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Wikipedia:Cauchy–Rassias stability#0	A classical problem of Stanislaw Ulam in the theory of functional equations is the following: When is it true that a function which approximately satisfies a functional equation E must be close to an exact solution of E? In 1941, Donald H. Hyers gave a partial affirmative answer to this question in the context of Banach spaces. This was the first significant breakthrough and a step towards more studies in this domain of research. Since then, a large number of papers have been published in connection with various generalizations of Ulam's problem and Hyers' theorem. In 1978, Themistocles M. Rassias succeeded in extending the Hyers' theorem by considering an unbounded Cauchy difference. He was the first to prove the stability of the linear mapping in Banach spaces. In 1950, T. Aoki had provided a proof of a special case of the Rassias' result when the given function is additive. For an extensive presentation of the stability of functional equations in the context of Ulam's problem, the interested reader is referred to the recent book of S.-M. Jung, published by Springer, New York, 2011 (see references below). Th. M. Rassias' theorem attracted a number of mathematicians who began to be stimulated to do research in stability theory of functional equations. By regarding the large influence of S. M. Ulam, D. H. Hyers and Th. M. Rassias on the study of stability problems of functional equations, this concept is called the Hyers–Ulam–Rassias stability. In the special case when Ulam's problem accepts a solution for Cauchy's functional equation f(x + y) = f(x) + f(y), the equation E is said to satisfy the Cauchy–Rassias stability. The name is referred to Augustin-Louis Cauchy and Themistocles M. Rassias. == References == P. M. Pardalos, P. G. Georgiev and H. M. Srivastava (eds.), Nonlinear Analysis. Stability, Approximation, and Inequalities. In honor of Themistocles M. Rassias on the occasion of his 60th birthday, Springer, New York, 2012. D. H. Hyers, On the stability of the linear functional Equation, Proc. Natl. Acad. Sci. USA, 27(1941), 222-224. Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proceedings of the American Mathematical Society 72(1978), 297-300. [Translated in Chinese and published in: Mathematical Advance in Translation, Chinese Academy of Sciences 4 (2009), 382-384.] Th. M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Applicandae Mathematicae, 62(1)(2000), 23-130. S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis, Springer, New York, 2011, ISBN 978-1-4419-9636-7. T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Jpn., 2(1950), 64-66. C.-G. Park, Generalized quadratic mappings in several variables, Nonlinear Anal., 57(2004), 713–722. J.-R. Lee and D.-Y. Shin, On the Cauchy-Rassias stability of a generalized additive functional equation, J. Math. Anal. Appl. 339(1)(2008), 372–383. C. Baak, Cauchy – Rassias stability of Cauchy-Jensen additive mappings in Banach spaces, Acta Math. Sinica (English Series), 15(1)(1999), 1-11. C.-G. Park, Homomorphisms between Lie JC- algebras and Cauchy – Rassias stability of Lie JC-algebra derivations, J. Lie Theory, 15(2005), 393–414. J.-R. Lee, D.-Y. Shin, On the Cauchy-Rassias stability of the Trif functional equation in C*-algebras. J. Math. Anal. Appl. 296(1)(2004), 351–363. C. Baak, H.- Y. Chu and M. S. Moslehian, On the Cauchy-Rassias inequality and linear n–inner product preserving mappings, Math. Inequal. Appl. 9(3)(2006), 453–464. C.-G. Park, M. Eshaghi Gordji and H. Khodaei, A fixed point approach to the Cauchy-Rassias stability of general Jensen type quadratic-quadratic mappings, Bull. Korean Math. Soc. 47(2010), no. 5, 987–996 A. Najati, Cauchy-Rassias stability of homomorphisms associated to a Pexiderized Cauchy-Jensen type functional equation, J. Math. Inequal. 3(2)(2009), 257-265. C.-G. Park and S. Y. Jang, Cauchy-Rassias stability of sesquilinear n-quadratic mappings in Banach modules, Rocky Mountain J. Math. 39(6)(2009), 2015–2027. Pl. Kannappan, Functional Equations and Inequalities with Applications, Springer, New York, 2009, ISBN 978-0-387-89491-1. P. K. Sahoo and Pl. Kannappan, Introduction to Functional Equations, CRC Press, Chapman & Hall Book, Florida, 2011, ISBN 978-1-4398-4111-2. Th. M. Rassias and J. Brzdek (eds.), Functional Equations in Mathematical Analysis, Springer, New York, 2012, ISBN 978-1-4614-0054-7.
Wikipedia:Cauchy–Schwarz inequality#0	The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) is an upper bound on the inner product between two vectors in an inner product space in terms of the product of the vector norms. It is considered one of the most important and widely used inequalities in mathematics. Inner products of vectors can describe finite sums (via finite-dimensional vector spaces), infinite series (via vectors in sequence spaces), and integrals (via vectors in Hilbert spaces). The inequality for sums was published by Augustin-Louis Cauchy (1821). The corresponding inequality for integrals was published by Viktor Bunyakovsky (1859) and Hermann Schwarz (1888). Schwarz gave the modern proof of the integral version. == Statement of the inequality == The Cauchy–Schwarz inequality states that for all vectors u {\displaystyle \mathbf {u} } and v {\displaystyle \mathbf {v} } of an inner product space where ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } is the inner product. Examples of inner products include the real and complex dot product; see the examples in inner product. Every inner product gives rise to a Euclidean ℓ 2 {\displaystyle \ell _{2}} norm, called the canonical or induced norm, where the norm of a vector u {\displaystyle \mathbf {u} } is denoted and defined by ‖ u ‖ := ⟨ u , u ⟩ , {\displaystyle \\|\mathbf {u} \\|:={\sqrt {\langle \mathbf {u} ,\mathbf {u} \rangle }},} where ⟨ u , u ⟩ {\displaystyle \langle \mathbf {u} ,\mathbf {u} \rangle } is always a non-negative real number (even if the inner product is complex-valued). By taking the square root of both sides of the above inequality, the Cauchy–Schwarz inequality can be written in its more familiar form in terms of the norm: Moreover, the two sides are equal if and only if u {\displaystyle \mathbf {u} } and v {\displaystyle \mathbf {v} } are linearly dependent. == Special cases == === Sedrakyan's lemma – positive real numbers === Sedrakyan's inequality, also known as Bergström's inequality, Engel's form, Titu's lemma (or the T2 lemma), states that for real numbers u 1 , u 2 , … , u n {\displaystyle u_{1},u_{2},\dots ,u_{n}} and positive real numbers v 1 , v 2 , … , v n {\displaystyle v_{1},v_{2},\dots ,v_{n}} : ( u 1 + u 2 + ⋯ + u n ) 2 v 1 + v 2 + ⋯ + v n ≤ u 1 2 v 1 + u 2 2 v 2 + ⋯ + u n 2 v n , {\displaystyle {\frac {\left(u_{1}+u_{2}+\cdots +u_{n}\right)^{2}}{v_{1}+v_{2}+\cdots +v_{n}}}\leq {\frac {u_{1}^{2}}{v_{1}}}+{\frac {u_{2}^{2}}{v_{2}}}+\cdots +{\frac {u_{n}^{2}}{v_{n}}},} or, using summation notation, ( ∑ i = 1 n u i ) 2 / ∑ i = 1 n v i ≤ ∑ i = 1 n u i 2 v i . {\displaystyle {\biggl (}\sum _{i=1}^{n}u_{i}{\biggr )}^{2}{\bigg /}\sum _{i=1}^{n}v_{i}\,\leq \,\sum _{i=1}^{n}{\frac {u_{i}^{2}}{v_{i}}}.} It is a direct consequence of the Cauchy–Schwarz inequality, obtained by using the dot product on R n {\displaystyle \mathbb {R} ^{n}} upon substituting u i ′ = u i v i t {\displaystyle u_{i}'={\frac {u_{i}}{\sqrt {v_{i}{\vphantom {t}}}}}} and v i ′ = v i t {\displaystyle v_{i}'={\textstyle {\sqrt {v_{i}{\vphantom {t}}}}}} . This form is especially helpful when the inequality involves fractions where the numerator is a perfect square. === R2 - The plane === The real vector space R 2 {\displaystyle \mathbb {R} ^{2}} denotes the 2-dimensional plane. It is also the 2-dimensional Euclidean space where the inner product is the dot product. If u = ( u 1 , u 2 ) {\displaystyle \mathbf {u} =(u_{1},u_{2})} and v = ( v 1 , v 2 ) {\displaystyle \mathbf {v} =(v_{1},v_{2})} then the Cauchy–Schwarz inequality becomes: ⟨ u , v ⟩ 2 = ( ‖ u ‖ ‖ v ‖ cos ⁡ θ ) 2 ≤ ‖ u ‖ 2 ‖ v ‖ 2 , {\displaystyle \langle \mathbf {u} ,\mathbf {v} \rangle ^{2}={\bigl (}\\|\mathbf {u} \\|\\|\mathbf {v} \\|\cos \theta {\bigr )}^{2}\leq \\|\mathbf {u} \\|^{2}\\|\mathbf {v} \\|^{2},} where θ {\displaystyle \theta } is the angle between u {\displaystyle \mathbf {u} } and v {\displaystyle \mathbf {v} } . The form above is perhaps the easiest in which to understand the inequality, since the square of the cosine can be at most 1, which occurs when the vectors are in the same or opposite directions. It can also be restated in terms of the vector coordinates u 1 {\displaystyle u_{1}} , u 2 {\displaystyle u_{2}} , v 1 {\displaystyle v_{1}} , and v 2 {\displaystyle v_{2}} as ( u 1 v 1 + u 2 v 2 ) 2 ≤ ( u 1 2 + u 2 2 ) ( v 1 2 + v 2 2 ) , {\displaystyle \left(u_{1}v_{1}+u_{2}v_{2}\right)^{2}\leq \left(u_{1}^{2}+u_{2}^{2}\right)\left(v_{1}^{2}+v_{2}^{2}\right),} where equality holds if and only if the vector ( u 1 , u 2 ) {\displaystyle \left(u_{1},u_{2}\right)} is in the same or opposite direction as the vector ( v 1 , v 2 ) {\displaystyle \left(v_{1},v_{2}\right)} , or if one of them is the zero vector. === Rn: n-dimensional Euclidean space === In Euclidean space R n {\displaystyle \mathbb {R} ^{n}} with the standard inner product, which is the dot product, the Cauchy–Schwarz inequality becomes: ( ∑ i = 1 n u i v i ) 2 ≤ ( ∑ i = 1 n u i 2 ) ( ∑ i = 1 n v i 2 ) . {\displaystyle {\biggl (}\sum _{i=1}^{n}u_{i}v_{i}{\biggr )}^{2}\leq {\biggl (}\sum _{i=1}^{n}u_{i}^{2}{\biggr )}{\biggl (}\sum _{i=1}^{n}v_{i}^{2}{\biggr )}.} The Cauchy–Schwarz inequality can be proved using only elementary algebra in this case by observing that the difference of the right and the left hand side is 1 2 ∑ i = 1 n ∑ j = 1 n ( u i v j − u j v i ) 2 ≥ 0 {\displaystyle {\tfrac {1}{2}}\sum _{i=1}^{n}\sum _{j=1}^{n}(u_{i}v_{j}-u_{j}v_{i})^{2}\geq 0} or by considering the following quadratic polynomial in x {\displaystyle x} ( u 1 x + v 1 ) 2 + ⋯ + ( u n x + v n ) 2 = ( ∑ i u i 2 ) x 2 + 2 ( ∑ i u i v i ) x + ∑ i v i 2 . {\displaystyle (u_{1}x+v_{1})^{2}+\cdots +(u_{n}x+v_{n})^{2}={\biggl (}\sum _{i}u_{i}^{2}{\biggr )}x^{2}+2{\biggl (}\sum _{i}u_{i}v_{i}{\biggr )}x+\sum _{i}v_{i}^{2}.} Since the latter polynomial is nonnegative, it has at most one real root, hence its discriminant is less than or equal to zero. That is, ( ∑ i u i v i ) 2 − ( ∑ i u i 2 ) ( ∑ i v i 2 ) ≤ 0. {\displaystyle {\biggl (}\sum _{i}u_{i}v_{i}{\biggr )}^{2}-{\biggl (}\sum _{i}{u_{i}^{2}}{\biggr )}{\biggl (}\sum _{i}{v_{i}^{2}}{\biggr )}\leq 0.} === Cn: n-dimensional complex space === If u , v ∈ C n {\displaystyle \mathbf {u} ,\mathbf {v} \in \mathbb {C} ^{n}} with u = ( u 1 , … , u n ) {\displaystyle \mathbf {u} =(u_{1},\ldots ,u_{n})} and v = ( v 1 , … , v n ) {\displaystyle \mathbf {v} =(v_{1},\ldots ,v_{n})} (where u 1 , … , u n ∈ C {\displaystyle u_{1},\ldots ,u_{n}\in \mathbb {C} } and v 1 , … , v n ∈ C {\displaystyle v_{1},\ldots ,v_{n}\in \mathbb {C} } ) and if the inner product on the vector space C n {\displaystyle \mathbb {C} ^{n}} is the canonical complex inner product (defined by ⟨ u , v ⟩ := u 1 v 1 ¯ + ⋯ + u n v n ¯ , {\displaystyle \langle \mathbf {u} ,\mathbf {v} \rangle :=u_{1}{\overline {v_{1}}}+\cdots +u_{n}{\overline {v_{n}}},} where the bar notation is used for complex conjugation), then the inequality may be restated more explicitly as follows: \| ⟨ u , v ⟩ \| 2 = \| ∑ k = 1 n u k v ¯ k \| 2 ≤ ⟨ u , u ⟩ ⟨ v , v ⟩ = ( ∑ k = 1 n u k u ¯ k ) ( ∑ k = 1 n v k v ¯ k ) = ∑ j = 1 n \| u j \| 2 ∑ k = 1 n \| v k \| 2 . {\displaystyle {\bigl \|}\langle \mathbf {u} ,\mathbf {v} \rangle {\bigr \|}^{2}={\Biggl \|}\sum _{k=1}^{n}u_{k}{\bar {v}}_{k}{\Biggr \|}^{2}\leq \langle \mathbf {u} ,\mathbf {u} \rangle \langle \mathbf {v} ,\mathbf {v} \rangle ={\biggl (}\sum _{k=1}^{n}u_{k}{\bar {u}}_{k}{\biggr )}{\biggl (}\sum _{k=1}^{n}v_{k}{\bar {v}}_{k}{\biggr )}=\sum _{j=1}^{n}\|u_{j}\|^{2}\sum _{k=1}^{n}\|v_{k}\|^{2}.} That is, \| u 1 v ¯ 1 + ⋯ + u n v ¯ n \| 2 ≤ ( \| u 1 \| 2 + ⋯ + \| u n \| 2 ) ( \| v 1 \| 2 + ⋯ + \| v n \| 2 ) . {\displaystyle {\bigl \|}u_{1}{\bar {v}}_{1}+\cdots +u_{n}{\bar {v}}_{n}{\bigr \|}^{2}\leq {\bigl (}\|u_{1}\|{}^{2}+\cdots +\|u_{n}\|{}^{2}{\bigr )}{\bigl (}\|v_{1}\|{}^{2}+\cdots +\|v_{n}\|{}^{2}{\bigr )}.} === L2 === For the inner product space of square-integrable complex-valued functions, the following inequality holds. \| ∫ R n f ( x ) g ( x ) ¯ d x \| 2 ≤ ∫ R n \| f ( x ) \| 2 d x ∫ R n \| g ( x ) \| 2 d x . {\displaystyle \left\|\int _{\mathbb {R} ^{n}}f(x){\overline {g(x)}}\,dx\right\|^{2}\leq \int _{\mathbb {R} ^{n}}{\bigl \|}f(x){\bigr \|}^{2}\,dx\int _{\mathbb {R} ^{n}}{\bigl \|}g(x){\bigr \|}^{2}\,dx.} The Hölder inequality is a generalization of this. == Applications == === Analysis === In any inner product space, the triangle inequality is a consequence of the Cauchy–Schwarz inequality, as is now shown: ‖ u + v ‖ 2 = ⟨ u + v , u + v ⟩ = ‖ u ‖ 2 + ⟨ u , v ⟩ + ⟨ v , u ⟩ + ‖ v ‖ 2 where ⟨ v , u ⟩ = ⟨ u , v ⟩ ¯ = ‖ u ‖ 2 + 2 Re ⁡ ⟨ u , v ⟩ + ‖ v ‖ 2 ≤ ‖ u ‖ 2 + 2 \| ⟨ u , v ⟩ \| + ‖ v ‖ 2 ≤ ‖ u ‖ 2 + 2 ‖ u ‖ ‖ v ‖ + ‖ v ‖ 2 using CS = ( ‖ u ‖ + ‖ v ‖ ) 2 . {\displaystyle {\begin{alignedat}{4}\\|\mathbf {u} +\mathbf {v} \\|^{2}&=\langle \mathbf {u} +\mathbf {v} ,\mathbf {u} +\mathbf {v} \rangle &&\\&=\\|\mathbf {u} \\|^{2}+\langle \mathbf {u} ,\mathbf {v} \rangle +\langle \mathbf {v} ,\mathbf {u} \rangle +\\|\mathbf {v} \\|^{2}~&&~{\text{ where }}\langle \mathbf {v} ,\mathbf {u} \rangle ={\overline {\langle \mathbf {u} ,\mathbf {v} \rangle }}\\&=\\|\mathbf {u} \\|^{2}+2\operatorname {Re} \langle \mathbf {u} ,\mathbf {v} \rangle +\\|\mathbf {v} \\|^{2}&&\\&\leq \\|\mathbf {u} \\|^{2}+2\|\langle \mathbf {u} ,\mathbf {v} \rangle \|+\\|\mathbf {v} \\|^{2}&&\\&\leq \\|\mathbf {u} \\|^{2}+2\\|\mathbf {u} \\|\\|\mathbf {v} \\|+\\|\mathbf {v} \\|^{2}~&&~{\text{ using CS}}\\&={\bigl (}\\|\mathbf {u} \\|+\\|\mathbf {v} \\|{\bigr )}^{2}.&&\end{alignedat}}} Taking square roots gives the triangle inequality: ‖ u + v ‖ ≤ ‖ u ‖ + ‖ v ‖ . {\displaystyle \\|\mathbf {u} +\mathbf {v} \\|\leq \\|\mathbf {u} \\|+\\|\mathbf {v} \\|.} The Cauchy–Schwarz inequality is used to prove that the inner product is a continuous function with respect to the topology induced by the inner product itself. === Geometry === The Cauchy–Schwarz inequality allows one to extend the notion of "angle between two vectors" to any real inner-product space by defining: cos ⁡ θ u v = ⟨ u , v ⟩ ‖ u ‖ ‖ v ‖ . {\displaystyle \cos \theta _{\mathbf {u} \mathbf {v} }={\frac {\langle \mathbf {u} ,\mathbf {v} \rangle }{\\|\mathbf {u} \\|\\|\mathbf {v} \\|}}.} The Cauchy–Schwarz inequality proves that this definition is sensible, by showing that the right-hand side lies in the interval [−1, 1] and justifies the notion that (real) Hilbert spaces are simply generalizations of the Euclidean space. It can also be used to define an angle in complex inner-product spaces, by taking the absolute value or the real part of the right-hand side, as is done when extracting a metric from quantum fidelity. === Probability theory === Let X {\displaystyle X} and Y {\displaystyle Y} be random variables. Then the covariance inequality is given by: Var ⁡ ( X ) ≥ Cov ⁡ ( X , Y ) 2 Var ⁡ ( Y ) . {\displaystyle \operatorname {Var} (X)\geq {\frac {\operatorname {Cov} (X,Y)^{2}}{\operatorname {Var} (Y)}}.} After defining an inner product on the set of random variables using the expectation of their product, ⟨ X , Y ⟩ := E ⁡ ( X Y ) , {\displaystyle \langle X,Y\rangle :=\operatorname {E} (XY),} the Cauchy–Schwarz inequality becomes \| E ⁡ ( X Y ) \| 2 ≤ E ⁡ ( X 2 ) E ⁡ ( Y 2 ) . {\displaystyle {\bigl \|}\operatorname {E} (XY){\bigr \|}^{2}\leq \operatorname {E} (X^{2})\operatorname {E} (Y^{2}).} To prove the covariance inequality using the Cauchy–Schwarz inequality, let μ = E ⁡ ( X ) {\displaystyle \mu =\operatorname {E} (X)} and ν = E ⁡ ( Y ) , {\displaystyle \nu =\operatorname {E} (Y),} then \| Cov ⁡ ( X , Y ) \| 2 = \| E ⁡ ( ( X − μ ) ( Y − ν ) ) \| 2 = \| ⟨ X − μ , Y − ν ⟩ \| 2 ≤ ⟨ X − μ , X − μ ⟩ ⟨ Y − ν , Y − ν ⟩ = E ⁡ ( ( X − μ ) 2 ) E ⁡ ( ( Y − ν ) 2 ) = Var ⁡ ( X ) Var ⁡ ( Y ) , {\displaystyle {\begin{aligned}{\bigl \|}\operatorname {Cov} (X,Y){\bigr \|}^{2}&={\bigl \|}\operatorname {E} ((X-\mu )(Y-\nu )){\bigr \|}^{2}\\&={\bigl \|}\langle X-\mu ,Y-\nu \rangle {\bigr \|}^{2}\\&\leq \langle X-\mu ,X-\mu \rangle \langle Y-\nu ,Y-\nu \rangle \\&=\operatorname {E} \left((X-\mu )^{2}\right)\operatorname {E} \left((Y-\nu )^{2}\right)\\&=\operatorname {Var} (X)\operatorname {Var} (Y),\end{aligned}}} where Var {\displaystyle \operatorname {Var} } denotes variance and Cov {\displaystyle \operatorname {Cov} } denotes covariance. == Proofs == There are many different proofs of the Cauchy–Schwarz inequality other than those given below. When consulting other sources, there are often two sources of confusion. First, some authors define ⟨⋅,⋅⟩ to be linear in the second argument rather than the first. Second, some proofs are only valid when the field is R {\displaystyle \mathbb {R} } and not C . {\displaystyle \mathbb {C} .} This section gives two proofs of the following theorem: In both of the proofs given below, the proof in the trivial case where at least one of the vectors is zero (or equivalently, in the case where ‖ u ‖ ‖ v ‖ = 0 {\displaystyle \\|\mathbf {u} \\|\\|\mathbf {v} \\|=0} ) is the same. It is presented immediately below only once to reduce repetition. It also includes the easy part of the proof of the Equality Characterization given above; that is, it proves that if u {\displaystyle \mathbf {u} } and v {\displaystyle \mathbf {v} } are linearly dependent then \| ⟨ u , v ⟩ \| = ‖ u ‖ ‖ v ‖ . {\displaystyle {\bigl \|}\langle \mathbf {u} ,\mathbf {v} \rangle {\bigr \|}=\\|\mathbf {u} \\|\\|\mathbf {v} \\|.} Consequently, the Cauchy–Schwarz inequality only needs to be proven only for non-zero vectors and also only the non-trivial direction of the Equality Characterization must be shown. === Proof via the Pythagorean theorem === The special case of v = 0 {\displaystyle \mathbf {v} =\mathbf {0} } was proven above so it is henceforth assumed that v ≠ 0 . {\displaystyle \mathbf {v} \neq \mathbf {0} .} Let z := u − ⟨ u , v ⟩ ⟨ v , v ⟩ v . {\displaystyle \mathbf {z} :=\mathbf {u} -{\frac {\langle \mathbf {u} ,\mathbf {v} \rangle }{\langle \mathbf {v} ,\mathbf {v} \rangle }}\mathbf {v} .} It follows from the linearity of the inner product in its first argument that: ⟨ z , v ⟩ = ⟨ u − ⟨ u , v ⟩ ⟨ v , v ⟩ v , v ⟩ = ⟨ u , v ⟩ − ⟨ u , v ⟩ ⟨ v , v ⟩ ⟨ v , v ⟩ = 0. {\displaystyle \langle \mathbf {z} ,\mathbf {v} \rangle =\left\langle \mathbf {u} -{\frac {\langle \mathbf {u} ,\mathbf {v} \rangle }{\langle \mathbf {v} ,\mathbf {v} \rangle }}\mathbf {v} ,\mathbf {v} \right\rangle =\langle \mathbf {u} ,\mathbf {v} \rangle -{\frac {\langle \mathbf {u} ,\mathbf {v} \rangle }{\langle \mathbf {v} ,\mathbf {v} \rangle }}\langle \mathbf {v} ,\mathbf {v} \rangle =0.} Therefore, z {\displaystyle \mathbf {z} } is a vector orthogonal to the vector v {\displaystyle \mathbf {v} } (Indeed, z {\displaystyle \mathbf {z} } is the projection of u {\displaystyle \mathbf {u} } onto the plane orthogonal to v . {\displaystyle \mathbf {v} .} ) We can thus apply the Pythagorean theorem to u = ⟨ u , v ⟩ ⟨ v , v ⟩ v + z {\displaystyle \mathbf {u} ={\frac {\langle \mathbf {u} ,\mathbf {v} \rangle }{\langle \mathbf {v} ,\mathbf {v} \rangle }}\mathbf {v} +\mathbf {z} } which gives ‖ u ‖ 2 = \| ⟨ u , v ⟩ ⟨ v , v ⟩ \| 2 ‖ v ‖ 2 + ‖ z ‖ 2 = \| ⟨ u , v ⟩ \| 2 ( ‖ v ‖ 2 ) 2 ‖ v ‖ 2 + ‖ z ‖ 2 = \| ⟨ u , v ⟩ \| 2 ‖ v ‖ 2 + ‖ z ‖ 2 ≥ \| ⟨ u , v ⟩ \| 2 ‖ v ‖ 2 . {\displaystyle \\|\mathbf {u} \\|^{2}=\left\|{\frac {\langle \mathbf {u} ,\mathbf {v} \rangle }{\langle \mathbf {v} ,\mathbf {v} \rangle }}\right\|^{2}\\|\mathbf {v} \\|^{2}+\\|\mathbf {z} \\|^{2}={\frac {\|\langle \mathbf {u} ,\mathbf {v} \rangle \|^{2}}{(\\|\mathbf {v} \\|^{2})^{2}}}\,\\|\mathbf {v} \\|^{2}+\\|\mathbf {z} \\|^{2}={\frac {\|\langle \mathbf {u} ,\mathbf {v} \rangle \|^{2}}{\\|\mathbf {v} \\|^{2}}}+\\|\mathbf {z} \\|^{2}\geq {\frac {\|\langle \mathbf {u} ,\mathbf {v} \rangle \|^{2}}{\\|\mathbf {v} \\|^{2}}}.} The Cauchy–Schwarz inequality follows by multiplying by ‖ v ‖ 2 {\displaystyle \\|\mathbf {v} \\|^{2}} and then taking the square root. Moreover, if the relation ≥ {\displaystyle \geq } in the above expression is actually an equality, then ‖ z ‖ 2 = 0 {\displaystyle \\|\mathbf {z} \\|^{2}=0} and hence z = 0 ; {\displaystyle \mathbf {z} =\mathbf {0} ;} the definition of z {\displaystyle \mathbf {z} } then establishes a relation of linear dependence between u {\displaystyle \mathbf {u} } and v . {\displaystyle \mathbf {v} .} The converse was proved at the beginning of this section, so the proof is complete. ◼ {\displaystyle \blacksquare } === Proof by analyzing a quadratic === Consider an arbitrary pair of vectors u , v {\displaystyle \mathbf {u} ,\mathbf {v} } . Define the function p : R → R {\displaystyle p:\mathbb {R} \to \mathbb {R} } defined by p ( t ) = ⟨ t α u + v , t α u + v ⟩ {\displaystyle p(t)=\langle t\alpha \mathbf {u} +\mathbf {v} ,t\alpha \mathbf {u} +\mathbf {v} \rangle } , where α {\displaystyle \alpha } is a complex number satisfying \| α \| = 1 {\displaystyle \|\alpha \|=1} and α ⟨ u , v ⟩ = \| ⟨ u , v ⟩ \| {\displaystyle \alpha \langle \mathbf {u} ,\mathbf {v} \rangle =\|\langle \mathbf {u} ,\mathbf {v} \rangle \|} . Such an α {\displaystyle \alpha } exists since if ⟨ u , v ⟩ = 0 {\displaystyle \langle \mathbf {u} ,\mathbf {v} \rangle =0} then α {\displaystyle \alpha } can be taken to be 1. Since the inner product is positive-definite, p ( t ) {\displaystyle p(t)} only takes non-negative real values. On the other hand, p ( t ) {\displaystyle p(t)} can be expanded using the bilinearity of the inner product: p ( t ) = ⟨ t α u , t α u ⟩ + ⟨ t α u , v ⟩ + ⟨ v , t α u ⟩ + ⟨ v , v ⟩ = t α t α ¯ ⟨ u , u ⟩ + t α ⟨ u , v ⟩ + t α ¯ ⟨ v , u ⟩ + ⟨ v , v ⟩ = ‖ u ‖ 2 t 2 + 2 \| ⟨ u , v ⟩ \| t + ‖ v ‖ 2 {\displaystyle {\begin{aligned}p(t)&=\langle t\alpha \mathbf {u} ,t\alpha \mathbf {u} \rangle +\langle t\alpha \mathbf {u} ,\mathbf {v} \rangle +\langle \mathbf {v} ,t\alpha \mathbf {u} \rangle +\langle \mathbf {v} ,\mathbf {v} \rangle \\&=t\alpha t{\overline {\alpha }}\langle \mathbf {u} ,\mathbf {u} \rangle +t\alpha \langle \mathbf {u} ,\mathbf {v} \rangle +t{\overline {\alpha }}\langle \mathbf {v} ,\mathbf {u} \rangle +\langle \mathbf {v} ,\mathbf {v} \rangle \\&=\lVert \mathbf {u} \rVert ^{2}t^{2}+2\|\langle \mathbf {u} ,\mathbf {v} \rangle \|t+\lVert \mathbf {v} \rVert ^{2}\end{aligned}}} Thus, p {\displaystyle p} is a polynomial of degree 2 {\displaystyle 2} (unless u = 0 , {\displaystyle \mathbf {u} =0,} which is a case that was checked earlier). Since the sign of p {\displaystyle p} does not change, the discriminant of this polynomial must be non-positive: Δ = 4 ( \| ⟨ u , v ⟩ \| 2 − ‖ u ‖ 2 ‖ v ‖ 2 ) ≤ 0. {\displaystyle \Delta =4{\bigl (}\,\|\langle \mathbf {u} ,\mathbf {v} \rangle \|^{2}-\Vert \mathbf {u} \Vert ^{2}\Vert \mathbf {v} \Vert ^{2}{\bigr )}\leq 0.} The conclusion follows. For the equality case, notice that Δ = 0 {\displaystyle \Delta =0} happens if and only if p ( t ) = ( t ‖ u ‖ + ‖ v ‖ ) 2 . {\displaystyle p(t)={\bigl (}t\Vert \mathbf {u} \Vert +\Vert \mathbf {v} \Vert {\bigr )}^{2}.} If t 0 = − ‖ v ‖ / ‖ u ‖ , {\displaystyle t_{0}=-\Vert \mathbf {v} \Vert /\Vert \mathbf {u} \Vert ,} then p ( t 0 ) = ⟨ t 0 α u + v , t 0 α u + v ⟩ = 0 , {\displaystyle p(t_{0})=\langle t_{0}\alpha \mathbf {u} +\mathbf {v} ,t_{0}\alpha \mathbf {u} +\mathbf {v} \rangle =0,} and hence v = − t 0 α u . {\displaystyle \mathbf {v} =-t_{0}\alpha \mathbf {u} .} == Generalizations == Various generalizations of the Cauchy–Schwarz inequality exist. Hölder's inequality generalizes it to L p {\displaystyle L^{p}} norms. More generally, it can be interpreted as a special case of the definition of the norm of a linear operator on a Banach space (Namely, when the space is a Hilbert space). Further generalizations are in the context of operator theory, e.g. for operator-convex functions and operator algebras, where the domain and/or range are replaced by a C-algebra or W-algebra. An inner product can be used to define a positive linear functional. For example, given a Hilbert space L 2 ( m ) , m {\displaystyle L^{2}(m),m} being a finite measure, the standard inner product gives rise to a positive functional φ {\displaystyle \varphi } by φ ( g ) = ⟨ g , 1 ⟩ . {\displaystyle \varphi (g)=\langle g,1\rangle .} Conversely, every positive linear functional φ {\displaystyle \varphi } on L 2 ( m ) {\displaystyle L^{2}(m)} can be used to define an inner product ⟨ f , g ⟩ φ := φ ( g ∗ f ) , {\displaystyle \langle f,g\rangle _{\varphi }:=\varphi \left(g^{}f\right),} where g ∗ {\displaystyle g^{}} is the pointwise complex conjugate of g . {\displaystyle g.} In this language, the Cauchy–Schwarz inequality becomes \| φ ( g ∗ f ) \| 2 ≤ φ ( f ∗ f ) φ ( g ∗ g ) , {\displaystyle {\bigl \|}\varphi (g^{}f){\bigr \|}^{2}\leq \varphi \left(f^{}f\right)\varphi \left(g^{}g\right),} which extends verbatim to positive functionals on C-algebras: The next two theorems are further examples in operator algebra. This extends the fact φ ( a ∗ a ) ⋅ 1 ≥ φ ( a ) ∗ φ ( a ) = \| φ ( a ) \| 2 , {\displaystyle \varphi \left(a^{}a\right)\cdot 1\geq \varphi (a)^{}\varphi (a)=\|\varphi (a)\|^{2},} when φ {\displaystyle \varphi } is a linear functional. The case when a {\displaystyle a} is self-adjoint, that is, a = a ∗ , {\displaystyle a=a^{*},} is sometimes known as Kadison's inequality. Another generalization is a refinement obtained by interpolating between both sides of the Cauchy–Schwarz inequality: This theorem can be deduced from Hölder's inequality. There are also non-commutative versions for operators and tensor products of matrices. Several matrix versions of the Cauchy–Schwarz inequality and Kantorovich inequality are applied to linear regression models. == See also == Bessel's inequality – Theorem on orthonormal sequences Hölder's inequality – Inequality between integrals in Lp spaces Jensen's inequality – Theorem of convex functions Kantorovich inequality Kunita–Watanabe inequality Minkowski inequality – Triangle inequality in Lp Paley–Zygmund inequality – Probability equation in mathematics == Notes == == Citations == == References == == External links == Earliest Uses: The entry on the Cauchy–Schwarz inequality has some historical information. Example of application of Cauchy–Schwarz inequality to determine Linearly Independent Vectors Tutorial and Interactive program.
Wikipedia:Cayley plane#0	In mathematics, the Cayley plane (or octonionic projective plane) P2(O) is a projective plane over the octonions. The Cayley plane was discovered in 1933 by Ruth Moufang, and is named after Arthur Cayley for his 1845 paper describing the octonions. == Properties == In the Cayley plane, lines and points may be defined in a natural way so that it becomes a 2-dimensional projective space, that is, a projective plane. It is a non-Desarguesian plane, where Desargues' theorem does not hold. More precisely, as of 2005, there are two objects called Cayley planes, namely the real and the complex Cayley plane. The real Cayley plane is the symmetric space F4/Spin(9), where F4 is a compact form of an exceptional Lie group and Spin(9) is the spin group of nine-dimensional Euclidean space (realized in F4). It admits a cell decomposition into three cells, of dimensions 0, 8 and 16. The complex Cayley plane is a homogeneous space under the complexification of the group E6 by a parabolic subgroup P1. It is the closed orbit in the projectivization of the minimal complex representation of E6. The complex Cayley plane consists of two complex F4-orbits: the closed orbit is a quotient of the complexified F4 by a parabolic subgroup, the open orbit is the complexification of the real Cayley plane, retracting to it. == See also == Rosenfeld projective plane == Notes == == References ==
Wikipedia:Cecilia Berdichevsky#0	Cecilia Berdichevsky or Berdichevski (née Tuwjasz; 30 March 1925 – 20 February 2010) was an Argentine computer scientist. She began her work in 1961 using the first Ferranti Mercury computer in that country. == Biography == She was born Mirjam Tuwjasz on 30 March 1925 to a Polish-Jewish family in Vidzy, at that time part of Poland, now Belarus. Because of growing hostilities toward the Jewish community, first her father and then her mother Hoda and her emigrated to Argentina when she was four years old, where she adopted the name Cecilia, and she spent her childhood years in Avellaneda, south of the Buenos Aires suburbs. Her father died within a few years of arriving in their new home and her mother remarried a rich man. Cecilia married Mario Berdichevsky, a physician from Avellaneda, in 1951. Despite having a good job as a practicing accountant for ten years, she was not happy there having experienced many frustrations. A friend, computer scientist Rebeca Guber, convinced her to go back to school, which changed her life. === Clementina === At the age of 31, Berdichevsky began her studies of mathematics at the University of Buenos Aires with Manuel Sadosky. There she had her first experience programming the new Ferranti Mercury computer, which became known by the nickname "Clementina" after someone programmed it to play the American song, "My darling Clementine." In 1961, when it arrived in Buenos Aires from England, Clementina was the most powerful computer in the country, cost $300,000 and measured 18 metres (59 ft) in length. It was the first large computer used for scientific purposes in the country (in that same year, an IBM 1401 was installed in Buenos Aires for business uses). The newly graduated Berdichevsky studied computing from the visiting English software engineer Cicely Popplewell (famous for having worked with Alan Turing in Manchester) and with the Spanish mathematician Ernesto García Camarero. Popplewell herself motivated Berdichevsky to write and run the first program for the new computer, which required multiple arithmetic calculations. A photoelectric device read a punched paper ribbon that was used to submit the data and Clementina produced the desired result in only seconds. Based on Berdichevsky's progress in Argentina, in 1962 she was one of two people awarded scholarships to continue studies at the University of London's Computer Unit for five months, followed by the same length of time at a French institution. She returned home the following year as an expert on the workings of Clementina. According to Berdichevsky, "Work with Mercury was defined by its resources and its characteristics, structure and operational capabilities, as well as by the languages, routines, stored libraries and facilities that it offered... Mercury could not perform more than one operation at the same time, and they were the three basic arithmetical operations: addition, subtraction, and multiplication." The computer's resources included: machine language, an assembler named Pig2; a high-level programming language (a compiler) called Autocode. Later another compiler called Comic replaced Autocode. In those days, compilers were unique and were written only for each computer; they were not developed for use on multiple models of computers until years later. In addition, Berdichevsky worked as Head of Practical Works of Numerical Calculus I, where the tenured professor was her mentor, Manuel Sadosky who was then vice-dean of the Faculty of Exact Sciences of the University of Buenos Aires from 1957 to 1966. === Coup d'état === Berdichevsky worked with Sadosky's institute until an Argentine coup d'état that installed a military dictatorship, which imposed government control over the workings of the previously autonomous state universities. This intrusion led to student/professor sit-ins that resulted in the violent Night of the Long Batons on 29 July 1966 when military troops physically beat and evicted the academic occupiers from the University of Buenos Aires and other institutions of higher learning. Many academics, including Sadosky, were forced into exile. Berdichevsky herself began working as an accountant. Between 1966 and 1970 she was also one of the directors of Scientific Technical Advisors (ACT), the company formed by her former academic associates, Manuel Sadosky, Rebeca Guber and Juan Chamero. === Later years === In 1984, Berdichevsky became Deputy General Manager of the Argentine savings bank Caja de Ahorro in charge of its computer center. She was also named the representative at the International Federation for Information Processing. After her retirement, she continued to work as a computer consultant and participated in important international projects and organizations such as United Nations Development Program. Cecilia Berdichevsky died in Avellaneda, Argentina, 28 February 2010. == Published work == Berdichevsky C. (2006) The Beginning of Computer Science in Argentina — Clementina – (1961–1966). In: Impagliazzo J. (eds) History of Computing and Education 2 (HCE2). IFIP Advances in Information and Communication Technology, vol 215. Springer, New York, NY (complete paper in English) == References ==
Wikipedia:Cecilia Wangechi Mwathi#0	Cecilia Wangechi Mwathi (15 May 1963 – 17 August 2011) was a Kenyan mathematician and union activist. She was the first woman in Kenya to become a mathematics professor, and was known both for her activism for higher education and for inspiring Kenyan girls to study science, technology, and mathematics. == Early life and education == Mwathi was born on 15 May 1963 in Kaigonde, a village in Kenya near Gichira, as the fifth of eight children in a poor family. She was educated in Gichira, having to walk 5km barefoot to reach the school. She then went to Mugoiri Girls High School and Chania High School before studying mathematics education at Kenyatta University, where she was awarded a bachelor's degree in 1987. In the next years, she worked as a high school teacher at Garissa High School and then Kenya High School. Returning to Kenyatta University in 1991, she earned a master's degree in mathematics in 1992. From 1995 through 1998, she was a doctoral student at the University of Zimbabwe, while working as a mathematics instructor in Kenya. She completed her Ph.D. at the University of Zimbabwe in 1998; her dissertation was Groups of Units in Algebraic Number Fields of Fourth and Eighth Degrees, and concerned algebraic number theory. == Academic career and later life == Mwathi joined the faculty of the Jomo Kenyatta University of Agriculture and Technology (JKUAT) as an assistant lecturer in 1992. She became a lecturer in 1995 and senior lecturer in 2000. In 2005 she became secretary general of the UASU-JKUAT faculty union chapter. In 2006, the university became embroiled in a crisis over its failure to pay its faculty in a timely manner, and Mwathi was a leading representative for the faculty in this issue. After the faculty went on strike in October 2006, Mwathi and another union leader, Moses Muchina, were fired from their faculty positions, and in 2008 the Kenyan courts upheld their firing. However, later in 2008, when Mabel Imbuga became vice-chancellor of JKUAT, she announced an amnesty on the issue and reinstated Mwathi to her professorship, conditioned on not pursuing further legal action. Mwathi was named associate professor in 2010. Mwathi was the hosting chair and convener of the Second Africa Regional Congress of the International Commission on Mathematics Instruction, held at JKUAT in 2007. She also served as editor in chief of the Journal of Agriculture, Science, and Technology, published by JKUAT. She died on 17 August 2011, in Nairobi Hospital, after a long illness, survived by three daughters and two foster children. JKUAT held a requiem mass in her honor on 24 August 2011. == References == == External links == Williams, Scott W., "Cecilia Wangechi Mwathi", Black Women in Mathematics, University at Buffalo
Wikipedia:Celia Hoyles#0	Dame Celia Mary Hoyles, (née French; born 18 May 1946) is a British mathematician, educationalist and Professor of Mathematics Education at University College London (UCL), in the Institute of Education (IoE). == Early life and education == Celia was born on 18 May 1946. She was educated at the University of Manchester where she graduated with a first class degree in mathematics from the Department of Mathematics in 1967. She subsequently completed a Postgraduate Certificate in Education (PGCE) in 1971, and a Master of Education degree (MEd) in 1973. She completed a Doctor of Philosophy (PhD) degree in 1980, with a thesis titled "Factors in school learning - the pupils' view: a study with particular reference to mathematics". All her degrees are from the University of London. == Career and research == Hoyles began her career as a secondary school teacher, later becoming an academic. In the late 1980s she was co-presenter of Fun and Games, a prime time television quiz show about mathematics. With Arthur Bakker, Phillip Kent, and Richard B. Noss she is the co-author of Improving Mathematics at Work: The Need for Techno-Mathematical Literacies. Hoyles served as president of the Institute of Mathematics and its Applications (IMA) from 2014 to 2015. She served as chief adviser for mathematics to the government of the United Kingdom from 2004 to 2007 and as director of the National Centre for Excellence in the Teaching of Mathematics (NCETM) from 2007 to 2013. == Awards and honours == In the 2004 New Year Honours, Hoyles was appointed Officer of the Order of the British Empire (OBE) 'for services to education'. In the 2014 New Year Honours, she was appointed Dame Commander of the Order of the British Empire (DBE) in recognition of her service as director of the National Centre for Excellence in the Teaching of Mathematics. She was elected a Fellow of the Academy of Social Sciences (FAcSS). In 2003, she was awarded the first Hans Freudenthal Medal by the International Commission on Mathematical Instruction (ICMI) in recognition of 'the outstanding contribution that [she] has made to research in the domain of technology and mathematics education'. In 2010, she was awarded the first Kavli Education Medal by the Royal Society 'in recognition of her outstanding contribution to research in mathematics education'. Hoyles has honorary degrees from the Open University (2006), Loughborough University (2008), Sheffield Hallam University (2011) and University of Bath (2019). == Personal life == Her first marriage was to Martin Hoyles: their marriage ended in divorce. In 1996, she married Richard Noss, Professor of Mathematics Education at University College London. Her second marriage brought two step children. == References ==
Wikipedia:Center (algebra)#0	The term center or centre is used in various contexts in abstract algebra to denote the set of all those elements that commute with all other elements. The center of a group G consists of all those elements x in G such that xg = gx for all g in G. This is a normal subgroup of G. The similarly named notion for a semigroup is defined likewise and it is a subsemigroup. The center of a ring (or an associative algebra) R is the subset of R consisting of all those elements x of R such that xr = rx for all r in R. The center is a commutative subring of R. The center of a Lie algebra L consists of all those elements x in L such that [x,a] = 0 for all a in L. This is an ideal of the Lie algebra L. == See also == Centralizer and normalizer Center (category theory) == References ==
Wikipedia:Center (ring theory)#0	In algebra, the center of a ring R is the subring consisting of the elements x such that xy = yx for all elements y in R. It is a commutative ring and is denoted as Z(R); 'Z' stands for the German word Zentrum, meaning "center". If R is a ring, then R is an associative algebra over its center. Conversely, if R is an associative algebra over a commutative subring S, then S is a subring of the center of R, and if S happens to be the center of R, then the algebra R is called a central algebra. == Examples == The center of a commutative ring R is R itself. The center of a skew-field is a field. The center of the (full) matrix ring with entries in a commutative ring R consists of R-scalar multiples of the identity matrix. Let F be a field extension of a field k, and R an algebra over k. Then Z(R ⊗k F) = Z(R) ⊗k F. The center of the universal enveloping algebra of a Lie algebra plays an important role in the representation theory of Lie algebras. For example, a Casimir element is an element of such a center that is used to analyze Lie algebra representations. See also: Harish-Chandra isomorphism. The center of a simple algebra is a field. == See also == Center of a group Central simple algebra Morita equivalence == Notes == == References ==
Wikipedia:Central differencing scheme#0	In applied mathematics, the central differencing scheme is a finite difference method that optimizes the approximation for the differential operator in the central node of the considered patch and provides numerical solutions to differential equations. It is one of the schemes used to solve the integrated convection–diffusion equation and to calculate the transported property Φ at the e and w faces, where e and w are short for east and west (compass directions being customarily used to indicate directions on computational grids). The method's advantages are that it is easy to understand and implement, at least for simple material relations; and that its convergence rate is faster than some other finite differencing methods, such as forward and backward differencing. The right side of the convection-diffusion equation, which basically highlights the diffusion terms, can be represented using central difference approximation. To simplify the solution and analysis, linear interpolation can be used logically to compute the cell face values for the left side of this equation, which is nothing but the convective terms. Therefore, cell face values of property for a uniform grid can be written as: Φ e = 1 2 ( Φ P + Φ E ) {\displaystyle \Phi _{e}={\tfrac {1}{2}}(\Phi _{P}+\Phi _{E})} Φ w = 1 2 ( Φ W + Φ P ) {\displaystyle \Phi _{w}={\tfrac {1}{2}}(\Phi _{W}+\Phi _{P})} == Steady-state convection diffusion equation == The convection–diffusion equation is a collective representation of diffusion and convection equations, and describes or explains every physical phenomenon involving convection and diffusion in the transference of particles, energy and other physical quantities inside a physical system: div ⁡ ( ρ u φ ) = div ⁡ ( Γ ∇ φ ) + S φ ; {\displaystyle \operatorname {div} (\rho u\varphi )=\operatorname {div} (\Gamma \nabla \varphi )+S_{\varphi };\,} where Г is diffusion coefficient and Φ is the property. == Formulation of steady-state convection diffusion equation == Formal integration of steady-state convection–diffusion equation over a control volume gives This equation represents flux balance in a control volume. The left side gives the net convective flux, and the right side contains the net diffusive flux and the generation or destruction of the property within the control volume. In the absence of source term equation, one becomes Continuity equation: Assuming a control volume and integrating equation 2 over control volume gives: Integration of equation 3 yields: It is convenient to define two variables to represent the convective mass flux per unit area and diffusion conductance at cell faces, for example: F = ρ u {\displaystyle F=\rho u} D = Γ / δ x {\displaystyle D=\Gamma /\delta x} Assuming A e = A w {\displaystyle A_{e}=A_{w}} , we can write integrated convection–diffusion equation as: F e φ e − F w φ w = D e ( φ E − φ P ) − D w ( φ P − φ W ) {\displaystyle F_{e}\varphi _{e}-F_{w}\varphi _{w}=D_{e}(\varphi _{E}-\varphi _{P})-D_{w}(\varphi _{P}-\varphi _{W})} And integrated continuity equation as: F e − F w = 0 {\displaystyle F_{e}-F_{w}=0} In a central differencing scheme, we try linear interpolation to compute cell face values for convection terms. For a uniform grid, we can write cell face values of property Φ as φ e = 1 2 ( φ E + φ P ) , φ w = 1 2 ( φ P + φ W ) {\displaystyle \varphi _{e}={\tfrac {1}{2}}(\varphi _{E}+\varphi _{P}),\quad \varphi _{w}={\tfrac {1}{2}}(\varphi _{P}+\varphi _{W})} On substituting this into integrated convection-diffusion equation, we obtain: F e φ E + φ P 2 − F w φ W + φ P 2 = D e ( φ E − φ P ) − D w ( φ P − φ W ) {\displaystyle F_{e}{\frac {\varphi _{E}+\varphi _{P}}{2}}-F_{w}{\frac {\varphi _{W}+\varphi _{P}}{2}}=D_{e}(\varphi _{E}-\varphi _{P})-D_{w}(\varphi _{P}-\varphi _{W})} And on rearranging: [ ( D w + F w 2 ) + ( D e − F e 2 ) + ( F e − F w ) ] φ P = ( D w + F w 2 ) φ W + ( D e − F e 2 ) φ E {\displaystyle \left[\left(D_{w}+{\frac {F_{w}}{2}}\right)+\left(D_{e}-{\frac {F_{e}}{2}}\right)+(F_{e}-F_{w})\right]\varphi _{P}=\left(D_{w}+{\frac {F_{w}}{2}}\right)\varphi _{W}+\left(D_{e}-{\frac {F_{e}}{2}}\right)\varphi _{E}} a P φ P = a W φ W + a E φ E {\displaystyle a_{P}\varphi _{P}=a_{W}\varphi _{W}+a_{E}\varphi _{E}} == Different aspects of central differencing scheme == === Conservativeness === Conservation is ensured in central differencing scheme since overall flux balance is obtained by summing the net flux through each control volume taking into account the boundary fluxes for the control volumes around nodes 1 and 4. Boundary flux for control volume around node 1 and 4 [ Γ e 1 ( φ 2 − φ 1 ) δ x − q A ] + [ Γ e 2 ( φ 3 − φ 2 ) δ x − Γ w 2 ( φ 2 − φ 1 ) δ x ] + [ Γ e 3 ( φ 4 − φ 3 ) δ x − Γ w 3 ( φ 3 − φ 2 ) δ x ] + [ q B − Γ w 4 ( φ 4 − φ 3 ) δ x ] = q B − q A {\displaystyle {\begin{aligned}&\left[{\frac {\Gamma _{e_{1}}(\varphi _{2}-\varphi _{1})}{\delta x}}-q_{A}\right]+\left[{\frac {\Gamma _{e_{2}}(\varphi _{3}-\varphi _{2})}{\delta x}}-{\frac {\Gamma _{w_{2}}(\varphi _{2}-\varphi _{1})}{\delta x}}\right]\\[10pt]+{}&\left[{\frac {\Gamma _{e_{3}}(\varphi _{4}-\varphi _{3})}{\delta x}}-{\frac {\Gamma _{w_{3}}(\varphi _{3}-\varphi _{2})}{\delta x}}\right]+\left[q_{B}-{\frac {\Gamma _{w_{4}}(\varphi _{4}-\varphi _{3})}{\delta x}}\right]=q_{B}-q_{A}\end{aligned}}} because Γ e 1 = Γ w 2 , Γ e 2 = Γ w 3 , Γ e 3 = Γ w 4 {\displaystyle \Gamma _{e_{1}}=\Gamma _{w_{2}},\Gamma _{e_{2}}=\Gamma _{w_{3}},\Gamma _{e_{3}}=\Gamma _{w_{4}}} === Boundedness === Central differencing scheme satisfies first condition of boundedness. Since F e − F w = 0 {\displaystyle F_{e}-F_{w}=0} from continuity equation, therefore; a P φ P = a W φ W + a E φ E {\displaystyle a_{P}\varphi _{P}=a_{W}\varphi _{W}+a_{E}\varphi _{E}} Another essential requirement for boundedness is that all coefficients of the discretised equations should have the same sign (usually all positive). But this is only satisfied when (peclet number) F e / D e < 2 {\displaystyle F_{e}/D_{e}<2} because for a unidirectional flow ( F e > 0 , F w > 0 {\displaystyle F_{e}>0,F_{w}>0} ) a E = ( D e − F e / 2 ) {\displaystyle a_{E}=(D_{e}-F_{e}/2)} is always positive if D e > F e / 2 {\displaystyle D_{e}>F_{e}/2} === Transportiveness === It requires that transportiveness changes according to magnitude of peclet number i.e. when pe is zero φ {\displaystyle \varphi } is spread in all directions equally and as Pe increases (convection > diffusion) φ {\displaystyle \varphi } at a point largely depends on upstream value and less on downstream value. But central differencing scheme does not possess transportiveness at higher pe since Φ at a point is average of neighbouring nodes for all Pe. === Accuracy === The Taylor series truncation error of the central differencing scheme is second order. Central differencing scheme will be accurate only if Pe < 2. Owing to this limitation, central differencing is not a suitable discretisation practice for general purpose flow calculations. == Applications of central differencing schemes == They are currently used on a regular basis in the solution of the Euler equations and Navier–Stokes equations. Results using central differencing approximation have shown noticeable improvements in accuracy in smooth regions. Shock wave representation and boundary-layer definition can be improved on coarse meshes. == Advantages == Simpler to program, requires less computer time per step, and works well with multigrid acceleration techniques Has a free parameter in conjunction with the fourth-difference dissipation, which is needed to approach a steady state. More accurate than the first-order upwind scheme if the Peclet number is less than 2. == Disadvantages == Somewhat more dissipative Leads to oscillations in the solution or divergence if the local Peclet number is larger than 2. == See also == Finite difference method Finite difference Taylor series Taylor theorem Convection–diffusion equation Diffusion Convection Peclet number Linear interpolation Symmetric derivative Upwind differencing scheme for convection == References == == Further reading == Computational Fluid Dynamics: The Basics with Applications – John D. Anderson, ISBN 0-07-001685-2 Computational Fluid Dynamics volume 1 – Klaus A. Hoffmann, Steve T. Chiang, ISBN 0-9623731-0-9 == External links == One-Dimensional_Steady-State_Convection_and_Diffusion#Central_Difference_Scheme Finite Differences Central Difference Methods Archived 5 November 2013 at the Wayback Machine A Conservative Finite Difference Scheme for Poisson–Nernst–Planck Equations
Wikipedia:Centrality#0	In graph theory and network analysis, indicators of centrality assign numbers or rankings to nodes within a graph corresponding to their network position. Applications include identifying the most influential person(s) in a social network, key infrastructure nodes in the Internet or urban networks, super-spreaders of disease, and brain networks. Centrality concepts were first developed in social network analysis, and many of the terms used to measure centrality reflect their sociological origin. == Definition and characterization of centrality indices == Centrality indices are answers to the question "What characterizes an important vertex?" The answer is given in terms of a real-valued function on the vertices of a graph, where the values produced are expected to provide a ranking which identifies the most important nodes. The word "importance" has a wide number of meanings, leading to many different definitions of centrality. Two categorization schemes have been proposed. "Importance" can be conceived in relation to a type of flow or transfer across the network. This allows centralities to be classified by the type of flow they consider important. "Importance" can alternatively be conceived as involvement in the cohesiveness of the network. This allows centralities to be classified based on how they measure cohesiveness. Both of these approaches divide centralities in distinct categories. A further conclusion is that a centrality which is appropriate for one category will often "get it wrong" when applied to a different category. Many, though not all, centrality measures effectively count the number of paths (also called walks) of some type going through a given vertex; the measures differ in how the relevant walks are defined and counted. Restricting consideration to this group allows for taxonomy which places many centralities on a spectrum from those concerned with walks of length one (degree centrality) to infinite walks (eigenvector centrality). Other centrality measures, such as betweenness centrality focus not just on overall connectedness but occupying positions that are pivotal to the network's connectivity. === Characterization by network flows === A network can be considered a description of the paths along which something flows. This allows a characterization based on the type of flow and the type of path encoded by the centrality. A flow can be based on transfers, where each indivisible item goes from one node to another, like a package delivery going from the delivery site to the client's house. A second case is serial duplication, in which an item is replicated so that both the source and the target have it. An example is the propagation of information through gossip, with the information being propagated in a private way and with both the source and the target nodes being informed at the end of the process. The last case is parallel duplication, with the item being duplicated to several links at the same time, like a radio broadcast which provides the same information to many listeners at once. Likewise, the type of path can be constrained to geodesics (shortest paths), paths (no vertex is visited more than once), trails (vertices can be visited multiple times, no edge is traversed more than once), or walks (vertices and edges can be visited/traversed multiple times). === Characterization by walk structure === An alternative classification can be derived from how the centrality is constructed. This again splits into two classes. Centralities are either radial or medial. Radial centralities count walks which start/end from the given vertex. The degree and eigenvalue centralities are examples of radial centralities, counting the number of walks of length one or length infinity. Medial centralities count walks which pass through the given vertex. The canonical example is Freeman's betweenness centrality, the number of shortest paths which pass through the given vertex. Likewise, the counting can capture either the volume or the length of walks. Volume is the total number of walks of the given type. The three examples from the previous paragraph fall into this category. Length captures the distance from the given vertex to the remaining vertices in the graph. Closeness centrality, the total geodesic distance from a given vertex to all other vertices, is the best known example. Note that this classification is independent of the type of walk counted (i.e. walk, trail, path, geodesic). Borgatti and Everett propose that this typology provides insight into how best to compare centrality measures. Centralities placed in the same box in this 2×2 classification are similar enough to make plausible alternatives; one can reasonably compare which is better for a given application. Measures from different boxes, however, are categorically distinct. Any evaluation of relative fitness can only occur within the context of predetermining which category is more applicable, rendering the comparison moot. === Radial-volume centralities exist on a spectrum === The characterization by walk structure shows that almost all centralities in wide use are radial-volume measures. These encode the belief that a vertex's centrality is a function of the centrality of the vertices it is associated with. Centralities distinguish themselves on how association is defined. Bonacich showed that if association is defined in terms of walks, then a family of centralities can be defined based on the length of walk considered. Degree centrality counts walks of length one, while eigenvalue centrality counts walks of length infinity. Alternative definitions of association are also reasonable. Alpha centrality allows vertices to have an external source of influence. Estrada's subgraph centrality proposes only counting closed paths (triangles, squares, etc.). The heart of such measures is the observation that powers of the graph's adjacency matrix gives the number of walks of length given by that power. Similarly, the matrix exponential is also closely related to the number of walks of a given length. An initial transformation of the adjacency matrix allows a different definition of the type of walk counted. Under either approach, the centrality of a vertex can be expressed as an infinite sum, either ∑ k = 0 ∞ A R k β k {\displaystyle \sum _{k=0}^{\infty }A_{R}^{k}\beta ^{k}} for matrix powers or ∑ k = 0 ∞ ( A R β ) k k ! {\displaystyle \sum _{k=0}^{\infty }{\frac {(A_{R}\beta )^{k}}{k!}}} for matrix exponentials, where k {\displaystyle k} is walk length, A R {\displaystyle A_{R}} is the transformed adjacency matrix, and β {\displaystyle \beta } is a discount parameter which ensures convergence of the sum. Bonacich's family of measures does not transform the adjacency matrix. Alpha centrality replaces the adjacency matrix with its resolvent. Subgraph centrality replaces the adjacency matrix with its trace. A startling conclusion is that regardless of the initial transformation of the adjacency matrix, all such approaches have common limiting behavior. As β {\displaystyle \beta } approaches zero, the indices converge to degree centrality. As β {\displaystyle \beta } approaches its maximal value, the indices converge to eigenvalue centrality. === Game-theoretic centrality === The common feature of most of the aforementioned standard measures is that they assess the importance of a node by focusing only on the role that a node plays by itself. However, in many applications such an approach is inadequate because of synergies that may occur if the functioning of nodes is considered in groups. For example, consider the problem of stopping an epidemic. Looking at above image of network, which nodes should we vaccinate? Based on previously described measures, we want to recognize nodes that are the most important in disease spreading. Approaches based only on centralities, that focus on individual features of nodes, may not be a good idea. Nodes in the red square, individually cannot stop disease spreading, but considering them as a group, we clearly see that they can stop disease if it has started in nodes v 1 {\displaystyle v_{1}} , v 4 {\displaystyle v_{4}} , and v 5 {\displaystyle v_{5}} . Game-theoretic centralities try to consult described problems and opportunities, using tools from game-theory. The approach proposed in uses the Shapley value. Because of the time-complexity hardness of the Shapley value calculation, most efforts in this domain are driven into implementing new algorithms and methods which rely on a peculiar topology of the network or a special character of the problem. Such an approach may lead to reducing time-complexity from exponential to polynomial. Similarly, the solution concept authority distribution () applies the Shapley-Shubik power index, rather than the Shapley value, to measure the bilateral direct influence between the players. The distribution is indeed a type of eigenvector centrality. It is used to sort big data objects in Hu (2020), such as ranking U.S. colleges. == Important limitations == Centrality indices have two important limitations, one obvious and the other subtle. The obvious limitation is that a centrality which is optimal for one application is often sub-optimal for a different application. Indeed, if this were not so, we would not need so many different centralities. An illustration of this phenomenon is provided by the Krackhardt kite graph, for which three different notions of centrality give three different choices of the most central vertex. The more subtle limitation is the commonly held fallacy that vertex centrality indicates the relative importance of vertices. Centrality indices are explicitly designed to produce a ranking which allows indication of the most important vertices. This they do well, under the limitation just noted. They are not designed to measure the influence of nodes in general. Recently, network physicists have begun developing node influence metrics to address this problem. The error is two-fold. Firstly, a ranking only orders vertices by importance, it does not quantify the difference in importance between different levels of the ranking. This may be mitigated by applying Freeman centralization to the centrality measure in question, which provide some insight to the importance of nodes depending on the differences of their centralization scores. Furthermore, Freeman centralization enables one to compare several networks by comparing their highest centralization scores. Secondly, the features which (correctly) identify the most important vertices in a given network/application do not necessarily generalize to the remaining vertices. For the majority of other network nodes the rankings may be meaningless. This explains why, for example, only the first few results of a Google image search appear in a reasonable order. The pagerank is a highly unstable measure, showing frequent rank reversals after small adjustments of the jump parameter. While the failure of centrality indices to generalize to the rest of the network may at first seem counter-intuitive, it follows directly from the above definitions. Complex networks have heterogeneous topology. To the extent that the optimal measure depends on the network structure of the most important vertices, a measure which is optimal for such vertices is sub-optimal for the remainder of the network. == Degree centrality == Historically first and conceptually simplest is degree centrality, which is defined as the number of links incident upon a node (i.e., the number of ties that a node has). The degree can be interpreted in terms of the immediate risk of a node for catching whatever is flowing through the network (such as a virus, or some information). In the case of a directed network (where ties have direction), we usually define two separate measures of degree centrality, namely indegree and outdegree. Accordingly, indegree is a count of the number of ties directed to the node and outdegree is the number of ties that the node directs to others. When ties are associated to some positive aspects such as friendship or collaboration, indegree is often interpreted as a form of popularity, and outdegree as gregariousness. The degree centrality of a vertex v {\displaystyle v} , for a given graph G := ( V , E ) {\displaystyle G:=(V,E)} with \| V \| {\displaystyle \|V\|} vertices and \| E \| {\displaystyle \|E\|} edges, is defined as C D ( v ) = deg ⁡ ( v ) {\displaystyle C_{D}(v)=\deg(v)} Calculating degree centrality for all the nodes in a graph takes Θ ( V 2 ) {\displaystyle \Theta (V^{2})} in a dense adjacency matrix representation of the graph, and for edges takes Θ ( E ) {\displaystyle \Theta (E)} in a sparse matrix representation. The definition of centrality on the node level can be extended to the whole graph, in which case we are speaking of graph centralization. Let v ∗ {\displaystyle v} be the node with highest degree centrality in G {\displaystyle G} . Let X := ( Y , Z ) {\displaystyle X:=(Y,Z)} be the \| Y \| {\displaystyle \|Y\|} -node connected graph that maximizes the following quantity (with y ∗ {\displaystyle y} being the node with highest degree centrality in X {\displaystyle X} ): H = ∑ j = 1 \| Y \| [ C D ( y ∗ ) − C D ( y j ) ] {\displaystyle H=\sum _{j=1}^{\|Y\|}[C_{D}(y)-C_{D}(y_{j})]} Correspondingly, the degree centralization of the graph G {\displaystyle G} is as follows: C D ( G ) = ∑ i = 1 \| V \| [ C D ( v ∗ ) − C D ( v i ) ] H {\displaystyle C_{D}(G)={\frac {\sum _{i=1}^{\|V\|}[C_{D}(v)-C_{D}(v_{i})]}{H}}} The value of H {\displaystyle H} is maximized when the graph X {\displaystyle X} contains one central node to which all other nodes are connected (a star graph), and in this case H = ( n − 1 ) ⋅ ( ( n − 1 ) − 1 ) = n 2 − 3 n + 2. {\displaystyle H=(n-1)\cdot ((n-1)-1)=n^{2}-3n+2.} So, for any graph G := ( V , E ) , {\displaystyle G:=(V,E),} C D ( G ) = ∑ i = 1 \| V \| [ C D ( v ∗ ) − C D ( v i ) ] \| V \| 2 − 3 \| V \| + 2 {\displaystyle C_{D}(G)={\frac {\sum _{i=1}^{\|V\|}[C_{D}(v)-C_{D}(v_{i})]}{\|V\|^{2}-3\|V\|+2}}} Also, a new extensive global measure for degree centrality named Tendency to Make Hub (TMH) defines as follows: TMH = ∑ i = 1 \| V \| deg ⁡ ( v ) 2 ∑ i = 1 \| V \| deg ⁡ ( v ) {\displaystyle {\text{TMH}}={\frac {\sum _{i=1}^{\|V\|}\deg(v)^{2}}{\sum _{i=1}^{\|V\|}\deg(v)}}} where TMH increases by appearance of degree centrality in the network. == Closeness centrality == In a connected graph, the normalized closeness centrality (or closeness) of a node is the average length of the shortest path between the node and all other nodes in the graph. Thus the more central a node is, the closer it is to all other nodes. Closeness was defined by Alex Bavelas (1950) as the reciprocal of the farness, that is C B ( v ) = ( ∑ u d ( u , v ) ) − 1 {\textstyle C_{B}(v)=(\sum _{u}d(u,v))^{-1}} where d ( u , v ) {\displaystyle d(u,v)} is the distance between vertices u and v. However, when speaking of closeness centrality, people usually refer to its normalized form, given by the previous formula multiplied by N − 1 {\displaystyle N-1} , where N {\displaystyle N} is the number of nodes in the graph C ( v ) = N − 1 ∑ u d ( u , v ) . {\displaystyle C(v)={\frac {N-1}{\sum _{u}d(u,v)}}.} This normalisation allows comparisons between nodes of graphs of different sizes. For many graphs, there is a strong correlation between the inverse of closeness and the logarithm of degree, ( C ( v ) ) − 1 ≈ − α ln ⁡ ( k v ) + β {\displaystyle (C(v))^{-1}\approx -\alpha \ln(k_{v})+\beta } where k v {\displaystyle k_{v}} is the degree of vertex v while α and β are constants for each network. Taking distances from or to all other nodes is irrelevant in undirected graphs, whereas it can produce totally different results in directed graphs (e.g. a website can have a high closeness centrality from an outgoing link, but low closeness centrality from incoming links). === Harmonic centrality === In a (not necessarily connected) graph, the harmonic centrality reverses the sum and reciprocal operations in the definition of closeness centrality: H ( v ) = ∑ u \| u ≠ v 1 d ( u , v ) {\displaystyle H(v)=\sum _{u\|u\neq v}{\frac {1}{d(u,v)}}} where 1 / d ( u , v ) = 0 {\displaystyle 1/d(u,v)=0} if there is no path from u to v. Harmonic centrality can be normalized by dividing by N − 1 {\displaystyle N-1} , where N {\displaystyle N} is the number of nodes in the graph. Harmonic centrality was proposed by Marchiori and Latora (2000) and then independently by Dekker (2005), using the name "valued centrality," and by Rochat (2009). == Betweenness centrality == Betweenness is a centrality measure of a vertex within a graph (there is also edge betweenness, which is not discussed here). Betweenness centrality quantifies the number of times a node acts as a bridge along the shortest path between two other nodes. It was introduced as a measure for quantifying the control of a human on the communication between other humans in a social network by Linton Freeman. In his conception, vertices that have a high probability to occur on a randomly chosen shortest path between two randomly chosen vertices have a high betweenness. The betweenness of a vertex v {\displaystyle v} in a graph G := ( V , E ) {\displaystyle G:=(V,E)} with V {\displaystyle V} vertices is computed as follows: For each pair of vertices (s,t), compute the shortest paths between them. For each pair of vertices (s,t), determine the fraction of shortest paths that pass through the vertex in question (here, vertex v). Sum this fraction over all pairs of vertices (s,t). More compactly the betweenness can be represented as: C B ( v ) = ∑ s ≠ v ≠ t ∈ V σ s t ( v ) σ s t {\displaystyle C_{B}(v)=\sum _{s\neq v\neq t\in V}{\frac {\sigma _{st}(v)}{\sigma _{st}}}} where σ s t {\displaystyle \sigma _{st}} is total number of shortest paths from node s {\displaystyle s} to node t {\displaystyle t} and σ s t ( v ) {\displaystyle \sigma _{st}(v)} is the number of those paths that pass through v {\displaystyle v} . The betweenness may be normalised by dividing through the number of pairs of vertices not including v, which for directed graphs is ( n − 1 ) ( n − 2 ) {\displaystyle (n-1)(n-2)} and for undirected graphs is ( n − 1 ) ( n − 2 ) / 2 {\displaystyle (n-1)(n-2)/2} . For example, in an undirected star graph, the center vertex (which is contained in every possible shortest path) would have a betweenness of ( n − 1 ) ( n − 2 ) / 2 {\displaystyle (n-1)(n-2)/2} (1, if normalised) while the leaves (which are contained in no shortest paths) would have a betweenness of 0. From a calculation aspect, both betweenness and closeness centralities of all vertices in a graph involve calculating the shortest paths between all pairs of vertices on a graph, which requires O ( V 3 ) {\displaystyle O(V^{3})} time with the Floyd–Warshall algorithm. However, on sparse graphs, Johnson's algorithm may be more efficient, taking O ( \| V \| \| E \| + \| V \| 2 log ⁡ \| V \| ) {\displaystyle O(\|V\|\|E\|+\|V\|^{2}\log \|V\|)} time. In the case of unweighted graphs the calculations can be done with Brandes' algorithm which takes O ( \| V \| \| E \| ) {\displaystyle O(\|V\|\|E\|)} time. Normally, these algorithms assume that graphs are undirected and connected with the allowance of loops and multiple edges. When specifically dealing with network graphs, often graphs are without loops or multiple edges to maintain simple relationships (where edges represent connections between two people or vertices). In this case, using Brandes' algorithm will divide final centrality scores by 2 to account for each shortest path being counted twice. == Eigenvector centrality == Eigenvector centrality (also called eigencentrality) is a measure of the influence of a node in a network. It assigns relative scores to all nodes in the network based on the concept that connections to high-scoring nodes contribute more to the score of the node in question than equal connections to low-scoring nodes. Google's PageRank and the Katz centrality are variants of the eigenvector centrality. === Using the adjacency matrix to find eigenvector centrality === For a given graph G := ( V , E ) {\displaystyle G:=(V,E)} with \| V \| {\displaystyle \|V\|} number of vertices let A = ( a v , t ) {\displaystyle A=(a_{v,t})} be the adjacency matrix, i.e. a v , t = 1 {\displaystyle a_{v,t}=1} if vertex v {\displaystyle v} is linked to vertex t {\displaystyle t} , and a v , t = 0 {\displaystyle a_{v,t}=0} otherwise. The relative centrality score x v {\displaystyle x_{v}} of vertex v {\displaystyle v} can be defined as the nonnegative solution over the set of vertices v ∈ V {\displaystyle v\in V} to the equations: x v = 1 λ ∑ t ∈ M ( v ) x t = 1 λ ∑ t ∈ G a v , t x t {\displaystyle x_{v}={\frac {1}{\lambda }}\sum _{t\in M(v)}x_{t}={\frac {1}{\lambda }}\sum _{t\in G}a_{v,t}x_{t}} where M ( v ) {\displaystyle M(v)} is a set of the neighbors of v {\displaystyle v} and λ {\displaystyle \lambda } is a constant. With a small rearrangement this can be rewritten in vector notation as the eigenvector equation A x = λ x {\displaystyle \mathbf {Ax} ={\lambda }\mathbf {x} } . In general, there will be many different eigenvalues λ {\displaystyle \lambda } for which a non-zero eigenvector solution exists. Since the entries in the adjacency matrix are non-negative, there is a unique largest eigenvalue, which is real and positive, by the Perron–Frobenius theorem. This greatest eigenvalue results in the desired centrality measure. The v t h {\displaystyle v^{th}} component of the related eigenvector then gives the relative centrality score of the vertex v {\displaystyle v} in the network. The eigenvector is only defined up to a common factor, so only the ratios of the centralities of the vertices are well defined. To define an absolute score one must normalise the eigenvector, e.g., such that the sum over all vertices is 1 or the total number of vertices n. Power iteration is one of many eigenvalue algorithms that may be used to find this dominant eigenvector. Furthermore, this can be generalized so that the entries in A can be real numbers representing connection strengths, as in a stochastic matrix. == Katz centrality == Katz centrality is a generalization of degree centrality. Degree centrality measures the number of direct neighbors, and Katz centrality measures the number of all nodes that can be connected through a path, while the contributions of distant nodes are penalized. Mathematically, it is defined as x i = ∑ k = 1 ∞ ∑ j = 1 N α k ( A k ) j i {\displaystyle x_{i}=\sum _{k=1}^{\infty }\sum _{j=1}^{N}\alpha ^{k}(A^{k})_{ji}} where α {\displaystyle \alpha } is an attenuation factor in ( 0 , 1 ) {\displaystyle (0,1)} . Katz centrality can be viewed as a variant of eigenvector centrality. Another form of Katz centrality is x i = α ∑ j = 1 N a i j ( x j + 1 ) . {\displaystyle x_{i}=\alpha \sum _{j=1}^{N}a_{ij}(x_{j}+1).} Compared to the expression of eigenvector centrality, x j {\displaystyle x_{j}} is replaced by x j + 1. {\displaystyle x_{j}+1.} It is shown that the principal eigenvector (associated with the largest eigenvalue of A {\displaystyle A} , the adjacency matrix) is the limit of Katz centrality as α {\displaystyle \alpha } approaches 1 λ {\displaystyle {\tfrac {1}{\lambda }}} from below. == PageRank centrality == PageRank satisfies the following equation x i = α ∑ j a j i x j L ( j ) + 1 − α N , {\displaystyle x_{i}=\alpha \sum _{j}a_{ji}{\frac {x_{j}}{L(j)}}+{\frac {1-\alpha }{N}},} where L ( j ) = ∑ i a j i {\displaystyle L(j)=\sum _{i}a_{ji}} is the number of neighbors of node j {\displaystyle j} (or number of outbound links in a directed graph). Compared to eigenvector centrality and Katz centrality, one major difference is the scaling factor L ( j ) {\displaystyle L(j)} . Another difference between PageRank and eigenvector centrality is that the PageRank vector is a left hand eigenvector (note the factor a j i {\displaystyle a_{ji}} has indices reversed). == Percolation centrality == A slew of centrality measures exist to determine the ‘importance’ of a single node in a complex network. However, these measures quantify the importance of a node in purely topological terms, and the value of the node does not depend on the ‘state’ of the node in any way. It remains constant regardless of network dynamics. This is true even for the weighted betweenness measures. However, a node may very well be centrally located in terms of betweenness centrality or another centrality measure, but may not be ‘centrally’ located in the context of a network in which there is percolation. Percolation of a ‘contagion’ occurs in complex networks in a number of scenarios. For example, viral or bacterial infection can spread over social networks of people, known as contact networks. The spread of disease can also be considered at a higher level of abstraction, by contemplating a network of towns or population centres, connected by road, rail or air links. Computer viruses can spread over computer networks. Rumours or news about business offers and deals can also spread via social networks of people. In all of these scenarios, a ‘contagion’ spreads over the links of a complex network, altering the ‘states’ of the nodes as it spreads, either recoverable or otherwise. For example, in an epidemiological scenario, individuals go from ‘susceptible’ to ‘infected’ state as the infection spreads. The states the individual nodes can take in the above examples could be binary (such as received/not received a piece of news), discrete (susceptible/infected/recovered), or even continuous (such as the proportion of infected people in a town), as the contagion spreads. The common feature in all these scenarios is that the spread of contagion results in the change of node states in networks. Percolation centrality (PC) was proposed with this in mind, which specifically measures the importance of nodes in terms of aiding the percolation through the network. This measure was proposed by Piraveenan et al. Percolation centrality is defined for a given node, at a given time, as the proportion of ‘percolated paths’ that go through that node. A ‘percolated path’ is a shortest path between a pair of nodes, where the source node is percolated (e.g., infected). The target node can be percolated or non-percolated, or in a partially percolated state. P C t ( v ) = 1 N − 2 ∑ s ≠ v ≠ r σ s r ( v ) σ s r x t s ∑ [ x t i ] − x t v {\displaystyle PC^{t}(v)={\frac {1}{N-2}}\sum _{s\neq v\neq r}{\frac {\sigma _{sr}(v)}{\sigma _{sr}}}{\frac {{x^{t}}_{s}}{{\sum {[{x^{t}}_{i}}]}-{x^{t}}_{v}}}} where σ s r {\displaystyle \sigma _{sr}} is the total number of shortest paths from node s {\displaystyle s} to node r {\displaystyle r} and σ s r ( v ) {\displaystyle \sigma _{sr}(v)} is the number of those paths that pass through v {\displaystyle v} . The percolation state of the node i {\displaystyle i} at time t {\displaystyle t} is denoted by x t i {\displaystyle {x^{t}}_{i}} and two special cases are when x t i = 0 {\displaystyle {x^{t}}_{i}=0} which indicates a non-percolated state at time t {\displaystyle t} whereas when x t i = 1 {\displaystyle {x^{t}}_{i}=1} which indicates a fully percolated state at time t {\displaystyle t} . The values in between indicate partially percolated states ( e.g., in a network of townships, this would be the percentage of people infected in that town). The attached weights to the percolation paths depend on the percolation levels assigned to the source nodes, based on the premise that the higher the percolation level of a source node is, the more important are the paths that originate from that node. Nodes which lie on shortest paths originating from highly percolated nodes are therefore potentially more important to the percolation. The definition of PC may also be extended to include target node weights as well. Percolation centrality calculations run in O ( N M ) {\displaystyle O(NM)} time with an efficient implementation adopted from Brandes' fast algorithm and if the calculation needs to consider target nodes weights, the worst case time is O ( N 3 ) {\displaystyle O(N^{3})} . == Cross-clique centrality == Cross-clique centrality of a single node in a complex graph determines the connectivity of a node to different cliques. A node with high cross-clique connectivity facilitates the propagation of information or disease in a graph. Cliques are subgraphs in which every node is connected to every other node in the clique. The cross-clique connectivity of a node v {\displaystyle v} for a given graph G := ( V , E ) {\displaystyle G:=(V,E)} with \| V \| {\displaystyle \|V\|} vertices and \| E \| {\displaystyle \|E\|} edges, is defined as X ( v ) {\displaystyle X(v)} where X ( v ) {\displaystyle X(v)} is the number of cliques to which vertex v {\displaystyle v} belongs. This measure was used by Faghani in 2013 but was first proposed by Everett and Borgatti in 1998 where they called it clique-overlap centrality. == Freeman centralization == The centralization of any network is a measure of how central its most central node is in relation to how central all the other nodes are. Centralization measures then (a) calculate the sum in differences in centrality between the most central node in a network and all other nodes; and (b) divide this quantity by the theoretically largest such sum of differences in any network of the same size. Thus, every centrality measure can have its own centralization measure. Defined formally, if C x ( p i ) {\displaystyle C_{x}(p_{i})} is any centrality measure of point i {\displaystyle i} , if C x ( p ∗ ) {\displaystyle C_{x}(p_{})} is the largest such measure in the network, and if: max ∑ i = 1 N ( C x ( p ∗ ) − C x ( p i ) ) {\displaystyle \max \sum _{i=1}^{N}(C_{x}(p_{})-C_{x}(p_{i}))} is the largest sum of differences in point centrality C x {\displaystyle C_{x}} for any graph with the same number of nodes, then the centralization of the network is: C x = ∑ i = 1 N ( C x ( p ∗ ) − C x ( p i ) ) max ∑ i = 1 N ( C x ( p ∗ ) − C x ( p i ) ) . {\displaystyle C_{x}={\frac {\sum _{i=1}^{N}(C_{x}(p_{})-C_{x}(p_{i}))}{\max \sum _{i=1}^{N}(C_{x}(p_{*})-C_{x}(p_{i}))}}.} The concept is due to Linton Freeman. == Dissimilarity-based centrality measures == In order to obtain better results in the ranking of the nodes of a given network, Alvarez-Socorro et. al. used dissimilarity measures (specific to the theory of classification and data mining) to enrich the centrality measures in complex networks. This is illustrated with eigenvector centrality, calculating the centrality of each node through the solution of the eigenvalue problem W c = λ c {\displaystyle W\mathbf {c} =\lambda \mathbf {c} } where W i j = A i j D i j {\displaystyle W_{ij}=A_{ij}D_{ij}} (coordinate-to-coordinate product) and D i j {\displaystyle D_{ij}} is an arbitrary dissimilarity matrix, defined through a dissimilarity measure, e.g., Jaccard dissimilarity given by D i j = 1 − \| V + ( i ) ∩ V + ( j ) \| \| V + ( i ) ∪ V + ( j ) \| {\displaystyle D_{ij}=1-{\dfrac {\|V^{+}(i)\cap V^{+}(j)\|}{\|V^{+}(i)\cup V^{+}(j)\|}}} Where this measure permits us to quantify the topological contribution (which is why is called contribution centrality) of each node to the centrality of a given node, having more weight/relevance those nodes with greater dissimilarity, since these allow to the given node access to nodes that which themselves can not access directly. Is noteworthy that W {\displaystyle W} is non-negative because A {\displaystyle A} and D {\displaystyle D} are non-negative matrices, so we can use the Perron–Frobenius theorem to ensure that the above problem has a unique solution for λ = λmax with c non-negative, allowing us to infer the centrality of each node in the network. Therefore, the centrality of the i-th node is c i = 1 n ∑ j = 1 n W i j c j , i = 1 , ⋯ , n {\displaystyle c_{i}={\dfrac {1}{n}}\sum _{j=1}^{n}W_{ij}c_{j},\,\,\,\,\,\,i=1,\cdots ,n} where n {\displaystyle n} is the number of the nodes in the network. Several dissimilarity measures and networks were tested in obtaining improved results in the studied cases. == Centrality measures used in transportation networks == Transportation networks such as road networks and railway networks are studied extensively in transportation science and urban planning. A number of recent studies have focused on using centrality measures to analyze transportation networks. While many of these studies simply use generic centrality measures such as Betweenness Centrality, custom centrality measures have also been defined specifically for transportation network analysis. Prominent among them is Transportation Centrality. Transportation centrality measures the summation of the proportions of paths from pairs of nodes in a network which go through the node under consideration. In this respect it is similar to Betweenness Centrality. However, unlike Betweenness Centrality which considers only shortest paths, Transportation Centrality considers all possible paths between a pair of nodes. Therefore, Transportation Centrality is a generic version of Betweenness Centrality, and under certain conditions, it indeed reduces to Betweenness Centrality. Transportation Centrality of a given node v is defined as: T C ( v ) = 1 / ( ( N − 1 ) ( N − 2 ) ) Σ s ≠ v ≠ t Σ i ∈ P s , t v e − β C s , t i Σ j ∈ P s , t v e − β C s , t j {\displaystyle TC(v)=1/((N-1)(N-2))\Sigma _{s\neq v\neq t}{\frac {\Sigma _{i\in P_{s,t}^{v}}e^{-\beta C_{s,t}^{i}}}{\Sigma _{j\in P_{s,t}^{v}}e^{-\beta C_{s,t}^{j}}}}} == See also == Alpha centrality Core–periphery structure Distance in graphs == Notes and references == == Further reading == Koschützki, D.; Lehmann, K. A.; Peeters, L.; Richter, S.; Tenfelde-Podehl, D. and Zlotowski, O. (2005) Centrality Indices. In Brandes, U. and Erlebach, T. (Eds.) Network Analysis: Methodological Foundations, pp. 16–61, LNCS 3418, Springer-Verlag.
