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Wikipedia:Contour set#0	In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and mentioned by German mathematician Georg Cantor in 1883. Through consideration of this set, Cantor and others helped lay the foundations of modern point-set topology. The most common construction is the Cantor ternary set, built by removing the middle third of a line segment and then repeating the process with the remaining shorter segments. Cantor mentioned this ternary construction only in passing, as an example of a perfect set that is nowhere dense. More generally, in topology, a Cantor space is a topological space homeomorphic to the Cantor ternary set (equipped with its subspace topology). The Cantor set is naturally homeomorphic to the countable product 2 _ N {\displaystyle {\underline {2}}^{\mathbb {N} }} of the discrete two-point space 2 _ {\displaystyle {\underline {2}}} . By a theorem of L. E. J. Brouwer, this is equivalent to being perfect, nonempty, compact, metrizable and zero-dimensional. == Construction and formula of the ternary set == The Cantor ternary set C {\displaystyle {\mathcal {C}}} is created by iteratively deleting the open middle third from a set of line segments. One starts by deleting the open middle third ( 1 3 , 2 3 ) {\textstyle \left({\frac {1}{3}},{\frac {2}{3}}\right)} from the interval [ 0 , 1 ] {\displaystyle \textstyle \left[0,1\right]} , leaving two line segments: [ 0 , 1 3 ] ∪ [ 2 3 , 1 ] {\textstyle \left[0,{\frac {1}{3}}\right]\cup \left[{\frac {2}{3}},1\right]} . Next, the open middle third of each of these remaining segments is deleted, leaving four line segments: [ 0 , 1 9 ] ∪ [ 2 9 , 1 3 ] ∪ [ 2 3 , 7 9 ] ∪ [ 8 9 , 1 ] {\textstyle \left[0,{\frac {1}{9}}\right]\cup \left[{\frac {2}{9}},{\frac {1}{3}}\right]\cup \left[{\frac {2}{3}},{\frac {7}{9}}\right]\cup \left[{\frac {8}{9}},1\right]} . The Cantor ternary set contains all points in the interval [ 0 , 1 ] {\displaystyle [0,1]} that are not deleted at any step in this infinite process. The same construction can be described recursively by setting C 0 := [ 0 , 1 ] {\displaystyle C_{0}:=[0,1]} and C n := C n − 1 3 ∪ ( 2 3 + C n − 1 3 ) = 1 3 ( C n − 1 ∪ ( 2 + C n − 1 ) ) {\displaystyle C_{n}:={\frac {C_{n-1}}{3}}\cup \left({\frac {2}{3}}+{\frac {C_{n-1}}{3}}\right)={\frac {1}{3}}{\bigl (}C_{n-1}\cup \left(2+C_{n-1}\right){\bigr )}} for n ≥ 1 {\displaystyle n\geq 1} , so that C := {\displaystyle {\mathcal {C}}:=} lim n → ∞ C n {\displaystyle {\color {Blue}\lim _{n\to \infty }C_{n}}} = ⋂ n = 0 ∞ C n = ⋂ n = m ∞ C n {\displaystyle =\bigcap _{n=0}^{\infty }C_{n}=\bigcap _{n=m}^{\infty }C_{n}} for any m ≥ 0 {\displaystyle m\geq 0} . The first six steps of this process are illustrated below. Using the idea of self-similar transformations, T L ( x ) = x / 3 , {\displaystyle T_{L}(x)=x/3,} T R ( x ) = ( 2 + x ) / 3 {\displaystyle T_{R}(x)=(2+x)/3} and C n = T L ( C n − 1 ) ∪ T R ( C n − 1 ) , {\displaystyle C_{n}=T_{L}(C_{n-1})\cup T_{R}(C_{n-1}),} the explicit closed formulas for the Cantor set are C = [ 0 , 1 ] ∖ ⋃ n = 0 ∞ ⋃ k = 0 3 n − 1 ( 3 k + 1 3 n + 1 , 3 k + 2 3 n + 1 ) , {\displaystyle {\mathcal {C}}=[0,1]\,\setminus \,\bigcup _{n=0}^{\infty }\bigcup _{k=0}^{3^{n}-1}\left({\frac {3k+1}{3^{n+1}}},{\frac {3k+2}{3^{n+1}}}\right)\!,} where every middle third is removed as the open interval ( 3 k + 1 3 n + 1 , 3 k + 2 3 n + 1 ) {\textstyle \left({\frac {3k+1}{3^{n+1}}},{\frac {3k+2}{3^{n+1}}}\right)} from the closed interval [ 3 k + 0 3 n + 1 , 3 k + 3 3 n + 1 ] = [ k + 0 3 n , k + 1 3 n ] {\textstyle \left[{\frac {3k+0}{3^{n+1}}},{\frac {3k+3}{3^{n+1}}}\right]=\left[{\frac {k+0}{3^{n}}},{\frac {k+1}{3^{n}}}\right]} surrounding it, or C = ⋂ n = 1 ∞ ⋃ k = 0 3 n − 1 − 1 ( [ 3 k + 0 3 n , 3 k + 1 3 n ] ∪ [ 3 k + 2 3 n , 3 k + 3 3 n ] ) , {\displaystyle {\mathcal {C}}=\bigcap _{n=1}^{\infty }\bigcup _{k=0}^{3^{n-1}-1}\left(\left[{\frac {3k+0}{3^{n}}},{\frac {3k+1}{3^{n}}}\right]\cup \left[{\frac {3k+2}{3^{n}}},{\frac {3k+3}{3^{n}}}\right]\right)\!,} where the middle third ( 3 k + 1 3 n , 3 k + 2 3 n ) {\textstyle \left({\frac {3k+1}{3^{n}}},{\frac {3k+2}{3^{n}}}\right)} of the foregoing closed interval [ k + 0 3 n − 1 , k + 1 3 n − 1 ] = [ 3 k + 0 3 n , 3 k + 3 3 n ] {\textstyle \left[{\frac {k+0}{3^{n-1}}},{\frac {k+1}{3^{n-1}}}\right]=\left[{\frac {3k+0}{3^{n}}},{\frac {3k+3}{3^{n}}}\right]} is removed by intersecting with [ 3 k + 0 3 n , 3 k + 1 3 n ] ∪ [ 3 k + 2 3 n , 3 k + 3 3 n ] . {\textstyle \left[{\frac {3k+0}{3^{n}}},{\frac {3k+1}{3^{n}}}\right]\cup \left[{\frac {3k+2}{3^{n}}},{\frac {3k+3}{3^{n}}}\right]\!.} This process of removing middle thirds is a simple example of a finite subdivision rule. The complement of the Cantor ternary set is an example of a fractal string. In arithmetical terms, the Cantor set consists of all real numbers of the unit interval [ 0 , 1 ] {\displaystyle [0,1]} that do not require the digit 1 in order to be expressed as a ternary (base 3) fraction. As the above diagram illustrates, each point in the Cantor set is uniquely located by a path through an infinitely deep binary tree, where the path turns left or right at each level according to which side of a deleted segment the point lies on. Representing each left turn with 0 and each right turn with 2 yields the ternary fraction for a point. === Mandelbrot's construction by "curdling" === In The Fractal Geometry of Nature, mathematician Benoit Mandelbrot provides a whimsical thought experiment to assist non-mathematical readers in imagining the construction of C {\displaystyle {\mathcal {C}}} . His narrative begins with imagining a bar, perhaps of lightweight metal, in which the bar's matter "curdles" by iteratively shifting towards its extremities. As the bar's segments become smaller, they become thin, dense slugs that eventually grow too small and faint to see.CURDLING: The construction of the Cantor bar results from the process I call curdling. It begins with a round bar. It is best to think of it as having a very low density. Then matter "curdles" out of this bar's middle third into the end thirds, so that the positions of the latter remain unchanged. Next matter curdles out of the middle third of each end third into its end thirds, and so on ad infinitum until one is left with an infinitely large number of infinitely thin slugs of infinitely high density. These slugs are spaced along the line in the very specific fashion induced by the generating process. In this illustration, curdling (which eventually requires hammering!) stops when both the printer's press and our eye cease to follow; the last line is indistinguishable from the last but one: each of its ultimate parts is seen as a gray slug rather than two parallel black slugs. == Composition == Since the Cantor set is defined as the set of points not excluded, the proportion (i.e., measure) of the unit interval remaining can be found by total length removed. This total is the geometric progression ∑ n = 0 ∞ 2 n 3 n + 1 = 1 3 + 2 9 + 4 27 + 8 81 + ⋯ = 1 3 ( 1 1 − 2 3 ) = 1. {\displaystyle \sum _{n=0}^{\infty }{\frac {2^{n}}{3^{n+1}}}={\frac {1}{3}}+{\frac {2}{9}}+{\frac {4}{27}}+{\frac {8}{81}}+\cdots ={\frac {1}{3}}\left({\frac {1}{1-{\frac {2}{3}}}}\right)=1.} So that the proportion left is 1 − 1 = 0 {\displaystyle 1-1=0} . This calculation suggests that the Cantor set cannot contain any interval of non-zero length. It may seem surprising that there should be anything left—after all, the sum of the lengths of the removed intervals is equal to the length of the original interval. However, a closer look at the process reveals that there must be something left, since removing the "middle third" of each interval involved removing open sets (sets that do not include their endpoints). So removing the line segment ( 1 3 , 2 3 ) {\textstyle \left({\frac {1}{3}},{\frac {2}{3}}\right)} from the original interval [ 0 , 1 ] {\displaystyle [0,1]} leaves behind the points ⁠1/3⁠ and ⁠2/3⁠. Subsequent steps do not remove these (or other) endpoints, since the intervals removed are always internal to the intervals remaining. So the Cantor set is not empty, and in fact contains an uncountably infinite number of points (as follows from the above description in terms of paths in an infinite binary tree). It may appear that only the endpoints of the construction segments are left, but that is not the case either. The number ⁠1/4⁠, for example, has the unique ternary form 0.020202... = 0.02. It is in the bottom third, and the top third of that third, and the bottom third of that top third, and so on. Since it is never in one of the middle segments, it is never removed. Yet it is also not an endpoint of any middle segment, because it is not a multiple of any power of ⁠1/3⁠. All endpoints of segments are terminating ternary fractions and are contained in the set { x ∈ [ 0 , 1 ] ∣ ∃ i ∈ N 0 : x 3 i ∈ Z } ( ⊂ N 0 3 − N 0 ) {\displaystyle \left\{x\in [0,1]\mid \exists i\in \mathbb {N} _{0}:x\,3^{i}\in \mathbb {Z} \right\}\qquad {\Bigl (}\subset \mathbb {N} _{0}\,3^{-\mathbb {N} _{0}}{\Bigr )}} which is a countably infinite set. As to cardinality, almost all elements of the Cantor set are not endpoints of intervals, nor rational points like ⁠1/4⁠. The whole Cantor set is in fact not countable. == Properties == === Cardinality === It can be shown that there are as many points left behind in this process as there were to begin with, and that therefore, the Cantor set is uncountable. To see this, we show that there is a function f from the Cantor set C {\displaystyle {\mathcal {C}}} to the closed interval [ 0 , 1 ] {\displaystyle [0,1]} that is surjective (i.e. f maps from C {\displaystyle {\mathcal {C}}} onto [ 0 , 1 ] {\displaystyle [0,1]} ) so that the cardinality of C {\displaystyle {\mathcal {C}}} is no less than that of [ 0 , 1 ] {\displaystyle [0,1]} . Since C {\displaystyle {\mathcal {C}}} is a subset of [ 0 , 1 ] {\displaystyle [0,1]} , its cardinality is also no greater, so the two cardinalities must in fact be equal, by the Cantor–Bernstein–Schröder theorem. To construct this function, consider the points in the [ 0 , 1 ] {\displaystyle [0,1]} interval in terms of base 3 (or ternary) notation. Recall that the proper ternary fractions, more precisely: the elements of ( Z ∖ { 0 } ) ⋅ 3 − N 0 {\displaystyle {\bigl (}\mathbb {Z} \setminus \{0\}{\bigr )}\cdot 3^{-\mathbb {N} _{0}}} , admit more than one representation in this notation, as for example ⁠1/3⁠, that can be written as 0.13 = 0.103, but also as 0.0222...3 = 0.023, and ⁠2/3⁠, that can be written as 0.23 = 0.203 but also as 0.1222...3 = 0.123. When we remove the middle third, this contains the numbers with ternary numerals of the form 0.1xxxxx...3 where xxxxx...3 is strictly between 00000...3 and 22222...3. So the numbers remaining after the first step consist of Numbers of the form 0.0xxxxx...3 (including 0.022222...3 = 1/3) Numbers of the form 0.2xxxxx...3 (including 0.222222...3 = 1) This can be summarized by saying that those numbers with a ternary representation such that the first digit after the radix point is not 1 are the ones remaining after the first step. The second step removes numbers of the form 0.01xxxx...3 and 0.21xxxx...3, and (with appropriate care for the endpoints) it can be concluded that the remaining numbers are those with a ternary numeral where neither of the first two digits is 1. Continuing in this way, for a number not to be excluded at step n, it must have a ternary representation whose nth digit is not 1. For a number to be in the Cantor set, it must not be excluded at any step, it must admit a numeral representation consisting entirely of 0s and 2s. It is worth emphasizing that numbers like 1, ⁠1/3⁠ = 0.13 and ⁠7/9⁠ = 0.213 are in the Cantor set, as they have ternary numerals consisting entirely of 0s and 2s: 1 = 0.222...3 = 0.23, ⁠1/3⁠ = 0.0222...3 = 0.023 and ⁠7/9⁠ = 0.20222...3 = 0.2023. All the latter numbers are "endpoints", and these examples are right limit points of C {\displaystyle {\mathcal {C}}} . The same is true for the left limit points of C {\displaystyle {\mathcal {C}}} , e.g. ⁠2/3⁠ = 0.1222...3 = 0.123 = 0.203 and ⁠8/9⁠ = 0.21222...3 = 0.2123 = 0.2203. All these endpoints are proper ternary fractions (elements of Z ⋅ 3 − N 0 {\displaystyle \mathbb {Z} \cdot 3^{-\mathbb {N} _{0}}} ) of the form ⁠p/q⁠, where denominator q is a power of 3 when the fraction is in its irreducible form. The ternary representation of these fractions terminates (i.e., is finite) or — recall from above that proper ternary fractions each have 2 representations — is infinite and "ends" in either infinitely many recurring 0s or infinitely many recurring 2s. Such a fraction is a left limit point of C {\displaystyle {\mathcal {C}}} if its ternary representation contains no 1's and "ends" in infinitely many recurring 0s. Similarly, a proper ternary fraction is a right limit point of C {\displaystyle {\mathcal {C}}} if it again its ternary expansion contains no 1's and "ends" in infinitely many recurring 2s. This set of endpoints is dense in C {\displaystyle {\mathcal {C}}} (but not dense in [ 0 , 1 ] {\displaystyle [0,1]} ) and makes up a countably infinite set. The numbers in C {\displaystyle {\mathcal {C}}} which are not endpoints also have only 0s and 2s in their ternary representation, but they cannot end in an infinite repetition of the digit 0, nor of the digit 2, because then it would be an endpoint. The function from C {\displaystyle {\mathcal {C}}} to [ 0 , 1 ] {\displaystyle [0,1]} is defined by taking the ternary numerals that do consist entirely of 0s and 2s, replacing all the 2s by 1s, and interpreting the sequence as a binary representation of a real number. In a formula, f ( ∑ k ∈ N a k 3 − k ) = ∑ k ∈ N a k 2 2 − k {\displaystyle f{\bigg (}\sum _{k\in \mathbb {N} }a_{k}3^{-k}{\bigg )}=\sum _{k\in \mathbb {N} }{\frac {a_{k}}{2}}2^{-k}} where ∀ k ∈ N : a k ∈ { 0 , 2 } . {\displaystyle \forall k\in \mathbb {N} :a_{k}\in \{0,2\}.} For any number y in [ 0 , 1 ] {\displaystyle [0,1]} , its binary representation can be translated into a ternary representation of a number x in C {\displaystyle {\mathcal {C}}} by replacing all the 1s by 2s. With this, f(x) = y so that y is in the range of f. For instance if y = ⁠3/5⁠ = 0.100110011001...2 = 0.1001, we write x = 0.2002 = 0.200220022002...3 = ⁠7/10⁠. Consequently, f is surjective. However, f is not injective — the values for which f(x) coincides are those at opposing ends of one of the middle thirds removed. For instance, take ⁠1/3⁠ = 0.023 (which is a right limit point of C {\displaystyle {\mathcal {C}}} and a left limit point of the middle third [⁠1/3⁠, ⁠2/3⁠]) and ⁠2/3⁠ = 0.203 (which is a left limit point of C {\displaystyle {\mathcal {C}}} and a right limit point of the middle third [⁠1/3⁠, ⁠2/3⁠]) so f ( 1 / 3 ) = f ( 0.0 2 ¯ 3 ) = 0.0 1 ¯ 2 = 0.1 2 = 0.1 0 ¯ 2 = f ( 0.2 0 ¯ 3 ) = f ( 2 / 3 ) . ∥ 1 / 2 {\displaystyle {\begin{array}{lcl}f{\bigl (}{}^{1}\!\!/\!_{3}{\bigr )}=f(0.0{\overline {2}}_{3})=0.0{\overline {1}}_{2}=\!\!&\!\!0.1_{2}\!\!&\!\!=0.1{\overline {0}}_{2}=f(0.2{\overline {0}}_{3})=f{\bigl (}{}^{2}\!\!/\!_{3}{\bigr )}.\\&\parallel \\&{}^{1}\!\!/\!_{2}\end{array}}} Thus there are as many points in the Cantor set as there are in the interval [ 0 , 1 ] {\displaystyle [0,1]} (which has the uncountable cardinality c = 2 ℵ 0 {\displaystyle {\mathfrak {c}}=2^{\aleph _{0}}} ). However, the set of endpoints of the removed intervals is countable, so there must be uncountably many numbers in the Cantor set which are not interval endpoints. As noted above, one example of such a number is ⁠1/4⁠, which can be written as 0.020202...3 = 0.02 in ternary notation. In fact, given any a ∈ [ − 1 , 1 ] {\displaystyle a\in [-1,1]} , there exist x , y ∈ C {\displaystyle x,y\in {\mathcal {C}}} such that a = y − x {\displaystyle a=y-x} . This was first demonstrated by Steinhaus in 1917, who proved, via a geometric argument, the equivalent assertion that { ( x , y ) ∈ R 2 ∣ y = x + a } ∩ ( C × C ) ≠ ∅ {\displaystyle \{(x,y)\in \mathbb {R} ^{2}\mid y=x+a\}\;\cap \;({\mathcal {C}}\times {\mathcal {C}})\neq \emptyset } for every a ∈ [ − 1 , 1 ] {\displaystyle a\in [-1,1]} . Since this construction provides an injection from [ − 1 , 1 ] {\displaystyle [-1,1]} to C × C {\displaystyle {\mathcal {C}}\times {\mathcal {C}}} , we have \| C × C \| ≥ \| [ − 1 , 1 ] \| = c {\displaystyle \|{\mathcal {C}}\times {\mathcal {C}}\|\geq \|[-1,1]\|={\mathfrak {c}}} as an immediate corollary. Assuming that \| A × A \| = \| A \| {\displaystyle \|A\times A\|=\|A\|} for any infinite set A {\displaystyle A} (a statement shown to be equivalent to the axiom of choice by Tarski), this provides another demonstration that \| C \| = c {\displaystyle \|{\mathcal {C}}\|={\mathfrak {c}}} . The Cantor set contains as many points as the interval from which it is taken, yet itself contains no interval of nonzero length. The irrational numbers have the same property, but the Cantor set has the additional property of being closed, so it is not even dense in any interval, unlike the irrational numbers which are dense in every interval. It has been conjectured that all algebraic irrational numbers are normal. Since members of the Cantor set are not normal in base 3, this would imply that all members of the Cantor set are either rational or transcendental. === Self-similarity === The Cantor set is the prototype of a fractal. It is self-similar, because it is equal to two copies of itself, if each copy is shrunk by a factor of 3 and translated. More precisely, the Cantor set is equal to the union of two functions, the left and right self-similarity transformations of itself, T L ( x ) = x / 3 {\displaystyle T_{L}(x)=x/3} and T R ( x ) = ( 2 + x ) / 3 {\displaystyle T_{R}(x)=(2+x)/3} , which leave the Cantor set invariant up to homeomorphism: T L ( C ) ≅ T R ( C ) ≅ C = T L ( C ) ∪ T R ( C ) . {\displaystyle T_{L}({\mathcal {C}})\cong T_{R}({\mathcal {C}})\cong {\mathcal {C}}=T_{L}({\mathcal {C}})\cup T_{R}({\mathcal {C}}).} Repeated iteration of T L {\displaystyle T_{L}} and T R {\displaystyle T_{R}} can be visualized as an infinite binary tree. That is, at each node of the tree, one may consider the subtree to the left or to the right. Taking the set { T L , T R } {\displaystyle \{T_{L},T_{R}\}} together with function composition forms a monoid, the dyadic monoid. The automorphisms of the binary tree are its hyperbolic rotations, and are given by the modular group. Thus, the Cantor set is a homogeneous space in the sense that for any two points x {\displaystyle x} and y {\displaystyle y} in the Cantor set C {\displaystyle {\mathcal {C}}} , there exists a homeomorphism h : C → C {\displaystyle h:{\mathcal {C}}\to {\mathcal {C}}} with h ( x ) = y {\displaystyle h(x)=y} . An explicit construction of h {\displaystyle h} can be described more easily if we see the Cantor set as a product space of countably many copies of the discrete space { 0 , 1 } {\displaystyle \{0,1\}} . Then the map h : { 0 , 1 } N → { 0 , 1 } N {\displaystyle h:\{0,1\}^{\mathbb {N} }\to \{0,1\}^{\mathbb {N} }} defined by h n ( u ) := u n + x n + y n mod 2 {\displaystyle h_{n}(u):=u_{n}+x_{n}+y_{n}\mod 2} is an involutive homeomorphism exchanging x {\displaystyle x} and y {\displaystyle y} . === Topological and analytical properties === Although "the" Cantor set typically refers to the original, middle-thirds Cantor set described above, topologists often talk about "a" Cantor set, which means any topological space that is homeomorphic (topologically equivalent) to it. As the above summation argument shows, the Cantor set is uncountable but has Lebesgue measure 0. Since the Cantor set is the complement of a union of open sets, it itself is a closed subset of the reals, and therefore a complete metric space. Since it is also totally bounded, the Heine–Borel theorem says that it must be compact. For any point in the Cantor set and any arbitrarily small neighborhood of the point, there is some other number with a ternary numeral of only 0s and 2s, as well as numbers whose ternary numerals contain 1s. Hence, every point in the Cantor set is an accumulation point (also called a cluster point or limit point) of the Cantor set, but none is an interior point. A closed set in which every point is an accumulation point is also called a perfect set in topology, while a closed subset of the interval with no interior points is nowhere dense in the interval. Every point of the Cantor set is also an accumulation point of the complement of the Cantor set. For any two points in the Cantor set, there will be some ternary digit where they differ — one will have 0 and the other 2. By splitting the Cantor set into "halves" depending on the value of this digit, one obtains a partition of the Cantor set into two closed sets that separate the original two points. In the relative topology on the Cantor set, the points have been separated by a clopen set. Consequently, the Cantor set is totally disconnected. As a compact totally disconnected Hausdorff space, the Cantor set is an example of a Stone space. As a topological space, the Cantor set is naturally homeomorphic to the product of countably many copies of the space { 0 , 1 } {\displaystyle \{0,1\}} , where each copy carries the discrete topology. This is the space of all sequences in two digits 2 N = { ( x n ) ∣ x n ∈ { 0 , 1 } for n ∈ N } , {\displaystyle 2^{\mathbb {N} }=\{(x_{n})\mid x_{n}\in \{0,1\}{\text{ for }}n\in \mathbb {N} \},} which can also be identified with the set of 2-adic integers. The basis for the open sets of the product topology are cylinder sets; the homeomorphism maps these to the subspace topology that the Cantor set inherits from the natural topology on the real line. This characterization of the Cantor space as a product of compact spaces gives a second proof that Cantor space is compact, via Tychonoff's theorem. From the above characterization, the Cantor set is homeomorphic to the p-adic integers, and, if one point is removed from it, to the p-adic numbers. The Cantor set is a subset of the reals, which are a metric space with respect to the ordinary distance metric; therefore the Cantor set itself is a metric space, by using that same metric. Alternatively, one can use the p-adic metric on 2 N {\displaystyle 2^{\mathbb {N} }} : given two sequences ( x n ) , ( y n ) ∈ 2 N {\displaystyle (x_{n}),(y_{n})\in 2^{\mathbb {N} }} , the distance between them is d ( ( x n ) , ( y n ) ) = 2 − k {\displaystyle d((x_{n}),(y_{n}))=2^{-k}} , where k {\displaystyle k} is the smallest index such that x k ≠ y k {\displaystyle x_{k}\neq y_{k}} ; if there is no such index, then the two sequences are the same, and one defines the distance to be zero. These two metrics generate the same topology on the Cantor set. We have seen above that the Cantor set is a totally disconnected perfect compact metric space. Indeed, in a sense it is the only one: every nonempty totally disconnected perfect compact metric space is homeomorphic to the Cantor set. See Cantor space for more on spaces homeomorphic to the Cantor set. The Cantor set is sometimes regarded as "universal" in the category of compact metric spaces, since any compact metric space is a continuous image of the Cantor set; however this construction is not unique and so the Cantor set is not universal in the precise categorical sense. The "universal" property has important applications in functional analysis, where it is sometimes known as the representation theorem for compact metric spaces. For any integer q ≥ 2, the topology on the group G = Zqω (the countable direct sum) is discrete. Although the Pontrjagin dual Γ is also Zqω, the topology of Γ is compact. One can see that Γ is totally disconnected and perfect - thus it is homeomorphic to the Cantor set. It is easiest to write out the homeomorphism explicitly in the case q = 2. (See Rudin 1962 p 40.) === Measure and probability === The Cantor set can be seen as the compact group of binary sequences, and as such, it is endowed with a natural Haar measure. When normalized so that the measure of the set is 1, it is a model of an infinite sequence of coin tosses. Furthermore, one can show that the usual Lebesgue measure on the interval is an image of the Haar measure on the Cantor set, while the natural injection into the ternary set is a canonical example of a singular measure. It can also be shown that the Haar measure is an image of any probability, making the Cantor set a universal probability space in some ways. In Lebesgue measure theory, the Cantor set is an example of a set which is uncountable and has zero measure. In contrast, the set has a Hausdorff measure of 1 {\displaystyle 1} in its dimension of log 3 ⁡ ( 2 ) {\displaystyle \log _{3}(2)} . === Cantor numbers === If we define a Cantor number as a member of the Cantor set, then Every real number in [ 0 , 2 ] {\displaystyle [0,2]} is the sum of two Cantor numbers. Between any two Cantor numbers there is a number that is not a Cantor number. === Descriptive set theory === The Cantor set is a meagre set (or a set of first category) as a subset of [ 0 , 1 ] {\displaystyle [0,1]} (although not as a subset of itself, since it is a Baire space). The Cantor set thus demonstrates that notions of "size" in terms of cardinality, measure, and (Baire) category need not coincide. Like the set Q ∩ [ 0 , 1 ] {\displaystyle \mathbb {Q} \cap [0,1]} , the Cantor set C {\displaystyle {\mathcal {C}}} is "small" in the sense that it is a null set (a set of measure zero) and it is a meagre subset of [ 0 , 1 ] {\displaystyle [0,1]} . However, unlike Q ∩ [ 0 , 1 ] {\displaystyle \mathbb {Q} \cap [0,1]} , which is countable and has a "small" cardinality, ℵ 0 {\displaystyle \aleph _{0}} , the cardinality of C {\displaystyle {\mathcal {C}}} is the same as that of [ 0 , 1 ] {\displaystyle [0,1]} , the continuum c {\displaystyle {\mathfrak {c}}} , and is "large" in the sense of cardinality. In fact, it is also possible to construct a subset of [ 0 , 1 ] {\displaystyle [0,1]} that is meagre but of positive measure and a subset that is non-meagre but of measure zero: By taking the countable union of "fat" Cantor sets C ( n ) {\displaystyle {\mathcal {C}}^{(n)}} of measure λ = ( n − 1 ) / n {\displaystyle \lambda =(n-1)/n} (see Smith–Volterra–Cantor set below for the construction), we obtain a set A := ⋃ n = 1 ∞ C ( n ) {\textstyle {\mathcal {A}}:=\bigcup _{n=1}^{\infty }{\mathcal {C}}^{(n)}} which has a positive measure (equal to 1) but is meagre in [0,1], since each C ( n ) {\displaystyle {\mathcal {C}}^{(n)}} is nowhere dense. Then consider the set A c = [ 0 , 1 ] ∖ ⋃ n = 1 ∞ C ( n ) {\textstyle {\mathcal {A}}^{\mathrm {c} }=[0,1]\setminus \bigcup _{n=1}^{\infty }{\mathcal {C}}^{(n)}} . Since A ∪ A c = [ 0 , 1 ] {\displaystyle {\mathcal {A}}\cup {\mathcal {A}}^{\mathrm {c} }=[0,1]} , A c {\displaystyle {\mathcal {A}}^{\mathrm {c} }} cannot be meagre, but since μ ( A ) = 1 {\displaystyle \mu ({\mathcal {A}})=1} , A c {\displaystyle {\mathcal {A}}^{\mathrm {c} }} must have measure zero. == Variants == === Smith–Volterra–Cantor set === Instead of repeatedly removing the middle third of every piece as in the Cantor set, we could also keep removing any other fixed percentage (other than 0% and 100%) from the middle. In the case where the middle ⁠8/10⁠ of the interval is removed, we get a remarkably accessible case — the set consists of all numbers in [0,1] that can be written as a decimal consisting entirely of 0s and 9s. If a fixed percentage is removed at each stage, then the limiting set will have measure zero, since the length of the remainder ( 1 − f ) n → 0 {\displaystyle (1-f)^{n}\to 0} as n → ∞ {\displaystyle n\to \infty } for any f {\displaystyle f} such that 0 < f ≤ 1 {\displaystyle 0<f\leq 1} . On the other hand, "fat Cantor sets" of positive measure can be generated by removal of smaller fractions of the middle of the segment in each iteration. Thus, one can construct sets homeomorphic to the Cantor set that have positive Lebesgue measure while still being nowhere dense. If an interval of length r n {\displaystyle r^{n}} ( r ≤ 1 / 3 {\displaystyle r\leq 1/3} ) is removed from the middle of each segment at the nth iteration, then the total length removed is ∑ n = 1 ∞ 2 n − 1 r n = r / ( 1 − 2 r ) {\textstyle \sum _{n=1}^{\infty }2^{n-1}r^{n}=r/(1-2r)} , and the limiting set will have a Lebesgue measure of λ = ( 1 − 3 r ) / ( 1 − 2 r ) {\displaystyle \lambda =(1-3r)/(1-2r)} . Thus, in a sense, the middle-thirds Cantor set is a limiting case with r = 1 / 3 {\displaystyle r=1/3} . If 0 < r < 1 / 3 {\displaystyle 0<r<1/3} , then the remainder will have positive measure with 0 < λ < 1 {\displaystyle 0<\lambda <1} . The case r = 1 / 4 {\displaystyle r=1/4} is known as the Smith–Volterra–Cantor set, which has a Lebesgue measure of 1 / 2 {\displaystyle 1/2} . === Cantor dust === Cantor dust is a multi-dimensional version of the Cantor set. It can be formed by taking a finite Cartesian product of the Cantor set with itself, making it a Cantor space. Like the Cantor set, Cantor dust has zero measure. A different 2D analogue of the Cantor set is the Sierpinski carpet, where a square is divided up into nine smaller squares, and the middle one removed. The remaining squares are then further divided into nine each and the middle removed, and so on ad infinitum. One 3D analogue of this is the Menger sponge. == Historical remarks == Cantor introduced what we call today the Cantor ternary set C {\displaystyle {\mathcal {C}}} as an example "of a perfect point-set, which is not everywhere-dense in any interval, however small." Cantor described C {\displaystyle {\mathcal {C}}} in terms of ternary expansions, as "the set of all real numbers given by the formula: z = c 1 / 3 + c 2 / 3 2 + ⋯ + c ν / 3 ν + ⋯ {\displaystyle z=c_{1}/3+c_{2}/3^{2}+\cdots +c_{\nu }/3^{\nu }+\cdots } where the coefficients c ν {\displaystyle c_{\nu }} arbitrarily take the two values 0 and 2, and the series can consist of a finite number or an infinite number of elements." A topological space P {\displaystyle P} is perfect if all its points are limit points or, equivalently, if it coincides with its derived set P ′ {\displaystyle P'} . Subsets of the real line, like C {\displaystyle {\mathcal {C}}} , can be seen as topological spaces under the induced subspace topology. Cantor was led to the study of derived sets by his results on uniqueness of trigonometric series. The latter did much to set him on the course for developing an abstract, general theory of infinite sets. Benoit Mandelbrot wrote much on Cantor dusts and their relation to natural fractals and statistical physics. He further reflected on the puzzling or even upsetting nature of such structures to those in the mathematics and physics community. In The Fractal Geometry of Nature, he described how "When I started on this topic in 1962, everyone was agreeing that Cantor dusts are at least as monstrous as the Koch and Peano curves," and added that "every self-respecting physicist was automatically turned off by a mention of Cantor, ready to run a mile from anyone claiming C {\displaystyle {\mathcal {C}}} to be interesting in science." == See also == The indicator function of the Cantor set Smith–Volterra–Cantor set Cantor function Cantor cube Antoine's necklace Koch snowflake Knaster–Kuratowski fan List of fractals by Hausdorff dimension Moser–de Bruijn sequence == Notes == == References == == External links == "Cantor set", Encyclopedia of Mathematics, EMS Press, 2001 [1994] Cantor Sets and Cantor Set and Function at cut-the-knot Cantor Set at Platonic Realms
Wikipedia:Contraction principle (large deviations theory)#0	In probability theory, the theory of large deviations concerns the asymptotic behaviour of remote tails of sequences of probability distributions. While some basic ideas of the theory can be traced to Laplace, the formalization started with insurance mathematics, namely ruin theory with Cramér and Lundberg. A unified formalization of large deviation theory was developed in 1966, in a paper by Varadhan. Large deviations theory formalizes the heuristic ideas of concentration of measures and widely generalizes the notion of convergence of probability measures. Roughly speaking, large deviations theory concerns itself with the exponential decline of the probability measures of certain kinds of extreme or tail events. == Introductory examples == Any large deviation is done in the least unlikely of all the unlikely ways! === An elementary example === Consider a sequence of independent tosses of a fair coin. The possible outcomes could be heads or tails. Let us denote the possible outcome of the i-th trial by X i {\displaystyle X_{i}} , where we encode head as 1 and tail as 0. Now let M N {\displaystyle M_{N}} denote the mean value after N {\displaystyle N} trials, namely M N = 1 N ∑ i = 1 N X i {\displaystyle M_{N}={\frac {1}{N}}\sum _{i=1}^{N}X_{i}} . Then M N {\displaystyle M_{N}} lies between 0 and 1. From the law of large numbers it follows that as N grows, the distribution of M N {\displaystyle M_{N}} converges to 0.5 = E ⁡ [ X ] {\displaystyle 0.5=\operatorname {E} [X]} (the expected value of a single coin toss). Moreover, by the central limit theorem, it follows that M N {\displaystyle M_{N}} is approximately normally distributed for large N {\displaystyle N} . The central limit theorem can provide more detailed information about the behavior of M N {\displaystyle M_{N}} than the law of large numbers. For example, we can approximately find a tail probability of M N {\displaystyle M_{N}} – the probability that M N {\displaystyle M_{N}} is greater than some value x {\displaystyle x} – for a fixed value of N {\displaystyle N} . However, the approximation by the central limit theorem may not be accurate if x {\displaystyle x} is far from E ⁡ [ X i ] {\displaystyle \operatorname {E} [X_{i}]} and N {\displaystyle N} is not sufficiently large. Also, it does not provide information about the convergence of the tail probabilities as N → ∞ {\displaystyle N\to \infty } . However, the large deviation theory can provide answers for such problems. Let us make this statement more precise. For a given value 0.5 < x < 1 {\displaystyle 0.5<x<1} , let us compute the tail probability P ( M N > x ) {\displaystyle P(M_{N}>x)} . Define I ( x ) = x ln ⁡ x + ( 1 − x ) ln ⁡ ( 1 − x ) + ln ⁡ 2 {\displaystyle I(x)=x\ln {x}+(1-x)\ln(1-x)+\ln {2}} . Note that the function I ( x ) {\displaystyle I(x)} is a convex, nonnegative function that is zero at x = 1 2 {\displaystyle x={\tfrac {1}{2}}} and increases as x {\displaystyle x} approaches 1 {\displaystyle 1} . It is the negative of the Bernoulli entropy with p = 1 2 {\displaystyle p={\tfrac {1}{2}}} ; that it's appropriate for coin tosses follows from the asymptotic equipartition property applied to a Bernoulli trial. Then by Chernoff's inequality, it can be shown that P ( M N > x ) < exp ⁡ ( − N I ( x ) ) {\displaystyle P(M_{N}>x)<\exp(-NI(x))} . This bound is rather sharp, in the sense that I ( x ) {\displaystyle I(x)} cannot be replaced with a larger number which would yield a strict inequality for all positive N {\displaystyle N} . (However, the exponential bound can still be reduced by a subexponential factor on the order of 1 / N {\displaystyle 1/{\sqrt {N}}} ; this follows from the Stirling approximation applied to the binomial coefficient appearing in the Bernoulli distribution.) Hence, we obtain the following result: P ( M N > x ) ≈ exp ⁡ ( − N I ( x ) ) {\displaystyle P(M_{N}>x)\approx \exp(-NI(x))} . The probability P ( M N > x ) {\displaystyle P(M_{N}>x)} decays exponentially as N → ∞ {\displaystyle N\to \infty } at a rate depending on x. This formula approximates any tail probability of the sample mean of i.i.d. variables and gives its convergence as the number of samples increases. === Large deviations for sums of independent random variables === In the above example of coin-tossing we explicitly assumed that each toss is an independent trial, and the probability of getting head or tail is always the same. Let X , X 1 , X 2 , … {\displaystyle X,X_{1},X_{2},\ldots } be independent and identically distributed (i.i.d.) random variables whose common distribution satisfies a certain growth condition. Then the following limit exists: lim N → ∞ 1 N ln ⁡ P ( M N > x ) = − I ( x ) {\displaystyle \lim _{N\to \infty }{\frac {1}{N}}\ln P(M_{N}>x)=-I(x)} . Here M N = 1 N ∑ i = 1 N X i {\displaystyle M_{N}={\frac {1}{N}}\sum _{i=1}^{N}X_{i}} , as before. Function I ( ⋅ ) {\displaystyle I(\cdot )} is called the "rate function" or "Cramér function" or sometimes the "entropy function". The above-mentioned limit means that for large N {\displaystyle N} , P ( M N > x ) ≈ exp ⁡ [ − N I ( x ) ] {\displaystyle P(M_{N}>x)\approx \exp[-NI(x)]} , which is the basic result of large deviations theory. If we know the probability distribution of X {\displaystyle X} , an explicit expression for the rate function can be obtained. This is given by a Legendre–Fenchel transformation, I ( x ) = sup θ > 0 [ θ x − λ ( θ ) ] {\displaystyle I(x)=\sup _{\theta >0}[\theta x-\lambda (\theta )]} , where λ ( θ ) = ln ⁡ E ⁡ [ exp ⁡ ( θ X ) ] {\displaystyle \lambda (\theta )=\ln \operatorname {E} [\exp(\theta X)]} is called the cumulant generating function (CGF) and E {\displaystyle \operatorname {E} } denotes the mathematical expectation. If X {\displaystyle X} follows a normal distribution, the rate function becomes a parabola with its apex at the mean of the normal distribution. If { X i } {\displaystyle \{X_{i}\}} is an irreducible and aperiodic Markov chain, the variant of the basic large deviations result stated above may hold. === Moderate deviations for sums of independent random variables === The previous example controlled the probability of the event [ M N > x ] {\displaystyle [M_{N}>x]} , that is, the concentration of the law of M N {\displaystyle M_{N}} on the compact set [ − x , x ] {\displaystyle [-x,x]} . It is also possible to control the probability of the event [ M N > x a N ] {\displaystyle [M_{N}>xa_{N}]} for some sequence a N → 0 {\displaystyle a_{N}\to 0} . The following is an example of a moderate deviations principle: In particular, the limit case a N = N {\displaystyle a_{N}={\sqrt {N}}} is the central limit theorem. == Formal definition == Given a Polish space X {\displaystyle {\mathcal {X}}} let { P N } {\displaystyle \{\mathbb {P} _{N}\}} be a sequence of Borel probability measures on X {\displaystyle {\mathcal {X}}} , let { a N } {\displaystyle \{a_{N}\}} be a sequence of positive real numbers such that lim N a N = ∞ {\displaystyle \lim _{N}a_{N}=\infty } , and finally let I : X → [ 0 , ∞ ] {\displaystyle I:{\mathcal {X}}\to [0,\infty ]} be a lower semicontinuous functional on X . {\displaystyle {\mathcal {X}}.} The sequence { P N } {\displaystyle \{\mathbb {P} _{N}\}} is said to satisfy a large deviation principle with speed { a n } {\displaystyle \{a_{n}\}} and rate I {\displaystyle I} if, and only if, for each Borel measurable set E ⊂ X {\displaystyle E\subset {\mathcal {X}}} , − inf x ∈ E ∘ I ( x ) ≤ lim _ N ⁡ a N − 1 log ⁡ ( P N ( E ) ) ≤ lim ¯ N ⁡ a N − 1 log ⁡ ( P N ( E ) ) ≤ − inf x ∈ E ¯ I ( x ) {\displaystyle -\inf _{x\in E^{\circ }}I(x)\leq \varliminf _{N}a_{N}^{-1}\log(\mathbb {P} _{N}(E))\leq \varlimsup _{N}a_{N}^{-1}\log(\mathbb {P} _{N}(E))\leq -\inf _{x\in {\overline {E}}}I(x)} , where E ¯ {\displaystyle {\overline {E}}} and E ∘ {\displaystyle E^{\circ }} denote respectively the closure and interior of E {\displaystyle E} . == Brief history == The first rigorous results concerning large deviations are due to the Swedish mathematician Harald Cramér, who applied them to model the insurance business. From the point of view of an insurance company, the earning is at a constant rate per month (the monthly premium) but the claims come randomly. For the company to be successful over a certain period of time (preferably many months), the total earning should exceed the total claim. Thus to estimate the premium you have to ask the following question: "What should we choose as the premium q {\displaystyle q} such that over N {\displaystyle N} months the total claim C = Σ X i {\displaystyle C=\Sigma X_{i}} should be less than N q {\displaystyle Nq} ?" This is clearly the same question asked by the large deviations theory. Cramér gave a solution to this question for i.i.d. random variables, where the rate function is expressed as a power series. A very incomplete list of mathematicians who have made important advances would include Petrov, Sanov, S.R.S. Varadhan (who has won the Abel prize for his contribution to the theory), D. Ruelle, O.E. Lanford, Mark Freidlin, Alexander D. Wentzell, Amir Dembo, and Ofer Zeitouni. == Applications == Principles of large deviations may be effectively applied to gather information out of a probabilistic model. Thus, theory of large deviations finds its applications in information theory and risk management. In physics, the best known application of large deviations theory arise in thermodynamics and statistical mechanics (in connection with relating entropy with rate function). === Large deviations and entropy === The rate function is related to the entropy in statistical mechanics. This can be heuristically seen in the following way. In statistical mechanics the entropy of a particular macro-state is related to the number of micro-states which corresponds to this macro-state. In our coin tossing example the mean value M N {\displaystyle M_{N}} could designate a particular macro-state. And the particular sequence of heads and tails which gives rise to a particular value of M N {\displaystyle M_{N}} constitutes a particular micro-state. Loosely speaking a macro-state having a higher number of micro-states giving rise to it, has higher entropy. And a state with higher entropy has a higher chance of being realised in actual experiments. The macro-state with mean value of 1/2 (as many heads as tails) has the highest number of micro-states giving rise to it and it is indeed the state with the highest entropy. And in most practical situations we shall indeed obtain this macro-state for large numbers of trials. The "rate function" on the other hand measures the probability of appearance of a particular macro-state. The smaller the rate function the higher is the chance of a macro-state appearing. In our coin-tossing the value of the "rate function" for mean value equal to 1/2 is zero. In this way one can see the "rate function" as the negative of the "entropy". There is a relation between the "rate function" in large deviations theory and the Kullback–Leibler divergence, the connection is established by Sanov's theorem (see Sanov and Novak, ch. 14.5). In a special case, large deviations are closely related to the concept of Gromov–Hausdorff limits. == See also == Large deviation principle Cramér's large deviation theorem Chernoff's inequality Sanov's theorem Contraction principle (large deviations theory), a result on how large deviations principles "push forward" Freidlin–Wentzell theorem, a large deviations principle for Itō diffusions Legendre transformation, Ensemble equivalence is based on this transformation. Laplace principle, a large deviations principle in Rd Laplace's method Schilder's theorem, a large deviations principle for Brownian motion Varadhan's lemma Extreme value theory Large deviations of Gaussian random functions == References == == Bibliography == Special invited paper: Large deviations by S. R. S. Varadhan The Annals of Probability 2008, Vol. 36, No. 2, 397–419 doi:10.1214/07-AOP348 A basic introduction to large deviations: Theory, applications, simulations, Hugo Touchette, arXiv:1106.4146. Entropy, Large Deviations and Statistical Mechanics by R.S. Ellis, Springer Publication. ISBN 3-540-29059-1 Large Deviations for Performance Analysis by Alan Weiss and Adam Shwartz. Chapman and Hall ISBN 0-412-06311-5 Large Deviations Techniques and Applications by Amir Dembo and Ofer Zeitouni. Springer ISBN 0-387-98406-2 A course on large deviations with an introduction to Gibbs measures by Firas Rassoul-Agha and Timo Seppäläinen. Grad. Stud. Math., 162. American Mathematical Society ISBN 978-0-8218-7578-0 Random Perturbations of Dynamical Systems by M.I. Freidlin and A.D. Wentzell. Springer ISBN 0-387-98362-7 "Large Deviations for Two Dimensional Navier-Stokes Equation with Multiplicative Noise", S. S. Sritharan and P. Sundar, Stochastic Processes and Their Applications, Vol. 116 (2006) 1636–1659.[2] "Large Deviations for the Stochastic Shell Model of Turbulence", U. Manna, S. S. Sritharan and P. Sundar, NoDEA Nonlinear Differential Equations Appl. 16 (2009), no. 4, 493–521.[3]
Wikipedia:Control variable#0	A control variable (or scientific constant) in scientific experimentation is an experimental element which is constant (controlled) and unchanged throughout the course of the investigation. Control variables could strongly influence experimental results were they not held constant during the experiment in order to test the relative relationship of the dependent variable (DV) and independent variable (IV). The control variables themselves are not of primary interest to the experimenter. "Good controls", also known as “confounders” or “deconfounders”, are variables which are theorized to be unaffected by the treatment and which are intended to eliminate omitted-variable bias. "Bad controls", on the other hand, are variables that could be affected by the treatment, might contribute to collider bias, and lead to erroneous results. == Usage == A variable in an experiment which is held constant in order to assess the relationship between multiple variables, is a control variable. A control variable is an element that is not changed throughout an experiment because its unchanging state allows better understanding of the relationship between the other variables being tested. In any system existing in a natural state, many variables may be interdependent, with each affecting the other. Scientific experiments test the relationship of an IV (or independent variable: that element that is manipulated by the experimenter) to the DV (or dependent variable: that element affected by the manipulation of the IV). Any additional independent variable can be a control variable. A control variable is an experimental condition or element that is kept the same throughout the experiment, and it is not of primary concern in the experiment, nor will it influence the outcome of the experiment. Any unexpected (e.g.: uncontrolled) change in a control variable during an experiment would invalidate the correlation of dependent variables (DV) to the independent variable (IV), thus skewing the results, and invalidating the working hypothesis. This indicates the presence of a spurious relationship existing within experimental parameters. Unexpected results may result from the presence of a confounding variable, thus requiring a re-working of the initial experimental hypothesis. Confounding variables are a threat to the internal validity of an experiment. This situation may be resolved by first identifying the confounding variable and then redesigning the experiment taking that information into consideration. One way to this is to control the confounding variable, thus making it a control variable. If, however, the spurious relationship cannot be identified, the working hypothesis may have to be abandoned. == Experimental examples == Take, for example, the well known combined gas law, which is stated mathematically as: P V T = k {\displaystyle \qquad {\frac {PV}{T}}=k} where: P is the pressure V is the volume T is the thermodynamic temperature measured in kelvins k is a constant (with units of energy divided by temperature). which shows that the ratio between the pressure-volume product and the temperature of a system remains constant. In an experimental verification of parts of the combined gas law, (P * V = T), where Pressure, Temperature, and Volume are all variables, to test the resultant changes to any of these variables requires at least one to be kept constant. This is in order to see comparable experimental results in the remaining variables. If Temperature is made the control variable and it is not allowed to change throughout the course of the experiment, the relationship between the dependent variables, Pressure, and Volume, can quickly be established by changing the value for one or the other, and this is Boyle's law. For instance, if the Pressure is raised then the Volume must decrease. If, however, Volume is made the control variable and it is not allowed to change throughout the course of the experiment, the relationship between dependent variables, Pressure, and Temperature, can quickly be established by changing the value for one or the other, and this is Gay-Lussac's law. For instance, if the Pressure is raised then the Temperature must increase. == Notes == == References == == External links == Definitions; Science Buddies – Science Fair Projects.
Wikipedia:Controlled invariant subspace#0	In control theory, a controlled invariant subspace of the state space representation of some system is a subspace. If the system's state is initially in the subspace, it can be controlled so that the state is always in the subspace. This concept was introduced by Giuseppe Basile and Giovanni Marro (Basile & Marro 1969). == Definition == Consider a linear system described by the differential equation x ˙ ( t ) = A x ( t ) + B u ( t ) . {\displaystyle {\dot {\mathbf {x} }}(t)=A\mathbf {x} (t)+B\mathbf {u} (t).} Here, x(t) ∈ Rn denotes the system's state, and u(t) ∈ Rp is the input. The matrices A and B have sizes n × n and n × p, respectively. A subspace V ⊂ Rn is a controlled invariant subspace if, for any x(0) ∈ V, there is an input u(t) such that x(t) ∈ V for all nonnegative t. == Properties == A subspace V ⊂ Rn is a controlled invariant subspace if and only if AV ⊂ V + Im B. If V is a controlled invariant subspace, then there exists a matrix K such that the input u(t) = Kx(t) keeps the state within V; this is a simple feedback control (Ghosh 1985, Thm 1.1). == References == Basile, Giuseppe; Marro, Giovanni (1969), "Controlled and conditioned invariant subspaces in linear system theory", Journal of Optimization Theory and Applications, 3 (5): 306–315, doi:10.1007/BF00931370, S2CID 120847885. Ghosh, Bijoy K. (1985), "Controlled invariant and feedback controlled invariant subspaces in the design of a generalized dynamical system", Proceedings of the 24th IEEE Conference on Decision and Control, IEEE, pp. 872–873, doi:10.1109/CDC.1985.268620, S2CID 9644586. Basile, Giuseppe; Marro, Giovanni (1992), Controlled and Conditioned Invariants in Linear System Theory, Englewood Cliffs : Prentice-Hall.
Wikipedia:Convergence proof techniques#0	Convergence proof techniques are canonical patterns of mathematical proofs that sequences or functions converge to a finite limit when the argument tends to infinity. There are many types of sequences and modes of convergence, and different proof techniques may be more appropriate than others for proving each type of convergence of each type of sequence. Below are some of the more common and typical examples. This article is intended as an introduction aimed to help practitioners explore appropriate techniques. The links below give details of necessary conditions and generalizations to more abstract settings. Proof techniques for the convergence of series, a particular type of sequences corresponding to sums of many terms, are covered in the article on convergence tests. == Convergence in Rn == It is common to want to prove convergence of a sequence f : N → R n {\displaystyle f:\mathbb {N} \rightarrow \mathbb {R} ^{n}} or function f : R → R n {\displaystyle f:\mathbb {R} \rightarrow \mathbb {R} ^{n}} , where N {\displaystyle \mathbb {N} } and R {\displaystyle \mathbb {R} } refer to the natural numbers and the real numbers, respectively, and convergence is with respect to the Euclidean norm, \| \| ⋅ \| \| 2 {\displaystyle \|\|\cdot \|\|_{2}} . Useful approaches for this are as follows. === First principles === The analytic definition of convergence of f {\displaystyle f} to a limit f ∞ {\displaystyle f_{\infty }} is that for all ϵ {\displaystyle \epsilon } there exists a k 0 ∈ N {\displaystyle k_{0}\in \mathbb {N} } such for all k > k 0 {\displaystyle k>k_{0}} , ‖ f ( k ) − f ∞ ‖ < ϵ {\displaystyle \\|f(k)-f_{\infty }\\|<\epsilon } . The most direct proof technique from this definition is to find such a k 0 {\displaystyle k_{0}} and prove the required inequality. If the value of f ∞ {\displaystyle f_{\infty }} is not known in advance, the techniques below may be useful. === Contraction mappings === In many cases, the function whose convergence is of interest has the form f ( k + 1 ) = T ( f ( k ) ) {\displaystyle f(k+1)=T(f(k))} for some transformation T {\displaystyle T} . For example, T {\displaystyle T} could map f ( k ) {\displaystyle f(k)} to f ( k + 1 ) = A f ( k ) {\displaystyle f(k+1)=Af(k)} for some conformable matrix A {\displaystyle A} , so that f ( k ) = A k f ( 0 ) {\displaystyle f(k)=A^{k}f(0)} , a matrix generalization of the geometric progression. Alternatively, T {\displaystyle T} may be an elementwise operation, such as replacing each element of f ( k ) {\displaystyle f(k)} by the square root of its magnitude. In such cases, if the problem satisfies the conditions of Banach fixed-point theorem (the domain is a non-empty complete metric space) then it is sufficient to prove convergence to prove that T {\displaystyle T} is a contraction mapping to prove that it has a fixed point. This requires that ‖ T ( x ) − T ( y ) ‖ < ‖ λ ( x − y ) ‖ {\displaystyle \\|T(x)-T(y)\\|<\\|\lambda (x-y)\\|} for some constant \| λ \| < 1 {\displaystyle \|\lambda \|<1} which is fixed for all x {\displaystyle x} and y {\displaystyle y} . The composition of two contraction mappings is a contraction mapping, so if T = T 1 ∘ T 2 {\displaystyle T=T_{1}\circ T_{2}} , then it is sufficient to show that T 1 {\displaystyle T_{1}} and T 2 {\displaystyle T_{2}} are both contraction mappings. ==== Example ==== Famous examples of applications of this approach include If T {\displaystyle T} has the form T ( x ) = A x + B {\displaystyle T(x)=Ax+B} for some matrices A {\displaystyle A} and B {\displaystyle B} , then T k ( x ) {\displaystyle T^{k}(x)} converges to ( I − A ) − 1 B {\displaystyle (I-A)^{-1}B} if the magnitudes of all eigenvalues of A {\displaystyle A} are less than 1. ==== Non-expansion mappings ==== If both above inequalities in the definition of a contraction mapping are weakened from "strictly less than" to "less than or equal to", the mapping is a non-expansion mapping. It is not sufficient to prove convergence to prove that T {\displaystyle T} is a non-expansion mapping. For example, T ( x ) = − x {\displaystyle T(x)=-x} is a non-expansion mapping, but the sequence T n ( x ) {\displaystyle T^{n}(x)} does not converge for any x ≠ 0 {\displaystyle x\neq 0} . However, the composition of a contraction mapping and a non-expansion mapping (or vice versa) is a contraction mapping. ==== Contraction mappings on limited domains ==== If T {\displaystyle T} is not a contraction mapping on its entire domain, but it is on its codomain (the image of the domain), that is also sufficient for convergence. This also applies for decompositions. For example, consider T ( x ) = cos ⁡ ( sin ⁡ ( x ) ) {\displaystyle T(x)=\cos(\sin(x))} . The function cos {\displaystyle \cos } is not a contraction mapping, but it is on the restricted domain [ − 1 , 1 ] {\displaystyle [-1,1]} , which is the codomain of sin {\displaystyle \sin } for real arguments. Since sin {\displaystyle \sin } is a non-expansion mapping, this implies T {\displaystyle T} is a contraction mapping. === Convergent subsequences === Every bounded sequence in R n {\displaystyle \mathbb {R} ^{n}} has a convergent subsequence, by the Bolzano–Weierstrass theorem. If these subsequences all have the same limit, then the original sequence also converges to that limit. If it can be shown that all of the subsequences of f {\displaystyle f} must have the same limit, such as by showing that there is a unique fixed point of the transformation T {\displaystyle T} and that there are no invariant sets of T {\displaystyle T} that contain no fixed points of T {\displaystyle T} , then the initial sequence must also converge to that limit. === Monotonicity (Lyapunov functions) === Every bounded monotonic sequence in R n {\displaystyle \mathbb {R} ^{n}} converges to a limit. This fact can be used directly and can also be used to prove the convergence of sequences that are not monotonic using techniques and theorems named for Aleksandr Lyapunov. In these cases, one defines a function V : R n → R {\displaystyle V:\mathbb {R} ^{n}\rightarrow \mathbb {R} } such that V ( f ( k ) ) {\displaystyle V(f(k))} is monotonic in k {\displaystyle k} and thus V ( f ( k ) ) {\displaystyle V(f(k))} converges. If V {\displaystyle V} satisfies the conditions to be a Lyapunov function then Lyapunov's theorem implies that f {\displaystyle f} is also convergent. Lyapunov's theorem is normally stated for ordinary differential equations, but it can also be applied to sequences of iterates by replacing derivatives with discrete differences. The basic requirements on V {\displaystyle V} to be a Lyapunov function are that V ( x ) > 0 {\displaystyle V(x)>0} for all x ≠ 0 {\displaystyle x\neq 0} and V ( 0 ) = 0 {\displaystyle V(0)=0} V ( f ( k + 1 ) ) − V ( f ( k ) ) < 0 {\displaystyle V(f(k+1))-V(f(k))<0} for f ( k ) ≠ 0 {\displaystyle f(k)\neq 0} (discrete case) or V ˙ ( x ) < 0 {\displaystyle {\dot {V}}(x)<0} for x ≠ 0 {\displaystyle x\neq 0} (continuous case) V {\displaystyle V} is "radially unbounded", i.e., that lim k → ∞ V ( f ( k ) ) = ∞ {\textstyle \lim _{k\rightarrow \infty }V(f(k))=\infty } for any sequence with lim k → ∞ \| \| f ( k ) \| \| = ∞ {\textstyle \lim _{k\rightarrow \infty }\|\|f(k)\|\|=\infty } . In many cases a quadratic Lyapunov function of the form V ( x ) = x T A x {\displaystyle V(x)=x^{T}Ax} can be found, although more complex forms are also common, for instance entropies in the study of convergence of probability distributions. For delay differential equations, a similar approach applies with Lyapunov functions replaced by Lyapunov functionals also called Lyapunov-Krasovskii functionals. If the inequality in the condition 2 is weak, LaSalle's invariance principle may be used. == Convergence of sequences of functions == To consider the convergence of sequences of functions, it is necessary to define a distance between functions to replace the Euclidean norm. These often include Convergence in the norm (strong convergence) -- a function norm, such as ‖ g ‖ f = ∫ x ∈ A ‖ g ( x ) ‖ d x {\textstyle \\|g\\|_{f}=\int _{x\in A}\\|g(x)\\|dx} is defined, and convergence occurs if \| \| f ( n ) − f ∞ \| \| f → 0 {\displaystyle \|\|f(n)-f_{\infty }\|\|_{f}\rightarrow 0} . For this case, all of the above techniques can be applied with this function norm. Pointwise convergence -- convergence occurs if for each x {\displaystyle x} , f n ( x ) → f ∞ ( x ) {\displaystyle f_{n}(x)\rightarrow f_{\infty }(x)} . For this case, the above techniques can be applied for each point x {\displaystyle x} with the norm appropriate for f ( x ) {\displaystyle f(x)} . uniform convergence -- In pointwise convergence, some (open) regions can converge arbitrarily slowly. With uniform convergence, there is a fixed convergence rate such that all points converge at least that fast. Formally, lim n → ∞ sup { \| f n ( x ) − f ∞ ( x ) \| : x ∈ A } = 0 , {\displaystyle \lim _{n\to \infty }\,\sup\{\,\left\|f_{n}(x)-f_{\infty }(x)\right\|:x\in A\,\}=0,} where A {\displaystyle A} is the domain of each f n {\displaystyle f_{n}} . See also Convergence of Fourier series == Convergence of random variables == Random variables are more complicated than simple elements of R n {\displaystyle \mathbb {R} ^{n}} . (Formally, a random variable is a mapping x : Ω → V {\displaystyle x:\Omega \rightarrow V} from an event space Ω {\displaystyle \Omega } to a value space V {\displaystyle V} . The value space may be R n {\displaystyle \mathbb {R} ^{n}} , such as the roll of a dice, and such a random variable is often spoken of informally as being in R n {\displaystyle \mathbb {R} ^{n}} , but convergence of sequence of random variables corresponds to convergence of the sequence of functions, or the distributions, rather than the sequence of values.) There are multiple types of convergence, depending on how the distance between functions is measured. Convergence in distribution -- pointwise convergence of the distribution functions of the random variables to the limit Convergence in probability Almost sure convergence -- pointwise convergence of the mappings x n : Ω → V {\displaystyle x_{n}:\Omega \rightarrow V} to the limit, except at a set in Ω {\displaystyle \Omega } with measure 0 in the limit. Convergence in the mean Each has its own proof techniques, which are beyond the current scope of this article. See also Dominated convergence Carleson's theorem establishing the pointwise (Lebesgue) almost everywhere convergence of Fourier series of L2 functions Doob's martingale convergence theorems a random variable analogue of the monotone convergence theorem == Topological convergence == For all of the above techniques, some form the basic analytic definition of convergence above applies. However, topology has its own definitions of convergence. For example, in a non-Hausdorff space, it is possible for a sequence to converge to multiple different limits. == References ==
Wikipedia:Convex combination#0	In convex geometry and vector algebra, a convex combination is a linear combination of points (which can be vectors, scalars, or more generally points in an affine space) where all coefficients are non-negative and sum to 1. In other words, the operation is equivalent to a standard weighted average, but whose weights are expressed as a percent of the total weight, instead of as a fraction of the count of the weights as in a standard weighted average. == Formal definition == More formally, given a finite number of points x 1 , x 2 , … , x n {\displaystyle x_{1},x_{2},\dots ,x_{n}} in a real vector space, a convex combination of these points is a point of the form α 1 x 1 + α 2 x 2 + ⋯ + α n x n {\displaystyle \alpha _{1}x_{1}+\alpha _{2}x_{2}+\cdots +\alpha _{n}x_{n}} where the real numbers α i {\displaystyle \alpha _{i}} satisfy α i ≥ 0 {\displaystyle \alpha _{i}\geq 0} and α 1 + α 2 + ⋯ + α n = 1. {\displaystyle \alpha _{1}+\alpha _{2}+\cdots +\alpha _{n}=1.} As a particular example, every convex combination of two points lies on the line segment between the points. A set is convex if it contains all convex combinations of its points. The convex hull of a given set of points is identical to the set of all their convex combinations. There exist subsets of a vector space that are not closed under linear combinations but are closed under convex combinations. For example, the interval [ 0 , 1 ] {\displaystyle [0,1]} is convex but generates the real-number line under linear combinations. Another example is the convex set of probability distributions, as linear combinations preserve neither nonnegativity nor affinity (i.e., having total integral one). == Other objects == A random variable X {\displaystyle X} is said to have an n {\displaystyle n} -component finite mixture distribution if its probability density function is a convex combination of n {\displaystyle n} so-called component densities. == Related constructions == A conical combination is a linear combination with nonnegative coefficients. When a point x {\displaystyle x} is to be used as the reference origin for defining displacement vectors, then x {\displaystyle x} is a convex combination of n {\displaystyle n} points x 1 , x 2 , … , x n {\displaystyle x_{1},x_{2},\dots ,x_{n}} if and only if the zero displacement is a non-trivial conical combination of their n {\displaystyle n} respective displacement vectors relative to x {\displaystyle x} . Weighted means are functionally the same as convex combinations, but they use a different notation. The coefficients (weights) in a weighted mean are not required to sum to 1; instead the weighted linear combination is explicitly divided by the sum of the weights. Affine combinations are like convex combinations, but the coefficients are not required to be non-negative. Hence affine combinations are defined in vector spaces over any field. == See also == Affine hull Carathéodory's theorem (convex hull) Simplex Barycentric coordinate system Convex space == References == == External links == Convex sum/combination with a triangle - interactive illustration Convex sum/combination with a hexagon - interactive illustration Convex sum/combination with a tetraeder - interactive illustration
Wikipedia:Convex cone#0	In linear algebra, a cone—sometimes called a linear cone to distinguish it from other sorts of cones—is a subset of a real vector space that is closed under positive scalar multiplication; that is, C {\displaystyle C} is a cone if x ∈ C {\displaystyle x\in C} implies s x ∈ C {\displaystyle sx\in C} for every positive scalar s {\displaystyle s} . This is a broad generalization of the standard cone in Euclidean space. A convex cone is a cone that is also closed under addition, or, equivalently, a subset of a vector space that is closed under linear combinations with positive coefficients. It follows that convex cones are convex sets. The definition of a convex cone makes sense in a vector space over any ordered field, although the field of real numbers is used most often. == Definition == A subset C {\displaystyle C} of a vector space is a cone if x ∈ C {\displaystyle x\in C} implies s x ∈ C {\displaystyle sx\in C} for every s > 0 {\displaystyle s>0} . Here s > 0 {\displaystyle s>0} refers to (strict) positivity in the scalar field. === Competing definitions === Some other authors require [ 0 , ∞ ) C ⊂ C {\displaystyle [0,\infty )C\subset C} or even 0 ∈ C {\displaystyle 0\in C} . Some require a cone to be convex and/or satisfy C ∩ − C ⊂ { 0 } {\displaystyle C\cap -C\subset \{0\}} . The conical hull of a set C {\displaystyle C} is defined as the smallest convex cone that contains C ∪ { 0 } {\displaystyle C\cup \{0\}} . Therefore, it need not be the smallest cone that contains C ∪ { 0 } {\displaystyle C\cup \{0\}} . Wedge may refer to what we call cones (when "cone" is reserved for something stronger), or just to a subset of them, depending on the author. === Cone: 0 or not === A subset C {\displaystyle C} of a vector space V {\displaystyle V} over an ordered field F {\displaystyle F} is a cone (or sometimes called a linear cone) if for each x {\displaystyle x} in C {\displaystyle C} and positive scalar α {\displaystyle \alpha } in F {\displaystyle F} , the product α x {\displaystyle \alpha x} is in C {\displaystyle C} . Note that some authors define cone with the scalar α {\displaystyle \alpha } ranging over all non-negative scalars (rather than all positive scalars, which does not include 0). Some authors even require 0 ∈ C {\displaystyle 0\in C} , thus excluding the empty set. Therefore, [ 0 , ∞ ) {\displaystyle [0,\infty )} is a cone, ∅ {\displaystyle \varnothing } is a cone only according to the 1st and 2nd definition above, and ( 0 , ∞ ) {\displaystyle (0,\infty )} is a cone only according to the 1st definition above. All of them are convex (see below). === Convex cone === A cone C {\displaystyle C} is a convex cone if α x + β y {\displaystyle \alpha x+\beta y} belongs to C {\displaystyle C} , for any positive scalars α {\displaystyle \alpha } , β {\displaystyle \beta } , and any x {\displaystyle x} , y {\displaystyle y} in C {\displaystyle C} . A cone C {\displaystyle C} is convex if and only if C + C ⊆ C {\displaystyle C+C\subseteq C} . This concept is meaningful for any vector space that allows the concept of "positive" scalar, such as spaces over the rational, algebraic, or (more commonly) the real numbers. Also note that the scalars in the definition are positive meaning that the origin does not have to belong to C {\displaystyle C} . Some authors use a definition that ensures the origin belongs to C {\displaystyle C} . Because of the scaling parameters α {\displaystyle \alpha } and β {\displaystyle \beta } , cones are infinite in extent and not bounded. If C {\displaystyle C} is a convex cone, then for any positive scalar α {\displaystyle \alpha } and any x {\displaystyle x} in C {\displaystyle C} the vector α x = α 2 x + α 2 x ∈ C . {\displaystyle \alpha x={\tfrac {\alpha }{2}}x+{\tfrac {\alpha }{2}}x\in C.} So a convex cone is a special case of a linear cone as defined above. It follows from the above property that a convex cone can also be defined as a linear cone that is closed under convex combinations, or just under addition. More succinctly, a set C {\displaystyle C} is a convex cone if and only if α C = C {\displaystyle \alpha C=C} for every positive scalar α {\displaystyle \alpha } and C + C = C {\displaystyle C+C=C} . === Face of a convex cone === A face of a convex cone C {\displaystyle C} is a subset F {\displaystyle F} of C {\displaystyle C} such that F {\displaystyle F} is also a convex cone, and for any vectors x , y {\displaystyle x,y} in C {\displaystyle C} with x + y {\displaystyle x+y} in F {\displaystyle F} , x {\displaystyle x} and y {\displaystyle y} must both be in F {\displaystyle F} . For example, C {\displaystyle C} itself is a face of C {\displaystyle C} . The origin { 0 } {\displaystyle \{0\}} is a face of C {\displaystyle C} if C {\displaystyle C} contains no line (so C {\displaystyle C} is "strictly convex", or "salient", as defined below). The origin and C {\displaystyle C} are sometimes called the trivial faces of C {\displaystyle C} . A ray (the set of nonnegative multiples of a nonzero vector) is called an extremal ray if it is a face of C {\displaystyle C} . Let C {\displaystyle C} be a closed, strictly convex cone in R n {\displaystyle \mathbb {R} ^{n}} . Suppose that C {\displaystyle C} is more than just the origin. Then C {\displaystyle C} is the convex hull of its extremal rays. == Examples == For a vector space V {\displaystyle V} , every linear subspace of V {\displaystyle V} is a convex cone. In particular, the space V {\displaystyle V} itself and the origin { 0 } {\displaystyle \{0\}} are convex cones in V {\displaystyle V} . For authors who do not require a convex cone to contain the origin, the empty set ∅ {\displaystyle \emptyset } is also a convex cone. The conical hull of a finite or infinite set of vectors in R n {\displaystyle \mathbb {R} ^{n}} is a convex cone. The tangent cones of a convex set are convex cones. The set { x ∈ R 2 ∣ x 2 ≥ 0 , x 1 = 0 } ∪ { x ∈ R 2 ∣ x 1 ≥ 0 , x 2 = 0 } {\displaystyle \left\{x\in \mathbb {R} ^{2}\mid x_{2}\geq 0,x_{1}=0\right\}\cup \left\{x\in \mathbb {R} ^{2}\mid x_{1}\geq 0,x_{2}=0\right\}} is a cone but not a convex cone. The norm cone C = { ( x , r ) ∈ R d + 1 ∣ ‖ x ‖ ≤ r } {\displaystyle C=\left\{(x,r)\in \mathbb {R} ^{d+1}\mid \\|x\\|\leq r\right\}} is a convex cone. (For d = 2 {\displaystyle d=2} , this is the round cone in the figure.) Each extremal ray of C {\displaystyle C} is spanned by a vector ( x , 1 ) {\displaystyle (x,1)} with ‖ x ‖ = 1 {\displaystyle \\|x\\|=1} (so x {\displaystyle x} is a point in the sphere S d − 1 {\displaystyle S^{d-1}} ). These rays are in fact the only nontrivial faces of C {\displaystyle C} . The intersection of two convex cones in the same vector space is again a convex cone, but their union may fail to be one. The class of convex cones is also closed under arbitrary linear maps. In particular, if C {\displaystyle C} is a convex cone, so is its opposite − C {\displaystyle -C} , and C ∩ − C {\displaystyle C\cap -C} is the largest linear subspace contained in C {\displaystyle C} . The set of positive semidefinite matrices. The set of nonnegative continuous functions is a convex cone. == Special examples == === Affine convex cones === An affine convex cone is the set resulting from applying an affine transformation to a convex cone. A common example is translating a convex cone by a point p: p + C. Technically, such transformations can produce non-cones. For example, unless p = 0, p + C is not a linear cone. However, it is still called an affine convex cone. === Half-spaces === A (linear) hyperplane is a set in the form { x ∈ V ∣ f ( x ) = c } {\displaystyle \{x\in V\mid f(x)=c\}} where f is a linear functional on the vector space V. A closed half-space is a set in the form { x ∈ V ∣ f ( x ) ≤ c } {\displaystyle \{x\in V\mid f(x)\leq c\}} or { x ∈ V ∣ f ( x ) ≥ c } , {\displaystyle \{x\in V\mid f(x)\geq c\},} and likewise an open half-space uses strict inequality. Half-spaces (open or closed) are affine convex cones. Moreover (in finite dimensions), any convex cone C that is not the whole space V must be contained in some closed half-space H of V; this is a special case of Farkas' lemma. === Polyhedral and finitely generated cones === Polyhedral cones are special kinds of cones that can be defined in several ways:: 256–257 A cone C {\displaystyle C} is polyhedral if it is the conical hull of finitely many vectors (this property is also called finitely-generated). I.e., there is a set of vectors { v 1 , … , v k } ⊂ R n {\displaystyle \{v_{1},\ldots ,v_{k}\}\subset \mathbb {R} ^{n}} so that C = { a 1 v 1 + ⋯ + a k v k ∣ a i ∈ R ≥ 0 } {\displaystyle C=\{a_{1}v_{1}+\cdots +a_{k}v_{k}\mid a_{i}\in \mathbb {R} _{\geq 0}\}} . A cone is polyhedral if it is the intersection of a finite number of half-spaces which have 0 on their boundary (the equivalence between these first two definitions was proved by Weyl in 1935). A cone C {\displaystyle C} is polyhedral if there is some matrix A ∈ R m × n {\displaystyle A\in \mathbb {R} ^{m\times n}} such that C = { x ∈ R n ∣ A x ≥ 0 } {\displaystyle C=\{x\in \mathbb {R} ^{n}\mid Ax\geq 0\}} . A cone is polyhedral if it is the solution set of a system of homogeneous linear inequalities. Algebraically, each inequality is defined by a row of the matrix A {\displaystyle A} . Geometrically, each inequality defines a halfspace that passes through the origin. Every finitely generated cone is a polyhedral cone, and every polyhedral cone is a finitely generated cone. Every polyhedral cone has a unique representation as a conical hull of its extremal generators, and a unique representation of intersections of halfspaces, given each linear form associated with the halfspaces also define a support hyperplane of a facet. Each face of a polyhedral cone is spanned by some subset of its extremal generators. As a result, a polyhedral cone has only finitely many faces. Polyhedral cones play a central role in the representation theory of polyhedra. For instance, the decomposition theorem for polyhedra states that every polyhedron can be written as the Minkowski sum of a convex polytope and a polyhedral cone. Polyhedral cones also play an important part in proving the related Finite Basis Theorem for polytopes which shows that every polytope is a polyhedron and every bounded polyhedron is a polytope. The two representations of a polyhedral cone - by inequalities and by vectors - may have very different sizes. For example, consider the cone of all non-negative n {\displaystyle n} -by- n {\displaystyle n} matrices with equal row and column sums. The inequality representation requires n 2 {\displaystyle n^{2}} inequalities and 2 n − 1 {\displaystyle 2n-1} equations, but the vector representation requires n ! {\displaystyle n!} vectors (see the Birkhoff-von Neumann Theorem). The opposite can also happen - the number of vectors may be polynomial while the number of inequalities is exponential.: 256 The two representations together provide an efficient way to decide whether a given vector is in the cone: to show that it is in the cone, it is sufficient to present it as a conic combination of the defining vectors; to show that it is not in the cone, it is sufficient to present a single defining inequality that it violates. This fact is known as Farkas' lemma. A subtle point in the representation by vectors is that the number of vectors may be exponential in the dimension, so the proof that a vector is in the cone might be exponentially long. Fortunately, Carathéodory's theorem guarantees that every vector in the cone can be represented by at most d {\displaystyle d} defining vectors, where d {\displaystyle d} is the dimension of the space. === Blunt, pointed, flat, salient, and proper cones === According to the above definition, if C is a convex cone, then C ∪ {0} is a convex cone, too. A convex cone is said to be pointed if 0 is in C, and blunt if 0 is not in C. Some authors use "pointed" for C ∩ − C = { 0 } {\displaystyle C\cap -C=\{0\}} or salient (see below). Blunt cones can be excluded from the definition of convex cone by substituting "non-negative" for "positive" in the condition of α, β. A cone is called flat if it contains some nonzero vector x and its opposite −x, meaning C contains a linear subspace of dimension at least one, and salient (or strictly convex) otherwise. A blunt convex cone is necessarily salient, but the converse is not necessarily true. A convex cone C is salient if and only if C ∩ −C ⊆ {0}. A cone C is said to be generating if C − C = { x − y ∣ x ∈ C , y ∈ C } {\displaystyle C-C=\{x-y\mid x\in C,y\in C\}} equals the whole vector space. Some authors require salient cones to be pointed. The term "pointed" is also often used to refer to a closed cone that contains no complete line (i.e., no nontrivial subspace of the ambient vector space V, or what is called a salient cone). The term proper (convex) cone is variously defined, depending on the context and author. It often means a cone that satisfies other properties like being convex, closed, pointed, salient, and full-dimensional. Because of these varying definitions, the context or source should be consulted for the definition of these terms. === Rational cones === A type of cone of particular interest to pure mathematicians is the partially ordered set of rational cones. "Rational cones are important objects in toric algebraic geometry, combinatorial commutative algebra, geometric combinatorics, integer programming.". This object arises when we study cones in R d {\displaystyle \mathbb {R} ^{d}} together with the lattice Z d {\displaystyle \mathbb {Z} ^{d}} . A cone is called rational (here we assume "pointed", as defined above) whenever its generators all have integer coordinates, i.e., if C {\displaystyle C} is a rational cone, then C = { a 1 v 1 + ⋯ + a k v k ∣ a i ∈ R + } {\displaystyle C=\{a_{1}v_{1}+\cdots +a_{k}v_{k}\mid a_{i}\in \mathbb {R} _{+}\}} for some v i ∈ Z d {\displaystyle v_{i}\in \mathbb {Z} ^{d}} . == Dual cone == Let C ⊂ V be a set, not necessarily a convex set, in a real vector space V equipped with an inner product. The (continuous or topological) dual cone to C is the set C ∗ = { v ∈ V ∣ ∀ w ∈ C , ⟨ w , v ⟩ ≥ 0 } , {\displaystyle C^{}=\{v\in V\mid \forall w\in C,\langle w,v\rangle \geq 0\},} which is always a convex cone. Here, ⟨ w , v ⟩ {\displaystyle \langle w,v\rangle } is the duality pairing between C and V, i.e. ⟨ w , v ⟩ = v ( w ) {\displaystyle \langle w,v\rangle =v(w)} . More generally, the (algebraic) dual cone to C ⊂ V in a linear space V is a subset of the dual space V defined by: C ∗ := { v ∈ V ∗ ∣ ∀ w ∈ C , v ( w ) ≥ 0 } . {\displaystyle C^{}:=\left\{v\in V^{}\mid \forall w\in C,v(w)\geq 0\right\}.} In other words, if V* is the algebraic dual space of V, C* is the set of linear functionals that are nonnegative on the primal cone C. If we take V* to be the continuous dual space then it is the set of continuous linear functionals nonnegative on C. This notion does not require the specification of an inner product on V. In finite dimensions, the two notions of dual cone are essentially the same because every finite dimensional linear functional is continuous, and every continuous linear functional in an inner product space induces a linear isomorphism (nonsingular linear map) from V* to V, and this isomorphism will take the dual cone given by the second definition, in V, onto the one given by the first definition; see the Riesz representation theorem. If C is equal to its dual cone, then C is called self-dual. A cone can be said to be self-dual without reference to any given inner product, if there exists an inner product with respect to which it is equal to its dual by the first definition. == Constructions == Given a closed, convex subset K of Hilbert space V, the outward normal cone to the set K at the point x in K is given by N K ( x ) = { p ∈ V : ∀ x ∗ ∈ K , ⟨ p , x ∗ − x ⟩ ≤ 0 } . {\displaystyle N_{K}(x)=\left\{p\in V\colon \forall x^{}\in K,\left\langle p,x^{*}-x\right\rangle \leq 0\right\}.} Given a closed, convex subset K of V, the tangent cone (or contingent cone) to the set K at the point x is given by T K ( x ) = ⋃ h > 0 K − x h ¯ . {\displaystyle T_{K}(x)={\overline {\bigcup _{h>0}{\frac {K-x}{h}}}}.} Given a closed, convex subset K of Hilbert space V, the tangent cone to the set K at the point x in K can be defined as polar cone to outwards normal cone N K ( x ) {\displaystyle N_{K}(x)} : T K ( x ) = N K o ( x ) = d e f { y ∈ V ∣ ∀ ξ ∈ N K ( x ) : ⟨ y , ξ ⟩ ⩽ 0 } {\displaystyle T_{K}(x)=N_{K}^{o}(x)\ {\overset {\underset {\mathrm {def} }{}}{=}}\ \{y\in V\mid \forall \xi \in N_{K}(x):\langle y,\xi \rangle \leqslant 0\}} Both the normal and tangent cone have the property of being closed and convex. They are important concepts in the fields of convex optimization, variational inequalities and projected dynamical systems. == Properties == If C is a non-empty convex cone in X, then the linear span of C is equal to C - C and the largest vector subspace of X contained in C is equal to C ∩ (−C). == Partial order defined by a convex cone == A pointed and salient convex cone C induces a partial ordering "≥" on V, defined so that x ≥ y {\displaystyle x\geq y} if and only if x − y ∈ C . {\displaystyle x-y\in C.} (If the cone is flat, the same definition gives merely a preorder.) Sums and positive scalar multiples of valid inequalities with respect to this order remain valid inequalities. A vector space with such an order is called an ordered vector space. Examples include the product order on real-valued vectors, R n , {\displaystyle \mathbb {R} ^{n},} and the Loewner order on positive semidefinite matrices. Such an ordering is commonly found in semidefinite programming. == See also == Cone (disambiguation) Cone (geometry) Cone (topology) Farkas' lemma Bipolar theorem Ordered vector space == Notes == == References == Bourbaki, Nicolas (1987). Topological Vector Spaces. Elements of Mathematics. Berlin, New York: Springer-Verlag. ISBN 978-3-540-13627-9. Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. Rockafellar, R. T. (1997) [1970]. Convex Analysis. Princeton, NJ: Princeton University Press. ISBN 1-4008-7317-7. Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. Zălinescu, C. (2002). Convex Analysis in General Vector Spaces. River Edge, NJ: World Scientific. ISBN 981-238-067-1. MR 1921556.
