex variable traditionally used in Dirichlet series. (For details see Hasse–Weil zeta function.) The global products of Z in the two cases used as examples in the previous section therefore come out as ζ ( s ) {\displaystyle \zeta (s)} and ζ ( s ) ζ ( s − 1 ) {\displaystyle \zeta (s)\zeta (s-1)} after letting q = p {\displaystyle q=p} . == Riemann hypothesis for curves over finite fields == For projective curves C over F that are non-singular, it can be shown that Z ( t ) = P ( t ) ( 1 − t ) ( 1 − q t ) , {\displaystyle Z(t)={\frac {P(t)}{(1-t)(1-qt)}}\ ,} with P(t) a polynomial, of degree 2g, where g is the genus of C. Rewriting P ( t ) = ∏ i = 1 2 g ( 1 − ω i t ) , {\displaystyle P(t)=\prod _{i=1}^{2g}(1-\omega _{i}t)\ ,} the Riemann hypothesis for curves over finite fields states \| ω i \| = q 1 / 2 . {\displaystyle \|\omega _{i}\|=q^{1/2}\ .} For example, for the elliptic curve case there are two roots, and it is easy to show the absolute values of the roots are q1/2. Hasse's theorem is that they have the same absolute value; and this has immediate consequences for the number of points. André Weil proved this for the general case, around 1940 (Comptes Rendus note, April 1940): he spent much time in the years after that writing up the algebraic geometry involved. This led him to the general Weil conjectures. Alexander Grothendieck developed scheme theory for the purpose of resolving these. A generation later Pierre Deligne completed the proof. (See étale cohomology for the basic formulae of the general theory.) == General formulas for the zeta function == It is a consequence of the Lefschetz trace formula for the Frobenius morphism that Z ( X , t ) = ∏ i = 0 2 dim ⁡ X det ( 1 − t Frob q \| H c i ( X ¯ , Q ℓ ) ) ( − 1 ) i + 1 . {\displaystyle Z(X,t)=\prod _{i=0}^{2\dim X}\det {\big (}1-t{\mbox{Frob}}_{q}\|H_{c}^{i}({\overline {X}},{\mathbb {Q} }_{\ell }){\big )}^{(-1)^{i+1}}.} Here X {\displaystyle X} is a separated scheme of finite type over the finite field F with q {\displaystyle q} elements, and Frobq is the geometric Frobenius acting on ℓ {\displaystyle \ell } -adic étale cohomology with compact supports of X ¯ {\displaystyle {\overline {X}}} , the lift of X {\displaystyle X} to the algebraic closure of the field F. This shows that the zeta function is a rational function of t {\displaystyle t} . An infinite product formula for Z ( X , t ) {\displaystyle Z(X,t)} is Z ( X , t ) = ∏ ( 1 − t deg ⁡ ( x ) ) − 1 . {\displaystyle Z(X,t)=\prod \ (1-t^{\deg(x)})^{-1}.} Here, the product ranges over all closed points x of X and deg(x) is the degree of x. The local zeta function Z(X, t) is viewed as a function of the complex variable s via the change of variables q−s. In the case where X is the variety V discussed above, the closed points are the equivalence classes x=[P] of points P on V ¯ {\displaystyle {\overline {V}}} , where two points are equivalent if they are conjugates over F. The degree of x is the degree of the field extension of F generated by the coordinates of P. The logarithmic derivative of the infinite product Z(X, t) is easily seen to be the generating function discussed above, namely N 1 + N 2 t 1 + N 3 t 2 + ⋯ {\displaystyle N_{1}+N_{2}t^{1}+N_{3}t^{2}+\cdots \,} . == See also == List of zeta functions Weil conjectures Elliptic curve == References ==
Wikipedia:Lodovico Ferrari#0	Lodovico de Ferrari (2 February 1522 – 5 October 1565) was an Italian mathematician best known today for solving the biquadratic equation. == Biography == Born in Bologna, Lodovico's grandfather, Bartolomeo Ferrari, was forced out of Milan to Bologna. Lodovico settled in Bologna, and he began his career as the servant of Gerolamo Cardano. He was extremely bright, so Cardano started teaching him mathematics. Ferrari aided Cardano on his solutions for biquadratic equations and cubic equations, and was mainly responsible for the solution of biquadratic equations that Cardano published. While still in his teens, Ferrari was able to obtain a prestigious teaching post in Rome after Cardano resigned from it and recommended him. Ferrari retired when young at 42 years old, and wealthy.: 300 He then moved back to his home town of Bologna where he lived with his widowed sister Maddalena to take up a professorship of mathematics at the University of Bologna in 1565. Shortly thereafter, he died of white arsenic poisoning, according to a legend, by his sister.: 18 == Cardano–Tartaglia formula == In 1545 a famous dispute erupted between Ferrari and Cardano's contemporary Niccolò Fontana Tartaglia, involving the solution to cubic equations. Widespread stories that Tartaglia devoted the rest of his life to ruining Ferrari's teacher and erstwhile master Cardano, however, appear to be fabricated. Mathematical historians now credit both Cardano and Tartaglia with the formula to solve cubic equations, referring to it as the "Cardano–Tartaglia formula". == References == == Further reading == Jayawardene, S. A. (1970–1980). "Ferrari, Lodovico". Dictionary of Scientific Biography. Vol. 4. New York: Charles Scribner's Sons. pp. 586–8. ISBN 978-0-684-10114-9. == External links == Media related to Lodovico Ferrari at Wikimedia Commons O'Connor, John J.; Robertson, Edmund F., "Lodovico Ferrari", MacTutor History of Mathematics Archive, University of St Andrews
Wikipedia:Loewner order#0	In mathematics, Loewner order is the partial order defined by the convex cone of positive semi-definite matrices. This order is usually employed to generalize the definitions of monotone and concave/convex scalar functions to monotone and concave/convex Hermitian valued functions. These functions arise naturally in matrix and operator theory and have applications in many areas of physics and engineering. == Definition == Let A and B be two Hermitian matrices of order n. We say that A ≥ B if A − B is positive semi-definite. Similarly, we say that A > B if A − B is positive definite. Although it is commonly discussed on matrices (as a finite-dimensional case), the Loewner order is also well-defined on operators (an infinite-dimensional case) in the analogous way. == Properties == When A and B are real scalars (i.e. n = 1), the Loewner order reduces to the usual ordering of R. Although some familiar properties of the usual order of R are also valid when n ≥ 2, several properties are no longer valid. For instance, the comparability of two matrices may no longer be valid. In fact, if A = [ 1 0 0 0 ] {\displaystyle A={\begin{bmatrix}1&0\\0&0\end{bmatrix}}\ } and B = [ 0 0 0 1 ] {\displaystyle B={\begin{bmatrix}0&0\\0&1\end{bmatrix}}\ } then neither A ≥ B or B ≥ A holds true. In other words, the Loewner order is a partial order, but not a total order. Moreover, since A and B are Hermitian matrices, their eigenvalues are all real numbers. If λ1(B) is the maximum eigenvalue of B and λn(A) the minimum eigenvalue of A, a sufficient criterion to have A ≥ B is that λn(A) ≥ λ1(B). If A or B is a multiple of the identity matrix, then this criterion is also necessary. The Loewner order does not have the least-upper-bound property, and therefore does not form a lattice. It is bounded: for any finite set S {\displaystyle S} of matrices, one can find an "upper-bound" matrix A that is greater than all of S. However, there will be multiple upper bounds. In a lattice, there would exist a unique maximum max ( S ) {\displaystyle \max(S)} such that any upper bound U on S {\displaystyle S} obeys max ( S ) {\displaystyle \max(S)} ≤ U. But in the Loewner order, one can have two upper bounds A and B that are both minimal (there is no element C < A that is also an upper bound) but that are incomparable (A - B is neither positive semidefinite nor negative semidefinite). == See also == Trace inequalities == References == Pukelsheim, Friedrich (2006). Optimal design of experiments. Society for Industrial and Applied Mathematics. pp. 11–12. ISBN 9780898716047. Bhatia, Rajendra (1997). Matrix Analysis. New York, NY: Springer. ISBN 9781461206538. Zhan, Xingzhi (2002). Matrix inequalities. Berlin: Springer. pp. 1–15. ISBN 9783540437987.
Wikipedia:Logarithmically concave function#0	In convex analysis, a non-negative function f : Rn → R+ is logarithmically concave (or log-concave for short) if its domain is a convex set, and if it satisfies the inequality f ( θ x + ( 1 − θ ) y ) ≥ f ( x ) θ f ( y ) 1 − θ {\displaystyle f(\theta x+(1-\theta )y)\geq f(x)^{\theta }f(y)^{1-\theta }} for all x,y ∈ dom f and 0 < θ < 1. If f is strictly positive, this is equivalent to saying that the logarithm of the function, log ∘ f, is concave; that is, log ⁡ f ( θ x + ( 1 − θ ) y ) ≥ θ log ⁡ f ( x ) + ( 1 − θ ) log ⁡ f ( y ) {\displaystyle \log f(\theta x+(1-\theta )y)\geq \theta \log f(x)+(1-\theta )\log f(y)} for all x,y ∈ dom f and 0 < θ < 1. Examples of log-concave functions are the 0-1 indicator functions of convex sets (which requires the more flexible definition), and the Gaussian function. Similarly, a function is log-convex if it satisfies the reverse inequality f ( θ x + ( 1 − θ ) y ) ≤ f ( x ) θ f ( y ) 1 − θ {\displaystyle f(\theta x+(1-\theta )y)\leq f(x)^{\theta }f(y)^{1-\theta }} for all x,y ∈ dom f and 0 < θ < 1. == Properties == A log-concave function is also quasi-concave. This follows from the fact that the logarithm is monotone implying that the superlevel sets of this function are convex. Every concave function that is nonnegative on its domain is log-concave. However, the reverse does not necessarily hold. An example is the Gaussian function f(x) = exp(−x2/2) which is log-concave since log f(x) = −x2/2 is a concave function of x. But f is not concave since the second derivative is positive for \|x\| > 1: f ″ ( x ) = e − x 2 2 ( x 2 − 1 ) ≰ 0 {\displaystyle f''(x)=e^{-{\frac {x^{2}}{2}}}(x^{2}-1)\nleq 0} From above two points, concavity ⇒ {\displaystyle \Rightarrow } log-concavity ⇒ {\displaystyle \Rightarrow } quasiconcavity. A twice differentiable, nonnegative function with a convex domain is log-concave if and only if for all x satisfying f(x) > 0, f ( x ) ∇ 2 f ( x ) ⪯ ∇ f ( x ) ∇ f ( x ) T {\displaystyle f(x)\nabla ^{2}f(x)\preceq \nabla f(x)\nabla f(x)^{T}} , i.e. f ( x ) ∇ 2 f ( x ) − ∇ f ( x ) ∇ f ( x ) T {\displaystyle f(x)\nabla ^{2}f(x)-\nabla f(x)\nabla f(x)^{T}} is negative semi-definite. For functions of one variable, this condition simplifies to f ( x ) f ″ ( x ) ≤ ( f ′ ( x ) ) 2 {\displaystyle f(x)f''(x)\leq (f'(x))^{2}} == Operations preserving log-concavity == Products: The product of log-concave functions is also log-concave. Indeed, if f and g are log-concave functions, then log f and log g are concave by definition. Therefore log f ( x ) + log g ( x ) = log ⁡ ( f ( x ) g ( x ) ) {\displaystyle \log \,f(x)+\log \,g(x)=\log(f(x)g(x))} is concave, and hence also f g is log-concave. Marginals: if f(x,y) : Rn+m → R is log-concave, then g ( x ) = ∫ f ( x , y ) d y {\displaystyle g(x)=\int f(x,y)dy} is log-concave (see Prékopa–Leindler inequality). This implies that convolution preserves log-concavity, since h(x,y) = f(x-y) g(y) is log-concave if f and g are log-concave, and therefore ( f ∗ g ) ( x ) = ∫ f ( x − y ) g ( y ) d y = ∫ h ( x , y ) d y {\displaystyle (f*g)(x)=\int f(x-y)g(y)dy=\int h(x,y)dy} is log-concave. == Log-concave distributions == Log-concave distributions are necessary for a number of algorithms, e.g. adaptive rejection sampling. Every distribution with log-concave density is a maximum entropy probability distribution with specified mean μ and Deviation risk measure D. As it happens, many common probability distributions are log-concave. Some examples: the normal distribution and multivariate normal distributions, the exponential distribution, the uniform distribution over any convex set, the logistic distribution, the extreme value distribution, the Laplace distribution, the chi distribution, the hyperbolic secant distribution, the Wishart distribution, if n ≥ p + 1, the Dirichlet distribution, if all parameters are ≥ 1, the gamma distribution if the shape parameter is ≥ 1, the chi-square distribution if the number of degrees of freedom is ≥ 2, the beta distribution if both shape parameters are ≥ 1, and the Weibull distribution if the shape parameter is ≥ 1. Note that all of the parameter restrictions have the same basic source: The exponent of non-negative quantity must be non-negative in order for the function to be log-concave. The following distributions are non-log-concave for all parameters: the Student's t-distribution, the Cauchy distribution, the Pareto distribution, the log-normal distribution, and the F-distribution. Note that the cumulative distribution function (CDF) of all log-concave distributions is also log-concave. However, some non-log-concave distributions also have log-concave CDF's: the log-normal distribution, the Pareto distribution, the Weibull distribution when the shape parameter < 1, and the gamma distribution when the shape parameter < 1. The following are among the properties of log-concave distributions: If a density is log-concave, so is its cumulative distribution function (CDF). If a multivariate density is log-concave, so is the marginal density over any subset of variables. The sum of two independent log-concave random variables is log-concave. This follows from the fact that the convolution of two log-concave functions is log-concave. The product of two log-concave functions is log-concave. This means that joint densities formed by multiplying two probability densities (e.g. the normal-gamma distribution, which always has a shape parameter ≥ 1) will be log-concave. This property is heavily used in general-purpose Gibbs sampling programs such as BUGS and JAGS, which are thereby able to use adaptive rejection sampling over a wide variety of conditional distributions derived from the product of other distributions. If a density is log-concave, so is its survival function. If a density is log-concave, it has a monotone hazard rate (MHR), and is a regular distribution since the derivative of the logarithm of the survival function is the negative hazard rate, and by concavity is monotone i.e. d d x log ⁡ ( 1 − F ( x ) ) = − f ( x ) 1 − F ( x ) {\displaystyle {\frac {d}{dx}}\log \left(1-F(x)\right)=-{\frac {f(x)}{1-F(x)}}} which is decreasing as it is the derivative of a concave function. == See also == logarithmically concave sequence logarithmically concave measure logarithmically convex function convex function == Notes == == References == Barndorff-Nielsen, Ole (1978). Information and exponential families in statistical theory. Wiley Series in Probability and Mathematical Statistics. Chichester: John Wiley \& Sons, Ltd. pp. ix+238 pp. ISBN 0-471-99545-2. MR 0489333. Dharmadhikari, Sudhakar; Joag-Dev, Kumar (1988). Unimodality, convexity, and applications. Probability and Mathematical Statistics. Boston, MA: Academic Press, Inc. pp. xiv+278. ISBN 0-12-214690-5. MR 0954608. Pfanzagl, Johann; with the assistance of R. Hamböker (1994). Parametric Statistical Theory. Walter de Gruyter. ISBN 3-11-013863-8. MR 1291393. Pečarić, Josip E.; Proschan, Frank; Tong, Y. L. (1992). Convex functions, partial orderings, and statistical applications. Mathematics in Science and Engineering. Vol. 187. Boston, MA: Academic Press, Inc. pp. xiv+467 pp. ISBN 0-12-549250-2. MR 1162312.
Wikipedia:Lokavibhaga#0	The Lokavibhāga (literally "division of the universe") is a 5th-century Sanskrit text by Rishi Simhasuri. Its manuscript was first discovered in an Indian temple of Karnataka by M.R.R. Narasimhachar. The Lokavibhaga consists of 11 chapters and a total of 1737 verses (shlokas) distributed over these chapters. The text has an incomplete colophon, which states it was completed in a village named Patalika near Kanchi (Tamil Nadu) in the 22nd year of Simhavarman's rule in Banarastra. The colophon includes astronomical observations along with a samvat date and year which together confirm the text was published by Rishi Simhasuri on 25 August 458 CE. The Lokavibhaga is notable as the oldest known text in the world that clearly uses three principles of positional decimal arithmetic system together – graphical signs and terms as numerals, assigning a value to the same numeral depending on the position it occupies in a number, and the use of fully operational zero. This Indian system contrasted with competing ancient arithmetic systems developed independently in Babylon, ancient Rome and China. The text presents Jain cosmology. It has been claimed by the Digambara tradition of Jainism to be a Sanskrit translation of an older Prakrit-language text Loyavibhaga by Muni Sarvanandi. The Lokavibhaga mentions Sarvanandi and others, but does not mention any text called "Loyavibhaga". No manuscript copy of the claimed older Prakrit "Loyavibhaga" has been found so far. The Lokavibhaga presents its cosmological ideas in a form that takes its mathematical system for granted. It is not a mathematical treatise, and it does not introduce principles of positional decimal arithmetic system. The arithmetic system used in Lokavibhaga text, state Jain and Dani, must have been invented earlier by someone else in some other context. That system of expressing numbers with positional decimal arithmetic was accepted and must have been in wide use in India by mid 5th-century to appear as it does in the Lokavibhaga text. The same Indian arithmetic system and operations appears in the mathematical treatise of Aryabhata published in 510 CE. The surviving manuscripts of the Lokavibhāga are listed in the New Catalogus Catalogorum. The published edition of the surviving Lokvibhaga manuscript is a palm leaf copy of the original Sanskrit text, likely from 11th or 12th century. The text was edited and translated in 1962 into the Hindi language by Balachandra Siddhanta-Shastri. == References ==
Wikipedia:Lokenath Debnath#0	Lokenath Debnath (September 30, 1935 – March 2, 2023) was an Indian-American mathematician. == Biography == Debnath was born on September 30, 1935, in India. He received both Masters and a doctorate degree from University of Calcutta in Pure Mathematics in 1965. He obtained a Ph.D. in Applied Mathematics at University of London in 1967. His doctoral advisor was Simon Rosenblat. He was a professor of mathematics at University of Texas Rio Grande Valley. He was a professor at University of Central Florida from 1983 to 2001. Debnath was a founder of the mathematical journal International Journal of Mathematics and Mathematical Sciences. He died on March 2, 2023, at the age of 87. == Publications == === Books === == References ==
Wikipedia:London Mathematical Society#0	The London Mathematical Society (LMS) is one of the United Kingdom's learned societies for mathematics (the others being the Royal Statistical Society (RSS), the Institute of Mathematics and its Applications (IMA), the Edinburgh Mathematical Society and the Operational Research Society (ORS). == History == The Society was established on 16 January 1865, the first president being Augustus De Morgan. The earliest meetings were held in University College, but the Society soon moved into Burlington House, Piccadilly. The initial activities of the Society included talks and publication of a journal. The LMS was used as a model for the establishment of the American Mathematical Society in 1888. Mary Cartwright was the first woman to be President of the LMS (in 1961–62). The Society was granted a royal charter in 1965, a century after its foundation. In 1998 the Society moved from rooms in Burlington House into De Morgan House (named after the society's first president), at 57–58 Russell Square, Bloomsbury, to accommodate an expansion of its staff. In 2015 the Society celebrated its 150th anniversary. During the year the anniversary was celebrated with a wide range of meetings, events, and other activities, highlighting the historical and continuing value and prevalence of mathematics in society, and in everyday life. == Membership == Membership is open to those who are interested in mathematics. Currently, there are four classes of membership, namely: (a) Ordinary, (b) Reciprocity, (c) Associate, and (d) Associate (undergraduate). In addition, Honorary Members of the Society are distinguished mathematicians who are not normally resident in the UK, who are proposed by the Society's Council for election to Membership at a Society Meeting. == LMS Activities == The Society publishes books and periodicals; organises mathematical conferences; provides funding to promote mathematics research and education; and awards a number of prizes and fellowships for excellence in mathematical research. === Grants === The Society supports mathematics in the UK through its grant schemes. These schemes provide support for mathematicians at different stages in their careers. The Society’s grants include research grants for mathematicians, early career researchers and computer scientists working at the interface of mathematics and computer science; education grants for teachers and other educators; travel grants to attend conferences; and grants for those with caring responsibilities. Awarding grants is one of the primary mechanisms through which the Society achieves its central purpose, namely to 'promote and extend mathematical knowledge’. === Fellowships === The Society also offers a range of Fellowships: LMS Early Career Fellowships; LMS Atiyah-Lebanon UK Fellowships; LMS Emmy Noether Fellowships and Grace Chisholm Young Fellowships. === Society lectures and meetings === The Society organises an annual programme of events and meetings. The programme provides meetings of interest to undergraduates, through early career researchers to established mathematicians. These include LMS-Bath Mathematical Symposia, Lecture Series (Aitken/Forder, Hardy, Invited), Research Schools, LMS Prospects in Mathematics Meeting, Public Lectures, Society Meetings, LMS Undergraduate Summer Schools and Women in Mathematics Days. == Publications == The Society's periodical publications include five journals: Bulletin of the London Mathematical Society (1969–present) Journal of the London Mathematical Society (1926–present) Proceedings of the London Mathematical Society (1865–present) Transactions of the London Mathematical Society (2014–present) Journal of Topology (2006 – present) It also publishes the journal Compositio Mathematica on behalf of its owning foundation, Mathematika on behalf of University College London and Nonlinearity with the Institute of Physics. === Books === The Society publishes two book series, the LMS Lecture Notes and LMS Student Texts. Previously it published a series of Monographs and (jointly with the American Mathematical Society) the History of Mathematics series. An electronic journal, the LMS Journal of Computation and Mathematics, ceased publication at the end of 2017. == Prizes == The named prizes are: De Morgan Medal (triennial) — the most prestigious Pólya Prize (two years out of three) Louis Bachelier Prize (biennial) Senior Berwick Prize Senior Whitehead Prize (biennial) Naylor Prize and Lectureship Berwick Prize Anne Bennett Prize Senior Anne Bennett Prize Fröhlich Prize (biennial) Shephard Prize Whitehead Prize (annual) Hirst Prize In addition, the Society jointly with the Institute of Mathematics and its Applications awards the David Crighton Medal and Christopher Zeeman Medal on alternating years. The LMS also awards the Emmy Noether Fellowships. == List of presidents == Source: == See also == Edinburgh Mathematical Society List of Mathematical Societies Council for the Mathematical Sciences BCS-FACS Specialist Group == References == Oakes, Susan Margaret; Pears, Alan Robson; Rice, Adrian Clifford (2005). The Book of Presidents 1865–1965. London Mathematical Society. ISBN 0-9502734-1-4. == External links == London Mathematical Society website A History of the London Mathematical Society MacTutor: The London Mathematical Society
Wikipedia:Loredana Lanzani#0	Loredana Lanzani (born 1965) is an Italian-American mathematician specializing in harmonic analysis, partial differential equations, and complex analysis. She is a professor of mathematics at Syracuse University. == Education and career == Lanzani earned a laurea from the University of Rome Tor Vergata in 1989, and completed a Ph.D. at Purdue University in 1997. Her dissertation, A New Perspective On The Cauchy Transform For Non-Smooth Domains In The Plane And Applications, was supervised by Steven R. Bell. She became an assistant professor at the University of Arkansas in 1997, and moved up the academic ranks there until becoming a full professor in 2008, also being given the Robert C. & Sandra Connor Endowed Faculty Fellowship in the same year. From 2011 to 2013 she was a program director for the National Science Foundation, and in 2014 she took her present position as a professor of mathematics at Syracuse University. == Recognition == Lanzani was named a Fellow of the American Mathematical Society, in the 2022 class of fellows, "for contributions to function theory in one and several complex variables". She became the first Syracuse University mathematician to win this honor. == References == == External links == Home page Loredana Lanzani publications indexed by Google Scholar
Wikipedia:Lorenzo Ramero#0	Lorenzo Ramero is an Italian mathematician living in France, specialized in algebraic and arithmetic geometry. He is currently a professor of mathematics at the University of Lille. Ramero obtained his Laurea in Matematica from the University of Pisa and his Diploma from the Scuola Normale Superiore di Pisa in 1989. He completed his Ph.D. at the Massachusetts Institute of Technology in 1994 under the supervision of Alexander Beilinson, with a thesis titled An ℓ {\displaystyle \ell } -adic Fourier transform over local fields. Together with Ofer Gabber, Ramero developed the algebraic geometry based on almost rings extending earlier ideas of Gerd Faltings on "almost mathematics". This theory extends already classical algebraic geometry formalism of Alexander Grothendieck's school in order to treat new phenomena in p-adic Hodge theory. This work is systematized in their monograph Almost ring theory. More foundational material was developed after the first book, and especially an extended theory of perfectoid rings and perfectoid spaces which generalizes the recent work of Peter Scholze. These aspects were recapitulated in the book "Foundations for Almost Ring Theory". Ramero has authored a textbook on commutative algebra, Grimoire d'Algèbre Commutative (in French), emphasizing the numerous connections and interactions between commutative algebra and other areas and branches of mathematics. == References == == External links == Personal website PhD thesis at MIT Lorenzo Ramero at the Mathematics Genealogy Project
Wikipedia:Lorna Stewart#0	Lorna Kay Stewart is a retired Canadian computer scientist and discrete mathematician whose research concerns algorithms in graph theory and special classes of graphs, including cographs, permutation graphs, interval graphs, comparability graphs and their complements, well-covered graphs, and asteroidal triple-free graphs. She earned her Ph.D. in 1985 at the University of Toronto under the supervision of Derek Corneil, and is a professor emerita at the University of Alberta. == Selected publications == Corneil, D. G.; Perl, Y.; Stewart, L. K. (1985), "A linear recognition algorithm for cographs", SIAM Journal on Computing, 14 (4): 926–934, doi:10.1137/0214065, MR 0807891, Zbl 0575.68065 Spinrad, Jeremy; Brandstädt, Andreas; Stewart, Lorna (1987), "Bipartite permutation graphs", Discrete Applied Mathematics, 18 (3): 279–292, doi:10.1016/0166-218X(87)90064-3, MR 0917130, Zbl 0628.05055 Sankaranarayana, Ramesh S.; Stewart, Lorna K. (1992), "Complexity results for well-covered graphs", Networks, 22 (3): 247–262, doi:10.1002/net.3230220304, MR 1161178, Zbl 0780.90104 Kratsch, Dieter; Stewart, Lorna (1993), "Domination on cocomparability graphs", SIAM Journal on Discrete Mathematics, 6 (3): 400–417, doi:10.1137/0406032, MR 1229694, Zbl 0780.05032 Corneil, Derek G.; Olariu, Stephan; Stewart, Lorna (1997), "Asteroidal triple-free graphs", SIAM Journal on Discrete Mathematics, 10 (3): 399–430, doi:10.1137/S0895480193250125, MR 1459947, Zbl 0884.05075 Corneil, Derek G.; Olariu, Stephan; Stewart, Lorna (October 2009), "The LBFS structure and recognition of interval graphs", SIAM Journal on Discrete Mathematics, 23 (4): 1905–1953, doi:10.1137/S0895480100373455, MR 2594964, Zbl 1207.05131 == References == == External links == Home page
Wikipedia:Lorna Swain#0	Lorna Mary Swain (22 March 1891 – 8 May 1936) was a British mathematician and college lecturer, known for being one of few female mathematicians to contribute their talents to the war effort in World War I, and for being one of few early female lecturers at University of Cambridge. Academically, she is known for her work in fluid dynamics as well as her deep desire to see more women pursue higher education and teaching in the field of mathematics. == Early life == Swain was born on 22 March 1891. She was the daughter of Edward Swain (born 1853) and Mary Isabella Swain (born 1865). Her father worked as a solicitor. == Education == After attending South Hampstead High School in London, Swain was awarded a scholarship to Newnham College, Cambridge in 1910. Following her graduation three years later with a bachelor's degree (First Class Honours) in mathematics, she was to take a position as assistant lecturer at Newnham College after a year's hiatus for research in Göttingen, Germany in 1914. == Career == Her plans to begin research in Göttingen, Germany in 1914 were scuttled by the outbreak of the First World War. With research in Germany untenable, Swain's specialization in fluid dynamics took her instead to Manchester where she began work alongside Horace Lamb with whom she co-published her first academic article. When she returned to Newnham after a year, as expected, the war temporarily focused her fluid dynamics research on the problem of propeller vibration in aircraft, a considerable problem for aircraft used in the First World War. As June Barrow-Green points out, Swain's work during this time, though it derailed her from planned postgraduate work in Germany, was not only practically useful, but also notable. According to Catherine Goldstein, Swain was "...one of the few women to have her name attached to an Advisory Committee for Aeronautics Report at that time." The resulting research was written up with colleague H.A. Webb in a Report of the Advisory Committee for Aeronautics. In 1923 after returning to Newnham, she would publish, with Arthur Berry, "On the Steady Motion of a Cylinder through Infinite Viscous Fluid" in the Proceedings of the Royal Society. She eventually get the opportunity to complete the planned research in Göttingen on sabbatical in 1928-1929. From this later period of research she produced work "On the Turbulent Wake Behind a Body of Revolution", also published in the Proceedings of the Royal Society in November 1929. By 1920 Swain was promoted to Director of Mathematics Studies at Newnham, and her research suffered under increased teaching and administrative roles. Despite this, Swain used the position to capitalize on her concern for education and teaching. Her teaching philosophy took into account various factors, not least of which was her concern that women were inadequately represented within her chosen field, mathematics. This was paired with a concern for the teaching of applied mathematics. Teaching, she believed, had the potential to stave off the injurious effects that tedious school work could have on the next generation working with applied mathematics. Promoted in 1926 to College Lecturer at Newnham, Swain returned to teaching and research, particularly teaching advanced courses on hydromechanics and dynamics. == Death == Swain died after a long-standing illness on 8 May 1936(1936-05-08) (aged 45), at Derby House, a nursing home in Parson Street, Hendon, in the Borough of Barnet. The funeral service was held on 13 May 1936 at Golders Green Crematorium. == References ==
Wikipedia:Lothar Collatz#0	Lothar Collatz (German: [ˈkɔlaʦ]; July 6, 1910 – September 26, 1990) was a German mathematician, born in Arnsberg, Westphalia. The "3x + 1" problem is also known as the Collatz conjecture, named after him and still unsolved. The Collatz–Wielandt formula for the Perron–Frobenius eigenvalue of a positive square matrix was also named after him. Collatz's 1957 paper with Ulrich Sinogowitz, who had been killed in the bombing of Darmstadt in World War II, founded the field of spectral graph theory. == Biography == Collatz studied at universities in Germany including the University of Greifswald and the University of Berlin, where he was supervised by Alfred Klose, receiving his doctorate in 1935 for a dissertation entitled Das Differenzenverfahren mit höherer Approximation für lineare Differentialgleichungen (The finite difference method with higher approximation for linear differential equations). He then worked as an assistant at the University of Berlin, before moving to the Technische Hochschule Karlsruhe (now Karlsruhe Institute of Technology) in 1935 where he remained through 1937. From 1938 to 1943, he worked as a Privatdozent in Karlsruhe. In the war years he worked with Alwin Walther at the Institute for Practical Mathematics of the Technische Hochschule Darmstadt. From 1943 to 1952, Collatz held a chair at the Technische Hochschule Hannover (now Leibniz University Hanover) . From 1952 until his retirement in 1978, Collatz worked at the University of Hamburg, where he founded the Institute of Applied Mathematics in 1953. After retirement as professor emeritus, he continued to be very active at mathematical conferences. For his many contributions to the field, Collatz had many honors bestowed upon him in his lifetime, including: election to the Academy of Sciences Leopoldina, the Academy of Sciences of the Institute of Bologna, and the Academy at Modena in Italy honorary member of the Hamburg Mathematical Society honorary degrees from the University of São Paulo, the Vienna University of Technology, the University of Dundee in Scotland, Brunel University in England, the University of Hannover in 1981, and the Technische Universität Dresden. He died unexpectedly from a heart attack in Varna, Bulgaria, while attending a mathematics conference. == Selected works == Das Differenzenverfahren mit höherer Approximation für lineare Differentialgleichungen (= Schriften des mathematischen Seminars und des Instituts für angewandte Mathematik der Universität Berlin – Band 3/Heft 1), Leipzig 1935 Eigenwertprobleme und ihre numerische Behandlung. Leipzig 1945 Eigenwertaufgaben mit technischen Anwendungen. Leipzig 1949, 1963 Numerische Behandlung von Differentialgleichungen. Berlin 1951, 1955 (Eng. trans. 1966) Differentialgleichungen für Ingenieure. Stuttgart 1960 with Wolfgang Wetterling: Optimierungsaufgaben Berlin 1966, 1971 (Eng. trans. 1975) Funktionalanalysis und Numerische Mathematik. Berlin 1964 Differentialgleichungen. Eine Einführung unter besonderer Berücksichtigung der Anwendungen. Stuttgart, Teubner Verlag, 1966, 7th edn. 1990 with Julius Albrecht: Aufgaben aus der angewandten Mathematik I. Gleichungen in einer und mehreren Variablen. Approximationen. Berlin 1972 Numerische Methoden der Approximationstheorie. vol. 2. Vortragsauszüge der Tagung über Numerische Methoden der Approximationstheorie vom 3.-9. Juni 1973 im Mathematischen Forschungsinstitut Oberwolfach, Stuttgart 1975 Approximationstheorie: Tschebyscheffsche Approximation und Anwendungen. Teubner 1973 == References == == Sources == Lothar Collatz (July 6, 1910 – September 26, 1990), Journal of Approximation Theory, vol. 65, issue 1, April 1991, page II by Günter Meinardus and Günther Nürnberger Collatz, Lothar (1942), "Einschließungssatz für die charakteristischen Zahlen von Matrizen", Mathematische Zeitschrift, 48 (1): 221–226, doi:10.1007/BF01180013, S2CID 120958677 == Further reading == J Albrecht, P Hagedorn and W Velte, Lothar Collatz (German), Numerical treatment of eigenvalue problems, vol. 5, Oberwolfach, 1990 (Birkhäuser, Basel, 1991), viii–ix. I Althoefer, Lothar Collatz zwischen 1933 und 1950 - Eine Teilbiographie (German), 3-Hirn-Verlag, Lage (Lippe), 2019. R Ansorge, Lothar Collatz (6 July 1910 – 26 September 1990) (German), Mitt. Ges. Angew. Math. Mech. No. 1 (1991), 4–9. U Eckhardt, Der Einfluss von Lothar Collatz auf die angewandte Mathematik, Numerical mathematics, Sympos., Inst. Appl. Math., Univ. Hamburg, Hamburg, 1979 (Birkhäuser, Basel-Boston, Mass., 1979), 9–23. L Elsner and K P Hadeler, Lothar Collatz – on the occasion of his 75th birthday, Linear Algebra Appl. 68 (1985), vi; 1–8. R B Guenther, Obituary : Lothar Collatz, 1910–1990, Aequationes Mathematicae 43 (2–3) (1992), 117–119. H Heinrich, Zum siebzigsten Geburtstag von Lothar Collatz, Z. Angew. Math. Mech. 60 (5) (1980), 274–275. G Meinardus, G Nürnberger, Th Riessinger and G Walz, In memoriam : the work of Lothar Collatz in approximation theory, J. Approx. Theory 67 (2) (1991), 119–128. G Meinardus and G Nürnberger, In memoriam : Lothar Collatz (July 6, 1910 – September 26, 1990), J. Approx. Theory 65 (1) (1991), i; 1–2. J R Whiteman, In memoriam : Lothar Collatz, Internat. J. Numer. Methods Engrg. 31 (8) (1991), 1475–1476. == External links == O'Connor, John J.; Robertson, Edmund F., "Lothar Collatz", MacTutor History of Mathematics Archive, University of St Andrews Lothar Collatz at the Mathematics Genealogy Project
Wikipedia:Lotka–Volterra equations#0	The Lotka–Volterra equations, also known as the Lotka–Volterra predator–prey model, are a pair of first-order nonlinear differential equations, frequently used to describe the dynamics of biological systems in which two species interact, one as a predator and the other as prey. The populations change through time according to the pair of equations: d x d t = α x − β x y , d y d t = − γ y + δ x y , {\displaystyle {\begin{aligned}{\frac {dx}{dt}}&=\alpha x-\beta xy,\\{\frac {dy}{dt}}&=-\gamma y+\delta xy,\end{aligned}}} where the variable x is the population density of prey (for example, the number of rabbits per square kilometre); the variable y is the population density of some predator (for example, the number of foxes per square kilometre); d y d t {\displaystyle {\tfrac {dy}{dt}}} and d x d t {\displaystyle {\tfrac {dx}{dt}}} represent the instantaneous growth rates of the two populations; t represents time; The prey's parameters, α and β, describe, respectively, the maximum prey per capita growth rate, and the effect of the presence of predators on the prey death rate. The predator's parameters, γ, δ, respectively describe the predator's per capita death rate, and the effect of the presence of prey on the predator's growth rate. All parameters are positive and real. The solution of the differential equations is deterministic and continuous. This, in turn, implies that the generations of both the predator and prey are continually overlapping. The Lotka–Volterra system of equations is an example of a Kolmogorov population model (not to be confused with the better known Kolmogorov equations), which is a more general framework that can model the dynamics of ecological systems with predator–prey interactions, competition, disease, and mutualism. == Biological interpretation and model assumptions == The prey are assumed to have an unlimited food supply and to reproduce exponentially, unless subject to predation; this exponential growth is represented in the equation above by the term αx. The rate of predation on the prey is assumed to be proportional to the rate at which the predators and the prey meet; this is represented above by βxy. If either x or y is zero, then there can be no predation. With these two terms the prey equation above can be interpreted as follows: the rate of change of the prey's population is given by its own growth rate minus the rate at which it is preyed upon. The term δxy represents the growth of the predator population. (Note the similarity to the predation rate; however, a different constant is used, as the rate at which the predator population grows is not necessarily equal to the rate at which it consumes the prey). The term γy represents the loss rate of the predators due to either natural death or emigration; it leads to an exponential decay in the absence of prey. Hence the equation expresses that the rate of change of the predator's population depends upon the rate at which it consumes prey, minus its intrinsic death rate. The Lotka–Volterra predator-prey model makes a number of assumptions about the environment and biology of the predator and prey populations: The prey population finds ample food at all times. The food supply of the predator population depends entirely on the size of the prey population. The rate of change of population is proportional to its size. During the process, the environment does not change in favour of one species, and genetic adaptation is inconsequential. Predators have limitless appetite. Both populations can be described by a single variable. This amounts to assuming that the populations do not have a spatial or age distribution that contributes to the dynamics. == Biological relevance of the model == None of the assumptions above are likely to hold for natural populations. Nevertheless, the Lotka–Volterra model shows two important properties of predator and prey populations and these properties often extend to variants of the model in which these assumptions are relaxed: Firstly, the dynamics of predator and prey populations have a tendency to oscillate. Fluctuating numbers of predators and prey have been observed in natural populations, such as the lynx and snowshoe hare data of the Hudson's Bay Company and the moose and wolf populations in Isle Royale National Park. Secondly, the population equilibrium of this model has the property that the prey equilibrium density (given by x = γ / δ {\displaystyle x=\gamma /\delta } ) depends on the predator's parameters, and the predator equilibrium density (given by y = α / β {\displaystyle y=\alpha /\beta } ) on the prey's parameters. This has as a consequence that an increase in, for instance, the prey growth rate, α {\displaystyle \alpha } , leads to an increase in the predator equilibrium density, but not the prey equilibrium density. Making the environment better for the prey benefits the predator, not the prey (this is related to the paradox of the pesticides and to the paradox of enrichment). A demonstration of this phenomenon is provided by the increased percentage of predatory fish caught had increased during the years of World War I (1914–18), when prey growth rate was increased due to a reduced fishing effort. A further example is provided by the experimental iron fertilization of the ocean. In several experiments large amounts of iron salts were dissolved in the ocean. The expectation was that iron, which is a limiting nutrient for phytoplankton, would boost growth of phytoplankton and that it would sequester carbon dioxide from the atmosphere. The addition of iron typically leads to a short bloom in phytoplankton, which is quickly consumed by other organisms (such as small fish or zooplankton) and limits the effect of enrichment mainly to increased predator density, which in turn limits the carbon sequestration. This is as predicted by the equilibrium population densities of the Lotka–Volterra predator-prey model, and is a feature that carries over to more elaborate models in which the restrictive assumptions of the simple model are relaxed. == Applications to economics and marketing == The Lotka–Volterra model has additional applications to areas such as economics and marketing. It can be used to describe the dynamics in a market with several competitors, complementary platforms and products, a sharing economy, and more. There are situations in which one of the competitors drives the other competitors out of the market and other situations in which the market reaches an equilibrium where each firm stabilizes on its market share. It is also possible to describe situations in which there are cyclical changes in the industry or chaotic situations with no equilibrium and changes are frequent and unpredictable. In economics, the Phillips curve, which shows the statistical relationship between unemployment and the rate of change in nominal wages, has been connected by the Goodwin model. This model reinterprets the dynamics of the biological prey-predator interaction, as described by the Lotka-Volterra model, in economic terms. The way the two species interact in this model led Goodwin to draw parallels with the Marxian class conflict. The Kolmogorov generalization of the prey-predator model, along with further developments of the Goodwin model, has extended these ideas. == History == The Lotka–Volterra predator–prey model was initially proposed by Alfred J. Lotka in the theory of autocatalytic chemical reactions in 1910. This was effectively the logistic equation, originally derived by Pierre François Verhulst. In 1920 Lotka extended the model, via Andrey Kolmogorov, to "organic systems" using a plant species and a herbivorous animal species as an example and in 1925 he used the equations to analyse predator–prey interactions in his book on biomathematics. The same set of equations was published in 1926 by Vito Volterra, a mathematician and physicist, who had become interested in mathematical biology. Volterra's enquiry was inspired through his interactions with the marine biologist Umberto D'Ancona, who was courting his daughter at the time and later was to become his son-in-law. D'Ancona studied the fish catches in the Adriatic Sea and had noticed that the percentage of predatory fish caught had increased during the years of World War I (1914–18). This puzzled him, as the fishing effort had been very much reduced during the war years and, as prey fish the preferred catch, one would intuitively expect this to increase of prey fish percentage. Volterra developed his model to explain D'Ancona's observation and did this independently from Alfred Lotka. He did credit Lotka's earlier work in his publication, after which the model has become known as the "Lotka-Volterra model". The model was later extended to include density-dependent prey growth and a functional response of the form developed by C. S. Holling; a model that has become known as the Rosenzweig–MacArthur model. Both the Lotka–Volterra and Rosenzweig–MacArthur models have been used to explain the dynamics of natural populations of predators and prey. In the late 1980s, an alternative to the Lotka–Volterra predator–prey model (and its common-prey-dependent generalizations) emerged, the ratio dependent or Arditi–Ginzburg model. The validity of prey- or ratio-dependent models has been much debated. The Lotka–Volterra equations have a long history of use in economic theory; their initial application is commonly credited to Richard Goodwin in 1965 or 1967. == Solutions to the equations == The equations have periodic solutions. These solutions do not have a simple expression in terms of the usual trigonometric functions, although they are quite tractable. If none of the non-negative parameters α, β, γ, δ vanishes, three can be absorbed into the normalization of variables to leave only one parameter: since the first equation is homogeneous in x, and the second one in y, the parameters β/α and δ/γ are absorbable in the normalizations of y and x respectively, and γ into the normalization of t, so that only α/γ remains arbitrary. It is the only parameter affecting the nature of the solutions. A linearization of the equations yields a solution similar to simple harmonic motion with the population of predators trailing that of prey by 90° in the cycle. === A simple example === Suppose there are two species of animals, a rabbit (prey) and a fox (predator). If the initial densities are 10 rabbits and 10 foxes per square kilometre, one can plot the progression of the two species over time; given the parameters that the growth and death rates of rabbits are 1.1 and 0.4 while that of foxes are 0.1 and 0.4 respectively. The choice of time interval is arbitrary. One may also plot solutions parametrically as orbits in phase space, without representing time, but with one axis representing the number of prey and the other axis representing the densities of predators for all times. This corresponds to eliminating time from the two differential equations above to produce a single differential equation d y d x = − y x δ x − γ β y − α {\displaystyle {\frac {dy}{dx}}=-{\frac {y}{x}}{\frac {\delta x-\gamma }{\beta y-\alpha }}} relating the variables x (predator) and y (prey). The solutions of this equation are closed curves. It is amenable to separation of variables: integrating β y − α y d y + δ x − γ x d x = 0 {\displaystyle {\frac {\beta y-\alpha }{y}}\,dy+{\frac {\delta x-\gamma }{x}}\,dx=0} yields the implicit relationship V = δ x − γ ln ⁡ ( x ) + β y − α ln ⁡ ( y ) , {\displaystyle V=\delta x-\gamma \ln(x)+\beta y-\alpha \ln(y),} where V is a constant quantity depending on the initial conditions and conserved on each curve. An aside: These graphs illustrate a serious potential limitation in the application as a biological model: for this specific choice of parameters, in each cycle, the rabbit population is reduced to extremely low numbers, yet recovers (while the fox population remains sizeable at the lowest rabbit density). In real-life situations, however, chance fluctuations of the discrete numbers of individuals might cause the rabbits to actually go extinct, and, by consequence, the foxes as well. This modelling problem has been called the "atto-fox problem", an atto-fox being a notional 10−18 of a fox. A density of 10−18 foxes per square kilometre equates to an average of approximately 5×10−10 foxes on the surface of the earth, which in practical terms means that foxes are extinct. === Hamiltonian structure of the system === Since the quantity V ( x , y ) {\displaystyle V(x,y)} is conserved over time, it plays role of a Hamiltonian function of the system. To see this we can define Poisson bracket as follows { f ( x , y ) , g ( x , y ) } = − x y ( ∂ f ∂ x ∂ g ∂ y − ∂ f ∂ y ∂ g ∂ x ) {\displaystyle \{f(x,y),g(x,y)\}=-xy\left({\frac {\partial f}{\partial x}}{\frac {\partial g}{\partial y}}-{\frac {\partial f}{\partial y}}{\frac {\partial g}{\partial x}}\right)} . Then Hamilton's equations read { x ˙ = { x , V } = α x − β x y , y ˙ = { y , V } = δ x y − γ y . {\displaystyle {\begin{cases}{\dot {x}}=\{x,V\}=\alpha x-\beta xy,\\{\dot {y}}=\{y,V\}=\delta xy-\gamma y.\end{cases}}} The variables x {\displaystyle x} and y {\displaystyle y} are not canonical, since { x , y } = − x y ≠ 1 {\displaystyle \{x,y\}=-xy\neq 1} . However, using transformations p = ln ⁡ ( x ) {\displaystyle p=\ln(x)} and q = ln ⁡ ( y ) {\displaystyle q=\ln(y)} we came up to a canonical form of the Hamilton's equations featuring the Hamiltonian H ( q , p ) = V ( x ( q , p ) , y ( q , p ) ) = δ e p − γ p + β e q − α q {\displaystyle H(q,p)=V(x(q,p),y(q,p))=\delta e^{p}-\gamma p+\beta e^{q}-\alpha q} : { q ˙ = ∂ H ∂ p = δ e p − γ , p ˙ = − ∂ H ∂ q = α − β e q . {\displaystyle {\begin{cases}{\dot {q}}={\frac {\partial H}{\partial p}}=\delta e^{p}-\gamma ,\\{\dot {p}}=-{\frac {\partial H}{\partial q}}=\alpha -\beta e^{q}.\end{cases}}} The Poisson bracket for the canonical variables ( q , p ) {\displaystyle (q,p)} now takes the standard form { F ( q , p ) , G ( q , p ) } = ( ∂ F ∂ q ∂ G ∂ p − ∂ F ∂ p ∂ G ∂ q ) {\displaystyle \{F(q,p),G(q,p)\}=\left({\frac {\partial F}{\partial q}}{\frac {\partial G}{\partial p}}-{\frac {\partial F}{\partial p}}{\frac {\partial G}{\partial q}}\right)} . === Phase-space plot of a further example === Another example covers: α = 2/3, β = 4/3, γ = 1 = δ. Assume x, y quantify thousands each. Circles represent prey and predator initial conditions from x = y = 0.9 to 1.8, in steps of 0.1. The fixed point is at (1, 1/2). == Dynamics of the system == In the model system, the predators thrive when prey is plentiful but, ultimately, outstrip their food supply and decline. As the predator population is low, the prey population will increase again. These dynamics continue in a population cycle of growth and decline. === Population equilibrium === Population equilibrium occurs in the model when neither of the population levels is changing, i.e. when both of the derivatives are equal to 0: x ( α − β y ) = 0 , {\displaystyle x(\alpha -\beta y)=0,} − y ( γ − δ x ) = 0. {\displaystyle -y(\gamma -\delta x)=0.} The above system of equations yields two solutions: { y = 0 , x = 0 } {\displaystyle \{y=0,\ \ x=0\}} and { y = α β , x = γ δ } . {\displaystyle \left\{y={\frac {\alpha }{\beta }},\ \ x={\frac {\gamma }{\delta }}\right\}.} Hence, there are two equilibria. The first solution effectively represents the extinction of both species. If both populations are at 0, then they will continue to be so indefinitely. The second solution represents a fixed point at which both populations sustain their current, non-zero numbers, and, in the simplified model, do so indefinitely. The levels of population at which this equilibrium is achieved depend on the chosen values of the parameters α, β, γ, and δ. === Stability of the fixed points === The stability of the fixed point at the origin can be determined by performing a linearization using partial derivatives. The Jacobian matrix of the predator–prey model is J ( x , y ) = [ α − β y − β x δ y δ x − γ ] . {\displaystyle J(x,y)={\begin{bmatrix}\alpha -\beta y&-\beta x\\\delta y&\delta x-\gamma \end{bmatrix}}.} and is known as the community matrix. ==== First fixed point (extinction) ==== When evaluated at the steady state of (0, 0), the Jacobian matrix J becomes J ( 0 , 0 ) = [ α 0 0 − γ ] . {\displaystyle J(0,0)={\begin{bmatrix}\alpha &0\\0&-\gamma \end{bmatrix}}.} The eigenvalues of this matrix are λ 1 = α , λ 2 = − γ . {\displaystyle \lambda _{1}=\alpha ,\quad \lambda _{2}=-\gamma .} In the model α and γ are always greater than zero, and as such the sign of the eigenvalues above will always differ. Hence the fixed point at the origin is a saddle point. The instability of this fixed point is of significance. If it were stable, non-zero populations might be attracted towards it, and as such the dynamics of the system might lead towards the extinction of both species for many cases of initial population levels. However, as the fixed point at the origin is a saddle point, and hence unstable, it follows that the extinction of both species is difficult in the model. (In fact, this could only occur if the prey were artificially completely eradicated, causing the predators to die of starvation. If the predators were eradicated, the prey population would grow without bound in this simple model.) The populations of prey and predator can get infinitesimally close to zero and still recover. ==== Second fixed point (oscillations) ==== Evaluating J at the second fixed point leads to J ( γ δ , α β ) = [ 0 − β γ δ α δ β 0 ] . {\displaystyle J\left({\frac {\gamma }{\delta }},{\frac {\alpha }{\beta }}\right)={\begin{bmatrix}0&-{\frac {\beta \gamma }{\delta }}\\{\frac {\alpha \delta }{\beta }}&0\end{bmatrix}}.} The eigenvalues of this matrix are λ 1 = i α γ , λ 2 = − i α γ . {\displaystyle \lambda _{1}=i{\sqrt {\alpha \gamma }},\quad \lambda _{2}=-i{\sqrt {\alpha \gamma }}.} As the eigenvalues are both purely imaginary and conjugate to each other, this fixed point must either be a center for closed orbits in the local vicinity or an attractive or repulsive spiral. In conservative systems, there must be closed orbits in the local vicinity of fixed points that exist at the minima and maxima of the conserved quantity. The conserved quantity is derived above to be V = δ x − γ ln ⁡ ( x ) + β y − α ln ⁡ ( y ) {\displaystyle V=\delta x-\gamma \ln(x)+\beta y-\alpha \ln(y)} on orbits. Thus orbits about the fixed point are closed and elliptic, so the solutions are periodic, oscillating on a small ellipse around the fixed point, with a frequency ω = λ 1 λ 2 = α γ {\displaystyle \omega ={\sqrt {\lambda _{1}\lambda _{2}}}={\sqrt {\alpha \gamma }}} and period T = 2 π / ( λ 1 λ 2 ) {\displaystyle T=2{\pi }/({\sqrt {\lambda _{1}\lambda _{2}}})} . As illustrated in the circulating oscillations in the figure above, the level curves are closed orbits surrounding the fixed point: the levels of the predator and prey populations cycle and oscillate without damping around the fixed point with frequency ω = α γ {\displaystyle \omega ={\sqrt {\alpha \gamma }}} . The value of the constant of motion V, or, equivalently, K = exp(−V), K = y α e − β y x γ e − δ x {\displaystyle K=y^{\alpha }e^{-\beta y}x^{\gamma }e^{-\delta x}} , can be found for the closed orbits near the fixed point. Increasing K moves a closed orbit closer to the fixed point. The largest value of the constant K is obtained by solving the optimization problem y α e − β y x γ e − δ x = y α x γ e δ x + β y ⟶ max x , y > 0 . {\displaystyle y^{\alpha }e^{-\beta y}x^{\gamma }e^{-\delta x}={\frac {y^{\alpha }x^{\gamma }}{e^{\delta x+\beta y}}}\longrightarrow \max _{x,y>0}.} The maximal value of K is thus attained at the stationary (fixed) point ( γ δ , α β ) {\displaystyle \left({\frac {\gamma }{\delta }},{\frac {\alpha }{\beta }}\right)} and amounts to K ∗ = ( α β e ) α ( γ δ e ) γ , {\displaystyle K^{*}=\left({\frac {\alpha }{\beta e}}\right)^{\alpha }\left({\frac {\gamma }{\delta e}}\right)^{\gamma },} where e is Euler's number. == See also == == Notes == == Further reading == Hofbauer, Josef; Sigmund, Karl (1998). "Dynamical Systems and Lotka–Volterra Equations". Evolutionary Games and Population Dynamics. New York: Cambridge University Press. pp. 1–54. ISBN 0-521-62570-X. Kaplan, Daniel; Glass, Leon (1995). Understanding Nonlinear Dynamics. New York: Springer. ISBN 978-0-387-94440-1. Leigh, E. R. (1968). "The ecological role of Volterra's equations". Some Mathematical Problems in Biology. – a modern discussion using Hudson's Bay Company data on lynx and hares in Canada from 1847 to 1903. Murray, J. D. (2003). Mathematical Biology I: An Introduction. New York: Springer. ISBN 978-0-387-95223-9.' Stefano Allesina's Community Ecology course lecture notes: https://stefanoallesina.github.io/Theoretical_Community_Ecology/ == External links == From the Wolfram Demonstrations Project — requires CDF player (free): Predator–Prey Equations Predator–Prey Model Predator–Prey Dynamics with Type-Two Functional Response Predator–Prey Ecosystem: A Real-Time Agent-Based Simulation Lotka-Volterra Algorithmic Simulation (Web simulation).