Wikipedia:Centralizer and normalizer#0	In mathematics, especially group theory, the centralizer (also called commutant) of a subset S in a group G is the set C G ⁡ ( S ) {\displaystyle \operatorname {C} _{G}(S)} of elements of G that commute with every element of S, or equivalently, the set of elements g ∈ G {\displaystyle g\in G} such that conjugation by g {\displaystyle g} leaves each element of S fixed. The normalizer of S in G is the set of elements N G ( S ) {\displaystyle \mathrm {N} _{G}(S)} of G that satisfy the weaker condition of leaving the set S ⊆ G {\displaystyle S\subseteq G} fixed under conjugation. The centralizer and normalizer of S are subgroups of G. Many techniques in group theory are based on studying the centralizers and normalizers of suitable subsets S. Suitably formulated, the definitions also apply to semigroups. In ring theory, the centralizer of a subset of a ring is defined with respect to the multiplication of the ring (a semigroup operation). The centralizer of a subset of a ring R is a subring of R. This article also deals with centralizers and normalizers in a Lie algebra. The idealizer in a semigroup or ring is another construction that is in the same vein as the centralizer and normalizer. == Definitions == === Group and semigroup === The centralizer of a subset S {\displaystyle S} of group (or semigroup) G is defined as C G ( S ) = { g ∈ G ∣ g s = s g for all s ∈ S } = { g ∈ G ∣ g s g − 1 = s for all s ∈ S } , {\displaystyle \mathrm {C} _{G}(S)=\left\{g\in G\mid gs=sg{\text{ for all }}s\in S\right\}=\left\{g\in G\mid gsg^{-1}=s{\text{ for all }}s\in S\right\},} where only the first definition applies to semigroups. If there is no ambiguity about the group in question, the G can be suppressed from the notation. When S = { a } {\displaystyle S=\{a\}} is a singleton set, we write CG(a) instead of CG({a}). Another less common notation for the centralizer is Z(a), which parallels the notation for the center. With this latter notation, one must be careful to avoid confusion between the center of a group G, Z(G), and the centralizer of an element g in G, Z(g). The normalizer of S in the group (or semigroup) G is defined as N G ( S ) = { g ∈ G ∣ g S = S g } = { g ∈ G ∣ g S g − 1 = S } , {\displaystyle \mathrm {N} _{G}(S)=\left\{g\in G\mid gS=Sg\right\}=\left\{g\in G\mid gSg^{-1}=S\right\},} where again only the first definition applies to semigroups. If the set S {\displaystyle S} is a subgroup of G {\displaystyle G} , then the normalizer N G ( S ) {\displaystyle N_{G}(S)} is the largest subgroup G ′ ⊆ G {\displaystyle G'\subseteq G} where S {\displaystyle S} is a normal subgroup of G ′ {\displaystyle G'} . The definitions of centralizer and normalizer are similar but not identical. If g is in the centralizer of S {\displaystyle S} and s is in S {\displaystyle S} , then it must be that gs = sg, but if g is in the normalizer, then gs = tg for some t in S {\displaystyle S} , with t possibly different from s. That is, elements of the centralizer of S {\displaystyle S} must commute pointwise with S {\displaystyle S} , but elements of the normalizer of S need only commute with S as a set. The same notational conventions mentioned above for centralizers also apply to normalizers. The normalizer should not be confused with the normal closure. Clearly C G ( S ) ⊆ N G ( S ) {\displaystyle C_{G}(S)\subseteq N_{G}(S)} and both are subgroups of G {\displaystyle G} . === Ring, algebra over a field, Lie ring, and Lie algebra === If R is a ring or an algebra over a field, and S {\displaystyle S} is a subset of R, then the centralizer of S {\displaystyle S} is exactly as defined for groups, with R in the place of G. If L {\displaystyle {\mathfrak {L}}} is a Lie algebra (or Lie ring) with Lie product [x, y], then the centralizer of a subset S {\displaystyle S} of L {\displaystyle {\mathfrak {L}}} is defined to be C L ( S ) = { x ∈ L ∣ [ x , s ] = 0 for all s ∈ S } . {\displaystyle \mathrm {C} _{\mathfrak {L}}(S)=\{x\in {\mathfrak {L}}\mid [x,s]=0{\text{ for all }}s\in S\}.} The definition of centralizers for Lie rings is linked to the definition for rings in the following way. If R is an associative ring, then R can be given the bracket product [x, y] = xy − yx. Of course then xy = yx if and only if [x, y] = 0. If we denote the set R with the bracket product as LR, then clearly the ring centralizer of S {\displaystyle S} in R is equal to the Lie ring centralizer of S {\displaystyle S} in LR. The normalizer of a subset S {\displaystyle S} of a Lie algebra (or Lie ring) L {\displaystyle {\mathfrak {L}}} is given by N L ( S ) = { x ∈ L ∣ [ x , s ] ∈ S for all s ∈ S } . {\displaystyle \mathrm {N} _{\mathfrak {L}}(S)=\{x\in {\mathfrak {L}}\mid [x,s]\in S{\text{ for all }}s\in S\}.} While this is the standard usage of the term "normalizer" in Lie algebra, this construction is actually the idealizer of the set S {\displaystyle S} in L {\displaystyle {\mathfrak {L}}} . If S {\displaystyle S} is an additive subgroup of L {\displaystyle {\mathfrak {L}}} , then N L ( S ) {\displaystyle \mathrm {N} _{\mathfrak {L}}(S)} is the largest Lie subring (or Lie subalgebra, as the case may be) in which S {\displaystyle S} is a Lie ideal. == Example == Consider the group G = S 3 = { [ 1 , 2 , 3 ] , [ 1 , 3 , 2 ] , [ 2 , 1 , 3 ] , [ 2 , 3 , 1 ] , [ 3 , 1 , 2 ] , [ 3 , 2 , 1 ] } {\displaystyle G=S_{3}=\{[1,2,3],[1,3,2],[2,1,3],[2,3,1],[3,1,2],[3,2,1]\}} (the symmetric group of permutations of 3 elements). Take a subset H {\displaystyle H} of the group G {\displaystyle G} : H = { [ 1 , 2 , 3 ] , [ 1 , 3 , 2 ] } . {\displaystyle H=\{[1,2,3],[1,3,2]\}.} Note that [ 1 , 2 , 3 ] {\displaystyle [1,2,3]} is the identity permutation in G {\displaystyle G} and retains the order of each element and [ 1 , 3 , 2 ] {\displaystyle [1,3,2]} is the permutation that fixes the first element and swaps the second and third element. The normalizer of H {\displaystyle H} with respect to the group G {\displaystyle G} are all elements of G {\displaystyle G} that yield the set H {\displaystyle H} (potentially permuted) when the element conjugates H {\displaystyle H} . Working out the example for each element of G {\displaystyle G} : [ 1 , 2 , 3 ] {\displaystyle [1,2,3]} when applied to H {\displaystyle H} : { [ 1 , 2 , 3 ] , [ 1 , 3 , 2 ] } = H {\displaystyle \{[1,2,3],[1,3,2]\}=H} ; therefore [ 1 , 2 , 3 ] {\displaystyle [1,2,3]} is in the normalizer N G ( H ) {\displaystyle N_{G}(H)} . [ 1 , 3 , 2 ] {\displaystyle [1,3,2]} when applied to H {\displaystyle H} : { [ 1 , 2 , 3 ] , [ 1 , 3 , 2 ] } = H {\displaystyle \{[1,2,3],[1,3,2]\}=H} ; therefore [ 1 , 3 , 2 ] {\displaystyle [1,3,2]} is in the normalizer N G ( H ) {\displaystyle N_{G}(H)} . [ 2 , 1 , 3 ] {\displaystyle [2,1,3]} when applied to H {\displaystyle H} : { [ 1 , 2 , 3 ] , [ 3 , 2 , 1 ] } ≠ H {\displaystyle \{[1,2,3],[3,2,1]\}\neq H} ; therefore [ 2 , 1 , 3 ] {\displaystyle [2,1,3]} is not in the normalizer N G ( H ) {\displaystyle N_{G}(H)} . [ 2 , 3 , 1 ] {\displaystyle [2,3,1]} when applied to H {\displaystyle H} : { [ 1 , 2 , 3 ] , [ 2 , 1 , 3 ] } ≠ H {\displaystyle \{[1,2,3],[2,1,3]\}\neq H} ; therefore [ 2 , 3 , 1 ] {\displaystyle [2,3,1]} is not in the normalizer N G ( H ) {\displaystyle N_{G}(H)} . [ 3 , 1 , 2 ] {\displaystyle [3,1,2]} when applied to H {\displaystyle H} : { [ 1 , 2 , 3 ] , [ 3 , 2 , 1 ] } ≠ H {\displaystyle \{[1,2,3],[3,2,1]\}\neq H} ; therefore [ 3 , 1 , 2 ] {\displaystyle [3,1,2]} is not in the normalizer N G ( H ) {\displaystyle N_{G}(H)} . [ 3 , 2 , 1 ] {\displaystyle [3,2,1]} when applied to H {\displaystyle H} : { [ 1 , 2 , 3 ] , [ 2 , 1 , 3 ] } ≠ H {\displaystyle \{[1,2,3],[2,1,3]\}\neq H} ; therefore [ 3 , 2 , 1 ] {\displaystyle [3,2,1]} is not in the normalizer N G ( H ) {\displaystyle N_{G}(H)} . Therefore, the normalizer N G ( H ) {\displaystyle N_{G}(H)} of H {\displaystyle H} in G {\displaystyle G} is { [ 1 , 2 , 3 ] , [ 1 , 3 , 2 ] } {\displaystyle \{[1,2,3],[1,3,2]\}} since both these group elements preserve the set H {\displaystyle H} under conjugation. The centralizer of the group G {\displaystyle G} is the set of elements that leave each element of H {\displaystyle H} unchanged by conjugation; that is, the set of elements that commutes with every element in H {\displaystyle H} . It's clear in this example that the only such element in S3 is H {\displaystyle H} itself ([1, 2, 3], [1, 3, 2]). == Properties == === Semigroups === Let S ′ {\displaystyle S'} denote the centralizer of S {\displaystyle S} in the semigroup A {\displaystyle A} ; i.e. S ′ = { x ∈ A ∣ s x = x s for every s ∈ S } . {\displaystyle S'=\{x\in A\mid sx=xs{\text{ for every }}s\in S\}.} Then S ′ {\displaystyle S'} forms a subsemigroup and S ′ = S ‴ = S ′′′′′ {\displaystyle S'=S'''=S'''''} ; i.e. a commutant is its own bicommutant. === Groups === Source: The centralizer and normalizer of S {\displaystyle S} are both subgroups of G. Clearly, CG(S) ⊆ NG(S). In fact, CG(S) is always a normal subgroup of NG(S), being the kernel of the homomorphism NG(S) → Bij(S) and the group NG(S)/CG(S) acts by conjugation as a group of bijections on S. E.g. the Weyl group of a compact Lie group G with a torus T is defined as W(G,T) = NG(T)/CG(T), and especially if the torus is maximal (i.e. CG(T) = T) it is a central tool in the theory of Lie groups. CG(CG(S)) contains S {\displaystyle S} , but CG(S) need not contain S {\displaystyle S} . Containment occurs exactly when S {\displaystyle S} is abelian. If H is a subgroup of G, then NG(H) contains H. If H is a subgroup of G, then the largest subgroup of G in which H is normal is the subgroup NG(H). If S {\displaystyle S} is a subset of G such that all elements of S commute with each other, then the largest subgroup of G whose center contains S {\displaystyle S} is the subgroup CG(S). A subgroup H of a group G is called a self-normalizing subgroup of G if NG(H) = H. The center of G is exactly CG(G) and G is an abelian group if and only if CG(G) = Z(G) = G. For singleton sets, CG(a) = NG(a). By symmetry, if S {\displaystyle S} and T are two subsets of G, T ⊆ CG(S) if and only if S ⊆ CG(T). For a subgroup H of group G, the N/C theorem states that the factor group NG(H)/CG(H) is isomorphic to a subgroup of Aut(H), the group of automorphisms of H. Since NG(G) = G and CG(G) = Z(G), the N/C theorem also implies that G/Z(G) is isomorphic to Inn(G), the subgroup of Aut(G) consisting of all inner automorphisms of G. If we define a group homomorphism T : G → Inn(G) by T(x)(g) = Tx(g) = xgx−1, then we can describe NG(S) and CG(S) in terms of the group action of Inn(G) on G: the stabilizer of S {\displaystyle S} in Inn(G) is T(NG(S)), and the subgroup of Inn(G) fixing S {\displaystyle S} pointwise is T(CG(S)). A subgroup H of a group G is said to be C-closed or self-bicommutant if H = CG(S) for some subset S ⊆ G. If so, then in fact, H = CG(CG(H)). === Rings and algebras over a field === Source: Centralizers in rings and in algebras over a field are subrings and subalgebras over a field, respectively; centralizers in Lie rings and in Lie algebras are Lie subrings and Lie subalgebras, respectively. The normalizer of S {\displaystyle S} in a Lie ring contains the centralizer of S {\displaystyle S} . CR(CR(S)) contains S {\displaystyle S} but is not necessarily equal. The double centralizer theorem deals with situations where equality occurs. If S {\displaystyle S} is an additive subgroup of a Lie ring A, then NA(S) is the largest Lie subring of A in which S {\displaystyle S} is a Lie ideal. If S {\displaystyle S} is a Lie subring of a Lie ring A, then S ⊆ NA(S). == See also == Commutator Multipliers and centralizers (Banach spaces) Stabilizer subgroup == Notes == == References == Isaacs, I. Martin (2009), Algebra: a graduate course, Graduate Studies in Mathematics, vol. 100 (reprint of the 1994 original ed.), Providence, RI: American Mathematical Society, doi:10.1090/gsm/100, ISBN 978-0-8218-4799-2, MR 2472787 Jacobson, Nathan (2009), Basic Algebra, vol. 1 (2 ed.), Dover Publications, ISBN 978-0-486-47189-1 Jacobson, Nathan (1979), Lie Algebras (republication of the 1962 original ed.), Dover Publications, ISBN 0-486-63832-4, MR 0559927
Wikipedia:Centre for Mathematical Sciences (Kerala)#0	Centre for Mathematical Sciences (CMS), with campuses at Thiruvananthapuram and Pala in Kerala, India, is a research level institution devoted to mathematics and other related disciplines like statistics, theoretical physics, computer and information sciences. The centre was incorporated in 1977 as a non-profit scientific research and training centre under the Travancore-Cochin Literary, Scientific and Charitable Societies Registration Act XII of 1955. The driving force behind the establishment of the centre was Prof. Aleyamma George, who had been Professor and Head of the Department of Statistics of University of Kerala. Since 2006, the centre is a Department of Science and Technology (India) (DST), Government of India, New Delhi Centre for Mathematical Sciences and is fully financed by DST, New Delhi. The centre is headed by a chairman, a position currently held by Dr. A Sukumaran Nair, a former vice-chancellor of Mahatma Gandhi University, Kottayam, and a Director a position now held by Dr. A.M. Mathai, emeritus professor of mathematics and statistics, McGill University, Canada. The activities of the Thiruvananthapuram Campus are coordinated by Dr. K. S. S. Nambooripad. The centre has started doing good work in its early years itself. At present, several research teams are operating in the centre like Astrophysics Research Group, Fractional Calculus Research Group, Special Functions Research Group, Statistical Distribution Theory Research Group, Geometrical Probability Research Group, Stochastic Process Research Group, and Discrete Mathematics in Chemistry Research Group. == History == Centre for Mathematical Sciences was established in 1977 in Trivandrum, Kerala, India. In 2002, the Pala Campus of the centre was established in a one-floor finished building donated by the Diocese of Palai in Kerala, India. In 2006, Hill Area Campus of the centre was established. The office, the library and most of the facilities are at the Pala Campus. Starting from 1985, Professor Dr. A.M. Mathai of McGill University, Canada is the director of the centre. In 2006-2007 the Department of Science and Technology (India) (DST) gave a development grant to the centre. Starting from December 2006, the centre is being developed as a DST Centre for Mathematical Sciences. DST has similar centres at three other locations in India. From 1977 to 2006, the activities at the centre were carried out by a group of researchers in Kerala, mostly retired professors, through voluntary service. Starting from 2007, DST created full-time salaried positions of three assistant professors, one full professor and one liaison officer. They are at Pala Campus. DST approved up to 15 junior research fellows (JRF) and one senior research fellow (SRF). They are the current PhD students at the Pala Campus. They would receive their PhD degrees from Mahatma Gandhi University, Kottayam, Kerala. == Publications == Publications Series (books, proceedings, collections of research papers, lecture notes etc.) Newsletter of two issues per year Modules Series (self-study books on basic topics; current number is 6) Mathematical Sciences for the General Public Series == Former chairmen == The following persons had held the position of chairman in the Centre for Mathematical Sciences. Mr. K.T. Chandy, former chairman, Kerala State Industrial Development Corporation. Prof. P.V. Sukhatme, statistician and nutritionist. Dr. K.R. Nair, former director of the Central Statistical Organization, Government of India and former director of the United Nations Asian Statistical Institute, Tokyo, Japan. Dr. G. Sankaranarayanan, former professor and head, Department of Mathematics and Statistics, Annamalai University. Dr. B.R. Bhat, a well-known probabilist. Dr. C.G. Ramachandran Nair, a former chairman of State Committee for Science, Technology and Environment (STEC) and a well-known chemist. Dr. A. Sukumaran Nair, former vice-chancellor of Mahatma Gandhi University, Kottayam, Kerala, India. == References ==
Wikipedia:Centrosymmetric matrix#0	In mathematics, especially in linear algebra and matrix theory, a centrosymmetric matrix is a matrix which is symmetric about its center. == Formal definition == An n × n matrix A = [Ai, j] is centrosymmetric when its entries satisfy A i , j = A n − i + 1 , n − j + 1 for all i , j ∈ { 1 , … , n } . {\displaystyle A_{i,\,j}=A_{n-i+1,\,n-j+1}\quad {\text{for all }}i,j\in \{1,\,\ldots ,\,n\}.} Alternatively, if J denotes the n × n exchange matrix with 1 on the antidiagonal and 0 elsewhere: J i , j = { 1 , i + j = n + 1 0 , i + j ≠ n + 1 {\displaystyle J_{i,\,j}={\begin{cases}1,&i+j=n+1\\0,&i+j\neq n+1\\\end{cases}}} then a matrix A is centrosymmetric if and only if AJ = JA. == Examples == All 2 × 2 centrosymmetric matrices have the form [ a b b a ] . {\displaystyle {\begin{bmatrix}a&b\\b&a\end{bmatrix}}.} All 3 × 3 centrosymmetric matrices have the form [ a b c d e d c b a ] . {\displaystyle {\begin{bmatrix}a&b&c\\d&e&d\\c&b&a\end{bmatrix}}.} Symmetric Toeplitz matrices are centrosymmetric. == Algebraic structure and properties == If A and B are n × n centrosymmetric matrices over a field F, then so are A + B and cA for any c in F. Moreover, the matrix product AB is centrosymmetric, since JAB = AJB = ABJ. Since the identity matrix is also centrosymmetric, it follows that the set of n × n centrosymmetric matrices over F forms a subalgebra of the associative algebra of all n × n matrices. If A is a centrosymmetric matrix with an m-dimensional eigenbasis, then its m eigenvectors can each be chosen so that they satisfy either x = J x or x = − J x where J is the exchange matrix. If A is a centrosymmetric matrix with distinct eigenvalues, then the matrices that commute with A must be centrosymmetric. The maximum number of unique elements in an m × m centrosymmetric matrix is m 2 + m mod 2 2 . {\displaystyle {\frac {m^{2}+m{\bmod {2}}}{2}}.} == Related structures == An n × n matrix A is said to be skew-centrosymmetric if its entries satisfy A i , j = − A n − i + 1 , n − j + 1 for all i , j ∈ { 1 , … , n } . {\displaystyle A_{i,\,j}=-A_{n-i+1,\,n-j+1}\quad {\text{for all }}i,j\in \{1,\,\ldots ,\,n\}.} Equivalently, A is skew-centrosymmetric if AJ = −JA, where J is the exchange matrix defined previously. The centrosymmetric relation AJ = JA lends itself to a natural generalization, where J is replaced with an involutory matrix K (i.e., K2 = I ) or, more generally, a matrix K satisfying Km = I for an integer m > 1. The inverse problem for the commutation relation AK = KA of identifying all involutory K that commute with a fixed matrix A has also been studied. Symmetric centrosymmetric matrices are sometimes called bisymmetric matrices. When the ground field is the real numbers, it has been shown that bisymmetric matrices are precisely those symmetric matrices whose eigenvalues remain the same aside from possible sign changes following pre- or post-multiplication by the exchange matrix. A similar result holds for Hermitian centrosymmetric and skew-centrosymmetric matrices. == References == == Further reading == Muir, Thomas (1960). A Treatise on the Theory of Determinants. Dover. p. 19. ISBN 0-486-60670-8. {{cite book}}: ISBN / Date incompatibility (help) Weaver, James R. (1985). "Centrosymmetric (cross-symmetric) matrices, their basic properties, eigenvalues, and eigenvectors". American Mathematical Monthly. 92 (10): 711–717. doi:10.2307/2323222. JSTOR 2323222. == External links == Centrosymmetric matrix on MathWorld.
Wikipedia:Ceyuan haijing#0	Mathematics emerged independently in China by the 11th century BCE. The Chinese independently developed a real number system that includes significantly large and negative numbers, more than one numeral system (binary and decimal), algebra, geometry, number theory and trigonometry. Since the Han dynasty, as diophantine approximation being a prominent numerical method, the Chinese made substantial progress on polynomial evaluation. Algorithms like regula falsi and expressions like simple continued fractions are widely used and have been well-documented ever since. They deliberately find the principal nth root of positive numbers and the roots of equations. The major texts from the period, The Nine Chapters on the Mathematical Art and the Book on Numbers and Computation gave detailed processes for solving various mathematical problems in daily life. All procedures were computed using a counting board in both texts, and they included inverse elements as well as Euclidean divisions. The texts provide procedures similar to that of Gaussian elimination and Horner's method for linear algebra. The achievement of Chinese algebra reached a zenith in the 13th century during the Yuan dynasty with the development of tian yuan shu. As a result of obvious linguistic and geographic barriers, as well as content, Chinese mathematics and the mathematics of the ancient Mediterranean world are presumed to have developed more or less independently up to the time when The Nine Chapters on the Mathematical Art reached its final form, while the Book on Numbers and Computation and Huainanzi are roughly contemporary with classical Greek mathematics. Some exchange of ideas across Asia through known cultural exchanges from at least Roman times is likely. Frequently, elements of the mathematics of early societies correspond to rudimentary results found later in branches of modern mathematics such as geometry or number theory. The Pythagorean theorem for example, has been attested to the time of the Duke of Zhou. Knowledge of Pascal's triangle has also been shown to have existed in China centuries before Pascal, such as the Song-era polymath Shen Kuo. == Pre-imperial era == Shang dynasty (1600–1050 BC). One of the oldest surviving mathematical works is the I Ching, which greatly influenced written literature during the Zhou dynasty (1050–256 BC). For mathematics, the book included a sophisticated use of hexagrams. Leibniz pointed out, the I Ching (Yi Jing) contained elements of binary numbers. Since the Shang period, the Chinese had already fully developed a decimal system. Since early times, Chinese understood basic arithmetic (which dominated far eastern history), algebra, equations, and negative numbers with counting rods. Although the Chinese were more focused on arithmetic and advanced algebra for astronomical uses, they were also the first to develop negative numbers, algebraic geometry, and the usage of decimals. Math was one of the Six Arts students were required to master during the Zhou dynasty (1122–256 BCE). Learning them all perfectly was required to be a perfect gentleman, comparable to the concept of a "renaissance man". Six Arts have their roots in the Confucian philosophy. The oldest existent work on geometry in China comes from the philosophical Mohist canon c. 330 BCE, compiled by the followers of Mozi (470–390 BCE). The Mo Jing described various aspects of many fields associated with physical science, and provided a small wealth of information on mathematics as well. It provided an 'atomic' definition of the geometric point, stating that a line is separated into parts, and the part which has no remaining parts (i.e. cannot be divided into smaller parts) and thus forms the extreme end of a line is a point. Much like Euclid's first and third definitions and Plato's 'beginning of a line', the Mo Jing stated that "a point may stand at the end (of a line) or at its beginning like a head-presentation in childbirth. (As to its invisibility) there is nothing similar to it." Similar to the atomists of Democritus, the Mo Jing stated that a point is the smallest unit, and cannot be cut in half, since 'nothing' cannot be halved." It stated that two lines of equal length will always finish at the same place," while providing definitions for the comparison of lengths and for parallels," along with principles of space and bounded space. It also described the fact that planes without the quality of thickness cannot be piled up since they cannot mutually touch. The book provided word recognition for circumference, diameter, and radius, along with the definition of volume. The history of mathematical development lacks some evidence. There are still debates about certain mathematical classics. For example, the Zhoubi Suanjing dates around 1200–1000 BC, yet many scholars believed it was written between 300 and 250 BCE. The Zhoubi Suanjing contains an in-depth proof of the Gougu Theorem (a special case of the Pythagorean theorem), but focuses more on astronomical calculations. However, the recent archaeological discovery of the Tsinghua Bamboo Slips, dated c. 305 BCE, has revealed some aspects of pre-Qin mathematics, such as the first known decimal multiplication table. The abacus was first mentioned in the second century BC, alongside 'calculation with rods' (suan zi) in which small bamboo sticks are placed in successive squares of a checkerboard. == Qin dynasty == Not much is known about Qin dynasty mathematics, or before, due to the burning of books and burying of scholars, circa 213–210 BC. Knowledge of this period can be determined from civil projects and historical evidence. The Qin dynasty created a standard system of weights. Civil projects of the Qin dynasty were significant feats of human engineering. Emperor Qin Shi Huang ordered many men to build large, life-sized statues for the palace tomb along with other temples and shrines, and the shape of the tomb was designed with geometric skills of architecture. It is certain that one of the greatest feats of human history, the Great Wall of China, required many mathematical techniques. All Qin dynasty buildings and grand projects used advanced computation formulas for volume, area and proportion. Qin bamboo cash purchased at the antiquarian market of Hong Kong by the Yuelu Academy, according to the preliminary reports, contains the earliest epigraphic sample of a mathematical treatise. == Han dynasty == In the Han dynasty, numbers were developed into a place value decimal system and used on a counting board with a set of counting rods called rod calculus, consisting of only nine symbols with a blank space on the counting board representing zero. Negative numbers and fractions were also incorporated into solutions of the great mathematical texts of the period. The mathematical texts of the time, the Book on Numbers and Computation and Jiuzhang suanshu solved basic arithmetic problems such as addition, subtraction, multiplication and division. Furthermore, they gave the processes for square and cubed root extraction, which eventually was applied to solving quadratic equations up to the third order. Both texts also made substantial progress in Linear Algebra, namely solving systems of equations with multiple unknowns. The value of pi is taken to be equal to three in both texts. However, the mathematicians Liu Xin (d. 23) and Zhang Heng (78–139) gave more accurate approximations for pi than Chinese of previous centuries had used. Mathematics was developed to solve practical problems in the time such as division of land or problems related to division of payment. The Chinese did not focus on theoretical proofs based on geometry or algebra in the modern sense of proving equations to find area or volume. The Book of Computations and The Nine Chapters on the Mathematical Art provide numerous practical examples that would be used in daily life. === Book on Numbers and Computation === The Book on Numbers and Computation is approximately seven thousand characters in length, written on 190 bamboo strips. It was discovered together with other writings in 1984 when archaeologists opened a tomb at Zhangjiashan in Hubei province. From documentary evidence this tomb is known to have been closed in 186 BC, early in the Western Han dynasty. While its relationship to the Nine Chapters is still under discussion by scholars, some of its contents are clearly paralleled there. The text of the Suan shu shu is however much less systematic than the Nine Chapters, and appears to consist of a number of more or less independent short sections of text drawn from a number of sources. The Book of Computations contains many perquisites to problems that would be expanded upon in The Nine Chapters on the Mathematical Art. An example of the elementary mathematics in the Suàn shù shū, the square root is approximated by using false position method which says to "combine the excess and deficiency as the divisor; (taking) the deficiency numerator multiplied by the excess denominator and the excess numerator times the deficiency denominator, combine them as the dividend." Furthermore, The Book of Computations solves systems of two equations and two unknowns using the same false position method. === The Nine Chapters on the Mathematical Art === The Nine Chapters on the Mathematical Art dates archeologically to 179 CE, though it is traditionally dated to 1000 BCE, but it was written perhaps as early as 300–200 BCE. Although the author(s) are unknown, they made a major contribution in the eastern world. Problems are set up with questions immediately followed by answers and procedure. There are no formal mathematical proofs within the text, just a step-by-step procedure. The commentary of Liu Hui provided geometrical and algebraic proofs to the problems given within the text. The Nine Chapters on the Mathematical Art was one of the most influential of all Chinese mathematical books and it is composed of 246 problems. It was later incorporated into The Ten Computational Canons, which became the core of mathematical education in later centuries. This book includes 246 problems on surveying, agriculture, partnerships, engineering, taxation, calculation, the solution of equations, and the properties of right triangles. The Nine Chapters made significant additions to solving quadratic equations in a way similar to Horner's method. It also made advanced contributions to fangcheng, or what is now known as linear algebra. Chapter seven solves system of linear equations with two unknowns using the false position method, similar to The Book of Computations. Chapter eight deals with solving determinate and indeterminate simultaneous linear equations using positive and negative numbers, with one problem dealing with solving four equations in five unknowns. The Nine Chapters solves systems of equations using methods similar to the modern Gaussian elimination and back substitution. The version of The Nine Chapters that has served as the foundation for modern renditions was a result of the efforts of the scholar Dai Zhen. Transcribing the problems directly from Yongle Encyclopedia, he then proceeded to make revisions to the original text, along with the inclusion his own notes explaining his reasoning behind the alterations. His finished work would be first published in 1774, but a new revision would be published in 1776 to correct various errors as well as include a version of The Nine Chapters from the Southern Song that contained the commentaries of Lui Hui and Li Chunfeng. The final version of Dai Zhen's work would come in 1777, titled Ripple Pavilion, with this final rendition being widely distributed and coming to serve as the standard for modern versions of The Nine Chapters. However, this version has come under scrutiny from Guo Shuchen, alleging that the edited version still contains numerous errors and that not all of the original amendments were done by Dai Zhen himself. === Calculation of pi === Problems in The Nine Chapters on the Mathematical Art take pi to be equal to three in calculating problems related to circles and spheres, such as spherical surface area. There is no explicit formula given within the text for the calculation of pi to be three, but it is used throughout the problems of both The Nine Chapters on the Mathematical Art and the Artificer's Record, which was produced in the same time period. Historians believe that this figure of pi was calculated using the 3:1 relationship between the circumference and diameter of a circle. Some Han mathematicians attempted to improve this number, such as Liu Xin, who is believed to have estimated pi to be 3.154. Later, Liu Hui attempted to improve the calculation by calculating pi to be 3.141024. Liu calculated this number by using polygons inside a hexagon as a lower limit compared to a circle. Zu Chongzhi later discovered the calculation of pi to be 3.1415926 < π < 3.1415927 by using polygons with 24,576 sides. This calculation would be discovered in Europe during the 16th century. There is no explicit method or record of how he calculated this estimate. === Division and root extraction === Basic arithmetic processes such as addition, subtraction, multiplication and division were present before the Han dynasty. The Nine Chapters on the Mathematical Art take these basic operations for granted and simply instruct the reader to perform them. Han mathematicians calculated square and cube roots in a similar manner as division, and problems on division and root extraction both occur in Chapter Four of The Nine Chapters on the Mathematical Art. Calculating the square and cube roots of numbers is done through successive approximation, the same as division, and often uses similar terms such as dividend (shi) and divisor (fa) throughout the process. This process of successive approximation was then extended to solving quadratics of the second and third order, such as x 2 + a = b {\displaystyle x^{2}+a=b} , using a method similar to Horner's method. The method was not extended to solve quadratics of the nth order during the Han dynasty; however, this method was eventually used to solve these equations. === Linear algebra === The Book of Computations is the first known text to solve systems of equations with two unknowns. There are a total of three sets of problems within The Book of Computations involving solving systems of equations with the false position method, which again are put into practical terms. Chapter Seven of The Nine Chapters on the Mathematical Art also deals with solving a system of two equations with two unknowns with the false position method. To solve for the greater of the two unknowns, the false position method instructs the reader to cross-multiply the minor terms or zi (which are the values given for the excess and deficit) with the major terms mu. To solve for the lesser of the two unknowns, simply add the minor terms together. Chapter Eight of The Nine Chapters on the Mathematical Art deals with solving infinite equations with infinite unknowns. This process is referred to as the "fangcheng procedure" throughout the chapter. Many historians chose to leave the term fangcheng untranslated due to conflicting evidence of what the term means. Many historians translate the word to linear algebra today. In this chapter, the process of Gaussian elimination and back-substitution are used to solve systems of equations with many unknowns. Problems were done on a counting board and included the use of negative numbers as well as fractions. The counting board was effectively a matrix, where the top line is the first variable of one equation and the bottom was the last. === Liu Hui's commentary on The Nine Chapters on the Mathematical Art === Liu Hui's commentary on The Nine Chapters on the Mathematical Art is the earliest edition of the original text available. Hui is believed by most to be a mathematician shortly after the Han dynasty. Within his commentary, Hui qualified and proved some of the problems from either an algebraic or geometrical standpoint. For instance, throughout The Nine Chapters on the Mathematical Art, the value of pi is taken to be equal to three in problems regarding circles or spheres. In his commentary, Liu Hui finds a more accurate estimation of pi using the method of exhaustion. The method involves creating successive polygons within a circle so that eventually the area of a higher-order polygon will be identical to that of the circle. From this method, Liu Hui asserted that the value of pi is about 3.14. Liu Hui also presented a geometric proof of square and cubed root extraction similar to the Greek method, which involved cutting a square or cube in any line or section and determining the square root through symmetry of the remaining rectangles. == Three Kingdoms, Jin, and Sixteen Kingdoms == In the third century Liu Hui wrote his commentary on the Nine Chapters and also wrote Haidao Suanjing which dealt with using Pythagorean theorem (already known by the 9 chapters), and triple, quadruple triangulation for surveying; his accomplishment in the mathematical surveying exceeded those accomplished in the west by a millennium. He was the first Chinese mathematician to calculate π=3.1416 with his π algorithm. He discovered the usage of Cavalieri's principle to find an accurate formula for the volume of a cylinder, and also developed elements of the infinitesimal calculus during the 3rd century CE. In the fourth century, another influential mathematician named Zu Chongzhi, introduced the Da Ming Li. This calendar was specifically calculated to predict many cosmological cycles that will occur in a period of time. Very little is really known about his life. Today, the only sources are found in Book of Sui, we now know that Zu Chongzhi was one of the generations of mathematicians. He used Liu Hui's pi-algorithm applied to a 12288-gon and obtained a value of pi to 7 accurate decimal places (between 3.1415926 and 3.1415927), which would remain the most accurate approximation of π available for the next 900 years. He also applied He Chengtian's interpolation for approximating irrational number with fraction in his astronomy and mathematical works, he obtained 355 113 {\displaystyle {\tfrac {355}{113}}} as a good fraction approximate for pi; Yoshio Mikami commented that neither the Greeks, nor the Hindus nor Arabs knew about this fraction approximation to pi, not until the Dutch mathematician Adrian Anthoniszoom rediscovered it in 1585, "the Chinese had therefore been possessed of this the most extraordinary of all fractional values over a whole millennium earlier than Europe". Along with his son, Zu Geng, Zu Chongzhi applied the Cavalieri's principle to find an accurate solution for calculating the volume of the sphere. Besides containing formulas for the volume of the sphere, his book also included formulas of cubic equations and the accurate value of pi. His work, Zhui Shu was discarded out of the syllabus of mathematics during the Song dynasty and lost. Many believed that Zhui Shu contains the formulas and methods for linear, matrix algebra, algorithm for calculating the value of π, formula for the volume of the sphere. The text should also associate with his astronomical methods of interpolation, which would contain knowledge, similar to our modern mathematics. A mathematical manual called Sunzi mathematical classic dated between 200 and 400 CE contained the most detailed step by step description of multiplication and division algorithm with counting rods. Intriguingly, Sunzi may have influenced the development of place-value systems and place-value systems and the associated Galley division in the West. European sources learned place-value techniques in the 13th century, from a Latin translation an early-9th-century work by Al-Khwarizmi. Khwarizmi's presentation is almost identical to the division algorithm in Sunzi, even regarding stylistic matters (for example, using blank spaces to represent trailing zeros); the similarity suggests that the results may not have been an independent discovery. Islamic commentators on Al-Khwarizmi's work believed that it primarily summarized Hindu knowledge; Al-Khwarizmi's failure to cite his sources makes it difficult to determine whether those sources had in turn learned the procedure from China. In the fifth century the manual called "Zhang Qiujian suanjing" discussed linear and quadratic equations. By this point the Chinese had the concept of negative numbers. == Tang dynasty == By the Tang dynasty study of mathematics was fairly standard in the great schools. The Ten Computational Canons was a collection of ten Chinese mathematical works, compiled by early Tang dynasty mathematician Li Chunfeng (李淳風 602–670), as the official mathematical texts for imperial examinations in mathematics. The Sui dynasty and Tang dynasty ran the "School of Computations". Wang Xiaotong was a great mathematician in the beginning of the Tang dynasty, and he wrote a book: Jigu Suanjing (Continuation of Ancient Mathematics), where numerical solutions which general cubic equations appear for the first time. The Tibetans obtained their first knowledge of mathematics (arithmetic) from China during the reign of Nam-ri srong btsan, who died in 630. The table of sines by the Indian mathematician, Aryabhata, were translated into the Chinese mathematical book of the Kaiyuan Zhanjing, compiled in 718 AD during the Tang dynasty. Although the Chinese excelled in other fields of mathematics such as solid geometry, binomial theorem, and complex algebraic formulas, early forms of trigonometry were not as widely appreciated as in the contemporary Indian and Islamic mathematics. Yi Xing, the mathematician and Buddhist monk was credited for calculating the tangent table. Instead, the early Chinese used an empirical substitute known as chong cha, while practical use of plane trigonometry in using the sine, the tangent, and the secant were known. Yi Xing was famed for his genius, and was known to have calculated the number of possible positions on a go board game (though without a symbol for zero he had difficulties expressing the number). == Song and Yuan dynasties == Northern Song dynasty mathematician Jia Xian developed an additive multiplicative method for extraction of square root and cubic root which implemented the "Horner" rule. Four outstanding mathematicians arose during the Song dynasty and Yuan dynasty, particularly in the twelfth and thirteenth centuries: Yang Hui, Qin Jiushao, Li Zhi (Li Ye), and Zhu Shijie. Yang Hui, Qin Jiushao, Zhu Shijie all used the Horner-Ruffini method six hundred years earlier to solve certain types of simultaneous equations, roots, quadratic, cubic, and quartic equations. Yang Hui was also the first person in history to discover and prove "Pascal's Triangle", along with its binomial proof (although the earliest mention of the Pascal's triangle in China exists before the eleventh century AD). Li Zhi on the other hand, investigated on a form of algebraic geometry based on tiān yuán shù. His book; Ceyuan haijing revolutionized the idea of inscribing a circle into triangles, by turning this geometry problem by algebra instead of the traditional method of using Pythagorean theorem. Guo Shoujing of this era also worked on spherical trigonometry for precise astronomical calculations. At this point of mathematical history, a lot of modern western mathematics were already discovered by Chinese mathematicians. Things grew quiet for a time until the thirteenth century Renaissance of Chinese math. This saw Chinese mathematicians solving equations with methods Europe would not know until the eighteenth century. The high point of this era came with Zhu Shijie's two books Suanxue qimeng and the Jade Mirror of the Four Unknowns. In one case he reportedly gave a method equivalent to Gauss's pivotal condensation. Qin Jiushao (c. 1202 – 1261) was the first to introduce the zero symbol into Chinese mathematics." Before this innovation, blank spaces were used instead of zeros in the system of counting rods. One of the most important contribution of Qin Jiushao was his method of solving high order numerical equations. Referring to Qin's solution of a 4th order equation, Yoshio Mikami put it: "Who can deny the fact of Horner's illustrious process being used in China at least nearly six long centuries earlier than in Europe?" Qin also solved a 10th order equation. Pascal's triangle was first illustrated in China by Yang Hui in his book Xiangjie Jiuzhang Suanfa (詳解九章算法), although it was described earlier around 1100 by Jia Xian. Although the Introduction to Computational Studies (算學啓蒙) written by Zhu Shijie (fl. 13th century) in 1299 contained nothing new in Chinese algebra, it had a great impact on the development of Japanese mathematics. === Algebra === ==== Ceyuan haijing ==== Ceyuan haijing (Chinese: 測圓海鏡; pinyin: Cèyuán Hǎijìng), or Sea-Mirror of the Circle Measurements, is a collection of 692 formula and 170 problems related to inscribed circle in a triangle, written by Li Zhi (or Li Ye) (1192–1272 AD). He used Tian yuan shu to convert intricated geometry problems into pure algebra problems. He then used fan fa, or Horner's method, to solve equations of degree as high as six, although he did not describe his method of solving equations. "Li Chih (or Li Yeh, 1192–1279), a mathematician of Peking who was offered a government post by Khublai Khan in 1206, but politely found an excuse to decline it. His Ts'e-yuan hai-ching (Sea-Mirror of the Circle Measurements) includes 170 problems dealing with[...]some of the problems leading to polynomial equations of sixth degree. Although he did not describe his method of solution of equations, it appears that it was not very different from that used by Chu Shih-chieh and Horner. Others who used the Horner method were Ch'in Chiu-shao (ca. 1202 – ca.1261) and Yang Hui (fl. ca. 1261–1275). ==== Jade Mirror of the Four Unknowns ==== The Jade Mirror of the Four Unknowns was written by Zhu Shijie in 1303 AD and marks the peak in the development of Chinese algebra. The four elements, called heaven, earth, man and matter, represented the four unknown quantities in his algebraic equations. It deals with simultaneous equations and with equations of degrees as high as fourteen. The author uses the method of fan fa, today called Horner's method, to solve these equations. There are many summation series equations given without proof in the Mirror. A few of the summation series are: 1 2 + 2 2 + 3 2 + ⋯ + n 2 = n ( n + 1 ) ( 2 n + 1 ) 3 ! {\displaystyle 1^{2}+2^{2}+3^{2}+\cdots +n^{2}={n(n+1)(2n+1) \over 3!}} 1 + 8 + 30 + 80 + ⋯ + n 2 ( n + 1 ) ( n + 2 ) 3 ! = n ( n + 1 ) ( n + 2 ) ( n + 3 ) ( 4 n + 1 ) 5 ! {\displaystyle 1+8+30+80+\cdots +{n^{2}(n+1)(n+2) \over 3!}={n(n+1)(n+2)(n+3)(4n+1) \over 5!}} ==== Mathematical Treatise in Nine Sections ==== The Mathematical Treatise in Nine Sections, was written by the wealthy governor and minister Ch'in Chiu-shao (c. 1202 – c. 1261) and with the invention of a method of solving simultaneous congruences, it marks the high point in Chinese indeterminate analysis. ==== Magic squares and magic circles ==== The earliest known magic squares of order greater than three are attributed to Yang Hui (fl. ca. 1261–1275), who worked with magic squares of order as high as ten. "The same "Horner" device was used by Yang Hui, about whose life almost nothing is known and who work has survived only in part. Among his contributions that are extant are the earliest Chinese magic squares of order greater than three, including two each of orders four through eight and one each of orders nine and ten." He also worked with magic circle. === Trigonometry === The embryonic state of trigonometry in China slowly began to change and advance during the Song dynasty (960–1279), where Chinese mathematicians began to express greater emphasis for the need of spherical trigonometry in calendar science and astronomical calculations. The polymath and official Shen Kuo (1031–1095) used trigonometric functions to solve mathematical problems of chords and arcs. Joseph W. Dauben notes that in Shen's "technique of intersecting circles" formula, he creates an approximation of the arc of a circle s by s = c + 2v2/d, where d is the diameter, v is the versine, c is the length of the chord c subtending the arc. Sal Restivo writes that Shen's work in the lengths of arcs of circles provided the basis for spherical trigonometry developed in the 13th century by the mathematician and astronomer Guo Shoujing (1231–1316). Gauchet and Needham state Guo used spherical trigonometry in his calculations to improve the Chinese calendar and astronomy. Along with a later 17th-century Chinese illustration of Guo's mathematical proofs, Needham writes: Guo used a quadrangular spherical pyramid, the basal quadrilateral of which consisted of one equatorial and one ecliptic arc, together with two meridian arcs, one of which passed through the summer solstice point...By such methods he was able to obtain the du lü (degrees of equator corresponding to degrees of ecliptic), the ji cha (values of chords for given ecliptic arcs), and the cha lü (difference between chords of arcs differing by 1 degree). Despite the achievements of Shen and Guo's work in trigonometry, another substantial work in Chinese trigonometry would not be published again until 1607, with the dual publication of Euclid's Elements by Chinese official and astronomer Xu Guangqi (1562–1633) and the Italian Jesuit Matteo Ricci (1552–1610). == Ming dynasty == After the overthrow of the Yuan dynasty, China became suspicious of Mongol-favored knowledge. The court turned away from math and physics in favor of botany and pharmacology. Imperial examinations included little mathematics, and what little they included ignored recent developments. Martzloff writes: At the end of the 16th century, Chinese autochthonous mathematics known by the Chinese themselves amounted to almost nothing, little more than calculation on the abacus, whilst in the 17th and 18th centuries nothing could be paralleled with the revolutionary progress in the theatre of European science. Moreover, at this same period, no one could report what had taken place in the more distant past, since the Chinese themselves only had a fragmentary knowledge of that. One should not forget that, in China itself, autochthonous mathematics was not rediscovered on a large scale prior to the last quarter of the 18th century. Correspondingly, scholars paid less attention to mathematics; preeminent mathematicians such as Gu Yingxiang and Tang Shunzhi appear to have been ignorant of the 'increase multiply' method. Without oral interlocutors to explicate them, the texts rapidly became incomprehensible; worse yet, most problems could be solved with more elementary methods. To the average scholar, then, tianyuan seemed numerology. When Wu Jing collated all the mathematical works of previous dynasties into The Annotations of Calculations in the Nine Chapters on the Mathematical Art, he omitted Tian yuan shu and the increase multiply method. Instead, mathematical progress became focused on computational tools. In 15 century, abacus came into its suan pan form. Easy to use and carry, both fast and accurate, it rapidly overtook rod calculus as the preferred form of computation. Zhusuan, the arithmetic calculation through abacus, inspired multiple new works. Suanfa Tongzong (General Source of Computational Methods), a 17-volume work published in 1592 by Cheng Dawei, remained in use for over 300 years. Zhu Zaiyu, Prince of Zheng used 81 position abacus to calculate the square root and cubic root of 2 to 25 figure accuracy, a precision that enabled his development of the equal-temperament system. In the late 16th century, Matteo Ricci decided to published Western scientific works in order to establish a position at the Imperial Court. With the assistance of Xu Guangqi, he was able to translate Euclid's Elements using the same techniques used to teach classical Buddhist texts. Other missionaries followed in his example, translating Western works on special functions (trigonometry and logarithms) that were neglected in the Chinese tradition. However, contemporary scholars found the emphasis on proofs — as opposed to solved problems — baffling, and most continued to work from classical texts alone. == Qing dynasty == Under the Kangxi Emperor, who learned Western mathematics from the Jesuits and was open to outside knowledge and ideas, Chinese mathematics enjoyed a brief period of official support. At Kangxi's direction, Mei Goucheng and three other outstanding mathematicians compiled a 53-volume work titled Shuli Jingyun ("The Essence of Mathematical Study") which was printed in 1723, and gave a systematic introduction to western mathematical knowledge. At the same time, Mei Goucheng also developed to Meishi Congshu Jiyang [The Compiled works of Mei]. Meishi Congshu Jiyang was an encyclopedic summary of nearly all schools of Chinese mathematics at that time, but it also included the cross-cultural works of Mei Wending (1633–1721), Goucheng's grandfather. The enterprise sought to alleviate the difficulties for Chinese mathematicians working on Western mathematics in tracking down citations. In 1773, the Qianlong Emperor decided to compile the Complete Library of the Four Treasuries (or Siku Quanshu). Dai Zhen (1724–1777) selected and proofread The Nine Chapters on the Mathematical Art from Yongle Encyclopedia and several other mathematical works from Han and Tang dynasties. The long-missing mathematical works from Song and Yuan dynasties such as Si-yüan yü-jian and Ceyuan haijing were also found and printed, which directly led to a wave of new research. The most annotated works were Jiuzhang suanshu xicaotushuo (The Illustrations of Calculation Process for The Nine Chapters on the Mathematical Art ) contributed by Li Huang and Siyuan yujian xicao (The Detailed Explanation of Si-yuan yu-jian) by Luo Shilin. == Western influences == In 1840, the First Opium War forced China to open its door and look at the outside world, which also led to an influx of western mathematical studies at a rate unrivaled in the previous centuries. In 1852, the Chinese mathematician Li Shanlan and the British missionary Alexander Wylie co-translated the later nine volumes of Elements and 13 volumes on Algebra. With the assistance of Joseph Edkins, more works on astronomy and calculus soon followed. Chinese scholars were initially unsure whether to approach the new works: was study of Western knowledge a form of submission to foreign invaders? But by the end of the century, it became clear that China could only begin to recover its sovereignty by incorporating Western works. Chinese scholars, taught in Western missionary schools, from (translated) Western texts, rapidly lost touch with the indigenous tradition. Those who were self-trained or in traditionalist circles nevertheless continued to work within the traditional framework of algorithmic mathematics without resorting to Western symbolism. Yet, as Martzloff notes, "from 1911 onwards, solely Western mathematics has been practised in China." === In modern China === Chinese mathematics experienced a great surge of revival following the establishment of a modern Chinese republic in 1912. Ever since then, modern Chinese mathematicians have made numerous achievements in various mathematical fields. Some famous modern ethnic Chinese mathematicians include: Shiing-Shen Chern was widely regarded as a leader in geometry and one of the greatest mathematicians of the 20th century and was awarded the Wolf Prize for his contributions to mathematics. Ky Fan made contributions to fixed point theory, in addition to influencing nonlinear functional analysis, which have found wide application in mathematical economics and game theory, potential theory, calculus of variations, and differential equations. Shing-Tung Yau, a Fields Medal laureate, has influenced both physics and mathematics, and he has been active at the interface between geometry and theoretical physics and subsequently awarded the for his contributions. Terence Tao, a Fields Medal laureate and child prodigy of Chinese heritage, was the youngest participant in the history of the International Mathematical Olympiad at the age of 10, winning a bronze, silver, and gold medal. He remains the youngest winner of each of the three medals in the Olympiad's history. Yitang Zhang, a number theorist who established the first finite bound on gaps between prime numbers. Chen Jingrun, a number theorist who proved that every sufficiently large even number can be written as the sum of either two primes, or a prime and a semiprime, which is now called Chen's theorem. His work was important for research of Goldbach's conjecture. == People's Republic of China == In 1949, at the beginning of the founding of the People's Republic of China, the government paid great attention to the cause of science although the country was in a predicament of lack of funds. The Chinese Academy of Sciences was established in November 1949. The Institute of Mathematics was formally established in July 1952. Then, the Chinese Mathematical Society and its founding journals restored and added other special journals. In the 18 years after 1949, the number of published papers accounted for more than three times the total number of articles before 1949. Many of them not only filled the gaps in China's past, but also reached the world's advanced level. During the chaos of the Cultural Revolution, the sciences declined. In the field of mathematics, in addition to Chen Jingrun, Hua Luogeng, Zhang Guanghou and other mathematicians struggling to continue their work. After the catastrophe, with the publication of Guo Moruo's literary "Spring of Science", Chinese sciences and mathematics experienced a revival. In 1977, a new mathematical development plan was formulated in Beijing, the work of the mathematics society was resumed, the journal was re-published, the academic journal was published, the mathematics education was strengthened, and basic theoretical research was strengthened. An important mathematical achievement of the Chinese mathematician in the direction of the power system is how Xia Zhihong proved the Painleve conjecture in 1988. When there are some initial states of N celestial bodies, one of the celestial bodies ran to infinity or speed in a limited time. Infinity is reached, that is, there are non-collision singularities. The Painleve conjecture is an important conjecture in the field of power systems proposed in 1895. A very important recent development for the 4-body problem is that Xue Jinxin and Dolgopyat proved a non-collision singularity in a simplified version of the 4-body system around 2013. In addition, in 2007, Shen Weixiao and Kozlovski, Van-Strien proved the Real Fatou conjecture: Real hyperbolic polynomials are dense in the space of real polynomials with fixed degree. This conjecture can be traced back to Fatou in the 1920s, and later Smale posed it in the 1960s. The proof of Real Fatou conjecture is one of the most important developments in conformal dynamics in the past decade. === IMO performance === In comparison to other participating countries at the International Mathematical Olympiad, China has highest team scores and has won the all-members-gold IMO with a full team the most number of times. == In education == The first reference to a book being used in learning mathematics in China is dated to the second century CE (Hou Hanshu: 24, 862; 35,1207). Ma Xu, who is a youth c. 110, and Zheng Xuan (127–200) both studied the Nine Chapters on Mathematical procedures. Christopher Cullen claims that mathematics, in a manner akin to medicine, was taught orally. The stylistics of the Suàn shù shū from Zhangjiashan suggest that the text was assembled from various sources and then underwent codification. == See also == Chinese astronomy History of mathematics Indian mathematics Islamic mathematics Japanese mathematics List of Chinese discoveries List of Chinese mathematicians Numbers in Chinese culture == References == === Citations === === Works cited === == External links == Early mathematics texts (Chinese) - Chinese Text Project Overview of Chinese mathematics Chinese Mathematics Through the Han Dynasty Primer of Mathematics by Zhu Shijie
Wikipedia:Chaim L. Pekeris#0	Chaim Leib Pekeris (Hebrew: חיים לייב פקריס; June 15, 1908 – February 24, 1993) was an Israeli-American physicist and mathematician. He made notable contributions to geophysics and the spectral theory of many-electron atoms, in particular the helium atom. He was also one of the designers of the first computer in Israel, WEIZAC. == Biography == Pekeris was born in Alytus, Vilna Governorate on June 15, 1908. With the assistance of his uncle, Pekeris and his two brothers emigrated to the United States around 1925. He entered the Massachusetts Institute of Technology in 1925 graduating in 1929 with a B.Sc. in meteorology. Pekeris also took his graduate studies at MIT, studying under Carl-Gustav Rossby. In January 1933 he married Lea Kaplan, who was also born in Lithuania. He graduated with his Sc.D. doctoral degree in 1934. In 1934 Pekeris joined the faculty at M.I.T. as an instructor in geophysics in the department of geology. He became a US citizen in 1938. Pekeris remained at M.I.T until 1941 when he moved to the Hudson Laboratories of Columbia University to conduct military research. In 1946 he joined the Institute for Advanced Study. Pekeris and his wife moved to Israel in 1948, where he joined the Weizmann Institute as head of its department of applied mathematics in 1949. During the 1948 Arab–Israeli War he was involved in a clandestine program in New York State developing munitions for the newborn State of Israel. He received the Gold Medal from the Royal Astronomical Society in 1980, and the Israel Prize from the State of Israel in 1981. Teddy Kollek, the mayor of Jerusalem from 1965 to 1993, said in 1990: "I have told you a lot about Chaim Pekeris tonight and there is much more that I could tell, but you will understand that there are reasons that I can’t. Let me simply say that Chaim Pekeris played a most significant role in the establishment of the State of Israel." He died in Rehovot, Israel on February 24, 1993. == Awards and honors == Rockefeller Fellow (1934) Fellow of the American Physical Society (1941) Guggenheim Fellowship (1946) Member of the American Philosophical Society (1971) Member of the National Academy of Sciences (1972) Vetlesen Prize (1974) Member of the American Philosophical Society (1974) Gold Medal of the Royal Astronomical Society (1980) Israel Prize, for physics (1980). == See also == List of Israel Prize recipients List of geophysicists == References == == Further reading == == External links == Biographical memoir by Freeman Gilbert
Wikipedia:Chain rule#0	In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions f and g in terms of the derivatives of f and g. More precisely, if h = f ∘ g {\displaystyle h=f\circ g} is the function such that h ( x ) = f ( g ( x ) ) {\displaystyle h(x)=f(g(x))} for every x, then the chain rule is, in Lagrange's notation, h ′ ( x ) = f ′ ( g ( x ) ) g ′ ( x ) . {\displaystyle h'(x)=f'(g(x))g'(x).} or, equivalently, h ′ = ( f ∘ g ) ′ = ( f ′ ∘ g ) ⋅ g ′ . {\displaystyle h'=(f\circ g)'=(f'\circ g)\cdot g'.} The chain rule may also be expressed in Leibniz's notation. If a variable z depends on the variable y, which itself depends on the variable x (that is, y and z are dependent variables), then z depends on x as well, via the intermediate variable y. In this case, the chain rule is expressed as d z d x = d z d y ⋅ d y d x , {\displaystyle {\frac {dz}{dx}}={\frac {dz}{dy}}\cdot {\frac {dy}{dx}},} and d z d x \| x = d z d y \| y ( x ) ⋅ d y d x \| x , {\displaystyle \left.{\frac {dz}{dx}}\right\|_{x}=\left.{\frac {dz}{dy}}\right\|_{y(x)}\cdot \left.{\frac {dy}{dx}}\right\|_{x},} for indicating at which points the derivatives have to be evaluated. In integration, the counterpart to the chain rule is the substitution rule. == Intuitive explanation == Intuitively, the chain rule states that knowing the instantaneous rate of change of z relative to y and that of y relative to x allows one to calculate the instantaneous rate of change of z relative to x as the product of the two rates of change. As put by George F. Simmons: "If a car travels twice as fast as a bicycle and the bicycle is four times as fast as a walking man, then the car travels 2 × 4 = 8 times as fast as the man." The relationship between this example and the chain rule is as follows. Let z, y and x be the (variable) positions of the car, the bicycle, and the walking man, respectively. The rate of change of relative positions of the car and the bicycle is d z d y = 2. {\textstyle {\frac {dz}{dy}}=2.} Similarly, d y d x = 4. {\textstyle {\frac {dy}{dx}}=4.} So, the rate of change of the relative positions of the car and the walking man is d z d x = d z d y ⋅ d y d x = 2 ⋅ 4 = 8. {\displaystyle {\frac {dz}{dx}}={\frac {dz}{dy}}\cdot {\frac {dy}{dx}}=2\cdot 4=8.} The rate of change of positions is the ratio of the speeds, and the speed is the derivative of the position with respect to the time; that is, d z d x = d z d t d x d t , {\displaystyle {\frac {dz}{dx}}={\frac {\frac {dz}{dt}}{\frac {dx}{dt}}},} or, equivalently, d z d t = d z d x ⋅ d x d t , {\displaystyle {\frac {dz}{dt}}={\frac {dz}{dx}}\cdot {\frac {dx}{dt}},} which is also an application of the chain rule. == History == The chain rule seems to have first been used by Gottfried Wilhelm Leibniz. He used it to calculate the derivative of a + b z + c z 2 {\displaystyle {\sqrt {a+bz+cz^{2}}}} as the composite of the square root function and the function a + b z + c z 2 {\displaystyle a+bz+cz^{2}\!} . He first mentioned it in a 1676 memoir (with a sign error in the calculation). The common notation of the chain rule is due to Leibniz. Guillaume de l'Hôpital used the chain rule implicitly in his Analyse des infiniment petits. The chain rule does not appear in any of Leonhard Euler's analysis books, even though they were written over a hundred years after Leibniz's discovery.. It is believed that the first "modern" version of the chain rule appears in Lagrange's 1797 Théorie des fonctions analytiques; it also appears in Cauchy's 1823 Résumé des Leçons données a L’École Royale Polytechnique sur Le Calcul Infinitesimal. == Statement == The simplest form of the chain rule is for real-valued functions of one real variable. It states that if g is a function that is differentiable at a point c (i.e. the derivative g′(c) exists) and f is a function that is differentiable at g(c), then the composite function f ∘ g {\displaystyle f\circ g} is differentiable at c, and the derivative is ( f ∘ g ) ′ ( c ) = f ′ ( g ( c ) ) ⋅ g ′ ( c ) . {\displaystyle (f\circ g)'(c)=f'(g(c))\cdot g'(c).} The rule is sometimes abbreviated as ( f ∘ g ) ′ = ( f ′ ∘ g ) ⋅ g ′ . {\displaystyle (f\circ g)'=(f'\circ g)\cdot g'.} If y = f(u) and u = g(x), then this abbreviated form is written in Leibniz notation as: d y d x = d y d u ⋅ d u d x . {\displaystyle {\frac {dy}{dx}}={\frac {dy}{du}}\cdot {\frac {du}{dx}}.} The points where the derivatives are evaluated may also be stated explicitly: d y d x \| x = c = d y d u \| u = g ( c ) ⋅ d u d x \| x = c . {\displaystyle \left.{\frac {dy}{dx}}\right\|_{x=c}=\left.{\frac {dy}{du}}\right\|_{u=g(c)}\cdot \left.{\frac {du}{dx}}\right\|_{x=c}.} Carrying the same reasoning further, given n functions f 1 , … , f n {\displaystyle f_{1},\ldots ,f_{n}\!} with the composite function f 1 ∘ ( f 2 ∘ ⋯ ( f n − 1 ∘ f n ) ) {\displaystyle f_{1}\circ (f_{2}\circ \cdots (f_{n-1}\circ f_{n}))\!} , if each function f i {\displaystyle f_{i}\!} is differentiable at its immediate input, then the composite function is also differentiable by the repeated application of Chain Rule, where the derivative is (in Leibniz's notation): d f 1 d x = d f 1 d f 2 d f 2 d f 3 ⋯ d f n d x . {\displaystyle {\frac {df_{1}}{dx}}={\frac {df_{1}}{df_{2}}}{\frac {df_{2}}{df_{3}}}\cdots {\frac {df_{n}}{dx}}.} == Applications == === Composites of more than two functions === The chain rule can be applied to composites of more than two functions. To take the derivative of a composite of more than two functions, notice that the composite of f, g, and h (in that order) is the composite of f with g ∘ h. The chain rule states that to compute the derivative of f ∘ g ∘ h, it is sufficient to compute the derivative of f and the derivative of g ∘ h. The derivative of f can be calculated directly, and the derivative of g ∘ h can be calculated by applying the chain rule again. For concreteness, consider the function y = e sin ⁡ ( x 2 ) . {\displaystyle y=e^{\sin(x^{2})}.} This can be decomposed as the composite of three functions: y = f ( u ) = e u , u = g ( v ) = sin ⁡ v , v = h ( x ) = x 2 . {\displaystyle {\begin{aligned}y&=f(u)=e^{u},\\u&=g(v)=\sin v,\\v&=h(x)=x^{2}.\end{aligned}}} So that y = f ( g ( h ( x ) ) ) {\displaystyle y=f(g(h(x)))} . Their derivatives are: d y d u = f ′ ( u ) = e u , d u d v = g ′ ( v ) = cos ⁡ v , d v d x = h ′ ( x ) = 2 x . {\displaystyle {\begin{aligned}{\frac {dy}{du}}&=f'(u)=e^{u},\\{\frac {du}{dv}}&=g'(v)=\cos v,\\{\frac {dv}{dx}}&=h'(x)=2x.\end{aligned}}} The chain rule states that the derivative of their composite at the point x = a is: ( f ∘ g ∘ h ) ′ ( a ) = f ′ ( ( g ∘ h ) ( a ) ) ⋅ ( g ∘ h ) ′ ( a ) = f ′ ( ( g ∘ h ) ( a ) ) ⋅ g ′ ( h ( a ) ) ⋅ h ′ ( a ) = ( f ′ ∘ g ∘ h ) ( a ) ⋅ ( g ′ ∘ h ) ( a ) ⋅ h ′ ( a ) . {\displaystyle {\begin{aligned}(f\circ g\circ h)'(a)&=f'((g\circ h)(a))\cdot (g\circ h)'(a)\\&=f'((g\circ h)(a))\cdot g'(h(a))\cdot h'(a)\\&=(f'\circ g\circ h)(a)\cdot (g'\circ h)(a)\cdot h'(a).\end{aligned}}} In Leibniz's notation, this is: d y d x = d y d u \| u = g ( h ( a ) ) ⋅ d u d v \| v = h ( a ) ⋅ d v d x \| x = a , {\displaystyle {\frac {dy}{dx}}=\left.{\frac {dy}{du}}\right\|_{u=g(h(a))}\cdot \left.{\frac {du}{dv}}\right\|_{v=h(a)}\cdot \left.{\frac {dv}{dx}}\right\|_{x=a},} or for short, d y d x = d y d u ⋅ d u d v ⋅ d v d x . {\displaystyle {\frac {dy}{dx}}={\frac {dy}{du}}\cdot {\frac {du}{dv}}\cdot {\frac {dv}{dx}}.} The derivative function is therefore: d y d x = e sin ⁡ ( x 2 ) ⋅ cos ⁡ ( x 2 ) ⋅ 2 x . {\displaystyle {\frac {dy}{dx}}=e^{\sin(x^{2})}\cdot \cos(x^{2})\cdot 2x.} Another way of computing this derivative is to view the composite function f ∘ g ∘ h as the composite of f ∘ g and h. Applying the chain rule in this manner would yield: ( f ∘ g ∘ h ) ′ ( a ) = ( f ∘ g ) ′ ( h ( a ) ) ⋅ h ′ ( a ) = f ′ ( g ( h ( a ) ) ) ⋅ g ′ ( h ( a ) ) ⋅ h ′ ( a ) . {\displaystyle {\begin{aligned}(f\circ g\circ h)'(a)&=(f\circ g)'(h(a))\cdot h'(a)\\&=f'(g(h(a)))\cdot g'(h(a))\cdot h'(a).\end{aligned}}} This is the same as what was computed above. This should be expected because (f ∘ g) ∘ h = f ∘ (g ∘ h). Sometimes, it is necessary to differentiate an arbitrarily long composition of the form f 1 ∘ f 2 ∘ ⋯ ∘ f n − 1 ∘ f n {\displaystyle f_{1}\circ f_{2}\circ \cdots \circ f_{n-1}\circ f_{n}\!} . In this case, define f a . . b = f a ∘ f a + 1 ∘ ⋯ ∘ f b − 1 ∘ f b {\displaystyle f_{a\,.\,.\,b}=f_{a}\circ f_{a+1}\circ \cdots \circ f_{b-1}\circ f_{b}} where f a . . a = f a {\displaystyle f_{a\,.\,.\,a}=f_{a}} and f a . . b ( x ) = x {\displaystyle f_{a\,.\,.\,b}(x)=x} when b < a {\displaystyle b<a} . Then the chain rule takes the form D f 1 . . n = ( D f 1 ∘ f 2 . . n ) ( D f 2 ∘ f 3 . . n ) ⋯ ( D f n − 1 ∘ f n . . n ) D f n = ∏ k = 1 n [ D f k ∘ f ( k + 1 ) . . n ] {\displaystyle {\begin{aligned}Df_{1\,.\,.\,n}&=(Df_{1}\circ f_{2\,.\,.\,n})(Df_{2}\circ f_{3\,.\,.\,n})\cdots (Df_{n-1}\circ f_{n\,.\,.\,n})Df_{n}\\&=\prod _{k=1}^{n}\left[Df_{k}\circ f_{(k+1)\,.\,.\,n}\right]\end{aligned}}} or, in the Lagrange notation, f 1 . . n ′ ( x ) = f 1 ′ ( f 2 . . n ( x ) ) f 2 ′ ( f 3 . . n ( x ) ) ⋯ f n − 1 ′ ( f n . . n ( x ) ) f n ′ ( x ) = ∏ k = 1 n f k ′ ( f ( k + 1 . . n ) ( x ) ) {\displaystyle {\begin{aligned}f_{1\,.\,.\,n}'(x)&=f_{1}'\left(f_{2\,.\,.\,n}(x)\right)\;f_{2}'\left(f_{3\,.\,.\,n}(x)\right)\cdots f_{n-1}'\left(f_{n\,.\,.\,n}(x)\right)\;f_{n}'(x)\\[1ex]&=\prod _{k=1}^{n}f_{k}'\left(f_{(k+1\,.\,.\,n)}(x)\right)\end{aligned}}} === Quotient rule === The chain rule can be used to derive some well-known differentiation rules. For example, the quotient rule is a consequence of the chain rule and the product rule. To see this, write the function f(x)/g(x) as the product f(x) · 1/g(x). First apply the product rule: d d x ( f ( x ) g ( x ) ) = d d x ( f ( x ) ⋅ 1 g ( x ) ) = f ′ ( x ) ⋅ 1 g ( x ) + f ( x ) ⋅ d d x ( 1 g ( x ) ) . {\displaystyle {\begin{aligned}{\frac {d}{dx}}\left({\frac {f(x)}{g(x)}}\right)&={\frac {d}{dx}}\left(f(x)\cdot {\frac {1}{g(x)}}\right)\\&=f'(x)\cdot {\frac {1}{g(x)}}+f(x)\cdot {\frac {d}{dx}}\left({\frac {1}{g(x)}}\right).\end{aligned}}} To compute the derivative of 1/g(x), notice that it is the composite of g with the reciprocal function, that is, the function that sends x to 1/x. The derivative of the reciprocal function is − 1 / x 2 {\displaystyle -1/x^{2}\!} . By applying the chain rule, the last expression becomes: f ′ ( x ) ⋅ 1 g ( x ) + f ( x ) ⋅ ( − 1 g ( x ) 2 ⋅ g ′ ( x ) ) = f ′ ( x ) g ( x ) − f ( x ) g ′ ( x ) g ( x ) 2 , {\displaystyle f'(x)\cdot {\frac {1}{g(x)}}+f(x)\cdot \left(-{\frac {1}{g(x)^{2}}}\cdot g'(x)\right)={\frac {f'(x)g(x)-f(x)g'(x)}{g(x)^{2}}},} which is the usual formula for the quotient rule. === Derivatives of inverse functions === Suppose that y = g(x) has an inverse function. Call its inverse function f so that we have x = f(y). There is a formula for the derivative of f in terms of the derivative of g. To see this, note that f and g satisfy the formula f ( g ( x ) ) = x . {\displaystyle f(g(x))=x.} And because the functions f ( g ( x ) ) {\displaystyle f(g(x))} and x are equal, their derivatives must be equal. The derivative of x is the constant function with value 1, and the derivative of f ( g ( x ) ) {\displaystyle f(g(x))} is determined by the chain rule. Therefore, we have that: f ′ ( g ( x ) ) g ′ ( x ) = 1. {\displaystyle f'(g(x))g'(x)=1.} To express f' as a function of an independent variable y, we substitute f ( y ) {\displaystyle f(y)} for x wherever it appears. Then we can solve for f'. f ′ ( g ( f ( y ) ) ) g ′ ( f ( y ) ) = 1 f ′ ( y ) g ′ ( f ( y ) ) = 1 f ′ ( y ) = 1 g ′ ( f ( y ) ) . {\displaystyle {\begin{aligned}f'(g(f(y)))g'(f(y))&=1\\f'(y)g'(f(y))&=1\\f'(y)={\frac {1}{g'(f(y))}}.\end{aligned}}} For example, consider the function g(x) = ex. It has an inverse f(y) = ln y. Because g′(x) = ex, the above formula says that d d y ln ⁡ y = 1 e ln ⁡ y = 1 y . {\displaystyle {\frac {d}{dy}}\ln y={\frac {1}{e^{\ln y}}}={\frac {1}{y}}.} This formula is true whenever g is differentiable and its inverse f is also differentiable. This formula can fail when one of these conditions is not true. For example, consider g(x) = x3. Its inverse is f(y) = y1/3, which is not differentiable at zero. If we attempt to use the above formula to compute the derivative of f at zero, then we must evaluate 1/g′(f(0)). Since f(0) = 0 and g′(0) = 0, we must evaluate 1/0, which is undefined. Therefore, the formula fails in this case. This is not surprising because f is not differentiable at zero. === Back propagation === The chain rule forms the basis of the back propagation algorithm, which is used in gradient descent of neural networks in deep learning (artificial intelligence). == Higher derivatives == Faà di Bruno's formula generalizes the chain rule to higher derivatives. Assuming that y = f(u) and u = g(x), then the first few derivatives are: d y d x = d y d u d u d x d 2 y d x 2 = d 2 y d u 2 ( d u d x ) 2 + d y d u d 2 u d x 2 d 3 y d x 3 = d 3 y d u 3 ( d u d x ) 3 + 3 d 2 y d u 2 d u d x d 2 u d x 2 + d y d u d 3 u d x 3 d 4 y d x 4 = d 4 y d u 4 ( d u d x ) 4 + 6 d 3 y d u 3 ( d u d x ) 2 d 2 u d x 2 + d 2 y d u 2 ( 4 d u d x d 3 u d x 3 + 3 ( d 2 u d x 2 ) 2 ) + d y d u d 4 u d x 4 . {\displaystyle {\begin{aligned}{\frac {dy}{dx}}&={\frac {dy}{du}}{\frac {du}{dx}}\\{\frac {d^{2}y}{dx^{2}}}&={\frac {d^{2}y}{du^{2}}}\left({\frac {du}{dx}}\right)^{2}+{\frac {dy}{du}}{\frac {d^{2}u}{dx^{2}}}\\{\frac {d^{3}y}{dx^{3}}}&={\frac {d^{3}y}{du^{3}}}\left({\frac {du}{dx}}\right)^{3}+3\,{\frac {d^{2}y}{du^{2}}}{\frac {du}{dx}}{\frac {d^{2}u}{dx^{2}}}+{\frac {dy}{du}}{\frac {d^{3}u}{dx^{3}}}\\{\frac {d^{4}y}{dx^{4}}}&={\frac {d^{4}y}{du^{4}}}\left({\frac {du}{dx}}\right)^{4}+6\,{\frac {d^{3}y}{du^{3}}}\left({\frac {du}{dx}}\right)^{2}{\frac {d^{2}u}{dx^{2}}}+{\frac {d^{2}y}{du^{2}}}\left(4\,{\frac {du}{dx}}{\frac {d^{3}u}{dx^{3}}}+3\,\left({\frac {d^{2}u}{dx^{2}}}\right)^{2}\right)+{\frac {dy}{du}}{\frac {d^{4}u}{dx^{4}}}.\end{aligned}}} == Proofs == === First proof === One proof of the chain rule begins by defining the derivative of the composite function f ∘ g, where we take the limit of the difference quotient for f ∘ g as x approaches a: ( f ∘ g ) ′ ( a ) = lim x → a f ( g ( x ) ) − f ( g ( a ) ) x − a . {\displaystyle (f\circ g)'(a)=\lim _{x\to a}{\frac {f(g(x))-f(g(a))}{x-a}}.} Assume for the moment that g ( x ) {\displaystyle g(x)\!} does not equal g ( a ) {\displaystyle g(a)} for any x {\displaystyle x} near a {\displaystyle a} . Then the previous expression is equal to the product of two factors: lim x → a f ( g ( x ) ) − f ( g ( a ) ) g ( x ) − g ( a ) ⋅ g ( x ) − g ( a ) x − a . {\displaystyle \lim _{x\to a}{\frac {f(g(x))-f(g(a))}{g(x)-g(a)}}\cdot {\frac {g(x)-g(a)}{x-a}}.} If g {\displaystyle g} oscillates near a, then it might happen that no matter how close one gets to a, there is always an even closer x such that g(x) = g(a). For example, this happens near a = 0 for the continuous function g defined by g(x) = 0 for x = 0 and g(x) = x2 sin(1/x) otherwise. Whenever this happens, the above expression is undefined because it involves division by zero. To work around this, introduce a function Q {\displaystyle Q} as follows: Q ( y ) = { f ( y ) − f ( g ( a ) ) y − g ( a ) , y ≠ g ( a ) , f ′ ( g ( a ) ) , y = g ( a ) . {\displaystyle Q(y)={\begin{cases}\displaystyle {\frac {f(y)-f(g(a))}{y-g(a)}},&y\neq g(a),\\f'(g(a)),&y=g(a).\end{cases}}} We will show that the difference quotient for f ∘ g is always equal to: Q ( g ( x ) ) ⋅ g ( x ) − g ( a ) x − a . {\displaystyle Q(g(x))\cdot {\frac {g(x)-g(a)}{x-a}}.} Whenever g(x) is not equal to g(a), this is clear because the factors of g(x) − g(a) cancel. When g(x) equals g(a), then the difference quotient for f ∘ g is zero because f(g(x)) equals f(g(a)), and the above product is zero because it equals f′(g(a)) times zero. So the above product is always equal to the difference quotient, and to show that the derivative of f ∘ g at a exists and to determine its value, we need only show that the limit as x goes to a of the above product exists and determine its value. To do this, recall that the limit of a product exists if the limits of its factors exist. When this happens, the limit of the product of these two factors will equal the product of the limits of the factors. The two factors are Q(g(x)) and (g(x) − g(a)) / (x − a). The latter is the difference quotient for g at a, and because g is differentiable at a by assumption, its limit as x tends to a exists and equals g′(a). As for Q(g(x)), notice that Q is defined wherever f is. Furthermore, f is differentiable at g(a) by assumption, so Q is continuous at g(a), by definition of the derivative. The function g is continuous at a because it is differentiable at a, and therefore Q ∘ g is continuous at a. So its limit as x goes to a exists and equals Q(g(a)), which is f′(g(a)). This shows that the limits of both factors exist and that they equal f′(g(a)) and g′(a), respectively. Therefore, the derivative of f ∘ g at a exists and equals f′(g(a))g′(a). === Second proof === Another way of proving the chain rule is to measure the error in the linear approximation determined by the derivative. This proof has the advantage that it generalizes to several variables. It relies on the following equivalent definition of differentiability at a point: A function g is differentiable at a if there exists a real number g′(a) and a function ε(h) that tends to zero as h tends to zero, and furthermore g ( a + h ) − g ( a ) = g ′ ( a ) h + ε ( h ) h . {\displaystyle g(a+h)-g(a)=g'(a)h+\varepsilon (h)h.} Here the left-hand side represents the true difference between the value of g at a and at a + h, whereas the right-hand side represents the approximation determined by the derivative plus an error term. In the situation of the chain rule, such a function ε exists because g is assumed to be differentiable at a. Again by assumption, a similar function also exists for f at g(a). Calling this function η, we have f ( g ( a ) + k ) − f ( g ( a ) ) = f ′ ( g ( a ) ) k + η ( k ) k . {\displaystyle f(g(a)+k)-f(g(a))=f'(g(a))k+\eta (k)k.} The above definition imposes no constraints on η(0), even though it is assumed that η(k) tends to zero as k tends to zero. If we set η(0) = 0, then η is continuous at 0. Proving the theorem requires studying the difference f(g(a + h)) − f(g(a)) as h tends to zero. The first step is to substitute for g(a + h) using the definition of differentiability of g at a: f ( g ( a + h ) ) − f ( g ( a ) ) = f ( g ( a ) + g ′ ( a ) h + ε ( h ) h ) − f ( g ( a ) ) . {\displaystyle f(g(a+h))-f(g(a))=f(g(a)+g'(a)h+\varepsilon (h)h)-f(g(a)).} The next step is to use the definition of differentiability of f at g(a). This requires a term of the form f(g(a) + k) for some k. In the above equation, the correct k varies with h. Set kh = g′(a) h + ε(h) h and the right hand side becomes f(g(a) + kh) − f(g(a)). Applying the definition of the derivative gives: f ( g ( a ) + k h ) − f ( g ( a ) ) = f ′ ( g ( a ) ) k h + η ( k h ) k h . {\displaystyle f(g(a)+k_{h})-f(g(a))=f'(g(a))k_{h}+\eta (k_{h})k_{h}.} To study the behavior of this expression as h tends to zero, expand kh. After regrouping the terms, the right-hand side becomes: f ′ ( g ( a ) ) g ′ ( a ) h + [ f ′ ( g ( a ) ) ε ( h ) + η ( k h ) g ′ ( a ) + η ( k h ) ε ( h ) ] h . {\displaystyle f'(g(a))g'(a)h+[f'(g(a))\varepsilon (h)+\eta (k_{h})g'(a)+\eta (k_{h})\varepsilon (h)]h.} Because ε(h) and η(kh) tend to zero as h tends to zero, the first two bracketed terms tend to zero as h tends to zero. Applying the same theorem on products of limits as in the first proof, the third bracketed term also tends zero. Because the above expression is equal to the difference f(g(a + h)) − f(g(a)), by the definition of the derivative f ∘ g is differentiable at a and its derivative is f′(g(a)) g′(a). The role of Q in the first proof is played by η in this proof. They are related by the equation: Q ( y ) = f ′ ( g ( a ) ) + η ( y − g ( a ) ) . {\displaystyle Q(y)=f'(g(a))+\eta (y-g(a)).} The need to define Q at g(a) is analogous to the need to define η at zero. === Third proof === Constantin Carathéodory's alternative definition of the differentiability of a function can be used to give an elegant proof of the chain rule. Under this definition, a function f is differentiable at a point a if and only if there is a function q, continuous at a and such that f(x) − f(a) = q(x)(x − a). There is at most one such function, and if f is differentiable at a then f ′(a) = q(a). Given the assumptions of the chain rule and the fact that differentiable functions and compositions of continuous functions are continuous, we have that there exist functions q, continuous at g(a), and r, continuous at a, and such that, f ( g ( x ) ) − f ( g ( a ) ) = q ( g ( x ) ) ( g ( x ) − g ( a ) ) {\displaystyle f(g(x))-f(g(a))=q(g(x))(g(x)-g(a))} and g ( x ) − g ( a ) = r ( x ) ( x − a ) . {\displaystyle g(x)-g(a)=r(x)(x-a).} Therefore, f ( g ( x ) ) − f ( g ( a ) ) = q ( g ( x ) ) r ( x ) ( x − a ) , {\displaystyle f(g(x))-f(g(a))=q(g(x))r(x)(x-a),} but the function given by h(x) = q(g(x))r(x) is continuous at a, and we get, for this a ( f ( g ( a ) ) ) ′ = q ( g ( a ) ) r ( a ) = f ′ ( g ( a ) ) g ′ ( a ) . {\displaystyle (f(g(a)))'=q(g(a))r(a)=f'(g(a))g'(a).} A similar approach works for continuously differentiable (vector-)functions of many variables. This method of factoring also allows a unified approach to stronger forms of differentiability, when the derivative is required to be Lipschitz continuous, Hölder continuous, etc. Differentiation itself can be viewed as the polynomial remainder theorem (the little Bézout theorem, or factor theorem), generalized to an appropriate class of functions. === Proof via infinitesimals === If y = f ( x ) {\displaystyle y=f(x)} and x = g ( t ) {\displaystyle x=g(t)} then choosing infinitesimal Δ t ≠ 0 {\displaystyle \Delta t\not =0} we compute the corresponding Δ x = g ( t + Δ t ) − g ( t ) {\displaystyle \Delta x=g(t+\Delta t)-g(t)} and then the corresponding Δ y = f ( x + Δ x ) − f ( x ) {\displaystyle \Delta y=f(x+\Delta x)-f(x)} , so that Δ y Δ t = Δ y Δ x Δ x Δ t {\displaystyle {\frac {\Delta y}{\Delta t}}={\frac {\Delta y}{\Delta x}}{\frac {\Delta x}{\Delta t}}} and applying the standard part we obtain d y d t = d y d x d x d t {\displaystyle {\frac {dy}{dt}}={\frac {dy}{dx}}{\frac {dx}{dt}}} which is the chain rule. == Multivariable case == The full generalization of the chain rule to multi-variable functions (such as f : R m → R n {\displaystyle f:\mathbb {R} ^{m}\to \mathbb {R} ^{n}} ) is rather technical. However, it is simpler to write in the case of functions of the form f ( g 1 ( x ) , … , g k ( x ) ) , {\displaystyle f(g_{1}(x),\dots ,g_{k}(x)),} where f : R k → R {\displaystyle f:\mathbb {R} ^{k}\to \mathbb {R} } , and g i : R → R {\displaystyle g_{i}:\mathbb {R} \to \mathbb {R} } for each i = 1 , 2 , … , k . {\displaystyle i=1,2,\dots ,k.} As this case occurs often in the study of functions of a single variable, it is worth describing it separately. === Case of scalar-valued functions with multiple inputs === Let f : R k → R {\displaystyle f:\mathbb {R} ^{k}\to \mathbb {R} } , and g i : R → R {\displaystyle g_{i}:\mathbb {R} \to \mathbb {R} } for each i = 1 , 2 , … , k . {\displaystyle i=1,2,\dots ,k.} To write the chain rule for the composition of functions x ↦ f ( g 1 ( x ) , … , g k ( x ) ) , {\displaystyle x\mapsto f(g_{1}(x),\dots ,g_{k}(x)),} one needs the partial derivatives of f with respect to its k arguments. The usual notations for partial derivatives involve names for the arguments of the function. As these arguments are not named in the above formula, it is simpler and clearer to use D-Notation, and to denote by D i f {\displaystyle D_{i}f} the partial derivative of f with respect to its ith argument, and by D i f ( z ) {\displaystyle D_{i}f(z)} the value of this derivative at z. With this notation, the chain rule is d d x f ( g 1 ( x ) , … , g k ( x ) ) = ∑ i = 1 k ( d d x g i ( x ) ) D i f ( g 1 ( x ) , … , g k ( x ) ) . {\displaystyle {\frac {d}{dx}}f(g_{1}(x),\dots ,g_{k}(x))=\sum _{i=1}^{k}\left({\frac {d}{dx}}{g_{i}}(x)\right)D_{i}f(g_{1}(x),\dots ,g_{k}(x)).} ==== Example: arithmetic operations ==== If the function f is addition, that is, if f ( u , v ) = u + v , {\displaystyle f(u,v)=u+v,} then D 1 f = ∂ f ∂ u = 1 {\textstyle D_{1}f={\frac {\partial f}{\partial u}}=1} and D 2 f = ∂ f ∂ v = 1 {\textstyle D_{2}f={\frac {\partial f}{\partial v}}=1} . Thus, the chain rule gives d d x ( g ( x ) + h ( x ) ) = ( d d x g ( x ) ) D 1 f + ( d d x h ( x ) ) D 2 f = d d x g ( x ) + d d x h ( x ) . {\displaystyle {\frac {d}{dx}}(g(x)+h(x))=\left({\frac {d}{dx}}g(x)\right)D_{1}f+\left({\frac {d}{dx}}h(x)\right)D_{2}f={\frac {d}{dx}}g(x)+{\frac {d}{dx}}h(x).} For multiplication f ( u , v ) = u v , {\displaystyle f(u,v)=uv,} the partials are D 1 f = v {\displaystyle D_{1}f=v} and D 2 f = u {\displaystyle D_{2}f=u} . Thus, d d x ( g ( x ) h ( x ) ) = h ( x ) d d x g ( x ) + g ( x ) d d x h ( x ) . {\displaystyle {\frac {d}{dx}}(g(x)h(x))=h(x){\frac {d}{dx}}g(x)+g(x){\frac {d}{dx}}h(x).} The case of exponentiation f ( u , v ) = u v {\displaystyle f(u,v)=u^{v}} is slightly more complicated, as D 1 f = v u v − 1 , {\displaystyle D_{1}f=vu^{v-1},} and, as u v = e v ln ⁡ u , {\displaystyle u^{v}=e^{v\ln u},} D 2 f = u v ln ⁡ u . {\displaystyle D_{2}f=u^{v}\ln u.} It follows that d d x ( g ( x ) h ( x ) ) = h ( x ) g ( x ) h ( x ) − 1 d d x g ( x ) + g ( x ) h ( x ) ln ⁡ g ( x ) d d x h ( x ) . {\displaystyle {\frac {d}{dx}}\left(g(x)^{h(x)}\right)=h(x)g(x)^{h(x)-1}{\frac {d}{dx}}g(x)+g(x)^{h(x)}\ln g(x)\,{\frac {d}{dx}}h(x).} === General rule: Vector-valued functions with multiple inputs === The simplest way for writing the chain rule in the general case is to use the total derivative, which is a linear transformation that captures all directional derivatives in a single formula. Consider differentiable functions f : Rm → Rk and g : Rn → Rm, and a point a in Rn. Let Da g denote the total derivative of g at a and Dg(a) f denote the total derivative of f at g(a). These two derivatives are linear transformations Rn → Rm and Rm → Rk, respectively, so they can be composed. The chain rule for total derivatives is that their composite is the total derivative of f ∘ g at a: D a ( f ∘ g ) = D g ( a ) f ∘ D a g , {\displaystyle D_{\mathbf {a} }(f\circ g)=D_{g(\mathbf {a} )}f\circ D_{\mathbf {a} }g,} or for short, D ( f ∘ g ) = D f ∘ D g . {\displaystyle D(f\circ g)=Df\circ Dg.} The higher-dimensional chain rule can be proved using a technique similar to the second proof given above. Because the total derivative is a linear transformation, the functions appearing in the formula can be rewritten as matrices. The matrix corresponding to a total derivative is called a Jacobian matrix, and the composite of two derivatives corresponds to the product of their Jacobian matrices. From this perspective the chain rule therefore says: J f ∘ g ( a ) = J f ( g ( a ) ) J g ( a ) , {\displaystyle J_{f\circ g}(\mathbf {a} )=J_{f}(g(\mathbf {a} ))J_{g}(\mathbf {a} ),} or for short, J f ∘ g = ( J f ∘ g ) J g . {\displaystyle J_{f\circ g}=(J_{f}\circ g)J_{g}.} That is, the Jacobian of a composite function is the product of the Jacobians of the composed functions (evaluated at the appropriate points). The higher-dimensional chain rule is a generalization of the one-dimensional chain rule. If k, m, and n are 1, so that f : R → R and g : R → R, then the Jacobian matrices of f and g are 1 × 1. Specifically, they are: J g ( a ) = ( g ′ ( a ) ) , J f ( g ( a ) ) = ( f ′ ( g ( a ) ) ) . {\displaystyle {\begin{aligned}J_{g}(a)&={\begin{pmatrix}g'(a)\end{pmatrix}},\\J_{f}(g(a))&={\begin{pmatrix}f'(g(a))\end{pmatrix}}.\end{aligned}}} The Jacobian of f ∘ g is the product of these 1 × 1 matrices, so it is f′(g(a))⋅g′(a), as expected from the one-dimensional chain rule. In the language of linear transformations, Da(g) is the function which scales a vector by a factor of g′(a) and Dg(a)(f) is the function which scales a vector by a factor of f′(g(a)). The chain rule says that the composite of these two linear transformations is the linear transformation Da(f ∘ g), and therefore it is the function that scales a vector by f′(g(a))⋅g′(a). Another way of writing the chain rule is used when f and g are expressed in terms of their components as y = f(u) = (f1(u), …, fk(u)) and u = g(x) = (g1(x), …, gm(x)). In this case, the above rule for Jacobian matrices is usually written as: ∂ ( y 1 , … , y k ) ∂ ( x 1 , … , x n ) = ∂ ( y 1 , … , y k ) ∂ ( u 1 , … , u m ) ∂ ( u 1 , … , u m ) ∂ ( x 1 , … , x n ) . {\displaystyle {\frac {\partial (y_{1},\ldots ,y_{k})}{\partial (x_{1},\ldots ,x_{n})}}={\frac {\partial (y_{1},\ldots ,y_{k})}{\partial (u_{1},\ldots ,u_{m})}}{\frac {\partial (u_{1},\ldots ,u_{m})}{\partial (x_{1},\ldots ,x_{n})}}.} The chain rule for total derivatives implies a chain rule for partial derivatives. Recall that when the total derivative exists, the partial derivative in the i-th coordinate direction is found by multiplying the Jacobian matrix by the i-th basis vector. By doing this to the formula above, we find: ∂ ( y 1 , … , y k ) ∂ x i = ∂ ( y 1 , … , y k ) ∂ ( u 1 , … , u m ) ∂ ( u 1 , … , u m ) ∂ x i . {\displaystyle {\frac {\partial (y_{1},\ldots ,y_{k})}{\partial x_{i}}}={\frac {\partial (y_{1},\ldots ,y_{k})}{\partial (u_{1},\ldots ,u_{m})}}{\frac {\partial (u_{1},\ldots ,u_{m})}{\partial x_{i}}}.} Since the entries of the Jacobian matrix are partial derivatives, we may simplify the above formula to get: ∂ ( y 1 , … , y k ) ∂ x i = ∑ ℓ = 1 m ∂ ( y 1 , … , y k ) ∂ u ℓ ∂ u ℓ ∂ x i . {\displaystyle {\frac {\partial (y_{1},\ldots ,y_{k})}{\partial x_{i}}}=\sum _{\ell =1}^{m}{\frac {\partial (y_{1},\ldots ,y_{k})}{\partial u_{\ell }}}{\frac {\partial u_{\ell }}{\partial x_{i}}}.} More conceptually, this rule expresses the fact that a change in the xi direction may change all of g1 through gm, and any of these changes may affect f. In the special case where k = 1, so that f is a real-valued function, then this formula simplifies even further: ∂ y ∂ x i = ∑ ℓ = 1 m ∂ y ∂ u ℓ ∂ u ℓ ∂ x i . {\displaystyle {\frac {\partial y}{\partial x_{i}}}=\sum _{\ell =1}^{m}{\frac {\partial y}{\partial u_{\ell }}}{\frac {\partial u_{\ell }}{\partial x_{i}}}.} This can be rewritten as a dot product. Recalling that u = (g1, …, gm), the partial derivative ∂u / ∂xi is also a vector, and the chain rule says that: ∂ y ∂ x i = ∇ y ⋅ ∂ u ∂ x i . {\displaystyle {\frac {\partial y}{\partial x_{i}}}=\nabla y\cdot {\frac {\partial \mathbf {u} }{\partial x_{i}}}.} ==== Example ==== Given u(x, y) = x2 + 2y where x(r, t) = r sin(t) and y(r,t) = sin2(t), determine the value of ∂u / ∂r and ∂u / ∂t using the chain rule. ∂ u ∂ r = ∂ u ∂ x ∂ x ∂ r + ∂ u ∂ y ∂ y ∂ r = ( 2 x ) ( sin ⁡ ( t ) ) + ( 2 ) ( 0 ) = 2 r sin 2 ⁡ ( t ) , {\displaystyle {\frac {\partial u}{\partial r}}={\frac {\partial u}{\partial x}}{\frac {\partial x}{\partial r}}+{\frac {\partial u}{\partial y}}{\frac {\partial y}{\partial r}}=(2x)(\sin(t))+(2)(0)=2r\sin ^{2}(t),} and ∂ u ∂ t = ∂ u ∂ x ∂ x ∂ t + ∂ u ∂ y ∂ y ∂ t = ( 2 x ) ( r cos ⁡ ( t ) ) + ( 2 ) ( 2 sin ⁡ ( t ) cos ⁡ ( t ) ) = ( 2 r sin ⁡ ( t ) ) ( r cos ⁡ ( t ) ) + 4 sin ⁡ ( t ) cos ⁡ ( t ) = 2 ( r 2 + 2 ) sin ⁡ ( t ) cos ⁡ ( t ) = ( r 2 + 2 ) sin ⁡ ( 2 t ) . {\displaystyle {\begin{aligned}{\frac {\partial u}{\partial t}}&={\frac {\partial u}{\partial x}}{\frac {\partial x}{\partial t}}+{\frac {\partial u}{\partial y}}{\frac {\partial y}{\partial t}}\\&=(2x)(r\cos(t))+(2)(2\sin(t)\cos(t))\\&=(2r\sin(t))(r\cos(t))+4\sin(t)\cos(t)\\&=2(r^{2}+2)\sin(t)\cos(t)\\&=(r^{2}+2)\sin(2t).\end{aligned}}} ==== Higher derivatives of multivariable functions ==== Faà di Bruno's formula for higher-order derivatives of single-variable functions generalizes to the multivariable case. If y = f(u) is a function of u = g(x) as above, then the second derivative of f ∘ g is: ∂ 2 y ∂ x i ∂ x j = ∑ k ( ∂ y ∂ u k ∂ 2 u k ∂ x i ∂ x j ) + ∑ k , ℓ ( ∂ 2 y ∂ u k ∂ u ℓ ∂ u k ∂ x i ∂ u ℓ ∂ x j ) . {\displaystyle {\frac {\partial ^{2}y}{\partial x_{i}\partial x_{j}}}=\sum _{k}\left({\frac {\partial y}{\partial u_{k}}}{\frac {\partial ^{2}u_{k}}{\partial x_{i}\partial x_{j}}}\right)+\sum _{k,\ell }\left({\frac {\partial ^{2}y}{\partial u_{k}\partial u_{\ell }}}{\frac {\partial u_{k}}{\partial x_{i}}}{\frac {\partial u_{\ell }}{\partial x_{j}}}\right).} == Further generalizations == All extensions of calculus have a chain rule. In most of these, the formula remains the same, though the meaning of that formula may be vastly different. One generalization is to manifolds. In this situation, the chain rule represents the fact that the derivative of f ∘ g is the composite of the derivative of f and the derivative of g. This theorem is an immediate consequence of the higher dimensional chain rule given above, and it has exactly the same formula. The chain rule is also valid for Fréchet derivatives in Banach spaces. The same formula holds as before. This case and the previous one admit a simultaneous generalization to Banach manifolds. In differential algebra, the derivative is interpreted as a morphism of modules of Kähler differentials. A ring homomorphism of commutative rings f : R → S determines a morphism of Kähler differentials Df : ΩR → ΩS which sends an element dr to d(f(r)), the exterior differential of f(r). The formula D(f ∘ g) = Df ∘ Dg holds in this context as well. The common feature of these examples is that they are expressions of the idea that the derivative is part of a functor. A functor is an operation on spaces and functions between them. It associates to each space a new space and to each function between two spaces a new function between the corresponding new spaces. In each of the above cases, the functor sends each space to its tangent bundle and it sends each function to its derivative. For example, in the manifold case, the derivative sends a Cr-manifold to a Cr−1-manifold (its tangent bundle) and a Cr-function to its total derivative. There is one requirement for this to be a functor, namely that the derivative of a composite must be the composite of the derivatives. This is exactly the formula D(f ∘ g) = Df ∘ Dg. There are also chain rules in stochastic calculus. One of these, Itō's lemma, expresses the composite of an Itō process (or more generally a semimartingale) dXt with a twice-differentiable function f. In Itō's lemma, the derivative of the composite function depends not only on dXt and the derivative of f but also on the second derivative of f. The dependence on the second derivative is a consequence of the non-zero quadratic variation of the stochastic process, which broadly speaking means that the process can move up and down in a very rough way. This variant of the chain rule is not an example of a functor because the two functions being composed are of different types. == See also == Automatic differentiation – Numerical calculations carrying along derivatives − a computational method that makes heavy use of the chain rule to compute exact numerical derivatives. Differentiation rules – Rules for computing derivatives of functions Integration by substitution – Technique in integral evaluation Leibniz integral rule – Differentiation under the integral sign formula Product rule – Formula for the derivative of a product Quotient rule – Formula for the derivative of a ratio of functions Triple product rule – Relation between relative derivatives of three variables == References == == External links == "Leibniz rule", Encyclopedia of Mathematics, EMS Press, 2001 [1994] Weisstein, Eric W. "Chain Rule". MathWorld.