Wikipedia:Conway base 13 function#0	The Conway base 13 function is a function created by British mathematician John H. Conway as a counterexample to the converse of the intermediate value theorem. In other words, it is a function that satisfies a particular intermediate-value property — on any interval ( a , b ) {\displaystyle (a,b)} , the function f {\displaystyle f} takes every value between f ( a ) {\displaystyle f(a)} and f ( b ) {\displaystyle f(b)} — but is not continuous. In 2018, a much simpler function with the property that every open set is mapped onto the full real line, was published by user Aksel Bergfeldt on the community question and answer site Mathematics Stack Exchange. This function is also nowhere continuous. == Purpose == The Conway base 13 function was created as part of a "produce" activity: in this case, the challenge was to produce a simple-to-understand function which takes on every real value in every interval, that is, it is an everywhere surjective function. It is thus discontinuous at every point. == Sketch of definition == Every real number x {\displaystyle x} can be represented in base 13 in a unique canonical way; such representations use the digits 0–9 plus three additional symbols, say {A, B, C}. For example, the number 54349589 has a base-13 representation B34C128. If instead of {A, B, C}, we judiciously choose the symbols {+, −, .}, some numbers in base 13 will have representations that look like well-formed decimals in base 10: for example, the number 54349589 has a base-13 representation of −34.128. Of course, most numbers will not be intelligible in this way; for example, the number 3629265 has the base-13 representation 9+0−−7. Conway's base-13 function takes in a real number x and considers its base-13 representation as a sequence of symbols {0, 1, ..., 9, +, −, .}. If from some position onward, the representation looks like a well-formed decimal number r, then f(x) = r. Otherwise, f(x) = 0. (Well-formed means that it starts with a + or − symbol, contains exactly one decimal-point symbol, and otherwise contains only the digits 0–9). For example, if a number x has the representation 8++2.19+0−−7+3.141592653..., then f(x) = +3.141592653.... == Definition == The Conway base-13 function is a function f : R → R {\displaystyle f:\mathbb {R} \to \mathbb {R} } defined as follows. Write the argument x {\displaystyle x} value as a tridecimal (a "decimal" in base 13) using 13 symbols as "digits": 0, 1, ..., 9, A, B, C; there should be no trailing C recurring. There may be a leading sign, and somewhere there will be a tridecimal point to separate the integer part from the fractional part; these should both be ignored in the sequel. These "digits" can be thought of as having the values 0 to 12 respectively; Conway originally used the digits "+", "−" and "." instead of A, B, C, and underlined all of the base-13 "digits" to clearly distinguish them from the usual base-10 digits and symbols. If from some point onwards, the tridecimal expansion of x {\displaystyle x} is of the form A x 1 x 2 … x n C y 1 y 2 … {\displaystyle Ax_{1}x_{2}\dots x_{n}Cy_{1}y_{2}\dots } where all the digits x i {\displaystyle x_{i}} and y j {\displaystyle y_{j}} are in { 0 , … , 9 } , {\displaystyle \{0,\dots ,9\},} then f ( x ) = x 1 … x n . y 1 y 2 … {\displaystyle f(x)=x_{1}\dots x_{n}.y_{1}y_{2}\dots } in usual base-10 notation. Similarly, if the tridecimal expansion of x {\displaystyle x} ends with B x 1 x 2 … x n C y 1 y 2 … , {\displaystyle Bx_{1}x_{2}\dots x_{n}Cy_{1}y_{2}\dots ,} then f ( x ) = − x 1 … x n . y 1 y 2 … . {\displaystyle f(x)=-x_{1}\dots x_{n}.y_{1}y_{2}\dots .} Otherwise, f ( x ) = 0. {\displaystyle f(x)=0.} For example: f ( 12345 A 3 C 14.159 … 13 ) = f ( A 3 C 14.159 … 13 ) = 3.14159 … , {\displaystyle f(\mathrm {12345A3C14.159} \dots _{13})=f(\mathrm {A3C14.159} \dots _{13})=3.14159\dots ,} f ( B 1 C 234 13 ) = − 1.234 , {\displaystyle f(\mathrm {B1C234} _{13})=-1.234,} f ( 1 C 234 A 567 13 ) = 0. {\displaystyle f(\mathrm {1C234A567} _{13})=0.} == Properties == According to the intermediate-value theorem, every continuous real function f {\displaystyle f} has the intermediate-value property: on every interval (a, b), the function f {\displaystyle f} passes through every point between f ( a ) {\displaystyle f(a)} and f ( b ) . {\displaystyle f(b).} The Conway base-13 function shows that the converse is false: it satisfies the intermediate-value property, but is not continuous. In fact, the Conway base-13 function satisfies a much stronger intermediate-value property—on every interval (a, b), the function f {\displaystyle f} passes through every real number. As a result, it satisfies a much stronger discontinuity property— it is discontinuous everywhere. From the above follows even more regarding the discontinuity of the function - its graph is dense in R 2 {\displaystyle \mathbb {R} ^{2}} . To prove that the Conway base-13 function satisfies this stronger intermediate property, let (a, b) be an interval, let c be a point in that interval, and let r be any real number. Create a base-13 encoding of r as follows: starting with the base-10 representation of r, replace the decimal point with C and indicate the sign of r by prepending either an A (if r is positive) or a B (if r is negative) to the beginning. By definition of the Conway base-13 function, the resulting string r ^ {\displaystyle {\hat {r}}} has the property that f ( r ^ ) = r . {\displaystyle f({\hat {r}})=r.} Moreover, any base-13 string that ends in r ^ {\displaystyle {\hat {r}}} will have this property. Thus, if we replace the tail end of c with r ^ , {\displaystyle {\hat {r}},} the resulting number will have f(c') = r. By introducing this modification sufficiently far along the tridecimal representation of c , {\displaystyle c,} you can ensure that the new number c ′ {\displaystyle c'} will still lie in the interval ( a , b ) . {\displaystyle (a,b).} This proves that for any number r, in every interval we can find a point c ′ {\displaystyle c'} such that f ( c ′ ) = r . {\displaystyle f(c')=r.} The Conway base-13 function is therefore discontinuous everywhere: a real function that is continuous at x must be locally bounded at x, i.e. it must be bounded on some interval around x. But as shown above, the Conway base-13 function is unbounded on every interval around every point; therefore it is not continuous anywhere. The Conway base-13 function maps almost all the reals in any interval to 0. == See also == Darboux function – All derivatives have the intermediate value propertyPages displaying short descriptions of redirect targets == References == Oman, Greg (2014). "The Converse of the Intermediate Value Theorem: From Conway to Cantor to Cosets and Beyond" (PDF). Missouri J. Math. Sci. 26 (2): 134–150. doi:10.35834/mjms/1418931955. Archived (PDF) from the original on 2016-08-20.
Wikipedia:Conway polynomial (finite fields)#0	In mathematics, the Conway polynomial Cp,n for the finite field Fpn is a particular irreducible polynomial of degree n over Fp that can be used to define a standard representation of Fpn as a splitting field of Cp,n. Conway polynomials were named after John H. Conway by Richard A. Parker, who was the first to define them and compute examples. Conway polynomials satisfy a certain compatibility condition that had been proposed by Conway between the representation of a field and the representations of its subfields. They are important in computer algebra where they provide portability among different mathematical databases and computer algebra systems. Since Conway polynomials are expensive to compute, they must be stored to be used in practice. Databases of Conway polynomials are available in the computer algebra systems GAP, Macaulay2, Magma, SageMath, at the web site of Frank Lübeck, and at the Online Encyclopedia of Integer Sequences. == Background == Elements of F p n {\displaystyle \mathbf {F} _{p^{n}}} may be represented as sums of the form a n − 1 β n − 1 + ⋯ + a 1 β + a 0 {\displaystyle a_{n-1}\beta ^{n-1}+\cdots +a_{1}\beta +a_{0}} where β {\displaystyle \beta } is a root of an irreducible polynomial of degree n {\displaystyle n} over F p {\displaystyle \mathbf {F} _{p}} and the a j {\displaystyle a_{j}} are elements of F p {\displaystyle \mathbf {F} _{p}} . Addition of field elements in this representation is simply vector addition. While there is a unique finite field of order p n {\displaystyle p^{n}} up to isomorphism, the representation of the field elements depends on the choice of irreducible polynomial. The Conway polynomial is a way of standardizing this choice. The non-zero elements of a finite field F {\displaystyle \mathbf {F} } form a cyclic group under multiplication, denoted F ∗ {\displaystyle \mathbf {F} ^{}} . A primitive element, a {\displaystyle a} , of F p n {\displaystyle \mathbf {F} _{p^{n}}} is an element that generates F p n ∗ {\displaystyle \mathbf {F} _{p^{n}}^{}} . Representing the non-zero field elements as powers of a {\displaystyle a} allows multiplication in the field to be performed efficiently. The primitive polynomial for a {\displaystyle a} is the monic polynomial of smallest possible degree with coefficients in F p {\displaystyle \mathbf {F} _{p}} that has a {\displaystyle a} as a root in F p n {\displaystyle \mathbf {F} _{p^{n}}} (the minimal polynomial for α {\displaystyle \alpha } ). It is necessarily irreducible. The Conway polynomial is chosen to be primitive, so that each of its roots generates the multiplicative group of the associated finite field. The field F p n {\displaystyle \mathbf {F} _{p^{n}}} contains a unique subfield isomorphic to F p m {\displaystyle \mathbf {F} _{p^{m}}} for each m {\displaystyle m} dividing n {\displaystyle n} , and this accounts for all the subfields of F p n {\displaystyle \mathbf {F} _{p^{n}}} . For any m {\displaystyle m} dividing n {\displaystyle n} the cyclic group F p n ∗ {\displaystyle \mathbf {F} _{p^{n}}^{}} contains a subgroup isomorphic to F p m ∗ {\displaystyle \mathbf {F} _{p^{m}}^{}} . If α {\displaystyle \alpha } generates F p n ∗ {\displaystyle \mathbf {F} _{p^{n}}^{*}} , then the smallest power of α {\displaystyle \alpha } that generates this subgroup is α r {\displaystyle \alpha ^{r}} where r = ( p n − 1 ) / ( p m − 1 ) {\displaystyle r=(p^{n}-1)/(p^{m}-1)} . If f n {\displaystyle f_{n}} is a primitive polynomial for F p n {\displaystyle \mathbf {F} _{p^{n}}} with root α {\displaystyle \alpha } and f m {\displaystyle f_{m}} is a primitive polynomial for F p m {\displaystyle \mathbf {F} _{p^{m}}} then, by Conway's definition, f m {\displaystyle f_{m}} and f n {\displaystyle f_{n}} are compatible if α r {\displaystyle \alpha ^{r}} is a root of f m {\displaystyle f_{m}} . This necessitates that f n ( x ) {\displaystyle f_{n}(x)} divide f m ( x r ) {\displaystyle f_{m}(x^{r})} . This notion of compatibility is called norm-compatibility by some authors. The Conway polynomial for a finite field is chosen so as to be compatible with the Conway polynomials of each of its subfields. That it is possible to make the choice in this way was proved by Werner Nickel. == Definition == The Conway polynomial Cp,n is defined as the lexicographically minimal monic primitive polynomial of degree n over Fp that is compatible with Cp,m for all m dividing n. This is an inductive definition on n: the base case is Cp,1(x) = x − α where α is the lexicographically minimal primitive element of Fp. The notion of lexicographical ordering used is the following: The elements of Fp are ordered 0 < 1 < 2 < … < p − 1. A polynomial of degree d in Fp[x] is written adxd − ad−1xd−1 + … + (−1)da0 (with terms alternately added and subtracted) and then expressed as the word ad ad−1 … a0. Two polynomials of degree d are ordered according to the lexicographical ordering of their corresponding words. Since there does not appear to be any natural mathematical criterion that would single out one monic primitive polynomial satisfying the compatibility conditions over all the others, the imposition of lexicographical ordering in the definition of the Conway polynomial should be regarded as a convention. == Table == Conway polynomials Cp,n for the lowest values of p and n are tabulated below. All of these were first computed by Richard Parker and were taken from the tables of Frank Luebeck. The calculations can be verified using the basic methods of the next section with the assistance of algebra software. == Examples == To illustrate the definition, let us compute the first six Conway polynomials over F5. By definition, a Conway polynomial is monic, primitive (which implies irreducible), and compatible with Conway polynomials of degree dividing its degree. The table below shows how imposing each of these conditions reduces the number of candidate polynomials. Degree 1. The primitive elements of F5 are 2 and 3. The two degree-1 polynomials with primitive roots are therefore x − 2 = x + 3 and x − 3 = x + 2, which correspond to the words 12 and 13, Since 12 is less than 13 in lexicographic ordering, C5,1(x) = x + 3. Degree 2. Since (52 − 1) / (51 − 1) = 6, compatibility requires that C5,2 be chosen so that C5,2(x) divides C5,1(x6) = x6 + 3. The latter factorizes into three degree-2 polynomials, irreducible over F5, namely x2 + 2, x2 + x + 2, and x2 + 4x + 2. Of these x2 + 2 is not primitive since it divides x8 − 1 implying that its roots have order at most 8, rather than the required 24. Both of the others are primitive and C5,2 is chosen to be the lexicographically lesser of the two. Now x2 + x + 2 = x2 − 4x + 2 corresponds to the word 142 and x2 + 4x + 2 = x2 − x + 2 corresponds to the word 112, the latter being lexicographically less than the former. Hence C5,2(x) = x2 + 4x + 2. Degree 3. Since (53 − 1) / (51 − 1) = 31, compatibility requires that C5,3(x) divide C5,1(x31) = x31 + 3, which factorizes as a degree-1 polynomial times the product of ten primitive degree-3 polynomials. Of these, two have no quadratic term, x3 + 3x + 3 = x3 − 0x2 + 3x − 2 and x3 + 4x + 3 = x3 − 0x2 + 4x − 2, which correspond to the words 1032 and 1042. As 1032 is lexicographically less than 1042, C5,3(x) = x3 + 3x + 3. Degree 4. The proper divisors of 4 are 1 and 2. Compute (54 − 1) / (52 − 1) = 26 and (54 − 1) / (51 − 1) = 156, and note that 156 / 26 = (52 − 1) / (52 − 1) = 6, the same exponent as appeared in the compatibility condition for degree 2. In degree 4, compatibility requires that C5,4 be chosen so that C5,4(x) divides both C5,2(x26) = x52 + 4x26 + 2 and C5,1(x156) = x156 + 3. The second condition is redundant, however, because of the compatibility condition imposed when choosing C5,2, which implies that C5,2(x26) divides C5,1(x156). In general, for composite degree d, the same reasoning implies that only the maximal proper divisors of d need be considered, that is, divisors of the form d / p, where p is a prime divisor of d. There are 13 factors of C5,2(x26), all of degree 4. All but one are primitive. Of the primitive ones, x4 + 4x2 + 4x + 2 is lexicographically minimal. Degree 5. The computation is similar to what was done in degrees 2 and 3: (55 − 1) / (51 − 1) = 781; C5,1(x781) = x781 + 3 has one factor of degree 1 and 156 factors of degree 5, of which 140 are primitive. The lexicographically least of the primitive factors is x5 + 4x + 3. Degree 6. Taking into consideration the discussion above in connection with degree 4, the two compatibility conditions that need to be considered are that C5,6(x) must divide C5,2(x651) = x1302 + 4x651 + 2 and C5,3(x126) = x378 + 3x126 + 3. It therefore must divide their greatest common divisor, x126 + x105 + 2x84 + 3x42 + 2, which factorizes into 21 degree-6 polynomials, 18 of which are primitive. The lexicographically least of these is x6 + x4 + 4x3 + x2 + 2. == Computation == Algorithms for computing Conway polynomials that are more efficient than brute-force search have been developed by Heath and Loehr. Lübeck indicates that their algorithm is a rediscovery of the method of Parker. == Notes == == References == Holt, Derek F.; Eick, Bettina; O'Brien, Eamonn A. (2005), Handbook of computational group theory, Discrete mathematics and its applications, vol. 24, CRC Press, ISBN 978-1-58488-372-2
Wikipedia:Coordinate singularity#0	In mathematics, a singularity is a point at which a given mathematical object is not defined, or a point where the mathematical object ceases to be well-behaved in some particular way, such as by lacking differentiability or analyticity. For example, the reciprocal function f ( x ) = 1 / x {\displaystyle f(x)=1/x} has a singularity at x = 0 {\displaystyle x=0} , where the value of the function is not defined, as involving a division by zero. The absolute value function g ( x ) = \| x \| {\displaystyle g(x)=\|x\|} also has a singularity at x = 0 {\displaystyle x=0} , since it is not differentiable there. The algebraic curve defined by { ( x , y ) : y 3 − x 2 = 0 } {\displaystyle \left\{(x,y):y^{3}-x^{2}=0\right\}} in the ( x , y ) {\displaystyle (x,y)} coordinate system has a singularity (called a cusp) at ( 0 , 0 ) {\displaystyle (0,0)} . For singularities in algebraic geometry, see singular point of an algebraic variety. For singularities in differential geometry, see singularity theory. == Real analysis == In real analysis, singularities are either discontinuities, or discontinuities of the derivative (sometimes also discontinuities of higher order derivatives). There are four kinds of discontinuities: type I, which has two subtypes, and type II, which can also be divided into two subtypes (though usually is not). To describe the way these two types of limits are being used, suppose that f ( x ) {\displaystyle f(x)} is a function of a real argument x {\displaystyle x} , and for any value of its argument, say c {\displaystyle c} , then the left-handed limit, f ( c − ) {\displaystyle f(c^{-})} , and the right-handed limit, f ( c + ) {\displaystyle f(c^{+})} , are defined by: f ( c − ) = lim x → c f ( x ) {\displaystyle f(c^{-})=\lim _{x\to c}f(x)} , constrained by x < c {\displaystyle x<c} and f ( c + ) = lim x → c f ( x ) {\displaystyle f(c^{+})=\lim _{x\to c}f(x)} , constrained by x > c {\displaystyle x>c} . The value f ( c − ) {\displaystyle f(c^{-})} is the value that the function f ( x ) {\displaystyle f(x)} tends towards as the value x {\displaystyle x} approaches c {\displaystyle c} from below, and the value f ( c + ) {\displaystyle f(c^{+})} is the value that the function f ( x ) {\displaystyle f(x)} tends towards as the value x {\displaystyle x} approaches c {\displaystyle c} from above, regardless of the actual value the function has at the point where x = c {\displaystyle x=c} . There are some functions for which these limits do not exist at all. For example, the function g ( x ) = sin ⁡ ( 1 x ) {\displaystyle g(x)=\sin \left({\frac {1}{x}}\right)} does not tend towards anything as x {\displaystyle x} approaches c = 0 {\displaystyle c=0} . The limits in this case are not infinite, but rather undefined: there is no value that g ( x ) {\displaystyle g(x)} settles in on. Borrowing from complex analysis, this is sometimes called an essential singularity. The possible cases at a given value c {\displaystyle c} for the argument are as follows. A point of continuity is a value of c {\displaystyle c} for which f ( c − ) = f ( c ) = f ( c + ) {\displaystyle f(c^{-})=f(c)=f(c^{+})} , as one expects for a smooth function. All the values must be finite. If c {\displaystyle c} is not a point of continuity, then a discontinuity occurs at c {\displaystyle c} . A type I discontinuity occurs when both f ( c − ) {\displaystyle f(c^{-})} and f ( c + ) {\displaystyle f(c^{+})} exist and are finite, but at least one of the following three conditions also applies: f ( c − ) ≠ f ( c + ) {\displaystyle f(c^{-})\neq f(c^{+})} ; f ( x ) {\displaystyle f(x)} is not defined for the case of x = c {\displaystyle x=c} ; or f ( c ) {\displaystyle f(c)} has a defined value, which, however, does not match the value of the two limits. Type I discontinuities can be further distinguished as being one of the following subtypes: A jump discontinuity occurs when f ( c − ) ≠ f ( c + ) {\displaystyle f(c^{-})\neq f(c^{+})} , regardless of whether f ( c ) {\displaystyle f(c)} is defined, and regardless of its value if it is defined. A removable discontinuity occurs when f ( c − ) = f ( c + ) {\displaystyle f(c^{-})=f(c^{+})} , also regardless of whether f ( c ) {\displaystyle f(c)} is defined, and regardless of its value if it is defined (but which does not match that of the two limits). A type II discontinuity occurs when either f ( c − ) {\displaystyle f(c^{-})} or f ( c + ) {\displaystyle f(c^{+})} does not exist (possibly both). This has two subtypes, which are usually not considered separately: An infinite discontinuity is the special case when either the left hand or right hand limit does not exist, specifically because it is infinite, and the other limit is either also infinite, or is some well defined finite number. In other words, the function has an infinite discontinuity when its graph has a vertical asymptote. An essential singularity is a term borrowed from complex analysis (see below). This is the case when either one or the other limits f ( c − ) {\displaystyle f(c^{-})} or f ( c + ) {\displaystyle f(c^{+})} does not exist, but not because it is an infinite discontinuity. Essential singularities approach no limit, not even if valid answers are extended to include ± ∞ {\displaystyle \pm \infty } . In real analysis, a singularity or discontinuity is a property of a function alone. Any singularities that may exist in the derivative of a function are considered as belonging to the derivative, not to the original function. === Coordinate singularities === A coordinate singularity occurs when an apparent singularity or discontinuity occurs in one coordinate frame, which can be removed by choosing a different frame. An example of this is the apparent singularity at the 90 degree latitude in spherical coordinates. An object moving due north (for example, along the line 0 degrees longitude) on the surface of a sphere will suddenly experience an instantaneous change in longitude at the pole (in the case of the example, jumping from longitude 0 to longitude 180 degrees). This discontinuity, however, is only apparent; it is an artifact of the coordinate system chosen, which is singular at the poles. A different coordinate system would eliminate the apparent discontinuity (e.g., by replacing the latitude/longitude representation with an n-vector representation). == Complex analysis == In complex analysis, there are several classes of singularities. These include the isolated singularities, the nonisolated singularities, and the branch points. === Isolated singularities === Suppose that f {\displaystyle f} is a function that is complex differentiable in the complement of a point a {\displaystyle a} in an open subset U {\displaystyle U} of the complex numbers C . {\displaystyle \mathbb {C} .} Then: The point a {\displaystyle a} is a removable singularity of f {\displaystyle f} if there exists a holomorphic function g {\displaystyle g} defined on all of U {\displaystyle U} such that f ( z ) = g ( z ) {\displaystyle f(z)=g(z)} for all z {\displaystyle z} in U ∖ { a } . {\displaystyle U\smallsetminus \{a\}.} The function g {\displaystyle g} is a continuous replacement for the function f . {\displaystyle f.} The point a {\displaystyle a} is a pole or non-essential singularity of f {\displaystyle f} if there exists a holomorphic function g {\displaystyle g} defined on U {\displaystyle U} with g ( a ) {\displaystyle g(a)} nonzero, and a natural number n {\displaystyle n} such that f ( z ) = g ( z ) ( z − a ) n {\displaystyle f(z)={\frac {g(z)}{(z-a)^{n}}}} for all z {\displaystyle z} in U ∖ { a } . {\displaystyle U\smallsetminus \{a\}.} The least such number n {\displaystyle n} is called the order of the pole. The derivative at a non-essential singularity itself has a non-essential singularity, with n {\displaystyle n} increased by 1 (except if n {\displaystyle n} is 0 so that the singularity is removable). The point a {\displaystyle a} is an essential singularity of f {\displaystyle f} if it is neither a removable singularity nor a pole. The point a {\displaystyle a} is an essential singularity if and only if the Laurent series has infinitely many powers of negative degree. === Nonisolated singularities === Other than isolated singularities, complex functions of one variable may exhibit other singular behaviour. These are termed nonisolated singularities, of which there are two types: Cluster points: limit points of isolated singularities. If they are all poles, despite admitting Laurent series expansions on each of them, then no such expansion is possible at its limit. Natural boundaries: any non-isolated set (e.g. a curve) on which functions cannot be analytically continued around (or outside them if they are closed curves in the Riemann sphere). === Branch points === Branch points are generally the result of a multi-valued function, such as z {\displaystyle {\sqrt {z}}} or log ⁡ ( z ) , {\displaystyle \log(z),} which are defined within a certain limited domain so that the function can be made single-valued within the domain. The cut is a line or curve excluded from the domain to introduce a technical separation between discontinuous values of the function. When the cut is genuinely required, the function will have distinctly different values on each side of the branch cut. The shape of the branch cut is a matter of choice, even though it must connect two different branch points (such as z = 0 {\displaystyle z=0} and z = ∞ {\displaystyle z=\infty } for log ⁡ ( z ) {\displaystyle \log(z)} ) which are fixed in place. == Finite-time singularity == A finite-time singularity occurs when one input variable is time, and an output variable increases towards infinity at a finite time. These are important in kinematics and Partial Differential Equations – infinites do not occur physically, but the behavior near the singularity is often of interest. Mathematically, the simplest finite-time singularities are power laws for various exponents of the form x − α , {\displaystyle x^{-\alpha },} of which the simplest is hyperbolic growth, where the exponent is (negative) 1: x − 1 . {\displaystyle x^{-1}.} More precisely, in order to get a singularity at positive time as time advances (so the output grows to infinity), one instead uses ( t 0 − t ) − α {\displaystyle (t_{0}-t)^{-\alpha }} (using t for time, reversing direction to − t {\displaystyle -t} so that time increases to infinity, and shifting the singularity forward from 0 to a fixed time t 0 {\displaystyle t_{0}} ). An example would be the bouncing motion of an inelastic ball on a plane. If idealized motion is considered, in which the same fraction of kinetic energy is lost on each bounce, the frequency of bounces becomes infinite, as the ball comes to rest in a finite time. Other examples of finite-time singularities include the various forms of the Painlevé paradox (for example, the tendency of a chalk to skip when dragged across a blackboard), and how the precession rate of a coin spun on a flat surface accelerates towards infinite—before abruptly stopping (as studied using the Euler's Disk toy). Hypothetical examples include Heinz von Foerster's facetious "Doomsday's equation" (simplistic models yield infinite human population in finite time). == Algebraic geometry and commutative algebra == In algebraic geometry, a singularity of an algebraic variety is a point of the variety where the tangent space may not be regularly defined. The simplest example of singularities are curves that cross themselves. But there are other types of singularities, like cusps. For example, the equation y2 − x3 = 0 defines a curve that has a cusp at the origin x = y = 0. One could define the x-axis as a tangent at this point, but this definition can not be the same as the definition at other points. In fact, in this case, the x-axis is a "double tangent." For affine and projective varieties, the singularities are the points where the Jacobian matrix has a rank which is lower than at other points of the variety. An equivalent definition in terms of commutative algebra may be given, which extends to abstract varieties and schemes: A point is singular if the local ring at this point is not a regular local ring. == See also == Catastrophe theory Defined and undefined Degeneracy (mathematics) Hyperbolic growth Movable singularity Pathological (mathematics) Regular singularity Singular solution == References ==
Wikipedia:Coordinate vector#0	In linear algebra, a coordinate vector is a representation of a vector as an ordered list of numbers (a tuple) that describes the vector in terms of a particular ordered basis. An easy example may be a position such as (5, 2, 1) in a 3-dimensional Cartesian coordinate system with the basis as the axes of this system. Coordinates are always specified relative to an ordered basis. Bases and their associated coordinate representations let one realize vector spaces and linear transformations concretely as column vectors, row vectors, and matrices; hence, they are useful in calculations. The idea of a coordinate vector can also be used for infinite-dimensional vector spaces, as addressed below. == Definition == Let V be a vector space of dimension n over a field F and let B = { b 1 , b 2 , … , b n } {\displaystyle B=\{b_{1},b_{2},\ldots ,b_{n}\}} be an ordered basis for V. Then for every v ∈ V {\displaystyle v\in V} there is a unique linear combination of the basis vectors that equals v {\displaystyle v} : v = α 1 b 1 + α 2 b 2 + ⋯ + α n b n . {\displaystyle v=\alpha _{1}b_{1}+\alpha _{2}b_{2}+\cdots +\alpha _{n}b_{n}.} The coordinate vector of v {\displaystyle v} relative to B is the sequence of coordinates [ v ] B = ( α 1 , α 2 , … , α n ) . {\displaystyle [v]_{B}=(\alpha _{1},\alpha _{2},\ldots ,\alpha _{n}).} This is also called the representation of v {\displaystyle v} with respect to B, or the B representation of v {\displaystyle v} . The α 1 , α 2 , … , α n {\displaystyle \alpha _{1},\alpha _{2},\ldots ,\alpha _{n}} are called the coordinates of v {\displaystyle v} . The order of the basis becomes important here, since it determines the order in which the coefficients are listed in the coordinate vector. Coordinate vectors of finite-dimensional vector spaces can be represented by matrices as column or row vectors. In the above notation, one can write [ v ] B = [ α 1 ⋮ α n ] {\displaystyle [v]_{B}={\begin{bmatrix}\alpha _{1}\\\vdots \\\alpha _{n}\end{bmatrix}}} and [ v ] B T = [ α 1 α 2 ⋯ α n ] {\displaystyle [v]_{B}^{T}={\begin{bmatrix}\alpha _{1}&\alpha _{2}&\cdots &\alpha _{n}\end{bmatrix}}} where [ v ] B T {\displaystyle [v]_{B}^{T}} is the transpose of the matrix [ v ] B {\displaystyle [v]_{B}} . == The standard representation == We can mechanize the above transformation by defining a function ϕ B {\displaystyle \phi _{B}} , called the standard representation of V with respect to B, that takes every vector to its coordinate representation: ϕ B ( v ) = [ v ] B {\displaystyle \phi _{B}(v)=[v]_{B}} . Then ϕ B {\displaystyle \phi _{B}} is a linear transformation from V to Fn. In fact, it is an isomorphism, and its inverse ϕ B − 1 : F n → V {\displaystyle \phi _{B}^{-1}:F^{n}\to V} is simply ϕ B − 1 ( α 1 , … , α n ) = α 1 b 1 + ⋯ + α n b n . {\displaystyle \phi _{B}^{-1}(\alpha _{1},\ldots ,\alpha _{n})=\alpha _{1}b_{1}+\cdots +\alpha _{n}b_{n}.} Alternatively, we could have defined ϕ B − 1 {\displaystyle \phi _{B}^{-1}} to be the above function from the beginning, realized that ϕ B − 1 {\displaystyle \phi _{B}^{-1}} is an isomorphism, and defined ϕ B {\displaystyle \phi _{B}} to be its inverse. == Examples == === Example 1 === Let P 3 {\displaystyle P_{3}} be the space of all the algebraic polynomials of degree at most 3 (i.e. the highest exponent of x can be 3). This space is linear and spanned by the following polynomials: B P = { 1 , x , x 2 , x 3 } {\displaystyle B_{P}=\left\{1,x,x^{2},x^{3}\right\}} matching 1 := [ 1 0 0 0 ] ; x := [ 0 1 0 0 ] ; x 2 := [ 0 0 1 0 ] ; x 3 := [ 0 0 0 1 ] {\displaystyle 1:={\begin{bmatrix}1\\0\\0\\0\end{bmatrix}};\quad x:={\begin{bmatrix}0\\1\\0\\0\end{bmatrix}};\quad x^{2}:={\begin{bmatrix}0\\0\\1\\0\end{bmatrix}};\quad x^{3}:={\begin{bmatrix}0\\0\\0\\1\end{bmatrix}}} then the coordinate vector corresponding to the polynomial p ( x ) = a 0 + a 1 x + a 2 x 2 + a 3 x 3 {\displaystyle p\left(x\right)=a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}} is [ a 0 a 1 a 2 a 3 ] . {\displaystyle {\begin{bmatrix}a_{0}\\a_{1}\\a_{2}\\a_{3}\end{bmatrix}}.} According to that representation, the differentiation operator d/dx which we shall mark D will be represented by the following matrix: D p ( x ) = P ′ ( x ) ; [ D ] = [ 0 1 0 0 0 0 2 0 0 0 0 3 0 0 0 0 ] {\displaystyle Dp(x)=P'(x);\quad [D]={\begin{bmatrix}0&1&0&0\\0&0&2&0\\0&0&0&3\\0&0&0&0\\\end{bmatrix}}} Using that method it is easy to explore the properties of the operator, such as: invertibility, Hermitian or anti-Hermitian or neither, spectrum and eigenvalues, and more. === Example 2 === The Pauli matrices, which represent the spin operator when transforming the spin eigenstates into vector coordinates. == Basis transformation matrix == Let B and C be two different bases of a vector space V, and let us mark with [ M ] C B {\displaystyle \lbrack M\rbrack _{C}^{B}} the matrix which has columns consisting of the C representation of basis vectors b1, b2, …, bn: [ M ] C B = [ [ b 1 ] C ⋯ [ b n ] C ] {\displaystyle \lbrack M\rbrack _{C}^{B}={\begin{bmatrix}\lbrack b_{1}\rbrack _{C}&\cdots &\lbrack b_{n}\rbrack _{C}\end{bmatrix}}} This matrix is referred to as the basis transformation matrix from B to C. It can be regarded as an automorphism over F n {\displaystyle F^{n}} . Any vector v represented in B can be transformed to a representation in C as follows: [ v ] C = [ M ] C B [ v ] B . {\displaystyle \lbrack v\rbrack _{C}=\lbrack M\rbrack _{C}^{B}\lbrack v\rbrack _{B}.} Under the transformation of basis, notice that the superscript on the transformation matrix, M, and the subscript on the coordinate vector, v, are the same, and seemingly cancel, leaving the remaining subscript. While this may serve as a memory aid, it is important to note that no such cancellation, or similar mathematical operation, is taking place. === Corollary === The matrix M is an invertible matrix and M−1 is the basis transformation matrix from C to B. In other words, Id = [ M ] C B [ M ] B C = [ M ] C C = [ M ] B C [ M ] C B = [ M ] B B {\displaystyle {\begin{aligned}\operatorname {Id} &=\lbrack M\rbrack _{C}^{B}\lbrack M\rbrack _{B}^{C}=\lbrack M\rbrack _{C}^{C}\\[3pt]&=\lbrack M\rbrack _{B}^{C}\lbrack M\rbrack _{C}^{B}=\lbrack M\rbrack _{B}^{B}\end{aligned}}} == Infinite-dimensional vector spaces == Suppose V is an infinite-dimensional vector space over a field F. If the dimension is κ, then there is some basis of κ elements for V. After an order is chosen, the basis can be considered an ordered basis. The elements of V are finite linear combinations of elements in the basis, which give rise to unique coordinate representations exactly as described before. The only change is that the indexing set for the coordinates is not finite. Since a given vector v is a finite linear combination of basis elements, the only nonzero entries of the coordinate vector for v will be the nonzero coefficients of the linear combination representing v. Thus the coordinate vector for v is zero except in finitely many entries. The linear transformations between (possibly) infinite-dimensional vector spaces can be modeled, analogously to the finite-dimensional case, with infinite matrices. The special case of the transformations from V into V is described in the full linear ring article. == See also == Change of basis Coordinate space == References ==
Wikipedia:Cora Sadosky#0	Cora Susana Sadosky de Goldstein (May 23, 1940 – December 3, 2010) was an Argentine mathematician and Professor of Mathematics at Howard University. == Early life and education == Sadosky was born in Buenos Aires, Argentina, the daughter of mathematicians Manuel Sadosky and Corina Eloísa "Cora" Ratto de Sadosky. At the age of 6, she moved with her parents to France and Italy. Sadosky began college at age 15 in the School of Science of the University of Buenos Aires, obtaining her degree of Licenciatura (comparable to a Bachelor's degree in the US nomenclature) in 1960. She earned her doctorate at the University of Chicago in 1965. == Career == After receiving her doctorate she returned to Argentina. She became an assistant professor of Mathematics at the University of Buenos Aires. She resigned her position in 1966, along with 400 other faculty members, in protest over a police assault on the School of Science. She taught for one semester at Uruguay National University and then became an assistant professor at Johns Hopkins. She returned to Argentina in 1968 but was unable to obtain an academic position there, instead working as a technical translator and editor. In 1974, due to political persecution, Sadosky left Argentina, relocating to Caracas to join the faculty of the Central University of Venezuela. At this time she wrote a graduate text on mathematics, Interpolation of Operators and Singular Integrals: An Introduction to Harmonic Analysis, which was published in the United States in 1979. She spent the academic year of 1978–1979 at the Institute for Advanced Study in Princeton, New Jersey. In 1980 she became an associate professor at Howard University. After spending a year as a visiting professor at the University of Buenos Aires, she returned to Howard University as a full professor in 1985. From 1995 to 1997, she served as an American Mathematical Society Council member at large. == Awards == She was appointed a visiting professorship for women (VPW) in science and technology from the National Science Foundation (NSF) for 1983–1984 and spent it at the Institute for Advanced Study. She received a second VPW in 1995 which she spent as visiting professor at the University of California at Berkeley. She received a Career Advancement Award from the NSF in 1987–1988 which allowed her to spend the year as a member of the classical analysis program at Mathematical Sciences Research Institute (MSRI), where she later returned as a research professor. She was elected president of the Association for Women in Mathematics (AWM) for 1993–1995. The Sadosky Prize of the AWM is named in her honor. == Research == Sadosky's research was in the field of analysis, particularly Fourier analysis and Operator Theory. Sadosky's doctoral thesis was on parabolic singular integrals, written under Alberto Pedro Calderón and Antoni Zygmund. Together with Mischa Cotlar, Sadosky wrote more than 30 articles as part of a collaborative research program. Their research included work on moments theory and lifting theorems for measures, Toeplitz forms, Hankel operators, and scattering systems, as well as their applications using weighted norm inequalities and functions of bounded mean oscillation. In addition to the above topics, Sadosky wrote extensively on harmonic analysis, particularly harmonic analysis on Euclidean space, the Hilbert transform, and other topics related to scattering and lifting techniques. == References == == External links == Estela A. Gavosto, Andrea R. Nahmod, María Cristina Pereyra, Gustavo Ponce, Rodolfo H. Torres, Wilfredo Urbina, "Remembering Cora Sadosky". Charlene Morrow and Teri Peri (eds), Notable Women in Mathematics, Greenwood Press, 1998, pp. 204–209. Larry Riddle, "Cora Sadosky", Biographies of Women Mathematicians, Cora Sadosky. María Cristina Pereyra, Stefania Marcantognini, Alexander M. Stokolos, Wilfredo Urbina (eds), Harmonic Analysis, Partial Differential Equations, Complex Analysis, Banach Spaces, and Operator Theory Volume 1 and Volume 2
Wikipedia:Coralie Colmez#0	Coralie Colmez is a French author and tutor in mathematics and mathematics education. == Early life and career == Coralie Colmez is the daughter of mathematicians Pierre Colmez and Leila Schneps. Colmez was raised in Paris. After completing her secondary education in Paris, Colmez moved to the United Kingdom and attended Gonville and Caius College of the University of Cambridge under a Cambridge European Trust scholarship, completing a first-class Bachelor of Arts in Mathematics and winning the Ryan Prize in Higher Mathematics. == Professional == Colmez worked for one year as a research assistant on Carol Vorderman's task force, commissioned by the UK government to study the state of mathematics education in the United Kingdom, and assisted with the presentation of the findings to the Joint Mathematics Council. She is now a co-director of unifrog, an organization that helps students discover future career pathways, apprenticeships and university courses, and teachers track their progress. == Writing == With her mother, mathematician Leila Schneps, Colmez co-authored Math on Trial: How Numbers Get Used and Abused in the Courtroom. This book, published in 2013 by Basic Books, targeted for a general audience, uses ten historical legal cases to show how mathematics, especially statistics, can affect the outcome of criminal proceedings, especially when incorrectly applied or interpreted. In 2022 Coralie published The Irrational Diary of Clara Valentine, a YA novel which includes high level mathematical concepts with the aim to introduce them to younger readers. == Public speaking == Since the publication of Math on Trial, Colmez has been an invited speaker at scientific education events in the UK. She has presented to the Conway Hall Ethical Society, the Cambridge Centre for Sixth-Form Studies, several shows for Maths Inspiration, including one at the University of Cambridge, and the 2014 QED conference. She has appeared on BBC Radio 4's Today Programme, discussing her book's subject of criminal trials in which math is used incorrectly or insufficiently, and on the BBC Radio 4 podcast, More or Less, discussing the same topic in relation to the Amanda Knox case. == References ==
Wikipedia:Corank#0	A rank is a position in a hierarchy. It can be formally recognized—for example, cardinal, chief executive officer, general, professor—or unofficial. == People == === Formal ranks === Academic rank Corporate title Diplomatic rank Hierarchy of the Catholic Church Imperial, royal and noble ranks Military rank Police rank === Unofficial ranks === Social class Social position Social status === Either === Seniority == Mathematics == Rank (differential topology) Rank (graph theory) Rank (linear algebra), the dimension of the vector space generated (or spanned) by a matrix's columns Rank (set theory) Rank (type theory) Rank of an abelian group, the cardinality of a maximal linearly independent subset Rank of a free module Rank of a greedoid, the maximal size of a feasible set Rank of a group, the smallest cardinality of a generating set for the group Rank of a Lie group – see Cartan subgroup Rank of a matroid, the maximal size of an independent set Rank of a partition, at least two definitions in number theory Rank of a tensor Rank of a vector bundle Rank statistics == Other == Taxonomic rank, in biology == See also == All pages with titles containing rank Ranking
Wikipedia:Corinna Ulcigrai#0	Corinna Ulcigrai (born 3 January 1980, Trieste) is an Italian mathematician working on dynamical systems. With Krzysztof Frączek in 2013, Ulcigrai is known for proving that in the Ehrenfest model (a mathematical abstraction of billiards with an infinite array of rectangular obstacles, used to model gas diffusion) most trajectories are not ergodic. == Education and career == Ulcigrai obtained her Ph.D. in 2007 from Princeton University with Yakov Sinai as her thesis advisor. She has worked at the University of Bristol, United Kingdom. and is currently a professor at the University of Zurich, Switzerland. == Recognition == Ulcigrai was awarded the European Mathematical Society Prize in 2012, and the Whitehead Prize in 2013. In 2020, Ulcigrai was the winner of the Michael Brin Prize in Dynamical Systems, "for her fundamental work on the ergodic theory of locally Hamiltonian flows on surfaces, of translation flows on periodic surfaces and wind-tree models, and her seminal work on higher genus generalizations of Markov and Lagrange spectra". == References == == External links == Home page Archived 2016-08-09 at the Wayback Machine Corinna Ulcigrai publications indexed by Google Scholar
Wikipedia:Cornelia Fabri#0	Cornelia Fabri (Ravenna, 9 September 1869 – Florence, 24 May 1915) was an Italian mathematician and the first woman to graduate in mathematics from University of Pisa (1891). == Life and work == Cornelia Fabri was born in Ravenna, Italy, into a noble family headed by Ruggero Fabri and Lucrezia Satanassi de Sordi. Her immediate family was well-schooled in math and science. Her grandfather, Santi Fabri, had been a mathematics graduate from the University of Bologna and taught at the College of Ravenna. Her father Ruggero Fabri, focused on scientific studies and graduated from the University of Rome in Physical and Mathematical Sciences. As a child, Cornelia demonstrated such an "uncommon ability" for scientific subjects that, with her father's approval, she enrolled in the city's technical institute becoming the only female in a class of males. She earned top marks, easily passed the entrance exam and was allowed to enroll in the Faculty of Physical, Mathematical and Natural Sciences at the University of Pisa. Again, she was the only woman and attended her lessons accompanied by her mother. She graduated in 1891. Her university teacher Vito Volterra, mathematical physicist and president of the Accademia dei Lincei, supervised her dissertation and followed Fabri's progress throughout her university years and remembered her as follows:"I have a very vivid memory of Signorina Cornelia Fabri, my student at the University of Pisa around 1880, the first, and perhaps the best, among the many students I subsequently had in Turin and Rome. I remember that her degree exam was an event for the University of Pisa, not only because it was the first time a woman had come there to get her doctorate, but also because the test was supported admirably by the candidate, who achieved full marks, absolutes and praise. On that occasion the Illustrious Dean of the Faculty of Science, Professor Antonio Pacinotti, uttered lofty and timely words, noting all the importance of the event, and foreseeing the opening of a new era with the entry into the field of science, of eminent female personalities."Fabri's scientific work focused primarily on hydraulics and was intense but brief. Her last academic work was published in 1895. In 1902, after the death of her mother, she left Pisa and returned to Ravenna to look after the family properties and her father. He died in 1904. She continued to keep in contact with Professor Volterra through detailed correspondence and the two mathematicians exchanged letters until 1902. They met for the last time in 1905. In Ravenna, Fabri dedicated herself to charities and charitable activities. Just three months before she died, Fabri used the language of science to describe to her confessor the reasons she was considering becoming a nun. "My heart has always been suspended between two equal and opposing forces, which balance each other and keep me in perfect blindness as to what my future will be." Fabri died at age 46 in Florence from pneumonia on 24 May 1915. == Selected works == Above are some general properties of functions that depend on other functions and lines. Note by Cornelia Fabri, Turin, Claudio Clausen, 1890. Brief considerations regarding the new regulations for the lock on the Montone river, Ravenna, Calderini, 1892. On vortex motions in perfect fluids. Memoir of Cornelia Fabri, Bologna, Gamberini and Parmeggiani typography, 1892. On the theory of vortex motions in incompressible fluids, Pisa, Mistri & C, 1892. Electric signal bell installed by Mr. Abbé Ravaglia in the port of Ravenna, Paris, A. Durand and Pedone-Lauriel Editeurs, 1893. Above hyperspace functions. Note by Cornelia Fabri, Proceedings of the Royal Veneto Institute of Letters, Science and Arts, 1893. Vortex motions of order higher than the 1st in relation to the equations for the movement of viscous fluids, Bologna, Tipografia Gamberini and Parmeggiani, 1894. Higher order vortex motions in relation to the equations for the movement of compressible viscous fluids, Il Nuovo Cimento, 1895. == References ==
Wikipedia:Correlation (projective geometry)#0	In projective geometry, a correlation is a transformation of a d-dimensional projective space that maps subspaces of dimension k to subspaces of dimension d − k − 1, reversing inclusion and preserving incidence. Correlations are also called reciprocities or reciprocal transformations. == In two dimensions == In the real projective plane, points and lines are dual to each other. As expressed by Coxeter, A correlation is a point-to-line and a line-to-point transformation that preserves the relation of incidence in accordance with the principle of duality. Thus it transforms ranges into pencils, pencils into ranges, [complete] quadrangles into [complete] quadrilaterals, and so on. Given a line m and P a point not on m, an elementary correlation is obtained as follows: for every Q on m form the line PQ. The inverse correlation starts with the pencil on P: for any line q in this pencil take the point m ∩ q. The composition of two correlations that share the same pencil is a perspectivity. == In three dimensions == In a 3-dimensional projective space a correlation maps a point to a plane. As stated in one textbook: If κ is such a correlation, every point P is transformed by it into a plane π′ = κP, and conversely, every point P arises from a unique plane π′ by the inverse transformation κ−1. Three-dimensional correlations also transform lines into lines, so they may be considered to be collineations of the two spaces. == In higher dimensions == In general n-dimensional projective space, a correlation takes a point to a hyperplane. This context was described by Paul Yale: A correlation of the projective space P(V) is an inclusion-reversing permutation of the proper subspaces of P(V). He proves a theorem stating that a correlation φ interchanges joins and intersections, and for any projective subspace W of P(V), the dimension of the image of W under φ is (n − 1) − dim W, where n is the dimension of the vector space V used to produce the projective space P(V). == Existence of correlations == Correlations can exist only if the space is self-dual. For dimensions 3 and higher, self-duality is easy to test: A coordinatizing skewfield exists and self-duality fails if and only if the skewfield is not isomorphic to its opposite. == Special types of correlations == === Polarity === If a correlation φ is an involution (that is, two applications of the correlation equals the identity: φ2(P) = P for all points P) then it is called a polarity. Polarities of projective spaces lead to polar spaces, which are defined by taking the collection of all subspace which are contained in their image under the polarity. === Natural correlation === There is a natural correlation induced between a projective space P(V) and its dual P(V∗) by the natural pairing ⟨⋅,⋅⟩ between the underlying vector spaces V and its dual V∗, where every subspace W of V∗ is mapped to its orthogonal complement W⊥ in V, defined as W⊥ = {v ∈ V \| ⟨w, v⟩ = 0, ∀w ∈ W}. Composing this natural correlation with an isomorphism of projective spaces induced by a semilinear map produces a correlation of P(V) to itself. In this way, every nondegenerate semilinear map V → V∗ induces a correlation of a projective space to itself. == References == Robert J. Bumcroft (1969), Modern Projective Geometry, Holt, Rinehart, and Winston, Chapter 4.5 Correlations p. 90 Robert A. Rosenbaum (1963), Introduction to Projective Geometry and Modern Algebra, Addison-Wesley, p. 198
Wikipedia:Correlation dimension#0	In chaos theory, the correlation dimension (denoted by ν) is a measure of the dimensionality of the space occupied by a set of random points, often referred to as a type of fractal dimension. For example, if we have a set of random points on the real number line between 0 and 1, the correlation dimension will be ν = 1, while if they are distributed on say, a triangle embedded in three-dimensional space (or m-dimensional space), the correlation dimension will be ν = 2. This is what we would intuitively expect from a measure of dimension. The real utility of the correlation dimension is in determining the (possibly fractional) dimensions of fractal objects. There are other methods of measuring dimension (e.g. the Hausdorff dimension, the box-counting dimension, and the information dimension) but the correlation dimension has the advantage of being straightforwardly and quickly calculated, of being less noisy when only a small number of points is available, and is often in agreement with other calculations of dimension. For any set of N points in an m-dimensional space x → ( i ) = [ x 1 ( i ) , x 2 ( i ) , … , x m ( i ) ] , i = 1 , 2 , … N {\displaystyle {\vec {x}}(i)=[x_{1}(i),x_{2}(i),\ldots ,x_{m}(i)],\qquad i=1,2,\ldots N} then the correlation integral C(ε) is calculated by: C ( ε ) = lim N → ∞ g N 2 {\displaystyle C(\varepsilon )=\lim _{N\rightarrow \infty }{\frac {g}{N^{2}}}} where g is the total number of pairs of points which have a distance between them that is less than distance ε (a graphical representation of such close pairs is the recurrence plot). As the number of points tends to infinity, and the distance between them tends to zero, the correlation integral, for small values of ε, will take the form: C ( ε ) ∼ ε ν {\displaystyle C(\varepsilon )\sim \varepsilon ^{\nu }} If the number of points is sufficiently large, and evenly distributed, a log-log graph of the correlation integral versus ε will yield an estimate of ν. This idea can be qualitatively understood by realizing that for higher-dimensional objects, there will be more ways for points to be close to each other, and so the number of pairs close to each other will rise more rapidly for higher dimensions. Grassberger and Procaccia introduced the technique in 1983; the article gives the results of such estimates for a number of fractal objects, as well as comparing the values to other measures of fractal dimension. The technique can be used to distinguish between (deterministic) chaotic and truly random behavior, although it may not be good at detecting deterministic behavior if the deterministic generating mechanism is very complex. As an example, in the "Sun in Time" article, the method was used to show that the number of sunspots on the sun, after accounting for the known cycles such as the daily and 11-year cycles, is very likely not random noise, but rather chaotic noise, with a low-dimensional fractal attractor. == See also == Takens's theorem Correlation integral Recurrence quantification analysis Approximate entropy == Notes ==
Wikipedia:Council for the Mathematical Sciences#0	The Council for the Mathematical Sciences (CMS) is an organisation that represents all types of British mathematicians at a national level. It is not a professional institution, but a collaboration of them. == History == It was established in 2001 by the Institute of Mathematics and its Applications, the London Mathematical Society and the Royal Statistical Society to provide a forum for mathematics. == Purpose == to represent the interests of mathematics to government, Research Councils and other public bodies; to promote good practice in the mathematics curriculum and its teaching and learning at all levels and in all sectors of education; to respond coherently and effectively to proposals from government and other public bodies which may affect the mathematical community; to work with other bodies such as the Joint Mathematical Council and HoDoMS. == Structure == It is situated off the A4200 in Russell Square, next to the University of London in the offices of the London Mathematical Society. It is accessed via the Russell Square tube station on the Piccadilly Line. == References == == External links == Web site
Wikipedia:Count On#0	Count (feminine: countess) is a historical title of nobility in certain European countries, varying in relative status, generally of middling rank in the hierarchy of nobility. Especially in earlier medieval periods the term often implied not only a certain status, but also that the count had specific responsibilities or offices. The etymologically related English term "county" denoted the territories associated with some countships, but not all. The title of count is typically not used in England or English-speaking countries, and the term earl is used instead. A female holder of the title is still referred to as a countess, however. == Origin of the term == The word count came into English from the French comte, itself from Latin comes—in its accusative form comitem. It meant "companion" or "attendant", and as a title it indicated that someone was delegated to represent the ruler. In the late Roman Empire, the Latin title comes denoted the high rank of various courtiers and provincial officials, either military or administrative. Before Anthemius became emperor in the West in 467, he was a military comes charged with strengthening defenses on the Danube frontier. In the Western Roman Empire, "count" came to indicate generically a military commander but was not a specific rank. In the Eastern Roman Empire, from about the seventh century, "count" was a specific rank indicating the commander of two centuriae (i.e., 200 men). The medieval title of comes was originally not hereditary. It was regarded as an administrative official dependent on the king, until the process of allodialisation during the 9th century in which such titles came to be private possessions of noble families. By virtue of their large estates, many counts could pass the title to their heirs—but not always. For instance, in Piast Poland, the position of komes was not hereditary, resembling the early Merovingian institution. The title had disappeared by the era of the Polish–Lithuanian Commonwealth, and the office had been replaced by others. Only after the Partitions of Poland did the title of "count" resurface in the title hrabia, derived from the German Graf. In the Frankish kingdoms in the early Middle Ages, a count might also be a count palatine, whose authority derived directly over a royal household, a palace in its original sense of the seat of power and administration. This other kind of count had vague antecedents in Late Antiquity too: the father of Cassiodorus held positions of trust with Theodoric, as comes rerum privatarum, in charge of the imperial lands, then as comes sacrarum largitionum ("count of the sacred doles"), concerned with the finances of the realm. In the United Kingdom, the title of earl is used instead of count. Although the exact reason is debated by historians and linguists, one of the more popular theories proposes that count fell into disuse because of its phonetic similarity to the vulgar slang word cunt. === Land attached to title === It is only after some time that the continental medieval title came to be strongly associated with the ownership of and jurisdiction over specific lands, which led to evolution of the term county to refer to specific regions. The English term county, used as an equivalent to the English term shire, is derived from the Old French conté or cunté which denoted the jurisdiction of a French count or viscount. The modern French is comté, and its equivalents in other languages are contea, contado, comtat, condado, Grafschaft, graafschap, etc. (cf. conte, comte, conde, Graf). The title of Count also continued to exist in cases which are not connected to any specific to a geographical "county". In the United Kingdom, the equivalent "Earl" can also be used as a courtesy title for the eldest son of a duke or marquess. In the Italian states, by contrast, all the sons of certain counts were little counts (contini). In Sweden there is a distinction between counts (Swedish: greve) created before and after 1809. All children in comital families elevated before 1809 were called count/countess. In families elevated after 1809, only the head of the family was called count, the rest have a status similar to barons and were called by the equivalent of "Mr/Ms/Mrs", before the recognition of titles of nobility was abolished. == Comital titles in different European languages == The following lists are originally based on a Glossary on Heraldica.org by Alexander Krischnig. The male form is followed by the female, and when available, by the territorial circumscription. === Etymological derivations from the Latin comes === === Etymological derivations from German Graf or Dutch Graaf === === Compound and related titles === Apart from all these, a few unusual titles have been of comital rank, not necessarily permanently. Dauphin (English: Dolphin; Spanish: Delfín; Italian: Delfino; Portuguese: Delfim; Latin: Delphinus) was a multiple (though rare) comital title in southern France, used by the Dauphins of Vienne and Auvergne, before 1349 when it became the title of the heir to the French throne. The Dauphin was the lord of the province still known as the région Dauphiné. Conde-Duque "Count-Duke" is a rare title used in Spain, notably by Gaspar de Guzmán, Count-Duke of Olivares. He had inherited the title of count of Olivares, but when created Duke of Sanlucar la Mayor by King Philip IV of Spain he begged permission to preserve his inherited title in combination with the new honour—according to a practice almost unique in Spanish history; logically the incumbent ranks as Duke (higher than Count) just as he would when simply concatenating both titles. Conde-Barão 'Count-Baron' is a rare title used in Portugal, notably by Dom Luís Lobo da Silveira, 7th Baron of Alvito, who received the title of Count of Oriola in 1653 from King John IV of Portugal. His palace in Lisbon still exists, located in a square named after him (Largo do Conde-Barão). Archcount is a very rare title, etymologically analogous to archduke, apparently never recognized officially, used by or for: the count of Flanders (an original pairie of the French realm in present Belgium, very rich, once expected to be raised to the rank of kingdom); the informal, rather descriptive use on account of the countship's de facto importance is rather analogous to the unofficial epithet Grand Duc de l'Occident (before Grand duke became a formal title) for the even wealthier Duke of Burgundy at least one Count of Burgundy (i.e. Freigraf of Franche-Comté) In German kingdoms, the title Graf was combined with the word for the jurisdiction or domain the nobleman was holding as a fief or as a conferred or inherited jurisdiction, such as Markgraf (see also Marquess), Landgraf, Freigraf ("free count"), Burggraf, where Burg signifies castle; see also Viscount, Pfalzgraf (translated both as "Count Palatine" and, historically, as "Palsgrave"), Raugraf ("Raugrave", see "Graf", and Waldgraf (comes nemoris), where Wald signifies a large forest) (from Latin nemus = grove). The German Graf and Dutch graaf (Latin: grafio) stem from the Byzantine-Greek γραφεύς grapheus meaning "he who calls a meeting [i.e. the court] together"). The Ottoman military title of Serdar was used in Montenegro and Serbia as a lesser noble title with the equivalent rank of a Count. These titles are not to be confused with various minor administrative titles containing the word -graf in various offices which are not linked to feudal nobility, such as the Dutch titles Pluimgraaf (a court sinecure, so usually held by noble courtiers, may even be rendered hereditary) and Dijkgraaf (to the present, in the Low Countries, a manager in the local or regional administration of watercourses through dykes, ditches, controls etc.; also in German Deichgraf, synonymous with Deichhauptmann, "dike captain"). == Lists of countships == === Territory of today's France === ==== Kingdom of the Western Franks ==== Since Louis VII (1137–80), the highest precedence amongst the vassals (Prince-bishops and secular nobility) of the French crown was enjoyed by those whose benefice or temporal fief was a pairie, i.e. carried the exclusive rank of pair; within the first (i.e. clerical) and second (noble) estates, the first three of the original twelve anciennes pairies were ducal, the next three comital comté-pairies: Bishop-counts of Beauvais (in Picardy) Bishop-counts of Châlons (in Champagne) Bishop-counts of Noyon (in Picardy) Count of Toulouse, until united to the crown in 1271 by marriage Count of Flanders (Flandres in French), which is in the Low countries and was confiscated in 1299, though returned in 1303 Count of Champagne, until united to the crown (in 1316 by marriage, conclusively in 1361) Later other countships (and duchies, even baronies) have been raised to this French peerage, but mostly as apanages (for members of the royal house) or for foreigners; after the 16th century all new peerages were always duchies and the medieval countship-peerages had died out, or were held by royal princes Other French countships of note included those of: Count of Angoulême, later Duke Count of Anjou, later Duke Count of Auvergne Count of Bar, later Duke Count of Blois Count of Boulogne Count of Foix Count of Montpensier Count of Poitiers ==== Parts of today's France long within other kingdoms of the Holy Roman Empire ==== Freigraf ("free count") of Burgundy (i.e. present Franche-Comté) The Dauphiné === The Holy Roman Empire === See also above for parts of present France ==== In Germany ==== A Graf ruled over a territory known as a Grafschaft ('county'). See also various comital and related titles; especially those actually reigning over a principality: Gefürsteter Graf, Landgraf, Reichsgraf; compare Markgraf, Burggraf, Pfalzgraf (see Imperial quaternions). ==== Northern Italian states ==== The title of Conte is very prolific on the peninsula. In the eleventh century, Conti like the Count of Savoy or the Norman Count of Apulia, were virtually sovereign lords of broad territories. Even apparently "lower"-sounding titles, like Viscount, could describe powerful dynasts, such as the House of Visconti which ruled a major city such as Milan. The essential title of a feudatory, introduced by the Normans, was signore, modeled on the French seigneur, used with the name of the fief. By the fourteenth century, conte and the Imperial title barone were virtually synonymous. Some titles of a count, according to the particulars of the patent, might be inherited by the eldest son of a Count. Younger brothers might be distinguished as "X dei conti di Y" ("X of the counts of Y"). However, if there is no male to inherit the title and the count has a daughter, in some regions she could inherit the title. Many Italian counts left their mark on Italian history as individuals, yet only a few contadi (countships; the word contadini for inhabitants of a "county" remains the Italian word for "peasant") were politically significant principalities, notably: Norman Count of Apulia Count of Savoy, later Duke (also partly in France and in Switzerland) Count of Asti Count of Montferrat (Monferrato) Count of Montefeltro Count of Tusculum ==== In Austria ==== The principalities tended to start out as margraviate or (promoted to) duchy, and became nominal archduchies within the Habsburg dynasty; noteworthy are: Count of Tyrol Count of Cilli Count of Schaumburg ==== In the Low Countries ==== Apart from various small ones, significant were : in presentday Belgium : Count of Flanders (Vlaanderen in Dutch), but only the small part east of the river Schelde remained within the empire; the far larger west, an original French comté-pairie became part of the French realm Count of Hainaut Count of Namur, later a margraviate Count of Leuven (Louvain), later a dukedom Count of Loon in the presentday Netherlands: Count of Guelders later Dukes of Guelders Count of Holland Count of Zeeland Count of Zutphen ==== In Switzerland ==== Count of Geneva Count of Neuchâtel Count of Toggenburg Count of Kyburg Count de Salis-Soglio (also in the UK, Canada and Australia) Count de Salis-Seewis Count of Panzutti === In other continental European countries === ==== Holy See ==== Count/Countess was one of the noble titles granted by the Pope as a temporal sovereign, and the title's holder was sometimes informally known as a papal count/papal countess or less so as a Roman count/Roman countess, but mostly as count/countess. The comital title, which could be for life or hereditary, was awarded in various forms by popes and Holy Roman Emperors since the Middle Ages, infrequently before the 14th century, and the pope continued to grant the comital and other noble titles even after 1870, it was largely discontinued in the mid 20th-century, on the accession of John XXIII. The Papacy and the Kingdom of the Two Sicilies might appoint counts palatine with no particular territorial fief. Until 1812 in some regions, the purchaser of land designated "feudal" was ennobled by the noble seat that he held and became a conte. This practice ceased with the formal abolition of feudalism in the various principalities of early-19th century Italy, last of all in the Papal States. ==== In Poland ==== Poland was notable throughout its history for not granting titles of nobility. This was on the premise that one could only be born into nobility, outside rare exceptions. Instead, it conferred non-hereditary courtly or civic roles. The noble titles that were in use on its territory were mostly of foreign provenance and usually subject to the process of indygenat, naturalisation. ==== In Hungary ==== Somewhat similar to the native privileged class of nobles found in Poland, Hungary also had a class of Conditional nobles. ==== On the Iberian peninsula ==== As opposed to the plethora of hollow "gentry" counts, only a few countships ever were important in medieval Iberia; most territory was firmly within the Reconquista kingdoms before counts could become important. However, during the 19th century, the title, having lost its high rank (equivalent to that of Duke), proliferated. ===== Portugal ===== Portugal itself started as a countship in 868, but became a kingdom in 1139 (see:County of Portugal). Throughout the history of Portugal, especially during the constitutional monarchy many other countships were created. ===== Spain ===== In Spain, no countships of wider importance exist, except in the former Spanish march. County of Barcelona, the initial core of the Principality of Catalonia, later one of the states of the Crown of Aragon, which became one of the two main components of the Spanish crown. Count of Aragon Count of Castile Count of Galicia Count of Lara Count Cassius, progenitor of the Banu Qasi County of Urgell, later integrated into the Principality of Catalonia. The other Catalan counties were much smaller and were absorbed early into the County of Barcelona (between parentheses the annexation year): County of Girona (897), County of Besalú, County of Osona, which included the nominal County of Manresa (1111), County of Berga and County of Conflent (1117) and County of Cerdanya (1118). From 1162 these counties, together with that of Barcelona, were merged into the Principality of Catalonia, a sovereign state that absorbed some other counties: County of Roussillon (1172), County of Pallars Jussà (1192), County of Empúries (1402), County of Urgell (1413) and County of Pallars Sobirà (1487), giving the Principality its definitive shape. === South Eastern Europe === ==== Bulgaria ==== In the First Bulgarian Empire, a komit was a hereditary provincial ruler under the tsar documented since the reign of Presian (836-852) The Cometopouli dynasty was named after its founder, the komit of Sredets. ==== Montenegro and Serbia ==== The title of Serdar was used in the Principality of Montenegro and the Principality of Serbia as a noble title below that of Voivode equivalent to that of Count. === Crusader states === Count of Edessa Count of Tripoli (1102–1288) === Scandinavia === In Denmark and historically in Denmark-Norway the title of count (greve) is the highest rank of nobility used in the modern period. Some Danish/Dano-Norwegian countships were associated with fiefs, and these counts were known as "feudal counts" (lensgreve). They rank above ordinary (titular) counts, and their position in the Danish aristocracy as the highest-ranking noblemen is broadly comparable to that of dukes in other European countries. With the first free Constitution of Denmark of 1849 came a complete abolition of the privileges of the nobility. Since then the title of count has been granted only to members of the Danish royal family, either as a replacement for a princely title when marrying a commoner, or in recent times, instead of that title in connection with divorce. Thus the first wife of Prince Joachim of Denmark, the younger son of Margrethe II of Denmark, became Alexandra, Countess of Frederiksborg on their divorce—initially retaining her title of princess, but losing it on her remarriage. In the Middle Ages the title of jarl (earl) was the highest title of nobility. The title was eventually replaced by the title of duke, but that title was abolished in Denmark and Norway as early as the Middle Ages. Titles were only reintroduced with the introduction of absolute monarchy in 1660, with count as the highest title. In Sweden the rank of count is the highest rank conferred upon nobles in the modern era and are, like their Danish and Norwegian counterparts, broadly comparable to that of dukes in other European countries. Unlike the rest of Scandinavia, the title of duke is still used in Sweden, but only by members of the royal family not considered part of the nobility. == Equivalents == Like other major Western noble titles, Count is sometimes used to render certain titles in non-western languages with their own traditions, even though they are as a rule historically unrelated and thus hard to compare, but which are considered "equivalent" in rank. This is the case with: the Chinese Bó (伯), or "Bojue" (伯爵), hereditary title of nobility ranking below Hóu (侯) and above Zĭ (子) earl of Britain the Japanese equivalent Hakushaku (伯爵), adapted during the Meiji restoration the Korean equivalent Baekjak (백작) or Poguk in Vietnam, it is rendered Bá, one of the lower titles reserved for male members of the Imperial clan, above Tử (Viscount), Nam (Baron) and Vinh phong (lowest noble title), but lower than—in ascending order—Hầu (Marquis), Công (Prince), Quận-Công (Duke/Duke of a commandery) and Quốc-Công (Grand Duke/Duke of the Nation), all under Vương (King) and Hoàng Đế (Emperor). the Indian Sardar, adopted by the Maratha Empire, additionally, Jagirdar and Deshmukh are close equivalents the Arabic equivalent Sheikh In traditional Sulu equivalent to Datu Sadja == In fiction == The title "Count" in fiction is commonly, though not always, given to evil characters, used as another word for prince or vampires: == See also == Czech nobility Icelandic nobility Romanian nobility Russian nobility Viscount Earl == References == == Sources == Labarre de Raillicourt: Les Comtes Romains Westermann, Großer Atlas zur Weltgeschichte (in German) == External links == Heraldica.org - here the French peerage Italian Titles of Nobility Archived 2012-05-27 at the Wayback Machine Webster's 1828 Dictionary
Wikipedia:Counting rods#0	Counting rods (筭) are small bars, typically 3–14 cm (1" to 6") long, that were used by mathematicians for calculation in ancient East Asia. They are placed either horizontally or vertically to represent any integer or rational number. The written forms based on them are called rod numerals. They are a true positional numeral system with digits for 1–9 and a blank for 0, from the Warring states period (circa 475 BCE) to the 16th century. == History == Chinese arithmeticians used counting rods well over two thousand years ago. In 1954, forty-odd counting rods of the Warring States period (5th century BCE to 221 BCE) were found in Zuǒjiāgōngshān (左家公山) Chu Grave No.15 in Changsha, Hunan. In 1973, archeologists unearthed a number of wood scripts from a tomb in Hubei dating from the period of the Han dynasty (206 BCE to 220 CE). On one of the wooden scripts was written: "当利二月定算𝍥". This is one of the earliest examples of using counting-rod numerals in writing. A square lacquer box, dating from c. 168 BCE, containing a square chess board with the TLV patterns, chessmen, counting rods, and other items, was excavated in 1972, from Mawangdui M3, Changsha, Hunan Province. In 1976, a bundle of Western Han-era (202 BCE to 9 CE) counting rods made of bones was unearthed from Qianyang County in Shaanxi. The use of counting rods must predate it; Sunzi (c. 544 to c. 496 BCE), a military strategist at the end of Spring and Autumn period of 771 BCE to 5th century BCE, mentions their use to make calculations to win wars before going into the battle; Laozi (died 531 BCE), writing in the Warring States period, said "a good calculator doesn't use counting rods". The Book of Han (finished 111 CE) recorded: "they calculate with bamboo, diameter one fen, length six cun, arranged into a hexagonal bundle of two hundred seventy one pieces". At first, calculating rods were round in cross-section, but by the time of the Sui dynasty (581 to 618 CE) mathematicians used triangular rods to represent positive numbers and rectangular rods for negative numbers. After the abacus flourished, counting rods were abandoned except in Japan, where rod numerals developed into a symbolic notation for algebra. == Using counting rods == Counting rods represent digits by the number of rods, and the perpendicular rod represents five. To avoid confusion, vertical and horizontal forms are alternately used. Generally, vertical rod numbers are used for the position for the units, hundreds, ten thousands, etc., while horizontal rod numbers are used for the tens, thousands, hundred thousands etc. It is written in Sunzi Suanjing that "one is vertical, ten is horizontal". Red rods represent positive numbers and black rods represent negative numbers. Ancient Chinese clearly understood negative numbers and zero (leaving a blank space for it), though they had no symbol for the latter. The Nine Chapters on the Mathematical Art, which was mainly composed in the first century CE, stated "(when using subtraction) subtract same signed numbers, add different signed numbers, subtract a positive number from zero to make a negative number, and subtract a negative number from zero to make a positive number". Later, a go stone was sometimes used to represent zero. This alternation of vertical and horizontal rod numeral form is very important to understanding written transcription of rod numerals on manuscripts correctly. For instance, in Licheng suanjin, 81 was transcribed as , and 108 was transcribed as ; it is clear that the latter clearly had a blank zero on the "counting board" (i.e., floor or mat), even though on the written transcription, there was no blank. In the same manuscript, 405 was transcribed as , with a blank space in between for obvious reasons, and could in no way be interpreted as "45". In other words, transcribed rod numerals may not be positional, but on the counting board, they are positional. is an exact image of the counting rod number 405 on a table top or floor. === Place value === The value of a number depends on its physical position on the counting board. A 9 at the rightmost position on the board stands for 9. Moving the batch of rods representing 9 to the left one position (i.e., to the tens place) gives 9[] or 90. Shifting left again to the third position (to the hundreds place) gives 9[][] or 900. Each time one shifts a number one position to the left, it is multiplied by 10. Each time one shifts a number one position to the right, it is divided by 10. This applies to single-digit numbers or multiple-digit numbers. Song dynasty mathematician Jia Xian used hand-written Chinese decimal orders 步十百千萬 as rod numeral place value, as evident from a facsimile from a page of Yongle Encyclopedia. He arranged 七萬一千八百二十四 as 七一八二四萬千百十步 He treated the Chinese order numbers as place value markers, and 七一八二四 became place value decimal number. He then wrote the rod numerals according to their place value: In Japan, mathematicians put counting rods on a counting board, a sheet of cloth with grids, and used only vertical forms relying on the grids. An 18th-century Japanese mathematics book has a checker counting board diagram, with the order of magnitude symbols "千百十一分厘毛" (thousand, hundred, ten, unit, tenth, hundredth, thousandth). Examples: == Rod numerals == Rod numerals are a positional numeral system made from shapes of counting rods. Positive numbers are written as they are and the negative numbers are written with a slant bar at the last digit. The vertical bar in the horizontal forms 6–9 are drawn shorter to have the same character height. A circle (〇) is used for 0. Many historians think it was imported from Indian numerals by Gautama Siddha in 718, but some think it was created from the Chinese text space filler "□", and others think that the Indians acquired it from China, because it resembles a Confucian philosophical symbol for "nothing". In the 13th century, Southern Song mathematicians changed digits for 4, 5, and 9 to reduce strokes. The new horizontal forms eventually transformed into Suzhou numerals. Japanese continued to use the traditional forms. Examples: In Japan, Seki Takakazu developed the rod numerals into symbolic notation for algebra and drastically improved Japanese mathematics. After his period, the positional numeral system using Chinese numeral characters was developed, and the rod numerals were used only for the plus and minus signs. == Fractions == A fraction was expressed with rod numerals as two rod numerals one on top of another (without any other symbol, like the modern horizontal bar). == Rod calculus == The method for using counting rods for mathematical calculation was called rod calculation or rod calculus (筹算). Rod calculus can be used for a wide range of calculations, including finding the value of π, finding square roots, cube roots, or higher order roots, and solving a system of linear equations. Before the introduction of a written zero, a space was used to indicate no units, and the rotation of the character in the subsequent unit column, by 90°, adopted, to help reduce the ambiguity in record values calculated on the rods. For example 107 (𝍠 𝍧) and 17 (𝍩𝍧) would be distinguished by rotation, though multiple zero units could lead to ambiguity, eg. 1007 (𝍩 𝍧) , and 10007 (𝍠 𝍧). Once written zero came into play, the rod numerals had become independent, and their use indeed outlives the counting rods, after its replacement by abacus. One variation of horizontal rod numerals, the Suzhou numerals is still in use for book-keeping and in herbal medicine prescription in Chinatowns in some parts of the world. == Unicode == Unicode 5.0 includes counting rod numerals in their own block in the Supplementary Multilingual Plane (SMP) from U+1D360 to U+1D37F. The code points for the horizontal digits 1–9 are U+1D360 to U+1D368 and those for the vertical digits 1–9 are U+1D369 to U+1D371. The former are called unit digits and the latter are called tens digits, which is opposite of the convention described above. The Unicode Standard states that the orientation of the Unicode characters follows Song dynasty convention, which differs from Han dynasty practice which represented digits as vertical lines, and tens as horizontal lines. Zero should be represented by U+3007 (〇, ideographic number zero) and the negative sign should be represented by U+20E5 (combining reverse solidus overlay). As these were recently added to the character set and since they are included in the SMP, font support may still be limited. == See also == == References == == External links == For a look of the ancient counting rods, and further explanation, you can visit the sites https://web.archive.org/web/20010217175749/http://www.math.sfu.ca/histmath/China/Beginning/Rod.html http://mathforum.org/library/drmath/view/52557.html Counting rods in China (in Chinese) (Translate to English: Google, Bing, Yandex) Counting rods and go stones of a Japanese mathematician around 1872 (in Japanese) (Translate to English: Google, Bing, Yandex)
Wikipedia:Covariant (invariant theory)#0	In invariant theory, a branch of algebra, given a group G, a covariant is a G-equivariant polynomial map V → W {\displaystyle V\to W} between linear representations V, W of G. It is a generalization of a classical convariant, which is a homogeneous polynomial map from the space of binary m-forms to the space of binary p-forms (over the complex numbers) that is S L 2 ( C ) {\displaystyle SL_{2}(\mathbb {C} )} -equivariant. == See also == Module of covariants Invariant of a binary form § Terminology Transvectant – method/process of constructing covariants == References == Procesi, Claudio (2007). Lie groups : an approach through invariants and representations. New York: Springer. ISBN 978-0-387-26040-2. OCLC 191464530. Kraft, Hanspeter; Procesi, Claudio (July 2016). "Classical Invariant Theory, a Primer". Department of Mathematics, IIT Bombay.