Wikipedia:Lotte Hollands#0	Lotte Hollands (born 1981) is a Dutch mathematician and mathematical physicist who studies quantum field theory, supersymmetric gauge theory, and string theory. She is an associate professor and Royal Society Dorothy Hodgkin Fellow in the Department of Mathematics at Heriot-Watt University. == Early life == Hollands was born in Maasbree, Netherlands. == Education == Hollands earned her PhD at the University of Amsterdam in 2009. Her dissertation, Topological Strings and Quantum Curves, was supervised by Robbert Dijkgraaf. Hollands did her postdoctoral research with Sergei Gukov at the California Institute of Technology. == Career == In 2013, Hollands became a research fellow at the University of Oxford. In 2015, Hollands became an associate professor at the Department of Math at Heriot-Watt University. == Recognition == In 2018 the London Mathematical Society gave her their Anne Bennett Prize "in recognition of her outstanding research at the interface between quantum theory and geometry and of her leadership in mathematical outreach activities". == References == == External links == Home page Lotte Hollands publications indexed by Google Scholar Lotte Hollands at the Mathematics Genealogy Project
Wikipedia:Lou van den Dries#0	Laurentius Petrus Dignus "Lou" van den Dries (born May 26, 1951) is a Dutch mathematician working in model theory. He is a professor emeritus of mathematics at the University of Illinois Urbana-Champaign. == Education == Van den Dries began his undergraduate studies in 1969 at Utrecht University, and in 1978 completed his PhD there under the supervision of Dirk van Dalen with a dissertation entitled Model Theory of Fields. == Career and research == Van den Dries was a member of the Institute for Advanced Study in the 1982–1983 academic year. He joined the faculty of the University of Illinois Urbana-Champaign in 1986 and became a professor in its Center for Advanced Study in 1998. In 2021, van den Dries retired and became a professor emeritus. Van den Dries is most known for his seminal work in o-minimality, but he has also made contributions to the model theory of p-adic fields, valued fields, and finite fields, and to the study of transseries. With Alex Wilkie, he improved Gromov's theorem on groups of polynomial growth using nonstandard methods. Van den Dries was an invited speaker at the International Congress of Mathematicians in 1990 and 2018, and delivered the Tarski Lectures at the University of California, Berkeley in 2017. == Awards and honours == Van den Dries has been a corresponding member of the Royal Netherlands Academy of Arts and Sciences since 1993. He was awarded the Shoenfield Prize from the Association for Symbolic Logic in 2016 for his chapter "Lectures on the Model Theory of Valued Fields" in Model Theory in Algebra, Analysis and Arithmetic, edited by Dugald Macpherson and Carlo Toffalori. Van den Dries was jointly awarded the 2018 Karp Prize with Matthias Aschenbrenner and Joris van der Hoeven "for their work in model theory, especially on asymptotic differential algebra and the model theory of transseries". == Ethics training == Since 2004, employees of the state of Illinois, including University of Illinois faculty, are required by the State Officials and Employees Ethics Act to complete ethics training annually. From 2006 to 2009, van den Dries refused to complete this training, arguing that mandatory ethics training for adults is an Orwellian concept and has no place in a civil and free society. It is Big Brother reducing us to the status of children. Symptoms: monitoring of the test taking, the 'award' of a diploma for passing the test. It betrays a totalitarian urge on those in power to infantilize the rest of us. An unfortunate byproduct of the computer revolution is that it has given new tools in the hands of unwise rulers to annoy us for no good reason. Rather than go meekly along, we should vigorously protest and resist whenever demeaning schemes like ethics training rear their ugly head. Eventually, van den Dries settled with the Illinois Executive Ethics Commission, which enforces the ethics act, for a $500 fine, noting that "while many of my colleagues agree that this ethics training is a big waste of time and money, they didn't really take the steps I took in trying to fight it. So without active support from my colleagues, it became too time consuming and costly (lawyers fees) to continue my resistance." Van den Dries was the first state employee to be fined by the Illinois Executive Ethics Commission for failing to complete the mandatory training. == Selected publications == M. Aschenbrenner; L. van den Dries; J. van der Hoeven (2017). Asymptotic Differential Algebra and Model Theory of Transseries. Annals of Mathematics Studies. Vol. 195. Princeton University Press. arXiv:1509.02588. doi:10.1515/9781400885411. ISBN 9781400885411. MR 3585498. Zbl 1430.12002. Z. Chatzidakis; L. van den Dries; A. Macintyre (1992). "Definable sets over finite fields". J. Reine Angew. Math. 1992 (427): 107–135. doi:10.1515/crll.1992.427.107. MR 1162433. S2CID 118058593. Zbl 0759.11045. J. Denef; L. van den Dries (1988). "p-adic and real subanalytic sets". Ann. of Math. Series 2. 128 (1): 79–138. doi:10.2307/1971463. JSTOR 1971463. MR 0951508. Zbl 0693.14012. L. van den Dries (1998). Tame topology and o-minimal structures. London Mathematical Society Lecture Notes. Vol. 248. Cambridge University Press. doi:10.1017/CBO9780511525919. ISBN 9780511525919. MR 1633348. Zbl 0953.03045. L. van den Dries (2014), "Lectures on the Model Theory of Valued Fields", in H. Dugald Macpherson; C. Toffalori (eds.), Model Theory in Algebra, Analysis and Arithmetic, Lecture Notes in Mathematics, vol. 2111, Springer-Verlag, pp. 55–157, doi:10.1007/978-3-642-54936-6_4, ISBN 978-3-642-54935-9, MR 3330198, Zbl 1347.03074 L. van den Dries; A. Macintyre; D. Marker (1994). "The elementary theory of restricted analytic fields with exponentiation". Ann. of Math. Series 2. 140 (1): 183–205. doi:10.2307/2118545. JSTOR 2118545. MR 1289495. S2CID 119703238. Zbl 0837.12006. L. van den Dries; C. Miller (1996). "Geometric categories and o-minimal structures". Duke Math. J. 84 (2): 497–540. doi:10.1215/S0012-7094-96-08416-1. MR 1404337. Zbl 0889.03025. L. van den Dries; K. Schmidt (1984). "Bounds in the theory of polynomial rings over fields. A nonstandard approach". Invent. Math. 76 (1): 77–91. Bibcode:1984InMat..76...77D. doi:10.1007/BF01388493. MR 0739626. S2CID 96471248. Zbl 0539.13011. L. van den Dries; A. Wilkie (1984). "Gromov's theorem of groups of polynomial growth and elementary logic". J. Algebra. 89 (2): 349–374. doi:10.1016/0021-8693(84)90223-0. MR 0751150. Zbl 0552.20017. == References ==
Wikipedia:Louay Bazzi#0	In computer science, the Akra–Bazzi method, or Akra–Bazzi theorem, is used to analyze the asymptotic behavior of the mathematical recurrences that appear in the analysis of divide and conquer algorithms where the sub-problems have substantially different sizes. It is a generalization of the master theorem for divide-and-conquer recurrences, which assumes that the sub-problems have equal size. It is named after mathematicians Mohamad Akra and Louay Bazzi. == Formulation == The Akra–Bazzi method applies to recurrence formulas of the form: T ( x ) = g ( x ) + ∑ i = 1 k a i T ( b i x + h i ( x ) ) for x ≥ x 0 . {\displaystyle T(x)=g(x)+\sum _{i=1}^{k}a_{i}T(b_{i}x+h_{i}(x))\qquad {\text{for }}x\geq x_{0}.} The conditions for usage are: sufficient base cases are provided a i {\displaystyle a_{i}} and b i {\displaystyle b_{i}} are constants for all i {\displaystyle i} a i > 0 {\displaystyle a_{i}>0} for all i {\displaystyle i} 0 < b i < 1 {\displaystyle 0<b_{i}<1} for all i {\displaystyle i} \| g ′ ( x ) \| ∈ O ( x c ) {\displaystyle \left\|g'(x)\right\|\in O(x^{c})} , where c is a constant and O notates Big O notation \| h i ( x ) \| ∈ O ( x ( log ⁡ x ) 2 ) {\displaystyle \left\|h_{i}(x)\right\|\in O\left({\frac {x}{(\log x)^{2}}}\right)} for all i {\displaystyle i} x 0 {\displaystyle x_{0}} is a constant The asymptotic behavior of T ( x ) {\displaystyle T(x)} is found by determining the value of p {\displaystyle p} for which ∑ i = 1 k a i b i p = 1 {\displaystyle \sum _{i=1}^{k}a_{i}b_{i}^{p}=1} and plugging that value into the equation: T ( x ) ∈ Θ ( x p ( 1 + ∫ 1 x g ( u ) u p + 1 d u ) ) {\displaystyle T(x)\in \Theta \left(x^{p}\left(1+\int _{1}^{x}{\frac {g(u)}{u^{p+1}}}du\right)\right)} (see Θ). Intuitively, h i ( x ) {\displaystyle h_{i}(x)} represents a small perturbation in the index of T {\displaystyle T} . By noting that ⌊ b i x ⌋ = b i x + ( ⌊ b i x ⌋ − b i x ) {\displaystyle \lfloor b_{i}x\rfloor =b_{i}x+(\lfloor b_{i}x\rfloor -b_{i}x)} and that the absolute value of ⌊ b i x ⌋ − b i x {\displaystyle \lfloor b_{i}x\rfloor -b_{i}x} is always between 0 and 1, h i ( x ) {\displaystyle h_{i}(x)} can be used to ignore the floor function in the index. Similarly, one can also ignore the ceiling function. For example, T ( n ) = n + T ( 1 2 n ) {\displaystyle T(n)=n+T\left({\frac {1}{2}}n\right)} and T ( n ) = n + T ( ⌊ 1 2 n ⌋ ) {\displaystyle T(n)=n+T\left(\left\lfloor {\frac {1}{2}}n\right\rfloor \right)} will, as per the Akra–Bazzi theorem, have the same asymptotic behavior. == Example == Suppose T ( n ) {\displaystyle T(n)} is defined as 1 for integers 0 ≤ n ≤ 3 {\displaystyle 0\leq n\leq 3} and n 2 + 7 4 T ( ⌊ 1 2 n ⌋ ) + T ( ⌈ 3 4 n ⌉ ) {\displaystyle n^{2}+{\frac {7}{4}}T\left(\left\lfloor {\frac {1}{2}}n\right\rfloor \right)+T\left(\left\lceil {\frac {3}{4}}n\right\rceil \right)} for integers n > 3 {\displaystyle n>3} . In applying the Akra–Bazzi method, the first step is to find the value of p {\displaystyle p} for which 7 4 ( 1 2 ) p + ( 3 4 ) p = 1 {\displaystyle {\frac {7}{4}}\left({\frac {1}{2}}\right)^{p}+\left({\frac {3}{4}}\right)^{p}=1} . In this example, p = 2 {\displaystyle p=2} . Then, using the formula, the asymptotic behavior can be determined as follows: T ( x ) ∈ Θ ( x p ( 1 + ∫ 1 x g ( u ) u p + 1 d u ) ) = Θ ( x 2 ( 1 + ∫ 1 x u 2 u 3 d u ) ) = Θ ( x 2 ( 1 + ln ⁡ x ) ) = Θ ( x 2 log ⁡ x ) . {\displaystyle {\begin{aligned}T(x)&\in \Theta \left(x^{p}\left(1+\int _{1}^{x}{\frac {g(u)}{u^{p+1}}}\,du\right)\right)\\&=\Theta \left(x^{2}\left(1+\int _{1}^{x}{\frac {u^{2}}{u^{3}}}\,du\right)\right)\\&=\Theta (x^{2}(1+\ln x))\\&=\Theta (x^{2}\log x).\end{aligned}}} == Significance == The Akra–Bazzi method is more useful than most other techniques for determining asymptotic behavior because it covers such a wide variety of cases. Its primary application is the approximation of the running time of many divide-and-conquer algorithms. For example, in the merge sort, the number of comparisons required in the worst case, which is roughly proportional to its runtime, is given recursively as T ( 1 ) = 0 {\displaystyle T(1)=0} and T ( n ) = T ( ⌊ 1 2 n ⌋ ) + T ( ⌈ 1 2 n ⌉ ) + n − 1 {\displaystyle T(n)=T\left(\left\lfloor {\frac {1}{2}}n\right\rfloor \right)+T\left(\left\lceil {\frac {1}{2}}n\right\rceil \right)+n-1} for integers n > 0 {\displaystyle n>0} , and can thus be computed using the Akra–Bazzi method to be Θ ( n log ⁡ n ) {\displaystyle \Theta (n\log n)} . == See also == Master theorem (analysis of algorithms) Asymptotic complexity == References == == External links == O Método de Akra-Bazzi na Resolução de Equações de Recorrência (in Portuguese)
Wikipedia:Louis Bertrand Castel#0	Louis Bertrand Castel (5 November 1688 – 11 January 1757) was a French mathematician born in Montpellier, who entered the order of the Jesuits in 1703. Having studied literature, he afterwards devoted himself entirely to mathematics and natural philosophy. After moving from Toulouse to Paris in 1720, at the behest of Bernard de Fontenelle, Castel acted as the science editor of the Jesuit Journal de Trévoux. He wrote several scientific works, that which attracted most attention at the time being his Optique des couleurs (1740), or treatise on the melody of colours. He also wrote Traité de physique sur la pesanteur universelle des corps (1724), Mathématique universelle (1728), and a critical account of the system of Sir Isaac Newton in 1743. == Philosophical approach == Castel wrote on areas as wide-ranging as physics, mathematics, morals, aesthetics, theology and history. His philosophical approach attempted to reconcile fields and viewpoints. Castel based much of his work on analogical thinking, seeking to understand the physical and moral worlds through the discovery of analogies. Castel's first major published work was his Traité de physique de la pesanteur universelle des corps (1724). He first attempted to systematise physical phenomena, through the mechanical action of universal gravity. He then considered a mechanistic world-view's shortcomings, from a theological and metaphysical perspective. He held humanity as central to natural philosophy, in that humans are embodied spirits whose actions, chosen with free will, affect the world around them and each other. In emphasising free will and the actions of mankind Castel attempted to counter deterministic views of man and nature. Castel considered that true science should focus on readily experienced and described phenomena. His emphasis on the description and analysis of the perceived world was consistent with analogic thinking and phenomenal explanation. Castel actively opposed the idea of a science based on experimental methods, instruments, speculation and theorising. == The Ocular Harpsichord == Early on, Castel illustrated his optical theories with a proposal for a Clavecin pour les yeux (Ocular Harpsichord, 1725). A new series of articles, published in the Mercure de France in 1735, gave his idea wider currency. In 1739 the German composer Telemann went to France to see Castel's Ocular Harpsichord for himself. He ended up composing several pieces for it, as well as writing a description of it. The ocular harpsichord had sixty small coloured glass panes, each with a curtain that opened when a key was struck. A second, improved model of the harpsichord was demonstrated for a small audience in December 1754. Pressing a key caused a small shaft to open, in turn allowing light to shine through a piece of stained glass. Castel thought of colour-music as akin to the lost language of paradise, where all men spoke alike, and he claimed that thanks to his instrument's capacity to paint sounds, even a deaf listener could enjoy music. == Criticism of Newton == It was in 1740 that Louis Bertrand Castel published a criticism of Newton's spectral description of prismatic colour in which he observed that the colours of white light split by a prism depended on the distance from the prism, and that Newton was looking at a special case. It was an argument that Goethe later developed in his Theory of Colours. == See also == Color organ == References == This article incorporates text from a publication now in the public domain: Chisholm, Hugh, ed. (1911). "Castel, Louis Bertrand". Encyclopædia Britannica (11th ed.). Cambridge University Press. == External links == "Musique Oculaire" in Edmé-Gilles Guyot, Nouvelles récréations physiques et mathématiques, Gueffier, Paris 1770, pp. 234–240.
Wikipedia:Louis Chen Hsiao Yun#0	Louis Chen Hsiao Yun (Chinese: 陈晓云; pinyin: Chen Xiaoyun; born 26 December 1940) is emeritus professor at the National University of Singapore. Chen earned his BSc (Honours) from University of Singapore in 1964 and completed his MSc as well as PhD at Stanford University in 1969 and 1971 respectively. In 1972, he joined the mathematics department of the then University of Singapore as a lecturer. He was promoted to senior lecturer in 1977, associate professor in 1981 and professor in 1989. == Academic career == Chen held several administrative appointments at the National University of Singapore (NUS). At NUS's Faculty of Science, he was head of the Department of Mathematics from July 1996 to June 2000 and head of the Department of Statistics and Applied Probability from July 2002 to June 2004. He was also director of NUS's Institute for Mathematical Sciences from July 2000 to December 2012. For his achievements in his field of study, he was appointed Tan Chin Tuan Centennial Professor from July 2006 to December 2012 and distinguished professor from January 2013 to June 2015. == Research, honours and awards == Chen's research interests include the Stein-Chen method of Poisson approximation which deals with the probability of rare events. This method and technique is widely applied in many areas ranging from molecular biology to computer science. He was the first Asian to be elected president of the Bernoulli Society for Mathematical Statistics and Probability from August 1997 to July 1999, and was the first East Asian to be elected president of the Institute of Mathematical Statistics from August 2004 to July 2005. He was also elected vice-president of the International Statistical Institute from August 2009 to July 2011. Among his numerous local and international awards and honours are his election as Fellow of TWAS, The World Academy of Sciences in 2000 and Fellow of the Singapore National Academy of Science in 2011; a Public Administration Medal (Silver) in 2002; and the conferment by the French Government of the title of Chevalier dans l'Ordre des Palmes Académiques in 2005 for his service to education. == Selected works == Chen, Louis H. Y., "Poisson approximation for dependent trials", Ann. Probability 3 (1975), no. 3, 534–545. Chen, Louis H. Y.; Goldstein, Larry; Shao, Qi-Man, "Normal approximation by Stein's method". Probability and its Applications (New York). Springer, Heidelberg, 2011. ISBN 978-3-642-15006-7. Barbour, A. D.; Chen, Louis H. Y.; Loh, Wei-Liem, "Compound Poisson approximation for nonnegative random variables via Stein's method", Ann. Probab. 20 (1992), no. 4, 1843–1866. Chen, Louis H. Y.; Röllin, Adrian; Xia, Aihua, “Palm theory, random measures and Stein couplings”, Ann. Appl. Probab. 31 (2021), no. 6, 2881-2923. Chen, Louis H. Y.; Shao, Qi-Man, "Stein's method for normal approximation". An introduction to Stein's method, 1–59, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., 4, Singapore Univ. Press, Singapore, 2005. == References == Leong, Y. K.; Lee, S. L.; Sun, Y. (2005). "Louis Hsiao-Yun Chen: Epitome of Rare Events". Cosmos. 01: 1–15. doi:10.1142/S0219607705000127. "Distinguished Science Alumni Awards 2004: CHEN Hsiao Yun Louis". National University of Singapore. Retrieved 28 March 2013. Chua, G. (2015). "A series of rare events". Singapore's Scientific Pioneers. Asian Scientist Publishing. pp. 30–33. == External links == Louis H. Y. CHEN, Department of Mathematics, National University of Singapore Louis Chen Hsiao Yun at the Mathematics Genealogy Project
Wikipedia:Louis Costaz#0	Louis, baron Costaz (17 March 1767, in Champagne-en-Valromey (Bugey – 15 February 1842, in Paris) was a French scientist and administrator. His brother Benoît Costaz (1761–1842) was bishop of Nancy. After studying mathematics, he taught at the military school at Thiron until 1793, then at the École polytechnique. A member of the Commission des Sciences et des Arts, he participated in the French invasion of Egypt, becoming secretary to the Institut d'Égypte and a member of the Privy Council of Egypt, as well as accompanying Bonaparte to Suez. On his return to France, he presided over the Tribunat (1801–1803) and was entrusted with organising a school of arts and crafts. Prefect of the Manche area (1804–1809) and a baron de l'Empire from 1809, he was intendant of crown buildings (1809–1813) before becoming director general of bridges and roads (1813–1814). Summoned to the Conseil d’État in 1813, he was made prefect of Nord (as extraordinary commissaire) during the Hundred Days and retired from public life shortly afterwards.
Wikipedia:Louis Couturat#0	Louis Couturat (French: [kutyʁa]; 17 January 1868 – 3 August 1914) was a French logician, mathematician, philosopher, and linguist. Couturat was a pioneer of the constructed language Ido. == Life and education == Born in Paris. In 1887 he entered École Normale Supérieure to study philosophy and mathematics. In 1895 he lectured in philosophy at the University of Toulouse and 1897 lectured in philosophy of mathematics at the University of Caen Normandy, taking a stand in favor of transfinite numbers. After a time in Hanover studying the writings of Leibniz, he became an assistant to Henri-Louis Bergson at the Collège de France in 1905. == Career == He was the French advocate of the symbolic logic that emerged in the years before World War I, thanks to the writings of Charles Sanders Peirce, Giuseppe Peano and his school, and especially to The Principles of Mathematics by Couturat's friend and correspondent Bertrand Russell. Like Russell, Couturat saw symbolic logic as a tool to advance both mathematics and the philosophy of mathematics. In this, he was opposed by Henri Poincaré, who took considerable exception to Couturat's efforts to interest the French in symbolic logic. With the benefit of hindsight, we can see that Couturat was in broad agreement with the logicism of Russell, while Poincaré anticipated Brouwer's intuitionism. His first major publication was De Platonicis mythis (1896). In 1901, he published La Logique de Leibniz, a detailed study of Leibniz the logician, based on his examination of the huge Leibniz Nachlass in Hanover. Even though Leibniz had died in 1716, his Nachlass was cataloged only in 1895. Only then was it possible to determine the extent of Leibniz's unpublished work on logic. In 1903, Couturat published much of that work in another large volume, his Opuscules et Fragments Inedits de Leibniz, containing many of the documents he had examined while writing La Logique. Couturat was thus the first to appreciate that Leibniz was the greatest logician during the more than 2000 years that separate Aristotle from George Boole and Augustus De Morgan. A significant part of the 20th century Leibniz revival is grounded in Couturat's editorial and exegetical efforts. This work on Leibniz attracted Russell, also the author of a 1900 book on Leibniz, and thus began their professional correspondence and friendship. In 1905, Couturat published a work on logic and the foundations of mathematics (with an appendix on Kant's philosophy of mathematics) that was originally conceived as a translation of Russell's Principles of Mathematics. In the same year, he published L'Algèbre de la logique, a classic introduction to Boolean algebra and the works of C.S. Peirce and Ernst Schröder. In 1907, Couturat helped found the constructed language Ido, an offshoot of Esperanto, and was Ido's principal advocate over the remainder of his life. By advocating a constructed international language, constructed along logical principles and with a vocabulary taken from existing European languages, Couturat was paralleling Peano's advocacy of Interlingua. By pushing Ido, Couturat walked in Leibniz's footsteps; Leibniz called for the creation a universal symbolic and conceptual language he named the characteristica universalis. Couturat, a confirmed pacifist, was killed when his car was hit by a car carrying orders for the mobilization of the French Army, in the first stage of World War I. He appears as a character in Joseph Skibell's 2010 novel, A Curable Romantic. == Works == 1896: De Platonicis mythis Thesim Facultati Litterarum Parisiensi proponebat Ludovicus Couturat, Scholae Normalae olim alumnus. Parisiis: Felix Alcan Bibliopola. 120 p. 1896: De l'Infini mathématique. Republished 1975, Georg Olms. 1901: La Logique de Leibniz. Republished 1961, Georg Olms. Donald Rutherford's English translation in progress. 1903: Opuscules et Fragments Inédits de Leibniz. Republished 1966, Georg Olms. 1903: (with Léopold Leau) Histoire de la langue universelle. Paris: Hachette. Republished 2001, Olms. 1905. Les Principes des Mathématiques: avec un appendice sur la philosophie des mathématiques de Kant. Republished 1965, Georg Olms. 1905: L'Algèbre de la logique. 1914: P. E. B. Jourdain translator, The Algebra of Logic, Open Court, from Project Gutenberg. 1906: ¨Pour la langue internationale, Päris 1907: (with Léopold Leau) Les nouvelles langues internationales. Paris: Hachette, republished 2001, Olms. 1910: Étude sur la dérivation dans la langue internationale. Paris: Delagrave. 100 p. 1910: (with Otto Jespersen, R. Lorenz, Wilhelm Ostwald and L.Pfaundler) International Language and Science: Considerations on the Introduction of an International Language into Science, Constable and Company Limited, London. 1915: (with Louis de Beaufront) Dictionnari Français-Ido. Paris: Chaix, 586 p. == References == == Sources == L'oeuvre de Louis Couturat. Presses de l'École Normale Supérieure. 1983. Proceedings of a conference. Grattan-Guinness, Ivor (2000). The Search for Mathematical Roots 1870-1940. Princeton University Press. Bibliography contains 27 items by Couturat. == External links == O'Connor, John J.; Robertson, Edmund F., "Louis Couturat", MacTutor History of Mathematics Archive, University of St Andrews Louis Couturat at the Mathematics Genealogy Project Auteur Couturat on French Wikisource Works by Louis Couturat at Project Gutenberg Works by or about Louis Couturat at the Internet Archive
Wikipedia:Louis Crelier#0	Louis Jacques Crelier (3 November 1873, Bure, Switzerland – 28 November 1935) was a Swiss mathematician. In 1886 he enrolled at l'Ecole normale in Porrentruy and then studied at the University of Berne, where he received his doctorate in 1895. He began his teaching career at the secondary school in Saint-Imier and then taught at the technical school (founded in 1873) in Biel/Bienne. He became in 1912 professor extraordinarius and in 1918 professor ordinarius at the University of Berne. Crelier served a two-year term from 1920 to 1921 as president of the Swiss Mathematical Society. He was an Invited Speaker of the ICM in 1924 at Toronto and in 1928 at Bologna. == Selected publications == Sur quelques propriétés des fonctions Besséliennes tirées de la théories des fractions continues. Milan: Imp. Bernardoni de C. Rebeschini et c.e. 1896. Systèmes cinématiques. Paris: Gauthier-Villars. 1911. "Puissance d'une droite par rapport à un cercle". Nouvelles annales de mathématiques. 17: 339–345. 1917. "Faisceaux de cercles relatifs à la puissance d'une droite". Nouvelles annales de mathématiques. 17: 290–297. 1917. == References == == External links == "A la mémoire de Louis Crelier, 1873-1935, docteur des sciences, professeur ordinaire à l'Université de Berne / offert par Mme Louis Crelier". helveticat.ch. (40 pages, List of publications by Louis Crelier, pp. 39–40)
Wikipedia:Louis Guttman#0	Louis Guttman (Hebrew: לואיס (אליהו) גוטמן; February 10, 1916 – October 25, 1987) was an American sociologist and Professor of Social and Psychological Assessment at the Hebrew University of Jerusalem, known primarily for his work in social statistics. == Biography == Louis (Eliyahu) Guttman was born in New York City and grew up in the Jewish community of Minneapolis, Minnesota. Guttman received both his BA in 1936 and MA in 1939 at the University of Minnesota, and his PhD in Social and Psychological Measurement in 1942. In 1947 Guttman and his wife Ruth immmigrated to Palestine. Guttman died on October 25, 1987, while on sabbatical in Minneapolis. == Academic career == From 1941 to 1947 Guttman was professor of sociology at Cornell University, while as part of the World War II effort, he also served as an Expert Consultant to the US Army's Research Branch. He founded and was the scientific director of the Israel Institute of Applied Social Research, later renamed the Guttman Institute before finally becoming the Guttman Center for Public Opinion and Policy Research. He was member of the Israel Academy of Sciences and Humanities, and foreign Honorary member of the American Academy of Arts and Sciences and President of the Psychometric Society. In 1956 he was elected a Fellow at the Center for Advanced Study in the Behavioral Sciences. In 1962 he received the Rothschild Prize. The development of scaling theory by Louis Guttman and Clyde Coombs has been recognized by Science as one of 62 major advances in the social sciences in the period 1900-1965. == Awards and recognition == 1974 Regents of the University of Minnesota - Outstanding Achievement Award 1978 Israel Prize in the social sciences 1984 Educational Testing Service Measurement Award from Princeton University. == Work == Guttman research interests were in the fields of scale and factor analysis, multidimensional scaling and facet theory. His mathematical and philosophical treatments of Factor analysis are among the important parts of his scientific legacy. His earlier work in scaling analysis produced what has become to be known as the Guttman scale. Later, searching for a more flexible scaling scheme, Guttman explored Partial Order Scalogram Analysis (POSA) and applied it in empirical studies. Notably, Guttman first proved several fundamental theorems in matrix algebra, as discussed in papers by Hubert, Meulman and Heiser (2000) and Takane and Yanai (2005). Several of Guttman's contributions, such as Smallest Space Analysis (SSA), have been incorporated into computer packages. Guttman was described as a brilliant innovator who "saw theory in method and method in theory", was "informed by high sophistication in mathematics, statistics, sociology and psychology", and one who "made a major contribution to democratic policy-making in the new state" and "was concerned with the 'well-being' of individuals, groups and society". Guttman published in numerous journals and books, including over 300 pages in Psychometrika. Many of his papers are still quoted in the scientific literature as being relevant and important to current statistical and mathematical advances. His innovative methodological work on attitudes was published in the 4th volume of Studies in Social Psychology in World War II (more popularly called The American Soldier series, after the title of the first two volumes). == References == == Further reading == Lingoes, James C.; Tucker, Ledyard R.; Shye, Samuel (June 1988). "Louis E. Guttman (1916–1987)". Psychometrika. 53 (2): 153–159. doi:10.1007/BF02294129. hdl:2027.42/45742. Shye, Samuel (March 1988). "Louis Guttman 1916-1987". Applied Psychological Measurement. 12 (1): 1–4. doi:10.1177/014662168801200101. == External links == Guttman Center
Wikipedia:Louis Paul Émile Richard#0	Louis Paul Émile Richard (31 March 1795 – 11 March 1849) was a French mathematician and teacher at the Collège Louis-le-Grand. Although he published nothing himself, he was known for his inspiring teaching and was one of the teachers of Évariste Galois. Richard was born in Rennes where his father was an artillery colonel. A physical injury in childhood prevented him from following a military career. This made him become a mathematics teacher for military aspirants, first at the lycée at Douai, then at the Collège de Pontivy (from 1815), Collège Saint-Louis (from 1820) and finally the Collège Louis-le-Grand (from 1822). Students from the college typically went into the École Polytechnique and École Normale Supérieure. His students included Evariste Galois, Urbain Le Verrier, Joseph Serret, and Charles Hermite. == References ==
Wikipedia:Louis Puissant#0	Louis Puissant (22 September 1769, in Le Châtelet-en-Brie – 10 January 1843, in Paris) was a French topographical engineer, geodesist, and mathematician. He was appointed an officer in the corps of topographical engineers (ingénieurs géographes) of l'armée des Pyrénées occidentales in 1792 and then a professor in l’école centrale d'Agen in 1796. From October 1802 to August 1803, he was in charge of geodesic triangulations on the island of Elba and then in 1803–1804 in Lombardy. He was elected a member of l'Société Philomathique de Paris in 1810 and a member of l'Académie des sciences in 1828. In addition to numerous scientific memoirs, he was the author of several books on geodesy and mathematics. His marriage to Françoise Coutet produced a son, Louis (1793–1836), who graduated from l'École polytechnique. == Principal publications == Recueil de diverses propositions de géométrie, résolues ou démontrées par l'analyse algébrique, suivant les principes de Monge et de Lacroix, à l'usage de ceux qui suivent le traité élémentaire d'application de l'algèbre à la géométrie de ce dernier (1801) Traité de géodésie ou exposition des méthodes astronomiques et trigonométriques, appliquées soit à la mesure de la terre, soit à la confection du canevas des cartes et des plans (1805) Traité de topographie, d'arpentage et de nivellement (1807) Cours de mathématiques à l'usage des écoles impériales militaires (1809) Supplément au second livre du Traité de topographie, contenant la théorie des projections des cartes (1810) Supplément au Traité de géodésie (1827) Nouvelle Description géométrique de la France (2 volumes, 1832–1840) Traité de géodésie (in French). Vol. 1. Paris: Charles Louis Étienne Bachelier. 1842. Traité de géodésie (in French). Vol. 2. Paris: Charles Louis Étienne Bachelier. 1842. == External links == O'Connor, John J.; Robertson, Edmund F., "Louis Puissant", MacTutor History of Mathematics Archive, University of St Andrews Éloge historique de Louis Puissant par M. Élie de Beaument, 14 juin 1869 PUISSANT louis – from Comité des travaux historiques et scientifiques LOUIS PUISSANT from Société d'histoire du Châtelet-en-Brie
Wikipedia:Louis de Branges de Bourcia#0	Louis de Branges de Bourcia (born August 21, 1932) is a French-American mathematician. He was the Edward C. Elliott Distinguished Professor of Mathematics at Purdue University in West Lafayette, Indiana, retiring in 2023. He is best known for proving the long-standing Bieberbach conjecture in 1984, now called de Branges's theorem. He claims to have proved several important conjectures in mathematics, including the generalized Riemann hypothesis. Born to American parents who lived in Paris, de Branges moved to the US in 1941 with his mother and sisters. His native language is French. He did his undergraduate studies at the Massachusetts Institute of Technology (1949–53), and received a PhD in mathematics from Cornell University (1953–57). His advisors were Wolfgang Fuchs and then-future Purdue colleague Harry Pollard. He spent two years (1959–60) at the Institute for Advanced Study and another two (1961–62) at the Courant Institute of Mathematical Sciences. He was appointed to Purdue in 1962. An analyst, de Branges has made incursions into real, functional, complex, harmonic (Fourier) and Diophantine analyses. As far as particular techniques and approaches are concerned, he is an expert in spectral and operator theories. == Works == De Branges' proof of the Bieberbach conjecture was not initially accepted by the mathematical community. Rumors of his proof began to circulate in March 1984, but many mathematicians were skeptical because de Branges had earlier announced some false (or inaccurate) results, including a claimed proof of the invariant subspace conjecture in 1964 (incidentally, in December 2008 he published a new claimed proof for this conjecture on his website). It took verification by a team of mathematicians at Steklov Institute of Mathematics in Leningrad to validate de Branges' proof, a process that took several months and led later to significant simplification of the main argument. The original proof uses hypergeometric functions and innovative tools from the theory of Hilbert spaces of entire functions, largely developed by de Branges. Actually, the correctness of the Bieberbach conjecture was not the only important consequence of de Branges' proof, which covers a more general problem, the Milin conjecture. == Controversial claims of solutions to unsolved problems == In June 2004, de Branges announced he had a proof of the Riemann hypothesis, often called the greatest unsolved problem in mathematics, and published the 124-page proof on his website. That original preprint suffered a number of revisions until it was replaced in December 2007 by a much more ambitious claim, which he had been developing for one year in the form of a parallel manuscript. He later released evolving versions of two claimed generalizations, following independent but complementary approaches, of his original argument. In the shortest of them (43 pages as of 2009), which he titles "Apology for the Proof of the Riemann Hypothesis" (using the word "apology" in the rarely used sense of apologia), he claims to use his tools from the theory of Hilbert spaces of entire functions to prove the Riemann hypothesis for Dirichlet L-functions (thus proving the generalized Riemann hypothesis) and a similar statement for the Euler zeta function, and even to be able to assert that zeros are simple. In the other one (57 pages), he claims to modify his earlier approach on the subject by means of spectral theory and harmonic analysis to obtain a proof of the Riemann hypothesis for Hecke L-functions, a group even more general than Dirichlet L-functions (which would imply an even more powerful result if his claim was shown to be correct). As of January 2016, his paper entitled "A proof of the Riemann Hypothesis" is 74 pages long, but does not conclude with a proof. A commentary on his attempt is available on the Internet. Mathematicians remain skeptical, and neither proof has been subjected to a serious analysis. The main objection to his approach comes from a 1998 paper (published two years later) by Brian Conrey and Xian-Jin Li, one of de Branges' former Ph.D. students and discoverer of Li's criterion, a notable equivalent statement of the Riemann hypothesis. Peter Sarnak also gave contributions to the central argument. The paper – which, contrarily to de Branges' claimed proof, was peer-reviewed and published in a scientific journal – gives numerical counterexamples and non-numerical counterclaims to some positivity conditions concerning Hilbert spaces which would, according to previous demonstrations by de Branges, imply the correctness of the Riemann hypothesis. Specifically, the authors proved that the positivity required of an analytic function F(z) which de Branges would use to construct his proof would also force it to assume certain inequalities that, according to them, the functions actually relevant to a proof do not satisfy. As their paper predates the current claimed proof by five years, and refers to work published in peer-reviewed journals by de Branges between 1986 and 1994, it remains to be seen whether de Branges has managed to circumvent their objections. He does not cite their paper in his preprints, but both of them cite a 1986 paper of his that was attacked by Li and Conrey. Journalist Karl Sabbagh, who in 2003 had written a book on the Riemann Hypothesis centered on de Branges, quoted Conrey as saying in 2005 that he still believed de Branges' approach was inadequate to tackling the conjecture, even though he acknowledged that it is a beautiful theory in many other ways. He gave no indication he had actually read the then current version of the purported proof (see reference 1). In a 2003 technical comment, Conrey said that he did not believe that the Riemann hypothesis was going to yield to functional analysis tools. De Branges, incidentally, also claims that his new proof represents a simplification of the arguments present in the removed paper on the classical Riemann hypothesis, and insists that number theorists will have no trouble checking it. Li and Conrey do not assert that de Branges' mathematics are wrong, only that the conclusions he drew from them in his original papers are, and that his tools are therefore inadequate to address the problems in question. Li released a claimed proof of the Riemann hypothesis in the arXiv in July 2008, but it was retracted a few days later, after several mainstream mathematicians exposed a crucial flaw, in a display of interest that his former advisor's claimed proofs have apparently not enjoyed so far. Meanwhile, the "apology" has become a diary of sorts, in which he also discusses the historical context of the Riemann hypothesis, and how his personal story is intertwined with the proofs. He signs his papers and preprints as "Louis de Branges", and is always cited this way. However, he does seem interested in his de Bourcia ancestors, and discusses the origins of both families in the Apology. The particular analysis tools he has developed, although largely successful in tackling the Bieberbach conjecture, have been mastered by only a handful of other mathematicians (many of whom have studied under de Branges). This poses another difficulty to verification of his current work, which is largely self-contained: most research papers de Branges chose to cite in his claimed proof of the Riemann hypothesis were written by himself over a period of forty years. During most of his working life he published articles as the sole author. The Riemann hypothesis is one of the deepest problems in all of mathematics. It is one of the six unsolved Millennium Prize Problems. A simple search in the arXiv yields several claims of proofs, some of them by mathematicians working at academic institutions, that remain unverified and are usually dismissed by mainstream scholars. A few of those have even cited de Branges' preprints in their references, which means that his work has not gone completely unnoticed. This shows that de Branges' apparent estrangement is not an isolated case, but he is probably the most renowned professional to have a current unverified claim. Two named concepts arose out of de Branges' work: an entire function satisfying a particular inequality is called a de Branges function; given a de Branges function, the set of all entire functions satisfying a particular relationship to that function, is called a de Branges space. He released another preprint on his Web site that claims to solve a measure problem due to Stefan Banach. == Awards and honors == In 1989, he was the first recipient of the Ostrowski Prize and in 1994 he was awarded the Leroy P. Steele Prize for Seminal Contribution to Research. In 2012, he became a fellow of the American Mathematical Society. == See also == Scattering theory – used by de Branges in his early approach to the Riemann hypothesis. Peter Lax == References == == External links == O'Connor, John J.; Robertson, Edmund F., "Louis de Branges de Bourcia", MacTutor History of Mathematics Archive, University of St Andrews Louis de Branges at the Mathematics Genealogy Project Papers by de Branges, including all his purported proofs (personal homepage, includes list of peer-reviewed publications).
Wikipedia:Louise Adelaide Wolf#0	Louise Adelaide Wolf (October 20, 1898 in Milwaukee – November 14, 1962 in Milwaukee) was an American mathematician and university professor. She was one of the few women to earn a math PhD in the United States before World War II. == Life and work == Wolf was the daughter of a German immigrant, Caroline Kupperian, and a Milwaukee-born streetcar conductor named John Theodore Wolf. Louise attended the 26th Avenue School and South Division High School in Milwaukee. From 1915 to 1916 she studied at Milwaukee-Downer College (now part of the University of Wisconsin). Over the next 12 years, she accepted positions that included working in a dentist's office and a public library in Milwaukee, and teaching for two years in Nevada and another two years in a Florida school. In 1928 she returned to study mathematics in Milwaukee at the Extension Division of the University of Wisconsin. In October of her senior year, mathematician Warren Weaver wrote to Dean Sellery to request an assistantship for Louise Wolf. He noted that Miss Wolf was a senior mathematics major with a straight A record in math. He said he ordinarily would not appoint a senior student to such a position but "Miss Wolf is an exceedingly capable woman, over 30 years of age, who has had experience in teaching and whose university education has been delayed by the fact that she is helping educate a younger sister." Wolf received her bachelor's degree in 1931 and her master's degree in 1933. She and her younger sister Margarete C. Wolf (Hopkins), born in November 1911, both remained at the University of Wisconsin and received their doctorates in 1935 as students of Mark Hoyt Ingraham. Louise's dissertation, entitled “Similarity of Matrices in Which the Elements are Real Quaternions,” was published in 1936 and subsequently was cited by other researchers. She immediately took a one-year faculty position at the University of Wisconsin's Extension Division, which involved circuit teaching in several Wisconsin towns. At the beginning of World War II, she became a faculty member at the University of Wisconsin–Madison. There she was a lecturer from 1936 to 1938, an assistant professor until 1951, and then an associate professor until her retirement in 1961. Louise A. Wolf was 64 when she died in Milwaukee in 1962. == Memberships == According to Judy Green, Wolf belonged to several professional societies. Mathematical Association of America American Mathematical Society American Association for the Advancement of Science National Council of Teachers of Mathematics Phi Beta Kappa Sigma Xi == Selected publications == 1936: Similarity of matrices in which the elements are real quaternions. Bull. Amer. Math. Soc. 42 1938: with M. C. Wolf: The linear equation in matrices with elements in a division algebra. Bull. Amer. Math. Soc. 44 == References ==
Wikipedia:Louise Doris Adams#0	Louise Doris Adams (2 July 1889 – 24 December 1965) was a British mathematics educator and school inspector (HMI) who wrote the 1953 book A Background to Primary School Mathematics (Oxford University Press) and became president of the Mathematical Association for 1959. == Life == Adams earned a degree from Bedford College, London, with second-class honours in mathematics in 1911. Her work as an inspector was centred on the West Country and particularly Bristol; she retired from the inspectorate in 1950. She joined the Mathematical Association in approximately 1915, and was a member for 51 years; she became a member of the Teaching Sub-Committee of the Mathematical Association in 1946, of which she became Chairman in 1954 and remained a member until her death. She was also a member of the Applications, Arithmetic and Secondary Modern Sub-Committees. When she became president of the Mathematical Association in 1959, she became only the second woman to hold that office since the association's founding in 1871, after Mary Cartwright in 1951, and the second HMI, after W. C. Fletcher in 1939. She died in 1965. == Contributions == Adams had "considerable experience as a teacher and inspector" and wrote her book, A Background to Primary School Mathematics (1953), on the basis of that experience. It was aimed at teachers of primary-school mathematics, and used case studies from approximately 80 students to advocate linking the teaching of mathematics to the individual experiences of the students. Her book "inspired many teachers" and prefigured a greater emphasis on play with mathematical tools over rote learning. As a member of the Teaching Sub-Committee of the Mathematical Association, Adams helped shift the association's focus "from teaching to learning" and from what should be taught to how it should be taught, and promoted the inclusion of primary as well as secondary education within the project's scope. Both her book and her presidential address to the Mathematical Association were a major impetus to the reform of mathematical education in the UK, as was the Teaching Sub-Committee's 1955 report The Teaching of Mathematics in Primary Schools, which she was instrumental in writing. == References ==
Wikipedia:Louise Duffield Cummings#0	Louise Duffield Cummings (21 November 1870 – 9 May 1947) was a Canadian-born American mathematician. She was born in Hamilton, Ontario. == Education and career == As a young child, Louise Duffield Cummings studied at the public schools and Collegiate Institute at Hamilton. Cummings received her B.A. in 1895 from the University of Toronto. She studied mathematics at the graduate level in 1895–1896 under the Professor DeLury at University of Toronto, in 1896–1897 at the University of Pennsylvania where she held a fellowship, in 1897–1898 at the University of Chicago, and in 1898–1900 at Bryn Mawr College. During 1900–1901 she taught at the Ontario Normal College and, while completing her A.M. at the University of Toronto, she taught at St. Margaret's College during 1901–1902. she worked with Henry White and Charlotte Scott was her supervisor. She returned to Bryn Mawr College in 1905 and 1912–1913. Cummings joined the faculty of Vassar in 1902 as an instructor; she worked with Henry White and Charlotte Scott was her supervisor. She finally received her Ph.D. from Bryn Mawr in 1914 with a thesis "On a Method of Comparison for Triple-Systems," published in the Transactions of the American Mathematical Society, Vol 15 (July 1914) . Her major subject was Pure Mathematics, and her minors were Applied Mathematics and Physics. before her retirement in 1936. She was promoted to assistant professor in 1915, to associate professor in 1919, and to full professor in 1927. She was an invited speaker at the International Congress of Mathematicians in 1924 at Toronto and again in 1932 at Zürich. == Selected publications == Cummings, L. D. (1913). "A note on the groups for triple-systems". Bull. Amer. Math. Soc. 19 (7): 355–356. doi:10.1090/s0002-9904-1913-02369-3. MR 1559362. Cummings, Louise D. (1914). "On a method of comparison for triple systems". Trans. Amer. Math. Soc. 15 (3): 311–327. doi:10.1090/s0002-9947-1914-1500982-1. hdl:2027/mdp.39015068190779. MR 1500982. (Ph.D. dissertation) with H. S. White: Cummings, Louise D.; White, H. S. (1915). "Groupless triad systems on fifteen elements". Bull. Amer. Math. Soc. 22: 12–16. doi:10.1090/s0002-9904-1915-02710-2. MR 1559701. Cummings, Louise D. (1918). "An undervalued Kirkman paper". Bull. Amer. Math. Soc. 24 (7): 336–339. doi:10.1090/s0002-9904-1918-03086-3. MR 1560081. Cummings, Louise D. (1919). "The trains for the 36 groupless triad systems on 15 elements". Bull. Amer. Math. Soc. 25 (7): 321–324. doi:10.1090/s0002-9904-1919-03192-9. MR 1560192. Cummings, Louise D. (1925). "A new type of double sextette closed under a binary (3,3) correspondence". Bull. Amer. Math. Soc. 31 (5): 266–274. doi:10.1090/s0002-9904-1925-04049-5. MR 1561036. Cummings, Louise D. (1932). "Hexagonal systems of seven lines in a plane". Bull. Amer. Math. Soc. 38 (2): 105–110. doi:10.1090/s0002-9904-1932-05337-x. MR 1562336. Cummings, Louise D. (1932). "Heptagonal systems of eight lines in a plane". Bull. Amer. Math. Soc. 38 (10): 700–702. doi:10.1090/s0002-9904-1932-05502-1. MR 1562491. Cummings, Louise D. (1933). "On a method of comparison for straight-line nets". Bull. Amer. Math. Soc. 39 (6): 411–416. doi:10.1090/s0002-9904-1933-05649-5. MR 1562638. == References == == External links == Works by or about Louise Duffield Cummings at the Internet Archive Media related to Louise Duffield Cummings (mathematician) at Wikimedia Commons
Wikipedia:Lovász conjecture#0	In graph theory, the Lovász conjecture (1969) is a classical problem on Hamiltonian paths in graphs. It says: Every finite connected vertex-transitive graph contains a Hamiltonian path. Originally László Lovász stated the problem in the opposite way, but this version became standard. In 1996, László Babai published a conjecture sharply contradicting this conjecture, but both conjectures remain widely open. It is not even known if a single counterexample would necessarily lead to a series of counterexamples. == Historical remarks == The problem of finding Hamiltonian paths in highly symmetric graphs is quite old. As Donald Knuth describes it in volume 4 of The Art of Computer Programming, the problem originated in British campanology (bell-ringing). Such Hamiltonian paths and cycles are also closely connected to Gray codes. In each case the constructions are explicit. == Variants of the Lovász conjecture == === Hamiltonian cycle === Another version of Lovász conjecture states that Every finite connected vertex-transitive graph contains a Hamiltonian cycle except the five known counterexamples. There are 5 known examples of vertex-transitive graphs with no Hamiltonian cycles (but with Hamiltonian paths): the complete graph K 2 {\displaystyle K_{2}} , the Petersen graph, the Coxeter graph and two graphs derived from the Petersen and Coxeter graphs by replacing each vertex with a triangle. === Cayley graphs === None of the 5 vertex-transitive graphs with no Hamiltonian cycles is a Cayley graph. This observation leads to a weaker version of the conjecture: Every finite connected Cayley graph contains a Hamiltonian cycle. The advantage of the Cayley graph formulation is that such graphs correspond to a finite group G {\displaystyle G} and a generating set S {\displaystyle S} . Thus one can ask for which G {\displaystyle G} and S {\displaystyle S} the conjecture holds rather than attack it in full generality. === Directed Cayley graph === For directed Cayley graphs (digraphs) the Lovász conjecture is false. Various counterexamples were obtained by Robert Alexander Rankin. Still, many of the below results hold in this restrictive setting. == Special cases == Every directed Cayley graph of an abelian group has a Hamiltonian path; however, every cyclic group whose order is not a prime power has a directed Cayley graph that does not have a Hamiltonian cycle. In 1986, D. Witte proved that the Lovász conjecture holds for the Cayley graphs of p-groups. It is open even for dihedral groups, although for special sets of generators some progress has been made. For the symmetric group S n {\displaystyle S_{n}} , there are many attractive generating sets. For example, the Lovász conjecture holds in the following cases of generating sets: a = ( 1 , 2 , … , n ) , b = ( 1 , 2 ) {\displaystyle a=(1,2,\dots ,n),b=(1,2)} (long cycle and a transposition). s 1 = ( 1 , 2 ) , s 2 = ( 2 , 3 ) , … , s n − 1 = ( n − 1 , n ) {\displaystyle s_{1}=(1,2),s_{2}=(2,3),\dots ,s_{n-1}=(n-1,n)} (Coxeter generators). In this case a Hamiltonian cycle is generated by the Steinhaus–Johnson–Trotter algorithm. any set of transpositions corresponding to a labelled tree on { 1 , 2 , . . , n } {\displaystyle \{1,2,..,n\}} . a = ( 1 , 2 ) , b = ( 1 , 2 ) ( 3 , 4 ) ⋯ , c = ( 2 , 3 ) ( 4 , 5 ) ⋯ {\displaystyle a=(1,2),b=(1,2)(3,4)\cdots ,c=(2,3)(4,5)\cdots } Stong has shown that the conjecture holds for the Cayley graph of the wreath product Zm wr Zn with the natural minimal generating set when m is either even or three. In particular this holds for the cube-connected cycles, which can be generated as the Cayley graph of the wreath product Z2 wr Zn. == General groups == For general finite groups, only a few results are known: S = { a , b } , ( a b ) 2 = 1 {\displaystyle S=\{a,b\},(ab)^{2}=1} (Rankin generators) S = { a , b , c } , a 2 = b 2 = c 2 = [ a , b ] = 1 {\displaystyle S=\{a,b,c\},a^{2}=b^{2}=c^{2}=[a,b]=1} (Rapaport–Strasser generators) S = { a , b , c } , a 2 = 1 , c = a − 1 b a {\displaystyle S=\{a,b,c\},a^{2}=1,c=a^{-1}ba} (Pak–Radoičić generators) S = { a , b } , a 2 = b s = ( a b ) 3 = 1 , {\displaystyle S=\{a,b\},a^{2}=b^{s}=(ab)^{3}=1,} where \| G \| , s = 2 m o d 4 {\displaystyle \|G\|,s=2~mod~4} (here we have (2,s,3)-presentation, Glover–Marušič theorem). Finally, it is known that for every finite group G {\displaystyle G} there exists a generating set of size at most log 2 ⁡ \| G \| {\displaystyle \log _{2}\|G\|} such that the corresponding Cayley graph is Hamiltonian (Pak-Radoičić). This result is based on classification of finite simple groups. The Lovász conjecture was also established for random generating sets of size Ω ( log 5 ⁡ \| G \| ) {\displaystyle \Omega (\log ^{5}\|G\|)} . == References ==
Wikipedia:Lowndean Professor of Astronomy and Geometry#0	The Lowndean chair of Astronomy and Geometry is one of the two major Professorships in Astronomy (alongside the Plumian Professorship) and a major Professorship in Mathematics at Cambridge University. It was founded in 1749 by Thomas Lowndes, an astronomer from Overton in Cheshire. The original bequest stated that the holder must give two courses of twenty lectures each year, one in astronomy, and the other in geometry, and spend at least six weeks making astronomical observations. Originally the holder was elected by a committee consisting of the Lord Chancellor, the Lord President of the Privy Council, the Lord Privy Seal, the Lord Steward of the Household, and the Lord High Treasurer of the First Lord of the Treasury. By the 20th century, the electors had changed to comprise the most senior scientists in the United Kingdom: the President of the Royal Society, the President of the Royal Astronomical Society, the Astronomer Royal, the Vice Chancellor of the University of Cambridge, and the Lucasian, Sadleirian, and Plumian Professors. Notwithstanding the title, a professor can be chosen who specializes solely or chiefly in only one, rather than both, of the subjects of astronomy and geometry. == Lowndean Professors == 1750–1771 Roger Long 1771–1795 John Smith 1795–1837 William Lax 1837–1859 George Peacock 1859–1892 John Couch Adams 1892–1913 Robert Stawell Ball 1914–1936 H. F. Baker 1936–1970 W. V. D. Hodge 1970–1989 J. Frank Adams 1990–1999 Graeme Segal 2000–2014 Burt Totaro 2015-present Mihalis Dafermos == References ==
Wikipedia:Loyiso Nongxa#0	Loyiso Nongxa is a South African mathematician, the current chairperson of the National Research Foundation of South Africa (NRF) and a former Vice-Chancellor and Principal of the University of the Witwatersrand, Johannesburg (Wits). == Early life and education == Nongxa was born on 22 October 1953 in Mhlanga near Lady Frere, Eastern Cape in what was then the Transkei. Both of his parents were qualified teachers, and his father was a school principal, although his mother remained at home to raise the family. Nongxa did well at school, and matriculated from Healdtown College with distinction as the top matric student in South Africa in 1972. Nongxa was accepted at the University of Fort Hare (UFH) thereafter, and obtained a BSc (Hons) in 1976. While at UFH, he also played for the university's "Baa-bas" rugby team. After obtaining a MSc from UFH in 1978, he became South Africa's first Black Rhodes scholar, and he obtained a D.Phil from Oxford University in 1982, where he holds the title of Honorary Fellow (Balliol College). == Academic career == Nongxa has lectured mathematics at UFH, the National University of Lesotho, the University of Natal and the University of the Western Cape (UWC). At UWC, he held the post of Professor of Mathematics, and he was later appointed dean of the Faculty of Natural Sciences. He had also been a visiting research scholar at the universities of Colorado, Harvard, Connecticut, Hawaii, and Baylor. He was appointed deputy vice-chancellor for Research at Wits in October 2000, and Vice-Principal in April 2002. Following the resignation of Professor Norma Reid Birley in November 2002, he became Acting Vice-Chancellor and principal of the university. The University Council then shortlisted him as one of the possible candidates for the position of vice-chancellor, and on 19 May 2003 he was appointed vice-chancellor by the council. Nongxa was the first black vice-chancellor of Wits. He was succeeded as vice-chancellor by Adam Habib on 1 June 2013. Besides his positions at universities, Nongxa has also served as a member of the Rhodes Scholarship Selection Committee, the SAUVCA Research Committee and various National Research Foundation of South Africa committees. Since leaving Wits, he has taken up to post of chairperson of the NRF. Nongxa was elected vice-president of the International Mathematical Union in July 2018. == References ==
Wikipedia:Luc Tartar#0	Luc C. Tartar is a French-American mathematician currently the University Professor of Mathematics, Emeritus at Carnegie Mellon University. == References ==
Wikipedia:Luc Vinet#0	Luc Vinet (born (1953-04-16)April 16, 1953) is a Canadian physicist and mathematician. He was former rector of the Université de Montréal between 2005 and 2010. He is the CEO of IVADO, created in 2015 since August 2021. == Biography == Born in Montreal, Quebec, Vinet holds a doctorate (3rd cycle) from the Université Pierre et Marie Curie and a PhD from the Université de Montréal, both in theoretical physics. After two years as Research Associate at MIT, he was appointed in the early 1980s as faculty member in the Physics Department at the Université de Montréal. He has held a number of visiting professorships at various universities. He is the author or co-author of ten books and more than three hundred scientific papers. His research areas include gauge field theories, supersymmetry, quantum algebra, integrable systems and combinatorics. At the Université de Montréal, Vinet held the position of director of the Centre de recherches mathématiques (CRM) from 1993 to 1999. During his term as director, the CRM succeeded in rallying the forces of quantitative research by forming a network of centers of excellence in computing from the association of seven major Montreal research centers (CERCA, CIRANO, CRIM, CRM, CRI, GERAD and INRS–Télécom) under the banner of the Network for Computing and Mathematical Modeling (NCM2). The research network provides "one-stop" access to expertise calculation and modeling for more than 20 partner enterprises. As president of NCM2 from 1996 to 1999, he was at the origin of two important research initiatives: the Bell University Laboratory, of which he became the first president and chief executive officer, and the Réseau québécois de calcul de haute performance (RQCHP), a high performance computing organization for which he presented the first grant application to the Canadian Foundation for Innovation (CFI). Vinet is also one of the founding members of and the MITACS Network of Centers of Excellence, which received an initial funding of 14 million dollars. In 1999, Vinet joined the ranks of McGill University where he held the positions of Vice-Principal (academic) and Provost. As McGill's Chief Academic Officer, he developed a renewal scheme of the professoriate and supervised the development of numerous campus infrastructures. It was at his initiative that the Government of Quebec and Quebec universities set up the Génome Québec organization. In June 2005, Vinet was appointed rector of the Université de Montréal. As rector, he set an inspired vision for this great institution, developing and executing an integrated strategic plan for the university. Among the many initiatives he realized are the development of a new campus and science pavilion project at the Outremont rail yards acquired in 2006, the creation of the School of Public Health, the establishment of the Cité du Savoir in Laval and the founding of the International Forum of Public Universities. In 2013, he became director of the Centre de recherches mathématiques (CRM) until 2021. Then, Vinet succeed Gilles Savard, in August 2021, as the CEO of IVADO, created in 2015. Vinet sits on or has sat on the boards of organizations including the Institut de finance mathématique de Montréal, the Canadian Institute for Telecommunications Research, and the Ouranos Consortium. He is the board chair for Fulbright. He has also participated in the research of a number of committees within organizations, among them Montréal International, the Chambre de commerce de Montréal, the Conférence des recteurs et des principaux des universités du Québec and the Association of American Universities. Honorary member of the Golden Key National Honor Society, he is also member of the Canadian Applied and Industrial Mathematics Society, the Canadian Association of Physicists, the Canadian Mathematical Society, the American Physical Society, the American Mathematical Society, the Association mathématique du Québec and the Society for Industrial and Applied Mathematics. == Visiting positions == Invited Professor at Université catholique de Louvain (1980–1982) Invited researcher at Massachusetts Institute of Technology (1987) Invited Professor at University of California, Los Angeles (UCLA) (1989–1990) Invited Professor at Shanghai Jiao Tong University (2013) Chair Visiting Professor at Shanghai Jiao Tong University (2014-2017) CNRS Invited Researcher at Université François Rabelais de Tours (2017) CNRS Invited Researcher at Savoy Mont Blanc University (2018) Invited Researcher at Savoy Mont Blanc University (2021) CNRS Invited Researcher at University of Tours (2021) Affiliate member Perimeter Institute (2021-) == Distinctions == Honorary Member of the Golden Key International Honour Society(2002) Ph.D. Honoris Causa from Université Claude-Bernard in Lyon (2006) Officer of the Ordre des Palmes Académiques de France (2009) Prix du Québec Armand-Frappier [1] (2009) CAP-CRM Prize in Theoretical and Mathematical Physics(2012) Officer of the National Order of Quebec (2017). Fellows of the American Mathematical Society(2017) Fellow of the Royal Society of Canada(2018) Fellow of the Canadian Mathematical Society(2019) Member, Order of Canada, The Governor General of Canada (2021) Prix Urgel Archambault, Acfas(2022) == See also == List of Université de Montréal people == References == == External links == Messages and communications by Rector Luc Vinet (Français) Luc Vinet's list of publications in Google Scholar Université de Montréal Le recteur Luc Vinet reçoit les Palmes académiques Past Rectors Université de Montréal Lettre ouverte dans The Gazette - Ideas, not military strength, are the currency of modern diplomacy L'archevêque, Maude (February 2005), "Un meneur discret", Quartier Libre, 12 (12), archived from the original on 10 October 2007, retrieved 31 October 2009.