Wikipedia:Chain rule (probability)#0	In probability theory, the chain rule (also called the general product rule) describes how to calculate the probability of the intersection of, not necessarily independent, events or the joint distribution of random variables respectively, using conditional probabilities. This rule allows one to express a joint probability in terms of only conditional probabilities. The rule is notably used in the context of discrete stochastic processes and in applications, e.g. the study of Bayesian networks, which describe a probability distribution in terms of conditional probabilities. == Chain rule for events == === Two events === For two events A {\displaystyle A} and B {\displaystyle B} , the chain rule states that P ( A ∩ B ) = P ( B ∣ A ) P ( A ) {\displaystyle \mathbb {P} (A\cap B)=\mathbb {P} (B\mid A)\mathbb {P} (A)} , where P ( B ∣ A ) {\displaystyle \mathbb {P} (B\mid A)} denotes the conditional probability of B {\displaystyle B} given A {\displaystyle A} . ==== Example ==== An Urn A has 1 black ball and 2 white balls and another Urn B has 1 black ball and 3 white balls. Suppose we pick an urn at random and then select a ball from that urn. Let event A {\displaystyle A} be choosing the first urn, i.e. P ( A ) = P ( A ¯ ) = 1 / 2 {\displaystyle \mathbb {P} (A)=\mathbb {P} ({\overline {A}})=1/2} , where A ¯ {\displaystyle {\overline {A}}} is the complementary event of A {\displaystyle A} . Let event B {\displaystyle B} be the chance we choose a white ball. The chance of choosing a white ball, given that we have chosen the first urn, is P ( B \| A ) = 2 / 3. {\displaystyle \mathbb {P} (B\|A)=2/3.} The intersection A ∩ B {\displaystyle A\cap B} then describes choosing the first urn and a white ball from it. The probability can be calculated by the chain rule as follows: P ( A ∩ B ) = P ( B ∣ A ) P ( A ) = 2 3 ⋅ 1 2 = 1 3 . {\displaystyle \mathbb {P} (A\cap B)=\mathbb {P} (B\mid A)\mathbb {P} (A)={\frac {2}{3}}\cdot {\frac {1}{2}}={\frac {1}{3}}.} === Finitely many events === For events A 1 , … , A n {\displaystyle A_{1},\ldots ,A_{n}} whose intersection has not probability zero, the chain rule states P ( A 1 ∩ A 2 ∩ … ∩ A n ) = P ( A n ∣ A 1 ∩ … ∩ A n − 1 ) P ( A 1 ∩ … ∩ A n − 1 ) = P ( A n ∣ A 1 ∩ … ∩ A n − 1 ) P ( A n − 1 ∣ A 1 ∩ … ∩ A n − 2 ) P ( A 1 ∩ … ∩ A n − 2 ) = P ( A n ∣ A 1 ∩ … ∩ A n − 1 ) P ( A n − 1 ∣ A 1 ∩ … ∩ A n − 2 ) ⋅ … ⋅ P ( A 3 ∣ A 1 ∩ A 2 ) P ( A 2 ∣ A 1 ) P ( A 1 ) = P ( A 1 ) P ( A 2 ∣ A 1 ) P ( A 3 ∣ A 1 ∩ A 2 ) ⋅ … ⋅ P ( A n ∣ A 1 ∩ ⋯ ∩ A n − 1 ) = ∏ k = 1 n P ( A k ∣ A 1 ∩ ⋯ ∩ A k − 1 ) = ∏ k = 1 n P ( A k \| ⋂ j = 1 k − 1 A j ) . {\displaystyle {\begin{aligned}\mathbb {P} \left(A_{1}\cap A_{2}\cap \ldots \cap A_{n}\right)&=\mathbb {P} \left(A_{n}\mid A_{1}\cap \ldots \cap A_{n-1}\right)\mathbb {P} \left(A_{1}\cap \ldots \cap A_{n-1}\right)\\&=\mathbb {P} \left(A_{n}\mid A_{1}\cap \ldots \cap A_{n-1}\right)\mathbb {P} \left(A_{n-1}\mid A_{1}\cap \ldots \cap A_{n-2}\right)\mathbb {P} \left(A_{1}\cap \ldots \cap A_{n-2}\right)\\&=\mathbb {P} \left(A_{n}\mid A_{1}\cap \ldots \cap A_{n-1}\right)\mathbb {P} \left(A_{n-1}\mid A_{1}\cap \ldots \cap A_{n-2}\right)\cdot \ldots \cdot \mathbb {P} (A_{3}\mid A_{1}\cap A_{2})\mathbb {P} (A_{2}\mid A_{1})\mathbb {P} (A_{1})\\&=\mathbb {P} (A_{1})\mathbb {P} (A_{2}\mid A_{1})\mathbb {P} (A_{3}\mid A_{1}\cap A_{2})\cdot \ldots \cdot \mathbb {P} (A_{n}\mid A_{1}\cap \dots \cap A_{n-1})\\&=\prod _{k=1}^{n}\mathbb {P} (A_{k}\mid A_{1}\cap \dots \cap A_{k-1})\\&=\prod _{k=1}^{n}\mathbb {P} \left(A_{k}\,{\Bigg \|}\,\bigcap _{j=1}^{k-1}A_{j}\right).\end{aligned}}} ==== Example 1 ==== For n = 4 {\displaystyle n=4} , i.e. four events, the chain rule reads P ( A 1 ∩ A 2 ∩ A 3 ∩ A 4 ) = P ( A 4 ∣ A 3 ∩ A 2 ∩ A 1 ) P ( A 3 ∩ A 2 ∩ A 1 ) = P ( A 4 ∣ A 3 ∩ A 2 ∩ A 1 ) P ( A 3 ∣ A 2 ∩ A 1 ) P ( A 2 ∩ A 1 ) = P ( A 4 ∣ A 3 ∩ A 2 ∩ A 1 ) P ( A 3 ∣ A 2 ∩ A 1 ) P ( A 2 ∣ A 1 ) P ( A 1 ) {\displaystyle {\begin{aligned}\mathbb {P} (A_{1}\cap A_{2}\cap A_{3}\cap A_{4})&=\mathbb {P} (A_{4}\mid A_{3}\cap A_{2}\cap A_{1})\mathbb {P} (A_{3}\cap A_{2}\cap A_{1})\\&=\mathbb {P} (A_{4}\mid A_{3}\cap A_{2}\cap A_{1})\mathbb {P} (A_{3}\mid A_{2}\cap A_{1})\mathbb {P} (A_{2}\cap A_{1})\\&=\mathbb {P} (A_{4}\mid A_{3}\cap A_{2}\cap A_{1})\mathbb {P} (A_{3}\mid A_{2}\cap A_{1})\mathbb {P} (A_{2}\mid A_{1})\mathbb {P} (A_{1})\end{aligned}}} . ==== Example 2 ==== We randomly draw 4 cards (one at a time) without replacement from deck with 52 cards. What is the probability that we have picked 4 aces? First, we set A n := { draw an ace in the n th try } {\textstyle A_{n}:=\left\{{\text{draw an ace in the }}n^{\text{th}}{\text{ try}}\right\}} . Obviously, we get the following probabilities P ( A 1 ) = 4 52 , P ( A 2 ∣ A 1 ) = 3 51 , P ( A 3 ∣ A 1 ∩ A 2 ) = 2 50 , P ( A 4 ∣ A 1 ∩ A 2 ∩ A 3 ) = 1 49 {\displaystyle \mathbb {P} (A_{1})={\frac {4}{52}},\qquad \mathbb {P} (A_{2}\mid A_{1})={\frac {3}{51}},\qquad \mathbb {P} (A_{3}\mid A_{1}\cap A_{2})={\frac {2}{50}},\qquad \mathbb {P} (A_{4}\mid A_{1}\cap A_{2}\cap A_{3})={\frac {1}{49}}} . Applying the chain rule, P ( A 1 ∩ A 2 ∩ A 3 ∩ A 4 ) = 4 52 ⋅ 3 51 ⋅ 2 50 ⋅ 1 49 = 24 6497400 {\displaystyle \mathbb {P} (A_{1}\cap A_{2}\cap A_{3}\cap A_{4})={\frac {4}{52}}\cdot {\frac {3}{51}}\cdot {\frac {2}{50}}\cdot {\frac {1}{49}}={\frac {24}{6497400}}} . === Statement of the theorem and proof === Let ( Ω , A , P ) {\displaystyle (\Omega ,{\mathcal {A}},\mathbb {P} )} be a probability space. Recall that the conditional probability of an A ∈ A {\displaystyle A\in {\mathcal {A}}} given B ∈ A {\displaystyle B\in {\mathcal {A}}} is defined as P ( A ∣ B ) := { P ( A ∩ B ) P ( B ) , P ( B ) > 0 , 0 P ( B ) = 0. {\displaystyle {\begin{aligned}\mathbb {P} (A\mid B):={\begin{cases}{\frac {\mathbb {P} (A\cap B)}{\mathbb {P} (B)}},&\mathbb {P} (B)>0,\\0&\mathbb {P} (B)=0.\end{cases}}\end{aligned}}} Then we have the following theorem. == Chain rule for discrete random variables == === Two random variables === For two discrete random variables X , Y {\displaystyle X,Y} , we use the events A := { X = x } {\displaystyle A:=\{X=x\}} and B := { Y = y } {\displaystyle B:=\{Y=y\}} in the definition above, and find the joint distribution as P ( X = x , Y = y ) = P ( X = x ∣ Y = y ) P ( Y = y ) , {\displaystyle \mathbb {P} (X=x,Y=y)=\mathbb {P} (X=x\mid Y=y)\mathbb {P} (Y=y),} or P ( X , Y ) ( x , y ) = P X ∣ Y ( x ∣ y ) P Y ( y ) , {\displaystyle \mathbb {P} _{(X,Y)}(x,y)=\mathbb {P} _{X\mid Y}(x\mid y)\mathbb {P} _{Y}(y),} where P X ( x ) := P ( X = x ) {\displaystyle \mathbb {P} _{X}(x):=\mathbb {P} (X=x)} is the probability distribution of X {\displaystyle X} and P X ∣ Y ( x ∣ y ) {\displaystyle \mathbb {P} _{X\mid Y}(x\mid y)} conditional probability distribution of X {\displaystyle X} given Y {\displaystyle Y} . === Finitely many random variables === Let X 1 , … , X n {\displaystyle X_{1},\ldots ,X_{n}} be random variables and x 1 , … , x n ∈ R {\displaystyle x_{1},\dots ,x_{n}\in \mathbb {R} } . By the definition of the conditional probability, P ( X n = x n , … , X 1 = x 1 ) = P ( X n = x n \| X n − 1 = x n − 1 , … , X 1 = x 1 ) P ( X n − 1 = x n − 1 , … , X 1 = x 1 ) {\displaystyle \mathbb {P} \left(X_{n}=x_{n},\ldots ,X_{1}=x_{1}\right)=\mathbb {P} \left(X_{n}=x_{n}\|X_{n-1}=x_{n-1},\ldots ,X_{1}=x_{1}\right)\mathbb {P} \left(X_{n-1}=x_{n-1},\ldots ,X_{1}=x_{1}\right)} and using the chain rule, where we set A k := { X k = x k } {\displaystyle A_{k}:=\{X_{k}=x_{k}\}} , we can find the joint distribution as P ( X 1 = x 1 , … X n = x n ) = P ( X 1 = x 1 ∣ X 2 = x 2 , … , X n = x n ) P ( X 2 = x 2 , … , X n = x n ) = P ( X 1 = x 1 ) P ( X 2 = x 2 ∣ X 1 = x 1 ) P ( X 3 = x 3 ∣ X 1 = x 1 , X 2 = x 2 ) ⋅ … ⋅ P ( X n = x n ∣ X 1 = x 1 , … , X n − 1 = x n − 1 ) {\displaystyle {\begin{aligned}\mathbb {P} \left(X_{1}=x_{1},\ldots X_{n}=x_{n}\right)&=\mathbb {P} \left(X_{1}=x_{1}\mid X_{2}=x_{2},\ldots ,X_{n}=x_{n}\right)\mathbb {P} \left(X_{2}=x_{2},\ldots ,X_{n}=x_{n}\right)\\&=\mathbb {P} (X_{1}=x_{1})\mathbb {P} (X_{2}=x_{2}\mid X_{1}=x_{1})\mathbb {P} (X_{3}=x_{3}\mid X_{1}=x_{1},X_{2}=x_{2})\cdot \ldots \\&\qquad \cdot \mathbb {P} (X_{n}=x_{n}\mid X_{1}=x_{1},\dots ,X_{n-1}=x_{n-1})\\\end{aligned}}} === Example === For n = 3 {\displaystyle n=3} , i.e. considering three random variables. Then, the chain rule reads P ( X 1 , X 2 , X 3 ) ( x 1 , x 2 , x 3 ) = P ( X 1 = x 1 , X 2 = x 2 , X 3 = x 3 ) = P ( X 3 = x 3 ∣ X 2 = x 2 , X 1 = x 1 ) P ( X 2 = x 2 , X 1 = x 1 ) = P ( X 3 = x 3 ∣ X 2 = x 2 , X 1 = x 1 ) P ( X 2 = x 2 ∣ X 1 = x 1 ) P ( X 1 = x 1 ) = P X 3 ∣ X 2 , X 1 ( x 3 ∣ x 2 , x 1 ) P X 2 ∣ X 1 ( x 2 ∣ x 1 ) P X 1 ( x 1 ) . {\displaystyle {\begin{aligned}\mathbb {P} _{(X_{1},X_{2},X_{3})}(x_{1},x_{2},x_{3})&=\mathbb {P} (X_{1}=x_{1},X_{2}=x_{2},X_{3}=x_{3})\\&=\mathbb {P} (X_{3}=x_{3}\mid X_{2}=x_{2},X_{1}=x_{1})\mathbb {P} (X_{2}=x_{2},X_{1}=x_{1})\\&=\mathbb {P} (X_{3}=x_{3}\mid X_{2}=x_{2},X_{1}=x_{1})\mathbb {P} (X_{2}=x_{2}\mid X_{1}=x_{1})\mathbb {P} (X_{1}=x_{1})\\&=\mathbb {P} _{X_{3}\mid X_{2},X_{1}}(x_{3}\mid x_{2},x_{1})\mathbb {P} _{X_{2}\mid X_{1}}(x_{2}\mid x_{1})\mathbb {P} _{X_{1}}(x_{1}).\end{aligned}}} == Bibliography == René L. Schilling (2021), Measure, Integral, Probability & Processes - Probab(ilistical)ly the Theoretical Minimum (1 ed.), Technische Universität Dresden, Germany, ISBN 979-8-5991-0488-9{{citation}}: CS1 maint: location missing publisher (link) William Feller (1968), An Introduction to Probability Theory and Its Applications, vol. I (3 ed.), New York / London / Sydney: Wiley, ISBN 978-0-471-25708-0 Russell, Stuart J.; Norvig, Peter (2003), Artificial Intelligence: A Modern Approach (2nd ed.), Upper Saddle River, New Jersey: Prentice Hall, ISBN 0-13-790395-2, p. 496. == References ==
Wikipedia:Chakravala method#0	The chakravala method (Sanskrit: चक्रवाल विधि) is a cyclic algorithm to solve indeterminate quadratic equations, including Pell's equation. It is commonly attributed to Bhāskara II, (c. 1114 – 1185 CE) although some attribute it to Jayadeva (c. 950 ~ 1000 CE). Jayadeva pointed out that Brahmagupta's approach to solving equations of this type could be generalized, and he then described this general method, which was later refined by Bhāskara II in his Bijaganita treatise. He called it the Chakravala method: chakra meaning "wheel" in Sanskrit, a reference to the cyclic nature of the algorithm. C.-O. Selenius held that no European performances at the time of Bhāskara, nor much later, exceeded its marvellous height of mathematical complexity. This method is also known as the cyclic method and contains traces of mathematical induction. == History == Chakra in Sanskrit means cycle. As per popular legend, Chakravala indicates a mythical range of mountains which orbits around the Earth like a wall and not limited by light and darkness. Brahmagupta in 628 CE studied indeterminate quadratic equations, including Pell's equation x 2 = N y 2 + 1 , {\displaystyle \,x^{2}=Ny^{2}+1,} for minimum integers x and y. Brahmagupta could solve it for several N, but not all. Jayadeva and Bhaskara offered the first complete solution to the equation, using the chakravala method to find for x 2 = 61 y 2 + 1 , {\displaystyle \,x^{2}=61y^{2}+1,} the solution x = 1766319049 , y = 226153980. {\displaystyle \,x=1766319049,y=226153980.} This case was notorious for its difficulty, and was first solved in Europe by Brouncker in 1657–58 in response to a challenge by Fermat, using continued fractions. A method for the general problem was first completely described rigorously by Lagrange in 1766. Lagrange's method, however, requires the calculation of 21 successive convergents of the simple continued fraction for the square root of 61, while the chakravala method is much simpler. Selenius, in his assessment of the chakravala method, states "The method represents a best approximation algorithm of minimal length that, owing to several minimization properties, with minimal effort and avoiding large numbers automatically produces the best solutions to the equation. The chakravala method anticipated the European methods by more than a thousand years. But no European performances in the whole field of algebra at a time much later than Bhaskara's, nay nearly equal up to our times, equalled the marvellous complexity and ingenuity of chakravala." Hermann Hankel calls the chakravala method "the finest thing achieved in the theory of numbers before Lagrange." == The method == From Brahmagupta's identity, we observe that for given N, ( x 1 x 2 + N y 1 y 2 ) 2 − N ( x 1 y 2 + x 2 y 1 ) 2 = ( x 1 2 − N y 1 2 ) ( x 2 2 − N y 2 2 ) {\displaystyle (x_{1}x_{2}+Ny_{1}y_{2})^{2}-N(x_{1}y_{2}+x_{2}y_{1})^{2}=(x_{1}^{2}-Ny_{1}^{2})(x_{2}^{2}-Ny_{2}^{2})} For the equation x 2 − N y 2 = k {\displaystyle x^{2}-Ny^{2}=k} , this allows the "composition" (samāsa) of two solution triples ( x 1 , y 1 , k 1 ) {\displaystyle (x_{1},y_{1},k_{1})} and ( x 2 , y 2 , k 2 ) {\displaystyle (x_{2},y_{2},k_{2})} into a new triple ( x 1 x 2 + N y 1 y 2 , x 1 y 2 + x 2 y 1 , k 1 k 2 ) . {\displaystyle (x_{1}x_{2}+Ny_{1}y_{2}\,,\,x_{1}y_{2}+x_{2}y_{1}\,,\,k_{1}k_{2}).} In the general method, the main idea is that any triple ( a , b , k ) {\displaystyle (a,b,k)} (that is, one which satisfies a 2 − N b 2 = k {\displaystyle a^{2}-Nb^{2}=k} ) can be composed with the trivial triple ( m , 1 , m 2 − N ) {\displaystyle (m,1,m^{2}-N)} to get the new triple ( a m + N b , a + b m , k ( m 2 − N ) ) {\displaystyle (am+Nb,a+bm,k(m^{2}-N))} for any m. Assuming we started with a triple for which gcd ( a , b ) = 1 {\displaystyle \gcd(a,b)=1} , this can be scaled down by k (this is Bhaskara's lemma): a 2 − N b 2 = k ⇒ ( a m + N b k ) 2 − N ( a + b m k ) 2 = m 2 − N k {\displaystyle a^{2}-Nb^{2}=k\Rightarrow \left({\frac {am+Nb}{k}}\right)^{2}-N\left({\frac {a+bm}{k}}\right)^{2}={\frac {m^{2}-N}{k}}} Since the signs inside the squares do not matter, the following substitutions are possible: a ← a m + N b \| k \| , b ← a + b m \| k \| , k ← m 2 − N k {\displaystyle a\leftarrow {\frac {am+Nb}{\|k\|}},b\leftarrow {\frac {a+bm}{\|k\|}},k\leftarrow {\frac {m^{2}-N}{k}}} When a positive integer m is chosen so that (a + bm)/k is an integer, so are the other two numbers in the triple. Among such m, the method chooses one that minimizes the absolute value of m2 − N and hence that of (m2 − N)/k. Then the substitution relations are applied for m equal to the chosen value. This results in a new triple (a, b, k). The process is repeated until a triple with k = 1 {\displaystyle k=1} is found. This method always terminates with a solution (proved by Lagrange in 1768). Optionally, we can stop when k is ±1, ±2, or ±4, as Brahmagupta's approach gives a solution for those cases. == Brahmagupta's composition method == In AD 628, Brahmagupta discovered a general way to find x {\displaystyle x} and y {\displaystyle y} of x 2 = N y 2 + 1 , {\displaystyle x^{2}=Ny^{2}+1,} when given a 2 = N b 2 + k {\displaystyle a^{2}=Nb^{2}+k} , when k is ±1, ±2, or ±4. === k = ±1 === Using Brahmagupta's identity to compose the triple ( a , b , k ) {\displaystyle (a,b,k)} with itself: ( a 2 + N b 2 ) 2 − N ( 2 a b ) 2 = k 2 {\displaystyle (a^{2}+Nb^{2})^{2}-N(2ab)^{2}=k^{2}} ⇒ {\displaystyle \Rightarrow } ( 2 a 2 − k ) 2 − N ( 2 a b ) 2 = k 2 {\displaystyle (2a^{2}-k)^{2}-N(2ab)^{2}=k^{2}} The new triple can be expressed as ( 2 a 2 − k , 2 a b , k 2 ) {\displaystyle (2a^{2}-k,2ab,k^{2})} . Substituting k = − 1 {\displaystyle k=-1} gives a solution: x = 2 a 2 + 1 , y = 2 a b {\displaystyle x=2a^{2}+1,y=2ab} For k = 1 {\displaystyle k=1} , the original ( a , b ) {\displaystyle (a,b)} was already a solution. Substituting k = 1 {\displaystyle k=1} yields a second: x = 2 a 2 − 1 , y = 2 a b {\displaystyle x=2a^{2}-1,y=2ab} === k = ±2 === Again using the equation, ( 2 a 2 − k ) 2 − N ( 2 a b ) 2 = k 2 {\displaystyle (2a^{2}-k)^{2}-N(2ab)^{2}=k^{2}} ⇒ {\displaystyle \Rightarrow } ( 2 a 2 − k k ) 2 − N ( 2 a b k ) 2 = 1 {\displaystyle \left({\frac {2a^{2}-k}{k}}\right)^{2}-N\left({\frac {2ab}{k}}\right)^{2}=1} Substituting k = 2 {\displaystyle k=2} , x = a 2 − 1 , y = a b {\displaystyle x=a^{2}-1,y=ab} Substituting k = − 2 {\displaystyle k=-2} , x = a 2 + 1 , y = a b {\displaystyle x=a^{2}+1,y=ab} === k = 4 === Substituting k = 4 {\displaystyle k=4} into the equation ( 2 a 2 − k k ) 2 − N ( 2 a b k ) 2 = 1 {\displaystyle ({\frac {2a^{2}-k}{k}})^{2}-N({\frac {2ab}{k}})^{2}=1} creates the triple ( a 2 − 2 2 , a b 2 , 1 ) {\displaystyle ({\frac {a^{2}-2}{2}},{\frac {ab}{2}},1)} . Which is a solution if a {\displaystyle a} is even: x = a 2 − 2 2 , y = a b 2 {\displaystyle x={\frac {a^{2}-2}{2}},y={\frac {ab}{2}}} If a is odd, start with the equations ( a 2 ) 2 − N ( b 2 ) 2 = 1 {\displaystyle ({\frac {a}{2}})^{2}-N({\frac {b}{2}})^{2}=1} and ( 2 a 2 − 4 4 ) 2 − N ( 2 a b 4 ) 2 = 1 {\displaystyle ({\frac {2a^{2}-4}{4}})^{2}-N({\frac {2ab}{4}})^{2}=1} . Leading to the triples ( a 2 , b 2 , 1 ) {\displaystyle ({\frac {a}{2}},{\frac {b}{2}},1)} and ( a 2 − 2 2 , a b 2 , 1 ) {\displaystyle ({\frac {a^{2}-2}{2}},{\frac {ab}{2}},1)} . Composing the triples gives ( a 2 ( a 2 − 3 ) ) 2 − N ( b 2 ( a 2 − 1 ) ) 2 = 1 {\displaystyle ({\frac {a}{2}}(a^{2}-3))^{2}-N({\frac {b}{2}}(a^{2}-1))^{2}=1} When a {\displaystyle a} is odd, x = a 2 ( a 2 − 3 ) , y = b 2 ( a 2 − 1 ) {\displaystyle x={\frac {a}{2}}(a^{2}-3),y={\frac {b}{2}}(a^{2}-1)} === k = -4 === When k = − 4 {\displaystyle k=-4} , then ( a 2 ) 2 − N ( b 2 ) 2 = − 1 {\displaystyle ({\frac {a}{2}})^{2}-N({\frac {b}{2}})^{2}=-1} . Composing with itself yields ( a 2 + N b 2 4 ) 2 − N ( a b 2 ) 2 = 1 {\displaystyle ({\frac {a^{2}+Nb^{2}}{4}})^{2}-N({\frac {ab}{2}})^{2}=1} ⇒ {\displaystyle \Rightarrow } ( a 2 + 2 2 ) 2 − N ( a b 2 ) 2 = 1 {\displaystyle ({\frac {a^{2}+2}{2}})^{2}-N({\frac {ab}{2}})^{2}=1} . Again composing itself yields ( ( a 2 + 2 ) 2 + N a 2 b 2 ) 4 ) 2 − N ( a b ( a 2 + 2 ) 2 ) 2 = 1 {\displaystyle ({\frac {(a^{2}+2)^{2}+Na^{2}b^{2})}{4}})^{2}-N({\frac {ab(a^{2}+2)}{2}})^{2}=1} ⇒ {\displaystyle \Rightarrow } ( a 4 + 4 a 2 + 2 2 ) 2 − N ( a b ( a 2 + 2 ) 2 ) 2 = 1 {\displaystyle ({\frac {a^{4}+4a^{2}+2}{2}})^{2}-N({\frac {ab(a^{2}+2)}{2}})^{2}=1} Finally, from the earlier equations, compose the triples ( a 2 + 2 2 , a b 2 , 1 ) {\displaystyle ({\frac {a^{2}+2}{2}},{\frac {ab}{2}},1)} and ( a 4 + 4 a 2 + 2 2 , a b ( a 2 + 2 ) 2 , 1 ) {\displaystyle ({\frac {a^{4}+4a^{2}+2}{2}},{\frac {ab(a^{2}+2)}{2}},1)} , to get ( ( a 2 + 2 ) ( a 4 + 4 a 2 + 2 ) + N a 2 b 2 ( a 2 + 2 ) 4 ) 2 − N ( a b ( a 4 + 4 a 2 + 3 ) 2 ) 2 = 1 {\displaystyle ({\frac {(a^{2}+2)(a^{4}+4a^{2}+2)+Na^{2}b^{2}(a^{2}+2)}{4}})^{2}-N({\frac {ab(a^{4}+4a^{2}+3)}{2}})^{2}=1} ⇒ {\displaystyle \Rightarrow } ( ( a 2 + 2 ) ( a 4 + 4 a 2 + 1 ) 2 ) 2 − N ( a b ( a 2 + 3 ) ( a 2 + 1 ) 2 ) 2 = 1 {\displaystyle ({\frac {(a^{2}+2)(a^{4}+4a^{2}+1)}{2}})^{2}-N({\frac {ab(a^{2}+3)(a^{2}+1)}{2}})^{2}=1} ⇒ {\displaystyle \Rightarrow } ( ( a 2 + 2 ) [ ( a 2 + 1 ) ( a 2 + 3 ) − 2 ) ] 2 ) 2 − N ( a b ( a 2 + 3 ) ( a 2 + 1 ) 2 ) 2 = 1 {\displaystyle ({\frac {(a^{2}+2)[(a^{2}+1)(a^{2}+3)-2)]}{2}})^{2}-N({\frac {ab(a^{2}+3)(a^{2}+1)}{2}})^{2}=1} . This give us the solutions x = ( a 2 + 2 ) [ ( a 2 + 1 ) ( a 2 + 3 ) − 2 ) ] 2 y = a b ( a 2 + 3 ) ( a 2 + 1 ) 2 {\displaystyle x={\frac {(a^{2}+2)[(a^{2}+1)(a^{2}+3)-2)]}{2}}y={\frac {ab(a^{2}+3)(a^{2}+1)}{2}}} (Note, k = − 4 {\displaystyle k=-4} is useful to find a solution to Pell's Equation, but it is not always the smallest integer pair. e.g. 36 2 − 52 ∗ 5 2 = − 4 {\displaystyle 36^{2}-52*5^{2}=-4} . The equation will give you x = 1093436498 , y = 151632270 {\displaystyle x=1093436498,y=151632270} , which when put into Pell's Equation yields 1195601955878350801 − 1195601955878350800 = 1 {\displaystyle 1195601955878350801-1195601955878350800=1} , which works, but so does x = 649 , y = 90 {\displaystyle x=649,y=90} for N = 52 {\displaystyle N=52} . == Examples == === n = 61 === The n = 61 case (determining an integer solution satisfying a 2 − 61 b 2 = 1 {\displaystyle a^{2}-61b^{2}=1} ), issued as a challenge by Fermat many centuries later, was given by Bhaskara as an example. We start with a solution a 2 − 61 b 2 = k {\displaystyle a^{2}-61b^{2}=k} for any k found by any means. In this case we can let b be 1, thus, since 8 2 − 61 ⋅ 1 2 = 3 {\displaystyle 8^{2}-61\cdot 1^{2}=3} , we have the triple ( a , b , k ) = ( 8 , 1 , 3 ) {\displaystyle (a,b,k)=(8,1,3)} . Composing it with ( m , 1 , m 2 − 61 ) {\displaystyle (m,1,m^{2}-61)} gives the triple ( 8 m + 61 , 8 + m , 3 ( m 2 − 61 ) ) {\displaystyle (8m+61,8+m,3(m^{2}-61))} , which is scaled down (or Bhaskara's lemma is directly used) to get: ( 8 m + 61 3 , 8 + m 3 , m 2 − 61 3 ) . {\displaystyle \left({\frac {8m+61}{3}},{\frac {8+m}{3}},{\frac {m^{2}-61}{3}}\right).} For 3 to divide 8 + m {\displaystyle 8+m} and \| m 2 − 61 \| {\displaystyle \|m^{2}-61\|} to be minimal, we choose m = 7 {\displaystyle m=7} , so that we have the triple ( 39 , 5 , − 4 ) {\displaystyle (39,5,-4)} . Now that k is −4, we can use Brahmagupta's idea: it can be scaled down to the rational solution ( 39 / 2 , 5 / 2 , − 1 ) {\displaystyle (39/2,5/2,-1)\,} , which composed with itself three times, with m = 7 , 11 , 9 {\displaystyle m={7,11,9}} respectively, when k becomes square and scaling can be applied, this gives ( 1523 / 2 , 195 / 2 , 1 ) {\displaystyle (1523/2,195/2,1)\,} . Finally, such procedure can be repeated until the solution is found (requiring 9 additional self-compositions and 4 additional square-scalings): ( 1766319049 , 226153980 , 1 ) {\displaystyle (1766319049,\,226153980,\,1)} . This is the minimal integer solution. === n = 67 === Suppose we are to solve x 2 − 67 y 2 = 1 {\displaystyle x^{2}-67y^{2}=1} for x and y. We start with a solution a 2 − 67 b 2 = k {\displaystyle a^{2}-67b^{2}=k} for any k found by any means; in this case we can let b be 1, thus producing 8 2 − 67 ⋅ 1 2 = − 3 {\displaystyle 8^{2}-67\cdot 1^{2}=-3} . At each step, we find an m > 0 such that k divides a + bm, and \|m2 − 67\| is minimal. We then update a, b, and k to a m + N b \| k \| , a + b m \| k \| {\displaystyle {\frac {am+Nb}{\|k\|}},{\frac {a+bm}{\|k\|}}} and m 2 − N k {\displaystyle {\frac {m^{2}-N}{k}}} respectively. First iteration We have ( a , b , k ) = ( 8 , 1 , − 3 ) {\displaystyle (a,b,k)=(8,1,-3)} . We want a positive integer m such that k divides a + bm, i.e. 3 divides 8 + m, and \|m2 − 67\| is minimal. The first condition implies that m is of the form 3t + 1 (i.e. 1, 4, 7, 10,… etc.), and among such m, the minimal value is attained for m = 7. Replacing (a, b, k) with ( a m + N b \| k \| , a + b m \| k \| , m 2 − N k ) {\displaystyle \left({\frac {am+Nb}{\|k\|}},{\frac {a+bm}{\|k\|}},{\frac {m^{2}-N}{k}}\right)} , we get the new values a = ( 8 ⋅ 7 + 67 ⋅ 1 ) / 3 = 41 , b = ( 8 + 1 ⋅ 7 ) / 3 = 5 , k = ( 7 2 − 67 ) / ( − 3 ) = 6 {\displaystyle a=(8\cdot 7+67\cdot 1)/3=41,b=(8+1\cdot 7)/3=5,k=(7^{2}-67)/(-3)=6} . That is, we have the new solution: 41 2 − 67 ⋅ ( 5 ) 2 = 6. {\displaystyle 41^{2}-67\cdot (5)^{2}=6.} At this point, one round of the cyclic algorithm is complete. Second iteration We now repeat the process. We have ( a , b , k ) = ( 41 , 5 , 6 ) {\displaystyle (a,b,k)=(41,5,6)} . We want an m > 0 such that k divides a + bm, i.e. 6 divides 41 + 5m, and \|m2 − 67\| is minimal. The first condition implies that m is of the form 6t + 5 (i.e. 5, 11, 17,… etc.), and among such m, \|m2 − 67\| is minimal for m = 5. This leads to the new solution a = (41⋅5 + 67⋅5)/6, etc.: 90 2 − 67 ⋅ 11 2 = − 7. {\displaystyle 90^{2}-67\cdot 11^{2}=-7.} Third iteration For 7 to divide 90 + 11m, we must have m = 2 + 7t (i.e. 2, 9, 16,… etc.) and among such m, we pick m = 9. 221 2 − 67 ⋅ 27 2 = − 2. {\displaystyle 221^{2}-67\cdot 27^{2}=-2.} Final solution At this point, we could continue with the cyclic method (and it would end, after seven iterations), but since the right-hand side is among ±1, ±2, ±4, we can also use Brahmagupta's observation directly. Composing the triple (221, 27, −2) with itself, we get ( 221 2 + 67 ⋅ 27 2 2 ) 2 − 67 ⋅ ( 221 ⋅ 27 ) 2 = 1 , {\displaystyle \left({\frac {221^{2}+67\cdot 27^{2}}{2}}\right)^{2}-67\cdot (221\cdot 27)^{2}=1,} that is, we have the integer solution: 48842 2 − 67 ⋅ 5967 2 = 1. {\displaystyle 48842^{2}-67\cdot 5967^{2}=1.} This equation approximates 67 {\displaystyle {\sqrt {67}}} as 48842 5967 {\displaystyle {\frac {48842}{5967}}} to within a margin of about 2 × 10 − 9 {\displaystyle 2\times 10^{-9}} . == Notes == == References == Florian Cajori (1918), Origin of the Name "Mathematical Induction", The American Mathematical Monthly 25 (5), p. 197-201. George Gheverghese Joseph, The Crest of the Peacock: Non-European Roots of Mathematics (1975). G. R. Kaye, "Indian Mathematics", Isis 2:2 (1919), p. 326–356. Clas-Olaf Selenius, "Rationale of the chakravala process of Jayadeva and Bhaskara II" Archived 2021-11-30 at the Wayback Machine, Historia Mathematica 2 (1975), pp. 167–184. Clas-Olaf Selenius, "Kettenbruchtheoretische Erklärung der zyklischen Methode zur Lösung der Bhaskara-Pell-Gleichung", Acta Acad. Abo. Math. Phys. 23 (10) (1963), pp. 1–44. Hoiberg, Dale & Ramchandani, Indu (2000). Students' Britannica India. Mumbai: Popular Prakashan. ISBN 0-85229-760-2 Goonatilake, Susantha (1998). Toward a Global Science: Mining Civilizational Knowledge. Indiana: Indiana University Press. ISBN 0-253-33388-1. Kumar, Narendra (2004). Science in Ancient India. Delhi: Anmol Publications Pvt Ltd. ISBN 81-261-2056-8 Ploker, Kim (2007) "Mathematics in India". The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook New Jersey: Princeton University Press. ISBN 0-691-11485-4 Edwards, Harold (1977). Fermat's Last Theorem. New York: Springer. ISBN 0-387-90230-9. == External links == Introduction to chakravala
Wikipedia:Chalkdust (magazine)#0	Hollis Urban Lester Liverpool, better known as Chalkdust or Chalkie (born 1940 in Chaguaramas, Trinidad), is a leading calypsonian from Trinidad and Tobago. He has been singing calypso since 1967 and has recorded more than 300 calypsos. == Awards == He is a nine-time winner of Trinidad's Calypso Monarch competition, most recently in 2017. Chalkdust's winning song performances for these most acclaimed Calypso Crowns are: 1976 *("No Smut For Me" and "Ah Put On Meh Guns Again"); 1977 ("My Way Of Protest" and "Shango Vision"); 1981 ("Ah Can't Make" and "My Kind of Worry"); 1989 ("Chauffeur Wanted" and "Carnival Is The Answer"); 1993 ("Kaiso Sick in de Hospital" and "Misconceptions"); 2004 ("Fish Monger" and "Trinidad in the Cemetery"); 2005 ("I in Town Too Long" and "Ah Doh Rhyme"); lastly in 2009 (Doh Touch My Heart). Before only the Mighty Sparrow had more wins in that competition, until 2009 when Chalkdust also achieved eight wins in the Calypso Monarch. Chalkdust also won Carifesta in 1976, the World Calypso King title in St. Thomas, US Virgin Islands, eight times, and the Calypso King of the World in New York City on the two occasions when that competition was held. === 1976 Calypso Monarch === An error has been perpetuated in which Chalkdust is listed in many fora as having sung ‘Three Blind Mice’ as one of his compositions in 1976, the first year he won the calypso monarch title. In fact, his other calypso was ‘No Smut For Me’. That competition was broadcast ‘live’ on Trinidad and Tobago Television. The judges score sheets at the National Archives on St Vincent Street, Port-of-Spain, record him as having sung – and being judged on – 'No Smut For Me' and 'Ah Put On Me Guns Again'; and Chalkdust himself has confirmed this since. In a piece in the series, ‘Interview With An Icon’, published in the online magazine, ‘Paradise Pulse’, which was done prior to him winning his ninth title in 2017, Chalkdust was asked the following: “You have been the Calypso Monarch of Trinidad and Tobago on eight occasions, a record which you share with the Mighty Sparrow. Which win was your most memorable and why?” He replied: “Tough question. All my wins mean the world to me. If you were to push bamboo under my nails, I would answer that it was my first win [in 1976] when I sang the songs ‘Ah Put On Meh Guns Again’ and ‘Why Smut’ [listed on the record as, ‘No Smut For Me’]. That put me on the map and people sat up and took me seriously.” This gives the lie to those listings that record him as having sung ‘Three Blind Mice’ that year. Similarly, in 1977, his offerings were,'Shango Vision' and 'My Way Of Protest'. 'Juba Dubai' was not sung at the 1977 calypso monarch competition. == Other activities == Chalkdust, who holds a Ph.D. in history and ethnomusicology from the University of Michigan, is an assistant professor of history at the University of the Virgin Islands, and frequently lectures and offers workshops on the history and culture of calypso music. He is the author of the books Rituals of Power and Rebellion (2001) and From the Horse's Mouth (2003), a socio-cultural history of calypso from 1900 to 2003. === Bibliography === Liverpool, Hollis (2001). Rituals of Power and Rebellion: The Carnival Tradition in Trinidad and Tobago, 1763-1962. Frontline Distribution International. ISBN 0948390808. Liverpool, Hollis (2003). From the Horse's Mouth: An Analysis of Certain Significant Aspects in the Development of the Calypso and Society as Gleaned from Personal Communication with Some Outstanding Calypsonians. Juba Publications. ISBN 9768194138. == References == == Sources == "Dr. Liverpool "Chalkdust" Favours Calypso In Schools". The Anguillan. 19 May 2006. == External links == "Mighty Chalkdust", bestoftrinidad.com == See also ==
Wikipedia:Chandler Davis#0	Horace Chandler Davis (August 12, 1926 – September 24, 2022) was an American-Canadian mathematician, writer, educator, and left-wing political activist. The socialist magazine Jacobin described Davis as "an internationally esteemed mathematician, a minor science fiction writer of note, and among the most celebrated political prisoners in the United States during the years of the high Cold War." == Background == Horace Chandler Davis, known as "Chan" by friends, was born on August 12, 1926, in Ithaca, New York, to parents Horace Bancroft Davis and Marian Rubins, both members of the Communist Party USA (CPUSA). He joined the Young Pioneers of America while in elementary school. Because of their politics, his parents moved frequently, so that Davis spent a year of his childhood in Brazil. In 1942, age 16, he received a Harvard National Scholarship. At Harvard, he joined the Astounding Science-Fiction Fanclub, whose members included John B. Michel, Frederik Pohl, Isaac Asimov, and Donald Wollheim. In 1943, Davis joined the Communist Party USA but left soon after so he could join the US Navy for officers training. In 1945, Davis graduated Harvard early and also received a commission from Naval Reserve Midshipman's School and spent a year in the US Navy as a minesweeper. In 1946, he returned to Harvard as a graduate student in Mathematics, rejoined the CPUSA, and joined the Federation of American Scientists, founded by former members of the Manhattan Project. In 1948, he supported Henry A. Wallace, Progressive Party candidate for the 1948 United States presidential election. In 1950, Davis received a doctorate in Mathematics from Harvard University. == Career == In 1950, Davis turned down an offer from the University of California Los Angeles (UCLA) due to loyalty oath requirements and accepted a position as instructor at the University of Michigan (UM). Davis left the CPUSA the following year. In 1953, Davis received a subpoena to appear before the House Un-American Activities Committee (HUAC). In 1954, UM suspended Davis, Clement Markert, and Mark Nickerson for refusing to cooperate with HUAC hearings held in Lansing, Michigan. Makert and Nickerson pled under the Fifth Amendment (right to avoid self-incrimination), while Davis pled the First Amendment (right to free speech). He hoped to establish a precedent that HUAC could not question witnesses on their political affiliations, but the US Supreme Court in 1959 refused to hear his case. After years of appeals, in 1960, Davis received a six-month jail sentence, served at a prison in Danbury, Connecticut. In 1962, Davis accepted a teaching position at the University of Toronto. He specialized in algebra and operator theory (a branch of functional analysis). In 1968, the Warsaw Pact invasion of Czechoslovakia ended Davis' CPUSA membership for good. He remained a political activist. For example, in 1971, he traveled to North Vietnam with other mathematicians including Laurent Schwartz. He also was an advocate for Palestinian rights. In July 2022, he publicly supported Russian mathematician Azat Miftakhov. === Mathematics === Davis' principal research investigations involved linear algebra and operator theory in Hilbert spaces. Furthermore, he made contributions to numerical analysis, geometry, and algebraic logic. He is one of the eponyms of the Davis–Kahan theorem and Bhatia–Davis inequality (along with Rajendra Bhatia). The Davis–Kahan–Weinberger dilation theorem is one of the landmark results in the dilation theory of Hilbert space operators and has found applications in many different areas. A PhD thesis titled "Backward Perturbation and Sensitivity Analysis of Structured Polynomial Eigenomial Eigenvalue Problem" is dedicated to this theorem. Davis wrote around eighty research papers in mathematics. Davis was a professor in the mathematics department of University of Michigan, working alongside Wilfred Kaplan. In the Mathematics Genealogy Project, he is listed as having 15 PhD (1964-2001), and 213 PhD descendants of his former doctoral students, with 107 being of them from his student John Benedetto (PhD 1964). He was one of the co-Editors-in-Chief of the Mathematical Intelligencer. In 2012 he became a fellow of the American Mathematical Society. He was part of the 2019 class of fellows of the Association for Women in Mathematics. === Fiction === Davis began his writing career in Astounding Science Fiction in 1946. From 1946 through 1962 he produced a spate of science fiction stories, mostly published there. One of the earliest, published May 1946, was The Nightmare, later the lead story in A Treasury of Science Fiction, edited by Groff Conklin; it argued for a national policy of decentralizing industry to evade nuclear attacks by terrorists. He also issued the fanzine "Blitherings" in the 1940s. He attended Torcon I, the 6th World Science Fiction Convention in 1948, appeared at the 2010 SFContario science fiction convention, and was Science Guest of Honor at the 2013 SFContario science fiction convention. === Politics === Davis came from a radical family and identified himself as a socialist and former member of the Communist Party of America. Davis—along with two other professors, Mark Nickerson and Clement Markert—refused to cooperate with the House Unamerican Activities Committee and was subsequently dismissed from the University of Michigan. Davis was then sentenced to a six-month prison term where he was able to do some research. A paper from this era has the following acknowledgement: Research supported in part by the Federal Prison System. Opinions expressed in this paper are the author's and are not necessarily those of the Bureau of Prisons. The Federal government released Davis from prison in 1960. After his release, Davis moved to Canada, where he worked at the University of Toronto. He also opposed the Vietnam War and was chair of the Toronto Anti-Draft Committee. In 1991, the University of Michigan Senate initiated the annual Davis, Markert, Nickerson Lecture on Academic and Intellectual Freedom. Recent speakers have included: Cass Sunstein (2008), Nadine Strossen (2007), Bill Keller (2006), Floyd Abrams (2005), and Noam Chomsky (2004). == Personal life and death == In 1948, Davis married Natalie Zemon Davis (author of the book and co-scriptwriter of the movie The Return of Martin Guerre); they had three children. H. Chandler Davis died on September 24, 2022. == Honors == Three mathematical theorems are named in Davis' honor: the Davis-Kahan theorem on how eigenspaces of an operator change under perturbation; the Bhatia-Davis inequality, bounding the variance of a given probability distribution on the real line; and the Davis–Kahan–Weinberger theorem on norm-preserving dilations of Hilbert space operators. A lecture in honor of his stand for his beliefs is now held at the University of Michigan, which had fired him. == Legacy == At his death, long-time friend Alan M. Wald wrote, "Chan Davis, who died last month at the age of 96, faced down McCarthyite blacklists and imprisonment to pursue a brilliant academic career. Davis knew how to change and learn from political experience, but he always remained loyal to his socialist principles." == Works == Books Edited Linear Algebra and Its Application Geometric Vein: The Coxeter Festschrift with Branko Grünbaum Coxeter Legacy: Reflections and Projections with Erich W. Ellers (2006) Shape of Content: Creative Writing in Mathematics and Science with Marjorie Wikler Senechal and Jan Zwicky (2008) Poetry Having Come This Far (1986) Prose It Walks in Beauty: Selected Prose of Chandler Davis (2010) Journals Edited The Mathematical Intelligencer == References == == Further reading == Batterson, Steve (2023-08-08). The Prosecution of Professor Chandler Davis: McCarthyism, Communism, and the Myth of Academic Freedom. New York: Monthly Review Press. ISBN 978-1-68590-036-6. Benedetto, John; Choi, Man-Duen; Kirkland, Stephen (2014). "Chandler Davis as Mentor". The Mathematical Intelligencer. 36 (1): 20–21. doi:10.1007/s00283-013-9431-3. ISSN 0343-6993. Holbrook, John (2014). "The Mathematics of Chandler Davis". The Mathematical Intelligencer. 36 (1): 6–12. doi:10.1007/s00283-013-9432-2. ISSN 0343-6993. == External sources == Biography at the University of Regina's Department of Mathematics & Statistics Chan Davis at the Internet Speculative Fiction Database Davis, Markert, Nickerson Lecture Series on Academic and Intellectual Freedom Chandler Davis at the Mathematics Genealogy Project Victim of McCarthy-Era Witch Hunt calls on U-Illinois not to Fire Critic of Israeli Policies Chandler Davis archival papers held at the University of Toronto Archives and Records Management Services
Wikipedia:Chandra Wickramasinghe#0	Nalin Chandra Wickramasinghe (born 20 January 1939) is a Sri Lankan-born British mathematician, astronomer and astrobiologist of Sinhalese ethnicity. His research interests include the interstellar medium, infrared astronomy, light scattering theory, applications of solid-state physics to astronomy, the early Solar System, comets, astrochemistry, the origin of life and astrobiology. A student and collaborator of Fred Hoyle, the pair worked jointly for over 40 years as influential proponents of panspermia. In 1974 they proposed the hypothesis that some dust in interstellar space was largely organic, later proven to be correct. Wickramasinghe has advanced numerous fringe claims, including the argument that various outbreaks of illnesses on Earth are of extraterrestrial origins, including the 1918 flu pandemic and certain outbreaks of polio and mad cow disease. For the 1918 flu pandemic they hypothesised that cometary dust brought the virus to Earth simultaneously at multiple locations—a view almost universally dismissed by experts on this pandemic. Claims connecting terrestrial disease and extraterrestrial pathogens have been rejected by the scientific community. Wickramasinghe has written more than 40 books about astrophysics and related topics; he has made appearances on radio, television and film, and he writes online blogs and articles. He has appeared on BBC Horizon, UK Channel 5 and the History Channel. He appeared on the 2013 Discovery Channel program "Red Rain". He has an association with Daisaku Ikeda, president of the Buddhist sect Soka Gakkai International, that led to the publication of a dialogue with him, first in Japanese and later in English, on the topic of Space and Eternal Life. == Education and career == Wickramasinghe studied at Royal College, Colombo, the University of Ceylon (where he graduated in 1960 with a BSc First Class Honours in mathematics), and at Trinity College and Jesus College, Cambridge, where he obtained his PhD and ScD degrees. Following his education, Wickramasinghe was a Fellow of Jesus College, Cambridge from 1963 to 1973, until he became professor of applied mathematics and astronomy at University College Cardiff. Wickramasinghe was a consultant and advisor to the President of Sri Lanka from 1982 to 1984, and played a key role in founding the Institute of Fundamental Studies in Sri Lanka. After fifteen years at University College Cardiff, Wickramasinghe took an equivalent position in the University of Cardiff, a post he held from 1990 until 2006. After retirement in 2006, he incubated the Cardiff Center for Astrobiology as a special project reporting to the president of the university. In 2011 the project closed down, losing its funding in a series of UK educational cut backs. After this event, Wickramasinghe was offered the opportunity to move to the University of Buckingham as Director of the Buckingham Centre for Astrobiology, University of Buckingham where he has been since 2011. He maintains his part-time position as a UK Professor at Cardiff University. In 2015 he was elected Visiting scholar, Churchill College, Cambridge, England 2015/16. He is a co-founder and board member of the Institute for the Study of Panspermia and Astroeconomics, set up in Japan in 2014, and the Editor-in-Chief of the Journal of Astrobiology & Outreach. He was a Visiting By-Fellow, Churchill College, Cambridge, England 2015/16; Professor and Director of the Buckingham Centre for Astrobiology at the University of Buckingham, a post he has held since 2011; Affiliated Visiting Professor, University of Peradeniya, Sri Lanka; and a board member and research director at the Institute for the Study of Panspermia and Astroeconomics, Ogaki-City, Gifu, Japan. In 2017, Professor Chandra Wickramasinghe was appointed adjunct professor in the Department of Physics, at the University of Ruhuna, Matara, Sri Lanka. == Research == In 1960 he commenced work in Cambridge on his PhD degree under the supervision of Fred Hoyle, and published his first scientific paper "On Graphite Particles as Interstellar Grains" in Monthly Notices of the Royal Astronomical Society in 1962. He was awarded a PhD degree in mathematics in 1963 and was elected a Fellow of Jesus College Cambridge in the same year. In the following year he was appointed a Staff Member of the Institute of Astronomy, Cambridge. Here he continued to work on the nature of interstellar dust, publishing many papers in this field, that led to a consideration of carbon-containing grains as well as the older silicate models. Wickramasinghe published the first definitive book on Interstellar Grains in 1967. He has made many contributions to this field, publishing over 350 papers in peer-reviewed journals, over 75 of which are in Nature. Hoyle and Wickramasinghe further proposed a radical kind of panspermia that included the claim that extraterrestrial life forms enter the Earth's atmosphere and were possibly responsible for epidemic outbreaks, new diseases, and genetic novelty that Hoyle and Wickramasinghe contended was necessary for macroevolution. Chandra Wickramasinghe had the longest-running collaboration with Fred Hoyle. Their publications on books and papers arguing for panspermia and a cosmic hypothesis of life are controversial and, in particular detail, essentially contra the scientific consensus in both astrophysics and biology. Several claims made by Hoyle and Wickramasinghe between 1977 and 1981, such as a report of having detected interstellar cellulose, were criticised by one author as pseudoscience. Phil Plait has described Wickramasinghe as a "fringe scientist" who "jumps on everything, with little or no evidence, and says it's from outer space". === Organic molecules in space === In 1974 Wickramasinghe first proposed the hypothesis that some dust in interstellar space was largely organic, and followed this up with other research confirming the hypothesis. Wickramasinghe also proposed and confirmed the existence of polymeric compounds based on the molecule formaldehyde (H2CO). Fred Hoyle and Wickramasinghe later proposed the identification of bicyclic aromatic compounds from an analysis of the ultraviolet extinction absorption at 2175A., thus demonstrating the existence of polycyclic aromatic hydrocarbon molecules in space. === Hoyle–Wickramasinghe model of panspermia === Throughout his career, Wickramasinghe, along with his collaborator Fred Hoyle, has advanced the panspermia hypothesis, that proposes that life on Earth is, at least in part, of extraterrestrial origin. The Hoyle–Wickramasinghe model of panspermia include the assumptions that dormant viruses and desiccated DNA and RNA can survive unprotected in space; that small bodies such as asteroids and comets can protect the "seeds of life", including DNA and RNA, living, fossilized, or dormant life, cellular or non-cellular; and that the collisions of asteroids, comets, and moons have the potential to spread these "seeds of life" throughout an individual star system and then onward to others. The most contentious issue around the Hoyle–Wickramasinghe model of the panspermia hypothesis is the corollary of their first two propositions that viruses and bacteria continue to enter the Earth's atmosphere from space, and are hence responsible for many major epidemics throughout history. Towards the end of their collaboration, Wickramasinghe and Hoyle hypothesised that abiogenesis occurred close to the Galactic Center before panspermia carried life throughout the Milky Way, and stated a belief that such a process could occur in many galaxies throughout the Universe. === Detection of living cells in the stratosphere === On 20 January 2001 the Indian Space Research Organisation (ISRO) conducted a balloon flight from Hyderabad, India to collect stratospheric dust from a height of 41 km (135,000 ft) with a view to testing for the presence of living cells. The collaborators on this project included a team of UK scientists led by Wickramasinghe. In a paper presented at a SPIE conference in San Diego in 2002 the detection of evidence for viable microorganisms from 41 km above the Earth's surface was presented. However, the experiment did not present evidence as to whether the findings are incoming microbes from space rather than microbes carried up to 41 km from the surface of the Earth. In 2005 the ISRO group carried out a second stratospheric sampling experiment from 41 km altitude and reported the isolation of three new species of bacteria including one that they named Janibacter hoylei sp.nov. in honour of Fred Hoyle. However, these facts do not prove that bacteria on Earth originated in the cosmic environment. Samplings of the stratosphere have also been carried out by Yang et al. (2005, 2009). During the experiment strains of highly radiation-resistant Deinococcus bacterium were detected at heights up to 35 km. Nevertheless, these authors have abstained from linking these discoveries to panspermia. Wickramasinghe was also involved in coordinating analyses of the red rain in Kerala in collaborations with Godfrey Louis. === Extraterrestrial pathogens === Hoyle and Wickramasinghe have advanced the argument that various outbreaks of illnesses on Earth are of extraterrestrial origins, including the 1918 flu pandemic and certain outbreaks of polio and mad cow disease. For the 1918 flu pandemic they hypothesised that cometary dust brought the virus to Earth simultaneously at multiple locations—a view almost universally dismissed by external experts on this pandemic. On 24 May 2003 The Lancet published a letter from Wickramasinghe, jointly signed by Milton Wainwright and Jayant Narlikar, in which they hypothesised that the virus that causes severe acute respiratory syndrome (SARS) could be extraterrestrial in origin instead of originating from chickens. The Lancet subsequently published three responses to this letter, showing that the hypothesis was not evidence-based, and casting doubts on the quality of the experiments referenced by Wickramasinghe in his letter. Claims connecting terrestrial disease and extraterrestrial pathogens have been rejected by the scientific community. In 2020, Wickramasinghe and colleagues published a paper claiming that Severe acute respiratory syndrome coronavirus 2, the virus responsible for the COVID-19 pandemic was also of extraterrestrial origin, the claim was criticised for lacking evidence. === Polonnaruwa === On 29 December 2012 a green fireball was observed in Polonnaruwa, Sri Lanka. It disintegrated into fragments that fell to the Earth near the villages of Aralaganwila and Dimbulagala and in a rice field near Dalukkane. Rock samples were submitted to the Medical Research Institute of the Ministry of Health in Colombo. The rocks were sent to the University of Cardiff in Wales for analysis, where Chandra Wickramasinghe's team analyzed them and claimed that they contained extraterrestrial diatoms. From January to March 2013, five papers were published in the fringe Journal of Cosmology outlining various results from teams in the United Kingdom, United States and Germany. However, independent experts in meteoritics stated that the object analyzed by Wickramasinghe's team was of terrestrial origin, a fulgurite created by lightning strikes on Earth. Experts in diatoms complemented the statement, saying that the organisms found in the rock represented a wide range of extant terrestrial taxa, confirming their earthly origin. Wickramasinghe and collaborators responded, using X-ray diffraction, oxygen isotope analysis, and scanning electron microscope observations, in a March 2013 paper asserting that the rocks they found were indeed meteorites, instead of being created by lightning strikes on Earth as stated by scientists from the University of Peradeniya. However, these claims were also criticised for not providing evidence that the rocks were actually meteorites. === Cephalopod alien origin === In 2018, Wickramasinghe and over 30 other authors published a paper in Progress in Biophysics and Molecular Biology entitled "Cause of Cambrian Explosion - Terrestrial or Cosmic?" which argued in favour of panspermia as the origin of the Cambrian explosion, and posited that cephalopods are alien lifeforms that originated from frozen eggs that were transported to earth via meteor. The claims gained widespread press coverage. Virologist Karin Mölling, in a companion commentary published in the same journal, stated that the claims "cannot be taken seriously". == Participation in the creation-evolution debate == Wickramasinghe and his mentor Fred Hoyle have also used their data to argue in favor of cosmic ancestry, and against the idea of life emerging from inanimate objects by abiogenesis. Once again the Universe gives the appearance of being biologically constructed, and on this occasion on a truly vast scale. Once again those who consider such thoughts to be too outlandish to be taken seriously will continue to do so. While we ourselves shall continue to take the view that those who believe they can match the complexities of the Universe by simple experiments in their laboratories will continue to be disappointed. Wickramasinghe attempts to present scientific evidence to support the notion of cosmic ancestry and "the possibility of high intelligence in the Universe and of many increasing levels of intelligence converging toward a God as an ideal limit." During the 1981 scientific creationist trial in Arkansas, Wickramasinghe was the only scientist testifying for the defense, which in turn was supporting creationism. In addition, he wrote that the Archaeopteryx fossil finding is a forgery, a charge that the scientific community considers an "absurd" and "ignorant" statement. == Honours and awards == Commonwealth Scholar at Trinity College, Cambridge, 1960-1963 Powell Prize for English Verse, Trinity College, 1961 Vidya Jyothi from the President of Sri Lanka, 1992 Honorary DLitt, Sōka University (Japan), 1996 Doctor of Science (honoris causa), University of Ruhuna, Sri Lanka, 2004 Visiting By-Fellowship, visiting scholar, Churchill College, Cambridge, England 2015/16 Ada Derana Sri Lankan of the Year 2017 - Global Scientist Wickramasinghe was appointed Member of the Order of the British Empire (MBE) in the 2022 New Year Honours for services to science, astronomy and astrobiology. == Books == Interstellar Grains (Chapman & Hall, London, 1967) Light Scattering Functions for Small Particles with Applications in Astronomy (Wiley, New York, 1973) Solid-State Astrophysics (ed. with D.J. Morgan) (D. Reidel, Boston, 1975) Interstellar Matter (with F.D. Khan & P.G. Mezger) (Swiss Society of Astronomy and Astrophysics, 1974) The Cosmic Laboratory (University College of Cardiff, 1975) Lifecloud: The Origin of Life in the Universe (with Fred Hoyle) (J.M. Dent, London, 1978) Diseases from Space (with Fred Hoyle) (J.M. Dent, London, 1979) Origin of Life (with Fred Hoyle) (University College Cardiff Press, 1979) Space Travellers: The Bringers of Life (with Fred Hoyle) (University College Cardiff Press, 1981) Evolution from Space (with Fred Hoyle) (J.M. Dent, London, 1981) ISBN 978-0-460-04535-3 Is Life an Astronomical Phenomenon? (University College Cardiff Press, 1982) ISBN 9780906449493 Why Neo-Darwinism Does Not Work (with Fred Hoyle) (University College Cardiff Press, 1982) ISBN 9780906449509 Proofs that Life is Cosmic (with Fred Hoyle) (Institute of Fundamental Studies, Sri Lanka, Memoirs no.1, 1982) From Grains to Bacteria (with Fred Hoyle) (University College Cardiff Press, 1984) ISBN 9780906449646 Fundamental Studies and the Future of Science (ed.) (University College Cardiff Press, 1984) ISBN 9780906449578 Living Comets (with Fred Hoyle) (University College Cardiff Press, 1985) ISBN 9780906449790 Archaeopteryx, the Primordial Bird: A Case of Fossil Forgery (with Fred Hoyle) (Christopher Davies, Swansea, 1986) ISBN 9780715406656 The Theory of Cosmic Grains (with Fred Hoyle) (Kluwer, Dordrecht, 1991) ISBN 9780792311898 Life on Mars? The Case for a Cosmic Heritage (with Fred Hoyle) (Clinical Press, Bristol, 1997) ISBN 9781854570413 Astronomical Origins of Life: Steps towards Panspermia (with Fred Hoyle) (Kluwer, Dordrecht, 2000) ISBN 9780792360810 Cosmic Dragons: Life and Death on Our Planet (Souvenir Press, London, 2001) ISBN 9780285636064 Fred Hoyle's Universe (ed. with G. Burbidge and J. Narlikar) (Kluwer, Dordrecht, 2003) ISBN 9781402014154 A Journey with Fred Hoyle (World Scientific, Singapore, 2005) ISBN 9789812565792 Comets and the Origin of Life (with J. Wickramasinghe and W. Napier) (World Scientific, Hackensack NJ, 2010) ISBN 9789812814005 A Journey with Fred Hoyle, Second Edition (World Scientific, Singapore, April 2013) ISBN 9789814436120 The search for our cosmic ancestry, World Scientific, New Jersey 2015, ISBN 978-981-461696-6. Walker, Theodore; Wickramasinghe, Chandra (2015). The Big Bang and God: An Astro-Theology. Palgrave Macmillan US. doi:10.1057/9781137535030. ISBN 978-1-349-57419-3. == Articles == Hoyle, F. and Wickramasinghe, N.C., 1962. On graphite particles as interstellar grains, Mon.Not.Roy.Astr.Soc. 124, 417-433 Hoyle, F.; Wickramasinghe, N.C. (1969). "Interstellar Grains". Nature. 223 (5205): 450–462. Bibcode:1969Natur.223..459H. doi:10.1038/223459a0. S2CID 4209522. Wickramasinghe, N.C., 1974. Formaldehyde polymers in interstellar space, Nature 252, 462-463 Wickramasinghe, N.C.; Hoyle, F.; Brooks, J.; Shaw, G. (1977). "Prebiotic polymers and infrared spectra of galactic sources". Nature. 269 (5630): 674–676. Bibcode:1977Natur.269..674W. doi:10.1038/269674a0. S2CID 4266722. Hoyle, F. and Wickramasinghe, N.C., 1977. Identification of the λ2,200A interstellar absorption feature, Nature 270, 323-324 F., Hoyle; N. C., Wickramasinghe (4 November 1976). "Primitive grain clumps and organic compounds in carbonaceous chondrites" (PDF). Nature. 264 (5581): 45–46. Bibcode:1976Natur.264...45H. doi:10.1038/264045a0. Retrieved 18 January 2013. Hoyle, F. and Wickramasinghe, N.C., 1977. Polysaccharides and infrared spectra of galactic sources, Nature 268, 610-612 Hoyle, F.; Wickramasinghe, N.C. (1979). "On the nature of interstellar grains". Astrophysics and Space Science. 66: 77–90. Bibcode:1999Ap&SS.268..249H. doi:10.1023/A:1002462602776. S2CID 189820472. Hoyle, F.; Wickramasinghe, N.C. (1979). "Biochemical chromophores and the interstellar extinction at ultraviolet wavelengths". Astrophysics and Space Science. 65 (1): 241–244. Bibcode:1979Ap&SS..65..241H. doi:10.1007/BF00643503. S2CID 120184918. Hoyle, F.; Wickramasinghe, N.C.; Al-Mufti, S.; et al. (1982). "Infrared spectroscopy over the 2.9-3.9 μm waveband in biochemistry and astronomy". Astrophysics and Space Science. 83: 405–409. Bibcode:1999Ap&SS.268..161H. doi:10.1023/A:1002417307802. Hoyle, F.; Wickramasinghe, N.C.; Al-Mufti, S. (1982). "Organo-siliceous biomolecules and the infrared spectrum of the Trapezium nebula". Astrophysics and Space Science. 86 (1): 63–69. Bibcode:1982Ap&SS..86...63H. doi:10.1007/BF00651830. S2CID 120249547. Hoyle, F.; Wickramasinghe, N.C. (1983). "Bacterial life in space". Nature. 306 (5942): 420. Bibcode:1983Natur.306..420H. doi:10.1038/306420a0. PMID 6646221. Hoyle, F. and Wickramasinghe, N.C., 1986. The case for life as a cosmic phenomenon, Nature 322, 509-511 Hoyle, F. and Wickramasinghe, N.C., 1990. Influenza – evidence against contagion, Journal of the Royal Society of Medicine 83. 258-261 Napier, W.M.; Wickramasinghe, J.T; Wickramasinghe, N.C. (2007). "The origin of life in comets". International Journal of Astrobiology. 6 (4): 321–323. Bibcode:2007IJAsB...6..321N. doi:10.1017/S1473550407003941. S2CID 121008660. Rauf, K.; Wickramasinghe, C. (2010). "Evidence for biodegradation products in the interstellar medium". International Journal of Astrobiology. 9 (1): 29–34. Bibcode:2010IJAsB...9...29R. CiteSeerX 10.1.1.643.9541. doi:10.1017/S1473550409990334. S2CID 17336375. Wickramasinghe, N. C. (2010). "The astrobiological case for our cosmic ancestry". International Journal of Astrobiology. 9 (2): 119–129. Bibcode:2010IJAsB...9..119W. doi:10.1017/S1473550409990413. S2CID 13978227. Wickramasinghe, N.C.; Wallis, J.; Wallis, D.H.; Schild, R.E.; Gibson, C.H. (2012). "Life-bearing planets in the solar vicinity". Astrophysics and Space Science. 341 (2): 295–9. Bibcode:2012Ap&SS.341..295W. doi:10.1007/s10509-012-1092-8. S2CID 120484953. Chandra Wickramasinghe, A Journey with Fred Hoyle: The Search for Cosmic Life, World Scientific Publishing, 2005, ISBN 981-238-912-1 Janaki Wickramasinghe, Chandra Wickramasinghe and William Napier, Comets and the Origin of Life, World Scientific Publishing, 2009, ISBN 981-256-635-X Chandra Wickramasinghe and Daisaku Ikeda, Space and Eternal Life, Journeyman Press, 1998, ISBN 1-85172-060-X == See also == Panspermia Red rain in Kerala Milton Wainwright == References == == External links == Professor Wickramasinghe's profile at the University of Buckingham Interviews Publication List Chandra Wickramasinghe@ Astrophysics Data System Prof Chandra Wickramasinghe in conversation with artist and poet, Himali Singh Soin, podcast, 2022
Wikipedia:Chandravakyas#0	Chandravākyas (IAST: Candravākyas) are a collection of numbers, arranged in the form of a list, related to the motion of the Moon in its orbit around the Earth. These numbers are couched in the katapayadi system of representation of numbers and so apparently appear like a list of words, or phrases or short sentences written in Sanskrit and hence the terminology Chandravākyas. In Sanskrit, Chandra is the Moon and vākya means a sentence. The term Chandravākyas could thus be translated as Moon-sentences. Vararuchi (c. 4th century CE), a legendary figure in the astronomical traditions of Kerala, is credited with the authorship of the collection of Chandravākyas. These were routinely made use of for computations of native almanacs and for predicting the position of the Moon. The work ascribed to Vararuchi is also known as Chandravākyāni, or Vararucivākyāni, or Pañcāṅgavākyāni. Madhava of Sangamagrama (c. 1350 – c. 1425), the founder of the Kerala school of astronomy and mathematics, had set forth a revised set of Chandravākyās, together with a method for computing them, in his work titled Venvaroha. Chandravākyas were also popular in Tamil Nadu region of South India. There, the astrologers and astronomers used these vākyās to construct almanacs. These almanacs were popularly referred to as the Vākya-pañcāṅgas. This is used in contrast to the modern mode computation of almanacs based on parameters derived from astronomical observations that are known as Dṛk Pañcāṅgas ( or Thirukanitha Pañcāṅgas). == Vākya tradition == The Parahita system of astronomical computations introduced by Haridatta (ca. 683 CE), though simplified the computational processes, required long tables of numbers for its effective implementation. For timely use of these numbers they had to be memorised in toto and probably the system of constructing astronomical Vākyas arose as an answer to this problem. The katapayadi system provided the most convenient medium for constructing easily memorable mnemonics for the numbers in these tables. Chandravākyās ascribed to Vararuci are the earliest example of such a set of mnemonics. The period of Vararuci of Kerala tradition has been determined as around fourth century CE and the year of the promulgation of the Parahita system is known to be 683 CE, Vararuci's Chandravākyās should have been around at the time of the institution of the Parahita system. Besides Vararuci's Vākyas, several other sets of Vākyas had been composed by astronomers and mathematicians of the Kerala school. While Vararuci's Vākyas contain a list of 248 numbers, another set of Vākyas relating to Moon's motion contains 3031 numbers. There is a set of 2075 Vākyas called Samudra-vākyas or Maṇḍala-vākyas or Kujādi-pañcagraha-mahāvākyas relating to the motion of the five planets Kuja (Mars), Budha (Mercury), Guru (Jupiter), Bhrigu (Venus) and Sani (Saturn). There are also lists of Vākyas encoding other mathematical tables like Madhava's sine table. == Vākya-pañcāṅga == The first known text to use these Chandravākyass is Haridatta's manual on his Parahita system, known as Graha-cāra-nibandhana. The next major work that makes use of the mnemonic system of the Vākyas which has come down to us is Vākya-karaṇa (karaṇa, or computations, utilising Vākyas). The authorship of this work is uncertain, but, is apocryphally assigned to Vararuci. The work is known to have been composed around 1300 CE. It has been extensively commented upon by Sundararaja (c.1500 CE) of Trichinopopy of Tamil Nadu. The almanac makers of Tamil Nadu fully make use of this Vākya-karaṇa for computing the almanacs. These almanacs are known as Vākya-pañcāṅgas. == Numbers encoded in Chandravākyās == The Moon's orbit approximates an ellipse rather than a circle. The orientation and the shape of this orbit is not fixed. In particular, the positions of the extreme points, the point of closest approach (perigee) and the point of farthest excursion (apogee), make a full circle in about nine years. It takes the Moon longer to return to the same position, perigee or apogee, because it moved ahead during one revolution. This longer period is called the anomalistic month, and has an average length of 27.554551 days (27 d 13 h 18 min 33.2 s). The apparent diameter of the Moon varies with this period. 9 anomalistic months constitute a period of approximately 248 days. The differences in the longitudes of the Moon on the successive days of a 248-day cycle constitute the Chandravākyas. Each set of Chandravākyas contains a list of 248 Vākyās or sentences. == See also == Indian astronomy Indian mathematics Vākyakaraṇa Vākyapañcāṅga Vararuchi == References == == Further reading == For details on Madhava's method of computation of Chandravakyas see : K. Chandra Hari (2003). "Computation of the true moon by Madhava of Sangamagrama" (PDF). Indian Journal of History of Science. 38 (3): 231–253. Archived from the original (PDF) on 16 March 2012. Retrieved 6 May 2010. For a discussion on the history of the 248-day schemes see : Jones, Alexander (March 1983). "The development and transmission of 248-day schemes for lunar motion in ancient astronomy". Archive for History of Exact Sciences. 29 (1). Berlin / Heidelberg: Springer: 1–36. Bibcode:1983AHES...29....1J. doi:10.1007/bf00535977. S2CID 121595932. For a discussion of the 248-day schemes in Babylonian astronomy see: Otto Neugebauer (1969). The exact sciences in antiquity. Courier Dover Publications. pp. 240. ISBN 978-0-486-22332-2. (Chapter II)
Wikipedia:Change of variables#0	In mathematics, a change of variables is a basic technique used to simplify problems in which the original variables are replaced with functions of other variables. The intent is that when expressed in new variables, the problem may become simpler, or equivalent to a better understood problem. Change of variables is an operation that is related to substitution. However these are different operations, as can be seen when considering differentiation (chain rule) or integration (integration by substitution). A very simple example of a useful variable change can be seen in the problem of finding the roots of the sixth-degree polynomial: x 6 − 9 x 3 + 8 = 0. {\displaystyle x^{6}-9x^{3}+8=0.} Sixth-degree polynomial equations are generally impossible to solve in terms of radicals (see Abel–Ruffini theorem). This particular equation, however, may be written ( x 3 ) 2 − 9 ( x 3 ) + 8 = 0 {\displaystyle (x^{3})^{2}-9(x^{3})+8=0} (this is a simple case of a polynomial decomposition). Thus the equation may be simplified by defining a new variable u = x 3 {\displaystyle u=x^{3}} . Substituting x by u 3 {\displaystyle {\sqrt[{3}]{u}}} into the polynomial gives u 2 − 9 u + 8 = 0 , {\displaystyle u^{2}-9u+8=0,} which is just a quadratic equation with the two solutions: u = 1 and u = 8. {\displaystyle u=1\quad {\text{and}}\quad u=8.} The solutions in terms of the original variable are obtained by substituting x3 back in for u, which gives x 3 = 1 and x 3 = 8. {\displaystyle x^{3}=1\quad {\text{and}}\quad x^{3}=8.} Then, assuming that one is interested only in real solutions, the solutions of the original equation are x = ( 1 ) 1 / 3 = 1 and x = ( 8 ) 1 / 3 = 2. {\displaystyle x=(1)^{1/3}=1\quad {\text{and}}\quad x=(8)^{1/3}=2.} == Simple example == Consider the system of equations x y + x + y = 71 {\displaystyle xy+x+y=71} x 2 y + x y 2 = 880 {\displaystyle x^{2}y+xy^{2}=880} where x {\displaystyle x} and y {\displaystyle y} are positive integers with x > y {\displaystyle x>y} . (Source: 1991 AIME) Solving this normally is not very difficult, but it may get a little tedious. However, we can rewrite the second equation as x y ( x + y ) = 880 {\displaystyle xy(x+y)=880} . Making the substitutions s = x + y {\displaystyle s=x+y} and t = x y {\displaystyle t=xy} reduces the system to s + t = 71 , s t = 880 {\displaystyle s+t=71,st=880} . Solving this gives ( s , t ) = ( 16 , 55 ) {\displaystyle (s,t)=(16,55)} and ( s , t ) = ( 55 , 16 ) {\displaystyle (s,t)=(55,16)} . Back-substituting the first ordered pair gives us x + y = 16 , x y = 55 , x > y {\displaystyle x+y=16,xy=55,x>y} , which gives the solution ( x , y ) = ( 11 , 5 ) . {\displaystyle (x,y)=(11,5).} Back-substituting the second ordered pair gives us x + y = 55 , x y = 16 , x > y {\displaystyle x+y=55,xy=16,x>y} , which gives no solutions. Hence the solution that solves the system is ( x , y ) = ( 11 , 5 ) {\displaystyle (x,y)=(11,5)} . == Formal introduction == Let A {\displaystyle A} , B {\displaystyle B} be smooth manifolds and let Φ : A → B {\displaystyle \Phi :A\rightarrow B} be a C r {\displaystyle C^{r}} -diffeomorphism between them, that is: Φ {\displaystyle \Phi } is a r {\displaystyle r} times continuously differentiable, bijective map from A {\displaystyle A} to B {\displaystyle B} with r {\displaystyle r} times continuously differentiable inverse from B {\displaystyle B} to A {\displaystyle A} . Here r {\displaystyle r} may be any natural number (or zero), ∞ {\displaystyle \infty } (smooth) or ω {\displaystyle \omega } (analytic). The map Φ {\displaystyle \Phi } is called a regular coordinate transformation or regular variable substitution, where regular refers to the C r {\displaystyle C^{r}} -ness of Φ {\displaystyle \Phi } . Usually one will write x = Φ ( y ) {\displaystyle x=\Phi (y)} to indicate the replacement of the variable x {\displaystyle x} by the variable y {\displaystyle y} by substituting the value of Φ {\displaystyle \Phi } in y {\displaystyle y} for every occurrence of x {\displaystyle x} . == Other examples == === Coordinate transformation === Some systems can be more easily solved when switching to polar coordinates. Consider for example the equation U ( x , y ) := ( x 2 + y 2 ) 1 − x 2 x 2 + y 2 = 0. {\displaystyle U(x,y):=(x^{2}+y^{2}){\sqrt {1-{\frac {x^{2}}{x^{2}+y^{2}}}}}=0.} This may be a potential energy function for some physical problem. If one does not immediately see a solution, one might try the substitution ( x , y ) = Φ ( r , θ ) {\displaystyle \displaystyle (x,y)=\Phi (r,\theta )} given by Φ ( r , θ ) = ( r cos ⁡ ( θ ) , r sin ⁡ ( θ ) ) . {\displaystyle \displaystyle \Phi (r,\theta )=(r\cos(\theta ),r\sin(\theta )).} Note that if θ {\displaystyle \theta } runs outside a 2 π {\displaystyle 2\pi } -length interval, for example, [ 0 , 2 π ] {\displaystyle [0,2\pi ]} , the map Φ {\displaystyle \Phi } is no longer bijective. Therefore, Φ {\displaystyle \Phi } should be limited to, for example ( 0 , ∞ ] × [ 0 , 2 π ) {\displaystyle (0,\infty ]\times [0,2\pi )} . Notice how r = 0 {\displaystyle r=0} is excluded, for Φ {\displaystyle \Phi } is not bijective in the origin ( θ {\displaystyle \theta } can take any value, the point will be mapped to (0, 0)). Then, replacing all occurrences of the original variables by the new expressions prescribed by Φ {\displaystyle \Phi } and using the identity sin 2 ⁡ x + cos 2 ⁡ x = 1 {\displaystyle \sin ^{2}x+\cos ^{2}x=1} , we get V ( r , θ ) = r 2 1 − r 2 cos 2 ⁡ θ r 2 = r 2 1 − cos 2 ⁡ θ = r 2 \| sin ⁡ θ \| . {\displaystyle V(r,\theta )=r^{2}{\sqrt {1-{\frac {r^{2}\cos ^{2}\theta }{r^{2}}}}}=r^{2}{\sqrt {1-\cos ^{2}\theta }}=r^{2}\left\|\sin \theta \right\|.} Now the solutions can be readily found: sin ⁡ ( θ ) = 0 {\displaystyle \sin(\theta )=0} , so θ = 0 {\displaystyle \theta =0} or θ = π {\displaystyle \theta =\pi } . Applying the inverse of Φ {\displaystyle \Phi } shows that this is equivalent to y = 0 {\displaystyle y=0} while x ≠ 0 {\displaystyle x\not =0} . Indeed, we see that for y = 0 {\displaystyle y=0} the function vanishes, except for the origin. Note that, had we allowed r = 0 {\displaystyle r=0} , the origin would also have been a solution, though it is not a solution to the original problem. Here the bijectivity of Φ {\displaystyle \Phi } is crucial. The function is always positive (for x , y ∈ R {\displaystyle x,y\in \mathbb {R} } ), hence the absolute values. === Differentiation === The chain rule is used to simplify complicated differentiation. For example, consider the problem of calculating the derivative d d x sin ⁡ ( x 2 ) . {\displaystyle {\frac {d}{dx}}\sin(x^{2}).} Let y = sin ⁡ u {\displaystyle y=\sin u} with u = x 2 . {\displaystyle u=x^{2}.} Then: d d x sin ⁡ ( x 2 ) = d y d x = d y d u d u d x This part is the chain rule. = ( d d u sin ⁡ u ) ( d d x x 2 ) = ( cos ⁡ u ) ( 2 x ) = ( cos ⁡ ( x 2 ) ) ( 2 x ) = 2 x cos ⁡ ( x 2 ) {\displaystyle {\begin{aligned}{\frac {d}{dx}}\sin(x^{2})&={\frac {dy}{dx}}\\[6pt]&={\frac {dy}{du}}{\frac {du}{dx}}&&{\text{This part is the chain rule.}}\\[6pt]&=\left({\frac {d}{du}}\sin u\right)\left({\frac {d}{dx}}x^{2}\right)\\[6pt]&=(\cos u)(2x)\\&=\left(\cos(x^{2})\right)(2x)\\&=2x\cos(x^{2})\end{aligned}}} === Integration === Difficult integrals may often be evaluated by changing variables; this is enabled by the substitution rule and is analogous to the use of the chain rule above. Difficult integrals may also be solved by simplifying the integral using a change of variables given by the corresponding Jacobian matrix and determinant. Using the Jacobian determinant and the corresponding change of variable that it gives is the basis of coordinate systems such as polar, cylindrical, and spherical coordinate systems. ==== Change of variables formula in terms of Lebesgue measure ==== The following theorem allows us to relate integrals with respect to Lebesgue measure to an equivalent integral with respect to the pullback measure under a parameterization G. The proof is due to approximations of the Jordan content. Suppose that Ω {\displaystyle \Omega } is an open subset of R n {\displaystyle \mathbb {R} ^{n}} and G : Ω → R n {\displaystyle G:\Omega \to \mathbb {R} ^{n}} is a C 1 {\displaystyle C^{1}} diffeomorphism. If f {\displaystyle f} is a Lebesgue measurable function on G ( Ω ) {\displaystyle G(\Omega )} , then f ∘ G {\displaystyle f\circ G} is Lebesgue measurable on Ω {\displaystyle \Omega } . If f ≥ 0 {\displaystyle f\geq 0} or f ∈ L 1 ( G ( Ω ) , m ) , {\displaystyle f\in L^{1}(G(\Omega ),m),} then ∫ G ( Ω ) f ( x ) d x = ∫ Ω f ∘ G ( x ) \| det D x G \| d x {\displaystyle \int _{G(\Omega )}f(x)dx=\int _{\Omega }f\circ G(x)\|{\text{det}}D_{x}G\|dx} . If E ⊂ Ω {\displaystyle E\subset \Omega } and E {\displaystyle E} is Lebesgue measurable, then G ( E ) {\displaystyle G(E)} is Lebesgue measurable, then m ( G ( E ) ) = ∫ E \| det D x G \| d x {\displaystyle m(G(E))=\int _{E}\|{\text{det}}D_{x}G\|dx} . As a corollary of this theorem, we may compute the Radon–Nikodym derivatives of both the pullback and pushforward measures of m {\displaystyle m} under T {\displaystyle T} . ===== Pullback measure and transformation formula ===== The pullback measure in terms of a transformation T {\displaystyle T} is defined as T ∗ μ := μ ( T ( A ) ) {\displaystyle T^{}\mu :=\mu (T(A))} . The change of variables formula for pullback measures is ∫ T ( Ω ) g d μ = ∫ Ω g ∘ T d T ∗ μ {\displaystyle \int _{T(\Omega )}gd\mu =\int _{\Omega }g\circ TdT^{}\mu } . Pushforward measure and transformation formula The pushforward measure in terms of a transformation T {\displaystyle T} , is defined as T ∗ μ := μ ( T − 1 ( A ) ) {\displaystyle T_{}\mu :=\mu (T^{-1}(A))} . The change of variables formula for pushforward measures is ∫ Ω g ∘ T d μ = ∫ T ( Ω ) g d T ∗ μ {\displaystyle \int _{\Omega }g\circ Td\mu =\int _{T(\Omega )}gdT_{}\mu } . As a corollary of the change of variables formula for Lebesgue measure, we have that Radon-Nikodym derivative of the pullback with respect to Lebesgue measure: d T ∗ m d m ( x ) = \| det D x T \| {\displaystyle {\frac {dT^{}m}{dm}}(x)=\|{\text{det}}D_{x}T\|} Radon-Nikodym derivative of the pushforward with respect to Lebesgue measure: d T ∗ m d m ( x ) = \| det D x T − 1 \| {\displaystyle {\frac {dT_{}m}{dm}}(x)=\|{\text{det}}D_{x}T^{-1}\|} From which we may obtain The change of variables formula for pullback measure: ∫ T ( Ω ) g d m = ∫ Ω g ∘ T d T ∗ m = ∫ Ω g ∘ T \| det D x T \| d m ( x ) {\displaystyle \int _{T(\Omega )}gdm=\int _{\Omega }g\circ TdT^{}m=\int _{\Omega }g\circ T\|{\text{det}}D_{x}T\|dm(x)} The change of variables formula for pushforward measure: ∫ Ω g d m = ∫ T ( Ω ) g ∘ T − 1 d T ∗ m = ∫ T ( Ω ) g ∘ T − 1 \| det D x T − 1 \| d m ( x ) {\displaystyle \int _{\Omega }gdm=\int _{T(\Omega )}g\circ T^{-1}dT_{}m=\int _{T(\Omega )}g\circ T^{-1}\|{\text{det}}D_{x}T^{-1}\|dm(x)} === Differential equations === Variable changes for differentiation and integration are taught in elementary calculus and the steps are rarely carried out in full. The very broad use of variable changes is apparent when considering differential equations, where the independent variables may be changed using the chain rule or the dependent variables are changed resulting in some differentiation to be carried out. Exotic changes, such as the mingling of dependent and independent variables in point and contact transformations, can be very complicated but allow much freedom. Very often, a general form for a change is substituted into a problem and parameters picked along the way to best simplify the problem. === Scaling and shifting === Probably the simplest change is the scaling and shifting of variables, that is replacing them with new variables that are "stretched" and "moved" by constant amounts. This is very common in practical applications to get physical parameters out of problems. For an nth order derivative, the change simply results in d n y d x n = y scale x scale n d n y ^ d x ^ n {\displaystyle {\frac {d^{n}y}{dx^{n}}}={\frac {y_{\text{scale}}}{x_{\text{scale}}^{n}}}{\frac {d^{n}{\hat {y}}}{d{\hat {x}}^{n}}}} where x = x ^ x scale + x shift {\displaystyle x={\hat {x}}x_{\text{scale}}+x_{\text{shift}}} y = y ^ y scale + y shift . {\displaystyle y={\hat {y}}y_{\text{scale}}+y_{\text{shift}}.} This may be shown readily through the chain rule and linearity of differentiation. This change is very common in practical applications to get physical parameters out of problems, for example, the boundary value problem μ d 2 u d y 2 = d p d x ; u ( 0 ) = u ( L ) = 0 {\displaystyle \mu {\frac {d^{2}u}{dy^{2}}}={\frac {dp}{dx}}\quad ;\quad u(0)=u(L)=0} describes parallel fluid flow between flat solid walls separated by a distance δ; μ is the viscosity and d p / d x {\displaystyle dp/dx} the pressure gradient, both constants. By scaling the variables the problem becomes d 2 u ^ d y ^ 2 = 1 ; u ^ ( 0 ) = u ^ ( 1 ) = 0 {\displaystyle {\frac {d^{2}{\hat {u}}}{d{\hat {y}}^{2}}}=1\quad ;\quad {\hat {u}}(0)={\hat {u}}(1)=0} where y = y ^ L and u = u ^ L 2 μ d p d x . {\displaystyle y={\hat {y}}L\qquad {\text{and}}\qquad u={\hat {u}}{\frac {L^{2}}{\mu }}{\frac {dp}{dx}}.} Scaling is useful for many reasons. It simplifies analysis both by reducing the number of parameters and by simply making the problem neater. Proper scaling may normalize variables, that is make them have a sensible unitless range such as 0 to 1. Finally, if a problem mandates numeric solution, the fewer the parameters the fewer the number of computations. === Momentum vs. velocity === Consider a system of equations m v ˙ = − ∂ H ∂ x m x ˙ = ∂ H ∂ v {\displaystyle {\begin{aligned}m{\dot {v}}&=-{\frac {\partial H}{\partial x}}\\[5pt]m{\dot {x}}&={\frac {\partial H}{\partial v}}\end{aligned}}} for a given function H ( x , v ) {\displaystyle H(x,v)} . The mass can be eliminated by the (trivial) substitution Φ ( p ) = 1 / m ⋅ p {\displaystyle \Phi (p)=1/m\cdot p} . Clearly this is a bijective map from R {\displaystyle \mathbb {R} } to R {\displaystyle \mathbb {R} } . Under the substitution v = Φ ( p ) {\displaystyle v=\Phi (p)} the system becomes p ˙ = − ∂ H ∂ x x ˙ = ∂ H ∂ p {\displaystyle {\begin{aligned}{\dot {p}}&=-{\frac {\partial H}{\partial x}}\\[5pt]{\dot {x}}&={\frac {\partial H}{\partial p}}\end{aligned}}} === Lagrangian mechanics === Given a force field φ ( t , x , v ) {\displaystyle \varphi (t,x,v)} , Newton's equations of motion are m x ¨ = φ ( t , x , v ) . {\displaystyle m{\ddot {x}}=\varphi (t,x,v).} Lagrange examined how these equations of motion change under an arbitrary substitution of variables x = Ψ ( t , y ) {\displaystyle x=\Psi (t,y)} , v = ∂ Ψ ( t , y ) ∂ t + ∂ Ψ ( t , y ) ∂ y ⋅ w . {\displaystyle v={\frac {\partial \Psi (t,y)}{\partial t}}+{\frac {\partial \Psi (t,y)}{\partial y}}\cdot w.} He found that the equations ∂ L ∂ y = d d t ∂ L ∂ w {\displaystyle {\frac {\partial {L}}{\partial y}}={\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial {L}}{\partial {w}}}} are equivalent to Newton's equations for the function L = T − V {\displaystyle L=T-V} , where T is the kinetic, and V the potential energy. In fact, when the substitution is chosen well (exploiting for example symmetries and constraints of the system) these equations are much easier to solve than Newton's equations in Cartesian coordinates. == See also == Change of variables (PDE) Change of variables for probability densities Substitution property of equality Universal instantiation == References ==
Wikipedia:Chantal David#0	Chantal David (born 1964) is a French Canadian mathematician who works as a professor of mathematics at Concordia University. Her interests include analytic number theory, arithmetic statistics, and random matrix theory, and she has shown interest in elliptic curves and Drinfeld modules. She is the 2013 winner of the Krieger–Nelson Prize, given annually by the Canadian Mathematical Society to an outstanding female researcher in mathematics. == Education and career == David completed her doctorate in mathematics in 1993 at McGill University, under the supervision of Ram Murty. Her thesis was entitled Supersingular Drinfeld Modules. In the same year, she joined the faculty at Concordia University. She became the deputy director of the Centre de Recherches Mathématiques in 2004. In 2008, David was an invited professor at Université Henri Poincaré. She spent September 2009 through April 2010 at the Institute for Advanced Study. From January through May 2017, she co-organized a program on analytic number theory at the Mathematical Sciences Research Institute. == Research == In 1999, David published a paper with Francesco Pappalardi which proved that the Lang–Trotter conjecture holds in most cases. She has shown that for several families of curves over finite fields, the zeroes of zeta functions are compatible with the Katz–Sarnak conjectures. She has also used random matrix theory to study the zeroes in families of elliptic curves. David and her collaborators have exhibited a new Cohen–Lenstra phenomenon for the group of points of elliptic curves over finite fields. == Awards and honors == David was awarded the Krieger-Nelson Prize by the Canadian Mathematical Society in 2013. == References == == External links == "Home Page Chantal David". mathstat.concordia.ca. Retrieved 2017-09-28.