Wikipedia:Cover (algebra)#0	In abstract algebra, a cover is one instance of some mathematical structure mapping onto another instance, such as a group (trivially) covering a subgroup. This should not be confused with the concept of a cover in topology. When some object X is said to cover another object Y, the cover is given by some surjective and structure-preserving map f : X → Y. The precise meaning of "structure-preserving" depends on the kind of mathematical structure of which X and Y are instances. In order to be interesting, the cover is usually endowed with additional properties, which are highly dependent on the context. == Examples == A classic result in semigroup theory due to D. B. McAlister states that every inverse semigroup has an E-unitary cover; besides being surjective, the homomorphism in this case is also idempotent separating, meaning that in its kernel an idempotent and non-idempotent never belong to the same equivalence class.; something slightly stronger has actually be shown for inverse semigroups: every inverse semigroup admits an F-inverse cover. McAlister's covering theorem generalizes to orthodox semigroups: every orthodox semigroup has a unitary cover. Examples from other areas of algebra include the Frattini cover of a profinite group and the universal cover of a Lie group. == Modules == If F is some family of modules over some ring R, then an F-cover of a module M is a homomorphism X→M with the following properties: X is in the family F X→M is surjective Any surjective map from a module in the family F to M factors through X Any endomorphism of X commuting with the map to M is an automorphism. In general an F-cover of M need not exist, but if it does exist then it is unique up to (non-unique) isomorphism. Examples include: Projective covers (always exist over perfect rings) flat covers (always exist) torsion-free covers (always exist over integral domains) injective covers == See also == Embedding == Notes == == References == Howie, John M. (1995). Fundamentals of Semigroup Theory. Clarendon Press. ISBN 0-19-851194-9.
Wikipedia:Craig S. Kaplan#0	Craig S. Kaplan is a Canadian computer scientist, mathematician, and mathematical artist. He is an editor of the Journal of Mathematics and the Arts (formerly chief editor), and an organizer of the Bridges Conference on mathematics and art. He is an associate professor of computer science at the University of Waterloo, Canada. Kaplan's work primarily focuses on applications of geometry and computer science to visual art and design. He was part of the team that proved that the tile discovered by hobbyist David Smith is a solution to the einstein problem, a single shape which aperiodically tiles the plane but cannot do so periodically. == Education == Kaplan received a BMath from the University of Waterloo in 1996. He went on to receive MSc and PhD degrees in computer science from University of Washington in 1998 and 2002, respectively. == Work == Kaplan's research work focuses on the application of computer graphics and mathematics in art and design. He is an expert on computational applications of tiling theory. === Exotic geometries in protein assembly === In 2019, Kaplan helped to apply the concepts of Archimedean solids to protein assembly, and together with an experimental team at RIKEN demonstrated that these exotic geometries lead to ultra-stable macromolecular cages. These new systems could have applications in targeted drug delivery systems or the design of new materials at the nanoscale. === Einstein problem === In 2023, Kaplan was part of the team that solved the einstein problem, a major open problem in tiling theory and Euclidean geometry. The problem is to find an "aperiodic monotile", a single geometric shape which can tesselate the plane aperiodically (without translational symmetry) but which cannot do so periodically. The discovery is under professional review and, upon confirmation, will be credited as solving a longstanding mathematical problem. In 2022, hobbyist David Smith discovered a shape constructed by gluing together eight kites (in this case, each kite is a sixth of a regular hexagon) which seemed from Smith's experiments to tile the plane but would not do so periodically. He contacted Kaplan for help analyzing the shape, which the two named the "hat". After Kaplan's computational tools also found the tiling to continue indefinitely, Kaplan and Smith recruited two other mathematicians, Joseph Samuel Myers and Chaim Goodman-Strauss to help prove they had found an aperiodic monotile. Smith also found a second tile, dubbed the "turtle", which seemed to have the same properties. In March 2023, the team of four announced their proof that the hat and turtle tiles, as well as an infinite family of other tiles interpolating the two, are aperiodic monotiles. Both the hat and turtle tiles require some reflected copies to tile the plane. After the initial preprint, Smith noticed that a tile related to the hat tile could tile the plane either periodically or aperiodically, with the aperiodic tiling not requiring reflections. A suitable manipulation of the edge prevents the periodic tiling. In May 2023 the team of Smith, Kaplan, Myers, and Goodman-Strauss posted a new preprint proving that the new shape, which Smith called a "spectre", is a strictly chiral aperiodic monotile: even if reflections are allowed, every tiling is non-periodic and uses only one chirality of the spectre. This new shape tiles a plane in a pattern that never repeats without the use of mirror images of the shape, hence been called a "vampire einstein". == References ==
Wikipedia:Crank–Nicolson method#0	In numerical analysis, the Crank–Nicolson method is a finite difference method used for numerically solving the heat equation and similar partial differential equations. It is a second-order method in time. It is implicit in time, can be written as an implicit Runge–Kutta method, and it is numerically stable. The method was developed by John Crank and Phyllis Nicolson in the 1940s. For diffusion equations (and many other equations), it can be shown the Crank–Nicolson method is unconditionally stable. However, the approximate solutions can still contain (decaying) spurious oscillations if the ratio of time step Δ t {\displaystyle \Delta t} times the thermal diffusivity to the square of space step, Δ x 2 {\displaystyle \Delta x^{2}} , is large (typically, larger than 1/2 per Von Neumann stability analysis). For this reason, whenever large time steps or high spatial resolution is necessary, the less accurate backward Euler method is often used, which is both stable and immune to oscillations. == Principle == The Crank–Nicolson method is based on the trapezoidal rule, giving second-order convergence in time. For linear equations, the trapezoidal rule is equivalent to the implicit midpoint method—the simplest example of a Gauss–Legendre implicit Runge–Kutta method—which also has the property of being a geometric integrator. For example, in one dimension, suppose the partial differential equation is ∂ u ∂ t = F ( u , x , t , ∂ u ∂ x , ∂ 2 u ∂ x 2 ) . {\displaystyle {\frac {\partial u}{\partial t}}=F\left(u,x,t,{\frac {\partial u}{\partial x}},{\frac {\partial ^{2}u}{\partial x^{2}}}\right).} Letting u ( i Δ x , n Δ t ) = u i n {\displaystyle u(i\Delta x,n\Delta t)=u_{i}^{n}} and F i n = F {\displaystyle F_{i}^{n}=F} evaluated for i , n {\displaystyle i,n} and u i n {\displaystyle u_{i}^{n}} , the equation for Crank–Nicolson method is a combination of the forward Euler method at n {\displaystyle n} and the backward Euler method at n + 1 {\displaystyle n+1} (note, however, that the method itself is not simply the average of those two methods, as the backward Euler equation has an implicit dependence on the solution): Note that this is an implicit method: to get the "next" value of u {\displaystyle u} in time, a system of algebraic equations must be solved. If the partial differential equation is nonlinear, the discretization will also be nonlinear, so that advancing in time will involve the solution of a system of nonlinear algebraic equations, though linearizations are possible. In many problems, especially linear diffusion, the algebraic problem is tridiagonal and may be efficiently solved with the tridiagonal matrix algorithm, which gives a fast O ( N ) {\displaystyle {\mathcal {O}}(N)} direct solution, as opposed to the usual O ( N 3 ) {\displaystyle {\mathcal {O}}(N^{3})} for a full matrix, in which N {\displaystyle N} indicates the matrix size. == Example: 1D diffusion == The Crank–Nicolson method is often applied to diffusion problems. As an example, for linear diffusion, ∂ u ∂ t = a ∂ 2 u ∂ x 2 , {\displaystyle {\frac {\partial u}{\partial t}}=a{\frac {\partial ^{2}u}{\partial x^{2}}},} applying a finite difference spatial discretization for the right-hand side, the Crank–Nicolson discretization is then u i n + 1 − u i n Δ t = a 2 ( Δ x ) 2 ( ( u i + 1 n + 1 − 2 u i n + 1 + u i − 1 n + 1 ) + ( u i + 1 n − 2 u i n + u i − 1 n ) ) {\displaystyle {\frac {u_{i}^{n+1}-u_{i}^{n}}{\Delta t}}={\frac {a}{2(\Delta x)^{2}}}\left((u_{i+1}^{n+1}-2u_{i}^{n+1}+u_{i-1}^{n+1})+(u_{i+1}^{n}-2u_{i}^{n}+u_{i-1}^{n})\right)} or, letting r = a Δ t 2 ( Δ x ) 2 {\displaystyle r={\frac {a\Delta t}{2(\Delta x)^{2}}}} , − r u i + 1 n + 1 + ( 1 + 2 r ) u i n + 1 − r u i − 1 n + 1 = r u i + 1 n + ( 1 − 2 r ) u i n + r u i − 1 n . {\displaystyle -ru_{i+1}^{n+1}+(1+2r)u_{i}^{n+1}-ru_{i-1}^{n+1}=ru_{i+1}^{n}+(1-2r)u_{i}^{n}+ru_{i-1}^{n}.} Given that the terms on the right-hand side of the equation are known, this is a tridiagonal problem, so that u i n + 1 {\displaystyle u_{i}^{n+1}} may be efficiently solved by using the tridiagonal matrix algorithm in favor over the much more costly matrix inversion. A quasilinear equation, such as (this is a minimalistic example and not general) ∂ u ∂ t = a ( u ) ∂ 2 u ∂ x 2 , {\displaystyle {\frac {\partial u}{\partial t}}=a(u){\frac {\partial ^{2}u}{\partial x^{2}}},} would lead to a nonlinear system of algebraic equations, which could not be easily solved as above; however, it is possible in some cases to linearize the problem by using the old value for a {\displaystyle a} , that is, a i n ( u ) {\displaystyle a_{i}^{n}(u)} instead of a i n + 1 ( u ) {\displaystyle a_{i}^{n+1}(u)} . Other times, it may be possible to estimate a i n + 1 ( u ) {\displaystyle a_{i}^{n+1}(u)} using an explicit method and maintain stability. == Example: 1D diffusion with advection for steady flow, with multiple channel connections == This is a solution usually employed for many purposes when there is a contamination problem in streams or rivers under steady flow conditions, but information is given in one dimension only. Often the problem can be simplified into a 1-dimensional problem and still yield useful information. Here we model the concentration of a solute contaminant in water. This problem is composed of three parts: the known diffusion equation ( D x {\displaystyle D_{x}} chosen as constant), an advective component (which means that the system is evolving in space due to a velocity field), which we choose to be a constant U x {\displaystyle U_{x}} , and a lateral interaction between longitudinal channels ( k {\displaystyle k} ): where C {\displaystyle C} is the concentration of the contaminant, and subscripts N {\displaystyle N} and M {\displaystyle M} correspond to previous and next channel. The Crank–Nicolson method (where i {\displaystyle i} represents position, and j {\displaystyle j} time) transforms each component of the PDE into the following: Now we create the following constants to simplify the algebra: λ = D x Δ t 2 Δ x 2 , {\displaystyle \lambda ={\frac {D_{x}\,\Delta t}{2\,\Delta x^{2}}},} α = U x Δ t 4 Δ x , {\displaystyle \alpha ={\frac {U_{x}\,\Delta t}{4\,\Delta x}},} β = k Δ t 2 , {\displaystyle \beta ={\frac {k\,\Delta t}{2}},} and substitute (2), (3), (4), (5), (6), (7), α {\displaystyle \alpha } , β {\displaystyle \beta } and λ {\displaystyle \lambda } into (1). We then put the new time terms on the left ( j + 1 {\displaystyle j+1} ) and the present time terms on the right ( j {\displaystyle j} ) to get − β C N i j + 1 − ( λ + α ) C i − 1 j + 1 + ( 1 + 2 λ + 2 β ) C i j + 1 − ( λ − α ) C i + 1 j + 1 − β C M i j + 1 = {\displaystyle -\beta C_{Ni}^{j+1}-(\lambda +\alpha )C_{i-1}^{j+1}+(1+2\lambda +2\beta )C_{i}^{j+1}-(\lambda -\alpha )C_{i+1}^{j+1}-\beta C_{Mi}^{j+1}={}} β C N i j + ( λ + α ) C i − 1 j + ( 1 − 2 λ − 2 β ) C i j + ( λ − α ) C i + 1 j + β C M i j . {\displaystyle \qquad \beta C_{Ni}^{j}+(\lambda +\alpha )C_{i-1}^{j}+(1-2\lambda -2\beta )C_{i}^{j}+(\lambda -\alpha )C_{i+1}^{j}+\beta C_{Mi}^{j}.} To model the first channel, we realize that it can only be in contact with the following channel ( M {\displaystyle M} ), so the expression is simplified to − ( λ + α ) C i − 1 j + 1 + ( 1 + 2 λ + β ) C i j + 1 − ( λ − α ) C i + 1 j + 1 − β C M i j + 1 = {\displaystyle -(\lambda +\alpha )C_{i-1}^{j+1}+(1+2\lambda +\beta )C_{i}^{j+1}-(\lambda -\alpha )C_{i+1}^{j+1}-\beta C_{Mi}^{j+1}={}} + ( λ + α ) C i − 1 j + ( 1 − 2 λ − β ) C i j + ( λ − α ) C i + 1 j + β C M i j . {\displaystyle \qquad {}+(\lambda +\alpha )C_{i-1}^{j}+(1-2\lambda -\beta )C_{i}^{j}+(\lambda -\alpha )C_{i+1}^{j}+\beta C_{Mi}^{j}.} In the same way, to model the last channel, we realize that it can only be in contact with the previous channel ( N {\displaystyle N} ), so the expression is simplified to − β C N i j + 1 − ( λ + α ) C i − 1 j + 1 + ( 1 + 2 λ + β ) C i j + 1 − ( λ − α ) C i + 1 j + 1 = {\displaystyle -\beta C_{Ni}^{j+1}-(\lambda +\alpha )C_{i-1}^{j+1}+(1+2\lambda +\beta )C_{i}^{j+1}-(\lambda -\alpha )C_{i+1}^{j+1}={}} β C N i j + ( λ + α ) C i − 1 j + ( 1 − 2 λ − β ) C i j + ( λ − α ) C i + 1 j . {\displaystyle \qquad \beta C_{Ni}^{j}+(\lambda +\alpha )C_{i-1}^{j}+(1-2\lambda -\beta )C_{i}^{j}+(\lambda -\alpha )C_{i+1}^{j}.} To solve this linear system of equations, we must now see that boundary conditions must be given first to the beginning of the channels: C 0 j {\displaystyle C_{0}^{j}} : boundary condition for the channel at present time step, C 0 j + 1 {\displaystyle C_{0}^{j+1}} : boundary condition for the channel at next time step, C N 0 j {\displaystyle C_{N0}^{j}} : boundary condition for the previous channel to the one analyzed at present time step, C M 0 j {\displaystyle C_{M0}^{j}} : boundary condition for the next channel to the one analyzed at present time step. For the last cell of the channels ( z {\displaystyle z} ), the most convenient condition becomes an adiabatic one, so ∂ C ∂ x \| x = z = C i + 1 − C i − 1 2 Δ x = 0. {\displaystyle \left.{\frac {\partial C}{\partial x}}\right\|_{x=z}={\frac {C_{i+1}-C_{i-1}}{2\,\Delta x}}=0.} This condition is satisfied if and only if (regardless of a null value) C i + 1 j + 1 = C i − 1 j + 1 . {\displaystyle C_{i+1}^{j+1}=C_{i-1}^{j+1}.} Let us solve this problem (in a matrix form) for the case of 3 channels and 5 nodes (including the initial boundary condition). We express this as a linear system problem: A A C j + 1 = B B C j + d , {\displaystyle \mathbf {AA} \,\mathbf {C^{j+1}} =\mathbf {BB} \,\mathbf {C^{j}} +\mathbf {d} ,} where C j + 1 = [ C 11 j + 1 C 12 j + 1 C 13 j + 1 C 14 j + 1 C 21 j + 1 C 22 j + 1 C 23 j + 1 C 24 j + 1 C 31 j + 1 C 32 j + 1 C 33 j + 1 C 34 j + 1 ] , C j = [ C 11 j C 12 j C 13 j C 14 j C 21 j C 22 j C 23 j C 24 j C 31 j C 32 j C 33 j C 34 j ] . {\displaystyle \mathbf {C^{j+1}} ={\begin{bmatrix}C_{11}^{j+1}\\C_{12}^{j+1}\\C_{13}^{j+1}\\C_{14}^{j+1}\\C_{21}^{j+1}\\C_{22}^{j+1}\\C_{23}^{j+1}\\C_{24}^{j+1}\\C_{31}^{j+1}\\C_{32}^{j+1}\\C_{33}^{j+1}\\C_{34}^{j+1}\end{bmatrix}},\quad \mathbf {C^{j}} ={\begin{bmatrix}C_{11}^{j}\\C_{12}^{j}\\C_{13}^{j}\\C_{14}^{j}\\C_{21}^{j}\\C_{22}^{j}\\C_{23}^{j}\\C_{24}^{j}\\C_{31}^{j}\\C_{32}^{j}\\C_{33}^{j}\\C_{34}^{j}\end{bmatrix}}.} Now we must realize that AA and BB should be arrays made of four different subarrays (remember that only three channels are considered for this example, but it covers the main part discussed above): A A = [ A A 1 A A 3 0 A A 3 A A 2 A A 3 0 A A 3 A A 1 ] , B B = [ B B 1 − A A 3 0 − A A 3 B B 2 − A A 3 0 − A A 3 B B 1 ] , {\displaystyle \mathbf {AA} ={\begin{bmatrix}AA1&AA3&0\\AA3&AA2&AA3\\0&AA3&AA1\end{bmatrix}},\quad \mathbf {BB} ={\begin{bmatrix}BB1&-AA3&0\\-AA3&BB2&-AA3\\0&-AA3&BB1\end{bmatrix}},} where the elements mentioned above correspond to the next arrays, and an additional 4×4 full of zeros. Please note that the sizes of AA and BB are 12×12: A A 1 = [ ( 1 + 2 λ + β ) − ( λ − α ) 0 0 − ( λ + α ) ( 1 + 2 λ + β ) − ( λ − α ) 0 0 − ( λ + α ) ( 1 + 2 λ + β ) − ( λ − α ) 0 0 − 2 λ ( 1 + 2 λ + β ) ] , {\displaystyle \mathbf {AA1} ={\begin{bmatrix}(1+2\lambda +\beta )&-(\lambda -\alpha )&0&0\\-(\lambda +\alpha )&(1+2\lambda +\beta )&-(\lambda -\alpha )&0\\0&-(\lambda +\alpha )&(1+2\lambda +\beta )&-(\lambda -\alpha )\\0&0&-2\lambda &(1+2\lambda +\beta )\end{bmatrix}},} A A 2 = [ ( 1 + 2 λ + 2 β ) − ( λ − α ) 0 0 − ( λ + α ) ( 1 + 2 λ + 2 β ) − ( λ − α ) 0 0 − ( λ + α ) ( 1 + 2 λ + 2 β ) − ( λ − α ) 0 0 − 2 λ ( 1 + 2 λ + 2 β ) ] , {\displaystyle \mathbf {AA2} ={\begin{bmatrix}(1+2\lambda +2\beta )&-(\lambda -\alpha )&0&0\\-(\lambda +\alpha )&(1+2\lambda +2\beta )&-(\lambda -\alpha )&0\\0&-(\lambda +\alpha )&(1+2\lambda +2\beta )&-(\lambda -\alpha )\\0&0&-2\lambda &(1+2\lambda +2\beta )\end{bmatrix}},} A A 3 = [ − β 0 0 0 0 − β 0 0 0 0 − β 0 0 0 0 − β ] , {\displaystyle \mathbf {AA3} ={\begin{bmatrix}-\beta &0&0&0\\0&-\beta &0&0\\0&0&-\beta &0\\0&0&0&-\beta \end{bmatrix}},} B B 1 = [ ( 1 − 2 λ − β ) ( λ − α ) 0 0 ( λ + α ) ( 1 − 2 λ − β ) ( λ − α ) 0 0 ( λ + α ) ( 1 − 2 λ − β ) ( λ − α ) 0 0 2 λ ( 1 − 2 λ − β ) ] , {\displaystyle \mathbf {BB1} ={\begin{bmatrix}(1-2\lambda -\beta )&(\lambda -\alpha )&0&0\\(\lambda +\alpha )&(1-2\lambda -\beta )&(\lambda -\alpha )&0\\0&(\lambda +\alpha )&(1-2\lambda -\beta )&(\lambda -\alpha )\\0&0&2\lambda &(1-2\lambda -\beta )\end{bmatrix}},} B B 2 = [ ( 1 − 2 λ − 2 β ) ( λ − α ) 0 0 ( λ + α ) ( 1 − 2 λ − 2 β ) ( λ − α ) 0 0 ( λ + α ) ( 1 − 2 λ − 2 β ) ( λ − α ) 0 0 2 λ ( 1 − 2 λ − 2 β ) ] . {\displaystyle \mathbf {BB2} ={\begin{bmatrix}(1-2\lambda -2\beta )&(\lambda -\alpha )&0&0\\(\lambda +\alpha )&(1-2\lambda -2\beta )&(\lambda -\alpha )&0\\0&(\lambda +\alpha )&(1-2\lambda -2\beta )&(\lambda -\alpha )\\0&0&2\lambda &(1-2\lambda -2\beta )\end{bmatrix}}.} The d vector here is used to hold the boundary conditions. In this example it is a 12×1 vector: d = [ ( λ + α ) ( C 10 j + 1 + C 10 j ) 0 0 0 ( λ + α ) ( C 20 j + 1 + C 20 j ) 0 0 0 ( λ + α ) ( C 30 j + 1 + C 30 j ) 0 0 0 ] . {\displaystyle \mathbf {d} ={\begin{bmatrix}(\lambda +\alpha )(C_{10}^{j+1}+C_{10}^{j})\\0\\0\\0\\(\lambda +\alpha )(C_{20}^{j+1}+C_{20}^{j})\\0\\0\\0\\(\lambda +\alpha )(C_{30}^{j+1}+C_{30}^{j})\\0\\0\\0\end{bmatrix}}.} To find the concentration at any time, one must iterate the following equation: C j + 1 = A A − 1 ( B B C j + d ) . {\displaystyle \mathbf {C^{j+1}} =\mathbf {AA} ^{-1}(\mathbf {BB} \,\mathbf {C^{j}} +\mathbf {d} ).} == Example: 2D diffusion == When extending into two dimensions on a uniform Cartesian grid, the derivation is similar and the results may lead to a system of band-diagonal equations rather than tridiagonal ones. The two-dimensional heat equation ∂ u ∂ t = a ∇ 2 u , {\displaystyle {\frac {\partial u}{\partial t}}=a\,\nabla ^{2}u,} ∂ u ∂ t = a ( ∂ 2 u ∂ x 2 + ∂ 2 u ∂ y 2 ) {\displaystyle {\frac {\partial u}{\partial t}}=a\left({\frac {\partial ^{2}u}{\partial x^{2}}}+{\frac {\partial ^{2}u}{\partial y^{2}}}\right)} can be solved with the Crank–Nicolson discretization of u i , j n + 1 = u i , j n + 1 2 a Δ t ( Δ x ) 2 [ ( u i + 1 , j n + 1 + u i − 1 , j n + 1 + u i , j + 1 n + 1 + u i , j − 1 n + 1 − 4 u i , j n + 1 ) + ( u i + 1 , j n + u i − 1 , j n + u i , j + 1 n + u i , j − 1 n − 4 u i , j n ) ] , {\displaystyle {\begin{aligned}u_{i,j}^{n+1}={}&u_{i,j}^{n}+{\frac {1}{2}}{\frac {a\Delta t}{(\Delta x)^{2}}}{\big [}(u_{i+1,j}^{n+1}+u_{i-1,j}^{n+1}+u_{i,j+1}^{n+1}+u_{i,j-1}^{n+1}-4u_{i,j}^{n+1})\\&+(u_{i+1,j}^{n}+u_{i-1,j}^{n}+u_{i,j+1}^{n}+u_{i,j-1}^{n}-4u_{i,j}^{n}){\big ]},\end{aligned}}} assuming that a square grid is used, so that Δ x = Δ y {\displaystyle \Delta x=\Delta y} . This equation can be simplified somewhat by rearranging terms and using the CFL number μ = a Δ t ( Δ x ) 2 . {\displaystyle \mu ={\frac {a\,\Delta t}{(\Delta x)^{2}}}.} For the Crank–Nicolson numerical scheme, a low CFL number is not required for stability, however, it is required for numerical accuracy. We can now write the scheme as ( 1 + 2 μ ) u i , j n + 1 − μ 2 ( u i + 1 , j n + 1 + u i − 1 , j n + 1 + u i , j + 1 n + 1 + u i , j − 1 n + 1 ) {\displaystyle (1+2\mu )u_{i,j}^{n+1}-{\frac {\mu }{2}}\left(u_{i+1,j}^{n+1}+u_{i-1,j}^{n+1}+u_{i,j+1}^{n+1}+u_{i,j-1}^{n+1}\right)} = ( 1 − 2 μ ) u i , j n + μ 2 ( u i + 1 , j n + u i − 1 , j n + u i , j + 1 n + u i , j − 1 n ) . {\displaystyle \qquad =(1-2\mu )u_{i,j}^{n}+{\frac {\mu }{2}}\left(u_{i+1,j}^{n}+u_{i-1,j}^{n}+u_{i,j+1}^{n}+u_{i,j-1}^{n}\right).} Solving such a linear system is costly. Hence an alternating-direction implicit method can be implemented to solve the numerical PDE, whereby one dimension is treated implicitly, and other dimension explicitly for half of the assigned time step and conversely for the remainder half of the time step. The benefit of this strategy is that the implicit solver only requires a tridiagonal matrix algorithm to be solved. The difference between the true Crank–Nicolson solution and ADI approximated solution has an order of accuracy of O ( Δ t 4 ) {\displaystyle O(\Delta t^{4})} and hence can be ignored with a sufficiently small time step. == Crank–Nicolson for nonlinear problems == Because the Crank–Nicolson method is implicit, it is generally impossible to solve exactly. Instead, an iterative technique should be used to converge to the solution. One option is to use Newton's method to converge on the prediction, but this requires the computation of the Jacobian. For a high-dimensional system like those in computational fluid dynamics or numerical relativity, it may be infeasible to compute this Jacobian. A Jacobian-free alternative is fixed-point iteration. If f {\displaystyle f} is the velocity of the system, then the Crank–Nicolson prediction will be a fixed point of the map Φ ( x ) = x 0 + h 2 [ f ( x 0 ) + f ( x ) ] . {\displaystyle \Phi (x)=x_{0}+{\frac {h}{2}}\left[f(x_{0})+f(x)\right].} If the map iteration x ( i + 1 ) = Φ ( x ( i ) ) {\displaystyle x^{(i+1)}=\Phi (x^{(i)})} does not converge, the parameterized map Θ ( x , α ) = α x + ( 1 − α ) Φ ( x ) {\displaystyle \Theta (x,\alpha )=\alpha x+(1-\alpha )\Phi (x)} , with α ∈ ( 0 , 1 ) {\displaystyle \alpha \in (0,1)} , may be better behaved. In expanded form, the update formula is x i + 1 = α x i + ( 1 − α ) [ x 0 + h 2 ( f ( x 0 ) + f ( x i ) ) ] , {\displaystyle x^{i+1}=\alpha x^{i}+(1-\alpha )\left[x_{0}+{\frac {h}{2}}\left(f(x_{0})+f(x^{i})\right)\right],} where x i {\displaystyle x^{i}} is the current guess and x i − 1 {\displaystyle x_{i-1}} is the previous time-step. Even for high-dimensional systems, iteration of this map can converge surprisingly quickly. == Application in financial mathematics == Because a number of other phenomena can be modeled with the heat equation (often called the diffusion equation in financial mathematics), the Crank–Nicolson method has been applied to those areas as well. Particularly, the Black–Scholes option pricing model's differential equation can be transformed into the heat equation, and thus numerical solutions for option pricing can be obtained with the Crank–Nicolson method. The importance of this for finance is that option pricing problems, when extended beyond the standard assumptions (e.g. incorporating changing dividends), cannot be solved in closed form, but can be solved using this method. Note however, that for non-smooth final conditions (which happen for most financial instruments), the Crank–Nicolson method is not satisfactory as numerical oscillations are not damped. For vanilla options, this results in oscillation in the gamma value around the strike price. Therefore, special damping initialization steps are necessary (e.g., fully implicit finite difference method). == See also == Financial mathematics Trapezoidal rule == References == == External links == Numerical PDE Techniques for Scientists and Engineers, open access Lectures and Codes for Numerical PDEs An example of how to apply and implement the Crank–Nicolson method for the Advection equation
Wikipedia:Cristian Calude#0	Cristian Sorin Calude (born 21 April 1952) is a Romanian-New Zealand mathematician and computer scientist. == Biography == After graduating from the Vasile Alecsandri National College in Galați, he studied at the University of Bucharest, where he was student of Grigore C. Moisil and Solomon Marcus. Calude received his Ph.D. in Mathematics from the University of Bucharest under the direction of Solomon Marcus in 1977. He is currently chair professor at the University of Auckland, New Zealand and also the founding director of the Centre for Discrete Mathematics and Theoretical Computer Science. Visiting professor in many universities in Europe, North and South America, Australasia, South Africa, including Monbusho Visiting professor, JAIST, 1999 and visiting professor ENS, Paris, 2009, École Polytechnique, Paris, 2011; visiting fellow, Isaac Newton Institute for Mathematical Sciences, 2012; guest professor, Sun Yat-sen University, Guangzhou, China, 2017–2020; visiting fellow ETH Zurich, 2019. Former professor at the University of Bucharest. Calude is author or co-author of more than 270 research articles and 8 books, and is cited by more than 550 authors. He is known for research in algorithmic information theory, quantum computing, discrete mathematics and the history and philosophy of computation. In 2017, together with Sanjay Jain, Bakhadyr Khoussainov, Wei Li, and Frank Stephan, he announced an algorithm for deciding parity games in quasipolynomial time. Their result was presented by Bakhadyr Khoussainov at the Symposium on Theory of Computing 2017 and won a Best Paper Award. Calude was awarded the National Order of Faithful Service in the degree of Knight by the President of Romania, Mr. Klaus Iohannis, in June 2019. In 2021, together with Sanjay Jain, Bakhadyr Khoussainov, Wei Li, and Frank Stephan, he won the EATCS Nerode Prize for their quasipolynomial time algorithm for deciding parity games. == Distinctions and prizes == "Computing Reviews Award", Association for Computing Machinery, New York City, 1986. "Gheorghe Lazăr" Mathematics Prize, Romanian Academy, Romania, 1988. Excellence in Research Award, University of Bucharest, Romania, 2007. Dean's Award for Excellence in Teaching, University of Auckland, 2007. Hood Fellow, 2008–2009. Member of the Academia Europaea, 2008. Romanian National Order of Faithful Service in the degree of Knight, June 2019. "EATCS-IPEC Nerode Prize", 2021. "Full Member SIGMA XI", 2024. == Selected bibliography == === Articles === Calude, Cristian S.; Svozil, Karl (2024). "Binary Quantum Random Number Generator Based on Value Indefinite Observables". Scientific Reports. 14. doi:10.1038/s41598-024-62566-2. ISSN 2045-2322. PMC 11150379. Agüero Trejo, José Manuel; Calude, Cristian S. (2023). "Photonic ternary quantum random number generators". Proc. R. Soc. A. 479 (2273). doi:10.1098/rspa.2022.0543. Calude, Cristian S.; Heidar, Shahrokh; Sifakis, Joseph (2023). "What perceptron neural networks are (not) good for?". Information Sciences. 621: IS–844–IS-188. doi:10.1016/j.ins.2022.11.083. Calude, Cristian S.; Jain, Sanjay; Khoussainov, Bakhadyr; Li, Wei; Stephan, Frank (2022). "Deciding Parity Games in Quasi-polynomial Time". SIAM Journal on Computing. 51 (2): STOC17–152–STOC17-188. doi:10.1137/17M1145288. hdl:2292/31757. ISSN 0097-5397. Abbott, Alastair A.; Calude, Cristian S.; Dinneen, Michael J.; Hua, Richard (2019). "A hybrid quantum-classical paradigm to mitigate embedding costs in quantum annealing". International Journal of Quantum Information. 17 (5): 1950042–1950453. arXiv:1803.04340. Bibcode:2019IJQI...1750042A. doi:10.1142/S0219749919500424. ISSN 0219-7499. Abbott, Alastair A; Calude, Cristian S; Dinneen, Michael J; Huang, Nan (1 April 2019). "Experimentally probing the algorithmic randomness and incomputability of quantum randomness". Physica Scripta. 94 (4): 045103. arXiv:1806.08762. Bibcode:2019PhyS...94d5103A. doi:10.1088/1402-4896/aaf36a. ISSN 0031-8949. Calude, Cristian S.; Dumitrescu, Monica (7 June 2018). "A probabilistic anytime algorithm for the halting problem". Computability. 7 (2–3): 259–271. doi:10.3233/COM-170073. Calude, Cristian S.; Staiger, Ludwig (2018). "Liouville, Computable, Borel Normal and Martin-Löf Random Numbers". Theory of Computing Systems. 62 (7): 1573–1585. doi:10.1007/s00224-017-9767-8. ISSN 1432-4350. Calude, Cristian S.; Staiger, Ludwig; Stephan, Frank (2016). "Finite state incompressible infinite sequences". Information and Computation. 247: 23–36. doi:10.1016/j.ic.2015.11.003. hdl:2292/21343. Calude, Cristian S.; Longo, Giuseppe (2017). "The Deluge of Spurious Correlations in Big Data" (PDF). Foundations of Science. 22 (3): 595–612. doi:10.1007/s10699-016-9489-4. ISSN 1233-1821. Abbott, Alastair A.; Calude, Cristian S.; Svozil, Karl (1 October 2015). "A variant of the Kochen-Specker theorem localising value indefiniteness". Journal of Mathematical Physics. 56 (10): 102201. arXiv:1503.01985. Bibcode:2015JMP....56j2201A. doi:10.1063/1.4931658. ISSN 0022-2488. Calude, Cristian S.; Calude, Elena; Dinneen, Michael J. (9 March 2015). "Guest Column: Adiabatic Quantum Computing Challenges". ACM SIGACT News. 46 (1): 40–61. doi:10.1145/2744447.2744459. ISSN 0163-5700. Abbott, Alastair A.; Calude, Cristian S.; Svozil, Karl (10 March 2014). "Value-indefinite observables are almost everywhere". Physical Review A. 89 (3): 032109-032116. arXiv:1309.7188. Bibcode:2014PhRvA..89c2109A. doi:10.1103/PhysRevA.89.032109. ISSN 1050-2947. Calude, Cristian S.; Dinneen, Michael J.; Dumitrescu, Monica; Svozil, Karl (6 August 2010). "Experimental evidence of quantum randomness incomputability". Physical Review A. 82 (2): 022102. arXiv:1004.1521. Bibcode:2010PhRvA..82b2102C. doi:10.1103/PhysRevA.82.022102. ISSN 1050-2947. Calude, Cristian S.; Stay, Michael A. (2008). "Most programs stop quickly or never halt". Advances in Applied Mathematics. 40 (3): 295–308. arXiv:cs/0610153. doi:10.1016/j.aam.2007.01.001. Calude, C. S.; Chaitin, G. J. (1999). "Randomness everywhere". Nature. 400 (6742): 319–320. doi:10.1038/22435. ISSN 0028-0836. === Books === C. S. Calude. To Halt or Not to Halt? That Is the Question, World Scientific, Singapore, 2024. doi:10.1142/12159. A. Bellow, C. S. Calude, T. Zamfirescu, (eds.) Mathematics Almost Everywhere: In Memory of Solomon Marcus, World Scientific, Singapore, 2018. doi:10.1142/10912. M. Burgin, C. S. Calude, (eds.) Information and Complexity World Scientific, Singapore, 2017. doi:10.1142/10017. C. S. Calude (ed.) The Human Face of Computing, Imperial College Press, London, 2015. 21st Annual Best of Computing, The Notable Books and Articles List for 2016, ACM Computing Reviews, July 2017. doi:10.1142/p992. C. S. Calude (ed.) Randomness & Complexity, From Leibniz to Chaitin, World Scientific, Singapore, 2007. doi:10.1142/6577, C. S. Calude. Information and Randomness: An Algorithmic Perspective, 2nd Edition, Revised and Extended, Springer-Verlag, Berlin, 2002. doi:10.1007/978-3-662-04978-5. C. S. Calude, G. Păun. Computing with Cells and Atoms, Taylor & Francis, London, 2001. ISBN 978-0-7484-0899-3. C. Calude. Theories of Computational Complexity, North-Holland, Amsterdam, 1988. ISBN 978-0-444-70356-9. == Notes == == External links == Official website Cristian Calude at Mathematics Genealogy Project Cristian Calude at DBLP Bibliography Server CDMTCS at the University of Auckland Cristian S. Calude member page at Academia Europaea "C. S. Calude" Mathematics Regional Contest, National College "Vasile Alecsandri", Galați, Romania "Cristian S. Calude 60th Birthday" Cristian S. Calude publications indexed by Google Scholar
Wikipedia:Cristiana De Filippis#0	Cristiana De Filippis (born 1992) is an Italian mathematician whose research concerns regularity theory for elliptic partial differential equations and parabolic partial differential equations. She is an associate professor at the University of Parma. == Education and career == De Filippis was born in Bari in 1992 and grew up in Matera. She earned a laurea in mathematics in 2014, at the University of Turin, and a laurea magistrale in 2016, at the University of Milano-Bicocca, the Italian equivalents of a bachelor's and master's degree, mentored by Susanna Terracini and Veronica Felli respectively. She completed her doctorate (DPhil) at the University of Oxford in England in 2020, with the dissertation Vectorial problems: sharp Lipschitz bounds and borderline regularity supervised by Jan Kristensen. After postdoctoral research at the University of Turin, she became an assistant professor at the University of Parma in 2021, earned a habilitation in 2023, and was promoted to associate professor in 2024. == Scientific activity == De Filippis' research is mainly devoted to problems from regularity theory in elliptic and parabolic partial differential equations, with special emphasis on those coming from the Calculus of Variations. Together with Giuseppe Mingione, she proved a Schauder type theory for nonuniformly elliptic equations and functionals. She made extensive use of nonlinear potential theoretic methods in the context of elliptic regularity. == Recognition == De Filippis was awarded a G-Research Ph.D. Prize in Oxford in 2019. She was the 2020 recipient of the Gioacchino Iapichino prize in Mathematical Analysis of the Accademia dei Lincei and one of two 2023 recipients of the Bartolozzi Prize. In 2024 she was awarded an EMS Prize, given for "outstanding contributions to elliptic regularity, in particular Schauder estimates for nonuniformly elliptic equations and non-differentiable variational integrals, and minima of quasiconvex integrals". In 2023, De Filippis was elected to the inaugural cohort of the European Mathematical Society Young Academy and the Italian edition of Forbes included her in the 2023 list of 100 successful Italian women. == References == == External links == Home page Cristiana De Filippis publications indexed by Google Scholar Video from Corriere della Sera Video of the University of Parma, English subtitles, YouTube
Wikipedia:Cristina Pereyra#0	María Cristina Pereyra (born 1964) is a Venezuelan mathematician. She is a professor of mathematics and statistics at the University of New Mexico, and the author of several books on wavelets and harmonic analysis. Pereyra was an American Mathematical Society (AMS) Council member at large from 2019 - 2021. == Education and employment == Pereyra was a member of the Venezuelan team for the 1981 and 1982 International Mathematical Olympiads. She earned a licenciado (the equivalent of a bachelor's degree) in mathematics in 1986 from the Central University of Venezuela. She went to Yale University for graduate studies, completing her Ph.D. there in 1993. Her dissertation, Sobolev Spaces On Lipschitz Curves: Paraproducts, Inverses And Some Related Operators, was supervised by Peter Jones. After working for three years as an instructor at Princeton University, she joined the University of New Mexico faculty in 1996. == Books == Pereyra is the author or editor of: Lecture Notes on Dyadic Harmonic Analysis (Second Summer School in Analysis and Mathematical Physics, Cuernavaca, 2000; Contemporary Mathematics 289, American Mathematical Society, 2001) Wavelets, Their Friends, and What They Can Do For You (with Martin Mohlenkamp, EMS Lecture Series in Mathematics, European Mathematical Society, 2008) Harmonic Analysis: from Fourier to Wavelets (with Lesley Ward, Student Mathematical Library 63, American Mathematical Society, 2012) Harmonic Analysis, Partial Differential Equations, Complex Analysis, Banach Spaces, and Operator Theory: Celebrating Cora Sadosky's Life, Vols. I, II (edited with S. Marcantognini, A. M. Stokolos, and W. Urbina, Association for Women in Mathematics Series, Springer, 2016 and 2017) == References == == External links == Home page Cristina Pereyra publications indexed by Google Scholar
Wikipedia:Cristina Toninelli#0	Cristina Toninelli is an Italian mathematician who works in France as a director of research for the French National Centre for Scientific Research (CNRS), at the Centre de recherche en mathématiques de la décision (Ceremade) of Paris Dauphine University. Her research concerns the probability theory and statistical mechanics of phase transitions in interacting particle systems, including bootstrap percolation, glass transitions, and jamming. She has also studied cellular automata and group testing. == Education and career == Toninelli studied physics at Sapienza University of Rome, earning a laurea in 2000 and completing a Ph.D. in 2004. Her doctoral dissertation, Kinetically constrained models for glassy dynamics, was supervised by Giovanni Jona-Lasinio. She became a postdoctoral researcher in the laboratory for theoretical physics (LPT) at the École normale supérieure (Paris) from 2003 to 2005 and then for shorter periods at Service de Physique Théorique in CEA Paris-Saclay and at the Laboratory of Theoretical Physics and Statistical Models in Paris-Saclay University, also visiting the Instituto Nacional de Matemática Pura e Aplicada in Brazil. In 2006 she began working as a researcher for the CNRS. She has been a director of research since 2018. == Recognition == The Société mathématique de France gave Toninelli the 2021 Marc Yor Prize, citing her work on the relaxation towards equilibrium of interacting particle systems. == References == == External links == Home page Cristina Toninelli publications indexed by Google Scholar
Wikipedia:Cristóbal de Losada y Puga#0	Cristóbal de Losada y Puga (14 April 1894 – 30 August 1961) was a Peruvian mathematician and mining engineer. He was Minister of Education of Peru in the government of José Luis Bustamante y Rivero and Director of the National Library of Peru between 1948 and 1961. == Biography == He was born in New York, son of Enrique Cristóbal de Losada Plissé and Amalia Natividad Puga y Puga. He was two years old when, in 1896, his father died so he was taken back to Peru and settled in Cajamarca, the land of his mother's family. There he attended his primary and secondary studies. In 1913 he went to Lima to study at the National School of Engineers (now the National University of Engineering), obtaining his title of Mining Engineer in 1919. His first professional work was with the Corps of Mining Engineers, until 1923. He was also admitted in the Faculty of Sciences of the National University of San Marcos. In 1922, he graduated with a bachelor's degree and later in 1923 he obtained his diploma as a doctor of mathematical sciences from this institution, with a thesis "On rolling curves". He dedicated himself to teaching. In the Chorrillos Military School he was professor of Arithmetic, Descriptive Geometry and Elemental Mechanics (1920–1926 and 1931–1940). In the Faculty of Sciences of San Marcos he was Professor of Differential and Integral Calculation (1924–1926), and of Calculation of Probabilities and Mathematical Physics (1935–1939). In the National School of Engineers he was professor of Rational Mechanics, Resistance of Materials and Infinitesimal Calculation (1930–1931), work that exerted until the closing of the School for political reasons. In 1924 he was a speaker at the International Congress of Mathematicians in Toronto. In 1931 he assumed the presidency of the National Society of Industries. In 1933 he became a professor at the Faculty of Science and Engineering at the Pontifical Catholic University of Peru, where he taught Analytical Geometry, Infinitesimal Calculation, Mechanics and Resistance of Materials, until 1953. He became dean of the Faculty (1939–1946 and 1948–1950. He was also director of the Magazine of the Catholic University (1938–1945) and even came to serve as prorector (1941–1946). President José Luis Bustamante y Rivero summoned him to serve as Minister of Public Education, he served in this role from January 12 to October 30, 1947. On July 12, 1948 he was appointed Director of the National Library of Peru, a position in which he remained until his death. During his long period at the head of that institution he directed the Fénix magazine. He was a member of the National Academy of Exact, Physical and Natural Sciences of Peru, of the Peruvian Association for the Progress of Science and of the Peruvian Academy of Language. He was also a member to the Royal Academy of Physical and Natural Sciences of Madrid, the Royal Spanish Mathematical Society, the French Physical Society and the American Mathematical Association of America. He married María Luisa Marrou y Correa, he was the father of five children. == Works == Curso de Análisis matemático (3 volumes, 1945-1954) Las anomalías de la gravedad: su interpretación geológica, sus aplicaciones mineras (1917; aumentada en 1920) Contribución a la teoría matemática de las clépsidras y de los filtros (1922) Sobre las curvas de rodadura (1923) Mecánica racional (1930) Curso de Cálculo Infinitesimal (1938) Teoría y técnica de la fotoelastisimetría (1941) Galileo (1942) Copérnico (1943) == References == == Bibliography == Basadre Grohmann, Jorge (2005). Historia de la República del Perú (1822–1933). 8 (in Spanish). Lima: Producciones Cantabria. pp. 179–180. ISBN 9972-205-70-3. Tauro del Pino, Alberto (2001). Enciclopedia Ilustrada del Perú. 10 (in Spanish) (3rd ed.). Lima: PEISA. ISBN 9972-40-159-6.
Wikipedia:Crystal Ball function#0	The Crystal Ball function, named after the Crystal Ball Collaboration (hence the capitalized initial letters), is a probability density function commonly used to model various lossy processes in high-energy physics. It consists of a Gaussian core portion and a power-law low-end tail, below a certain threshold. The function itself and its first derivative are both continuous. The Crystal Ball function is given by: f ( x ; α , n , x ¯ , σ ) = N ⋅ { exp ⁡ ( − ( x − x ¯ ) 2 2 σ 2 ) , for x − x ¯ σ > − α A ⋅ ( B − x − x ¯ σ ) − n , for x − x ¯ σ ⩽ − α {\displaystyle f(x;\alpha ,n,{\bar {x}},\sigma )=N\cdot {\begin{cases}\exp(-{\frac {(x-{\bar {x}})^{2}}{2\sigma ^{2}}}),&{\mbox{for }}{\frac {x-{\bar {x}}}{\sigma }}>-\alpha \\A\cdot (B-{\frac {x-{\bar {x}}}{\sigma }})^{-n},&{\mbox{for }}{\frac {x-{\bar {x}}}{\sigma }}\leqslant -\alpha \end{cases}}} where A = ( n \| α \| ) n ⋅ exp ⁡ ( − \| α \| 2 2 ) {\displaystyle A=\left({\frac {n}{\left\|\alpha \right\|}}\right)^{n}\cdot \exp \left(-{\frac {\left\|\alpha \right\|^{2}}{2}}\right)} , B = n \| α \| − \| α \| {\displaystyle B={\frac {n}{\left\|\alpha \right\|}}-\left\|\alpha \right\|} , N = 1 σ ( C + D ) {\displaystyle N={\frac {1}{\sigma (C+D)}}} , C = n \| α \| ⋅ 1 n − 1 ⋅ exp ⁡ ( − \| α \| 2 2 ) {\displaystyle C={\frac {n}{\left\|\alpha \right\|}}\cdot {\frac {1}{n-1}}\cdot \exp \left(-{\frac {\left\|\alpha \right\|^{2}}{2}}\right)} , D = π 2 ( 1 + erf ⁡ ( \| α \| 2 ) ) {\displaystyle D={\sqrt {\frac {\pi }{2}}}\left(1+\operatorname {erf} \left({\frac {\left\|\alpha \right\|}{\sqrt {2}}}\right)\right)} . N {\displaystyle N} (Skwarnicki 1986) is a normalization factor and α {\displaystyle \alpha } , n {\displaystyle n} , x ¯ {\displaystyle {\bar {x}}} and σ {\displaystyle \sigma } are parameters which are fitted with the data. erf is the error function. == External links == J. E. Gaiser, Appendix-F Charmonium Spectroscopy from Radiative Decays of the J/Psi and Psi-Prime, Ph.D. Thesis, SLAC-R-255 (1982). (This is a 205-page document in .pdf form – the function is defined on p. 178.) M. J. Oreglia, A Study of the Reactions psi prime --> gamma gamma psi, Ph.D. Thesis, SLAC-R-236 (1980), Appendix D. T. Skwarnicki, A study of the radiative CASCADE transitions between the Upsilon-Prime and Upsilon resonances, Ph.D Thesis, DESY F31-86-02(1986), Appendix E.
Wikipedia:Crystallographic restriction theorem#0	The crystallographic restriction theorem in its basic form was based on the observation that the rotational symmetries of a crystal are usually limited to 2-fold, 3-fold, 4-fold, and 6-fold. However, quasicrystals can occur with other diffraction pattern symmetries, such as 5-fold; these were not discovered until 1982 by Dan Shechtman. Crystals are modeled as discrete lattices, generated by a list of independent finite translations (Coxeter 1989). Because discreteness requires that the spacings between lattice points have a lower bound, the group of rotational symmetries of the lattice at any point must be a finite group (alternatively, the point is the only system allowing for infinite rotational symmetry). The strength of the theorem is that not all finite groups are compatible with a discrete lattice; in any dimension, we will have only a finite number of compatible groups. == Dimensions 2 and 3 == The special cases of 2D (wallpaper groups) and 3D (space groups) are most heavily used in applications, and they can be treated together. === Lattice proof === A rotation symmetry in dimension 2 or 3 must move a lattice point to a succession of other lattice points in the same plane, generating a regular polygon of coplanar lattice points. We now confine our attention to the plane in which the symmetry acts (Scherrer 1946), illustrated with lattice vectors in the figure. Now consider an 8-fold rotation, and the displacement vectors between adjacent points of the polygon. If a displacement exists between any two lattice points, then that same displacement is repeated everywhere in the lattice. So collect all the edge displacements to begin at a single lattice point. The edge vectors become radial vectors, and their 8-fold symmetry implies a regular octagon of lattice points around the collection point. But this is impossible, because the new octagon is about 80% as large as the original. The significance of the shrinking is that it is unlimited. The same construction can be repeated with the new octagon, and again and again until the distance between lattice points is as small as we like; thus no discrete lattice can have 8-fold symmetry. The same argument applies to any k-fold rotation, for k greater than 6. A shrinking argument also eliminates 5-fold symmetry. Consider a regular pentagon of lattice points. If it exists, then we can take every other edge displacement and (head-to-tail) assemble a 5-point star, with the last edge returning to the starting point. The vertices of such a star are again vertices of a regular pentagon with 5-fold symmetry, but about 60% smaller than the original. Thus the theorem is proved. The existence of quasicrystals and Penrose tilings shows that the assumption of a linear translation is necessary. Penrose tilings may have 5-fold rotational symmetry and a discrete lattice, and any local neighborhood of the tiling is repeated infinitely many times, but there is no linear translation for the tiling as a whole. And without the discrete lattice assumption, the above construction not only fails to reach a contradiction, but produces a (non-discrete) counterexample. Thus 5-fold rotational symmetry cannot be eliminated by an argument missing either of those assumptions. A Penrose tiling of the whole (infinite) plane can only have exact 5-fold rotational symmetry (of the whole tiling) about a single point, however, whereas the 4-fold and 6-fold lattices have infinitely many centres of rotational symmetry. === Trigonometry proof === Consider two lattice points A and B separated by a translation vector r. Consider an angle α such that a rotation of angle α about any lattice point is a symmetry of the lattice. Rotating about point B by α maps point A to a new point A'. Similarly, rotating about point A by α maps B to a point B'. Since both rotations mentioned are symmetry operations, A' and B' must both be lattice points. Due to periodicity of the crystal, the new vector r' which connects them must be equal to an integer multiple of r: r ′ = m r {\displaystyle \mathbf {r} '=m\mathbf {r} } with m {\displaystyle m} integer. The four translation vectors, three of length r = \| r \| {\displaystyle r=\|\mathbf {r} \|} and one, connecting A' and B', of length r ′ = \| r ′ \| {\displaystyle r'=\|\mathbf {r} '\|} , form a trapezium. Therefore, the length of r' is also given by: r ′ = 2 r cos ⁡ α − r . {\displaystyle r'=2r\cos \alpha -r.} Combining the two equations gives: cos ⁡ α = m + 1 2 = M 2 {\displaystyle \cos \alpha ={\frac {m+1}{2}}={\frac {M}{2}}} where M = m + 1 {\displaystyle M=m+1} is also an integer. Bearing in mind that \| cos ⁡ α \| ≤ 1 {\displaystyle \|\cos \alpha \|\leq 1} we have allowed integers M ∈ { − 2 , − 1 , 0 , 1 , 2 } {\displaystyle M\in \{-2,-1,0,1,2\}} . Solving for possible values of α {\displaystyle \alpha } reveals that the only values in the 0° to 180° range are 0°, 60°, 90°, 120°, and 180°. In radians, the only allowed rotations consistent with lattice periodicity are given by 2π/n, where n = 1, 2, 3, 4, 6. This corresponds to 1-, 2-, 3-, 4-, and 6-fold symmetry, respectively, and therefore excludes the possibility of 5-fold or greater than 6-fold symmetry. === Short trigonometry proof === Consider a line of atoms A-O-B, separated by distance a. Rotate the entire row by θ = +2π/n and θ = −2π/n, with point O kept fixed. After the rotation by +2π/n, A is moved to the lattice point C and after the rotation by -2π/n, B is moved to the lattice point D. Due to the assumed periodicity of the lattice, the two lattice points C and D will be also in a line directly below the initial row; moreover C and D will be separated by r = ma, with m an integer. But by trigonometry, the separation between these points is: 2 a cos ⁡ θ = 2 a cos ⁡ 2 π n {\displaystyle 2a\cos {\theta }=2a\cos {\frac {2\pi }{n}}} . Equating the two relations gives: 2 cos ⁡ 2 π n = m {\displaystyle 2\cos {\frac {2\pi }{n}}=m} This is satisfied by only n = 1, 2, 3, 4, 6. === Matrix proof === For an alternative proof, consider matrix properties. The sum of the diagonal elements of a matrix is called the trace of the matrix. In 2D and 3D every rotation is a planar rotation, and the trace is a function of the angle alone. For a 2D rotation, the trace is 2 cos θ; for a 3D rotation, 1 + 2 cos θ. Examples Consider a 60° (6-fold) rotation matrix with respect to an orthonormal basis in 2D. [ 1 / 2 − 3 / 2 3 / 2 1 / 2 ] {\displaystyle {\begin{bmatrix}{1/2}&-{{\sqrt {3}}/2}\\{{\sqrt {3}}/2}&{1/2}\end{bmatrix}}} The trace is precisely 1, an integer. Consider a 45° (8-fold) rotation matrix. [ 1 / 2 − 1 / 2 1 / 2 1 / 2 ] {\displaystyle {\begin{bmatrix}{1/{\sqrt {2}}}&-{1/{\sqrt {2}}}\\{1/{\sqrt {2}}}&{1/{\sqrt {2}}}\end{bmatrix}}} The trace is 2/√2, not an integer. Selecting a basis formed from vectors that spans the lattice, neither orthogonality nor unit length is guaranteed, only linear independence. However the trace of the rotation matrix is the same with respect to any basis. The trace is a similarity invariant under linear transformations. In the lattice basis, the rotation operation must map every lattice point into an integer number of lattice vectors, so the entries of the rotation matrix in the lattice basis – and hence the trace – are necessarily integers. Similar as in other proofs, this implies that the only allowed rotational symmetries correspond to 1,2,3,4 or 6-fold invariance. For example, wallpapers and crystals cannot be rotated by 45° and remain invariant, the only possible angles are: 360°, 180°, 120°, 90° or 60°. Example Consider a 60° (360°/6) rotation matrix with respect to the oblique lattice basis for a tiling by equilateral triangles. [ 0 − 1 1 1 ] {\displaystyle {\begin{bmatrix}0&-1\\1&1\end{bmatrix}}} The trace is still 1. The determinant (always +1 for a rotation) is also preserved. The general crystallographic restriction on rotations does not guarantee that a rotation will be compatible with a specific lattice. For example, a 60° rotation will not work with a square lattice; nor will a 90° rotation work with a rectangular lattice. == Higher dimensions == When the dimension of the lattice rises to four or more, rotations need no longer be planar; the 2D proof is inadequate. However, restrictions still apply, though more symmetries are permissible. For example, the hypercubic lattice has an eightfold rotational symmetry, corresponding to an eightfold rotational symmetry of the hypercube. This is of interest, not just for mathematics, but for the physics of quasicrystals under the cut-and-project theory. In this view, a 3D quasicrystal with 8-fold rotation symmetry might be described as the projection of a slab cut from a 4D lattice. The following 4D rotation matrix is the aforementioned eightfold symmetry of the hypercube (and the cross-polytope): A = [ 0 0 0 − 1 1 0 0 0 0 − 1 0 0 0 0 − 1 0 ] . {\displaystyle A={\begin{bmatrix}0&0&0&-1\\1&0&0&0\\0&-1&0&0\\0&0&-1&0\end{bmatrix}}.} Transforming this matrix to the new coordinates given by B = [ − 1 / 2 0 − 1 / 2 2 / 2 1 / 2 2 / 2 − 1 / 2 0 − 1 / 2 0 − 1 / 2 − 2 / 2 − 1 / 2 2 / 2 1 / 2 0 ] {\displaystyle B={\begin{bmatrix}-1/2&0&-1/2&{\sqrt {2}}/2\\1/2&{\sqrt {2}}/2&-1/2&0\\-1/2&0&-1/2&-{\sqrt {2}}/2\\-1/2&{\sqrt {2}}/2&1/2&0\end{bmatrix}}} will produce: B A B − 1 = [ 2 / 2 2 / 2 0 0 − 2 / 2 2 / 2 0 0 0 0 − 2 / 2 2 / 2 0 0 − 2 / 2 − 2 / 2 ] . {\displaystyle BAB^{-1}={\begin{bmatrix}{\sqrt {2}}/2&{\sqrt {2}}/2&0&0\\-{\sqrt {2}}/2&{\sqrt {2}}/2&0&0\\0&0&-{\sqrt {2}}/2&{\sqrt {2}}/2\\0&0&-{\sqrt {2}}/2&-{\sqrt {2}}/2\end{bmatrix}}.} This third matrix then corresponds to a rotation both by 45° (in the first two dimensions) and by 135° (in the last two). Projecting a slab of hypercubes along the first two dimensions of the new coordinates produces an Ammann–Beenker tiling (another such tiling is produced by projecting along the last two dimensions), which therefore also has 8-fold rotational symmetry on average. The A4 lattice and F4 lattice have order 10 and order 12 rotational symmetries, respectively. To state the restriction for all dimensions, it is convenient to shift attention away from rotations alone and concentrate on the integer matrices (Bamberg, Cairns & Kilminster 2003). We say that a matrix A has order k when its k-th power (but no lower), Ak, equals the identity. Thus a 6-fold rotation matrix in the equilateral triangle basis is an integer matrix with order 6. Let OrdN denote the set of integers that can be the order of an N×N integer matrix. For example, Ord2 = {1, 2, 3, 4, 6}. We wish to state an explicit formula for OrdN. Define a function ψ based on Euler's totient function φ; it will map positive integers to non-negative integers. For an odd prime, p, and a positive integer, k, set ψ(pk) equal to the totient function value, φ(pk), which in this case is pk−pk−1. Do the same for ψ(2k) when k > 1. Set ψ(2) and ψ(1) to 0. Using the fundamental theorem of arithmetic, we can write any other positive integer uniquely as a product of prime powers, m = Πα pαk α; set ψ(m) = Σα ψ(pαk α). This differs from the totient itself, because it is a sum instead of a product. The crystallographic restriction in general form states that OrdN consists of those positive integers m such that ψ(m) ≤ N. For m>2, the values of ψ(m) are equal to twice the algebraic degree of cos(2π/m); therefore, ψ(m) is strictly less than m and reaches this maximum value if and only if m is a prime. These additional symmetries do not allow a planar slice to have, say, 8-fold rotation symmetry. In the plane, the 2D restrictions still apply. Thus the cuts used to model quasicrystals necessarily have thickness. Integer matrices are not limited to rotations; for example, a reflection is also a symmetry of order 2. But by insisting on determinant +1, we can restrict the matrices to proper rotations. == Formulation in terms of isometries == The crystallographic restriction theorem can be formulated in terms of isometries of Euclidean space. A set of isometries can form a group. By a discrete isometry group we will mean an isometry group that maps each point to a discrete subset of RN, i.e. the orbit of any point is a set of isolated points. With this terminology, the crystallographic restriction theorem in two and three dimensions can be formulated as follows. For every discrete isometry group in two- and three-dimensional space which includes translations spanning the whole space, all isometries of finite order are of order 1, 2, 3, 4 or 6. Isometries of order n include, but are not restricted to, n-fold rotations. The theorem also excludes S8, S12, D4d, and D6d (see point groups in three dimensions), even though they have 4- and 6-fold rotational symmetry only. Rotational symmetry of any order about an axis is compatible with translational symmetry along that axis. The result in the table above implies that for every discrete isometry group in four- and five-dimensional space which includes translations spanning the whole space, all isometries of finite order are of order 1, 2, 3, 4, 5, 6, 8, 10, or 12. All isometries of finite order in six- and seven-dimensional space are of order 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 24 or 30 . == See also == Crystallographic point group Crystallography == Notes == == References == Bamberg, John; Cairns, Grant; Kilminster, Devin (March 2003), "The crystallographic restriction, permutations, and Goldbach's conjecture" (PDF), American Mathematical Monthly, 110 (3): 202–209, CiteSeerX 10.1.1.124.8582, doi:10.2307/3647934, JSTOR 3647934 Elliott, Stephen (1998), The Physics and Chemistry of Solids, Wiley, ISBN 978-0-471-98194-7 Coxeter, H. S. M. (1989), Introduction to Geometry (2nd ed.), Wiley, ISBN 978-0-471-50458-0 Scherrer, W. (1946), "Die Einlagerung eines regulären Vielecks in ein Gitter", Elemente der Mathematik, 1 (6): 97–98 Shechtman, D.; Blech, I.; Gratias, D.; Cahn, JW (1984), "Metallic phase with long-range orientational order and no translational symmetry", Physical Review Letters, 53 (20): 1951–1953, Bibcode:1984PhRvL..53.1951S, doi:10.1103/PhysRevLett.53.1951 == External links == The crystallographic restriction The crystallographic restriction theorem by CSIC
Wikipedia:Cube root#0	In mathematics, a cube root of a number x is a number y that has the given number as its third power; that is y 3 = x . {\displaystyle y^{3}=x.} The number of cube roots of a number depends on the number system that is considered. Every real number x has exactly one real cube root that is denoted x 3 {\textstyle {\sqrt[{3}]{x}}} and called the real cube root of x or simply the cube root of x in contexts where complex numbers are not considered. For example, the real cube roots of 8 and −8 are respectively 2 and −2. The real cube root of an integer or of a rational number is generally not a rational number, neither a constructible number. Every nonzero real or complex number has exactly three cube roots that are complex numbers. If the number is real, one of the cube roots is real and the two other are nonreal complex conjugate numbers. Otherwise, the three cube roots are all nonreal. For example, the real cube root of 8 is 2 and the other cube roots of 8 are − 1 + i 3 {\displaystyle -1+i{\sqrt {3}}} and − 1 − i 3 {\displaystyle -1-i{\sqrt {3}}} . The three cube roots of −27i are 3 i , 3 3 2 − 3 2 i , {\displaystyle 3i,{\tfrac {3{\sqrt {3}}}{2}}-{\tfrac {3}{2}}i,} and − 3 3 2 − 3 2 i . {\displaystyle -{\tfrac {3{\sqrt {3}}}{2}}-{\tfrac {3}{2}}i.} The number zero has a unique cube root, which is zero itself. The cube root is a multivalued function. The principal cube root is its principal value, that is a unique cube root that has been chosen once for all. The principal cube root is the cube root with the largest real part. In the case of negative real numbers, the largest real part is shared by the two nonreal cube roots, and the principal cube root is the one with positive imaginary part. So, for negative real numbers, the real cube root is not the principal cube root. For positive real numbers, the principal cube root is the real cube root. If y is any cube root of the complex number x, the other cube roots are y − 1 + i 3 2 {\displaystyle y\,{\tfrac {-1+i{\sqrt {3}}}{2}}} and y − 1 − i 3 2 . {\displaystyle y\,{\tfrac {-1-i{\sqrt {3}}}{2}}.} In an algebraically closed field of characteristic different from three, every nonzero element has exactly three cube roots, which can be obtained from any of them by multiplying it by either root of the polynomial x 2 + x + 1. {\displaystyle x^{2}+x+1.} In an algebraically closed field of characteristic three, every element has exactly one cube root. In other number systems or other algebraic structures, a number or element may have more than three cube roots. For example, in the quaternions, a real number has infinitely many cube roots. == Definition == The cube roots of a number x are the numbers y which satisfy the equation y 3 = x . {\displaystyle y^{3}=x.\ } == Properties == === Real numbers === For any real number x, there is exactly one real number y such that y 3 = x {\displaystyle y^{3}=x} . Indeed, the cube function is increasing, so it does not give the same result for two different inputs, and covers all real numbers. In other words, it is a bijection or one-to-one correspondence. That is, one can define the cube root of a real number as its unique cube root that is also real. With this definition, the cube root of a negative number is a negative number. However this definition may be confusing when real numbers are considered as specific complex numbers, since, in this case the cube root is generally defined as the principal cube root, and the principal cube root of a negative real number is not real. If x and y are allowed to be complex, then there are three solutions (if x is non-zero) and so x has three cube roots. A real number has one real cube root and two further cube roots which form a complex conjugate pair. For instance, the cube roots of 1 are: 1 , − 1 2 + 3 2 i , − 1 2 − 3 2 i . {\displaystyle 1,\quad -{\frac {1}{2}}+{\frac {\sqrt {3}}{2}}i,\quad -{\frac {1}{2}}-{\frac {\sqrt {3}}{2}}i.} The last two of these roots lead to a relationship between all roots of any real or complex number. If a number is one cube root of a particular real or complex number, the other two cube roots can be found by multiplying that cube root by one or the other of the two complex cube roots of 1. === Complex numbers === For complex numbers, the principal cube root is usually defined as the cube root that has the greatest real part, or, equivalently, the cube root whose argument has the least absolute value. It is related to the principal value of the natural logarithm by the formula x 1 / 3 = exp ⁡ ( 1 3 ln ⁡ x ) . {\displaystyle x^{1/3}=\exp \left({\frac {1}{3}}\ln {x}\right).} If we write x as x = r exp ⁡ ( i θ ) {\displaystyle x=r\exp(i\theta )\,} where r is a non-negative real number and θ {\displaystyle \theta } lies in the range − π < θ ≤ π {\displaystyle -\pi <\theta \leq \pi } , then the principal complex cube root is x 3 = r 3 exp ⁡ ( i θ 3 ) . {\displaystyle {\sqrt[{3}]{x}}={\sqrt[{3}]{r}}\exp \left({\frac {i\theta }{3}}\right).} This means that in polar coordinates, we are taking the cube root of the radius and dividing the polar angle by three in order to define a cube root. With this definition, the principal cube root of a negative number is a complex number, and for instance − 8 3 {\displaystyle {\sqrt[{3}]{-8}}} will not be −2, but rather 1 + i 3 {\displaystyle 1+i{\sqrt {3}}} This difficulty can also be solved by considering the cube root as a multivalued function: if we write the original complex number x in three equivalent forms, namely x = { r exp ⁡ ( i θ ) , r exp ⁡ ( i θ + 2 i π ) , r exp ⁡ ( i θ − 2 i π ) . {\displaystyle x={\begin{cases}r\exp(i\theta ),\\[3px]r\exp(i\theta +2i\pi ),\\[3px]r\exp(i\theta -2i\pi ).\end{cases}}} The principal complex cube roots of these three forms are then respectively x 3 = { r 3 exp ⁡ ( i θ 3 ) , r 3 exp ⁡ ( i θ 3 + 2 i π 3 ) , r 3 exp ⁡ ( i θ 3 − 2 i π 3 ) . {\displaystyle {\sqrt[{3}]{x}}={\begin{cases}{\sqrt[{3}]{r}}\exp \left({\frac {i\theta }{3}}\right),\\{\sqrt[{3}]{r}}\exp \left({\frac {i\theta }{3}}+{\frac {2i\pi }{3}}\right),\\{\sqrt[{3}]{r}}\exp \left({\frac {i\theta }{3}}-{\frac {2i\pi }{3}}\right).\end{cases}}} Unless x = 0, these three complex numbers are distinct, even though the three representations of x were equivalent. For example, − 8 3 {\displaystyle {\sqrt[{3}]{-8}}} may then be calculated to be −2, 1 + i 3 {\displaystyle 1+i{\sqrt {3}}} , or 1 − i 3 {\displaystyle 1-i{\sqrt {3}}} . This is related with the concept of monodromy: if one follows by continuity the function cube root along a closed path around zero, after a turn the value of the cube root is multiplied (or divided) by e 2 i π / 3 . {\displaystyle e^{2i\pi /3}.} == Impossibility of compass-and-straightedge construction == Cube roots arise in the problem of finding an angle whose measure is one third that of a given angle (angle trisection) and in the problem of finding the edge of a cube whose volume is twice that of a cube with a given edge (doubling the cube). In 1837 Pierre Wantzel proved that neither of these can be done with a compass-and-straightedge construction. == Numerical methods == Newton's method is an iterative method that can be used to calculate the cube root. For real floating-point numbers this method reduces to the following iterative algorithm to produce successively better approximations of the cube root of a: x n + 1 = 1 3 ( a x n 2 + 2 x n ) . {\displaystyle x_{n+1}={\frac {1}{3}}\left({\frac {a}{x_{n}^{2}}}+2x_{n}\right).} The method is simply averaging three factors chosen such that x n × x n × a x n 2 = a {\displaystyle x_{n}\times x_{n}\times {\frac {a}{x_{n}^{2}}}=a} at each iteration. Halley's method improves upon this with an algorithm that converges more quickly with each iteration, albeit with more work per iteration: x n + 1 = x n ( x n 3 + 2 a 2 x n 3 + a ) . {\displaystyle x_{n+1}=x_{n}\left({\frac {x_{n}^{3}+2a}{2x_{n}^{3}+a}}\right).} This converges cubically, so two iterations do as much work as three iterations of Newton's method. Each iteration of Newton's method costs two multiplications, one addition and one division, assuming that ⁠1/3⁠a is precomputed, so three iterations plus the precomputation require seven multiplications, three additions, and three divisions. Each iteration of Halley's method requires three multiplications, three additions, and one division, so two iterations cost six multiplications, six additions, and two divisions. Thus, Halley's method has the potential to be faster if one division is more expensive than three additions. With either method a poor initial approximation of x0 can give very poor algorithm performance, and coming up with a good initial approximation is somewhat of a black art. Some implementations manipulate the exponent bits of the floating-point number; i.e. they arrive at an initial approximation by dividing the exponent by 3. Also useful is this generalized continued fraction, based on the nth root method: If x is a good first approximation to the cube root of a and y = a − x 3 {\displaystyle y=a-x^{3}} , then: a 3 = x 3 + y 3 = x + y 3 x 2 + 2 y 2 x + 4 y 9 x 2 + 5 y 2 x + 7 y 15 x 2 + 8 y 2 x + ⋱ {\displaystyle {\sqrt[{3}]{a}}={\sqrt[{3}]{x^{3}+y}}=x+{\cfrac {y}{3x^{2}+{\cfrac {2y}{2x+{\cfrac {4y}{9x^{2}+{\cfrac {5y}{2x+{\cfrac {7y}{15x^{2}+{\cfrac {8y}{2x+\ddots }}}}}}}}}}}}} = x + 2 x ⋅ y 3 ( 2 x 3 + y ) − y − 2 ⋅ 4 y 2 9 ( 2 x 3 + y ) − 5 ⋅ 7 y 2 15 ( 2 x 3 + y ) − 8 ⋅ 10 y 2 21 ( 2 x 3 + y ) − ⋱ . {\displaystyle =x+{\cfrac {2x\cdot y}{3(2x^{3}+y)-y-{\cfrac {2\cdot 4y^{2}}{9(2x^{3}+y)-{\cfrac {5\cdot 7y^{2}}{15(2x^{3}+y)-{\cfrac {8\cdot 10y^{2}}{21(2x^{3}+y)-\ddots }}}}}}}}.} The second equation combines each pair of fractions from the first into a single fraction, thus doubling the speed of convergence. == Appearance in solutions of third and fourth degree equations == Cubic equations, which are polynomial equations of the third degree (meaning the highest power of the unknown is 3) can always be solved for their three solutions in terms of cube roots and square roots (although simpler expressions only in terms of square roots exist for all three solutions, if at least one of them is a rational number). If two of the solutions are complex numbers, then all three solution expressions involve the real cube root of a real number, while if all three solutions are real numbers then they may be expressed in terms of the complex cube root of a complex number. Quartic equations can also be solved in terms of cube roots and square roots. == History == The calculation of cube roots can be traced back to Babylonian mathematicians from as early as 1800 BCE. In the fourth century BCE Plato posed the problem of doubling the cube, which required a compass-and-straightedge construction of the edge of a cube with twice the volume of a given cube; this required the construction, now known to be impossible, of the length 2 3 {\displaystyle {\sqrt[{3}]{2}}} . A method for extracting cube roots appears in The Nine Chapters on the Mathematical Art, a Chinese mathematical text compiled around the second century BCE and commented on by Liu Hui in the third century CE. The Greek mathematician Hero of Alexandria devised a method for calculating cube roots in the first century CE. His formula is again mentioned by Eutokios in a commentary on Archimedes. In 499 CE Aryabhata, a mathematician-astronomer from the classical age of Indian mathematics and Indian astronomy, gave a method for finding the cube root of numbers having many digits in the Aryabhatiya (section 2.5). == See also == Methods of computing square roots List of polynomial topics Nth root Square root Nested radical Root of unity == References == == External links == Cube root calculator reduces any number to simplest radical form Computing the Cube Root, Ken Turkowski, Apple Technical Report #KT-32, 1998. Includes C source code. Weisstein, Eric W. "Cube Root". MathWorld.