Wikipedia:Lucasian Professor of Mathematics#0	The Lucasian Chair of Mathematics () is a mathematics professorship in the University of Cambridge, England; its holder is known as the Lucasian Professor. The post was founded in 1663 by Henry Lucas, who was Cambridge University's Member of Parliament in 1639–1640, and it was officially established by King Charles II on 18 January 1664. It has been called the most celebrated professorship in the world, and the most famous academic chair in the world due to the prestige of many of its holders, and the groundbreaking work done by them. It was said by The Daily Telegraph to be one of the most prestigious academic posts in the world. Since its establishment, the professorship has been held by, among others, Isaac Newton, Charles Babbage, George Stokes, Joseph Larmor, Paul Dirac and Stephen Hawking. == History == Henry Lucas, in his will, bequeathed his library of 4,000 volumes to the university and left instructions for the purchase of land whose yielding should provide £100 a year for the founding of a professorship. It is the third oldest chair of mathematics in Great Britain, after the Gresham Professor of Geometry at Gresham College and the Savilian Professor of Geometry at the University of Oxford. In the 1800s and following decades, the Lucasian professors "played important roles in making Britain the preeminent scientific state and in changing the university from a ‘gentleman’s club’ to a research institution." Babbage applied for the vacancy in 1826, after Turton, but Airy was appointed. William Whewell (who considered applying, but preferred both Herschel and Babbage to himself) remarked that he would be the best professor, but that the heads of the colleges would not see that. Nonetheless, Babbage was appointed when the chair became free again two years later. The current (19th) Lucasian Professor is Michael Cates, starting from 1 July 2015. The previous holder of the post was theoretical physicist Michael Green who was a fellow in Clare Hall. He was appointed in October 2009, succeeding Stephen Hawking, who himself retired in September 2009, in the year of his 67th birthday, as required by the university. Green holds the position of Emeritus Lucasian Professor of Mathematics. == List of Lucasian professors == == Cultural references == In the final episode of the science-fiction television series Star Trek: The Next Generation, one of the main characters, the android Data, holds the Lucasian Chair in the late 24th century, albeit in an alternate reality. == References == == Further reading == Kevin Knox and Richard Noakes, From Newton to Hawking: A History of Cambridge University's Lucasian Professors of Mathematics ISBN 0-521-66310-5
Wikipedia:Luchezar Avramov#0	Luchezar L. Avramov (Bulgarian: Лъчезар Л. Аврамов) is a Bulgarian-American mathematician who works in commutative algebra. He held the Dale M. Jensen Chair in Mathematics at the University of Nebraska, and is now an Emeritus. == Career == Avramov was educated at Moscow State University, earning a master's degree in 1970, a Ph.D. in 1975 (under the supervision of Evgeny Golod), and a D.Sc. in 1986. He worked for the Bulgarian Academy of Sciences in 1970–1981 and 1989–1990, and Sofia University in 1981–1989, before moving to the United States in 1991 to become a professor at Purdue University. He moved again to the University of Nebraska in 2002. == Awards and honors == In 2012, he became a Fellow of the American Mathematical Society in its inaugural class. == See also == Coherent duality Ext functor Tor functor == References == == External links == Luchezar Avramov publications indexed by Google Scholar
Wikipedia:Lucia Caporaso#0	Lucia Caporaso is an Italian mathematician, holding a professorship in mathematics at Roma Tre University. She was born in Rome, Italy, on May 22, 1965. Her research includes work in algebraic geometry, arithmetic geometry, tropical geometry and enumerative geometry. == Education and career == Caporaso earned a laurea from Sapienza University of Rome in 1989. She completed her Ph.D. at Harvard University in 1993. Her dissertation, On a Compactification of the Universal Picard Variety over the Moduli Space of Stable Curves, was supervised by Joe Harris. She became a Benjamin Pierce Assistant Professor of Mathematics at Harvard, a researcher at the University of Rome Tor Vergata, an assistant professor at the Massachusetts Institute of Technology, and an associate professor at the University of Sannio, before moving to Roma Tre as a professor in 2001. From 2013 to 2018, she has headed the Department of Mathematics and Physics at Roma Tre. == Recognition == Caporaso was the 1998 winner of the Bartolozzi Prize. She is an invited speaker at the 2018 International Congress of Mathematicians, speaking in the section on algebraic and complex geometry. == References == == External links == Home page
Wikipedia:Luciano Orlando#0	Luciano Orlando (13 May 1887 – 21 August 1915) was an Italian mathematician and military engineer. == Biography == Orlando was born in Caronia, Messina. In 1903 he received his laurea from the University of Messina, where he was a student of Bagnera and Marcolongo. After a year of graduate study at the University of Pisa, he became an assistant and libero docente at the University of Messina. After the 1908 Messina earthquake, he moved to Rome, where he taught at the Istituto superiore di Magistero and at the Aeronautical School of Engineering of the Sapienza University of Rome. He took part in some university competitions, but was unsuccessful. In 1915, when he went into military action, some of his friends warned him that they thought his courage might quickly lead to his death. He died that year in Isonzo as Captain of Military Engineers, leading an action of his company of demolition specialists against the bridge of St. Daniel near Tolmin. (Half of the entire Italian WWI casualties occurred in the Battles of the Isonzo.) He was an invited speaker of the International Congress of Mathematicians in 1908 in Rome. Orlando's most important publications deal with mathematical physics, especially the theory of elasticity and the theory of integral equations. He was one of the first to recognize the importance of Pincherle-Goursat kernels, which are an important special case of Fredholm kernels. Also noteworthy is some of Orlando's algebraic research, inspired by his teacher Bagnera. He was the father of the journalist, writer and politician Ruggero Orlando. == Selected publications == "Sulla deformazione di un triedro trirettangolo e di una lastra indefinita, elastici, isotropi." Rendiconti del Circolo Matematico di Palermo (1884–1940) 17, no. 2 (1903): 335–352. Relazione fra i minori d'ordine p d'una matrice quadrata di caratteristica p. Giornale di matematiche di Battaglini 40 (1902): 233–245. "Sulla riduzione delle quadriche a forma canonica Giornale di matematiche di Battaglini 41 (1903): 222–224. "Sulla sviluppo della funzione (1–z) exp(z + (z2)/2 + ... + (zp–1)/(p–1)) Giornale di matematiche di Battaglini 41 (1903): 377–378. "Sulla funzione nma di Green per la sfera". Giornale di matematiche di Battaglini 42 (1904): 292–296. "Sulla deformazione del suolo elastico isotropo." Rendiconti del Circolo Matematico di Palermo (1884–1940) 18, no. 1 (1904): 311–317. "Sopra alcune funzioni analoghe alla funzione di green per un parallelepipedo rettangolo." Rendiconti del Circolo Matematico di Palermo (1884–1940) 19, no. 1 (1905): 62–65. "Sulla deformazione di un solido isotropo limitato da due piani paralleli, per tensioni superficiali date." Rendiconti del Circolo Matematico di Palermo (1884–1940) 19, no. 1 (1905): 66–77. "Sull’integrazione della Δ4 in un parallelepipedo rettangolo." Rendiconti del Circolo Matematico di Palermo (1884–1940) 21, no. 1 (1906): 316–318. "Nuove osservazioni sulla formula integrale di Fourier." Rendiconti Accademia Lincei (5) vol. 18 (1909): 343–348. "Sulla risoluzione delle equazioni integrali." Tip. della R. Accad. dei Lincei, 1909. "Sopra alcuni problemi di aerodinamica." Il Nuovo Cimento (1901–1910) 20, no. 1 (1910): 46–51. "Sopra un brevetto crocco relativo all’attacco delle ali di un aeroplano." Il Nuovo Cimento (1901–1910) 20, no. 1 (1910): 52–57. "Effetto dell’attacco elastico sul rollio d’un aeroplano." Il Nuovo Cimento (1901-1910) 20, no. 1 (1910): 58–63. "Modo d’intensifigare gli effetti dell’attacco elastico in un aeroplano." Il Nuovo Cimento (1901–1910) 20, no. 1 (1910): 69–73. "Sulla caratteristica del risultante di Sylvester." Rendiconti Accademia Lincei (5) vol. 19 (1910): 257–269. "Nuove osservazioni sul problema di Hurwitz." Rendiconti Accademia Lincei (5) vol. 19 (1910): 317–321. "Sull'equazione alle semisomme e sul teorema di Hurwitz." Rendiconti Accademia Lincei (5) vol. 19 (1910): 390–393. "Sopra alcune questioni relative al problema di Hurwitz." Rendiconti Accademia Lincei (5) vol. 19 (1910): 430–434. "Sulla dimonstrazione elementare del teorema di Hurwitz." Rendiconti Accademia Lincei (5) vol. 20 (1911): 742–745. "Sul problema di Hurwitz relativo alle parti reali delle radici di un'equazione algebrica." Mathematische Annalen 71, no. 2 (1911): 233–245. == References ==
Wikipedia:Lucy Campbell (mathematician)#0	Lucy Jean Campbell is an applied mathematician and numerical analyst from Barbados, Jamaica, Ghana, and Canada, specializing in the applications of fluid dynamics to modeling the Earth's atmosphere and oceans. Beyond fluid dynamics, she has also investigated methods for tracing the sources of greenhouse gas emissions. She is an associate professor in the School of Mathematics and Statistics at Carleton University. == Early life and education == Campbell was born in Barbados. Her father, Merville O'Neale Campbell, was a mathematician at the University of the West Indies and was the first person from Barbados to earn a doctorate in mathematics. Her mother, a teacher, was from Ghana, where her father had taught prior to his 1964 return to Barbados. In 1967, the family moved to another campus of the University of the West Indies in Jamaica, where Campbell grew up. She writes that she always wanted to become an educator, aiming more specifically for mathematics after finding it to be her best subject already as a preschooler. Campbell did her undergraduate studies in mathematics at the Kwame Nkrumah University of Science and Technology in Ghana, graduating with first class honours. After additional study at the University of Cambridge, she moved to McGill University in Canada, where she worked with Sherwin Maslowe. She earned a master's degree at McGill in 1996, with a master's thesis on Forced Rossby Wave Packets in a Zonal Shear Flow in the Presence of Critical Layers. In 2000 she completed her Ph.D. there. Her dissertation was Nonlinear Critical Layer Development of Forced Wave Packets in Geophysical Shear Flows. == Later career == After completing her doctorate, Campbell did postdoctoral research in atmospheric physics at the University of Toronto, working with Ted Shepherd. She joined the Carleton faculty in 2003. == Recognition == In 2019, the Canadian Applied and Industrial Mathematics Society gave Campbell their Arthur Beaumont Distinguished Service Award, in recognition of her service as an organizer of fluid dynamics meetings and as an officer of the society. == References == == External links == Home page
Wikipedia:Ludomir Newelski#0	Ludomir Newelski (born 27 November 1960, Wrocław) is a Polish mathematician, specializing in model theory, set theory, foundations of mathematics, and universal algebra. He attended the 14th High School in Wrocław, where in April 1977, as a second-year student, he became one of the first laureates of the Polish Mathematical Olympiad in this school. He studied and graduated in mathematics at the University of Wrocław and then worked at the Mathematical Institute of the Polish Academy of Sciences (PAN). At PAN he received his PhD in 1987 and habilitated in 1991. He worked at PAN until 1994 and then moved to the University of Wrocław, where he now works. He obtained the rank of full professor in 1998. From 2007 to 2016 Newelski was the director of the Mathematical Institute of the University of Wrocław. Newelski was an Invited Speaker at the International Congress of Mathematicians in 1998 in Berlin. He was the winner of the Prize of the Foundation for Polish Science in the field of exact sciences in 2001 "for work in the field of mathematical logic constituting a breakthrough in model theory and algebra". == Selected publications == Newelski, Ludomir; Pawlikowski, Janusz; Seredyński, Witold (1987). "Infinite free set for small measure set mappings". Proceedings of the American Mathematical Society. 100 (2): 335. doi:10.1090/S0002-9939-1987-0884475-3. Newelski, Ludomir (1987). "On the number of squares in a group". Proceedings of the American Mathematical Society. 99 (2): 213. doi:10.1090/S0002-9939-1987-0870773-6. Newelski, L.; Rosłanowski, A. (1993). "The ideal determined by the unsymmetric game". Proceedings of the American Mathematical Society. 117 (3): 823. doi:10.1090/S0002-9939-1993-1112500-6. Newelski, Ludomir (1994). "On U {\displaystyle U} -rank 2 {\displaystyle 2} types". Transactions of the American Mathematical Society. 344 (2): 553–581. doi:10.1090/S0002-9947-1994-1142779-1. Newelski, Ludomir (1996). "On the prime model property". Proceedings of the American Mathematical Society. 124 (8): 2519–2525. doi:10.1090/S0002-9939-96-03311-4. Newelski, Ludomir (2002). "Small profinite structures". Transactions of the American Mathematical Society. 354 (3): 925–944. doi:10.1090/S0002-9947-01-02854-9. == References ==
Wikipedia:Ludwig Stickelberger#0	Ludwig Stickelberger (18 May 1850 – 11 April 1936) was a Swiss mathematician who made important contributions to linear algebra (theory of elementary divisors) and algebraic number theory (Stickelberger relation in the theory of cyclotomic fields). == Short biography == Stickelberger was born in Buch in the canton of Schaffhausen into a family of a pastor. He graduated from a gymnasium in 1867 and studied next in the University of Heidelberg. In 1874 he received a doctorate in Berlin under the direction of Karl Weierstrass for his work on the transformation of quadratic forms to a diagonal form. In the same year, he obtained his Habilitation from Polytechnicum in Zurich (now ETH Zurich). In 1879 he became an extraordinary professor in the Albert Ludwigs University of Freiburg. From 1896 to 1919 he worked there as a full professor, and from 1919 until his return to Basel in 1924 he held the title of a distinguished professor ("ordentlicher Honorarprofessor"). He was married in 1895, but his wife and son both died in 1918. Stickelberger died on 11 April 1936 and was buried next to his wife and son in Freiburg. == Mathematical contributions == Stickelberger's obituary lists the total of 14 publications: his thesis (in Latin), 8 further papers that he authored which appeared during his lifetime, 4 joint papers with Georg Frobenius and a posthumously published paper written circa 1915. Despite this modest output, he is characterized there as "one of the sharpest among the pupils of Weierstrass" and a "mathematician of high rank". Stickelberger's thesis and several later papers streamline and complete earlier investigations of various authors, in a direct and elegant way. === Linear algebra === Stickelberger's work on the classification of pairs of bilinear and quadratic forms filled in important gaps in the theory earlier developed by Weierstrass and Darboux. Augmented with the contemporaneous work of Frobenius, it set the theory of elementary divisors upon a rigorous foundation. An important 1878 paper of Stickelberger and Frobenius gave the first complete treatment of the classification of finitely generated abelian groups and sketched the relation with the theory of modules that had just been developed by Dedekind. === Number theory === Three joint papers with Frobenius deal with the theory of elliptic functions. Today Stickelberger's name is most closely associated with his 1890 paper that established the Stickelberger relation for cyclotomic Gaussian sums. This generalized earlier work of Jacobi and Kummer and was later used by Hilbert in his formulation of the reciprocity laws in algebraic number fields. The Stickelberger relation also yields information about the structure of the class group of a cyclotomic field as a module over its abelian Galois group (cf Iwasawa theory). == References == Lothar Heffter, Ludwig Stickelberger, Jahresbericht der Deutschen Matematische Vereinigung, XLVII (1937), pp. 79–86 Ludwig Stickelberger, Ueber eine Verallgemeinerung der Kreistheilung, Mathematische Annalen 37 (1890), pp. 321–367 == External links == Works by or about Ludwig Stickelberger at the Internet Archive
Wikipedia:Luis Caffarelli#0	Luis Ángel Caffarelli (Spanish pronunciation: [ˈlwis 'anxel kafaˈɾeli]; born December 8, 1948) is an Argentine-American mathematician. He studies partial differential equations and their applications. Caffarelli is a professor of mathematics at the University of Texas at Austin, and the winner of the 2023 Abel Prize. == Career == Caffarelli was born and grew up in Buenos Aires. He obtained his Masters of Science (1968) and Ph.D. (1972) at the University of Buenos Aires. His Ph.D. advisor was Calixto Calderón. He currently holds the Sid Richardson Chair at the University of Texas at Austin and is core faculty at the Oden Institute for Computational Engineering and Sciences. He also has been a professor at the University of Minnesota, the University of Chicago, and the Courant Institute of Mathematical Sciences at New York University. From 1986 to 1996 he was a professor at the Institute for Advanced Study in Princeton. == Research == Caffarelli published "The regularity of free boundaries in higher dimensions" in 1977 in Acta Mathematica. One of his most cited results regards the Partial regularity of suitable weak solutions of the Navier–Stokes equations; it was obtained in 1982 in collaboration with Louis Nirenberg and Robert V. Kohn. == Awards and recognition == In 1991 he was elected to the U.S. National Academy of Sciences. He was awarded honorary doctorates by the École Normale Supérieure, Paris, the University of Notre Dame, the Universidad Autónoma de Madrid, and the Universidad de La Plata, Argentina. He received the Bôcher Memorial Prize in 1984. He is listed as an ISI highly cited researcher. In 2003 Konex Foundation from Argentina granted him the Diamond Konex Award, one of the most prestigious awards in Argentina, as the most important Scientist of his country in the last decade. In 2005, he received the prestigious Rolf Schock Prize of the Royal Swedish Academy of Sciences "for his important contributions to the theory of nonlinear partial differential equations". He also received the Leroy P. Steele Prize for Lifetime Achievement in Mathematics in 2009. In 2012 he was awarded the Wolf Prize in Mathematics (jointly with Michael Aschbacher) and became a fellow of the American Mathematical Society. In 2017 he gave the Łojasiewicz Lecture (on "Some models of segregation") at the Jagiellonian University in Kraków. In 2018, he was named a SIAM Fellow and he received the Shaw Prize in Mathematics. In 2023, he was awarded the Abel Prize "for his seminal contributions to regularity theory for nonlinear partial differential equations including free-boundary problems and the Monge–Ampère equation". == Bibliography == Caffarelli has coauthored two books: Fully Nonlinear Elliptic Equations by Luis Caffarelli and Xavier Cabré (1995), American Mathematical Society. ISBN 0-8218-0437-5 A Geometric Approach to Free Boundary Problems by Luis Caffarelli and Sandro Salsa (2005), American Mathematical Society. ISBN 0-8218-3784-2 == References == == External links == Luis Caffarelli at the Mathematics Genealogy Project Home page Biographical data
Wikipedia:Luis Fernando Alday#0	Luis Fernando Alday is presently Rouse Ball Professor of Mathematics at the University of Oxford and Head of the Mathematical Physics Group. His research interests are bootstrap approach to conformal field theories and string theory, several aspects of the AdS/CFT duality, four-dimensional N=2 super-symmetric theories and their relation to conformal field theories and exact computation of observables in super-symmetric gauge theories. Alday was elected a Fellow of the Royal Society in May 2022. == References == == External links == Luis Alday
Wikipedia:Luis Huergo#0	Luis Augusto Huergo (November 1, 1837 – November 4, 1913) was an Argentine engineer prominent in the development of his country's ports. == Life and times == === Early career === Luis Huergo was born in Buenos Aires, in 1837, to a family of prosperous retailers. He was sent to the Jesuit Mount St. Mary's University previously known as Mount St. Mary's College, where he obtained his secondary education from 1852 to 1857. Returning to Argentina, he assisted urbanist Pedro Benoit plan the first road to Ensenada (a harbor town 56 km (35 mi) south of Buenos Aires) and earned a degree as a surveyor from the Topography and Geodesics School of Buenos Aires, in 1862. Huergo was among the first class to enroll at the School of Engineering created by the Rector of the University of Buenos Aires, Juan María Gutiérrez, in 1866, and four years later, his thesis on the value of roads earned him the school's first engineering degree. Huergo designed flood control projects for the torrential Tercero River and other Córdoba Province waterways. He also designed 120 railroad bridges during his early career, as well as the harbor of the city of San Fernando. Huergo co-founded the Argentine Scientific Society in 1872 and the Argentine Geographic Institute, in 1879. He taught at the newly created School of Mathematics of the University of Buenos Aires from 1874, and was designated its dean in 1881. === The port === Huergo's plans to build a house at the mouth of the Riachuelo River flowing along Buenos Aires' industrial southside earned him the appointment of Director of the Riachuelo Works Bureau in 1876. This powerful post enabled him to develop the Port of La Boca, the first modern port in Buenos Aires. The port's opening in 1880 coincided with a sudden economic boom in Argentina, and the Provincial Legislature awarded him a generous budget for improvements, including a breakwater and the dredging of the silty Riachuelo mouth to 6.5 m (21 ft). His ambitious proposal for a massive, new port north of the one at La Boca received initial support in the Argentine Congress, though the backing of Argentina's main financier (Barings Bank) for a proposal put forth by local import-export mogul Eduardo Madero helped sway congre ssional support away from Huergo's proposal. Madero's project was approved by Congress in 1882. Huergo appealed the decision on the grounds that it would be uneconomical to build and difficult to modify, once new, larger freighters began to arrive. Madero's project was signed into law by President Julio Roca in 1884, however, and in 1886, Huergo resigned his post at the Riachuelo Bureau. === New projects and a new port === Huergo continued to campaign against the costly Puerto Madero works in his capacity as Dean of the School of Exact Sciences at Buenos Aires, while also accepting new projects. Returning to Córdoba Province, he designed San Roque Reservoir in 1888 as a means to prevent flooding along the Suquia and Cosquín Rivers. He was appointed Minister of Public Works for the important Province of Buenos Aires and designed Puerto Belgrano, the Argentine Navy's first deep-water port. Huergo designed a canal from the city of Córdoba to the Paraná River, over 320 km (200 mi) to the east, and the Port of Asunción, Paraguay. The accomplished engineer was named director of the oil field discovered at Comodoro Rivadavia, in 1907. The nation's first such discovery, the office served as a precursor to YPF, the state oil concern established in 1922. Another reflection of the booming Argentine economy of the time, maritime shipping, had increased dramatically since Huergo's proposal for the Port of Buenos Aires had been passed over in 1882. The tonnage of vessels arriving at the port jumped from 4 million to 20 million by 1907, and in September of that year, Congress approved the construction of a new port. The facility would be built north of Madero's nearly-obsolete docks, and the design would be Huergo's. Work started in 1911 on the massive, new port, which included six open inner harbors, separated by cargo piers and protected by two breakwaters. Luis Huergo did not live to see the new port's completion, and he died in 1913, at age 76. His port design solved the very limitations he had anticipated for the former facility, and his many and early contributions to his country's infrastructure made him Argentina's "first engineer." == References and external links ==
Wikipedia:Luis Nunes Vicente#0	Luis Nunes Vicente (born 1967) is an applied mathematician and optimizer who is known for his research work in Continuous Optimization and particularly in Derivative-Free Optimization. He is the Timothy J. Wilmott '80 Endowed Chair Professor and Department Chair of the Department of Industrial and Systems Engineering of Lehigh University. == Education == Luis Nunes Vicente was born in Coimbra, Portugal in 1967. He obtained a B.S. in Mathematics and Operations Research at the University of Coimbra in 1990. He continued his studies at Rice University, USA, earning a Ph.D. in Applied Mathematics in 1996. His Ph.D. dissertation, titled Trust-Region Interior-Point Algorithms for a Class of Nonlinear Programming Problems, was supervised by John Dennis. == Career == From 1996 to 2018, Luis Nunes Vicente was a faculty member at the Department of Mathematics of the University of Coimbra, Portugal, becoming full professor in 2009. He held several visiting positions, namely at the IBM T.J. Watson Research Center and the University of Minnesota in 2002–2003, at the Courant Institute of Mathematical Sciences, New York University and the Université Paul Verlaine of Metz in 2009–2010, and at Roma/Sapienza and Rice University in 2016–2017. He was visiting Chercheur Sénior of the Fondation de Coopération Sciences et Technologies pour l'Aéronautique et l'Espace at CERFACS and IPN Toulouse, during 2010–2015. He has served on numerous editorial boards, including SIAM Journal on Optimization (2009–2017), EURO Journal on Computational Optimization, and Optimization Methods and Software (2010–2018). He was Editor-in-Chief of Portugaliae Mathematica during 2013–2018. == Book == Luis Nunes Vicente co-authored the book Introduction to Derivative-Free Optimization, MPS-SIAM Series on Optimization, SIAM, Philadelphia, 2009, with Katya Scheinberg and Andrew R. Conn. == Recognition == Luis Nunes Vicente was awarded the Lagrange Prize of SIAM (Society for Industrial and Applied Mathematics) and MOS (Mathematical Optimization Society) for the co-authorship of the book Introduction to Derivative-Free Optimization. With his Ph.D. dissertation, he received the Ralph Budd Thesis Award from Rice University in 1996 and was one of the three finalists of the 94-96 A. W. Tucker Prize of MOS. He is also the winner of the Lagrange Award. == References == == External links == Personal web page at Lehigh University Personal web page at the University of Coimbra Publications form Google Scholar Luis Nunes Vicente at the Mathematics Genealogy Project
Wikipedia:Lulu smoothing#0	In signal processing, Lulu smoothing is a nonlinear mathematical technique for removing impulsive noise from a data sequence such as a time series. It is a nonlinear equivalent to taking a moving average (or other smoothing technique) of a time series, and is similar to other nonlinear smoothing techniques, such as Tukey or median smoothing. LULU smoothers are compared in detail to median smoothers by Jankowitz and found to be superior in some aspects, particularly in mathematical properties like idempotence. == Properties == Lulu operators have a number of attractive mathematical properties, among them idempotence – meaning that repeated application of the operator yields the same result as a single application – and co-idempotence. An interpretation of idempotence is that: 'Idempotence means that there is no “noise” left in the smoothed data and co-idempotence means that there is no “signal” left in the residual.' When studying smoothers there are four properties that are useful to optimize: Effectiveness Consistency Stability Efficiency The operators can also be used to decompose a signal into various subcomponents similar to wavelet or Fourier decomposition. == History == Lulu smoothers were discovered by C. H. Rohwer and have been studied for the last 30 years. Their exact and asymptotic distributions have been derived. == Operation == Applying a Lulu smoother consists of repeated applications of the min and max operators over a given subinterval of the data. As with other smoothers, a width or interval must be specified. The Lulu smoothers are composed of repeated applications of the L (lower) and U (Upper) operators, which are defined as follows: === L operator === For an L operator of width n over an infinite sequence of xs (..., xj, xj+1,...), the operation on xj is calculated as follows: Firstly we create (n + 1) mini-sequences of length (n + 1) each. Each of these mini-sequences contains the element xj. For example, for width 1, we create 2 mini-sequences of length 2 each. For width 1 these mini sequences are (xj−1, xj) and (xj, xj+1). For width 2, the mini-sequences are (xj−2, xj−1, xj), (xj−1, xj, xj+1) and (xj, xj+1, xj+2). For width 2, we refer to these mini-sequences as seq−1, seq0 and seq+1 Then we take the minimum of each of the mini sequences. Again for width 2 this gives: (Min(seq−1), Min(seq0), Min(seq+1)). This gives us (n + 1) numbers for each point. Lastly we take the maximum of (the minimums of the mini sequences), or Max(Min(seq−1), Min(seq0), Min(seq+1)) and this becomes L(xj) Thus for width 2, the L operator is: L(xj) = Max(Min(seq−1), Min(seq0), Min(seq+1)) === U Operator === This is identical to the L operator, except that the order of Min and Max is reversed, i.e. for width 2: U(xj) = Min(Max(seq−1), Max(seq0), Max(seq+1)) === Examples === Examples of the U and L operators, as well as combined UL and LU operators on a sample data set are shown in the following figures. It can be seen that the results of the UL and LU operators can be different. The combined operators are very effective at removing impulsive noise, the only cases where the noise is not removed effectively is where we get multiple noise signals very close together, in which case the filter 'sees' the multiple noises as part of the signal. == References ==
Wikipedia:Luminița Vese#0	Luminița Aura Vese is a Romanian professor of mathematics at the University of California, Los Angeles, known for her research in image processing, including work on active contour models, level-set methods, image segmentation, and inpainting. == Contributions == The Chan–Vese method of image segmentation using active contours is named after her and Tony F. Chan; Chan and Vese published the method in 2001. The Vese–Osher and Osher–Solé–Vese models are optimization problems used for noise reduction of images, by decomposing an image into a sum of signal and noise in a way that optimizes a combination of measures of the smoothness of the image and the total amount of noise. They are again named after Vese, and her co-authors Stanley Osher and Andrés Solé on two papers published in 2003. With Carole Le Guyader, Vese is the author of the book Variational Methods in Image Processing (CRC Press, 2016). == Education and career == Vese earned bachelor's and master's degrees in 1992 at the West University of Timișoara in Romania. She then moved to the University of Nice Sophia Antipolis in France, earning a second master's degree in 1992 and completing her doctorate in 1997. Her dissertation, Problèmes variationnels et EDP pour l’analyse d’image et l’évolution des courbes, was jointly supervised by Gilles Aubert and Michel Rascle. After taking a temporary position at Paris Dauphine University, she joined the University of California, Los Angeles faculty in 2000. She received a Sloan Research Fellowship in 2003. == References == == External links == Home page Luminița Vese publications indexed by Google Scholar
Wikipedia:Lune of Hippocrates#0	In geometry, the lune of Hippocrates, named after Hippocrates of Chios, is a lune bounded by arcs of two circles, the smaller of which has as its diameter a chord spanning a right angle on the larger circle. Equivalently, it is a non-convex plane region bounded by one 180-degree circular arc and one 90-degree circular arc. It was the first curved figure to have its exact area calculated mathematically. == History == Hippocrates wanted to solve the classic problem of squaring the circle, i.e. constructing a square by means of straightedge and compass, having the same area as a given circle. He proved that the lune bounded by the arcs labeled E and F in the figure has the same area as triangle ABO. This afforded some hope of solving the circle-squaring problem, since the lune is bounded only by arcs of circles. Heath concludes that, in proving his result, Hippocrates was also the first to prove that the area of a circle is proportional to the square of its diameter. Hippocrates' book on geometry in which this result appears, Elements, has been lost, but may have formed the model for Euclid's Elements. Hippocrates' proof was preserved through the History of Geometry compiled by Eudemus of Rhodes, which has also not survived, but which was excerpted by Simplicius of Cilicia in his commentary on Aristotle's Physics. Not until 1882, with Ferdinand von Lindemann's proof of the transcendence of π, was squaring the circle proved to be impossible. == Proof == Hippocrates' result can be proved as follows: The center of the circle on which the arc AEB lies is the point D, which is the midpoint of the hypotenuse of the isosceles right triangle ABO. Therefore, the diameter AC of the larger circle ABC is ⁠ 2 {\displaystyle {\sqrt {2}}} ⁠ times the diameter of the smaller circle on which the arc AEB lies. Consequently, the smaller circle has half the area of the larger circle, and therefore the quarter circle AFBOA is equal in area to the semicircle AEBDA. Subtracting the crescent-shaped area AFBDA from the quarter circle gives triangle ABO and subtracting the same crescent from the semicircle gives the lune. Since the triangle and lune are both formed by subtracting equal areas from equal area, they are themselves equal in area. == Generalizations == Using a similar proof to the one above, the Arab mathematician Hasan Ibn al-Haytham (Latinized name Alhazen, c. 965 – c. 1040) showed that where two lunes are formed, on the two sides of a right triangle, whose outer boundaries are semicircles and whose inner boundaries are formed by the circumcircle of the triangle, then the areas of these two lunes added together are equal to the area of the triangle. The lunes formed in this way from a right triangle are known as the lunes of Alhazen. The quadrature of the lune of Hippocrates is the special case of this result for an isosceles right triangle. All lunes constructable by compass and straight-edge can be specified by the two angles formed by the inner and outer arcs on their respective circles; in this notation, for instance, the lune of Hippocrates would have the inner and outer angles (90°, 180°) with ratio 1:2. Hippocrates found two other squarable concave lunes, with angles approximately (107.2°, 160.9°) with ratio 2:3 and (68.5°, 205.6°) with ratio 1:3. Two more squarable concave lunes, with angles approximately (46.9°, 234.4°) with ratio 1:5 and (100.8°, 168.0°) with ratio 3:5 were found in 1766 by Martin Johan Wallenius and again in 1840 by Thomas Clausen. In the mid-20th century, two Russian mathematicians, Nikolai Chebotaryov and his student Anatoly Dorodnov, completely classified the lunes that are constructible by compass and straightedge and that have equal area to a given square. As Chebotaryov and Dorodnov showed, these five pairs of angles give the only constructible squarable lunes; in particular, there are no other constructible squarable lunes. == References ==
Wikipedia:Lute of Pythagoras#0	The lute of Pythagoras is a self-similar geometric figure made from a sequence of pentagrams. == Constructions == The lute may be drawn from a sequence of pentagrams. The centers of the pentagrams lie on a line and (except for the first and largest of them) each shares two vertices with the next larger one in the sequence. An alternative construction is based on the golden triangle, an isosceles triangle with base angles of 72° and apex angle 36°. Two smaller copies of the same triangle may be drawn inside the given triangle, having the base of the triangle as one of their sides. The two new edges of these two smaller triangles, together with the base of the original golden triangle, form three of the five edges of the polygon. Adding a segment between the endpoints of these two new edges cuts off a smaller golden triangle, within which the construction can be repeated. Some sources add another pentagram, inscribed within the inner pentagon of the largest pentagram of the figure. The other pentagons of the figure do not have inscribed pentagrams. == Properties == The convex hull of the lute is a kite shape with three 108° angles and one 36° angle. The sizes of any two consecutive pentagrams in the sequence are in the golden ratio to each other, and many other instances of the golden ratio appear within the lute. == History == The lute is named after the ancient Greek mathematician Pythagoras, but its origins are unclear. An early reference to it is in a 1990 book on the golden ratio by Boles and Newman. == See also == Spidron == References ==
Wikipedia:Luz de Teresa#0	María de la Luz (Lucero) Jimena de Teresa de Oteyza (born 1965) is a Mexican and Spanish mathematician specializing in the control theory of parabolic partial differential equations. She is a researcher in the Institute of Mathematics at the National Autonomous University of Mexico (UNAM), and a former president of the Mexican Mathematical Society. == Education and career == De Teresa was born in Mexico City on 14 June 1965; and is a citizen of both Mexico and Spain. Her father was a physicist who encouraged her to do what made her happiest; she decided that not having integrals in her life would be a horrible absence. She became an undergraduate at UNAM, graduating in 1990. Next, she studied applied mathematics at the Complutense University of Madrid, completing her PhD in 1995. Her dissertation, Control de algunas ecuaciones de la Física-Matemática: Ecuación de ondas, del calor y sistema de la termoelasticidad, was supervised by Enrique Zuazua. She has been a researcher in the Institute of Mathematics at UNAM since 1995, and was president of the Mexican Mathematical Society for the 2018–2020 term. In 2020 she was named to the governing board of UNAM's university council. == Recognition == De Teresa was elected to the Mexican Academy of Sciences in 2011. She was named an honorary member of the Royal Spanish Mathematical Society in 2018. UNAM gave her their Reconocimiento Sor Juana Inés de la Cruz award in 2009. == References == == External links == Home page Luz de Teresa publications indexed by Google Scholar
Wikipedia:Lyapunov fractal#0	In mathematics, Lyapunov fractals (also known as Markus–Lyapunov fractals) are bifurcational fractals derived from an extension of the logistic map in which the degree of the growth of the population, r, periodically switches between two values A and B. A Lyapunov fractal is constructed by mapping the regions of stability and chaotic behaviour (measured using the Lyapunov exponent λ {\displaystyle \lambda } ) in the a−b plane for given periodic sequences of a and b. In the images, yellow corresponds to λ < 0 {\displaystyle \lambda <0} (stability), and blue corresponds to λ > 0 {\displaystyle \lambda >0} (chaos). Lyapunov fractals were discovered in the late 1980s by the Germano-Chilean physicist Mario Markus from the Max Planck Institute of Molecular Physiology. They were introduced to a large public by a science popularization article on recreational mathematics published in Scientific American in 1991. == Properties == Lyapunov fractals are generally drawn for values of A and B in the interval [ 0 , 4 ] {\displaystyle [0,4]} . For larger values, the interval [0,1] is no longer stable, and the sequence is likely to be attracted by infinity, although convergent cycles of finite values continue to exist for some parameters. For all iteration sequences, the diagonal a = b is always the same as for the standard one parameter logistic function. The sequence is usually started at the value 0.5, which is a critical point of the iterative function. The other (even complex valued) critical points of the iterative function during one entire round are those that pass through the value 0.5 in the first round. A convergent cycle must attract at least one critical point. Therefore, all convergent cycles can be obtained by just shifting the iteration sequence, and keeping the starting value 0.5. In practice, shifting this sequence leads to changes in the fractal, as some branches get covered by others. For instance, the Lyapunov fractal for the iteration sequence AB (see top figure on the right) is not perfectly symmetric with respect to a and b. == Algorithm == The algorithm for computing Lyapunov fractals works as follows: Choose a string of As and Bs of any nontrivial length (e.g., AABAB). Construct the sequence S {\displaystyle S} formed by successive terms in the string, repeated as many times as necessary. Choose a point ( a , b ) ∈ [ 0 , 4 ] × [ 0 , 4 ] {\displaystyle (a,b)\in [0,4]\times [0,4]} . Define the function r n = a {\displaystyle r_{n}=a} if S n = A {\displaystyle S_{n}=A} , and r n = b {\displaystyle r_{n}=b} if S n = B {\displaystyle S_{n}=B} . Let x 0 = 0.5 {\displaystyle x_{0}=0.5} , and compute the iterates x n + 1 = r n x n ( 1 − x n ) {\displaystyle x_{n+1}=r_{n}x_{n}(1-x_{n})} . Compute the Lyapunov exponent: λ = lim N → ∞ 1 N ∑ n = 1 N log ⁡ \| d x n + 1 d x n \| = lim N → ∞ 1 N ∑ n = 1 N log ⁡ \| r n ( 1 − 2 x n ) \| {\displaystyle \lambda =\lim _{N\rightarrow \infty }{1 \over N}\sum _{n=1}^{N}\log \left\|{dx_{n+1} \over dx_{n}}\right\|=\lim _{N\rightarrow \infty }{1 \over N}\sum _{n=1}^{N}\log \|r_{n}(1-2x_{n})\|} In practice, λ {\displaystyle \lambda } is approximated by choosing a suitably large N {\displaystyle N} and dropping the first summand as r 0 ( 1 − 2 x 0 ) = r n ⋅ 0 = 0 {\displaystyle r_{0}(1-2x_{0})=r_{n}\cdot 0=0} for x 0 = 0.5 {\displaystyle x_{0}=0.5} . Color the point ( a , b ) {\displaystyle (a,b)} according to the value of λ {\displaystyle \lambda } obtained. Repeat steps (3–7) for each point in the image plane. == More Iterations == == More dimensions == Lyapunov fractals can be calculated in more than two dimensions. The sequence string for a n-dimensional fractal has to be built from an alphabet with n characters, e.g. "ABBBCA" for a 3D fractal, which can be visualized either as 3D object or as an animation showing a "slice" in the C direction for each animation frame, like the example given here. == Notes == == References == == External links == EFG's Fractals and Chaos – Lyapunov Exponents Elert, Glenn. "Lyapunov Space". The Chaos Hypertextbook.
Wikipedia:Lydia Bieri#0	Lydia Rosina Bieri (born 1972) is a Swiss-American applied mathematician, geometric analyst, mathematical physicist, cosmologist, and historian of science whose research concerns general relativity, gravity waves, and gravitational memory effects. She is a professor of mathematics and director of the Michigan Center for Applied and Interdisciplinary Mathematics at the University of Michigan. == Early life == Bieri originally from Sempach, in Switzerland, with her parents and younger brother. Always around nature, Bieri grew accustomed to inquiry and, with the guidance of her parents, learned early on to question others beliefs to further structure her own. Her journey with mathematics and physics started when she was young reading about astronomy. == Education and career == Bieri studied mathematics at ETH Zurich, earning a diploma (the equivalent of a master's degree) in 2001. She completed a doctorate (Dr. sc.) at ETH Zurich in 2007, with the support of a Swiss National Funds Fellowship. Her dissertation, An Extension of the Stability Theorem of the Minkowski Space in General Relativity, was supervised by Demetrios Christodoulou, and jointly promoted by Michael Struwe. After postdoctoral research as a Benjamin Peirce Fellow in mathematics at Harvard University from 2007 to 2010, Bieri became an assistant professor of mathematics at the University of Michigan in 2010. She became associate professor in 2015, director of the Michigan Center for Applied and Interdisciplinary Mathematics in 2019, and full professor in 2021. == Books == With Harry Nussbaumer of ETH Zurich, Bieri is the coauthor of a general-audience book on cosmology and its history, Discovering the Expanding Universe (Cambridge University Press, 2009), She is also the coauthor of a research monograph with Nina Zipser, Extensions of the Stability Theorem of the Minkowski Space in General Relativity (AMS/IP Studies in Advanced Mathematics, American Mathematical Society, 2009). == Research == Lydia Bieri has made profound contributions to understanding the mathematical structures and dynamics of the universe, with a primary focus on general relativity. Her research delves into the intricate interplay of geometry and physics as encapsulated in Einstein's equations, which describe the geometry of spacetime and the phenomenon of gravitation—the fundamental laws governing the universe. Bieri develops advanced geometric-analytic frameworks that not only solve significant questions in physics but also have independent mathematical implications, extending their application to diverse fields such as economics and biology. A key avenue of her work explores gravitational waves, ripples in spacetime resulting from cosmic events like black hole mergers, neutron star collisions, or supernovae. Since the groundbreaking detection of gravitational waves by the Advanced LIGO experiment in 2015, Bieri's investigations have illuminated how these phenomena provide unprecedented insights into otherwise inaccessible regions of the universe. Her studies encompass the memory effect of gravitational waves—lasting imprints on the curvature of spacetime—linking rigorous mathematical theory to experimental findings. Additionally, Bieri's research addresses foundational questions in astrophysics and cosmology, including the formation and stability of black holes and galaxies, as well as the profound mysteries connecting large-scale cosmological structures with quantum phenomena. These efforts aim to bridge the gap between general relativity and quantum field theory, tackling one of modern physics' greatest challenges: understanding the relationship between dark energy and vacuum energy. Through her work on nonlinear partial differential equations and geometric analysis, Bieri continues to unravel the complex synergy between mathematics and physics, advancing knowledge of the universe while opening new paths for interdisciplinary applications. == Recognition == Bieri won an NSF CAREER Award in 2013 and was named a Simons Fellow in Mathematics in 2018. She was named a Fellow of the American Physical Society (APS) in 2021, after a nomination from the APS Division of Gravitational Physics, "for fundamental results on the global existence of solutions of the Einstein field equations, and many contributions to the understanding of gravitational wave memory". She was named to the 2023 class of Fellows of the American Mathematical Society, "for contributions to mathematical general relativity and geometric analysis". == References == == External links == Home page
Wikipedia:Lynn Batten#0	Lynn Margaret Batten (1948 – 28 July 2022) was a Canadian-Australian mathematician known for her books about finite geometry and cryptography, and for her research on the classification of malware. == Education and career == Batten earned her Ph.D. at the University of Waterloo in 1977. Her dissertation was D-Partition Geometries. Formerly the Associate Dean for Academic and Industrial Research at the University of Manitoba, she moved to Deakin University, Australia in 2000, where she held the Deakin Chair in Mathematics, and directed the Information Security Group. She was involved in the founding of the Australian Mathematical Sciences Institute (AMSI) from 2001. == Books == Combinatorics of Finite Geometries. Cambridge; New York: Cambridge University Press. 31 March 1986. ISBN 978-0-521-31857-0. With Albrecht Beutelspacher: The Theory of Finite Linear Spaces. Cambridge: Cambridge University Press. 9 April 2009. ISBN 978-0-521-11418-9. Public Key Cryptography. Hoboken, N.J: John Wiley & Sons. 22 January 2013. ISBN 978-1-118-31712-9. == References == == External links == Home page Lynn Batten publications indexed by Google Scholar
Wikipedia:László Fuchs#0	László Fuchs (born June 24, 1924) is a Hungarian-born American mathematician, the Evelyn and John G. Phillips Distinguished Professor Emeritus in Mathematics at Tulane University. He is known for his research and textbooks in group theory and abstract algebra. == Biography == Fuchs was born on June 24, 1924, in Budapest, into an academic family: his father was a linguist and a member of the Hungarian Academy of Sciences. He earned a bachelor's degree in 1946 and a doctorate in 1947 from Eötvös Loránd University. After teaching high school mathematics for two years, and then holding positions at Eötvös Loránd, the Mathematical Research Institute of the Hungarian Academy of Sciences, and the University of Miami, he joined the Tulane faculty in 1968. At Tulane, Fuchs chaired the mathematics department from 1977 to 1979. He retired in 2004. In 2004, Fuchs was honored at the Hungarian Academy of Sciences 80th anniversary as one of the "big five" most distinguished Hungarian mathematicians. The other honorees included John Horvath, János Aczél, Ákos Császár and Steven Gaal. Fuchs has nearly 100 academic descendants, many of them through his student at Eötvös Loránd, George Grätzer. He was treasurer of the János Bolyai Mathematical Society from 1949 until 1963, and secretary-general of the society from 1963 to 1966. Fuchs turned 100 on June 24, 2024. == Books == Fuchs, L. (1958), Abelian Groups, Publishing House of the Hungarian Academy of Sciences, Budapest, 367pp, MR 0106942. Reprinted by Pergamon Press, International Series of Monographs on Pure and Applied Mathematics, 1960. Fuchs, L. (1963), Partially ordered algebraic systems, Oxford: Pergamon Press, 229pp, MR 0171864. Translated into Russian and German. Fuchs, László (1970), Infinite abelian groups. Vol. I, Pure and Applied Mathematics, Vol. 36, New York: Academic Press, 290pp, MR 0255673. Fuchs, László (1973), Infinite abelian groups. Vol. II, New York: Academic Press, 363pp, MR 0349869. Fuchs, László (1980), Abelian p-Groups and Mixed Groups, Séminaire de Mathématiques Supérieures [Seminar on Higher Mathematics], vol. 70, Montreal, Que.: Presses de l'Université de Montréal, 139pp, ISBN 2-7606-0468-3, MR 0569744. Fuchs, László (1983), Modules over valuation domains, Vorlesungen aus dem Fachbereich Mathematik der Universität Essen [Lecture Notes in Mathematics at the University of Essen], vol. 9, Essen: Universität Essen Fachbereich Mathematik, 133pp, MR 0709258. Fuchs, László; Salce, Luigi (1985), Modules over valuation domains, Lecture Notes in Pure and Applied Mathematics, vol. 97, New York: Marcel Dekker Inc., 317pp, ISBN 0-8247-7326-8, MR 0786121. Fuchs, László; Salce, Luigi (2001), Modules over non-Noetherian domains, Mathematical Surveys and Monographs, vol. 84, Providence, RI: American Mathematical Society, 613pp, ISBN 0-8218-1963-1, MR 1794715. == Awards and honors == Fuchs won the Kossuth Prize in 1953. He is a foreign member of the Hungarian Academy of Sciences. Two conferences were dedicated to him on the occasion of his 70th birthday, and another on his 75th. At Tulane University, Fuchs held the W. R. Irby Professorship from 1979 to 1992, and the Evelyn and John G. Phillips Distinguished Professorship from then until his retirement. In 2012 he became a fellow of the American Mathematical Society. == References ==
Wikipedia:László Rédei#0	László Rédei (15 November 1900 – 21 November 1980) was a Hungarian mathematician. Rédei graduated from the University of Budapest and initially worked as a schoolteacher. In 1940 he was appointed professor in the University of Szeged and in 1967 moved to the Mathematical Institute of the Hungarian Academy of Sciences in Budapest. His mathematical work was in algebraic number theory and abstract algebra, especially group theory. He proved that every finite tournament contains an odd number of Hamiltonian paths. He gave several proofs of the theorem on quadratic reciprocity. He proved important results concerning the invariants of the class groups of quadratic number fields. In several cases, he determined if the ring of integers of the real quadratic field Q(√d) is Euclidean or not. He successfully generalized Hajós's theorem. This led him to the investigations of lacunary polynomials over finite fields, which he eventually published in a book. This work on lacunary polynomials has had a big influence in the field of finite geometry where it plays an important role in the theory of blocking sets. He introduced a very general notion of skew product of groups, of which both the Schreier-extension and the Zappa–Szép product are special case. He explicitly determined those finite noncommutative groups whose all proper subgroups were commutative (1947). This is one of the very early results which eventually led to the classification of all finite simple groups. Rédei was the president of the János Bolyai Mathematical Society (1947–1949). He was awarded the Kossuth Prize twice. He was elected corresponding member (1949), full member (1955) of the Hungarian Academy of Sciences. == Books == 1959: Algebra. Erster Teil, Mathematik und ihre Anwendungen in Physik und Technik, Reihe A, 26, Teil 1 Akademische Verlagsgesellschaft, Geest & Portig, K.-G., Leipzig, xv+797 pp. 1967: English translation, Algebra, volume 1, Pergamon Press 1963: Theorie der endlich erzeugbaren kommutativen Halbgruppen, Hamburger Mathematische Einzelschriften, 41, Physica-Verlag, Würzburg 228 pp. 1968: Foundation of Euclidean and non-Euclidean geometries according to F. Klein, Pergamon Press, 404 pp. 1970: Lückenhafte Polynome über endlichen Körpern, Lehrbücher und Monographien aus dem Gebiete der exakten Wissenschaften, Mathematische Reihe, 42, Birkhäuser Verlag, Basel-Stuttgart, 271 pp. 1973: English translation: I. Földes: Lacunary Polynomials over Finite Fields North--Holland, London and Amsterdam, American Elsevier, New York, ISBN 0-7204-2050-4 (Europe) ISBN 0-444-10400-3 (US) 1989: Endliche p-Gruppen, Akadémiai Kiadó, Budapest, 304 pp. ISBN 963-05-4660-4 == References == 1981: László Rédei, Acta Scientiarum Mathematicarum, 43: 1–2 L. Márki (1985) "A tribute to L. Rédei", Semigroup Forum, 32, 1–21. == External links == O'Connor, John J.; Robertson, Edmund F., "László Rédei", MacTutor History of Mathematics Archive, University of St Andrews László Rédei at the Mathematics Genealogy Project
Wikipedia:Lévy flight#0	A Lévy flight is a random walk in which the step-lengths have a stable distribution, a probability distribution that is heavy-tailed. When defined as a walk in a space of dimension greater than one, the steps made are in isotropic random directions. Later researchers have extended the use of the term "Lévy flight" to also include cases where the random walk takes place on a discrete grid rather than on a continuous space. The term "Lévy flight" was coined after Paul Lévy by Benoît Mandelbrot, who used this for one specific definition of the distribution of step sizes. He used the term Cauchy flight for the case where the distribution of step sizes is a Cauchy distribution, and Rayleigh flight for when the distribution is a normal distribution (which is not an example of a heavy-tailed probability distribution). The particular case for which Mandelbrot used the term "Lévy flight" is defined by the survival function of the distribution of step-sizes, U, being Pr ( U > u ) = { 1 : u < 1 , u − D : u ≥ 1. {\displaystyle \Pr(U>u)={\begin{cases}1&:\ u<1,\\u^{-D}&:\ u\geq 1.\end{cases}}} Here D is a parameter related to the fractal dimension and the distribution is a particular case of the Pareto distribution. == Properties == Lévy flights are, by construction, Markov processes. For general distributions of the step-size, satisfying the power-like condition, the distance from the origin of the random walk tends, after a large number of steps, to a stable distribution due to the generalized central limit theorem, enabling many processes to be modeled using Lévy flights. The probability densities for particles undergoing a Levy flight can be modeled using a generalized version of the Fokker–Planck equation, which is usually used to model Brownian motion. The equation requires the use of fractional derivatives. For jump lengths which have a symmetric probability distribution, the equation takes a simple form in terms of the Riesz fractional derivative. In one dimension, the equation reads as ∂ φ ( x , t ) ∂ t = − ∂ ∂ x f ( x , t ) φ ( x , t ) + γ ∂ α φ ( x , t ) ∂ \| x \| α {\displaystyle {\frac {\partial \varphi (x,t)}{\partial t}}=-{\frac {\partial }{\partial x}}f(x,t)\varphi (x,t)+\gamma {\frac {\partial ^{\alpha }\varphi (x,t)}{\partial \|x\|^{\alpha }}}} where γ is a constant akin to the diffusion constant, α is the stability parameter and f(x,t) is the potential. The Riesz derivative can be understood in terms of its Fourier Transform. F k [ ∂ α φ ( x , t ) ∂ \| x \| α ] = − \| k \| α F k [ φ ( x , t ) ] {\displaystyle F_{k}\left[{\frac {\partial ^{\alpha }\varphi (x,t)}{\partial \|x\|^{\alpha }}}\right]=-\|k\|^{\alpha }F_{k}[\varphi (x,t)]} This can be easily extended to multiple dimensions. Another important property of the Lévy flight is that of diverging variances in all cases except that of α = 2, i.e. Brownian motion. In general, the θ fractional moment of the distribution diverges if α ≤ θ. Also, ⟨ \| x \| θ ⟩ ∝ t θ / α if θ < α . {\displaystyle \left\langle \|x\|^{\theta }\right\rangle \propto t^{\theta /\alpha }\quad {\text{if }}\theta <\alpha .} The exponential scaling of the step lengths gives Lévy flights a scale invariant property, and they are used to model data that exhibits clustering. == Applications == The definition of a Lévy flight stems from the mathematics related to chaos theory and is useful in stochastic measurement and simulations for random or pseudo-random natural phenomena. Examples include earthquake data analysis, financial mathematics, cryptography, signals analysis as well as many applications in astronomy, biology, and physics. It has been found that jumping between climate states observed in the paleoclimatic record can be described as a Lévy flight or an alpha-stable process Another application is the Lévy flight foraging hypothesis. When sharks and other ocean predators cannot find food, they abandon the Brownian motion, the random motion seen in swirling gas molecules, for the Lévy flight — a mix of long trajectories and short, random movements found in turbulent fluids. Researchers analyzed over 12 million movements recorded over 5,700 days in 55 data-logger-tagged animals from 14 ocean predator species in the Atlantic and Pacific Oceans, including silky sharks, yellowfin tuna, blue marlin and swordfish. The data showed that Lévy flights interspersed with Brownian motion can describe the animals' hunting patterns. Birds and other animals (including humans) follow paths that have been modeled using Lévy flight (e.g. when searching for food). An example of an animal, specifically a beetle, that uses Lévy flight patterns is Pterostichus melanarius. When the beetles are hungry and food is scarce, they avoid searching for prey in locations that other individuals of P. melanarius have visited. This behavior is optimal for widely dispersed prey that may not always be fully consumed at one time, such as slugs. Additionally, biological flight can also apparently be mimicked by other models such as composite correlated random walks, which grow across scales to converge on optimal Lévy walks. Composite Brownian walks can be finely tuned to theoretically optimal Lévy walks but they are not as efficient as Lévy search across most landscapes types, suggesting selection pressure for Lévy walk characteristics is more likely than multi-scaled normal diffusive patterns. Furthermore, it has been shown that Lévy walk appears in high-energy particle physics as well. Observations indicate that Lévy-processes occur in high-energy heavy-ion collisions. Here, hadronic scattering and decays after a high-energy heavy-ion collision lead to power-law tailed spatial particle creation (hadron freeze-out from the quark-gluon plasma) distributions. Efficient routing in a network can be performed by links having a Levy flight length distribution with specific values of alpha. == See also == Anomalous diffusion Fat-tailed distribution Heavy-tailed distribution Lévy process Lévy alpha-stable distribution Lévy flight foraging hypothesis == Notes == == References == Mandelbrot, Benoit B. (1982). The Fractal Geometry of Nature (Updated and augm. ed.). New York: W. H. Freeman. ISBN 0-7167-1186-9. OCLC 7876824. == Further reading == Cheng, Z.; Savit, R. (1987). "Fractal and nonfractal behavior in Levy flights" (PDF). Journal of Mathematical Physics. 28 (3): 592. Bibcode:1987JMP....28..592C. doi:10.1063/1.527644. hdl:2027.42/70735. Shlesinger, Michael F.; Klafter, Joseph; Zumofen, Gert (December 1999). "Above, below and beyond Brownian motion" (PDF). American Journal of Physics. 67 (12): 1253–1259. Bibcode:1999AmJPh..67.1253S. doi:10.1119/1.19112. Archived from the original (PDF) on 2012-03-28. == External links == A comparison of the paintings of Jackson Pollock to a Lévy flight model
Wikipedia:Lévy–Steinitz theorem#0	In mathematics, the Lévy–Steinitz theorem identifies the set of values to which sums of rearrangements of an infinite series of vectors in Rn can converge. It was proved by Paul Lévy in his first published paper when he was 19 years old. In 1913 Ernst Steinitz filled in a gap in Lévy's proof and also proved the result by a different method. In an expository article, Peter Rosenthal stated the theorem in the following way. The set of all sums of rearrangements of a given series of vectors in a finite-dimensional real Euclidean space is either the empty set or a translate of a linear subspace (i.e., a set of the form v + M, where v is a given vector and M is a linear subspace). == See also == Riemann series theorem == References == Banaszczyk, Wojciech (1991). Additive Subgroups of Topological Vector Spaces. Lecture Notes in Mathematics. Vol. 1466. Berlin: Springer-Verlag. pp. 93–109. doi:10.1007/BFb0089147. ISBN 3-540-53917-4. MR 1119302. Zbl 0743.46002. Kadets, V. M.; Kadets, M. I. (1991). Rearrangements of series in Banach spaces. Translations of Mathematical Monographs. Vol. 86 (Translated by Harold H. McFaden from the Russian-language (Tartu) 1988 ed.). Providence, RI: American Mathematical Society. pp. iv+123. ISBN 0-8218-4546-2. MR 1108619. Kadets, Mikhail I.; Kadets, Vladimir M. (1997). Series in Banach spaces: Conditional and unconditional convergence. Operator Theory: Advances and Applications. Vol. 94. Translated by Andrei Iacob from the Russian-language. Basel: Birkhäuser Verlag. pp. viii+156. ISBN 3-7643-5401-1. MR 1442255.