Wikipedia:Chaos game#0	In mathematics, the term chaos game originally referred to a method of creating a fractal, using a polygon and an initial point selected at random inside it. The fractal is created by iteratively creating a sequence of points, starting with the initial random point, in which each point in the sequence is a given fraction of the distance between the previous point and one of the vertices of the polygon; the vertex is chosen at random in each iteration. Repeating this iterative process a large number of times, selecting the vertex at random on each iteration, and throwing out the first few points in the sequence, will often (but not always) produce a fractal shape. Using a regular triangle and the factor 1/2 will result in the Sierpinski triangle, while creating the proper arrangement with four points and a factor 1/2 will create a display of a "Sierpinski Tetrahedron", the three-dimensional analogue of the Sierpinski triangle. As the number of points is increased to a number N, the arrangement forms a corresponding (N-1)-dimensional Sierpinski Simplex. The term has been generalized to refer to a method of generating the attractor, or the fixed point, of any iterated function system (IFS). Starting with any point x0, successive iterations are formed as xk+1 = fr(xk), where fr is a member of the given IFS randomly selected for each iteration. The iterations converge to the fixed point of the IFS. Whenever x0 belongs to the attractor of the IFS, all iterations xk stay inside the attractor and, with probability 1, form a dense set in the latter. The "chaos game" method plots points in random order all over the attractor. This is in contrast to other methods of drawing fractals, which test each pixel on the screen to see whether it belongs to the fractal. The general shape of a fractal can be plotted quickly with the "chaos game" method, but it may be difficult to plot some areas of the fractal in detail. With the aid of the "chaos game" a new fractal can be made and while making the new fractal some parameters can be obtained. These parameters are useful for applications of fractal theory such as classification and identification. The new fractal is self-similar to the original in some important features such as fractal dimension. == Optimal value of r for every regular polygon == At each iteration of the chaos game, the point xk+1 can be placed anywhere along the line connecting the point xk and the vertex of choice, v. Defining r as the ratio between the two distances d(xk,xk+1) and d(xk,v), it is possible to find the optimal value of r, i.e., ropt, for every N-sided regular polygon, that produces a fractal with optimal packing, i.e., the subscale polygons are in contact but do not overlap. The value of ropt can be calculated as the ratio between the length of the side of the first subscale polygon and the side of the original polygon. This ratio can be calculated geometrically: r o p t = ( 1 + 2 a ) ( 2 + 2 a ) {\displaystyle r_{opt}={\frac {(1+2a)}{(2+2a)}}} In which a is calculated as: a = ∑ i = 1 n cos ⁡ [ i ( π − θ ) ] {\displaystyle a=\sum _{i=1}^{n}\cos[i(\pi -\theta )]} Where θ is the internal angle of the polygon and n is the index of the most protruding vertex, counted starting from the base, i.e. n = ⌊ N 4 ⌋ {\displaystyle n=\left\lfloor {\frac {N}{4}}\right\rfloor } where ⌊ ⌋ {\displaystyle \lfloor \rfloor } represents the integral part of the fraction. The optimal ratio r o p t {\displaystyle r_{opt}} has also been called the kissing ratio K n {\displaystyle K_{n}} . Abdulaziz & Said showed that K n {\displaystyle K_{n}} is 1 1 + tan ⁡ π n if n ≡ 0 mod 4 {\displaystyle {\frac {1}{1+\tan {\frac {\pi }{n}}}}\qquad {\text{if}}\quad n\equiv 0{\bmod {4}}} 1 1 + 2 sin ⁡ π 2 n if n ≡ 1 , 3 mod 4 {\displaystyle {\frac {1}{1+2\sin {\frac {\pi }{2n}}}}\quad {\text{if}}\quad n\equiv 1,3{\bmod {4}}} 1 1 + sin ⁡ π n if n ≡ 2 mod 4 {\displaystyle {\frac {1}{1+\sin {\frac {\pi }{n}}}}\qquad {\text{if}}\quad n\equiv 2{\bmod {4}}} == Expansion of the chaos game for values of r greater than 1 == While an optimally packed fractal appears only for a defined value of r, i.e., ropt, it is possible to play the chaos game using other values as well. If r>1 (the point xk+1 jumps at a greater distance than the distance between the point xk and the vertex v), the generated figure extends outside the initial polygon. When r=2, the algorithm enters in a meta-stable state and generates quasi-symmetric figures. For values of r>2, the points are placed further and further from the center of the initial polygon at each iteration, the algorithm becomes unstable and no figure is generated. == Restricted chaos game == If the chaos game is run with a square, the fractal is not visible and the interior of the square fills evenly with points. However, if restrictions are placed on the choice of vertices, fractals will appear in the square. For example, if the current vertex cannot be chosen in the next iteration, this fractal appears: If the current vertex cannot be one place away (anti-clockwise) from the previously chosen vertex, this fractal appears: If the point is prevented from landing on a particular region of the square, the shape of that region will be reproduced as a fractal in other and apparently unrestricted parts of the square. == Jumps other than 1/2 == When the length of the jump towards a vertex or another point is not 1/2, the chaos game generates other fractals, some of them very well-known. For example, when the jump is 2/3 and the point can also jump towards the center of the square, the chaos game generates the Vicsek fractal: When the jump is 2/3 and the point can also jump towards the midpoints of the four sides, the chaos game generates the Sierpinski carpet: == Chaos game used to represent sequences == With minor modifications to the game rules, it is possible to use the chaos game algorithm to represent any well-defined sequence, i.e., a sequence composed by the repetition of a limited number of distinct elements. In fact, for a sequence with a number N of distinct elements, it is possible to play the chaos game on an N-sided polygon, assigning each element to a vertex and playing the game choosing the vertices following the progression of the sequence (instead of choosing a random vertex). In this version of the game, the generated image is a unique representation of the sequence. This method was applied to the representation of genes (N=4, r=0.5) and proteins (N=20, r=0.863). Additionally, the representations of protein sequences was used to instruct ML models to predict protein features. The expansion of the chaos game using r=2 can be useful to magnify small mutations in the comparison between two (or more) sequences. == See also == Chaos theory == External links == Simulations of chaos games made with Scratch. Explanation of the chaos game at beltoforion.de. == References ==
Wikipedia:Chaplygin's theorem#0	In mathematical theory of differential equations the Chaplygin's theorem (Chaplygin's method) states about existence and uniqueness of the solution to an initial value problem for the first order explicit ordinary differential equation. This theorem was stated by Sergey Chaplygin. It is one of many comparison theorems. == Important definitions == Consider an initial value problem: differential equation y ′ ( t ) = f ( t , y ( t ) ) {\displaystyle y'\left(t\right)=f\left(t,y\left(t\right)\right)} in t ∈ [ t 0 ; α ] {\displaystyle t\in \left[t_{0};\alpha \right]} , α > t 0 {\displaystyle \alpha >t_{0}} with an initial condition y ( t 0 ) = y 0 {\displaystyle y\left(t_{0}\right)=y_{0}} . For the initial value problem described above the upper boundary solution and the lower boundary solution are the functions z ¯ ( t ) {\displaystyle {\overline {z}}\left(t\right)} and z _ ( t ) {\displaystyle {\underline {z}}\left(t\right)} respectively, both of which are smooth in t ∈ ( t 0 ; α ] {\displaystyle t\in \left(t_{0};\alpha \right]} and continuous in t ∈ [ t 0 ; α ] {\displaystyle t\in \left[t_{0};\alpha \right]} , such as the following inequalities are true: z _ ( t 0 ) < y ( t 0 ) < z ¯ ( t 0 ) {\displaystyle {\underline {z}}\left(t_{0}\right)<y\left(t_{0}\right)<{\overline {z}}\left(t_{0}\right)} ; z _ ′ ( t ) < f ( t , z _ ( t ) ) {\displaystyle {\underline {z}}'\left(t\right)<f(t,{\underline {z}}\left(t\right))} and z ¯ ′ ( t ) > f ( t , z ¯ ( t ) ) {\displaystyle {\overline {z}}\ '\left(t\right)>f(t,{\overline {z}}\left(t\right))} for t ∈ ( t 0 ; α ] {\displaystyle t\in \left(t_{0};\alpha \right]} . == Statement == Source: Given the aforementioned initial value problem and respective upper boundary solution z ¯ ( t ) {\displaystyle {\overline {z}}\left(t\right)} and lower boundary solution z _ ( t ) {\displaystyle {\underline {z}}\left(t\right)} for t ∈ [ t 0 ; α ] {\displaystyle t\in \left[t_{0};\alpha \right]} . If the right part f ( t , y ( t ) ) {\displaystyle f\left(t,y\left(t\right)\right)} is continuous in t ∈ [ t 0 ; α ] {\displaystyle t\in \left[t_{0};\alpha \right]} , y ( t ) ∈ [ z _ ( t ) ; z ¯ ( t ) ] {\displaystyle y\left(t\right)\in \left[{\underline {z}}\left(t\right);{\overline {z}}\left(t\right)\right]} ; satisfies the Lipschitz condition over variable y {\displaystyle y} between functions z ¯ ( t ) {\displaystyle {\overline {z}}\left(t\right)} and z _ ( t ) {\displaystyle {\underline {z}}\left(t\right)} : there exists constant K > 0 {\displaystyle K>0} such as for every t ∈ [ t 0 ; α ] {\displaystyle t\in \left[t_{0};\alpha \right]} , y 1 ( t ) ∈ [ z _ ( t ) ; z ¯ ( t ) ] {\displaystyle y_{1}\left(t\right)\in \left[{\underline {z}}\left(t\right);{\overline {z}}\left(t\right)\right]} , y 2 ( t ) ∈ [ z _ ( t ) ; z ¯ ( t ) ] {\displaystyle y_{2}\left(t\right)\in \left[{\underline {z}}\left(t\right);{\overline {z}}\left(t\right)\right]} the inequality \| f ( t , y 1 ( t ) ) − f ( t , y 2 ( t ) ) \| ≤ K \| y 1 ( t ) − y 2 ( t ) \| {\displaystyle \left\vert f\left(t,y_{1}\left(t\right)\right)-f\left(t,y_{2}\left(t\right)\right)\right\vert \leq K\left\vert y_{1}\left(t\right)-y_{2}\left(t\right)\right\vert } holds, then in t ∈ [ t 0 ; α ] {\displaystyle t\in \left[t_{0};\alpha \right]} there exists one and only one solution y ( t ) {\displaystyle y\left(t\right)} for the given initial value problem and moreover for all t ∈ [ t 0 ; α ] {\displaystyle t\in \left[t_{0};\alpha \right]} z _ ( t ) < y ( t ) < z ¯ ( t ) {\displaystyle {\underline {z}}\left(t\right)<y\left(t\right)<{\overline {z}}\left(t\right)} . == Remarks == Source: === Weakening inequalities === Inside inequalities within both of definitions of the upper boundary solution and the lower boundary solution signs of inequalities (all at once) can be altered to unstrict. As a result, inequalities sings at Chaplygin's theorem concusion would change to unstrict by z ¯ ( t ) {\displaystyle {\overline {z}}\left(t\right)} and z _ ( t ) {\displaystyle {\underline {z}}\left(t\right)} respectively. In particular, any of z ¯ ( t ) = y ( t ) {\displaystyle {\overline {z}}\left(t\right)=y\left(t\right)} , z _ ( t ) = y ( t ) {\displaystyle {\underline {z}}\left(t\right)=y\left(t\right)} could be chosen. === Proving inequality only === If y ( t ) {\displaystyle y\left(t\right)} is already known to be an existent solution for the initial value problem in t ∈ [ t 0 ; α ] {\displaystyle t\in \left[t_{0};\alpha \right]} , the Lipschitz condition requirement can be omitted entirely for proving the resulting inequality. There exists applications for this method while researching whether the solution is stable or not ( pp. 7–9). This is often called "Differential inequality method" in literature and, for example, Grönwall's inequality can be proven using this technique. === Continuation of the solution towards positive infinity === Chaplygin's theorem answers the question about existence and uniqueness of the solution in t ∈ [ t 0 ; α ] {\displaystyle t\in \left[t_{0};\alpha \right]} and the constant K {\displaystyle K} from the Lipschitz condition is, generally speaking, dependent on α {\displaystyle \alpha } : K = K ( α ) {\displaystyle K=K\left(\alpha \right)} . If for t ∈ [ t 0 ; + ∞ ) {\displaystyle t\in \left[t_{0};+\infty \right)} both functions z ¯ ( t ) {\displaystyle {\overline {z}}\left(t\right)} and z _ ( t ) {\displaystyle {\underline {z}}\left(t\right)} retain their smoothness and for α ∈ ( t 0 ; + ∞ ) {\displaystyle \alpha \in \left(t_{0};+\infty \right)} a set { K ( α ) } {\displaystyle \left\{K\left(\alpha \right)\right\}} is bounded, the theorem holds for all t ∈ [ t 0 ; + ∞ ) {\displaystyle t\in \left[t_{0};+\infty \right)} . == References == == Further reading == Komlenko, Yuriy (1967-09-01). "Chaplygin's theorem for a second-order linear differential equation with lagging argument". Mathematical Notes of the Academy of Sciences of the USSR. 2 (3): 666–669. doi:10.1007/BF01094057. ISSN 1573-8876.
Wikipedia:Characteristic polynomial#0	In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The characteristic polynomial of an endomorphism of a finite-dimensional vector space is the characteristic polynomial of the matrix of that endomorphism over any basis (that is, the characteristic polynomial does not depend on the choice of a basis). The characteristic equation, also known as the determinantal equation, is the equation obtained by equating the characteristic polynomial to zero. In spectral graph theory, the characteristic polynomial of a graph is the characteristic polynomial of its adjacency matrix. == Motivation == In linear algebra, eigenvalues and eigenvectors play a fundamental role, since, given a linear transformation, an eigenvector is a vector whose direction is not changed by the transformation, and the corresponding eigenvalue is the measure of the resulting change of magnitude of the vector. More precisely, suppose the transformation is represented by a square matrix A . {\displaystyle A.} Then an eigenvector v {\displaystyle \mathbf {v} } and the corresponding eigenvalue λ {\displaystyle \lambda } must satisfy the equation A v = λ v , {\displaystyle A\mathbf {v} =\lambda \mathbf {v} ,} or, equivalently (since λ v = λ I v {\displaystyle \lambda \mathbf {v} =\lambda I\mathbf {v} } ), ( λ I − A ) v = 0 {\displaystyle (\lambda I-A)\mathbf {v} =\mathbf {0} } where I {\displaystyle I} is the identity matrix, and v ≠ 0 {\displaystyle \mathbf {v} \neq \mathbf {0} } (although the zero vector satisfies this equation for every λ , {\displaystyle \lambda ,} it is not considered an eigenvector). It follows that the matrix ( λ I − A ) {\displaystyle (\lambda I-A)} must be singular, and its determinant det ( λ I − A ) = 0 {\displaystyle \det(\lambda I-A)=0} must be zero. In other words, the eigenvalues of A are the roots of det ( x I − A ) , {\displaystyle \det(xI-A),} which is a monic polynomial in x of degree n if A is a n×n matrix. This polynomial is the characteristic polynomial of A. == Formal definition == Consider an n × n {\displaystyle n\times n} matrix A . {\displaystyle A.} The characteristic polynomial of A , {\displaystyle A,} denoted by p A ( t ) , {\displaystyle p_{A}(t),} is the polynomial defined by p A ( t ) = det ( t I − A ) {\displaystyle p_{A}(t)=\det(tI-A)} where I {\displaystyle I} denotes the n × n {\displaystyle n\times n} identity matrix. Some authors define the characteristic polynomial to be det ( A − t I ) . {\displaystyle \det(A-tI).} That polynomial differs from the one defined here by a sign ( − 1 ) n , {\displaystyle (-1)^{n},} so it makes no difference for properties like having as roots the eigenvalues of A {\displaystyle A} ; however the definition above always gives a monic polynomial, whereas the alternative definition is monic only when n {\displaystyle n} is even. == Examples == To compute the characteristic polynomial of the matrix A = ( 2 1 − 1 0 ) . {\displaystyle A={\begin{pmatrix}2&1\\-1&0\end{pmatrix}}.} the determinant of the following is computed: t I − A = ( t − 2 − 1 1 t − 0 ) {\displaystyle tI-A={\begin{pmatrix}t-2&-1\\1&t-0\end{pmatrix}}} and found to be ( t − 2 ) t − 1 ( − 1 ) = t 2 − 2 t + 1 , {\displaystyle (t-2)t-1(-1)=t^{2}-2t+1\,\!,} the characteristic polynomial of A . {\displaystyle A.} Another example uses hyperbolic functions of a hyperbolic angle φ. For the matrix take A = ( cosh ⁡ ( φ ) sinh ⁡ ( φ ) sinh ⁡ ( φ ) cosh ⁡ ( φ ) ) . {\displaystyle A={\begin{pmatrix}\cosh(\varphi )&\sinh(\varphi )\\\sinh(\varphi )&\cosh(\varphi )\end{pmatrix}}.} Its characteristic polynomial is det ( t I − A ) = ( t − cosh ⁡ ( φ ) ) 2 − sinh 2 ⁡ ( φ ) = t 2 − 2 t cosh ⁡ ( φ ) + 1 = ( t − e φ ) ( t − e − φ ) . {\displaystyle \det(tI-A)=(t-\cosh(\varphi ))^{2}-\sinh ^{2}(\varphi )=t^{2}-2t\ \cosh(\varphi )+1=(t-e^{\varphi })(t-e^{-\varphi }).} == Properties == The characteristic polynomial p A ( t ) {\displaystyle p_{A}(t)} of a n × n {\displaystyle n\times n} matrix is monic (its leading coefficient is 1 {\displaystyle 1} ) and its degree is n . {\displaystyle n.} The most important fact about the characteristic polynomial was already mentioned in the motivational paragraph: the eigenvalues of A {\displaystyle A} are precisely the roots of p A ( t ) {\displaystyle p_{A}(t)} (this also holds for the minimal polynomial of A , {\displaystyle A,} but its degree may be less than n {\displaystyle n} ). All coefficients of the characteristic polynomial are polynomial expressions in the entries of the matrix. In particular its constant coefficient of t 0 {\displaystyle t^{0}} is det ( − A ) = ( − 1 ) n det ( A ) , {\displaystyle \det(-A)=(-1)^{n}\det(A),} the coefficient of t n {\displaystyle t^{n}} is one, and the coefficient of t n − 1 {\displaystyle t^{n-1}} is tr(−A) = −tr(A), where tr(A) is the trace of A . {\displaystyle A.} (The signs given here correspond to the formal definition given in the previous section; for the alternative definition these would instead be det ( A ) {\displaystyle \det(A)} and (−1)n – 1 tr(A) respectively.) For a 2 × 2 {\displaystyle 2\times 2} matrix A , {\displaystyle A,} the characteristic polynomial is thus given by t 2 − tr ⁡ ( A ) t + det ( A ) . {\displaystyle t^{2}-\operatorname {tr} (A)t+\det(A).} Using the language of exterior algebra, the characteristic polynomial of an n × n {\displaystyle n\times n} matrix A {\displaystyle A} may be expressed as p A ( t ) = ∑ k = 0 n t n − k ( − 1 ) k tr ⁡ ( ⋀ k A ) {\displaystyle p_{A}(t)=\sum _{k=0}^{n}t^{n-k}(-1)^{k}\operatorname {tr} \left(\textstyle \bigwedge ^{k}A\right)} where tr ⁡ ( ⋀ k A ) {\textstyle \operatorname {tr} \left(\bigwedge ^{k}A\right)} is the trace of the k {\displaystyle k} th exterior power of A , {\displaystyle A,} which has dimension ( n k ) . {\textstyle {\binom {n}{k}}.} This trace may be computed as the sum of all principal minors of A {\displaystyle A} of size k . {\displaystyle k.} The recursive Faddeev–LeVerrier algorithm computes these coefficients more efficiently . When the characteristic of the field of the coefficients is 0 , {\displaystyle 0,} each such trace may alternatively be computed as a single determinant, that of the k × k {\displaystyle k\times k} matrix, tr ⁡ ( ⋀ k A ) = 1 k ! \| tr ⁡ A k − 1 0 ⋯ 0 tr ⁡ A 2 tr ⁡ A k − 2 ⋯ 0 ⋮ ⋮ ⋱ ⋮ tr ⁡ A k − 1 tr ⁡ A k − 2 ⋯ 1 tr ⁡ A k tr ⁡ A k − 1 ⋯ tr ⁡ A \| . {\displaystyle \operatorname {tr} \left(\textstyle \bigwedge ^{k}A\right)={\frac {1}{k!}}{\begin{vmatrix}\operatorname {tr} A&k-1&0&\cdots &0\\\operatorname {tr} A^{2}&\operatorname {tr} A&k-2&\cdots &0\\\vdots &\vdots &&\ddots &\vdots \\\operatorname {tr} A^{k-1}&\operatorname {tr} A^{k-2}&&\cdots &1\\\operatorname {tr} A^{k}&\operatorname {tr} A^{k-1}&&\cdots &\operatorname {tr} A\end{vmatrix}}~.} The Cayley–Hamilton theorem states that replacing t {\displaystyle t} by A {\displaystyle A} in the characteristic polynomial (interpreting the resulting powers as matrix powers, and the constant term c {\displaystyle c} as c {\displaystyle c} times the identity matrix) yields the zero matrix. Informally speaking, every matrix satisfies its own characteristic equation. This statement is equivalent to saying that the minimal polynomial of A {\displaystyle A} divides the characteristic polynomial of A . {\displaystyle A.} Two similar matrices have the same characteristic polynomial. The converse however is not true in general: two matrices with the same characteristic polynomial need not be similar. The matrix A {\displaystyle A} and its transpose have the same characteristic polynomial. A {\displaystyle A} is similar to a triangular matrix if and only if its characteristic polynomial can be completely factored into linear factors over K {\displaystyle K} (the same is true with the minimal polynomial instead of the characteristic polynomial). In this case A {\displaystyle A} is similar to a matrix in Jordan normal form. == Characteristic polynomial of a product of two matrices == If A {\displaystyle A} and B {\displaystyle B} are two square n × n {\displaystyle n\times n} matrices then characteristic polynomials of A B {\displaystyle AB} and B A {\displaystyle BA} coincide: p A B ( t ) = p B A ( t ) . {\displaystyle p_{AB}(t)=p_{BA}(t).\,} When A {\displaystyle A} is non-singular this result follows from the fact that A B {\displaystyle AB} and B A {\displaystyle BA} are similar: B A = A − 1 ( A B ) A . {\displaystyle BA=A^{-1}(AB)A.} For the case where both A {\displaystyle A} and B {\displaystyle B} are singular, the desired identity is an equality between polynomials in t {\displaystyle t} and the coefficients of the matrices. Thus, to prove this equality, it suffices to prove that it is verified on a non-empty open subset (for the usual topology, or, more generally, for the Zariski topology) of the space of all the coefficients. As the non-singular matrices form such an open subset of the space of all matrices, this proves the result. More generally, if A {\displaystyle A} is a matrix of order m × n {\displaystyle m\times n} and B {\displaystyle B} is a matrix of order n × m , {\displaystyle n\times m,} then A B {\displaystyle AB} is m × m {\displaystyle m\times m} and B A {\displaystyle BA} is n × n {\displaystyle n\times n} matrix, and one has p B A ( t ) = t n − m p A B ( t ) . {\displaystyle p_{BA}(t)=t^{n-m}p_{AB}(t).\,} To prove this, one may suppose n > m , {\displaystyle n>m,} by exchanging, if needed, A {\displaystyle A} and B . {\displaystyle B.} Then, by bordering A {\displaystyle A} on the bottom by n − m {\displaystyle n-m} rows of zeros, and B {\displaystyle B} on the right, by, n − m {\displaystyle n-m} columns of zeros, one gets two n × n {\displaystyle n\times n} matrices A ′ {\displaystyle A^{\prime }} and B ′ {\displaystyle B^{\prime }} such that B ′ A ′ = B A {\displaystyle B^{\prime }A^{\prime }=BA} and A ′ B ′ {\displaystyle A^{\prime }B^{\prime }} is equal to A B {\displaystyle AB} bordered by n − m {\displaystyle n-m} rows and columns of zeros. The result follows from the case of square matrices, by comparing the characteristic polynomials of A ′ B ′ {\displaystyle A^{\prime }B^{\prime }} and A B . {\displaystyle AB.} == Characteristic polynomial of Ak == If λ {\displaystyle \lambda } is an eigenvalue of a square matrix A {\displaystyle A} with eigenvector v , {\displaystyle \mathbf {v} ,} then λ k {\displaystyle \lambda ^{k}} is an eigenvalue of A k {\displaystyle A^{k}} because A k v = A k − 1 A v = λ A k − 1 v = ⋯ = λ k v . {\displaystyle A^{k}{\textbf {v}}=A^{k-1}A{\textbf {v}}=\lambda A^{k-1}{\textbf {v}}=\dots =\lambda ^{k}{\textbf {v}}.} The multiplicities can be shown to agree as well, and this generalizes to any polynomial in place of x k {\displaystyle x^{k}} : That is, the algebraic multiplicity of λ {\displaystyle \lambda } in f ( A ) {\displaystyle f(A)} equals the sum of algebraic multiplicities of λ ′ {\displaystyle \lambda '} in A {\displaystyle A} over λ ′ {\displaystyle \lambda '} such that f ( λ ′ ) = λ . {\displaystyle f(\lambda ')=\lambda .} In particular, tr ⁡ ( f ( A ) ) = ∑ i = 1 n f ( λ i ) {\displaystyle \operatorname {tr} (f(A))=\textstyle \sum _{i=1}^{n}f(\lambda _{i})} and det ⁡ ( f ( A ) ) = ∏ i = 1 n f ( λ i ) . {\displaystyle \operatorname {det} (f(A))=\textstyle \prod _{i=1}^{n}f(\lambda _{i}).} Here a polynomial f ( t ) = t 3 + 1 , {\displaystyle f(t)=t^{3}+1,} for example, is evaluated on a matrix A {\displaystyle A} simply as f ( A ) = A 3 + I . {\displaystyle f(A)=A^{3}+I.} The theorem applies to matrices and polynomials over any field or commutative ring. However, the assumption that p A ( t ) {\displaystyle p_{A}(t)} has a factorization into linear factors is not always true, unless the matrix is over an algebraically closed field such as the complex numbers. == Secular function and secular equation == === Secular function === The term secular function has been used for what is now called characteristic polynomial (in some literature the term secular function is still used). The term comes from the fact that the characteristic polynomial was used to calculate secular perturbations (on a time scale of a century, that is, slow compared to annual motion) of planetary orbits, according to Lagrange's theory of oscillations. === Secular equation === Secular equation may have several meanings. In linear algebra it is sometimes used in place of characteristic equation. In astronomy it is the algebraic or numerical expression of the magnitude of the inequalities in a planet's motion that remain after the inequalities of a short period have been allowed for. In molecular orbital calculations relating to the energy of the electron and its wave function it is also used instead of the characteristic equation. == For general associative algebras == The above definition of the characteristic polynomial of a matrix A ∈ M n ( F ) {\displaystyle A\in M_{n}(F)} with entries in a field F {\displaystyle F} generalizes without any changes to the case when F {\displaystyle F} is just a commutative ring. Garibaldi (2004) defines the characteristic polynomial for elements of an arbitrary finite-dimensional (associative, but not necessarily commutative) algebra over a field F {\displaystyle F} and proves the standard properties of the characteristic polynomial in this generality. == See also == Characteristic equation (disambiguation) Invariants of tensors Companion matrix Faddeev–LeVerrier algorithm Cayley–Hamilton theorem Samuelson–Berkowitz algorithm == References == T.S. Blyth & E.F. Robertson (1998) Basic Linear Algebra, p 149, Springer ISBN 3-540-76122-5 . John B. Fraleigh & Raymond A. Beauregard (1990) Linear Algebra 2nd edition, p 246, Addison-Wesley ISBN 0-201-11949-8 . Garibaldi, Skip (2004), "The characteristic polynomial and determinant are not ad hoc constructions", American Mathematical Monthly, 111 (9): 761–778, arXiv:math/0203276, doi:10.2307/4145188, JSTOR 4145188, MR 2104048 Werner Greub (1974) Linear Algebra 4th edition, pp 120–5, Springer, ISBN 0-387-90110-8 . Paul C. Shields (1980) Elementary Linear Algebra 3rd edition, p 274, Worth Publishers ISBN 0-87901-121-1 . Gilbert Strang (1988) Linear Algebra and Its Applications 3rd edition, p 246, Brooks/Cole ISBN 0-15-551005-3 .
Wikipedia:Characteristic variety#0	In mathematical analysis, the characteristic variety of a microdifferential operator P is an algebraic variety that is the zero set of the principal symbol of P in the cotangent bundle. It is invariant under a quantized contact transformation. The notion is also defined more generally in commutative algebra. A basic theorem says a characteristic variety is involutive. == References == M. Sato, T. Kawai, and M. Kashiwara: Microfunctions and Pseudo-differential Equations. Lecture note in Math., No. 287, Springer, Berlin-Heidelberg-New York, pp. 265–529 (1973)
Wikipedia:Characterizations of the exponential function#0	In mathematics, the exponential function can be characterized in many ways. This article presents some common characterizations, discusses why each makes sense, and proves that they are all equivalent. The exponential function occurs naturally in many branches of mathematics. Walter Rudin called it "the most important function in mathematics". It is therefore useful to have multiple ways to define (or characterize) it. Each of the characterizations below may be more or less useful depending on context. The "product limit" characterization of the exponential function was discovered by Leonhard Euler. == Characterizations == The six most common definitions of the exponential function exp ⁡ ( x ) = e x {\displaystyle \exp(x)=e^{x}} for real values x ∈ R {\displaystyle x\in \mathbb {R} } are as follows. Product limit. Define e x {\displaystyle e^{x}} by the limit: e x = lim n → ∞ ( 1 + x n ) n . {\displaystyle e^{x}=\lim _{n\to \infty }\left(1+{\frac {x}{n}}\right)^{n}.} Power series. Define ex as the value of the infinite series e x = ∑ n = 0 ∞ x n n ! = 1 + x + x 2 2 ! + x 3 3 ! + x 4 4 ! + ⋯ {\displaystyle e^{x}=\sum _{n=0}^{\infty }{x^{n} \over n!}=1+x+{\frac {x^{2}}{2!}}+{\frac {x^{3}}{3!}}+{\frac {x^{4}}{4!}}+\cdots } (Here n! denotes the factorial of n. One proof that e is irrational uses a special case of this formula.) Inverse of logarithm integral. Define e x {\displaystyle e^{x}} to be the unique number y > 0 such that ∫ 1 y d t t = x . {\displaystyle \int _{1}^{y}{\frac {dt}{t}}=x.} That is, e x {\displaystyle e^{x}} is the inverse of the natural logarithm function x = ln ⁡ ( y ) {\displaystyle x=\ln(y)} , which is defined by this integral. Differential equation. Define y ( x ) = e x {\displaystyle y(x)=e^{x}} to be the unique solution to the differential equation with initial value: y ′ = y , y ( 0 ) = 1 , {\displaystyle y'=y,\quad y(0)=1,} where y ′ = d y d x {\displaystyle y'={\tfrac {dy}{dx}}} denotes the derivative of y. Functional equation. The exponential function e x {\displaystyle e^{x}} is the unique function f with the multiplicative property f ( x + y ) = f ( x ) f ( y ) {\displaystyle f(x+y)=f(x)f(y)} for all x , y {\displaystyle x,y} and f ′ ( 0 ) = 1 {\displaystyle f'(0)=1} . The condition f ′ ( 0 ) = 1 {\displaystyle f'(0)=1} can be replaced with f ( 1 ) = e {\displaystyle f(1)=e} together with any of the following regularity conditions: For the uniqueness, one must impose some regularity condition, since other functions satisfying f ( x + y ) = f ( x ) f ( y ) {\displaystyle f(x+y)=f(x)f(y)} can be constructed using a basis for the real numbers over the rationals, as described by Hewitt and Stromberg. Elementary definition by powers. Define the exponential function with base a > 0 {\displaystyle a>0} to be the continuous function a x {\displaystyle a^{x}} whose value on integers x = n {\displaystyle x=n} is given by repeated multiplication or division of a {\displaystyle a} , and whose value on rational numbers x = n / m {\displaystyle x=n/m} is given by a n / m = A 2 a n m {\displaystyle a^{n/m}=\ \ {\sqrt[{m}]{{\vphantom {A^{2}}}a^{n}}}} . Then define e x {\displaystyle e^{x}} to be the exponential function whose base a = e {\displaystyle a=e} is the unique positive real number satisfying: lim h → 0 e h − 1 h = 1. {\displaystyle \lim _{h\to 0}{\frac {e^{h}-1}{h}}=1.} == Larger domains == One way of defining the exponential function over the complex numbers is to first define it for the domain of real numbers using one of the above characterizations, and then extend it as an analytic function, which is characterized by its values on any infinite domain set. Also, characterisations (1), (2), and (4) for e x {\displaystyle e^{x}} apply directly for x {\displaystyle x} a complex number. Definition (3) presents a problem because there are non-equivalent paths along which one could integrate; but the equation of (3) should hold for any such path modulo 2 π i {\displaystyle 2\pi i} . As for definition (5), the additive property together with the complex derivative f ′ ( 0 ) = 1 {\displaystyle f'(0)=1} are sufficient to guarantee f ( x ) = e x {\displaystyle f(x)=e^{x}} . However, the initial value condition f ( 1 ) = e {\displaystyle f(1)=e} together with the other regularity conditions are not sufficient. For example, for real x and y, the function f ( x + i y ) = e x ( cos ⁡ ( 2 y ) + i sin ⁡ ( 2 y ) ) = e x + 2 i y {\displaystyle f(x+iy)=e^{x}(\cos(2y)+i\sin(2y))=e^{x+2iy}} satisfies the three listed regularity conditions in (5) but is not equal to exp ⁡ ( x + i y ) {\displaystyle \exp(x+iy)} . A sufficient condition is that f ( 1 ) = e {\displaystyle f(1)=e} and that f {\displaystyle f} is a conformal map at some point; or else the two initial values f ( 1 ) = e {\displaystyle f(1)=e} and f ( i ) = cos ⁡ ( 1 ) + i sin ⁡ ( 1 ) {\textstyle f(i)=\cos(1)+i\sin(1)} together with the other regularity conditions. One may also define the exponential on other domains, such as matrices and other algebras. Definitions (1), (2), and (4) all make sense for arbitrary Banach algebras. == Proof that each characterization makes sense == Some of these definitions require justification to demonstrate that they are well-defined. For example, when the value of the function is defined as the result of a limiting process (i.e. an infinite sequence or series), it must be demonstrated that such a limit always exists. === Characterization 1 === The error of the product limit expression is described by: ( 1 + x n ) n = e x ( 1 − x 2 2 n + x 3 ( 8 + 3 x ) 24 n 2 + ⋯ ) , {\displaystyle \left(1+{\frac {x}{n}}\right)^{n}=e^{x}\left(1-{\frac {x^{2}}{2n}}+{\frac {x^{3}(8+3x)}{24n^{2}}}+\cdots \right),} where the polynomial's degree (in x) in the term with denominator nk is 2k. === Characterization 2 === Since lim n → ∞ \| x n + 1 / ( n + 1 ) ! x n / n ! \| = lim n → ∞ \| x n + 1 \| = 0 < 1. {\displaystyle \lim _{n\to \infty }\left\|{\frac {x^{n+1}/(n+1)!}{x^{n}/n!}}\right\|=\lim _{n\to \infty }\left\|{\frac {x}{n+1}}\right\|=0<1.} it follows from the ratio test that ∑ n = 0 ∞ x n n ! {\textstyle \sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}} converges for all x. === Characterization 3 === Since the integrand is an integrable function of t, the integral expression is well-defined. It must be shown that the function from R + {\displaystyle \mathbb {R} ^{+}} to R {\displaystyle \mathbb {R} } defined by x ↦ ∫ 1 x d t t {\displaystyle x\mapsto \int _{1}^{x}{\frac {dt}{t}}} is a bijection. Since 1/t is positive for positive t, this function is strictly increasing, hence injective. If the two integrals ∫ 1 ∞ d t t = ∞ ∫ 1 0 d t t = − ∞ {\displaystyle {\begin{aligned}\int _{1}^{\infty }{\frac {dt}{t}}&=\infty \\[8pt]\int _{1}^{0}{\frac {dt}{t}}&=-\infty \end{aligned}}} hold, then it is surjective as well. Indeed, these integrals do hold; they follow from the integral test and the divergence of the harmonic series. === Characterization 6 === The definition depends on the unique positive real number a = e {\displaystyle a=e} satisfying: lim h → 0 a h − 1 h = 1. {\displaystyle \lim _{h\to 0}{\frac {a^{h}-1}{h}}=1.} This limit can be shown to exist for any a {\displaystyle a} , and it defines a continuous increasing function f ( a ) = ln ⁡ ( a ) {\displaystyle f(a)=\ln(a)} with f ( 1 ) = 0 {\displaystyle f(1)=0} and lim a → ∞ f ( a ) = ∞ {\displaystyle \lim _{a\to \infty }f(a)=\infty } , so the Intermediate value theorem guarantees the existence of such a value a = e {\displaystyle a=e} . == Equivalence of the characterizations == The following arguments demonstrate the equivalence of the above characterizations for the exponential function. === Characterization 1 ⇔ characterization 2 === The following argument is adapted from Rudin, theorem 3.31, p. 63–65. Let x ≥ 0 {\displaystyle x\geq 0} be a fixed non-negative real number. Define t n = ( 1 + x n ) n , s n = ∑ k = 0 n x k k ! , e x = lim n → ∞ s n . {\displaystyle t_{n}=\left(1+{\frac {x}{n}}\right)^{n},\qquad s_{n}=\sum _{k=0}^{n}{\frac {x^{k}}{k!}},\qquad e^{x}=\lim _{n\to \infty }s_{n}.} By the binomial theorem, t n = ∑ k = 0 n ( n k ) x k n k = 1 + x + ∑ k = 2 n n ( n − 1 ) ( n − 2 ) ⋯ ( n − ( k − 1 ) ) x k k ! n k = 1 + x + x 2 2 ! ( 1 − 1 n ) + x 3 3 ! ( 1 − 1 n ) ( 1 − 2 n ) + ⋯ ⋯ + x n n ! ( 1 − 1 n ) ⋯ ( 1 − n − 1 n ) ≤ s n {\displaystyle {\begin{aligned}t_{n}&=\sum _{k=0}^{n}{n \choose k}{\frac {x^{k}}{n^{k}}}=1+x+\sum _{k=2}^{n}{\frac {n(n-1)(n-2)\cdots (n-(k-1))x^{k}}{k!\,n^{k}}}\\[8pt]&=1+x+{\frac {x^{2}}{2!}}\left(1-{\frac {1}{n}}\right)+{\frac {x^{3}}{3!}}\left(1-{\frac {1}{n}}\right)\left(1-{\frac {2}{n}}\right)+\cdots \\[8pt]&{}\qquad \cdots +{\frac {x^{n}}{n!}}\left(1-{\frac {1}{n}}\right)\cdots \left(1-{\frac {n-1}{n}}\right)\leq s_{n}\end{aligned}}} (using x ≥ 0 to obtain the final inequality) so that: lim sup n → ∞ t n ≤ lim sup n → ∞ s n = e x {\displaystyle \limsup _{n\to \infty }t_{n}\leq \limsup _{n\to \infty }s_{n}=e^{x}} One must use lim sup because it is not known if tn converges. For the other inequality, by the above expression for tn, if 2 ≤ m ≤ n, we have: 1 + x + x 2 2 ! ( 1 − 1 n ) + ⋯ + x m m ! ( 1 − 1 n ) ( 1 − 2 n ) ⋯ ( 1 − m − 1 n ) ≤ t n . {\displaystyle 1+x+{\frac {x^{2}}{2!}}\left(1-{\frac {1}{n}}\right)+\cdots +{\frac {x^{m}}{m!}}\left(1-{\frac {1}{n}}\right)\left(1-{\frac {2}{n}}\right)\cdots \left(1-{\frac {m-1}{n}}\right)\leq t_{n}.} Fix m, and let n approach infinity. Then s m = 1 + x + x 2 2 ! + ⋯ + x m m ! ≤ lim inf n → ∞ t n {\displaystyle s_{m}=1+x+{\frac {x^{2}}{2!}}+\cdots +{\frac {x^{m}}{m!}}\leq \liminf _{n\to \infty }\ t_{n}} (again, one must use lim inf because it is not known if tn converges). Now, take the above inequality, let m approach infinity, and put it together with the other inequality to obtain: lim sup n → ∞ t n ≤ e x ≤ lim inf n → ∞ t n {\displaystyle \limsup _{n\to \infty }t_{n}\leq e^{x}\leq \liminf _{n\to \infty }t_{n}} so that lim n → ∞ t n = e x . {\displaystyle \lim _{n\to \infty }t_{n}=e^{x}.} This equivalence can be extended to the negative real numbers by noting ( 1 − r n ) n ( 1 + r n ) n = ( 1 − r 2 n 2 ) n {\textstyle \left(1-{\frac {r}{n}}\right)^{n}\left(1+{\frac {r}{n}}\right)^{n}=\left(1-{\frac {r^{2}}{n^{2}}}\right)^{n}} and taking the limit as n goes to infinity. === Characterization 1 ⇔ characterization 3 === Here, the natural logarithm function is defined in terms of a definite integral as above. By the first part of fundamental theorem of calculus, d d x ln ⁡ x = d d x ∫ 1 x 1 t d t = 1 x . {\displaystyle {\frac {d}{dx}}\ln x={\frac {d}{dx}}\int _{1}^{x}{\frac {1}{t}}\,dt={\frac {1}{x}}.} Besides, ln ⁡ 1 = ∫ 1 1 d t t = 0 {\textstyle \ln 1=\int _{1}^{1}{\frac {dt}{t}}=0} Now, let x be any fixed real number, and let y = lim n → ∞ ( 1 + x n ) n . {\displaystyle y=\lim _{n\to \infty }\left(1+{\frac {x}{n}}\right)^{n}.} Ln(y) = x, which implies that y = ex, where ex is in the sense of definition 3. We have ln ⁡ y = ln ⁡ lim n → ∞ ( 1 + x n ) n = lim n → ∞ ln ⁡ ( 1 + x n ) n . {\displaystyle \ln y=\ln \lim _{n\to \infty }\left(1+{\frac {x}{n}}\right)^{n}=\lim _{n\to \infty }\ln \left(1+{\frac {x}{n}}\right)^{n}.} Here, the continuity of ln(y) is used, which follows from the continuity of 1/t: ln ⁡ y = lim n → ∞ n ln ⁡ ( 1 + x n ) = lim n → ∞ x ln ⁡ ( 1 + ( x / n ) ) ( x / n ) . {\displaystyle \ln y=\lim _{n\to \infty }n\ln \left(1+{\frac {x}{n}}\right)=\lim _{n\to \infty }{\frac {x\ln \left(1+(x/n)\right)}{(x/n)}}.} Here, the result lnan = nlna has been used. This result can be established for n a natural number by induction, or using integration by substitution. (The extension to real powers must wait until ln and exp have been established as inverses of each other, so that ab can be defined for real b as eb lna.) = x ⋅ lim h → 0 ln ⁡ ( 1 + h ) h where h = x n {\displaystyle =x\cdot \lim _{h\to 0}{\frac {\ln \left(1+h\right)}{h}}\quad {\text{ where }}h={\frac {x}{n}}} = x ⋅ lim h → 0 ln ⁡ ( 1 + h ) − ln ⁡ 1 h {\displaystyle =x\cdot \lim _{h\to 0}{\frac {\ln \left(1+h\right)-\ln 1}{h}}} = x ⋅ d d t ln ⁡ t \| t = 1 {\displaystyle =x\cdot {\frac {d}{dt}}\ln t{\Bigg \|}_{t=1}} = x . {\displaystyle \!\,=x.} === Characterization 1 ⇔ characterization 4 === Let y ( t ) {\displaystyle y(t)} denote the solution to the initial value problem y ′ = y , y ( 0 ) = 1 {\displaystyle y'=y,\ y(0)=1} . Applying the simplest form of Euler's method with increment Δ t = x n {\displaystyle \Delta t={\frac {x}{n}}} and sample points t = 0 , Δ t , 2 Δ t , … , n Δ t {\displaystyle t\ =\ 0,\ \Delta t,\ 2\Delta t,\ldots ,\ n\Delta t} gives the recursive formula: y ( t + Δ t ) ≈ y ( t ) + y ′ ( t ) Δ t = y ( t ) + y ( t ) Δ t = y ( t ) ( 1 + Δ t ) . {\displaystyle y(t+\Delta t)\ \approx \ y(t)+y'(t)\Delta t\ =\ y(t)+y(t)\Delta t\ =\ y(t)\,(1+\Delta t).} This recursion is immediately solved to give the approximate value y ( x ) = y ( n Δ t ) ≈ ( 1 + Δ t ) n {\displaystyle y(x)=y(n\Delta t)\approx (1+\Delta t)^{n}} , and since Euler's Method is known to converge to the exact solution, we have: y ( x ) = lim n → ∞ ( 1 + x n ) n . {\displaystyle y(x)=\lim _{n\to \infty }\left(1+{\frac {x}{n}}\right)^{n}.} === Characterization 2 ⇔ characterization 4 === Let n be a non-negative integer. In the sense of definition 4 and by induction, d n y d x n = y {\displaystyle {\frac {d^{n}y}{dx^{n}}}=y} . Therefore d n y d x n \| x = 0 = y ( 0 ) = 1. {\displaystyle {\frac {d^{n}y}{dx^{n}}}{\Bigg \|}_{x=0}=y(0)=1.} Using Taylor series, y = ∑ n = 0 ∞ f ( n ) ( 0 ) n ! x n = ∑ n = 0 ∞ 1 n ! x n = ∑ n = 0 ∞ x n n ! . {\displaystyle y=\sum _{n=0}^{\infty }{\frac {f^{(n)}(0)}{n!}}\,x^{n}=\sum _{n=0}^{\infty }{\frac {1}{n!}}\,x^{n}=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}.} This shows that definition 4 implies definition 2. In the sense of definition 2, d d x e x = d d x ( 1 + ∑ n = 1 ∞ x n n ! ) = ∑ n = 1 ∞ n x n − 1 n ! = ∑ n = 1 ∞ x n − 1 ( n − 1 ) ! = ∑ k = 0 ∞ x k k ! , where k = n − 1 = e x {\displaystyle {\begin{aligned}{\frac {d}{dx}}e^{x}&={\frac {d}{dx}}\left(1+\sum _{n=1}^{\infty }{\frac {x^{n}}{n!}}\right)=\sum _{n=1}^{\infty }{\frac {nx^{n-1}}{n!}}=\sum _{n=1}^{\infty }{\frac {x^{n-1}}{(n-1)!}}\\[6pt]&=\sum _{k=0}^{\infty }{\frac {x^{k}}{k!}},{\text{ where }}k=n-1\\[6pt]&=e^{x}\end{aligned}}} Besides, e 0 = 1 + 0 + 0 2 2 ! + 0 3 3 ! + ⋯ = 1. {\textstyle e^{0}=1+0+{\frac {0^{2}}{2!}}+{\frac {0^{3}}{3!}}+\cdots =1.} This shows that definition 2 implies definition 4. === Characterization 2 ⇒ characterization 5 === In the sense of definition 2, the equation exp ⁡ ( x + y ) = exp ⁡ ( x ) exp ⁡ ( y ) {\displaystyle \exp(x+y)=\exp(x)\exp(y)} follows from the term-by-term manipulation of power series justified by uniform convergence, and the resulting equality of coefficients is just the Binomial theorem. Furthermore: exp ′ ⁡ ( 0 ) = lim h → 0 e h − 1 h = lim h → 0 1 h ( ( 1 + h + h 2 2 ! + h 3 3 ! + h 4 4 ! + ⋯ ) − 1 ) = lim h → 0 ( 1 + h 2 ! + h 2 3 ! + h 3 4 ! + ⋯ ) = 1. {\displaystyle {\begin{aligned}\exp '(0)&=\lim _{h\to 0}{\frac {e^{h}-1}{h}}\\&=\lim _{h\to 0}{\frac {1}{h}}\left(\left(1+h+{\frac {h^{2}}{2!}}+{\frac {h^{3}}{3!}}+{\frac {h^{4}}{4!}}+\cdots \right)-1\right)\\&=\lim _{h\to 0}\left(1+{\frac {h}{2!}}+{\frac {h^{2}}{3!}}+{\frac {h^{3}}{4!}}+\cdots \right)\ =\ 1.\\\end{aligned}}} === Characterization 3 ⇔ characterization 4 === Characterisation 3 first defines the natural logarithm: log ⁡ x = def ∫ 1 x d t t , {\displaystyle \log x\ \ {\stackrel {\text{def}}{=}}\ \int _{1}^{x}\!{\frac {dt}{t}},} then exp {\displaystyle \exp } as the inverse function with x = log ⁡ ( exp ⁡ x ) {\textstyle x=\log(\exp x)} . Then by the Chain rule: 1 = d d x [ log ⁡ ( exp ⁡ ( x ) ) ] = log ′ ⁡ ( exp ⁡ ( x ) ) ⋅ exp ′ ⁡ ( x ) = exp ′ ⁡ ( x ) exp ⁡ ( x ) , {\displaystyle 1={\frac {d}{dx}}[\log(\exp(x))]=\log '(\exp(x))\cdot \exp '(x)={\frac {\exp '(x)}{\exp(x)}},} i.e. exp ′ ⁡ ( x ) = exp ⁡ ( x ) {\displaystyle \exp '(x)=\exp(x)} . Finally, log ⁡ ( 1 ) = 0 {\displaystyle \log(1)=0} , so exp ′ ⁡ ( 0 ) = exp ⁡ ( 0 ) = 1 {\displaystyle \exp '(0)=\exp(0)=1} . That is, y = exp ⁡ ( x ) {\displaystyle y=\exp(x)} is the unique solution of the initial value problem d y d x = y {\displaystyle {\frac {dy}{dx}}=y} , y ( 0 ) = 1 {\displaystyle y(0)=1} of characterization 4. Conversely, assume y = exp ⁡ ( x ) {\displaystyle y=\exp(x)} has exp ′ ⁡ ( x ) = exp ⁡ ( x ) {\displaystyle \exp '(x)=\exp(x)} and exp ⁡ ( 0 ) = 1 {\displaystyle \exp(0)=1} , and define log ⁡ ( x ) {\displaystyle \log(x)} as its inverse function with x = exp ⁡ ( log ⁡ x ) {\displaystyle x=\exp(\log x)} and log ⁡ ( 1 ) = 0 {\displaystyle \log(1)=0} . Then: 1 = d d x [ exp ⁡ ( log ⁡ ( x ) ) ] = exp ′ ⁡ ( log ⁡ ( x ) ) ⋅ log ′ ⁡ ( x ) = exp ⁡ ( log ⁡ ( x ) ) ⋅ log ′ ⁡ ( x ) = x ⋅ log ′ ⁡ ( x ) , {\displaystyle 1={\frac {d}{dx}}[\exp(\log(x))]=\exp '(\log(x))\cdot \log '(x)=\exp(\log(x))\cdot \log '(x)=x\cdot \log '(x),} i.e. log ′ ⁡ ( x ) = 1 x {\displaystyle \log '(x)={\frac {1}{x}}} . By the Fundamental theorem of calculus, ∫ 1 x 1 t d t = log ⁡ ( x ) − log ⁡ ( 1 ) = log ⁡ ( x ) . {\displaystyle \int _{1}^{x}{\frac {1}{t}}\,dt=\log(x)-\log(1)=\log(x).} === Characterization 5 ⇒ characterization 4 === The conditions f'(0) = 1 and f(x + y) = f(x) f(y) imply both conditions in characterization 4. Indeed, one gets the initial condition f(0) = 1 by dividing both sides of the equation f ( 0 ) = f ( 0 + 0 ) = f ( 0 ) f ( 0 ) {\displaystyle f(0)=f(0+0)=f(0)f(0)} by f(0), and the condition that f′(x) = f(x) follows from the condition that f′(0) = 1 and the definition of the derivative as follows: f ′ ( x ) = lim h → 0 f ( x + h ) − f ( x ) h = lim h → 0 f ( x ) f ( h ) − f ( x ) h = lim h → 0 f ( x ) f ( h ) − 1 h = f ( x ) lim h → 0 f ( h ) − 1 h = f ( x ) lim h → 0 f ( 0 + h ) − f ( 0 ) h = f ( x ) f ′ ( 0 ) = f ( x ) . {\displaystyle {\begin{array}{rcccccc}f'(x)&=&\lim \limits _{h\to 0}{\frac {f(x+h)-f(x)}{h}}&=&\lim \limits _{h\to 0}{\frac {f(x)f(h)-f(x)}{h}}&=&\lim \limits _{h\to 0}f(x){\frac {f(h)-1}{h}}\\[1em]&=&f(x)\lim \limits _{h\to 0}{\frac {f(h)-1}{h}}&=&f(x)\lim \limits _{h\to 0}{\frac {f(0+h)-f(0)}{h}}&=&f(x)f'(0)=f(x).\end{array}}} === Characterization 5 ⇒ characterization 4 === Assum characterization 5, the multiplicative property together with the initial condition exp ′ ⁡ ( 0 ) = 1 {\displaystyle \exp '(0)=1} imply that: d d x exp ⁡ ( x ) = lim h → 0 exp ⁡ ( x + h ) − exp ⁡ ( x ) h = exp ⁡ ( x ) ⋅ lim h → 0 exp ⁡ ( h ) − 1 h = exp ⁡ ( x ) exp ′ ⁡ ( 0 ) = exp ⁡ ( x ) . {\displaystyle {\begin{array}{rcl}{\frac {d}{dx}}\exp(x)&=&\lim _{h\to 0}{\frac {\exp(x{+}h)-\exp(x)}{h}}\\&=&\exp(x)\cdot \lim _{h\to 0}{\frac {\exp(h)-1}{h}}\\&=&\exp(x)\exp '(0)=\exp(x).\end{array}}} === Characterization 5 ⇔ characterization 6 === By inductively applying the multiplication rule, we get: f ( n m ) m = f ( n m + ⋯ + n m ) = f ( n ) = f ( 1 ) n , {\displaystyle f\left({\frac {n}{m}}\right)^{m}=f\left({\frac {n}{m}}+\cdots +{\frac {n}{m}}\right)=f(n)=f(1)^{n},} and thus f ( n m ) = f ( 1 ) n m = def a n / m {\displaystyle f\left({\frac {n}{m}}\right)={\sqrt[{m}]{f(1)^{n}}}\ {\stackrel {\text{def}}{=}}\ a^{n/m}} for a = f ( 1 ) {\displaystyle a=f(1)} . Then the condition f ′ ( 0 ) = 1 {\displaystyle f'(0)=1} means that lim h → 0 a h − 1 h = 1 {\displaystyle \lim _{h\to 0}{\tfrac {a^{h}-1}{h}}=1} , so a = e {\displaystyle a=e} by definition. Also, any of the regularity conditions of definition 5 imply that f ( x ) {\displaystyle f(x)} is continuous at all real x {\displaystyle x} (see below). The converse is similar. === Characterization 5 ⇒ characterization 6 === Let f ( x ) {\displaystyle f(x)} be a Lebesgue-integrable non-zero function satisfying the mulitiplicative property f ( x + y ) = f ( x ) f ( y ) {\displaystyle f(x+y)=f(x)f(y)} with f ( 1 ) = e {\displaystyle f(1)=e} . Following Hewitt and Stromberg, exercise 18.46, we will prove that Lebesgue-integrability implies continuity. This is sufficient to imply f ( x ) = e x {\displaystyle f(x)=e^{x}} according to characterization 6, arguing as above. First, a few elementary properties: If f ( x ) {\displaystyle f(x)} is nonzero anywhere (say at x = y {\displaystyle x=y} ), then it is non-zero everywhere. Proof: f ( y ) = f ( x ) f ( y − x ) ≠ 0 {\displaystyle f(y)=f(x)f(y-x)\neq 0} implies f ( x ) ≠ 0 {\displaystyle f(x)\neq 0} . f ( 0 ) = 1 {\displaystyle f(0)=1} . Proof: f ( x ) = f ( x + 0 ) = f ( x ) f ( 0 ) {\displaystyle f(x)=f(x+0)=f(x)f(0)} and f ( x ) {\displaystyle f(x)} is non-zero. f ( − x ) = 1 / f ( x ) {\displaystyle f(-x)=1/f(x)} . Proof: 1 = f ( 0 ) = f ( x − x ) = f ( x ) f ( − x ) {\displaystyle 1=f(0)=f(x-x)=f(x)f(-x)} . If f ( x ) {\displaystyle f(x)} is continuous anywhere (say at x = y {\displaystyle x=y} ), then it is continuous everywhere. Proof: f ( x + δ ) − f ( x ) = f ( x − y ) [ f ( y + δ ) − f ( y ) ] → 0 {\displaystyle f(x+\delta )-f(x)=f(x-y)[f(y+\delta )-f(y)]\to 0} as δ → 0 {\displaystyle \delta \to 0} by continuity at y {\displaystyle y} . The second and third properties mean that it is sufficient to prove f ( x ) = e x {\displaystyle f(x)=e^{x}} for positive x. Since f ( x ) {\displaystyle f(x)} is a Lebesgue-integrable function, then we may define g ( x ) = ∫ 0 x f ( t ) d t {\textstyle g(x)=\int _{0}^{x}f(t)\,dt} . It then follows that g ( x + y ) − g ( x ) = ∫ x x + y f ( t ) d t = ∫ 0 y f ( x + t ) d t = f ( x ) g ( y ) . {\displaystyle g(x+y)-g(x)=\int _{x}^{x+y}f(t)\,dt=\int _{0}^{y}f(x+t)\,dt=f(x)g(y).} Since f ( x ) {\displaystyle f(x)} is nonzero, some y can be chosen such that g ( y ) ≠ 0 {\displaystyle g(y)\neq 0} and solve for f ( x ) {\displaystyle f(x)} in the above expression. Therefore: f ( x + δ ) − f ( x ) = [ g ( x + δ + y ) − g ( x + δ ) ] − [ g ( x + y ) − g ( x ) ] g ( y ) = [ g ( x + y + δ ) − g ( x + y ) ] − [ g ( x + δ ) − g ( x ) ] g ( y ) = f ( x + y ) g ( δ ) − f ( x ) g ( δ ) g ( y ) = g ( δ ) f ( x + y ) − f ( x ) g ( y ) . {\displaystyle {\begin{aligned}f(x+\delta )-f(x)&={\frac {[g(x+\delta +y)-g(x+\delta )]-[g(x+y)-g(x)]}{g(y)}}\\&={\frac {[g(x+y+\delta )-g(x+y)]-[g(x+\delta )-g(x)]}{g(y)}}\\&={\frac {f(x+y)g(\delta )-f(x)g(\delta )}{g(y)}}=g(\delta ){\frac {f(x+y)-f(x)}{g(y)}}.\end{aligned}}} The final expression must go to zero as δ → 0 {\displaystyle \delta \to 0} since g ( 0 ) = 0 {\displaystyle g(0)=0} and g ( x ) {\displaystyle g(x)} is continuous. It follows that f ( x ) {\displaystyle f(x)} is continuous. == References == Walter Rudin, Principles of Mathematical Analysis, 3rd edition (McGraw–Hill, 1976), chapter 8. Edwin Hewitt and Karl Stromberg, Real and Abstract Analysis (Springer, 1965).