Wikipedia:Cubic equation#0	In algebra, a cubic equation in one variable is an equation of the form a x 3 + b x 2 + c x + d = 0 {\displaystyle ax^{3}+bx^{2}+cx+d=0} in which a is not zero. The solutions of this equation are called roots of the cubic function defined by the left-hand side of the equation. If all of the coefficients a, b, c, and d of the cubic equation are real numbers, then it has at least one real root (this is true for all odd-degree polynomial functions). All of the roots of the cubic equation can be found by the following means: algebraically: more precisely, they can be expressed by a cubic formula involving the four coefficients, the four basic arithmetic operations, square roots, and cube roots. (This is also true of quadratic (second-degree) and quartic (fourth-degree) equations, but not for higher-degree equations, by the Abel–Ruffini theorem.) trigonometrically numerical approximations of the roots can be found using root-finding algorithms such as Newton's method. The coefficients do not need to be real numbers. Much of what is covered below is valid for coefficients in any field with characteristic other than 2 and 3. The solutions of the cubic equation do not necessarily belong to the same field as the coefficients. For example, some cubic equations with rational coefficients have roots that are irrational (and even non-real) complex numbers. == History == Cubic equations were known to the ancient Babylonians, Greeks, Chinese, Indians, and Egyptians. Babylonian (20th to 16th centuries BC) cuneiform tablets have been found with tables for calculating cubes and cube roots. The Babylonians could have used the tables to solve cubic equations, but no evidence exists to confirm that they did. The problem of doubling the cube involves the simplest and oldest studied cubic equation, and one for which the ancient Egyptians did not believe a solution existed. In the 5th century BC, Hippocrates reduced this problem to that of finding two mean proportionals between one line and another of twice its length, but could not solve this with a compass and straightedge construction, a task which is now known to be impossible. Methods for solving cubic equations appear in The Nine Chapters on the Mathematical Art, a Chinese mathematical text compiled around the 2nd century BC and commented on by Liu Hui in the 3rd century. In the 3rd century AD, the Greek mathematician Diophantus found integer or rational solutions for some bivariate cubic equations (Diophantine equations). Hippocrates, Menaechmus and Archimedes are believed to have come close to solving the problem of doubling the cube using intersecting conic sections, though historians such as Reviel Netz dispute whether the Greeks were thinking about cubic equations or just problems that can lead to cubic equations. Some others like T. L. Heath, who translated all of Archimedes's works, disagree, putting forward evidence that Archimedes really solved cubic equations using intersections of two conics, but also discussed the conditions where the roots are 0, 1 or 2. In the 7th century, the Tang dynasty astronomer mathematician Wang Xiaotong in his mathematical treatise titled Jigu Suanjing systematically established and solved numerically 25 cubic equations of the form x3 + px2 + qx = N, 23 of them with p, q ≠ 0, and two of them with q = 0. In the 11th century, the Persian poet-mathematician, Omar Khayyam (1048–1131), made significant progress in the theory of cubic equations. In an early paper, he discovered that a cubic equation can have more than one solution and stated that it cannot be solved using compass and straightedge constructions. He also found a geometric solution. In his later work, the Treatise on Demonstration of Problems of Algebra, he wrote a complete classification of cubic equations with general geometric solutions found by means of intersecting conic sections. Khayyam made an attempt to come up with an algebraic formula for extracting cubic roots. He wrote: “We have tried to express these roots by algebra but have failed. It may be, however, that men who come after us will succeed.” In the 12th century, the Indian mathematician Bhaskara II attempted the solution of cubic equations without general success. However, he gave one example of a cubic equation: x3 + 12x = 6x2 + 35. In the 12th century, another Persian mathematician, Sharaf al-Dīn al-Tūsī (1135–1213), wrote the Al-Muʿādalāt (Treatise on Equations), which dealt with eight types of cubic equations with positive solutions and five types of cubic equations which may not have positive solutions. He used what would later be known as the Horner–Ruffini method to numerically approximate the root of a cubic equation. He also used the concepts of maxima and minima of curves in order to solve cubic equations which may not have positive solutions. He understood the importance of the discriminant of the cubic equation to find algebraic solutions to certain types of cubic equations. In his book Flos, Leonardo de Pisa, also known as Fibonacci (1170–1250), was able to closely approximate the positive solution to the cubic equation x3 + 2x2 + 10x = 20. Writing in Babylonian numerals he gave the result as 1,22,7,42,33,4,40 (equivalent to 1 + 22/60 + 7/602 + 42/603 + 33/604 + 4/605 + 40/606), which has a relative error of about 10−9. In the early 16th century, the Italian mathematician Scipione del Ferro (1465–1526) found a method for solving a class of cubic equations, namely those of the form x3 + mx = n. In fact, all cubic equations can be reduced to this form if one allows m and n to be negative, but negative numbers were not known to him at that time. Del Ferro kept his achievement secret until just before his death, when he told his student Antonio Fior about it. In 1535, Niccolò Tartaglia (1500–1557) received two problems in cubic equations from Zuanne da Coi and announced that he could solve them. He was soon challenged by Fior, which led to a famous contest between the two. Each contestant had to put up a certain amount of money and to propose a number of problems for his rival to solve. Whoever solved more problems within 30 days would get all the money. Tartaglia received questions in the form x3 + mx = n, for which he had worked out a general method. Fior received questions in the form x3 + mx2 = n, which proved to be too difficult for him to solve, and Tartaglia won the contest. Later, Tartaglia was persuaded by Gerolamo Cardano (1501–1576) to reveal his secret for solving cubic equations. In 1539, Tartaglia did so only on the condition that Cardano would never reveal it and that if he did write a book about cubics, he would give Tartaglia time to publish. Some years later, Cardano learned about del Ferro's prior work and published del Ferro's method in his book Ars Magna in 1545, meaning Cardano gave Tartaglia six years to publish his results (with credit given to Tartaglia for an independent solution). Cardano's promise to Tartaglia said that he would not publish Tartaglia's work, and Cardano felt he was publishing del Ferro's, so as to get around the promise. Nevertheless, this led to a challenge to Cardano from Tartaglia, which Cardano denied. The challenge was eventually accepted by Cardano's student Lodovico Ferrari (1522–1565). Ferrari did better than Tartaglia in the competition, and Tartaglia lost both his prestige and his income. Cardano noticed that Tartaglia's method sometimes required him to extract the square root of a negative number. He even included a calculation with these complex numbers in Ars Magna, but he did not really understand it. Rafael Bombelli studied this issue in detail and is therefore often considered as the discoverer of complex numbers. François Viète (1540–1603) independently derived the trigonometric solution for the cubic with three real roots, and René Descartes (1596–1650) extended the work of Viète. == Factorization == If the coefficients of a cubic equation are rational numbers, one can obtain an equivalent equation with integer coefficients, by multiplying all coefficients by a common multiple of their denominators. Such an equation a x 3 + b x 2 + c x + d = 0 , {\displaystyle ax^{3}+bx^{2}+cx+d=0,} with integer coefficients, is said to be reducible if the polynomial on the left-hand side is the product of polynomials of lower degrees. By Gauss's lemma, if the equation is reducible, one can suppose that the factors have integer coefficients. Finding the roots of a reducible cubic equation is easier than solving the general case. In fact, if the equation is reducible, one of the factors must have degree one, and thus have the form q x − p , {\displaystyle qx-p,} with q and p being coprime integers. The rational root test allows finding q and p by examining a finite number of cases (because q must be a divisor of a, and p must be a divisor of d). Thus, one root is x 1 = p q , {\displaystyle \textstyle x_{1}={\frac {p}{q}},} and the other roots are the roots of the other factor, which can be found by polynomial long division. This other factor is a q x 2 + b q + a p q 2 x + c q 2 + b p q + a p 2 q 3 . {\displaystyle {\frac {a}{q}}\,x^{2}+{\frac {bq+ap}{q^{2}}}\,x+{\frac {cq^{2}+bpq+ap^{2}}{q^{3}}}.} (The coefficients seem not to be integers, but must be integers if ⁠ p / q {\displaystyle p/q} ⁠ is a root.) Then, the other roots are the roots of this quadratic polynomial and can be found by using the quadratic formula. == Depressed cubic == Cubics of the form t 3 + p t + q {\displaystyle t^{3}+pt+q} are said to be depressed. They are much simpler than general cubics, but are fundamental, because the study of any cubic may be reduced by a simple change of variable to that of a depressed cubic. Let a x 3 + b x 2 + c x + d = 0 {\displaystyle ax^{3}+bx^{2}+cx+d=0} be a cubic equation. The change of variable x = t − b 3 a {\displaystyle x=t-{\frac {b}{3a}}} gives a cubic (in t) that has no term in t2. After dividing by a one gets the depressed cubic equation t 3 + p t + q = 0 , {\displaystyle t^{3}+pt+q=0,} with t = x + b 3 a p = 3 a c − b 2 3 a 2 q = 2 b 3 − 9 a b c + 27 a 2 d 27 a 3 . {\displaystyle {\begin{aligned}t={}&x+{\frac {b}{3a}}\\p={}&{\frac {3ac-b^{2}}{3a^{2}}}\\q={}&{\frac {2b^{3}-9abc+27a^{2}d}{27a^{3}}}.\end{aligned}}} The roots x 1 , x 2 , x 3 {\displaystyle x_{1},x_{2},x_{3}} of the original equation are related to the roots t 1 , t 2 , t 3 {\displaystyle t_{1},t_{2},t_{3}} of the depressed equation by the relations x i = t i − b 3 a , {\displaystyle x_{i}=t_{i}-{\frac {b}{3a}},} for i = 1 , 2 , 3 {\displaystyle i=1,2,3} . == Discriminant and nature of the roots == The nature (real or not, distinct or not) of the roots of a cubic can be determined without computing them explicitly, by using the discriminant. === Discriminant === The discriminant of a polynomial is a function of its coefficients that is zero if and only if the polynomial has a multiple root, or, if it is divisible by the square of a non-constant polynomial. In other words, the discriminant is nonzero if and only if the polynomial is square-free. If r1, r2, r3 are the three roots (not necessarily distinct nor real) of the cubic a x 3 + b x 2 + c x + d , {\displaystyle ax^{3}+bx^{2}+cx+d,} then the discriminant is a 4 ( r 1 − r 2 ) 2 ( r 1 − r 3 ) 2 ( r 2 − r 3 ) 2 . {\displaystyle a^{4}(r_{1}-r_{2})^{2}(r_{1}-r_{3})^{2}(r_{2}-r_{3})^{2}.} The discriminant of the depressed cubic t 3 + p t + q {\displaystyle t^{3}+pt+q} is − ( 4 p 3 + 27 q 2 ) . {\displaystyle -\left(4\,p^{3}+27\,q^{2}\right).} The discriminant of the general cubic a x 3 + b x 2 + c x + d {\displaystyle ax^{3}+bx^{2}+cx+d} is 18 a b c d − 4 b 3 d + b 2 c 2 − 4 a c 3 − 27 a 2 d 2 . {\displaystyle 18\,abcd-4\,b^{3}d+b^{2}c^{2}-4\,ac^{3}-27\,a^{2}d^{2}.} It is the product of a 4 {\displaystyle a^{4}} and the discriminant of the corresponding depressed cubic. Using the formula relating the general cubic and the associated depressed cubic, this implies that the discriminant of the general cubic can be written as 4 ( b 2 − 3 a c ) 3 − ( 2 b 3 − 9 a b c + 27 a 2 d ) 2 27 a 2 . {\displaystyle {\frac {4(b^{2}-3ac)^{3}-(2b^{3}-9abc+27a^{2}d)^{2}}{27a^{2}}}.} It follows that one of these two discriminants is zero if and only if the other is also zero, and, if the coefficients are real, the two discriminants have the same sign. In summary, the same information can be deduced from either one of these two discriminants. To prove the preceding formulas, one can use Vieta's formulas to express everything as polynomials in r1, r2, r3, and a. The proof then results in the verification of the equality of two polynomials. === Nature of the roots === If the coefficients of a polynomial are real numbers, and its discriminant Δ {\displaystyle \Delta } is not zero, there are two cases: If Δ > 0 , {\displaystyle \Delta >0,} the cubic has three distinct real roots If Δ < 0 , {\displaystyle \Delta <0,} the cubic has one real root and two non-real complex conjugate roots. This can be proved as follows. First, if r is a root of a polynomial with real coefficients, then its complex conjugate is also a root. So the non-real roots, if any, occur as pairs of complex conjugate roots. As a cubic polynomial has three roots (not necessarily distinct) by the fundamental theorem of algebra, at least one root must be real. As stated above, if r1, r2, r3 are the three roots of the cubic a x 3 + b x 2 + c x + d {\displaystyle ax^{3}+bx^{2}+cx+d} , then the discriminant is Δ = a 4 ( r 1 − r 2 ) 2 ( r 1 − r 3 ) 2 ( r 2 − r 3 ) 2 {\displaystyle \Delta =a^{4}(r_{1}-r_{2})^{2}(r_{1}-r_{3})^{2}(r_{2}-r_{3})^{2}} If the three roots are real and distinct, the discriminant is a product of positive reals, that is Δ > 0. {\displaystyle \Delta >0.} If only one root, say r1, is real, then r2 and r3 are complex conjugates, which implies that r2 − r3 is a purely imaginary number, and thus that (r2 − r3)2 is real and negative. On the other hand, r1 − r2 and r1 − r3 are complex conjugates, and their product is real and positive. Thus the discriminant is the product of a single negative number and several positive ones. That is Δ < 0. {\displaystyle \Delta <0.} === Multiple root === If the discriminant of a cubic is zero, the cubic has a multiple root. If furthermore its coefficients are real, then all of its roots are real. The discriminant of the depressed cubic t 3 + p t + q {\displaystyle t^{3}+pt+q} is zero if 4 p 3 + 27 q 2 = 0. {\displaystyle 4p^{3}+27q^{2}=0.} If p is also zero, then p = q = 0 , and 0 is a triple root of the cubic. If 4 p 3 + 27 q 2 = 0 , {\displaystyle 4p^{3}+27q^{2}=0,} and p ≠ 0 , then the cubic has a simple root t 1 = 3 q p {\displaystyle t_{1}={\frac {3q}{p}}} and a double root t 2 = t 3 = − 3 q 2 p . {\displaystyle t_{2}=t_{3}=-{\frac {3q}{2p}}.} In other words, t 3 + p t + q = ( t − 3 q p ) ( t + 3 q 2 p ) 2 . {\displaystyle t^{3}+pt+q=\left(t-{\frac {3q}{p}}\right)\left(t+{\frac {3q}{2p}}\right)^{2}.} This result can be proved by expanding the latter product or retrieved by solving the rather simple system of equations resulting from Vieta's formulas. By using the reduction of a depressed cubic, these results can be extended to the general cubic. This gives: If the discriminant of the cubic a x 3 + b x 2 + c x + d {\displaystyle ax^{3}+bx^{2}+cx+d} is zero, then either, if b 2 = 3 a c , {\displaystyle b^{2}=3ac,} the cubic has a triple root x 1 = x 2 = x 3 = − b 3 a , {\displaystyle x_{1}=x_{2}=x_{3}=-{\frac {b}{3a}},} and a x 3 + b x 2 + c x + d = a ( x + b 3 a ) 3 {\displaystyle ax^{3}+bx^{2}+cx+d=a\left(x+{\frac {b}{3a}}\right)^{3}} or, if b 2 ≠ 3 a c , {\displaystyle b^{2}\neq 3ac,} the cubic has a double root x 2 = x 3 = 9 a d − b c 2 ( b 2 − 3 a c ) , {\displaystyle x_{2}=x_{3}={\frac {9ad-bc}{2(b^{2}-3ac)}},} and a simple root, x 1 = 4 a b c − 9 a 2 d − b 3 a ( b 2 − 3 a c ) . {\displaystyle x_{1}={\frac {4abc-9a^{2}d-b^{3}}{a(b^{2}-3ac)}}.} and thus a x 3 + b x 2 + c x + d = a ( x − x 1 ) ( x − x 2 ) 2 . {\displaystyle ax^{3}+bx^{2}+cx+d=a(x-x_{1})(x-x_{2})^{2}.} === Characteristic 2 and 3 === The above results are valid when the coefficients belong to a field of characteristic other than 2 or 3, but must be modified for characteristic 2 or 3, because of the involved divisions by 2 and 3. The reduction to a depressed cubic works for characteristic 2, but not for characteristic 3. However, in both cases, it is simpler to establish and state the results for the general cubic. The main tool for that is the fact that a multiple root is a common root of the polynomial and its formal derivative. In these characteristics, if the derivative is not a constant, it is a linear polynomial in characteristic 3, and is the square of a linear polynomial in characteristic 2. Therefore, for either characteristic 2 or 3, the derivative has only one root. This allows computing the multiple root, and the third root can be deduced from the sum of the roots, which is provided by Vieta's formulas. A difference with other characteristics is that, in characteristic 2, the formula for a double root involves a square root, and, in characteristic 3, the formula for a triple root involves a cube root. == Cardano's formula == Gerolamo Cardano is credited with publishing the first formula for solving cubic equations, attributing it to Scipione del Ferro and Niccolo Fontana Tartaglia. The formula applies to depressed cubics, but, as shown in § Depressed cubic, it allows solving all cubic equations. Cardano's result is that if t 3 + p t + q = 0 {\displaystyle t^{3}+pt+q=0} is a cubic equation such that p and q are real numbers such that q 2 4 + p 3 27 {\displaystyle {\frac {q^{2}}{4}}+{\frac {p^{3}}{27}}} is positive (this implies that the discriminant of the equation is negative) then the equation has the real root u 1 3 + u 2 3 , {\displaystyle {\sqrt[{3}]{u_{1}}}+{\sqrt[{3}]{u_{2}}},} where u 1 {\displaystyle u_{1}} and u 2 {\displaystyle u_{2}} are the two numbers − q 2 + q 2 4 + p 3 27 {\displaystyle -{\frac {q}{2}}+{\sqrt {{\frac {q^{2}}{4}}+{\frac {p^{3}}{27}}}}} and − q 2 − q 2 4 + p 3 27 . {\displaystyle -{\frac {q}{2}}-{\sqrt {{\frac {q^{2}}{4}}+{\frac {p^{3}}{27}}}}.} See § Derivation of the roots, below, for several methods for getting this result. As shown in § Nature of the roots, the two other roots are non-real complex conjugate numbers, in this case. It was later shown (Cardano did not know complex numbers) that the two other roots are obtained by multiplying one of the cube roots by the primitive cube root of unity ε 1 = − 1 + i 3 2 , {\displaystyle \varepsilon _{1}={\frac {-1+i{\sqrt {3}}}{2}},} and the other cube root by the other primitive cube root of the unity ε 2 = ε 1 2 = − 1 − i 3 2 . {\displaystyle \varepsilon _{2}=\varepsilon _{1}^{2}={\frac {-1-i{\sqrt {3}}}{2}}.} That is, the other roots of the equation are ε 1 u 1 3 + ε 2 u 2 3 {\displaystyle \varepsilon _{1}{\sqrt[{3}]{u_{1}}}+\varepsilon _{2}{\sqrt[{3}]{u_{2}}}} and ε 2 u 1 3 + ε 1 u 2 3 . {\displaystyle \varepsilon _{2}{\sqrt[{3}]{u_{1}}}+\varepsilon _{1}{\sqrt[{3}]{u_{2}}}.} If 4 p 3 + 27 q 2 < 0 , {\displaystyle 4p^{3}+27q^{2}<0,} there are three real roots, but Galois theory allows proving that, if there is no rational root, the roots cannot be expressed by an algebraic expression involving only real numbers. Therefore, the equation cannot be solved in this case with the knowledge of Cardano's time. This case has thus been called casus irreducibilis, meaning irreducible case in Latin. In casus irreducibilis, Cardano's formula can still be used, but some care is needed in the use of cube roots. A first method is to define the symbols {\displaystyle {\sqrt {{~}^{~}}}} and 3 {\displaystyle {\sqrt[{3}]{{~}^{~}}}} as representing the principal values of the root function (that is the root that has the largest real part). With this convention Cardano's formula for the three roots remains valid, but is not purely algebraic, as the definition of a principal part is not purely algebraic, since it involves inequalities for comparing real parts. Also, the use of principal cube root may give a wrong result if the coefficients are non-real complex numbers. Moreover, if the coefficients belong to another field, the principal cube root is not defined in general. The second way for making Cardano's formula always correct, is to remark that the product of the two cube roots must be −p / 3. It results that a root of the equation is C − p 3 C with C = − q 2 + q 2 4 + p 3 27 3 . {\displaystyle C-{\frac {p}{3C}}\quad {\text{with}}\quad C={\sqrt[{3}]{-{\frac {q}{2}}+{\sqrt {{\frac {q^{2}}{4}}+{\frac {p^{3}}{27}}}}}}.} In this formula, the symbols {\displaystyle {\sqrt {{~}^{~}}}} and 3 {\displaystyle {\sqrt[{3}]{{~}^{~}}}} denote any square root and any cube root. The other roots of the equation are obtained either by changing of cube root or, equivalently, by multiplying the cube root by a primitive cube root of unity, that is − 1 ± − 3 2 . {\displaystyle \textstyle {\frac {-1\pm {\sqrt {-3}}}{2}}.} This formula for the roots is always correct except when p = q = 0, with the proviso that if p = 0, the square root is chosen so that C ≠ 0. However, Cardano's formula is useless if p = 0 , {\displaystyle p=0,} as the roots are the cube roots of − q . {\displaystyle -q.} Similarly, the formula is also useless in the cases where no cube root is needed, that is when the cubic polynomial is not irreducible; this includes the case 4 p 3 + 27 q 2 = 0. {\displaystyle 4p^{3}+27q^{2}=0.} This formula is also correct when p and q belong to any field of characteristic other than 2 or 3. == General cubic formula == A cubic formula for the roots of the general cubic equation (with a ≠ 0) a x 3 + b x 2 + c x + d = 0 {\displaystyle ax^{3}+bx^{2}+cx+d=0} can be deduced from every variant of Cardano's formula by reduction to a depressed cubic. The variant that is presented here is valid not only for complex coefficients, but also for coefficients a, b, c, d belonging to any algebraically closed field of characteristic other than 2 or 3. If the coefficients are real numbers, the formula covers all complex solutions, not just real ones. The formula being rather complicated, it is worth splitting it in smaller formulas. Let Δ 0 = b 2 − 3 a c , Δ 1 = 2 b 3 − 9 a b c + 27 a 2 d . {\displaystyle {\begin{aligned}\Delta _{0}&=b^{2}-3ac,\\\Delta _{1}&=2b^{3}-9abc+27a^{2}d.\end{aligned}}} (Both Δ 0 {\displaystyle \Delta _{0}} and Δ 1 {\displaystyle \Delta _{1}} can be expressed as resultants of the cubic and its derivatives: Δ 1 {\displaystyle \Delta _{1}} is ⁠−1/8a⁠ times the resultant of the cubic and its second derivative, and Δ 0 {\displaystyle \Delta _{0}} is ⁠−1/12a⁠ times the resultant of the first and second derivatives of the cubic polynomial.) Then let C = Δ 1 ± Δ 1 2 − 4 Δ 0 3 2 3 , {\displaystyle C={\sqrt[{3}]{\frac {\Delta _{1}\pm {\sqrt {\Delta _{1}^{2}-4\Delta _{0}^{3}}}}{2}}},} where the symbols {\displaystyle {\sqrt {{~}^{~}}}} and 3 {\displaystyle {\sqrt[{3}]{{~}^{~}}}} are interpreted as any square root and any cube root, respectively (every nonzero complex number has two square roots and three cubic roots). The sign "±" before the square root is either "+" or "–"; the choice is almost arbitrary, and changing it amounts to choosing a different square root. However, if a choice yields C = 0 (this occurs if Δ 0 = 0 {\displaystyle \Delta _{0}=0} ), then the other sign must be selected instead. If both choices yield C = 0, that is, if Δ 0 = Δ 1 = 0 , {\displaystyle \Delta _{0}=\Delta _{1}=0,} a fraction ⁠0/0⁠ occurs in following formulas; this fraction must be interpreted as equal to zero (see the end of this section). With these conventions, one of the roots is x = − 1 3 a ( b + C + Δ 0 C ) . {\displaystyle x=-{\frac {1}{3a}}\left(b+C+{\frac {\Delta _{0}}{C}}\right).} The other two roots can be obtained by changing the choice of the cube root in the definition of C, or, equivalently by multiplying C by a primitive cube root of unity, that is ⁠–1 ± √–3/2⁠. In other words, the three roots are x k = − 1 3 a ( b + ξ k C + Δ 0 ξ k C ) , k ∈ { 0 , 1 , 2 } , {\displaystyle x_{k}=-{\frac {1}{3a}}\left(b+\xi ^{k}C+{\frac {\Delta _{0}}{\xi ^{k}C}}\right),\qquad k\in \{0,1,2\}{\text{,}}} where ξ = ⁠–1 + √–3/2⁠. As for the special case of a depressed cubic, this formula applies but is useless when the roots can be expressed without cube roots. In particular, if Δ 0 = Δ 1 = 0 , {\displaystyle \Delta _{0}=\Delta _{1}=0,} the formula gives that the three roots equal − b 3 a , {\displaystyle {\frac {-b}{3a}},} which means that the cubic polynomial can be factored as a ( x + b 3 a ) 3 . {\displaystyle \textstyle a(x+{\frac {b}{3a}})^{3}.} A straightforward computation allows verifying that the existence of this factorization is equivalent with Δ 0 = Δ 1 = 0. {\displaystyle \Delta _{0}=\Delta _{1}=0.} == Trigonometric and hyperbolic solutions == === Trigonometric solution for three real roots === When a cubic equation with real coefficients has three real roots, the formulas expressing these roots in terms of radicals involve complex numbers. Galois theory allows proving that when the three roots are real, and none is rational (casus irreducibilis), one cannot express the roots in terms of real radicals. Nevertheless, purely real expressions of the solutions may be obtained using trigonometric functions, specifically in terms of cosines and arccosines. More precisely, the roots of the depressed cubic t 3 + p t + q = 0 {\displaystyle t^{3}+pt+q=0} are t k = 2 − p 3 cos ⁡ [ 1 3 arccos ⁡ ( 3 q 2 p − 3 p ) − 2 π k 3 ] for k = 0 , 1 , 2. {\displaystyle t_{k}=2\,{\sqrt {-{\frac {p}{3}}}}\,\cos \left[\,{\frac {1}{3}}\arccos \left({\frac {3q}{2p}}{\sqrt {\frac {-3}{p}}}\,\right)-{\frac {2\pi k}{3}}\,\right]\qquad {\text{for }}k=0,1,2.} This formula is due to François Viète. It is purely real when the equation has three real roots (that is 4 p 3 + 27 q 2 < 0 {\displaystyle 4p^{3}+27q^{2}<0} ). Otherwise, it is still correct but involves complex cosines and arccosines when there is only one real root, and it is nonsensical (division by zero) when p = 0. This formula can be straightforwardly transformed into a formula for the roots of a general cubic equation, using the back-substitution described in § Depressed cubic. The formula can be proved as follows: Starting from the equation t3 + pt + q = 0, let us set t = u cos θ. The idea is to choose u to make the equation coincide with the identity 4 cos 3 ⁡ θ − 3 cos ⁡ θ − cos ⁡ ( 3 θ ) = 0. {\displaystyle 4\cos ^{3}\theta -3\cos \theta -\cos(3\theta )=0.} For this, choose u = 2 − p 3 , {\displaystyle u=2\,{\sqrt {-{\frac {p}{3}}}}\,,} and divide the equation by u 3 4 . {\displaystyle {\frac {u^{3}}{4}}.} This gives 4 cos 3 ⁡ θ − 3 cos ⁡ θ − 3 q 2 p − 3 p = 0. {\displaystyle 4\cos ^{3}\theta -3\cos \theta -{\frac {3q}{2p}}\,{\sqrt {\frac {-3}{p}}}=0.} Combining with the above identity, one gets cos ⁡ ( 3 θ ) = 3 q 2 p − 3 p , {\displaystyle \cos(3\theta )={\frac {3q}{2p}}{\sqrt {\frac {-3}{p}}}\,,} and the roots are thus t k = 2 − p 3 cos ⁡ [ 1 3 arccos ⁡ ( 3 q 2 p − 3 p ) − 2 π k 3 ] for k = 0 , 1 , 2. {\displaystyle t_{k}=2\,{\sqrt {-{\frac {p}{3}}}}\,\cos \left[{\frac {1}{3}}\arccos \left({\frac {3q}{2p}}{\sqrt {\frac {-3}{p}}}\right)-{\frac {2\pi k}{3}}\right]\qquad {\text{for }}k=0,1,2.} === Hyperbolic solution for one real root === When there is only one real root (and p ≠ 0), this root can be similarly represented using hyperbolic functions, as t 0 = − 2 \| q \| q − p 3 cosh ⁡ [ 1 3 arcosh ⁡ ( − 3 \| q \| 2 p − 3 p ) ] if 4 p 3 + 27 q 2 > 0 and p < 0 , t 0 = − 2 p 3 sinh ⁡ [ 1 3 arsinh ⁡ ( 3 q 2 p 3 p ) ] if p > 0. {\displaystyle {\begin{aligned}t_{0}&=-2{\frac {\|q\|}{q}}{\sqrt {-{\frac {p}{3}}}}\cosh \left[{\frac {1}{3}}\operatorname {arcosh} \left({\frac {-3\|q\|}{2p}}{\sqrt {\frac {-3}{p}}}\right)\right]\qquad {\text{if }}~4p^{3}+27q^{2}>0~{\text{ and }}~p<0,\\t_{0}&=-2{\sqrt {\frac {p}{3}}}\sinh \left[{\frac {1}{3}}\operatorname {arsinh} \left({\frac {3q}{2p}}{\sqrt {\frac {3}{p}}}\right)\right]\qquad {\text{if }}~p>0.\end{aligned}}} If p ≠ 0 and the inequalities on the right are not satisfied (the case of three real roots), the formulas remain valid but involve complex quantities. When p = ±3, the above values of t0 are sometimes called the Chebyshev cube root. More precisely, the values involving cosines and hyperbolic cosines define, when p = −3, the same analytic function denoted C1/3(q), which is the proper Chebyshev cube root. The value involving hyperbolic sines is similarly denoted S1/3(q), when p = 3. == Geometric solutions == === Omar Khayyám's solution === For solving the cubic equation x3 + m2x = n where n > 0, Omar Khayyám constructed the parabola y = x2/m, the circle that has as a diameter the line segment [0, n/m2] on the positive x-axis, and a vertical line through the point where the circle and the parabola intersect above the x-axis. The solution is given by the length of the horizontal line segment from the origin to the intersection of the vertical line and the x-axis (see the figure). A simple modern proof is as follows. Multiplying the equation by x/m2 and regrouping the terms gives x 4 m 2 = x ( n m 2 − x ) . {\displaystyle {\frac {x^{4}}{m^{2}}}=x\left({\frac {n}{m^{2}}}-x\right).} The left-hand side is the value of y2 on the parabola. The equation of the circle being y2 + x(x − ⁠n/m2⁠) = 0, the right hand side is the value of y2 on the circle. === Solution with angle trisector === A cubic equation with real coefficients can be solved geometrically using compass, straightedge, and an angle trisector if and only if it has three real roots.: Thm. 1 A cubic equation can be solved by compass-and-straightedge construction (without trisector) if and only if it has a rational root. This implies that the old problems of angle trisection and doubling the cube, set by ancient Greek mathematicians, cannot be solved by compass-and-straightedge construction. == Geometric interpretation of the roots == === Three real roots === Viète's trigonometric expression of the roots in the three-real-roots case lends itself to a geometric interpretation in terms of a circle. When the cubic is written in depressed form (2), t3 + pt + q = 0, as shown above, the solution can be expressed as t k = 2 − p 3 cos ⁡ ( 1 3 arccos ⁡ ( 3 q 2 p − 3 p ) − k 2 π 3 ) for k = 0 , 1 , 2 . {\displaystyle t_{k}=2{\sqrt {-{\frac {p}{3}}}}\cos \left({\frac {1}{3}}\arccos \left({\frac {3q}{2p}}{\sqrt {\frac {-3}{p}}}\right)-k{\frac {2\pi }{3}}\right)\quad {\text{for}}\quad k=0,1,2\,.} Here arccos ⁡ ( 3 q 2 p − 3 p ) {\displaystyle \arccos \left({\frac {3q}{2p}}{\sqrt {\frac {-3}{p}}}\right)} is an angle in the unit circle; taking ⁠1/3⁠ of that angle corresponds to taking a cube root of a complex number; adding −k⁠2π/3⁠ for k = 1, 2 finds the other cube roots; and multiplying the cosines of these resulting angles by 2 − p 3 {\displaystyle 2{\sqrt {-{\frac {p}{3}}}}} corrects for scale. For the non-depressed case (1) (shown in the accompanying graph), the depressed case as indicated previously is obtained by defining t such that x = t − ⁠b/3a⁠ so t = x + ⁠b/3a⁠. Graphically this corresponds to simply shifting the graph horizontally when changing between the variables t and x, without changing the angle relationships. This shift moves the point of inflection and the centre of the circle onto the y-axis. Consequently, the roots of the equation in t sum to zero. === One real root === ==== In the Cartesian plane ==== When the graph of a cubic function is plotted in the Cartesian plane, if there is only one real root, it is the abscissa (x-coordinate) of the horizontal intercept of the curve (point R on the figure). Further, if the complex conjugate roots are written as g ± hi, then the real part g is the abscissa of the tangency point H of the tangent line to cubic that passes through x-intercept R of the cubic (that is the signed length OM, negative on the figure). The imaginary parts ±h are the square roots of the tangent of the angle between this tangent line and the horizontal axis. ==== In the complex plane ==== With one real and two complex roots, the three roots can be represented as points in the complex plane, as can the two roots of the cubic's derivative. There is an interesting geometrical relationship among all these roots. The points in the complex plane representing the three roots serve as the vertices of an isosceles triangle. (The triangle is isosceles because one root is on the horizontal (real) axis and the other two roots, being complex conjugates, appear symmetrically above and below the real axis.) Marden's theorem says that the points representing the roots of the derivative of the cubic are the foci of the Steiner inellipse of the triangle—the unique ellipse that is tangent to the triangle at the midpoints of its sides. If the angle at the vertex on the real axis is less than ⁠π/3⁠ then the major axis of the ellipse lies on the real axis, as do its foci and hence the roots of the derivative. If that angle is greater than ⁠π/3⁠, the major axis is vertical and its foci, the roots of the derivative, are complex conjugates. And if that angle is ⁠π/3⁠, the triangle is equilateral, the Steiner inellipse is simply the triangle's incircle, its foci coincide with each other at the incenter, which lies on the real axis, and hence the derivative has duplicate real roots. == Galois group == Given a cubic irreducible polynomial over a field K of characteristic different from 2 and 3, the Galois group over K is the group of the field automorphisms that fix K of the smallest extension of K (splitting field). As these automorphisms must permute the roots of the polynomials, this group is either the group S3 of all six permutations of the three roots, or the group A3 of the three circular permutations. The discriminant Δ of the cubic is the square of Δ = a 2 ( r 1 − r 2 ) ( r 1 − r 3 ) ( r 2 − r 3 ) , {\displaystyle {\sqrt {\Delta }}=a^{2}(r_{1}-r_{2})(r_{1}-r_{3})(r_{2}-r_{3}),} where a is the leading coefficient of the cubic, and r1, r2 and r3 are the three roots of the cubic. As Δ {\displaystyle {\sqrt {\Delta }}} changes of sign if two roots are exchanged, Δ {\displaystyle {\sqrt {\Delta }}} is fixed by the Galois group only if the Galois group is A3. In other words, the Galois group is A3 if and only if the discriminant is the square of an element of K. As most integers are not squares, when working over the field Q of the rational numbers, the Galois group of most irreducible cubic polynomials is the group S3 with six elements. An example of a Galois group A3 with three elements is given by p(x) = x3 − 3x − 1, whose discriminant is 81 = 92. == Derivation of the roots == This section regroups several methods for deriving Cardano's formula. === Cardano's method === This method is due to Scipione del Ferro and Tartaglia, but is named after Gerolamo Cardano who first published it in his book Ars Magna (1545). This method applies to a depressed cubic t3 + pt + q = 0. The idea is to introduce two variables u and v {\displaystyle v} such that u + v = t {\displaystyle u+v=t} and to substitute this in the depressed cubic, giving u 3 + v 3 + ( 3 u v + p ) ( u + v ) + q = 0. {\displaystyle u^{3}+v^{3}+(3uv+p)(u+v)+q=0.} At this point Cardano imposed the condition 3 u v + p = 0. {\displaystyle 3uv+p=0.} This removes the third term in the previous equality, leading to the system of equations u 3 + v 3 = − q u v = − p 3 . {\displaystyle {\begin{aligned}u^{3}+v^{3}&=-q\\uv&=-{\frac {p}{3}}.\end{aligned}}} Knowing the sum and the product of u3 and v 3 , {\displaystyle v^{3},} one deduces that they are the two solutions of the quadratic equation 0 = ( x − u 3 ) ( x − v 3 ) = x 2 − ( u 3 + v 3 ) x + u 3 v 3 = x 2 − ( u 3 + v 3 ) x + ( u v ) 3 {\displaystyle {\begin{aligned}0&=(x-u^{3})(x-v^{3})\\&=x^{2}-(u^{3}+v^{3})x+u^{3}v^{3}\\&=x^{2}-(u^{3}+v^{3})x+(uv)^{3}\end{aligned}}} so x 2 + q x − p 3 27 = 0. {\displaystyle x^{2}+qx-{\frac {p^{3}}{27}}=0.} The discriminant of this equation is Δ = q 2 + 4 p 3 27 {\displaystyle \Delta =q^{2}+{\frac {4p^{3}}{27}}} , and assuming it is positive, real solutions to this equation are (after folding division by 4 under the square root): − q 2 ± q 2 4 + p 3 27 . {\displaystyle -{\frac {q}{2}}\pm {\sqrt {{\frac {q^{2}}{4}}+{\frac {p^{3}}{27}}}}.} So (without loss of generality in choosing u or v {\displaystyle v} ): u = − q 2 + q 2 4 + p 3 27 3 . {\displaystyle u={\sqrt[{3}]{-{\frac {q}{2}}+{\sqrt {{\frac {q^{2}}{4}}+{\frac {p^{3}}{27}}}}}}.} v = − q 2 − q 2 4 + p 3 27 3 . {\displaystyle v={\sqrt[{3}]{-{\frac {q}{2}}-{\sqrt {{\frac {q^{2}}{4}}+{\frac {p^{3}}{27}}}}}}.} As u + v = t , {\displaystyle u+v=t,} the sum of the cube roots of these solutions is a root of the equation. That is t = − q 2 + q 2 4 + p 3 27 3 + − q 2 − q 2 4 + p 3 27 3 {\displaystyle t={\sqrt[{3}]{-{q \over 2}+{\sqrt {{q^{2} \over 4}+{p^{3} \over 27}}}}}+{\sqrt[{3}]{-{q \over 2}-{\sqrt {{q^{2} \over 4}+{p^{3} \over 27}}}}}} is a root of the equation; this is Cardano's formula. This works well when 4 p 3 + 27 q 2 > 0 , {\displaystyle 4p^{3}+27q^{2}>0,} but, if 4 p 3 + 27 q 2 < 0 , {\displaystyle 4p^{3}+27q^{2}<0,} the square root appearing in the formula is not real. As a complex number has three cube roots, using Cardano's formula without care would provide nine roots, while a cubic equation cannot have more than three roots. This was clarified first by Rafael Bombelli in his book L'Algebra (1572). The solution is to use the fact that u v = − p 3 , {\displaystyle uv=-{\frac {p}{3}},} that is, v = − p 3 u . {\displaystyle v={\frac {-p}{3u}}.} This means that only one cube root needs to be computed, and leads to the second formula given in § Cardano's formula. The other roots of the equation can be obtained by changing of cube root, or, equivalently, by multiplying the cube root by each of the two primitive cube roots of unity, which are − 1 ± − 3 2 . {\displaystyle {\frac {-1\pm {\sqrt {-3}}}{2}}.} === Vieta's substitution === Vieta's substitution is a method introduced by François Viète (Vieta is his Latin name) in a text published posthumously in 1615, which provides directly the second formula of § Cardano's method, and avoids the problem of computing two different cube roots. Starting from the depressed cubic t3 + pt + q = 0, Vieta's substitution is t = w − ⁠p/3w⁠. The substitution t = w – ⁠p/3w⁠ transforms the depressed cubic into w 3 + q − p 3 27 w 3 = 0. {\displaystyle w^{3}+q-{\frac {p^{3}}{27w^{3}}}=0.} Multiplying by w3, one gets a quadratic equation in w3: ( w 3 ) 2 + q ( w 3 ) − p 3 27 = 0. {\displaystyle (w^{3})^{2}+q(w^{3})-{\frac {p^{3}}{27}}=0.} Let W = − q 2 ± p 3 27 + q 2 4 {\displaystyle W=-{\frac {q}{2}}\pm {\sqrt {{\frac {p^{3}}{27}}+{\frac {q^{2}}{4}}}}} be any nonzero root of this quadratic equation. If w1, w2 and w3 are the three cube roots of W, then the roots of the original depressed cubic are w1 − ⁠p/3w1⁠, w2 − ⁠p/3w2⁠, and w3 − ⁠p/3w3⁠. The other root of the quadratic equation is − p 3 27 W . {\displaystyle \textstyle -{\frac {p^{3}}{27W}}.} This implies that changing the sign of the square root exchanges wi and − ⁠p/3wi⁠ for i = 1, 2, 3, and therefore does not change the roots. This method only fails when both roots of the quadratic equation are zero, that is when p = q = 0, in which case the only root of the depressed cubic is 0. === Lagrange's method === In his paper Réflexions sur la résolution algébrique des équations ("Thoughts on the algebraic solving of equations"), Joseph Louis Lagrange introduced a new method to solve equations of low degree in a uniform way, with the hope that he could generalize it for higher degrees. This method works well for cubic and quartic equations, but Lagrange did not succeed in applying it to a quintic equation, because it requires solving a resolvent polynomial of degree at least six. Apart from the fact that nobody had previously succeeded, this was the first indication of the non-existence of an algebraic formula for degrees 5 and higher; as was later proved by the Abel–Ruffini theorem. Nevertheless, modern methods for solving solvable quintic equations are mainly based on Lagrange's method. In the case of cubic equations, Lagrange's method gives the same solution as Cardano's. Lagrange's method can be applied directly to the general cubic equation ax3 + bx2 + cx + d = 0, but the computation is simpler with the depressed cubic equation, t3 + pt + q = 0. Lagrange's main idea was to work with the discrete Fourier transform of the roots instead of with the roots themselves. More precisely, let ξ be a primitive third root of unity, that is a number such that ξ3 = 1 and ξ2 + ξ + 1 = 0 (when working in the space of complex numbers, one has ξ = − 1 ± i 3 2 = e 2 i π / 3 , {\displaystyle \textstyle \xi ={\frac {-1\pm i{\sqrt {3}}}{2}}=e^{2i\pi /3},} but this complex interpretation is not used here). Denoting x0, x1 and x2 the three roots of the cubic equation to be solved, let s 0 = x 0 + x 1 + x 2 , s 1 = x 0 + ξ x 1 + ξ 2 x 2 , s 2 = x 0 + ξ 2 x 1 + ξ x 2 , {\displaystyle {\begin{aligned}s_{0}&=x_{0}+x_{1}+x_{2},\\s_{1}&=x_{0}+\xi x_{1}+\xi ^{2}x_{2},\\s_{2}&=x_{0}+\xi ^{2}x_{1}+\xi x_{2},\end{aligned}}} be the discrete Fourier transform of the roots. If s0, s1 and s2 are known, the roots may be recovered from them with the inverse Fourier transform consisting of inverting this linear transformation; that is, x 0 = 1 3 ( s 0 + s 1 + s 2 ) , x 1 = 1 3 ( s 0 + ξ 2 s 1 + ξ s 2 ) , x 2 = 1 3 ( s 0 + ξ s 1 + ξ 2 s 2 ) . {\displaystyle {\begin{aligned}x_{0}&={\tfrac {1}{3}}(s_{0}+s_{1}+s_{2}),\\x_{1}&={\tfrac {1}{3}}(s_{0}+\xi ^{2}s_{1}+\xi s_{2}),\\x_{2}&={\tfrac {1}{3}}(s_{0}+\xi s_{1}+\xi ^{2}s_{2}).\end{aligned}}} By Vieta's formulas, s0 is known to be zero in the case of a depressed cubic, and −⁠b/a⁠ for the general cubic. So, only s1 and s2 need to be computed. They are not symmetric functions of the roots (exchanging x1 and x2 exchanges also s1 and s2), but some simple symmetric functions of s1 and s2 are also symmetric in the roots of the cubic equation to be solved. Thus these symmetric functions can be expressed in terms of the (known) coefficients of the original cubic, and this allows eventually expressing the si as roots of a polynomial with known coefficients. This works well for every degree, but, in degrees higher than four, the resulting polynomial that has the si as roots has a degree higher than that of the initial polynomial, and is therefore unhelpful for solving. This is the reason for which Lagrange's method fails in degrees five and higher. In the case of a cubic equation, P = s 1 s 2 , {\displaystyle P=s_{1}s_{2},} and S = s 1 3 + s 2 3 {\displaystyle S=s_{1}^{3}+s_{2}^{3}} are such symmetric polynomials (see below). It follows that s 1 3 {\displaystyle s_{1}^{3}} and s 2 3 {\displaystyle s_{2}^{3}} are the two roots of the quadratic equation z 2 − S z + P 3 = 0. {\displaystyle z^{2}-Sz+P^{3}=0.} Thus the resolution of the equation may be finished exactly as with Cardano's method, with s 1 {\displaystyle s_{1}} and s 2 {\displaystyle s_{2}} in place of u and v . {\displaystyle v.} In the case of the depressed cubic, one has x 0 = 1 3 ( s 1 + s 2 ) {\displaystyle x_{0}={\tfrac {1}{3}}(s_{1}+s_{2})} and s 1 s 2 = − 3 p , {\displaystyle s_{1}s_{2}=-3p,} while in Cardano's method we have set x 0 = u + v {\displaystyle x_{0}=u+v} and u v = − 1 3 p . {\displaystyle uv=-{\tfrac {1}{3}}p.} Thus, up to the exchange of u and v , {\displaystyle v,} we have s 1 = 3 u {\displaystyle s_{1}=3u} and s 2 = 3 v . {\displaystyle s_{2}=3v.} In other words, in this case, Cardano's method and Lagrange's method compute exactly the same things, up to a factor of three in the auxiliary variables, the main difference being that Lagrange's method explains why these auxiliary variables appear in the problem. ==== Computation of S and P ==== A straightforward computation using the relations ξ3 = 1 and ξ2 + ξ + 1 = 0 gives P = s 1 s 2 = x 0 2 + x 1 2 + x 2 2 − ( x 0 x 1 + x 1 x 2 + x 2 x 0 ) , S = s 1 3 + s 2 3 = 2 ( x 0 3 + x 1 3 + x 2 3 ) − 3 ( x 0 2 x 1 + x 1 2 x 2 + x 2 2 x 0 + x 0 x 1 2 + x 1 x 2 2 + x 2 x 0 2 ) + 12 x 0 x 1 x 2 . {\displaystyle {\begin{aligned}P&=s_{1}s_{2}=x_{0}^{2}+x_{1}^{2}+x_{2}^{2}-(x_{0}x_{1}+x_{1}x_{2}+x_{2}x_{0}),\\S&=s_{1}^{3}+s_{2}^{3}=2(x_{0}^{3}+x_{1}^{3}+x_{2}^{3})-3(x_{0}^{2}x_{1}+x_{1}^{2}x_{2}+x_{2}^{2}x_{0}+x_{0}x_{1}^{2}+x_{1}x_{2}^{2}+x_{2}x_{0}^{2})+12x_{0}x_{1}x_{2}.\end{aligned}}} This shows that P and S are symmetric functions of the roots. Using Newton's identities, it is straightforward to express them in terms of the elementary symmetric functions of the roots, giving P = e 1 2 − 3 e 2 , S = 2 e 1 3 − 9 e 1 e 2 + 27 e 3 , {\displaystyle {\begin{aligned}P&=e_{1}^{2}-3e_{2},\\S&=2e_{1}^{3}-9e_{1}e_{2}+27e_{3},\end{aligned}}} with e1 = 0, e2 = p and e3 = −q in the case of a depressed cubic, and e1 = −⁠b/a⁠, e2 = ⁠c/a⁠ and e3 = −⁠d/a⁠, in the general case. == Applications == Cubic equations arise in various other contexts. === In mathematics === Angle trisection and doubling the cube are two ancient problems of geometry that have been proved to not be solvable by straightedge and compass construction, because they are equivalent to solving a cubic equation. Marden's theorem states that the foci of the Steiner inellipse of any triangle can be found by using the cubic function whose roots are the coordinates in the complex plane of the triangle's three vertices. The roots of the first derivative of this cubic are the complex coordinates of those foci. The area of a regular heptagon can be expressed in terms of the roots of a cubic. Further, the ratios of the long diagonal to the side, the side to the short diagonal, and the negative of the short diagonal to the long diagonal all satisfy a particular cubic equation. In addition, the ratio of the inradius to the circumradius of a heptagonal triangle is one of the solutions of a cubic equation. The values of trigonometric functions of angles related to 2 π / 7 {\displaystyle 2\pi /7} satisfy cubic equations. Given the cosine (or other trigonometric function) of an arbitrary angle, the cosine of one-third of that angle is one of the roots of a cubic. The solution of the general quartic equation relies on the solution of its resolvent cubic. The eigenvalues of a 3×3 matrix are the roots of a cubic polynomial which is the characteristic polynomial of the matrix. The characteristic equation of a third-order constant coefficients or Cauchy–Euler (equidimensional variable coefficients) linear differential equation or difference equation is a cubic equation. Intersection points of cubic Bézier curve and straight line can be computed using direct cubic equation representing Bézier curve. Critical points of a quartic function are found by solving a cubic equation (the derivative set equal to zero). Inflection points of a quintic function are the solution of a cubic equation (the second derivative set equal to zero). === In other sciences === In analytical chemistry, the Charlot equation, which can be used to find the pH of buffer solutions, can be solved using a cubic equation. In thermodynamics, equations of state (which relate pressure, volume, and temperature of a substances), e.g. the Van der Waals equation of state, are cubic in the volume. Kinematic equations involving linear rates of acceleration are cubic. The speed of seismic Rayleigh waves is a solution of the Rayleigh wave cubic equation. The steady state speed of a vehicle moving on a slope with air friction for a given input power is solved by a depressed cubic equation. Kepler's third law of planetary motion is cubic in the semi-major axis. == See also == Quartic equation Quintic equation Tschirnhaus transformation Principal equation form == Notes == == References == Guilbeau, Lucye (1930), "The History of the Solution of the Cubic Equation", Mathematics News Letter, 5 (4): 8–12, doi:10.2307/3027812, JSTOR 3027812 == Further reading == Anglin, W. S.; Lambek, Joachim (1995), "Mathematics in the Renaissance", The Heritage of Thales, Springers, pp. 125–131, ISBN 978-0-387-94544-6 Ch. 24. Dence, T. (November 1997), "Cubics, chaos and Newton's method", Mathematical Gazette, 81 (492), Mathematical Association: 403–408, doi:10.2307/3619617, ISSN 0025-5572, JSTOR 3619617, S2CID 125196796 Dunnett, R. (November 1994), "Newton–Raphson and the cubic", Mathematical Gazette, 78 (483), Mathematical Association: 347–348, doi:10.2307/3620218, ISSN 0025-5572, JSTOR 3620218, S2CID 125643035 Jacobson, Nathan (2009), Basic algebra, vol. 1 (2nd ed.), Dover, ISBN 978-0-486-47189-1 Mitchell, D. W. (November 2007), "Solving cubics by solving triangles", Mathematical Gazette, 91, Mathematical Association: 514–516, doi:10.1017/S0025557200182178, ISSN 0025-5572, S2CID 124710259 Mitchell, D. W. (November 2009), "Powers of φ as roots of cubics", Mathematical Gazette, 93, Mathematical Association, doi:10.1017/S0025557200185237, ISSN 0025-5572, S2CID 126286653 Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. (2007), "Section 5.6 Quadratic and Cubic Equations", Numerical Recipes: The Art of Scientific Computing (3rd ed.), New York: Cambridge University Press, ISBN 978-0-521-88068-8 Rechtschaffen, Edgar (July 2008), "Real roots of cubics: Explicit formula for quasi-solutions", Mathematical Gazette, 92, Mathematical Association: 268–276, doi:10.1017/S0025557200183147, ISSN 0025-5572, S2CID 125870578 Zucker, I. J. (July 2008), "The cubic equation – a new look at the irreducible case", Mathematical Gazette, 92, Mathematical Association: 264–268, doi:10.1017/S0025557200183135, ISSN 0025-5572, S2CID 125986006 == External links == "Cardano formula", Encyclopedia of Mathematics, EMS Press, 2001 [1994] History of quadratic, cubic and quartic equations on MacTutor archive. 500 years of NOT teaching THE CUBIC FORMULA. What is it they think you can't handle? – YouTube video by Mathologer about the history of cubic equations and Cardano's solution, as well as Ferrari's solution to quartic equations
Wikipedia:Curl (mathematics)#0	In vector calculus, the curl, also known as rotor, is a vector operator that describes the infinitesimal circulation of a vector field in three-dimensional Euclidean space. The curl at a point in the field is represented by a vector whose length and direction denote the magnitude and axis of the maximum circulation. The curl of a field is formally defined as the circulation density at each point of the field. A vector field whose curl is zero is called irrotational. The curl is a form of differentiation for vector fields. The corresponding form of the fundamental theorem of calculus is Stokes' theorem, which relates the surface integral of the curl of a vector field to the line integral of the vector field around the boundary curve. The notation curl F is more common in North America. In the rest of the world, particularly in 20th century scientific literature, the alternative notation rot F is traditionally used, which comes from the "rate of rotation" that it represents. To avoid confusion, modern authors tend to use the cross product notation with the del (nabla) operator, as in ∇ × F {\displaystyle \nabla \times \mathbf {F} } , which also reveals the relation between curl (rotor), divergence, and gradient operators. Unlike the gradient and divergence, curl as formulated in vector calculus does not generalize simply to other dimensions; some generalizations are possible, but only in three dimensions is the geometrically defined curl of a vector field again a vector field. This deficiency is a direct consequence of the limitations of vector calculus; on the other hand, when expressed as an antisymmetric tensor field via the wedge operator of geometric calculus, the curl generalizes to all dimensions. The circumstance is similar to that attending the 3-dimensional cross product, and indeed the connection is reflected in the notation ∇ × {\displaystyle \nabla \times } for the curl. The name "curl" was first suggested by James Clerk Maxwell in 1871 but the concept was apparently first used in the construction of an optical field theory by James MacCullagh in 1839. == Definition == The curl of a vector field F, denoted by curl F, or ∇ × F {\displaystyle \nabla \times \mathbf {F} } , or rot F, is an operator that maps Ck functions in R3 to Ck−1 functions in R3, and in particular, it maps continuously differentiable functions R3 → R3 to continuous functions R3 → R3. It can be defined in several ways, to be mentioned below: One way to define the curl of a vector field at a point is implicitly through its components along various axes passing through the point: if u ^ {\displaystyle \mathbf {\hat {u}} } is any unit vector, the component of the curl of F along the direction u ^ {\displaystyle \mathbf {\hat {u}} } may be defined to be the limiting value of a closed line integral in a plane perpendicular to u ^ {\displaystyle \mathbf {\hat {u}} } divided by the area enclosed, as the path of integration is contracted indefinitely around the point. More specifically, the curl is defined at a point p as ( ∇ × F ) ( p ) ⋅ u ^ = d e f lim A → 0 1 \| A \| ∮ C ( p ) F ⋅ d r {\displaystyle (\nabla \times \mathbf {F} )(p)\cdot \mathbf {\hat {u}} \ {\overset {\underset {\mathrm {def} }{}}{{}={}}}\lim _{A\to 0}{\frac {1}{\|A\|}}\oint _{C(p)}\mathbf {F} \cdot \mathrm {d} \mathbf {r} } where the line integral is calculated along the boundary C of the area A containing point p, \|A\| being the magnitude of the area. This equation defines the component of the curl of F along the direction u ^ {\displaystyle \mathbf {\hat {u}} } . The infinitesimal surfaces bounded by C have u ^ {\displaystyle \mathbf {\hat {u}} } as their normal. C is oriented via the right-hand rule. The above formula means that the component of the curl of a vector field along a certain axis is the infinitesimal area density of the circulation of the field in a plane perpendicular to that axis. This formula does not a priori define a legitimate vector field, for the individual circulation densities with respect to various axes a priori need not relate to each other in the same way as the components of a vector do; that they do indeed relate to each other in this precise manner must be proven separately. To this definition fits naturally the Kelvin–Stokes theorem, as a global formula corresponding to the definition. It equates the surface integral of the curl of a vector field to the above line integral taken around the boundary of the surface. Another way one can define the curl vector of a function F at a point is explicitly as the limiting value of a vector-valued surface integral around a shell enclosing p divided by the volume enclosed, as the shell is contracted indefinitely around p. More specifically, the curl may be defined by the vector formula ( ∇ × F ) ( p ) = d e f lim V → 0 1 \| V \| ∮ S n ^ × F d S {\displaystyle (\nabla \times \mathbf {F} )(p){\overset {\underset {\mathrm {def} }{}}{{}={}}}\lim _{V\to 0}{\frac {1}{\|V\|}}\oint _{S}\mathbf {\hat {n}} \times \mathbf {F} \ \mathrm {d} S} where the surface integral is calculated along the boundary S of the volume V, \|V\| being the magnitude of the volume, and n ^ {\displaystyle \mathbf {\hat {n}} } pointing outward from the surface S perpendicularly at every point in S. In this formula, the cross product in the integrand measures the tangential component of F at each point on the surface S, and points along the surface at right angles to the tangential projection of F. Integrating this cross product over the whole surface results in a vector whose magnitude measures the overall circulation of F around S, and whose direction is at right angles to this circulation. The above formula says that the curl of a vector field at a point is the infinitesimal volume density of this "circulation vector" around the point. To this definition fits naturally another global formula (similar to the Kelvin-Stokes theorem) which equates the volume integral of the curl of a vector field to the above surface integral taken over the boundary of the volume. Whereas the above two definitions of the curl are coordinate free, there is another "easy to memorize" definition of the curl in curvilinear orthogonal coordinates, e.g. in Cartesian coordinates, spherical, cylindrical, or even elliptical or parabolic coordinates: ( curl ⁡ F ) 1 = 1 h 2 h 3 ( ∂ ( h 3 F 3 ) ∂ u 2 − ∂ ( h 2 F 2 ) ∂ u 3 ) , ( curl ⁡ F ) 2 = 1 h 3 h 1 ( ∂ ( h 1 F 1 ) ∂ u 3 − ∂ ( h 3 F 3 ) ∂ u 1 ) , ( curl ⁡ F ) 3 = 1 h 1 h 2 ( ∂ ( h 2 F 2 ) ∂ u 1 − ∂ ( h 1 F 1 ) ∂ u 2 ) . {\displaystyle {\begin{aligned}&(\operatorname {curl} \mathbf {F} )_{1}={\frac {1}{h_{2}h_{3}}}\left({\frac {\partial (h_{3}F_{3})}{\partial u_{2}}}-{\frac {\partial (h_{2}F_{2})}{\partial u_{3}}}\right),\\[5pt]&(\operatorname {curl} \mathbf {F} )_{2}={\frac {1}{h_{3}h_{1}}}\left({\frac {\partial (h_{1}F_{1})}{\partial u_{3}}}-{\frac {\partial (h_{3}F_{3})}{\partial u_{1}}}\right),\\[5pt]&(\operatorname {curl} \mathbf {F} )_{3}={\frac {1}{h_{1}h_{2}}}\left({\frac {\partial (h_{2}F_{2})}{\partial u_{1}}}-{\frac {\partial (h_{1}F_{1})}{\partial u_{2}}}\right).\end{aligned}}} The equation for each component (curl F)k can be obtained by exchanging each occurrence of a subscript 1, 2, 3 in cyclic permutation: 1 → 2, 2 → 3, and 3 → 1 (where the subscripts represent the relevant indices). If (x1, x2, x3) are the Cartesian coordinates and (u1, u2, u3) are the orthogonal coordinates, then h i = ( ∂ x 1 ∂ u i ) 2 + ( ∂ x 2 ∂ u i ) 2 + ( ∂ x 3 ∂ u i ) 2 {\displaystyle h_{i}={\sqrt {\left({\frac {\partial x_{1}}{\partial u_{i}}}\right)^{2}+\left({\frac {\partial x_{2}}{\partial u_{i}}}\right)^{2}+\left({\frac {\partial x_{3}}{\partial u_{i}}}\right)^{2}}}} is the length of the coordinate vector corresponding to ui. The remaining two components of curl result from cyclic permutation of indices: 3,1,2 → 1,2,3 → 2,3,1. == Usage == In practice, the two coordinate-free definitions described above are rarely used because in virtually all cases, the curl operator can be applied using some set of curvilinear coordinates, for which simpler representations have been derived. The notation ∇ × F {\displaystyle \nabla \times \mathbf {F} } has its origins in the similarities to the 3-dimensional cross product, and it is useful as a mnemonic in Cartesian coordinates if ∇ {\displaystyle \nabla } is taken as a vector differential operator del. Such notation involving operators is common in physics and algebra. Expanded in 3-dimensional Cartesian coordinates (see Del in cylindrical and spherical coordinates for spherical and cylindrical coordinate representations), ∇ × F {\displaystyle \nabla \times \mathbf {F} } is, for F {\displaystyle \mathbf {F} } composed of [ F x , F y , F z ] {\displaystyle [F_{x},F_{y},F_{z}]} (where the subscripts indicate the components of the vector, not partial derivatives): ∇ × F = \| ı ^ ȷ ^ k ^ ∂ ∂ x ∂ ∂ y ∂ ∂ z F x F y F z \| {\displaystyle \nabla \times \mathbf {F} ={\begin{vmatrix}{\boldsymbol {\hat {\imath }}}&{\boldsymbol {\hat {\jmath }}}&{\boldsymbol {\hat {k}}}\\[5mu]{\dfrac {\partial }{\partial x}}&{\dfrac {\partial }{\partial y}}&{\dfrac {\partial }{\partial z}}\\[5mu]F_{x}&F_{y}&F_{z}\end{vmatrix}}} where i, j, and k are the unit vectors for the x-, y-, and z-axes, respectively. This expands as follows: ∇ × F = ( ∂ F z ∂ y − ∂ F y ∂ z ) ı ^ + ( ∂ F x ∂ z − ∂ F z ∂ x ) ȷ ^ + ( ∂ F y ∂ x − ∂ F x ∂ y ) k ^ {\displaystyle \nabla \times \mathbf {F} =\left({\frac {\partial F_{z}}{\partial y}}-{\frac {\partial F_{y}}{\partial z}}\right){\boldsymbol {\hat {\imath }}}+\left({\frac {\partial F_{x}}{\partial z}}-{\frac {\partial F_{z}}{\partial x}}\right){\boldsymbol {\hat {\jmath }}}+\left({\frac {\partial F_{y}}{\partial x}}-{\frac {\partial F_{x}}{\partial y}}\right){\boldsymbol {\hat {k}}}} Although expressed in terms of coordinates, the result is invariant under proper rotations of the coordinate axes but the result inverts under reflection. In a general coordinate system, the curl is given by ( ∇ × F ) k = 1 g ε k ℓ m ∇ ℓ F m {\displaystyle (\nabla \times \mathbf {F} )^{k}={\frac {1}{\sqrt {g}}}\varepsilon ^{k\ell m}\nabla _{\ell }F_{m}} where ε denotes the Levi-Civita tensor, ∇ the covariant derivative, g {\displaystyle g} is the determinant of the metric tensor and the Einstein summation convention implies that repeated indices are summed over. Due to the symmetry of the Christoffel symbols participating in the covariant derivative, this expression reduces to the partial derivative: ( ∇ × F ) = 1 g R k ε k ℓ m ∂ ℓ F m {\displaystyle (\nabla \times \mathbf {F} )={\frac {1}{\sqrt {g}}}\mathbf {R} _{k}\varepsilon ^{k\ell m}\partial _{\ell }F_{m}} where Rk are the local basis vectors. Equivalently, using the exterior derivative, the curl can be expressed as: ∇ × F = ( ⋆ ( d F ♭ ) ) ♯ {\displaystyle \nabla \times \mathbf {F} =\left(\star {\big (}{\mathrm {d} }\mathbf {F} ^{\flat }{\big )}\right)^{\sharp }} Here ♭ and ♯ are the musical isomorphisms, and ★ is the Hodge star operator. This formula shows how to calculate the curl of F in any coordinate system, and how to extend the curl to any oriented three-dimensional Riemannian manifold. Since this depends on a choice of orientation, curl is a chiral operation. In other words, if the orientation is reversed, then the direction of the curl is also reversed. == Examples == === Example 1 === Suppose the vector field describes the velocity field of a fluid flow (such as a large tank of liquid or gas) and a small ball is located within the fluid or gas (the center of the ball being fixed at a certain point). If the ball has a rough surface, the fluid flowing past it will make it rotate. The rotation axis (oriented according to the right hand rule) points in the direction of the curl of the field at the center of the ball, and the angular speed of the rotation is half the magnitude of the curl at this point. The curl of the vector field at any point is given by the rotation of an infinitesimal area in the xy-plane (for z-axis component of the curl), zx-plane (for y-axis component of the curl) and yz-plane (for x-axis component of the curl vector). This can be seen in the examples below. === Example 2 === The vector field F ( x , y , z ) = y ı ^ − x ȷ ^ {\displaystyle \mathbf {F} (x,y,z)=y{\boldsymbol {\hat {\imath }}}-x{\boldsymbol {\hat {\jmath }}}} can be decomposed as F x = y , F y = − x , F z = 0. {\displaystyle F_{x}=y,F_{y}=-x,F_{z}=0.} Upon visual inspection, the field can be described as "rotating". If the vectors of the field were to represent a linear force acting on objects present at that point, and an object were to be placed inside the field, the object would start to rotate clockwise around itself. This is true regardless of where the object is placed. Calculating the curl: ∇ × F = 0 ı ^ + 0 ȷ ^ + ( ∂ ∂ x ( − x ) − ∂ ∂ y y ) k ^ = − 2 k ^ {\displaystyle \nabla \times \mathbf {F} =0{\boldsymbol {\hat {\imath }}}+0{\boldsymbol {\hat {\jmath }}}+\left({\frac {\partial }{\partial x}}(-x)-{\frac {\partial }{\partial y}}y\right){\boldsymbol {\hat {k}}}=-2{\boldsymbol {\hat {k}}}} The resulting vector field describing the curl would at all points be pointing in the negative z direction. The results of this equation align with what could have been predicted using the right-hand rule using a right-handed coordinate system. Being a uniform vector field, the object described before would have the same rotational intensity regardless of where it was placed. === Example 3 === For the vector field F ( x , y , z ) = − x 2 ȷ ^ {\displaystyle \mathbf {F} (x,y,z)=-x^{2}{\boldsymbol {\hat {\jmath }}}} the curl is not as obvious from the graph. However, taking the object in the previous example, and placing it anywhere on the line x = 3, the force exerted on the right side would be slightly greater than the force exerted on the left, causing it to rotate clockwise. Using the right-hand rule, it can be predicted that the resulting curl would be straight in the negative z direction. Inversely, if placed on x = −3, the object would rotate counterclockwise and the right-hand rule would result in a positive z direction. Calculating the curl: ∇ × F = 0 ı ^ + 0 ȷ ^ + ∂ ∂ x ( − x 2 ) k ^ = − 2 x k ^ . {\displaystyle {\nabla }\times \mathbf {F} =0{\boldsymbol {\hat {\imath }}}+0{\boldsymbol {\hat {\jmath }}}+{\frac {\partial }{\partial x}}\left(-x^{2}\right){\boldsymbol {\hat {k}}}=-2x{\boldsymbol {\hat {k}}}.} The curl points in the negative z direction when x is positive and vice versa. In this field, the intensity of rotation would be greater as the object moves away from the plane x = 0. === Further examples === In a vector field describing the linear velocities of each part of a rotating disk in uniform circular motion, the curl has the same value at all points, and this value turns out to be exactly two times the vectorial angular velocity of the disk (oriented as usual by the right-hand rule). More generally, for any flowing mass, the linear velocity vector field at each point of the mass flow has a curl (the vorticity of the flow at that point) equal to exactly two times the local vectorial angular velocity of the mass about the point. For any solid object subject to an external physical force (such as gravity or the electromagnetic force), one may consider the vector field representing the infinitesimal force-per-unit-volume contributions acting at each of the points of the object. This force field may create a net torque on the object about its center of mass, and this torque turns out to be directly proportional and vectorially parallel to the (vector-valued) integral of the curl of the force field over the whole volume. Of the four Maxwell's equations, two—Faraday's law and Ampère's law—can be compactly expressed using curl. Faraday's law states that the curl of an electric field is equal to the opposite of the time rate of change of the magnetic field, while Ampère's law relates the curl of the magnetic field to the current and the time rate of change of the electric field. == Identities == In general curvilinear coordinates (not only in Cartesian coordinates), the curl of a cross product of vector fields v and F can be shown to be ∇ × ( v × F ) = ( ( ∇ ⋅ F ) + F ⋅ ∇ ) v − ( ( ∇ ⋅ v ) + v ⋅ ∇ ) F . {\displaystyle \nabla \times \left(\mathbf {v\times F} \right)={\Big (}\left(\mathbf {\nabla \cdot F} \right)+\mathbf {F\cdot \nabla } {\Big )}\mathbf {v} -{\Big (}\left(\mathbf {\nabla \cdot v} \right)+\mathbf {v\cdot \nabla } {\Big )}\mathbf {F} \ .} Interchanging the vector field v and ∇ operator, we arrive at the cross product of a vector field with curl of a vector field: v × ( ∇ × F ) = ∇ F ( v ⋅ F ) − ( v ⋅ ∇ ) F , {\displaystyle \mathbf {v\ \times } \left(\mathbf {\nabla \times F} \right)=\nabla _{\mathbf {F} }\left(\mathbf {v\cdot F} \right)-\left(\mathbf {v\cdot \nabla } \right)\mathbf {F} \ ,} where ∇F is the Feynman subscript notation, which considers only the variation due to the vector field F (i.e., in this case, v is treated as being constant in space). Another example is the curl of a curl of a vector field. It can be shown that in general coordinates ∇ × ( ∇ × F ) = ∇ ( ∇ ⋅ F ) − ∇ 2 F , {\displaystyle \nabla \times \left(\mathbf {\nabla \times F} \right)=\mathbf {\nabla } (\mathbf {\nabla \cdot F} )-\nabla ^{2}\mathbf {F} \ ,} and this identity defines the vector Laplacian of F, symbolized as ∇2F. The curl of the gradient of any scalar field φ is always the zero vector field ∇ × ( ∇ φ ) = 0 {\displaystyle \nabla \times (\nabla \varphi )={\boldsymbol {0}}} which follows from the antisymmetry in the definition of the curl, and the symmetry of second derivatives. The divergence of the curl of any vector field is equal to zero: ∇ ⋅ ( ∇ × F ) = 0. {\displaystyle \nabla \cdot (\nabla \times \mathbf {F} )=0.} If φ is a scalar valued function and F is a vector field, then ∇ × ( φ F ) = ∇ φ × F + φ ∇ × F {\displaystyle \nabla \times (\varphi \mathbf {F} )=\nabla \varphi \times \mathbf {F} +\varphi \nabla \times \mathbf {F} } == Generalizations == The vector calculus operations of grad, curl, and div are most easily generalized in the context of differential forms, which involves a number of steps. In short, they correspond to the derivatives of 0-forms, 1-forms, and 2-forms, respectively. The geometric interpretation of curl as rotation corresponds to identifying bivectors (2-vectors) in 3 dimensions with the special orthogonal Lie algebra s o ( 3 ) {\displaystyle {\mathfrak {so}}(3)} of infinitesimal rotations (in coordinates, skew-symmetric 3 × 3 matrices), while representing rotations by vectors corresponds to identifying 1-vectors (equivalently, 2-vectors) and s o ( 3 ) {\displaystyle {\mathfrak {so}}(3)} , these all being 3-dimensional spaces. === Differential forms === In 3 dimensions, a differential 0-form is a real-valued function f ( x , y , z ) {\displaystyle f(x,y,z)} ; a differential 1-form is the following expression, where the coefficients are functions: a 1 d x + a 2 d y + a 3 d z ; {\displaystyle a_{1}\,dx+a_{2}\,dy+a_{3}\,dz;} a differential 2-form is the formal sum, again with function coefficients: a 12 d x ∧ d y + a 13 d x ∧ d z + a 23 d y ∧ d z ; {\displaystyle a_{12}\,dx\wedge dy+a_{13}\,dx\wedge dz+a_{23}\,dy\wedge dz;} and a differential 3-form is defined by a single term with one function as coefficient: a 123 d x ∧ d y ∧ d z . {\displaystyle a_{123}\,dx\wedge dy\wedge dz.} (Here the a-coefficients are real functions of three variables; the wedge products, e.g. d x ∧ d y {\displaystyle {\text{d}}x\wedge {\text{d}}y} , can be interpreted as oriented plane segments, d x ∧ d y = − d y ∧ d x {\displaystyle {\text{d}}x\wedge {\text{d}}y=-{\text{d}}y\wedge {\text{d}}x} , etc.) The exterior derivative of a k-form in R3 is defined as the (k + 1)-form from above—and in Rn if, e.g., ω ( k ) = ∑ 1 ≤ i 1 < i 2 < ⋯ < i k ≤ n a i 1 , … , i k d x i 1 ∧ ⋯ ∧ d x i k , {\displaystyle \omega ^{(k)}=\sum _{1\leq i_{1}<i_{2}<\cdots <i_{k}\leq n}a_{i_{1},\ldots ,i_{k}}\,dx_{i_{1}}\wedge \cdots \wedge dx_{i_{k}},} then the exterior derivative d leads to d ω ( k ) = ∑ j = 1 i 1 < ⋯ < i k n ∂ a i 1 , … , i k ∂ x j d x j ∧ d x i 1 ∧ ⋯ ∧ d x i k . {\displaystyle d\omega ^{(k)}=\sum _{\scriptstyle {j=1} \atop \scriptstyle {i_{1}<\cdots <i_{k}}}^{n}{\frac {\partial a_{i_{1},\ldots ,i_{k}}}{\partial x_{j}}}\,dx_{j}\wedge dx_{i_{1}}\wedge \cdots \wedge dx_{i_{k}}.} The exterior derivative of a 1-form is therefore a 2-form, and that of a 2-form is a 3-form. On the other hand, because of the interchangeability of mixed derivatives, ∂ 2 ∂ x i ∂ x j = ∂ 2 ∂ x j ∂ x i , {\displaystyle {\frac {\partial ^{2}}{\partial x_{i}\,\partial x_{j}}}={\frac {\partial ^{2}}{\partial x_{j}\,\partial x_{i}}},} and antisymmetry, d x i ∧ d x j = − d x j ∧ d x i {\displaystyle dx_{i}\wedge dx_{j}=-dx_{j}\wedge dx_{i}} the twofold application of the exterior derivative yields 0 {\displaystyle 0} (the zero k + 2 {\displaystyle k+2} -form). Thus, denoting the space of k-forms by Ω k ( R 3 ) {\displaystyle \Omega ^{k}(\mathbb {R} ^{3})} and the exterior derivative by d one gets a sequence: 0 ⟶ d Ω 0 ( R 3 ) ⟶ d Ω 1 ( R 3 ) ⟶ d Ω 2 ( R 3 ) ⟶ d Ω 3 ( R 3 ) ⟶ d 0. {\displaystyle 0\,{\overset {d}{\longrightarrow }}\;\Omega ^{0}\left(\mathbb {R} ^{3}\right)\,{\overset {d}{\longrightarrow }}\;\Omega ^{1}\left(\mathbb {R} ^{3}\right)\,{\overset {d}{\longrightarrow }}\;\Omega ^{2}\left(\mathbb {R} ^{3}\right)\,{\overset {d}{\longrightarrow }}\;\Omega ^{3}\left(\mathbb {R} ^{3}\right)\,{\overset {d}{\longrightarrow }}\,0.} Here Ω k ( R n ) {\displaystyle \Omega ^{k}(\mathbb {R} ^{n})} is the space of sections of the exterior algebra Λ k ( R n ) {\displaystyle \Lambda ^{k}(\mathbb {R} ^{n})} vector bundle over Rn, whose dimension is the binomial coefficient ( n k ) {\displaystyle {\binom {n}{k}}} ; note that Ω k ( R 3 ) = 0 {\displaystyle \Omega ^{k}(\mathbb {R} ^{3})=0} for k > 3 {\displaystyle k>3} or k < 0 {\displaystyle k<0} . Writing only dimensions, one obtains a row of Pascal's triangle: 0 → 1 → 3 → 3 → 1 → 0 ; {\displaystyle 0\rightarrow 1\rightarrow 3\rightarrow 3\rightarrow 1\rightarrow 0;} the 1-dimensional fibers correspond to scalar fields, and the 3-dimensional fibers to vector fields, as described below. Modulo suitable identifications, the three nontrivial occurrences of the exterior derivative correspond to grad, curl, and div. Differential forms and the differential can be defined on any Euclidean space, or indeed any manifold, without any notion of a Riemannian metric. On a Riemannian manifold, or more generally pseudo-Riemannian manifold, k-forms can be identified with k-vector fields (k-forms are k-covector fields, and a pseudo-Riemannian metric gives an isomorphism between vectors and covectors), and on an oriented vector space with a nondegenerate form (an isomorphism between vectors and covectors), there is an isomorphism between k-vectors and (n − k)-vectors; in particular on (the tangent space of) an oriented pseudo-Riemannian manifold. Thus on an oriented pseudo-Riemannian manifold, one can interchange k-forms, k-vector fields, (n − k)-forms, and (n − k)-vector fields; this is known as Hodge duality. Concretely, on R3 this is given by: 1-forms and 1-vector fields: the 1-form ax dx + ay dy + az dz corresponds to the vector field (ax, ay, az). 1-forms and 2-forms: one replaces dx by the dual quantity dy ∧ dz (i.e., omit dx), and likewise, taking care of orientation: dy corresponds to dz ∧ dx = −dx ∧ dz, and dz corresponds to dx ∧ dy. Thus the form ax dx + ay dy + az dz corresponds to the "dual form" az dx ∧ dy + ay dz ∧ dx + ax dy ∧ dz. Thus, identifying 0-forms and 3-forms with scalar fields, and 1-forms and 2-forms with vector fields: grad takes a scalar field (0-form) to a vector field (1-form); curl takes a vector field (1-form) to a pseudovector field (2-form); div takes a pseudovector field (2-form) to a pseudoscalar field (3-form) On the other hand, the fact that d2 = 0 corresponds to the identities ∇ × ( ∇ f ) = 0 {\displaystyle \nabla \times (\nabla f)=\mathbf {0} } for any scalar field f, and ∇ ⋅ ( ∇ × v ) = 0 {\displaystyle \nabla \cdot (\nabla \times \mathbf {v} )=0} for any vector field v. Grad and div generalize to all oriented pseudo-Riemannian manifolds, with the same geometric interpretation, because the spaces of 0-forms and n-forms at each point are always 1-dimensional and can be identified with scalar fields, while the spaces of 1-forms and (n − 1)-forms are always fiberwise n-dimensional and can be identified with vector fields. Curl does not generalize in this way to 4 or more dimensions (or down to 2 or fewer dimensions); in 4 dimensions the dimensions are so the curl of a 1-vector field (fiberwise 4-dimensional) is a 2-vector field, which at each point belongs to 6-dimensional vector space, and so one has ω ( 2 ) = ∑ i < k = 1 , 2 , 3 , 4 a i , k d x i ∧ d x k , {\displaystyle \omega ^{(2)}=\sum _{i<k=1,2,3,4}a_{i,k}\,dx_{i}\wedge dx_{k},} which yields a sum of six independent terms, and cannot be identified with a 1-vector field. Nor can one meaningfully go from a 1-vector field to a 2-vector field to a 3-vector field (4 → 6 → 4), as taking the differential twice yields zero (d2 = 0). Thus there is no curl function from vector fields to vector fields in other dimensions arising in this way. However, one can define a curl of a vector field as a 2-vector field in general, as described below. === Curl geometrically === 2-vectors correspond to the exterior power Λ2V; in the presence of an inner product, in coordinates these are the skew-symmetric matrices, which are geometrically considered as the special orthogonal Lie algebra s o {\displaystyle {\mathfrak {so}}} (V) of infinitesimal rotations. This has (n2) = ⁠1/2⁠n(n − 1) dimensions, and allows one to interpret the differential of a 1-vector field as its infinitesimal rotations. Only in 3 dimensions (or trivially in 0 dimensions) we have n = ⁠1/2⁠n(n − 1), which is the most elegant and common case. In 2 dimensions the curl of a vector field is not a vector field but a function, as 2-dimensional rotations are given by an angle (a scalar – an orientation is required to choose whether one counts clockwise or counterclockwise rotations as positive); this is not the div, but is rather perpendicular to it. In 3 dimensions the curl of a vector field is a vector field as is familiar (in 1 and 0 dimensions the curl of a vector field is 0, because there are no non-trivial 2-vectors), while in 4 dimensions the curl of a vector field is, geometrically, at each point an element of the 6-dimensional Lie algebra s o ( 4 ) {\displaystyle {\mathfrak {so}}(4)} . The curl of a 3-dimensional vector field which only depends on 2 coordinates (say x and y) is simply a vertical vector field (in the z direction) whose magnitude is the curl of the 2-dimensional vector field, as in the examples on this page. Considering curl as a 2-vector field (an antisymmetric 2-tensor) has been used to generalize vector calculus and associated physics to higher dimensions. == Inverse == In the case where the divergence of a vector field V is zero, a vector field W exists such that V = curl(W). This is why the magnetic field, characterized by zero divergence, can be expressed as the curl of a magnetic vector potential. If W is a vector field with curl(W) = V, then adding any gradient vector field grad(f) to W will result in another vector field W + grad(f) such that curl(W + grad(f)) = V as well. This can be summarized by saying that the inverse curl of a three-dimensional vector field can be obtained up to an unknown irrotational field with the Biot–Savart law. == See also == Helmholtz decomposition Hiptmair–Xu preconditioner Del in cylindrical and spherical coordinates Vorticity == References == == Further reading == Korn, Granino Arthur and Theresa M. Korn (January 2000). Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review. New York: Dover Publications. pp. 157–160. ISBN 0-486-41147-8. Schey, H. M. (1997). Div, Grad, Curl, and All That: An Informal Text on Vector Calculus. New York: Norton. ISBN 0-393-96997-5. == External links == "Curl", Encyclopedia of Mathematics, EMS Press, 2001 [1994] "Multivariable calculus". mathinsight.org. Retrieved February 12, 2022. "Divergence and Curl: The Language of Maxwell's Equations, Fluid Flow, and More". June 21, 2018. Archived from the original on 2021-11-24 – via YouTube.