Wikipedia:Līlāvatī#0	Līlāvatī is a treatise by Indian mathematician Bhāskara II on mathematics, written in 1150 AD. It is the first volume of his main work, the Siddhānta Shiromani, alongside the Bijaganita, the Grahaganita and the Golādhyāya. == Name == Bhaskara II's book on arithmetic is the subject of interesting legends that assert that it was written for his daughter, Lilavati. As the story goes, the author had studied Lilavati's horoscope and predicted that she would remain both childless and unmarried. To avoid this fate, he ascertained an auspicious moment for his daughter's wedding. To alert his daughter at the correct time, he placed a cup with a small hole at the bottom of a vessel filled with water, arranged so that the cup would sink at the beginning of the propitious hour. He put the device in a room with a warning to Lilavati to not go near it. In her curiosity, though, she went to look at the device. A pearl from her bridal dress accidentally dropped into it, thus upsetting it. The auspicious moment for the wedding thus passed unnoticed leaving Bhaskara II devastated. Thus, he promised his daughter to write a book in her name, one that would remain till the end of time as a good name is akin to a second life. Many of the problems are addressed to Līlāvatī herself, who must have been a very bright young woman. For example "Oh Līlāvatī, intelligent girl, if you understand addition and subtraction, tell me the sum of the amounts 2, 5, 32, 193, 18, 10, and 100, as well as [the remainder of] those when subtracted from 10000." and "Fawn-eyed child Līlāvatī, tell me, how much is the number [resulting from] 135 multiplied by 12, if you understand multiplication by separate parts and by separate digits. And tell [me], beautiful one, how much is that product divided by the same multiplier?" The word Līlāvatī itself means playful or one possessing play (from Sanskrit, Līlā = play, -vatī = female possessing the quality). == Contents == The book contains thirteen chapters, mainly definitions, arithmetical terms, interest computation, arithmetical and geometrical progressions, plane geometry, solid geometry, the shadow of the gnomon, the Kuṭṭaka - a method to solve indeterminate equations, and combinations. Bhaskara II gives the value of pi as 22/7 in the book but suggest a more accurate ratio of 3927/1250 for use in astronomical calculations. Also according to the book, the largest number is the parardha equal to one hundred thousand billion. Lilavati includes a number of methods of computing numbers such as multiplications, squares, and progressions, with examples using kings and elephants, objects which a common man could understand. Excerpt from Lilavati (Appears as an additional problem attached to stanza 54, Chapter 3. Translated by T N Colebrook) Bhaskaracharya's conclusion to Lilavati states: Joy and happiness is indeed ever increasing in this world for those who have Lilavati clasped to their throats, decorated as the members are with neat reduction of fractions, multiplication and involution, pure and perfect as are the solutions, and tasteful as is the speech which is exemplified. == Translations == The translations or editions of the Lilavati into English and other languages include: 1816. John Taylor, Lilawati: or A Treatise on Arithmetic or Geometry by Bhascara Acharya 11th century: Eluganti Pedana ( ఎలుగంటి పెద్దన) translated Lilavati into Telugu. The work is called Prakīrna Ganitamu( ప్రకిర్ణ గణితము). 1817. Henry Thomas Colebrooke, Algebra, with Arithmetic and mensuration, from the Sanscrit of Brahmegupta and Bháscara, Page 24, chap 2/3 1842. Amichandra Shravaga of Jaipur translated Lilavati into Hindi. 1936. Pidaparti Krishnamurti Sastry translated the work into Telugu language and it was published by Srividya press, Vizianagaram. 1975. K. V. Sarma, Līlāvatī of Bhāskarācārya with Kriyā-kramakarī, Hoshiarpur: VVBIS & IS, Panjab University 2001. K. S. Patwardhan, S. A. Naimpally and S. L. Singh. Līlāvatī of Bhāskarācārya: a treatise of mathematics of Vedic tradition : with rationale in terms of modern mathematics largely based on N.H. Phadke's Marāthī translation of Līlāvatī Bhaskaracharya's work 'Lilavati' was translated into Persian(फारसी) by-( Abul Faizi-in 1587 ). Bakul Kayastha from medieval Assam (1400CE) made Assamese rendering of Lilavati. == See also == Indian mathematics Timeline of algebra and geometry == References == === Works cited === Plofker, Kim (2009), Mathematics in India, Princeton University Press, ISBN 9780691120676 Poulose, K. G., ed. (1991), Scientific heritage of India, mathematics, Ravivarma Samskr̥ta granthāvali, vol. 22, Govt. Sanskrit College (Tripunithura, India) == External links == Bhaskaracharya's Lilavathi - Book Review Bhaskara Khagol Mandal
Wikipedia:Lương Thế Vinh#0	Lương Thế Vinh (1441–1496) was a prominent Vietnamese scholar and mathematician of the fifteenth century. == Life == Lương Thế Vinh was born in the district of Vụ Bản, Nam Định Province, and during the mid-15th century. He obtained doctorate in 1463 or 1478, during the reign of Le Thanh Tong, the golden era of Vietnamese scholarship. He co-worked with Vũ Hựu, another scholar, and introduced Chinese mathematical methods into Vietnam. == Works == Math: Great Compendium of Mathematical Methods–Đại thành Toán pháp (edited). Khải minh Toán học Chèo: Hý phường phả lục Buddhism: Thiền môn Khoa giáo . Advanced methods of Mathematical Calculus . “Các phương pháp tính toán nâng cao” == References == === Citations === === Sources === Volkov, Alexei (2002), "On the Origins of the Toan phap dai thanh", in Samplonius, Yvonne Dold; Dauben, Josephn W. (eds.), From China to Paris: 2000 Years Transmission of Mathematical Ideas, Franz Steiner Verlag, pp. 369–410, ISBN 978-3-515-08223-5 Volkov, Alexei (2009), "Mathematics and Mathematics Education in Traditional Vietnam", in Robson, Eleanor; Stedall, Jacqueline (eds.), The Oxford Handbook of the History of Mathematics, Oxford: Oxford University Press, pp. 153–76, ISBN 978-0-19-921312-2 Volkov, Alexei (2016), "Mathematics in Vietnam", in Selin, Helaine (ed.), Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures (3 ed.), Berlin: Springer-Verlag, pp. 2818–2833, ISBN 978-94-007-7748-4
Wikipedia:M. Ram Murty#0	Maruti Ram Pedaprolu Murty, FRSC (born 16 October 1953) is an Indo-Canadian mathematician at Queen's University, where he holds a Queen's Research Chair in mathematics. == Biography == M. Ram Murty is the brother of mathematician V. Kumar Murty. Murty graduated with a B.Sc. from Carleton University in 1976. He received his Ph.D. in 1980 from the Massachusetts Institute of Technology, supervised by Harold Stark and Dorian Goldfeld. He was on the faculty of McGill University from 1982 until 1996, when he joined Queen's University. Murty is also cross-appointed as a professor of philosophy at Queen's, specialising in Indian philosophy. == Research == Specializing in number theory, Murty is a researcher in the areas of modular forms, elliptic curves, and sieve theory. Murty has Erdős number 1 and frequently collaborates with his brother, V. Kumar Murty. == Awards == Murty received the Coxeter–James Prize in 1988. He was elected a Fellow of the Royal Society of Canada in 1990, was elected to the Indian National Science Academy (INSA) in 2008, and became a fellow of the American Mathematical Society in 2012. == Selected publications == Cojocaru, Alina Carmen; Murty, M. Ram (2006). An introduction to sieve methods and their applications. London Mathematical Society Student Texts. Vol. 66. Cambridge University Press. ISBN 0-521-84816-4. MR 2200366.. Murty, M. Ram; Murty, V. Kumar (1991). "Mean values of derivatives of modular L-series". Annals of Mathematics. Second Series. 133 (3). Annals of Mathematics: 447–475. doi:10.2307/2944316. JSTOR 2944316. MR 1109350.. == References == == External links == Murty's home page at Queen's
Wikipedia:M. Salah Baouendi#0	Mohammed Salah Baouendi (Arabic: محمد صالح باوندي; October 12, 1937 in Tunis – December 24, 2011 in La Jolla, California) was a Tunisian-American mathematician who worked as a Distinguished Professor of Mathematics at the University of California, San Diego. His research concerned partial differential equations and the theory of several complex variables. == Education and career == Baouendi moved from Tunis to France after finishing his Baccalauréat in the high school Sadiki College, and earned a licence (a French undergraduate degree) in 1961 from the Sorbonne. In this he had the assistance of a scholarship from the Tunisian government, which however demanded that he return to Tunis afterwards to become a schoolteacher. After the intervention of Laurent Schwartz, Baouendi was allowed to return to France for his graduate studies. He completed a doctorate in 1967 from the University of Paris-Sud (then part of the University of Paris), under the supervision of Bernard Malgrange, with a dissertation concerning elliptic operators. Schwartz attempted to secure for him a suitable academic position in Tunis, in which he would be allowed to conduct research and collaborate with mathematicians from other countries, and Baouendi became an associate professor at Tunis University in 1968, but his administrative struggles there were too much, and he left in 1970. After a short stay at the University of Nice, Baouendi moved in 1971 to the USA. Baouendi's first American faculty position was at Purdue University. During his tenure at Purdue, he was promoted to full professor in 1973, became department chair from 1980 to 1987, and also held visiting positions at Pierre and Marie Curie University, the University of Chicago, and Rutgers University. He became a Distinguished Professor at UCSD in 1988 (giving up his Purdue professorship in 1990). He was a co-founder of two journals, Communications in Partial Differential Equations and Mathematical Research Letters. == Awards and honors == Baouendi was given the Prix d'Aumale of the French Academy of Sciences in 1969. He was an invited speaker at the International Congress of Mathematicians in 1974. He and his wife, mathematician Linda Preiss Rothschild, were jointly awarded the Stefan Bergman Prize of the American Mathematical Society in 2003. In 2005 he became a fellow of the American Academy of Arts and Sciences. == See also == Moungi Bawendi (his son), winner of the 2023 Nobel Prize in Chemistry. == References == == External links == Salah Baouendi's Doctoral Dissertation Paris 1967(in French, 7.9MB, pdf), "On a Class of Degenerate Elliptical Operators", on Rothschild's homepage
Wikipedia:MATLAB#0	MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other languages. Although MATLAB is intended primarily for numeric computing, an optional toolbox uses the MuPAD symbolic engine allowing access to symbolic computing abilities. An additional package, Simulink, adds graphical multi-domain simulation and model-based design for dynamic and embedded systems. As of 2020, MATLAB has more than four million users worldwide. They come from various backgrounds of engineering, science, and economics. As of 2017, more than 5000 global colleges and universities use MATLAB to support instruction and research. == History == === Origins === MATLAB was invented by mathematician and computer programmer Cleve Moler. The idea for MATLAB was based on his 1960s PhD thesis. Moler became a math professor at the University of New Mexico and started developing MATLAB for his students as a hobby. He developed MATLAB's initial linear algebra programming in 1967 with his one-time thesis advisor, George Forsythe. This was followed by Fortran code for linear equations in 1971. Before version 1.0, MATLAB "was not a programming language; it was a simple interactive matrix calculator. There were no programs, no toolboxes, no graphics. And no ODEs or FFTs." The first early version of MATLAB was completed in the late 1970s. The software was disclosed to the public for the first time in February 1979 at the Naval Postgraduate School in California. Early versions of MATLAB were simple matrix calculators with 71 pre-built functions. At the time, MATLAB was distributed for free to universities. Moler would leave copies at universities he visited and the software developed a strong following in the math departments of university campuses.: 5 In the 1980s, Cleve Moler met John N. Little. They decided to reprogram MATLAB in C and market it for the IBM desktops that were replacing mainframe computers at the time. John Little and programmer Steve Bangert re-programmed MATLAB in C, created the MATLAB programming language, and developed features for toolboxes. === Commercial development === MATLAB was first released as a commercial product in 1984 at the Automatic Control Conference in Las Vegas. MathWorks, Inc. was founded to develop the software and the MATLAB programming language was released. The first MATLAB sale was the following year, when Nick Trefethen from the Massachusetts Institute of Technology bought ten copies. By the end of the 1980s, several hundred copies of MATLAB had been sold to universities for student use. The software was popularized largely thanks to toolboxes created by experts in various fields for performing specialized mathematical tasks. Many of the toolboxes were developed as a result of Stanford students that used MATLAB in academia, then brought the software with them to the private sector. Over time, MATLAB was re-written for early operating systems created by Digital Equipment Corporation, VAX, Sun Microsystems, and for Unix PCs. Version 3 was released in 1987. The first MATLAB compiler was developed by Stephen C. Johnson in the 1990s. In 2000, MathWorks added a Fortran-based library for linear algebra in MATLAB 6, replacing the software's original LINPACK and EISPACK subroutines that were in C. MATLAB's Parallel Computing Toolbox was released at the 2004 Supercomputing Conference and support for graphics processing units (GPUs) was added to it in 2010. === Recent history === Some especially large changes to the software were made with version 8 in 2012. The user interface was reworked and Simulink's functionality was expanded. By 2016, MATLAB had introduced several technical and user interface improvements, including the MATLAB Live Editor notebook, and other features. == Release history == For a complete list of changes of both MATLAB an official toolboxes, check MATLAB previous releases. == Syntax == The MATLAB application is built around the MATLAB programming language. Common usage of the MATLAB application involves using the "Command Window" as an interactive mathematical shell or executing text files containing MATLAB code. === "Hello, world!" example === An example of a "Hello, world!" program exists in MATLAB. It displays like so: === Variables === Variables are defined using the assignment operator, =. MATLAB is a weakly typed programming language because types are implicitly converted. It is an inferred typed language because variables can be assigned without declaring their type, except if they are to be treated as symbolic objects, and that their type can change. Values can come from constants, from computation involving values of other variables, or from the output of a function. For example: === Vectors and matrices === A simple array is defined using the colon syntax: initial:increment:terminator. For instance: defines a variable named array (or assigns a new value to an existing variable with the name array) which is an array consisting of the values 1, 3, 5, 7, and 9. That is, the array starts at 1 (the initial value), increments with each step from the previous value by 2 (the increment value), and stops once it reaches (or is about to exceed) 9 (the terminator value). The increment value can actually be left out of this syntax (along with one of the colons), to use a default value of 1. assigns to the variable named ari an array with the values 1, 2, 3, 4, and 5, since the default value of 1 is used as the increment. Indexing is one-based, which is the usual convention for matrices in mathematics, unlike zero-based indexing commonly used in other programming languages such as C, C++, and Java. Matrices can be defined by separating the elements of a row with blank space or comma and using a semicolon to separate the rows. The list of elements should be surrounded by square brackets []. Parentheses () are used to access elements and subarrays (they are also used to denote a function argument list). Sets of indices can be specified by expressions such as 2:4, which evaluates to [2, 3, 4]. For example, a submatrix taken from rows 2 through 4 and columns 3 through 4 can be written as: A square identity matrix of size n can be generated using the function eye, and matrices of any size with zeros or ones can be generated with the functions zeros and ones, respectively. Transposing a vector or a matrix is done either by the function transpose or by adding dot-prime after the matrix (without the dot, prime will perform conjugate transpose for complex arrays): Most functions accept arrays as input and operate element-wise on each element. For example, mod(2*J,n) will multiply every element in J by 2, and then reduce each element modulo n. MATLAB does include standard for and while loops, but (as in other similar applications such as APL and R), using the vectorized notation is encouraged and is often faster to execute. The following code, excerpted from the function magic.m, creates a magic square M for odd values of n (MATLAB function meshgrid is used here to generate square matrices I and J containing ⁠ 1 : n {\displaystyle 1:n} ⁠): === Structures === MATLAB supports structure data types. Since all variables in MATLAB are arrays, a more adequate name is "structure array", where each element of the array has the same field names. In addition, MATLAB supports dynamic field names (field look-ups by name, field manipulations, etc.). === Functions === When creating a MATLAB function, the name of the file should match the name of the first function in the file. Valid function names begin with an alphabetic character, and can contain letters, numbers, or underscores. Variables and functions are case sensitive. === Function handles === MATLAB supports elements of lambda calculus by introducing function handles, or function references, which are implemented either in .m files or anonymous/nested functions. === Classes and object-oriented programming === MATLAB supports object-oriented programming including classes, inheritance, virtual dispatch, packages, pass-by-value semantics, and pass-by-reference semantics. However, the syntax and calling conventions are significantly different from other languages. MATLAB has value classes and reference classes, depending on whether the class has handle as a super-class (for reference classes) or not (for value classes). Method call behavior is different between value and reference classes. For example, a call to a method: can alter any member of object only if object is an instance of a reference class, otherwise value class methods must return a new instance if it needs to modify the object. An example of a simple class is provided below: When put into a file named hello.m, this can be executed with the following commands: == Graphics and graphical user interface programming == MATLAB has tightly integrated graph-plotting features. For example, the function plot can be used to produce a graph from two vectors x and y. The code: produces the following figure of the sine function: MATLAB supports three-dimensional graphics as well: MATLAB supports developing graphical user interface (GUI) applications. UIs can be generated either programmatically or using visual design environments such as GUIDE and App Designer. == MATLAB and other languages == MATLAB can call functions and subroutines written in the programming languages C or Fortran. A wrapper function is created allowing MATLAB data types to be passed and returned. MEX files (MATLAB executables) are the dynamically loadable object files created by compiling such functions. Since 2014 increasing two-way interfacing with Python was being added. Libraries written in Perl, Java, ActiveX or .NET can be directly called from MATLAB, and many MATLAB libraries (for example XML or SQL support) are implemented as wrappers around Java or ActiveX libraries. Calling MATLAB from Java is more complicated, but can be done with a MATLAB toolbox which is sold separately by MathWorks, or using an undocumented mechanism called JMI (Java-to-MATLAB Interface), (which should not be confused with the unrelated Java Metadata Interface that is also called JMI). Official MATLAB API for Java was added in 2016. As alternatives to the MuPAD based Symbolic Math Toolbox available from MathWorks, MATLAB can be connected to Maple or Mathematica. Libraries also exist to import and export MathML. == Relations to US sanctions == In 2020, MATLAB withdrew services from two Chinese universities as a result of US sanctions. The universities said this will be responded to by increased use of open-source alternatives and by developing domestic alternatives. == See also == Comparison of numerical-analysis software List of numerical-analysis software == Notes == == Further reading == == External links == Official website
Wikipedia:Maarten Solleveld#0	Maarten Solleveld is a Dutch chess grandmaster and mathematician. == Chess career == He won the 2000 Gouda Open ahead of IM Johan van Mil. He achieved his GM norms at the: Dutch Team Championship in June 2003 North Sea Cup in July 2005 Dutch Team Competition in April 2012 He has been inactive in chess since June 2017. == Mathematics career == He obtained his PhD in mathematics from the University of Amsterdam in 2007 under Eric Opdam. He has held teaching positions at University of Amsterdam, Hausdorff Center for Mathematics, and University of Göttingen. In 2011, he became a professor of mathematics at Radboud University Nijmegen. == References ==
Wikipedia:Mac Lane's planarity criterion#0	In graph theory, Mac Lane's planarity criterion is a characterisation of planar graphs in terms of their cycle spaces, named after Saunders Mac Lane who published it in 1937. It states that a finite undirected graph is planar if and only if the cycle space of the graph (taken modulo 2) has a cycle basis in which each edge of the graph participates in at most two basis vectors. == Statement == For any cycle c in a graph G on m edges, one can form an m-dimensional 0-1 vector that has a 1 in the coordinate positions corresponding to edges in c and a 0 in the remaining coordinate positions. The cycle space C(G) of the graph is the vector space formed by all possible linear combinations of vectors formed in this way. In Mac Lane's characterization, C(G) is a vector space over the finite field GF(2) with two elements; that is, in this vector space, vectors are added coordinatewise modulo two. A 2-basis of G is a basis of C(G) with the property that, for each edge e in G, at most two basis vectors have nonzero coordinates in the position corresponding to e. Then, stated more formally, Mac Lane's characterization is that the planar graphs are exactly the graphs that have a 2-basis. == Existence of a 2-basis for planar graphs == One direction of the characterisation states that every planar graph has a 2-basis. Such a basis may be found as the collection of boundaries of the bounded faces of a planar embedding of the given graph G. If an edge is a bridge of G, it appears twice on a single face boundary and therefore has a zero coordinate in the corresponding vector. Thus, the only edges that have nonzero coordinates are the ones that separate two different faces; these edges appear either once (if one of the faces is the unbounded one) or twice in the collection of boundaries of bounded faces. It remains to prove that these cycles form a basis. One way to prove this by induction. As a base case, G is a tree, then it has no bounded faces and C(G) is zero-dimensional and has an empty basis. Otherwise, removing an edge from the unbounded face of G reduces both the dimension of the cycle space and the number of bounded faces by one and the induction follows. Alternatively, it is possible to use Euler's formula to show that the number of cycles in this collection equals the circuit rank of G, which is the dimension of the cycle space. Each nonempty subset of cycles has a vector sum that represents the boundary of the union of the bounded faces in the subset, which cannot be empty (the union includes at least one bounded face and excludes the unbounded face, so there must be some edges separating them). Therefore, there is no subset of cycles whose vectors sum to zero, which means that all the cycles are linearly independent. As a linearly independent set of the same size as the dimension of the space, this collection of cycles must form a basis. == Necessity of planarity when a 2-basis exists == O'Neil (1973) provided the following simple argument for the other direction of the characterization, based on Wagner's theorem characterizing the planar graphs by forbidden minors. As O'Neill observes, the property of having a 2-basis is preserved under graph minors: if one contracts an edge, the same contraction may be performed in the basis vectors, if one removes an edge that has a nonzero coordinate in a single basis vector, then that vector may be removed from the basis, and if one removes an edge that has a nonzero coordinate in two basis vectors, then those two vectors may be replaced by their sum (modulo two). Additionally, if C(G) is a cycle basis for any graph, then it must cover some edges exactly once, for otherwise its sum would be zero (impossible for a basis), and so C(G) can be augmented by one more cycle consisting of these singly-covered edges while preserving the property that every edge is covered at most twice. However, the complete graph K5 has no 2-basis: C(G) is six-dimensional, each nontrivial vector in C(G) has nonzero coordinates for at least three edges, and so any augmented basis would have at least 21 nonzeros, exceeding the 20 nonzeros that would be allowed if each of the ten edges were nonzero in at most two basis vectors. By similar reasoning, the complete bipartite graph K3,3 has no 2-basis: C(G) is four-dimensional, and each nontrivial vector in C(G) has nonzero coordinates for at least four edges, so any augmented basis would have at least 20 nonzeros, exceeding the 18 nonzeros that would be allowed if each of the nine edges were nonzero in at most two basis vectors. Since the property of having a 2-basis is minor-closed and is not true of the two minor-minimal nonplanar graphs K5 and K3,3, it is also not true of any other nonplanar graph. Lefschetz (1965) provided another proof, based on algebraic topology. He uses a slightly different formulation of the planarity criterion, according to which a graph is planar if and only if it has a set of (not necessarily simple) cycles covering every edge exactly twice, such that the only nontrivial relation among these cycles in C(G) is that their sum be zero. If this is the case, then leaving any one of the cycles out produces a basis satisfying Mac Lane's formulation of the criterion. If a planar graph is embedded on a sphere, its face cycles clearly satisfy Lefschetz's property. Conversely, as Lefschetz shows, whenever a graph G has a set of cycles with this property, they necessarily form the face cycles of an embedding of the graph onto the sphere. == Application == Ja'Ja' & Simon (1982) used Mac Lane's planarity criterion as part of a parallel algorithm for testing graph planarity and finding planar embeddings. Their algorithm partitions the graph into triconnected components, after which there is a unique planar embedding (up to the choice of the outer face) and the cycles in a 2-basis can be assumed to be all the peripheral cycles of the graph. Ja'Ja' and Simon start with a fundamental cycle basis of the graph (a cycle basis generated from a spanning tree by forming a cycle for each possible combination of a path in the tree and an edge outside the tree) and transform it into a 2-basis of peripheral cycles. These cycles form the faces of a planar embedding of the given graph. Mac Lane's planarity criterion allows the number of bounded face cycles in a planar graph to be counted easily, as the circuit rank of the graph. This property is used in defining the meshedness coefficient of the graph, a normalized variant of the number of bounded face cycles that is computed by dividing the circuit rank by 2n − 5, the maximum possible number of bounded faces of a planar graph with the same vertex set (Buhl et al. 2004). == References == Buhl, J.; Gautrais, J.; Sole, R.V.; Kuntz, P.; Valverde, S.; Deneubourg, J.L.; Theraulaz, G. (2004), "Efficiency and robustness in ant networks of galleries", The European Physical Journal B, 42 (1), Springer-Verlag: 123–129, Bibcode:2004EPJB...42..123B, doi:10.1140/epjb/e2004-00364-9, S2CID 14975826. Ja'Ja', Joseph; Simon, Janos (1982), "Parallel algorithms in graph theory: planarity testing", SIAM Journal on Computing, 11 (2): 314–328, doi:10.1137/0211024, MR 0652905. Lefschetz, Solomon (1965), "Planar graphs and related topics", Proceedings of the National Academy of Sciences of the United States of America, 54 (6): 1763–1765, Bibcode:1965PNAS...54.1763L, doi:10.1073/pnas.54.6.1763, JSTOR 72818, MR 0189011, PMC 300546, PMID 16591326. Mac Lane, S. (1937), "A combinatorial condition for planar graphs" (PDF), Fundamenta Mathematicae, 28: 22–32, doi:10.4064/fm-28-1-22-32. O'Neil, P. V. (1973), "A short proof of Mac Lane's planarity theorem", Proceedings of the American Mathematical Society, 37 (2): 617–618, doi:10.1090/S0002-9939-1973-0313098-X, hdl:2060/19720020939, JSTOR 2039496, MR 0313098.
Wikipedia:Macaulay brackets#0	Macaulay brackets are a notation used to describe the ramp function { x } = { 0 , x < 0 x , x ≥ 0. {\displaystyle \{x\}={\begin{cases}0,&x<0\\x,&x\geq 0.\end{cases}}} A popular alternative transcription uses angle brackets, viz. ⟨ x ⟩ {\displaystyle \langle x\rangle } . Another commonly used notation is x {\displaystyle x} + or ( x ) {\displaystyle (x)} + for the positive part of x {\displaystyle x} , which avoids conflicts with { . . . } {\displaystyle \{...\}} for set notation. == In engineering == Macaulay's notation is commonly used in the static analysis of bending moments of a beam. This is useful because shear forces applied on a member render the shear and moment diagram discontinuous. Macaulay's notation also provides an easy way of integrating these discontinuous curves to give bending moments, angular deflection, and so on. For engineering purposes, angle brackets are often used to denote the use of Macaulay's method. ⟨ x − a ⟩ n = { 0 , x < a ( x − a ) n , x ≥ a . {\displaystyle \langle x-a\rangle ^{n}={\begin{cases}0,&x<a\\(x-a)^{n},&x\geq a.\end{cases}}} ( n ≥ 0 ) {\displaystyle (n\geq 0)} The above example simply states that the function takes the value ( x − a ) n {\displaystyle (x-a)^{n}} for all x values larger than a. With this, all the forces acting on a beam can be added, with their respective points of action being the value of a. A particular case is the unit step function, ⟨ x − a ⟩ 0 ≡ { x − a } 0 = { 0 , x < a 1 , x > a . {\displaystyle \langle x-a\rangle ^{0}\equiv \{x-a\}^{0}={\begin{cases}0,&x<a\\1,&x>a.\end{cases}}} == See also == Singularity function == References ==
Wikipedia:Macdonald identities#0	In mathematics, the Macdonald identities are some infinite product identities associated to affine root systems, introduced by Ian Macdonald (1972). They include as special cases the Jacobi triple product identity, Watson's quintuple product identity, several identities found by Dyson (1972), and a 10-fold product identity found by Winquist (1969). Kac (1974) and Moody (1975) pointed out that the Macdonald identities are the analogs of the Weyl denominator formula for affine Kac–Moody algebras and superalgebras. == References == Demazure, Michel (1977), "Identités de Macdonald", Séminaire Bourbaki, 28e année (1975/1976), Exp. No. 483, Lecture Notes in Math, vol. 567, Berlin, New York: Springer-Verlag, pp. 191–201, MR 0476815 Dyson, Freeman J. (1972), "Missed opportunities", Bulletin of the American Mathematical Society, 78: 635–652, doi:10.1090/S0002-9904-1972-12971-9, ISSN 0002-9904, MR 0522147 Kac, Victor G (1974), "Infinite-dimensional Lie algebras, and the Dedekind η-function", Akademija Nauk SSSR. Funkcionalnyi Analiz i ego Priloženija, 8 (1): 77–78, doi:10.1007/BF02028313, ISSN 0374-1990, MR 0374210 Moody, R. V. (1975), "Macdonald identities and Euclidean Lie algebras", Proceedings of the American Mathematical Society, 48: 43–52, doi:10.2307/2040690, ISSN 0002-9939, JSTOR 2040690, MR 0442048 Macdonald, I. G. (1972), "Affine root systems and Dedekind's η-function", Inventiones Mathematicae, 15: 91–143, doi:10.1007/BF01418931, ISSN 0020-9910, MR 0357528 Winquist, Lasse (1969), "An elementary proof of p(11m+6) ≡ 0 mod 11", Journal of Combinatorial Theory, 6: 56–59, doi:10.1016/s0021-9800(69)80105-5, MR 0236136
Wikipedia:Maciej Zworski#0	Maciej Zworski is a Polish-Canadian mathematician, currently a professor of mathematics at the University of California, Berkeley. His mathematical interests include microlocal analysis, scattering theory, and partial differential equations. He was an invited speaker at International Congress of Mathematicians in Beijing in 2002, and a plenary speaker at the conference Dynamics, Equations and Applications in Kraków in 2019. == Selected publications == === Articles === Zworski M (1988). "Decomposition of normal currents". Proc. Amer. Math. Soc. 102 (4): 831–839. doi:10.1090/s0002-9939-1988-0934852-8. MR 0934852. Sjöstrand Johannes (1991). "Complex scaling and the distribution of scattering poles". J. Amer. Math. Soc. 4: 729–769. doi:10.2307/2939287. JSTOR 2939287. MR 1115789. Guillopé Laurent (1997). "Scattering Asymptotics for Riemann Surfaces". Annals of Mathematics. Second Series. 145 (3): 597–660. doi:10.2307/2951846. JSTOR 2951846. "Resonances in Physics and Geometry" (PDF). Notices of the AMS. 46 (3): 319–328. 1999. Zworksi M (2001). "A remark on a paper of E. B. Davies". Proc. Amer. Math. Soc. 129 (10): 2955–2957. doi:10.1090/s0002-9939-01-05909-3. MR 1840099. Wunsch Jared (2001). "The FBI transform on compact C∞ manifolds". Trans. Amer. Math. Soc. 353: 1151–1167. doi:10.1090/s0002-9947-00-02751-3. MR 1804416. robin Graham With C (2003). "Scattering matrix in conformal geometry". Inventiones Mathematicae. 152 (1): 89–118. arXiv:math/0109089. Bibcode:2003InMat.152...89G. doi:10.1007/s00222-002-0268-1. S2CID 18310849. Burq Nicolas (2004). "Geometric control in the presence of a black box". J. Amer. Math. Soc. 17 (2): 443–471. doi:10.1090/s0894-0347-04-00452-7. MR 2051618. === Books === with Richard Melrose and Antônio Sá Barreto: Semi-linear diffraction of conormal waves, Astérisque, vol. 240, Societé Mathématique de France, 1996 abstract Semiclassical analysis, American Mathematical Society 2012 as editor with Plamen Stefanov and András Vasy: Inverse Problems and Applications. Contemporary Mathematics, vol. 615. American Mathematical Society. 2014. ISBN 9781470410797. with Semyon Dyatlov: Mathematical Theory of Scattering Resonances. Graduate Studies in Mathematics, vol. 200. American Mathematical Society. 2019. ISBN 9781470443665. == References == == External links == Professor Zworski's webpage Maciej Zworski at the Mathematics Genealogy Project
Wikipedia:Maclaurin series#0	In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor series are named after Brook Taylor, who introduced them in 1715. A Taylor series is also called a Maclaurin series when 0 is the point where the derivatives are considered, after Colin Maclaurin, who made extensive use of this special case of Taylor series in the 18th century. The partial sum formed by the first n + 1 terms of a Taylor series is a polynomial of degree n that is called the nth Taylor polynomial of the function. Taylor polynomials are approximations of a function, which become generally more accurate as n increases. Taylor's theorem gives quantitative estimates on the error introduced by the use of such approximations. If the Taylor series of a function is convergent, its sum is the limit of the infinite sequence of the Taylor polynomials. A function may differ from the sum of its Taylor series, even if its Taylor series is convergent. A function is analytic at a point x if it is equal to the sum of its Taylor series in some open interval (or open disk in the complex plane) containing x. This implies that the function is analytic at every point of the interval (or disk). == Definition == The Taylor series of a real or complex-valued function f (x), that is infinitely differentiable at a real or complex number a, is the power series f ( a ) + f ′ ( a ) 1 ! ( x − a ) + f ″ ( a ) 2 ! ( x − a ) 2 + ⋯ = ∑ n = 0 ∞ f ( n ) ( a ) n ! ( x − a ) n . {\displaystyle f(a)+{\frac {f'(a)}{1!}}(x-a)+{\frac {f''(a)}{2!}}(x-a)^{2}+\cdots =\sum _{n=0}^{\infty }{\frac {f^{(n)}(a)}{n!}}(x-a)^{n}.} Here, n! denotes the factorial of n. The function f(n)(a) denotes the nth derivative of f evaluated at the point a. The derivative of order zero of f is defined to be f itself and (x − a)0 and 0! are both defined to be 1. This series can be written by using sigma notation, as in the right side formula. With a = 0, the Maclaurin series takes the form: f ( 0 ) + f ′ ( 0 ) 1 ! x + f ″ ( 0 ) 2 ! x 2 + ⋯ = ∑ n = 0 ∞ f ( n ) ( 0 ) n ! x n . {\displaystyle f(0)+{\frac {f'(0)}{1!}}x+{\frac {f''(0)}{2!}}x^{2}+\cdots =\sum _{n=0}^{\infty }{\frac {f^{(n)}(0)}{n!}}x^{n}.} == Examples == The Taylor series of any polynomial is the polynomial itself. The Maclaurin series of ⁠1/1 − x⁠ is the geometric series 1 + x + x 2 + x 3 + ⋯ . {\displaystyle 1+x+x^{2}+x^{3}+\cdots .} So, by substituting x for 1 − x, the Taylor series of ⁠1/x⁠ at a = 1 is 1 − ( x − 1 ) + ( x − 1 ) 2 − ( x − 1 ) 3 + ⋯ . {\displaystyle 1-(x-1)+(x-1)^{2}-(x-1)^{3}+\cdots .} By integrating the above Maclaurin series, we find the Maclaurin series of ln(1 − x), where ln denotes the natural logarithm: − x − 1 2 x 2 − 1 3 x 3 − 1 4 x 4 − ⋯ . {\displaystyle -x-{\tfrac {1}{2}}x^{2}-{\tfrac {1}{3}}x^{3}-{\tfrac {1}{4}}x^{4}-\cdots .} The corresponding Taylor series of ln x at a = 1 is ( x − 1 ) − 1 2 ( x − 1 ) 2 + 1 3 ( x − 1 ) 3 − 1 4 ( x − 1 ) 4 + ⋯ , {\displaystyle (x-1)-{\tfrac {1}{2}}(x-1)^{2}+{\tfrac {1}{3}}(x-1)^{3}-{\tfrac {1}{4}}(x-1)^{4}+\cdots ,} and more generally, the corresponding Taylor series of ln x at an arbitrary nonzero point a is: ln ⁡ a + 1 a ( x − a ) − 1 a 2 ( x − a ) 2 2 + ⋯ . {\displaystyle \ln a+{\frac {1}{a}}(x-a)-{\frac {1}{a^{2}}}{\frac {\left(x-a\right)^{2}}{2}}+\cdots .} The Maclaurin series of the exponential function ex is ∑ n = 0 ∞ x n n ! = x 0 0 ! + x 1 1 ! + x 2 2 ! + x 3 3 ! + x 4 4 ! + x 5 5 ! + ⋯ = 1 + x + x 2 2 + x 3 6 + x 4 24 + x 5 120 + ⋯ . {\displaystyle {\begin{aligned}\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}&={\frac {x^{0}}{0!}}+{\frac {x^{1}}{1!}}+{\frac {x^{2}}{2!}}+{\frac {x^{3}}{3!}}+{\frac {x^{4}}{4!}}+{\frac {x^{5}}{5!}}+\cdots \\&=1+x+{\frac {x^{2}}{2}}+{\frac {x^{3}}{6}}+{\frac {x^{4}}{24}}+{\frac {x^{5}}{120}}+\cdots .\end{aligned}}} The above expansion holds because the derivative of ex with respect to x is also ex, and e0 equals 1. This leaves the terms (x − 0)n in the numerator and n! in the denominator of each term in the infinite sum. == History == The ancient Greek philosopher Zeno of Elea considered the problem of summing an infinite series to achieve a finite result, but rejected it as an impossibility; the result was Zeno's paradox. Later, Aristotle proposed a philosophical resolution of the paradox, but the mathematical content was apparently unresolved until taken up by Archimedes, as it had been prior to Aristotle by the Presocratic Atomist Democritus. It was through Archimedes's method of exhaustion that an infinite number of progressive subdivisions could be performed to achieve a finite result. Liu Hui independently employed a similar method a few centuries later. In the 14th century, the earliest examples of specific Taylor series (but not the general method) were given by Indian mathematician Madhava of Sangamagrama. Though no record of his work survives, writings of his followers in the Kerala school of astronomy and mathematics suggest that he found the Taylor series for the trigonometric functions of sine, cosine, and arctangent (see Madhava series). During the following two centuries his followers developed further series expansions and rational approximations. In late 1670, James Gregory was shown in a letter from John Collins several Maclaurin series ( sin ⁡ x , {\textstyle \sin x,} cos ⁡ x , {\textstyle \cos x,} arcsin ⁡ x , {\textstyle \arcsin x,} and x cot ⁡ x {\textstyle x\cot x} ) derived by Isaac Newton, and told that Newton had developed a general method for expanding functions in series. Newton had in fact used a cumbersome method involving long division of series and term-by-term integration, but Gregory did not know it and set out to discover a general method for himself. In early 1671 Gregory discovered something like the general Maclaurin series and sent a letter to Collins including series for arctan ⁡ x , {\textstyle \arctan x,} tan ⁡ x , {\textstyle \tan x,} sec ⁡ x , {\textstyle \sec x,} ln sec ⁡ x {\textstyle \ln \,\sec x} (the integral of tan {\displaystyle \tan } ), ln tan ⁡ 1 2 ( 1 2 π + x ) {\textstyle \ln \,\tan {\tfrac {1}{2}}{{\bigl (}{\tfrac {1}{2}}\pi +x{\bigr )}}} (the integral of sec, the inverse Gudermannian function), arcsec ⁡ ( 2 e x ) , {\textstyle \operatorname {arcsec} {\bigl (}{\sqrt {2}}e^{x}{\bigr )},} and 2 arctan ⁡ e x − 1 2 π {\textstyle 2\arctan e^{x}-{\tfrac {1}{2}}\pi } (the Gudermannian function). However, thinking that he had merely redeveloped a method by Newton, Gregory never described how he obtained these series, and it can only be inferred that he understood the general method by examining scratch work he had scribbled on the back of another letter from 1671. In 1691–1692, Isaac Newton wrote down an explicit statement of the Taylor and Maclaurin series in an unpublished version of his work De Quadratura Curvarum. However, this work was never completed and the relevant sections were omitted from the portions published in 1704 under the title Tractatus de Quadratura Curvarum. It was not until 1715 that a general method for constructing these series for all functions for which they exist was finally published by Brook Taylor, after whom the series are now named. The Maclaurin series was named after Colin Maclaurin, a Scottish mathematician, who published a special case of the Taylor result in the mid-18th century. == Analytic functions == If f (x) is given by a convergent power series in an open disk centred at b in the complex plane (or an interval in the real line), it is said to be analytic in this region. Thus for x in this region, f is given by a convergent power series f ( x ) = ∑ n = 0 ∞ a n ( x − b ) n . {\displaystyle f(x)=\sum _{n=0}^{\infty }a_{n}(x-b)^{n}.} Differentiating by x the above formula n times, then setting x = b gives: f ( n ) ( b ) n ! = a n {\displaystyle {\frac {f^{(n)}(b)}{n!}}=a_{n}} and so the power series expansion agrees with the Taylor series. Thus a function is analytic in an open disk centered at b if and only if its Taylor series converges to the value of the function at each point of the disk. If f (x) is equal to the sum of its Taylor series for all x in the complex plane, it is called entire. The polynomials, exponential function ex, and the trigonometric functions sine and cosine, are examples of entire functions. Examples of functions that are not entire include the square root, the logarithm, the trigonometric function tangent, and its inverse, arctan. For these functions the Taylor series do not converge if x is far from b. That is, the Taylor series diverges at x if the distance between x and b is larger than the radius of convergence. The Taylor series can be used to calculate the value of an entire function at every point, if the value of the function, and of all of its derivatives, are known at a single point. Uses of the Taylor series for analytic functions include: The partial sums (the Taylor polynomials) of the series can be used as approximations of the function. These approximations are good if sufficiently many terms are included. Differentiation and integration of power series can be performed term by term and is hence particularly easy. An analytic function is uniquely extended to a holomorphic function on an open disk in the complex plane. This makes the machinery of complex analysis available. The (truncated) series can be used to compute function values numerically, (often by recasting the polynomial into the Chebyshev form and evaluating it with the Clenshaw algorithm). Algebraic operations can be done readily on the power series representation; for instance, Euler's formula follows from Taylor series expansions for trigonometric and exponential functions. This result is of fundamental importance in such fields as harmonic analysis. Approximations using the first few terms of a Taylor series can make otherwise unsolvable problems possible for a restricted domain; this approach is often used in physics. == Approximation error and convergence == Pictured is an accurate approximation of sin x around the point x = 0. The pink curve is a polynomial of degree seven: sin ⁡ x ≈ x − x 3 3 ! + x 5 5 ! − x 7 7 ! . {\displaystyle \sin {x}\approx x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}.\!} The error in this approximation is no more than \|x\|9 / 9!. For a full cycle centered at the origin (−π < x < π) the error is less than 0.08215. In particular, for −1 < x < 1, the error is less than 0.000003. In contrast, also shown is a picture of the natural logarithm function ln(1 + x) and some of its Taylor polynomials around a = 0. These approximations converge to the function only in the region −1 < x ≤ 1; outside of this region the higher-degree Taylor polynomials are worse approximations for the function. The error incurred in approximating a function by its nth-degree Taylor polynomial is called the remainder or residual and is denoted by the function Rn(x). Taylor's theorem can be used to obtain a bound on the size of the remainder. In general, Taylor series need not be convergent at all. In fact, the set of functions with a convergent Taylor series is a meager set in the Fréchet space of smooth functions. Even if the Taylor series of a function f does converge, its limit need not be equal to the value of the function f (x). For example, the function f ( x ) = { e − 1 / x 2 if x ≠ 0 0 if x = 0 {\displaystyle f(x)={\begin{cases}e^{-1/x^{2}}&{\text{if }}x\neq 0\\[3mu]0&{\text{if }}x=0\end{cases}}} is infinitely differentiable at x = 0, and has all derivatives zero there. Consequently, the Taylor series of f (x) about x = 0 is identically zero. However, f (x) is not the zero function, so does not equal its Taylor series around the origin. Thus, f (x) is an example of a non-analytic smooth function. In real analysis, this example shows that there are infinitely differentiable functions f (x) whose Taylor series are not equal to f (x) even if they converge. By contrast, the holomorphic functions studied in complex analysis always possess a convergent Taylor series, and even the Taylor series of meromorphic functions, which might have singularities, never converge to a value different from the function itself. The complex function e−1/z2, however, does not approach 0 when z approaches 0 along the imaginary axis, so it is not continuous in the complex plane and its Taylor series is undefined at 0. More generally, every sequence of real or complex numbers can appear as coefficients in the Taylor series of an infinitely differentiable function defined on the real line, a consequence of Borel's lemma. As a result, the radius of convergence of a Taylor series can be zero. There are even infinitely differentiable functions defined on the real line whose Taylor series have a radius of convergence 0 everywhere. A function cannot be written as a Taylor series centred at a singularity; in these cases, one can often still achieve a series expansion if one allows also negative powers of the variable x; see Laurent series. For example, f (x) = e−1/x2 can be written as a Laurent series. === Generalization === The generalization of the Taylor series does converge to the value of the function itself for any bounded continuous function on (0,∞), and this can be done by using the calculus of finite differences. Specifically, the following theorem, due to Einar Hille, that for any t > 0, lim h → 0 + ∑ n = 0 ∞ t n n ! Δ h n f ( a ) h n = f ( a + t ) . {\displaystyle \lim _{h\to 0^{+}}\sum _{n=0}^{\infty }{\frac {t^{n}}{n!}}{\frac {\Delta _{h}^{n}f(a)}{h^{n}}}=f(a+t).} Here Δnh is the nth finite difference operator with step size h. The series is precisely the Taylor series, except that divided differences appear in place of differentiation: the series is formally similar to the Newton series. When the function f is analytic at a, the terms in the series converge to the terms of the Taylor series, and in this sense generalizes the usual Taylor series. In general, for any infinite sequence ai, the following power series identity holds: ∑ n = 0 ∞ u n n ! Δ n a i = e − u ∑ j = 0 ∞ u j j ! a i + j . {\displaystyle \sum _{n=0}^{\infty }{\frac {u^{n}}{n!}}\Delta ^{n}a_{i}=e^{-u}\sum _{j=0}^{\infty }{\frac {u^{j}}{j!}}a_{i+j}.} So in particular, f ( a + t ) = lim h → 0 + e − t / h ∑ j = 0 ∞ f ( a + j h ) ( t / h ) j j ! . {\displaystyle f(a+t)=\lim _{h\to 0^{+}}e^{-t/h}\sum _{j=0}^{\infty }f(a+jh){\frac {(t/h)^{j}}{j!}}.} The series on the right is the expected value of f (a + X), where X is a Poisson-distributed random variable that takes the value jh with probability e−t/h·⁠(t/h)j/j!⁠. Hence, f ( a + t ) = lim h → 0 + ∫ − ∞ ∞ f ( a + x ) d P t / h , h ( x ) . {\displaystyle f(a+t)=\lim _{h\to 0^{+}}\int _{-\infty }^{\infty }f(a+x)dP_{t/h,h}(x).} The law of large numbers implies that the identity holds. == List of Maclaurin series of some common functions == Several important Maclaurin series expansions follow. All these expansions are valid for complex arguments x. === Exponential function === The exponential function e x {\displaystyle e^{x}} (with base e) has Maclaurin series e x = ∑ n = 0 ∞ x n n ! = 1 + x + x 2 2 ! + x 3 3 ! + ⋯ . {\displaystyle e^{x}=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}=1+x+{\frac {x^{2}}{2!}}+{\frac {x^{3}}{3!}}+\cdots .} It converges for all x. The exponential generating function of the Bell numbers is the exponential function of the predecessor of the exponential function: exp ⁡ ( exp ⁡ x − 1 ) = ∑ n = 0 ∞ B n n ! x n {\displaystyle \exp(\exp {x}-1)=\sum _{n=0}^{\infty }{\frac {B_{n}}{n!}}x^{n}} === Natural logarithm === The natural logarithm (with base e) has Maclaurin series ln ⁡ ( 1 − x ) = − ∑ n = 1 ∞ x n n = − x − x 2 2 − x 3 3 − ⋯ , ln ⁡ ( 1 + x ) = ∑ n = 1 ∞ ( − 1 ) n + 1 x n n = x − x 2 2 + x 3 3 − ⋯ . {\displaystyle {\begin{aligned}\ln(1-x)&=-\sum _{n=1}^{\infty }{\frac {x^{n}}{n}}=-x-{\frac {x^{2}}{2}}-{\frac {x^{3}}{3}}-\cdots ,\\\ln(1+x)&=\sum _{n=1}^{\infty }(-1)^{n+1}{\frac {x^{n}}{n}}=x-{\frac {x^{2}}{2}}+{\frac {x^{3}}{3}}-\cdots .\end{aligned}}} The last series is known as Mercator series, named after Nicholas Mercator (since it was published in his 1668 treatise Logarithmotechnia). Both of these series converge for \| x \| < 1 {\displaystyle \|x\|<1} . (In addition, the series for ln(1 − x) converges for x = −1, and the series for ln(1 + x) converges for x = 1.) === Geometric series === The geometric series and its derivatives have Maclaurin series 1 1 − x = ∑ n = 0 ∞ x n 1 ( 1 − x ) 2 = ∑ n = 1 ∞ n x n − 1 1 ( 1 − x ) 3 = ∑ n = 2 ∞ ( n − 1 ) n 2 x n − 2 . {\displaystyle {\begin{aligned}{\frac {1}{1-x}}&=\sum _{n=0}^{\infty }x^{n}\\{\frac {1}{(1-x)^{2}}}&=\sum _{n=1}^{\infty }nx^{n-1}\\{\frac {1}{(1-x)^{3}}}&=\sum _{n=2}^{\infty }{\frac {(n-1)n}{2}}x^{n-2}.\end{aligned}}} All are convergent for \| x \| < 1 {\displaystyle \|x\|<1} . These are special cases of the binomial series given in the next section. === Binomial series === The binomial series is the power series ( 1 + x ) α = ∑ n = 0 ∞ ( α n ) x n {\displaystyle (1+x)^{\alpha }=\sum _{n=0}^{\infty }{\binom {\alpha }{n}}x^{n}} whose coefficients are the generalized binomial coefficients ( α n ) = ∏ k = 1 n α − k + 1 k = α ( α − 1 ) ⋯ ( α − n + 1 ) n ! . {\displaystyle {\binom {\alpha }{n}}=\prod _{k=1}^{n}{\frac {\alpha -k+1}{k}}={\frac {\alpha (\alpha -1)\cdots (\alpha -n+1)}{n!}}.} (If n = 0, this product is an empty product and has value 1.) It converges for \| x \| < 1 {\displaystyle \|x\|<1} for any real or complex number α. When α = −1, this is essentially the infinite geometric series mentioned in the previous section. The special cases α = ⁠1/2⁠ and α = −⁠1/2⁠ give the square root function and its inverse: ( 1 + x ) 1 2 = 1 + 1 2 x − 1 8 x 2 + 1 16 x 3 − 5 128 x 4 + 7 256 x 5 − ⋯ = ∑ n = 0 ∞ ( − 1 ) n − 1 ( 2 n ) ! 4 n ( n ! ) 2 ( 2 n − 1 ) x n , ( 1 + x ) − 1 2 = 1 − 1 2 x + 3 8 x 2 − 5 16 x 3 + 35 128 x 4 − 63 256 x 5 + ⋯ = ∑ n = 0 ∞ ( − 1 ) n ( 2 n ) ! 4 n ( n ! ) 2 x n . {\displaystyle {\begin{aligned}(1+x)^{\frac {1}{2}}&=1+{\frac {1}{2}}x-{\frac {1}{8}}x^{2}+{\frac {1}{16}}x^{3}-{\frac {5}{128}}x^{4}+{\frac {7}{256}}x^{5}-\cdots &=\sum _{n=0}^{\infty }{\frac {(-1)^{n-1}(2n)!}{4^{n}(n!)^{2}(2n-1)}}x^{n},\\(1+x)^{-{\frac {1}{2}}}&=1-{\frac {1}{2}}x+{\frac {3}{8}}x^{2}-{\frac {5}{16}}x^{3}+{\frac {35}{128}}x^{4}-{\frac {63}{256}}x^{5}+\cdots &=\sum _{n=0}^{\infty }{\frac {(-1)^{n}(2n)!}{4^{n}(n!)^{2}}}x^{n}.\end{aligned}}} When only the linear term is retained, this simplifies to the binomial approximation. === Trigonometric functions === The usual trigonometric functions and their inverses have the following Maclaurin series: sin ⁡ x = ∑ n = 0 ∞ ( − 1 ) n ( 2 n + 1 ) ! x 2 n + 1 = x − x 3 3 ! + x 5 5 ! − ⋯ for all x cos ⁡ x = ∑ n = 0 ∞ ( − 1 ) n ( 2 n ) ! x 2 n = 1 − x 2 2 ! + x 4 4 ! − ⋯ for all x tan ⁡ x = ∑ n = 1 ∞ B 2 n ( − 4 ) n ( 1 − 4 n ) ( 2 n ) ! x 2 n − 1 = x + x 3 3 + 2 x 5 15 + ⋯ for \| x \| < π 2 sec ⁡ x = ∑ n = 0 ∞ ( − 1 ) n E 2 n ( 2 n ) ! x 2 n = 1 + x 2 2 + 5 x 4 24 + ⋯ for \| x \| < π 2 arcsin ⁡ x = ∑ n = 0 ∞ ( 2 n ) ! 4 n ( n ! ) 2 ( 2 n + 1 ) x 2 n + 1 = x + x 3 6 + 3 x 5 40 + ⋯ for \| x \| ≤ 1 arccos ⁡ x = π 2 − arcsin ⁡ x = π 2 − ∑ n = 0 ∞ ( 2 n ) ! 4 n ( n ! ) 2 ( 2 n + 1 ) x 2 n + 1 = π 2 − x − x 3 6 − 3 x 5 40 − ⋯ for \| x \| ≤ 1 arctan ⁡ x = ∑ n = 0 ∞ ( − 1 ) n 2 n + 1 x 2 n + 1 = x − x 3 3 + x 5 5 − ⋯ for \| x \| ≤ 1 , x ≠ ± i {\displaystyle {\begin{aligned}\sin x&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)!}}x^{2n+1}&&=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-\cdots &&{\text{for all }}x\\[6pt]\cos x&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n)!}}x^{2n}&&=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-\cdots &&{\text{for all }}x\\[6pt]\tan x&=\sum _{n=1}^{\infty }{\frac {B_{2n}(-4)^{n}\left(1-4^{n}\right)}{(2n)!}}x^{2n-1}&&=x+{\frac {x^{3}}{3}}+{\frac {2x^{5}}{15}}+\cdots &&{\text{for }}\|x\|<{\frac {\pi }{2}}\\[6pt]\sec x&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}E_{2n}}{(2n)!}}x^{2n}&&=1+{\frac {x^{2}}{2}}+{\frac {5x^{4}}{24}}+\cdots &&{\text{for }}\|x\|<{\frac {\pi }{2}}\\[6pt]\arcsin x&=\sum _{n=0}^{\infty }{\frac {(2n)!}{4^{n}(n!)^{2}(2n+1)}}x^{2n+1}&&=x+{\frac {x^{3}}{6}}+{\frac {3x^{5}}{40}}+\cdots &&{\text{for }}\|x\|\leq 1\\[6pt]\arccos x&={\frac {\pi }{2}}-\arcsin x\\&={\frac {\pi }{2}}-\sum _{n=0}^{\infty }{\frac {(2n)!}{4^{n}(n!)^{2}(2n+1)}}x^{2n+1}&&={\frac {\pi }{2}}-x-{\frac {x^{3}}{6}}-{\frac {3x^{5}}{40}}-\cdots &&{\text{for }}\|x\|\leq 1\\[6pt]\arctan x&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{2n+1}}x^{2n+1}&&=x-{\frac {x^{3}}{3}}+{\frac {x^{5}}{5}}-\cdots &&{\text{for }}\|x\|\leq 1,\ x\neq \pm i\end{aligned}}} All angles are expressed in radians. The numbers Bk appearing in the expansions of tan x are the Bernoulli numbers. The Ek in the expansion of sec x are Euler numbers. === Hyperbolic functions === The hyperbolic functions have Maclaurin series closely related to the series for the corresponding trigonometric functions: sinh ⁡ x = ∑ n = 0 ∞ x 2 n + 1 ( 2 n + 1 ) ! = x + x 3 3 ! + x 5 5 ! + ⋯ for all x cosh ⁡ x = ∑ n = 0 ∞ x 2 n ( 2 n ) ! = 1 + x 2 2 ! + x 4 4 ! + ⋯ for all x tanh ⁡ x = ∑ n = 1 ∞ B 2 n 4 n ( 4 n − 1 ) ( 2 n ) ! x 2 n − 1 = x − x 3 3 + 2 x 5 15 − 17 x 7 315 + ⋯ for \| x \| < π 2 arsinh ⁡ x = ∑ n = 0 ∞ ( − 1 ) n ( 2 n ) ! 4 n ( n ! ) 2 ( 2 n + 1 ) x 2 n + 1 = x − x 3 6 + 3 x 5 40 − ⋯ for \| x \| ≤ 1 artanh ⁡ x = ∑ n = 0 ∞ x 2 n + 1 2 n + 1 = x + x 3 3 + x 5 5 + ⋯ for \| x \| ≤ 1 , x ≠ ± 1 {\displaystyle {\begin{aligned}\sinh x&=\sum _{n=0}^{\infty }{\frac {x^{2n+1}}{(2n+1)!}}&&=x+{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}+\cdots &&{\text{for all }}x\\[6pt]\cosh x&=\sum _{n=0}^{\infty }{\frac {x^{2n}}{(2n)!}}&&=1+{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}+\cdots &&{\text{for all }}x\\[6pt]\tanh x&=\sum _{n=1}^{\infty }{\frac {B_{2n}4^{n}\left(4^{n}-1\right)}{(2n)!}}x^{2n-1}&&=x-{\frac {x^{3}}{3}}+{\frac {2x^{5}}{15}}-{\frac {17x^{7}}{315}}+\cdots &&{\text{for }}\|x\|<{\frac {\pi }{2}}\\[6pt]\operatorname {arsinh} x&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}(2n)!}{4^{n}(n!)^{2}(2n+1)}}x^{2n+1}&&=x-{\frac {x^{3}}{6}}+{\frac {3x^{5}}{40}}-\cdots &&{\text{for }}\|x\|\leq 1\\[6pt]\operatorname {artanh} x&=\sum _{n=0}^{\infty }{\frac {x^{2n+1}}{2n+1}}&&=x+{\frac {x^{3}}{3}}+{\frac {x^{5}}{5}}+\cdots &&{\text{for }}\|x\|\leq 1,\ x\neq \pm 1\end{aligned}}} The numbers Bk appearing in the series for tanh x are the Bernoulli numbers. === Polylogarithmic functions === The polylogarithms have these defining identities: Li 2 ( x ) = ∑ n = 1 ∞ 1 n 2 x n Li 3 ( x ) = ∑ n = 1 ∞ 1 n 3 x n {\displaystyle {\begin{aligned}{\text{Li}}_{2}(x)&=\sum _{n=1}^{\infty }{\frac {1}{n^{2}}}x^{n}\\{\text{Li}}_{3}(x)&=\sum _{n=1}^{\infty }{\frac {1}{n^{3}}}x^{n}\end{aligned}}} The Legendre chi functions are defined as follows: χ 2 ( x ) = ∑ n = 0 ∞ 1 ( 2 n + 1 ) 2 x 2 n + 1 χ 3 ( x ) = ∑ n = 0 ∞ 1 ( 2 n + 1 ) 3 x 2 n + 1 {\displaystyle {\begin{aligned}\chi _{2}(x)&=\sum _{n=0}^{\infty }{\frac {1}{(2n+1)^{2}}}x^{2n+1}\\\chi _{3}(x)&=\sum _{n=0}^{\infty }{\frac {1}{(2n+1)^{3}}}x^{2n+1}\end{aligned}}} And the formulas presented below are called inverse tangent integrals: Ti 2 ( x ) = ∑ n = 0 ∞ ( − 1 ) n ( 2 n + 1 ) 2 x 2 n + 1 Ti 3 ( x ) = ∑ n = 0 ∞ ( − 1 ) n ( 2 n + 1 ) 3 x 2 n + 1 {\displaystyle {\begin{aligned}{\text{Ti}}_{2}(x)&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)^{2}}}x^{2n+1}\\{\text{Ti}}_{3}(x)&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)^{3}}}x^{2n+1}\end{aligned}}} In statistical thermodynamics these formulas are of great importance. === Elliptic functions === The complete elliptic integrals of first kind K and of second kind E can be defined as follows: 2 π K ( x ) = ∑ n = 0 ∞ [ ( 2 n ) ! ] 2 16 n ( n ! ) 4 x 2 n 2 π E ( x ) = ∑ n = 0 ∞ [ ( 2 n ) ! ] 2 ( 1 − 2 n ) 16 n ( n ! ) 4 x 2 n {\displaystyle {\begin{aligned}{\frac {2}{\pi }}K(x)&=\sum _{n=0}^{\infty }{\frac {[(2n)!]^{2}}{16^{n}(n!)^{4}}}x^{2n}\\{\frac {2}{\pi }}E(x)&=\sum _{n=0}^{\infty }{\frac {[(2n)!]^{2}}{(1-2n)16^{n}(n!)^{4}}}x^{2n}\end{aligned}}} The Jacobi theta functions describe the world of the elliptic modular functions and they have these Taylor series: ϑ 00 ( x ) = 1 + 2 ∑ n = 1 ∞ x n 2 ϑ 01 ( x ) = 1 + 2 ∑ n = 1 ∞ ( − 1 ) n x n 2 {\displaystyle {\begin{aligned}\vartheta _{00}(x)&=1+2\sum _{n=1}^{\infty }x^{n^{2}}\\\vartheta _{01}(x)&=1+2\sum _{n=1}^{\infty }(-1)^{n}x^{n^{2}}\end{aligned}}} The regular partition number sequence P(n) has this generating function: ϑ 00 ( x ) − 1 / 6 ϑ 01 ( x ) − 2 / 3 [ ϑ 00 ( x ) 4 − ϑ 01 ( x ) 4 16 x ] − 1 / 24 = ∑ n = 0 ∞ P ( n ) x n = ∏ k = 1 ∞ 1 1 − x k {\displaystyle \vartheta _{00}(x)^{-1/6}\vartheta _{01}(x)^{-2/3}{\biggl [}{\frac {\vartheta _{00}(x)^{4}-\vartheta _{01}(x)^{4}}{16\,x}}{\biggr ]}^{-1/24}=\sum _{n=0}^{\infty }P(n)x^{n}=\prod _{k=1}^{\infty }{\frac {1}{1-x^{k}}}} The strict partition number sequence Q(n) has that generating function: ϑ 00 ( x ) 1 / 6 ϑ 01 ( x ) − 1 / 3 [ ϑ 00 ( x ) 4 − ϑ 01 ( x ) 4 16 x ] 1 / 24 = ∑ n = 0 ∞ Q ( n ) x n = ∏ k = 1 ∞ 1 1 − x 2 k − 1 {\displaystyle \vartheta _{00}(x)^{1/6}\vartheta _{01}(x)^{-1/3}{\biggl [}{\frac {\vartheta _{00}(x)^{4}-\vartheta _{01}(x)^{4}}{16\,x}}{\biggr ]}^{1/24}=\sum _{n=0}^{\infty }Q(n)x^{n}=\prod _{k=1}^{\infty }{\frac {1}{1-x^{2k-1}}}} == Calculation of Taylor series == Several methods exist for the calculation of Taylor series of a large number of functions. One can attempt to use the definition of the Taylor series, though this often requires generalizing the form of the coefficients according to a readily apparent pattern. Alternatively, one can use manipulations such as substitution, multiplication or division, addition or subtraction of standard Taylor series to construct the Taylor series of a function, by virtue of Taylor series being power series. In some cases, one can also derive the Taylor series by repeatedly applying integration by parts. Particularly convenient is the use of computer algebra systems to calculate Taylor series. === First example === In order to compute the 7th degree Maclaurin polynomial for the function f ( x ) = ln ⁡ ( cos ⁡ x ) , x ∈ ( − π 2 , π 2 ) , {\displaystyle f(x)=\ln(\cos x),\quad x\in {\bigl (}{-{\tfrac {\pi }{2}}},{\tfrac {\pi }{2}}{\bigr )},} one may first rewrite the function as f ( x ) = ln ( 1 + ( cos ⁡ x − 1 ) ) , {\displaystyle f(x)={\ln }{\bigl (}1+(\cos x-1){\bigr )},} the composition of two functions x ↦ ln ⁡ ( 1 + x ) {\displaystyle x\mapsto \ln(1+x)} and x ↦ cos ⁡ x − 1. {\displaystyle x\mapsto \cos x-1.} The Taylor series for the natural logarithm is (using big O notation) ln ⁡ ( 1 + x ) = x − x 2 2 + x 3 3 + O ( x 4 ) {\displaystyle \ln(1+x)=x-{\frac {x^{2}}{2}}+{\frac {x^{3}}{3}}+O{\left(x^{4}\right)}} and for the cosine function cos ⁡ x − 1 = − x 2 2 + x 4 24 − x 6 720 + O ( x 8 ) . {\displaystyle \cos x-1=-{\frac {x^{2}}{2}}+{\frac {x^{4}}{24}}-{\frac {x^{6}}{720}}+O{\left(x^{8}\right)}.} The first several terms from the second series can be substituted into each term of the first series. Because the first term in the second series has degree 2, three terms of the first series suffice to give a 7th-degree polynomial: f ( x ) = ln ⁡ ( 1 + ( cos ⁡ x − 1 ) ) = ( cos ⁡ x − 1 ) − 1 2 ( cos ⁡ x − 1 ) 2 + 1 3 ( cos ⁡ x − 1 ) 3 + O ( ( cos ⁡ x − 1 ) 4 ) = − x 2 2 − x 4 12 − x 6 45 + O ( x 8 ) . {\displaystyle {\begin{aligned}f(x)&=\ln {\bigl (}1+(\cos x-1){\bigr )}\\&=(\cos x-1)-{\tfrac {1}{2}}(\cos x-1)^{2}+{\tfrac {1}{3}}(\cos x-1)^{3}+O{\left((\cos x-1)^{4}\right)}\\&=-{\frac {x^{2}}{2}}-{\frac {x^{4}}{12}}-{\frac {x^{6}}{45}}+O{\left(x^{8}\right)}.\end{aligned}}\!} Since the cosine is an even function, the coefficients for all the odd powers are zero. === Second example === Suppose we want the Taylor series at 0 of the function g ( x ) = e x cos ⁡ x . {\displaystyle g(x)={\frac {e^{x}}{\cos x}}.\!} The Taylor series for the exponential function is e x = 1 + x + x 2 2 ! + x 3 3 ! + x 4 4 ! + ⋯ , {\displaystyle e^{x}=1+x+{\frac {x^{2}}{2!}}+{\frac {x^{3}}{3!}}+{\frac {x^{4}}{4!}}+\cdots ,} and the series for cosine is cos ⁡ x = 1 − x 2 2 ! + x 4 4 ! − ⋯ . {\displaystyle \cos x=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-\cdots .} Assume the series for their quotient is e x cos ⁡ x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + c 4 x 4 + ⋯ {\displaystyle {\frac {e^{x}}{\cos x}}=c_{0}+c_{1}x+c_{2}x^{2}+c_{3}x^{3}+c_{4}x^{4}+\cdots } Multiplying both sides by the denominator cos ⁡ x {\displaystyle \cos x} and then expanding it as a series yields e x = ( c 0 + c 1 x + c 2 x 2 + c 3 x 3 + c 4 x 4 + ⋯ ) ( 1 − x 2 2 ! + x 4 4 ! − ⋯ ) = c 0 + c 1 x + ( c 2 − c 0 2 ) x 2 + ( c 3 − c 1 2 ) x 3 + ( c 4 − c 2 2 + c 0 4 ! ) x 4 + ⋯ {\displaystyle {\begin{aligned}e^{x}&=\left(c_{0}+c_{1}x+c_{2}x^{2}+c_{3}x^{3}+c_{4}x^{4}+\cdots \right)\left(1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-\cdots \right)\\[5mu]&=c_{0}+c_{1}x+\left(c_{2}-{\frac {c_{0}}{2}}\right)x^{2}+\left(c_{3}-{\frac {c_{1}}{2}}\right)x^{3}+\left(c_{4}-{\frac {c_{2}}{2}}+{\frac {c_{0}}{4!}}\right)x^{4}+\cdots \end{aligned}}} Comparing the coefficients of g ( x ) cos ⁡ x {\displaystyle g(x)\cos x} with the coefficients of e x , {\displaystyle e^{x},} c 0 = 1 , c 1 = 1 , c 2 − 1 2 c 0 = 1 2 , c 3 − 1 2 c 1 = 1 6 , c 4 − 1 2 c 2 + 1 24 c 0 = 1 24 , … . {\displaystyle c_{0}=1,\ \ c_{1}=1,\ \ c_{2}-{\tfrac {1}{2}}c_{0}={\tfrac {1}{2}},\ \ c_{3}-{\tfrac {1}{2}}c_{1}={\tfrac {1}{6}},\ \ c_{4}-{\tfrac {1}{2}}c_{2}+{\tfrac {1}{24}}c_{0}={\tfrac {1}{24}},\ \ldots .} The coefficients c i {\displaystyle c_{i}} of the series for g ( x ) {\displaystyle g(x)} can thus be computed one at a time, amounting to long division of the series for e x {\displaystyle e^{x}} and cos ⁡ x {\displaystyle \cos x} : e x cos ⁡ x = 1 + x + x 2 + 2 3 x 3 + 1 2 x 4 + ⋯ . {\displaystyle {\frac {e^{x}}{\cos x}}=1+x+x^{2}+{\tfrac {2}{3}}x^{3}+{\tfrac {1}{2}}x^{4}+\cdots .} === Third example === Here we employ a method called "indirect expansion" to expand the given function. This method uses the known Taylor expansion of the exponential function. In order to expand (1 + x)ex as a Taylor series in x, we use the known Taylor series of function ex: e x = ∑ n = 0 ∞ x n n ! = 1 + x + x 2 2 ! + x 3 3 ! + x 4 4 ! + ⋯ . {\displaystyle e^{x}=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}=1+x+{\frac {x^{2}}{2!}}+{\frac {x^{3}}{3!}}+{\frac {x^{4}}{4!}}+\cdots .} Thus, ( 1 + x ) e x = e x + x e x = ∑ n = 0 ∞ x n n ! + ∑ n = 0 ∞ x n + 1 n ! = 1 + ∑ n = 1 ∞ x n n ! + ∑ n = 0 ∞ x n + 1 n ! = 1 + ∑ n = 1 ∞ x n n ! + ∑ n = 1 ∞ x n ( n − 1 ) ! = 1 + ∑ n = 1 ∞ ( 1 n ! + 1 ( n − 1 ) ! ) x n = 1 + ∑ n = 1 ∞ n + 1 n ! x n = ∑ n = 0 ∞ n + 1 n ! x n . {\displaystyle {\begin{aligned}(1+x)e^{x}&=e^{x}+xe^{x}=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}+\sum _{n=0}^{\infty }{\frac {x^{n+1}}{n!}}=1+\sum _{n=1}^{\infty }{\frac {x^{n}}{n!}}+\sum _{n=0}^{\infty }{\frac {x^{n+1}}{n!}}\\&=1+\sum _{n=1}^{\infty }{\frac {x^{n}}{n!}}+\sum _{n=1}^{\infty }{\frac {x^{n}}{(n-1)!}}=1+\sum _{n=1}^{\infty }\left({\frac {1}{n!}}+{\frac {1}{(n-1)!}}\right)x^{n}\\&=1+\sum _{n=1}^{\infty }{\frac {n+1}{n!}}x^{n}\\&=\sum _{n=0}^{\infty }{\frac {n+1}{n!}}x^{n}.\end{aligned}}} == Taylor series as definitions == Classically, algebraic functions are defined by an algebraic equation, and transcendental functions (including those discussed above) are defined by some property that holds for them, such as a differential equation. For example, the exponential function is the function which is equal to its own derivative everywhere, and assumes the value 1 at the origin. However, one may equally well define an analytic function by its Taylor series. Taylor series are used to define functions and "operators" in diverse areas of mathematics. In particular, this is true in areas where the classical definitions of functions break down. For example, using Taylor series, one may extend analytic functions to sets of matrices and operators, such as the matrix exponential or matrix logarithm. In other areas, such as formal analysis, it is more convenient to work directly with the power series themselves. Thus one may define a solution of a differential equation as a power series which, one hopes to prove, is the Taylor series of the desired solution. == Taylor series in several variables == The Taylor series may also be generalized to functions of more than one variable with T ( x 1 , … , x d ) = ∑ n 1 = 0 ∞ ⋯ ∑ n d = 0 ∞ ( x 1 − a 1 ) n 1 ⋯ ( x d − a d ) n d n 1 ! ⋯ n d ! ( ∂ n 1 + ⋯ + n d f ∂ x 1 n 1 ⋯ ∂ x d n d ) ( a 1 , … , a d ) = f ( a 1 , … , a d ) + ∑ j = 1 d ∂ f ( a 1 , … , a d ) ∂ x j ( x j − a j ) + 1 2 ! ∑ j = 1 d ∑ k = 1 d ∂ 2 f ( a 1 , … , a d ) ∂ x j ∂ x k ( x j − a j ) ( x k − a k ) + 1 3 ! ∑ j = 1 d ∑ k = 1 d ∑ l = 1 d ∂ 3 f ( a 1 , … , a d ) ∂ x j ∂ x k ∂ x l ( x j − a j ) ( x k − a k ) ( x l − a l ) + ⋯ {\displaystyle {\begin{aligned}T(x_{1},\ldots ,x_{d})&=\sum _{n_{1}=0}^{\infty }\cdots \sum _{n_{d}=0}^{\infty }{\frac {(x_{1}-a_{1})^{n_{1}}\cdots (x_{d}-a_{d})^{n_{d}}}{n_{1}!\cdots n_{d}!}}\,\left({\frac {\partial ^{n_{1}+\cdots +n_{d}}f}{\partial x_{1}^{n_{1}}\cdots \partial x_{d}^{n_{d}}}}\right)(a_{1},\ldots ,a_{d})\\&=f(a_{1},\ldots ,a_{d})+\sum _{j=1}^{d}{\frac {\partial f(a_{1},\ldots ,a_{d})}{\partial x_{j}}}(x_{j}-a_{j})+{\frac {1}{2!}}\sum _{j=1}^{d}\sum _{k=1}^{d}{\frac {\partial ^{2}f(a_{1},\ldots ,a_{d})}{\partial x_{j}\partial x_{k}}}(x_{j}-a_{j})(x_{k}-a_{k})\\&\qquad \qquad +{\frac {1}{3!}}\sum _{j=1}^{d}\sum _{k=1}^{d}\sum _{l=1}^{d}{\frac {\partial ^{3}f(a_{1},\ldots ,a_{d})}{\partial x_{j}\partial x_{k}\partial x_{l}}}(x_{j}-a_{j})(x_{k}-a_{k})(x_{l}-a_{l})+\cdots \end{aligned}}} For example, for a function f ( x , y ) {\displaystyle f(x,y)} that depends on two variables, x and y, the Taylor series to second order about the point (a, b) is f ( a , b ) + ( x − a ) f x ( a , b ) + ( y − b ) f y ( a , b ) + 1 2 ! ( ( x − a ) 2 f x x ( a , b ) + 2 ( x − a ) ( y − b ) f x y ( a , b ) + ( y − b ) 2 f y y ( a , b ) ) {\displaystyle f(a,b)+(x-a)f_{x}(a,b)+(y-b)f_{y}(a,b)+{\frac {1}{2!}}{\Big (}(x-a)^{2}f_{xx}(a,b)+2(x-a)(y-b)f_{xy}(a,b)+(y-b)^{2}f_{yy}(a,b){\Big )}} where the subscripts denote the respective partial derivatives. === Second-order Taylor series in several variables === A second-order Taylor series expansion of a scalar-valued function of more than one variable can be written compactly as T ( x ) = f ( a ) + ( x − a ) T D f ( a ) + 1 2 ! ( x − a ) T { D 2 f ( a ) } ( x − a ) + ⋯ , {\displaystyle T(\mathbf {x} )=f(\mathbf {a} )+(\mathbf {x} -\mathbf {a} )^{\mathsf {T}}Df(\mathbf {a} )+{\frac {1}{2!}}(\mathbf {x} -\mathbf {a} )^{\mathsf {T}}\left\{D^{2}f(\mathbf {a} )\right\}(\mathbf {x} -\mathbf {a} )+\cdots ,} where D f (a) is the gradient of f evaluated at x = a and D2 f (a) is the Hessian matrix. Applying the multi-index notation the Taylor series for several variables becomes T ( x ) = ∑ \| α \| ≥ 0 ( x − a ) α α ! ( ∂ α f ) ( a ) , {\displaystyle T(\mathbf {x} )=\sum _{\|\alpha \|\geq 0}{\frac {(\mathbf {x} -\mathbf {a} )^{\alpha }}{\alpha !}}\left({\mathrm {\partial } ^{\alpha }}f\right)(\mathbf {a} ),} which is to be understood as a still more abbreviated multi-index version of the first equation of this paragraph, with a full analogy to the single variable case. === Example === In order to compute a second-order Taylor series expansion around point (a, b) = (0, 0) of the function f ( x , y ) = e x ln ⁡ ( 1 + y ) , {\displaystyle f(x,y)=e^{x}\ln(1+y),} one first computes all the necessary partial derivatives: f x = e x ln ⁡ ( 1 + y ) f y = e x 1 + y f x x = e x ln ⁡ ( 1 + y ) f y y = − e x ( 1 + y ) 2 f x y = f y x = e x 1 + y . {\displaystyle {\begin{aligned}f_{x}&=e^{x}\ln(1+y)\\[6pt]f_{y}&={\frac {e^{x}}{1+y}}\\[6pt]f_{xx}&=e^{x}\ln(1+y)\\[6pt]f_{yy}&=-{\frac {e^{x}}{(1+y)^{2}}}\\[6pt]f_{xy}&=f_{yx}={\frac {e^{x}}{1+y}}.\end{aligned}}} Evaluating these derivatives at the origin gives the Taylor coefficients f x ( 0 , 0 ) = 0 f y ( 0 , 0 ) = 1 f x x ( 0 , 0 ) = 0 f y y ( 0 , 0 ) = − 1 f x y ( 0 , 0 ) = f y x ( 0 , 0 ) = 1. {\displaystyle {\begin{aligned}f_{x}(0,0)&=0\\f_{y}(0,0)&=1\\f_{xx}(0,0)&=0\\f_{yy}(0,0)&=-1\\f_{xy}(0,0)&=f_{yx}(0,0)=1.\end{aligned}}} Substituting these values in to the general formula T ( x , y ) = f ( a , b ) + ( x − a ) f x ( a , b ) + ( y − b ) f y ( a , b ) + 1 2 ! ( ( x − a ) 2 f x x ( a , b ) + 2 ( x − a ) ( y − b ) f x y ( a , b ) + ( y − b ) 2 f y y ( a , b ) ) + ⋯ {\displaystyle {\begin{aligned}T(x,y)=&f(a,b)+(x-a)f_{x}(a,b)+(y-b)f_{y}(a,b)\\&{}+{\frac {1}{2!}}\left((x-a)^{2}f_{xx}(a,b)+2(x-a)(y-b)f_{xy}(a,b)+(y-b)^{2}f_{yy}(a,b)\right)+\cdots \end{aligned}}} produces T ( x , y ) = 0 + 0 ( x − 0 ) + 1 ( y − 0 ) + 1 2 ( 0 ( x − 0 ) 2 + 2 ( x − 0 ) ( y − 0 ) + ( − 1 ) ( y − 0 ) 2 ) + ⋯ = y + x y − 1 2 y 2 + ⋯ {\displaystyle {\begin{aligned}T(x,y)&=0+0(x-0)+1(y-0)+{\frac {1}{2}}{\big (}0(x-0)^{2}+2(x-0)(y-0)+(-1)(y-0)^{2}{\big )}+\cdots \\&=y+xy-{\tfrac {1}{2}}y^{2}+\cdots \end{aligned}}} Since ln(1 + y) is analytic in \|y\| < 1, we have e x ln ⁡ ( 1 + y ) = y + x y − 1 2 y 2 + ⋯ , \| y \| < 1. {\displaystyle e^{x}\ln(1+y)=y+xy-{\tfrac {1}{2}}y^{2}+\cdots ,\qquad \|y\|<1.} == Comparison with Fourier series == The trigonometric Fourier series enables one to express a periodic function (or a function defined on a closed interval [a,b]) as an infinite sum of trigonometric functions (sines and cosines). In this sense, the Fourier series is analogous to Taylor series, since the latter allows one to express a function as an infinite sum of powers. Nevertheless, the two series differ from each other in several relevant issues: The finite truncations of the Taylor series of f (x) about the point x = a are all exactly equal to f at a. In contrast, the Fourier series is computed by integrating over an entire interval, so there is generally no such point where all the finite truncations of the series are exact. The computation of Taylor series requires the knowledge of the function on an arbitrary small neighbourhood of a point, whereas the computation of the Fourier series requires knowing the function on its whole domain interval. In a certain sense one could say that the Taylor series is "local" and the Fourier series is "global". The Taylor series is defined for a function which has infinitely many derivatives at a single point, whereas the Fourier series is defined for any integrable function. In particular, the function could be nowhere differentiable. (For example, f (x) could be a Weierstrass function.) The convergence of both series has very different properties. Even if the Taylor series has positive convergence radius, the resulting series may not coincide with the function; but if the function is analytic then the series converges pointwise to the function, and uniformly on every compact subset of the convergence interval. Concerning the Fourier series, if the function is square-integrable then the series converges in quadratic mean, but additional requirements are needed to ensure the pointwise or uniform convergence (for instance, if the function is periodic and of class C1 then the convergence is uniform). Finally, in practice one wants to approximate the function with a finite number of terms, say with a Taylor polynomial or a partial sum of the trigonometric series, respectively. In the case of the Taylor series the error is very small in a neighbourhood of the point where it is computed, while it may be very large at a distant point. In the case of the Fourier series the error is distributed along the domain of the function. == See also == Asymptotic expansion Newton polynomial Padé approximant – best approximation by a rational function Puiseux series – Power series with rational exponents Approximation theory Function approximation == Notes == == References == == External links == "Taylor series", Encyclopedia of Mathematics, EMS Press, 2001 [1994] Weisstein, Eric W. "Taylor Series". MathWorld.