Wikipedia:Charles Angas Hurst#0	Charles Angas Hurst AM DSc FAA (22 September 1923 – 19 October 2011) was an Australian mathematical physicist noted for his work in lattice models, quantum field theory, asymptotic expansions and Lie groups. He was appointed a Member of the Order of Australia in 2003, elected a Fellow of the Australian Academy of Science in 1972, and awarded the Centenary Medal and an Hon DSc (Melb). His PhD was a seminal work on quantum field theory, developing asymptotic expansions for perturbation expansions. In 1952 Hurst represented Australia in the inaugural International Mathematical Union. Hurst's work with Herbert Green on lattice problems and the Ising model led to the Free fermion field model, which contained all known properties of Fermions at the time of its publication. Hurst's work with Thirring (Thirring model) found the simplest non-linear field and is still used as a test model for perturbation theory. == References == == External links == Hurst's mathematical genealogy chemphys.adelaide.edu.au Interviews with Scientists Trove Obituary, austms.org.au Encyclopedia of Australian Science
Wikipedia:Charles Castonguay#0	Charles Castonguay (born 1940) is a retired associate professor of Mathematics and Statistics at the University of Ottawa. == Biography == A native English speaker, Castonguay was sent by his parents to a French Catholic primary school. He took his first English courses in high school. Enrolled in the Canadian Armed Forces to pursue university-level studies, he obtained a masters of mathematics from the University of Ottawa. During the three years of his military service, he was posted to National Defence headquarters in Ottawa as counsellor in mathematics and also taught young officers at the Collège militaire royal de Saint-Jean. After his service, he began teaching mathematics and statistics at the University of Ottawa and registered at McGill University to study the philosophy of mathematics and epistemology. The subject of his doctoral thesis was "meaning" and "existence" in mathematics. He obtained his Ph.D. in 1971. The thesis was published in 1972 and 1973. In 1970, he attended a meeting of the Parti Québécois in the Laurier riding. René Lévesque was the speaker at this meeting, designed to inform English speakers of the party's Sovereignty-Association project. After that experience, he campaigned for the party until the election of 1976. Partly to understand himself as a francized English speaker, he took great interest in the analysis of the linguistic behaviours of populations and language policies. He became a specialist on the subject of language shifts and completed several studies on behalf of the Office québécois de la langue française. On January 25, 2001, he took an active part in a symposium held by the commission of the Estates-General on the Situation and Future of the French Language in Quebec (Les enjeux démographiques et l'intégration des immigrants). Thereafter, he co-authored a book criticizing its final report. Since 2000, he contributes to the Dossier linguistique in the newspaper L'aut'journal. He participated in the foundation of the Institut de recherche sur le français en Amérique in 2008. == Works == === In English === Thesis Meaning and Existence in Mathematics, New York: Springer-Verlag, 1972, 159 pages (also New York: Springer, 1973) Articles "Nation Building and Anglicization in Canada's Capital Region", in Inroads Journal, Issue 11, 2002, pp. 71–86 "French is on the ropes. Why won’t Ottawa admit it ?", in Policy Options, volume 20, issue 8, 1999, pp. 39–50 "Getting the facts straight on French : Reflections following the 1996 Census", in Inroads Journal, Issue 8, 1999, pages 57 to 77 "The Fading Canadian Duality", in Language in Canada (ed. John R. Edwards), pp. 36–60, Cambridge: Cambridge University Press, 1998, 520 pages "Assimilation Trends among Official-Language Minorities. 1971-1991", in Towards the Twenty-First Century: Emerging Socio-Demographic Trends and Policy Issues in Canada, pp. 201–205, Federation of Canadian Demographers, Ottawa, 1996 "The Anglicization of Canada, 1971-1981", in Language Problems and Language Planning, vol. 11, issue 1, Spring 1987 "Intermarriage and Language Shift in Canada, 1971 and 1976", in Canadian Journal of Sociology, Vol. 7, No. 3 (Summer, 1982), pp. 263–277 "The Economic Context of Bilingualism and Language Transfer in the Montreal Metropolitan Area", in The Canadian Journal of Economics, Vol. 12, No. 3 (Aug., 1979), pp. 468–479 (with Calvin J. Veltman, Jac-Andre Boulet) "Why Hide the Facts? The Federalist Approach to the Language Crisis in Canada", in Canadian Public Policy, 1979 "Opportunities for the Study of Language Transfer in the 1971 Census", in Paul Lamy (ed.), Language Maintenance and Language Shift in Canada: New Dimensions in the Use of Census Language Data, Ottawa: Ottawa University Press, 1977, p. 63-73. "An Analysis of the Canadian Bilingual Districts Policy", in American Review of Canadian Studies, 1976 Other Transcript of a Standing Joint Committee on Official Languages hearing, recorded on April 28, 1998 === In French === Studies Incidence du sous-dénombrement et des changements apportés aux questions de recensement sur l'évolution de la composition linguistique de la population du Québec entre 1991 et 2001 (Étude 3), Montréal: Office québécois de la langue française, September 26, 2005, 29 pages Les indicateurs généraux de vitalité des langues au Québec : comparabilité et tendances 1971-2001 (Étude 1), Montréal : Office québécois de la langue française, May 26, 2005, 48 pages La langue parlée au foyer : signification pour l'avenir du français et tendances récentes, Oral presentation given to a symposium held by the commission of the Estates-General on the Situation and Future of the French Language in Quebec, January 25, 2001 L'indicateur de développement humain de l'ONU : le concept et son usage, Sainte-Foy: Publications du Québec, 1995, 68 pages L'assimilation linguistique : mesure et évolution 1971-1986, Québec: Conseil de la langue française, 1994, 243 pages Exogamie et anglicisation dans les régions de Montréal, Hull, Ottawa et Sudbury, Québec: Centre international de recherche sur le bilinguisme, 1981 == Notes == == References == André Langlois, «Pensée obsessive et minorités francophones : Quand l’obsession remplace la raison», Recherches sociographiques, XLIII-2, May–August 2002 : 381-387. Richard Marcoux, Book Review of Charles Castonguay, Les indicateurs généraux de vitalité des langues au Québec : comparabilité et tendances 1971-2001, in Recherches sociographiques, XLVII-2, May–August 2006 : 394-397. Ginette Leroux. "Le mathématicien de la langue Archived 2011-07-21 at the Wayback Machine", in L'aut'journal sur le Web, Issue 208, April 2002
Wikipedia:Charles Cook (academic)#0	Charles Henry Herbert Cook (30 September 1843 – 21 May 1910) was an English-born, Australian-raised, New Zealand-based mathematician. He was born in Kentish Town, Middlesex, England, on 30 September 1843, but educated in Melbourne, Australia, where he got a BA and an LLB from University of Melbourne. He then went to St John's College, Cambridge, initially to train for the English Bar but became interested in mathematics. In 1874, a year before being due to be called to the bar, Cook was appointed founding head of mathematics at Canterbury College, University of New Zealand (now Canterbury University). He joined co-founders John Macmillan Brown and Alexander Bickerton in Christchurch, New Zealand, and initially focused on Latin and mathematics. He was also involved in promoting the establishment of the University's engineering school. He is remembered primarily for his teaching; Nobel Prize–winning physicist Ernest Rutherford cited him as an influence: Cook was evidently a sound mathematician and an excellent teacher along orthodox lines, with no marked tendency to stray from those lines. He was involved in secondary education, acting as an examiner for the New Zealand Department of Education and holding a fellowship at the Anglican Christ's College, Christchurch from 1891 to 1908. Cook was a member of the Royal Commission on Higher Education 1878–1800 and a member of the senate of the University of New Zealand. In 1903, Cook appeared in the vanity press The Cyclopedia of New Zealand, with a photo and short article. He died in Marton, New Zealand, on 21 May 1910. == Cook Memorial Prize == After his death in 1910 a memorial prize was established, with Ernest Rutherford among the contributors. == References ==
Wikipedia:Charles Dunnett#0	Charles William Dunnett (24 August 1921 – May 18, 2007) was a Canadian statistician. He was the Statistical Society of Canada 1986 Gold Medalist and Professor Emeritus of the Departments of Mathematics, Statistics, Clinical Epidemiology, and Biostatistics of McMaster University. Two of his papers are listed among the top 25 most-cited papers in statistics (numbers 14 and 21 in the list). In 1965 he was elected as a Fellow of the American Statistical Association. Dunnett died on 18 May 2007 from lymphoma. == See also == Dunnett's test == References == == External links == A Conversation with Charles Dunnett at the SSC
Wikipedia:Charles Fox (mathematician)#0	Charles Fox (17 March 1897, in London – 30 April 1977, in Montreal) was the English mathematician who introduced the Fox–Wright function and the Fox H-function. In 1976, he received an honorary doctorate from Concordia University. == References == == External links == O'Connor, John J.; Robertson, Edmund F., "Charles Fox (mathematician)", MacTutor History of Mathematics Archive, University of St Andrews Charles Fox at the Mathematics Genealogy Project Charles Fox at the MacTutor History of Mathematics archive
Wikipedia:Charles Hellaby#0	Charles William Hellaby is a South African mathematician who is an associate professor of applied mathematics at the University of Cape Town, South Africa, working in the field of cosmology. He is a member of the International Astronomical Union and a member of the Baháʼí Faith. == Life == Hellaby was born to Rev. William Allen Meldrum Hellaby and Emily Madeline Hellaby. His twin brother, Mark Edwin Hellaby, pursued a career in literature while his younger brother, Julian Meldrum Hellaby, took to music as a career. He obtained a BSc (Physics & Astronomy) at the University of St Andrews, Scotland in 1977. He completed his MSc (Relativity) at Queen's University, Kingston, Ontario in 1981 and his PhD (Relativity) at Queen's University in 1985. From 1985 to 1988 he was a Post Doctoral Researcher at the University of Cape Town under George Ellis. In 1989 he was appointed a lecturer at the University of Cape Town. Hellaby is a member of the International Astronomical Union (Division J Galaxies and Cosmology), having previously been a member of Division VIII Galaxies & the Universe and subsequently Commission 47 Cosmology. == Research == His research interests include: Inhomogeneous cosmology. Standard cosmology assumes a smooth homogeneous universe, but the real universe is very lumpy Inhomogeneous cosmological models - their evolution, geometry and singularities Non-linear structure formation in the universe Extracting the geometry of the cosmos from observations The Lemaitre–Tolman cosmological model The Szekeres cosmological model Junction conditions in GR Dense black holes Local inhomogeneities and the Swiss cheese model He has also worked on The models of Vaidya, Schwarzschild–Kruskal–Szekeres & Kinnersley Classical signature change Cosmic strings Gravitational collapse Hellaby co-authored Structures in the Universe by Exact Methods: Formation, Evolution, Interactions in which applications of inhomogenous solutions to Albert Einstein's field equations of cosmology are reviewed. The structure of galaxy clusters, galaxies with central black holes and supernovae dimming can be studied with the aid of inhomogenous models. == References == == External links == Publications by Charles Hellaby at ResearchGate
Wikipedia:Charles Loewner#0	Charles Loewner (29 May 1893 – 8 January 1968) was an American mathematician. His name was Karel Löwner in Czech and Karl Löwner in German. == Early life and career == Karl Loewner was born into a Jewish family in Lany, about 30 km from Prague, where his father Sigmund Löwner was a store owner. Loewner received his Ph.D. from the University of Prague in 1917 under supervision of Georg Pick. One of his central mathematical contributions is the proof of the Bieberbach conjecture in the first highly nontrivial case of the third coefficient. The technique he introduced, the Loewner differential equation, has had far-reaching implications in geometric function theory; it was used in the final solution of the Bieberbach conjecture by Louis de Branges in 1985. Loewner worked at the University of Berlin, University of Prague, University of Louisville, Brown University, Syracuse University and eventually at Stanford University. His students include Lipman Bers, Roger Horn, Adriano Garsia, and P. M. Pu. == Loewner's torus inequality == In 1949 Loewner proved his torus inequality, to the effect that every metric on the 2-torus satisfies the optimal inequality sys 2 ≤ 2 3 area ⁡ ( T 2 ) , {\displaystyle \operatorname {sys} ^{2}\leq {\frac {2}{\sqrt {3}}}\operatorname {area} (\mathbb {T} ^{2}),} where sys is its systole. The boundary case of equality is attained if and only if the metric is flat and homothetic to the so-called equilateral torus, i.e. torus whose group of deck transformations is precisely the hexagonal lattice spanned by the cube roots of unity in C {\displaystyle \mathbb {C} } . == Loewner matrix theorem == The Loewner matrix (in linear algebra) is a square matrix or, more specifically, a linear operator (of real C 1 {\displaystyle C^{1}} functions) associated with 2 input parameters consisting of (1) a real continuously differentiable function on a subinterval of the real numbers and (2) an n {\displaystyle n} -dimensional vector with elements chosen from the subinterval; the 2 input parameters are assigned an output parameter consisting of an n × n {\displaystyle n\times n} matrix. Let f {\displaystyle f} be a real-valued function that is continuously differentiable on the open interval ( a , b ) {\displaystyle (a,b)} . For any s , t ∈ ( a , b ) {\displaystyle s,t\in (a,b)} define the divided difference of f {\displaystyle f} at s , t {\displaystyle s,t} as f [ 1 ] ( s , t ) = { f ( s ) − f ( t ) s − t , if s ≠ t f ′ ( s ) , if s = t {\displaystyle f^{[1]}(s,t)={\begin{cases}\displaystyle {\frac {f(s)-f(t)}{s-t}},&{\text{if }}s\neq t\\f'(s),&{\text{if }}s=t\end{cases}}} . Given t 1 , … , t n ∈ ( a , b ) {\displaystyle t_{1},\ldots ,t_{n}\in (a,b)} , the Loewner matrix L f ( t 1 , … , t n ) {\displaystyle L_{f}(t_{1},\ldots ,t_{n})} associated with f {\displaystyle f} for ( t 1 , … , t n ) {\displaystyle (t_{1},\ldots ,t_{n})} is defined as the n × n {\displaystyle n\times n} matrix whose ( i , j ) {\displaystyle (i,j)} -entry is f [ 1 ] ( t i , t j ) {\displaystyle f^{[1]}(t_{i},t_{j})} . In his fundamental 1934 paper, Loewner proved that for each positive integer n {\displaystyle n} , f {\displaystyle f} is n {\displaystyle n} -monotone on ( a , b ) {\displaystyle (a,b)} if and only if L f ( t 1 , … , t n ) {\displaystyle L_{f}(t_{1},\ldots ,t_{n})} is positive semidefinite for any choice of t 1 , … , t n ∈ ( a , b ) {\displaystyle t_{1},\ldots ,t_{n}\in (a,b)} . Most significantly, using this equivalence, he proved that f {\displaystyle f} is n {\displaystyle n} -monotone on ( a , b ) {\displaystyle (a,b)} for all n {\displaystyle n} if and only if f {\displaystyle f} is real analytic with an analytic continuation to the upper half plane that has a positive imaginary part on the upper plane. See Operator monotone function. == Continuous groups == "During [Loewner's] 1955 visit to Berkeley he gave a course on continuous groups, and his lectures were reproduced in the form of duplicated notes. Loewner planned to write a detailed book on continuous groups based on these lecture notes, but the project was still in the formative stage at the time of his death." Harley Flanders and Murray H. Protter "decided to revise and correct the original lecture notes and make them available in permanent form." Charles Loewner: Theory of Continuous Groups (1971) was published by The MIT Press, and re-issued in 2008. In Loewner's terminology, if x ∈ S {\displaystyle x\in S} and a group action is performed on S {\displaystyle S} , then x {\displaystyle x} is called a quantity (page 10). The distinction is made between an abstract group g , {\displaystyle {\mathfrak {g}},} and a realization of g , {\displaystyle {\mathfrak {g}},} in terms of linear transformations that yield a group representation. These linear transformations are Jacobians denoted J ( v u ) {\displaystyle J({\overset {u}{v}})} (page 41). The term invariant density is used for the Haar measure, which Loewner attributes to Adolph Hurwitz (page 46). Loewner proves that compact groups have equal left and right invariant densities (page 48). A reviewer said, "The reader is helped by illuminating examples and comments on relations with analysis and geometry." == See also == Löwner-John ellipsoid Schramm–Loewner evolution Loop-erased random walk Systoles of surfaces == References == Berger, Marcel: À l'ombre de Loewner. (French) Ann. Sci. École Norm. Sup. (4) 5 (1972), 241–260. Loewner, Charles; Nirenberg, Louis: Partial differential equations invariant under conformal or projective transformations. Contributions to analysis (a collection of papers dedicated to Lipman Bers), pp. 245–272. Academic Press, New York, 1974. == External links == Stanford memorial resolution O'Connor, John J.; Robertson, Edmund F., "Charles Loewner", MacTutor History of Mathematics Archive, University of St Andrews
Wikipedia:Charles Macdonald (professor)#0	Charles Macdonald (19 July 1828 – 11 March 1901) was a Scottish-Canadian mathematician and educator. Born in Aberdeen, Scotland, Macdonald studied at King's College, Aberdeen, earning degrees in the arts and divinity. The Church of Scotland named Macdonald the chair in mathematics at Dalhousie College in Halifax, Nova Scotia, which he held until his death in 1901. He was an advocate for education reform in Nova Scotia, and was a significant presence for Dalhousie in Halifax. Dalhousie's first library, Macdonald Memorial Library, was named in his honour by former students who raised money to build it. == Early life and education == MacDonald attended King's College, Aberdeen and studied arts and divinity. In 1850 he was awarded the Hutton Prize, which was reserved for the student who had performed best in the arts curriculum. He earned a Master of Arts, and then studied divinity, earning a licentiate in the Church of Scotland. == Career == After earning his licentiate, MacDonald became an educator. He was teaching at Aberdeen Grammar School when in 1863 he was named by the Church of Scotland to hold the chair of mathematics at Dalhousie College in Halifax, Nova Scotia. During his first lecture at Dalhousie, Macdonald criticized the existing system in Nova Scotia of different colleges run by Christian denominations, and called for a unified university for Nova Scotia. Macdonald's speech sparked a debate in Nova Scotia in the 1870s about education reform, and was cited in the 1920s as Nova Scotia began a new education reform process. Macdonald was considered the most popular member of the faculty at the time. He was the strictest professor with the students, but was known for his wit. He was a frequent public speaker in Halifax, and he increased the presence of Dalhousie in what was fundamentally at the time a garrison town. == Personal life == Macdonald married Maryanne Stairs in 1882 in Halifax. She died in 1883 during childbirth, and Macdonald lived alone with their son, not remarrying. == Death and legacy == Macdonald caught a cold and died of pneumonia on 11 March 1901. He was still teaching five days before his death. Macdonald left Dalhousie $2,000 to buy books for its library. Following his gift, alumni who had been his students raised funds to build a library at the university to honour him, and in 1914, the cornerstone for Macdonald Memorial Library was laid. It would serve as the primary library for Dalhousie until the 1970s. == References ==
Wikipedia:Charles Paul Narcisse Moreau#0	Colonel Charles Paul Narcisse Moreau (14 September 1837, in Paris – 6 July 1916) was a French soldier and mathematician. He served in the artillery and as an officier of the French Legion of Honor. He introduced Moreau's necklace-counting function into mathematics, and achieved the worst result ever recorded in an international chess tournament. == Military service == Colonel Moreau's military career is given by documents on the Legion of Honor website as follows. He was promoted to lieutenant on 1 October 1861. He served in Mexico from 23 May 1863 to 22 March 1867 during the French intervention in Mexico and was named Chevalier de l'Ordre Impérial de la Guadeloupe on 16 September 1866 and was awarded the Commemorative medal of the Mexico Expedition. On 10 August 1868 he was promoted to captain. He served in Africa from 27 January 1869 to 3 August 1870, when he returned to take part in the Franco-Prussian war. He participated in the battle of Sedan on 1 September 1870, after which he was taken prisoner until 4 June 1871. He served again in Africa (Algeria) from 5 August 1871 until 20 November 1873, during which time he was made a chevalier of the French Legion of Honor on 20 November 1872. On 8 July 1886 he was promoted to lieutenant colonel, and on 15 April 1890 was promoted to colonel. He was made an officier of the French Legion of Honor on 5 July 1893. == Mathematics and chess == Spinrad (2008a, 2008b) identified Moreau as the chess player "Colonel Moreau" who set a record for the worst-ever performance in an international tournament by losing all his 26 games in the 1903 Monte Carlo chess tournament. It is unclear why someone that weak was playing in an international tournament. He is sometimes said to have been a last-minute substitute for Mikhail Chigorin, who was apparently dropped after a dispute with the organizer Prince André Dadian, but Spinrad pointed out that this is unlikely because Moreau and Chigorin were both listed among the 14 competitors in a newspaper story in The New York Sun and Salt Lake Herald from 21 December 1902, several weeks before the tournament started on 10 February 1903. Moreau was on the tournament organizing committee for the 1902 Monte Carlo tournament. Spinrad also pointed out that Moreau published several mathematical papers. In particular Moreau (1872) introduced Moreau's necklace-counting function, and Lucas (1891b, pp. 501–503) described a variation of this that he credited to Moreau. Moreau (1873) pointed out a counterexample to a lemma used by Adrien-Marie Legendre in his attempt to prove Dirichlet's theorem on arithmetic progressions. Lucas (1891, pp. 181–195) describes Moreau's analysis of the mathematical game "red and black" invented by Arnous de Rivière. Laisant (1891, p. 106) mentions Moreau's unpublished solution to a combinatorial problem involving rooks on a chessboard. == Publications == Moreau, C. (1872), "Sur les permutations circulaires distinctes (On distinct circular permutations)", Nouvelles annales de mathématiques, journal des candidats aux écoles polytechnique et normale, Sér. 2 (in French), 11: 309–314, JFM 04.0086.01 Moreau, C. (1873), "Correspondence", Nouvelles annales de mathématiques, journal des candidats aux écoles polytechnique et normale, Sér. 2 (in French), 12: 322–324 Moreau, C. (1875), "Propositions sur les nombres", Nouvelles annales de mathématiques, journal des candidats aux écoles polytechnique et normale, Sér. 2 (in French), 14: 274–275 Moreau, C. (1875), "Questions", Nouvelles annales de mathématiques, journal des candidats aux écoles polytechnique et normale, Sér. 2 (in French), 14: 527–528 Moreau, C. (1898), "Sur quelques théorèmes d'arithmétique", Nouvelles annales de mathématiques, journal des candidats aux écoles polytechnique et normale, Sér. 3 (in French), 17: 293–307, JFM 29.0146.02 Moreau, C. (1902), "Solution d'un problème de probabilités", Archiv der Mathematik und Physik, Third Series, 4: 184–189 Moreau also published several notes titled "Solution de la question ...." in volumes XI to XVI of the journal Nouvelles annales de mathématiques giving solutions to questions asked in it. == References == Laisant (1891), "Sur deux problèmes de permutations", Bulletin de la Société Mathématique de France, 19: 105–108 Lucas, Édouard (1891), Récréations mathématiques, Paris: Gauthier-Villars et fils Lucas, Édouard (1891b), Théorie des nombres, Gauthier-Villars Spinrad, Jeremy P. (2008a), The mystery of Colonel Moreau, part 1 (PDF), Chess Cafe Spinrad, Jeremy P. (2008b), The mystery of Colonel Moreau, part 2 (PDF), Chess Cafe == External links == The Legion of honor site has several documents with details about Colonel Moreau's military career, awards, and his dates of birth and death. Chess games of Colonel Moreau "The Colonel Moreau Chess Mystery" by Edward Winter Papers by C.Moreau at JFM
Wikipedia:Charles Pierre Trémaux#0	In graph theory, a Trémaux tree of an undirected graph G {\displaystyle G} is a type of spanning tree, generalizing depth-first search trees. They are defined by the property that every edge of G {\displaystyle G} connects an ancestor–descendant pair in the tree. Trémaux trees are named after Charles Pierre Trémaux, a 19th-century French author who used a form of depth-first search as a strategy for solving mazes. They have also been called normal spanning trees, especially in the context of infinite graphs. All depth-first search trees and all Hamiltonian paths are Trémaux trees. In finite graphs, every Trémaux tree is a depth-first search tree, but although depth-first search itself is inherently sequential, Trémaux trees can be constructed by a randomized parallel algorithm in the complexity class RNC. They can be used to define the tree-depth of a graph, and as part of the left-right planarity test for testing whether a graph is a planar graph. A characterization of Trémaux trees in the monadic second-order logic of graphs allows graph properties involving orientations to be recognized efficiently for graphs of bounded treewidth using Courcelle's theorem. Not every infinite connected graph has a Trémaux tree, and not every infinite Trémaux tree is a depth-first search tree. The graphs that have Trémaux trees can be characterized by forbidden minors. An infinite Trémaux tree must have exactly one infinite path for each end of the graph, and the existence of a Trémaux tree characterizes the graphs whose topological completions, formed by adding a point at infinity for each end, are metric spaces. == Definition and examples == A Trémaux tree, for an undirected graph G {\displaystyle G} , is a spanning tree T {\displaystyle T} with the property that, for every edge u v {\displaystyle uv} in G {\displaystyle G} , one of the two endpoints u {\displaystyle u} and v {\displaystyle v} is an ancestor of the other. To be a spanning tree, it must only use edges of G {\displaystyle G} , and include every vertex, with a unique finite path between every pair of vertices. Additionally, to define the ancestor–descendant relation in this tree, one of its vertices must be designated as its root. If a finite graph has a Hamiltonian path, then rooting that path at one of its two endpoints produces a Trémaux tree. For such a path, every pair of vertices is an ancestor–descendant pair. In the graph shown below, the tree with edges 1–3, 2–3, and 3–4 is a Trémaux tree when it is rooted at vertex 1 or vertex 2: every edge of the graph belongs to the tree except for the edge 1–2, which (for these choices of root) connects an ancestor-descendant pair. However, rooting the same tree at vertex 3 or vertex 4 produces a rooted tree that is not a Trémaux tree, because with this root 1 and 2 are no longer an ancestor and descendant of each other. == In finite graphs == === Existence === Every finite connected undirected graph has at least one Trémaux tree. One can construct such a tree by performing a depth-first search and connecting each vertex (other than the starting vertex of the search) to the earlier vertex from which it was discovered. The tree constructed in this way is known as a depth-first search tree. If u v {\displaystyle uv} is an arbitrary edge in the graph, and u {\displaystyle u} is the earlier of the two vertices to be reached by the search, then v {\displaystyle v} must belong to the subtree descending from u {\displaystyle u} in the depth-first search tree, because the search will necessarily discover v {\displaystyle v} while it is exploring this subtree, either from one of the other vertices in the subtree or, failing that, from u {\displaystyle u} directly. Every finite Trémaux tree can be generated as a depth-first search tree: If T {\displaystyle T} is a Trémaux tree of a finite graph, and a depth-first search explores the children in T {\displaystyle T} of each vertex prior to exploring any other vertices, it will necessarily generate T {\displaystyle T} as its depth-first search tree. === Parallel construction === It is P-complete to find the Trémaux tree that would be found by a sequential depth-first search algorithm, in which the neighbors of each vertex are searched in order by their identities. Nevertheless, it is possible to find a different Trémaux tree by a randomized parallel algorithm, showing that the construction of Trémaux trees belongs to the complexity class RNC. The algorithm is based on another randomized parallel algorithm, for finding minimum-weight perfect matchings in 0-1-weighted graphs. As of 1997, it remained unknown whether Trémaux tree construction could be performed by a deterministic parallel algorithm, in the complexity class NC. If matchings can be found in NC, then so can Trémaux trees. === Logical expression === It is possible to express the property that a set T {\displaystyle T} of edges with a choice of root vertex r {\displaystyle r} forms a Trémaux tree, in the monadic second-order logic of graphs, and more specifically in the form of this logic called MSO2, which allows quantification over both vertex and edge sets. This property can be expressed as the conjunction of the following properties: The graph is connected by the edges in T {\displaystyle T} . This can be expressed logically as the statement that, for every non-empty proper subset of the graph's vertices, there exists an edge in T {\displaystyle T} with exactly one endpoint in the given subset. T {\displaystyle T} is acyclic. This can be expressed logically as the statement that there does not exist a nonempty subset C {\displaystyle C} of T {\displaystyle T} for which each vertex is incident to either zero or two edges of C {\displaystyle C} . Every edge e {\displaystyle e} not in T {\displaystyle T} connects an ancestor-descendant pair of vertices in T {\displaystyle T} . This is true when both endpoints of e {\displaystyle e} belong to a path in T {\displaystyle T} . It can be expressed logically as the statement that, for all edges e {\displaystyle e} , there exists a subset P {\displaystyle P} of T {\displaystyle T} such that exactly two vertices, one of them r {\displaystyle r} , are incident to a single edge of P {\displaystyle P} , and such that both endpoints of e {\displaystyle e} are incident to at least one edge of P {\displaystyle P} . Once a Trémaux tree has been identified in this way, one can describe an orientation of the given graph, also in monadic second-order logic, by specifying the set of edges whose orientation is from the ancestral endpoint to the descendant endpoint. The remaining edges outside this set must be oriented in the other direction. This technique allows graph properties involving orientations to be specified in monadic second order logic, allowing these properties to be tested efficiently on graphs of bounded treewidth using Courcelle's theorem. === Related properties === If a graph has a Hamiltonian path, then that path (rooted at one of its endpoints) is also a Trémaux tree. The undirected graphs for which every Trémaux tree has this form are the cycle graphs, complete graphs, and balanced complete bipartite graphs. Trémaux trees are closely related to the concept of tree-depth. The tree-depth of a graph G {\displaystyle G} can be defined as the smallest number d {\displaystyle d} for which there exist a graph H {\displaystyle H} , with a Trémaux tree T {\displaystyle T} of height d {\displaystyle d} , such that G {\displaystyle G} is a subgraph of H {\displaystyle H} . Bounded tree-depth, in a family of graphs, is equivalent to the existence of a path that cannot occur as a graph minor of the graphs in the family. Many hard computational problems on graphs have algorithms that are fixed-parameter tractable when parameterized by the tree-depth of their inputs. Trémaux trees also play a key role in the Fraysseix–Rosenstiehl planarity criterion for testing whether a given graph is planar. According to this criterion, a graph G {\displaystyle G} is planar if, for a given Trémaux tree T {\displaystyle T} of G {\displaystyle G} , the remaining edges can be placed in a consistent way to the left or the right of the tree, subject to constraints that prevent edges with the same placement from crossing each other. == In infinite graphs == === Existence === Not every infinite graph has a normal spanning tree. For instance, a complete graph on an uncountable set of vertices does not have one: a normal spanning tree in a complete graph can only be a path, but a path has only a countable number of vertices. However, every connected graph on a countable set of vertices does have a normal spanning tree. Even in countable graphs, a depth-first search might not succeed in eventually exploring the entire graph, and not every normal spanning tree can be generated by a depth-first search: to be a depth-first search tree, a countable normal spanning tree must have only one infinite path or one node with infinitely many children (and not both). === Minors === If an infinite graph G {\displaystyle G} has a normal spanning tree, so does every connected graph minor of G {\displaystyle G} . It follows from this that the graphs that have normal spanning trees have a characterization by forbidden minors. One of the two classes of forbidden minors consists of bipartite graphs in which one side of the bipartition is countable, the other side is uncountable, and every vertex has infinite degree. The other class of forbidden minors consists of certain graphs derived from Aronszajn trees. The details of this characterization depend on the choice of set-theoretic axiomatization used to formalize mathematics. In particular, in models of set theory for which Martin's axiom is true and the continuum hypothesis is false, the class of bipartite graphs in this characterization can be replaced by a single forbidden minor. However, for models in which the continuum hypothesis is true, this class contains graphs which are incomparable with each other in the minor ordering. === Ends and metrizability === Normal spanning trees are also closely related to the ends of an infinite graph, equivalence classes of infinite paths that, intuitively, go to infinity in the same direction. If a graph has a normal spanning tree, this tree must have exactly one infinite path for each of the graph's ends. An infinite graph can be used to form a topological space by viewing the graph itself as a simplicial complex and adding a point at infinity for each end of the graph. With this topology, a graph has a normal spanning tree if and only if its set of vertices can be decomposed into a countable union of closed sets. Additionally, this topological space can be represented by a metric space if and only if the graph has a normal spanning tree. == References ==
Wikipedia:Charles Read (mathematician)#0	Charles John Read (16 February 1958 – 14 August 2015) was a British mathematician known for his work in functional analysis. In operator theory, he is best known for his work in the 1980s on the invariant subspace problem, where he constructed operators with only trivial invariant subspaces on particular Banach spaces, especially on ℓ 1 {\displaystyle \ell _{1}} . He won the 1985 Junior Berwick Prize for his work on the invariant subspace problem. Read has also published on Banach algebras and hypercyclicity; in particular, he constructed the first example of an amenable, commutative, radical Banach algebra. == Education and career == Read won a scholarship to study mathematics at Trinity College, Cambridge in October 1975, and was awarded a first-class degree in Mathematics in 1978. He completed his PhD thesis entitled Some Problems in the Geometry of Banach Spaces at the University of Cambridge under the supervision of Béla Bollobás. He spent the year 1981–82 at Louisiana State University. From 2000 until his death, he was a Professor of Pure Mathematics at the University of Leeds after having been a fellow of Trinity College for several years. == Personal life == === Christianity === On his personal website, formerly hosted on a server at the University of Leeds, Read described himself first and foremost as a Born-Again Christian. Some biographical details could be found in what he described as his "Christian Testimony" on that site, where he described his conversion process. He described losing his father to cancer in 1970 when he was 11 years old, and that this loss prompted him to ask questions about whether, and in what form, we might continue to live after we die - and that consciousness may be independent of the body. He came to the conclusion that the conscious mind must survive after death. This also led him to believe that since we are "immortal beings" that we must always try to "do the right thing". Some time later the article described an incident where he had pushed a smaller boy out of the way in a queue at a sweet shop. He later interpreted his later sense of remorse at having done something wrong as the "Classical Christian conviction of sin", and claimed to have had a religious experience on a London Underground train where he felt a sense of joy at being forgiven, and simultaneously bursting into tears. Read also claimed to have taken part in a miracle of Christian healing at a Christian meeting run by John Wimber, organiser of the Vineyard Movement. ==== Controversy over Read's Christian Testimony ==== An article in The Gryphon, the Leeds University Student Union newspaper, in February 2015 stated that Read had "sparked controversy" by stating at the end of his testimony that "I strongly urge you to seek the truth as a researcher, not trusting anyone else to do your basic investigations for you. That’s right, Jesus is the Way. But you have to find that out for yourself. For those who seek find, but those who can’t be bothered, or who think they’re too cool, end in a very dark place. It won't be cool in Hell." The article was prompted by a third-year Maths student who had expressed the opinion that "I don’t think his university webpage should be showing his personal opinions about faith and religion as it doesn’t have anything to do with someone’s ability to learn maths". Read subsequently displayed on his website a scanned image of the original article, under which a handwritten comment notes that "Leeds Gryphon, 13-2-15 notices Christian Testimony of CJR after several years!" === Cave diving === Read was also a devotee of solo cave diving and wrote extensively about it on his website. == Death == Read died in Winnipeg in August 2015 while on a research visit at the University of Manitoba. == References == == External links == Charles Read's Homepage Charles John Read at the Mathematics Genealogy Project
Wikipedia:Charles-Michel Marle#0	Charles-Michel Marle (born 26 November 1934 in Guelma, Algeria) is a French engineer and mathematician, currently a Professor Emeritus at Pierre and Marie Curie University. == Biography == Charles-Michel Marle completed in 1951 his primary and secondary education in Constantine, Algeria. He was a pupil of the preparatory classes for the grandes écoles at the Lycée Bugeaud in Algiers: higher mathematics in 1951-1952, then special mathematics in 1952-1953. He was admitted to the École Polytechnique in 1953. When he left this school in 1955, he opted for the Corps des mines. He did his military service as a sub-lieutenant at the Engineering School in Angers from October 1955 to February 1956, then in Algeria during the war until 30 December 1956. In 1957 he began attending the École Nationale Supérieure des Mines in Paris and from October 1957 to September 1958 he attended the École nationale supérieure du pétrole et des moteurs and completed various internships in the oil industry in France and Algeria. Returning to the École des mines in October 1958, his last year of study was interrupted in 1959 by the decision of the Minister of Industry, to send all junior civil servants of category A to Algeria to participate in the Constantine Plan. He was then attached to the short-lived Common organisation of the Saharan regions and worked in Algiers, the Sahara and Paris on various industrial projects. Between 1959 and 1969 he was seconded by the Corps des Mines to the French Institute of Petroleum (IFP), where he was research engineer, head of department and director of division. While working at this Institute he obtained a degree in mathematics and in 1968 he defended a doctoral thesis under the supervision of André Lichnerowicz. In 1969 he changed his career and entered higher education, becoming a lecturer at the University of Besançon. In 1975 he moved to Pierre and Marie Curie University in Paris, and was appointed professor at this university in 1977. In 1983 he was elected corresponding member of the French Academy of sciences. In 1989, he was appointed visiting professor at the University of California at Berkeley, for a semester of scientific courses, collaborating in particular with professors Jerrold E. Marsden and Alan Weinstein. He retired in September 2000 and since then he is Professor Emeritus. Charles-Michel Marle is the great-great-grandson of the grammarian L. C. Marle (1799-1860), author of an attempt at spelling reform around 1840. == Scientific work == While working at the French Institute of Petroleum, Marle's research focused on fluid flows in porous media, which are being investigated for applications in hydrocarbon field development. He also published a book on the subject, developing a course that he taught at the École nationale du pétrole et des moteurs. Transitioning from applied to pure mathematics, in his PhD thesis he worked on fluid dynamics and the Boltzmann relativistic equation. Since the early 1970s he has worked mainly in the field of differential geometry, notably on Hamiltonian group actions and Poisson geometry, and its applications to mechanics. With his colleague Paulette Libermann (1919-2007) he published in 1987 a research-level book on symplectic geometry and geometric mechanics. He has recently published another book, taking up part of the previous one, exposing recent results obtained in this field since 1987. == Honours and awards == French Academy of sciences laboratory prize for his thesis work, 1973. Member of the Mathematical Society of France, the French Physical Society and the American Mathematical Society. == References ==
Wikipedia:Charles-René de Fourcroy#0	Charles-René de Fourcroy de Ramecourt (1715–1791) was a French officer of the Royal Engineers Corps. He is known for having published the first synthetic map of urban geography in his Essai d'une table poléométrique (1782). == Biography == Grandson of Nicolas de Fourcroy, King's Councillor at the Bailiwick and Royal Provost of the town of Clermont, and son of Charles de Fourcroy, lawyer, Charles-René de Fourcroy was born in Paris on 19 January 1715. He entered the Corps du Génie in 1735 or 1737, became a member in 1740 and then captain in 1744. He became director of fortifications for the Minister of War in December 1774, then maréchal de camp on 1 March 1780. He married Marie Marguerite Lemaistre (1732-1772) on 6 November 1755 in Andonville and they had two daughters: Charlotte Marie Louise Cornélie (1758-?) and Charlotte Marie Françoise (1762-1765). He was a member of the Royal Academy of Sciences, appointed correspondent of Abbé Nollet on 25 November 1767, then of Pingré on 20 June 1770. He was created Grand Cross of the Order of Saint-Louis in 1781. A bust portrait was made in 1781 by René Descarsin (now the property of the Conservatoire du portrait du dix-huitième siècle, CPDHS). In 1835, an engraving was made by L. Lorin and lithographed by Langlumé. == Work == Fourcroy was the author of Essai d'une table poléométrique, a treatise on engineering and civil construction, published in 1782, which is remarkable for its period in its use of graphs to list the achievements of civil engineers of bridges and roads from 1740 to 1780 and its cross-sectional and mathematical analysis of the growth of urban areas. === Essai d’une table poléométrique, 1782 === In 1782 Fourcroy published his Essai d'une table poléométrique, ou amusement d'un amateur de plans sur les grandeurs de quelques villes; Avec une Carte, ou Tableau qui offre la comparaison de ces Villes par une même échelle. This book gives an analysis of the urban growth of European cities, which are graphically compared in a diagram, called Table poléométrique or Poleometric Table. In his work Fourcroy (1782) explained: "If we have the surfaces (i.e., areas) of all the cities/villages in the table, or the proportions representing these cities, transformed into cross-sections, each of equal extension, and each on the same scale; then if we successively put one on the other, from the largest to the smallest, and joined all by one of their angles; these squares would overlap relative to their size, and the whole would form a kind of table that visually represents an idea of the actual proportion that can be found between the surfaces of these different cities. We could as well find in this table two cities of equal size, cut their squares diagonally, and have the table represent only half of each; which essentially means the same. Such is the attached figure, which requires no further explanation." A 1782 review of this work by Élie Catherine Fréron noticed that the Table poléométrique is plausible for 230 cities, foreign and domestic. More recently Jacques Bertin in his Semiology of Graphics (1983) further explained the work, which was first published anonymously. In the French National library the work is still listed by its publisher. Bertin wrote "The 'Poleometric Table' (see figure), published in 1782 by Dupain-Triel, is one of the oldest proportional representations of human phenomena which is currently known. François de Dainville has demonstrated... that the author was Charles de Fourcroy, a Director of Fortification. [in the diagram] each city is represented by a square whose area is proportional to the geographic area occupied by the city (and for the smallest cities, by a half square only, divided by the diagonal line. When superimposed, the squares are classed automatically. This results in visual groupings which lead the author to propose an 'urban classification.' This example allows us to appreciate the evolution of graphic representation and the efficiency of more recent solutions, based on the standard construction." Following Fourcroy's map, a not so far comparison diagram was published in 1785 by the German economist and statistician August Friedrich Wilhelm Crome, was entitled "Groessen Karte von Europa." This map made a comparison of European states, instead of a comparison of cities in the Poleometric Table. Palsky (1996) concluded, that "the Table established by Fourcroy signals a fundamental moment in the evolution of the graphical method. We see the passage to the abstract, to fictitious features. By these proportional triangles, the author constructs an image that does not return/relate to its original existence." The attribution of authorship of the manuscript was carried out in 1958 by François de Dainville with the help of internal validations (choice of towns, often places of battle; access to the Galerie des plans en relief du Roi) or external validations (an annotated copy of the work preserved in the Bibliothèque de l'Inspection du Génie bears the handwritten inscription at the bottom of the frontispiece: "made by Mr de Fourcroy, chief of the Génie, and given by him"). == Selected publications == Fourcroy, Charles de. Essai d’une table poléométrique, ou amusement d’un amateur de plans sur la grandeur de quelques villes. Dupain-Triel, Paris (1782) == References ==
Wikipedia:Charlotte Wedell#0	Charlotte Bolette Sophie, Baroness Wedell-Wedellsborg (27 January 1862 – 22 July 1953) was one of four women mathematicians to attend the inaugural International Congress of Mathematicians, held in Zurich in 1897. Wedell was originally from Denmark, the daughter of Vilhelm Ferdinand, Baron Wedell-Wedellsborg (of the Wedel noble family) and Louise Marie Sophie, Countess Schulin, and the granddaughter of Johan Sigismund Schulin (1808–1880). At the time of the Congress, in 1897, she had just completed a doctorate at the University of Lausanne in Switzerland, with Adolf Hurwitz as an unofficial mentor. The subject of her dissertation was the application of elliptic functions to the construction of the Malfatti circles. At the congress, Wedell was listed as being affiliated with the University of Göttingen. The other three women at the congress were Iginia Massarini, Vera von Schiff, and Charlotte Scott. None were speakers; the first Congress with a woman as a speaker was in 1912. Wedell married engineer Eugène Tomasini in Copenhagen in 1898; they divorced in 1909. == References ==
Wikipedia:Chartered Mathematician#0	Chartered Mathematician (CMath) is a professional qualification in Mathematics awarded to professional practising mathematicians by the Institute of Mathematics and its Applications (IMA) in the United Kingdom. Chartered Mathematician is the IMA's highest professional qualification; achieving it is done through a rigorous peer-reviewed process. The required standard for Chartered Mathematician registration is typically an accredited UK MMath degree, at least five years of peer-reviewed professional practise of advanced Mathematics, attainment of a senior-level of technical standing, and an ongoing commitment to Continuing Professional Development. A Chartered Mathematician is entitled to use the post-nominal letters CMath, in accordance with the Royal Charter granted to the IMA by the Privy Council. The profession of Chartered Mathematician is a 'regulated profession' under the European professional qualification directives. == See also == Institute of Mathematics and its Applications == References == == External links == Institute of Mathematics and its Applications website
Wikipedia:Chebyshev–Markov–Stieltjes inequalities#0	Andrey Andreyevich Markov (14 June 1856 – 20 July 1922) was a Russian mathematician best known for his work on stochastic processes. A primary subject of his research later became known as the Markov chain. He was also a strong, close to master-level, chess player. Markov and his younger brother Vladimir Andreyevich Markov (1871–1897) proved the Markov brothers' inequality. His son, another Andrey Andreyevich Markov (1903–1979), was also a notable mathematician, making contributions to constructive mathematics and recursive function theory. == Biography == Andrey Markov was born on 14 June 1856 in Russia. He attended the St. Petersburg Grammar School, where some teachers saw him as a rebellious student. In his academics he performed poorly in most subjects other than mathematics. Later in life he attended Saint Petersburg Imperial University (now Saint Petersburg State University). Among his teachers were Yulian Sokhotski (differential calculus, higher algebra), Konstantin Posse (analytic geometry), Yegor Zolotarev (integral calculus), Pafnuty Chebyshev (number theory and probability theory), Aleksandr Korkin (ordinary and partial differential equations), Mikhail Okatov (mechanism theory), Osip Somov (mechanics), and Nikolai Budajev (descriptive and higher geometry). He completed his studies at the university and was later asked if he would like to stay and have a career as a mathematician. He later taught at high schools and continued his own mathematical studies. In this time he found a practical use for his mathematical skills. He figured out that he could use chains to model the alliteration of vowels and consonants in Russian literature. He also contributed to many other mathematical aspects in his time. He died at age 66 on 20 July 1922. == Timeline == In 1877, Markov was awarded a gold medal for his outstanding solution of the problem About Integration of Differential Equations by Continued Fractions with an Application to the Equation ( 1 + x 2 ) d y d x = n ( 1 + y 2 ) {\displaystyle (1+x^{2}){\frac {dy}{dx}}=n(1+y^{2})} . During the following year, he passed the candidate's examinations, and he remained at the university to prepare for a lecturer's position. In April 1880, Markov defended his master's thesis "On the Binary Square Forms with Positive Determinant", which was directed by Aleksandr Korkin and Yegor Zolotarev. Four years later in 1884, he defended his doctoral thesis titled "On Certain Applications of the Algebraic Continuous Fractions". His pedagogical work began after the defense of his master's thesis in autumn 1880. As a privatdozent he lectured on differential and integral calculus. Later he lectured alternately on "introduction to analysis", probability theory (succeeding Chebyshev, who had left the university in 1882) and the calculus of differences. From 1895 through 1905 he also lectured in differential calculus. One year after the defense of his doctoral thesis, Markov was appointed extraordinary professor (1886) and in the same year he was elected adjunct to the Academy of Sciences. In 1890, after the death of Viktor Bunyakovsky, Markov became an extraordinary member of the academy. His promotion to an ordinary professor of St. Petersburg University followed in the fall of 1894. In 1896, Markov was elected an ordinary member of the academy as the successor of Chebyshev. In 1905, he was appointed merited professor and was granted the right to retire, which he did immediately. Until 1910, however, he continued to lecture in the calculus of differences. In connection with student riots in 1908, professors and lecturers of St. Petersburg University were ordered to monitor their students. Markov refused to accept this decree, and he wrote an explanation in which he declined to be an "agent of the governance". Markov was removed from further teaching duties at St. Petersburg University, and hence he decided to retire from the university. Markov was an atheist. In 1912, he responded to Leo Tolstoy's excommunication from the Russian Orthodox Church by requesting his own excommunication. The Church complied with his request. In 1913, the council of St. Petersburg elected nine scientists honorary members of the university. Markov was among them, but his election was not affirmed by the minister of education. The affirmation only occurred four years later, after the February Revolution in 1917. Markov then resumed his teaching activities and lectured on probability theory and the calculus of differences until his death in 1922. == See also == == Notes == == References == == Further reading == Karl-Georg Steffens (28 July 2007). The History of Approximation Theory: From Euler to Bernstein. Springer Science & Business Media. pp. 98–105. ISBN 978-0-8176-4475-8. А. А. Марков. "Распространение закона больших чисел на величины, зависящие друг от друга". "Известия Физико-математического общества при Казанском университете", 2-я серия, том 15, с. 135–156, 1906. A. A. Markov. "Extension of the limit theorems of probability theory to a sum of variables connected in a chain". reprinted in Appendix B of: R. Howard. Dynamic Probabilistic Systems, volume 1: Markov Chains. John Wiley and Sons, 1971. Pavlyk, Oleksandr (4 February 2013). "Centennial of Markov Chains". Wolfram Blog. == External links == Andrei Andreyevich Markov at the Mathematics Genealogy Project
Wikipedia:Chehrzad Shakiban#0	Chehrzad "Cheri" Shakiban (born 1951) is an Iranian and American mathematician, the first Iranian woman to receive a Ph.D. in mathematics and the first Iranian woman to become a full professor of mathematics. She is retired after working for 37 years as a professor of mathematics at the University of St. Thomas (Minnesota), where she was the first female full professor; she is also a former director of the Institute for Mathematics and its Applications. She is the author of a textbook on applied linear algebra, and has published highly cited work on the use of differential invariants in image recognition. == Early life and education == Shakiban was born in 1951 in Tehran, to a family of the Baháʼí Faith. As a high school student, she came to the US for her final year of high school study through the AFS Intercultural Programs, at a high school in St. Louis, Missouri. After receiving a US high school diploma and returning to Iran, she took a job at Pakistan's Embassy to Iran and studied at night, eventually earning an Iranian high school diploma by examination in 1970 rather than returning to school. She became an undergraduate mathematics student at the National University of Iran, mentored in number theory there by Ahmad Mirbagheri and completing her degree program in three years. In the last year of her studies, a conference brought Paul Erdős, Paul Halmos, and Garrett Birkhoff to Tehran, and she served as their guide and translator. At the invitation of Birkhoff, and with the support of the Iranian government, she went to Harvard University in the US as a special student from 1973 to 1975, receiving a master's degree with a thesis in the calculus of variations. She continued her studies at Brown University from 1975 to 1979, working there with Wendell Fleming. During this time, in 1976, she married mathematician Peter J. Olver, a student of Birkhoff. She followed her husband to the University of Oxford in England in 1978, continuing her studies at Brown remotely. She became a refugee from Iran after the Iranian revolution in 1978 and 1979, in which her brother was killed. Pregnant with her first child, she successfully defended her dissertation, The Euler Operator in the Formal Calculus of Variations, in 1979, becoming the first Iranian woman to complete a Ph.D. in mathematics. == Career and later life == After receiving her doctorate, Shakiban became a tutor in Somerville College, Oxford in 1979. In 1980, her husband took a position at the University of Minnesota and she moved with him, taking a teaching position at St. Catherine University. She moved to the University of St. Thomas (Minnesota) in 1983. In 1996, she was promoted to full professor, the first woman to become a full professor at the University of St. Thomas, and the first Iranian woman to become a full professor of mathematics. She chaired the Department of Mathematics at the University of St. Thomas from 1996 to 2004, and directed the Institute for Mathematics and its Applications from 2006 to 2008, later becoming its associate director for diversity. She retired in 2020, but continues as a senior fellow in the university's Center for Common Good. As an expatriate Iranian Baháʼí, she has taught online courses aimed at Iranian members of the Baháʼí Faith, who have been blocked from accessing higher education in Iran, and has protested the treatment of Baháʼí in Iran. == Recognition == In 2024, Heriot Watt University in Scotland gave Shakiban an honorary doctorate, recognizing her as "a renowned international figure in higher education, teaching and inspiring generations of mathematicians for over four decades". == Selected publications == Calabi, Eugenio; Olver, Peter J.; Shakiban, Chehrzad; Tannenbaum, Allen R.; Haker, Steven (1998), "Differential and numerically invariant signature curves applied to object recognition", International Journal of Computer Vision, 26 (2): 107–135, doi:10.1023/A:1007992709392 Olver, Peter J.; Shakiban, Chehrzad (2006), Applied Linear Algebra, Upper Saddle River, New Jersey: Pearson Prentice Hall, ISBN 0-13-147382-4, MR 2127863; 2nd ed., Springer, 2018 == References == == External links == Home page
Wikipedia:Chen Mufa#0	Chen Mufa is a Chinese professor of mathematics at Beijing Normal University. He is a member of the Chinese Academy of Sciences and the World Academy of Sciences. In addition to his work on probability theory, as a mathematician, Chen contributed to mathematical physics. Chen is a faculty member at Beijing Normal University and a member of the advisory committee for the Beijing International Center for Mathematics. He is Jiangsu Province's first academician of mathematical science. == Education and career == Chen obtained a doctorate degree in 1983 from Beijing Normal University. He joined Beijing Normal University in 1980 and became a full professor in 1985. In 2003, he was elected a member of the Chinese Academy of Sciences, and in 2009, he was elected a member of the World Academy of Sciences. He is a fellow of the American Mathematical Society. == References ==
Wikipedia:Chen's theorem#0	In number theory, Chen's theorem states that every sufficiently large even number can be written as the sum of either two primes, or a prime and a semiprime (the product of two primes). It is a weakened form of Goldbach's conjecture, which states that every even number is the sum of two primes. == History == The theorem was first stated by Chinese mathematician Chen Jingrun in 1966, with further details of the proof in 1973. His original proof was much simplified by P. M. Ross in 1975. Chen's theorem is a significant step towards Goldbach's conjecture, and a celebrated application of sieve methods. Chen's theorem represents the strengthening of a previous result due to Alfréd Rényi, who in 1947 had shown there exists a finite K such that any even number can be written as the sum of a prime number and the product of at most K primes. == Variations == Chen's 1973 paper stated two results with nearly identical proofs.: 158 His Theorem I, on the Goldbach conjecture, was stated above. His Theorem II is a result on the twin prime conjecture. It states that if h is a positive even integer, there are infinitely many primes p such that p + h is either prime or the product of two primes. Ying Chun Cai proved the following in 2002: In 2025, Daniel R. Johnston, Matteo Bordignon, and Valeriia Starichkova provided an explicit version of Chen's theorem: which refined upon an earlier result by Tomohiro Yamada. Also in 2024, Bordignon and Starichkova showed that the bound can be lowered to e e 14 ≈ 2.5 ⋅ 10 522284 {\displaystyle e^{e^{14}}\approx 2.5\cdot 10^{522284}} assuming the Generalized Riemann hypothesis (GRH) for Dirichlet L-functions. In 2019, Huixi Li gave a version of Chen's theorem for odd numbers. In particular, Li proved that every sufficiently large odd integer N {\displaystyle N} can be represented as N = p + 2 a , {\displaystyle N=p+2a,} where p {\displaystyle p} is prime and a {\displaystyle a} has at most 2 prime factors. Here, the factor of 2 is necessitated since every prime (except for 2) is odd, causing N − p {\displaystyle N-p} to be even. Li's result can be viewed as an approximation to Lemoine's conjecture. == References == === Citations === === Books === Nathanson, Melvyn B. (1996). Additive Number Theory: the Classical Bases. Graduate Texts in Mathematics. Vol. 164. Springer-Verlag. ISBN 0-387-94656-X. Chapter 10. Wang, Yuan (1984). Goldbach conjecture. World Scientific. ISBN 9971-966-09-3. == External links == Jean-Claude Evard, Almost twin primes and Chen's theorem Weisstein, Eric W. "Chen's Theorem". MathWorld.
Wikipedia:Chen-Bo Zhu#0	Chen-Bo Zhu, also known as Zhu Chengbo (Chinese: 朱程波), is a Chinese-born Singaporean mathematician working in representation theory of Lie groups. He was Head of the Department of Mathematics at the National University of Singapore from 2014 to 2020. Zhu served as President of the Singapore Mathematical Society from 2009 to 2012 and Vice President of the Southeast Asian Mathematical Society from 2012 to 2013. == Biography == Zhu was born in September 1964 in Yin County (鄞县, current Yinzhou District), Ningbo, Zhejiang Province in China, Zhu attended Yinzhou High School from 1978 to 1980 and studied mathematics as an undergraduate at Zhejiang University from 1980 to 1984. In 1984, Zhu was among the 15 students nationwide selected by the Government of China and by a joint AMS-SIAM committee for PhD study in the U.S. (also known as the Shiing-Shen Chern Program). Subsequently, he went to Yale University in 1985 and obtained his PhD in 1990, under the direction of Roger Howe. Zhu joined the Department of Mathematics at NUS in 1991, and became a Singapore citizen in 1995. == Contributions == In representation theory, Zhu’s work is focused on classical groups and their smooth representations. Jointly with Sun Binyong, he proved multiplicity at most one for the branching (also called strong Gelfand pair property) of irreducible Casselman-Wallach representations of classical groups in the Archimedean case, and the conservation relation conjecture of Stephen S. Kudla and Stephen Rallis. He has also applied Howe correspondence to the structural study of degenerate representations and to the understanding of singularities for infinite-dimensional representations. == Selected works == Sun, Binyong; Zhu, Chen-Bo (2012), "Multiplicity one theorems: the Archimedean case", Annals of Mathematics, 175 (1): 23–44, arXiv:0903.1413, doi:10.4007/annals.2012.175.1.2 Sun, Binyong; Zhu, Chen-Bo (2015), "Conservation relations for local theta correspondence", Journal of the American Mathematical Society, 28 (4): 939–983, arXiv:1204.2969, doi:10.1090/s0894-0347-2014-00817-1, S2CID 5936119 Lee, Soo Teck; Zhu, Chen-Bo (1998), "Degenerate principal series and local theta correspondence", Transactions of the American Mathematical Society, 350 (12): 5017–5046, doi:10.1090/s0002-9947-98-02036-4 Gomez, Raul; Zhu, Chen-Bo (2014), "Local theta lifting of generalized Whittaker models associated to nilpotent orbits", Geometric and Functional Analysis, 24 (3): 796–853, arXiv:1302.3744, doi:10.1007/s00039-014-0276-5, S2CID 119596509 == Awards and honors == Zhu has been a Fellow of the Singapore National Academy of Science since 2014. == References == == External links == Homepage of Chen-Bo Zhu The Mathematics Genealogy Project – Chen-Bo Zhu
Wikipedia:Cheng's eigenvalue comparison theorem#0	In Riemannian geometry, Cheng's eigenvalue comparison theorem states in general terms that when a domain is large, the first Dirichlet eigenvalue of its Laplace–Beltrami operator is small. This general characterization is not precise, in part because the notion of "size" of the domain must also account for its curvature. The theorem is due to Cheng (1975b) by Shiu-Yuen Cheng. Using geodesic balls, it can be generalized to certain tubular domains (Lee 1990). == Theorem == Let M be a Riemannian manifold with dimension n, and let BM(p, r) be a geodesic ball centered at p with radius r less than the injectivity radius of p ∈ M. For each real number k, let N(k) denote the simply connected space form of dimension n and constant sectional curvature k. Cheng's eigenvalue comparison theorem compares the first eigenvalue λ1(BM(p, r)) of the Dirichlet problem in BM(p, r) with the first eigenvalue in BN(k)(r) for suitable values of k. There are two parts to the theorem: Suppose that KM, the sectional curvature of M, satisfies K M ≤ k . {\displaystyle K_{M}\leq k.} Then λ 1 ( B N ( k ) ( r ) ) ≤ λ 1 ( B M ( p , r ) ) . {\displaystyle \lambda _{1}\left(B_{N(k)}(r)\right)\leq \lambda _{1}\left(B_{M}(p,r)\right).} The second part is a comparison theorem for the Ricci curvature of M: Suppose that the Ricci curvature of M satisfies, for every vector field X, Ric ⁡ ( X , X ) ≥ k ( n − 1 ) \| X \| 2 . {\displaystyle \operatorname {Ric} (X,X)\geq k(n-1)\|X\|^{2}.} Then, with the same notation as above, λ 1 ( B N ( k ) ( r ) ) ≥ λ 1 ( B M ( p , r ) ) . {\displaystyle \lambda _{1}\left(B_{N(k)}(r)\right)\geq \lambda _{1}\left(B_{M}(p,r)\right).} S.Y. Cheng used Barta's theorem to derive the eigenvalue comparison theorem. As a special case, if k = −1 and inj(p) = ∞, Cheng’s inequality becomes λ(N) ≥ λ(H n(−1)) which is McKean’s inequality. == See also == Comparison theorem Eigenvalue comparison theorem == References == === Citations === === Bibliography === Bessa, G. P.; Montenegro, J. F. (2008), "On Cheng's eigenvalue comparison theorem", Mathematical Proceedings of the Cambridge Philosophical Society, 144 (3): 673–682, arXiv:math/0507318, doi:10.1017/s0305004107000965, ISSN 0305-0041. Chavel, Isaac (1984), Eigenvalues in Riemannian geometry, Pure and Applied Mathematics, vol. 115, Academic Press. Cheng, Shiu Yuen (1975a), "Eigenfunctions and eigenvalues of Laplacian", Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII, Stanford Univ., Stanford, Calif., 1973), Part 2, Providence, R.I.: American Mathematical Society, pp. 185–193, MR 0378003 Cheng, Shiu Yuen (1975b), "Eigenvalue Comparison Theorems and its Geometric Applications", Math. Z., 143: 289–297, doi:10.1007/BF01214381. Lee, Jeffrey M. (1990), "Eigenvalue Comparison for Tubular Domains", Proceedings of the American Mathematical Society, 109 (3), American Mathematical Society: 843–848, doi:10.2307/2048228, JSTOR 2048228. McKean, Henry (1970), "An upper bound for the spectrum of △ on a manifold of negative curvature", Journal of Differential Geometry, 4: 359–366. Lee, Jeffrey M.; Richardson, Ken (1998), "Riemannian foliations and eigenvalue comparison", Ann. Global Anal. Geom., 16: 497–525, doi:10.1023/A:1006573301591/
Wikipedia:Chennai Mathematical Institute#0	Chennai Mathematical Institute (CMI) is a higher education and research institute in Chennai, India. It was founded in 1989 by the SPIC Science Foundation, and offers undergraduate and postgraduate programmes in physics, mathematics and computer science. CMI is noted for its research in algebraic geometry, in particular in the area of moduli of bundles. CMI was at first located in T. Nagar in the heart of Chennai in an office complex. It moved to a new 5-acre (20,000 m2) campus in Siruseri in October 2005. In December 2006, CMI was recognized as a university under Section 3 of the University Grants Commission (UGC) Act 1956, making it a deemed university. Until then, the teaching program was offered in association with Bhoj Open University, as it offered more flexibility. == History == CMI began as the School of Mathematics, SPIC Science Foundation, in 1989. The SPIC Science Foundation was set up in 1986 by Southern Petrochemical Industries Corporation (SPIC) Ltd., one of the major industrial houses in India, to foster the growth of science and technology in the country. In 1996, the School of Mathematics became an independent institution and changed its name to SPIC Mathematical Institute. In 1998, in order to better reflect the emerging role of the institute, it was renamed the Chennai Mathematical Institute (CMI). From its inception, the institute has had a Ph.D. programme in Mathematics and Computer Science. In the initial years, the Ph.D. programme was affiliated to the BITS, Pilani and the University of Madras. In December 2006, CMI was recognized as a university under Section 3 of the UGC Act 1956. In 1998, CMI took the initiative to bridge the gap between teaching and research by starting B.Sc.(Hons.) and M.Sc. programmes in Mathematics and allied subjects. In 2001, the B.Sc. programme was extended to incorporate two courses with research components, leading to an M.Sc. degree in mathematics and an M.Sc. degree in Computer Science. In 2003, a new undergraduate course was added, leading to a B.Sc. degree in physics. In 2010, CMI launched a summer fellowship programme whereby they invited about 30 students from all over India to work under the faculty at CMI on various research projects. Later, in 2012, the B.Sc. degree in Physics was restructured as an integrated B.Sc. degree in Mathematics and Physics. == Campus == CMI moved into its new campus on 5 acres (20,000 m2) of land at the SIPCOT Information Technology Park in Siruseri in October, 2005. The campus is located along the Old Mahabalipuram Road, which is developing as the IT corridor to the south of the city. The library block and the student's hostel were completed in late 2006 and become operational from January 2007. In 2006, CMI implemented a grey water recycling system on its campus. The system was designed for CMI by Sultan Ahmed Ismail to treat waste water produced after cleaning, washing and bathing to be used for gardening or ground water recharge. Construction is underway for a new building that will house an auditorium, accommodation for guests, as well as additional academic space - faculty offices, library and lecture halls. This construction is funded by a grant from the Ministry of Human Resource Development through the University Grants Commission. == Organisation and administration == === Director === The founding director of CMI was C. S. Seshadri, who was known for his work in the area of algebraic geometry, especially moduli problems and algebraic groups. He stepped down in 2010, after which Rajeeva L. Karandikar was appointed as director. Prof. Seshadri continued as Emeritus Director until his demise on 17 July 2020. Professor Madhavan Mukund was appointed as the new director in 2021. === Funding === CMI's funding comes from both private and government sources. ==== Government funding ==== DAE: CMI receives support for its teaching programme from the Department of Atomic Energy, through the National Board of Higher Mathematics. ISRO: The Indian Space Research Organization also funds CMI substantially. DST and DRDO: For some of its projects, CMI receives funding from the Department of Science and Technology as well as the Defence Research and Development Organization. ==== Private funding ==== The Southern Petrochemical Industries Corporation (SPIC) was an important founder of CMI during its initial years. In fact, CMI started as the Spic Mathematical Institute. Sriram Group of Companies is an important funder and also arranges for other funding for CMI. Matrix Laboratories has made a major contribution to the new campus at the SIPCOT IT Park. Infosys Foundation recently donated a large corpus to CMI to enhance faculty compensation and fellowships for students. == Academics == === Academic programmes === CMI has Ph.D. programmes in Computer Sciences, Mathematics and Physics. Recently, CMI has introduced the possibility of students pursuing a part-time Ph.D. at the institute. Since 1998, CMI has offered a B.Sc.(Hons) degree in Mathematics and Computer Science. This three-year program also includes courses in Humanities and Physics. Many students, after completion of the B.Sc. degree, have pursued higher studies in Mathematics and Computer Sciences from universities both in India and abroad. Some students also go into industry while others take up subjects such as finance. In 2001, CMI began separate M.Sc. programmes in Mathematics and in Computer Science. In 2009, CMI began to offer a new programme M.Sc. in Applied Mathematics, which is scheduled to be replaced with a new M.Sc. in Data Science programme in 2018 [1] Archived 1 November 2020 at the Wayback Machine. In 2003, CMI introduced a new three-year programme in the form of a B.Sc.(Hons) degree in physics. The course topics are largely in theoretical physics. CMI now has its own physics laboratory. From the academic year 2007–2008, the Physics students are having regular lab courses right from the first year. In the academic year 2005–2006, lab sessions for third-year students were conducted at IIT Madras based on an agreement. In the summer following their first year, physics students go to HBCSE (under TIFR) in Mumbai for practical sessions and in the second year, they go to IGCAR, Kalpakkam. However, in 2012, the B.Sc. degree in Physics was restructured as an integrated B.Sc. degree in Mathematics and Physics. Degrees for the B.Sc. and M.Sc. programmes were earlier offered by MPBOU, the Madhya Pradesh Bhoj Open University and doctoral degrees by Madras University. After CMI became a deemed university, it gives its own degrees. CMI awarded its first official degrees in August 2007. The batch sizes typically vary from 10 to 50 and the overall strength of CMI is about 150–200 students and 40–50 faculty members. Nearly all the CMI programmes are run in conjunction and coordination with programmes at IMSc, an institute for research in Mathematics, Theoretical Computer Science and Theoretical Physics, located in Chennai. === Admission criteria === The entrance to each of these courses is based on a nationwide entrance test. The advertisement for this entrance test appears around the end of February or the beginning of March. The entrance test is held in the end of May and is usually scheduled so as not to clash with major entrance examinations. Results are given to students by the end of June. Students who have passed the Indian National Mathematics Olympiad get direct admission to the programme B.Sc.(Hons.) in Mathematics and Computer Science, and those who have passed the Indian National Physics Olympiad are offered direct entry to the B.Sc.(Hons.) in Physics programme. However, these students are also advised to fill in and send the application form some time in March. Students who pass the Indian National Olympiad in Informatics may be granted admission to the B.Sc.(Hons.) Mathematics and Computer Science programme. Admission is not guaranteed but is decided on a case-by-case basis by the admissions committee. === Fee structure and other payments === Till 2018, all students, including undergraduate students, were given a monthly stipend, subject to academic performance. The tuition was waived for students in good academic standing. In 2018, the fees were increased for all students with waivers available for students in good academic standing. === Arrangements with other Institutes === Till 2006, students received their B.Sc. and M.Sc. degrees from MPBOU and their Ph.D. degrees from Madras University. CMI conducts its academic programmes in conjunction with IMSc, so students from either institute can take courses at the other. CMI has agreements with TIFR (Tata Institute of Fundamental Research) and with the Indian Statistical Institutes in Delhi, Bangalore, Chennai and Kolkata, for cooperation on the furtherance of mathematical sciences. The physics programmes are run in conjunction with IMSc and IGCAR. The physics students spend one summer in HBCSE (under TIFR) in Mumbai and another in IGCAR, Kalpakkam, garnering practical experience. CMI has a memorandum of understanding with the École Normale Supérieure in Paris. Under this memorandum, research scholars from the ENS spend a semester in CMI. In exchange, three B.Sc. Mathematics students, at the end of their third year, go to the ENS for two months. The institute has a similar arrangement with École Polytechnique in Paris, whereby top-ranking senior B.Sc. Physics students spend the summer in Paris working with the faculty at École Polytechnique. CMI has a memorandum of understanding with IFMR, the Institute of Financial Management and Research, located in Nungambakkam, Chennai. Students from CMI getting a CGPA of more than 8.50 are offered direct admission to IFMR's one-year programme in Financial Mathematics, which is sponsored by ICICI Bank. Faculty from CMI are involved in teaching this programme. In exchange, CMI gets its Economics professor from IFMR. CMI is a part of ReLaX, an Indo-French joint research unit dedicated to research in theoretical computer science, its applications and its interactions with mathematics. This collaboration allows for collaborative work in computer science, academic visits for professors and graduate students, summer internship programs for students, and organizing conferences in the subject. CMI has had two sponsored research projects with Honeywell Technology Solutions, Bangalore, both in the area of formal verification. == Research == In mathematics, the main areas of research activity have been in algebraic geometry, representation theory, operator algebra, commutative algebra, harmonic analysis, control theory and game theory. Research work includes stratification of binary forms in representation theory, the Donaldson-Uhlenbeck compactification in algebraic geometry, stochastic games, inductive algebras of harmonic analysis, etc. The research activity in theoretical computer science at CMI has been primarily in computational complexity theory, specification and verification of timed and distributed systems and analysis of security protocols. A computer scientist at CMI extended the deterministic isolation technique for reachability in planar graphs to obtain better complexity upper bounds for planar bipartite matching. In theoretical physics, research is being carried out mainly in string theory, quantum field theory and mathematical physics. In mathematical physics, research included developing a path integral approach to quantum entanglement. CMI string theorists study problems such as Big Bang like cosmological singularities, embeddings of BKL cosmology, dyons in super-Yang–Mills theories etc. == Student life == === Hostel === The on-campus hostel was opened in January 2007. All students are expected to stay in the hostel. Security guards are posted in the campus round the clock, including women security guards for the girls wing on all floors. Washing machines are installed in the hostel which is managed by the students. A vehicle is parked in the campus at night for emergencies. Wi-fi facility is available in the campus. === Annual Inter-Collegiate Student Festival === The students of CMI organize an annual inter-collegiate student festival Tessellate (earlier known as Fiesta, renamed in 2018), supported by corporations, IT companies, and local businesses. Tessellate enjoys participation from students of several Chennai-based colleges including IIT Madras, SRM Institute of Science and Technology, Sathyabama University, KCG College of Technology. Tessellate comprises academic, cultural, technical and sports events. From 2018, Tessellate also includes a social initiative for the benefit of the underprivileged in Chennai. === S.T.E.M.S. === Also, as a part of Tessellate, a nationwide contest called S.T.E.M.S. (Scholastic Test of Excellence in Mathematical Sciences) is organised for students from 8th grade to Final year UG in different categories for the subjects: Mathematics, Physics and Computer Science. Toppers get to win exciting prizes and are invited to a fully funded 3-day camp at CMI. === Other activities === CMI faculty coordinate the training and selection of students to represent India at the International Olympiad in Informatics through the Indian Association for Research in Computing Science (IARCS). CMI hosts the official IARCS website. From September 2004, a monthly online programming competition has been conducted by the CMI faculty via the IARCS website. Two of CMI's faculty members, Madhavan Mukund and Narayan Kumar, lead the Indian team to the International Olympiad in Informatics (IOI). Madhavan Mukund is also the National Coordinator for the Indian Computing Olympiad. == Notes == == External links == Media related to Chennai Mathematical Institute at Wikimedia Commons Official website Homepage for the annual inter-collegiate student festival Tessellate News story: CMI gets its own campus Cover story for India Today: "From preaching to practice" CMI in SIPCOT IT Park map == References == A tribute to C.S. Seshadri: a collection of articles on geometry and representation theory. Birkhäuser. 2003. ISBN 978-3-7643-0444-7.
Wikipedia:Chennas Narayanan Namboodiripad#0	Chennas Narayanan Namboodiripad (born 1428) was a 15th-century mathematician and Tantra ritualist from Kerala, India. Narayanan Namboodiripad was considered to be an authority in the fields of Vaasthusaastram (Indian Architecture), Mathematics and Tantram. He authored a book titled Thanthra Samuchayam which is still considered as the authentic reference manual in the field of temple architecture and rituals. Other contributions to mathematics include: A method of arriving at a circle starting with a square, and successively making it a regular octagon, a regular 16-sided, a 32-sided, 64-sided polygons, etc. Co-ordinate system of fixing points in a plane. Converting a square to a regular hexagon having approximately equal area. Finding the width of a regular octagon, given the perimeter. == References == https://www.namboothiri.com/articles/thanthram.htm == See also == Indian Mathematics
Wikipedia:Chern class#0	In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since become fundamental concepts in many branches of mathematics and physics, such as string theory, Chern–Simons theory, knot theory, and Gromov–Witten invariants. Chern classes were introduced by Shiing-Shen Chern (1946). == Geometric approach == === Basic idea and motivation === Chern classes are characteristic classes. They are topological invariants associated with vector bundles on a smooth manifold. The question of whether two ostensibly different vector bundles are the same can be quite hard to answer. The Chern classes provide a simple test: if the Chern classes of a pair of vector bundles do not agree, then the vector bundles are different. The converse, however, is not true. In topology, differential geometry, and algebraic geometry, it is often important to count how many linearly independent sections a vector bundle has. The Chern classes offer some information about this through, for instance, the Riemann–Roch theorem and the Atiyah–Singer index theorem. Chern classes are also feasible to calculate in practice. In differential geometry (and some types of algebraic geometry), the Chern classes can be expressed as polynomials in the coefficients of the curvature form. === Construction === There are various ways of approaching the subject, each of which focuses on a slightly different flavor of Chern class. The original approach to Chern classes was via algebraic topology: the Chern classes arise via homotopy theory which provides a mapping associated with a vector bundle to a classifying space (an infinite Grassmannian in this case). For any complex vector bundle V over a manifold M, there exists a map f from M to the classifying space such that the bundle V is equal to the pullback, by f, of a universal bundle over the classifying space, and the Chern classes of V can therefore be defined as the pullback of the Chern classes of the universal bundle. In turn, these universal Chern classes can be explicitly written down in terms of Schubert cycles. It can be shown that for any two maps f, g from M to the classifying space whose pullbacks are the same bundle V, the maps must be homotopic. Therefore, the pullback by either f or g of any universal Chern class to a cohomology class of M must be the same class. This shows that the Chern classes of V are well-defined. Chern's approach used differential geometry, via the curvature approach described predominantly in this article. He showed that the earlier definition was in fact equivalent to his. The resulting theory is known as the Chern–Weil theory. There is also an approach of Alexander Grothendieck showing that axiomatically one need only define the line bundle case. Chern classes arise naturally in algebraic geometry. The generalized Chern classes in algebraic geometry can be defined for vector bundles (or more precisely, locally free sheaves) over any nonsingular variety. Algebro-geometric Chern classes do not require the underlying field to have any special properties. In particular, the vector bundles need not necessarily be complex. Regardless of the particular paradigm, the intuitive meaning of the Chern class concerns 'required zeroes' of a section of a vector bundle: for example the theorem saying one can't comb a hairy ball flat (hairy ball theorem). Although that is strictly speaking a question about a real vector bundle (the "hairs" on a ball are actually copies of the real line), there are generalizations in which the hairs are complex (see the example of the complex hairy ball theorem below), or for 1-dimensional projective spaces over many other fields. See Chern–Simons theory for more discussion. == The Chern class of line bundles == (Let X be a topological space having the homotopy type of a CW complex.) An important special case occurs when V is a line bundle. Then the only nontrivial Chern class is the first Chern class, which is an element of the second cohomology group of X. As it is the top Chern class, it equals the Euler class of the bundle. The first Chern class turns out to be a complete invariant with which to classify complex line bundles, topologically speaking. That is, there is a bijection between the isomorphism classes of line bundles over X and the elements of H 2 ( X ; Z ) {\displaystyle H^{2}(X;\mathbb {Z} )} , which associates to a line bundle its first Chern class. Moreover, this bijection is a group homomorphism (thus an isomorphism): c 1 ( L ⊗ L ′ ) = c 1 ( L ) + c 1 ( L ′ ) ; {\displaystyle c_{1}(L\otimes L')=c_{1}(L)+c_{1}(L');} the tensor product of complex line bundles corresponds to the addition in the second cohomology group. In algebraic geometry, this classification of (isomorphism classes of) complex line bundles by the first Chern class is a crude approximation to the classification of (isomorphism classes of) holomorphic line bundles by linear equivalence classes of divisors. For complex vector bundles of dimension greater than one, the Chern classes are not a complete invariant. == Constructions == === Via the Chern–Weil theory === Given a complex hermitian vector bundle V of complex rank n over a smooth manifold M, representatives of each Chern class (also called a Chern form) c k ( V ) {\displaystyle c_{k}(V)} of V are given as the coefficients of the characteristic polynomial of the curvature form Ω {\displaystyle \Omega } of V. det ( i t Ω 2 π + I ) = ∑ k c k ( V ) t k {\displaystyle \det \left({\frac {it\Omega }{2\pi }}+I\right)=\sum _{k}c_{k}(V)t^{k}} The determinant is over the ring of n × n {\displaystyle n\times n} matrices whose entries are polynomials in t with coefficients in the commutative algebra of even complex differential forms on M. The curvature form Ω {\displaystyle \Omega } of V is defined as Ω = d ω + 1 2 [ ω , ω ] {\displaystyle \Omega =d\omega +{\frac {1}{2}}[\omega ,\omega ]} with ω the connection form and d the exterior derivative, or via the same expression in which ω is a gauge field for the gauge group of V. The scalar t is used here only as an indeterminate to generate the sum from the determinant, and I denotes the n × n identity matrix. To say that the expression given is a representative of the Chern class indicates that 'class' here means up to addition of an exact differential form. That is, Chern classes are cohomology classes in the sense of de Rham cohomology. It can be shown that the cohomology classes of the Chern forms do not depend on the choice of connection in V. If follows from the matrix identity t r ( ln ⁡ ( X ) ) = ln ⁡ ( det ( X ) ) {\displaystyle \mathrm {tr} (\ln(X))=\ln(\det(X))} that det ( X ) = exp ⁡ ( t r ( ln ⁡ ( X ) ) ) {\displaystyle \det(X)=\exp(\mathrm {tr} (\ln(X)))} . Now applying the Maclaurin series for ln ⁡ ( X + I ) {\displaystyle \ln(X+I)} , we get the following expression for the Chern forms: ∑ k c k ( V ) t k = [ 1 + i t r ( Ω ) 2 π t + t r ( Ω 2 ) − t r ( Ω ) 2 8 π 2 t 2 + i − 2 t r ( Ω 3 ) + 3 t r ( Ω 2 ) t r ( Ω ) − t r ( Ω ) 3 48 π 3 t 3 + ⋯ ] . {\displaystyle \sum _{k}c_{k}(V)t^{k}=\left[1+i{\frac {\mathrm {tr} (\Omega )}{2\pi }}t+{\frac {\mathrm {tr} (\Omega ^{2})-\mathrm {tr} (\Omega )^{2}}{8\pi ^{2}}}t^{2}+i{\frac {-2\mathrm {tr} (\Omega ^{3})+3\mathrm {tr} (\Omega ^{2})\mathrm {tr} (\Omega )-\mathrm {tr} (\Omega )^{3}}{48\pi ^{3}}}t^{3}+\cdots \right].} === Via an Euler class === One can define a Chern class in terms of an Euler class. This is the approach in the book by Milnor and Stasheff, and emphasizes the role of an orientation of a vector bundle. The basic observation is that a complex vector bundle comes with a canonical orientation, ultimately because GL n ⁡ ( C ) {\displaystyle \operatorname {GL} _{n}(\mathbb {C} )} is connected. Hence, one simply defines the top Chern class of the bundle to be its Euler class (the Euler class of the underlying real vector bundle) and handles lower Chern classes in an inductive fashion. The precise construction is as follows. The idea is to do base change to get a bundle of one-less rank. Let π : E → B {\displaystyle \pi \colon E\to B} be a complex vector bundle over a paracompact space B. Thinking of B as being embedded in E as the zero section, let B ′ = E ∖ B {\displaystyle B'=E\setminus B} and define the new vector bundle: E ′ → B ′ {\displaystyle E'\to B'} such that each fiber is the quotient of a fiber F of E by the line spanned by a nonzero vector v in F (a point of B′ is specified by a fiber F of E and a nonzero vector on F.) Then E ′ {\displaystyle E'} has rank one less than that of E. From the Gysin sequence for the fiber bundle π \| B ′ : B ′ → B {\displaystyle \pi \|_{B'}\colon B'\to B} : ⋯ → H k ⁡ ( B ; Z ) → π \| B ′ ∗ H k ⁡ ( B ′ ; Z ) → ⋯ , {\displaystyle \cdots \to \operatorname {H} ^{k}(B;\mathbb {Z} ){\overset {\pi \|_{B'}^{}}{\to }}\operatorname {H} ^{k}(B';\mathbb {Z} )\to \cdots ,} we see that π \| B ′ ∗ {\displaystyle \pi \|_{B'}^{}} is an isomorphism for k < 2 n − 1 {\displaystyle k<2n-1} . Let c k ( E ) = { π \| B ′ ∗ − 1 c k ( E ′ ) k < n e ( E R ) k = n 0 k > n {\displaystyle c_{k}(E)={\begin{cases}{\pi \|_{B'}^{}}^{-1}c_{k}(E')&k<n\\e(E_{\mathbb {R} })&k=n\\0&k>n\end{cases}}} It then takes some work to check the axioms of Chern classes are satisfied for this definition. See also: The Thom isomorphism. == Examples == === The complex tangent bundle of the Riemann sphere === Let C P 1 {\displaystyle \mathbb {CP} ^{1}} be the Riemann sphere: 1-dimensional complex projective space. Suppose that z is a holomorphic local coordinate for the Riemann sphere. Let V = T C P 1 {\displaystyle V=T\mathbb {CP} ^{1}} be the bundle of complex tangent vectors having the form a ∂ / ∂ z {\displaystyle a\partial /\partial z} at each point, where a is a complex number. We prove the complex version of the hairy ball theorem: V has no section which is everywhere nonzero. For this, we need the following fact: the first Chern class of a trivial bundle is zero, i.e., c 1 ( C P 1 × C ) = 0. {\displaystyle c_{1}(\mathbb {CP} ^{1}\times \mathbb {C} )=0.} This is evinced by the fact that a trivial bundle always admits a flat connection. So, we shall show that c 1 ( V ) ≠ 0. {\displaystyle c_{1}(V)\not =0.} Consider the Kähler metric h = d z d z ¯ ( 1 + \| z \| 2 ) 2 . {\displaystyle h={\frac {dzd{\bar {z}}}{(1+\|z\|^{2})^{2}}}.} One readily shows that the curvature 2-form is given by Ω = 2 d z ∧ d z ¯ ( 1 + \| z \| 2 ) 2 . {\displaystyle \Omega ={\frac {2dz\wedge d{\bar {z}}}{(1+\|z\|^{2})^{2}}}.} Furthermore, by the definition of the first Chern class c 1 = [ i 2 π tr ⁡ Ω ] . {\displaystyle c_{1}=\left[{\frac {i}{2\pi }}\operatorname {tr} \Omega \right].} We must show that this cohomology class is non-zero. It suffices to compute its integral over the Riemann sphere: ∫ c 1 = i π ∫ d z ∧ d z ¯ ( 1 + \| z \| 2 ) 2 = 2 {\displaystyle \int c_{1}={\frac {i}{\pi }}\int {\frac {dz\wedge d{\bar {z}}}{(1+\|z\|^{2})^{2}}}=2} after switching to polar coordinates. By Stokes' theorem, an exact form would integrate to 0, so the cohomology class is nonzero. This proves that T C P 1 {\displaystyle T\mathbb {CP} ^{1}} is not a trivial vector bundle. === Complex projective space === There is an exact sequence of sheaves/bundles: 0 → O C P n → O C P n ( 1 ) ⊕ ( n + 1 ) → T C P n → 0 {\displaystyle 0\to {\mathcal {O}}_{\mathbb {CP} ^{n}}\to {\mathcal {O}}_{\mathbb {CP} ^{n}}(1)^{\oplus (n+1)}\to T\mathbb {CP} ^{n}\to 0} where O C P n {\displaystyle {\mathcal {O}}_{\mathbb {CP} ^{n}}} is the structure sheaf (i.e., the trivial line bundle), O C P n ( 1 ) {\displaystyle {\mathcal {O}}_{\mathbb {CP} ^{n}}(1)} is Serre's twisting sheaf (i.e., the hyperplane bundle) and the last nonzero term is the tangent sheaf/bundle. There are two ways to get the above sequence: By the additivity of total Chern class c = 1 + c 1 + c 2 + ⋯ {\displaystyle c=1+c_{1}+c_{2}+\cdots } (i.e., the Whitney sum formula), c ( C P n ) = d e f c ( T C P n ) = c ( O C P n ( 1 ) ) n + 1 = ( 1 + a ) n + 1 , {\displaystyle c(\mathbb {C} \mathbb {P} ^{n}){\overset {\mathrm {def} }{=}}c(T\mathbb {CP} ^{n})=c({\mathcal {O}}_{\mathbb {C} \mathbb {P} ^{n}}(1))^{n+1}=(1+a)^{n+1},} where a is the canonical generator of the cohomology group H 2 ( C P n , Z ) {\displaystyle H^{2}(\mathbb {C} \mathbb {P} ^{n},\mathbb {Z} )} ; i.e., the negative of the first Chern class of the tautological line bundle O C P n ( − 1 ) {\displaystyle {\mathcal {O}}_{\mathbb {C} \mathbb {P} ^{n}}(-1)} (note: c 1 ( E ∗ ) = − c 1 ( E ) {\displaystyle c_{1}(E^{})=-c_{1}(E)} when E ∗ {\displaystyle E^{}} is the dual of E.) In particular, for any k ≥ 0 {\displaystyle k\geq 0} , c k ( C P n ) = ( n + 1 k ) a k . {\displaystyle c_{k}(\mathbb {C} \mathbb {P} ^{n})={\binom {n+1}{k}}a^{k}.} == Chern polynomial == A Chern polynomial is a convenient way to handle Chern classes and related notions systematically. By definition, for a complex vector bundle E, the Chern polynomial ct of E is given by: c t ( E ) = 1 + c 1 ( E ) t + ⋯ + c n ( E ) t n . {\displaystyle c_{t}(E)=1+c_{1}(E)t+\cdots +c_{n}(E)t^{n}.} This is not a new invariant: the formal variable t simply keeps track of the degree of ck(E). In particular, c t ( E ) {\displaystyle c_{t}(E)} is completely determined by the total Chern class of E: c ( E ) = 1 + c 1 ( E ) + ⋯ + c n ( E ) {\displaystyle c(E)=1+c_{1}(E)+\cdots +c_{n}(E)} and conversely. The Whitney sum formula, one of the axioms of Chern classes (see below), says that ct is additive in the sense: c t ( E ⊕ E ′ ) = c t ( E ) c t ( E ′ ) . {\displaystyle c_{t}(E\oplus E')=c_{t}(E)c_{t}(E').} Now, if E = L 1 ⊕ ⋯ ⊕ L n {\displaystyle E=L_{1}\oplus \cdots \oplus L_{n}} is a direct sum of (complex) line bundles, then it follows from the sum formula that: c t ( E ) = ( 1 + a 1 ( E ) t ) ⋯ ( 1 + a n ( E ) t ) {\displaystyle c_{t}(E)=(1+a_{1}(E)t)\cdots (1+a_{n}(E)t)} where a i ( E ) = c 1 ( L i ) {\displaystyle a_{i}(E)=c_{1}(L_{i})} are the first Chern classes. The roots a i ( E ) {\displaystyle a_{i}(E)} , called the Chern roots of E, determine the coefficients of the polynomial: i.e., c k ( E ) = σ k ( a 1 ( E ) , … , a n ( E ) ) {\displaystyle c_{k}(E)=\sigma _{k}(a_{1}(E),\ldots ,a_{n}(E))} where σk are elementary symmetric polynomials. In other words, thinking of ai as formal variables, ck "are" σk. A basic fact on symmetric polynomials is that any symmetric polynomial in, say, ti's is a polynomial in elementary symmetric polynomials in ti's. Either by splitting principle or by ring theory, any Chern polynomial c t ( E ) {\displaystyle c_{t}(E)} factorizes into linear factors after enlarging the cohomology ring; E need not be a direct sum of line bundles in the preceding discussion. The conclusion is Example: We have polynomials sk t 1 k + ⋯ + t n k = s k ( σ 1 ( t 1 , … , t n ) , … , σ k ( t 1 , … , t n ) ) {\displaystyle t_{1}^{k}+\cdots +t_{n}^{k}=s_{k}(\sigma _{1}(t_{1},\ldots ,t_{n}),\ldots ,\sigma _{k}(t_{1},\ldots ,t_{n}))} with s 1 = σ 1 , s 2 = σ 1 2 − 2 σ 2 {\displaystyle s_{1}=\sigma _{1},s_{2}=\sigma _{1}^{2}-2\sigma _{2}} and so on (cf. Newton's identities). The sum ch ⁡ ( E ) = e a 1 ( E ) + ⋯ + e a n ( E ) = ∑ s k ( c 1 ( E ) , … , c n ( E ) ) / k ! {\displaystyle \operatorname {ch} (E)=e^{a_{1}(E)}+\cdots +e^{a_{n}(E)}=\sum s_{k}(c_{1}(E),\ldots ,c_{n}(E))/k!} is called the Chern character of E, whose first few terms are: (we drop E from writing.) ch ⁡ ( E ) = rk + c 1 + 1 2 ( c 1 2 − 2 c 2 ) + 1 6 ( c 1 3 − 3 c 1 c 2 + 3 c 3 ) + ⋯ . {\displaystyle \operatorname {ch} (E)=\operatorname {rk} +c_{1}+{\frac {1}{2}}(c_{1}^{2}-2c_{2})+{\frac {1}{6}}(c_{1}^{3}-3c_{1}c_{2}+3c_{3})+\cdots .} Example: The Todd class of E is given by: td ⁡ ( E ) = ∏ 1 n a i 1 − e − a i = 1 + 1 2 c 1 + 1 12 ( c 1 2 + c 2 ) + ⋯ . {\displaystyle \operatorname {td} (E)=\prod _{1}^{n}{a_{i} \over 1-e^{-a_{i}}}=1+{1 \over 2}c_{1}+{1 \over 12}(c_{1}^{2}+c_{2})+\cdots .} Remark: The observation that a Chern class is essentially an elementary symmetric polynomial can be used to "define" Chern classes. Let Gn be the infinite Grassmannian of n-dimensional complex vector spaces. This space is equipped with a tautologous vector bundle of rank n {\displaystyle n} , say E n → G n {\displaystyle E_{n}\to G_{n}} . G n {\displaystyle G_{n}} is called the classifying space for rank- n {\displaystyle n} vector bundles because given any complex vector bundle E of rank n over X, there is a continuous map f E : X → G n {\displaystyle f_{E}:X\to G_{n}} such that the pullback of E n {\displaystyle E_{n}} to X {\displaystyle X} along f E {\displaystyle f_{E}} is isomorphic to E {\displaystyle E} , and this map f E {\displaystyle f_{E}} is unique up to homotopy. Borel's theorem says the cohomology ring of Gn is exactly the ring of symmetric polynomials, which are polynomials in elementary symmetric polynomials σk; so, the pullback of fE reads: f E ∗ : Z [ σ 1 , … , σ n ] → H ∗ ( X , Z ) . {\displaystyle f_{E}^{}:\mathbb {Z} [\sigma _{1},\ldots ,\sigma _{n}]\to H^{}(X,\mathbb {Z} ).} One then puts: c k ( E ) = f E ∗ ( σ k ) . {\displaystyle c_{k}(E)=f_{E}^{}(\sigma _{k}).} Remark: Any characteristic class is a polynomial in Chern classes, for the reason as follows. Let Vect n C {\displaystyle \operatorname {Vect} _{n}^{\mathbb {C} }} be the contravariant functor that, to a CW complex X, assigns the set of isomorphism classes of complex vector bundles of rank n over X and, to a map, its pullback. By definition, a characteristic class is a natural transformation from Vect n C = [ − , G n ] {\displaystyle \operatorname {Vect} _{n}^{\mathbb {C} }=[-,G_{n}]} to the cohomology functor H ∗ ( − , Z ) . {\displaystyle H^{}(-,\mathbb {Z} ).} Characteristic classes form a ring because of the ring structure of cohomology ring. Yoneda's lemma says this ring of characteristic classes is exactly the cohomology ring of Gn: Nat ⁡ ( [ − , G n ] , H ∗ ( − , Z ) ) = H ∗ ( G n , Z ) = Z [ σ 1 , … , σ n ] . {\displaystyle \operatorname {Nat} ([-,G_{n}],H^{}(-,\mathbb {Z} ))=H^{}(G_{n},\mathbb {Z} )=\mathbb {Z} [\sigma _{1},\ldots ,\sigma _{n}].} == Computation formulae == Let E be a vector bundle of rank r and c t ( E ) = ∑ i = 0 r c i ( E ) t i {\displaystyle c_{t}(E)=\sum _{i=0}^{r}c_{i}(E)t^{i}} the Chern polynomial of it. For the dual bundle E ∗ {\displaystyle E^{}} of E {\displaystyle E} , c i ( E ∗ ) = ( − 1 ) i c i ( E ) {\displaystyle c_{i}(E^{})=(-1)^{i}c_{i}(E)} . If L is a line bundle, then c t ( E ⊗ L ) = ∑ i = 0 r c i ( E ) c t ( L ) r − i t i {\displaystyle c_{t}(E\otimes L)=\sum _{i=0}^{r}c_{i}(E)c_{t}(L)^{r-i}t^{i}} and so c i ( E ⊗ L ) , i = 1 , 2 , … , r {\displaystyle c_{i}(E\otimes L),i=1,2,\dots ,r} are c 1 ( E ) + r c 1 ( L ) , … , ∑ j = 0 i ( r − i + j j ) c i − j ( E ) c 1 ( L ) j , … , ∑ j = 0 r c r − j ( E ) c 1 ( L ) j . {\displaystyle c_{1}(E)+rc_{1}(L),\dots ,\sum _{j=0}^{i}{\binom {r-i+j}{j}}c_{i-j}(E)c_{1}(L)^{j},\dots ,\sum _{j=0}^{r}c_{r-j}(E)c_{1}(L)^{j}.} For the Chern roots α 1 , … , α r {\displaystyle \alpha _{1},\dots ,\alpha _{r}} of E {\displaystyle E} , c t ( Sym p ⁡ E ) = ∏ i 1 ≤ ⋯ ≤ i p ( 1 + ( α i 1 + ⋯ + α i p ) t ) , c t ( ∧ p E ) = ∏ i 1 < ⋯ < i p ( 1 + ( α i 1 + ⋯ + α i p ) t ) . {\displaystyle {\begin{aligned}c_{t}(\operatorname {Sym} ^{p}E)&=\prod _{i_{1}\leq \cdots \leq i_{p}}(1+(\alpha _{i_{1}}+\cdots +\alpha _{i_{p}})t),\\c_{t}(\wedge ^{p}E)&=\prod _{i_{1}<\cdots <i_{p}}(1+(\alpha _{i_{1}}+\cdots +\alpha _{i_{p}})t).\end{aligned}}} In particular, c 1 ( ∧ r E ) = c 1 ( E ) . {\displaystyle c_{1}(\wedge ^{r}E)=c_{1}(E).} For example, for c i = c i ( E ) {\displaystyle c_{i}=c_{i}(E)} , when r = 2 {\displaystyle r=2} , c ( Sym 2 ⁡ E ) = 1 + 3 c 1 + 2 c 1 2 + 4 c 2 + 4 c 1 c 2 , {\displaystyle c(\operatorname {Sym} ^{2}E)=1+3c_{1}+2c_{1}^{2}+4c_{2}+4c_{1}c_{2},} when r = 3 {\displaystyle r=3} , c ( Sym 2 ⁡ E ) = 1 + 4 c 1 + 5 c 1 2 + 5 c 2 + 2 c 1 3 + 11 c 1 c 2 + 7 c 3 . {\displaystyle c(\operatorname {Sym} ^{2}E)=1+4c_{1}+5c_{1}^{2}+5c_{2}+2c_{1}^{3}+11c_{1}c_{2}+7c_{3}.} (cf. Segre class#Example 2.) === Applications of formulae === We can use these abstract properties to compute the rest of the chern classes of line bundles on C P 1 {\displaystyle \mathbb {CP} ^{1}} . Recall that O ( − 1 ) ∗ ≅ O ( 1 ) {\displaystyle {\mathcal {O}}(-1)^{}\cong {\mathcal {O}}(1)} showing c 1 ( O ( 1 ) ) = 1 ∈ H 2 ( C P 1 ; Z ) {\displaystyle c_{1}({\mathcal {O}}(1))=1\in H^{2}(\mathbb {CP} ^{1};\mathbb {Z} )} . Then using tensor powers, we can relate them to the chern classes of c 1 ( O ( n ) ) = n {\displaystyle c_{1}({\mathcal {O}}(n))=n} for any integer. == Properties == Given a complex vector bundle E over a topological space X, the Chern classes of E are a sequence of elements of the cohomology of X. The k-th Chern class of E, which is usually denoted ck(E), is an element of H 2 k ( X ; Z ) , {\displaystyle H^{2k}(X;\mathbb {Z} ),} the cohomology of X with integer coefficients. One can also define the total Chern class c ( E ) = c 0 ( E ) + c 1 ( E ) + c 2 ( E ) + ⋯ . {\displaystyle c(E)=c_{0}(E)+c_{1}(E)+c_{2}(E)+\cdots .} Since the values are in integral cohomology groups, rather than cohomology with real coefficients, these Chern classes are slightly more refined than those in the Riemannian example. === Classical axiomatic definition === The Chern classes satisfy the following four axioms: c 0 ( E ) = 1 {\displaystyle c_{0}(E)=1} for all E. Naturality: If f : Y → X {\displaystyle f:Y\to X} is continuous and fE is the vector bundle pullback of E, then c k ( f ∗ E ) = f ∗ c k ( E ) {\displaystyle c_{k}(f^{}E)=f^{}c_{k}(E)} . Whitney sum formula: If F → X {\displaystyle F\to X} is another complex vector bundle, then the Chern classes of the direct sum E ⊕ F {\displaystyle E\oplus F} are given by c ( E ⊕ F ) = c ( E ) ⌣ c ( F ) ; {\displaystyle c(E\oplus F)=c(E)\smile c(F);} that is, c k ( E ⊕ F ) = ∑ i = 0 k c i ( E ) ⌣ c k − i ( F ) . {\displaystyle c_{k}(E\oplus F)=\sum _{i=0}^{k}c_{i}(E)\smile c_{k-i}(F).} Normalization: The total Chern class of the tautological line bundle over C P k {\displaystyle \mathbb {CP} ^{k}} is 1−H, where H is Poincaré dual to the hyperplane C P k − 1 ⊆ C P k {\displaystyle \mathbb {CP} ^{k-1}\subseteq \mathbb {CP} ^{k}} . === Grothendieck axiomatic approach === Alternatively, Alexander Grothendieck (1958) replaced these with a slightly smaller set of axioms: Naturality: (Same as above) Additivity: If 0 → E ′ → E → E ″ → 0 {\displaystyle 0\to E'\to E\to E''\to 0} is an exact sequence of vector bundles, then c ( E ) = c ( E ′ ) ⌣ c ( E ″ ) {\displaystyle c(E)=c(E')\smile c(E'')} . Normalization: If E is a line bundle, then c ( E ) = 1 + e ( E R ) {\displaystyle c(E)=1+e(E_{\mathbb {R} })} where e ( E R ) {\displaystyle e(E_{\mathbb {R} })} is the Euler class of the underlying real vector bundle. He shows using the Leray–Hirsch theorem that the total Chern class of an arbitrary finite rank complex vector bundle can be defined in terms of the first Chern class of a tautologically-defined line bundle. Namely, introducing the projectivization P ( E ) {\displaystyle \mathbb {P} (E)} of the rank n complex vector bundle E → B as the fiber bundle on B whose fiber at any point b ∈ B {\displaystyle b\in B} is the projective space of the fiber Eb. The total space of this bundle P ( E ) {\displaystyle \mathbb {P} (E)} is equipped with its tautological complex line bundle, that we denote τ {\displaystyle \tau } , and the first Chern class c 1 ( τ ) =: − a {\displaystyle c_{1}(\tau )=:-a} restricts on each fiber P ( E b ) {\displaystyle \mathbb {P} (E_{b})} to minus the (Poincaré-dual) class of the hyperplane, that spans the cohomology of the fiber, in view of the cohomology of complex projective spaces. The classes 1 , a , a 2 , … , a n − 1 ∈ H ∗ ( P ( E ) ) {\displaystyle 1,a,a^{2},\ldots ,a^{n-1}\in H^{}(\mathbb {P} (E))} therefore form a family of ambient cohomology classes restricting to a basis of the cohomology of the fiber. The Leray–Hirsch theorem then states that any class in H ∗ ( P ( E ) ) {\displaystyle H^{}(\mathbb {P} (E))} can be written uniquely as a linear combination of the 1, a, a2, ..., an−1 with classes on the base as coefficients. In particular, one may define the Chern classes of E in the sense of Grothendieck, denoted c 1 ( E ) , … c n ( E ) {\displaystyle c_{1}(E),\ldots c_{n}(E)} by expanding this way the class − a n {\displaystyle -a^{n}} , with the relation: − a n = c 1 ( E ) ⋅ a n − 1 + ⋯ + c n − 1 ( E ) ⋅ a + c n ( E ) . {\displaystyle -a^{n}=c_{1}(E)\cdot a^{n-1}+\cdots +c_{n-1}(E)\cdot a+c_{n}(E).} One then may check that this alternative definition coincides with whatever other definition one may favor, or use the previous axiomatic characterization. === The top Chern class === In fact, these properties uniquely characterize the Chern classes. They imply, among other things: If n is the complex rank of V, then c k ( V ) = 0 {\displaystyle c_{k}(V)=0} for all k > n. Thus the total Chern class terminates. The top Chern class of V (meaning c n ( V ) {\displaystyle c_{n}(V)} , where n is the rank of V) is always equal to the Euler class of the underlying real vector bundle. == In algebraic geometry == === Axiomatic description === There is another construction of Chern classes which take values in the algebrogeometric analogue of the cohomology ring, the Chow ring. Let X {\displaystyle X} be a nonsingular quasi-projective variety of dimension n {\displaystyle n} . It can be shown that there is a unique theory of Chern classes which assigns an algebraic vector bundle E → X {\displaystyle E\to X} to elements c i ( E ) ∈ A i ( X ) {\displaystyle c_{i}(E)\in A^{i}(X)} called Chern classes, with Chern polynomial c t ( E ) = c 0 ( E ) + c 1 ( E ) t + ⋯ + c n ( E ) t n {\displaystyle c_{t}(E)=c_{0}(E)+c_{1}(E)t+\cdots +c_{n}(E)t^{n}} , satisfying the following (similar to Grothendieck's axiomatic approach). If for a Cartier divisor D {\displaystyle D} , we have E ≅ O X ( D ) {\displaystyle E\cong {\mathcal {O}}_{X}(D)} , then c t ( E ) = 1 + D t {\displaystyle c_{t}(E)=1+Dt} . If f : X ′ → X {\displaystyle f:X'\to X} is a morphism, then c i ( f ∗ E ) = f ∗ c i ( E ) {\displaystyle c_{i}(f^{}E)=f^{*}c_{i}(E)} . If 0 → E ′ → E → E ″ → 0 {\displaystyle 0\to E'\to E\to E''\to 0} is an exact sequence of vector bundles on X {\displaystyle X} , the Whitney sum formula holds: c t ( E ) = c t ( E ′ ) c t ( E ″ ) {\displaystyle c_{t}(E)=c_{t}(E')c_{t}(E'')} . === Normal sequence === Computing the characteristic classes for projective space forms the basis for many characteristic class computations since for any smooth projective subvariety X ⊂ P n {\displaystyle X\subset \mathbb {P} ^{n}} there is the short exact sequence 0 → T X → T P n \| X → N X / P n → 0 {\displaystyle 0\to {\mathcal {T}}_{X}\to {\mathcal {T}}_{\mathbb {P} ^{n}}\|_{X}\to {\mathcal {N}}_{X/\mathbb {P} ^{n}}\to 0} ==== Quintic threefold ==== For example, consider a nonsingular quintic threefold in P 4 {\displaystyle \mathbb {P} ^{4}} . Its normal bundle is given by O X ( 5 ) {\displaystyle {\mathcal {O}}_{X}(5)} and we have the short exact sequence 0 → T X → T P 4 \| X → O X ( 5 ) → 0 {\displaystyle 0\to {\mathcal {T}}_{X}\to {\mathcal {T}}_{\mathbb {P} ^{4}}\|_{X}\to {\mathcal {O}}_{X}(5)\to 0} Let h {\displaystyle h} denote the hyperplane class in A ∙ ( X ) {\displaystyle A^{\bullet }(X)} . Then the Whitney sum formula gives us that c ( T X ) c ( O X ( 5 ) ) = ( 1 + h ) 5 = 1 + 5 h + 10 h 2 + 10 h 3 {\displaystyle c({\mathcal {T}}_{X})c({\mathcal {O}}_{X}(5))=(1+h)^{5}=1+5h+10h^{2}+10h^{3}} Since the Chow ring of a hypersurface is difficult to compute, we will consider this sequence as a sequence of coherent sheaves in P 4 {\displaystyle \mathbb {P} ^{4}} . This gives us that c ( T X ) = 1 + 5 h + 10 h 2 + 10 h 3 1 + 5 h = ( 1 + 5 h + 10 h 2 + 10 h 3 ) ( 1 − 5 h + 25 h 2 − 125 h 3 ) = 1 + 10 h 2 − 40 h 3 {\displaystyle {\begin{aligned}c({\mathcal {T}}_{X})&={\frac {1+5h+10h^{2}+10h^{3}}{1+5h}}\\&=\left(1+5h+10h^{2}+10h^{3}\right)\left(1-5h+25h^{2}-125h^{3}\right)\\&=1+10h^{2}-40h^{3}\end{aligned}}} Using the Gauss-Bonnet theorem we can integrate the class c 3 ( T X ) {\displaystyle c_{3}({\mathcal {T}}_{X})} to compute the Euler characteristic. Traditionally this is called the Euler class. This is ∫ [ X ] c 3 ( T X ) = ∫ [ X ] − 40 h 3 = − 200 {\displaystyle \int _{[X]}c_{3}({\mathcal {T}}_{X})=\int _{[X]}-40h^{3}=-200} since the class of h 3 {\displaystyle h^{3}} can be represented by five points (by Bézout's theorem). The Euler characteristic can then be used to compute the Betti numbers for the cohomology of X {\displaystyle X} by using the definition of the Euler characteristic and using the Lefschetz hyperplane theorem. ==== Degree d hypersurfaces ==== If X ⊂ P 3 {\displaystyle X\subset \mathbb {P} ^{3}} is a degree d {\displaystyle d} smooth hypersurface, we have the short exact sequence 0 → T X → T P 3 \| X → O X ( d ) → 0 {\displaystyle 0\to {\mathcal {T}}_{X}\to {\mathcal {T}}_{\mathbb {P} ^{3}}\|_{X}\to {\mathcal {O}}_{X}(d)\to 0} giving the relation c ( T X ) = c ( T P 3 \| X ) c ( O X ( d ) ) {\displaystyle c({\mathcal {T}}_{X})={\frac {c({\mathcal {T}}_{\mathbb {P} ^{3}\|_{X}})}{c({\mathcal {O}}_{X}(d))}}} we can then calculate this as c ( T X ) = ( 1 + [ H ] ) 4 ( 1 + d [ H ] ) = ( 1 + 4 [ H ] + 6 [ H ] 2 ) ( 1 − d [ H ] + d 2 [ H ] 2 ) = 1 + ( 4 − d ) [ H ] + ( 6 − 4 d + d 2 ) [ H ] 2 {\displaystyle {\begin{aligned}c({\mathcal {T}}_{X})&={\frac {(1+[H])^{4}}{(1+d[H])}}\\&=(1+4[H]+6[H]^{2})(1-d[H]+d^{2}[H]^{2})\\&=1+(4-d)[H]+(6-4d+d^{2})[H]^{2}\end{aligned}}} Giving the total chern class. In particular, we can find X {\displaystyle X} is a spin 4-manifold if 4 − d {\displaystyle 4-d} is even, so every smooth hypersurface of degree 2 k {\displaystyle 2k} is a spin manifold. == Proximate notions == === The Chern character === Chern classes can be used to construct a homomorphism of rings from the topological K-theory of a space to (the completion of) its rational cohomology. For a line bundle L, the Chern character ch is defined by ch ⁡ ( L ) = exp ⁡ ( c 1 ( L ) ) := ∑ m = 0 ∞ c 1 ( L ) m m ! . {\displaystyle \operatorname {ch} (L)=\exp(c_{1}(L)):=\sum _{m=0}^{\infty }{\frac {c_{1}(L)^{m}}{m!}}.} More generally, if V = L 1 ⊕ ⋯ ⊕ L n {\displaystyle V=L_{1}\oplus \cdots \oplus L_{n}} is a direct sum of line bundles, with first Chern classes x i = c 1 ( L i ) , {\displaystyle x_{i}=c_{1}(L_{i}),} the Chern character is defined additively ch ⁡ ( V ) = e x 1 + ⋯ + e x n := ∑ m = 0 ∞ 1 m ! ( x 1 m + ⋯ + x n m ) . {\displaystyle \operatorname {ch} (V)=e^{x_{1}}+\cdots +e^{x_{n}}:=\sum _{m=0}^{\infty }{\frac {1}{m!}}(x_{1}^{m}+\cdots +x_{n}^{m}).} This can be rewritten as: ch ⁡ ( V ) = rk ⁡ ( V ) + c 1 ( V ) + 1 2 ( c 1 ( V ) 2 − 2 c 2 ( V ) ) + 1 6 ( c 1 ( V ) 3 − 3 c 1 ( V ) c 2 ( V ) + 3 c 3 ( V ) ) + ⋯ . {\displaystyle \operatorname {ch} (V)=\operatorname {rk} (V)+c_{1}(V)+{\frac {1}{2}}(c_{1}(V)^{2}-2c_{2}(V))+{\frac {1}{6}}(c_{1}(V)^{3}-3c_{1}(V)c_{2}(V)+3c_{3}(V))+\cdots .} This last expression, justified by invoking the splitting principle, is taken as the definition ch(V) for arbitrary vector bundles V. If a connection is used to define the Chern classes when the base is a manifold (i.e., the Chern–Weil theory), then the explicit form of the Chern character is ch ⁡ ( V ) = [ tr ⁡ ( exp ⁡ ( i Ω 2 π ) ) ] {\displaystyle \operatorname {ch} (V)=\left[\operatorname {tr} \left(\exp \left({\frac {i\Omega }{2\pi }}\right)\right)\right]} where Ω is the curvature of the connection. The Chern character is useful in part because it facilitates the computation of the Chern class of a tensor product. Specifically, it obeys the following identities: ch ⁡ ( V ⊕ W ) = ch ⁡ ( V ) + ch ⁡ ( W ) {\displaystyle \operatorname {ch} (V\oplus W)=\operatorname {ch} (V)+\operatorname {ch} (W)} ch ⁡ ( V ⊗ W ) = ch ⁡ ( V ) ch ⁡ ( W ) . {\displaystyle \operatorname {ch} (V\otimes W)=\operatorname {ch} (V)\operatorname {ch} (W).} As stated above, using the Grothendieck additivity axiom for Chern classes, the first of these identities can be generalized to state that ch is a homomorphism of abelian groups from the K-theory K(X) into the rational cohomology of X. The second identity establishes the fact that this homomorphism also respects products in K(X), and so ch is a homomorphism of rings. The Chern character is used in the Hirzebruch–Riemann–Roch theorem. === Chern numbers === If we work on an oriented manifold of dimension 2 n {\displaystyle 2n} , then any product of Chern classes of total degree 2 n {\displaystyle 2n} (i.e., the sum of indices of the Chern classes in the product should be n {\displaystyle n} ) can be paired with the orientation homology class (or "integrated over the manifold") to give an integer, a Chern number of the vector bundle. For example, if the manifold has dimension 6, there are three linearly independent Chern numbers, given by c 1 3 {\displaystyle c_{1}^{3}} , c 1 c 2 {\displaystyle c_{1}c_{2}} , and c 3 {\displaystyle c_{3}} . In general, if the manifold has dimension 2 n {\displaystyle 2n} , the number of possible independent Chern numbers is the number of partitions of n {\displaystyle n} . The Chern numbers of the tangent bundle of a complex (or almost complex) manifold are called the Chern numbers of the manifold, and are important invariants. === Generalized cohomology theories === There is a generalization of the theory of Chern classes, where ordinary cohomology is replaced with a generalized cohomology theory. The theories for which such generalization is possible are called complex orientable. The formal properties of the Chern classes remain the same, with one crucial difference: the rule which computes the first Chern class of a tensor product of line bundles in terms of first Chern classes of the factors is not (ordinary) addition, but rather a formal group law. === Algebraic geometry === In algebraic geometry there is a similar theory of Chern classes of vector bundles. There are several variations depending on what groups the Chern classes lie in: For complex varieties the Chern classes can take values in ordinary cohomology, as above. For varieties over general fields, the Chern classes can take values in cohomology theories such as etale cohomology or l-adic cohomology. For varieties V over general fields the Chern classes can also take values in homomorphisms of Chow groups CH(V): for example, the first Chern class of a line bundle over a variety V is a homomorphism from CH(V) to CH(V) reducing degrees by 1. This corresponds to the fact that the Chow groups are a sort of analog of homology groups, and elements of cohomology groups can be thought of as homomorphisms of homology groups using the cap product. === Manifolds with structure === The theory of Chern classes gives rise to cobordism invariants for almost complex manifolds. If M is an almost complex manifold, then its tangent bundle is a complex vector bundle. The Chern classes of M are thus defined to be the Chern classes of its tangent bundle. If M is also compact and of dimension 2d, then each monomial of total degree 2d in the Chern classes can be paired with the fundamental class of M, giving an integer, a Chern number of M. If M′ is another almost complex manifold of the same dimension, then it is cobordant to M if and only if the Chern numbers of M′ coincide with those of M. The theory also extends to real symplectic vector bundles, by the intermediation of compatible almost complex structures. In particular, symplectic manifolds have a well-defined Chern class. === Arithmetic schemes and Diophantine equations === (See Arakelov geometry) == See also == Pontryagin class Stiefel–Whitney class Euler class Segre class Schubert calculus Quantum Hall effect Localized Chern class == Notes == == References == Chern, Shiing-Shen (1946), "Characteristic classes of Hermitian Manifolds", Annals of Mathematics, Second Series, 47 (1): 85–121, doi:10.2307/1969037, ISSN 0003-486X, JSTOR 1969037 Fulton, W. (29 June 2013). Intersection Theory. Springer Science & Business Media. ISBN 978-3-662-02421-8. Grothendieck, Alexander (1958), "La théorie des classes de Chern", Bulletin de la Société Mathématique de France, 86: 137–154, doi:10.24033/bsmf.1501, ISSN 0037-9484, MR 0116023 Hartshorne, Robin (29 June 2013). Algebraic Geometry. Springer Science & Business Media. ISBN 978-1-4757-3849-0. Jost, Jürgen (2005), Riemannian Geometry and Geometric Analysis (4th ed.), Springer-Verlag, ISBN 978-3-540-25907-7 (Provides a very short, introductory review of Chern classes). May, J. Peter (1999), A Concise Course in Algebraic Topology, University of Chicago Press, ISBN 9780226511832 Milnor, John Willard; Stasheff, James D. (1974), Characteristic classes, Annals of Mathematics Studies, vol. 76, Princeton University Press; University of Tokyo Press, ISBN 978-0-691-08122-9 Rubei, Elena (2014), Algebraic Geometry, a concise dictionary, Walter De Gruyter, ISBN 978-3-11-031622-3 == External links == Vector Bundles & K-Theory – A downloadable book-in-progress by Allen Hatcher. Contains a chapter about characteristic classes. Dieter Kotschick, Chern numbers of algebraic varieties
Wikipedia:Chevalley basis#0	In mathematics, a Chevalley basis for a simple complex Lie algebra is a basis constructed by Claude Chevalley with the property that all structure constants are integers. Chevalley used these bases to construct analogues of Lie groups over finite fields, called Chevalley groups. The Chevalley basis is the Cartan-Weyl basis, but with a different normalization. The generators of a Lie group are split into the generators H and E indexed by simple roots and their negatives ± α i {\displaystyle \pm \alpha _{i}} . The Cartan-Weyl basis may be written as [ H i , H j ] = 0 {\displaystyle [H_{i},H_{j}]=0} [ H i , E α ] = α i E α {\displaystyle [H_{i},E_{\alpha }]=\alpha _{i}E_{\alpha }} Defining the dual root or coroot of α {\displaystyle \alpha } as α ∨ = 2 α ( α , α ) {\displaystyle \alpha ^{\vee }={\frac {2\alpha }{(\alpha ,\alpha )}}} where ( ⋅ , ⋅ ) {\displaystyle (\cdot ,\cdot )} is the euclidean inner product. One may perform a change of basis to define H α i = ( α i ∨ , H ) {\displaystyle H_{\alpha _{i}}=(\alpha _{i}^{\vee },H)} The Cartan integers are A i j = ( α i , α j ∨ ) {\displaystyle A_{ij}=(\alpha _{i},\alpha _{j}^{\vee })} The resulting relations among the generators are the following: [ H α i , H α j ] = 0 {\displaystyle [H_{\alpha _{i}},H_{\alpha _{j}}]=0} [ H α i , E α j ] = A j i E α j {\displaystyle [H_{\alpha _{i}},E_{\alpha _{j}}]=A_{ji}E_{\alpha _{j}}} [ E − α i , E α i ] = H α i {\displaystyle [E_{-\alpha _{i}},E_{\alpha _{i}}]=H_{\alpha _{i}}} [ E β , E γ ] = ± ( p + 1 ) E β + γ {\displaystyle [E_{\beta },E_{\gamma }]=\pm (p+1)E_{\beta +\gamma }} where in the last relation p {\displaystyle p} is the greatest positive integer such that γ − p β {\displaystyle \gamma -p\beta } is a root and we consider E β + γ = 0 {\displaystyle E_{\beta +\gamma }=0} if β + γ {\displaystyle \beta +\gamma } is not a root. For determining the sign in the last relation one fixes an ordering of roots which respects addition, i.e., if β ≺ γ {\displaystyle \beta \prec \gamma } then β + α ≺ γ + α {\displaystyle \beta +\alpha \prec \gamma +\alpha } provided that all four are roots. We then call ( β , γ ) {\displaystyle (\beta ,\gamma )} an extraspecial pair of roots if they are both positive and β {\displaystyle \beta } is minimal among all β 0 {\displaystyle \beta _{0}} that occur in pairs of positive roots ( β 0 , γ 0 ) {\displaystyle (\beta _{0},\gamma _{0})} satisfying β 0 + γ 0 = β + γ {\displaystyle \beta _{0}+\gamma _{0}=\beta +\gamma } . The sign in the last relation can be chosen arbitrarily whenever ( β , γ ) {\displaystyle (\beta ,\gamma )} is an extraspecial pair of roots. This then determines the signs for all remaining pairs of roots. == References == Carter, Roger W. (1993). Finite Groups of Lie Type: Conjugacy Classes and Complex Characters. Wiley Classics Library. Chichester: Wiley. ISBN 978-0-471-94109-5. Chevalley, Claude (1955). "Sur certains groupes simples". Tohoku Mathematical Journal (in French). 7 (1–2): 14–66. doi:10.2748/tmj/1178245104. MR 0073602. Zbl 0066.01503. Tits, Jacques (1966). "Sur les constantes de structure et le théorème d'existence des algèbres de Lie semi-simples". Publications Mathématiques de l'IHÉS (in French). 31: 21–58. doi:10.1007/BF02684801. MR 0214638. Zbl 0145.25804.