Wikipedia:Curl operator#0	In vector calculus, the curl, also known as rotor, is a vector operator that describes the infinitesimal circulation of a vector field in three-dimensional Euclidean space. The curl at a point in the field is represented by a vector whose length and direction denote the magnitude and axis of the maximum circulation. The curl of a field is formally defined as the circulation density at each point of the field. A vector field whose curl is zero is called irrotational. The curl is a form of differentiation for vector fields. The corresponding form of the fundamental theorem of calculus is Stokes' theorem, which relates the surface integral of the curl of a vector field to the line integral of the vector field around the boundary curve. The notation curl F is more common in North America. In the rest of the world, particularly in 20th century scientific literature, the alternative notation rot F is traditionally used, which comes from the "rate of rotation" that it represents. To avoid confusion, modern authors tend to use the cross product notation with the del (nabla) operator, as in ∇ × F {\displaystyle \nabla \times \mathbf {F} } , which also reveals the relation between curl (rotor), divergence, and gradient operators. Unlike the gradient and divergence, curl as formulated in vector calculus does not generalize simply to other dimensions; some generalizations are possible, but only in three dimensions is the geometrically defined curl of a vector field again a vector field. This deficiency is a direct consequence of the limitations of vector calculus; on the other hand, when expressed as an antisymmetric tensor field via the wedge operator of geometric calculus, the curl generalizes to all dimensions. The circumstance is similar to that attending the 3-dimensional cross product, and indeed the connection is reflected in the notation ∇ × {\displaystyle \nabla \times } for the curl. The name "curl" was first suggested by James Clerk Maxwell in 1871 but the concept was apparently first used in the construction of an optical field theory by James MacCullagh in 1839. == Definition == The curl of a vector field F, denoted by curl F, or ∇ × F {\displaystyle \nabla \times \mathbf {F} } , or rot F, is an operator that maps Ck functions in R3 to Ck−1 functions in R3, and in particular, it maps continuously differentiable functions R3 → R3 to continuous functions R3 → R3. It can be defined in several ways, to be mentioned below: One way to define the curl of a vector field at a point is implicitly through its components along various axes passing through the point: if u ^ {\displaystyle \mathbf {\hat {u}} } is any unit vector, the component of the curl of F along the direction u ^ {\displaystyle \mathbf {\hat {u}} } may be defined to be the limiting value of a closed line integral in a plane perpendicular to u ^ {\displaystyle \mathbf {\hat {u}} } divided by the area enclosed, as the path of integration is contracted indefinitely around the point. More specifically, the curl is defined at a point p as ( ∇ × F ) ( p ) ⋅ u ^ = d e f lim A → 0 1 \| A \| ∮ C ( p ) F ⋅ d r {\displaystyle (\nabla \times \mathbf {F} )(p)\cdot \mathbf {\hat {u}} \ {\overset {\underset {\mathrm {def} }{}}{{}={}}}\lim _{A\to 0}{\frac {1}{\|A\|}}\oint _{C(p)}\mathbf {F} \cdot \mathrm {d} \mathbf {r} } where the line integral is calculated along the boundary C of the area A containing point p, \|A\| being the magnitude of the area. This equation defines the component of the curl of F along the direction u ^ {\displaystyle \mathbf {\hat {u}} } . The infinitesimal surfaces bounded by C have u ^ {\displaystyle \mathbf {\hat {u}} } as their normal. C is oriented via the right-hand rule. The above formula means that the component of the curl of a vector field along a certain axis is the infinitesimal area density of the circulation of the field in a plane perpendicular to that axis. This formula does not a priori define a legitimate vector field, for the individual circulation densities with respect to various axes a priori need not relate to each other in the same way as the components of a vector do; that they do indeed relate to each other in this precise manner must be proven separately. To this definition fits naturally the Kelvin–Stokes theorem, as a global formula corresponding to the definition. It equates the surface integral of the curl of a vector field to the above line integral taken around the boundary of the surface. Another way one can define the curl vector of a function F at a point is explicitly as the limiting value of a vector-valued surface integral around a shell enclosing p divided by the volume enclosed, as the shell is contracted indefinitely around p. More specifically, the curl may be defined by the vector formula ( ∇ × F ) ( p ) = d e f lim V → 0 1 \| V \| ∮ S n ^ × F d S {\displaystyle (\nabla \times \mathbf {F} )(p){\overset {\underset {\mathrm {def} }{}}{{}={}}}\lim _{V\to 0}{\frac {1}{\|V\|}}\oint _{S}\mathbf {\hat {n}} \times \mathbf {F} \ \mathrm {d} S} where the surface integral is calculated along the boundary S of the volume V, \|V\| being the magnitude of the volume, and n ^ {\displaystyle \mathbf {\hat {n}} } pointing outward from the surface S perpendicularly at every point in S. In this formula, the cross product in the integrand measures the tangential component of F at each point on the surface S, and points along the surface at right angles to the tangential projection of F. Integrating this cross product over the whole surface results in a vector whose magnitude measures the overall circulation of F around S, and whose direction is at right angles to this circulation. The above formula says that the curl of a vector field at a point is the infinitesimal volume density of this "circulation vector" around the point. To this definition fits naturally another global formula (similar to the Kelvin-Stokes theorem) which equates the volume integral of the curl of a vector field to the above surface integral taken over the boundary of the volume. Whereas the above two definitions of the curl are coordinate free, there is another "easy to memorize" definition of the curl in curvilinear orthogonal coordinates, e.g. in Cartesian coordinates, spherical, cylindrical, or even elliptical or parabolic coordinates: ( curl ⁡ F ) 1 = 1 h 2 h 3 ( ∂ ( h 3 F 3 ) ∂ u 2 − ∂ ( h 2 F 2 ) ∂ u 3 ) , ( curl ⁡ F ) 2 = 1 h 3 h 1 ( ∂ ( h 1 F 1 ) ∂ u 3 − ∂ ( h 3 F 3 ) ∂ u 1 ) , ( curl ⁡ F ) 3 = 1 h 1 h 2 ( ∂ ( h 2 F 2 ) ∂ u 1 − ∂ ( h 1 F 1 ) ∂ u 2 ) . {\displaystyle {\begin{aligned}&(\operatorname {curl} \mathbf {F} )_{1}={\frac {1}{h_{2}h_{3}}}\left({\frac {\partial (h_{3}F_{3})}{\partial u_{2}}}-{\frac {\partial (h_{2}F_{2})}{\partial u_{3}}}\right),\\[5pt]&(\operatorname {curl} \mathbf {F} )_{2}={\frac {1}{h_{3}h_{1}}}\left({\frac {\partial (h_{1}F_{1})}{\partial u_{3}}}-{\frac {\partial (h_{3}F_{3})}{\partial u_{1}}}\right),\\[5pt]&(\operatorname {curl} \mathbf {F} )_{3}={\frac {1}{h_{1}h_{2}}}\left({\frac {\partial (h_{2}F_{2})}{\partial u_{1}}}-{\frac {\partial (h_{1}F_{1})}{\partial u_{2}}}\right).\end{aligned}}} The equation for each component (curl F)k can be obtained by exchanging each occurrence of a subscript 1, 2, 3 in cyclic permutation: 1 → 2, 2 → 3, and 3 → 1 (where the subscripts represent the relevant indices). If (x1, x2, x3) are the Cartesian coordinates and (u1, u2, u3) are the orthogonal coordinates, then h i = ( ∂ x 1 ∂ u i ) 2 + ( ∂ x 2 ∂ u i ) 2 + ( ∂ x 3 ∂ u i ) 2 {\displaystyle h_{i}={\sqrt {\left({\frac {\partial x_{1}}{\partial u_{i}}}\right)^{2}+\left({\frac {\partial x_{2}}{\partial u_{i}}}\right)^{2}+\left({\frac {\partial x_{3}}{\partial u_{i}}}\right)^{2}}}} is the length of the coordinate vector corresponding to ui. The remaining two components of curl result from cyclic permutation of indices: 3,1,2 → 1,2,3 → 2,3,1. == Usage == In practice, the two coordinate-free definitions described above are rarely used because in virtually all cases, the curl operator can be applied using some set of curvilinear coordinates, for which simpler representations have been derived. The notation ∇ × F {\displaystyle \nabla \times \mathbf {F} } has its origins in the similarities to the 3-dimensional cross product, and it is useful as a mnemonic in Cartesian coordinates if ∇ {\displaystyle \nabla } is taken as a vector differential operator del. Such notation involving operators is common in physics and algebra. Expanded in 3-dimensional Cartesian coordinates (see Del in cylindrical and spherical coordinates for spherical and cylindrical coordinate representations), ∇ × F {\displaystyle \nabla \times \mathbf {F} } is, for F {\displaystyle \mathbf {F} } composed of [ F x , F y , F z ] {\displaystyle [F_{x},F_{y},F_{z}]} (where the subscripts indicate the components of the vector, not partial derivatives): ∇ × F = \| ı ^ ȷ ^ k ^ ∂ ∂ x ∂ ∂ y ∂ ∂ z F x F y F z \| {\displaystyle \nabla \times \mathbf {F} ={\begin{vmatrix}{\boldsymbol {\hat {\imath }}}&{\boldsymbol {\hat {\jmath }}}&{\boldsymbol {\hat {k}}}\\[5mu]{\dfrac {\partial }{\partial x}}&{\dfrac {\partial }{\partial y}}&{\dfrac {\partial }{\partial z}}\\[5mu]F_{x}&F_{y}&F_{z}\end{vmatrix}}} where i, j, and k are the unit vectors for the x-, y-, and z-axes, respectively. This expands as follows: ∇ × F = ( ∂ F z ∂ y − ∂ F y ∂ z ) ı ^ + ( ∂ F x ∂ z − ∂ F z ∂ x ) ȷ ^ + ( ∂ F y ∂ x − ∂ F x ∂ y ) k ^ {\displaystyle \nabla \times \mathbf {F} =\left({\frac {\partial F_{z}}{\partial y}}-{\frac {\partial F_{y}}{\partial z}}\right){\boldsymbol {\hat {\imath }}}+\left({\frac {\partial F_{x}}{\partial z}}-{\frac {\partial F_{z}}{\partial x}}\right){\boldsymbol {\hat {\jmath }}}+\left({\frac {\partial F_{y}}{\partial x}}-{\frac {\partial F_{x}}{\partial y}}\right){\boldsymbol {\hat {k}}}} Although expressed in terms of coordinates, the result is invariant under proper rotations of the coordinate axes but the result inverts under reflection. In a general coordinate system, the curl is given by ( ∇ × F ) k = 1 g ε k ℓ m ∇ ℓ F m {\displaystyle (\nabla \times \mathbf {F} )^{k}={\frac {1}{\sqrt {g}}}\varepsilon ^{k\ell m}\nabla _{\ell }F_{m}} where ε denotes the Levi-Civita tensor, ∇ the covariant derivative, g {\displaystyle g} is the determinant of the metric tensor and the Einstein summation convention implies that repeated indices are summed over. Due to the symmetry of the Christoffel symbols participating in the covariant derivative, this expression reduces to the partial derivative: ( ∇ × F ) = 1 g R k ε k ℓ m ∂ ℓ F m {\displaystyle (\nabla \times \mathbf {F} )={\frac {1}{\sqrt {g}}}\mathbf {R} _{k}\varepsilon ^{k\ell m}\partial _{\ell }F_{m}} where Rk are the local basis vectors. Equivalently, using the exterior derivative, the curl can be expressed as: ∇ × F = ( ⋆ ( d F ♭ ) ) ♯ {\displaystyle \nabla \times \mathbf {F} =\left(\star {\big (}{\mathrm {d} }\mathbf {F} ^{\flat }{\big )}\right)^{\sharp }} Here ♭ and ♯ are the musical isomorphisms, and ★ is the Hodge star operator. This formula shows how to calculate the curl of F in any coordinate system, and how to extend the curl to any oriented three-dimensional Riemannian manifold. Since this depends on a choice of orientation, curl is a chiral operation. In other words, if the orientation is reversed, then the direction of the curl is also reversed. == Examples == === Example 1 === Suppose the vector field describes the velocity field of a fluid flow (such as a large tank of liquid or gas) and a small ball is located within the fluid or gas (the center of the ball being fixed at a certain point). If the ball has a rough surface, the fluid flowing past it will make it rotate. The rotation axis (oriented according to the right hand rule) points in the direction of the curl of the field at the center of the ball, and the angular speed of the rotation is half the magnitude of the curl at this point. The curl of the vector field at any point is given by the rotation of an infinitesimal area in the xy-plane (for z-axis component of the curl), zx-plane (for y-axis component of the curl) and yz-plane (for x-axis component of the curl vector). This can be seen in the examples below. === Example 2 === The vector field F ( x , y , z ) = y ı ^ − x ȷ ^ {\displaystyle \mathbf {F} (x,y,z)=y{\boldsymbol {\hat {\imath }}}-x{\boldsymbol {\hat {\jmath }}}} can be decomposed as F x = y , F y = − x , F z = 0. {\displaystyle F_{x}=y,F_{y}=-x,F_{z}=0.} Upon visual inspection, the field can be described as "rotating". If the vectors of the field were to represent a linear force acting on objects present at that point, and an object were to be placed inside the field, the object would start to rotate clockwise around itself. This is true regardless of where the object is placed. Calculating the curl: ∇ × F = 0 ı ^ + 0 ȷ ^ + ( ∂ ∂ x ( − x ) − ∂ ∂ y y ) k ^ = − 2 k ^ {\displaystyle \nabla \times \mathbf {F} =0{\boldsymbol {\hat {\imath }}}+0{\boldsymbol {\hat {\jmath }}}+\left({\frac {\partial }{\partial x}}(-x)-{\frac {\partial }{\partial y}}y\right){\boldsymbol {\hat {k}}}=-2{\boldsymbol {\hat {k}}}} The resulting vector field describing the curl would at all points be pointing in the negative z direction. The results of this equation align with what could have been predicted using the right-hand rule using a right-handed coordinate system. Being a uniform vector field, the object described before would have the same rotational intensity regardless of where it was placed. === Example 3 === For the vector field F ( x , y , z ) = − x 2 ȷ ^ {\displaystyle \mathbf {F} (x,y,z)=-x^{2}{\boldsymbol {\hat {\jmath }}}} the curl is not as obvious from the graph. However, taking the object in the previous example, and placing it anywhere on the line x = 3, the force exerted on the right side would be slightly greater than the force exerted on the left, causing it to rotate clockwise. Using the right-hand rule, it can be predicted that the resulting curl would be straight in the negative z direction. Inversely, if placed on x = −3, the object would rotate counterclockwise and the right-hand rule would result in a positive z direction. Calculating the curl: ∇ × F = 0 ı ^ + 0 ȷ ^ + ∂ ∂ x ( − x 2 ) k ^ = − 2 x k ^ . {\displaystyle {\nabla }\times \mathbf {F} =0{\boldsymbol {\hat {\imath }}}+0{\boldsymbol {\hat {\jmath }}}+{\frac {\partial }{\partial x}}\left(-x^{2}\right){\boldsymbol {\hat {k}}}=-2x{\boldsymbol {\hat {k}}}.} The curl points in the negative z direction when x is positive and vice versa. In this field, the intensity of rotation would be greater as the object moves away from the plane x = 0. === Further examples === In a vector field describing the linear velocities of each part of a rotating disk in uniform circular motion, the curl has the same value at all points, and this value turns out to be exactly two times the vectorial angular velocity of the disk (oriented as usual by the right-hand rule). More generally, for any flowing mass, the linear velocity vector field at each point of the mass flow has a curl (the vorticity of the flow at that point) equal to exactly two times the local vectorial angular velocity of the mass about the point. For any solid object subject to an external physical force (such as gravity or the electromagnetic force), one may consider the vector field representing the infinitesimal force-per-unit-volume contributions acting at each of the points of the object. This force field may create a net torque on the object about its center of mass, and this torque turns out to be directly proportional and vectorially parallel to the (vector-valued) integral of the curl of the force field over the whole volume. Of the four Maxwell's equations, two—Faraday's law and Ampère's law—can be compactly expressed using curl. Faraday's law states that the curl of an electric field is equal to the opposite of the time rate of change of the magnetic field, while Ampère's law relates the curl of the magnetic field to the current and the time rate of change of the electric field. == Identities == In general curvilinear coordinates (not only in Cartesian coordinates), the curl of a cross product of vector fields v and F can be shown to be ∇ × ( v × F ) = ( ( ∇ ⋅ F ) + F ⋅ ∇ ) v − ( ( ∇ ⋅ v ) + v ⋅ ∇ ) F . {\displaystyle \nabla \times \left(\mathbf {v\times F} \right)={\Big (}\left(\mathbf {\nabla \cdot F} \right)+\mathbf {F\cdot \nabla } {\Big )}\mathbf {v} -{\Big (}\left(\mathbf {\nabla \cdot v} \right)+\mathbf {v\cdot \nabla } {\Big )}\mathbf {F} \ .} Interchanging the vector field v and ∇ operator, we arrive at the cross product of a vector field with curl of a vector field: v × ( ∇ × F ) = ∇ F ( v ⋅ F ) − ( v ⋅ ∇ ) F , {\displaystyle \mathbf {v\ \times } \left(\mathbf {\nabla \times F} \right)=\nabla _{\mathbf {F} }\left(\mathbf {v\cdot F} \right)-\left(\mathbf {v\cdot \nabla } \right)\mathbf {F} \ ,} where ∇F is the Feynman subscript notation, which considers only the variation due to the vector field F (i.e., in this case, v is treated as being constant in space). Another example is the curl of a curl of a vector field. It can be shown that in general coordinates ∇ × ( ∇ × F ) = ∇ ( ∇ ⋅ F ) − ∇ 2 F , {\displaystyle \nabla \times \left(\mathbf {\nabla \times F} \right)=\mathbf {\nabla } (\mathbf {\nabla \cdot F} )-\nabla ^{2}\mathbf {F} \ ,} and this identity defines the vector Laplacian of F, symbolized as ∇2F. The curl of the gradient of any scalar field φ is always the zero vector field ∇ × ( ∇ φ ) = 0 {\displaystyle \nabla \times (\nabla \varphi )={\boldsymbol {0}}} which follows from the antisymmetry in the definition of the curl, and the symmetry of second derivatives. The divergence of the curl of any vector field is equal to zero: ∇ ⋅ ( ∇ × F ) = 0. {\displaystyle \nabla \cdot (\nabla \times \mathbf {F} )=0.} If φ is a scalar valued function and F is a vector field, then ∇ × ( φ F ) = ∇ φ × F + φ ∇ × F {\displaystyle \nabla \times (\varphi \mathbf {F} )=\nabla \varphi \times \mathbf {F} +\varphi \nabla \times \mathbf {F} } == Generalizations == The vector calculus operations of grad, curl, and div are most easily generalized in the context of differential forms, which involves a number of steps. In short, they correspond to the derivatives of 0-forms, 1-forms, and 2-forms, respectively. The geometric interpretation of curl as rotation corresponds to identifying bivectors (2-vectors) in 3 dimensions with the special orthogonal Lie algebra s o ( 3 ) {\displaystyle {\mathfrak {so}}(3)} of infinitesimal rotations (in coordinates, skew-symmetric 3 × 3 matrices), while representing rotations by vectors corresponds to identifying 1-vectors (equivalently, 2-vectors) and s o ( 3 ) {\displaystyle {\mathfrak {so}}(3)} , these all being 3-dimensional spaces. === Differential forms === In 3 dimensions, a differential 0-form is a real-valued function f ( x , y , z ) {\displaystyle f(x,y,z)} ; a differential 1-form is the following expression, where the coefficients are functions: a 1 d x + a 2 d y + a 3 d z ; {\displaystyle a_{1}\,dx+a_{2}\,dy+a_{3}\,dz;} a differential 2-form is the formal sum, again with function coefficients: a 12 d x ∧ d y + a 13 d x ∧ d z + a 23 d y ∧ d z ; {\displaystyle a_{12}\,dx\wedge dy+a_{13}\,dx\wedge dz+a_{23}\,dy\wedge dz;} and a differential 3-form is defined by a single term with one function as coefficient: a 123 d x ∧ d y ∧ d z . {\displaystyle a_{123}\,dx\wedge dy\wedge dz.} (Here the a-coefficients are real functions of three variables; the wedge products, e.g. d x ∧ d y {\displaystyle {\text{d}}x\wedge {\text{d}}y} , can be interpreted as oriented plane segments, d x ∧ d y = − d y ∧ d x {\displaystyle {\text{d}}x\wedge {\text{d}}y=-{\text{d}}y\wedge {\text{d}}x} , etc.) The exterior derivative of a k-form in R3 is defined as the (k + 1)-form from above—and in Rn if, e.g., ω ( k ) = ∑ 1 ≤ i 1 < i 2 < ⋯ < i k ≤ n a i 1 , … , i k d x i 1 ∧ ⋯ ∧ d x i k , {\displaystyle \omega ^{(k)}=\sum _{1\leq i_{1}<i_{2}<\cdots <i_{k}\leq n}a_{i_{1},\ldots ,i_{k}}\,dx_{i_{1}}\wedge \cdots \wedge dx_{i_{k}},} then the exterior derivative d leads to d ω ( k ) = ∑ j = 1 i 1 < ⋯ < i k n ∂ a i 1 , … , i k ∂ x j d x j ∧ d x i 1 ∧ ⋯ ∧ d x i k . {\displaystyle d\omega ^{(k)}=\sum _{\scriptstyle {j=1} \atop \scriptstyle {i_{1}<\cdots <i_{k}}}^{n}{\frac {\partial a_{i_{1},\ldots ,i_{k}}}{\partial x_{j}}}\,dx_{j}\wedge dx_{i_{1}}\wedge \cdots \wedge dx_{i_{k}}.} The exterior derivative of a 1-form is therefore a 2-form, and that of a 2-form is a 3-form. On the other hand, because of the interchangeability of mixed derivatives, ∂ 2 ∂ x i ∂ x j = ∂ 2 ∂ x j ∂ x i , {\displaystyle {\frac {\partial ^{2}}{\partial x_{i}\,\partial x_{j}}}={\frac {\partial ^{2}}{\partial x_{j}\,\partial x_{i}}},} and antisymmetry, d x i ∧ d x j = − d x j ∧ d x i {\displaystyle dx_{i}\wedge dx_{j}=-dx_{j}\wedge dx_{i}} the twofold application of the exterior derivative yields 0 {\displaystyle 0} (the zero k + 2 {\displaystyle k+2} -form). Thus, denoting the space of k-forms by Ω k ( R 3 ) {\displaystyle \Omega ^{k}(\mathbb {R} ^{3})} and the exterior derivative by d one gets a sequence: 0 ⟶ d Ω 0 ( R 3 ) ⟶ d Ω 1 ( R 3 ) ⟶ d Ω 2 ( R 3 ) ⟶ d Ω 3 ( R 3 ) ⟶ d 0. {\displaystyle 0\,{\overset {d}{\longrightarrow }}\;\Omega ^{0}\left(\mathbb {R} ^{3}\right)\,{\overset {d}{\longrightarrow }}\;\Omega ^{1}\left(\mathbb {R} ^{3}\right)\,{\overset {d}{\longrightarrow }}\;\Omega ^{2}\left(\mathbb {R} ^{3}\right)\,{\overset {d}{\longrightarrow }}\;\Omega ^{3}\left(\mathbb {R} ^{3}\right)\,{\overset {d}{\longrightarrow }}\,0.} Here Ω k ( R n ) {\displaystyle \Omega ^{k}(\mathbb {R} ^{n})} is the space of sections of the exterior algebra Λ k ( R n ) {\displaystyle \Lambda ^{k}(\mathbb {R} ^{n})} vector bundle over Rn, whose dimension is the binomial coefficient ( n k ) {\displaystyle {\binom {n}{k}}} ; note that Ω k ( R 3 ) = 0 {\displaystyle \Omega ^{k}(\mathbb {R} ^{3})=0} for k > 3 {\displaystyle k>3} or k < 0 {\displaystyle k<0} . Writing only dimensions, one obtains a row of Pascal's triangle: 0 → 1 → 3 → 3 → 1 → 0 ; {\displaystyle 0\rightarrow 1\rightarrow 3\rightarrow 3\rightarrow 1\rightarrow 0;} the 1-dimensional fibers correspond to scalar fields, and the 3-dimensional fibers to vector fields, as described below. Modulo suitable identifications, the three nontrivial occurrences of the exterior derivative correspond to grad, curl, and div. Differential forms and the differential can be defined on any Euclidean space, or indeed any manifold, without any notion of a Riemannian metric. On a Riemannian manifold, or more generally pseudo-Riemannian manifold, k-forms can be identified with k-vector fields (k-forms are k-covector fields, and a pseudo-Riemannian metric gives an isomorphism between vectors and covectors), and on an oriented vector space with a nondegenerate form (an isomorphism between vectors and covectors), there is an isomorphism between k-vectors and (n − k)-vectors; in particular on (the tangent space of) an oriented pseudo-Riemannian manifold. Thus on an oriented pseudo-Riemannian manifold, one can interchange k-forms, k-vector fields, (n − k)-forms, and (n − k)-vector fields; this is known as Hodge duality. Concretely, on R3 this is given by: 1-forms and 1-vector fields: the 1-form ax dx + ay dy + az dz corresponds to the vector field (ax, ay, az). 1-forms and 2-forms: one replaces dx by the dual quantity dy ∧ dz (i.e., omit dx), and likewise, taking care of orientation: dy corresponds to dz ∧ dx = −dx ∧ dz, and dz corresponds to dx ∧ dy. Thus the form ax dx + ay dy + az dz corresponds to the "dual form" az dx ∧ dy + ay dz ∧ dx + ax dy ∧ dz. Thus, identifying 0-forms and 3-forms with scalar fields, and 1-forms and 2-forms with vector fields: grad takes a scalar field (0-form) to a vector field (1-form); curl takes a vector field (1-form) to a pseudovector field (2-form); div takes a pseudovector field (2-form) to a pseudoscalar field (3-form) On the other hand, the fact that d2 = 0 corresponds to the identities ∇ × ( ∇ f ) = 0 {\displaystyle \nabla \times (\nabla f)=\mathbf {0} } for any scalar field f, and ∇ ⋅ ( ∇ × v ) = 0 {\displaystyle \nabla \cdot (\nabla \times \mathbf {v} )=0} for any vector field v. Grad and div generalize to all oriented pseudo-Riemannian manifolds, with the same geometric interpretation, because the spaces of 0-forms and n-forms at each point are always 1-dimensional and can be identified with scalar fields, while the spaces of 1-forms and (n − 1)-forms are always fiberwise n-dimensional and can be identified with vector fields. Curl does not generalize in this way to 4 or more dimensions (or down to 2 or fewer dimensions); in 4 dimensions the dimensions are so the curl of a 1-vector field (fiberwise 4-dimensional) is a 2-vector field, which at each point belongs to 6-dimensional vector space, and so one has ω ( 2 ) = ∑ i < k = 1 , 2 , 3 , 4 a i , k d x i ∧ d x k , {\displaystyle \omega ^{(2)}=\sum _{i<k=1,2,3,4}a_{i,k}\,dx_{i}\wedge dx_{k},} which yields a sum of six independent terms, and cannot be identified with a 1-vector field. Nor can one meaningfully go from a 1-vector field to a 2-vector field to a 3-vector field (4 → 6 → 4), as taking the differential twice yields zero (d2 = 0). Thus there is no curl function from vector fields to vector fields in other dimensions arising in this way. However, one can define a curl of a vector field as a 2-vector field in general, as described below. === Curl geometrically === 2-vectors correspond to the exterior power Λ2V; in the presence of an inner product, in coordinates these are the skew-symmetric matrices, which are geometrically considered as the special orthogonal Lie algebra s o {\displaystyle {\mathfrak {so}}} (V) of infinitesimal rotations. This has (n2) = ⁠1/2⁠n(n − 1) dimensions, and allows one to interpret the differential of a 1-vector field as its infinitesimal rotations. Only in 3 dimensions (or trivially in 0 dimensions) we have n = ⁠1/2⁠n(n − 1), which is the most elegant and common case. In 2 dimensions the curl of a vector field is not a vector field but a function, as 2-dimensional rotations are given by an angle (a scalar – an orientation is required to choose whether one counts clockwise or counterclockwise rotations as positive); this is not the div, but is rather perpendicular to it. In 3 dimensions the curl of a vector field is a vector field as is familiar (in 1 and 0 dimensions the curl of a vector field is 0, because there are no non-trivial 2-vectors), while in 4 dimensions the curl of a vector field is, geometrically, at each point an element of the 6-dimensional Lie algebra s o ( 4 ) {\displaystyle {\mathfrak {so}}(4)} . The curl of a 3-dimensional vector field which only depends on 2 coordinates (say x and y) is simply a vertical vector field (in the z direction) whose magnitude is the curl of the 2-dimensional vector field, as in the examples on this page. Considering curl as a 2-vector field (an antisymmetric 2-tensor) has been used to generalize vector calculus and associated physics to higher dimensions. == Inverse == In the case where the divergence of a vector field V is zero, a vector field W exists such that V = curl(W). This is why the magnetic field, characterized by zero divergence, can be expressed as the curl of a magnetic vector potential. If W is a vector field with curl(W) = V, then adding any gradient vector field grad(f) to W will result in another vector field W + grad(f) such that curl(W + grad(f)) = V as well. This can be summarized by saying that the inverse curl of a three-dimensional vector field can be obtained up to an unknown irrotational field with the Biot–Savart law. == See also == Helmholtz decomposition Hiptmair–Xu preconditioner Del in cylindrical and spherical coordinates Vorticity == References == == Further reading == Korn, Granino Arthur and Theresa M. Korn (January 2000). Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review. New York: Dover Publications. pp. 157–160. ISBN 0-486-41147-8. Schey, H. M. (1997). Div, Grad, Curl, and All That: An Informal Text on Vector Calculus. New York: Norton. ISBN 0-393-96997-5. == External links == "Curl", Encyclopedia of Mathematics, EMS Press, 2001 [1994] "Multivariable calculus". mathinsight.org. Retrieved February 12, 2022. "Divergence and Curl: The Language of Maxwell's Equations, Fluid Flow, and More". June 21, 2018. Archived from the original on 2021-11-24 – via YouTube.
Wikipedia:Cycle basis#0	In graph theory, a branch of mathematics, a cycle basis of an undirected graph is a set of simple cycles that forms a basis of the cycle space of the graph. That is, it is a minimal set of cycles that allows every even-degree subgraph to be expressed as a symmetric difference of basis cycles. A fundamental cycle basis may be formed from any spanning tree or spanning forest of the given graph, by selecting the cycles formed by the combination of a path in the tree and a single edge outside the tree. Alternatively, if the edges of the graph have positive weights, the minimum weight cycle basis may be constructed in polynomial time. In planar graphs, the set of bounded cycles of an embedding of the graph forms a cycle basis. The minimum weight cycle basis of a planar graph corresponds to the Gomory–Hu tree of the dual graph. == Definitions == A spanning subgraph of a given graph G has the same set of vertices as G itself but, possibly, fewer edges. A graph G, or one of its subgraphs, is said to be Eulerian if each of its vertices has even degree (its number of incident edges). Every simple cycle in a graph is an Eulerian subgraph, but there may be others. The cycle space of a graph is the collection of its Eulerian subgraphs. It forms a vector space over the two-element finite field. The vector addition operation is the symmetric difference of two or more subgraphs, which forms another subgraph consisting of the edges that appear an odd number of times in the arguments to the symmetric difference operation. A cycle basis is a basis of this vector space in which each basis vector represents a simple cycle. It consists of a set of cycles that can be combined, using symmetric differences, to form every Eulerian subgraph, and that is minimal with this property. Every cycle basis of a given graph has the same number of cycles, which equals the dimension of its cycle space. This number is called the circuit rank of the graph, and it equals m − n + c {\displaystyle m-n+c} where m {\displaystyle m} is the number of edges in the graph, n {\displaystyle n} is the number of vertices, and c {\displaystyle c} is the number of connected components. == Special cycle bases == Several special types of cycle bases have been studied, including the fundamental cycle bases, weakly fundamental cycle bases, sparse (or 2-) cycle bases, and integral cycle bases. === Induced cycles === Every graph has a cycle basis in which every cycle is an induced cycle. In a 3-vertex-connected graph, there always exists a basis consisting of peripheral cycles, cycles whose removal does not separate the remaining graph. In any graph other than one formed by adding one edge to a cycle, a peripheral cycle must be an induced cycle. === Fundamental cycles === If T {\displaystyle T} is a spanning tree or spanning forest of a given graph G {\displaystyle G} , and e {\displaystyle e} is an edge that does not belong to T {\displaystyle T} , then the fundamental cycle C e {\displaystyle C_{e}} defined by e {\displaystyle e} is the simple cycle consisting of e {\displaystyle e} together with the path in T {\displaystyle T} connecting the endpoints of e {\displaystyle e} . There are exactly m − n + c {\displaystyle m-n+c} fundamental cycles, one for each edge that does not belong to T {\displaystyle T} . Each of them is linearly independent from the remaining cycles, because it includes an edge e {\displaystyle e} that is not present in any other fundamental cycle. Therefore, the fundamental cycles form a basis for the cycle space. A cycle basis constructed in this way is called a fundamental cycle basis or strongly fundamental cycle basis. It is also possible to characterize fundamental cycle bases without specifying the tree for which they are fundamental. There exists a tree for which a given cycle basis is fundamental if and only if each cycle contains an edge that is not included in any other basis cycle, that is, each cycle is independent of others. It follows that a collection of cycles is a fundamental cycle basis of G {\displaystyle G} if and only if it has the independence property and has the correct number of cycles to be a basis of G {\displaystyle G} . === Weakly fundamental cycles === A cycle basis is called weakly fundamental if its cycles can be placed into a linear ordering such that each cycle includes at least one edge that is not included in any earlier cycle. A fundamental cycle basis is automatically weakly fundamental (for any edge ordering). If every cycle basis of a graph is weakly fundamental, the same is true for every minor of the graph. Based on this property, the class of graphs (and multigraphs) for which every cycle basis is weakly fundamental can be characterized by five forbidden minors: the graph of the square pyramid, the multigraph formed by doubling all edges of a four-vertex cycle, two multigraphs formed by doubling two edges of a tetrahedron, and the multigraph formed by tripling the edges of a triangle. === Face cycles === If a connected finite planar graph is embedded into the plane, each face of the embedding is bounded by a cycle of edges. One face is necessarily unbounded (it includes points arbitrarily far from the vertices of the graph) and the remaining faces are bounded. By Euler's formula for planar graphs, there are exactly m − n + 1 {\displaystyle m-n+1} bounded faces. The symmetric difference of any set of face cycles is the boundary of the corresponding set of faces, and different sets of bounded faces have different boundaries, so it is not possible to represent the same set as a symmetric difference of face cycles in more than one way; this means that the set of face cycles is linearly independent. As a linearly independent set of enough cycles, it necessarily forms a cycle basis. It is always a weakly fundamental cycle basis, and is fundamental if and only if the embedding of the graph is outerplanar. For graphs properly embedded onto other surfaces so that all faces of the embedding are topological disks, it is not in general true that there exists a cycle basis using only face cycles. The face cycles of these embeddings generate a proper subset of all Eulerian subgraphs. The homology group H 2 ( S , Z 2 ) {\displaystyle H_{2}(S,\mathbb {Z} _{2})} of the given surface S {\displaystyle S} characterizes the Eulerian subgraphs that cannot be represented as the boundary of a set of faces. Mac Lane's planarity criterion uses this idea to characterize the planar graphs in terms of the cycle bases: a finite undirected graph is planar if and only if it has a sparse cycle basis or 2-basis, a basis in which each edge of the graph participates in at most two basis cycles. In a planar graph, the cycle basis formed by the set of bounded faces is necessarily sparse, and conversely, a sparse cycle basis of any graph necessarily forms the set of bounded faces of a planar embedding of its graph. === Integral bases === The cycle space of a graph may be interpreted using the theory of homology as the homology group H 1 ( G , Z 2 ) {\displaystyle H_{1}(G,\mathbb {Z} _{2})} of a simplicial complex with a point for each vertex of the graph and a line segment for each edge of the graph. This construction may be generalized to the homology group H 1 ( G , R ) {\displaystyle H_{1}(G,R)} over an arbitrary ring R {\displaystyle R} . An important special case is the ring of integers, for which the homology group H 1 ( G , Z ) {\displaystyle H_{1}(G,\mathbb {Z} )} is a free abelian group, a subgroup of the free abelian group generated by the edges of the graph. Less abstractly, this group can be constructed by assigning an arbitrary orientation to the edges of the given graph; then the elements of H 1 ( G , Z ) {\displaystyle H_{1}(G,\mathbb {Z} )} are labelings of the edges of the graph by integers with the property that, at each vertex, the sum of the incoming edge labels equals the sum of the outgoing edge labels. The group operation is addition of these vectors of labels. An integral cycle basis is a set of simple cycles that generates this group. == Minimum weight == If the edges of a graph are given real number weights, the weight of a subgraph may be computed as the sum of the weights of its edges. The minimum weight basis of the cycle space is necessarily a cycle basis: by Veblen's theorem, every Eulerian subgraph that is not itself a simple cycle can be decomposed into multiple simple cycles, which necessarily have smaller weight. By standard properties of bases in vector spaces and matroids, the minimum weight cycle basis not only minimizes the sum of the weights of its cycles, it also minimizes any other monotonic combination of the cycle weights. For instance, it is the cycle basis that minimizes the weight of its longest cycle. === Polynomial time algorithms === In any vector space, and more generally in any matroid, a minimum weight basis may be found by a greedy algorithm that considers potential basis elements one at a time, in sorted order by their weights, and that includes an element in the basis when it is linearly independent of the previously chosen basis elements. Testing for linear independence can be done by Gaussian elimination. However, an undirected graph may have an exponentially large set of simple cycles, so it would be computationally infeasible to generate and test all such cycles. Horton (1987) provided the first polynomial time algorithm for finding a minimum weight basis, in graphs for which every edge weight is positive. His algorithm uses this generate-and-test approach, but restricts the generated cycles to a small set of O ( m n ) {\displaystyle O(mn)} cycles, called Horton cycles. A Horton cycle is a fundamental cycle of a shortest path tree of the given graph. There are at most n different shortest path trees (one for each starting vertex) and each has fewer than m fundamental cycles, giving the bound on the total number of Horton cycles. As Horton showed, every cycle in the minimum weight cycle basis is a Horton cycle. Using Dijkstra's algorithm to find each shortest path tree and then using Gaussian elimination to perform the testing steps of the greedy basis algorithm leads to a polynomial time algorithm for the minimum weight cycle basis. Subsequent researchers have developed improved algorithms for this problem, reducing the worst-case time complexity for finding a minimum weight cycle basis in a graph with m {\displaystyle m} edges and n {\displaystyle n} vertices to O ( m 2 n / log ⁡ n ) {\displaystyle O(m^{2}n/\log n)} . === NP-hardness === Finding the fundamental basis with the minimum possible weight is closely related to the problem of finding a spanning tree that minimizes the average of the pairwise distances; both are NP-hard. Finding a minimum weight weakly fundamental basis is also NP-hard, and approximating it is MAXSNP-hard. If negative weights and negatively weighted cycles are allowed, then finding a minimum cycle basis (without restriction) is also NP-hard, as it can be used to find a Hamiltonian cycle: if a graph is Hamiltonian, and all edges are given weight −1, then a minimum weight cycle basis necessarily includes at least one Hamiltonian cycle. === In planar graphs === The minimum weight cycle basis for a planar graph is not necessarily the same as the basis formed by its bounded faces: it can include cycles that are not faces, and some faces may not be included as cycles in the minimum weight cycle basis. However, there exists a minimum weight cycle basis in which no two cycles cross each other: for every two cycles in the basis, either the cycles enclose disjoint subsets of the bounded faces, or one of the two cycles encloses the other one. This set of cycles corresponds, in the dual graph of the given planar graph, to a set of cuts that form a Gomory–Hu tree of the dual graph, the minimum weight basis of its cut space. Based on this duality, an implicit representation of the minimum weight cycle basis in a planar graph can be constructed in time O ( n log 3 ⁡ n ) {\displaystyle O(n\log ^{3}n)} . == Applications == Cycle bases have been used for solving periodic scheduling problems, such as the problem of determining the schedule for a public transportation system. In this application, the cycles of a cycle basis correspond to variables in an integer program for solving the problem. In the theory of structural rigidity and kinematics, cycle bases are used to guide the process of setting up a system of non-redundant equations that can be solved to predict the rigidity or motion of a structure. In this application, minimum or near-minimum weight cycle bases lead to simpler systems of equations. In distributed computing, cycle bases have been used to analyze the number of steps needed for an algorithm to stabilize. In bioinformatics, cycle bases have been used to determine haplotype information from genome sequence data. Cycle bases have also been used to analyze the tertiary structure of RNA. The minimum weight cycle basis of a nearest neighbor graph of points sampled from a three-dimensional surface can be used to obtain a reconstruction of the surface. In cheminformatics, the minimal cycle basis of a molecular graph is referred to as the smallest set of smallest rings. == References ==
Wikipedia:Cycle detection#0	In computer science, cycle detection or cycle finding is the algorithmic problem of finding a cycle in a sequence of iterated function values. For any function f that maps a finite set S to itself, and any initial value x0 in S, the sequence of iterated function values x 0 , x 1 = f ( x 0 ) , x 2 = f ( x 1 ) , … , x i = f ( x i − 1 ) , … {\displaystyle x_{0},\ x_{1}=f(x_{0}),\ x_{2}=f(x_{1}),\ \dots ,\ x_{i}=f(x_{i-1}),\ \dots } must eventually use the same value twice: there must be some pair of distinct indices i and j such that xi = xj. Once this happens, the sequence must continue periodically, by repeating the same sequence of values from xi to xj − 1. Cycle detection is the problem of finding i and j, given f and x0. Several algorithms are known for finding cycles quickly and with little memory. Robert W. Floyd's tortoise and hare algorithm moves two pointers at different speeds through the sequence of values until they both point to equal values. Alternatively, Brent's algorithm is based on the idea of exponential search. Both Floyd's and Brent's algorithms use only a constant number of memory cells, and take a number of function evaluations that is proportional to the distance from the start of the sequence to the first repetition. Several other algorithms trade off larger amounts of memory for fewer function evaluations. The applications of cycle detection include testing the quality of pseudorandom number generators and cryptographic hash functions, computational number theory algorithms, detection of infinite loops in computer programs and periodic configurations in cellular automata, automated shape analysis of linked list data structures, and detection of deadlocks for transactions management in DBMS. == Example == The figure shows a function f that maps the set S = {0,1,2,3,4,5,6,7,8} to itself. If one starts from x0 = 2 and repeatedly applies f, one sees the sequence of values 2, 0, 6, 3, 1, 6, 3, 1, 6, 3, 1, .... The cycle in this value sequence is 6, 3, 1. == Definitions == Let S be any finite set, f be any endofunction from S to itself, and x0 be any element of S. For any i > 0, let xi = f(xi − 1). Let μ be the smallest index such that the value xμ reappears infinitely often within the sequence of values xi, and let λ (the loop length) be the smallest positive integer such that xμ = xλ + μ. The cycle detection problem is the task of finding λ and μ. One can view the same problem graph-theoretically, by constructing a functional graph (that is, a directed graph in which each vertex has a single outgoing edge) the vertices of which are the elements of S and the edges of which map an element to the corresponding function value, as shown in the figure. The set of vertices reachable from starting vertex x0 form a subgraph with a shape resembling the Greek letter rho (ρ): a path of length μ from x0 to a cycle of λ vertices. Practical cycle-detection algorithms do not find λ and μ exactly. They usually find lower and upper bounds μl ≤ μ ≤ μh for the start of the cycle, and a more detailed search of the range must be performed if the exact value of μ is needed. Also, most algorithms do not guarantee to find λ directly, but may find some multiple kλ < μ + λ. (Continuing the search for an additional kλ/q steps, where q is the smallest prime divisor of kλ, will either find the true λ or prove that k = 1.) == Computer representation == Except in toy examples like the above, f will not be specified as a table of values. Such a table implies O(\|S\|) space complexity, and if that is permissible, an associative array mapping xi to i will detect the first repeated value. Rather, a cycle detection algorithm is given a black box for generating the sequence xi, and the task is to find λ and μ using very little memory. The black box might consist of an implementation of the recurrence function f, but it might also store additional internal state to make the computation more efficient. Although xi = f(xi−1) must be true in principle, this might be expensive to compute directly; the function could be defined in terms of the discrete logarithm of xi−1 or some other difficult-to-compute property which can only be practically computed in terms of additional information. In such cases, the number of black boxes required becomes a figure of merit distinguishing the algorithms. A second reason to use one of these algorithms is that they are pointer algorithms which do no operations on elements of S other than testing for equality. An associative array implementation requires computing a hash function on the elements of S, or ordering them. But cycle detection can be applied in cases where neither of these are possible. The classic example is Pollard's rho algorithm for integer factorization, which searches for a factor p of a given number n by looking for values xi and xi+λ which are equal modulo p without knowing p in advance. This is done by computing the greatest common divisor of the difference xi − xi+λ with a known multiple of p, namely n. If the gcd is non-trivial (neither 1 nor n), then the value is a proper factor of n, as desired. If n is not prime, it must have at least one factor p ≤ √n, and by the birthday paradox, a random function f has an expected cycle length (modulo p) of √p ≤ 4√n. == Algorithms == If the input is given as a subroutine for calculating f, the cycle detection problem may be trivially solved using only λ + μ function applications, simply by computing the sequence of values xi and using a data structure such as a hash table to store these values and test whether each subsequent value has already been stored. However, the space complexity of this algorithm is proportional to λ + μ, unnecessarily large. Additionally, to implement this method as a pointer algorithm would require applying the equality test to each pair of values, resulting in quadratic time overall. Thus, research in this area has concentrated on two goals: using less space than this naive algorithm, and finding pointer algorithms that use fewer equality tests. === Floyd's tortoise and hare === Floyd's cycle-finding algorithm is a pointer algorithm that uses only two pointers, which move through the sequence at different speeds. It is also called the "tortoise and the hare algorithm", alluding to Aesop's fable of The Tortoise and the Hare. The algorithm is named after Robert W. Floyd, who was credited with its invention by Donald Knuth. However, the algorithm does not appear in Floyd's published work, and this may be a misattribution: Floyd describes algorithms for listing all simple cycles in a directed graph in a 1967 paper, but this paper does not describe the cycle-finding problem in functional graphs that is the subject of this article. In fact, Knuth's statement (in 1969), attributing it to Floyd, without citation, is the first known appearance in print, and it thus may be a folk theorem, not attributable to a single individual. The key insight in the algorithm is as follows. If there is a cycle, then, for any integers i ≥ μ and k ≥ 0, xi = xi + kλ, where λ is the length of the loop to be found, μ is the index of the first element of the cycle, and k is a whole integer representing the number of loops. Based on this, it can then be shown that i = kλ ≥ μ for some k if and only if xi = x2i (if xi = x2i in the cycle, then there exists some k such that 2i = i + kλ, which implies that i = kλ; and if there are some i and k such that i = kλ, then 2i = i + kλ and x2i = xi + kλ). Thus, the algorithm only needs to check for repeated values of this special form, one twice as far from the start of the sequence as the other, to find a period ν of a repetition that is a multiple of λ. Once ν is found, the algorithm retraces the sequence from its start to find the first repeated value xμ in the sequence, using the fact that λ divides ν and therefore that xμ = xμ + v. Finally, once the value of μ is known it is trivial to find the length λ of the shortest repeating cycle, by searching for the first position μ + λ for which xμ + λ = xμ. The algorithm thus maintains two pointers into the given sequence, one (the tortoise) at xi, and the other (the hare) at x2i. At each step of the algorithm, it increases i by one, moving the tortoise one step forward and the hare two steps forward in the sequence, and then compares the sequence values at these two pointers. The smallest value of i > 0 for which the tortoise and hare point to equal values is the desired value ν. The following Python code shows how this idea may be implemented as an algorithm. This code only accesses the sequence by storing and copying pointers, function evaluations, and equality tests; therefore, it qualifies as a pointer algorithm. The algorithm uses O(λ + μ) operations of these types, and O(1) storage space. === Brent's algorithm === Richard P. Brent described an alternative cycle detection algorithm that, like the tortoise and hare algorithm, requires only two pointers into the sequence. However, it is based on a different principle: searching for the smallest power of two 2i that is larger than both λ and μ. For i = 0, 1, 2, ..., the algorithm compares x2i−1 with each subsequent sequence value up to the next power of two, stopping when it finds a match. It has two advantages compared to the tortoise and hare algorithm: it finds the correct length λ of the cycle directly, rather than needing to search for it in a subsequent stage, and its steps involve only one evaluation of the function f rather than three. The following Python code shows how this technique works in more detail. Like the tortoise and hare algorithm, this is a pointer algorithm that uses O(λ + μ) tests and function evaluations and O(1) storage space. It is not difficult to show that the number of function evaluations can never be higher than for Floyd's algorithm. Brent claims that, on average, his cycle finding algorithm runs around 36% more quickly than Floyd's and that it speeds up the Pollard rho algorithm by around 24%. He also performs an average case analysis for a randomized version of the algorithm in which the sequence of indices traced by the slower of the two pointers is not the powers of two themselves, but rather a randomized multiple of the powers of two. Although his main intended application was in integer factorization algorithms, Brent also discusses applications in testing pseudorandom number generators. === Gosper's algorithm === R. W. Gosper's algorithm finds the period λ {\displaystyle \lambda } , and the lower and upper bound of the starting point, μ l {\displaystyle \mu _{l}} and μ u {\displaystyle \mu _{u}} , of the first cycle. The difference between the lower and upper bound is of the same order as the period, i.e. μ l + λ ≈ μ h {\displaystyle \mu _{l}+\lambda \approx \mu _{h}} . The algorithm maintains an array of tortoises T j {\displaystyle T_{j}} . For each x i {\displaystyle x_{i}} : For each 0 ≤ j ≤ log 2 ⁡ i , {\displaystyle 0\leq j\leq \log _{2}i,} compare x i {\displaystyle x_{i}} to T j {\displaystyle T_{j}} . If x i = T j {\displaystyle x_{i}=T_{j}} , a cycle has been detected, of length λ = ( i − 2 j ) mod 2 j + 1 + 1. {\displaystyle \lambda =(i-2^{j}){\bmod {2}}^{j+1}+1.} If no match is found, set T k ← x i {\displaystyle T_{k}\leftarrow x_{i}} , where k {\displaystyle k} is the number of trailing zeros in the binary representation of i + 1 {\displaystyle i+1} . I.e. the greatest power of 2 which divides i + 1 {\displaystyle i+1} . If it is inconvenient to vary the number of comparisons as i {\displaystyle i} increases, you may initialize all of the T j = x 0 {\displaystyle T_{j}=x_{0}} , but must then return λ = i {\displaystyle \lambda =i} if x i = T j {\displaystyle x_{i}=T_{j}} while i < 2 j {\displaystyle i<2^{j}} . ==== Advantages ==== The main features of Gosper's algorithm are that it is economical in space, very economical in evaluations of the generator function, and always finds the exact cycle length (never a multiple). The cost is a large number of equality comparisons. It could be roughly described as a concurrent version of Brent's algorithm. While Brent's algorithm uses a single tortoise, repositioned every time the hare passes a power of two, Gosper's algorithm uses several tortoises (several previous values are saved), which are roughly exponentially spaced. According to the note in HAKMEM item 132, this algorithm will detect repetition before the third occurrence of any value, i.e. the cycle will be iterated at most twice. HAKMEM also states that it is sufficient to store ⌈ log 2 ⁡ λ ⌉ {\displaystyle \lceil \log _{2}\lambda \rceil } previous values; however, this only offers a saving if we know a priori that λ {\displaystyle \lambda } is significantly smaller than μ {\displaystyle \mu } . The standard implementations store ⌈ log 2 ⁡ ( μ + 2 λ ) ⌉ {\displaystyle \lceil \log _{2}(\mu +2\lambda )\rceil } values. For example, assume the function values are 32-bit integers, so μ + λ ≤ 2 32 {\displaystyle \mu +\lambda \leq 2^{32}} and μ + 2 λ ≤ 2 33 . {\displaystyle \mu +2\lambda \leq 2^{33}.} Then Gosper's algorithm will find the cycle after less than μ + 2 λ {\displaystyle \mu +2\lambda } function evaluations (in fact, the most possible is 3 ⋅ 2 31 − 1 {\displaystyle 3\cdot 2^{31}-1} ), while consuming the space of 33 values (each value being a 32-bit integer). ==== Complexity ==== Upon the i {\displaystyle i} -th evaluation of the generator function, the algorithm compares the generated value with log 2 ⁡ i {\displaystyle \log _{2}i} previous values; observe that i {\displaystyle i} goes up to at least μ + λ {\displaystyle \mu +\lambda } and at most μ + 2 λ {\displaystyle \mu +2\lambda } . Therefore, the time complexity of this algorithm is O ( ( μ + λ ) ⋅ log ⁡ ( μ + λ ) ) {\displaystyle O((\mu +\lambda )\cdot \log(\mu +\lambda ))} . Since it stores log 2 ⁡ ( μ + 2 λ ) {\displaystyle \log _{2}(\mu +2\lambda )} values, its space complexity is Θ ( log ⁡ ( μ + λ ) ) {\displaystyle \Theta (\log(\mu +\lambda ))} . This is under the usual transdichotomous model, assumed throughout this article, in which the size of the function values is constant. Without this assumption, we know it requires Ω ( log ⁡ ( μ + λ ) ) {\displaystyle \Omega (\log(\mu +\lambda ))} space to store μ + λ {\displaystyle \mu +\lambda } distinct values, so the overall space complexity is Ω ( log 2 ⁡ ( μ + λ ) ) . {\displaystyle \Omega (\log ^{2}(\mu +\lambda )).} === Time–space tradeoffs === A number of authors have studied techniques for cycle detection that use more memory than Floyd's and Brent's methods, but detect cycles more quickly. In general these methods store several previously-computed sequence values, and test whether each new value equals one of the previously-computed values. In order to do so quickly, they typically use a hash table or similar data structure for storing the previously-computed values, and therefore are not pointer algorithms: in particular, they usually cannot be applied to Pollard's rho algorithm. Where these methods differ is in how they determine which values to store. Following Nivasch, we survey these techniques briefly. Brent already describes variations of his technique in which the indices of saved sequence values are powers of a number R other than two. By choosing R to be a number close to one, and storing the sequence values at indices that are near a sequence of consecutive powers of R, a cycle detection algorithm can use a number of function evaluations that is within an arbitrarily small factor of the optimum λ + μ. Sedgewick, Szymanski, and Yao provide a method that uses M memory cells and requires in the worst case only ( λ + μ ) ( 1 + c M − 1 / 2 ) {\displaystyle (\lambda +\mu )(1+cM^{-1/2})} function evaluations, for some constant c, which they show to be optimal. The technique involves maintaining a numerical parameter d, storing in a table only those positions in the sequence that are multiples of d, and clearing the table and doubling d whenever too many values have been stored. Several authors have described distinguished point methods that store function values in a table based on a criterion involving the values, rather than (as in the method of Sedgewick et al.) based on their positions. For instance, values equal to zero modulo some value d might be stored. More simply, Nivasch credits D. P. Woodruff with the suggestion of storing a random sample of previously seen values, making an appropriate random choice at each step so that the sample remains random. Nivasch describes an algorithm that does not use a fixed amount of memory, but for which the expected amount of memory used (under the assumption that the input function is random) is logarithmic in the sequence length. An item is stored in the memory table, with this technique, when no later item has a smaller value. As Nivasch shows, the items with this technique can be maintained using a stack data structure, and each successive sequence value need be compared only to the top of the stack. The algorithm terminates when the repeated sequence element with smallest value is found. Running the same algorithm with multiple stacks, using random permutations of the values to reorder the values within each stack, allows a time–space tradeoff similar to the previous algorithms. However, even the version of this algorithm with a single stack is not a pointer algorithm, due to the comparisons needed to determine which of two values is smaller. Any cycle detection algorithm that stores at most M values from the input sequence must perform at least ( λ + μ ) ( 1 + 1 M − 1 ) {\displaystyle (\lambda +\mu )\left(1+{\frac {1}{M-1}}\right)} function evaluations. == Applications == Cycle detection has been used in many applications. Determining the cycle length of a pseudorandom number generator is one measure of its strength. This is the application cited by Knuth in describing Floyd's method. Brent describes the results of testing a linear congruential generator in this fashion; its period turned out to be significantly smaller than advertised. For more complex generators, the sequence of values in which the cycle is to be found may not represent the output of the generator, but rather its internal state. Several number-theoretic algorithms are based on cycle detection, including Pollard's rho algorithm for integer factorization and his related kangaroo algorithm for the discrete logarithm problem. In cryptographic applications, the ability to find two distinct values xμ−1 and xλ+μ−1 mapped by some cryptographic function ƒ to the same value xμ may indicate a weakness in ƒ. For instance, Quisquater and Delescaille apply cycle detection algorithms in the search for a message and a pair of Data Encryption Standard keys that map that message to the same encrypted value; Kaliski, Rivest, and Sherman also use cycle detection algorithms to attack DES. The technique may also be used to find a collision in a cryptographic hash function. Cycle detection may be helpful as a way of discovering infinite loops in certain types of computer programs. Periodic configurations in cellular automaton simulations may be found by applying cycle detection algorithms to the sequence of automaton states. Shape analysis of linked list data structures is a technique for verifying the correctness of an algorithm using those structures. If a node in the list incorrectly points to an earlier node in the same list, the structure will form a cycle that can be detected by these algorithms. In Common Lisp, the S-expression printer, under control of the print-circle variable, detects circular list structure and prints it compactly. Teske describes applications in computational group theory: determining the structure of an Abelian group from a set of its generators. The cryptographic algorithms of Kaliski et al. may also be viewed as attempting to infer the structure of an unknown group. Fich (1981) briefly mentions an application to computer simulation of celestial mechanics, which she attributes to William Kahan. In this application, cycle detection in the phase space of an orbital system may be used to determine whether the system is periodic to within the accuracy of the simulation. In Mandelbrot Set fractal generation some performance techniques are used to speed up the image generation. One of them is called "period checking", which consists of finding the cycles in a point orbit. This article describes the "period checking" technique. You can find another explanation here. Some cycle detection algorithms have to be implemented in order to implement this technique. == References == == External links == Gabriel Nivasch, The Cycle Detection Problem and the Stack Algorithm Tortoise and Hare, Portland Pattern Repository Floyd's Cycle Detection Algorithm (The Tortoise and the Hare) Brent's Cycle Detection Algorithm (The Teleporting Turtle)
Wikipedia:Cycle graph (algebra)#0	In group theory, a subfield of abstract algebra, a cycle graph of a group is an undirected graph that illustrates the various cycles of that group, given a set of generators for the group. Cycle graphs are particularly useful in visualizing the structure of small finite groups. A cycle is the set of powers of a given group element a, where an, the n-th power of an element a, is defined as the product of a multiplied by itself n times. The element a is said to generate the cycle. In a finite group, some non-zero power of a must be the group identity, which we denote either as e or 1; the lowest such power is the order of the element a, the number of distinct elements in the cycle that it generates. In a cycle graph, the cycle is represented as a polygon, with its vertices representing the group elements and its edges indicating how they are linked together to form the cycle. == Definition == Each group element is represented by a node in the cycle graph, and enough cycles are represented as polygons in the graph so that every node lies on at least one cycle. All of those polygons pass through the node representing the identity, and some other nodes may also lie on more than one cycle. Suppose that a group element a generates a cycle of order 6 (has order 6), so that the nodes a, a2, a3, a4, a5, and a6 = e are the vertices of a hexagon in the cycle graph. The element a2 then has order 3; but making the nodes a2, a4, and e be the vertices of a triangle in the graph would add no new information. So, only the primitive cycles need be considered, those that are not subsets of another cycle. Also, the node a5, which also has order 6, generates the same cycle as does a itself; so we have at least two choices for which element to use in generating a cycle --- often more. To build a cycle graph for a group, we start with a node for each group element. For each primitive cycle, we then choose some element a that generates that cycle, and we connect the node for e to the one for a, a to a2, ..., ak−1 to ak, etc., until returning to e. The result is a cycle graph for the group. When a group element a has order 2 (so that multiplication by a is an involution), the rule above would connect e to a by two edges, one going out and the other coming back. Except when the intent is to emphasize the two edges of such a cycle, it is typically drawn as a single line between the two elements. Note that this correspondence between groups and graphs is not one-to-one in either direction: Two different groups can have the same cycle graph, and two different graphs can be cycle graphs for a single group. We give examples of each in the non-uniqueness section. == Example and properties == As an example of a group cycle graph, consider the dihedral group Dih4. The multiplication table for this group is shown on the left, and the cycle graph is shown on the right, with e specifying the identity element. Notice the cycle {e, a, a2, a3} in the multiplication table, with a4 = e. The inverse a−1 = a3 is also a generator of this cycle: (a3)2 = a2, (a3)3 = a, and (a3)4 = e. Similarly, any cycle in any group has at least two generators, and may be traversed in either direction. More generally, the number of generators of a cycle with n elements is given by the Euler φ function of n, and any of these generators may be written as the first node in the cycle (next to the identity e); or more commonly the nodes are left unmarked. Two distinct cycles cannot intersect in a generator. Cycles that contain a non-prime number of elements have cyclic subgroups that are not shown in the graph. For the group Dih4 above, we could draw a line between a2 and e since (a2)2 = e, but since a2 is part of a larger cycle, this is not an edge of the cycle graph. There can be ambiguity when two cycles share a non-identity element. For example, the 8-element quaternion group has cycle graph shown at right. Each of the elements in the middle row when multiplied by itself gives −1 (where 1 is the identity element). In this case we may use different colors to keep track of the cycles, although symmetry considerations will work as well. As noted earlier, the two edges of a 2-element cycle are typically represented as a single line. The inverse of an element is the node symmetric to it in its cycle, with respect to the reflection which fixes the identity. == Non-uniqueness == The cycle graph of a group is not uniquely determined up to graph isomorphism; nor does it uniquely determine the group up to group isomorphism. That is, the graph obtained depends on the set of generators chosen, and two different groups (with chosen sets of generators) can generate the same cycle graph. === A single group can have different cycle graphs === For some groups, choosing different elements to generate the various primitive cycles of that group can lead to different cycle graphs. There is an example of this for the abelian group C 5 × C 2 × C 2 {\displaystyle C_{5}\times C_{2}\times C_{2}} , which has order 20. We denote an element of that group as a triple of numbers ( i ; j , k ) {\displaystyle (i;j,k)} , where 0 ≤ i < 5 {\displaystyle 0\leq i<5} and each of j {\displaystyle j} and k {\displaystyle k} is either 0 or 1. The triple ( 0 ; 0 , 0 ) {\displaystyle (0;0,0)} is the identity element. In the drawings below, i {\displaystyle i} is shown above j {\displaystyle j} and k {\displaystyle k} . This group has three primitive cycles, each of order 10. In the first cycle graph, we choose, as the generators of those three cycles, the nodes ( 1 ; 1 , 0 ) {\displaystyle (1;1,0)} , ( 1 ; 0 , 1 ) {\displaystyle (1;0,1)} , and ( 1 ; 1 , 1 ) {\displaystyle (1;1,1)} . In the second, we generate the third of those cycles --- the blue one --- by starting instead with ( 2 ; 1 , 1 ) {\displaystyle (2;1,1)} . The two resulting graphs are not isomorphic because they have diameters 5 and 4 respectively. === Different groups can have the same cycle graph === Two different (non-isomorphic) groups can have cycle graphs that are isomorphic, where the latter isomorphism ignores the labels on the nodes of the graphs. It follows that the structure of a group is not uniquely determined by its cycle graph. There is an example of this already for groups of order 16, the two groups being C 8 × C 2 {\displaystyle C_{8}\times C_{2}} and C 8 ⋊ 5 C 2 {\displaystyle C_{8}\rtimes _{5}C_{2}} . The abelian group C 8 × C 2 {\displaystyle C_{8}\times C_{2}} is the direct product of the cyclic groups of orders 8 and 2. The non-abelian group C 8 ⋊ 5 C 2 {\displaystyle C_{8}\rtimes _{5}C_{2}} is that semidirect product of C 8 {\displaystyle C_{8}} and C 2 {\displaystyle C_{2}} in which the non-identity element of C 2 {\displaystyle C_{2}} maps to the multiply-by-5 automorphism of C 8 {\displaystyle C_{8}} . In drawing the cycle graphs of those two groups, we take C 8 × C 2 {\displaystyle C_{8}\times C_{2}} to be generated by elements s and t with s 8 = t 2 = 1 and t s = s t , {\displaystyle s^{8}=t^{2}=1\quad {\text{and}}\quad ts=st,} where that latter relation makes C 8 × C 2 {\displaystyle C_{8}\times C_{2}} abelian. And we take C 8 ⋊ 5 C 2 {\displaystyle C_{8}\rtimes _{5}C_{2}} to be generated by elements 𝜎 and 𝜏 with σ 8 = τ 2 = 1 and τ σ = σ 5 τ . {\displaystyle \sigma ^{8}=\tau ^{2}=1\quad {\text{and}}\quad \tau \sigma =\sigma ^{5}\tau .} Here are cycle graphs for those two groups, where we choose s t {\displaystyle st} to generate the green cycle on the left and σ τ {\displaystyle \sigma \tau } to generate that cycle on the right: In the right-hand graph, the green cycle, after moving from 1 to σ τ {\displaystyle \sigma \tau } , moves next to σ 6 , {\displaystyle \sigma ^{6},} because ( σ τ ) ( σ τ ) = σ ( τ σ ) τ = σ ( σ 5 τ ) τ = σ 6 . {\displaystyle (\sigma \tau )(\sigma \tau )=\sigma (\tau \sigma )\tau =\sigma (\sigma ^{5}\tau )\tau =\sigma ^{6}.} == History == Cycle graphs were investigated by the number theorist Daniel Shanks in the early 1950s as a tool to study multiplicative groups of residue classes. Shanks first published the idea in the 1962 first edition of his book Solved and Unsolved Problems in Number Theory. In the book, Shanks investigates which groups have isomorphic cycle graphs and when a cycle graph is planar. In the 1978 second edition, Shanks reflects on his research on class groups and the development of the baby-step giant-step method: The cycle graphs have proved to be useful when working with finite Abelian groups; and I have used them frequently in finding my way around an intricate structure [77, p. 852], in obtaining a wanted multiplicative relation [78, p. 426], or in isolating some wanted subgroup [79]. Cycle graphs are used as a pedagogical tool in Nathan Carter's 2009 introductory textbook Visual Group Theory. == Graph characteristics of particular group families == Certain group types give typical graphs: Cyclic groups Zn, order n, is a single cycle graphed simply as an n-sided polygon with the elements at the vertices: When n is a prime number, groups of the form (Zn)m will have (nm − 1)/(n − 1) n-element cycles sharing the identity element: Dihedral groups Dihn, order 2n consists of an n-element cycle and n 2-element cycles: Dicyclic groups, Dicn = Q4n, order 4n: Other direct products: Symmetric groups – The symmetric group Sn contains, for any group of order n, a subgroup isomorphic to that group. Thus the cycle graph of every group of order n will be found in the cycle graph of Sn. See example: Subgroups of S4 == Extended example: Subgroups of the full octahedral group == The full octahedral group is the direct product S 4 × Z 2 {\displaystyle S_{4}\times Z_{2}} of the symmetric group S4 and the cyclic group Z2. Its order is 48, and it has subgroups of every order that divides 48. In the examples below nodes that are related to each other are placed next to each other, so these are not the simplest possible cycle graphs for these groups (like those on the right). Like all graphs a cycle graph can be represented in different ways to emphasize different properties. The two representations of the cycle graph of S4 are an example of that. == See also == List of small groups Cayley graph == References == Skiena, S. (1990). Cycles, Stars, and Wheels. Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica (pp. 144-147). Shanks, Daniel (1978) [1962], Solved and Unsolved Problems in Number Theory (2nd ed.), New York: Chelsea Publishing Company, ISBN 0-8284-0297-3 Pemmaraju, S., & Skiena, S. (2003). Cycles, Stars, and Wheels. Computational Discrete Mathematics: Combinatorics and Graph Theory with Mathematica (pp. 248-249). Cambridge University Press. == External links == Weisstein, Eric W. "Group Cycle Graph". MathWorld.
Wikipedia:Cycle space#0	In graph theory, a branch of mathematics, the (binary) cycle space of an undirected graph is the set of its even-degree subgraphs. This set of subgraphs can be described algebraically as a vector space over the two-element finite field. The dimension of this space is the circuit rank of the graph. The same space can also be described in terms from algebraic topology as the first homology group of the graph. Using homology theory, the binary cycle space may be generalized to cycle spaces over arbitrary rings. == Definitions == The cycle space of a graph can be described with increasing levels of mathematical sophistication as a set of subgraphs, as a binary vector space, or as a homology group. === Graph theory === A spanning subgraph of a given graph G may be defined from any subset S of the edges of G. The subgraph has the same set of vertices as G itself (this is the meaning of the word "spanning") but has the elements of S as its edges. Thus, a graph G with m edges has 2m spanning subgraphs, including G itself as well as the empty graph on the same set of vertices as G. The collection of all spanning subgraphs of a graph G forms the edge space of G. A graph G, or one of its subgraphs, is said to be Eulerian if each of its vertices has an even number of incident edges (this number is called the degree of the vertex). This property is named after Leonhard Euler who proved in 1736, in his work on the Seven Bridges of Königsberg, that a connected graph has a tour that visits each edge exactly once if and only if it is Eulerian. However, for the purposes of defining cycle spaces, an Eulerian subgraph does not need to be connected; for instance, the empty graph, in which all vertices are disconnected from each other, is Eulerian in this sense. The cycle space of a graph is the collection of its Eulerian spanning subgraphs. === Algebra === If one applies any set operation such as union or intersection of sets to two spanning subgraphs of a given graph, the result will again be a subgraph. In this way, the edge space of an arbitrary graph can be interpreted as a Boolean algebra. The cycle space, also, has an algebraic structure, but a more restrictive one. The union or intersection of two Eulerian subgraphs may fail to be Eulerian. However, the symmetric difference of two Eulerian subgraphs (the graph consisting of the edges that belong to exactly one of the two given graphs) is again Eulerian. This follows from the fact that the symmetric difference of two sets with an even number of elements is also even. Applying this fact separately to the neighbourhoods of each vertex shows that the symmetric difference operator preserves the property of being Eulerian. A family of sets closed under the symmetric difference operation can be described algebraically as a vector space over the two-element finite field Z 2 {\displaystyle \mathbb {Z} _{2}} . This field has two elements, 0 and 1, and its addition and multiplication operations can be described as the familiar addition and multiplication of integers, taken modulo 2. A vector space consists of a set of elements together with an addition and scalar multiplication operation satisfying certain properties generalizing the properties of the familiar real vector spaces. For the cycle space, the elements of the vector space are the Eulerian subgraphs, the addition operation is symmetric differencing, multiplication by the scalar 1 is the identity operation, and multiplication by the scalar 0 takes every element to the empty graph, which forms the additive identity element for the cycle space. The edge space is also a vector space over Z 2 {\displaystyle \mathbb {Z} _{2}} with the symmetric difference as addition. As vector spaces, the cycle space and the cut space of the graph (the family of edge sets that span the cuts of the graph) are the orthogonal complements of each other within the edge space. This means that a set S {\displaystyle S} of edges in a graph forms a cut if and only if every Eulerian subgraph has an even number of edges in common with S {\displaystyle S} , and S {\displaystyle S} forms an Eulerian subgraph if and only if every cut has an even number of edges in common with S {\displaystyle S} . Although these two spaces are orthogonal complements, some graphs have nonempty subgraphs that belong to both of them. Such a subgraph (an Eulerian cut) exists as part of a graph G {\displaystyle G} if and only if G {\displaystyle G} has an even number of spanning forests. === Topology === An undirected graph may be viewed as a simplicial complex with its vertices as zero-dimensional simplices and the edges as one-dimensional simplices. The chain complex of this topological space consists of its edge space and vertex space (the Boolean algebra of sets of vertices), connected by a boundary operator that maps any spanning subgraph (an element of the edge space) to its set of odd-degree vertices (an element of the vertex space). The homology group H 1 ( G , Z 2 ) {\displaystyle H_{1}(G,\mathbb {Z} _{2})} consists of the elements of the edge space that map to the zero element of the vertex space; these are exactly the Eulerian subgraphs. Its group operation is the symmetric difference operation on Eulerian subgraphs. Replacing Z 2 {\displaystyle \mathbb {Z} _{2}} in this construction by an arbitrary ring allows the definition of cycle spaces to be extended to cycle spaces with coefficients in the given ring, that form modules over the ring. In particular, the integral cycle space is the space H 1 ( G , Z ) . {\displaystyle H_{1}(G,\mathbb {Z} ).} It can be defined in graph-theoretic terms by choosing an arbitrary orientation of the graph, and defining an integral cycle of a graph G {\displaystyle G} to be an assignment of integers to the edges of G {\displaystyle G} (an element of the free abelian group over the edges) with the property that, at each vertex, the sum of the numbers assigned to incoming edges equals the sum of the numbers assigned to outgoing edges. A member of H 1 ( G , Z ) {\displaystyle H_{1}(G,\mathbb {Z} )} or of H 1 ( G , Z k ) {\displaystyle H_{1}(G,\mathbb {Z} _{k})} (the cycle space modulo k {\displaystyle k} ) with the additional property that all of the numbers assigned to the edges are nonzero is called a nowhere-zero flow or a nowhere-zero k {\displaystyle k} -flow respectively. == Circuit rank == As a vector space, the dimension of the cycle space of a graph with n {\displaystyle n} vertices, m {\displaystyle m} edges, and c {\displaystyle c} connected components is m − n + c {\displaystyle m-n+c} . This number can be interpreted topologically as the first Betti number of the graph. In graph theory, it is known as the circuit rank, cyclomatic number, or nullity of the graph. Combining this formula for the rank with the fact that the cycle space is a vector space over the two-element field shows that the total number of elements in the cycle space is exactly 2 m − n + c {\displaystyle 2^{m-n+c}} . == Cycle bases == A basis of a vector space is a minimal subset of the elements with the property that all other elements can be written as a linear combination of basis elements. Every basis of a finite-dimensional space has the same number of elements, which equals the dimension of the space. In the case of the cycle space, a basis is a family of exactly m − n + c {\displaystyle m-n+c} Eulerian subgraphs, with the property that every Eulerian subgraph can be written as the symmetric difference of a family of basis elements. === Existence === By Veblen's theorem, every Eulerian subgraph of a given graph can be decomposed into simple cycles, subgraphs in which all vertices have degree zero or two and in which the degree-two vertices form a connected set. Therefore, it is always possible to find a basis in which the basis elements are themselves all simple cycles. Such a basis is called a cycle basis of the given graph. More strongly, it is always possible to find a basis in which the basis elements are induced cycles or even (in a 3-vertex-connected graph) non-separating induced cycles. === Fundamental and weakly fundamental bases === One way of constructing a cycle basis is to form a maximal forest of the graph, and then for each edge e {\displaystyle e} that does not belong to the forest, form a cycle C e {\displaystyle C_{e}} consisting of e {\displaystyle e} together with the path in the forest connecting the endpoints of e {\displaystyle e} . The cycles C e {\displaystyle C_{e}} formed in this way are linearly independent (each one contains an edge e {\displaystyle e} that does not belong to any of the other cycles) and has the correct size m − n + c {\displaystyle m-n+c} to be a basis, so it necessarily is a basis. A basis formed in this way is called a fundamental cycle basis (with respect to the chosen forest). If there exists a linear ordering of the cycles in a cycle basis such that each cycle includes at least one edge that is not part of any previous cycle, then the cycle basis is called weakly fundamental. Every fundamental cycle basis is weakly fundamental (for all linear orderings) but not necessarily vice versa. There exist graphs, and cycle bases for those graphs, that are not weakly fundamental. === Minimum weight bases === If the edges of a graph are given real number weights, the weight of a subgraph may be computed as the sum of the weights of its edges. The minimum weight basis of the cycle space is necessarily a cycle basis, and can be constructed in polynomial time. The minimum weight basis is not always weakly fundamental, and when it is not it is NP-hard to find the weakly fundamental basis with the minimum possible weight. == Planar graphs == === Homology === If a planar graph is embedded into the plane, its chain complex of edges and vertices may be embedded into a higher dimensional chain complex that also includes the sets of faces of the graph. The boundary map of this chain complex takes any 2-chain (a set of faces) to the set of edges that belong to an odd number of faces in the 2-chain. The boundary of a 2-chain is necessarily an Eulerian subgraph, and every Eulerian subgraph can be generated in this way from exactly two different 2-chains (each of which is the complement of the other). It follows from this that the set of bounded faces of the embedding forms a cycle basis for the planar graph: removing the unbounded face from this set of cycles reduces the number of ways each Eulerian subgraph can be generated from two to exactly one. === Mac Lane's planarity criterion === Mac Lane's planarity criterion, named after Saunders Mac Lane, characterizes planar graphs in terms of their cycle spaces and cycle bases. It states that a finite undirected graph is planar if and only if the graph has a cycle basis in which each edge of the graph participates in at most two basis cycles. In a planar graph, a cycle basis formed by the set of bounded faces of an embedding necessarily has this property: each edge participates only in the basis cycles for the two faces it separates. Conversely, if a cycle basis has at most two cycles per edge, then its cycles can be used as the set of bounded faces of a planar embedding of its graph. === Duality === The cycle space of a planar graph is the cut space of its dual graph, and vice versa. The minimum weight cycle basis for a planar graph is not necessarily the same as the basis formed by its bounded faces: it can include cycles that are not faces, and some faces may not be included as cycles in the minimum weight cycle basis. There exists a minimum weight cycle basis in which no two cycles cross each other: for every two cycles in the basis, either the cycles enclose disjoint subsets of the bounded faces, or one of the two cycles encloses the other one. Following the duality between cycle spaces and cut spaces, this basis for a planar graph corresponds to a Gomory–Hu tree of the dual graph, a minimum weight basis for its cut space. === Nowhere-zero flows === In planar graphs, colorings with k {\displaystyle k} distinct colors are dual to nowhere-zero flows over the ring Z k {\displaystyle \mathbb {Z} _{k}} of integers modulo k {\displaystyle k} . In this duality, the difference between the colors of two adjacent regions is represented by a flow value across the edge separating the regions. In particular, the existence of nowhere-zero 4-flows is equivalent to the four color theorem. The snark theorem generalizes this result to nonplanar graphs. == References ==
Wikipedia:Cycles and fixed points#0	In mathematics, the cycles of a permutation π of a finite set S correspond bijectively to the orbits of the subgroup generated by π acting on S. These orbits are subsets of S that can be written as { c1, ..., cn }, such that π(ci) = ci + 1 for i = 1, ..., n − 1, and π(cn) = c1. The corresponding cycle of π is written as ( c1 c2 ... cn ); this expression is not unique since c1 can be chosen to be any element of the orbit. The size n of the orbit is called the length of the corresponding cycle; when n = 1, the single element in the orbit is called a fixed point of the permutation. A permutation is determined by giving an expression for each of its cycles, and one notation for permutations consist of writing such expressions one after another in some order. For example, let π = ( 1 6 7 2 5 4 8 3 2 8 7 4 5 3 6 1 ) = ( 1 2 3 4 5 6 7 8 2 4 1 3 5 8 7 6 ) {\displaystyle \pi ={\begin{pmatrix}1&6&7&2&5&4&8&3\\2&8&7&4&5&3&6&1\end{pmatrix}}={\begin{pmatrix}1&2&3&4&5&6&7&8\\2&4&1&3&5&8&7&6\end{pmatrix}}} be a permutation that maps 1 to 2, 6 to 8, etc. Then one may write π = ( 1 2 4 3 ) ( 5 ) ( 6 8 ) (7) = (7) ( 1 2 4 3 ) ( 6 8 ) ( 5 ) = ( 4 3 1 2 ) ( 8 6 ) ( 5 ) (7) = ... Here 5 and 7 are fixed points of π, since π(5) = 5 and π(7)=7. It is typical, but not necessary, to not write the cycles of length one in such an expression. Thus, π = (1 2 4 3)(6 8), would be an appropriate way to express this permutation. There are different ways to write a permutation as a list of its cycles, but the number of cycles and their contents are given by the partition of S into orbits, and these are therefore the same for all such expressions. == Counting permutations by number of cycles == The unsigned Stirling number of the first kind, s(k, j) counts the number of permutations of k elements with exactly j disjoint cycles. === Properties === (1) For every k > 0 : s(k, k) = 1. (2) For every k > 0 : s(k, 1) = (k − 1)!. (3) For every k > j > 1, s(k, j) = s(k − 1,j − 1) + s(k − 1, j)·(k − 1) === Reasons for properties === (1) There is only one way to construct a permutation of k elements with k cycles: Every cycle must have length 1 so every element must be a fixed point. (2.a) Every cycle of length k may be written as permutation of the number 1 to k; there are k! of these permutations. (2.b) There are k different ways to write a given cycle of length k, e.g. ( 1 2 4 3 ) = ( 2 4 3 1 ) = ( 4 3 1 2 ) = ( 3 1 2 4 ). (2.c) Finally: s(k, 1) = k!/k = (k − 1)!. (3) There are two different ways to construct a permutation of k elements with j cycles: (3.a) If we want element k to be a fixed point we may choose one of the s(k − 1, j − 1) permutations with k − 1 elements and j − 1 cycles and add element k as a new cycle of length 1. (3.b) If we want element k not to be a fixed point we may choose one of the s(k − 1, j ) permutations with k − 1 elements and j cycles and insert element k in an existing cycle in front of one of the k − 1 elements. === Some values === == Counting permutations by number of fixed points == The value f(k, j) counts the number of permutations of k elements with exactly j fixed points. For the main article on this topic, see rencontres numbers. === Properties === (1) For every j < 0 or j > k : f(k, j) = 0. (2) f(0, 0) = 1. (3) For every k > 1 and k ≥ j ≥ 0, f(k, j) = f(k − 1, j − 1) + f(k − 1, j)·(k − 1 − j) + f(k − 1, j + 1)·(j + 1) === Reasons for properties === (3) There are three different methods to construct a permutation of k elements with j fixed points: (3.a) We may choose one of the f(k − 1, j − 1) permutations with k − 1 elements and j − 1 fixed points and add element k as a new fixed point. (3.b) We may choose one of the f(k − 1, j) permutations with k − 1 elements and j fixed points and insert element k in an existing cycle of length > 1 in front of one of the (k − 1) − j elements. (3.c) We may choose one of the f(k − 1, j + 1) permutations with k − 1 elements and j + 1 fixed points and join element k with one of the j + 1 fixed points to a cycle of length 2. === Some values === === Alternate calculations === == See also == Cyclic permutation Cycle notation == Notes == == References == Brualdi, Richard A. (2010), Introductory Combinatorics (5th ed.), Prentice-Hall, ISBN 978-0-13-602040-0 Cameron, Peter J. (1994), Combinatorics: Topics, Techniques, Algorithms, Cambridge University Press, ISBN 0-521-45761-0 Gerstein, Larry J. (1987), Discrete Mathematics and Algebraic Structures, W.H. Freeman and Co., ISBN 0-7167-1804-9
Wikipedia:Cyclic algebra#0	In algebra, a cyclic division algebra is one of the basic examples of a division algebra over a field and plays a key role in the theory of central simple algebras. == Definition == Let A be a finite-dimensional central simple algebra over a field F. Then A is said to be cyclic if it contains a strictly maximal subfield E such that E/F is a cyclic field extension (i.e., the Galois group is a cyclic group). == See also == Factor system § Cyclic algebras – cyclic algebras described by factor systems. Brauer group § Cyclic algebras – cyclic algebras are representative of Brauer classes. == References == Pierce, Richard S. (1982). Associative Algebras. Graduate Texts in Mathematics, volume 88. Springer-Verlag. ISBN 978-0-387-90693-5. OCLC 249353240. Weil, André (1995). Basic Number Theory (third ed.). Springer. ISBN 978-3-540-58655-5. OCLC 32381827.