Wikipedia:Maclaurin's inequality#0	In mathematics, Maclaurin's inequality, named after Colin Maclaurin, is a refinement of the inequality of arithmetic and geometric means. Let a 1 , a 2 , … , a n {\displaystyle a_{1},a_{2},\ldots ,a_{n}} be non-negative real numbers, and for k = 1 , 2 , … , n {\displaystyle k=1,2,\ldots ,n} , define the averages S k {\displaystyle S_{k}} as follows: S k = ∑ 1 ≤ i 1 < ⋯ < i k ≤ n a i 1 a i 2 ⋯ a i k ( n k ) . {\displaystyle S_{k}={\frac {\displaystyle \sum _{1\leq i_{1}<\cdots <i_{k}\leq n}a_{i_{1}}a_{i_{2}}\cdots a_{i_{k}}}{\displaystyle {n \choose k}}}.} The numerator of this fraction is the elementary symmetric polynomial of degree k {\displaystyle k} in the n {\displaystyle n} variables a 1 , a 2 , … , a n {\displaystyle a_{1},a_{2},\ldots ,a_{n}} , that is, the sum of all products of k {\displaystyle k} of the numbers a 1 , a 2 , … , a n {\displaystyle a_{1},a_{2},\ldots ,a_{n}} with the indices in increasing order. The denominator is the number of terms in the numerator, the binomial coefficient ( n k ) . {\displaystyle {\tbinom {n}{k}}.} Maclaurin's inequality is the following chain of inequalities: S 1 ≥ S 2 ≥ S 3 3 ≥ ⋯ ≥ S n n {\textstyle S_{1}\geq {\sqrt {S_{2}}}\geq {\sqrt[{3}]{S_{3}}}\geq \cdots \geq {\sqrt[{n}]{S_{n}}}} , with equality if and only if all the a i {\displaystyle a_{i}} are equal. For n = 2 {\displaystyle n=2} , this gives the usual inequality of arithmetic and geometric means of two non-negative numbers. Maclaurin's inequality is well illustrated by the case n = 4 {\displaystyle n=4} : a 1 + a 2 + a 3 + a 4 4 ≥ a 1 a 2 + a 1 a 3 + a 1 a 4 + a 2 a 3 + a 2 a 4 + a 3 a 4 6 ≥ a 1 a 2 a 3 + a 1 a 2 a 4 + a 1 a 3 a 4 + a 2 a 3 a 4 4 3 ≥ a 1 a 2 a 3 a 4 4 . {\begin{aligned}&\quad {\frac {a_{1}+a_{2}+a_{3}+a_{4}}{4}}\\[8pt]&\geq {\sqrt {\frac {a_{1}a_{2}+a_{1}a_{3}+a_{1}a_{4}+a_{2}a_{3}+a_{2}a_{4}+a_{3}a_{4}}{6}}}\\[8pt]&\geq {\sqrt[{3}]{\frac {a_{1}a_{2}a_{3}+a_{1}a_{2}a_{4}+a_{1}a_{3}a_{4}+a_{2}a_{3}a_{4}}{4}}}\\[8pt]&\geq {\sqrt[{4}]{a_{1}a_{2}a_{3}a_{4}}}.\end{aligned}} Maclaurin's inequality can be proved using Newton's inequalities or a generalised version of Bernoulli's inequality. == See also == Newton's inequalities Muirhead's inequality Generalized mean inequality Bernoulli's inequality == References == Biler, Piotr; Witkowski, Alfred (1990). Problems in mathematical analysis. New York, N.Y.: M. Dekker. ISBN 0-8247-8312-3. This article incorporates material from MacLaurin's Inequality on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
Wikipedia:Madeleine Chaumont#0	Madeleine Chaumont (8 April 1896 – 27 July 1973) was a French mathematics teacher, who was notable as one of the first 41 women to be admitted to the École normale supérieure, and the second woman to be awarded the male agrégation in mathematics. Throughout her life, her teaching career was disrupted by various health problems. == Life == Chaumont was the daughter of Alfred Chaumont, director of the Chaumont Frères distillery, and Hélène Chaumont, a pianist. Having contracted several ear infections as a child, she suffered from hearing problems all her life. A student at the Collège Sévigné, she obtained her Baccalauréat in mathematics and philosophy in 1912 and 1913. After a brief spell in preparatory classes at the Lycée Chaptal, she obtained a degree in mathematics. Encouraged by her former teacher at Chaptal, Alexandre Bernheim, and by the success of Marguerite Rouvière and Georgette Parize in the competitive examination for the École Normale Supérieure, she decided to apply in 1919; she was accepted, but had to make do with the status of bachelor's scholarship holder. It was not until 1927 that a decree issued by Édouard Herriot granted her the title of former student of the École normale supérieure. She was thus one of the 41 female students at the École before the competitive examination was banned in 1939. She stayed at the school for only one year, to prepare for the men's agrégation in mathematics in 1920. Chaumont obtained it in 1920, in first place; she was the first female laureate since Liouba Bortniker in 1885. She demanded that "young girls should not have the right, but the obligation, to take the agrégation in boys' high schools", and she spoke out in favour of abolishing the women's agrégation, which was not achieved until 1976, after her death. == Career == In September 1920, she was assigned to the girls' high school in Reims, where she demanded equal treatment with her male colleagues. Unanimously praised for her pedagogical qualities, she was nevertheless regularly absent due to health problems. In 1927, she moved to the Lycée de Jeunes Filles de Versailles, where she prepared for the competitive examination for the École Normale Supérieure de Jeunes Filles, and then in 1933 to the Lycée Fénelon. Her pupils' applications were regularly successful. Chaumont moved to Limoges in 1939–40, she was excluded from teaching in October 1941 in application of the Second law on the status of Jews and had to wear the yellow star. She was replaced by her former colleague from the Ecole Normale Supérieure, François Deschamps, who sent her pupils for private lessons. She was reinstated at Fénelon in October 1944, and regularly received praise from the Inspectors General and her headmistress; again, many of her students were successful. From 1955 onwards, however, it had to face the reservations of the new headmistress and competition from the new special mathematics class at the Lycée Jules-Ferry. After a drastic fall in its numbers, the Fénelon class was closed in 1956. Appointed to the Centre national d'enseignement à distance because of her health problems, she prepared for the CAPES and the agrégation, but fell victim to overwork and missed the contact with her students1. In 1958, she obtained an appointment in elementary mathematics at the Lycée Claude-Monet, but the number of pupils fell again and she provoked an outcry. After much pressure, she retired in October of the same year. However, she continued to teach a course at the Institut Catholique de Paris until 1963. In 1971, she again gave lessons to a candidate at the École Polytechnique. All in all, during her career, she played "a decisive role in the access of women to quality scientific education". Chaumont was woman of culture, who remained close to her cousin Geneviève Cahn, wife of Germain Debré, and a pianist like her own mother. Having entered a retirement home in Châtenay-Malabry, she died on 27 July 1973 at the hospital in Châtenay-Malabry, aged 77, following a fracture of the neck of the femur. She was cremated in the Père-Lachaise cemetery and her ashes were placed in the tomb of her sister and brother-in-law in the Montparnasse cemetery. == References ==
Wikipedia:Madeline Early#0	Madeline Levin Early (April 1, 1912 – January 20, 2001) was an American mathematician and university professor. She was one of the few American women to be awarded a PhD in math before World War II. == Biography == Madeline Levin was born April 1, 1912 in Brooklyn, New York, the youngest child of Dora (Siegal) and Hyman Levin. She was the Russian immigrants' youngest of seven children. She attended public schools in New York City and studied mathematics at Hunter College, where she was a member of Pi Mu Epsilon, graduating magna cum laude. She received her bachelor 's degree there in 1932 and then studied at Bryn Mawr College where she earned a master's degree in 1933. She received her doctorate in 1936 under William Welch Flexner with the dissertation An Extension of the Lefschetz Intersectional Theory. After Madeline received her doctorate, she returned to Hunter College as an instructor. During the early years of World War II, she took a military leave of absence to attend the Hunter-Bellevue School of Nursing (Brookdale Campus) and earned a bachelor's degree in nursing from the affiliated New York University. In March 1945, she was appointed to the rank of ensign in the Naval Reserve and from 1945 to 1947, she served in the Navy at the U.S. Naval Hospital St. Albans on Long Island and later on the island of Guam. After her discharge from the Navy in March 1947, she used the benefits she received from the G.I. Bill, to do post-doctoral study at the University of Michigan during the 1947–1948 school year, and there she met her future husband Harold C. Early. In 1956, Madeline became an assistant professor at Eastern Michigan University in Ypsilanti, and was promoted to associate professor in 1959, and full professor in 1967. She retired in 1975. Madeline Early died January 20, 2001 in Ann Arbor, Michigan, and was buried in the Fort Custer National Cemetery in Battle Creek, Michigan. == Personal life == In 1949, Madeline Levin married fellow mathematician Harold C. Early, and changed her name to Madeline Early. They had a son, Robert Early, in 1951. The couple divorced in 1958. == Selected publications == 1934: Levin, M.; Flexner, W. W.: The intersection of arbitrary chains and its boundary. Proc. Natl. Acad. Sci. USA 20. 1937: Levin, M.: An extension of the Lefschetz intersection theory. Rev. Cienc. (Univ. Nac. Mayor San Marcos, Lima) 39. == Memberships == Organizational affiliations, according to Green. American Mathematical Society American Association of University Professors Phi Beta Kappa Pi Mu Epsilon == References ==
Wikipedia:Madhava of Sangamagrama#0	Mādhava of Sangamagrāma (Mādhavan) (c. 1340 – c. 1425) was an Indian mathematician and astronomer who is considered to be the founder of the Kerala school of astronomy and mathematics in the Late Middle Ages. Madhava made pioneering contributions to the study of infinite series, calculus, trigonometry, geometry and algebra. He was the first to use infinite series approximations for a range of trigonometric functions, which has been called the "decisive step onward from the finite procedures of ancient mathematics to treat their limit-passage to infinity". == Biography == Little is known about Madhava's life with certainty. However, from scattered references to Madhava found in diverse manuscripts, historians of Kerala school have pieced together information about the mathematician. In a manuscript preserved in the Oriental Institute, Baroda, Madhava has been referred to as Mādhavan vēṇvārōhādīnām karttā ... Mādhavan Ilaññippaḷḷi Emprān. It has been noted that the epithet 'Emprān' refers to the Emprāntiri community, to which Madhava might have belonged. The term "Ilaññippaḷḷi" has been identified as a reference to the residence of Madhava. This is corroborated by Madhava himself. In his short work on the moon's positions titled Veṇvāroha, Madhava says that he was born in a house named bakuḷādhiṣṭhita . . . vihāra. This is clearly Sanskrit for Ilaññippaḷḷi. Ilaññi is the Malayalam name of the evergreen tree Mimusops elengi and the Sanskrit name for the same is Bakuḷa. Palli is a term for village. The Sanskrit house name bakuḷādhiṣṭhita . . . vihāra has also been interpreted as a reference to the Malayalam house name Iraññi ninna ppaḷḷi and some historians have tried to identify it with one of two currently existing houses with names Iriññanavaḷḷi and Iriññārapaḷḷi both of which are located near Irinjalakuda town in central Kerala. This identification is far fetched because both names have neither phonetic similarity nor semantic equivalence to the word "Ilaññippaḷḷi". Most of the writers of astronomical and mathematical works who lived after Madhava's period have referred to Madhava as "Sangamagrama Madhava" and as such it is important that the real import of the word "Sangamagrama" be made clear. The general view among many scholars is that Sangamagrama is the town of Irinjalakuda some 70 kilometers south of the Nila river and about 70 kilometers south of Cochin. It seems that there is not much concrete ground for this belief except perhaps the fact that the presiding deity of an early medieval temple in the town, the Koodalmanikyam Temple, is worshiped as Sangameswara meaning the Lord of the Samgama and so Samgamagrama can be interpreted as the village of Samgameswara. But there are several places in Karnataka with samgama or its equivalent kūḍala in their names and with a temple dedicated to Samgamḗsvara, the lord of the confluence. (Kudalasangama in Bagalkot district is one such place with a celebrated temple dedicated to the Lord of the Samgama.) There is a small town on the southern banks of the Nila river, around 10 kilometers upstream from Tirunavaya, called Kūḍallūr. The exact literal Sanskrit translation of this place name is Samgamagram: kūṭal in Malayalam means a confluence (which in Sanskrit is samgama) and ūr means a village (which in Sanskrit is grama). Also the place is at the confluence of the Nila river and its most important tributary, namely, the Kunti river. (There is no confluence of rivers near Irinjalakuada.) Incidentally there is still existing a Nambudiri (Malayali Brahmin) family by name Kūtallūr Mana a few kilometers away from the Kudallur village. The family has its origins in Kudallur village itself. For many generations this family hosted a great Gurukulam specialising in Vedanga. That the only available manuscript of Sphuṭacandrāpti, a book authored by Madhava, was obtained from the manuscript collection of Kūtallūr Mana might strengthen the conjecture that Madhava might have had some association with Kūtallūr Mana. Thus the most plausible possibility is that the forefathers of Madhava migrated from the Tulu land or thereabouts to settle in Kudallur village, which is situated on the southern banks of the Nila river not far from Tirunnavaya, a generation or two before his birth and lived in a house known as Ilaññippaḷḷi whose present identity is unknown. === Date === There are also no definite evidences to pinpoint the period during which Madhava flourished. In his Venvaroha, Madhava gives a date in 1400 CE as the epoch. Madhava's pupil Parameshvara Nambudiri, the only known direct pupil of Madhava, is known to have completed his seminal work Drigganita in 1430 and the Paramesvara's date has been determined as c. 1360-1455. From such circumstantial evidences historians have assigned the date c. 1340 – c. 1425 to Madhava. == Historiography == Although there is some evidence of mathematical work in Kerala prior to Madhava (e.g., Sadratnamala c. 1300, a set of fragmentary results), it is clear from citations that Madhava provided the creative impulse for the development of a rich mathematical tradition in medieval Kerala. However, except for a couple, most of Madhava's original works have been lost. He is referred to in the work of subsequent Kerala mathematicians, particularly in Nilakantha Somayaji's Tantrasangraha (c. 1500), as the source for several infinite series expansions, including sin θ and arctan θ. The 16th-century text Mahajyānayana prakāra (Method of Computing Great Sines) cites Madhava as the source for several series derivations for π. In Jyeṣṭhadeva's Yuktibhāṣā (c. 1530), written in Malayalam, these series are presented with proofs in terms of the Taylor series expansions for polynomials like 1/(1+x2), with x = tan θ, etc. Thus, what is explicitly Madhava's work is a source of some debate. The Yukti-dipika (also called the Tantrasangraha-vyakhya), possibly composed by Sankara Variar, a student of Jyeṣṭhadeva, presents several versions of the series expansions for sin θ, cos θ, and arctan θ, as well as some products with radius and arclength, most versions of which appear in Yuktibhāṣā. For those that do not, Rajagopal and Rangachari have argued, quoting extensively from the original Sanskrit, that since some of these have been attributed by Nilakantha to Madhava, some of the other forms might also be the work of Madhava. Others have speculated that the early text Karanapaddhati (c. 1375–1475), or the Mahajyānayana prakāra was written by Madhava, but this is unlikely. Karanapaddhati, along with the even earlier Keralite mathematics text Sadratnamala, as well as the Tantrasangraha and Yuktibhāṣā, were considered in an 1834 article by C. M. Whish, which was the first to draw attention to their priority over Newton in discovering the Fluxion (Newton's name for differentials). In the mid-20th century, the Russian scholar Jushkevich revisited the legacy of Madhava, and a comprehensive look at the Kerala school was provided by Sarma in 1972. === Lineage === There are several known astronomers who preceded Madhava, including Kǖṭalur Kizhār (2nd century), Vararuci (4th century), and Śaṅkaranārāyaṇa (866 AD). It is possible that other unknown figures preceded him. However, we have a clearer record of the tradition after Madhava. Parameshvara was a direct disciple. According to a palm leaf manuscript of a Malayalam commentary on the Surya Siddhanta, Parameswara's son Damodara (c. 1400–1500) had Nilakantha Somayaji as one of his disciples. Jyeshtadeva was a disciple of Nilakantha. Achyutha Pisharadi of Trikkantiyur is mentioned as a disciple of Jyeṣṭhadeva, and the grammarian Melpathur Narayana Bhattathiri as his disciple. == Contributions == If we consider mathematics as a progression from finite processes of algebra to considerations of the infinite, then the first steps towards this transition typically come with infinite series expansions. It is this transition to the infinite series that is attributed to Madhava. In Europe, the first such series were developed by James Gregory in 1667. Madhava's work is notable for the series, but what is truly remarkable is his estimate of an error term (or correction term). This implies that he understood very well the limit nature of the infinite series. Thus, Madhava may have invented the ideas underlying infinite series expansions of functions, power series, trigonometric series, and rational approximations of infinite series. However, as stated above, which results are precisely Madhava's and which are those of his successors is difficult to determine. The following presents a summary of results that have been attributed to Madhava by various scholars. === Infinite series === Among his many contributions, he discovered infinite series for the trigonometric functions of sine, cosine, arctangent, and many methods for calculating the circumference of a circle. One of Madhava's series is known from the text Yuktibhāṣā, which contains the derivation and proof of the power series for inverse tangent, discovered by Madhava. In the text, Jyeṣṭhadeva describes the series in the following manner: The first term is the product of the given sine and radius of the desired arc divided by the cosine of the arc. The succeeding terms are obtained by a process of iteration when the first term is repeatedly multiplied by the square of the sine and divided by the square of the cosine. All the terms are then divided by the odd numbers 1, 3, 5, .... The arc is obtained by adding and subtracting respectively the terms of odd rank and those of even rank. It is laid down that the sine of the arc or that of its complement whichever is the smaller should be taken here as the given sine. Otherwise the terms obtained by this above iteration will not tend to the vanishing magnitude. This yields: r θ = r sin ⁡ θ cos ⁡ θ − ( 1 / 3 ) r ( sin ⁡ θ ) 3 ( cos ⁡ θ ) 3 + ( 1 / 5 ) r ( sin ⁡ θ ) 5 ( cos ⁡ θ ) 5 − ( 1 / 7 ) r ( sin ⁡ θ ) 7 ( cos ⁡ θ ) 7 + ⋯ {\displaystyle r\theta ={\frac {r\sin \theta }{\cos \theta }}-(1/3)\,r\,{\frac {\left(\sin \theta \right)^{3}}{\left(\cos \theta \right)^{3}}}+(1/5)\,r\,{\frac {\left(\sin \theta \right)^{5}}{\left(\cos \theta \right)^{5}}}-(1/7)\,r\,{\frac {\left(\sin \theta \right)^{7}}{\left(\cos \theta \right)^{7}}}+\cdots } or equivalently: θ = tan ⁡ θ − tan 3 ⁡ θ 3 + tan 5 ⁡ θ 5 − tan 7 ⁡ θ 7 + ⋯ {\displaystyle \theta =\tan \theta -{\frac {\tan ^{3}\theta }{3}}+{\frac {\tan ^{5}\theta }{5}}-{\frac {\tan ^{7}\theta }{7}}+\cdots } This series is Gregory's series (named after James Gregory, who rediscovered it three centuries after Madhava). Even if we consider this particular series as the work of Jyeṣṭhadeva, it would pre-date Gregory by a century, and certainly other infinite series of a similar nature had been worked out by Madhava. Today, it is referred to as the Madhava-Gregory-Leibniz series. === Trigonometry === Madhava composed an accurate table of sines. Madhava's values are accurate to the seventh decimal place. Marking a quarter circle at twenty-four equal intervals, he gave the lengths of the half-chord (sines) corresponding to each of them. It is believed that he may have computed these values based on the series expansions: sin q = q − q3/3! + q5/5! − q7/7! + ... cos q = 1 − q2/2! + q4/4! − q6/6! + ... === The value of π (pi) === Madhava's work on the value of the mathematical constant Pi is cited in the Mahajyānayana prakāra ("Methods for the great sines"). While some scholars such as Sarma feel that this book may have been composed by Madhava himself, it is more likely the work of a 16th-century successor. This text attributes most of the expansions to Madhava, and gives the following infinite series expansion of π, now known as the Madhava-Leibniz series: π 4 = 1 − 1 3 + 1 5 − 1 7 + ⋯ = ∑ n = 1 ∞ ( − 1 ) n − 1 2 n − 1 , {\displaystyle {\frac {\pi }{4}}=1-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+\cdots =\sum _{n=1}^{\infty }{\frac {(-1)^{n-1}}{2n-1}},} which he obtained from the power-series expansion of the arc-tangent function. However, what is most impressive is that he also gave a correction term Rn for the error after computing the sum up to n terms, namely: Rn = (−1)n / (4n), or Rn = (−1)n⋅n / (4n2 + 1), or Rn = (−1)n⋅(n2 + 1) / (4n3 + 5n), where the third correction leads to highly accurate computations of π. It has long been speculated how Madhava found these correction terms. They are the first three convergents of a finite continued fraction, which, when combined with the original Madhava's series evaluated to n terms, yields about 3n/2 correct digits: π 4 ≈ 1 − 1 3 + 1 5 − 1 7 + ⋯ + ( − 1 ) n − 1 2 n − 1 + ( − 1 ) n 4 n + 1 2 n + 2 2 4 n + 3 2 n + 4 2 ⋯ + … ⋯ + n 2 n [ 4 − 3 ( n mod 2 ) ] . {\displaystyle {\frac {\pi }{4}}\approx 1-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+\cdots +{\frac {(-1)^{n-1}}{2n-1}}+{\cfrac {(-1)^{n}}{4n+{\cfrac {1^{2}}{n+{\cfrac {2^{2}}{4n+{\cfrac {3^{2}}{n+{\cfrac {4^{2}}{\dots +{\cfrac {\dots }{\dots +{\cfrac {n^{2}}{n[4-3(n{\bmod {2}})]}}}}}}}}}}}}}}.} The absolute value of the correction term in next higher order is \|Rn\| = (4n3 + 13n) / (16n4 + 56n2 + 9). He also gave a more rapidly converging series by transforming the original infinite series of π, obtaining the infinite series π = 12 ( 1 − 1 3 ⋅ 3 + 1 5 ⋅ 3 2 − 1 7 ⋅ 3 3 + ⋯ ) . {\displaystyle \pi ={\sqrt {12}}\left(1-{\frac {1}{3\cdot 3}}+{\frac {1}{5\cdot 3^{2}}}-{\frac {1}{7\cdot 3^{3}}}+\cdots \right).} By using the first 21 terms to compute an approximation of π, he obtains a value correct to 11 decimal places (3.14159265359). The value of 3.1415926535898, correct to 13 decimals, is sometimes attributed to Madhava, but may be due to one of his followers. These were the most accurate approximations of π given since the 5th century (see History of numerical approximations of π). The text Sadratnamala appears to give the astonishingly accurate value of π = 3.14159265358979324 (correct to 17 decimal places). Based on this, R. Gupta has suggested that this text was also composed by Madhava. Madhava also carried out investigations into other series for arc lengths and the associated approximations to rational fractions of π. === Calculus === Madhava developed the power series expansion for some trigonometry functions which were further developed by his successors at the Kerala school of astronomy and mathematics. (Certain ideas of calculus were known to earlier mathematicians.) Madhava also extended some results found in earlier works, including those of Bhāskara II. However, they did not combine many differing ideas under the two unifying themes of the derivative and the integral, show the connection between the two, or turn calculus into the powerful problem-solving tool we have today. == Madhava's works == K. V. Sarma has identified Madhava as the author of the following works: Golavada Madhyamanayanaprakara Mahajyanayanaprakara (Method of Computing Great Sines) Lagnaprakarana (लग्नप्रकरण) Venvaroha (वेण्वारोह) Sphuṭacandrāpti (स्फुटचन्द्राप्ति) Aganita-grahacara (अगणित-ग्रहचार) Chandravakyani (चन्द्रवाक्यानि) (Table of Moon-mnemonics) == Kerala School of Astronomy and Mathematics == The Kerala school of astronomy and mathematics, founded by Madhava, flourished between the 14th and 16th centuries, and included among its members Parameshvara, Neelakanta Somayaji, Jyeshtadeva, Achyuta Pisharati, Melpathur Narayana Bhattathiri and Achyuta Panikkar. The group is known for series expansion of three trigonometric functions of sine, cosine and arctant and proofs of their results where later given in the Yuktibhasa. The group also did much other work in astronomy: more pages are devoted to astronomical computations than purely mathematical results. The Kerala school also contributed to linguistics (the relation between language and mathematics is an ancient Indian tradition, see Kātyāyana). The ayurvedic and poetic traditions of Kerala can be traced back to this school. The famous poem, Narayaniyam, was composed by Narayana Bhattathiri. == Influence == Madhava has been called "the greatest mathematician-astronomer of medieval India", some of his discoveries in this field show him to have possessed extraordinary intuition". O'Connor and Robertson state that a fair assessment of Madhava is that he took the decisive step towards modern classical analysis. === Possible propagation to Europe === The Kerala school was well known in the 15th and 16th centuries, in the period of the first contact with European navigators in the Malabar Coast. At the time, the port of Muziris, near Sangamagrama, was a major center for maritime trade, and a number of Jesuit missionaries and traders were active in this region. Given the fame of the Kerala school, and the interest shown by some of the Jesuit groups during this period in local scholarship, some scholars, including G. Joseph of the U. Manchester have suggested that the writings of the Kerala school may have also been transmitted to Europe around this time, which was still about a century before Newton. However, there is no direct evidence by way of relevant manuscripts that such a transmission actually took place. According to David Bressoud, "there is no evidence that the Indian work of series was known beyond India, or even outside of Kerala, until the nineteenth century." == See also == == References == == External links == Biography on MacTutor
Wikipedia:Madhava series#0	In mathematics, a Madhava series is one of the three Taylor series expansions for the sine, cosine, and arctangent functions discovered in 14th or 15th century in Kerala, India by the mathematician and astronomer Madhava of Sangamagrama (c. 1350 – c. 1425) or his followers in the Kerala school of astronomy and mathematics. Using modern notation, these series are: sin ⁡ θ = θ − θ 3 3 ! + θ 5 5 ! − θ 7 7 ! + ⋯ = ∑ k = 0 ∞ ( − 1 ) k ( 2 k + 1 ) ! θ 2 k + 1 , cos ⁡ θ = 1 − θ 2 2 ! + θ 4 4 ! − θ 6 6 ! + ⋯ = ∑ k = 0 ∞ ( − 1 ) k ( 2 k ) ! θ 2 k , arctan ⁡ x = x − x 3 3 + x 5 5 − x 7 7 + ⋯ = ∑ k = 0 ∞ ( − 1 ) k 2 k + 1 x 2 k + 1 where \| x \| ≤ 1. {\displaystyle {\begin{alignedat}{3}\sin \theta &=\theta -{\frac {\theta ^{3}}{3!}}+{\frac {\theta ^{5}}{5!}}-{\frac {\theta ^{7}}{7!}}+\cdots &&=\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{(2k+1)!}}\theta ^{2k+1},\\[10mu]\cos \theta &=1-{\frac {\theta ^{2}}{2!}}+{\frac {\theta ^{4}}{4!}}-{\frac {\theta ^{6}}{6!}}+\cdots &&=\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{(2k)!}}\theta ^{2k},\\[10mu]\arctan x&=x-{\frac {x^{3}}{3}}+{\frac {x^{5}}{5}}-{\frac {x^{7}}{7}}+\cdots &&=\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{2k+1}}x^{2k+1}\quad {\text{where }}\|x\|\leq 1.\end{alignedat}}} All three series were later independently discovered in 17th century Europe. The series for sine and cosine were rediscovered by Isaac Newton in 1669, and the series for arctangent was rediscovered by James Gregory in 1671 and Gottfried Leibniz in 1673, and is conventionally called Gregory's series. The specific value arctan ⁡ 1 = π 4 {\textstyle \arctan 1={\tfrac {\pi }{4}}} can be used to calculate the circle constant π, and the arctangent series for 1 is conventionally called Leibniz's series. In recognition of Madhava's priority, in recent literature these series are sometimes called the Madhava–Newton series, Madhava–Gregory series, or Madhava–Leibniz series (among other combinations). No surviving works of Madhava contain explicit statements regarding the expressions which are now referred to as Madhava series. However, in the writing of later Kerala school mathematicians Nilakantha Somayaji (1444 – 1544) and Jyeshthadeva (c. 1500 – c. 1575) one can find unambiguous attribution of these series to Madhava. These later works also include proofs and commentary which suggest how Madhava may have arrived at the series. The translations of the relevant verses as given in the Yuktidipika commentary of Tantrasamgraha (also known as Tantrasamgraha-vyakhya) by Sankara Variar (circa. 1500 - 1560 CE) are reproduced below. These are then rendered in current mathematical notations. == Madhava's sine series == === In Madhava's own words === Madhava's sine series is stated in verses 2.440 and 2.441 in Yukti-dipika commentary (Tantrasamgraha-vyakhya) by Sankara Variar. A translation of the verses follows. Multiply the arc by the square of the arc, and take the result of repeating that (any number of times). Divide by the squares of the successive even numbers (such that current is multiplied by previous) increased by that number and multiplied by the square of the radius. Place the arc and the successive results so obtained one below the other, and subtract each from the one above. These together give the jiva [sine], as collected together in the verse beginning with "vidvan" etc. === Rendering in modern notations === Let r denote the radius of the circle and s the arc-length. The following numerators are formed first: s ⋅ s 2 , s ⋅ s 2 ⋅ s 2 , s ⋅ s 2 ⋅ s 2 ⋅ s 2 , ⋯ {\displaystyle s\cdot s^{2},\qquad s\cdot s^{2}\cdot s^{2},\qquad s\cdot s^{2}\cdot s^{2}\cdot s^{2},\qquad \cdots } These are then divided by quantities specified in the verse. s ⋅ s 2 ( 2 2 + 2 ) r 2 , s ⋅ s 2 ( 2 2 + 2 ) r 2 ⋅ s 2 ( 4 2 + 4 ) r 2 , s ⋅ s 2 ( 2 2 + 2 ) r 2 ⋅ s 2 ( 4 2 + 4 ) r 2 ⋅ s 2 ( 6 2 + 6 ) r 2 , ⋯ {\displaystyle s\cdot {\frac {s^{2}}{(2^{2}+2)r^{2}}},\qquad s\cdot {\frac {s^{2}}{(2^{2}+2)r^{2}}}\cdot {\frac {s^{2}}{(4^{2}+4)r^{2}}},\qquad s\cdot {\frac {s^{2}}{(2^{2}+2)r^{2}}}\cdot {\frac {s^{2}}{(4^{2}+4)r^{2}}}\cdot {\frac {s^{2}}{(6^{2}+6)r^{2}}},\qquad \cdots } Place the arc and the successive results so obtained one below the other, and subtract each from the one above to get jiva: jiva = s − [ s ⋅ s 2 ( 2 2 + 2 ) r 2 − [ s ⋅ s 2 ( 2 2 + 2 ) r 2 ⋅ s 2 ( 4 2 + 4 ) r 2 − [ s ⋅ s 2 ( 2 2 + 2 ) r 2 ⋅ s 2 ( 4 2 + 4 ) r 2 ⋅ s 2 ( 6 2 + 6 ) r 2 − ⋯ ] ] ] {\displaystyle {\text{jiva}}=s-\left[s\cdot {\frac {s^{2}}{(2^{2}+2)r^{2}}}-\left[s\cdot {\frac {s^{2}}{(2^{2}+2)r^{2}}}\cdot {\frac {s^{2}}{(4^{2}+4)r^{2}}}-\left[s\cdot {\frac {s^{2}}{(2^{2}+2)r^{2}}}\cdot {\frac {s^{2}}{(4^{2}+4)r^{2}}}\cdot {\frac {s^{2}}{(6^{2}+6)r^{2}}}-\cdots \right]\right]\right]} === Transformation to current notation === Let θ be the angle subtended by the arc s at the centre of the circle. Then s = r θ and jiva = r sin θ. Substituting these in the last expression and simplifying we get sin ⁡ θ = θ − θ 3 3 ! + θ 5 5 ! − θ 7 7 ! + ⋯ {\displaystyle \sin \theta =\theta -{\frac {\theta ^{3}}{3!}}+{\frac {\theta ^{5}}{5!}}-{\frac {\theta ^{7}}{7!}}+\quad \cdots } which is the infinite power series expansion of the sine function. === Madhava's reformulation for numerical computation === The last line in the verse ′as collected together in the verse beginning with "vidvan" etc.′ is a reference to a reformulation of the series introduced by Madhava himself to make it convenient for easy computations for specified values of the arc and the radius. For such a reformulation, Madhava considers a circle one quarter of which measures 5400 minutes (say C minutes) and develops a scheme for the easy computations of the jiva′s of the various arcs of such a circle. Let R be the radius of a circle one quarter of which measures C. Madhava had already computed the value of π using his series formula for π. Using this value of π, namely 3.1415926535922, the radius R is computed as follows: Then R = 2 × 5400 / π = 3437.74677078493925 = 3437 arcminutes 44 arcseconds 48 sixtieths of an arcsecond = 3437′ 44′′ 48′′′. Madhava's expression for jiva corresponding to any arc s of a circle of radius R is equivalent to the following: jiva = s − s 3 R 2 ( 2 2 + 2 ) + s 5 R 4 ( 2 2 + 2 ) ( 4 2 + 4 ) − ⋯ = s − ( s C ) 3 [ R ( π 2 ) 3 3 ! − ( s C ) 2 [ R ( π 2 ) 5 5 ! − ( s C ) 2 [ R ( π 2 ) 7 7 ! − ⋯ ] ] ] . {\displaystyle {\begin{aligned}{\text{jiva }}&=s-{\frac {s^{3}}{R^{2}(2^{2}+2)}}+{\frac {s^{5}}{R^{4}(2^{2}+2)(4^{2}+4)}}-\cdots \\[6pt]&=s-\left({\frac {s}{C}}\right)^{3}\left[{\frac {R\left({\frac {\pi }{2}}\right)^{3}}{3!}}-\left({\frac {s}{C}}\right)^{2}\left[{\frac {R\left({\frac {\pi }{2}}\right)^{5}}{5!}}-\left({\frac {s}{C}}\right)^{2}\left[{\frac {R\left({\frac {\pi }{2}}\right)^{7}}{7!}}-\cdots \right]\right]\right].\end{aligned}}} Madhava now computes the following values: The jiva can now be computed using the following scheme: jiva = s − ( s C ) 3 [ ( R ( π 2 ) 3 3 ! ) − ( s C ) 2 [ ( R ( π 2 ) 5 5 ! ) − ( s C ) 2 [ ( R ( π 2 ) 7 7 ! ) − ( s C ) 2 [ ( R ( π 2 ) 9 9 ! ) − ( s C ) 2 ( R ( π 2 ) 11 11 ! ) ] ] ] ] . {\displaystyle {\text{jiva }}=s-\left({\frac {s}{C}}\right)^{3}\left[\left({\frac {R({\frac {\pi }{2}})^{3}}{3!}}\right)-\left({\frac {s}{C}}\right)^{2}\left[\left({\frac {R({\frac {\pi }{2}})^{5}}{5!}}\right)-\left({\frac {s}{C}}\right)^{2}\left[\left({\frac {R({\frac {\pi }{2}})^{7}}{7!}}\right)-\left({\frac {s}{C}}\right)^{2}\left[\left({\frac {R({\frac {\pi }{2}})^{9}}{9!}}\right)-\left({\frac {s}{C}}\right)^{2}\left({\frac {R({\frac {\pi }{2}})^{11}}{11!}}\right)\right]\right]\right]\right].} This gives an approximation of jiva by its Taylor polynomial of the 11'th order. It involves one division, six multiplications and five subtractions only. Madhava prescribes this numerically efficient computational scheme in the following words (translation of verse 2.437 in Yukti-dipika): vi-dvān, tu-nna-ba-la, ka-vī-śa-ni-ca-ya, sa-rvā-rtha-śī-la-sthi-ro, ni-rvi-ddhā-nga-na-rē-ndra-rung . Successively multiply these five numbers in order by the square of the arc divided by the quarter of the circumference (5400′), and subtract from the next number. (Continue this process with the result so obtained and the next number.) Multiply the final result by the cube of the arc divided by quarter of the circumference and subtract from the arc. == Madhava's cosine series == === In Madhava's own words === Madhava's cosine series is stated in verses 2.442 and 2.443 in Yukti-dipika commentary (Tantrasamgraha-vyakhya) by Sankara Variar. A translation of the verses follows. Multiply the square of the arc by the unit (i.e. the radius) and take the result of repeating that (any number of times). Divide (each of the above numerators) by the square of the successive even numbers decreased by that number and multiplied by the square of the radius. But the first term is (now)(the one which is) divided by twice the radius. Place the successive results so obtained one below the other and subtract each from the one above. These together give the śara as collected together in the verse beginning with stena, stri, etc. === Rendering in modern notations === Let r denote the radius of the circle and s the arc-length. The following numerators are formed first: r ⋅ s 2 , r ⋅ s 2 ⋅ s 2 , r ⋅ s 2 ⋅ s 2 ⋅ s 2 , ⋯ {\displaystyle r\cdot s^{2},\qquad r\cdot s^{2}\cdot s^{2},\qquad r\cdot s^{2}\cdot s^{2}\cdot s^{2},\qquad \cdots } These are then divided by quantities specified in the verse. r ⋅ s 2 ( 2 2 − 2 ) r 2 , r ⋅ s 2 ( 2 2 − 2 ) r 2 ⋅ s 2 ( 4 2 − 4 ) r 2 , r ⋅ s 2 ( 2 2 − 2 ) r 2 ⋅ s 2 ( 4 2 − 4 ) r 2 ⋅ s 2 ( 6 2 − 6 ) r 2 , ⋯ {\displaystyle r\cdot {\frac {s^{2}}{(2^{2}-2)r^{2}}},\qquad r\cdot {\frac {s^{2}}{(2^{2}-2)r^{2}}}\cdot {\frac {s^{2}}{(4^{2}-4)r^{2}}},\qquad r\cdot {\frac {s^{2}}{(2^{2}-2)r^{2}}}\cdot {\frac {s^{2}}{(4^{2}-4)r^{2}}}\cdot {\frac {s^{2}}{(6^{2}-6)r^{2}}},\qquad \cdots } Place the arc and the successive results so obtained one below the other, and subtract each from the one above to get śara: sara = r ⋅ s 2 ( 2 2 − 2 ) r 2 − [ r ⋅ s 2 ( 2 2 − 2 ) r 2 ⋅ s 2 ( 4 2 − 4 ) r 2 − [ r ⋅ s 2 ( 2 2 − 2 ) r 2 ⋅ s 2 ( 4 2 − 4 ) r 2 ⋅ s 2 ( 6 2 − 6 ) r 2 − ⋯ ] ] {\displaystyle {\text{sara}}=r\cdot {\frac {s^{2}}{(2^{2}-2)r^{2}}}-\left[r\cdot {\frac {s^{2}}{(2^{2}-2)r^{2}}}\cdot {\frac {s^{2}}{(4^{2}-4)r^{2}}}-\left[r\cdot {\frac {s^{2}}{(2^{2}-2)r^{2}}}\cdot {\frac {s^{2}}{(4^{2}-4)r^{2}}}\cdot {\frac {s^{2}}{(6^{2}-6)r^{2}}}-\cdots \right]\right]} === Transformation to current notation === Let θ be the angle subtended by the arc s at the centre of the circle. Then s = rθ and śara = r(1 − cos θ). Substituting these in the last expression and simplifying we get 1 − cos ⁡ θ = θ 2 2 ! − θ 4 4 ! + θ 6 6 ! + ⋯ {\displaystyle 1-\cos \theta ={\frac {\theta ^{2}}{2!}}-{\frac {\theta ^{4}}{4!}}+{\frac {\theta ^{6}}{6!}}+\quad \cdots } which gives the infinite power series expansion of the cosine function. === Madhava's reformulation for numerical computation === The last line in the verse ′as collected together in the verse beginning with stena, stri, etc.′ is a reference to a reformulation introduced by Madhava himself to make the series convenient for easy computations for specified values of the arc and the radius. As in the case of the sine series, Madhava considers a circle one quarter of which measures 5400 minutes (say C minutes) and develops a scheme for the easy computations of the śara′s of the various arcs of such a circle. Let R be the radius of a circle one quarter of which measures C. Then, as in the case of the sine series, Madhava gets R = 3437′ 44′′ 48′′′. Madhava's expression for śara corresponding to any arc s of a circle of radius R is equivalent to the following: jiva = R ⋅ s 2 R 2 ( 2 2 − 2 ) − R ⋅ s 4 R 4 ( 2 2 − 2 ) ( 4 2 − 4 ) − ⋯ = ( s C ) 2 [ R ( π 2 ) 2 2 ! − ( s C ) 2 [ R ( π 2 ) 4 4 ! − ( s C ) 2 [ R ( π 2 ) 6 6 ! − ⋯ ] ] ] {\displaystyle {\begin{aligned}{\text{jiva }}&=R\cdot {\frac {s^{2}}{R^{2}(2^{2}-2)}}-R\cdot {\frac {s^{4}}{R^{4}(2^{2}-2)(4^{2}-4)}}-\cdots \\[6pt]&=\left({\frac {s}{C}}\right)^{2}\left[{\frac {R\left({\frac {\pi }{2}}\right)^{2}}{2!}}-\left({\frac {s}{C}}\right)^{2}\left[{\frac {R\left({\frac {\pi }{2}}\right)^{4}}{4!}}-\left({\frac {s}{C}}\right)^{2}\left[{\frac {R\left({\frac {\pi }{2}}\right)^{6}}{6!}}-\cdots \right]\right]\right]\end{aligned}}} Madhava now computes the following values: The śara can now be computed using the following scheme: sara = ( s C ) 2 [ ( R ( π 2 ) 2 2 ! ) − ( s C ) 2 [ ( R ( π 2 ) 4 4 ! ) − ( s C ) 2 [ ( R ( π 2 ) 6 6 ! ) − ( s C ) 2 [ ( R ( π 2 ) 8 8 ! ) − ( s C ) 2 [ ( R ( π 2 ) 10 10 ! ) − ( s C ) 2 ( R ( π 2 ) 12 12 ! ) ] ] ] ] ] {\textstyle {\text{sara }}=\left({\frac {s}{C}}\right)^{2}\left[\left({\frac {R({\frac {\pi }{2}})^{2}}{2!}}\right)-\left({\frac {s}{C}}\right)^{2}\left[\left({\frac {R({\frac {\pi }{2}})^{4}}{4!}}\right)-\left({\frac {s}{C}}\right)^{2}\left[\left({\frac {R({\frac {\pi }{2}})^{6}}{6!}}\right)-\left({\frac {s}{C}}\right)^{2}\left[\left({\frac {R({\frac {\pi }{2}})^{8}}{8!}}\right)-\left({\frac {s}{C}}\right)^{2}\left[\left({\frac {R({\frac {\pi }{2}})^{10}}{10!}}\right)-\left({\frac {s}{C}}\right)^{2}\left({\frac {R({\frac {\pi }{2}})^{12}}{12!}}\right)\right]\right]\right]\right]\right]} This gives an approximation of śara by its Taylor polynomial of the 12'th order. This also involves one division, six multiplications and five subtractions only. Madhava prescribes this numerically efficient computational scheme in the following words (translation of verse 2.438 in Yukti-dipika): The six stena, strīpiśuna, sugandhinaganud, bhadrāngabhavyāsana, mīnāngonarasimha, unadhanakrtbhureva. Multiply by the square of the arc divided by the quarter of the circumference and subtract from the next number. (Continue with the result and the next number.) Final result will be utkrama-jya (R versed sign). == Madhava's arctangent series == === In Madhava's own words === Madhava's arctangent series is stated in verses 2.206 – 2.209 in Yukti-dipika commentary (Tantrasamgraha-vyakhya) by Sankara Variar. A translation of the verses is given below. Jyesthadeva has also given a description of this series in Yuktibhasa. Now, by just the same argument, the determination of the arc of a desired sine can be (made). That is as follows: The first result is the product of the desired sine and the radius divided by the cosine of the arc. When one has made the square of the sine the multiplier and the square of the cosine the divisor, now a group of results is to be determined from the (previous) results beginning from the first. When these are divided in order by the odd numbers 1, 3, and so forth, and when one has subtracted the sum of the even(-numbered) results from the sum of the odd (ones), that should be the arc. Here the smaller of the sine and cosine is required to be considered as the desired (sine). Otherwise, there would be no termination of results even if repeatedly (computed). By means of the same argument, the circumference can be computed in another way too. That is as (follows): The first result should by the square root of the square of the diameter multiplied by twelve. From then on, the result should be divided by three (in) each successive (case). When these are divided in order by the odd numbers, beginning with 1, and when one has subtracted the (even) results from the sum of the odd, (that) should be the circumference. === Rendering in modern notations === Let s be the arc of the desired sine (jya or jiva) y. Let r be the radius and x be the cosine (kotijya). The first result is y ⋅ r x {\displaystyle {\tfrac {y\cdot r}{x}}} . Form the multiplier and divisor y 2 x 2 {\displaystyle {\tfrac {y^{2}}{x^{2}}}} . Form the group of results: y ⋅ r x ⋅ y 2 x 2 , y ⋅ r x ⋅ y 2 x 2 ⋅ y 2 x 2 , ⋯ {\displaystyle {\frac {y\cdot r}{x}}\cdot {\frac {y^{2}}{x^{2}}},\qquad {\frac {y\cdot r}{x}}\cdot {\frac {y^{2}}{x^{2}}}\cdot {\frac {y^{2}}{x^{2}}},\qquad \cdots } These are divided in order by the numbers 1, 3, and so forth: 1 1 y ⋅ r x , 1 3 y ⋅ r x ⋅ y 2 x 2 , 1 5 y ⋅ r x ⋅ y 2 x 2 ⋅ y 2 x 2 , ⋯ {\displaystyle {\frac {1}{1}}{\frac {y\cdot r}{x}},\qquad {\frac {1}{3}}{\frac {y\cdot r}{x}}\cdot {\frac {y^{2}}{x^{2}}},\qquad {\frac {1}{5}}{\frac {y\cdot r}{x}}\cdot {\frac {y^{2}}{x^{2}}}\cdot {\frac {y^{2}}{x^{2}}},\qquad \cdots } Sum of odd-numbered results: 1 1 y ⋅ r x + 1 5 y ⋅ r x ⋅ y 2 x 2 ⋅ y 2 x 2 + ⋯ {\displaystyle {\frac {1}{1}}{\frac {y\cdot r}{x}}+{\frac {1}{5}}{\frac {y\cdot r}{x}}\cdot {\frac {y^{2}}{x^{2}}}\cdot {\frac {y^{2}}{x^{2}}}+\cdots } Sum of even-numbered results: 1 3 y ⋅ r x ⋅ y 2 x 2 + 1 7 y ⋅ r x ⋅ y 2 x 2 ⋅ y 2 x 2 ⋅ y 2 x 2 + ⋯ {\displaystyle {\frac {1}{3}}{\frac {y\cdot r}{x}}\cdot {\frac {y^{2}}{x^{2}}}+{\frac {1}{7}}{\frac {y\cdot r}{x}}\cdot {\frac {y^{2}}{x^{2}}}\cdot {\frac {y^{2}}{x^{2}}}\cdot {\frac {y^{2}}{x^{2}}}+\cdots } The arc is now given by s = ( 1 1 y ⋅ r x + 1 5 y ⋅ r x ⋅ y 2 x 2 ⋅ y 2 x 2 + ⋯ ) − ( 1 3 y ⋅ r x ⋅ y 2 x 2 + 1 7 y ⋅ r x ⋅ y 2 x 2 ⋅ y 2 x 2 ⋅ y 2 x 2 + ⋯ ) {\displaystyle s=\left({\frac {1}{1}}{\frac {y\cdot r}{x}}+{\frac {1}{5}}{\frac {y\cdot r}{x}}\cdot {\frac {y^{2}}{x^{2}}}\cdot {\frac {y^{2}}{x^{2}}}+\cdots \right)-\left({\frac {1}{3}}{\frac {y\cdot r}{x}}\cdot {\frac {y^{2}}{x^{2}}}+{\frac {1}{7}}{\frac {y\cdot r}{x}}\cdot {\frac {y^{2}}{x^{2}}}\cdot {\frac {y^{2}}{x^{2}}}\cdot {\frac {y^{2}}{x^{2}}}+\cdots \right)} === Transformation to current notation === Let θ be the angle subtended by the arc s at the centre of the circle. Then s = rθ, x = kotijya = r cos θ and y = jya = r sin θ. Then y / x = tan θ. Substituting these in the last expression and simplifying we get θ = tan ⁡ θ − tan 3 ⁡ θ 3 + tan 5 ⁡ θ 5 − tan 7 ⁡ θ 7 + ⋯ {\displaystyle \theta =\tan \theta -{\frac {\tan ^{3}\theta }{3}}+{\frac {\tan ^{5}\theta }{5}}-{\frac {\tan ^{7}\theta }{7}}+\quad \cdots } . Letting tan θ = q we finally have tan − 1 ⁡ q = q − q 3 3 + q 5 5 − q 7 7 + ⋯ {\displaystyle \tan ^{-1}q=q-{\frac {q^{3}}{3}}+{\frac {q^{5}}{5}}-{\frac {q^{7}}{7}}+\quad \cdots } === Another formula for the circumference of a circle === The second part of the quoted text specifies another formula for the computation of the circumference c of a circle having diameter d. This is as follows. c = 12 d 2 − 12 d 2 3 ⋅ 3 + 12 d 2 3 2 ⋅ 5 − 12 d 2 3 3 ⋅ 7 + ⋯ {\displaystyle c={\sqrt {12d^{2}}}-{\frac {\sqrt {12d^{2}}}{3\cdot 3}}+{\frac {\sqrt {12d^{2}}}{3^{2}\cdot 5}}-{\frac {\sqrt {12d^{2}}}{3^{3}\cdot 7}}+\quad \cdots } Since c = π d this can be reformulated as a formula to compute π as follows. π = 12 ( 1 − 1 3 ⋅ 3 + 1 3 2 ⋅ 5 − 1 3 3 ⋅ 7 + ⋯ ) {\displaystyle \pi ={\sqrt {12}}\left(1-{\frac {1}{3\cdot 3}}+{\frac {1}{3^{2}\cdot 5}}-{\frac {1}{3^{3}\cdot 7}}+\quad \cdots \right)} This is obtained by substituting q = 1 / 3 {\displaystyle 1/{\sqrt {3}}} (therefore θ = π / 6) in the power series expansion for tan−1 q above. == Comparison of convergence of various infinite series for π == == See also == Madhava of Sangamagrama Madhava's sine table Madhava's correction term Padé approximant Taylor series Laurent series Puiseux series == Notes == == References ==
Wikipedia:Madhava's correction term#0	Madhava's correction term is a mathematical expression attributed to Madhava of Sangamagrama (c. 1340 – c. 1425), the founder of the Kerala school of astronomy and mathematics, that can be used to give a better approximation to the value of the mathematical constant π (pi) than the partial sum approximation obtained by truncating the Madhava–Leibniz infinite series for π. The Madhava–Leibniz infinite series for π is π 4 = 1 − 1 3 + 1 5 − 1 7 + ⋯ {\displaystyle {\frac {\pi }{4}}=1-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+\cdots } Taking the partial sum of the first n {\displaystyle n} terms we have the following approximation to π: π 4 ≈ 1 − 1 3 + 1 5 − 1 7 + ⋯ + ( − 1 ) n − 1 1 2 n − 1 {\displaystyle {\frac {\pi }{4}}\approx 1-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+\cdots +(-1)^{n-1}{\frac {1}{2n-1}}} Denoting the Madhava correction term by F ( n ) {\displaystyle F(n)} , we have the following better approximation to π: π 4 ≈ 1 − 1 3 + 1 5 − 1 7 + ⋯ + ( − 1 ) n − 1 1 2 n − 1 + ( − 1 ) n F ( n ) {\displaystyle {\frac {\pi }{4}}\approx 1-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+\cdots +(-1)^{n-1}{\frac {1}{2n-1}}+(-1)^{n}F(n)} Three different expressions have been attributed to Madhava as possible values of F ( n ) {\displaystyle F(n)} , namely, F 1 ( n ) = 1 4 n {\displaystyle F_{1}(n)={\frac {1}{4n}}} F 2 ( n ) = n 4 n 2 + 1 {\displaystyle F_{2}(n)={\frac {n}{4n^{2}+1}}} F 3 ( n ) = n 2 + 1 4 n 3 + 5 n {\displaystyle F_{3}(n)={\frac {n^{2}+1}{4n^{3}+5n}}} In the extant writings of the mathematicians of the Kerala school there are some indications regarding how the correction terms F 1 ( n ) {\displaystyle F_{1}(n)} and F 2 ( n ) {\displaystyle F_{2}(n)} have been obtained, but there are no indications on how the expression F 3 ( n ) {\displaystyle F_{3}(n)} has been obtained. This has led to a lot of speculative work on how the formulas might have been derived. == Correction terms as given in Kerala texts == The expressions for F 2 ( n ) {\displaystyle F_{2}(n)} and F 3 ( n ) {\displaystyle F_{3}(n)} are given explicitly in the Yuktibhasha, a major treatise on mathematics and astronomy authored by the Indian astronomer Jyesthadeva of the Kerala school of mathematics around 1530, but that for F 1 ( n ) {\displaystyle F_{1}(n)} appears there only as a step in the argument leading to the derivation of F 2 ( n ) {\displaystyle F_{2}(n)} . The Yuktidipika–Laghuvivrthi commentary of Tantrasangraha, a treatise written by Nilakantha Somayaji an astronomer/mathematician belonging to the Kerala school of astronomy and mathematics and completed in 1501, presents the second correction term in the following verses (Chapter 2: Verses 271–274): English translation of the verses: "To the diameter multiplied by 4 alternately add and subtract in order the diameter multiplied by 4 and divided separately by the odd numbers 3, 5, etc. That odd number at which this process ends, four times the diameter should be multiplied by the next even number, halved and [then] divided by one added to that [even] number squared. The result is to be added or subtracted according as the last term was subtracted or added. This gives the circumference more accurately than would be obtained by going on with that process." In modern notations this can be stated as follows (where d {\displaystyle d} is the diameter of the circle): Circumference = 4 d − 4 d 3 + 4 d 5 − ⋯ ± 4 d p ∓ 4 d ( p + 1 ) / 2 1 + ( p + 1 ) 2 {\displaystyle =4d-{\frac {4d}{3}}+{\frac {4d}{5}}-\cdots \pm {\frac {4d}{p}}\mp {\frac {4d\left(p+1\right)/2}{1+(p+1)^{2}}}} If we set p = 2 n − 1 {\displaystyle p=2n-1} , the last term in the right hand side of the above equation reduces to 4 d F 2 ( n ) {\displaystyle 4dF_{2}(n)} . The same commentary also gives the correction term F 3 ( n ) {\displaystyle F_{3}(n)} in the following verses (Chapter 2: Verses 295–296): English translation of the verses: "A subtler method, with another correction. [Retain] the first procedure involving division of four times the diameter by the odd numbers, 3, 5, etc. [But] then add or subtract it [four times the diameter] multiplied by one added to the next even number halved and squared, and divided by one added to four times the preceding multiplier [with this] multiplied by the even number halved." In modern notations, this can be stated as follows: Circumference = 4 d − 4 d 3 + 4 d 5 − ⋯ ± 4 d p ∓ 4 d m ( 1 + 4 m ) ( p + 1 ) / 2 , {\displaystyle {\text{Circumference}}=4d-{\frac {4d}{3}}+{\frac {4d}{5}}-\cdots \pm {\frac {4d}{p}}\mp {\frac {4dm}{\left(1+4m\right)(p+1)/2}},} where the "multiplier" m = 1 + ( ( p + 1 ) / 2 ) 2 . {\textstyle m=1+\left((p+1)/2\right)^{2}.} If we set p = 2 n − 1 {\displaystyle p=2n-1} , the last term in the right hand side of the above equation reduces to 4 d F 3 ( n ) {\displaystyle 4dF_{3}(n)} . == Accuracy of the correction terms == Let s i = 1 − 1 3 + 1 5 − 1 7 + ⋯ + ( − 1 ) n − 1 1 2 n − 1 + ( − 1 ) n F i ( n ) {\displaystyle s_{i}=1-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+\cdots +(-1)^{n-1}{\frac {1}{2n-1}}+(-1)^{n}F_{i}(n)} . Then, writing p = 2 n + 1 {\displaystyle p=2n+1} , the errors \| π 4 − s i ( n ) \| {\displaystyle \left\|{\frac {\pi }{4}}-s_{i}(n)\right\|} have the following bounds: 1 p 3 − p − 1 ( p + 2 ) 3 − ( p + 2 ) < \| π 4 − s 1 ( n ) \| < 1 p 3 − p , 4 p 5 + 4 p − 4 ( p + 2 ) 5 + 4 ( p + 2 ) < \| π 4 − s 2 ( n ) \| < 4 p 5 + 4 p , 36 p 7 + 7 p 5 + 28 p 3 − 36 p − 36 ( p + 2 ) 7 + 7 ( p + 2 ) 5 + 28 ( p + 2 ) 3 − 36 ( p + 2 ) ⋯ 4 p 5 + 4 p − 4 ( p + 2 ) 5 + 4 ( p + 2 ) < \| π 4 − s 3 ( n ) \| < 36 p 7 + 7 p 5 + 28 p 3 − 36 p . {\displaystyle {\begin{aligned}&{\begin{aligned}{\frac {1}{p^{3}-p}}-{\frac {1}{(p+2)^{3}-(p+2)}}&<\left\|{\frac {\pi }{4}}-s_{1}(n)\right\|<{\frac {1}{p^{3}-p}},\\[10mu]{\frac {4}{p^{5}+4p}}-{\frac {4}{(p+2)^{5}+4(p+2)}}&<\left\|{\frac {\pi }{4}}-s_{2}(n)\right\|<{\frac {4}{p^{5}+4p}},\end{aligned}}\\[20mu]&{\begin{aligned}&{\frac {36}{p^{7}+7p^{5}+28p^{3}-36p}}-{\frac {36}{(p+2)^{7}+7(p+2)^{5}+28(p+2)^{3}-36(p+2)}}\cdots \\[10mu]&{\phantom {{\frac {4}{p^{5}+4p}}-{\frac {4}{(p+2)^{5}+4(p+2)}}}}<\left\|{\frac {\pi }{4}}-s_{3}(n)\right\|<{\frac {36}{p^{7}+7p^{5}+28p^{3}-36p}}.\end{aligned}}\end{aligned}}} === Numerical values of the errors in the computation of π === The errors in using these approximations in computing the value of π are E ( n ) = π − 4 ( 1 − 1 3 + 1 5 − 1 7 + ⋯ + ( − 1 ) n − 1 1 2 n − 1 ) {\displaystyle E(n)=\pi -4\left(1-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+\cdots +(-1)^{n-1}{\frac {1}{2n-1}}\right)} E i ( n ) = E ( n ) − 4 × ( − 1 ) n F i ( n ) {\displaystyle E_{i}(n)=E(n)-4\times (-1)^{n}F_{i}(n)} The following table gives the values of these errors for a few selected values of n {\displaystyle n} . == Continued fraction expressions for the correction terms == It has been noted that the correction terms F 1 ( n ) , F 2 ( n ) , F 3 ( n ) {\displaystyle F_{1}(n),F_{2}(n),F_{3}(n)} are the first three convergents of the following continued fraction expressions: 1 4 n + 1 n + 1 n + ⋯ {\displaystyle {\cfrac {1}{4n+{\cfrac {1}{n+{\cfrac {1}{n+\cdots }}}}}}} 1 4 n + 1 2 n + 2 2 4 n + 3 2 n + ⋯ ⋯ + r 2 n [ 4 − 3 ( r mod 2 ) ] + ⋯ = 1 4 n + 2 2 4 n + 4 2 4 n + 6 2 4 n + 8 2 4 n + ⋯ {\displaystyle {\cfrac {1}{4n+{\cfrac {1^{2}}{n+{\cfrac {2^{2}}{4n+{\cfrac {3^{2}}{n+{\cfrac {\cdots }{\cdots +{\cfrac {r^{2}}{n[4-3(r{\bmod {2}})]+\cdots }}}}}}}}}}}}={\cfrac {1}{4n+{\cfrac {2^{2}}{4n+{\cfrac {4^{2}}{4n+{\cfrac {6^{2}}{4n+{\cfrac {8^{2}}{4n+\cdots }}}}}}}}}}} The function f ( n ) {\displaystyle f(n)} that renders the equation π 4 = 1 − 1 3 + 1 5 − ⋯ ± 1 n ∓ f ( n + 1 ) {\displaystyle {\frac {\pi }{4}}=1-{\frac {1}{3}}+{\frac {1}{5}}-\cdots \pm {\frac {1}{n}}\mp f(n+1)} exact can be expressed in the following form: f ( n ) = 1 2 × 1 n + 1 2 n + 2 2 n + 3 2 n + ⋯ {\displaystyle f(n)={\frac {1}{2}}\times {\cfrac {1}{n+{\cfrac {1^{2}}{n+{\cfrac {2^{2}}{n+{\cfrac {3^{2}}{n+\cdots }}}}}}}}} The first three convergents of this infinite continued fraction are precisely the correction terms of Madhava. Also, this function f ( n ) {\displaystyle f(n)} has the following property: f ( 2 n ) = 1 4 n + 2 2 4 n + 4 2 4 n + 6 2 4 n + 8 2 4 n + ⋯ {\displaystyle f(2n)={\cfrac {1}{4n+{\cfrac {2^{2}}{4n+{\cfrac {4^{2}}{4n+{\cfrac {6^{2}}{4n+{\cfrac {8^{2}}{4n+\cdots }}}}}}}}}}} == Speculative derivation by Hayashi et al. == In a paper published in 1990, a group of three Japanese researchers proposed an ingenious method by which Madhava might have obtained the three correction terms. Their proposal was based on two assumptions: Madhava used 355 / 113 {\displaystyle 355/113} as the value of π and he used the Euclidean algorithm for division. Writing S ( n ) = \| 1 − 1 3 + 1 5 − 1 7 + ⋯ + ( − 1 ) n − 1 2 n − 1 − π 4 \| {\displaystyle S(n)=\left\|1-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+\cdots +{\frac {(-1)^{n-1}}{2n-1}}-{\frac {\pi }{4}}\right\|} and taking π = 355 / 113 , {\displaystyle \pi =355/113,} compute the values S ( n ) , {\displaystyle S(n),} express them as a fraction with 1 as numerator, and finally ignore the fractional parts in the denominator to obtain approximations: S ( 1 ) = 97 452 = 1 4 + 64 97 ≈ 1 4 , S ( 2 ) = 161 1356 = 1 8 + 68 161 ≈ 1 8 , S ( 3 ) = 551 6780 = 1 12 + 168 551 ≈ 1 12 , S ( 4 ) = 2923 47460 = 1 16 + 692 2923 ≈ 1 16 , S ( 5 ) = 21153 427140 = 1 20 + 4080 21153 ≈ 1 20 . {\displaystyle {\begin{alignedat}{3}S(1)&=\ \ \,{\frac {97}{452}}&&=\ \ \ {\frac {1}{4+{\frac {64}{97}}}}&&\approx {\frac {1}{4}},\\[6mu]S(2)&=\ \ {\frac {161}{1356}}&&=\ \ \,{\frac {1}{8+{\frac {68}{161}}}}&&\approx {\frac {1}{8}},\\[6mu]S(3)&=\ \ {\frac {551}{6780}}&&=\ \,{\frac {1}{12+{\frac {168}{551}}}}&&\approx {\frac {1}{12}},\\[6mu]S(4)&=\ {\frac {2923}{47460}}&&=\ {\frac {1}{16+{\frac {692}{2923}}}}&&\approx {\frac {1}{16}},\\[6mu]S(5)&={\frac {21153}{427140}}&&={\frac {1}{20+{\frac {4080}{21153}}}}&&\approx {\frac {1}{20}}.\end{alignedat}}} This suggests the following first approximation to S ( n ) {\displaystyle S(n)} which is the correction term F 1 ( n ) {\displaystyle F_{1}(n)} talked about earlier. S ( n ) ≈ 1 4 n {\displaystyle S(n)\approx {\frac {1}{4n}}} The fractions that were ignored can then be expressed with 1 as numerator, with the fractional parts in the denominators ignored to obtain the next approximation. Two such steps are: 64 97 = 1 1 + 33 64 ≈ 1 1 , 33 64 = 1 1 + 31 33 ≈ 1 1 , 68 161 = 1 2 + 25 68 ≈ 1 2 , 25 68 = 1 2 + 18 25 ≈ 1 2 , 168 551 = 1 3 + 47 168 ≈ 1 3 , 47 168 = 1 3 + 27 47 ≈ 1 3 , 692 2923 = 1 4 + 155 692 ≈ 1 4 , 155 692 = 1 4 + 72 155 ≈ 1 4 , 4080 21153 = 1 5 + 753 4080 ≈ 1 5 , 753 4080 = 1 5 + 315 753 ≈ 1 5 . {\displaystyle {\begin{alignedat}{5}{\frac {64}{97}}&=\ \,{\frac {1}{1+{\frac {33}{64}}}}&&\approx {\frac {1}{1}},&{\frac {33}{64}}&=\,{\frac {1}{1+{\frac {31}{33}}}}&&\approx {\frac {1}{1}},\\[6mu]{\frac {68}{161}}&=\ \,{\frac {1}{2+{\frac {25}{68}}}}&&\approx {\frac {1}{2}},&{\frac {25}{68}}&=\,{\frac {1}{2+{\frac {18}{25}}}}&&\approx {\frac {1}{2}},\\[6mu]{\frac {168}{551}}&=\ {\frac {1}{3+{\frac {47}{168}}}}&&\approx {\frac {1}{3}},&{\frac {47}{168}}&=\,{\frac {1}{3+{\frac {27}{47}}}}&&\approx {\frac {1}{3}},\\[6mu]{\frac {692}{2923}}&={\frac {1}{4+{\frac {155}{692}}}}&&\approx {\frac {1}{4}},&{\frac {155}{692}}&={\frac {1}{4+{\frac {72}{155}}}}&&\approx {\frac {1}{4}},\\[6mu]{\frac {4080}{21153}}&={\frac {1}{5+{\frac {753}{4080}}}}&&\approx {\frac {1}{5}},&\quad {\frac {753}{4080}}&={\frac {1}{5+{\frac {315}{753}}}}&&\approx {\frac {1}{5}}.\end{alignedat}}} This yields the next two approximations to S ( n ) , {\displaystyle S(n),} exactly the same as the correction terms F 2 ( n ) , {\displaystyle F_{2}(n),} S ( n ) ≈ 1 4 n + 1 n = n 4 n 2 + 1 , {\displaystyle S(n)\approx {\frac {1}{4n+{\dfrac {1}{n}}}}={\frac {n}{4n^{2}+1}},} and F 3 ( n ) , {\displaystyle F_{3}(n),} S ( n ) ≈ 1 4 n + 1 n + 1 n = n 2 + 1 n ( 4 n 2 + 5 ) , {\displaystyle S(n)\approx {\dfrac {1}{4n+{\dfrac {1}{n+{\dfrac {1}{n}}}}}}={\frac {n^{2}+1}{n(4n^{2}+5)}},} attributed to Madhava. == See also == Madhava series Madhava's sine table == References == == Additional reading == C. T. Rajagopal and M. S. Rangachari (1986). "On Medieval Kerala Mathematics". Archive for History of Exact Sciences. 35 (2): 91–99. doi:10.1007/BF00357622. JSTOR 41133779. S2CID 121678430. P. Rajasekhar (June 2011). "Derivation of remainder term for the Series expansion of π as depicted in Yukthibhasa and its modern Interpretation". Bulletin of Kerala Mathematics Association. 8 (1): 17–39. Ranjan Roy (13 June 2011). "Power Series in Fifteenth-Century Kerala", except from Sources in the Development of Mathematics: Infinite Series and Products from the Fifteenth to the Twenty-first Century. Cambridge University Press. ISBN 978-0-521-11470-7.