Wikipedia:Chevalley–Warning theorem#0	In number theory, the Chevalley–Warning theorem implies that certain polynomial equations in sufficiently many variables over a finite field have solutions. It was proved by Ewald Warning (1935) and a slightly weaker form of the theorem, known as Chevalley's theorem, was proved by Chevalley (1935). Chevalley's theorem implied Artin's and Dickson's conjecture that finite fields are quasi-algebraically closed fields (Artin 1982, page x). == Statement of the theorems == Let F {\displaystyle \mathbb {F} } be a finite field and { f j } j = 1 r ⊆ F [ X 1 , … , X n ] {\displaystyle \{f_{j}\}_{j=1}^{r}\subseteq \mathbb {F} [X_{1},\ldots ,X_{n}]} be a set of polynomials such that the number of variables satisfies n > ∑ j = 1 r d j {\displaystyle n>\sum _{j=1}^{r}d_{j}} where d j {\displaystyle d_{j}} is the total degree of f j {\displaystyle f_{j}} . The theorems are statements about the solutions of the following system of polynomial equations f j ( x 1 , … , x n ) = 0 for j = 1 , … , r . {\displaystyle f_{j}(x_{1},\dots ,x_{n})=0\quad {\text{for}}\,j=1,\ldots ,r.} The Chevalley–Warning theorem states that the number of common solutions ( a 1 , … , a n ) ∈ F n {\displaystyle (a_{1},\dots ,a_{n})\in \mathbb {F} ^{n}} is divisible by the characteristic p {\displaystyle p} of F {\displaystyle \mathbb {F} } . Or in other words, the cardinality of the vanishing set of { f j } j = 1 r {\displaystyle \{f_{j}\}_{j=1}^{r}} is 0 {\displaystyle 0} modulo p {\displaystyle p} . The Chevalley theorem states that if the system has the trivial solution ( 0 , … , 0 ) ∈ F n {\displaystyle (0,\dots ,0)\in \mathbb {F} ^{n}} , that is, if the polynomials have no constant terms, then the system also has a non-trivial solution ( a 1 , … , a n ) ∈ F n ∖ { ( 0 , … , 0 ) } {\displaystyle (a_{1},\dots ,a_{n})\in \mathbb {F} ^{n}\backslash \{(0,\dots ,0)\}} . Chevalley's theorem is an immediate consequence of the Chevalley–Warning theorem since p {\displaystyle p} is at least 2. Both theorems are best possible in the sense that, given any n {\displaystyle n} , the list f j = x j , j = 1 , … , n {\displaystyle f_{j}=x_{j},j=1,\dots ,n} has total degree n {\displaystyle n} and only the trivial solution. Alternatively, using just one polynomial, we can take f1 to be the degree n polynomial given by the norm of x1a1 + ... + xnan where the elements a form a basis of the finite field of order pn. Warning proved another theorem, known as Warning's second theorem, which states that if the system of polynomial equations has the trivial solution, then it has at least q n − d {\displaystyle q^{n-d}} solutions where q {\displaystyle q} is the size of the finite field and d := d 1 + ⋯ + d r {\displaystyle d:=d_{1}+\dots +d_{r}} . Chevalley's theorem also follows directly from this. == Proof of Warning's theorem == Remark: If i < q − 1 {\displaystyle i<q-1} then ∑ x ∈ F x i = 0 {\displaystyle \sum _{x\in \mathbb {F} }x^{i}=0} so the sum over F n {\displaystyle \mathbb {F} ^{n}} of any polynomial in x 1 , … , x n {\displaystyle x_{1},\ldots ,x_{n}} of degree less than n ( q − 1 ) {\displaystyle n(q-1)} also vanishes. The total number of common solutions modulo p {\displaystyle p} of f 1 , … , f r = 0 {\displaystyle f_{1},\ldots ,f_{r}=0} is equal to ∑ x ∈ F n ( 1 − f 1 q − 1 ( x ) ) ⋅ … ⋅ ( 1 − f r q − 1 ( x ) ) {\displaystyle \sum _{x\in \mathbb {F} ^{n}}(1-f_{1}^{q-1}(x))\cdot \ldots \cdot (1-f_{r}^{q-1}(x))} because each term is 1 for a solution and 0 otherwise. If the sum of the degrees of the polynomials f i {\displaystyle f_{i}} is less than n then this vanishes by the remark above. == Artin's conjecture == It is a consequence of Chevalley's theorem that finite fields are quasi-algebraically closed. This had been conjectured by Emil Artin in 1935. The motivation behind Artin's conjecture was his observation that quasi-algebraically closed fields have trivial Brauer group, together with the fact that finite fields have trivial Brauer group by Wedderburn's theorem. == The Ax–Katz theorem == The Ax–Katz theorem, named after James Ax and Nicholas Katz, determines more accurately a power q b {\displaystyle q^{b}} of the cardinality q {\displaystyle q} of F {\displaystyle \mathbb {F} } dividing the number of solutions; here, if d {\displaystyle d} is the largest of the d j {\displaystyle d_{j}} , then the exponent b {\displaystyle b} can be taken as the ceiling function of n − ∑ j d j d . {\displaystyle {\frac {n-\sum _{j}d_{j}}{d}}.} The Ax–Katz result has an interpretation in étale cohomology as a divisibility result for the (reciprocals of) the zeroes and poles of the local zeta-function. Namely, the same power of q {\displaystyle q} divides each of these algebraic integers. == See also == Combinatorial Nullstellensatz == References == Artin, Emil (1982), Lang, Serge.; Tate, John (eds.), Collected papers, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90686-7, MR 0671416 Ax, James (1964), "Zeros of polynomials over finite fields", American Journal of Mathematics, 86: 255–261, doi:10.2307/2373163, MR 0160775 Chevalley, Claude (1935), "Démonstration d'une hypothèse de M. Artin", Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg (in French), 11: 73–75, doi:10.1007/BF02940714, JFM 61.1043.01, Zbl 0011.14504 Katz, Nicholas M. (1971), "On a theorem of Ax", Amer. J. Math., 93 (2): 485–499, doi:10.2307/2373389 Warning, Ewald (1935), "Bemerkung zur vorstehenden Arbeit von Herrn Chevalley", Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg (in German), 11: 76–83, doi:10.1007/BF02940715, JFM 61.1043.02, Zbl 0011.14601 Serre, Jean-Pierre (1973), A course in arithmetic, pp. 5–6, ISBN 0-387-90040-3 == External links == "Proofs of the Chevalley-Warning theorem".
Wikipedia:Chi-Wang Shu#0	Chi-Wang Shu (Chinese: 舒其望, born 1 January 1957) is the Theodore B. Stowell University Professor of Applied Mathematics at Brown University. He is known for his research in the fields of computational fluid dynamics, numerical solutions of conservation laws and Hamilton–Jacobi type equations. Shu has been listed as an ISI Highly Cited Author in Mathematics by the ISI Web of Knowledge. == Career == He received his B.S. in Mathematics from the University of Science and Technology of China, Hefei, in 1982 and his Ph.D. in Mathematics from the University of California at Los Angeles in 1986. His Ph.D. thesis advisor was Stanley Osher. He started his academic career in 1987 as an assistant professor in the Division of Applied Mathematics at Brown University. He was an associate professor from 1992 to 1996 and became full professor in 1996. == Honors and awards == He is the 2021 recipient of the John von Neumann Lecture Prize, the highest honor and flagship lecture of Society for Industrial and Applied Mathematics (SIAM). The prize recognizes his fundamental contributions to the numerical solution of partial differential equations: "His work on finite difference essentially non-oscillatory (ENO) methods, weighted ENO (WENO) methods, finite element discontinuous Galerkin methods, and spectral methods has had a major impact on scientific computing." The Association for Women in Mathematics has included him in the 2020 class of AWM Fellows for "his exceptional dedication and contribution to mentoring, supporting, and advancing women in the mathematical sciences; for his incredible role in supervising many women Ph.D.s, bringing them into the world of research to which he has made fundamental contributions, and nurturing their professional success". In 2012 he became a fellow of the American Mathematical Society. In 2009, he was selected as one of the first 183 Fellows of the Society for Industrial and Applied Mathematics (SIAM). SIAM/ACM Prize in Computational Science and Engineering (SIAM/ACM CSE Prize), 2007. He received the prize "for the development of numerical methods that have had a great impact on scientific computing, including TVD temporal discretizations, ENO and WENO finite difference schemes, discontinuous Galerkin methods, and spectral methods." Feng Kang Prize of Scientific Computing by the Chinese Academy of Sciences, 1995 NASA Public Service Group Achievement Award for Pioneering Work in Computational Fluid Dynamics by NASA Langley Research Center, 1992 == References == == External links == Chi-Wang Shu at the Mathematics Genealogy Project
Wikipedia:Chicago movement#0	The Chicago movement was an educational reform initiative in Illinois during the early 20th century that attempted to create a unified mathematics curriculum in secondary schools. The movement represented one of the earliest systematic attempts to integrate different branches of mathematics in American education. == Overview == The Chicago movement emerged in Illinois secondary schools between 1890 and 1930, advocating for an integrated approach to teaching mathematics rather than maintaining traditional divisions between subjects like algebra and geometry. Proponents sought to demonstrate the interconnections between different mathematical topics and their practical applications. The movement reached its peak in Illinois secondary schools during the early 20th century, where it became known specifically as the "Chicago movement" due to its concentration in that area. According to Malaty, the movement faced criticism "because it did not care about the need for continuity of studying each branch, especially geometry, as a structure" and was ultimately considered "just a temporary fashion." The primary concern of the Chicago movement was its attempt to break down the traditional compartmentalization of mathematical subjects in secondary education. Instead of teaching algebra, geometry, and other mathematical branches as separate disciplines, the movement promoted a more unified curriculum that emphasized the relationships between different mathematical concepts. The movement faced opposition from mathematics educators who argued that it undermined the systematic study of individual mathematical disciplines. Critics were particularly concerned about its impact on geometry education, contending that studying each branch as a coherent structure was essential for proper mathematical understanding. The Chicago movement was relatively short-lived, with historians describing it as a "temporary fashion" in mathematics education. However, it raised important questions about mathematics curriculum integration that would resurface during the New Math movement of the 1950s and 1960s. The fundamental question it posed—how to balance integrated mathematical understanding with systematic study of individual branches—remains relevant to contemporary discussions of mathematics education reform. == See also == Computer-based mathematics education Mathematics education Mathematics education in the United States New Math Education reform == References ==
Wikipedia:Chinese Mathematical Society#0	The Chinese Mathematical Society (CMS, Chinese: 中国数学会) is an academic organization for Chinese mathematicians, with the official website www.cms.org.cn. It is a member of China Association of Science and Technology. == History == The Chinese Mathematical Society (CMS) was founded in July 1935 in Shanghai. The inaugural conference was held in the library of Shanghai Jiao Tong University on July 25, and 33 people attended the meeting. Its founding members included Hu Dunfu, Feng Zuxun, Zhou Meiquan, Jiang Lifu, Xiong Qinglai, Chen Jiangong, Gu Deng, Su Buqing, Jiang Zehan, Qian Baozong, and Fu Zhongsun. Hu Dunfu served as its first president. The society published Journal of Chinese Mathematical Society, and a math promoting magazine, Mathematics Magazine. In 1952 and 1953, these two journals was renamed Acta Mathematica Sinica, and Mathematics Letters. The CMS was originally located at the China Science Society at 533 Albert Road (now South Shaanxi Road) in Shanghai. After establishment of the People's Republic of China in 1949, it was moved to the Institute of Mathematics of the Chinese Academy of Sciences in Beijing. Currently, CMS is affiliated to the Academy of Mathematics and Systems Science (AMSS) of the CAS. In the PRC era, the Chinese Mathematical Society held its 1st to 4th national conferences in August 1951, February 1960, November 1978, and October 1983, in the cities of Beijing, Shanghai, Chengdu and Wuhan, respectively. Hua Luogeng was the president of first three conferences. In the 4th conference, Hua Luogeng, Su Buqing, Jiang Zehan, Wu Daren and Ke Zhao were elected honorary presidents. The presidents of 4th to 8th conferences were Wu Wenjun, Wang Yuan, Yang Le, Zhang Gongqing and Ma Zhi-ming, respectively. The 50th anniversary conference of CMS was held in Shanghai in December 1985. Zhou Peiyuan and Zhou Guangzhao attended and delivered speeches. World-renowned mathematicians Shiing-shen Chern and Henri Cartan were also invited. In May 1995, the CMS 7th National Conference & 60 Anniversary Conference was held in Beijing. Zhu Guangya and Lu Yongxiang attended, and Shiing-shen Chern and Shing-Tung Yau were invited to give research talks. The President of the 13th Council of the Chinese Mathematical Society was Gang Tian 田刚, Vice Presidents included Yuguang Shi 史宇光, and Coucil Members included mathematicians Jun Hu 胡俊, Chunwei Song 宋春伟 and Mingyao Ai 艾明要. The Society has over 50,000 members. == References == == External links == Official website
Wikipedia:Chinese hypothesis#0	In number theory, the Chinese hypothesis is a disproven conjecture stating that an integer n is prime if and only if it satisfies the condition that 2 n − 2 {\displaystyle 2^{n}-2} is divisible by n—in other words, that an integer n is prime if and only if 2 n ≡ 2 mod n {\displaystyle 2^{n}\equiv 2{\bmod {n}}} . It is true that if n is prime, then 2 n ≡ 2 mod n {\displaystyle 2^{n}\equiv 2{\bmod {n}}} (this is a special case of Fermat's little theorem), however the converse (if 2 n ≡ 2 mod n {\displaystyle 2^{n}\equiv 2{\bmod {n}}} then n is prime) is false, and therefore the hypothesis as a whole is false. The smallest counterexample is n = 341 = 11×31. Composite numbers n for which 2 n − 2 {\displaystyle 2^{n}-2} is divisible by n are called Poulet numbers. They are a special class of Fermat pseudoprimes. == History == Once, and sometimes still, mistakenly thought to be of ancient Chinese origin, the Chinese hypothesis actually originates in the mid-19th century from the work of Qing dynasty mathematician Li Shanlan (1811–1882). He was later made aware his statement was incorrect and removed it from his subsequent work but it was not enough to prevent the false proposition from appearing elsewhere under his name; a later mistranslation in the 1898 work of Jeans dated the conjecture to Confucian times and gave birth to the ancient origin myth. == References == === Bibliography === Dickson, Leonard Eugene (2005), History of the Theory of Numbers, Vol. 1: Divisibility and Primality, New York: Dover, ISBN 0-486-44232-2 Erdős, Paul (1949), "On the Converse of Fermat's Theorem", American Mathematical Monthly, 56 (9): 623–624, doi:10.2307/2304732, JSTOR 2304732 Honsberger, Ross (1973), "An Old Chinese Theorem and Pierre de Fermat", Mathematical Gems, vol. I, Washington, DC: Math. Assoc. Amer., pp. 1–9 Jeans, James H. (1898), "The converse of Fermat's theorem", Messenger of Mathematics, 27: 174 Needham, Joseph (1959), "Ch. 19", Science and Civilisation in China, Vol. 3: Mathematics and the Sciences of the Heavens and the Earth, Cambridge, England: Cambridge University Press Han Qi (1991), Transmission of Western Mathematics during the Kangxi Kingdom and Its Influence Over Chinese Mathematics, Beijing: Ph.D. thesis Ribenboim, Paulo (1996), The New Book of Prime Number Records, New York: Springer-Verlag, pp. 103–105, ISBN 0-387-94457-5 Shanks, Daniel (1993), Solved and Unsolved Problems in Number Theory (4th ed.), New York: Chelsea, pp. 19–20, ISBN 0-8284-1297-9 Li Yan; Du Shiran (1987), Chinese Mathematics: A Concise History, Translated by John N. Crossley and Anthony W.-C. Lun, Oxford, England: Clarendon Press, ISBN 0-19-858181-5
Wikipedia:Chinese mathematics#0	Mathematics emerged independently in China by the 11th century BCE. The Chinese independently developed a real number system that includes significantly large and negative numbers, more than one numeral system (binary and decimal), algebra, geometry, number theory and trigonometry. Since the Han dynasty, as diophantine approximation being a prominent numerical method, the Chinese made substantial progress on polynomial evaluation. Algorithms like regula falsi and expressions like simple continued fractions are widely used and have been well-documented ever since. They deliberately find the principal nth root of positive numbers and the roots of equations. The major texts from the period, The Nine Chapters on the Mathematical Art and the Book on Numbers and Computation gave detailed processes for solving various mathematical problems in daily life. All procedures were computed using a counting board in both texts, and they included inverse elements as well as Euclidean divisions. The texts provide procedures similar to that of Gaussian elimination and Horner's method for linear algebra. The achievement of Chinese algebra reached a zenith in the 13th century during the Yuan dynasty with the development of tian yuan shu. As a result of obvious linguistic and geographic barriers, as well as content, Chinese mathematics and the mathematics of the ancient Mediterranean world are presumed to have developed more or less independently up to the time when The Nine Chapters on the Mathematical Art reached its final form, while the Book on Numbers and Computation and Huainanzi are roughly contemporary with classical Greek mathematics. Some exchange of ideas across Asia through known cultural exchanges from at least Roman times is likely. Frequently, elements of the mathematics of early societies correspond to rudimentary results found later in branches of modern mathematics such as geometry or number theory. The Pythagorean theorem for example, has been attested to the time of the Duke of Zhou. Knowledge of Pascal's triangle has also been shown to have existed in China centuries before Pascal, such as the Song-era polymath Shen Kuo. == Pre-imperial era == Shang dynasty (1600–1050 BC). One of the oldest surviving mathematical works is the I Ching, which greatly influenced written literature during the Zhou dynasty (1050–256 BC). For mathematics, the book included a sophisticated use of hexagrams. Leibniz pointed out, the I Ching (Yi Jing) contained elements of binary numbers. Since the Shang period, the Chinese had already fully developed a decimal system. Since early times, Chinese understood basic arithmetic (which dominated far eastern history), algebra, equations, and negative numbers with counting rods. Although the Chinese were more focused on arithmetic and advanced algebra for astronomical uses, they were also the first to develop negative numbers, algebraic geometry, and the usage of decimals. Math was one of the Six Arts students were required to master during the Zhou dynasty (1122–256 BCE). Learning them all perfectly was required to be a perfect gentleman, comparable to the concept of a "renaissance man". Six Arts have their roots in the Confucian philosophy. The oldest existent work on geometry in China comes from the philosophical Mohist canon c. 330 BCE, compiled by the followers of Mozi (470–390 BCE). The Mo Jing described various aspects of many fields associated with physical science, and provided a small wealth of information on mathematics as well. It provided an 'atomic' definition of the geometric point, stating that a line is separated into parts, and the part which has no remaining parts (i.e. cannot be divided into smaller parts) and thus forms the extreme end of a line is a point. Much like Euclid's first and third definitions and Plato's 'beginning of a line', the Mo Jing stated that "a point may stand at the end (of a line) or at its beginning like a head-presentation in childbirth. (As to its invisibility) there is nothing similar to it." Similar to the atomists of Democritus, the Mo Jing stated that a point is the smallest unit, and cannot be cut in half, since 'nothing' cannot be halved." It stated that two lines of equal length will always finish at the same place," while providing definitions for the comparison of lengths and for parallels," along with principles of space and bounded space. It also described the fact that planes without the quality of thickness cannot be piled up since they cannot mutually touch. The book provided word recognition for circumference, diameter, and radius, along with the definition of volume. The history of mathematical development lacks some evidence. There are still debates about certain mathematical classics. For example, the Zhoubi Suanjing dates around 1200–1000 BC, yet many scholars believed it was written between 300 and 250 BCE. The Zhoubi Suanjing contains an in-depth proof of the Gougu Theorem (a special case of the Pythagorean theorem), but focuses more on astronomical calculations. However, the recent archaeological discovery of the Tsinghua Bamboo Slips, dated c. 305 BCE, has revealed some aspects of pre-Qin mathematics, such as the first known decimal multiplication table. The abacus was first mentioned in the second century BC, alongside 'calculation with rods' (suan zi) in which small bamboo sticks are placed in successive squares of a checkerboard. == Qin dynasty == Not much is known about Qin dynasty mathematics, or before, due to the burning of books and burying of scholars, circa 213–210 BC. Knowledge of this period can be determined from civil projects and historical evidence. The Qin dynasty created a standard system of weights. Civil projects of the Qin dynasty were significant feats of human engineering. Emperor Qin Shi Huang ordered many men to build large, life-sized statues for the palace tomb along with other temples and shrines, and the shape of the tomb was designed with geometric skills of architecture. It is certain that one of the greatest feats of human history, the Great Wall of China, required many mathematical techniques. All Qin dynasty buildings and grand projects used advanced computation formulas for volume, area and proportion. Qin bamboo cash purchased at the antiquarian market of Hong Kong by the Yuelu Academy, according to the preliminary reports, contains the earliest epigraphic sample of a mathematical treatise. == Han dynasty == In the Han dynasty, numbers were developed into a place value decimal system and used on a counting board with a set of counting rods called rod calculus, consisting of only nine symbols with a blank space on the counting board representing zero. Negative numbers and fractions were also incorporated into solutions of the great mathematical texts of the period. The mathematical texts of the time, the Book on Numbers and Computation and Jiuzhang suanshu solved basic arithmetic problems such as addition, subtraction, multiplication and division. Furthermore, they gave the processes for square and cubed root extraction, which eventually was applied to solving quadratic equations up to the third order. Both texts also made substantial progress in Linear Algebra, namely solving systems of equations with multiple unknowns. The value of pi is taken to be equal to three in both texts. However, the mathematicians Liu Xin (d. 23) and Zhang Heng (78–139) gave more accurate approximations for pi than Chinese of previous centuries had used. Mathematics was developed to solve practical problems in the time such as division of land or problems related to division of payment. The Chinese did not focus on theoretical proofs based on geometry or algebra in the modern sense of proving equations to find area or volume. The Book of Computations and The Nine Chapters on the Mathematical Art provide numerous practical examples that would be used in daily life. === Book on Numbers and Computation === The Book on Numbers and Computation is approximately seven thousand characters in length, written on 190 bamboo strips. It was discovered together with other writings in 1984 when archaeologists opened a tomb at Zhangjiashan in Hubei province. From documentary evidence this tomb is known to have been closed in 186 BC, early in the Western Han dynasty. While its relationship to the Nine Chapters is still under discussion by scholars, some of its contents are clearly paralleled there. The text of the Suan shu shu is however much less systematic than the Nine Chapters, and appears to consist of a number of more or less independent short sections of text drawn from a number of sources. The Book of Computations contains many perquisites to problems that would be expanded upon in The Nine Chapters on the Mathematical Art. An example of the elementary mathematics in the Suàn shù shū, the square root is approximated by using false position method which says to "combine the excess and deficiency as the divisor; (taking) the deficiency numerator multiplied by the excess denominator and the excess numerator times the deficiency denominator, combine them as the dividend." Furthermore, The Book of Computations solves systems of two equations and two unknowns using the same false position method. === The Nine Chapters on the Mathematical Art === The Nine Chapters on the Mathematical Art dates archeologically to 179 CE, though it is traditionally dated to 1000 BCE, but it was written perhaps as early as 300–200 BCE. Although the author(s) are unknown, they made a major contribution in the eastern world. Problems are set up with questions immediately followed by answers and procedure. There are no formal mathematical proofs within the text, just a step-by-step procedure. The commentary of Liu Hui provided geometrical and algebraic proofs to the problems given within the text. The Nine Chapters on the Mathematical Art was one of the most influential of all Chinese mathematical books and it is composed of 246 problems. It was later incorporated into The Ten Computational Canons, which became the core of mathematical education in later centuries. This book includes 246 problems on surveying, agriculture, partnerships, engineering, taxation, calculation, the solution of equations, and the properties of right triangles. The Nine Chapters made significant additions to solving quadratic equations in a way similar to Horner's method. It also made advanced contributions to fangcheng, or what is now known as linear algebra. Chapter seven solves system of linear equations with two unknowns using the false position method, similar to The Book of Computations. Chapter eight deals with solving determinate and indeterminate simultaneous linear equations using positive and negative numbers, with one problem dealing with solving four equations in five unknowns. The Nine Chapters solves systems of equations using methods similar to the modern Gaussian elimination and back substitution. The version of The Nine Chapters that has served as the foundation for modern renditions was a result of the efforts of the scholar Dai Zhen. Transcribing the problems directly from Yongle Encyclopedia, he then proceeded to make revisions to the original text, along with the inclusion his own notes explaining his reasoning behind the alterations. His finished work would be first published in 1774, but a new revision would be published in 1776 to correct various errors as well as include a version of The Nine Chapters from the Southern Song that contained the commentaries of Lui Hui and Li Chunfeng. The final version of Dai Zhen's work would come in 1777, titled Ripple Pavilion, with this final rendition being widely distributed and coming to serve as the standard for modern versions of The Nine Chapters. However, this version has come under scrutiny from Guo Shuchen, alleging that the edited version still contains numerous errors and that not all of the original amendments were done by Dai Zhen himself. === Calculation of pi === Problems in The Nine Chapters on the Mathematical Art take pi to be equal to three in calculating problems related to circles and spheres, such as spherical surface area. There is no explicit formula given within the text for the calculation of pi to be three, but it is used throughout the problems of both The Nine Chapters on the Mathematical Art and the Artificer's Record, which was produced in the same time period. Historians believe that this figure of pi was calculated using the 3:1 relationship between the circumference and diameter of a circle. Some Han mathematicians attempted to improve this number, such as Liu Xin, who is believed to have estimated pi to be 3.154. Later, Liu Hui attempted to improve the calculation by calculating pi to be 3.141024. Liu calculated this number by using polygons inside a hexagon as a lower limit compared to a circle. Zu Chongzhi later discovered the calculation of pi to be 3.1415926 < π < 3.1415927 by using polygons with 24,576 sides. This calculation would be discovered in Europe during the 16th century. There is no explicit method or record of how he calculated this estimate. === Division and root extraction === Basic arithmetic processes such as addition, subtraction, multiplication and division were present before the Han dynasty. The Nine Chapters on the Mathematical Art take these basic operations for granted and simply instruct the reader to perform them. Han mathematicians calculated square and cube roots in a similar manner as division, and problems on division and root extraction both occur in Chapter Four of The Nine Chapters on the Mathematical Art. Calculating the square and cube roots of numbers is done through successive approximation, the same as division, and often uses similar terms such as dividend (shi) and divisor (fa) throughout the process. This process of successive approximation was then extended to solving quadratics of the second and third order, such as x 2 + a = b {\displaystyle x^{2}+a=b} , using a method similar to Horner's method. The method was not extended to solve quadratics of the nth order during the Han dynasty; however, this method was eventually used to solve these equations. === Linear algebra === The Book of Computations is the first known text to solve systems of equations with two unknowns. There are a total of three sets of problems within The Book of Computations involving solving systems of equations with the false position method, which again are put into practical terms. Chapter Seven of The Nine Chapters on the Mathematical Art also deals with solving a system of two equations with two unknowns with the false position method. To solve for the greater of the two unknowns, the false position method instructs the reader to cross-multiply the minor terms or zi (which are the values given for the excess and deficit) with the major terms mu. To solve for the lesser of the two unknowns, simply add the minor terms together. Chapter Eight of The Nine Chapters on the Mathematical Art deals with solving infinite equations with infinite unknowns. This process is referred to as the "fangcheng procedure" throughout the chapter. Many historians chose to leave the term fangcheng untranslated due to conflicting evidence of what the term means. Many historians translate the word to linear algebra today. In this chapter, the process of Gaussian elimination and back-substitution are used to solve systems of equations with many unknowns. Problems were done on a counting board and included the use of negative numbers as well as fractions. The counting board was effectively a matrix, where the top line is the first variable of one equation and the bottom was the last. === Liu Hui's commentary on The Nine Chapters on the Mathematical Art === Liu Hui's commentary on The Nine Chapters on the Mathematical Art is the earliest edition of the original text available. Hui is believed by most to be a mathematician shortly after the Han dynasty. Within his commentary, Hui qualified and proved some of the problems from either an algebraic or geometrical standpoint. For instance, throughout The Nine Chapters on the Mathematical Art, the value of pi is taken to be equal to three in problems regarding circles or spheres. In his commentary, Liu Hui finds a more accurate estimation of pi using the method of exhaustion. The method involves creating successive polygons within a circle so that eventually the area of a higher-order polygon will be identical to that of the circle. From this method, Liu Hui asserted that the value of pi is about 3.14. Liu Hui also presented a geometric proof of square and cubed root extraction similar to the Greek method, which involved cutting a square or cube in any line or section and determining the square root through symmetry of the remaining rectangles. == Three Kingdoms, Jin, and Sixteen Kingdoms == In the third century Liu Hui wrote his commentary on the Nine Chapters and also wrote Haidao Suanjing which dealt with using Pythagorean theorem (already known by the 9 chapters), and triple, quadruple triangulation for surveying; his accomplishment in the mathematical surveying exceeded those accomplished in the west by a millennium. He was the first Chinese mathematician to calculate π=3.1416 with his π algorithm. He discovered the usage of Cavalieri's principle to find an accurate formula for the volume of a cylinder, and also developed elements of the infinitesimal calculus during the 3rd century CE. In the fourth century, another influential mathematician named Zu Chongzhi, introduced the Da Ming Li. This calendar was specifically calculated to predict many cosmological cycles that will occur in a period of time. Very little is really known about his life. Today, the only sources are found in Book of Sui, we now know that Zu Chongzhi was one of the generations of mathematicians. He used Liu Hui's pi-algorithm applied to a 12288-gon and obtained a value of pi to 7 accurate decimal places (between 3.1415926 and 3.1415927), which would remain the most accurate approximation of π available for the next 900 years. He also applied He Chengtian's interpolation for approximating irrational number with fraction in his astronomy and mathematical works, he obtained 355 113 {\displaystyle {\tfrac {355}{113}}} as a good fraction approximate for pi; Yoshio Mikami commented that neither the Greeks, nor the Hindus nor Arabs knew about this fraction approximation to pi, not until the Dutch mathematician Adrian Anthoniszoom rediscovered it in 1585, "the Chinese had therefore been possessed of this the most extraordinary of all fractional values over a whole millennium earlier than Europe". Along with his son, Zu Geng, Zu Chongzhi applied the Cavalieri's principle to find an accurate solution for calculating the volume of the sphere. Besides containing formulas for the volume of the sphere, his book also included formulas of cubic equations and the accurate value of pi. His work, Zhui Shu was discarded out of the syllabus of mathematics during the Song dynasty and lost. Many believed that Zhui Shu contains the formulas and methods for linear, matrix algebra, algorithm for calculating the value of π, formula for the volume of the sphere. The text should also associate with his astronomical methods of interpolation, which would contain knowledge, similar to our modern mathematics. A mathematical manual called Sunzi mathematical classic dated between 200 and 400 CE contained the most detailed step by step description of multiplication and division algorithm with counting rods. Intriguingly, Sunzi may have influenced the development of place-value systems and place-value systems and the associated Galley division in the West. European sources learned place-value techniques in the 13th century, from a Latin translation an early-9th-century work by Al-Khwarizmi. Khwarizmi's presentation is almost identical to the division algorithm in Sunzi, even regarding stylistic matters (for example, using blank spaces to represent trailing zeros); the similarity suggests that the results may not have been an independent discovery. Islamic commentators on Al-Khwarizmi's work believed that it primarily summarized Hindu knowledge; Al-Khwarizmi's failure to cite his sources makes it difficult to determine whether those sources had in turn learned the procedure from China. In the fifth century the manual called "Zhang Qiujian suanjing" discussed linear and quadratic equations. By this point the Chinese had the concept of negative numbers. == Tang dynasty == By the Tang dynasty study of mathematics was fairly standard in the great schools. The Ten Computational Canons was a collection of ten Chinese mathematical works, compiled by early Tang dynasty mathematician Li Chunfeng (李淳風 602–670), as the official mathematical texts for imperial examinations in mathematics. The Sui dynasty and Tang dynasty ran the "School of Computations". Wang Xiaotong was a great mathematician in the beginning of the Tang dynasty, and he wrote a book: Jigu Suanjing (Continuation of Ancient Mathematics), where numerical solutions which general cubic equations appear for the first time. The Tibetans obtained their first knowledge of mathematics (arithmetic) from China during the reign of Nam-ri srong btsan, who died in 630. The table of sines by the Indian mathematician, Aryabhata, were translated into the Chinese mathematical book of the Kaiyuan Zhanjing, compiled in 718 AD during the Tang dynasty. Although the Chinese excelled in other fields of mathematics such as solid geometry, binomial theorem, and complex algebraic formulas, early forms of trigonometry were not as widely appreciated as in the contemporary Indian and Islamic mathematics. Yi Xing, the mathematician and Buddhist monk was credited for calculating the tangent table. Instead, the early Chinese used an empirical substitute known as chong cha, while practical use of plane trigonometry in using the sine, the tangent, and the secant were known. Yi Xing was famed for his genius, and was known to have calculated the number of possible positions on a go board game (though without a symbol for zero he had difficulties expressing the number). == Song and Yuan dynasties == Northern Song dynasty mathematician Jia Xian developed an additive multiplicative method for extraction of square root and cubic root which implemented the "Horner" rule. Four outstanding mathematicians arose during the Song dynasty and Yuan dynasty, particularly in the twelfth and thirteenth centuries: Yang Hui, Qin Jiushao, Li Zhi (Li Ye), and Zhu Shijie. Yang Hui, Qin Jiushao, Zhu Shijie all used the Horner-Ruffini method six hundred years earlier to solve certain types of simultaneous equations, roots, quadratic, cubic, and quartic equations. Yang Hui was also the first person in history to discover and prove "Pascal's Triangle", along with its binomial proof (although the earliest mention of the Pascal's triangle in China exists before the eleventh century AD). Li Zhi on the other hand, investigated on a form of algebraic geometry based on tiān yuán shù. His book; Ceyuan haijing revolutionized the idea of inscribing a circle into triangles, by turning this geometry problem by algebra instead of the traditional method of using Pythagorean theorem. Guo Shoujing of this era also worked on spherical trigonometry for precise astronomical calculations. At this point of mathematical history, a lot of modern western mathematics were already discovered by Chinese mathematicians. Things grew quiet for a time until the thirteenth century Renaissance of Chinese math. This saw Chinese mathematicians solving equations with methods Europe would not know until the eighteenth century. The high point of this era came with Zhu Shijie's two books Suanxue qimeng and the Jade Mirror of the Four Unknowns. In one case he reportedly gave a method equivalent to Gauss's pivotal condensation. Qin Jiushao (c. 1202 – 1261) was the first to introduce the zero symbol into Chinese mathematics." Before this innovation, blank spaces were used instead of zeros in the system of counting rods. One of the most important contribution of Qin Jiushao was his method of solving high order numerical equations. Referring to Qin's solution of a 4th order equation, Yoshio Mikami put it: "Who can deny the fact of Horner's illustrious process being used in China at least nearly six long centuries earlier than in Europe?" Qin also solved a 10th order equation. Pascal's triangle was first illustrated in China by Yang Hui in his book Xiangjie Jiuzhang Suanfa (詳解九章算法), although it was described earlier around 1100 by Jia Xian. Although the Introduction to Computational Studies (算學啓蒙) written by Zhu Shijie (fl. 13th century) in 1299 contained nothing new in Chinese algebra, it had a great impact on the development of Japanese mathematics. === Algebra === ==== Ceyuan haijing ==== Ceyuan haijing (Chinese: 測圓海鏡; pinyin: Cèyuán Hǎijìng), or Sea-Mirror of the Circle Measurements, is a collection of 692 formula and 170 problems related to inscribed circle in a triangle, written by Li Zhi (or Li Ye) (1192–1272 AD). He used Tian yuan shu to convert intricated geometry problems into pure algebra problems. He then used fan fa, or Horner's method, to solve equations of degree as high as six, although he did not describe his method of solving equations. "Li Chih (or Li Yeh, 1192–1279), a mathematician of Peking who was offered a government post by Khublai Khan in 1206, but politely found an excuse to decline it. His Ts'e-yuan hai-ching (Sea-Mirror of the Circle Measurements) includes 170 problems dealing with[...]some of the problems leading to polynomial equations of sixth degree. Although he did not describe his method of solution of equations, it appears that it was not very different from that used by Chu Shih-chieh and Horner. Others who used the Horner method were Ch'in Chiu-shao (ca. 1202 – ca.1261) and Yang Hui (fl. ca. 1261–1275). ==== Jade Mirror of the Four Unknowns ==== The Jade Mirror of the Four Unknowns was written by Zhu Shijie in 1303 AD and marks the peak in the development of Chinese algebra. The four elements, called heaven, earth, man and matter, represented the four unknown quantities in his algebraic equations. It deals with simultaneous equations and with equations of degrees as high as fourteen. The author uses the method of fan fa, today called Horner's method, to solve these equations. There are many summation series equations given without proof in the Mirror. A few of the summation series are: 1 2 + 2 2 + 3 2 + ⋯ + n 2 = n ( n + 1 ) ( 2 n + 1 ) 3 ! {\displaystyle 1^{2}+2^{2}+3^{2}+\cdots +n^{2}={n(n+1)(2n+1) \over 3!}} 1 + 8 + 30 + 80 + ⋯ + n 2 ( n + 1 ) ( n + 2 ) 3 ! = n ( n + 1 ) ( n + 2 ) ( n + 3 ) ( 4 n + 1 ) 5 ! {\displaystyle 1+8+30+80+\cdots +{n^{2}(n+1)(n+2) \over 3!}={n(n+1)(n+2)(n+3)(4n+1) \over 5!}} ==== Mathematical Treatise in Nine Sections ==== The Mathematical Treatise in Nine Sections, was written by the wealthy governor and minister Ch'in Chiu-shao (c. 1202 – c. 1261) and with the invention of a method of solving simultaneous congruences, it marks the high point in Chinese indeterminate analysis. ==== Magic squares and magic circles ==== The earliest known magic squares of order greater than three are attributed to Yang Hui (fl. ca. 1261–1275), who worked with magic squares of order as high as ten. "The same "Horner" device was used by Yang Hui, about whose life almost nothing is known and who work has survived only in part. Among his contributions that are extant are the earliest Chinese magic squares of order greater than three, including two each of orders four through eight and one each of orders nine and ten." He also worked with magic circle. === Trigonometry === The embryonic state of trigonometry in China slowly began to change and advance during the Song dynasty (960–1279), where Chinese mathematicians began to express greater emphasis for the need of spherical trigonometry in calendar science and astronomical calculations. The polymath and official Shen Kuo (1031–1095) used trigonometric functions to solve mathematical problems of chords and arcs. Joseph W. Dauben notes that in Shen's "technique of intersecting circles" formula, he creates an approximation of the arc of a circle s by s = c + 2v2/d, where d is the diameter, v is the versine, c is the length of the chord c subtending the arc. Sal Restivo writes that Shen's work in the lengths of arcs of circles provided the basis for spherical trigonometry developed in the 13th century by the mathematician and astronomer Guo Shoujing (1231–1316). Gauchet and Needham state Guo used spherical trigonometry in his calculations to improve the Chinese calendar and astronomy. Along with a later 17th-century Chinese illustration of Guo's mathematical proofs, Needham writes: Guo used a quadrangular spherical pyramid, the basal quadrilateral of which consisted of one equatorial and one ecliptic arc, together with two meridian arcs, one of which passed through the summer solstice point...By such methods he was able to obtain the du lü (degrees of equator corresponding to degrees of ecliptic), the ji cha (values of chords for given ecliptic arcs), and the cha lü (difference between chords of arcs differing by 1 degree). Despite the achievements of Shen and Guo's work in trigonometry, another substantial work in Chinese trigonometry would not be published again until 1607, with the dual publication of Euclid's Elements by Chinese official and astronomer Xu Guangqi (1562–1633) and the Italian Jesuit Matteo Ricci (1552–1610). == Ming dynasty == After the overthrow of the Yuan dynasty, China became suspicious of Mongol-favored knowledge. The court turned away from math and physics in favor of botany and pharmacology. Imperial examinations included little mathematics, and what little they included ignored recent developments. Martzloff writes: At the end of the 16th century, Chinese autochthonous mathematics known by the Chinese themselves amounted to almost nothing, little more than calculation on the abacus, whilst in the 17th and 18th centuries nothing could be paralleled with the revolutionary progress in the theatre of European science. Moreover, at this same period, no one could report what had taken place in the more distant past, since the Chinese themselves only had a fragmentary knowledge of that. One should not forget that, in China itself, autochthonous mathematics was not rediscovered on a large scale prior to the last quarter of the 18th century. Correspondingly, scholars paid less attention to mathematics; preeminent mathematicians such as Gu Yingxiang and Tang Shunzhi appear to have been ignorant of the 'increase multiply' method. Without oral interlocutors to explicate them, the texts rapidly became incomprehensible; worse yet, most problems could be solved with more elementary methods. To the average scholar, then, tianyuan seemed numerology. When Wu Jing collated all the mathematical works of previous dynasties into The Annotations of Calculations in the Nine Chapters on the Mathematical Art, he omitted Tian yuan shu and the increase multiply method. Instead, mathematical progress became focused on computational tools. In 15 century, abacus came into its suan pan form. Easy to use and carry, both fast and accurate, it rapidly overtook rod calculus as the preferred form of computation. Zhusuan, the arithmetic calculation through abacus, inspired multiple new works. Suanfa Tongzong (General Source of Computational Methods), a 17-volume work published in 1592 by Cheng Dawei, remained in use for over 300 years. Zhu Zaiyu, Prince of Zheng used 81 position abacus to calculate the square root and cubic root of 2 to 25 figure accuracy, a precision that enabled his development of the equal-temperament system. In the late 16th century, Matteo Ricci decided to published Western scientific works in order to establish a position at the Imperial Court. With the assistance of Xu Guangqi, he was able to translate Euclid's Elements using the same techniques used to teach classical Buddhist texts. Other missionaries followed in his example, translating Western works on special functions (trigonometry and logarithms) that were neglected in the Chinese tradition. However, contemporary scholars found the emphasis on proofs — as opposed to solved problems — baffling, and most continued to work from classical texts alone. == Qing dynasty == Under the Kangxi Emperor, who learned Western mathematics from the Jesuits and was open to outside knowledge and ideas, Chinese mathematics enjoyed a brief period of official support. At Kangxi's direction, Mei Goucheng and three other outstanding mathematicians compiled a 53-volume work titled Shuli Jingyun ("The Essence of Mathematical Study") which was printed in 1723, and gave a systematic introduction to western mathematical knowledge. At the same time, Mei Goucheng also developed to Meishi Congshu Jiyang [The Compiled works of Mei]. Meishi Congshu Jiyang was an encyclopedic summary of nearly all schools of Chinese mathematics at that time, but it also included the cross-cultural works of Mei Wending (1633–1721), Goucheng's grandfather. The enterprise sought to alleviate the difficulties for Chinese mathematicians working on Western mathematics in tracking down citations. In 1773, the Qianlong Emperor decided to compile the Complete Library of the Four Treasuries (or Siku Quanshu). Dai Zhen (1724–1777) selected and proofread The Nine Chapters on the Mathematical Art from Yongle Encyclopedia and several other mathematical works from Han and Tang dynasties. The long-missing mathematical works from Song and Yuan dynasties such as Si-yüan yü-jian and Ceyuan haijing were also found and printed, which directly led to a wave of new research. The most annotated works were Jiuzhang suanshu xicaotushuo (The Illustrations of Calculation Process for The Nine Chapters on the Mathematical Art ) contributed by Li Huang and Siyuan yujian xicao (The Detailed Explanation of Si-yuan yu-jian) by Luo Shilin. == Western influences == In 1840, the First Opium War forced China to open its door and look at the outside world, which also led to an influx of western mathematical studies at a rate unrivaled in the previous centuries. In 1852, the Chinese mathematician Li Shanlan and the British missionary Alexander Wylie co-translated the later nine volumes of Elements and 13 volumes on Algebra. With the assistance of Joseph Edkins, more works on astronomy and calculus soon followed. Chinese scholars were initially unsure whether to approach the new works: was study of Western knowledge a form of submission to foreign invaders? But by the end of the century, it became clear that China could only begin to recover its sovereignty by incorporating Western works. Chinese scholars, taught in Western missionary schools, from (translated) Western texts, rapidly lost touch with the indigenous tradition. Those who were self-trained or in traditionalist circles nevertheless continued to work within the traditional framework of algorithmic mathematics without resorting to Western symbolism. Yet, as Martzloff notes, "from 1911 onwards, solely Western mathematics has been practised in China." === In modern China === Chinese mathematics experienced a great surge of revival following the establishment of a modern Chinese republic in 1912. Ever since then, modern Chinese mathematicians have made numerous achievements in various mathematical fields. Some famous modern ethnic Chinese mathematicians include: Shiing-Shen Chern was widely regarded as a leader in geometry and one of the greatest mathematicians of the 20th century and was awarded the Wolf Prize for his contributions to mathematics. Ky Fan made contributions to fixed point theory, in addition to influencing nonlinear functional analysis, which have found wide application in mathematical economics and game theory, potential theory, calculus of variations, and differential equations. Shing-Tung Yau, a Fields Medal laureate, has influenced both physics and mathematics, and he has been active at the interface between geometry and theoretical physics and subsequently awarded the for his contributions. Terence Tao, a Fields Medal laureate and child prodigy of Chinese heritage, was the youngest participant in the history of the International Mathematical Olympiad at the age of 10, winning a bronze, silver, and gold medal. He remains the youngest winner of each of the three medals in the Olympiad's history. Yitang Zhang, a number theorist who established the first finite bound on gaps between prime numbers. Chen Jingrun, a number theorist who proved that every sufficiently large even number can be written as the sum of either two primes, or a prime and a semiprime, which is now called Chen's theorem. His work was important for research of Goldbach's conjecture. == People's Republic of China == In 1949, at the beginning of the founding of the People's Republic of China, the government paid great attention to the cause of science although the country was in a predicament of lack of funds. The Chinese Academy of Sciences was established in November 1949. The Institute of Mathematics was formally established in July 1952. Then, the Chinese Mathematical Society and its founding journals restored and added other special journals. In the 18 years after 1949, the number of published papers accounted for more than three times the total number of articles before 1949. Many of them not only filled the gaps in China's past, but also reached the world's advanced level. During the chaos of the Cultural Revolution, the sciences declined. In the field of mathematics, in addition to Chen Jingrun, Hua Luogeng, Zhang Guanghou and other mathematicians struggling to continue their work. After the catastrophe, with the publication of Guo Moruo's literary "Spring of Science", Chinese sciences and mathematics experienced a revival. In 1977, a new mathematical development plan was formulated in Beijing, the work of the mathematics society was resumed, the journal was re-published, the academic journal was published, the mathematics education was strengthened, and basic theoretical research was strengthened. An important mathematical achievement of the Chinese mathematician in the direction of the power system is how Xia Zhihong proved the Painleve conjecture in 1988. When there are some initial states of N celestial bodies, one of the celestial bodies ran to infinity or speed in a limited time. Infinity is reached, that is, there are non-collision singularities. The Painleve conjecture is an important conjecture in the field of power systems proposed in 1895. A very important recent development for the 4-body problem is that Xue Jinxin and Dolgopyat proved a non-collision singularity in a simplified version of the 4-body system around 2013. In addition, in 2007, Shen Weixiao and Kozlovski, Van-Strien proved the Real Fatou conjecture: Real hyperbolic polynomials are dense in the space of real polynomials with fixed degree. This conjecture can be traced back to Fatou in the 1920s, and later Smale posed it in the 1960s. The proof of Real Fatou conjecture is one of the most important developments in conformal dynamics in the past decade. === IMO performance === In comparison to other participating countries at the International Mathematical Olympiad, China has highest team scores and has won the all-members-gold IMO with a full team the most number of times. == In education == The first reference to a book being used in learning mathematics in China is dated to the second century CE (Hou Hanshu: 24, 862; 35,1207). Ma Xu, who is a youth c. 110, and Zheng Xuan (127–200) both studied the Nine Chapters on Mathematical procedures. Christopher Cullen claims that mathematics, in a manner akin to medicine, was taught orally. The stylistics of the Suàn shù shū from Zhangjiashan suggest that the text was assembled from various sources and then underwent codification. == See also == Chinese astronomy History of mathematics Indian mathematics Islamic mathematics Japanese mathematics List of Chinese discoveries List of Chinese mathematicians Numbers in Chinese culture == References == === Citations === === Works cited === == External links == Early mathematics texts (Chinese) - Chinese Text Project Overview of Chinese mathematics Chinese Mathematics Through the Han Dynasty Primer of Mathematics by Zhu Shijie
Wikipedia:Chinese multiplication table#0	The Chinese multiplication table is the first requisite for using the Rod calculus for carrying out multiplication, division, the extraction of square roots, and the solving of equations based on place value decimal notation. It was known in China as early as the Spring and Autumn period, and survived through the age of the abacus; pupils in elementary school today still must memorise it. The Chinese multiplication table consists of eighty-one terms. It was often called the nine-nine table, or simply nine-nine, because in ancient times, the nine nine table started with 9 × 9: nine nines beget eighty-one, eight nines beget seventy-two ... seven nines beget sixty three, etc. two ones beget two. In the opinion of Wang Guowei, a noted scholar, the nine-nine table probably started with nine because of the "worship of nine" in ancient China; the emperor was considered the "nine five supremacy" in the Book of Change. See also Numbers in Chinese culture § Nine. It is also known as nine-nine song (or poem), as the table consists of eighty-one lines with four or five Chinese characters per lines; this thus created a constant metre and render the multiplication table as a poem. For example, 9 × 9 = 81 would be rendered as "九九八十一", or "nine nine eighty one", with the world for "begets" "得" implied. This makes it easy to learn by heart. A shorter version of the table consists of only forty-five sentences, as terms such as "nine eights beget seventy-two" are identical to "eight nines beget seventy-two" so there is no need to learn them twice. When the abacus replaced the counting rods in the Ming dynasty, many authors on the abacus advocated the use of the full table instead of the shorter one. They claimed that memorising it without needing a moment of thinking makes abacus calculation much faster. The existence of the Chinese multiplication table is evidence of an early positional decimal system: otherwise a much larger multiplication table would be needed with terms beyond 9×9. == The Nine-nine song text in Chinese == It can be read in either row-major or column-major order. == The Nine-nine table in Chinese literature == Many Chinese classics make reference to the nine-nine table: Zhoubi Suanjing: "nine nine eighty one" Guan Zi has sentences of the form "three eights beget twenty four, three sevens beget twenty-one" The Nine Chapters on the Mathematical Art: "Fu Xi invented the art of nine-nine". In Huainanzi, there were eight sentences: "nine nines beget eighty one", "eight nines beget seventy two", all the way to "two nines beget eighteen". A nine-nine table manuscript was discovered in Dun Huang. Xia Houyang's Computational Canons: "To learn the art of multiplication and division, one must understand nine-nine". The Song dynasty author Hong Zhai's Notebooks said: "three threes as nine, three fours as twelve, two eights as sixteen, four fours as sixteen, three nines as twenty seven, four nines as thirty six, six sixes as thirty six, five eights as forty, five nines as forty five, seven nines as sixty three, eight nines as seventy two, nine nines as eighty one". This suggests that the table has begun with the smallest term since the Song dynasty. Song dynasty mathematician Yang Hui's mathematics text book: Suan fa tong bian ben mo, meaning "You must learn nine nine song from one one equals one to nine nine eighty one, in small to large order" Yuan dynasty mathematician Zhu Shijie's Suanxue qimeng (Elementary mathematics): "one one equals one, two by two equals four, one by three equals three, two by three equals six, three by three equals nine, one by four equals four... nine by nine equals eight one" == Archeological artifacts == At the end of the 19th century, archeologists unearthed pieces of written bamboo script from the Han dynasty in Xin Jiang. One such Han dynasty bamboo script, from Liusha, is a remnant of the nine-nine table. It starts with nine: nine nine eighty one, eight nine seventy two, seven nine sixty three, eight eight sixty four, seven eight fifty six, six eight forty eight, ... two two gets four, altogether 1100 Chinese words. In 2002, Chinese archeologists unearthed a written wood script from a two-thousand-year-old site from the Warring States, on which was written: "four eight thirty two, five eight forty, six eight forty eight." This is the earliest artifact of the nine-nine table that has been unearthed, indicating that the nine-nine table, as well as a positional decimal system, had appeared by the Warring States period. Tsinghua Bamboo Slips Calculation Table, is an ancient calculator artifact from the Warring States period in 305 BC. It is included in the "Tsinghua University Collection of Warring States Bamboo Slips (Part IV)," predating the previously discovered Liye Qin Bamboo Slips and Zhangjiashan Han Bamboo Slips nine-nine tables by a century. In 2023, a bamboo slip from the 4th century BC, containing a multiplication formula, was found in a Jingzhou tomb in Hubei Province, China. The formula was deciphered using infrared scanning, revealing calculations such as "five times seven is thirty plus five, four times seven is twenty plus eight, three times seven is twenty plus one." As of December 2023, this represents the earliest discovery of a nine-nine table artifact. The nine-nine table was transmitted to Japan, and appeared in a Japanese primary mathematics book in the 10th century, beginning with 9×9. == References ==
Wikipedia:Chinese numerals#0	Chinese numerals are words and characters used to denote numbers in written Chinese. Today, speakers of Chinese languages use three written numeral systems: the system of Arabic numerals used worldwide, and two indigenous systems. The more familiar indigenous system is based on Chinese characters that correspond to numerals in the spoken language. These may be shared with other languages of the Chinese cultural sphere such as Korean, Japanese, and Vietnamese. Most people and institutions in China primarily use the Arabic or mixed Arabic-Chinese systems for convenience, with traditional Chinese numerals used in finance, mainly for writing amounts on cheques, banknotes, some ceremonial occasions, some boxes, and on commercials. The other indigenous system consists of the Suzhou numerals, or huama, a positional system, the only surviving form of the rod numerals. These were once used by Chinese mathematicians, and later by merchants in Chinese markets, such as those in Hong Kong until the 1990s, but were gradually supplanted by Arabic numerals. == Basic counting in Chinese == The Chinese character numeral system consists of the Chinese characters used by the Chinese written language to write spoken numerals. Similar to spelling-out numbers in English (e.g., "one thousand nine hundred forty-five"), it is not an independent system per se. Since it reflects spoken language, it does not use the positional system as in Arabic numerals, in the same way that spelling out numbers in English does not. === Ordinary numerals === There are characters representing the numbers zero through nine, and other characters representing larger numbers such as tens, hundreds, thousands, ten thousands and hundred millions. There are two sets of characters for Chinese numerals: one for everyday writing, known as xiǎoxiě (小寫; 小写; 'small writing'), and one for use in commercial, accounting or financial contexts, known as dàxiě (大寫; 大写; 'big writing' or 'capital numbers'). The latter were developed by Wu Zetian (fl. 690–705) and were further refined by the Hongwu Emperor (fl. 1328–1398). They arose because the characters used for writing numerals are geometrically simple, so simply using those numerals cannot prevent forgeries in the same way spelling numbers out in English would. A forger could easily change the everyday characters 三十 (30) to 五千 (5000) just by adding a few strokes. That would not be possible when writing using the financial characters 參拾 (30) and 伍仟 (5000). They are also referred to as "banker's numerals" or "anti-fraud numerals". For the same reason, rod numerals were never used in commercial records. 1. ^ Wugniu is a pan-Wu romanization scheme, but the exact romanization depends on the variety. The romanization listed here is specifically for Shanghainese. === Regional usage === == Powers of 10 == === Large numbers === For numbers larger than 10,000, similarly to the long and short scales in the West, there have been four systems in ancient and modern usage. The original one, with unique names for all powers of ten up to the 14th, is ascribed to the Yellow Emperor in the 6th century book by Zhen Luan, Wujing suanshu; 'Arithmetic in Five Classics'. In modern Chinese, only the second system is used, in which the same ancient names are used, but each represents a myriad, 萬; wàn times the previous: In practice, this situation does not lead to ambiguity, with the exception of 兆; zhào, which means 1012 according to the system in common usage throughout the Chinese communities as well as in Japan and Korea, but has also been used for 106 in recent years (especially in mainland China for megabyte). To avoid problems arising from the ambiguity, the PRC government never uses this character in official documents, but uses 万亿; wànyì) or 太; tài; 'tera-' instead. Partly due to this, combinations of 万 and 亿 are often used instead of the larger units of the traditional system as well, for example 亿亿; yìyì instead of 京. The ROC government in Taiwan uses 兆; zhào to mean 1012 in official documents. === Large numbers from Buddhism === Numerals beyond 載 zǎi come from Buddhist texts in Sanskrit, but are mostly found in ancient texts. Some of the following words are still being used today, but may have transferred meanings. === Small numbers === The following are characters used to denote small order of magnitude in Chinese historically. With the introduction of SI units, some of them have been incorporated as SI prefixes, while the rest have fallen into disuse. === Small numbers from Buddhism === === SI prefixes === In the People's Republic of China, the early translation for the SI prefixes in 1981 was different from those used today. The larger (兆, 京, 垓, 秭, 穰) and smaller Chinese numerals (微, 纖, 沙, 塵, 渺) were defined as translation for the SI prefixes as mega, giga, tera, peta, exa, micro, nano, pico, femto, atto, resulting in the creation of yet more values for each numeral. The Republic of China (Taiwan) defined 百萬 as the translation for mega and 兆 as the translation for tera. This translation is widely used in official documents, academic communities, informational industries, etc. However, the civil broadcasting industries sometimes use 兆赫 to represent "megahertz". Today, the governments of both China and Taiwan use phonetic transliterations for the SI prefixes. However, the governments have each chosen different Chinese characters for certain prefixes. The following table lists the two different standards together with the early translation. == Reading and transcribing numbers == === Whole numbers === Multiple-digit numbers are constructed using a multiplicative principle; first the digit itself (from 1 to 9), then the place (such as 10 or 100); then the next digit. In Mandarin, the multiplier 兩 (liǎng) is often used rather than 二; èr for all numbers 200 and greater with the "2" numeral (although as noted earlier this varies from dialect to dialect and person to person). Use of both 兩; liǎng or 二; èr are acceptable for the number 200. When writing in the Cantonese dialect, 二; yi6 is used to represent the "2" numeral for all numbers. In the southern Min dialect of Chaozhou (Teochew), 兩 (no6) is used to represent the "2" numeral in all numbers from 200 onwards. Thus: For the numbers 11 through 19, the leading 'one' (一; yī) is usually omitted. In some dialects, like Shanghainese, when there are only two significant digits in the number, the leading 'one' and the trailing zeroes are omitted. Sometimes, the one before "ten" in the middle of a number, such as 213, is omitted. Thus: Notes: Nothing is ever omitted in large and more complicated numbers such as this. In certain older texts like the Protestant Bible, or in poetic usage, numbers such as 114 may be written as [100] [10] [4] (百十四). Outside of Taiwan, digits are sometimes grouped by myriads instead of thousands. Hence it is more convenient to think of numbers here as in groups of four, thus 1,234,567,890 is regrouped here as 12,3456,7890. Larger than a myriad, each number is therefore four zeroes longer than the one before it, thus 10000 × 萬; wàn = 億; yì. If one of the numbers is between 10 and 19, the leading 'one' is omitted as per the above point. Hence (numbers in parentheses indicate that the number has been written as one number rather than expanded): In Taiwan, pure Arabic numerals are officially always and only grouped by thousands. Unofficially, they are often not grouped, particularly for numbers below 100,000. Mixed Arabic-Chinese numerals are often used in order to denote myriads. This is used both officially and unofficially, and come in a variety of styles: Interior zeroes before the unit position (as in 1002) must be spelt explicitly. The reason for this is that trailing zeroes (as in 1200) are often omitted as shorthand, so ambiguity occurs. One zero is sufficient to resolve the ambiguity. Where the zero is before a digit other than the units digit, the explicit zero is not ambiguous and is therefore optional, but preferred. Thus: === Fractional values === To construct a fraction, the denominator is written first, followed by 分; fēn; 'part', then the literary possessive particle 之; zhī; 'of this', and lastly the numerator. This is the opposite of how fractions are read in English, which is numerator first. Each half of the fraction is written the same as a whole number. For example, to express "two thirds", the structure "three parts of-this two" is used. Mixed numbers are written with the whole-number part first, followed by 又; yòu; 'and', then the fractional part. Percentages are constructed similarly, using 百; bǎi; '100' as the denominator. (The number 100 is typically expressed as 一百; yībǎi; 'one hundred', like the English 'one hundred'. However, for percentages, 百 is used on its own.) Because percentages and other fractions are formulated the same, Chinese are more likely than not to express 10%, 20% etc. as 'parts of 10' (or 1⁄10, 2⁄10, etc. i.e. 十分之一; shí fēnzhī yī, 十分之二; shí fēnzhī èr, etc.) rather than "parts of 100" (or 10⁄100, 20⁄100, etc. i.e. 百分之十; bǎi fēnzhī shí, 百分之二十; bǎi fēnzhī èrshí, etc.) In Taiwan, the most common formation of percentages in the spoken language is the number per hundred followed by the word 趴; pā, a contraction of the Japanese パーセント; pāsento, itself taken from 'percent'. Thus 25% is 二十五趴; èrshíwǔ pā. Decimal numbers are constructed by first writing the whole number part, then inserting a point (点; 點; diǎn), and finally the fractional part. The fractional part is expressed using only the numbers for 0 to 9, similarly to English. 半; bàn; 'half' functions as a number and therefore requires a measure word. For example: 半杯水; bàn bēi shuǐ; 'half a glass of water'. === Ordinal numbers === Ordinal numbers are formed by adding 第; dì; 'sequence' before the number. The Heavenly Stems are a traditional Chinese ordinal system. === Negative numbers === Negative numbers are formed by adding 负; 負; fù before the number. === Usage === Chinese grammar requires the use of classifiers (measure words) when a numeral is used together with a noun to express a quantity. For example, "three people" is expressed as 三个人; 三個人; sān ge rén, "three (ge particle) person", where 个/個 ge is a classifier. There exist many different classifiers, for use with different sets of nouns, although 个/個 is the most common, and may be used informally in place of other classifiers. Chinese uses cardinal numbers in certain situations in which English would use ordinals. For example, 三楼/三樓; sān lóu (literally "three story/storey") means "third floor" ("second floor" in British § Numbering). Likewise, 二十一世纪/二十一世紀; èrshí yī shìjì (literally "twenty-one century") is used for "21st century". Numbers of years are commonly spoken as a sequence of digits, as in 二零零一; èr líng líng yī ("two zero zero one") for the year 2001. Names of months and days (in the Western system) are also expressed using numbers: 一月; yīyuè ("one month") for January, etc.; and 星期一; xīngqīyī ("week one") for Monday, etc. There is only one exception: Sunday is 星期日; xīngqīrì, or informally 星期天; xīngqītiān, both literally "week day". When meaning "week", "星期" xīngqī and "禮拜; 礼拜" lǐbài are interchangeable. "禮拜天" lǐbàitiān or "禮拜日" lǐbàirì means "day of worship". Chinese Catholics call Sunday "主日" zhǔrì, "Lord's day". Full dates are usually written in the format 2001年1月20日 for January 20, 2001 (using 年; nián "year", 月; yuè "month", and 日; rì "day") – all the numbers are read as cardinals, not ordinals, with no leading zeroes, and the year is read as a sequence of digits. For brevity the nián, yuè and rì may be dropped to give a date composed of just numbers. For example "6-4" in Chinese is "six-four", short for "month six, day four" i.e. June Fourth, a common Chinese shorthand for the 1989 Tiananmen Square protests (because of the violence that occurred on June 4). For another example 67, in Chinese is sixty seven, short for year nineteen sixty seven, a common Chinese shorthand for the Hong Kong 1967 leftist riots. == Counting rod and Suzhou numerals == In the same way that Roman numerals were standard in ancient and medieval Europe for mathematics and commerce, the Chinese formerly used the rod numerals, which is a positional system. The Suzhou numerals (simplified Chinese: 苏州花码; traditional Chinese: 蘇州花碼; pinyin: Sūzhōu huāmǎ) system is a variation of the Southern Song rod numerals. Nowadays, the huāmǎ system is only used for displaying prices in Chinese markets or on traditional handwritten invoices. == Hand gestures == There is a common method of using of one hand to signify the numbers one to ten. While the five digits on one hand can easily express the numbers one to five, six to ten have special signs that can be used in commerce or day-to-day communication. == Historical use of numerals in China == Most Chinese numerals of later periods were descendants of the Shang dynasty oracle numerals of the 14th century BC. The oracle bone script numerals were found on tortoise shell and animal bones. In early civilizations, the Shang were able to express any numbers, however large, with only nine symbols and a counting board though it was still not positional. Some of the bronze script numerals such as 1, 2, 3, 4, 10, 11, 12, and 13 became part of the system of rod numerals. In this system, 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". The counting rod numerals system has place value and decimal numerals for computation, and was used widely by Chinese merchants, mathematicians and astronomers from the Han dynasty to the 16th century. Alexander Wylie, Christian missionary to China, in 1853 already refuted the notion that "the Chinese numbers were written in words at length", and stated that in ancient China, calculation was carried out by means of counting rods, and "the written character is evidently a rude presentation of these". After being introduced to the rod numerals, he said "Having thus obtained a simple but effective system of figures, we find the Chinese in actual use of a method of notation depending on the theory of local value [i.e. place-value], several centuries before such theory was understood in Europe, and while yet the science of numbers had scarcely dawned among the Arabs." During the Ming and Qing dynasties (after Arabic numerals were introduced into China), some Chinese mathematicians used Chinese numeral characters as positional system digits. After the Qing period, both the Chinese numeral characters and the Suzhou numerals were replaced by Arabic numerals in mathematical writings. == Cultural influences == Traditional Chinese numeric characters are also used in Japan and Korea and were used in Vietnam before the 20th century. In vertical text (that is, read top to bottom), using characters for numbers is the norm, while in horizontal text, Arabic numerals are most common. Chinese numeric characters are also used in much the same formal or decorative fashion that Roman numerals are in Western cultures. Chinese numerals may appear together with Arabic numbers on the same sign or document. == See also == Numbers in Chinese culture Celestial stem == Notes == == References ==