Wikipedia:Cyclic subspace#0	In mathematics, in linear algebra and functional analysis, a cyclic subspace is a certain special subspace of a vector space associated with a vector in the vector space and a linear transformation of the vector space. The cyclic subspace associated with a vector v in a vector space V and a linear transformation T of V is called the T-cyclic subspace generated by v. The concept of a cyclic subspace is a basic component in the formulation of the cyclic decomposition theorem in linear algebra. == Definition == Let T : V → V {\displaystyle T:V\rightarrow V} be a linear transformation of a vector space V {\displaystyle V} and let v {\displaystyle v} be a vector in V {\displaystyle V} . The T {\displaystyle T} -cyclic subspace of V {\displaystyle V} generated by v {\displaystyle v} , denoted Z ( v ; T ) {\displaystyle Z(v;T)} , is the subspace of V {\displaystyle V} generated by the set of vectors { v , T ( v ) , T 2 ( v ) , … , T r ( v ) , … } {\displaystyle \{v,T(v),T^{2}(v),\ldots ,T^{r}(v),\ldots \}} . In the case when V {\displaystyle V} is a topological vector space, v {\displaystyle v} is called a cyclic vector for T {\displaystyle T} if Z ( v ; T ) {\displaystyle Z(v;T)} is dense in V {\displaystyle V} . For the particular case of finite-dimensional spaces, this is equivalent to saying that Z ( v ; T ) {\displaystyle Z(v;T)} is the whole space V {\displaystyle V} . There is another equivalent definition of cyclic spaces. Let T : V → V {\displaystyle T:V\rightarrow V} be a linear transformation of a topological vector space over a field F {\displaystyle F} and v {\displaystyle v} be a vector in V {\displaystyle V} . The set of all vectors of the form g ( T ) v {\displaystyle g(T)v} , where g ( x ) {\displaystyle g(x)} is a polynomial in the ring F [ x ] {\displaystyle F[x]} of all polynomials in x {\displaystyle x} over F {\displaystyle F} , is the T {\displaystyle T} -cyclic subspace generated by v {\displaystyle v} . The subspace Z ( v ; T ) {\displaystyle Z(v;T)} is an invariant subspace for T {\displaystyle T} , in the sense that T Z ( v ; T ) ⊂ Z ( v ; T ) {\displaystyle TZ(v;T)\subset Z(v;T)} . === Examples === For any vector space V {\displaystyle V} and any linear operator T {\displaystyle T} on V {\displaystyle V} , the T {\displaystyle T} -cyclic subspace generated by the zero vector is the zero-subspace of V {\displaystyle V} . If I {\displaystyle I} is the identity operator then every I {\displaystyle I} -cyclic subspace is one-dimensional. Z ( v ; T ) {\displaystyle Z(v;T)} is one-dimensional if and only if v {\displaystyle v} is a characteristic vector (eigenvector) of T {\displaystyle T} . Let V {\displaystyle V} be the two-dimensional vector space and let T {\displaystyle T} be the linear operator on V {\displaystyle V} represented by the matrix [ 0 1 0 0 ] {\displaystyle {\begin{bmatrix}0&1\\0&0\end{bmatrix}}} relative to the standard ordered basis of V {\displaystyle V} . Let v = [ 0 1 ] {\displaystyle v={\begin{bmatrix}0\\1\end{bmatrix}}} . Then T v = [ 1 0 ] , T 2 v = 0 , … , T r v = 0 , … {\displaystyle Tv={\begin{bmatrix}1\\0\end{bmatrix}},\quad T^{2}v=0,\ldots ,T^{r}v=0,\ldots } . Therefore { v , T ( v ) , T 2 ( v ) , … , T r ( v ) , … } = { [ 0 1 ] , [ 1 0 ] } {\displaystyle \{v,T(v),T^{2}(v),\ldots ,T^{r}(v),\ldots \}=\left\{{\begin{bmatrix}0\\1\end{bmatrix}},{\begin{bmatrix}1\\0\end{bmatrix}}\right\}} and so Z ( v ; T ) = V {\displaystyle Z(v;T)=V} . Thus v {\displaystyle v} is a cyclic vector for T {\displaystyle T} . == Companion matrix == Let T : V → V {\displaystyle T:V\rightarrow V} be a linear transformation of a n {\displaystyle n} -dimensional vector space V {\displaystyle V} over a field F {\displaystyle F} and v {\displaystyle v} be a cyclic vector for T {\displaystyle T} . Then the vectors B = { v 1 = v , v 2 = T v , v 3 = T 2 v , … v n = T n − 1 v } {\displaystyle B=\{v_{1}=v,v_{2}=Tv,v_{3}=T^{2}v,\ldots v_{n}=T^{n-1}v\}} form an ordered basis for V {\displaystyle V} . Let the characteristic polynomial for T {\displaystyle T} be p ( x ) = c 0 + c 1 x + c 2 x 2 + ⋯ + c n − 1 x n − 1 + x n {\displaystyle p(x)=c_{0}+c_{1}x+c_{2}x^{2}+\cdots +c_{n-1}x^{n-1}+x^{n}} . Then T v 1 = v 2 T v 2 = v 3 T v 3 = v 4 ⋮ T v n − 1 = v n T v n = − c 0 v 1 − c 1 v 2 − ⋯ c n − 1 v n {\displaystyle {\begin{aligned}Tv_{1}&=v_{2}\\Tv_{2}&=v_{3}\\Tv_{3}&=v_{4}\\\vdots &\\Tv_{n-1}&=v_{n}\\Tv_{n}&=-c_{0}v_{1}-c_{1}v_{2}-\cdots c_{n-1}v_{n}\end{aligned}}} Therefore, relative to the ordered basis B {\displaystyle B} , the operator T {\displaystyle T} is represented by the matrix [ 0 0 0 ⋯ 0 − c 0 1 0 0 … 0 − c 1 0 1 0 … 0 − c 2 ⋮ 0 0 0 … 1 − c n − 1 ] {\displaystyle {\begin{bmatrix}0&0&0&\cdots &0&-c_{0}\\1&0&0&\ldots &0&-c_{1}\\0&1&0&\ldots &0&-c_{2}\\\vdots &&&&&\\0&0&0&\ldots &1&-c_{n-1}\end{bmatrix}}} This matrix is called the companion matrix of the polynomial p ( x ) {\displaystyle p(x)} . == See also == Companion matrix Krylov subspace == External links == PlanetMath: cyclic subspace == References ==
Wikipedia:Cyclic vector#0	In the mathematics of operator theory, an operator A on an (infinite-dimensional) Banach space or Hilbert space H has a cyclic vector f if the vectors f, Af, A2f,... span H. Equivalently, f is a cyclic vector for A in case the set of all vectors of the form p(A)f, where p varies over all polynomials, is dense in H. == See also == Cyclic and separating vector == References ==
Wikipedia:Cyclical monotonicity#0	In mathematics, cyclical monotonicity is a generalization of the notion of monotonicity to the case of vector-valued function. == Definition == Let ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } denote the inner product on an inner product space X {\displaystyle X} and let U {\displaystyle U} be a nonempty subset of X {\displaystyle X} . A correspondence f : U ⇉ X {\displaystyle f:U\rightrightarrows X} is called cyclically monotone if for every set of points x 1 , … , x m + 1 ∈ U {\displaystyle x_{1},\dots ,x_{m+1}\in U} with x m + 1 = x 1 {\displaystyle x_{m+1}=x_{1}} it holds that ∑ k = 1 m ⟨ x k + 1 , f ( x k + 1 ) − f ( x k ) ⟩ ≥ 0. {\displaystyle \sum _{k=1}^{m}\langle x_{k+1},f(x_{k+1})-f(x_{k})\rangle \geq 0.} == Properties == For the case of scalar functions of one variable the definition above is equivalent to usual monotonicity. Gradients of convex functions are cyclically monotone. In fact, the converse is true. Suppose U {\displaystyle U} is convex and f : U ⇉ R n {\displaystyle f:U\rightrightarrows \mathbb {R} ^{n}} is a correspondence with nonempty values. Then if f {\displaystyle f} is cyclically monotone, there exists an upper semicontinuous convex function F : U → R {\displaystyle F:U\to \mathbb {R} } such that f ( x ) ⊂ ∂ F ( x ) {\displaystyle f(x)\subset \partial F(x)} for every x ∈ U {\displaystyle x\in U} , where ∂ F ( x ) {\displaystyle \partial F(x)} denotes the subgradient of F {\displaystyle F} at x {\displaystyle x} . == See also == Absolutely and completely monotonic functions and sequences == References ==
Wikipedia:Cyclically reduced word#0	In mathematics, cyclically reduced word is a concept of combinatorial group theory. Let F(X) be a free group. Then a word in F(X) is said to be cyclically reduced if and only if every cyclic permutation of the word is reduced. == Properties == Every cyclic shift and the inverse of a cyclically reduced word are cyclically reduced again. Every word is conjugate to a cyclically reduced word. The cyclically reduced words are minimal-length representatives of the conjugacy classes in the free group. This representative is not uniquely determined, but it is unique up to cyclic shifts (since every cyclic shift is a conjugate element). == References == Solitar, Donald; Magnus, Wilhelm; Karrass, Abraham (1976), Combinatorial group theory: presentations of groups in terms of generators and relations, New York: Dover, pp. 33, 188, 212, ISBN 0-486-63281-4
Wikipedia:Cyclotomic identity#0	In mathematics, the cyclotomic identity states that 1 1 − α z = ∏ j = 1 ∞ ( 1 1 − z j ) M ( α , j ) {\displaystyle {1 \over 1-\alpha z}=\prod _{j=1}^{\infty }\left({1 \over 1-z^{j}}\right)^{M(\alpha ,j)}} where M is Moreau's necklace-counting function, M ( α , n ) = 1 n ∑ d \| n μ ( n d ) α d , {\displaystyle M(\alpha ,n)={1 \over n}\sum _{d\,\|\,n}\mu \left({n \over d}\right)\alpha ^{d},} and μ is the classic Möbius function of number theory. The name comes from the denominator, 1 − z j, which is the product of cyclotomic polynomials. The left hand side of the cyclotomic identity is the generating function for the free associative algebra on α generators, and the right hand side is the generating function for the universal enveloping algebra of the free Lie algebra on α generators. The cyclotomic identity witnesses the fact that these two algebras are isomorphic. There is also a symmetric generalization of the cyclotomic identity found by Strehl: ∏ j = 1 ∞ ( 1 1 − α z j ) M ( β , j ) = ∏ j = 1 ∞ ( 1 1 − β z j ) M ( α , j ) {\displaystyle \prod _{j=1}^{\infty }\left({1 \over 1-\alpha z^{j}}\right)^{M(\beta ,j)}=\prod _{j=1}^{\infty }\left({1 \over 1-\beta z^{j}}\right)^{M(\alpha ,j)}} == References == Metropolis, N.; Rota, Gian-Carlo (1984), "The cyclotomic identity", in Greene, Curtis (ed.), Combinatorics and algebra (Boulder, Colo., 1983). Proceedings of the AMS-IMS-SIAM joint summer research conference held at the University of Colorado, Boulder, Colo., June 5–11, 1983., Contemp. Math., vol. 34, Providence, R.I.: American Mathematical Society, pp. 19–27, ISBN 978-0-8218-5029-9, MR 0777692
Wikipedia:Cyclotomic polynomial#0	In mathematics, the nth cyclotomic polynomial, for any positive integer n, is the unique irreducible polynomial with integer coefficients that is a divisor of x n − 1 {\displaystyle x^{n}-1} and is not a divisor of x k − 1 {\displaystyle x^{k}-1} for any k < n. Its roots are all nth primitive roots of unity e 2 i π k n {\displaystyle e^{2i\pi {\frac {k}{n}}}} , where k runs over the positive integers less than n and coprime to n (and i is the imaginary unit). In other words, the nth cyclotomic polynomial is equal to Φ n ( x ) = ∏ gcd ( k , n ) = 1 1 ≤ k ≤ n ( x − e 2 i π k n ) . {\displaystyle \Phi _{n}(x)=\prod _{\stackrel {1\leq k\leq n}{\gcd(k,n)=1}}\left(x-e^{2i\pi {\frac {k}{n}}}\right).} It may also be defined as the monic polynomial with integer coefficients that is the minimal polynomial over the field of the rational numbers of any primitive nth-root of unity ( e 2 i π / n {\displaystyle e^{2i\pi /n}} is an example of such a root). An important relation linking cyclotomic polynomials and primitive roots of unity is ∏ d ∣ n Φ d ( x ) = x n − 1 , {\displaystyle \prod _{d\mid n}\Phi _{d}(x)=x^{n}-1,} showing that x {\displaystyle x} is a root of x n − 1 {\displaystyle x^{n}-1} if and only if it is a d th primitive root of unity for some d that divides n. == Examples == If n is a prime number, then Φ n ( x ) = 1 + x + x 2 + ⋯ + x n − 1 = ∑ k = 0 n − 1 x k . {\displaystyle \Phi _{n}(x)=1+x+x^{2}+\cdots +x^{n-1}=\sum _{k=0}^{n-1}x^{k}.} If n = 2p where p is a prime number other than 2, then Φ 2 p ( x ) = 1 − x + x 2 − ⋯ + x p − 1 = ∑ k = 0 p − 1 ( − x ) k . {\displaystyle \Phi _{2p}(x)=1-x+x^{2}-\cdots +x^{p-1}=\sum _{k=0}^{p-1}(-x)^{k}.} For n up to 30, the cyclotomic polynomials are: Φ 1 ( x ) = x − 1 Φ 2 ( x ) = x + 1 Φ 3 ( x ) = x 2 + x + 1 Φ 4 ( x ) = x 2 + 1 Φ 5 ( x ) = x 4 + x 3 + x 2 + x + 1 Φ 6 ( x ) = x 2 − x + 1 Φ 7 ( x ) = x 6 + x 5 + x 4 + x 3 + x 2 + x + 1 Φ 8 ( x ) = x 4 + 1 Φ 9 ( x ) = x 6 + x 3 + 1 Φ 10 ( x ) = x 4 − x 3 + x 2 − x + 1 Φ 11 ( x ) = x 10 + x 9 + x 8 + x 7 + x 6 + x 5 + x 4 + x 3 + x 2 + x + 1 Φ 12 ( x ) = x 4 − x 2 + 1 Φ 13 ( x ) = x 12 + x 11 + x 10 + x 9 + x 8 + x 7 + x 6 + x 5 + x 4 + x 3 + x 2 + x + 1 Φ 14 ( x ) = x 6 − x 5 + x 4 − x 3 + x 2 − x + 1 Φ 15 ( x ) = x 8 − x 7 + x 5 − x 4 + x 3 − x + 1 Φ 16 ( x ) = x 8 + 1 Φ 17 ( x ) = x 16 + x 15 + x 14 + x 13 + x 12 + x 11 + x 10 + x 9 + x 8 + x 7 + x 6 + x 5 + x 4 + x 3 + x 2 + x + 1 Φ 18 ( x ) = x 6 − x 3 + 1 Φ 19 ( x ) = x 18 + x 17 + x 16 + x 15 + x 14 + x 13 + x 12 + x 11 + x 10 + x 9 + x 8 + x 7 + x 6 + x 5 + x 4 + x 3 + x 2 + x + 1 Φ 20 ( x ) = x 8 − x 6 + x 4 − x 2 + 1 Φ 21 ( x ) = x 12 − x 11 + x 9 − x 8 + x 6 − x 4 + x 3 − x + 1 Φ 22 ( x ) = x 10 − x 9 + x 8 − x 7 + x 6 − x 5 + x 4 − x 3 + x 2 − x + 1 Φ 23 ( x ) = x 22 + x 21 + x 20 + x 19 + x 18 + x 17 + x 16 + x 15 + x 14 + x 13 + x 12 + x 11 + x 10 + x 9 + x 8 + x 7 + x 6 + x 5 + x 4 + x 3 + x 2 + x + 1 Φ 24 ( x ) = x 8 − x 4 + 1 Φ 25 ( x ) = x 20 + x 15 + x 10 + x 5 + 1 Φ 26 ( x ) = x 12 − x 11 + x 10 − x 9 + x 8 − x 7 + x 6 − x 5 + x 4 − x 3 + x 2 − x + 1 Φ 27 ( x ) = x 18 + x 9 + 1 Φ 28 ( x ) = x 12 − x 10 + x 8 − x 6 + x 4 − x 2 + 1 Φ 29 ( x ) = x 28 + x 27 + x 26 + x 25 + x 24 + x 23 + x 22 + x 21 + x 20 + x 19 + x 18 + x 17 + x 16 + x 15 + x 14 + x 13 + x 12 + x 11 + x 10 + x 9 + x 8 + x 7 + x 6 + x 5 + x 4 + x 3 + x 2 + x + 1 Φ 30 ( x ) = x 8 + x 7 − x 5 − x 4 − x 3 + x + 1. {\displaystyle {\begin{aligned}\Phi _{1}(x)&=x-1\\\Phi _{2}(x)&=x+1\\\Phi _{3}(x)&=x^{2}+x+1\\\Phi _{4}(x)&=x^{2}+1\\\Phi _{5}(x)&=x^{4}+x^{3}+x^{2}+x+1\\\Phi _{6}(x)&=x^{2}-x+1\\\Phi _{7}(x)&=x^{6}+x^{5}+x^{4}+x^{3}+x^{2}+x+1\\\Phi _{8}(x)&=x^{4}+1\\\Phi _{9}(x)&=x^{6}+x^{3}+1\\\Phi _{10}(x)&=x^{4}-x^{3}+x^{2}-x+1\\\Phi _{11}(x)&=x^{10}+x^{9}+x^{8}+x^{7}+x^{6}+x^{5}+x^{4}+x^{3}+x^{2}+x+1\\\Phi _{12}(x)&=x^{4}-x^{2}+1\\\Phi _{13}(x)&=x^{12}+x^{11}+x^{10}+x^{9}+x^{8}+x^{7}+x^{6}+x^{5}+x^{4}+x^{3}+x^{2}+x+1\\\Phi _{14}(x)&=x^{6}-x^{5}+x^{4}-x^{3}+x^{2}-x+1\\\Phi _{15}(x)&=x^{8}-x^{7}+x^{5}-x^{4}+x^{3}-x+1\\\Phi _{16}(x)&=x^{8}+1\\\Phi _{17}(x)&=x^{16}+x^{15}+x^{14}+x^{13}+x^{12}+x^{11}+x^{10}+x^{9}+x^{8}+x^{7}+x^{6}+x^{5}+x^{4}+x^{3}+x^{2}+x+1\\\Phi _{18}(x)&=x^{6}-x^{3}+1\\\Phi _{19}(x)&=x^{18}+x^{17}+x^{16}+x^{15}+x^{14}+x^{13}+x^{12}+x^{11}+x^{10}+x^{9}+x^{8}+x^{7}+x^{6}+x^{5}+x^{4}+x^{3}+x^{2}+x+1\\\Phi _{20}(x)&=x^{8}-x^{6}+x^{4}-x^{2}+1\\\Phi _{21}(x)&=x^{12}-x^{11}+x^{9}-x^{8}+x^{6}-x^{4}+x^{3}-x+1\\\Phi _{22}(x)&=x^{10}-x^{9}+x^{8}-x^{7}+x^{6}-x^{5}+x^{4}-x^{3}+x^{2}-x+1\\\Phi _{23}(x)&=x^{22}+x^{21}+x^{20}+x^{19}+x^{18}+x^{17}+x^{16}+x^{15}+x^{14}+x^{13}+x^{12}\\&\qquad \quad +x^{11}+x^{10}+x^{9}+x^{8}+x^{7}+x^{6}+x^{5}+x^{4}+x^{3}+x^{2}+x+1\\\Phi _{24}(x)&=x^{8}-x^{4}+1\\\Phi _{25}(x)&=x^{20}+x^{15}+x^{10}+x^{5}+1\\\Phi _{26}(x)&=x^{12}-x^{11}+x^{10}-x^{9}+x^{8}-x^{7}+x^{6}-x^{5}+x^{4}-x^{3}+x^{2}-x+1\\\Phi _{27}(x)&=x^{18}+x^{9}+1\\\Phi _{28}(x)&=x^{12}-x^{10}+x^{8}-x^{6}+x^{4}-x^{2}+1\\\Phi _{29}(x)&=x^{28}+x^{27}+x^{26}+x^{25}+x^{24}+x^{23}+x^{22}+x^{21}+x^{20}+x^{19}+x^{18}+x^{17}+x^{16}+x^{15}\\&\qquad \quad +x^{14}+x^{13}+x^{12}+x^{11}+x^{10}+x^{9}+x^{8}+x^{7}+x^{6}+x^{5}+x^{4}+x^{3}+x^{2}+x+1\\\Phi _{30}(x)&=x^{8}+x^{7}-x^{5}-x^{4}-x^{3}+x+1.\end{aligned}}} The case of the 105th cyclotomic polynomial is interesting because 105 is the least positive integer that is the product of three distinct odd prime numbers (3×5×7) and this polynomial is the first one that has a coefficient other than 1, 0, or −1: Φ 105 ( x ) = x 48 + x 47 + x 46 − x 43 − x 42 − 2 x 41 − x 40 − x 39 + x 36 + x 35 + x 34 + x 33 + x 32 + x 31 − x 28 − x 26 − x 24 − x 22 − x 20 + x 17 + x 16 + x 15 + x 14 + x 13 + x 12 − x 9 − x 8 − 2 x 7 − x 6 − x 5 + x 2 + x + 1. {\displaystyle {\begin{aligned}\Phi _{105}(x)={}&x^{48}+x^{47}+x^{46}-x^{43}-x^{42}-2x^{41}-x^{40}-x^{39}+x^{36}+x^{35}+x^{34}\\&{}+x^{33}+x^{32}+x^{31}-x^{28}-x^{26}-x^{24}-x^{22}-x^{20}+x^{17}+x^{16}+x^{15}\\&{}+x^{14}+x^{13}+x^{12}-x^{9}-x^{8}-2x^{7}-x^{6}-x^{5}+x^{2}+x+1.\end{aligned}}} == Properties == === Fundamental tools === The cyclotomic polynomials are monic polynomials with integer coefficients that are irreducible over the field of the rational numbers. Except for n equal to 1 or 2, they are palindromes of even degree. The degree of Φ n {\displaystyle \Phi _{n}} , or in other words the number of nth primitive roots of unity, is φ ( n ) {\displaystyle \varphi (n)} , where φ {\displaystyle \varphi } is Euler's totient function. The fact that Φ n {\displaystyle \Phi _{n}} is an irreducible polynomial of degree φ ( n ) {\displaystyle \varphi (n)} in the ring Z [ x ] {\displaystyle \mathbb {Z} [x]} is a nontrivial result due to Gauss. Depending on the chosen definition, it is either the value of the degree or the irreducibility which is a nontrivial result. The case of prime n is easier to prove than the general case, thanks to Eisenstein's criterion. A fundamental relation involving cyclotomic polynomials is x n − 1 = ∏ 1 ⩽ k ⩽ n ( x − e 2 i π k n ) = ∏ d ∣ n ∏ 1 ⩽ k ⩽ n gcd ( k , n ) = d ( x − e 2 i π k n ) = ∏ d ∣ n Φ n d ( x ) = ∏ d ∣ n Φ d ( x ) . {\displaystyle {\begin{aligned}x^{n}-1&=\prod _{1\leqslant k\leqslant n}\left(x-e^{2i\pi {\frac {k}{n}}}\right)\\&=\prod _{d\mid n}\prod _{1\leqslant k\leqslant n \atop \gcd(k,n)=d}\left(x-e^{2i\pi {\frac {k}{n}}}\right)\\&=\prod _{d\mid n}\Phi _{\frac {n}{d}}(x)=\prod _{d\mid n}\Phi _{d}(x).\end{aligned}}} which means that each n-th root of unity is a primitive d-th root of unity for a unique d dividing n. The Möbius inversion formula allows Φ n ( x ) {\displaystyle \Phi _{n}(x)} to be expressed as an explicit rational fraction: Φ n ( x ) = ∏ d ∣ n ( x d − 1 ) μ ( n d ) , {\displaystyle \Phi _{n}(x)=\prod _{d\mid n}(x^{d}-1)^{\mu \left({\frac {n}{d}}\right)},} where μ {\displaystyle \mu } is the Möbius function. This provides a recursive formula for the cyclotomic polynomial Φ n ( x ) {\displaystyle \Phi _{n}(x)} , which may be computed by dividing x n − 1 {\displaystyle x^{n}-1} by the cyclotomic polynomials Φ d ( x ) {\displaystyle \Phi _{d}(x)} for the proper divisors d dividing n, starting from Φ 1 ( x ) = x − 1 {\displaystyle \Phi _{1}(x)=x-1} : Φ n ( x ) = x n − 1 ∏ d < n d \| n Φ d ( x ) . {\displaystyle \Phi _{n}(x)={\frac {x^{n}-1}{\prod _{\stackrel {d\|n}{{}_{d<n}}}\Phi _{d}(x)}}.} This gives an algorithm for computing any Φ n ( x ) {\displaystyle \Phi _{n}(x)} , provided integer factorization and division of polynomials are available. Many computer algebra systems, such as SageMath, Maple, Mathematica, and PARI/GP, have a built-in function to compute the cyclotomic polynomials. === Easy cases for computation === As noted above, if n = p is a prime number, then Φ p ( x ) = 1 + x + x 2 + ⋯ + x p − 1 = ∑ k = 0 p − 1 x k . {\displaystyle \Phi _{p}(x)=1+x+x^{2}+\cdots +x^{p-1}=\sum _{k=0}^{p-1}x^{k}\;.} If n is an odd integer greater than one, then Φ 2 n ( x ) = Φ n ( − x ) . {\displaystyle \Phi _{2n}(x)=\Phi _{n}(-x)\;.} In particular, if n = 2p is twice an odd prime, then (as noted above) Φ 2 p ( x ) = 1 − x + x 2 − ⋯ + x p − 1 = ∑ k = 0 p − 1 ( − x ) k . {\displaystyle \Phi _{2p}(x)=1-x+x^{2}-\cdots +x^{p-1}=\sum _{k=0}^{p-1}(-x)^{k}\;.} If n = pm is a prime power (where p is prime), then Φ p m ( x ) = Φ p ( x p m − 1 ) = ∑ k = 0 p − 1 x k p m − 1 . {\displaystyle \Phi _{p^{m}}(x)=\Phi _{p}(x^{p^{m-1}})=\sum _{k=0}^{p-1}x^{kp^{m-1}}\;.} More generally, if n = pmr with r relatively prime to p, then Φ p m r ( x ) = Φ p r ( x p m − 1 ) . {\displaystyle \Phi _{p^{m}r}(x)=\Phi _{pr}(x^{p^{m-1}})\;.} These formulas may be applied repeatedly to get a simple expression for any cyclotomic polynomial Φ n ( x ) {\displaystyle \Phi _{n}(x)} in terms of a cyclotomic polynomial of square free index: If q is the product of the prime divisors of n (its radical), then Φ n ( x ) = Φ q ( x n / q ) . {\displaystyle \Phi _{n}(x)=\Phi _{q}(x^{n/q})\;.} This allows formulas to be given for the nth cyclotomic polynomial when n has at most one odd prime factor: If p is an odd prime number, and ⁠ ℓ {\displaystyle \ell } ⁠ and m are positive integers, then Φ 2 m ( x ) = x 2 m − 1 + 1 , {\displaystyle \Phi _{2^{m}}(x)=x^{2^{m-1}}+1\;,} Φ p m ( x ) = ∑ j = 0 p − 1 x j p m − 1 , {\displaystyle \Phi _{p^{m}}(x)=\sum _{j=0}^{p-1}x^{jp^{m-1}}\;,} Φ 2 ℓ p m ( x ) = ∑ j = 0 p − 1 ( − 1 ) j x j 2 ℓ − 1 p m − 1 . {\displaystyle \Phi _{2^{\ell }p^{m}}(x)=\sum _{j=0}^{p-1}(-1)^{j}x^{j2^{\ell -1}p^{m-1}}\;.} For other values of n, the computation of the nth cyclotomic polynomial is similarly reduced to that of Φ q ( x ) , {\displaystyle \Phi _{q}(x),} where q is the product of the distinct odd prime divisors of n. To deal with this case, one has that, for p prime and not dividing n, Φ n p ( x ) = Φ n ( x p ) / Φ n ( x ) . {\displaystyle \Phi _{np}(x)=\Phi _{n}(x^{p})/\Phi _{n}(x)\;.} === Integers appearing as coefficients === The problem of bounding the magnitude of the coefficients of the cyclotomic polynomials has been the object of a number of research papers. If n has at most two distinct odd prime factors, then Migotti showed that the coefficients of Φ n {\displaystyle \Phi _{n}} are all in the set {1, −1, 0}. The first cyclotomic polynomial for a product of three different odd prime factors is Φ 105 ( x ) ; {\displaystyle \Phi _{105}(x);} it has a coefficient −2 (see above). The converse is not true: Φ 231 ( x ) = Φ 3 × 7 × 11 ( x ) {\displaystyle \Phi _{231}(x)=\Phi _{3\times 7\times 11}(x)} only has coefficients in {1, −1, 0}. If n is a product of more different odd prime factors, the coefficients may increase to very high values. E.g., Φ 15015 ( x ) = Φ 3 × 5 × 7 × 11 × 13 ( x ) {\displaystyle \Phi _{15015}(x)=\Phi _{3\times 5\times 7\times 11\times 13}(x)} has coefficients running from −22 to 23; also Φ 255255 ( x ) = Φ 3 × 5 × 7 × 11 × 13 × 17 ( x ) {\displaystyle \Phi _{255255}(x)=\Phi _{3\times 5\times 7\times 11\times 13\times 17}(x)} , the smallest n with 6 different odd primes, has coefficients of magnitude up to 532. Let A(n) denote the maximum absolute value of the coefficients of Φ n ( x ) {\displaystyle \Phi _{n}(x)} . It is known that for any positive k, the number of n up to x with A(n) > nk is at least c(k)⋅x for a positive c(k) depending on k and x sufficiently large. In the opposite direction, for any function ψ(n) tending to infinity with n we have A(n) bounded above by nψ(n) for almost all n. A combination of theorems of Bateman and Vaughan states that: 10 on the one hand, for every ε > 0 {\displaystyle \varepsilon >0} , we have A ( n ) < e ( n ( log ⁡ 2 + ε ) / ( log ⁡ log ⁡ n ) ) {\displaystyle A(n)<e^{\left(n^{(\log 2+\varepsilon )/(\log \log n)}\right)}} for all sufficiently large positive integers n {\displaystyle n} , and on the other hand, we have A ( n ) > e ( n ( log ⁡ 2 ) / ( log ⁡ log ⁡ n ) ) {\displaystyle A(n)>e^{\left(n^{(\log 2)/(\log \log n)}\right)}} for infinitely many positive integers n {\displaystyle n} . This implies in particular that univariate polynomials (concretely x n − 1 {\displaystyle x^{n}-1} for infinitely many positive integers n {\displaystyle n} ) can have factors (like Φ n {\displaystyle \Phi _{n}} ) whose coefficients are superpolynomially larger than the original coefficients. This is not too far from the general Landau-Mignotte bound. === Gauss's formula === Let n be odd, square-free, and greater than 3. Then: 4 Φ n ( z ) = A n 2 ( z ) − ( − 1 ) n − 1 2 n z 2 B n 2 ( z ) {\displaystyle 4\Phi _{n}(z)=A_{n}^{2}(z)-(-1)^{\frac {n-1}{2}}nz^{2}B_{n}^{2}(z)} for certain polynomials An(z) and Bn(z) with integer coefficients, An(z) of degree φ(n)/2, and Bn(z) of degree φ(n)/2 − 2. Furthermore, An(z) is palindromic when its degree is even; if its degree is odd it is antipalindromic. Similarly, Bn(z) is palindromic unless n is composite and n ≡ 3 (mod 4), in which case it is antipalindromic. The first few cases are 4 Φ 5 ( z ) = 4 ( z 4 + z 3 + z 2 + z + 1 ) = ( 2 z 2 + z + 2 ) 2 − 5 z 2 4 Φ 7 ( z ) = 4 ( z 6 + z 5 + z 4 + z 3 + z 2 + z + 1 ) = ( 2 z 3 + z 2 − z − 2 ) 2 + 7 z 2 ( z + 1 ) 2 4 Φ 11 ( z ) = 4 ( z 10 + z 9 + z 8 + z 7 + z 6 + z 5 + z 4 + z 3 + z 2 + z + 1 ) = ( 2 z 5 + z 4 − 2 z 3 + 2 z 2 − z − 2 ) 2 + 11 z 2 ( z 3 + 1 ) 2 {\displaystyle {\begin{aligned}4\Phi _{5}(z)&=4(z^{4}+z^{3}+z^{2}+z+1)\\&=(2z^{2}+z+2)^{2}-5z^{2}\\[6pt]4\Phi _{7}(z)&=4(z^{6}+z^{5}+z^{4}+z^{3}+z^{2}+z+1)\\&=(2z^{3}+z^{2}-z-2)^{2}+7z^{2}(z+1)^{2}\\[6pt]4\Phi _{11}(z)&=4(z^{10}+z^{9}+z^{8}+z^{7}+z^{6}+z^{5}+z^{4}+z^{3}+z^{2}+z+1)\\&=(2z^{5}+z^{4}-2z^{3}+2z^{2}-z-2)^{2}+11z^{2}(z^{3}+1)^{2}\end{aligned}}} === Lucas's formula === Let n be odd, square-free and greater than 3. Then Φ n ( z ) = U n 2 ( z ) − ( − 1 ) n − 1 2 n z V n 2 ( z ) {\displaystyle \Phi _{n}(z)=U_{n}^{2}(z)-(-1)^{\frac {n-1}{2}}nzV_{n}^{2}(z)} for certain polynomials Un(z) and Vn(z) with integer coefficients, Un(z) of degree φ(n)/2, and Vn(z) of degree φ(n)/2 − 1. This can also be written Φ n ( ( − 1 ) n − 1 2 z ) = C n 2 ( z ) − n z D n 2 ( z ) . {\displaystyle \Phi _{n}\left((-1)^{\frac {n-1}{2}}z\right)=C_{n}^{2}(z)-nzD_{n}^{2}(z).} If n is even, square-free and greater than 2 (this forces n/2 to be odd), Φ n 2 ( − z 2 ) = Φ 2 n ( z ) = C n 2 ( z ) − n z D n 2 ( z ) {\displaystyle \Phi _{\frac {n}{2}}(-z^{2})=\Phi _{2n}(z)=C_{n}^{2}(z)-nzD_{n}^{2}(z)} for Cn(z) and Dn(z) with integer coefficients, Cn(z) of degree φ(n), and Dn(z) of degree φ(n) − 1. Cn(z) and Dn(z) are both palindromic. The first few cases are: Φ 3 ( − z ) = Φ 6 ( z ) = z 2 − z + 1 = ( z + 1 ) 2 − 3 z Φ 5 ( z ) = z 4 + z 3 + z 2 + z + 1 = ( z 2 + 3 z + 1 ) 2 − 5 z ( z + 1 ) 2 Φ 6 / 2 ( − z 2 ) = Φ 12 ( z ) = z 4 − z 2 + 1 = ( z 2 + 3 z + 1 ) 2 − 6 z ( z + 1 ) 2 {\displaystyle {\begin{aligned}\Phi _{3}(-z)&=\Phi _{6}(z)=z^{2}-z+1\\&=(z+1)^{2}-3z\\[6pt]\Phi _{5}(z)&=z^{4}+z^{3}+z^{2}+z+1\\&=(z^{2}+3z+1)^{2}-5z(z+1)^{2}\\[6pt]\Phi _{6/2}(-z^{2})&=\Phi _{12}(z)=z^{4}-z^{2}+1\\&=(z^{2}+3z+1)^{2}-6z(z+1)^{2}\end{aligned}}} === Sister Beiter conjecture === The Sister Beiter conjecture is concerned with the maximal size (in absolute value) A ( p q r ) {\displaystyle A(pqr)} of coefficients of ternary cyclotomic polynomials Φ p q r ( x ) {\displaystyle \Phi _{pqr}(x)} where p ≤ q ≤ r {\displaystyle p\leq q\leq r} are three odd primes. == Cyclotomic polynomials over a finite field and over the p-adic integers == Over a finite field with a prime number p of elements, for any integer n that is not a multiple of p, the cyclotomic polynomial Φ n {\displaystyle \Phi _{n}} factorizes into φ ( n ) d {\displaystyle {\frac {\varphi (n)}{d}}} irreducible polynomials of degree d, where φ ( n ) {\displaystyle \varphi (n)} is Euler's totient function and d is the multiplicative order of p modulo n. In particular, Φ n {\displaystyle \Phi _{n}} is irreducible if and only if p is a primitive root modulo n, that is, p does not divide n, and its multiplicative order modulo n is φ ( n ) {\displaystyle \varphi (n)} , the degree of Φ n {\displaystyle \Phi _{n}} . These results are also true over the p-adic integers, since Hensel's lemma allows lifting a factorization over the field with p elements to a factorization over the p-adic integers. == Polynomial values == If x takes any real value, then Φ n ( x ) > 0 {\displaystyle \Phi _{n}(x)>0} for every n ≥ 3 (this follows from the fact that the roots of a cyclotomic polynomial are all non-real, for n ≥ 3). For studying the values that a cyclotomic polynomial may take when x is given an integer value, it suffices to consider only the case n ≥ 3, as the cases n = 1 and n = 2 are trivial (one has Φ 1 ( x ) = x − 1 {\displaystyle \Phi _{1}(x)=x-1} and Φ 2 ( x ) = x + 1 {\displaystyle \Phi _{2}(x)=x+1} ). For n ≥ 2, one has Φ n ( 0 ) = 1 , {\displaystyle \Phi _{n}(0)=1,} Φ n ( 1 ) = 1 {\displaystyle \Phi _{n}(1)=1} if n is not a prime power, Φ n ( 1 ) = p {\displaystyle \Phi _{n}(1)=p} if n = p k {\displaystyle n=p^{k}} is a prime power with k ≥ 1. The values that a cyclotomic polynomial Φ n ( x ) {\displaystyle \Phi _{n}(x)} may take for other integer values of x is strongly related with the multiplicative order modulo a prime number. More precisely, given a prime number p and an integer b coprime with p, the multiplicative order of b modulo p, is the smallest positive integer n such that p is a divisor of b n − 1. {\displaystyle b^{n}-1.} For b > 1, the multiplicative order of b modulo p is also the shortest period of the representation of 1/p in the numeral base b (see Unique prime; this explains the notation choice). The definition of the multiplicative order implies that, if n is the multiplicative order of b modulo p, then p is a divisor of Φ n ( b ) . {\displaystyle \Phi _{n}(b).} The converse is not true, but one has the following. If n > 0 is a positive integer and b > 1 is an integer, then (see below for a proof) Φ n ( b ) = 2 k g h , {\displaystyle \Phi _{n}(b)=2^{k}gh,} where k is a non-negative integer, always equal to 0 when b is even. (In fact, if n is neither 1 nor 2, then k is either 0 or 1. Besides, if n is not a power of 2, then k is always equal to 0) g is 1 or the largest odd prime factor of n. h is odd, coprime with n, and its prime factors are exactly the odd primes p such that n is the multiplicative order of b modulo p. This implies that, if p is an odd prime divisor of Φ n ( b ) , {\displaystyle \Phi _{n}(b),} then either n is a divisor of p − 1 or p is a divisor of n. In the latter case, p 2 {\displaystyle p^{2}} does not divide Φ n ( b ) . {\displaystyle \Phi _{n}(b).} Zsigmondy's theorem implies that the only cases where b > 1 and h = 1 are Φ 1 ( 2 ) = 1 Φ 2 ( 2 k − 1 ) = 2 k k > 0 Φ 6 ( 2 ) = 3 {\displaystyle {\begin{aligned}\Phi _{1}(2)&=1\\\Phi _{2}\left(2^{k}-1\right)&=2^{k}&&k>0\\\Phi _{6}(2)&=3\end{aligned}}} It follows from above factorization that the odd prime factors of Φ n ( b ) gcd ( n , Φ n ( b ) ) {\displaystyle {\frac {\Phi _{n}(b)}{\gcd(n,\Phi _{n}(b))}}} are exactly the odd primes p such that n is the multiplicative order of b modulo p. This fraction may be even only when b is odd. In this case, the multiplicative order of b modulo 2 is always 1. There are many pairs (n, b) with b > 1 such that Φ n ( b ) {\displaystyle \Phi _{n}(b)} is prime. In fact, Bunyakovsky conjecture implies that, for every n, there are infinitely many b > 1 such that Φ n ( b ) {\displaystyle \Phi _{n}(b)} is prime. See OEIS: A085398 for the list of the smallest b > 1 such that Φ n ( b ) {\displaystyle \Phi _{n}(b)} is prime (the smallest b > 1 such that Φ n ( b ) {\displaystyle \Phi _{n}(b)} is prime is about γ ⋅ φ ( n ) {\displaystyle \gamma \cdot \varphi (n)} , where γ {\displaystyle \gamma } is Euler–Mascheroni constant, and φ {\displaystyle \varphi } is Euler's totient function). See also OEIS: A206864 for the list of the smallest primes of the form Φ n ( b ) {\displaystyle \Phi _{n}(b)} with n > 2 and b > 1, and, more generally, OEIS: A206942, for the smallest positive integers of this form. == Applications == Using Φ n {\displaystyle \Phi _{n}} , one can give an elementary proof for the infinitude of primes congruent to 1 modulo n, which is a special case of Dirichlet's theorem on arithmetic progressions. === Periodic recursive sequences === The constant-coefficient linear recurrences which are periodic are precisely the power series coefficients of rational functions whose denominators are products of cyclotomic polynomials. In the theory of combinatorial generating functions, the denominator of a rational function determines a linear recurrence for its power series coefficients. For example, the Fibonacci sequence has generating function F ( x ) = F 1 x + F 2 x 2 + F 3 x 3 + ⋯ = x 1 − x − x 2 , {\displaystyle F(x)=F_{1}x+F_{2}x^{2}+F_{3}x^{3}+\cdots ={\frac {x}{1-x-x^{2}}},} and equating coefficients on both sides of F ( x ) ( 1 − x − x 2 ) = x {\displaystyle F(x)(1-x-x^{2})=x} gives F n − F n − 1 − F n − 2 = 0 {\displaystyle F_{n}-F_{n-1}-F_{n-2}=0} for n ≥ 2 {\displaystyle n\geq 2} . Any rational function whose denominator is a divisor of x n − 1 {\displaystyle x^{n}-1} has a recursive sequence of coefficients which is periodic with period at most n. For example, P ( x ) = − 1 + 2 x Φ 6 ( x ) = 1 + 2 x 1 − x + x 2 = ∑ n ≥ 0 P n x n = 1 + 3 x + 2 x 2 − x 3 − 3 x 4 − 2 x 5 + x 6 + 3 x 7 + 2 x 8 + ⋯ {\displaystyle P(x)=-{\frac {1+2x}{\Phi _{6}(x)}}={\frac {1+2x}{1-x+x^{2}}}=\sum _{n\geq 0}P_{n}x^{n}=1+3x+2x^{2}-x^{3}-3x^{4}-2x^{5}+x^{6}+3x^{7}+2x^{8}+\cdots } has coefficients defined by the recurrence P n − P n − 1 + P n − 2 = 0 {\displaystyle P_{n}-P_{n-1}+P_{n-2}=0} for n ≥ 2 {\displaystyle n\geq 2} , starting from P 0 = 1 , P 1 = 3 {\displaystyle P_{0}=1,P_{1}=3} . But 1 − x 6 = Φ 6 ( x ) Φ 3 ( x ) Φ 2 ( x ) Φ 1 ( x ) {\displaystyle 1-x^{6}=\Phi _{6}(x)\Phi _{3}(x)\Phi _{2}(x)\Phi _{1}(x)} , so we may write P ( x ) = ( 1 + 2 x ) Φ 3 ( x ) Φ 2 ( x ) Φ 1 ( x ) 1 − x 6 = 1 + 3 x + 2 x 2 − x 3 − 3 x 4 − 2 x 5 1 − x 6 , {\displaystyle P(x)={\frac {(1+2x)\Phi _{3}(x)\Phi _{2}(x)\Phi _{1}(x)}{1-x^{6}}}={\frac {1+3x+2x^{2}-x^{3}-3x^{4}-2x^{5}}{1-x^{6}}},} which means P n − P n − 6 = 0 {\displaystyle P_{n}-P_{n-6}=0} for n ≥ 6 {\displaystyle n\geq 6} , and the sequence has period 6 with initial values given by the coefficients of the numerator. == See also == Cyclotomic field Aurifeuillean factorization Root of unity == References == == Further reading == Gauss's book Disquisitiones Arithmeticae [Arithmetical Investigations] has been translated from Latin into French, German, and English. The German edition includes all of his papers on number theory: all the proofs of quadratic reciprocity, the determination of the sign of the Gauss sum, the investigations into biquadratic reciprocity, and unpublished notes. Gauss, Carl Friedrich (1801), Disquisitiones Arithmeticae (in Latin), Leipzig: Gerh. Fleischer Gauss, Carl Friedrich (1807) [1801], Recherches Arithmétiques (in French), translated by Poullet-Delisle, A.-C.-M., Paris: Courcier Gauss, Carl Friedrich (1889) [1801], Carl Friedrich Gauss' Untersuchungen über höhere Arithmetik (in German), translated by Maser, H., Berlin: Springer; Reprinted 1965, New York: Chelsea, ISBN 0-8284-0191-8 Gauss, Carl Friedrich (1966) [1801], Disquisitiones Arithmeticae, translated by Clarke, Arthur A., New Haven: Yale, doi:10.12987/9780300194258, ISBN 978-0-300-09473-2; Corrected ed. 1986, New York: Springer, doi:10.1007/978-1-4939-7560-0, ISBN 978-0-387-96254-2 Lemmermeyer, Franz (2000), Reciprocity Laws: from Euler to Eisenstein, Berlin: Springer, doi:10.1007/978-3-662-12893-0, ISBN 978-3-642-08628-1 == External links == Weisstein, Eric W., "Cyclotomic polynomial", MathWorld "Cyclotomic polynomials", Encyclopedia of Mathematics, EMS Press, 2001 [1994] OEIS sequence A013595 (Triangle of coefficients of cyclotomic polynomial Phi_n(x) (exponents in increasing order)) OEIS sequence A013594 (Smallest order of cyclotomic polynomial containing n or −n as a coefficient)
Wikipedia:Cylinder (algebra)#0	In mathematics, the notion of cylindric algebra, developed by Alfred Tarski, arises naturally in the algebraization of first-order logic with equality. This is comparable to the role Boolean algebras play for propositional logic. Cylindric algebras are Boolean algebras equipped with additional cylindrification operations that model quantification and equality. They differ from polyadic algebras in that the latter do not model equality. The cylindric algebra should not be confused with the measure theoretic concept cylindrical algebra that arises in the study of cylinder set measures and the cylindrical σ-algebra. == Definition of a cylindric algebra == A cylindric algebra of dimension α {\displaystyle \alpha } (where α {\displaystyle \alpha } is any ordinal number) is an algebraic structure ( A , + , ⋅ , − , 0 , 1 , c κ , d κ λ ) κ , λ < α {\displaystyle (A,+,\cdot ,-,0,1,c_{\kappa },d_{\kappa \lambda })_{\kappa ,\lambda <\alpha }} such that ( A , + , ⋅ , − , 0 , 1 ) {\displaystyle (A,+,\cdot ,-,0,1)} is a Boolean algebra, c κ {\displaystyle c_{\kappa }} a unary operator on A {\displaystyle A} for every κ {\displaystyle \kappa } (called a cylindrification), and d κ λ {\displaystyle d_{\kappa \lambda }} a distinguished element of A {\displaystyle A} for every κ {\displaystyle \kappa } and λ {\displaystyle \lambda } (called a diagonal), such that the following hold: (C1) c κ 0 = 0 {\displaystyle c_{\kappa }0=0} (C2) x ≤ c κ x {\displaystyle x\leq c_{\kappa }x} (C3) c κ ( x ⋅ c κ y ) = c κ x ⋅ c κ y {\displaystyle c_{\kappa }(x\cdot c_{\kappa }y)=c_{\kappa }x\cdot c_{\kappa }y} (C4) c κ c λ x = c λ c κ x {\displaystyle c_{\kappa }c_{\lambda }x=c_{\lambda }c_{\kappa }x} (C5) d κ κ = 1 {\displaystyle d_{\kappa \kappa }=1} (C6) If κ ∉ { λ , μ } {\displaystyle \kappa \notin \{\lambda ,\mu \}} , then d λ μ = c κ ( d λ κ ⋅ d κ μ ) {\displaystyle d_{\lambda \mu }=c_{\kappa }(d_{\lambda \kappa }\cdot d_{\kappa \mu })} (C7) If κ ≠ λ {\displaystyle \kappa \neq \lambda } , then c κ ( d κ λ ⋅ x ) ⋅ c κ ( d κ λ ⋅ − x ) = 0 {\displaystyle c_{\kappa }(d_{\kappa \lambda }\cdot x)\cdot c_{\kappa }(d_{\kappa \lambda }\cdot -x)=0} Assuming a presentation of first-order logic without function symbols, the operator c κ x {\displaystyle c_{\kappa }x} models existential quantification over variable κ {\displaystyle \kappa } in formula x {\displaystyle x} while the operator d κ λ {\displaystyle d_{\kappa \lambda }} models the equality of variables κ {\displaystyle \kappa } and λ {\displaystyle \lambda } . Hence, reformulated using standard logical notations, the axioms read as (C1) ∃ κ . f a l s e ⟺ f a l s e {\displaystyle \exists \kappa .{\mathit {false}}\iff {\mathit {false}}} (C2) x ⟹ ∃ κ . x {\displaystyle x\implies \exists \kappa .x} (C3) ∃ κ . ( x ∧ ∃ κ . y ) ⟺ ( ∃ κ . x ) ∧ ( ∃ κ . y ) {\displaystyle \exists \kappa .(x\wedge \exists \kappa .y)\iff (\exists \kappa .x)\wedge (\exists \kappa .y)} (C4) ∃ κ ∃ λ . x ⟺ ∃ λ ∃ κ . x {\displaystyle \exists \kappa \exists \lambda .x\iff \exists \lambda \exists \kappa .x} (C5) κ = κ ⟺ t r u e {\displaystyle \kappa =\kappa \iff {\mathit {true}}} (C6) If κ {\displaystyle \kappa } is a variable different from both λ {\displaystyle \lambda } and μ {\displaystyle \mu } , then λ = μ ⟺ ∃ κ . ( λ = κ ∧ κ = μ ) {\displaystyle \lambda =\mu \iff \exists \kappa .(\lambda =\kappa \wedge \kappa =\mu )} (C7) If κ {\displaystyle \kappa } and λ {\displaystyle \lambda } are different variables, then ∃ κ . ( κ = λ ∧ x ) ∧ ∃ κ . ( κ = λ ∧ ¬ x ) ⟺ f a l s e {\displaystyle \exists \kappa .(\kappa =\lambda \wedge x)\wedge \exists \kappa .(\kappa =\lambda \wedge \neg x)\iff {\mathit {false}}} == Cylindric set algebras == A cylindric set algebra of dimension α {\displaystyle \alpha } is an algebraic structure ( A , ∪ , ∩ , − , ∅ , X α , c κ , d κ λ ) κ , λ < α {\displaystyle (A,\cup ,\cap ,-,\emptyset ,X^{\alpha },c_{\kappa },d_{\kappa \lambda })_{\kappa ,\lambda <\alpha }} such that ⟨ X α , A ⟩ {\displaystyle \langle X^{\alpha },A\rangle } is a field of sets, c κ S {\displaystyle c_{\kappa }S} is given by { y ∈ X α ∣ ∃ x ∈ S ∀ β ≠ κ y ( β ) = x ( β ) } {\displaystyle \{y\in X^{\alpha }\mid \exists x\in S\ \forall \beta \neq \kappa \ y(\beta )=x(\beta )\}} , and d κ λ {\displaystyle d_{\kappa \lambda }} is given by { x ∈ X α ∣ x ( κ ) = x ( λ ) } {\displaystyle \{x\in X^{\alpha }\mid x(\kappa )=x(\lambda )\}} . It necessarily validates the axioms C1–C7 of a cylindric algebra, with ∪ {\displaystyle \cup } instead of + {\displaystyle +} , ∩ {\displaystyle \cap } instead of ⋅ {\displaystyle \cdot } , set complement for complement, empty set as 0, X α {\displaystyle X^{\alpha }} as the unit, and ⊆ {\displaystyle \subseteq } instead of ≤ {\displaystyle \leq } . The set X is called the base. A representation of a cylindric algebra is an isomorphism from that algebra to a cylindric set algebra. Not every cylindric algebra has a representation as a cylindric set algebra. It is easier to connect the semantics of first-order predicate logic with cylindric set algebra. (For more details, see § Further reading.) == Generalizations == Cylindric algebras have been generalized to the case of many-sorted logic (Caleiro and Gonçalves 2006), which allows for a better modeling of the duality between first-order formulas and terms. == Relation to monadic Boolean algebra == When α = 1 {\displaystyle \alpha =1} and κ , λ {\displaystyle \kappa ,\lambda } are restricted to being only 0, then c κ {\displaystyle c_{\kappa }} becomes ∃ {\displaystyle \exists } , the diagonals can be dropped out, and the following theorem of cylindric algebra (Pinter 1973): c κ ( x + y ) = c κ x + c κ y {\displaystyle c_{\kappa }(x+y)=c_{\kappa }x+c_{\kappa }y} turns into the axiom ∃ ( x + y ) = ∃ x + ∃ y {\displaystyle \exists (x+y)=\exists x+\exists y} of monadic Boolean algebra. The axiom (C4) drops out (becomes a tautology). Thus monadic Boolean algebra can be seen as a restriction of cylindric algebra to the one variable case. == See also == Abstract algebraic logic Lambda calculus and Combinatory logic—other approaches to modelling quantification and eliminating variables Hyperdoctrines are a categorical formulation of cylindric algebras Relation algebras (RA) Polyadic algebra Cylindrical algebraic decomposition == Notes == == References == Charles Pinter (1973). "A Simple Algebra of First Order Logic". Notre Dame Journal of Formal Logic. XIV: 361–366. Leon Henkin, J. Donald Monk, and Alfred Tarski (1971) Cylindric Algebras, Part I. North-Holland. ISBN 978-0-7204-2043-2. Leon Henkin, J. Donald Monk, and Alfred Tarski (1985) Cylindric Algebras, Part II. North-Holland. Robin Hirsch and Ian Hodkinson (2002) Relation algebras by games Studies in logic and the foundations of mathematics, North-Holland Carlos Caleiro, Ricardo Gonçalves (2006). "On the algebraization of many-sorted logics" (PDF). In J. Fiadeiro and P.-Y. Schobbens (ed.). Proc. 18th int. conf. on Recent trends in algebraic development techniques (WADT). LNCS. Vol. 4409. Springer. pp. 21–36. ISBN 978-3-540-71997-7. == Further reading == Imieliński, T.; Lipski, W. (1984). "The relational model of data and cylindric algebras". Journal of Computer and System Sciences. 28: 80–102. doi:10.1016/0022-0000(84)90077-1. == External links == example of cylindrical algebra by CWoo on planetmath.org
Wikipedia:Cylindrical algebraic decomposition#0	In mathematics, cylindrical algebraic decomposition (CAD) is a notion, along with an algorithm to compute it, that is fundamental for computer algebra and real algebraic geometry. Given a set S of polynomials in Rn, a cylindrical algebraic decomposition is a decomposition of Rn into connected semialgebraic sets called cells, on which each polynomial has constant sign, either +, − or 0. To be cylindrical, this decomposition must satisfy the following condition: If 1 ≤ k < n and π is the projection from Rn onto Rn−k consisting in removing the last k coordinates, then for every pair of cells c and d, one has either π(c) = π(d) or π(c) ∩ π(d) = ∅. This implies that the images by π of the cells define a cylindrical decomposition of Rn−k. The notion was introduced by George E. Collins in 1975, together with an algorithm for computing it. Collins' algorithm has a computational complexity that is double exponential in n. This is an upper bound, which is reached on most entries. There are also examples for which the minimal number of cells is doubly exponential, showing that every general algorithm for cylindrical algebraic decomposition has a double exponential complexity. CAD provides an effective version of quantifier elimination over the reals that has a much better computational complexity than that resulting from the original proof of Tarski–Seidenberg theorem. It is efficient enough to be implemented on a computer. It is one of the most important algorithms of computational real algebraic geometry. Searching to improve Collins' algorithm, or to provide algorithms that have a better complexity for subproblems of general interest, is an active field of research. == Implementations == Mathematica: CylindricalDecomposition QEPCAD -- Quantifier Elimination by Partial Cylindrical Algebraic Decomposition redlog Maple: The RegularChains Library and ProjectionCAD == References == Basu, Saugata; Pollack, Richard; Roy, Marie-Françoise Algorithms in real algebraic geometry. Second edition. Algorithms and Computation in Mathematics, 10. Springer-Verlag, Berlin, 2006. x+662 pp. ISBN 978-3-540-33098-1; 3-540-33098-4 Strzebonski, Adam. Cylindrical Algebraic Decomposition from MathWorld. Cylindrical Algebraic Decomposition in Chapter 6 ("Combinatorial Motion Planning") of Planning algorithms by Steven M. LaValle. Accessed 8 February 2023 Caviness, Bob; Johnson, Jeremy; Quantifier Elimination and Cylindrical Algebraic Decomposition. Texts and Monographs in Symbolic Computation. Springer-Verlag, Berlin, 1998. Collins, George E.: Quantifier elimination for the elementary theory of real closed fields by cylindrical algebraic decomposition, Second GI Conf. Automata Theory and Formal Languages, Springer LNCS 33, 1975. Davenport, James H.; Heintz, Joos: Real quantifier elimination is doubly exponential, Journal of Symbolic Computation, 1988. Volume 5, Issues 1–2, ISSN 0747-7171,
Wikipedia:Cynthia Vinzant#0	Cynthia Vinzant is an American mathematician specializing in real algebraic geometry; her research has also involved algebraic combinatorics, matroid theory, Hermitian matrices, and spectrahedra in convex optimization. She is an associate professor of mathematics at the University of Washington. == Education and career == Vinzant is a 2007 graduate of Oberlin College in Ohio, where she studied mathematics and neuroscience. She completed her Ph.D. in mathematics in 2011 at the University of California, Berkeley, with the dissertation Real Algebraic Geometry in Convex Optimization supervised by Bernd Sturmfels. After working as a Hildebrandt Assistant Professor at the University of Michigan and as a postdoctoral researcher at the Simons Institute for the Theory of Computing in Berkeley, Vinzant obtained a tenure-track assistant professor position at North Carolina State University in 2015. She moved to the University of Washington in 2021 and was promoted to associate professor in 2023. == Recognition == In 2020, Vinzant was named as a Sloan Research Fellow in mathematics and as a von Neumann Fellow at the Institute for Advanced Study. She was elected as a Fellow of the American Mathematical Society in the 2024 class of fellows. Vinzant and colleagues will receive the 2025 Michael and Sheila Held Prize from the National Academy of Sciences. == References == == External links == Home page Cynthia Vinzant publications indexed by Google Scholar
Wikipedia:Cynthia Y. Young#0	Cynthia Yvonne Young (also published as Cynthia Y. Hopen) is an American applied mathematician, textbook author, and academic administrator. Her research has included mathematical modeling of the effects of atmospheric turbulence on electromagnetic radiation with applications to laser-based communication with satellites. She is also the author of a series of textbooks on high school mathematics. She is the founding dean of the Clemson University College of Science. == Education == Young majored in mathematics education at the University of North Carolina at Chapel Hill. After earning a master's degree in mathematical science from the University of Central Florida, she continued her studies as a doctoral student at the University of Washington, where she earned a second master's degree in electrical engineering, and completed a Ph.D. in applied mathematics. Her 1996 doctoral dissertation, The two-frequency mutual coherence function of a Gaussian beam pulse in weak turbulence, was supervised by Akira Ishimaru. == Career == She returned to the University of Central Florida as an assistant professor of mathematics in 1997. At Clemson, her efforts as a professor also included the creation of programs for encouraging mathematics students from underrepresented groups and for mentoring new faculty. She was named as Pegasus Professor of Mathematics in 2015 and, in 2016, as the university's Vice Provost for Faculty Excellence and International Affairs and Global Strategies. She moved to Clemson University to become founding dean of the College of Science in 2017. == Awards and honors == Young was named as an Office of Naval Research Young Investigator in 2001. She was named a Fellow of SPIE in 2007. == Books == Young's books include: Laser Beam Scintillation with Applications (with Larry C. Andrews and Ronald L. Phillips, SPIE Press, 2001) Intermediate Algebra (2nd ed., Wiley, 2009) Algebra and Trigonometry (4th ed. Wiley, 2016; 5th ed., 2021) College Algebra (4th ed. Wiley, 2016; 5th ed., 2021) Precalculus (3rd ed., Wiley, 2017; 4th ed., 2023) Trigonometry (4th ed., Wiley, 2017; 5th ed., 2021) == References ==
Wikipedia:Czesław Olech#0	Czesław Olech (22 May 1931 – 1 July 2015) was a Polish mathematician. He was a representative of the Kraków school of mathematics, especially the differential equations school of Tadeusz Ważewski. == Education and career == In 1954 he completed his mathematical studies at the Jagiellonian University, in Kraków obtained his doctorate at the Institute of Mathematical Sciences in 1958, habilitation in 1962, the title of associate professor in 1966, and the title of professor in 1973. 1970–1986: director of The Institute of Mathematics, Polish Academy of Sciences. 1972–1991: director of Stefan Banach International Mathematical Center in Warsaw. 1979–1986: member of the Executive Committee, International Mathematical Union. 1982–1983: president of the Organizing Committee, International Congress of Mathematicians in Warsaw, 1987–1989: president of the Board of Mathematics, Polish Academy of Sciences. 1990–2002: president of the Scientific Council, Institute of Mathematics of the Polish Academy of Sciences. Czeslaw Olech, often as a visiting professor, was invited by the world's leading mathematical centers in the United States, USSR (later Russia), Canada and many European countries. He cooperated with Solomon Lefschetz, Sergey Nikolsky, Philip Hartman and Roberto Conti, the most distinguished mathematicians involved in the theory of differential equations. Based on joint work with Hartman, he proved the Olech theorem. Lefschetz highly valued Ważewski's school, and especially the retract method, which Olech applied by developing, among other things, control theory. He supervised nine doctoral dissertations, and reviewed a number of theses and dissertations. == Main fields of research interest == Contributions to ordinary differential equations: various applications of Tadeusz Ważewski topological method in studying asymptotic behaviour of solutions; exact estimates of exponential growth of solution of second-order linear differential equations with bounded coefficients; theorems concerning global asymptotic stability of the autonomous system on the plane with stable Jacobian matrix at each point of the plane, results establishing relation between question of global asymptotic stability of an autonomous system and that of global one-to-oneness of a differentiable map; contribution to the question whether unicity condition implies convergence of successive approximation to solutions of ordinary differential equations. Contribution to optimal control theory: establishing a most general version of the so-called bang-bang principle for linear control problem by detailed study of the integral of set valued map; existence theorems for optimal control problem with unbounded controls and multidimensional cost functions; existence of solution of differential inclusions with nonconvex right-hand side; characterization of controllability of convex processes. == Recognition == Honorary doctorates: Vilnius University 1989 Jagiellonian University in Kraków 2006 AGH University of Science and Technology in Kraków 2009. Membership of: PAN Polish Academy of Sciences (member of the Presidium), PAU Polish Academy of Arts when he was there he mastered the skill “I need the maxween Pontifical Academy of Sciences Russian Academy of Sciences Polish Mathematical Society European Mathematical Society American Mathematical Society Awards and honours: State Prize of Poland 1st Class The Commander's Cross of the Order of Polonia Restituta Marin Drinov Golden Medal, Bulgarian Academy of Sciences Bernard Bolzano Golden Medal, Czechoslovak Academy of Sciences Stefan Banach Medal, Polish Academy of Sciences Mikołaj Kopernik Medal, Polish Academy of Sciences == Publications == Szafraniec, F.; Olech, Cz.; Janas, J. (1997). "Włodzimierz Mlak (1931-1994)". Annales Polonici Mathematici. 66: 1–9. doi:10.4064/ap-66-1-1-9. Meisters, Gary H.; Olech, Czesław (1993). "Power-exact, nilpotent, homogeneous matrices". Linear and Multilinear Algebra. 35 (3–4): 225–236. doi:10.1080/03081089308818260. S2CID 121224054. Meisters, Gary H.; Olech, Czeslaw (1991). "Strong nilpotence holds in dimensions up to five only∗". Linear and Multilinear Algebra. 30 (4): 231–255. doi:10.1080/03081089108818109. Olech, C.; Parthasarathy, T.; Ravindran, G. (1991). "Almost N-matrices and linear complementarity". Linear Algebra and Its Applications. 145: 107–125. doi:10.1016/0024-3795(91)90290-D. Olech, Czesław; Meisters, Gary H. (1990). "A Jacobian condition for injectivity of differentiable plane maps". Annales Polonici Mathematici. 51: 249–254. doi:10.4064/ap-51-1-249-254. Olech, Czesław; Lasota, Andrzej (1990). "Zdzisław Opial---a mathematician 1930--1974". Annales Polonici Mathematici. 51: 7–13. doi:10.4064/ap-51-1-7-13. Olech, Czesraw (1990). "The Lyapunov Theorem: Its extensions and applications". Methods of Nonconvex Analysis. Lecture Notes in Mathematics. Vol. 1446. pp. 84–103. doi:10.1007/BFb0084932. ISBN 978-3-540-53120-3. Olech, Czeslaw (1987). "Onn-dimensional extensions of Fatou's lemma". Zeitschrift für Angewandte Mathematik und Physik. 38 (2): 266–272. Bibcode:1987ZaMP...38..266O. doi:10.1007/BF00945411. S2CID 119939965. Aubin, Jean-Pierre; Frankowska, Halina; Olech, Czesław (1986). "Controllability of Convex Processes" (PDF). SIAM Journal on Control and Optimization. 24 (6): 1192–1211. doi:10.1137/0324072. Olech, Czesław (1984). "Decomposability as a substitute for convexity". Multifunctions and Integrands. Lecture Notes in Mathematics. Vol. 1091. pp. 193–205. doi:10.1007/BFb0098812. ISBN 978-3-540-13882-2. Frankowska, Halina; Olech, Czeslaw (1982). "Boundary solutions of differential inclusion". Journal of Differential Equations. 44 (2): 156–165. Bibcode:1982JDE....44..156F. doi:10.1016/0022-0396(82)90011-0. Olech, C. (1976). "Weak lower semicontinuity of integral functionals". Journal of Optimization Theory and Applications. 19: 3–16. doi:10.1007/BF00934048. S2CID 121143492. Olech, Czeslaw (1975). "Existence Theory in Optimal Control Problems - the Underlying Ideas". International Conference on Differential Equations. pp. 612–635. doi:10.1016/B978-0-12-059650-8.50050-8. ISBN 9780120596508. Olech, Czeslaw (1974). "The Characterization of the Weak Closure of Certain Sets of Integrable Functions". SIAM Journal on Control. 12 (2): 311–318. doi:10.1137/0312024. Olech, Czełsaw (1969). "Existence theorems for optimal control problems involving multiple integrals". Journal of Differential Equations. 6 (3): 512–526. Bibcode:1969JDE.....6..512O. doi:10.1016/0022-0396(69)90007-2. Olech, Czesław (1969). "Existence theorems for optimal problems with vector-valued cost function". Transactions of the American Mathematical Society. 136: 159. doi:10.1090/S0002-9947-1969-0234338-5. hdl:2060/19680002338. Olech, Czesław (1968). "On the range of an unbounded vector-valued measure". Mathematical Systems Theory. 2 (3): 251–256. doi:10.1007/BF01694009. hdl:2060/19680006853. S2CID 29086435. Olech, Czesław (1968). "Approximation of set-valued functions by continuous functions". Colloquium Mathematicum. 19 (2): 285–293. doi:10.4064/cm-19-2-285-293. Szegö, G. P.; Olech, C.; Cellina, A. (1968). "On the stability properties of a third order system". Annali di Matematica Pura ed Applicata. 78: 91–103. doi:10.1007/BF02415111. S2CID 119973869. Klee, Victor; Olech, Czeslaw (1967). "Characterizations of a Class of Convex Sets". Mathematica Scandinavica. 20: 290. doi:10.7146/math.scand.a-10839. Pliś, A.; Olech, C. (1967). "Monotonicity assumption in uniqueness criteria for differential equations". Colloquium Mathematicum. 18: 43–58. doi:10.4064/cm-18-1-43-58. Olech, C. (1967). "On a system of integral inequalities". Colloquium Mathematicum. 16: 137–139. doi:10.4064/cm-16-1-137-139. Olech, C.; Lasota, A. (1966). "An optimal solution of Nicoletti's boundary value problem". Annales Polonici Mathematici. 18 (2): 131–139. doi:10.4064/ap-18-2-131-139. Olech, Czesław (1966). "Extremal solutions of a control system". Journal of Differential Equations. 2 (1): 74–101. Bibcode:1966JDE.....2...74O. doi:10.1016/0022-0396(66)90064-7. Olech, Czeslaw (1964). "Global phase- portrait of a plane autonomous system". Annales de l'Institut Fourier. 14: 87–97. doi:10.5802/aif.164. Olech, C.; Mlak, W. (1963). "Integration of infinite systems of differential inequalities". Annales Polonici Mathematici. 13: 105–112. doi:10.4064/ap-13-1-105-112. Meisters, G. H.; Olech, C. (1963). "Schlicht functions". Duke Mathematical Journal. 30: 63–80. doi:10.1215/S0012-7094-63-03008-4. Hartman, Philip; Olech, Czeslaw (1962). "On Global Asymptotic Stability of Solutions of Differential Equations". Transactions of the American Mathematical Society. 104 (1): 154–178. doi:10.2307/1993939. JSTOR 1993939. Olech, C. (1962). "A connection between two certain methods of successive approximations in differential equations". Annales Polonici Mathematici. 11 (3): 237–245. doi:10.4064/ap-11-3-237-245. Olech, C. (1961). "On the asymptotic coincidence of sets filled up by integrals of two systems of ordinary differential equations". Annales Polonici Mathematici. 11: 49–74. doi:10.4064/ap-11-1-49-74. Olech, C. (1960). "A simple proof of a certain result of Z. Opial". Annales Polonici Mathematici. 8: 61–63. doi:10.4064/ap-8-1-61-63. Opial, Z.; Olech, C. (1960). "Sur une inégalité différentielle". Annales Polonici Mathematici. 7 (3): 247–254. doi:10.4064/ap-7-3-247-254. Olech, Czeslaw (1957). "Sur un problème de M. G. Sansone lié à la théorie du synchrotrone". Annali di Matematica Pura ed Applicata. 44: 317–329. doi:10.1007/BF02415206. S2CID 170536734. Olech, C. (1957). "Sur certaines propriétés des intégrales de l'équation y'=f(x,y) dont le second membre est doublement périodique". Annales Polonici Mathematici. 3 (2): 189–199. doi:10.4064/ap-3-2-189-199. Olech, C.; Gołąb, Stanisław (1955). "Contribution à la théorie de la formule simpsonienne des quadratures approchées". Annales Polonici Mathematici. 1: 176–183. doi:10.4064/ap-1-1-176-183. == Notes and references == == External links == Media related to Czesław Olech at Wikimedia Commons
Wikipedia:Czesław Ryll-Nardzewski#0	In functional analysis, a branch of mathematics, the Ryll-Nardzewski fixed-point theorem states that if E {\displaystyle E} is a normed vector space and K {\displaystyle K} is a nonempty convex subset of E {\displaystyle E} that is compact under the weak topology, then every group (or equivalently: every semigroup) of affine isometries of K {\displaystyle K} has at least one fixed point. (Here, a fixed point of a set of maps is a point that is fixed by each map in the set.) This theorem was announced by Czesław Ryll-Nardzewski. Later Namioka and Asplund gave a proof based on a different approach. Ryll-Nardzewski himself gave a complete proof in the original spirit. == Applications == The Ryll-Nardzewski theorem yields the existence of a Haar measure on compact groups. == See also == Fixed-point theorems Fixed-point theorems in infinite-dimensional spaces Markov-Kakutani fixed-point theorem - abelian semigroup of continuous affine self-maps on compact convex set in a topological vector space has a fixed point == References == Andrzej Granas and James Dugundji, Fixed Point Theory (2003) Springer-Verlag, New York, ISBN 0-387-00173-5. A proof written by J. Lurie
Wikipedia:Cândido Lima da Silva Dias#0	Cândido Lima da Silva Dias (Mococa, December 31, 1913 – São Paulo, September 15, 1998) was a Brazilian mathematician and one of the first graduates in mathematics from the Faculty of Philosophy, Sciences and Letters of the University of São Paulo (FFCL). He was an important figure in the creation of the Institute of Mathematics and Statistics at the University of São Paulo. == Biography == Cândido was born in 1913 in the city of Mococa. He was the son of Gabriel Antonio da Silva Dias, an electrical engineer who graduated from the Polytechnic School of the University of São Paulo in 1905, and Adília Lima da Silva Dias, a housewife. Encouraged by his father, Cândido began to learn and play with numbers, as well as arithmetic. To continue his studies, he moved to São Paulo, where he studied at the Colégio Franco Brasileiro. He enrolled at the Polytechnic School in February 1932, where he graduated as a surveyor in 1934. In the same year, he entered the mathematics course at the Faculty of Philosophy, Sciences and Letters of the University of São Paulo. In 1937, Cândido married Odila Leite Ribeiro and had four children, all of whom became university professors. == Career == After graduating in 1936, Cândido was appointed second scientific assistant to the Italian mathematician Luigi Fantappiè, a visiting professor at the University of São Paulo who was working on mathematical analysis. In 1937, he was promoted to first assistant, where he worked on theory and practice. Also in 1937, at a seminar, he was introduced to the theory of algebra developed by Gaetano Scorza, another Italian mathematician. Between 1939 and 1941, he was responsible for the applied mathematics course. Throughout his career, he was a visiting researcher at Harvard and the University of Chicago, and in 1949 he was elected President of the Mathematical Society of São Paulo. He was crucial in founding the Institute of Mathematics and Statistics at the University of São Paulo, of which he was its first director, and the National Institute for Pure and Applied Mathematics. == Retirement and death == Cândido retired from the University of São Paulo in 1978 and the following year became a professor at the Mathematics Institute of the Federal University of São Carlos, from where he retired in 1990. He died in the city of São Paulo on September 15, 1998, at the age of 84. == See also == Maria Laura Moura Mouzinho Leite Lopes Carlos Benjamin de Lyra == References ==
Wikipedia:D. B. Singh#0	Dinesh Bahadur Singh is a career India civil servant, mathematician and a scholar who formerly served as Secretary of Rajya Sabha and Rajya Sabha Secretariat, Parliament of India, i.e. the Upper House in the Indian Parliament (similar to the House of Lords but different as most Rajya Sabha members are elected by people's representatives, unlike most of the members of the House of Lords who have peerages bestowed upon them). He was appointed as Advisor in Rajya Sabha in 2014 and previously served as Additional Secretary and Joint Secretary in Rajya Sabha. He is a 1981 batch Central Secretariat Service officer. == Early life and education == D.B. Singh has degrees in Bachelor of Science, Master of Science and Doctor of Philosophy in Mathematics from Indian Institute of Technology Kanpur. He also has a Bachelor of Laws from Faculty of Law, University of Delhi and a Master of Business Administration from the United Kingdom. == Career == Singh joined the Central Secretariat Service in 1981 after qualifying through the Civil Services Examination. He has undergone professional trainings in prestigious institutes of USA, UK, Japan, Spain and Thailand. He has led delegation of officers of Rajya Sabha to US Congress, Australian Parliament and South African Parliament. He has also led Indian delegation for negotiation of bi-lateral investment and promotion treaties to various countries. He has previously served as Joint Secretary of Rajya Sabha and has also worked in Department of Economic Affairs, Constitution Review Commission and Ministry of Law and Justice. He also served as Officer on Special Duty to Suresh Pachouri, then Minister of State in the Ministry of Personnel, Public Grievances and Pensions. == Recognition == He is the first non Indian Administrative Service officer and the first retired civil servant to hold the position of Secretary in Rajya Sabha. == References == == External links == "Dr. DB Singh, Secretary of Rajya Sabha". Rajya Sabha, Parliament of India. Archived from the original on 19 April 2016. Rajya Sabha Secretariat Complete list of officials at Rajya Sabha "Vice President of India provokes editors: '24X7 agitation' putting pressures". The Tribune (Chandigarh). Archived from the original on 19 April 2016. Appointment of DB Singh as Secretary at Rajya Sabha Dr. DB Singh Blog News
Wikipedia:D. N. Mishra#0	Devendra Nath Mishra was an Indian mathematician, and academic administrator. He was the 19th Vice-Chancellor of Banaras Hindu University from February 1994 to June 1995. Mishra died in 2020. == References ==
Wikipedia:D. S. Malik#0	Davender Singh Malik (14 August 1958 – 13 May 2025) was an Indian-American mathematician and professor of mathematics and computer science at Creighton University. == Education == Malik attended the University of Delhi in New Delhi, India, receiving his bachelor's and master's degrees in mathematics, where he won the Prof. Ram Behari Gold Medal in 1980 for his high marks. Then at the University of Waterloo in Ontario, Canada, he received a master's degree in pure mathematics. In the United States, Malik went to Ohio University, earning an M.S. in computer science, and a Ph.D. in mathematics in 1985, writing his dissertation on "A Study of Q-Hypercyclic Rings." == Career == In 1985, Malik joined the faculty of Creighton University, teaching in the mathematics department. In 2013 he became the first holder of the Frederick H. and Anna K. Scheerer Endowed Chair in Mathematics. His research focused on ring theory, abstract algebra, information science, and fuzzy mathematics, including fuzzy automata theory, fuzzy logic, and applications of fuzzy set theory in other disciplines. In the academic community, Malik was a member of the American Mathematical Society and Phi Kappa Phi. Within his community, he co-created a Creighton program in which faculty help area high school students pursue scientific research, to be published in their own student journal. Malik published more than 45 papers and 18 books. He created a computer science line of textbooks that includes extensive and complete programming examples, exercises, and case studies throughout using programming languages such as C++ and Java. Malik died on 13 May 2025, at the age of 66. == Books == The books he wrote include: Programming C++ Programming: From Problem Analysis to Program Design (1st ed., 2002; 8th ed. 2017) C++ Programming: Program Design Including Data Structures (1st ed., 2002; 8th ed. 2017) Data Structures Using C++ (1st ed., 2003; 2nd ed. 2010) Data Structures Using Java (2003) Java programming: From Problem Analysis to Program Design (1st ed., 2003; 5th ed. 2012) Java programming: Program Design including Data structures (2006) Java programming: Guided Learning With Early Objects (2009) Introduction to C++ Programming, Brief Edition (2009) Mathematics Fundamentals of Abstract Algebra (1997) Fuzzy Commutative Algebra (1998) Fuzzy Discrete Structures (2000) Fuzzy Mathematics in Medicine (2000) Fuzzy Automata and Languages: Theory and Applications (2002) Fuzzy Semigroups (2003) Application of Fuzzy Logic to Social Choice Theory (2015) == References == == External links == Faculty webpage at Creighton University