Wikipedia:Madhava's sine table#0	Madhava's sine table is the table of trigonometric sines constructed by the 14th century Kerala mathematician-astronomer Madhava of Sangamagrama (c. 1340 – c. 1425). The table lists the jya-s or Rsines of the twenty-four angles from 3.75° to 90° in steps of 3.75° (1/24 of a right angle, 90°). Rsine is just the sine multiplied by a selected radius and given as an integer. In this table, as in Aryabhata's earlier table, R is taken as 21600 ÷ 2π ≈ 3437.75. The table is encoded in the letters of the Sanskrit alphabet using the Katapayadi system, giving entries the appearance of the verses of a poem. Madhava's original work containing the table has not been found. The table is reproduced in the Aryabhatiyabhashya of Nilakantha Somayaji (1444–1544) and also in the Yuktidipika/Laghuvivrti commentary of Tantrasamgraha by Sankara Variar (circa. 1500–1560).: 114–123 The verses below are given as in Cultural foundations of mathematics by C.K. Raju.: 114–123 They are also given in the Malayalam Commentary of Karanapaddhati by P.K. Koru but slightly differently. == The table == The verses are: The quarters of the first six verses represent entries for the twenty-four angles from 3.75° to 90° in steps of 3.75° (first column). The second column contains the Rsine values encoded as Sanskrit words (in Devanagari). The third column contains the same in ISO 15919 transliterations. The fourth column contains the numbers decoded into arcminutes, arcseconds, and arcthirds in modern numerals. The modern values scaled by the traditional “radius” (21600 ÷ 2π, with the modern value of π with two decimals in the arcthirds are given in the fifth column. The last verse means: “These are the great R-sines as said by Madhava, comprising arcminutes, seconds and thirds. Subtracting from each the previous will give the R-sine-differences.” By comparing, one can note that Madhava's values are accurately given rounded to the declared precision of thirds except for Rsin(15°) where one feels he should have rounded up to 889′45″16‴ instead. Note that in the Katapayadi system the digits are written in the reverse order, so for example the literal entry corresponding to 15° is 51549880 which is reversed and then read as 0889′45″15‴. Note that the 0 does not carry a value but is used for the metre of the poem alone. == A simple way of understanding the table == Without going into the philosophy of why the value of R = 21600 ÷ 2π was chosen etc, the simplest way to relate the jya tables to our modern concept of sine tables is as follows: Even today sine tables are given as decimals to a certain precision. If sin(15°) is given as 0.1736, it means the rational 1736 ÷ 10000 is a good approximation of the actual infinite precision number. The only difference is that in the earlier days they had not standardized on decimal values (or powers of ten as denominator) for fractions. Hence they used other denominators based on other considerations (which are not discussed here). Hence the sine values represented in the tables may simply be taken as approximated by the given integer values divided by the R chosen for the table. Another possible confusion point is the usage of angle measures like arcminute etc in expressing the R-sines. Modern sines are unitless ratios. Jya-s or R-sines are the same multiplied by a measure of length or distance. However, since these tables were mostly used for astronomy, and distance on the celestial sphere is expressed in angle measures, these values are also given likewise. However, the unit is not really important and need not be taken too seriously, as the value will anyhow be used as part of a rational and the unit will cancel out. However, this also leads to the usage of sexagesimal subdivisions in Madhava's refining the earlier table of Aryabhata. Instead of choosing a larger R, he gave the extra precision determined by him on top of the earlier given minutes by using seconds and thirds. As before, these may simply be taken as a different way of expressing fractions and not necessarily as angle measures. == Another (more difficult) way to understand the values == Consider some angle whose measure is A. Consider a circle of unit radius and center O. Let the arc PQ of the circle subtend an angle A at the center O. Drop the perpendicular QR from Q to OP; then the length of the line segment RQ is the value of the trigonometric sine of the angle A. Let PS be an arc of the circle whose length is equal to the length of the segment RQ. For various angles A, Madhava's table gives the measures of the corresponding angles ∠ {\displaystyle \angle } POS in arcminutes, arcseconds and sixtieths of an arcsecond. As an example, let A be an angle whose measure is 22.50°. In Madhava's table, the entry corresponding to 22.50° is the measure in arcminutes, arcseconds and sixtieths of an arcsecond of the angle whose radian measure is the value of sin 22.50°, which is 0.3826834; multiply 0.3826834 radians by 180/π to convert to 21.92614 degrees, which is 1315 arcminutes 34 arcseconds 07 sixtieths of an arcsecond, abbreviated 13153407. For an angle whose measure is A, let ∠ P O S = m arcminutes, s arcseconds, t sixtieths of an arcsecond {\displaystyle \angle POS=m{\text{ arcminutes, }}s{\text{ arcseconds, }}t{\text{ sixtieths of an arcsecond}}} Then: sin ⁡ ( A ) = R Q = length of arc P S = ∠ P O S in radians {\displaystyle {\begin{aligned}\sin(A)&=RQ\\&={\text{length of arc }}PS\\&=\angle POS{\text{ in radians}}\\\end{aligned}}} == Derivation of trigonometric sines from the table == Each of the lines in the table specifies eight digits. Let the digits corresponding to angle A (read from left to right) be: d 1 d 2 d 3 d 4 d 5 d 6 d 7 d 8 {\displaystyle d_{1}\quad d_{2}\quad d_{3}\quad d_{4}\quad d_{5}\quad d_{6}\quad d_{7}\quad d_{8}} Then according to the rules of the Katapayadi system they should be taken from right to left and we have: m = d 8 × 1000 + d 7 × 100 + d 6 × 10 + d 5 s = d 4 × 10 + d 3 t = d 2 × 10 + d 1 {\displaystyle {\begin{aligned}m&=d_{8}\times 1000+d_{7}\times 100+d_{6}\times 10+d_{5}\\s&=d_{4}\times 10+d_{3}\\t&=d_{2}\times 10+d_{1}\end{aligned}}} B = m ′ s ′ ′ t ′ ′ ′ = 1 ∘ 60 ( m + s 60 + t 60 × 60 ) {\displaystyle B=m^{\prime }s^{\prime \prime }t^{\prime \prime \prime }={\frac {1^{\circ }}{60}}\left(m+{\frac {s}{60}}+{\frac {t}{60\times 60}}\right)} The value of the above angle B expressed in radians will correspond to the sine value of A. sin ⁡ A = π 180 B {\displaystyle \sin A={\frac {\pi }{180}}B} As said earlier, this is the same as dividing the encoded value by the taken R value: sin ⁡ A = B 21600 ′ 2 π {\displaystyle \sin A={\frac {B}{\frac {21600^{\prime }}{2\pi }}}} == Example == The table lists the following digits corresponding to the angle A = 45.00°: 5 1 1 5 0 3 4 2 {\displaystyle 5\quad 1\quad 1\quad 5\quad 0\quad 3\quad 4\quad 2} This yields the angle with measure: m = 2 × 1000 + 4 × 100 + 3 × 10 + 0 arcminutes = 2430 arcminutes s = 5 × 10 + 1 arcseconds = 51 arcseconds t = 1 × 10 + 5 sixtieths of an arcsecond = 15 sixtieths of an arcsecond {\displaystyle {\begin{aligned}m&=2\times 1000+4\times 100+3\times 10+0{\text{ arcminutes}}\\&=2430{\text{ arcminutes}}\\s&=5\times 10+1{\text{ arcseconds}}\\&=51{\text{ arcseconds}}\\t&=1\times 10+5{\text{ sixtieths of an arcsecond}}\\&=15{\text{ sixtieths of an arcsecond}}\end{aligned}}} From which we get: B = 1 ∘ 60 ( 2430 + 51 60 + 15 60 × 60 ) = 116681 2880 {\displaystyle B={\frac {1^{\circ }}{60}}\left(2430+{\frac {51}{60}}+{\frac {15}{60\times 60}}\right)={\frac {116681}{2880}}} The value of the sine of A = 45.00° as given in Madhava's table is then just B converted to radians: sin ⁡ 45 ∘ = π 180 B = π 180 × 116681 2880 {\displaystyle \sin 45^{\circ }={\frac {\pi }{180}}B={\frac {\pi }{180}}\times {\frac {116681}{2880}}} Evaluating the above, one can find that sin 45° is 0.70710681… This is accurate to 6 decimal places. == Madhava's method of computation == No work of Madhava detailing the methods used by him for the computation of the sine table has survived. However from the writings of later Kerala mathematicians including Nilakantha Somayaji (Tantrasangraha) and Jyeshtadeva (Yuktibhāṣā) that give ample references to Madhava's accomplishments, it is conjectured that Madhava computed his sine table using the power series expansion of sin x: sin ⁡ x = x − x 3 3 ! + x 5 5 ! − x 7 7 ! + ⋯ {\displaystyle \sin x=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots } == See also == Madhava series Madhava's correction term Madhava's value of π Āryabhaṭa's sine table Ptolemy's table of chords == References == == Further references == Bag, A.K. (1976). "Madhava's sine and cosine series" (PDF). Indian Journal of History of Science. 11 (1). Indian National Academy of Science: 54–57. Archived from the original (PDF) on 5 July 2015. Retrieved 21 August 2016. For an account of Madhava's computation of the sine table see : Van Brummelen, Glen (2009). The mathematics of the heavens and the earth : the early history of trigonometry. Princeton: Princeton University Press. pp. 113–120. ISBN 978-0-691-12973-0. For a thorough discussion of the computation of Madhava's sine table with historical references : C.K. Raju (2007). Cultural foundations of mathematics: The nature of mathematical proof and the transmission of calculus from India to Europe in the 16 thc. CE. History of Philosophy, Science and Culture in Indian Civilization. Vol. X Part 4. Delhi: Centre for Studies in Civilizations. pp. 114–123.
Wikipedia:Magda Peligrad#0	Magda Peligrad is a Romanian mathematician and mathematical statistician known for her research in probability theory, and particularly on central limit theorems and stochastic processes. She works at the University of Cincinnati, where she is Distinguished Charles Phelps Taft Professor of Mathematical Sciences. == Education and career == Peligrad obtained her Ph.D. in 1980 from the Center of Statistics of the Romanian Academy. By 1983 she was working at the Sapienza University of Rome and by 1984 she had arrived at Cincinnati, where since 1988 she has supervised the dissertations of seven doctoral students. With Florence Merlevède and Sergey Utev, she is coauthor of the book Functional Gaussian Approximation for Dependent Structures (Oxford University Press, 2019). == Recognition == In 1995, Peligrad was elected as a Fellow of the Institute of Mathematical Statistics, which she had served in 1990 as the Institute's representative to the Joint Committee on Women in Mathematical Sciences, an umbrella organization for women in eight societies of mathematics and statistics. A conference on "limit theorems for dependent data and applications" was organized in her honor in Paris in 2010, celebrating her 60th birthday, by the researchers at four Parisian universities. She was named Taft professor in 2004. == References ==
Wikipedia:Magdalena Mouján#0	Magdalena Araceli Mouján Otaño (1926–2005) was an Argentine mathematician of Basque descent, a pioneer of Argentine computer science, operations research, and nuclear physics, and an award-winning science fiction author. == Life == Mouján was born on March 26, 1926, in Pehuajó (Buenos Aires Province), the granddaughter of Basque writer Pedro Mari Otaño. After studying mathematics at the National University of La Plata, she completed a doctorate in 1950. She went on to hold teaching positions at the Catholic University of La Plata, the National University of Córdoba, the National University of Comahue and the National University of Luján, with a temporary hiatus beginning in 1966 because of the Argentine Revolution. She died on July 17, 2005, in Mar del Plata. == Research == In 1957, Mouján became one of four founding members of an operations research group funded by the Argentine Army and led by mathematician Agustín Durañona y Vedia. In the 1960s, she joined the National Atomic Energy Commission and began using the Clementina computer, the first scientific computer in Argentina, at the University of Buenos Aires. Her calculations were used to help build the RA-1 Enrico Fermi nuclear reactor. == Writing == Mouján began writing science fiction in the early 1960s under a pseudonym, "Inge Matquim". A science fiction story by Mouján, "Los Huáqueros", won joint first prize at Mardelcon, the 1968 Argentine science fiction convention. Another of her stories, "Gu ta Gutarrak" (Basque for "we and ours"), was written in homage to her grandfather's 1899 poem of the same title, and as "a satire of the Basque nationalist myth of the antiquity and purity of the Basque race". It describes the adventures of a time-traveling Basque family who return to their homeland in the time of their ancestors. The story was accepted for a 1970 issue of the Spanish science fiction magazine Nueva Dimensión, but its publication was blocked by the Franco regime as being contrary to the ideals of Spanish unity. The story was translated into multiple languages, and finally republished by Nueva Dimensión in 1979, after Franco's death. == References ==
Wikipedia:Magdalena Toda#0	Magdalena Daniela Toda is a Romanian-American mathematician, a professor of mathematics and the chair of the Department of Mathematics & Statistics at Texas Tech University. Her research focuses on the curvature of surfaces, geometric flow, the geometry of timelike surfaces, and the uses of differential geometry and partial differential equations in scientific and engineering applications. == Education and career == Toda did her early education in Romania, but is currently a US citizen. She earned a licenciate (effectively a combined bachelor's and master's degree) in mathematics from the University of Bucharest in 1991. After moving to the US, she continued her education at the University of Kansas, where she earned a second master's degree in 1997 and completed her Ph.D. in 2000, also receiving at the same time a Ph.D. from the Politehnica University of Bucharest. Her dissertation, Pseudospherical Surfaces via Moving Frames and Loop Groups, was jointly supervised by Josef Dorfmeister at the University of Kansas, and Constantin Udriște of the Politehnica University of Bucharest. She became an assistant professor at Ball State University in 2000, and moved to her present position at Texas Tech in 2001. There, she was tenured as an associate professor in 2008 and promoted to full professor in 2014. She became interim department chair in 2015 and permanent chair since 2016, with a leave in 2022–2023 to serve as program director for applied mathematics at the National Science Foundation. == Recognition == The Association for Women in Mathematics named Toda to their 2025 Class of AWM Fellows, "for her outstanding leadership in supporting women and girls in mathematics, most notably through the longstanding Emmy Noether High School Days, as well as her service to AWM and local and national committees working towards equity". == Books == Toda is the editor of the research monograph Willmore Energy and Willmore Conjecture (CRC Press / Chapman & Hall 2017). She was added as a coauthor to the 2013 6th edition of the textbook Calculus, by Gerald L. Bradley and Karl J. Smith. == References == == External links == Home page Magdalena Toda publications indexed by Google Scholar
Wikipedia:Magic circle (mathematics)#0	Magic circles were invented by the Song dynasty (960–1279) Chinese mathematician Yang Hui (c. 1238–1298). It is the arrangement of natural numbers on circles where the sum of the numbers on each circle and the sum of numbers on diameters are identical. One of his magic circles was constructed from the natural numbers from 1 to 33 arranged on four concentric circles, with 9 at the center. == Yang Hui magic circles == Yang Hui's magic circle series was published in his Xugu Zhaiqi Suanfa《續古摘奇算法》(Sequel to Excerpts of Mathematical Wonders) of 1275. His magic circle series includes: magic 5 circles in square, 6 circles in ring, magic eight circle in square magic concentric circles, magic 9 circles in square. === Yang Hui magic concentric circle === Yang Hui's magic concentric circle has the following properties The sum of the numbers on four diameters = 147, 28 + 5 + 11 + 25 + 9 + 7 + 19 + 31 + 12 = 147 The sum of 8 numbers plus 9 at the center = 147; 28 + 27 + 20 + 33 + 12 + 4 + 6 + 8 + 9 = 147 The sum of eight radius without 9 = magic number 69: such as 27 + 15 + 3 + 24 = 69 The sum of all numbers on each circle (not including 9) = 2 × 69 There exist 8 semicircles, where the sum of numbers = magic number 69; there are 16 line segments (semicircles and radii) with magic number 69, more than a 6 order magic square with only 12 magic numbers. === Yang Hui magic eight circles in a square === 64 numbers (1–64) are arranged in eight circles, each with eight numbers; each circle sums to 260. The total sum of all numbers is 2080 (=8×260). The circles are arranged in a 3×3 square grid with the center area open in a way that also makes the horizontal / vertical sum along the central columns and rows is 260, and the total sum of the numbers along both diagonals is 520. === Yang Hui magic nine circles in a square === 72 numbers from 1 to 72, arranged in nine circles of eight numbers in a square; with neighbouring numbers forming four additional eight number circles: thus making a total of 13 eight number circles: Extra circle x1 contains numbers from circles NW, N, C, and W; x2 contains numbers from N, NE, E, and C; x3 contains numbers from W, C, S, and SW; x4 contains numbers from C, E, SE, and S. Total sum of 72 numbers = 2628; sum of numbers in any eight number circle = 292; sums of three circles along horizontal lines = 876; sum of three circles along vertical lines = 876; sum of three circles along the diagonals = 876. == Ding Yidong magic circles == Ding Yidong was a mathematician contemporary with Yang Hui. In his magic circle with 6 rings, the unit numbers of the 5 outer rings, combined with the unit number of the center ring, form the following magic square: Method of construction: Let radial group 1 =1,11,21,31,41 Let radial group 2=2,12,22,32,42 Let radial group 3=3,13,23,33,43 Let radial group 4=4,14,24,34,44 Let radial group 6=6,16,26,36,46 Let radial group 7=7,17,27,37,47 Let radial group 8=8,18,28,38,48 Let radial group 9=9,19,29,39,49 Let center group =5,15,25,35,45 Arrange group 1,2,3,4,6,7,9 radially such that each number occupies one position on circle alternate the direction such that one radial has smallest number at the outside, the adjacent radial has largest number outside. Each group occupies the radial position corresponding to the number on the Luoshu magic square, i.e., group 1 at 1 position, group 2 at 2 position etc. Finally arrange center group at the center circle, such that number 5 on group 1 radial number 10 on group 2 radial number 15 on group 3 radial ... number 45 on group 9 radial == Cheng Dawei magic circles == Cheng Dawei, a mathematician in the Ming dynasty, in his book Suanfa Tongzong listed several magic circles == Extension to higher dimensions == In 1917, W. S. Andrews published an arrangement of numbers 1, 2, 3, and 62 in eleven circles of twelve numbers each on a sphere representing the parallels and meridians of the Earth, such that each circle has 12 numbers totalling 378. == Relationship with magic squares == A magic circle can be derived from one or more magic squares by putting a number at each intersection of a circle and a spoke. Additional spokes can be added by replicating the columns of the magic square. In the example in the figure, the following 4 × 4 most-perfect magic square was copied into the upper part of the magic circle. Each number, with 16 added, was placed at the intersection symmetric about the centre of the circles. This results in a magic circle containing numbers 1 to 32, with each circle and diameter totalling 132. == References == Lam Lay Yong: A Critical Study of Hang Hui Suan Fa 《杨辉算法》 Singapore University Press 1977 Wu Wenjun (editor in chief), Grand Series of History of Chinese Mathematics, Vol 6, Part 6 Yang Hui, section 2 Magic circle (吴文俊主编沈康身执笔《中国数学史大系》第六卷第六篇《杨辉》第二节《幻圆》) ISBN 7-303-04926-6/O
Wikipedia:Magic square#0	In mathematics, especially historical and recreational mathematics, a square array of numbers, usually positive integers, is called a magic square if the sums of the numbers in each row, each column, and both main diagonals are the same. The "order" of the magic square is the number of integers along one side (n), and the constant sum is called the "magic constant". If the array includes just the positive integers 1 , 2 , . . . , n 2 {\displaystyle 1,2,...,n^{2}} , the magic square is said to be "normal". Some authors take "magic square" to mean "normal magic square". Magic squares that include repeated entries do not fall under this definition and are referred to as "trivial". Some well-known examples, including the Sagrada Família magic square and the Parker square, are trivial in this sense. When all the rows and columns but not both diagonals sum to the magic constant, this gives a semimagic square (sometimes called orthomagic square). The mathematical study of a magic square typically deals with its construction, classification, and enumeration. Although completely general methods for producing all the magic squares of all orders do not exist, historically three general techniques have been discovered: by bordering method, by making composite magic squares, and by adding two preliminary squares. There are also more specific strategies like the continuous enumeration method that reproduces specific patterns. Magic squares are generally classified according to their order n as: odd if n is odd, evenly even (also referred to as "doubly even") if n is a multiple of 4, oddly even (also known as "singly even") if n is any other even number. This classification is based on different techniques required to construct odd, evenly even, and oddly even squares. Beside this, depending on further properties, magic squares are also classified as associative magic squares, pandiagonal magic squares, most-perfect magic squares, and so on. More challengingly, attempts have also been made to classify all the magic squares of a given order as transformations of a smaller set of squares. Except for n ≤ 5, the enumeration of higher order magic squares is still an open challenge. The enumeration of most-perfect magic squares of any order was only accomplished in the late 20th century. Magic squares have a long history, dating back to at least 190 BCE in China. At various times they have acquired occult or mythical significance, and have appeared as symbols in works of art. In modern times they have been generalized a number of ways, including using extra or different constraints, multiplying instead of adding cells, using alternate shapes or more than two dimensions, and replacing numbers with shapes and addition with geometric operations. == History == The third-order magic square was known to Chinese mathematicians as early as 190 BCE, and explicitly given by the first century of the common era. The first dateable instance of the fourth-order magic square occurred in 587 CE in India. Specimens of magic squares of order 3 to 9 appear in an encyclopedia from Baghdad c. 983, the Encyclopedia of the Brethren of Purity (Rasa'il Ikhwan al-Safa). By the end of the 12th century, the general methods for constructing magic squares were well established. Around this time, some of these squares were increasingly used in conjunction with magic letters, as in Shams Al-ma'arif, for occult purposes. In India, all the fourth-order pandiagonal magic squares were enumerated by Narayana in 1356. Magic squares were made known to Europe through translation of Arabic sources as occult objects during the Renaissance, and the general theory had to be re-discovered independent of prior developments in China, India, and Middle East. Also notable are the ancient cultures with a tradition of mathematics and numerology that did not discover the magic squares: Greeks, Babylonians, Egyptians, and Pre-Columbian Americans. === Chinese === While ancient references to the pattern of even and odd numbers in the 3×3 magic square appear in the I Ching, the first unequivocal instance of this magic square appears in the chapter called Mingtang (Bright Hall) of a 1st-century book Da Dai Liji (Record of Rites by the Elder Dai), which purported to describe ancient Chinese rites of the Zhou dynasty. These numbers also occur in a possibly earlier mathematical text called Shushu jiyi (Memoir on Some Traditions of Mathematical Art), said to be written in 190 BCE. This is the earliest appearance of a magic square on record; and it was mainly used for divination and astrology. The 3×3 magic square was referred to as the "Nine Halls" by earlier Chinese mathematicians. The identification of the 3×3 magic square to the legendary Luoshu chart was only made in the 12th century, after which it was referred to as the Luoshu square. The oldest surviving Chinese treatise that displays magic squares of order larger than 3 is Yang Hui's Xugu zheqi suanfa (Continuation of Ancient Mathematical Methods for Elucidating the Strange) written in 1275. The contents of Yang Hui's treatise were collected from older works, both native and foreign; and he only explains the construction of third and fourth-order magic squares, while merely passing on the finished diagrams of larger squares. He gives a magic square of order 3, two squares for each order of 4 to 8, one of order nine, and one semi-magic square of order 10. He also gives six magic circles of varying complexity. The above magic squares of orders 3 to 9 are taken from Yang Hui's treatise, in which the Luo Shu principle is clearly evident. The order 5 square is a bordered magic square, with central 3×3 square formed according to Luo Shu principle. The order 9 square is a composite magic square, in which the nine 3×3 sub squares are also magic. After Yang Hui, magic squares frequently occur in Chinese mathematics such as in Ding Yidong's Dayan suoyin (c. 1300), Cheng Dawei's Suanfa tongzong (1593), Fang Zhongtong's Shuduyan (1661) which contains magic circles, cubes and spheres, Zhang Chao's Xinzhai zazu (c. 1650), who published China's first magic square of order ten, and lastly Bao Qishou's Binaishanfang ji (c. 1880), who gave various three dimensional magic configurations. However, despite being the first to discover the magic squares and getting a head start by several centuries, the Chinese development of the magic squares are much inferior compared to the Indian, Middle Eastern, or European developments. The high point of Chinese mathematics that deals with the magic squares seems to be contained in the work of Yang Hui; but even as a collection of older methods, this work is much more primitive, lacking general methods for constructing magic squares of any order, compared to a similar collection written around the same time by the Byzantine scholar Manuel Moschopoulos. This is possibly because of the Chinese scholars' enthralment with the Lo Shu principle, which they tried to adapt to solve higher squares; and after Yang Hui and the fall of Yuan dynasty, their systematic purging of the foreign influences in Chinese mathematics. === Japan === Japan and China have similar mathematical traditions and have repeatedly influenced each other in the history of magic squares. The Japanese interest in magic squares began after the dissemination of Chinese works—Yang Hui's Suanfa and Cheng Dawei's Suanfa tongzong—in the 17th century, and as a result, almost all the wasans devoted their time to its study. In the 1660 edition of Ketsugi-sho, Isomura Kittoku gave both odd and even ordered bordered magic squares as well as magic circles; while the 1684 edition of the same book contained a large section on magic squares, demonstrating that he had a general method for constructing bordered magic squares. In Jinko-ki (1665) by Muramatsu Kudayu Mosei, both magic squares and magic circles are displayed. The largest square Mosei constructs is of 19th order. Various magic squares and magic circles were also published by Nozawa Teicho in Dokai-sho (1666), Sato Seiko in Kongenki (1666), and Hosino Sanenobu in Ko-ko-gen Sho (1673). One of Seki Takakazu's Seven Books (Hojin Yensan) (1683) is devoted completely to magic squares and circles. This is the first Japanese book to give a general treatment of magic squares in which the algorithms for constructing odd, singly even and doubly even bordered magic squares are clearly described. In 1694 and 1695, Yueki Ando gave different methods to create the magic squares and displayed squares of order 3 to 30. A fourth-order magic cube was constructed by Yoshizane Tanaka (1651–1719) in Rakusho-kikan (1683). The study of magic squares was continued by Seki's pupils, notably by Katahiro Takebe, whose squares were displayed in the fourth volume of Ichigen Kappo by Shukei Irie, Yoshisuke Matsunaga in Hojin-Shin-jutsu, Yoshihiro Kurushima in Kyushi Iko who rediscovered a method to produce the odd squares given by Agrippa, and Naonobu Ajima. Thus by the beginning of the 18th century, the Japanese mathematicians were in possession of methods to construct magic squares of arbitrary order. After this, attempts at enumerating the magic squares was initiated by Nushizumi Yamaji. === India === The 3×3 magic square first appears in India in Gargasamhita by Garga, who recommends its use to pacify the nine planets (navagraha). The oldest version of this text dates from 100 CE, but the passage on planets could not have been written earlier than 400 CE. The first dateable instance of 3×3 magic square in India occur in a medical text Siddhayog (c. 900 CE) by Vrnda, which was prescribed to women in labor in order to have easy delivery. The oldest dateable fourth order magic square in the world is found in an encyclopaedic work written by Varahamihira around 587 CE called Brhat Samhita. The magic square is constructed for the purpose of making perfumes using 4 substances selected from 16 different substances. Each cell of the square represents a particular ingredient, while the number in the cell represents the proportion of the associated ingredient, such that the mixture of any four combination of ingredients along the columns, rows, diagonals, and so on, gives the total volume of the mixture to be 18. Although the book is mostly about divination, the magic square is given as a matter of combinatorial design, and no magical properties are attributed to it. The special features of this magic square were commented on by Bhattotpala (c. 966 CE) The square of Varahamihira as given above has sum of 18. Here the numbers 1 to 8 appear twice in the square. It is a pan-diagonal magic square. Four different magic squares can be obtained by adding 8 to one of the two sets of 1 to 8 sequence. The sequence is selected such that the number 8 is added exactly twice in each row, each column and each of the main diagonals. One of the possible magic squares shown in the right side. This magic square is remarkable in that it is a 90 degree rotation of a magic square that appears in the 13th century Islamic world as one of the most popular magic squares. The construction of 4th-order magic square is detailed in a work titled Kaksaputa, composed by the alchemist Nagarjuna around 10th century CE. All of the squares given by Nagarjuna are 4×4 magic squares, and one of them is called Nagarjuniya after him. Nagarjuna gave a method of constructing 4×4 magic square using a primary skeleton square, given an odd or even magic sum. The Nagarjuniya square is given below, and has the sum total of 100. The Nagarjuniya square is a pan-diagonal magic square. The Nagarjuniya square is made up of two arithmetic progressions starting from 6 and 16 with eight terms each, with a common difference between successive terms as 4. When these two progressions are reduced to the normal progression of 1 to 8, the adjacent square is obtained. Around 12th-century, a 4×4 magic square was inscribed on the wall of Parshvanath temple in Khajuraho, India. Several Jain hymns teach how to make magic squares, although they are undateable. As far as is known, the first systematic study of magic squares in India was conducted by Thakkar Pheru, a Jain scholar, in his Ganitasara Kaumudi (c. 1315). This work contains a small section on magic squares which consists of nine verses. Here he gives a square of order four, and alludes to its rearrangement; classifies magic squares into three (odd, evenly even, and oddly even) according to its order; gives a square of order six; and prescribes one method each for constructing even and odd squares. For the even squares, Pheru divides the square into component squares of order four, and puts the numbers into cells according to the pattern of a standard square of order four. For odd squares, Pheru gives the method using horse move or knight's move. Although algorithmically different, it gives the same square as the De la Loubere's method. The next comprehensive work on magic squares was taken up by Narayana Pandit, who in the fourteenth chapter of his Ganita Kaumudi (1356) gives general methods for their construction, along with the principles governing such constructions. It consists of 55 verses for rules and 17 verses for examples. Narayana gives a method to construct all the pan-magic squares of fourth order using knight's move; enumerates the number of pan-diagonal magic squares of order four, 384, including every variation made by rotation and reflection; three general methods for squares having any order and constant sum when a standard square of the same order is known; two methods each for constructing evenly even, oddly even, and of squares when the sum is given. While Narayana describes one older method for each species of square, he claims the method of superposition for evenly even and odd squares and a method of interchange for oddly even squares to be his own invention. The superposition method was later re-discovered by De la Hire in Europe. In the last section, he conceives of other figures, such as circles, rectangles, and hexagons, in which the numbers may be arranged to possess properties similar to those of magic squares. Below are some of the magic squares constructed by Narayana: The order 8 square is interesting in itself since it is an instance of the most-perfect magic square. Incidentally, Narayana states that the purpose of studying magic squares is to construct yantra, to destroy the ego of bad mathematicians, and for the pleasure of good mathematicians. The subject of magic squares is referred to as bhadraganita and Narayana states that it was first taught to men by god Shiva. === Middle East, North Africa, Muslim Iberia === Although the early history of magic squares in Persia and Arabia is not known, it has been suggested that they were known in pre-Islamic times. It is clear, however, that the study of magic squares was common in medieval Islam, and it was thought to have begun after the introduction of chess into the region. The first dateable appearance of a magic square of order 3 occurs in Jābir ibn Hayyān's (fl. c. 721 – c. 815) Kitab al-mawazin al-Saghir (The Small Book of Balances) where the magic square and its related numerology is associated with alchemy. While it is known that treatises on magic squares were written in the 9th century, the earliest extant treaties date from the 10th-century: one by Abu'l-Wafa al-Buzjani (c. 998) and another by Ali b. Ahmad al-Antaki (c. 987). These early treatises were purely mathematical, and the Arabic designation for magic squares used is wafq al-a'dad, which translates as harmonious disposition of the numbers. By the end of 10th century, the two treatises by Buzjani and Antaki makes it clear that the Middle Eastern mathematicians had understood how to construct bordered squares of any order as well as simple magic squares of small orders (n ≤ 6) which were used to make composite magic squares. A specimen of magic squares of orders 3 to 9 devised by Middle Eastern mathematicians appear in an encyclopedia from Baghdad c. 983, the Rasa'il Ikhwan al-Safa (the Encyclopedia of the Brethren of Purity). The squares of order 3 to 7 from Rasa'il are given below: The 11th century saw the finding of several ways to construct simple magic squares for odd and evenly even orders; the more difficult case of oddly even case (n = 4k + 2) was solved by Ibn al-Haytham with k even (c. 1040), and completely by the beginning of 12th century, if not already in the latter half of the 11th century. Around the same time, pandiagonal squares were being constructed. Treaties on magic squares were numerous in the 11th and 12th century. These later developments tended to be improvements on or simplifications of existing methods. From the 13th century, magic squares were increasingly put to occult purposes. However, much of these later texts written for occult purposes merely depict certain magic squares and mention their attributes, without describing their principle of construction, with only some authors keeping the general theory alive. One such occultist was the Algerian Ahmad al-Buni (c. 1225), who gave general methods on constructing bordered magic squares; some others were the 17th century Egyptian Shabramallisi and the 18th century Nigerian al-Kishnawi. The magic square of order three was described as a child-bearing charm since its first literary appearances in the alchemical works of Jābir ibn Hayyān (fl. c. 721 – c. 815) and al-Ghazālī (1058–1111) and it was preserved in the tradition of the planetary tables. The earliest occurrence of the association of seven magic squares to the virtues of the seven heavenly bodies appear in Andalusian scholar Ibn Zarkali's (known as Azarquiel in Europe) (1029–1087) Kitāb tadbīrāt al-kawākib (Book on the Influences of the Planets). A century later, the Algerian scholar Ahmad al-Buni attributed mystical properties to magic squares in his highly influential book Shams al-Ma'arif (The Book of the Sun of Gnosis and the Subtleties of Elevated Things), which also describes their construction. This tradition about a series of magic squares from order three to nine, which are associated with the seven planets, survives in Greek, Arabic, and Latin versions. There are also references to the use of magic squares in astrological calculations, a practice that seems to have originated with the Arabs. === Latin Europe === Unlike in Persia and Arabia, better documentation exists of how the magic squares were transmitted to Europe. Around 1315, influenced by Arab sources, the Greek Byzantine scholar Manuel Moschopoulos wrote a mathematical treatise on the subject of magic squares, leaving out the mysticism of his Middle Eastern predecessors, where he gave two methods for odd squares and two methods for evenly even squares. Moschopoulos was essentially unknown to the Latin Europe until the late 17th century, when Philippe de la Hire rediscovered his treatise in the Royal Library of Paris. However, he was not the first European to have written on magic squares; and the magic squares were disseminated to rest of Europe through Spain and Italy as occult objects. The early occult treaties that displayed the squares did not describe how they were constructed. Thus the entire theory had to be rediscovered. Magic squares had first appeared in Europe in Kitāb tadbīrāt al-kawākib (Book on the Influences of the Planets) written by Ibn Zarkali of Toledo, Al-Andalus, as planetary squares by 11th century. The magic square of three was discussed in numerological manner in early 12th century by Jewish scholar Abraham ibn Ezra of Toledo, which influenced later Kabbalists. Ibn Zarkali's work was translated as Libro de Astromagia in the 1280s, due to Alfonso X of Castille. In the Alfonsine text, magic squares of different orders are assigned to the respective planets, as in the Islamic literature; unfortunately, of all the squares discussed, the Mars magic square of order five is the only square exhibited in the manuscript. Magic squares surface again in Florence, Italy in the 14th century. A 6×6 and a 9×9 square are exhibited in a manuscript of the Trattato d'Abbaco (Treatise of the Abacus) by Paolo Dagomari. It is interesting to observe that Paolo Dagomari, like Pacioli after him, refers to the squares as a useful basis for inventing mathematical questions and games, and does not mention any magical use. Incidentally, though, he also refers to them as being respectively the Sun's and the Moon's squares, and mentions that they enter astrological calculations that are not better specified. As said, the same point of view seems to motivate the fellow Florentine Luca Pacioli, who describes 3×3 to 9×9 squares in his work De Viribus Quantitatis by the end of 15th century. === Europe after 15th century === The planetary squares had disseminated into northern Europe by the end of the 15th century. For instance, the Cracow manuscript of Picatrix from Poland displays magic squares of orders 3 to 9. The same set of squares as in the Cracow manuscript later appears in the writings of Paracelsus in Archidoxa Magica (1567), although in highly garbled form. In 1514 Albrecht Dürer immortalized a 4×4 square in his famous engraving Melencolia I. Paracelsus' contemporary Heinrich Cornelius Agrippa von Nettesheim published his famous three volume book De occulta philosophia in 1531, where he devoted Chapter 22 of Book II to the planetary squares shown below. The same set of squares given by Agrippa reappear in 1539 in Practica Arithmetice by Girolamo Cardano, where he explains the construction of the odd ordered squares using "diamond method", which was later reproduced by Bachet. The tradition of planetary squares was continued into the 17th century by Athanasius Kircher in Oedipi Aegyptici (1653). In Germany, mathematical treaties concerning magic squares were written in 1544 by Michael Stifel in Arithmetica Integra, who rediscovered the bordered squares, and Adam Riese, who rediscovered the continuous numbering method to construct odd ordered squares published by Agrippa. However, due to the religious upheavals of that time, these works were unknown to the rest of Europe. In 1624 France, Claude Gaspard Bachet described the "diamond method" for constructing Agrippa's odd ordered squares in his book Problèmes Plaisants. During 1640 Bernard Frenicle de Bessy and Pierre Fermat exchanged letters on magic squares and cubes, and in one of the letters Fermat boasts of being able to construct 1,004,144,995,344 magic squares of order 8 by his method. An early account on the construction of bordered squares was given by Antoine Arnauld in his Nouveaux éléments de géométrie (1667). In the two treatise Des quarrez ou tables magiques and Table générale des quarrez magiques de quatre de côté, published posthumously in 1693, twenty years after his death, Bernard Frenicle de Bessy demonstrated that there were exactly 880 distinct magic squares of order four. Frenicle gave methods to construct magic square of any odd and even order, where the even ordered squares were constructed using borders. He also showed that interchanging rows and columns of a magic square produced new magic squares. In 1691, Simon de la Loubère described the Indian continuous method of constructing odd ordered magic squares in his book Du Royaume de Siam, which he had learned while returning from a diplomatic mission to Siam, which was faster than Bachet's method. In an attempt to explain its working, de la Loubere used the primary numbers and root numbers, and rediscovered the method of adding two preliminary squares. This method was further investigated by Abbe Poignard in Traité des quarrés sublimes (1704), by Philippe de La Hire in Mémoires de l'Académie des Sciences for the Royal Academy (1705), and by Joseph Sauveur in Construction des quarrés magiques (1710). Concentric bordered squares were also studied by De la Hire in 1705, while Sauveur introduced magic cubes and lettered squares, which was taken up later by Euler in 1776, who is often credited for devising them. In 1750 d'Ons-le-Bray rediscovered the method of constructing doubly even and singly even squares using bordering technique; while in 1767 Benjamin Franklin published a semi-magic square that had the properties of eponymous Franklin square. By this time the earlier mysticism attached to the magic squares had completely vanished, and the subject was treated as a part of recreational mathematics. In the 19th century, Bernard Violle gave a comprehensive treatment of magic squares in his three volume Traité complet des carrés magiques (1837–1838), which also described magic cubes, parallelograms, parallelopipeds, and circles. Pandiagonal squares were extensively studied by Andrew Hollingworth Frost, who learned it while in the town of Nasik, India, (thus calling them Nasik squares) in a series of articles: On the knight's path (1877), On the General Properties of Nasik Squares (1878), On the General Properties of Nasik Cubes (1878), On the construction of Nasik Squares of any order (1896). He showed that it is impossible to have normal singly-even pandiagonal magic squares. Frederick A.P. Barnard constructed inlaid magic squares and other three dimensional magic figures like magic spheres and magic cylinders in Theory of magic squares and of magic cubes (1888). In 1897, Emroy McClintock published On the most perfect form of magic squares, coining the words pandiagonal square and most perfect square, which had previously been referred to as perfect, or diabolic, or Nasik. == Some famous magic squares == === Luo Shu magic square === Legends dating from as early as 650 BCE tell the story of the Lo Shu (洛書) or "scroll of the river Lo". According to the legend, there was at one time in ancient China a huge flood. While the great king Yu was trying to channel the water out to sea, a turtle emerged from it with a curious pattern on its shell: a 3×3 grid in which circular dots of numbers were arranged, such that the sum of the numbers in each row, column and diagonal was the same: 15. According to the legend, thereafter people were able to use this pattern in a certain way to control the river and protect themselves from floods. The Lo Shu Square, as the magic square on the turtle shell is called, is the unique normal magic square of order three in which 1 is at the bottom and 2 is in the upper right corner. Every normal magic square of order three is obtained from the Lo Shu by rotation or reflection. === Magic square in Parshavnath temple === There is a well-known 12th-century 4×4 normal magic square inscribed on the wall of the Parshvanath temple in Khajuraho, India. This is known as the Chautisa Yantra (Chautisa, 34; Yantra, lit. "device"), since its magic sum is 34. It is one of the three 4×4 pandiagonal magic squares and is also an instance of the most-perfect magic square. The study of this square led to the appreciation of pandiagonal squares by European mathematicians in the late 19th century. Pandiagonal squares were referred to as Nasik squares or Jain squares in older English literature. === Albrecht Dürer's magic square === The order four normal magic square Albrecht Dürer immortalized in his 1514 engraving Melencolia I, referred to above, is believed to be the first seen in European art. The square associated with Jupiter appears as a talisman used to drive away melancholy. It is very similar to Yang Hui's square, which was created in China about 250 years before Dürer's time. As with every order 4 normal magic square, the magic sum is 34. But in the Durer square this sum is also found in each of the quadrants, in the center four squares, and in the corner squares (of the 4×4 as well as the four contained 3×3 grids). This sum can also be found in the four outer numbers clockwise from the corners (3+8+14+9) and likewise the four counter-clockwise (the locations of four queens in the two solutions of the 4 queens puzzle), the two sets of four symmetrical numbers (2+8+9+15 and 3+5+12+14), the sum of the middle two entries of the two outer columns and rows (5+9+8+12 and 3+2+15+14), and in four kite or cross shaped quartets (3+5+11+15, 2+10+8+14, 3+9+7+15, and 2+6+12+14). The two numbers in the middle of the bottom row give the date of the engraving: 1514. It has been speculated that the numbers 4,1 bordering the publication date correspond to Durer's initials D,A. But if that had been his intention, he could have inverted the order of columns 1 and 4 to achieve "A1514D" without compromising the square's properties. Dürer's magic square can also be extended to a magic cube. === Sagrada Família magic square === The Passion façade of the Sagrada Família church in Barcelona, conceptualized by Antoni Gaudí and designed by sculptor Josep Subirachs, features a trivial order 4 magic square: The magic constant of the square is 33, the age of Jesus at the time of the Passion. Structurally, it is very similar to the Melancholia magic square, but it has had the numbers in four of the cells reduced by 1. Trivial squares such as this one are not generally mathematically interesting and only have historical significance. Lee Sallows has pointed out that, due to Subirachs's ignorance of magic square theory, the renowned sculptor made a needless blunder, and supports this assertion by giving several examples of non-trivial 4 × 4 magic squares showing the desired magic constant of 33. Similarly to Dürer's magic square, the Sagrada Familia's magic square can also be extended to a magic cube. === Parker square === The Parker square, named after recreational mathematician and maths youtuber Matt Parker, is an attempt to create a 3 × 3 magic square of squares — a prized unsolved problem since Euler. Discovered in 2016, the Parker square is a trivial semimagic square, since it uses some numbers more than once, and the diagonal 232 + 372 + 472 sums to 4107, not 3051 as for all the other rows and columns, and the other diagonal. A true 3 × 3 magic square of square numbers has to date not been discovered (despite computer searches). In a Numberphile video from June 2023, Mathematician Tony Várilly-Alvarado used mathematics to speculate that the existence of such a square is "probably impossible". In February 2025, Parker upped his years-old bounty of US$1,000 to $10,000 to find a fully magic 3 × 3 square using square numbers. === Gardner square === The Gardner square, named after recreational mathematician Martin Gardner, similar to the Parker square, is given as a problem to determine a, b, c and d. This solution for a = 74, b = 113, c = 94 and d = 97 gives a semimagic square; the diagonal 1272 + b2 + d2 sums to 38307, not 21609 as for all the other rows and columns, and the other diagonal. == Properties of magic squares == === Magic constant === The constant that is the sum of any row, or column, or diagonal is called the magic constant or magic sum, M. Every normal magic square has a constant dependent on the order n, calculated by the formula M = n ( n 2 + 1 ) / 2 {\displaystyle M=n(n^{2}+1)/2} . This can be demonstrated by noting that the sum of 1 , 2 , . . . , n 2 {\displaystyle 1,2,...,n^{2}} is n 2 ( n 2 + 1 ) / 2 {\displaystyle n^{2}(n^{2}+1)/2} . Since the sum of each row is M {\displaystyle M} , the sum of n {\displaystyle n} rows is n M = n 2 ( n 2 + 1 ) / 2 {\displaystyle nM=n^{2}(n^{2}+1)/2} , which when divided by the order n yields the magic constant as M = n ( n 2 + 1 ) / 2 {\displaystyle M=n(n^{2}+1)/2} . For normal magic squares of orders n = 3, 4, 5, 6, 7, and 8, the magic constants are, respectively: 15, 34, 65, 111, 175, and 260 (sequence A006003 in the OEIS). === Magic square of order 1 is trivial === The 1×1 magic square, with only one cell containing the number 1, is called trivial, because it is typically not under consideration when discussing magic squares; but it is indeed a magic square by definition, if a single cell is regarded as a square of order one. === Magic square of order 2 cannot be constructed === Normal magic squares of all sizes can be constructed except 2×2 (that is, where order n = 2). === Center of mass === If the numbers in the magic square are seen as masses located in various cells, then the center of mass of a magic square coincides with its geometric center. === Moment of inertia === The moment of inertia of a magic square has been defined as the sum over all cells of the number in the cell times the squared distance from the center of the cell to the center of the square; here the unit of measurement is the width of one cell. (Thus for example a corner cell of a 3×3 square has a distance of 2 , {\displaystyle {\sqrt {2}},} a non-corner edge cell has a distance of 1, and the center cell has a distance of 0.) Then all magic squares of a given order have the same moment of inertia as each other. For the order-3 case the moment of inertia is always 60, while for the order-4 case the moment of inertia is always 340. In general, for the n×n case the moment of inertia is n 2 ( n 4 − 1 ) / 12. {\displaystyle n^{2}(n^{4}-1)/12.} === Birkhoff–von Neumann decomposition === Dividing each number of the magic square by the magic constant will yield a doubly stochastic matrix, whose row sums and column sums equal to unity. However, unlike the doubly stochastic matrix, the diagonal sums of such matrices will also equal to unity. Thus, such matrices constitute a subset of doubly stochastic matrix. The Birkhoff–von Neumann theorem states that for any doubly stochastic matrix A {\displaystyle A} , there exists real numbers θ 1 , … , θ k ≥ 0 {\displaystyle \theta _{1},\ldots ,\theta _{k}\geq 0} , where ∑ i = 1 k θ i = 1 {\displaystyle \sum _{i=1}^{k}\theta _{i}=1} and permutation matrices P 1 , … , P k {\displaystyle P_{1},\ldots ,P_{k}} such that A = θ 1 P 1 + ⋯ + θ k P k . {\displaystyle A=\theta _{1}P_{1}+\cdots +\theta _{k}P_{k}.} This representation may not be unique in general. By Marcus-Ree theorem, however, there need not be more than k ≤ n 2 − 2 n + 2 {\displaystyle k\leq n^{2}-2n+2} terms in any decomposition. Clearly, this decomposition carries over to magic squares as well, since a magic square can be recovered from a doubly stochastic matrix by multiplying it by the magic constant. == Classification of magic squares == While the classification of magic squares can be done in many ways, some useful categories are given below. An n×n square array of integers 1, 2, ..., n2 is called: Semi-magic square when its rows and columns sum to give the magic constant. Simple magic square when its rows, columns, and two diagonals sum to give magic constant and no more. They are also known as ordinary magic squares or normal magic squares. Self-complementary magic square when it is a magic square which when complemented (i.e. each number subtracted from n2 + 1) will give a rotated or reflected version of the original magic square. Associative magic square when it is a magic square with a further property that every number added to the number equidistant, in a straight line, from the center gives n2 + 1. They are also called symmetric magic squares. Associative magic squares do not exist for squares of singly even order. All associative magic square are self-complementary magic squares as well. Pandiagonal magic square when it is a magic square with a further property that the broken diagonals sum to the magic constant. They are also called panmagic squares, perfect squares, diabolic squares, Jain squares, or Nasik squares. Panmagic squares do not exist for singly even orders. However, singly even non-normal squares can be panmagic. Ultra magic square when it is both associative and pandiagonal magic square. Ultra magic square exist only for orders n ≥ 5. Bordered magic square when it is a magic square and it remains magic when the rows and columns on the outer edge are removed. They are also called concentric bordered magic squares if removing a border of a square successively gives another smaller bordered magic square. Bordered magic square do not exist for order 4. Composite magic square when it is a magic square that is created by "multiplying" (in some sense) smaller magic squares, such that the order of the composite magic square is a multiple of the order of the smaller squares. Such squares can usually be partitioned into smaller non-overlapping magic sub-squares. Inlaid magic square when it is a magic square inside which a magic sub-square is embedded, regardless of construction technique. The embedded magic sub-squares are themselves referred to as inlays. Most-perfect magic square when it is a pandiagonal magic square with two further properties (i) each 2×2 subsquare add to 1/k of the magic constant where n = 4k, and (ii) all pairs of integers distant n/2 along any diagonal (major or broken) are complementary (i.e. they sum to n2 + 1). The first property is referred to as compactness, while the second property is referred to as completeness. Most-perfect magic squares exist only for squares of doubly even order. All the pandiagonal squares of order 4 are also most perfect. Franklin magic square when it is a doubly even magic square with three further properties (i) every bent diagonal adds to the magic constant, (ii) every half row and half column starting at an outside edge adds to half the magic constant, and (iii) the square is compact. Multimagic square when it is a magic square that remains magic even if all its numbers are replaced by their k-th power for 1 ≤ k ≤ P. They are also known as P-multimagic square or satanic squares. They are also referred to as bimagic squares, trimagic squares, tetramagic squares, and pentamagic squares when the value of P is 2, 3, 4, and 5 respectively. == Enumeration of magic squares == Low-order squares There is only one (trivial) magic square of order 1 and no magic square of order 2. As mentioned above, the set of normal squares of order three constitutes a single equivalence class-all equivalent to the Lo Shu square. Thus there is basically just one normal magic square of order 3. The number of different n × n magic squares for n from 1 to 6, not counting rotations and reflections is: 1, 0, 1, 880, 275305224, 17753889197660635632. (sequence A006052 in the OEIS) The number for n = 6 had previously been estimated to be (1.7745 ± 0.0016) × 1019. Magic tori Cross-referenced to the above sequence, a new classification enumerates the magic tori that display these magic squares. The number of magic tori of order n from 1 to 5, is: 1, 0, 1, 255, 251449712 (sequence A270876 in the OEIS). Higher-order squares and tori The number of distinct normal magic squares rapidly increases for higher orders. The 880 magic squares of order 4 are displayed on 255 magic tori of order 4 and the 275,305,224 squares of order 5 are displayed on 251,449,712 magic tori of order 5. The numbers of magic tori and distinct normal squares are not yet known for orders beyond 5 and 6, respectively. Algorithms tend to only generate magic squares of a certain type or classification, making counting all possible magic squares quite difficult. Since traditional counting methods have proven unsuccessful, statistical analysis using the Monte Carlo method has been applied. The basic principle applied to magic squares is to randomly generate n × n matrices of elements 1 to n2 and check if the result is a magic square. The probability that a randomly generated matrix of numbers is a magic square is then used to approximate the number of magic squares. More intricate versions of the Monte Carlo method, such as the exchange Monte Carlo, and Monte Carlo backtracking have produced even more accurate estimations. Using these methods it has been shown that the probability of magic squares decreases rapidly as n increases. Using fitting functions give the curves seen to the right. == Transformations that preserve the magic property == === For any magic square === The sum of any two magic squares of the same order by matrix addition is a magic square. A magic square remains magic when all of its numbers undergo the same linear transformation (i.e., a function of the form f(x) = m x + b). For example, a magic square remains magic when its numbers are multiplied by any constant. Moreover, a magic square remains magic when a constant is added or subtracted to its numbers, or if its numbers are subtracted from a constant. In particular, if every element in a normal magic square of order n {\displaystyle n} is subtracted from n 2 + 1 {\displaystyle n^{2}+1} , the complement of the original square is obtained. In the example below, each element of the magic square on the left is subtracted from 17 to obtain the complement magic square on the right. A magic square remains magic when transformed by any element of D4, the symmetry group of a square (see Dihedral group of order 8 § The symmetry group of a square: dihedral group of order 8). Every combination of one or more rotations of 90 degrees, reflections, or both produce eight trivially distinct squares which are generally considered equivalent. The eight such squares are said to make up a single equivalence class. The eight equivalent magic squares for the 3×3 magic square are shown below: A magic square of order n {\displaystyle n} remains magic when both its rows and columns are symmetrically permuted by p {\displaystyle p} such that p ( i ) + p ( n + 1 − i ) = n + 1 {\displaystyle p(i)+p(n+1-i)=n+1} for 1 ≤ i ≤ n {\displaystyle 1\leq i\leq n} . Every permutation of the rows or columns preserves all row and column sums, but generally not the two diagonal sums. If the same permutation p {\displaystyle p} is applied to both the rows and columns, then diagonal element in row i {\displaystyle i} and column i {\displaystyle i} is mapped to row p ( i ) {\displaystyle p(i)} and column p ( i ) {\displaystyle p(i)} which is on the same diagonal; therefore, applying the same permutation to rows and columns preserves the main (upper left to lower right) diagonal sum. If the permutation is symmetric as described, then the diagonal element in row i {\displaystyle i} and column n + 1 − i {\displaystyle n+1-i} is mapped to row p ( i ) {\displaystyle p(i)} and column p ( n + 1 − i ) = n + 1 − p ( i ) {\displaystyle p(n+1-i)=n+1-p(i)} which is on the same diagonal; therefore, applying the same symmetric permutation to both rows and columns preserves both diagonal sums. For even n {\displaystyle n} , there are 2 n 2 ( n 2 ) ! {\displaystyle 2^{\frac {n}{2}}\left({\frac {n}{2}}\right)!} such symmetric permutations, and 2 n − 1 2 ( n − 1 2 ) ! {\displaystyle 2^{\frac {n-1}{2}}\left({\frac {n-1}{2}}\right)!} for n {\displaystyle n} odd. In the example below, the original magic square on the left has its rows and columns symmetrically permuted by ( 4 , 6 , 5 , 2 , 1 , 3 ) {\displaystyle (4,6,5,2,1,3)} resulting in the magic square on the right. A magic square of order n {\displaystyle n} remains magic when rows i {\displaystyle i} and ( n + 1 − i ) {\displaystyle (n+1-i)} are exchanged and columns i {\displaystyle i} and ( n + 1 − i ) {\displaystyle (n+1-i)} are exchanged because this is a symmetric permutation of the form described above. In the example below, the square on the right is obtained by interchanging the 1st and 4th rows and columns of the original square on the left. A magic square of order n {\displaystyle n} remains magic when rows i {\displaystyle i} and j {\displaystyle j} are exchanged, rows ( n + 1 − i ) {\displaystyle (n+1-i)} and ( n + 1 − j ) {\displaystyle (n+1-j)} are exchanged, columns i {\displaystyle i} and j {\displaystyle j} are exchanged, and columns ( n + 1 − i ) {\displaystyle (n+1-i)} and ( n + 1 − j ) {\displaystyle (n+1-j)} are exchanged where i < j < n + 1 2 {\displaystyle i<j<{\frac {n+1}{2}}} because this is another symmetric permutation of the form described above. In the example below, the left square is the original square, while the right square is the new square obtained by this transformation. In the middle square, rows 1 and 2 and rows 3 and 4 have been swapped. The final square on the right is obtained by interchanging columns 1 and 2 and columns 3 and 4 of the middle square. In this particular example, this transform rotates the quadrants 180 degrees. The middle square is also magic because the original square is associative. A magic square remains magic when its quadrants are diagonally interchanged because this is another symmetric permutation of the form described above. For even-order n {\displaystyle n} , permute the rows and columns by permutation p {\displaystyle p} where p ( i ) = i + n 2 {\displaystyle p(i)=i+{\frac {n}{2}}} for i ≤ n 2 {\displaystyle i\leq {\frac {n}{2}}} , and p ( i ) = i − n 2 {\displaystyle p(i)=i-{\frac {n}{2}}} for i > n 2 {\displaystyle i>{\frac {n}{2}}} . For odd-order n {\displaystyle n} , permute rows and columns by permutation p {\displaystyle p} where p ( i ) = i + n + 1 2 {\displaystyle p(i)=i+{\frac {n+1}{2}}} for i < n + 1 2 {\displaystyle i<{\frac {n+1}{2}}} , and p ( i ) = i − n + 1 2 {\displaystyle p(i)=i-{\frac {n+1}{2}}} for i > n + 1 2 {\displaystyle i>{\frac {n+1}{2}}} . For odd ordered square, the halves of the central row and column are also interchanged. Examples for order 4 and 5 magic squares are given below: === For associative magic squares === An associative magic square remains associative when two rows or columns equidistant from the center are interchanged. For an even square, there are n/2 pairs of rows or columns that can be interchanged; thus 2n/2 × 2n/2 = 2n equivalent magic squares by combining such interchanges can be obtained. For odd square, there are (n − 1)/2 pairs of rows or columns that can be interchanged; and 2n−1 equivalent magic squares obtained by combining such interchanges. Interchanging all the rows flips the square vertically (i.e. reflected along the horizontal axis), while interchanging all the columns flips the square horizontally (i.e. reflected along the vertical axis). In the example below, a 4×4 associative magic square on the left is transformed into a square on the right by interchanging the second and third row, yielding the famous Durer's magic square. An associative magic square remains associative when two same sided rows (or columns) are interchanged along with corresponding other sided rows (or columns). For an even square, since there are n/2 same sided rows (or columns), there are n(n − 2)/8 pairs of such rows (or columns) that can be interchanged. Thus, 2n(n − 2)/8 × 2n(n − 2)/8 = 2n(n − 2)/4 equivalent magic squares can be obtained by combining such interchanges. For odd square, since there are (n − 1)/2 same sided rows or columns, there are (n − 1)(n − 3)/8 pairs of such rows or columns that can be interchanged. Thus, there are 2(n − 1)(n − 3)/8 × 2(n − 1)(n − 3)/8 = 2(n − 1)(n − 3)/4 equivalent magic squares obtained by combining such interchanges. Interchanging all the same sided rows flips each quadrants of the square vertically, while interchanging all the same sided columns flips each quadrant of the square horizontally. In the example below, the original square is on the left, whose rows 1 and 2 are interchanged with each other, along with rows 3 and 4, to obtain the transformed square on the right. An associative magic square remains associative when its entries are replaced with corresponding numbers from a set of s arithmetic progressions with the same common difference among r terms, such that r × s = n2, and whose initial terms are also in arithmetic progression, to obtain a non-normal magic square. Here either s or r should be a multiple of n. Let us have s arithmetic progressions given by a a + c a + 2 c ⋯ a + ( r − 1 ) c a + d a + c + d a + 2 c + d ⋯ a + ( r − 1 ) c + d a + 2 d a + c + 2 d a + 2 c + 2 d ⋯ a + ( r − 1 ) c + 2 d ⋯ ⋯ ⋯ ⋯ ⋯ a + ( s − 1 ) d a + c + ( s − 1 ) d a + 2 c + ( s − 1 ) d ⋯ a + ( r − 1 ) c + ( s − 1 ) d {\displaystyle {\begin{array}{lllll}a&a+c&a+2c&\cdots &a+(r-1)c\\a+d&a+c+d&a+2c+d&\cdots &a+(r-1)c+d\\a+2d&a+c+2d&a+2c+2d&\cdots &a+(r-1)c+2d\\\cdots &\cdots &\cdots &\cdots &\cdots \\a+(s-1)d&a+c+(s-1)d&a+2c+(s-1)d&\cdots &a+(r-1)c+(s-1)d\\\end{array}}} where a is the initial term, c is the common difference of the arithmetic progressions, and d is the common difference among the initial terms of each progression. The new magic constant will be M = n a + n 2 [ ( r − 1 ) c + ( s − 1 ) d ] . {\displaystyle M=na+{\frac {n}{2}}{\big [}(r-1)c+(s-1)d{\big ]}.} If s = r = n, then follows the simplification M = n a + n 2 ( n − 1 ) ( c + d ) . {\displaystyle M=na+{\frac {n}{2}}(n-1)(c+d).