Wikipedia:Chinese remainder theorem#0	In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer n by several integers, then one can determine uniquely the remainder of the division of n by the product of these integers, under the condition that the divisors are pairwise coprime (no two divisors share a common factor other than 1). The theorem is sometimes called Sunzi's theorem. Both names of the theorem refer to its earliest known statement that appeared in Sunzi Suanjing, a Chinese manuscript written during the 3rd to 5th century CE. This first statement was restricted to the following example: If one knows that the remainder of n divided by 3 is 2, the remainder of n divided by 5 is 3, and the remainder of n divided by 7 is 2, then with no other information, one can determine the remainder of n divided by 105 (the product of 3, 5, and 7) without knowing the value of n. In this example, the remainder is 23. Moreover, this remainder is the only possible positive value of n that is less than 105. The Chinese remainder theorem is widely used for computing with large integers, as it allows replacing a computation for which one knows a bound on the size of the result by several similar computations on small integers. The Chinese remainder theorem (expressed in terms of congruences) is true over every principal ideal domain. It has been generalized to any ring, with a formulation involving two-sided ideals. == History == The earliest known statement of the problem appears in the 5th-century book Sunzi Suanjing by the Chinese mathematician Sunzi: There are certain things whose number is unknown. If we count them by threes, we have two left over; by fives, we have three left over; and by sevens, two are left over. How many things are there? Sunzi's work would not be considered a theorem by modern standards; it only gives one particular problem, without showing how to solve it, much less any proof about the general case or a general algorithm for solving it. An algorithm for solving this problem was described by Aryabhata (6th century). Special cases of the Chinese remainder theorem were also known to Brahmagupta (7th century) and appear in Fibonacci's Liber Abaci (1202). The result was later generalized with a complete solution called Da-yan-shu (大衍術) in Qin Jiushao's 1247 Mathematical Treatise in Nine Sections which was translated into English in early 19th century by British missionary Alexander Wylie. The notion of congruences was first introduced and used by Carl Friedrich Gauss in his Disquisitiones Arithmeticae of 1801. Gauss illustrates the Chinese remainder theorem on a problem involving calendars, namely, "to find the years that have a certain period number with respect to the solar and lunar cycle and the Roman indiction." Gauss introduces a procedure for solving the problem that had already been used by Leonhard Euler but was in fact an ancient method that had appeared several times. == Statement == Let n1, ..., nk be integers greater than 1, which are often called moduli or divisors. Let us denote by N the product of the ni. The Chinese remainder theorem asserts that if the ni are pairwise coprime, and if a1, ..., ak are integers such that 0 ≤ ai < ni for every i, then there is one and only one integer x, such that 0 ≤ x < N and the remainder of the Euclidean division of x by ni is ai for every i. This may be restated as follows in terms of congruences: If the n i {\displaystyle n_{i}} are pairwise coprime, and if a1, ..., ak are any integers, then the system x ≡ a 1 ( mod n 1 ) ⋮ x ≡ a k ( mod n k ) , {\displaystyle {\begin{aligned}x&\equiv a_{1}{\pmod {n_{1}}}\\&\,\,\,\vdots \\x&\equiv a_{k}{\pmod {n_{k}}},\end{aligned}}} has a solution, and any two solutions, say x1 and x2, are congruent modulo N, that is, x1 ≡ x2 (mod N ). In abstract algebra, the theorem is often restated as: if the ni are pairwise coprime, the map x mod N ↦ ( x mod n 1 , … , x mod n k ) {\displaystyle x{\bmod {N}}\;\mapsto \;(x{\bmod {n}}_{1},\,\ldots ,\,x{\bmod {n}}_{k})} defines a ring isomorphism Z / N Z ≅ Z / n 1 Z × ⋯ × Z / n k Z {\displaystyle \mathbb {Z} /N\mathbb {Z} \cong \mathbb {Z} /n_{1}\mathbb {Z} \times \cdots \times \mathbb {Z} /n_{k}\mathbb {Z} } between the ring of integers modulo N and the direct product of the rings of integers modulo the ni. This means that for doing a sequence of arithmetic operations in Z / N Z , {\displaystyle \mathbb {Z} /N\mathbb {Z} ,} one may do the same computation independently in each Z / n i Z {\displaystyle \mathbb {Z} /n_{i}\mathbb {Z} } and then get the result by applying the isomorphism (from the right to the left). This may be much faster than the direct computation if N and the number of operations are large. This is widely used, under the name multi-modular computation, for linear algebra over the integers or the rational numbers. The theorem can also be restated in the language of combinatorics as the fact that the infinite arithmetic progressions of integers form a Helly family. == Proof == The existence and the uniqueness of the solution may be proven independently. However, the first proof of existence, given below, uses this uniqueness. === Uniqueness === Suppose that x and y are both solutions to all the congruences. As x and y give the same remainder, when divided by ni, their difference x − y is a multiple of each ni. As the ni are pairwise coprime, their product N also divides x − y, and thus x and y are congruent modulo N. If x and y are supposed to be non-negative and less than N (as in the first statement of the theorem), then their difference may be a multiple of N only if x = y. === Existence (first proof) === The map x mod N ↦ ( x mod n 1 , … , x mod n k ) {\displaystyle x{\bmod {N}}\mapsto (x{\bmod {n}}_{1},\ldots ,x{\bmod {n}}_{k})} maps congruence classes modulo N to sequences of congruence classes modulo ni. The proof of uniqueness shows that this map is injective. As the domain and the codomain of this map have the same number of elements, the map is also surjective, which proves the existence of the solution. This proof is very simple but does not provide any direct way for computing a solution. Moreover, it cannot be generalized to other situations where the following proof can. === Existence (constructive proof) === Existence may be established by an explicit construction of x. This construction may be split into two steps, first solving the problem in the case of two moduli, and then extending this solution to the general case by induction on the number of moduli. ==== Case of two moduli ==== We want to solve the system: x ≡ a 1 ( mod n 1 ) x ≡ a 2 ( mod n 2 ) , {\displaystyle {\begin{aligned}x&\equiv a_{1}{\pmod {n_{1}}}\\x&\equiv a_{2}{\pmod {n_{2}}},\end{aligned}}} where n 1 {\displaystyle n_{1}} and n 2 {\displaystyle n_{2}} are coprime. Bézout's identity asserts the existence of two integers m 1 {\displaystyle m_{1}} and m 2 {\displaystyle m_{2}} such that m 1 n 1 + m 2 n 2 = 1. {\displaystyle m_{1}n_{1}+m_{2}n_{2}=1.} The integers m 1 {\displaystyle m_{1}} and m 2 {\displaystyle m_{2}} may be computed by the extended Euclidean algorithm. A solution is given by x = a 1 m 2 n 2 + a 2 m 1 n 1 . {\displaystyle x=a_{1}m_{2}n_{2}+a_{2}m_{1}n_{1}.} Indeed, x = a 1 m 2 n 2 + a 2 m 1 n 1 = a 1 ( 1 − m 1 n 1 ) + a 2 m 1 n 1 = a 1 + ( a 2 − a 1 ) m 1 n 1 , {\displaystyle {\begin{aligned}x&=a_{1}m_{2}n_{2}+a_{2}m_{1}n_{1}\\&=a_{1}(1-m_{1}n_{1})+a_{2}m_{1}n_{1}\\&=a_{1}+(a_{2}-a_{1})m_{1}n_{1},\end{aligned}}} implying that x ≡ a 1 ( mod n 1 ) . {\displaystyle x\equiv a_{1}{\pmod {n_{1}}}.} The second congruence is proved similarly, by exchanging the subscripts 1 and 2. ==== General case ==== Consider a sequence of congruence equations: x ≡ a 1 ( mod n 1 ) ⋮ x ≡ a k ( mod n k ) , {\displaystyle {\begin{aligned}x&\equiv a_{1}{\pmod {n_{1}}}\\&\vdots \\x&\equiv a_{k}{\pmod {n_{k}}},\end{aligned}}} where the n i {\displaystyle n_{i}} are pairwise coprime. The two first equations have a solution a 1 , 2 {\displaystyle a_{1,2}} provided by the method of the previous section. The set of the solutions of these two first equations is the set of all solutions of the equation x ≡ a 1 , 2 ( mod n 1 n 2 ) . {\displaystyle x\equiv a_{1,2}{\pmod {n_{1}n_{2}}}.} As the other n i {\displaystyle n_{i}} are coprime with n 1 n 2 , {\displaystyle n_{1}n_{2},} this reduces solving the initial problem of k equations to a similar problem with k − 1 {\displaystyle k-1} equations. Iterating the process, one gets eventually the solutions of the initial problem. === Existence (direct construction) === For constructing a solution, it is not necessary to make an induction on the number of moduli. However, such a direct construction involves more computation with large numbers, which makes it less efficient and less used. Nevertheless, Lagrange interpolation is a special case of this construction, applied to polynomials instead of integers. Let N i = N / n i {\displaystyle N_{i}=N/n_{i}} be the product of all moduli but one. As the n i {\displaystyle n_{i}} are pairwise coprime, N i {\displaystyle N_{i}} and n i {\displaystyle n_{i}} are coprime. Thus Bézout's identity applies, and there exist integers M i {\displaystyle M_{i}} and m i {\displaystyle m_{i}} such that M i N i + m i n i = 1. {\displaystyle M_{i}N_{i}+m_{i}n_{i}=1.} A solution of the system of congruences is x = ∑ i = 1 k a i M i N i . {\displaystyle x=\sum _{i=1}^{k}a_{i}M_{i}N_{i}.} In fact, as N j {\displaystyle N_{j}} is a multiple of n i {\displaystyle n_{i}} for i ≠ j , {\displaystyle i\neq j,} we have x ≡ a i M i N i ≡ a i ( 1 − m i n i ) ≡ a i ( mod n i ) , {\displaystyle x\equiv a_{i}M_{i}N_{i}\equiv a_{i}(1-m_{i}n_{i})\equiv a_{i}{\pmod {n_{i}}},} for every i . {\displaystyle i.} == Computation == Consider a system of congruences: x ≡ a 1 ( mod n 1 ) ⋮ x ≡ a k ( mod n k ) , {\displaystyle {\begin{aligned}x&\equiv a_{1}{\pmod {n_{1}}}\\&\vdots \\x&\equiv a_{k}{\pmod {n_{k}}},\\\end{aligned}}} where the n i {\displaystyle n_{i}} are pairwise coprime, and let N = n 1 n 2 ⋯ n k . {\displaystyle N=n_{1}n_{2}\cdots n_{k}.} In this section several methods are described for computing the unique solution for x {\displaystyle x} , such that 0 ≤ x < N , {\displaystyle 0\leq x<N,} and these methods are applied on the example x ≡ 0 ( mod 3 ) x ≡ 3 ( mod 4 ) x ≡ 4 ( mod 5 ) . {\displaystyle {\begin{aligned}x&\equiv 0{\pmod {3}}\\x&\equiv 3{\pmod {4}}\\x&\equiv 4{\pmod {5}}.\end{aligned}}} Several methods of computation are presented. The two first ones are useful for small examples, but become very inefficient when the product n 1 ⋯ n k {\displaystyle n_{1}\cdots n_{k}} is large. The third one uses the existence proof given in § Existence (constructive proof). It is the most convenient when the product n 1 ⋯ n k {\displaystyle n_{1}\cdots n_{k}} is large, or for computer computation. === Systematic search === It is easy to check whether a value of x is a solution: it suffices to compute the remainder of the Euclidean division of x by each ni. Thus, to find the solution, it suffices to check successively the integers from 0 to N until finding the solution. Although very simple, this method is very inefficient. For the simple example considered here, 40 integers (including 0) have to be checked for finding the solution, which is 39. This is an exponential time algorithm, as the size of the input is, up to a constant factor, the number of digits of N, and the average number of operations is of the order of N. Therefore, this method is rarely used, neither for hand-written computation nor on computers. === Search by sieving === The search of the solution may be made dramatically faster by sieving. For this method, we suppose, without loss of generality, that 0 ≤ a i < n i {\displaystyle 0\leq a_{i}<n_{i}} (if it were not the case, it would suffice to replace each a i {\displaystyle a_{i}} by the remainder of its division by n i {\displaystyle n_{i}} ). This implies that the solution belongs to the arithmetic progression a 1 , a 1 + n 1 , a 1 + 2 n 1 , … {\displaystyle a_{1},a_{1}+n_{1},a_{1}+2n_{1},\ldots } By testing the values of these numbers modulo n 2 , {\displaystyle n_{2},} one eventually finds a solution x 2 {\displaystyle x_{2}} of the two first congruences. Then the solution belongs to the arithmetic progression x 2 , x 2 + n 1 n 2 , x 2 + 2 n 1 n 2 , … {\displaystyle x_{2},x_{2}+n_{1}n_{2},x_{2}+2n_{1}n_{2},\ldots } Testing the values of these numbers modulo n 3 , {\displaystyle n_{3},} and continuing until every modulus has been tested eventually yields the solution. This method is faster if the moduli have been ordered by decreasing value, that is if n 1 > n 2 > ⋯ > n k . {\displaystyle n_{1}>n_{2}>\cdots >n_{k}.} For the example, this gives the following computation. We consider first the numbers that are congruent to 4 modulo 5 (the largest modulus), which are 4, 9 = 4 + 5, 14 = 9 + 5, ... For each of them, compute the remainder by 4 (the second largest modulus) until getting a number congruent to 3 modulo 4. Then one can proceed by adding 20 = 5 × 4 at each step, and computing only the remainders by 3. This gives 4 mod 4 → 0. Continue 4 + 5 = 9 mod 4 →1. Continue 9 + 5 = 14 mod 4 → 2. Continue 14 + 5 = 19 mod 4 → 3. OK, continue by considering remainders modulo 3 and adding 5 × 4 = 20 each time 19 mod 3 → 1. Continue 19 + 20 = 39 mod 3 → 0. OK, this is the result. This method works well for hand-written computation with a product of moduli that is not too big. However, it is much slower than other methods, for very large products of moduli. Although dramatically faster than the systematic search, this method also has an exponential time complexity and is therefore not used on computers. === Using the existence construction === The constructive existence proof shows that, in the case of two moduli, the solution may be obtained by the computation of the Bézout coefficients of the moduli, followed by a few multiplications, additions and reductions modulo n 1 n 2 {\displaystyle n_{1}n_{2}} (for getting a result in the interval ( 0 , n 1 n 2 − 1 ) {\displaystyle (0,n_{1}n_{2}-1)} ). As the Bézout's coefficients may be computed with the extended Euclidean algorithm, the whole computation, at most, has a quadratic time complexity of O ( ( s 1 + s 2 ) 2 ) , {\displaystyle O((s_{1}+s_{2})^{2}),} where s i {\displaystyle s_{i}} denotes the number of digits of n i . {\displaystyle n_{i}.} For more than two moduli, the method for two moduli allows the replacement of any two congruences by a single congruence modulo the product of the moduli. Iterating this process provides eventually the solution with a complexity, which is quadratic in the number of digits of the product of all moduli. This quadratic time complexity does not depend on the order in which the moduli are regrouped. One may regroup the two first moduli, then regroup the resulting modulus with the next one, and so on. This strategy is the easiest to implement, but it also requires more computation involving large numbers. Another strategy consists in partitioning the moduli in pairs whose product have comparable sizes (as much as possible), applying, in parallel, the method of two moduli to each pair, and iterating with a number of moduli approximatively divided by two. This method allows an easy parallelization of the algorithm. Also, if fast algorithms (that is, algorithms working in quasilinear time) are used for the basic operations, this method provides an algorithm for the whole computation that works in quasilinear time. On the current example (which has only three moduli), both strategies are identical and work as follows. Bézout's identity for 3 and 4 is 1 × 4 + ( − 1 ) × 3 = 1. {\displaystyle 1\times 4+(-1)\times 3=1.} Putting this in the formula given for proving the existence gives 0 × 1 × 4 + 3 × ( − 1 ) × 3 = − 9 {\displaystyle 0\times 1\times 4+3\times (-1)\times 3=-9} for a solution of the two first congruences, the other solutions being obtained by adding to −9 any multiple of 3 × 4 = 12. One may continue with any of these solutions, but the solution 3 = −9 +12 is smaller (in absolute value) and thus leads probably to an easier computation Bézout identity for 5 and 3 × 4 = 12 is 5 × 5 + ( − 2 ) × 12 = 1. {\displaystyle 5\times 5+(-2)\times 12=1.} Applying the same formula again, we get a solution of the problem: 5 × 5 × 3 + 12 × ( − 2 ) × 4 = − 21. {\displaystyle 5\times 5\times 3+12\times (-2)\times 4=-21.} The other solutions are obtained by adding any multiple of 3 × 4 × 5 = 60, and the smallest positive solution is −21 + 60 = 39. === As a linear Diophantine system === The system of congruences solved by the Chinese remainder theorem may be rewritten as a system of linear Diophantine equations: x = a 1 + x 1 n 1 ⋮ x = a k + x k n k , {\displaystyle {\begin{aligned}x&=a_{1}+x_{1}n_{1}\\&\vdots \\x&=a_{k}+x_{k}n_{k},\end{aligned}}} where the unknown integers are x {\displaystyle x} and the x i . {\displaystyle x_{i}.} Therefore, every general method for solving such systems may be used for finding the solution of Chinese remainder theorem, such as the reduction of the matrix of the system to Smith normal form or Hermite normal form. However, as usual when using a general algorithm for a more specific problem, this approach is less efficient than the method of the preceding section, based on a direct use of Bézout's identity. == Over principal ideal domains == In § Statement, the Chinese remainder theorem has been stated in three different ways: in terms of remainders, of congruences, and of a ring isomorphism. The statement in terms of remainders does not apply, in general, to principal ideal domains, as remainders are not defined in such rings. However, the two other versions make sense over a principal ideal domain R: it suffices to replace "integer" by "element of the domain" and Z {\displaystyle \mathbb {Z} } by R. These two versions of the theorem are true in this context, because the proofs (except for the first existence proof), are based on Euclid's lemma and Bézout's identity, which are true over every principal domain. However, in general, the theorem is only an existence theorem and does not provide any way for computing the solution, unless one has an algorithm for computing the coefficients of Bézout's identity. == Over univariate polynomial rings and Euclidean domains == The statement in terms of remainders given in § Theorem statement cannot be generalized to any principal ideal domain, but its generalization to Euclidean domains is straightforward. The univariate polynomials over a field is the typical example of a Euclidean domain which is not the integers. Therefore, we state the theorem for the case of the ring R = K [ X ] {\displaystyle R=K[X]} for a field K . {\displaystyle K.} For getting the theorem for a general Euclidean domain, it suffices to replace the degree by the Euclidean function of the Euclidean domain. The Chinese remainder theorem for polynomials is thus: Let P i ( X ) {\displaystyle P_{i}(X)} (the moduli) be, for i = 1 , … , k {\displaystyle i=1,\dots ,k} , pairwise coprime polynomials in R = K [ X ] {\displaystyle R=K[X]} . Let d i = deg ⁡ P i {\displaystyle d_{i}=\deg P_{i}} be the degree of P i ( X ) {\displaystyle P_{i}(X)} , and D {\displaystyle D} be the sum of the d i . {\displaystyle d_{i}.} If A i ( X ) , … , A k ( X ) {\displaystyle A_{i}(X),\ldots ,A_{k}(X)} are polynomials such that A i ( X ) = 0 {\displaystyle A_{i}(X)=0} or deg ⁡ A i < d i {\displaystyle \deg A_{i}<d_{i}} for every i, then, there is one and only one polynomial P ( X ) {\displaystyle P(X)} , such that deg ⁡ P < D {\displaystyle \deg P<D} and the remainder of the Euclidean division of P ( X ) {\displaystyle P(X)} by P i ( X ) {\displaystyle P_{i}(X)} is A i ( X ) {\displaystyle A_{i}(X)} for every i. The construction of the solution may be done as in § Existence (constructive proof) or § Existence (direct proof). However, the latter construction may be simplified by using, as follows, partial fraction decomposition instead of the extended Euclidean algorithm. Thus, we want to find a polynomial P ( X ) {\displaystyle P(X)} , which satisfies the congruences P ( X ) ≡ A i ( X ) ( mod P i ( X ) ) , {\displaystyle P(X)\equiv A_{i}(X){\pmod {P_{i}(X)}},} for i = 1 , … , k . {\displaystyle i=1,\ldots ,k.} Consider the polynomials Q ( X ) = ∏ i = 1 k P i ( X ) Q i ( X ) = Q ( X ) P i ( X ) . {\displaystyle {\begin{aligned}Q(X)&=\prod _{i=1}^{k}P_{i}(X)\\Q_{i}(X)&={\frac {Q(X)}{P_{i}(X)}}.\end{aligned}}} The partial fraction decomposition of 1 / Q ( X ) {\displaystyle 1/Q(X)} gives k polynomials S i ( X ) {\displaystyle S_{i}(X)} with degrees deg ⁡ S i ( X ) < d i , {\displaystyle \deg S_{i}(X)<d_{i},} such that 1 Q ( X ) = ∑ i = 1 k S i ( X ) P i ( X ) , {\displaystyle {\frac {1}{Q(X)}}=\sum _{i=1}^{k}{\frac {S_{i}(X)}{P_{i}(X)}},} and thus 1 = ∑ i = 1 k S i ( X ) Q i ( X ) . {\displaystyle 1=\sum _{i=1}^{k}S_{i}(X)Q_{i}(X).} Then a solution of the simultaneous congruence system is given by the polynomial ∑ i = 1 k A i ( X ) S i ( X ) Q i ( X ) . {\displaystyle \sum _{i=1}^{k}A_{i}(X)S_{i}(X)Q_{i}(X).} In fact, we have ∑ i = 1 k A i ( X ) S i ( X ) Q i ( X ) = A i ( X ) + ∑ j = 1 k ( A j ( X ) − A i ( X ) ) S j ( X ) Q j ( X ) ≡ A i ( X ) ( mod P i ( X ) ) , {\displaystyle \sum _{i=1}^{k}A_{i}(X)S_{i}(X)Q_{i}(X)=A_{i}(X)+\sum _{j=1}^{k}(A_{j}(X)-A_{i}(X))S_{j}(X)Q_{j}(X)\equiv A_{i}(X){\pmod {P_{i}(X)}},} for 1 ≤ i ≤ k . {\displaystyle 1\leq i\leq k.} This solution may have a degree larger than D = ∑ i = 1 k d i . {\displaystyle D=\sum _{i=1}^{k}d_{i}.} The unique solution of degree less than D {\displaystyle D} may be deduced by considering the remainder B i ( X ) {\displaystyle B_{i}(X)} of the Euclidean division of A i ( X ) S i ( X ) {\displaystyle A_{i}(X)S_{i}(X)} by P i ( X ) . {\displaystyle P_{i}(X).} This solution is P ( X ) = ∑ i = 1 k B i ( X ) Q i ( X ) . {\displaystyle P(X)=\sum _{i=1}^{k}B_{i}(X)Q_{i}(X).} === Lagrange interpolation === A special case of Chinese remainder theorem for polynomials is Lagrange interpolation. For this, consider k monic polynomials of degree one: P i ( X ) = X − x i . {\displaystyle P_{i}(X)=X-x_{i}.} They are pairwise coprime if the x i {\displaystyle x_{i}} are all different. The remainder of the division by P i ( X ) {\displaystyle P_{i}(X)} of a polynomial P ( X ) {\displaystyle P(X)} is P ( x i ) {\displaystyle P(x_{i})} , by the polynomial remainder theorem. Now, let A 1 , … , A k {\displaystyle A_{1},\ldots ,A_{k}} be constants (polynomials of degree 0) in K . {\displaystyle K.} Both Lagrange interpolation and Chinese remainder theorem assert the existence of a unique polynomial P ( X ) , {\displaystyle P(X),} of degree less than k {\displaystyle k} such that P ( x i ) = A i , {\displaystyle P(x_{i})=A_{i},} for every i . {\displaystyle i.} Lagrange interpolation formula is exactly the result, in this case, of the above construction of the solution. More precisely, let Q ( X ) = ∏ i = 1 k ( X − x i ) Q i ( X ) = Q ( X ) X − x i . {\displaystyle {\begin{aligned}Q(X)&=\prod _{i=1}^{k}(X-x_{i})\\[6pt]Q_{i}(X)&={\frac {Q(X)}{X-x_{i}}}.\end{aligned}}} The partial fraction decomposition of 1 Q ( X ) {\displaystyle {\frac {1}{Q(X)}}} is 1 Q ( X ) = ∑ i = 1 k 1 Q i ( x i ) ( X − x i ) . {\displaystyle {\frac {1}{Q(X)}}=\sum _{i=1}^{k}{\frac {1}{Q_{i}(x_{i})(X-x_{i})}}.} In fact, reducing the right-hand side to a common denominator one gets ∑ i = 1 k 1 Q i ( x i ) ( X − x i ) = 1 Q ( X ) ∑ i = 1 k Q i ( X ) Q i ( x i ) , {\displaystyle \sum _{i=1}^{k}{\frac {1}{Q_{i}(x_{i})(X-x_{i})}}={\frac {1}{Q(X)}}\sum _{i=1}^{k}{\frac {Q_{i}(X)}{Q_{i}(x_{i})}},} and the numerator is equal to one, as being a polynomial of degree less than k , {\displaystyle k,} which takes the value one for k {\displaystyle k} different values of X . {\displaystyle X.} Using the above general formula, we get the Lagrange interpolation formula: P ( X ) = ∑ i = 1 k A i Q i ( X ) Q i ( x i ) . {\displaystyle P(X)=\sum _{i=1}^{k}A_{i}{\frac {Q_{i}(X)}{Q_{i}(x_{i})}}.} === Hermite interpolation === Hermite interpolation is an application of the Chinese remainder theorem for univariate polynomials, which may involve moduli of arbitrary degrees (Lagrange interpolation involves only moduli of degree one). The problem consists of finding a polynomial of the least possible degree, such that the polynomial and its first derivatives take given values at some fixed points. More precisely, let x 1 , … , x k {\displaystyle x_{1},\ldots ,x_{k}} be k {\displaystyle k} elements of the ground field K , {\displaystyle K,} and, for i = 1 , … , k , {\displaystyle i=1,\ldots ,k,} let a i , 0 , a i , 1 , … , a i , r i − 1 {\displaystyle a_{i,0},a_{i,1},\ldots ,a_{i,r_{i}-1}} be the values of the first r i {\displaystyle r_{i}} derivatives of the sought polynomial at x i {\displaystyle x_{i}} (including the 0th derivative, which is the value of the polynomial itself). The problem is to find a polynomial P ( X ) {\displaystyle P(X)} such that its j th derivative takes the value a i , j {\displaystyle a_{i,j}} at x i , {\displaystyle x_{i},} for i = 1 , … , k {\displaystyle i=1,\ldots ,k} and j = 0 , … , r j . {\displaystyle j=0,\ldots ,r_{j}.} Consider the polynomial P i ( X ) = ∑ j = 0 r i − 1 a i , j j ! ( X − x i ) j . {\displaystyle P_{i}(X)=\sum _{j=0}^{r_{i}-1}{\frac {a_{i,j}}{j!}}(X-x_{i})^{j}.} This is the Taylor polynomial of order r i − 1 {\displaystyle r_{i}-1} at x i {\displaystyle x_{i}} , of the unknown polynomial P ( X ) . {\displaystyle P(X).} Therefore, we must have P ( X ) ≡ P i ( X ) ( mod ( X − x i ) r i ) . {\displaystyle P(X)\equiv P_{i}(X){\pmod {(X-x_{i})^{r_{i}}}}.} Conversely, any polynomial P ( X ) {\displaystyle P(X)} that satisfies these k {\displaystyle k} congruences, in particular verifies, for any i = 1 , … , k {\displaystyle i=1,\ldots ,k} P ( X ) = P i ( X ) + o ( X − x i ) r i − 1 {\displaystyle P(X)=P_{i}(X)+o(X-x_{i})^{r_{i}-1}} therefore P i ( X ) {\displaystyle P_{i}(X)} is its Taylor polynomial of order r i − 1 {\displaystyle r_{i}-1} at x i {\displaystyle x_{i}} , that is, P ( X ) {\displaystyle P(X)} solves the initial Hermite interpolation problem. The Chinese remainder theorem asserts that there exists exactly one polynomial of degree less than the sum of the r i , {\displaystyle r_{i},} which satisfies these k {\displaystyle k} congruences. There are several ways for computing the solution P ( X ) . {\displaystyle P(X).} One may use the method described at the beginning of § Over univariate polynomial rings and Euclidean domains. One may also use the constructions given in § Existence (constructive proof) or § Existence (direct proof). == Generalization to non-coprime moduli == The Chinese remainder theorem can be generalized to non-coprime moduli. Let m , n , a , b {\displaystyle m,n,a,b} be any integers, let g = gcd ( m , n ) {\displaystyle g=\gcd(m,n)} ; M = lcm ⁡ ( m , n ) {\displaystyle M=\operatorname {lcm} (m,n)} , and consider the system of congruences: x ≡ a ( mod m ) x ≡ b ( mod n ) , {\displaystyle {\begin{aligned}x&\equiv a{\pmod {m}}\\x&\equiv b{\pmod {n}},\end{aligned}}} If a ≡ b ( mod g ) {\displaystyle a\equiv b{\pmod {g}}} , then this system has a unique solution modulo M = m n / g {\displaystyle M=mn/g} . Otherwise, it has no solutions. If one uses Bézout's identity to write g = u m + v n {\displaystyle g=um+vn} , then the solution is given by x = a v n + b u m g . {\displaystyle x={\frac {avn+bum}{g}}.} This defines an integer, as g divides both m and n. Otherwise, the proof is very similar to that for coprime moduli. == Generalization to arbitrary rings == The Chinese remainder theorem can be generalized to any ring, by using coprime ideals (also called comaximal ideals). Two ideals I and J are coprime if there are elements i ∈ I {\displaystyle i\in I} and j ∈ J {\displaystyle j\in J} such that i + j = 1. {\displaystyle i+j=1.} This relation plays the role of Bézout's identity in the proofs related to this generalization, which otherwise are very similar. The generalization may be stated as follows. Let I1, ..., Ik be two-sided ideals of a ring R {\displaystyle R} and let I be their intersection. If the ideals are pairwise coprime, we have the isomorphism: R / I → ( R / I 1 ) × ⋯ × ( R / I k ) x mod I ↦ ( x mod I 1 , … , x mod I k ) , {\displaystyle {\begin{aligned}R/I&\to (R/I_{1})\times \cdots \times (R/I_{k})\\x{\bmod {I}}&\mapsto (x{\bmod {I}}_{1},\,\ldots ,\,x{\bmod {I}}_{k}),\end{aligned}}} between the quotient ring R / I {\displaystyle R/I} and the direct product of the R / I i , {\displaystyle R/I_{i},} where " x mod I {\displaystyle x{\bmod {I}}} " denotes the image of the element x {\displaystyle x} in the quotient ring defined by the ideal I . {\displaystyle I.} Moreover, if R {\displaystyle R} is commutative, then the ideal intersection of pairwise coprime ideals is equal to their product; that is I = I 1 ∩ I 2 ∩ ⋯ ∩ I k = I 1 I 2 ⋯ I k , {\displaystyle I=I_{1}\cap I_{2}\cap \cdots \cap I_{k}=I_{1}I_{2}\cdots I_{k},} if Ii and Ij are coprime for all i ≠ j. === Interpretation in terms of idempotents === Let I 1 , I 2 , … , I k {\displaystyle I_{1},I_{2},\dots ,I_{k}} be pairwise coprime two-sided ideals with ⋂ i = 1 k I i = 0 , {\displaystyle \bigcap _{i=1}^{k}I_{i}=0,} and φ : R → ( R / I 1 ) × ⋯ × ( R / I k ) {\displaystyle \varphi :R\to (R/I_{1})\times \cdots \times (R/I_{k})} be the isomorphism defined above. Let f i = ( 0 , … , 1 , … , 0 ) {\displaystyle f_{i}=(0,\ldots ,1,\ldots ,0)} be the element of ( R / I 1 ) × ⋯ × ( R / I k ) {\displaystyle (R/I_{1})\times \cdots \times (R/I_{k})} whose components are all 0 except the i th which is 1, and e i = φ − 1 ( f i ) . {\displaystyle e_{i}=\varphi ^{-1}(f_{i}).} The e i {\displaystyle e_{i}} are central idempotents that are pairwise orthogonal; this means, in particular, that e i 2 = e i {\displaystyle e_{i}^{2}=e_{i}} and e i e j = e j e i = 0 {\displaystyle e_{i}e_{j}=e_{j}e_{i}=0} for every i and j. Moreover, one has e 1 + ⋯ + e n = 1 , {\textstyle e_{1}+\cdots +e_{n}=1,} and I i = R ( 1 − e i ) . {\displaystyle I_{i}=R(1-e_{i}).} In summary, this generalized Chinese remainder theorem is the equivalence between giving pairwise coprime two-sided ideals with a zero intersection, and giving central and pairwise orthogonal idempotents that sum to 1. == Applications == === Sequence numbering === The Chinese remainder theorem has been used to construct a Gödel numbering for sequences, which is involved in the proof of Gödel's incompleteness theorems. === Fast Fourier transform === The prime-factor FFT algorithm (also called Good-Thomas algorithm) uses the Chinese remainder theorem for reducing the computation of a fast Fourier transform of size n 1 n 2 {\displaystyle n_{1}n_{2}} to the computation of two fast Fourier transforms of smaller sizes n 1 {\displaystyle n_{1}} and n 2 {\displaystyle n_{2}} (providing that n 1 {\displaystyle n_{1}} and n 2 {\displaystyle n_{2}} are coprime). === Encryption === Most implementations of RSA use the Chinese remainder theorem during signing of HTTPS certificates and during decryption. The Chinese remainder theorem can also be used in secret sharing, which consists of distributing a set of shares among a group of people who, all together (but no one alone), can recover a certain secret from the given set of shares. Each of the shares is represented in a congruence, and the solution of the system of congruences using the Chinese remainder theorem is the secret to be recovered. Secret sharing using the Chinese remainder theorem uses, along with the Chinese remainder theorem, special sequences of integers that guarantee the impossibility of recovering the secret from a set of shares with less than a certain cardinality. === Range ambiguity resolution === The range ambiguity resolution techniques used with medium pulse repetition frequency radar can be seen as a special case of the Chinese remainder theorem. === Decomposition of surjections of finite abelian groups === Given a surjection Z / n → Z / m {\displaystyle \mathbb {Z} /n\to \mathbb {Z} /m} of finite abelian groups, we can use the Chinese remainder theorem to give a complete description of any such map. First of all, the theorem gives isomorphisms Z / n ≅ Z / p n 1 a 1 × ⋯ × Z / p n i a i Z / m ≅ Z / p m 1 b 1 × ⋯ × Z / p m j b j {\displaystyle {\begin{aligned}\mathbb {Z} /n&\cong \mathbb {Z} /p_{n_{1}}^{a_{1}}\times \cdots \times \mathbb {Z} /p_{n_{i}}^{a_{i}}\\\mathbb {Z} /m&\cong \mathbb {Z} /p_{m_{1}}^{b_{1}}\times \cdots \times \mathbb {Z} /p_{m_{j}}^{b_{j}}\end{aligned}}} where { p m 1 , … , p m j } ⊆ { p n 1 , … , p n i } {\displaystyle \{p_{m_{1}},\ldots ,p_{m_{j}}\}\subseteq \{p_{n_{1}},\ldots ,p_{n_{i}}\}} . In addition, for any induced map Z / p n k a k → Z / p m l b l {\displaystyle \mathbb {Z} /p_{n_{k}}^{a_{k}}\to \mathbb {Z} /p_{m_{l}}^{b_{l}}} from the original surjection, we have a k ≥ b l {\displaystyle a_{k}\geq b_{l}} and p n k = p m l , {\displaystyle p_{n_{k}}=p_{m_{l}},} since for a pair of primes p , q {\displaystyle p,q} , the only non-zero surjections Z / p a → Z / q b {\displaystyle \mathbb {Z} /p^{a}\to \mathbb {Z} /q^{b}} can be defined if p = q {\displaystyle p=q} and a ≥ b {\displaystyle a\geq b} . These observations are pivotal for constructing the ring of profinite integers, which is given as an inverse limit of all such maps. === Dedekind's theorem === Dedekind's theorem on the linear independence of characters. Let M be a monoid and k an integral domain, viewed as a monoid by considering the multiplication on k. Then any finite family ( fi )i∈I of distinct monoid homomorphisms fi : M → k is linearly independent. In other words, every family (αi)i∈I of elements αi ∈ k satisfying ∑ i ∈ I α i f i = 0 {\displaystyle \sum _{i\in I}\alpha _{i}f_{i}=0} must be equal to the family (0)i∈I. Proof. First assume that k is a field, otherwise, replace the integral domain k by its quotient field, and nothing will change. We can linearly extend the monoid homomorphisms fi : M → k to k-algebra homomorphisms Fi : k[M] → k, where k[M] is the monoid ring of M over k. Then, by linearity, the condition ∑ i ∈ I α i f i = 0 , {\displaystyle \sum _{i\in I}\alpha _{i}f_{i}=0,} yields ∑ i ∈ I α i F i = 0. {\displaystyle \sum _{i\in I}\alpha _{i}F_{i}=0.} Next, for i, j ∈ I; i ≠ j the two k-linear maps Fi : k[M] → k and Fj : k[M] → k are not proportional to each other. Otherwise fi and fj would also be proportional, and thus equal since as monoid homomorphisms they satisfy: fi (1) = 1 = fj (1), which contradicts the assumption that they are distinct. Therefore, the kernels Ker Fi and Ker Fj are distinct. Since k[M]/Ker Fi ≅ Fi (k[M]) = k is a field, Ker Fi is a maximal ideal of k[M] for every i in I. Because they are distinct and maximal the ideals Ker Fi and Ker Fj are coprime whenever i ≠ j. The Chinese Remainder Theorem (for general rings) yields an isomorphism: ϕ : k [ M ] / K → ∏ i ∈ I k [ M ] / K e r F i ϕ ( x + K ) = ( x + K e r F i ) i ∈ I {\displaystyle {\begin{aligned}\phi :k[M]/K&\to \prod _{i\in I}k[M]/\mathrm {Ker} F_{i}\\\phi (x+K)&=\left(x+\mathrm {Ker} F_{i}\right)_{i\in I}\end{aligned}}} where K = ∏ i ∈ I K e r F i = ⋂ i ∈ I K e r F i . {\displaystyle K=\prod _{i\in I}\mathrm {Ker} F_{i}=\bigcap _{i\in I}\mathrm {Ker} F_{i}.} Consequently, the map Φ : k [ M ] → ∏ i ∈ I k [ M ] / K e r F i Φ ( x ) = ( x + K e r F i ) i ∈ I {\displaystyle {\begin{aligned}\Phi :k[M]&\to \prod _{i\in I}k[M]/\mathrm {Ker} F_{i}\\\Phi (x)&=\left(x+\mathrm {Ker} F_{i}\right)_{i\in I}\end{aligned}}} is surjective. Under the isomorphisms k[M]/Ker Fi → Fi (k[M]) = k, the map Φ corresponds to: ψ : k [ M ] → ∏ i ∈ I k ψ ( x ) = [ F i ( x ) ] i ∈ I {\displaystyle {\begin{aligned}\psi :k[M]&\to \prod _{i\in I}k\\\psi (x)&=\left[F_{i}(x)\right]_{i\in I}\end{aligned}}} Now, ∑ i ∈ I α i F i = 0 {\displaystyle \sum _{i\in I}\alpha _{i}F_{i}=0} yields ∑ i ∈ I α i u i = 0 {\displaystyle \sum _{i\in I}\alpha _{i}u_{i}=0} for every vector (ui)i∈I in the image of the map ψ. Since ψ is surjective, this means that ∑ i ∈ I α i u i = 0 {\displaystyle \sum _{i\in I}\alpha _{i}u_{i}=0} for every vector ( u i ) i ∈ I ∈ ∏ i ∈ I k . {\displaystyle \left(u_{i}\right)_{i\in I}\in \prod _{i\in I}k.} Consequently, (αi)i∈I = (0)i∈I. QED. == See also == Covering system Hasse principle Residue number system == Notes == == References == Dauben, Joseph W. (2007), "Chapter 3: Chinese Mathematics", in Katz, Victor J. (ed.), The Mathematics of Egypt, Mesopotamia, China, India and Islam : A Sourcebook, Princeton University Press, pp. 187–384, ISBN 978-0-691-11485-9 Dence, Joseph B.; Dence, Thomas P. (1999), Elements of the Theory of Numbers, Academic Press, ISBN 9780122091308 Duchet, Pierre (1995), "Hypergraphs", in Graham, R. L.; Grötschel, M.; Lovász, L. (eds.), Handbook of combinatorics, Vol. 1, 2, Amsterdam: Elsevier, pp. 381–432, MR 1373663. See in particular Section 2.5, "Helly Property", pp. 393–394. Gauss, Carl Friedrich (1986), Disquisitiones Arithemeticae, translated by Clarke, Arthur A. (Second, corrected ed.), New York: Springer, ISBN 978-0-387-96254-2 Ireland, Kenneth; Rosen, Michael (1990), A Classical Introduction to Modern Number Theory (2nd ed.), Springer-Verlag, ISBN 0-387-97329-X Kak, Subhash (1986), "Computational aspects of the Aryabhata algorithm" (PDF), Indian Journal of History of Science, 21 (1): 62–71 Katz, Victor J. (1998), A History of Mathematics / An Introduction (2nd ed.), Addison Wesley Longman, ISBN 978-0-321-01618-8 Libbrecht, Ulrich (1973), Chinese Mathematics in the Thirteenth Century: the "Shu-shu Chiu-chang" of Ch'in Chiu-shao, Dover Publications Inc, ISBN 978-0-486-44619-6 Ore, Øystein (1952), "The general Chinese remainder theorem", The American Mathematical Monthly, 59 (6): 365–370, doi:10.2307/2306804, JSTOR 2306804, MR 0048481 Ore, Oystein (1988) [1948], Number Theory and Its History, Dover, ISBN 978-0-486-65620-5 Pisano, Leonardo (2002), Fibonacci's Liber Abaci, translated by Sigler, Laurence E., Springer-Verlag, pp. 402–403, ISBN 0-387-95419-8 Rosen, Kenneth H. (1993), Elementary Number Theory and its Applications (3rd ed.), Addison-Wesley, ISBN 978-0201-57889-8 Sengupta, Ambar N. (2012), Representing Finite Groups, A Semisimple Introduction, Springer, ISBN 978-1-4614-1232-8 Bourbaki, N. (1989), Algebra I, Springer, ISBN 3-540-64243-9 == Further reading == Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L.; Stein, Clifford (2001), Introduction to Algorithms (Second ed.), MIT Press and McGraw-Hill, ISBN 0-262-03293-7. See Section 31.5: The Chinese remainder theorem, pp. 873–876. Ding, Cunsheng; Pei, Dingyi; Salomaa, Arto (1996), Chinese Remainder Theorem: Applications in Computing, Coding, Cryptography, World Scientific Publishing, pp. 1–213, ISBN 981-02-2827-9 Hungerford, Thomas W. (1974), Algebra, Graduate Texts in Mathematics, Vol. 73, Springer-Verlag, pp. 131–132, ISBN 978-1-4612-6101-8 Knuth, Donald (1997), The Art of Computer Programming, vol. 2: Seminumerical Algorithms (Third ed.), Addison-Wesley, ISBN 0-201-89684-2. See Section 4.3.2 (pp. 286–291), exercise 4.6.2–3 (page 456). == External links == "Chinese remainder theorem", Encyclopedia of Mathematics, EMS Press, 2001 [1994] Weisstein, Eric W., "Chinese Remainder Theorem", MathWorld Chinese Remainder Theorem at PlanetMath. Full text of the Sun-tzu Suan-ching (Chinese) – Chinese Text Project
Wikipedia:Chiral Lie algebra#0	In algebra, a chiral Lie algebra is a D-module on a curve with a certain structure of Lie algebra. It is related to an E 2 {\displaystyle {\mathcal {E}}_{2}} -algebra via the Riemann–Hilbert correspondence. == See also == Chiral algebra Chiral homology == References == Francis, John; Gaitsgory, Dennis (2012). "Chiral Koszul duality". Sel. Math. New Series. 18 (1): 27–87. arXiv:1103.5803. doi:10.1007/s00029-011-0065-z.
Wikipedia:Chiral algebra#0	In mathematics, a chiral algebra is an algebraic structure introduced by Beilinson & Drinfeld (2004) as a rigorous version of the rather vague concept of a chiral algebra in physics. In Chiral Algebras, Beilinson and Drinfeld introduced the notion of chiral algebra, which based on the pseudo-tensor category of D-modules. They give a 'coordinate independent' notion of vertex algebras, which are based on formal power series. Chiral algebras on curves are essentially conformal vertex algebras. == Definition == A chiral algebra on a smooth algebraic curve X {\displaystyle X} is a right D-module A {\displaystyle {\mathcal {A}}} , equipped with a D-module homomorphism μ : A ⊠ A ( ∞ Δ ) → Δ ! A {\displaystyle \mu :{\mathcal {A}}\boxtimes {\mathcal {A}}(\infty \Delta )\rightarrow \Delta _{!}{\mathcal {A}}} on X 2 {\displaystyle X^{2}} and with an embedding Ω ↪ A {\displaystyle \Omega \hookrightarrow {\mathcal {A}}} , satisfying the following conditions μ = − σ 12 ∘ μ ∘ σ 12 {\displaystyle \mu =-\sigma _{12}\circ \mu \circ \sigma _{12}} (Skew-symmetry) μ 1 { 23 } = μ { 12 } 3 + μ 2 { 13 } {\displaystyle \mu _{1\{23\}}=\mu _{\{12\}3}+\mu _{2\{13\}}} (Jacobi identity) The unit map is compatible with the homomorphism μ Ω : Ω ⊠ Ω ( ∞ Δ ) → Δ ! Ω {\displaystyle \mu _{\Omega }:\Omega \boxtimes \Omega (\infty \Delta )\rightarrow \Delta _{!}\Omega } ; that is, the following diagram commutes Ω ⊠ A ( ∞ Δ ) → A ⊠ A ( ∞ Δ ) ↓ ↓ Δ ! A → Δ ! A {\displaystyle {\begin{array}{lcl}&\Omega \boxtimes {\mathcal {A}}(\infty \Delta )&\rightarrow &{\mathcal {A}}\boxtimes {\mathcal {A}}(\infty \Delta )&\\&\downarrow &&\downarrow \\&\Delta _{!}{\mathcal {A}}&\rightarrow &\Delta _{!}{\mathcal {A}}&\\\end{array}}} Where, for sheaves M , N {\displaystyle {\mathcal {M}},{\mathcal {N}}} on X {\displaystyle X} , the sheaf M ⊠ N ( ∞ Δ ) {\displaystyle {\mathcal {M}}\boxtimes {\mathcal {N}}(\infty \Delta )} is the sheaf on X 2 {\displaystyle X^{2}} whose sections are sections of the external tensor product M ⊠ N {\displaystyle {\mathcal {M}}\boxtimes {\mathcal {N}}} with arbitrary poles on the diagonal: M ⊠ N ( ∞ Δ ) = lim → ⁡ M ⊠ N ( n Δ ) , {\displaystyle {\mathcal {M}}\boxtimes {\mathcal {N}}(\infty \Delta )=\varinjlim {\mathcal {M}}\boxtimes {\mathcal {N}}(n\Delta ),} Ω {\displaystyle \Omega } is the canonical bundle, and the 'diagonal extension by delta-functions' Δ ! {\displaystyle \Delta _{!}} is Δ ! M = Ω ⊠ M ( ∞ Δ ) Ω ⊠ M . {\displaystyle \Delta _{!}{\mathcal {M}}={\frac {\Omega \boxtimes {\mathcal {M}}(\infty \Delta )}{\Omega \boxtimes {\mathcal {M}}}}.} == Relation to other algebras == === Vertex algebra === The category of vertex algebras as defined by Borcherds or Kac is equivalent to the category of chiral algebras on X = A 1 {\displaystyle X=\mathbb {A} ^{1}} equivariant with respect to the group T {\displaystyle T} of translations. === Factorization algebra === Chiral algebras can also be reformulated as factorization algebras. == See also == Chiral homology Chiral Lie algebra == References == Beilinson, Alexander; Drinfeld, Vladimir (2004), Chiral algebras, American Mathematical Society Colloquium Publications, vol. 51, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-3528-9, MR 2058353 == Further reading == Francis, John; Gaitsgory, Dennis (2012). "Chiral Koszul duality". Sel. Math. New Series. 18 (1): 27–87. arXiv:1103.5803. doi:10.1007/s00029-011-0065-z. S2CID 8316715.
Wikipedia:Chisini mean#0	In mathematics, a function f of n variables x1, ..., xn leads to a Chisini mean M if, for every vector ⟨x1, ..., xn⟩, there exists a unique M such that f(M,M, ..., M) = f(x1,x2, ..., xn). The arithmetic, harmonic, geometric, generalised, Heronian and quadratic means are all Chisini means, as are their weighted variants. While Oscar Chisini was arguably the first to deal with "substitution means" in some depth in 1929, the idea of defining a mean as above is quite old, appearing (for example) in early works of Augustus De Morgan. == See also == Fréchet mean Generalized mean Jensen's inequality Quasi-arithmetic mean Stolarsky mean == References ==
Wikipedia:Chitrabhanu (mathematician)#0	Chitrabhanu (IAST: Citrabhānu; fl. 16th century) was a mathematician of the Kerala school and a student of Nilakantha Somayaji. He was a Nambudiri brahmin from the town of Covvaram near present day Trissur. He is noted for a karaṇa, a concise astronomical manual, dated to 1530, an algebraic treatise, and a commentary on a poetic text. Nilakantha and he were both teachers of Shankara Variyar. == Contributions == He gave integer solutions to 21 types of systems of two simultaneous Diophantine equations in two unknowns. These types are all the possible pairs of equations of the following seven forms: x + y = a , x − y = b , x y = c , x 2 + y 2 = d , x 2 − y 2 = e , x 3 + y 3 = f , x 3 − y 3 = g {\displaystyle \ x+y=a,x-y=b,xy=c,x^{2}+y^{2}=d,x^{2}-y^{2}=e,x^{3}+y^{3}=f,x^{3}-y^{3}=g} For each case, Chitrabhanu gave an explanation and justification of his rule as well as an example. Some of his explanations are algebraic, while others are geometric. == References ==
Wikipedia:Chiungtze C. Tsen#0	Chiungtze C. Tsen (Chinese: 曾炯之; pinyin: Zēng Jiǒngzhī; Wade–Giles: Tseng Chiung-chih; Chang-Du Gan: [tsɛn˦˨ tɕjuŋ˨˩˧ tsɹ̩˦˨], April 2, 1898 – October 1, 1940), given name Chiung (Chinese: 炯; pinyin: Jiǒng), was a Chinese mathematician born in Nanchang, Jiangxi. He is known for his work in algebra. He was one of Emmy Noether's students at the University of Göttingen, Germany. One of his research interests was quasi-algebraic closure. In that area he proved a fundamental result which is now called Tsen's theorem. == Biography == Tsen was born in a poor fisherman's family in Xinjian Country, Nanchang, Jiangxi Province. His father Tschu-Wun Tsen (曾祖文 Zeng Zuwen) had two sons and several daughters, and Tsen was the eldest son. His uncle Lei Heng (雷恒), who was a jinshi and a member of the Hanlin Academy, persuaded Tsen's father to send Tsen to school. Due to poverty, Tsen had to take leaves from school intermittently to work. After leaving primary school, he worked in a coal mine while self-studying. In 1917, he passed the entrance examination and was admitted to Jiangxi Provincial First Normal College in Nanchang. He was subsidised by Lei Heng's son Tsebu S. Lee (雷子布 Lei Zibu, given name 宣 Xuan), who was studying in Japan on government scholarship. After graduation in 1920, Tsen taught in primary school for two years. In 1922, Tsen entered National Wuchang Senior Normal College, later National Wuchang University, to study undergraduate mathematics, and he graduated in 1926. After graduation, he worked as teacher in high schools for two years to perform the mandatory teaching service of his degree. In 1927, when Kuomintang split with the Chinese Communist Party, Tsen and some teachers and students protested against the breakup and called for alliance. Several of them including Tsen were beaten up and were hospitalized. Guo Moruo, then serving as director of the political department of the National Revolutionary Army, visited them in the hospital. In 1928, Tsen passed the Jiangxi provincial government scholarship examination for studying in Europe and America. He went to Berlin University for language training for a year, and then he started studying mathematics at University of Göttingen in the summer semester of 1929. He studied algebra under Emmy Noether. Tsen received his doctoral degree in February 1934 under the supervision of Emmy Noether and Friedrich Karl Schmidt, and he dedicated his dissertation to his elder cousin Tsebu S. Lee. Having fled to the US, Noether evaluated the dissertation in a letter as "sehr gut" (very good). As a research fellow sponsored by the China Foundation for the Promotion of Education and Culture, Tsen did a postdoctoral research with Emil Artin at Hamburg University for a year. There he became friend with Shiing-Shen Chern, who was a graduate student back then. Chern remembered him as a cordial and open-minded person well-liked by everyone. Tsen returned to China in July 1935 and was invited by Chen Jiangong to National Chekiang University in Hangzhou as professor in the area of algebra. Chen was Tsen's teacher at Wuchang Senior Normal College and had encouraged Tsen to study in Germany. Tsen taught a course on algebra and a course on group theory based on the German textbooks of van der Waerden and Andreas Speiser respectively. As the books were in German, it was not easy for the students to understand, so he edited the notes taken by his student Chuan-Chih Hsiung and printed out for the students. In 1936, Tsen published his third paper in the journal of the new Chinese Mathematical Society. The paper contained the work that he had done in Hamburg, and he dedicated it to the memory of his advisor Noether, who died in the previous year. The paper was hardly known outside China before 1970s, and the results therein were rediscovered by Serge Lang in his dissertation. Ernst Witt, who was Tsen's friend and had also been a student of Noether, always talked about Tsen's results in his algebra lectures and would correct others if they attributed them to Lang but not Tsen, thus helped bring attention to this paper. Tsen and Chen fell out because of a failed matching of Tsen and Chen's younger sister for marriage. In 1937, Tsen left Chekiang University and was invited by National Beiyang Institute of Technology to become a professor. That year, the full-scale Japanese invasion of China started, and the school was evacuated from Tianjin to Xi'an. Tsen went to Xi'an to take up his post. The school merged with some other evacuated universities to form National Xi'an Provisory University. The new university moved to Hanzhong and was renamed to National Northwestern Associated University, and it moved again to Chenggu. The university soon split into several schools, one of which was National Northwestern Institute of Technology, and Tsen became a professor of this school. He married a high-school chemistry teacher Qin Hesui (秦禾穗) in Nanchang in 1937. His wife suffered a miscarriage on the long and difficult journey over mountainous terrains to Xichang. They had no children. He adopted a nephew as his son. He bought a lot of mathematics books while in Germany, and he brought the books and his manuscripts back to China in seven full metal trunks. After the start of war, he kept them in his relative's home at a village in Xinjian, Jiangxi. Unfortunately, when the village had fallen, all his seven trunks of books and manuscripts were burnt by the Japanese invaders. In 1939, Shu-tien Li, former president of Beiyang Institute of Technology and the president of the newly-founded National Xikang Institute of Technology, invited Tsen to be a professor at the new school. The campus of the school was temples scattered on Mount Lu in the suburb of Xichang in Xikang Province. Tsen had a chronic stomach problem, and his condition was made worse by poor living condition and shortage of medical supplies in time of war. Tsen died of a stomach ulcer in Xichang, Xikang on October 1, 1940, and the school held a memorial service for him on November 18, 1940. == Publications == Tsen, Chiungtze C. Divisionsalgebren über Funktionenkörpern. Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl. I, No.44, II, No.48, 335–339 (1933). Tsen, Chiungtze C. Algebren über Funktionenkörpern. Göttingen: Diss. 19 S. (1934). Tsen, Chiungtze C. Zur Stufentheorie der quasialgebraisch-Abgeschlossenheit kommutativer Körper. J. Chin. Math. Soc. 1, 81–92 (1936). A Chinese translation of these three papers was published in a book in memory of Tsen. === Short articles === Some short articles written by Tsen in Chinese that can be found: 曾烱 Zeng Jiong (1924). "Faa de Bruno's Theorem". 國立武昌師範大學數理化雜誌 Guoli Wuchang Shifan Daxue Shulihua Zazhi (12): 38–41. 曾烱 Zeng Jiong (1930). "實數表示法及其應用一題問 Shishu Biaoshifa Ji Qi Yingyong Yiti Wen". 留德學誌 Liu de Xuezhi (1): 25–30. 曾烱 Zeng Jiong (1933). "數學討論（一） Shuxue Taolun (Yi)". 山東大學科學叢刊 Shandong Daxue Kexue Congkan. 1 (1): 24–26. == See also == Tsen rank == Notes == == References == Chiungtze C. Tsen at the Mathematics Genealogy Project
Wikipedia:Choi's theorem on completely positive maps#0	In mathematics, Choi's theorem on completely positive maps is a result that classifies completely positive maps between finite-dimensional (matrix) C-algebras. The theorem is due to Man-Duen Choi. An infinite-dimensional algebraic generalization of Choi's theorem is known as Belavkin's "Radon–Nikodym" theorem for completely positive maps. == Statement == Choi's theorem. Let Φ : C n × n → C m × m {\displaystyle \Phi :\mathbb {C} ^{n\times n}\to \mathbb {C} ^{m\times m}} be a linear map. The following are equivalent: (i) Φ is n-positive (i.e. ( id n ⊗ Φ ) ( A ) ∈ C n × n ⊗ C m × m {\displaystyle \left(\operatorname {id} _{n}\otimes \Phi \right)(A)\in \mathbb {C} ^{n\times n}\otimes \mathbb {C} ^{m\times m}} is positive whenever A ∈ C n × n ⊗ C n × n {\displaystyle A\in \mathbb {C} ^{n\times n}\otimes \mathbb {C} ^{n\times n}} is positive). (ii) The matrix with operator entries C Φ = ( id n ⊗ Φ ) ( ∑ i j E i j ⊗ E i j ) = ∑ i j E i j ⊗ Φ ( E i j ) ∈ C n m × n m {\displaystyle C_{\Phi }=\left(\operatorname {id} _{n}\otimes \Phi \right)\left(\sum _{ij}E_{ij}\otimes E_{ij}\right)=\sum _{ij}E_{ij}\otimes \Phi (E_{ij})\in \mathbb {C} ^{nm\times nm}} is positive, where E i j ∈ C n × n {\displaystyle E_{ij}\in \mathbb {C} ^{n\times n}} is the matrix with 1 in the ij-th entry and 0s elsewhere. (The matrix CΦ is sometimes called the Choi matrix of Φ.) (iii) Φ is completely positive. == Proof == === (i) implies (ii) === We observe that if E = ∑ i j E i j ⊗ E i j , {\displaystyle E=\sum _{ij}E_{ij}\otimes E_{ij},} then E=E and E2=nE, so E=n−1EE* which is positive. Therefore CΦ =(In ⊗ Φ)(E) is positive by the n-positivity of Φ. === (iii) implies (i) === This holds trivially. === (ii) implies (iii) === This mainly involves chasing the different ways of looking at Cnm×nm: C n m × n m ≅ C n m ⊗ ( C n m ) ∗ ≅ C n ⊗ C m ⊗ ( C n ⊗ C m ) ∗ ≅ C n ⊗ ( C n ) ∗ ⊗ C m ⊗ ( C m ) ∗ ≅ C n × n ⊗ C m × m . {\displaystyle \mathbb {C} ^{nm\times nm}\cong \mathbb {C} ^{nm}\otimes (\mathbb {C} ^{nm})^{}\cong \mathbb {C} ^{n}\otimes \mathbb {C} ^{m}\otimes (\mathbb {C} ^{n}\otimes \mathbb {C} ^{m})^{}\cong \mathbb {C} ^{n}\otimes (\mathbb {C} ^{n})^{}\otimes \mathbb {C} ^{m}\otimes (\mathbb {C} ^{m})^{}\cong \mathbb {C} ^{n\times n}\otimes \mathbb {C} ^{m\times m}.} Let the eigenvector decomposition of CΦ be C Φ = ∑ i = 1 n m λ i v i v i ∗ , {\displaystyle C_{\Phi }=\sum _{i=1}^{nm}\lambda _{i}v_{i}v_{i}^{},} where the vectors v i {\displaystyle v_{i}} lie in Cnm . By assumption, each eigenvalue λ i {\displaystyle \lambda _{i}} is non-negative so we can absorb the eigenvalues in the eigenvectors and redefine v i {\displaystyle v_{i}} so that C Φ = ∑ i = 1 n m v i v i ∗ . {\displaystyle \;C_{\Phi }=\sum _{i=1}^{nm}v_{i}v_{i}^{}.} The vector space Cnm can be viewed as the direct sum ⊕ i = 1 n C m {\displaystyle \textstyle \oplus _{i=1}^{n}\mathbb {C} ^{m}} compatibly with the above identification C n m ≅ C n ⊗ C m {\displaystyle \textstyle \mathbb {C} ^{nm}\cong \mathbb {C} ^{n}\otimes \mathbb {C} ^{m}} and the standard basis of Cn. If Pk ∈ Cm × nm is projection onto the k-th copy of Cm, then Pk* ∈ Cnm×m is the inclusion of Cm as the k-th summand of the direct sum and Φ ( E k l ) = P k ⋅ C Φ ⋅ P l ∗ = ∑ i = 1 n m P k v i ( P l v i ) ∗ . {\displaystyle \;\Phi (E_{kl})=P_{k}\cdot C_{\Phi }\cdot P_{l}^{}=\sum _{i=1}^{nm}P_{k}v_{i}(P_{l}v_{i})^{}.} Now if the operators Vi ∈ Cm×n are defined on the k-th standard basis vector ek of Cn by V i e k = P k v i , {\displaystyle \;V_{i}e_{k}=P_{k}v_{i},} then Φ ( E k l ) = ∑ i = 1 n m P k v i ( P l v i ) ∗ = ∑ i = 1 n m V i e k e l ∗ V i ∗ = ∑ i = 1 n m V i E k l V i ∗ . {\displaystyle \Phi (E_{kl})=\sum _{i=1}^{nm}P_{k}v_{i}(P_{l}v_{i})^{}=\sum _{i=1}^{nm}V_{i}e_{k}e_{l}^{}V_{i}^{}=\sum _{i=1}^{nm}V_{i}E_{kl}V_{i}^{}.} Extending by linearity gives us Φ ( A ) = ∑ i = 1 n m V i A V i ∗ {\displaystyle \Phi (A)=\sum _{i=1}^{nm}V_{i}AV_{i}^{}} for any A ∈ Cn×n. Any map of this form is manifestly completely positive: the map A → V i A V i ∗ {\displaystyle A\to V_{i}AV_{i}^{}} is completely positive, and the sum (across i {\displaystyle i} ) of completely positive operators is again completely positive. Thus Φ {\displaystyle \Phi } is completely positive, the desired result. The above is essentially Choi's original proof. Alternative proofs have also been known. == Consequences == === Kraus operators === In the context of quantum information theory, the operators {Vi} are called the Kraus operators (after Karl Kraus) of Φ. Notice, given a completely positive Φ, its Kraus operators need not be unique. For example, any "square root" factorization of the Choi matrix CΦ = B∗B gives a set of Kraus operators. Let B ∗ = [ b 1 , … , b n m ] , {\displaystyle B^{}=[b_{1},\ldots ,b_{nm}],} where bi's are the row vectors of B, then C Φ = ∑ i = 1 n m b i b i ∗ . {\displaystyle C_{\Phi }=\sum _{i=1}^{nm}b_{i}b_{i}^{}.} The corresponding Kraus operators can be obtained by exactly the same argument from the proof. When the Kraus operators are obtained from the eigenvector decomposition of the Choi matrix, because the eigenvectors form an orthogonal set, the corresponding Kraus operators are also orthogonal in the Hilbert–Schmidt inner product. This is not true in general for Kraus operators obtained from square root factorizations. (Positive semidefinite matrices do not generally have a unique square-root factorizations.) If two sets of Kraus operators {Ai}1nm and {Bi}1nm represent the same completely positive map Φ, then there exists a unitary operator matrix { U i j } i j ∈ C n m 2 × n m 2 such that A i = ∑ j = 1 U i j B j . {\displaystyle \{U_{ij}\}_{ij}\in \mathbb {C} ^{nm^{2}\times nm^{2}}\quad {\text{such that}}\quad A_{i}=\sum _{j=1}U_{ij}B_{j}.} This can be viewed as a special case of the result relating two minimal Stinespring representations. Alternatively, there is an isometry scalar matrix {uij}ij ∈ Cnm × nm such that A i = ∑ j = 1 u i j B j . {\displaystyle A_{i}=\sum _{j=1}u_{ij}B_{j}.} This follows from the fact that for two square matrices M and N, M M = N N* if and only if M = N U for some unitary U. === Completely copositive maps === It follows immediately from Choi's theorem that Φ is completely copositive if and only if it is of the form Φ ( A ) = ∑ i V i A T V i ∗ . {\displaystyle \Phi (A)=\sum _{i}V_{i}A^{T}V_{i}^{}.} === Hermitian-preserving maps === Choi's technique can be used to obtain a similar result for a more general class of maps. Φ is said to be Hermitian-preserving if A is Hermitian implies Φ(A) is also Hermitian. One can show Φ is Hermitian-preserving if and only if it is of the form Φ ( A ) = ∑ i = 1 n m λ i V i A V i ∗ {\displaystyle \Phi (A)=\sum _{i=1}^{nm}\lambda _{i}V_{i}AV_{i}^{}} where λi are real numbers, the eigenvalues of CΦ, and each Vi corresponds to an eigenvector of CΦ. Unlike the completely positive case, CΦ may fail to be positive. Since Hermitian matrices do not admit factorizations of the form B*B in general, the Kraus representation is no longer possible for a given Φ. == See also == Stinespring factorization theorem Quantum operation Holevo's theorem == References == M.-D. Choi, Completely Positive Linear Maps on Complex Matrices, Linear Algebra and its Applications, 10, 285–290 (1975). V. P. Belavkin, P. Staszewski, Radon-Nikodym Theorem for Completely Positive Maps, Reports on Mathematical Physics, v.24, No 1, 49–55 (1986). J. de Pillis, Linear Transformations Which Preserve Hermitian and Positive Semidefinite Operators, Pacific Journal of Mathematics, 23, 129–137 (1967).
Wikipedia:Chong Chi Tat#0	Chong Chi Tat (Chinese: 莊志達; pinyin: Zhuāng Zhìdá) is university professor and director of the Institute for Mathematical Sciences at the National University of Singapore (NUS). His research interests are in the areas of recursion/computability theory. == Academic career == Chong received his BSc with distinction from Iowa State University and his PhD from Yale University. He began his career in University of Singapore, the predecessor of NUS, as a lecturer in 1974. He was subsequently promoted to senior lecturer (1980–1985), associate professor (1985–1989), and professor (1989–2004). Chong held many administrative leadership positions, including vice dean of science (1985–1996), department head of information systems and computer science (1993–1996), deputy vice chancellor, deputy president and provost (1996–2004); and department head of mathematics (2006–2012). In 2004, Chong was named the second University professor of NUS, the highest honor bestowed upon a very small number of its tenured faculty members. == Selected works == Chi Tat Chong, Theodore A Slaman and Yue Yang, The inductive strength of Ramsey's Theorem for Pairs, Advances in Mathematics 308 (2017), 121–141. Chi Tat Chong, Theodore A Slaman and Yue Yang, The metamathematics of stable Ramsey’s theorem for pairs, Journal of the American Mathematical Society 27 (2014), 863-892. Chi Tat Chong, Theodore A Slaman and Yue Yang, Pi^1_1 conservation of combinatorial principles weaker than Ramsey’s theorem for pairs, Advances in Mathematics 230 (2012), 1060-1071. Chi Tat Chong and Yue Yang, The jump of a Sigma_n cut, Journal of the London Mathematical Society 75 (2007), 690—704. Chi Tat Chong and Yue Yang, Σ_2 induction and infinite injury priority argument. I. Maximal sets and the jump operator. Journal of Symbolic Logic 63 (1998), no. 3, 797–814. == Awards and honors == Chong received the Pingat Pentadbiran Awam - Emas (Public Administration Medal - Gold), in Singapore's National Day Awards 2002. He has been a Fellow of the Singapore National Academy of Science since 2011. == References == == External links == One Page CV List of Recent Publications
Wikipedia:Chow group#0	In algebraic geometry, the Chow groups (named after Wei-Liang Chow by Claude Chevalley (1958)) of an algebraic variety over any field are algebro-geometric analogs of the homology of a topological space. The elements of the Chow group are formed out of subvarieties (so-called algebraic cycles) in a similar way to how simplicial or cellular homology groups are formed out of subcomplexes. When the variety is smooth, the Chow groups can be interpreted as cohomology groups (compare Poincaré duality) and have a multiplication called the intersection product. The Chow groups carry rich information about an algebraic variety, and they are correspondingly hard to compute in general. == Rational equivalence and Chow groups == For what follows, define a variety over a field k {\displaystyle k} to be an integral scheme of finite type over k {\displaystyle k} . For any scheme X {\displaystyle X} of finite type over k {\displaystyle k} , an algebraic cycle on X {\displaystyle X} means a finite linear combination of subvarieties of X {\displaystyle X} with integer coefficients. (Here and below, subvarieties are understood to be closed in X {\displaystyle X} , unless stated otherwise.) For a natural number i {\displaystyle i} , the group Z i ( X ) {\displaystyle Z_{i}(X)} of i {\displaystyle i} -dimensional cycles (or i {\displaystyle i} -cycles, for short) on X {\displaystyle X} is the free abelian group on the set of i {\displaystyle i} -dimensional subvarieties of X {\displaystyle X} . For a variety W {\displaystyle W} of dimension i + 1 {\displaystyle i+1} and any rational function f {\displaystyle f} on W {\displaystyle W} which is not identically zero, the divisor of f {\displaystyle f} is the i {\displaystyle i} -cycle ( f ) = ∑ Z ord Z ⁡ ( f ) Z , {\displaystyle (f)=\sum _{Z}\operatorname {ord} _{Z}(f)Z,} where the sum runs over all i {\displaystyle i} -dimensional subvarieties Z {\displaystyle Z} of W {\displaystyle W} and the integer ord Z ⁡ ( f ) {\displaystyle \operatorname {ord} _{Z}(f)} denotes the order of vanishing of f {\displaystyle f} along Z {\displaystyle Z} . (Thus ord Z ⁡ ( f ) {\displaystyle \operatorname {ord} _{Z}(f)} is negative if f {\displaystyle f} has a pole along Z {\displaystyle Z} .) The definition of the order of vanishing requires some care for W {\displaystyle W} singular. For a scheme X {\displaystyle X} of finite type over k {\displaystyle k} , the group of i {\displaystyle i} -cycles rationally equivalent to zero is the subgroup of Z i ( X ) {\displaystyle Z_{i}(X)} generated by the cycles ( f ) {\displaystyle (f)} for all ( i + 1 ) {\displaystyle (i+1)} -dimensional subvarieties W {\displaystyle W} of X {\displaystyle X} and all nonzero rational functions f {\displaystyle f} on W {\displaystyle W} . The Chow group C H i ( X ) {\displaystyle CH_{i}(X)} of i {\displaystyle i} -dimensional cycles on X {\displaystyle X} is the quotient group of Z i ( X ) {\displaystyle Z_{i}(X)} by the subgroup of cycles rationally equivalent to zero. Sometimes one writes [ Z ] {\displaystyle [Z]} for the class of a subvariety Z {\displaystyle Z} in the Chow group, and if two subvarieties Z {\displaystyle Z} and W {\displaystyle W} have [ Z ] = [ W ] {\displaystyle [Z]=[W]} , then Z {\displaystyle Z} and W {\displaystyle W} are said to be rationally equivalent. For example, when X {\displaystyle X} is a variety of dimension n {\displaystyle n} , the Chow group C H n − 1 ( X ) {\displaystyle CH_{n-1}(X)} is the divisor class group of X {\displaystyle X} . When X {\displaystyle X} is smooth over k {\displaystyle k} (or more generally, a locally Noetherian normal factorial scheme ), this is isomorphic to the Picard group of line bundles on X {\displaystyle X} . === Examples of Rational Equivalence === ==== Rational Equivalence on Projective Space ==== Rationally equivalent cycles defined by hypersurfaces are easy to construct on projective space because they can all be constructed as the vanishing loci of the same vector bundle. For example, given two homogeneous polynomials of degree d {\displaystyle d} , so f , g ∈ H 0 ( P n , O ( d ) ) {\displaystyle f,g\in H^{0}(\mathbb {P} ^{n},{\mathcal {O}}(d))} , we can construct a family of hypersurfaces defined as the vanishing locus of s f + t g {\displaystyle sf+tg} . Schematically, this can be constructed as X = Proj ( C [ s , t ] [ x 0 , … , x n ] ( s f + t g ) ) ↪ P 1 × P n {\displaystyle X={\text{Proj}}\left({\frac {\mathbb {C} [s,t][x_{0},\ldots ,x_{n}]}{(sf+tg)}}\right)\hookrightarrow \mathbb {P} ^{1}\times \mathbb {P} ^{n}} using the projection π 1 : X → P 1 {\displaystyle \pi _{1}:X\to \mathbb {P} ^{1}} we can see the fiber over a point [ s 0 : t 0 ] {\displaystyle [s_{0}:t_{0}]} is the projective hypersurface defined by s 0 f + t 0 g {\displaystyle s_{0}f+t_{0}g} . This can be used to show that the cycle class of every hypersurface of degree d {\displaystyle d} is rationally equivalent to d [ P n − 1 ] {\displaystyle d[\mathbb {P} ^{n-1}]} , since s f + t x 0 d {\displaystyle sf+tx_{0}^{d}} can be used to establish a rational equivalence. Notice that the locus of x 0 d = 0 {\displaystyle x_{0}^{d}=0} is P n − 1 {\displaystyle \mathbb {P} ^{n-1}} and it has multiplicity d {\displaystyle d} , which is the coefficient of its cycle class. ==== Rational Equivalence of Cycles on a Curve ==== If we take two distinct line bundles L , L ′ ∈ Pic ⁡ ( C ) {\displaystyle L,L'\in \operatorname {Pic} (C)} of a smooth projective curve C {\displaystyle C} , then the vanishing loci of a generic section of both line bundles defines non-equivalent cycle classes in C H ( C ) {\displaystyle CH(C)} . This is because Div ⁡ ( C ) ≅ Pic ⁡ ( C ) {\displaystyle \operatorname {Div} (C)\cong \operatorname {Pic} (C)} for smooth varieties, so the divisor classes of s ∈ H 0 ( C , L ) {\displaystyle s\in H^{0}(C,L)} and s ′ ∈ H 0 ( C , L ′ ) {\displaystyle s'\in H^{0}(C,L')} define inequivalent classes. == The Chow ring == When the scheme X {\displaystyle X} is smooth over a field k {\displaystyle k} , the Chow groups form a ring, not just a graded abelian group. Namely, when X {\displaystyle X} is smooth over k {\displaystyle k} , define C H i ( X ) {\displaystyle CH^{i}(X)} to be the Chow group of codimension- i {\displaystyle i} cycles on X {\displaystyle X} . (When X {\displaystyle X} is a variety of dimension n {\displaystyle n} , this just means that C H i ( X ) = C H n − i ( X ) {\displaystyle CH^{i}(X)=CH_{n-i}(X)} .) Then the groups C H ∗ ( X ) {\displaystyle CH^{}(X)} form a commutative graded ring with the product: C H i ( X ) × C H j ( X ) → C H i + j ( X ) . {\displaystyle CH^{i}(X)\times CH^{j}(X)\rightarrow CH^{i+j}(X).} The product arises from intersecting algebraic cycles. For example, if Y {\displaystyle Y} and Z {\displaystyle Z} are smooth subvarieties of X {\displaystyle X} of codimension i {\displaystyle i} and j {\displaystyle j} respectively, and if Y {\displaystyle Y} and Z {\displaystyle Z} intersect transversely, then the product [ Y ] [ Z ] {\displaystyle [Y][Z]} in C H i + j ( X ) {\displaystyle CH^{i+j}(X)} is the sum of the irreducible components of the intersection Y ∩ Z {\displaystyle Y\cap Z} , which all have codimension i + j {\displaystyle i+j} . More generally, in various cases, intersection theory constructs an explicit cycle that represents the product [ Y ] [ Z ] {\displaystyle [Y][Z]} in the Chow ring. For example, if Y {\displaystyle Y} and Z {\displaystyle Z} are subvarieties of complementary dimension (meaning that their dimensions sum to the dimension of X {\displaystyle X} ) whose intersection has dimension zero, then [ Y ] [ Z ] {\displaystyle [Y][Z]} is equal to the sum of the points of the intersection with coefficients called intersection numbers. For any subvarieties Y {\displaystyle Y} and Z {\displaystyle Z} of a smooth scheme X {\displaystyle X} over k {\displaystyle k} , with no assumption on the dimension of the intersection, William Fulton and Robert MacPherson's intersection theory constructs a canonical element of the Chow groups of Y ∩ Z {\displaystyle Y\cap Z} whose image in the Chow groups of X {\displaystyle X} is the product [ Y ] [ Z ] {\displaystyle [Y][Z]} . == Examples == === Projective space === The Chow ring of projective space P n {\displaystyle \mathbb {P} ^{n}} over any field k {\displaystyle k} is the ring C H ∗ ( P n ) ≅ Z [ H ] / ( H n + 1 ) , {\displaystyle CH^{}(\mathbb {P} ^{n})\cong \mathbf {Z} [H]/(H^{n+1}),} where H {\displaystyle H} is the class of a hyperplane (the zero locus of a single linear function). Furthermore, any subvariety Y {\displaystyle Y} of degree d {\displaystyle d} and codimension a {\displaystyle a} in projective space is rationally equivalent to d H a {\displaystyle dH^{a}} . It follows that for any two subvarieties Y {\displaystyle Y} and Z {\displaystyle Z} of complementary dimension in P n {\displaystyle \mathbb {P} ^{n}} and degrees a {\displaystyle a} , b {\displaystyle b} , respectively, their product in the Chow ring is simply [ Y ] ⋅ [ Z ] = a b H n {\displaystyle [Y]\cdot [Z]=a\,b\,H^{n}} where H n {\displaystyle H^{n}} is the class of a k {\displaystyle k} -rational point in P n {\displaystyle \mathbb {P} ^{n}} . For example, if Y {\displaystyle Y} and Z {\displaystyle Z} intersect transversely, it follows that Y ∩ Z {\displaystyle Y\cap Z} is a zero-cycle of degree a b {\displaystyle ab} . If the base field k {\displaystyle k} is algebraically closed, this means that there are exactly a b {\displaystyle ab} points of intersection; this is a version of Bézout's theorem, a classic result of enumerative geometry. === Projective bundle formula === Given a vector bundle E → X {\displaystyle E\to X} of rank r {\displaystyle r} over a smooth proper scheme X {\displaystyle X} over a field, the Chow ring of the associated projective bundle P ( E ) {\displaystyle \mathbb {P} (E)} can be computed using the Chow ring of X {\displaystyle X} and the Chern classes of E {\displaystyle E} . If we let ζ = c 1 ( O P ( E ) ( 1 ) ) {\displaystyle \zeta =c_{1}({\mathcal {O}}_{\mathbb {P} (E)}(1))} and c 1 , … , c r {\displaystyle c_{1},\ldots ,c_{r}} the Chern classes of E {\displaystyle E} , then there is an isomorphism of rings C H ∙ ( P ( E ) ) ≅ C H ∙ ( X ) [ ζ ] ζ r + c 1 ζ r − 1 + c 2 ζ r − 2 + ⋯ + c r {\displaystyle CH^{\bullet }(\mathbb {P} (E))\cong {\frac {CH^{\bullet }(X)[\zeta ]}{\zeta ^{r}+c_{1}\zeta ^{r-1}+c_{2}\zeta ^{r-2}+\cdots +c_{r}}}} ==== Hirzebruch surfaces ==== For example, the Chow ring of a Hirzebruch surface can be readily computed using the projective bundle formula. Recall that it is constructed as F a = P ( O ⊕ O ( a ) ) {\displaystyle F_{a}=\mathbb {P} ({\mathcal {O}}\oplus {\mathcal {O}}(a))} over P 1 {\displaystyle \mathbb {P} ^{1}} . Then, the only non-trivial Chern class of this vector bundle is c 1 = a H {\displaystyle c_{1}=aH} . This implies that the Chow ring is isomorphic to C H ∙ ( F a ) ≅ C H ∙ ( P 1 ) [ ζ ] ( ζ 2 + a H ζ ) ≅ Z [ H , ζ ] ( H 2 , ζ 2 + a H ζ ) {\displaystyle CH^{\bullet }(F_{a})\cong {\frac {CH^{\bullet }(\mathbb {P} ^{1})[\zeta ]}{(\zeta ^{2}+aH\zeta )}}\cong {\frac {\mathbf {Z} [H,\zeta ]}{(H^{2},\zeta ^{2}+aH\zeta )}}} === Remarks === For other algebraic varieties, Chow groups can have richer behavior. For example, let X {\displaystyle X} be an elliptic curve over a field k {\displaystyle k} . Then the Chow group of zero-cycles on X {\displaystyle X} fits into an exact sequence 0 → X ( k ) → C H 0 ( X ) → Z → 0. {\displaystyle 0\rightarrow X(k)\rightarrow CH_{0}(X)\rightarrow \mathbf {Z} \rightarrow 0.} Thus the Chow group of an elliptic curve X {\displaystyle X} is closely related to the group X ( k ) {\displaystyle X(k)} of k {\displaystyle k} -rational points of X {\displaystyle X} . When k {\displaystyle k} is a number field, X ( k ) {\displaystyle X(k)} is called the Mordell–Weil group of X {\displaystyle X} , and some of the deepest problems in number theory are attempts to understand this group. When k {\displaystyle k} is the complex numbers, the example of an elliptic curve shows that Chow groups can be uncountable abelian groups. == Functoriality == For a proper morphism f : X → Y {\displaystyle f:X\to Y} of schemes over k {\displaystyle k} , there is a pushforward homomorphism f ∗ : C H i ( X ) → C H i ( Y ) {\displaystyle f_{}:CH_{i}(X)\to CH_{i}(Y)} for each integer i {\displaystyle i} . For example, for a proper scheme X {\displaystyle X} over k {\displaystyle k} , this gives a homomorphism C H 0 ( X ) → Z {\displaystyle CH_{0}(X)\to \mathbf {Z} } , which takes a closed point in X {\displaystyle X} to its degree over k {\displaystyle k} . (A closed point in X {\displaystyle X} has the form Spec ⁡ ( E ) {\displaystyle \operatorname {Spec} (E)} for a finite extension field E {\displaystyle E} of k {\displaystyle k} , and its degree means the degree of the field E {\displaystyle E} over k {\displaystyle k} .) For a flat morphism f : X → Y {\displaystyle f:X\to Y} of schemes over k {\displaystyle k} with fibers of dimension r {\displaystyle r} (possibly empty), there is a homomorphism f ∗ : C H i ( Y ) → C H i + r ( X ) {\displaystyle f^{}:CH_{i}(Y)\to CH_{i+r}(X)} . A key computational tool for Chow groups is the localization sequence, as follows. For a scheme X {\displaystyle X} over a field k {\displaystyle k} and a closed subscheme Z {\displaystyle Z} of X {\displaystyle X} , there is an exact sequence C H i ( Z ) → C H i ( X ) → C H i ( X − Z ) → 0 , {\displaystyle CH_{i}(Z)\rightarrow CH_{i}(X)\rightarrow CH_{i}(X-Z)\rightarrow 0,} where the first homomorphism is the pushforward associated to the proper morphism Z → X {\displaystyle Z\to X} , and the second homomorphism is pullback with respect to the flat morphism X − Z → X {\displaystyle X-Z\to X} . The localization sequence can be extended to the left using a generalization of Chow groups, (Borel–Moore) motivic homology groups, also known as higher Chow groups. For any morphism f : X → Y {\displaystyle f:X\to Y} of smooth schemes over k {\displaystyle k} , there is a pullback homomorphism f ∗ : C H i ( Y ) → C H i ( X ) {\displaystyle f^{}:CH^{i}(Y)\to CH^{i}(X)} , which is in fact a ring homomorphism C H ∗ ( Y ) → C H ∗ ( X ) {\displaystyle CH^{}(Y)\to CH^{}(X)} . === Examples of flat pullbacks === Note that non-examples can be constructed using blowups; for example, if we take the blowup of the origin in A 2 {\displaystyle \mathbb {A} ^{2}} then the fiber over the origin is isomorphic to P 1 {\displaystyle \mathbb {P} ^{1}} . ==== Branched coverings of curves ==== Consider the branched covering of curves f : Spec ⁡ ( C [ x , y ] ( f ( x ) − g ( x , y ) ) ) → A x 1 {\displaystyle f:\operatorname {Spec} \left({\frac {\mathbb {C} [x,y]}{(f(x)-g(x,y))}}\right)\to \mathbb {A} _{x}^{1}} Since the morphism ramifies whenever f ( α ) = 0 {\displaystyle f(\alpha )=0} we get a factorization g ( α , y ) = ( y − a 1 ) e 1 ⋯ ( y − a k ) e k {\displaystyle g(\alpha ,y)=(y-a_{1})^{e_{1}}\cdots (y-a_{k})^{e_{k}}} where one of the e i > 1 {\displaystyle e_{i}>1} . This implies that the points { α 1 , … , α k } = f − 1 ( α ) {\displaystyle \{\alpha _{1},\ldots ,\alpha _{k}\}=f^{-1}(\alpha )} have multiplicities e 1 , … , e k {\displaystyle e_{1},\ldots ,e_{k}} respectively. The flat pullback of the point α {\displaystyle \alpha } is then f ∗ [ α ] = e 1 [ α ] + ⋯ + e k [ α k ] {\displaystyle f^{}[\alpha ]=e_{1}[\alpha ]+\cdots +e_{k}[\alpha _{k}]} ==== Flat family of varieties ==== Consider a flat family of varieties X → S {\displaystyle X\to S} and a subvariety S ′ ⊂ S {\displaystyle S'\subset S} . Then, using the cartesian square S ′ × S X → X ↓ ↓ S ′ → S {\displaystyle {\begin{matrix}S'\times _{S}X&\to &X\\\downarrow &&\downarrow \\S'&\to &S\end{matrix}}} we see that the image of S ′ × S X {\displaystyle S'\times _{S}X} is a subvariety of X {\displaystyle X} . Therefore, we have f ∗ [ S ′ ] = [ S ′ × S X ] {\displaystyle f^{*}[S']=[S'\times _{S}X]} == Cycle maps == There are several homomorphisms (known as cycle maps) from Chow groups to more computable theories. First, for a scheme X over the complex numbers, there is a homomorphism from Chow groups to Borel–Moore homology: C H i ( X ) → H 2 i B M ( X , Z ) . {\displaystyle {\mathit {CH}}_{i}(X)\rightarrow H_{2i}^{BM}(X,\mathbf {Z} ).} The factor of 2 appears because an i-dimensional subvariety of X has real dimension 2i. When X is smooth over the complex numbers, this cycle map can be rewritten using Poincaré duality as a homomorphism C H j ( X ) → H 2 j ( X , Z ) . {\displaystyle {\mathit {CH}}^{j}(X)\rightarrow H^{2j}(X,\mathbf {Z} ).} In this case (X smooth over C), these homomorphisms form a ring homomorphism from the Chow ring to the cohomology ring. Intuitively, this is because the products in both the Chow ring and the cohomology ring describe the intersection of cycles. For a smooth complex projective variety, the cycle map from the Chow ring to ordinary cohomology factors through a richer theory, Deligne cohomology. This incorporates the Abel–Jacobi map from cycles homologically equivalent to zero to the intermediate Jacobian. The exponential sequence shows that CH1(X) maps isomorphically to Deligne cohomology, but that fails for CHj(X) with j > 1. For a scheme X over an arbitrary field k, there is an analogous cycle map from Chow groups to (Borel–Moore) etale homology. When X is smooth over k, this homomorphism can be identified with a ring homomorphism from the Chow ring to etale cohomology. == Relation to K-theory == An (algebraic) vector bundle E on a smooth scheme X over a field has Chern classes ci(E) in CHi(X), with the same formal properties as in topology. The Chern classes give a close connection between vector bundles and Chow groups. Namely, let K0(X) be the Grothendieck group of vector bundles on X. As part of the Grothendieck–Riemann–Roch theorem, Grothendieck showed that the Chern character gives an isomorphism K 0 ( X ) ⊗ Z Q ≅ ∏ i C H i ( X ) ⊗ Z Q . {\displaystyle K_{0}(X)\otimes _{\mathbf {Z} }\mathbf {Q} \cong \prod _{i}{\mathit {CH}}^{i}(X)\otimes _{\mathbf {Z} }\mathbf {Q} .} This isomorphism shows the importance of rational equivalence, compared to any other adequate equivalence relation on algebraic cycles. == Conjectures == Some of the deepest conjectures in algebraic geometry and number theory are attempts to understand Chow groups. For example: The Mordell–Weil theorem implies that the divisor class group CHn-1(X) is finitely generated for any variety X of dimension n over a number field. It is an open problem whether all Chow groups are finitely generated for every variety over a number field. The Bloch–Kato conjecture on values of L-functions predicts that these groups are finitely generated. Moreover, the rank of the group of cycles modulo homological equivalence, and also of the group of cycles homologically equivalent to zero, should be equal to the order of vanishing of an L-function of the given variety at certain integer points. Finiteness of these ranks would also follow from the Bass conjecture in algebraic K-theory. For a smooth complex projective variety X, the Hodge conjecture predicts the image (tensored with the rationals Q) of the cycle map from the Chow groups to singular cohomology. For a smooth projective variety over a finitely generated field (such as a finite field or number field), the Tate conjecture predicts the image (tensored with Ql) of the cycle map from Chow groups to l-adic cohomology. For a smooth projective variety X over any field, the Bloch–Beilinson conjecture predicts a filtration on the Chow groups of X (tensored with the rationals) with strong properties. The conjecture would imply a tight connection between the singular or etale cohomology of X and the Chow groups of X. For example, let X be a smooth complex projective surface. The Chow group of zero-cycles on X maps onto the integers by the degree homomorphism; let K be the kernel. If the geometric genus h0(X, Ω2) is not zero, Mumford showed that K is "infinite-dimensional" (not the image of any finite-dimensional family of zero-cycles on X). The Bloch–Beilinson conjecture would imply a satisfying converse, Bloch's conjecture on zero-cycles: for a smooth complex projective surface X with geometric genus zero, K should be finite-dimensional; more precisely, it should map isomorphically to the group of complex points of the Albanese variety of X. == Variants == === Bivariant theory === Fulton and MacPherson extended the Chow ring to singular varieties by defining the "operational Chow ring" and more generally a bivariant theory associated to any morphism of schemes. A bivariant theory is a pair of covariant and contravariant functors that assign to a map a group and a ring respectively. It generalizes a cohomology theory, which is a contravariant functor that assigns to a space a ring, namely a cohomology ring. The name "bivariant" refers to the fact that the theory contains both covariant and contravariant functors. This is in a sense the most elementary extension of the Chow ring to singular varieties; other theories such as motivic cohomology map to the operational Chow ring. === Other variants === Arithmetic Chow groups are an amalgamation of Chow groups of varieties over Q together with a component encoding Arakelov-theoretical information, that is, differential forms on the associated complex manifold. The theory of Chow groups of schemes of finite type over a field extends easily to that of algebraic spaces. The key advantage of this extension is that it is easier to form quotients in the latter category and thus it is more natural to consider equivariant Chow groups of algebraic spaces. A much more formidable extension is that of Chow group of a stack, which has been constructed only in some special case and which is needed in particular to make sense of a virtual fundamental class. == History == Rational equivalence of divisors (known as linear equivalence) was studied in various forms during the 19th century, leading to the ideal class group in number theory and the Jacobian variety in the theory of algebraic curves. For higher-codimension cycles, rational equivalence was introduced by Francesco Severi in the 1930s. In 1956, Wei-Liang Chow gave an influential proof that the intersection product is well-defined on cycles modulo rational equivalence for a smooth quasi-projective variety, using Chow's moving lemma. Starting in the 1970s, Fulton and MacPherson gave the current standard foundation for Chow groups, working with singular varieties wherever possible. In their theory, the intersection product for smooth varieties is constructed by deformation to the normal cone. == See also == Intersection theory Grothendieck–Riemann–Roch theorem Hodge conjecture Motive (algebraic geometry) == References == === Citations === === Introductory === Eisenbud, David; Harris, Joe, 3264 and All That: A Second Course in Algebraic Geometry === Advanced === Bloch, Spencer (1986), "Algebraic cycles and higher K-theory", Advances in Mathematics, 61 (3): 267–304, doi:10.1016/0001-8708(86)90081-2, ISSN 0001-8708, MR 0852815 Claude, Chevalley (1958), "Les classes d'équivalence rationnelle, I", Anneaux de Chow et applications, Séminaire Claude Chevalley, vol. 3, pp. 1–14 Claude, Chevalley (1958), "Les classes d'équivalence rationnelle, II", Anneaux de Chow et applications, Séminaire Claude Chevalley, vol. 3, pp. 1–18 Chow, Wei-Liang (1956), "On equivalence classes of cycles in an algebraic variety", Annals of Mathematics, 64 (3): 450–479, doi:10.2307/1969596, ISSN 0003-486X, JSTOR 1969596, MR 0082173 Deligne, Pierre (1977), Cohomologie Etale (SGA 4 1/2), Springer-Verlag, ISBN 978-3-540-08066-4, MR 0463174 Fulton, William (1998), Intersection Theory, Berlin, New York: Springer-Verlag, ISBN 978-0-387-98549-7, MR 1644323 Severi, Francesco (1932), "La serie canonica e la teoria delle serie principali di gruppi di punti sopra una superficie algebrica", Commentarii Mathematici Helvetici, 4: 268–326, doi:10.1007/bf01202721, JFM 58.1229.01 Voevodsky, Vladimir (2000), "Triangulated categories of motives over a field", Cycles, Transfers, and Motivic Homology Theories, Princeton University Press, pp. 188–238, ISBN 9781400837120, MR 1764202 Voisin, Claire (2002), Hodge Theory and Complex Algebraic Geometry (2 vols.), Cambridge University Press, ISBN 978-0-521-71801-1, MR 1997577
Wikipedia:Chow's lemma#0	Chow's lemma, named after Wei-Liang Chow, is one of the foundational results in algebraic geometry. It roughly says that a proper morphism is fairly close to being a projective morphism. More precisely, a version of it states the following: If X {\displaystyle X} is a scheme that is proper over a noetherian base S {\displaystyle S} , then there exists a projective S {\displaystyle S} -scheme X ′ {\displaystyle X'} and a surjective S {\displaystyle S} -morphism f : X ′ → X {\displaystyle f:X'\to X} that induces an isomorphism f − 1 ( U ) ≃ U {\displaystyle f^{-1}(U)\simeq U} for some dense open U ⊆ X . {\displaystyle U\subseteq X.} == Proof == The proof here is a standard one. === Reduction to the case of === X {\displaystyle X} irreducible We can first reduce to the case where X {\displaystyle X} is irreducible. To start, X {\displaystyle X} is noetherian since it is of finite type over a noetherian base. Therefore it has finitely many irreducible components X i {\displaystyle X_{i}} , and we claim that for each X i {\displaystyle X_{i}} there is an irreducible proper S {\displaystyle S} -scheme Y i {\displaystyle Y_{i}} so that Y i → X {\displaystyle Y_{i}\to X} has set-theoretic image X i {\displaystyle X_{i}} and is an isomorphism on the open dense subset X i ∖ ∪ j ≠ i X j {\displaystyle X_{i}\setminus \cup _{j\neq i}X_{j}} of X i {\displaystyle X_{i}} . To see this, define Y i {\displaystyle Y_{i}} to be the scheme-theoretic image of the open immersion X ∖ ∪ j ≠ i X j → X . {\displaystyle X\setminus \cup _{j\neq i}X_{j}\to X.} Since X ∖ ∪ j ≠ i X j {\displaystyle X\setminus \cup _{j\neq i}X_{j}} is set-theoretically noetherian for each i {\displaystyle i} , the map X ∖ ∪ j ≠ i X j → X {\displaystyle X\setminus \cup _{j\neq i}X_{j}\to X} is quasi-compact and we may compute this scheme-theoretic image affine-locally on X {\displaystyle X} , immediately proving the two claims. If we can produce for each Y i {\displaystyle Y_{i}} a projective S {\displaystyle S} -scheme Y i ′ {\displaystyle Y_{i}'} as in the statement of the theorem, then we can take X ′ {\displaystyle X'} to be the disjoint union ∐ Y i ′ {\displaystyle \coprod Y_{i}'} and f {\displaystyle f} to be the composition ∐ Y i ′ → ∐ Y i → X {\displaystyle \coprod Y_{i}'\to \coprod Y_{i}\to X} : this map is projective, and an isomorphism over a dense open set of X {\displaystyle X} , while ∐ Y i ′ {\displaystyle \coprod Y_{i}'} is a projective S {\displaystyle S} -scheme since it is a finite union of projective S {\displaystyle S} -schemes. Since each Y i {\displaystyle Y_{i}} is proper over S {\displaystyle S} , we've completed the reduction to the case X {\displaystyle X} irreducible. === === X {\displaystyle X} can be covered by finitely many quasi-projective S {\displaystyle S} -schemes Next, we will show that X {\displaystyle X} can be covered by a finite number of open subsets U i {\displaystyle U_{i}} so that each U i {\displaystyle U_{i}} is quasi-projective over S {\displaystyle S} . To do this, we may by quasi-compactness first cover S {\displaystyle S} by finitely many affine opens S j {\displaystyle S_{j}} , and then cover the preimage of each S j {\displaystyle S_{j}} in X {\displaystyle X} by finitely many affine opens X j k {\displaystyle X_{jk}} each with a closed immersion in to A S j n {\displaystyle \mathbb {A} _{S_{j}}^{n}} since X → S {\displaystyle X\to S} is of finite type and therefore quasi-compact. Composing this map with the open immersions A S j n → P S j n {\displaystyle \mathbb {A} _{S_{j}}^{n}\to \mathbb {P} _{S_{j}}^{n}} and P S j n → P S n {\displaystyle \mathbb {P} _{S_{j}}^{n}\to \mathbb {P} _{S}^{n}} , we see that each X i j {\displaystyle X_{ij}} is a closed subscheme of an open subscheme of P S n {\displaystyle \mathbb {P} _{S}^{n}} . As P S n {\displaystyle \mathbb {P} _{S}^{n}} is noetherian, every closed subscheme of an open subscheme is also an open subscheme of a closed subscheme, and therefore each X i j {\displaystyle X_{ij}} is quasi-projective over S {\displaystyle S} . === Construction of === X ′ {\displaystyle X'} and f : X ′ → X {\displaystyle f:X'\to X} Now suppose { U i } {\displaystyle \{U_{i}\}} is a finite open cover of X {\displaystyle X} by quasi-projective S {\displaystyle S} -schemes, with ϕ i : U i → P i {\displaystyle \phi _{i}:U_{i}\to P_{i}} an open immersion in to a projective S {\displaystyle S} -scheme. Set U = ∩ i U i {\displaystyle U=\cap _{i}U_{i}} , which is nonempty as X {\displaystyle X} is irreducible. The restrictions of the ϕ i {\displaystyle \phi _{i}} to U {\displaystyle U} define a morphism ϕ : U → P = P 1 × S ⋯ × S P n {\displaystyle \phi :U\to P=P_{1}\times _{S}\cdots \times _{S}P_{n}} so that U → U i → P i = U → ϕ P → p i P i {\displaystyle U\to U_{i}\to P_{i}=U{\stackrel {\phi }{\to }}P{\stackrel {p_{i}}{\to }}P_{i}} , where U → U i {\displaystyle U\to U_{i}} is the canonical injection and p i : P → P i {\displaystyle p_{i}:P\to P_{i}} is the projection. Letting j : U → X {\displaystyle j:U\to X} denote the canonical open immersion, we define ψ = ( j , ϕ ) S : U → X × S P {\displaystyle \psi =(j,\phi )_{S}:U\to X\times _{S}P} , which we claim is an immersion. To see this, note that this morphism can be factored as the graph morphism U → U × S P {\displaystyle U\to U\times _{S}P} (which is a closed immersion as P → S {\displaystyle P\to S} is separated) followed by the open immersion U × S P → X × S P {\displaystyle U\times _{S}P\to X\times _{S}P} ; as X × S P {\displaystyle X\times _{S}P} is noetherian, we can apply the same logic as before to see that we can swap the order of the open and closed immersions. Now let X ′ {\displaystyle X'} be the scheme-theoretic image of ψ {\displaystyle \psi } , and factor ψ {\displaystyle \psi } as ψ : U → ψ ′ X ′ → h X × S P {\displaystyle \psi :U{\stackrel {\psi '}{\to }}X'{\stackrel {h}{\to }}X\times _{S}P} where ψ ′ {\displaystyle \psi '} is an open immersion and h {\displaystyle h} is a closed immersion. Let q 1 : X × S P → X {\displaystyle q_{1}:X\times _{S}P\to X} and q 2 : X × S P → P {\displaystyle q_{2}:X\times _{S}P\to P} be the canonical projections. Set f : X ′ → h X × S P → q 1 X , {\displaystyle f:X'{\stackrel {h}{\to }}X\times _{S}P{\stackrel {q_{1}}{\to }}X,} g : X ′ → h X × S P → q 2 P . {\displaystyle g:X'{\stackrel {h}{\to }}X\times _{S}P{\stackrel {q_{2}}{\to }}P.} We will show that X ′ {\displaystyle X'} and f {\displaystyle f} satisfy the conclusion of the theorem. === Verification of the claimed properties of === X ′ {\displaystyle X'} and f {\displaystyle f} To show f {\displaystyle f} is surjective, we first note that it is proper and therefore closed. As its image contains the dense open set U ⊂ X {\displaystyle U\subset X} , we see that f {\displaystyle f} must be surjective. It is also straightforward to see that f {\displaystyle f} induces an isomorphism on U {\displaystyle U} : we may just combine the facts that f − 1 ( U ) = h − 1 ( U × S P ) {\displaystyle f^{-1}(U)=h^{-1}(U\times _{S}P)} and ψ {\displaystyle \psi } is an isomorphism on to its image, as ψ {\displaystyle \psi } factors as the composition of a closed immersion followed by an open immersion U → U × S P → X × S P {\displaystyle U\to U\times _{S}P\to X\times _{S}P} . It remains to show that X ′ {\displaystyle X'} is projective over S {\displaystyle S} . We will do this by showing that g : X ′ → P {\displaystyle g:X'\to P} is an immersion. We define the following four families of open subschemes: V i = ϕ i ( U i ) ⊂ P i {\displaystyle V_{i}=\phi _{i}(U_{i})\subset P_{i}} W i = p i − 1 ( V i ) ⊂ P {\displaystyle W_{i}=p_{i}^{-1}(V_{i})\subset P} U i ′ = f − 1 ( U i ) ⊂ X ′ {\displaystyle U_{i}'=f^{-1}(U_{i})\subset X'} U i ″ = g − 1 ( W i ) ⊂ X ′ . {\displaystyle U_{i}''=g^{-1}(W_{i})\subset X'.} As the U i {\displaystyle U_{i}} cover X {\displaystyle X} , the U i ′ {\displaystyle U_{i}'} cover X ′ {\displaystyle X'} , and we wish to show that the U i ″ {\displaystyle U_{i}''} also cover X ′ {\displaystyle X'} . We will do this by showing that U i ′ ⊂ U i ″ {\displaystyle U_{i}'\subset U_{i}''} for all i {\displaystyle i} . It suffices to show that p i ∘ g \| U i ′ : U i ′ → P i {\displaystyle p_{i}\circ g\|_{U_{i}'}:U_{i}'\to P_{i}} is equal to ϕ i ∘ f \| U i ′ : U i ′ → P i {\displaystyle \phi _{i}\circ f\|_{U_{i}'}:U_{i}'\to P_{i}} as a map of topological spaces. Replacing U i ′ {\displaystyle U_{i}'} by its reduction, which has the same underlying topological space, we have that the two morphisms ( U i ′ ) r e d → P i {\displaystyle (U_{i}')_{red}\to P_{i}} are both extensions of the underlying map of topological space U → U i → P i {\displaystyle U\to U_{i}\to P_{i}} , so by the reduced-to-separated lemma they must be equal as U {\displaystyle U} is topologically dense in U i {\displaystyle U_{i}} . Therefore U i ′ ⊂ U i ″ {\displaystyle U_{i}'\subset U_{i}''} for all i {\displaystyle i} and the claim is proven. The upshot is that the W i {\displaystyle W_{i}} cover g ( X ′ ) {\displaystyle g(X')} , and we can check that g {\displaystyle g} is an immersion by checking that g \| U i ″ : U i ″ → W i {\displaystyle g\|_{U_{i}''}:U_{i}''\to W_{i}} is an immersion for all i {\displaystyle i} . For this, consider the morphism u i : W i → p i V i → ϕ i − 1 U i → X . {\displaystyle u_{i}:W_{i}{\stackrel {p_{i}}{\to }}V_{i}{\stackrel {\phi _{i}^{-1}}{\to }}U_{i}\to X.} Since X → S {\displaystyle X\to S} is separated, the graph morphism Γ u i : W i → X × S W i {\displaystyle \Gamma _{u_{i}}:W_{i}\to X\times _{S}W_{i}} is a closed immersion and the graph T i = Γ u i ( W i ) {\displaystyle T_{i}=\Gamma _{u_{i}}(W_{i})} is a closed subscheme of X × S W i {\displaystyle X\times _{S}W_{i}} ; if we show that U → X × S W i {\displaystyle U\to X\times _{S}W_{i}} factors through this graph (where we consider U ⊂ X ′ {\displaystyle U\subset X'} via our observation that f {\displaystyle f} is an isomorphism over f − 1 ( U ) {\displaystyle f^{-1}(U)} from earlier), then the map from U i ″ {\displaystyle U_{i}''} must also factor through this graph by construction of the scheme-theoretic image. Since the restriction of q 2 {\displaystyle q_{2}} to T i {\displaystyle T_{i}} is an isomorphism onto W i {\displaystyle W_{i}} , the restriction of g {\displaystyle g} to U i ″ {\displaystyle U_{i}''} will be an immersion into W i {\displaystyle W_{i}} , and our claim will be proven. Let v i {\displaystyle v_{i}} be the canonical injection U ⊂ X ′ → X × S W i {\displaystyle U\subset X'\to X\times _{S}W_{i}} ; we have to show that there is a morphism w i : U ⊂ X ′ → W i {\displaystyle w_{i}:U\subset X'\to W_{i}} so that v i = Γ u i ∘ w i {\displaystyle v_{i}=\Gamma _{u_{i}}\circ w_{i}} . By the definition of the fiber product, it suffices to prove that q 1 ∘ v i = u i ∘ q 2 ∘ v i {\displaystyle q_{1}\circ v_{i}=u_{i}\circ q_{2}\circ v_{i}} , or by identifying U ⊂ X {\displaystyle U\subset X} and U ⊂ X ′ {\displaystyle U\subset X'} , that q 1 ∘ ψ = u i ∘ q 2 ∘ ψ {\displaystyle q_{1}\circ \psi =u_{i}\circ q_{2}\circ \psi } . But q 1 ∘ ψ = j {\displaystyle q_{1}\circ \psi =j} and q 2 ∘ ψ = ϕ {\displaystyle q_{2}\circ \psi =\phi } , so the desired conclusion follows from the definition of ϕ : U → P {\displaystyle \phi :U\to P} and g {\displaystyle g} is an immersion. Since X ′ → S {\displaystyle X'\to S} is proper, any S {\displaystyle S} -morphism out of X ′ {\displaystyle X'} is closed, and thus g : X ′ → P {\displaystyle g:X'\to P} is a closed immersion, so X ′ {\displaystyle X'} is projective. ◼ {\displaystyle \blacksquare } == Additional statements == In the statement of Chow's lemma, if X {\displaystyle X} is reduced, irreducible, or integral, we can assume that the same holds for X ′ {\displaystyle X'} . If both X {\displaystyle X} and X ′ {\displaystyle X'} are irreducible, then f : X ′ → X {\displaystyle f:X'\to X} is a birational morphism. == References == == Bibliography == Grothendieck, Alexandre; Dieudonné, Jean (1961). "Éléments de géométrie algébrique: II. Étude globale élémentaire de quelques classes de morphismes". Publications Mathématiques de l'IHÉS. 8. doi:10.1007/bf02699291. MR 0217084. Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, vol. 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157
Wikipedia:Chow's moving lemma#0	In algebraic geometry, Chow's moving lemma, proved by Wei-Liang Chow (1956), states: given algebraic cycles Y, Z on a nonsingular quasi-projective variety X, there is another algebraic cycle Z' which is rationally equivalent to Z on X, such that Y and Z' intersect properly. The lemma is one of the key ingredients in developing intersection theory and the Chow ring, as it is used to show the uniqueness of the theory. Even if Z is an effective cycle, it is not, in general, possible to choose Z' to be effective. == References == Chow, Wei-Liang (1956), "On equivalence classes of cycles in an algebraic variety", Annals of Mathematics, 64 (3): 450–479, doi:10.2307/1969596, ISSN 0003-486X, JSTOR 1969596, MR 0082173 Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, vol. 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157 Roberts, Joel (1972). "Chow's moving lemma. Appendix 2 to: "Motives" by Steven L. Kleiman.". Algebraic geometry, Oslo 1970 (Proc. Fifth Nordic Summer School in Math.). Groningen, Wolters-Noordhoff. pp. 89–96. ISBN 9001670806. MR 0382269. OCLC 579160.