Wikipedia:Da Ruan#0	The ruan (Chinese: 阮; pinyin: ruǎn) is a traditional Chinese plucked string instrument. It is a lute with a fretted neck, a circular body, and four strings. Its four strings were formerly made of silk but since the 20th century they have been made of steel (flatwound for the lower strings). The modern ruan has 24 frets with 12 semitones on each string, which has greatly expanded its range from a previous 13 frets. The frets are commonly made of ivory or in recent times of metal mounted on wood. The metal frets produce a brighter tone as compared to the ivory frets. It is sometimes called ruanqin, particularly in Taiwan. == Sizes == The ruan comes in a family of five sizes: soprano: gaoyinruan (高音阮, lit. "high pitched ruan"; tuning: G3-D4-G4-D5) alto: xiaoruan (小阮, lit. "small ruan"; tuning: D3-A3-D4-A4) tenor: zhongruan (中阮, lit. "medium ruan"; tuning: G2-D3-G3-D4) bass: daruan (大阮, lit. "large ruan"; tuning: D2-A2-D3-A3) contrabass: diyinruan (低音阮, lit. "low pitched ruan"; tuning: G1-D2-G2-D3) The ruan is now most commonly used in Chinese opera and the Chinese orchestra, where it belongs to the plucked string (弹拨乐 or chordophone) section. == Playing techniques and usage == The instrument can be played using a plectrum similar to a guitar pick (formerly made of animal horn, but today often plastic), or using a set of two or five acrylic nails that are affixed to the fingers with adhesive tape. Mainstream ruan players use plectrums, though there are some schools which teach the fingernail technique, similar to that of the pipa. Pipa players who play ruan as a second instrument also often use their fingernails. Plectrums produce a louder and more clear tone, while fingernails allow the performance of polyphonic solo music. The instrument produces a mellow tone. In Chinese orchestras, only the zhongruan and daruan are commonly used, to fill in the tenor and bass section of the plucked string section. Occasionally the gaoyinruan is used to substitute the high-pitched liuqin. Daruan soloists generally use the D-A-D-A tuning, as it allows for the easy performance of diatonic chords. Some orchestral players tune to C-G-D-A, which is exactly the same as cello tuning. The advantage of using C-G-D-A in orchestras is so that the daruan can easily double the cello part. A ruan ensemble (重奏) consists of two or more members of the ruan family, for instance, an ensemble of the xiaoruan, zhongruan and daruan. The wide range covered by the ruan, its easily blended tone quality, and the variety of soprano, alto, tenor, bass, and contrabass instruments all make ruan ensembles very effective in playing polyphonic music. == History == Ruan may have a history of over 2,000 years, the earliest form may be the qin pipa (秦琵琶), which was then developed into ruanxian (named after Ruan Xian, 阮咸), shortened to ruan (阮). In old Chinese texts from the Han to the Tang dynasty, the term pipa was used as a generic term for a number plucked chordophones, including ruan, therefore does not necessarily mean the same as the modern usage of pipa which refers only to the pear-shaped instrument. According to the Pipa Annals 《琵琶赋》 by Fu Xuan (傅玄) of the Western Jin dynasty, the pipa was designed after revision of other Chinese plucked string instruments of the day such as the Chinese zither, zheng (筝) and zhu (筑), or konghou (箜篌), the Chinese harp. However, it is believed that ruan may have been descended from an instrument called xiantao (弦鼗) which was constructed by labourers on the Great Wall of China during the late Qin dynasty (hence the name Qin pipa) using strings stretched over a pellet drum. The antecedent of ruan in the Qin dynasty (221 BC – 206 BC), i.e. the Qin pipa, had a long, straight neck with a round sound box in contrast to the pear-shape of pipa of later dynasties. The name of "pipa" is associated with "tantiao" (彈挑), a right hand techniques of playing a plucked string instrument. "Pi" (琵), which means "tan" (彈), is the downward movement of plucking the string. "Pa" (琶), which means "tiao" (挑), is the upward movement of plucking the string. The present name of the Qin pipa, which is "ruan", was not given until the Tang dynasty (8th century). During the reign of Empress Wu Zetian (武則天) (about 684–704 AD), a copper instrument that looked like the Qin pipa was discovered in an ancient tomb in Sichuan (四川). It had 13 frets and a round sound box. It was believed that it was the instrument which the Eastern Jin (東晉) musician Ruan Xian (阮咸) loved to play. Ruan Xian was a scholar in the Three Kingdoms Eastern Jin (三國東晉) dynasty period (3rd century). He and six other scholars disliked the corrupt government, so they gathered in a bamboo grove in Shanyang (山陽, now in Henan [河南] province). They drank, wrote poems, played music and enjoyed the simple life. The group was known as the Seven Sages of the Bamboo Grove (竹林七賢). Since Ruan Xian was an expert and famous in playing an instrument that looked like the Qin pipa, the instrument was named after him as ruanxian (阮咸) when the copper Qin pipa was found in a tomb during the Tang dynasty. Today it is shortened to ruan (阮). Also during the Tang dynasty, a ruanxian was brought to Japan from China. Now this ruanxian is still stored in Shosoin of the Nara National Museum in Japan. The ruanxian was made of red sandalwood and decorated with mother of pearl inlay. The ancient ruanxian shows that the look of today's ruan has not changed much since the 8th century. Nowadays, although the ruan was never as popular as the pipa, the ruan has been divided into several smaller and better-known instruments within the recent few centuries, such as yueqin ("moon" lute, 月琴) and qinqin (Qin [dynasty] lute, 秦琴) . The short-necked yueqin, with no sound holes, is now used primarily in Beijing opera accompaniment. The long-necked qinqin is a member of both Cantonese (廣東) and Chaozhou (潮州) ensembles. The famed Tang poet Bai Juyi (白居易) once penned a poem about the ruan, entitled "Having a Little Drink and Listening to the Ruanxian with the Deputy Minister of Linghu" 《和令狐仆射小饮听阮咸》 (He Linghu Puye Xiao Yin Ting Ruanxian): 《和令狐仆射小饮听阮咸》 Having a Little Drink and Listening to the Ruanxian with the Deputy Minister of Linghu (He Linghu Puye Xiao Yin Ting Ruanxian) 作者：白居易（唐） by Bai Juyi (Tang dynasty, 772–846) 掩抑复凄清，非琴不是筝。 Gloom and melancholy compounded with misery and desolation; It's not a qin, and neither is it a zheng. 还弹乐府曲，别占阮家名。 It still plays yuefu songs, And also bears the Ruan family name. 古调何人识，初闻满座惊。 Of ancient melodies, who [today] knows them? [Yet] upon first listen, all those in attendance are left in awe. 落盘珠历历，摇佩玉琤琤。 Pearls fall on a platter, one by one; Shaken pendants of jade jangle. 似劝杯中物，如含林下情。 As if to urge [listeners to drain] the contents of their winecups, Or to harbor emotions [such as one might feel while lying] beneath a grove [of flowering plum trees]. 时移音律改，岂是昔时声。 As the times change, so too does music; Can this be the sound of former times? == Ruan and Pipa == A small pipa was found in murals of tombs in Liaoning (遼寧) province in northeastern China. The date of these tombs is about late Eastern Han (東漢) or Wei (魏) period (220–265 AD). However, the pear-shaped pipa was not brought to China from Dunhuang (敦煌, now in northwestern China) until the Northern Wei period (386–524 AD) when ancient China traded with the western countries through the Silk Road (絲綢之路). Evidence was shown on the Dunhuang Caves frescoes that the frescoes contain a large number of pipa, and they date to 4th to 5th century. During the Han period (206 BC-220 AD), Lady Wang Zhaojun (王昭君, known as one of the Four Beauties [四大美人] in ancient China) departed mainland to the west and married the Grand Khan of the Huns. The marriage was meant to maintain peace between the two ancient countries. On her way to the west, she carried a pipa on the horse. Looking back today, her pipa must have been a ruan-type instrument with a round sound box, since the pear-shaped pipa was not brought to China until the Northern Wei dynasty after the Han dynasty. However, in almost all the portraits and dramas, Lady Zhaojun's pipa is displayed inaccurately. The pipa is usually shown with a pear-shaped sound box (as in today's pipa), rather than a round sound box. Note that the frets on all Chinese lutes are high so that the fingers never touch the actual body—distinctively different from western fretted instruments. This allows for a greater control over timbre and intonation than their western counterparts, but makes chordal playing more difficult. == Laruan (bowed ruan) == In addition to the plucked ruan instruments mentioned above, there also exist a family of bowed string instruments called lāruǎn and dalaruan (literally "bowed ruan" and "large bowed ruan"). Both are bowed bass register instruments designed as alternatives to the gehu and diyingehu in large orchestras of Chinese traditional instruments. These instruments correspond to the cello and double bass in range. Chinese orchestras currently using the laruan and dalaruan include the China National Traditional Orchestra and Central Broadcasting National Orchestra, the latter formerly conducted by the late maestro Peng Xiuwen (彭修文). == Repertoire == A famous work in the zhongruan repertoire is the zhongruan concerto "Reminiscences of Yunnan" 《云南回忆》 by Liu Xing (刘星, b. China, 1962), the first full-scale concerto for the zhongruan and the Chinese orchestra. This work finally established the zhongruan as an instrument capable of playing solo with the Chinese orchestra. Some works for the ruan: 《满江红》 Red Fills the River – zhongruan concerto 《汉琵琶情》 Love of the Han Pipa – zhongruan concerto 《玉关引》 Narration of Yuguan – ruan quartet 《山韵》 Mountain Tune – zhongruan concerto 《塞外音诗》 Sound Poem Beyond The Great Wall- zhongruan concerto 《泼水节》The Water Festival- Ruan Tecerto 《睡莲》 Water Lilies- zhongruan solo 《火把节之夜》 Night of the Torch Festival- zhongruan solo 吴俊生* – Fernwood "Nightingale" 《翠华山的传说》 Some of Lin Jiliang's compositions for the ruan: 《石头韵》《凤凰花开》 Flowers Open in Fenghuang Translation from MDBG.net 《满江红》《侗歌》《草原抒怀》《牧马人之歌》《石林夜曲》 Some of Liu Xing's compositions for the ruan: 《云南回忆》 Reminiscences of Yunnan, zhongruan concerto 《第二中阮协奏曲》Second Zhongruan Concerto 《山歌》, zhongruan solo 《月光》, zhongruan solo 《孤芳自赏》, zhongruan solo 《天地之间》, zhongruan solo 《第六号-异想天开》, zhongruan duet 《第七号- 夜长梦多》, zhongruan solo 《第十一号-心不在焉》, zhongruan solo 《流连忘返》, zhongruan solo 《随心所欲》, zhongruan solo 《回心转意》, zhongruan solo 《来日方长》, zhongruan solo 《无所事事》, zhongruan solo 《水到渠成》, zhongruan solo 《心旷神怡》, zhongruan solo Some of Ning Yong's compositions for the ruan: 《拍鼓翔龙》 Flying Dragons in Drum Beats, zhongruan solo (composed with Lin Jiliang) 《丝路驼铃》 Camel Bells on the Silk Road, zhongruan/ daruan solo 《篮关雪》 Snow at Lan Guan, zhongruan solo 《终南古韵》 Ancient Tune of Zhongnan, zhongruan/ daruan solo 《望秦川》 zhongruan solo == Notable players and composers == Cui Jun Miao (崔军淼) Ding Xiaoyan (丁晓燕) Fei Jian Rong(费剑蓉) Feng Mantian (冯满天) Lin Jiliang (林吉良) Liu Bo (刘波) Liu Xing (刘星) Miao Xiaoyun (苗晓芸) Ning Yong (宁勇) NiNi Music Ruan Shi Chun (阮仕春) Shen Fei (沈非) Su Handa (苏涵达) Tan Su-Min, Clara(陈素敏) Wang Zhong Bing (王仲丙) Wei Wei(魏蔚) Wei Yuru (魏育茹) Wu Qiang (吴强) Xu Yang (徐阳) Zhang Rong Hui (张蓉晖) == Makers == === Beijing === Xinghai (星海) === Shanghai === Dunhuang (敦煌) === Suzhou === Huqiu (虎丘) == See also == Zhongruan Đàn nguyệt == References == == External links == === More information === Seven Sages of the Bamboo Grove Ruan at MelodyofChina.com An introduction to the ruan on the Chinese Culture Channel in traditional Chinese Ruan Yahoo Group Ruan photographs (fifth, sixth, and seventh rows) === Listening === Audio of zhongruan
Wikipedia:Dainton Report#0	The Dainton Report was a 1968 British government report on secondary schools in the UK, also known as The Swing away from Science. == History == The report was produced in March 1968 by Frederick Dainton, Baron Dainton FRS, who was the Vice-Chancellor of the University of Nottingham. In October 1966 there were 1,600 vacant places in the science and technology faculties of British universities. Sheffield-born Fred Dainton was a Professor of Chemistry for fifteen years, researching radiation chemistry, and was later knighted in 1971. He died in December 1997. The report was published by the Dainton Committee of the Department for Education and Science. The Dainton Committee was formed in 1965 by the Council for Scientific Policy. == Content == The report found that there was a reduction in the numbers of people entering science and engineering at university. The report was also known as the Enquiry into the Flow of Candidates in Science and Technology into Higher Education. Around 40,000 of those at sixth-form were studying science in 1964; he predicted this would lower to around 30,000 in 1971, when the total numbers in sixth-form would rise from 107,000 to 130,000. One fifth of those taking science at sixth-form were female. In 1962 around 42% were opting for science at school, but by 1967 it was 31%; it was predicted this would lower to 25% by 1971. 45.9% of university admissions were to science courses, which had lowered to 40.6% in 1967. At the same time, those opting to study social sciences at university was rapidly increasing, doubling in proportion from 1962 to 1967. The report wanted to improve the position of science, technology, and engineering in the education system, and society. The report wanted to increase the supply of scientists and technologists, which the report claimed, was limited by the immature and misinformed choices of 13 and 14 year olds at school, making premature decisions, often unfavourable to a career in science or technology. The report claimed that fewer people were choosing science, because of schools making 13 or 14 year olds decide between either arts or sciences. The report wanted to create less irrevocable decisions at school, where science was not part of that decision. 14 year olds were deciding not to study science, which was largely irrevocable. The report had found that heavy factual content of science courses had deterred 14 year olds from choosing science. The cut-off point of studying science for many at school was the age of 14. === Recommendations === All should study Mathematics until the end of secondary school, and should study arts and sciences. Those in the sixth form should study five subjects, not just two or three. The five subjects should include Mathematics, a science, a social study and a language. University entrance should be based on five subjects, not three. Universities should not know the individual A-level results of each candidate, but receive a grade as a whole for all the examinations taken - an overall grade. Universities should provide refresher courses for teachers in the latest know-how, and should not wait for teachers to come to them, but actively visit schools. Science teaching should contain less arid rote learning and dreary experiments, and should be more relevant to human experience. Mathematics should be seen as not only preparation for becoming a scientist, but for application in other walks of life, such as decision making and with organisation. == Effect == The Schools Council had also recommended to universities that entrance should be two GCE A-level passes, with four elective courses chosen by individual schools. Universities were asked to reply to the report later in 1968. == See also == Education in England Making Mathematics Count Secondary education in Scotland Science and Technology Select Committee == References ==
Wikipedia:Dale W. Lick#0	Dale Wesley Lick (born June 1, 1933) is an American mathematician, professor and college president at three universities. == Early life and education == Born in Marlette, Michigan, Lick was raised in the heart of the rural Michigan Thumb. His father was a farmer and he was the younger of two brothers. He graduated from Lapeer High School in 1955 and married his sweetheart Marilyn Foster the following year. He earned his bachelor's degree in mathematics at Michigan State University in 1958. He was a research and teaching assistant at MSU while completing his master's degree in 1959, also in mathematics. == Career == Lick's first job after graduation was across the state near his family home at Port Huron Junior College (now St. Clair County Community College). He was an instructor and chairman of the department of mathematics for a year before moving to California to begin work on his doctorate and teach at the University of Redlands from 1961 to 1963. He subsequently contributed during the 1964 and 1965 academic year as a mathematics TA at the University of California, Riverside, departing after receiving his doctorate to join the faculty at the University of Tennessee as an assistant professor of mathematics for the 1965–66 academic year. Lick completed a postdoctoral fellowship at the Brookhaven National Laboratory from 1967 until 1968, and returned to Tennessee as an associate professor for a year. While at UT, Lick also served as a scientific consultant to the US Atomic Energy Commission in Oak Ridge, Tennessee. Lick became an associate professor at Drexel University and was named head of the mathematics department before departing for Russell Sage College in 1972. He was vice president of academic affairs until 1974 when he became dean of sciences and health professions at Old Dominion University. === Georgia === Lick was inducted as president at Georgia Southern College (now Georgia Southern University) in 1978. In a recent interview, he related the three things he accomplished at Georgia Southern that he was most proud of. The first was establishing a Nursing School. When exploring this goal, his vice-chancellor referred him to a woman named Em Olivia Bevis. She was known in nursing circles throughout Georgia as an influential nurse educator. She worked with the Georgia Southern faculty to develop a BSN program with an emphasis on rural nursing. The second was restarting football. The Georgia Normal School in the mid-1920s had a successful football team until the start of World War II, when football was discontinued. Lick knew he would need an exceptional head coach to be successful, and everyone he talked to told him Erk Russell would be great. "Erk" was a great athlete at Auburn University, lettering in four sports. He was a popular and charismatic defensive coordinator at the University of Georgia for 17 years. Lick gave Georgia president Fred Davison a courtesy call. In disbelief, Davison told Lick to speak to head coach Vince Dooley. Lick said, "Vince laughed when I told him." Coach Russell visited Statesboro three times. His biggest attraction was the opportunity to build a program from the ground up. The college was able to provide an acceptable compensation package, the community pledged their financial support, and the coach signed on. The third was related to the second: he insisted on a marching band to go with football. Lick simply announced that part of the funds raised for football would support the band. After nine years in Statesboro he sought new challenges as the president of the University of Maine. === Maine === While there, he led the group that created the American University in Bulgaria. === Florida === Lick was invested as president of Florida State University in 1991 and during his tenure, he is credited with beginning construction of the University Center, the installation of the National High Magnetic Field Laboratory, establishment of the FSU London Study Center and a revived emphasis on diversity and women in hiring. Unfortunately, his presidency ended early due to controversy. == Controversies == A quote from Lick in 1989 would return to haunt him: "A black athlete can actually outjump a white athlete on the average, so they're better at the game." === Maine === When Lick was president of the University of Maine in 1989, he was in a meeting and was asked how to increase minority involvement in Maine university sports. A journalist from the student newspaper tape-recorded the conversation. His infamous reply referred to basketball. He made the above quote and followed it up with: "All you need to do is turn to the NCAA playoffs in basketball to see that the bulk of the players on those outstanding teams are black." Students, faculty, the chancellor and other officials criticized the president, saying his comments were "insensitive and showed poor judgment". Lick apologized and stated that he thought he was making a statement of science, not opinion. "The last thing I wanted to do was offend people." The controversy eventually died down and Lick faced no punishment. === Michigan === After serving two years as president of FSU, the presidency of Lick's alma mater, Michigan State University opened in 1993. Lick claimed he was recruited and could not pass the opportunity. Initially considered the leading candidate, his popularity took a nosedive when the Detroit Free Press repeated Licks statements from 1989. After discussion and deliberation, the presidential search was restarted and Lick returned to FSU. Eventually, the job was offered to and accepted by M. Peter McPherson. === Florida === Upon his return to Tallahassee, Lick found his relationship with some influential alumni and faculty had chilled. When Lick was hired, his primary focus was to be fund-raising. They were puzzled why he considered leaving immediately before FSU began the biggest donor campaign in the school's history. His loyalty was questioned. Lick had failed to notify the University Chancellor Charles Reed before flying to Michigan twice within one week to interview. The Florida Board of Regents were unaware of Lick's interest until the media revealed that Lick was a finalist for the job. Upon Lick's return, he was in limbo for three weeks. Both the Chancellor and a committee of Regents were tasked with evaluating Lick's performance and his standing with faculty, alumni and boosters. Lick attempted to keep his job by visiting constituents on campus and off; asking people to write letters. Prominent FSU alumnus Jim King, a State Representative from Jacksonville, stated that many alumni were sensitive to being considered Licks second-choice, and the fact that he was rejected by his alma mater. King added that another problem with Lick was that "he was never able to establish a constituency." Al Lawson, a State Representative from Tallahassee, insisted that Lick "might not have fit in with some of the good ol' boys network, but you couldn't ask for a person of more quality." He called the regent's actions a "hatchet job". Evaluating committee members worked individually to avoid the requirements for public meetings. Each interviewed people from FSU before talking to Lick. Tom Petway was committee chairman who also chaired the search committee that selected Lick. After meeting Lick for lunch, Petway returned to Jacksonville and Lick cancelled his afternoon appointments. One newspaper noted the irony that Lick was rejected at Michigan State by black members of the community who viewed the 1989 remarks as racist. Lick's biggest supporters at Florida State were African-American alumni, students, faculty and staff. Prior to his hire in 1991, Lick's 1989 comments were discussed at length, but were not significant in his selection. Upon his return to Florida, the "racist" remarks weren't even discussed. Everyone insisted that resigning was Lick's decision, but it was apparent that he was told that did not have the support to remain president. Reed said they spoke to negotiate the terms of a resignation. Reed noted that "He (Lick) said he came to the conclusion it would be in the best interest of FSU and wanted to talk to me about how to make the transition." Lick resigned as FSU president effective August 31, 1993. During a formal statement, Lick stated, "Obviously, it is personally painful and difficult for me to leave the presidency. Controversy surrounding myself as an individual, however, cannot be allowed to hinder progress for Florida State University. While I am disappointed that I will no longer be able to lead all its efforts, I am pleased that I will remain at Florida State in a faculty position and continue to contribute to its future." Lick paused before continuing, "I'm not feeling very good about the circumstances and where I am, but maybe it's a blessing in disguise. I tend to believe that things have a way of working out for the best, whatever happens, if you try hard to make it so. And I'm going to try very hard to make it so." === Post presidency === Lick became a tenured professor with appointments his academic field, mathematics, and educational policy. He was also a contributor to the FSU Learning Systems Institute. His $165,000 salary was cut by 10 percent on September 1, 1993, and another 10 percent in 1994. His contract was changed to nine months in 1995 with a reduction of 20 percent in pay. Lick and family remained in the president's house until January 1, 1994. He went on sabbatical for the fall of 1993 and returned to teach the spring 1994 semester. == Later years == As of 2006, Lick had served as chairman of the board for the technology company, Hylighter LLC, and has been a member on the board of trustees at Graceland University since 2016. His name is included in more than 50 international and national biography listings and published over 100 research articles, 285 newspaper columns and eight books. === Family === As of 2023, Lick was residing in Tallahassee with his wife of 67 years, Marilyn. She is an ordained minister in the Community of Christ and has stayed active in both church and community. They enjoy square dancing and returning to Michigan to see family and friends. The couple have two children, Ron and Diana. While her father Dale was FSU president, Diana completed a MSW degree and Ron was a golf professional. The couple also enjoy their three grandchildren. === Honors === Lick was the recipient of the 2010 International Peace Prize from the United Cultural Convention, honored by Michigan State University with the 2006 Distinguished Alumni Award; given the One of 40 Alumni Who Make a Difference award by University of California, Riverside and in 2016 recognized by the non-alumni emeritus award, the Distinguished Service Award at FSU. A bronze statue and plaque were placed in front of the Stone College of Education building on the Tallahassee campus in honor of his years as president, part of Legacy Walk III. == Books == Fundamentals of Algebra (1970) Whole-Faculty Study Groups: A Powerful Way to Change Schools and Enhance Learning (1998) New Directions in Mentoring: Creating a Culture of Synergy (1999) Whole-Faculty Study Groups: Creating Student-Based Professional Development (2001) Whole-Faculty Study Groups: Creating Professional Learning Communities That Target Student Learning (2005) Whole-Faculty Study Groups Fieldbook: Lessens Learned and Best Practices From Classrooms, Districts, and Schools (2007) Schoolwide Action Research for Professional Learning Communities: Improving Student Learning Through The Whole-Faculty Study Groups Approach (2008) Schools Can Change: A Step-By-Step Change Creation System for Building Innovative Schools and Increasing Student Learning (2012) == See also == List of Florida State University people List of presidents of Florida State University == References == == External links == FSU Office of the President History
Wikipedia:Damir Filipović#0	Damir Filipović (born 1970 in Switzerland) is a Swiss mathematician specializing in quantitative finance. He holds the Swissquote Chair in Quantitative Finance and is the director of the Swiss Finance Institute at EPFL (École Polytechnique Fédérale de Lausanne). == Career == Filipović studied mathematics at ETH Zurich and earned his Master's degree in 1995. He joined Freddy Delbaen as PhD student and graduated in 2000 with thesis on mathematical finance titled "Consistency problems for HJM interest rate models". As a postdoctoral research fellow he joined Vienna University of Technology (2000), Stanford University (2001) and Princeton University (2001) to work on consistency problems for Heath-Jarrow-Morton interest rate models and affine processes and applications in finance. From 2002 to 2003 he was an assistant professor at Princeton University's Department of Operations Research and Financial Engineering. As scientific consultant for solvency testing and risk analysis in insurance (Swiss Solvency Test) he joined the Swiss Federal Office of Private Insurance (BPV) in 2003. In 2004, he became full professor on the Chair of Financial and Insurance Mathematics at the Ludwig Maximilian University of Munich. In 2007, he was appointed as director of the Vienna Institute of Finance and full professor at the University of Vienna. Since 2010 he has been the Swissquote Chair in Quantitative Finance and director of the Swiss Finance Institute at EPFL. == Research == Filipović's research focuses on quantitative finance by drawing in an interdisciplinary manner on fields such as quantitative finance, quantitative risk management, and machine learning in finance. It aims at both the advancement of theoretical understanding of financial engineering, and its implementation in the financial industry and governmental policies. His research interests encompass polynomial processes and applications in finance, systemic risk in financial networks, Interest rates, credit risk, stochastic volatility, Stochastic processes, quantitative risk management and regulation, and machine learning in finance. He also teaches a MOOC on "Interest Rate Models" on Coursera. == Distinctions == Filipović is the 2016 recipient of the Louis Bachelier Prize awarded by the London Mathematical Society, the Natixis Foundation for Quantitative Research and the Société de Mathématiques Appliquées et Industrielles. He is a member and former president (2016–2017) of the Bachelier finance society. He has been an associate editor at academic journals such as Mathematics and Financial Economics, Stochastics, SIAM Journal on Financial Mathematics, Mathematical Finance, and Finance and Stochastics. == Selected works == Duffie, D.; Filipović, D.; Schachermayer, W. (1 August 2003). "Affine processes and applications in finance". The Annals of Applied Probability. 13 (3). doi:10.1214/aoap/1060202833. ISSN 1050-5164. S2CID 6340845. Cheridito, Patrick; Filipović, Damir; Yor, Marc (1 August 2005). "Equivalent and absolutely continuous measure changes for jump-diffusion processes". The Annals of Applied Probability. 15 (3). arXiv:math/0508450. doi:10.1214/105051605000000197. ISSN 1050-5164. S2CID 2504454. Cheridito, Patrick; Filipović, Damir; Kimmel, Robert L. (January 2007). "Market price of risk specifications for affine models: Theory and evidence☆". Journal of Financial Economics. 83 (1): 123–170. doi:10.1016/j.jfineco.2005.09.008. ISSN 0304-405X. Filipovic, Damir (2009). Term-Structure Models. doi:10.1007/978-3-540-68015-4. ISBN 978-3-540-09726-6. Filipović, Damir; Overbeck, Ludger; Schmidt, Thorsten (22 September 2010). "Dynamic Cdo Term Structure Modeling". Mathematical Finance. 21 (1): 53–71. doi:10.1111/j.1467-9965.2010.00421.x. ISSN 0960-1627. S2CID 14323028. Cuchiero, Christa; Filipović, Damir; Mayerhofer, Eberhard; Teichmann, Josef (1 April 2011). "Affine processes on positive semidefinite matrices". The Annals of Applied Probability. 21 (2). arXiv:0910.0137. doi:10.1214/10-aap710. ISSN 1050-5164. S2CID 15944588. Filipović, Damir; Trolle, Anders B. (September 2013). "The term structure of interbank risk". Journal of Financial Economics. 109 (3): 707–733. doi:10.1016/j.jfineco.2013.03.014. ISSN 0304-405X. Filipović, Damir; Mayerhofer, Eberhard; Schneider, Paul (October 2013). "Density approximations for multivariate affine jump-diffusion processes". Journal of Econometrics. 176 (2): 93–111. arXiv:1104.5326. doi:10.1016/j.jeconom.2012.12.003. ISSN 0304-4076. S2CID 122805766. Filipović, Damir; Kremslehner, Robert; Muermann, Alexander (16 January 2014). "Optimal Investment and Premium Policies Under Risk Shifting and Solvency Regulation". Journal of Risk and Insurance. 82 (2): 261–288. arXiv:1103.1729. doi:10.1111/jori.12021. ISSN 0022-4367. S2CID 340316. Cambou, Mathieu; Filipović, Damir (19 June 2015). "Model Uncertainty and Scenario Aggregation". Mathematical Finance. 27 (2): 534–567. doi:10.1111/mafi.12097. ISSN 0960-1627. S2CID 157683249. Filipović, Damir; Gourier, Elise; Mancini, Loriano (January 2016). "Quadratic variance swap models". Journal of Financial Economics. 119 (1): 44–68. doi:10.1016/j.jfineco.2015.08.015. ISSN 0304-405X. FILIPOVIĆ, DAMIR; LARSSON, MARTIN; TROLLE, ANDERS B. (21 March 2017). "Linear-Rational Term Structure Models". The Journal of Finance. 72 (2): 655–704. doi:10.1111/jofi.12488. ISSN 0022-1082. Cambou, Mathieu; Filipović, Damir (13 November 2017). "Replicating portfolio approach to capital calculation". Finance and Stochastics. 22 (1): 181–203. doi:10.1007/s00780-017-0347-1. ISSN 0949-2984. S2CID 508079. Ackerer, Damien; Filipović, Damir; Pulido, Sergio (18 May 2018). "The Jacobi stochastic volatility model". Finance and Stochastics. 22 (3): 667–700. arXiv:1605.07099. doi:10.1007/s00780-018-0364-8. ISSN 0949-2984. S2CID 49415504. Filipović, Damir; Larsson, Martin; Trolle, Anders B. (1 October 2018). "On the relation between linearity-generating processes and linear-rational models". Mathematical Finance. 29 (3): 804–826. arXiv:1806.03153. doi:10.1111/mafi.12198. ISSN 0960-1627. S2CID 158476820. Ackerer, Damien; Filipović, Damir (4 October 2019). "Linear credit risk models". Finance and Stochastics. 24 (1): 169–214. arXiv:1605.07419. doi:10.1007/s00780-019-00409-z. ISSN 0949-2984. S2CID 209509242. Boudabsa, Lotfi; Filipovic, Damir (2019). "Machine Learning With Kernels for Portfolio Valuation and Risk Management". SSRN Electronic Journal. arXiv:1906.03726. doi:10.2139/ssrn.3401539. ISSN 1556-5068. S2CID 182952325. Amini, Hamed; Filipović, Damir; Minca, Andreea (January 2020). "Systemic Risk in Networks with a Central Node". SIAM Journal on Financial Mathematics. 11 (1): 60–98. doi:10.1137/18m1184667. ISSN 1945-497X. S2CID 214014802. Fernandez Arjona, Lucio; Filipovic, Damir (2020). "A machine learning approach to portfolio pricing and risk management for high-dimensional problems". SSRN Electronic Journal. arXiv:2004.14149. doi:10.2139/ssrn.3588376. ISSN 1556-5068. S2CID 216641606. == References == == External links == Damir Filipović publications indexed by Google Scholar Website of the Swissquote Chair in Quantitative Finance MOOC on Interest Rate Models
Wikipedia:Damodara#0	Vatasseri Damodara Nambudiri was an astronomer-mathematician of the Kerala school of astronomy and mathematics who flourished during the fifteenth century CE. He was a son of Paramesvara (1360–1425) who developed the drigganita system of astronomical computations. The family home of Paramesvara was Vatasseri (sometimes called Vatasreni) in the village of Alathiyur, Tirur in Kerala. Damodara was a teacher of Nilakantha Somayaji. As a teacher he initiated Nilakantha into the science of astronomy and taught him the basic principles in mathematical computations. == See also == List of astronomers and mathematicians of the Kerala school == References ==
Wikipedia:Damping#0	In physical systems, damping is the loss of energy of an oscillating system by dissipation. Damping is an influence within or upon an oscillatory system that has the effect of reducing or preventing its oscillation. Examples of damping include viscous damping in a fluid (see viscous drag), surface friction, radiation, resistance in electronic oscillators, and absorption and scattering of light in optical oscillators. Damping not based on energy loss can be important in other oscillating systems such as those that occur in biological systems and bikes (ex. Suspension (mechanics)). Damping is not to be confused with friction, which is a type of dissipative force acting on a system. Friction can cause or be a factor of damping. Many systems exhibit oscillatory behavior when they are disturbed from their position of static equilibrium. A mass suspended from a spring, for example, might, if pulled and released, bounce up and down. On each bounce, the system tends to return to its equilibrium position, but overshoots it. Sometimes losses (e.g. frictional) damp the system and can cause the oscillations to gradually decay in amplitude towards zero or attenuate. The damping ratio is a dimensionless measure, amongst other measures, that characterises how damped a system is. It is denoted by ζ ("zeta") and varies from undamped (ζ = 0), underdamped (ζ < 1) through critically damped (ζ = 1) to overdamped (ζ > 1). The behaviour of oscillating systems is often of interest in a diverse range of disciplines that include control engineering, chemical engineering, mechanical engineering, structural engineering, and electrical engineering. The physical quantity that is oscillating varies greatly, and could be the swaying of a tall building in the wind, or the speed of an electric motor, but a normalised, or non-dimensionalised approach can be convenient in describing common aspects of behavior. == Oscillation cases == Depending on the amount of damping present, a system exhibits different oscillatory behaviors and speeds. Where the spring–mass system is completely lossless, the mass would oscillate indefinitely, with each bounce of equal height to the last. This hypothetical case is called undamped. If the system contained high losses, for example if the spring–mass experiment were conducted in a viscous fluid, the mass could slowly return to its rest position without ever overshooting. This case is called overdamped. Commonly, the mass tends to overshoot its starting position, and then return, overshooting again. With each overshoot, some energy in the system is dissipated, and the oscillations die towards zero. This case is called underdamped. Between the overdamped and underdamped cases, there exists a certain level of damping at which the system will just fail to overshoot and will not make a single oscillation. This case is called critical damping. The key difference between critical damping and overdamping is that, in critical damping, the system returns to equilibrium in the minimum amount of time. == Damped sine wave == A damped sine wave or damped sinusoid is a sinusoidal function whose amplitude approaches zero as time increases. It corresponds to the underdamped case of damped second-order systems, or underdamped second-order differential equations. Damped sine waves are commonly seen in science and engineering, wherever a harmonic oscillator is losing energy faster than it is being supplied. A true sine wave starting at time = 0 begins at the origin (amplitude = 0). A cosine wave begins at its maximum value due to its phase difference from the sine wave. A given sinusoidal waveform may be of intermediate phase, having both sine and cosine components. The term "damped sine wave" describes all such damped waveforms, whatever their initial phase. The most common form of damping, which is usually assumed, is the form found in linear systems. This form is exponential damping, in which the outer envelope of the successive peaks is an exponential decay curve. That is, when you connect the maximum point of each successive curve, the result resembles an exponential decay function. The general equation for an exponentially damped sinusoid may be represented as: y ( t ) = A e − λ t cos ⁡ ( ω t − φ ) {\displaystyle y(t)=Ae^{-\lambda t}\cos(\omega t-\varphi )} where: y ( t ) {\displaystyle y(t)} is the instantaneous amplitude at time t; A {\displaystyle A} is the initial amplitude of the envelope; λ {\displaystyle \lambda } is the decay rate, in the reciprocal of the time units of the independent variable t; φ {\displaystyle \varphi } is the phase angle at t = 0; ω {\displaystyle \omega } is the angular frequency. Other important parameters include: Frequency: f = ω / ( 2 π ) {\displaystyle f=\omega /(2\pi )} , the number of cycles per time unit. It is expressed in inverse time units t − 1 {\displaystyle t^{-1}} , or hertz. Time constant: τ = 1 / λ {\displaystyle \tau =1/\lambda } , the time for the amplitude to decrease by the factor of e. Half-life is the time it takes for the exponential amplitude envelope to decrease by a factor of 2. It is equal to ln ⁡ ( 2 ) / λ {\displaystyle \ln(2)/\lambda } which is approximately 0.693 / λ {\displaystyle 0.693/\lambda } . Damping ratio: ζ {\displaystyle \zeta } is a non-dimensional characterization of the decay rate relative to the frequency, approximately ζ = λ / ω {\displaystyle \zeta =\lambda /\omega } , or exactly ζ = λ / λ 2 + ω 2 < 1 {\displaystyle \zeta =\lambda /{\sqrt {\lambda ^{2}+\omega ^{2}}}<1} . Q factor: Q = 1 / ( 2 ζ ) {\displaystyle Q=1/(2\zeta )} is another non-dimensional characterization of the amount of damping; high Q indicates slow damping relative to the oscillation. == Damping ratio == The damping ratio is a dimensionless parameter, usually denoted by ζ (Greek letter zeta), that characterizes the extent of damping in a second-order ordinary differential equation. It is particularly important in the study of control theory. It is also important in the harmonic oscillator. The greater the damping ratio, the more damped a system is. Undamped systems have a damping ratio of 0. Underdamped systems have a value of less than one. Critically damped systems have a damping ratio of 1. Overdamped systems have a damping ratio greater than 1. The damping ratio expresses the level of damping in a system relative to critical damping and can be defined using the damping coefficient: ζ = c c c = actual damping critical damping , {\displaystyle \zeta ={\frac {c}{c_{c}}}={\frac {\text{actual damping}}{\text{critical damping}}},} The damping ratio is dimensionless, being the ratio of two coefficients of identical units. Taking the simple example of a mass-spring-damper model with mass m, damping coefficient c, and spring constant k, where x {\displaystyle x} represents the degree of freedom, the system's equation of motion is given by: m x ¨ + c x ˙ + k x = 0 {\displaystyle m{\ddot {x}}+c{\dot {x}}+kx=0} . The corresponding critical damping coefficient is: c c = 2 k m {\displaystyle c_{c}=2{\sqrt {km}}} and the natural frequency of the system is: ω n = k m {\displaystyle \omega _{n}={\sqrt {\frac {k}{m}}}} Using these definitions, the equation of motion can then be expressed as: x ¨ + 2 ζ ω n x ˙ + ω n 2 x = 0. {\displaystyle {\ddot {x}}+2\zeta \omega _{n}{\dot {x}}+\omega _{n}^{2}x=0.} This equation is more general than just the mass-spring-damper system and applies to electrical circuits and to other domains. It can be solved with the approach x ( t ) = C e s t , {\displaystyle x(t)=Ce^{st},} where C and s are both complex constants, with s satisfying s = − ω n ( ζ ± i 1 − ζ 2 ) . {\displaystyle s=-\omega _{n}\left(\zeta \pm i{\sqrt {1-\zeta ^{2}}}\right).} Two such solutions, for the two values of s satisfying the equation, can be combined to make the general real solutions, with oscillatory and decaying properties in several regimes: Undamped Is the case where ζ = 0 {\displaystyle \zeta =0} corresponds to the undamped simple harmonic oscillator, and in that case the solution looks like exp ⁡ ( i ω n t ) {\displaystyle \exp(i\omega _{n}t)} , as expected. This case is extremely rare in the natural world with the closest examples being cases where friction was purposefully reduced to minimal values. Underdamped If s is a pair of complex values, then each complex solution term is a decaying exponential combined with an oscillatory portion that looks like exp ⁡ ( i ω n 1 − ζ 2 t ) {\textstyle \exp \left(i\omega _{n}{\sqrt {1-\zeta ^{2}}}t\right)} . This case occurs for 0 ≤ ζ < 1 {\displaystyle \ 0\leq \zeta <1} , and is referred to as underdamped (e.g., bungee cable). Overdamped If s is a pair of real values, then the solution is simply a sum of two decaying exponentials with no oscillation. This case occurs for ζ > 1 {\displaystyle \zeta >1} , and is referred to as overdamped. Situations where overdamping is practical tend to have tragic outcomes if overshooting occurs, usually electrical rather than mechanical. For example, landing a plane in autopilot: if the system overshoots and releases landing gear too late, the outcome would be a disaster. Critically damped The case where ζ = 1 {\displaystyle \zeta =1} is the border between the overdamped and underdamped cases, and is referred to as critically damped. This turns out to be a desirable outcome in many cases where engineering design of a damped oscillator is required (e.g., a door closing mechanism). == Q factor and decay rate == The Q factor, damping ratio ζ, and exponential decay rate α are related such that ζ = 1 2 Q = α ω n . {\displaystyle \zeta ={\frac {1}{2Q}}={\alpha \over \omega _{n}}.} When a second-order system has ζ < 1 {\displaystyle \zeta <1} (that is, when the system is underdamped), it has two complex conjugate poles that each have a real part of − α {\displaystyle -\alpha } ; that is, the decay rate parameter represents the rate of exponential decay of the oscillations. A lower damping ratio implies a lower decay rate, and so very underdamped systems oscillate for long times. For example, a high quality tuning fork, which has a very low damping ratio, has an oscillation that lasts a long time, decaying very slowly after being struck by a hammer. == Logarithmic decrement == For underdamped vibrations, the damping ratio is also related to the logarithmic decrement δ {\displaystyle \delta } . The damping ratio can be found for any two peaks, even if they are not adjacent. For adjacent peaks: ζ = δ δ 2 + ( 2 π ) 2 {\displaystyle \zeta ={\frac {\delta }{\sqrt {\delta ^{2}+\left(2\pi \right)^{2}}}}} where δ = ln ⁡ x 0 x 1 {\displaystyle \delta =\ln {\frac {x_{0}}{x_{1}}}} where x0 and x1 are amplitudes of any two successive peaks. As shown in the right figure: δ = ln ⁡ x 1 x 3 = ln ⁡ x 2 x 4 = ln ⁡ x 1 − x 2 x 3 − x 4 {\displaystyle \delta =\ln {\frac {x_{1}}{x_{3}}}=\ln {\frac {x_{2}}{x_{4}}}=\ln {\frac {x_{1}-x_{2}}{x_{3}-x_{4}}}} where x 1 {\displaystyle x_{1}} , x 3 {\displaystyle x_{3}} are amplitudes of two successive positive peaks and x 2 {\displaystyle x_{2}} , x 4 {\displaystyle x_{4}} are amplitudes of two successive negative peaks. == Percentage overshoot == In control theory, overshoot refers to an output exceeding its final, steady-state value. For a step input, the percentage overshoot (PO) is the maximum value minus the step value divided by the step value. In the case of the unit step, the overshoot is just the maximum value of the step response minus one. The percentage overshoot (PO) is related to damping ratio (ζ) by: P O = 100 exp ⁡ ( − ζ π 1 − ζ 2 ) {\displaystyle \mathrm {PO} =100\exp \left({-{\frac {\zeta \pi }{\sqrt {1-\zeta ^{2}}}}}\right)} Conversely, the damping ratio (ζ) that yields a given percentage overshoot is given by: ζ = − ln ⁡ ( P O 100 ) π 2 + ln 2 ⁡ ( P O 100 ) {\displaystyle \zeta ={\frac {-\ln \left({\frac {\rm {PO}}{100}}\right)}{\sqrt {\pi ^{2}+\ln ^{2}\left({\frac {\rm {PO}}{100}}\right)}}}} == Examples and applications == === Viscous drag === When an object is falling through the air, the only force opposing its freefall is air resistance. An object falling through water or oil would slow down at a greater rate, until eventually reaching a steady-state velocity as the drag force comes into equilibrium with the force from gravity. This is the concept of viscous drag, which for example is applied in automatic doors or anti-slam doors. === Damping in electrical systems === Electrical systems that operate with alternating current (AC) use resistors to damp LC resonant circuits. === Magnetic damping === Kinetic energy that causes oscillations is dissipated as heat by electric eddy currents which are induced by passing through a magnet's poles, either by a coil or aluminum plate. Eddy currents are a key component of electromagnetic induction where they set up a magnetic flux directly opposing the oscillating movement, creating a resistive force. In other words, the resistance caused by magnetic forces slows a system down. An example of this concept being applied is the brakes on roller coasters. === Magnetorheological damping === Magnetorheological dampers (MR Dampers) use Magnetorheological fluid, which changes viscosity when subjected to a magnetic field. In this case, Magnetorheological damping may be considered an interdisciplinary form of damping with both viscous and magnetic damping mechanisms. === Material damping === Materials have varying degrees of internal damping properties due to microstructural mechanisms within them. This property is sometimes known as damping capacity. In metals, this arises due to movements of dislocations, as demonstrated nicely in this video: Metals, as well as ceramics and glass, are known for having very light material damping. By contrast, polymers have a much higher material damping that arises from the energy loss required to contiually break and reform the Van der Waals forces between polymer chains. The cross-linking in thermoset plastics causes less movement of the polymer chains and so the damping is less. Material damping is best characterized by the loss factor η {\displaystyle \eta } , given by the equation below for the case of very light damping, such as in metals or ceramics: η = 2 ζ ω ω n {\displaystyle \eta =2\zeta {\frac {\omega }{\omega _{n}}}} This is because many microstructural processes that contribute to material damping are not well modelled by viscous damping, and so the damping ratio varies with frequency. Adding the frequency ratio as a factor typically makes the loss factor constant over a wide frequency range. == References ==
Wikipedia:Dan Laksov#0	Dan Laksov (10 July 1940 – 25 October 2013) was a Norwegian-Swedish mathematician and human rights activist. He was primarily active within the field of algebraic geometry. == Biography == Laksov was born in Oslo in 1940, the same year that Norway was occupied by Nazi Germany. He was a son of Amalie Laksov (née Scheer) and Håkon Laksov (ne Laks), both born 1911; the family were Jews. The ancestors on both sides had immigrated from Russia via the Baltics to Norway in the late 19th century. Håkon Laksov was a lawyer and active in the Jewish community. In the book I slik en natt. Historien om deportasjonen av jøder fra Norge by Kristian Ottosen, the escape of Amalie and Dan from Norway in November 1942 is chronicled. Håkon and Amalie's four brothers were all captured in October 1942 as a part of the arresting of all Norwegian Jews, shipped on SS Donau to Auschwitz in November 1942 and perished there sometime in early 1943. Amalie had been tipped off ahead of the next wave of arrests and managed to hide together with her young son at various addresses in Oslo before being able to flee to Sweden, where they reunited with Amalie's mother and two aunts and spent the rest of the war in Norrköping. The family's apartment was usurped by the family of a leading Young Nazi leader, Bjørn Østring, but retrieved after the war. In 1945 Dan returned to Oslo where he lived with his grandparents, while Amalie commuted to Bergen. After finishing secondary school, he studied one year at a commercial high school before entering University of Bergen in 1960 where he studied mathematics. He graduated in 1964 and after one year of non-armed conscription service, he travelled to Paris on a scholarship to study at Institut Henri Poincaré. In Paris he encountered Steven Kleiman and in 1967 Laksov became one of Kleiman's Ph.D. students at Columbia University, and when Kleiman moved to Massachusetts Institute of Technology (MIT) in 1968 Laksov followed him. Laksov took his Ph.D. from MIT in 1972 and wrote a thesis with the title The Structure of Schubert Schemes and Schubert Cycles. He remained one year at MIT as a postdoc. During the next couple of years he mostly alternated between Oslo and Stockholm. 1978–1981 he was head of algebraic geometry at the Mittag-Leffler Institute in Stockholm. 1981–1984 he was a senior lecturer at the Stockholm University and 1984–1986 he was professor of mathematics at Uppsala University. From 1986 to his retirement he was professor of mathematics at the Royal Institute of Technology in Stockholm . He also served as a director of the Mittag-Leffler Institute during the period 1986–1994 and was editor of the institute's journal Acta Mathematica. His main contributions were in algebra, algebraic geometry and Schubert calculus. He was a foreign member of the Royal Swedish Academy of Sciences and a fellow of the Norwegian Academy of Science and Letters. In 2008 he received an honorary degree at the University of Bergen. In 1983, his mother Amalie Laksov created a foundation for human rights, Amalie Laksovs Minnefond. Dan Laksov was a board member, and member of the committee that awarded the Laksov Prize for human rights. Amalie Laksov died in 2008, aged 97, while Dan Laksov died in Stockholm in October 2013. == References ==
Wikipedia:Dan-Virgil Voiculescu#0	Dan-Virgil Voiculescu (Romanian pronunciation: [dan virˈd͡ʒil vojkuˈlesku]; born 14 June 1949) is a Romanian professor of mathematics at the University of California, Berkeley. He has worked in single operator theory, operator K-theory and von Neumann algebras. More recently, he developed free probability theory. == Education and career == Voiculescu studied at the University of Bucharest, receiving his PhD in 1977 under the direction of Ciprian Foias. He was an assistant at the University of Bucharest (1972–1973), a researcher at the Institute of Mathematics of the Romanian Academy (1973–1975), and a researcher at INCREST (1975–1986). He came to Berkeley in 1986 for the International Congress of Mathematicians, and stayed on as visiting professor. Voiculescu was appointed professor at Berkeley in 1987. == Awards and honors == He received the 2004 NAS Award in Mathematics from the National Academy of Sciences (NAS) for “the theory of free probability, in particular, using random matrices and a new concept of entropy to solve several hitherto intractable problems in von Neumann algebras.” Voiculescu was elected to the National Academy of Sciences in 2006. In 2012 he became a fellow of the American Mathematical Society. == References == == External links == Berkeley page Notes on Free probability aspects of random matrices Dan-Virgil Voiculescu: visionary operator algebraist and creator of free probability theory
Wikipedia:Daniel Abibi#0	Daniel Abibi (born 1942) is a Congolese politician, mathematician and diplomat. During the 1980s, he served in the government of Congo-Brazzaville as Minister of Information and as Minister of Secondary and Higher Education. Later, during the 1990s, he was Congo-Brazzaville's Permanent Representative to the United Nations. == Life and career == Abibi obtained his doctorate in mathematics in 1970 from the University of Grenoble in France. He was amongst the first Central Africans to receive doctoral degrees in Mathematics. Abibi was one of the Northern political activists educated in France that ensured support for Marien Ngouabi in the Congolese Students Association (AEC) in 1972. The rapprochement between AEC and the regime was, however, fiercely rejected by cadres of the Congolese Party of Labour (PCT). The PCT cadres kidnapped Abibi, forcing Ngouabi to order his release. Politically, Abibi espoused a Marxist ideological line that was heavily influenced by radical African nationalism. He served as rector of the Marien Ngouabi University. Becoming a confidant of Denis Sassou Nguesso (espousing an ideological line to the liking of Sassou) Abibi was named Minister of Information, Posts and Telecommunications in 1983, replacing captain Florent Ntsiba. In 1984 Abibi was included in the Central Committee of the PCT. He was put in-charge of the international relations of the party. Also in 1984, he was moved from his post as Minister of Information to the post of Minister of Secondary and Higher Education. He lost his cabinet seat in a December 1986 reshuffle. Abibi chaired the Congolese Anti-Apartheid Committee, and in 1989 he was named chairman of the African Anti-Apartheid Committee. Also in 1989, he was included in the PCT Politburo and assigned responsibility for education, ideology, and political and civic training. In the 1990s he joined the Pan-African Union for Social Democracy of Pascal Lissouba. He also served as Permanent Representative of the Congo to the United Nations during this decade. Following the June–October 1997 civil war, in which Lissouba was ousted and Sassou Nguesso returned to power, Abibi was absent from Congolese politics for years. Eventually, however, he rejoined Sassou Nguesso's party, the PCT. In October 2011, he was elected to the Senate of Congo-Brazzaville as a PCT candidate in Sangha Department. In the indirect Senate election, he received 61 votes from the electors in Sangha, 87.14% of the total; this placed Abibi in a three-way tie for first place and secured him one of the six available seats for Sangha. == References ==
Wikipedia:Daniel Afedzi Akyeampong#0	Daniel Afedzi Akyeampong (24 November 1938 – 7 March 2015) was a Ghanaian academic. He was the first Ghanaian to attain full professorship status in mathematics at the University of Ghana, Legon. In 1966, Daniel Akyeampong and Francis Allotey became the first Ghanaians to obtain a doctorate in mathematical sciences. He was the Pro-Vice Chancellor of the University of Ghana from 1983 to 1985. == Early life and education == Akyeampong was born on 24 November 1938 at Senya Beraku in the Gold Coast colony (now Ghana). He was the youngest child of his father, Peter Napoleon Akyeampong, and his mother, Charity Afful. He was a pupil at the Senya Beraku Local Council School in 1945 when he was six years old and completed in 1953. In 1954, he entered Mfantsipim School, Cape Coast for his secondary education and was a member of Balmer-Acquah House. Upon matriculating at Mfantsipim, his intellect became apparent after only a few weeks - he skipped his first year entirely and was quickly promoted to the second year. Akyeampong received prizes in Mathematics and Physical Sciences upon his graduation in 1959. The following year, he gained admission to study at the University of Ghana, where he was a resident of the all-male Commonwealth Hall. He graduated in 1963 with a Bachelor of Science degree in Mathematics. After his undergraduate studies, Akyeampong proceeded to the United Kingdom and enrolled at the University of London for his postgraduate research. He was admitted to Imperial College London in 1963 to complete the foundational diploma course in Mathematical Physics prior to his doctoral work. He later recalled the unexpected turns in his academic journey: "Professor Abdus Salam taught the class in group theory. His lectures were very popular and one had to go early to find a seat. Following my successful completion of the coursework in the summer of 1964, he invited me to his office one day, and told me about a new international centre for theoretical physics to be established in Trieste, Italy, which he was going to direct in the autumn of the same year, and the reason why he expected me to join him there. That was how in October 1964, Jimmy Boyce, Ray Rivers and myself became the first postgraduate students of Salam in Trieste, and so had the honour of joining the post doctoral colleagues Bob Delbourgo and John Strathdee to become members of the group that was christened by Ms. Jean Bouckley and Ms Miriam Lewis as "the Salam Boys"."In 1965, Akyeampong became one of the first Fellows of the International Centre for Theoretical Physics, Trieste. About the vibrant scholarly culture at the Trieste institution, Akyeampong recounted:"Salam worked hard to get the Centre known world-wide and we were naturally infected by his ceaseless dedication. We had our lunches together at the Mensa dei Ferrovieri with him at the table head, John and Ray at one side of the table and Bob and Jimmy on the other, with me deciding when to sit next to Ray or to Jimmy. These became physics working lunches - with paper napkins serving as writing equipment - and each session usually ended with several suggestions from Salam or Bob or John for us graduate students to pursue later. The excitement and enthusiasm Salam displayed were infectious; and later when I had the privilege to collaborate with Delbourgo, who was my unofficial supervisor, it became clear that he was indeed a chip off the old block."In September 1966, Akyeampong returned to London and a month later, the University of London awarded him a doctorate in Mathematical Physics. His thesis was entitled, Applications of higher symmetry groups to particle physics. His secondary advisor was the University of London-based theoretical physicist, Paul Taunton Matthews. He was also awarded a Diploma of Imperial College(DIC) in Mathematical Physics in November 1966. == Career == Akyeampong returned to Ghana and joined the University of Ghana's Department of Mathematics as lecturer in December 1966. While he was on the faculty, he became an Associate (1967–75) of the International Centre for Theoretical Physics, Trieste, from 1967 to 1975 and later a Senior Associate (1976–93). In 1972, he became a Senior Lecturer in 1972 and an associate professor in 1976. That same year, he was appointed Head of the Mathematics Department. He worked in this capacity until 1983 and two years later, his was reappointed to hold this office for three more years. In 1994, he was reappointed to this position for a third time and this time he served for five years. In 1982, while serving as the departmental chairman, he was promoted to the rank of full Professor of Mathematics - the first Ghanaian to achieve that distinction at the university. Prior to his second tenure as the mathematics departmental head, he served as the university's Pro-Vice Chancellor. This was during the period between his first and second appointment as Head of department. === Committees and Boards === He held several concurrent appointments outside the university: Between 1972 and 1980, he was a Member of the National Council for Higher Education He also served as a Member of the Council for Scientific and Industrial Research from 1975 to 1978 and from 1992 to 1997 He was also the Honorary Secretary of the Ghana Academy of Arts and Sciences from 1975 to 1978 and between 1995 and 1998, he served as the vice-president of the academy He was the President of the Mathematical Association of Ghana from 1988 to 1994 Between 1989 and 1992, he became a member of the National Implementation Committee on Tertiary Education Reforms until 1992 when He was made a Representative of University of Ghana on the Ghana National Committee of the West African Examinations Council from 1992 to 1996 From 1993 to 2004, he was a member of the National Council for Tertiary Education and the Chairman of the National Accreditation Board from 1994 to 2004 In 1998, became the Chairman of the Committee on Evaluation of National Policy Objectives on Tertiary Education, Ministry of Education. He was the Chairman of the committee to Review the Grading System for the Basic Education Certificate Examination in 1999, and the Chairman of the Country Selection Committee for the Ford Foundation International Fellowship Program from 2000 to 2002 In 2002, he was made the Chairman of West African Examinations Council Committee to examine malpractice, specifically the leakage of the 2002 Basic Education Certificate Examination. That same year, he was made a member of the President's Committee of the Review of Education Reforms in Ghana, and from 2002 to 2003, a Member of the Board of Trustees of the Ghana Education Trust Fund. A member of the African Mathematical Union, he served as an assistant editor of its journal, Afrika Matematica. He was on the editorial board of a journal of the African Academy of Sciences, Discovery and Innovation. From 1993 to 1999 he served as a member of the executive board of the International Council for Science, Paris, France, where he was elected the body's vice president in 1996 and serving in that capacity until 1999. Due to his contributions to theoretical physics, symmetry groups in physics, and grand unified theories, he was also elected to the Mathematical Sciences Section of the World Academy of Sciences in 1991. == Publications == Akyeampong published about 26 papers in leading journals and conference proceedings, and also published the book which was the text of his speech at the silver jubilee edition of the J. B. Danquah Memorial Lectures: The Two Cultures Revisited: Interactions of Science and Culture, published by the Ghana Academy of Arts and Sciences. == Personal life == Akyeampong married Charlotte Sally Newton on 11 April 1970. Together, they had two sons and one daughter. == Death == On Sunday 21 December 2014, Akyeampong sustained an injury leading to surgery at the Korle-Bu Teaching Hospital . He was successfully treated and released on the 1 January 2015. He suffered a septic shock and was admitted once again to Korle-Bu Teaching Hospital. Akyeampong died a month and a half later, on 7 March 2015, aged 76. He was survived by his wife, three children and nine grandchildren. == References ==
Wikipedia:Daniel B. Szyld#0	Daniel B. Szyld (born 1955 in Buenos Aires) is an Argentinian and American mathematician who is a professor at Temple University in Philadelphia. He has made contributions to numerical and applied linear algebra as well as matrix theory. == Education == He was admitted without an undergraduate degree to the graduate school at New York University, where he defended his PhD in 1983. While there, he worked as a research assistant for Wassily Leontief. == International awards and appointments == He was named as a SIAM Fellow and as a fellow of the American Mathematical Society in 2017. In 2020, he was elected president of the International Linear Algebra Society. He was editor-in-chief for the Electronic Transactions on Numerical Analysis from 2005 to 2013 and SIAM Journal on Matrix Analysis and Applications from 2015 to 2020 and is on the editorial boards of several journals, including the Electronic Journal of Linear Algebra (ELA), the Electronic Transactions on Numerical Analysis (ETNA), Linear Algebra and its Applications, Mathematics of Computation, Numerical Linear Algebra with Applications, and Journal of Numerical Analysis and Approximation Theory. A conference in honor of his 65th birthday was held in 2022 == Books and edited proceedings == Klapper, Isaac; Szyld, Daniel B.; Zhao, Kai (2021). Metabolic Networks, Elementary Flux Modes, and Polyhedral Cones. Philadelphia: Other Titles in Applied Mathematics series, vol. 171, SIAM. doi:10.1137/1.9781611976533. ISBN 978-1-61197-652-6. MR 4279642. S2CID 237914557. Brenner, Susanne C.; Shparlinski, Igor; Shu, Chi-Wang; Szyld, Daniel B. (2020). 75 Years of Mathematics of Computation. Contemporary Mathematics. Vol. 754. Providence, R.I.: Contemporary Mathematics series vol. 754, American Mathematical Society. doi:10.1090/conm/754. ISBN 9781470451639. MR 4132114. S2CID 226239867. == Selected papers == Leontief, Wassily; Duchin, Faye; Szyld, Daniel B. (1985). "New Approaches in Economic Analysis". Science. 228 (4698): 419–422. Bibcode:1985Sci...228..419L. doi:10.1126/science.228.4698.419. PMID 17746871. S2CID 40269573. Benzi, Michele; Szyld, Daniel B.; van Duin, Arno (1999). "Orderings for Incomplete Factorization Preconditioning of Nonsymmetric Problems". SIAM Journal on Scientific Computing. 20 (5): 1652–1670. Bibcode:1999SJSC...20.1652B. doi:10.1137/S1064827597326845. MR 1694677. Frommer, Andreas; Szyld, Daniel B. (2000). "On Asynchronous Iterations". Journal of Computational and Applied Mathematics. 123 (1–2): 201–216. Bibcode:2000JCoAM.123..201F. doi:10.1016/S0377-0427(00)00409-X. MR 1798526. Frommer, Andreas; Szyld, Daniel B. (2001). "An Algebraic Convergence Theory for Restricted Additive Schwarz Methods Using Weighted Max Norms". SIAM Journal on Numerical Analysis. 39 (2): 463–479. doi:10.1137/S0036142900370824. MR 1860268. Simoncini, Valeria; Szyld, Daniel B. (2003). "Theory of Inexact Krylov Subspace Methods and Applications to Scientific Computing". SIAM Journal on Scientific Computing. 25 (2): 454–477. Bibcode:2003SJSC...25..454S. doi:10.1137/S1064827502406415. MR 2058070. Simoncini, Valeria; Szyld, Daniel B. (2005). "On the Occurrence of Superlinear convergence of exact and inexact Krylov subspace methods". SIAM Review. 47 (2): 247–272. Bibcode:2005SIAMR..47..247S. doi:10.1137/S0036144503424439. MR 2179897. Simoncini, Valeria; Szyld, Daniel B. (2007). "Recent computational developments in Krylov Subspace Methods for linear systems". Numerical Linear Algebra with Applications. 14: 1–59. CiteSeerX 10.1.1.67.1300. doi:10.1002/nla.499. MR 2289520. S2CID 4012149. == References ==
Wikipedia:Daniel Bernoulli#0	Daniel Bernoulli ( bur-NOO-lee; Swiss Standard German: [ˈdaːni̯eːl bɛrˈnʊli]; 8 February [O.S. 29 January] 1700 – 27 March 1782) was a Swiss-French mathematician and physicist and was one of the many prominent mathematicians in the Bernoulli family from Basel. He is particularly remembered for his applications of mathematics to mechanics, especially fluid mechanics, and for his pioneering work in probability and statistics. His name is commemorated in the Bernoulli's principle, a particular example of the conservation of energy, which describes the mathematics of the mechanism underlying the operation of two important technologies of the 20th century: the carburetor and the aeroplane wing. == Early life == Daniel Bernoulli was born in Groningen, in the Netherlands, into a family of distinguished mathematicians. The Bernoulli family came originally from Antwerp, at that time in the Spanish Netherlands, but emigrated to escape the Spanish persecution of the Protestants. After a brief period in Frankfurt the family moved to Basel, in Switzerland. Daniel was the son of Johann Bernoulli (one of the early developers of calculus) and a nephew of Jacob Bernoulli (an early researcher in probability theory and the discoverer of the mathematical constant e). He had two brothers, Niklaus and Johann II. Daniel Bernoulli was described by W. W. Rouse Ball as "by far the ablest of the younger Bernoullis". He is said to have had a bad relationship with his father. Both of them entered and tied for first place in a scientific contest at the University of Paris. Johann banned Daniel from his house, allegedly being unable to bear the "shame" of Daniel being considered his equal. Johann allegedly plagiarized key ideas from Daniel's book Hydrodynamica in his book Hydraulica and backdated them to before Hydrodynamica. Daniel's attempts at reconciliation with his father were unsuccessful. When he was in school, Johann encouraged Daniel to study business citing poor financial compensation for mathematicians. Daniel initially refused but later relented and studied both business and medicine at his father's behest under the condition that his father would teach him mathematics privately. Daniel studied medicine at Basel, Heidelberg, and Strasbourg, and earned a PhD in anatomy and botany in 1721. He was a contemporary and close friend of Leonhard Euler. He went to St. Petersburg in 1724 as professor of mathematics, but was very unhappy there. A temporary illness together with the censorship by the Russian Orthodox Church and disagreements over his salary gave him an excuse for leaving St. Petersburg in 1733. He returned to the University of Basel, where he successively held the chairs of medicine, metaphysics, and natural philosophy until his death. In May 1750 he was elected a Fellow of the Royal Society. == Mathematical work == His earliest mathematical work was the Exercitationes (Mathematical Exercises), published in 1724 with the help of Goldbach. Two years later he pointed out for the first time the frequent desirability of resolving a compound motion into motions of translation and motion of rotation. In 1729, he published a polynomial root-finding algorithm which became known as Bernoulli's method. His chief work is Hydrodynamica, published in 1738. It resembles Joseph Louis Lagrange's Mécanique Analytique in being arranged so that all the results are consequences of a single principle, namely, the conservation of vis viva, and early version of the conservation of energy. This was followed by a memoir on the theory of the tides, to which, conjointly with the memoirs by Euler and Colin Maclaurin, a prize was awarded by the French Academy: these three memoirs contain all that was done on this subject between the publication of Isaac Newton's Philosophiae Naturalis Principia Mathematica and the investigations of Pierre-Simon Laplace. Bernoulli also wrote a large number of papers on various mechanical questions, especially on problems connected with vibrating strings, and the solutions given by Brook Taylor and by Jean le Rond d'Alembert. == Economics and statistics == In his 1738 book Specimen theoriae novae de mensura sortis (Exposition of a New Theory on the Measurement of Risk), Bernoulli offered a solution to the St. Petersburg paradox as the basis of the economic theory of risk aversion, risk premium, and utility. Bernoulli often noticed that when making decisions that involved some uncertainty, people did not always try to maximize their possible monetary gain, but rather tried to maximize "utility", an economic term encompassing their personal satisfaction and benefit. Bernoulli realized that for humans, there is a direct relationship between money gained and utility, but that it diminishes as the money gained increases. For example, to a person whose income is $10,000 per year, an additional $100 in income will provide more utility than it would to a person whose income is $50,000 per year. One of the earliest attempts to analyze a statistical problem involving censored data was Bernoulli's 1766 analysis of smallpox morbidity and mortality data to demonstrate the efficacy of inoculation. == Physics == In Hydrodynamica (1738) he laid the basis for the kinetic theory of gases, and applied the idea to explain Boyle's law. He worked with Euler on elasticity and the development of the Euler–Bernoulli beam equation. Bernoulli's principle is of critical use in aerodynamics. According to Léon Brillouin, the principle of superposition was first stated by Daniel Bernoulli in 1753: "The general motion of a vibrating system is given by a superposition of its proper vibrations." == Works == Pieces qui ont remporté le Prix double de l'Academie royale des sciences en 1737 (in French). Paris: Imprimerie Royale. 1737. == Legacy == In 2002, Bernoulli was inducted into the International Air & Space Hall of Fame at the San Diego Air & Space Museum. == See also == Hydrodynamica Mathematical modelling of infectious diseases List of second-generation Mathematicians == References == === Footnotes === === Works cited === == External links == "Bernoulli Daniel". Mathematik.ch. Archived from the original on 23 October 2015. Retrieved 7 September 2007. Rothbard, Murray. Daniel Bernoulli and the Founding of Mathematical Economics Archived 28 July 2013 at the Wayback Machine, Mises Institute (excerpted from An Austrian Perspective on the History of Economic Thought) Weisstein, Eric Wolfgang (ed.). "Bernoulli, Daniel (1700–1782)". ScienceWorld. Works by Daniel Bernoulli at Project Gutenberg Works by or about Daniel Bernoulli at the Internet Archive
Wikipedia:Daniel Brélaz#0	Daniel Brélaz (born 4 January 1950, in Lausanne) is a Swiss mathematician and politician, member of the Green Party of Switzerland and 93th mayor of Lausanne between 2001 and 2016. In 1979, Daniel Brélaz became the first green representative elected to sit in a national parliament. == Biography == Brélaz received a degree in mathematics from École polytechnique fédérale de Lausanne (EPFL) in 1975, and afterwards taught mathematics. He is responsible for a well-known approximation algorithm for graph colouring. In 1975, he joined the Group for the Protection of the Environment in Lausanne. In 1978 he was one of the first environmentalists elected to parliament, in the Grand Council of Vaud, and re-elected in 1982–1983. From 1979 to 1989, Brélaz was the first environmentalist elected to sit in a national parliament, in the National Council of Switzerland. In 1989, he was elected to the City Council of Lausanne where he was responsible for industrial services. On 25 November 2001, he became the trustee responsible for finance, and was re-elected in the first round of Vaud elections on 12 March 2006. In 2007, he was elected again to the National Council. He resigned his seat on the Grand Council of Vaud but remained a trustee in Lausanne, and was criticized for maintaining this dual mandate. On 13 March 2011, he was re-elected in the first round of the Lausanne municipal elections with 11,503 votes in his favour. Brélaz was elected in 2015 Swiss federal election and re-elected in 2019. A month after his election, he announced that he would retire from the National Council in 2022. His term officially ended on 17 February 2022 and was succeeded by Raphaël Mahaim. Brélaz is vice-president of the Administrative Council for Public Transportation in the Region of Lausanne. == References == == Bibliography == Brélaz, Daniel (1979). "New methods to color the vertices of a graph". Communications of the ACM. 22 (4): 251–256. doi:10.1145/359094.359101. Brélaz, Daniel (2019). L'avenir est plus que jamais notre affaire: l'impact des grandes disruptions. Lausanne: Favre. ISBN 978-2-8289-1803-3. == External links == Media related to Daniel Brélaz at Wikimedia Commons Biography of Daniel Brélaz on the website of the Swiss Parliament.