} With a = c = 1 and d = n, the usual M = n(n2+1)/2 is obtained. For given M the required a, c, and d can be found by solving the linear Diophantine equation. In the examples below, there are order 4 normal magic squares on the left most side. The second square is a corresponding non-normal magic square with r = 8, s = 2, a = 1, c = 1, and d = 10 such that the new magic constant is M = 38. The third square is an order 5 normal magic square, which is a 90 degree clockwise rotated version of the square generated by De la Loubere method. On the right most side is a corresponding non-normal magic square with a = 4, c = 1, and d = 6 such that the new magic constant is M = 90. === For pan-diagonal magic squares === A pan-diagonal magic square remains a pan-diagonal magic square under cyclic shifting of rows or of columns or both. This allows us to position a given number in any one of the n2 cells of an n order square. Thus, for a given pan-magic square, there are n2 equivalent pan-magic squares. In the example below, the original square on the left is transformed by shifting the first row to the bottom to obtain a new pan-magic square in the middle. Next, the 1st and 2nd column of the middle pan-magic square is circularly shifted to the right to obtain a new pan-magic square on the right. === For bordered magic squares === A bordered magic square remains a bordered magic square after permuting the border cells in the rows or columns, together with their corresponding complementary terms, keeping the corner cells fixed. Since the cells in each row and column of every concentric border can be permuted independently, when the order n ≥ 5 is odd, there are ( ( n − 2 ) ! ( n − 4 ) ! ⋯ ⋅ 3 ! ) 2 {\displaystyle ((n-2)!(n-4)!\dots \cdot 3!)^{2}} equivalent bordered squares. When n ≥ 6 is even, there are ( ( n − 2 ) ! ( n − 4 ) ! ⋯ ⋅ 4 ! ) 2 {\displaystyle ((n-2)!(n-4)!\dots \cdot 4!)^{2}} equivalent bordered squares. In the example below, a square of order 5 is given whose border row has been permuted and (3!)2 = 36 such equivalent squares can be obtained. A bordered magic square remains a bordered magic square after each of its concentric borders are independently rotated or reflected with respect to the central core magic square. If there are b borders, then this transform will yield 8b equivalent squares. In the example below of the 5×5 magic square, the border has been rotated 90 degrees anti-clockwise. === For composite magic squares === A composite magic square remains a composite magic square when the embedded magic squares undergo transformations that do not disturb the magic property (e.g. rotation, reflection, shifting of rows and columns, and so on). == Special methods of construction == Over the millennia, many ways to construct magic squares have been discovered. These methods can be classified as general methods and special methods, in the sense that general methods allow us to construct more than a single magic square of a given order, whereas special methods allow us to construct just one magic square of a given order. Special methods are specific algorithms whereas general methods may require some trial-and-error. Special methods are the most simple ways to construct magic squares. They follow certain algorithms which generate regular patterns of numbers in a square. The correctness of these special methods can be proved using one of the general methods given in later sections. After a magic square has been constructed using a special method, the transformations described in the previous section can be applied to yield further magic squares. Special methods are usually referred to using the name of the author(s) (if known) who described the method, for e.g. De la Loubere's method, Starchey's method, Bachet's method, etc. Magic squares are believed to exist for all orders, except for order 2. Magic squares can be classified according to their order as odd, doubly even (n divisible by four), and singly even (n even, but not divisible by four). This classification is based on the fact that entirely different techniques need to be employed to construct these different species of squares. Odd and doubly even magic squares are easy to generate; the construction of singly even magic squares is more difficult but several methods exist, including John Horton Conway's LUX method for magic squares and the Strachey method for magic squares. === A method for constructing a magic square of order 3 === In the 19th century, Édouard Lucas devised the general formula for order 3 magic squares. Consider the following table made up of positive integers a, b and c: These nine numbers will be distinct positive integers forming a magic square with the magic constant 3c so long as 0 < a < b < c − a and b ≠ 2a. Moreover, every 3×3 magic square of distinct positive integers is of this form. In 1997 Lee Sallows discovered that leaving aside rotations and reflections, then every distinct parallelogram drawn on the Argand diagram defines a unique 3×3 magic square, and vice versa, a result that had never previously been noted. === A method for constructing a magic square of odd order === A method for constructing magic squares of odd order was published by the French diplomat de la Loubère in his book, A new historical relation of the kingdom of Siam (Du Royaume de Siam, 1693), in the chapter entitled The problem of the magical square according to the Indians. The method operates as follows: The method prescribes starting in the central column of the first row with the number 1. After that, the fundamental movement for filling the squares is diagonally up and right, one step at a time. If a square is filled with a multiple of the order n, one moves vertically down one square instead, then continues as before. When an "up and to the right" move would leave the square, it is wrapped around to the last row or first column, respectively. Starting from other squares rather than the central column of the first row is possible, but then only the row and column sums will be identical and result in a magic sum, whereas the diagonal sums will differ. The result will thus be a semimagic square and not a true magic square. Moving in directions other than north east can also result in magic squares. === A method of constructing a magic square of doubly even order === Doubly even means that n is an even multiple of an even integer; or 4p (e.g. 4, 8, 12), where p is an integer. Generic pattern All the numbers are written in order from left to right across each row in turn, starting from the top left hand corner. Numbers are then either retained in the same place or interchanged with their diametrically opposite numbers in a certain regular pattern. In the magic square of order four, the numbers in the four central squares and one square at each corner are retained in the same place and the others are interchanged with their diametrically opposite numbers. A construction of a magic square of order 4 Starting from top left, go left to right through each row of the square, counting each cell from 1 to 16 and filling the cells along the diagonals with its corresponding number. Once the bottom right cell is reached, continue by going right to left, starting from the bottom right of the table through each row, and fill in the non-diagonal cells counting up from 1 to 16 with its corresponding number. As shown below: An extension of the above example for Orders 8 and 12 First generate a pattern table, where a '1' indicates selecting from the square where the numbers are written in order 1 to n2 (left-to-right, top-to-bottom), and a '0' indicates selecting from the square where the numbers are written in reverse order n2 to 1. For M = 4, the pattern table is as shown below (third matrix from left). With the unaltered cells (cells with '1') shaded, a criss-cross pattern is obtained. The patterns are a) there are equal number of '1's and '0's in each row and column; b) each row and each column are "palindromic"; c) the left- and right-halves are mirror images; and d) the top- and bottom-halves are mirror images (c and d imply b). The pattern table can be denoted using hexadecimals as (9, 6, 6, 9) for simplicity (1-nibble per row, 4 rows). The simplest method of generating the required pattern for higher ordered doubly even squares is to copy the generic pattern for the fourth-order square in each four-by-four sub-squares. For M = 8, possible choices for the pattern are (99, 66, 66, 99, 99, 66, 66, 99); (3C, 3C, C3, C3, C3, C3, 3C, 3C); (A5, 5A, A5, 5A, 5A, A5, 5A, A5) (2-nibbles per row, 8 rows). For M = 12, the pattern table (E07, E07, E07, 1F8, 1F8, 1F8, 1F8, 1F8, 1F8, E07, E07, E07) yields a magic square (3-nibbles per row, 12 rows.) It is possible to count the number of choices one has based on the pattern table, taking rotational symmetries into account. == Method of superposition == The earliest discovery of the superposition method was made by the Indian mathematician Narayana in the 14th century. The same method was later re-discovered and studied in early 18th century Europe by de la Loubere, Poignard, de La Hire, and Sauveur; and the method is usually referred to as de la Hire's method. Although Euler's work on magic square was unoriginal, he famously conjectured the impossibility of constructing the evenly odd ordered mutually orthogonal Graeco-Latin squares. This conjecture was disproved in the mid 20th century. For clarity of exposition, two important variations of this method can be distinguished. === Euler's method === This method consists in constructing two preliminary squares, which when added together gives the magic square. As a running example, a 3×3 magic square is considered. Each number of the 3×3 natural square by a pair of numbers can be labeled as where every pair of Greek and Latin alphabets, e.g. αa, are meant to be added together, i.e. αa = α + a. Here, (α, β, γ) = (0, 3, 6) and (a, b, c) = (1, 2, 3). The numbers 0, 3, and 6 are referred to as the root numbers while the numbers 1, 2, and 3 are referred to as the primary numbers. An important general constraint here is a Greek letter is paired with a Latin letter only once. Thus, the original square can now be split into two simpler squares: The lettered squares are referred to as Greek square or Latin square if they are filled with Greek or Latin letters, respectively. A magic square can be constructed by ensuring that the Greek and Latin squares are magic squares too. The converse of this statement is also often, but not always (e.g. bordered magic squares), true: A magic square can be decomposed into a Greek and a Latin square, which are themselves magic squares. Thus the method is useful for both synthesis as well as analysis of a magic square. Lastly, by examining the pattern in which the numbers are laid out in the finished square, it is often possible to come up with a faster algorithm to construct higher order squares that replicate the given pattern, without the necessity of creating the preliminary Greek and Latin squares. During the construction of the 3×3 magic square, the Greek and Latin squares with just three unique terms are much easier to deal with than the original square with nine different terms. The row sum and the column sum of the Greek square will be the same, α + β + γ, if each letter appears exactly once in a given column or a row. This can be achieved by cyclic permutation of α, β, and γ. Satisfaction of these two conditions ensures that the resulting square is a semi-magic square; and such Greek and Latin squares are said to be mutually orthogonal to each other. For a given order n, there are at most n − 1 squares in a set of mutually orthogonal squares, not counting the variations due to permutation of the symbols. This upper bound is exact when n is a prime number. In order to construct a magic square, we should also ensure that the diagonals sum to magic constant. For this, we have a third condition: either all the letters should appear exactly once in both the diagonals; or in case of odd ordered squares, one of the diagonals should consist entirely of the middle term, while the other diagonal should have all the letters exactly once. The mutually orthogonal Greek and Latin squares that satisfy the first part of the third condition (that all letters appear in both the diagonals) are said to be mutually orthogonal doubly diagonal Graeco-Latin squares. Odd squares: For the 3×3 odd square, since α, β, and γ are in arithmetic progression, their sum is equal to the product of the square's order and the middle term, i.e. α + β + γ = 3 β. Thus, the diagonal sums will be equal if we have βs in the main diagonal and α, β, γ in the skew diagonal. Similarly, for the Latin square. The resulting Greek and Latin squares and their combination will be as below. The Latin square is just a 90 degree anti-clockwise rotation of the Greek square (or equivalently, flipping about the vertical axis) with the corresponding letters interchanged. Substituting the values of the Greek and Latin letters will give the 3×3 magic square. For the odd squares, this method explains why the Siamese method (method of De la Loubere) and its variants work. This basic method can be used to construct odd ordered magic squares of higher orders. To summarise: For odd ordered squares, to construct Greek square, place the middle term along the main diagonal, and place the rest of the terms along the skew diagonal. The remaining empty cells are filled by diagonal moves. The Latin square can be constructed by rotating or flipping the Greek square, and replacing the corresponding alphabets. The magic square is obtained by adding the Greek and Latin squares. A peculiarity of the construction method given above for the odd magic squares is that the middle number (n2 + 1)/2 will always appear at the center cell of the magic square. Since there are (n − 1)! ways to arrange the skew diagonal terms, we can obtain (n − 1)! Greek squares this way; same with the Latin squares. Also, since each Greek square can be paired with (n − 1)! Latin squares, and since for each of Greek square the middle term may be arbitrarily placed in the main diagonal or the skew diagonal (and correspondingly along the skew diagonal or the main diagonal for the Latin squares), we can construct a total of 2 × (n − 1)! × (n − 1)! magic squares using this method. For n = 3, 5, and 7, this will give 8, 1152, and 1,036,800 different magic squares, respectively. Dividing by 8 to neglect equivalent squares due to rotation and reflections, we obtain 1, 144, and 129,600 essentially different magic squares, respectively. As another example, the construction of 5×5 magic square is given. Numbers are directly written in place of alphabets. The numbered squares are referred to as primary square or root square if they are filled with primary numbers or root numbers, respectively. The numbers are placed about the skew diagonal in the root square such that the middle column of the resulting root square has 0, 5, 10, 15, 20 (from bottom to top). The primary square is obtained by rotating the root square counter-clockwise by 90 degrees, and replacing the numbers. The resulting square is an associative magic square, in which every pair of numbers symmetrically opposite to the center sum up to the same value, 26. For e.g., 16+10, 3+23, 6+20, etc. In the finished square, 1 is placed at center cell of bottom row, and successive numbers are placed via elongated knight's move (two cells right, two cells down), or equivalently, bishop's move (two cells diagonally down right). When a collision occurs, the break move is to move one cell up. All the odd numbers occur inside the central diamond formed by 1, 5, 25 and 21, while the even numbers are placed at the corners. The occurrence of the even numbers can be deduced by copying the square to the adjacent sides. The even numbers from four adjacent squares will form a cross. A variation of the above example, where the skew diagonal sequence is taken in different order, is given below. The resulting magic square is the flipped version of the famous Agrippa's Mars magic square. It is an associative magic square and is the same as that produced by Moschopoulos's method. Here the resulting square starts with 1 placed in the cell which is to the right of the centre cell, and proceeds as De la Loubere's method, with downwards-right move. When a collision occurs, the break move is to shift two cells to the right. In the previous examples, for the Greek square, the second row can be obtained from the first row by circularly shifting it to the right by one cell. Similarly, the third row is a circularly shifted version of the second row by one cell to the right; and so on. Likewise, the rows of the Latin square is circularly shifted to the left by one cell. The row shifts for the Greek and Latin squares are in mutually opposite direction. It is possible to circularly shift the rows by more than one cell to create the Greek and Latin square. For odd ordered squares, whose order is not divisible by three, we can create the Greek squares by shifting a row by two places to the left or to the right to form the next row. The Latin square is made by flipping the Greek square along the main diagonal and interchanging the corresponding letters. This gives us a Latin square whose rows are created by shifting the row in the direction opposite to that of the Greek square. A Greek square and a Latin square should be paired such that their row shifts are in mutually opposite direction. The magic square is obtained by adding the Greek and Latin squares. When the order also happens to be a prime number, this method always creates pandiagonal magic square. This essentially re-creates the knight's move. All the letters will appear in both the diagonals, ensuring correct diagonal sum. Since there are n! permutations of the Greek letters by which we can create the first row of the Greek square, there are thus n! Greek squares that can be created by shifting the first row in one direction. Likewise, there are n! such Latin squares created by shifting the first row in the opposite direction. Since a Greek square can be combined with any Latin square with opposite row shifts, there are n! × n! such combinations. Lastly, since the Greek square can be created by shifting the rows either to the left or to the right, there are a total of 2 × n! × n! magic squares that can be formed by this method. For n = 5 and 7, since they are prime numbers, this method creates 28,800 and 50,803,200 pandiagonal magic squares. Dividing by 8 to neglect equivalent squares due to rotation and reflections, we obtain 3,600 and 6,350,400 equivalent squares. Further dividing by n2 to neglect equivalent panmagic squares due to cyclic shifting of rows or columns, we obtain 144 and 129,600 essentially different panmagic squares. For order 5 squares, these are the only panmagic square there are. The condition that the square's order not be divisible by 3 means that we cannot construct squares of orders 9, 15, 21, 27, and so on, by this method. In the example below, the square has been constructed such that 1 is at the center cell. In the finished square, the numbers can be continuously enumerated by the knight's move (two cells up, one cell right). When collision occurs, the break move is to move one cell up, one cell left. The resulting square is a pandiagonal magic square. This square also has a further diabolical property that any five cells in quincunx pattern formed by any odd sub-square, including wrap around, sum to the magic constant, 65. For e.g., 13+7+1+20+24, 23+1+9+15+17, 13+21+10+19+2 etc. Also the four corners of any 5×5 square and the central cell, as well as the middle cells of each side together with the central cell, including wrap around, give the magic sum: 13+10+19+22+1 and 20+24+12+8+1. Lastly the four rhomboids that form elongated crosses also give the magic sum: 23+1+9+24+8, 15+1+17+20+12, 14+1+18+13+19, 7+1+25+22+10. Such squares with 1 at the center cell are also called God's magic squares in Islamic amulet design, where the center cell is either left blank or filled with God's name. We can also combine the Greek and Latin squares constructed by different methods. In the example below, the primary square is made using knight's move. We have re-created the magic square obtained by De la Loubere's method. As before, we can form 8 × (n − 1)! × n! magic squares by this combination. For n = 5 and 7, this will create 23,040 and 29,030,400 magic squares. After dividing by 8 in order to neglect equivalent squares due to rotation and reflection, we get 2,880 and 3,628,800 squares. For order 5 squares, these three methods give a complete census of the number of magic squares that can be constructed by the method of superposition. Neglecting the rotation and reflections, the total number of magic squares of order 5 produced by the superposition method is 144 + 3,600 + 2,880 = 6,624. Even squares: We can also construct even ordered squares in this fashion. Since there is no middle term among the Greek and Latin alphabets for even ordered squares, in addition to the first two constraint, for the diagonal sums to yield the magic constant, all the letters in the alphabet should appear in the main diagonal and in the skew diagonal. An example of a 4×4 square is given below. For the given diagonal and skew diagonal in the Greek square, the rest of the cells can be filled using the condition that each letter appear only once in a row and a column. Using these two Graeco-Latin squares, we can construct 2 × 4! × 4! = 1,152 magic squares. Dividing by 8 to eliminate equivalent squares due to rotation and reflections, we get 144 essentially different magic squares of order 4. These are the only magic squares constructible by the Euler method, since there are only two mutually orthogonal doubly diagonal Graeco-Latin squares of order 4. Similarly, an 8×8 magic square can be constructed as below. Here the order of appearance of the numbers is not important; however the quadrants imitate the layout pattern of the 4×4 Graeco-Latin squares. Euler's method has given rise to the study of Graeco-Latin squares. Euler's method for constructing magic squares is valid for any order except 2 and 6. Variations: Magic squares constructed from mutually orthogonal doubly diagonal Graeco-Latin squares are interesting in themselves since the magic property emerges from the relative position of the alphabets in the square, and not due to any arithmetic property of the value assigned to them. This means that we can assign any value to the alphabets of such squares and still obtain a magic square. This is the basis for constructing squares that display some information (e.g. birthdays, years, etc.) in the square and for creating "reversible squares". For example, we can display the number π ≈ 3.141592 at the bottom row of a 4×4 magic square using the Graeco-Latin square given above by assigning (α, β, γ, δ) = (10, 0, 90, 15) and (a, b, c, d) = (0, 2, 3, 4). We will obtain the following non-normal magic square with the magic sum 124: === Narayana-De la Hire's method for even orders === Narayana-De la Hire's method for odd square is the same as that of Euler's. However, for even squares, we drop the second requirement that each Greek and Latin letter appear only once in a given row or column. This allows us to take advantage of the fact that the sum of an arithmetic progression with an even number of terms is equal to the sum of two opposite symmetric terms multiplied by half the total number of terms. Thus, when constructing the Greek or Latin squares, for even ordered squares, a letter can appear n/2 times in a column but only once in a row, or vice versa. As a running example, if we take a 4×4 square, where the Greek and Latin terms have the values (α, β, γ, δ) = (0, 4, 8, 12) and (a, b, c, d) = (1, 2, 3, 4), respectively, then we have α + β + γ + δ = 2 (α + δ) = 2 (β + γ). Similarly, a + b + c + d = 2 (a + d) = 2 (b + c). This means that the complementary pair α and δ (or β and γ) can appear twice in a column (or a row) and still give the desired magic sum. Thus, we can construct: For even ordered squares, the Greek magic square is made by first placing the Greek alphabets along the main diagonal in some order. The skew diagonal is then filled in the same order or by picking the terms that are complementary to the terms in the main diagonal. Finally, the remaining cells are filled column wise. Given a column, we use the complementary terms in the diagonal cells intersected by that column, making sure that they appear only once in a given row but n/2 times in the given column. The Latin square is obtained by flipping or rotating the Greek square and interchanging the corresponding alphabets. The final magic square is obtained by adding the Greek and Latin squares. In the example given below, the main diagonal (from top left to bottom right) is filled with sequence ordered as α, β, γ, δ, while the skew diagonal (from bottom left to top right) filled in the same order. The remaining cells are then filled column wise such that the complementary letters appears only once within a row, but twice within a column. In the first column, since α appears on the 1st and 4th row, the remaining cells are filled with its complementary term δ. Similarly, the empty cells in the 2nd column are filled with γ; in 3rd column β; and 4th column α. Each Greek letter appears only once along the rows, but twice along the columns. As such, the row sums are α + β + γ + δ while the column sums are either 2 (α + δ) or 2 (β + γ). Likewise for the Latin square, which is obtained by flipping the Greek square along the main diagonal and interchanging the corresponding letters. The above example explains why the "criss-cross" method for doubly even magic square works. Another possible 4×4 magic square, which is also pan-diagonal as well as most-perfect, is constructed below using the same rule. However, the diagonal sequence is chosen such that all four letters α, β, γ, δ appear inside the central 2×2 sub-square. Remaining cells are filled column wise such that each letter appears only once within a row. In the 1st column, the empty cells need to be filled with one of the letters selected from the complementary pair α and δ. Given the 1st column, the entry in the 2nd row can only be δ since α is already there in the 2nd row; while, in the 3rd row the entry can only be α since δ is already present in the 3rd row. We proceed similarly until all cells are filled. The Latin square given below has been obtained by flipping the Greek square along the main diagonal and replacing the Greek alphabets with corresponding Latin alphabets. We can use this approach to construct singly even magic squares as well. However, we have to be more careful in this case since the criteria of pairing the Greek and Latin alphabets uniquely is not automatically satisfied. Violation of this condition leads to some missing numbers in the final square, while duplicating others. Thus, here is an important proviso: For singly even squares, in the Greek square, check the cells of the columns which is vertically paired to its complement. In such a case, the corresponding cell of the Latin square must contain the same letter as its horizontally paired cell. Below is a construction of a 6×6 magic square, where the numbers are directly given, rather than the alphabets. The second square is constructed by flipping the first square along the main diagonal. Here in the first column of the root square the 3rd cell is paired with its complement in the 4th cells. Thus, in the primary square, the numbers in the 1st and 6th cell of the 3rd row are same. Likewise, with other columns and rows. In this example the flipped version of the root square satisfies this proviso. As another example of a 6×6 magic square constructed this way is given below. Here the diagonal entries are arranged differently. The primary square is constructed by flipping the root square about the main diagonal. In the second square the proviso for singly even square is not satisfied, leading to a non-normal magic square (third square) where the numbers 3, 13, 24, and 34 are duplicated while missing the numbers 4, 18, 19, and 33. The last condition is a bit arbitrary and may not always need to be invoked, as in this example, where in the root square each cell is vertically paired with its complement: As one more example, we have generated an 8×8 magic square. Unlike the criss-cross pattern of the earlier section for evenly even square, here we have a checkered pattern for the altered and unaltered cells. Also, in each quadrant the odd and even numbers appear in alternating columns. Variations: A number of variations of the basic idea are possible: a complementary pair can appear n/2 times or less in a column. That is, a column of a Greek square can be constructed using more than one complementary pair. This method allows us to imbue the magic square with far richer properties. The idea can also be extended to the diagonals too. An example of an 8×8 magic square is given below. In the finished square each of four quadrants are pan-magic squares as well, each quadrant with same magic constant 130. == Method of borders == === Bordering method for order 3 === In this method, the objective is to wrap a border around a smaller magic square which serves as a core. Consider the 3×3 square for example. Subtracting the middle number 5 from each number 1, 2, ..., 9, we obtain 0, ±1, ±2, ±3, and ±4, which we will, for lack of better words, following S. Harry White, refer to as bone numbers. The magic constant of a magic square, which we will refer to as the skeleton square, made by these bone numbers will be zero since adding all the rows of a magic square will give nM = Σ k = 0; thus M = 0. It is not difficult to argue that the middle number should be placed at the center cell: let x be the number placed in the middle cell, then the sum of the middle column, middle row, and the two diagonals give Σ k + 3 x = 4 M. Since Σ k = 3 M, we have x = M / 3. Here M = 0, so x = 0. Putting the middle number 0 in the center cell, we want to construct a border such that the resulting square is magic. Let the border be given by: Since the sum of each row, column, and diagonals must be a constant (which is zero), we have a + a∗ = 0, b + b∗ = 0, u + u∗ = 0, v + v∗ = 0. Now, if we have chosen a, b, u, and v, then we have a∗ = −a, b∗ = −b, u∗ = −u, and v∗ = −v. This means that if we assign a given number to a variable, say a = 1, then its complement will be assigned to a∗, i.e. a∗ = −1. Thus out of eight unknown variables, it is sufficient to specify the value of only four variables. We will consider a, b, u, and v as independent variables, while a∗, b∗, u∗, and v∗ as dependent variables. This allows us to consider a bone number ±x as a single number regardless of sign because (1) its assignment to a given variable, say a, will automatically imply that the same number of opposite sign will be shared with its complement a∗, and (2) two independent variables, say a and b, cannot be assigned the same bone number. But how should we choose a, b, u, and v? We have the sum of the top row and the sum of the right column as u + a + v = 0, v + b + u∗ = 0. Since 0 is an even number, there are only two ways that the sum of three integers will yield an even number: 1) if all three were even, or 2) if two were odd and one was even. Since in our choice of numbers we only have two even non-zero number (±2 and ±4), the first statement is false. Hence, it must be the case that the second statement is true: that two of the numbers are odd and one even. The only way that both the above two equations can satisfy this parity condition simultaneously, and still be consistent with the set of numbers we have, is when u and v are odd. For on the contrary, if we had assumed u and a to be odd and v to be even in the first equation, then u∗ = −u will be odd in the second equation, making b odd as well, in order to satisfy the parity condition. But this requires three odd numbers (u, a, and b), contradicting the fact that we only have two odd numbers (±1 and ±3) which we can use. This proves that the odd bone numbers occupy the corners cells. When converted to normal numbers by adding 5, this implies that the corners of a 3×3 magic square are all occupied by even numbers. Thus, taking u = 1 and v = 3, we have a = −4 and b = −2. Hence, the finished skeleton square will be as in the left. Adding 5 to each number, we get the finished magic square. Similar argument can be used to construct larger squares. Since there does not exist a 2×2 magic square around which we can wrap a border to construct a 4×4 magic square, the next smallest order for which we can construct bordered square is the order 5. === Bordering method for order 5 === Consider the fifth-order square. For this, we have a 3×3 magic core, around which we will wrap a magic border. The bone numbers to be used will be ±5, ±6, ±7, ±8, ±9, ±10, ±11, and ±12. Disregarding the signs, we have 8 bone numbers, 4 of which are even and 4 of which are odd. In general, for a square of any order n, there will be 4(n − 1) border cells, which are to be filled using 2(n − 1) bone numbers. Let the magic border be given as As before, we should place a bone number and its complement opposite to each other, so that the magic sum will be zero. It is sufficient to determine the numbers u, v, a, b, c, d, e, f to describe the magic border. As before, we have the two constraint equations for the top row and right column: u + a + b + c + v = 0 v + d + e + f + u* = 0. Multiple solutions are possible. The standard procedure is to first try to determine the corner cells, after which we will try to determine the rest of the border. There are 28 ways of choosing two numbers from the set of 8 bone numbers for the corner cells u and v. However, not all pairs are admissible. Among the 28 pairs, 16 pairs are made of an even and an odd number, 6 pairs have both as even numbers, while 6 pairs have them both as odd numbers. We can prove that the corner cells u and v cannot have an even and an odd number. This is because if this were so, then the sums u + v and v + u∗ will be odd, and since 0 is an even number, the sums a + b + c and d + e + f should be odd as well. The only way that the sum of three integers will result in an odd number is when 1) two of them are even and one is odd, or 2) when all three are odd. Since the corner cells are assumed to be odd and even, neither of these two statements are compatible with the fact that we only have 3 even and 3 odd bone numbers at our disposal. This proves that u and v cannot have different parity. This eliminates 16 possibilities. Using similar type reasoning we can also draw some conclusions about the sets {a, b, c} and {d, e, f}. If u and v are both even, then both the sets should have two odd numbers and one even number. If u and v are both odd, then one of the sets should have three even numbers while the other set should have one even number and two odd numbers. As a running example, consider the case when both u and v are even. The 6 possible pairs are: (6, 8), (6, 10), (6, 12), (8, 10), (8, 12), and (10, 12). Since the sums u + v and v + u∗ are even, the sums a + b + c and d + e + f should be even as well. The only way that the sum of three integers will result in an even number is when 1) two of them are odd and one is even, or 2) when all three are even. The fact that the two corner cells are even means that we have only 2 even numbers at our disposal. Thus, the second statement is not compatible with this fact. Hence, it must be the case that the first statement is true: two of the three numbers should be odd, while one be even. Now let a, b, d, e be odd numbers while c and f be even numbers. Given the odd bone numbers at our disposal: ±5, ±7, ±9, and ±11, their differences range from D = {±2, ±4, ±6} while their sums range from S = {±12, ±14, ±16, ±18, ±20}. It is also useful to have a table of their sum and differences for later reference. Now, given the corner cells (u, v), we can check its admissibility by checking if the sums u + v + c and v + u∗ + f fall within the set D or S. The admissibility of the corner numbers is a necessary but not a sufficient condition for the solution to exist. For example, if we consider the pair (u, v) = (8, 12), then u + v = 20 and v + u* = 6; and we will have ±6 and ±10 even bone numbers at our disposal. Taking c = ±6, we have the sum u + v + c to be 26 and 14, depending on the sign of ±6 taken, both of which do not fall within the sets D or S. Likewise, taking c = ±10, we have the sum u + v + c to be 30 and 10, both of which again do not fall within the sets D or S. Thus, the pair (8, 12) is not admissible. By similar process of reasoning, we can also rule out the pair (6, 12). As another example, if we consider the pair (u, v) = (10, 12), then u + v = 22 and v + u∗ = 2; and we will have ± 6 and ± 8 even bone numbers at our disposal. Taking c = ±6, we have the sum u + v + c to be 28 and 16. While 28 does not fall within the sets D or S, 16 falls in set S. By inspection, we find that if (a, b) = (−7, − 9), then a + b = −16; and it will satisfy the first constraint equation. Also, taking f = ± 8, we have the sum v + u∗ + f to be 10 and -6. While 10 does not fall within the sets D or S, −6 falls in set D. Since −7 and −9 have already been assigned to a and b, clearly (d, e) = (-5, 11) so that d + e = 6; and it will satisfy the second constraint equation. Likewise, taking c = ±8, we have the sum u + v + c to be 30 and 14. While 30 does not fall within the sets D or S, 14 falls in set S. By inspection, we find that if (a, b) = (−5, −9), then a + b = −14. Also, taking f = ± 6, we have the sum v + u∗ + f to be 8 and -4. While 8 does not fall within the sets D or S, −4 falls in set D. Clearly, (d, e) = (−7, 11) so that d + e = 4, and the second constraint equation will be satisfied. Hence the corner pair (u, v) = (10, 12) is admissible; and it admits two solutions: ⁠ ( a , b , c , d , e , f ) = ( − 7 , − 9 , − 6 , − 5 , 11 , − 8 ) {\displaystyle (a,b,c,d,e,f)=(-7,-9,-6,-5,11,-8)} ⁠ and ⁠ ( a , b , c , d , e , f ) = ( − 5 , − 9 , − 8 , − 7 , 11 , − 6 ) {\displaystyle (a,b,c,d,e,f)=(-5,-9,-8,-7,11,-6)} ⁠. The finished skeleton squares are given below. The magic square is obtained by adding 13 to each cells. Using similar process of reasoning, we can construct the following table for the values of u, v, a, b, c, d, e, f expressed as bone numbers as given below. There are only 6 possible choices for the corner cells, which leads to 10 possible border solutions. Given this group of 10 borders, we can construct 10×8×(3!)2 = 2880 essentially different bordered magic squares. Here the bone numbers ±5, ..., ±12 were consecutive. More bordered squares can be constructed if the numbers are not consecutive. If non-consecutive bone numbers were also used, then there are a total of 605 magic borders. Thus, the total number of order 5 essentially different bordered magic squares (with consecutive and non-consecutive numbers) is 174,240. See history. The number of fifth-order magic squares constructible via the bordering method is about 26 times larger than via the superposition method. === Continuous enumeration methods === Exhaustive enumeration of all the borders of a magic square of a given order, as done previously, is very tedious. As such a structured solution is often desirable, which allows us to construct a border for a square of any order. Below we give three algorithms for constructing border for odd, doubly even, and singly even squares. These continuous enumeration algorithms were discovered in 10th century by Arab scholars; and their earliest surviving exposition comes from the two treatises by al-Buzjani and al-Antaki, although they themselves were not the discoverers. Since then many more such algorithms have been discovered. Odd-ordered squares: The following is the algorithm given by al-Buzjani to construct a border for odd squares. A peculiarity of this method is that for order n square, the two adjacent corners are numbers ⁠ n − 1 {\displaystyle n-1} ⁠ and ⁠ n + 1 {\displaystyle n+1} ⁠. Starting from the cell above the lower left corner, we put the numbers alternately in left column and bottom row until we arrive at the middle cell. The next number is written in the middle cell of the bottom row just reached, after which we fill the cell in the upper left corner, then the middle cell of the right column, then the upper right corner. After this, starting from the cell above middle cell of the right column already filled, we resume the alternate placement of the numbers in the right column and the top row. Once half of the border cells are filled, the other half are filled by numbers complementary to opposite cells. The subsequent inner borders is filled in the same manner, until the square of order 3 is filled. Below is an example for 9th-order square. Doubly even order: The following is the method given by al-Antaki. Consider an empty border of order n = 4k with k ≥ 3. The peculiarity of this algorithm is that the adjacent corner cells are occupied by numbers n and ⁠ n − 1 {\displaystyle n-1} ⁠. Starting at the upper left corner cell, we put the successive numbers by groups of four, the first one next to the corner, the second and the third on the bottom, and the fourth at the top, and so on until there remains in the top row (excluding the corners) six empty cells. We then write the next two numbers above and the next four below. We then fill the upper corners, first left then right. We place the next number below the upper right corner in the right column, the next number on the other side in the left column. We then resume placing groups of four consecutive numbers in the two columns as before. Once half of the border cells are filled, the other half are filled by numbers complementary to opposite cells. The example below gives the border for order 16 square. For order 8 square, we just begin directly with the six cells. Singly even order: For singly even order, we have the algorithm given by al-Antaki. Here the corner cells are occupied by n and n − 1. Below is an example of 10th-order square. Start by placing 1 at the bottom row next to the left corner cell, then place 2 in the top row. After this, place 3 at the bottom row and turn around the border in anti-clockwise direction placing the next numbers, until n − 2 is reached on the right column. The next two numbers are placed in the upper corners (n − 1 in upper left corner and n in upper right corner). Then, the next two numbers are placed on the left column, then we resume the cyclic placement of the numbers until half of all the border cells are filled. Once half of the border cells are filled, the other half are filled by numbers complementary to opposite cells. == Method of composition == === For squares of order m × n where m, n > 2 === This is a method reminiscent of the Kronecker product of two matrices, that builds an nm × nm magic square from an n × n magic square and an m × m magic square. The "product" of two magic squares creates a magic square of higher order than the two multiplicands. Let the two magic squares be of orders m and n. The final square will be of order m × n. Divide the square of order m × n into m × m sub-squares, such that there are a total of n2 such sub-squares. In the square of order n, reduce by 1 the value of all the numbers. Multiply these reduced values by m2, and place the results in the corresponding sub-squares of the m × n whole square. The squares of order m are added n2 times to the sub-squares of the final square. The peculiarity of this construction method is that each magic subsquare will have different magic sums. The square made of such magic sums from each magic subsquare will again be a magic square. The smallest composite magic square of order 9, composed of two order 3 squares is given below. Since each of the 3×3 sub-squares can be independently rotated and reflected into 8 different squares, from this single 9×9 composite square we can derive 89 = 134,217,728 essentially different 9×9 composite squares. Plenty more composite magic squares can also be derived if we select non-consecutive numbers in the magic sub-squares, like in Yang Hui's version of the 9×9 composite magic square. The next smallest composite magic squares of order 12, composed of magic squares of order 3 and 4 are given below. For the base squares, there is only one essentially different 3rd order square, while there 880 essentially different 4th-order squares that we can choose from. Each pairing can produce two different composite squares. Since each magic sub-squares in each composite square can be expressed in 8 different forms due to rotations and reflections, there can be 1×880×89 + 880×1×816 ≈ 2.476×1017 essentially different 12×12 composite magic squares created this way, with consecutive numbers in each sub-square. In general, if there are cm and cn essentially different magic squares of order m and n, then we can form cm × cn × ( 8m2 + 8n2) composite squares of order mn, provided m ≠ n. If m = n, then we can form (cm)2 × 8m2 composite squares of order m2. === For squares of doubly even order === When the squares are of doubly even order, we can construct a composite magic square in a manner more elegant than the above process, in the sense that every magic subsquare will have the same magic constant. Let n be the order of the main square and m the order of the equal subsquares. The subsquares are filled one by one, in any order, with a continuous sequence of m2/2 smaller numbers (i.e. numbers less than or equal to n2/2) together with their complements to n2 + 1. Each subsquare as a whole will yield the same magic sum. The advantage of this type of composite square is that each subsquare is filled in the same way and their arrangement is arbitrary. Thus, the knowledge of a single construction of even order will suffice to fill the whole square. Furthermore, if the subsquares are filled in the natural sequence, then the resulting square will be pandiagonal. The magic sum of the subsquares is related to the magic sum of the whole square by M m = M n k {\displaystyle M_{m}={\frac {M_{n}}{k}}} where n = km. In the examples below, we have divided the order 12 square into nine subsquares of order 4 filled each with eight smaller numbers and, in the corresponding bishop's cells (two cells diagonally across, including wrap arounds, in the 4×4 subsquare), their complements to n2 + 1 = 145. Each subsquare is pandiagonal with magic constant 290; while the whole square on the left is also pandiagonal with magic constant 870. In another example below, we have divided the order 12 square into four order 6 squares. Each of the order 6 squares are filled with eighteen small numbers and their complements using bordering technique given by al-Antaki. If we remove the shaded borders of the order 6 subsquares and form an order 8 square, then this order 8 square is again a magic square. In its full generality, we can take any m2/2 smaller numbers together with their complements to n2 + 1 to fill the subsquares, not necessarily in continuous sequence. === Medjig-method for squares of even order 2n, where n > 2 === In this method a magic square is "multiplied" with a medjig square to create a larger magic square. The namesake of this method derives from mathematical game called medjig created by Willem Barink in 2006, although the method itself is much older. An early instance of a magic square constructed using this method occurs in Yang Hui's text for order 6 magic square. The LUX method to construct singly even magic squares is a special case of the medjig method, where only 3 out of 24 patterns are used to construct the medjig square. The pieces of the medjig puzzle are 2×2 squares on which the numbers 0, 1, 2 and 3 are placed. There are three basic patterns by which the numbers 0, 1, 2 and 3 can be placed in a 2×2 square, where 0 is at the top left corner: Each pattern can be reflected and rotated to obtain 8 equivalent patterns, giving us a total of 3×8 = 24 patterns. The aim of the puzzle is to take n2 medjig pieces and arrange them in an n × n medjig square in such a way that each row, column, along with the two long diagonals, formed by the medjig square sums to 3n, the magic constant of the medjig square. An n × n medjig square can create a 2n × 2n magic square where n > 2. Given an n×n medjig square and an n×n magic square base, a magic square of order 2n×2n can be constructed as follows: Each cell of an n×n magic square is associated with a corresponding 2×2 subsquare of the medjig square Fill each 2×2 subsquares of the medjig square with the four numbers from 1 to 4n2 that equal the original number modulo n2, i.e. x+n2y where x is the corresponding number from the magic square and y is a number from 0 to 3 in the 2×2 subsquares. Assuming that we have an initial magic square base, the challenge lies in constructing a medjig square. For reference, the sums of each medjig piece along the rows, columns and diagonals, denoted in italics, are: Doubly even squares: The smallest even ordered medjig square is of order 2 with magic constant 6. While it is possible to construct a 2×2 medjig square, we cannot construct a 4×4 magic square from it since 2×2 magic squares required to "multiply" it does not exist. Nevertheless, it is worth constructing these 2×2 medjig squares. The magic constant 6 can be partitioned into two parts in three ways as 6 = 5 + 1 = 4 + 2 = 3 + 3. There exist 96 such 2×2 medjig squares. In the examples below, each 2×2 medjig square is made by combining different orientations of a single medjig piece. We can use the 2×2 medjig squares to construct larger even ordered medjig squares. One possible approach is to simply combine the 2×2 medjig squares together. Another possibility is to wrap a smaller medjig square core with a medjig border. The pieces of a 2×2 medjig square can form the corner pieces of the border. Yet another possibility is to append a row and a column to an odd ordered medjig square. An example of an 8×8 magic square is constructed below by combining four copies of the left most 2×2 medjig square given above: The next example is constructed by bordering a 2×2 medjig square core. Singly even squares: Medjig square of order 1 does not exist. As such, the smallest odd ordered medjig square is of order 3, with magic constant 9. There are only 7 ways of partitioning the integer 9, our magic constant, into three parts. If these three parts correspond to three of the medjig pieces in a row, column or diagonal, then the relevant partitions for us are: 9 = 1 + 3 + 5 = 1 + 4 + 4 = 2 + 3 + 4 = 2 + 2 + 5 = 3 + 3 + 3. A 3×3 medjig square can be constructed with some trial-and-error, as in the left most square below. Another approach is to add a row and a column to a 2×2 medjig square. In the middle square below, a left column and bottom row has been added, creating an L-shaped medjig border, to a 2×2 medjig square given previously. The right most square below is essentially same as the middle square, except that the row and column has been added in the middle to form a cross while the pieces of 2×2 medjig square are placed at the corners. Once a 3×3 medjig square has been constructed, it can be converted into a 6×6 magic square. For example, using the left most 3×3 medjig square given above: There are 1,740,800 such 3×3 medjig squares. An easy approach to construct higher order odd medjig square is by wrapping a smaller odd ordered medjig square with a medjig border, just as with even ordered medjig squares. Another approach is to append a row and a column to an even ordered medjig square. Approaches such as the LUX method can also be used. In the example below, a 5×5 medjig square is created by wrapping a medjig border around a 3×3 medjig square given previously: == Solving partially completed magic squares == Solving partially completed magic squares is a popular mathematical pastime. The techniques needed are similar to those used in Sudoku or KenKen puzzles, and involve deducing the values of unfilled squares using logic and permutation group theory (Sudoku grids are not magic squares but are based on a related idea called Graeco-Latin squares). == Variations of the magic square == === Extra constraints === Certain extra restrictions can be imposed on magic squares. If raising each number to the nth power yields another magic square, the result is a bimagic (n = 2), a trimagic (n = 3), or, in general, a multimagic square. A magic square in which the number of letters in the name of each number in the square generates another magic square is called an alphamagic square. There are magic squares consisting entirely of primes. Rudolf Ondrejka (1928–2001) discovered the following 3×3 magic square of primes, in this case nine Chen primes: The Green–Tao theorem implies that there are arbitrarily large magic squares consisting of primes. The following "reversible magic square" has a magic constant of 264 both upside down and right way up: When the extra constraint is to display some date, especially a birth date, then such magic squares are called birthday magic square. An early instance of such birthday magic square was created by Srinivasa Ramanujan. He created a 4×4 square in which he entered his date of birth in D–M–C-Y format in the top row and the magic happened with additions and subtractions of numbers in squares. Not only do the rows, columns, and diagonals add up to the same number, but the four corners, the four middle squares (17, 9, 24, 89), the first and last rows two middle numbers (12, 18, 86, 23), and the first and last columns two middle numbers (88, 10, 25, 16) all add up to the sum of 139. === Multiplicative magic squares === Instead of adding the numbers in each row, column and diagonal, one can apply some other operation. For example, a multiplicative magic square has a constant product of numbers. A multiplicative magic square can be derived from an additive magic square by raising 2 (or any other integer) to the power of each element, because the logarithm of the product of 2 numbers is the sum of logarithm of each. Alternatively, if any 3 numbers in a line are 2a, 2b and 2c, their product is 2a+b+c, which is constant if a+b+c is constant, as they would be if a, b and c were taken from ordinary (additive) magic square. For example, the original Lo-Shu magic square becomes: Other examples of multiplicative magic squares include: === Multiplicative magic squares of complex numbers === Still using Ali Skalli's non iterative method, it is possible to produce an infinity of multiplicative magic squares of complex numbers belonging to C {\displaystyle \mathbb {C} } set. On the example below, the real and imaginary parts are integer numbers, but they can also belong to the entire set of real numbers R {\displaystyle \mathbb {R} } . The product is: −352,507,340,640 − 400,599,719,520 i. === Additive-multiplicative magic and semimagic squares === Additive-multiplicative magic squares and semimagic squares satisfy properties of both ordinary and multiplicative magic squares and semimagic squares, respectively. It is unknown if any additive-multiplicative magic squares smaller than 7×7 exist, but it has been proven that no 3×3 or 4×4 additive-multiplicative magic squares and no 3×3 additive-multiplicative semimagic squares exist. === Geometric magic squares === Magic squares may be constructed which contain geometric shapes instead of numbers. Such squares, known as geometric magic squares, were invented and named by Lee Sallows in 2001. In the example shown the shapes appearing are two dimensional. It was Sallows' discovery that all magic squares are geometric, the numbers that appear in numerical magic squares can be interpreted as a shorthand notation which indicates the lengths of straight line segments that are the geometric 'shapes' occurring in the square. That is, numerical magic squares are that special case of a geometric magic square using one dimensional shapes. === Area magic squares === In 2017, following initial ideas of William Walkington and Inder Taneja, the first linear area magic square (L-AMS) was constructed by Walter Trump. === Other magic shapes === Other two dimensional shapes than squares can be considered. The general case is to consider a design with N parts to be magic if the N parts are labeled with the numbers 1 through N and a number of identical sub-designs give the same sum. Examples include magic circles, magic rectangles, magic triangles magic stars, magic hexagons, magic diamonds. Going up in dimension results in magic spheres, magic cylinders, magic cubes, magic parallelepiped, magic solids, and other magic hypercubes. Possible magic shapes are constrained by the number of equal-sized, equal-sum subsets of the chosen set of labels. For example, if one proposes to form a magic shape labeling the parts with {1, 2, 3, 4}, the sub-designs will have to be labeled with {1,4} and {2,3}. == Related problems == === n-Queens problem === In 1992, Demirörs, Rafraf, and Tanik published a method for converting some magic squares into n-queens solutions, and vice versa. == Magic squares in occultism == Magic squares of order 3 through 9, assigned to the seven planets, and described as means to attract the influence of planets and their angels (or demons) during magical practices, can be found in several manuscripts all around Europe starting at least since the 15th century. Among the best known, the Liber de Angelis, a magical handbook written around 1440, is included in Cambridge Univ. Lib. MS Dd.xi.45. The text of the Liber de Angelis is very close to that of De septem quadraturis planetarum seu quadrati magici, another handbook of planetary image magic contained in the Codex 793 of the Biblioteka Jagiellońska (Ms BJ 793). The magical operations involve engraving the appropriate square on a plate made with the metal assigned to the corresponding planet, as well as performing a variety of rituals. For instance, the 3×3 square, that belongs to Saturn, has to be inscribed on a lead plate. It will, in particular, help women during a difficult childbirth. In about 1510 Heinrich Cornelius Agrippa wrote De Occulta Philosophia, drawing on the Hermetic and magical works of Marsilio Ficino and Pico della Mirandola. In its 1531 edition, he expounded on the magical virtues of the seven magical squares of orders 3 to 9, each associated with one of the astrological planets, much in the same way as the older texts did. This book was very influential throughout Europe until the Counter-Reformation, and Agrippa's magic squares, sometimes called kameas, continue to be used within modern ceremonial magic in much the same way as he first prescribed. The most common use for these kameas is to provide a pattern upon which to construct the sigils of spirits, angels or demons; the letters of the entity's name are converted into numbers, and lines are traced through the pattern that these successive numbers make on the kamea. In a magical context, the term magic square is also applied to a variety of word squares or number squares found in magical grimoires, including some that do not follow any obvious pattern, and even those with differing numbers of rows and columns. They are generally intended for use as talismans. For instance the following squares are: The Sator square, one of the most famous magic squares found in a number of grimoires including the Key of Solomon; a square "to overcome envy", from The Book of Power; and two squares from The Book of the Sacred Magic of Abramelin the Mage, the first to cause the illusion of a superb palace to appear, and the second to be worn on the head of a child during an angelic invocation: == Magic squares in popular culture == In Goethe's Faust, the witch's spell used to make a youth elixir for Faust, the Hexen-Einmal-Eins, has been interpreted as a construction of a magic square. The English composer Peter Maxwell Davies has used magic squares to structure many of his compositions. For example, his 1975 Ave Maris Stella uses the 9×9 magic square of Moon while his 1977 A Mirror of Whitening Light uses the 8×8 magic square of Mercury to create the entire set of notes and durations for the piece. His other works that employ magic squares include The Lighthouse (1979), Resurrection (1987), Strathclyde Concerto No. 3 for Horn and Trumpet (1989), as well as many of his symphonies. According to Davies' own account: A magic square in a musical composition is not a block of numbers – it is a generating principle, to be learned and known intimately, perceived inwardly as a multi-dimensional projection into that vast (chaotic!) area of the internal ear – the space/time crucible – where music is conceived. ... Projected onto the page, a magic square is a dead, black conglomeration of digits; tune in, and one hears a powerful, orbiting dynamo of musical images, glowing with numen and lumen. Magic squares, including Benjamin Franklin's, appear as clues to the mystery in Katherine Neville's novels The Eight and The Fire. Magic squares play a role in Steve Martin's 2003 novel The Pleasure of My Company. Dürer's magic square and his Melencolia I both also played large roles in Dan Brown's 2009 novel, The Lost Symbol. In the 2011 Korean television drama Deep Rooted Tree, King Sejong is shown attempting to construct a 33×33 magic square using lunch boxes. He ultimately discovers the "pyramid method" and completes the magic square with the help of an army of court attendants. This inspires him to create a more just form of government ruled by reason and words rather than military might. On October 9, 2014, the post office of Macao in the People's Republic of China issued a series of stamps based on magic squares. The figure below shows the stamps featuring the nine magic squares chosen to be in this collection. The metallic artifact at the center of The X-Files episode "Biogenesis" is alleged by Chuck Burks to be a magic square. Mathematician Matt Parker attempted to create a 3×3 magic square using square numbers in a YouTube video on the Numberphile channel. His failed attempt is known as the Parker square. The first season Stargate Atlantis episode "Brotherhood" involves completing a magic square as part of a puzzle guarding a powerful Ancient artefact. Magic Squares are also featured in the 2019 Spanish film Vivir dos veces. == See also == == Notes == == References == == Further reading ==
Wikipedia:Magnhild Lien#0	Magnhild Lien is a Norwegian mathematician specializing in knot theory. She is a professor emeritus of mathematics at California State University, Northridge, and the former executive director of the Association for Women in Mathematics. == Education and career == Lien was born in Arendal, a town on the southern Norwegian island of Tromøya, where she grew up as the youngest of six children and the only girl in her family. She earned a bachelor's degree from McGill University in 1979, and completed her Ph.D. in 1984 from the University of Iowa. Her dissertation, Construction of High Dimensional Knot Groups from Classical Knot Groups, was supervised by Jonathan Kalman Simon. She joined the faculty at California State University, Northridge in 1987, and served as department chair of mathematics there from 1998 to 2006. She was executive director of the Association for Women in Mathematics from 2012 to 2018. == Contributions == As well as publishing her own mathematical research on knot theory, Lien has written about women in mathematics in collaboration with her husband, sociologist Harvey Rich. == Recognition == Lien was included in the 2019 class of fellows of the Association for Women in Mathematics "for extraordinary leadership and service devoted to advancing and supporting women in the mathematical sciences, as AWM executive director and, for a quarter century, as initiator, director, and fundraiser of programs for women". == References ==
Wikipedia:Maharam algebra#0	In mathematics, a Maharam algebra is a complete Boolean algebra with a continuous submeasure (defined below). They were introduced by Dorothy Maharam (1947). == Definitions == A continuous submeasure or Maharam submeasure on a Boolean algebra is a real-valued function m such that m ( 0 ) = 0 , m ( 1 ) = 1 , {\displaystyle m(0)=0,m(1)=1,} and m ( x ) > 0 {\displaystyle m(x)>0} if x ≠ 0 {\displaystyle x\neq 0} . If x ≤ y {\displaystyle x\leq y} , then m ( x ) ≤ m ( y ) {\displaystyle m(x)\leq m(y)} . m ( x ∨ y ) ≤ m ( x ) + m ( y ) − m ( x ∧ y ) {\displaystyle m(x\vee y)\leq m(x)+m(y)-m(x\wedge y)} . If x n {\displaystyle x_{n}} is a decreasing sequence with greatest lower bound 0, then the sequence m ( x n ) {\displaystyle m(x_{n})} has limit 0. A Maharam algebra is a complete Boolean algebra with a continuous submeasure. == Examples == Every probability measure is a continuous submeasure, so as the corresponding Boolean algebra of measurable sets modulo measure zero sets is complete, it is a Maharam algebra. Michel Talagrand (2008) solved a long-standing problem by constructing a Maharam algebra that is not a measure algebra, i.e., that does not admit any countably additive strictly positive finite measure. == References == Balcar, Bohuslav; Jech, Thomas (2006), "Weak distributivity, a problem of von Neumann and the mystery of measurability", Bulletin of Symbolic Logic, 12 (2): 241–266, doi:10.2178/bsl/1146620061, MR 2223923, Zbl 1120.03028 Maharam, Dorothy (1947), "An algebraic characterization of measure algebras", Annals of Mathematics, Second Series, 48 (1): 154–167, doi:10.2307/1969222, JSTOR 1969222, MR 0018718, Zbl 0029.20401 Talagrand, Michel (2008), "Maharam's Problem", Annals of Mathematics, Second Series, 168 (3): 981–1009, doi:10.4007/annals.2008.168.981, JSTOR 40345433, MR 2456888, Zbl 1185.28002 Velickovic, Boban (2005), "CCC forcing and splitting reals", Israel Journal of Mathematics, 147: 209–220, doi:10.1007/BF02785365, MR 2166361, Zbl 1118.03046
Wikipedia:Mahler's theorem#0	In mathematics, Mahler's theorem, introduced by Kurt Mahler (1958), expresses any continuous p-adic function as an infinite series of certain special polynomials. It is the p-adic counterpart to the Stone-Weierstrass theorem for continuous real-valued functions on a closed interval. == Statement == Let ( Δ f ) ( x ) = f ( x + 1 ) − f ( x ) {\displaystyle (\Delta f)(x)=f(x+1)-f(x)} be the forward difference operator. Then for any p-adic function f : Z p → Q p {\displaystyle f:\mathbb {Z} _{p}\to \mathbb {Q} _{p}} , Mahler's theorem states that f {\displaystyle f} is continuous if and only if its Newton series converges everywhere to f {\displaystyle f} , so that for all x ∈ Z p {\displaystyle x\in \mathbb {Z} _{p}} we have f ( x ) = ∑ n = 0 ∞ ( Δ n f ) ( 0 ) ( x n ) , {\displaystyle f(x)=\sum _{n=0}^{\infty }(\Delta ^{n}f)(0){x \choose n},} where ( x n ) = x ( x − 1 ) ( x − 2 ) ⋯ ( x − n + 1 ) n ! {\displaystyle {x \choose n}={\frac {x(x-1)(x-2)\cdots (x-n+1)}{n!}}} is the n {\displaystyle n} th binomial coefficient polynomial. Here, the n {\displaystyle n} th forward difference is computed by the binomial transform, so that ( Δ n f ) ( 0 ) = ∑ k = 0 n ( − 1 ) n − k ( n k ) f ( k ) . {\displaystyle (\Delta ^{n}f)(0)=\sum _{k=0}^{n}(-1)^{n-k}{\binom {n}{k}}f(k).} Moreover, we have that f {\displaystyle f} is continuous if and only if the coefficients ( Δ n f ) ( 0 ) → 0 {\displaystyle (\Delta ^{n}f)(0)\to 0} in Q p {\displaystyle \mathbb {Q} _{p}} as n → ∞ {\displaystyle n\to \infty } . It is remarkable that as weak an assumption as continuity is enough in the p-adic setting to establish convergence of Newton series. By contrast, Newton series on the field of complex numbers are far more tightly constrained, and require Carlson's theorem to hold. == References == Mahler, K. (1958), "An interpolation series for continuous functions of a p-adic variable", Journal für die reine und angewandte Mathematik, 1958 (199): 23–34, doi:10.1515/crll.1958.199.23, ISSN 0075-4102, MR 0095821, S2CID 199546556
Wikipedia:Mahlon Marsh Day#0	Mahlon Marsh Day (1913–1992) was an American mathematician, who specialized in functional analysis, geometry of linear spaces and amenable semigroups. == Career == In 1939 he graduated from Brown University. He became a member of the Institute for Advanced Study in the years 1939-40 and later in 1948–49. In most of his career, between the years 1940–83, he was a professor of mathematics in University of Illinois at Urbana-Champaign. In June 1983, a conference named "the Geometry of Normed Linear Spaces" was held in honor of Day at the University of Illinois at Urbana-Champaign. A proceedings issue to the conference was published in Contemporary Mathematics. In the preface for this proceedings issue, Day was described as "the first American mathematician to study normed spaces from a geometric standpoint". His monograph "Normed Linear Spaces" from 1973 is highly cited and considered to be classical in the field. In the field of amenable semigroups, his work under this name, is highly cited and considered fundamental to the field. He served as an editor of Illinois Journal of Mathematics in the years 1968-73 and 1981–85. == Selected publications == Day, M. M. (1957). Amenable semigroups. Illinois Journal of Mathematics, 1(4), 509–544. Day, M. M. (1973). Normed linear spaces. In Normed Linear Spaces (pp. 27–52). Springer, Berlin, Heidelberg. R. G. Bartle, N. T. Peck, A. L. Peressini, J. J. Uhl (Editors). Geometry of Normed Linear Spaces. Contemporary Mathematics. Volume: 52; 1986; 171 pp. Book front matter. == References == == External links == Mahlon M. Day Papers, 1934-93, University of Illinois Archives Profile at Institute for Advanced Study "Mahlon Marsh Day". Institute for Advanced Study. 9 December 2019.