Wikipedia:Chris Brink#0	Chris Brink, CBE, FRSSAf (born 31 January 1951) is a South African mathematician and academic. He was the Vice-Chancellor of Newcastle University between 2007 and December 2016. == Career == After graduating with a degree in maths and computer science from Rand Afrikaans University, Brink undertook post-graduate study at Rhodes University and the University of Cambridge. He became professor and head of mathematics and applied mathematics at the University of Cape Town in 1995, pro-vice-chancellor (research) at the University of Wollongong in 1999 and rector and vice-chancellor of Stellenbosch University in 2002 before being appointed vice-chancellor of Newcastle University in 2007. In the 1980s Chris Brink was a senior research fellow at the Australian National University. In 1994 he joined with Gunther Schmidt to organize at Dagstuhl the initial RAMiCS conference on relation algebra. In 1996 The Foundation for Research Development in South Africa rated Chris Brink in category A. He is a fellow of the Royal Society of South Africa, a former president of the South African Mathematical Society, a founder member of the Academy of Science of South Africa and a former chair of the Advisory Board of the African Institute of Mathematical Sciences. He chaired the Student Policy Network (part of Universities UK) and the N8 Research Partnership, a group of eight research-intensive universities in the North of England. Nationally he has served on the Board of the Equality Challenge Unit (including two years as a Co-Chair), the Board of the Quality Assurance Agency (and its Advisory Committee on Degree-Awarding Powers), and the Advisory Committee on Leadership, Governance and Management of the Higher Education Funding Council for England. In November 2015, it was announced that Brink would be retiring in December 2016. He was succeeded by Professor Chris Day, the Pro-Vice-Chancellor for the Faculty of Medical Sciences at Newcastle, in January 2017. Since 2017 Brink has served on the University Grants Committee (Hong Kong), where he convened the 2020 Research Assessment Exercise Group and currently convenes the Research Group. His book The Soul of a University – Why excellence is not enough was published by Bristol University Press in July 2018. It deals with the role of universities in society, and elaborates on the two key questions he became known for while at Newcastle University: ‘What are we good at?’, and ‘What are we good for?’ He was appointed a CBE in the Queen's Birthday Honours List in 2018. In 2021 Brink published an edited volume The Responsive University and the Crisis in South Africa, with Brill/Sense Publishers. The book argues that beyond the now-standard practice of universities’ engagement with society, the more pressing question is how they actually respond to societal challenges. Half the contributing authors are from South Africa and the other half from around the world. == Work in mathematics == Chris Brink developed the study of Boolean modules over relation algebras. He focused on formal aspects of computer science with emphasis on program semantics and Popper's concept of verisimilitude and on the universal-algebraic concept of power structures. == References == == External links == Official website of the Rector, Stellenbosch University Personal website
Wikipedia:Chris Caldwell (mathematician)#0	The PrimePages is a website about prime numbers originally created by Chris Caldwell at the University of Tennessee at Martin who maintained it from 1994 to 2023. The site maintains the list of the "5,000 largest known primes", selected smaller primes of special forms, and many "top twenty" lists for primes of various forms. The PrimePages has articles on primes and primality testing. It includes "The Prime Glossary" with articles on hundreds of glosses related to primes, and "Prime Curios!" with thousands of curios about specific numbers. The database started as a list of "titanic primes" (primes with at least 1000 decimal digits) by Samuel Yates in 1984. On March 11, 2023, the PrimePages moved from primes.utm.edu to t5k.org, and is no longer maintained by Caldwell. == See also == List of largest known primes and probable primes List of prime numbers == References == == External links == Official website
Wikipedia:Chris Soteros#0	Christine Elaine Soteros is a Canadian applied mathematician. She is a professor in the Department of Mathematics and Statistics at the University of Saskatchewan and was the University's Site Director for the Pacific Institute for the Mathematical Sciences from 2015 to 2024. Her research involves the folding and packing behavior of DNA, proteins, and other string-like biomolecules, and the knot theory of random space curves. Soteros graduated from the University of Windsor in 1980. She completed her Ph.D. in chemical engineering at Princeton University in 1988. Her dissertation, Studies of Metal Hydride Phase Transitions Using the Cluster Variation Method, was supervised by Carol K. Hall. After postdoctoral research at the University of Toronto, working with Stuart Whittington and De Witt Sumners, she became a faculty member at the University of Saskatchewan in 1989. == References == == External links == Home page
Wikipedia:Christiaan Heij#0	Christiaan Heij (born 1950s) is a Dutch mathematician, an assistant professor in statistics and econometrics at the Econometric Institute at the Erasmus University Rotterdam. He is known for his work in the field of mathematical systems theory, and econometrics. == Life and work == Heij did his PhD research at the University of Groningen in the 1980s among other young system theorists, such as Hans Nieuwenhuis, Pieter Otter, Jan Camiel Willems, and Dirk T. Tempelaar. In 1988 he graduated under Willems, Professor of Systems and Control and Nieuwenhuis with the thesis "Deterministic Identification of Dynamical Systems," which was published the next year by Springer in the "Lecture Notes in Control and Information Sciences" series. In the 1990s Heij continued his research at the Econometric Institute of the Erasmus University Rotterdam, and wrote a series of books on systems theory, modelling, dynamics systems, and econometrics. With Jan Camiel Willems he supervised the promotion of Berend Roorda, who graduated in 1995 with the thesis, entitled "Deterministic Identification of Dynamical Systems." Heij is credited extending "The behavioral approach to system theory put forward by Willems," and for presenting a new total least squares algorithms, specifically for system identification. His most cited work is the 2004 textbook "Econometric methods with applications in business and economics," co-authored with Philip Hans Franses, Teun Kloek, and Herman K. van Dijk, and published by the Oxford University Press. == Selected publications == === Books === Heij C., Deterministic Identification of Dynamical Systems, PhD Thesis University of Groningen, Lecture Notes in Control and Information Sciences, vol. 127, Springer, 1989. Heij, C., Schumacher, J. M., Hanzon, B., & Praagman, C. System Dynamics in Economic and Financial Models, 1997. Heij, C., De Boer, P., Franses, P. H., Kloek, T., & Van Dijk, H. K., Econometric Methods with Applications in Business and Economics, Oxford University Press, 2004. Christiaan Heij, André C.M. Ran and F. van Schagen. Introduction to Mathematical Systems Theory: Linear Systems, Identification and Control, Birkhäuser, 2006; 2nd ed., 2021. === Articles === Heij, Christiaan. "Exact modelling and identifiability of linear systems." Automatica 28.2 (1992): 325–344. Roorda, Berend, and Christiaan Heij. "Global total least squares modeling of multivariable time series." Automatic Control, IEEE Transactions on 40.1 (1995): 50–63. Heij, Christiaan, and Wolfgang Scherrer. "System identification by dynamic factor models." SIAM Journal on Control and Optimization 35.6 (1997): 1924–1951. Heij, Christiaan, Patrick JF Groenen, and Dick van Dijk. "Forecast comparison of principal component regression and principal covariate regression." Computational statistics & data analysis 51.7 (2007): 3612–3625. == References == == External links == Christiaan Heij, profile at eur.nl. Works by Christiaan Heij at narcis.nl.
Wikipedia:Christian Genest#0	Christian Genest (; born January 11, 1957, in Chicoutimi, Quebec) is a professor in the Department of Mathematics and Statistics at McGill University (Montréal, Canada), where he holds a Canada Research Chair. He is the author of numerous research papers in multivariate analysis, nonparametric statistics, extreme-value theory, and multiple-criteria decision analysis. He is a recipient of the Statistical Society of Canada's Gold Medal for Research and was elected a Fellow of the Royal Society of Canada in 2015. == Contributions == Genest is best known for developing models and statistical inference techniques for studying the dependence between variables through the concept of copula. He has designed, among others, various techniques for selecting, estimating and validating copula-based models through rank-based methods. His methodological contributions in multivariate analysis and extreme-value theory found numerous practical applications in finance, insurance, and hydrology. Throughout his career, Genest also made significant contributions to the development of techniques for the reconciliation and use of expert opinions and pairwise comparison methods used to establish priorities in multiple-criteria decision analysis. He is the author or co-author of over 250 scientific publications, about half of which appeared in peer-reviewed journals. Part of his work is also concerned with the history of statistics and scientometrics. Christian Genest has given over 300 invited talks, including 75+ presentations for a general audience. == Birthplace and education == Christian Genest was born on January 11, 1957, in Chicoutimi (Québec, Canada). He was trained as a mathematician at the Université du Québec à Chicoutimi (B.Sp.Sc., 1977) and at the Université de Montréal (M.Sc., 1978) before completing graduate studies in statistics at the University of British Columbia (Ph.D., 1983). His thesis, entitled "Towards a Consensus of Opinion", was written under the supervision of James V. Zidek and earned him the Pierre Robillard Award from the Statistical Society of Canada (SSC) in 1984. == Academic career == After completing his PhD, Christian Genest was a postdoctoral fellow and visiting assistant professor at Carnegie Mellon University (Pittsburgh, Pennsylvania) in 1983–84. From 1984 to 1987, he was an assistant professor in the Department of Statistics and Actuarial Science at the University of Waterloo (Waterloo, ON). He was then hired by Université Laval (Québec, QC), where he was promoted to the ranks of associate in 1989 and professor in 1993. He joined McGill University (Montréal, QC) in 2010, where he holds a Canada Research Chair in Stochastic Dependence Modeling. == Honors and prizes == Christian Genest was the first recipient of the CRM-SSC Prize in 1999. He received the SUMMA Research Award from Université Laval the same year. In 2011, the Statistical Society of Canada awarded him its most prestigious distinction, the gold medal, "in recognition of his remarkable contributions to multivariate analysis and nonparametric statistics, notably through the development of models and methods of inference for studying stochastic dependence, synthesizing expert judgments and multi-criteria decision making, as well as for his applications thereof in various fields such as insurance, finance, and hydrology." Christian Genest is a fellow of the American Statistical Association since 1996, a fellow of the Institute of Mathematical Statistics since 1997, and an honorary member of the Association des statisticiennes et statisticiens du Québec since 2012. He was elected a Fellow of the Royal Society of Canada in 2015 and received a Humboldt Research Prize from the Alexander von Humboldt Foundation in 2019. He was the first statistician to receive the John L. Synge Award in 2020 and the CRM-Fields-PIMS prize in 2023. == Community service == Christian Genest has served the mathematical and statistical communities in many ways. Among others, he was director of the Institut des sciences mathématiques du Québec (2012–15), president of the Statistical Society of Canada (2007–08) and president of the Association des statisticiennes et statisticiens du Québec (2005–08). He served on Statistics Canada's Advisory Committee on Statistical Methods for several years, and on the editorial board of various peer-review journals, including The Canadian Journal of Statistics (1988–2003), the Journal de la Société française de statistique (1999–2008) and the Journal of Multivariate Analysis (2003–2015). He was also editor in chief of The Canadian Journal of Statistics (1998–2000) and guest editor for various books and special issues, including two for Insurance: Mathematics and Economics (2005, 2009). From September 2015 to May 2019, he was editor in chief of the Journal of Multivariate Analysis. His many contributions earned him the Distinguished Service Award from the Statistical Society of Canada as early as 1997. == Others == Christian Genest is married to Johanna G. Nešlehová, professor of statistics at McGill University. One of his sisters, Sylvie Genest, is a professor in the Faculty of Arts at the Université du Québec à Montréal. Christian has four children (Marianne, Arnaud, Vincent, Richard). Vincent Genest is himself a researcher and the author of many papers in mathematical physics. == References == == External links == Personal Webpage Christian Genest publications indexed by Google Scholar
Wikipedia:Christian Krattenthaler#0	Christian Friedrich Krattenthaler (born 8 October 1958 in Vienna) is an Austrian mathematician. He is a retired professor of discrete mathematics (with a focus on combinatorics). From 2016 to 2020 he has been the Dean of the Faculty of Mathematics at the University of Vienna. He received his doctoral degree sub auspiciis Praesidentis rei publicae at the University of Vienna in 1983 under Johann Cigler with the dissertation Lagrangeformel und inverse Relationen (Lagrange formula and inverse relations). Krattenthaler worked at various universities, including the University of California, San Diego, the Mathematical Sciences Research Institute in Berkeley, California, the University of Strasbourg, and the Claude Bernard University Lyon 1 before being appointed to a professorship at the University of Vienna in 2005. He took his retirement in 2024. His area of specialization is the problems of combinatorial enumeration, such as those in algebra, algebraic geometry, number theory, computer science, or statistical physics. Krattenthaler won in 1990 the Prize of the Austrian Mathematical Society and in 2007 the Wittgenstein Award of the Austrian Science Fund. He was elected in 2005 a corresponding member of the Austrian Academy of Sciences, in 2011 a full member of the Academia Europaea, and in 2012 a Fellow of the American Mathematical Society. In 2015 he received a Docteur honoris causa from the Université Sorbonne Paris Nord. Krattenthaler is also a trained concert pianist. He studied piano at the (then) Hochschule für Musik und Darstellende Kunst Wien (today University of Music and Performing Arts Vienna) with Hans Graf. He completed his studies in 1986 with the concert diploma. Until 1991, he performed as soloist and chamber musician. Frequent chamber music partners were Bernhard Biberauer (violin), Alfred Hertel (oboe), Peter Siakala (violoncello), Anton Straka (violin), Herwig Tachezi (violoncello) and Thomas C. Wolf (violin). Krattenthaler terminated his concert activities in 1991 because of an irreversible chronic medical condition in both hands. == References == == External links == "Christian Krattenthaler – Determinants and Pfaffians in Enumerative Combinatorics (2011)". YouTube. 17 November 2017. "Christian Krattenthaler – Combinatorics of Discrete Lattice Models (2012)". YouTube. 24 August 2015. "UF – Alladi 60 – Christian Krattenthaler". YouTube. 10 June 2016. (2016 International Conference of Number Theory in honor of Krishna Alladi's 60th birthday)
Wikipedia:Christian Lantuéjoul#0	Christian Lantuéjoul (born 1950) is a French mathematician. Lantuéjoul was selected to receive Georges Matheron Lectureship Award – 2018 from the International Association for Mathematical Geosciences. Lantuéjoul serves as Director of Research at School of Mines ParisTech. == Education == PhD School of Mines, Nancy == Selected book == Christian Lantuéjoul, Geostatistical Simulation. Models and Algorithms (2002), Springer-Verlag, 256 pages == References ==
Wikipedia:Christian Lubich#0	Christian Lubich (born 29 July 1959) is an Austrian mathematician, specializing in numerical analysis. == Education and career == After secondary education at the Bundesrealgymnasium in Innsbruck, Lubich studied mathematics at the University of Innsbruck from 1977 to graduation with Magister degree in 1981. He was from 1979 to 1981 a student assistant in Innsbruck and from 1981 to 1983 a research associate in the Sonderforschungsbereich 123 Stochastische mathematische Modelle at the University of Heidelberg. He received his doctorate in 1983 from the University of Innsbruck with dissertation Zur Stabilität linearer Mehrschrittverfahren für Volterra-Gleichungen (On stability of linear multistep methods for Volterra equations) under Ernst Hairer and habilitated there in 1987. Lubich was from 1983 to 1987 a university assistant in Innsbruck, in 1986/87 an assistant at the University of Geneva, in 1987/88 a visiting professor at the IRMAR at the University of Rennes, and in 1988 a visiting professor at the University of Geneva. He was from 1991 to 1992 an assistant professor at ETH Zurich and from 1992 to 1994 a professor of applied mathematics at University of Würzburg. He is since 1994 a professor of numerical mathematics at the University of Tübingen. Lubich received in 2001 the Dahlquist Prize of the Society for Industrial and Applied Mathematics (SIAM) and in 1985 the Research Prize of the city of Innsbruck. In 2018 at the International Congress of Mathematicians (ICM) in Rio de Janeiro, he was a plenary speaker with talk Dynamics, numerical analysis, and some geometry, written jointly with his former doctoral student Ludwig Gauckler and with Ernst Hairer. Lubich has been a member of the editorial board of Numerische Mathematik since 1995, of Ricerche di Matematica and of the IMA Journal of Numerical Analysis since 2006, of BIT Numerical Mathematics since 1996, and of SIAM Journal on Scientific Computing from 1996 to 2001. == Selected publications == === Articles === Hairer, E.; Lubich, Ch.; Schlichte, M. (1985). "Fast Numerical Solution of Nonlinear Volterra Convolution Equations". SIAM Journal on Scientific and Statistical Computing. 6 (3): 532–541. doi:10.1137/0906037. Lubich, Christian; Nowak, Ulrich; Pöhle, Uwe; Engstler, Christian (1993). "An overview of MEXX: Numerical Software for the Integration of Multibody Systems". Advanced Multibody System Dynamics. Solid Mechanics and Its Applications. Vol. 20. pp. 421–426. doi:10.1007/978-94-017-0625-4_29. ISBN 978-90-481-4253-8. ISSN 0925-0042. MEXX — Numerical Software for the Integration of Constrained Mechanical Multibody Systems, Preprint SC 92–12 (December 1992) Hairer, E.; Lubich, C. (1997). "The life-span of backward error analysis for numerical integrators". Numerische Mathematik. 76 (4): 441–462. doi:10.1007/s002110050271. S2CID 13877070. Hairer, Ernst; Lubich, Christian (2000). "Long-Time Energy Conservation of Numerical Methods for Oscillatory Differential Equations". SIAM Journal on Numerical Analysis. 38 (2): 414–441. doi:10.1137/S0036142999353594. S2CID 13174771. Hairer, Ernst; Lubich, Christian; Wanner, Gerhard (2003). "Geometric numerical integration illustrated by the Störmer–Verlet method". Acta Numerica. 12: 399–450. doi:10.1017/S0962492902000144. Cohen, David; Jahnke, Tobias; Lorenz, Katina; Lubich, Christian (2006). "Numerical Integrators for Highly Oscillatory Hamiltonian Systems: A Review". Analysis, Modeling and Simulation of Multiscale Problems. pp. 553–576. doi:10.1007/3-540-35657-6_20. ISBN 978-3-540-35656-1. Lubich, Christian; Oseledets, Ivan V.; Vandereycken, Bart (2015). "Time Integration of Tensor Trains". SIAM Journal on Numerical Analysis. 53 (2): 917–941. arXiv:1407.2042. doi:10.1137/140976546. ISSN 0036-1429. S2CID 13110671. Ernst Hairer; Christian Lubich (2015). "IV.12 Numerical solution of ordinary differential equations". In Nicholas J. Higham; Mark R. Dennis; Paul Glendinning; Paul A. Martin; Fadil Santosa; Jared Tanner (eds.). The Princeton Companion to Applied Mathematics. Princeton, NJ: Princeton University Press. pp. 293–305. ISBN 978-0-691-15039-0. MR 3380576; preprint === Books === Hairer, Ernst; Lubich, Christian; Roche, Michel (2006-11-14). The Numerical Solution of Differential-Algebraic Systems by Runge-Kutta Methods. Springer. ISBN 978-3-540-46832-5. (pbk reprint of 1989 original) Hairer, Ernst; Lubich, Christian; Wanner, Gerhard (2002). Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations. Springer. ISBN 978-3-540-43003-2.; Hairer, Ernst; Lubich, Christian; Wanner, Gerhard (2006-05-18). 2nd edition. Springer. ISBN 978-3-540-30666-5. Lubich, Christian (2008). From Quantum to Classical Molecular Dynamics: Reduced Models and Numerical Analysis. European Mathematical Society. ISBN 978-3-03719-067-8. == References == == External links == Homepage in Tübingen "Dynamics, numerical analysis and some geometry – Christian Lubich – ICM2018". YouTube. Rio ICM2018. 19 September 2018.
Wikipedia:Christian Pommerenke#0	Christian Pommerenke (17 December 1933 – 18 August 2024) was a German mathematician known for his work in complex analysis. == Life and career == Pommerenke studied at the University of Göttingen (1954–1958), achieving diploma in mathematics (1957), Ph.D. (1959) on the dissertation Über die Gleichverteilung von Gitterpunkten auf m-dimensionalen Ellipsoiden (1959) and habilitation (1963). Pommerenke subsequently joined the faculty as Assistant (1958–1964) and Privatdozent (1964–1966). Around the same time he served as assistant professor at the University of Michigan in Ann Arbor (1961–1962), was at Harvard University (1962–1963) and was guest lecturer and reader at Imperial College in London (1965–1967). Since 1967 he was professor in complex analysis at the mathematics department of Technische Universität Berlin. He was later an emeritus. His doctoral students include Herbert Robert Stahl, known for proving the Bessis-Moussa-Villani (BMV) conjecture. Pommerenke died on 18 August 2024, at the age of 90. == Books == Boundary Behaviour of Conformal Maps. Springer-Verlag. 1992. ISBN 978-3-540-54751-8. Boundary Behaviour of Conformal Maps. Springer Science & Business Media. 9 April 2013. ISBN 978-3-662-02770-7(Reprint of 1992 original){{cite book}}: CS1 maint: postscript (link) Univalent Functions. Vandenhoeck & Ruprecht. 1975. ISBN 978-3525401330. With Gerd Jensen == References ==
Wikipedia:Christian of Prachatice#0	Christian of Prachatice (Czech: Křišťan z Prachatic) (1360–1368, Prachatice, Kingdom of Bohemia – 4 September 1439, Prague, Kingdom of Bohemia) was a medieval Bohemian astronomer, mathematician and former Catholic priest who converted to the Hussite movement. He was the author of several books about medicine and herbs, and contributed to the field of astronomy with many papers and data recordings. == Biography == Christian of Prachatice was born in the 1360s, perhaps 1366 or 1368. In 1386 he matriculated at Charles University, where he earned a bachelor's degree two years later and a master's degree in liberal arts in 1390. He later taught at the university and counted Jan Hus among his students. In 1403 he served as dean of the Faculty of Arts and 1405 as rector of the university. In 1405 he was appointed pastor of The Church of St. Michael the Archangel in Prague's Old Town; it is not known where he was ordained. Together with Johannes Cardinalis von Bergreichenstein, he attended the Council of Pisa in 1409. In 1415 he defended Jan Hus at the Council of Constance. Returning to Prague, he served as dean of the Faculty of Philosophy from 1417. Ten years later, in 1427, he was forced to flee from the radical Hussites. He returned two years later and in 1434 he converted back to Catholicism was once again elected rector. Christian of Prachitice died on 4 September 1439, a victim of the plague epidemic. == Writings == Christian of Prachatice produced numerous treatises, primarily in the fields of astronomy, mathematics, and medicine. His theological writings survive only in a few fragments. His works survive in manuscript copies at the Bayerische Staatsbibliothek, the library of Saint Peter's Abbey, Salzburg, and the Königsberg State and University Library, as well as the Beinecke Rare Book and Manuscript Library. === Latin === De composicione astrolabii, on the composition of the astrolabe; De utilitate (usu) astrolabii, on the use of the astrolabe; Regula ad fixanda festa mobilia; Algorismus prosaycus, on translating Roman numerals into Arabic numerals; Computus chirometralis, an aid to counting on fingers; Antidotar; Herbularium, an herbal; De sanguinis minucione, treatise on bloodletting. === Czech === Lékařské knížky; Knihy o mocech rozličného kořenie, an herbal. === German === Theriak-Pest-Traktat; Arzneibüchlein des Magisters Christian von Prachatitz. == References == == External links == (in Czech)MacTutor Entry (in Czech)Extensive Biography
Wikipedia:Christina Goldschmidt#0	Christina Anna Goldschmidt is a British probabilist known for her work in probability theory including coalescent theory, random minimum spanning trees, and the theory of random graphs. She is professor of probability in the department of statistics, University of Oxford and a fellow of Lady Margaret Hall, Oxford. == Education and career == Goldschmidt read mathematics at New Hall, Cambridge, and continued at the statistical laboratory of Cambridge for her Ph.D. Her 2004 dissertation, Large Random Hypergraphs, was supervised by James R. Norris. She did postdoctoral research with Jean Bertoin at Pierre and Marie Curie University, as a Stokes fellow at Pembroke College, Cambridge, and as an EPSRC postdoctoral fellow at Oxford, before becoming an assistant professor in 2009 at the University of Warwick. She returned to Oxford in 2011 and was promoted to full professor in 2017. == Recognition == Goldschmidt was a Medallion Lecturer of the Institute of Mathematical Statistics in 2016. In 2019 she was chosen to become a fellow of the Institute of Mathematical Statistics, "for fundamental contributions to the fields of coalescence and fragmentation theory, and to continuum limits for random trees and graphs". == References == == External links == Home page Christina Goldschmidt publications indexed by Google Scholar Goldschmidt describes her work on random minimum spanning trees, Oxford Mathematical Institute
Wikipedia:Christina Tønnesen-Friedman#0	Christina Wiis Tønnesen-Friedman is a Danish-American mathematician specializing in Riemannian geometry, especially of Kähler manifolds and Sasakian manifolds. She is Marie Louise Bailey Professor of Mathematics at Union College in Schenectady, New York. == Education == Tønnesen-Friedman studied mathematics and chemistry at Odense University, earning a candidate degree in 1995 and completing her Ph.D. in mathematics in 1997. Her doctoral dissertation, Extremal Kähler Metrics on Ruled Surfaces, was co-advised by Claude LeBrun and Henrik Laurberg Pedersen. == Career == Tønnesen-Friedman became a research assistant professor at Aarhus University in 1997. In 2001, she moved to Union College as an assistant professor of mathematics. She was tenured as an associate professor in 2007, promoted to full professor in 2012, and chaired the mathematics department at Union College from 2017 to 2021. == References == == External links == Home page
Wikipedia:Christine Ayoub#0	Christine Sykes Williams Ayoub (1922–2024) was a Canadian and American mathematician specializing in commutative algebra and a professor of mathematics at Pennsylvania State University. A Quaker and descendant of Quakers, she also edited a book of biographies of Quakers. == Early life and education == Ayoub was the daughter of William Lloyd Garrison Williams, also a Canadian and American mathematician, and his wife, pianist Anne Sykes. She was born on February 7, 1922, in Cincinnati. Although her father was working at Cornell University in Ithaca, New York at the time, her mother, originally from Cincinnati, went to her family home in Cincinnati for the births of both Ayoub and her older sister, Hester. In 1924, her father moved to McGill University in Montreal, Canada, and she grew up in Montreal. Her first school, in 1928, was "an Italian school in Rome", where her mother was wintering; the family trip to Italy also included the 1928 International Congress of Mathematicians in Bologna. Later, she attended both English-language and French-language schools in Montreal, including the Trafalgar School for Girls, from which she graduated young, in 1938. Intent on studying mathematics, but avoiding her father's department at McGill despite earning top admission scores there, she entered Bryn Mawr College in 1939, where (after the 1935 death of Emmy Noether) the mathematics department was headed by Anna Johnson Pell Wheeler. After graduating "at the top of her class", she did a master's degree program at Radcliffe College. There, she took courses with Saunders Mac Lane and Hassler Whitney and, inspired by Mac Lane, decided to focus on algebra, despite Wheeler's preference for mathematical analysis. Because her Radcliffe master's degree did not have a thesis, she returned to McGill University for a second master's degree, in 1945, the year that her father was starting the Canadian Mathematical Congress. She completed a Ph.D. in 1947, at Yale University, with the dissertation A Theory of Normal Chains. The Mathematics Genealogy Project lists this work as jointly supervised by Reinhold Baer and Nathan Jacobson. In a 2014 interview, she stated that it would have been directed by Øystein Ore had he not been on leave that year, that instead it was directed by Baer, whom she had visited at the University of Illinois, and that Jacobson, newly arrived at Yale, served as her outside examiner. == Career and later life == She became a postdoctoral research fellow and member of the School of Mathematics at the Institute for Advanced Study in Princeton, New Jersey from 1947 to 1948, with the support of a fellowship from the Office of Naval Research. Although she hoped to work there with Emil Artin at Princeton University, he turned out to be uninterested in working with women. (Artin had worked with Noether, but famously remarked that "she wasn't a woman".) After an interview at the University of Michigan, whose chair Theophil Henry Hildebrandt told her that a man in her position would have been hired without an interview, she was hired by the Cornell University mathematics department as an instructor in 1947, later learning that there had been a big fight among the Cornell mathematics faculty over whether to hire a woman. In 1950, she married Raymond Ayoub, a Canadian mathematician of Lebanese descent who had been a student of her father. She continued teaching at Cornell until 1951, and in 1951–1952 was a postdoctoral fellow at Harvard University, supported by the National Science Foundation. In 1952, both Ayoubs moved to the mathematics department at Pennsylvania State University, in State College, Pennsylvania. At Penn State, she became the first algebraist in a new program, and for many years was the only woman faculty member in the entire College of Science. She and her husband both directed large graduate programs, encompassing "more theses than all the rest of the department put together". She also had two daughters, Cynthia in 1953 and Daphne in 1956. After retiring in 1984, she became a professor emerita. Ayoub was "was descended from generations of Quakers", active in the State College Meeting of the Quakers, and a leader in the meeting's oral history project. After retiring, she and her husband helped found a Quaker retirement community in State College, which they moved into in 1997, and regularly traveled to teach in the Middle East. She published a book on Quaker biography, Memories of the Quaker Past: Stories of Thirty-seven Senior Quakers, in 2014. Her husband died in 2013, and Ayoub died on July 18, 2024, in State College. == References == == Further reading == Doll, Keely (March 6, 2023), "Centre County centenarians share their favorite memories, advice and hope for the future", Centre Daily Times, retrieved 2025-04-30
Wikipedia:Christine Bessenrodt#0	Christine Bessenrodt (1958–2022) was a German mathematician who was for many years the Chair of Algebra and Number Theory at Leibniz University Hannover. Her research involved representation theory, algebraic combinatorics, and additive number theory. She was also known for her advocacy of women in mathematics, including founding the Emmy Noether Lecture program of the German Mathematical Society. == Early life and education == Bessenrodt was born on 18 March 1958, in Ahlten, the daughter of two physicists. After undergraduate study in mathematics at Heinrich Heine University Düsseldorf, and graduate study at the University of Essen (now part of the University of Duisburg-Essen), she earned a doctorate in 1980. Her dissertation, Unzerlegbare Gitter in Blöcken mit zyklischen Defektgruppen [Indecomposable lattices in blocks with cyclic defect groups], concerned the representation theory of finite groups, and was supervised by Gerhard O. Michler. == Career and later life == From 1980 to 1993, Bessenrodt did postdoctoral research at the University of Essen, University of Duisburg, and University of Illinois Urbana–Champaign, supported in part by a Heisenberg grant. In 1993 she obtained a professorship in algebra at Otto von Guericke University Magdeburg, in East Germany, soon after the German reunion. The following year she became vice dean of the faculty of mathematics. She moved to Leibniz University Hannover in 2002, taking the Chair of Algebra and Number Theory there. She remained in Hannover for the rest of her career, later becoming director of the newly formed Institut für Algebra, Zahlentheorie und Diskrete Mathematik. She died on 25 January 2022. == Recognition == A colloquium and conference in memory of Bessenrodt was held at Leibniz University Hannover in July 2022. == References ==
Wikipedia:Christine De Mol#0	Christine De Mol (born 23 April 1954) is a Belgian applied mathematician and mathematical physicist interested in inverse problems, regularization, wavelets, and machine learning, and known for her work on proximal gradient methods and the application of proximal gradient methods for learning. She is a professor of mathematics at the Université libre de Bruxelles, and the former chair of the SIAM Activity Group on Imaging Science. == Education == De Mol was educated at the Université libre de Bruxelles, earning a licence in physics in 1975 and a Ph.D. in 1979, with a dissertation Sur la régularisation des problèmes inverses linéaires under the joint supervision of Jean Reignier and Mario Bertero. == Career == De Mol became a researcher for the Belgian National Fund for Scientific Research (FNRS), obtaining a permanent position there in 1981 and becoming a director of research in 1996. Meanwhile, she had obtained a habilitation from the Université libre de Bruxelles; her habilitation thesis was Super-résolution en microscopie confocale. In 1998 she gave up her position with the FNRS, becoming an honorary researcher with them, to become a full professor at the Université libre de Bruxelles. She was head of the mathematics department at the university for 2009–2010, and chair of the Society for Industrial and Applied Mathematics Activity Group on Imaging Science for 2012–2013. == References == == External links == Christine De Mol publications indexed by Google Scholar
Wikipedia:Christine O'Keefe#0	Christine Margaret O'Keefe is an Australian mathematician and computer scientist whose research has included work in finite geometry, information security, and data privacy. She is a researcher at CSIRO, and was the lead author of a 2017 report from the Office of the Australian Information Commissioner on best practices for de-identification of personally identifying data. == Education and career == O'Keefe has a bachelor's degree from the University of Adelaide, initially intending to study medicine but earning first-class honours in mathematics there in 1982. She returned to Adelaide for doctoral study in 1985, and completed her Ph.D. in 1988. Her dissertation, Concerning t {\displaystyle t} -spreads of P G ( ( s + 1 ) ( t + 1 ) − 1 , q ) {\displaystyle PG((s+1)(t+1)-1,q)} , was supervised by Rey Casse. She was a lecturer and research fellow at the University of Western Australia from 1999 to 2001, when she returned to the University of Adelaide. At Adelaide, she worked as a lecturer, senior lecturer, Queen Elizabeth II Fellow, and senior research fellow. Her research interests shifted from finite geometry to information security and to effect that shift she moved in 2000 from Adelaide to CSIRO. At CSIRO, she founded the Information Security and Privacy Group in 2002, became head of the Health Informatics Group in 2004, became Theme Leader for Health Data and Information in 2006, and Strategic Operations Director for Preventative Health National Research in 2008. While doing this, she studied for an MBA at Australian National University, finishing in 2008. She became Director of the Population Health Research Network Centre and Professor of Health Sciences at Curtin University from 2009 to 2010 before returning to CSIRO as Science Leader for Privacy and Confidentiality in the CSIRO Department of Mathematics, Informatics and Statistics. == Recognition == O'Keefe has been a Fellow of the Institute of Combinatorics and its Applications since 1991. In 1996, O'Keefe won the Hall Medal of the Institute of Combinatorics and its Applications for her work in finite geometry. She won the Australian Mathematical Society Medal in 2000, the first woman to win the medal, and in the same year became a Fellow of the Australian Mathematical Society. Although the Medal citation primarily discussed O'Keefe's work in finite geometry, such as the discovery of new hyperovals, it included a paragraph on her research using geometry in secret sharing, a precursor to her later work on information security. == References == == External links ==
Wikipedia:Christine Riedtmann#0	Christine Riedtmann (born 1952) is a Swiss mathematician specializing in abstract algebra. She earned her PhD in 1978 from the University of Zurich under the supervision of Pierre Gabriel, and is a professor emeritus (since 2016) at the University of Bern. In 2012–2013 she was president of the Swiss Mathematical Society. == Selected publications == Gabriel, P.; Riedtmann, Ch. (1979), "Group representations without groups", Commentarii Mathematici Helvetici, 54 (2): 240–287, doi:10.1007/BF02566271, MR 0535058, S2CID 120707173 Riedtmann, Christine (1980), "Representation-finite selfinjective algebras of class An", Representation theory, II (Proc. Second Internat. Conf., Carleton Univ., Ottawa, Ont., 1979), Lecture Notes in Mathematics, vol. 832, Berlin: Springer, pp. 449–520, doi:10.1007/BFb0088479, MR 0607169 Riedtmann, C. (1980), "Algebren, Darstellungsköcher, Überlagerungen und zurück", Commentarii Mathematici Helvetici (in German), 55 (2): 199–224, doi:10.1007/BF02566682, MR 0576602, S2CID 125072194 Reiten, Idun; Riedtmann, Christine (1985), "Skew group algebras in the representation theory of Artin algebras", Journal of Algebra, 92 (1): 224–282, doi:10.1016/0021-8693(85)90156-5, MR 0772481 == References == == Further reading == Interview with Riedtmann in the Berner Zeitung, February 19, 2015 (in German)
Wikipedia:Christoffer Dybvad#0	Christoffer Dybvad (1578–1622) was a Danish mathematician. He was born in Copenhagen, the son of Professor Jørgen Dybvad. He adapted Simon Stevin's De Thiende into Danish. == References ==
Wikipedia:Christopher Budd (mathematician)#0	Christopher John Budd (born 15 February 1960) is a British mathematician known especially for his contribution to non-linear differential equations and their applications in industry. He is currently Professor of Applied Mathematics at the University of Bath, and was Professor of Geometry at Gresham College from 2016 to 2020. Budd gained his Bachelor's degree in mathematics at St John's College, Cambridge, where he was senior wrangler. He went on to be awarded a D.Phil. from Oxford University, studying numerical methods for nonlinear elliptic partial differential equations under the supervision of John Norbury. He spent three years as a fellow of St John's College, Oxford, working in numerical analysis at the Oxford University Computing Laboratory and as a fellow sponsored by the CEGB developing numerical methods for third-order partial differential equations. He went on to a permanent post as a lecturer in numerical analysis at the University of Bristol before gaining a position as Professor of Applied Mathematics at the University of Bath in 1995. He was appointed the Professor of Geometry at Gresham College in 2016, where he delivered a series of public lectures on Mathematics and the Making of the Modern World. His research interests involve the analysis, application and numerical analysis of the solution of nonlinear differential equations with a particular emphasis on problems which arise in industry. His recent work has been in geometric integration which aims to develop numerical methods which reproduce qualitative structures in differential equations. He is co-director of the interdisciplinary Centre for Nonlinear Mechanics at the University of Bath and is active in promoting interdisciplinary collaboration both nationally and internationally. Budd is a passionate populariser of mathematics, reflected in his appointment as Chair of Mathematics of the Royal Institution of Great Britain in 2000. He works on a number of projects with schools and has written a book, "Mathematics Galore", based on his series of popular talks. He has also made numerous guest appearances on national radio and television, such as on the BBC's The One Show[1] and popular science panel comedy game show It's Only a Theory[2]. He won the Leslie Fox Prize for Numerical Analysis in 1991. In 1999 he was one of ten scientists awarded the title of "Scientist for the new century" by the Royal Institution. In 2001 he was one of 20 lecturers in the UK to be awarded an ILT Teaching Fellowship, and he was nominated the LMS popular lecturer in applied mathematics. He was awarded the Order of the British Empire (OBE) in the Queen's Birthday Honours List in 2015 for services to science and maths education. He has supervised at least 9 students for a PhD. == Bibliography == Christopher Budd and Christopher Sangwin, Mathematics Galore!: Masterclasses, Workshops and Team Projects in Mathematics and Its Applications, Oxford University Press (2001) ISBN 0-19-850770-4. == See also == Leslie Fox Prize for Numerical Analysis == References == == External links == Christopher Budd at the Mathematics Genealogy Project Home page at Bath
Wikipedia:Christ–Kiselev maximal inequality#0	In mathematics, the Christ–Kiselev maximal inequality is a maximal inequality for filtrations, named for mathematicians Michael Christ and Alexander Kiselev. == Continuous filtrations == A continuous filtration of ( M , μ ) {\displaystyle (M,\mu )} is a family of measurable sets { A α } α ∈ R {\displaystyle \{A_{\alpha }\}_{\alpha \in \mathbb {R} }} such that A α ↗ M {\displaystyle A_{\alpha }\nearrow M} , ⋂ α ∈ R A α = ∅ {\displaystyle \bigcap _{\alpha \in \mathbb {R} }A_{\alpha }=\emptyset } , and μ ( A β ∖ A α ) < ∞ {\displaystyle \mu (A_{\beta }\setminus A_{\alpha })<\infty } for all β > α {\displaystyle \beta >\alpha } (stratific) lim ε → 0 + μ ( A α + ε ∖ A α ) = lim ε → 0 + μ ( A α ∖ A α + ε ) = 0 {\displaystyle \lim _{\varepsilon \to 0^{+}}\mu (A_{\alpha +\varepsilon }\setminus A_{\alpha })=\lim _{\varepsilon \to 0^{+}}\mu (A_{\alpha }\setminus A_{\alpha +\varepsilon })=0} (continuity) For example, R = M {\displaystyle \mathbb {R} =M} with measure μ {\displaystyle \mu } that has no pure points and A α := { { \| x \| ≤ α } , α > 0 , ∅ , α ≤ 0. {\displaystyle A_{\alpha }:={\begin{cases}\{\|x\|\leq \alpha \},&\alpha >0,\\\emptyset ,&\alpha \leq 0.\end{cases}}} is a continuous filtration. == Continuum version == Let 1 ≤ p < q ≤ ∞ {\displaystyle 1\leq p<q\leq \infty } and suppose T : L p ( M , μ ) → L q ( N , ν ) {\displaystyle T:L^{p}(M,\mu )\to L^{q}(N,\nu )} is a bounded linear operator for σ − {\displaystyle \sigma -} finite ( M , μ ) , ( N , ν ) {\displaystyle (M,\mu ),(N,\nu )} . Define the Christ–Kiselev maximal function T ∗ f := sup α \| T ( f χ α ) \| , {\displaystyle T^{}f:=\sup _{\alpha }\|T(f\chi _{\alpha })\|,} where χ α := χ A α {\displaystyle \chi _{\alpha }:=\chi _{A_{\alpha }}} . Then T ∗ : L p ( M , μ ) → L q ( N , ν ) {\displaystyle T^{}:L^{p}(M,\mu )\to L^{q}(N,\nu )} is a bounded operator, and ‖ T ∗ f ‖ q ≤ 2 − ( p − 1 − q − 1 ) ( 1 − 2 − ( p − 1 − q − 1 ) ) − 1 ‖ T ‖ ‖ f ‖ p . {\displaystyle \\|T^{}f\\|_{q}\leq 2^{-(p^{-1}-q^{-1})}(1-2^{-(p^{-1}-q^{-1})})^{-1}\\|T\\|\\|f\\|_{p}.} == Discrete version == Let 1 ≤ p < q ≤ ∞ {\displaystyle 1\leq p<q\leq \infty } , and suppose W : ℓ p ( Z ) → L q ( N , ν ) {\displaystyle W:\ell ^{p}(\mathbb {Z} )\to L^{q}(N,\nu )} is a bounded linear operator for σ − {\displaystyle \sigma -} finite ( M , μ ) , ( N , ν ) {\displaystyle (M,\mu ),(N,\nu )} . Define, for a ∈ ℓ p ( Z ) {\displaystyle a\in \ell ^{p}(\mathbb {Z} )} , ( χ n a ) := { a k , \| k \| ≤ n 0 , otherwise . {\displaystyle (\chi _{n}a):={\begin{cases}a_{k},&\|k\|\leq n\\0,&{\text{otherwise}}.\end{cases}}} and sup n ∈ Z ≥ 0 \| W ( χ n a ) \| =: W ∗ ( a ) {\displaystyle \sup _{n\in \mathbb {Z} ^{\geq 0}}\|W(\chi _{n}a)\|=:W^{}(a)} . Then W ∗ : ℓ p ( Z ) → L q ( N , ν ) {\displaystyle W^{*}:\ell ^{p}(\mathbb {Z} )\to L^{q}(N,\nu )} is a bounded operator. Here, A α = { [ − α , α ] , α > 0 ∅ , α ≤ 0 {\displaystyle A_{\alpha }={\begin{cases}[-\alpha ,\alpha ],&\alpha >0\\\emptyset ,&\alpha \leq 0\end{cases}}} . The discrete version can be proved from the continuum version through constructing T : L p ( R , d x ) → L q ( N , ν ) {\displaystyle T:L^{p}(\mathbb {R} ,dx)\to L^{q}(N,\nu )} . == Applications == The Christ–Kiselev maximal inequality has applications to the Fourier transform and convergence of Fourier series, as well as to the study of Schrödinger operators. == References ==
Wikipedia:Chromatic symmetric function#0	The chromatic symmetric function is a symmetric function invariant of graphs studied in algebraic graph theory, a branch of mathematics. It is the weight generating function for proper graph colorings, and was originally introduced by Richard Stanley as a generalization of the chromatic polynomial of a graph. == Definition == For a finite graph G = ( V , E ) {\displaystyle G=(V,E)} with vertex set V = { v 1 , v 2 , … , v n } {\displaystyle V=\{v_{1},v_{2},\ldots ,v_{n}\}} , a vertex coloring is a function κ : V → C {\displaystyle \kappa :V\to C} where C {\displaystyle C} is a set of colors. A vertex coloring is called proper if all adjacent vertices are assigned distinct colors (i.e., { i , j } ∈ E ⟹ κ ( i ) ≠ κ ( j ) {\displaystyle \{i,j\}\in E\implies \kappa (i)\neq \kappa (j)} ). The chromatic symmetric function denoted X G ( x 1 , x 2 , … ) {\displaystyle X_{G}(x_{1},x_{2},\ldots )} is defined to be the weight generating function of proper vertex colorings of G {\displaystyle G} : X G ( x 1 , x 2 , … ) := ∑ κ : V → N proper x κ ( v 1 ) x κ ( v 2 ) ⋯ x κ ( v n ) {\displaystyle X_{G}(x_{1},x_{2},\ldots ):=\sum _{\underset {\text{proper}}{\kappa :V\to \mathbb {N} }}x_{\kappa (v_{1})}x_{\kappa (v_{2})}\cdots x_{\kappa (v_{n})}} == Examples == For λ {\displaystyle \lambda } a partition, let m λ {\displaystyle m_{\lambda }} be the monomial symmetric polynomial associated to λ {\displaystyle \lambda } . === Example 1: complete graphs === Consider the complete graph K n {\displaystyle K_{n}} on n {\displaystyle n} vertices: There are n ! {\displaystyle n!} ways to color K n {\displaystyle K_{n}} with exactly n {\displaystyle n} colors yielding the term n ! x 1 ⋯ x n {\displaystyle n!x_{1}\cdots x_{n}} Since every pair of vertices in K n {\displaystyle K_{n}} is adjacent, it can be properly colored with no fewer than n {\displaystyle n} colors. Thus, X K n ( x 1 , … , x n ) = n ! x 1 ⋯ x n = n ! m ( 1 , … , 1 ) {\displaystyle X_{K_{n}}(x_{1},\ldots ,x_{n})=n!x_{1}\cdots x_{n}=n!m_{(1,\ldots ,1)}} === Example 2: a path graph === Consider the path graph P 3 {\displaystyle P_{3}} of length 3 {\displaystyle 3} : There are 3 ! {\displaystyle 3!} ways to color P 3 {\displaystyle P_{3}} with exactly 3 {\displaystyle 3} colors, yielding the term 6 x 1 x 2 x 3 {\displaystyle 6x_{1}x_{2}x_{3}} For each pair of colors, there are 2 {\displaystyle 2} ways to color P 3 {\displaystyle P_{3}} yielding the terms x i 2 x j {\displaystyle x_{i}^{2}x_{j}} and x i x j 2 {\displaystyle x_{i}x_{j}^{2}} for i ≠ j {\displaystyle i\neq j} Altogether, the chromatic symmetric function of P 3 {\displaystyle P_{3}} is then given by: X P 3 ( x 1 , x 2 , x 3 ) = 6 x 1 x 2 x 3 + x 1 2 x 2 + x 1 x 2 2 + x 1 2 x 3 + x 1 x 3 2 + x 2 2 x 3 + x 2 x 3 2 = 6 m ( 1 , 1 , 1 ) + m ( 1 , 2 ) {\displaystyle X_{P_{3}}(x_{1},x_{2},x_{3})=6x_{1}x_{2}x_{3}+x_{1}^{2}x_{2}+x_{1}x_{2}^{2}+x_{1}^{2}x_{3}+x_{1}x_{3}^{2}+x_{2}^{2}x_{3}+x_{2}x_{3}^{2}=6m_{(1,1,1)}+m_{(1,2)}} == Properties == Let χ G {\displaystyle \chi _{G}} be the chromatic polynomial of G {\displaystyle G} , so that χ G ( k ) {\displaystyle \chi _{G}(k)} is equal to the number of proper vertex colorings of G {\displaystyle G} using at most k {\displaystyle k} distinct colors. The values of χ G {\displaystyle \chi _{G}} can then be computed by specializing the chromatic symmetric function, setting the first k {\displaystyle k} variables x i {\displaystyle x_{i}} equal to 1 {\displaystyle 1} and the remaining variables equal to 0 {\displaystyle 0} : X G ( 1 k ) = X G ( 1 , … , 1 , 0 , 0 , … ) = χ G ( k ) {\displaystyle X_{G}(1^{k})=X_{G}(1,\ldots ,1,0,0,\ldots )=\chi _{G}(k)} If G ⨿ H {\displaystyle G\amalg H} is the disjoint union of two graphs, then the chromatic symmetric function for G ⨿ H {\displaystyle G\amalg H} can be written as a product of the corresponding functions for G {\displaystyle G} and H {\displaystyle H} : X G ⨿ H = X G ⋅ X H {\displaystyle X_{G\amalg H}=X_{G}\cdot X_{H}} A stable partition π {\displaystyle \pi } of G {\displaystyle G} is defined to be a set partition of vertices V {\displaystyle V} such that each block of π {\displaystyle \pi } is an independent set in G {\displaystyle G} . The type of a stable partition type ( π ) {\displaystyle {\text{type}}(\pi )} is the partition consisting of parts equal to the sizes of the connected components of the vertex induced subgraphs. For a partition λ ⊢ n {\displaystyle \lambda \vdash n} , let z λ {\displaystyle z_{\lambda }} be the number of stable partitions of G {\displaystyle G} with type ( π ) = λ = ⟨ 1 r 1 2 r 2 … ⟩ {\displaystyle {\text{type}}(\pi )=\lambda =\langle 1^{r_{1}}2^{r2}\ldots \rangle } . Then, X G {\displaystyle X_{G}} expands into the augmented monomial symmetric functions, m ~ λ := r 1 ! r 2 ! ⋯ m λ {\displaystyle {\tilde {m}}_{\lambda }:=r_{1}!r_{2}!\cdots m_{\lambda }} with coefficients given by the number of stable partitions of G {\displaystyle G} : X G = ∑ λ ⊢ n z λ m ~ λ {\displaystyle X_{G}=\sum _{\lambda \vdash n}z_{\lambda }{\tilde {m}}_{\lambda }} Let p λ {\displaystyle p_{\lambda }} be the power-sum symmetric function associated to a partition λ {\displaystyle \lambda } . For S ⊆ E {\displaystyle S\subseteq E} , let λ ( S ) {\displaystyle \lambda (S)} be the partition whose parts are the vertex sizes of the connected components of the edge-induced subgraph of G {\displaystyle G} specified by S {\displaystyle S} . The chromatic symmetric function can be expanded in the power-sum symmetric functions via the following formula: X G = ∑ S ⊆ E ( − 1 ) \| S \| p λ ( S ) {\displaystyle X_{G}=\sum _{S\subseteq E}(-1)^{\|S\|}p_{\lambda (S)}} Let X G = ∑ λ ⊢ n c λ e λ {\textstyle X_{G}=\sum _{\lambda \vdash n}c_{\lambda }e_{\lambda }} be the expansion of X G {\displaystyle X_{G}} in the basis of elementary symmetric functions e λ {\displaystyle e_{\lambda }} . Let sink ( G , s ) {\displaystyle {\text{sink}}(G,s)} be the number of acyclic orientations on the graph G {\displaystyle G} which contain exactly s {\displaystyle s} sinks. Then we have the following formula for the number of sinks: sink ( G , s ) = ∑ λ ⊢ n l ( λ ) = s c λ {\displaystyle {\text{sink}}(G,s)=\sum _{\underset {l(\lambda )=s}{\lambda \vdash n}}c_{\lambda }} == Open problems == There are a number of outstanding questions regarding the chromatic symmetric function which have received substantial attention in the literature surrounding them. === (3+1)-free conjecture === For a partition λ {\displaystyle \lambda } , let e λ {\displaystyle e_{\lambda }} be the elementary symmetric function associated to λ {\displaystyle \lambda } . A partially ordered set P {\displaystyle P} is called ( 3 + 1 ) {\displaystyle (3+1)} -free if it does not contain a subposet isomorphic to the direct sum of the 3 {\displaystyle 3} element chain and the 1 {\displaystyle 1} element chain. The incomparability graph inc ( P ) {\displaystyle {\text{inc}}(P)} of a poset P {\displaystyle P} is the graph with vertices given by the elements of P {\displaystyle P} which includes an edge between two vertices if and only if their corresponding elements in P {\displaystyle P} are incomparable. Conjecture (Stanley–Stembridge) Let G {\displaystyle G} be the incomparability graph of a ( 3 + 1 ) {\textstyle (3+1)} -free poset, then X G {\textstyle X_{G}} is e {\displaystyle e} -positive. A weaker positivity result is known for the case of expansions into the basis of Schur functions. Theorem (Gasharov) Let G {\displaystyle G} be the incomparability graph of a ( 3 + 1 ) {\textstyle (3+1)} -free poset, then X G {\textstyle X_{G}} is s {\displaystyle s} -positive. In the proof of the theorem above, there is a combinatorial formula for the coefficients of the Schur expansion given in terms of P {\displaystyle P} -tableaux which are a generalization of semistandard Young tableaux instead labelled with the elements of P {\displaystyle P} . == Generalizations == There are a number of generalizations of the chromatic symmetric function: There is a categorification of the invariant into a homology theory which is called chromatic symmetric homology. This homology theory is known to be a stronger invariant than the chromatic symmetric function alone. The chromatic symmetric function can also be defined for vertex-weighted graphs, where it satisfies a deletion-contraction property analogous to that of the chromatic polynomial. If the theory of chromatic symmetric homology is generalized to vertex-weighted graphs as well, this deletion-contraction property lifts to a long exact sequence of the corresponding homology theory. There is also a quasisymmetric refinement of the chromatic symmetric function which can be used to refine the formulae expressing X G {\displaystyle X_{G}} in terms of Gessel's basis of fundamental quasisymmetric functions and the expansion in the basis of Schur functions. Fixing an order for the set of vertices, the ascent set of a proper coloring κ {\displaystyle \kappa } is defined to be asc ( κ ) = { { i , j } ∈ E : i < j and κ ( i ) < κ ( j ) } {\displaystyle {\text{asc}}(\kappa )=\{\{i,j\}\in E:i<j{\text{ and }}\kappa (i)<\kappa (j)\}} . The chromatic quasisymmetric function of a graph G {\displaystyle G} is then defined to be: X G ( x 1 , x 2 , … ; t ) := ∑ κ : V → N proper t \| a s c ( κ ) \| x κ ( v 1 ) ⋯ x κ ( v n ) {\displaystyle X_{G}(x_{1},x_{2},\ldots ;t):=\sum _{\underset {\text{proper}}{\kappa :V\to \mathbb {N} }}t^{\|asc(\kappa )\|}x_{\kappa (v_{1})}\cdots x_{\kappa (v_{n})}} == See also == Chromatic polynomial Symmetric function == References == == Further reading == Blasiak, Jonah; Eriksson, Holden; Pylyavskyy, Pavlo; Siegl, Isaiah (2022). "Noncommutative Schur functions for posets". arXiv:2211.03967 [math.CO]. Chow, Timothy Y. (1999). "Descents, Quasi-Symmetric Functions, Robinson-Schensted for Posets, and the Chromatic Symmetric Function". Journal of Algebraic Combinatorics. 10 (3): 227–240. doi:10.1023/A:1018719315718. Harada, Megumi; Precup, Martha E. (2019). "The cohomology of abelian Hessenberg varieties and the Stanley–Stembridge conjecture". Algebraic Combinatorics. 2 (6): 1059–1108. arXiv:1709.06736. doi:10.5802/alco.76. Hwang, Byung-Hak (2024). "Chromatic quasisymmetric functions and noncommutative P {\displaystyle P} -symmetric functions". Transactions of the American Mathematical Society. 377 (4): 2855–2896. arXiv:2208.09857. doi:10.1090/tran/9096. Shareshian, John; Wachs, Michelle L. (2012). "Chromatic quasisymmetric functions and Hessenberg varieties". Configuration Spaces. pp. 433–460. arXiv:1106.4287. doi:10.1007/978-88-7642-431-1_20. ISBN 978-88-7642-430-4.
Wikipedia:Churchill Professor of Mathematics of Information#0	The Churchill Professorship of Mathematics of Information (known until November 2022 as the Churchill Professorship of Mathematics for Operational Research) is a professorship in the mathematics of information at the University of Cambridge. It was established in 1966 by a benefaction from Esso in memory of Sir Winston Churchill, who died the previous year, for the promotion of the study of operations research. This was the second professorship established within the Cambridge Statistical Laboratory (the first being the Professorship of Mathematical Statistics). == List of Churchill Professors == 1967–1994 Peter Whittle 1994–2017 Richard Weber 2020– Ioannis Kontoyiannis == References ==
Wikipedia:Cicely Popplewell#0	Cicely Mary Williams (née Popplewell; 29 October 1920 – 20 June 1995) was a British software engineer who worked with Alan Turing on the Manchester Mark 1 computer. == Early life and education == Popplewell was born on 29 October 1920 in Bramhall, Stockport, England. Her parents were Bessie (née Fazakerley) and Alfred Popplewell, a chartered accountant. She attended Sherbrook Private Girls School at Greaves Hall in Lancashire. She studied the Mathematical Tripos at the University of Cambridge where she worked with statistics in the form of punched cards. She was considered an expert in the Brunsviga desk calculator. She graduated with a Bachelor of Arts degree in 1942, which was converted to a Master of Arts degree in 1949 from Girton College, Cambridge. == Career == In 1943 she was a Technical Assistant in the Experimental Department at Rolls-Royce Ltd. and joined the Women's Engineering Society. In 1949 Popplewell joined Alan Turing in the Computer Machine Learning department at the University of Manchester to help with the programming of a prototype computer. At first she shared an office with Turing and Audrey Bates, a University of Manchester mathematics graduate. Her first role was to create a library for the prototype Manchester Mark 1. This included input/output routines and mathematical functions, and a reciprocal square root routine. She worked on ray tracing. She wrote the first versions of sections of the subroutines for functions like COSINE. Together they designed the programming language for the Ferranti Mark 1. She wrote the Programmers Handbook for the Ferranti Mark 1 in 1951, reworking Turing's programming manual to make it comprehensible. Whilst Turing worked on Scheme A, an early operating system, Popplewell proposed Scheme B, which allowed for decimal numbers, in 1952. Popplewell went on to become an advisor and administrator in the newly formed University of Manchester Computing Service where she was remembered as a 'universally liked' mother-figure. She left the Service in the late 1960s shortly before her marriage. Popplewell taught the first ever programming class in Argentina at the University of Buenos Aires in 1961. Her class there included the computer scientist Cecilia Berdichevsky. She was supported by the British Council. Popplewell published the textbook Information Processing in 1962. Her life was documented in Jonathan Swinton's 2019 book Alan Turing’s Manchester. == Personal life == In 1969 Popplewell married George Keith Williams in Chapel-en-le-Frith. She died on 20 June 1995 at Stockport Infirmary, Stockport. The funeral service was held on 27 June 1995 at St John's church, Buxton, followed by a private cremation. == References ==
Wikipedia:Cindy Greenwood#0	Priscilla E. (Cindy) Greenwood (born 1937) is a Canadian mathematician who is a professor emeritus of mathematics at the University of British Columbia. She is known for her research in probability theory. == Education and career == Greenwood graduated from Duke University with a B.A. in 1959. She began her graduate studies in operations research at the Massachusetts Institute of Technology, where she became exposed to probability theory through a course on stochastic processes offered in 1960 by Henry McKean. Soon afterwards, she switched to the University of Wisconsin–Madison, where she completed her Ph.D. in 1963 under the supervision of Joshua Chover. She taught for two years at North Carolina College before moving to the University of British Columbia in 1966. She has also been associated with Arizona State University, as a visiting professor from 2000 to 2003 and since 2004 as a research professor. == Research == Greenwood's research in the 1970s concerned Brownian motion, Lévy processes, and Wiener–Hopf factorization. During this method she developed the theory of the martintote, a process similar to a martingale used to study asymptotic properties of processes. In the 1980s Greenwood began working with Ed Perkins on nonstandard analysis, which they used to study local time and excursions. In this timeframe she also began working on set-indexed processes, a topic that would lead her to the theory of random fields, and on semimartingales. She traveled to Russia, and wrote a monograph on chi-squared tests with Mikhail Nikulin. In 1990 she and Igor Evstigneev wrote a second monograph, on random fields. Her research in this period also concerned metric entropy and asymptotic efficiency. She began her work in biostatistics, involving studies of different mammalian populations, and led a major study on statistical estimation near critical points of a parameter. Beginning in 2000, at Arizona State, she studied pink noise and stochastic resonance, which she applied to epidemic models in biostatistics as well as to the firing patterns of neurons. == Awards and honours == Greenwood was elected as a fellow of the Institute of Mathematical Statistics in 1985. She won the Krieger–Nelson Prize of the Canadian Mathematical Society in 2002. == Books == Contiguity and the statistical invariance principle (with A. N. Shiryayev, Gordon & Breach, 1985) Markov fields over countable partially ordered sets: extrema and splitting (with I. V. Evstigneev, Memoirs of the American Mathematical Society 112, American Mathematical Society, 1994) A guide to chi-squared testing (with Mikhail S. Nikulin, Wiley, 1996) Stochastic neuron models (with Lawrence M. Ward, Mathematical Biosciences Institute Lecture Series, Springer, 2016) == References == == External links == Home page
Wikipedia:Cis (mathematics)#0	cis is a mathematical notation defined by cis x = cos x + i sin x, where cos is the cosine function, i is the imaginary unit and sin is the sine function. x is the argument of the complex number (angle between line to point and x-axis in polar form). The notation is less commonly used in mathematics than Euler's formula, eix, which offers an even shorter notation for cos x + i sin x, but cis(x) is widely used as a name for this function in software libraries. == Overview == The cis notation is a shorthand for the combination of functions on the right-hand side of Euler's formula: e i x = cos ⁡ x + i sin ⁡ x , {\displaystyle e^{ix}=\cos x+i\sin x,} where i2 = −1. So, cis ⁡ x = cos ⁡ x + i sin ⁡ x , {\displaystyle \operatorname {cis} x=\cos x+i\sin x,} i.e. "cis" is an acronym for "Cos i Sin". It connects trigonometric functions with exponential functions in the complex plane via Euler's formula. While the domain of definition is usually x ∈ R {\displaystyle x\in \mathbb {R} } , complex values z ∈ C {\displaystyle z\in \mathbb {C} } are possible as well: cis ⁡ z = cos ⁡ z + i sin ⁡ z , {\displaystyle \operatorname {cis} z=\cos z+i\sin z,} so the cis function can be used to extend Euler's formula to a more general complex version. The function is mostly used as a convenient shorthand notation to simplify some expressions, for example in conjunction with Fourier and Hartley transforms, or when exponential functions shouldn't be used for some reason in math education. In information technology, the function sees dedicated support in various high-performance math libraries (such as Intel's Math Kernel Library (MKL) or MathCW), available for many compilers and programming languages (including C, C++, Common Lisp, D, Haskell, Julia, and Rust). Depending on the platform, the fused operation is about twice as fast as calling the sine and cosine functions individually. == Mathematical identities == === Derivative === d d z cis ⁡ z = i cis ⁡ z = i e i z {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} z}}\operatorname {cis} z=i\operatorname {cis} z=ie^{iz}} === Integral === ∫ cis ⁡ z d z = − i cis ⁡ z = − i e i z {\displaystyle \int \operatorname {cis} z\,\mathrm {d} z=-i\operatorname {cis} z=-ie^{iz}} === Other properties === These follow directly from Euler's formula. cos ⁡ ( x ) = cis ⁡ ( x ) + cis ⁡ ( − x ) 2 = e i x + e − i x 2 {\displaystyle \cos(x)={\frac {\operatorname {cis} (x)+\operatorname {cis} (-x)}{2}}={\frac {e^{ix}+e^{-ix}}{2}}} sin ⁡ ( x ) = cis ⁡ ( x ) − cis ⁡ ( − x ) 2 i = e i x − e − i x 2 i {\displaystyle \sin(x)={\frac {\operatorname {cis} (x)-\operatorname {cis} (-x)}{2i}}={\frac {e^{ix}-e^{-ix}}{2i}}} cis ⁡ ( x + y ) = cis ⁡ x cis ⁡ y {\displaystyle \operatorname {cis} (x+y)=\operatorname {cis} x\,\operatorname {cis} y} cis ⁡ ( x − y ) = cis ⁡ x cis ⁡ y {\displaystyle \operatorname {cis} (x-y)={\operatorname {cis} x \over \operatorname {cis} y}} The identities above hold if x and y are any complex numbers. If x and y are real, then \| cis ⁡ x − cis ⁡ y \| ≤ \| x − y \| . {\displaystyle \|\operatorname {cis} x-\operatorname {cis} y\|\leq \|x-y\|.} == History == The cis notation was first coined by William Rowan Hamilton in Elements of Quaternions (1866) and subsequently used by Irving Stringham (who also called it "sector of x") in works such as Uniplanar Algebra (1893), James Harkness and Frank Morley in their Introduction to the Theory of Analytic Functions (1898), or by George Ashley Campbell (who also referred to it as "cisoidal oscillation") in his works on transmission lines (1901) and Fourier integrals (1928). In 1942, inspired by the cis notation, Ralph V. L. Hartley introduced the cas (for cosine-and-sine) function for the real-valued Hartley kernel, a meanwhile established shortcut in conjunction with Hartley transforms: cas ⁡ x = cos ⁡ x + sin ⁡ x . {\displaystyle \operatorname {cas} x=\cos x+\sin x.} == Motivation == The cis notation is sometimes used to emphasize one method of viewing and dealing with a problem over another. The mathematics of trigonometry and exponentials are related but not exactly the same; exponential notation emphasizes the whole, whereas cis x and cos x + i sin x notations emphasize the parts. This can be rhetorically useful to mathematicians and engineers when discussing this function, and further serve as a mnemonic (for cos + i sin). The cis notation is convenient for math students whose knowledge of trigonometry and complex numbers permit this notation, but whose conceptual understanding does not yet permit the notation eix. The usual proof that cis x = eix requires calculus, which the student may not have studied before encountering the expression cos x + i sin x. This notation was more common when typewriters were used to convey mathematical expressions. == See also == De Moivre's formula Euler's formula Complex number Ptolemy's theorem Phasor Versor == Notes == == References ==
Wikipedia:Cish (mathematics)#0	cis is a mathematical notation defined by cis x = cos x + i sin x, where cos is the cosine function, i is the imaginary unit and sin is the sine function. x is the argument of the complex number (angle between line to point and x-axis in polar form). The notation is less commonly used in mathematics than Euler's formula, eix, which offers an even shorter notation for cos x + i sin x, but cis(x) is widely used as a name for this function in software libraries. == Overview == The cis notation is a shorthand for the combination of functions on the right-hand side of Euler's formula: e i x = cos ⁡ x + i sin ⁡ x , {\displaystyle e^{ix}=\cos x+i\sin x,} where i2 = −1. So, cis ⁡ x = cos ⁡ x + i sin ⁡ x , {\displaystyle \operatorname {cis} x=\cos x+i\sin x,} i.e. "cis" is an acronym for "Cos i Sin". It connects trigonometric functions with exponential functions in the complex plane via Euler's formula. While the domain of definition is usually x ∈ R {\displaystyle x\in \mathbb {R} } , complex values z ∈ C {\displaystyle z\in \mathbb {C} } are possible as well: cis ⁡ z = cos ⁡ z + i sin ⁡ z , {\displaystyle \operatorname {cis} z=\cos z+i\sin z,} so the cis function can be used to extend Euler's formula to a more general complex version. The function is mostly used as a convenient shorthand notation to simplify some expressions, for example in conjunction with Fourier and Hartley transforms, or when exponential functions shouldn't be used for some reason in math education. In information technology, the function sees dedicated support in various high-performance math libraries (such as Intel's Math Kernel Library (MKL) or MathCW), available for many compilers and programming languages (including C, C++, Common Lisp, D, Haskell, Julia, and Rust). Depending on the platform, the fused operation is about twice as fast as calling the sine and cosine functions individually. == Mathematical identities == === Derivative === d d z cis ⁡ z = i cis ⁡ z = i e i z {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} z}}\operatorname {cis} z=i\operatorname {cis} z=ie^{iz}} === Integral === ∫ cis ⁡ z d z = − i cis ⁡ z = − i e i z {\displaystyle \int \operatorname {cis} z\,\mathrm {d} z=-i\operatorname {cis} z=-ie^{iz}} === Other properties === These follow directly from Euler's formula. cos ⁡ ( x ) = cis ⁡ ( x ) + cis ⁡ ( − x ) 2 = e i x + e − i x 2 {\displaystyle \cos(x)={\frac {\operatorname {cis} (x)+\operatorname {cis} (-x)}{2}}={\frac {e^{ix}+e^{-ix}}{2}}} sin ⁡ ( x ) = cis ⁡ ( x ) − cis ⁡ ( − x ) 2 i = e i x − e − i x 2 i {\displaystyle \sin(x)={\frac {\operatorname {cis} (x)-\operatorname {cis} (-x)}{2i}}={\frac {e^{ix}-e^{-ix}}{2i}}} cis ⁡ ( x + y ) = cis ⁡ x cis ⁡ y {\displaystyle \operatorname {cis} (x+y)=\operatorname {cis} x\,\operatorname {cis} y} cis ⁡ ( x − y ) = cis ⁡ x cis ⁡ y {\displaystyle \operatorname {cis} (x-y)={\operatorname {cis} x \over \operatorname {cis} y}} The identities above hold if x and y are any complex numbers. If x and y are real, then \| cis ⁡ x − cis ⁡ y \| ≤ \| x − y \| . {\displaystyle \|\operatorname {cis} x-\operatorname {cis} y\|\leq \|x-y\|.} == History == The cis notation was first coined by William Rowan Hamilton in Elements of Quaternions (1866) and subsequently used by Irving Stringham (who also called it "sector of x") in works such as Uniplanar Algebra (1893), James Harkness and Frank Morley in their Introduction to the Theory of Analytic Functions (1898), or by George Ashley Campbell (who also referred to it as "cisoidal oscillation") in his works on transmission lines (1901) and Fourier integrals (1928). In 1942, inspired by the cis notation, Ralph V. L. Hartley introduced the cas (for cosine-and-sine) function for the real-valued Hartley kernel, a meanwhile established shortcut in conjunction with Hartley transforms: cas ⁡ x = cos ⁡ x + sin ⁡ x . {\displaystyle \operatorname {cas} x=\cos x+\sin x.} == Motivation == The cis notation is sometimes used to emphasize one method of viewing and dealing with a problem over another. The mathematics of trigonometry and exponentials are related but not exactly the same; exponential notation emphasizes the whole, whereas cis x and cos x + i sin x notations emphasize the parts. This can be rhetorically useful to mathematicians and engineers when discussing this function, and further serve as a mnemonic (for cos + i sin). The cis notation is convenient for math students whose knowledge of trigonometry and complex numbers permit this notation, but whose conceptual understanding does not yet permit the notation eix. The usual proof that cis x = eix requires calculus, which the student may not have studied before encountering the expression cos x + i sin x. This notation was more common when typewriters were used to convey mathematical expressions. == See also == De Moivre's formula Euler's formula Complex number Ptolemy's theorem Phasor Versor == Notes == == References ==
Wikipedia:Cissoid of Diocles#0	In geometry, the cissoid of Diocles (from Ancient Greek κισσοειδής (kissoeidēs) 'ivy-shaped'; named for Diocles) is a cubic plane curve notable for the property that it can be used to construct two mean proportionals to a given ratio. In particular, it can be used to double a cube. It can be defined as the cissoid of a circle and a line tangent to it with respect to the point on the circle opposite to the point of tangency. In fact, the curve family of cissoids is named for this example and some authors refer to it simply as the cissoid. It has a single cusp at the pole, and is symmetric about the diameter of the circle which is the line of tangency of the cusp. The line is an asymptote. It is a member of the conchoid of de Sluze family of curves and in form it resembles a tractrix. == Construction and equations == Let the radius of C be a. By translation and rotation, we may take O to be the origin and the center of the circle to be (a, 0), so A is (2a, 0). Then the polar equations of L and C are: r = 2 a sec ⁡ θ r = 2 a cos ⁡ θ . {\displaystyle {\begin{aligned}&r=2a\sec \theta \\&r=2a\cos \theta .\end{aligned}}} By construction, the distance from the origin to a point on the cissoid is equal to the difference between the distances between the origin and the corresponding points on L and C. In other words, the polar equation of the cissoid is r = 2 a sec ⁡ θ − 2 a cos ⁡ θ = 2 a ( sec ⁡ θ − cos ⁡ θ ) . {\displaystyle r=2a\sec \theta -2a\cos \theta =2a(\sec \theta -\cos \theta ).} Applying some trigonometric identities, this is equivalent to r = 2 a sin 2 θ / cos ⁡ θ = 2 a sin ⁡ θ tan ⁡ θ . {\displaystyle r=2a\sin ^{2}\!\theta \mathbin {/} \cos \theta =2a\sin \theta \tan \theta .} Let t = tan θ in the above equation. Then x = r cos ⁡ θ = 2 a sin 2 θ = 2 a tan 2 θ sec 2 θ = 2 a t 2 1 + t 2 y = t x = 2 a t 3 1 + t 2 {\displaystyle {\begin{aligned}&x=r\cos \theta =2a\sin ^{2}\!\theta ={\frac {2a\tan ^{2}\!\theta }{\sec ^{2}\!\theta }}={\frac {2at^{2}}{1+t^{2}}}\\&y=tx={\frac {2at^{3}}{1+t^{2}}}\end{aligned}}} are parametric equations for the cissoid. Converting the polar form to Cartesian coordinates produces ( x 2 + y 2 ) x = 2 a y 2 {\displaystyle (x^{2}+y^{2})x=2ay^{2}} === Construction by double projection === A compass-and-straightedge construction of various points on the cissoid proceeds as follows. Given a line L and a point O not on L, construct the line L' through O parallel to L. Choose a variable point P on L, and construct Q, the orthogonal projection of P on L', then R, the orthogonal projection of Q on OP. Then the cissoid is the locus of points R. To see this, let O be the origin and L the line x = 2a as above. Let P be the point (2a, 2at); then Q is (0, 2at) and the equation of the line OP is y = tx. The line through Q perpendicular to OP is t ( y − 2 a t ) + x = 0. {\displaystyle t(y-2at)+x=0.} To find the point of intersection R, set y = tx in this equation to get t ( t x − 2 a t ) + x = 0 , x ( t 2 + 1 ) = 2 a t 2 , x = 2 a t 2 t 2 + 1 y = t x = 2 a t 3 t 2 + 1 {\displaystyle {\begin{aligned}&t(tx-2at)+x=0,\ x(t^{2}+1)=2at^{2},\ x={\frac {2at^{2}}{t^{2}+1}}\\&y=tx={\frac {2at^{3}}{t^{2}+1}}\end{aligned}}} which are the parametric equations given above. While this construction produces arbitrarily many points on the cissoid, it cannot trace any continuous segment of the curve. === Newton's construction === The following construction was given by Isaac Newton. Let J be a line and B a point not on J. Let ∠BST be a right angle which moves so that ST equals the distance from B to J and T remains on J, while the other leg BS slides along B. Then the midpoint P of ST describes the curve. To see this, let the distance between B and J be 2a. By translation and rotation, take B = (–a, 0) and J the line x = a. Let P = (x, y) and let ψ be the angle between SB and the x-axis; this is equal to the angle between ST and J. By construction, PT = a, so the distance from P to J is a sin ψ. In other words a – x = a sin ψ. Also, SP = a is the y-coordinate of (x, y) if it is rotated by angle ψ, so a = (x + a) sin ψ + y cos ψ. After simplification, this produces parametric equations x = a ( 1 − sin ⁡ ψ ) , y = a ( 1 − sin ⁡ ψ ) 2 cos ⁡ ψ . {\displaystyle x=a(1-\sin \psi ),\,y=a{\frac {(1-\sin \psi )^{2}}{\cos \psi }}.} Change parameters by replacing ψ with its complement to get x = a ( 1 − cos ⁡ ψ ) , y = a ( 1 − cos ⁡ ψ ) 2 sin ⁡ ψ {\displaystyle x=a(1-\cos \psi ),\,y=a{\frac {(1-\cos \psi )^{2}}{\sin \psi }}} or, applying double angle formulas, x = 2 a sin 2 ⁡ ψ 2 , y = a 4 sin 4 ⁡ ψ 2 2 sin ⁡ ψ 2 cos ⁡ ψ 2 = 2 a sin 3 ⁡ ψ 2 cos ⁡ ψ 2 . {\displaystyle x=2a\sin ^{2}{\psi \over 2},\,y=a{\frac {4\sin ^{4}{\psi \over 2}}{2\sin {\psi \over 2}\cos {\psi \over 2}}}=2a{\frac {\sin ^{3}{\psi \over 2}}{\cos {\psi \over 2}}}.} But this is polar equation r = 2 a sin 2 ⁡ θ cos ⁡ θ {\displaystyle r=2a{\frac {\sin ^{2}\theta }{\cos \theta }}} given above with θ = ψ/2. Note that, as with the double projection construction, this can be adapted to produce a mechanical device that generates the curve. == Delian problem == The Greek geometer Diocles used the cissoid to obtain two mean proportionals to a given ratio. This means that given lengths a and b, the curve can be used to find u and v so that a is to u as u is to v as v is to b, i.e. a/u = u/v = v/b, as discovered by Hippocrates of Chios. As a special case, this can be used to solve the Delian problem: how much must the length of a cube be increased in order to double its volume? Specifically, if a is the side of a cube, and b = 2a, then the volume of a cube of side u is u 3 = a 3 ( u a ) 3 = a 3 ( u a ) ( v u ) ( b v ) = a 3 ( b a ) = 2 a 3 {\displaystyle u^{3}=a^{3}\left({\frac {u}{a}}\right)^{3}=a^{3}\left({\frac {u}{a}}\right)\left({\frac {v}{u}}\right)\left({\frac {b}{v}}\right)=a^{3}\left({\frac {b}{a}}\right)=2a^{3}} so u is the side of a cube with double the volume of the original cube. Note however that this solution does not fall within the rules of compass and straightedge construction since it relies on the existence of the cissoid. Let a and b be given. It is required to find u so that u3 = a2b, giving u and v = u2/a as the mean proportionals. Let the cissoid ( x 2 + y 2 ) x = 2 a y 2 {\displaystyle (x^{2}+y^{2})x=2ay^{2}} be constructed as above, with O the origin, A the point (2a, 0), and J the line x = a, also as given above. Let C be the point of intersection of J with OA. From the given length b, mark B on J so that CB = b. Draw BA and let P = (x, y) be the point where it intersects the cissoid. Draw OP and let it intersect J at U. Then u = CU is the required length. To see this, rewrite the equation of the curve as y 2 = x 3 2 a − x {\displaystyle y^{2}={\frac {x^{3}}{2a-x}}} and let N = (x, 0), so PN is the perpendicular to OA through P. From the equation of the curve, P N ¯ 2 = O N ¯ 3 N A ¯ . {\displaystyle {\overline {PN}}^{2}={\frac {{\overline {ON}}^{3}}{\overline {NA}}}.} From this, P N ¯ 3 O N ¯ 3 = P N ¯ N A ¯ . {\displaystyle {\frac {{\overline {PN}}^{3}}{{\overline {ON}}^{3}}}={\frac {\overline {PN}}{\overline {NA}}}.} By similar triangles PN/ON = UC/OC and PN/NA = BC/CA. So the equation becomes U C ¯ 3 O C ¯ 3 = B C ¯ C A ¯ , {\displaystyle {\frac {{\overline {UC}}^{3}}{{\overline {OC}}^{3}}}={\frac {\overline {BC}}{\overline {CA}}},} so u 3 a 3 = b a , u 3 = a 2 b {\displaystyle {\frac {u^{3}}{a^{3}}}={\frac {b}{a}},\,u^{3}=a^{2}b} as required. Diocles did not really solve the Delian problem. The reason is that the cissoid of Diocles cannot be constructed perfectly, at least not with compass and straightedge. To construct the cissoid of Diocles, one would construct a finite number of its individual points, then connect all these points to form a curve. (An example of this construction is shown on the right.) The problem is that there is no well-defined way to connect the points. If they are connected by line segments, then the construction will be well-defined, but it will not be an exact cissoid of Diocles, but only an approximation. Likewise, if the dots are connected with circular arcs, the construction will be well-defined, but incorrect. Or one could simply draw a curve directly, trying to eyeball the shape of the curve, but the result would only be imprecise guesswork. Once the finite set of points on the cissoid have been drawn, then line PC will probably not intersect one of these points exactly, but will pass between them, intersecting the cissoid of Diocles at some point whose exact location has not been constructed, but has only been approximated. An alternative is to keep adding constructed points to the cissoid which get closer and closer to the intersection with line PC, but the number of steps may very well be infinite, and the Greeks did not recognize approximations as limits of infinite steps (so they were very puzzled by Zeno's paradoxes). One could also construct a cissoid of Diocles by means of a mechanical tool specially designed for that purpose, but this violates the rule of only using compass and straightedge. This rule was established for reasons of logical — axiomatic — consistency. Allowing construction by new tools would be like adding new axioms, but axioms are supposed to be simple and self-evident, but such tools are not. So by the rules of classical, synthetic geometry, Diocles did not solve the Delian problem, which actually can not be solved by such means. == As a pedal curve == The pedal curve of a parabola with respect to its vertex is a cissoid of Diocles. The geometrical properties of pedal curves in general produce several alternate methods of constructing the cissoid. It is the envelopes of circles whose centers lie on a parabola and which pass through the vertex of the parabola. Also, if two congruent parabolas are set vertex-to-vertex and one is rolled along the other; the vertex of the rolling parabola will trace the cissoid. == Inversion == The cissoid of Diocles can also be defined as the inverse curve of a parabola with the center of inversion at the vertex. To see this, take the parabola to be x = y2, in polar coordinate r cos ⁡ θ = ( r sin ⁡ θ ) 2 {\displaystyle r\cos \theta =(r\sin \theta )^{2}} or: r = cos ⁡ θ sin 2 θ . {\displaystyle r={\frac {\cos \theta }{\sin ^{2}\!\theta }}\,.} The inverse curve is thus: r = sin 2 θ cos ⁡ θ = sin ⁡ θ tan ⁡ θ , {\displaystyle r={\frac {\sin ^{2}\!\theta }{\cos \theta }}=\sin \theta \tan \theta ,} which agrees with the polar equation of the cissoid above. == References == J. Dennis Lawrence (1972). A catalog of special plane curves. Dover Publications. pp. 95, 98–100. ISBN 0-486-60288-5. Weisstein, Eric W. "Cissoid of Diocles". MathWorld. "Cissoid of Diocles" at Visual Dictionary Of Special Plane Curves "Cissoid of Diocles" at MacTutor's Famous Curves Index "Cissoid" on 2dcurves.com "Cissoïde de Dioclès ou Cissoïde Droite" at Encyclopédie des Formes Mathématiques Remarquables (in French) "The Cissoid" An elementary treatise on cubic and quartic curves Alfred Barnard Basset (1901) Cambridge pp. 85ff

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