Wikipedia:Daniel Hershkowitz#0	Daniel Hershkowitz (Hebrew: דניאל הרשקוביץ; born 2 January 1953 in Haifa, Israel) is an Israeli politician, mathematician, and Orthodox rabbi. Since 2018, he has headed the Israel Civil Service Commission. He is professor emeritus of mathematics at the Technion, and is also rabbi of the Ahuza neighborhood in Haifa. He was president of Bar-Ilan University from 2013-17. == Early life == Hershkowitz was born in Haifa. His parents were Holocaust survivors from Hungary, and his father was wounded in the 1948 Arab-Israeli War. Hershkowitz studied at a religious high school, and graduated at age 16. He served for five years in the Intelligence Corps of the Israel Defense Forces, reaching the rank of Major. Hershkowitz earned his BSc in mathematics in 1973, MSc in 1976, and DSc in 1982, all from the Technion – Israel Institute of Technology. His yeshiva studies were conducted at Mercaz HaRav; he received his Semikha (ordination) in 1995 from Rabbis She'ar Yashuv Cohen, Shlomo Chelouche, and Nehemyah Roth, as well as an additional ordination "Rabbi of the City" from the Chief Rabbinate of Israel (2001). == Academia == He has published over 80 mathematics articles in academic journals. He was President of the International Linear Algebra Society (2002-2008), and was previously a Professor of Mathematics at the University of Wisconsin–Madison. In 1982, he was awarded the Landau Research Prize in Mathematics; in 1990, the New England Academic Award for Excellence in Research; in 1990, the Technion's Award for Excellence in Teaching; and in 1991, the Henri Gutwirth Award for Promotion of Research. == Political career == In 2009, he was elected to the Knesset as the leader of the Jewish Home, and was appointed Minister of Science and Technology after joining Benjamin Netanyahu's government. He did not contest the 2013 elections, and subsequently left the Knesset. Since September 2018, he is the Head of the Civil Service Commission under the office of the Prime Minister of Israel. == Bar-Ilan University == He was president of Bar-Ilan University from 2013 to 2017, succeeding Moshe Kaveh and followed by Arie Zaban. == References == == External links == Daniel Hershkowitz on the Knesset website Prof. Daniel Hershkowitz, MK, Israeli Ministry of Foreign Affairs Daniel Hershkowitz's homepage at the Technion Mathematics Department Daniel Hershkowitz at the Mathematics Genealogy Project Rabbinic homepage Biography, borhatorah.org
Wikipedia:Daniel Kráľ#0	Daniel Kráľ (born June 30, 1978) is a Czech mathematician and computer scientist who works as a professor of mathematics and computer science at the Masaryk University. His research primarily concerns graph theory and graph algorithms. == Education and career == He obtained his Ph.D. from Charles University in Prague in 2004, under the supervision of Jan Kratochvíl. After short-term positions at TU Berlin, Charles University, and the Georgia Institute of Technology, he returned to Charles University as a researcher in 2006, and became a tenured associate professor there in 2010. He was awarded the degree of Doctor of Science by the Academy of Sciences of the Czech Republic in 2012, and in the same year moved to a professorship at the University of Warwick. In 2018, Kráľ moved back to the Czech Republic and started working at Faculty of Informatics, Masaryk University, accepting the Donald Knuth professorship chair. == Contributions == In the 1970s, Michael D. Plummer and László Lovász conjectured that every bridgeless cubic graph has an exponential number of perfect matchings, strengthening Petersen's theorem that at least one perfect matching exists. In a pair of papers with different sets of co-authors, Kráľ was able to show that this conjecture is true. == Recognition == Kráľ won first place and a gold medal at the International Olympiad in Informatics in 1996. In 2011, Kráľ won the European Prize in Combinatorics for his work in graph theory, particularly citing his solution to the Plummer–Lovász conjecture and his results on graph coloring. In 2014, he won a Philip Leverhulme Prize in Mathematics and Statistics; the award citation again included Kráľ's research on the Plummer–Lovász conjecture, as well as other publications of Kráľ on pseudorandom permutations and systems of equations. He was elected as a Fellow of the American Mathematical Society in the 2020 Class, for "contributions to extremal combinatorics and graph theory, and for service to the profession". == References == == External links == Home page Google scholar profile
Wikipedia:Daniel Makinde#0	Oluwole Daniel Makinde is a Nigerian professor of Theoretical and Applied Physics, the Secretary General of African Mathematical Union (AMU), General Secretary and Vice President of Southern Africa Mathematical Science Association (SAMSA) and the Director of the Institute for Advanced Research in Mathematical Modeling and Computations (IARMMC) at Cape Peninsula University of Technology, South Africa. == Education == In 1987, Daniel Makinde obtained his first degree from Obafemi Awolowo University, Ile Ife in the field of Mathematics. In 1990, He also obtained his MSc degree in Applied and Computational Mathematics from the same Alma mater and in 1996, he bagged his doctorate degree from the University of Bristol, United Kingdom. == Membership and fellowship == He is an advisory board member of the Pan African Centre of Mathematics (PACM) based in Tanzania between 2003 and 2005. A board member of the Centre for Applied Research in Mathematical Sciences (CARMS) at Inmore University in Kenya and an associate member of the National Institute of Theoretical Physics (NITheP) in South Africa. In 2012, he became a fellow of the African Academy of Sciences and in 2013, he became a fellow of the Papua New Guinea Mathematical Society in recognition of his outreach contributions. == Award and honours == In 2003, he received the African Mathematics Union (AMU) and the International Conference of Mathematical Sciences (ICMS) Young African Medal Award. In 2012, he the African Academy of Sciences (AAS) fellowship award. In 2014, he received the Nigerian National Honour Award, MFR, for his numerous contributions to mathematics and in 2011, he won the African Union Kwame Nkrumah Award for Scientific Excellence at the Continental level for his excellent contributions to basic science in Africa. In 2021, he won Obada Prize (International) Distinguished Researcher Award. == References ==
Wikipedia:Daniel Revuz#0	Daniel Revuz (born 1936 in Paris) is a French mathematician specializing in probability theory, particularly in functional analysis applied to stochastic processes. He is the author of several reference works on Brownian motion, Markov chains, and martingales. == Family and early life == Revuz is the son of mathematicians Germaine and André Revuz, and is one of six children. His family spent parts of his childhood in Poitiers and Istanbul before settling in Paris in 1950. == Education and career == Revuz graduated from Polytechnique in 1956 and received his doctorate from the Sorbonne in 1969 under Jacques Neveu and Paul-André Meyer. He taught at Paris Diderot University at the Laboratory for Probability Theory of the Institut Mathématique de Jussieu. === Research === From his doctoral thesis work Revuz published two articles in 1970, in which he established a theory of one-to-one correspondence between positive Markov additive functionals and associated measures. This theory and the associated measures are now known respectively as "Revuz correspondence" and "Revuz measures." In 1991 Revuz co-authored a research monograph with Marc Yor on stochastic processes and stochastic analysis called "Continuous Martingales and Brownian Motion". The book was highly praised upon its publication. Wilfrid Kendall called it "the book for a capable graduate student starting out on research in probability." == References ==
Wikipedia:Daniela di Serafino#0	Daniela di Serafino (8 April 1966 – 22 August 2022) was an Italian applied mathematician and numerical analyst whose research involved numerical linear algebra, gradient descent methods for nonlinear optimization, and applications in scientific computing. She was a professor of numerical analysis in the Department of Mathematics and Applications at the University of Naples Federico II. == Early life and education == Di Serafino was born in Naples, on 8 April 1966. She was an undergraduate at the University of Naples Federico II, where she earned a master's degree in mathematics in 1989. After two years working as a researcher in the Center of Research for Parallel Computing and Supercomputers of the National Research Council (Italy) in Naples, she returned to the University of Naples for doctoral study in applied mathematics and computer science, completing her Ph.D. in 1995. == Career and later life == After finishing her doctorate, di Serafino worked as an assistant professor at the Università degli Studi della Campania Luigi Vanvitelli (originally the "Second University of Naples") from 1995 to 2004, and as an associate professor there from 2005 to 2018. In this time frame she also held positions as a researcher at the Institute for High-Performance Computing and Networking of the National Research Council. In 2014, she earned a habilitation as a professor of numerical analysis, and in 2018 she obtained a full professorship at the University of Campagna. In 2020 she returned to the University of Naples Federico II as a full professor. She died on 22 August 2022. == Recognition == A 2010 paper by di Serafino with Marco D'Apuzzo and Valentina De Simone, "On mutual impact of numerical linear algebra and large-scale optimization with focus on interior point methods", received the Computational Optimization and Applications Best Paper Award. == References == == External links == Home page Daniela di Serafino publications indexed by Google Scholar
Wikipedia:Daniele Cassani#0	Daniele Cassani is an Italian mathematician. He is a full professor of Mathematical Analysis at Università degli Studi dell'Insubria. == Education == Cassani completed his PhD in pure Mathematics in 2006 at University of Milan, supervised by B. Ruf. == Career == From 2006 to 2007, he undertook a postdoctoral position at the Pacific Institute for the Mathematical Sciences, University of British Columbia. This postdoc was directed by Ivar Ekeland and supervised by Nassif Ghoussoub. Between 2007 and 2011, Cassani held a postdoctoral position at University of Milan, concurrently serving as a lecturer at Polytechnic University of Milan. In 2012, he became a senior research fellow at the Department of Science and High Technology, Università degli Studi dell'Insubria. In 2017, he was appointed as an associate professor, and in 2023, he was promoted to the position of Full Professor of Mathematical Analysis. Cassani has been invited to serve as a visiting professor and speaker at international conferences hosted by prestigious institutions in China, Europe, North and South America, and Japan. He actively contributes as a reviewer for numerous internationally recognized journals. Since 2019, he has also held the role of Springer associate editor for the Milan Journal of Mathematics. Since 2016, Cassani has held the position of President at the Riemann International School of Mathematics. From 2018 to 2023, he served as a member of the Board of Directors of the University of Insubria, and from 2023 onwards, he is a member of the Board of Directors of the Foundation University of Insubria. Since 2023, Cassani is the CEO of Foundation University of Insubria and Editor of Advances in Nonlinear Analysis. == Research work == Cassani's primary research interests encompass Nonlinear Analysis and Calculus of Variations, Partial Differential Equations and Inequalities, Systems of PDE and Applications to MEMS, Solitons Field Theory, Best constants in functional inequalities, Maximum principle, Inverse problems, and Image processing. == References ==
Wikipedia:Danilo Blanuša#0	Danilo Blanuša (7 December 1903 – 8 August 1987) was a Croatian mathematician, physicist, engineer and a professor at the University of Zagreb. == Biography == Blanuša was born in Osijek, Austria-Hungary (today Croatia). He attended elementary school in Vienna and Steyr in Austria and gymnasium in Osijek and Zagreb. He studied engineering in both Zagreb and Vienna and also mathematics and physics. His career started in Zagreb, where he started to work and lecture. His student Mileva Prvanović completed her doctorate in 1955, the first in geometry in Serbia. Blanuša was the dean of the Faculty of Electrical Engineering, Zagreb in the 1957–58 school year. He received the Ruđer Bošković prize in 1960. He died in Zagreb. == Mathematics == In mathematics, Blanuša became known for discovering the second and third known snarks in 1946 (the Petersen graph was the first), triggering a new area of graph theory. The study of snarks had its origin in the 1880 work of P. G. Tait, who at that time had proved that the four color theorem is equivalent to the statement that no snark is planar. Snarks were so named later by the American mathematician Martin Gardner in 1976, after the mysterious and elusive object of Lewis Carroll's poem The Hunting of the Snark. Blanuša's most important works were related to isometric immersions of pseudo-Riemannian manifolds in differential geometry. In particular, in his most cited work he has exhibited an embedding of a hyperbolic (Lobachevsky) two-dimensional plane into 6-dimensional Euclidean space and another construction, for all natural numbers n ≥ 2 {\displaystyle n\geq 2} , of an n {\displaystyle n} -dimensional hyperbolic space into 6 n − 5 {\displaystyle 6n-5} -dimensional Euclidean space. In an earlier work he has exhibited embeddings of n {\displaystyle n} -dimensional hyperbolic spaces into a separated (infinite-dimensional) Hilbert space. His other important works are in the theory of the special functions (Bessel functions) and in graph theory. Some of his results are included in the Japanese mathematical encyclopedia Sugaku jiten in Tokyo, published by Iwanami Shoten in 1962. == Physics == His works were mostly related to the theory of relativity. He discovered a mistake in relations for absolute heat Q and temperature T in relativistic phenomenological thermodynamics, published by Max Planck in Annalen der Physik in 1908. Q0 and T0 are the corresponding classical values, and a=(1-v2/c2)1/2 in the relation → Q=Q0a, T=T0a really should be → Q=Q0/a, T=T0/a This correction was published in Glasnik, the journal relating to mathematics, physics and astronomy in 1947 in the article "Sur les paradoxes de la notion d'énergie". It was rediscovered in 1960, and the correction is still wrongly attributed to H. Ott in the mainstream scientific literature. == See also == Blanuša snarks == References == == External links == Blanuša snarks in the logo of the Croatian Mathematical Society Blanuša snark on a Croatian postage stamp Blanuša Snarks
Wikipedia:Danskin's theorem#0	In convex analysis, Danskin's theorem is a theorem which provides information about the derivatives of a function of the form f ( x ) = max z ∈ Z ϕ ( x , z ) . {\displaystyle f(x)=\max _{z\in Z}\phi (x,z).} The theorem has applications in optimization, where it sometimes is used to solve minimax problems. The original theorem given by J. M. Danskin in his 1967 monograph provides a formula for the directional derivative of the maximum of a (not necessarily convex) directionally differentiable function. An extension to more general conditions was proven 1971 by Dimitri Bertsekas. == Statement == The following version is proven in "Nonlinear programming" (1991). Suppose ϕ ( x , z ) {\displaystyle \phi (x,z)} is a continuous function of two arguments, ϕ : R n × Z → R {\displaystyle \phi :\mathbb {R} ^{n}\times Z\to \mathbb {R} } where Z ⊂ R m {\displaystyle Z\subset \mathbb {R} ^{m}} is a compact set. Under these conditions, Danskin's theorem provides conclusions regarding the convexity and differentiability of the function f ( x ) = max z ∈ Z ϕ ( x , z ) . {\displaystyle f(x)=\max _{z\in Z}\phi (x,z).} To state these results, we define the set of maximizing points Z 0 ( x ) {\displaystyle Z_{0}(x)} as Z 0 ( x ) = { z ¯ : ϕ ( x , z ¯ ) = max z ∈ Z ϕ ( x , z ) } . {\displaystyle Z_{0}(x)=\left\{{\overline {z}}:\phi (x,{\overline {z}})=\max _{z\in Z}\phi (x,z)\right\}.} Danskin's theorem then provides the following results. Convexity f ( x ) {\displaystyle f(x)} is convex if ϕ ( x , z ) {\displaystyle \phi (x,z)} is convex in x {\displaystyle x} for every z ∈ Z {\displaystyle z\in Z} . Directional semi-differential The semi-differential of f ( x ) {\displaystyle f(x)} in the direction y {\displaystyle y} , denoted ∂ y f ( x ) , {\displaystyle \partial _{y}\ f(x),} is given by ∂ y f ( x ) = max z ∈ Z 0 ( x ) ϕ ′ ( x , z ; y ) , {\displaystyle \partial _{y}f(x)=\max _{z\in Z_{0}(x)}\phi '(x,z;y),} where ϕ ′ ( x , z ; y ) {\displaystyle \phi '(x,z;y)} is the directional derivative of the function ϕ ( ⋅ , z ) {\displaystyle \phi (\cdot ,z)} at x {\displaystyle x} in the direction y . {\displaystyle y.} Derivative f ( x ) {\displaystyle f(x)} is differentiable at x {\displaystyle x} if Z 0 ( x ) {\displaystyle Z_{0}(x)} consists of a single element z ¯ {\displaystyle {\overline {z}}} . In this case, the derivative of f ( x ) {\displaystyle f(x)} (or the gradient of f ( x ) {\displaystyle f(x)} if x {\displaystyle x} is a vector) is given by ∂ f ∂ x = ∂ ϕ ( x , z ¯ ) ∂ x . {\displaystyle {\frac {\partial f}{\partial x}}={\frac {\partial \phi (x,{\overline {z}})}{\partial x}}.} === Example of no directional derivative === In the statement of Danskin, it is important to conclude semi-differentiability of f {\displaystyle f} and not directional-derivative as explains this simple example. Set Z = { − 1 , + 1 } , ϕ ( x , z ) = z x {\displaystyle Z=\{-1,+1\},\ \phi (x,z)=zx} , we get f ( x ) = \| x \| {\displaystyle f(x)=\|x\|} which is semi-differentiable with ∂ − f ( 0 ) = − 1 , ∂ + f ( 0 ) = + 1 {\displaystyle \partial _{-}f(0)=-1,\partial _{+}f(0)=+1} but has not a directional derivative at x = 0 {\displaystyle x=0} . === Subdifferential === If ϕ ( x , z ) {\displaystyle \phi (x,z)} is differentiable with respect to x {\displaystyle x} for all z ∈ Z , {\displaystyle z\in Z,} and if ∂ ϕ / ∂ x {\displaystyle \partial \phi /\partial x} is continuous with respect to z {\displaystyle z} for all x {\displaystyle x} , then the subdifferential of f ( x ) {\displaystyle f(x)} is given by ∂ f ( x ) = c o n v { ∂ ϕ ( x , z ) ∂ x : z ∈ Z 0 ( x ) } {\displaystyle \partial f(x)=\mathrm {conv} \left\{{\frac {\partial \phi (x,z)}{\partial x}}:z\in Z_{0}(x)\right\}} where c o n v {\displaystyle \mathrm {conv} } indicates the convex hull operation. == Extension == The 1971 Ph.D. Thesis by Dimitri P. Bertsekas (Proposition A.22) proves a more general result, which does not require that ϕ ( ⋅ , z ) {\displaystyle \phi (\cdot ,z)} is differentiable. Instead it assumes that ϕ ( ⋅ , z ) {\displaystyle \phi (\cdot ,z)} is an extended real-valued closed proper convex function for each z {\displaystyle z} in the compact set Z , {\displaystyle Z,} that int ⁡ ( dom ⁡ ( f ) ) , {\displaystyle \operatorname {int} (\operatorname {dom} (f)),} the interior of the effective domain of f , {\displaystyle f,} is nonempty, and that ϕ {\displaystyle \phi } is continuous on the set int ⁡ ( dom ⁡ ( f ) ) × Z . {\displaystyle \operatorname {int} (\operatorname {dom} (f))\times Z.} Then for all x {\displaystyle x} in int ⁡ ( dom ⁡ ( f ) ) , {\displaystyle \operatorname {int} (\operatorname {dom} (f)),} the subdifferential of f {\displaystyle f} at x {\displaystyle x} is given by ∂ f ( x ) = conv ⁡ { ∂ ϕ ( x , z ) : z ∈ Z 0 ( x ) } {\displaystyle \partial f(x)=\operatorname {conv} \left\{\partial \phi (x,z):z\in Z_{0}(x)\right\}} where ∂ ϕ ( x , z ) {\displaystyle \partial \phi (x,z)} is the subdifferential of ϕ ( ⋅ , z ) {\displaystyle \phi (\cdot ,z)} at x {\displaystyle x} for any z {\displaystyle z} in Z . {\displaystyle Z.} == See also == Maximum theorem Envelope theorem Hotelling's lemma == References ==
Wikipedia:Danuta Przeworska-Rolewicz#0	Danuta Przeworska-Rolewicz (25 May 1931 – 23 June 2012), was a Polish professor of mathematics and long-time employee of the Institute of Mathematics of the Polish Academy of Sciences. During World War II, as a child, she was a resistance fighter. == Life and work == Danuta Przeworska was born in Warsaw into the family of two archaeologists Stefan Przeworski and his wife Janina. Initially, Danuta wanted to pursue archaeology but switched to mathematics instead. With the outbreak of World War II, she participated in the resistance movement as a child and fought in the Warsaw Uprising of 1944, for which she was awarded the Warsaw Uprising Cross. === Academic work === After the secondary school, she began studies at the Faculty of Mathematics of the University of Warsaw, where in 1956 she obtained a master's degree. Then she started research work at the Mathematical Institute of the Polish Academy of Sciences. In 1958 she obtained the title of doctor (with her dissertation titled On systems of strongly-singular integral equations, under the supervision of Witold Pogorzelski), and in 1964 she successfully defended her habilitation thesis. From 1954 to 1960 she worked as an assistant and lecturer at the Warsaw University of Technology, from 1960 she lectured at the Mathematical Institute of the Polish Academy of Sciences. In 1973, she lectured for a year at the Cybernetics Department of the Military University of Technology where she taught algebraic analysis classes using an experimental method of her own invention. In 1974, she obtained the title of professor of mathematical sciences and she went on to supervise nine PhD students. Przeworska-Rolewicz's fields of interest included singular integral equations, algebraic methods in analysis / operational calculus, functional analysis and others. Her scientific achievements include more than 200 scientific papers and four textbooks. Danuta Przeworska-Rolewicz was the organizer of the international conferences organized in Warsaw entitled "Functional-differential systems and related systems," which took place in 1979, 1981, 1983 and 1985, and then "Various aspects of differentiability" in 1993 and 1995. The conferences attempted to build collaborative research relationships among the attendees. She was among the founders of the Fractional Calculus and Applied Analysis journal, and she was also a member of the editorial boards of Demonstratio Mathematica (since 1987), Scientiae Mathematicae (since 1997) and Matematica Japonica (since 1998). === Personal life === She married the Polish mathematician and colleague Stefan Henryk Rolewicz (1932–2015) in January 1952 and they had two children On 23 August 1980, she joined the appeal of 64 scholars, writers and journalists to the communist authorities for dialogue with striking workers. Danuta Przeworska-Rolewicz died in Warsaw in June 2012 and is buried at the Powązki Military Cemetery in Warsaw (section G-2-2/3). == Awards == Warsaw Uprising Cross (1982) Gold Cross of Merit (1985) Stefan-Banach Prize of the Polish Mathematical Society (1969) Prize of the Polish Academy of Sciences (1972) Prize of the German Academy of Sciences and the Polish Academy of Sciences (1978) == References ==
Wikipedia:Dany Leviatan#0	Dany Leviatan (Hebrew: דני לויתן; born 21 February 1942) is an Israeli mathematician and former rector of Tel Aviv University. == Biography == Dany Leviatan completed his B.Sc. and M.Sc. at the Hebrew University of Jerusalem. A participant in the Academic Atuda program, Leviatan served as a mathematician in the Israel Air Force while working on his doctorate at the Hebrew University, which he completed in 1966. == Academic career == He worked as visiting professor at the University of Illinois Urbana-Champaign from 1967 to 1970 through the Fulbright Scholarship Program, and became associate professor at Tel Aviv University in 1972. Leviatan served as head of the university's Department of Mathematics from 1972 to 1974, and dean of the Faculty of Exact Sciences from 1976 to 1980. He became head of the recently established School of Mathematics in 1982. Leviatan was appointed rector of Tel Aviv University on 16 August 2005, a position he kept until 2010. He briefly served as acting president of the university following the resignation of Zvi Galil in July 2009. == Selected works == Cohen, Albert; Davenport, Mark A.; Leviatan, Dany (2013). "On the Stability and Accuracy of Least Squares Approximations". Foundations of Computational Mathematics. 13 (5). Springer Nature: 819–834. arXiv:1111.4422. doi:10.1007/s10208-013-9142-3. ISSN 1615-3375. S2CID 12757269. Leviatan, D.; Temlyakov, Vladimir N. (2006). "Simultaneous approximation by greedy algorithms". Advances in Computational Mathematics. 25 (1–3). Springer Nature: 73–90. CiteSeerX 10.1.1.377.936. doi:10.1007/s10444-004-7613-4. ISSN 1019-7168. S2CID 9429496. DeVore, Ronald A.; Leviatan, Dany; Yu, Xiang Ming (1992). "Polynomial approximation in L p {\displaystyle L_{p}} ( 0 < p < 1 {\displaystyle 0<p<1} )". Constructive Approximation. 8 (2): 187–201. doi:10.1007/BF01238268. S2CID 122111054. Jakimovski, Amnon; Leviatan, Dany (1969). "Generalized Szász operators for the approximation in the finite interval". Mathematica (Cluj). 11 (34): 97–103. == See also == Dany Leviatan at the Mathematics Genealogy Project == References ==
Wikipedia:Darboux's formula#0	In mathematical analysis, Darboux's formula is a formula introduced by Gaston Darboux (1876) for summing infinite series by using integrals or evaluating integrals using infinite series. It is a generalization to the complex plane of the Euler–Maclaurin summation formula, which is used for similar purposes and derived in a similar manner (by repeated integration by parts of a particular choice of integrand). Darboux's formula can also be used to derive the Taylor series from calculus. == Statement == If φ(t) is a polynomial of degree n and f an analytic function then ∑ m = 0 n ( − 1 ) m ( z − a ) m [ φ ( n − m ) ( 1 ) f ( m ) ( z ) − φ ( n − m ) ( 0 ) f ( m ) ( a ) ] = ( − 1 ) n ( z − a ) n + 1 ∫ 0 1 φ ( t ) f ( n + 1 ) [ a + t ( z − a ) ] d t . {\displaystyle {\begin{aligned}&\sum _{m=0}^{n}(-1)^{m}(z-a)^{m}\left[\varphi ^{(n-m)}(1)f^{(m)}(z)-\varphi ^{(n-m)}(0)f^{(m)}(a)\right]\\={}&(-1)^{n}(z-a)^{n+1}\int _{0}^{1}\varphi (t)f^{(n+1)}\left[a+t(z-a)\right]\,dt.\end{aligned}}} The formula can be proved by repeated integration by parts. == Special cases == Taking φ to be a Bernoulli polynomial in Darboux's formula gives the Euler–Maclaurin summation formula. Taking φ to be (t − 1)n gives the formula for a Taylor series. == References == Darboux (1876), "Sur les développements en série des fonctions d'une seule variable", Journal de Mathématiques Pures et Appliquées, 3 (II): 291–312 Whittaker, E. T. and Watson, G. N. "A Formula Due to Darboux." §7.1 in A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, p. 125, 1990. [1] == External links == Darboux's formula at MathWorld
Wikipedia:David A. Sánchez#0	David A. Sánchez (born January 13, 1933) is a Mexican-American university and research administrator, mathematician, educator, and author. He held the posts of provost at Lehigh University, assistant director for the National Science Foundation and Los Alamos National Laboratory, and assistant Chancellor for the Texas A&M University (TAMU) system; As a mathematician, his areas of focus include ordinary differential equations and biomathematics. He has been a researcher and professor at several universities, including the University of Wisconsin's Mathematics Research Lab, UCLA, and the University of New Mexico. He published his memoir, Don't Forget the Accent Mark: A Memoir, in 2011. == Biography == === Early life and education === David Alan Sánchez was born on January 13, 1933 in San Francisco, California to Berta Sánchez. At the age of three, he moved to Mission Hills in San Diego and was adopted by his grandparents, Cecilio and Concepcion Sánchez. Sánchez earned his Bachelor of Science (B.S.) in 1955 from the University of New Mexico. After graduating, he served the U.S. Marine Corps for three years. Sánchez continued with his graduate studies at the University of Michigan (UM), where he worked at their Institute of Science and Technology in the Radar Laboratory. His research involved battlefield simulations for the Army. He received his Master of Science (M.S.) in 1960. He wrote his doctoral dissertation, Calculus of Variations for Integrals Depending on a Convolution Product under the direction Lamberto Cesari; receiving his doctorate (Ph.D.) in 1964 from UM. === Personal life === In 1957, Sánchez married Joan Thomas. They have two children - a son and a daughter. After he retired from the University of New Mexico, he made his home in Corrales, New Mexico. == Career == === Academia === After completing his Ph.D, Sánchez spent two years at the University of Chicago as an instructor, before traveling to Manchester, England as a visiting professor at Manchester University. UCLA (1966-1977)In 1966, Sánchez became an assistant professor at the University of California, Los Angeles (UCLA). While at UCLA, he presented papers at the American Mathematical Societies 1967 meeting, and at the Conference on Qualitative Theory of Nonlinear Differential and Integral Equations in 1968, on extremals of composite functions and Ricotti equations, respectively. After four years, spent a year at Brown University as a visiting professor (1973-1974); and then returned to UCLA as an associate professor. He became a full professor at UCLA in 1976. University of New Mexico (1977-1986) In 1977, Sánchez returned to his undergraduate alma mater, the University of New Mexico, as a full professor in the Department of Mathematics and Statistics. He served as the department chairman from 1983-1986). He took a sabbatical in 1982 to teach at the University of Wales in Aberystwyth, Ceredigion, Wales. In 1986, Sánchez took a nine-year detour in his academic career, into administrative roles. Final posts - Texas A&M and University of New Mexico In 1995, after serving as assistant Chancellor for Texas A&M, Sánchez returned to a teaching and research role at the university. Sánchez' returned to the University of New Mexico (UNM) for his final university position before retiring. At UNM, he served as chairman of the Department of Mathematics and Statistics. === Administrative career (1986-1995) === In 1986, Sánchez took a detour from his traditional academic path to a new career path, accepting a position as the vice president and provost at Lehigh University in Bethlehem, Pennsylvania. While at Lehigh, he restructured the promotion and tenure process; and, chaired a commission on racial diversity. He remained at Lehigh until 1990, when he moved to the National Science Foundation (NSF) to take a two-year term as an assistant director. At the NSF, he headed the Mathematics and Physical Sciences Directorate, the largest division at the NSF, "setting national priorities on scientific research." At the end of his term at the NSF, he took a position as an assistant director for the Los Alamos National Laboratory. In November 1993, Sánchez moved to the Texas A&M University System as Vice Chancellor for Academic Affairs, to take on long-range planning. His foray into administration lasted for nine years, when he returned to teaching at Texas A&M. == Selected publications == Sánchez is the author of three books on mathematics and has written over fifty journal articles in the fields of optimization, biomathematics, differential equations, and numerical analysis.Here is a selected list of his publications: === Books === Sánchez, David A.; Lakin, William D. (1982). Topics in Ordinary Differential Equations. Dover Publications. ISBN 0486616061. Sánchez, David A. (1983). Differential Equations:An Introduction (1st ed.). Addison Wesley Publishing Company. ISBN 0201077604. Sánchez, David A. (1988). Differential Equations (2nd ed.). Addison Wesley Publishing Company. ISBN 0201154072. Sánchez, David A. (2002). Ordinary Differential Equations: A Brief Eclectic Tour. Classroom Resource Materials. Vol. 19. Washington, DC: Mathematical Association of America. ISBN 0-88385-723-5. (a brief guide to concepts of ordinary differential equations) Sánchez, David A. (2012). Ordinary Differential Equations and Stability Theory: An Introduction. Dover Books on Mathematics. Dover Publications. ISBN 978-0486638287. === Journals === Sánchez, D. A. (1964). "Total Variation and Uniform Convergence". The American Mathematical Monthly, 71(5), 537–539. Sánchez, D. A. (1966). "On Composite Variational Problems". SIAM Journal on Applied Mathematics, 14(1), 60–64. Sánchez, D. A. (1968). "On Extremals of Composite Variational Problems". Proceedings of the American Mathematical Society, 19(3), 555–559. Sánchez, D. A. (1969). "A Note on Periodic Solutions of Riccati-Type Equations." SIAM Journal on Applied Mathematics, 17(5), 957–959. Sánchez, D. A. (1975). "The Green’s Function and Determining Equations". The American Mathematical Monthly, 82(7), 747–749. Sánchez, David (1975). "Constant Rate Population Harvesting: Equilibrium and Stability". Theoretical population biology, 8, 12-30. Lehmer, E., & Sánchez, D. A. (1980). "Problems Dedicated to Emory P. Starke: S27-S28". The American Mathematical Monthly, 87(3), 218–218. Sánchez, D. A. (1998). "Review of Ordinary Differential Equations Texts". The American Mathematical Monthly, 105(4), 377–383. Sánchez, D. A. (2001). "An Alternative to the Shooting Method for a Certain Class of Boundary Value Problems". The American Mathematical Monthly, 108(6), 552–555. https://doi.org/10.2307/2695711 Littleton, P., & Sánchez, D. A. (2001). "Dipsticks for Cylindrical Storage Tanks--Exact and Approximate". The College Mathematics Journal, 32(5), 352–358. https://doi.org/10.2307/2687307 Sánchez, David. (2008). "Periodic environments and periodic harvesting". Natural Resource Modeling, 16, 233-244. == See also == Richard A. Tapia - American mathematician, mentored by Sánchez == Notes == == References ==
Wikipedia:David Benney#0	David John Benney (8 April 1930 – 9 October 2015) was a New Zealand applied mathematician, known for work on the nonlinear partial differential equations of fluid dynamics. == Education and early life == Born in Wellington, New Zealand, on 8 April 1930 to Cecil Henry (Matt) Benney and Phyllis Marjorie Jenkins, Benney was educated at Wellington College. He graduated BSc from Victoria University College in 1950, and MSc from the same institution in 1951. He then went to Emmanuel College, Cambridge, from where he graduated BA in the Mathematical Tripos in 1954. He was at Canterbury University College for two years as a lecturer, before taking leave of absence in August 1957 to undertake doctoral studies at Massachusetts Institute of Technology (MIT), graduating PhD in 1959. == Career and research == Benney joined the mathematics faculty at MIT in 1960. He spent the rest of his career there, as a prolific researcher in fluid dynamics and supervisor of students, becoming emeritus professor. He received a Guggenheim Fellowship in 1964. == Notes == == References == Petersen, George Conrad (1971). Who's Who in New Zealand, 1971 (10th ed.). Wellington: A.H & A.W. Reed.
Wikipedia:David Borwein#0	David Borwein (March 24, 1924 – September 3, 2021) was a Lithuanian-born Canadian mathematician, known for his research in the summability theory of series and integrals. He also did work in measure theory and probability theory, number theory, and approximate subgradients and coderivatives. He latterly collaborated with his son, Jonathan Borwein, and with B.A. Mares Jr. on the properties of single-variable and many-variable sinc integrals. == Biography == Borwein was born in March 1924 in Lithuania to an Ashkenazi Jewish family. He formerly resided and worked in St. Andrews, Scotland, before moving to London, Ontario where he eventually became Head of Mathematics at the University of Western Ontario. He was also the president of the Canadian Mathematical Society (CMS). The David Borwein Distinguished Career Award given out by the CMS is named after him. He was an active researcher in summability theory, classical analysis, inequalities, matrix transformations, and was professor emeritus at the University of Western Ontario, department of Mathematics. His wife of over 60 years, Bessie Borwein, is a prominent anatomist, and is professor emerita of anatomy at the University of Western Ontario. In 2018 the Canadian Mathematical Society listed him in their inaugural class of fellows. Borwein died in his sleep on September 3, 2021, at the age of 97. == See also == Borwein integral Jonathan Borwein (son and co-researcher) Peter Borwein (son and mathematician) == References == == External links == Curriculum Vitae, University of Western Ontario The David Borwein Distinguished Career Award David Borwein at the Mathematics Genealogy Project
Wikipedia:David Cheriton#0	David Ross Cheriton (born March 29, 1951) is a Canadian computer scientist, businessman, philanthropist, and venture capitalist. He is a computer science professor at Stanford University, where he founded and leads the Distributed Systems Group. He is a distributed computing and computer networking expert, with insight into identifying big market opportunities and building the architectures needed to address such opportunities. He has founded and invested in technology companies, including Google, where he was among the first angel investors; VMware, where he was an early investor; and Arista, where he was cofounder and chief scientist. He has funded at least 20 companies. As of 2025, Forbes estimated Cheriton's net worth at US$19.8 billion while Maclean's estimates his worth at $18.64 billion. He has made contributions to education, with a $25 million donation to support graduate studies and research in the School of Computer Science (subsequently renamed David R. Cheriton School of Computer Science) at the University of Waterloo, a $7.5 million donation to the University of British Columbia, and a $12 million endowment in 2016 to Stanford University to support Computer Science faculty, graduate fellowships, and undergraduate scholarships. == Education == Born in Vancouver, British Columbia, Canada, Cheriton attended public schools in the Highlands neighborhood of Edmonton, Alberta, Canada. He briefly attended the University of Alberta where he had applied for both mathematics and music. He was rejected by the music program, and then went on to study mathematics and received his Bachelor of Science (B.S.) degree from the University of British Columbia in 1973. Cheriton received his Master of Science (M.S.) and Doctor of Philosophy (Ph.D.) degrees in computer science from the University of Waterloo in 1974 and 1978, respectively. He spent three years as an assistant professor at his alma mater, the University of British Columbia, before moving to Stanford. == Research == Cheriton was involved in creating three microkernel operating systems (OSes). He was one of the early principal developers of Thoth, a real-time operating system, and then the Verex kernel. He then founded and led the Distributed Systems Group at Stanford University, which developed a microkernel OS named V. He has published profusely in the areas of distributed computing and computer networking. He won the prestigious SIGCOMM award in 2003, in recognition for his lifetime contribution to the field of telecommunications networks. Cheriton was the mentor and advisor of students such as: Sergey Brin and Larry Page (founders of Google), Kenneth Duda (founder of Arista Networks), Hugh Holbrook (VP Software Engineering at Arista Networks), Sandeep Singhal (was GM at Microsoft, now at Google), and Kieran Harty (CTO and founder of Tintri). As of 2016, Cheriton is working with Stanford students on transactional memory, making memory systems that are resilient to failures. In-memory processing leads to dramatically faster computers – in some cases speeding up applications by a factor of 100,000. It changes the complete nature of how a business can run. We’re trying to lower the cost and to fit these systems in existing memory structures and reduce the number of components to make them more reliable and more secure. == Industry == Cheriton cofounded Granite Systems with Andy Bechtolsheim. The company developed gigabit Ethernet products. It was acquired by Cisco Systems in 1996. In August 1998, Stanford students Sergey Brin and Larry Page met Bechtolsheim on Cheriton's front porch. At the meeting, Bechtolsheim wrote the first cheque to fund their company, Google, and Cheriton joined him as an angel investor with a $200,000 investment. Cheriton was also an early investor in compute virtualization leader VMware, which was later acquired for $625M by EMC in 2004. VMware had a successful public offering in 2007. In 2001 Cheriton and Bechtolsheim founded another start-up company, Palo Alto based Kealia. Kealia designed a high-capacity streaming video server; Galaxy, a range of servers based on AMD's Opteron microprocessor; and Thumper, an enterprise-grade network attached storage system. Kealia was bought by Sun Microsystems in 2004, with Thumper becoming the Sun Fire X4500. In 2004, Cheriton cofounded (again with Bechtolsheim) and was chief scientist of Arista Networks, where he worked on the foundations of the Extensible Operating System (EOS). Arista had a successful public offering in 2014. Cheriton is an investor in and advisory board member for frontline data warehouse company Aster Data Systems, which was acquired by Teradata in 2011 for $263M. Cheriton is also one of the earliest investors in Tintri, a storage virtualization company founded by his student Kieran Harty. Cheriton was also an early investor in in-video advertising company Zunavision, and he founded OptumSoft. In 2014, Cheriton cofounded and invested in Apstra, Inc. In 2015, he cofounded and invested in BrainofT, Inc. (Caspar). He currently serves as the Chief Data Center Scientist at Juniper Networks. == Lifestyle == Although the Google investment alone would be worth over US$1 billion, Cheriton has a reputation for a frugal lifestyle, avoiding costly cars or large houses. He was once included in a list of "cheapskate billionaires". On November 18, 2005, the University of Waterloo announced that Cheriton had donated $25 million to support graduate studies and research in its School of Computer Science. In recognition of his contribution, the school was renamed the David R. Cheriton School of Computer Science. In 2009, he donated $2 million to the University of British Columbia, which will go to fund the Carl Wieman Science Education Initiative (CWSEI). He more recently donated $7.5M to fund a new chair in computing, and a new course on computational thinking. Cheriton has also funded two graduate student fellowships and one undergrad fellowship at Stanford, and donated several millions of dollars to Stanford to fund research. He campaigned against Asynchronous Transfer Mode (ATM) that was favored by telephone carriers, preferring Ethernet, which he saw as a simpler, proven option. Ethernet gradually superseded alternatives. == Personal life == In 1980, Cheriton married Iris Fraser. They divorced in 1994. According to public record, Cheriton has made donations to Republican causes including the party, candidate PACs, senators, and made a total of over $5,000 donations to the presidential candidate Donald Trump. == See also == List of University of Waterloo people == References == == External links == Official website Stanford University Distributed Systems Group Founder of Intent Based Networking Company Apstra David Cheriton speaks at the Open Networking User Group (ONUG) Great Debate
Wikipedia:David Gans#0	David Gans (Hebrew: דָּוִד בֶּן שְׁלֹמֹה גנז; ‎1541–1613), also known as Rabbi Dovid Solomon Ganz, was a German-Jewish chronicler, mathematician, historian, astronomer and astrologer. He is the author of "Tzemach David" (1592) and therefore also known by this title, the צמח דוד. == Biography == David was born in Lippstadt, in what is now North Rhine-Westphalia, Germany. His father, Shlomo, was a moneylender. He studied rabbinical literature in Bonn and Frankfurt am Main, then in Kraków under Moses Isserles. Later he attended the lectures of the Maharal of Prague and of his brother, Rabbi Sinai. They introduced philosophy, mathematics, and astronomy into the circle of their studies, and from them Gans received the impulse to devote himself to these branches of science. He lived for a time in Nordheim (where he studied Euclid), passed several years in his native city of Lippstadt, and then in about 1564 settled in Prague. There he came into contact with Kepler and Tycho Brahe, and took part for three consecutive days in astronomical observations at the Prague observatory. He was charged with the translation of the Alfonsine Tables from Hebrew into German by Tycho Brahe. Gans was the first influential Jewish writer to reconcile Copernicanism. Gans wrote on a variety of liberal arts and scientific topics, making him unique among the Ashkenazi for his production of secular scholarship. His grave in the Old Jewish Cemetery in Prague is marked with a Star of David and a goose (Gans' last name and the Yiddish word for goose, גאַנדז, are homophones). The star of David, in Hebrew called a "Magen David", alludes to his work titled Magen David. == Writings == Among Gans's works the most widely known is his history entitled Tzemach David, published first in Prague in 1592. It is divided into two parts, the first containing the annals of Jewish history, the second those of general history. The author consulted the writings of Cyriacus Spangenberg, Laurentius Faustus, Hubertus Holtzius, Georg Cassino, and Martin Borisk for the second part of his work. Gans's annals are memorable as the first work of this kind among the German Jews. In his preface to the second volume the author tries to justify writing about a "profane" subject like general history, and demonstrated that it was permitted to read history on Shabbat. Tzemach David was published in many editions. To the 1692 edition published in Frankfurt, David ben Moses Rheindorf added a third part containing the annals of that century, which addition has been retained in later editions of the Tzemach. The first part of Gans's work, and extracts from the second, were translated into Latin by Wilhelm Heinrich Vorst (Leyden, 1644). It was translated also into Yiddish by Solomon Hanau (Frankfurt, 1692). modern day translations also include English (1966, Hornier and Czech (2016, Šedinová) Gans was also the author of: Gebulat ha-Eretz, a work on cosmography, which is in all probability identical to the Zurat ha-Eretz, published in Constantinople under the name of "David Avazi" ("avaz" means "goose" in Hebrew, a reference to the surname "Gans", which means "goose" in Yiddish); Magen David, an astronomical treatise, a part of which is included in the Nechmad ve'naim mentioned below; the mathematical works Ma'or ha-Ḳatan, Migdal David, and Prozdor, which are no longer in existence; Nechmad ve'naim dealing with astronomy and mathematical geography, published with additions by Joel ben Jekuthiel of Glogau at Jessnitz, 1743. This work is divided into 12 chapters and 305 paragraphs. In the introduction the author gives a historical survey of the development of astronomy and mathematical geography among the nations. Although acquainted with the work of Copernicus, Gans followed the Ptolemaic system, attributing the Copernican system to the Pythagoreans. He also ventures to assert that the prophet Daniel made a mistake in computation. A Latin translation of the introduction, and a résumé made by Hebenstreit, are appended to the Nechmad ve'naim. Gans claims that Abraham and Solomon knew astronomy, and that the Egyptians learned astronomy from Jacob, which they then taught to the Greeks. Gans took inspiration from Josippon and Maimonides. Gans' work is a hybrid of two parallel stories of world and Jewish history. While not as cutting-edge a historian as his contemporary, Azariah de Rossi, his books introduced historiography to the Ashkenazi audience, making him a forerunner of subsequent developments in Jewish culture. Gans' work can be seen as a defense of the traditional dissemination of knowledge. == See also == Judah Loew ben Bezalel == Reference notes == == External links == Jewish Encyclopedia: "Gans, David ben Solomon ben Seligman" by Joseph Jacobs & Isaac Broydé (1906). Zemach David:A Chronicle of Jewish and World History
Wikipedia:David Gauld (mathematician)#0	David Barry Gauld (born 28 June 1942) is a New Zealand mathematician. He is a professor of mathematics at the University of Auckland. == Biography == Within mathematics, Gauld works in set-theoretic topology with emphasis on applications to non-metrisable manifolds and topological properties of manifolds close to metrisability. Gauld has authored two monographs and over 70 research papers. Gauld was born on 28 June 1942 in Inglewood and grew up there. He was educated at Wanganui Technical College, Inglewood High School and New Plymouth Boys’ High School, and later obtained his BSc and MSc degrees with first-class honours in mathematics from the University of Auckland. Awarded a Fulbright Grant, he completed his PhD in topology, in the University of California, Los Angeles, supervised by Robion Kirby. He was Head of the Department of Mathematics for 15 years and Assistant Vice-Chancellor (Research) for two-and-a-half years at the University of Auckland. Notable students of Gauld include Sina Greenwood. == Honours == In the years 1981–1982, Gauld served as president of the New Zealand Mathematical Society. He was the founding secretary of the New Zealand Mathematics Research Institute, and served in this position for 13 years, retiring in 2011. In 1997, he was awarded a New Zealand Science and Technology Medal by the Royal Society of New Zealand. In 2015, he became an honorary life member of the New Zealand Mathematical Society. In the 2016 New Year Honours, Gauld was appointed an Officer of the New Zealand Order of Merit for services to mathematics. == References ==
Wikipedia:David Gruen (economist)#0	David William Gruen (born 31 August 1954) is an Australian statistician and mathematician. He is the current Australian Statistician at the Australian Bureau of Statistics serving since 11 December 2019. He previously served as Deputy Secretary, Economic and Australia's G20 Sherpa at the Department of the Prime Minister and Cabinet. He previously held office as the executive director of the Macroeconomic group of Australian Treasury and was the head of the Economic Research Department of the Reserve Bank of Australia from 1998 to 2002. Gruen was appointed an Officer of the Order of Australia in the 2022 Australia Day Honours. He was elected a Fellow of the Academy of the Social Sciences in Australia in 2023. == References ==
Wikipedia:David Holcman#0	David Holcman is a computational neurobiologist, applied mathematician and biophysicist at École Normale Supérieure in Paris. He is recognized for his pioneering work in several areas of the sciences, showing that data modeling in biology can lead to predictions, quantifications and understanding, while developing computational approaches. Narrow escape problem: to estimate escape times of stochastic particles from confined domains, Holcman, Schuss and Singer developed asymptotic methods based on the Laplace equation. The theory has been validated by physical experiments and is used in cell biology to estimate time scales of molecular activation. Redundancy principle in biology: He developed extreme statistics in the context of Narrow escape to demonstrate how biological systems leverage redundancy to maintain cell function despite stochastic fluctuations. Neurobiological and Biophysical Modeling: His research encompasses the modeling of receptors, ions, and molecular trafficking in neurobiology, including studies of diffusion and electrodiffusion in nanodomains such as dendritic spines, as well as the analysis and simulations of neuronal networks dynamics (e.g., Up and Down states in electrophysiology). Modeling developmental biology and neuronal navigation: through the modeling of morphogen gradients, intracellular trafficking, and axon guidance. In collaboration with Alain Prochiantz, he developed quantitative models of morphogen signaling, challenging classical views of transcription factor action. Holcman introduced novel mathematical tools to study how cells interpret spatial cues during development. A landmark contribution is the concept of triangulation sensing, which explains how cells localize signal sources using spatially distributed receptors. Holcman's models combine stochastic processes, diffusion theory, and complex geometry. Data science of single particle trajectories, Multiscale Methods and Polymer Physics: He developed multiscale methods, simulation techniques for analyzing extensive molecular super-resolution trajectory data and polymer physics models to study cell nucleus organization. Reconstruction Algorithms of astrocyte networks within neural tissue. He introduced several software such as AstroNet, a data-driven algorithm that utilizes two-photon calcium imaging to map temporal correlations in astrocyte activation. This method revealed distinct connectivity patterns in the hippocampus and motor cortex, providing new insights into the functional organization of astrocytic networks in the brain. In general his computational models of astrocyte signaling offer a deeper understanding of how these glial cells maintain neural homeostasis and modulate synaptic function. EEG Analysis and real-time Anesthesia Monitoring: The development of adaptive algorithms for analyzing real-time EEG data during general anesthesia allowed for dynamic prediction of brain state transitions. By integrating time-frequency analysis with statistical methods, his work has significantly improved the precision of EEG monitoring, leading to optimized decision in anesthesia dosing and enhanced patient safety. AI and Spatial Statistics Applications in cell biology :appling AI-based techniques to extract and interpret complex spatial patterns from neurophysiological data, has deepening our understanding of brain connectivity and the neural effects of anesthetic agents. These interdisciplinary contributions effectively merge advanced computational methods with clinical neuroscience, paving the way for innovative research tools and practical medical applications. These computational approaches have led to several experimentally verified predictions in the life sciences, including the nanocolumn organization of synapses, astrocytic protrusion penetrating neuronal synapses, and insights into the organization of the endoplasmic reticulum and topologically associated domains, where multiple boundary types have been found. == Works == Holcman's research interests include Computational Neuroscience, Data Modeling, Computational Methods, Mathematical Biology, Stochastic Processes, stochastic simulations, theory of cellular microworld, neuronal networks, computational biology and neuroscience, asymptotic approaches in partial differential equations, predictive medicine, electroencephalography (EEG) analysis, and modeling organelles in cells. His contributions also extend to methods for analyzing single particle trajectories, calcium dynamics in dendritic spines, AI-based statistical methods, polymer models, and simulations for chromatin and nucleus organization. His recent work has focused on predicting brain state transitions during general anesthesia by analyzing real-time multidimensional dynamics, including time-frequency patterns and signal suppressions == Publications == Holcman has published over 250 journal articles and holds two patents. He is also co-author or editor of several influential books: He is the co-author of the books: David Holcman and Zeev Schuss, Stochastic Narrow Escape in Molecular and Cellular Biology: Analysis and Applications, 2015-09-08, ISBN 978-1-4939-3102-6 David Holcman (editor), Stochastic Processes, Multiscale Modeling, and Numerical Methods for Computational Cellular Biology, 2017-10-04, ISBN 978-3-319-62626-0 David Holcman and Zeev Schuss, Asymptotics of Elliptic and Parabolic PDEs: and their Applications in Statistical Physics, Computational Neuroscience, and Biophysics, 2018-05-25 ISBN 978-3-319-76894-6 IVAN KUPKA LEGACY: A Tour Through Controlled Dynamics. By Bernard Bonnard, Monique Chyba, David Holcman and Emmanuel Trélat (Eds.) ISBN-10: 1-60133-026-X == Press coverage == To celebrate the first winners of the Europeran Research council (ERC) in 2007, an international meeting was organized in Paris, so that they could discuss their vision and research plan for the future. The narrow escape theory has inspired the Fargo TV series in 2017. The novel nanoscale molecular organization underlying calcium dynamics in synapses, revealed by combining multidisciplinary approaches (live cell imaging, modeling, simulation, super-resolution) and published in 2021 brought novel concepts to the basis of memory and memory architecture. During the year 2019–2022, the work on Electro-encephalogram (EEG) analysis led to several applications to better monitor and control anesthesia doses, popularized in "Pour La science". The work on computing the time for spermatozoa to reach an egg in the uterus received the Pineapple Science Award (Math Prize), the Chinese equivalent of the Ig Noble Prize in 2018. The notion of time for living organism can be defined as the first time the shortest telomere reaches a minimal threshold value: it is a random variable, controlled by the extreme statistics associated to telomere dynamics. "How cells are counting time?": this work was popularized in the viewpoint article in 2013: The Life and Death of Cells. The discovery reported in 2014, that astrocytes could invade the synaptic cleft under some specific conditions was recognized as a key result for controlling synaptic function. In 2011, mathematical modeling was at a turning point as it was becoming predictive for molecular and cellular: this moment was summarized in an interview with the CNRS journal. In 2022, the adaptative algorithm to predict the sensitivity to general anesthesesia developed by the Holcman's group gained interest from the national French newspaper Le Monde. == Awards == Holcman has received several awards, including a Sloan-Keck fellowship award (2002) a Marie-Curie Award (2013), and a Simons Fellowship. He is also recipient of 2 ERCs: an ERC Starting Grant in mathematics (2007) and an ERC-Advanced Grant in computational biology (2019) and a grant Proofs of Concept 2024 == References ==
Wikipedia:David J. Thomson#0	David J. Thomson is a professor in the Department of Mathematics and Statistics at Queen's University in Ontario and a Canada Research Chair in statistics and signal processing, formerly a member of the technical staff at Bell Labs. He is a professional engineer in the province of Ontario, a fellow of the IEEE and a chartered statistician. He holds memberships of the Royal Statistical Society, the American Statistical Association, the Statistical Society of Canada and the American Geophysical Union and, in 2009, received a Killam Research Fellowship (administered through the Canada Council for the Arts). In 2010, he was made a fellow of the Royal Society of Canada. In 2013, he was awarded the Statistical Society of Canada impact award. He is best known for creation of the multitaper method of spectral estimation, first published in complete form in 1982 in a special issue of Proceedings of the IEEE. Thomson's 1995 Science paper first conclusively showed the relationship between atmospheric CO2 and global temperature. Thomson and Bell Labs colleagues Carol G. Maclennan and Louis J. Lanzerotti authored a 1995 Nature paper in which they showed evidence that the magnetic signatures of the Sun's normal modes permeate the interplanetary magnetic field as far as Jupiter. He has written over 100 other peer-reviewed journal articles in the fields of statistics, space physics, climatology and paleoclimatology, and seismology. == Career == Thomson joined the Technical Staff at Bell Labs in 1965, where he was assigned to work on the WT4 Millimeter Waveguide System and the Advanced Mobile Phone Service project. In 1983, he was reassigned to the Communications Analysis Research Department where he remained as a Distinguished Member until his retirement in 2001. During this time, he was a Member of the Panel on Sensors and Electron Devices of the Army Research Laboratory Technical Assessment Board chairman of Commission C of USNC-URSI associate editor for Radio Science associate editor for Communications Theory and for Detection and Estimation of the IEEE Transactions on Information Theory adjunct professor in the Graduate Department of Scripps Institution of Oceanography consulted at the Neurological Institute of Columbia University visiting professor at Princeton University (statistical inference) visiting professor at Stanford University (time series) guest lecturer at Massachusetts Institute of Technology (the Houghton lectures) participant at the Isaac Newton Institute at the University of Cambridge On retirement from Bell Labs, Thomson took a Canada Research Chair at Queen's University at Kingston, where he has remained to this date. == References == == External links == Personal academic web page Spotlight David J. Thomson at the Mathematics Genealogy Project
Wikipedia:David Kazhdan#0	David Kazhdan (Hebrew: דוד קשדן), born Dmitry Aleksandrovich Kazhdan (Russian: Дми́трий Александро́вич Кажда́н), is a Soviet and Israeli mathematician known for work in representation theory. Kazhdan is a 1990 MacArthur Fellow. == Biography == Kazhdan was born on 20 June 1946 in Moscow, USSR. His father is Alexander Kazhdan. He earned a doctorate under Alexandre Kirillov in 1969 and was a member of Israel Gelfand's school of mathematics. He is Jewish, and emigrated from the Soviet Union to take a position at Harvard University in 1975. He changed his name from Dmitri Aleksandrovich to David and became an Orthodox Jew around that time. In 2002, he immigrated to Israel and is now a professor at the Hebrew University of Jerusalem as well as a professor emeritus at Harvard. On October 6, 2013, Kazhdan was critically injured in a car accident while riding a bicycle in Jerusalem. Kazhdan has four children. His son, Eli Kazhdan, was general director of Natan Sharansky's Yisrael BaAliyah political party (now merged with Likud). == Research == He is known for collaboration with Israel Gelfand, Victor Kac, George Lusztig (on the Kazhdan–Lusztig conjecture on Verma modules), with Grigory Margulis (Kazhdan–Margulis theorem), with Yuval Flicker and S. J. Patterson on the representations of metaplectic groups. Kazhdan's property (T) is widely used in representation theory. Kazhdan held a MacArthur Fellowship from 1990 to 1995. He was the doctoral advisor of Vladimir Voevodsky, a recipient of the Fields Medal, one of the highest awards in mathematics. Kazhdan has been a member of United States National Academy of Sciences since 1990, of the Israel Academy of Sciences since 2006, and of the American Academy of Arts and Sciences since 2008. In 2012, he was awarded the Israel Prize, the country's highest academic honor, for mathematics and computer science. In 2020 he received the Shaw Prize in Mathematics. == Selected publications == Quantum fields and strings: a course for mathematicians. Vol. 1, 2. Material from the Special Year on Quantum Field Theory held at the Institute for Advanced Study, Princeton, NJ, 1996–1997. Edited by Pierre Deligne, Pavel Etingof, Daniel S. Freed, Lisa C. Jeffrey, David Kazhdan, John W. Morgan, David R. Morrison and Edward Witten. American Mathematical Society, Providence, RI; Institute for Advanced Study (IAS), Princeton, NJ, 1999. Vol. 1: xxii+723 pp.; Vol. 2: pp. i--xxiv and 727–1501. ISBN 0-8218-1198-3, 81-06 (81T30 81Txx) == References == American Academy of Arts and Sciences, Class of 2008 == External links == Official Harvard home page Archived 2015-07-04 at the Wayback Machine Official Hebrew University home page David Kazhdan at the Mathematics Genealogy Project
Wikipedia:David Lovelock#0	David Lovelock (born 1938) is a British theoretical physicist and mathematician. He is known for the Lovelock theory of gravity and Lovelock's theorem. == Notes == == Books == Lovelock, David; Rund, Hanno (1989), Tensors, Differential Forms, and Variational Principles, Dover, ISBN 978-0-486-65840-7 == External links == David Lovelock at the Mathematics Genealogy Project David Lovelock Personal Home Page
Wikipedia:David M. Jackson#0	David M.R. Jackson is a professor at the University of Waterloo in the department of combinatorics and optimization. He graduated from Cambridge University in 1969. Jackson has been responsible for many developments in enumerative combinatorics in his career, as well as being a mathematical consultant to the Oxford English Dictionary Project. He is a Fellow of the Royal Society of Canada and a Member of the Academy of Mathematical and Physical Sciences. With Ian Goulden, Jackson published the book Combinatorial Enumeration. == Selected publications == Goulden, I. P.; Jackson, D. M. (2004). Combinatorial Enumeration. ISBN 0486435970. Jackson, David M.; Goulden, I. P.; Vakil, R. (2005). "Towards the Geometry of Double Hurwitz Numbers". Advances in Mathematics. 198: 43–92. arXiv:math/0309440. doi:10.1016/j.aim.2005.01.008. S2CID 8872816. Jackson, D. M.; Goulden, I. P.; Vainshtein, A (2000). "The number of ramified coverings of the sphere by the torus and surfaces of higher genera". Annals of Combinatorics. 4: 27–46. arXiv:math/9902125. doi:10.1007/PL00001274. S2CID 16725623. Jackson, D. M.; Goulden, I. P. (1997). "Transitive factorisations into transpositions and holomorphic mappings on the sphere" (PDF). Proceedings of the American Mathematical Society. 125 (1): 381–436. JSTOR 2161793. == See also == List of University of Waterloo people == References == == External links == David M. Jackson at the Mathematics Genealogy Project
Wikipedia:David Mayne#0	David Quinn Mayne (23 April 1930 – 27 May 2024) was a South African-born British academic, engineer, teacher and author. His pioneering and lasting contribution is in the field of control systems engineering. His research interests centred on optimization and optimization-based design, nonlinear control, control of constrained systems, model predictive control and adaptive control. == Career == Having obtained his BSc.(Eng) at the University of the Witwatersrand David Mayne began his career in 1950 as a lecturer there (1950–54; 1957–59). In 1954 he took up a two year post working as an electrical engineer at the British Thomson-Houston Company, Rugby, England. At the end of 1956 he returned to his academic post at the University of Witwatersrand to develop a new course in automatic control and gaining a MSc.(Eng). He next applied for a research position at Imperial College London. Impressed by his MSc thesis, Arnold Tustin and John Westcott, appointed him as lecturer. He lectured at Imperial College London from 1959-67 and in 1967 obtained his DSc (Eng) and PhD at the University of London under John Westcott. He was a Research Fellow at Harvard (1971). At Imperial College he was Professor of Control theory (1971–91) as well as concurrently heading the Department of Electrical Engineering (1984–88). He was subsequently a professor in the Dept. of Electrical and Computer Engineering at University of California, Davis from 1989-96. In 1996 he became Professor Emeritus and Senior Research Investigator in the Control and Power Research Group of the Department of Electrical and Electronic Engineering at Imperial College London. He was named honorary professor at Beihang University in Beijing in 2006. His students included Peter Caines. == Contribution to science == Mayne's research work is regarded as not only having had a lasting impact on the development of control theory, but his leadership style has inspired generations of new researchers. Among his many breakthroughs, arguably his most important contribution was his development of a rigorous mathematical method for analysing Model predictive control algorithms (MPC). It is currently used in tens of thousands of applications and is a core part of the advanced control technology by hundreds of process control producers. MPC's major strength is its capacity to deal with nonlinearities and hard constraints in a simple and intuitive fashion. His work underpins a class of algorithms that are provably correct, heuristically explainable, and yield control system designs which meet practically important objectives. Parisini and Astolfi consider that, "Mayne is also responsible for developing the first two-filter solution to the smoothing problem. This opened the door to substantial developments and is recognised as a pivotal contribution and precursor of the so-called particle filtering. Another cutting-edge contribution was his work on optimization-based design. He was an early user of exact penalty functions for optimization using sequential quadratic programming. The exact penalty method overcomes the widely referenced Maratos effect, identified by one of Mayne’s Ph.D. students. He also contributed to the early development of algorithms for non-differentiable and semi-infinite optimization problems". == Personal life and death == David Quinn Mayne was born in Germiston, South Africa. He completed his education up to Master's level at the University of the Witwatersrand. Early in his career he married fellow South African, Josephine. They had three daughters. The family moved to the UK in the 1950s where Mayne continued his research. Mayne died in Oxford on 27 May 2024, at the age of 94. == Awards and affiliations == Giorgio Quazza Medal, 2014 IEEE Control Systems Award, 2009 Fellow, International Federation of Automatic Control, 2006 Honorary Fellow Imperial College London, 2000 Hon. DTech Lund University Sweden, 1995 Fellow of the Royal Academy of Engineering 1987 Sir Harold Hartley Medal 1986 Royal Society Fellow 1985 Institute of Electrical and Electronics Engineers FIEEE, Fellow 1981 Foreign Member of Academia Nacional de Ingeniera, Mexico 1981 Institution of Engineering and Technology, Fellow 1980 Institution of Engineering and Technology IET Heaviside Premium (1979 & 1984) Engineering and Physical Sciences Research Council EPSRC Senior Research Fellow (1979-1980) == Selected publications == Differential Dynamic Programming ISBN 9780444000705 (1970) D. Q. Mayne and R. W. Brockett (editors), Geometric Methods in System Theory, D. Reidel Publishing Co., (1973). Mayne, David Quinn (2015). "John Hugh Westcott 3 November 1920 — 10 October 2014". Biographical Memoirs of Fellows of the Royal Society. 61. Royal Society publishing: 541–554. doi:10.1098/rsbm.2015.0017. ISSN 0080-4606. Rawlings, James B.; Mayne, David Q.; and Diehl, Moritz M.; Model Predictive Control: Theory, Computation, and Design (2nd Ed.), Nob Hill Publishing, LLC, ISBN 978-0975937730 (Oct. 2017) === Papers === D. Q. Mayne, Optimal Non-Stationary Filters, Chapter 7 in An Exposition of Adaptive Control, Pergamon Press, 1962. D. Q. Mayne, Optimal Non-Stationary Estimation of the Parameters of a Linear System with Gaussian Inputs, Journal of Electronics and Control, 14(1): 101--112, 1963. D. Q. Mayne, Parameter Estimation, Automatica, 3(3/4):245--256, 1966. D. Q. Mayne, A Gradient Method for Determining Optimal Control of Nonlinear Stochastic Systems, Proceedings of IFAC Symposium, Theory of Self-Adaptive Control Systems, editor P. H. Hammond, Plenum Press, 19--27, 1965. D. Q. Mayne, A Solution of the Smoothing Problem for Linear Dynamic Systems", Automatica, 4:73--92, 1966. D. Q. Mayne, A Second-Order Gradient Method for Determining Optimal Trajectories of Nonlinear Discrete-Time Systems, International Journal of Control, 3:85--95, 1966. G. F. Bryant and D. Q. Mayne, A Minimum Principle for a Class of Discrete-Time Stochastic Systems, IEEE Transactions Automatic Control, 14(4):401--403, 1969. J. E. Handschin and D. Q. Mayne, Monte Carlo Techniques to Estimate the Conditional Expectation in Multistage Nonlinear Filtering, International Journal of Control, 9(5):547--559, 1966. D. Q. Mayne, Differential Dynamic Programming---a Unified Approach to Optimal Control, in Advances in Control Systems, editor C. T. Leondes, Academic Press, 10: 179--254, 1973. G. F. Bryant and D. Q. Mayne, The Maximum Principle, International Journal of Control, 20(6):1021--1054, 1974. Mayne, David Q.; Michalska, Hannah (1990). "Receding horizon control of nonlinear systems". IEEE Transactions on Automatic Control. 35 (7): 814–824. doi:10.1109/9.57020. Mayne, David Q.; Rawlings, James B.; Rao, Christopher V.; Scokaert, Pierre O. M. (2000). "Constrained model predictive control: stability and optimality". Automatica. 36 (6): 789–814. doi:10.1016/S0005-1098(99)00214-9. === Papers on optimization and optimal control === Mayne, D. Q. and Polak, E., First Order, Strong Variations Algorithms for Optimal Control, Journal of Optimization Theory and Applications, 16(3/4):277--301, 1975. D. Q. Mayne and E. Polak, Feasible Directions Algorithms for Optimization Problems with Equality and Inequality Constraints, Mathematical Programming, 11(1):67--80, 1976. Polak, E. and Mayne, D. Q., An Algorithm for Optimization Problems with Functional Inequality Constraints, IEEE Transactions on Automatic Control, 21(2):184--193, 1976. D. Q. Mayne, Sufficient Conditions for a Control to be a Strong Minimum, Journal of Optimization and Applications, 21(3):339--352, 1977. D. Q. Mayne, E. Polak and R. Trahan, An Outer Approximations Algorithm for Computer Aided Design Problems, Journal of Optimization and Applications, 28(3):231--352, 1979. Mayne, D. Q. and Polak, E., An Exact Penalty Function Algorithm for Control Problems with Control and Terminal Equality Constraints---Part 1, Journal of Optimization and Applications, 32(2):211--246, 1980. Mayne, D. Q. and Polak, E., An Exact Penalty Function Algorithm for Control Problems with Control and Terminal Equality Constraints---Part 2, Journal of Optimization and Applications, 32(3):345--363, 1980. Polak, E. and Mayne, D. Q., On the Solution of Singular Value Inequalities over a Continuum of Frequencies, IEEE Transactions on Automatic Control, 26(3):690--695, 1981. Polak, E. and Mayne, D. Q., Design of Nonlinear Feedback Controllers, IEEE Transactions on Automatic Control, 26(3):730--733, 1981. Mayne, D. Q., Polak, E. and Voreadis, A., A Cut Map Algorithm for Design Problems with Tolerances, IEEE Transactions on Circuits and Systems, 29(1):35--46, 1982. Mayne, D. Q. and Polak, E., Nondifferentiable Optimization via Adaptive Smoothing, Journal of Optimization and Applications, 43(4):601--613, 1984. D. Q. Mayne and E. Polak, A Superlinearly Convergent Algorithm for Constrained Optimization Problems, Mathematical Programming Studies, 16:45--61, 1982. Polak, E., Mayne, D. Q. and Stimler, D. M., Control System Design via Semi-Infinite Optimization, Proceedings of the IEEE, 72(12):1777--1795, 1984. E. Polak and D. Q. Mayne, Algorithm Models for Non-Differentiable Optimization, SIAM Journal of Control and Optimization, 23:477--491, 1985. Pantoja, J. F. A. de O. and D. Q. Mayne, A Sequential Quadratic Programming Algorithm for Discrete Optimal Control Problems with Control Inequality Constraints, International Journal of Control, 53(4):823--836, 1991. E. Polak, T. H. Yang and D. Q. Mayne, A Method of Centers Based on Barrier Functions for Solving Optimal Control Problems with Continuum State and Control Constraints, in New Trends in System Theory, editors G. Conte, A. M. Perdon and B. Wyman, Birkhauser, 591--598, 1991. == References == == External links == Biodata at Debrett's Control Global website David Q. Mayne publications (with Saša V. Raković, and others) David Q. Mayne aided the creation of software that he contributed to algorithmically David Mayne at the Mathematics Genealogy Project
Wikipedia:David Nualart#0	David Nualart (born 21 March 1951) is a Spanish mathematician working in the field of probability theory, in particular on aspects of stochastic processes and stochastic analysis. He is retired as Black-Babcock Distinguished Professor of Mathematics at the University of Kansas. == Education and career == Nualart obtained his PhD titled "Contribución al estudio de la integral estocástica" in 1975 at the University of Barcelona under the supervision of Francesc d'Assís Sales Vallès. After positions at the University of Barcelona and the Polytechnique University of Barcelona he took up a professorship at the University of Kansas and was the Black-Babcock Distinguished Professor in its Mathematics Department from 2012 to 2022. He retired in 2022. He was the Chief Editor of Electronic Communications in Probability from 2006 to 2008. == Recognition == He has been elected a Fellow of the Institute of Mathematical Statistics in 1997. He received a Doctor Honoris Causa by the Université Blaise Pascal of Clermond-Ferrand in 1998. He received the Prize IBERDROLA de Ciencia y Tecnologia in 1999. He has been a Corresponding Member of the Real Academia de Ciencias Exactas Fisicas y Naturales of Madrid since 2003. He has been a member of the Reial Academia de Ciencies i Arts of Barcelona since 2003. He received the Research Prize of the Real Academia de Ciencias de Madrid in 1991. In March 2011 the International Conference on Malliavin Calculus and Stochastic Analysis in honor of David Nualart took place at University of Kansas. The results were published in 2013 as a festschrift, Malliavin Calculus and Stochastic Analysis: A Festschrift in Honor of David Nualart. He was named to the 2023 class of Fellows of the American Mathematical Society, "for contributions to Malliavin calculus, stochastic PDE's, and fractional Brownian motion". == References == == External links == Home page
Wikipedia:David Orrell#0	David John Orrell is a Canadian writer and mathematician. He received his doctorate in mathematics from the University of Oxford. His work in the prediction of complex systems such as the weather, genetics and the economy has been featured in New Scientist, the Financial Times, The Economist, Adbusters, BBC Radio, Russia-1, and CBC TV. He now conducts research and writes in the areas of systems biology and economics, and runs a mathematical consultancy Systems Forecasting. He is the son of theatre historian and English professor John Orrell. His books have been translated into over ten languages. Apollo's Arrow: The Science of Prediction and the Future of Everything was a national bestseller and finalist for the 2007 Canadian Science Writers' Award. Economyths: Ten Ways Economics Gets It Wrong was a finalist for the 2011 National Business Book Award. == Criticism of use of mathematical models == A consistent topic in Orrell’s work is the limitations of mathematical models, and the need to acknowledge these limitations if we are to understand the causes of forecast error. He argues for example that errors in weather prediction are caused primarily by model error, rather than the butterfly effect. Economic models are seen as particularly unrealistic. In Truth or Beauty: Science and the Quest for Order, he suggests that many such theories, along with areas of physics such as string theory, are motivated largely by the desire to conform with a traditional scientific aesthetic, that is currently being subverted by developments in complexity science. == Quantum theory of money and value == Orrell is considered a leading proponent of quantum finance and quantum economics. In The Evolution of Money (coauthored with journalist Roman Chlupatý) and a series of articles he proposed a quantum theory of money and value, which states that money has dualistic properties because it combines the properties of an owned and valued thing, with those of abstract number. The fact that these two sides of money are incompatible leads to its complex and often unpredictable behavior. In Quantum Economics: The New Science of Money he argued that these dualistic properties feed up to affect the economy as a whole. == Books == Orrell, David (2023). Quantum Economics and Finance: An Applied Mathematics Introduction (3rd ed.). Panda Ohana. ISBN 978-1916081611. Orrell, David (2022). Money, Magic, and How to Dismantle a Financial Bomb: Quantum Economics for the Real World. Icon. ISBN 978-1785788284. Orrell, David (2021). Instant Economics: Key Thinkers, Theories, Discoveries and Concepts. Welbeck. ISBN 978-1787394193. Orrell, David (2021). Behavioural Economics: Psychology, Neuroscience, and the Human Side of Economics. Icon. ISBN 978-1785786440. Orrell, David (2020). A Brief History of Money: 4,000 Years of Markets, Currencies, Debt and Crisis. Welbeck. ISBN 978-1785788284. Orrell, David (2018). Quantum Economics: The New Science of Money. Icon. ISBN 978-1785782299. Orrell, David (2017). Economyths: 11 Ways Economics Gets It Wrong. Icon. ISBN 978-1848311480. Revised and extended edition of 2010 book. Wilmott, Paul; Orrell, David (2017). The Money Formula: Dodgy Finance, Pseudo Science, and How Mathematicians Took Over the Markets. Wiley. ISBN 978-1119358619. Orrell, David; Chlupatý, Roman (2016). The Evolution of Money. Columbia University Press. ISBN 978-0231173728. Orrell, David (2012). Truth or Beauty: Science and the Quest for Order. Yale University Press. ISBN 978-0300186611. Sedlacek, Tomas; Orrell, David; Chlupatý, Roman (2012). Soumrak Homo Economicus. 65th Square. ISBN 978-8087506073. Orrell, David; Van Loon, Borin (2011). Introducing Economics: A Graphic Guide. Icon. ISBN 978-1848312159. Orrell, David (2010). Economyths: Ten Ways Economics Gets It Wrong. Icon. ISBN 978-1848311480. Orrell, David (2008). The Other Side of the Coin: The Emerging Vision of Economics and Our Place in The World. Key Porter. ISBN 978-1552639818. Orrell, David (2008). Gaia. Lulu.com. ISBN 978-1409255178. Orrell, David (2007). Apollo's Arrow: The Science of Prediction and the Future of Everything. HarperCollins. ISBN 978-0002007405. Published in the U.S. as The Future of Everything: The Science of Prediction. == See also == Anticipatory science forecasts Complex systems Mathematical model Computer model Chaos theory Systems biology Quantum economics == References == == External links == David Orrell's homepage Systems Forecasting Video of talk on money given for Marshall McLuhan lecture, Berlin 2015 Video of talk on prediction given at TEDx Park Kultury, Moscow in 2012 ABC News - Good Morning America (excerpts from Apollo's Arrow) National Post's review of Apollo's Arrow Sunday Times review of Truth or Beauty David Orrell interview with E-International Relations
Wikipedia:David Ruelle#0	David Pierre Ruelle (French: [david pjɛʁ ʁɥɛl]; born 20 August 1935) is a Belgian and naturalized French mathematical physicist. He has worked on statistical physics and dynamical systems. With Floris Takens, Ruelle coined the term strange attractor, and developed a new theory of turbulence. == Biography == Ruelle studied physics at the Free University of Brussels, obtaining a PhD degree in 1959 under the supervision of Res Jost. He spent two years (1960–1962) at the ETH Zurich, and another two years (1962–1964) at the Institute for Advanced Study in Princeton, New Jersey. In 1964, he became professor at the Institut des Hautes Études Scientifiques in Bures-sur-Yvette, France. Since 2000, he has been an emeritus professor at IHES and distinguished visiting professor at Rutgers University. David Ruelle made fundamental contributions in various aspects of mathematical physics. In quantum field theory, the most important contribution is the rigorous formulation of scattering processes based on Wightman's axiomatic theory. This approach is known as the Haag–Ruelle scattering theory. Later Ruelle helped to create a rigorous theory of statistical mechanics of equilibrium, that includes the study of the thermodynamic limit, the equivalence of ensembles, and the convergence of Mayer's series. A further result is the Asano-Ruelle lemma, which allows the study of the zeros of certain polynomial functions that are recurrent in statistical mechanics. The study of infinite systems led to the local definition of Gibbs states or to the global definition of equilibrium states. Ruelle demonstrated with Roland L. Dobrushin and Oscar E. Lanford that translationally invariant Gibbs states are precisely the equilibrium states. Together with Floris Takens, he proposed the description of hydrodynamic turbulence based on strange attractors with chaotic properties of hyperbolic dynamics. == Honors and awards == Since 1985 David Ruelle has been a member of the French Academy of Sciences and in 1988 he was Josiah Willard Gibbs Lecturer in Atlanta, Georgia. Since 1992 he has been an international honorary member of the American Academy of Arts and Sciences and since 1993 ordinary member of the Academia Europaea. Since 2002 he has been an international member of the United States National Academy of Sciences and since 2003 a foreign member of the Accademia Nazionale dei Lincei. Since 2012 he has been a fellow of the American Mathematical Society. In 1985 David Ruelle was awarded the Dannie Heineman Prize for Mathematical Physics and in 1986 he received the Boltzmann Medal for his outstanding contributions to statistical mechanics. In 1993 he won the Holweck Prize and in 2004 he received the Matteucci Medal. In 2006 he was awarded the Henri Poincaré Prize and in 2014 he was honored with the prestigious Max Planck Medal for his achievements in theoretical physics. In 2022, Ruelle was awarded the ICTP's Dirac Medal for Mathematical Physics, along with Elliott H. Lieb and Joel Lebowitz, "for groundbreaking and mathematically rigorous contributions to the understanding of the statistical mechanics of classical and quantum physical systems". == Selected publications == == See also == == References == == External links == David Ruelle at the Mathematics Genealogy Project Literature by and about David Ruelle in the German National Library catalogue. David Ruelle at zbMATH. Takens, Floris. "Laudatio for David Ruelle" (PDF). math.leidenuniv.nl. Archived from the original (PDF) on 6 March 2009. Gallavotti, Giovanni. "Laudatio on the occasion of the Henri Poincaré Prize". International Association of Mathematical Physics. Retrieved 12 March 2021. Ruelle, David. "'What is a... Strange Attractor?" (PDF). American Mathematical Society. Retrieved 12 March 2021. "Biography from his website". ihes.fr. Retrieved 12 March 2021. "David Ruelle \| April 7, 2022 \| A Natural Limitation for Properly Human Scientific Progress". YouTube. Mathematical Picture Language. 7 April 2022.
Wikipedia:David Rytz#0	David Rytz von Brugg (1 April 1801, in Bucheggberg – 25 March 1868, in Aarau) was a Swiss mathematician and teacher. == Life == Rytz von Brugg was son of a priest and studied mathematics at Göttingen and Leipzig. He had teaching positions at various cities, one of them 1835 until 1862 at Aarau, where he was „Professor der Mathematik an der Gewerbeschule zu Aarau“. == Merits == Rytz von Brugg is famous for a geometrical method which is known as Rytz’s axis construction. This classical procedure retrieves the semi-axes of an Ellipse from any pair of conjugate diameters. This method is known since 1845, when it was published within a paper by Leopold Moosbrugger. == Sources == Siegfried Gottwald, Hans-Joachim Ilgauds und Karl-Heinz Schlote, ed. (1990), Lexikon bedeutender Mathematiker (in German), Thun: Verlag Harri Deutsch, p. 407, ISBN 3-8171-1164-9 MR1089881 Hans Honsberg (1971), Analytische Geometrie: Mit Anhang "Einführung in die Vektorrechnung", Mathematik für Gymnasien (in German) (3. ed.), München: Bayerischer Schulbuch-Verlag, p. 96, ISBN 3-7627-0677-8 Emil Müller, Erwin Kruppa (1961), Lehrbuch der darstellenden Geometrie: Unveränderter Neudruck der fünften Auflage (in German) (6. ed.), Wien: Springer Verlag, p. 98 Alexander Ostermann, Gerhard Wanner (2012), Geometry by Its History, Undergraduate Texts in Mathematics. Readings in Mathematics (in German), Heidelberg, New York, Dordrecht, London: Springer Verlag, p. 69, doi:10.1007/978-3-642-29163-0, ISBN 978-3-642-29162-3 MR2918594 Guido Walz (Red.) (2002), Lexikon der Mathematik in sechs Bänden: Vierter Band (in German), Heidelberg, Berlin: Spektrum Akademischer Verlag, p. 448, ISBN 3-8274-0436-3 == References ==
Wikipedia:David Schmeidler#0	David Schmeidler (Hebrew: דוד שמידלר; 1939 – 17 March 2022) was an Israeli mathematician and economic theorist. He was a Professor Emeritus at Tel Aviv University and the Ohio State University. == Biography == David Schmeidler was born in 1939 in Kraków, Poland. He spent the war years in Russia and moved back to Poland at the end of the war and to Israel in 1949. From 1960 to 1969 he studied mathematics at the Hebrew University of Jerusalem (BSc, MSc, and PhD), the advanced degrees under the supervision of Robert Aumann. He visited the Catholic University of Louvain and University of California at Berkeley before joining Tel-Aviv University in 1971, holding professorships in statistics, economics, and management. He held a part-time position as professor of economics at the Ohio State University since 1987. Schmeidler died on 17 March 2022. == Main contributions == Schmeidler's early contributions were in game theory and general equilibrium theory. He suggested a new approach to solving cooperative games – the nucleolus – based on equity as well as feasibility considerations. This concept, originating from Schmeidler's PhD dissertation, was used to resolve a 2000 years old problem. Robert Aumann and Michael Maschler, in a paper published in 1985, showed that a conundrum from the Babylonian Talmud, which defied scholars’ attempts at comprehension over two millennia, was naturally resolved when applying the concept of the nucleolus. Schmeidler also pioneered the study of non-atomic strategic games, in which each player has negligible impact on the play of the game, as well as the related concept of “congestion games”, where a player's payoff only depends on the distribution of the other players’ strategic choices (and not on individual choices). Schmeidler has made many other contributions, ranging from conceptual issues in implementation theory, to mathematical results in measure theory. But his most influential contribution is probably in decision theory. Schmeidler was the first to propose a general-purpose, axiomatically-based decision theoretic model that deviated from the Bayesian dictum, according to which any uncertainty can and should be quantified by probabilities. He suggested and axiomatized Choquet Expected Utility, according to which uncertainty is modeled by a capacity (not-necessarily-additive set function) and expectation is computed by the Choquet integral. While this approach can be used to explain commonly observed behavior in Ellsberg's experiments, Schmeidler's motivation was not to explain psychological findings. Rather, along the lines attributed to Frank Knight and John Maynard Keynes, the argument is normative, suggesting that it is not necessarily more rational to be Bayesian than not. While in the experiments, drawing balls from urns, one may adopt a probabilistic belief, in real life one often couldn't find a natural candidate for one's beliefs. With Elisha Pazner, he introduced the notion of egalitarian equivalence - a criterion for fair division of homogeneous resources, that has advantages over the previously studied criterion of envy-freeness. With his student, Itzhak Gilboa, David Schmeidler also developed the theory maxmin expected utility and case-based decision theory. He has also served as the advisor of Peter Wakker, Shiri Alon, and Xiangyu Qu. == Selected works == 1969: "The nucleolus of a characteristic function game", SIAM Journal on Applied Mathematics 17: 1163–1170. 1973: "Equilibrium points of non-atomic games", Journal of Statistical Physics 7: 295–301. 1986: "Integral representation without additivity", Proceedings of the American Mathematical Society 97: 255–261. 1989: "Subjective probability and expected utility without additivity", Econometrica 57: 571–587. 1989: (with Itzhak Gilboa) "Maximin expected utility with a non-unique prior", Journal of Mathematical Economics 18: 141–153. 1995: (with Itzhak Gilboa) "Case-based decision theory", Quarterly Journal of Economics 110: 605–639. 2001: (with Itzhak Gilboa) A Theory of Case-Based Decisions, Cambridge University Press 2015: (with Itzhak Gilboa & Larry Samuelson) Analogies and Theories: Formal Models of Reasoning, Oxford University Press ISBN 978-0-19-873802-2 MR3362708 == Honors == David Schmeidler was a Fellow of the Econometric Society, Honorary Foreign Member of the American Academy of Arts and Sciences, and a Member of the Israeli Academy of Sciences and Humanities. He served as the President of the Game Theory Society (2014–2016). == References == == External links == Personal website David Schmeidler at the Mathematics Genealogy Project

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