Wikipedia:Mahmud Salohiddinov#0	Mahmud Salohiddinovich Salohiddinov (Uzbek Cyrillic: Маҳмуд Салоҳиддинович Салоҳиддинов, Russian: Махмуд Салахитдинович Салахитдинов, 23 November 1933 — 27 April 2018) was a Soviet-Uzbek mathematician, academic, and politician. From 1988 to 1994 he was the head of the Academy of Sciences of Uzbekistan. == Biography == He was born in Namangan on 23 November 1993. His father was the director of a local canning factory, and his mother was a housewife. He had three siblings. In 1958 he graduated from the Mathematics Department of Central Asia State University. In 1959 he began working at the V.I. Romanovsky Institute of Mathematics as a junior research fellow, and he became the director of it 1967. He was admitted to the Communist Party in 1962. He became a corresponding member of the Academy of Sciences of the Uzbek SSR in 1968 and a full member in 1974. From 1985 to 1988, he was the minister of higher and secondary specialized education of the Uzbek SSR. In 1988 he became the head of the Academy of Sciences of the Uzbek SSR, and he continued to serve in the position after Uzbekistan gained independence until 1994. He died on 27 April 2018 at the age of 85. == Awards == Biruni State Prize (1974) Order of the Badge of Honor (1976) Order of Outstanding Merit (27 August 2007) Order of Labor Glory Order "For selfless services" == References ==
Wikipedia:Mahmut Bajraktarević#0	Mahmut Bajraktarević (22 December 1909 in Sarajevo – 13 April 1985 in Bugojno) was a Bosnian mathematician and academician. He graduated from the University of Belgrade in 1933 and received his doctorate from the Sorbonne in 1953 with the dissertation Sur certaines suites itérées. Bajraktarević was a professor at the University of Sarajevo and had a great influence on the development of mathematics in Bosnia and Herzegovina. He contributed to the research areas of functional equations, iterative sequences and summability theory. == References == == Further reading == Maravić, Manojlo; Perić, Veselin; Vajzović, Fikret (1980), "Life and work of Academician Mahmut Bajraktarević", Akademija Nauka i Umjetnosti Bosne i Hercegovine (19): 5–17, MR 0590293.
Wikipedia:Mahyar Amouzegar#0	Mahyar A. Amouzegar is an Iranian-American mathematician, engineer, policy analyst, author, and academic. He is the 18th President of New Mexico Tech. Amouzegar research encompasses modeling and simulation, optimization, logistics and supply chain management, organizational studies and national security policy analysis. Amouzegar is a Fellow of the Institute of Mathematics and Its Applications, and Institute of Combinatorics and Its Applications. He served as Editor-in-Chief for the Journal of Applied Mathematics and Decision Sciences and is an Associate Editor for the International Journal of Applied Decision Sciences. == Early life == Born in Iran to a bookkeeper, Amouzegar grew up surrounded by literature. Each night, his father brought home a new book from the publishing house where he worked, filling their home with a variety of works. At age fourteen, in 1978, he left Tehran to live with his older sisters in San Francisco. Due to the political unrest leading to the Iranian Revolution, his parents were unable to join him in the United States for over five years. During this period, he developed a writing habit, balancing his interest in literature with his aptitude for math. == Education == Amouzegar pursued his higher education, earning a Bachelor of Science in Applied Mathematics from San Francisco State University in 1983. He then proceeded to obtain a Master of Science in Electrical Engineering at the University of California, Los Angeles (UCLA) in 1989, concurrently completing an Engineering Degree. During his master's studies, he focused on operations research to address policy issues later completing his doctoral studies in Operations Research at UCLA, with a specialization in Nonconvex Optimization. == Career == After obtaining his PhD, Amouzegar began his career as an Assistant Professor (Lecturer) at Massey University in 1995, teaching operations research and developing models and algorithms for nonconvex optimization problems. He then moved to California State University, Long Beach, serving as Associate Dean for Research and Graduate Studies from 2005 to 2011. From 2003 to 2006, he was a Senior Honorary Visiting Fellow at Cass Business School, City University of London, and later served as President of the Western Decision Sciences Institute. Afterward, he was the Dean of Engineering and Professor of Systems Engineering at California State Polytechnic University-Pomona from 2011 to 2017. Concurrently, he was a Senior National Security Policy Analyst for the RAND Corporation from 1998 to 2023. At the University of New Orleans, he served as Provost and Senior Vice President for Academic Affairs from 2017 to 2024. He was appointed by the Governor to the Louisiana STEM Council in 2017 and served as the Freeport McMoRan Distinguished Professor. Since 2019, Amouzegar has held the Hancock-Whitney Distinguished Professorship of Finance while also serving as Vice President for International Relations at Alpha Iota Delta and as a member of the Jefferson Chamber Education and Workforce Development Committee. In 2024, he became President of the New Mexico Institute of Mining and Technology. == Creative writing == Amouzegar's writing is influenced by his love of old movies, particularly 1940s screwball comedies and film noir, as well as novels like The Godfather. He developed an appreciation for Ernest Hemingway and Graham Greene, often writing to the music of operas like “Aida”. His first two novels, A Dark Sunny Afternoon and Pisgah Road, explored the lives of two men navigating past love and drama. Hanya Yanagihara of Kirkus Reviews praised "Pisgah Road" saying the "plot sing with nostalgia and regret, beautifully capturing the narrator's struggles with his own vulnerability. The author's touch is light as his characters deny their feelings, to themselves and to one another, until the right circumstances finally allow them to speak… A capable portrait of grief, longing, and second chances." Among other works, he delved into the complexities of intimate human relationships through the reminiscences of protagonist Donte, offering an exploration of how personal history shapes identity in Dinner At 10:32. His 2021 novel The Hubris of an Empty Hand presented eight stories on the themes of love, sacrifice, and the desire for remembrance and recognition, receiving numerous reviews. == Research == Amouzegar's research has centered on optimization techniques and their application in solving complex problems across various domains, ranging from environmental management to military logistics and policy analysis. === Bilevel optimization === Amouzegar has conducted research on linear bilevel programming and environmental optimization offering methodologies and strategic decision models. He introduced a method for generating test problems for linear bilevel programming, systematically selecting a distant vertex from the solution of the relaxed linear programming problem. His collaboration with Khosrow Moshirvaziri led to the development of two optimization models for hazardous waste capacity planning and treatment facility locations. Building upon this, he addressed environmental challenges through bilevel programming, particularly focusing on the solid waste management deficit in California's San Francisco Bay Area. === Efficiency in combat and air support === Amouzegar's research efforts have provided insights into the development of agile combat support systems. While examining Agile Combat Support (ACS) and mobility system design and evaluation, he alongside other researchers addressed trade-offs, resource requirements, and support options across scenarios, to establish a robust planning framework for future ACS. He then utilized an optimization model to evaluate different global forward support location (FSL) options for storing war reserve materiel, balancing land and sea-based FSLs to minimize peacetime costs while ensuring sufficient support for training. In related research, his publication, "Evaluation of Options for Overseas Combat Support Basing," presented an analytic framework and model for assessing overseas combat support basing options to ensure swift and efficient support for US forces in diverse deployment scenarios. He was also part of a 2015 technical report at the Rand Corporation which delved into the challenges faced by the US Air Force in maintaining secure air bases in the Pacific region due to potential threats from near-peer powers highlighting significant impacts on combat support and logistics requirements. == Awards and honors == 1998 – Fellow, Institute of Combinatorics and Its Applications 2000 – Fellow, Institute of Mathematics and Its Applications == Bibliography == === Selected books === A Dark Sunny Afternoon (2016) ISBN 978-1628681611 Pisgah Road (2017) ISBN 978-1628682069 Dinner at 10:32 (2020) ISBN 978-1608011810 The Hubris of an Empty Hand (2021) ISBN 978-1608012213 === Selected articles === Amouzegar, M. A. (1999). A global optimization method for nonlinear bilevel programming problems. IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), 29(6), 771-777. Amouzegar, M. A., & Moshirvaziri, K. (2001). Strategic management decision support system: An analysis of the environmental policy issues. Environmental Modeling & Assessment, 6, 297-306. Amouzegar, M. et. al. (2010). A simulation model for the analysis of end-to-end support of unmanned aerial vehicles. International Journal of Applied Decision Sciences (IJADS), Volume 3(2). Amouzegar, M., & Moshirvaziri K. (2011). A deep cutting plane technique for reverse convex optimization. IEEE Transactions, Systems, Man, Cybernetics B, Volume 41(4), 2011 Amouzegar, M. A., Defence Practice: Military Logistics in Routledge Handbook of Defence Studies, D. J. Galbreath and J. Deni, Editors. Taylor & Francis Group Publishing, U.K., 2018 Amouzegar, M. A., Bodine-Baron, E., & Snyder, D. (2019). Cyber mission thread analysis, a prototype framework for assessing impact to missions from cyber attacks to weapon systems, Rand. == References ==
Wikipedia:Mai Gehrke#0	Mai Gehrke (born 10 May 1964) is a Danish mathematician who studies the theory of lattices and their applications to mathematical logic and theoretical computer science. She is a director of research for the French Centre national de la recherche scientifique (CNRS), affiliated with the Laboratoire J. A. Dieudonné (LJAD) at the University of Nice Sophia Antipolis. == Education == As a child, Gehrke was educated at a French school in Algiers, which used a Bourbakist and very abstract mathematics curriculum. As a high school student in Denmark, she spent a year as an exchange student in Houston studying painting, but was brought back to mathematics by a Polish mathematics teacher who taught her point-set topology according to the Moore method. She earned her Ph.D. from the University of Houston in 1987. Her dissertation, Order Structure of Stone Spaces and the TD-axiom, was supervised by Klaus Hermann Kaiser. == Career == After postdoctoral study at Vanderbilt University, Gehrke joined the faculty of New Mexico State University in 1990. She moved to Radboud University Nijmegen in 2007, and to CNRS in 2011. From 2011 to 2017 her work for CNRS was associated with the Laboratoire d'Informatique Algorithmique: Fondements et Applications (LIAFA) at Paris Diderot University; in 2017 she moved to LJAD in Sophia Antipolis. == References == == External links == Home page Mai Gehrke publications indexed by Google Scholar
Wikipedia:Majid Rasulov#0	Majid Latif Rasulov (Azerbaijani: Məcid Lətif oğlu Rəsulov; 6 July 1916, Nukha – 11 February 1993, Baku) was a Soviet and Azerbaijani mathematician, academician (1983), physics-mathematics PhD (1960). == Achievements == Majid Rasulov graduated from Azerbaijan State University (Baku State University), which was postgraduate. He was a prominent specialist about mathematical-physics equations and worked at functional analysis sections. His researches are divided into 4 directions: First direction consist of equation which is differential equation which gives special solution of Koshi issues, contour integral. At spectral theory works is concerned second direction. There he proved new formulas for differential equation, contour integral. In third direction-To keep linear functional norm which is determined in Banach space, its continuation was proved by solitariness condition. Lastly in fourth direction, normality condition of linear differential operators was extracted. In 1960, Rasulov received the Doctor of Sciences degree in the Scientific Council of Mathematics University. "Method of contour integration" monograph was written by Majid Rasulov and it was published in 1964 in Moscow. Prof. A.I. Ivanov was editor of monograph. He wrote in this review: "M. Rasulov's monograph is exceptional event. There isn't like as this book in Earth press." In 1967 "Method of contour integration" monograph was translated to English language by order of England Mathematics Society and it was published in Canada, USA, Netherlands. == Awards and honors == Medal "For the Defence of the Caucasus" (1944) Medal "For the Victory over Germany in the Great Patriotic War 1941–1945" (1945) Jubilee Medal "In Commemoration of the 100th Anniversary of the Birth of Vladimir Ilyich Lenin" (1970) Medal "Veteran of Labour" (1979) Order of the Red Banner of Labour (1980) Order of the Patriotic War, 2nd class (1985) == References ==
Wikipedia:Majorization#0	In mathematics, majorization is a preorder on vectors of real numbers. For two such vectors, x , y ∈ R n {\displaystyle \mathbf {x} ,\ \mathbf {y} \in \mathbb {R} ^{n}} , we say that x {\displaystyle \mathbf {x} } weakly majorizes (or dominates) y {\displaystyle \mathbf {y} } from below, commonly denoted x ≻ w y , {\displaystyle \mathbf {x} \succ _{w}\mathbf {y} ,} when ∑ i = 1 k x i ↓ ≥ ∑ i = 1 k y i ↓ {\displaystyle \sum _{i=1}^{k}x_{i}^{\downarrow }\geq \sum _{i=1}^{k}y_{i}^{\downarrow }} for all k = 1 , … , n {\displaystyle k=1,\,\dots ,\,n} , where x i ↓ {\displaystyle x_{i}^{\downarrow }} denotes i {\displaystyle i} th largest entry of x {\displaystyle x} . If x , y {\displaystyle \mathbf {x} ,\mathbf {y} } further satisfy ∑ i = 1 n x i = ∑ i = 1 n y i {\displaystyle \sum _{i=1}^{n}x_{i}=\sum _{i=1}^{n}y_{i}} , we say that x {\displaystyle \mathbf {x} } majorizes (or dominates) y {\displaystyle \mathbf {y} } , commonly denoted x ≻ y {\displaystyle \mathbf {x} \succ \mathbf {y} } . Both weak majorization and majorization are partial orders for vectors whose entries are non-decreasing, but only a preorder for general vectors, since majorization is agnostic to the ordering of the entries in vectors, e.g., the statement ( 1 , 2 ) ≺ ( 0 , 3 ) {\displaystyle (1,2)\prec (0,3)} is simply equivalent to ( 2 , 1 ) ≺ ( 3 , 0 ) {\displaystyle (2,1)\prec (3,0)} . Specifically, x ≻ y ∧ y ≻ x {\displaystyle \mathbf {x} \succ \mathbf {y} \wedge \mathbf {y} \succ \mathbf {x} } if and only if x , y {\displaystyle \mathbf {x} ,\mathbf {y} } are permutations of each other. Similarly for ≻ w {\displaystyle \succ _{w}} . Majorizing also sometimes refers to entrywise ordering, e.g. the real-valued function f majorizes the real-valued function g when f ( x ) ≥ g ( x ) {\displaystyle f(x)\geq g(x)} for all x {\displaystyle x} in the domain, or other technical definitions, such as majorizing measures in probability theory. == Equivalent conditions == === Geometric definition === For x , y ∈ R n , {\displaystyle \mathbf {x} ,\ \mathbf {y} \in \mathbb {R} ^{n},} we have x ≺ y {\displaystyle \mathbf {x} \prec \mathbf {y} } if and only if x {\displaystyle \mathbf {x} } is in the convex hull of all vectors obtained by permuting the coordinates of y {\displaystyle \mathbf {y} } . This is equivalent to saying that x = D y {\displaystyle \mathbf {x} =\mathbf {D} \mathbf {y} } for some doubly stochastic matrix D {\displaystyle \mathbf {D} } .: Thm. 2.1 In particular, x {\displaystyle \mathbf {x} } can be written as a convex combination of n {\displaystyle n} permutations of y {\displaystyle \mathbf {y} } . In other words, x {\displaystyle \mathbf {x} } is in the permutahedron of y {\displaystyle \mathbf {y} } . Figure 1 displays the convex hull in 2D for the vector y = ( 3 , 1 ) {\displaystyle \mathbf {y} =(3,\,1)} . Notice that the center of the convex hull, which is an interval in this case, is the vector x = ( 2 , 2 ) {\displaystyle \mathbf {x} =(2,\,2)} . This is the "smallest" vector satisfying x ≺ y {\displaystyle \mathbf {x} \prec \mathbf {y} } for this given vector y {\displaystyle \mathbf {y} } . Figure 2 shows the convex hull in 3D. The center of the convex hull, which is a 2D polygon in this case, is the "smallest" vector x {\displaystyle \mathbf {x} } satisfying x ≺ y {\displaystyle \mathbf {x} \prec \mathbf {y} } for this given vector y {\displaystyle \mathbf {y} } . === Other definitions === Each of the following statements is true if and only if x ≻ y {\displaystyle \mathbf {x} \succ \mathbf {y} } . From x {\displaystyle \mathbf {x} } we can produce y {\displaystyle \mathbf {y} } by a finite sequence of "Robin Hood operations" where we replace two elements x i {\displaystyle x_{i}} and x j < x i {\displaystyle x_{j}<x_{i}} with x i − ε {\displaystyle x_{i}-\varepsilon } and x j + ε {\displaystyle x_{j}+\varepsilon } , respectively, for some ε ∈ ( 0 , x i − x j ) {\displaystyle \varepsilon \in (0,x_{i}-x_{j})} .: 11 For every convex function h : R → R {\displaystyle h:\mathbb {R} \to \mathbb {R} } , ∑ i = 1 d h ( x i ) ≥ ∑ i = 1 d h ( y i ) {\displaystyle \sum _{i=1}^{d}h(x_{i})\geq \sum _{i=1}^{d}h(y_{i})} .: Thm. 2.9 In fact, a special case suffices: ∑ i x i = ∑ i y i {\displaystyle \sum _{i}{x_{i}}=\sum _{i}{y_{i}}} and, for every t, ∑ i = 1 d max ( 0 , x i − t ) ≥ ∑ i = 1 d max ( 0 , y i − t ) {\displaystyle \sum _{i=1}^{d}\max(0,x_{i}-t)\geq \sum _{i=1}^{d}\max(0,y_{i}-t)} . For every t ∈ R {\displaystyle t\in \mathbb {R} } , ∑ j = 1 d \| x j − t \| ≥ ∑ j = 1 d \| y j − t \| {\displaystyle \sum _{j=1}^{d}\|x_{j}-t\|\geq \sum _{j=1}^{d}\|y_{j}-t\|} .: Exercise 12.17 Each vector x {\displaystyle \mathbf {x} } can be plotted as a concave curve by connecting ( 0 , 0 ) , ( 1 , x 1 ↓ ) , ( 2 , x 1 ↓ + x 2 ↓ ) , … , ( n , x 1 ↓ + x 2 ↓ + ⋯ + x n ↓ ) {\displaystyle (0,0),(1,x_{1}^{\downarrow }),(2,x_{1}^{\downarrow }+x_{2}^{\downarrow }),\dots ,(n,x_{1}^{\downarrow }+x_{2}^{\downarrow }+\dots +x_{n}^{\downarrow })} . Then x ≻ y {\displaystyle \mathbf {x} \succ \mathbf {y} } is equivalent to the curve of x {\displaystyle \mathbf {x} } being higher than that of y {\displaystyle \mathbf {y} } . == Examples == Among non-negative vectors with three components, ( 1 , 0 , 0 ) {\displaystyle (1,0,0)} and permutations of it majorize all other vectors ( p 1 , p 2 , p 3 ) {\displaystyle (p_{1},p_{2},p_{3})} such that p 1 + p 2 + p 3 = 1 {\displaystyle p_{1}+p_{2}+p_{3}=1} . For example, ( 1 , 0 , 0 ) ≻ ( 1 / 2 , 0 , 1 / 2 ) {\displaystyle (1,0,0)\succ (1/2,0,1/2)} . Similarly, ( 1 / 3 , 1 / 3 , 1 / 3 ) {\displaystyle (1/3,1/3,1/3)} is majorized by all other such vectors, so ( 1 / 2 , 0 , 1 / 2 ) ≻ ( 1 / 3 , 1 / 3 , 1 / 3 ) {\displaystyle (1/2,0,1/2)\succ (1/3,1/3,1/3)} . This behavior extends to general-length probability vectors: the singleton vector majorizes all other probability vectors, and the uniform distribution is majorized by all probability vectors. == Schur convexity == A function f : R n → R {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} } is said to be Schur convex when x ≻ y {\displaystyle \mathbf {x} \succ \mathbf {y} } implies f ( x ) ≥ f ( y ) {\displaystyle f(\mathbf {x} )\geq f(\mathbf {y} )} . Hence, Schur-convex functions translate the ordering of vectors to a standard ordering in R {\displaystyle \mathbb {R} } . Similarly, f ( x ) {\displaystyle f(\mathbf {x} )} is Schur concave when x ≻ y {\displaystyle \mathbf {x} \succ \mathbf {y} } implies f ( x ) ≤ f ( y ) . {\displaystyle f(\mathbf {x} )\leq f(\mathbf {y} ).} An example of a Schur-convex function is the max function, max ( x ) = x 1 ↓ {\displaystyle \max(\mathbf {x} )=x_{1}^{\downarrow }} . Schur convex functions are necessarily symmetric that the entries of it argument can be switched without modifying the value of the function. Therefore, linear functions, which are convex, are not Schur-convex unless they are symmetric. If a function is symmetric and convex, then it is Schur-convex. == Generalizations == Majorization can be generalized to the Lorenz ordering, a partial order on distribution functions. For example, a wealth distribution is Lorenz-greater than another if its Lorenz curve lies below the other. As such, a Lorenz-greater wealth distribution has a higher Gini coefficient, and has more income disparity. The majorization preorder can be naturally extended to density matrices in the context of quantum information. In particular, ρ ≻ ρ ′ {\displaystyle \rho \succ \rho '} exactly when s p e c [ ρ ] ≻ s p e c [ ρ ′ ] {\displaystyle \mathrm {spec} [\rho ]\succ \mathrm {spec} [\rho ']} (where s p e c {\displaystyle \mathrm {spec} } denotes the state's spectrum). Similarly, one can say a Hermitian operator, H {\displaystyle \mathbf {H} } , majorizes another, M {\displaystyle \mathbf {M} } , if the set of eigenvalues of H {\displaystyle \mathbf {H} } majorizes that of M {\displaystyle \mathbf {M} } . == See also == Muirhead's inequality Karamata's Inequality Schur-convex function Schur–Horn theorem relating diagonal entries of a matrix to its eigenvalues. For positive integer numbers, weak majorization is called Dominance order. Leximin order == Notes == == References == J. Karamata. "Sur une inegalite relative aux fonctions convexes." Publ. Math. Univ. Belgrade 1, 145–158, 1932. G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities, 2nd edition, 1952, Cambridge University Press, London. Inequalities: Theory of Majorization and Its Applications Albert W. Marshall, Ingram Olkin, Barry Arnold, Second edition. Springer Series in Statistics. Springer, New York, 2011. ISBN 978-0-387-40087-7 A tribute to Marshall and Olkin's book "Inequalities: Theory of Majorization and its Applications" Matrix Analysis (1996) Rajendra Bhatia, Springer, ISBN 978-0-387-94846-1 Topics in Matrix Analysis (1994) Roger A. Horn and Charles R. Johnson, Cambridge University Press, ISBN 978-0-521-46713-1 Majorization and Matrix Monotone Functions in Wireless Communications (2007) Eduard Jorswieck and Holger Boche, Now Publishers, ISBN 978-1-60198-040-3 The Cauchy Schwarz Master Class (2004) J. Michael Steele, Cambridge University Press, ISBN 978-0-521-54677-5 == External links == Majorization in MathWorld Majorization in PlanetMath == Software == OCTAVE/MATLAB code to check majorization
Wikipedia:Making Mathematics Count#0	Making Mathematics Count is the title of a report on mathematics education in the United Kingdom (U.K.). The report was written by Adrian Smith as leader of an "Inquiry into Post–14 Mathematics Education", which was commissioned by the UK Government in 2002. The report recommended an increase in mathematics schooling and for statistics to be taught as part of the natural sciences rather than as a part of the mathematics curriculum. == Inquiry and report == Making Mathematics Count is the title of a report on mathematics education in the United Kingdom (U.K.). The report was written by Adrian Smith as leader of an "Inquiry into Post–14 Mathematics Education", which was commissioned by the UK Government in 2002. The purpose of the Inquiry was: "To make recommendations on changes to the curriculum, qualifications and pedagogy for those aged 14 and over in schools, colleges and higher education institutions to enable those students to acquire the mathematical knowledge and skills necessary to meet the requirements of employers and of further and higher education." Publication of the report was followed two years later by a conference of 241 delegates, who included mathematics teachers, college lecturers, as well as university mathematicians, head teachers, local authority consultants and advisers, and other mathematics professionals. There is a report of the conclusions of this conference, which was intended to bring together policymakers and practitioners to share information and discuss ways in which changes in mathematics education could be implemented to benefit schools, teachers and students. == Influence == The Smith report has influenced debate on U.K. educational policy. A particular concern of the report was where and how statistics should be taught: the report recommended that statistics should be embedded in application subjects and taught by teachers of those subjects where it is applied. The government decision was that statistics teaching should remain within the mathematics curriculum. A more recent report for the Royal Statistical Society, The Future of Statistics in our Schools and Colleges retains this view. == Predecessor reports == The report's title recalls the Cockcroft report Mathematics Counts which addressed some of the same issues but was compiled 2 decades earlier, instigated by Callaghan and submitted under the Thatcher government. == Notes == == References == Smith, Adrian (2004). Making mathematics count: The report of Professor Adrian Smith's inquiry into post-14 mathematics education. London, England: The Stationery Office.
Wikipedia:Makoto Matsumoto (mathematician)#0	Makoto Matsumoto (松本眞, born February 18, 1965) is a Japanese mathematician principally known as the inventor of the Mersenne Twister, a widely used pseudorandom number generator. He is also the author of the CryptMT stream cipher. == Career == In Jan 1998, while an associate professor at Keio University, he invented the Mersenne Twister along with Takuji Nishimura. Two years later, he completed his Ph.D. on random number generators. Until his retirement in Aug 2023, he was a professor at the Department of Mathematics, Graduate School of Science, Hiroshima University. == Awards == Kirkman Medal - 1997 IBM Japan Science Prize - 1999 Japan Society for the Promotion of Science - 2008 == References ==
Wikipedia:Maksym Radziwill#0	Maksym Radziwill (born 24 February 1988) is a Polish-Canadian mathematician specializing in number theory. He is currently a professor of mathematics at the Northwestern University. == Life == He was born in Moscow in 1988. His family moved to Poland in 1991 where he graduated from high school and in 2006 to Canada. Radziwill graduated from McGill University in Montreal in 2009, and in 2013 earned a PhD under Kannan Soundararajan at Stanford University in California. In 2013–2014, he was at the Institute for Advanced Study in Princeton, New Jersey as a visiting member, and in 2014 became a Hill assistant professor at Rutgers University. In 2016, he became an assistant professor at McGill. In 2018, he became Professor of Mathematics at California Institute of Technology, and in 2022 he moved to the University of Texas at Austin. In 2023, Radziwill joined Northwestern University as the Wayne and Elizabeth Jones Professor of Mathematics. == Honors and awards == In 2016, along with Kaisa Matomäki of the University of Turku, Radziwill was awarded the SASTRA Ramanujan Prize. In February 2017, Maksym Radziwill was awarded the prestigious Sloan Fellowship. In 2018, he was awarded the Coxeter–James Prize by the Canadian Mathematical Society. In 2018 he was invited with Matomäki to present their work at the International Congress of Mathematicians. With Matomäki, he is one of five winners of the 2019 New Horizons Prize for Early-Career Achievement in Mathematics, associated with the Breakthrough Prize in Mathematics. In the same year he was awarded the Stefan Banach Prize (2018) of the Polish Mathematical Society. For 2023 he received the Cole Prize in Number Theory of the AMS. In 2023, he was also invited to give a Łojasiewicz Lecture by the Jagiellonian University. == References ==
Wikipedia:Malabika Pramanik#0	Malabika Pramanik is a Canadian mathematician who works as a professor of mathematics at the University of British Columbia. Her interests include harmonic analysis, complex variables, and partial differential equations. == Education and career == Pramanik studied statistics at the Indian Statistical Institute, earning a bachelor's degree in 1993 and a master's in 1995. She then moved to the University of California, Berkeley, where she completed a doctorate in mathematics in 2001. Her dissertation, Weighted Integrals in R 2 {\displaystyle \mathbb {R} ^{2}} and the Maximal Conjugated Calderon–Zygmund Operator, was supervised by F. Michael Christ. After short-term positions at the University of Wisconsin, University of Rochester, and California Institute of Technology, she joined the UBC faculty in 2006. She was appointed director of the Banff International Research Station in 2020. In 2025, she was elected vice president of the American Mathematical Society. == Recognition == Pramanik is the 2015–2016 winner of the Ruth I. Michler Memorial Prize of the Association for Women in Mathematics, and the 2016 winner of the Krieger–Nelson Prize, given annually by the Canadian Mathematical Society to an outstanding female researcher in mathematics. In 2018 the Canadian Mathematical Society listed her in their inaugural class of fellows. She was named a Fellow of the American Mathematical Society, in the 2022 "for contributions to complex and harmonic analysis and mentoring and support for the participation of under-represented groups in mathematics". Pramanik was an invited speaker at the International Congress of Mathematicians in 2022. == References == == External links == Home page
Wikipedia:Malcev algebra#0	In mathematics, a Malcev algebra (or Maltsev algebra or Moufang–Lie algebra) over a field is a nonassociative algebra that is antisymmetric, so that x y = − y x {\displaystyle xy=-yx} and satisfies the Malcev identity ( x y ) ( x z ) = ( ( x y ) z ) x + ( ( y z ) x ) x + ( ( z x ) x ) y . {\displaystyle (xy)(xz)=((xy)z)x+((yz)x)x+((zx)x)y.} They were first defined by Anatoly Maltsev (1955). Malcev algebras play a role in the theory of Moufang loops that generalizes the role of Lie algebras in the theory of groups. Namely, just as the tangent space of the identity element of a Lie group forms a Lie algebra, the tangent space of the identity of a smooth Moufang loop forms a Malcev algebra. Moreover, just as a Lie group can be recovered from its Lie algebra under certain supplementary conditions, a smooth Moufang loop can be recovered from its Malcev algebra if certain supplementary conditions hold. For example, this is true for a connected, simply connected real-analytic Moufang loop. == Examples == Any Lie algebra is a Malcev algebra. Any alternative algebra may be made into a Malcev algebra by defining the Malcev product to be xy − yx. The 7-sphere may be given the structure of a smooth Moufang loop by identifying it with the unit octonions. The tangent space of the identity of this Moufang loop may be identified with the 7-dimensional space of imaginary octonions. The imaginary octonions form a Malcev algebra with the Malcev product xy − yx. == Kernel == In the case of Malcev algebras, this construction can be simplified. Every Malcev algebra has a special neutral element (the zero vector in the case of vector spaces, the identity element in the case of commutative groups, and the zero element in the case of rings or modules). The characteristic feature of a Malcev algebra is that we can recover the entire equivalence relation ker f from the equivalence class of the neutral element. To be specific, let A and B be Malcev algebraic structures of a given type and let f be a homomorphism of that type from A to B. If eB is the neutral element of B, then the kernel of f is the preimage of the singleton set {eB}; that is, the subset of A consisting of all those elements of A that are mapped by f to the element eB. The kernel is usually denoted ker f (or a variation). In symbols: ker ⁡ f = { a ∈ A : f ( a ) = e B } . {\displaystyle \operatorname {ker} f=\{a\in A:f(a)=e_{B}\}.} Since a Malcev algebra homomorphism preserves neutral elements, the identity element eA of A must belong to the kernel. The homomorphism f is injective if and only if its kernel is only the singleton set {eA}. The notion of ideal generalises to any Malcev algebra (as linear subspace in the case of vector spaces, normal subgroup in the case of groups, two-sided ideals in the case of rings, and submodule in the case of modules). It turns out that ker f is not a subalgebra of A, but it is an ideal. Then it makes sense to speak of the quotient algebra G / (ker f). The first isomorphism theorem for Malcev algebras states that this quotient algebra is naturally isomorphic to the image of f (which is a subalgebra of B). The connection between this and the congruence relation for more general types of algebras is as follows. First, the kernel-as-an-ideal is the equivalence class of the neutral element eA under the kernel-as-a-congruence. For the converse direction, we need the notion of quotient in the Mal'cev algebra (which is division on either side for groups and subtraction for vector spaces, modules, and rings). Using this, elements a and b of A are equivalent under the kernel-as-a-congruence if and only if their quotient a/b is an element of the kernel-as-an-ideal. == See also == Malcev-admissible algebra == Notes == == References ==
Wikipedia:Malena Español#0	Malena Inés Español is an Argentine-American applied mathematician. Her research involves computational methods such as ridge regression in scientific computing, especially as applied to materials science and image processing. She is an associate professor in the Arizona State University School of Mathematical and Statistical Sciences. == Education and career == Español attended the Colegio Nacional de Buenos Aires (National School of Buenos Aires), a public high school in Buenos Aires, Argentina, affiliated with the University of Buenos Aires. She was an undergraduate at the University of Buenos Aires, where she earned a licenciate in applied mathematics in 2003. She moved to Tufts University in the US for graduate study in mathematics, earning a master's degree in 2005 and completing her Ph.D. in 2009. Her doctoral dissertation, Multilevel Methods for Discrete Ill-Posed Problems: Application to Deblurring, was supervised by Misha Kilmer. After postdoctoral research with Michael Ortiz at the California Institute of Technology, Español took an assistant professorship at the University of Akron in 2012, and was tenured as an associate professor there in 2018. She moved to Arizona State University in 2019, returning to the assistant professor rank, and was again promoted to associate professor in 2024. == Recognition == The EDGE Foundation named Español as the 2022 Karen EDGE Fellow. She is a 2024 recipient of the Deborah and Franklin Haimo Award for Distinguished College or University Teaching of Mathematics. == References == == External links == Home page Malena Español publications indexed by Google Scholar Malena Español, Lathisms 2017 Malena Español, Women Do Math, Association for Women in Mathematics
Wikipedia:Malgrange preparation theorem#0	In mathematics, the Malgrange preparation theorem is an analogue of the Weierstrass preparation theorem for smooth functions. It was conjectured by René Thom and proved by B. Malgrange (1962–1963, 1964, 1967). == Statement of Malgrange preparation theorem == Suppose that f(t,x) is a smooth complex function of t∈R and x∈Rn near the origin, and let k be the smallest integer such that f ( 0 , 0 ) = 0 , ∂ f ∂ t ( 0 , 0 ) = 0 , … , ∂ k − 1 f ∂ t k − 1 ( 0 , 0 ) = 0 , ∂ k f ∂ t k ( 0 , 0 ) ≠ 0. {\displaystyle f(0,0)=0,{\partial f \over \partial t}(0,0)=0,\dots ,{\partial ^{k-1}f \over \partial t^{k-1}}(0,0)=0,{\partial ^{k}f \over \partial t^{k}}(0,0)\neq 0.} Then one form of the preparation theorem states that near the origin f can be written as the product of a smooth function c that is nonzero at the origin and a smooth function that as a function of t is a polynomial of degree k. In other words, f ( t , x ) = c ( t , x ) ( t k + a k − 1 ( x ) t k − 1 + ⋯ + a 0 ( x ) ) {\displaystyle f(t,x)=c(t,x)\left(t^{k}+a_{k-1}(x)t^{k-1}+\cdots +a_{0}(x)\right)} where the functions c and a are smooth and c is nonzero at the origin. A second form of the theorem, occasionally called the Mather division theorem, is a sort of "division with remainder" theorem: it says that if f and k satisfy the conditions above and g is a smooth function near the origin, then we can write g = q f + r {\displaystyle g=qf+r} where q and r are smooth, and as a function of t, r is a polynomial of degree less than k. This means that r ( x ) = ∑ 0 ≤ j < k t j r j ( x ) {\displaystyle r(x)=\sum _{0\leq j<k}t^{j}r_{j}(x)} for some smooth functions rj(x). The two forms of the theorem easily imply each other: the first form is the special case of the "division with remainder" form where g is tk, and the division with remainder form follows from the first form of the theorem as we may assume that f as a function of t is a polynomial of degree k. If the functions f and g are real, then the functions c, a, q, and r can also be taken to be real. In the case of the Weierstrass preparation theorem these functions are uniquely determined by f and g, but uniqueness no longer holds for the Malgrange preparation theorem. == Proof of Malgrange preparation theorem == The Malgrange preparation theorem can be deduced from the Weierstrass preparation theorem. The obvious way of doing this does not work: although smooth functions have a formal power series expansion at the origin, and the Weierstrass preparation theorem applies to formal power series, the formal power series will not usually converge to smooth functions near the origin. Instead one can use the idea of decomposing a smooth function as a sum of analytic functions by applying a partition of unity to its Fourier transform. For a proof along these lines see (Mather 1968) or (Hörmander 1983a, section 7.5) == Algebraic version of the Malgrange preparation theorem == The Malgrange preparation theorem can be restated as a theorem about modules over rings of smooth, real-valued germs. If X is a manifold, with p∈X, let C∞p(X) denote the ring of real-valued germs of smooth functions at p on X. Let Mp(X) denote the unique maximal ideal of C∞p(X), consisting of germs which vanish at p. Let A be a C∞p(X)-module, and let f:X → Y be a smooth function between manifolds. Let q = f(p). f induces a ring homomorphism f*:C∞q(Y) → C∞p(X) by composition on the right with f. Thus we can view A as a C∞q(Y)-module. Then the Malgrange preparation theorem says that if A is a finitely-generated C∞p(X)-module, then A is a finitely-generated C∞q(Y)-module if and only if A/Mq(Y)A is a finite-dimensional real vector space. == References == Golubitsky, Martin; Guillemin, Victor (1973), Stable Mappings and Their Singularities, Graduate Texts in mathematics 14, Springer-Verlag, ISBN 0-387-90073-X Hörmander, L. (1983a), The analysis of linear partial differential operators I, Grundl. Math. Wissenschaft., vol. 256, Springer, ISBN 978-3-540-00662-6 Malgrange, Bernard (1962–1963), Le théorème de préparation en géométrie différentiable I–IV, Séminaire Henri Cartan, 1962/63, vol. 11–14, Secrétariat mathématique, Paris, MR 0160234 Malgrange, Bernard (1964), The preparation theorem for differentiable functions. 1964 Differential Analysis, Bombay Colloq., London: Oxford Univ. Press, pp. 203–208, MR 0182695 Malgrange, Bernard (1967), Ideals of differentiable functions, Tata Institute of Fundamental Research Studies in Mathematics, vol. 3, London: Oxford University Press, pp. vii+106, MR 0212575 Mather, John N. (1968), "Stability of C∞ mappings. I. The division theorem.", Ann. of Math., 2, 87 (1), The Annals of Mathematics, Vol. 87, No. 1: 89–104, doi:10.2307/1970595, JSTOR 1970595, MR 0232401
Wikipedia:Malgrange–Ehrenpreis theorem#0	In mathematics, the Malgrange–Ehrenpreis theorem states that every non-zero linear differential operator with constant coefficients has a Green's function. It was first proved independently by Leon Ehrenpreis (1954, 1955) and Bernard Malgrange (1955–1956). This means that the differential equation P ( ∂ ∂ x 1 , … , ∂ ∂ x ℓ ) u ( x ) = δ ( x ) , {\displaystyle P\left({\frac {\partial }{\partial x_{1}}},\ldots ,{\frac {\partial }{\partial x_{\ell }}}\right)u(\mathbf {x} )=\delta (\mathbf {x} ),} where P {\displaystyle P} is a polynomial in several variables and δ {\displaystyle \delta } is the Dirac delta function, has a distributional solution u {\displaystyle u} . It can be used to show that P ( ∂ ∂ x 1 , … , ∂ ∂ x ℓ ) u ( x ) = f ( x ) {\displaystyle P\left({\frac {\partial }{\partial x_{1}}},\ldots ,{\frac {\partial }{\partial x_{\ell }}}\right)u(\mathbf {x} )=f(\mathbf {x} )} has a solution for any compactly supported distribution f {\displaystyle f} . The solution is not unique in general. The analogue for differential operators whose coefficients are polynomials (rather than constants) is false: see Lewy's example. == Proofs == The original proofs of Malgrange and Ehrenpreis did not use explicit constructions as they used the Hahn–Banach theorem. Since then several constructive proofs have been found. There is a very short proof using the Fourier transform and the Bernstein–Sato polynomial, as follows. By taking Fourier transforms the Malgrange–Ehrenpreis theorem is equivalent to the fact that every non-zero polynomial P {\displaystyle P} has a distributional inverse. By replacing P {\displaystyle P} by the product with its complex conjugate, one can also assume that P {\displaystyle P} is non-negative. For non-negative polynomials P {\displaystyle P} the existence of a distributional inverse follows from the existence of the Bernstein–Sato polynomial, which implies that P s {\displaystyle P^{s}} can be analytically continued as a meromorphic distribution-valued function of the complex variable s {\displaystyle s} ; the constant term of the Laurent expansion of P s {\displaystyle P^{s}} at s = − 1 {\displaystyle s=-1} is then a distributional inverse of P {\displaystyle P} . Other proofs, often giving better bounds on the growth of a solution, are given in (Hörmander 1983a, Theorem 7.3.10), (Reed & Simon 1975, Theorem IX.23, p. 48) and (Rosay 1991). (Hörmander 1983b, chapter 10) gives a detailed discussion of the regularity properties of the fundamental solutions. A short constructive proof was presented in (Wagner 2009, Proposition 1, p. 458): E = 1 P m ( 2 η ) ¯ ∑ j = 0 m a j e λ j η x F ξ − 1 ( P ( i ξ + λ j η ) ¯ P ( i ξ + λ j η ) ) {\displaystyle E={\frac {1}{\overline {P_{m}(2\eta )}}}\sum _{j=0}^{m}a_{j}e^{\lambda _{j}\eta x}{\mathcal {F}}_{\xi }^{-1}\left({\frac {\overline {P(i\xi +\lambda _{j}\eta )}}{P(i\xi +\lambda _{j}\eta )}}\right)} is a fundamental solution of P ( ∂ ) {\displaystyle P(\partial )} , i.e., P ( ∂ ) E = δ {\displaystyle P(\partial )E=\delta } , if P m {\displaystyle P_{m}} is the principal part of P {\displaystyle P} , η ∈ R n {\displaystyle \eta \in \mathbb {R} ^{n}} with P m ( η ) ≠ 0 {\displaystyle P_{m}(\eta )\neq 0} , the real numbers λ 0 , … , λ m {\displaystyle \lambda _{0},\ldots ,\lambda _{m}} are pairwise different, and a j = ∏ k = 0 , k ≠ j m ( λ j − λ k ) − 1 . {\displaystyle a_{j}=\prod _{k=0,k\neq j}^{m}(\lambda _{j}-\lambda _{k})^{-1}.} == References == Ehrenpreis, Leon (1954), "Solution of some problems of division. I. Division by a polynomial of derivation.", Amer. J. Math., 76 (4): 883–903, doi:10.2307/2372662, JSTOR 2372662, MR 0068123 Ehrenpreis, Leon (1955), "Solution of some problems of division. II. Division by a punctual distribution", Amer. J. Math., 77 (2): 286–292, doi:10.2307/2372532, JSTOR 2372532, MR 0070048 Hörmander, L. (1983a), The analysis of linear partial differential operators I, Grundl. Math. Wissenschaft., vol. 256, Springer, doi:10.1007/978-3-642-96750-4, ISBN 978-3-540-12104-6, MR 0717035 Hörmander, L. (1983b), The analysis of linear partial differential operators II, Grundl. Math. Wissenschaft., vol. 257, Springer, doi:10.1007/978-3-642-96750-4, ISBN 978-3-540-12139-8, MR 0705278 Malgrange, Bernard (1955–1956), "Existence et approximation des solutions des équations aux dérivées partielles et des équations de convolution", Annales de l'Institut Fourier, 6: 271–355, doi:10.5802/aif.65, MR 0086990 Reed, Michael; Simon, Barry (1975), Methods of modern mathematical physics. II. Fourier analysis, self-adjointness, New York-London: Academic Press Harcourt Brace Jovanovich, Publishers, pp. xv+361, ISBN 978-0-12-585002-5, MR 0493420 Rosay, Jean-Pierre (1991), "A very elementary proof of the Malgrange-Ehrenpreis theorem", Amer. Math. Monthly, 98 (6): 518–523, doi:10.2307/2324871, JSTOR 2324871, MR 1109574 Rosay, Jean-Pierre (2001) [1994], "Malgrange–Ehrenpreis theorem", Encyclopedia of Mathematics, EMS Press Wagner, Peter (2009), "A new constructive proof of the Malgrange-Ehrenpreis theorem", Amer. Math. Monthly, 116 (5): 457–462, CiteSeerX 10.1.1.488.6651, doi:10.4169/193009709X470362, MR 2510844
Wikipedia:Malmquist's theorem#0	In mathematics, Malmquist's theorem, is the name of any of the three theorems proved by Axel Johannes Malmquist (1913, 1920, 1941). These theorems restrict the forms of first order algebraic differential equations which have transcendental meromorphic or algebroid solutions. == Statement of the theorems == Theorem (1913). If the differential equation d w d z = R ( z , w ) {\displaystyle {\frac {dw}{dz}}=R(z,w)} where R(z,w) is a rational function, has a transcendental meromorphic solution, then R is a polynomial of degree at most 2 with respect to w; in other words the differential equation is a Riccati equation, or linear. Theorem (1920). If an irreducible differential equation F ( d w d z , w , z ) = 0 {\displaystyle F\left({\frac {dw}{dz}},w,z\right)=0} where F is a polynomial, has a transcendental meromorphic solution, then the equation has no movable singularities. Moreover, it can be algebraically reduced either to a Riccati equation or to ( d w d z ) 2 = a ( z ) P ( z , w ) , {\displaystyle \left({\frac {dw}{dz}}\right)^{2}=a(z)P(z,w),} where P is a polynomial of degree 3 with respect to w. Theorem (1941). If an irreducible differential equation F ( d w d z , w , z ) = 0 , {\displaystyle F\left({\frac {dw}{dz}},w,z\right)=0,} where F is a polynomial, has a transcendental algebroid solution, then it can be algebraically reduced to an equation that has no movable singularities. A modern account of theorems 1913, 1920 is given in the paper of A. Eremenko(1982) == References == Malmquist, J. (1913), "Sur les fonctions à un nombre fini de branches définies par les équations différentielles du premier ordre", Acta Mathematica, 36 (1): 297–343, doi:10.1007/BF02422385 Malmquist, J. (1920), "Sur les fonctions à un nombre fini de branches satisfaisant à une équation différentielle du premier ordre" (PDF), Acta Mathematica, 42 (1): 317–325, doi:10.1007/BF02404413 Malmquist, J. (1941), "Sur les fonchillotions à un nombre fini de branches satisfaisant à une équation différentielle du premier ordre", Acta Mathematica, 74 (1): 175–196, doi:10.1007/BF02392253, MR 0005974 Eremenko, A. (1982), "Meromorphic solutions of algebraic differential equations", Russian Mathematical Surveys, 37 (4): 61–95, Bibcode:1982RuMaS..37...61E, doi:10.1070/rm1982v037yeahn04abeh003967, MR 0667974, S2CID 250879409

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